Accepted Manuscript

Stochastics and Statistics

Optimal Selection Of Process Mean For A Stochastic Inventory Model

M.A. Darwish, F. Abdulmalek, M. Alkhedher

PII: S0377-2217(12)00870-3

DOI: http://dx.doi.org/10.1016/j.ejor.2012.11.022

Reference: EOR 11378

To appear in: European Journal of Operational Research

Received Date: 19 December 2011

Accepted Date: 15 November 2012

Please cite this article as: Darwish, M.A., Abdulmalek, F., Alkhedher, M., Optimal Selection Of Process Mean For

A Stochastic Inventory Model, European Journal of Operational Research (2012), doi: http://dx.doi.org/10.1016/

j.ejor.2012.11.022

This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers

we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and

review of the resulting proof before it is published in its final form. Please note that during the production process

errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

OPTIMAL SELECTION OF PROCESS MEAN FOR A STOCHASTIC INVENTORY MODEL

M. A. Darwish, F. Abdulmalek, M. Alkhedher

Department of Industrial and Management Systems Engineering College of Engineering and Petroleum, Kuwait University

P.O. Box 5969, 13060 Safat, Kuwait

e-mail: m.darwish@ku.edu.kw ABSTRACT It is very common to assume deterministic demand in the literature of integrated targeting - inventory models. However, if variability in demand is high, there may be significant disruptions from using the deterministic solution in probabilistic environment. Thus, the model would not be applicable to real world situations and adjustment must be made. The purpose of this paper is to develop a model for integrated targeting - inventory problem when the demand is a random variable. In particular, the proposed model jointly determines the optimal process mean, lot size and reorder point in (Q, R) continuous review model. In order to investigate the effect of uncertainty in demand, the proposed model is compared with three baseline cases. The first of which considers a hierarchical model where the producer determines the process mean and lot-sizing decisions separately. This hierarchical model is used to show the effect of integrating the process targeting with production/inventory decisions. Another baseline case is the deterministic 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 case

1

demonstrates the sensitivity of the total cost with respect to the variation in the process output. Also, a procedure is developed to determine the optimal solution for the proposed models. Empirical results show 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 its variation. KEYWORDS: Quality control, targeting problem, production, demand uncertainty.

2

1. INTRODUCTION The problem of selecting the optimum process mean (targeting problem) is one of the areas in economics of quality control, which has received a lot of interest from researchers in the recent times. Process targeting is often a difficult decision in production processes such as canning/filling, metal plating, grinding, glass industry, fiber industry, and steel industry (Park et al. 2011, Shao et al. 2000, and Hariga and Al-Fawzan 2005). Usually, producers set specification limits on a quality performance measure (weight, volume, concentration, thickness, length, etc.) and an item is classified as conforming when the quality performance measure is within specification limits. Otherwise, the item is classified as nonconforming and may be sold at a reduced price, reprocessed or scrapped (Roan et al. 2000). In USA, federal agencies examined the practice of process mean selection and reported that it is a common trend that many manufacturers select high process mean in order to conform to specifications. This strategy leads to a “give away” cost (Roan et al. 2000). On the other hand, a tight process setting will have less production cost, but high costs of rejection, recycling and reprocessing (Al-Sultan and Pulak 2000). Thus, the targeting problem is a trade-off between material cost and the cost associated with producing nonconforming items. Traditionally, the process mean selection and production/inventory decisions are determined separately. However, the choice of process mean affects the probability that a given produced item is nonconforming. Hence, the process mean determines the production yield rate which, in turn, influences other important production and inventory decisions, in particular, the production lot size and reorder point. Consequently, the process mean and production/inventory decisions should be determined jointly in order 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 customer due to the deviation from the optimum target value, as well as, and perhaps most importantly, providing better products at reduced cost for customers (Darwish 2009). The organization of the paper is as follows; motivation and contribution is discussed in the next section. In section 3, we review the literature. Model assumptions and notation are introduced in section 4 followed by model

3

development. In section 6, solution method is outlined. Then we investigate some baseline cases in section 7. In section 8, we present the empirical results and section 9 concludes the paper.

2. MOTIVATION AND CONTRIBUTION

It is very common to make simplifying assumptions in modeling real-world phenomenon. This is important in order to render the mathematics tractable. On the other hand, we cannot make too many simplifying assumptions, for then our conclusions, obtained from the mathematical model would not be applicable to real world situations (Ross, 2007). In integrated targeting-inventory models, it is usually assumed that demand is deterministic. In reality however, deterministic models are only approximations and the goodness of these approximations depends on the degree of uncertainty in the demand pattern (Nahmias 2005). Thus, there exists a need to develop a model that addresses the limitations imposed by this assumption. Unlike the models developed in the integrated targeting-inventory literature, in this paper, the targeting problem is integrated with inventory decisions when randomness is introduced into the demand pattern. The proposed model simultaneously determines the optimal decisions regarding process mean, production lot size, and reorder point in (Q, R) continuous review model. In general, there are three levels of decisions related to production processes: (1) long-term (strategic) (2) Intermediate-term (tactical), and (3) short-term (operational). Usually, these decision levels are optimized independently due to the nature of their varying time horizon. However, these decision levels affect and are affected by each other. For instance, the selection of process mean (which is operational decision) affects the conformance rate which determines number of conforming/nonconforming items produced. Therefore, the determination of production lot size (which is tactical decision) should compensate for the nonconforming items produced during a production run. Consequently, integrating the process mean targeting and production planning and inventory control decisions is expected to reduce the total cost. Another important aspect of this integration is that it jointly addresses quality control issues through process targeting and production planning and inventory control issues.

4

After deriving the proposed model, it is compared with its deterministic counterpart by studying three baseline cases. The first of which considers a hierarchical model where the producer determines the process mean and lot-sizing decisions separately. This hierarchical model is used to show the effect of integrating the process targeting with production/inventory decisions. The other baseline case is for deterministic demand which is used to explore the effect of variation in demand on the optimal solution. Then we use the baseline case where variation in the filling amount can be neglected. This case shows the sensitivity of the total cost with respect to the variation in the process output. The proposed models show that ignoring randomness in demand may lead to underestimating 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 investigated and discussed for more than 60 years. Springer (1951) was the first to consider the problem of process targeting, in which the process mean that minimizes the total cost is obtained. Bettes (1962) presented a similar model by determining the optimal process mean and upper specification limit simultaneously. In this model, the nonconforming items are scrapped with no salvage value. This model is extended by Hunter and Kartha (1977) by considering the problem of selecting the optimum target value when nonconforming items are sold at a secondary market. Then, Bisgaard et al. (1984) modified Hunter and Kartha's model to a situation where nonconforming products are sold at a price proportional to the amount of material used. Golhar (1987) considered the optimal determination of the optimum process mean when conforming items are sold at a fixed price and nonconforming cans are emptied and refilled at a reprocessing cost. Boucher and Jafari (1991) extended this line of research by introducing a sampling plan as opposed to 100% inspection; in this model the effect of single sampling plan on the optimal set point of a filling operation is examined. Al-Sultan (1994) extended the model developed by Boucher and Jafari (1991) for two machines in series using a sampling inspection plan. Tang and Lo (1993), Lee and Jang (1997) and Lee et al. (2001) used a surrogate variable in inspection and determined the optimum process mean and the screening limits. Duffuaa and Siddiqui (2003) discussed a model for a multi class

