This article was downloaded by: [Gazi University]On: 03 October 2014, At: 04:56Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office:Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK

International Journal of Production ResearchPublication details, including instructions for authors and subscriptioninformation:http://www.tandfonline.com/loi/tprs20

The impact of quality considerations onmaterial flow in two-stage inventory systemsMoutaz Khouja aa Business Information Systems and Operations Management Department,The Belk College of Business Administration , The University of NorthCarolina at Charlotte , Charlotte, NC, 28223, USA E-mail:Published online: 14 Nov 2010.

To cite this article: Moutaz Khouja (2003) The impact of quality considerations on material flow intwo-stage inventory systems, International Journal of Production Research, 41:7, 1533-1547, DOI:10.1080/0020754031000069616

To link to this article: http://dx.doi.org/10.1080/0020754031000069616

PLEASE SCROLL DOWN FOR ARTICLE

Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”)contained in the publications on our platform. However, Taylor & Francis, our agents, and ourlicensors make no representations or warranties whatsoever as to the accuracy, completeness, orsuitability for any purpose of the Content. Any opinions and views expressed in this publicationare the opinions and views of the authors, and are not the views of or endorsed by Taylor &Francis. The accuracy of the Content should not be relied upon and should be independentlyverified with primary sources of information. Taylor and Francis shall not be liable for anylosses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilitieswhatsoever or howsoever caused arising directly or indirectly in connection with, in relation to orarising out of the use of the Content.

This article may be used for research, teaching, and private study purposes. Any substantialor systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, ordistribution in any form to anyone is expressly forbidden. Terms & Conditions of access and usecan be found at http://www.tandfonline.com/page/terms-and-conditions

int. j. prod. res., 2003, vol. 41, no. 7, 1533–1547

The impact of quality considerations on material flow in two-stage

inventory systems

MOUTAZ KHOUJAy

Supply chain management literature calls for coordination between the differentmembers of the chain. Inventory models achieve this coordination along a supplychain by making the lot size at an upstream entity an integer multiplier of the lotsize at the adjacent downstream entity. Such models assume that all componentsproduced are of acceptable quality and may cause suppliers to produce largerquantities than what is optimal. In this paper, we formulate and solve two-stagesupply chain inventory models in which the proportion of defective productsincreases with increased production lot sizes. We show that quality considerationscan lead to significant reduction in production lot sizes. In addition, the modelsshow that most benefits to the supply chain are attained from the suppliersproducing on a just-in-time basis rather than delivering to their customers just-in-time. We derive closed-form expressions for the optimal lot sizes for a two-stage supply chain under deterministic and then stochastic demand and illustratethe models with numerical examples.

1. Introduction

Efficient and effective management of material flows across a supply chain is

critical to the chain’s success (Handfield and Nichols 1999, Simchi-Levi et al.

2000). To coordinate material flow in supply chains, inventory models make the

lot size at an upstream entity of a supply chain an integer multiplier of the lot size

at the adjacent downstream entity. This allows upstream stages to enjoy the cost

savings of the larger cycle times justified by their small unit holding cost of the low

value items at that stage (Sharafali and Co 2000). Such models assume that all

components produced are of acceptable quality. However, the assumption of perfect

quality is unrealistic for many manufacturing systems (Hall 1987). Inman (1994)

surveyed 114 manufacturing firms and found that reductions in lot sizes were

accompanied by a reduction in scrap and rework. Kekre and Mukhopadhyay

(1992) also empirically asserted the negative relationship between levels of inventory

and quality in their study of the impact of electronic data interchange (EDI) on US

manufacturing companies. Urban (1998), based on a review of the literature,

described two reasons for why product quality may improve with smaller lot sizes.

(1) Quicker identification of defects. Smaller lot sizes implies that inventory isprocessed faster and if there are quality problems they are corrected earlier(Schonberger 1982).

International Journal of Production Research ISSN 0020–7543 print/ISSN 1366–588X online # 2003 Taylor & Francis Ltd

http://www.tandf.co.uk/journals

DOI: 10.1080/0020754031000069616

Revision received September 2002.{Business Information Systems and Operations Management Department, The Belk

College of Business Administration, The University of North Carolina at Charlotte,Charlotte, NC 28223, USA. e-mail: mjkhouja@email.uncc.edu

Dow

nloa

ded

by [

Gaz

i Uni

vers

ity]

at 0

4:56

03

Oct

ober

201

4

(2) Learning curve effects. Smaller lot sizes lead to shorter durations betweenproduction runs and less need for recall on the part of the workers of how toperform tasks (Fogarty et al. 1991).

Rosenblatt and Lee (1986) and Porteus (1986) added a third reason—an increased

probability of the system going out of control as it is operated for longer duration

without adjustment. The increased probability can be a result of tool wear, operator

error, calibration measurement errors, etc.

The need to incorporate quality considerations into lot sizing decisions has been

a central theme in just-in-time (JIT) inventory management. For example,

Schonberger (1982), in his discussion of JIT’s benefits, states:

The advantage may seem small—some savings on inventory carrying costs, since

you produce and carry smaller lots. But the Japanese have found that the main

benefits are in quality, worker, motivation, and productivity.

