- Email: [email protected]
Sole sourcing in EOQ models with Binomial yield
Journal of Purchasing & Supply Management 16 (2010) 163–170
Contents lists available at ScienceDirect
Journal of Purchasing & Supply Management journal homepage: www.elsevier.com/locate/pursup
Sole sourcing in EOQ models with Binomial yield M. Mahdi Tajbakhsh a,, Chi-Guhn Lee b, Saeed Zolfaghari c a b c
Department of Industrial Engineering, Dalhousie University, P.O. Box 1000, Halifax, Nova Scotia, Canada B3J 2X4 Department of Mechanical and Industrial Engineering, University of Toronto, 5 King’s College Road, Toronto, Ontario, Canada M5S 3G8 Department of Mechanical and Industrial Engineering, Ryerson University, 350 Victoria Street, Toronto, Ontario, Canada M5B 2K3
a r t i c l e in fo
abstract
Article history: Received 16 April 2009 Received in revised form 28 November 2009 Accepted 8 December 2009
We develop two EOQ-based inventory models in which an inventory manager must determine the order size as well as the supply reliability level in the presence of uncertainty in the quality and/or quantity of supply. The number of acceptable units in the order is captured by the Binomial yield model, and reliability is increased both by increasing the order setup cost and by increasing the unit price. For each developed model, we present an equation of which a solution is the optimal reliability level and a closed form solution for the optimal order size given the optimal reliability level. We then provide a comparison of the two models. & 2009 Elsevier Ltd. All rights reserved.
Keywords: Inventory management Supply management Binomial yield
1. Introduction Uncertainty in supply quantity and/or quality has been one of the major concerns in procurement decisions (Treleven and Schweikhart, 1988). This type of uncertainty, also known as random yield, affects lot sizing decisions in production control and inventory management. Applications are numerous (Yano and Lee, 1995) and include electronic fabrication and assembly, chemical processes, and any procurement system with suppliers that may produce defective products. In inventory systems, random yield usually refers to one of two cases:
The quantity delivered may contain defective units (quality uncertainty).
The quantity supplied may not be equal to the quantity ordered (quantity uncertainty). Random yield has been studied extensively in the literature on production system control and inventory management. Yano and Lee (1995) present five main approaches to modeling yield uncertainty in production and inventory systems, and discuss their advantages and disadvantages. Only three models are relevant to pure inventory management: Binomial yield, stochastically proportional yield, and random capacity. In the Binomial yield model, the acceptable quantity is Binomially distributed Corresponding author. Tel.: + 1 902 494 6173.
E-mail addresses: [email protected] (M.M. Tajbakhsh), [email protected] (C.-G. Lee), [email protected] (S. Zolfaghari). 1478-4092/$ - see front matter & 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.pursup.2009.12.005
with two parameters: the order quantity and the probability of an individual item being non-defective. On the other hand, in the stochastic proportional yield model the acceptable quantity is a multiplication of the yield rate, which is a random variable, and the order quantity. In the random capacity model, the received quantity is the minimum of the order quantity and the random capacity. The Binomial yield model has been used in discrete production systems to describe random yield by some researchers (GrosfeldNir and Ronen, 1993; Barad and Braha, 1996; Grosfeld-Nir, 2005). As Yano and Lee (1995) explain, the Binomial yield model assumes that the supplier’s production process is stationary and that the generation of one unit (at the supplier’s facility) is independent of the generation of other units. This representation is suitable when the (supplier’s) production system is in control for long durations. Examples include high-tech products (Grosfeld-Nir and Ronen, 1993), and electronic or mechanical products (Barad and Braha, 1996). Although random yield has been studied extensively, most researchers assume that the pattern of yield randomness, or the yield distribution, is given and focus on determining optimal lot sizes. However, as argued by Gerchak and Parlar (1990), a firm may be able to improve yield by implementing a tighter quality assurance program in its supply base at the expense of increased unit purchasing cost and/or order setup cost. Gerchak and Parlar (1990) consider an EOQ model with stochastically proportional yield, where the expected number of acceptable units in an order is the order size times the mean yield rate (i.e., expected fraction acceptable) and the variance of the acceptable units is the square of order size times the variance of the yield rate. Assuming that
164
M.M. Tajbakhsh et al. / Journal of Purchasing & Supply Management 16 (2010) 163–170
the yield rate variance is a control variable but not the mean yield rate, they analyzed two cases: the Cost-Per-Order model and the Unit-Cost model. In the former, when the order setup cost is a decreasing, convex function of the standard deviation of the yield rate and the unit cost is fixed, Gerchak and Parlar (1990) show that the optimal order size is decreasing in the standard deviation of the yield rate (this relationship has also been observed by Silver, 1976 in a simpler model). They also show that the optimal order size is decreasing in the mean yield rate for a fixed standard deviation, and that the optimal value for the standard deviation does not depend on the demand rate and unit holding cost. In the latter, it is the cost per unit ordered that depends on yield variability. Using a logarithmic functional form for the cost per unit, they show that the optimal value of the standard deviation is increasing in the demand rate and decreasing in the unit holding cost. Cheng (1991) considers an EOQ model in which the expected fraction acceptable (mean yield rate) can be changed through quality assurance efforts. While the yield variance (variability) is a decision variable in Gerchak and Parlar (1990), it is implicitly ignored by Cheng (1991). Assuming that the unit production cost is an increasing power function of the acceptable fraction, Cheng (1991) obtains closed-form expressions for the optimal production run and the optimal fraction acceptable. Moreover, he shows that the higher the quality, the smaller the lot size. Tripathy