5

screening when measurement error existed and Darwish and Duffuaa (2010) developed a model that determines the optimal process mean and sampling plan's parameters. The model maximizes producer expected profit while protecting the consumer through a constraint on the probability of accepting lots with low incoming quality. Moreover, Elsayed and Chen (1993) determined optimum levels of process parameters for items with multiple characteristics. Arcelus and Rahim (1994) provided joint optimal settings for variable and attribute target means. Cheng and Chung (1996) considered a model for determining the optimal process mean and measuring precision level for a production process. Further, Hong and ElSayed (1999) developed a model for jointly determining the optimum process mean and cutoff value on the observed characteristic when measurement error was present. Further, Pfeifer (1999) used electronic spreadsheet program to find the solution. Hong et al. (1999) examined the case when there are several markets with different price/cost structures. Recently, the optimal target value and variance using Taguchi loss function is found by Rahim and Shaibu (2000) and Lee et al. (2004). The effect of variance reduction and process capability on the optimal target value is explored by Kim et al. (2000). Moreover, Williams et al. (2000) examined process improvement alternatives for a container-filling process. They considered reducing the process setup cost, reducing the frequency of the out-of-control signals and reducing the process variation. Teeravaraprug and Cho (2002) designed the most economical process target levels for multiple performance variables. Rahim et al. (2002) obtained the optimum target mean and variance for a continuous production process. Lee and Elsayed (2002) determined the optimum process mean and screening limits of a surrogate variable associated with product quality under a two-stage screening procedure. Shao et al. (2000) developed strategies for determining the optimal process mean for industrial processes when rejected goods can be held and sold to other customers in the same market at a later time. Recently, Bowling et al. (2004) studied the targeting problem in the context of multi-stage serial production process. Lee et al. (2005) and Hong et al. (2006) investigated the optimal process mean for variety of production processes. Al-Sultan and Pulak (1997) presented a model for finding the optimal mean of a filling process under rectifying inspection. Rahim and Al-Sultan (2000) considered the problem of joint determination of the optimal target mean and variance. Other researchers including Gibra (1974), Arcelus and Banerjee (1985), Rahim and Banerjee (1988), Al-Sultan and Al-Fawzan (1997), considered

6

the targeting problem when the process mean is time dependent. A reverse programming routine that identifies the relationship between the process mean and the settings within an experimental factor space was established by Goethals and Cho (2011). Chen and Lai (2007) developed a model that finds the economic process mean based on quadratic quality loss function and rectifying inspection plan. Also, Based on quality loss function, Chen and Kao (2009) determined the optimal process mean and screening limits. Hong, S.H., and Cho, B. R. (2007) found the optimal process mean and tolerance limits with measurement errors under multi-decision alternatives. Integrating the targeting problem with inventory/production decisions has also attracted the attention of many researchers for years. For example, Gong et al. (1988) developed integrated targeting-inventory model where the process mean is constant during the production cycle. This model is generalized by Al-Fawzan and Hariga (2002) by considering time-dependent process mean. Roan et al. (2000) integrated the issues of production lot size and raw material procurement policy with the targeting problem. Hariga and Al-Fawzan (2005) determined simultaneously the optimal production lot size and process mean for container-filling processes with multiple markets. Lee et al. (2007) considered the targeting-inventory problem for a production process where multiple products are processed. Moreover, Darwish (2004a and 2004b) integrated the targeting problem with single-vendor single-buyer problem. He assumed an equal size shipment policy and nonconforming items are scraped with no salvage value. Chen and Lai (2007) developed a model that finds the optimal process mean, specification limits, and manufacturing quantity under rectifying inspection plan. Chen and Khoo (2009) addressed targeting-inventory for serial production systems under quality loss and rectifying inspection plan. Also, Darwish (2009) developed an integrated targeting-inventory model for a two-layer supply chain. In this model, unequal size shipment policy is used and nonconforming containers are reprocessed at a certain cost. Recently, Alkhedher and Darwish (2012) developed a model that determines the optimal process mean for a stochastic inventory model under service level constraint. Park et al. (2011) established a profit model that determined the optimal common process mean and screening limits for a production process with multiple products. 4. MODEL ASSUMPTIONS AND NOTATION

7

Consider a manufacturer that orders raw material from a supplier and uses it to produce a product. The quality characteristic X is a measure of the amount of raw material used in the manufacturing of the end item. It is assumed that the performance variable X is a “large-is-better” variable. That is an item is classified as conforming if X ≥ L where L is a lower specification limit, otherwise, the item is rejected and scrapped with no salvage value. The practice of setting only a lower specification limit on a quality characteristic can be found in many industries. For example, in glass manufacturing, raw materials are melted and rolled into sheets with certain thickness. It is common that customers set a lower specification limit for the thickness of the sheet and any excess thickness is acceptable (Al-Sultan and Pullak 2000). Another example is gold plating of silver which is used in the manufacture of jewelry. The time until the surface of an item is tarnished depends on the thickness of the gold layer. Evidently, the expectation of a customer is exceeded when the thickness of gold layer is larger. In this case, the quality of the item is defined in terms of a lower specification limit. In the literature, one can find more practical examples for industries that require only lower specification limit on a quality characteristic (for instance, steel galvanization industry (Shao et al. 2000) and fiber industry (Hariga and Al-Fawzan 2005)). It is also assumed that the performance variable X follows a normal distribution with an adjustable mean, µ, and a constant variance, σ2. We have to mention that µ is not time dependent, that is, drift in process mean is not considered. Let φ denote the standard normal probability density function, thus the probability of producing a conforming item is given by

[ ] ( )L

Lp P X L P Z z dzμ

σ

μ φσ

∞

−

−⎡ ⎤= ≥ = ≥ =⎢ ⎥⎣ ⎦ ∫

and the yield rate of the process is rp=λ , where r is the production rate. Suppose that the inventory level of the item is monitored continuously and that the policy is to start producing a lot of size Q when the inventory level drops to a reorder point R. It is worth mentioning that the probability of producing a conforming item (p), yield rate (λ), and lot size (Q) are not random variables because they depend on process mean (µ) which is not a random variable. Assume that the demand in any given interval of time is a random variable whose probability distribution is stationary. Let τ be the

8

production lead time and Y be a random variable represents the lead time demand with probability distribution f(y). In addition, let A denote the fixed setup cost and h denote the cost of carrying a finished item for one unit of time. In developing the proposed model, the following assumptions are used:

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

(2) There is never more than a single order outstanding and the mean rate of demand (E[Λ]) is constant over infinite horizon. It is shown in (Hadley and Whitin, 1963, p. 178) that the expected number of orders per year is E[Λ]/Q.

