Research ArticleAn Extended EPQ-Based Problem with a Discontinuous DeliveryPolicy Scrap Rate and Random Breakdown

Singa Wang Chiu1 Hong-Dar Lin2 Ming-Syuan Song2 Hsin-Mei Chen2 and Yuan-Shyi PChiu2

1Department of Business Administration Chaoyang University of Technology Taichung 413 Taiwan2Department of Industrial Engineering amp Management Chaoyang University of Technology Wufong District Taichung 413 Taiwan

Correspondence should be addressed to Hong-Dar Lin hdlincyutedutw

Received 7 August 2014 Accepted 26 November 2014

Academic Editor Hsueh-Ming Wang

Copyright copy 2015 Singa Wang Chiu 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

In real supply chain environments the discontinuous multidelivery policy is often used when finished products need to betransported to retailers or customers outside the production units To address this real-life production-shipment situation thisstudy extends recent work using an economic production quantity- (EPQ-) based inventory model with a continuous inventoryissuing policy defective items and machine breakdown by incorporating a multiple delivery policy into the model to replace thecontinuous policy and investigates the effect on the optimal run time decision for this specific EPQmodel Next we further expandthe scope of the problem to combine the retailerrsquos stock holding cost into our study This enhanced EPQ-based model can be usedto reflect the situation found in contemporary manufacturing firms in which finished products are delivered to the producerrsquosown retail stores and stocked there for sale A second model is developed and studied With the help of mathematical modelingand optimization techniques the optimal run times that minimize the expected total system costs comprising costs incurred inproduction units transportation and retail stores are derived for both models Numerical examples are provided to demonstratethe applicability of our research results

1 Introduction

This paper focuses on optimizing a producer-retailer inte-grated economic production quantity- (EPQ-) based problemwith a discontinuous delivery policy scrap rate and randombreakdown The EPQ model was first introduced by Taft[1] and its concept has since been frequently implementedby manufacturing firms to determine the most economicreplenishment batch sizes for the products that need to beproduced in-house [2 3] Although the traditional EPQmodel assumes a perfect condition in each production runin real manufacturing environments due to process dete-rioration or other uncontrollable factors generation of defec-tive items and random breakdown are inevitable [4]Widmerand Solot [5] applied a queuing network theory to the study ofa breakdown and maintenance operation problem A simpleway of modeling these perturbations was proposed to takeinto account the performances evaluation of the flexible

manufacturing system (FMS) (including the production rateand machine utilization) The analytical and simulationresults were compared in order to demonstrate the accuracyof their modeling technique Yu and Bricker [6] presentedan informative application of Markovrsquos chain analysis to amultistage manufacturing problem They also pointed outan error in the literature that had been undetected formany years Groenevelt et al [7] investigated the effects ofbreakdowns and corrective maintenance on the economicbatch sizing decisions Two inventory control policies wereexamined in the case of a breakdown namely the no-resumption (NR) and abort-resume (AR) The NR controlpolicy assumes that after a breakdown situation is handledthe production of the interrupted lots is not resumed whilethe AR control policy assumes that if the current on-handinventory is below a certain threshold level then the pro-duction is immediately resumed after a breakdown situationis taken care of Their research results indicated that this

Hindawi Publishing Corporatione Scientific World JournalVolume 2015 Article ID 621978 13 pageshttpdxdoiorg1011552015621978

2 The Scientific World Journal

control structure is optimal among all stationary policies andoffered the exact optimal and closed form approximate lotsizing formulas and bounds on average cost per unit timefor the approximations Widyadana and Wee [8] developeddeteriorating items production inventory models with ran-dom breakdown and stochastic machine repair time Therepair time is assumed to be independent of the breakdownrate They applied the classical optimization technique tothe problem and derived an optimal solution Through anumerical example and sensitivity analysis they showed thatthe production and demand rates are the most sensitiveparameters to the optimal uptime and the demand rate is themost sensitive variable to the system costs for the stochasticmodel with exponential distribution repair time Chiu etal [9] determined the optimal replenishment run time foran EPQ-based inventory model with nonconforming itemsand breakdown Their model assumes that after a Poissondistributed breakdownoccurs themachine goes under repairinstantly and the production of the interrupted lot resumesimmediately when the machine is fixed and restored Thesystem also considers a uniformly distributed scrap rateassociated with the production process A mathematicalmodel along with a recursive searching algorithm is usedin their study to derive the optimal replenishment policythat minimizes the total system costs A numerical examplewas provided to demonstrate the practical application andbetter cost efficiency of the proposed policy compared toa breakdown that occurs under a no-resumption policyAdditional studies relating to the issues of product qualityunreliable production equipment and their consequencequality assurance can be found in [10ndash17]

Unlike the assumption of a continuous inventory issuingpolicy of the traditional EPQ model in real supply chainenvironments the discontinuousmultidelivery policy is oftenused when finished products need to be transported toretailers or customers outside the production units Schwarzet al [18] determined the fill-rate of a one-warehouseN-identical retailer distribution system An approximationmodel was adopted from a prior work to maximize thesystem fill-rate subjected to a constraint on safety stockThe properties of the fill-rate policy were used to providemanagerial insights into system optimization Sarker andKhan [19] examined a manufacturing system that procuresraw materials in a lot from suppliers and processes theminto the finished products that are subsequently shipped tooutside customers at fixed points in time The system costfunction for the model was formulated by including bothraw materials and finished products A solution procedurewas developed to determine an optimal ordering policy forthe procurement of raw materials and the production batchsize that minimizes the total system costs Comez et al [20]considered a centralized inventory sharing system of tworetailers that are replenished periodicallyThey assumed thatbetween two replenishments a unit can be transshipped to astocked-out retailer from the other retailer Whenever thereis an absence of transshipments the backorder costs areincurred until the next replenishment The objective of theirstudy is to minimize the long-run average costs comprisingthe replenishment holding backorder and transshipment

costs They discussed the challenges associated with positivereplenishment time and developed upper and lower boundsof average costs in such situations Other studies [21ndash29]focused on various aspects of periodic or multiple deliveryissues in the vendor-buyer integrated supply chains

With the purpose of addressing the real-life production-shipment situation this study extends a recent work [9] byincorporating a multiple delivery policy into their modelto replace the continuous policy and investigates the effecton the optimal run time decision for this specific EPQmodel Next we further expand the scope of the problemto combine the retailerrsquos stock holding cost into our studyThis enhanced EPQ-based model can be used to reflect thesituation found in contemporary manufacturing firms inwhich finished products are delivered to the producerrsquos ownretail stores and stocked there for sale The objectives are todetermine the optimal run times that minimize the expectedtotal system costs comprising costs incurred in productionunits transportation and retail stores for both models Aslittle attention has been paid to this specific area the presentstudy is intended to fill this gap

2 Statement and Optimization ofProposed Model 1

In real supply chain environments the discontinuous mul-tidelivery policy is often used when finished products needto be transported to retail stores or customers outside theproduction units To explicitly address this realistic situationthe first proposedmodel in this study incorporates a multipledelivery policy into an EPQ-based inventory model withscrap items and breakdown [9] to replace the continuousissuing policy and investigates the effect on the optimalmanufacturing run time decision

Summary of assumptions (features) considered in theproposed multi-item EPQ-based model are as follows (1) arandom machine breakdown rate (2) a random scrap rate inproduction and (3) a discontinuous multidelivery policy forfinished products The details of the proposed model can bedescribed as follows Suppose a product can bemanufacturedat an annual rate 119875

1and its demand is 120582 units per year All

items produced must pass a quality conformation check andthe unit screening cost is included in the unit manufacturingcost 119862 A random 119909 proportion of the products producedis defective and will be scrapped at the end of the regularproduction process Hence the production rate of scrapsis 1198891and 119889

1= 1198751119909 Under regular operations (ie to

avoid a stock-out situation) (1198751minus 1198891minus 120582) gt 0 must be

satisfied Upon the completion of the production processthe acceptable quality (finished) products are transported tothe outside retail store or customer under a discontinuousmultidelivery policy while fixed quantity 119899 installments ofthe finished items are shipped to retail store at a fixed intervalof time in 1199051015840

2(Figure 1) The proposed model assumes that

during the production uptime a Poisson distributedmachinebreakdownmay occur and an abortresume (AR) inventorycontrol policy is employed when a breakdown happensUnder such an AR policy the machine goes under repair

The Scientific World Journal 3

I(t)

H998400

H9984001

P1 minus d1

P1 minus d1

t tr

tn

t1 minus t

T998400

t9984002

Time

Figure 1 Inventory level of finished items in the proposed manufacturing run time problem

immediately and a constant repair time is assumed Uponthe completion of the repair the interrupted lot is instantlyresumed (Figure 1)

Additional cost-related parameters used in this studyare the machine repairing cost 119872 setup cost per cycle 119870holding cost per item at the producerrsquos side ℎ disposal costper scrapped item 119862

119878 fixed delivery cost per shipment 119870

1

variable delivery cost per item119862119879 unit holding cost for safety

stock at the producerrsquos side ℎ3 and holding cost per item at

the retailerrsquos side ℎ2 Other notations used in the modeling

and analysis also include the following

119905 production time before a random machine break-down takes place1198671015840

1 on-hand inventory level in units when a random

machine breakdown takes place120573 number of machine breakdowns per unit time (ieyear) assumed to be a random variable that followsthe Poisson distribution119905119903 machine repair time

1199051 production uptime the decision variable of the

proposed manufacturing run time model1198671015840 maximum on-hand inventory level in units when

the regular production process ends (in the case of abreakdown)1199051015840

2 time required to deliver all finished items produced

in a cycle (in the case of a breakdown)1198791015840 production cycle length (in the case of a break-

down)119876 lot size for each production cycleTC1(1199051) total production-inventory-delivery costs

per cycle (in the case of a breakdown)

119864[TC1(1199051)] the expected production-inventory-

delivery costs per cycle (in the case of a breakdown)1199052 time required to deliver all finished items produced

(in the case of no breakdown)119867 on-hand inventory level in units when the regularproduction process ends (in the case of no break-down)119879 cycle length (in the case of no breakdown)119868(119905) on-hand inventory level of finished items at time119905119868119904(119905) on-hand inventory level of scrap items at time 119905

TC2(1199051) total production-inventory-delivery costs

per cycle (in the case of no breakdown)119864[TC

2(1199051)] the expected production-inventory-

delivery costs per cycle (in the case of no breakdown)TCU(119905

1) total production-inventory-delivery costs

per unit timewhether or not a breakdown takes place119864[TCU(119905

1)] the long-run expected production-

inventory-delivery costs per unit time whether or nota breakdown takes placeT the cycle length whether or not a machine break-down takes place

Since a machine breakdown may randomly take place atproduction uptime 119905

1 the following two distinct cases must

be examined

21 Case 1 A Random Machine Breakdown Takes Place atUptime 119905

1 In such a situation 119905 lt 119905

1 Under theAR inventory

control policy the machine goes under repair immediatelyand once it is fixed and restored the interrupted lot isinstantly resumed (Figure 1) Since 119909 proportion of scrap

4 The Scientific World Journal

Id(t)

t9984002

Time

t

T998400

d1t

d1t1

d1

d1

d1

tr t1 minus t

Figure 2 Inventory level of scrap items in the proposed manufacturing run time problem

products is produced the maximum number of scraps in acycle is 119909119876 (or 119889

11199051) and the on-hand inventory of scrap

items in the proposed manufacturing run time problem is asillustrated in Figure 2

The production cycle time 1198791015840 can be seen as (1) fromFigure 1

1198791015840= 1199051+ 119905119903+ 1199051015840

2 (1)

The total production-inventory-delivery cost per cycleTC1(1199051) is comprised of (1) the variable production cost (2)

the setup cost (3) the disposal cost for scraps (4) themachinerepair cost (5) fixed and variable product delivery costs (6)holding cost for safety stocks and (7) the producerrsquos inventoryholding costs in the entire production cycle Thus TC

1(1199051) is

TC1(1199051) = 119862 (119875

11199051) + 119870 + 119862

119878(11990511198751119909) +119872 + 119899119870

1

+ 119862119879[11990511198751] + ℎ3(120582119905119903) 1198791015840

+ ℎ[1198671015840+ 11988911199051

21199051+ (1198671015840

1+ 1198891119905) 119905119903+119899 minus 1

211989911986710158401199051015840

2]

(2)

Since 119909 is assumed to be a random variable with a knownprobability density function the expected values of119909 are usedin our analysis to take the randomness of 119909 into account Bysubstituting all related system parameters into (2) [9] withfurther derivations 119864[TC

1(1199051)] becomes (see the appendix

for more details)

119864 [TC1(1199051)]

= 119870 + 1198991198701+119872 + ℎ119905119875

1119892

+ [1198621198751+ 1198621198781198751119864 [119909] + 1198621198791198751 (1 minus 119864 [119909])

+ ℎ31198751119892 (1 minus 119864 [119909]) minus

ℎ1198751119892 (1 minus 119864 [119909])

2(1 minus

1

119899)] sdot 1199051

+ [ℎ1198751119864 [119909]

2+ℎ1198752

1

2120582(1 minus 119864 [119909])

2(1 minus

1

119899)

+ℎ1198751

2119899(1 minus 119864 [119909])] 119905

2

1

(3)

22 Case 2 No Breakdown Takes Place at Uptime 1199051 In such

a situation 119905 gt 1199051 The inventory level of finished items in

this case is depicted in Figure 3 and 119879 = 1199051+ 1199052 The total

production-inventory-delivery cost per cycle TC2(1199051) is as

displayed inTC2(1199051) = 119862 (119875

11199051) + 119870 + 119862

119878(11990911990511198751) + 119899119870

1

+ 119862119879[11990511198751(1 minus 119909)] + ℎ

3(120582119905119903) 119879

+ ℎ [119867 + 119889

11199051

21199051+119899 minus 1

21198991198671199052]

(4)

Again to take the randomness of 119909 into account andsubstitute all related parameters into (4) with further deriva-tions 119864[TC

2(1199051)] becomes [12]

119864 [TC2(1199051)] = 119870 + 119899119870

1

+ [1198621198751+ 119862119878119864 [119909] 1198751 + 1198621198791198751 (1 minus 119864 [119909])

+ ℎ31198751119892 (1 minus 119864 [119909])] 1199051

+ [ℎ1198751119864 [119909]

2+ℎ1198752

1

2120582(1 minus 119864 [119909])

2(1 minus

1

119899)

+ℎ1198751

2119899(1 minus 119864 [119909])] 119905

2

1

(5)

The Scientific World Journal 5

H

P1 minus d1

T

Time

minus120582

tn

t1 t1t2

I(t)

Figure 3 Inventory level of finished items in a manufacturing run time problem with no breakdown defective rate and discontinuousdelivery policy

23 Integration of the Proposed Run TimeModels withwithoutBreakdown Amachine breakdownmay take place randomlyand it follows a Poisson distribution with mean equal to120573 per year Let 119891(119905) be the probability density function ofrandom time 119905 before a breakdown takes place and 119865(119905)

represents the cumulative density function of 119905 Hence thelong-run expected system costs per unit time 119864[TCU(119905

1)]

are

119864 [TCU (1199051)]

=

int1199051

0119864 [TC

1(1199051)] 119891 (119905) 119889119905 + int

infin

1199051

119864 [TC2(1199051)] 119891 (119905) 119889119905

119864 [T]

(6)

where

119864 [T] = int1199051

0

119864 [1198791015840] 119891 (119905) 119889119905 + int

infin

1199051

119864 [119879] 119891 (119905) 119889119905

=11990511198751(1 minus 119864 [119909])

120582

(7)

From Figures 1 and 3 it can be seen that 1198791015840 and 119879 aredifferent in length (1198791015840 is longer than 119879 since it containsmachine repairing time) and because a breakdown can occurrandomly it is necessary to use the integration (ie equation(7)) to derive the expected cycle length

It is also noted that the time between breakdowns obeysthe exponential distribution with density function 119891(119905) =

120573119890minus120573119905 and cumulative density function 119865(119905) = 1 minus 119890

minus120573119905 Bysubstituting 119864[TC

1(1199051)] 119864[TC

2(1199051)] and 119864[T] into (6) and

solving the integration of the mean time to breakdown in119864[TCU(119905

1)] we obtain

119864 [TCU (1199051)]

=120582

(1 minus 119864 [119909])

sdot (119870 + 119899119870

1)

11990511198751

+ 1205741+12057421199051

2+ [

119872

1198751

+ℎ119892

120573](

1 minus 119890minus1205731199051

1199051

)

minusℎ119892 (119890minus1205731199051) minus

ℎ119892 (1 minus 119864 [119909])

2(1 minus

1

119899) (1 minus 119890

minus1205731199051)

(8)

where

1205741= [119862 + 119862

119878119864 [119909] + 119862

119879(1 minus 119864 [119909]) + ℎ

3119892 (1 minus 119864 [119909])]

(9)

1205742= [

ℎ1198751

120582(1 minus 119864 [119909])

2(1 minus

1

119899) + ℎ119864 [119909] +

ℎ

119899(1 minus 119864 [119909])]

(10)

24 Derivation of the Optimal Production Run Time In orderto derive the optimal production run time 119905lowast

1 we first have

to prove that 119864[TCU(1199051)] is convex Let 120585(119905

1) represent the

following

120585 (1199051) =

2 (119870 + 1198991198701) 120573 + 2 (1 minus 119890

minus1205731199051) 1205744

[1199052

1119875112057321205743+ 1205744(2 + 120573119905

1)] 120573119890minus1205731199051

(11)

6 The Scientific World Journal

Theorem 1 (119864[TCU(1199051)] is convex if 0 lt 119905

1lt 120585(119905

1)) The

second derivative of 119864[119879119862119880(1199051)] with respect to 119905

1is

1198892119864 [119879119862119880 (119905

1)]

11988921199052

1

=120582

(1 minus 119864 [119909])

sdot [2 (119870 + 119899119870

1)

1199053

11198751

minus ℎ119892 [1 minus(1 minus 119864 [119909])

2(1 minus

1

119899)] (120573

2119890minus1205731199051)

+ [119872

1198751

+ℎ119892

120573](

2 (1 minus 119890minus1205731199051)

1199053

1

minus2120573119890minus1205731199051

1199052

1

minus1205732119890minus1205731199051

1199051

)]

(12)

It is noted that because annual demand 120582 gt 0 the firstterm in the right-hand size (RHS) of (12) is positive Hence weobtain

1198892119864 [119879119862119880 (119905

1)]

1198891199051

2gt 0

119894119891 [2 (119870 + 119899119870

1)

1199053

11198751

minus ℎ119892 [1 minus(1 minus 119864 [119909])

2(1 minus

1

119899)] (120573

2119890minus1205731199051)

+ [119872

1198751

+ℎ119892

120573](

2 (1 minus 119890minus1205731199051)

1199053

1

minus2120573119890minus1205731199051

1199052

1

minus1205732119890minus1205731199051

1199051

)]

gt 0

(13)

The RHS of (13) can be further derived as

119894119891 [2 (119870 + 1198991198701) 120573 minus 119905

3

11198751120573ℎ119892

sdot [1 minus(1 minus 119864 [119909])

2(1 minus

1

119899)] (120573

2119890minus1205731199051)

+ (119872120573 + ℎ1198921198751)

times [2 (1 minus 119890minus1205731199051) minus 2119905

1120573119890minus1205731199051 minus 12057321199052

1119890minus1205731199051]]

gt 0

(14)

Let

1205743= ℎ119892 [1 minus

(1 minus 119864 [119909])

2(1 minus

1

119899)]

1205744= (119872120573 + ℎ119892119875

1)

(15)

then (13) can be rewritten as

1198892119864 [119879119862119880 (119905

1)]

1198891199051

2gt 0

119894119891 [2 (119870 + 1198991198701) 120573 + 2 (1 minus 119890

minus1205731199051) 1205744

minus 1199051[1199052

1119875112057321205743+ 1205744(2 + 120573119905

1)] 120573119890minus1205731199051]

gt 0

(16)

or

1198892119864 [119879119862119880 (119905

1)]

1198891199051

2gt 0

119894119891 0 lt 1199051lt

2 (119870 + 1198991198701) 120573 + 2 (1 minus 119890

minus1205731199051) 1205744

[1199052

1119875112057321205743+ 1205744(2 + 120573119905

1)] 120573119890minus1205731199051

= 120585 (1199051)

(17)

If 119864[119879119862119880(1199051)] is a convex function then the minimum

point exists In order to locate the optimal production run time119905lowast

1that minimizes 119864[119879119862119880(119905

1)] we can set the first derivative of

119864[119879119862119880(1199051)] equal to zero and solve 119905lowast

1

119889119864 [119879119862119880 (1199051)]

1198891199051

=120582

(1 minus 119864 [119909])

sdot minus (119870 + 119899119870

1)

1199052

1119875

+1205742

2+ 1205743(120573119890minus1205731199051)

+ [119872

1198751

+ℎ119892

120573](

minus (1 minus 119890minus1205731199051)

1199052

1

+120573119890minus1205731199051

1199051

)

(18)

In the RHS of (18) it can be seen that the first term ispositive so the second term is equal to zero Let

119905lowast

1119880= radic

2 (1205744+ 119870120573 + 119899119870

1120573)

12057421198751120573

(19)

119905lowast

1119871= 119905ℎ119890 119901119900119904119894119905119894V119890 119903119900119900119905 119900119891

minus1205744plusmn radic1205744

2 + 21198751(1205742+ 2120573120574

3) (119870 + 119899119870

1)

1198751(1205742+ 2120573120574

3)

(20)

Theorem 2 (119905lowast1119871

lt 119905lowast

1lt 119905lowast

1119880) Because the proof of 119905lowast

1falls

within the upper and lower bounds we can multiply the secondterm of (18) by (2120573119875

11199052

1) and obtain

(11987511205731205742+ 2119875112057321205743119890minus1205731199051) 1199052

1+ (21205744120573119890minus1205731199051) 1199051

minus2 [120573 (119870 + 1198991198701) + 1205744(1 minus 119890

minus1205731199051)] = 0

(21)

The Scientific World Journal 7

Producerrsquosproduction-shipment system

Extended toProductionunit

Productionunit

Multideliverypolicy

Multideliverypolicy

Optimization OptimizationProducer-retailer integratedproduction-shipment system

Customer

CustomerRetailer storeor sales office

Figure 4 Extension to a producer-retailer integrated production-shipment system

Thus119905lowast

1

= 119905ℎ119890 119901119900119904119894119905119894V119890 119903119900119900119905 119900119891

(minus (21205744120573119890minus1205731199051)

plusmn ((21205744120573119890minus1205731199051)2

minus [4 (11987511205731205742+ 2119875112057321205743119890minus1205731199051)

times [minus2 [120573 (119870 + 1198991198701) + 1205744(1 minus 119890

minus1205731199051)]]])

12

)

times (2 (12057421198751120573 + 2119875

112057321205743119890minus1205731199051))minus1

(22)

Equation (21) can be rearranged as

2 [1198751120573212057431199052

1+ 12057441205731199051+ 1205744] (119890minus1205731199051)

= 2 [120573 (119870 + 1198991198701) + 1205744] minus (119875

112057312057421199052

1)

(23)

or

119890minus1205731199051 =

2 [120573 (119870 + 1198991198701) + 1205744] minus (119875

112057312057421199052

1)

2 [1198751120573212057431199052

1+ 12057441205731199051+ 1205744]

(24)

where 119890minus1205731199051 is the complement of the cumulative density

function 119865(1199051) = 1 minus 119890

minus1205731199051 As 0 le 119865(119905

1) le 1 0 le 119890

minus1205731199051 le 1

Let 119890minus1205731199051 = 0 and 119890minus1205731199051 = 1 be the upper and lower bounds for119890minus1205731199051 respectively By substituting them into (22) we obtain

119905lowast

1119880= radic

2 [120573 (119870 + 1198991198701) + 1205744]

12057421198751120573

119905lowast

1119871= 119905ℎ119890 119901119900119904119894119905119894V119890 119903119900119900119905 119900119891

minus1205744plusmn radic1205744

2 + 21198751(1205742+ 2120573120574

3) (119870 + 119899119870

1)

1198751(1205742+ 2120573120574

3)

(25)

and 119905lowast1119871lt 119905lowast

1lt 119905lowast

1119880

It is noted that although the optimal production run time119905lowast

1cannot be presented in a closed form it does fall within the

bounds 119905lowast1can be locatedwith the use of a proposed recursive

searching algorithm Let

120596 (1199051) = 119890minus1205731199051 =

2 [120573 (119870 + 1198991198701) + 1205744] minus (119875

112057312057421199052

1)

2 [1198751120573212057431199052

1+ 12057441205731199051+ 1205744]

there4 0 le 120596 (1199051) le 1

(26)

In order to locate 119905lowast1 we can use the following recursive

searching algorithm

(1) Let 120596(1199051) = 0 and 120596(119905

1) = 1 initially and calculate

the upper and lower bounds for 119905lowast1 respectively (ie

to obtain the initial values of [119905lowast1119871 119905lowast

1119880])

(2) Substitute the current values of [119905lowast1119871 119905lowast

1119880] into 119890minus1205731199051 and

compute the new bounds expressed as 120596119871and 120596

119880for

119890minus1205731199051 Hence 120596

119871lt 120596(1199051) lt 120596119880

(3) Let 120596(1199051) = 120596119871and 120596(119905

1) = 120596119880and update the upper

and lower bounds for 119905lowast1 respectively (ie to obtain

the new values of [119905lowast1119871 119905lowast

1119880])

(4) Repeat steps (2) and (3) until there is no signifi-cant difference between 119905lowast

1119871and 119905lowast1119880

(or there is nosignificant difference in terms of their effects on119864[TCU(119905lowast

1)])

(5) Stop 119905lowast1is found

3 Extension to a Producer-Retailer IntegratedEPQ-Based System (Model 2)

31 Enhanced Model Description and Formulation In thissection we further extend the scope of the problem to incor-porate the retailerrsquos stock holding cost into our study Thenew model can be considered a producer-retailer integratedsystem because in the present-day manufacturing sectorsome producers of consumer goods may own and operateretail stores or regional sales offices to promote and sell theirend products to customers (see Figure 4) With the intentionof addressing such a real-life intrasupply chain situation thesecond model of this study incorporates the retailerrsquos stockholding cost into the first model and investigates its effect onthe optimal production run time decision

8 The Scientific World Journal

Dminus 120582tn = I

D

I

t1 + tr tn

Time

t1 + tr

Ic(t)

t9984002

T998400

Figure 5 Inventory level of finished products on the retailerrsquos side in the proposed manufacturing run time problem with breakdown

In the proposed study the retailerrsquos stock holding posi-tions are illustrated in Figure 5

Extra parameters used in this enhanced model includethe following

ℎ2 holding cost per product stored on the retailerrsquos

side

119868119888(119905) on-hand inventory levels in units on the retail-

errsquos side end at time 119905

119863 number of finished products (a fixed quantity)transported to the retail store per shipment

119868 number of left-over products in 119905119899after satisfying

the demand in 119905119899

TC3(1199051) total production-inventory-delivery costs

per cycle of this enhanced model (in the case of abreakdown)

TC4(1199051) total production-inventory-delivery costs

per cycle of this enhanced model (in the case of nobreakdown)

119864[TC3(1199051)] the expected production-inventory-

delivery costs per cycle of this enhanced model (inthe case of a breakdown)

119864[TC4(1199051)] the expected production-inventory-

delivery costs per cycle of this enhanced model (inthe case of no breakdown)

119864[TCU119890(1199051)] the long-run expected production-

inventory-delivery costs per unit time in thisenhanced model whether or not a breakdown takesplace

Since the demand on the retailerrsquos side in time interval119905119899is 120582119905119899 after satisfying the demand the number of left-over

items (see Figure 5) in each 119905119899is

119868 = 119863 minus 120582119905119899 (27)

Total inventory holding costs on retailerrsquos side with andwithout breakdown are shown respectively in

ℎ2[119899(119863 minus 119868)

2119905119899+119899 (119899 + 1)

2119868119905119899+119899119868

2(1199051+ 119905119903)] (28)

ℎ2[119899(119863 minus 119868)

2119905119899+119899 (119899 + 1)

2119868119905119899+119899119868

2(1199051)] (29)

To incorporate the retailerrsquos holding costs into the originalmodels with and without breakdown respectively we obtain

TC3(1199051) = TC

1(1199051)

+ ℎ2[119899(119863 minus 119868)

2119905119899+119899 (119899 + 1)

2119868119905119899+119899119868

2(1199051+ 119905119903)]

TC4(1199051)

= TC2(1199051) + ℎ2[119899(119863 minus 119868)

2119905119899+119899 (119899 + 1)

2119868119905119899+119899119868

2(1199051)]

(30)

To take the randomness of defective rate 119909 into accountand substitute all related variables into (30) with furtherderivations 119864[TC

3(1199051)] and 119864[TC

4(1199051)] can be obtained as

follows

119864 [TC3(1199051)] = 119864 [TC

1(1199051)] + ℎ

2(1198751119892 (1 minus 119864 [119909])

2)

sdot (1 minus1

119899) 1199051+ ℎ2(1198751(1 minus 119864 [119909])

2)

sdot [1198751(1 minus 119864 [119909])

120582119899+ (1 minus

1

119899)] 1199052

1

119864 [TC4(1199051)] = 119864 [TC

2(1199051)] + ℎ

2(1198751(1 minus 119864 [119909])

2)

sdot [1198751(1 minus 119864 [119909])

120582119899+ (1 minus

1

119899)] 1199052

1

(31)

The Scientific World Journal 9

32 Integration of Enhanced Model withwithout BreakdownThe mean time to breakdowns obeys the exponential distri-bution with 119891(119905) = 120573119890minus120573119905 Therefore 119864[TCU

2(1199051)] is

119864 [TCU2(1199051)]

=

int1199051

0119864 [TC

3(1199051)] 119891 (119905) 119889119905 + int

infin

1199051

119864 [TC4(1199051)] 119891 (119905) 119889119905

119864 [T]

(32)

Substituting 119864[TCU3(1199051)] 119864[TC

4(1199051)] and 119864[T] into (32)

and resolving 119864[TCU2(1199051)] we obtain

119864 [TCU2(1199051)]

=120582

(1 minus 119864 [119909])

sdot (119870 + 119899119870

1)

11990511198751

+ 1205741+12057421199051

2+ 12057451199051

+ [119872

1198751

+ℎ119892

120573](

1 minus 119890minus1205731199051

1199051

) minus ℎ119892 (119890minus1205731199051)

minus (ℎ minus ℎ2) [119892 (1 minus 119864 [119909])

2(1 minus

1

119899)] (1 minus 119890

minus1205731199051)

(33)

where

1205745=ℎ2(1 minus 119864 [119909])

2[1198751(1 minus 119864 [119909])

120582119899+ (1 minus

1

119899)] (34)

33 Determining the Optimal Run Time Let 120595(1199051) stand for

the following

120595 (1199051) =

2 (119870 + 1198991198701) 120573 + 2120574

4(1 minus 119890

minus1205731199051)

[1199052

1119875112057321205746+ 1205744(2 + 120573119905

1)] 120573119890minus1205731199051

(35)

Theorem 3 (119864[TCU2(1199051)] is convex if 0 lt 119905

1lt 120595(119905

1)) The

second derivative of 119864[1198791198621198802(1199051)] with respect to 119905

1is

1198892119864 [119879119862119880

2(1199051)]

11988921199052

1

=120582

(1 minus 119864 [119909])[2 (119870 + 119899119870

1)

1199053

11198751

+ (ℎ minus ℎ2)119892 (1 minus 119864 [119909])

2

sdot (1 minus1

119899) (1205732119890minus1205731199051) minus ℎ119892 (120573

2119890minus1205731199051)

+ [119872

1198751

+ℎ119892

120573](

2 (1 minus 119890minus1205731199051)

1199053

1

minus2120573119890minus1205731199051

1199052

1

minus1205732119890minus1205731199051

1199051

)]

(36)

Since annual demand 120582 gt 0 the first term in the RHS of(36) is positive and

119894119891 [2 (119870 + 119899119870

1)

1199053

11198751

+ (ℎ minus ℎ2) [119892 (1 minus 119864 [119909])

2(1 minus

1

119899)]

sdot (1205732119890minus1205731199051) minus ℎ119892 (120573

2119890minus1205731199051) + [

119872

1198751

+ℎ119892

120573]

sdot(

2 (1 minus 119890minus1205731199051)

1199053

1

minus2120573119890minus1205731199051

1199052

1

minus1205732119890minus1205731199051

1199051

)] gt 0

119905ℎ1198901198991198892119864 [119879119862119880

2(1199051)]

1198891199051

2gt 0

(37)

With further derivations the left-hand side (LHS) of (37)becomes

119894119891 [2 (119870 + 1198991198701) 120573 minus 119905

3

11198751120573 (ℎ minus ℎ

2) [119892 (1 minus 119864 [119909])

2(1 minus

1

119899)]

sdot (1205732119890minus1205731199051) minus 1199053

11198751120573ℎ119892 (120573

2119890minus1205731199051)

+ (119872120573 + ℎ1198921198751) [2 (1 minus 119890

minus1205731199051) minus 2119905

1120573119890minus1205731199051 minus 12057321199052

1119890minus1205731199051] ]

gt 0

(38)

Let

1205746= (ℎ minus ℎ

2) [119892 (1 minus 119864 [119909])

2(1 minus

1

119899)] + ℎ119892 (39)

then (37) becomes

119894119891 [2 (119870 + 1198991198701) 120573 + 2120574

4(1 minus 119890

minus1205731199051)

minus1199051[1199052

1119875112057321205746+ 1205744(2 + 120573119905

1)] 120573119890minus1205731199051] gt 0

119905ℎ1198901198991198892119864 [119879119862119880

2(1199051)]

1198891199051

2gt 0

(40)

or

1198892119864 [119879119862119880

2(1199051)]

1198891199051

2gt 0

119894119891 0 lt 1199051lt

2 (119870 + 1198991198701) 120573 + 2120574

4(1 minus 119890

minus1205731199051)

[1199052

1119875112057321205746+ 1205744(2 + 120573119905

1)] 120573119890minus1205731199051

= 120595 (1199051)

(41)

10 The Scientific World Journal

Once 119864[1198791198621198802(1199051)] is proven to be convex the optimal

run time 119905lowast1can be solved by setting the first derivative of

119864[1198791198621198802(1199051)] = 0

119889119864 [1198791198621198802(1199051)]

1198891199051

=120582

(1 minus 119864 [119909])sdot minus (119870 + 119899119870

1)

1199052

1119875

+ (1205742

2+ 1205745) + 1205746(120573119890minus1205731199051)

+ [119872

1198751

+ℎ119892

120573](

minus (1 minus 119890minus1205731199051)

1199052

1

+120573119890minus1205731199051

1199051

)

= 0

(42)

It can be seen that the first term in the RHS of (42) ispositive so the second term is equal to zero In order to findthe bounds for 119905lowast

1 let

119905lowast

1119880= radic

2 [120573 (119870 + 1198991198701) + 1205744]

1198751120573 (1205742+ 21205745)

(43)

119905lowast

1119871= 119905ℎ119890 119901119900119904119894119905119894V119890 119903119900119900119905 119900119891

minus1205744plusmn radic1205742

4+ 21198751(119870 + 119899119870

1) (1205742+ 21205745+ 21205746120573)

1198751(1205742+ 21205745+ 21205746120573)

(44)

Theorem4 (119905lowast1119871lt 119905lowast

1lt 119905lowast

1119880) For the proof ofTheorem 4 please

refer to the proof for Theorem 2 in Section 2

Once we are certain that 119905lowast1falls within the aforemen-

tioned upper and lower bounds in order to find 119905lowast1 we can

first multiply the second term of (42) by (211987511199052

1120573) and obtain

the following

[1198751120573 (1205742+ 21205745) + 2119875

112057321205746119890minus1205731199051] 1199052

1+ (21205744120573119890minus1205731199051) 1199051

minus2 [120573 (119870 + 1198991198701) + 1205744(1 minus 119890

minus1205731199051)]

= 0

(45)

Equation (45) can be rearranged as

119890minus1205731199051 =

2 [120573 (119870 + 1198991198701) + 1205744] minus [119875

1120573 (1205742+ 21205745)] 1199052

1

2 [1198751120573212057461199052

1+ 1205744(1 + 120573119905

1)]

(46)

where 119890minus1205731199051 is the complement of the cumulative densityfunction 119865(119905

1) = 1minus119890

minus1205731199051 As 0 le 119865(119905

1) le 1 0 le 119890minus1205731199051 le 1 Let

119890minus1205731199051 = 0 and 119890minus1205731199051 = 1 be the initial upper and lower bounds

of 119890minus1205731199051 respectively Then by using the proposed recursivesearching algorithm given at the end of Section 2 we can findthe optimal production run time 119905lowast

1

4 Numerical Example

In order to relieve the comparison efforts for readers thissection adopts the same numerical example as in [9] For

0225 0265 0305 0345 0385 0425 0465 0505

E[T

CU(t

1)]

t1

tlowast1L = 03411 t

lowast1 = 03748

tlowast1U = 05183

$10600

$10800

$11000

$11200

$11400

$11600

$11006

$11800

Figure 6 The behavior of 119864[TCU(1199051)] in connection with the

production run time 1199051in the proposed model 1

a demonstration of the proposed EPQ-based model 1 thefollowing system parameters are used

1198751 production rate 10000 products per year120582 demand rate 4000 products per year119909 random scrap rate which follows uniformly distribu-

tion over the interval [0 02]120573 Poisson breakdown rate 05 average times per year119892 constant machine repair time 119905

119903 0018 year per repair

119872 machine repair cost $500 for each breakdown119870 setup cost $450 per production run119862 manufacturing cost $2 per item119862119878 disposal cost $03 per scrap itemℎ holding cost $06 per item per unit time1198701 fixed delivery cost $90 per shipment119899 number of deliveries 4 per cycle

119862119879 variable delivery cost $0001 per item

First we use both upper and lower bounds of 119905lowast1to

test for the convexity of 119864[TCU(1199051)] (see Theorem 1) The

computation results of (19) (20) and (11) indicate that 119905lowast1119880

=05183 lt 120585(119905

lowast

1119880) = 28723 and 119905lowast

1119871= 03411 lt 120585(119905

lowast

1119871) = 26261

Hence [TCU(1199051)] is convex (Figure 6)

In order to find the optimal 119905lowast1 we first substitute the

upper and lower bounds of 119905lowast1in (8) and obtain 119864[TCU(119905lowast

1119880)]

= $1160163 and 119864[TCU(119905lowast1119871)] = $1101450 respectively

Because the optimal run time 119905lowast1falls within the interval

of [119905lowast1119871 119905lowast

1119880] we apply the proposed recursive searching

algorithm stated at the end of Section 2 and find 119905lowast1= 03748

years Accordingly the optimal expected system costs perunit time 119864[TCU(119905lowast

1)] = $1100641 (Figure 6) Table 1 shows

the step-by-step iterations of the algorithmIn this specific studied model we focus on incorporating

a discontinuous multidelivery policy into a prior work [9]and consider a fixed transportation cost associated witheach delivery Applying the research result we can easilyinvestigate the effects of different fixed transportation costs

The Scientific World Journal 11

Table 1 Iterations of the recursive searching algorithm for locating 119905lowast1

120573 Step 119905lowast

1119880120596119880= 119890minus1205731199051119880 119905

lowast

1119871120596119871= 119890minus1205731199051119871

Differencebetween 119905lowast

1119880and

119905lowast

1119871

[119880]

119864[TCU(119905lowast1119880)]

[119871]

119864[TCU(119905lowast1119871)]

Differencebetween [119880]

and [119871]05 Initial 000000 100000 $1160163 $1101450 $58713

1st 05183 07717 03411 08432 01772 $1110330 $1101450 $88802nd 03857 08246 03721 08302 00136 $1100716 $1100646 $0703rd 03756 08287 03746 08292 00010 $1100642 $1100641 $0014th 03749 08291 03748 08291 00001 $1100641 $1100641 $00045th 03748 08291 03748 08291 000000 $1100641 $1100641 $0000

Table 2 Variations of the fixed delivery cost1198701effects on the optimal production run time 119905lowast

1

1198701119870 005 02 04 06 08 1 12 14 16 18 2 22

1198701

$225 $90 $180 $270 $360 $450 $540 $630 $720 $810 $900 $990119864[TCU(119905lowast

1)] $10721 $11067 $11448 $11773 $12062 $12325 $12567 $12793 $13006 $13208 $13399 $13583

119905lowast

103116 03816 04586 05244 05828 06359 06848 07305 07735 08142 08529 08900

Table 3 Variations of the unit retailerrsquos holding cost ℎ2and their effects on 119864[TCU(119905lowast

1)]

ℎ2ℎ 05 075 1 125 15 175 2 225 25 275 3 325

ℎ2

03 045 06 075 09 105 12 135 15 165 18 195119864 [TCU (119905lowast

1)] $11282 $11407 $11526 $11638 $11746 $11850 $11949 $12045 $12138 $12229 $12316 $12401

119905lowast

103256 03074 02918 02785 02668 02564 02472 02389 02314 02246 02183 02125

1198701on the optimal system cost 119864[TCU(119905lowast

1)] and on the

optimal production run time 119905lowast1(see Table 2) It can be seen

from Table 2 that as the ratio of1198701119870 increases the expected

system costs per unit time119864[TCU(119905lowast1119871)] increase significantly

It is also noted that as 1198701increases optimal production run

time 119905lowast1also increases significantly

41 Numerical Example for the Producer-Retailer IntegratedEPQ System (Model 2) In order to demonstrate the researchresult of the producer-retailer integrated EPQ-based modelan additional system variable ℎ

2= $150 per item stored at the

retailerrsquos side is includedAgain one can use the upper and lower bounds of 119905lowast

1

(equations (43) and (44)) to test for convexity of 119864[TCU(1199051)]

(Theorem 3 and equation (35)) The results reveal that 119905lowast1119880

=03213 lt 120595(119905lowast

1119880) = 26462 and 119905lowast

1119871= 02186 lt 120595(119905lowast

1119871) = 24876

Therefore the expected cost [TCU(1199051)] is convex

Next by applying the proposed recursive searching algo-rithm we can calculate that the optimal run time 119905lowast

1=

02314 years and the optimal 119864[TCU(119905lowast1)] = $1213849 It is

noted that the computation time for reaching the optimal 119905lowast1

solution is 21 seconds (usingExcel software in a desktop com-puter Intel CPU G850 with 294GB RAM and 289GHz)

Figure 7 illustrates the behavior of 119864[TCU(1199051)] with

regard to production run time It is noted that without theresearch result from the second model the management ofsuch a producer-retailer integrated system would probablyuse 1199051= 03748 years (from the result of model 1) for their

run time decision Further analysis (see Figure 7) shows cost

0130 0170 0210 0250 0290 0330 0370 0410

t1

E[T

CU(t

1)]

11600

11800

12000

12200

12400

12600

12800

tlowast1 = 02314

t1 = 0374812138

(from model 1)

E[TCU( t1)] = 12489

$$

$

$

$

$

$

$

$

Figure 7 The behavior of 119864[TCU(1199051)] with respect to production

run time 1199051in the proposed model 2

savings of $351 (or 29 over the total system costs) simply byapplying our research result

The effects of the unit retailerrsquos holding cost ℎ2on the

expected system cost 119864[TCU(119905lowast1)] and on the optimal run

time 119905lowast1are shown in Table 3 respectively

It can be seen that as ℎ2or the ratio of ℎ

2ℎ increases

the expected cost 119864[TCU(119905lowast1)] increases but the optimal

production run time 119905lowast1decreases In decision-making these

sensitivity analyses results can provide the management ofa producer-retailer integrated system with valuable informa-tion and insights into the effects of various stock holding costsin different retailersrsquo locations

12 The Scientific World Journal

5 Concluding Remarks

Two exact models for an extended EPQ-based problem witha discontinuous delivery policy scrap rate and randombreakdown are developed in this study They specificallyaddress different real-life situations in production end-item delivery and intrasupply chains such as a producer-retailer integrated systemMathematicalmodeling alongwithoptimization techniques is used to determine the optimalproduction run times that minimize the expected systemcosts per unit time Without in-depth investigations onthese separate models the optimal production run time andother important information related to the systemparameterscannot be revealed The proposed real-life EPQ models withrandom machine breakdown discontinuous product distri-bution policies and quality assurance must be specificallystudied in order to (1) obtain the joint effects of breakdowndiscontinuous distribution policies and quality assurance onthe optimal production run time (2) get to know the effectsof different policy and scope of supply chains managementon the optimal run time and overall system costs and (3)gain the insight with regard to various systemrsquos parametersof all particular EPQ-based models Since little attention hasbeen paid to the investigation of joint effects of these practicalproduction situations on the optimal run time this researchis intended to bridge the gap An interesting area for futurestudy is the examination of the effect of variable productionrates on these models

Appendix

Derivations of (3) are as followsRecall (2) as follows

TC1(1199051) = 119862 (119875

11199051) + 119870 + 119862

119878(11990511198751119909) +119872 + 119899119870

1

+ 119862119879[11990511198751] + ℎ3(120582119905119903) 1198791015840

+ ℎ[1198671015840+ 11988911199051

21199051+ (1198671015840

1+ 1198891119905) 119905119903+119899 minus 1

211989911986710158401199051015840

2]

(A1)

Substituting all related system parameters into (2) (pleaserefer to the basic formulations and solution process in [9])the TC

1(1199051) can be obtained as

TC1(1199051) = 119870 +119872

+[1198621198751+ 1198621198781198751119909 + 119862

1198791198751(1 minus 119909)+ ℎ

31198751119892 (1 minus 119909)] 119905

1

+ 1198991198701+ ℎ119901119892119905 minus[

ℎ1198751119892 (1 minus 119909)

2minusℎ1198751119892 (1 minus 119909)

2119899] 1199051

+ 1199052

1[ℎ1198751

2+ℎ1198752

1

2120582(1 minus 119909)

2minusℎ1198751

2(1 minus 119909)

minusℎ1198752

1

2120582119899(1 minus 119909)

2+ℎ1198751

2119899(1 minus 119909)]

(A2)

To take the randomness of 119909 into account by using theexpected values of 119909 with further derivations119864[TC

1(1199051)] can

be derived as follows (ie equation (3))119864 [TC

1(1199051)]

= 119870 + 1198991198701+119872 + ℎ119905119875

1119892

+ [1198621198751+ 1198621198781198751119864 [119909] + 1198621198791198751 (1 minus 119864 [119909])

+ ℎ31198751119892 (1 minus 119864 [119909]) minus

ℎ1198751119892 (1 minus 119864 [119909])

2(1 minus

1

119899)] sdot 1199051

+ [ℎ1198751119864 [119909]

2+ℎ1198752

1

2120582(1 minus 119864 [119909])

2(1 minus

1

119899)

+ℎ1198751

2119899(1 minus 119864 [119909])] 119905

2

1

(A3)

Conflict of Interests

The authors of the paper declare that there is no conflict ofinterests regarding the publication of this paper

Acknowledgment

The authors greatly appreciate the National Science Council(NSC) of Taiwan for supporting this research underGrant noNSC 102-2410-H-324-005

References

[1] E W Taft ldquoThemost economical production lotrdquo Iron Age vol101 pp 1410ndash1412 1918

[2] G Hadley and T M Whitin ldquoAn optimal final inventorymodelrdquoManagement Science vol 7 pp 179ndash183 1961

[3] E A Silver D F Pyke and R Peterson Inventory Managementand Production Planning and Scheduling John Wiley amp SonsNew York NY USA 1998

[4] S Nahmias Production amp Operations Analysis McGraw-HillNew York NY USA 2009

[5] MWidmer and P Solot ldquoDo not forget the breakdowns and themaintenance operations in FMSdesign problemsrdquo InternationalJournal of Production Research vol 28 pp 421ndash430 1990

[6] K-Y C Yu and D L Bricker ldquoAnalysis of a markov chainmodel of a multistage manufacturing system with inspectionrejection and reworkrdquo IIE Transactions vol 25 no 1 pp 109ndash112 1993

[7] H Groenevelt L Pintelon and A Seidmann ldquoProduction lotsizing with machine breakdownsrdquoManagement Science vol 38no 1 pp 104ndash123 1992

[8] G A Widyadana and H M Wee ldquoOptimal deterioratingitems production inventory models with random machinebreakdown and stochastic repair timerdquo Applied MathematicalModelling vol 35 no 7 pp 3495ndash3508 2011

[9] S W Chiu C-L Chou and W-K Wu ldquoOptimizing replenish-ment policy in an EPQ-based inventory model with noncon-forming items and breakdownrdquo EconomicModelling vol 35 pp330ndash337 2013

The Scientific World Journal 13

[10] A M Zargar ldquoEffect of rework strategies on cycle timerdquoComputers amp Industrial Engineering vol 29 no 1ndash4 pp 239ndash243 1995

[11] P Biswas and B R Sarker ldquoOptimal batch quantity modelsfor a lean production system with in-cycle rework and scraprdquoInternational Journal of Production Research vol 46 no 23 pp6585ndash6610 2008

[12] S W Chiu H-D Lin C-B Cheng and C-L Chung ldquoOptimalproduction-shipment decisions for the finite production ratemodel with scraprdquo International Journal for Engineering Mod-elling vol 22 no 1ndash4 pp 25ndash34 2009

[13] Y-S P Chiu K-K Chen and C-K Ting ldquoReplenishment runtime problem with machine breakdown and failure in reworkrdquoExpert Systems with Applications vol 39 no 1 pp 1291ndash12972012