et al. (2003) generalize the model and the results of Cheng (1991). In their model, the unit production cost is a power function of both the acceptable fraction and the demand rate, increasing in the former and decreasing in the latter. Thus, in addition to the lot size and acceptable fraction, the demand rate is a decision variable. Tripathy et al. (2003) obtain closed-form expressions for the optimal production run, the optimal fraction acceptable, and the optimal demand rate. Moreover, they show that higher quality may lead to both a smaller lot size and a lower demand rate. Porteus (1986) has also shown a significant relationship between lot sizing and quality, where he modifies the EOQ model as follows. The production process begins in the ‘‘in-control’’ state after each setup. While producing any single unit in the lot, the production process (machine) can go ‘‘out-of-control’’ with a given probability. Once out of control, the process produces defective units throughout the current lot and until the next setup (which returns the process to the in-control state). Here, the probability of going out of control while producing each unit is constant, and thus, the time to failure is geometrically distributed. Moreover, it is assumed that each defective unit can be repaired (reworked) instantaneously at a cost. Porteus (1986) provides three options for investing in quality improvements: (a) reducing the probability that the process goes out of control; (b) reducing the setup cost; and (c) using the two previous options simultaneously. He shows that the optimal lot size is smaller than the classical EOQ. By assuming a logarithmic form of the investment cost function, he also shows that the optimal process quality level (i.e., probability that the process goes out of control) is decreasing in the demand rate, increasing in the unit holding cost, and decreasing in the unit repair cost. One should note that there are two main differences between the work of Porteus (1986) and the other papers reviewed thus far. First, Porteus (1986) assumes that defective items are repaired while the previous papers do not consider repair or rework. Second, Porteus (1986) assumes a constant setup cost and unit production cost to obtain the optimal lot size, and then, introduces an investment cost to change (and thereby, optimize) the probability of producing a defective unit or the setup cost. The previous papers, however, define the setup cost or unit cost as a function of random yield parameters, and optimize both the order quantity and the yield distribution simultaneously.
The main objective of our paper is to develop inventory models to simultaneously optimize the order quantity and the reliability level of an unreliable supplier, when the unreliability is captured by the Binomial yield model. To this end, we study two EOQ models when the acceptable (usable) quantity is Binomially distributed with two parameters: the order quantity and the reliability level. In the first model called the return model, the supplier assumes responsibility for the supply unreliability, whereas in the second model called the repair model, the buyer absorbs the unreliability risks at a cost. The major contribution of this study is twofold. Firstly, we allow both the order setup cost and the unit cost to simultaneously change with unreliability level, whereas Gerchak and Parlar (1990) allow both but not simultaneously, Cheng (1991) allows only the latter, and Porteus (1986) allows neither. Secondly, and more importantly, by investigating the return and repair models in a unified framework, we can compare these two modeling approaches and highlight their implications on the obtained results, depending on whether the supplier or the buyer takes responsibility for supply unreliability. The previous papers in the literature, however, assume that either the supplier (similar to our return model) or the buyer (similar to our repair model) absorbs the unreliability risks, and thus, have not compared the results of these two different assumptions. The assumption that the order setup cost and unit variable cost are functions of unreliability level deserves more explanation. Consider a company that uses a production process with a certain level of capability to manufacture a product, and suppose that a quality assurance program is in place to monitor the quality of the items. The capability of the process and the effectiveness of the quality assurance program depend upon various factors such as production technology, machine capability, work methods, use of online monitoring systems, skill level of workers, and inspection policies (Cheng, 1991). Enforcing a high level of process capability and stringent quality assurance will evidently result in highquality products with lower defectives (i.e., higher reliability). This, however, requires substantial investment in plant, machinery, equipment, and employee training, thereby increasing the unit production cost. Furthermore, adopting advanced production technology necessitates the use of more skilled labor and more expensive tools to initiate production runs. This in turn implies higher setup costs. It is worth mentioning that Zipkin (2000) has also studied the EOQ setting with imperfect quality. Specifically, the return and repair models in our paper are similar to, respectively, the ‘‘immediate detection, reimbursement’’ and ‘‘immediate detection, no reimbursement’’ cases in Section 3.6 of Zipkin (2000). However, there are three main points that distinguish our work from his. First, the unreliability level in our models is a decision variable while it is a given parameter in his models, i.e., he only focuses on determining optimal order quantities. Second, the order setup cost and the unit cost are functions of the unreliability level in our models whereas they are fixed in his models. Third, the way he has modeled unreliability differs from ours. Zipkin assumes that each batch contains a fixed fraction of defective items, irrespective of the batch size, and thereby, he does not take yield variability into account (similar to Cheng, 1991). Using the Binomial yield model, however, we are able to incorporate the yield variance into our models. The organization of the paper is as follows. In the next section, we explain the main assumptions and notation of our models. Then, we analyze the return and repair models where determination of supply reliability is taken into consideration. Next, we compare these two models. The last section provides concluding remarks.