(3) The expected number of orders per year and the expected number of backorders incurred per unit time are both independent of the demand pattern provided that the stochastic process generating demand does not change with time. In other words, using typical cycle approach is valid in this case, that is

QE ][cycleper cost Expectedunit timeper cost Expected Λ

×=

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

The following notation will be used in developing the proposed model: X a random variable represents the amount of raw material an item µ process mean σ standard deviation of the amount of raw material used in an item, L lower specification limit, Λ A random variable represents the demand per unit time, E[Λ] expected value of the demand per unit time, σΛ standard deviation of the demand per unit time, Δ constant component of lead time, Q lot size,

R reorder point, r production rate, p probability of producing a conforming item, λ yield rate of process (λ = rp) τ production lead time (τ =Q/λ), Y a random variable represents the lead time demand (Y = Λ(τ+Δ)), 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[Λ](τ+Δ)), σY standard deviation of lead time demand,

9

5. MODEL DEVELOPMENT In this section, we present an integrated targeting-inventory model where the process mean and inventory decisions are determined jointly. In deriving the costs in this model, we use typical cycle approach as indicated in previous section. Thus, the costs associated with this model are itemized as follows: 1. Setup Cost: A cost A is incurred each production run. On the average, the number of setups per unit time is E[Λ]/Q. Therefore, the setup cost per unit time is

[ ]AESC

QΛ

= (1)

2. Holding Cost: Figure 1 depicts the evolution of net inventory over time. In depletion period [0, T], the expected net inventory at the beginning of a

φ standard probability density function of the normal distribution, Φ standard cumulative probability function of the normal distribution, A setup cost, Ar ordering cost of raw material, h holding cost for the vendor per item per unit time, hr holding cost for the raw material per unit per unit time, π fixed penalty cost, b fixed production cost (b>0), α value-added factor (α ≥ 1), c unit material cost, T inventory cycle length, NL(u) normal loss function, S a random variable represents safety stock, s expected safety stock (s = E[S] ), N A random variable represents number of shortages, n expected number of shortages (n = E[N]), TC expected total cost per unit time of the integrated model, TCHR expected total cost per unit time of the hierarchical model, TCPP expected total cost per unit time of the perfect filling process, TCDD expected total cost per unit time of the deterministic demand model.

10

cycle is S+Q and S at the end of a cycle, where S is the safety stock. Note that these are also the expected values of the on hand inventory when the expected number of backorders can be neglected. In this situation, since the expected demand rate is constant, the expected on hand inventory changes linearly from S+Q to S, and is Q/2 + S, hence the expected inventory cost per unit time is h(Q/2 + s) where s is the expected safety stock. We have to indicate that this approximation can be found in many textbooks (for example, Hadley and Whitin (1963)) and it is included here so that the paper is self-contained. This approximation depends on the assumption that there is no overshooting (crossing the reorder point) at the beginning of inventory cycle (see for example Montgomery and Johnson, 1974, p. 60). In addition, in the inventory build-up period [T-τ, T], the inventory level increases at a rate λ, building inventory, until Q conforming units are produced which then will be used to fulfill demand (Darwish et al. 2011). The expected inventory cost per unit time in the inventory build-up period during production is hQE[Λ]/2λ. Hence, the total expected holding cost per unit time is given by

[ ]1

2Q EHC h s

λ⎡ ⎤Λ⎛ ⎞= + +⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

(2)

We have to point out that the computation of the expected safety stock s in Equation (2) depends on the model assumption regarding whether the shortage is eventually satisfied, partially satisfied or lost. When the shortage is completely backordered, the safety stock S, which is a random variable (because it is a function of Y), is unrestricted in sign, that is S = R – Y. Hence, the expected value of the safety stock is R – E[Y], where E[Y] is expected lead time demand and is equal to (Δ+Q/λ) E[Λ]. Thus,

[ ] [ ]Qs E S R Eλ

⎛ ⎞= = − Δ + Λ⎜ ⎟⎝ ⎠

(3)

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

[ ]1 [ ]2Q EHC h R E

λ⎡ ⎤Λ⎛ ⎞= − + −Δ Λ⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

(4)

11

____________________________________ Please insert Figure 1 about here

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

0Y R y R

Ny R

− >⎧= ⎨ ≤⎩

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

[ ]( ) ( ) ( ) Y

YR

R E Yn R y R f y dy NLσσ

∞ ⎛ ⎞−= − = ⎜ ⎟

⎝ ⎠∫ (5)

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

( ) ( ) ( ) ( ) (1 ( ))u

NL u z u z dz u u uφ φ∞

= − = − −Φ∫

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

[ ] [ ]Y

Y

E R E YSHC NLQ

πσσ

⎛ ⎞Λ −= ⎜ ⎟

⎝ ⎠ (6)

4. Inventory Control Costs Related to Raw Material: The costs under this category include raw material ordering and holding costs. When the producer orders raw material from a supplier, he incurs an ordering cost Ar. In addition, since the vendor is charged for this raw material as soon as it is stocked in his warehouse, he incurs a holding cost for this raw material. The inventory profile of the raw material at the vendor premises as a function of time is depicted in Figure 1. It can be easily verified that the inventory of raw material at a given time t (t ≤ τ) is

12

( ) ( )rI t r tμ τ= − (7) Using Equation (7) and τ = Q/λ, the holding cost of raw material per cycle is

2

0

( )2

rr r r

hHC h I t dt Qp

τ μλ

= =∫

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

[ ] [ ]2

r rR

A E h EC QQ p

μλ

Λ Λ= + (8)

5. Direct Production Cost: This category involves cost of producing items. As in Al-Fawzan and Hariga (2002), Roan et al. (2000) and Gong et al. (1988), we assume that the direct production cost is a linear function of the item’s material cost and is given by g(X) = b + αcX, where α ≥ 1. We have to indicate that α is included in g(X) so that the model is more general, however, α can be set to 1 whenever appropriate. Consequently, the per-

cycle cost of producing Q items, DPC, is 0

( )r g dtτ

μ∫ which yields

( )QDPC b cp

μα= +

As shown in Figure 1, the amount of raw material in the vendor’s warehouse at t =0 is Ir(0). Thus, the amount of raw material that the vendor receives from the supplier in one cycle is Ir(0), it follows therefor that the acquisition cost, AC, per cycle is cIr(0) and can be setout as follows

cQACpμ

=

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

( )[ ] [ ] ( ( 1) )P

E EC AC DPC b cQ p

α μΛ Λ= + = + + (9)