[14] H-D Lin F-Y Pai and S W Chiu ldquoA note on ldquointra-supply chain system with multiple sales locations and qualityassurancerdquordquo Expert Systems with Applications vol 40 no 11 pp4730ndash4732 2013

[15] B Kamsu-Foguem F Rigal and FMauget ldquoMining associationrules for the quality improvement of the production processrdquoExpert Systems with Applications vol 40 no 4 pp 1034ndash10452013

[16] Y Li H Guo L Wang and J Fu ldquoA hybrid genetic-simulatedannealing algorithm for the location-inventory- routing prob-lem considering returns under E-supply chain environmentrdquoThe Scientific World Journal vol 2013 Article ID 125893 10pages 2013

[17] Y-S P Chiu K-K Chen F-T Cheng and C-K Ting ldquoReex-amination of ldquocombining an alternative multi-delivery policyinto economic production lot size problemwith partial reworkrdquousing an alternative approachrdquo Journal of Applied Research andTechnology vol 11 no 3 pp 317ndash323 2013

[18] L B Schwarz B L Deuermeyer and R D Badinelli ldquoFill-rateoptimization in a one-warehouse 119873-identical retailer distribu-tion systemrdquo Management Science vol 31 no 4 pp 488ndash4981985

[19] R A Sarker and L R Khan ldquoOptimal batch size for a pro-duction system operating under periodic delivery policyrdquoCom-puters amp Industrial Engineering vol 37 no 4 pp 711ndash730 1999

[20] N Comez K E Stecke and M Cakanyildirim ldquoMultiple in-cycle transshipments with positive delivery timesrdquo Productionand Operations Management vol 21 no 2 pp 378ndash395 2012

[21] S W Chiu F-Y Pai and W K Wu ldquoAlternative approach todetermine the common cycle time for a multi-item productionsystem with discontinuous deliveries and failure in reworkrdquoEconomic Modelling vol 35 pp 593ndash596 2013

[22] Y-S P Chiu C-C Huang M-F Wu and H-H Chang ldquoJointdetermination of rotation cycle time and number of shipmentsfor a multi-item EPQ model with random defective raterdquoEconomic Modelling vol 35 pp 112ndash117 2013

[23] M A Hoque ldquoSynchronization in the single-manufacturermulti-buyer integrated inventory supply chainrdquo European Jour-nal of Operational Research vol 188 no 3 pp 811ndash825 2008

[24] S W Chiu L-W Lin K-K Chen and C-L Chou ldquoDetermin-ing production-shipment policy for a vendor-buyer integratedsystem with rework and an amending multi-delivery schedulerdquoEconomic Modelling vol 33 pp 668ndash675 2013

[25] M Cedillo-Campos and C Sanchez-Ramırez ldquoDynamic self-assessment of supply chains performance an emerging marketapproachrdquo Journal of Applied Research and Technology vol 11no 3 pp 338ndash347 2013

[26] Y-S P Chiu H-D Lin F-T Cheng and M-H Hwang ldquoOpti-mal common cycle time for a multi-item production systemwith discontinuous delivery policy and failure in reworkrdquoJournal of Scientific and Industrial Research vol 72 no 7 pp435ndash440 2013

[27] HHishamuddin R A Sarker andD Essam ldquoA recoverymech-anism for a two echelon supply chain system under supplydisruptionrdquo Economic Modelling vol 38 pp 555ndash563 2014

[28] M Murugan and V Selladurai ldquoProductivity improvement inmanufacturing submersible pump diffuser housing using leanmanufacturing systemrdquo Journal of Engineering Research vol 2no 1 pp 164ndash182 2014

[29] L Wang H Qu S Liu and C-X Dun ldquoModeling and opti-mization of the multiobjective stochastic joint replenishmentand delivery problem under supply chain environmentrdquo TheScientific World Journal vol 2013 Article ID 916057 11 pages2013

Submit your manuscripts athttpwwwhindawicom

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Stochastic AnalysisInternational Journal of

2 The Scientific World Journal

control structure is optimal among all stationary policies andoffered the exact optimal and closed form approximate lotsizing formulas and bounds on average cost per unit timefor the approximations Widyadana and Wee [8] developeddeteriorating items production inventory models with ran-dom breakdown and stochastic machine repair time Therepair time is assumed to be independent of the breakdownrate They applied the classical optimization technique tothe problem and derived an optimal solution Through anumerical example and sensitivity analysis they showed thatthe production and demand rates are the most sensitiveparameters to the optimal uptime and the demand rate is themost sensitive variable to the system costs for the stochasticmodel with exponential distribution repair time Chiu etal [9] determined the optimal replenishment run time foran EPQ-based inventory model with nonconforming itemsand breakdown Their model assumes that after a Poissondistributed breakdownoccurs themachine goes under repairinstantly and the production of the interrupted lot resumesimmediately when the machine is fixed and restored Thesystem also considers a uniformly distributed scrap rateassociated with the production process A mathematicalmodel along with a recursive searching algorithm is usedin their study to derive the optimal replenishment policythat minimizes the total system costs A numerical examplewas provided to demonstrate the practical application andbetter cost efficiency of the proposed policy compared toa breakdown that occurs under a no-resumption policyAdditional studies relating to the issues of product qualityunreliable production equipment and their consequencequality assurance can be found in [10ndash17]

Unlike the assumption of a continuous inventory issuingpolicy of the traditional EPQ model in real supply chainenvironments the discontinuousmultidelivery policy is oftenused when finished products need to be transported toretailers or customers outside the production units Schwarzet al [18] determined the fill-rate of a one-warehouseN-identical retailer distribution system An approximationmodel was adopted from a prior work to maximize thesystem fill-rate subjected to a constraint on safety stockThe properties of the fill-rate policy were used to providemanagerial insights into system optimization Sarker andKhan [19] examined a manufacturing system that procuresraw materials in a lot from suppliers and processes theminto the finished products that are subsequently shipped tooutside customers at fixed points in time The system costfunction for the model was formulated by including bothraw materials and finished products A solution procedurewas developed to determine an optimal ordering policy forthe procurement of raw materials and the production batchsize that minimizes the total system costs Comez et al [20]considered a centralized inventory sharing system of tworetailers that are replenished periodicallyThey assumed thatbetween two replenishments a unit can be transshipped to astocked-out retailer from the other retailer Whenever thereis an absence of transshipments the backorder costs areincurred until the next replenishment The objective of theirstudy is to minimize the long-run average costs comprisingthe replenishment holding backorder and transshipment

costs They discussed the challenges associated with positivereplenishment time and developed upper and lower boundsof average costs in such situations Other studies [21ndash29]focused on various aspects of periodic or multiple deliveryissues in the vendor-buyer integrated supply chains

With the purpose of addressing the real-life production-shipment situation this study extends a recent work [9] byincorporating a multiple delivery policy into their modelto replace the continuous policy and investigates the effecton the optimal run time decision for this specific EPQmodel Next we further expand the scope of the problemto combine the retailerrsquos stock holding cost into our studyThis enhanced EPQ-based model can be used to reflect thesituation found in contemporary manufacturing firms inwhich finished products are delivered to the producerrsquos ownretail stores and stocked there for sale The objectives are todetermine the optimal run times that minimize the expectedtotal system costs comprising costs incurred in productionunits transportation and retail stores for both models Aslittle attention has been paid to this specific area the presentstudy is intended to fill this gap

2 Statement and Optimization ofProposed Model 1

In real supply chain environments the discontinuous mul-tidelivery policy is often used when finished products needto be transported to retail stores or customers outside theproduction units To explicitly address this realistic situationthe first proposedmodel in this study incorporates a multipledelivery policy into an EPQ-based inventory model withscrap items and breakdown [9] to replace the continuousissuing policy and investigates the effect on the optimalmanufacturing run time decision

Summary of assumptions (features) considered in theproposed multi-item EPQ-based model are as follows (1) arandom machine breakdown rate (2) a random scrap rate inproduction and (3) a discontinuous multidelivery policy forfinished products The details of the proposed model can bedescribed as follows Suppose a product can bemanufacturedat an annual rate 119875

1and its demand is 120582 units per year All

items produced must pass a quality conformation check andthe unit screening cost is included in the unit manufacturingcost 119862 A random 119909 proportion of the products producedis defective and will be scrapped at the end of the regularproduction process Hence the production rate of scrapsis 1198891and 119889

1= 1198751119909 Under regular operations (ie to

avoid a stock-out situation) (1198751minus 1198891minus 120582) gt 0 must be

satisfied Upon the completion of the production processthe acceptable quality (finished) products are transported tothe outside retail store or customer under a discontinuousmultidelivery policy while fixed quantity 119899 installments ofthe finished items are shipped to retail store at a fixed intervalof time in 1199051015840

2(Figure 1) The proposed model assumes that

during the production uptime a Poisson distributedmachinebreakdownmay occur and an abortresume (AR) inventorycontrol policy is employed when a breakdown happensUnder such an AR policy the machine goes under repair

The Scientific World Journal 3

I(t)

H998400

H9984001

P1 minus d1

P1 minus d1

t tr

tn

t1 minus t

T998400

t9984002

Time

Figure 1 Inventory level of finished items in the proposed manufacturing run time problem

immediately and a constant repair time is assumed Uponthe completion of the repair the interrupted lot is instantlyresumed (Figure 1)

Additional cost-related parameters used in this studyare the machine repairing cost 119872 setup cost per cycle 119870holding cost per item at the producerrsquos side ℎ disposal costper scrapped item 119862

119878 fixed delivery cost per shipment 119870

1

variable delivery cost per item119862119879 unit holding cost for safety

stock at the producerrsquos side ℎ3 and holding cost per item at

the retailerrsquos side ℎ2 Other notations used in the modeling

and analysis also include the following

119905 production time before a random machine break-down takes place1198671015840

1 on-hand inventory level in units when a random

machine breakdown takes place120573 number of machine breakdowns per unit time (ieyear) assumed to be a random variable that followsthe Poisson distribution119905119903 machine repair time

1199051 production uptime the decision variable of the

proposed manufacturing run time model1198671015840 maximum on-hand inventory level in units when

the regular production process ends (in the case of abreakdown)1199051015840

2 time required to deliver all finished items produced

in a cycle (in the case of a breakdown)1198791015840 production cycle length (in the case of a break-

down)119876 lot size for each production cycleTC1(1199051) total production-inventory-delivery costs

per cycle (in the case of a breakdown)

119864[TC1(1199051)] the expected production-inventory-

delivery costs per cycle (in the case of a breakdown)1199052 time required to deliver all finished items produced

(in the case of no breakdown)119867 on-hand inventory level in units when the regularproduction process ends (in the case of no break-down)119879 cycle length (in the case of no breakdown)119868(119905) on-hand inventory level of finished items at time119905119868119904(119905) on-hand inventory level of scrap items at time 119905

TC2(1199051) total production-inventory-delivery costs

per cycle (in the case of no breakdown)119864[TC

2(1199051)] the expected production-inventory-

delivery costs per cycle (in the case of no breakdown)TCU(119905

1) total production-inventory-delivery costs

per unit timewhether or not a breakdown takes place119864[TCU(119905

1)] the long-run expected production-

inventory-delivery costs per unit time whether or nota breakdown takes placeT the cycle length whether or not a machine break-down takes place

Since a machine breakdown may randomly take place atproduction uptime 119905

1 the following two distinct cases must

be examined

21 Case 1 A Random Machine Breakdown Takes Place atUptime 119905

1 In such a situation 119905 lt 119905

1 Under theAR inventory

control policy the machine goes under repair immediatelyand once it is fixed and restored the interrupted lot isinstantly resumed (Figure 1) Since 119909 proportion of scrap

4 The Scientific World Journal

Id(t)

t9984002

Time

t

T998400

d1t

d1t1

d1

d1

d1

tr t1 minus t

Figure 2 Inventory level of scrap items in the proposed manufacturing run time problem

products is produced the maximum number of scraps in acycle is 119909119876 (or 119889

11199051) and the on-hand inventory of scrap

items in the proposed manufacturing run time problem is asillustrated in Figure 2

The production cycle time 1198791015840 can be seen as (1) fromFigure 1

1198791015840= 1199051+ 119905119903+ 1199051015840

2 (1)

The total production-inventory-delivery cost per cycleTC1(1199051) is comprised of (1) the variable production cost (2)

the setup cost (3) the disposal cost for scraps (4) themachinerepair cost (5) fixed and variable product delivery costs (6)holding cost for safety stocks and (7) the producerrsquos inventoryholding costs in the entire production cycle Thus TC

1(1199051) is

TC1(1199051) = 119862 (119875

11199051) + 119870 + 119862

119878(11990511198751119909) +119872 + 119899119870

1

+ 119862119879[11990511198751] + ℎ3(120582119905119903) 1198791015840

+ ℎ[1198671015840+ 11988911199051

21199051+ (1198671015840

1+ 1198891119905) 119905119903+119899 minus 1

211989911986710158401199051015840

2]

(2)

Since 119909 is assumed to be a random variable with a knownprobability density function the expected values of119909 are usedin our analysis to take the randomness of 119909 into account Bysubstituting all related system parameters into (2) [9] withfurther derivations 119864[TC

1(1199051)] becomes (see the appendix

for more details)

119864 [TC1(1199051)]

= 119870 + 1198991198701+119872 + ℎ119905119875

1119892

+ [1198621198751+ 1198621198781198751119864 [119909] + 1198621198791198751 (1 minus 119864 [119909])

+ ℎ31198751119892 (1 minus 119864 [119909]) minus

ℎ1198751119892 (1 minus 119864 [119909])

2(1 minus

1

119899)] sdot 1199051

+ [ℎ1198751119864 [119909]

2+ℎ1198752

1

2120582(1 minus 119864 [119909])

2(1 minus

1

119899)

+ℎ1198751

2119899(1 minus 119864 [119909])] 119905

2

1

(3)

22 Case 2 No Breakdown Takes Place at Uptime 1199051 In such

a situation 119905 gt 1199051 The inventory level of finished items in

this case is depicted in Figure 3 and 119879 = 1199051+ 1199052 The total

production-inventory-delivery cost per cycle TC2(1199051) is as

displayed inTC2(1199051) = 119862 (119875

11199051) + 119870 + 119862

119878(11990911990511198751) + 119899119870

1

+ 119862119879[11990511198751(1 minus 119909)] + ℎ

3(120582119905119903) 119879

+ ℎ [119867 + 119889

11199051

21199051+119899 minus 1

21198991198671199052]

(4)

Again to take the randomness of 119909 into account andsubstitute all related parameters into (4) with further deriva-tions 119864[TC

2(1199051)] becomes [12]

119864 [TC2(1199051)] = 119870 + 119899119870

1

+ [1198621198751+ 119862119878119864 [119909] 1198751 + 1198621198791198751 (1 minus 119864 [119909])

+ ℎ31198751119892 (1 minus 119864 [119909])] 1199051

+ [ℎ1198751119864 [119909]

2+ℎ1198752

1

2120582(1 minus 119864 [119909])

2(1 minus

1

119899)

+ℎ1198751

2119899(1 minus 119864 [119909])] 119905

2

1

(5)

The Scientific World Journal 5

H

P1 minus d1

T

Time

minus120582

tn

t1 t1t2

I(t)

Figure 3 Inventory level of finished items in a manufacturing run time problem with no breakdown defective rate and discontinuousdelivery policy

23 Integration of the Proposed Run TimeModels withwithoutBreakdown Amachine breakdownmay take place randomlyand it follows a Poisson distribution with mean equal to120573 per year Let 119891(119905) be the probability density function ofrandom time 119905 before a breakdown takes place and 119865(119905)

represents the cumulative density function of 119905 Hence thelong-run expected system costs per unit time 119864[TCU(119905

1)]

are

119864 [TCU (1199051)]

=

int1199051

0119864 [TC

1(1199051)] 119891 (119905) 119889119905 + int

infin

1199051

119864 [TC2(1199051)] 119891 (119905) 119889119905

119864 [T]

(6)

where

119864 [T] = int1199051

0

119864 [1198791015840] 119891 (119905) 119889119905 + int

infin

1199051

119864 [119879] 119891 (119905) 119889119905

=11990511198751(1 minus 119864 [119909])

120582

(7)

From Figures 1 and 3 it can be seen that 1198791015840 and 119879 aredifferent in length (1198791015840 is longer than 119879 since it containsmachine repairing time) and because a breakdown can occurrandomly it is necessary to use the integration (ie equation(7)) to derive the expected cycle length

It is also noted that the time between breakdowns obeysthe exponential distribution with density function 119891(119905) =

120573119890minus120573119905 and cumulative density function 119865(119905) = 1 minus 119890

minus120573119905 Bysubstituting 119864[TC

1(1199051)] 119864[TC

2(1199051)] and 119864[T] into (6) and

solving the integration of the mean time to breakdown in119864[TCU(119905

1)] we obtain

119864 [TCU (1199051)]

=120582

(1 minus 119864 [119909])

sdot (119870 + 119899119870

1)

11990511198751

+ 1205741+12057421199051

2+ [

119872

1198751

+ℎ119892

120573](

1 minus 119890minus1205731199051

1199051

)

minusℎ119892 (119890minus1205731199051) minus

ℎ119892 (1 minus 119864 [119909])

2(1 minus

1

119899) (1 minus 119890

minus1205731199051)

(8)

where

1205741= [119862 + 119862

119878119864 [119909] + 119862

119879(1 minus 119864 [119909]) + ℎ

3119892 (1 minus 119864 [119909])]

(9)

1205742= [

ℎ1198751

120582(1 minus 119864 [119909])

2(1 minus

1

119899) + ℎ119864 [119909] +

ℎ

119899(1 minus 119864 [119909])]

(10)

24 Derivation of the Optimal Production Run Time In orderto derive the optimal production run time 119905lowast

1 we first have

to prove that 119864[TCU(1199051)] is convex Let 120585(119905

1) represent the

following

120585 (1199051) =

2 (119870 + 1198991198701) 120573 + 2 (1 minus 119890

minus1205731199051) 1205744

[1199052

1119875112057321205743+ 1205744(2 + 120573119905

1)] 120573119890minus1205731199051

(11)

6 The Scientific World Journal

Theorem 1 (119864[TCU(1199051)] is convex if 0 lt 119905

1lt 120585(119905

1)) The

second derivative of 119864[119879119862119880(1199051)] with respect to 119905

1is

1198892119864 [119879119862119880 (119905

1)]

11988921199052

1

=120582

(1 minus 119864 [119909])

sdot [2 (119870 + 119899119870

1)

1199053

11198751

minus ℎ119892 [1 minus(1 minus 119864 [119909])

2(1 minus

1

119899)] (120573

2119890minus1205731199051)

+ [119872

1198751

+ℎ119892

120573](

2 (1 minus 119890minus1205731199051)

1199053

1

minus2120573119890minus1205731199051

1199052

1

minus1205732119890minus1205731199051

1199051

)]

(12)

It is noted that because annual demand 120582 gt 0 the firstterm in the right-hand size (RHS) of (12) is positive Hence weobtain

1198892119864 [119879119862119880 (119905

1)]

1198891199051

2gt 0

119894119891 [2 (119870 + 119899119870

1)

1199053

11198751

minus ℎ119892 [1 minus(1 minus 119864 [119909])

2(1 minus

1

119899)] (120573

2119890minus1205731199051)

+ [119872

1198751

+ℎ119892

120573](

2 (1 minus 119890minus1205731199051)

1199053

1

minus2120573119890minus1205731199051

1199052

1

minus1205732119890minus1205731199051

1199051

)]

gt 0

(13)

The RHS of (13) can be further derived as

119894119891 [2 (119870 + 1198991198701) 120573 minus 119905

3

11198751120573ℎ119892

sdot [1 minus(1 minus 119864 [119909])

2(1 minus

1

119899)] (120573

2119890minus1205731199051)

+ (119872120573 + ℎ1198921198751)

times [2 (1 minus 119890minus1205731199051) minus 2119905

1120573119890minus1205731199051 minus 12057321199052

1119890minus1205731199051]]

gt 0

(14)

Let

1205743= ℎ119892 [1 minus

(1 minus 119864 [119909])

2(1 minus

1

119899)]

1205744= (119872120573 + ℎ119892119875

1)

(15)

then (13) can be rewritten as

1198892119864 [119879119862119880 (119905

1)]

1198891199051

2gt 0

119894119891 [2 (119870 + 1198991198701) 120573 + 2 (1 minus 119890

minus1205731199051) 1205744

minus 1199051[1199052

1119875112057321205743+ 1205744(2 + 120573119905

1)] 120573119890minus1205731199051]

gt 0

(16)

or

1198892119864 [119879119862119880 (119905

1)]

1198891199051

2gt 0

119894119891 0 lt 1199051lt

2 (119870 + 1198991198701) 120573 + 2 (1 minus 119890

minus1205731199051) 1205744

[1199052

1119875112057321205743+ 1205744(2 + 120573119905

1)] 120573119890minus1205731199051

= 120585 (1199051)

(17)

If 119864[119879119862119880(1199051)] is a convex function then the minimum

point exists In order to locate the optimal production run time119905lowast

1that minimizes 119864[119879119862119880(119905

1)] we can set the first derivative of

119864[119879119862119880(1199051)] equal to zero and solve 119905lowast

1

119889119864 [119879119862119880 (1199051)]

1198891199051

=120582

(1 minus 119864 [119909])

sdot minus (119870 + 119899119870

1)

1199052

1119875

+1205742

2+ 1205743(120573119890minus1205731199051)

+ [119872

1198751

+ℎ119892

120573](

minus (1 minus 119890minus1205731199051)

1199052

1

+120573119890minus1205731199051

1199051

)

(18)

In the RHS of (18) it can be seen that the first term ispositive so the second term is equal to zero Let

119905lowast

1119880= radic

2 (1205744+ 119870120573 + 119899119870

1120573)

12057421198751120573

(19)

119905lowast

1119871= 119905ℎ119890 119901119900119904119894119905119894V119890 119903119900119900119905 119900119891

minus1205744plusmn radic1205744

2 + 21198751(1205742+ 2120573120574

3) (119870 + 119899119870

1)

1198751(1205742+ 2120573120574

3)

(20)

Theorem 2 (119905lowast1119871

lt 119905lowast

1lt 119905lowast

1119880) Because the proof of 119905lowast

1falls

within the upper and lower bounds we can multiply the secondterm of (18) by (2120573119875

11199052

1) and obtain

(11987511205731205742+ 2119875112057321205743119890minus1205731199051) 1199052

1+ (21205744120573119890minus1205731199051) 1199051

minus2 [120573 (119870 + 1198991198701) + 1205744(1 minus 119890

minus1205731199051)] = 0

(21)

The Scientific World Journal 7

Producerrsquosproduction-shipment system

Extended toProductionunit

Productionunit

Multideliverypolicy

Multideliverypolicy

Optimization OptimizationProducer-retailer integratedproduction-shipment system

Customer

CustomerRetailer storeor sales office

Figure 4 Extension to a producer-retailer integrated production-shipment system

Thus119905lowast

1

= 119905ℎ119890 119901119900119904119894119905119894V119890 119903119900119900119905 119900119891

(minus (21205744120573119890minus1205731199051)

plusmn ((21205744120573119890minus1205731199051)2

minus [4 (11987511205731205742+ 2119875112057321205743119890minus1205731199051)

times [minus2 [120573 (119870 + 1198991198701) + 1205744(1 minus 119890

minus1205731199051)]]])

12

)

times (2 (12057421198751120573 + 2119875

112057321205743119890minus1205731199051))minus1

(22)

Equation (21) can be rearranged as

2 [1198751120573212057431199052

1+ 12057441205731199051+ 1205744] (119890minus1205731199051)

= 2 [120573 (119870 + 1198991198701) + 1205744] minus (119875

112057312057421199052

1)

(23)

or

119890minus1205731199051 =

2 [120573 (119870 + 1198991198701) + 1205744] minus (119875

112057312057421199052

1)

2 [1198751120573212057431199052

1+ 12057441205731199051+ 1205744]

(24)

where 119890minus1205731199051 is the complement of the cumulative density

function 119865(1199051) = 1 minus 119890

minus1205731199051 As 0 le 119865(119905

1) le 1 0 le 119890

minus1205731199051 le 1

Let 119890minus1205731199051 = 0 and 119890minus1205731199051 = 1 be the upper and lower bounds for119890minus1205731199051 respectively By substituting them into (22) we obtain

119905lowast

1119880= radic

2 [120573 (119870 + 1198991198701) + 1205744]

12057421198751120573

119905lowast

1119871= 119905ℎ119890 119901119900119904119894119905119894V119890 119903119900119900119905 119900119891

minus1205744plusmn radic1205744

2 + 21198751(1205742+ 2120573120574

3) (119870 + 119899119870

1)

1198751(1205742+ 2120573120574

3)

(25)

and 119905lowast1119871lt 119905lowast

1lt 119905lowast

1119880

It is noted that although the optimal production run time119905lowast

1cannot be presented in a closed form it does fall within the

bounds 119905lowast1can be locatedwith the use of a proposed recursive

searching algorithm Let

120596 (1199051) = 119890minus1205731199051 =

2 [120573 (119870 + 1198991198701) + 1205744] minus (119875

112057312057421199052

1)

2 [1198751120573212057431199052

1+ 12057441205731199051+ 1205744]

there4 0 le 120596 (1199051) le 1

(26)

In order to locate 119905lowast1 we can use the following recursive

searching algorithm

(1) Let 120596(1199051) = 0 and 120596(119905

1) = 1 initially and calculate

the upper and lower bounds for 119905lowast1 respectively (ie

to obtain the initial values of [119905lowast1119871 119905lowast

1119880])

(2) Substitute the current values of [119905lowast1119871 119905lowast

1119880] into 119890minus1205731199051 and

compute the new bounds expressed as 120596119871and 120596

119880for

119890minus1205731199051 Hence 120596

119871lt 120596(1199051) lt 120596119880

(3) Let 120596(1199051) = 120596119871and 120596(119905

1) = 120596119880and update the upper

and lower bounds for 119905lowast1 respectively (ie to obtain

the new values of [119905lowast1119871 119905lowast

1119880])

(4) Repeat steps (2) and (3) until there is no signifi-cant difference between 119905lowast

1119871and 119905lowast1119880

(or there is nosignificant difference in terms of their effects on119864[TCU(119905lowast

1)])

(5) Stop 119905lowast1is found

3 Extension to a Producer-Retailer IntegratedEPQ-Based System (Model 2)

31 Enhanced Model Description and Formulation In thissection we further extend the scope of the problem to incor-porate the retailerrsquos stock holding cost into our study Thenew model can be considered a producer-retailer integratedsystem because in the present-day manufacturing sectorsome producers of consumer goods may own and operateretail stores or regional sales offices to promote and sell theirend products to customers (see Figure 4) With the intentionof addressing such a real-life intrasupply chain situation thesecond model of this study incorporates the retailerrsquos stockholding cost into the first model and investigates its effect onthe optimal production run time decision

8 The Scientific World Journal

Dminus 120582tn = I

D

I

t1 + tr tn

Time

t1 + tr

Ic(t)

t9984002

T998400

Figure 5 Inventory level of finished products on the retailerrsquos side in the proposed manufacturing run time problem with breakdown

In the proposed study the retailerrsquos stock holding posi-tions are illustrated in Figure 5

Extra parameters used in this enhanced model includethe following

ℎ2 holding cost per product stored on the retailerrsquos

side

119868119888(119905) on-hand inventory levels in units on the retail-

errsquos side end at time 119905

119863 number of finished products (a fixed quantity)transported to the retail store per shipment

119868 number of left-over products in 119905119899after satisfying

the demand in 119905119899

TC3(1199051) total production-inventory-delivery costs

per cycle of this enhanced model (in the case of abreakdown)

TC4(1199051) total production-inventory-delivery costs

per cycle of this enhanced model (in the case of nobreakdown)

119864[TC3(1199051)] the expected production-inventory-

delivery costs per cycle of this enhanced model (inthe case of a breakdown)

119864[TC4(1199051)] the expected production-inventory-

delivery costs per cycle of this enhanced model (inthe case of no breakdown)

119864[TCU119890(1199051)] the long-run expected production-

inventory-delivery costs per unit time in thisenhanced model whether or not a breakdown takesplace

Since the demand on the retailerrsquos side in time interval119905119899is 120582119905119899 after satisfying the demand the number of left-over

items (see Figure 5) in each 119905119899is

119868 = 119863 minus 120582119905119899 (27)

Total inventory holding costs on retailerrsquos side with andwithout breakdown are shown respectively in

ℎ2[119899(119863 minus 119868)

2119905119899+119899 (119899 + 1)

2119868119905119899+119899119868

2(1199051+ 119905119903)] (28)

ℎ2[119899(119863 minus 119868)

2119905119899+119899 (119899 + 1)

2119868119905119899+119899119868

2(1199051)] (29)

To incorporate the retailerrsquos holding costs into the originalmodels with and without breakdown respectively we obtain

TC3(1199051) = TC

1(1199051)

+ ℎ2[119899(119863 minus 119868)

2119905119899+119899 (119899 + 1)

2119868119905119899+119899119868

2(1199051+ 119905119903)]

TC4(1199051)

= TC2(1199051) + ℎ2[119899(119863 minus 119868)

2119905119899+119899 (119899 + 1)

2119868119905119899+119899119868

2(1199051)]

(30)

To take the randomness of defective rate 119909 into accountand substitute all related variables into (30) with furtherderivations 119864[TC

3(1199051)] and 119864[TC

4(1199051)] can be obtained as

follows

119864 [TC3(1199051)] = 119864 [TC

1(1199051)] + ℎ

2(1198751119892 (1 minus 119864 [119909])

2)

sdot (1 minus1

119899) 1199051+ ℎ2(1198751(1 minus 119864 [119909])

2)

sdot [1198751(1 minus 119864 [119909])

120582119899+ (1 minus

1

119899)] 1199052

1

119864 [TC4(1199051)] = 119864 [TC

2(1199051)] + ℎ

2(1198751(1 minus 119864 [119909])

2)

sdot [1198751(1 minus 119864 [119909])

120582119899+ (1 minus

1

119899)] 1199052

1

(31)

The Scientific World Journal 9

32 Integration of Enhanced Model withwithout BreakdownThe mean time to breakdowns obeys the exponential distri-bution with 119891(119905) = 120573119890minus120573119905 Therefore 119864[TCU

2(1199051)] is

119864 [TCU2(1199051)]

=

int1199051

0119864 [TC

3(1199051)] 119891 (119905) 119889119905 + int

infin

1199051

119864 [TC4(1199051)] 119891 (119905) 119889119905

119864 [T]

(32)

Substituting 119864[TCU3(1199051)] 119864[TC

4(1199051)] and 119864[T] into (32)

and resolving 119864[TCU2(1199051)] we obtain

119864 [TCU2(1199051)]

=120582

(1 minus 119864 [119909])

sdot (119870 + 119899119870

1)

11990511198751

+ 1205741+12057421199051

2+ 12057451199051

+ [119872

1198751

+ℎ119892

120573](

1 minus 119890minus1205731199051

1199051

) minus ℎ119892 (119890minus1205731199051)

minus (ℎ minus ℎ2) [119892 (1 minus 119864 [119909])

2(1 minus

1

119899)] (1 minus 119890

minus1205731199051)

(33)

where

1205745=ℎ2(1 minus 119864 [119909])

2[1198751(1 minus 119864 [119909])

120582119899+ (1 minus

1

119899)] (34)

33 Determining the Optimal Run Time Let 120595(1199051) stand for

the following

120595 (1199051) =

2 (119870 + 1198991198701) 120573 + 2120574

4(1 minus 119890

minus1205731199051)

[1199052

1119875112057321205746+ 1205744(2 + 120573119905

1)] 120573119890minus1205731199051

(35)

Theorem 3 (119864[TCU2(1199051)] is convex if 0 lt 119905

1lt 120595(119905

1)) The

second derivative of 119864[1198791198621198802(1199051)] with respect to 119905

1is

1198892119864 [119879119862119880

2(1199051)]

11988921199052

1

=120582

(1 minus 119864 [119909])[2 (119870 + 119899119870

1)

1199053

11198751

+ (ℎ minus ℎ2)119892 (1 minus 119864 [119909])

2

sdot (1 minus1

119899) (1205732119890minus1205731199051) minus ℎ119892 (120573

2119890minus1205731199051)

+ [119872

1198751

+ℎ119892

120573](

2 (1 minus 119890minus1205731199051)

1199053

1

minus2120573119890minus1205731199051

1199052

1

minus1205732119890minus1205731199051

1199051

)]

(36)

Since annual demand 120582 gt 0 the first term in the RHS of(36) is positive and

119894119891 [2 (119870 + 119899119870

1)

1199053

11198751

+ (ℎ minus ℎ2) [119892 (1 minus 119864 [119909])

2(1 minus

1

119899)]

sdot (1205732119890minus1205731199051) minus ℎ119892 (120573

2119890minus1205731199051) + [

119872

1198751

+ℎ119892

120573]

sdot(

2 (1 minus 119890minus1205731199051)

1199053

1

minus2120573119890minus1205731199051

1199052

1

minus1205732119890minus1205731199051

1199051

)] gt 0

119905ℎ1198901198991198892119864 [119879119862119880

2(1199051)]

1198891199051

2gt 0

(37)

With further derivations the left-hand side (LHS) of (37)becomes

119894119891 [2 (119870 + 1198991198701) 120573 minus 119905

3

11198751120573 (ℎ minus ℎ

2) [119892 (1 minus 119864 [119909])

2(1 minus

1

119899)]

sdot (1205732119890minus1205731199051) minus 1199053

11198751120573ℎ119892 (120573

2119890minus1205731199051)

+ (119872120573 + ℎ1198921198751) [2 (1 minus 119890

minus1205731199051) minus 2119905

1120573119890minus1205731199051 minus 12057321199052

1119890minus1205731199051] ]

gt 0

(38)

Let

1205746= (ℎ minus ℎ

2) [119892 (1 minus 119864 [119909])

2(1 minus

1

119899)] + ℎ119892 (39)

then (37) becomes

119894119891 [2 (119870 + 1198991198701) 120573 + 2120574

4(1 minus 119890

minus1205731199051)

minus1199051[1199052

1119875112057321205746+ 1205744(2 + 120573119905

1)] 120573119890minus1205731199051] gt 0

119905ℎ1198901198991198892119864 [119879119862119880

2(1199051)]

1198891199051

2gt 0

(40)

or

1198892119864 [119879119862119880

2(1199051)]

1198891199051

2gt 0

119894119891 0 lt 1199051lt

2 (119870 + 1198991198701) 120573 + 2120574

4(1 minus 119890

minus1205731199051)

[1199052

1119875112057321205746+ 1205744(2 + 120573119905

1)] 120573119890minus1205731199051

= 120595 (1199051)

(41)

10 The Scientific World Journal

Once 119864[1198791198621198802(1199051)] is proven to be convex the optimal

run time 119905lowast1can be solved by setting the first derivative of

119864[1198791198621198802(1199051)] = 0

119889119864 [1198791198621198802(1199051)]

1198891199051

=120582

(1 minus 119864 [119909])sdot minus (119870 + 119899119870

1)

1199052

1119875

+ (1205742

2+ 1205745) + 1205746(120573119890minus1205731199051)

+ [119872

1198751

+ℎ119892

120573](

minus (1 minus 119890minus1205731199051)

1199052

1

+120573119890minus1205731199051

1199051

)

= 0

(42)

It can be seen that the first term in the RHS of (42) ispositive so the second term is equal to zero In order to findthe bounds for 119905lowast

1 let

119905lowast

1119880= radic

2 [120573 (119870 + 1198991198701) + 1205744]

1198751120573 (1205742+ 21205745)

(43)

119905lowast

1119871= 119905ℎ119890 119901119900119904119894119905119894V119890 119903119900119900119905 119900119891

minus1205744plusmn radic1205742

4+ 21198751(119870 + 119899119870

1) (1205742+ 21205745+ 21205746120573)

1198751(1205742+ 21205745+ 21205746120573)

(44)

Theorem4 (119905lowast1119871lt 119905lowast

1lt 119905lowast

1119880) For the proof ofTheorem 4 please

refer to the proof for Theorem 2 in Section 2

Once we are certain that 119905lowast1falls within the aforemen-

tioned upper and lower bounds in order to find 119905lowast1 we can

first multiply the second term of (42) by (211987511199052

1120573) and obtain

the following

[1198751120573 (1205742+ 21205745) + 2119875

112057321205746119890minus1205731199051] 1199052

1+ (21205744120573119890minus1205731199051) 1199051

minus2 [120573 (119870 + 1198991198701) + 1205744(1 minus 119890

minus1205731199051)]

= 0

(45)

Equation (45) can be rearranged as

119890minus1205731199051 =

2 [120573 (119870 + 1198991198701) + 1205744] minus [119875

1120573 (1205742+ 21205745)] 1199052

1

2 [1198751120573212057461199052

1+ 1205744(1 + 120573119905

1)]

(46)

where 119890minus1205731199051 is the complement of the cumulative densityfunction 119865(119905

1) = 1minus119890

minus1205731199051 As 0 le 119865(119905

1) le 1 0 le 119890minus1205731199051 le 1 Let

119890minus1205731199051 = 0 and 119890minus1205731199051 = 1 be the initial upper and lower bounds

of 119890minus1205731199051 respectively Then by using the proposed recursivesearching algorithm given at the end of Section 2 we can findthe optimal production run time 119905lowast

1

4 Numerical Example

In order to relieve the comparison efforts for readers thissection adopts the same numerical example as in [9] For

0225 0265 0305 0345 0385 0425 0465 0505

E[T

CU(t

1)]

t1

tlowast1L = 03411 t

lowast1 = 03748

tlowast1U = 05183

$10600

$10800

$11000

$11200

$11400

$11600

$11006

$11800

Figure 6 The behavior of 119864[TCU(1199051)] in connection with the

production run time 1199051in the proposed model 1

a demonstration of the proposed EPQ-based model 1 thefollowing system parameters are used

1198751 production rate 10000 products per year120582 demand rate 4000 products per year119909 random scrap rate which follows uniformly distribu-

tion over the interval [0 02]120573 Poisson breakdown rate 05 average times per year119892 constant machine repair time 119905

119903 0018 year per repair

119872 machine repair cost $500 for each breakdown119870 setup cost $450 per production run119862 manufacturing cost $2 per item119862119878 disposal cost $03 per scrap itemℎ holding cost $06 per item per unit time1198701 fixed delivery cost $90 per shipment119899 number of deliveries 4 per cycle

119862119879 variable delivery cost $0001 per item

First we use both upper and lower bounds of 119905lowast1to

test for the convexity of 119864[TCU(1199051)] (see Theorem 1) The

computation results of (19) (20) and (11) indicate that 119905lowast1119880

=05183 lt 120585(119905

lowast

1119880) = 28723 and 119905lowast

1119871= 03411 lt 120585(119905

lowast

1119871) = 26261

Hence [TCU(1199051)] is convex (Figure 6)

In order to find the optimal 119905lowast1 we first substitute the

upper and lower bounds of 119905lowast1in (8) and obtain 119864[TCU(119905lowast

1119880)]

= $1160163 and 119864[TCU(119905lowast1119871)] = $1101450 respectively

Because the optimal run time 119905lowast1falls within the interval

of [119905lowast1119871 119905lowast

1119880] we apply the proposed recursive searching

algorithm stated at the end of Section 2 and find 119905lowast1= 03748

years Accordingly the optimal expected system costs perunit time 119864[TCU(119905lowast

1)] = $1100641 (Figure 6) Table 1 shows

the step-by-step iterations of the algorithmIn this specific studied model we focus on incorporating

a discontinuous multidelivery policy into a prior work [9]and consider a fixed transportation cost associated witheach delivery Applying the research result we can easilyinvestigate the effects of different fixed transportation costs

The Scientific World Journal 11

Table 1 Iterations of the recursive searching algorithm for locating 119905lowast1

120573 Step 119905lowast

1119880120596119880= 119890minus1205731199051119880 119905

lowast

1119871120596119871= 119890minus1205731199051119871

Differencebetween 119905lowast

1119880and

119905lowast

1119871

[119880]

119864[TCU(119905lowast1119880)]

[119871]

119864[TCU(119905lowast1119871)]

Differencebetween [119880]

and [119871]05 Initial 000000 100000 $1160163 $1101450 $58713

1st 05183 07717 03411 08432 01772 $1110330 $1101450 $88802nd 03857 08246 03721 08302 00136 $1100716 $1100646 $0703rd 03756 08287 03746 08292 00010 $1100642 $1100641 $0014th 03749 08291 03748 08291 00001 $1100641 $1100641 $00045th 03748 08291 03748 08291 000000 $1100641 $1100641 $0000

Table 2 Variations of the fixed delivery cost1198701effects on the optimal production run time 119905lowast

1

1198701119870 005 02 04 06 08 1 12 14 16 18 2 22

1198701

$225 $90 $180 $270 $360 $450 $540 $630 $720 $810 $900 $990119864[TCU(119905lowast

1)] $10721 $11067 $11448 $11773 $12062 $12325 $12567 $12793 $13006 $13208 $13399 $13583

119905lowast

103116 03816 04586 05244 05828 06359 06848 07305 07735 08142 08529 08900

Table 3 Variations of the unit retailerrsquos holding cost ℎ2and their effects on 119864[TCU(119905lowast

1)]

ℎ2ℎ 05 075 1 125 15 175 2 225 25 275 3 325

ℎ2

03 045 06 075 09 105 12 135 15 165 18 195119864 [TCU (119905lowast

1)] $11282 $11407 $11526 $11638 $11746 $11850 $11949 $12045 $12138 $12229 $12316 $12401

119905lowast

103256 03074 02918 02785 02668 02564 02472 02389 02314 02246 02183 02125

1198701on the optimal system cost 119864[TCU(119905lowast

1)] and on the

optimal production run time 119905lowast1(see Table 2) It can be seen

from Table 2 that as the ratio of1198701119870 increases the expected

system costs per unit time119864[TCU(119905lowast1119871)] increase significantly

It is also noted that as 1198701increases optimal production run

time 119905lowast1also increases significantly

41 Numerical Example for the Producer-Retailer IntegratedEPQ System (Model 2) In order to demonstrate the researchresult of the producer-retailer integrated EPQ-based modelan additional system variable ℎ

2= $150 per item stored at the

retailerrsquos side is includedAgain one can use the upper and lower bounds of 119905lowast

1

(equations (43) and (44)) to test for convexity of 119864[TCU(1199051)]

(Theorem 3 and equation (35)) The results reveal that 119905lowast1119880

=03213 lt 120595(119905lowast

1119880) = 26462 and 119905lowast

1119871= 02186 lt 120595(119905lowast

1119871) = 24876

Therefore the expected cost [TCU(1199051)] is convex

Next by applying the proposed recursive searching algo-rithm we can calculate that the optimal run time 119905lowast

1=

02314 years and the optimal 119864[TCU(119905lowast1)] = $1213849 It is

noted that the computation time for reaching the optimal 119905lowast1

solution is 21 seconds (usingExcel software in a desktop com-puter Intel CPU G850 with 294GB RAM and 289GHz)

Figure 7 illustrates the behavior of 119864[TCU(1199051)] with

regard to production run time It is noted that without theresearch result from the second model the management ofsuch a producer-retailer integrated system would probablyuse 1199051= 03748 years (from the result of model 1) for their

run time decision Further analysis (see Figure 7) shows cost

0130 0170 0210 0250 0290 0330 0370 0410

t1

E[T

CU(t

1)]

11600

11800

12000

12200

12400

12600

12800

tlowast1 = 02314

t1 = 0374812138

(from model 1)

E[TCU( t1)] = 12489

$$

$

$

$

$

$

$

$

Figure 7 The behavior of 119864[TCU(1199051)] with respect to production

run time 1199051in the proposed model 2

savings of $351 (or 29 over the total system costs) simply byapplying our research result

The effects of the unit retailerrsquos holding cost ℎ2on the

expected system cost 119864[TCU(119905lowast1)] and on the optimal run

time 119905lowast1are shown in Table 3 respectively

It can be seen that as ℎ2or the ratio of ℎ

2ℎ increases

the expected cost 119864[TCU(119905lowast1)] increases but the optimal

production run time 119905lowast1decreases In decision-making these

sensitivity analyses results can provide the management ofa producer-retailer integrated system with valuable informa-tion and insights into the effects of various stock holding costsin different retailersrsquo locations

12 The Scientific World Journal

5 Concluding Remarks

Two exact models for an extended EPQ-based problem witha discontinuous delivery policy scrap rate and randombreakdown are developed in this study They specificallyaddress different real-life situations in production end-item delivery and intrasupply chains such as a producer-retailer integrated systemMathematicalmodeling alongwithoptimization techniques is used to determine the optimalproduction run times that minimize the expected systemcosts per unit time Without in-depth investigations onthese separate models the optimal production run time andother important information related to the systemparameterscannot be revealed The proposed real-life EPQ models withrandom machine breakdown discontinuous product distri-bution policies and quality assurance must be specificallystudied in order to (1) obtain the joint effects of breakdowndiscontinuous distribution policies and quality assurance onthe optimal production run time (2) get to know the effectsof different policy and scope of supply chains managementon the optimal run time and overall system costs and (3)gain the insight with regard to various systemrsquos parametersof all particular EPQ-based models Since little attention hasbeen paid to the investigation of joint effects of these practicalproduction situations on the optimal run time this researchis intended to bridge the gap An interesting area for futurestudy is the examination of the effect of variable productionrates on these models

Appendix

Derivations of (3) are as followsRecall (2) as follows

TC1(1199051) = 119862 (119875

11199051) + 119870 + 119862

119878(11990511198751119909) +119872 + 119899119870

1

+ 119862119879[11990511198751] + ℎ3(120582119905119903) 1198791015840

+ ℎ[1198671015840+ 11988911199051

21199051+ (1198671015840

1+ 1198891119905) 119905119903+119899 minus 1

211989911986710158401199051015840

2]

(A1)

Substituting all related system parameters into (2) (pleaserefer to the basic formulations and solution process in [9])the TC

1(1199051) can be obtained as

TC1(1199051) = 119870 +119872

+[1198621198751+ 1198621198781198751119909 + 119862

1198791198751(1 minus 119909)+ ℎ

31198751119892 (1 minus 119909)] 119905

1

+ 1198991198701+ ℎ119901119892119905 minus[

ℎ1198751119892 (1 minus 119909)

2minusℎ1198751119892 (1 minus 119909)

2119899] 1199051

+ 1199052

1[ℎ1198751

2+ℎ1198752

1

2120582(1 minus 119909)

2minusℎ1198751

2(1 minus 119909)

minusℎ1198752

1

2120582119899(1 minus 119909)

2+ℎ1198751

2119899(1 minus 119909)]

(A2)

To take the randomness of 119909 into account by using theexpected values of 119909 with further derivations119864[TC

1(1199051)] can

be derived as follows (ie equation (3))119864 [TC

1(1199051)]

= 119870 + 1198991198701+119872 + ℎ119905119875

1119892

+ [1198621198751+ 1198621198781198751119864 [119909] + 1198621198791198751 (1 minus 119864 [119909])

+ ℎ31198751119892 (1 minus 119864 [119909]) minus

ℎ1198751119892 (1 minus 119864 [119909])

2(1 minus

1

119899)] sdot 1199051

+ [ℎ1198751119864 [119909]

2+ℎ1198752

1

2120582(1 minus 119864 [119909])

2(1 minus

1

119899)

+ℎ1198751

2119899(1 minus 119864 [119909])] 119905

2

1

(A3)

Conflict of Interests

The authors of the paper declare that there is no conflict ofinterests regarding the publication of this paper

Acknowledgment

The authors greatly appreciate the National Science Council(NSC) of Taiwan for supporting this research underGrant noNSC 102-2410-H-324-005

References

[1] E W Taft ldquoThemost economical production lotrdquo Iron Age vol101 pp 1410ndash1412 1918

[2] G Hadley and T M Whitin ldquoAn optimal final inventorymodelrdquoManagement Science vol 7 pp 179ndash183 1961

[3] E A Silver D F Pyke and R Peterson Inventory Managementand Production Planning and Scheduling John Wiley amp SonsNew York NY USA 1998

[4] S Nahmias Production amp Operations Analysis McGraw-HillNew York NY USA 2009

[5] MWidmer and P Solot ldquoDo not forget the breakdowns and themaintenance operations in FMSdesign problemsrdquo InternationalJournal of Production Research vol 28 pp 421ndash430 1990

[6] K-Y C Yu and D L Bricker ldquoAnalysis of a markov chainmodel of a multistage manufacturing system with inspectionrejection and reworkrdquo IIE Transactions vol 25 no 1 pp 109ndash112 1993

[7] H Groenevelt L Pintelon and A Seidmann ldquoProduction lotsizing with machine breakdownsrdquoManagement Science vol 38no 1 pp 104ndash123 1992

[8] G A Widyadana and H M Wee ldquoOptimal deterioratingitems production inventory models with random machinebreakdown and stochastic repair timerdquo Applied MathematicalModelling vol 35 no 7 pp 3495ndash3508 2011

[9] S W Chiu C-L Chou and W-K Wu ldquoOptimizing replenish-ment policy in an EPQ-based inventory model with noncon-forming items and breakdownrdquo EconomicModelling vol 35 pp330ndash337 2013