The above relationship between lot size and quality seems to be largely ignored in

two-stage inventory models. Furthermore, empirical studies indicate that it is also

ignored by many retailers and manufacturers as they have pushed their suppliers to

stockpile inventory at their facilities and deliver it in small lots (Landry et al. 1998).

In a study of the auto industry, Helper (1991) found that only 48% of suppliers used

JIT production with their JIT delivery. The rest of the suppliers stockpiled inventory

to meet their customers’ JIT requirements. This approach to JIT, in which suppliers

carry large amounts of inventory, was evident in a survey in which more than half of

all suppliers questioned in the United States agreed with the statement that JIT only

transfers inventory responsibility from customers to suppliers (Helper and Sako

1995). Therefore, ignoring the relationship between quality and lot size has

contributed to overestimating the optimal production lot sizes in inventory

models. Furthermore, it may have led to an industrial practice that has the appear-

ance of JIT, owing to frequent small shipments from suppliers to retailers, but fails

to realize the full benefits JIT provides.

Erenguc et al. (1999) describe a successful JIT implementation as one in which

both the buyers’ and suppliers’ inventory holdings are reduced and suppliers become

more competitive as JIT suppliers. For the full benefits of JIT to be realized, JIT

deliveries should be matched with JIT production rather than producers stockpiling

inventory to meet their customers’ JIT delivery requirements. The main thesis behind

having suppliers become JIT producers is that inventory hides problems and, by

reducing inventory, quality problems are discovered and solved (Trevino et al. 1993,

Cordon 1995). Given the above arguments asserting the negative relationship

between inventory and quality, it is essential that supply chain inventory models

incorporate this relationship.

The main measure of quality that has been used in the literature, and will be used

here, is quality of conformance—which measures the degree to which a product

meets the design specifications. When a product does not meet the specifications it

is considered defective and needs rework or becomes scrap (Porteus 1986, Rosenblatt

and Lee 1986). Therefore, we use the number of conforming products in a lot as a

measure of quality.

To analyse the effects of imperfect quality on inventory decisions in supply

chains, we focus on a buyer–vendor system in which the buyer is a retailer and

the vendor is a producer. Thomas and Griffin (1996), in their review of coordination

1534 M. Khouja

Dow

nloa

ded

by [

Gaz

i Uni

vers

ity]

at 0

4:56

03

Oct

ober

201

4

in supply chains, classify supply chain operational models into three categories: (1)buyer–vendor coordination, (2) production–distribution coordination, and (3)inventory–distribution coordination. The buyer–vendor focus enables us to analysethe model from the perspective of one entity that performs the production (i.e. theproducer), and one that only sells the product (i.e. the retailer).

The goal of this paper is to formulate and solve a two-stage supply chaininventory model in which the quality of the output deteriorates with increased lotsizes. The number of acceptable quality items in a lot is used to measure quality. Theproposed models show that the inclusion of rework cost can lead to a large reductionin production lot sizes and can cause the suppliers to produce according to JIT ratherthan only deliver to the retailer just-in-time. Therefore, this research makes adistinction between JIT delivery and JIT suppliers. JIT delivery means the supplierdelivers the inventory to the retailer on JIT basis, but does not mean the supplierproduces on JIT basis. He/she may hold the extra inventory. A JIT supplier producesthe product on a JIT basis and does not carry extra inventory, which constitutes atrue JIT system. Another interesting result from the model is that the incorporationof quality into the lot decision may cause the retailer to order in larger quantities.While this seems to contradict JIT, a closer look shows that this increase at theretailer’s end is accompanied by a decreased lot size (i.e. smaller integer multiplier)at the supplier’s end and less average inventory in the system.

The remainder of this paper is organized into five sections. In section 2, weformulate and solve a producer–retailer model under quality deterioration using atwo-state Markovian production process and deterministic demand. In section 3, weuse a constant failure rate production process. In section 4, we deal with the case ofstochastic demand. Section 5 contains a discussion, and section 6 has some conclu-sions and suggestions for future research.

2. Producer-retailer with a two-state Markov production process

The following assumptions are made:

(a) the producer produces the product on a single machine,(b) production and usage rates are deterministic and constant,(c) both the producer and the retailer incur linear holding costs on inventory,

and(d) materials are inspected and reworked if needed at the producer’s site.

Several assumptions have been made in the literature with regard to the beha-viour of producers and retailers. Banerjee (1986) assumed that the order quantity ofthe retailer and the production lot size of the producer are equal and found the jointoptimal lot size. Goyal (1988) allowed the producer’s lot size to be an integer multi-plier of the retailer’s order quantity, assumed that the whole lot is produced beforethe first shipment is made to the retailer, and the shipments are of equal size. Lu(1995) allowed the first shipment to be made before the whole lot is produced at theproducer but kept the equal shipment size assumption. Goyal (1995) also allowed thefirst shipment to be made before the whole lot is produced, and incorporated a policyin which the size of successive shipments from the producer to the retailer within aproduction cycle increases by a factor equal to the ratio of the production to thedemand rates. Hill (1997) showed that neither the equal shipment size nor theincreasing by a factor of the ratio of the production to the demand rates arealways optimal. Rather, these are two extremes on a continuum. Hill provided a

1535Quality considerations on material flow

Dow

nloa

ded

by [

Gaz

i Uni

vers

ity]

at 0

4:56

03

Oct

ober

201

4

generalized policy for the factor by which to increase shipment sizes. Viswanathan

(1998) identified problem parameters under which each policy extreme is optimal.

