affisco2002

Production, Manufacturing and Logistics

Quality improvement and setup reduction in the jointeconomic lot size model

John F. Affisco *, M. Javad Paknejad, Farrokh Nasri

Department of BCIS/QM, Frank G. Zarb School of Business, Hofstra University, Hempstead, NY 11549-1340, USA

Received 13 April 2001; accepted 9 August 2001

Abstract

The co-maker concept has become accepted practice in many successful global business organizations. This fact

has resulted in a class of inventory models known as joint economic lot size (JELS) models. Heretofore such models

assumed perfect quality production on the part of the vendor. This paper relaxes this assumption and proposes a

quality-adjusted JELS model. In addition, classical optimization methods are used to derive models for the cases of

setup cost reduction, quality improvement, and simultaneous setup cost reduction and quality improvement for the

quality-adjusted JELS. Numerical results are presented for each of these models. Comparisons are made to the basic

quality-adjusted model. Results indicate that all three policies exhibit significantly reduced total cost. However, the

simultaneous model results in the lowest cost overall and the smallest lot size. This suggests a synergistic impact of

continuous improvement programs that focus on both setup and quality improvement of the vendor’s production

process. Sensitivity analysis indicates that the simultaneous model is robust and representative of practice.

� 2002 Elsevier Science B.V. All rights reserved.

Keywords: Inventory; Quality; Joint economic lot size models

1. Introduction

The co-maker concept has become acceptedpractice in many successful global business orga-nizations. The basic tenet of this philosophy is thatvendor (supplier) and purchaser are value chainpartners in manufacturing and delivering a highquality product to the purchaser’s customers. Thisviewpoint has led to the development of a class of

inventory models known as integrated or jointeconomic lot size (JELS) models. These modelsconsist of lot size formulas based on the joint op-timization of vendor and purchaser costs.

This idea of the joint optimization of vendorand purchaser costs was first initiated by Goyal(1977). In this work Goyal developed a model foran integrated lot size in which the supplier’s lotsize is an integer multiple of the customer’s orderquantity. Trial and error is used to arrive at thisinteger value. This approach to integrated lot siz-ing was revisited by Banerjee, among others.

The term JELS was coined by Banerjee (1986)who used classical optimization to derive the JELS

European Journal of Operational Research 142 (2002) 497–508

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*Corresponding author. Tel.: +1-516-463-5362; fax: +1-516-

463-4834.

E-mail address: [email protected] (J.F. Affisco).

0377-2217/02/$ - see front matter � 2002 Elsevier Science B.V. All rights reserved.

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formula which is a function of demand, the annualinventory carrying charge, the vendor’s annualproduction rate, setup cost, and unit productioncost, and the purchaser’s order cost and unit pur-chase cost. The JELS, in general, is not the optimallot size for either the purchaser or vendor operat-ing independently. Thus, some cost is involved onboth parties’ part to operate at this mutuallybeneficial level. Banerjee investigates the cost-tradeoffs involved in adopting the JELS from boththe purchaser’s and vendor’s points of view. Es-sentially, one party will be at a disadvantage if theJELS is adopted. This situation can be amelioratedby the advantaged party offering some price con-cession to the other party. The JELS model pre-sented in Banerjee (1986) assumes that the vendorproduces on a lot-for-lot basis in response to or-ders from a single purchaser, demand is deter-ministic, and the vendor is the sole supplier.

This initial work by Banerjee motivated otherresearchers to more deeply investigate the JELSconcept. One JELS research thrust revolves aroundrelaxing Banerjee’s lot-for-lot assumption. Goyal(1988) developed a joint total relevant cost modelfor a single-vendor–single-buyer production in-ventory system assuming that the vendor’s lot sizeis an integer multiple of the purchaser’s order size.Goyal’s model was derived based on the impliedassumption that the vendor can supply to thepurchaser only after completing the entire lot. Lu(1995) relaxed Goyal’s assumption and assumedthat the vendor can supply the purchaser evenbefore completing the entire lot. That is, Lu’s workallows lot splitting. Further, the article considersthe case of multiple buyers from a single vendor.As to the applicability of Lu’s model, the authorstates ‘‘our new model will be suitable when thevendor has an advantage over the buyer in thepurchasing negotiation. This situation is not un-usual when the vendor is the sole supplier of anitem, and the buyer lacks the economic power todemand a price discount’’.