The Scientific World Journal 13

[10] A M Zargar ldquoEffect of rework strategies on cycle timerdquoComputers amp Industrial Engineering vol 29 no 1ndash4 pp 239ndash243 1995

[11] P Biswas and B R Sarker ldquoOptimal batch quantity modelsfor a lean production system with in-cycle rework and scraprdquoInternational Journal of Production Research vol 46 no 23 pp6585ndash6610 2008

[12] S W Chiu H-D Lin C-B Cheng and C-L Chung ldquoOptimalproduction-shipment decisions for the finite production ratemodel with scraprdquo International Journal for Engineering Mod-elling vol 22 no 1ndash4 pp 25ndash34 2009

[13] Y-S P Chiu K-K Chen and C-K Ting ldquoReplenishment runtime problem with machine breakdown and failure in reworkrdquoExpert Systems with Applications vol 39 no 1 pp 1291ndash12972012

[14] H-D Lin F-Y Pai and S W Chiu ldquoA note on ldquointra-supply chain system with multiple sales locations and qualityassurancerdquordquo Expert Systems with Applications vol 40 no 11 pp4730ndash4732 2013

[15] B Kamsu-Foguem F Rigal and FMauget ldquoMining associationrules for the quality improvement of the production processrdquoExpert Systems with Applications vol 40 no 4 pp 1034ndash10452013

[16] Y Li H Guo L Wang and J Fu ldquoA hybrid genetic-simulatedannealing algorithm for the location-inventory- routing prob-lem considering returns under E-supply chain environmentrdquoThe Scientific World Journal vol 2013 Article ID 125893 10pages 2013

[17] Y-S P Chiu K-K Chen F-T Cheng and C-K Ting ldquoReex-amination of ldquocombining an alternative multi-delivery policyinto economic production lot size problemwith partial reworkrdquousing an alternative approachrdquo Journal of Applied Research andTechnology vol 11 no 3 pp 317ndash323 2013

[18] L B Schwarz B L Deuermeyer and R D Badinelli ldquoFill-rateoptimization in a one-warehouse 119873-identical retailer distribu-tion systemrdquo Management Science vol 31 no 4 pp 488ndash4981985

[19] R A Sarker and L R Khan ldquoOptimal batch size for a pro-duction system operating under periodic delivery policyrdquoCom-puters amp Industrial Engineering vol 37 no 4 pp 711ndash730 1999

[20] N Comez K E Stecke and M Cakanyildirim ldquoMultiple in-cycle transshipments with positive delivery timesrdquo Productionand Operations Management vol 21 no 2 pp 378ndash395 2012

[21] S W Chiu F-Y Pai and W K Wu ldquoAlternative approach todetermine the common cycle time for a multi-item productionsystem with discontinuous deliveries and failure in reworkrdquoEconomic Modelling vol 35 pp 593ndash596 2013

[22] Y-S P Chiu C-C Huang M-F Wu and H-H Chang ldquoJointdetermination of rotation cycle time and number of shipmentsfor a multi-item EPQ model with random defective raterdquoEconomic Modelling vol 35 pp 112ndash117 2013

[23] M A Hoque ldquoSynchronization in the single-manufacturermulti-buyer integrated inventory supply chainrdquo European Jour-nal of Operational Research vol 188 no 3 pp 811ndash825 2008

[24] S W Chiu L-W Lin K-K Chen and C-L Chou ldquoDetermin-ing production-shipment policy for a vendor-buyer integratedsystem with rework and an amending multi-delivery schedulerdquoEconomic Modelling vol 33 pp 668ndash675 2013

[25] M Cedillo-Campos and C Sanchez-Ramırez ldquoDynamic self-assessment of supply chains performance an emerging marketapproachrdquo Journal of Applied Research and Technology vol 11no 3 pp 338ndash347 2013

[26] Y-S P Chiu H-D Lin F-T Cheng and M-H Hwang ldquoOpti-mal common cycle time for a multi-item production systemwith discontinuous delivery policy and failure in reworkrdquoJournal of Scientific and Industrial Research vol 72 no 7 pp435ndash440 2013

[27] HHishamuddin R A Sarker andD Essam ldquoA recoverymech-anism for a two echelon supply chain system under supplydisruptionrdquo Economic Modelling vol 38 pp 555ndash563 2014

[28] M Murugan and V Selladurai ldquoProductivity improvement inmanufacturing submersible pump diffuser housing using leanmanufacturing systemrdquo Journal of Engineering Research vol 2no 1 pp 164ndash182 2014

[29] L Wang H Qu S Liu and C-X Dun ldquoModeling and opti-mization of the multiobjective stochastic joint replenishmentand delivery problem under supply chain environmentrdquo TheScientific World Journal vol 2013 Article ID 916057 11 pages2013

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

The Scientific World Journal 3

I(t)

H998400

H9984001

P1 minus d1

P1 minus d1

t tr

tn

t1 minus t

T998400

t9984002

Time

Figure 1 Inventory level of finished items in the proposed manufacturing run time problem

immediately and a constant repair time is assumed Uponthe completion of the repair the interrupted lot is instantlyresumed (Figure 1)

Additional cost-related parameters used in this studyare the machine repairing cost 119872 setup cost per cycle 119870holding cost per item at the producerrsquos side ℎ disposal costper scrapped item 119862

119878 fixed delivery cost per shipment 119870

1

variable delivery cost per item119862119879 unit holding cost for safety

stock at the producerrsquos side ℎ3 and holding cost per item at

the retailerrsquos side ℎ2 Other notations used in the modeling

and analysis also include the following

119905 production time before a random machine break-down takes place1198671015840

1 on-hand inventory level in units when a random

machine breakdown takes place120573 number of machine breakdowns per unit time (ieyear) assumed to be a random variable that followsthe Poisson distribution119905119903 machine repair time

1199051 production uptime the decision variable of the

proposed manufacturing run time model1198671015840 maximum on-hand inventory level in units when

the regular production process ends (in the case of abreakdown)1199051015840

2 time required to deliver all finished items produced

in a cycle (in the case of a breakdown)1198791015840 production cycle length (in the case of a break-

down)119876 lot size for each production cycleTC1(1199051) total production-inventory-delivery costs

per cycle (in the case of a breakdown)

119864[TC1(1199051)] the expected production-inventory-

delivery costs per cycle (in the case of a breakdown)1199052 time required to deliver all finished items produced

(in the case of no breakdown)119867 on-hand inventory level in units when the regularproduction process ends (in the case of no break-down)119879 cycle length (in the case of no breakdown)119868(119905) on-hand inventory level of finished items at time119905119868119904(119905) on-hand inventory level of scrap items at time 119905

TC2(1199051) total production-inventory-delivery costs

per cycle (in the case of no breakdown)119864[TC

2(1199051)] the expected production-inventory-

delivery costs per cycle (in the case of no breakdown)TCU(119905

1) total production-inventory-delivery costs

per unit timewhether or not a breakdown takes place119864[TCU(119905

1)] the long-run expected production-

inventory-delivery costs per unit time whether or nota breakdown takes placeT the cycle length whether or not a machine break-down takes place

Since a machine breakdown may randomly take place atproduction uptime 119905

1 the following two distinct cases must

be examined

21 Case 1 A Random Machine Breakdown Takes Place atUptime 119905

1 In such a situation 119905 lt 119905

1 Under theAR inventory

control policy the machine goes under repair immediatelyand once it is fixed and restored the interrupted lot isinstantly resumed (Figure 1) Since 119909 proportion of scrap

4 The Scientific World Journal

Id(t)

t9984002

Time

t

T998400

d1t

d1t1

d1

d1

d1

tr t1 minus t

Figure 2 Inventory level of scrap items in the proposed manufacturing run time problem

products is produced the maximum number of scraps in acycle is 119909119876 (or 119889

11199051) and the on-hand inventory of scrap

items in the proposed manufacturing run time problem is asillustrated in Figure 2

The production cycle time 1198791015840 can be seen as (1) fromFigure 1

1198791015840= 1199051+ 119905119903+ 1199051015840

2 (1)

The total production-inventory-delivery cost per cycleTC1(1199051) is comprised of (1) the variable production cost (2)

the setup cost (3) the disposal cost for scraps (4) themachinerepair cost (5) fixed and variable product delivery costs (6)holding cost for safety stocks and (7) the producerrsquos inventoryholding costs in the entire production cycle Thus TC

1(1199051) is

TC1(1199051) = 119862 (119875

11199051) + 119870 + 119862

119878(11990511198751119909) +119872 + 119899119870

1

+ 119862119879[11990511198751] + ℎ3(120582119905119903) 1198791015840

+ ℎ[1198671015840+ 11988911199051

21199051+ (1198671015840

1+ 1198891119905) 119905119903+119899 minus 1

211989911986710158401199051015840

2]

(2)

Since 119909 is assumed to be a random variable with a knownprobability density function the expected values of119909 are usedin our analysis to take the randomness of 119909 into account Bysubstituting all related system parameters into (2) [9] withfurther derivations 119864[TC

1(1199051)] becomes (see the appendix

for more details)

119864 [TC1(1199051)]

= 119870 + 1198991198701+119872 + ℎ119905119875

1119892

+ [1198621198751+ 1198621198781198751119864 [119909] + 1198621198791198751 (1 minus 119864 [119909])

+ ℎ31198751119892 (1 minus 119864 [119909]) minus

ℎ1198751119892 (1 minus 119864 [119909])

2(1 minus

1

119899)] sdot 1199051

+ [ℎ1198751119864 [119909]

2+ℎ1198752

1

2120582(1 minus 119864 [119909])

2(1 minus

1

119899)

+ℎ1198751

2119899(1 minus 119864 [119909])] 119905

2

1

(3)

22 Case 2 No Breakdown Takes Place at Uptime 1199051 In such

a situation 119905 gt 1199051 The inventory level of finished items in

this case is depicted in Figure 3 and 119879 = 1199051+ 1199052 The total

production-inventory-delivery cost per cycle TC2(1199051) is as

displayed inTC2(1199051) = 119862 (119875

11199051) + 119870 + 119862

119878(11990911990511198751) + 119899119870

1

+ 119862119879[11990511198751(1 minus 119909)] + ℎ

3(120582119905119903) 119879

+ ℎ [119867 + 119889

11199051

21199051+119899 minus 1

21198991198671199052]

(4)

Again to take the randomness of 119909 into account andsubstitute all related parameters into (4) with further deriva-tions 119864[TC

2(1199051)] becomes [12]

119864 [TC2(1199051)] = 119870 + 119899119870

1

+ [1198621198751+ 119862119878119864 [119909] 1198751 + 1198621198791198751 (1 minus 119864 [119909])

+ ℎ31198751119892 (1 minus 119864 [119909])] 1199051

+ [ℎ1198751119864 [119909]

2+ℎ1198752

1

2120582(1 minus 119864 [119909])

2(1 minus

1

119899)

+ℎ1198751

2119899(1 minus 119864 [119909])] 119905

2

1

(5)

The Scientific World Journal 5

H

P1 minus d1

T

Time

minus120582

tn

t1 t1t2

I(t)

Figure 3 Inventory level of finished items in a manufacturing run time problem with no breakdown defective rate and discontinuousdelivery policy

23 Integration of the Proposed Run TimeModels withwithoutBreakdown Amachine breakdownmay take place randomlyand it follows a Poisson distribution with mean equal to120573 per year Let 119891(119905) be the probability density function ofrandom time 119905 before a breakdown takes place and 119865(119905)

represents the cumulative density function of 119905 Hence thelong-run expected system costs per unit time 119864[TCU(119905

1)]

are

119864 [TCU (1199051)]

=

int1199051

0119864 [TC

1(1199051)] 119891 (119905) 119889119905 + int

infin

1199051

119864 [TC2(1199051)] 119891 (119905) 119889119905

119864 [T]

(6)

where

119864 [T] = int1199051

0

119864 [1198791015840] 119891 (119905) 119889119905 + int

infin

1199051

119864 [119879] 119891 (119905) 119889119905

=11990511198751(1 minus 119864 [119909])

120582

(7)

From Figures 1 and 3 it can be seen that 1198791015840 and 119879 aredifferent in length (1198791015840 is longer than 119879 since it containsmachine repairing time) and because a breakdown can occurrandomly it is necessary to use the integration (ie equation(7)) to derive the expected cycle length

It is also noted that the time between breakdowns obeysthe exponential distribution with density function 119891(119905) =

120573119890minus120573119905 and cumulative density function 119865(119905) = 1 minus 119890

minus120573119905 Bysubstituting 119864[TC

1(1199051)] 119864[TC

2(1199051)] and 119864[T] into (6) and

solving the integration of the mean time to breakdown in119864[TCU(119905

1)] we obtain

119864 [TCU (1199051)]

=120582

(1 minus 119864 [119909])

sdot (119870 + 119899119870

1)

11990511198751

+ 1205741+12057421199051

2+ [

119872

1198751

+ℎ119892

120573](

1 minus 119890minus1205731199051

1199051

)

minusℎ119892 (119890minus1205731199051) minus

ℎ119892 (1 minus 119864 [119909])

2(1 minus

1

119899) (1 minus 119890

minus1205731199051)

(8)

where

1205741= [119862 + 119862

119878119864 [119909] + 119862

119879(1 minus 119864 [119909]) + ℎ

3119892 (1 minus 119864 [119909])]

(9)

1205742= [

ℎ1198751

120582(1 minus 119864 [119909])

2(1 minus

1

119899) + ℎ119864 [119909] +

ℎ

119899(1 minus 119864 [119909])]

(10)

24 Derivation of the Optimal Production Run Time In orderto derive the optimal production run time 119905lowast

1 we first have

to prove that 119864[TCU(1199051)] is convex Let 120585(119905

1) represent the

following

120585 (1199051) =

2 (119870 + 1198991198701) 120573 + 2 (1 minus 119890

minus1205731199051) 1205744

[1199052

1119875112057321205743+ 1205744(2 + 120573119905

1)] 120573119890minus1205731199051

(11)

6 The Scientific World Journal

Theorem 1 (119864[TCU(1199051)] is convex if 0 lt 119905

1lt 120585(119905

1)) The

second derivative of 119864[119879119862119880(1199051)] with respect to 119905

1is

1198892119864 [119879119862119880 (119905

1)]

11988921199052

1

=120582

(1 minus 119864 [119909])

sdot [2 (119870 + 119899119870

1)

1199053

11198751

minus ℎ119892 [1 minus(1 minus 119864 [119909])

2(1 minus

1

119899)] (120573

2119890minus1205731199051)

+ [119872

1198751

+ℎ119892

120573](

2 (1 minus 119890minus1205731199051)

1199053

1

minus2120573119890minus1205731199051

1199052

1

minus1205732119890minus1205731199051

1199051

)]

(12)

It is noted that because annual demand 120582 gt 0 the firstterm in the right-hand size (RHS) of (12) is positive Hence weobtain

1198892119864 [119879119862119880 (119905

1)]

1198891199051

2gt 0

119894119891 [2 (119870 + 119899119870

1)

1199053

11198751

minus ℎ119892 [1 minus(1 minus 119864 [119909])

2(1 minus

1

119899)] (120573

2119890minus1205731199051)

+ [119872

1198751

+ℎ119892

120573](

2 (1 minus 119890minus1205731199051)

1199053

1

minus2120573119890minus1205731199051

1199052

1

minus1205732119890minus1205731199051

1199051

)]

gt 0

(13)

The RHS of (13) can be further derived as

119894119891 [2 (119870 + 1198991198701) 120573 minus 119905

3

11198751120573ℎ119892

sdot [1 minus(1 minus 119864 [119909])

2(1 minus

1

119899)] (120573

2119890minus1205731199051)

+ (119872120573 + ℎ1198921198751)

times [2 (1 minus 119890minus1205731199051) minus 2119905

1120573119890minus1205731199051 minus 12057321199052

1119890minus1205731199051]]

gt 0

(14)

Let

1205743= ℎ119892 [1 minus

(1 minus 119864 [119909])

2(1 minus

1

119899)]

1205744= (119872120573 + ℎ119892119875

1)

(15)

then (13) can be rewritten as

1198892119864 [119879119862119880 (119905

1)]

1198891199051

2gt 0

119894119891 [2 (119870 + 1198991198701) 120573 + 2 (1 minus 119890

minus1205731199051) 1205744

minus 1199051[1199052

1119875112057321205743+ 1205744(2 + 120573119905

1)] 120573119890minus1205731199051]

gt 0

(16)

or

1198892119864 [119879119862119880 (119905

1)]

1198891199051

2gt 0

119894119891 0 lt 1199051lt

2 (119870 + 1198991198701) 120573 + 2 (1 minus 119890

minus1205731199051) 1205744

[1199052

1119875112057321205743+ 1205744(2 + 120573119905

1)] 120573119890minus1205731199051

= 120585 (1199051)

(17)

If 119864[119879119862119880(1199051)] is a convex function then the minimum

point exists In order to locate the optimal production run time119905lowast

1that minimizes 119864[119879119862119880(119905

1)] we can set the first derivative of

119864[119879119862119880(1199051)] equal to zero and solve 119905lowast

1

119889119864 [119879119862119880 (1199051)]

1198891199051

=120582

(1 minus 119864 [119909])

sdot minus (119870 + 119899119870

1)

1199052

1119875

+1205742

2+ 1205743(120573119890minus1205731199051)

+ [119872

1198751

+ℎ119892

120573](

minus (1 minus 119890minus1205731199051)

1199052

1

+120573119890minus1205731199051

1199051

)

(18)

In the RHS of (18) it can be seen that the first term ispositive so the second term is equal to zero Let

119905lowast

1119880= radic

2 (1205744+ 119870120573 + 119899119870

1120573)

12057421198751120573

(19)

119905lowast

1119871= 119905ℎ119890 119901119900119904119894119905119894V119890 119903119900119900119905 119900119891

minus1205744plusmn radic1205744

2 + 21198751(1205742+ 2120573120574

3) (119870 + 119899119870

1)

1198751(1205742+ 2120573120574

3)

(20)

Theorem 2 (119905lowast1119871

lt 119905lowast

1lt 119905lowast

1119880) Because the proof of 119905lowast

1falls

within the upper and lower bounds we can multiply the secondterm of (18) by (2120573119875

11199052

1) and obtain

(11987511205731205742+ 2119875112057321205743119890minus1205731199051) 1199052

1+ (21205744120573119890minus1205731199051) 1199051

minus2 [120573 (119870 + 1198991198701) + 1205744(1 minus 119890

minus1205731199051)] = 0

(21)

The Scientific World Journal 7

Producerrsquosproduction-shipment system

Extended toProductionunit

Productionunit

Multideliverypolicy

Multideliverypolicy

Optimization OptimizationProducer-retailer integratedproduction-shipment system

Customer

CustomerRetailer storeor sales office

Figure 4 Extension to a producer-retailer integrated production-shipment system

Thus119905lowast

1

= 119905ℎ119890 119901119900119904119894119905119894V119890 119903119900119900119905 119900119891

(minus (21205744120573119890minus1205731199051)

plusmn ((21205744120573119890minus1205731199051)2

minus [4 (11987511205731205742+ 2119875112057321205743119890minus1205731199051)

times [minus2 [120573 (119870 + 1198991198701) + 1205744(1 minus 119890

minus1205731199051)]]])

12

)

times (2 (12057421198751120573 + 2119875

112057321205743119890minus1205731199051))minus1

(22)

Equation (21) can be rearranged as

2 [1198751120573212057431199052

1+ 12057441205731199051+ 1205744] (119890minus1205731199051)

= 2 [120573 (119870 + 1198991198701) + 1205744] minus (119875

112057312057421199052

1)

(23)

or

119890minus1205731199051 =

2 [120573 (119870 + 1198991198701) + 1205744] minus (119875

112057312057421199052

1)

2 [1198751120573212057431199052

1+ 12057441205731199051+ 1205744]

(24)

where 119890minus1205731199051 is the complement of the cumulative density

function 119865(1199051) = 1 minus 119890

minus1205731199051 As 0 le 119865(119905

1) le 1 0 le 119890

minus1205731199051 le 1

Let 119890minus1205731199051 = 0 and 119890minus1205731199051 = 1 be the upper and lower bounds for119890minus1205731199051 respectively By substituting them into (22) we obtain

119905lowast

1119880= radic

2 [120573 (119870 + 1198991198701) + 1205744]

12057421198751120573

119905lowast

1119871= 119905ℎ119890 119901119900119904119894119905119894V119890 119903119900119900119905 119900119891

minus1205744plusmn radic1205744

2 + 21198751(1205742+ 2120573120574

3) (119870 + 119899119870

1)

1198751(1205742+ 2120573120574

3)

(25)

and 119905lowast1119871lt 119905lowast

1lt 119905lowast

1119880

It is noted that although the optimal production run time119905lowast

1cannot be presented in a closed form it does fall within the

bounds 119905lowast1can be locatedwith the use of a proposed recursive

searching algorithm Let

120596 (1199051) = 119890minus1205731199051 =

2 [120573 (119870 + 1198991198701) + 1205744] minus (119875

112057312057421199052

1)

2 [1198751120573212057431199052

1+ 12057441205731199051+ 1205744]

there4 0 le 120596 (1199051) le 1

(26)

In order to locate 119905lowast1 we can use the following recursive

searching algorithm

(1) Let 120596(1199051) = 0 and 120596(119905

1) = 1 initially and calculate

the upper and lower bounds for 119905lowast1 respectively (ie

to obtain the initial values of [119905lowast1119871 119905lowast

1119880])

(2) Substitute the current values of [119905lowast1119871 119905lowast

1119880] into 119890minus1205731199051 and

compute the new bounds expressed as 120596119871and 120596

119880for

119890minus1205731199051 Hence 120596

119871lt 120596(1199051) lt 120596119880

(3) Let 120596(1199051) = 120596119871and 120596(119905

1) = 120596119880and update the upper

and lower bounds for 119905lowast1 respectively (ie to obtain

the new values of [119905lowast1119871 119905lowast

1119880])

(4) Repeat steps (2) and (3) until there is no signifi-cant difference between 119905lowast

1119871and 119905lowast1119880

(or there is nosignificant difference in terms of their effects on119864[TCU(119905lowast

1)])

(5) Stop 119905lowast1is found

3 Extension to a Producer-Retailer IntegratedEPQ-Based System (Model 2)

31 Enhanced Model Description and Formulation In thissection we further extend the scope of the problem to incor-porate the retailerrsquos stock holding cost into our study Thenew model can be considered a producer-retailer integratedsystem because in the present-day manufacturing sectorsome producers of consumer goods may own and operateretail stores or regional sales offices to promote and sell theirend products to customers (see Figure 4) With the intentionof addressing such a real-life intrasupply chain situation thesecond model of this study incorporates the retailerrsquos stockholding cost into the first model and investigates its effect onthe optimal production run time decision

8 The Scientific World Journal

Dminus 120582tn = I

D

I

t1 + tr tn

Time

t1 + tr

Ic(t)

t9984002

T998400

Figure 5 Inventory level of finished products on the retailerrsquos side in the proposed manufacturing run time problem with breakdown

In the proposed study the retailerrsquos stock holding posi-tions are illustrated in Figure 5

Extra parameters used in this enhanced model includethe following

ℎ2 holding cost per product stored on the retailerrsquos

side

119868119888(119905) on-hand inventory levels in units on the retail-

errsquos side end at time 119905

119863 number of finished products (a fixed quantity)transported to the retail store per shipment

119868 number of left-over products in 119905119899after satisfying

the demand in 119905119899

TC3(1199051) total production-inventory-delivery costs

per cycle of this enhanced model (in the case of abreakdown)

TC4(1199051) total production-inventory-delivery costs

per cycle of this enhanced model (in the case of nobreakdown)

119864[TC3(1199051)] the expected production-inventory-

delivery costs per cycle of this enhanced model (inthe case of a breakdown)

119864[TC4(1199051)] the expected production-inventory-

delivery costs per cycle of this enhanced model (inthe case of no breakdown)

119864[TCU119890(1199051)] the long-run expected production-

inventory-delivery costs per unit time in thisenhanced model whether or not a breakdown takesplace

Since the demand on the retailerrsquos side in time interval119905119899is 120582119905119899 after satisfying the demand the number of left-over

items (see Figure 5) in each 119905119899is

119868 = 119863 minus 120582119905119899 (27)

Total inventory holding costs on retailerrsquos side with andwithout breakdown are shown respectively in

ℎ2[119899(119863 minus 119868)

2119905119899+119899 (119899 + 1)

2119868119905119899+119899119868

2(1199051+ 119905119903)] (28)

ℎ2[119899(119863 minus 119868)

2119905119899+119899 (119899 + 1)

2119868119905119899+119899119868

2(1199051)] (29)

To incorporate the retailerrsquos holding costs into the originalmodels with and without breakdown respectively we obtain

TC3(1199051) = TC

1(1199051)

+ ℎ2[119899(119863 minus 119868)

2119905119899+119899 (119899 + 1)

2119868119905119899+119899119868

2(1199051+ 119905119903)]

TC4(1199051)

= TC2(1199051) + ℎ2[119899(119863 minus 119868)

2119905119899+119899 (119899 + 1)

2119868119905119899+119899119868

2(1199051)]

(30)

To take the randomness of defective rate 119909 into accountand substitute all related variables into (30) with furtherderivations 119864[TC

3(1199051)] and 119864[TC

4(1199051)] can be obtained as

follows

119864 [TC3(1199051)] = 119864 [TC

1(1199051)] + ℎ

2(1198751119892 (1 minus 119864 [119909])

2)

sdot (1 minus1

119899) 1199051+ ℎ2(1198751(1 minus 119864 [119909])

2)

sdot [1198751(1 minus 119864 [119909])

120582119899+ (1 minus

1

119899)] 1199052

1

119864 [TC4(1199051)] = 119864 [TC

2(1199051)] + ℎ

2(1198751(1 minus 119864 [119909])

2)

sdot [1198751(1 minus 119864 [119909])

120582119899+ (1 minus

1

119899)] 1199052

1

(31)

The Scientific World Journal 9

32 Integration of Enhanced Model withwithout BreakdownThe mean time to breakdowns obeys the exponential distri-bution with 119891(119905) = 120573119890minus120573119905 Therefore 119864[TCU

2(1199051)] is

119864 [TCU2(1199051)]

=

int1199051

0119864 [TC

3(1199051)] 119891 (119905) 119889119905 + int

infin

1199051

119864 [TC4(1199051)] 119891 (119905) 119889119905

119864 [T]

(32)

Substituting 119864[TCU3(1199051)] 119864[TC

4(1199051)] and 119864[T] into (32)

and resolving 119864[TCU2(1199051)] we obtain

119864 [TCU2(1199051)]

=120582

(1 minus 119864 [119909])

sdot (119870 + 119899119870

1)

11990511198751

+ 1205741+12057421199051

2+ 12057451199051

+ [119872

1198751

+ℎ119892

120573](

1 minus 119890minus1205731199051

1199051

) minus ℎ119892 (119890minus1205731199051)

minus (ℎ minus ℎ2) [119892 (1 minus 119864 [119909])

2(1 minus

1

119899)] (1 minus 119890

minus1205731199051)

(33)

where

1205745=ℎ2(1 minus 119864 [119909])

2[1198751(1 minus 119864 [119909])

120582119899+ (1 minus

1

119899)] (34)

33 Determining the Optimal Run Time Let 120595(1199051) stand for

the following

120595 (1199051) =

2 (119870 + 1198991198701) 120573 + 2120574

4(1 minus 119890

minus1205731199051)

[1199052

1119875112057321205746+ 1205744(2 + 120573119905

1)] 120573119890minus1205731199051

(35)

Theorem 3 (119864[TCU2(1199051)] is convex if 0 lt 119905

1lt 120595(119905

1)) The

second derivative of 119864[1198791198621198802(1199051)] with respect to 119905

1is

1198892119864 [119879119862119880

2(1199051)]

11988921199052

1

=120582

(1 minus 119864 [119909])[2 (119870 + 119899119870

1)

1199053

11198751

+ (ℎ minus ℎ2)119892 (1 minus 119864 [119909])

2

sdot (1 minus1

119899) (1205732119890minus1205731199051) minus ℎ119892 (120573

2119890minus1205731199051)

+ [119872

1198751

+ℎ119892

120573](

2 (1 minus 119890minus1205731199051)

1199053

1

minus2120573119890minus1205731199051

1199052

1

minus1205732119890minus1205731199051

1199051

)]

(36)

Since annual demand 120582 gt 0 the first term in the RHS of(36) is positive and

119894119891 [2 (119870 + 119899119870

1)

1199053

11198751

+ (ℎ minus ℎ2) [119892 (1 minus 119864 [119909])

2(1 minus

1

119899)]

sdot (1205732119890minus1205731199051) minus ℎ119892 (120573

2119890minus1205731199051) + [

119872

1198751

+ℎ119892

120573]

sdot(

2 (1 minus 119890minus1205731199051)

1199053

1

minus2120573119890minus1205731199051

1199052

1

minus1205732119890minus1205731199051

1199051

)] gt 0

119905ℎ1198901198991198892119864 [119879119862119880

2(1199051)]

1198891199051

2gt 0

(37)

With further derivations the left-hand side (LHS) of (37)becomes

119894119891 [2 (119870 + 1198991198701) 120573 minus 119905

3

11198751120573 (ℎ minus ℎ

2) [119892 (1 minus 119864 [119909])

2(1 minus

1

119899)]

sdot (1205732119890minus1205731199051) minus 1199053

11198751120573ℎ119892 (120573

2119890minus1205731199051)

+ (119872120573 + ℎ1198921198751) [2 (1 minus 119890

minus1205731199051) minus 2119905

1120573119890minus1205731199051 minus 12057321199052

1119890minus1205731199051] ]

gt 0

(38)

Let

1205746= (ℎ minus ℎ

2) [119892 (1 minus 119864 [119909])

2(1 minus

1

119899)] + ℎ119892 (39)

then (37) becomes

119894119891 [2 (119870 + 1198991198701) 120573 + 2120574

4(1 minus 119890

minus1205731199051)

minus1199051[1199052

1119875112057321205746+ 1205744(2 + 120573119905

1)] 120573119890minus1205731199051] gt 0

119905ℎ1198901198991198892119864 [119879119862119880

2(1199051)]

1198891199051

2gt 0

(40)

or

1198892119864 [119879119862119880

2(1199051)]

1198891199051

2gt 0

119894119891 0 lt 1199051lt

2 (119870 + 1198991198701) 120573 + 2120574

4(1 minus 119890

minus1205731199051)

[1199052

1119875112057321205746+ 1205744(2 + 120573119905

1)] 120573119890minus1205731199051

= 120595 (1199051)

(41)

10 The Scientific World Journal

Once 119864[1198791198621198802(1199051)] is proven to be convex the optimal

run time 119905lowast1can be solved by setting the first derivative of

119864[1198791198621198802(1199051)] = 0

119889119864 [1198791198621198802(1199051)]

1198891199051

=120582

(1 minus 119864 [119909])sdot minus (119870 + 119899119870

1)

1199052

1119875

+ (1205742

2+ 1205745) + 1205746(120573119890minus1205731199051)

+ [119872

1198751

+ℎ119892

120573](

minus (1 minus 119890minus1205731199051)

1199052

1

+120573119890minus1205731199051

1199051

)

= 0

(42)

It can be seen that the first term in the RHS of (42) ispositive so the second term is equal to zero In order to findthe bounds for 119905lowast

1 let

119905lowast

1119880= radic

2 [120573 (119870 + 1198991198701) + 1205744]

1198751120573 (1205742+ 21205745)

(43)

119905lowast

1119871= 119905ℎ119890 119901119900119904119894119905119894V119890 119903119900119900119905 119900119891

minus1205744plusmn radic1205742

4+ 21198751(119870 + 119899119870

1) (1205742+ 21205745+ 21205746120573)

1198751(1205742+ 21205745+ 21205746120573)

(44)

Theorem4 (119905lowast1119871lt 119905lowast

1lt 119905lowast

1119880) For the proof ofTheorem 4 please

refer to the proof for Theorem 2 in Section 2

Once we are certain that 119905lowast1falls within the aforemen-

tioned upper and lower bounds in order to find 119905lowast1 we can

first multiply the second term of (42) by (211987511199052

1120573) and obtain

the following

[1198751120573 (1205742+ 21205745) + 2119875

112057321205746119890minus1205731199051] 1199052

1+ (21205744120573119890minus1205731199051) 1199051

minus2 [120573 (119870 + 1198991198701) + 1205744(1 minus 119890

minus1205731199051)]

= 0

(45)

Equation (45) can be rearranged as

119890minus1205731199051 =

2 [120573 (119870 + 1198991198701) + 1205744] minus [119875

1120573 (1205742+ 21205745)] 1199052

1

2 [1198751120573212057461199052

1+ 1205744(1 + 120573119905

1)]

(46)

where 119890minus1205731199051 is the complement of the cumulative densityfunction 119865(119905

1) = 1minus119890

minus1205731199051 As 0 le 119865(119905

1) le 1 0 le 119890minus1205731199051 le 1 Let

119890minus1205731199051 = 0 and 119890minus1205731199051 = 1 be the initial upper and lower bounds

of 119890minus1205731199051 respectively Then by using the proposed recursivesearching algorithm given at the end of Section 2 we can findthe optimal production run time 119905lowast

1

4 Numerical Example

In order to relieve the comparison efforts for readers thissection adopts the same numerical example as in [9] For

0225 0265 0305 0345 0385 0425 0465 0505

E[T

CU(t

1)]

t1

tlowast1L = 03411 t

lowast1 = 03748

tlowast1U = 05183

$10600

$10800

$11000

$11200

$11400

$11600

$11006

$11800

Figure 6 The behavior of 119864[TCU(1199051)] in connection with the

production run time 1199051in the proposed model 1

a demonstration of the proposed EPQ-based model 1 thefollowing system parameters are used

1198751 production rate 10000 products per year120582 demand rate 4000 products per year119909 random scrap rate which follows uniformly distribu-

tion over the interval [0 02]120573 Poisson breakdown rate 05 average times per year119892 constant machine repair time 119905

119903 0018 year per repair

119872 machine repair cost $500 for each breakdown119870 setup cost $450 per production run119862 manufacturing cost $2 per item119862119878 disposal cost $03 per scrap itemℎ holding cost $06 per item per unit time1198701 fixed delivery cost $90 per shipment119899 number of deliveries 4 per cycle

119862119879 variable delivery cost $0001 per item

First we use both upper and lower bounds of 119905lowast1to

test for the convexity of 119864[TCU(1199051)] (see Theorem 1) The

computation results of (19) (20) and (11) indicate that 119905lowast1119880

=05183 lt 120585(119905

lowast

1119880) = 28723 and 119905lowast

1119871= 03411 lt 120585(119905

lowast

1119871) = 26261

Hence [TCU(1199051)] is convex (Figure 6)

In order to find the optimal 119905lowast1 we first substitute the

upper and lower bounds of 119905lowast1in (8) and obtain 119864[TCU(119905lowast

1119880)]

= $1160163 and 119864[TCU(119905lowast1119871)] = $1101450 respectively

Because the optimal run time 119905lowast1falls within the interval

of [119905lowast1119871 119905lowast

1119880] we apply the proposed recursive searching

algorithm stated at the end of Section 2 and find 119905lowast1= 03748

years Accordingly the optimal expected system costs perunit time 119864[TCU(119905lowast

1)] = $1100641 (Figure 6) Table 1 shows

the step-by-step iterations of the algorithmIn this specific studied model we focus on incorporating

a discontinuous multidelivery policy into a prior work [9]and consider a fixed transportation cost associated witheach delivery Applying the research result we can easilyinvestigate the effects of different fixed transportation costs

The Scientific World Journal 11

Table 1 Iterations of the recursive searching algorithm for locating 119905lowast1

120573 Step 119905lowast

1119880120596119880= 119890minus1205731199051119880 119905

lowast

1119871120596119871= 119890minus1205731199051119871

Differencebetween 119905lowast

1119880and

119905lowast

1119871

[119880]

119864[TCU(119905lowast1119880)]

[119871]

119864[TCU(119905lowast1119871)]

Differencebetween [119880]

and [119871]05 Initial 000000 100000 $1160163 $1101450 $58713

1st 05183 07717 03411 08432 01772 $1110330 $1101450 $88802nd 03857 08246 03721 08302 00136 $1100716 $1100646 $0703rd 03756 08287 03746 08292 00010 $1100642 $1100641 $0014th 03749 08291 03748 08291 00001 $1100641 $1100641 $00045th 03748 08291 03748 08291 000000 $1100641 $1100641 $0000

Table 2 Variations of the fixed delivery cost1198701effects on the optimal production run time 119905lowast

1

1198701119870 005 02 04 06 08 1 12 14 16 18 2 22

1198701

$225 $90 $180 $270 $360 $450 $540 $630 $720 $810 $900 $990119864[TCU(119905lowast

1)] $10721 $11067 $11448 $11773 $12062 $12325 $12567 $12793 $13006 $13208 $13399 $13583

119905lowast

103116 03816 04586 05244 05828 06359 06848 07305 07735 08142 08529 08900

Table 3 Variations of the unit retailerrsquos holding cost ℎ2and their effects on 119864[TCU(119905lowast

1)]

ℎ2ℎ 05 075 1 125 15 175 2 225 25 275 3 325

ℎ2

03 045 06 075 09 105 12 135 15 165 18 195119864 [TCU (119905lowast

1)] $11282 $11407 $11526 $11638 $11746 $11850 $11949 $12045 $12138 $12229 $12316 $12401

119905lowast

103256 03074 02918 02785 02668 02564 02472 02389 02314 02246 02183 02125

1198701on the optimal system cost 119864[TCU(119905lowast

1)] and on the

optimal production run time 119905lowast1(see Table 2) It can be seen

from Table 2 that as the ratio of1198701119870 increases the expected

system costs per unit time119864[TCU(119905lowast1119871)] increase significantly

It is also noted that as 1198701increases optimal production run

time 119905lowast1also increases significantly

41 Numerical Example for the Producer-Retailer IntegratedEPQ System (Model 2) In order to demonstrate the researchresult of the producer-retailer integrated EPQ-based modelan additional system variable ℎ

2= $150 per item stored at the

retailerrsquos side is includedAgain one can use the upper and lower bounds of 119905lowast

1

(equations (43) and (44)) to test for convexity of 119864[TCU(1199051)]

(Theorem 3 and equation (35)) The results reveal that 119905lowast1119880

=03213 lt 120595(119905lowast

1119880) = 26462 and 119905lowast

1119871= 02186 lt 120595(119905lowast

1119871) = 24876

Therefore the expected cost [TCU(1199051)] is convex

Next by applying the proposed recursive searching algo-rithm we can calculate that the optimal run time 119905lowast

1=

02314 years and the optimal 119864[TCU(119905lowast1)] = $1213849 It is

noted that the computation time for reaching the optimal 119905lowast1

solution is 21 seconds (usingExcel software in a desktop com-puter Intel CPU G850 with 294GB RAM and 289GHz)

Figure 7 illustrates the behavior of 119864[TCU(1199051)] with

regard to production run time It is noted that without theresearch result from the second model the management ofsuch a producer-retailer integrated system would probablyuse 1199051= 03748 years (from the result of model 1) for their

run time decision Further analysis (see Figure 7) shows cost

0130 0170 0210 0250 0290 0330 0370 0410

t1

E[T

CU(t

1)]

11600

11800

12000

12200

12400

12600

12800

tlowast1 = 02314

t1 = 0374812138

(from model 1)

E[TCU( t1)] = 12489

$$

$

$

$

$

$

$

$

Figure 7 The behavior of 119864[TCU(1199051)] with respect to production

run time 1199051in the proposed model 2

savings of $351 (or 29 over the total system costs) simply byapplying our research result

The effects of the unit retailerrsquos holding cost ℎ2on the

expected system cost 119864[TCU(119905lowast1)] and on the optimal run

time 119905lowast1are shown in Table 3 respectively

It can be seen that as ℎ2or the ratio of ℎ

2ℎ increases

the expected cost 119864[TCU(119905lowast1)] increases but the optimal

production run time 119905lowast1decreases In decision-making these

sensitivity analyses results can provide the management ofa producer-retailer integrated system with valuable informa-tion and insights into the effects of various stock holding costsin different retailersrsquo locations

12 The Scientific World Journal

5 Concluding Remarks

Two exact models for an extended EPQ-based problem witha discontinuous delivery policy scrap rate and randombreakdown are developed in this study They specificallyaddress different real-life situations in production end-item delivery and intrasupply chains such as a producer-retailer integrated systemMathematicalmodeling alongwithoptimization techniques is used to determine the optimalproduction run times that minimize the expected systemcosts per unit time Without in-depth investigations onthese separate models the optimal production run time andother important information related to the systemparameterscannot be revealed The proposed real-life EPQ models withrandom machine breakdown discontinuous product distri-bution policies and quality assurance must be specificallystudied in order to (1) obtain the joint effects of breakdowndiscontinuous distribution policies and quality assurance onthe optimal production run time (2) get to know the effectsof different policy and scope of supply chains managementon the optimal run time and overall system costs and (3)gain the insight with regard to various systemrsquos parametersof all particular EPQ-based models Since little attention hasbeen paid to the investigation of joint effects of these practicalproduction situations on the optimal run time this researchis intended to bridge the gap An interesting area for futurestudy is the examination of the effect of variable productionrates on these models

Appendix

Derivations of (3) are as followsRecall (2) as follows

TC1(1199051) = 119862 (119875

11199051) + 119870 + 119862

119878(11990511198751119909) +119872 + 119899119870

1

+ 119862119879[11990511198751] + ℎ3(120582119905119903) 1198791015840

+ ℎ[1198671015840+ 11988911199051

21199051+ (1198671015840

1+ 1198891119905) 119905119903+119899 minus 1

211989911986710158401199051015840

2]

(A1)

Substituting all related system parameters into (2) (pleaserefer to the basic formulations and solution process in [9])the TC

1(1199051) can be obtained as

TC1(1199051) = 119870 +119872

+[1198621198751+ 1198621198781198751119909 + 119862

1198791198751(1 minus 119909)+ ℎ

31198751119892 (1 minus 119909)] 119905

1

+ 1198991198701+ ℎ119901119892119905 minus[

ℎ1198751119892 (1 minus 119909)

2minusℎ1198751119892 (1 minus 119909)

2119899] 1199051

+ 1199052

1[ℎ1198751

2+ℎ1198752

1

2120582(1 minus 119909)

2minusℎ1198751

2(1 minus 119909)

minusℎ1198752

1

2120582119899(1 minus 119909)

2+ℎ1198751

2119899(1 minus 119909)]

(A2)

To take the randomness of 119909 into account by using theexpected values of 119909 with further derivations119864[TC

1(1199051)] can

be derived as follows (ie equation (3))119864 [TC

1(1199051)]

= 119870 + 1198991198701+119872 + ℎ119905119875

1119892

+ [1198621198751+ 1198621198781198751119864 [119909] + 1198621198791198751 (1 minus 119864 [119909])

+ ℎ31198751119892 (1 minus 119864 [119909]) minus

ℎ1198751119892 (1 minus 119864 [119909])

2(1 minus

1

119899)] sdot 1199051

+ [ℎ1198751119864 [119909]

2+ℎ1198752

1

2120582(1 minus 119864 [119909])

2(1 minus

1

119899)

+ℎ1198751

2119899(1 minus 119864 [119909])] 119905

2

1

(A3)

Conflict of Interests

The authors of the paper declare that there is no conflict ofinterests regarding the publication of this paper

Acknowledgment

The authors greatly appreciate the National Science Council(NSC) of Taiwan for supporting this research underGrant noNSC 102-2410-H-324-005

References

[1] E W Taft ldquoThemost economical production lotrdquo Iron Age vol101 pp 1410ndash1412 1918

[2] G Hadley and T M Whitin ldquoAn optimal final inventorymodelrdquoManagement Science vol 7 pp 179ndash183 1961

[3] E A Silver D F Pyke and R Peterson Inventory Managementand Production Planning and Scheduling John Wiley amp SonsNew York NY USA 1998

[4] S Nahmias Production amp Operations Analysis McGraw-HillNew York NY USA 2009

[5] MWidmer and P Solot ldquoDo not forget the breakdowns and themaintenance operations in FMSdesign problemsrdquo InternationalJournal of Production Research vol 28 pp 421ndash430 1990

[6] K-Y C Yu and D L Bricker ldquoAnalysis of a markov chainmodel of a multistage manufacturing system with inspectionrejection and reworkrdquo IIE Transactions vol 25 no 1 pp 109ndash112 1993

[7] H Groenevelt L Pintelon and A Seidmann ldquoProduction lotsizing with machine breakdownsrdquoManagement Science vol 38no 1 pp 104ndash123 1992

[8] G A Widyadana and H M Wee ldquoOptimal deterioratingitems production inventory models with random machinebreakdown and stochastic repair timerdquo Applied MathematicalModelling vol 35 no 7 pp 3495ndash3508 2011

[9] S W Chiu C-L Chou and W-K Wu ldquoOptimizing replenish-ment policy in an EPQ-based inventory model with noncon-forming items and breakdownrdquo EconomicModelling vol 35 pp330ndash337 2013

The Scientific World Journal 13

[10] A M Zargar ldquoEffect of rework strategies on cycle timerdquoComputers amp Industrial Engineering vol 29 no 1ndash4 pp 239ndash243 1995

[11] P Biswas and B R Sarker ldquoOptimal batch quantity modelsfor a lean production system with in-cycle rework and scraprdquoInternational Journal of Production Research vol 46 no 23 pp6585ndash6610 2008

[12] S W Chiu H-D Lin C-B Cheng and C-L Chung ldquoOptimalproduction-shipment decisions for the finite production ratemodel with scraprdquo International Journal for Engineering Mod-elling vol 22 no 1ndash4 pp 25ndash34 2009

[13] Y-S P Chiu K-K Chen and C-K Ting ldquoReplenishment runtime problem with machine breakdown and failure in reworkrdquoExpert Systems with Applications vol 39 no 1 pp 1291ndash12972012

[14] H-D Lin F-Y Pai and S W Chiu ldquoA note on ldquointra-supply chain system with multiple sales locations and qualityassurancerdquordquo Expert Systems with Applications vol 40 no 11 pp4730ndash4732 2013

[15] B Kamsu-Foguem F Rigal and FMauget ldquoMining associationrules for the quality improvement of the production processrdquoExpert Systems with Applications vol 40 no 4 pp 1034ndash10452013

[16] Y Li H Guo L Wang and J Fu ldquoA hybrid genetic-simulatedannealing algorithm for the location-inventory- routing prob-lem considering returns under E-supply chain environmentrdquoThe Scientific World Journal vol 2013 Article ID 125893 10pages 2013

[17] Y-S P Chiu K-K Chen F-T Cheng and C-K Ting ldquoReex-amination of ldquocombining an alternative multi-delivery policyinto economic production lot size problemwith partial reworkrdquousing an alternative approachrdquo Journal of Applied Research andTechnology vol 11 no 3 pp 317ndash323 2013

[18] L B Schwarz B L Deuermeyer and R D Badinelli ldquoFill-rateoptimization in a one-warehouse 119873-identical retailer distribu-tion systemrdquo Management Science vol 31 no 4 pp 488ndash4981985

[19] R A Sarker and L R Khan ldquoOptimal batch size for a pro-duction system operating under periodic delivery policyrdquoCom-puters amp Industrial Engineering vol 37 no 4 pp 711ndash730 1999

[20] N Comez K E Stecke and M Cakanyildirim ldquoMultiple in-cycle transshipments with positive delivery timesrdquo Productionand Operations Management vol 21 no 2 pp 378ndash395 2012

[21] S W Chiu F-Y Pai and W K Wu ldquoAlternative approach todetermine the common cycle time for a multi-item productionsystem with discontinuous deliveries and failure in reworkrdquoEconomic Modelling vol 35 pp 593ndash596 2013

[22] Y-S P Chiu C-C Huang M-F Wu and H-H Chang ldquoJointdetermination of rotation cycle time and number of shipmentsfor a multi-item EPQ model with random defective raterdquoEconomic Modelling vol 35 pp 112ndash117 2013

[23] M A Hoque ldquoSynchronization in the single-manufacturermulti-buyer integrated inventory supply chainrdquo European Jour-nal of Operational Research vol 188 no 3 pp 811ndash825 2008

[24] S W Chiu L-W Lin K-K Chen and C-L Chou ldquoDetermin-ing production-shipment policy for a vendor-buyer integratedsystem with rework and an amending multi-delivery schedulerdquoEconomic Modelling vol 33 pp 668ndash675 2013

[25] M Cedillo-Campos and C Sanchez-Ramırez ldquoDynamic self-assessment of supply chains performance an emerging marketapproachrdquo Journal of Applied Research and Technology vol 11no 3 pp 338ndash347 2013

[26] Y-S P Chiu H-D Lin F-T Cheng and M-H Hwang ldquoOpti-mal common cycle time for a multi-item production systemwith discontinuous delivery policy and failure in reworkrdquoJournal of Scientific and Industrial Research vol 72 no 7 pp435ndash440 2013

[27] HHishamuddin R A Sarker andD Essam ldquoA recoverymech-anism for a two echelon supply chain system under supplydisruptionrdquo Economic Modelling vol 38 pp 555ndash563 2014

[28] M Murugan and V Selladurai ldquoProductivity improvement inmanufacturing submersible pump diffuser housing using leanmanufacturing systemrdquo Journal of Engineering Research vol 2no 1 pp 164ndash182 2014

