Research ArticleAn Integrated Multiechelon Logistics Model with UncertainDelivery Lead Time and Quality Unreliability

Ming-Feng Yang1 Yi Lin2 Li Hsing Ho3 and Wei Feng Kao4

1Department of Transportation Science National Taiwan Ocean University No 2 Beining Road Keelung City 202 Taiwan2Graduate Institute of Industrial and Business Management National Taipei University of Technology No 1 Section 3Zhongxiao E Road Taipei City 106 Taiwan3Department of Technology Management Chung Hua University No 707 Section 2 Wufu Road Hsinchu 30012 Taiwan4PhD Program of Technology Management Chung Hua University No 707 Section 2 Wufu Road Hsinchu 30012 Taiwan

Correspondence should be addressed to Yi Lin iwc3706yahoocomtw

Received 29 November 2015 Revised 28 January 2016 Accepted 16 March 2016

Academic Editor Lijun Ma

Copyright copy 2016 Ming-Feng Yang et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Nowadays in order to achieve advantages in supply chainmanagement how to keep inventory in adequate level and how to enhancecustomer service level are two critical practices for decision makers Generally uncertain lead time and defective products havemuch to do with inventory and service levelTherefore this study mainly aims at developing a multiechelon integrated just-in-timeinventory model with uncertain lead time and imperfect quality to enhance the benefits of the logistics model In addition the AntColony Algorithm (ACA) is established to determine the optimal solutions Moreover based on our proposed model and analysisthe ACA is more efficient than Particle Swarm Optimization (PSO) and Lingo in SMEIJI model An example is provided in thisstudy to illustrate how production run and defective rate have an effect on system costs Finally the results of our research couldprovide some managerial insights which support decision makers in real-world operations

1 Introduction

The EPQEOQ model has been researched for a long timevendors can estimate the optimal economic production byusing EPQ model and buyers estimate the order quantity byusing EOQ model However it will not lead to a win-winsituation if vendors only focus on the economic productionor buyers only care about the order quantity In 1977 anintegrated vendor-buyer concept has been proposed byGoyal[1] he was the first to consider integrated inventory modelIn his model economic production and order quantitycould be determined to minimize the joint total cost Healso concluded that the optimal order time interval andproduction cycle time could be obtained by supposing thatthe supplierrsquos production cycle time was an integer of buyerrsquosorder time interval In 1986 Banerjee [2] relaxed Goyalrsquos [1]integrated model to develop a joint economic lot size (JELS)model it indicated how a lot-for-lot policy works Goyal [3]indicated an integrated model that produces a lower-joint

total cost by loosening Banerjeersquos [2] lot-for-lot assumptionAfterwards many researchers built their own two-echeloninventory model by combining JELS model and Goyalrsquos [3]model

Seo [4] proposed an improved reorder decision policy forcontrolling general multiechelon distribution systems Thissystem utilizes shared stock information Chiu and Huang[5] addressed a multiechelon integrated JIT inventory modelwith a randomdelivery lead time Sadeghi et al [6] developeda constrained multivendor multiretailer single-warehouse(MV-MR-SW) supply chain in which both the space andthe annual number of orders of the central warehouse werelimited He would like to find the order quantities along withthe number of shipments received by retailers and vendorssuch that the total inventory cost of the chain wasminimizedSana [7] proposed an integrated production-inventorymodelfor supplier manufacturer and retailer considering perfectand imperfect quality itemsThismodel discussed the impactof business strategies such as optimal order size of raw

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 8494268 13 pageshttpdxdoiorg10115520168494268

2 Mathematical Problems in Engineering

materials production rate and unit production cost and idletimes in different sectors on collaborating marketing systemSoni and Patel [8] investigated an integrated inventory modelwith variable production rate and price-sensitive demandrate under two-level trade credit This study attempted tooffer a best policy for retail price replenishment cycle andthe number of shipments from the supplier to the retailerin one production run An algorithm was also designed todetermine the optimum solution of their proposed model

Lead time usually consists of the following componentsorder preparation order transit supplier lead time deliverytime and set-up time (Tersine [9]) this makes lead timedifficult to estimate precisely In real life firms cannot keepinventory in adequate level if lead time is uncertain it maycause unnecessary costs for firms Furthermore uncertainlead time also increases the probability of shortage firms cannot satisfy customers immediately if shortage occurs Henceuncertain lead time has much to do with inventory leveland customer service level Liao and Shyu [10] presented aprobabilistic inventorymodel they assumed that the demandfollows normal distribution and the lead time consists of 119899

components each having a different cost for reduced leadtime Ben-Daya and Raouf [11] considered both the leadtime and the order quantity as decision variables based onLiao and Shyursquos [10] model Mirzapour Al-e-Hashem et al[12] assumed an interrelationship between lead time andtransportationmode the shorter the lead times are the moreexpensive transportationwill be Accordingly they developeda stochastic programming approach to solve a multiperiodmultiproduct multisite aggregate production planning prob-lem in a green supply chain for a medium-term planninghorizon under the assumption of demand uncertainty andflexible lead times Chandra and Grabis [13] indicated thatshort lead time could enhance the service level and lowerinventory level effectively but short lead time also causedhighly order cost

However demands during the lead time of the differentcustomers are not identical and the demand distribution foreach customer is not the same either Therefore we cannotapply single distribution to describe the demands during thelead time Accordingly Wu and Tsai [14] considered the leadtime demand with the mixture of normal distributions andthe fixed back-order rate Hoque [15] developed a vendor-buyer integrated production-inventory model following nor-mal distribution of lead time Then he derived an optimalsolution technique to themodel to obtainminimumexpectedjoint total cost that follows development of the solutionalgorithm

In real life supply chains always have many membersin multiechelon however multiechelon inventory problemsare too complicated to solve by using traditional methodsHence we would like to discuss some heuristic algorithm inour proposed model to solve the multiechelon problems anddetermine the optimal solutions Altiparmak [16] proposed anew solution procedure based on genetic algorithms to findthe set of Pareto-optimal solutions for multiobjective SCNdesign problem To deal with multiobjective design problemand enable the decision makers to evaluate a greater numberof alternative solutions two different weight approaches were

implemented in the proposed solution procedure Sadeghiet al [17] developed a biobjective vendor-managed inventorymodel in a supply chain with one vendor (producer) andseveral retailers The aim was to find the order size thereplenishment frequency of retailers optimal traveling tourfrom the vendor to retailers and the number of machinesso as the total chain cost was minimized while the systemreliability of producing the item was maximized Li et al[18] studied the dynamic lot-sizing problem with productreturns and remanufacturing (DLRR)which is to determine aproduction schedule of manufacturing new products andorremanufacturing returns such that demand in each periodwas satisfied and the total cost (set-up cost plus holdingcost of inventory) was minimized To generate a good initialsolution they used a block-chain based method where theplanning horizon was split into a chain of blocks A blockmay contain either a string of manufacturing set-ups or astring of remanufacturing set-ups or both Given the costof each block an initial solution corresponding to a bestcombination of blocks is found by solving a shortest-pathproblem Chen and Sarker [19] established an integratedoptimal inventory lot-sizing and vehicle-routing model for amultisupplier single-assembler systemwith JIT delivery theyapplied ant colony optimization to solve their model Nia etal [20] built a multi-item economic order quantity modelwith shortage under vendor-managed inventory policy in asingle vendor single buyer supply chain they proposed anew modeling to the fuzzy VMI problem with multi-itemsand shortage and employed three metaheuristic algorithms(ACA GA and DE) to solve a FNIP problem Chen andSarker [19] andNia et al [20] both considered the Ant ColonyAlgorithm as a metaheuristic algorithm

In recent years many researchers dedicated themselvesto improving the integrated logistic model with the issues ofuncertainties Some related researches are listed belowHatefiand Jolai [21] proposed a robust and reliable model for anintegrated forward-reverse logistics network design whichsimultaneously takes uncertain parameters and facility dis-ruptions into account Ma [22] constructed the mathematicalprogramming model and proposed two-stage heuristic algo-rithm In addition the taboo search algorithm was designedto improve the initial solution Hashim et al [23] presentedthe study whichmainly investigated amultiobjective supplierselection planning problem in fuzzy environment and theuncertain model is converted into deterministic form bythe expected value measure Q-M Hu and Z-H Hu [24]proposed a reliable closed-loop supply chain network designmodel which accounts for both partial and complete facilitydisruptions as well as the uncertainty in the critical inputdata Torabi et al [25] considered the characteristics of theuncertainty flows And then a stochastic mixed-integer linearprogrammingmodel for designing hub-and-spoke network isestablished based on the capacities of spokes

With the above discussion we would like to establishan integrated vendor-buyer inventory model To fit the reallife we expanded the basic integrated model to a SMEIJImodel with uncertain lead time and imperfect quality Ourproposed model was too complicated to solve if we usetraditional method hence we applied Ant Colony Algorithm

Mathematical Problems in Engineering 3

to solve our SMEIJI model In given example we would liketo compare ACAwith other traditional methods and observehow system cost works

2 Fundamental Assumptions and Notations

A serial inventory system with 119878-echelon where echelon 1 ispurchasing only and echelon 119878 is manufacturing only wasconsidered In the inventory system we proposed there isonly one member in each echelon that plays the purchaserand manufacturer roles simultaneously except echelon 1 andechelon 119878 Notation and assumptions are defined as follows

For member 119894 engaged in purchasing activities where 119894 =

1 2 119878 minus 119897 we have the following

Decision variables are as follows

119873119901119894 number of purchase orders during 119879119898119894 forpurchasing 119894119899119894 number of deliveries per purchase order forpurchasing 119894

119898119871119894 the maximum allowable planned deliverylead time that would not cause shortage 119898119894 =120583119871119863119894+ safety lead time (a real decision variable)

Parameters are as follows

119878 number of echelons in a serial supply chain119879119901119894 length of a purchasing time interval forpurchasing 119894 (in years)119863119894 demand rate of purchasing 119894 (unitsyear)119876119901119894 lot size per purchase order for purchasing 119894

(unitsorder)119902119894 delivery lot size per shipment for purchasing119894 (unitsdelivery)119862119904119894 fixed ordering cost for purchasing 119894

($order)119867119901119894 holding cost per unit of purchased goodsper year for purchasing 119894 ($unityear)119862119890119894 ordering cost of emergency borrowing forpurchasing 119894 ($borrowing)120573119894 borrowing cost per unit per year for purchas-ing 119894 ($unityear)119865119894 fixed delivery cost for purchasing 119894 ($deliv-ery)119871119889119894 delivery lead time from manufacturing 119894 +

1 to purchasing 119894 a nonnegative random vari-able following a probability distribution withexpected delivery lead time 120583119871

119863119894and standard

deviation 120590119871119863119894119905119894 time interval between two adjacent deliveriesfor purchasing 119894119903119894 redelivery point of purchasing 119894 that is analo-gous to a reorder point if stock drops to119877119894 thena delivery notice is issued to manufacturer 119894 + 1119909119894 screening time of 119902119894 amount materials thatwere received by purchaser 119894 (in years)

0119894 defective rate of a shipment that was receivedby purchaser 119894

For member 119894 engaged in manufacturing activities where 119894 =

1 2 119878 minus 119897 we have the following

Decision variables are as follows

119873119898119894 number of production runs during 119879119898119894minus1

for manufacturing 119894119870119894 number of deliveries per production run formanufacturing 119894

Parameters are as follows

119879119898119894 length of a manufacturing time interval formanufacturing 119894 119879119898119894 = 119876119898119894119901119894 (in years)119875119894 production rate for manufacturing 119894

(unitsyear)119876119898119894 lot size per production run formanufactur-ing 119894 (unitsrun)119862119898119894 fixed set-up cost formanufacturing 119894 ($set-up)119867119898119894 holding cost per unit of produced goodsper year for manufacturing 119894 ($unityear)

Additional notations will be introduced if necessary Theassumptions set in this research are as follows

(1) We consider an integrated multiechelon supply chainwhich only contains a member in each echelon Themember in echelon 119894 is denoted as member 119894 where119894 = 1 2 3 119878 A single product is produced by thesupply chain

(2) Except echelons 1 and 119878 purchaser and manufacturerare the two roles that each member played Member1 purchasing activities only is the lowest echelon inthis serial supply chain Member 119878 involves manufac-turing activities only

(3) Member 119894 purchases requiredmaterials frommember119894 + 1 and produces goods to member 119894 minus 1 during 119879119901119894

(4) The demand and production rate for the product areconstant over time The demand rates of member 119894

should be less than the production rate of member119894 + 1 Member 119894 + 1 produces the necessary quantityto satisfy the demand of member 119894

(5) The delivery lead time (119871119889119894) for member 119894 is assumedto follow a probability distribution with a densityfunction of 119871119889119894 = 119891(119871119889119894) where 0 le 120582119871119894 le 119871119889119894 le 120574119871119894

and 119894 = 1 2 119878 minus 1(6) The time interval between two adjacent deliveries

for member 119894 (119905119894) should be longer than or equalto 120574119871119894 The delivery lot crossing makes the problemintractable and it can be prevented by this assump-tion

(7) When a delivery delay occurs an emergency borrow-ing action should be touched off bymember 119894 becausethe shortages are not allowed in JIT environment

4 Mathematical Problems in Engineering

Time

Inventory level ti ti

Di

qi

ri

0iqi

mLi mLi mLixi + Ldi minus mLi

Ldi Ldixi mLi + ti minus Ldi

0 le 120582Li le Ldi le mLi mLi le Ldi le 120574Li le ti

(1 minus 0i)qi

(B)

(A)

Figure 1 Delivery cases for the member who implements purchasing

(8) The required time of borrowing goods from nearbysupplier is ignored

(9) A finite planning horizon for the whole serial supplychain that denoted 119879119901119894 is considered

(10) Defective items in a shipment of materials arereceived by purchaser 119894 with defective rate 0119894 Thenumber of goodmaterialsmust bemore than or equalto the required quantity during the screening time

(11) The screening process is followed by the arrival ofshipments with screening time 119909119894 The length ofscreening time is proportional to the number ofreceived materials

(12) The accumulated number of defective items that weredelivered to the buyer during a production run mustbe less than the quantity of one delivery

(13) When emergency borrowing happened the materialsthat were borrowed and returned are all good items

3 Formulation of the Model

In this integrated multiechelon JIT inventory model thedecision variables are 119899119894 (number of deliveries per purchase)119873119901119894 (number of purchase orders during 119879119901119894) and 119898119871119894 (themaximum allowable planned delivery lead time that wouldnot cause shortage) where 119894 = 1 119878minus1119873119898119894 is the numberof production runs during 119879119898119894 and 119870119894 are delivery times perproduction runs for manufacturer 119894 where 119894 = 2 119878 Thevalue of these decision variables should be determined inorder to minimize the joint total cost

31 The Cost of Member 119894 in Purchasing Activities Fromechelon1 to echelon 119878 minus 1 members adopt JIT purchasing toreplenish goodsThe time buffer policy used on planned lead

time (119898119871119894) and emergency borrowing policy used to deal withuncertain delivery lead time Each order lot size119876119901119894 has beentaken apart into 119899119894 small lots for delivery frequently Duringthe planning horizon 119879119901119894 the total number of deliveriesfor member 119894 is 119899119894 times 119873119901119894 The delivery lead time follows aprobability density distribution 119891(119871119889119894) The 120574119871119894 and 120582119871119894 arethe upper bound and the lower bound of 119871119889119894

The costs are related to purchasing activities of member119894 during 119879119901119894 where 119894 = 1 119878 minus 1 including ordering costholding cost delivery cost transportation carrying cost andemergency borrowing cost The transportation carrying costwould not be considered in this model since it is constant anddoes not affect any decision variables

Owing to the uncertain delivery lead time there are threedifferent delivery cases early arrival delay arrival and arrivalon time The screening process would be implemented whenthe purchaser receives a shipment of goods The defectiveitem would be deducted from the inventory in a singlebatch at the end of the purchaserrsquos 100 screening processFigure 1 has shown two delivery cases for the member whoimplements purchasing

On the left side of Figure 1 0 le 120582119871119894 le 119871119889119894 le 119898119871119894means theearly arrival occurs and results in unexpected holding costThe shadow area (A) presents the unexpected extra inventoryIn this case the holding cost (HCA119894) of member 119894 during 119905119894

is shownThe shadow area (A) presents the unexpected extrainventory caused by early delivery and 119905119894119902119894(1minus0119894)2+(119909119894+119871119889119894minus

119898119871119894)1199021198940119894 represents the expected inventoryIn the right side of Figure 1 119898119871119894 le 119871119889119894 le 120574119871119894 le 119905119894

presents the delay arrival situation When it occurs member119894 should adopt emergency borrowing policy The shadowarea (B) presents the quantity of emergency borrowing Theborrowing quantity from other nearby suppliers outside thesupply chain is 119863119894(120574119871119894minus119898119871119894) units After the delay arrivallot is received the remand quantity of goods 119863119894(120574119871119894 minus 119871119889119894)

Mathematical Problems in Engineering 5

should be given back immediately The borrowed and theremanded units would not be screened and the screeningtime is ((119902119894 minus119863119894(120574119871119894 minus119898119871119894))119902119894)119909119894 during119898119871119894 le 119871119889119894 le 120574119871119894 le 119905119894The holding cost (HCB119894) and the expected borrowing cost(EBC119894) of member 119894 during 119905119894 are shown The holding cost(HCB119894) involves the normal holding cost and the emergencyborrowing holding cost

The unexpected extra inventory of shadow area (A) dueto the early arrivals is as follows

HCA119894 = 119867119901119894 [119902119894 (119898119871119894 minus 119871119889119894) +

119905119894119902119894 (1 minus 0119894)

2

+ (119909119894 + 119871119889119894 minus 119898119871119894) 1199021198940119894] = 119863119894119867119901119894 [119905119894 (119898119871119894 minus 119871119889119894)

+

1199051198941199091198940119894

1 minus 0119894

+

119905119894

2

2

]

(1)

The emergency borrowing holding cost due to delay arrivalsis as follows

119863119894119867119901119894 [

[(119903119871119894 minus 119871119889119894) + (119903119871119894 minus 119898119871119894)] (119871119889119894 minus 119898119871119894)

2

] (2)

Normal holding cost without early or delay arrivals is asfollows

119863119894119867119901119894 [

(119905119894 + 119871119871119894

minus 119871119889119894)2

2

+ (

119905119894 minus (119903119871119894 minus 119898119871119894) (1 minus 0119894)

1 minus 0119894

)1199091198940119894]

HCB119894 = 119867119901119894 [(119861) + (119862)] = 119863119894119867119901119894 [(119903119871119894 minus 119898119871119894)

+ (119871119889119894 minus 119898119871119894)] + 119905119894 (119898119871119894 minus 119871119889119894) +

119905119894

2

2

minus (119903119871119894 minus 119898119871119894)

sdot (1199091198940119894) +

1199051198941199091198940119894

1 minus 0119894

(3)

Nowwe combine (1) and (3) to form the expected holdingcost (EHC119894) of member 119894 as follows

EHC119894 = int

119898119871119894

120582119871119894

HCA119894119891 (119871119889119894) 119889119871119889119894

+ int

119903119871119894

119898119871119894

HCB119894119891 (119871119889119894) 119889119871119889119894 = 119863119894119867119901119894 [119905119894 (119898119871119894

minus 119871119889119894) +

119905119894

2

2

+

1199051198941199091198940119894

1 minus 0119894

+ (119903119871119894 minus 119898119871119894)

sdot [int

119903119871119894

119898119871119894

(119871119889119894 minus 119898119871119894) 119891 (119871119889119894) 119889119871119889119894

minus (1199091198940119894) int

119903119871119894

119898119871119894

119891 (119871119889119894) 119889119871119889119894]]

(4)

The expected borrowing cost (EBC119894) is as follows

EBC119894 = 119862119890119894 int

119903119871119894

119898119871119894

119891 (119871119889119894) 119889119871119889119894

+ 120573119894 int

119903119871119894

119898119871119894

119863119894 (119903119871119894 minus 119898119871119894) 119891 (119871119889119894) 119889119871119889119894

= [119862119890119894 + 120573119894119863119894 (119903119871119894 minus 119898119871119894)] int

119903119871119894

119898119871119894

119891 (119871119889119894) 119889119871119889119894

(5)

The EBC119894 only occurs at time interval [120574119871119894 119898119871119894]119862119890119894 int119903119871119894

119898119871119894119891(119871119889119894)119889119871119889119894 means the ordering cost of emergency

borrowing and 120573119894 int119903119871119894

119898119871119894119863119894(119903119871119894 minus 119898119871119894)119891(119871119889119894)119889119871119889119894 presents the

excepted cost of borrowed unitsDuring the purchasing time interval (119879119901119894) of each mem-

ber it has 119873119901119894 purchasing orders 119899119894119873119901119894 delivery receivingtimes 119899119894119873119901119894119865119894 delivery cost and 119873119901119894119862119904119894 ordering cost Thusthe expected cost (EP119888119894) of member 119894 in purchasing activitiesis shown as follows

EC119901119894 (119899119894 119873119901119894 119898119871119894) = holding cost

+ emergency borrowing cost + ordering cost

+ delivery cost = 119899119894119873119901119894 [EHC119894 + EBC119894] + 119873119901119894119862119904119894

+ 119899119894119873119901119894119865119894 = 119899119894119873119901119894119863119894119867119901119894 [119905119894 (119898119871119894 minus 119871119889119894) +

1199051198942

2

+

1199051198941199091198940119894

1 minus 0119894

+ (119903119871119894 minus 119898119871119894)]

sdot [int

119903119871119894

119898119871119894

(119871119889119894 minus 119898119871119894) 119891 (119871119889119894) 119889119871119889119894

minus (1199091198940119894) int

119903119871119894

119898119871119894

119891 (119871119889119894) 119889119871119889119894] + 119899119894119873119901119894 [119862119890119894

+ 120573119894119863119894 (119903119871119894 minus 119898119871119894)] int

119903119871119894

119898119871119894

119891 (119871119889119894) 119889119871119889119894 + 119865119894 + 119873119901119894119862119904119894

(6)

32 The Cost of Member 119894 Related to Manufacturing ActivitiesIn this model member 119894 + 1 adopts JIT manufacturing toproduce goods to member 119894 There are 119873119898119894+1 productionruns during 119879119901119894 The start time of each production can bedetermined by counting backward 119902119894minus1119875119894 time units whenthe first delivery lot is delivered The average inventory ofgoods produced bymember 119894per production run is illustratedby Figure 2 and we can determine this by calculatingthe cumulative time-weighted production quantity (1) (thetrapezoid area) minus the cumulative time-weighted deliveryquantity (2) (the ladder area)

Calculating the cumulative time-weighted productionquantity of member 119894 per production run is equal to calcu-lating a square measure of trapezoid area Another way tocalculate the cumulative time-weighted production quantity

6 Mathematical Problems in Engineering

Time

Cumulative quantity

(2)(2)

(1) (1)

ti

Tpi Tpi

Pi

Tpiminus1NmiTpiminus1Nmi

Qmi

Tpiminus1

Figure 2 Average inventory of goods produced by member 119894 perproduction run

(1) is subtracting the triangle area from the rectangle area asfollows

(1) = [

119879119901119894minus1

119873119898119894

minus (119905119894minus1 minus

119902119894minus1

119901119894

)](

119863119894minus1119879119901119894minus1

119873119898119894

)

minus

1

2

(

119863119894minus1119879119901119894

119875119894119873119898119894

)(

119863119894minus1119879119901119894minus1

119873119898119894

) = (

119879119901119894minus1

119873119898119894

)

2

119863119894minus1 [1

2

+

0119894minus1

119870119894 (1 minus 0119894)

+ (

1

2

minus

1

119870119894 (1 minus 0119894minus1)

)(1 minus

119863119894minus1

119875119894

)]

(7)

During each production run of member i there are 119870119894

deliveries and 119902119894 units of goods delivered per shipment Thetime interval between two adjacent deliveries for member119894 (119905119894) should be longer than or equal to 120574119871119894 Hence thecumulative time-weighted delivery quantity (2) of member 119894

per production run is

(2) = (

119870119894minus1

sum

119897

119897) 119905119894minus1119902119894minus1 = (1 + 2 + sdot sdot sdot + (119870119894minus1)) 119905119894minus1119902119894minus1

= (

119879119901119894minus1

119873119898119894

)

2

119863119894minus1 [1

2

119870119894minus1

119870119894 (1 minus 0119894minus1)

]

(8)

Consequently the average inventory (AI119894) of member 119894

during 119879119901119894minus1 can be calculated by subtracting area (2) fromarea (1) as follows

AI119894 = 119873119898119894 (1) minus (2) = 119873119898119894 (

119879119901119894minus1

119873119898119894

)

2

119863119894minus1 [1

2

+

0119894

119870119894 (1 minus 0119894)

+ (

1

2

minus

1

119870119894 (1 minus 0119894minus1)

)(1 minus

119863119894

119875119894

)]

minus

119879119901119894minus1

119873119898119894

119863119894minus1 [1

2

119870119894minus1

119870119894 (1 minus 0119894minus1)

] = (

119879119901119894minus12

119873119898119894

)

sdot 119863119894minus1 [1

2

(

119870119894 + 0119894minus1 minus 1

119870119894 (1 minus 0119894)

) + (

1

2

minus

1

119870119894 (1 minus 0119894minus1)

)

sdot (1 minus

119863119894minus1

119875119894

)]

(9)

Thus the production cost of member 119894 (EC119898119894) during 119879119901119894 is

EC119898119894 (119873119898119894119870119894) = 119873119898119894119862119898119894 + 119867119898119894AI119894 = 119873119898119894119862119898119894

+ 119867119898119894 (

119879119901119894minus12

119873119898119894

)119863119894minus1 [1

2

(

119870119894 + 0119894minus1 minus 1

119870119894 (1 minus 0119894)

)

+ (

1

2

minus

1

119870119894 (1 minus 0119894minus1)

)(1 minus

119863119894minus1

119875119894

)]

(10)

33 The Joint Cost of Member 119894 in Purchasing Activities andMember 119894 + 1 in Manufacturing Activities With the abovediscussion we have inferred the expected cost of member 119894

in purchasing andmanufacturing respectivelyTherefore thejoint cost of member 119894 in purchasing activities and member119894 + 1 in manufacturing activities can be obtained easily Wesubstituted 119899119894119873119901119894 = 119870119894+1119873119898119894+1 into (6) and (10) and gainedthe joint cost of members 119894 and 119894 + 1 as

119862119894119894+1 (119873119901119894 119898119871119894 119873119898119894+1 119870119894+1) = EC119901119894 + EC119898119894+1

= 119873119898119894+1119870119894+1 [EHC119894 + EBC119894] + 119873119901119894119862119901119894

+ 119873119898119894+1119862119898119894+1 + 119867119898119894+1AI119894+1

= 119873119898119894+1119870119894+1 119863119894119867119901119894 [119905119894 (119898119871119894 minus 120583119871119889119894)] +

1199051198942

2

+

1199051198941199091198940119894

1 minus 0119894

+ (119903119871119894 minus 119898119871119894)

sdot [int

119903119871119894

119898119871119894

(119871119889119894 minus 119898119871119894) 119891 (119871119889119894) 119889119871119889119894

minus (1199091198940119894) int

119903119871119894

119898119871119894

119891 (119871119889119894) 119889119871119889119894] + [119862119890119894 + 120573119894119863119894 (119903119871119894

minus 119898119871119894)] [int

119903119871119894

119898119871119894

119891 (119871119889119894) 119889119871119889119894 + 119865119894] + 119873119901119894119862119904119894

+ 119873119898119894119862119898119894 + 119867119898119894 (

119879119901119894minus1

2

119873119898119894

)

sdot 119863119894minus1 [1

2

(1 minus

119870119894 + 0119894minus1 minus 1

119870119894 (1 minus 0119894)

)

+ (

1

2

minus

1

119870119894 (1 minus 0119894minus1)

)(1 minus

119863119894minus1

119875119894

)]

(11a)

Mathematical Problems in Engineering 7

Substitute 119905119894 = 119879119901119894119870119894+1119873119898119894+1 into (11a) Consider

119862119894119894+1 (119873119901119894 119898119871119894 119873119898119894+1 119870119894+1) = V119894119879119901119894 (119898119871119894 minus 120583119871119889119894)

+ V119894119879119901119894119909119894120588119894 +119879119901119894

2

119870119894+1119873119898119894+1

(Ω119894 minus

119867119898119894+1119863119894119870119894+1120588119894

2

)

+

119879119901119894

2

119873119898119894+1

120579119894 + 119870119894+1119873119898119894+1 [(119903119871119894 minus 119898119871119894)

sdot V119894 int119903119871119894

119898119871119894

(119871119889119894 minus 119898119871119894) 119891 (119871119889119894) 119889119871119889119894

+ 120576119894 int

119903119871119894

119898119871119894

119891 (119871119889119894) 119889119871119889119894 + 119865119894] + 119873119901119894119862119904119894

+ 119873119898119894+1119862119898119894+1

(11b)

where

V119894 = 119867119901119894119863119894

120588119894 =0119894

1 minus 0119894

Ω119894

=

119863119894

2

[119867119901119894 + 119867119898119894+1 (1 minus (

2

1 minus 0119894

)(1 minus

119863119894

119875119894+1

))]

120576119894 = 119862119890119894 + (120573119894119863119894 minus V1198941199091198940119894) (119903119871119894 minus 119898119871119894)

120579119894 =

119867119898119894+1119863119894

2

(1 minus

119863119894

119875119894+1

)

(12)

Since 119879119901119894 = 119876119898119894minus1119875119894 = (119863119894minus1119873119898119894+1119875119894)119879119901119894minus1 119879119901119894 can beexpressed further as

119879119901119894 = (

120593119894

prod119894

119897=1119873119898119897

)1198791199011 (13)

where

120593119894 =

1 if 119894 = 1

119894

prod

119897=2

(

119863119894minus1

119875119894

) if 119894 = 2 3 119878 minus 1

(14)

All the 119879119901119894 in function (11b) could be replaced as follows

119862119894119894+1 (119873119901119894 119898119871119894 119873119898119894+1 119870119894+1) = V119894(120593119894

prod119894

119897=1119873119898119897

)

sdot 1198791199011 (119898119871119894 minus 120583119871119889119894) + V119894(120593119894

prod119894

119897=1119873119898119897

)1198791199011119909119894120588119894

+

(120593119894prod119894

119897=1119873119898119897) 1198791199011

2

119870119894+1119873119898119894+1

(Ω119894 minus

119867119898119894+1119863119894119870119894+1120588119894

2

)

+

(120593119894prod119894

119897=1119873119898119897) 1198791199011

2

119873119898119894+1

120579119894

+ 119870119894+1119873119898119894+1 [(119903119871119894 minus 119898119871119894)

sdot V119894 int119903119871119894

119898119871119894

(119871119889119894 minus 119898119871119894) 119891 (119871119889119894) 119889119871119889119894

+ 120576119894 int

119903119871119894

119898119871119894

119891 (119871119889119894) 119889119871119889119894 + 119865119894] + 119873119901119894119862119904119894

+ 119873119898119894+1119862119898119894+1

(11c)

There are119873119898119894+1 purchasing times of member 119894+1 during119879119901119894 According to (11c) the joint total cost function includingpurchasing and manufacturing activities of the serial 119878-echelon integrated JIT inventory model is presented in

119878minus1

sum

119894=1

V1198941205931198941198791199011 (119898119871119894 minus 120583119871119889119894) + V1198941205931198941198791199011119909119894120588119894

+

(1205931198941198791199011)

2

119870119894+1prod119894

119897=1119873119898119897

(Ω119894 minus

119867119898119894+1119863119894119870119894+1120588119894

2

)

+

1198791199011

2

prod119894

119897=1119873119898119897

120579119894 + (

119894

prod

119897=1

119873119898119897)119870119894+1119873119898119894+1 [(119903119871119894 minus 119898119871119894)

sdot V119894 int119903119871119894

119898119871119894

(119871119889119894 minus 119898119871119894) 119891 (119871119889119894) 119889119871119889119894

+ 120576119894 int

119903119871119894

119898119871119894

119891 (119871119889119894) 119889119871119889119894 + 119865119894] + (

119894

prod

119897=1

119873119898119897)119873119901119894119862119904119894

+ (

119894

prod

119897=1

119873119898119897)119862119898119894+1

(15)

The delivery time constrains 0 le 120582119871119894 le 119898119871119894 le 120574119871119894 le

119905119894 119894 = 1 2 119878 minus 1 and variables constrains are consideredin the inventory model Eventually the 119878-echelon integratedJIT model can be expressed as

Minimize 1198691198621119878 (119873119901 119898119871 119873119898 119870)

=

119878minus1

sum

119894=1

V1198941205931198941198791199011 (119898119871119894 minus 120583119871119889119894) + V1198941205931198941198791199011119909119894120588119894 +(1205931198941198791199011)

2

119870119894+1prod119894

119897=1119873119898119897

(Ω119894 minus

119867119898119894+1119863119894119870119894+1120588119894

2

) +

11987911990112

prod119894

119897=1119873119898119897

120579119894

8 Mathematical Problems in Engineering

+(

119894

prod

119897=1

119873119898119897)119870119894+1119873119898119894+1 [(119903119871119894 minus 119898119871119894) V119894 int119903119871119894

119898119871119894

(119871119889119894 minus 119898119871119894) 119891 (119871119889119894) 119889119871119889119894 + 120576119894 int

119903119871119894

119898119871119894

119891 (119871119889119894) 119889119871119889119894 + 119865119894]

+(

119894

prod

119897=1

119873119898119897)119873119901119894119862119904119894 + (

119894

prod

119897=1

119873119898119897)119862119898119894+1

Subject to 0 le 120582119871119894 le 119898119871119894 le 120574119871119894 le 119905119894 for 119894 = 1 2 119878 minus 1

119870119894+1 le1

0119894

minus 1

119873119901119894 119870119894+1 119873119898119894+1 are all integers for 119894 = 1 2 119878 minus 1

(16)

4 Numeral Experiment and Analysis

In this section a numeral experiment is proposed based onproper parameter settings All results in this research wereperformed on a PC Programs were written in Matlab 2012bAn efficiency comparison between Lingo PSO algorithmand Ant Colony Algorithm was proposed as well Further-more we would like to discuss how production runs anddefective rate influences whole system cost

41 Experiment Results of Ant Colony Algorithm ParticleSwarm Optimization and Lingo With appropriate param-eters settings we may build a 7-echelon supply chain inte-grated JIT inventory model with uncertain delivery leadtime and unreliable quality of received items Assume thepurchasing time interval for first purchaser in years (1198791199011) is12 years The relevant data of purchasing and manufacturingactivities are shown in Tables 1 and 2

The objective function and constraint function are shownbelow Consider

1198691198621119878 (119873119901 119898119871 119873119898 119870)

=

119878minus1

sum

119894=1

V1198941205931198941198791199011 (119898119871119894 minus 120583119871119889119894) + V1198941205931198941198791199011119909119894120588119894 +(1205931198941198791199011)

2

119870119894+1prod119894

119897=1119873119898119897

(Ω119894 minus

119867119898119894+1119863119894119870119894+1120588119894

2

) +

1198791199011

2

prod119894

119897=1119873119898119897

120579119894

+(

119894

prod

119897=1

119873119898119897)119870119894+1119873119898119894+1 [(119903119871119894 minus 119898119871119894) V119894 int119903119871119894

119898119871119894

(119871119889119894 minus 119898119871119894) 119891 (119871119889119894) 119889119871119889119894 + 120576119894 int

119903119871119894

119898119871119894

119891 (119871119889119894) 119889119871119889119894 + 119865119894]

+(

119894

prod

119897=1

119873119898119897)119873119901119894119862119904119894 + (

119894

prod

119897=1

119873119898119897)119862119898119894+1

Subject to 0 le 120582119871119894 le 119898119871119894 le 120574119871119894 le 119905119894 for 119894 = 1 2 119878 minus 1

119870119894+1 le1

0119894

minus 1

119873119901119894 119870119894+1 119873119898119894+1 are all integers for 119894 = 1 2 119878 minus 1

(17)

The goal of this research is to find out the optimal solutionof the 7-echelon inventory model via the above functions andACA and compare the efficiency between Lingo ACA andPSO

As Table 3 shows though all the three methods werecapable of finding the optimal solution for SMEIJI problemACA was obviously better than PSO and Lingo in efficiencyACA has shown fastest CPU searching time and least averageiterations among three methods

42 Experiments of Performance and 119878-Echelon In this sec-tion we would like to discuss the relationship between CPU

searching time and echelon number among three methodswe also discussed the relationship between average iterationsand echelon number

As Figure 3 shows the CPU searching time wouldincrease with the number of echelons due to the compli-cated calculation However because of the well-organizedconstraint function and good initial solution ACA still keptlower CPU searching time than the other two methods Onthe other hand Figure 4 shows that ACArsquos average iterationswere the least among three methods At the beginning of themultiechelon all the three algorithms need a great amountof iterations to search the optimal solution After calculating

Mathematical Problems in Engineering 9

Table 1 The relevant data of purchasing activities

Purchasing activityEchelon 119863119894 119862119904119894 119867119901119894 119862119890119894 120573119894 120582119871119894 120574119871119894 120583119871119889119894 120590119871119889119894 119865119894 119883119894 0119894

1 18 w 25 28 29 32 0005 0015 0009 0002 70 0005 0032 20w 28 22 32 26 0004 001 00064 00012 50 0003 0013 24w 32 17 38 21 0003 0009 00054 00012 48 0003 0044 30w 36 12 42 16 0004 0012 00072 00016 55 0004 0055 35w 35 08 37 12 0006 0016 001 0002 80 0005 0016 40w 38 06 40 09 0004 0014 0008 0002 65 0005 0027 mdash mdash mdash mdash mdash mdash mdash mdash mdash mdash mdash mdash

Table 2 The relevant data of manufacturing activities

Manufacturing activitiesEchelon 119875119894 119862119898119894 119867119898119894

1 mdash mdash mdash2 200000 280 253 240000 200 194 300000 320 145 350000 360 106 400000 380 097 450000 420 05

1 2 3 4 5 6 70

05

1

15

2

25

3

ACAPSO

Lingo

Figure 3 Experiment results of CPU time and echelon number

repeatedly the trend of the average iterations would come flatwithout extreme variety

Nevertheless although ACA seems better than the othertwo algorithms we could not ensure that the performancewould still be great when applied to higher echelon inventorymodel Hence a 15-echelon model has proposed to observethe performance again The following table displayed theparameters of 15-echelon model

To evaluate the performance of ACA PSO and Lingo forthe 15-echelon inventorymodel we applied the parameters inTables 4 and 5 to solve the 15-echelon inventory problem

According to Table 6 all three algorithms were still ableto find out the optimal solution of the 15-echelon inventorymodel Once the echelon increases the complexity of the

1 2 3 4 5 6 70

500

1000

1500

2000

2500

3000

3500

4000

ACAPSO

Lingo

Figure 4 Experiment results of average iterations and echelonnumber

model increases as well However the performance of ACAwas still better than the other two algorithms

There are several advantages of ACA as follows

(1) Applying the positive and negative feedback enablesthe process of searching optimization to converge atthe end which means the results would close to theoptimal solutions gradually

(2) Every single ant is able to change the surroundingenvironment through releasing pheromone and beingconscious of the variety of environment which makesthem communicate with each other

(3) Due to distributed computing all the individuals startto calculate simultaneously during the searching pro-cess which enhances the efficiency of the calculationability

(4) For the proposed algorithm it is difficult to stick inthe partial optimization instead it is always easy tofind the best solution

To sum up ACA was proposed to solve the complicatedmathematical problem we also have proved that ACA wasmore efficient than PSO and Lingo These are the new majorcontributions of the present paper

43 The Sensitive Analysis of SMEIJI In this section thenumber of production runs 119873119898119894 and the defective rate of

10 Mathematical Problems in Engineering

Table 3 Comparisons between three methods in SMEIJI problem

Echelon ACA PSO Lingo119873119901119894 119898119871119894 119870119894 119873119898119894 119873119901119894 119898119871119894 119870119894 119873119898119894 119873119901119894 119898119871119894 119870119894 119873119898119894

1 1 00147 mdash mdash 1 00147 mdash mdash 1 00147 mdash mdash2 1 000987 11 1 1 000987 11 1 1 000987 11 13 1 000889 13 1 1 000889 13 1 1 000889 13 14 1 001184 11 1 1 001184 11 1 1 001184 11 15 1 001571 8 1 1 001571 8 1 1 001571 8 16 1 001378 5 1 1 001378 5 1 1 001378 5 17 mdash mdash 5 1 mdash mdash 5 1 mdash mdash 5 1Optimal cost 89166 89166 89166CPU time 0655 1744 23Average iterations 350 1278 3758

Table 4 The relevant data of purchasing activities of 15-echelon model

Purchasing activityEchelon 119863119894 119862119904119894 119867119901119894 119862119890119894 120573119894 120582119871119894 120574119871119894 120583119871119889119894 120590119871119889119894 119865119894 119883119894 0119894

1 18 w 25 28 29 32 0005 0015 0009 0002 70 0005 0032 20w 28 22 32 26 0004 001 00064 00012 50 0003 0013 24w 32 17 38 21 0003 0009 00054 00012 48 0003 0044 30w 36 12 42 16 0004 0012 00072 00016 55 0004 0055 35w 35 08 37 12 0006 0016 001 0002 80 0005 0016 40w 38 06 40 09 0004 0014 0008 0002 65 0005 0027 42w 40 05 28 12 0005 0013 001 0002 70 0005 0038 45w 42 07 35 15 0004 0009 0009 00018 50 0004 0029 50w 44 10 42 10 0003 0010 0005 00024 55 0004 00210 52w 46 15 35 22 0005 0012 00062 00025 60 0005 00211 56w 48 20 49 25 0006 0009 001 00026 65 0003 00312 60w 50 22 40 14 0004 0016 0012 0002 48 0005 00413 62w 55 17 35 15 0007 0015 0009 00018 72 0004 00414 65w 56 12 32 16 0003 0014 00078 0002 70 0006 00315 mdash mdash mdash mdash mdash mdash mdash mdash mdash mdash mdash mdash

Table 5The relevant data of manufacturing activities of 15-echelonmodel

Manufacturing activitiesEchelon 119875119894 119862119898119894 119867119898119894

1 mdash mdash mdash2 200000 280 253 240000 200 194 300000 320 145 350000 360 106 400000 380 097 450000 420 058 480000 440 109 500000 420 1210 520000 400 1511 540000 450 2012 580000 440 2213 620000 420 1814 640000 460 1515 660000 380 16

receiving shipment 0119894 were considered to make sensitiveanalysis and observe the influence on the optimal total jointcost The defective rate and production runs were changedby 0119894(1 + 119877) and 119873119898119894(1 + 119877) where the values of 119877 are0 50 100 200 300 400

Figure 5 illustrated that defective rate was more sensitivethan production runs under the situation that productionruns were one-time production except first echelon In thatcase the higher the defective ratewas the higher the total costwas Accordingly inspecting goods cautiously before deliveryand reducing the defective rate would lower total cost of thewhole supply chainHowever if relaxing that assumption thissensitive analysis would result in different outcomes

Figure 6 has shown that production runs were more sen-sitive than defective rate when relaxing one-time productionassumption In this case the higher the production runswerethe higher the total cost was Hence reducing productionruns should play an essential role in the whole supply chainOnce the one batch production policy was able to satisfy thedemand from the buyer the total cost of whole supply chainwill go down extremely

Mathematical Problems in Engineering 11

Table 6 Comparisons between three ways of 15-echelon inventory model

ACA PSO LingoEchelon 119873119901119894 119898119871119894 119870119894 119873119898119894 119873119901119894 119898119871119894 119870119894 119873119898119894 119873119901119894 119898119871119894 119870119894 119873119898119894

1 1 00147 mdash mdash 1 00147 mdash mdash 1 00147 mdash mdash2 1 000987 11 1 1 000987 11 1 1 000987 11 13 1 000889 13 1 1 000889 13 1 1 000889 13 14 1 001184 11 1 1 001184 11 1 1 001184 11 15 1 001571 8 1 1 001571 8 1 1 001571 8 16 1 001378 5 1 1 001378 5 1 1 001378 5 17 1 001451 5 1 1 001451 5 1 1 001451 5 18 1 001287 5 1 1 001287 5 1 1 001287 5 19 1 00154 7 1 1 00154 7 1 1 00154 7 110 1 001432 9 1 1 001432 9 1 1 001432 9 111 1 001352 7 1 1 001352 7 1 1 001352 7 112 1 00162 8 1 1 00162 8 1 1 00162 8 113 1 001571 10 1 1 001571 10 1 1 001571 10 114 1 001611 5 1 1 001611 5 1 1 001611 5 115 mdash mdash 5 1 mdash mdash 5 1 mdash mdash 5 1Optimal cost ($) 231254 231254 231254CPU time (s) 1725 6621 58215Average iterations 700 2883 8241

Defective rateProduction runs

400100 200 300 5000 500

20000

40000

60000

80000

100000

120000

140000

160000

180000

Figure 5 The sensitive analysis between defective rate and produc-tion runs

5 Conclusions

In summary this research proposed an integrated JIT inven-tory model to solve the partial optimization problem ofpurchaser andmanufacturer finding the optimal solution forthe whole supply chain Furthermore a multiechelon supplychainwith uncertain delivery lead time and unreliable qualityhas been considered to fit the real supply chain environ-ment Besides ACA was proposed to solve the complicatedmathematical problem we also have proved that ACA was

0 50 100 200 300 400 500

Defective rateProduction runs

0

50000

100000

150000

200000

250000

300000

350000

400000

Figure 6 The sensitive analysis between defective rate and produc-tions runs with relaxing assumption

more efficient than PSO and Lingo These are the new majorcontributions of the present paper

Finally there are a great deal of helpful directions forenterprises and researchers (1)The variety of conditions thatoccur in real supply chain can be considered in the modelFor instance time value of cash problem and deterioration(2)The warehousing and storage can be considered in supplychain which make the model more complex to solve (3)The model contains only one member in each echelon amultimember in each echelon should be considered to fit

12 Mathematical Problems in Engineering

real situation in supply chain (4) Some real cases andordata can be applied to our proposed model in the followingworks

Futureworkwill hopefully involve the real-world concernto our research which will be our main target In addition we

do hope our research can be extended intomore considerableareas and more considerable variables

Appendix

The Hessian matrix 119865 of 1198691198621119878(119873119901 119898119871 119873119898 119870) for given 119873119901119894

equal to 1 can be shown as

119865 =

[

[

[

[

[

[

[

[

[

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

1205971198731198981198942

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119873119898119894120597119898119871119894

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119873119898119894120597119870119894

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119898119871119894120597119873119898119894

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

1205971198981198711198942

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119898119871119894120597119870119894

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119870119894120597119873119898119894

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119870119894120597119898119871119894

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

1205971198701198942

]

]

]

]

]

]

]

]

]

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

1205971198731198981198942

=

2

1198731198981198943(120579120593119879

2+

1205931198792(1198671198631198701205932 + Ω)

119870

)

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

1205971198981198711198942

= 119870119873[(

119890minus119898120583

1205832

+ 2 (119890minus119898120583

minus 119890minus120574120583

) +

119890(minus119898120583)(minus119898+120583)

120583

) 120592]

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

1205971198701198942

=

2

1198701198943(120579120593119879

2+

1205931198792(1198671198631198701205932 + Ω)