Goyal (1995) presents an improved solutionmethodology for the model proposed in Lu (1995).Hill (1997) generalizes the model presented in Lu(1995) and modified in Goyal (1995). Lu’s model,in addition to developing heuristics for the single-vendor–multiple-buyer problem, gave an optimal

solution to the single-vendor–single-buyer prob-lem based on the assumption that a single batchcan be split into an integral number of equalshipments. Goyal showed that an alternativepolicy involving successive shipments within a pro-duction batch, increasing by a factor equal to theproduction rate divided by the demand rate, gen-erates lower total joint relevant costs. Hill’s papershows that neither of these policies is optimal.They can ‘‘be thought of as limiting extremes of amore general class of policy for which successiveshipments within a production batch increase by afixed factor’’.

Of greater significance to this present work isresearch that investigates the impact of reduced lotsizes resulting from investment in setup cost re-duction on the JELS. Affisco et al. (1988) integratethe concepts of JELS and vendor setup cost re-duction. For the case of a single vendor and pur-chaser and assuming a logarithmic investmentfunction, they derive relationships for the optimalJELS, optimal vendor’s setup cost, and the opti-mal joint total cost per year. Results of a numer-ical example indicate that significant savings injoint total cost can accrue from investing in de-creased setup costs on the part of the vendor. As inthe case of the JELS with constant setup cost,adoption of the joint economic lot size includinginvestment results in one of the parties being at acost disadvantage and a major question is themethod by which joint cost savings may be equi-tably distributed. Further work by Affisco et al.(1991) extends this approach to the case of one-vendor and many-nonidentical-purchasers. Muchthe same results are achieved as in the single ven-dor and purchaser case.

Nasri et al. (1991) investigated the impact ofsimultaneous investment in setup cost and ordercost reduction on Banerjee’s JELS model. Resultsindicate that significant savings in joint total costcan accrue from such simultaneous investment.Further, both the vendor and purchaser realizesignificant savings when compared to the basicJELS model.

After reviewing the state of the JELS literature,Joglekar and Tharthare (1990) built a more real-istic model, the refined JELS model, by relaxingthe lot-for-lot assumption, and separating the

498 J.F. Affisco et al. / European Journal of Operational Research 142 (2002) 497–508

traditional setup cost into two independent costs.The first is the standard manufacturing setup costper production run, and the second is the vendor’scost of handling and processing an order from apurchaser. Based on these changes, they develop arefined JELS model for both the one-vendor,many-identical-purchasers and the more generalone-vendor, many-nonidentical-purchasers situa-tions.

Finally, Affisco et al. (1993b) investigated theone-vendor, many-nonidentical-purchasers JELSmodel with vendor setup cost reduction and pur-chaser order cost reduction. The results indicatethat there are significant cost savings for the JELSover independent optimization when such invest-ments are made. This suggests that when an en-vironment of cooperation between the parties hasbeen established the JELS is a superior policy.

The impact of quality on lot sizing decisions isanother major thread of inventory research. Anumber of authors have investigated the effect ofquality on lot size for the case of independentoptimization for the vendor. Rosenblatt and Lee(1986) investigated the effect of process quality onlot size in the classical economic manufacturingquantity (EMQ) model. Porteus (1986) introduceda modified EMQ model that indicates a significantrelationship between quality and lot size. In bothof these works, the optimal lot size is shown to besmaller than that of the EMQ model. Paknejadet al. (1995) extend this work to consider stochasticdemand and constant lead time in the continuousreview ðs;QÞ model.

Cheng (1991) develops a model that integratesquality considerations with the EPQ. The authorassumes that unit production cost increases withincreases in process capability and quality assur-ance expenses. Classical optimization results inclosed forms for the optimal lot size and optimalexpected fraction acceptable. The optimal lot size isintuitively appealing since it indicates an inverserelationship between lot size and process capability.

It should be noted that a good survey of theliterature on integrating lot size and quality con-trol policies is given in Goyal et al. (1993).

Schonberger (1982, 1986) presents some basicfoundations of JIT purchasing as expressed in theco-maker concept. They include, among others:

1. Reduce suppliers (vendors) for an item to a few,and in some cases only one.

2. Establish long-term relationships with such sup-pliers. In order to improve the viability and re-liability of such long-term relationships, engagein a supplier development program aimed at im-proving supplier performance by assisting themin improving the quality of their processes, andultimately their products.

3. Work to decrease lot sizes because smaller lotsizes lead to improved manufacturing perfor-mance.

Therefore, it is obvious that quality is a crucialissue for a successful co-maker relationship. Infact, the big three US automobile makers requirethat all suppliers achieve QS-9000 quality certifi-cation. Further, suppliers are continuously moni-tored to assure that they maintain practices thatare consistent with this level of quality.

Given this importance of quality to the co-maker concept, it is evident that the next phase inthe development of JELS models is to investigatethe impact of quality on their performance.