Goyal (2000) suggested an improvement over Hill’s generalized policy. Goyal and

Nebebe (2000) introduced a policy in which the first shipment has a small size and

the next ðn� 1Þ shipments are equal to the size of the first shipment multiplied by the

ratio of production to the demand rate. To analyse the impact of incorporating

quality deterioration into the supply chain, we focus on the simpler policy in

which the entire lot is produced before the first shipment of an n equal-size shipments

is made (Goyal 1988).

One of the earliest models incorporating the relationship between quality and

inventory in a single-stage inventory system was proposed by Porteus (1986). The

model focused on a single producer and used the economic order quantity (EOQ)

framework. Porteus assumed the production process to be functioning perfectly at

the start of production. With the production of each unit, the process may shift out-

of-control with a constant known transition probability, and start producing all

defective units. Once the process is out of control, it stays that way while the

remainder of the lot is produced. The production system is restored to perfect quality

when it is set up again. Based on this assumption, the number of conforming units in

a lot is a random variable that depends on the transition probability and the lot size.

The author then derived the optimal lot size. The reasons the process may shift out

of control can be tool wear, operator error, calibration measurement errors, etc. The

longer the process is operated without readjusting (i.e. the larger the lot size), the

more likely that the shift will occur during the production cycle. Porteus gave three

interpretations for the above assumption on process quality and the inspection

process. The first interpretation is that the firm inspects only the first and last

pieces of a component, as suggested by Hall (1983), and if the last piece is good

then the entire lot is judged to be good. The second interpretation is that the type of

defect cannot be identified until the product is processed on the next station and the

entire lot is produced before it is moved to the next station. The third interpretation

is that even though the process is continuously monitored, there is a delay between

the time the process shifts out of control and the time the operators can determine

the process is out of control. The monitoring is still useful in determining which

components need rework. Rosenblatt and Lee (1986) assumed that the elapsed time

until the process shifts out of control to be exponentially distributed and once the

process shifts out-of-control, it starts to produce � percent of defective products. The

authors also examined the cases of linear and exponential deterioration in terms of

the percentage of defectives and found the constant case to be a good approxima-

tion. Cheng (1991) solved an extension to the economic order quantity model in

which demand exceeds supply, quality is imperfect, and the amount of demand to be

satisfied is a decision variable. Khouja and Mehrez (1994) formulated an economic

production lot size model that treats production rates as decision variables and

assumes the percentage of conforming components in a lot decreases as the

production rate increases. The authors used the quality assumptions of Rosenblatt

and Lee (1986). Cordon (1995) analysed the interactions between quality defaults

and work-in-process inventory in a system with two uncapacitated stations in

tandem, with the first station going out of control with a certain probability and

with Markovian arrivals. Khouja (1999) examined the effect of imperfect quality,

which deteriorates with increased lot sizes and production rates within a family

1536 M. Khouja

Dow

nloa

ded

by [

Gaz

i Uni

vers

ity]

at 0

4:56

03

Oct

ober

201

4

production context. A review of the literature on the relationship between inventoryand quality can be found in Wright and Mehrez (1998).

We consider a simple supply chain consisting of a supplier who produces aproduct and delivers it to a retailer who in turn sells it to the final customer.

Notation

D annual demand,P annual production rate,Ss set-up cost at the producer/supplier,Sr ordering cost for the retailer,hs annual inventory holding cost per unit for the producer,hr annual inventory holding cost per unit for the retailer,W cost to rework one unit for the producer,Qs the optimal production lot size for the producer,Qr the optimal order quantity for the retailer,n an integer multiplier,

TCs the total annual cost for the producer,TCr the total annual cost at for retailer, andTCj ¼ TCs þ TCr; the total annual cost for both producer and retailer.

Under the assumption of perfect quality, Goyal (1988) showed that the total annualcosts are:

TCr ¼D

Qr

Sr þQr

2hr; ð1Þ

TCs ¼D

nQr

Ss þQr

2hs

�n

�1 þD

P

�� 1

�; ð2Þ

TCj ¼D

Qr

ðSr þ Ss=nÞ þQr

2

�hr � hs þ hsn

�1 þD

P

��ð3Þ

and the joint optimal solution satisfies:

n*ðn* � 1Þ � hr � hsSrhsð1 þD=PÞ � n*ðn* þ 1Þ; ð4Þ

Qr*ðnÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2DðnSr þ SsÞ

nðhr � hs þ nhsð1 þD=PÞÞ

s: ð5Þ

We begin by using the model assumptions of Porteus (1986). For small values ofthe transition probability q , Porteus (1986) showed that EðUÞ ¼ Q� �ð1 � �QÞ=q ,where U is the number of defectives per lot of size Q and � ¼ 1 � q . Porteus showedthat EðUÞ is a strictly increasing, strictly convex function of Q . The expectedproportion of defectives is given by EðUÞ=Q ¼ ðQ� �ð1 � �QÞ=qÞ=Q and thedirect rework cost per unit time is DW ½ðQ� �ð1 � �QÞ=qÞ=Q . If q is close tozero, then Porteus (1986) showed that ðQ� �ð1 � �QÞ=qÞ=Q 1