13

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

( ) [ ] [ ]( , , ) 1 [ ]2

[ ] [ ][ ] [ ] ( ( 1) )2

r

Y r

Y

A A E Q ETC Q R h R EQ

E h ER E Y ENL Q b cQ p p

μλ

πσ μ α μσ λ

+ Λ ⎡ ⎤Λ⎛ ⎞= + − + − Δ Λ +⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦⎛ ⎞Λ Λ− Λ

+ + + +⎜ ⎟⎝ ⎠

(10)

6. SOLUTION METHOD Usually models for targeting problem are highly nonlinear and difficult to solve analytically and the proposed model is not an exception. This difficulty is mainly due to the probability of conformance (p) which depends on the decision variable µ. In this section, a brief description of the computational method is outlined. The decision variables in the model presented in Equation (10) are the process mean μ, the lot size Q, and the reorder point R. The objective is to find the values of Q, R and μ that minimize TC(Q, R, μ). It is important to indicate that the quantities p and λ in Equation (10) are not constants but function of the process mean μ. The value of Q which minimizes TC (obtained by differentiating TC with respect to Q and setting the result to 0) is

[ ]

2 [ ][ ][ ]1

r YY

r

R E YA A NLQ E

h EEhp

πσσ

μλ λ

⎛ ⎞−+ + ⎜ ⎟

⎝ ⎠= ΛΛΛ⎛ ⎞− +⎜ ⎟

⎝ ⎠

(11)

Similarly, the optimal value of R is given by

[ ] 1[ ]Y

R E Y hQEσ π

⎛ ⎞−Φ = −⎜ ⎟ Λ⎝ ⎠

(12)

where [ ]

Y

R E Yσ

⎛ ⎞−Φ⎜ ⎟⎝ ⎠

is the probability that the lead time demand is less than

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

14

( )2 [ ] / 1 [ ] /EPQ E A h E r= Λ − Λ . We then find R0 from Equation (12). The values Q0 and R0 are used in the model in Equation (10) to find the optimal value of process mean μ0. That value of μ0 is used to compute p and λ which are substituted into Equation (11) to find Q1, which is then substituted into Equation (12) to find R1, and so on. The computations should be continued until the difference in two successive values of TC(Q, R, μ) are within a certain level of accuracy δ. Convergence generally occurs within six iterations. The statement of the algorithm is as follows:

Step 1: Let i = 1, TC0 → ∞, and ( )rEhAEQ /][1/][21 Λ−Λ= .

Step 2: If 1][≥

ΛEhQi

π, find Ri from Equation (12). Otherwise, Ri = E[Y].

Step 3: Determine the value of μi which minimizes TC in Equation (10). Step 4: Let TCi = TC. Step 5: If TCi < TCi-1, set Q* = Qi, R* = Ri, and μ* = μi. Otherwise, go

to Step 6. Step 6: If | TCi - TCi-1| > δ, then increase i by 1, go to Step 7.

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

It is important to point out that in Step 3, a one-dimensional direct search procedure, such as the uniform, Dichotomous, or golden-section search method, can be used to find the optimal process mean. In this paper, interval halving method is utilized because of its simplicity and efficiency (Bazaraa et al. 1993). The range for the search for μ is [ L, L+zσ ], where z is a predetermined real number. The reason L is used as the lower bound is that, when μ = L, the probability 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 model presented in (10), a hierarchical model is developed where the producer sets the process

15

mean independently of lot sizing decisions. This hierarchical model is used as a baseline case to show the impact of integrating the process mean with the production/inventory decisions. Determination of Optimal Process Mean for the Hierarchical Model When the process mean is set in isolation of other decisions, the probability of producing a conforming item increases for high process mean and consequently the loss due to scrap is reduced. On the other hand, a large process mean will increase the raw material requirement for producing an item. Thus, the optimal process mean is a balance point between raw material costs and the cost of producing non-conforming items. Thus, the costs included in this model are the cost of producing items, and raw material costs. These costs are presented in Section 5, therein; we referred to their sum as the total production costs CP. Using Equation (9), the average total cost per unit time is

))1((][)(1 μαμ ++Λ

= cbp

EC (13)

The decision variable in C1 is the process mean μ and its optimal value is determined numerically because it is not easy to find it in closed form expression (note that the quantity p, in Equation (13), is not a constant, rather, it is a function of μ). After determining optimal value of μ, the production lot size and reorder point are found accordingly, this is shown in the next section. Determination of Optimal Lot Size and Reorder Point for the Hierarchical Model The costs in this case are setup cost, holding cost of finished items, and raw material ordering and holding costs. For a fixed value of μ, one can use Equations (1), (4), (6), and (8) to find the expected total cost per unit time C2 as follows

16

( )2

[ ] [ ][ ] [ ]( , ) 12

[ ]2

r Y

Y

r

A A E EQ E R E YC Q R h R NLQ Q

h E Qp

πσλ σ

μλ

+ Λ ⎛ ⎞⎡ ⎤ ΛΛ −⎛ ⎞= + − + + +⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦ ⎝ ⎠Λ

(14)

The optimal values of lot size and reorder point are respectively given as follows

[ ]

2 [ ][ ][ ]1

r YY

r

R E YA A NLQ E

h EEhp

πσσ

μλ λ

⎛ ⎞−+ + ⎜ ⎟

⎝ ⎠= ΛΛΛ⎛ ⎞− +⎜ ⎟

⎝ ⎠

(15)

and

[ ] 1[ ]Y

R E Y hQEσ π

⎛ ⎞−Φ = −⎜ ⎟ Λ⎝ ⎠

(16)

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

* *HR 1 2TC C C= + (17)

where *

1C and *2C are the minimum values of C1 and C2 respectively. One

obvious drawback of the hierarchical model is that the production/inventory decisions are not considered in determining the process mean, and consequently it may lead to infeasible solution if the production yield rate is smaller than the demand rate (λ<D). In order to find the optimal solution for the hierarchical model, we first determine the optimal process mean μ by minimizing C1 in Equation (13), we then determine the value of λ. If λ ≥ D, we find the optimal values of Q and R from Equations (15) and (16) respectively. On the other hand, if λ < D, then there is no feasible solution. 7.2 Perfect Filling-Process In this case, the variation in the amount of material in a produced can is neglected, that is σ = 0. In this situation, it is optimal that the amount filled

17

in the cans is equal to the lower specification limit (μ* = L) and the probability of conformance is 1. This reduces the raw material requirement and at the same time no cost associated with the nonconforming items is incurred because all containers are conforming. The model in (10) becomes