2. The basic models

We begin our investigation of this new thread ofJELS research with the seminal work of Banerjee.Consider a system in which a single vendor pro-duces on a lot-for-lot basis in response to ordersfrom a single purchaser, demand is deterministic,and the vendor is the sole supplier. Under theseconditions the JELS may be obtained by mini-mizing the joint total relevant cost given byBanerjee (1986) as

JTRCðQÞ ¼ ðD=QÞðS þ AÞ

þ ðQ=2Þr½ðD=RÞCv þ Cp�; ð1Þ

whereD¼ annual demand or usage of the item,R¼ vendor’s annual production rate for this

item,A¼ purchaser’s ordering cost per order,S¼ vendor’s setup cost per setup,r¼ annual inventory carrying charge, expressed

as fraction of dollar value,

J.F. Affisco et al. / European Journal of Operational Research 142 (2002) 497–508 499

Cv ¼ unit production cost incurred by the ven-dor,

Cp ¼ unit purchase cost paid by the purchaser,Q¼ order or production lot size in units,

and RPD, Cv 6Cp.The result of classical optimization yields the fol-lowing formula for the JELS:

Q�j ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2DðS þ AÞ=r½ðD=RÞCv þ Cp�

q: ð2Þ

Implicit in this derivation is that all units producedby the vendor, in response to the purchaser’s or-der, are of acceptable quality. Now, assume thatthis is not the case.

Let us look at a situation where the vendoroperates a process that is in statistical control.That is, the process generates a known, constantproportion of defectives, p. (Such an assumptionhas been made previously in the literature byCheng (1991).) The vendor ships on a lot-for-lotbasis to the purchaser. Under these circumstanceson receiving the parts from the vendor, Deming(1981) (also detailed in Gitlow et al. (1995) andPapadakis (1985)) proves that the purchasershould consider only two inspection policies – zeroinspection or 100% inspection. Full inspection ispreferred when the cost of inspecting an incomingitem is small when compared to the cost of a de-fective item being released to the purchaser’s pro-duction process. We assume this to be the situationfor the present research. To be consistent withDeming’s ideas, we further assume that the pur-chaser’s inspection process is perfect, and that allrejected parts are replaced by the vendor at hisexpense. Of course, it is likely that the vendor willrecover some of these costs from the purchasereither directly or indirectly.

Based on this scenario, we now adjust the JELSmodel for the quality factor. The joint total rele-vant cost including quality may be written as

JTRC�ppðQ; �ppÞ ¼ DðS þ AÞ=�ppQþ ðQ=2Þ�pprCþ ðD=�ppÞ½ð1� �ppÞCM þ CN�; ð3Þ

where�pp ¼ 1� p ¼ the proportion of good items pro-

duced by the vendor’s process,

C ¼ ½ðD=RÞCv þ Cp�,

CM ¼ vendor’s cost of repair and replacement ofdefective items per unit,

CN ¼ purchaser’s cost of inspection per unit.Implicit in the structure of the cost function (3) isthe fact that defective units are detected beforethey go into the purchaser’s inventory and areimmediately returned to the vendor. The result ofclassical optimization yields the following formulafor the quality-adjusted JELS:

Q�j�pp ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2DðS þ AÞ=rC�pp2

p: ð4Þ

Note that in Eq. (4), if �pp ¼ 1 then quality is perfectand the quality-adjusted JELS simply reduces tothe basic JELS expressed as Eq. (2). Also note thatQ�j�pp is inversely related to �pp. This is intuitively ev-

ident because the required joint lot size will besmaller if more items are acceptable in the lot.

3. Quality improvement

In this section we consider the option of in-vesting to improve the quality of the vendor’sproduction process. Quality at the source is a basicplank of any co-maker relationship. Supplier de-velopment programs that aim to improve thequality of supplier processes, and therefore prod-ucts, are prevalent in most industries. In the autoindustry, for example, manufacturers work closelywith suppliers to improve their production pro-cesses. The ultimate aim being supplier certifica-tion according to widely accepted standards suchas QS 9000. Generally, one result of such certifi-cation is that purchaser’s are able to eliminateincoming inspection. This results in improved qual-ity, increased throughput, and lower costs.

We now consider �pp to be a decision variable andpursue the objective of minimizing the sum of theinvestment cost for increasing �pp and the quality-adjusted joint total relevant cost. Specifically, weseek to minimize

f ðQ; �ppÞ ¼ ia�ppð�ppÞ þ JTRC�ppðQ; �ppÞ ð5Þsubject to

�pp0 < �pp6 1; ð6Þwhere i is the cost of capital, �pp0 ¼ 1� p0 is theoriginal proportion of good units produced by the


vendor’s process, a�ppð�ppÞ is a convex and strictlyincreasing function of �pp, JTRC�ppðQ; �ppÞ is the jointtotal relevant cost given by Eq. (3), and �pp0 is theoriginal proportion of good items before any in-vestment is made.