2qQ . Thus, the

total expected annual cost, including rework cost, for the producer becomes:

1537Quality considerations on material flow

Dow

nloa

ded

by [

Gaz

i Uni

vers

ity]

at 0

4:56

03

Oct

ober

201

4

TCs ¼D

nQr

Ss þQr

2hs

�n

�1 þD

P

�� 1

�þ 1

2qnQrDW ; ð6Þ

and the expected total annual cost for the producer and retailer is

TCj ¼D

Qr

ðSr þ Ss=nÞ þQr

2

�hr � hs þ hsn

�1 þD

P

��þ 1

2qnQrDW : ð7Þ

If the producer and retailer minimize their total annual costs independently then theoptimal order quantity and minimum total annual cost for the retailer are:

Qr* ¼

ffiffiffiffiffiffiffiffiffiffi2DA

hr

s; ð8Þ

TCr* ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffi2DAhr

p: ð9Þ

Using equation (8) in (6) and minimizing gives the following optimality condition forthe producer:

n*ðn* � 1Þ � hrSsSrðDWqþ ð1 þD=PÞhsÞ

� n*ðn* þ 1Þ: ð10Þ

If the expected total annual cost for the whole supply chain is minimized (i.e. jointoptimization), then the optimality conditions are given by:

n*ðn* � 1Þ � ðhr � hsÞSsSrðDWqþ ð1 þD=PÞhsÞ

� n*ðn* þ 1Þ; ð11Þ

Qr*ðNÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ðnSr þ SsÞ

nðnWqþ hr=Dþ ðn=Pþ ðn� 1Þ=DÞhSÞ

s: ð12Þ

Numerical example 1

Consider the example used by Banerjee (1986) and Goyal (1988) in whichD ¼ 1000 units/year, P ¼ 3200 units/year, Sr ¼ $25 /order, Ss ¼ $400/set-up,hr ¼ $5/unit/year, and hs ¼ $4/unit/year. In addition, suppose that the rework costand the transition probability are W ¼ $20/unit and q ¼ 0:001, repectively. Table 1shows three solutions to the problem. Solution 1 optimizes the whole supply chainbut without incorporating quality (equations (4) and (5)). In solution 2, quality is

1538 M. Khouja

Solution 1:Joint optimization

perfect quality

Solution 2:Independent optimization

imperfect quality

Solution 3:Joint optimizationimperfect quality

Qr* 197.8 100 244.2

n* 2 3 1Qs

* 395.6 300 244.2TCr, $ 620.9 500 712.9TCs, $(excluding rework cost)

1653.9 1920.8 1790.4

Annual rework cost, $ 1582.5 1200.0 976.9TCj , $ 3857.38 3620.8 3480.2

Table 1. Solution to numerical example under different scenarios and deterministic demand.

Dow

nloa

ded

by [

Gaz

i Uni

vers

ity]

at 0

4:56

03

Oct

ober

201

4

incorporated, but each member independently optimizes its own position (equations(8) and (10)). In solution 3, quality is incorporated and the whole chain is optimized(equations (11) and (12)). The incorporation of quality has a significant impact onboth independent optimization and joint optimization. Incorporating quality insolution 3 causes the producer to become a true JIT producer producing what isneeded, in the quantity it is needed, when it is needed.

3. Producer-retailer with a constant failure rate

In this section, we use the assumptions of Rosenblatt and Lee (1986) in which theelapsed time until the process shifts out of control is exponentially distributed withmean of 1/� and once the process shifts out-of-control, it starts to produce � percentof defective products. Using McClaurin series expansion, the authors show that theexpected number of defectives per lot of size Q is ��Q2=2P . Therefore, for a supplierlot size of Qs ¼ nQr the expected number of defectives is ��ðnQrÞ2=2P and theexpected total annual cost for both the producer and retailer is:

TCj ¼D

Qr

ðSr þ Ss=nÞ þQr

2

�hr � hs þ hsn

�1 þD

P

��þD��WnQr

2P: ð13Þ

If the expected total annual cost for the whole supply chain is minimized (i.e. jointoptimization), then the optimality conditions are given by:

n*ðn* � 1Þ � ðhr � hsÞSsSrðDW��=Pþ ð1 þD=PÞhsÞ

� n*ðn* þ 1Þ; ð14Þ

Qr*ðnÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ðnSr þ SsÞ

nðnW��=Pþ hr=Dþ ðn=Pþ ðn� 1Þ=DÞhsÞ

s: ð15Þ

Returning to the previous example but using � ¼ 3 and � ¼ 0:25 givesQr

* ¼ Qs* ¼ 278 with TCj* ¼ $3049/year. Ignoring rework in the analysis gives

Qr* ¼ 198 , n ¼ 2 , Qs

* ¼ 396 and TCj* ¼ $2275/year. However, there is an additional$927/year in rework cost resulting in an actual expected total annual cost of $3202/year.