[29] L Wang H Qu S Liu and C-X Dun ldquoModeling and opti-mization of the multiobjective stochastic joint replenishmentand delivery problem under supply chain environmentrdquo TheScientific World Journal vol 2013 Article ID 916057 11 pages2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Mathematical PhysicsAdvances in

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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

International Journal of Mathematics and Mathematical Sciences

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 The Scientific World Journal

Id(t)

t9984002

Time

t

T998400

d1t

d1t1

d1

d1

d1

tr t1 minus t

Figure 2 Inventory level of scrap items in the proposed manufacturing run time problem

products is produced the maximum number of scraps in acycle is 119909119876 (or 119889

11199051) and the on-hand inventory of scrap

items in the proposed manufacturing run time problem is asillustrated in Figure 2

The production cycle time 1198791015840 can be seen as (1) fromFigure 1

1198791015840= 1199051+ 119905119903+ 1199051015840

2 (1)

The total production-inventory-delivery cost per cycleTC1(1199051) is comprised of (1) the variable production cost (2)

the setup cost (3) the disposal cost for scraps (4) themachinerepair cost (5) fixed and variable product delivery costs (6)holding cost for safety stocks and (7) the producerrsquos inventoryholding costs in the entire production cycle Thus TC

1(1199051) is

TC1(1199051) = 119862 (119875

11199051) + 119870 + 119862

119878(11990511198751119909) +119872 + 119899119870

1

+ 119862119879[11990511198751] + ℎ3(120582119905119903) 1198791015840

+ ℎ[1198671015840+ 11988911199051

21199051+ (1198671015840

1+ 1198891119905) 119905119903+119899 minus 1

211989911986710158401199051015840

2]

(2)

Since 119909 is assumed to be a random variable with a knownprobability density function the expected values of119909 are usedin our analysis to take the randomness of 119909 into account Bysubstituting all related system parameters into (2) [9] withfurther derivations 119864[TC

1(1199051)] becomes (see the appendix

for more details)

119864 [TC1(1199051)]

= 119870 + 1198991198701+119872 + ℎ119905119875

1119892

+ [1198621198751+ 1198621198781198751119864 [119909] + 1198621198791198751 (1 minus 119864 [119909])

+ ℎ31198751119892 (1 minus 119864 [119909]) minus

ℎ1198751119892 (1 minus 119864 [119909])

2(1 minus

1

119899)] sdot 1199051

+ [ℎ1198751119864 [119909]

2+ℎ1198752

1

2120582(1 minus 119864 [119909])

2(1 minus

1

119899)

+ℎ1198751

2119899(1 minus 119864 [119909])] 119905

2

1

(3)

22 Case 2 No Breakdown Takes Place at Uptime 1199051 In such

a situation 119905 gt 1199051 The inventory level of finished items in

this case is depicted in Figure 3 and 119879 = 1199051+ 1199052 The total

production-inventory-delivery cost per cycle TC2(1199051) is as

displayed inTC2(1199051) = 119862 (119875

11199051) + 119870 + 119862

119878(11990911990511198751) + 119899119870

1

+ 119862119879[11990511198751(1 minus 119909)] + ℎ

3(120582119905119903) 119879

+ ℎ [119867 + 119889

11199051

21199051+119899 minus 1

21198991198671199052]

(4)

Again to take the randomness of 119909 into account andsubstitute all related parameters into (4) with further deriva-tions 119864[TC

2(1199051)] becomes [12]

119864 [TC2(1199051)] = 119870 + 119899119870

1

+ [1198621198751+ 119862119878119864 [119909] 1198751 + 1198621198791198751 (1 minus 119864 [119909])

+ ℎ31198751119892 (1 minus 119864 [119909])] 1199051

+ [ℎ1198751119864 [119909]

2+ℎ1198752

1

2120582(1 minus 119864 [119909])

2(1 minus

1

119899)

+ℎ1198751

2119899(1 minus 119864 [119909])] 119905

2

1

(5)

The Scientific World Journal 5

H

P1 minus d1

T

Time

minus120582

tn

t1 t1t2

I(t)

Figure 3 Inventory level of finished items in a manufacturing run time problem with no breakdown defective rate and discontinuousdelivery policy

23 Integration of the Proposed Run TimeModels withwithoutBreakdown Amachine breakdownmay take place randomlyand it follows a Poisson distribution with mean equal to120573 per year Let 119891(119905) be the probability density function ofrandom time 119905 before a breakdown takes place and 119865(119905)

represents the cumulative density function of 119905 Hence thelong-run expected system costs per unit time 119864[TCU(119905

1)]

are

119864 [TCU (1199051)]

=

int1199051

0119864 [TC

1(1199051)] 119891 (119905) 119889119905 + int

infin

1199051

119864 [TC2(1199051)] 119891 (119905) 119889119905

119864 [T]

(6)

where

119864 [T] = int1199051

0

119864 [1198791015840] 119891 (119905) 119889119905 + int

infin

1199051

119864 [119879] 119891 (119905) 119889119905

=11990511198751(1 minus 119864 [119909])

120582

(7)

From Figures 1 and 3 it can be seen that 1198791015840 and 119879 aredifferent in length (1198791015840 is longer than 119879 since it containsmachine repairing time) and because a breakdown can occurrandomly it is necessary to use the integration (ie equation(7)) to derive the expected cycle length

It is also noted that the time between breakdowns obeysthe exponential distribution with density function 119891(119905) =

120573119890minus120573119905 and cumulative density function 119865(119905) = 1 minus 119890

minus120573119905 Bysubstituting 119864[TC

1(1199051)] 119864[TC

2(1199051)] and 119864[T] into (6) and

solving the integration of the mean time to breakdown in119864[TCU(119905

1)] we obtain

119864 [TCU (1199051)]

=120582

(1 minus 119864 [119909])

sdot (119870 + 119899119870

1)

11990511198751

+ 1205741+12057421199051

2+ [

119872

1198751

+ℎ119892

120573](

1 minus 119890minus1205731199051

1199051

)

minusℎ119892 (119890minus1205731199051) minus

ℎ119892 (1 minus 119864 [119909])

2(1 minus

1

119899) (1 minus 119890

minus1205731199051)

(8)

where

1205741= [119862 + 119862

119878119864 [119909] + 119862

119879(1 minus 119864 [119909]) + ℎ

3119892 (1 minus 119864 [119909])]

(9)

1205742= [

ℎ1198751

120582(1 minus 119864 [119909])

2(1 minus

1

119899) + ℎ119864 [119909] +

ℎ

119899(1 minus 119864 [119909])]

(10)

24 Derivation of the Optimal Production Run Time In orderto derive the optimal production run time 119905lowast

1 we first have

to prove that 119864[TCU(1199051)] is convex Let 120585(119905

1) represent the

following

120585 (1199051) =

2 (119870 + 1198991198701) 120573 + 2 (1 minus 119890

minus1205731199051) 1205744

[1199052

1119875112057321205743+ 1205744(2 + 120573119905

1)] 120573119890minus1205731199051

(11)

6 The Scientific World Journal

Theorem 1 (119864[TCU(1199051)] is convex if 0 lt 119905

1lt 120585(119905

1)) The

second derivative of 119864[119879119862119880(1199051)] with respect to 119905

1is

1198892119864 [119879119862119880 (119905

1)]

11988921199052

1

=120582

(1 minus 119864 [119909])

sdot [2 (119870 + 119899119870

1)

1199053

11198751

minus ℎ119892 [1 minus(1 minus 119864 [119909])

2(1 minus

1

119899)] (120573

2119890minus1205731199051)

+ [119872

1198751

+ℎ119892

120573](

2 (1 minus 119890minus1205731199051)

1199053

1

minus2120573119890minus1205731199051

1199052

1

minus1205732119890minus1205731199051

1199051

)]

(12)

It is noted that because annual demand 120582 gt 0 the firstterm in the right-hand size (RHS) of (12) is positive Hence weobtain

1198892119864 [119879119862119880 (119905

1)]

1198891199051

2gt 0

119894119891 [2 (119870 + 119899119870

1)

1199053

11198751

minus ℎ119892 [1 minus(1 minus 119864 [119909])

2(1 minus

1

119899)] (120573

2119890minus1205731199051)

+ [119872

1198751

+ℎ119892

120573](

2 (1 minus 119890minus1205731199051)

1199053

1

minus2120573119890minus1205731199051

1199052

1

minus1205732119890minus1205731199051

1199051

)]

gt 0

(13)

The RHS of (13) can be further derived as

119894119891 [2 (119870 + 1198991198701) 120573 minus 119905

3

11198751120573ℎ119892

sdot [1 minus(1 minus 119864 [119909])

2(1 minus

1

119899)] (120573

2119890minus1205731199051)

+ (119872120573 + ℎ1198921198751)

times [2 (1 minus 119890minus1205731199051) minus 2119905

1120573119890minus1205731199051 minus 12057321199052

1119890minus1205731199051]]

gt 0

(14)

Let

1205743= ℎ119892 [1 minus

(1 minus 119864 [119909])

2(1 minus

1

119899)]

1205744= (119872120573 + ℎ119892119875

1)

(15)

then (13) can be rewritten as

1198892119864 [119879119862119880 (119905

1)]

1198891199051

2gt 0

119894119891 [2 (119870 + 1198991198701) 120573 + 2 (1 minus 119890

minus1205731199051) 1205744

minus 1199051[1199052

1119875112057321205743+ 1205744(2 + 120573119905

1)] 120573119890minus1205731199051]

gt 0

(16)

or

1198892119864 [119879119862119880 (119905

1)]

1198891199051

2gt 0

119894119891 0 lt 1199051lt

2 (119870 + 1198991198701) 120573 + 2 (1 minus 119890

minus1205731199051) 1205744

[1199052

1119875112057321205743+ 1205744(2 + 120573119905

1)] 120573119890minus1205731199051

= 120585 (1199051)

(17)

If 119864[119879119862119880(1199051)] is a convex function then the minimum

point exists In order to locate the optimal production run time119905lowast

1that minimizes 119864[119879119862119880(119905

1)] we can set the first derivative of

119864[119879119862119880(1199051)] equal to zero and solve 119905lowast

1

119889119864 [119879119862119880 (1199051)]

1198891199051

=120582

(1 minus 119864 [119909])

sdot minus (119870 + 119899119870

1)

1199052

1119875

+1205742

2+ 1205743(120573119890minus1205731199051)

+ [119872

1198751

+ℎ119892

120573](

minus (1 minus 119890minus1205731199051)

1199052

1

+120573119890minus1205731199051

1199051

)

(18)

In the RHS of (18) it can be seen that the first term ispositive so the second term is equal to zero Let

119905lowast

1119880= radic

2 (1205744+ 119870120573 + 119899119870

1120573)

12057421198751120573

(19)

119905lowast

1119871= 119905ℎ119890 119901119900119904119894119905119894V119890 119903119900119900119905 119900119891

minus1205744plusmn radic1205744

2 + 21198751(1205742+ 2120573120574

3) (119870 + 119899119870

1)

1198751(1205742+ 2120573120574

3)

(20)

Theorem 2 (119905lowast1119871

lt 119905lowast

1lt 119905lowast

1119880) Because the proof of 119905lowast

1falls

within the upper and lower bounds we can multiply the secondterm of (18) by (2120573119875

11199052

1) and obtain

(11987511205731205742+ 2119875112057321205743119890minus1205731199051) 1199052

1+ (21205744120573119890minus1205731199051) 1199051

minus2 [120573 (119870 + 1198991198701) + 1205744(1 minus 119890

minus1205731199051)] = 0

(21)

The Scientific World Journal 7

Producerrsquosproduction-shipment system

Extended toProductionunit

Productionunit

Multideliverypolicy

Multideliverypolicy

Optimization OptimizationProducer-retailer integratedproduction-shipment system

Customer

CustomerRetailer storeor sales office

Figure 4 Extension to a producer-retailer integrated production-shipment system

Thus119905lowast

1

= 119905ℎ119890 119901119900119904119894119905119894V119890 119903119900119900119905 119900119891

(minus (21205744120573119890minus1205731199051)

plusmn ((21205744120573119890minus1205731199051)2

minus [4 (11987511205731205742+ 2119875112057321205743119890minus1205731199051)

times [minus2 [120573 (119870 + 1198991198701) + 1205744(1 minus 119890

minus1205731199051)]]])

12

)

times (2 (12057421198751120573 + 2119875

112057321205743119890minus1205731199051))minus1

(22)

Equation (21) can be rearranged as

2 [1198751120573212057431199052

1+ 12057441205731199051+ 1205744] (119890minus1205731199051)

= 2 [120573 (119870 + 1198991198701) + 1205744] minus (119875

112057312057421199052

1)

(23)

or

119890minus1205731199051 =

2 [120573 (119870 + 1198991198701) + 1205744] minus (119875

112057312057421199052

1)

2 [1198751120573212057431199052

1+ 12057441205731199051+ 1205744]

(24)

where 119890minus1205731199051 is the complement of the cumulative density

function 119865(1199051) = 1 minus 119890

minus1205731199051 As 0 le 119865(119905

1) le 1 0 le 119890

minus1205731199051 le 1

Let 119890minus1205731199051 = 0 and 119890minus1205731199051 = 1 be the upper and lower bounds for119890minus1205731199051 respectively By substituting them into (22) we obtain

119905lowast

1119880= radic

2 [120573 (119870 + 1198991198701) + 1205744]

12057421198751120573

119905lowast

1119871= 119905ℎ119890 119901119900119904119894119905119894V119890 119903119900119900119905 119900119891

minus1205744plusmn radic1205744

2 + 21198751(1205742+ 2120573120574

3) (119870 + 119899119870

1)

1198751(1205742+ 2120573120574

3)

(25)

and 119905lowast1119871lt 119905lowast

1lt 119905lowast

1119880

It is noted that although the optimal production run time119905lowast

1cannot be presented in a closed form it does fall within the

bounds 119905lowast1can be locatedwith the use of a proposed recursive

searching algorithm Let

120596 (1199051) = 119890minus1205731199051 =

2 [120573 (119870 + 1198991198701) + 1205744] minus (119875

112057312057421199052

1)

2 [1198751120573212057431199052

1+ 12057441205731199051+ 1205744]

there4 0 le 120596 (1199051) le 1

(26)

In order to locate 119905lowast1 we can use the following recursive

searching algorithm

(1) Let 120596(1199051) = 0 and 120596(119905

1) = 1 initially and calculate

the upper and lower bounds for 119905lowast1 respectively (ie

to obtain the initial values of [119905lowast1119871 119905lowast

1119880])

(2) Substitute the current values of [119905lowast1119871 119905lowast

1119880] into 119890minus1205731199051 and

compute the new bounds expressed as 120596119871and 120596

119880for

119890minus1205731199051 Hence 120596

119871lt 120596(1199051) lt 120596119880

(3) Let 120596(1199051) = 120596119871and 120596(119905

1) = 120596119880and update the upper

and lower bounds for 119905lowast1 respectively (ie to obtain

the new values of [119905lowast1119871 119905lowast

1119880])

(4) Repeat steps (2) and (3) until there is no signifi-cant difference between 119905lowast

1119871and 119905lowast1119880

(or there is nosignificant difference in terms of their effects on119864[TCU(119905lowast

1)])

(5) Stop 119905lowast1is found

3 Extension to a Producer-Retailer IntegratedEPQ-Based System (Model 2)

31 Enhanced Model Description and Formulation In thissection we further extend the scope of the problem to incor-porate the retailerrsquos stock holding cost into our study Thenew model can be considered a producer-retailer integratedsystem because in the present-day manufacturing sectorsome producers of consumer goods may own and operateretail stores or regional sales offices to promote and sell theirend products to customers (see Figure 4) With the intentionof addressing such a real-life intrasupply chain situation thesecond model of this study incorporates the retailerrsquos stockholding cost into the first model and investigates its effect onthe optimal production run time decision

8 The Scientific World Journal

Dminus 120582tn = I

D

I

t1 + tr tn

Time

t1 + tr

Ic(t)

t9984002

T998400

Figure 5 Inventory level of finished products on the retailerrsquos side in the proposed manufacturing run time problem with breakdown

In the proposed study the retailerrsquos stock holding posi-tions are illustrated in Figure 5

Extra parameters used in this enhanced model includethe following

ℎ2 holding cost per product stored on the retailerrsquos

side

119868119888(119905) on-hand inventory levels in units on the retail-

errsquos side end at time 119905

119863 number of finished products (a fixed quantity)transported to the retail store per shipment

119868 number of left-over products in 119905119899after satisfying

the demand in 119905119899

TC3(1199051) total production-inventory-delivery costs

per cycle of this enhanced model (in the case of abreakdown)

TC4(1199051) total production-inventory-delivery costs

per cycle of this enhanced model (in the case of nobreakdown)

119864[TC3(1199051)] the expected production-inventory-

delivery costs per cycle of this enhanced model (inthe case of a breakdown)

119864[TC4(1199051)] the expected production-inventory-

delivery costs per cycle of this enhanced model (inthe case of no breakdown)

119864[TCU119890(1199051)] the long-run expected production-

inventory-delivery costs per unit time in thisenhanced model whether or not a breakdown takesplace

Since the demand on the retailerrsquos side in time interval119905119899is 120582119905119899 after satisfying the demand the number of left-over

items (see Figure 5) in each 119905119899is

119868 = 119863 minus 120582119905119899 (27)

Total inventory holding costs on retailerrsquos side with andwithout breakdown are shown respectively in

ℎ2[119899(119863 minus 119868)

2119905119899+119899 (119899 + 1)

2119868119905119899+119899119868

2(1199051+ 119905119903)] (28)

ℎ2[119899(119863 minus 119868)

2119905119899+119899 (119899 + 1)

2119868119905119899+119899119868

2(1199051)] (29)

To incorporate the retailerrsquos holding costs into the originalmodels with and without breakdown respectively we obtain

TC3(1199051) = TC

1(1199051)

+ ℎ2[119899(119863 minus 119868)

2119905119899+119899 (119899 + 1)

2119868119905119899+119899119868

2(1199051+ 119905119903)]

TC4(1199051)

= TC2(1199051) + ℎ2[119899(119863 minus 119868)

2119905119899+119899 (119899 + 1)

2119868119905119899+119899119868

2(1199051)]

(30)

To take the randomness of defective rate 119909 into accountand substitute all related variables into (30) with furtherderivations 119864[TC

3(1199051)] and 119864[TC

4(1199051)] can be obtained as

follows

119864 [TC3(1199051)] = 119864 [TC

1(1199051)] + ℎ

2(1198751119892 (1 minus 119864 [119909])

2)

sdot (1 minus1

119899) 1199051+ ℎ2(1198751(1 minus 119864 [119909])

2)

sdot [1198751(1 minus 119864 [119909])

120582119899+ (1 minus

1

119899)] 1199052

1

119864 [TC4(1199051)] = 119864 [TC

2(1199051)] + ℎ

2(1198751(1 minus 119864 [119909])

2)

sdot [1198751(1 minus 119864 [119909])

120582119899+ (1 minus

1

119899)] 1199052

1

(31)

The Scientific World Journal 9

32 Integration of Enhanced Model withwithout BreakdownThe mean time to breakdowns obeys the exponential distri-bution with 119891(119905) = 120573119890minus120573119905 Therefore 119864[TCU

2(1199051)] is

119864 [TCU2(1199051)]

=

int1199051

0119864 [TC

3(1199051)] 119891 (119905) 119889119905 + int

infin

1199051

119864 [TC4(1199051)] 119891 (119905) 119889119905

119864 [T]

(32)

Substituting 119864[TCU3(1199051)] 119864[TC

4(1199051)] and 119864[T] into (32)

and resolving 119864[TCU2(1199051)] we obtain

119864 [TCU2(1199051)]

=120582

(1 minus 119864 [119909])

sdot (119870 + 119899119870

1)

11990511198751

+ 1205741+12057421199051

2+ 12057451199051

+ [119872

1198751

+ℎ119892

120573](

1 minus 119890minus1205731199051

1199051

) minus ℎ119892 (119890minus1205731199051)

minus (ℎ minus ℎ2) [119892 (1 minus 119864 [119909])

2(1 minus

1

119899)] (1 minus 119890

minus1205731199051)

(33)

where

1205745=ℎ2(1 minus 119864 [119909])

2[1198751(1 minus 119864 [119909])

120582119899+ (1 minus

1

119899)] (34)

33 Determining the Optimal Run Time Let 120595(1199051) stand for

the following

120595 (1199051) =

2 (119870 + 1198991198701) 120573 + 2120574

4(1 minus 119890

minus1205731199051)

[1199052

1119875112057321205746+ 1205744(2 + 120573119905

1)] 120573119890minus1205731199051

(35)

Theorem 3 (119864[TCU2(1199051)] is convex if 0 lt 119905

1lt 120595(119905

1)) The

second derivative of 119864[1198791198621198802(1199051)] with respect to 119905

1is

1198892119864 [119879119862119880

2(1199051)]

11988921199052

1

=120582

(1 minus 119864 [119909])[2 (119870 + 119899119870

1)

1199053

11198751

+ (ℎ minus ℎ2)119892 (1 minus 119864 [119909])

2

sdot (1 minus1

119899) (1205732119890minus1205731199051) minus ℎ119892 (120573

2119890minus1205731199051)

+ [119872

1198751

+ℎ119892

120573](

2 (1 minus 119890minus1205731199051)

1199053

1

minus2120573119890minus1205731199051

1199052

1

minus1205732119890minus1205731199051

1199051

)]

(36)

Since annual demand 120582 gt 0 the first term in the RHS of(36) is positive and

119894119891 [2 (119870 + 119899119870

1)

1199053

11198751

+ (ℎ minus ℎ2) [119892 (1 minus 119864 [119909])

2(1 minus

1

119899)]

sdot (1205732119890minus1205731199051) minus ℎ119892 (120573

2119890minus1205731199051) + [

119872

1198751

+ℎ119892

120573]

sdot(

2 (1 minus 119890minus1205731199051)

1199053

1

minus2120573119890minus1205731199051

1199052

1

minus1205732119890minus1205731199051

1199051

)] gt 0

119905ℎ1198901198991198892119864 [119879119862119880

2(1199051)]

1198891199051

2gt 0

(37)

With further derivations the left-hand side (LHS) of (37)becomes

119894119891 [2 (119870 + 1198991198701) 120573 minus 119905

3

11198751120573 (ℎ minus ℎ

2) [119892 (1 minus 119864 [119909])

2(1 minus

1

119899)]

sdot (1205732119890minus1205731199051) minus 1199053

11198751120573ℎ119892 (120573

2119890minus1205731199051)

+ (119872120573 + ℎ1198921198751) [2 (1 minus 119890

minus1205731199051) minus 2119905

1120573119890minus1205731199051 minus 12057321199052

1119890minus1205731199051] ]

gt 0

(38)

Let

1205746= (ℎ minus ℎ

2) [119892 (1 minus 119864 [119909])

2(1 minus

1

119899)] + ℎ119892 (39)

then (37) becomes

119894119891 [2 (119870 + 1198991198701) 120573 + 2120574

4(1 minus 119890

minus1205731199051)

minus1199051[1199052

1119875112057321205746+ 1205744(2 + 120573119905

1)] 120573119890minus1205731199051] gt 0

119905ℎ1198901198991198892119864 [119879119862119880

2(1199051)]

1198891199051

2gt 0

(40)

or

1198892119864 [119879119862119880

2(1199051)]

1198891199051

2gt 0

119894119891 0 lt 1199051lt

2 (119870 + 1198991198701) 120573 + 2120574

4(1 minus 119890

minus1205731199051)

[1199052

1119875112057321205746+ 1205744(2 + 120573119905

1)] 120573119890minus1205731199051

= 120595 (1199051)

(41)

10 The Scientific World Journal

Once 119864[1198791198621198802(1199051)] is proven to be convex the optimal

run time 119905lowast1can be solved by setting the first derivative of

119864[1198791198621198802(1199051)] = 0

119889119864 [1198791198621198802(1199051)]

1198891199051

=120582

(1 minus 119864 [119909])sdot minus (119870 + 119899119870

1)

1199052

1119875

+ (1205742

2+ 1205745) + 1205746(120573119890minus1205731199051)

+ [119872

1198751

+ℎ119892

120573](

minus (1 minus 119890minus1205731199051)

1199052

1

+120573119890minus1205731199051

1199051

)

= 0

(42)

It can be seen that the first term in the RHS of (42) ispositive so the second term is equal to zero In order to findthe bounds for 119905lowast

1 let

119905lowast

1119880= radic

2 [120573 (119870 + 1198991198701) + 1205744]

1198751120573 (1205742+ 21205745)

(43)

119905lowast

1119871= 119905ℎ119890 119901119900119904119894119905119894V119890 119903119900119900119905 119900119891

minus1205744plusmn radic1205742

4+ 21198751(119870 + 119899119870

1) (1205742+ 21205745+ 21205746120573)

1198751(1205742+ 21205745+ 21205746120573)

(44)

Theorem4 (119905lowast1119871lt 119905lowast

1lt 119905lowast

1119880) For the proof ofTheorem 4 please

refer to the proof for Theorem 2 in Section 2

Once we are certain that 119905lowast1falls within the aforemen-

tioned upper and lower bounds in order to find 119905lowast1 we can

first multiply the second term of (42) by (211987511199052

1120573) and obtain

the following

[1198751120573 (1205742+ 21205745) + 2119875

112057321205746119890minus1205731199051] 1199052

1+ (21205744120573119890minus1205731199051) 1199051

minus2 [120573 (119870 + 1198991198701) + 1205744(1 minus 119890

minus1205731199051)]

= 0

(45)

Equation (45) can be rearranged as

119890minus1205731199051 =

2 [120573 (119870 + 1198991198701) + 1205744] minus [119875

1120573 (1205742+ 21205745)] 1199052

1

2 [1198751120573212057461199052

1+ 1205744(1 + 120573119905

1)]

(46)

where 119890minus1205731199051 is the complement of the cumulative densityfunction 119865(119905

1) = 1minus119890

minus1205731199051 As 0 le 119865(119905

1) le 1 0 le 119890minus1205731199051 le 1 Let

119890minus1205731199051 = 0 and 119890minus1205731199051 = 1 be the initial upper and lower bounds

of 119890minus1205731199051 respectively Then by using the proposed recursivesearching algorithm given at the end of Section 2 we can findthe optimal production run time 119905lowast

1

4 Numerical Example

In order to relieve the comparison efforts for readers thissection adopts the same numerical example as in [9] For

0225 0265 0305 0345 0385 0425 0465 0505

E[T

CU(t

1)]

t1

tlowast1L = 03411 t

lowast1 = 03748

tlowast1U = 05183

$10600

$10800

$11000

$11200

$11400

$11600

$11006

$11800

Figure 6 The behavior of 119864[TCU(1199051)] in connection with the

production run time 1199051in the proposed model 1

a demonstration of the proposed EPQ-based model 1 thefollowing system parameters are used

1198751 production rate 10000 products per year120582 demand rate 4000 products per year119909 random scrap rate which follows uniformly distribu-

tion over the interval [0 02]120573 Poisson breakdown rate 05 average times per year119892 constant machine repair time 119905

119903 0018 year per repair

119872 machine repair cost $500 for each breakdown119870 setup cost $450 per production run119862 manufacturing cost $2 per item119862119878 disposal cost $03 per scrap itemℎ holding cost $06 per item per unit time1198701 fixed delivery cost $90 per shipment119899 number of deliveries 4 per cycle

119862119879 variable delivery cost $0001 per item

First we use both upper and lower bounds of 119905lowast1to

test for the convexity of 119864[TCU(1199051)] (see Theorem 1) The

computation results of (19) (20) and (11) indicate that 119905lowast1119880

=05183 lt 120585(119905

lowast

1119880) = 28723 and 119905lowast

1119871= 03411 lt 120585(119905

lowast

1119871) = 26261

Hence [TCU(1199051)] is convex (Figure 6)

In order to find the optimal 119905lowast1 we first substitute the

upper and lower bounds of 119905lowast1in (8) and obtain 119864[TCU(119905lowast

1119880)]

= $1160163 and 119864[TCU(119905lowast1119871)] = $1101450 respectively

Because the optimal run time 119905lowast1falls within the interval

of [119905lowast1119871 119905lowast

1119880] we apply the proposed recursive searching

algorithm stated at the end of Section 2 and find 119905lowast1= 03748

years Accordingly the optimal expected system costs perunit time 119864[TCU(119905lowast

1)] = $1100641 (Figure 6) Table 1 shows

the step-by-step iterations of the algorithmIn this specific studied model we focus on incorporating

a discontinuous multidelivery policy into a prior work [9]and consider a fixed transportation cost associated witheach delivery Applying the research result we can easilyinvestigate the effects of different fixed transportation costs

The Scientific World Journal 11

Table 1 Iterations of the recursive searching algorithm for locating 119905lowast1

120573 Step 119905lowast

1119880120596119880= 119890minus1205731199051119880 119905

lowast

1119871120596119871= 119890minus1205731199051119871

Differencebetween 119905lowast

1119880and

119905lowast

1119871

[119880]

119864[TCU(119905lowast1119880)]

[119871]

119864[TCU(119905lowast1119871)]

Differencebetween [119880]

and [119871]05 Initial 000000 100000 $1160163 $1101450 $58713

1st 05183 07717 03411 08432 01772 $1110330 $1101450 $88802nd 03857 08246 03721 08302 00136 $1100716 $1100646 $0703rd 03756 08287 03746 08292 00010 $1100642 $1100641 $0014th 03749 08291 03748 08291 00001 $1100641 $1100641 $00045th 03748 08291 03748 08291 000000 $1100641 $1100641 $0000

Table 2 Variations of the fixed delivery cost1198701effects on the optimal production run time 119905lowast

1

1198701119870 005 02 04 06 08 1 12 14 16 18 2 22

1198701

$225 $90 $180 $270 $360 $450 $540 $630 $720 $810 $900 $990119864[TCU(119905lowast

1)] $10721 $11067 $11448 $11773 $12062 $12325 $12567 $12793 $13006 $13208 $13399 $13583

119905lowast

103116 03816 04586 05244 05828 06359 06848 07305 07735 08142 08529 08900

Table 3 Variations of the unit retailerrsquos holding cost ℎ2and their effects on 119864[TCU(119905lowast

1)]

ℎ2ℎ 05 075 1 125 15 175 2 225 25 275 3 325

ℎ2

03 045 06 075 09 105 12 135 15 165 18 195119864 [TCU (119905lowast

1)] $11282 $11407 $11526 $11638 $11746 $11850 $11949 $12045 $12138 $12229 $12316 $12401

119905lowast

103256 03074 02918 02785 02668 02564 02472 02389 02314 02246 02183 02125

1198701on the optimal system cost 119864[TCU(119905lowast

1)] and on the

optimal production run time 119905lowast1(see Table 2) It can be seen

from Table 2 that as the ratio of1198701119870 increases the expected

system costs per unit time119864[TCU(119905lowast1119871)] increase significantly

It is also noted that as 1198701increases optimal production run

time 119905lowast1also increases significantly

41 Numerical Example for the Producer-Retailer IntegratedEPQ System (Model 2) In order to demonstrate the researchresult of the producer-retailer integrated EPQ-based modelan additional system variable ℎ

2= $150 per item stored at the

retailerrsquos side is includedAgain one can use the upper and lower bounds of 119905lowast

1

(equations (43) and (44)) to test for convexity of 119864[TCU(1199051)]

(Theorem 3 and equation (35)) The results reveal that 119905lowast1119880

=03213 lt 120595(119905lowast

1119880) = 26462 and 119905lowast

1119871= 02186 lt 120595(119905lowast

1119871) = 24876

Therefore the expected cost [TCU(1199051)] is convex

Next by applying the proposed recursive searching algo-rithm we can calculate that the optimal run time 119905lowast

1=

02314 years and the optimal 119864[TCU(119905lowast1)] = $1213849 It is

noted that the computation time for reaching the optimal 119905lowast1

solution is 21 seconds (usingExcel software in a desktop com-puter Intel CPU G850 with 294GB RAM and 289GHz)

Figure 7 illustrates the behavior of 119864[TCU(1199051)] with

regard to production run time It is noted that without theresearch result from the second model the management ofsuch a producer-retailer integrated system would probablyuse 1199051= 03748 years (from the result of model 1) for their

run time decision Further analysis (see Figure 7) shows cost

0130 0170 0210 0250 0290 0330 0370 0410

t1

E[T

CU(t

1)]

11600

11800

12000

12200

12400

12600

12800

tlowast1 = 02314

t1 = 0374812138

(from model 1)

E[TCU( t1)] = 12489

$$

$

$

$

$

$

$

$

Figure 7 The behavior of 119864[TCU(1199051)] with respect to production

run time 1199051in the proposed model 2

savings of $351 (or 29 over the total system costs) simply byapplying our research result

The effects of the unit retailerrsquos holding cost ℎ2on the

expected system cost 119864[TCU(119905lowast1)] and on the optimal run

time 119905lowast1are shown in Table 3 respectively

It can be seen that as ℎ2or the ratio of ℎ

2ℎ increases

the expected cost 119864[TCU(119905lowast1)] increases but the optimal

production run time 119905lowast1decreases In decision-making these

sensitivity analyses results can provide the management ofa producer-retailer integrated system with valuable informa-tion and insights into the effects of various stock holding costsin different retailersrsquo locations

12 The Scientific World Journal

5 Concluding Remarks

Two exact models for an extended EPQ-based problem witha discontinuous delivery policy scrap rate and randombreakdown are developed in this study They specificallyaddress different real-life situations in production end-item delivery and intrasupply chains such as a producer-retailer integrated systemMathematicalmodeling alongwithoptimization techniques is used to determine the optimalproduction run times that minimize the expected systemcosts per unit time Without in-depth investigations onthese separate models the optimal production run time andother important information related to the systemparameterscannot be revealed The proposed real-life EPQ models withrandom machine breakdown discontinuous product distri-bution policies and quality assurance must be specificallystudied in order to (1) obtain the joint effects of breakdowndiscontinuous distribution policies and quality assurance onthe optimal production run time (2) get to know the effectsof different policy and scope of supply chains managementon the optimal run time and overall system costs and (3)gain the insight with regard to various systemrsquos parametersof all particular EPQ-based models Since little attention hasbeen paid to the investigation of joint effects of these practicalproduction situations on the optimal run time this researchis intended to bridge the gap An interesting area for futurestudy is the examination of the effect of variable productionrates on these models

Appendix

Derivations of (3) are as followsRecall (2) as follows

TC1(1199051) = 119862 (119875

11199051) + 119870 + 119862

119878(11990511198751119909) +119872 + 119899119870

1

+ 119862119879[11990511198751] + ℎ3(120582119905119903) 1198791015840

+ ℎ[1198671015840+ 11988911199051

21199051+ (1198671015840

1+ 1198891119905) 119905119903+119899 minus 1

211989911986710158401199051015840

2]

(A1)

Substituting all related system parameters into (2) (pleaserefer to the basic formulations and solution process in [9])the TC

1(1199051) can be obtained as

TC1(1199051) = 119870 +119872

+[1198621198751+ 1198621198781198751119909 + 119862

1198791198751(1 minus 119909)+ ℎ

31198751119892 (1 minus 119909)] 119905

1

+ 1198991198701+ ℎ119901119892119905 minus[

ℎ1198751119892 (1 minus 119909)

2minusℎ1198751119892 (1 minus 119909)

2119899] 1199051

+ 1199052

1[ℎ1198751

2+ℎ1198752

1

2120582(1 minus 119909)

2minusℎ1198751

2(1 minus 119909)

minusℎ1198752

1

2120582119899(1 minus 119909)

2+ℎ1198751

2119899(1 minus 119909)]

(A2)

To take the randomness of 119909 into account by using theexpected values of 119909 with further derivations119864[TC

1(1199051)] can

be derived as follows (ie equation (3))119864 [TC

1(1199051)]

= 119870 + 1198991198701+119872 + ℎ119905119875

1119892

+ [1198621198751+ 1198621198781198751119864 [119909] + 1198621198791198751 (1 minus 119864 [119909])

+ ℎ31198751119892 (1 minus 119864 [119909]) minus

ℎ1198751119892 (1 minus 119864 [119909])

2(1 minus

1

119899)] sdot 1199051

+ [ℎ1198751119864 [119909]

2+ℎ1198752

1

2120582(1 minus 119864 [119909])

2(1 minus

1

119899)

+ℎ1198751

2119899(1 minus 119864 [119909])] 119905

2

1

(A3)

Conflict of Interests

The authors of the paper declare that there is no conflict ofinterests regarding the publication of this paper

Acknowledgment

The authors greatly appreciate the National Science Council(NSC) of Taiwan for supporting this research underGrant noNSC 102-2410-H-324-005

References

[1] E W Taft ldquoThemost economical production lotrdquo Iron Age vol101 pp 1410ndash1412 1918

[2] G Hadley and T M Whitin ldquoAn optimal final inventorymodelrdquoManagement Science vol 7 pp 179ndash183 1961

[3] E A Silver D F Pyke and R Peterson Inventory Managementand Production Planning and Scheduling John Wiley amp SonsNew York NY USA 1998

[4] S Nahmias Production amp Operations Analysis McGraw-HillNew York NY USA 2009

[5] MWidmer and P Solot ldquoDo not forget the breakdowns and themaintenance operations in FMSdesign problemsrdquo InternationalJournal of Production Research vol 28 pp 421ndash430 1990

[6] K-Y C Yu and D L Bricker ldquoAnalysis of a markov chainmodel of a multistage manufacturing system with inspectionrejection and reworkrdquo IIE Transactions vol 25 no 1 pp 109ndash112 1993

[7] H Groenevelt L Pintelon and A Seidmann ldquoProduction lotsizing with machine breakdownsrdquoManagement Science vol 38no 1 pp 104ndash123 1992

[8] G A Widyadana and H M Wee ldquoOptimal deterioratingitems production inventory models with random machinebreakdown and stochastic repair timerdquo Applied MathematicalModelling vol 35 no 7 pp 3495ndash3508 2011

[9] S W Chiu C-L Chou and W-K Wu ldquoOptimizing replenish-ment policy in an EPQ-based inventory model with noncon-forming items and breakdownrdquo EconomicModelling vol 35 pp330ndash337 2013

The Scientific World Journal 13

[10] A M Zargar ldquoEffect of rework strategies on cycle timerdquoComputers amp Industrial Engineering vol 29 no 1ndash4 pp 239ndash243 1995

[11] P Biswas and B R Sarker ldquoOptimal batch quantity modelsfor a lean production system with in-cycle rework and scraprdquoInternational Journal of Production Research vol 46 no 23 pp6585ndash6610 2008

[12] S W Chiu H-D Lin C-B Cheng and C-L Chung ldquoOptimalproduction-shipment decisions for the finite production ratemodel with scraprdquo International Journal for Engineering Mod-elling vol 22 no 1ndash4 pp 25ndash34 2009

[13] Y-S P Chiu K-K Chen and C-K Ting ldquoReplenishment runtime problem with machine breakdown and failure in reworkrdquoExpert Systems with Applications vol 39 no 1 pp 1291ndash12972012

[14] H-D Lin F-Y Pai and S W Chiu ldquoA note on ldquointra-supply chain system with multiple sales locations and qualityassurancerdquordquo Expert Systems with Applications vol 40 no 11 pp4730ndash4732 2013

[15] B Kamsu-Foguem F Rigal and FMauget ldquoMining associationrules for the quality improvement of the production processrdquoExpert Systems with Applications vol 40 no 4 pp 1034ndash10452013

[16] Y Li H Guo L Wang and J Fu ldquoA hybrid genetic-simulatedannealing algorithm for the location-inventory- routing prob-lem considering returns under E-supply chain environmentrdquoThe Scientific World Journal vol 2013 Article ID 125893 10pages 2013

[17] Y-S P Chiu K-K Chen F-T Cheng and C-K Ting ldquoReex-amination of ldquocombining an alternative multi-delivery policyinto economic production lot size problemwith partial reworkrdquousing an alternative approachrdquo Journal of Applied Research andTechnology vol 11 no 3 pp 317ndash323 2013

[18] L B Schwarz B L Deuermeyer and R D Badinelli ldquoFill-rateoptimization in a one-warehouse 119873-identical retailer distribu-tion systemrdquo Management Science vol 31 no 4 pp 488ndash4981985

[19] R A Sarker and L R Khan ldquoOptimal batch size for a pro-duction system operating under periodic delivery policyrdquoCom-puters amp Industrial Engineering vol 37 no 4 pp 711ndash730 1999

[20] N Comez K E Stecke and M Cakanyildirim ldquoMultiple in-cycle transshipments with positive delivery timesrdquo Productionand Operations Management vol 21 no 2 pp 378ndash395 2012

[21] S W Chiu F-Y Pai and W K Wu ldquoAlternative approach todetermine the common cycle time for a multi-item productionsystem with discontinuous deliveries and failure in reworkrdquoEconomic Modelling vol 35 pp 593ndash596 2013

[22] Y-S P Chiu C-C Huang M-F Wu and H-H Chang ldquoJointdetermination of rotation cycle time and number of shipmentsfor a multi-item EPQ model with random defective raterdquoEconomic Modelling vol 35 pp 112ndash117 2013

[23] M A Hoque ldquoSynchronization in the single-manufacturermulti-buyer integrated inventory supply chainrdquo European Jour-nal of Operational Research vol 188 no 3 pp 811ndash825 2008

[24] S W Chiu L-W Lin K-K Chen and C-L Chou ldquoDetermin-ing production-shipment policy for a vendor-buyer integratedsystem with rework and an amending multi-delivery schedulerdquoEconomic Modelling vol 33 pp 668ndash675 2013

[25] M Cedillo-Campos and C Sanchez-Ramırez ldquoDynamic self-assessment of supply chains performance an emerging marketapproachrdquo Journal of Applied Research and Technology vol 11no 3 pp 338ndash347 2013

[26] Y-S P Chiu H-D Lin F-T Cheng and M-H Hwang ldquoOpti-mal common cycle time for a multi-item production systemwith discontinuous delivery policy and failure in reworkrdquoJournal of Scientific and Industrial Research vol 72 no 7 pp435ndash440 2013

[27] HHishamuddin R A Sarker andD Essam ldquoA recoverymech-anism for a two echelon supply chain system under supplydisruptionrdquo Economic Modelling vol 38 pp 555ndash563 2014

[28] M Murugan and V Selladurai ldquoProductivity improvement inmanufacturing submersible pump diffuser housing using leanmanufacturing systemrdquo Journal of Engineering Research vol 2no 1 pp 164ndash182 2014

[29] L Wang H Qu S Liu and C-X Dun ldquoModeling and opti-mization of the multiobjective stochastic joint replenishmentand delivery problem under supply chain environmentrdquo TheScientific World Journal vol 2013 Article ID 916057 11 pages2013

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

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The Scientific World Journal 5

H

P1 minus d1

T

Time

minus120582

tn

t1 t1t2

I(t)

Figure 3 Inventory level of finished items in a manufacturing run time problem with no breakdown defective rate and discontinuousdelivery policy

23 Integration of the Proposed Run TimeModels withwithoutBreakdown Amachine breakdownmay take place randomlyand it follows a Poisson distribution with mean equal to120573 per year Let 119891(119905) be the probability density function ofrandom time 119905 before a breakdown takes place and 119865(119905)

represents the cumulative density function of 119905 Hence thelong-run expected system costs per unit time 119864[TCU(119905

1)]

are

119864 [TCU (1199051)]

=

int1199051

0119864 [TC

1(1199051)] 119891 (119905) 119889119905 + int

infin

1199051

119864 [TC2(1199051)] 119891 (119905) 119889119905

119864 [T]

(6)

where

119864 [T] = int1199051

0

119864 [1198791015840] 119891 (119905) 119889119905 + int

infin

1199051

119864 [119879] 119891 (119905) 119889119905

=11990511198751(1 minus 119864 [119909])

120582

(7)

From Figures 1 and 3 it can be seen that 1198791015840 and 119879 aredifferent in length (1198791015840 is longer than 119879 since it containsmachine repairing time) and because a breakdown can occurrandomly it is necessary to use the integration (ie equation(7)) to derive the expected cycle length

It is also noted that the time between breakdowns obeysthe exponential distribution with density function 119891(119905) =

120573119890minus120573119905 and cumulative density function 119865(119905) = 1 minus 119890

minus120573119905 Bysubstituting 119864[TC

1(1199051)] 119864[TC

2(1199051)] and 119864[T] into (6) and

solving the integration of the mean time to breakdown in119864[TCU(119905

1)] we obtain

119864 [TCU (1199051)]

=120582

(1 minus 119864 [119909])

sdot (119870 + 119899119870

1)

11990511198751

+ 1205741+12057421199051

2+ [

119872

1198751

+ℎ119892

120573](

1 minus 119890minus1205731199051

1199051

)

minusℎ119892 (119890minus1205731199051) minus

ℎ119892 (1 minus 119864 [119909])

2(1 minus

1

119899) (1 minus 119890

minus1205731199051)

(8)

where

1205741= [119862 + 119862

119878119864 [119909] + 119862

119879(1 minus 119864 [119909]) + ℎ

3119892 (1 minus 119864 [119909])]

(9)

1205742= [

ℎ1198751

120582(1 minus 119864 [119909])

2(1 minus

1

119899) + ℎ119864 [119909] +

ℎ

119899(1 minus 119864 [119909])]

(10)

24 Derivation of the Optimal Production Run Time In orderto derive the optimal production run time 119905lowast

1 we first have

to prove that 119864[TCU(1199051)] is convex Let 120585(119905

1) represent the

following

120585 (1199051) =

2 (119870 + 1198991198701) 120573 + 2 (1 minus 119890

minus1205731199051) 1205744

[1199052

1119875112057321205743+ 1205744(2 + 120573119905

1)] 120573119890minus1205731199051

(11)

6 The Scientific World Journal

Theorem 1 (119864[TCU(1199051)] is convex if 0 lt 119905

1lt 120585(119905

1)) The

second derivative of 119864[119879119862119880(1199051)] with respect to 119905

1is

1198892119864 [119879119862119880 (119905

1)]

11988921199052

1

=120582

(1 minus 119864 [119909])

sdot [2 (119870 + 119899119870

1)

1199053

11198751

minus ℎ119892 [1 minus(1 minus 119864 [119909])

2(1 minus

1

119899)] (120573

2119890minus1205731199051)

+ [119872

1198751

+ℎ119892

120573](

2 (1 minus 119890minus1205731199051)

1199053

1

minus2120573119890minus1205731199051

1199052

1

minus1205732119890minus1205731199051

1199051

)]

(12)

It is noted that because annual demand 120582 gt 0 the firstterm in the right-hand size (RHS) of (12) is positive Hence weobtain

1198892119864 [119879119862119880 (119905

1)]

1198891199051

2gt 0

119894119891 [2 (119870 + 119899119870

1)

1199053

11198751

minus ℎ119892 [1 minus(1 minus 119864 [119909])

2(1 minus

1

119899)] (120573

2119890minus1205731199051)

+ [119872

1198751

+ℎ119892

120573](

2 (1 minus 119890minus1205731199051)

1199053

1

minus2120573119890minus1205731199051

1199052

1

minus1205732119890minus1205731199051

1199051

)]

gt 0

(13)

The RHS of (13) can be further derived as

119894119891 [2 (119870 + 1198991198701) 120573 minus 119905

3

11198751120573ℎ119892

sdot [1 minus(1 minus 119864 [119909])

2(1 minus

1

119899)] (120573

2119890minus1205731199051)

+ (119872120573 + ℎ1198921198751)

times [2 (1 minus 119890minus1205731199051) minus 2119905

1120573119890minus1205731199051 minus 12057321199052

1119890minus1205731199051]]

gt 0

(14)

Let

1205743= ℎ119892 [1 minus

(1 minus 119864 [119909])

2(1 minus

1

119899)]

1205744= (119872120573 + ℎ119892119875

1)

(15)

then (13) can be rewritten as

1198892119864 [119879119862119880 (119905

1)]

1198891199051

2gt 0

119894119891 [2 (119870 + 1198991198701) 120573 + 2 (1 minus 119890

minus1205731199051) 1205744

minus 1199051[1199052

1119875112057321205743+ 1205744(2 + 120573119905

1)] 120573119890minus1205731199051]

gt 0

(16)

or

1198892119864 [119879119862119880 (119905

1)]

1198891199051

2gt 0

119894119891 0 lt 1199051lt

2 (119870 + 1198991198701) 120573 + 2 (1 minus 119890

minus1205731199051) 1205744

[1199052

1119875112057321205743+ 1205744(2 + 120573119905

1)] 120573119890minus1205731199051

= 120585 (1199051)

(17)

If 119864[119879119862119880(1199051)] is a convex function then the minimum

point exists In order to locate the optimal production run time119905lowast

1that minimizes 119864[119879119862119880(119905

1)] we can set the first derivative of

119864[119879119862119880(1199051)] equal to zero and solve 119905lowast

1

119889119864 [119879119862119880 (1199051)]

1198891199051

=120582

(1 minus 119864 [119909])

sdot minus (119870 + 119899119870

1)

1199052

1119875

+1205742

2+ 1205743(120573119890minus1205731199051)

+ [119872

1198751

+ℎ119892

120573](

minus (1 minus 119890minus1205731199051)

1199052

1

+120573119890minus1205731199051

1199051

)

(18)

In the RHS of (18) it can be seen that the first term ispositive so the second term is equal to zero Let

119905lowast

1119880= radic

2 (1205744+ 119870120573 + 119899119870

1120573)

12057421198751120573

(19)