To obtain the JELS including investment inquality improvement we minimize (5) over Q and �ppby classical optimization techniques. Of course, ifthe optimal quality proportion does not satisfy therestriction (6), we should not make any invest-ment, and the results of (4) hold.

3.1. The logarithmic investment function case

We seek to improve quality by investing to in-crease, �pp, the proportion of good units producedby the vendor’s production process. We use thefollowing exponential growth function of invest-ment:

�pp ¼ �pp0eDa�pp ð7Þsubject to

�pp0 < �pp6 1; ð8Þwhere �pp0 is the original proportion of good unitsand D is the percentage increase in �pp per dollarincrease in a�pp. The restriction (8) induces a com-panion restriction on the admissible amount ofinvestment,

06 a�pp 6 ln½1=�pp0�=D:Taking log on both sides of (7) gives

a�pp ¼ a ln �pp þ b; ð9Þwhere a ¼ 1=D and b ¼ � ln �pp0=D.

We are now ready to prove Theorem 1 whena�ppð�ppÞ as represented by (9) is used in (5).

Theorem 1. If 06 �pp0 < 1 and D is strictly positive,then the following hold:

(a) f ðQ; �ppÞ is strictly convex if 2DðS þ AÞia > 0.(b) The optimal proportion of good parts pro-

duced by the vendor’s process and optimal JELS aregiven by

�pp�� ¼ maxf�pp0; �pp�Ig;

Q�� ¼ minfQ�j�pp;Q

�j�ppIg;

where

�pp0 ¼ original proportion of good parts before

investment;

�pp�I ¼ ½ðCM þ CN ÞD=ia�; ð10Þ

Q�j�pp ¼ ½2DðS þ AÞ=rC�pp2�1=2; ð11Þ

Q�j�ppI ¼ ½2DðS þ AÞ=rC�1=2½ia=ðCM þ CNÞD�: ð12Þ

(c) The resulting optimal total cost per unit timeis given by

f ðQ��; �pp��Þ ¼ minfJTRC�ppðQ�j�ppÞ; JTRCIðQ�

j�ppI; �pp�I Þg;

where

JTRC�ppðQ�j�ppÞ ¼ ½DðS þ AÞ=�ppQ�

j�pp�þ ½Q�

j�pp�pprC=2�þ ½ðD=�ppÞðð1� �ppÞCM þ CNÞ� ð13Þ

and Q�j�pp is given by Eq. (11),

JTRCIðQ�j�ppI; �pp

�I Þ ¼ i a ln �pp�I

hþ b

iþ DðS þ AÞ=�pp�IQ�

j�ppI

þ Q�j�ppI�pp

�I rC=2

þ ðD=�pp�I Þðð1h

� �pp�I ÞCM þ CNÞi

and Q�j�ppI is given by Eq. (12), �pp

�I is given by Eq. (10).

Proof. (a) Let f ðQ; �ppÞ ¼ ia�ppð�ppÞ þ JTRC�ppðQ; �ppÞ for06 �pp0 < 1, where JTRC�ppðQ; �ppÞ is given by Eq. (3).f ðQ; �ppÞ is strictly convex if the principal minors ofits Hessian determinant are strictly positive. Weproceed by computing the principal minors

jH11j ¼ 2DðS þ AÞ=�ppQ3 > 0;

jH22j ¼ 2DðS þ AÞia > 0:

It is clear that both principal minors are strictlypositive. Hence, part (a) holds.

(b) The optimal values of the decision variablesmay be found by solving the two simultaneousequations given by

of =oQ ¼ of =o�pp ¼ 0:

The solution to these equations yields Eqs. (10)and (12). The stationary point ðQ�

j�ppI; �pp�I Þ is a rela-

tive minimum if it satisfies the convexity condition


of part (a). Beginning with Eq. (12) and after somealgebraic manipulation it can be shown that

Q�2j�ppI rCðCM

hþ CNÞ2D2=ia

i¼ 2DðS þ AÞia:

Therefore,

Q�j�ppI > 0:

Since all variables that compose Q�j�ppI are strictly

positive, this condition is satisfied and we have alocal minimum at ðQ�

j�ppI; �pp�I Þ, and part (b) holds.

(c) The proof of this part results from substi-tuting the optimal values Q�

j�ppI and �pp�I into the ap-propriate joint relevant cost function. �

4. Simultaneous quality improvement and setup cost

reduction

In this section we consider the option of si-multaneous investment in vendor quality improve-ment and setup cost reduction. The synergisticeffects of such a policy, as part of a supplier de-velopment program, have been widely recognizedin the literature and in practice. See for example,Schonberger (1982, 1986), Hall (1983) and Affiscoet al. (1993a).

