European Journal of Operational Research 226 (2013) 481–490

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European Journal of Operational Research

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Stochastics and Statistics

Optimal selection of process mean for a stochastic inventory model

M.A. Darwish ⇑, F. Abdulmalek, M. AlkhedherDepartment of Industrial and Management Systems Engineering, College of Engineering and Petroleum, Kuwait University, P.O. Box 5969, 13060 Safat, Kuwait

a r t i c l e i n f o a b s t r a c t

Article history:Received 19 December 2011Accepted 15 November 2012Available online 29 November 2012

Keywords:Quality controlTargeting problemProductionDemand uncertainty

0377-2217/$ - see front matter � 2012 Elsevier B.V. Ahttp://dx.doi.org/10.1016/j.ejor.2012.11.022

⇑ Corresponding author. Tel.: +965 38601163; fax:E-mail address: m.darwish@ku.edu.kw (M.A. Darw

It is very common to assume deterministic demand in the literature of integrated targeting – inventorymodels. However, if variability in demand is high, there may be significant disruptions from using thedeterministic solution in probabilistic environment. Thus, the model would not be applicable to realworld situations and adjustment must be made. The purpose of this paper is to develop a model for inte-grated targeting – inventory problem when the demand is a random variable. In particular, the proposedmodel jointly determines the optimal process mean, lot size and reorder point in (Q,R) continuous reviewmodel. In order to investigate the effect of uncertainty in demand, the proposed model is compared withthree baseline cases. The first of which considers a hierarchical model where the producer determines theprocess mean and lot-sizing decisions separately. This hierarchical model is used to show the effect ofintegrating the process targeting with production/inventory decisions. Another baseline case is the deter-ministic demand case which is used to show the effect of variation in demand on the optimal solution.The last baseline case is for the situation where the variation in the filling amount is negligible. This casedemonstrates the sensitivity of the total cost with respect to the variation in the process output. Also, aprocedure is developed to determine the optimal solution for the proposed models. Empirical resultsshow that ignoring randomness in the demand pattern leads to underestimating the expected total cost.Moreover, the results indicate that performance of a process can be improved significantly by reducing itsvariation.

� 2012 Elsevier B.V. All rights reserved.

1. Introduction

The problem of selecting the optimum process mean (targetingproblem) is one of the areas in economics of quality control, whichhas received a lot of interest from researchers in the recent times.Process targeting is often a difficult decision in productionprocesses such as canning/filling, metal plating, grinding, glassindustry, fiber industry, and steel industry (Park et al., 2011; Shaoet al., 2000; Hariga and Al-Fawzan, 2005). Usually, producers setspecification limits on a quality performance measure (weight, vol-ume, concentration, thickness, length, etc.) and an item is classifiedas conforming when the quality performance measure is withinspecification limits. Otherwise, the item is classified as noncon-forming and may be sold at a reduced price, reprocessed orscrapped (Roan et al., 2000). In USA, federal agencies examinedthe practice of process mean selection and reported that it is acommon trend that many manufacturers select high process meanin order to conform to specifications. This strategy leads to a ‘‘giveaway’’ cost (Roan et al., 2000). On the other hand, a tight processsetting will have less production cost, but high costs of rejection,

ll rights reserved.

+965 38604426.ish).

recycling and reprocessing (Al-Sultan and Pulak, 2000). Thus, thetargeting problem is a trade-off between material cost and the costassociated with producing nonconforming items.

Traditionally, the process mean selection and production/inven-tory decisions are determined separately. However, the choice ofprocess mean affects the probability that a given produced itemis nonconforming. Hence, the process mean determines theproduction yield rate which, in turn, influences other importantproduction and inventory decisions, in particular, the productionlot size and reorder point. Consequently, the process mean andproduction/inventory decisions should be determined jointly in or-der to control the total cost associated with production processes.This integration leads to higher conforming and yield rates,reduction in scrap or reprocessing cost, minimal loss to customerdue to the deviation from the optimum target value, as well as,and perhaps most importantly, providing better products atreduced cost for customers (Darwish, 2009).

The organization of the paper is as follows; motivation and con-tribution is discussed in the next section. In Section 3, we reviewthe literature. Model assumptions and notation are introduced inSection 4 followed by model development. In Section 6, solutionmethod is outlined. Then we investigate some baseline cases inSection 7. In Section 8, we present the empirical results and Sec-tion 9 concludes the paper.

482 M.A. Darwish et al. / European Journal of Operational Research 226 (2013) 481–490

2. Motivation and contribution

It is very common to make simplifying assumptions in model-ing real-world phenomenon. This is important in order to renderthe mathematics tractable. On the other hand, we cannot maketoo many simplifying assumptions, for then our conclusions, ob-tained from the mathematical model would not be applicable toreal world situations (Ross, 2007). In integrated targeting-inven-tory models, it is usually assumed that demand is deterministic.In reality however, deterministic models are only approximationsand the goodness of these approximations depends on the degreeof uncertainty in the demand pattern (Nahmias, 2005). Thus, thereexists a need to develop a model that addresses the limitationsimposed by this assumption.

Unlike the models developed in the integrated targeting-inven-tory literature, in this paper, the targeting problem is integratedwith inventory decisions when randomness is introduced intothe demand pattern. The proposed model simultaneously deter-mines the optimal decisions regarding process mean, productionlot size, and reorder point in (Q,R) continuous review model.

In general, there are three levels of decisions related to produc-tion processes: (1) long-term (strategic) (2) Intermediate-term(tactical), and (3) short-term (operational). Usually, these decisionlevels are optimized independently due to the nature of their vary-ing time horizon. However, these decision levels affect and are af-fected by each other. For instance, the selection of process mean(which is operational decision) affects the conformance rate whichdetermines number of conforming/nonconforming items pro-duced. Therefore, the determination of production lot size (whichis tactical decision) should compensate for the nonconformingitems produced during a production run. Consequently, integratingthe process mean targeting and production planning and inventorycontrol decisions is expected to reduce the total cost. Anotherimportant aspect of this integration is that it jointly addressesquality control issues through process targeting and productionplanning and inventory control issues.