119873

)

12059721198691198621119904 (119873119898119894 119898119871119894 119870119894)

120597119898119871119894120597119873119898119894

=

12059721198691198621119904 (119873119898119894 119898119871119894 119870119894)

120597119873119898119894120597119898119871119894

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119870119894120597119873119898119894

=

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119873119898119894120597119870119894

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119870119894120597119898119871119894

=

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119898119871119894120597119870119894

100381610038161003816100381611986511

1003816100381610038161003816=

1205972119865

1205971198731198981198942

=

2

1198731198981198943(120579120593119879

2+

1205931198792(1198671198631198701205932 + Ω)

119870

) gt 0

100381610038161003816100381611986522

1003816100381610038161003816= (

1205972119865

1205971198731198981198942)(

1205972119865

1205971198981198711198942) minus [

1205972119865

120597119873119898119894120597119898119871119894

]

2

=

2

1198731198981198943(120579120593119879

2+

1205931198792(1198671198631198701205932 + Ω)

119870

)

1205972119865

1205971198981198711198942minus (

1205972119865

120597119873119898119894120597119898119871119894

)

2

2

1198733119898119894

(1205791205931198792+

119861

119870

)[119870119873(

119890minus119898120576120583

1205832

+ (2119909 +

119890(minus119898120583)119909119910

120583

120592))] + 1198702[

119890minus119898120576120583

120583

+ 120592 (119909119910 minus 119860)] gt 0

100381610038161003816100381611986533

1003816100381610038161003816= (

1205972119865

1205971198731198981198942

1205972119865

1205971198981198711198942

1205972119865

1205971198701198942) minus [(

1205972119865

120597119873119898119894120597119870119894

)

2

(

1205972119865

1205971198981198711198942)]

=

2

1198731198981198943(120579120593119879

2+

119861

119870

)[119870119873(

119890minus119898120576120583

1205832

+ 2(119909 +

119890(minus119898120603)119909119910

120583

) 120592)]

2

1198703

119861

119873

minus [119865 + 119909120576 + 119860120592119910 minus

1205791205931198792

1198732

+

119861

2 (119870119873)2]

[119870119873(

119890minus119898120576120583

1205832

+ (2119909 +

119890minus119898120576120583

120583

) 120592)] gt 0

(A1)

Mathematical Problems in Engineering 13

As a result the matrix F has proved to be a positivedefinite matrix where |11986511| |11986522| and |11986533| are all positiveTherefore theminimumvalue exists in 1198691198621119878(119873119901119898119871 119873119898 119870)Furthermore we can find the optimization in the proposedmodel

Competing Interests

The authors declare that they have no competing interests

References

[1] S K Goyal ldquoAn integrated inventory model for a singlesupplier-single customer problemrdquo International Journal ofProduction Research vol 15 no 1 pp 107ndash111 1977

[2] A Banerjee ldquoA joint economic-lot-sizemodel for purchaser andvendorrdquo Decision Sciences vol 17 no 3 pp 292ndash311 1986

[3] S K Goyal ldquoldquoA joint economic-lot-size model for purchaserand vendorrdquo a commentrdquo Decision Sciences vol 19 no 1 pp236ndash241 1988

[4] Y Seo ldquoControlling general multi-echelon distribution supplychainswith improved reorder decision policy utilizing real-timeshared stock informationrdquo Computers amp Industrial Engineeringvol 51 no 2 pp 229ndash246 2006

[5] H N Chiu and H L Huang ldquoA Multi-echelon integratedJIT inventory model using the time buffer and emergencyborrowing policies to deal with random delivery lead timesrdquoInternational Journal of Production Research vol 41 no 13 pp2911ndash2931 2003

[6] J Sadeghi S M Mousavi S T A Niaki and S SadeghildquoOptimizing a multi-vendor multi-retailer vendor managedinventory problem two tuned meta-heuristic algorithmsrdquoKnowledge-Based Systems vol 50 pp 159ndash170 2013

[7] S S Sana ldquoA production-inventory model of imperfect qualityproducts in a three-layer supply chainrdquo Decision Support Sys-tems vol 50 no 2 pp 539ndash547 2011

[8] H N Soni and K A Patel ldquoOptimal strategy for an integratedinventory system involving variable production and defectiveitems under retailer partial trade credit policyrdquoDecision SupportSystems vol 54 no 1 pp 235ndash247 2012

[9] R J Tersine Principles of Inventory andMaterials ManagementNorthHolland AmsterdamTheNetherlands 3rd edition 1988

[10] C J Liao and C H Shyu ldquoAn analytical determination of leadtime with normal demandrdquo International Journal of Operationsamp Production Management vol 11 no 9 pp 72ndash78 1991

[11] M Ben-Daya and A Raouf ldquoInventory models involving leadtime as a decision variablerdquo Journal of the Operational ResearchSociety vol 45 no 5 pp 579ndash582 1994

[12] S M J Mirzapour Al-e-Hashem A Baboli and Z Sazvar ldquoAstochastic aggregate production planningmodel in a green sup-ply chain considering flexible lead times nonlinear purchaseand shortage cost functionsrdquo European Journal of OperationalResearch vol 230 no 1 pp 26ndash41 2013

[13] C Chandra and J Grabis ldquoInventorymanagementwith variablelead-time dependent procurement costrdquo Omega vol 36 no 5pp 877ndash887 2008

[14] J-W Wu and H-Y Tsai ldquoMixture inventory model with backorders and lost sales for variable lead time demand withthe mixtures of normal distributionrdquo International Journal ofSystems Science Principles and Applications of Systems andIntegration vol 32 no 2 pp 259ndash268 2001

[15] M A Hoque ldquoA vendorndashbuyer integrated productionndashinventory model with normal distribution of lead timerdquoInternational Journal of Production Economics vol 144 no 2pp 409ndash417 2013

[16] F Altiparmak M Gen L Lin and T Paksoy ldquoA geneticalgorithm approach for multi-objective optimization of supplychain networksrdquo Computers and Industrial Engineering vol 51no 1 pp 196ndash215 2006

[17] J Sadeghi S Sadeghi and S T Niaki ldquoA hybrid vendormanaged inventory and redundancy allocation optimizationproblem in supply chain management an NSGA-II with tunedparametersrdquo Computers and Operations Research vol 41 pp53ndash64 2014

[18] X Li F Baki P Tian and B A Chaouch ldquoA robust block-chainbased tabu search algorithm for the dynamic lot sizing problemwith product returns and remanufacturingrdquoOmega vol 42 no1 pp 75ndash87 2014

[19] Z Chen and B R Sarker ldquoAn integrated optimal inventorylot-sizing and vehicle-routing model for a multisupplier single-assembler system with JIT deliveryrdquo International Journal ofProduction Research vol 52 no 17 pp 5086ndash5114 2014

[20] A R Nia M Hemmati Far and S T A Niaki ldquoA fuzzy vendormanaged inventory of multi-item economic order quantitymodel under shortage an ant colony optimization algorithmrdquoInternational Journal of Production Economics vol 155 pp 259ndash271 2014

[21] S M Hatefi and F Jolai ldquoRobust and reliable forward-reverselogistics network design under demand uncertainty and facilitydisruptionsrdquo Applied Mathematical Modelling vol 38 no 9-10pp 2630ndash2647 2014

[22] R H Ma ldquoCross warehouse scheduling of logistics distributionand cycle re-claimer based on heuristics algorithmrdquo AppliedMechanics and Materials vol 644ndash650 pp 2606ndash2610 2014

[23] M Hashim L Yao A H Nadeem M Nazim and M NazamldquoMulti-objective optimization model for supplier selectionproblem in fuzzy environmentrdquo Advances in Intelligent Systemsand Computing vol 281 pp 1201ndash1213 2014

[24] Q-M Hu and Z-H Hu ldquoA stochastic programming modelfor hub-and-spoke network with uncertain flowsrdquo InternationalJournal of Industrial and Systems Engineering vol 21 no 3 pp302ndash319 2015

[25] S A Torabi J Namdar S M Hatefi and F Jolai ldquoAn enhancedpossibilistic programming approach for reliable closed-loopsupply chain network designrdquo International Journal of Produc-tion Research vol 54 no 5 pp 1358ndash1387 2016

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

materials production rate and unit production cost and idletimes in different sectors on collaborating marketing systemSoni and Patel [8] investigated an integrated inventory modelwith variable production rate and price-sensitive demandrate under two-level trade credit This study attempted tooffer a best policy for retail price replenishment cycle andthe number of shipments from the supplier to the retailerin one production run An algorithm was also designed todetermine the optimum solution of their proposed model

Lead time usually consists of the following componentsorder preparation order transit supplier lead time deliverytime and set-up time (Tersine [9]) this makes lead timedifficult to estimate precisely In real life firms cannot keepinventory in adequate level if lead time is uncertain it maycause unnecessary costs for firms Furthermore uncertainlead time also increases the probability of shortage firms cannot satisfy customers immediately if shortage occurs Henceuncertain lead time has much to do with inventory leveland customer service level Liao and Shyu [10] presented aprobabilistic inventorymodel they assumed that the demandfollows normal distribution and the lead time consists of 119899

components each having a different cost for reduced leadtime Ben-Daya and Raouf [11] considered both the leadtime and the order quantity as decision variables based onLiao and Shyursquos [10] model Mirzapour Al-e-Hashem et al[12] assumed an interrelationship between lead time andtransportationmode the shorter the lead times are the moreexpensive transportationwill be Accordingly they developeda stochastic programming approach to solve a multiperiodmultiproduct multisite aggregate production planning prob-lem in a green supply chain for a medium-term planninghorizon under the assumption of demand uncertainty andflexible lead times Chandra and Grabis [13] indicated thatshort lead time could enhance the service level and lowerinventory level effectively but short lead time also causedhighly order cost

However demands during the lead time of the differentcustomers are not identical and the demand distribution foreach customer is not the same either Therefore we cannotapply single distribution to describe the demands during thelead time Accordingly Wu and Tsai [14] considered the leadtime demand with the mixture of normal distributions andthe fixed back-order rate Hoque [15] developed a vendor-buyer integrated production-inventory model following nor-mal distribution of lead time Then he derived an optimalsolution technique to themodel to obtainminimumexpectedjoint total cost that follows development of the solutionalgorithm

In real life supply chains always have many membersin multiechelon however multiechelon inventory problemsare too complicated to solve by using traditional methodsHence we would like to discuss some heuristic algorithm inour proposed model to solve the multiechelon problems anddetermine the optimal solutions Altiparmak [16] proposed anew solution procedure based on genetic algorithms to findthe set of Pareto-optimal solutions for multiobjective SCNdesign problem To deal with multiobjective design problemand enable the decision makers to evaluate a greater numberof alternative solutions two different weight approaches were

implemented in the proposed solution procedure Sadeghiet al [17] developed a biobjective vendor-managed inventorymodel in a supply chain with one vendor (producer) andseveral retailers The aim was to find the order size thereplenishment frequency of retailers optimal traveling tourfrom the vendor to retailers and the number of machinesso as the total chain cost was minimized while the systemreliability of producing the item was maximized Li et al[18] studied the dynamic lot-sizing problem with productreturns and remanufacturing (DLRR)which is to determine aproduction schedule of manufacturing new products andorremanufacturing returns such that demand in each periodwas satisfied and the total cost (set-up cost plus holdingcost of inventory) was minimized To generate a good initialsolution they used a block-chain based method where theplanning horizon was split into a chain of blocks A blockmay contain either a string of manufacturing set-ups or astring of remanufacturing set-ups or both Given the costof each block an initial solution corresponding to a bestcombination of blocks is found by solving a shortest-pathproblem Chen and Sarker [19] established an integratedoptimal inventory lot-sizing and vehicle-routing model for amultisupplier single-assembler systemwith JIT delivery theyapplied ant colony optimization to solve their model Nia etal [20] built a multi-item economic order quantity modelwith shortage under vendor-managed inventory policy in asingle vendor single buyer supply chain they proposed anew modeling to the fuzzy VMI problem with multi-itemsand shortage and employed three metaheuristic algorithms(ACA GA and DE) to solve a FNIP problem Chen andSarker [19] andNia et al [20] both considered the Ant ColonyAlgorithm as a metaheuristic algorithm

In recent years many researchers dedicated themselvesto improving the integrated logistic model with the issues ofuncertainties Some related researches are listed belowHatefiand Jolai [21] proposed a robust and reliable model for anintegrated forward-reverse logistics network design whichsimultaneously takes uncertain parameters and facility dis-ruptions into account Ma [22] constructed the mathematicalprogramming model and proposed two-stage heuristic algo-rithm In addition the taboo search algorithm was designedto improve the initial solution Hashim et al [23] presentedthe study whichmainly investigated amultiobjective supplierselection planning problem in fuzzy environment and theuncertain model is converted into deterministic form bythe expected value measure Q-M Hu and Z-H Hu [24]proposed a reliable closed-loop supply chain network designmodel which accounts for both partial and complete facilitydisruptions as well as the uncertainty in the critical inputdata Torabi et al [25] considered the characteristics of theuncertainty flows And then a stochastic mixed-integer linearprogrammingmodel for designing hub-and-spoke network isestablished based on the capacities of spokes

With the above discussion we would like to establishan integrated vendor-buyer inventory model To fit the reallife we expanded the basic integrated model to a SMEIJImodel with uncertain lead time and imperfect quality Ourproposed model was too complicated to solve if we usetraditional method hence we applied Ant Colony Algorithm

Mathematical Problems in Engineering 3

to solve our SMEIJI model In given example we would liketo compare ACAwith other traditional methods and observehow system cost works

2 Fundamental Assumptions and Notations

A serial inventory system with 119878-echelon where echelon 1 ispurchasing only and echelon 119878 is manufacturing only wasconsidered In the inventory system we proposed there isonly one member in each echelon that plays the purchaserand manufacturer roles simultaneously except echelon 1 andechelon 119878 Notation and assumptions are defined as follows

For member 119894 engaged in purchasing activities where 119894 =

1 2 119878 minus 119897 we have the following

Decision variables are as follows

119873119901119894 number of purchase orders during 119879119898119894 forpurchasing 119894119899119894 number of deliveries per purchase order forpurchasing 119894

119898119871119894 the maximum allowable planned deliverylead time that would not cause shortage 119898119894 =120583119871119863119894+ safety lead time (a real decision variable)

Parameters are as follows

119878 number of echelons in a serial supply chain119879119901119894 length of a purchasing time interval forpurchasing 119894 (in years)119863119894 demand rate of purchasing 119894 (unitsyear)119876119901119894 lot size per purchase order for purchasing 119894

(unitsorder)119902119894 delivery lot size per shipment for purchasing119894 (unitsdelivery)119862119904119894 fixed ordering cost for purchasing 119894

($order)119867119901119894 holding cost per unit of purchased goodsper year for purchasing 119894 ($unityear)119862119890119894 ordering cost of emergency borrowing forpurchasing 119894 ($borrowing)120573119894 borrowing cost per unit per year for purchas-ing 119894 ($unityear)119865119894 fixed delivery cost for purchasing 119894 ($deliv-ery)119871119889119894 delivery lead time from manufacturing 119894 +

1 to purchasing 119894 a nonnegative random vari-able following a probability distribution withexpected delivery lead time 120583119871

119863119894and standard

deviation 120590119871119863119894119905119894 time interval between two adjacent deliveriesfor purchasing 119894119903119894 redelivery point of purchasing 119894 that is analo-gous to a reorder point if stock drops to119877119894 thena delivery notice is issued to manufacturer 119894 + 1119909119894 screening time of 119902119894 amount materials thatwere received by purchaser 119894 (in years)

0119894 defective rate of a shipment that was receivedby purchaser 119894

For member 119894 engaged in manufacturing activities where 119894 =

1 2 119878 minus 119897 we have the following

Decision variables are as follows

119873119898119894 number of production runs during 119879119898119894minus1

for manufacturing 119894119870119894 number of deliveries per production run formanufacturing 119894

Parameters are as follows

119879119898119894 length of a manufacturing time interval formanufacturing 119894 119879119898119894 = 119876119898119894119901119894 (in years)119875119894 production rate for manufacturing 119894

(unitsyear)119876119898119894 lot size per production run formanufactur-ing 119894 (unitsrun)119862119898119894 fixed set-up cost formanufacturing 119894 ($set-up)119867119898119894 holding cost per unit of produced goodsper year for manufacturing 119894 ($unityear)

Additional notations will be introduced if necessary Theassumptions set in this research are as follows

(1) We consider an integrated multiechelon supply chainwhich only contains a member in each echelon Themember in echelon 119894 is denoted as member 119894 where119894 = 1 2 3 119878 A single product is produced by thesupply chain

(2) Except echelons 1 and 119878 purchaser and manufacturerare the two roles that each member played Member1 purchasing activities only is the lowest echelon inthis serial supply chain Member 119878 involves manufac-turing activities only

(3) Member 119894 purchases requiredmaterials frommember119894 + 1 and produces goods to member 119894 minus 1 during 119879119901119894

(4) The demand and production rate for the product areconstant over time The demand rates of member 119894

should be less than the production rate of member119894 + 1 Member 119894 + 1 produces the necessary quantityto satisfy the demand of member 119894

(5) The delivery lead time (119871119889119894) for member 119894 is assumedto follow a probability distribution with a densityfunction of 119871119889119894 = 119891(119871119889119894) where 0 le 120582119871119894 le 119871119889119894 le 120574119871119894

and 119894 = 1 2 119878 minus 1(6) The time interval between two adjacent deliveries

for member 119894 (119905119894) should be longer than or equalto 120574119871119894 The delivery lot crossing makes the problemintractable and it can be prevented by this assump-tion

(7) When a delivery delay occurs an emergency borrow-ing action should be touched off bymember 119894 becausethe shortages are not allowed in JIT environment

4 Mathematical Problems in Engineering

Time

Inventory level ti ti

Di

qi

ri

0iqi

mLi mLi mLixi + Ldi minus mLi

Ldi Ldixi mLi + ti minus Ldi

0 le 120582Li le Ldi le mLi mLi le Ldi le 120574Li le ti

(1 minus 0i)qi

(B)

(A)

Figure 1 Delivery cases for the member who implements purchasing

(8) The required time of borrowing goods from nearbysupplier is ignored

(9) A finite planning horizon for the whole serial supplychain that denoted 119879119901119894 is considered

(10) Defective items in a shipment of materials arereceived by purchaser 119894 with defective rate 0119894 Thenumber of goodmaterialsmust bemore than or equalto the required quantity during the screening time

(11) The screening process is followed by the arrival ofshipments with screening time 119909119894 The length ofscreening time is proportional to the number ofreceived materials

(12) The accumulated number of defective items that weredelivered to the buyer during a production run mustbe less than the quantity of one delivery

(13) When emergency borrowing happened the materialsthat were borrowed and returned are all good items

3 Formulation of the Model

In this integrated multiechelon JIT inventory model thedecision variables are 119899119894 (number of deliveries per purchase)119873119901119894 (number of purchase orders during 119879119901119894) and 119898119871119894 (themaximum allowable planned delivery lead time that wouldnot cause shortage) where 119894 = 1 119878minus1119873119898119894 is the numberof production runs during 119879119898119894 and 119870119894 are delivery times perproduction runs for manufacturer 119894 where 119894 = 2 119878 Thevalue of these decision variables should be determined inorder to minimize the joint total cost

31 The Cost of Member 119894 in Purchasing Activities Fromechelon1 to echelon 119878 minus 1 members adopt JIT purchasing toreplenish goodsThe time buffer policy used on planned lead

time (119898119871119894) and emergency borrowing policy used to deal withuncertain delivery lead time Each order lot size119876119901119894 has beentaken apart into 119899119894 small lots for delivery frequently Duringthe planning horizon 119879119901119894 the total number of deliveriesfor member 119894 is 119899119894 times 119873119901119894 The delivery lead time follows aprobability density distribution 119891(119871119889119894) The 120574119871119894 and 120582119871119894 arethe upper bound and the lower bound of 119871119889119894

The costs are related to purchasing activities of member119894 during 119879119901119894 where 119894 = 1 119878 minus 1 including ordering costholding cost delivery cost transportation carrying cost andemergency borrowing cost The transportation carrying costwould not be considered in this model since it is constant anddoes not affect any decision variables

Owing to the uncertain delivery lead time there are threedifferent delivery cases early arrival delay arrival and arrivalon time The screening process would be implemented whenthe purchaser receives a shipment of goods The defectiveitem would be deducted from the inventory in a singlebatch at the end of the purchaserrsquos 100 screening processFigure 1 has shown two delivery cases for the member whoimplements purchasing

On the left side of Figure 1 0 le 120582119871119894 le 119871119889119894 le 119898119871119894means theearly arrival occurs and results in unexpected holding costThe shadow area (A) presents the unexpected extra inventoryIn this case the holding cost (HCA119894) of member 119894 during 119905119894

is shownThe shadow area (A) presents the unexpected extrainventory caused by early delivery and 119905119894119902119894(1minus0119894)2+(119909119894+119871119889119894minus

119898119871119894)1199021198940119894 represents the expected inventoryIn the right side of Figure 1 119898119871119894 le 119871119889119894 le 120574119871119894 le 119905119894

presents the delay arrival situation When it occurs member119894 should adopt emergency borrowing policy The shadowarea (B) presents the quantity of emergency borrowing Theborrowing quantity from other nearby suppliers outside thesupply chain is 119863119894(120574119871119894minus119898119871119894) units After the delay arrivallot is received the remand quantity of goods 119863119894(120574119871119894 minus 119871119889119894)

Mathematical Problems in Engineering 5

should be given back immediately The borrowed and theremanded units would not be screened and the screeningtime is ((119902119894 minus119863119894(120574119871119894 minus119898119871119894))119902119894)119909119894 during119898119871119894 le 119871119889119894 le 120574119871119894 le 119905119894The holding cost (HCB119894) and the expected borrowing cost(EBC119894) of member 119894 during 119905119894 are shown The holding cost(HCB119894) involves the normal holding cost and the emergencyborrowing holding cost

The unexpected extra inventory of shadow area (A) dueto the early arrivals is as follows

HCA119894 = 119867119901119894 [119902119894 (119898119871119894 minus 119871119889119894) +

119905119894119902119894 (1 minus 0119894)

2

+ (119909119894 + 119871119889119894 minus 119898119871119894) 1199021198940119894] = 119863119894119867119901119894 [119905119894 (119898119871119894 minus 119871119889119894)

+

1199051198941199091198940119894

1 minus 0119894

+

119905119894

2

2

]

(1)

The emergency borrowing holding cost due to delay arrivalsis as follows

119863119894119867119901119894 [

[(119903119871119894 minus 119871119889119894) + (119903119871119894 minus 119898119871119894)] (119871119889119894 minus 119898119871119894)

2

] (2)

Normal holding cost without early or delay arrivals is asfollows

119863119894119867119901119894 [

(119905119894 + 119871119871119894

minus 119871119889119894)2

2

+ (

119905119894 minus (119903119871119894 minus 119898119871119894) (1 minus 0119894)

1 minus 0119894

)1199091198940119894]

HCB119894 = 119867119901119894 [(119861) + (119862)] = 119863119894119867119901119894 [(119903119871119894 minus 119898119871119894)

+ (119871119889119894 minus 119898119871119894)] + 119905119894 (119898119871119894 minus 119871119889119894) +

119905119894

2

2

minus (119903119871119894 minus 119898119871119894)

sdot (1199091198940119894) +

1199051198941199091198940119894

1 minus 0119894

(3)

Nowwe combine (1) and (3) to form the expected holdingcost (EHC119894) of member 119894 as follows

EHC119894 = int

119898119871119894

120582119871119894

HCA119894119891 (119871119889119894) 119889119871119889119894

+ int

119903119871119894

119898119871119894

HCB119894119891 (119871119889119894) 119889119871119889119894 = 119863119894119867119901119894 [119905119894 (119898119871119894

minus 119871119889119894) +

119905119894

2

2

+

1199051198941199091198940119894

1 minus 0119894

+ (119903119871119894 minus 119898119871119894)

sdot [int

119903119871119894

119898119871119894

(119871119889119894 minus 119898119871119894) 119891 (119871119889119894) 119889119871119889119894

minus (1199091198940119894) int

119903119871119894

119898119871119894

119891 (119871119889119894) 119889119871119889119894]]

(4)

The expected borrowing cost (EBC119894) is as follows

EBC119894 = 119862119890119894 int

119903119871119894

119898119871119894

119891 (119871119889119894) 119889119871119889119894

+ 120573119894 int

119903119871119894

119898119871119894

119863119894 (119903119871119894 minus 119898119871119894) 119891 (119871119889119894) 119889119871119889119894

= [119862119890119894 + 120573119894119863119894 (119903119871119894 minus 119898119871119894)] int

119903119871119894

119898119871119894

119891 (119871119889119894) 119889119871119889119894

(5)

The EBC119894 only occurs at time interval [120574119871119894 119898119871119894]119862119890119894 int119903119871119894

119898119871119894119891(119871119889119894)119889119871119889119894 means the ordering cost of emergency

borrowing and 120573119894 int119903119871119894

119898119871119894119863119894(119903119871119894 minus 119898119871119894)119891(119871119889119894)119889119871119889119894 presents the

excepted cost of borrowed unitsDuring the purchasing time interval (119879119901119894) of each mem-

ber it has 119873119901119894 purchasing orders 119899119894119873119901119894 delivery receivingtimes 119899119894119873119901119894119865119894 delivery cost and 119873119901119894119862119904119894 ordering cost Thusthe expected cost (EP119888119894) of member 119894 in purchasing activitiesis shown as follows

EC119901119894 (119899119894 119873119901119894 119898119871119894) = holding cost

+ emergency borrowing cost + ordering cost

+ delivery cost = 119899119894119873119901119894 [EHC119894 + EBC119894] + 119873119901119894119862119904119894

+ 119899119894119873119901119894119865119894 = 119899119894119873119901119894119863119894119867119901119894 [119905119894 (119898119871119894 minus 119871119889119894) +

1199051198942

2

+

1199051198941199091198940119894

1 minus 0119894

+ (119903119871119894 minus 119898119871119894)]

sdot [int

119903119871119894

119898119871119894

(119871119889119894 minus 119898119871119894) 119891 (119871119889119894) 119889119871119889119894

minus (1199091198940119894) int

119903119871119894

119898119871119894

119891 (119871119889119894) 119889119871119889119894] + 119899119894119873119901119894 [119862119890119894

+ 120573119894119863119894 (119903119871119894 minus 119898119871119894)] int

119903119871119894

119898119871119894

119891 (119871119889119894) 119889119871119889119894 + 119865119894 + 119873119901119894119862119904119894

(6)

32 The Cost of Member 119894 Related to Manufacturing ActivitiesIn this model member 119894 + 1 adopts JIT manufacturing toproduce goods to member 119894 There are 119873119898119894+1 productionruns during 119879119901119894 The start time of each production can bedetermined by counting backward 119902119894minus1119875119894 time units whenthe first delivery lot is delivered The average inventory ofgoods produced bymember 119894per production run is illustratedby Figure 2 and we can determine this by calculatingthe cumulative time-weighted production quantity (1) (thetrapezoid area) minus the cumulative time-weighted deliveryquantity (2) (the ladder area)

Calculating the cumulative time-weighted productionquantity of member 119894 per production run is equal to calcu-lating a square measure of trapezoid area Another way tocalculate the cumulative time-weighted production quantity

6 Mathematical Problems in Engineering

Time

Cumulative quantity

(2)(2)

(1) (1)

ti

Tpi Tpi

Pi

Tpiminus1NmiTpiminus1Nmi

Qmi

Tpiminus1

Figure 2 Average inventory of goods produced by member 119894 perproduction run

(1) is subtracting the triangle area from the rectangle area asfollows

(1) = [

119879119901119894minus1

119873119898119894

minus (119905119894minus1 minus

119902119894minus1

119901119894

)](

119863119894minus1119879119901119894minus1

119873119898119894

)

minus

1

2

(

119863119894minus1119879119901119894

119875119894119873119898119894

)(

119863119894minus1119879119901119894minus1

119873119898119894

) = (

119879119901119894minus1

119873119898119894

)

2

119863119894minus1 [1

2

+

0119894minus1

119870119894 (1 minus 0119894)

+ (

1

2

minus

1

119870119894 (1 minus 0119894minus1)

)(1 minus

119863119894minus1

119875119894

)]

(7)

During each production run of member i there are 119870119894

deliveries and 119902119894 units of goods delivered per shipment Thetime interval between two adjacent deliveries for member119894 (119905119894) should be longer than or equal to 120574119871119894 Hence thecumulative time-weighted delivery quantity (2) of member 119894

per production run is

(2) = (

119870119894minus1

sum

119897

119897) 119905119894minus1119902119894minus1 = (1 + 2 + sdot sdot sdot + (119870119894minus1)) 119905119894minus1119902119894minus1

= (

119879119901119894minus1

119873119898119894

)

2

119863119894minus1 [1

2

119870119894minus1

119870119894 (1 minus 0119894minus1)

]

(8)

Consequently the average inventory (AI119894) of member 119894

during 119879119901119894minus1 can be calculated by subtracting area (2) fromarea (1) as follows

AI119894 = 119873119898119894 (1) minus (2) = 119873119898119894 (

119879119901119894minus1

119873119898119894

)

2

119863119894minus1 [1

2

+

0119894

119870119894 (1 minus 0119894)

+ (

1

2

minus

1

119870119894 (1 minus 0119894minus1)

)(1 minus

119863119894

119875119894

)]

minus

119879119901119894minus1

119873119898119894

119863119894minus1 [1

2

119870119894minus1

119870119894 (1 minus 0119894minus1)

] = (

119879119901119894minus12

119873119898119894

)

sdot 119863119894minus1 [1

2

(

119870119894 + 0119894minus1 minus 1

119870119894 (1 minus 0119894)

) + (

1

2

minus

1

119870119894 (1 minus 0119894minus1)

)

sdot (1 minus

119863119894minus1

119875119894

)]

(9)

Thus the production cost of member 119894 (EC119898119894) during 119879119901119894 is

EC119898119894 (119873119898119894119870119894) = 119873119898119894119862119898119894 + 119867119898119894AI119894 = 119873119898119894119862119898119894

+ 119867119898119894 (

119879119901119894minus12

119873119898119894

)119863119894minus1 [1

2

(

119870119894 + 0119894minus1 minus 1

119870119894 (1 minus 0119894)

)

+ (

1

2

minus

1

119870119894 (1 minus 0119894minus1)

)(1 minus

119863119894minus1

119875119894

)]

(10)

33 The Joint Cost of Member 119894 in Purchasing Activities andMember 119894 + 1 in Manufacturing Activities With the abovediscussion we have inferred the expected cost of member 119894

in purchasing andmanufacturing respectivelyTherefore thejoint cost of member 119894 in purchasing activities and member119894 + 1 in manufacturing activities can be obtained easily Wesubstituted 119899119894119873119901119894 = 119870119894+1119873119898119894+1 into (6) and (10) and gainedthe joint cost of members 119894 and 119894 + 1 as

119862119894119894+1 (119873119901119894 119898119871119894 119873119898119894+1 119870119894+1) = EC119901119894 + EC119898119894+1

= 119873119898119894+1119870119894+1 [EHC119894 + EBC119894] + 119873119901119894119862119901119894

+ 119873119898119894+1119862119898119894+1 + 119867119898119894+1AI119894+1

= 119873119898119894+1119870119894+1 119863119894119867119901119894 [119905119894 (119898119871119894 minus 120583119871119889119894)] +

1199051198942

2

+

1199051198941199091198940119894

1 minus 0119894

+ (119903119871119894 minus 119898119871119894)

sdot [int

119903119871119894

119898119871119894

(119871119889119894 minus 119898119871119894) 119891 (119871119889119894) 119889119871119889119894

minus (1199091198940119894) int

119903119871119894

119898119871119894

119891 (119871119889119894) 119889119871119889119894] + [119862119890119894 + 120573119894119863119894 (119903119871119894

minus 119898119871119894)] [int

119903119871119894

119898119871119894

119891 (119871119889119894) 119889119871119889119894 + 119865119894] + 119873119901119894119862119904119894

+ 119873119898119894119862119898119894 + 119867119898119894 (

119879119901119894minus1

2

119873119898119894

)

sdot 119863119894minus1 [1

2

(1 minus

119870119894 + 0119894minus1 minus 1

119870119894 (1 minus 0119894)

)

+ (

1

2

minus

1

119870119894 (1 minus 0119894minus1)

)(1 minus

119863119894minus1

119875119894

)]

(11a)

Mathematical Problems in Engineering 7

Substitute 119905119894 = 119879119901119894119870119894+1119873119898119894+1 into (11a) Consider

119862119894119894+1 (119873119901119894 119898119871119894 119873119898119894+1 119870119894+1) = V119894119879119901119894 (119898119871119894 minus 120583119871119889119894)

+ V119894119879119901119894119909119894120588119894 +119879119901119894

2

119870119894+1119873119898119894+1

(Ω119894 minus

119867119898119894+1119863119894119870119894+1120588119894

2

)

+

119879119901119894

2

119873119898119894+1

120579119894 + 119870119894+1119873119898119894+1 [(119903119871119894 minus 119898119871119894)

sdot V119894 int119903119871119894

119898119871119894

(119871119889119894 minus 119898119871119894) 119891 (119871119889119894) 119889119871119889119894

+ 120576119894 int

119903119871119894

119898119871119894

119891 (119871119889119894) 119889119871119889119894 + 119865119894] + 119873119901119894119862119904119894

+ 119873119898119894+1119862119898119894+1

(11b)

where

V119894 = 119867119901119894119863119894

120588119894 =0119894

1 minus 0119894

Ω119894

=

119863119894

2

[119867119901119894 + 119867119898119894+1 (1 minus (

2

1 minus 0119894

)(1 minus

119863119894

119875119894+1

))]

120576119894 = 119862119890119894 + (120573119894119863119894 minus V1198941199091198940119894) (119903119871119894 minus 119898119871119894)

120579119894 =

119867119898119894+1119863119894

2

(1 minus

119863119894

119875119894+1

)

(12)

Since 119879119901119894 = 119876119898119894minus1119875119894 = (119863119894minus1119873119898119894+1119875119894)119879119901119894minus1 119879119901119894 can beexpressed further as

119879119901119894 = (

120593119894

prod119894

119897=1119873119898119897

)1198791199011 (13)

where

120593119894 =

1 if 119894 = 1

119894

prod

119897=2

(

119863119894minus1

119875119894

) if 119894 = 2 3 119878 minus 1

(14)

All the 119879119901119894 in function (11b) could be replaced as follows

119862119894119894+1 (119873119901119894 119898119871119894 119873119898119894+1 119870119894+1) = V119894(120593119894

prod119894

119897=1119873119898119897

)

sdot 1198791199011 (119898119871119894 minus 120583119871119889119894) + V119894(120593119894

prod119894

119897=1119873119898119897

)1198791199011119909119894120588119894

+

(120593119894prod119894

119897=1119873119898119897) 1198791199011

2

119870119894+1119873119898119894+1

(Ω119894 minus

119867119898119894+1119863119894119870119894+1120588119894

2

)

+

(120593119894prod119894

119897=1119873119898119897) 1198791199011

2

119873119898119894+1

120579119894

+ 119870119894+1119873119898119894+1 [(119903119871119894 minus 119898119871119894)

sdot V119894 int119903119871119894

119898119871119894

(119871119889119894 minus 119898119871119894) 119891 (119871119889119894) 119889119871119889119894

+ 120576119894 int

119903119871119894

119898119871119894

119891 (119871119889119894) 119889119871119889119894 + 119865119894] + 119873119901119894119862119904119894

+ 119873119898119894+1119862119898119894+1

(11c)

There are119873119898119894+1 purchasing times of member 119894+1 during119879119901119894 According to (11c) the joint total cost function includingpurchasing and manufacturing activities of the serial 119878-echelon integrated JIT inventory model is presented in

119878minus1

sum

119894=1

V1198941205931198941198791199011 (119898119871119894 minus 120583119871119889119894) + V1198941205931198941198791199011119909119894120588119894

+

(1205931198941198791199011)

2

119870119894+1prod119894

119897=1119873119898119897

(Ω119894 minus

119867119898119894+1119863119894119870119894+1120588119894

2

)

+

1198791199011

2

prod119894

119897=1119873119898119897

120579119894 + (

119894

prod

119897=1

119873119898119897)119870119894+1119873119898119894+1 [(119903119871119894 minus 119898119871119894)

sdot V119894 int119903119871119894

119898119871119894

(119871119889119894 minus 119898119871119894) 119891 (119871119889119894) 119889119871119889119894

+ 120576119894 int

119903119871119894

119898119871119894

119891 (119871119889119894) 119889119871119889119894 + 119865119894] + (

119894

prod

119897=1

119873119898119897)119873119901119894119862119904119894

+ (

119894

prod

119897=1

119873119898119897)119862119898119894+1

(15)

The delivery time constrains 0 le 120582119871119894 le 119898119871119894 le 120574119871119894 le

119905119894 119894 = 1 2 119878 minus 1 and variables constrains are consideredin the inventory model Eventually the 119878-echelon integratedJIT model can be expressed as

Minimize 1198691198621119878 (119873119901 119898119871 119873119898 119870)

=

119878minus1

sum

119894=1

V1198941205931198941198791199011 (119898119871119894 minus 120583119871119889119894) + V1198941205931198941198791199011119909119894120588119894 +(1205931198941198791199011)

2

119870119894+1prod119894

119897=1119873119898119897

(Ω119894 minus

119867119898119894+1119863119894119870119894+1120588119894

2

) +

11987911990112

prod119894

119897=1119873119898119897

120579119894

8 Mathematical Problems in Engineering

+(

119894

prod

119897=1

119873119898119897)119870119894+1119873119898119894+1 [(119903119871119894 minus 119898119871119894) V119894 int119903119871119894

119898119871119894

(119871119889119894 minus 119898119871119894) 119891 (119871119889119894) 119889119871119889119894 + 120576119894 int

119903119871119894

119898119871119894

119891 (119871119889119894) 119889119871119889119894 + 119865119894]

+(

119894

prod

119897=1

119873119898119897)119873119901119894119862119904119894 + (

119894

prod

119897=1

119873119898119897)119862119898119894+1

Subject to 0 le 120582119871119894 le 119898119871119894 le 120574119871119894 le 119905119894 for 119894 = 1 2 119878 minus 1

119870119894+1 le1

0119894

minus 1

119873119901119894 119870119894+1 119873119898119894+1 are all integers for 119894 = 1 2 119878 minus 1

(16)

4 Numeral Experiment and Analysis

In this section a numeral experiment is proposed based onproper parameter settings All results in this research wereperformed on a PC Programs were written in Matlab 2012bAn efficiency comparison between Lingo PSO algorithmand Ant Colony Algorithm was proposed as well Further-more we would like to discuss how production runs anddefective rate influences whole system cost

41 Experiment Results of Ant Colony Algorithm ParticleSwarm Optimization and Lingo With appropriate param-eters settings we may build a 7-echelon supply chain inte-grated JIT inventory model with uncertain delivery leadtime and unreliable quality of received items Assume thepurchasing time interval for first purchaser in years (1198791199011) is12 years The relevant data of purchasing and manufacturingactivities are shown in Tables 1 and 2

The objective function and constraint function are shownbelow Consider

1198691198621119878 (119873119901 119898119871 119873119898 119870)

=

119878minus1

sum

119894=1

V1198941205931198941198791199011 (119898119871119894 minus 120583119871119889119894) + V1198941205931198941198791199011119909119894120588119894 +(1205931198941198791199011)

2

119870119894+1prod119894

119897=1119873119898119897

(Ω119894 minus

119867119898119894+1119863119894119870119894+1120588119894

2

) +

1198791199011

2

prod119894

119897=1119873119898119897

120579119894

+(

119894

prod

119897=1

119873119898119897)119870119894+1119873119898119894+1 [(119903119871119894 minus 119898119871119894) V119894 int119903119871119894

119898119871119894

(119871119889119894 minus 119898119871119894) 119891 (119871119889119894) 119889119871119889119894 + 120576119894 int

119903119871119894

119898119871119894

119891 (119871119889119894) 119889119871119889119894 + 119865119894]

+(

119894

prod

119897=1

119873119898119897)119873119901119894119862119904119894 + (

119894

prod

119897=1

119873119898119897)119862119898119894+1

Subject to 0 le 120582119871119894 le 119898119871119894 le 120574119871119894 le 119905119894 for 119894 = 1 2 119878 minus 1

119870119894+1 le1

0119894

minus 1

119873119901119894 119870119894+1 119873119898119894+1 are all integers for 119894 = 1 2 119878 minus 1

(17)

The goal of this research is to find out the optimal solutionof the 7-echelon inventory model via the above functions andACA and compare the efficiency between Lingo ACA andPSO

As Table 3 shows though all the three methods werecapable of finding the optimal solution for SMEIJI problemACA was obviously better than PSO and Lingo in efficiencyACA has shown fastest CPU searching time and least averageiterations among three methods

42 Experiments of Performance and 119878-Echelon In this sec-tion we would like to discuss the relationship between CPU

searching time and echelon number among three methodswe also discussed the relationship between average iterationsand echelon number

As Figure 3 shows the CPU searching time wouldincrease with the number of echelons due to the compli-cated calculation However because of the well-organizedconstraint function and good initial solution ACA still keptlower CPU searching time than the other two methods Onthe other hand Figure 4 shows that ACArsquos average iterationswere the least among three methods At the beginning of themultiechelon all the three algorithms need a great amountof iterations to search the optimal solution After calculating

Mathematical Problems in Engineering 9

Table 1 The relevant data of purchasing activities

Purchasing activityEchelon 119863119894 119862119904119894 119867119901119894 119862119890119894 120573119894 120582119871119894 120574119871119894 120583119871119889119894 120590119871119889119894 119865119894 119883119894 0119894

1 18 w 25 28 29 32 0005 0015 0009 0002 70 0005 0032 20w 28 22 32 26 0004 001 00064 00012 50 0003 0013 24w 32 17 38 21 0003 0009 00054 00012 48 0003 0044 30w 36 12 42 16 0004 0012 00072 00016 55 0004 0055 35w 35 08 37 12 0006 0016 001 0002 80 0005 0016 40w 38 06 40 09 0004 0014 0008 0002 65 0005 0027 mdash mdash mdash mdash mdash mdash mdash mdash mdash mdash mdash mdash

Table 2 The relevant data of manufacturing activities

Manufacturing activitiesEchelon 119875119894 119862119898119894 119867119898119894

1 mdash mdash mdash2 200000 280 253 240000 200 194 300000 320 145 350000 360 106 400000 380 097 450000 420 05

1 2 3 4 5 6 70

05

1

15

2

25

3

ACAPSO

Lingo

Figure 3 Experiment results of CPU time and echelon number

repeatedly the trend of the average iterations would come flatwithout extreme variety

Nevertheless although ACA seems better than the othertwo algorithms we could not ensure that the performancewould still be great when applied to higher echelon inventorymodel Hence a 15-echelon model has proposed to observethe performance again The following table displayed theparameters of 15-echelon model

To evaluate the performance of ACA PSO and Lingo forthe 15-echelon inventorymodel we applied the parameters inTables 4 and 5 to solve the 15-echelon inventory problem

According to Table 6 all three algorithms were still ableto find out the optimal solution of the 15-echelon inventorymodel Once the echelon increases the complexity of the

1 2 3 4 5 6 70

500

1000

1500

2000

2500

3000

3500

4000

ACAPSO

Lingo

Figure 4 Experiment results of average iterations and echelonnumber

model increases as well However the performance of ACAwas still better than the other two algorithms

There are several advantages of ACA as follows

(1) Applying the positive and negative feedback enablesthe process of searching optimization to converge atthe end which means the results would close to theoptimal solutions gradually

(2) Every single ant is able to change the surroundingenvironment through releasing pheromone and beingconscious of the variety of environment which makesthem communicate with each other

(3) Due to distributed computing all the individuals startto calculate simultaneously during the searching pro-cess which enhances the efficiency of the calculationability

(4) For the proposed algorithm it is difficult to stick inthe partial optimization instead it is always easy tofind the best solution

To sum up ACA was proposed to solve the complicatedmathematical problem we also have proved that ACA wasmore efficient than PSO and Lingo These are the new majorcontributions of the present paper

43 The Sensitive Analysis of SMEIJI In this section thenumber of production runs 119873119898119894 and the defective rate of

10 Mathematical Problems in Engineering

Table 3 Comparisons between three methods in SMEIJI problem

Echelon ACA PSO Lingo119873119901119894 119898119871119894 119870119894 119873119898119894 119873119901119894 119898119871119894 119870119894 119873119898119894 119873119901119894 119898119871119894 119870119894 119873119898119894

1 1 00147 mdash mdash 1 00147 mdash mdash 1 00147 mdash mdash2 1 000987 11 1 1 000987 11 1 1 000987 11 13 1 000889 13 1 1 000889 13 1 1 000889 13 14 1 001184 11 1 1 001184 11 1 1 001184 11 15 1 001571 8 1 1 001571 8 1 1 001571 8 16 1 001378 5 1 1 001378 5 1 1 001378 5 17 mdash mdash 5 1 mdash mdash 5 1 mdash mdash 5 1Optimal cost 89166 89166 89166CPU time 0655 1744 23Average iterations 350 1278 3758

Table 4 The relevant data of purchasing activities of 15-echelon model

Purchasing activityEchelon 119863119894 119862119904119894 119867119901119894 119862119890119894 120573119894 120582119871119894 120574119871119894 120583119871119889119894 120590119871119889119894 119865119894 119883119894 0119894

1 18 w 25 28 29 32 0005 0015 0009 0002 70 0005 0032 20w 28 22 32 26 0004 001 00064 00012 50 0003 0013 24w 32 17 38 21 0003 0009 00054 00012 48 0003 0044 30w 36 12 42 16 0004 0012 00072 00016 55 0004 0055 35w 35 08 37 12 0006 0016 001 0002 80 0005 0016 40w 38 06 40 09 0004 0014 0008 0002 65 0005 0027 42w 40 05 28 12 0005 0013 001 0002 70 0005 0038 45w 42 07 35 15 0004 0009 0009 00018 50 0004 0029 50w 44 10 42 10 0003 0010 0005 00024 55 0004 00210 52w 46 15 35 22 0005 0012 00062 00025 60 0005 00211 56w 48 20 49 25 0006 0009 001 00026 65 0003 00312 60w 50 22 40 14 0004 0016 0012 0002 48 0005 00413 62w 55 17 35 15 0007 0015 0009 00018 72 0004 00414 65w 56 12 32 16 0003 0014 00078 0002 70 0006 00315 mdash mdash mdash mdash mdash mdash mdash mdash mdash mdash mdash mdash

Table 5The relevant data of manufacturing activities of 15-echelonmodel

Manufacturing activitiesEchelon 119875119894 119862119898119894 119867119898119894

1 mdash mdash mdash2 200000 280 253 240000 200 194 300000 320 145 350000 360 106 400000 380 097 450000 420 058 480000 440 109 500000 420 1210 520000 400 1511 540000 450 2012 580000 440 2213 620000 420 1814 640000 460 1515 660000 380 16

receiving shipment 0119894 were considered to make sensitiveanalysis and observe the influence on the optimal total jointcost The defective rate and production runs were changedby 0119894(1 + 119877) and 119873119898119894(1 + 119877) where the values of 119877 are0 50 100 200 300 400