Assuming a logarithmic investment function,Affisco et al. (1988) show that the JELS includingvendor setup cost reduction is

Q�jSI ¼ ½ibþ ði2b2 þ 2DArCÞ1=2�=rC ð14Þ

with optimal setup cost of

S�jSI ¼ ½i2b2 þ ibði2b2 þ 2DArCÞ1=2�=DrC: ð15Þ

This results from the minimization of the followingcost function:

f ðQ; SÞ ¼ iða� b ln SÞ þ JTRCðQÞ;where a ¼ ln S0=d and b ¼ 1=d.

Of course, this again assumes that quality isperfect. To adjust (14) and (15) for the effects ofquality, one simply is required to minimize

f ðQ�pp; SÞ ¼ iða� b ln SÞ þ JTRC�ppðQ; �ppÞ: ð16Þ

This results in

Q�j�ppSI ¼ ib

hþ ði2b2 þ 2DArCÞ1=2

i=�pprC ð17Þ

and

S�j�ppSI ¼ i2b2h

þ ibði2b2 þ 2DArCÞ1=2i=DrC: ð18Þ

Note that in (17) when �pp ¼ 1, indicating perfectquality, the quality-adjusted JELS with investmentin vendor setup cost reduction reduces to (14), theJELS with vendor setup cost reduction. Interest-ingly, the optimal setup cost is the same for bothcases. That is, it is independent of the quality level.

Now we turn our attention to the simultaneousinvestment case. First we define two logarithmicinvestment functions. The quality improvementfunction is identical to that presented as Eq. (9).The setup cost reduction function is

aSðSÞ ¼ a� b ln S for 0 < S6 S0; ð19Þ

where a ¼ ln S0=d, b ¼ 1=d, and S0 is the originalsetup cost.

We are now ready to prove Theorem 2 whena�ppð�ppÞ is as represented in (9) and aSðSÞ as repre-sented in (19) are used in the following costequation:

f ðQ; �pp; SÞ ¼ a�ppð�ppÞ þ aSðSÞ þ JTRC�ppðQ; �ppÞ: ð20Þ

Theorem 2. If 06 �pp0 < 1 and D is strictly positive,and if S0 and d are strictly positive, then the fol-lowing hold:

(a) f ðQ; �pp; SÞ is strictly convex iff D < 8ðSþAÞ3i2b2=S4rC.

(b) The optimal proportion of good parts pro-duced by the vendor’s process, the optimal vendor’ssetup cost, and the optimal JELS are given by

p�� ¼ max �pp0; �pp�IIn o

;

S�� ¼ min S0; S�j�ppIIn o

;

Q�� ¼ min Q�j�pp;Q

�j�ppII

n o;

where

�pp0 ¼ original proportion of good parts before

investment;

�pp�II ¼ ðCM þ CNÞD=ia;


S0 ¼ original vendor0s setup cost before investment;

S�j�ppII ¼ S�j�ppSI;

Q�j�pp is as presented in Eq. (4),

Q�j�ppII ¼ 2ia½i2b2 þ ibði2b2 þ 2DArCÞ1=2

þ DArC�1=2=DrCðCM þ CNÞ:

(c) The resulting optimal total cost per unit timeis given by

f ðQ��; �pp��; S��Þ ¼ min JTRC�ppðQ�j�ppÞ;

nJTRCIIðQ�

j�ppII; �pp�II; S

�j�ppIIÞ

o;

where

JTRC�ppðQ�j�ppÞ is given as Eq: ð13Þ;

JTRCII Q�j�ppII; �pp

�II; S

�j�ppII

� ¼ i a ln �pp�II

�hþ b

þ a�

� b ln S�j�ppIIi

þ D S�j�ppII�

þ A=�pp�IIQ

�j�ppII þ Q�

j�ppII�pp�IIrC=2

þ D=�pp�II�

1�h

� �pp�IICM þ CN

i:

Proof. (a) Let f ðQ; �pp; SÞ ¼ ia�ppð�ppÞ þ iaSðSÞþJTRCðQ; �ppÞ for 06 �pp0 < 1 and 0 < S6 S0, whereJTRCðQ; �ppÞ is given by Eq. (3). f ðQ; �pp; SÞ is strictlyconvex iff the principal minors of its Hessian de-terminant are strictly positive. We proceed bycomputing the principal minors

jH11j ¼ 2DðS þ AÞ=�ppQ3 > 0;

jH22j ¼ 2DðS þ AÞia > 0;

jH33j ¼ ½2DðS þ AÞ=rC�1=2½2ðS þ AÞib=S2� � D > 0:

It can be seen that jH11j and jH22j are strictlypositive, and that jH33j > 0 if and only if theconvexity condition of part (a) of Theorem 2holds.