M.M. Tajbakhsh et al. / Journal of Purchasing & Supply Management 16 (2010) 163–170
2. Preliminaries In this paper, we consider inventory problems with a single unreliable supplier, in which both the optimal order quantity and the optimal unreliability level must be determined simultaneously. That is, in addition to determining the optimal order quantity, the inventory manager can choose the best unreliability level, noting that as supply reliability increases, the item becomes more expensive and the fixed order cost may increase. The fact that unreliability level is a decision variable affecting the unit cost and fixed order cost may have one of the following two interpretations: 1. When dealing with a predetermined supplier, the inventory manager may ask for a higher level of supply reliability and may be willing to pay more. 2. When selecting a supplier from a supplier pool, the inventory manager can choose a more reliable supplier with a possibly higher fixed order cost and acquire the product at a higher price. Notice that supplier selection is generally based on minimizing total costs (e.g., supplier management, transportation, and inventory costs). Thus, one might reduce the total costs by procuring the product from an unreliable supplier and saving on the purchase price and order cost. This is obviously applicable to items that are not critical. For more discussion, the reader is referred to Micheli (2008) and the references therein.
165
They are general enough to capture common functional forms in the literature, yet they allow us to obtain analytical results for our models. One should note that the log-convexity assumption for AðyÞ is not too strong. For example, the power b function AðyÞ ¼ a y ; a; b4 0, and the logarithmic function AðyÞ ¼ ab ln y; a Z b 40, are log-convex. These two functional forms have been used extensively in the literature to model the order setup cost as a function of uncertainty or variability measures (e.g., see Gerchak and Parlar, 1990; Porteus, 1986). 2.2. Binomial yield In this paper, we use the Binomial yield model: When an order of size Q is placed with a supplier, the number of usable items supplied or the quantity delivered, YQ , is a Binomial random variable with parameters Q and 1y, i.e., ! Q Q y PrfYQ ¼ yg ¼ ; y ¼ 0; 1; . . . ; Q : ð1yÞy y y Thus, we can write the mean and variance of random yield, respectively, as follows: E½YQ ¼ Q ð1yÞ;
ð1Þ
VarðYQ Þ ¼ Q yð1yÞ:
ð2Þ
As observed, Binomial yield can be used to model either of the following two cases:
Quality uncertainty: A proportion of any quantity supplied 2.1. Notation and assumptions We modify the classical EOQ model and use the following notation:
D ¼ Constant demand rate (units/unit time). h ¼ Holding cost per unit per unit time ($/unit/unit time). Q ¼ Order quantity. y ¼ Unreliability level: Probability that a unit is defective or is not delivered. AðyÞ ¼ Fixed order cost ($/replenishment). cðyÞ ¼ Unit variable cost of the item ($/unit). YQ ¼ Random yield: usable items supplied or quantity delivered by a supplier (random variable). ETðQ ; yÞ ¼ ðExpectedÞ length of a cycle (a cycle is defined as the time between two consecutive replenishments). ECðQ ; yÞ ¼ Expected total cost per cycle. GðQ ; yÞ ¼ Long run average cost per unit time ($/unit time).
Our assumptions are as follows: Assumption S1: Q and y are assumed to be continuous decision variables. Assumption S2: cðyÞ is assumed to be a non-increasing and convex function of y. Assumption S3: AðyÞ is assumed to be a non-increasing and logconvex function of y. Assumption S4: We assume that 0 r y r ymax o1, where ymax is the highest unreliability level acceptable to management. In fact, ymax shows the maximum risk that management is willing to accept in the supply process, and is expected to be moderate in practice. That cðyÞ and AðyÞ are non-increasing in y is to establish that higher reliability comes at a price; that is, as y decreases (reliability improves), the unit variable cost and the fixed order cost will go up. The convexity assumption on cðyÞ and the log-convexity assumption on AðyÞ are technical assumptions.
may be defective, i.e., each unit supplied is defective with probability y and non-defective with probability 1y. Here, y is called the defective rate. Quantity uncertainty: A proportion of any quantity ordered may not be delivered, i.e., any item ordered may not be delivered with probability y. Here, 1y is called the yield ratio.