Figure 5 illustrated that defective rate was more sensitivethan production runs under the situation that productionruns were one-time production except first echelon In thatcase the higher the defective ratewas the higher the total costwas Accordingly inspecting goods cautiously before deliveryand reducing the defective rate would lower total cost of thewhole supply chainHowever if relaxing that assumption thissensitive analysis would result in different outcomes

Figure 6 has shown that production runs were more sen-sitive than defective rate when relaxing one-time productionassumption In this case the higher the production runswerethe higher the total cost was Hence reducing productionruns should play an essential role in the whole supply chainOnce the one batch production policy was able to satisfy thedemand from the buyer the total cost of whole supply chainwill go down extremely

Mathematical Problems in Engineering 11

Table 6 Comparisons between three ways of 15-echelon inventory model

ACA PSO LingoEchelon 119873119901119894 119898119871119894 119870119894 119873119898119894 119873119901119894 119898119871119894 119870119894 119873119898119894 119873119901119894 119898119871119894 119870119894 119873119898119894

1 1 00147 mdash mdash 1 00147 mdash mdash 1 00147 mdash mdash2 1 000987 11 1 1 000987 11 1 1 000987 11 13 1 000889 13 1 1 000889 13 1 1 000889 13 14 1 001184 11 1 1 001184 11 1 1 001184 11 15 1 001571 8 1 1 001571 8 1 1 001571 8 16 1 001378 5 1 1 001378 5 1 1 001378 5 17 1 001451 5 1 1 001451 5 1 1 001451 5 18 1 001287 5 1 1 001287 5 1 1 001287 5 19 1 00154 7 1 1 00154 7 1 1 00154 7 110 1 001432 9 1 1 001432 9 1 1 001432 9 111 1 001352 7 1 1 001352 7 1 1 001352 7 112 1 00162 8 1 1 00162 8 1 1 00162 8 113 1 001571 10 1 1 001571 10 1 1 001571 10 114 1 001611 5 1 1 001611 5 1 1 001611 5 115 mdash mdash 5 1 mdash mdash 5 1 mdash mdash 5 1Optimal cost ($) 231254 231254 231254CPU time (s) 1725 6621 58215Average iterations 700 2883 8241

Defective rateProduction runs

400100 200 300 5000 500

20000

40000

60000

80000

100000

120000

140000

160000

180000

Figure 5 The sensitive analysis between defective rate and produc-tion runs

5 Conclusions

In summary this research proposed an integrated JIT inven-tory model to solve the partial optimization problem ofpurchaser andmanufacturer finding the optimal solution forthe whole supply chain Furthermore a multiechelon supplychainwith uncertain delivery lead time and unreliable qualityhas been considered to fit the real supply chain environ-ment Besides ACA was proposed to solve the complicatedmathematical problem we also have proved that ACA was

0 50 100 200 300 400 500

Defective rateProduction runs

0

50000

100000

150000

200000

250000

300000

350000

400000

Figure 6 The sensitive analysis between defective rate and produc-tions runs with relaxing assumption

more efficient than PSO and Lingo These are the new majorcontributions of the present paper

Finally there are a great deal of helpful directions forenterprises and researchers (1)The variety of conditions thatoccur in real supply chain can be considered in the modelFor instance time value of cash problem and deterioration(2)The warehousing and storage can be considered in supplychain which make the model more complex to solve (3)The model contains only one member in each echelon amultimember in each echelon should be considered to fit

12 Mathematical Problems in Engineering

real situation in supply chain (4) Some real cases andordata can be applied to our proposed model in the followingworks

Futureworkwill hopefully involve the real-world concernto our research which will be our main target In addition we

do hope our research can be extended intomore considerableareas and more considerable variables

Appendix

The Hessian matrix 119865 of 1198691198621119878(119873119901 119898119871 119873119898 119870) for given 119873119901119894

equal to 1 can be shown as

119865 =

[

[

[

[

[

[

[

[

[

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

1205971198731198981198942

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119873119898119894120597119898119871119894

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119873119898119894120597119870119894

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119898119871119894120597119873119898119894

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

1205971198981198711198942

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119898119871119894120597119870119894

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119870119894120597119873119898119894

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119870119894120597119898119871119894

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

1205971198701198942

]

]

]

]

]

]

]

]

]

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

1205971198731198981198942

=

2

1198731198981198943(120579120593119879

2+

1205931198792(1198671198631198701205932 + Ω)

119870

)

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

1205971198981198711198942

= 119870119873[(

119890minus119898120583

1205832

+ 2 (119890minus119898120583

minus 119890minus120574120583

) +

119890(minus119898120583)(minus119898+120583)

120583

) 120592]

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

1205971198701198942

=

2

1198701198943(120579120593119879

2+

1205931198792(1198671198631198701205932 + Ω)

119873

)

12059721198691198621119904 (119873119898119894 119898119871119894 119870119894)

120597119898119871119894120597119873119898119894

=

12059721198691198621119904 (119873119898119894 119898119871119894 119870119894)

120597119873119898119894120597119898119871119894

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119870119894120597119873119898119894

=

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119873119898119894120597119870119894

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119870119894120597119898119871119894

=

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119898119871119894120597119870119894

100381610038161003816100381611986511

1003816100381610038161003816=

1205972119865

1205971198731198981198942

=

2

1198731198981198943(120579120593119879

2+

1205931198792(1198671198631198701205932 + Ω)

119870

) gt 0

100381610038161003816100381611986522

1003816100381610038161003816= (

1205972119865

1205971198731198981198942)(

1205972119865

1205971198981198711198942) minus [

1205972119865

120597119873119898119894120597119898119871119894

]

2

=

2

1198731198981198943(120579120593119879

2+

1205931198792(1198671198631198701205932 + Ω)

119870

)

1205972119865

1205971198981198711198942minus (

1205972119865

120597119873119898119894120597119898119871119894

)

2

2

1198733119898119894

(1205791205931198792+

119861

119870

)[119870119873(

119890minus119898120576120583

1205832

+ (2119909 +

119890(minus119898120583)119909119910

120583

120592))] + 1198702[

119890minus119898120576120583

120583

+ 120592 (119909119910 minus 119860)] gt 0

100381610038161003816100381611986533

1003816100381610038161003816= (

1205972119865

1205971198731198981198942

1205972119865

1205971198981198711198942

1205972119865

1205971198701198942) minus [(

1205972119865

120597119873119898119894120597119870119894

)

2

(

1205972119865

1205971198981198711198942)]

=

2

1198731198981198943(120579120593119879

2+

119861

119870

)[119870119873(

119890minus119898120576120583

1205832

+ 2(119909 +

119890(minus119898120603)119909119910

120583

) 120592)]

2

1198703

119861

119873

minus [119865 + 119909120576 + 119860120592119910 minus

1205791205931198792

1198732

+

119861

2 (119870119873)2]

[119870119873(

119890minus119898120576120583

1205832

+ (2119909 +

119890minus119898120576120583

120583

) 120592)] gt 0

(A1)

Mathematical Problems in Engineering 13

As a result the matrix F has proved to be a positivedefinite matrix where |11986511| |11986522| and |11986533| are all positiveTherefore theminimumvalue exists in 1198691198621119878(119873119901119898119871 119873119898 119870)Furthermore we can find the optimization in the proposedmodel

Competing Interests

The authors declare that they have no competing interests

References

[1] S K Goyal ldquoAn integrated inventory model for a singlesupplier-single customer problemrdquo International Journal ofProduction Research vol 15 no 1 pp 107ndash111 1977

[2] A Banerjee ldquoA joint economic-lot-sizemodel for purchaser andvendorrdquo Decision Sciences vol 17 no 3 pp 292ndash311 1986

[3] S K Goyal ldquoldquoA joint economic-lot-size model for purchaserand vendorrdquo a commentrdquo Decision Sciences vol 19 no 1 pp236ndash241 1988

[4] Y Seo ldquoControlling general multi-echelon distribution supplychainswith improved reorder decision policy utilizing real-timeshared stock informationrdquo Computers amp Industrial Engineeringvol 51 no 2 pp 229ndash246 2006

[5] H N Chiu and H L Huang ldquoA Multi-echelon integratedJIT inventory model using the time buffer and emergencyborrowing policies to deal with random delivery lead timesrdquoInternational Journal of Production Research vol 41 no 13 pp2911ndash2931 2003

[6] J Sadeghi S M Mousavi S T A Niaki and S SadeghildquoOptimizing a multi-vendor multi-retailer vendor managedinventory problem two tuned meta-heuristic algorithmsrdquoKnowledge-Based Systems vol 50 pp 159ndash170 2013

[7] S S Sana ldquoA production-inventory model of imperfect qualityproducts in a three-layer supply chainrdquo Decision Support Sys-tems vol 50 no 2 pp 539ndash547 2011

[8] H N Soni and K A Patel ldquoOptimal strategy for an integratedinventory system involving variable production and defectiveitems under retailer partial trade credit policyrdquoDecision SupportSystems vol 54 no 1 pp 235ndash247 2012

[9] R J Tersine Principles of Inventory andMaterials ManagementNorthHolland AmsterdamTheNetherlands 3rd edition 1988

[10] C J Liao and C H Shyu ldquoAn analytical determination of leadtime with normal demandrdquo International Journal of Operationsamp Production Management vol 11 no 9 pp 72ndash78 1991

[11] M Ben-Daya and A Raouf ldquoInventory models involving leadtime as a decision variablerdquo Journal of the Operational ResearchSociety vol 45 no 5 pp 579ndash582 1994

[12] S M J Mirzapour Al-e-Hashem A Baboli and Z Sazvar ldquoAstochastic aggregate production planningmodel in a green sup-ply chain considering flexible lead times nonlinear purchaseand shortage cost functionsrdquo European Journal of OperationalResearch vol 230 no 1 pp 26ndash41 2013

[13] C Chandra and J Grabis ldquoInventorymanagementwith variablelead-time dependent procurement costrdquo Omega vol 36 no 5pp 877ndash887 2008

[14] J-W Wu and H-Y Tsai ldquoMixture inventory model with backorders and lost sales for variable lead time demand withthe mixtures of normal distributionrdquo International Journal ofSystems Science Principles and Applications of Systems andIntegration vol 32 no 2 pp 259ndash268 2001

[15] M A Hoque ldquoA vendorndashbuyer integrated productionndashinventory model with normal distribution of lead timerdquoInternational Journal of Production Economics vol 144 no 2pp 409ndash417 2013

[16] F Altiparmak M Gen L Lin and T Paksoy ldquoA geneticalgorithm approach for multi-objective optimization of supplychain networksrdquo Computers and Industrial Engineering vol 51no 1 pp 196ndash215 2006

[17] J Sadeghi S Sadeghi and S T Niaki ldquoA hybrid vendormanaged inventory and redundancy allocation optimizationproblem in supply chain management an NSGA-II with tunedparametersrdquo Computers and Operations Research vol 41 pp53ndash64 2014

[18] X Li F Baki P Tian and B A Chaouch ldquoA robust block-chainbased tabu search algorithm for the dynamic lot sizing problemwith product returns and remanufacturingrdquoOmega vol 42 no1 pp 75ndash87 2014

[19] Z Chen and B R Sarker ldquoAn integrated optimal inventorylot-sizing and vehicle-routing model for a multisupplier single-assembler system with JIT deliveryrdquo International Journal ofProduction Research vol 52 no 17 pp 5086ndash5114 2014

[20] A R Nia M Hemmati Far and S T A Niaki ldquoA fuzzy vendormanaged inventory of multi-item economic order quantitymodel under shortage an ant colony optimization algorithmrdquoInternational Journal of Production Economics vol 155 pp 259ndash271 2014

[21] S M Hatefi and F Jolai ldquoRobust and reliable forward-reverselogistics network design under demand uncertainty and facilitydisruptionsrdquo Applied Mathematical Modelling vol 38 no 9-10pp 2630ndash2647 2014

[22] R H Ma ldquoCross warehouse scheduling of logistics distributionand cycle re-claimer based on heuristics algorithmrdquo AppliedMechanics and Materials vol 644ndash650 pp 2606ndash2610 2014

[23] M Hashim L Yao A H Nadeem M Nazim and M NazamldquoMulti-objective optimization model for supplier selectionproblem in fuzzy environmentrdquo Advances in Intelligent Systemsand Computing vol 281 pp 1201ndash1213 2014

[24] Q-M Hu and Z-H Hu ldquoA stochastic programming modelfor hub-and-spoke network with uncertain flowsrdquo InternationalJournal of Industrial and Systems Engineering vol 21 no 3 pp302ndash319 2015

[25] S A Torabi J Namdar S M Hatefi and F Jolai ldquoAn enhancedpossibilistic programming approach for reliable closed-loopsupply chain network designrdquo International Journal of Produc-tion Research vol 54 no 5 pp 1358ndash1387 2016

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

to solve our SMEIJI model In given example we would liketo compare ACAwith other traditional methods and observehow system cost works

2 Fundamental Assumptions and Notations

A serial inventory system with 119878-echelon where echelon 1 ispurchasing only and echelon 119878 is manufacturing only wasconsidered In the inventory system we proposed there isonly one member in each echelon that plays the purchaserand manufacturer roles simultaneously except echelon 1 andechelon 119878 Notation and assumptions are defined as follows

For member 119894 engaged in purchasing activities where 119894 =

1 2 119878 minus 119897 we have the following

Decision variables are as follows

119873119901119894 number of purchase orders during 119879119898119894 forpurchasing 119894119899119894 number of deliveries per purchase order forpurchasing 119894

119898119871119894 the maximum allowable planned deliverylead time that would not cause shortage 119898119894 =120583119871119863119894+ safety lead time (a real decision variable)

Parameters are as follows

119878 number of echelons in a serial supply chain119879119901119894 length of a purchasing time interval forpurchasing 119894 (in years)119863119894 demand rate of purchasing 119894 (unitsyear)119876119901119894 lot size per purchase order for purchasing 119894

(unitsorder)119902119894 delivery lot size per shipment for purchasing119894 (unitsdelivery)119862119904119894 fixed ordering cost for purchasing 119894

($order)119867119901119894 holding cost per unit of purchased goodsper year for purchasing 119894 ($unityear)119862119890119894 ordering cost of emergency borrowing forpurchasing 119894 ($borrowing)120573119894 borrowing cost per unit per year for purchas-ing 119894 ($unityear)119865119894 fixed delivery cost for purchasing 119894 ($deliv-ery)119871119889119894 delivery lead time from manufacturing 119894 +

1 to purchasing 119894 a nonnegative random vari-able following a probability distribution withexpected delivery lead time 120583119871

119863119894and standard

deviation 120590119871119863119894119905119894 time interval between two adjacent deliveriesfor purchasing 119894119903119894 redelivery point of purchasing 119894 that is analo-gous to a reorder point if stock drops to119877119894 thena delivery notice is issued to manufacturer 119894 + 1119909119894 screening time of 119902119894 amount materials thatwere received by purchaser 119894 (in years)

0119894 defective rate of a shipment that was receivedby purchaser 119894

For member 119894 engaged in manufacturing activities where 119894 =

1 2 119878 minus 119897 we have the following

Decision variables are as follows

119873119898119894 number of production runs during 119879119898119894minus1

for manufacturing 119894119870119894 number of deliveries per production run formanufacturing 119894

Parameters are as follows

119879119898119894 length of a manufacturing time interval formanufacturing 119894 119879119898119894 = 119876119898119894119901119894 (in years)119875119894 production rate for manufacturing 119894

(unitsyear)119876119898119894 lot size per production run formanufactur-ing 119894 (unitsrun)119862119898119894 fixed set-up cost formanufacturing 119894 ($set-up)119867119898119894 holding cost per unit of produced goodsper year for manufacturing 119894 ($unityear)

Additional notations will be introduced if necessary Theassumptions set in this research are as follows

(1) We consider an integrated multiechelon supply chainwhich only contains a member in each echelon Themember in echelon 119894 is denoted as member 119894 where119894 = 1 2 3 119878 A single product is produced by thesupply chain

(2) Except echelons 1 and 119878 purchaser and manufacturerare the two roles that each member played Member1 purchasing activities only is the lowest echelon inthis serial supply chain Member 119878 involves manufac-turing activities only

(3) Member 119894 purchases requiredmaterials frommember119894 + 1 and produces goods to member 119894 minus 1 during 119879119901119894

(4) The demand and production rate for the product areconstant over time The demand rates of member 119894

should be less than the production rate of member119894 + 1 Member 119894 + 1 produces the necessary quantityto satisfy the demand of member 119894

(5) The delivery lead time (119871119889119894) for member 119894 is assumedto follow a probability distribution with a densityfunction of 119871119889119894 = 119891(119871119889119894) where 0 le 120582119871119894 le 119871119889119894 le 120574119871119894

and 119894 = 1 2 119878 minus 1(6) The time interval between two adjacent deliveries

for member 119894 (119905119894) should be longer than or equalto 120574119871119894 The delivery lot crossing makes the problemintractable and it can be prevented by this assump-tion

(7) When a delivery delay occurs an emergency borrow-ing action should be touched off bymember 119894 becausethe shortages are not allowed in JIT environment

4 Mathematical Problems in Engineering

Time

Inventory level ti ti

Di

qi

ri

0iqi

mLi mLi mLixi + Ldi minus mLi

Ldi Ldixi mLi + ti minus Ldi

0 le 120582Li le Ldi le mLi mLi le Ldi le 120574Li le ti

(1 minus 0i)qi

(B)

(A)

Figure 1 Delivery cases for the member who implements purchasing

(8) The required time of borrowing goods from nearbysupplier is ignored

(9) A finite planning horizon for the whole serial supplychain that denoted 119879119901119894 is considered

(10) Defective items in a shipment of materials arereceived by purchaser 119894 with defective rate 0119894 Thenumber of goodmaterialsmust bemore than or equalto the required quantity during the screening time

(11) The screening process is followed by the arrival ofshipments with screening time 119909119894 The length ofscreening time is proportional to the number ofreceived materials

(12) The accumulated number of defective items that weredelivered to the buyer during a production run mustbe less than the quantity of one delivery

(13) When emergency borrowing happened the materialsthat were borrowed and returned are all good items

3 Formulation of the Model

In this integrated multiechelon JIT inventory model thedecision variables are 119899119894 (number of deliveries per purchase)119873119901119894 (number of purchase orders during 119879119901119894) and 119898119871119894 (themaximum allowable planned delivery lead time that wouldnot cause shortage) where 119894 = 1 119878minus1119873119898119894 is the numberof production runs during 119879119898119894 and 119870119894 are delivery times perproduction runs for manufacturer 119894 where 119894 = 2 119878 Thevalue of these decision variables should be determined inorder to minimize the joint total cost

31 The Cost of Member 119894 in Purchasing Activities Fromechelon1 to echelon 119878 minus 1 members adopt JIT purchasing toreplenish goodsThe time buffer policy used on planned lead

time (119898119871119894) and emergency borrowing policy used to deal withuncertain delivery lead time Each order lot size119876119901119894 has beentaken apart into 119899119894 small lots for delivery frequently Duringthe planning horizon 119879119901119894 the total number of deliveriesfor member 119894 is 119899119894 times 119873119901119894 The delivery lead time follows aprobability density distribution 119891(119871119889119894) The 120574119871119894 and 120582119871119894 arethe upper bound and the lower bound of 119871119889119894

The costs are related to purchasing activities of member119894 during 119879119901119894 where 119894 = 1 119878 minus 1 including ordering costholding cost delivery cost transportation carrying cost andemergency borrowing cost The transportation carrying costwould not be considered in this model since it is constant anddoes not affect any decision variables

Owing to the uncertain delivery lead time there are threedifferent delivery cases early arrival delay arrival and arrivalon time The screening process would be implemented whenthe purchaser receives a shipment of goods The defectiveitem would be deducted from the inventory in a singlebatch at the end of the purchaserrsquos 100 screening processFigure 1 has shown two delivery cases for the member whoimplements purchasing

On the left side of Figure 1 0 le 120582119871119894 le 119871119889119894 le 119898119871119894means theearly arrival occurs and results in unexpected holding costThe shadow area (A) presents the unexpected extra inventoryIn this case the holding cost (HCA119894) of member 119894 during 119905119894

is shownThe shadow area (A) presents the unexpected extrainventory caused by early delivery and 119905119894119902119894(1minus0119894)2+(119909119894+119871119889119894minus

119898119871119894)1199021198940119894 represents the expected inventoryIn the right side of Figure 1 119898119871119894 le 119871119889119894 le 120574119871119894 le 119905119894

presents the delay arrival situation When it occurs member119894 should adopt emergency borrowing policy The shadowarea (B) presents the quantity of emergency borrowing Theborrowing quantity from other nearby suppliers outside thesupply chain is 119863119894(120574119871119894minus119898119871119894) units After the delay arrivallot is received the remand quantity of goods 119863119894(120574119871119894 minus 119871119889119894)

Mathematical Problems in Engineering 5

should be given back immediately The borrowed and theremanded units would not be screened and the screeningtime is ((119902119894 minus119863119894(120574119871119894 minus119898119871119894))119902119894)119909119894 during119898119871119894 le 119871119889119894 le 120574119871119894 le 119905119894The holding cost (HCB119894) and the expected borrowing cost(EBC119894) of member 119894 during 119905119894 are shown The holding cost(HCB119894) involves the normal holding cost and the emergencyborrowing holding cost

The unexpected extra inventory of shadow area (A) dueto the early arrivals is as follows

HCA119894 = 119867119901119894 [119902119894 (119898119871119894 minus 119871119889119894) +

119905119894119902119894 (1 minus 0119894)

2

+ (119909119894 + 119871119889119894 minus 119898119871119894) 1199021198940119894] = 119863119894119867119901119894 [119905119894 (119898119871119894 minus 119871119889119894)

+

1199051198941199091198940119894

1 minus 0119894

+

119905119894

2

2

]

(1)

The emergency borrowing holding cost due to delay arrivalsis as follows

119863119894119867119901119894 [

[(119903119871119894 minus 119871119889119894) + (119903119871119894 minus 119898119871119894)] (119871119889119894 minus 119898119871119894)

2

] (2)

Normal holding cost without early or delay arrivals is asfollows

119863119894119867119901119894 [

(119905119894 + 119871119871119894

minus 119871119889119894)2

2

+ (

119905119894 minus (119903119871119894 minus 119898119871119894) (1 minus 0119894)

1 minus 0119894

)1199091198940119894]

HCB119894 = 119867119901119894 [(119861) + (119862)] = 119863119894119867119901119894 [(119903119871119894 minus 119898119871119894)

+ (119871119889119894 minus 119898119871119894)] + 119905119894 (119898119871119894 minus 119871119889119894) +

119905119894

2

2

minus (119903119871119894 minus 119898119871119894)

sdot (1199091198940119894) +

1199051198941199091198940119894

1 minus 0119894

(3)

Nowwe combine (1) and (3) to form the expected holdingcost (EHC119894) of member 119894 as follows

EHC119894 = int

119898119871119894

120582119871119894

HCA119894119891 (119871119889119894) 119889119871119889119894

+ int

119903119871119894

119898119871119894

HCB119894119891 (119871119889119894) 119889119871119889119894 = 119863119894119867119901119894 [119905119894 (119898119871119894

minus 119871119889119894) +

119905119894

2

2

+

1199051198941199091198940119894

1 minus 0119894

+ (119903119871119894 minus 119898119871119894)

sdot [int

119903119871119894

119898119871119894

(119871119889119894 minus 119898119871119894) 119891 (119871119889119894) 119889119871119889119894

minus (1199091198940119894) int

119903119871119894

119898119871119894

119891 (119871119889119894) 119889119871119889119894]]

(4)

The expected borrowing cost (EBC119894) is as follows

EBC119894 = 119862119890119894 int

119903119871119894

119898119871119894

119891 (119871119889119894) 119889119871119889119894

+ 120573119894 int

119903119871119894

119898119871119894

119863119894 (119903119871119894 minus 119898119871119894) 119891 (119871119889119894) 119889119871119889119894

= [119862119890119894 + 120573119894119863119894 (119903119871119894 minus 119898119871119894)] int

119903119871119894

119898119871119894

119891 (119871119889119894) 119889119871119889119894

(5)

The EBC119894 only occurs at time interval [120574119871119894 119898119871119894]119862119890119894 int119903119871119894

119898119871119894119891(119871119889119894)119889119871119889119894 means the ordering cost of emergency

borrowing and 120573119894 int119903119871119894

119898119871119894119863119894(119903119871119894 minus 119898119871119894)119891(119871119889119894)119889119871119889119894 presents the

excepted cost of borrowed unitsDuring the purchasing time interval (119879119901119894) of each mem-

ber it has 119873119901119894 purchasing orders 119899119894119873119901119894 delivery receivingtimes 119899119894119873119901119894119865119894 delivery cost and 119873119901119894119862119904119894 ordering cost Thusthe expected cost (EP119888119894) of member 119894 in purchasing activitiesis shown as follows

EC119901119894 (119899119894 119873119901119894 119898119871119894) = holding cost

+ emergency borrowing cost + ordering cost

+ delivery cost = 119899119894119873119901119894 [EHC119894 + EBC119894] + 119873119901119894119862119904119894

+ 119899119894119873119901119894119865119894 = 119899119894119873119901119894119863119894119867119901119894 [119905119894 (119898119871119894 minus 119871119889119894) +

1199051198942

2

+

1199051198941199091198940119894

1 minus 0119894

+ (119903119871119894 minus 119898119871119894)]

sdot [int

119903119871119894

119898119871119894

(119871119889119894 minus 119898119871119894) 119891 (119871119889119894) 119889119871119889119894

minus (1199091198940119894) int

119903119871119894

119898119871119894

119891 (119871119889119894) 119889119871119889119894] + 119899119894119873119901119894 [119862119890119894

+ 120573119894119863119894 (119903119871119894 minus 119898119871119894)] int

119903119871119894

119898119871119894

119891 (119871119889119894) 119889119871119889119894 + 119865119894 + 119873119901119894119862119904119894

(6)

32 The Cost of Member 119894 Related to Manufacturing ActivitiesIn this model member 119894 + 1 adopts JIT manufacturing toproduce goods to member 119894 There are 119873119898119894+1 productionruns during 119879119901119894 The start time of each production can bedetermined by counting backward 119902119894minus1119875119894 time units whenthe first delivery lot is delivered The average inventory ofgoods produced bymember 119894per production run is illustratedby Figure 2 and we can determine this by calculatingthe cumulative time-weighted production quantity (1) (thetrapezoid area) minus the cumulative time-weighted deliveryquantity (2) (the ladder area)

Calculating the cumulative time-weighted productionquantity of member 119894 per production run is equal to calcu-lating a square measure of trapezoid area Another way tocalculate the cumulative time-weighted production quantity

6 Mathematical Problems in Engineering

Time

Cumulative quantity

(2)(2)

(1) (1)

ti

Tpi Tpi

Pi

Tpiminus1NmiTpiminus1Nmi

Qmi

Tpiminus1

Figure 2 Average inventory of goods produced by member 119894 perproduction run

(1) is subtracting the triangle area from the rectangle area asfollows

(1) = [

119879119901119894minus1

119873119898119894

minus (119905119894minus1 minus

119902119894minus1

119901119894

)](

119863119894minus1119879119901119894minus1

119873119898119894

)

minus

1

2

(

119863119894minus1119879119901119894

119875119894119873119898119894

)(

119863119894minus1119879119901119894minus1

119873119898119894

) = (

119879119901119894minus1

119873119898119894

)

2

119863119894minus1 [1

2

+

0119894minus1

119870119894 (1 minus 0119894)

+ (

1

2

minus

1

119870119894 (1 minus 0119894minus1)

)(1 minus

119863119894minus1

119875119894

)]

(7)

During each production run of member i there are 119870119894

deliveries and 119902119894 units of goods delivered per shipment Thetime interval between two adjacent deliveries for member119894 (119905119894) should be longer than or equal to 120574119871119894 Hence thecumulative time-weighted delivery quantity (2) of member 119894

per production run is

(2) = (

119870119894minus1

sum

119897

119897) 119905119894minus1119902119894minus1 = (1 + 2 + sdot sdot sdot + (119870119894minus1)) 119905119894minus1119902119894minus1

= (

119879119901119894minus1

119873119898119894

)

2

119863119894minus1 [1

2

119870119894minus1

119870119894 (1 minus 0119894minus1)

]

(8)

Consequently the average inventory (AI119894) of member 119894

during 119879119901119894minus1 can be calculated by subtracting area (2) fromarea (1) as follows

AI119894 = 119873119898119894 (1) minus (2) = 119873119898119894 (

119879119901119894minus1

119873119898119894

)

2

119863119894minus1 [1

2

+

0119894

119870119894 (1 minus 0119894)

+ (

1

2

minus

1

119870119894 (1 minus 0119894minus1)

)(1 minus

119863119894

119875119894

)]

minus

119879119901119894minus1

119873119898119894

119863119894minus1 [1

2

119870119894minus1

119870119894 (1 minus 0119894minus1)

] = (

119879119901119894minus12

119873119898119894

)

sdot 119863119894minus1 [1

2

(

119870119894 + 0119894minus1 minus 1

119870119894 (1 minus 0119894)

) + (

1

2

minus

1

119870119894 (1 minus 0119894minus1)

)

sdot (1 minus

119863119894minus1

119875119894

)]

(9)

Thus the production cost of member 119894 (EC119898119894) during 119879119901119894 is

EC119898119894 (119873119898119894119870119894) = 119873119898119894119862119898119894 + 119867119898119894AI119894 = 119873119898119894119862119898119894

+ 119867119898119894 (

119879119901119894minus12

119873119898119894

)119863119894minus1 [1

2

(

119870119894 + 0119894minus1 minus 1

119870119894 (1 minus 0119894)

)

+ (

1

2

minus

1

119870119894 (1 minus 0119894minus1)

)(1 minus

119863119894minus1

119875119894

)]

(10)

33 The Joint Cost of Member 119894 in Purchasing Activities andMember 119894 + 1 in Manufacturing Activities With the abovediscussion we have inferred the expected cost of member 119894

in purchasing andmanufacturing respectivelyTherefore thejoint cost of member 119894 in purchasing activities and member119894 + 1 in manufacturing activities can be obtained easily Wesubstituted 119899119894119873119901119894 = 119870119894+1119873119898119894+1 into (6) and (10) and gainedthe joint cost of members 119894 and 119894 + 1 as

119862119894119894+1 (119873119901119894 119898119871119894 119873119898119894+1 119870119894+1) = EC119901119894 + EC119898119894+1

= 119873119898119894+1119870119894+1 [EHC119894 + EBC119894] + 119873119901119894119862119901119894

+ 119873119898119894+1119862119898119894+1 + 119867119898119894+1AI119894+1

= 119873119898119894+1119870119894+1 119863119894119867119901119894 [119905119894 (119898119871119894 minus 120583119871119889119894)] +

1199051198942

2

+

1199051198941199091198940119894

1 minus 0119894

+ (119903119871119894 minus 119898119871119894)

sdot [int

119903119871119894

119898119871119894

(119871119889119894 minus 119898119871119894) 119891 (119871119889119894) 119889119871119889119894

minus (1199091198940119894) int

119903119871119894

119898119871119894

119891 (119871119889119894) 119889119871119889119894] + [119862119890119894 + 120573119894119863119894 (119903119871119894

minus 119898119871119894)] [int

119903119871119894

119898119871119894

119891 (119871119889119894) 119889119871119889119894 + 119865119894] + 119873119901119894119862119904119894

+ 119873119898119894119862119898119894 + 119867119898119894 (

119879119901119894minus1

2

119873119898119894

)

sdot 119863119894minus1 [1

2

(1 minus

119870119894 + 0119894minus1 minus 1

119870119894 (1 minus 0119894)

)

+ (

1

2

minus

1

119870119894 (1 minus 0119894minus1)

)(1 minus

119863119894minus1

119875119894

)]

(11a)

Mathematical Problems in Engineering 7

Substitute 119905119894 = 119879119901119894119870119894+1119873119898119894+1 into (11a) Consider

119862119894119894+1 (119873119901119894 119898119871119894 119873119898119894+1 119870119894+1) = V119894119879119901119894 (119898119871119894 minus 120583119871119889119894)

+ V119894119879119901119894119909119894120588119894 +119879119901119894

2

119870119894+1119873119898119894+1

(Ω119894 minus

119867119898119894+1119863119894119870119894+1120588119894

2

)

+

119879119901119894

2

119873119898119894+1

120579119894 + 119870119894+1119873119898119894+1 [(119903119871119894 minus 119898119871119894)

sdot V119894 int119903119871119894

119898119871119894

(119871119889119894 minus 119898119871119894) 119891 (119871119889119894) 119889119871119889119894

+ 120576119894 int

119903119871119894

119898119871119894

119891 (119871119889119894) 119889119871119889119894 + 119865119894] + 119873119901119894119862119904119894

+ 119873119898119894+1119862119898119894+1

(11b)

where

V119894 = 119867119901119894119863119894

120588119894 =0119894

1 minus 0119894

Ω119894

=

119863119894

2

[119867119901119894 + 119867119898119894+1 (1 minus (

2

1 minus 0119894

)(1 minus

119863119894

119875119894+1

))]

120576119894 = 119862119890119894 + (120573119894119863119894 minus V1198941199091198940119894) (119903119871119894 minus 119898119871119894)

120579119894 =

119867119898119894+1119863119894

2

(1 minus

119863119894

119875119894+1

)

(12)

Since 119879119901119894 = 119876119898119894minus1119875119894 = (119863119894minus1119873119898119894+1119875119894)119879119901119894minus1 119879119901119894 can beexpressed further as

119879119901119894 = (

120593119894

prod119894

119897=1119873119898119897

)1198791199011 (13)

where

120593119894 =

1 if 119894 = 1

119894

prod

119897=2

(

119863119894minus1

119875119894

) if 119894 = 2 3 119878 minus 1

(14)

All the 119879119901119894 in function (11b) could be replaced as follows

119862119894119894+1 (119873119901119894 119898119871119894 119873119898119894+1 119870119894+1) = V119894(120593119894

prod119894

119897=1119873119898119897

)

sdot 1198791199011 (119898119871119894 minus 120583119871119889119894) + V119894(120593119894

prod119894

119897=1119873119898119897

)1198791199011119909119894120588119894

+

(120593119894prod119894

119897=1119873119898119897) 1198791199011

2

119870119894+1119873119898119894+1

(Ω119894 minus

119867119898119894+1119863119894119870119894+1120588119894

2

)

+

(120593119894prod119894

119897=1119873119898119897) 1198791199011

2

119873119898119894+1

120579119894

+ 119870119894+1119873119898119894+1 [(119903119871119894 minus 119898119871119894)

sdot V119894 int119903119871119894

119898119871119894

(119871119889119894 minus 119898119871119894) 119891 (119871119889119894) 119889119871119889119894

+ 120576119894 int

119903119871119894

119898119871119894

119891 (119871119889119894) 119889119871119889119894 + 119865119894] + 119873119901119894119862119904119894

+ 119873119898119894+1119862119898119894+1

(11c)

There are119873119898119894+1 purchasing times of member 119894+1 during119879119901119894 According to (11c) the joint total cost function includingpurchasing and manufacturing activities of the serial 119878-echelon integrated JIT inventory model is presented in

119878minus1

sum

119894=1

V1198941205931198941198791199011 (119898119871119894 minus 120583119871119889119894) + V1198941205931198941198791199011119909119894120588119894

+

(1205931198941198791199011)

2

119870119894+1prod119894

119897=1119873119898119897

(Ω119894 minus

119867119898119894+1119863119894119870119894+1120588119894

2

)

+

1198791199011

2

prod119894

119897=1119873119898119897

120579119894 + (

119894

prod

119897=1

119873119898119897)119870119894+1119873119898119894+1 [(119903119871119894 minus 119898119871119894)

sdot V119894 int119903119871119894

119898119871119894

(119871119889119894 minus 119898119871119894) 119891 (119871119889119894) 119889119871119889119894

+ 120576119894 int

119903119871119894

119898119871119894

119891 (119871119889119894) 119889119871119889119894 + 119865119894] + (

119894

prod

119897=1

119873119898119897)119873119901119894119862119904119894

+ (

119894

prod

119897=1

119873119898119897)119862119898119894+1

(15)

The delivery time constrains 0 le 120582119871119894 le 119898119871119894 le 120574119871119894 le

119905119894 119894 = 1 2 119878 minus 1 and variables constrains are consideredin the inventory model Eventually the 119878-echelon integratedJIT model can be expressed as

Minimize 1198691198621119878 (119873119901 119898119871 119873119898 119870)

=

119878minus1

sum

119894=1

V1198941205931198941198791199011 (119898119871119894 minus 120583119871119889119894) + V1198941205931198941198791199011119909119894120588119894 +(1205931198941198791199011)

2

119870119894+1prod119894

119897=1119873119898119897

(Ω119894 minus

119867119898119894+1119863119894119870119894+1120588119894

2

) +

11987911990112

prod119894

119897=1119873119898119897

120579119894

8 Mathematical Problems in Engineering

+(

119894

prod

119897=1

119873119898119897)119870119894+1119873119898119894+1 [(119903119871119894 minus 119898119871119894) V119894 int119903119871119894

119898119871119894

(119871119889119894 minus 119898119871119894) 119891 (119871119889119894) 119889119871119889119894 + 120576119894 int

119903119871119894

119898119871119894

119891 (119871119889119894) 119889119871119889119894 + 119865119894]

+(

119894

prod

119897=1

119873119898119897)119873119901119894119862119904119894 + (

119894

prod

119897=1

119873119898119897)119862119898119894+1

Subject to 0 le 120582119871119894 le 119898119871119894 le 120574119871119894 le 119905119894 for 119894 = 1 2 119878 minus 1

119870119894+1 le1

0119894

minus 1

119873119901119894 119870119894+1 119873119898119894+1 are all integers for 119894 = 1 2 119878 minus 1

(16)

4 Numeral Experiment and Analysis

In this section a numeral experiment is proposed based onproper parameter settings All results in this research wereperformed on a PC Programs were written in Matlab 2012bAn efficiency comparison between Lingo PSO algorithmand Ant Colony Algorithm was proposed as well Further-more we would like to discuss how production runs anddefective rate influences whole system cost

41 Experiment Results of Ant Colony Algorithm ParticleSwarm Optimization and Lingo With appropriate param-eters settings we may build a 7-echelon supply chain inte-grated JIT inventory model with uncertain delivery leadtime and unreliable quality of received items Assume thepurchasing time interval for first purchaser in years (1198791199011) is12 years The relevant data of purchasing and manufacturingactivities are shown in Tables 1 and 2

The objective function and constraint function are shownbelow Consider

1198691198621119878 (119873119901 119898119871 119873119898 119870)

=

119878minus1

sum

119894=1

V1198941205931198941198791199011 (119898119871119894 minus 120583119871119889119894) + V1198941205931198941198791199011119909119894120588119894 +(1205931198941198791199011)

2

119870119894+1prod119894

119897=1119873119898119897

(Ω119894 minus

119867119898119894+1119863119894119870119894+1120588119894

2

) +

1198791199011

2

prod119894

119897=1119873119898119897

120579119894

+(

119894

prod

119897=1

119873119898119897)119870119894+1119873119898119894+1 [(119903119871119894 minus 119898119871119894) V119894 int119903119871119894

119898119871119894

(119871119889119894 minus 119898119871119894) 119891 (119871119889119894) 119889119871119889119894 + 120576119894 int

119903119871119894

119898119871119894

119891 (119871119889119894) 119889119871119889119894 + 119865119894]

+(

119894

prod

119897=1

119873119898119897)119873119901119894119862119904119894 + (

119894

prod

119897=1

119873119898119897)119862119898119894+1

Subject to 0 le 120582119871119894 le 119898119871119894 le 120574119871119894 le 119905119894 for 119894 = 1 2 119878 minus 1

119870119894+1 le1

0119894

minus 1

119873119901119894 119870119894+1 119873119898119894+1 are all integers for 119894 = 1 2 119878 minus 1

(17)

The goal of this research is to find out the optimal solutionof the 7-echelon inventory model via the above functions andACA and compare the efficiency between Lingo ACA andPSO

As Table 3 shows though all the three methods werecapable of finding the optimal solution for SMEIJI problemACA was obviously better than PSO and Lingo in efficiencyACA has shown fastest CPU searching time and least averageiterations among three methods

42 Experiments of Performance and 119878-Echelon In this sec-tion we would like to discuss the relationship between CPU

searching time and echelon number among three methodswe also discussed the relationship between average iterationsand echelon number

As Figure 3 shows the CPU searching time wouldincrease with the number of echelons due to the compli-cated calculation However because of the well-organizedconstraint function and good initial solution ACA still keptlower CPU searching time than the other two methods Onthe other hand Figure 4 shows that ACArsquos average iterationswere the least among three methods At the beginning of themultiechelon all the three algorithms need a great amountof iterations to search the optimal solution After calculating

Mathematical Problems in Engineering 9

Table 1 The relevant data of purchasing activities

Purchasing activityEchelon 119863119894 119862119904119894 119867119901119894 119862119890119894 120573119894 120582119871119894 120574119871119894 120583119871119889119894 120590119871119889119894 119865119894 119883119894 0119894

1 18 w 25 28 29 32 0005 0015 0009 0002 70 0005 0032 20w 28 22 32 26 0004 001 00064 00012 50 0003 0013 24w 32 17 38 21 0003 0009 00054 00012 48 0003 0044 30w 36 12 42 16 0004 0012 00072 00016 55 0004 0055 35w 35 08 37 12 0006 0016 001 0002 80 0005 0016 40w 38 06 40 09 0004 0014 0008 0002 65 0005 0027 mdash mdash mdash mdash mdash mdash mdash mdash mdash mdash mdash mdash

Table 2 The relevant data of manufacturing activities

Manufacturing activitiesEchelon 119875119894 119862119898119894 119867119898119894

1 mdash mdash mdash2 200000 280 253 240000 200 194 300000 320 145 350000 360 106 400000 380 097 450000 420 05

1 2 3 4 5 6 70

05

1

15

2

25

3

ACAPSO

Lingo

Figure 3 Experiment results of CPU time and echelon number

repeatedly the trend of the average iterations would come flatwithout extreme variety

Nevertheless although ACA seems better than the othertwo algorithms we could not ensure that the performancewould still be great when applied to higher echelon inventorymodel Hence a 15-echelon model has proposed to observethe performance again The following table displayed theparameters of 15-echelon model

To evaluate the performance of ACA PSO and Lingo forthe 15-echelon inventorymodel we applied the parameters inTables 4 and 5 to solve the 15-echelon inventory problem

According to Table 6 all three algorithms were still ableto find out the optimal solution of the 15-echelon inventorymodel Once the echelon increases the complexity of the

1 2 3 4 5 6 70

500

1000

1500

2000

2500

3000

3500

4000

ACAPSO

Lingo

Figure 4 Experiment results of average iterations and echelonnumber

model increases as well However the performance of ACAwas still better than the other two algorithms

There are several advantages of ACA as follows

(1) Applying the positive and negative feedback enablesthe process of searching optimization to converge atthe end which means the results would close to theoptimal solutions gradually

(2) Every single ant is able to change the surroundingenvironment through releasing pheromone and beingconscious of the variety of environment which makesthem communicate with each other

(3) Due to distributed computing all the individuals startto calculate simultaneously during the searching pro-cess which enhances the efficiency of the calculationability

(4) For the proposed algorithm it is difficult to stick inthe partial optimization instead it is always easy tofind the best solution

To sum up ACA was proposed to solve the complicatedmathematical problem we also have proved that ACA wasmore efficient than PSO and Lingo These are the new majorcontributions of the present paper

43 The Sensitive Analysis of SMEIJI In this section thenumber of production runs 119873119898119894 and the defective rate of

10 Mathematical Problems in Engineering

Table 3 Comparisons between three methods in SMEIJI problem

Echelon ACA PSO Lingo119873119901119894 119898119871119894 119870119894 119873119898119894 119873119901119894 119898119871119894 119870119894 119873119898119894 119873119901119894 119898119871119894 119870119894 119873119898119894

1 1 00147 mdash mdash 1 00147 mdash mdash 1 00147 mdash mdash2 1 000987 11 1 1 000987 11 1 1 000987 11 13 1 000889 13 1 1 000889 13 1 1 000889 13 14 1 001184 11 1 1 001184 11 1 1 001184 11 15 1 001571 8 1 1 001571 8 1 1 001571 8 16 1 001378 5 1 1 001378 5 1 1 001378 5 17 mdash mdash 5 1 mdash mdash 5 1 mdash mdash 5 1Optimal cost 89166 89166 89166CPU time 0655 1744 23Average iterations 350 1278 3758

Table 4 The relevant data of purchasing activities of 15-echelon model

Purchasing activityEchelon 119863119894 119862119904119894 119867119901119894 119862119890119894 120573119894 120582119871119894 120574119871119894 120583119871119889119894 120590119871119889119894 119865119894 119883119894 0119894

1 18 w 25 28 29 32 0005 0015 0009 0002 70 0005 0032 20w 28 22 32 26 0004 001 00064 00012 50 0003 0013 24w 32 17 38 21 0003 0009 00054 00012 48 0003 0044 30w 36 12 42 16 0004 0012 00072 00016 55 0004 0055 35w 35 08 37 12 0006 0016 001 0002 80 0005 0016 40w 38 06 40 09 0004 0014 0008 0002 65 0005 0027 42w 40 05 28 12 0005 0013 001 0002 70 0005 0038 45w 42 07 35 15 0004 0009 0009 00018 50 0004 0029 50w 44 10 42 10 0003 0010 0005 00024 55 0004 00210 52w 46 15 35 22 0005 0012 00062 00025 60 0005 00211 56w 48 20 49 25 0006 0009 001 00026 65 0003 00312 60w 50 22 40 14 0004 0016 0012 0002 48 0005 00413 62w 55 17 35 15 0007 0015 0009 00018 72 0004 00414 65w 56 12 32 16 0003 0014 00078 0002 70 0006 00315 mdash mdash mdash mdash mdash mdash mdash mdash mdash mdash mdash mdash

Table 5The relevant data of manufacturing activities of 15-echelonmodel

Manufacturing activitiesEchelon 119875119894 119862119898119894 119867119898119894

1 mdash mdash mdash2 200000 280 253 240000 200 194 300000 320 145 350000 360 106 400000 380 097 450000 420 058 480000 440 109 500000 420 1210 520000 400 1511 540000 450 2012 580000 440 2213 620000 420 1814 640000 460 1515 660000 380 16

receiving shipment 0119894 were considered to make sensitiveanalysis and observe the influence on the optimal total jointcost The defective rate and production runs were changedby 0119894(1 + 119877) and 119873119898119894(1 + 119877) where the values of 119877 are0 50 100 200 300 400

Figure 5 illustrated that defective rate was more sensitivethan production runs under the situation that productionruns were one-time production except first echelon In thatcase the higher the defective ratewas the higher the total costwas Accordingly inspecting goods cautiously before deliveryand reducing the defective rate would lower total cost of thewhole supply chainHowever if relaxing that assumption thissensitive analysis would result in different outcomes

Figure 6 has shown that production runs were more sen-sitive than defective rate when relaxing one-time productionassumption In this case the higher the production runswerethe higher the total cost was Hence reducing productionruns should play an essential role in the whole supply chainOnce the one batch production policy was able to satisfy thedemand from the buyer the total cost of whole supply chainwill go down extremely