After deriving the proposed model, it is compared with itsdeterministic counterpart by studying three baseline cases. Thefirst of which considers a hierarchical model where the producerdetermines the process mean and lot-sizing decisions separately.This hierarchical model is used to show the effect of integratingthe process targeting with production/inventory decisions. Theother baseline case is for deterministic demand which is used toexplore the effect of variation in demand on the optimal solution.Then we use the baseline case where variation in the filling amountcan be neglected. This case shows the sensitivity of the total costwith respect to the variation in the process output. The proposedmodels show that ignoring randomness in demand may lead tounderestimating the expected total cost significantly. Furthermore,a procedure is devised to find the optimal solution of the model.

3. Literature review

The optimal determination of process mean has been investi-gated and discussed for more than 60 years. Springer (1951) wasthe first to consider the problem of process targeting, in whichthe process mean that minimizes the total cost is obtained. Bettes(1962) presented a similar model by determining the optimalprocess mean and upper specification limit simultaneously. In thismodel, the nonconforming items are scrapped with no salvage va-lue. This model is extended by Hunter and Kartha (1977) by consid-ering the problem of selecting the optimum target value whennonconforming items are sold at a secondary market. Then, Bisg-aard et al. (1984) modified Hunter and Kartha’s model to a situationwhere nonconforming products are sold at a price proportional to

the amount of material used. Golhar (1987) considered the optimaldetermination of the optimum process mean when conformingitems are sold at a fixed price and nonconforming cans are emptiedand refilled at a reprocessing cost. Boucher and Jafari (1991) ex-tended this line of research by introducing a sampling plan as op-posed to 100% inspection; in this model the effect of singlesampling plan on the optimal set point of a filling operation isexamined. Al-Sultan (1994) extended the model developed byBoucher and Jafari (1991) for two machines in series using a samplinginspection plan. Tang and Lo (1993), Lee and Jang (1997) and Leeet al. (2001) used a surrogate variable in inspection and determinedthe optimum process mean and the screening limits. Duffuaa andSiddiqui (2003) discussed a model for a multi class screening whenmeasurement error existed and Darwish and Duffuaa (2010) devel-oped a model that determines the optimal process mean and sam-pling plan’s parameters. The model maximizes producer expectedprofit while protecting the consumer through a constraint on theprobability of accepting lots with low incoming quality. Moreover,Elsayed and Chen (1993) determined optimum levels of processparameters for items with multiple characteristics. Arcelus andRahim (1994) provided joint optimal settings for variable and attri-bute target means. Chen and Chung (1996) considered a model fordetermining the optimal process mean and measuring precision le-vel for a production process. Further, Hong and Elsayed (1999)developed a model for jointly determining the optimum processmean and cutoff value on the observed characteristic when mea-surement error was present. Further, Pfeifer (1999) used electronicspreadsheet program to find the solution. Hong et al. (1999) exam-ined the case when there are several markets with different price/cost structures. Recently, the optimal target value and varianceusing Taguchi loss function is found by Rahim and Shaibu (2000)and Lee et al. (2004). The effect of variance reduction and processcapability on the optimal target value is explored by Kim et al.(2000). Moreover, Williams et al. (2000) examined processimprovement alternatives for a container-filling process. They con-sidered reducing the process setup cost, reducing the frequency ofthe out-of-control signals and reducing the process variation. Teer-avaraprug and Cho (2002) designed the most economical processtarget levels for multiple performance variables. Rahim et al.,2002) obtained the optimum target mean and variance for a contin-uous production process. Lee and Elsayed (2002) determined theoptimum process mean and screening limits of a surrogate variableassociated with product quality under a two-stage screening proce-dure. Shao et al. (2000) developed strategies for determining theoptimal process mean for industrial processes when rejected goodscan be held and sold to other customers in the same market at a la-ter time. Recently, Bowling et al. (2004) studied the targeting prob-lem in the context of multi-stage serial production process. Leeet al., 2005) and Hong et al. (2006) investigated the optimal processmean for variety of production processes. Al-Sultan and Pulak(1997) presented a model for finding the optimal mean of a fillingprocess under rectifying inspection. Rahim and Al-Sultan (2000)considered the problem of joint determination of the optimal targetmean and variance. Other researchers including Gibra (1974), Arce-lus and Banerjee (1985), Rahim and Banerjee (1988), and Al-Sultanand Al-Fawzan (1997), considered the targeting problem when theprocess mean is time dependent. A reverse programming routinethat identifies the relationship between the process mean and thesettings within an experimental factor space was established byGoethals and Cho (2011). Chen and Lai (2007) developed a modelthat finds the economic process mean based on quadratic qualityloss function and rectifying inspection plan. Also, Based on qualityloss function, Chen and Kao (2009) determined the optimal processmean and screening limits. Hong and Cho (2007) found the optimalprocess mean and tolerance limits with measurement errors undermulti-decision alternatives.