119905lowast

1119871= 119905ℎ119890 119901119900119904119894119905119894V119890 119903119900119900119905 119900119891

minus1205744plusmn radic1205744

2 + 21198751(1205742+ 2120573120574

3) (119870 + 119899119870

1)

1198751(1205742+ 2120573120574

3)

(20)

Theorem 2 (119905lowast1119871

lt 119905lowast

1lt 119905lowast

1119880) Because the proof of 119905lowast

1falls

within the upper and lower bounds we can multiply the secondterm of (18) by (2120573119875

11199052

1) and obtain

(11987511205731205742+ 2119875112057321205743119890minus1205731199051) 1199052

1+ (21205744120573119890minus1205731199051) 1199051

minus2 [120573 (119870 + 1198991198701) + 1205744(1 minus 119890

minus1205731199051)] = 0

(21)

The Scientific World Journal 7

Producerrsquosproduction-shipment system

Extended toProductionunit

Productionunit

Multideliverypolicy

Multideliverypolicy

Optimization OptimizationProducer-retailer integratedproduction-shipment system

Customer

CustomerRetailer storeor sales office

Figure 4 Extension to a producer-retailer integrated production-shipment system

Thus119905lowast

1

= 119905ℎ119890 119901119900119904119894119905119894V119890 119903119900119900119905 119900119891

(minus (21205744120573119890minus1205731199051)

plusmn ((21205744120573119890minus1205731199051)2

minus [4 (11987511205731205742+ 2119875112057321205743119890minus1205731199051)

times [minus2 [120573 (119870 + 1198991198701) + 1205744(1 minus 119890

minus1205731199051)]]])

12

)

times (2 (12057421198751120573 + 2119875

112057321205743119890minus1205731199051))minus1

(22)

Equation (21) can be rearranged as

2 [1198751120573212057431199052

1+ 12057441205731199051+ 1205744] (119890minus1205731199051)

= 2 [120573 (119870 + 1198991198701) + 1205744] minus (119875

112057312057421199052

1)

(23)

or

119890minus1205731199051 =

2 [120573 (119870 + 1198991198701) + 1205744] minus (119875

112057312057421199052

1)

2 [1198751120573212057431199052

1+ 12057441205731199051+ 1205744]

(24)

where 119890minus1205731199051 is the complement of the cumulative density

function 119865(1199051) = 1 minus 119890

minus1205731199051 As 0 le 119865(119905

1) le 1 0 le 119890

minus1205731199051 le 1

Let 119890minus1205731199051 = 0 and 119890minus1205731199051 = 1 be the upper and lower bounds for119890minus1205731199051 respectively By substituting them into (22) we obtain

119905lowast

1119880= radic

2 [120573 (119870 + 1198991198701) + 1205744]

12057421198751120573

119905lowast

1119871= 119905ℎ119890 119901119900119904119894119905119894V119890 119903119900119900119905 119900119891

minus1205744plusmn radic1205744

2 + 21198751(1205742+ 2120573120574

3) (119870 + 119899119870

1)

1198751(1205742+ 2120573120574

3)

(25)

and 119905lowast1119871lt 119905lowast

1lt 119905lowast

1119880

It is noted that although the optimal production run time119905lowast

1cannot be presented in a closed form it does fall within the

bounds 119905lowast1can be locatedwith the use of a proposed recursive

searching algorithm Let

120596 (1199051) = 119890minus1205731199051 =

2 [120573 (119870 + 1198991198701) + 1205744] minus (119875

112057312057421199052

1)

2 [1198751120573212057431199052

1+ 12057441205731199051+ 1205744]

there4 0 le 120596 (1199051) le 1

(26)

In order to locate 119905lowast1 we can use the following recursive

searching algorithm

(1) Let 120596(1199051) = 0 and 120596(119905

1) = 1 initially and calculate

the upper and lower bounds for 119905lowast1 respectively (ie

to obtain the initial values of [119905lowast1119871 119905lowast

1119880])

(2) Substitute the current values of [119905lowast1119871 119905lowast

1119880] into 119890minus1205731199051 and

compute the new bounds expressed as 120596119871and 120596

119880for

119890minus1205731199051 Hence 120596

119871lt 120596(1199051) lt 120596119880

(3) Let 120596(1199051) = 120596119871and 120596(119905

1) = 120596119880and update the upper

and lower bounds for 119905lowast1 respectively (ie to obtain

the new values of [119905lowast1119871 119905lowast

1119880])

(4) Repeat steps (2) and (3) until there is no signifi-cant difference between 119905lowast

1119871and 119905lowast1119880

(or there is nosignificant difference in terms of their effects on119864[TCU(119905lowast

1)])

(5) Stop 119905lowast1is found

3 Extension to a Producer-Retailer IntegratedEPQ-Based System (Model 2)

31 Enhanced Model Description and Formulation In thissection we further extend the scope of the problem to incor-porate the retailerrsquos stock holding cost into our study Thenew model can be considered a producer-retailer integratedsystem because in the present-day manufacturing sectorsome producers of consumer goods may own and operateretail stores or regional sales offices to promote and sell theirend products to customers (see Figure 4) With the intentionof addressing such a real-life intrasupply chain situation thesecond model of this study incorporates the retailerrsquos stockholding cost into the first model and investigates its effect onthe optimal production run time decision

8 The Scientific World Journal

Dminus 120582tn = I

D

I

t1 + tr tn

Time

t1 + tr

Ic(t)

t9984002

T998400

Figure 5 Inventory level of finished products on the retailerrsquos side in the proposed manufacturing run time problem with breakdown

In the proposed study the retailerrsquos stock holding posi-tions are illustrated in Figure 5

Extra parameters used in this enhanced model includethe following

ℎ2 holding cost per product stored on the retailerrsquos

side

119868119888(119905) on-hand inventory levels in units on the retail-

errsquos side end at time 119905

119863 number of finished products (a fixed quantity)transported to the retail store per shipment

119868 number of left-over products in 119905119899after satisfying

the demand in 119905119899

TC3(1199051) total production-inventory-delivery costs

per cycle of this enhanced model (in the case of abreakdown)

TC4(1199051) total production-inventory-delivery costs

per cycle of this enhanced model (in the case of nobreakdown)

119864[TC3(1199051)] the expected production-inventory-

delivery costs per cycle of this enhanced model (inthe case of a breakdown)

119864[TC4(1199051)] the expected production-inventory-

delivery costs per cycle of this enhanced model (inthe case of no breakdown)

119864[TCU119890(1199051)] the long-run expected production-

inventory-delivery costs per unit time in thisenhanced model whether or not a breakdown takesplace

Since the demand on the retailerrsquos side in time interval119905119899is 120582119905119899 after satisfying the demand the number of left-over

items (see Figure 5) in each 119905119899is

119868 = 119863 minus 120582119905119899 (27)

Total inventory holding costs on retailerrsquos side with andwithout breakdown are shown respectively in

ℎ2[119899(119863 minus 119868)

2119905119899+119899 (119899 + 1)

2119868119905119899+119899119868

2(1199051+ 119905119903)] (28)

ℎ2[119899(119863 minus 119868)

2119905119899+119899 (119899 + 1)

2119868119905119899+119899119868

2(1199051)] (29)

To incorporate the retailerrsquos holding costs into the originalmodels with and without breakdown respectively we obtain

TC3(1199051) = TC

1(1199051)

+ ℎ2[119899(119863 minus 119868)

2119905119899+119899 (119899 + 1)

2119868119905119899+119899119868

2(1199051+ 119905119903)]

TC4(1199051)

= TC2(1199051) + ℎ2[119899(119863 minus 119868)

2119905119899+119899 (119899 + 1)

2119868119905119899+119899119868

2(1199051)]

(30)

To take the randomness of defective rate 119909 into accountand substitute all related variables into (30) with furtherderivations 119864[TC

3(1199051)] and 119864[TC

4(1199051)] can be obtained as

follows

119864 [TC3(1199051)] = 119864 [TC

1(1199051)] + ℎ

2(1198751119892 (1 minus 119864 [119909])

2)

sdot (1 minus1

119899) 1199051+ ℎ2(1198751(1 minus 119864 [119909])

2)

sdot [1198751(1 minus 119864 [119909])

120582119899+ (1 minus

1

119899)] 1199052

1

119864 [TC4(1199051)] = 119864 [TC

2(1199051)] + ℎ

2(1198751(1 minus 119864 [119909])

2)

sdot [1198751(1 minus 119864 [119909])

120582119899+ (1 minus

1

119899)] 1199052

1

(31)

The Scientific World Journal 9

32 Integration of Enhanced Model withwithout BreakdownThe mean time to breakdowns obeys the exponential distri-bution with 119891(119905) = 120573119890minus120573119905 Therefore 119864[TCU

2(1199051)] is

119864 [TCU2(1199051)]

=

int1199051

0119864 [TC

3(1199051)] 119891 (119905) 119889119905 + int

infin

1199051

119864 [TC4(1199051)] 119891 (119905) 119889119905

119864 [T]

(32)

Substituting 119864[TCU3(1199051)] 119864[TC

4(1199051)] and 119864[T] into (32)

and resolving 119864[TCU2(1199051)] we obtain

119864 [TCU2(1199051)]

=120582

(1 minus 119864 [119909])

sdot (119870 + 119899119870

1)

11990511198751

+ 1205741+12057421199051

2+ 12057451199051

+ [119872

1198751

+ℎ119892

120573](

1 minus 119890minus1205731199051

1199051

) minus ℎ119892 (119890minus1205731199051)

minus (ℎ minus ℎ2) [119892 (1 minus 119864 [119909])

2(1 minus

1

119899)] (1 minus 119890

minus1205731199051)

(33)

where

1205745=ℎ2(1 minus 119864 [119909])

2[1198751(1 minus 119864 [119909])

120582119899+ (1 minus

1

119899)] (34)

33 Determining the Optimal Run Time Let 120595(1199051) stand for

the following

120595 (1199051) =

2 (119870 + 1198991198701) 120573 + 2120574

4(1 minus 119890

minus1205731199051)

[1199052

1119875112057321205746+ 1205744(2 + 120573119905

1)] 120573119890minus1205731199051

(35)

Theorem 3 (119864[TCU2(1199051)] is convex if 0 lt 119905

1lt 120595(119905

1)) The

second derivative of 119864[1198791198621198802(1199051)] with respect to 119905

1is

1198892119864 [119879119862119880

2(1199051)]

11988921199052

1

=120582

(1 minus 119864 [119909])[2 (119870 + 119899119870

1)

1199053

11198751

+ (ℎ minus ℎ2)119892 (1 minus 119864 [119909])

2

sdot (1 minus1

119899) (1205732119890minus1205731199051) minus ℎ119892 (120573

2119890minus1205731199051)

+ [119872

1198751

+ℎ119892

120573](

2 (1 minus 119890minus1205731199051)

1199053

1

minus2120573119890minus1205731199051

1199052

1

minus1205732119890minus1205731199051

1199051

)]

(36)

Since annual demand 120582 gt 0 the first term in the RHS of(36) is positive and

119894119891 [2 (119870 + 119899119870

1)

1199053

11198751

+ (ℎ minus ℎ2) [119892 (1 minus 119864 [119909])

2(1 minus

1

119899)]

sdot (1205732119890minus1205731199051) minus ℎ119892 (120573

2119890minus1205731199051) + [

119872

1198751

+ℎ119892

120573]

sdot(

2 (1 minus 119890minus1205731199051)

1199053

1

minus2120573119890minus1205731199051

1199052

1

minus1205732119890minus1205731199051

1199051

)] gt 0

119905ℎ1198901198991198892119864 [119879119862119880

2(1199051)]

1198891199051

2gt 0

(37)

With further derivations the left-hand side (LHS) of (37)becomes

119894119891 [2 (119870 + 1198991198701) 120573 minus 119905

3

11198751120573 (ℎ minus ℎ

2) [119892 (1 minus 119864 [119909])

2(1 minus

1

119899)]

sdot (1205732119890minus1205731199051) minus 1199053

11198751120573ℎ119892 (120573

2119890minus1205731199051)

+ (119872120573 + ℎ1198921198751) [2 (1 minus 119890

minus1205731199051) minus 2119905

1120573119890minus1205731199051 minus 12057321199052

1119890minus1205731199051] ]

gt 0

(38)

Let

1205746= (ℎ minus ℎ

2) [119892 (1 minus 119864 [119909])

2(1 minus

1

119899)] + ℎ119892 (39)

then (37) becomes

119894119891 [2 (119870 + 1198991198701) 120573 + 2120574

4(1 minus 119890

minus1205731199051)

minus1199051[1199052

1119875112057321205746+ 1205744(2 + 120573119905

1)] 120573119890minus1205731199051] gt 0

119905ℎ1198901198991198892119864 [119879119862119880

2(1199051)]

1198891199051

2gt 0

(40)

or

1198892119864 [119879119862119880

2(1199051)]

1198891199051

2gt 0

119894119891 0 lt 1199051lt

2 (119870 + 1198991198701) 120573 + 2120574

4(1 minus 119890

minus1205731199051)

[1199052

1119875112057321205746+ 1205744(2 + 120573119905

1)] 120573119890minus1205731199051

= 120595 (1199051)

(41)

10 The Scientific World Journal

Once 119864[1198791198621198802(1199051)] is proven to be convex the optimal

run time 119905lowast1can be solved by setting the first derivative of

119864[1198791198621198802(1199051)] = 0

119889119864 [1198791198621198802(1199051)]

1198891199051

=120582

(1 minus 119864 [119909])sdot minus (119870 + 119899119870

1)

1199052

1119875

+ (1205742

2+ 1205745) + 1205746(120573119890minus1205731199051)

+ [119872

1198751

+ℎ119892

120573](

minus (1 minus 119890minus1205731199051)

1199052

1

+120573119890minus1205731199051

1199051

)

= 0

(42)

It can be seen that the first term in the RHS of (42) ispositive so the second term is equal to zero In order to findthe bounds for 119905lowast

1 let

119905lowast

1119880= radic

2 [120573 (119870 + 1198991198701) + 1205744]

1198751120573 (1205742+ 21205745)

(43)

119905lowast

1119871= 119905ℎ119890 119901119900119904119894119905119894V119890 119903119900119900119905 119900119891

minus1205744plusmn radic1205742

4+ 21198751(119870 + 119899119870

1) (1205742+ 21205745+ 21205746120573)

1198751(1205742+ 21205745+ 21205746120573)

(44)

Theorem4 (119905lowast1119871lt 119905lowast

1lt 119905lowast

1119880) For the proof ofTheorem 4 please

refer to the proof for Theorem 2 in Section 2

Once we are certain that 119905lowast1falls within the aforemen-

tioned upper and lower bounds in order to find 119905lowast1 we can

first multiply the second term of (42) by (211987511199052

1120573) and obtain

the following

[1198751120573 (1205742+ 21205745) + 2119875

112057321205746119890minus1205731199051] 1199052

1+ (21205744120573119890minus1205731199051) 1199051

minus2 [120573 (119870 + 1198991198701) + 1205744(1 minus 119890

minus1205731199051)]

= 0

(45)

Equation (45) can be rearranged as

119890minus1205731199051 =

2 [120573 (119870 + 1198991198701) + 1205744] minus [119875

1120573 (1205742+ 21205745)] 1199052

1

2 [1198751120573212057461199052

1+ 1205744(1 + 120573119905

1)]

(46)

where 119890minus1205731199051 is the complement of the cumulative densityfunction 119865(119905

1) = 1minus119890

minus1205731199051 As 0 le 119865(119905

1) le 1 0 le 119890minus1205731199051 le 1 Let

119890minus1205731199051 = 0 and 119890minus1205731199051 = 1 be the initial upper and lower bounds

of 119890minus1205731199051 respectively Then by using the proposed recursivesearching algorithm given at the end of Section 2 we can findthe optimal production run time 119905lowast

1

4 Numerical Example

In order to relieve the comparison efforts for readers thissection adopts the same numerical example as in [9] For

0225 0265 0305 0345 0385 0425 0465 0505

E[T

CU(t

1)]

t1

tlowast1L = 03411 t

lowast1 = 03748

tlowast1U = 05183

$10600

$10800

$11000

$11200

$11400

$11600

$11006

$11800

Figure 6 The behavior of 119864[TCU(1199051)] in connection with the

production run time 1199051in the proposed model 1

a demonstration of the proposed EPQ-based model 1 thefollowing system parameters are used

1198751 production rate 10000 products per year120582 demand rate 4000 products per year119909 random scrap rate which follows uniformly distribu-

tion over the interval [0 02]120573 Poisson breakdown rate 05 average times per year119892 constant machine repair time 119905

119903 0018 year per repair

119872 machine repair cost $500 for each breakdown119870 setup cost $450 per production run119862 manufacturing cost $2 per item119862119878 disposal cost $03 per scrap itemℎ holding cost $06 per item per unit time1198701 fixed delivery cost $90 per shipment119899 number of deliveries 4 per cycle

119862119879 variable delivery cost $0001 per item

First we use both upper and lower bounds of 119905lowast1to

test for the convexity of 119864[TCU(1199051)] (see Theorem 1) The

computation results of (19) (20) and (11) indicate that 119905lowast1119880

=05183 lt 120585(119905

lowast

1119880) = 28723 and 119905lowast

1119871= 03411 lt 120585(119905

lowast

1119871) = 26261

Hence [TCU(1199051)] is convex (Figure 6)

In order to find the optimal 119905lowast1 we first substitute the

upper and lower bounds of 119905lowast1in (8) and obtain 119864[TCU(119905lowast

1119880)]

= $1160163 and 119864[TCU(119905lowast1119871)] = $1101450 respectively

Because the optimal run time 119905lowast1falls within the interval

of [119905lowast1119871 119905lowast

1119880] we apply the proposed recursive searching

algorithm stated at the end of Section 2 and find 119905lowast1= 03748

years Accordingly the optimal expected system costs perunit time 119864[TCU(119905lowast

1)] = $1100641 (Figure 6) Table 1 shows

the step-by-step iterations of the algorithmIn this specific studied model we focus on incorporating

a discontinuous multidelivery policy into a prior work [9]and consider a fixed transportation cost associated witheach delivery Applying the research result we can easilyinvestigate the effects of different fixed transportation costs

The Scientific World Journal 11

Table 1 Iterations of the recursive searching algorithm for locating 119905lowast1

120573 Step 119905lowast

1119880120596119880= 119890minus1205731199051119880 119905

lowast

1119871120596119871= 119890minus1205731199051119871

Differencebetween 119905lowast

1119880and

119905lowast

1119871

[119880]

119864[TCU(119905lowast1119880)]

[119871]

119864[TCU(119905lowast1119871)]

Differencebetween [119880]

and [119871]05 Initial 000000 100000 $1160163 $1101450 $58713

1st 05183 07717 03411 08432 01772 $1110330 $1101450 $88802nd 03857 08246 03721 08302 00136 $1100716 $1100646 $0703rd 03756 08287 03746 08292 00010 $1100642 $1100641 $0014th 03749 08291 03748 08291 00001 $1100641 $1100641 $00045th 03748 08291 03748 08291 000000 $1100641 $1100641 $0000

Table 2 Variations of the fixed delivery cost1198701effects on the optimal production run time 119905lowast

1

1198701119870 005 02 04 06 08 1 12 14 16 18 2 22

1198701

$225 $90 $180 $270 $360 $450 $540 $630 $720 $810 $900 $990119864[TCU(119905lowast

1)] $10721 $11067 $11448 $11773 $12062 $12325 $12567 $12793 $13006 $13208 $13399 $13583

119905lowast

103116 03816 04586 05244 05828 06359 06848 07305 07735 08142 08529 08900

Table 3 Variations of the unit retailerrsquos holding cost ℎ2and their effects on 119864[TCU(119905lowast

1)]

ℎ2ℎ 05 075 1 125 15 175 2 225 25 275 3 325

ℎ2

03 045 06 075 09 105 12 135 15 165 18 195119864 [TCU (119905lowast

1)] $11282 $11407 $11526 $11638 $11746 $11850 $11949 $12045 $12138 $12229 $12316 $12401

119905lowast

103256 03074 02918 02785 02668 02564 02472 02389 02314 02246 02183 02125

1198701on the optimal system cost 119864[TCU(119905lowast

1)] and on the

optimal production run time 119905lowast1(see Table 2) It can be seen

from Table 2 that as the ratio of1198701119870 increases the expected

system costs per unit time119864[TCU(119905lowast1119871)] increase significantly

It is also noted that as 1198701increases optimal production run

time 119905lowast1also increases significantly

41 Numerical Example for the Producer-Retailer IntegratedEPQ System (Model 2) In order to demonstrate the researchresult of the producer-retailer integrated EPQ-based modelan additional system variable ℎ

2= $150 per item stored at the

retailerrsquos side is includedAgain one can use the upper and lower bounds of 119905lowast

1

(equations (43) and (44)) to test for convexity of 119864[TCU(1199051)]

(Theorem 3 and equation (35)) The results reveal that 119905lowast1119880

=03213 lt 120595(119905lowast

1119880) = 26462 and 119905lowast

1119871= 02186 lt 120595(119905lowast

1119871) = 24876

Therefore the expected cost [TCU(1199051)] is convex

Next by applying the proposed recursive searching algo-rithm we can calculate that the optimal run time 119905lowast

1=

02314 years and the optimal 119864[TCU(119905lowast1)] = $1213849 It is

noted that the computation time for reaching the optimal 119905lowast1

solution is 21 seconds (usingExcel software in a desktop com-puter Intel CPU G850 with 294GB RAM and 289GHz)

Figure 7 illustrates the behavior of 119864[TCU(1199051)] with

regard to production run time It is noted that without theresearch result from the second model the management ofsuch a producer-retailer integrated system would probablyuse 1199051= 03748 years (from the result of model 1) for their

run time decision Further analysis (see Figure 7) shows cost

0130 0170 0210 0250 0290 0330 0370 0410

t1

E[T

CU(t

1)]

11600

11800

12000

12200

12400

12600

12800

tlowast1 = 02314

t1 = 0374812138

(from model 1)

E[TCU( t1)] = 12489

$$

$

$

$

$

$

$

$

Figure 7 The behavior of 119864[TCU(1199051)] with respect to production

run time 1199051in the proposed model 2

savings of $351 (or 29 over the total system costs) simply byapplying our research result

The effects of the unit retailerrsquos holding cost ℎ2on the

expected system cost 119864[TCU(119905lowast1)] and on the optimal run

time 119905lowast1are shown in Table 3 respectively

It can be seen that as ℎ2or the ratio of ℎ

2ℎ increases

the expected cost 119864[TCU(119905lowast1)] increases but the optimal

production run time 119905lowast1decreases In decision-making these

sensitivity analyses results can provide the management ofa producer-retailer integrated system with valuable informa-tion and insights into the effects of various stock holding costsin different retailersrsquo locations

12 The Scientific World Journal

5 Concluding Remarks

Two exact models for an extended EPQ-based problem witha discontinuous delivery policy scrap rate and randombreakdown are developed in this study They specificallyaddress different real-life situations in production end-item delivery and intrasupply chains such as a producer-retailer integrated systemMathematicalmodeling alongwithoptimization techniques is used to determine the optimalproduction run times that minimize the expected systemcosts per unit time Without in-depth investigations onthese separate models the optimal production run time andother important information related to the systemparameterscannot be revealed The proposed real-life EPQ models withrandom machine breakdown discontinuous product distri-bution policies and quality assurance must be specificallystudied in order to (1) obtain the joint effects of breakdowndiscontinuous distribution policies and quality assurance onthe optimal production run time (2) get to know the effectsof different policy and scope of supply chains managementon the optimal run time and overall system costs and (3)gain the insight with regard to various systemrsquos parametersof all particular EPQ-based models Since little attention hasbeen paid to the investigation of joint effects of these practicalproduction situations on the optimal run time this researchis intended to bridge the gap An interesting area for futurestudy is the examination of the effect of variable productionrates on these models

Appendix

Derivations of (3) are as followsRecall (2) as follows

TC1(1199051) = 119862 (119875

11199051) + 119870 + 119862

119878(11990511198751119909) +119872 + 119899119870

1

+ 119862119879[11990511198751] + ℎ3(120582119905119903) 1198791015840

+ ℎ[1198671015840+ 11988911199051

21199051+ (1198671015840

1+ 1198891119905) 119905119903+119899 minus 1

211989911986710158401199051015840

2]

(A1)

Substituting all related system parameters into (2) (pleaserefer to the basic formulations and solution process in [9])the TC

1(1199051) can be obtained as

TC1(1199051) = 119870 +119872

+[1198621198751+ 1198621198781198751119909 + 119862

1198791198751(1 minus 119909)+ ℎ

31198751119892 (1 minus 119909)] 119905

1

+ 1198991198701+ ℎ119901119892119905 minus[

ℎ1198751119892 (1 minus 119909)

2minusℎ1198751119892 (1 minus 119909)

2119899] 1199051

+ 1199052

1[ℎ1198751

2+ℎ1198752

1

2120582(1 minus 119909)

2minusℎ1198751

2(1 minus 119909)

minusℎ1198752

1

2120582119899(1 minus 119909)

2+ℎ1198751

2119899(1 minus 119909)]

(A2)

To take the randomness of 119909 into account by using theexpected values of 119909 with further derivations119864[TC

1(1199051)] can

be derived as follows (ie equation (3))119864 [TC

1(1199051)]

= 119870 + 1198991198701+119872 + ℎ119905119875

1119892

+ [1198621198751+ 1198621198781198751119864 [119909] + 1198621198791198751 (1 minus 119864 [119909])

+ ℎ31198751119892 (1 minus 119864 [119909]) minus

ℎ1198751119892 (1 minus 119864 [119909])

2(1 minus

1

119899)] sdot 1199051

+ [ℎ1198751119864 [119909]

2+ℎ1198752

1

2120582(1 minus 119864 [119909])

2(1 minus

1

119899)

+ℎ1198751

2119899(1 minus 119864 [119909])] 119905

2

1

(A3)

Conflict of Interests

The authors of the paper declare that there is no conflict ofinterests regarding the publication of this paper

Acknowledgment

The authors greatly appreciate the National Science Council(NSC) of Taiwan for supporting this research underGrant noNSC 102-2410-H-324-005

References

[1] E W Taft ldquoThemost economical production lotrdquo Iron Age vol101 pp 1410ndash1412 1918

[2] G Hadley and T M Whitin ldquoAn optimal final inventorymodelrdquoManagement Science vol 7 pp 179ndash183 1961

[3] E A Silver D F Pyke and R Peterson Inventory Managementand Production Planning and Scheduling John Wiley amp SonsNew York NY USA 1998

[4] S Nahmias Production amp Operations Analysis McGraw-HillNew York NY USA 2009

[5] MWidmer and P Solot ldquoDo not forget the breakdowns and themaintenance operations in FMSdesign problemsrdquo InternationalJournal of Production Research vol 28 pp 421ndash430 1990

[6] K-Y C Yu and D L Bricker ldquoAnalysis of a markov chainmodel of a multistage manufacturing system with inspectionrejection and reworkrdquo IIE Transactions vol 25 no 1 pp 109ndash112 1993

[7] H Groenevelt L Pintelon and A Seidmann ldquoProduction lotsizing with machine breakdownsrdquoManagement Science vol 38no 1 pp 104ndash123 1992

[8] G A Widyadana and H M Wee ldquoOptimal deterioratingitems production inventory models with random machinebreakdown and stochastic repair timerdquo Applied MathematicalModelling vol 35 no 7 pp 3495ndash3508 2011

[9] S W Chiu C-L Chou and W-K Wu ldquoOptimizing replenish-ment policy in an EPQ-based inventory model with noncon-forming items and breakdownrdquo EconomicModelling vol 35 pp330ndash337 2013

The Scientific World Journal 13

[10] A M Zargar ldquoEffect of rework strategies on cycle timerdquoComputers amp Industrial Engineering vol 29 no 1ndash4 pp 239ndash243 1995

[11] P Biswas and B R Sarker ldquoOptimal batch quantity modelsfor a lean production system with in-cycle rework and scraprdquoInternational Journal of Production Research vol 46 no 23 pp6585ndash6610 2008

[12] S W Chiu H-D Lin C-B Cheng and C-L Chung ldquoOptimalproduction-shipment decisions for the finite production ratemodel with scraprdquo International Journal for Engineering Mod-elling vol 22 no 1ndash4 pp 25ndash34 2009

[13] Y-S P Chiu K-K Chen and C-K Ting ldquoReplenishment runtime problem with machine breakdown and failure in reworkrdquoExpert Systems with Applications vol 39 no 1 pp 1291ndash12972012

[14] H-D Lin F-Y Pai and S W Chiu ldquoA note on ldquointra-supply chain system with multiple sales locations and qualityassurancerdquordquo Expert Systems with Applications vol 40 no 11 pp4730ndash4732 2013

[15] B Kamsu-Foguem F Rigal and FMauget ldquoMining associationrules for the quality improvement of the production processrdquoExpert Systems with Applications vol 40 no 4 pp 1034ndash10452013

[16] Y Li H Guo L Wang and J Fu ldquoA hybrid genetic-simulatedannealing algorithm for the location-inventory- routing prob-lem considering returns under E-supply chain environmentrdquoThe Scientific World Journal vol 2013 Article ID 125893 10pages 2013

[17] Y-S P Chiu K-K Chen F-T Cheng and C-K Ting ldquoReex-amination of ldquocombining an alternative multi-delivery policyinto economic production lot size problemwith partial reworkrdquousing an alternative approachrdquo Journal of Applied Research andTechnology vol 11 no 3 pp 317ndash323 2013

[18] L B Schwarz B L Deuermeyer and R D Badinelli ldquoFill-rateoptimization in a one-warehouse 119873-identical retailer distribu-tion systemrdquo Management Science vol 31 no 4 pp 488ndash4981985

[19] R A Sarker and L R Khan ldquoOptimal batch size for a pro-duction system operating under periodic delivery policyrdquoCom-puters amp Industrial Engineering vol 37 no 4 pp 711ndash730 1999

[20] N Comez K E Stecke and M Cakanyildirim ldquoMultiple in-cycle transshipments with positive delivery timesrdquo Productionand Operations Management vol 21 no 2 pp 378ndash395 2012

[21] S W Chiu F-Y Pai and W K Wu ldquoAlternative approach todetermine the common cycle time for a multi-item productionsystem with discontinuous deliveries and failure in reworkrdquoEconomic Modelling vol 35 pp 593ndash596 2013

[22] Y-S P Chiu C-C Huang M-F Wu and H-H Chang ldquoJointdetermination of rotation cycle time and number of shipmentsfor a multi-item EPQ model with random defective raterdquoEconomic Modelling vol 35 pp 112ndash117 2013

[23] M A Hoque ldquoSynchronization in the single-manufacturermulti-buyer integrated inventory supply chainrdquo European Jour-nal of Operational Research vol 188 no 3 pp 811ndash825 2008

[24] S W Chiu L-W Lin K-K Chen and C-L Chou ldquoDetermin-ing production-shipment policy for a vendor-buyer integratedsystem with rework and an amending multi-delivery schedulerdquoEconomic Modelling vol 33 pp 668ndash675 2013

[25] M Cedillo-Campos and C Sanchez-Ramırez ldquoDynamic self-assessment of supply chains performance an emerging marketapproachrdquo Journal of Applied Research and Technology vol 11no 3 pp 338ndash347 2013

[26] Y-S P Chiu H-D Lin F-T Cheng and M-H Hwang ldquoOpti-mal common cycle time for a multi-item production systemwith discontinuous delivery policy and failure in reworkrdquoJournal of Scientific and Industrial Research vol 72 no 7 pp435ndash440 2013

[27] HHishamuddin R A Sarker andD Essam ldquoA recoverymech-anism for a two echelon supply chain system under supplydisruptionrdquo Economic Modelling vol 38 pp 555ndash563 2014

[28] M Murugan and V Selladurai ldquoProductivity improvement inmanufacturing submersible pump diffuser housing using leanmanufacturing systemrdquo Journal of Engineering Research vol 2no 1 pp 164ndash182 2014

[29] L Wang H Qu S Liu and C-X Dun ldquoModeling and opti-mization of the multiobjective stochastic joint replenishmentand delivery problem under supply chain environmentrdquo TheScientific World Journal vol 2013 Article ID 916057 11 pages2013

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 The Scientific World Journal

Theorem 1 (119864[TCU(1199051)] is convex if 0 lt 119905

1lt 120585(119905

1)) The

second derivative of 119864[119879119862119880(1199051)] with respect to 119905

1is

1198892119864 [119879119862119880 (119905

1)]

11988921199052

1

=120582

(1 minus 119864 [119909])

sdot [2 (119870 + 119899119870

1)

1199053

11198751

minus ℎ119892 [1 minus(1 minus 119864 [119909])

2(1 minus

1

119899)] (120573

2119890minus1205731199051)

+ [119872

1198751

+ℎ119892

120573](

2 (1 minus 119890minus1205731199051)

1199053

1

minus2120573119890minus1205731199051

1199052

1

minus1205732119890minus1205731199051

1199051

)]

(12)

It is noted that because annual demand 120582 gt 0 the firstterm in the right-hand size (RHS) of (12) is positive Hence weobtain

1198892119864 [119879119862119880 (119905

1)]

1198891199051

2gt 0

119894119891 [2 (119870 + 119899119870

1)

1199053

11198751

minus ℎ119892 [1 minus(1 minus 119864 [119909])

2(1 minus

1

119899)] (120573

2119890minus1205731199051)

+ [119872

1198751

+ℎ119892

120573](

2 (1 minus 119890minus1205731199051)

1199053

1

minus2120573119890minus1205731199051

1199052

1

minus1205732119890minus1205731199051

1199051

)]

gt 0

(13)

The RHS of (13) can be further derived as

119894119891 [2 (119870 + 1198991198701) 120573 minus 119905

3

11198751120573ℎ119892

sdot [1 minus(1 minus 119864 [119909])

2(1 minus

1

119899)] (120573

2119890minus1205731199051)

+ (119872120573 + ℎ1198921198751)

times [2 (1 minus 119890minus1205731199051) minus 2119905

1120573119890minus1205731199051 minus 12057321199052

1119890minus1205731199051]]

gt 0

(14)

Let

1205743= ℎ119892 [1 minus

(1 minus 119864 [119909])

2(1 minus

1

119899)]

1205744= (119872120573 + ℎ119892119875

1)

(15)

then (13) can be rewritten as

1198892119864 [119879119862119880 (119905

1)]

1198891199051

2gt 0

119894119891 [2 (119870 + 1198991198701) 120573 + 2 (1 minus 119890

minus1205731199051) 1205744

minus 1199051[1199052

1119875112057321205743+ 1205744(2 + 120573119905

1)] 120573119890minus1205731199051]

gt 0

(16)

or

1198892119864 [119879119862119880 (119905

1)]

1198891199051

2gt 0

119894119891 0 lt 1199051lt

2 (119870 + 1198991198701) 120573 + 2 (1 minus 119890

minus1205731199051) 1205744

[1199052

1119875112057321205743+ 1205744(2 + 120573119905

1)] 120573119890minus1205731199051

= 120585 (1199051)

(17)

If 119864[119879119862119880(1199051)] is a convex function then the minimum

point exists In order to locate the optimal production run time119905lowast

1that minimizes 119864[119879119862119880(119905

1)] we can set the first derivative of

119864[119879119862119880(1199051)] equal to zero and solve 119905lowast

1

119889119864 [119879119862119880 (1199051)]

1198891199051

=120582

(1 minus 119864 [119909])

sdot minus (119870 + 119899119870

1)

1199052

1119875

+1205742

2+ 1205743(120573119890minus1205731199051)

+ [119872

1198751

+ℎ119892

120573](

minus (1 minus 119890minus1205731199051)

1199052

1

+120573119890minus1205731199051

1199051

)

(18)

In the RHS of (18) it can be seen that the first term ispositive so the second term is equal to zero Let

119905lowast

1119880= radic

2 (1205744+ 119870120573 + 119899119870

1120573)

12057421198751120573

(19)

119905lowast

1119871= 119905ℎ119890 119901119900119904119894119905119894V119890 119903119900119900119905 119900119891

minus1205744plusmn radic1205744

2 + 21198751(1205742+ 2120573120574

3) (119870 + 119899119870

1)

1198751(1205742+ 2120573120574

3)

(20)

Theorem 2 (119905lowast1119871

lt 119905lowast

1lt 119905lowast

1119880) Because the proof of 119905lowast

1falls

within the upper and lower bounds we can multiply the secondterm of (18) by (2120573119875

11199052

1) and obtain

(11987511205731205742+ 2119875112057321205743119890minus1205731199051) 1199052

1+ (21205744120573119890minus1205731199051) 1199051

minus2 [120573 (119870 + 1198991198701) + 1205744(1 minus 119890

minus1205731199051)] = 0

(21)

The Scientific World Journal 7

Producerrsquosproduction-shipment system

Extended toProductionunit

Productionunit

Multideliverypolicy

Multideliverypolicy

Optimization OptimizationProducer-retailer integratedproduction-shipment system

Customer

CustomerRetailer storeor sales office

Figure 4 Extension to a producer-retailer integrated production-shipment system

Thus119905lowast

1

= 119905ℎ119890 119901119900119904119894119905119894V119890 119903119900119900119905 119900119891

(minus (21205744120573119890minus1205731199051)

plusmn ((21205744120573119890minus1205731199051)2

minus [4 (11987511205731205742+ 2119875112057321205743119890minus1205731199051)

times [minus2 [120573 (119870 + 1198991198701) + 1205744(1 minus 119890

minus1205731199051)]]])

12

)

times (2 (12057421198751120573 + 2119875

112057321205743119890minus1205731199051))minus1

(22)

Equation (21) can be rearranged as

2 [1198751120573212057431199052

1+ 12057441205731199051+ 1205744] (119890minus1205731199051)

= 2 [120573 (119870 + 1198991198701) + 1205744] minus (119875

112057312057421199052

1)

(23)

or

119890minus1205731199051 =

2 [120573 (119870 + 1198991198701) + 1205744] minus (119875

112057312057421199052

1)

2 [1198751120573212057431199052

1+ 12057441205731199051+ 1205744]

(24)

where 119890minus1205731199051 is the complement of the cumulative density

function 119865(1199051) = 1 minus 119890

minus1205731199051 As 0 le 119865(119905

1) le 1 0 le 119890

minus1205731199051 le 1

Let 119890minus1205731199051 = 0 and 119890minus1205731199051 = 1 be the upper and lower bounds for119890minus1205731199051 respectively By substituting them into (22) we obtain

119905lowast

1119880= radic

2 [120573 (119870 + 1198991198701) + 1205744]

12057421198751120573

119905lowast

1119871= 119905ℎ119890 119901119900119904119894119905119894V119890 119903119900119900119905 119900119891

minus1205744plusmn radic1205744

2 + 21198751(1205742+ 2120573120574

3) (119870 + 119899119870

1)

1198751(1205742+ 2120573120574

3)

(25)

and 119905lowast1119871lt 119905lowast

1lt 119905lowast

1119880

It is noted that although the optimal production run time119905lowast

1cannot be presented in a closed form it does fall within the

bounds 119905lowast1can be locatedwith the use of a proposed recursive

searching algorithm Let

120596 (1199051) = 119890minus1205731199051 =

2 [120573 (119870 + 1198991198701) + 1205744] minus (119875

112057312057421199052

1)

2 [1198751120573212057431199052

1+ 12057441205731199051+ 1205744]

there4 0 le 120596 (1199051) le 1

(26)

In order to locate 119905lowast1 we can use the following recursive

searching algorithm

(1) Let 120596(1199051) = 0 and 120596(119905

1) = 1 initially and calculate

the upper and lower bounds for 119905lowast1 respectively (ie

to obtain the initial values of [119905lowast1119871 119905lowast

1119880])

(2) Substitute the current values of [119905lowast1119871 119905lowast

1119880] into 119890minus1205731199051 and

compute the new bounds expressed as 120596119871and 120596

119880for

119890minus1205731199051 Hence 120596

119871lt 120596(1199051) lt 120596119880

(3) Let 120596(1199051) = 120596119871and 120596(119905

1) = 120596119880and update the upper

and lower bounds for 119905lowast1 respectively (ie to obtain

the new values of [119905lowast1119871 119905lowast

1119880])

(4) Repeat steps (2) and (3) until there is no signifi-cant difference between 119905lowast

1119871and 119905lowast1119880

(or there is nosignificant difference in terms of their effects on119864[TCU(119905lowast

1)])

(5) Stop 119905lowast1is found

3 Extension to a Producer-Retailer IntegratedEPQ-Based System (Model 2)

31 Enhanced Model Description and Formulation In thissection we further extend the scope of the problem to incor-porate the retailerrsquos stock holding cost into our study Thenew model can be considered a producer-retailer integratedsystem because in the present-day manufacturing sectorsome producers of consumer goods may own and operateretail stores or regional sales offices to promote and sell theirend products to customers (see Figure 4) With the intentionof addressing such a real-life intrasupply chain situation thesecond model of this study incorporates the retailerrsquos stockholding cost into the first model and investigates its effect onthe optimal production run time decision

8 The Scientific World Journal

Dminus 120582tn = I

D

I

t1 + tr tn

Time

t1 + tr

Ic(t)

t9984002

T998400

Figure 5 Inventory level of finished products on the retailerrsquos side in the proposed manufacturing run time problem with breakdown

In the proposed study the retailerrsquos stock holding posi-tions are illustrated in Figure 5

Extra parameters used in this enhanced model includethe following

ℎ2 holding cost per product stored on the retailerrsquos

side

119868119888(119905) on-hand inventory levels in units on the retail-

errsquos side end at time 119905

119863 number of finished products (a fixed quantity)transported to the retail store per shipment

119868 number of left-over products in 119905119899after satisfying

the demand in 119905119899

TC3(1199051) total production-inventory-delivery costs

per cycle of this enhanced model (in the case of abreakdown)

TC4(1199051) total production-inventory-delivery costs

per cycle of this enhanced model (in the case of nobreakdown)

119864[TC3(1199051)] the expected production-inventory-

delivery costs per cycle of this enhanced model (inthe case of a breakdown)

119864[TC4(1199051)] the expected production-inventory-

delivery costs per cycle of this enhanced model (inthe case of no breakdown)

119864[TCU119890(1199051)] the long-run expected production-

inventory-delivery costs per unit time in thisenhanced model whether or not a breakdown takesplace

Since the demand on the retailerrsquos side in time interval119905119899is 120582119905119899 after satisfying the demand the number of left-over

items (see Figure 5) in each 119905119899is

119868 = 119863 minus 120582119905119899 (27)

Total inventory holding costs on retailerrsquos side with andwithout breakdown are shown respectively in

ℎ2[119899(119863 minus 119868)

2119905119899+119899 (119899 + 1)

2119868119905119899+119899119868

2(1199051+ 119905119903)] (28)

ℎ2[119899(119863 minus 119868)

2119905119899+119899 (119899 + 1)

2119868119905119899+119899119868

2(1199051)] (29)

To incorporate the retailerrsquos holding costs into the originalmodels with and without breakdown respectively we obtain

TC3(1199051) = TC

1(1199051)

+ ℎ2[119899(119863 minus 119868)

2119905119899+119899 (119899 + 1)

2119868119905119899+119899119868

2(1199051+ 119905119903)]

TC4(1199051)

= TC2(1199051) + ℎ2[119899(119863 minus 119868)

2119905119899+119899 (119899 + 1)

2119868119905119899+119899119868

2(1199051)]

(30)

To take the randomness of defective rate 119909 into accountand substitute all related variables into (30) with furtherderivations 119864[TC

3(1199051)] and 119864[TC

4(1199051)] can be obtained as

follows

119864 [TC3(1199051)] = 119864 [TC

1(1199051)] + ℎ

2(1198751119892 (1 minus 119864 [119909])

2)

sdot (1 minus1

119899) 1199051+ ℎ2(1198751(1 minus 119864 [119909])

2)

sdot [1198751(1 minus 119864 [119909])

120582119899+ (1 minus

1

119899)] 1199052

1

119864 [TC4(1199051)] = 119864 [TC

2(1199051)] + ℎ

2(1198751(1 minus 119864 [119909])

2)

sdot [1198751(1 minus 119864 [119909])

120582119899+ (1 minus

1

119899)] 1199052

1

(31)

The Scientific World Journal 9

32 Integration of Enhanced Model withwithout BreakdownThe mean time to breakdowns obeys the exponential distri-bution with 119891(119905) = 120573119890minus120573119905 Therefore 119864[TCU

2(1199051)] is

119864 [TCU2(1199051)]

=

int1199051

0119864 [TC

3(1199051)] 119891 (119905) 119889119905 + int

infin

1199051

119864 [TC4(1199051)] 119891 (119905) 119889119905

119864 [T]

(32)

Substituting 119864[TCU3(1199051)] 119864[TC

4(1199051)] and 119864[T] into (32)

and resolving 119864[TCU2(1199051)] we obtain

119864 [TCU2(1199051)]

=120582

(1 minus 119864 [119909])

sdot (119870 + 119899119870

1)

11990511198751

+ 1205741+12057421199051

2+ 12057451199051

+ [119872

1198751

+ℎ119892

120573](

1 minus 119890minus1205731199051

1199051

) minus ℎ119892 (119890minus1205731199051)

minus (ℎ minus ℎ2) [119892 (1 minus 119864 [119909])

2(1 minus

1

119899)] (1 minus 119890

minus1205731199051)

(33)

where

1205745=ℎ2(1 minus 119864 [119909])

2[1198751(1 minus 119864 [119909])

120582119899+ (1 minus

1

119899)] (34)

33 Determining the Optimal Run Time Let 120595(1199051) stand for

the following

120595 (1199051) =

2 (119870 + 1198991198701) 120573 + 2120574

4(1 minus 119890

minus1205731199051)

[1199052

1119875112057321205746+ 1205744(2 + 120573119905

1)] 120573119890minus1205731199051

(35)

Theorem 3 (119864[TCU2(1199051)] is convex if 0 lt 119905

1lt 120595(119905

1)) The

second derivative of 119864[1198791198621198802(1199051)] with respect to 119905

1is

1198892119864 [119879119862119880

2(1199051)]

11988921199052

1

=120582

(1 minus 119864 [119909])[2 (119870 + 119899119870

1)

1199053

11198751

+ (ℎ minus ℎ2)119892 (1 minus 119864 [119909])

2

sdot (1 minus1

119899) (1205732119890minus1205731199051) minus ℎ119892 (120573

2119890minus1205731199051)

+ [119872

1198751

+ℎ119892

120573](

2 (1 minus 119890minus1205731199051)

1199053

1

minus2120573119890minus1205731199051

1199052

1

minus1205732119890minus1205731199051

1199051

)]

(36)

Since annual demand 120582 gt 0 the first term in the RHS of(36) is positive and

119894119891 [2 (119870 + 119899119870

1)

1199053

11198751

+ (ℎ minus ℎ2) [119892 (1 minus 119864 [119909])

2(1 minus

1

119899)]

sdot (1205732119890minus1205731199051) minus ℎ119892 (120573

2119890minus1205731199051) + [

119872

1198751

+ℎ119892

120573]

sdot(

2 (1 minus 119890minus1205731199051)

1199053

1

minus2120573119890minus1205731199051

1199052

1

minus1205732119890minus1205731199051

1199051

)] gt 0

119905ℎ1198901198991198892119864 [119879119862119880

2(1199051)]

1198891199051

2gt 0

(37)

With further derivations the left-hand side (LHS) of (37)becomes

119894119891 [2 (119870 + 1198991198701) 120573 minus 119905

3

11198751120573 (ℎ minus ℎ

2) [119892 (1 minus 119864 [119909])

2(1 minus

1

119899)]

sdot (1205732119890minus1205731199051) minus 1199053

11198751120573ℎ119892 (120573

2119890minus1205731199051)

+ (119872120573 + ℎ1198921198751) [2 (1 minus 119890

minus1205731199051) minus 2119905

1120573119890minus1205731199051 minus 12057321199052

1119890minus1205731199051] ]

gt 0

(38)

Let

1205746= (ℎ minus ℎ

2) [119892 (1 minus 119864 [119909])

2(1 minus

1

119899)] + ℎ119892 (39)

then (37) becomes

119894119891 [2 (119870 + 1198991198701) 120573 + 2120574

4(1 minus 119890

minus1205731199051)

minus1199051[1199052

1119875112057321205746+ 1205744(2 + 120573119905

1)] 120573119890minus1205731199051] gt 0

119905ℎ1198901198991198892119864 [119879119862119880

2(1199051)]

1198891199051

2gt 0

(40)

or

1198892119864 [119879119862119880

2(1199051)]

1198891199051

2gt 0

119894119891 0 lt 1199051lt

2 (119870 + 1198991198701) 120573 + 2120574

4(1 minus 119890

minus1205731199051)

[1199052

1119875112057321205746+ 1205744(2 + 120573119905

1)] 120573119890minus1205731199051

= 120595 (1199051)

(41)

10 The Scientific World Journal

Once 119864[1198791198621198802(1199051)] is proven to be convex the optimal

run time 119905lowast1can be solved by setting the first derivative of

119864[1198791198621198802(1199051)] = 0

119889119864 [1198791198621198802(1199051)]

1198891199051

=120582

(1 minus 119864 [119909])sdot minus (119870 + 119899119870

1)

1199052

1119875

+ (1205742

2+ 1205745) + 1205746(120573119890minus1205731199051)

+ [119872

1198751

+ℎ119892

120573](

minus (1 minus 119890minus1205731199051)