Mathematical Problems in Engineering 11

Table 6 Comparisons between three ways of 15-echelon inventory model

ACA PSO LingoEchelon 119873119901119894 119898119871119894 119870119894 119873119898119894 119873119901119894 119898119871119894 119870119894 119873119898119894 119873119901119894 119898119871119894 119870119894 119873119898119894

1 1 00147 mdash mdash 1 00147 mdash mdash 1 00147 mdash mdash2 1 000987 11 1 1 000987 11 1 1 000987 11 13 1 000889 13 1 1 000889 13 1 1 000889 13 14 1 001184 11 1 1 001184 11 1 1 001184 11 15 1 001571 8 1 1 001571 8 1 1 001571 8 16 1 001378 5 1 1 001378 5 1 1 001378 5 17 1 001451 5 1 1 001451 5 1 1 001451 5 18 1 001287 5 1 1 001287 5 1 1 001287 5 19 1 00154 7 1 1 00154 7 1 1 00154 7 110 1 001432 9 1 1 001432 9 1 1 001432 9 111 1 001352 7 1 1 001352 7 1 1 001352 7 112 1 00162 8 1 1 00162 8 1 1 00162 8 113 1 001571 10 1 1 001571 10 1 1 001571 10 114 1 001611 5 1 1 001611 5 1 1 001611 5 115 mdash mdash 5 1 mdash mdash 5 1 mdash mdash 5 1Optimal cost ($) 231254 231254 231254CPU time (s) 1725 6621 58215Average iterations 700 2883 8241

Defective rateProduction runs

400100 200 300 5000 500

20000

40000

60000

80000

100000

120000

140000

160000

180000

Figure 5 The sensitive analysis between defective rate and produc-tion runs

5 Conclusions

In summary this research proposed an integrated JIT inven-tory model to solve the partial optimization problem ofpurchaser andmanufacturer finding the optimal solution forthe whole supply chain Furthermore a multiechelon supplychainwith uncertain delivery lead time and unreliable qualityhas been considered to fit the real supply chain environ-ment Besides ACA was proposed to solve the complicatedmathematical problem we also have proved that ACA was

0 50 100 200 300 400 500

Defective rateProduction runs

0

50000

100000

150000

200000

250000

300000

350000

400000

Figure 6 The sensitive analysis between defective rate and produc-tions runs with relaxing assumption

more efficient than PSO and Lingo These are the new majorcontributions of the present paper

Finally there are a great deal of helpful directions forenterprises and researchers (1)The variety of conditions thatoccur in real supply chain can be considered in the modelFor instance time value of cash problem and deterioration(2)The warehousing and storage can be considered in supplychain which make the model more complex to solve (3)The model contains only one member in each echelon amultimember in each echelon should be considered to fit

12 Mathematical Problems in Engineering

real situation in supply chain (4) Some real cases andordata can be applied to our proposed model in the followingworks

Futureworkwill hopefully involve the real-world concernto our research which will be our main target In addition we

do hope our research can be extended intomore considerableareas and more considerable variables

Appendix

The Hessian matrix 119865 of 1198691198621119878(119873119901 119898119871 119873119898 119870) for given 119873119901119894

equal to 1 can be shown as

119865 =

[

[

[

[

[

[

[

[

[

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

1205971198731198981198942

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119873119898119894120597119898119871119894

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119873119898119894120597119870119894

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119898119871119894120597119873119898119894

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

1205971198981198711198942

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119898119871119894120597119870119894

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119870119894120597119873119898119894

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119870119894120597119898119871119894

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

1205971198701198942

]

]

]

]

]

]

]

]

]

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

1205971198731198981198942

=

2

1198731198981198943(120579120593119879

2+

1205931198792(1198671198631198701205932 + Ω)

119870

)

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

1205971198981198711198942

= 119870119873[(

119890minus119898120583

1205832

+ 2 (119890minus119898120583

minus 119890minus120574120583

) +

119890(minus119898120583)(minus119898+120583)

120583

) 120592]

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

1205971198701198942

=

2

1198701198943(120579120593119879

2+

1205931198792(1198671198631198701205932 + Ω)

119873

)

12059721198691198621119904 (119873119898119894 119898119871119894 119870119894)

120597119898119871119894120597119873119898119894

=

12059721198691198621119904 (119873119898119894 119898119871119894 119870119894)

120597119873119898119894120597119898119871119894

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119870119894120597119873119898119894

=

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119873119898119894120597119870119894

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119870119894120597119898119871119894

=

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119898119871119894120597119870119894

100381610038161003816100381611986511

1003816100381610038161003816=

1205972119865

1205971198731198981198942

=

2

1198731198981198943(120579120593119879

2+

1205931198792(1198671198631198701205932 + Ω)

119870

) gt 0

100381610038161003816100381611986522

1003816100381610038161003816= (

1205972119865

1205971198731198981198942)(

1205972119865

1205971198981198711198942) minus [

1205972119865

120597119873119898119894120597119898119871119894

]

2

=

2

1198731198981198943(120579120593119879

2+

1205931198792(1198671198631198701205932 + Ω)

119870

)

1205972119865

1205971198981198711198942minus (

1205972119865

120597119873119898119894120597119898119871119894

)

2

2

1198733119898119894

(1205791205931198792+

119861

119870

)[119870119873(

119890minus119898120576120583

1205832

+ (2119909 +

119890(minus119898120583)119909119910

120583

120592))] + 1198702[

119890minus119898120576120583

120583

+ 120592 (119909119910 minus 119860)] gt 0

100381610038161003816100381611986533

1003816100381610038161003816= (

1205972119865

1205971198731198981198942

1205972119865

1205971198981198711198942

1205972119865

1205971198701198942) minus [(

1205972119865

120597119873119898119894120597119870119894

)

2

(

1205972119865

1205971198981198711198942)]

=

2

1198731198981198943(120579120593119879

2+

119861

119870

)[119870119873(

119890minus119898120576120583

1205832

+ 2(119909 +

119890(minus119898120603)119909119910

120583

) 120592)]

2

1198703

119861

119873

minus [119865 + 119909120576 + 119860120592119910 minus

1205791205931198792

1198732

+

119861

2 (119870119873)2]

[119870119873(

119890minus119898120576120583

1205832

+ (2119909 +

119890minus119898120576120583

120583

) 120592)] gt 0

(A1)

Mathematical Problems in Engineering 13

As a result the matrix F has proved to be a positivedefinite matrix where |11986511| |11986522| and |11986533| are all positiveTherefore theminimumvalue exists in 1198691198621119878(119873119901119898119871 119873119898 119870)Furthermore we can find the optimization in the proposedmodel

Competing Interests

The authors declare that they have no competing interests

References

[1] S K Goyal ldquoAn integrated inventory model for a singlesupplier-single customer problemrdquo International Journal ofProduction Research vol 15 no 1 pp 107ndash111 1977

[2] A Banerjee ldquoA joint economic-lot-sizemodel for purchaser andvendorrdquo Decision Sciences vol 17 no 3 pp 292ndash311 1986

[3] S K Goyal ldquoldquoA joint economic-lot-size model for purchaserand vendorrdquo a commentrdquo Decision Sciences vol 19 no 1 pp236ndash241 1988

[4] Y Seo ldquoControlling general multi-echelon distribution supplychainswith improved reorder decision policy utilizing real-timeshared stock informationrdquo Computers amp Industrial Engineeringvol 51 no 2 pp 229ndash246 2006

[5] H N Chiu and H L Huang ldquoA Multi-echelon integratedJIT inventory model using the time buffer and emergencyborrowing policies to deal with random delivery lead timesrdquoInternational Journal of Production Research vol 41 no 13 pp2911ndash2931 2003

[6] J Sadeghi S M Mousavi S T A Niaki and S SadeghildquoOptimizing a multi-vendor multi-retailer vendor managedinventory problem two tuned meta-heuristic algorithmsrdquoKnowledge-Based Systems vol 50 pp 159ndash170 2013

[7] S S Sana ldquoA production-inventory model of imperfect qualityproducts in a three-layer supply chainrdquo Decision Support Sys-tems vol 50 no 2 pp 539ndash547 2011

[8] H N Soni and K A Patel ldquoOptimal strategy for an integratedinventory system involving variable production and defectiveitems under retailer partial trade credit policyrdquoDecision SupportSystems vol 54 no 1 pp 235ndash247 2012

[9] R J Tersine Principles of Inventory andMaterials ManagementNorthHolland AmsterdamTheNetherlands 3rd edition 1988

[10] C J Liao and C H Shyu ldquoAn analytical determination of leadtime with normal demandrdquo International Journal of Operationsamp Production Management vol 11 no 9 pp 72ndash78 1991

[11] M Ben-Daya and A Raouf ldquoInventory models involving leadtime as a decision variablerdquo Journal of the Operational ResearchSociety vol 45 no 5 pp 579ndash582 1994

[12] S M J Mirzapour Al-e-Hashem A Baboli and Z Sazvar ldquoAstochastic aggregate production planningmodel in a green sup-ply chain considering flexible lead times nonlinear purchaseand shortage cost functionsrdquo European Journal of OperationalResearch vol 230 no 1 pp 26ndash41 2013

[13] C Chandra and J Grabis ldquoInventorymanagementwith variablelead-time dependent procurement costrdquo Omega vol 36 no 5pp 877ndash887 2008

[14] J-W Wu and H-Y Tsai ldquoMixture inventory model with backorders and lost sales for variable lead time demand withthe mixtures of normal distributionrdquo International Journal ofSystems Science Principles and Applications of Systems andIntegration vol 32 no 2 pp 259ndash268 2001

[15] M A Hoque ldquoA vendorndashbuyer integrated productionndashinventory model with normal distribution of lead timerdquoInternational Journal of Production Economics vol 144 no 2pp 409ndash417 2013

[16] F Altiparmak M Gen L Lin and T Paksoy ldquoA geneticalgorithm approach for multi-objective optimization of supplychain networksrdquo Computers and Industrial Engineering vol 51no 1 pp 196ndash215 2006

[17] J Sadeghi S Sadeghi and S T Niaki ldquoA hybrid vendormanaged inventory and redundancy allocation optimizationproblem in supply chain management an NSGA-II with tunedparametersrdquo Computers and Operations Research vol 41 pp53ndash64 2014

[18] X Li F Baki P Tian and B A Chaouch ldquoA robust block-chainbased tabu search algorithm for the dynamic lot sizing problemwith product returns and remanufacturingrdquoOmega vol 42 no1 pp 75ndash87 2014

[19] Z Chen and B R Sarker ldquoAn integrated optimal inventorylot-sizing and vehicle-routing model for a multisupplier single-assembler system with JIT deliveryrdquo International Journal ofProduction Research vol 52 no 17 pp 5086ndash5114 2014

[20] A R Nia M Hemmati Far and S T A Niaki ldquoA fuzzy vendormanaged inventory of multi-item economic order quantitymodel under shortage an ant colony optimization algorithmrdquoInternational Journal of Production Economics vol 155 pp 259ndash271 2014

[21] S M Hatefi and F Jolai ldquoRobust and reliable forward-reverselogistics network design under demand uncertainty and facilitydisruptionsrdquo Applied Mathematical Modelling vol 38 no 9-10pp 2630ndash2647 2014

[22] R H Ma ldquoCross warehouse scheduling of logistics distributionand cycle re-claimer based on heuristics algorithmrdquo AppliedMechanics and Materials vol 644ndash650 pp 2606ndash2610 2014

[23] M Hashim L Yao A H Nadeem M Nazim and M NazamldquoMulti-objective optimization model for supplier selectionproblem in fuzzy environmentrdquo Advances in Intelligent Systemsand Computing vol 281 pp 1201ndash1213 2014

[24] Q-M Hu and Z-H Hu ldquoA stochastic programming modelfor hub-and-spoke network with uncertain flowsrdquo InternationalJournal of Industrial and Systems Engineering vol 21 no 3 pp302ndash319 2015

[25] S A Torabi J Namdar S M Hatefi and F Jolai ldquoAn enhancedpossibilistic programming approach for reliable closed-loopsupply chain network designrdquo International Journal of Produc-tion Research vol 54 no 5 pp 1358ndash1387 2016

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

Time

Inventory level ti ti

Di

qi

ri

0iqi

mLi mLi mLixi + Ldi minus mLi

Ldi Ldixi mLi + ti minus Ldi

0 le 120582Li le Ldi le mLi mLi le Ldi le 120574Li le ti

(1 minus 0i)qi

(B)

(A)

Figure 1 Delivery cases for the member who implements purchasing

(8) The required time of borrowing goods from nearbysupplier is ignored

(9) A finite planning horizon for the whole serial supplychain that denoted 119879119901119894 is considered

(10) Defective items in a shipment of materials arereceived by purchaser 119894 with defective rate 0119894 Thenumber of goodmaterialsmust bemore than or equalto the required quantity during the screening time

(11) The screening process is followed by the arrival ofshipments with screening time 119909119894 The length ofscreening time is proportional to the number ofreceived materials

(12) The accumulated number of defective items that weredelivered to the buyer during a production run mustbe less than the quantity of one delivery

(13) When emergency borrowing happened the materialsthat were borrowed and returned are all good items

3 Formulation of the Model

In this integrated multiechelon JIT inventory model thedecision variables are 119899119894 (number of deliveries per purchase)119873119901119894 (number of purchase orders during 119879119901119894) and 119898119871119894 (themaximum allowable planned delivery lead time that wouldnot cause shortage) where 119894 = 1 119878minus1119873119898119894 is the numberof production runs during 119879119898119894 and 119870119894 are delivery times perproduction runs for manufacturer 119894 where 119894 = 2 119878 Thevalue of these decision variables should be determined inorder to minimize the joint total cost

31 The Cost of Member 119894 in Purchasing Activities Fromechelon1 to echelon 119878 minus 1 members adopt JIT purchasing toreplenish goodsThe time buffer policy used on planned lead

time (119898119871119894) and emergency borrowing policy used to deal withuncertain delivery lead time Each order lot size119876119901119894 has beentaken apart into 119899119894 small lots for delivery frequently Duringthe planning horizon 119879119901119894 the total number of deliveriesfor member 119894 is 119899119894 times 119873119901119894 The delivery lead time follows aprobability density distribution 119891(119871119889119894) The 120574119871119894 and 120582119871119894 arethe upper bound and the lower bound of 119871119889119894

The costs are related to purchasing activities of member119894 during 119879119901119894 where 119894 = 1 119878 minus 1 including ordering costholding cost delivery cost transportation carrying cost andemergency borrowing cost The transportation carrying costwould not be considered in this model since it is constant anddoes not affect any decision variables

Owing to the uncertain delivery lead time there are threedifferent delivery cases early arrival delay arrival and arrivalon time The screening process would be implemented whenthe purchaser receives a shipment of goods The defectiveitem would be deducted from the inventory in a singlebatch at the end of the purchaserrsquos 100 screening processFigure 1 has shown two delivery cases for the member whoimplements purchasing

On the left side of Figure 1 0 le 120582119871119894 le 119871119889119894 le 119898119871119894means theearly arrival occurs and results in unexpected holding costThe shadow area (A) presents the unexpected extra inventoryIn this case the holding cost (HCA119894) of member 119894 during 119905119894

is shownThe shadow area (A) presents the unexpected extrainventory caused by early delivery and 119905119894119902119894(1minus0119894)2+(119909119894+119871119889119894minus

119898119871119894)1199021198940119894 represents the expected inventoryIn the right side of Figure 1 119898119871119894 le 119871119889119894 le 120574119871119894 le 119905119894

presents the delay arrival situation When it occurs member119894 should adopt emergency borrowing policy The shadowarea (B) presents the quantity of emergency borrowing Theborrowing quantity from other nearby suppliers outside thesupply chain is 119863119894(120574119871119894minus119898119871119894) units After the delay arrivallot is received the remand quantity of goods 119863119894(120574119871119894 minus 119871119889119894)

Mathematical Problems in Engineering 5

should be given back immediately The borrowed and theremanded units would not be screened and the screeningtime is ((119902119894 minus119863119894(120574119871119894 minus119898119871119894))119902119894)119909119894 during119898119871119894 le 119871119889119894 le 120574119871119894 le 119905119894The holding cost (HCB119894) and the expected borrowing cost(EBC119894) of member 119894 during 119905119894 are shown The holding cost(HCB119894) involves the normal holding cost and the emergencyborrowing holding cost

The unexpected extra inventory of shadow area (A) dueto the early arrivals is as follows

HCA119894 = 119867119901119894 [119902119894 (119898119871119894 minus 119871119889119894) +

119905119894119902119894 (1 minus 0119894)

2

+ (119909119894 + 119871119889119894 minus 119898119871119894) 1199021198940119894] = 119863119894119867119901119894 [119905119894 (119898119871119894 minus 119871119889119894)

+

1199051198941199091198940119894

1 minus 0119894

+

119905119894

2

2

]

(1)

The emergency borrowing holding cost due to delay arrivalsis as follows

119863119894119867119901119894 [

[(119903119871119894 minus 119871119889119894) + (119903119871119894 minus 119898119871119894)] (119871119889119894 minus 119898119871119894)

2

] (2)

Normal holding cost without early or delay arrivals is asfollows

119863119894119867119901119894 [

(119905119894 + 119871119871119894

minus 119871119889119894)2

2

+ (

119905119894 minus (119903119871119894 minus 119898119871119894) (1 minus 0119894)

1 minus 0119894

)1199091198940119894]

HCB119894 = 119867119901119894 [(119861) + (119862)] = 119863119894119867119901119894 [(119903119871119894 minus 119898119871119894)

+ (119871119889119894 minus 119898119871119894)] + 119905119894 (119898119871119894 minus 119871119889119894) +

119905119894

2

2

minus (119903119871119894 minus 119898119871119894)

sdot (1199091198940119894) +

1199051198941199091198940119894

1 minus 0119894

(3)

Nowwe combine (1) and (3) to form the expected holdingcost (EHC119894) of member 119894 as follows

EHC119894 = int

119898119871119894

120582119871119894

HCA119894119891 (119871119889119894) 119889119871119889119894

+ int

119903119871119894

119898119871119894

HCB119894119891 (119871119889119894) 119889119871119889119894 = 119863119894119867119901119894 [119905119894 (119898119871119894

minus 119871119889119894) +

119905119894

2

2

+

1199051198941199091198940119894

1 minus 0119894

+ (119903119871119894 minus 119898119871119894)

sdot [int

119903119871119894

119898119871119894

(119871119889119894 minus 119898119871119894) 119891 (119871119889119894) 119889119871119889119894

minus (1199091198940119894) int

119903119871119894

119898119871119894

119891 (119871119889119894) 119889119871119889119894]]

(4)

The expected borrowing cost (EBC119894) is as follows

EBC119894 = 119862119890119894 int

119903119871119894

119898119871119894

119891 (119871119889119894) 119889119871119889119894

+ 120573119894 int

119903119871119894

119898119871119894

119863119894 (119903119871119894 minus 119898119871119894) 119891 (119871119889119894) 119889119871119889119894

= [119862119890119894 + 120573119894119863119894 (119903119871119894 minus 119898119871119894)] int

119903119871119894

119898119871119894

119891 (119871119889119894) 119889119871119889119894

(5)

The EBC119894 only occurs at time interval [120574119871119894 119898119871119894]119862119890119894 int119903119871119894

119898119871119894119891(119871119889119894)119889119871119889119894 means the ordering cost of emergency

borrowing and 120573119894 int119903119871119894

119898119871119894119863119894(119903119871119894 minus 119898119871119894)119891(119871119889119894)119889119871119889119894 presents the

excepted cost of borrowed unitsDuring the purchasing time interval (119879119901119894) of each mem-

ber it has 119873119901119894 purchasing orders 119899119894119873119901119894 delivery receivingtimes 119899119894119873119901119894119865119894 delivery cost and 119873119901119894119862119904119894 ordering cost Thusthe expected cost (EP119888119894) of member 119894 in purchasing activitiesis shown as follows

EC119901119894 (119899119894 119873119901119894 119898119871119894) = holding cost

+ emergency borrowing cost + ordering cost

+ delivery cost = 119899119894119873119901119894 [EHC119894 + EBC119894] + 119873119901119894119862119904119894

+ 119899119894119873119901119894119865119894 = 119899119894119873119901119894119863119894119867119901119894 [119905119894 (119898119871119894 minus 119871119889119894) +

1199051198942

2

+

1199051198941199091198940119894

1 minus 0119894

+ (119903119871119894 minus 119898119871119894)]

sdot [int

119903119871119894

119898119871119894

(119871119889119894 minus 119898119871119894) 119891 (119871119889119894) 119889119871119889119894

minus (1199091198940119894) int

119903119871119894

119898119871119894

119891 (119871119889119894) 119889119871119889119894] + 119899119894119873119901119894 [119862119890119894

+ 120573119894119863119894 (119903119871119894 minus 119898119871119894)] int

119903119871119894

119898119871119894

119891 (119871119889119894) 119889119871119889119894 + 119865119894 + 119873119901119894119862119904119894

(6)

32 The Cost of Member 119894 Related to Manufacturing ActivitiesIn this model member 119894 + 1 adopts JIT manufacturing toproduce goods to member 119894 There are 119873119898119894+1 productionruns during 119879119901119894 The start time of each production can bedetermined by counting backward 119902119894minus1119875119894 time units whenthe first delivery lot is delivered The average inventory ofgoods produced bymember 119894per production run is illustratedby Figure 2 and we can determine this by calculatingthe cumulative time-weighted production quantity (1) (thetrapezoid area) minus the cumulative time-weighted deliveryquantity (2) (the ladder area)

Calculating the cumulative time-weighted productionquantity of member 119894 per production run is equal to calcu-lating a square measure of trapezoid area Another way tocalculate the cumulative time-weighted production quantity

6 Mathematical Problems in Engineering

Time

Cumulative quantity

(2)(2)

(1) (1)

ti

Tpi Tpi

Pi

Tpiminus1NmiTpiminus1Nmi

Qmi

Tpiminus1

Figure 2 Average inventory of goods produced by member 119894 perproduction run

(1) is subtracting the triangle area from the rectangle area asfollows

(1) = [

119879119901119894minus1

119873119898119894

minus (119905119894minus1 minus

119902119894minus1

119901119894

)](

119863119894minus1119879119901119894minus1

119873119898119894

)

minus

1

2

(

119863119894minus1119879119901119894

119875119894119873119898119894

)(

119863119894minus1119879119901119894minus1

119873119898119894

) = (

119879119901119894minus1

119873119898119894

)

2

119863119894minus1 [1

2

+

0119894minus1

119870119894 (1 minus 0119894)

+ (

1

2

minus

1

119870119894 (1 minus 0119894minus1)

)(1 minus

119863119894minus1

119875119894

)]

(7)

During each production run of member i there are 119870119894

deliveries and 119902119894 units of goods delivered per shipment Thetime interval between two adjacent deliveries for member119894 (119905119894) should be longer than or equal to 120574119871119894 Hence thecumulative time-weighted delivery quantity (2) of member 119894

per production run is

(2) = (

119870119894minus1

sum

119897

119897) 119905119894minus1119902119894minus1 = (1 + 2 + sdot sdot sdot + (119870119894minus1)) 119905119894minus1119902119894minus1

= (

119879119901119894minus1

119873119898119894

)

2

119863119894minus1 [1

2

119870119894minus1

119870119894 (1 minus 0119894minus1)

]

(8)

Consequently the average inventory (AI119894) of member 119894

during 119879119901119894minus1 can be calculated by subtracting area (2) fromarea (1) as follows

AI119894 = 119873119898119894 (1) minus (2) = 119873119898119894 (

119879119901119894minus1

119873119898119894

)

2

119863119894minus1 [1

2

+

0119894

119870119894 (1 minus 0119894)

+ (

1

2

minus

1

119870119894 (1 minus 0119894minus1)

)(1 minus

119863119894

119875119894

)]

minus

119879119901119894minus1

119873119898119894

119863119894minus1 [1

2

119870119894minus1

119870119894 (1 minus 0119894minus1)

] = (

119879119901119894minus12

119873119898119894

)

sdot 119863119894minus1 [1

2

(

119870119894 + 0119894minus1 minus 1

119870119894 (1 minus 0119894)

) + (

1

2

minus

1

119870119894 (1 minus 0119894minus1)

)

sdot (1 minus

119863119894minus1

119875119894

)]

(9)

Thus the production cost of member 119894 (EC119898119894) during 119879119901119894 is

EC119898119894 (119873119898119894119870119894) = 119873119898119894119862119898119894 + 119867119898119894AI119894 = 119873119898119894119862119898119894

+ 119867119898119894 (

119879119901119894minus12

119873119898119894

)119863119894minus1 [1

2

(

119870119894 + 0119894minus1 minus 1

119870119894 (1 minus 0119894)

)

+ (

1

2

minus

1

119870119894 (1 minus 0119894minus1)

)(1 minus

119863119894minus1

119875119894

)]

(10)

33 The Joint Cost of Member 119894 in Purchasing Activities andMember 119894 + 1 in Manufacturing Activities With the abovediscussion we have inferred the expected cost of member 119894

in purchasing andmanufacturing respectivelyTherefore thejoint cost of member 119894 in purchasing activities and member119894 + 1 in manufacturing activities can be obtained easily Wesubstituted 119899119894119873119901119894 = 119870119894+1119873119898119894+1 into (6) and (10) and gainedthe joint cost of members 119894 and 119894 + 1 as

119862119894119894+1 (119873119901119894 119898119871119894 119873119898119894+1 119870119894+1) = EC119901119894 + EC119898119894+1

= 119873119898119894+1119870119894+1 [EHC119894 + EBC119894] + 119873119901119894119862119901119894

+ 119873119898119894+1119862119898119894+1 + 119867119898119894+1AI119894+1

= 119873119898119894+1119870119894+1 119863119894119867119901119894 [119905119894 (119898119871119894 minus 120583119871119889119894)] +

1199051198942

2

+

1199051198941199091198940119894

1 minus 0119894

+ (119903119871119894 minus 119898119871119894)

sdot [int

119903119871119894

119898119871119894

(119871119889119894 minus 119898119871119894) 119891 (119871119889119894) 119889119871119889119894

minus (1199091198940119894) int

119903119871119894

119898119871119894

119891 (119871119889119894) 119889119871119889119894] + [119862119890119894 + 120573119894119863119894 (119903119871119894

minus 119898119871119894)] [int

119903119871119894

119898119871119894

119891 (119871119889119894) 119889119871119889119894 + 119865119894] + 119873119901119894119862119904119894

+ 119873119898119894119862119898119894 + 119867119898119894 (

119879119901119894minus1

2

119873119898119894

)

sdot 119863119894minus1 [1

2

(1 minus

119870119894 + 0119894minus1 minus 1

119870119894 (1 minus 0119894)

)

+ (

1

2

minus

1

119870119894 (1 minus 0119894minus1)

)(1 minus

119863119894minus1

119875119894

)]

(11a)

Mathematical Problems in Engineering 7

Substitute 119905119894 = 119879119901119894119870119894+1119873119898119894+1 into (11a) Consider

119862119894119894+1 (119873119901119894 119898119871119894 119873119898119894+1 119870119894+1) = V119894119879119901119894 (119898119871119894 minus 120583119871119889119894)

+ V119894119879119901119894119909119894120588119894 +119879119901119894

2

119870119894+1119873119898119894+1

(Ω119894 minus

119867119898119894+1119863119894119870119894+1120588119894

2

)

+

119879119901119894

2

119873119898119894+1

120579119894 + 119870119894+1119873119898119894+1 [(119903119871119894 minus 119898119871119894)

sdot V119894 int119903119871119894

119898119871119894

(119871119889119894 minus 119898119871119894) 119891 (119871119889119894) 119889119871119889119894

+ 120576119894 int

119903119871119894

119898119871119894

119891 (119871119889119894) 119889119871119889119894 + 119865119894] + 119873119901119894119862119904119894

+ 119873119898119894+1119862119898119894+1

(11b)

where

V119894 = 119867119901119894119863119894

120588119894 =0119894

1 minus 0119894

Ω119894

=

119863119894

2

[119867119901119894 + 119867119898119894+1 (1 minus (

2

1 minus 0119894

)(1 minus

119863119894

119875119894+1

))]

120576119894 = 119862119890119894 + (120573119894119863119894 minus V1198941199091198940119894) (119903119871119894 minus 119898119871119894)

120579119894 =

119867119898119894+1119863119894

2

(1 minus

119863119894

119875119894+1

)

(12)

Since 119879119901119894 = 119876119898119894minus1119875119894 = (119863119894minus1119873119898119894+1119875119894)119879119901119894minus1 119879119901119894 can beexpressed further as

119879119901119894 = (

120593119894

prod119894

119897=1119873119898119897

)1198791199011 (13)

where

120593119894 =

1 if 119894 = 1

119894

prod

119897=2

(

119863119894minus1

119875119894

) if 119894 = 2 3 119878 minus 1

(14)

All the 119879119901119894 in function (11b) could be replaced as follows

119862119894119894+1 (119873119901119894 119898119871119894 119873119898119894+1 119870119894+1) = V119894(120593119894

prod119894

119897=1119873119898119897

)

sdot 1198791199011 (119898119871119894 minus 120583119871119889119894) + V119894(120593119894

prod119894

119897=1119873119898119897

)1198791199011119909119894120588119894

+

(120593119894prod119894

119897=1119873119898119897) 1198791199011

2

119870119894+1119873119898119894+1

(Ω119894 minus

119867119898119894+1119863119894119870119894+1120588119894

2

)

+

(120593119894prod119894

119897=1119873119898119897) 1198791199011

2

119873119898119894+1

120579119894

+ 119870119894+1119873119898119894+1 [(119903119871119894 minus 119898119871119894)

sdot V119894 int119903119871119894

119898119871119894

(119871119889119894 minus 119898119871119894) 119891 (119871119889119894) 119889119871119889119894

+ 120576119894 int

119903119871119894

119898119871119894

119891 (119871119889119894) 119889119871119889119894 + 119865119894] + 119873119901119894119862119904119894

+ 119873119898119894+1119862119898119894+1

(11c)

There are119873119898119894+1 purchasing times of member 119894+1 during119879119901119894 According to (11c) the joint total cost function includingpurchasing and manufacturing activities of the serial 119878-echelon integrated JIT inventory model is presented in

119878minus1

sum

119894=1

V1198941205931198941198791199011 (119898119871119894 minus 120583119871119889119894) + V1198941205931198941198791199011119909119894120588119894

+

(1205931198941198791199011)

2

119870119894+1prod119894

119897=1119873119898119897

(Ω119894 minus

119867119898119894+1119863119894119870119894+1120588119894

2

)

+

1198791199011

2

prod119894

119897=1119873119898119897

120579119894 + (

119894

prod

119897=1

119873119898119897)119870119894+1119873119898119894+1 [(119903119871119894 minus 119898119871119894)

sdot V119894 int119903119871119894

119898119871119894

(119871119889119894 minus 119898119871119894) 119891 (119871119889119894) 119889119871119889119894

+ 120576119894 int

119903119871119894

119898119871119894

119891 (119871119889119894) 119889119871119889119894 + 119865119894] + (

119894

prod

119897=1

119873119898119897)119873119901119894119862119904119894

+ (

119894

prod

119897=1

119873119898119897)119862119898119894+1

(15)

The delivery time constrains 0 le 120582119871119894 le 119898119871119894 le 120574119871119894 le

119905119894 119894 = 1 2 119878 minus 1 and variables constrains are consideredin the inventory model Eventually the 119878-echelon integratedJIT model can be expressed as

Minimize 1198691198621119878 (119873119901 119898119871 119873119898 119870)

=

119878minus1

sum

119894=1

V1198941205931198941198791199011 (119898119871119894 minus 120583119871119889119894) + V1198941205931198941198791199011119909119894120588119894 +(1205931198941198791199011)

2

119870119894+1prod119894

119897=1119873119898119897

(Ω119894 minus

119867119898119894+1119863119894119870119894+1120588119894

2

) +

11987911990112

prod119894

119897=1119873119898119897

120579119894

8 Mathematical Problems in Engineering

+(

119894

prod

119897=1

119873119898119897)119870119894+1119873119898119894+1 [(119903119871119894 minus 119898119871119894) V119894 int119903119871119894

119898119871119894

(119871119889119894 minus 119898119871119894) 119891 (119871119889119894) 119889119871119889119894 + 120576119894 int

119903119871119894

119898119871119894

119891 (119871119889119894) 119889119871119889119894 + 119865119894]

+(

119894

prod

119897=1

119873119898119897)119873119901119894119862119904119894 + (

119894

prod

119897=1

119873119898119897)119862119898119894+1

Subject to 0 le 120582119871119894 le 119898119871119894 le 120574119871119894 le 119905119894 for 119894 = 1 2 119878 minus 1

119870119894+1 le1

0119894

minus 1

119873119901119894 119870119894+1 119873119898119894+1 are all integers for 119894 = 1 2 119878 minus 1

(16)

4 Numeral Experiment and Analysis

In this section a numeral experiment is proposed based onproper parameter settings All results in this research wereperformed on a PC Programs were written in Matlab 2012bAn efficiency comparison between Lingo PSO algorithmand Ant Colony Algorithm was proposed as well Further-more we would like to discuss how production runs anddefective rate influences whole system cost

41 Experiment Results of Ant Colony Algorithm ParticleSwarm Optimization and Lingo With appropriate param-eters settings we may build a 7-echelon supply chain inte-grated JIT inventory model with uncertain delivery leadtime and unreliable quality of received items Assume thepurchasing time interval for first purchaser in years (1198791199011) is12 years The relevant data of purchasing and manufacturingactivities are shown in Tables 1 and 2

The objective function and constraint function are shownbelow Consider

1198691198621119878 (119873119901 119898119871 119873119898 119870)

=

119878minus1

sum

119894=1

V1198941205931198941198791199011 (119898119871119894 minus 120583119871119889119894) + V1198941205931198941198791199011119909119894120588119894 +(1205931198941198791199011)

2

119870119894+1prod119894

119897=1119873119898119897

(Ω119894 minus

119867119898119894+1119863119894119870119894+1120588119894

2

) +

1198791199011

2

prod119894

119897=1119873119898119897

120579119894

+(

119894

prod

119897=1

119873119898119897)119870119894+1119873119898119894+1 [(119903119871119894 minus 119898119871119894) V119894 int119903119871119894

119898119871119894

(119871119889119894 minus 119898119871119894) 119891 (119871119889119894) 119889119871119889119894 + 120576119894 int

119903119871119894

119898119871119894

119891 (119871119889119894) 119889119871119889119894 + 119865119894]

+(

119894

prod

119897=1

119873119898119897)119873119901119894119862119904119894 + (

119894

prod

119897=1

119873119898119897)119862119898119894+1

Subject to 0 le 120582119871119894 le 119898119871119894 le 120574119871119894 le 119905119894 for 119894 = 1 2 119878 minus 1

119870119894+1 le1

0119894

minus 1

119873119901119894 119870119894+1 119873119898119894+1 are all integers for 119894 = 1 2 119878 minus 1

(17)

The goal of this research is to find out the optimal solutionof the 7-echelon inventory model via the above functions andACA and compare the efficiency between Lingo ACA andPSO

As Table 3 shows though all the three methods werecapable of finding the optimal solution for SMEIJI problemACA was obviously better than PSO and Lingo in efficiencyACA has shown fastest CPU searching time and least averageiterations among three methods

42 Experiments of Performance and 119878-Echelon In this sec-tion we would like to discuss the relationship between CPU

searching time and echelon number among three methodswe also discussed the relationship between average iterationsand echelon number

As Figure 3 shows the CPU searching time wouldincrease with the number of echelons due to the compli-cated calculation However because of the well-organizedconstraint function and good initial solution ACA still keptlower CPU searching time than the other two methods Onthe other hand Figure 4 shows that ACArsquos average iterationswere the least among three methods At the beginning of themultiechelon all the three algorithms need a great amountof iterations to search the optimal solution After calculating

Mathematical Problems in Engineering 9

Table 1 The relevant data of purchasing activities

Purchasing activityEchelon 119863119894 119862119904119894 119867119901119894 119862119890119894 120573119894 120582119871119894 120574119871119894 120583119871119889119894 120590119871119889119894 119865119894 119883119894 0119894

1 18 w 25 28 29 32 0005 0015 0009 0002 70 0005 0032 20w 28 22 32 26 0004 001 00064 00012 50 0003 0013 24w 32 17 38 21 0003 0009 00054 00012 48 0003 0044 30w 36 12 42 16 0004 0012 00072 00016 55 0004 0055 35w 35 08 37 12 0006 0016 001 0002 80 0005 0016 40w 38 06 40 09 0004 0014 0008 0002 65 0005 0027 mdash mdash mdash mdash mdash mdash mdash mdash mdash mdash mdash mdash

Table 2 The relevant data of manufacturing activities

Manufacturing activitiesEchelon 119875119894 119862119898119894 119867119898119894

1 mdash mdash mdash2 200000 280 253 240000 200 194 300000 320 145 350000 360 106 400000 380 097 450000 420 05

1 2 3 4 5 6 70

05

1

15

2

25

3

ACAPSO

Lingo

Figure 3 Experiment results of CPU time and echelon number

repeatedly the trend of the average iterations would come flatwithout extreme variety

Nevertheless although ACA seems better than the othertwo algorithms we could not ensure that the performancewould still be great when applied to higher echelon inventorymodel Hence a 15-echelon model has proposed to observethe performance again The following table displayed theparameters of 15-echelon model

To evaluate the performance of ACA PSO and Lingo forthe 15-echelon inventorymodel we applied the parameters inTables 4 and 5 to solve the 15-echelon inventory problem

According to Table 6 all three algorithms were still ableto find out the optimal solution of the 15-echelon inventorymodel Once the echelon increases the complexity of the

1 2 3 4 5 6 70

500

1000

1500

2000

2500

3000

3500

4000

ACAPSO

Lingo

Figure 4 Experiment results of average iterations and echelonnumber

model increases as well However the performance of ACAwas still better than the other two algorithms

There are several advantages of ACA as follows

(1) Applying the positive and negative feedback enablesthe process of searching optimization to converge atthe end which means the results would close to theoptimal solutions gradually

(2) Every single ant is able to change the surroundingenvironment through releasing pheromone and beingconscious of the variety of environment which makesthem communicate with each other

(3) Due to distributed computing all the individuals startto calculate simultaneously during the searching pro-cess which enhances the efficiency of the calculationability

(4) For the proposed algorithm it is difficult to stick inthe partial optimization instead it is always easy tofind the best solution

To sum up ACA was proposed to solve the complicatedmathematical problem we also have proved that ACA wasmore efficient than PSO and Lingo These are the new majorcontributions of the present paper

43 The Sensitive Analysis of SMEIJI In this section thenumber of production runs 119873119898119894 and the defective rate of

10 Mathematical Problems in Engineering

Table 3 Comparisons between three methods in SMEIJI problem

Echelon ACA PSO Lingo119873119901119894 119898119871119894 119870119894 119873119898119894 119873119901119894 119898119871119894 119870119894 119873119898119894 119873119901119894 119898119871119894 119870119894 119873119898119894

1 1 00147 mdash mdash 1 00147 mdash mdash 1 00147 mdash mdash2 1 000987 11 1 1 000987 11 1 1 000987 11 13 1 000889 13 1 1 000889 13 1 1 000889 13 14 1 001184 11 1 1 001184 11 1 1 001184 11 15 1 001571 8 1 1 001571 8 1 1 001571 8 16 1 001378 5 1 1 001378 5 1 1 001378 5 17 mdash mdash 5 1 mdash mdash 5 1 mdash mdash 5 1Optimal cost 89166 89166 89166CPU time 0655 1744 23Average iterations 350 1278 3758

Table 4 The relevant data of purchasing activities of 15-echelon model

Purchasing activityEchelon 119863119894 119862119904119894 119867119901119894 119862119890119894 120573119894 120582119871119894 120574119871119894 120583119871119889119894 120590119871119889119894 119865119894 119883119894 0119894

1 18 w 25 28 29 32 0005 0015 0009 0002 70 0005 0032 20w 28 22 32 26 0004 001 00064 00012 50 0003 0013 24w 32 17 38 21 0003 0009 00054 00012 48 0003 0044 30w 36 12 42 16 0004 0012 00072 00016 55 0004 0055 35w 35 08 37 12 0006 0016 001 0002 80 0005 0016 40w 38 06 40 09 0004 0014 0008 0002 65 0005 0027 42w 40 05 28 12 0005 0013 001 0002 70 0005 0038 45w 42 07 35 15 0004 0009 0009 00018 50 0004 0029 50w 44 10 42 10 0003 0010 0005 00024 55 0004 00210 52w 46 15 35 22 0005 0012 00062 00025 60 0005 00211 56w 48 20 49 25 0006 0009 001 00026 65 0003 00312 60w 50 22 40 14 0004 0016 0012 0002 48 0005 00413 62w 55 17 35 15 0007 0015 0009 00018 72 0004 00414 65w 56 12 32 16 0003 0014 00078 0002 70 0006 00315 mdash mdash mdash mdash mdash mdash mdash mdash mdash mdash mdash mdash

Table 5The relevant data of manufacturing activities of 15-echelonmodel

Manufacturing activitiesEchelon 119875119894 119862119898119894 119867119898119894

1 mdash mdash mdash2 200000 280 253 240000 200 194 300000 320 145 350000 360 106 400000 380 097 450000 420 058 480000 440 109 500000 420 1210 520000 400 1511 540000 450 2012 580000 440 2213 620000 420 1814 640000 460 1515 660000 380 16

receiving shipment 0119894 were considered to make sensitiveanalysis and observe the influence on the optimal total jointcost The defective rate and production runs were changedby 0119894(1 + 119877) and 119873119898119894(1 + 119877) where the values of 119877 are0 50 100 200 300 400

Figure 5 illustrated that defective rate was more sensitivethan production runs under the situation that productionruns were one-time production except first echelon In thatcase the higher the defective ratewas the higher the total costwas Accordingly inspecting goods cautiously before deliveryand reducing the defective rate would lower total cost of thewhole supply chainHowever if relaxing that assumption thissensitive analysis would result in different outcomes

Figure 6 has shown that production runs were more sen-sitive than defective rate when relaxing one-time productionassumption In this case the higher the production runswerethe higher the total cost was Hence reducing productionruns should play an essential role in the whole supply chainOnce the one batch production policy was able to satisfy thedemand from the buyer the total cost of whole supply chainwill go down extremely

Mathematical Problems in Engineering 11

Table 6 Comparisons between three ways of 15-echelon inventory model

ACA PSO LingoEchelon 119873119901119894 119898119871119894 119870119894 119873119898119894 119873119901119894 119898119871119894 119870119894 119873119898119894 119873119901119894 119898119871119894 119870119894 119873119898119894

1 1 00147 mdash mdash 1 00147 mdash mdash 1 00147 mdash mdash2 1 000987 11 1 1 000987 11 1 1 000987 11 13 1 000889 13 1 1 000889 13 1 1 000889 13 14 1 001184 11 1 1 001184 11 1 1 001184 11 15 1 001571 8 1 1 001571 8 1 1 001571 8 16 1 001378 5 1 1 001378 5 1 1 001378 5 17 1 001451 5 1 1 001451 5 1 1 001451 5 18 1 001287 5 1 1 001287 5 1 1 001287 5 19 1 00154 7 1 1 00154 7 1 1 00154 7 110 1 001432 9 1 1 001432 9 1 1 001432 9 111 1 001352 7 1 1 001352 7 1 1 001352 7 112 1 00162 8 1 1 00162 8 1 1 00162 8 113 1 001571 10 1 1 001571 10 1 1 001571 10 114 1 001611 5 1 1 001611 5 1 1 001611 5 115 mdash mdash 5 1 mdash mdash 5 1 mdash mdash 5 1Optimal cost ($) 231254 231254 231254CPU time (s) 1725 6621 58215Average iterations 700 2883 8241

Defective rateProduction runs

400100 200 300 5000 500

20000

40000

60000

80000

100000

120000

140000

160000

180000

Figure 5 The sensitive analysis between defective rate and produc-tion runs

5 Conclusions

In summary this research proposed an integrated JIT inven-tory model to solve the partial optimization problem ofpurchaser andmanufacturer finding the optimal solution forthe whole supply chain Furthermore a multiechelon supplychainwith uncertain delivery lead time and unreliable qualityhas been considered to fit the real supply chain environ-ment Besides ACA was proposed to solve the complicatedmathematical problem we also have proved that ACA was

0 50 100 200 300 400 500

Defective rateProduction runs

0

50000

100000

150000

200000

250000

300000

350000

400000

Figure 6 The sensitive analysis between defective rate and produc-tions runs with relaxing assumption

more efficient than PSO and Lingo These are the new majorcontributions of the present paper

Finally there are a great deal of helpful directions forenterprises and researchers (1)The variety of conditions thatoccur in real supply chain can be considered in the modelFor instance time value of cash problem and deterioration(2)The warehousing and storage can be considered in supplychain which make the model more complex to solve (3)The model contains only one member in each echelon amultimember in each echelon should be considered to fit

12 Mathematical Problems in Engineering

real situation in supply chain (4) Some real cases andordata can be applied to our proposed model in the followingworks

Futureworkwill hopefully involve the real-world concernto our research which will be our main target In addition we

do hope our research can be extended intomore considerableareas and more considerable variables

Appendix

The Hessian matrix 119865 of 1198691198621119878(119873119901 119898119871 119873119898 119870) for given 119873119901119894

equal to 1 can be shown as

119865 =

[

[

[

[

[

[

[

[

[

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

1205971198731198981198942

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119873119898119894120597119898119871119894

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119873119898119894120597119870119894

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119898119871119894120597119873119898119894

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

1205971198981198711198942

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119898119871119894120597119870119894

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119870119894120597119873119898119894

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119870119894120597119898119871119894

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

1205971198701198942

]

]

]

]

]

]

]

]

]

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

1205971198731198981198942

=

2

1198731198981198943(120579120593119879

2+

1205931198792(1198671198631198701205932 + Ω)

119870

)

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

1205971198981198711198942

= 119870119873[(

119890minus119898120583

1205832

+ 2 (119890minus119898120583

minus 119890minus120574120583

) +

119890(minus119898120583)(minus119898+120583)

120583

) 120592]

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

1205971198701198942

=

2

1198701198943(120579120593119879

2+

1205931198792(1198671198631198701205932 + Ω)

119873

)

12059721198691198621119904 (119873119898119894 119898119871119894 119870119894)

120597119898119871119894120597119873119898119894

=

12059721198691198621119904 (119873119898119894 119898119871119894 119870119894)

120597119873119898119894120597119898119871119894

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119870119894120597119873119898119894

=

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119873119898119894120597119870119894

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119870119894120597119898119871119894

=

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119898119871119894120597119870119894

100381610038161003816100381611986511

1003816100381610038161003816=

1205972119865

1205971198731198981198942

=

2

1198731198981198943(120579120593119879

2+

1205931198792(1198671198631198701205932 + Ω)

119870

) gt 0

100381610038161003816100381611986522

1003816100381610038161003816= (

1205972119865

1205971198731198981198942)(

1205972119865

1205971198981198711198942) minus [

1205972119865

120597119873119898119894120597119898119871119894

]

2

=

2

1198731198981198943(120579120593119879

2+

1205931198792(1198671198631198701205932 + Ω)

119870

)