M.A. Darwish et al. / European Journal of Operational Research 226 (2013) 481–490 483

Integrating the targeting problem with inventory/productiondecisions has also attracted the attention of many researchers foryears. For example, Gong et al. (1988) developed integrated target-ing-inventory model where the process mean is constant duringthe production cycle. This model is generalized by Al-Fawzan andHariga (2002) by considering time-dependent process mean. Roanet al. (2000) integrated the issues of production lot size and rawmaterial procurement policy with the targeting problem. Harigaand Al-Fawzan (2005) determined simultaneously the optimal pro-duction lot size and process mean for container-filling processeswith multiple markets. Lee et al. (2007) considered the targeting-inventory problem for a production process where multiple prod-ucts are processed. Moreover, Darwish (2004a) and Darwish(2004b) integrated the targeting problem with single-vendor sin-gle-buyer problem. He assumed an equal size shipment policyand nonconforming items are scraped with no salvage value. Chenand Lai (2007) developed a model that finds the optimal processmean, specification limits, and manufacturing quantity under rec-tifying inspection plan. Chen and Khoo (2009) addressed targeting-inventory for serial production systems under quality loss and rec-tifying inspection plan. Also, Darwish (2009) developed an inte-grated targeting-inventory model for a two-layer supply chain. Inthis model, unequal size shipment policy is used and nonconform-ing containers are reprocessed at a certain cost. Recently, Alkhed-her and Darwish (2013) developed a model that determines theoptimal process mean for a stochastic inventory model under ser-vice level constraint. Park et al., 2011) established a profit modelthat determined the optimal common process mean and screeninglimits for a production process with multiple products.

4. Model assumptions and notation

Consider a manufacturer that orders raw material from a sup-plier and uses it to produce a product. The quality characteristic Xis a measure of the amount of raw material used in the manufactur-ing of the end item. It is assumed that the performance variable X isa ‘‘large-is-better’’ variable. That is an item is classified as conform-ing if X P L where L is a lower specification limit, otherwise, theitem is rejected and scrapped with no salvage value. The practiceof setting only a lower specification limit on a quality characteristiccan be found in many industries. For example, in glass manufactur-ing, raw materials are melted and rolled into sheets with certainthickness. It is common that customers set a lower specificationlimit for the thickness of the sheet and any excess thickness isacceptable (Al-Sultan and Pulak, 2000). Another example is goldplating of silver which is used in the manufacture of jewelry. Thetime until the surface of an item is tarnished depends on the thick-ness of the gold layer. Evidently, the expectation of a customer isexceeded when the thickness of gold layer is larger. In this case,the quality of the item is defined in terms of a lower specificationlimit. In the literature, one can find more practical examples forindustries that require only lower specification limit on a qualitycharacteristic (for instance, steel galvanization industry (Shaoet al., 2000) and fiber industry (Hariga and Al-Fawzan, 2005)).

It is also assumed that the performance variable X follows a nor-mal distribution with an adjustable mean, l, and a constant vari-ance, r2. We have to mention that l is not time dependent, thatis, drift in process mean is not considered. Let / denote the stan-dard normal probability density function, thus the probability ofproducing a conforming item is given by

p ¼ P½X P L� ¼ P Z PL� l

r

� �¼Z 1

L�lr

/ðzÞdz

and the yield rate of the process isk = rp, where r is the productionrate. Suppose that the inventory level of the item is monitored con-

tinuously and that the policy is to start producing a lot of sizeQwhen the inventory level drops to a reorder point R. It is worthmentioning that the probability of producing a conforming item(p), yield rate (k), and lot size (Q) are not random variables becausethey depend on process mean (l) which is not a random variable.Assume that the demand in any given interval of time is a randomvariable whose probability distribution is stationary. Let s be theproduction lead time and Y be a random variable represents thelead time demand with probability distribution f(y). In addition,let A denote the fixed setup cost and h denote the cost of carryinga finished item for one unit of time. In developing the proposedmodel, the following assumptions are used:

(1) Shortage is backordered, this situation corresponds to a cap-tive market and at the wholesale-retail link of some distribu-tion systems (Silver et al., 1998, p. 234).

(2) There is never more than a single order outstanding and themean rate of demand (E[K]) is constant over infinite hori-zon. It is shown in (Hadley and Whitin, 1963, p. 178) thatthe expected number of orders per year is E[K]/Q.

(3) The expected number of orders per year and the expectednumber of backorders incurred per unit time are both inde-pendent of the demand pattern provided that the stochasticprocess generating demand does not change with time. Inother words, using typical cycle approach is valid in thiscase, that is

Expected cost per unit time ¼ Expected cost per cycle� E½K�Q

(4) The demand pattern is modeled by a normal probability dis-tribution. In fact, this is a common assumption in stochasticinventory models. This is because empirically the normaldistribution is adequate for most situations (Silver et al.,1998, p. 273).

The following notation will be used in developing the proposedmodel:

X

A random variable represents the amount of rawmaterial an item receives

l

Process mean r Standard deviation of the amount of raw material

used in an item

L Lower specification limit K A random variable represents the demand per unit

time

E[K] Expected value of the demand per unit time rK Standard deviation of the demand per unit time D Constant component of lead time Q Lot size R Reorder point r Production rate p Probability of producing a conforming item k Yield rate of process (k = rp) s Production lead time (s = Q/k) Y A random variable represents the lead time demand

(Y = K(s + D))

f(y)dy Probability that the lead time demand is between y

and y + dy

E[Y] Expected value of lead time demand

(E[Y] = E[K](s + D))

rY Standard deviation of lead time demand / Standard probability density function of the normal

(continued on next page)

484 M.A. Darwish et al. / European Journal of Operational Research 226 (2013) 481–490

distribution

U Standard cumulative probability function of the

normal distribution

of

A Setup cost

vel

(t)

Ar Ordering cost of raw material

y le s, I

Q λ

h Holding cost for the vendor per item per unit time

tor

item

hr

R

nven

ed

Holding cost for the raw material per unit per unittime

et i

nish

p Fixed penalty cost

N fi

b Fixed production cost (b > 0) a Value-added factor (a P 1)

S

c Unit material cost T Inventory cycle length

TimeΔ τ

NL(u) Normal loss function

T

S A random variable represents safety stock

s

Expected safety stock (s = E[S])

raw

N A random variable represents number of shortages

of

n Expected number of shortages (n = E[N])

evel

(t)

TC

-r

ry l

l,I r μ

Expected total cost per unit time of the integratedmodel

ento

eria

TCHR

Inv

mat

Expected total cost per unit time of the hierarchicalmodel

TCPP

Time

Expected total cost per unit time of the perfect fillingprocess

TCDD

Fig. 1. Evolution of inventory over time for finished items and raw material.