(b) The optimal values of the decision variablesmay be found by solving the three simultaneousequations

of =oQ ¼ of =o�pp ¼ of =oS ¼ 0:

The solution to these equations yields Q�j�ppII, S

�j�ppII,

and �pp�II of part (b) of Theorem 2. To prove thestationary point ðQ�

j�ppII; S�j�ppII; �pp

�IIÞ is a relative mini-

mum, it is sufficient to show that it satisfies theconvexity condition of part (a). Using the optimalvalues of Q, �pp, and S, the convexity condition ofpart (a), after some manipulation, will be reducedto

Q�j�ppII > �2DA=ib�pp�II

since D, A, i, b, and �pp�II are all strictly positive, theconvexity condition is satisfied at point ðQ�

j�ppII; S�j�ppII;

�pp�IIÞ. Therefore, part (b) follows.(c) The proof of this part results from substi-

tuting the optimal values Q�j�ppII, S

�j�ppII, and �pp�II into

the appropriate joint relevant cost function. �

5. Numerical examples

Consider the case of an inventory item pro-duced to order by a vendor on a lot-for-lot basis.A single purchaser periodically orders and buys abatch of this item from the vendor, who is thebuyer’s sole source for this item. The vendor andpurchaser have agreed to cooperate in accordancewith the results of the JELS model. The followingparameters are known: D¼ 1000 units/year, R¼3200 units/year, Cv ¼ $20/unit, Cp ¼ $25/unit, S0 ¼$400/setup, A¼ $100/order, r ¼ 0:2, �pp ¼ 0:75,CM ¼ $12/unit, and CN ¼ $5/unit. Further, thevendor may invest in reducing setup cost accord-ing to a logarithmic investment function with pa-rameters i ¼ 0:10 and d ¼ 0:002. Investment mayalso be made in improving the quality of thevendor’s manufacturing process according to alogarithmic investment function with i ¼ 0:10 andD ¼ 5:75 10�6.

Table 1 presents the results of calculations forthe quality-adjusted JELS (JELS�pp), the quality-improved JELS (JELSI�pp), the quality-adjustedJELS with setup cost reduction (JELSI�ppS), and thequality-adjusted JELS with simultaneous invest-ment in quality improvement and setup cost re-duction (JELSII�pp). It is interesting to note that thequality-adjusted JELS is an upper bound for lotsize and total cost. In all cases when investment is


not warranted, the results reduce to those of thismodel.

As Table 1 indicates, when a policy of investingin quality improvement is adopted the optimal lotsize is reduced by 11.4% and the optimal processquality is improved by 30.35%. This is accompa-nied by a decrease of $3725.04 or 93.13% invendor replacement costs, a decrease of $1552.10or 23.28 in purchaser inspection costs, and a de-crease of 5.28% in total cost. These reductionscome at the added expense of $4606.03 in qualityimprovement investment costs. In the situationwhere investment is made solely in setup reduc-tion, the optimal lot size is reduced by 46% whileno impact on process quality is realized. For thiscase, we experience a 97.6% reduction in setupcost and an 8.86% decrease in total cost whilequality related costs remain the same as forJELS�pp. The investment cost for setup reduction is$187.80. Finally, when a simultaneous investmentpolicy is in force, the optimal lot size is reducedby 58.57% and the optimal process quality isimproved by 30.35%. The vendor and the pur-chaser experience the identical quality cost savingsas in the JELSI�pp case and identical setup cost

savings as in the JELSI�ppS case. Total cost savingsof $1840.09 or 13.95% are also realized. The in-vestment costs are the same as those for the twoindependent investment models.

At this point a word about the relative amountsof the investment costs is in order. The investmentcost for setup reduction is significantly less thanthat for quality improvement. This is wholly con-sistent with what is experienced in practice. Setupreduction is essentially a localized activity. Affiscoet al. (1993a) discuss how programs for setup re-duction have become somewhat standardized.Existing setup operations are closely monitored;then internal and external activities are separatedafter which as many internal activities are con-verted to external activities as possible. These ac-tivities are usually conducted on a single machineor work center at a time. By way of contrast,quality improvement programs are wide-rangingand continuous in nature. They may include train-ing in statistical process control, product designand specification, scheduling, preventive mainte-nance, etc. Essentially all the factors that have animpact on process quality may be part of theprogram. As such, a larger and more consistent

Table 1

Comparative results for JELS models

Variable Quality-adjusted

JELS (JELS�pp)