1205972119865

1205971198981198711198942minus (

1205972119865

120597119873119898119894120597119898119871119894

)

2

2

1198733119898119894

(1205791205931198792+

119861

119870

)[119870119873(

119890minus119898120576120583

1205832

+ (2119909 +

119890(minus119898120583)119909119910

120583

120592))] + 1198702[

119890minus119898120576120583

120583

+ 120592 (119909119910 minus 119860)] gt 0

100381610038161003816100381611986533

1003816100381610038161003816= (

1205972119865

1205971198731198981198942

1205972119865

1205971198981198711198942

1205972119865

1205971198701198942) minus [(

1205972119865

120597119873119898119894120597119870119894

)

2

(

1205972119865

1205971198981198711198942)]

=

2

1198731198981198943(120579120593119879

2+

119861

119870

)[119870119873(

119890minus119898120576120583

1205832

+ 2(119909 +

119890(minus119898120603)119909119910

120583

) 120592)]

2

1198703

119861

119873

minus [119865 + 119909120576 + 119860120592119910 minus

1205791205931198792

1198732

+

119861

2 (119870119873)2]

[119870119873(

119890minus119898120576120583

1205832

+ (2119909 +

119890minus119898120576120583

120583

) 120592)] gt 0

(A1)

Mathematical Problems in Engineering 13

As a result the matrix F has proved to be a positivedefinite matrix where |11986511| |11986522| and |11986533| are all positiveTherefore theminimumvalue exists in 1198691198621119878(119873119901119898119871 119873119898 119870)Furthermore we can find the optimization in the proposedmodel

Competing Interests

The authors declare that they have no competing interests

References

[1] S K Goyal ldquoAn integrated inventory model for a singlesupplier-single customer problemrdquo International Journal ofProduction Research vol 15 no 1 pp 107ndash111 1977

[2] A Banerjee ldquoA joint economic-lot-sizemodel for purchaser andvendorrdquo Decision Sciences vol 17 no 3 pp 292ndash311 1986

[3] S K Goyal ldquoldquoA joint economic-lot-size model for purchaserand vendorrdquo a commentrdquo Decision Sciences vol 19 no 1 pp236ndash241 1988

[4] Y Seo ldquoControlling general multi-echelon distribution supplychainswith improved reorder decision policy utilizing real-timeshared stock informationrdquo Computers amp Industrial Engineeringvol 51 no 2 pp 229ndash246 2006

[5] H N Chiu and H L Huang ldquoA Multi-echelon integratedJIT inventory model using the time buffer and emergencyborrowing policies to deal with random delivery lead timesrdquoInternational Journal of Production Research vol 41 no 13 pp2911ndash2931 2003

[6] J Sadeghi S M Mousavi S T A Niaki and S SadeghildquoOptimizing a multi-vendor multi-retailer vendor managedinventory problem two tuned meta-heuristic algorithmsrdquoKnowledge-Based Systems vol 50 pp 159ndash170 2013

[7] S S Sana ldquoA production-inventory model of imperfect qualityproducts in a three-layer supply chainrdquo Decision Support Sys-tems vol 50 no 2 pp 539ndash547 2011

[8] H N Soni and K A Patel ldquoOptimal strategy for an integratedinventory system involving variable production and defectiveitems under retailer partial trade credit policyrdquoDecision SupportSystems vol 54 no 1 pp 235ndash247 2012

[9] R J Tersine Principles of Inventory andMaterials ManagementNorthHolland AmsterdamTheNetherlands 3rd edition 1988

[10] C J Liao and C H Shyu ldquoAn analytical determination of leadtime with normal demandrdquo International Journal of Operationsamp Production Management vol 11 no 9 pp 72ndash78 1991

[11] M Ben-Daya and A Raouf ldquoInventory models involving leadtime as a decision variablerdquo Journal of the Operational ResearchSociety vol 45 no 5 pp 579ndash582 1994

[12] S M J Mirzapour Al-e-Hashem A Baboli and Z Sazvar ldquoAstochastic aggregate production planningmodel in a green sup-ply chain considering flexible lead times nonlinear purchaseand shortage cost functionsrdquo European Journal of OperationalResearch vol 230 no 1 pp 26ndash41 2013

[13] C Chandra and J Grabis ldquoInventorymanagementwith variablelead-time dependent procurement costrdquo Omega vol 36 no 5pp 877ndash887 2008

[14] J-W Wu and H-Y Tsai ldquoMixture inventory model with backorders and lost sales for variable lead time demand withthe mixtures of normal distributionrdquo International Journal ofSystems Science Principles and Applications of Systems andIntegration vol 32 no 2 pp 259ndash268 2001

[15] M A Hoque ldquoA vendorndashbuyer integrated productionndashinventory model with normal distribution of lead timerdquoInternational Journal of Production Economics vol 144 no 2pp 409ndash417 2013

[16] F Altiparmak M Gen L Lin and T Paksoy ldquoA geneticalgorithm approach for multi-objective optimization of supplychain networksrdquo Computers and Industrial Engineering vol 51no 1 pp 196ndash215 2006

[17] J Sadeghi S Sadeghi and S T Niaki ldquoA hybrid vendormanaged inventory and redundancy allocation optimizationproblem in supply chain management an NSGA-II with tunedparametersrdquo Computers and Operations Research vol 41 pp53ndash64 2014

[18] X Li F Baki P Tian and B A Chaouch ldquoA robust block-chainbased tabu search algorithm for the dynamic lot sizing problemwith product returns and remanufacturingrdquoOmega vol 42 no1 pp 75ndash87 2014

[19] Z Chen and B R Sarker ldquoAn integrated optimal inventorylot-sizing and vehicle-routing model for a multisupplier single-assembler system with JIT deliveryrdquo International Journal ofProduction Research vol 52 no 17 pp 5086ndash5114 2014

[20] A R Nia M Hemmati Far and S T A Niaki ldquoA fuzzy vendormanaged inventory of multi-item economic order quantitymodel under shortage an ant colony optimization algorithmrdquoInternational Journal of Production Economics vol 155 pp 259ndash271 2014

[21] S M Hatefi and F Jolai ldquoRobust and reliable forward-reverselogistics network design under demand uncertainty and facilitydisruptionsrdquo Applied Mathematical Modelling vol 38 no 9-10pp 2630ndash2647 2014

[22] R H Ma ldquoCross warehouse scheduling of logistics distributionand cycle re-claimer based on heuristics algorithmrdquo AppliedMechanics and Materials vol 644ndash650 pp 2606ndash2610 2014

[23] M Hashim L Yao A H Nadeem M Nazim and M NazamldquoMulti-objective optimization model for supplier selectionproblem in fuzzy environmentrdquo Advances in Intelligent Systemsand Computing vol 281 pp 1201ndash1213 2014

[24] Q-M Hu and Z-H Hu ldquoA stochastic programming modelfor hub-and-spoke network with uncertain flowsrdquo InternationalJournal of Industrial and Systems Engineering vol 21 no 3 pp302ndash319 2015

[25] S A Torabi J Namdar S M Hatefi and F Jolai ldquoAn enhancedpossibilistic programming approach for reliable closed-loopsupply chain network designrdquo International Journal of Produc-tion Research vol 54 no 5 pp 1358ndash1387 2016

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

should be given back immediately The borrowed and theremanded units would not be screened and the screeningtime is ((119902119894 minus119863119894(120574119871119894 minus119898119871119894))119902119894)119909119894 during119898119871119894 le 119871119889119894 le 120574119871119894 le 119905119894The holding cost (HCB119894) and the expected borrowing cost(EBC119894) of member 119894 during 119905119894 are shown The holding cost(HCB119894) involves the normal holding cost and the emergencyborrowing holding cost

The unexpected extra inventory of shadow area (A) dueto the early arrivals is as follows

HCA119894 = 119867119901119894 [119902119894 (119898119871119894 minus 119871119889119894) +

119905119894119902119894 (1 minus 0119894)

2

+ (119909119894 + 119871119889119894 minus 119898119871119894) 1199021198940119894] = 119863119894119867119901119894 [119905119894 (119898119871119894 minus 119871119889119894)

+

1199051198941199091198940119894

1 minus 0119894

+

119905119894

2

2

]

(1)

The emergency borrowing holding cost due to delay arrivalsis as follows

119863119894119867119901119894 [

[(119903119871119894 minus 119871119889119894) + (119903119871119894 minus 119898119871119894)] (119871119889119894 minus 119898119871119894)

2

] (2)

Normal holding cost without early or delay arrivals is asfollows

119863119894119867119901119894 [

(119905119894 + 119871119871119894

minus 119871119889119894)2

2

+ (

119905119894 minus (119903119871119894 minus 119898119871119894) (1 minus 0119894)

1 minus 0119894

)1199091198940119894]

HCB119894 = 119867119901119894 [(119861) + (119862)] = 119863119894119867119901119894 [(119903119871119894 minus 119898119871119894)

+ (119871119889119894 minus 119898119871119894)] + 119905119894 (119898119871119894 minus 119871119889119894) +

119905119894

2

2

minus (119903119871119894 minus 119898119871119894)

sdot (1199091198940119894) +

1199051198941199091198940119894

1 minus 0119894

(3)

Nowwe combine (1) and (3) to form the expected holdingcost (EHC119894) of member 119894 as follows

EHC119894 = int

119898119871119894

120582119871119894

HCA119894119891 (119871119889119894) 119889119871119889119894

+ int

119903119871119894

119898119871119894

HCB119894119891 (119871119889119894) 119889119871119889119894 = 119863119894119867119901119894 [119905119894 (119898119871119894

minus 119871119889119894) +

119905119894

2

2

+

1199051198941199091198940119894

1 minus 0119894

+ (119903119871119894 minus 119898119871119894)

sdot [int

119903119871119894

119898119871119894

(119871119889119894 minus 119898119871119894) 119891 (119871119889119894) 119889119871119889119894

minus (1199091198940119894) int

119903119871119894

119898119871119894

119891 (119871119889119894) 119889119871119889119894]]

(4)

The expected borrowing cost (EBC119894) is as follows

EBC119894 = 119862119890119894 int

119903119871119894

119898119871119894

119891 (119871119889119894) 119889119871119889119894

+ 120573119894 int

119903119871119894

119898119871119894

119863119894 (119903119871119894 minus 119898119871119894) 119891 (119871119889119894) 119889119871119889119894

= [119862119890119894 + 120573119894119863119894 (119903119871119894 minus 119898119871119894)] int

119903119871119894

119898119871119894

119891 (119871119889119894) 119889119871119889119894

(5)

The EBC119894 only occurs at time interval [120574119871119894 119898119871119894]119862119890119894 int119903119871119894

119898119871119894119891(119871119889119894)119889119871119889119894 means the ordering cost of emergency

borrowing and 120573119894 int119903119871119894

119898119871119894119863119894(119903119871119894 minus 119898119871119894)119891(119871119889119894)119889119871119889119894 presents the

excepted cost of borrowed unitsDuring the purchasing time interval (119879119901119894) of each mem-

ber it has 119873119901119894 purchasing orders 119899119894119873119901119894 delivery receivingtimes 119899119894119873119901119894119865119894 delivery cost and 119873119901119894119862119904119894 ordering cost Thusthe expected cost (EP119888119894) of member 119894 in purchasing activitiesis shown as follows

EC119901119894 (119899119894 119873119901119894 119898119871119894) = holding cost

+ emergency borrowing cost + ordering cost

+ delivery cost = 119899119894119873119901119894 [EHC119894 + EBC119894] + 119873119901119894119862119904119894

+ 119899119894119873119901119894119865119894 = 119899119894119873119901119894119863119894119867119901119894 [119905119894 (119898119871119894 minus 119871119889119894) +

1199051198942

2

+

1199051198941199091198940119894

1 minus 0119894

+ (119903119871119894 minus 119898119871119894)]

sdot [int

119903119871119894

119898119871119894

(119871119889119894 minus 119898119871119894) 119891 (119871119889119894) 119889119871119889119894

minus (1199091198940119894) int

119903119871119894

119898119871119894

119891 (119871119889119894) 119889119871119889119894] + 119899119894119873119901119894 [119862119890119894

+ 120573119894119863119894 (119903119871119894 minus 119898119871119894)] int

119903119871119894

119898119871119894

119891 (119871119889119894) 119889119871119889119894 + 119865119894 + 119873119901119894119862119904119894

(6)

32 The Cost of Member 119894 Related to Manufacturing ActivitiesIn this model member 119894 + 1 adopts JIT manufacturing toproduce goods to member 119894 There are 119873119898119894+1 productionruns during 119879119901119894 The start time of each production can bedetermined by counting backward 119902119894minus1119875119894 time units whenthe first delivery lot is delivered The average inventory ofgoods produced bymember 119894per production run is illustratedby Figure 2 and we can determine this by calculatingthe cumulative time-weighted production quantity (1) (thetrapezoid area) minus the cumulative time-weighted deliveryquantity (2) (the ladder area)

Calculating the cumulative time-weighted productionquantity of member 119894 per production run is equal to calcu-lating a square measure of trapezoid area Another way tocalculate the cumulative time-weighted production quantity

6 Mathematical Problems in Engineering

Time

Cumulative quantity

(2)(2)

(1) (1)

ti

Tpi Tpi

Pi

Tpiminus1NmiTpiminus1Nmi

Qmi

Tpiminus1

Figure 2 Average inventory of goods produced by member 119894 perproduction run

(1) is subtracting the triangle area from the rectangle area asfollows

(1) = [

119879119901119894minus1

119873119898119894

minus (119905119894minus1 minus

119902119894minus1

119901119894

)](

119863119894minus1119879119901119894minus1

119873119898119894

)

minus

1

2

(

119863119894minus1119879119901119894

119875119894119873119898119894

)(

119863119894minus1119879119901119894minus1

119873119898119894

) = (

119879119901119894minus1

119873119898119894

)

2

119863119894minus1 [1

2

+

0119894minus1

119870119894 (1 minus 0119894)

+ (

1

2

minus

1

119870119894 (1 minus 0119894minus1)

)(1 minus

119863119894minus1

119875119894

)]

(7)

During each production run of member i there are 119870119894

deliveries and 119902119894 units of goods delivered per shipment Thetime interval between two adjacent deliveries for member119894 (119905119894) should be longer than or equal to 120574119871119894 Hence thecumulative time-weighted delivery quantity (2) of member 119894

per production run is

(2) = (

119870119894minus1

sum

119897

119897) 119905119894minus1119902119894minus1 = (1 + 2 + sdot sdot sdot + (119870119894minus1)) 119905119894minus1119902119894minus1

= (

119879119901119894minus1

119873119898119894

)

2

119863119894minus1 [1

2

119870119894minus1

119870119894 (1 minus 0119894minus1)

]

(8)

Consequently the average inventory (AI119894) of member 119894

during 119879119901119894minus1 can be calculated by subtracting area (2) fromarea (1) as follows

AI119894 = 119873119898119894 (1) minus (2) = 119873119898119894 (

119879119901119894minus1

119873119898119894

)

2

119863119894minus1 [1

2

+

0119894

119870119894 (1 minus 0119894)

+ (

1

2

minus

1

119870119894 (1 minus 0119894minus1)

)(1 minus

119863119894

119875119894

)]

minus

119879119901119894minus1

119873119898119894

119863119894minus1 [1

2

119870119894minus1

119870119894 (1 minus 0119894minus1)

] = (

119879119901119894minus12

119873119898119894

)

sdot 119863119894minus1 [1

2

(

119870119894 + 0119894minus1 minus 1

119870119894 (1 minus 0119894)

) + (

1

2

minus

1

119870119894 (1 minus 0119894minus1)

)

sdot (1 minus

119863119894minus1

119875119894

)]

(9)

Thus the production cost of member 119894 (EC119898119894) during 119879119901119894 is

EC119898119894 (119873119898119894119870119894) = 119873119898119894119862119898119894 + 119867119898119894AI119894 = 119873119898119894119862119898119894

+ 119867119898119894 (

119879119901119894minus12

119873119898119894

)119863119894minus1 [1

2

(

119870119894 + 0119894minus1 minus 1

119870119894 (1 minus 0119894)

)

+ (

1

2

minus

1

119870119894 (1 minus 0119894minus1)

)(1 minus

119863119894minus1

119875119894

)]

(10)

33 The Joint Cost of Member 119894 in Purchasing Activities andMember 119894 + 1 in Manufacturing Activities With the abovediscussion we have inferred the expected cost of member 119894

in purchasing andmanufacturing respectivelyTherefore thejoint cost of member 119894 in purchasing activities and member119894 + 1 in manufacturing activities can be obtained easily Wesubstituted 119899119894119873119901119894 = 119870119894+1119873119898119894+1 into (6) and (10) and gainedthe joint cost of members 119894 and 119894 + 1 as

119862119894119894+1 (119873119901119894 119898119871119894 119873119898119894+1 119870119894+1) = EC119901119894 + EC119898119894+1

= 119873119898119894+1119870119894+1 [EHC119894 + EBC119894] + 119873119901119894119862119901119894

+ 119873119898119894+1119862119898119894+1 + 119867119898119894+1AI119894+1

= 119873119898119894+1119870119894+1 119863119894119867119901119894 [119905119894 (119898119871119894 minus 120583119871119889119894)] +

1199051198942

2

+

1199051198941199091198940119894

1 minus 0119894

+ (119903119871119894 minus 119898119871119894)

sdot [int

119903119871119894

119898119871119894

(119871119889119894 minus 119898119871119894) 119891 (119871119889119894) 119889119871119889119894

minus (1199091198940119894) int

119903119871119894

119898119871119894

119891 (119871119889119894) 119889119871119889119894] + [119862119890119894 + 120573119894119863119894 (119903119871119894

minus 119898119871119894)] [int

119903119871119894

119898119871119894

119891 (119871119889119894) 119889119871119889119894 + 119865119894] + 119873119901119894119862119904119894

+ 119873119898119894119862119898119894 + 119867119898119894 (

119879119901119894minus1

2

119873119898119894

)

sdot 119863119894minus1 [1

2

(1 minus

119870119894 + 0119894minus1 minus 1

119870119894 (1 minus 0119894)

)

+ (

1

2

minus

1

119870119894 (1 minus 0119894minus1)

)(1 minus

119863119894minus1

119875119894

)]

(11a)

Mathematical Problems in Engineering 7

Substitute 119905119894 = 119879119901119894119870119894+1119873119898119894+1 into (11a) Consider

119862119894119894+1 (119873119901119894 119898119871119894 119873119898119894+1 119870119894+1) = V119894119879119901119894 (119898119871119894 minus 120583119871119889119894)

+ V119894119879119901119894119909119894120588119894 +119879119901119894

2

119870119894+1119873119898119894+1

(Ω119894 minus

119867119898119894+1119863119894119870119894+1120588119894

2

)

+

119879119901119894

2

119873119898119894+1

120579119894 + 119870119894+1119873119898119894+1 [(119903119871119894 minus 119898119871119894)

sdot V119894 int119903119871119894

119898119871119894

(119871119889119894 minus 119898119871119894) 119891 (119871119889119894) 119889119871119889119894

+ 120576119894 int

119903119871119894

119898119871119894

119891 (119871119889119894) 119889119871119889119894 + 119865119894] + 119873119901119894119862119904119894

+ 119873119898119894+1119862119898119894+1

(11b)

where

V119894 = 119867119901119894119863119894

120588119894 =0119894

1 minus 0119894

Ω119894

=

119863119894

2

[119867119901119894 + 119867119898119894+1 (1 minus (

2

1 minus 0119894

)(1 minus

119863119894

119875119894+1

))]

120576119894 = 119862119890119894 + (120573119894119863119894 minus V1198941199091198940119894) (119903119871119894 minus 119898119871119894)

120579119894 =

119867119898119894+1119863119894

2

(1 minus

119863119894

119875119894+1

)

(12)

Since 119879119901119894 = 119876119898119894minus1119875119894 = (119863119894minus1119873119898119894+1119875119894)119879119901119894minus1 119879119901119894 can beexpressed further as

119879119901119894 = (

120593119894

prod119894

119897=1119873119898119897

)1198791199011 (13)

where

120593119894 =

1 if 119894 = 1

119894

prod

119897=2

(

119863119894minus1

119875119894

) if 119894 = 2 3 119878 minus 1

(14)

All the 119879119901119894 in function (11b) could be replaced as follows

119862119894119894+1 (119873119901119894 119898119871119894 119873119898119894+1 119870119894+1) = V119894(120593119894

prod119894

119897=1119873119898119897

)

sdot 1198791199011 (119898119871119894 minus 120583119871119889119894) + V119894(120593119894

prod119894

119897=1119873119898119897

)1198791199011119909119894120588119894

+

(120593119894prod119894

119897=1119873119898119897) 1198791199011

2

119870119894+1119873119898119894+1

(Ω119894 minus

119867119898119894+1119863119894119870119894+1120588119894

2

)

+

(120593119894prod119894

119897=1119873119898119897) 1198791199011

2

119873119898119894+1

120579119894

+ 119870119894+1119873119898119894+1 [(119903119871119894 minus 119898119871119894)

sdot V119894 int119903119871119894

119898119871119894

(119871119889119894 minus 119898119871119894) 119891 (119871119889119894) 119889119871119889119894

+ 120576119894 int

119903119871119894

119898119871119894

119891 (119871119889119894) 119889119871119889119894 + 119865119894] + 119873119901119894119862119904119894

+ 119873119898119894+1119862119898119894+1

(11c)

There are119873119898119894+1 purchasing times of member 119894+1 during119879119901119894 According to (11c) the joint total cost function includingpurchasing and manufacturing activities of the serial 119878-echelon integrated JIT inventory model is presented in

119878minus1

sum

119894=1

V1198941205931198941198791199011 (119898119871119894 minus 120583119871119889119894) + V1198941205931198941198791199011119909119894120588119894

+

(1205931198941198791199011)

2

119870119894+1prod119894

119897=1119873119898119897

(Ω119894 minus

119867119898119894+1119863119894119870119894+1120588119894

2

)

+

1198791199011

2

prod119894

119897=1119873119898119897

120579119894 + (

119894

prod

119897=1

119873119898119897)119870119894+1119873119898119894+1 [(119903119871119894 minus 119898119871119894)

sdot V119894 int119903119871119894

119898119871119894

(119871119889119894 minus 119898119871119894) 119891 (119871119889119894) 119889119871119889119894

+ 120576119894 int

119903119871119894

119898119871119894

119891 (119871119889119894) 119889119871119889119894 + 119865119894] + (

119894

prod

119897=1

119873119898119897)119873119901119894119862119904119894

+ (

119894

prod

119897=1

119873119898119897)119862119898119894+1

(15)

The delivery time constrains 0 le 120582119871119894 le 119898119871119894 le 120574119871119894 le

119905119894 119894 = 1 2 119878 minus 1 and variables constrains are consideredin the inventory model Eventually the 119878-echelon integratedJIT model can be expressed as

Minimize 1198691198621119878 (119873119901 119898119871 119873119898 119870)

=

119878minus1

sum

119894=1

V1198941205931198941198791199011 (119898119871119894 minus 120583119871119889119894) + V1198941205931198941198791199011119909119894120588119894 +(1205931198941198791199011)

2

119870119894+1prod119894

119897=1119873119898119897

(Ω119894 minus

119867119898119894+1119863119894119870119894+1120588119894

2

) +

11987911990112

prod119894

119897=1119873119898119897

120579119894

8 Mathematical Problems in Engineering

+(

119894

prod

119897=1

119873119898119897)119870119894+1119873119898119894+1 [(119903119871119894 minus 119898119871119894) V119894 int119903119871119894

119898119871119894

(119871119889119894 minus 119898119871119894) 119891 (119871119889119894) 119889119871119889119894 + 120576119894 int

119903119871119894

119898119871119894

119891 (119871119889119894) 119889119871119889119894 + 119865119894]

+(

119894

prod

119897=1

119873119898119897)119873119901119894119862119904119894 + (

119894

prod

119897=1

119873119898119897)119862119898119894+1

Subject to 0 le 120582119871119894 le 119898119871119894 le 120574119871119894 le 119905119894 for 119894 = 1 2 119878 minus 1

119870119894+1 le1

0119894

minus 1

119873119901119894 119870119894+1 119873119898119894+1 are all integers for 119894 = 1 2 119878 minus 1

(16)

4 Numeral Experiment and Analysis

In this section a numeral experiment is proposed based onproper parameter settings All results in this research wereperformed on a PC Programs were written in Matlab 2012bAn efficiency comparison between Lingo PSO algorithmand Ant Colony Algorithm was proposed as well Further-more we would like to discuss how production runs anddefective rate influences whole system cost

41 Experiment Results of Ant Colony Algorithm ParticleSwarm Optimization and Lingo With appropriate param-eters settings we may build a 7-echelon supply chain inte-grated JIT inventory model with uncertain delivery leadtime and unreliable quality of received items Assume thepurchasing time interval for first purchaser in years (1198791199011) is12 years The relevant data of purchasing and manufacturingactivities are shown in Tables 1 and 2

The objective function and constraint function are shownbelow Consider

1198691198621119878 (119873119901 119898119871 119873119898 119870)

=

119878minus1

sum

119894=1

V1198941205931198941198791199011 (119898119871119894 minus 120583119871119889119894) + V1198941205931198941198791199011119909119894120588119894 +(1205931198941198791199011)

2

119870119894+1prod119894

119897=1119873119898119897

(Ω119894 minus

119867119898119894+1119863119894119870119894+1120588119894

2

) +

1198791199011

2

prod119894

119897=1119873119898119897

120579119894

+(

119894

prod

119897=1

119873119898119897)119870119894+1119873119898119894+1 [(119903119871119894 minus 119898119871119894) V119894 int119903119871119894

119898119871119894

(119871119889119894 minus 119898119871119894) 119891 (119871119889119894) 119889119871119889119894 + 120576119894 int

119903119871119894

119898119871119894

119891 (119871119889119894) 119889119871119889119894 + 119865119894]

+(

119894

prod

119897=1

119873119898119897)119873119901119894119862119904119894 + (

119894

prod

119897=1

119873119898119897)119862119898119894+1

Subject to 0 le 120582119871119894 le 119898119871119894 le 120574119871119894 le 119905119894 for 119894 = 1 2 119878 minus 1

119870119894+1 le1

0119894

minus 1

119873119901119894 119870119894+1 119873119898119894+1 are all integers for 119894 = 1 2 119878 minus 1

(17)

The goal of this research is to find out the optimal solutionof the 7-echelon inventory model via the above functions andACA and compare the efficiency between Lingo ACA andPSO

As Table 3 shows though all the three methods werecapable of finding the optimal solution for SMEIJI problemACA was obviously better than PSO and Lingo in efficiencyACA has shown fastest CPU searching time and least averageiterations among three methods

42 Experiments of Performance and 119878-Echelon In this sec-tion we would like to discuss the relationship between CPU

searching time and echelon number among three methodswe also discussed the relationship between average iterationsand echelon number

As Figure 3 shows the CPU searching time wouldincrease with the number of echelons due to the compli-cated calculation However because of the well-organizedconstraint function and good initial solution ACA still keptlower CPU searching time than the other two methods Onthe other hand Figure 4 shows that ACArsquos average iterationswere the least among three methods At the beginning of themultiechelon all the three algorithms need a great amountof iterations to search the optimal solution After calculating

Mathematical Problems in Engineering 9

Table 1 The relevant data of purchasing activities

Purchasing activityEchelon 119863119894 119862119904119894 119867119901119894 119862119890119894 120573119894 120582119871119894 120574119871119894 120583119871119889119894 120590119871119889119894 119865119894 119883119894 0119894

1 18 w 25 28 29 32 0005 0015 0009 0002 70 0005 0032 20w 28 22 32 26 0004 001 00064 00012 50 0003 0013 24w 32 17 38 21 0003 0009 00054 00012 48 0003 0044 30w 36 12 42 16 0004 0012 00072 00016 55 0004 0055 35w 35 08 37 12 0006 0016 001 0002 80 0005 0016 40w 38 06 40 09 0004 0014 0008 0002 65 0005 0027 mdash mdash mdash mdash mdash mdash mdash mdash mdash mdash mdash mdash

Table 2 The relevant data of manufacturing activities

Manufacturing activitiesEchelon 119875119894 119862119898119894 119867119898119894

1 mdash mdash mdash2 200000 280 253 240000 200 194 300000 320 145 350000 360 106 400000 380 097 450000 420 05

1 2 3 4 5 6 70

05

1

15

2

25

3

ACAPSO

Lingo

Figure 3 Experiment results of CPU time and echelon number

repeatedly the trend of the average iterations would come flatwithout extreme variety

Nevertheless although ACA seems better than the othertwo algorithms we could not ensure that the performancewould still be great when applied to higher echelon inventorymodel Hence a 15-echelon model has proposed to observethe performance again The following table displayed theparameters of 15-echelon model

To evaluate the performance of ACA PSO and Lingo forthe 15-echelon inventorymodel we applied the parameters inTables 4 and 5 to solve the 15-echelon inventory problem

According to Table 6 all three algorithms were still ableto find out the optimal solution of the 15-echelon inventorymodel Once the echelon increases the complexity of the

1 2 3 4 5 6 70

500

1000

1500

2000

2500

3000

3500

4000

ACAPSO

Lingo

Figure 4 Experiment results of average iterations and echelonnumber

model increases as well However the performance of ACAwas still better than the other two algorithms

There are several advantages of ACA as follows

(1) Applying the positive and negative feedback enablesthe process of searching optimization to converge atthe end which means the results would close to theoptimal solutions gradually

(2) Every single ant is able to change the surroundingenvironment through releasing pheromone and beingconscious of the variety of environment which makesthem communicate with each other

(3) Due to distributed computing all the individuals startto calculate simultaneously during the searching pro-cess which enhances the efficiency of the calculationability

(4) For the proposed algorithm it is difficult to stick inthe partial optimization instead it is always easy tofind the best solution

To sum up ACA was proposed to solve the complicatedmathematical problem we also have proved that ACA wasmore efficient than PSO and Lingo These are the new majorcontributions of the present paper

43 The Sensitive Analysis of SMEIJI In this section thenumber of production runs 119873119898119894 and the defective rate of

10 Mathematical Problems in Engineering

Table 3 Comparisons between three methods in SMEIJI problem

Echelon ACA PSO Lingo119873119901119894 119898119871119894 119870119894 119873119898119894 119873119901119894 119898119871119894 119870119894 119873119898119894 119873119901119894 119898119871119894 119870119894 119873119898119894

1 1 00147 mdash mdash 1 00147 mdash mdash 1 00147 mdash mdash2 1 000987 11 1 1 000987 11 1 1 000987 11 13 1 000889 13 1 1 000889 13 1 1 000889 13 14 1 001184 11 1 1 001184 11 1 1 001184 11 15 1 001571 8 1 1 001571 8 1 1 001571 8 16 1 001378 5 1 1 001378 5 1 1 001378 5 17 mdash mdash 5 1 mdash mdash 5 1 mdash mdash 5 1Optimal cost 89166 89166 89166CPU time 0655 1744 23Average iterations 350 1278 3758

Table 4 The relevant data of purchasing activities of 15-echelon model

Purchasing activityEchelon 119863119894 119862119904119894 119867119901119894 119862119890119894 120573119894 120582119871119894 120574119871119894 120583119871119889119894 120590119871119889119894 119865119894 119883119894 0119894

1 18 w 25 28 29 32 0005 0015 0009 0002 70 0005 0032 20w 28 22 32 26 0004 001 00064 00012 50 0003 0013 24w 32 17 38 21 0003 0009 00054 00012 48 0003 0044 30w 36 12 42 16 0004 0012 00072 00016 55 0004 0055 35w 35 08 37 12 0006 0016 001 0002 80 0005 0016 40w 38 06 40 09 0004 0014 0008 0002 65 0005 0027 42w 40 05 28 12 0005 0013 001 0002 70 0005 0038 45w 42 07 35 15 0004 0009 0009 00018 50 0004 0029 50w 44 10 42 10 0003 0010 0005 00024 55 0004 00210 52w 46 15 35 22 0005 0012 00062 00025 60 0005 00211 56w 48 20 49 25 0006 0009 001 00026 65 0003 00312 60w 50 22 40 14 0004 0016 0012 0002 48 0005 00413 62w 55 17 35 15 0007 0015 0009 00018 72 0004 00414 65w 56 12 32 16 0003 0014 00078 0002 70 0006 00315 mdash mdash mdash mdash mdash mdash mdash mdash mdash mdash mdash mdash

Table 5The relevant data of manufacturing activities of 15-echelonmodel

Manufacturing activitiesEchelon 119875119894 119862119898119894 119867119898119894

1 mdash mdash mdash2 200000 280 253 240000 200 194 300000 320 145 350000 360 106 400000 380 097 450000 420 058 480000 440 109 500000 420 1210 520000 400 1511 540000 450 2012 580000 440 2213 620000 420 1814 640000 460 1515 660000 380 16

receiving shipment 0119894 were considered to make sensitiveanalysis and observe the influence on the optimal total jointcost The defective rate and production runs were changedby 0119894(1 + 119877) and 119873119898119894(1 + 119877) where the values of 119877 are0 50 100 200 300 400

Figure 5 illustrated that defective rate was more sensitivethan production runs under the situation that productionruns were one-time production except first echelon In thatcase the higher the defective ratewas the higher the total costwas Accordingly inspecting goods cautiously before deliveryand reducing the defective rate would lower total cost of thewhole supply chainHowever if relaxing that assumption thissensitive analysis would result in different outcomes

Figure 6 has shown that production runs were more sen-sitive than defective rate when relaxing one-time productionassumption In this case the higher the production runswerethe higher the total cost was Hence reducing productionruns should play an essential role in the whole supply chainOnce the one batch production policy was able to satisfy thedemand from the buyer the total cost of whole supply chainwill go down extremely

Mathematical Problems in Engineering 11

Table 6 Comparisons between three ways of 15-echelon inventory model

ACA PSO LingoEchelon 119873119901119894 119898119871119894 119870119894 119873119898119894 119873119901119894 119898119871119894 119870119894 119873119898119894 119873119901119894 119898119871119894 119870119894 119873119898119894

1 1 00147 mdash mdash 1 00147 mdash mdash 1 00147 mdash mdash2 1 000987 11 1 1 000987 11 1 1 000987 11 13 1 000889 13 1 1 000889 13 1 1 000889 13 14 1 001184 11 1 1 001184 11 1 1 001184 11 15 1 001571 8 1 1 001571 8 1 1 001571 8 16 1 001378 5 1 1 001378 5 1 1 001378 5 17 1 001451 5 1 1 001451 5 1 1 001451 5 18 1 001287 5 1 1 001287 5 1 1 001287 5 19 1 00154 7 1 1 00154 7 1 1 00154 7 110 1 001432 9 1 1 001432 9 1 1 001432 9 111 1 001352 7 1 1 001352 7 1 1 001352 7 112 1 00162 8 1 1 00162 8 1 1 00162 8 113 1 001571 10 1 1 001571 10 1 1 001571 10 114 1 001611 5 1 1 001611 5 1 1 001611 5 115 mdash mdash 5 1 mdash mdash 5 1 mdash mdash 5 1Optimal cost ($) 231254 231254 231254CPU time (s) 1725 6621 58215Average iterations 700 2883 8241

Defective rateProduction runs

400100 200 300 5000 500

20000

40000

60000

80000

100000

120000

140000

160000

180000

Figure 5 The sensitive analysis between defective rate and produc-tion runs

5 Conclusions

In summary this research proposed an integrated JIT inven-tory model to solve the partial optimization problem ofpurchaser andmanufacturer finding the optimal solution forthe whole supply chain Furthermore a multiechelon supplychainwith uncertain delivery lead time and unreliable qualityhas been considered to fit the real supply chain environ-ment Besides ACA was proposed to solve the complicatedmathematical problem we also have proved that ACA was

0 50 100 200 300 400 500

Defective rateProduction runs

0

50000

100000

150000

200000

250000

300000

350000

400000

Figure 6 The sensitive analysis between defective rate and produc-tions runs with relaxing assumption

more efficient than PSO and Lingo These are the new majorcontributions of the present paper

Finally there are a great deal of helpful directions forenterprises and researchers (1)The variety of conditions thatoccur in real supply chain can be considered in the modelFor instance time value of cash problem and deterioration(2)The warehousing and storage can be considered in supplychain which make the model more complex to solve (3)The model contains only one member in each echelon amultimember in each echelon should be considered to fit

12 Mathematical Problems in Engineering

real situation in supply chain (4) Some real cases andordata can be applied to our proposed model in the followingworks

Futureworkwill hopefully involve the real-world concernto our research which will be our main target In addition we

do hope our research can be extended intomore considerableareas and more considerable variables

Appendix

The Hessian matrix 119865 of 1198691198621119878(119873119901 119898119871 119873119898 119870) for given 119873119901119894

equal to 1 can be shown as

119865 =

[

[

[

[

[

[

[

[

[

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

1205971198731198981198942

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119873119898119894120597119898119871119894

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119873119898119894120597119870119894

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119898119871119894120597119873119898119894

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

1205971198981198711198942

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119898119871119894120597119870119894

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119870119894120597119873119898119894

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119870119894120597119898119871119894

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

1205971198701198942

]

]

]

]

]

]

]

]

]

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

1205971198731198981198942

=

2

1198731198981198943(120579120593119879

2+

1205931198792(1198671198631198701205932 + Ω)

119870

)

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

1205971198981198711198942

= 119870119873[(

119890minus119898120583

1205832

+ 2 (119890minus119898120583

minus 119890minus120574120583

) +

119890(minus119898120583)(minus119898+120583)

120583

) 120592]

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

1205971198701198942

=

2

1198701198943(120579120593119879

2+

1205931198792(1198671198631198701205932 + Ω)

119873

)

12059721198691198621119904 (119873119898119894 119898119871119894 119870119894)

120597119898119871119894120597119873119898119894

=

12059721198691198621119904 (119873119898119894 119898119871119894 119870119894)

120597119873119898119894120597119898119871119894

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119870119894120597119873119898119894

=

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119873119898119894120597119870119894

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119870119894120597119898119871119894

=

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119898119871119894120597119870119894

100381610038161003816100381611986511

1003816100381610038161003816=

1205972119865

1205971198731198981198942

=

2

1198731198981198943(120579120593119879

2+

1205931198792(1198671198631198701205932 + Ω)

119870

) gt 0

100381610038161003816100381611986522

1003816100381610038161003816= (

1205972119865

1205971198731198981198942)(

1205972119865

1205971198981198711198942) minus [

1205972119865

120597119873119898119894120597119898119871119894

]

2

=

2

1198731198981198943(120579120593119879

2+

1205931198792(1198671198631198701205932 + Ω)

119870

)

1205972119865

1205971198981198711198942minus (

1205972119865

120597119873119898119894120597119898119871119894

)

2

2

1198733119898119894

(1205791205931198792+

119861

119870

)[119870119873(

119890minus119898120576120583

1205832

+ (2119909 +

119890(minus119898120583)119909119910

120583

120592))] + 1198702[

119890minus119898120576120583

120583

+ 120592 (119909119910 minus 119860)] gt 0

100381610038161003816100381611986533

1003816100381610038161003816= (

1205972119865

1205971198731198981198942

1205972119865

1205971198981198711198942

1205972119865

1205971198701198942) minus [(

1205972119865

120597119873119898119894120597119870119894

)

2

(

1205972119865

1205971198981198711198942)]

=

2

1198731198981198943(120579120593119879

2+

119861

119870

)[119870119873(

119890minus119898120576120583

1205832

+ 2(119909 +

119890(minus119898120603)119909119910

120583

) 120592)]

2

1198703

119861

119873

minus [119865 + 119909120576 + 119860120592119910 minus

1205791205931198792

1198732

+

119861

2 (119870119873)2]

[119870119873(

119890minus119898120576120583

1205832

+ (2119909 +

119890minus119898120576120583

120583

) 120592)] gt 0

(A1)

Mathematical Problems in Engineering 13

As a result the matrix F has proved to be a positivedefinite matrix where |11986511| |11986522| and |11986533| are all positiveTherefore theminimumvalue exists in 1198691198621119878(119873119901119898119871 119873119898 119870)Furthermore we can find the optimization in the proposedmodel

Competing Interests

The authors declare that they have no competing interests

References

[1] S K Goyal ldquoAn integrated inventory model for a singlesupplier-single customer problemrdquo International Journal ofProduction Research vol 15 no 1 pp 107ndash111 1977

[2] A Banerjee ldquoA joint economic-lot-sizemodel for purchaser andvendorrdquo Decision Sciences vol 17 no 3 pp 292ndash311 1986

[3] S K Goyal ldquoldquoA joint economic-lot-size model for purchaserand vendorrdquo a commentrdquo Decision Sciences vol 19 no 1 pp236ndash241 1988

[4] Y Seo ldquoControlling general multi-echelon distribution supplychainswith improved reorder decision policy utilizing real-timeshared stock informationrdquo Computers amp Industrial Engineeringvol 51 no 2 pp 229ndash246 2006

[5] H N Chiu and H L Huang ldquoA Multi-echelon integratedJIT inventory model using the time buffer and emergencyborrowing policies to deal with random delivery lead timesrdquoInternational Journal of Production Research vol 41 no 13 pp2911ndash2931 2003

[6] J Sadeghi S M Mousavi S T A Niaki and S SadeghildquoOptimizing a multi-vendor multi-retailer vendor managedinventory problem two tuned meta-heuristic algorithmsrdquoKnowledge-Based Systems vol 50 pp 159ndash170 2013

[7] S S Sana ldquoA production-inventory model of imperfect qualityproducts in a three-layer supply chainrdquo Decision Support Sys-tems vol 50 no 2 pp 539ndash547 2011

[8] H N Soni and K A Patel ldquoOptimal strategy for an integratedinventory system involving variable production and defectiveitems under retailer partial trade credit policyrdquoDecision SupportSystems vol 54 no 1 pp 235ndash247 2012

[9] R J Tersine Principles of Inventory andMaterials ManagementNorthHolland AmsterdamTheNetherlands 3rd edition 1988

[10] C J Liao and C H Shyu ldquoAn analytical determination of leadtime with normal demandrdquo International Journal of Operationsamp Production Management vol 11 no 9 pp 72ndash78 1991

[11] M Ben-Daya and A Raouf ldquoInventory models involving leadtime as a decision variablerdquo Journal of the Operational ResearchSociety vol 45 no 5 pp 579ndash582 1994

[12] S M J Mirzapour Al-e-Hashem A Baboli and Z Sazvar ldquoAstochastic aggregate production planningmodel in a green sup-ply chain considering flexible lead times nonlinear purchaseand shortage cost functionsrdquo European Journal of OperationalResearch vol 230 no 1 pp 26ndash41 2013

[13] C Chandra and J Grabis ldquoInventorymanagementwith variablelead-time dependent procurement costrdquo Omega vol 36 no 5pp 877ndash887 2008

[14] J-W Wu and H-Y Tsai ldquoMixture inventory model with backorders and lost sales for variable lead time demand withthe mixtures of normal distributionrdquo International Journal ofSystems Science Principles and Applications of Systems andIntegration vol 32 no 2 pp 259ndash268 2001

[15] M A Hoque ldquoA vendorndashbuyer integrated productionndashinventory model with normal distribution of lead timerdquoInternational Journal of Production Economics vol 144 no 2pp 409ndash417 2013

[16] F Altiparmak M Gen L Lin and T Paksoy ldquoA geneticalgorithm approach for multi-objective optimization of supplychain networksrdquo Computers and Industrial Engineering vol 51no 1 pp 196ndash215 2006

[17] J Sadeghi S Sadeghi and S T Niaki ldquoA hybrid vendormanaged inventory and redundancy allocation optimizationproblem in supply chain management an NSGA-II with tunedparametersrdquo Computers and Operations Research vol 41 pp53ndash64 2014

[18] X Li F Baki P Tian and B A Chaouch ldquoA robust block-chainbased tabu search algorithm for the dynamic lot sizing problemwith product returns and remanufacturingrdquoOmega vol 42 no1 pp 75ndash87 2014

[19] Z Chen and B R Sarker ldquoAn integrated optimal inventorylot-sizing and vehicle-routing model for a multisupplier single-assembler system with JIT deliveryrdquo International Journal ofProduction Research vol 52 no 17 pp 5086ndash5114 2014

[20] A R Nia M Hemmati Far and S T A Niaki ldquoA fuzzy vendormanaged inventory of multi-item economic order quantitymodel under shortage an ant colony optimization algorithmrdquoInternational Journal of Production Economics vol 155 pp 259ndash271 2014

[21] S M Hatefi and F Jolai ldquoRobust and reliable forward-reverselogistics network design under demand uncertainty and facilitydisruptionsrdquo Applied Mathematical Modelling vol 38 no 9-10pp 2630ndash2647 2014

[22] R H Ma ldquoCross warehouse scheduling of logistics distributionand cycle re-claimer based on heuristics algorithmrdquo AppliedMechanics and Materials vol 644ndash650 pp 2606ndash2610 2014

[23] M Hashim L Yao A H Nadeem M Nazim and M NazamldquoMulti-objective optimization model for supplier selectionproblem in fuzzy environmentrdquo Advances in Intelligent Systemsand Computing vol 281 pp 1201ndash1213 2014

[24] Q-M Hu and Z-H Hu ldquoA stochastic programming modelfor hub-and-spoke network with uncertain flowsrdquo InternationalJournal of Industrial and Systems Engineering vol 21 no 3 pp302ndash319 2015

[25] S A Torabi J Namdar S M Hatefi and F Jolai ldquoAn enhancedpossibilistic programming approach for reliable closed-loopsupply chain network designrdquo International Journal of Produc-tion Research vol 54 no 5 pp 1358ndash1387 2016

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

Time

Cumulative quantity

(2)(2)

(1) (1)

ti

Tpi Tpi

Pi

Tpiminus1NmiTpiminus1Nmi

Qmi

Tpiminus1

Figure 2 Average inventory of goods produced by member 119894 perproduction run

(1) is subtracting the triangle area from the rectangle area asfollows

(1) = [

119879119901119894minus1

119873119898119894

minus (119905119894minus1 minus

119902119894minus1

119901119894

)](

119863119894minus1119879119901119894minus1

119873119898119894

)

minus

1

2

(

119863119894minus1119879119901119894

119875119894119873119898119894

)(

119863119894minus1119879119901119894minus1

119873119898119894

) = (

119879119901119894minus1

119873119898119894

)

2

119863119894minus1 [1

2

+

0119894minus1

119870119894 (1 minus 0119894)

+ (

1

2

minus

1

119870119894 (1 minus 0119894minus1)

)(1 minus

119863119894minus1

119875119894

)]

(7)

During each production run of member i there are 119870119894

deliveries and 119902119894 units of goods delivered per shipment Thetime interval between two adjacent deliveries for member119894 (119905119894) should be longer than or equal to 120574119871119894 Hence thecumulative time-weighted delivery quantity (2) of member 119894

per production run is

(2) = (

119870119894minus1

sum

119897

119897) 119905119894minus1119902119894minus1 = (1 + 2 + sdot sdot sdot + (119870119894minus1)) 119905119894minus1119902119894minus1

= (

119879119901119894minus1

119873119898119894

)

2

119863119894minus1 [1

2

119870119894minus1

119870119894 (1 minus 0119894minus1)

]

(8)

Consequently the average inventory (AI119894) of member 119894

during 119879119901119894minus1 can be calculated by subtracting area (2) fromarea (1) as follows