Expected total cost per unit time of the deterministicdemand model

5. Model development

In this section, we present an integrated targeting-inventorymodel where the process mean and inventory decisions are deter-mined jointly. In deriving the costs in this model, we use typicalcycle approach as indicated in previous section. Thus, the costsassociated with this model are itemized as follows:

1. Setup Cost: A cost A is incurred each production run. On theaverage, the number of setups per unit time is E[K]/Q. There-fore, the setup cost per unit time is

SC ¼ AE½K�Q

ð1Þ

2. Holding Cost: Fig. 1 depicts the evolution of net inventory overtime. In depletion period [0,T], the expected net inventory atthe beginning of a cycle is S + Q and S at the end of a cycle,where S is the safety stock. Note that these are also the expectedvalues of the on hand inventory when the expected number ofbackorders can be neglected. In this situation, since theexpected demand rate is constant, the expected on hand inven-tory changes linearly from S + Q to S, and is Q/2 + S, hence theexpected inventory cost per unit time is h(Q/2 + s) where s isthe expected safety stock. We have to indicate that this approx-imation can be found in many textbooks (for example, Hadleyand Whitin (1963)) and it is included here so that the paper isself-contained. This approximation depends on the assumptionthat there is no overshooting (crossing the reorder point) at thebeginning of inventory cycle (see for example Montgomery andJohnson, 1974, p. 60). In addition, in the inventory build-up per-iod [T � s,T], the inventory level increases at a rate k, buildinginventory, until Q conforming units are produced which thenwill be used to fulfill demand (Darwish et al., 2012). Theexpected inventory cost per unit time in the inventory build-up period during production is hQE[K]/2k. Hence, the totalexpected holding cost per unit time is given by

� �� �

HC ¼ h

Q2

1þ E½K�k

þ s ð2Þ

We have to point out that the computation of the expected safetystock s in Eq. (2) depends on the model assumption regardingwhether the shortage is eventually satisfied, partially satisfied orlost. When the shortage is completely backordered, the safety stockS, which is a random variable (because it is a function of Y), is unre-stricted in sign, that is S = R � Y. Hence, the expected value of thesafety stock is R – E[Y], where E[Y] is expected lead time demandand is equal to (D + Q/k)E[K]. Thus,

s ¼ E½S� ¼ R� Dþ Qk

� �E½K� ð3Þ

Hence, the holding cost per unit time can be found from Eqs. (2) and(3) as follows

HC ¼ hQ2

1� E½K�k

� �þ R� DE½K�

� �ð4Þ

3. Shortage Cost: A shortage can occur only if the demand during alead time exceeds the reorder point. Thus, the shortage quan-tity, N, at the end of a cycle is a random variable given by

N ¼Y � R y > R

0 y 6 R

�

and the expected shortage quantity per cycle, n(R), is as follows

nðRÞ ¼Z 1

Rðy� RÞf ðyÞdy ¼ rY NL

R� E½Y �rY

� �ð5Þ

where NL(u) is the standardized normal loss function and is givenby

NLðuÞ ¼Z 1

uðz� uÞ/ðzÞdz ¼ /ðuÞ � uð1�UðuÞÞ

The function NL(u) can be solved numerically and its tables can befound in many texts, for example (Nahmias, 2005). Therefore, theexpected shortage cost per unit time can be setout as

M.A. Darwish et al. / European Journal of Operational Research 226 (2013) 481–490 485

SHC ¼ prY E½K�Q

NLR� E½Y�

rY

� �ð6Þ

4. Inventory Control Costs Related to Raw Material: The costs underthis category include raw material ordering and holding costs.When the producer orders raw material from a supplier, heincurs an ordering cost Ar. In addition, since the vendor ischarged for this raw material as soon as it is stocked in hiswarehouse, he incurs a holding cost for this raw material. Theinventory profile of the raw material at the vendor premisesas a function of time is depicted in Fig. 1. It can be easily verifiedthat the inventory of raw material at a given time t (t 6 s) is

IrðtÞ ¼ rlðs� tÞ ð7Þ

Using Eq. (7) and s = Q/k, the holding cost of raw material per cycleis

Z HCr ¼

s

0hrIrðtÞdt ¼ hrl

2kpQ 2

Hence, the average total cost related to inventory control of rawmaterial per unit time, CR, is given by

CR ¼ArE½K�

Qþ hrlE½K�

2kpQ ð8Þ

5. Direct Production Cost: This category involves cost of producingitems. As in Al-Fawzan and Hariga (2002), Roan et al. (2000))and Gong et al. (1988)), we assume that the direct productioncost is a linear function of the item’s material cost and is givenby g(X) = b + acX, where a P 1. We have to indicate that a isincluded in g(X) so that the model is more general, however, acan be set to 1 whenever appropriate. Consequently, the per-cycle cost of producing Q items, DPC, is r

R s0 gðlÞdt which yields

DPC ¼ Qpðbþ claÞ

As shown in Fig. 1, the amount of raw material in the vendor’s ware-house at t = 0 is Ir(0). Thus, the amount of raw material that the ven-dor receives from the supplier in one cycle is Ir(0), it follows thereforthat the acquisition cost,AC, per cycle is cIr(0) and can be setout asfollows

AC ¼ cQlp

Hence, the total production cost per unit time is as follows:

CP ¼E½K�

QðAC þ DPCÞ ¼ E½K�

pðbþ cðaþ 1ÞlÞ ð9Þ

The expected total cost is the sum of SC, HC, SHC, CR, and CP in Eqs. 1,4, 6, 8 and 9 respectively and is established as follows