In order to capture both the quality uncertainty and the quantity uncertainty cases, we refer to y as unreliability level. Clearly, as y increases, supply reliability decreases.
3. The return model In this section, we consider a pure inventory model (procurement environment) where each unit produced at the supplier’s facility is defective with probability y and non-defective with probability 1y. The model is called the return model and is interpreted in two ways:
The buyer performs 100% inspection after receiving
the quantity ordered, and returns defective units to the supplier for refund. Here, the buyer encounters quality uncertainty. The supplier performs 100% inspection before dispatching the buyer’s order, and removes imperfect units from the batch. Here, the buyer experiences quantity uncertainty. For instance, this is the case when the supplier is not able to replace the defectives due to a lack of enough inventory for immediate replacement. Outsourcing the item, the supplier requires some time to receive new orders; manufacturing the item, the supplier needs time to produce new items.
In the return model, implicit assumptions are that the quality imperfection is the supplier’s responsibility, and that the defective items identified in a batch cannot be immediately reworked or
166
M.M. Tajbakhsh et al. / Journal of Purchasing & Supply Management 16 (2010) 163–170
replaced by non-defective items. Therefore, the buyer does not pay for the units defective or not delivered, i.e., cðyÞ is the cost per (non-defective) unit received.
00
The function uðyÞ is convex in y (i.e., u ðyÞ Z0) if and only if 00
2AðyÞA ðyÞ½A0 ðyÞ2 Z 0()
½AðyÞ2 A ðyÞ AðyÞA ðyÞ½A ðyÞ2 þ Z 0() Z0: AðyÞ ½AðyÞ2 00
3.1. Model formulation In the return model, the expected total cost per cycle and the expected cycle length, respectively, are ECðQ ; yÞ ¼ AðyÞ þcðyÞ E½YQ þ h ETðQ ; yÞ ¼
E½YQ2 2D
AðyÞA ðyÞ½A0 ðyÞ2 00
¼
d A0 ðyÞ d d d2 ¼ ln AðyÞ ¼ ln AðyÞ: 2 dy AðyÞ dy dy dy
Thus, uðyÞ is convex in y if and only if 00
A ðyÞ d2 þ 2 ln AðyÞ Z 0: AðyÞ dy
In accordance with the Renewal Reward Theorem, we can write the long-run average cost per unit time as 2 ECðQ ; yÞ AðyÞD h E½YQ GðQ ; yÞ ¼ ¼ cðyÞD þ þ ; ETðQ ; yÞ E½YQ 2 E½YQ
AðyÞ D h þ ½Q ð1yÞ þ y: Q ð1yÞ 2
d2 2
ln AðyÞ Z 0; 00
moreover, A ðyÞ Z 0, because a log-convex function is also a convex function. Therefore, Inequality (4) is satisfied, implying that uðyÞ is a convex function of y. & ð3Þ
The optimization problem is represented as fminGðQ ; yÞjQ 4 0; 0 r y r ymax g: 3.2. Model analysis
In the proof of Lemma 1, we notice that the log-convexity of AðyÞ is a sufficient (not necessary) condition for the convexity of uðyÞ. Using Lemma 1, we observe that kðyÞ is the sum of three convex functions, and hence, is a convex function of y. Now, we have A0 ðyÞD
Examining the second partial derivatives of GðQ ; yÞ with respect to Q and y, one can easily verify that GðQ ; yÞ is strictly convex in Q but is not convex in y. Thus, GðQ ; yÞ is not jointly convex in ðQ ; yÞ. A reasonable approach to the optimization problem is to fix y, optimize over Q to obtain Q ðyÞ, and then optimize over y. Therefore, to obtain the optimal values of Q and y, we proceed as follows. First, for a fixed y, we obtain the optimal order quantity, Q ðyÞ, by setting @GðQ ; yÞ=@Q ¼ 0, i.e., qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi AðyÞD hð1yÞ 1 ¼ 0 ) Q ðyÞ ¼ 2AðyÞD=h: þ 2 1y Q 2 ð1yÞ If AðyÞ ¼ A (i.e., the fixed order cost is a constant independent of y), then Q ðyÞ is increasing in y. In this case, improved quality (low value of y) corresponds to small order quantities. Gerchak and Parlar (1990) and Cheng (1991) have also pointed out this behavior. Next, we substitute Q ðyÞ in GðQ ; yÞ to obtain kðyÞ as follows: AðyÞD h þ ½Q ðyÞð1yÞ þ y ¼ cðyÞD kðyÞ :¼ GðQ ðyÞ; yÞ ¼ cðyÞD þ Q ðyÞð1yÞ 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi AðyÞD h hy pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ 2AðyÞDh: þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ ½ 2AðyÞD=h þ y ¼ cðyÞD þ 2 2AðyÞD=h 2