AI119894 = 119873119898119894 (1) minus (2) = 119873119898119894 (

119879119901119894minus1

119873119898119894

)

2

119863119894minus1 [1

2

+

0119894

119870119894 (1 minus 0119894)

+ (

1

2

minus

1

119870119894 (1 minus 0119894minus1)

)(1 minus

119863119894

119875119894

)]

minus

119879119901119894minus1

119873119898119894

119863119894minus1 [1

2

119870119894minus1

119870119894 (1 minus 0119894minus1)

] = (

119879119901119894minus12

119873119898119894

)

sdot 119863119894minus1 [1

2

(

119870119894 + 0119894minus1 minus 1

119870119894 (1 minus 0119894)

) + (

1

2

minus

1

119870119894 (1 minus 0119894minus1)

)

sdot (1 minus

119863119894minus1

119875119894

)]

(9)

Thus the production cost of member 119894 (EC119898119894) during 119879119901119894 is

EC119898119894 (119873119898119894119870119894) = 119873119898119894119862119898119894 + 119867119898119894AI119894 = 119873119898119894119862119898119894

+ 119867119898119894 (

119879119901119894minus12

119873119898119894

)119863119894minus1 [1

2

(

119870119894 + 0119894minus1 minus 1

119870119894 (1 minus 0119894)

)

+ (

1

2

minus

1

119870119894 (1 minus 0119894minus1)

)(1 minus

119863119894minus1

119875119894

)]

(10)

33 The Joint Cost of Member 119894 in Purchasing Activities andMember 119894 + 1 in Manufacturing Activities With the abovediscussion we have inferred the expected cost of member 119894

in purchasing andmanufacturing respectivelyTherefore thejoint cost of member 119894 in purchasing activities and member119894 + 1 in manufacturing activities can be obtained easily Wesubstituted 119899119894119873119901119894 = 119870119894+1119873119898119894+1 into (6) and (10) and gainedthe joint cost of members 119894 and 119894 + 1 as

119862119894119894+1 (119873119901119894 119898119871119894 119873119898119894+1 119870119894+1) = EC119901119894 + EC119898119894+1

= 119873119898119894+1119870119894+1 [EHC119894 + EBC119894] + 119873119901119894119862119901119894

+ 119873119898119894+1119862119898119894+1 + 119867119898119894+1AI119894+1

= 119873119898119894+1119870119894+1 119863119894119867119901119894 [119905119894 (119898119871119894 minus 120583119871119889119894)] +

1199051198942

2

+

1199051198941199091198940119894

1 minus 0119894

+ (119903119871119894 minus 119898119871119894)

sdot [int

119903119871119894

119898119871119894

(119871119889119894 minus 119898119871119894) 119891 (119871119889119894) 119889119871119889119894

minus (1199091198940119894) int

119903119871119894

119898119871119894

119891 (119871119889119894) 119889119871119889119894] + [119862119890119894 + 120573119894119863119894 (119903119871119894

minus 119898119871119894)] [int

119903119871119894

119898119871119894

119891 (119871119889119894) 119889119871119889119894 + 119865119894] + 119873119901119894119862119904119894

+ 119873119898119894119862119898119894 + 119867119898119894 (

119879119901119894minus1

2

119873119898119894

)

sdot 119863119894minus1 [1

2

(1 minus

119870119894 + 0119894minus1 minus 1

119870119894 (1 minus 0119894)

)

+ (

1

2

minus

1

119870119894 (1 minus 0119894minus1)

)(1 minus

119863119894minus1

119875119894

)]

(11a)

Mathematical Problems in Engineering 7

Substitute 119905119894 = 119879119901119894119870119894+1119873119898119894+1 into (11a) Consider

119862119894119894+1 (119873119901119894 119898119871119894 119873119898119894+1 119870119894+1) = V119894119879119901119894 (119898119871119894 minus 120583119871119889119894)

+ V119894119879119901119894119909119894120588119894 +119879119901119894

2

119870119894+1119873119898119894+1

(Ω119894 minus

119867119898119894+1119863119894119870119894+1120588119894

2

)

+

119879119901119894

2

119873119898119894+1

120579119894 + 119870119894+1119873119898119894+1 [(119903119871119894 minus 119898119871119894)

sdot V119894 int119903119871119894

119898119871119894

(119871119889119894 minus 119898119871119894) 119891 (119871119889119894) 119889119871119889119894

+ 120576119894 int

119903119871119894

119898119871119894

119891 (119871119889119894) 119889119871119889119894 + 119865119894] + 119873119901119894119862119904119894

+ 119873119898119894+1119862119898119894+1

(11b)

where

V119894 = 119867119901119894119863119894

120588119894 =0119894

1 minus 0119894

Ω119894

=

119863119894

2

[119867119901119894 + 119867119898119894+1 (1 minus (

2

1 minus 0119894

)(1 minus

119863119894

119875119894+1

))]

120576119894 = 119862119890119894 + (120573119894119863119894 minus V1198941199091198940119894) (119903119871119894 minus 119898119871119894)

120579119894 =

119867119898119894+1119863119894

2

(1 minus

119863119894

119875119894+1

)

(12)

Since 119879119901119894 = 119876119898119894minus1119875119894 = (119863119894minus1119873119898119894+1119875119894)119879119901119894minus1 119879119901119894 can beexpressed further as

119879119901119894 = (

120593119894

prod119894

119897=1119873119898119897

)1198791199011 (13)

where

120593119894 =

1 if 119894 = 1

119894

prod

119897=2

(

119863119894minus1

119875119894

) if 119894 = 2 3 119878 minus 1

(14)

All the 119879119901119894 in function (11b) could be replaced as follows

119862119894119894+1 (119873119901119894 119898119871119894 119873119898119894+1 119870119894+1) = V119894(120593119894

prod119894

119897=1119873119898119897

)

sdot 1198791199011 (119898119871119894 minus 120583119871119889119894) + V119894(120593119894

prod119894

119897=1119873119898119897

)1198791199011119909119894120588119894

+

(120593119894prod119894

119897=1119873119898119897) 1198791199011

2

119870119894+1119873119898119894+1

(Ω119894 minus

119867119898119894+1119863119894119870119894+1120588119894

2

)

+

(120593119894prod119894

119897=1119873119898119897) 1198791199011

2

119873119898119894+1

120579119894

+ 119870119894+1119873119898119894+1 [(119903119871119894 minus 119898119871119894)

sdot V119894 int119903119871119894

119898119871119894

(119871119889119894 minus 119898119871119894) 119891 (119871119889119894) 119889119871119889119894

+ 120576119894 int

119903119871119894

119898119871119894

119891 (119871119889119894) 119889119871119889119894 + 119865119894] + 119873119901119894119862119904119894

+ 119873119898119894+1119862119898119894+1

(11c)

There are119873119898119894+1 purchasing times of member 119894+1 during119879119901119894 According to (11c) the joint total cost function includingpurchasing and manufacturing activities of the serial 119878-echelon integrated JIT inventory model is presented in

119878minus1

sum

119894=1

V1198941205931198941198791199011 (119898119871119894 minus 120583119871119889119894) + V1198941205931198941198791199011119909119894120588119894

+

(1205931198941198791199011)

2

119870119894+1prod119894

119897=1119873119898119897

(Ω119894 minus

119867119898119894+1119863119894119870119894+1120588119894

2

)

+

1198791199011

2

prod119894

119897=1119873119898119897

120579119894 + (

119894

prod

119897=1

119873119898119897)119870119894+1119873119898119894+1 [(119903119871119894 minus 119898119871119894)

sdot V119894 int119903119871119894

119898119871119894

(119871119889119894 minus 119898119871119894) 119891 (119871119889119894) 119889119871119889119894

+ 120576119894 int

119903119871119894

119898119871119894

119891 (119871119889119894) 119889119871119889119894 + 119865119894] + (

119894

prod

119897=1

119873119898119897)119873119901119894119862119904119894

+ (

119894

prod

119897=1

119873119898119897)119862119898119894+1

(15)

The delivery time constrains 0 le 120582119871119894 le 119898119871119894 le 120574119871119894 le

119905119894 119894 = 1 2 119878 minus 1 and variables constrains are consideredin the inventory model Eventually the 119878-echelon integratedJIT model can be expressed as

Minimize 1198691198621119878 (119873119901 119898119871 119873119898 119870)

=

119878minus1

sum

119894=1

V1198941205931198941198791199011 (119898119871119894 minus 120583119871119889119894) + V1198941205931198941198791199011119909119894120588119894 +(1205931198941198791199011)

2

119870119894+1prod119894

119897=1119873119898119897

(Ω119894 minus

119867119898119894+1119863119894119870119894+1120588119894

2

) +

11987911990112

prod119894

119897=1119873119898119897

120579119894

8 Mathematical Problems in Engineering

+(

119894

prod

119897=1

119873119898119897)119870119894+1119873119898119894+1 [(119903119871119894 minus 119898119871119894) V119894 int119903119871119894

119898119871119894

(119871119889119894 minus 119898119871119894) 119891 (119871119889119894) 119889119871119889119894 + 120576119894 int

119903119871119894

119898119871119894

119891 (119871119889119894) 119889119871119889119894 + 119865119894]

+(

119894

prod

119897=1

119873119898119897)119873119901119894119862119904119894 + (

119894

prod

119897=1

119873119898119897)119862119898119894+1

Subject to 0 le 120582119871119894 le 119898119871119894 le 120574119871119894 le 119905119894 for 119894 = 1 2 119878 minus 1

119870119894+1 le1

0119894

minus 1

119873119901119894 119870119894+1 119873119898119894+1 are all integers for 119894 = 1 2 119878 minus 1

(16)

4 Numeral Experiment and Analysis

In this section a numeral experiment is proposed based onproper parameter settings All results in this research wereperformed on a PC Programs were written in Matlab 2012bAn efficiency comparison between Lingo PSO algorithmand Ant Colony Algorithm was proposed as well Further-more we would like to discuss how production runs anddefective rate influences whole system cost

41 Experiment Results of Ant Colony Algorithm ParticleSwarm Optimization and Lingo With appropriate param-eters settings we may build a 7-echelon supply chain inte-grated JIT inventory model with uncertain delivery leadtime and unreliable quality of received items Assume thepurchasing time interval for first purchaser in years (1198791199011) is12 years The relevant data of purchasing and manufacturingactivities are shown in Tables 1 and 2

The objective function and constraint function are shownbelow Consider

1198691198621119878 (119873119901 119898119871 119873119898 119870)

=

119878minus1

sum

119894=1

V1198941205931198941198791199011 (119898119871119894 minus 120583119871119889119894) + V1198941205931198941198791199011119909119894120588119894 +(1205931198941198791199011)

2

119870119894+1prod119894

119897=1119873119898119897

(Ω119894 minus

119867119898119894+1119863119894119870119894+1120588119894

2

) +

1198791199011

2

prod119894

119897=1119873119898119897

120579119894

+(

119894

prod

119897=1

119873119898119897)119870119894+1119873119898119894+1 [(119903119871119894 minus 119898119871119894) V119894 int119903119871119894

119898119871119894

(119871119889119894 minus 119898119871119894) 119891 (119871119889119894) 119889119871119889119894 + 120576119894 int

119903119871119894

119898119871119894

119891 (119871119889119894) 119889119871119889119894 + 119865119894]

+(

119894

prod

119897=1

119873119898119897)119873119901119894119862119904119894 + (

119894

prod

119897=1

119873119898119897)119862119898119894+1

Subject to 0 le 120582119871119894 le 119898119871119894 le 120574119871119894 le 119905119894 for 119894 = 1 2 119878 minus 1

119870119894+1 le1

0119894

minus 1

119873119901119894 119870119894+1 119873119898119894+1 are all integers for 119894 = 1 2 119878 minus 1

(17)

The goal of this research is to find out the optimal solutionof the 7-echelon inventory model via the above functions andACA and compare the efficiency between Lingo ACA andPSO

As Table 3 shows though all the three methods werecapable of finding the optimal solution for SMEIJI problemACA was obviously better than PSO and Lingo in efficiencyACA has shown fastest CPU searching time and least averageiterations among three methods

42 Experiments of Performance and 119878-Echelon In this sec-tion we would like to discuss the relationship between CPU

searching time and echelon number among three methodswe also discussed the relationship between average iterationsand echelon number

As Figure 3 shows the CPU searching time wouldincrease with the number of echelons due to the compli-cated calculation However because of the well-organizedconstraint function and good initial solution ACA still keptlower CPU searching time than the other two methods Onthe other hand Figure 4 shows that ACArsquos average iterationswere the least among three methods At the beginning of themultiechelon all the three algorithms need a great amountof iterations to search the optimal solution After calculating

Mathematical Problems in Engineering 9

Table 1 The relevant data of purchasing activities

Purchasing activityEchelon 119863119894 119862119904119894 119867119901119894 119862119890119894 120573119894 120582119871119894 120574119871119894 120583119871119889119894 120590119871119889119894 119865119894 119883119894 0119894

1 18 w 25 28 29 32 0005 0015 0009 0002 70 0005 0032 20w 28 22 32 26 0004 001 00064 00012 50 0003 0013 24w 32 17 38 21 0003 0009 00054 00012 48 0003 0044 30w 36 12 42 16 0004 0012 00072 00016 55 0004 0055 35w 35 08 37 12 0006 0016 001 0002 80 0005 0016 40w 38 06 40 09 0004 0014 0008 0002 65 0005 0027 mdash mdash mdash mdash mdash mdash mdash mdash mdash mdash mdash mdash

Table 2 The relevant data of manufacturing activities

Manufacturing activitiesEchelon 119875119894 119862119898119894 119867119898119894

1 mdash mdash mdash2 200000 280 253 240000 200 194 300000 320 145 350000 360 106 400000 380 097 450000 420 05

1 2 3 4 5 6 70

05

1

15

2

25

3

ACAPSO

Lingo

Figure 3 Experiment results of CPU time and echelon number

repeatedly the trend of the average iterations would come flatwithout extreme variety

Nevertheless although ACA seems better than the othertwo algorithms we could not ensure that the performancewould still be great when applied to higher echelon inventorymodel Hence a 15-echelon model has proposed to observethe performance again The following table displayed theparameters of 15-echelon model

To evaluate the performance of ACA PSO and Lingo forthe 15-echelon inventorymodel we applied the parameters inTables 4 and 5 to solve the 15-echelon inventory problem

According to Table 6 all three algorithms were still ableto find out the optimal solution of the 15-echelon inventorymodel Once the echelon increases the complexity of the

1 2 3 4 5 6 70

500

1000

1500

2000

2500

3000

3500

4000

ACAPSO

Lingo

Figure 4 Experiment results of average iterations and echelonnumber

model increases as well However the performance of ACAwas still better than the other two algorithms

There are several advantages of ACA as follows

(1) Applying the positive and negative feedback enablesthe process of searching optimization to converge atthe end which means the results would close to theoptimal solutions gradually

(2) Every single ant is able to change the surroundingenvironment through releasing pheromone and beingconscious of the variety of environment which makesthem communicate with each other

(3) Due to distributed computing all the individuals startto calculate simultaneously during the searching pro-cess which enhances the efficiency of the calculationability

(4) For the proposed algorithm it is difficult to stick inthe partial optimization instead it is always easy tofind the best solution

To sum up ACA was proposed to solve the complicatedmathematical problem we also have proved that ACA wasmore efficient than PSO and Lingo These are the new majorcontributions of the present paper

43 The Sensitive Analysis of SMEIJI In this section thenumber of production runs 119873119898119894 and the defective rate of

10 Mathematical Problems in Engineering

Table 3 Comparisons between three methods in SMEIJI problem

Echelon ACA PSO Lingo119873119901119894 119898119871119894 119870119894 119873119898119894 119873119901119894 119898119871119894 119870119894 119873119898119894 119873119901119894 119898119871119894 119870119894 119873119898119894

1 1 00147 mdash mdash 1 00147 mdash mdash 1 00147 mdash mdash2 1 000987 11 1 1 000987 11 1 1 000987 11 13 1 000889 13 1 1 000889 13 1 1 000889 13 14 1 001184 11 1 1 001184 11 1 1 001184 11 15 1 001571 8 1 1 001571 8 1 1 001571 8 16 1 001378 5 1 1 001378 5 1 1 001378 5 17 mdash mdash 5 1 mdash mdash 5 1 mdash mdash 5 1Optimal cost 89166 89166 89166CPU time 0655 1744 23Average iterations 350 1278 3758

Table 4 The relevant data of purchasing activities of 15-echelon model

Purchasing activityEchelon 119863119894 119862119904119894 119867119901119894 119862119890119894 120573119894 120582119871119894 120574119871119894 120583119871119889119894 120590119871119889119894 119865119894 119883119894 0119894

1 18 w 25 28 29 32 0005 0015 0009 0002 70 0005 0032 20w 28 22 32 26 0004 001 00064 00012 50 0003 0013 24w 32 17 38 21 0003 0009 00054 00012 48 0003 0044 30w 36 12 42 16 0004 0012 00072 00016 55 0004 0055 35w 35 08 37 12 0006 0016 001 0002 80 0005 0016 40w 38 06 40 09 0004 0014 0008 0002 65 0005 0027 42w 40 05 28 12 0005 0013 001 0002 70 0005 0038 45w 42 07 35 15 0004 0009 0009 00018 50 0004 0029 50w 44 10 42 10 0003 0010 0005 00024 55 0004 00210 52w 46 15 35 22 0005 0012 00062 00025 60 0005 00211 56w 48 20 49 25 0006 0009 001 00026 65 0003 00312 60w 50 22 40 14 0004 0016 0012 0002 48 0005 00413 62w 55 17 35 15 0007 0015 0009 00018 72 0004 00414 65w 56 12 32 16 0003 0014 00078 0002 70 0006 00315 mdash mdash mdash mdash mdash mdash mdash mdash mdash mdash mdash mdash

Table 5The relevant data of manufacturing activities of 15-echelonmodel

Manufacturing activitiesEchelon 119875119894 119862119898119894 119867119898119894

1 mdash mdash mdash2 200000 280 253 240000 200 194 300000 320 145 350000 360 106 400000 380 097 450000 420 058 480000 440 109 500000 420 1210 520000 400 1511 540000 450 2012 580000 440 2213 620000 420 1814 640000 460 1515 660000 380 16

receiving shipment 0119894 were considered to make sensitiveanalysis and observe the influence on the optimal total jointcost The defective rate and production runs were changedby 0119894(1 + 119877) and 119873119898119894(1 + 119877) where the values of 119877 are0 50 100 200 300 400

Figure 5 illustrated that defective rate was more sensitivethan production runs under the situation that productionruns were one-time production except first echelon In thatcase the higher the defective ratewas the higher the total costwas Accordingly inspecting goods cautiously before deliveryand reducing the defective rate would lower total cost of thewhole supply chainHowever if relaxing that assumption thissensitive analysis would result in different outcomes

Figure 6 has shown that production runs were more sen-sitive than defective rate when relaxing one-time productionassumption In this case the higher the production runswerethe higher the total cost was Hence reducing productionruns should play an essential role in the whole supply chainOnce the one batch production policy was able to satisfy thedemand from the buyer the total cost of whole supply chainwill go down extremely

Mathematical Problems in Engineering 11

Table 6 Comparisons between three ways of 15-echelon inventory model

ACA PSO LingoEchelon 119873119901119894 119898119871119894 119870119894 119873119898119894 119873119901119894 119898119871119894 119870119894 119873119898119894 119873119901119894 119898119871119894 119870119894 119873119898119894

1 1 00147 mdash mdash 1 00147 mdash mdash 1 00147 mdash mdash2 1 000987 11 1 1 000987 11 1 1 000987 11 13 1 000889 13 1 1 000889 13 1 1 000889 13 14 1 001184 11 1 1 001184 11 1 1 001184 11 15 1 001571 8 1 1 001571 8 1 1 001571 8 16 1 001378 5 1 1 001378 5 1 1 001378 5 17 1 001451 5 1 1 001451 5 1 1 001451 5 18 1 001287 5 1 1 001287 5 1 1 001287 5 19 1 00154 7 1 1 00154 7 1 1 00154 7 110 1 001432 9 1 1 001432 9 1 1 001432 9 111 1 001352 7 1 1 001352 7 1 1 001352 7 112 1 00162 8 1 1 00162 8 1 1 00162 8 113 1 001571 10 1 1 001571 10 1 1 001571 10 114 1 001611 5 1 1 001611 5 1 1 001611 5 115 mdash mdash 5 1 mdash mdash 5 1 mdash mdash 5 1Optimal cost ($) 231254 231254 231254CPU time (s) 1725 6621 58215Average iterations 700 2883 8241

Defective rateProduction runs

400100 200 300 5000 500

20000

40000

60000

80000

100000

120000

140000

160000

180000

Figure 5 The sensitive analysis between defective rate and produc-tion runs

5 Conclusions

In summary this research proposed an integrated JIT inven-tory model to solve the partial optimization problem ofpurchaser andmanufacturer finding the optimal solution forthe whole supply chain Furthermore a multiechelon supplychainwith uncertain delivery lead time and unreliable qualityhas been considered to fit the real supply chain environ-ment Besides ACA was proposed to solve the complicatedmathematical problem we also have proved that ACA was

0 50 100 200 300 400 500

Defective rateProduction runs

0

50000

100000

150000

200000

250000

300000

350000

400000

Figure 6 The sensitive analysis between defective rate and produc-tions runs with relaxing assumption

more efficient than PSO and Lingo These are the new majorcontributions of the present paper

Finally there are a great deal of helpful directions forenterprises and researchers (1)The variety of conditions thatoccur in real supply chain can be considered in the modelFor instance time value of cash problem and deterioration(2)The warehousing and storage can be considered in supplychain which make the model more complex to solve (3)The model contains only one member in each echelon amultimember in each echelon should be considered to fit

12 Mathematical Problems in Engineering

real situation in supply chain (4) Some real cases andordata can be applied to our proposed model in the followingworks

Futureworkwill hopefully involve the real-world concernto our research which will be our main target In addition we

do hope our research can be extended intomore considerableareas and more considerable variables

Appendix

The Hessian matrix 119865 of 1198691198621119878(119873119901 119898119871 119873119898 119870) for given 119873119901119894

equal to 1 can be shown as

119865 =

[

[

[

[

[

[

[

[

[

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

1205971198731198981198942

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119873119898119894120597119898119871119894

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119873119898119894120597119870119894

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119898119871119894120597119873119898119894

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

1205971198981198711198942

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119898119871119894120597119870119894

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119870119894120597119873119898119894

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119870119894120597119898119871119894

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

1205971198701198942

]

]

]

]

]

]

]

]

]

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

1205971198731198981198942

=

2

1198731198981198943(120579120593119879

2+

1205931198792(1198671198631198701205932 + Ω)

119870

)

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

1205971198981198711198942

= 119870119873[(

119890minus119898120583

1205832

+ 2 (119890minus119898120583

minus 119890minus120574120583

) +

119890(minus119898120583)(minus119898+120583)

120583

) 120592]

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

1205971198701198942

=

2

1198701198943(120579120593119879

2+

1205931198792(1198671198631198701205932 + Ω)

119873

)

12059721198691198621119904 (119873119898119894 119898119871119894 119870119894)

120597119898119871119894120597119873119898119894

=

12059721198691198621119904 (119873119898119894 119898119871119894 119870119894)

120597119873119898119894120597119898119871119894

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119870119894120597119873119898119894

=

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119873119898119894120597119870119894

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119870119894120597119898119871119894

=

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119898119871119894120597119870119894

100381610038161003816100381611986511

1003816100381610038161003816=

1205972119865

1205971198731198981198942

=

2

1198731198981198943(120579120593119879

2+

1205931198792(1198671198631198701205932 + Ω)

119870

) gt 0

100381610038161003816100381611986522

1003816100381610038161003816= (

1205972119865

1205971198731198981198942)(

1205972119865

1205971198981198711198942) minus [

1205972119865

120597119873119898119894120597119898119871119894

]

2

=

2

1198731198981198943(120579120593119879

2+

1205931198792(1198671198631198701205932 + Ω)

119870

)

1205972119865

1205971198981198711198942minus (

1205972119865

120597119873119898119894120597119898119871119894

)

2

2

1198733119898119894

(1205791205931198792+

119861

119870

)[119870119873(

119890minus119898120576120583

1205832

+ (2119909 +

119890(minus119898120583)119909119910

120583

120592))] + 1198702[

119890minus119898120576120583

120583

+ 120592 (119909119910 minus 119860)] gt 0

100381610038161003816100381611986533

1003816100381610038161003816= (

1205972119865

1205971198731198981198942

1205972119865

1205971198981198711198942

1205972119865

1205971198701198942) minus [(

1205972119865

120597119873119898119894120597119870119894

)

2

(

1205972119865

1205971198981198711198942)]

=

2

1198731198981198943(120579120593119879

2+

119861

119870

)[119870119873(

119890minus119898120576120583

1205832

+ 2(119909 +

119890(minus119898120603)119909119910

120583

) 120592)]

2

1198703

119861

119873

minus [119865 + 119909120576 + 119860120592119910 minus

1205791205931198792

1198732

+

119861

2 (119870119873)2]

[119870119873(

119890minus119898120576120583

1205832

+ (2119909 +

119890minus119898120576120583

120583

) 120592)] gt 0

(A1)

Mathematical Problems in Engineering 13

As a result the matrix F has proved to be a positivedefinite matrix where |11986511| |11986522| and |11986533| are all positiveTherefore theminimumvalue exists in 1198691198621119878(119873119901119898119871 119873119898 119870)Furthermore we can find the optimization in the proposedmodel

Competing Interests

The authors declare that they have no competing interests

References

[1] S K Goyal ldquoAn integrated inventory model for a singlesupplier-single customer problemrdquo International Journal ofProduction Research vol 15 no 1 pp 107ndash111 1977

[2] A Banerjee ldquoA joint economic-lot-sizemodel for purchaser andvendorrdquo Decision Sciences vol 17 no 3 pp 292ndash311 1986

[3] S K Goyal ldquoldquoA joint economic-lot-size model for purchaserand vendorrdquo a commentrdquo Decision Sciences vol 19 no 1 pp236ndash241 1988

[4] Y Seo ldquoControlling general multi-echelon distribution supplychainswith improved reorder decision policy utilizing real-timeshared stock informationrdquo Computers amp Industrial Engineeringvol 51 no 2 pp 229ndash246 2006

[5] H N Chiu and H L Huang ldquoA Multi-echelon integratedJIT inventory model using the time buffer and emergencyborrowing policies to deal with random delivery lead timesrdquoInternational Journal of Production Research vol 41 no 13 pp2911ndash2931 2003

[6] J Sadeghi S M Mousavi S T A Niaki and S SadeghildquoOptimizing a multi-vendor multi-retailer vendor managedinventory problem two tuned meta-heuristic algorithmsrdquoKnowledge-Based Systems vol 50 pp 159ndash170 2013

[7] S S Sana ldquoA production-inventory model of imperfect qualityproducts in a three-layer supply chainrdquo Decision Support Sys-tems vol 50 no 2 pp 539ndash547 2011

[8] H N Soni and K A Patel ldquoOptimal strategy for an integratedinventory system involving variable production and defectiveitems under retailer partial trade credit policyrdquoDecision SupportSystems vol 54 no 1 pp 235ndash247 2012

[9] R J Tersine Principles of Inventory andMaterials ManagementNorthHolland AmsterdamTheNetherlands 3rd edition 1988

[10] C J Liao and C H Shyu ldquoAn analytical determination of leadtime with normal demandrdquo International Journal of Operationsamp Production Management vol 11 no 9 pp 72ndash78 1991

[11] M Ben-Daya and A Raouf ldquoInventory models involving leadtime as a decision variablerdquo Journal of the Operational ResearchSociety vol 45 no 5 pp 579ndash582 1994

[12] S M J Mirzapour Al-e-Hashem A Baboli and Z Sazvar ldquoAstochastic aggregate production planningmodel in a green sup-ply chain considering flexible lead times nonlinear purchaseand shortage cost functionsrdquo European Journal of OperationalResearch vol 230 no 1 pp 26ndash41 2013

[13] C Chandra and J Grabis ldquoInventorymanagementwith variablelead-time dependent procurement costrdquo Omega vol 36 no 5pp 877ndash887 2008

[14] J-W Wu and H-Y Tsai ldquoMixture inventory model with backorders and lost sales for variable lead time demand withthe mixtures of normal distributionrdquo International Journal ofSystems Science Principles and Applications of Systems andIntegration vol 32 no 2 pp 259ndash268 2001

[15] M A Hoque ldquoA vendorndashbuyer integrated productionndashinventory model with normal distribution of lead timerdquoInternational Journal of Production Economics vol 144 no 2pp 409ndash417 2013

[16] F Altiparmak M Gen L Lin and T Paksoy ldquoA geneticalgorithm approach for multi-objective optimization of supplychain networksrdquo Computers and Industrial Engineering vol 51no 1 pp 196ndash215 2006

[17] J Sadeghi S Sadeghi and S T Niaki ldquoA hybrid vendormanaged inventory and redundancy allocation optimizationproblem in supply chain management an NSGA-II with tunedparametersrdquo Computers and Operations Research vol 41 pp53ndash64 2014

[18] X Li F Baki P Tian and B A Chaouch ldquoA robust block-chainbased tabu search algorithm for the dynamic lot sizing problemwith product returns and remanufacturingrdquoOmega vol 42 no1 pp 75ndash87 2014

[19] Z Chen and B R Sarker ldquoAn integrated optimal inventorylot-sizing and vehicle-routing model for a multisupplier single-assembler system with JIT deliveryrdquo International Journal ofProduction Research vol 52 no 17 pp 5086ndash5114 2014

[20] A R Nia M Hemmati Far and S T A Niaki ldquoA fuzzy vendormanaged inventory of multi-item economic order quantitymodel under shortage an ant colony optimization algorithmrdquoInternational Journal of Production Economics vol 155 pp 259ndash271 2014

[21] S M Hatefi and F Jolai ldquoRobust and reliable forward-reverselogistics network design under demand uncertainty and facilitydisruptionsrdquo Applied Mathematical Modelling vol 38 no 9-10pp 2630ndash2647 2014

[22] R H Ma ldquoCross warehouse scheduling of logistics distributionand cycle re-claimer based on heuristics algorithmrdquo AppliedMechanics and Materials vol 644ndash650 pp 2606ndash2610 2014

[23] M Hashim L Yao A H Nadeem M Nazim and M NazamldquoMulti-objective optimization model for supplier selectionproblem in fuzzy environmentrdquo Advances in Intelligent Systemsand Computing vol 281 pp 1201ndash1213 2014

[24] Q-M Hu and Z-H Hu ldquoA stochastic programming modelfor hub-and-spoke network with uncertain flowsrdquo InternationalJournal of Industrial and Systems Engineering vol 21 no 3 pp302ndash319 2015

[25] S A Torabi J Namdar S M Hatefi and F Jolai ldquoAn enhancedpossibilistic programming approach for reliable closed-loopsupply chain network designrdquo International Journal of Produc-tion Research vol 54 no 5 pp 1358ndash1387 2016

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

Substitute 119905119894 = 119879119901119894119870119894+1119873119898119894+1 into (11a) Consider

119862119894119894+1 (119873119901119894 119898119871119894 119873119898119894+1 119870119894+1) = V119894119879119901119894 (119898119871119894 minus 120583119871119889119894)

+ V119894119879119901119894119909119894120588119894 +119879119901119894

2

119870119894+1119873119898119894+1

(Ω119894 minus

119867119898119894+1119863119894119870119894+1120588119894

2

)

+

119879119901119894

2

119873119898119894+1

120579119894 + 119870119894+1119873119898119894+1 [(119903119871119894 minus 119898119871119894)

sdot V119894 int119903119871119894

119898119871119894

(119871119889119894 minus 119898119871119894) 119891 (119871119889119894) 119889119871119889119894

+ 120576119894 int

119903119871119894

119898119871119894

119891 (119871119889119894) 119889119871119889119894 + 119865119894] + 119873119901119894119862119904119894

+ 119873119898119894+1119862119898119894+1

(11b)

where

V119894 = 119867119901119894119863119894

120588119894 =0119894

1 minus 0119894

Ω119894

=

119863119894

2

[119867119901119894 + 119867119898119894+1 (1 minus (

2

1 minus 0119894

)(1 minus

119863119894

119875119894+1

))]

120576119894 = 119862119890119894 + (120573119894119863119894 minus V1198941199091198940119894) (119903119871119894 minus 119898119871119894)

120579119894 =

119867119898119894+1119863119894

2

(1 minus

119863119894

119875119894+1

)

(12)

Since 119879119901119894 = 119876119898119894minus1119875119894 = (119863119894minus1119873119898119894+1119875119894)119879119901119894minus1 119879119901119894 can beexpressed further as

119879119901119894 = (

120593119894

prod119894

119897=1119873119898119897

)1198791199011 (13)

where

120593119894 =

1 if 119894 = 1

119894

prod

119897=2

(

119863119894minus1

119875119894

) if 119894 = 2 3 119878 minus 1

(14)

All the 119879119901119894 in function (11b) could be replaced as follows

119862119894119894+1 (119873119901119894 119898119871119894 119873119898119894+1 119870119894+1) = V119894(120593119894

prod119894

119897=1119873119898119897

)

sdot 1198791199011 (119898119871119894 minus 120583119871119889119894) + V119894(120593119894

prod119894

119897=1119873119898119897

)1198791199011119909119894120588119894

+

(120593119894prod119894

119897=1119873119898119897) 1198791199011

2

119870119894+1119873119898119894+1

(Ω119894 minus

119867119898119894+1119863119894119870119894+1120588119894

2

)

+

(120593119894prod119894

119897=1119873119898119897) 1198791199011

2

119873119898119894+1

120579119894

+ 119870119894+1119873119898119894+1 [(119903119871119894 minus 119898119871119894)

sdot V119894 int119903119871119894

119898119871119894

(119871119889119894 minus 119898119871119894) 119891 (119871119889119894) 119889119871119889119894

+ 120576119894 int

119903119871119894

119898119871119894

119891 (119871119889119894) 119889119871119889119894 + 119865119894] + 119873119901119894119862119904119894

+ 119873119898119894+1119862119898119894+1

(11c)

There are119873119898119894+1 purchasing times of member 119894+1 during119879119901119894 According to (11c) the joint total cost function includingpurchasing and manufacturing activities of the serial 119878-echelon integrated JIT inventory model is presented in

119878minus1

sum

119894=1

V1198941205931198941198791199011 (119898119871119894 minus 120583119871119889119894) + V1198941205931198941198791199011119909119894120588119894

+

(1205931198941198791199011)

2

119870119894+1prod119894

119897=1119873119898119897

(Ω119894 minus

119867119898119894+1119863119894119870119894+1120588119894

2

)

+

1198791199011

2

prod119894

119897=1119873119898119897

120579119894 + (

119894

prod

119897=1

119873119898119897)119870119894+1119873119898119894+1 [(119903119871119894 minus 119898119871119894)

sdot V119894 int119903119871119894

119898119871119894

(119871119889119894 minus 119898119871119894) 119891 (119871119889119894) 119889119871119889119894

+ 120576119894 int

119903119871119894

119898119871119894

119891 (119871119889119894) 119889119871119889119894 + 119865119894] + (

119894

prod

119897=1

119873119898119897)119873119901119894119862119904119894

+ (

119894

prod

119897=1

119873119898119897)119862119898119894+1

(15)

The delivery time constrains 0 le 120582119871119894 le 119898119871119894 le 120574119871119894 le

119905119894 119894 = 1 2 119878 minus 1 and variables constrains are consideredin the inventory model Eventually the 119878-echelon integratedJIT model can be expressed as

Minimize 1198691198621119878 (119873119901 119898119871 119873119898 119870)

=

119878minus1

sum

119894=1

V1198941205931198941198791199011 (119898119871119894 minus 120583119871119889119894) + V1198941205931198941198791199011119909119894120588119894 +(1205931198941198791199011)

2

119870119894+1prod119894

119897=1119873119898119897

(Ω119894 minus

119867119898119894+1119863119894119870119894+1120588119894

2

) +

11987911990112

prod119894

119897=1119873119898119897

120579119894

8 Mathematical Problems in Engineering

+(

119894

prod

119897=1

119873119898119897)119870119894+1119873119898119894+1 [(119903119871119894 minus 119898119871119894) V119894 int119903119871119894

119898119871119894

(119871119889119894 minus 119898119871119894) 119891 (119871119889119894) 119889119871119889119894 + 120576119894 int

119903119871119894

119898119871119894

119891 (119871119889119894) 119889119871119889119894 + 119865119894]

+(

119894

prod

119897=1

119873119898119897)119873119901119894119862119904119894 + (

119894

prod

119897=1

119873119898119897)119862119898119894+1

Subject to 0 le 120582119871119894 le 119898119871119894 le 120574119871119894 le 119905119894 for 119894 = 1 2 119878 minus 1

119870119894+1 le1

0119894

minus 1

119873119901119894 119870119894+1 119873119898119894+1 are all integers for 119894 = 1 2 119878 minus 1

(16)

4 Numeral Experiment and Analysis

In this section a numeral experiment is proposed based onproper parameter settings All results in this research wereperformed on a PC Programs were written in Matlab 2012bAn efficiency comparison between Lingo PSO algorithmand Ant Colony Algorithm was proposed as well Further-more we would like to discuss how production runs anddefective rate influences whole system cost

41 Experiment Results of Ant Colony Algorithm ParticleSwarm Optimization and Lingo With appropriate param-eters settings we may build a 7-echelon supply chain inte-grated JIT inventory model with uncertain delivery leadtime and unreliable quality of received items Assume thepurchasing time interval for first purchaser in years (1198791199011) is12 years The relevant data of purchasing and manufacturingactivities are shown in Tables 1 and 2

The objective function and constraint function are shownbelow Consider

1198691198621119878 (119873119901 119898119871 119873119898 119870)

=

119878minus1

sum

119894=1

V1198941205931198941198791199011 (119898119871119894 minus 120583119871119889119894) + V1198941205931198941198791199011119909119894120588119894 +(1205931198941198791199011)

2

119870119894+1prod119894

119897=1119873119898119897

(Ω119894 minus

119867119898119894+1119863119894119870119894+1120588119894

2

) +

1198791199011

2

prod119894

119897=1119873119898119897

120579119894

+(

119894

prod

119897=1

119873119898119897)119870119894+1119873119898119894+1 [(119903119871119894 minus 119898119871119894) V119894 int119903119871119894

119898119871119894

(119871119889119894 minus 119898119871119894) 119891 (119871119889119894) 119889119871119889119894 + 120576119894 int

119903119871119894

119898119871119894

119891 (119871119889119894) 119889119871119889119894 + 119865119894]

+(

119894

prod

119897=1

119873119898119897)119873119901119894119862119904119894 + (

119894

prod

119897=1

119873119898119897)119862119898119894+1

Subject to 0 le 120582119871119894 le 119898119871119894 le 120574119871119894 le 119905119894 for 119894 = 1 2 119878 minus 1

119870119894+1 le1

0119894

minus 1

119873119901119894 119870119894+1 119873119898119894+1 are all integers for 119894 = 1 2 119878 minus 1

(17)

The goal of this research is to find out the optimal solutionof the 7-echelon inventory model via the above functions andACA and compare the efficiency between Lingo ACA andPSO

As Table 3 shows though all the three methods werecapable of finding the optimal solution for SMEIJI problemACA was obviously better than PSO and Lingo in efficiencyACA has shown fastest CPU searching time and least averageiterations among three methods

42 Experiments of Performance and 119878-Echelon In this sec-tion we would like to discuss the relationship between CPU

searching time and echelon number among three methodswe also discussed the relationship between average iterationsand echelon number

As Figure 3 shows the CPU searching time wouldincrease with the number of echelons due to the compli-cated calculation However because of the well-organizedconstraint function and good initial solution ACA still keptlower CPU searching time than the other two methods Onthe other hand Figure 4 shows that ACArsquos average iterationswere the least among three methods At the beginning of themultiechelon all the three algorithms need a great amountof iterations to search the optimal solution After calculating

Mathematical Problems in Engineering 9

Table 1 The relevant data of purchasing activities

Purchasing activityEchelon 119863119894 119862119904119894 119867119901119894 119862119890119894 120573119894 120582119871119894 120574119871119894 120583119871119889119894 120590119871119889119894 119865119894 119883119894 0119894

1 18 w 25 28 29 32 0005 0015 0009 0002 70 0005 0032 20w 28 22 32 26 0004 001 00064 00012 50 0003 0013 24w 32 17 38 21 0003 0009 00054 00012 48 0003 0044 30w 36 12 42 16 0004 0012 00072 00016 55 0004 0055 35w 35 08 37 12 0006 0016 001 0002 80 0005 0016 40w 38 06 40 09 0004 0014 0008 0002 65 0005 0027 mdash mdash mdash mdash mdash mdash mdash mdash mdash mdash mdash mdash

Table 2 The relevant data of manufacturing activities

Manufacturing activitiesEchelon 119875119894 119862119898119894 119867119898119894

1 mdash mdash mdash2 200000 280 253 240000 200 194 300000 320 145 350000 360 106 400000 380 097 450000 420 05

1 2 3 4 5 6 70

05

1

15

2

25

3

ACAPSO

Lingo

Figure 3 Experiment results of CPU time and echelon number

repeatedly the trend of the average iterations would come flatwithout extreme variety

Nevertheless although ACA seems better than the othertwo algorithms we could not ensure that the performancewould still be great when applied to higher echelon inventorymodel Hence a 15-echelon model has proposed to observethe performance again The following table displayed theparameters of 15-echelon model

To evaluate the performance of ACA PSO and Lingo forthe 15-echelon inventorymodel we applied the parameters inTables 4 and 5 to solve the 15-echelon inventory problem

According to Table 6 all three algorithms were still ableto find out the optimal solution of the 15-echelon inventorymodel Once the echelon increases the complexity of the

1 2 3 4 5 6 70

500

1000

1500

2000

2500

3000

3500

4000

ACAPSO

Lingo

Figure 4 Experiment results of average iterations and echelonnumber

model increases as well However the performance of ACAwas still better than the other two algorithms

There are several advantages of ACA as follows

(1) Applying the positive and negative feedback enablesthe process of searching optimization to converge atthe end which means the results would close to theoptimal solutions gradually

(2) Every single ant is able to change the surroundingenvironment through releasing pheromone and beingconscious of the variety of environment which makesthem communicate with each other

(3) Due to distributed computing all the individuals startto calculate simultaneously during the searching pro-cess which enhances the efficiency of the calculationability

(4) For the proposed algorithm it is difficult to stick inthe partial optimization instead it is always easy tofind the best solution

To sum up ACA was proposed to solve the complicatedmathematical problem we also have proved that ACA wasmore efficient than PSO and Lingo These are the new majorcontributions of the present paper

43 The Sensitive Analysis of SMEIJI In this section thenumber of production runs 119873119898119894 and the defective rate of

10 Mathematical Problems in Engineering

Table 3 Comparisons between three methods in SMEIJI problem

Echelon ACA PSO Lingo119873119901119894 119898119871119894 119870119894 119873119898119894 119873119901119894 119898119871119894 119870119894 119873119898119894 119873119901119894 119898119871119894 119870119894 119873119898119894

1 1 00147 mdash mdash 1 00147 mdash mdash 1 00147 mdash mdash2 1 000987 11 1 1 000987 11 1 1 000987 11 13 1 000889 13 1 1 000889 13 1 1 000889 13 14 1 001184 11 1 1 001184 11 1 1 001184 11 15 1 001571 8 1 1 001571 8 1 1 001571 8 16 1 001378 5 1 1 001378 5 1 1 001378 5 17 mdash mdash 5 1 mdash mdash 5 1 mdash mdash 5 1Optimal cost 89166 89166 89166CPU time 0655 1744 23Average iterations 350 1278 3758

Table 4 The relevant data of purchasing activities of 15-echelon model

Purchasing activityEchelon 119863119894 119862119904119894 119867119901119894 119862119890119894 120573119894 120582119871119894 120574119871119894 120583119871119889119894 120590119871119889119894 119865119894 119883119894 0119894

1 18 w 25 28 29 32 0005 0015 0009 0002 70 0005 0032 20w 28 22 32 26 0004 001 00064 00012 50 0003 0013 24w 32 17 38 21 0003 0009 00054 00012 48 0003 0044 30w 36 12 42 16 0004 0012 00072 00016 55 0004 0055 35w 35 08 37 12 0006 0016 001 0002 80 0005 0016 40w 38 06 40 09 0004 0014 0008 0002 65 0005 0027 42w 40 05 28 12 0005 0013 001 0002 70 0005 0038 45w 42 07 35 15 0004 0009 0009 00018 50 0004 0029 50w 44 10 42 10 0003 0010 0005 00024 55 0004 00210 52w 46 15 35 22 0005 0012 00062 00025 60 0005 00211 56w 48 20 49 25 0006 0009 001 00026 65 0003 00312 60w 50 22 40 14 0004 0016 0012 0002 48 0005 00413 62w 55 17 35 15 0007 0015 0009 00018 72 0004 00414 65w 56 12 32 16 0003 0014 00078 0002 70 0006 00315 mdash mdash mdash mdash mdash mdash mdash mdash mdash mdash mdash mdash

Table 5The relevant data of manufacturing activities of 15-echelonmodel

Manufacturing activitiesEchelon 119875119894 119862119898119894 119867119898119894

1 mdash mdash mdash2 200000 280 253 240000 200 194 300000 320 145 350000 360 106 400000 380 097 450000 420 058 480000 440 109 500000 420 1210 520000 400 1511 540000 450 2012 580000 440 2213 620000 420 1814 640000 460 1515 660000 380 16

receiving shipment 0119894 were considered to make sensitiveanalysis and observe the influence on the optimal total jointcost The defective rate and production runs were changedby 0119894(1 + 119877) and 119873119898119894(1 + 119877) where the values of 119877 are0 50 100 200 300 400

Figure 5 illustrated that defective rate was more sensitivethan production runs under the situation that productionruns were one-time production except first echelon In thatcase the higher the defective ratewas the higher the total costwas Accordingly inspecting goods cautiously before deliveryand reducing the defective rate would lower total cost of thewhole supply chainHowever if relaxing that assumption thissensitive analysis would result in different outcomes

Figure 6 has shown that production runs were more sen-sitive than defective rate when relaxing one-time productionassumption In this case the higher the production runswerethe higher the total cost was Hence reducing productionruns should play an essential role in the whole supply chainOnce the one batch production policy was able to satisfy thedemand from the buyer the total cost of whole supply chainwill go down extremely

Mathematical Problems in Engineering 11

Table 6 Comparisons between three ways of 15-echelon inventory model

ACA PSO LingoEchelon 119873119901119894 119898119871119894 119870119894 119873119898119894 119873119901119894 119898119871119894 119870119894 119873119898119894 119873119901119894 119898119871119894 119870119894 119873119898119894