TCðQ ;R;lÞ ¼ ðAþ ArÞE½K�Q

þ hQ2

1� E½K�k

� �þ R� DE½K�

� �

þ prY E½K�Q

NLR� E½Y �

rY

� �þ hrlE½K�

2kpQ þ E½K�

pðbþ cðaþ 1ÞlÞ

ð10Þ

6. Solution method

Usually models for targeting problem are highly nonlinear anddifficult to solve analytically and the proposed model is not anexception. This difficulty is mainly due to the probability of confor-mance (p) which depends on the decision variable l. In this sec-tion, a brief description of the computational method is outlined.The decision variables in the model presented in Eq. (10) are theprocess mean l, the lot size Q, and the reorder point R. The objec-

tive is to find the values of Q, R and l that minimize TC(Q,R,l). It isimportant to indicate that the quantities p and k in Eq. (10) are notconstants but function of the process mean l.

The value of Q which minimizes TC (obtained by differentiatingTC with respect to Q and setting the result to 0) is

Q ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2E½K�

Aþ Ar þ prY NL R�E½Y �rY

� h 1� E½K�

k

� þ hrlE½K�

kp

vuuut ð11Þ

Similarly, the optimal value of R is given by

UR� E½Y�

rY

� �¼ 1� hQ

pE½K� ð12Þ

where U R�E½Y �rY

� is the probability that the lead time demand is less

than the reorder point. The procedure is started by using Q0 = EPQ,where EPQ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2E½K�A=hð1� E½K�=rÞ

p. We then find R0 from Eq.

(12). The values Q0 and R0 are used in the model in Eq. (10) to findthe optimal value of process mean l0. That value of l0 is used tocompute p and k which are substituted into Eq. (11) to find Q1,which is then substituted into Eq. (12) to find R1, and so on. Thecomputations should be continued until the difference in two suc-cessive values of TC(Q, R, l) are within a certain level of accuracyd. Convergence generally occurs within six iterations. The statementof the algorithm is as follows:

Step 1: Let i = 1, TC0 ?1, and Q 1 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2E½K�A=hð1� E½K�=rÞ

p.

Step 2: If hQipE½K� P 1, find Ri from Eq. (12). Otherwise, Ri = E[Y].

Step 3: Determine the value of li which minimizes TC in Eq. (10).Step 4: Let TCi = TC.Step 5: If TCi < TCi�1, set Q⁄ = Qi, R⁄ = Ri, and l⁄ = li. Otherwise, go

to Step 6.Step 6: If jTCi � TCi�1j > d, then increase i by 1, go to Step 7. Other-

wise, go to Step 8.Step 7: Determine Qi from Eq. (11) and go to Step 2.Step 8: Stop.

It is important to point out that in Step 3, a one-dimensional di-rect search procedure, such as the uniform, Dichotomous, or gold-en-section search method, can be used to find the optimal processmean. In this paper, interval halving method is utilized because ofits simplicity and efficiency (Bazaraa et al., 1993). The range for thesearch for l is [L,L + zr], where z is a predetermined real number.The reason L is used as the lower bound is that, when l = L, theprobability that the process produces a conforming item is 50%,which is very low in most realistic applications (Roan et al., 2000).

7. Baseline Cases

7.1. Hierarchical model

In order to study the advantages of using the integrated modelpresented in (10), a hierarchical model is developed where the pro-ducer sets the process mean independently of lot sizing decisions.This hierarchical model is used as a baseline case to show the im-pact of integrating the process mean with the production/inven-tory decisions.

7.1.1. Determination of optimal process mean for the hierarchicalmodel

When the process mean is set in isolation of other decisions, theprobability of producing a conforming item increases for high pro-cess mean and consequently the loss due to scrap is reduced. Onthe other hand, a large process mean will increase the raw materialrequirement for producing an item. Thus, the optimal process

486 M.A. Darwish et al. / European Journal of Operational Research 226 (2013) 481–490

mean is a balance point between raw material costs and the cost ofproducing non-conforming items. Thus, the costs included in thismodel are the cost of producing items, and raw material costs.These costs are presented in Section 5, therein; we referred to theirsum as the total production costs CP. Using Eq. (9), the average totalcost per unit time is

C1ðlÞ ¼E½K�

pðbþ cðaþ 1ÞlÞ ð13Þ

The decision variable in C1 is the process mean l and its optimal va-lue is determined numerically because it is not easy to find it inclosed form expression (note that the quantity p, in Eq. (13), isnot a constant, rather, it is a function of l). After determining opti-mal value of l, the production lot size and reorder point are foundaccordingly, this is shown in the next section.

7.1.2. Determination of optimal lot size and reorder point for thehierarchical model

The costs in this case are setup cost, holding cost of finisheditems, and raw material ordering and holding costs. For a fixed va-lue of l, one can use Eqs. 1, 4, 6, and 8 to find the expected totalcost per unit time C2 as follows

C2ðQ ;RÞ ¼ðAþ ArÞE½K�

Qþ h

Q2

1� E½K�k

� �þ R

� �

þ prY E½K�Q

NLR� E½Y�

rY

� �þ hrlE½K�

2kpQ ð14Þ

The optimal values of lot size and reorder point are respectively gi-ven as follows

Q ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2E½K�

Aþ Ar þ prY NL R�E½Y �rY

� h 1� E½K�

k

� þ hrlE½K�

kp

vuuut ð15Þ

and

UR� E½Y�

rY

� �¼ 1� hQ

pE½K� ð16Þ

One can now find the average total cost per unit time of the hierar-chical model, TCHR, as follows

TCHR ¼ C�1 þ C�2 ð17Þ

where C�1 and C�2 are the minimum values of C1 and C2 respectively.One obvious drawback of the hierarchical model is that the produc-tion/inventory decisions are not considered in determining the pro-cess mean, and consequently it may lead to infeasible solution if theproduction yield rate is smaller than the demand rate (k < D). In or-der to find the optimal solution for the hierarchical model, we firstdetermine the optimal process mean l by minimizing C1 in Eq. (13),we then determine the value of k. If k P D, we find the optimal val-ues of Q and R from Eqs. (15) and (16) respectively. On the otherhand, if k < D, then there is no feasible solution.