fminkðyÞj0 r y r ymax g: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Letting uðyÞ :¼ 2AðyÞDh, we obtain the following result. Lemma 1. uðyÞ is a convex function of y. Proof. We have rffiffiffiffiffiffiffi! Dh A0 ðyÞ 0 pffiffiffiffiffiffiffiffiffi u ðyÞ ¼ 2 AðyÞ and ) rffiffiffiffiffiffiffi( 00 0 Dh A ðyÞ 1 ½A ðyÞ2 pffiffiffiffiffiffiffiffiffi : u ðyÞ ¼ 2 2 ½AðyÞ3=2 AðyÞ
h 2
k0 ðyÞ ¼ c0 ðyÞD þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ : 2AðyÞD=h
Now the optimization problem reduces to
ð4Þ
Since AðyÞ is assumed to be log-convex in y, we have dy
which, using (1) and (2), yields
00
0
Now, we can write ½AðyÞ
E½YQ : D
GðQ ; yÞ ¼ cðyÞD þ
00
2
;
2AðyÞA ðyÞ½A0 ðyÞ2 00
By setting k0 ðyÞ ¼ 0, we get f ðyÞ ¼
h ; 2D
ð5Þ
where A0 ðyÞ f ðyÞ :¼ c0 ðyÞ þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 2AðyÞD=h Assume that y^ is a solution to Eq. (5), i.e., f ðy^ Þ ¼ h=2D. Moreover, let y denote the optimal unreliability level. Due to convexity of kðyÞ in y, we must consider the following three cases:
If y^ o 0, then y ¼ 0. If 0 r y^ r ymax , then y ¼ y^ . If y^ 4 ymax , then y ¼ ymax . Moreover, because cðyÞ and uðyÞ are convex functions of y, then c0 ðyÞ and u0 ðyÞ=D are both non-decreasing in y. This implies that f ðyÞ is a non-decreasing function of y. Thus, using Eq. (5), we observe that, if either the fixed order cost or the unit variable cost is a constant independent of y (i.e., either A0 ðyÞ ¼ 0 or c0 ðyÞ ¼ 0), then y is non-decreasing in D and non-increasing in h. Using the above results, we now ascertain the following theorem. Theorem 2. If y^ is a solution to f ðyÞ ¼ h=2D, then the following hold:
(a) The optimal 8 > > <0 y ¼ y^ > > :y max
unreliability level, y , is if y^ o 0; if 0 r y^ r ymax ; if y^ 4 ymax :
(b) If either AðyÞ or cðyÞ is a constant independent of y, then y is a non-decreasing function of D and a non-increasing function of h.
M.M. Tajbakhsh et al. / Journal of Purchasing & Supply Management 16 (2010) 163–170
(c) The optimal order quantity, Q , is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 Q ¼ 2Aðy ÞD=h: 1y (d) The optimal average cost per unit time, GðQ ; y Þ, is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi GðQ ; y Þ ¼ cðy ÞD þ 2Aðy ÞDh þhy =2: Here, y is assumed to be continuous. Because kðyÞ is convex in y, Part (a) can be easily modified to address the case of discrete unreliability level. To do so, assume that y is to be selected from a set of discrete values represented by Y :¼ fy0 ; y1 ; y2 ; . . . ; ym j0 r yi o yi þ 1 ; i ¼ 0; 1; . . . ; m1g, where ym ¼ ymax . Then, Theorem 2(a) can be modified as follows: 8 > if y^ o y0 ; > < y0 y ¼ argminfkðyi Þ; kðyi þ 1 Þg if yi r y^ o yi þ 1 ; i ¼ 0; 1; . . . ; m1; > > :y if y^ Z ym : m The discrete case can be used in supplier selection where a supplier is to be chosen from a set of suppliers with different prices, order costs, and unreliability levels and the goal is to minimize the average cost per unit time. Here, yi refers to the unreliability level of supplier i. Part (b) shows that as the demand rate increases and/or the unit holding cost decreases, no improvement in supply reliability should be carried out (i.e., it may be economical to choose a less reliable supplier). In particular, assume that either AðyÞ is strictly log-convex in y or cðyÞ is strictly convex in y, so that f ðyÞ is strictly increasing in y. Our observations then suggest that as the market grows in medium ranges, one should rely on less reliable suppliers, and that as storage cost increases in medium ranges, more reliable suppliers should be chosen. As mentioned before, Gerchak and Parlar (1990) obtain similar results in their Unit-Cost model. Namely, with a logarithmic unit-cost function, they find that the optimal yield variability is increasing in D and decreasing in h. In general, Eq. (5) may not have a solution at all or may have more than one solution. Here, we present examples in which a unique, closed-form solution to Eq. (5) can be found. Example 1. Let AðyÞ ¼ A. We now consider two functional forms for the unit variable cost:
Porteus (1985) to denote investment costs of reducing the order setup cost. We can easily verify that AðyÞ is a decreasing, logconvex function of y. We now consider two cases for the unit variable cost: 1. Letting cðyÞ ¼ c, we can write Eq. (5) as rffiffiffiffiffiffiffi! ah h b=21 ; ¼ by 2D 2D which yields rffiffiffiffiffiffiffiffiffi!2=ðb þ 2Þ 2aD y^ ¼ b : h Thus, we get 9 8 rffiffiffiffiffiffiffiffiffi! 2=ðb þ 2Þ = < 2aD y ¼ min b ; ymax : ; : h b=2
2. Letting cðyÞ ¼ cy ; c 4 0, we can write Eq. (5) as rffiffiffiffiffiffiffi! bc b=21 ah h b=21 ; y ¼ by 2 2D 2D which gives " pffiffiffiffiffiffiffiffiffiffiffiffiffiffi #2=ðb þ 2Þ bDðc þ 2ah=DÞ y^ ¼ : h Hence, we get 8" 9 #2=ðb þ 2Þ < bDðc þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi = 2ah=DÞ y ¼ min ; ymax : : ; h
4. The repair model In this section, we consider a production-inventory model where each unit produced (at the supplier’s facility) is defective with probability y and non-defective with probability 1y. The model is called the repair model and is interpreted in two ways:
The buyer performs 100% inspection after receiving the
1. Logarithmic function. Let cðyÞ ¼ ab ln y;
a; b 40:
This functional form has been used by Gerchak and Parlar (1990) to describe the unit variable cost as a function of yield variability. One can easily verify that cðyÞ is strictly decreasing and convex in y. Now, using Eq. (5), we get b=y ¼ h=2D, which yields y^ ¼ 2bD=h. Finally, we have y ¼ minf2bD=h; ymax g. 2. Power function. Let b
cðyÞ ¼ ay
;
a; b 40:
This functional form has been used by Cheng (1991) to describe the unit production cost as a function of fraction acceptable. One can easily verify that cðyÞ is strictly decreasing b1 and convex in y. Now, using Eq. (5), we get aby ¼ h=2D, 1=ðb þ 1Þ ^ which yields y ¼ ð2abD=hÞ . Finally, we have y ¼ minfð2abD=hÞ1=ðb þ 1Þ ; ymax g. b
Example 2. Let AðyÞ ¼ ay ; a; b 40 (power function). This functional form has been used by Gerchak and Parlar (1990) to describe the order cost as a function of yield variability, and by
167
quantity ordered, and repairs (reworks) defective units at an additional cost. Alternatively, this can represent a manufacturing environment in which the production process is unreliable and imperfect units are repaired (reworked). The supplier performs 100% inspection before dispatching the buyer’s order, and replaces the defective units with nondefective units at an additional cost. As mentioned by Chen et al. (2001), this represents a new feature that has not been addressed in other random yield models: At a premium (in addition to the purchase price) per unit, the supplier guarantees to replace the defective units and delivers the entire order.
In the former, we let cR denote the repair (or rework) cost of each defective item ($/unit). In the latter, ycR denotes the additional premium per unit ordered, which is paid to guarantee the delivery. In the repair model, implicit assumptions are that the quality imperfection is the buyer’s responsibility, and that the defective items identified in a batch can be instantaneously reworked or replaced by non-defective items. Therefore, the buyer also pays for the defective units, i.e., cðyÞ is the cost per unit ordered. Gerchak and Parlar (1990), in their Unit-Cost model, also assume
168
M.M. Tajbakhsh et al. / Journal of Purchasing & Supply Management 16 (2010) 163–170
that the unit variable cost is the cost per unit ordered, which is more appropriate in manufacturing environments (see Yano and Lee, 1995 for more discussion).