4. Producer-retailer with a two-state Markov process and stochastic demand

To investigate the effect of quality consideration under stochastic demand, weincorporate rework into a continuous-review (Q;R) inventory model (Silver et al.1998). Under (Q;R) policy, a retailer orders Qr units when inventory position falls toR units. The Qr units arrive after a fixed deterministic lead-time has elapsed.Shortages during lead-time are treated as lost sales, backorders to be satisfiedwhen the order arrives, or a combination of both (Silver et al. 1998). We assumeshortages are backordered and that the per unit backorder cost is independent of thelength of time the backordered units are held on the books before they are filled. Thisassumption simplifies the analysis and allows us to focus on the effects of quality.The assumption is realistic when lead-time is short and shortages do not occur earlyin the lead time.

Let B be the per unit backorder cost. Then the expected total annual cost for theretailer is:

1539Quality considerations on material flow

Dow

nloa

ded

by [

Gaz

i Uni

vers

ity]

at 0

4:56

03

Oct

ober

201

4

TCr ¼D

Qr

ðSr þ BEfUgÞ þ hr

�Qr

2þ R� �

�: ð16Þ

The expected total annual cost for the producer is given by equation (6) and the

expected total annual cost for the chain is:

TCj ¼D

Qr

�Sr þ

Ssnþ BEfUg

�þQr

2

�hr � hs þ hsn

�1 þD

P

��

þ hrðR� �Þ þ 1

2qnQrDW : ð17Þ

Suppose lead-time demand, denoted by x, is exponentially distributed with mean

� ¼ 1=�. The expected number of units that will be backordered is

EfUg ¼Rþ1R

ðx� RÞf ðxÞdx ¼ e�R�=� and using the McClaurin series expansion,

EfUg can be approximated as:

EfUg ¼ ½ð1 � R�þ ðR�Þ2=2=�: ð18Þ

Substituting from equation (18) into (17) and simplifying gives:

TCj ¼DSsnQr

þ�Qr

2þ R� �

�hr þ

D

�Sr þ B

�1

�þ 1

2RðR�� 2Þ

��Q

þ 1

2

�K þDK

P� 1

�Qhs þ

1

2qnQrDW ð19Þ

where K is lead time. If the producer and retailer minimize their expected total

annual costs independently, then the optimal solution is given by:

Qr ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2DBð1 þ Sr�=B� R�ð1 � R�=2Þ

�hr

s; ð20Þ

R ¼1 þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffihrðBþ 2Sr�ÞBðDB�� hrÞ

s

�; ð21Þ

L* ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2PSshrðDB�� hrÞ

DBðBþ 2Sr�ÞðDPWqþ ðDþ PÞhrÞ

s; ð22Þ

n* ¼ L*b cor L*d e: ð23Þ

Both values of n* in equation (23) and the corresponding TCs* must be computed to

determine the optimal solution. If the expected total annual cost for the whole supply

chain is minimized (i.e. joint optimization), then the optimality conditions are given

by:

1540 M. Khouja

Dow

nloa

ded

by [

Gaz

i Uni

vers

ity]

at 0

4:56

03

Oct

ober

201

4

L* ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2PSshrðDB�� hr �DB�hs=hrÞ

DBðBþ 2Sr�ÞðDPWqþ ðDþ PÞhrÞ

s; ð24Þ

n* ¼ L*b c or L*d e; ð25Þ

Qr* ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiD2BPðnBþ 2ðSrnþ SsÞ�Þ

KðDBP�hr � Ph2r þDB�ðDnPWqþ ðDnþ ðn� 1ÞPÞhsÞÞ

s; ð26Þ

R* ¼ DB �QrhrDB�

: ð27Þ

Both values of n* in equation (25) and the corresponding Qr* , R* and TCj* must be

computed to determine the optimal solution.

Numerical example 2

Consider an example with D ¼ 5000 units/year, P ¼ 20000 units/year, Sr ¼ $100/

order, Ss ¼ $1000/set-up, hr ¼ $3/unit/year, hs ¼ $6/unit/year, W ¼ $6/unit and

q ¼ 0:0002. In addition, suppose that the lead time is one week, which implies

that lead time demand, which is exponentially distributed, has a mean of

� ¼ 96:15 units. Table 2 shows three solutions to the problem. Solution 1 optimizes

the whole supply chain but without incorporating quality (equations (24)–(27) with

q ¼ 0). In solution 2, quality is incorporated, but each member independently

optimizes its own position (equations (20)–(23)). In solution 3, quality is incorpo-

rated and the whole chain is optimized (equations (24)–(27) with q ¼ 0:0002). The

inclusion of quality causes the producer’s lot size to decrease for both solutions 2 and

3. However, for solution 3, the reduction is larger because, although the retailer

orders a larger quantity, the producer becomes true a JIT supplier producing what

is needed, in the quantity it is needed, when it is needed. Interestingly, the retailer’s

order quantity increases, which seems to contradict JIT. However, the average

amount of inventory in the system decreases, which is the true measure of JIT

implementation in the system.

1541Quality considerations on material flow

Solution 1:Joint optimization

perfect quality

Solution 2:Independent optimization

imperfect quality

Solution 3:Joint optimizationimperfect quality

Qr* 828.01 589.02 982.01

R 48.38 130.13 39.5n* 2 2 1Qs

* 1656 1178.04 982.01TCr, $ 3525 3738 3775TCs, $(excluding rework cost)

4882 5569 5460

Annual rework cost, $ 4968 3534 2946TCj , $ 13375 12841 12181

Table 2. Solution to numerical example under different scenarios and stochastic demand.