1199052

1

+120573119890minus1205731199051

1199051

)

= 0

(42)

It can be seen that the first term in the RHS of (42) ispositive so the second term is equal to zero In order to findthe bounds for 119905lowast

1 let

119905lowast

1119880= radic

2 [120573 (119870 + 1198991198701) + 1205744]

1198751120573 (1205742+ 21205745)

(43)

119905lowast

1119871= 119905ℎ119890 119901119900119904119894119905119894V119890 119903119900119900119905 119900119891

minus1205744plusmn radic1205742

4+ 21198751(119870 + 119899119870

1) (1205742+ 21205745+ 21205746120573)

1198751(1205742+ 21205745+ 21205746120573)

(44)

Theorem4 (119905lowast1119871lt 119905lowast

1lt 119905lowast

1119880) For the proof ofTheorem 4 please

refer to the proof for Theorem 2 in Section 2

Once we are certain that 119905lowast1falls within the aforemen-

tioned upper and lower bounds in order to find 119905lowast1 we can

first multiply the second term of (42) by (211987511199052

1120573) and obtain

the following

[1198751120573 (1205742+ 21205745) + 2119875

112057321205746119890minus1205731199051] 1199052

1+ (21205744120573119890minus1205731199051) 1199051

minus2 [120573 (119870 + 1198991198701) + 1205744(1 minus 119890

minus1205731199051)]

= 0

(45)

Equation (45) can be rearranged as

119890minus1205731199051 =

2 [120573 (119870 + 1198991198701) + 1205744] minus [119875

1120573 (1205742+ 21205745)] 1199052

1

2 [1198751120573212057461199052

1+ 1205744(1 + 120573119905

1)]

(46)

where 119890minus1205731199051 is the complement of the cumulative densityfunction 119865(119905

1) = 1minus119890

minus1205731199051 As 0 le 119865(119905

1) le 1 0 le 119890minus1205731199051 le 1 Let

119890minus1205731199051 = 0 and 119890minus1205731199051 = 1 be the initial upper and lower bounds

of 119890minus1205731199051 respectively Then by using the proposed recursivesearching algorithm given at the end of Section 2 we can findthe optimal production run time 119905lowast

1

4 Numerical Example

In order to relieve the comparison efforts for readers thissection adopts the same numerical example as in [9] For

0225 0265 0305 0345 0385 0425 0465 0505

E[T

CU(t

1)]

t1

tlowast1L = 03411 t

lowast1 = 03748

tlowast1U = 05183

$10600

$10800

$11000

$11200

$11400

$11600

$11006

$11800

Figure 6 The behavior of 119864[TCU(1199051)] in connection with the

production run time 1199051in the proposed model 1

a demonstration of the proposed EPQ-based model 1 thefollowing system parameters are used

1198751 production rate 10000 products per year120582 demand rate 4000 products per year119909 random scrap rate which follows uniformly distribu-

tion over the interval [0 02]120573 Poisson breakdown rate 05 average times per year119892 constant machine repair time 119905

119903 0018 year per repair

119872 machine repair cost $500 for each breakdown119870 setup cost $450 per production run119862 manufacturing cost $2 per item119862119878 disposal cost $03 per scrap itemℎ holding cost $06 per item per unit time1198701 fixed delivery cost $90 per shipment119899 number of deliveries 4 per cycle

119862119879 variable delivery cost $0001 per item

First we use both upper and lower bounds of 119905lowast1to

test for the convexity of 119864[TCU(1199051)] (see Theorem 1) The

computation results of (19) (20) and (11) indicate that 119905lowast1119880

=05183 lt 120585(119905

lowast

1119880) = 28723 and 119905lowast

1119871= 03411 lt 120585(119905

lowast

1119871) = 26261

Hence [TCU(1199051)] is convex (Figure 6)

In order to find the optimal 119905lowast1 we first substitute the

upper and lower bounds of 119905lowast1in (8) and obtain 119864[TCU(119905lowast

1119880)]

= $1160163 and 119864[TCU(119905lowast1119871)] = $1101450 respectively

Because the optimal run time 119905lowast1falls within the interval

of [119905lowast1119871 119905lowast

1119880] we apply the proposed recursive searching

algorithm stated at the end of Section 2 and find 119905lowast1= 03748

years Accordingly the optimal expected system costs perunit time 119864[TCU(119905lowast

1)] = $1100641 (Figure 6) Table 1 shows

the step-by-step iterations of the algorithmIn this specific studied model we focus on incorporating

a discontinuous multidelivery policy into a prior work [9]and consider a fixed transportation cost associated witheach delivery Applying the research result we can easilyinvestigate the effects of different fixed transportation costs

The Scientific World Journal 11

Table 1 Iterations of the recursive searching algorithm for locating 119905lowast1

120573 Step 119905lowast

1119880120596119880= 119890minus1205731199051119880 119905

lowast

1119871120596119871= 119890minus1205731199051119871

Differencebetween 119905lowast

1119880and

119905lowast

1119871

[119880]

119864[TCU(119905lowast1119880)]

[119871]

119864[TCU(119905lowast1119871)]

Differencebetween [119880]

and [119871]05 Initial 000000 100000 $1160163 $1101450 $58713

1st 05183 07717 03411 08432 01772 $1110330 $1101450 $88802nd 03857 08246 03721 08302 00136 $1100716 $1100646 $0703rd 03756 08287 03746 08292 00010 $1100642 $1100641 $0014th 03749 08291 03748 08291 00001 $1100641 $1100641 $00045th 03748 08291 03748 08291 000000 $1100641 $1100641 $0000

Table 2 Variations of the fixed delivery cost1198701effects on the optimal production run time 119905lowast

1

1198701119870 005 02 04 06 08 1 12 14 16 18 2 22

1198701

$225 $90 $180 $270 $360 $450 $540 $630 $720 $810 $900 $990119864[TCU(119905lowast

1)] $10721 $11067 $11448 $11773 $12062 $12325 $12567 $12793 $13006 $13208 $13399 $13583

119905lowast

103116 03816 04586 05244 05828 06359 06848 07305 07735 08142 08529 08900

Table 3 Variations of the unit retailerrsquos holding cost ℎ2and their effects on 119864[TCU(119905lowast

1)]

ℎ2ℎ 05 075 1 125 15 175 2 225 25 275 3 325

ℎ2

03 045 06 075 09 105 12 135 15 165 18 195119864 [TCU (119905lowast

1)] $11282 $11407 $11526 $11638 $11746 $11850 $11949 $12045 $12138 $12229 $12316 $12401

119905lowast

103256 03074 02918 02785 02668 02564 02472 02389 02314 02246 02183 02125

1198701on the optimal system cost 119864[TCU(119905lowast

1)] and on the

optimal production run time 119905lowast1(see Table 2) It can be seen

from Table 2 that as the ratio of1198701119870 increases the expected

system costs per unit time119864[TCU(119905lowast1119871)] increase significantly

It is also noted that as 1198701increases optimal production run

time 119905lowast1also increases significantly

41 Numerical Example for the Producer-Retailer IntegratedEPQ System (Model 2) In order to demonstrate the researchresult of the producer-retailer integrated EPQ-based modelan additional system variable ℎ

2= $150 per item stored at the

retailerrsquos side is includedAgain one can use the upper and lower bounds of 119905lowast

1

(equations (43) and (44)) to test for convexity of 119864[TCU(1199051)]

(Theorem 3 and equation (35)) The results reveal that 119905lowast1119880

=03213 lt 120595(119905lowast

1119880) = 26462 and 119905lowast

1119871= 02186 lt 120595(119905lowast

1119871) = 24876

Therefore the expected cost [TCU(1199051)] is convex

Next by applying the proposed recursive searching algo-rithm we can calculate that the optimal run time 119905lowast

1=

02314 years and the optimal 119864[TCU(119905lowast1)] = $1213849 It is

noted that the computation time for reaching the optimal 119905lowast1

solution is 21 seconds (usingExcel software in a desktop com-puter Intel CPU G850 with 294GB RAM and 289GHz)

Figure 7 illustrates the behavior of 119864[TCU(1199051)] with

regard to production run time It is noted that without theresearch result from the second model the management ofsuch a producer-retailer integrated system would probablyuse 1199051= 03748 years (from the result of model 1) for their

run time decision Further analysis (see Figure 7) shows cost

0130 0170 0210 0250 0290 0330 0370 0410

t1

E[T

CU(t

1)]

11600

11800

12000

12200

12400

12600

12800

tlowast1 = 02314

t1 = 0374812138

(from model 1)

E[TCU( t1)] = 12489

$$

$

$

$

$

$

$

$

Figure 7 The behavior of 119864[TCU(1199051)] with respect to production

run time 1199051in the proposed model 2

savings of $351 (or 29 over the total system costs) simply byapplying our research result

The effects of the unit retailerrsquos holding cost ℎ2on the

expected system cost 119864[TCU(119905lowast1)] and on the optimal run

time 119905lowast1are shown in Table 3 respectively

It can be seen that as ℎ2or the ratio of ℎ

2ℎ increases

the expected cost 119864[TCU(119905lowast1)] increases but the optimal

production run time 119905lowast1decreases In decision-making these

sensitivity analyses results can provide the management ofa producer-retailer integrated system with valuable informa-tion and insights into the effects of various stock holding costsin different retailersrsquo locations

12 The Scientific World Journal

5 Concluding Remarks

Two exact models for an extended EPQ-based problem witha discontinuous delivery policy scrap rate and randombreakdown are developed in this study They specificallyaddress different real-life situations in production end-item delivery and intrasupply chains such as a producer-retailer integrated systemMathematicalmodeling alongwithoptimization techniques is used to determine the optimalproduction run times that minimize the expected systemcosts per unit time Without in-depth investigations onthese separate models the optimal production run time andother important information related to the systemparameterscannot be revealed The proposed real-life EPQ models withrandom machine breakdown discontinuous product distri-bution policies and quality assurance must be specificallystudied in order to (1) obtain the joint effects of breakdowndiscontinuous distribution policies and quality assurance onthe optimal production run time (2) get to know the effectsof different policy and scope of supply chains managementon the optimal run time and overall system costs and (3)gain the insight with regard to various systemrsquos parametersof all particular EPQ-based models Since little attention hasbeen paid to the investigation of joint effects of these practicalproduction situations on the optimal run time this researchis intended to bridge the gap An interesting area for futurestudy is the examination of the effect of variable productionrates on these models

Appendix

Derivations of (3) are as followsRecall (2) as follows

TC1(1199051) = 119862 (119875

11199051) + 119870 + 119862

119878(11990511198751119909) +119872 + 119899119870

1

+ 119862119879[11990511198751] + ℎ3(120582119905119903) 1198791015840

+ ℎ[1198671015840+ 11988911199051

21199051+ (1198671015840

1+ 1198891119905) 119905119903+119899 minus 1

211989911986710158401199051015840

2]

(A1)

Substituting all related system parameters into (2) (pleaserefer to the basic formulations and solution process in [9])the TC

1(1199051) can be obtained as

TC1(1199051) = 119870 +119872

+[1198621198751+ 1198621198781198751119909 + 119862

1198791198751(1 minus 119909)+ ℎ

31198751119892 (1 minus 119909)] 119905

1

+ 1198991198701+ ℎ119901119892119905 minus[

ℎ1198751119892 (1 minus 119909)

2minusℎ1198751119892 (1 minus 119909)

2119899] 1199051

+ 1199052

1[ℎ1198751

2+ℎ1198752

1

2120582(1 minus 119909)

2minusℎ1198751

2(1 minus 119909)

minusℎ1198752

1

2120582119899(1 minus 119909)

2+ℎ1198751

2119899(1 minus 119909)]

(A2)

To take the randomness of 119909 into account by using theexpected values of 119909 with further derivations119864[TC

1(1199051)] can

be derived as follows (ie equation (3))119864 [TC

1(1199051)]

= 119870 + 1198991198701+119872 + ℎ119905119875

1119892

+ [1198621198751+ 1198621198781198751119864 [119909] + 1198621198791198751 (1 minus 119864 [119909])

+ ℎ31198751119892 (1 minus 119864 [119909]) minus

ℎ1198751119892 (1 minus 119864 [119909])

2(1 minus

1

119899)] sdot 1199051

+ [ℎ1198751119864 [119909]

2+ℎ1198752

1

2120582(1 minus 119864 [119909])

2(1 minus

1

119899)

+ℎ1198751

2119899(1 minus 119864 [119909])] 119905

2

1

(A3)

Conflict of Interests

The authors of the paper declare that there is no conflict ofinterests regarding the publication of this paper

Acknowledgment

The authors greatly appreciate the National Science Council(NSC) of Taiwan for supporting this research underGrant noNSC 102-2410-H-324-005

References

[1] E W Taft ldquoThemost economical production lotrdquo Iron Age vol101 pp 1410ndash1412 1918

[2] G Hadley and T M Whitin ldquoAn optimal final inventorymodelrdquoManagement Science vol 7 pp 179ndash183 1961

[3] E A Silver D F Pyke and R Peterson Inventory Managementand Production Planning and Scheduling John Wiley amp SonsNew York NY USA 1998

[4] S Nahmias Production amp Operations Analysis McGraw-HillNew York NY USA 2009

[5] MWidmer and P Solot ldquoDo not forget the breakdowns and themaintenance operations in FMSdesign problemsrdquo InternationalJournal of Production Research vol 28 pp 421ndash430 1990

[6] K-Y C Yu and D L Bricker ldquoAnalysis of a markov chainmodel of a multistage manufacturing system with inspectionrejection and reworkrdquo IIE Transactions vol 25 no 1 pp 109ndash112 1993

[7] H Groenevelt L Pintelon and A Seidmann ldquoProduction lotsizing with machine breakdownsrdquoManagement Science vol 38no 1 pp 104ndash123 1992

[8] G A Widyadana and H M Wee ldquoOptimal deterioratingitems production inventory models with random machinebreakdown and stochastic repair timerdquo Applied MathematicalModelling vol 35 no 7 pp 3495ndash3508 2011

[9] S W Chiu C-L Chou and W-K Wu ldquoOptimizing replenish-ment policy in an EPQ-based inventory model with noncon-forming items and breakdownrdquo EconomicModelling vol 35 pp330ndash337 2013

The Scientific World Journal 13

[10] A M Zargar ldquoEffect of rework strategies on cycle timerdquoComputers amp Industrial Engineering vol 29 no 1ndash4 pp 239ndash243 1995

[11] P Biswas and B R Sarker ldquoOptimal batch quantity modelsfor a lean production system with in-cycle rework and scraprdquoInternational Journal of Production Research vol 46 no 23 pp6585ndash6610 2008

[12] S W Chiu H-D Lin C-B Cheng and C-L Chung ldquoOptimalproduction-shipment decisions for the finite production ratemodel with scraprdquo International Journal for Engineering Mod-elling vol 22 no 1ndash4 pp 25ndash34 2009

[13] Y-S P Chiu K-K Chen and C-K Ting ldquoReplenishment runtime problem with machine breakdown and failure in reworkrdquoExpert Systems with Applications vol 39 no 1 pp 1291ndash12972012

[14] H-D Lin F-Y Pai and S W Chiu ldquoA note on ldquointra-supply chain system with multiple sales locations and qualityassurancerdquordquo Expert Systems with Applications vol 40 no 11 pp4730ndash4732 2013

[15] B Kamsu-Foguem F Rigal and FMauget ldquoMining associationrules for the quality improvement of the production processrdquoExpert Systems with Applications vol 40 no 4 pp 1034ndash10452013

[16] Y Li H Guo L Wang and J Fu ldquoA hybrid genetic-simulatedannealing algorithm for the location-inventory- routing prob-lem considering returns under E-supply chain environmentrdquoThe Scientific World Journal vol 2013 Article ID 125893 10pages 2013

[17] Y-S P Chiu K-K Chen F-T Cheng and C-K Ting ldquoReex-amination of ldquocombining an alternative multi-delivery policyinto economic production lot size problemwith partial reworkrdquousing an alternative approachrdquo Journal of Applied Research andTechnology vol 11 no 3 pp 317ndash323 2013

[18] L B Schwarz B L Deuermeyer and R D Badinelli ldquoFill-rateoptimization in a one-warehouse 119873-identical retailer distribu-tion systemrdquo Management Science vol 31 no 4 pp 488ndash4981985

[19] R A Sarker and L R Khan ldquoOptimal batch size for a pro-duction system operating under periodic delivery policyrdquoCom-puters amp Industrial Engineering vol 37 no 4 pp 711ndash730 1999

[20] N Comez K E Stecke and M Cakanyildirim ldquoMultiple in-cycle transshipments with positive delivery timesrdquo Productionand Operations Management vol 21 no 2 pp 378ndash395 2012

[21] S W Chiu F-Y Pai and W K Wu ldquoAlternative approach todetermine the common cycle time for a multi-item productionsystem with discontinuous deliveries and failure in reworkrdquoEconomic Modelling vol 35 pp 593ndash596 2013

[22] Y-S P Chiu C-C Huang M-F Wu and H-H Chang ldquoJointdetermination of rotation cycle time and number of shipmentsfor a multi-item EPQ model with random defective raterdquoEconomic Modelling vol 35 pp 112ndash117 2013

[23] M A Hoque ldquoSynchronization in the single-manufacturermulti-buyer integrated inventory supply chainrdquo European Jour-nal of Operational Research vol 188 no 3 pp 811ndash825 2008

[24] S W Chiu L-W Lin K-K Chen and C-L Chou ldquoDetermin-ing production-shipment policy for a vendor-buyer integratedsystem with rework and an amending multi-delivery schedulerdquoEconomic Modelling vol 33 pp 668ndash675 2013

[25] M Cedillo-Campos and C Sanchez-Ramırez ldquoDynamic self-assessment of supply chains performance an emerging marketapproachrdquo Journal of Applied Research and Technology vol 11no 3 pp 338ndash347 2013

[26] Y-S P Chiu H-D Lin F-T Cheng and M-H Hwang ldquoOpti-mal common cycle time for a multi-item production systemwith discontinuous delivery policy and failure in reworkrdquoJournal of Scientific and Industrial Research vol 72 no 7 pp435ndash440 2013

[27] HHishamuddin R A Sarker andD Essam ldquoA recoverymech-anism for a two echelon supply chain system under supplydisruptionrdquo Economic Modelling vol 38 pp 555ndash563 2014

[28] M Murugan and V Selladurai ldquoProductivity improvement inmanufacturing submersible pump diffuser housing using leanmanufacturing systemrdquo Journal of Engineering Research vol 2no 1 pp 164ndash182 2014

[29] L Wang H Qu S Liu and C-X Dun ldquoModeling and opti-mization of the multiobjective stochastic joint replenishmentand delivery problem under supply chain environmentrdquo TheScientific World Journal vol 2013 Article ID 916057 11 pages2013

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

The Scientific World Journal 7

Producerrsquosproduction-shipment system

Extended toProductionunit

Productionunit

Multideliverypolicy

Multideliverypolicy

Optimization OptimizationProducer-retailer integratedproduction-shipment system

Customer

CustomerRetailer storeor sales office

Figure 4 Extension to a producer-retailer integrated production-shipment system

Thus119905lowast

1

= 119905ℎ119890 119901119900119904119894119905119894V119890 119903119900119900119905 119900119891

(minus (21205744120573119890minus1205731199051)

plusmn ((21205744120573119890minus1205731199051)2

minus [4 (11987511205731205742+ 2119875112057321205743119890minus1205731199051)

times [minus2 [120573 (119870 + 1198991198701) + 1205744(1 minus 119890

minus1205731199051)]]])

12

)

times (2 (12057421198751120573 + 2119875

112057321205743119890minus1205731199051))minus1

(22)

Equation (21) can be rearranged as

2 [1198751120573212057431199052

1+ 12057441205731199051+ 1205744] (119890minus1205731199051)

= 2 [120573 (119870 + 1198991198701) + 1205744] minus (119875

112057312057421199052

1)

(23)

or

119890minus1205731199051 =

2 [120573 (119870 + 1198991198701) + 1205744] minus (119875

112057312057421199052

1)

2 [1198751120573212057431199052

1+ 12057441205731199051+ 1205744]

(24)

where 119890minus1205731199051 is the complement of the cumulative density

function 119865(1199051) = 1 minus 119890

minus1205731199051 As 0 le 119865(119905

1) le 1 0 le 119890

minus1205731199051 le 1

Let 119890minus1205731199051 = 0 and 119890minus1205731199051 = 1 be the upper and lower bounds for119890minus1205731199051 respectively By substituting them into (22) we obtain

119905lowast

1119880= radic

2 [120573 (119870 + 1198991198701) + 1205744]

12057421198751120573

119905lowast

1119871= 119905ℎ119890 119901119900119904119894119905119894V119890 119903119900119900119905 119900119891

minus1205744plusmn radic1205744

2 + 21198751(1205742+ 2120573120574

3) (119870 + 119899119870

1)

1198751(1205742+ 2120573120574

3)

(25)

and 119905lowast1119871lt 119905lowast

1lt 119905lowast

1119880

It is noted that although the optimal production run time119905lowast

1cannot be presented in a closed form it does fall within the

bounds 119905lowast1can be locatedwith the use of a proposed recursive

searching algorithm Let

120596 (1199051) = 119890minus1205731199051 =

2 [120573 (119870 + 1198991198701) + 1205744] minus (119875

112057312057421199052

1)

2 [1198751120573212057431199052

1+ 12057441205731199051+ 1205744]

there4 0 le 120596 (1199051) le 1

(26)

In order to locate 119905lowast1 we can use the following recursive

searching algorithm

(1) Let 120596(1199051) = 0 and 120596(119905

1) = 1 initially and calculate

the upper and lower bounds for 119905lowast1 respectively (ie

to obtain the initial values of [119905lowast1119871 119905lowast

1119880])

(2) Substitute the current values of [119905lowast1119871 119905lowast

1119880] into 119890minus1205731199051 and

compute the new bounds expressed as 120596119871and 120596

119880for

119890minus1205731199051 Hence 120596

119871lt 120596(1199051) lt 120596119880

(3) Let 120596(1199051) = 120596119871and 120596(119905

1) = 120596119880and update the upper

and lower bounds for 119905lowast1 respectively (ie to obtain

the new values of [119905lowast1119871 119905lowast

1119880])

(4) Repeat steps (2) and (3) until there is no signifi-cant difference between 119905lowast

1119871and 119905lowast1119880

(or there is nosignificant difference in terms of their effects on119864[TCU(119905lowast

1)])

(5) Stop 119905lowast1is found

3 Extension to a Producer-Retailer IntegratedEPQ-Based System (Model 2)

31 Enhanced Model Description and Formulation In thissection we further extend the scope of the problem to incor-porate the retailerrsquos stock holding cost into our study Thenew model can be considered a producer-retailer integratedsystem because in the present-day manufacturing sectorsome producers of consumer goods may own and operateretail stores or regional sales offices to promote and sell theirend products to customers (see Figure 4) With the intentionof addressing such a real-life intrasupply chain situation thesecond model of this study incorporates the retailerrsquos stockholding cost into the first model and investigates its effect onthe optimal production run time decision

8 The Scientific World Journal

Dminus 120582tn = I

D

I

t1 + tr tn

Time

t1 + tr

Ic(t)

t9984002

T998400

Figure 5 Inventory level of finished products on the retailerrsquos side in the proposed manufacturing run time problem with breakdown

In the proposed study the retailerrsquos stock holding posi-tions are illustrated in Figure 5

Extra parameters used in this enhanced model includethe following

ℎ2 holding cost per product stored on the retailerrsquos

side

119868119888(119905) on-hand inventory levels in units on the retail-

errsquos side end at time 119905

119863 number of finished products (a fixed quantity)transported to the retail store per shipment

119868 number of left-over products in 119905119899after satisfying

the demand in 119905119899

TC3(1199051) total production-inventory-delivery costs

per cycle of this enhanced model (in the case of abreakdown)

TC4(1199051) total production-inventory-delivery costs

per cycle of this enhanced model (in the case of nobreakdown)

119864[TC3(1199051)] the expected production-inventory-

delivery costs per cycle of this enhanced model (inthe case of a breakdown)

119864[TC4(1199051)] the expected production-inventory-

delivery costs per cycle of this enhanced model (inthe case of no breakdown)

119864[TCU119890(1199051)] the long-run expected production-

inventory-delivery costs per unit time in thisenhanced model whether or not a breakdown takesplace

Since the demand on the retailerrsquos side in time interval119905119899is 120582119905119899 after satisfying the demand the number of left-over

items (see Figure 5) in each 119905119899is

119868 = 119863 minus 120582119905119899 (27)

Total inventory holding costs on retailerrsquos side with andwithout breakdown are shown respectively in

ℎ2[119899(119863 minus 119868)

2119905119899+119899 (119899 + 1)

2119868119905119899+119899119868

2(1199051+ 119905119903)] (28)

ℎ2[119899(119863 minus 119868)

2119905119899+119899 (119899 + 1)

2119868119905119899+119899119868

2(1199051)] (29)

To incorporate the retailerrsquos holding costs into the originalmodels with and without breakdown respectively we obtain

TC3(1199051) = TC

1(1199051)

+ ℎ2[119899(119863 minus 119868)

2119905119899+119899 (119899 + 1)

2119868119905119899+119899119868

2(1199051+ 119905119903)]

TC4(1199051)

= TC2(1199051) + ℎ2[119899(119863 minus 119868)

2119905119899+119899 (119899 + 1)

2119868119905119899+119899119868

2(1199051)]

(30)

To take the randomness of defective rate 119909 into accountand substitute all related variables into (30) with furtherderivations 119864[TC

3(1199051)] and 119864[TC

4(1199051)] can be obtained as

follows

119864 [TC3(1199051)] = 119864 [TC

1(1199051)] + ℎ

2(1198751119892 (1 minus 119864 [119909])

2)

sdot (1 minus1

119899) 1199051+ ℎ2(1198751(1 minus 119864 [119909])

2)

sdot [1198751(1 minus 119864 [119909])

120582119899+ (1 minus

1

119899)] 1199052

1

119864 [TC4(1199051)] = 119864 [TC

2(1199051)] + ℎ

2(1198751(1 minus 119864 [119909])

2)

sdot [1198751(1 minus 119864 [119909])

120582119899+ (1 minus

1

119899)] 1199052

1

(31)

The Scientific World Journal 9

32 Integration of Enhanced Model withwithout BreakdownThe mean time to breakdowns obeys the exponential distri-bution with 119891(119905) = 120573119890minus120573119905 Therefore 119864[TCU

2(1199051)] is

119864 [TCU2(1199051)]

=

int1199051

0119864 [TC

3(1199051)] 119891 (119905) 119889119905 + int

infin

1199051

119864 [TC4(1199051)] 119891 (119905) 119889119905

119864 [T]

(32)

Substituting 119864[TCU3(1199051)] 119864[TC

4(1199051)] and 119864[T] into (32)

and resolving 119864[TCU2(1199051)] we obtain

119864 [TCU2(1199051)]

=120582

(1 minus 119864 [119909])

sdot (119870 + 119899119870

1)

11990511198751

+ 1205741+12057421199051

2+ 12057451199051

+ [119872

1198751

+ℎ119892

120573](

1 minus 119890minus1205731199051

1199051

) minus ℎ119892 (119890minus1205731199051)

minus (ℎ minus ℎ2) [119892 (1 minus 119864 [119909])

2(1 minus

1

119899)] (1 minus 119890

minus1205731199051)

(33)

where

1205745=ℎ2(1 minus 119864 [119909])

2[1198751(1 minus 119864 [119909])

120582119899+ (1 minus

1

119899)] (34)

33 Determining the Optimal Run Time Let 120595(1199051) stand for

the following

120595 (1199051) =

2 (119870 + 1198991198701) 120573 + 2120574

4(1 minus 119890

minus1205731199051)

[1199052

1119875112057321205746+ 1205744(2 + 120573119905

1)] 120573119890minus1205731199051

(35)

Theorem 3 (119864[TCU2(1199051)] is convex if 0 lt 119905

1lt 120595(119905

1)) The

second derivative of 119864[1198791198621198802(1199051)] with respect to 119905

1is

1198892119864 [119879119862119880

2(1199051)]

11988921199052

1

=120582

(1 minus 119864 [119909])[2 (119870 + 119899119870

1)

1199053

11198751

+ (ℎ minus ℎ2)119892 (1 minus 119864 [119909])

2

sdot (1 minus1

119899) (1205732119890minus1205731199051) minus ℎ119892 (120573

2119890minus1205731199051)

+ [119872

1198751

+ℎ119892

120573](

2 (1 minus 119890minus1205731199051)

1199053

1

minus2120573119890minus1205731199051

1199052

1

minus1205732119890minus1205731199051

1199051

)]

(36)

Since annual demand 120582 gt 0 the first term in the RHS of(36) is positive and

119894119891 [2 (119870 + 119899119870

1)

1199053

11198751

+ (ℎ minus ℎ2) [119892 (1 minus 119864 [119909])

2(1 minus

1

119899)]

sdot (1205732119890minus1205731199051) minus ℎ119892 (120573

2119890minus1205731199051) + [

119872

1198751

+ℎ119892

120573]

sdot(

2 (1 minus 119890minus1205731199051)

1199053

1

minus2120573119890minus1205731199051

1199052

1

minus1205732119890minus1205731199051

1199051

)] gt 0

119905ℎ1198901198991198892119864 [119879119862119880

2(1199051)]

1198891199051

2gt 0

(37)

With further derivations the left-hand side (LHS) of (37)becomes

119894119891 [2 (119870 + 1198991198701) 120573 minus 119905

3

11198751120573 (ℎ minus ℎ

2) [119892 (1 minus 119864 [119909])

2(1 minus

1

119899)]

sdot (1205732119890minus1205731199051) minus 1199053

11198751120573ℎ119892 (120573

2119890minus1205731199051)

+ (119872120573 + ℎ1198921198751) [2 (1 minus 119890

minus1205731199051) minus 2119905

1120573119890minus1205731199051 minus 12057321199052

1119890minus1205731199051] ]

gt 0

(38)

Let

1205746= (ℎ minus ℎ

2) [119892 (1 minus 119864 [119909])

2(1 minus

1

119899)] + ℎ119892 (39)

then (37) becomes

119894119891 [2 (119870 + 1198991198701) 120573 + 2120574

4(1 minus 119890

minus1205731199051)

minus1199051[1199052

1119875112057321205746+ 1205744(2 + 120573119905

1)] 120573119890minus1205731199051] gt 0

119905ℎ1198901198991198892119864 [119879119862119880

2(1199051)]

1198891199051

2gt 0

(40)

or

1198892119864 [119879119862119880

2(1199051)]

1198891199051

2gt 0

119894119891 0 lt 1199051lt

2 (119870 + 1198991198701) 120573 + 2120574

4(1 minus 119890

minus1205731199051)

[1199052

1119875112057321205746+ 1205744(2 + 120573119905

1)] 120573119890minus1205731199051

= 120595 (1199051)

(41)

10 The Scientific World Journal

Once 119864[1198791198621198802(1199051)] is proven to be convex the optimal

run time 119905lowast1can be solved by setting the first derivative of

119864[1198791198621198802(1199051)] = 0

119889119864 [1198791198621198802(1199051)]

1198891199051

=120582

(1 minus 119864 [119909])sdot minus (119870 + 119899119870

1)

1199052

1119875

+ (1205742

2+ 1205745) + 1205746(120573119890minus1205731199051)

+ [119872

1198751

+ℎ119892

120573](

minus (1 minus 119890minus1205731199051)

1199052

1

+120573119890minus1205731199051

1199051

)

= 0

(42)

It can be seen that the first term in the RHS of (42) ispositive so the second term is equal to zero In order to findthe bounds for 119905lowast

1 let

119905lowast

1119880= radic

2 [120573 (119870 + 1198991198701) + 1205744]

1198751120573 (1205742+ 21205745)

(43)

119905lowast

1119871= 119905ℎ119890 119901119900119904119894119905119894V119890 119903119900119900119905 119900119891

minus1205744plusmn radic1205742

4+ 21198751(119870 + 119899119870

1) (1205742+ 21205745+ 21205746120573)

1198751(1205742+ 21205745+ 21205746120573)

(44)

Theorem4 (119905lowast1119871lt 119905lowast

1lt 119905lowast

1119880) For the proof ofTheorem 4 please

refer to the proof for Theorem 2 in Section 2

Once we are certain that 119905lowast1falls within the aforemen-

tioned upper and lower bounds in order to find 119905lowast1 we can

first multiply the second term of (42) by (211987511199052

1120573) and obtain

the following

[1198751120573 (1205742+ 21205745) + 2119875

112057321205746119890minus1205731199051] 1199052

1+ (21205744120573119890minus1205731199051) 1199051

minus2 [120573 (119870 + 1198991198701) + 1205744(1 minus 119890

minus1205731199051)]

= 0

(45)

Equation (45) can be rearranged as

119890minus1205731199051 =

2 [120573 (119870 + 1198991198701) + 1205744] minus [119875

1120573 (1205742+ 21205745)] 1199052

1

2 [1198751120573212057461199052

1+ 1205744(1 + 120573119905

1)]

(46)

where 119890minus1205731199051 is the complement of the cumulative densityfunction 119865(119905

1) = 1minus119890

minus1205731199051 As 0 le 119865(119905

1) le 1 0 le 119890minus1205731199051 le 1 Let

119890minus1205731199051 = 0 and 119890minus1205731199051 = 1 be the initial upper and lower bounds

of 119890minus1205731199051 respectively Then by using the proposed recursivesearching algorithm given at the end of Section 2 we can findthe optimal production run time 119905lowast

1

4 Numerical Example

In order to relieve the comparison efforts for readers thissection adopts the same numerical example as in [9] For

0225 0265 0305 0345 0385 0425 0465 0505

E[T

CU(t

1)]

t1

tlowast1L = 03411 t

lowast1 = 03748

tlowast1U = 05183

$10600

$10800

$11000

$11200

$11400

$11600

$11006

$11800

Figure 6 The behavior of 119864[TCU(1199051)] in connection with the

production run time 1199051in the proposed model 1

a demonstration of the proposed EPQ-based model 1 thefollowing system parameters are used

1198751 production rate 10000 products per year120582 demand rate 4000 products per year119909 random scrap rate which follows uniformly distribu-

tion over the interval [0 02]120573 Poisson breakdown rate 05 average times per year119892 constant machine repair time 119905

119903 0018 year per repair

119872 machine repair cost $500 for each breakdown119870 setup cost $450 per production run119862 manufacturing cost $2 per item119862119878 disposal cost $03 per scrap itemℎ holding cost $06 per item per unit time1198701 fixed delivery cost $90 per shipment119899 number of deliveries 4 per cycle

119862119879 variable delivery cost $0001 per item

First we use both upper and lower bounds of 119905lowast1to

test for the convexity of 119864[TCU(1199051)] (see Theorem 1) The

computation results of (19) (20) and (11) indicate that 119905lowast1119880

=05183 lt 120585(119905

lowast

1119880) = 28723 and 119905lowast

1119871= 03411 lt 120585(119905

lowast

1119871) = 26261

Hence [TCU(1199051)] is convex (Figure 6)

In order to find the optimal 119905lowast1 we first substitute the

upper and lower bounds of 119905lowast1in (8) and obtain 119864[TCU(119905lowast

1119880)]

= $1160163 and 119864[TCU(119905lowast1119871)] = $1101450 respectively

Because the optimal run time 119905lowast1falls within the interval

of [119905lowast1119871 119905lowast

1119880] we apply the proposed recursive searching

algorithm stated at the end of Section 2 and find 119905lowast1= 03748

years Accordingly the optimal expected system costs perunit time 119864[TCU(119905lowast

1)] = $1100641 (Figure 6) Table 1 shows

the step-by-step iterations of the algorithmIn this specific studied model we focus on incorporating

a discontinuous multidelivery policy into a prior work [9]and consider a fixed transportation cost associated witheach delivery Applying the research result we can easilyinvestigate the effects of different fixed transportation costs

The Scientific World Journal 11

Table 1 Iterations of the recursive searching algorithm for locating 119905lowast1

120573 Step 119905lowast

1119880120596119880= 119890minus1205731199051119880 119905

lowast

1119871120596119871= 119890minus1205731199051119871

Differencebetween 119905lowast

1119880and

119905lowast

1119871

[119880]

119864[TCU(119905lowast1119880)]

[119871]

119864[TCU(119905lowast1119871)]

Differencebetween [119880]

and [119871]05 Initial 000000 100000 $1160163 $1101450 $58713

1st 05183 07717 03411 08432 01772 $1110330 $1101450 $88802nd 03857 08246 03721 08302 00136 $1100716 $1100646 $0703rd 03756 08287 03746 08292 00010 $1100642 $1100641 $0014th 03749 08291 03748 08291 00001 $1100641 $1100641 $00045th 03748 08291 03748 08291 000000 $1100641 $1100641 $0000

Table 2 Variations of the fixed delivery cost1198701effects on the optimal production run time 119905lowast

1

1198701119870 005 02 04 06 08 1 12 14 16 18 2 22

1198701

$225 $90 $180 $270 $360 $450 $540 $630 $720 $810 $900 $990119864[TCU(119905lowast

1)] $10721 $11067 $11448 $11773 $12062 $12325 $12567 $12793 $13006 $13208 $13399 $13583

119905lowast

103116 03816 04586 05244 05828 06359 06848 07305 07735 08142 08529 08900

Table 3 Variations of the unit retailerrsquos holding cost ℎ2and their effects on 119864[TCU(119905lowast

1)]

ℎ2ℎ 05 075 1 125 15 175 2 225 25 275 3 325

ℎ2

03 045 06 075 09 105 12 135 15 165 18 195119864 [TCU (119905lowast

1)] $11282 $11407 $11526 $11638 $11746 $11850 $11949 $12045 $12138 $12229 $12316 $12401

119905lowast

103256 03074 02918 02785 02668 02564 02472 02389 02314 02246 02183 02125

1198701on the optimal system cost 119864[TCU(119905lowast

1)] and on the

optimal production run time 119905lowast1(see Table 2) It can be seen

from Table 2 that as the ratio of1198701119870 increases the expected

system costs per unit time119864[TCU(119905lowast1119871)] increase significantly

It is also noted that as 1198701increases optimal production run

time 119905lowast1also increases significantly

41 Numerical Example for the Producer-Retailer IntegratedEPQ System (Model 2) In order to demonstrate the researchresult of the producer-retailer integrated EPQ-based modelan additional system variable ℎ

2= $150 per item stored at the

retailerrsquos side is includedAgain one can use the upper and lower bounds of 119905lowast

1

(equations (43) and (44)) to test for convexity of 119864[TCU(1199051)]

(Theorem 3 and equation (35)) The results reveal that 119905lowast1119880

=03213 lt 120595(119905lowast

1119880) = 26462 and 119905lowast

1119871= 02186 lt 120595(119905lowast

1119871) = 24876

Therefore the expected cost [TCU(1199051)] is convex

Next by applying the proposed recursive searching algo-rithm we can calculate that the optimal run time 119905lowast

1=

02314 years and the optimal 119864[TCU(119905lowast1)] = $1213849 It is

noted that the computation time for reaching the optimal 119905lowast1

solution is 21 seconds (usingExcel software in a desktop com-puter Intel CPU G850 with 294GB RAM and 289GHz)

Figure 7 illustrates the behavior of 119864[TCU(1199051)] with

regard to production run time It is noted that without theresearch result from the second model the management ofsuch a producer-retailer integrated system would probablyuse 1199051= 03748 years (from the result of model 1) for their

run time decision Further analysis (see Figure 7) shows cost

0130 0170 0210 0250 0290 0330 0370 0410

t1

E[T

CU(t

1)]

11600

11800

12000

12200

12400

12600

12800

tlowast1 = 02314

t1 = 0374812138

(from model 1)

E[TCU( t1)] = 12489

$$

$

$

$

$

$

$

$

Figure 7 The behavior of 119864[TCU(1199051)] with respect to production

run time 1199051in the proposed model 2

savings of $351 (or 29 over the total system costs) simply byapplying our research result

The effects of the unit retailerrsquos holding cost ℎ2on the

expected system cost 119864[TCU(119905lowast1)] and on the optimal run

time 119905lowast1are shown in Table 3 respectively

It can be seen that as ℎ2or the ratio of ℎ

2ℎ increases

the expected cost 119864[TCU(119905lowast1)] increases but the optimal

production run time 119905lowast1decreases In decision-making these

sensitivity analyses results can provide the management ofa producer-retailer integrated system with valuable informa-tion and insights into the effects of various stock holding costsin different retailersrsquo locations

12 The Scientific World Journal

5 Concluding Remarks

Two exact models for an extended EPQ-based problem witha discontinuous delivery policy scrap rate and randombreakdown are developed in this study They specificallyaddress different real-life situations in production end-item delivery and intrasupply chains such as a producer-retailer integrated systemMathematicalmodeling alongwithoptimization techniques is used to determine the optimalproduction run times that minimize the expected systemcosts per unit time Without in-depth investigations onthese separate models the optimal production run time andother important information related to the systemparameterscannot be revealed The proposed real-life EPQ models withrandom machine breakdown discontinuous product distri-bution policies and quality assurance must be specificallystudied in order to (1) obtain the joint effects of breakdowndiscontinuous distribution policies and quality assurance onthe optimal production run time (2) get to know the effectsof different policy and scope of supply chains managementon the optimal run time and overall system costs and (3)gain the insight with regard to various systemrsquos parametersof all particular EPQ-based models Since little attention hasbeen paid to the investigation of joint effects of these practicalproduction situations on the optimal run time this researchis intended to bridge the gap An interesting area for futurestudy is the examination of the effect of variable productionrates on these models

Appendix

Derivations of (3) are as followsRecall (2) as follows

TC1(1199051) = 119862 (119875

11199051) + 119870 + 119862

119878(11990511198751119909) +119872 + 119899119870

1

+ 119862119879[11990511198751] + ℎ3(120582119905119903) 1198791015840

+ ℎ[1198671015840+ 11988911199051

21199051+ (1198671015840

1+ 1198891119905) 119905119903+119899 minus 1

211989911986710158401199051015840

2]

(A1)

Substituting all related system parameters into (2) (pleaserefer to the basic formulations and solution process in [9])the TC

1(1199051) can be obtained as

TC1(1199051) = 119870 +119872

+[1198621198751+ 1198621198781198751119909 + 119862

1198791198751(1 minus 119909)+ ℎ

31198751119892 (1 minus 119909)] 119905

1

+ 1198991198701+ ℎ119901119892119905 minus[

ℎ1198751119892 (1 minus 119909)

2minusℎ1198751119892 (1 minus 119909)

2119899] 1199051

+ 1199052

1[ℎ1198751

2+ℎ1198752

1

2120582(1 minus 119909)

2minusℎ1198751

2(1 minus 119909)

minusℎ1198752

1

2120582119899(1 minus 119909)

2+ℎ1198751

2119899(1 minus 119909)]

(A2)

To take the randomness of 119909 into account by using theexpected values of 119909 with further derivations119864[TC

1(1199051)] can

be derived as follows (ie equation (3))119864 [TC

1(1199051)]

= 119870 + 1198991198701+119872 + ℎ119905119875

1119892

+ [1198621198751+ 1198621198781198751119864 [119909] + 1198621198791198751 (1 minus 119864 [119909])

+ ℎ31198751119892 (1 minus 119864 [119909]) minus

ℎ1198751119892 (1 minus 119864 [119909])

2(1 minus

1

119899)] sdot 1199051

+ [ℎ1198751119864 [119909]

2+ℎ1198752

1

2120582(1 minus 119864 [119909])

2(1 minus

1

119899)

+ℎ1198751

2119899(1 minus 119864 [119909])] 119905

2

1

(A3)

Conflict of Interests

The authors of the paper declare that there is no conflict ofinterests regarding the publication of this paper

Acknowledgment

The authors greatly appreciate the National Science Council(NSC) of Taiwan for supporting this research underGrant noNSC 102-2410-H-324-005

References

[1] E W Taft ldquoThemost economical production lotrdquo Iron Age vol101 pp 1410ndash1412 1918

[2] G Hadley and T M Whitin ldquoAn optimal final inventorymodelrdquoManagement Science vol 7 pp 179ndash183 1961

[3] E A Silver D F Pyke and R Peterson Inventory Managementand Production Planning and Scheduling John Wiley amp SonsNew York NY USA 1998

[4] S Nahmias Production amp Operations Analysis McGraw-HillNew York NY USA 2009

[5] MWidmer and P Solot ldquoDo not forget the breakdowns and themaintenance operations in FMSdesign problemsrdquo InternationalJournal of Production Research vol 28 pp 421ndash430 1990

[6] K-Y C Yu and D L Bricker ldquoAnalysis of a markov chainmodel of a multistage manufacturing system with inspectionrejection and reworkrdquo IIE Transactions vol 25 no 1 pp 109ndash112 1993

[7] H Groenevelt L Pintelon and A Seidmann ldquoProduction lotsizing with machine breakdownsrdquoManagement Science vol 38no 1 pp 104ndash123 1992

[8] G A Widyadana and H M Wee ldquoOptimal deterioratingitems production inventory models with random machinebreakdown and stochastic repair timerdquo Applied MathematicalModelling vol 35 no 7 pp 3495ndash3508 2011

[9] S W Chiu C-L Chou and W-K Wu ldquoOptimizing replenish-ment policy in an EPQ-based inventory model with noncon-forming items and breakdownrdquo EconomicModelling vol 35 pp330ndash337 2013

The Scientific World Journal 13

[10] A M Zargar ldquoEffect of rework strategies on cycle timerdquoComputers amp Industrial Engineering vol 29 no 1ndash4 pp 239ndash243 1995

[11] P Biswas and B R Sarker ldquoOptimal batch quantity modelsfor a lean production system with in-cycle rework and scraprdquoInternational Journal of Production Research vol 46 no 23 pp6585ndash6610 2008

[12] S W Chiu H-D Lin C-B Cheng and C-L Chung ldquoOptimalproduction-shipment decisions for the finite production ratemodel with scraprdquo International Journal for Engineering Mod-elling vol 22 no 1ndash4 pp 25ndash34 2009

[13] Y-S P Chiu K-K Chen and C-K Ting ldquoReplenishment runtime problem with machine breakdown and failure in reworkrdquoExpert Systems with Applications vol 39 no 1 pp 1291ndash12972012

[14] H-D Lin F-Y Pai and S W Chiu ldquoA note on ldquointra-supply chain system with multiple sales locations and qualityassurancerdquordquo Expert Systems with Applications vol 40 no 11 pp4730ndash4732 2013

[15] B Kamsu-Foguem F Rigal and FMauget ldquoMining associationrules for the quality improvement of the production processrdquoExpert Systems with Applications vol 40 no 4 pp 1034ndash10452013

[16] Y Li H Guo L Wang and J Fu ldquoA hybrid genetic-simulatedannealing algorithm for the location-inventory- routing prob-lem considering returns under E-supply chain environmentrdquoThe Scientific World Journal vol 2013 Article ID 125893 10pages 2013

[17] Y-S P Chiu K-K Chen F-T Cheng and C-K Ting ldquoReex-amination of ldquocombining an alternative multi-delivery policyinto economic production lot size problemwith partial reworkrdquousing an alternative approachrdquo Journal of Applied Research andTechnology vol 11 no 3 pp 317ndash323 2013

[18] L B Schwarz B L Deuermeyer and R D Badinelli ldquoFill-rateoptimization in a one-warehouse 119873-identical retailer distribu-tion systemrdquo Management Science vol 31 no 4 pp 488ndash4981985

[19] R A Sarker and L R Khan ldquoOptimal batch size for a pro-duction system operating under periodic delivery policyrdquoCom-puters amp Industrial Engineering vol 37 no 4 pp 711ndash730 1999

[20] N Comez K E Stecke and M Cakanyildirim ldquoMultiple in-cycle transshipments with positive delivery timesrdquo Productionand Operations Management vol 21 no 2 pp 378ndash395 2012

[21] S W Chiu F-Y Pai and W K Wu ldquoAlternative approach todetermine the common cycle time for a multi-item productionsystem with discontinuous deliveries and failure in reworkrdquoEconomic Modelling vol 35 pp 593ndash596 2013

[22] Y-S P Chiu C-C Huang M-F Wu and H-H Chang ldquoJointdetermination of rotation cycle time and number of shipmentsfor a multi-item EPQ model with random defective raterdquoEconomic Modelling vol 35 pp 112ndash117 2013

[23] M A Hoque ldquoSynchronization in the single-manufacturermulti-buyer integrated inventory supply chainrdquo European Jour-nal of Operational Research vol 188 no 3 pp 811ndash825 2008

[24] S W Chiu L-W Lin K-K Chen and C-L Chou ldquoDetermin-ing production-shipment policy for a vendor-buyer integratedsystem with rework and an amending multi-delivery schedulerdquoEconomic Modelling vol 33 pp 668ndash675 2013

[25] M Cedillo-Campos and C Sanchez-Ramırez ldquoDynamic self-assessment of supply chains performance an emerging marketapproachrdquo Journal of Applied Research and Technology vol 11no 3 pp 338ndash347 2013

[26] Y-S P Chiu H-D Lin F-T Cheng and M-H Hwang ldquoOpti-mal common cycle time for a multi-item production systemwith discontinuous delivery policy and failure in reworkrdquoJournal of Scientific and Industrial Research vol 72 no 7 pp435ndash440 2013