( )PP

[ ] [ ]( , ) 12

[ ] [ ][ ] [ ]( ( 1) )2

r

Y r

Y

A A E Q ETC Q R h RQ r

E h LER E YNL Q E b c LQ r

πσ ασ

+ Λ ⎡ ⎤Λ⎛ ⎞= + − + +⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦⎛ ⎞Λ Λ−

+ + Λ + +⎜ ⎟⎝ ⎠

[ ]

2 [ ][ ][ ]1

r YY

r

R E YA A NLQ E

h LEEhr r

πσσ

⎛ ⎞−+ + ⎜ ⎟

⎝ ⎠= ΛΛΛ⎛ ⎞− +⎜ ⎟

⎝ ⎠

(18)

Similarly, the optimal value of R is given by

[ ] 1[ ]Y

R E Y hQEσ π

⎛ ⎞−Φ = −⎜ ⎟ Λ⎝ ⎠

(19)

Equations (18) and (19) can be solved iteratively to find the optimal lot size and reorder point. 7.3 Deterministic Demand In this case, the randomness in the demand pattern is ignored, that is, 0σΛ = . The expected total cost per unit time is obtained by substituting 0YR σ= = into expression (10)

( )DD

[ ] [ ][ ]( , ) 12 2

[ ] ( ( 1) )

r rA A E h EhQ ETC Q QQ p

E b cp

μμλ λ

α μ

+ Λ ΛΛ⎛ ⎞= + − + +⎜ ⎟⎝ ⎠

Λ+ +

(20)

18

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

( )2 [ ][ ][ ]1

r

r

A A EQ

h EEhp

μλ λ

+ Λ=

ΛΛ⎛ ⎞− +⎜ ⎟⎝ ⎠

(21)

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

( )DD[ ][ ] [ ]( ) 2 [ ] 1 ( ( 1) )r

rh EE ETC A A E h b c

p pμμ α μ

λ λ⎡ ⎤ΛΛ Λ⎛ ⎞= + Λ − + + + +⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

(22)

It is worthwhile to indicate that the total cost in (22) is a function of single variable μ. As a result, a one dimensional search method can be used to find the optimal process mean. 8. SENSITIVITY ANALYSIS The effects of filling process variation, demand variation, expected demand rate, raw material cost, holding cost of end items, and process setup cost are investigated in this section. The benefit of using the integrated model in (10) is investigated by comparing its performance with that of the hierarchical model where the vendor sets the process mean in isolation of lot sizing decisions. We define the percent benefit (saving) of the integrated model over the hierarchical model as follows:

%100×−

=HR

HR

TCTCTCε

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

____________________________________ Please insert Table 1 about here

____________________________________

8.1 Sensitivity Analysis on Process Parameters 8.1.1 Effect of Process Variation

19

In order to investigate the effect of variation of filling process on the optimal solution, the expected total cost of the integrated model when variation exists is compared with perfect filling process where variation is neglected. We, therefore, define the percentage increase in the expected total cost due to variation as follows:

%100×

−=Ω

TCTCTC PP

Table 2 shows that as the filling process standard deviation (σ) increases, the benefit of using the integrated model (ε) increases. This is true because for high values of σ the likelihood of producing a nonconforming item increases, this leads to high cost of scrap resulting in high total cost. To overcome the high cost of scrap, the process mean may be set at a high target value, however, this leads to a higher material cost. Therefore, high variation in the filling process leads to higher total cost and the need for integration is more crucial. For example in the seventh run when σ = 1.2, the total cost for the integrated model is almost $2,116 whereas the total cost for the hierarchical model is approximately $2,303 with a benefit of almost 8%. This trend is evident because in the integrated model the decisions are taken simultaneously with respect to μ, Q, and R as oppose to the hierarchical model where μ is found first and then Q and R are determined accordingly. It is also important to mention here that when σ is larger than 1.6, the solution given by the hierarchical model is infeasible. This is true because the probability of producing a conforming item (p) at that level is not high enough to give high yield rate (λ) that is sufficient to satisfy demand, that is λ < E[Λ].

____________________________________ Please insert Table 2 about here

____________________________________ Finally, it can be also observed in Table 2 that Ω increases with σ. For example, ignoring the variation in the filling process when σ is actually 1, can cause the total cost to be underestimated by approximately 30%. This indicates that, when the effect of variation in filling process is ignored, the producer significantly underestimates the actual cost of the produced items. Thus, it is important to consider the filling process variation in determining

20

the inventory decisions and the integrated model is more appropriate than the hierarchical model. 8.1.2 Effect of Demand Variation In order to study the effect of demand standard deviation on the total cost of the integrated model, we compare the performance of the integrated model presented in (10) with that in (22) where the demand is deterministic. The impact of the randomness in demand is quantified by the parameter ψ which is defined as follows:

DD 100%TC TCTC

ψ −= ×

The value of ψ measures the magnitude of the underestimation in the expected total cost if the randomness in demand is ignored. From Table 3, one would observe that if demand standard deviation (σΛ) is high, ψ is significantly high. For example, in run number 9 (Table 3), if the actual standard deviation σΛ = 200, then the actual expected total cost is TC = 2840.10. However, when ignoring the variation in demand, the total cost, TCDD = 1762.69 is obtained from Equations 20, 21 and 22. Therefore, ignoring uncertainty in demand can cause the total cost to be underestimated by approximately 37.9%. Hence, there is significant disruption from using the deterministic solution in probabilistic environment and therefore, adjustment must be made. Moreover, it is well known that high demand variation results in high shortages and consequently high penalty cost. Also, when the variation in demand σΛ rises, the production lot size Q must be high enough to cover fluctuations in demand which gives high inventory cost. Therefore, increasing demand standard deviation leads to increased total cost in both hierarchal and integrated models. In addition, the benefit of the integrated model over hierarchal model (ε) is significant for high values of σΛ as shown in Table 3. For instance, when σΛ = 140, the total cost for the integrated model is roughly $2,224 while the total cost for the hierarchical model is approximately $2,785 with a reduction in total cost of almost 20%.

21

The results also show that demand variation affects the process mean (µ). This is evident because shortages depend not only on demand uncertainty but also on the probability of producing a conforming item. Therefore, increasing demand standard deviation leads to higher process mean as shown in Table 3. Finally, we have to point out that for σΛ more than 160, the hierarchical model gives no feasible solution since λ < E[Λ].

____________________________________ Please insert Table 3 about here

____________________________________ 8.1.3 Effect of Expected Demand Rate To study the effect of expected demand rate, we obtained optimal solutions for some selected values of E[Λ] ranging from 200 to 400 with an increment of 20. The results are reported in Table 4. It is expected that, as the expected demand increases, the total cost in both integrated and hierarchal model will increase. This is due in part to the need for a higher yield rate to cover the increasing demand which will directly influence the production cost and consequently the total cost, this observation is shown in Table 4. Moreover, when expected demand rate is large, the process mean increases to raise the process yield rate and, in turn, satisfying the demand. Another important observation is that the benefit of integration (ε) is larger as expected demand rate is closer to production rate. As a result, the importance of the integrated model is manifested as one gets close to the production rate (r = 500). It is also worth mentioning that when the demand rate is larger than 400, the solution given by the hierarchal model is infeasible because the yield rate is less than the expected demand rate.