7.2. Perfect filling-process

In this case, the variation in the amount of material in a pro-duced can is neglected, that is r = 0. In this situation, it is optimalthat the amount filled in the cans is equal to the lower specificationlimit (l⁄ = L) and the probability of conformance is 1. This reducesthe raw material requirement and at the same time no cost associ-ated with the nonconforming items is incurred because all contain-ers are conforming. The model in (10) becomes

TCPPðQ ;RÞ¼ðAþArÞE½K�

Qþh

Q2

1�E½K�r

� �þR

� �

þprY E½K�Q

NLR�E½Y�

rY

� �þhrLE½K�

2rQ þE½K�ðbþcðaþ1ÞLÞ

Q ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2E½K�

Aþ Ar þ prY NL R�E½Y�rY

� h 1� E½K�

r

� þ hr LE½K�

r

vuuut ð18Þ

Similarly, the optimal value of R is given by

UR� E½Y�

rY

� �¼ 1� hQ

pE½K� ð19Þ

Eqs. (18) and (19) can be solved iteratively to find the optimal lotsize and reorder point.

7.3. Deterministic demand

In this case, the randomness in the demand pattern is ignored,that is, rK = 0. The expected total cost per unit time is obtainedby substituting R = rY = 0 into expression (10)

TCDDðQ ;lÞ ¼ðAþ ArÞE½K�

Qþ hQ

21� E½K�

k

� �þ hrlE½K�

2kpQ

þ E½K�pðbþ cðaþ 1ÞlÞ ð20Þ

The optimal Q for a given l can be found by solving @TCDD/@Q = 0.The result is stated as follows.

Q ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2ðAþ ArÞE½K�h 1� E½K�

k

� þ hrlE½K�

kp

vuut ð21Þ

Substituting Q given by (21) into (20), the following expression forthe expected total cost is obtained.

TCDDðlÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ðAþ ArÞE½K� h 1� E½K�

k

� �þ hrlE½K�

kp

� �s

þ E½K�pðbþ cðaþ 1ÞlÞ ð22Þ

It is worthwhile to indicate that the total cost in (22) is a function ofsingle variable l. As a result, a one dimensional search method canbe used to find the optimal process mean.

8. Sensitivity analysis

The effects of filling process variation, demand variation, ex-pected demand rate, raw material cost, holding cost of end items,and process setup cost are investigated in this section. The benefitof using the integrated model in (10) is investigated by comparingits performance with that of the hierarchical model where the ven-dor sets the process mean in isolation of lot sizing decisions. Wedefine the percent benefit (saving) of the integrated model overthe hierarchical model as follows:

e ¼ TCHR � TCTCHR

� 100%

The data in Table 1 is used as the basis for all results unless specifiedotherwise.

8.1. Sensitivity analysis on process parameters

8.1.1. Effect of process variationIn order to investigate the effect of variation of filling process on

the optimal solution, the expected total cost of the integrated

Table 1Basic data.

Lower specification limit L = 1.00Standard deviation of the amount of raw material used in an

itemr = 1.00

Expected value of the demand per unit time E[K] = 300.00Standard deviation of the demand per unit time rK = 100.00Production rate r = 500.00Setup cost A = 50.00Ordering cost of raw material Ar = 20.00Holding cost for the vendor per item per unit time h = 4.00Holding cost for the raw material per unit per unit time hr = 1.00Fixed penalty cost p = 7.00Fixed production cost b = 1.00Value-added factor a = 1.00Unit material cost c = 0.50Constant component of lead time D = 0.01

M.A. Darwish et al. / European Journal of Operational Research 226 (2013) 481–490 487

model when variation exists is compared with perfect filling pro-cess where variation is neglected. We, therefore, define the per-centage increase in the expected total cost due to variation asfollows:

X ¼ TC � TCPP

TC� 100%

Table 2 shows that as the filling process standard deviation (r)increases, the benefit of using the integrated model (e) increases.This is true because for high values of r the likelihood of producinga nonconforming item increases, this leads to high cost of scrapresulting in high total cost. To overcome the high cost of scrap,the process mean may be set at a high target value, however, thisleads to a higher material cost. Therefore, high variation in the fill-ing process leads to higher total cost and the need for integration ismore crucial. For example in the seventh run when r = 1.2, the to-tal cost for the integrated model is almost $2,116 whereas the totalcost for the hierarchical model is approximately $2,303 with a ben-efit of almost 8%. This trend is evident because in the integratedmodel the decisions are taken simultaneously with respect to l,Q, and R as oppose to the hierarchical model where l is found firstand then Q and R are determined accordingly. It is also importantto mention here that when r is larger than 1.6, the solution givenby the hierarchical model is infeasible. This is true because theprobability of producing a conforming item (p) at that level isnot high enough to give high yield rate (k) that is sufficient to sat-isfy demand, that is k < E[K].

Finally, it can be also observed in Table 2 that X increases withr. For example, ignoring the variation in the filling process when ris actually 1, can cause the total cost to be underestimated byapproximately 30%. This indicates that, when the effect of variationin filling process is ignored, the producer significantly underesti-mates the actual cost of the produced items. Thus, it is important

Table 2Effect of filling process variation.