Quality-improved

JELS (JELSI�pp)

Quality-adjusted JELS

with setup reduction

(JELSI�ppS)

Quality-adjusted JELS

with simultaneous

investment (JELSII�pp)

Q (units) 461.88 409.16 249.41 191.35

(% reduction) – (11.4) (46.0) (58.57)

S ($) 400 400 9.35 9.35

(% reduction) – – (97.66) (97.66)

�pp 0.75 0.9776 0.75 0.9776

(% increase) – (30.35) – (30.35)

Vendor replacement costs ($) 4000.00 274.96 4000.00 274.96

$ Savings (% savings) – 3725.04 (93.13) – 3725.04 (93.13)

Purchaser inspection costs ($) 6666.67 5114.57 6666.67 51114.57

$ Savings (% savings) – 1552.10 (23.28) – 1552.10 (23.28)

Quality improvement costs ($) – 4606.03 – 4606.03

Setup reduction costs ($) – – 187.70 187.70

Total cost ($) 13192.58 12495.56 12023.36 11352.49

$ Savings (% savings) – 697.02 (5.28) 1169.22 (8.86) 1840.09 (13.95)

S0 ¼ 400, �pp0 ¼ 0:75.


investment is likely to be required if such a pro-gram is to be successful.

The overall results indicate that, consistent withpractice, there are synergistic benefits to be gainedfrom supplier improvement programs that em-phasize process improvement including bothquality and setup elements. The total quality re-lated cost savings for the purchaser and the vendoris $5277.14. This is accompanied by a setup costsavings to the vendor of $390.65. Thus, the totalsystem savings are $5667.79. These total savingsare achieved at a total investment cost of $4793.83,resulting in a net saving of $873.96. The savingsare disproportionate in favor of the vendor –$4115.69 versus $1552.10. As is the case for alljoint lot size models, some negotiation between theparties will be required to determine their relativecontribution to the improvement efforts. However,it is not unreasonable to assume that the vendorwould be willing to make a larger immediate fi-nancial contribution as consideration for a long-term relationship with the purchaser. In fact, thisis the basis for many existing supplier certificationprograms. From the purchaser’s point of viewthere is a significant prospect for further savings.The optimal solution indicates that the purchaserpays a total of $5114.57 in inspection costs. Withthe vendor’s process producing 97.76% good units,it is clear that, by far, the largest amount of thesecosts is devoted to the inspection of good items. Atsome point in time it is likely that the qualityof the vendor’s process will be improved to thepoint that s/he can become a certified supplier.When this occurs, it is likely that the purchaser willforgo all incoming inspection and eliminate relatedcosts. Finally, the savings discussed above areconservative when viewed in the context of whatimproved quality and setup mean to overall op-erations.

For example, the vendor should realize ex-panded effective capacity that would allow him toseek additional work from other customers. Thepurchaser should realize improved operations fromdirect-to-floor shipments from the vendor madepossible by the elimination of the inspection stage.Ultimately, Schonberger (1982) indicates suchprograms should lead to increased productivityand improved market response.

6. Sensitivity analysis

In this section we turn our attention to an in-vestigation of the conditions under which simul-taneous investment in quality improvement andsetup cost reduction is worthwhile. This investi-gation begins with the derivation of critical (in-difference) points for the simultaneous investmentmodel. Under this scenario investment is warrantedif and only if S�j�ppII < S0, that is, the optimal setupcost is less than the original setup cost, and�pp�II > �pp0, that is, the optimal proportion of goodunits produced by the vendor’s process exceeds theoriginal proportion. By substituting Eq. (18) forS�j�ppII in the former relationship and Eq. (10) for �pp�IIin the latter relationship, we may solve for criticalpoints for various parameters of interest in orderto perform sensitivity analysis. These derived re-lationships can provide managers with a yardstickto determine whether investment would be worth-while.

Following the procedure outlined above, criti-cal points for demand, interest rate, and for thetwo technology coefficients were developed and arepresented in Table 2. For demand, D, and interestrate, i, two restrictions are developed, one eachbased on the setup cost constraint and the qualityproportion constraint. For each of the technologycoefficients, d and D, a single restriction results. Atthis juncture it is worth noting that critical pointsfor the individual investment models, JELSI�pp andJELSI�ppS, may be derived by working with thecorresponding single restriction on quality pro-portion and setup cost, respectively. The resultingcritical points are equivalent to the restrictionsdeveloped for the simultaneous investment case.