____________________________________ Please insert Table 4 about here

____________________________________

8.2 Sensitivity Analysis on Cost Parameters At the outset, we derive the rate of change of the total cost with respect to different cost parameters. The partial derivatives of the cost function with respect to raw material unit cost, holding cost of end items, and process setup cost are as follows:

22

[ ]( )

pE

cTC μα 1+Λ

=∂∂ (23)

[ ] [ ]⎥⎦

⎤⎢⎣

⎡ΛΔ−+⎟

⎠⎞

⎜⎝⎛ Λ−=

∂∂ EREQ

hTC

λ1

2 (24)

[ ]Q

EA

TC Λ=

∂∂ (25)

As can be seen from Equations (23), (24), and (25), the rate of change of the total cost with raw material unit cost depends on the targeting problem decision (µ) and the inventory side of the problem through the expected demand rate. Further, the rate of change of the objective function with respect the holding cost depends on targeting problem parameters through λ and Q (note that Q depends on µ). It also depends on the production/inventory control side through Q, R, Δ and E[Λ]. Similarly, the change in TC with respect to A is function of targeting (through Q and E[Λ]) and production/inventory control (through Q). Thus, the interaction between the targeting and the production/inventory problems exist. Consequently, the choice of one decision will affect the other decision. In the next three subsections, we numerically study the effect of c, h and A on the optimal total cost. 8.2.1 Effect of Raw Material Unit Cost We determined optimal solutions for selected values of raw material unit cost, c, ranging from 0.2 to 1.1 with an increment of 0.1 and the results are presented in Table 5. When raw material unit cost is high, the total material cost per unit time becomes a major component of the expected total cost function TC. Hence, it is more economical to fill less content into a container to reduce material cost, this leads to a lower optimal process mean as shown by 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 integrated model increases for high material unit cost.

____________________________________ Please insert Table 5 about here

____________________________________

23

8.2.2 Effect of End Items Holding Cost As the holding cost of end items (h) increases, the total holding cost per unit time becomes a significant part of the total cost. Relatively speaking, this makes raw material cost less important and consequently an increase in the optimal target value is expected. This is demonstrated by Table 6. Also, Table 6 shows the total costs of the integrated and hierarchical models increase with h. Furthermore, the benefits of the integrated model are more apparent when the holding cost h increases.

____________________________________ Please insert Table 6 about here

____________________________________

8.2.3 Effect of Process Setup Cost When the setup cost A is large, expected total costs of the integrated and hierarchical models increase. This observation is shown in Table 7. In addition, the optimal process mean increases with the setup cost. This is because the raw material cost becomes less significant compared to setup cost. Another important observation is that the savings increase due to using the integrated model over the hierarchical model.

____________________________________ Please insert Table 7 about here

____________________________________

9. CONCLUSION In this paper, the classical stochastic continuous review inventory control (Q, R) model is integrated with process targeting problem to determine the optimal lot size, reorder point, and process mean. In the proposed model, the performance variable of the product has a lower specification limit, and items that do not conform to specification limit are scrapped with no salvage value. It is assumed that the performance variable follows a normal distribution with an adjustable mean and constant variance. A procedure is developed to find the optimal solution of the proposed model. The impact of considering the randomness in demand pattern is investigated

24

by comparing the proposed model with three baseline models. The first of which considers a hierarchical model where the vendor determines the process mean in isolation of inventory decisions. This hierarchical model is used to show the effect of integrating the process mean with production/inventory decisions when the demand is a random variable. The other baseline case is for the deterministic demand. This special case is used to explore the effect of variation in demand on the optimal solution. The last baseline case is for the situation when the variation in the filling amount is neglected. Empirical results confirm that ignoring the randomness in demand or filling amount may lead to invalid results. In particular, using deterministic demand leads to significantly underestimating the expected total cost, thus, the model would not be applicable to real world situations and adjustment to stochastic model must be made. Moreover, the results show that the variation in the process is key parameter in controlling the total cost, and the performance of a process can be improved considerably by reducing its variation. ACKNOWLEDGMENT: The authors are grateful to the anonymous referees for their constructive comments and helpful suggestions. They would like also to acknowledge Kuwait University. REFERENCES Al-Fawzan, M.A., and Hariga, M. (2002). An integrated inventory-targeting problem with time dependent process mean. Production Planning and Control, 13(1), 11-16. Alkhedher, M. and Darwish, M. A. (2012). Optimal process mean for a stochastic inventory model under service level constraint. International Journal of Operational Research. In press. Al-Sultan, K.S. (1994). An algorithm for the determination of the optimal target values for two machines in series with quality sampling plans, International Journal of Production Research, 12(1), 37-45.

25

Al-Sultan, K.S., and Al-Fawzan, M.A. (1997). An extension of Rahim and Banerjee’s model for a process with upper and lower specification limits. International Journal of Production Economics, 3, 265-280. Al-Sultan K.S., and Pulak M.F. (1997). Process improvement by variance reduction for a single filling operation with rectifying inspection. Production Planning and Control, 8(5), 431-436. Al-Sultan, K.S., and Pulak M.F.S. (2000). Optimum target values for two machines in series with 100% inspection. European Journal of Operational Research, 120, 181-189. Arcelus, F.J., and Banerjee, P.K. (1985). Selection of the most economical production plan is a tool-wear process. Technometrics, 27(4), 433-437. Arcelus, F.J., and Rahim, M.A. (1994). Simultaneous economic selection of a variables and an attribute target mean, Journal of Quality Technology, 26(2), 125-133. Bazaraa, M., Sherali, H., and Shetty, C. (1993). Nonlinear Programming: Theory and Algorithms. John Wiley & Sons, 2nd edition. Bettes, D.C. (1962). Finding an optimum target value in relation to a fixed lower limit and an arbitrary upper limit. Applied Statistics, 11, 202–210. Bisgaard, S., Hunter, W.G., and Pallesen, L. (1984). Economic selection of quality of manufactured products. Technometrics, 26, 9-18. Boucher, T.O., and Jafari, M. (1991). The optimum target value for single filling operations with quality sampling plans. Journal of Quality Technology, 23(1), 47-48. Bowling, S.R., Khasawneh, M.T., Kaewkuekool, S., and Cho, B.R. (2004). A Markovian approach to determining optimum process target levels for a multi-stage serial production system, European Journal of Operational Research, 159, 636–650.