Dow

nloa

ded

by [

Gaz

i Uni

vers

ity]

at 0

4:56

03

Oct

ober

201

4

5. Discussion

To analyse the effects of incorporating poor quality into the two-stage model wederive an expression for the ratio of the average inventory for a system with perfectquality to the average inventory for a system with imperfect quality. To betterunderstand the effects of quality, we treat n as a continuous decision variable. Theaverage inventory in the two-stage system is:

AVINVj ¼ AVINVr þ AVINVs: ð28Þ

Substituting the appropriate values for AVINVr and AVINVs in equation (28) gives:

AVINV ¼ Q

2þQ

2

�n

�1 þD

P

�� 1

�: ð29Þ

For both perfect and imperfect quality cases, solving dTCj=dn ¼ 0 gives an expres-sion for the optimal n and substituting that expression into TCj then solvingdTCj=dQ ¼ 0 gives the expression for the optimal Q . Substituting the expressionsfor the optimal Q and n into AVINVj for both systems and taking the ratio gives:

AVINVPQ

AVINVIQ

¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 þ DP

Dþ P

W

hsq

s; ð30Þ

where the subscripts PQ and IQ denote perfect and imperfect quality, respectively.From equation (30), the average inventory for an imperfect quality system is sub-stantially smaller relative to a perfect quality system, for systems with:

(1) a large rework cost relative to a supplier’s holding cost. For these systems,ignoring quality cost causes suppliers to produce large quantities because oftheir small holding cost. When the high cost of poor quality is considered,there is a large drop in the production lot size;

(2) large production rates of suppliers. Under the assumption that no shipmentfrom a lot is made to the retailer until the whole lot is produced, largeproduction rates cause the production lot size to be large since the lot isnot held for long production durations at the supplier. When the cost ofpoor quality is considered, this cost advantage of larger production lot size issmaller and, therefore, the optimal production lot size decreases;

(3) high transition probability, which indicates the system may shift out ofcontrol sooner and therefore large production lot sizes incur higher reworkcost.

To examine the effects of imperfect quality on different problem parameters, weplot the ratio of the average inventory computed in equation (30) versus demand andtransition probability for the parameter combinations shown in table 3. In the table,

1542 M. Khouja

P=D

Low (1.5) High (5)

Low (1) LL LHW=hs High (4) HL HH

Table 3. Parameter combinations used in examining effects of poor quality on averageinventory.

Dow

nloa

ded

by [

Gaz

i Uni

vers

ity]

at 0

4:56

03

Oct

ober

201

4

the ratio of the production rate to the demand rate is set to two levels, low

(P=D ¼ 1:5) and high (P=D ¼ 5). The ratio of the unit rework cost to the unit

holding cost is also set to two levels, low (W=hs ¼ 1) and high (W=hs ¼ 4).

For both figures 1 and 2, larger values of AVINVPQ=AVINVIQ implies smaller

average inventory for a system with imperfect quality relative to a system assumed to

have perfect quality. From figure 1, the decrease in average inventory brought about

by incorporating poor quality becomes larger as demand increases. The decrease in

1543Quality considerations on material flow

1000 2000 3000 4000 5000Demand

2

4

6

8

IQ

PQ

AVINV

AVINV

LL

LH

HL

HH

Figure 1. Ratio of average inventory of perfect to imperfect quality systems versus demand,q ¼ 0:0005.

0.0001 0.0002 0.0003 0.0004 0.0005

2

4

6

8

IQ

PQ

AVINV

AVINV

Transition

Probability

LL

LH

HL

HH

Figure 2. Ratio of average inventory of perfect to imperfect quality systems versustransition probability, D ¼ 50; 000 units/year.

Dow

nloa

ded

by [

Gaz

i Uni

vers

ity]

at 0

4:56

03

Oct

ober

201

4

average inventory is more significant for cases where the rework cost is high and thesupplier holding cost is low (the HL curve) with the average inventory of an imper-fect quality system being as low as one sixth of the average inventory of a perfectquality system for annual demand of about 30 000 units. For both high ratios of theproduction rate to the demand rate and the unit rework cost to the unit holding cost(the HH curve), the average inventory of an imperfect quality system is as low as oneeighth of the average inventory of a perfect quality system for annual demand ofabout 40 000 units.

From figure 2, the decrease in average inventory brought about by incorporatingpoor quality becomes larger as the transition probability increases. Similar to figure1, the decrease in average inventory is more significant for systems in which therework cost is high and the supplier holding cost is low (the HL curve). For bothhigh ratios of the production rate to the demand rate and the unit rework cost to theunit holding cost (the HH curve), the average inventory of an imperfect qualitysystem is as low as one quarter of the average inventory of a perfect qualitysystem for a small transition probability of 0.0001.