For each of the critical points presented inTable 2, we examine their sensitivity to a singleparameter, holding all others constant. The resultsof this sensitivity analysis are given in Table 3.These results must be analyzed in concert with therestriction on product quality (8) and the inducedcompanion restriction on the admissible amountof investment. The first portion of Table 3 looks atthe changes in optimal quality proportion, interestrate, and the technology coefficients as a functionof changes in demand. Note that as annual de-mand increases from 800 to 1025, the optimal


quality proportion �pp�II increases from 0.7821 to itsupper bound 1.00; the upper bound on the interestrate one would be willing to pay for the investmentincreases from 10.43% to 13.36%; the lower boundon the technology coefficient d for which invest-ment in setup cost reduction is desirable decreasesfrom 0.0001118 to 0.0000988; and the lower boundon the technology coefficient D for which invest-ment in quality improvement is warranted de-creases from 0.0000055 to 0.0000043. In thisscenario it appears that as demand increases to

1025 units, the most effective utilization of the$50,000 investment in quality improvement isachieved. This suggests that in developing anysupplier development program that has a majorfocus on quality improvement, the purchaser mustconsider the volume of parts that will be requiredfrom the vendor. Further, as demand increasesboth technology coefficients, which are in the formof percent improvement per dollar of investment,decrease suggesting that at higher levels of demandthe marginal percentage improvement required forsuccessful investment is less. This indicates thatwhen demand is large and more defective units areproduced generating higher quality costs, even arelatively small improvement is desirable. This isobviously consistent with the continuous improve-ment principle of Total Quality Management. Thesame can be said for setup cost reduction. Finally,the upper bound on interest rate increases from10.43% to 13.36% with the increase in demand.These clearly are acceptable rates in the presentday financial environment.

The second portion of Table 3 shows that as theoriginal setup cost per setup increases from $100 to$1000, the lower bound on the technology coeffi-cient d decreases from 0.000253 to 0.0000593. Thisis an indication that diminishing returns existssince when setup cost is high it is worthwhile toinvest even with a relatively smaller marginal re-

Table 2

Critical points for simultaneous investment model

Parameter Simultaneous model

i < MinS0b

DrC2ðAþ S0Þ

�1=2;ðCM þ CNÞD

a�pp0

( )

D > Max2i2b2ðAþ S0Þ

S20 rC;

ia�pp0ðCM þ CNÞ

( )

d >iS0

2ðAþ S0ÞDrC

�1=2

D >i�pp0

ðCM þ CNÞD

Table 3

Results of sensitivity analysis

Variables Demand (D)

800 900 1000 1025

�pp�II 0.7821 0.8798 0.9776 1.0020

i 0.1043 0.1173 0.1303 0.1336

d 0.0001118 0.0001054 0.0001 0.0000988

D 0.0000055 0.0000049 0.0000044 0.0000043

Variable Original setup cost (S0)

100 400 700 1000

d 0.000253 0.0001 0.0000723 0.0000593

Variable Original proportion of good units ð�pp0Þ0.5 0.6 0.7 0.8 0.9 0.95

D 0.00000294 0.00000353 0.00000412 0.00000471 0.00000529 0.00000559


turn. At lower setup costs one must experience agreater marginal return to invest since furthersetup reduction efforts are likely to be more diffi-cult. Much the same may be said for the qualityimprovement case as is exhibited in the thirdportion of Table 3. As the original quality levelincreases from 0.5 to 0.95, the technology coeffi-cient D increases from 0.00000294 to 0.00000559.Thus, as original quality approaches zero defects,the marginal percent improvement in quality mustbe greater to warrant further investment. Thisagain is due to the fact that there are decreasingmarginal returns. That is, the last few percent ofimprovement in quality is more difficult to attainthan earlier improvements. This is, in fact, what isexperienced in practice. On balance, the results ofthe simultaneous investment model must be clas-sified as robust and representative.

7. Conclusion

This paper presents a JELS model adjusted forthe quality factor. Specifically, the vendor operatesa process that is in statistical control that generatesa known constant proportion of defectives. Basedon this model we investigate the options of in-vesting in quality improvement, investing in setupcost reduction in the quality-adjusted model, andsimultaneously investing in quality improvementand setup cost reduction. For all three cases, clas-sical optimization is used to derive closed formsfor the decision variables. Results of a numericalexample indicate that a significant reduction intotal cost over the quality-adjusted JELS isachieved by each of the models. The simultaneousmodel generates the largest cost reduction of thethree. The simultaneous investment model alsoexhibits a synergistic effect on lot size as a result ofa supplier development program focused on bothquality and setup improvement. These resultssuggest that a program of continuous improve-ment should view quality and setup improvementto be complementary and that they should bepursued concurrently. Additional results of sensi-tivity analysis show the simultaneous model to berobust and representative of practice.

Acknowledgements

This research was supported by a Frank G.Zarb School of Business Summer Research Grant.

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