Therefore, if either A0 ðyÞ ¼ 0 or A0 ðyÞ o 0, then y is nonincreasing in D and non-decreasing in h. Using the above results, we now ascertain the following theorem. Theorem 4. If y^ is a solution to f ðyÞ ¼ cR , then the following hold:
4.1. Model formulation
Using (1), the mean of random yield is Q ð1yÞ, and thus, the expected number of defective items in a lot of size Q is Q y. Therefore, the expected total cost per cycle is 2
ECðQ ; yÞ ¼ AðyÞ þcðyÞ Q þ
hQ þcR Q y: 2D
Since the cycle length, ETðQ Þ, is ETðQ Þ ¼
Q ; D
we can write the long-run average cost per unit time as GðQ ; yÞ ¼
ECðQ ; yÞ AðyÞD hQ ¼ cðyÞD þ þ þ cR Dy: ETðQ Þ Q 2
ð6Þ
(b) If AðyÞ is either a constant independent of y or strictly decreasing in y, then y is a non-increasing function of D and a nondecreasing function of h. (c) y is a non-increasing function of cR . (d) The optimal order quantity, Q , is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Q ¼ 2Aðy ÞD=h:
(e) The optimal average cost per unit time, GðQ ; y Þ, is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi GðQ ; y Þ ¼ cðy ÞD þ 2Aðy ÞDh þ cR Dy :
As before, the optimization problem is represented as fminGðQ ; yÞjQ 4 0; 0 r y r ymax g: 4.2. Model analysis As we observe, the cost function per unit time in the repair model is very similar to that in the return model. Indeed, one could obtain Expression (6) by making the following transformations in Expression (3): h=2 to cR D in the linear term hy=2, and Q ð1yÞ to Q. Using this relationship, we can shorten the analysis of the repair model. For a fixed y, we obtain the optimal order quantity Q ðyÞ as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Q ðyÞ ¼ 2AðyÞD=h: Thus, in the repair model, Q ðyÞ is a non-increasing function of y. That is, improved quality implies large order quantities, which has also been observed by Porteus (1986). Next, substituting Q ðyÞ in GðQ ; yÞ, we obtain kðyÞ as pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi kðyÞ ¼ GðQ ðyÞ; yÞ ¼ cðyÞD þ 2AðyÞDh þcR Dy: By setting k0 ðyÞ ¼ 0, we get f ðyÞ ¼ cR ;
(a) The optimal unreliability level, y , is 8 > if y^ o 0; > <0 y ¼ y^ if 0 r y^ r ymax ; > > :y if y^ 4 ymax : max
ð7Þ
where A0 ðyÞ f ðyÞ ¼ c0 ðyÞ þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 2AðyÞD=h Assuming that f ðy^ Þ ¼ cR , one can obtain y in the same way as in the return model (see Theorem 2(a)). We have shown that f ðyÞ is a non-decreasing function of y. Thus, using Eq. (7), we observe that y is non-increasing in cR . Obviously, if AðyÞ ¼ A, then y is independent of D and h.
As shown for Theorem 2(a), we can easily modify Part (a) to handle the case of discrete unreliability level. Part (b) shows that as the demand rate decreases and/or the unit holding cost increases, no improvement in supply reliability should be carried out (i.e., it may be economical to choose a less reliable supplier). In particular, assume that AðyÞ is strictly log-convex in y or cðyÞ is strictly convex in y, so that f ðyÞ is strictly increasing in y. Our observations then suggest that as the market grows in medium ranges, one should rely on more reliable suppliers, and that as storage cost increases in medium ranges, less reliable suppliers should be chosen. Part (c) implies that as the unit repair cost increases in medium ranges, more reliable suppliers should be selected. As mentioned before, Porteus (1986) obtains similar results. Namely, with a logarithmic investment cost function, he shows that the optimal process quality probability (i.e., probability that the production process becomes out-of-control) is decreasing in D, increasing in h, and decreasing in cR . Similar to Eq. (5), Eq. (7) may not have a solution at all or may have more than one solution. We now present examples in which a unique, closed-form solution to Eq. (7) can be found. Example 3. Once again, we let AðyÞ ¼ A, and consider two functional forms for the unit variable cost: the logarithmic and power forms: 1. Logarithmic function. Let cðyÞ ¼ ab ln y;
a; b4 0:
We use Eq. (7) to get y^ ¼ b=cR . Therefore, y ¼ minfb=cR ; ymax g. 2. Power function. Let
Lemma 3. If AðyÞ is strictly decreasing in y, then the following hold:
cðyÞ ¼ ay
(a) y^ is strictly decreasing in D. (b) y^ is strictly increasing in h.
We use Eq. (7) to get y ¼ minfðab=cR Þ1=ðb þ 1Þ ; ymax g.
Proof. (a) y^ is a solution to Eq. (7). Moreover, since AðyÞ is strictly decreasing in y (i.e., A0 ðyÞ o0), f ðyÞ is strictly increasing in D. Thus, as D increases, f ðy^ Þ increases. Since f ðyÞ is a non-decreasing function of y, a smaller value of y^ is needed to bring f ðy^ Þ back down to cR . This means that as D increases, y^ decreases. (b) The argument is very similar to that of Part (a). The only difference is that f ðyÞ is strictly decreasing in h. &
b
;
a; b 4 0:
b
y^ ¼ ðab=cR Þ1=ðb þ 1Þ .