For the supplier to be a true JIT supplier producing only the quantity needed,only when it is needed, implies n* ¼ 1 , which is true if the middle term of inequality(11) is less than or equal to 2. Setting the middle term in equation (11) to 2 yields thefollowing solution in terms of q:

qI ¼Sshr � ð2Að1 þD=PÞ þ SrÞhs

2ADW: ð31Þ

Therefore, if q � qI then n* ¼ 1 and the supply chain is completely synchronized.Dividing the middle term of inequality (4) by its counterpart in inequality (11) gives:

t ¼ 1 þ DWq

ð1 þD=PÞhs: ð32Þ

The larger the value of t, the more significant the change in the optimal integermultiplier n. If q ¼ 0 (i.e. perfect quality) then conditions (4) and (11) becomeidentical. Equation (32) indicates that the optimal integer multiplier n is morelikely to decrease when quality is considered, for problems with:

(1) large values of the rework cost and transition probability;(2) small values of the holding cost at the producer. Small values of hs result in

large optimal order quantity at the producer (which are, in part, indicated bylarge nÞ . Incorporating quality effects leads to a large rework cost because ofthe large lot size, which in turns leads to a significant reduction in theproduction lot size;

(3) large production rates at the producer.

The incorporation of poor quality does not always cause a decrease in theretailer’s order quantity. For cases in which the optimal value of the integer multi-plier decreases when poor quality is incorporated, the retailer order quantity mayincrease. To show how such an increase may occur, the expression for the ratio of theoptimal order quantity for a perfect quality system given by equation (5) to that ofan imperfect quality system given by equation (12) can be used:

QPQ

QIQ

¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffinIQðPhr þDPWqnIQ þ hsððDþ PÞnIQ � PÞÞðSs þ SrnPQÞ

nPQðSs þ SrnPQÞðPhr þ hsððDþ PÞnPQ � PÞÞ

s: ð33Þ

1544 M. Khouja

Dow

nloa

ded

by [

Gaz

i Uni

vers

ity]

at 0

4:56

03

Oct

ober

201

4

If the optimal integer multiplier is the same for both cases, then substitutingnPQ ¼ nIQ into equation (33) and simplifying gives:

QPQ

QIQ

¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 þ

DPWqnIQ

Phr þ hsððDþ PÞnPQ � PÞ

s; ð34Þ

which implies that if the integer multiplier remains the same, the inventory reductionis obtained by reducing the retailer’s order quantity, which also leads to a reductionin the supplier production lot size. For the case in which there is a reduction in theoptimal integer multiplier, we examine a scenario in which nIQ ¼ nPQ � 1 andnPQ ¼ 2, which when substituted in equation (33) give:

QPQ

QIQ

¼ 0:707

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 þ Sr

Sr þ Ss

s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 þDPWq� hsðDþ PÞ

Phr þ hsð2Dþ PÞ

s: ð35Þ

For many realistic values of problem parameters, equation (35) yields values lessthan one. Therefore, the retailer’s order quantity is larger when poor quality is takeninto account. This behaviour is explained by the fact that the retailer is increasing theorder quantity to synchronize with the supplier whose reduction in the productionquantity (due to the decrease in n) offsets the retailer’s increase in such a way that theinventory in the whole system is reduced as shown by equation (30).

6. Conclusions and suggestions for future research

In this paper we formulated and solved a two-stage supply chain inventory modelin which quality of the output deteriorates with increased lot sizes. The number ofacceptable quality items in a lot is used to measure quality. Quality deterioration wasmodelled using a two-state Markovian production process and a constant failurerate production process. Models were solved under both deterministic and stochasticdemand.

The models show that the inclusion of rework cost can lead to large reduction inthe production lot sizes of suppliers and may lead to complete synchronization of thesupply chain. These results show that the practices of shifting inventory to thesuppliers common in industry (Landry et al. 1998), while having the appearanceof JIT, do not reap the full benefits JIT can provide. To reap full JIT benefitssuppliers should not only be delivering to retailers just-in-time but also matchingthis delivery with JIT production.

The model also indicates that ignoring quality may lead to smaller retailer orderquantity, while incorporating quality leads to an increase in the retailer orderquantity. Therefore, the incorporation of quality may seem to move the systemaway from JIT, while in reality the opposite is true. The increase in the orderquantity of the retailer is accompanied by a decrease in the production lot size ofthe supplier (because of a smaller integer multiplier). The decrease in the productionlot size of the supplier is large enough to cause a decrease in the average inventoryfor the whole system. Therefore, two-stage models that ignore quality considerationsmay seem to have the system operating on JIT bases because of the frequent smallshipments to the retailer while, in reality, they cause an increase in total inventory inthe system.

The incorporation of quality is most important for firms fitting the descriptiongiven by Porteus with regard to inspection or product characteristics: (1) firms that

1545Quality considerations on material flow

Dow

nloa

ded

by [

Gaz

i Uni

vers

ity]

at 0

4:56

03

Oct

ober

201

4

inspect only the first and last pieces of a component, and if the last piece is good then

the entire lot is judged to be good; (2) the type of defect cannot be identified until the

product is processed on the next station and the entire lot is produced before it is

moved to the next station; or (3) even though the process is continuously monitored,

there is a delay between the time the process shifts out of control and the time the

operators can determine the process is out of control. The monitoring is still useful in

determining which components need rework.

Future research may focus on multi-stage, multi-product supply chains. Another

interesting area of research may be the case in which defectives are not discovered

until the product reaches the retailer, where the products cannot be reworked, and

the effect of such a delayed discovery on customers’ service levels.