Run # r la p TC TCHR e X

1 0.0 1.00 1.00 1566.50 1578.33 0.75 0.002 0.2 1.40 0.98 1675.51 1705.85 1.78 6.963 0.4 1.66 0.95 1775.12 1821.71 2.56 13.324 0.6 1.87 0.93 1867.77 1931.96 3.32 19.235 0.8 2.03 0.90 1954.84 2041.77 4.26 24.796 1.0 2.15 0.88 2037.36 2158.77 5.62 30.067 1.2 2.24 0.85 2116.17 2302.99 8.11 35.098 1.4 2.29 0.82 2192.16 2609.84 16.00 39.949 1.6 2.30 0.79 2266.64 –a –a 44.69

10 1.8 2.26 0.76 2342.14 –a –a 49.51

a Denotes that the solution given by the hierarchical model is not feasiblebecause k < E[K].

to consider the filling process variation in determining theinventory decisions and the integrated model is more appropriatethan the hierarchical model.

8.1.2. Effect of demand variationIn order to study the effect of demand standard deviation on the

total cost of the integrated model, we compare the performance ofthe integrated model presented in (10) with that in (22) where thedemand is deterministic. The impact of the randomness in demandis quantified by the parameter w which is defined as follows:

w ¼ TC � TCDD

TC� 100%

The value of w measures the magnitude of the underestimationin the expected total cost if the randomness in demand is ignored.From Table 3, one would observe that if demand standard devia-tion (rK) is high, w is significantly high. For example, in run num-ber nine (Table 3), if the actual standard deviation rK = 200, thenthe actual expected total cost is TC = 2840.10. However, whenignoring the variation in demand, the total cost, TCDD = 1762.69 isobtained from Eqs. (20)–(22). Therefore, ignoring uncertainty indemand can cause the total cost to be underestimated by approx-imately 37.9%. Hence, there is significant disruption from using thedeterministic solution in probabilistic environment and therefore,adjustment must be made.

Moreover, it is well known that high demand variation resultsin high shortages and consequently high penalty cost. Also, whenthe variation in demand rK rises, the production lot size Q mustbe high enough to cover fluctuations in demand which gives highinventory cost. Therefore, increasing demand standard deviationleads to increased total cost in both hierarchal and integrated mod-els. In addition, the benefit of the integrated model over hierarchalmodel (e) is significant for high values of rK as shown in Table 3.For instance, when rK = 140, the total cost for the integrated modelis roughly $2,224 while the total cost for the hierarchical model isapproximately $2,785 with a reduction in total cost of almost 20%.

The results also show that demand variation affects the processmean (l). This is evident because shortages depend not only on de-mand uncertainty but also on the probability of producing a con-forming item. Therefore, increasing demand standard deviationleads to higher process mean as shown in Table 3. Finally, we haveto point out that for rK more than 160, the hierarchical modelgives no feasible solution since k < E[K].

8.1.3. Effect of expected demand rateTo study the effect of expected demand rate, we obtained opti-

mal solutions for some selected values of E[K] ranging from 200 to400 with an increment of 20. The results are reported in Table 4. Itis expected that, as the expected demand increases, the total costin both integrated and hierarchal model will increase. This is due

Table 3Effect of demand standard deviation.

Run # rK la p TC TCHR e w

1 20 2.05 0.85 1806.45 1842.08 1.93 2.422 40 2.07 0.86 1855.52 1901.40 2.41 5.003 60 2.10 0.86 1909.19 1969.81 3.08 7.674 80 2.13 0.87 1968.98 2052.07 4.05 10.485 100 2.15 0.88 2037.36 2158.77 5.62 13.486 120 2.18 0.88 2118.93 2324.29 8.84 16.817 140 2.21 0.89 2224.19 2785.33 20.15 20.758 160 2.25 0.89 2394.95 –a –a 26.409 180 2.26 0.90 2709.64 –a –a 34.95

10 200 2.28 0.90 2840.10 –a –a 37.94

a Denotes that the solution given by the hierarchical model is not feasiblebecause k < E[K].

Table 4Effect of expected demand rate.

Run # E[K] la p TC TCHR e

1 200 2.13 0.87 1368.97 1418.92 3.522 220 2.13 0.87 1498.86 1558.72 3.843 240 2.14 0.87 1630.39 1701.93 4.204 260 2.14 0.87 1763.84 1849.16 4.615 280 2.15 0.87 1899.43 2001.15 5.086 300 2.15 0.88 2037.36 2158.77 5.627 320 2.16 0.88 2177.92 2323.18 6.258 340 2.16 0.88 2321.36 2495.98 7.009 360 2.17 0.88 2467.96 2680.28 7.92

10 380 2.17 0.88 2618.04 2926.36 10.5411 400 2.18 0.88 2771.93 –a –a

a Denotes that the solution given by the hierarchical model is not feasiblebecause k < E[K].

Table 5Effect of raw material unit cost.

Run # c l⁄ p TC TCHR e

1 0.2 2.450 0.93 1407.02 1448.00 2.832 0.3 2.310 0.90 1575.63 1632.98 3.513 0.4 2.207 0.89 1738.29 1812.40 4.094 0.5 2.125 0.87 1896.93 1989.14 4.645 0.6 2.058 0.85 2052.61 2165.14 5.206 0.7 2.001 0.84 2206.04 2342.18 5.817 0.8 1.951 0.83 2357.71 2522.24 6.528 0.9 1.907 0.82 2507.98 2708.03 7.399 1.0 1.869 0.81 2657.09 2904.43 8.52

10 1.1 1.833 0.80 2805.30 3121.86 10.14

Table 6Effect of end items holding cost.

Run # h l⁄ p TC TCHR e

1 4.2 2.132 0.871 1923.82 2028.62 5.172 4.4 2.140 0.873 1950.77 2070.09 5.763 4.6 2.147 0.874 1977.82 2114.15 6.454 4.8 2.154 0.876 2005.05 2161.66 7.245 5.0 2.161 0.877 2032.52 2213.92 8.196 5.2 2.168 0.879 2060.29 2272.96 9.367 5.4 2.174 0.880 2088.48 2342.23 10.838 5.6 2.181 0.881 2117.22 2427.96 12.809 5.8 2.188 0.882 2146.56 2542.69 15.58

10 6.0 2.194 0.884 2176.80 2715.56 19.84

Table 7Effect of setup cost.