1 1 00147 mdash mdash 1 00147 mdash mdash 1 00147 mdash mdash2 1 000987 11 1 1 000987 11 1 1 000987 11 13 1 000889 13 1 1 000889 13 1 1 000889 13 14 1 001184 11 1 1 001184 11 1 1 001184 11 15 1 001571 8 1 1 001571 8 1 1 001571 8 16 1 001378 5 1 1 001378 5 1 1 001378 5 17 1 001451 5 1 1 001451 5 1 1 001451 5 18 1 001287 5 1 1 001287 5 1 1 001287 5 19 1 00154 7 1 1 00154 7 1 1 00154 7 110 1 001432 9 1 1 001432 9 1 1 001432 9 111 1 001352 7 1 1 001352 7 1 1 001352 7 112 1 00162 8 1 1 00162 8 1 1 00162 8 113 1 001571 10 1 1 001571 10 1 1 001571 10 114 1 001611 5 1 1 001611 5 1 1 001611 5 115 mdash mdash 5 1 mdash mdash 5 1 mdash mdash 5 1Optimal cost ($) 231254 231254 231254CPU time (s) 1725 6621 58215Average iterations 700 2883 8241

Defective rateProduction runs

400100 200 300 5000 500

20000

40000

60000

80000

100000

120000

140000

160000

180000

Figure 5 The sensitive analysis between defective rate and produc-tion runs

5 Conclusions

In summary this research proposed an integrated JIT inven-tory model to solve the partial optimization problem ofpurchaser andmanufacturer finding the optimal solution forthe whole supply chain Furthermore a multiechelon supplychainwith uncertain delivery lead time and unreliable qualityhas been considered to fit the real supply chain environ-ment Besides ACA was proposed to solve the complicatedmathematical problem we also have proved that ACA was

0 50 100 200 300 400 500

Defective rateProduction runs

0

50000

100000

150000

200000

250000

300000

350000

400000

Figure 6 The sensitive analysis between defective rate and produc-tions runs with relaxing assumption

more efficient than PSO and Lingo These are the new majorcontributions of the present paper

Finally there are a great deal of helpful directions forenterprises and researchers (1)The variety of conditions thatoccur in real supply chain can be considered in the modelFor instance time value of cash problem and deterioration(2)The warehousing and storage can be considered in supplychain which make the model more complex to solve (3)The model contains only one member in each echelon amultimember in each echelon should be considered to fit

12 Mathematical Problems in Engineering

real situation in supply chain (4) Some real cases andordata can be applied to our proposed model in the followingworks

Futureworkwill hopefully involve the real-world concernto our research which will be our main target In addition we

do hope our research can be extended intomore considerableareas and more considerable variables

Appendix

The Hessian matrix 119865 of 1198691198621119878(119873119901 119898119871 119873119898 119870) for given 119873119901119894

equal to 1 can be shown as

119865 =

[

[

[

[

[

[

[

[

[

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

1205971198731198981198942

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119873119898119894120597119898119871119894

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119873119898119894120597119870119894

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119898119871119894120597119873119898119894

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

1205971198981198711198942

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119898119871119894120597119870119894

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119870119894120597119873119898119894

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119870119894120597119898119871119894

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

1205971198701198942

]

]

]

]

]

]

]

]

]

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

1205971198731198981198942

=

2

1198731198981198943(120579120593119879

2+

1205931198792(1198671198631198701205932 + Ω)

119870

)

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

1205971198981198711198942

= 119870119873[(

119890minus119898120583

1205832

+ 2 (119890minus119898120583

minus 119890minus120574120583

) +

119890(minus119898120583)(minus119898+120583)

120583

) 120592]

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

1205971198701198942

=

2

1198701198943(120579120593119879

2+

1205931198792(1198671198631198701205932 + Ω)

119873

)

12059721198691198621119904 (119873119898119894 119898119871119894 119870119894)

120597119898119871119894120597119873119898119894

=

12059721198691198621119904 (119873119898119894 119898119871119894 119870119894)

120597119873119898119894120597119898119871119894

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119870119894120597119873119898119894

=

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119873119898119894120597119870119894

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119870119894120597119898119871119894

=

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119898119871119894120597119870119894

100381610038161003816100381611986511

1003816100381610038161003816=

1205972119865

1205971198731198981198942

=

2

1198731198981198943(120579120593119879

2+

1205931198792(1198671198631198701205932 + Ω)

119870

) gt 0

100381610038161003816100381611986522

1003816100381610038161003816= (

1205972119865

1205971198731198981198942)(

1205972119865

1205971198981198711198942) minus [

1205972119865

120597119873119898119894120597119898119871119894

]

2

=

2

1198731198981198943(120579120593119879

2+

1205931198792(1198671198631198701205932 + Ω)

119870

)

1205972119865

1205971198981198711198942minus (

1205972119865

120597119873119898119894120597119898119871119894

)

2

2

1198733119898119894

(1205791205931198792+

119861

119870

)[119870119873(

119890minus119898120576120583

1205832

+ (2119909 +

119890(minus119898120583)119909119910

120583

120592))] + 1198702[

119890minus119898120576120583

120583

+ 120592 (119909119910 minus 119860)] gt 0

100381610038161003816100381611986533

1003816100381610038161003816= (

1205972119865

1205971198731198981198942

1205972119865

1205971198981198711198942

1205972119865

1205971198701198942) minus [(

1205972119865

120597119873119898119894120597119870119894

)

2

(

1205972119865

1205971198981198711198942)]

=

2

1198731198981198943(120579120593119879

2+

119861

119870

)[119870119873(

119890minus119898120576120583

1205832

+ 2(119909 +

119890(minus119898120603)119909119910

120583

) 120592)]

2

1198703

119861

119873

minus [119865 + 119909120576 + 119860120592119910 minus

1205791205931198792

1198732

+

119861

2 (119870119873)2]

[119870119873(

119890minus119898120576120583

1205832

+ (2119909 +

119890minus119898120576120583

120583

) 120592)] gt 0

(A1)

Mathematical Problems in Engineering 13

As a result the matrix F has proved to be a positivedefinite matrix where |11986511| |11986522| and |11986533| are all positiveTherefore theminimumvalue exists in 1198691198621119878(119873119901119898119871 119873119898 119870)Furthermore we can find the optimization in the proposedmodel

Competing Interests

The authors declare that they have no competing interests

References

[1] S K Goyal ldquoAn integrated inventory model for a singlesupplier-single customer problemrdquo International Journal ofProduction Research vol 15 no 1 pp 107ndash111 1977

[2] A Banerjee ldquoA joint economic-lot-sizemodel for purchaser andvendorrdquo Decision Sciences vol 17 no 3 pp 292ndash311 1986

[3] S K Goyal ldquoldquoA joint economic-lot-size model for purchaserand vendorrdquo a commentrdquo Decision Sciences vol 19 no 1 pp236ndash241 1988

[4] Y Seo ldquoControlling general multi-echelon distribution supplychainswith improved reorder decision policy utilizing real-timeshared stock informationrdquo Computers amp Industrial Engineeringvol 51 no 2 pp 229ndash246 2006

[5] H N Chiu and H L Huang ldquoA Multi-echelon integratedJIT inventory model using the time buffer and emergencyborrowing policies to deal with random delivery lead timesrdquoInternational Journal of Production Research vol 41 no 13 pp2911ndash2931 2003

[6] J Sadeghi S M Mousavi S T A Niaki and S SadeghildquoOptimizing a multi-vendor multi-retailer vendor managedinventory problem two tuned meta-heuristic algorithmsrdquoKnowledge-Based Systems vol 50 pp 159ndash170 2013

[7] S S Sana ldquoA production-inventory model of imperfect qualityproducts in a three-layer supply chainrdquo Decision Support Sys-tems vol 50 no 2 pp 539ndash547 2011

[8] H N Soni and K A Patel ldquoOptimal strategy for an integratedinventory system involving variable production and defectiveitems under retailer partial trade credit policyrdquoDecision SupportSystems vol 54 no 1 pp 235ndash247 2012

[9] R J Tersine Principles of Inventory andMaterials ManagementNorthHolland AmsterdamTheNetherlands 3rd edition 1988

[10] C J Liao and C H Shyu ldquoAn analytical determination of leadtime with normal demandrdquo International Journal of Operationsamp Production Management vol 11 no 9 pp 72ndash78 1991

[11] M Ben-Daya and A Raouf ldquoInventory models involving leadtime as a decision variablerdquo Journal of the Operational ResearchSociety vol 45 no 5 pp 579ndash582 1994

[12] S M J Mirzapour Al-e-Hashem A Baboli and Z Sazvar ldquoAstochastic aggregate production planningmodel in a green sup-ply chain considering flexible lead times nonlinear purchaseand shortage cost functionsrdquo European Journal of OperationalResearch vol 230 no 1 pp 26ndash41 2013

[13] C Chandra and J Grabis ldquoInventorymanagementwith variablelead-time dependent procurement costrdquo Omega vol 36 no 5pp 877ndash887 2008

[14] J-W Wu and H-Y Tsai ldquoMixture inventory model with backorders and lost sales for variable lead time demand withthe mixtures of normal distributionrdquo International Journal ofSystems Science Principles and Applications of Systems andIntegration vol 32 no 2 pp 259ndash268 2001

[15] M A Hoque ldquoA vendorndashbuyer integrated productionndashinventory model with normal distribution of lead timerdquoInternational Journal of Production Economics vol 144 no 2pp 409ndash417 2013

[16] F Altiparmak M Gen L Lin and T Paksoy ldquoA geneticalgorithm approach for multi-objective optimization of supplychain networksrdquo Computers and Industrial Engineering vol 51no 1 pp 196ndash215 2006

[17] J Sadeghi S Sadeghi and S T Niaki ldquoA hybrid vendormanaged inventory and redundancy allocation optimizationproblem in supply chain management an NSGA-II with tunedparametersrdquo Computers and Operations Research vol 41 pp53ndash64 2014

[18] X Li F Baki P Tian and B A Chaouch ldquoA robust block-chainbased tabu search algorithm for the dynamic lot sizing problemwith product returns and remanufacturingrdquoOmega vol 42 no1 pp 75ndash87 2014

[19] Z Chen and B R Sarker ldquoAn integrated optimal inventorylot-sizing and vehicle-routing model for a multisupplier single-assembler system with JIT deliveryrdquo International Journal ofProduction Research vol 52 no 17 pp 5086ndash5114 2014

[20] A R Nia M Hemmati Far and S T A Niaki ldquoA fuzzy vendormanaged inventory of multi-item economic order quantitymodel under shortage an ant colony optimization algorithmrdquoInternational Journal of Production Economics vol 155 pp 259ndash271 2014

[21] S M Hatefi and F Jolai ldquoRobust and reliable forward-reverselogistics network design under demand uncertainty and facilitydisruptionsrdquo Applied Mathematical Modelling vol 38 no 9-10pp 2630ndash2647 2014

[22] R H Ma ldquoCross warehouse scheduling of logistics distributionand cycle re-claimer based on heuristics algorithmrdquo AppliedMechanics and Materials vol 644ndash650 pp 2606ndash2610 2014

[23] M Hashim L Yao A H Nadeem M Nazim and M NazamldquoMulti-objective optimization model for supplier selectionproblem in fuzzy environmentrdquo Advances in Intelligent Systemsand Computing vol 281 pp 1201ndash1213 2014

[24] Q-M Hu and Z-H Hu ldquoA stochastic programming modelfor hub-and-spoke network with uncertain flowsrdquo InternationalJournal of Industrial and Systems Engineering vol 21 no 3 pp302ndash319 2015

[25] S A Torabi J Namdar S M Hatefi and F Jolai ldquoAn enhancedpossibilistic programming approach for reliable closed-loopsupply chain network designrdquo International Journal of Produc-tion Research vol 54 no 5 pp 1358ndash1387 2016

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

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Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

+(

119894

prod

119897=1

119873119898119897)119870119894+1119873119898119894+1 [(119903119871119894 minus 119898119871119894) V119894 int119903119871119894

119898119871119894

(119871119889119894 minus 119898119871119894) 119891 (119871119889119894) 119889119871119889119894 + 120576119894 int

119903119871119894

119898119871119894

119891 (119871119889119894) 119889119871119889119894 + 119865119894]

+(

119894

prod

119897=1

119873119898119897)119873119901119894119862119904119894 + (

119894

prod

119897=1

119873119898119897)119862119898119894+1

Subject to 0 le 120582119871119894 le 119898119871119894 le 120574119871119894 le 119905119894 for 119894 = 1 2 119878 minus 1

119870119894+1 le1

0119894

minus 1

119873119901119894 119870119894+1 119873119898119894+1 are all integers for 119894 = 1 2 119878 minus 1

(16)

4 Numeral Experiment and Analysis

In this section a numeral experiment is proposed based onproper parameter settings All results in this research wereperformed on a PC Programs were written in Matlab 2012bAn efficiency comparison between Lingo PSO algorithmand Ant Colony Algorithm was proposed as well Further-more we would like to discuss how production runs anddefective rate influences whole system cost

41 Experiment Results of Ant Colony Algorithm ParticleSwarm Optimization and Lingo With appropriate param-eters settings we may build a 7-echelon supply chain inte-grated JIT inventory model with uncertain delivery leadtime and unreliable quality of received items Assume thepurchasing time interval for first purchaser in years (1198791199011) is12 years The relevant data of purchasing and manufacturingactivities are shown in Tables 1 and 2

The objective function and constraint function are shownbelow Consider

1198691198621119878 (119873119901 119898119871 119873119898 119870)

=

119878minus1

sum

119894=1

V1198941205931198941198791199011 (119898119871119894 minus 120583119871119889119894) + V1198941205931198941198791199011119909119894120588119894 +(1205931198941198791199011)

2

119870119894+1prod119894

119897=1119873119898119897

(Ω119894 minus

119867119898119894+1119863119894119870119894+1120588119894

2

) +

1198791199011

2

prod119894

119897=1119873119898119897

120579119894

+(

119894

prod

119897=1

119873119898119897)119870119894+1119873119898119894+1 [(119903119871119894 minus 119898119871119894) V119894 int119903119871119894

119898119871119894

(119871119889119894 minus 119898119871119894) 119891 (119871119889119894) 119889119871119889119894 + 120576119894 int

119903119871119894

119898119871119894

119891 (119871119889119894) 119889119871119889119894 + 119865119894]

+(

119894

prod

119897=1

119873119898119897)119873119901119894119862119904119894 + (

119894

prod

119897=1

119873119898119897)119862119898119894+1

Subject to 0 le 120582119871119894 le 119898119871119894 le 120574119871119894 le 119905119894 for 119894 = 1 2 119878 minus 1

119870119894+1 le1

0119894

minus 1

119873119901119894 119870119894+1 119873119898119894+1 are all integers for 119894 = 1 2 119878 minus 1

(17)

The goal of this research is to find out the optimal solutionof the 7-echelon inventory model via the above functions andACA and compare the efficiency between Lingo ACA andPSO

As Table 3 shows though all the three methods werecapable of finding the optimal solution for SMEIJI problemACA was obviously better than PSO and Lingo in efficiencyACA has shown fastest CPU searching time and least averageiterations among three methods

42 Experiments of Performance and 119878-Echelon In this sec-tion we would like to discuss the relationship between CPU

searching time and echelon number among three methodswe also discussed the relationship between average iterationsand echelon number

As Figure 3 shows the CPU searching time wouldincrease with the number of echelons due to the compli-cated calculation However because of the well-organizedconstraint function and good initial solution ACA still keptlower CPU searching time than the other two methods Onthe other hand Figure 4 shows that ACArsquos average iterationswere the least among three methods At the beginning of themultiechelon all the three algorithms need a great amountof iterations to search the optimal solution After calculating

Mathematical Problems in Engineering 9

Table 1 The relevant data of purchasing activities

Purchasing activityEchelon 119863119894 119862119904119894 119867119901119894 119862119890119894 120573119894 120582119871119894 120574119871119894 120583119871119889119894 120590119871119889119894 119865119894 119883119894 0119894

1 18 w 25 28 29 32 0005 0015 0009 0002 70 0005 0032 20w 28 22 32 26 0004 001 00064 00012 50 0003 0013 24w 32 17 38 21 0003 0009 00054 00012 48 0003 0044 30w 36 12 42 16 0004 0012 00072 00016 55 0004 0055 35w 35 08 37 12 0006 0016 001 0002 80 0005 0016 40w 38 06 40 09 0004 0014 0008 0002 65 0005 0027 mdash mdash mdash mdash mdash mdash mdash mdash mdash mdash mdash mdash

Table 2 The relevant data of manufacturing activities

Manufacturing activitiesEchelon 119875119894 119862119898119894 119867119898119894

1 mdash mdash mdash2 200000 280 253 240000 200 194 300000 320 145 350000 360 106 400000 380 097 450000 420 05

1 2 3 4 5 6 70

05

1

15

2

25

3

ACAPSO

Lingo

Figure 3 Experiment results of CPU time and echelon number

repeatedly the trend of the average iterations would come flatwithout extreme variety

Nevertheless although ACA seems better than the othertwo algorithms we could not ensure that the performancewould still be great when applied to higher echelon inventorymodel Hence a 15-echelon model has proposed to observethe performance again The following table displayed theparameters of 15-echelon model

To evaluate the performance of ACA PSO and Lingo forthe 15-echelon inventorymodel we applied the parameters inTables 4 and 5 to solve the 15-echelon inventory problem

According to Table 6 all three algorithms were still ableto find out the optimal solution of the 15-echelon inventorymodel Once the echelon increases the complexity of the

1 2 3 4 5 6 70

500

1000

1500

2000

2500

3000

3500

4000

ACAPSO

Lingo

Figure 4 Experiment results of average iterations and echelonnumber

model increases as well However the performance of ACAwas still better than the other two algorithms

There are several advantages of ACA as follows

(1) Applying the positive and negative feedback enablesthe process of searching optimization to converge atthe end which means the results would close to theoptimal solutions gradually

(2) Every single ant is able to change the surroundingenvironment through releasing pheromone and beingconscious of the variety of environment which makesthem communicate with each other

(3) Due to distributed computing all the individuals startto calculate simultaneously during the searching pro-cess which enhances the efficiency of the calculationability

(4) For the proposed algorithm it is difficult to stick inthe partial optimization instead it is always easy tofind the best solution

To sum up ACA was proposed to solve the complicatedmathematical problem we also have proved that ACA wasmore efficient than PSO and Lingo These are the new majorcontributions of the present paper

43 The Sensitive Analysis of SMEIJI In this section thenumber of production runs 119873119898119894 and the defective rate of

10 Mathematical Problems in Engineering

Table 3 Comparisons between three methods in SMEIJI problem

Echelon ACA PSO Lingo119873119901119894 119898119871119894 119870119894 119873119898119894 119873119901119894 119898119871119894 119870119894 119873119898119894 119873119901119894 119898119871119894 119870119894 119873119898119894

1 1 00147 mdash mdash 1 00147 mdash mdash 1 00147 mdash mdash2 1 000987 11 1 1 000987 11 1 1 000987 11 13 1 000889 13 1 1 000889 13 1 1 000889 13 14 1 001184 11 1 1 001184 11 1 1 001184 11 15 1 001571 8 1 1 001571 8 1 1 001571 8 16 1 001378 5 1 1 001378 5 1 1 001378 5 17 mdash mdash 5 1 mdash mdash 5 1 mdash mdash 5 1Optimal cost 89166 89166 89166CPU time 0655 1744 23Average iterations 350 1278 3758

Table 4 The relevant data of purchasing activities of 15-echelon model

Purchasing activityEchelon 119863119894 119862119904119894 119867119901119894 119862119890119894 120573119894 120582119871119894 120574119871119894 120583119871119889119894 120590119871119889119894 119865119894 119883119894 0119894

1 18 w 25 28 29 32 0005 0015 0009 0002 70 0005 0032 20w 28 22 32 26 0004 001 00064 00012 50 0003 0013 24w 32 17 38 21 0003 0009 00054 00012 48 0003 0044 30w 36 12 42 16 0004 0012 00072 00016 55 0004 0055 35w 35 08 37 12 0006 0016 001 0002 80 0005 0016 40w 38 06 40 09 0004 0014 0008 0002 65 0005 0027 42w 40 05 28 12 0005 0013 001 0002 70 0005 0038 45w 42 07 35 15 0004 0009 0009 00018 50 0004 0029 50w 44 10 42 10 0003 0010 0005 00024 55 0004 00210 52w 46 15 35 22 0005 0012 00062 00025 60 0005 00211 56w 48 20 49 25 0006 0009 001 00026 65 0003 00312 60w 50 22 40 14 0004 0016 0012 0002 48 0005 00413 62w 55 17 35 15 0007 0015 0009 00018 72 0004 00414 65w 56 12 32 16 0003 0014 00078 0002 70 0006 00315 mdash mdash mdash mdash mdash mdash mdash mdash mdash mdash mdash mdash

Table 5The relevant data of manufacturing activities of 15-echelonmodel

Manufacturing activitiesEchelon 119875119894 119862119898119894 119867119898119894

1 mdash mdash mdash2 200000 280 253 240000 200 194 300000 320 145 350000 360 106 400000 380 097 450000 420 058 480000 440 109 500000 420 1210 520000 400 1511 540000 450 2012 580000 440 2213 620000 420 1814 640000 460 1515 660000 380 16

receiving shipment 0119894 were considered to make sensitiveanalysis and observe the influence on the optimal total jointcost The defective rate and production runs were changedby 0119894(1 + 119877) and 119873119898119894(1 + 119877) where the values of 119877 are0 50 100 200 300 400

Figure 5 illustrated that defective rate was more sensitivethan production runs under the situation that productionruns were one-time production except first echelon In thatcase the higher the defective ratewas the higher the total costwas Accordingly inspecting goods cautiously before deliveryand reducing the defective rate would lower total cost of thewhole supply chainHowever if relaxing that assumption thissensitive analysis would result in different outcomes

Figure 6 has shown that production runs were more sen-sitive than defective rate when relaxing one-time productionassumption In this case the higher the production runswerethe higher the total cost was Hence reducing productionruns should play an essential role in the whole supply chainOnce the one batch production policy was able to satisfy thedemand from the buyer the total cost of whole supply chainwill go down extremely

Mathematical Problems in Engineering 11

Table 6 Comparisons between three ways of 15-echelon inventory model

ACA PSO LingoEchelon 119873119901119894 119898119871119894 119870119894 119873119898119894 119873119901119894 119898119871119894 119870119894 119873119898119894 119873119901119894 119898119871119894 119870119894 119873119898119894

1 1 00147 mdash mdash 1 00147 mdash mdash 1 00147 mdash mdash2 1 000987 11 1 1 000987 11 1 1 000987 11 13 1 000889 13 1 1 000889 13 1 1 000889 13 14 1 001184 11 1 1 001184 11 1 1 001184 11 15 1 001571 8 1 1 001571 8 1 1 001571 8 16 1 001378 5 1 1 001378 5 1 1 001378 5 17 1 001451 5 1 1 001451 5 1 1 001451 5 18 1 001287 5 1 1 001287 5 1 1 001287 5 19 1 00154 7 1 1 00154 7 1 1 00154 7 110 1 001432 9 1 1 001432 9 1 1 001432 9 111 1 001352 7 1 1 001352 7 1 1 001352 7 112 1 00162 8 1 1 00162 8 1 1 00162 8 113 1 001571 10 1 1 001571 10 1 1 001571 10 114 1 001611 5 1 1 001611 5 1 1 001611 5 115 mdash mdash 5 1 mdash mdash 5 1 mdash mdash 5 1Optimal cost ($) 231254 231254 231254CPU time (s) 1725 6621 58215Average iterations 700 2883 8241

Defective rateProduction runs

400100 200 300 5000 500

20000

40000

60000

80000

100000

120000

140000

160000

180000

Figure 5 The sensitive analysis between defective rate and produc-tion runs

5 Conclusions

In summary this research proposed an integrated JIT inven-tory model to solve the partial optimization problem ofpurchaser andmanufacturer finding the optimal solution forthe whole supply chain Furthermore a multiechelon supplychainwith uncertain delivery lead time and unreliable qualityhas been considered to fit the real supply chain environ-ment Besides ACA was proposed to solve the complicatedmathematical problem we also have proved that ACA was

0 50 100 200 300 400 500

Defective rateProduction runs

0

50000

100000

150000

200000

250000

300000

350000

400000

Figure 6 The sensitive analysis between defective rate and produc-tions runs with relaxing assumption

more efficient than PSO and Lingo These are the new majorcontributions of the present paper

Finally there are a great deal of helpful directions forenterprises and researchers (1)The variety of conditions thatoccur in real supply chain can be considered in the modelFor instance time value of cash problem and deterioration(2)The warehousing and storage can be considered in supplychain which make the model more complex to solve (3)The model contains only one member in each echelon amultimember in each echelon should be considered to fit

12 Mathematical Problems in Engineering

real situation in supply chain (4) Some real cases andordata can be applied to our proposed model in the followingworks

Futureworkwill hopefully involve the real-world concernto our research which will be our main target In addition we

do hope our research can be extended intomore considerableareas and more considerable variables

Appendix

The Hessian matrix 119865 of 1198691198621119878(119873119901 119898119871 119873119898 119870) for given 119873119901119894

equal to 1 can be shown as

119865 =

[

[

[

[

[

[

[

[

[

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

1205971198731198981198942

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119873119898119894120597119898119871119894

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119873119898119894120597119870119894

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119898119871119894120597119873119898119894

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

1205971198981198711198942

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119898119871119894120597119870119894

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119870119894120597119873119898119894

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119870119894120597119898119871119894

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

1205971198701198942

]

]

]

]

]

]

]

]

]

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

1205971198731198981198942

=

2

1198731198981198943(120579120593119879

2+

1205931198792(1198671198631198701205932 + Ω)

119870

)

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

1205971198981198711198942

= 119870119873[(

119890minus119898120583

1205832

+ 2 (119890minus119898120583

minus 119890minus120574120583

) +

119890(minus119898120583)(minus119898+120583)

120583

) 120592]

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

1205971198701198942

=

2

1198701198943(120579120593119879

2+

1205931198792(1198671198631198701205932 + Ω)

119873

)

12059721198691198621119904 (119873119898119894 119898119871119894 119870119894)

120597119898119871119894120597119873119898119894

=

12059721198691198621119904 (119873119898119894 119898119871119894 119870119894)

120597119873119898119894120597119898119871119894

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119870119894120597119873119898119894

=

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119873119898119894120597119870119894

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119870119894120597119898119871119894

=

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119898119871119894120597119870119894

100381610038161003816100381611986511

1003816100381610038161003816=

1205972119865

1205971198731198981198942

=

2

1198731198981198943(120579120593119879

2+

1205931198792(1198671198631198701205932 + Ω)

119870

) gt 0

100381610038161003816100381611986522

1003816100381610038161003816= (

1205972119865

1205971198731198981198942)(

1205972119865

1205971198981198711198942) minus [

1205972119865

120597119873119898119894120597119898119871119894

]

2

=

2

1198731198981198943(120579120593119879

2+

1205931198792(1198671198631198701205932 + Ω)

119870

)

1205972119865

1205971198981198711198942minus (

1205972119865

120597119873119898119894120597119898119871119894

)

2

2

1198733119898119894

(1205791205931198792+

119861

119870

)[119870119873(

119890minus119898120576120583

1205832

+ (2119909 +

119890(minus119898120583)119909119910

120583

120592))] + 1198702[

119890minus119898120576120583

120583

+ 120592 (119909119910 minus 119860)] gt 0

100381610038161003816100381611986533

1003816100381610038161003816= (

1205972119865

1205971198731198981198942

1205972119865

1205971198981198711198942

1205972119865

1205971198701198942) minus [(

1205972119865

120597119873119898119894120597119870119894

)

2

(

1205972119865

1205971198981198711198942)]

=

2

1198731198981198943(120579120593119879

2+

119861

119870

)[119870119873(

119890minus119898120576120583

1205832

+ 2(119909 +

119890(minus119898120603)119909119910

120583

) 120592)]

2

1198703

119861

119873

minus [119865 + 119909120576 + 119860120592119910 minus

1205791205931198792

1198732

+

119861

2 (119870119873)2]

[119870119873(

119890minus119898120576120583

1205832

+ (2119909 +

119890minus119898120576120583

120583

) 120592)] gt 0

(A1)

Mathematical Problems in Engineering 13

As a result the matrix F has proved to be a positivedefinite matrix where |11986511| |11986522| and |11986533| are all positiveTherefore theminimumvalue exists in 1198691198621119878(119873119901119898119871 119873119898 119870)Furthermore we can find the optimization in the proposedmodel

Competing Interests

The authors declare that they have no competing interests

References

[1] S K Goyal ldquoAn integrated inventory model for a singlesupplier-single customer problemrdquo International Journal ofProduction Research vol 15 no 1 pp 107ndash111 1977

[2] A Banerjee ldquoA joint economic-lot-sizemodel for purchaser andvendorrdquo Decision Sciences vol 17 no 3 pp 292ndash311 1986

[3] S K Goyal ldquoldquoA joint economic-lot-size model for purchaserand vendorrdquo a commentrdquo Decision Sciences vol 19 no 1 pp236ndash241 1988

[4] Y Seo ldquoControlling general multi-echelon distribution supplychainswith improved reorder decision policy utilizing real-timeshared stock informationrdquo Computers amp Industrial Engineeringvol 51 no 2 pp 229ndash246 2006

[5] H N Chiu and H L Huang ldquoA Multi-echelon integratedJIT inventory model using the time buffer and emergencyborrowing policies to deal with random delivery lead timesrdquoInternational Journal of Production Research vol 41 no 13 pp2911ndash2931 2003

[6] J Sadeghi S M Mousavi S T A Niaki and S SadeghildquoOptimizing a multi-vendor multi-retailer vendor managedinventory problem two tuned meta-heuristic algorithmsrdquoKnowledge-Based Systems vol 50 pp 159ndash170 2013

[7] S S Sana ldquoA production-inventory model of imperfect qualityproducts in a three-layer supply chainrdquo Decision Support Sys-tems vol 50 no 2 pp 539ndash547 2011

[8] H N Soni and K A Patel ldquoOptimal strategy for an integratedinventory system involving variable production and defectiveitems under retailer partial trade credit policyrdquoDecision SupportSystems vol 54 no 1 pp 235ndash247 2012

[9] R J Tersine Principles of Inventory andMaterials ManagementNorthHolland AmsterdamTheNetherlands 3rd edition 1988

[10] C J Liao and C H Shyu ldquoAn analytical determination of leadtime with normal demandrdquo International Journal of Operationsamp Production Management vol 11 no 9 pp 72ndash78 1991

[11] M Ben-Daya and A Raouf ldquoInventory models involving leadtime as a decision variablerdquo Journal of the Operational ResearchSociety vol 45 no 5 pp 579ndash582 1994

[12] S M J Mirzapour Al-e-Hashem A Baboli and Z Sazvar ldquoAstochastic aggregate production planningmodel in a green sup-ply chain considering flexible lead times nonlinear purchaseand shortage cost functionsrdquo European Journal of OperationalResearch vol 230 no 1 pp 26ndash41 2013

[13] C Chandra and J Grabis ldquoInventorymanagementwith variablelead-time dependent procurement costrdquo Omega vol 36 no 5pp 877ndash887 2008

[14] J-W Wu and H-Y Tsai ldquoMixture inventory model with backorders and lost sales for variable lead time demand withthe mixtures of normal distributionrdquo International Journal ofSystems Science Principles and Applications of Systems andIntegration vol 32 no 2 pp 259ndash268 2001

[15] M A Hoque ldquoA vendorndashbuyer integrated productionndashinventory model with normal distribution of lead timerdquoInternational Journal of Production Economics vol 144 no 2pp 409ndash417 2013

[16] F Altiparmak M Gen L Lin and T Paksoy ldquoA geneticalgorithm approach for multi-objective optimization of supplychain networksrdquo Computers and Industrial Engineering vol 51no 1 pp 196ndash215 2006

[17] J Sadeghi S Sadeghi and S T Niaki ldquoA hybrid vendormanaged inventory and redundancy allocation optimizationproblem in supply chain management an NSGA-II with tunedparametersrdquo Computers and Operations Research vol 41 pp53ndash64 2014

[18] X Li F Baki P Tian and B A Chaouch ldquoA robust block-chainbased tabu search algorithm for the dynamic lot sizing problemwith product returns and remanufacturingrdquoOmega vol 42 no1 pp 75ndash87 2014

[19] Z Chen and B R Sarker ldquoAn integrated optimal inventorylot-sizing and vehicle-routing model for a multisupplier single-assembler system with JIT deliveryrdquo International Journal ofProduction Research vol 52 no 17 pp 5086ndash5114 2014

[20] A R Nia M Hemmati Far and S T A Niaki ldquoA fuzzy vendormanaged inventory of multi-item economic order quantitymodel under shortage an ant colony optimization algorithmrdquoInternational Journal of Production Economics vol 155 pp 259ndash271 2014

[21] S M Hatefi and F Jolai ldquoRobust and reliable forward-reverselogistics network design under demand uncertainty and facilitydisruptionsrdquo Applied Mathematical Modelling vol 38 no 9-10pp 2630ndash2647 2014

[22] R H Ma ldquoCross warehouse scheduling of logistics distributionand cycle re-claimer based on heuristics algorithmrdquo AppliedMechanics and Materials vol 644ndash650 pp 2606ndash2610 2014

[23] M Hashim L Yao A H Nadeem M Nazim and M NazamldquoMulti-objective optimization model for supplier selectionproblem in fuzzy environmentrdquo Advances in Intelligent Systemsand Computing vol 281 pp 1201ndash1213 2014

[24] Q-M Hu and Z-H Hu ldquoA stochastic programming modelfor hub-and-spoke network with uncertain flowsrdquo InternationalJournal of Industrial and Systems Engineering vol 21 no 3 pp302ndash319 2015

[25] S A Torabi J Namdar S M Hatefi and F Jolai ldquoAn enhancedpossibilistic programming approach for reliable closed-loopsupply chain network designrdquo International Journal of Produc-tion Research vol 54 no 5 pp 1358ndash1387 2016

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

Table 1 The relevant data of purchasing activities

Purchasing activityEchelon 119863119894 119862119904119894 119867119901119894 119862119890119894 120573119894 120582119871119894 120574119871119894 120583119871119889119894 120590119871119889119894 119865119894 119883119894 0119894

1 18 w 25 28 29 32 0005 0015 0009 0002 70 0005 0032 20w 28 22 32 26 0004 001 00064 00012 50 0003 0013 24w 32 17 38 21 0003 0009 00054 00012 48 0003 0044 30w 36 12 42 16 0004 0012 00072 00016 55 0004 0055 35w 35 08 37 12 0006 0016 001 0002 80 0005 0016 40w 38 06 40 09 0004 0014 0008 0002 65 0005 0027 mdash mdash mdash mdash mdash mdash mdash mdash mdash mdash mdash mdash

Table 2 The relevant data of manufacturing activities

Manufacturing activitiesEchelon 119875119894 119862119898119894 119867119898119894

1 mdash mdash mdash2 200000 280 253 240000 200 194 300000 320 145 350000 360 106 400000 380 097 450000 420 05

1 2 3 4 5 6 70

05

1

15

2

25

3

ACAPSO

Lingo

Figure 3 Experiment results of CPU time and echelon number

repeatedly the trend of the average iterations would come flatwithout extreme variety

Nevertheless although ACA seems better than the othertwo algorithms we could not ensure that the performancewould still be great when applied to higher echelon inventorymodel Hence a 15-echelon model has proposed to observethe performance again The following table displayed theparameters of 15-echelon model

To evaluate the performance of ACA PSO and Lingo forthe 15-echelon inventorymodel we applied the parameters inTables 4 and 5 to solve the 15-echelon inventory problem

According to Table 6 all three algorithms were still ableto find out the optimal solution of the 15-echelon inventorymodel Once the echelon increases the complexity of the

1 2 3 4 5 6 70

500

1000

1500

2000

2500

3000

3500

4000

ACAPSO

Lingo

Figure 4 Experiment results of average iterations and echelonnumber

model increases as well However the performance of ACAwas still better than the other two algorithms

There are several advantages of ACA as follows

(1) Applying the positive and negative feedback enablesthe process of searching optimization to converge atthe end which means the results would close to theoptimal solutions gradually

(2) Every single ant is able to change the surroundingenvironment through releasing pheromone and beingconscious of the variety of environment which makesthem communicate with each other

(3) Due to distributed computing all the individuals startto calculate simultaneously during the searching pro-cess which enhances the efficiency of the calculationability

(4) For the proposed algorithm it is difficult to stick inthe partial optimization instead it is always easy tofind the best solution

To sum up ACA was proposed to solve the complicatedmathematical problem we also have proved that ACA wasmore efficient than PSO and Lingo These are the new majorcontributions of the present paper

43 The Sensitive Analysis of SMEIJI In this section thenumber of production runs 119873119898119894 and the defective rate of

10 Mathematical Problems in Engineering

Table 3 Comparisons between three methods in SMEIJI problem

Echelon ACA PSO Lingo119873119901119894 119898119871119894 119870119894 119873119898119894 119873119901119894 119898119871119894 119870119894 119873119898119894 119873119901119894 119898119871119894 119870119894 119873119898119894

1 1 00147 mdash mdash 1 00147 mdash mdash 1 00147 mdash mdash2 1 000987 11 1 1 000987 11 1 1 000987 11 13 1 000889 13 1 1 000889 13 1 1 000889 13 14 1 001184 11 1 1 001184 11 1 1 001184 11 15 1 001571 8 1 1 001571 8 1 1 001571 8 16 1 001378 5 1 1 001378 5 1 1 001378 5 17 mdash mdash 5 1 mdash mdash 5 1 mdash mdash 5 1Optimal cost 89166 89166 89166CPU time 0655 1744 23Average iterations 350 1278 3758

Table 4 The relevant data of purchasing activities of 15-echelon model

Purchasing activityEchelon 119863119894 119862119904119894 119867119901119894 119862119890119894 120573119894 120582119871119894 120574119871119894 120583119871119889119894 120590119871119889119894 119865119894 119883119894 0119894

1 18 w 25 28 29 32 0005 0015 0009 0002 70 0005 0032 20w 28 22 32 26 0004 001 00064 00012 50 0003 0013 24w 32 17 38 21 0003 0009 00054 00012 48 0003 0044 30w 36 12 42 16 0004 0012 00072 00016 55 0004 0055 35w 35 08 37 12 0006 0016 001 0002 80 0005 0016 40w 38 06 40 09 0004 0014 0008 0002 65 0005 0027 42w 40 05 28 12 0005 0013 001 0002 70 0005 0038 45w 42 07 35 15 0004 0009 0009 00018 50 0004 0029 50w 44 10 42 10 0003 0010 0005 00024 55 0004 00210 52w 46 15 35 22 0005 0012 00062 00025 60 0005 00211 56w 48 20 49 25 0006 0009 001 00026 65 0003 00312 60w 50 22 40 14 0004 0016 0012 0002 48 0005 00413 62w 55 17 35 15 0007 0015 0009 00018 72 0004 00414 65w 56 12 32 16 0003 0014 00078 0002 70 0006 00315 mdash mdash mdash mdash mdash mdash mdash mdash mdash mdash mdash mdash

Table 5The relevant data of manufacturing activities of 15-echelonmodel

Manufacturing activitiesEchelon 119875119894 119862119898119894 119867119898119894

1 mdash mdash mdash2 200000 280 253 240000 200 194 300000 320 145 350000 360 106 400000 380 097 450000 420 058 480000 440 109 500000 420 1210 520000 400 1511 540000 450 2012 580000 440 2213 620000 420 1814 640000 460 1515 660000 380 16

receiving shipment 0119894 were considered to make sensitiveanalysis and observe the influence on the optimal total jointcost The defective rate and production runs were changedby 0119894(1 + 119877) and 119873119898119894(1 + 119877) where the values of 119877 are0 50 100 200 300 400

Figure 5 illustrated that defective rate was more sensitivethan production runs under the situation that productionruns were one-time production except first echelon In thatcase the higher the defective ratewas the higher the total costwas Accordingly inspecting goods cautiously before deliveryand reducing the defective rate would lower total cost of thewhole supply chainHowever if relaxing that assumption thissensitive analysis would result in different outcomes

Figure 6 has shown that production runs were more sen-sitive than defective rate when relaxing one-time productionassumption In this case the higher the production runswerethe higher the total cost was Hence reducing productionruns should play an essential role in the whole supply chainOnce the one batch production policy was able to satisfy thedemand from the buyer the total cost of whole supply chainwill go down extremely

Mathematical Problems in Engineering 11

Table 6 Comparisons between three ways of 15-echelon inventory model

ACA PSO LingoEchelon 119873119901119894 119898119871119894 119870119894 119873119898119894 119873119901119894 119898119871119894 119870119894 119873119898119894 119873119901119894 119898119871119894 119870119894 119873119898119894

1 1 00147 mdash mdash 1 00147 mdash mdash 1 00147 mdash mdash2 1 000987 11 1 1 000987 11 1 1 000987 11 13 1 000889 13 1 1 000889 13 1 1 000889 13 14 1 001184 11 1 1 001184 11 1 1 001184 11 15 1 001571 8 1 1 001571 8 1 1 001571 8 16 1 001378 5 1 1 001378 5 1 1 001378 5 17 1 001451 5 1 1 001451 5 1 1 001451 5 18 1 001287 5 1 1 001287 5 1 1 001287 5 19 1 00154 7 1 1 00154 7 1 1 00154 7 110 1 001432 9 1 1 001432 9 1 1 001432 9 111 1 001352 7 1 1 001352 7 1 1 001352 7 112 1 00162 8 1 1 00162 8 1 1 00162 8 113 1 001571 10 1 1 001571 10 1 1 001571 10 114 1 001611 5 1 1 001611 5 1 1 001611 5 115 mdash mdash 5 1 mdash mdash 5 1 mdash mdash 5 1Optimal cost ($) 231254 231254 231254CPU time (s) 1725 6621 58215Average iterations 700 2883 8241

Defective rateProduction runs

400100 200 300 5000 500

20000

40000

60000

80000

100000

120000

140000

160000

180000

Figure 5 The sensitive analysis between defective rate and produc-tion runs

5 Conclusions

In summary this research proposed an integrated JIT inven-tory model to solve the partial optimization problem ofpurchaser andmanufacturer finding the optimal solution forthe whole supply chain Furthermore a multiechelon supplychainwith uncertain delivery lead time and unreliable qualityhas been considered to fit the real supply chain environ-ment Besides ACA was proposed to solve the complicatedmathematical problem we also have proved that ACA was

0 50 100 200 300 400 500

Defective rateProduction runs

0

50000

100000

150000

200000

250000

300000

350000

400000

Figure 6 The sensitive analysis between defective rate and produc-tions runs with relaxing assumption

more efficient than PSO and Lingo These are the new majorcontributions of the present paper

Finally there are a great deal of helpful directions forenterprises and researchers (1)The variety of conditions thatoccur in real supply chain can be considered in the modelFor instance time value of cash problem and deterioration(2)The warehousing and storage can be considered in supplychain which make the model more complex to solve (3)The model contains only one member in each echelon amultimember in each echelon should be considered to fit

12 Mathematical Problems in Engineering

real situation in supply chain (4) Some real cases andordata can be applied to our proposed model in the followingworks

Futureworkwill hopefully involve the real-world concernto our research which will be our main target In addition we

do hope our research can be extended intomore considerableareas and more considerable variables

Appendix

The Hessian matrix 119865 of 1198691198621119878(119873119901 119898119871 119873119898 119870) for given 119873119901119894

equal to 1 can be shown as

119865 =

[

[

[

[

[

[

[

[

[

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

1205971198731198981198942

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119873119898119894120597119898119871119894

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119873119898119894120597119870119894

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119898119871119894120597119873119898119894

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

1205971198981198711198942

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119898119871119894120597119870119894

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119870119894120597119873119898119894

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119870119894120597119898119871119894

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

1205971198701198942

]

]

]

]

]

]

]

]

]

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

1205971198731198981198942

=

2

1198731198981198943(120579120593119879

2+

1205931198792(1198671198631198701205932 + Ω)

119870

)

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

1205971198981198711198942

= 119870119873[(

119890minus119898120583

1205832

+ 2 (119890minus119898120583

minus 119890minus120574120583

) +

119890(minus119898120583)(minus119898+120583)

120583

) 120592]

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

1205971198701198942

=

2

1198701198943(120579120593119879

2+

1205931198792(1198671198631198701205932 + Ω)

119873

)

12059721198691198621119904 (119873119898119894 119898119871119894 119870119894)

120597119898119871119894120597119873119898119894

=

12059721198691198621119904 (119873119898119894 119898119871119894 119870119894)

120597119873119898119894120597119898119871119894

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119870119894120597119873119898119894

=

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119873119898119894120597119870119894

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119870119894120597119898119871119894

=

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119898119871119894120597119870119894

100381610038161003816100381611986511

1003816100381610038161003816=

1205972119865

1205971198731198981198942

=

2

1198731198981198943(120579120593119879

2+

1205931198792(1198671198631198701205932 + Ω)

119870

) gt 0

100381610038161003816100381611986522

1003816100381610038161003816= (

1205972119865

1205971198731198981198942)(

1205972119865

1205971198981198711198942) minus [

1205972119865

120597119873119898119894120597119898119871119894

]

2

=

2

1198731198981198943(120579120593119879

2+

1205931198792(1198671198631198701205932 + Ω)

119870

)

1205972119865

1205971198981198711198942minus (

1205972119865

120597119873119898119894120597119898119871119894

)

2

2

1198733119898119894

(1205791205931198792+

119861

119870

)[119870119873(

119890minus119898120576120583

1205832

+ (2119909 +

119890(minus119898120583)119909119910

120583

120592))] + 1198702[

119890minus119898120576120583

120583

+ 120592 (119909119910 minus 119860)] gt 0

100381610038161003816100381611986533

1003816100381610038161003816= (

1205972119865

1205971198731198981198942

1205972119865

1205971198981198711198942

1205972119865

1205971198701198942) minus [(

1205972119865

120597119873119898119894120597119870119894

)

2

(

1205972119865

1205971198981198711198942)]

=

2

1198731198981198943(120579120593119879

2+

119861

119870

)[119870119873(

119890minus119898120576120583

1205832

+ 2(119909 +

119890(minus119898120603)119909119910

120583

) 120592)]

2

1198703

119861

119873

minus [119865 + 119909120576 + 119860120592119910 minus

1205791205931198792

1198732

+

119861

2 (119870119873)2]

[119870119873(

119890minus119898120576120583

1205832

+ (2119909 +

119890minus119898120576120583

120583

) 120592)] gt 0

(A1)

Mathematical Problems in Engineering 13

As a result the matrix F has proved to be a positivedefinite matrix where |11986511| |11986522| and |11986533| are all positiveTherefore theminimumvalue exists in 1198691198621119878(119873119901119898119871 119873119898 119870)Furthermore we can find the optimization in the proposedmodel

Competing Interests

The authors declare that they have no competing interests

References

[1] S K Goyal ldquoAn integrated inventory model for a singlesupplier-single customer problemrdquo International Journal ofProduction Research vol 15 no 1 pp 107ndash111 1977

[2] A Banerjee ldquoA joint economic-lot-sizemodel for purchaser andvendorrdquo Decision Sciences vol 17 no 3 pp 292ndash311 1986

[3] S K Goyal ldquoldquoA joint economic-lot-size model for purchaserand vendorrdquo a commentrdquo Decision Sciences vol 19 no 1 pp236ndash241 1988

[4] Y Seo ldquoControlling general multi-echelon distribution supplychainswith improved reorder decision policy utilizing real-timeshared stock informationrdquo Computers amp Industrial Engineeringvol 51 no 2 pp 229ndash246 2006