26

Chen, Chung-Ho and Khoo, Michael B.C. (2009).Optimum process mean and manufacturing quantity settings for serial production system under the quality loss and rectifying inspection plan, Computers and Industrial Engineering, 57, 1080-1088. Chen, Chung-Ho and Kao, Hui-Sung (2009). The determination of optimum process mean and screening limits based on quality loss function, Expert Systems with Applications, 36, 7332-7335. Chen, Chung-Ho and Lai, Min-Tsai (2007). Determination of optimum process mean based on quadratic quality loss function and rectifying inspection plan, European Journal of Operational Research, 182, 755-763. Chen, Chung-Ho and Lai, Min-Tsai (2007). Economic manufacturing quantity, optimum process mean, and economic specification limits setting under rectifying inspection plan, European Journal of Operational Research, 183, 336-344. Chen, S.L., and Chung, K.J. (1996). Selection of the optimal precision level and target value for a production process: the lower specification limit case, IIE Transactions, 28, 979–985. Darwish, M.A. (2009). Economic Selection of Process Mean for Single-Vendor Single-Buyer Supply Chain. European Journal of Operational Research, 199(1), 162-169. Darwish, M.A. (2004a). Determination of the optimal process mean for a single vendor-single buyer-problem. International Journal of Operations & Quantitative Management, 10(3), 1- 13. Darwish, M.A. (2004b). An integrated single-vendor single-buyer targeting problem”. 2nd International Industrial Engineering Conference, Riyadh, Saudi Arabia. Darwish, M.A., and Duffuaa, S.O. (2010). A Mathematical Model for the Joint Determination of Optimal Process and Sampling Plan Parameters. Journal of Quality in Maintenance Engineering, 16(2), 181-189.

27

Darwish, M.A., Goyal, S.K., and Alenezi, Abdulrahman (2011). Stochastic Inventory Model with Finite Production Rate and Partial Backorders. International Journal of Services and Operations Management, In press. Duffuaa, S.O. and Siddiqui, A.W. (2003). Process targeting with multi-class screening and measurements error. International Journal of Production Research, 41(7), 1373-1391. Elsayed, E.A., and Chen, A. (1993). Optimal levels of process parameters for products with multiple characteristics, International Journal of Production Research, 31, 1117–1132. Gibra, I.N. (1974). Optimal production runs of processes subject to systematic trends. International Journal of Production Research, 12(4), 511-517. Goethals, P. L. and Cho, B. R. (2011). Reverse Programming the optimal process mean problem to identify a factor space profile, European Journal of Operational Research, (In press). Golhar, D.Y. (1987). Determination of the best mean contents for a canning problem. Journal of Quality Technology, 19(2), 82-84. Gong, L., Roan, J., and Tang, K., (1988). Process mean determination with quantity discounts in raw material cost. Decision Sciences, 29(1), 271-302. Hadley, G., and Whitin, T. M., 1963. Analysis of Inventory Systems. Prentice Hall, New Jersey, USA. Hariga, M.A., and Al-Fawzan, M.A. (2005). Joint determination of target value and production run for a process with multiple markets. International Journal of Production Economics, 96(2), 201-212. Hong, S.H., and Cho, B. R. (2007). Joint optimization of process target mean and tolerance limits with measurement errors under multi-decision alternatives, European Journal of Operational Research, 183, 327-335.

28

Hong, S.H., and Elsayed, E.A. (1999). The optimum mean for processes with normally distributed measurement error, Journal of Quality Technology, 31 (4), 338–344. Hong, S.H., Elsayed, E.A., and Lee, M.K. (1999). Optimum mean value and screening limits for production processes with multi-class screening, International Journal of Production Research, 37(2), 155–163. Hong, S.H., Kwon, H.M., Lee, M.K., and Cho, B.R. (2006). Joint optimization in process target and tolerance limit for L-type quality characteristics. International Journal of Production Research, 44(15), 3051–3060. Hunter, W.G., and Kartha, C.P. (1977). Determining the most profitable target value for a production process. Journal of Quality Technology 9(3), 176-181. Kim, Y.J., Cho, B.R., and Phillips, M.D. (2000). Determination of the optimal process mean with the consideration of variance reduction and process capability. Quality Engineering, 13(3), 251–260. Lee, M.K., and Elsayed, E.A. (2002). Process mean and screening limits for filling processes under two-stage screening procedure. European Journal of Operational Research, 138(1), 118-126. Lee, M.K., Hong, S.H., and Elsayed, E.A. (2001). The optimum target value under single and two-stage screenings. Journal of Quality Technology, 33(4),506–514. Lee, M.K., and Jang, J.S. (1997). The optimum target values for a production process with three-class screening. International Journal of Production Economics, 49, 91–99. Lee, M.K., Kim, S.B., Kwon, H.M., and Hong, S.H. (2004) Economic selection of mean value for a filling process under quadratic quality loss, International Journal of Reliability, Quality and Safety Engineering, 11(1), 506–514.

29

Lee, M.K., Kwon, H.M., Hong, S.H., and Kim, Y.J. (2007). Determination of the optimum target value for a production process with multiple products. International Journal of Production Economics, 107(1), 173-178. Lee, M.K., Kwon, H.M., Kim, Y.J., and Bae, J.H. (2005). Determination of Optimum Target Values for a Production Process Based on Two Surrogate Variables, Lecture Notes in Computer Science vol. 3483, Springer, Berlin, 232–240. Montgomery, D., Johnson, L., 1974. Operations Research in Production Planning, Scheduling, and Inventory Control. Wiley, New York. Nahmias, S., 2005. Production and Operations Analysis. Fifth edition, Irwin, Homewood, IL, USA. Park, T., Kwon, H. M., Hong, Sung-Hoon, and Lee, M. K., (2011).The optimum common process mean and screening limits for a production process with multiple products. Computers and Industrial Engineering 60, 158-163. Pfeifer, P.E. (1999). A general piecewise linear canning problem model. Journal of Quality Technology, 31 (4), 326–337. Rahim, M.A., and Al-Sultan, K.S. (2000). Joint determination of the optimum target mean and variance of a process. Journal of Quality in Maintenance Engineering, 6(3), 192 – 199. Rahim, M.A., and Banerjee, P.K. (1988). Optimum production run for a process with random linear drift. Omega, 16(4), 317-351. Rahim, M.A., Bhadury, J., and Al-Sultan, K.S. (2002) Joint economic selection of target mean and variance. Engineering Optimization, 34(1), 1–14. Rahim, M.A., and Shaibu, A.B. (2000). Economic selection of optimal target values. Process Control and Quality, 11(5), 369–381.