Run # A l⁄ p TC TCHR e

1 40.0 2.140 0.873 1970.13 2076.73 5.132 50.0 2.152 0.875 2037.36 2158.77 5.623 60.0 2.163 0.878 2099.92 2236.88 6.124 70.0 2.173 0.880 2158.79 2312.33 6.645 80.0 2.181 0.881 2214.58 2386.24 7.196 90.0 2.189 0.883 2267.87 2459.77 7.807 100.0 2.197 0.884 2319.03 2534.28 8.498 110.0 2.203 0.886 2368.42 2611.73 9.329 120.0 2.210 0.887 2416.32 2695.32 10.35

10 130.0 2.216 0.888 2462.99 2791.34 11.76

488 M.A. Darwish et al. / European Journal of Operational Research 226 (2013) 481–490

in part to the need for a higher yield rate to cover the increasingdemand which will directly influence the production cost and con-sequently the total cost, this observation is shown in Table 4.Moreover, when expected demand rate is large, the process meanincreases to raise the process yield rate and, in turn, satisfying thedemand. Another important observation is that the benefit ofintegration (e) is larger as expected demand rate is closer to pro-duction rate. As a result, the importance of the integrated modelis manifested as one gets close to the production rate (r = 500). Itis also worth mentioning that when the demand rate is larger than400, the solution given by the hierarchal model is infeasible be-cause the yield rate is less than the expected demand rate.

8.2. Sensitivity analysis on cost parameters

At the outset, we derive the rate of change of the total cost withrespect to different cost parameters. The partial derivatives of thecost function with respect to raw material unit cost, holding costof end items, and process setup cost are as follows:

@TC@c¼ E½K�ðaþ 1Þl

pð23Þ

@TC@h¼ Q

21� E½K�

k

� �þ R� DE½K�

� �ð24Þ

@TC@A¼ E½K�

Qð25Þ

As can be seen from Eqs. (23)–(25), the rate of change of the totalcost with raw material unit cost depends on the targeting problemdecision (l) and the inventory side of the problem through the ex-pected demand rate. Further, the rate of change of the objectivefunction with respect the holding cost depends on targeting prob-lem parameters through k and Q (note that Q depends on l). It alsodepends on the production/inventory control side through Q, R, Dand E[K]. Similarly, the change in TC with respect to A is functionof targeting (through Q and E[K]) and production/inventory control(through Q). Thus, the interaction between the targeting and theproduction/inventory problems exist. Consequently, the choice ofone decision will affect the other decision. In the next three subsec-tions, we numerically study the effect of c, h and A on the optimaltotal cost.

8.2.1. Effect of raw material unit costWe determined optimal solutions for selected values of raw

material unit cost, c, ranging from 0.2 to 1.1 with an incrementof 0.1 and the results are presented in Table 5. When raw materialunit cost is high, the total material cost per unit time becomes amajor component of the expected total cost function TC. Hence, itis more economical to fill less content into a container to reducematerial cost, this leads to a lower optimal process mean as shownby Table 5. It is evident also that the expected total cost increases

for large values of c. We also observe that the benefit of the inte-grated model increases for high material unit cost.

8.2.2. Effect of end items holding costAs the holding cost of end items (h) increases, the total holding

cost per unit time becomes a significant part of the total cost. Rel-atively speaking, this makes raw material cost less important andconsequently an increase in the optimal target value is expected.This is demonstrated by Table 6. Also, Table 6 shows the total costsof the integrated and hierarchical models increase with h. Further-more, the benefits of the integrated model are more apparentwhen the holding cost h increases.

8.2.3. Effect of process setup costWhen the setup cost A is large, expected total costs of the inte-

grated and hierarchical models increase. This observation is shownin Table 7. In addition, the optimal process mean increases with thesetup cost. This is because the raw material cost becomes less sig-nificant compared to setup cost. Another important observation isthat the savings increase due to using the integrated model overthe hierarchical model.

M.A. Darwish et al. / European Journal of Operational Research 226 (2013) 481–490 489

9. Conclusion

In this paper, the classical stochastic continuous review inven-tory control (Q,R) model is integrated with process targeting prob-lem to determine the optimal lot size, reorder point, and processmean. In the proposed model, the performance variable of theproduct has a lower specification limit, and items that do not con-form to specification limit are scrapped with no salvage value. It isassumed that the performance variable follows a normal distribu-tion with an adjustable mean and constant variance.

A procedure is developed to find the optimal solution of theproposed model. The impact of considering the randomness in de-mand pattern is investigated by comparing the proposed modelwith three baseline models. The first of which considers a hierar-chical model where the vendor determines the process mean inisolation of inventory decisions. This hierarchical model is usedto show the effect of integrating the process mean with produc-tion/inventory decisions when the demand is a random variable.The other baseline case is for the deterministic demand. This spe-cial case is used to explore the effect of variation in demand on theoptimal solution. The last baseline case is for the situation whenthe variation in the filling amount is neglected.

Empirical results confirm that ignoring the randomness in de-mand or filling amount may lead to invalid results. In particular,using deterministic demand leads to significantly underestimatingthe expected total cost, thus, the model would not be applicable toreal world situations and adjustment to stochastic model must bemade. Moreover, the results show that the variation in the processis key parameter in controlling the total cost, and the performanceof a process can be improved considerably by reducing itsvariation.

Acknowledgments

The authors are grateful to the anonymous referees for theirconstructive comments and helpful suggestions. They would likealso to acknowledge Kuwait University.

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