[5] H N Chiu and H L Huang ldquoA Multi-echelon integratedJIT inventory model using the time buffer and emergencyborrowing policies to deal with random delivery lead timesrdquoInternational Journal of Production Research vol 41 no 13 pp2911ndash2931 2003

[6] J Sadeghi S M Mousavi S T A Niaki and S SadeghildquoOptimizing a multi-vendor multi-retailer vendor managedinventory problem two tuned meta-heuristic algorithmsrdquoKnowledge-Based Systems vol 50 pp 159ndash170 2013

[7] S S Sana ldquoA production-inventory model of imperfect qualityproducts in a three-layer supply chainrdquo Decision Support Sys-tems vol 50 no 2 pp 539ndash547 2011

[8] H N Soni and K A Patel ldquoOptimal strategy for an integratedinventory system involving variable production and defectiveitems under retailer partial trade credit policyrdquoDecision SupportSystems vol 54 no 1 pp 235ndash247 2012

[9] R J Tersine Principles of Inventory andMaterials ManagementNorthHolland AmsterdamTheNetherlands 3rd edition 1988

[10] C J Liao and C H Shyu ldquoAn analytical determination of leadtime with normal demandrdquo International Journal of Operationsamp Production Management vol 11 no 9 pp 72ndash78 1991

[11] M Ben-Daya and A Raouf ldquoInventory models involving leadtime as a decision variablerdquo Journal of the Operational ResearchSociety vol 45 no 5 pp 579ndash582 1994

[12] S M J Mirzapour Al-e-Hashem A Baboli and Z Sazvar ldquoAstochastic aggregate production planningmodel in a green sup-ply chain considering flexible lead times nonlinear purchaseand shortage cost functionsrdquo European Journal of OperationalResearch vol 230 no 1 pp 26ndash41 2013

[13] C Chandra and J Grabis ldquoInventorymanagementwith variablelead-time dependent procurement costrdquo Omega vol 36 no 5pp 877ndash887 2008

[14] J-W Wu and H-Y Tsai ldquoMixture inventory model with backorders and lost sales for variable lead time demand withthe mixtures of normal distributionrdquo International Journal ofSystems Science Principles and Applications of Systems andIntegration vol 32 no 2 pp 259ndash268 2001

[15] M A Hoque ldquoA vendorndashbuyer integrated productionndashinventory model with normal distribution of lead timerdquoInternational Journal of Production Economics vol 144 no 2pp 409ndash417 2013

[16] F Altiparmak M Gen L Lin and T Paksoy ldquoA geneticalgorithm approach for multi-objective optimization of supplychain networksrdquo Computers and Industrial Engineering vol 51no 1 pp 196ndash215 2006

[17] J Sadeghi S Sadeghi and S T Niaki ldquoA hybrid vendormanaged inventory and redundancy allocation optimizationproblem in supply chain management an NSGA-II with tunedparametersrdquo Computers and Operations Research vol 41 pp53ndash64 2014

[18] X Li F Baki P Tian and B A Chaouch ldquoA robust block-chainbased tabu search algorithm for the dynamic lot sizing problemwith product returns and remanufacturingrdquoOmega vol 42 no1 pp 75ndash87 2014

[19] Z Chen and B R Sarker ldquoAn integrated optimal inventorylot-sizing and vehicle-routing model for a multisupplier single-assembler system with JIT deliveryrdquo International Journal ofProduction Research vol 52 no 17 pp 5086ndash5114 2014

[20] A R Nia M Hemmati Far and S T A Niaki ldquoA fuzzy vendormanaged inventory of multi-item economic order quantitymodel under shortage an ant colony optimization algorithmrdquoInternational Journal of Production Economics vol 155 pp 259ndash271 2014

[21] S M Hatefi and F Jolai ldquoRobust and reliable forward-reverselogistics network design under demand uncertainty and facilitydisruptionsrdquo Applied Mathematical Modelling vol 38 no 9-10pp 2630ndash2647 2014

[22] R H Ma ldquoCross warehouse scheduling of logistics distributionand cycle re-claimer based on heuristics algorithmrdquo AppliedMechanics and Materials vol 644ndash650 pp 2606ndash2610 2014

[23] M Hashim L Yao A H Nadeem M Nazim and M NazamldquoMulti-objective optimization model for supplier selectionproblem in fuzzy environmentrdquo Advances in Intelligent Systemsand Computing vol 281 pp 1201ndash1213 2014

[24] Q-M Hu and Z-H Hu ldquoA stochastic programming modelfor hub-and-spoke network with uncertain flowsrdquo InternationalJournal of Industrial and Systems Engineering vol 21 no 3 pp302ndash319 2015

[25] S A Torabi J Namdar S M Hatefi and F Jolai ldquoAn enhancedpossibilistic programming approach for reliable closed-loopsupply chain network designrdquo International Journal of Produc-tion Research vol 54 no 5 pp 1358ndash1387 2016

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

Table 3 Comparisons between three methods in SMEIJI problem

Echelon ACA PSO Lingo119873119901119894 119898119871119894 119870119894 119873119898119894 119873119901119894 119898119871119894 119870119894 119873119898119894 119873119901119894 119898119871119894 119870119894 119873119898119894

1 1 00147 mdash mdash 1 00147 mdash mdash 1 00147 mdash mdash2 1 000987 11 1 1 000987 11 1 1 000987 11 13 1 000889 13 1 1 000889 13 1 1 000889 13 14 1 001184 11 1 1 001184 11 1 1 001184 11 15 1 001571 8 1 1 001571 8 1 1 001571 8 16 1 001378 5 1 1 001378 5 1 1 001378 5 17 mdash mdash 5 1 mdash mdash 5 1 mdash mdash 5 1Optimal cost 89166 89166 89166CPU time 0655 1744 23Average iterations 350 1278 3758

Table 4 The relevant data of purchasing activities of 15-echelon model

Purchasing activityEchelon 119863119894 119862119904119894 119867119901119894 119862119890119894 120573119894 120582119871119894 120574119871119894 120583119871119889119894 120590119871119889119894 119865119894 119883119894 0119894

1 18 w 25 28 29 32 0005 0015 0009 0002 70 0005 0032 20w 28 22 32 26 0004 001 00064 00012 50 0003 0013 24w 32 17 38 21 0003 0009 00054 00012 48 0003 0044 30w 36 12 42 16 0004 0012 00072 00016 55 0004 0055 35w 35 08 37 12 0006 0016 001 0002 80 0005 0016 40w 38 06 40 09 0004 0014 0008 0002 65 0005 0027 42w 40 05 28 12 0005 0013 001 0002 70 0005 0038 45w 42 07 35 15 0004 0009 0009 00018 50 0004 0029 50w 44 10 42 10 0003 0010 0005 00024 55 0004 00210 52w 46 15 35 22 0005 0012 00062 00025 60 0005 00211 56w 48 20 49 25 0006 0009 001 00026 65 0003 00312 60w 50 22 40 14 0004 0016 0012 0002 48 0005 00413 62w 55 17 35 15 0007 0015 0009 00018 72 0004 00414 65w 56 12 32 16 0003 0014 00078 0002 70 0006 00315 mdash mdash mdash mdash mdash mdash mdash mdash mdash mdash mdash mdash

Table 5The relevant data of manufacturing activities of 15-echelonmodel

Manufacturing activitiesEchelon 119875119894 119862119898119894 119867119898119894

1 mdash mdash mdash2 200000 280 253 240000 200 194 300000 320 145 350000 360 106 400000 380 097 450000 420 058 480000 440 109 500000 420 1210 520000 400 1511 540000 450 2012 580000 440 2213 620000 420 1814 640000 460 1515 660000 380 16

receiving shipment 0119894 were considered to make sensitiveanalysis and observe the influence on the optimal total jointcost The defective rate and production runs were changedby 0119894(1 + 119877) and 119873119898119894(1 + 119877) where the values of 119877 are0 50 100 200 300 400

Figure 5 illustrated that defective rate was more sensitivethan production runs under the situation that productionruns were one-time production except first echelon In thatcase the higher the defective ratewas the higher the total costwas Accordingly inspecting goods cautiously before deliveryand reducing the defective rate would lower total cost of thewhole supply chainHowever if relaxing that assumption thissensitive analysis would result in different outcomes

Figure 6 has shown that production runs were more sen-sitive than defective rate when relaxing one-time productionassumption In this case the higher the production runswerethe higher the total cost was Hence reducing productionruns should play an essential role in the whole supply chainOnce the one batch production policy was able to satisfy thedemand from the buyer the total cost of whole supply chainwill go down extremely

Mathematical Problems in Engineering 11

Table 6 Comparisons between three ways of 15-echelon inventory model

ACA PSO LingoEchelon 119873119901119894 119898119871119894 119870119894 119873119898119894 119873119901119894 119898119871119894 119870119894 119873119898119894 119873119901119894 119898119871119894 119870119894 119873119898119894

1 1 00147 mdash mdash 1 00147 mdash mdash 1 00147 mdash mdash2 1 000987 11 1 1 000987 11 1 1 000987 11 13 1 000889 13 1 1 000889 13 1 1 000889 13 14 1 001184 11 1 1 001184 11 1 1 001184 11 15 1 001571 8 1 1 001571 8 1 1 001571 8 16 1 001378 5 1 1 001378 5 1 1 001378 5 17 1 001451 5 1 1 001451 5 1 1 001451 5 18 1 001287 5 1 1 001287 5 1 1 001287 5 19 1 00154 7 1 1 00154 7 1 1 00154 7 110 1 001432 9 1 1 001432 9 1 1 001432 9 111 1 001352 7 1 1 001352 7 1 1 001352 7 112 1 00162 8 1 1 00162 8 1 1 00162 8 113 1 001571 10 1 1 001571 10 1 1 001571 10 114 1 001611 5 1 1 001611 5 1 1 001611 5 115 mdash mdash 5 1 mdash mdash 5 1 mdash mdash 5 1Optimal cost ($) 231254 231254 231254CPU time (s) 1725 6621 58215Average iterations 700 2883 8241

Defective rateProduction runs

400100 200 300 5000 500

20000

40000

60000

80000

100000

120000

140000

160000

180000

Figure 5 The sensitive analysis between defective rate and produc-tion runs

5 Conclusions

In summary this research proposed an integrated JIT inven-tory model to solve the partial optimization problem ofpurchaser andmanufacturer finding the optimal solution forthe whole supply chain Furthermore a multiechelon supplychainwith uncertain delivery lead time and unreliable qualityhas been considered to fit the real supply chain environ-ment Besides ACA was proposed to solve the complicatedmathematical problem we also have proved that ACA was

0 50 100 200 300 400 500

Defective rateProduction runs

0

50000

100000

150000

200000

250000

300000

350000

400000

Figure 6 The sensitive analysis between defective rate and produc-tions runs with relaxing assumption

more efficient than PSO and Lingo These are the new majorcontributions of the present paper

Finally there are a great deal of helpful directions forenterprises and researchers (1)The variety of conditions thatoccur in real supply chain can be considered in the modelFor instance time value of cash problem and deterioration(2)The warehousing and storage can be considered in supplychain which make the model more complex to solve (3)The model contains only one member in each echelon amultimember in each echelon should be considered to fit

12 Mathematical Problems in Engineering

real situation in supply chain (4) Some real cases andordata can be applied to our proposed model in the followingworks

Futureworkwill hopefully involve the real-world concernto our research which will be our main target In addition we

do hope our research can be extended intomore considerableareas and more considerable variables

Appendix

The Hessian matrix 119865 of 1198691198621119878(119873119901 119898119871 119873119898 119870) for given 119873119901119894

equal to 1 can be shown as

119865 =

[

[

[

[

[

[

[

[

[

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

1205971198731198981198942

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119873119898119894120597119898119871119894

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119873119898119894120597119870119894

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119898119871119894120597119873119898119894

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

1205971198981198711198942

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119898119871119894120597119870119894

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119870119894120597119873119898119894

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119870119894120597119898119871119894

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

1205971198701198942

]

]

]

]

]

]

]

]

]

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

1205971198731198981198942

=

2

1198731198981198943(120579120593119879

2+

1205931198792(1198671198631198701205932 + Ω)

119870

)

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

1205971198981198711198942

= 119870119873[(

119890minus119898120583

1205832

+ 2 (119890minus119898120583

minus 119890minus120574120583

) +

119890(minus119898120583)(minus119898+120583)

120583

) 120592]

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

1205971198701198942

=

2

1198701198943(120579120593119879

2+

1205931198792(1198671198631198701205932 + Ω)

119873

)

12059721198691198621119904 (119873119898119894 119898119871119894 119870119894)

120597119898119871119894120597119873119898119894

=

12059721198691198621119904 (119873119898119894 119898119871119894 119870119894)

120597119873119898119894120597119898119871119894

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119870119894120597119873119898119894

=

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119873119898119894120597119870119894

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119870119894120597119898119871119894

=

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119898119871119894120597119870119894

100381610038161003816100381611986511

1003816100381610038161003816=

1205972119865

1205971198731198981198942

=

2

1198731198981198943(120579120593119879

2+

1205931198792(1198671198631198701205932 + Ω)

119870

) gt 0

100381610038161003816100381611986522

1003816100381610038161003816= (

1205972119865

1205971198731198981198942)(

1205972119865

1205971198981198711198942) minus [

1205972119865

120597119873119898119894120597119898119871119894

]

2

=

2

1198731198981198943(120579120593119879

2+

1205931198792(1198671198631198701205932 + Ω)

119870

)

1205972119865

1205971198981198711198942minus (

1205972119865

120597119873119898119894120597119898119871119894

)

2

2

1198733119898119894

(1205791205931198792+

119861

119870

)[119870119873(

119890minus119898120576120583

1205832

+ (2119909 +

119890(minus119898120583)119909119910

120583

120592))] + 1198702[

119890minus119898120576120583

120583

+ 120592 (119909119910 minus 119860)] gt 0

100381610038161003816100381611986533

1003816100381610038161003816= (

1205972119865

1205971198731198981198942

1205972119865

1205971198981198711198942

1205972119865

1205971198701198942) minus [(

1205972119865

120597119873119898119894120597119870119894

)

2

(

1205972119865

1205971198981198711198942)]

=

2

1198731198981198943(120579120593119879

2+

119861

119870

)[119870119873(

119890minus119898120576120583

1205832

+ 2(119909 +

119890(minus119898120603)119909119910

120583

) 120592)]

2

1198703

119861

119873

minus [119865 + 119909120576 + 119860120592119910 minus

1205791205931198792

1198732

+

119861

2 (119870119873)2]

[119870119873(

119890minus119898120576120583

1205832

+ (2119909 +

119890minus119898120576120583

120583

) 120592)] gt 0

(A1)

Mathematical Problems in Engineering 13

As a result the matrix F has proved to be a positivedefinite matrix where |11986511| |11986522| and |11986533| are all positiveTherefore theminimumvalue exists in 1198691198621119878(119873119901119898119871 119873119898 119870)Furthermore we can find the optimization in the proposedmodel

Competing Interests

The authors declare that they have no competing interests

References

[1] S K Goyal ldquoAn integrated inventory model for a singlesupplier-single customer problemrdquo International Journal ofProduction Research vol 15 no 1 pp 107ndash111 1977

[2] A Banerjee ldquoA joint economic-lot-sizemodel for purchaser andvendorrdquo Decision Sciences vol 17 no 3 pp 292ndash311 1986

[3] S K Goyal ldquoldquoA joint economic-lot-size model for purchaserand vendorrdquo a commentrdquo Decision Sciences vol 19 no 1 pp236ndash241 1988

[4] Y Seo ldquoControlling general multi-echelon distribution supplychainswith improved reorder decision policy utilizing real-timeshared stock informationrdquo Computers amp Industrial Engineeringvol 51 no 2 pp 229ndash246 2006

[5] H N Chiu and H L Huang ldquoA Multi-echelon integratedJIT inventory model using the time buffer and emergencyborrowing policies to deal with random delivery lead timesrdquoInternational Journal of Production Research vol 41 no 13 pp2911ndash2931 2003

[6] J Sadeghi S M Mousavi S T A Niaki and S SadeghildquoOptimizing a multi-vendor multi-retailer vendor managedinventory problem two tuned meta-heuristic algorithmsrdquoKnowledge-Based Systems vol 50 pp 159ndash170 2013

[7] S S Sana ldquoA production-inventory model of imperfect qualityproducts in a three-layer supply chainrdquo Decision Support Sys-tems vol 50 no 2 pp 539ndash547 2011

[8] H N Soni and K A Patel ldquoOptimal strategy for an integratedinventory system involving variable production and defectiveitems under retailer partial trade credit policyrdquoDecision SupportSystems vol 54 no 1 pp 235ndash247 2012

[9] R J Tersine Principles of Inventory andMaterials ManagementNorthHolland AmsterdamTheNetherlands 3rd edition 1988

[10] C J Liao and C H Shyu ldquoAn analytical determination of leadtime with normal demandrdquo International Journal of Operationsamp Production Management vol 11 no 9 pp 72ndash78 1991

[11] M Ben-Daya and A Raouf ldquoInventory models involving leadtime as a decision variablerdquo Journal of the Operational ResearchSociety vol 45 no 5 pp 579ndash582 1994

[12] S M J Mirzapour Al-e-Hashem A Baboli and Z Sazvar ldquoAstochastic aggregate production planningmodel in a green sup-ply chain considering flexible lead times nonlinear purchaseand shortage cost functionsrdquo European Journal of OperationalResearch vol 230 no 1 pp 26ndash41 2013

[13] C Chandra and J Grabis ldquoInventorymanagementwith variablelead-time dependent procurement costrdquo Omega vol 36 no 5pp 877ndash887 2008

[14] J-W Wu and H-Y Tsai ldquoMixture inventory model with backorders and lost sales for variable lead time demand withthe mixtures of normal distributionrdquo International Journal ofSystems Science Principles and Applications of Systems andIntegration vol 32 no 2 pp 259ndash268 2001

[15] M A Hoque ldquoA vendorndashbuyer integrated productionndashinventory model with normal distribution of lead timerdquoInternational Journal of Production Economics vol 144 no 2pp 409ndash417 2013

[16] F Altiparmak M Gen L Lin and T Paksoy ldquoA geneticalgorithm approach for multi-objective optimization of supplychain networksrdquo Computers and Industrial Engineering vol 51no 1 pp 196ndash215 2006

[17] J Sadeghi S Sadeghi and S T Niaki ldquoA hybrid vendormanaged inventory and redundancy allocation optimizationproblem in supply chain management an NSGA-II with tunedparametersrdquo Computers and Operations Research vol 41 pp53ndash64 2014

[18] X Li F Baki P Tian and B A Chaouch ldquoA robust block-chainbased tabu search algorithm for the dynamic lot sizing problemwith product returns and remanufacturingrdquoOmega vol 42 no1 pp 75ndash87 2014

[19] Z Chen and B R Sarker ldquoAn integrated optimal inventorylot-sizing and vehicle-routing model for a multisupplier single-assembler system with JIT deliveryrdquo International Journal ofProduction Research vol 52 no 17 pp 5086ndash5114 2014

[20] A R Nia M Hemmati Far and S T A Niaki ldquoA fuzzy vendormanaged inventory of multi-item economic order quantitymodel under shortage an ant colony optimization algorithmrdquoInternational Journal of Production Economics vol 155 pp 259ndash271 2014

[21] S M Hatefi and F Jolai ldquoRobust and reliable forward-reverselogistics network design under demand uncertainty and facilitydisruptionsrdquo Applied Mathematical Modelling vol 38 no 9-10pp 2630ndash2647 2014

[22] R H Ma ldquoCross warehouse scheduling of logistics distributionand cycle re-claimer based on heuristics algorithmrdquo AppliedMechanics and Materials vol 644ndash650 pp 2606ndash2610 2014

[23] M Hashim L Yao A H Nadeem M Nazim and M NazamldquoMulti-objective optimization model for supplier selectionproblem in fuzzy environmentrdquo Advances in Intelligent Systemsand Computing vol 281 pp 1201ndash1213 2014

[24] Q-M Hu and Z-H Hu ldquoA stochastic programming modelfor hub-and-spoke network with uncertain flowsrdquo InternationalJournal of Industrial and Systems Engineering vol 21 no 3 pp302ndash319 2015

[25] S A Torabi J Namdar S M Hatefi and F Jolai ldquoAn enhancedpossibilistic programming approach for reliable closed-loopsupply chain network designrdquo International Journal of Produc-tion Research vol 54 no 5 pp 1358ndash1387 2016

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 11

Table 6 Comparisons between three ways of 15-echelon inventory model

ACA PSO LingoEchelon 119873119901119894 119898119871119894 119870119894 119873119898119894 119873119901119894 119898119871119894 119870119894 119873119898119894 119873119901119894 119898119871119894 119870119894 119873119898119894

1 1 00147 mdash mdash 1 00147 mdash mdash 1 00147 mdash mdash2 1 000987 11 1 1 000987 11 1 1 000987 11 13 1 000889 13 1 1 000889 13 1 1 000889 13 14 1 001184 11 1 1 001184 11 1 1 001184 11 15 1 001571 8 1 1 001571 8 1 1 001571 8 16 1 001378 5 1 1 001378 5 1 1 001378 5 17 1 001451 5 1 1 001451 5 1 1 001451 5 18 1 001287 5 1 1 001287 5 1 1 001287 5 19 1 00154 7 1 1 00154 7 1 1 00154 7 110 1 001432 9 1 1 001432 9 1 1 001432 9 111 1 001352 7 1 1 001352 7 1 1 001352 7 112 1 00162 8 1 1 00162 8 1 1 00162 8 113 1 001571 10 1 1 001571 10 1 1 001571 10 114 1 001611 5 1 1 001611 5 1 1 001611 5 115 mdash mdash 5 1 mdash mdash 5 1 mdash mdash 5 1Optimal cost ($) 231254 231254 231254CPU time (s) 1725 6621 58215Average iterations 700 2883 8241

Defective rateProduction runs

400100 200 300 5000 500

20000

40000

60000

80000

100000

120000

140000

160000

180000

Figure 5 The sensitive analysis between defective rate and produc-tion runs

5 Conclusions

In summary this research proposed an integrated JIT inven-tory model to solve the partial optimization problem ofpurchaser andmanufacturer finding the optimal solution forthe whole supply chain Furthermore a multiechelon supplychainwith uncertain delivery lead time and unreliable qualityhas been considered to fit the real supply chain environ-ment Besides ACA was proposed to solve the complicatedmathematical problem we also have proved that ACA was

0 50 100 200 300 400 500

Defective rateProduction runs

0

50000

100000

150000

200000

250000

300000

350000

400000

Figure 6 The sensitive analysis between defective rate and produc-tions runs with relaxing assumption

more efficient than PSO and Lingo These are the new majorcontributions of the present paper

Finally there are a great deal of helpful directions forenterprises and researchers (1)The variety of conditions thatoccur in real supply chain can be considered in the modelFor instance time value of cash problem and deterioration(2)The warehousing and storage can be considered in supplychain which make the model more complex to solve (3)The model contains only one member in each echelon amultimember in each echelon should be considered to fit

12 Mathematical Problems in Engineering

real situation in supply chain (4) Some real cases andordata can be applied to our proposed model in the followingworks

Futureworkwill hopefully involve the real-world concernto our research which will be our main target In addition we

do hope our research can be extended intomore considerableareas and more considerable variables

Appendix

The Hessian matrix 119865 of 1198691198621119878(119873119901 119898119871 119873119898 119870) for given 119873119901119894

equal to 1 can be shown as

119865 =

[

[

[

[

[

[

[

[

[

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

1205971198731198981198942

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119873119898119894120597119898119871119894

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119873119898119894120597119870119894

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119898119871119894120597119873119898119894

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

1205971198981198711198942

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119898119871119894120597119870119894

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119870119894120597119873119898119894

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119870119894120597119898119871119894

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

1205971198701198942

]

]

]

]

]

]

]

]

]

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

1205971198731198981198942

=

2

1198731198981198943(120579120593119879

2+

1205931198792(1198671198631198701205932 + Ω)

119870

)

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

1205971198981198711198942

= 119870119873[(

119890minus119898120583

1205832

+ 2 (119890minus119898120583

minus 119890minus120574120583

) +

119890(minus119898120583)(minus119898+120583)

120583

) 120592]

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

1205971198701198942

=

2

1198701198943(120579120593119879

2+

1205931198792(1198671198631198701205932 + Ω)

119873

)

12059721198691198621119904 (119873119898119894 119898119871119894 119870119894)

120597119898119871119894120597119873119898119894

=

12059721198691198621119904 (119873119898119894 119898119871119894 119870119894)

120597119873119898119894120597119898119871119894

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119870119894120597119873119898119894

=

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119873119898119894120597119870119894

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119870119894120597119898119871119894

=

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119898119871119894120597119870119894

100381610038161003816100381611986511

1003816100381610038161003816=

1205972119865

1205971198731198981198942

=

2

1198731198981198943(120579120593119879

2+

1205931198792(1198671198631198701205932 + Ω)

119870

) gt 0

100381610038161003816100381611986522

1003816100381610038161003816= (

1205972119865

1205971198731198981198942)(

1205972119865

1205971198981198711198942) minus [

1205972119865

120597119873119898119894120597119898119871119894

]

2

=

2

1198731198981198943(120579120593119879

2+

1205931198792(1198671198631198701205932 + Ω)

119870

)

1205972119865

1205971198981198711198942minus (

1205972119865

120597119873119898119894120597119898119871119894

)

2

2

1198733119898119894

(1205791205931198792+

119861

119870

)[119870119873(

119890minus119898120576120583

1205832

+ (2119909 +

119890(minus119898120583)119909119910

120583

120592))] + 1198702[

119890minus119898120576120583

120583

+ 120592 (119909119910 minus 119860)] gt 0

100381610038161003816100381611986533

1003816100381610038161003816= (

1205972119865

1205971198731198981198942

1205972119865

1205971198981198711198942

1205972119865

1205971198701198942) minus [(

1205972119865

120597119873119898119894120597119870119894

)

2

(

1205972119865

1205971198981198711198942)]

=

2

1198731198981198943(120579120593119879

2+

119861

119870

)[119870119873(

119890minus119898120576120583

1205832

+ 2(119909 +

119890(minus119898120603)119909119910

120583

) 120592)]

2

1198703

119861

119873

minus [119865 + 119909120576 + 119860120592119910 minus

1205791205931198792

1198732

+

119861

2 (119870119873)2]

[119870119873(

119890minus119898120576120583

1205832

+ (2119909 +

119890minus119898120576120583

120583

) 120592)] gt 0

(A1)

Mathematical Problems in Engineering 13

As a result the matrix F has proved to be a positivedefinite matrix where |11986511| |11986522| and |11986533| are all positiveTherefore theminimumvalue exists in 1198691198621119878(119873119901119898119871 119873119898 119870)Furthermore we can find the optimization in the proposedmodel

Competing Interests

The authors declare that they have no competing interests

References

[1] S K Goyal ldquoAn integrated inventory model for a singlesupplier-single customer problemrdquo International Journal ofProduction Research vol 15 no 1 pp 107ndash111 1977

[2] A Banerjee ldquoA joint economic-lot-sizemodel for purchaser andvendorrdquo Decision Sciences vol 17 no 3 pp 292ndash311 1986

[3] S K Goyal ldquoldquoA joint economic-lot-size model for purchaserand vendorrdquo a commentrdquo Decision Sciences vol 19 no 1 pp236ndash241 1988

[4] Y Seo ldquoControlling general multi-echelon distribution supplychainswith improved reorder decision policy utilizing real-timeshared stock informationrdquo Computers amp Industrial Engineeringvol 51 no 2 pp 229ndash246 2006

[5] H N Chiu and H L Huang ldquoA Multi-echelon integratedJIT inventory model using the time buffer and emergencyborrowing policies to deal with random delivery lead timesrdquoInternational Journal of Production Research vol 41 no 13 pp2911ndash2931 2003

[6] J Sadeghi S M Mousavi S T A Niaki and S SadeghildquoOptimizing a multi-vendor multi-retailer vendor managedinventory problem two tuned meta-heuristic algorithmsrdquoKnowledge-Based Systems vol 50 pp 159ndash170 2013

[7] S S Sana ldquoA production-inventory model of imperfect qualityproducts in a three-layer supply chainrdquo Decision Support Sys-tems vol 50 no 2 pp 539ndash547 2011

[8] H N Soni and K A Patel ldquoOptimal strategy for an integratedinventory system involving variable production and defectiveitems under retailer partial trade credit policyrdquoDecision SupportSystems vol 54 no 1 pp 235ndash247 2012

[9] R J Tersine Principles of Inventory andMaterials ManagementNorthHolland AmsterdamTheNetherlands 3rd edition 1988

[10] C J Liao and C H Shyu ldquoAn analytical determination of leadtime with normal demandrdquo International Journal of Operationsamp Production Management vol 11 no 9 pp 72ndash78 1991

[11] M Ben-Daya and A Raouf ldquoInventory models involving leadtime as a decision variablerdquo Journal of the Operational ResearchSociety vol 45 no 5 pp 579ndash582 1994

[12] S M J Mirzapour Al-e-Hashem A Baboli and Z Sazvar ldquoAstochastic aggregate production planningmodel in a green sup-ply chain considering flexible lead times nonlinear purchaseand shortage cost functionsrdquo European Journal of OperationalResearch vol 230 no 1 pp 26ndash41 2013

[13] C Chandra and J Grabis ldquoInventorymanagementwith variablelead-time dependent procurement costrdquo Omega vol 36 no 5pp 877ndash887 2008

[14] J-W Wu and H-Y Tsai ldquoMixture inventory model with backorders and lost sales for variable lead time demand withthe mixtures of normal distributionrdquo International Journal ofSystems Science Principles and Applications of Systems andIntegration vol 32 no 2 pp 259ndash268 2001

[15] M A Hoque ldquoA vendorndashbuyer integrated productionndashinventory model with normal distribution of lead timerdquoInternational Journal of Production Economics vol 144 no 2pp 409ndash417 2013

[16] F Altiparmak M Gen L Lin and T Paksoy ldquoA geneticalgorithm approach for multi-objective optimization of supplychain networksrdquo Computers and Industrial Engineering vol 51no 1 pp 196ndash215 2006

[17] J Sadeghi S Sadeghi and S T Niaki ldquoA hybrid vendormanaged inventory and redundancy allocation optimizationproblem in supply chain management an NSGA-II with tunedparametersrdquo Computers and Operations Research vol 41 pp53ndash64 2014

[18] X Li F Baki P Tian and B A Chaouch ldquoA robust block-chainbased tabu search algorithm for the dynamic lot sizing problemwith product returns and remanufacturingrdquoOmega vol 42 no1 pp 75ndash87 2014

[19] Z Chen and B R Sarker ldquoAn integrated optimal inventorylot-sizing and vehicle-routing model for a multisupplier single-assembler system with JIT deliveryrdquo International Journal ofProduction Research vol 52 no 17 pp 5086ndash5114 2014

[20] A R Nia M Hemmati Far and S T A Niaki ldquoA fuzzy vendormanaged inventory of multi-item economic order quantitymodel under shortage an ant colony optimization algorithmrdquoInternational Journal of Production Economics vol 155 pp 259ndash271 2014

[21] S M Hatefi and F Jolai ldquoRobust and reliable forward-reverselogistics network design under demand uncertainty and facilitydisruptionsrdquo Applied Mathematical Modelling vol 38 no 9-10pp 2630ndash2647 2014

[22] R H Ma ldquoCross warehouse scheduling of logistics distributionand cycle re-claimer based on heuristics algorithmrdquo AppliedMechanics and Materials vol 644ndash650 pp 2606ndash2610 2014

[23] M Hashim L Yao A H Nadeem M Nazim and M NazamldquoMulti-objective optimization model for supplier selectionproblem in fuzzy environmentrdquo Advances in Intelligent Systemsand Computing vol 281 pp 1201ndash1213 2014

[24] Q-M Hu and Z-H Hu ldquoA stochastic programming modelfor hub-and-spoke network with uncertain flowsrdquo InternationalJournal of Industrial and Systems Engineering vol 21 no 3 pp302ndash319 2015

[25] S A Torabi J Namdar S M Hatefi and F Jolai ldquoAn enhancedpossibilistic programming approach for reliable closed-loopsupply chain network designrdquo International Journal of Produc-tion Research vol 54 no 5 pp 1358ndash1387 2016

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 Mathematical Problems in Engineering

real situation in supply chain (4) Some real cases andordata can be applied to our proposed model in the followingworks

Futureworkwill hopefully involve the real-world concernto our research which will be our main target In addition we

do hope our research can be extended intomore considerableareas and more considerable variables

Appendix

The Hessian matrix 119865 of 1198691198621119878(119873119901 119898119871 119873119898 119870) for given 119873119901119894

equal to 1 can be shown as

119865 =

[

[

[

[

[

[

[

[

[

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

1205971198731198981198942

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119873119898119894120597119898119871119894

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119873119898119894120597119870119894

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119898119871119894120597119873119898119894

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

1205971198981198711198942

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119898119871119894120597119870119894

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119870119894120597119873119898119894

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119870119894120597119898119871119894

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

1205971198701198942

]

]

]

]

]

]

]

]

]

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

1205971198731198981198942

=

2

1198731198981198943(120579120593119879

2+

1205931198792(1198671198631198701205932 + Ω)

119870

)

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

1205971198981198711198942

= 119870119873[(

119890minus119898120583

1205832

+ 2 (119890minus119898120583

minus 119890minus120574120583

) +

119890(minus119898120583)(minus119898+120583)

120583

) 120592]

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

1205971198701198942

=

2

1198701198943(120579120593119879

2+

1205931198792(1198671198631198701205932 + Ω)

119873

)

12059721198691198621119904 (119873119898119894 119898119871119894 119870119894)

120597119898119871119894120597119873119898119894

=

12059721198691198621119904 (119873119898119894 119898119871119894 119870119894)

120597119873119898119894120597119898119871119894

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119870119894120597119873119898119894

=

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119873119898119894120597119870119894

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119870119894120597119898119871119894

=

12059721198691198621119878 (119873119898119894 119898119871119894 119870119894)

120597119898119871119894120597119870119894

100381610038161003816100381611986511

1003816100381610038161003816=

1205972119865

1205971198731198981198942

=

2

1198731198981198943(120579120593119879

2+

1205931198792(1198671198631198701205932 + Ω)

119870

) gt 0

100381610038161003816100381611986522

1003816100381610038161003816= (

1205972119865

1205971198731198981198942)(

1205972119865

1205971198981198711198942) minus [

1205972119865

120597119873119898119894120597119898119871119894

]

2

=

2

1198731198981198943(120579120593119879

2+

1205931198792(1198671198631198701205932 + Ω)

119870

)

1205972119865

1205971198981198711198942minus (

1205972119865

120597119873119898119894120597119898119871119894

)

2

2

1198733119898119894

(1205791205931198792+

119861

119870

)[119870119873(

119890minus119898120576120583

1205832

+ (2119909 +

119890(minus119898120583)119909119910

120583

120592))] + 1198702[

119890minus119898120576120583

120583

+ 120592 (119909119910 minus 119860)] gt 0

100381610038161003816100381611986533

1003816100381610038161003816= (

1205972119865

1205971198731198981198942

1205972119865

1205971198981198711198942

1205972119865

1205971198701198942) minus [(

1205972119865

120597119873119898119894120597119870119894

)

2

(

1205972119865

1205971198981198711198942)]

=

2

1198731198981198943(120579120593119879

2+

119861

119870

)[119870119873(

119890minus119898120576120583

1205832

+ 2(119909 +

119890(minus119898120603)119909119910

120583

) 120592)]

2

1198703

119861

119873

minus [119865 + 119909120576 + 119860120592119910 minus

1205791205931198792

1198732

+

119861

2 (119870119873)2]

[119870119873(

119890minus119898120576120583

1205832

+ (2119909 +

119890minus119898120576120583

120583

) 120592)] gt 0

(A1)

Mathematical Problems in Engineering 13

As a result the matrix F has proved to be a positivedefinite matrix where |11986511| |11986522| and |11986533| are all positiveTherefore theminimumvalue exists in 1198691198621119878(119873119901119898119871 119873119898 119870)Furthermore we can find the optimization in the proposedmodel

Competing Interests

The authors declare that they have no competing interests

References

[1] S K Goyal ldquoAn integrated inventory model for a singlesupplier-single customer problemrdquo International Journal ofProduction Research vol 15 no 1 pp 107ndash111 1977

[2] A Banerjee ldquoA joint economic-lot-sizemodel for purchaser andvendorrdquo Decision Sciences vol 17 no 3 pp 292ndash311 1986

[3] S K Goyal ldquoldquoA joint economic-lot-size model for purchaserand vendorrdquo a commentrdquo Decision Sciences vol 19 no 1 pp236ndash241 1988

[4] Y Seo ldquoControlling general multi-echelon distribution supplychainswith improved reorder decision policy utilizing real-timeshared stock informationrdquo Computers amp Industrial Engineeringvol 51 no 2 pp 229ndash246 2006

[5] H N Chiu and H L Huang ldquoA Multi-echelon integratedJIT inventory model using the time buffer and emergencyborrowing policies to deal with random delivery lead timesrdquoInternational Journal of Production Research vol 41 no 13 pp2911ndash2931 2003

[6] J Sadeghi S M Mousavi S T A Niaki and S SadeghildquoOptimizing a multi-vendor multi-retailer vendor managedinventory problem two tuned meta-heuristic algorithmsrdquoKnowledge-Based Systems vol 50 pp 159ndash170 2013

[7] S S Sana ldquoA production-inventory model of imperfect qualityproducts in a three-layer supply chainrdquo Decision Support Sys-tems vol 50 no 2 pp 539ndash547 2011

[8] H N Soni and K A Patel ldquoOptimal strategy for an integratedinventory system involving variable production and defectiveitems under retailer partial trade credit policyrdquoDecision SupportSystems vol 54 no 1 pp 235ndash247 2012

[9] R J Tersine Principles of Inventory andMaterials ManagementNorthHolland AmsterdamTheNetherlands 3rd edition 1988

[10] C J Liao and C H Shyu ldquoAn analytical determination of leadtime with normal demandrdquo International Journal of Operationsamp Production Management vol 11 no 9 pp 72ndash78 1991

[11] M Ben-Daya and A Raouf ldquoInventory models involving leadtime as a decision variablerdquo Journal of the Operational ResearchSociety vol 45 no 5 pp 579ndash582 1994

[12] S M J Mirzapour Al-e-Hashem A Baboli and Z Sazvar ldquoAstochastic aggregate production planningmodel in a green sup-ply chain considering flexible lead times nonlinear purchaseand shortage cost functionsrdquo European Journal of OperationalResearch vol 230 no 1 pp 26ndash41 2013

[13] C Chandra and J Grabis ldquoInventorymanagementwith variablelead-time dependent procurement costrdquo Omega vol 36 no 5pp 877ndash887 2008

[14] J-W Wu and H-Y Tsai ldquoMixture inventory model with backorders and lost sales for variable lead time demand withthe mixtures of normal distributionrdquo International Journal ofSystems Science Principles and Applications of Systems andIntegration vol 32 no 2 pp 259ndash268 2001

[15] M A Hoque ldquoA vendorndashbuyer integrated productionndashinventory model with normal distribution of lead timerdquoInternational Journal of Production Economics vol 144 no 2pp 409ndash417 2013

[16] F Altiparmak M Gen L Lin and T Paksoy ldquoA geneticalgorithm approach for multi-objective optimization of supplychain networksrdquo Computers and Industrial Engineering vol 51no 1 pp 196ndash215 2006

[17] J Sadeghi S Sadeghi and S T Niaki ldquoA hybrid vendormanaged inventory and redundancy allocation optimizationproblem in supply chain management an NSGA-II with tunedparametersrdquo Computers and Operations Research vol 41 pp53ndash64 2014

[18] X Li F Baki P Tian and B A Chaouch ldquoA robust block-chainbased tabu search algorithm for the dynamic lot sizing problemwith product returns and remanufacturingrdquoOmega vol 42 no1 pp 75ndash87 2014

[19] Z Chen and B R Sarker ldquoAn integrated optimal inventorylot-sizing and vehicle-routing model for a multisupplier single-assembler system with JIT deliveryrdquo International Journal ofProduction Research vol 52 no 17 pp 5086ndash5114 2014

[20] A R Nia M Hemmati Far and S T A Niaki ldquoA fuzzy vendormanaged inventory of multi-item economic order quantitymodel under shortage an ant colony optimization algorithmrdquoInternational Journal of Production Economics vol 155 pp 259ndash271 2014

[21] S M Hatefi and F Jolai ldquoRobust and reliable forward-reverselogistics network design under demand uncertainty and facilitydisruptionsrdquo Applied Mathematical Modelling vol 38 no 9-10pp 2630ndash2647 2014

[22] R H Ma ldquoCross warehouse scheduling of logistics distributionand cycle re-claimer based on heuristics algorithmrdquo AppliedMechanics and Materials vol 644ndash650 pp 2606ndash2610 2014

[23] M Hashim L Yao A H Nadeem M Nazim and M NazamldquoMulti-objective optimization model for supplier selectionproblem in fuzzy environmentrdquo Advances in Intelligent Systemsand Computing vol 281 pp 1201ndash1213 2014

[24] Q-M Hu and Z-H Hu ldquoA stochastic programming modelfor hub-and-spoke network with uncertain flowsrdquo InternationalJournal of Industrial and Systems Engineering vol 21 no 3 pp302ndash319 2015

[25] S A Torabi J Namdar S M Hatefi and F Jolai ldquoAn enhancedpossibilistic programming approach for reliable closed-loopsupply chain network designrdquo International Journal of Produc-tion Research vol 54 no 5 pp 1358ndash1387 2016

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 13

As a result the matrix F has proved to be a positivedefinite matrix where |11986511| |11986522| and |11986533| are all positiveTherefore theminimumvalue exists in 1198691198621119878(119873119901119898119871 119873119898 119870)Furthermore we can find the optimization in the proposedmodel

Competing Interests

The authors declare that they have no competing interests

References

[1] S K Goyal ldquoAn integrated inventory model for a singlesupplier-single customer problemrdquo International Journal ofProduction Research vol 15 no 1 pp 107ndash111 1977

[2] A Banerjee ldquoA joint economic-lot-sizemodel for purchaser andvendorrdquo Decision Sciences vol 17 no 3 pp 292ndash311 1986

[3] S K Goyal ldquoldquoA joint economic-lot-size model for purchaserand vendorrdquo a commentrdquo Decision Sciences vol 19 no 1 pp236ndash241 1988

[4] Y Seo ldquoControlling general multi-echelon distribution supplychainswith improved reorder decision policy utilizing real-timeshared stock informationrdquo Computers amp Industrial Engineeringvol 51 no 2 pp 229ndash246 2006

[5] H N Chiu and H L Huang ldquoA Multi-echelon integratedJIT inventory model using the time buffer and emergencyborrowing policies to deal with random delivery lead timesrdquoInternational Journal of Production Research vol 41 no 13 pp2911ndash2931 2003

[6] J Sadeghi S M Mousavi S T A Niaki and S SadeghildquoOptimizing a multi-vendor multi-retailer vendor managedinventory problem two tuned meta-heuristic algorithmsrdquoKnowledge-Based Systems vol 50 pp 159ndash170 2013

[7] S S Sana ldquoA production-inventory model of imperfect qualityproducts in a three-layer supply chainrdquo Decision Support Sys-tems vol 50 no 2 pp 539ndash547 2011

[8] H N Soni and K A Patel ldquoOptimal strategy for an integratedinventory system involving variable production and defectiveitems under retailer partial trade credit policyrdquoDecision SupportSystems vol 54 no 1 pp 235ndash247 2012

[9] R J Tersine Principles of Inventory andMaterials ManagementNorthHolland AmsterdamTheNetherlands 3rd edition 1988

[10] C J Liao and C H Shyu ldquoAn analytical determination of leadtime with normal demandrdquo International Journal of Operationsamp Production Management vol 11 no 9 pp 72ndash78 1991

[11] M Ben-Daya and A Raouf ldquoInventory models involving leadtime as a decision variablerdquo Journal of the Operational ResearchSociety vol 45 no 5 pp 579ndash582 1994

[12] S M J Mirzapour Al-e-Hashem A Baboli and Z Sazvar ldquoAstochastic aggregate production planningmodel in a green sup-ply chain considering flexible lead times nonlinear purchaseand shortage cost functionsrdquo European Journal of OperationalResearch vol 230 no 1 pp 26ndash41 2013

[13] C Chandra and J Grabis ldquoInventorymanagementwith variablelead-time dependent procurement costrdquo Omega vol 36 no 5pp 877ndash887 2008

[14] J-W Wu and H-Y Tsai ldquoMixture inventory model with backorders and lost sales for variable lead time demand withthe mixtures of normal distributionrdquo International Journal ofSystems Science Principles and Applications of Systems andIntegration vol 32 no 2 pp 259ndash268 2001

[15] M A Hoque ldquoA vendorndashbuyer integrated productionndashinventory model with normal distribution of lead timerdquoInternational Journal of Production Economics vol 144 no 2pp 409ndash417 2013

[16] F Altiparmak M Gen L Lin and T Paksoy ldquoA geneticalgorithm approach for multi-objective optimization of supplychain networksrdquo Computers and Industrial Engineering vol 51no 1 pp 196ndash215 2006

[17] J Sadeghi S Sadeghi and S T Niaki ldquoA hybrid vendormanaged inventory and redundancy allocation optimizationproblem in supply chain management an NSGA-II with tunedparametersrdquo Computers and Operations Research vol 41 pp53ndash64 2014

[18] X Li F Baki P Tian and B A Chaouch ldquoA robust block-chainbased tabu search algorithm for the dynamic lot sizing problemwith product returns and remanufacturingrdquoOmega vol 42 no1 pp 75ndash87 2014

[19] Z Chen and B R Sarker ldquoAn integrated optimal inventorylot-sizing and vehicle-routing model for a multisupplier single-assembler system with JIT deliveryrdquo International Journal ofProduction Research vol 52 no 17 pp 5086ndash5114 2014

[20] A R Nia M Hemmati Far and S T A Niaki ldquoA fuzzy vendormanaged inventory of multi-item economic order quantitymodel under shortage an ant colony optimization algorithmrdquoInternational Journal of Production Economics vol 155 pp 259ndash271 2014

[21] S M Hatefi and F Jolai ldquoRobust and reliable forward-reverselogistics network design under demand uncertainty and facilitydisruptionsrdquo Applied Mathematical Modelling vol 38 no 9-10pp 2630ndash2647 2014

[22] R H Ma ldquoCross warehouse scheduling of logistics distributionand cycle re-claimer based on heuristics algorithmrdquo AppliedMechanics and Materials vol 644ndash650 pp 2606ndash2610 2014

[23] M Hashim L Yao A H Nadeem M Nazim and M NazamldquoMulti-objective optimization model for supplier selectionproblem in fuzzy environmentrdquo Advances in Intelligent Systemsand Computing vol 281 pp 1201ndash1213 2014

[24] Q-M Hu and Z-H Hu ldquoA stochastic programming modelfor hub-and-spoke network with uncertain flowsrdquo InternationalJournal of Industrial and Systems Engineering vol 21 no 3 pp302ndash319 2015

[25] S A Torabi J Namdar S M Hatefi and F Jolai ldquoAn enhancedpossibilistic programming approach for reliable closed-loopsupply chain network designrdquo International Journal of Produc-tion Research vol 54 no 5 pp 1358ndash1387 2016

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of