[27] HHishamuddin R A Sarker andD Essam ldquoA recoverymech-anism for a two echelon supply chain system under supplydisruptionrdquo Economic Modelling vol 38 pp 555ndash563 2014

[28] M Murugan and V Selladurai ldquoProductivity improvement inmanufacturing submersible pump diffuser housing using leanmanufacturing systemrdquo Journal of Engineering Research vol 2no 1 pp 164ndash182 2014

[29] L Wang H Qu S Liu and C-X Dun ldquoModeling and opti-mization of the multiobjective stochastic joint replenishmentand delivery problem under supply chain environmentrdquo TheScientific World Journal vol 2013 Article ID 916057 11 pages2013

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

8 The Scientific World Journal

Dminus 120582tn = I

D

I

t1 + tr tn

Time

t1 + tr

Ic(t)

t9984002

T998400

Figure 5 Inventory level of finished products on the retailerrsquos side in the proposed manufacturing run time problem with breakdown

In the proposed study the retailerrsquos stock holding posi-tions are illustrated in Figure 5

Extra parameters used in this enhanced model includethe following

ℎ2 holding cost per product stored on the retailerrsquos

side

119868119888(119905) on-hand inventory levels in units on the retail-

errsquos side end at time 119905

119863 number of finished products (a fixed quantity)transported to the retail store per shipment

119868 number of left-over products in 119905119899after satisfying

the demand in 119905119899

TC3(1199051) total production-inventory-delivery costs

per cycle of this enhanced model (in the case of abreakdown)

TC4(1199051) total production-inventory-delivery costs

per cycle of this enhanced model (in the case of nobreakdown)

119864[TC3(1199051)] the expected production-inventory-

delivery costs per cycle of this enhanced model (inthe case of a breakdown)

119864[TC4(1199051)] the expected production-inventory-

delivery costs per cycle of this enhanced model (inthe case of no breakdown)

119864[TCU119890(1199051)] the long-run expected production-

inventory-delivery costs per unit time in thisenhanced model whether or not a breakdown takesplace

Since the demand on the retailerrsquos side in time interval119905119899is 120582119905119899 after satisfying the demand the number of left-over

items (see Figure 5) in each 119905119899is

119868 = 119863 minus 120582119905119899 (27)

Total inventory holding costs on retailerrsquos side with andwithout breakdown are shown respectively in

ℎ2[119899(119863 minus 119868)

2119905119899+119899 (119899 + 1)

2119868119905119899+119899119868

2(1199051+ 119905119903)] (28)

ℎ2[119899(119863 minus 119868)

2119905119899+119899 (119899 + 1)

2119868119905119899+119899119868

2(1199051)] (29)

To incorporate the retailerrsquos holding costs into the originalmodels with and without breakdown respectively we obtain

TC3(1199051) = TC

1(1199051)

+ ℎ2[119899(119863 minus 119868)

2119905119899+119899 (119899 + 1)

2119868119905119899+119899119868

2(1199051+ 119905119903)]

TC4(1199051)

= TC2(1199051) + ℎ2[119899(119863 minus 119868)

2119905119899+119899 (119899 + 1)

2119868119905119899+119899119868

2(1199051)]

(30)

To take the randomness of defective rate 119909 into accountand substitute all related variables into (30) with furtherderivations 119864[TC

3(1199051)] and 119864[TC

4(1199051)] can be obtained as

follows

119864 [TC3(1199051)] = 119864 [TC

1(1199051)] + ℎ

2(1198751119892 (1 minus 119864 [119909])

2)

sdot (1 minus1

119899) 1199051+ ℎ2(1198751(1 minus 119864 [119909])

2)

sdot [1198751(1 minus 119864 [119909])

120582119899+ (1 minus

1

119899)] 1199052

1

119864 [TC4(1199051)] = 119864 [TC

2(1199051)] + ℎ

2(1198751(1 minus 119864 [119909])

2)

sdot [1198751(1 minus 119864 [119909])

120582119899+ (1 minus

1

119899)] 1199052

1

(31)

The Scientific World Journal 9

32 Integration of Enhanced Model withwithout BreakdownThe mean time to breakdowns obeys the exponential distri-bution with 119891(119905) = 120573119890minus120573119905 Therefore 119864[TCU

2(1199051)] is

119864 [TCU2(1199051)]

=

int1199051

0119864 [TC

3(1199051)] 119891 (119905) 119889119905 + int

infin

1199051

119864 [TC4(1199051)] 119891 (119905) 119889119905

119864 [T]

(32)

Substituting 119864[TCU3(1199051)] 119864[TC

4(1199051)] and 119864[T] into (32)

and resolving 119864[TCU2(1199051)] we obtain

119864 [TCU2(1199051)]

=120582

(1 minus 119864 [119909])

sdot (119870 + 119899119870

1)

11990511198751

+ 1205741+12057421199051

2+ 12057451199051

+ [119872

1198751

+ℎ119892

120573](

1 minus 119890minus1205731199051

1199051

) minus ℎ119892 (119890minus1205731199051)

minus (ℎ minus ℎ2) [119892 (1 minus 119864 [119909])

2(1 minus

1

119899)] (1 minus 119890

minus1205731199051)

(33)

where

1205745=ℎ2(1 minus 119864 [119909])

2[1198751(1 minus 119864 [119909])

120582119899+ (1 minus

1

119899)] (34)

33 Determining the Optimal Run Time Let 120595(1199051) stand for

the following

120595 (1199051) =

2 (119870 + 1198991198701) 120573 + 2120574

4(1 minus 119890

minus1205731199051)

[1199052

1119875112057321205746+ 1205744(2 + 120573119905

1)] 120573119890minus1205731199051

(35)

Theorem 3 (119864[TCU2(1199051)] is convex if 0 lt 119905

1lt 120595(119905

1)) The

second derivative of 119864[1198791198621198802(1199051)] with respect to 119905

1is

1198892119864 [119879119862119880

2(1199051)]

11988921199052

1

=120582

(1 minus 119864 [119909])[2 (119870 + 119899119870

1)

1199053

11198751

+ (ℎ minus ℎ2)119892 (1 minus 119864 [119909])

2

sdot (1 minus1

119899) (1205732119890minus1205731199051) minus ℎ119892 (120573

2119890minus1205731199051)

+ [119872

1198751

+ℎ119892

120573](

2 (1 minus 119890minus1205731199051)

1199053

1

minus2120573119890minus1205731199051

1199052

1

minus1205732119890minus1205731199051

1199051

)]

(36)

Since annual demand 120582 gt 0 the first term in the RHS of(36) is positive and

119894119891 [2 (119870 + 119899119870

1)

1199053

11198751

+ (ℎ minus ℎ2) [119892 (1 minus 119864 [119909])

2(1 minus

1

119899)]

sdot (1205732119890minus1205731199051) minus ℎ119892 (120573

2119890minus1205731199051) + [

119872

1198751

+ℎ119892

120573]

sdot(

2 (1 minus 119890minus1205731199051)

1199053

1

minus2120573119890minus1205731199051

1199052

1

minus1205732119890minus1205731199051

1199051

)] gt 0

119905ℎ1198901198991198892119864 [119879119862119880

2(1199051)]

1198891199051

2gt 0

(37)

With further derivations the left-hand side (LHS) of (37)becomes

119894119891 [2 (119870 + 1198991198701) 120573 minus 119905

3

11198751120573 (ℎ minus ℎ

2) [119892 (1 minus 119864 [119909])

2(1 minus

1

119899)]

sdot (1205732119890minus1205731199051) minus 1199053

11198751120573ℎ119892 (120573

2119890minus1205731199051)

+ (119872120573 + ℎ1198921198751) [2 (1 minus 119890

minus1205731199051) minus 2119905

1120573119890minus1205731199051 minus 12057321199052

1119890minus1205731199051] ]

gt 0

(38)

Let

1205746= (ℎ minus ℎ

2) [119892 (1 minus 119864 [119909])

2(1 minus

1

119899)] + ℎ119892 (39)

then (37) becomes

119894119891 [2 (119870 + 1198991198701) 120573 + 2120574

4(1 minus 119890

minus1205731199051)

minus1199051[1199052

1119875112057321205746+ 1205744(2 + 120573119905

1)] 120573119890minus1205731199051] gt 0

119905ℎ1198901198991198892119864 [119879119862119880

2(1199051)]

1198891199051

2gt 0

(40)

or

1198892119864 [119879119862119880

2(1199051)]

1198891199051

2gt 0

119894119891 0 lt 1199051lt

2 (119870 + 1198991198701) 120573 + 2120574

4(1 minus 119890

minus1205731199051)

[1199052

1119875112057321205746+ 1205744(2 + 120573119905

1)] 120573119890minus1205731199051

= 120595 (1199051)

(41)

10 The Scientific World Journal

Once 119864[1198791198621198802(1199051)] is proven to be convex the optimal

run time 119905lowast1can be solved by setting the first derivative of

119864[1198791198621198802(1199051)] = 0

119889119864 [1198791198621198802(1199051)]

1198891199051

=120582

(1 minus 119864 [119909])sdot minus (119870 + 119899119870

1)

1199052

1119875

+ (1205742

2+ 1205745) + 1205746(120573119890minus1205731199051)

+ [119872

1198751

+ℎ119892

120573](

minus (1 minus 119890minus1205731199051)

1199052

1

+120573119890minus1205731199051

1199051

)

= 0

(42)

It can be seen that the first term in the RHS of (42) ispositive so the second term is equal to zero In order to findthe bounds for 119905lowast

1 let

119905lowast

1119880= radic

2 [120573 (119870 + 1198991198701) + 1205744]

1198751120573 (1205742+ 21205745)

(43)

119905lowast

1119871= 119905ℎ119890 119901119900119904119894119905119894V119890 119903119900119900119905 119900119891

minus1205744plusmn radic1205742

4+ 21198751(119870 + 119899119870

1) (1205742+ 21205745+ 21205746120573)

1198751(1205742+ 21205745+ 21205746120573)

(44)

Theorem4 (119905lowast1119871lt 119905lowast

1lt 119905lowast

1119880) For the proof ofTheorem 4 please

refer to the proof for Theorem 2 in Section 2

Once we are certain that 119905lowast1falls within the aforemen-

tioned upper and lower bounds in order to find 119905lowast1 we can

first multiply the second term of (42) by (211987511199052

1120573) and obtain

the following

[1198751120573 (1205742+ 21205745) + 2119875

112057321205746119890minus1205731199051] 1199052

1+ (21205744120573119890minus1205731199051) 1199051

minus2 [120573 (119870 + 1198991198701) + 1205744(1 minus 119890

minus1205731199051)]

= 0

(45)

Equation (45) can be rearranged as

119890minus1205731199051 =

2 [120573 (119870 + 1198991198701) + 1205744] minus [119875

1120573 (1205742+ 21205745)] 1199052

1

2 [1198751120573212057461199052

1+ 1205744(1 + 120573119905

1)]

(46)

where 119890minus1205731199051 is the complement of the cumulative densityfunction 119865(119905

1) = 1minus119890

minus1205731199051 As 0 le 119865(119905

1) le 1 0 le 119890minus1205731199051 le 1 Let

119890minus1205731199051 = 0 and 119890minus1205731199051 = 1 be the initial upper and lower bounds

of 119890minus1205731199051 respectively Then by using the proposed recursivesearching algorithm given at the end of Section 2 we can findthe optimal production run time 119905lowast

1

4 Numerical Example

In order to relieve the comparison efforts for readers thissection adopts the same numerical example as in [9] For

0225 0265 0305 0345 0385 0425 0465 0505

E[T

CU(t

1)]

t1

tlowast1L = 03411 t

lowast1 = 03748

tlowast1U = 05183

$10600

$10800

$11000

$11200

$11400

$11600

$11006

$11800

Figure 6 The behavior of 119864[TCU(1199051)] in connection with the

production run time 1199051in the proposed model 1

a demonstration of the proposed EPQ-based model 1 thefollowing system parameters are used

1198751 production rate 10000 products per year120582 demand rate 4000 products per year119909 random scrap rate which follows uniformly distribu-

tion over the interval [0 02]120573 Poisson breakdown rate 05 average times per year119892 constant machine repair time 119905

119903 0018 year per repair

119872 machine repair cost $500 for each breakdown119870 setup cost $450 per production run119862 manufacturing cost $2 per item119862119878 disposal cost $03 per scrap itemℎ holding cost $06 per item per unit time1198701 fixed delivery cost $90 per shipment119899 number of deliveries 4 per cycle

119862119879 variable delivery cost $0001 per item

First we use both upper and lower bounds of 119905lowast1to

test for the convexity of 119864[TCU(1199051)] (see Theorem 1) The

computation results of (19) (20) and (11) indicate that 119905lowast1119880

=05183 lt 120585(119905

lowast

1119880) = 28723 and 119905lowast

1119871= 03411 lt 120585(119905

lowast

1119871) = 26261

Hence [TCU(1199051)] is convex (Figure 6)

In order to find the optimal 119905lowast1 we first substitute the

upper and lower bounds of 119905lowast1in (8) and obtain 119864[TCU(119905lowast

1119880)]

= $1160163 and 119864[TCU(119905lowast1119871)] = $1101450 respectively

Because the optimal run time 119905lowast1falls within the interval

of [119905lowast1119871 119905lowast

1119880] we apply the proposed recursive searching

algorithm stated at the end of Section 2 and find 119905lowast1= 03748

years Accordingly the optimal expected system costs perunit time 119864[TCU(119905lowast

1)] = $1100641 (Figure 6) Table 1 shows

the step-by-step iterations of the algorithmIn this specific studied model we focus on incorporating

a discontinuous multidelivery policy into a prior work [9]and consider a fixed transportation cost associated witheach delivery Applying the research result we can easilyinvestigate the effects of different fixed transportation costs

The Scientific World Journal 11

Table 1 Iterations of the recursive searching algorithm for locating 119905lowast1

120573 Step 119905lowast

1119880120596119880= 119890minus1205731199051119880 119905

lowast

1119871120596119871= 119890minus1205731199051119871

Differencebetween 119905lowast

1119880and

119905lowast

1119871

[119880]

119864[TCU(119905lowast1119880)]

[119871]

119864[TCU(119905lowast1119871)]

Differencebetween [119880]

and [119871]05 Initial 000000 100000 $1160163 $1101450 $58713

1st 05183 07717 03411 08432 01772 $1110330 $1101450 $88802nd 03857 08246 03721 08302 00136 $1100716 $1100646 $0703rd 03756 08287 03746 08292 00010 $1100642 $1100641 $0014th 03749 08291 03748 08291 00001 $1100641 $1100641 $00045th 03748 08291 03748 08291 000000 $1100641 $1100641 $0000

Table 2 Variations of the fixed delivery cost1198701effects on the optimal production run time 119905lowast

1

1198701119870 005 02 04 06 08 1 12 14 16 18 2 22

1198701

$225 $90 $180 $270 $360 $450 $540 $630 $720 $810 $900 $990119864[TCU(119905lowast

1)] $10721 $11067 $11448 $11773 $12062 $12325 $12567 $12793 $13006 $13208 $13399 $13583

119905lowast

103116 03816 04586 05244 05828 06359 06848 07305 07735 08142 08529 08900

Table 3 Variations of the unit retailerrsquos holding cost ℎ2and their effects on 119864[TCU(119905lowast

1)]

ℎ2ℎ 05 075 1 125 15 175 2 225 25 275 3 325

ℎ2

03 045 06 075 09 105 12 135 15 165 18 195119864 [TCU (119905lowast

1)] $11282 $11407 $11526 $11638 $11746 $11850 $11949 $12045 $12138 $12229 $12316 $12401

119905lowast

103256 03074 02918 02785 02668 02564 02472 02389 02314 02246 02183 02125

1198701on the optimal system cost 119864[TCU(119905lowast

1)] and on the

optimal production run time 119905lowast1(see Table 2) It can be seen

from Table 2 that as the ratio of1198701119870 increases the expected

system costs per unit time119864[TCU(119905lowast1119871)] increase significantly

It is also noted that as 1198701increases optimal production run

time 119905lowast1also increases significantly

41 Numerical Example for the Producer-Retailer IntegratedEPQ System (Model 2) In order to demonstrate the researchresult of the producer-retailer integrated EPQ-based modelan additional system variable ℎ

2= $150 per item stored at the

retailerrsquos side is includedAgain one can use the upper and lower bounds of 119905lowast

1

(equations (43) and (44)) to test for convexity of 119864[TCU(1199051)]

(Theorem 3 and equation (35)) The results reveal that 119905lowast1119880

=03213 lt 120595(119905lowast

1119880) = 26462 and 119905lowast

1119871= 02186 lt 120595(119905lowast

1119871) = 24876

Therefore the expected cost [TCU(1199051)] is convex

Next by applying the proposed recursive searching algo-rithm we can calculate that the optimal run time 119905lowast

1=

02314 years and the optimal 119864[TCU(119905lowast1)] = $1213849 It is

noted that the computation time for reaching the optimal 119905lowast1

solution is 21 seconds (usingExcel software in a desktop com-puter Intel CPU G850 with 294GB RAM and 289GHz)

Figure 7 illustrates the behavior of 119864[TCU(1199051)] with

regard to production run time It is noted that without theresearch result from the second model the management ofsuch a producer-retailer integrated system would probablyuse 1199051= 03748 years (from the result of model 1) for their

run time decision Further analysis (see Figure 7) shows cost

0130 0170 0210 0250 0290 0330 0370 0410

t1

E[T

CU(t

1)]

11600

11800

12000

12200

12400

12600

12800

tlowast1 = 02314

t1 = 0374812138

(from model 1)

E[TCU( t1)] = 12489

$$

$

$

$

$

$

$

$

Figure 7 The behavior of 119864[TCU(1199051)] with respect to production

run time 1199051in the proposed model 2

savings of $351 (or 29 over the total system costs) simply byapplying our research result

The effects of the unit retailerrsquos holding cost ℎ2on the

expected system cost 119864[TCU(119905lowast1)] and on the optimal run

time 119905lowast1are shown in Table 3 respectively

It can be seen that as ℎ2or the ratio of ℎ

2ℎ increases

the expected cost 119864[TCU(119905lowast1)] increases but the optimal

production run time 119905lowast1decreases In decision-making these

sensitivity analyses results can provide the management ofa producer-retailer integrated system with valuable informa-tion and insights into the effects of various stock holding costsin different retailersrsquo locations

12 The Scientific World Journal

5 Concluding Remarks

Two exact models for an extended EPQ-based problem witha discontinuous delivery policy scrap rate and randombreakdown are developed in this study They specificallyaddress different real-life situations in production end-item delivery and intrasupply chains such as a producer-retailer integrated systemMathematicalmodeling alongwithoptimization techniques is used to determine the optimalproduction run times that minimize the expected systemcosts per unit time Without in-depth investigations onthese separate models the optimal production run time andother important information related to the systemparameterscannot be revealed The proposed real-life EPQ models withrandom machine breakdown discontinuous product distri-bution policies and quality assurance must be specificallystudied in order to (1) obtain the joint effects of breakdowndiscontinuous distribution policies and quality assurance onthe optimal production run time (2) get to know the effectsof different policy and scope of supply chains managementon the optimal run time and overall system costs and (3)gain the insight with regard to various systemrsquos parametersof all particular EPQ-based models Since little attention hasbeen paid to the investigation of joint effects of these practicalproduction situations on the optimal run time this researchis intended to bridge the gap An interesting area for futurestudy is the examination of the effect of variable productionrates on these models

Appendix

Derivations of (3) are as followsRecall (2) as follows

TC1(1199051) = 119862 (119875

11199051) + 119870 + 119862

119878(11990511198751119909) +119872 + 119899119870

1

+ 119862119879[11990511198751] + ℎ3(120582119905119903) 1198791015840

+ ℎ[1198671015840+ 11988911199051

21199051+ (1198671015840

1+ 1198891119905) 119905119903+119899 minus 1

211989911986710158401199051015840

2]

(A1)

Substituting all related system parameters into (2) (pleaserefer to the basic formulations and solution process in [9])the TC

1(1199051) can be obtained as

TC1(1199051) = 119870 +119872

+[1198621198751+ 1198621198781198751119909 + 119862

1198791198751(1 minus 119909)+ ℎ

31198751119892 (1 minus 119909)] 119905

1

+ 1198991198701+ ℎ119901119892119905 minus[

ℎ1198751119892 (1 minus 119909)

2minusℎ1198751119892 (1 minus 119909)

2119899] 1199051

+ 1199052

1[ℎ1198751

2+ℎ1198752

1

2120582(1 minus 119909)

2minusℎ1198751

2(1 minus 119909)

minusℎ1198752

1

2120582119899(1 minus 119909)

2+ℎ1198751

2119899(1 minus 119909)]

(A2)

To take the randomness of 119909 into account by using theexpected values of 119909 with further derivations119864[TC

1(1199051)] can

be derived as follows (ie equation (3))119864 [TC

1(1199051)]

= 119870 + 1198991198701+119872 + ℎ119905119875

1119892

+ [1198621198751+ 1198621198781198751119864 [119909] + 1198621198791198751 (1 minus 119864 [119909])

+ ℎ31198751119892 (1 minus 119864 [119909]) minus

ℎ1198751119892 (1 minus 119864 [119909])

2(1 minus

1

119899)] sdot 1199051

+ [ℎ1198751119864 [119909]

2+ℎ1198752

1

2120582(1 minus 119864 [119909])

2(1 minus

1

119899)

+ℎ1198751

2119899(1 minus 119864 [119909])] 119905

2

1

(A3)

Conflict of Interests

The authors of the paper declare that there is no conflict ofinterests regarding the publication of this paper

Acknowledgment

The authors greatly appreciate the National Science Council(NSC) of Taiwan for supporting this research underGrant noNSC 102-2410-H-324-005

References

[1] E W Taft ldquoThemost economical production lotrdquo Iron Age vol101 pp 1410ndash1412 1918

[2] G Hadley and T M Whitin ldquoAn optimal final inventorymodelrdquoManagement Science vol 7 pp 179ndash183 1961

[3] E A Silver D F Pyke and R Peterson Inventory Managementand Production Planning and Scheduling John Wiley amp SonsNew York NY USA 1998

[4] S Nahmias Production amp Operations Analysis McGraw-HillNew York NY USA 2009

[5] MWidmer and P Solot ldquoDo not forget the breakdowns and themaintenance operations in FMSdesign problemsrdquo InternationalJournal of Production Research vol 28 pp 421ndash430 1990

[6] K-Y C Yu and D L Bricker ldquoAnalysis of a markov chainmodel of a multistage manufacturing system with inspectionrejection and reworkrdquo IIE Transactions vol 25 no 1 pp 109ndash112 1993

[7] H Groenevelt L Pintelon and A Seidmann ldquoProduction lotsizing with machine breakdownsrdquoManagement Science vol 38no 1 pp 104ndash123 1992

[8] G A Widyadana and H M Wee ldquoOptimal deterioratingitems production inventory models with random machinebreakdown and stochastic repair timerdquo Applied MathematicalModelling vol 35 no 7 pp 3495ndash3508 2011

[9] S W Chiu C-L Chou and W-K Wu ldquoOptimizing replenish-ment policy in an EPQ-based inventory model with noncon-forming items and breakdownrdquo EconomicModelling vol 35 pp330ndash337 2013

The Scientific World Journal 13

[10] A M Zargar ldquoEffect of rework strategies on cycle timerdquoComputers amp Industrial Engineering vol 29 no 1ndash4 pp 239ndash243 1995

[11] P Biswas and B R Sarker ldquoOptimal batch quantity modelsfor a lean production system with in-cycle rework and scraprdquoInternational Journal of Production Research vol 46 no 23 pp6585ndash6610 2008

[12] S W Chiu H-D Lin C-B Cheng and C-L Chung ldquoOptimalproduction-shipment decisions for the finite production ratemodel with scraprdquo International Journal for Engineering Mod-elling vol 22 no 1ndash4 pp 25ndash34 2009

[13] Y-S P Chiu K-K Chen and C-K Ting ldquoReplenishment runtime problem with machine breakdown and failure in reworkrdquoExpert Systems with Applications vol 39 no 1 pp 1291ndash12972012

[14] H-D Lin F-Y Pai and S W Chiu ldquoA note on ldquointra-supply chain system with multiple sales locations and qualityassurancerdquordquo Expert Systems with Applications vol 40 no 11 pp4730ndash4732 2013

[15] B Kamsu-Foguem F Rigal and FMauget ldquoMining associationrules for the quality improvement of the production processrdquoExpert Systems with Applications vol 40 no 4 pp 1034ndash10452013

[16] Y Li H Guo L Wang and J Fu ldquoA hybrid genetic-simulatedannealing algorithm for the location-inventory- routing prob-lem considering returns under E-supply chain environmentrdquoThe Scientific World Journal vol 2013 Article ID 125893 10pages 2013

[17] Y-S P Chiu K-K Chen F-T Cheng and C-K Ting ldquoReex-amination of ldquocombining an alternative multi-delivery policyinto economic production lot size problemwith partial reworkrdquousing an alternative approachrdquo Journal of Applied Research andTechnology vol 11 no 3 pp 317ndash323 2013

[18] L B Schwarz B L Deuermeyer and R D Badinelli ldquoFill-rateoptimization in a one-warehouse 119873-identical retailer distribu-tion systemrdquo Management Science vol 31 no 4 pp 488ndash4981985

[19] R A Sarker and L R Khan ldquoOptimal batch size for a pro-duction system operating under periodic delivery policyrdquoCom-puters amp Industrial Engineering vol 37 no 4 pp 711ndash730 1999

[20] N Comez K E Stecke and M Cakanyildirim ldquoMultiple in-cycle transshipments with positive delivery timesrdquo Productionand Operations Management vol 21 no 2 pp 378ndash395 2012

[21] S W Chiu F-Y Pai and W K Wu ldquoAlternative approach todetermine the common cycle time for a multi-item productionsystem with discontinuous deliveries and failure in reworkrdquoEconomic Modelling vol 35 pp 593ndash596 2013

[22] Y-S P Chiu C-C Huang M-F Wu and H-H Chang ldquoJointdetermination of rotation cycle time and number of shipmentsfor a multi-item EPQ model with random defective raterdquoEconomic Modelling vol 35 pp 112ndash117 2013

[23] M A Hoque ldquoSynchronization in the single-manufacturermulti-buyer integrated inventory supply chainrdquo European Jour-nal of Operational Research vol 188 no 3 pp 811ndash825 2008

[24] S W Chiu L-W Lin K-K Chen and C-L Chou ldquoDetermin-ing production-shipment policy for a vendor-buyer integratedsystem with rework and an amending multi-delivery schedulerdquoEconomic Modelling vol 33 pp 668ndash675 2013

[25] M Cedillo-Campos and C Sanchez-Ramırez ldquoDynamic self-assessment of supply chains performance an emerging marketapproachrdquo Journal of Applied Research and Technology vol 11no 3 pp 338ndash347 2013

[26] Y-S P Chiu H-D Lin F-T Cheng and M-H Hwang ldquoOpti-mal common cycle time for a multi-item production systemwith discontinuous delivery policy and failure in reworkrdquoJournal of Scientific and Industrial Research vol 72 no 7 pp435ndash440 2013

[27] HHishamuddin R A Sarker andD Essam ldquoA recoverymech-anism for a two echelon supply chain system under supplydisruptionrdquo Economic Modelling vol 38 pp 555ndash563 2014

[28] M Murugan and V Selladurai ldquoProductivity improvement inmanufacturing submersible pump diffuser housing using leanmanufacturing systemrdquo Journal of Engineering Research vol 2no 1 pp 164ndash182 2014

[29] L Wang H Qu S Liu and C-X Dun ldquoModeling and opti-mization of the multiobjective stochastic joint replenishmentand delivery problem under supply chain environmentrdquo TheScientific World Journal vol 2013 Article ID 916057 11 pages2013

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

The Scientific World Journal 9

32 Integration of Enhanced Model withwithout BreakdownThe mean time to breakdowns obeys the exponential distri-bution with 119891(119905) = 120573119890minus120573119905 Therefore 119864[TCU

2(1199051)] is

119864 [TCU2(1199051)]

=

int1199051

0119864 [TC

3(1199051)] 119891 (119905) 119889119905 + int

infin

1199051

119864 [TC4(1199051)] 119891 (119905) 119889119905

119864 [T]

(32)

Substituting 119864[TCU3(1199051)] 119864[TC

4(1199051)] and 119864[T] into (32)

and resolving 119864[TCU2(1199051)] we obtain

119864 [TCU2(1199051)]

=120582

(1 minus 119864 [119909])

sdot (119870 + 119899119870

1)

11990511198751

+ 1205741+12057421199051

2+ 12057451199051

+ [119872

1198751

+ℎ119892

120573](

1 minus 119890minus1205731199051

1199051

) minus ℎ119892 (119890minus1205731199051)

minus (ℎ minus ℎ2) [119892 (1 minus 119864 [119909])

2(1 minus

1

119899)] (1 minus 119890

minus1205731199051)

(33)

where

1205745=ℎ2(1 minus 119864 [119909])

2[1198751(1 minus 119864 [119909])

120582119899+ (1 minus

1

119899)] (34)

33 Determining the Optimal Run Time Let 120595(1199051) stand for

the following

120595 (1199051) =

2 (119870 + 1198991198701) 120573 + 2120574

4(1 minus 119890

minus1205731199051)

[1199052

1119875112057321205746+ 1205744(2 + 120573119905

1)] 120573119890minus1205731199051

(35)

Theorem 3 (119864[TCU2(1199051)] is convex if 0 lt 119905

1lt 120595(119905

1)) The

second derivative of 119864[1198791198621198802(1199051)] with respect to 119905

1is

1198892119864 [119879119862119880

2(1199051)]

11988921199052

1

=120582

(1 minus 119864 [119909])[2 (119870 + 119899119870

1)

1199053

11198751

+ (ℎ minus ℎ2)119892 (1 minus 119864 [119909])

2

sdot (1 minus1

119899) (1205732119890minus1205731199051) minus ℎ119892 (120573

2119890minus1205731199051)

+ [119872

1198751

+ℎ119892

120573](

2 (1 minus 119890minus1205731199051)

1199053

1

minus2120573119890minus1205731199051

1199052

1

minus1205732119890minus1205731199051

1199051

)]

(36)

Since annual demand 120582 gt 0 the first term in the RHS of(36) is positive and

119894119891 [2 (119870 + 119899119870

1)

1199053

11198751

+ (ℎ minus ℎ2) [119892 (1 minus 119864 [119909])

2(1 minus

1

119899)]

sdot (1205732119890minus1205731199051) minus ℎ119892 (120573

2119890minus1205731199051) + [

119872

1198751

+ℎ119892

120573]

sdot(

2 (1 minus 119890minus1205731199051)

1199053

1

minus2120573119890minus1205731199051

1199052

1

minus1205732119890minus1205731199051

1199051

)] gt 0

119905ℎ1198901198991198892119864 [119879119862119880

2(1199051)]

1198891199051

2gt 0

(37)

With further derivations the left-hand side (LHS) of (37)becomes

119894119891 [2 (119870 + 1198991198701) 120573 minus 119905

3

11198751120573 (ℎ minus ℎ

2) [119892 (1 minus 119864 [119909])

2(1 minus

1

119899)]

sdot (1205732119890minus1205731199051) minus 1199053

11198751120573ℎ119892 (120573

2119890minus1205731199051)

+ (119872120573 + ℎ1198921198751) [2 (1 minus 119890

minus1205731199051) minus 2119905

1120573119890minus1205731199051 minus 12057321199052

1119890minus1205731199051] ]

gt 0

(38)

Let

1205746= (ℎ minus ℎ

2) [119892 (1 minus 119864 [119909])

2(1 minus

1

119899)] + ℎ119892 (39)

then (37) becomes

119894119891 [2 (119870 + 1198991198701) 120573 + 2120574

4(1 minus 119890

minus1205731199051)

minus1199051[1199052

1119875112057321205746+ 1205744(2 + 120573119905

1)] 120573119890minus1205731199051] gt 0

119905ℎ1198901198991198892119864 [119879119862119880

2(1199051)]

1198891199051

2gt 0

(40)

or

1198892119864 [119879119862119880

2(1199051)]

1198891199051

2gt 0

119894119891 0 lt 1199051lt

2 (119870 + 1198991198701) 120573 + 2120574

4(1 minus 119890

minus1205731199051)

[1199052

1119875112057321205746+ 1205744(2 + 120573119905

1)] 120573119890minus1205731199051

= 120595 (1199051)

(41)

10 The Scientific World Journal

Once 119864[1198791198621198802(1199051)] is proven to be convex the optimal

run time 119905lowast1can be solved by setting the first derivative of

119864[1198791198621198802(1199051)] = 0

119889119864 [1198791198621198802(1199051)]

1198891199051

=120582

(1 minus 119864 [119909])sdot minus (119870 + 119899119870

1)

1199052

1119875

+ (1205742

2+ 1205745) + 1205746(120573119890minus1205731199051)

+ [119872

1198751

+ℎ119892

120573](

minus (1 minus 119890minus1205731199051)

1199052

1

+120573119890minus1205731199051

1199051

)

= 0

(42)

It can be seen that the first term in the RHS of (42) ispositive so the second term is equal to zero In order to findthe bounds for 119905lowast

1 let

119905lowast

1119880= radic

2 [120573 (119870 + 1198991198701) + 1205744]

1198751120573 (1205742+ 21205745)

(43)

119905lowast

1119871= 119905ℎ119890 119901119900119904119894119905119894V119890 119903119900119900119905 119900119891

minus1205744plusmn radic1205742

4+ 21198751(119870 + 119899119870

1) (1205742+ 21205745+ 21205746120573)

1198751(1205742+ 21205745+ 21205746120573)

(44)

Theorem4 (119905lowast1119871lt 119905lowast

1lt 119905lowast

1119880) For the proof ofTheorem 4 please

refer to the proof for Theorem 2 in Section 2

Once we are certain that 119905lowast1falls within the aforemen-

tioned upper and lower bounds in order to find 119905lowast1 we can

first multiply the second term of (42) by (211987511199052

1120573) and obtain

the following

[1198751120573 (1205742+ 21205745) + 2119875

112057321205746119890minus1205731199051] 1199052

1+ (21205744120573119890minus1205731199051) 1199051

minus2 [120573 (119870 + 1198991198701) + 1205744(1 minus 119890

minus1205731199051)]

= 0

(45)

Equation (45) can be rearranged as

119890minus1205731199051 =

2 [120573 (119870 + 1198991198701) + 1205744] minus [119875

1120573 (1205742+ 21205745)] 1199052

1

2 [1198751120573212057461199052

1+ 1205744(1 + 120573119905

1)]

(46)

where 119890minus1205731199051 is the complement of the cumulative densityfunction 119865(119905

1) = 1minus119890

minus1205731199051 As 0 le 119865(119905

1) le 1 0 le 119890minus1205731199051 le 1 Let

119890minus1205731199051 = 0 and 119890minus1205731199051 = 1 be the initial upper and lower bounds

of 119890minus1205731199051 respectively Then by using the proposed recursivesearching algorithm given at the end of Section 2 we can findthe optimal production run time 119905lowast

1

4 Numerical Example

In order to relieve the comparison efforts for readers thissection adopts the same numerical example as in [9] For

0225 0265 0305 0345 0385 0425 0465 0505

E[T

CU(t

1)]

t1

tlowast1L = 03411 t

lowast1 = 03748

tlowast1U = 05183

$10600

$10800

$11000

$11200

$11400

$11600

$11006

$11800

Figure 6 The behavior of 119864[TCU(1199051)] in connection with the

production run time 1199051in the proposed model 1

a demonstration of the proposed EPQ-based model 1 thefollowing system parameters are used

1198751 production rate 10000 products per year120582 demand rate 4000 products per year119909 random scrap rate which follows uniformly distribu-

tion over the interval [0 02]120573 Poisson breakdown rate 05 average times per year119892 constant machine repair time 119905

119903 0018 year per repair

119872 machine repair cost $500 for each breakdown119870 setup cost $450 per production run119862 manufacturing cost $2 per item119862119878 disposal cost $03 per scrap itemℎ holding cost $06 per item per unit time1198701 fixed delivery cost $90 per shipment119899 number of deliveries 4 per cycle

119862119879 variable delivery cost $0001 per item

First we use both upper and lower bounds of 119905lowast1to

test for the convexity of 119864[TCU(1199051)] (see Theorem 1) The

computation results of (19) (20) and (11) indicate that 119905lowast1119880

=05183 lt 120585(119905

lowast

1119880) = 28723 and 119905lowast

1119871= 03411 lt 120585(119905

lowast

1119871) = 26261

Hence [TCU(1199051)] is convex (Figure 6)

In order to find the optimal 119905lowast1 we first substitute the

upper and lower bounds of 119905lowast1in (8) and obtain 119864[TCU(119905lowast

1119880)]

= $1160163 and 119864[TCU(119905lowast1119871)] = $1101450 respectively

Because the optimal run time 119905lowast1falls within the interval

of [119905lowast1119871 119905lowast

1119880] we apply the proposed recursive searching

algorithm stated at the end of Section 2 and find 119905lowast1= 03748

years Accordingly the optimal expected system costs perunit time 119864[TCU(119905lowast

1)] = $1100641 (Figure 6) Table 1 shows

the step-by-step iterations of the algorithmIn this specific studied model we focus on incorporating

a discontinuous multidelivery policy into a prior work [9]and consider a fixed transportation cost associated witheach delivery Applying the research result we can easilyinvestigate the effects of different fixed transportation costs

The Scientific World Journal 11

Table 1 Iterations of the recursive searching algorithm for locating 119905lowast1

120573 Step 119905lowast

1119880120596119880= 119890minus1205731199051119880 119905

lowast

1119871120596119871= 119890minus1205731199051119871

Differencebetween 119905lowast

1119880and

119905lowast

1119871

[119880]

119864[TCU(119905lowast1119880)]

[119871]

119864[TCU(119905lowast1119871)]

Differencebetween [119880]

and [119871]05 Initial 000000 100000 $1160163 $1101450 $58713

1st 05183 07717 03411 08432 01772 $1110330 $1101450 $88802nd 03857 08246 03721 08302 00136 $1100716 $1100646 $0703rd 03756 08287 03746 08292 00010 $1100642 $1100641 $0014th 03749 08291 03748 08291 00001 $1100641 $1100641 $00045th 03748 08291 03748 08291 000000 $1100641 $1100641 $0000

Table 2 Variations of the fixed delivery cost1198701effects on the optimal production run time 119905lowast

1

1198701119870 005 02 04 06 08 1 12 14 16 18 2 22

1198701

$225 $90 $180 $270 $360 $450 $540 $630 $720 $810 $900 $990119864[TCU(119905lowast

1)] $10721 $11067 $11448 $11773 $12062 $12325 $12567 $12793 $13006 $13208 $13399 $13583

119905lowast

103116 03816 04586 05244 05828 06359 06848 07305 07735 08142 08529 08900

Table 3 Variations of the unit retailerrsquos holding cost ℎ2and their effects on 119864[TCU(119905lowast

1)]

ℎ2ℎ 05 075 1 125 15 175 2 225 25 275 3 325

ℎ2

03 045 06 075 09 105 12 135 15 165 18 195119864 [TCU (119905lowast

1)] $11282 $11407 $11526 $11638 $11746 $11850 $11949 $12045 $12138 $12229 $12316 $12401

119905lowast

103256 03074 02918 02785 02668 02564 02472 02389 02314 02246 02183 02125

1198701on the optimal system cost 119864[TCU(119905lowast

1)] and on the

optimal production run time 119905lowast1(see Table 2) It can be seen

from Table 2 that as the ratio of1198701119870 increases the expected

system costs per unit time119864[TCU(119905lowast1119871)] increase significantly

It is also noted that as 1198701increases optimal production run

time 119905lowast1also increases significantly

41 Numerical Example for the Producer-Retailer IntegratedEPQ System (Model 2) In order to demonstrate the researchresult of the producer-retailer integrated EPQ-based modelan additional system variable ℎ

2= $150 per item stored at the

retailerrsquos side is includedAgain one can use the upper and lower bounds of 119905lowast

1

(equations (43) and (44)) to test for convexity of 119864[TCU(1199051)]

(Theorem 3 and equation (35)) The results reveal that 119905lowast1119880

=03213 lt 120595(119905lowast

1119880) = 26462 and 119905lowast

1119871= 02186 lt 120595(119905lowast

1119871) = 24876

Therefore the expected cost [TCU(1199051)] is convex

Next by applying the proposed recursive searching algo-rithm we can calculate that the optimal run time 119905lowast

1=

02314 years and the optimal 119864[TCU(119905lowast1)] = $1213849 It is

noted that the computation time for reaching the optimal 119905lowast1

solution is 21 seconds (usingExcel software in a desktop com-puter Intel CPU G850 with 294GB RAM and 289GHz)

Figure 7 illustrates the behavior of 119864[TCU(1199051)] with

regard to production run time It is noted that without theresearch result from the second model the management ofsuch a producer-retailer integrated system would probablyuse 1199051= 03748 years (from the result of model 1) for their

run time decision Further analysis (see Figure 7) shows cost

0130 0170 0210 0250 0290 0330 0370 0410

t1

E[T

CU(t

1)]

11600

11800

12000

12200

12400

12600

12800

tlowast1 = 02314

t1 = 0374812138

(from model 1)

E[TCU( t1)] = 12489

$$

$

$

$

$

$

$

$

Figure 7 The behavior of 119864[TCU(1199051)] with respect to production

run time 1199051in the proposed model 2

savings of $351 (or 29 over the total system costs) simply byapplying our research result

The effects of the unit retailerrsquos holding cost ℎ2on the

expected system cost 119864[TCU(119905lowast1)] and on the optimal run

time 119905lowast1are shown in Table 3 respectively

It can be seen that as ℎ2or the ratio of ℎ

2ℎ increases

the expected cost 119864[TCU(119905lowast1)] increases but the optimal

production run time 119905lowast1decreases In decision-making these

sensitivity analyses results can provide the management ofa producer-retailer integrated system with valuable informa-tion and insights into the effects of various stock holding costsin different retailersrsquo locations

12 The Scientific World Journal

5 Concluding Remarks

Two exact models for an extended EPQ-based problem witha discontinuous delivery policy scrap rate and randombreakdown are developed in this study They specificallyaddress different real-life situations in production end-item delivery and intrasupply chains such as a producer-retailer integrated systemMathematicalmodeling alongwithoptimization techniques is used to determine the optimalproduction run times that minimize the expected systemcosts per unit time Without in-depth investigations onthese separate models the optimal production run time andother important information related to the systemparameterscannot be revealed The proposed real-life EPQ models withrandom machine breakdown discontinuous product distri-bution policies and quality assurance must be specificallystudied in order to (1) obtain the joint effects of breakdowndiscontinuous distribution policies and quality assurance onthe optimal production run time (2) get to know the effectsof different policy and scope of supply chains managementon the optimal run time and overall system costs and (3)gain the insight with regard to various systemrsquos parametersof all particular EPQ-based models Since little attention hasbeen paid to the investigation of joint effects of these practicalproduction situations on the optimal run time this researchis intended to bridge the gap An interesting area for futurestudy is the examination of the effect of variable productionrates on these models

Appendix

Derivations of (3) are as followsRecall (2) as follows

TC1(1199051) = 119862 (119875

11199051) + 119870 + 119862

119878(11990511198751119909) +119872 + 119899119870

1

+ 119862119879[11990511198751] + ℎ3(120582119905119903) 1198791015840

+ ℎ[1198671015840+ 11988911199051

21199051+ (1198671015840

1+ 1198891119905) 119905119903+119899 minus 1

211989911986710158401199051015840

2]

(A1)

Substituting all related system parameters into (2) (pleaserefer to the basic formulations and solution process in [9])the TC

1(1199051) can be obtained as

TC1(1199051) = 119870 +119872

+[1198621198751+ 1198621198781198751119909 + 119862

1198791198751(1 minus 119909)+ ℎ

31198751119892 (1 minus 119909)] 119905

1

+ 1198991198701+ ℎ119901119892119905 minus[

ℎ1198751119892 (1 minus 119909)

2minusℎ1198751119892 (1 minus 119909)

2119899] 1199051

+ 1199052

1[ℎ1198751

2+ℎ1198752

1

2120582(1 minus 119909)

2minusℎ1198751

2(1 minus 119909)

minusℎ1198752

1

2120582119899(1 minus 119909)

2+ℎ1198751

2119899(1 minus 119909)]

(A2)

To take the randomness of 119909 into account by using theexpected values of 119909 with further derivations119864[TC

1(1199051)] can

be derived as follows (ie equation (3))119864 [TC

1(1199051)]

= 119870 + 1198991198701+119872 + ℎ119905119875

1119892

+ [1198621198751+ 1198621198781198751119864 [119909] + 1198621198791198751 (1 minus 119864 [119909])

+ ℎ31198751119892 (1 minus 119864 [119909]) minus

ℎ1198751119892 (1 minus 119864 [119909])

2(1 minus

1

119899)] sdot 1199051

+ [ℎ1198751119864 [119909]

2+ℎ1198752

1

2120582(1 minus 119864 [119909])

2(1 minus

1

119899)

+ℎ1198751

2119899(1 minus 119864 [119909])] 119905

2

1

(A3)

Conflict of Interests

The authors of the paper declare that there is no conflict ofinterests regarding the publication of this paper

Acknowledgment

The authors greatly appreciate the National Science Council(NSC) of Taiwan for supporting this research underGrant noNSC 102-2410-H-324-005

References

[1] E W Taft ldquoThemost economical production lotrdquo Iron Age vol101 pp 1410ndash1412 1918

[2] G Hadley and T M Whitin ldquoAn optimal final inventorymodelrdquoManagement Science vol 7 pp 179ndash183 1961

[3] E A Silver D F Pyke and R Peterson Inventory Managementand Production Planning and Scheduling John Wiley amp SonsNew York NY USA 1998

[4] S Nahmias Production amp Operations Analysis McGraw-HillNew York NY USA 2009

[5] MWidmer and P Solot ldquoDo not forget the breakdowns and themaintenance operations in FMSdesign problemsrdquo InternationalJournal of Production Research vol 28 pp 421ndash430 1990

[6] K-Y C Yu and D L Bricker ldquoAnalysis of a markov chainmodel of a multistage manufacturing system with inspectionrejection and reworkrdquo IIE Transactions vol 25 no 1 pp 109ndash112 1993

[7] H Groenevelt L Pintelon and A Seidmann ldquoProduction lotsizing with machine breakdownsrdquoManagement Science vol 38no 1 pp 104ndash123 1992

[8] G A Widyadana and H M Wee ldquoOptimal deterioratingitems production inventory models with random machinebreakdown and stochastic repair timerdquo Applied MathematicalModelling vol 35 no 7 pp 3495ndash3508 2011

[9] S W Chiu C-L Chou and W-K Wu ldquoOptimizing replenish-ment policy in an EPQ-based inventory model with noncon-forming items and breakdownrdquo EconomicModelling vol 35 pp330ndash337 2013

The Scientific World Journal 13

[10] A M Zargar ldquoEffect of rework strategies on cycle timerdquoComputers amp Industrial Engineering vol 29 no 1ndash4 pp 239ndash243 1995

[11] P Biswas and B R Sarker ldquoOptimal batch quantity modelsfor a lean production system with in-cycle rework and scraprdquoInternational Journal of Production Research vol 46 no 23 pp6585ndash6610 2008

[12] S W Chiu H-D Lin C-B Cheng and C-L Chung ldquoOptimalproduction-shipment decisions for the finite production ratemodel with scraprdquo International Journal for Engineering Mod-elling vol 22 no 1ndash4 pp 25ndash34 2009

[13] Y-S P Chiu K-K Chen and C-K Ting ldquoReplenishment runtime problem with machine breakdown and failure in reworkrdquoExpert Systems with Applications vol 39 no 1 pp 1291ndash12972012

[14] H-D Lin F-Y Pai and S W Chiu ldquoA note on ldquointra-supply chain system with multiple sales locations and qualityassurancerdquordquo Expert Systems with Applications vol 40 no 11 pp4730ndash4732 2013

[15] B Kamsu-Foguem F Rigal and FMauget ldquoMining associationrules for the quality improvement of the production processrdquoExpert Systems with Applications vol 40 no 4 pp 1034ndash10452013

[16] Y Li H Guo L Wang and J Fu ldquoA hybrid genetic-simulatedannealing algorithm for the location-inventory- routing prob-lem considering returns under E-supply chain environmentrdquoThe Scientific World Journal vol 2013 Article ID 125893 10pages 2013

[17] Y-S P Chiu K-K Chen F-T Cheng and C-K Ting ldquoReex-amination of ldquocombining an alternative multi-delivery policyinto economic production lot size problemwith partial reworkrdquousing an alternative approachrdquo Journal of Applied Research andTechnology vol 11 no 3 pp 317ndash323 2013

[18] L B Schwarz B L Deuermeyer and R D Badinelli ldquoFill-rateoptimization in a one-warehouse 119873-identical retailer distribu-tion systemrdquo Management Science vol 31 no 4 pp 488ndash4981985

[19] R A Sarker and L R Khan ldquoOptimal batch size for a pro-duction system operating under periodic delivery policyrdquoCom-puters amp Industrial Engineering vol 37 no 4 pp 711ndash730 1999

[20] N Comez K E Stecke and M Cakanyildirim ldquoMultiple in-cycle transshipments with positive delivery timesrdquo Productionand Operations Management vol 21 no 2 pp 378ndash395 2012