References

Banerjee, A., 1986, A joint economic-lot-size model for purchaser and vendor. DecisionSciences, 17, 292–311.

Cheng, T. C. E., 1991, An economic order quantity model with demand-dependent unitproduction cost and imperfect production processes. IIE Transactions, 23, 23–28.

Cordon, C., 1995, Quality defaults and work-in-process inventory. European Journal ofOperational Research, 80, 240–251.

Erenguc, E, S. S., Simpson, N. C. and Vakharia, A. J., 1999, Integrated production/dis-tribution planning in supply chains: an invited review. European Journal of OperationalResearch, 115, 219–236.

Fogarty, R. W., Blackstone, J. H. and Hoffmann, T. R., 1991, Production and InventoryManagement, 2nd ed (Cincinnati, OH: South-Western).

Fogarty, D. W., Blackstone, Jr., J. H. and Hoffmann, T. R., 1991, Production &Inventory Management (Cincinnati, OH: South-Western Publishing).

Goyal, S. K., 1988, A joint economic-lot-size model for purchaser and vendor: a comment.Decision Sciences, 19, 236–241.

Goyal, S. K., 1995, A one-vendor multi-buyer integrated inventory model: a comment.European Journal of Operational Research, 82, 209–210.

Goyal, S. K., 2000, On improving the single-vendor single-buyer integrated productioninventory model with a generalized policy. European Journal of Operational Research,125, 429–430.

Goyal, S. K. and Nebebe, F., 2000, Determination of economic production–shipment policyfor a single-vendor-single-buyer system. European Journal of Operational Research, 121,175–178.

Hall, R., 1983, Zero Inventories (Homewood, IL: Dow Jones).Hall, R. W., 1987, Attaining manufacturing excellence: just-in-time, total quality, total people

involvement (Homewood, ILL: Dow Jones-Irwin).Handfield, R. B. and Nichols, Jr. E. L., 1999, Introduction to Supply Chain Management

(Upper Saddle River, NJ: Prentice Hall).Helper, S., 1991, How much has really changed between U.S. automakers and their suppliers?

Sloan Management Review, 32(4), 15–28.Helper, S. and Sako, M., 1995, Supplier relations in Japan and the United States: are they

converging? Sloan Management Review, 36(3), 77–84.Hill, R. M., 1997, The single-vendor single-buyer integrated production-inventory model

with a generalised policy. European Journal of Operational Research, 97, 493–499.Inman, R. A., 1994, The impact of lot-size reduction on quality. Production and Inventory

Management Journal, 35, 5–8.Kekre, S. and Mukhopadhyay, T., 1992, Impact of electronic data interchange technology

on quality improvement and inventory reduction programs: a field study. InternationalJournal of Production Economics, 28, 265–281.

Khouja, M., 1999, A note on ‘deliberately slowing down output in a family productioncontext’. International Journal of Production Research, 37, 4067–4077.

1546 M. Khouja

Dow

nloa

ded

by [

Gaz

i Uni

vers

ity]

at 0

4:56

03

Oct

ober

201

4

Khouja, M. and Mehrez, A., 1994, An economic production lot size model with imperfectquality and variable production rate. Journal of the Operational Research Society, 45,1405–1417.

Landry, S., Trudel, Y. and Diorio, M. O., 1998, Just-in-time supply: cooperation,competition, and abuse. Competitiveness Review, 8(1), 37–45.

Lu, L., 1995, A one-vendor multi-buyer integrated inventory model. European Journal ofOperational Research, 81, 312–323.

Porteus, E. L., 1986, Optimal lot sizing, process quality improvement, and setup cost reduc-tion. Operations Research, 34, 137–144.

Rosenblatt, M. J. and Lee, H. L., 1986, Economic production cycles with imperfect produc-tion processes. IIE Transactions, 17, 48–54.

Schonberger, R. J., 1982, Japanese Manufacturing Techniques (New York: Free Press).Sharafali,M. and Co, H. C., 2000, Some models for understanding the cooperation between

the supplier and the buyer. International Journal of Production Research, 38, 3425–3449.Silver, E. A., Pyke, D. F. and Peterson, R. P., 1998, Inventory Management and Production

Planning and Scheduling, 3rd edn (New York: Wiley).Simchi-Levi, D., Kaminsky, P. and Simchi-Levi, E., 2000, Designing and Managing the

Supply Chain (New York: Irwin McGraw-Hill).Thomas, D. J. and Griffin, P. M., 1996, Coordinated supply chain management. European

Journal of Operational Research, 94, 1–15.Trevino, J., Hurley, B. J. and Friedrich,W., 1993, A mathematical model for the economic

justification of setup time reduction. International Journal of Production Research, 31,191–202.

Urban, T. L., 1998, Analysis of production systems when run length influences productquality. International Journal of Production Research, 36, 3085–3094.

Viswanathan, S., 1998, Optimal strategy for the integrated vendor–buyer inventory model.European Journal of Operational Research, 105, 38–42.

Wright, C.M. and Mehrez, A., 1998, An overview of representative research of the relation-ships between quality and inventory. Omega, 26, 29–47.

1547Quality considerations on material flow

Dow

nloa

ded

by [

Gaz

i Uni

vers

ity]

at 0

4:56

03

Oct

ober

201

4