30

Roan, J., Gong, L., and Tang, K. (2000). Joint determination of process mean, production run size and material order quantity for a container-filling process. International Journal of Production Economics, 63, 303-317. Ross, S.M. (2007). Introduction to probability models. Ninth edition, Academic Press, Burlington, MA, USA. Shao, Y.E., Fowler, J.W., and Runger, G.C. (2000). Determining the optimal target for a process with multiple markets and variable holding costs. International Journal of Production Economics, 65(3), 229-242. Silver, E.A, Pyke, D.F., Peterson, R. (1998). Inventory management and production planning and scheduling. Third edition, John Wiley & Sons, New York, USA. Springer C. (1951). A method for determining the most economic position of a process mean. Industrial Quality Control, 8(1), 36-39. Tang, K., and Lo, J. (1993). Determination of the process mean when inspection is based on a correlated variable, IIE Transactions 25(3), 66-72. Teeravaraprug, J., and Cho, B.R. (2002). Designing the optimal process target levels for multiple quality characteristics, International Journal of Production Research, 40(1), 37–54. Williams, W.W., Tang, K., and Gong, L. (2000). Process improvement for a container-filling process with random shifts. International Journal of Production Economics, 66(1), 23-31.

31

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

R

Net

inve

ntor

y le

vel o

f fin

ishe

d ite

ms,

I(t)

Timeτ

Q

S

λ

T

-rμ

Time

Inve

ntor

y le

vel o

f raw

m

ater

ial ,

I r(t)

Δ

32

Table 1: Basic data

lower specification limit, L = 1.00standard deviation of the amount of raw material used i i

σ = 1.00expected value of the demand per unit time, E[Λ] = 300.00standard deviation of the demand per unit time, σΛ = 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 i

hr = 1.00fixed penalty cost, π = 7.00fixed production cost, b = 1.00value-added factor, α = 1.00unit material cost, c = 0.50constant component of lead time, Δ = 0.01

Table 2: Effect of filling process variation

Run # σ µ* p TC TCHR ε Ω

1 0.0 1.00 1.00 1566.50 1578.33 0.75 0.00 2 0.2 1.40 0.98 1675.51 1705.85 1.78 6.96 3 0.4 1.66 0.95 1775.12 1821.71 2.56 13.32 4 0.6 1.87 0.93 1867.77 1931.96 3.32 19.23 5 0.8 2.03 0.90 1954.84 2041.77 4.26 24.79 6 1.0 2.15 0.88 2037.36 2158.77 5.62 30.06 7 1.2 2.24 0.85 2116.17 2302.99 8.11 35.09 8 1.4 2.29 0.82 2192.16 2609.84 16.00 39.94 9 1.6 2.30 0.79 2266.64 ―* ―* 44.69

10 1.8 2.26 0.76 2342.14 ―* ―* 49.51

33

* ― denotes that the solution given by the hierarchical model is not feasible because λ < E[Λ].

34

Table 3: Effect of Demand Standard Deviation

Run # σΛ µ* p TC TCHR ε ψ

1 20 2.05 0.85 1806.45 1842.08 1.93 2.42 2 40 2.07 0.86 1855.52 1901.40 2.41 5.00 3 60 2.10 0.86 1909.19 1969.81 3.08 7.67 4 80 2.13 0.87 1968.98 2052.07 4.05 10.48 5 100 2.15 0.88 2037.36 2158.77 5.62 13.48 6 120 2.18 0.88 2118.93 2324.29 8.84 16.81 7 140 2.21 0.89 2224.19 2785.33 20.15 20.75 8 160 2.25 0.89 2394.95 ―* ―* 26.40 9 180 2.26 0.90 2709.64 ―* ―* 34.95

10 200 2.28 0.90 2840.10 ―* ―* 37.94 * ― denotes that the solution given by the hierarchical model is not feasible because λ < E[Λ].

Table 4: Effect of expected demand rate

Run # E[Λ] µ* p TC TCHR ε

1 200 2.13 0.87 1368.97 1418.92 3.52 2 220 2.13 0.87 1498.86 1558.72 3.84 3 240 2.14 0.87 1630.39 1701.93 4.20 4 260 2.14 0.87 1763.84 1849.16 4.61 5 280 2.15 0.87 1899.43 2001.15 5.08 6 300 2.15 0.88 2037.36 2158.77 5.62 7 320 2.16 0.88 2177.92 2323.18 6.25 8 340 2.16 0.88 2321.36 2495.98 7.00 9 360 2.17 0.88 2467.96 2680.28 7.92

10 380 2.17 0.88 2618.04 2926.36 10.54 11 400 2.18 0.88 2771.93 ―* ―*

* ― denotes that the solution given by the hierarchical model is not feasible because λ < E[Λ].

36

Table 5: Effect of raw material unit cost

Run # c µ* p TC TCHR ε 1 0.2 2.450 0.93 1407.02 1448.00 2.83 2 0.3 2.310 0.90 1575.63 1632.98 3.51 3 0.4 2.207 0.89 1738.29 1812.40 4.09 4 0.5 2.125 0.87 1896.93 1989.14 4.64 5 0.6 2.058 0.85 2052.61 2165.14 5.20 6 0.7 2.001 0.84 2206.04 2342.18 5.81 7 0.8 1.951 0.83 2357.71 2522.24 6.52 8 0.9 1.907 0.82 2507.98 2708.03 7.39 9 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 6: Effect of End Items Holding Cost

Run # h µ* p TC TCHR ε

1 4.2 2.132 0.871 1923.82 2028.62 5.17 2 4.4 2.140 0.873 1950.77 2070.09 5.76 3 4.6 2.147 0.874 1977.82 2114.15 6.45 4 4.8 2.154 0.876 2005.05 2161.66 7.24 5 5.0 2.161 0.877 2032.52 2213.92 8.19 6 5.2 2.168 0.879 2060.29 2272.96 9.36 7 5.4 2.174 0.880 2088.48 2342.23 10.83 8 5.6 2.181 0.881 2117.22 2427.96 12.80 9 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

37

Table 7: Effect of setup cost

Run # A µ* p TC TCHR ε

1 40.0 2.140 0.873 1970.13 2076.73 5.13 2 50.0 2.152 0.875 2037.36 2158.77 5.62 3 60.0 2.163 0.878 2099.92 2236.88 6.12 4 70.0 2.173 0.880 2158.79 2312.33 6.64 5 80.0 2.181 0.881 2214.58 2386.24 7.19 6 90.0 2.189 0.883 2267.87 2459.77 7.80 7 100.0 2.197 0.884 2319.03 2534.28 8.49 8 110.0 2.203 0.886 2368.42 2611.73 9.32 9 120.0 2.210 0.887 2416.32 2695.32 10.35

10 130.0 2.216 0.888 2462.99 2791.34 11.76

38

Highlights:

• We develop a model for integrated targeting - inventory problem when the demand is a random variable.

• We investigate the significance of neglecting the uncertainty in demand on targeting - inventory system by considering three baseline cases.

• Empirical results show that ignoring randomness in the demand pattern leads to underestimating the expected total cost significantly.