[21] S W Chiu F-Y Pai and W K Wu ldquoAlternative approach todetermine the common cycle time for a multi-item productionsystem with discontinuous deliveries and failure in reworkrdquoEconomic Modelling vol 35 pp 593ndash596 2013

[22] Y-S P Chiu C-C Huang M-F Wu and H-H Chang ldquoJointdetermination of rotation cycle time and number of shipmentsfor a multi-item EPQ model with random defective raterdquoEconomic Modelling vol 35 pp 112ndash117 2013

[23] M A Hoque ldquoSynchronization in the single-manufacturermulti-buyer integrated inventory supply chainrdquo European Jour-nal of Operational Research vol 188 no 3 pp 811ndash825 2008

[24] S W Chiu L-W Lin K-K Chen and C-L Chou ldquoDetermin-ing production-shipment policy for a vendor-buyer integratedsystem with rework and an amending multi-delivery schedulerdquoEconomic Modelling vol 33 pp 668ndash675 2013

[25] M Cedillo-Campos and C Sanchez-Ramırez ldquoDynamic self-assessment of supply chains performance an emerging marketapproachrdquo Journal of Applied Research and Technology vol 11no 3 pp 338ndash347 2013

[26] Y-S P Chiu H-D Lin F-T Cheng and M-H Hwang ldquoOpti-mal common cycle time for a multi-item production systemwith discontinuous delivery policy and failure in reworkrdquoJournal of Scientific and Industrial Research vol 72 no 7 pp435ndash440 2013

[27] HHishamuddin R A Sarker andD Essam ldquoA recoverymech-anism for a two echelon supply chain system under supplydisruptionrdquo Economic Modelling vol 38 pp 555ndash563 2014

[28] M Murugan and V Selladurai ldquoProductivity improvement inmanufacturing submersible pump diffuser housing using leanmanufacturing systemrdquo Journal of Engineering Research vol 2no 1 pp 164ndash182 2014

[29] L Wang H Qu S Liu and C-X Dun ldquoModeling and opti-mization of the multiobjective stochastic joint replenishmentand delivery problem under supply chain environmentrdquo TheScientific World Journal vol 2013 Article ID 916057 11 pages2013

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

10 The Scientific World Journal

Once 119864[1198791198621198802(1199051)] is proven to be convex the optimal

run time 119905lowast1can be solved by setting the first derivative of

119864[1198791198621198802(1199051)] = 0

119889119864 [1198791198621198802(1199051)]

1198891199051

=120582

(1 minus 119864 [119909])sdot minus (119870 + 119899119870

1)

1199052

1119875

+ (1205742

2+ 1205745) + 1205746(120573119890minus1205731199051)

+ [119872

1198751

+ℎ119892

120573](

minus (1 minus 119890minus1205731199051)

1199052

1

+120573119890minus1205731199051

1199051

)

= 0

(42)

It can be seen that the first term in the RHS of (42) ispositive so the second term is equal to zero In order to findthe bounds for 119905lowast

1 let

119905lowast

1119880= radic

2 [120573 (119870 + 1198991198701) + 1205744]

1198751120573 (1205742+ 21205745)

(43)

119905lowast

1119871= 119905ℎ119890 119901119900119904119894119905119894V119890 119903119900119900119905 119900119891

minus1205744plusmn radic1205742

4+ 21198751(119870 + 119899119870

1) (1205742+ 21205745+ 21205746120573)

1198751(1205742+ 21205745+ 21205746120573)

(44)

Theorem4 (119905lowast1119871lt 119905lowast

1lt 119905lowast

1119880) For the proof ofTheorem 4 please

refer to the proof for Theorem 2 in Section 2

Once we are certain that 119905lowast1falls within the aforemen-

tioned upper and lower bounds in order to find 119905lowast1 we can

first multiply the second term of (42) by (211987511199052

1120573) and obtain

the following

[1198751120573 (1205742+ 21205745) + 2119875

112057321205746119890minus1205731199051] 1199052

1+ (21205744120573119890minus1205731199051) 1199051

minus2 [120573 (119870 + 1198991198701) + 1205744(1 minus 119890

minus1205731199051)]

= 0

(45)

Equation (45) can be rearranged as

119890minus1205731199051 =

2 [120573 (119870 + 1198991198701) + 1205744] minus [119875

1120573 (1205742+ 21205745)] 1199052

1

2 [1198751120573212057461199052

1+ 1205744(1 + 120573119905

1)]

(46)

where 119890minus1205731199051 is the complement of the cumulative densityfunction 119865(119905

1) = 1minus119890

minus1205731199051 As 0 le 119865(119905

1) le 1 0 le 119890minus1205731199051 le 1 Let

119890minus1205731199051 = 0 and 119890minus1205731199051 = 1 be the initial upper and lower bounds

of 119890minus1205731199051 respectively Then by using the proposed recursivesearching algorithm given at the end of Section 2 we can findthe optimal production run time 119905lowast

1

4 Numerical Example

In order to relieve the comparison efforts for readers thissection adopts the same numerical example as in [9] For

0225 0265 0305 0345 0385 0425 0465 0505

E[T

CU(t

1)]

t1

tlowast1L = 03411 t

lowast1 = 03748

tlowast1U = 05183

$10600

$10800

$11000

$11200

$11400

$11600

$11006

$11800

Figure 6 The behavior of 119864[TCU(1199051)] in connection with the

production run time 1199051in the proposed model 1

a demonstration of the proposed EPQ-based model 1 thefollowing system parameters are used

1198751 production rate 10000 products per year120582 demand rate 4000 products per year119909 random scrap rate which follows uniformly distribu-

tion over the interval [0 02]120573 Poisson breakdown rate 05 average times per year119892 constant machine repair time 119905

119903 0018 year per repair

119872 machine repair cost $500 for each breakdown119870 setup cost $450 per production run119862 manufacturing cost $2 per item119862119878 disposal cost $03 per scrap itemℎ holding cost $06 per item per unit time1198701 fixed delivery cost $90 per shipment119899 number of deliveries 4 per cycle

119862119879 variable delivery cost $0001 per item

First we use both upper and lower bounds of 119905lowast1to

test for the convexity of 119864[TCU(1199051)] (see Theorem 1) The

computation results of (19) (20) and (11) indicate that 119905lowast1119880

=05183 lt 120585(119905

lowast

1119880) = 28723 and 119905lowast

1119871= 03411 lt 120585(119905

lowast

1119871) = 26261

Hence [TCU(1199051)] is convex (Figure 6)

In order to find the optimal 119905lowast1 we first substitute the

upper and lower bounds of 119905lowast1in (8) and obtain 119864[TCU(119905lowast

1119880)]

= $1160163 and 119864[TCU(119905lowast1119871)] = $1101450 respectively

Because the optimal run time 119905lowast1falls within the interval

of [119905lowast1119871 119905lowast

1119880] we apply the proposed recursive searching

algorithm stated at the end of Section 2 and find 119905lowast1= 03748

years Accordingly the optimal expected system costs perunit time 119864[TCU(119905lowast

1)] = $1100641 (Figure 6) Table 1 shows

the step-by-step iterations of the algorithmIn this specific studied model we focus on incorporating

a discontinuous multidelivery policy into a prior work [9]and consider a fixed transportation cost associated witheach delivery Applying the research result we can easilyinvestigate the effects of different fixed transportation costs

The Scientific World Journal 11

Table 1 Iterations of the recursive searching algorithm for locating 119905lowast1

120573 Step 119905lowast

1119880120596119880= 119890minus1205731199051119880 119905

lowast

1119871120596119871= 119890minus1205731199051119871

Differencebetween 119905lowast

1119880and

119905lowast

1119871

[119880]

119864[TCU(119905lowast1119880)]

[119871]

119864[TCU(119905lowast1119871)]

Differencebetween [119880]

and [119871]05 Initial 000000 100000 $1160163 $1101450 $58713

1st 05183 07717 03411 08432 01772 $1110330 $1101450 $88802nd 03857 08246 03721 08302 00136 $1100716 $1100646 $0703rd 03756 08287 03746 08292 00010 $1100642 $1100641 $0014th 03749 08291 03748 08291 00001 $1100641 $1100641 $00045th 03748 08291 03748 08291 000000 $1100641 $1100641 $0000

Table 2 Variations of the fixed delivery cost1198701effects on the optimal production run time 119905lowast

1

1198701119870 005 02 04 06 08 1 12 14 16 18 2 22

1198701

$225 $90 $180 $270 $360 $450 $540 $630 $720 $810 $900 $990119864[TCU(119905lowast

1)] $10721 $11067 $11448 $11773 $12062 $12325 $12567 $12793 $13006 $13208 $13399 $13583

119905lowast

103116 03816 04586 05244 05828 06359 06848 07305 07735 08142 08529 08900

Table 3 Variations of the unit retailerrsquos holding cost ℎ2and their effects on 119864[TCU(119905lowast

1)]

ℎ2ℎ 05 075 1 125 15 175 2 225 25 275 3 325

ℎ2

03 045 06 075 09 105 12 135 15 165 18 195119864 [TCU (119905lowast

1)] $11282 $11407 $11526 $11638 $11746 $11850 $11949 $12045 $12138 $12229 $12316 $12401

119905lowast

103256 03074 02918 02785 02668 02564 02472 02389 02314 02246 02183 02125

1198701on the optimal system cost 119864[TCU(119905lowast

1)] and on the

optimal production run time 119905lowast1(see Table 2) It can be seen

from Table 2 that as the ratio of1198701119870 increases the expected

system costs per unit time119864[TCU(119905lowast1119871)] increase significantly

It is also noted that as 1198701increases optimal production run

time 119905lowast1also increases significantly

41 Numerical Example for the Producer-Retailer IntegratedEPQ System (Model 2) In order to demonstrate the researchresult of the producer-retailer integrated EPQ-based modelan additional system variable ℎ

2= $150 per item stored at the

retailerrsquos side is includedAgain one can use the upper and lower bounds of 119905lowast

1

(equations (43) and (44)) to test for convexity of 119864[TCU(1199051)]

(Theorem 3 and equation (35)) The results reveal that 119905lowast1119880

=03213 lt 120595(119905lowast

1119880) = 26462 and 119905lowast

1119871= 02186 lt 120595(119905lowast

1119871) = 24876

Therefore the expected cost [TCU(1199051)] is convex

Next by applying the proposed recursive searching algo-rithm we can calculate that the optimal run time 119905lowast

1=

02314 years and the optimal 119864[TCU(119905lowast1)] = $1213849 It is

noted that the computation time for reaching the optimal 119905lowast1

solution is 21 seconds (usingExcel software in a desktop com-puter Intel CPU G850 with 294GB RAM and 289GHz)

Figure 7 illustrates the behavior of 119864[TCU(1199051)] with

regard to production run time It is noted that without theresearch result from the second model the management ofsuch a producer-retailer integrated system would probablyuse 1199051= 03748 years (from the result of model 1) for their

run time decision Further analysis (see Figure 7) shows cost

0130 0170 0210 0250 0290 0330 0370 0410

t1

E[T

CU(t

1)]

11600

11800

12000

12200

12400

12600

12800

tlowast1 = 02314

t1 = 0374812138

(from model 1)

E[TCU( t1)] = 12489

$$

$

$

$

$

$

$

$

Figure 7 The behavior of 119864[TCU(1199051)] with respect to production

run time 1199051in the proposed model 2

savings of $351 (or 29 over the total system costs) simply byapplying our research result

The effects of the unit retailerrsquos holding cost ℎ2on the

expected system cost 119864[TCU(119905lowast1)] and on the optimal run

time 119905lowast1are shown in Table 3 respectively

It can be seen that as ℎ2or the ratio of ℎ

2ℎ increases

the expected cost 119864[TCU(119905lowast1)] increases but the optimal

production run time 119905lowast1decreases In decision-making these

sensitivity analyses results can provide the management ofa producer-retailer integrated system with valuable informa-tion and insights into the effects of various stock holding costsin different retailersrsquo locations

12 The Scientific World Journal

5 Concluding Remarks

Two exact models for an extended EPQ-based problem witha discontinuous delivery policy scrap rate and randombreakdown are developed in this study They specificallyaddress different real-life situations in production end-item delivery and intrasupply chains such as a producer-retailer integrated systemMathematicalmodeling alongwithoptimization techniques is used to determine the optimalproduction run times that minimize the expected systemcosts per unit time Without in-depth investigations onthese separate models the optimal production run time andother important information related to the systemparameterscannot be revealed The proposed real-life EPQ models withrandom machine breakdown discontinuous product distri-bution policies and quality assurance must be specificallystudied in order to (1) obtain the joint effects of breakdowndiscontinuous distribution policies and quality assurance onthe optimal production run time (2) get to know the effectsof different policy and scope of supply chains managementon the optimal run time and overall system costs and (3)gain the insight with regard to various systemrsquos parametersof all particular EPQ-based models Since little attention hasbeen paid to the investigation of joint effects of these practicalproduction situations on the optimal run time this researchis intended to bridge the gap An interesting area for futurestudy is the examination of the effect of variable productionrates on these models

Appendix

Derivations of (3) are as followsRecall (2) as follows

TC1(1199051) = 119862 (119875

11199051) + 119870 + 119862

119878(11990511198751119909) +119872 + 119899119870

1

+ 119862119879[11990511198751] + ℎ3(120582119905119903) 1198791015840

+ ℎ[1198671015840+ 11988911199051

21199051+ (1198671015840

1+ 1198891119905) 119905119903+119899 minus 1

211989911986710158401199051015840

2]

(A1)

Substituting all related system parameters into (2) (pleaserefer to the basic formulations and solution process in [9])the TC

1(1199051) can be obtained as

TC1(1199051) = 119870 +119872

+[1198621198751+ 1198621198781198751119909 + 119862

1198791198751(1 minus 119909)+ ℎ

31198751119892 (1 minus 119909)] 119905

1

+ 1198991198701+ ℎ119901119892119905 minus[

ℎ1198751119892 (1 minus 119909)

2minusℎ1198751119892 (1 minus 119909)

2119899] 1199051

+ 1199052

1[ℎ1198751

2+ℎ1198752

1

2120582(1 minus 119909)

2minusℎ1198751

2(1 minus 119909)

minusℎ1198752

1

2120582119899(1 minus 119909)

2+ℎ1198751

2119899(1 minus 119909)]

(A2)

To take the randomness of 119909 into account by using theexpected values of 119909 with further derivations119864[TC

1(1199051)] can

be derived as follows (ie equation (3))119864 [TC

1(1199051)]

= 119870 + 1198991198701+119872 + ℎ119905119875

1119892

+ [1198621198751+ 1198621198781198751119864 [119909] + 1198621198791198751 (1 minus 119864 [119909])

+ ℎ31198751119892 (1 minus 119864 [119909]) minus

ℎ1198751119892 (1 minus 119864 [119909])

2(1 minus

1

119899)] sdot 1199051

+ [ℎ1198751119864 [119909]

2+ℎ1198752

1

2120582(1 minus 119864 [119909])

2(1 minus

1

119899)

+ℎ1198751

2119899(1 minus 119864 [119909])] 119905

2

1

(A3)

Conflict of Interests

The authors of the paper declare that there is no conflict ofinterests regarding the publication of this paper

Acknowledgment

The authors greatly appreciate the National Science Council(NSC) of Taiwan for supporting this research underGrant noNSC 102-2410-H-324-005

References

[1] E W Taft ldquoThemost economical production lotrdquo Iron Age vol101 pp 1410ndash1412 1918

[2] G Hadley and T M Whitin ldquoAn optimal final inventorymodelrdquoManagement Science vol 7 pp 179ndash183 1961

[3] E A Silver D F Pyke and R Peterson Inventory Managementand Production Planning and Scheduling John Wiley amp SonsNew York NY USA 1998

[4] S Nahmias Production amp Operations Analysis McGraw-HillNew York NY USA 2009

[5] MWidmer and P Solot ldquoDo not forget the breakdowns and themaintenance operations in FMSdesign problemsrdquo InternationalJournal of Production Research vol 28 pp 421ndash430 1990

[6] K-Y C Yu and D L Bricker ldquoAnalysis of a markov chainmodel of a multistage manufacturing system with inspectionrejection and reworkrdquo IIE Transactions vol 25 no 1 pp 109ndash112 1993

[7] H Groenevelt L Pintelon and A Seidmann ldquoProduction lotsizing with machine breakdownsrdquoManagement Science vol 38no 1 pp 104ndash123 1992

[8] G A Widyadana and H M Wee ldquoOptimal deterioratingitems production inventory models with random machinebreakdown and stochastic repair timerdquo Applied MathematicalModelling vol 35 no 7 pp 3495ndash3508 2011

[9] S W Chiu C-L Chou and W-K Wu ldquoOptimizing replenish-ment policy in an EPQ-based inventory model with noncon-forming items and breakdownrdquo EconomicModelling vol 35 pp330ndash337 2013

The Scientific World Journal 13

[10] A M Zargar ldquoEffect of rework strategies on cycle timerdquoComputers amp Industrial Engineering vol 29 no 1ndash4 pp 239ndash243 1995

[11] P Biswas and B R Sarker ldquoOptimal batch quantity modelsfor a lean production system with in-cycle rework and scraprdquoInternational Journal of Production Research vol 46 no 23 pp6585ndash6610 2008

[12] S W Chiu H-D Lin C-B Cheng and C-L Chung ldquoOptimalproduction-shipment decisions for the finite production ratemodel with scraprdquo International Journal for Engineering Mod-elling vol 22 no 1ndash4 pp 25ndash34 2009

[13] Y-S P Chiu K-K Chen and C-K Ting ldquoReplenishment runtime problem with machine breakdown and failure in reworkrdquoExpert Systems with Applications vol 39 no 1 pp 1291ndash12972012

[14] H-D Lin F-Y Pai and S W Chiu ldquoA note on ldquointra-supply chain system with multiple sales locations and qualityassurancerdquordquo Expert Systems with Applications vol 40 no 11 pp4730ndash4732 2013

[15] B Kamsu-Foguem F Rigal and FMauget ldquoMining associationrules for the quality improvement of the production processrdquoExpert Systems with Applications vol 40 no 4 pp 1034ndash10452013

[16] Y Li H Guo L Wang and J Fu ldquoA hybrid genetic-simulatedannealing algorithm for the location-inventory- routing prob-lem considering returns under E-supply chain environmentrdquoThe Scientific World Journal vol 2013 Article ID 125893 10pages 2013

[17] Y-S P Chiu K-K Chen F-T Cheng and C-K Ting ldquoReex-amination of ldquocombining an alternative multi-delivery policyinto economic production lot size problemwith partial reworkrdquousing an alternative approachrdquo Journal of Applied Research andTechnology vol 11 no 3 pp 317ndash323 2013

[18] L B Schwarz B L Deuermeyer and R D Badinelli ldquoFill-rateoptimization in a one-warehouse 119873-identical retailer distribu-tion systemrdquo Management Science vol 31 no 4 pp 488ndash4981985

[19] R A Sarker and L R Khan ldquoOptimal batch size for a pro-duction system operating under periodic delivery policyrdquoCom-puters amp Industrial Engineering vol 37 no 4 pp 711ndash730 1999

[20] N Comez K E Stecke and M Cakanyildirim ldquoMultiple in-cycle transshipments with positive delivery timesrdquo Productionand Operations Management vol 21 no 2 pp 378ndash395 2012

[21] S W Chiu F-Y Pai and W K Wu ldquoAlternative approach todetermine the common cycle time for a multi-item productionsystem with discontinuous deliveries and failure in reworkrdquoEconomic Modelling vol 35 pp 593ndash596 2013

[22] Y-S P Chiu C-C Huang M-F Wu and H-H Chang ldquoJointdetermination of rotation cycle time and number of shipmentsfor a multi-item EPQ model with random defective raterdquoEconomic Modelling vol 35 pp 112ndash117 2013

[23] M A Hoque ldquoSynchronization in the single-manufacturermulti-buyer integrated inventory supply chainrdquo European Jour-nal of Operational Research vol 188 no 3 pp 811ndash825 2008

[24] S W Chiu L-W Lin K-K Chen and C-L Chou ldquoDetermin-ing production-shipment policy for a vendor-buyer integratedsystem with rework and an amending multi-delivery schedulerdquoEconomic Modelling vol 33 pp 668ndash675 2013

[25] M Cedillo-Campos and C Sanchez-Ramırez ldquoDynamic self-assessment of supply chains performance an emerging marketapproachrdquo Journal of Applied Research and Technology vol 11no 3 pp 338ndash347 2013

[26] Y-S P Chiu H-D Lin F-T Cheng and M-H Hwang ldquoOpti-mal common cycle time for a multi-item production systemwith discontinuous delivery policy and failure in reworkrdquoJournal of Scientific and Industrial Research vol 72 no 7 pp435ndash440 2013

[27] HHishamuddin R A Sarker andD Essam ldquoA recoverymech-anism for a two echelon supply chain system under supplydisruptionrdquo Economic Modelling vol 38 pp 555ndash563 2014

[28] M Murugan and V Selladurai ldquoProductivity improvement inmanufacturing submersible pump diffuser housing using leanmanufacturing systemrdquo Journal of Engineering Research vol 2no 1 pp 164ndash182 2014

[29] L Wang H Qu S Liu and C-X Dun ldquoModeling and opti-mization of the multiobjective stochastic joint replenishmentand delivery problem under supply chain environmentrdquo TheScientific World Journal vol 2013 Article ID 916057 11 pages2013

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

The Scientific World Journal 11

Table 1 Iterations of the recursive searching algorithm for locating 119905lowast1

120573 Step 119905lowast

1119880120596119880= 119890minus1205731199051119880 119905

lowast

1119871120596119871= 119890minus1205731199051119871

Differencebetween 119905lowast

1119880and

119905lowast

1119871

[119880]

119864[TCU(119905lowast1119880)]

[119871]

119864[TCU(119905lowast1119871)]

Differencebetween [119880]

and [119871]05 Initial 000000 100000 $1160163 $1101450 $58713

1st 05183 07717 03411 08432 01772 $1110330 $1101450 $88802nd 03857 08246 03721 08302 00136 $1100716 $1100646 $0703rd 03756 08287 03746 08292 00010 $1100642 $1100641 $0014th 03749 08291 03748 08291 00001 $1100641 $1100641 $00045th 03748 08291 03748 08291 000000 $1100641 $1100641 $0000

Table 2 Variations of the fixed delivery cost1198701effects on the optimal production run time 119905lowast

1

1198701119870 005 02 04 06 08 1 12 14 16 18 2 22

1198701

$225 $90 $180 $270 $360 $450 $540 $630 $720 $810 $900 $990119864[TCU(119905lowast

1)] $10721 $11067 $11448 $11773 $12062 $12325 $12567 $12793 $13006 $13208 $13399 $13583

119905lowast

103116 03816 04586 05244 05828 06359 06848 07305 07735 08142 08529 08900

Table 3 Variations of the unit retailerrsquos holding cost ℎ2and their effects on 119864[TCU(119905lowast

1)]

ℎ2ℎ 05 075 1 125 15 175 2 225 25 275 3 325

ℎ2

03 045 06 075 09 105 12 135 15 165 18 195119864 [TCU (119905lowast

1)] $11282 $11407 $11526 $11638 $11746 $11850 $11949 $12045 $12138 $12229 $12316 $12401

119905lowast

103256 03074 02918 02785 02668 02564 02472 02389 02314 02246 02183 02125

1198701on the optimal system cost 119864[TCU(119905lowast

1)] and on the

optimal production run time 119905lowast1(see Table 2) It can be seen

from Table 2 that as the ratio of1198701119870 increases the expected

system costs per unit time119864[TCU(119905lowast1119871)] increase significantly

It is also noted that as 1198701increases optimal production run

time 119905lowast1also increases significantly

41 Numerical Example for the Producer-Retailer IntegratedEPQ System (Model 2) In order to demonstrate the researchresult of the producer-retailer integrated EPQ-based modelan additional system variable ℎ

2= $150 per item stored at the

retailerrsquos side is includedAgain one can use the upper and lower bounds of 119905lowast

1

(equations (43) and (44)) to test for convexity of 119864[TCU(1199051)]

(Theorem 3 and equation (35)) The results reveal that 119905lowast1119880

=03213 lt 120595(119905lowast

1119880) = 26462 and 119905lowast

1119871= 02186 lt 120595(119905lowast

1119871) = 24876

Therefore the expected cost [TCU(1199051)] is convex

Next by applying the proposed recursive searching algo-rithm we can calculate that the optimal run time 119905lowast

1=

02314 years and the optimal 119864[TCU(119905lowast1)] = $1213849 It is

noted that the computation time for reaching the optimal 119905lowast1

solution is 21 seconds (usingExcel software in a desktop com-puter Intel CPU G850 with 294GB RAM and 289GHz)

Figure 7 illustrates the behavior of 119864[TCU(1199051)] with

regard to production run time It is noted that without theresearch result from the second model the management ofsuch a producer-retailer integrated system would probablyuse 1199051= 03748 years (from the result of model 1) for their

run time decision Further analysis (see Figure 7) shows cost

0130 0170 0210 0250 0290 0330 0370 0410

t1

E[T

CU(t

1)]

11600

11800

12000

12200

12400

12600

12800

tlowast1 = 02314

t1 = 0374812138

(from model 1)

E[TCU( t1)] = 12489

$$

$

$

$

$

$

$

$

Figure 7 The behavior of 119864[TCU(1199051)] with respect to production

run time 1199051in the proposed model 2

savings of $351 (or 29 over the total system costs) simply byapplying our research result

The effects of the unit retailerrsquos holding cost ℎ2on the

expected system cost 119864[TCU(119905lowast1)] and on the optimal run

time 119905lowast1are shown in Table 3 respectively

It can be seen that as ℎ2or the ratio of ℎ

2ℎ increases

the expected cost 119864[TCU(119905lowast1)] increases but the optimal

production run time 119905lowast1decreases In decision-making these

sensitivity analyses results can provide the management ofa producer-retailer integrated system with valuable informa-tion and insights into the effects of various stock holding costsin different retailersrsquo locations

12 The Scientific World Journal

5 Concluding Remarks

Two exact models for an extended EPQ-based problem witha discontinuous delivery policy scrap rate and randombreakdown are developed in this study They specificallyaddress different real-life situations in production end-item delivery and intrasupply chains such as a producer-retailer integrated systemMathematicalmodeling alongwithoptimization techniques is used to determine the optimalproduction run times that minimize the expected systemcosts per unit time Without in-depth investigations onthese separate models the optimal production run time andother important information related to the systemparameterscannot be revealed The proposed real-life EPQ models withrandom machine breakdown discontinuous product distri-bution policies and quality assurance must be specificallystudied in order to (1) obtain the joint effects of breakdowndiscontinuous distribution policies and quality assurance onthe optimal production run time (2) get to know the effectsof different policy and scope of supply chains managementon the optimal run time and overall system costs and (3)gain the insight with regard to various systemrsquos parametersof all particular EPQ-based models Since little attention hasbeen paid to the investigation of joint effects of these practicalproduction situations on the optimal run time this researchis intended to bridge the gap An interesting area for futurestudy is the examination of the effect of variable productionrates on these models

Appendix

Derivations of (3) are as followsRecall (2) as follows

TC1(1199051) = 119862 (119875

11199051) + 119870 + 119862

119878(11990511198751119909) +119872 + 119899119870

1

+ 119862119879[11990511198751] + ℎ3(120582119905119903) 1198791015840

+ ℎ[1198671015840+ 11988911199051

21199051+ (1198671015840

1+ 1198891119905) 119905119903+119899 minus 1

211989911986710158401199051015840

2]

(A1)

Substituting all related system parameters into (2) (pleaserefer to the basic formulations and solution process in [9])the TC

1(1199051) can be obtained as

TC1(1199051) = 119870 +119872

+[1198621198751+ 1198621198781198751119909 + 119862

1198791198751(1 minus 119909)+ ℎ

31198751119892 (1 minus 119909)] 119905

1

+ 1198991198701+ ℎ119901119892119905 minus[

ℎ1198751119892 (1 minus 119909)

2minusℎ1198751119892 (1 minus 119909)

2119899] 1199051

+ 1199052

1[ℎ1198751

2+ℎ1198752

1

2120582(1 minus 119909)

2minusℎ1198751

2(1 minus 119909)

minusℎ1198752

1

2120582119899(1 minus 119909)

2+ℎ1198751

2119899(1 minus 119909)]

(A2)

To take the randomness of 119909 into account by using theexpected values of 119909 with further derivations119864[TC

1(1199051)] can

be derived as follows (ie equation (3))119864 [TC

1(1199051)]

= 119870 + 1198991198701+119872 + ℎ119905119875

1119892

+ [1198621198751+ 1198621198781198751119864 [119909] + 1198621198791198751 (1 minus 119864 [119909])

+ ℎ31198751119892 (1 minus 119864 [119909]) minus

ℎ1198751119892 (1 minus 119864 [119909])

2(1 minus

1

119899)] sdot 1199051

+ [ℎ1198751119864 [119909]

2+ℎ1198752

1

2120582(1 minus 119864 [119909])

2(1 minus

1

119899)

+ℎ1198751

2119899(1 minus 119864 [119909])] 119905

2

1

(A3)

Conflict of Interests

The authors of the paper declare that there is no conflict ofinterests regarding the publication of this paper

Acknowledgment

The authors greatly appreciate the National Science Council(NSC) of Taiwan for supporting this research underGrant noNSC 102-2410-H-324-005

References

[1] E W Taft ldquoThemost economical production lotrdquo Iron Age vol101 pp 1410ndash1412 1918

[2] G Hadley and T M Whitin ldquoAn optimal final inventorymodelrdquoManagement Science vol 7 pp 179ndash183 1961

[3] E A Silver D F Pyke and R Peterson Inventory Managementand Production Planning and Scheduling John Wiley amp SonsNew York NY USA 1998

[4] S Nahmias Production amp Operations Analysis McGraw-HillNew York NY USA 2009

[5] MWidmer and P Solot ldquoDo not forget the breakdowns and themaintenance operations in FMSdesign problemsrdquo InternationalJournal of Production Research vol 28 pp 421ndash430 1990

[6] K-Y C Yu and D L Bricker ldquoAnalysis of a markov chainmodel of a multistage manufacturing system with inspectionrejection and reworkrdquo IIE Transactions vol 25 no 1 pp 109ndash112 1993

[7] H Groenevelt L Pintelon and A Seidmann ldquoProduction lotsizing with machine breakdownsrdquoManagement Science vol 38no 1 pp 104ndash123 1992

[8] G A Widyadana and H M Wee ldquoOptimal deterioratingitems production inventory models with random machinebreakdown and stochastic repair timerdquo Applied MathematicalModelling vol 35 no 7 pp 3495ndash3508 2011

[9] S W Chiu C-L Chou and W-K Wu ldquoOptimizing replenish-ment policy in an EPQ-based inventory model with noncon-forming items and breakdownrdquo EconomicModelling vol 35 pp330ndash337 2013

The Scientific World Journal 13

[10] A M Zargar ldquoEffect of rework strategies on cycle timerdquoComputers amp Industrial Engineering vol 29 no 1ndash4 pp 239ndash243 1995

[11] P Biswas and B R Sarker ldquoOptimal batch quantity modelsfor a lean production system with in-cycle rework and scraprdquoInternational Journal of Production Research vol 46 no 23 pp6585ndash6610 2008

[12] S W Chiu H-D Lin C-B Cheng and C-L Chung ldquoOptimalproduction-shipment decisions for the finite production ratemodel with scraprdquo International Journal for Engineering Mod-elling vol 22 no 1ndash4 pp 25ndash34 2009

[13] Y-S P Chiu K-K Chen and C-K Ting ldquoReplenishment runtime problem with machine breakdown and failure in reworkrdquoExpert Systems with Applications vol 39 no 1 pp 1291ndash12972012

[14] H-D Lin F-Y Pai and S W Chiu ldquoA note on ldquointra-supply chain system with multiple sales locations and qualityassurancerdquordquo Expert Systems with Applications vol 40 no 11 pp4730ndash4732 2013

[15] B Kamsu-Foguem F Rigal and FMauget ldquoMining associationrules for the quality improvement of the production processrdquoExpert Systems with Applications vol 40 no 4 pp 1034ndash10452013

[16] Y Li H Guo L Wang and J Fu ldquoA hybrid genetic-simulatedannealing algorithm for the location-inventory- routing prob-lem considering returns under E-supply chain environmentrdquoThe Scientific World Journal vol 2013 Article ID 125893 10pages 2013

[17] Y-S P Chiu K-K Chen F-T Cheng and C-K Ting ldquoReex-amination of ldquocombining an alternative multi-delivery policyinto economic production lot size problemwith partial reworkrdquousing an alternative approachrdquo Journal of Applied Research andTechnology vol 11 no 3 pp 317ndash323 2013

[18] L B Schwarz B L Deuermeyer and R D Badinelli ldquoFill-rateoptimization in a one-warehouse 119873-identical retailer distribu-tion systemrdquo Management Science vol 31 no 4 pp 488ndash4981985

[19] R A Sarker and L R Khan ldquoOptimal batch size for a pro-duction system operating under periodic delivery policyrdquoCom-puters amp Industrial Engineering vol 37 no 4 pp 711ndash730 1999

[20] N Comez K E Stecke and M Cakanyildirim ldquoMultiple in-cycle transshipments with positive delivery timesrdquo Productionand Operations Management vol 21 no 2 pp 378ndash395 2012

[21] S W Chiu F-Y Pai and W K Wu ldquoAlternative approach todetermine the common cycle time for a multi-item productionsystem with discontinuous deliveries and failure in reworkrdquoEconomic Modelling vol 35 pp 593ndash596 2013

[22] Y-S P Chiu C-C Huang M-F Wu and H-H Chang ldquoJointdetermination of rotation cycle time and number of shipmentsfor a multi-item EPQ model with random defective raterdquoEconomic Modelling vol 35 pp 112ndash117 2013

[23] M A Hoque ldquoSynchronization in the single-manufacturermulti-buyer integrated inventory supply chainrdquo European Jour-nal of Operational Research vol 188 no 3 pp 811ndash825 2008

[24] S W Chiu L-W Lin K-K Chen and C-L Chou ldquoDetermin-ing production-shipment policy for a vendor-buyer integratedsystem with rework and an amending multi-delivery schedulerdquoEconomic Modelling vol 33 pp 668ndash675 2013

[25] M Cedillo-Campos and C Sanchez-Ramırez ldquoDynamic self-assessment of supply chains performance an emerging marketapproachrdquo Journal of Applied Research and Technology vol 11no 3 pp 338ndash347 2013

[26] Y-S P Chiu H-D Lin F-T Cheng and M-H Hwang ldquoOpti-mal common cycle time for a multi-item production systemwith discontinuous delivery policy and failure in reworkrdquoJournal of Scientific and Industrial Research vol 72 no 7 pp435ndash440 2013

[27] HHishamuddin R A Sarker andD Essam ldquoA recoverymech-anism for a two echelon supply chain system under supplydisruptionrdquo Economic Modelling vol 38 pp 555ndash563 2014

[28] M Murugan and V Selladurai ldquoProductivity improvement inmanufacturing submersible pump diffuser housing using leanmanufacturing systemrdquo Journal of Engineering Research vol 2no 1 pp 164ndash182 2014

[29] L Wang H Qu S Liu and C-X Dun ldquoModeling and opti-mization of the multiobjective stochastic joint replenishmentand delivery problem under supply chain environmentrdquo TheScientific World Journal vol 2013 Article ID 916057 11 pages2013

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 The Scientific World Journal

5 Concluding Remarks

Two exact models for an extended EPQ-based problem witha discontinuous delivery policy scrap rate and randombreakdown are developed in this study They specificallyaddress different real-life situations in production end-item delivery and intrasupply chains such as a producer-retailer integrated systemMathematicalmodeling alongwithoptimization techniques is used to determine the optimalproduction run times that minimize the expected systemcosts per unit time Without in-depth investigations onthese separate models the optimal production run time andother important information related to the systemparameterscannot be revealed The proposed real-life EPQ models withrandom machine breakdown discontinuous product distri-bution policies and quality assurance must be specificallystudied in order to (1) obtain the joint effects of breakdowndiscontinuous distribution policies and quality assurance onthe optimal production run time (2) get to know the effectsof different policy and scope of supply chains managementon the optimal run time and overall system costs and (3)gain the insight with regard to various systemrsquos parametersof all particular EPQ-based models Since little attention hasbeen paid to the investigation of joint effects of these practicalproduction situations on the optimal run time this researchis intended to bridge the gap An interesting area for futurestudy is the examination of the effect of variable productionrates on these models

Appendix

Derivations of (3) are as followsRecall (2) as follows

TC1(1199051) = 119862 (119875

11199051) + 119870 + 119862

119878(11990511198751119909) +119872 + 119899119870

1

+ 119862119879[11990511198751] + ℎ3(120582119905119903) 1198791015840

+ ℎ[1198671015840+ 11988911199051

21199051+ (1198671015840

1+ 1198891119905) 119905119903+119899 minus 1

211989911986710158401199051015840

2]

(A1)

Substituting all related system parameters into (2) (pleaserefer to the basic formulations and solution process in [9])the TC

1(1199051) can be obtained as

TC1(1199051) = 119870 +119872

+[1198621198751+ 1198621198781198751119909 + 119862

1198791198751(1 minus 119909)+ ℎ

31198751119892 (1 minus 119909)] 119905

1

+ 1198991198701+ ℎ119901119892119905 minus[

ℎ1198751119892 (1 minus 119909)

2minusℎ1198751119892 (1 minus 119909)

2119899] 1199051

+ 1199052

1[ℎ1198751

2+ℎ1198752

1

2120582(1 minus 119909)

2minusℎ1198751

2(1 minus 119909)

minusℎ1198752

1

2120582119899(1 minus 119909)

2+ℎ1198751

2119899(1 minus 119909)]

(A2)

To take the randomness of 119909 into account by using theexpected values of 119909 with further derivations119864[TC

1(1199051)] can

be derived as follows (ie equation (3))119864 [TC

1(1199051)]

= 119870 + 1198991198701+119872 + ℎ119905119875

1119892

+ [1198621198751+ 1198621198781198751119864 [119909] + 1198621198791198751 (1 minus 119864 [119909])

+ ℎ31198751119892 (1 minus 119864 [119909]) minus

ℎ1198751119892 (1 minus 119864 [119909])

2(1 minus

1

119899)] sdot 1199051

+ [ℎ1198751119864 [119909]

2+ℎ1198752

1

2120582(1 minus 119864 [119909])

2(1 minus

1

119899)

+ℎ1198751

2119899(1 minus 119864 [119909])] 119905

2

1

(A3)

Conflict of Interests

The authors of the paper declare that there is no conflict ofinterests regarding the publication of this paper

Acknowledgment

The authors greatly appreciate the National Science Council(NSC) of Taiwan for supporting this research underGrant noNSC 102-2410-H-324-005

References

[1] E W Taft ldquoThemost economical production lotrdquo Iron Age vol101 pp 1410ndash1412 1918

[2] G Hadley and T M Whitin ldquoAn optimal final inventorymodelrdquoManagement Science vol 7 pp 179ndash183 1961

[3] E A Silver D F Pyke and R Peterson Inventory Managementand Production Planning and Scheduling John Wiley amp SonsNew York NY USA 1998

[4] S Nahmias Production amp Operations Analysis McGraw-HillNew York NY USA 2009

[5] MWidmer and P Solot ldquoDo not forget the breakdowns and themaintenance operations in FMSdesign problemsrdquo InternationalJournal of Production Research vol 28 pp 421ndash430 1990

[6] K-Y C Yu and D L Bricker ldquoAnalysis of a markov chainmodel of a multistage manufacturing system with inspectionrejection and reworkrdquo IIE Transactions vol 25 no 1 pp 109ndash112 1993

[7] H Groenevelt L Pintelon and A Seidmann ldquoProduction lotsizing with machine breakdownsrdquoManagement Science vol 38no 1 pp 104ndash123 1992

[8] G A Widyadana and H M Wee ldquoOptimal deterioratingitems production inventory models with random machinebreakdown and stochastic repair timerdquo Applied MathematicalModelling vol 35 no 7 pp 3495ndash3508 2011

[9] S W Chiu C-L Chou and W-K Wu ldquoOptimizing replenish-ment policy in an EPQ-based inventory model with noncon-forming items and breakdownrdquo EconomicModelling vol 35 pp330ndash337 2013

The Scientific World Journal 13

[10] A M Zargar ldquoEffect of rework strategies on cycle timerdquoComputers amp Industrial Engineering vol 29 no 1ndash4 pp 239ndash243 1995

[11] P Biswas and B R Sarker ldquoOptimal batch quantity modelsfor a lean production system with in-cycle rework and scraprdquoInternational Journal of Production Research vol 46 no 23 pp6585ndash6610 2008

[12] S W Chiu H-D Lin C-B Cheng and C-L Chung ldquoOptimalproduction-shipment decisions for the finite production ratemodel with scraprdquo International Journal for Engineering Mod-elling vol 22 no 1ndash4 pp 25ndash34 2009

[13] Y-S P Chiu K-K Chen and C-K Ting ldquoReplenishment runtime problem with machine breakdown and failure in reworkrdquoExpert Systems with Applications vol 39 no 1 pp 1291ndash12972012

[14] H-D Lin F-Y Pai and S W Chiu ldquoA note on ldquointra-supply chain system with multiple sales locations and qualityassurancerdquordquo Expert Systems with Applications vol 40 no 11 pp4730ndash4732 2013

[15] B Kamsu-Foguem F Rigal and FMauget ldquoMining associationrules for the quality improvement of the production processrdquoExpert Systems with Applications vol 40 no 4 pp 1034ndash10452013

[16] Y Li H Guo L Wang and J Fu ldquoA hybrid genetic-simulatedannealing algorithm for the location-inventory- routing prob-lem considering returns under E-supply chain environmentrdquoThe Scientific World Journal vol 2013 Article ID 125893 10pages 2013

[17] Y-S P Chiu K-K Chen F-T Cheng and C-K Ting ldquoReex-amination of ldquocombining an alternative multi-delivery policyinto economic production lot size problemwith partial reworkrdquousing an alternative approachrdquo Journal of Applied Research andTechnology vol 11 no 3 pp 317ndash323 2013

[18] L B Schwarz B L Deuermeyer and R D Badinelli ldquoFill-rateoptimization in a one-warehouse 119873-identical retailer distribu-tion systemrdquo Management Science vol 31 no 4 pp 488ndash4981985

[19] R A Sarker and L R Khan ldquoOptimal batch size for a pro-duction system operating under periodic delivery policyrdquoCom-puters amp Industrial Engineering vol 37 no 4 pp 711ndash730 1999

[20] N Comez K E Stecke and M Cakanyildirim ldquoMultiple in-cycle transshipments with positive delivery timesrdquo Productionand Operations Management vol 21 no 2 pp 378ndash395 2012

[21] S W Chiu F-Y Pai and W K Wu ldquoAlternative approach todetermine the common cycle time for a multi-item productionsystem with discontinuous deliveries and failure in reworkrdquoEconomic Modelling vol 35 pp 593ndash596 2013

[22] Y-S P Chiu C-C Huang M-F Wu and H-H Chang ldquoJointdetermination of rotation cycle time and number of shipmentsfor a multi-item EPQ model with random defective raterdquoEconomic Modelling vol 35 pp 112ndash117 2013

[23] M A Hoque ldquoSynchronization in the single-manufacturermulti-buyer integrated inventory supply chainrdquo European Jour-nal of Operational Research vol 188 no 3 pp 811ndash825 2008

[24] S W Chiu L-W Lin K-K Chen and C-L Chou ldquoDetermin-ing production-shipment policy for a vendor-buyer integratedsystem with rework and an amending multi-delivery schedulerdquoEconomic Modelling vol 33 pp 668ndash675 2013

[25] M Cedillo-Campos and C Sanchez-Ramırez ldquoDynamic self-assessment of supply chains performance an emerging marketapproachrdquo Journal of Applied Research and Technology vol 11no 3 pp 338ndash347 2013

[26] Y-S P Chiu H-D Lin F-T Cheng and M-H Hwang ldquoOpti-mal common cycle time for a multi-item production systemwith discontinuous delivery policy and failure in reworkrdquoJournal of Scientific and Industrial Research vol 72 no 7 pp435ndash440 2013

[27] HHishamuddin R A Sarker andD Essam ldquoA recoverymech-anism for a two echelon supply chain system under supplydisruptionrdquo Economic Modelling vol 38 pp 555ndash563 2014

[28] M Murugan and V Selladurai ldquoProductivity improvement inmanufacturing submersible pump diffuser housing using leanmanufacturing systemrdquo Journal of Engineering Research vol 2no 1 pp 164ndash182 2014

[29] L Wang H Qu S Liu and C-X Dun ldquoModeling and opti-mization of the multiobjective stochastic joint replenishmentand delivery problem under supply chain environmentrdquo TheScientific World Journal vol 2013 Article ID 916057 11 pages2013

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

The Scientific World Journal 13

[10] A M Zargar ldquoEffect of rework strategies on cycle timerdquoComputers amp Industrial Engineering vol 29 no 1ndash4 pp 239ndash243 1995

[11] P Biswas and B R Sarker ldquoOptimal batch quantity modelsfor a lean production system with in-cycle rework and scraprdquoInternational Journal of Production Research vol 46 no 23 pp6585ndash6610 2008

[12] S W Chiu H-D Lin C-B Cheng and C-L Chung ldquoOptimalproduction-shipment decisions for the finite production ratemodel with scraprdquo International Journal for Engineering Mod-elling vol 22 no 1ndash4 pp 25ndash34 2009

[13] Y-S P Chiu K-K Chen and C-K Ting ldquoReplenishment runtime problem with machine breakdown and failure in reworkrdquoExpert Systems with Applications vol 39 no 1 pp 1291ndash12972012

[14] H-D Lin F-Y Pai and S W Chiu ldquoA note on ldquointra-supply chain system with multiple sales locations and qualityassurancerdquordquo Expert Systems with Applications vol 40 no 11 pp4730ndash4732 2013

[15] B Kamsu-Foguem F Rigal and FMauget ldquoMining associationrules for the quality improvement of the production processrdquoExpert Systems with Applications vol 40 no 4 pp 1034ndash10452013

[16] Y Li H Guo L Wang and J Fu ldquoA hybrid genetic-simulatedannealing algorithm for the location-inventory- routing prob-lem considering returns under E-supply chain environmentrdquoThe Scientific World Journal vol 2013 Article ID 125893 10pages 2013

[17] Y-S P Chiu K-K Chen F-T Cheng and C-K Ting ldquoReex-amination of ldquocombining an alternative multi-delivery policyinto economic production lot size problemwith partial reworkrdquousing an alternative approachrdquo Journal of Applied Research andTechnology vol 11 no 3 pp 317ndash323 2013

[18] L B Schwarz B L Deuermeyer and R D Badinelli ldquoFill-rateoptimization in a one-warehouse 119873-identical retailer distribu-tion systemrdquo Management Science vol 31 no 4 pp 488ndash4981985

[19] R A Sarker and L R Khan ldquoOptimal batch size for a pro-duction system operating under periodic delivery policyrdquoCom-puters amp Industrial Engineering vol 37 no 4 pp 711ndash730 1999

[20] N Comez K E Stecke and M Cakanyildirim ldquoMultiple in-cycle transshipments with positive delivery timesrdquo Productionand Operations Management vol 21 no 2 pp 378ndash395 2012

[21] S W Chiu F-Y Pai and W K Wu ldquoAlternative approach todetermine the common cycle time for a multi-item productionsystem with discontinuous deliveries and failure in reworkrdquoEconomic Modelling vol 35 pp 593ndash596 2013

[22] Y-S P Chiu C-C Huang M-F Wu and H-H Chang ldquoJointdetermination of rotation cycle time and number of shipmentsfor a multi-item EPQ model with random defective raterdquoEconomic Modelling vol 35 pp 112ndash117 2013

[23] M A Hoque ldquoSynchronization in the single-manufacturermulti-buyer integrated inventory supply chainrdquo European Jour-nal of Operational Research vol 188 no 3 pp 811ndash825 2008

[24] S W Chiu L-W Lin K-K Chen and C-L Chou ldquoDetermin-ing production-shipment policy for a vendor-buyer integratedsystem with rework and an amending multi-delivery schedulerdquoEconomic Modelling vol 33 pp 668ndash675 2013

[25] M Cedillo-Campos and C Sanchez-Ramırez ldquoDynamic self-assessment of supply chains performance an emerging marketapproachrdquo Journal of Applied Research and Technology vol 11no 3 pp 338ndash347 2013

[26] Y-S P Chiu H-D Lin F-T Cheng and M-H Hwang ldquoOpti-mal common cycle time for a multi-item production systemwith discontinuous delivery policy and failure in reworkrdquoJournal of Scientific and Industrial Research vol 72 no 7 pp435ndash440 2013

[27] HHishamuddin R A Sarker andD Essam ldquoA recoverymech-anism for a two echelon supply chain system under supplydisruptionrdquo Economic Modelling vol 38 pp 555ndash563 2014

[28] M Murugan and V Selladurai ldquoProductivity improvement inmanufacturing submersible pump diffuser housing using leanmanufacturing systemrdquo Journal of Engineering Research vol 2no 1 pp 164ndash182 2014

[29] L Wang H Qu S Liu and C-X Dun ldquoModeling and opti-mization of the multiobjective stochastic joint replenishmentand delivery problem under supply chain environmentrdquo TheScientific World Journal vol 2013 Article ID 916057 11 pages2013

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