Hence,
Example 4. Let AðyÞ ¼ ay ; a; b4 0 (power function). We now consider two cases for the unit variable cost: 1. Let cðyÞ ¼ c. Using Eq. (7) yields
y^ ¼
b cR
rffiffiffiffiffiffiffi!2=ðb þ 2Þ ah : 2D
M.M. Tajbakhsh et al. / Journal of Purchasing & Supply Management 16 (2010) 163–170
Thus,
8 9 ffi!2=ðb þ 2Þ < b rffiffiffiffiffiffi = ah y ¼ min ; ymax : : cR 2D ; b=2
; c 4 0. Using Eq. (7) yields 2. Let cðyÞ ¼ cy " pffiffiffiffiffiffiffiffiffiffiffiffiffiffi #2=ðb þ 2Þ bðc þ 2ah=DÞ y^ ¼ : 2cR Hence,
8" 9 #2=ðb þ 2Þ < bðc þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi = 2ah=DÞ y ¼ min ; ymax : : ; 2cR 5. Comparison of the return and repair models As mentioned before, the return model represents a procurement environment in which the buyer returns defective items to the supplier for refund, whereas the repair model represents a production-inventory environment in which the buyer repairs the defective items. The firm may own the production facility in the repair model. We also pointed out that as the demand rate (or the unit holding cost) changes, the optimal unreliability level in the return model and in the repair model may show opposite behavior. This is an interesting observation that implies, depending on whether the buyer or the supplier assumes the risk of quality imperfection, the buyer should adopt completely different strategies in supplier selection. Despite their different implications, the return and repair models turn out to be very similar in their costs. Comparing the results of the previous two sections, we find out that the average unit holding cost, h=ð2DÞ, in the return model plays the same role as the unit repair cost, cR , in the repair model. To provide an intuitive interpretation of this equivalence, we assume that y is given and then compare our models with the classic EOQ model. The return model requires a larger order quantity than the EOQ model does. This is intuitive because in the return model, the defectives cannot be repaired, and thereby, one has to order, on average, Dy extra units per unit time to receive the required number of non-defective items (D units per unit time). This additional inventory would then translate into an additional average holding cost of ½h=ð2DÞ ðDyÞ per unit time. Thus, as shown before, the minimum total cost per unit time for the return model is equal to that of the EOQ model plus hy=2, i.e., cðyÞD þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi hy : 2AðyÞDh þ 2
On the other hand, the repair model requires the same order quantity as the EOQ model. This is so because in the repair model,
169
the defective items are repaired and ultimately become usable. In this case, one has to repair, on average, Dy units per unit time out of the D units received, which results in an average repair cost of cR ðDyÞ per unit time. Hence, as seen before, the minimum total cost per unit time for the repair model becomes equal to that of the EOQ model plus cR Dy, i.e., cðyÞD þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2AðyÞDh þcR Dy:
Following the above argument and comparing the minimum cost expressions, we can easily see that h=ð2DÞ and cR have exactly the same role: They both represent the cost imposed on the inventory system by a defective unit. Before closing this section, we provide a numerical example in which we illustrate the practical relevance of our models and results. Illustration: Consider a firm that procures an item from a single supplier. After receiving each batch, the firm inspects the items to identify and remove the defectives. The number of usable items in a batch is believed to follow a Binomial distribution. The firm may ask the supplier for higher quality (i.e., a lower defective rate) but high-quality items cost more in terms of both the unit price and the order cost. Moreover, the firm may accept defective rates up to 5%, i.e., ymax ¼ 0:05. The unit price and fixed order cost are functions pof ffiffiffi the defective rate, respectively, as follows: cðyÞ ¼ 0:5= y and AðyÞ ¼ 1=y. These two functions are shown in Fig. 1. A defective rate of 5%, for instance, translates into a unit price of $2.24/unit and a fixed order cost of $20/order. To get a defective rate of 1%, the firm has to pay much more: $5/unit and $100/order. The annual demand for the item is D ¼ 1000 units, the holding cost is h ¼ $1=unit=year, and the repair cost is cR ¼ $1=unit. If the firm follows the return policy where defective units are returned to the supplier for refund, then the optimal defective rate is y ¼ 0:05 (using Example 2), the optimal order quantity is Q ¼ 211 (by Theorem 2(c)), and the average annual cost is $2436.09 (by Theorem 2(d)). If, on the other hand, the firm decides to apply the repair policy where defective units are repaired, then y ¼ 0:05 (using Example 4), Q ¼ 200 (by Theorem 4(d)), and the average annual cost is $2486.07 (by Theorem 4(e)). In this example, implementing either policy would require that the firm purchase low-quality items. However, the firm is better off returning the defectives to the supplier for refund rather than repairing the defectives itself. Unless the firm is able to cut the unit repair cost cR to below $0.0005 ½ ¼ h=ð2DÞ, the return policy will dominate the repair policy. To obtain more insights into the problem, we now perform sensitivity analysis on the parameters h, cR , and ymax . We change the parameter values as follows: h ¼ 1; 10; 100; cR ¼ 0:01; 0:1; 1; and ymax ¼ 0:05; 0:1. Table 1 presents the average annual cost for the return and repair
Fig. 1. Unit price and order cost as functions of defective rate.
170
M.M. Tajbakhsh et al. / Journal of Purchasing & Supply Management 16 (2010) 163–170
Table 1 Return vs. repair model. Parameters
ymax ¼ 0:05
ymax ¼ 0:1
h ($/unit/year)
cR ($/unit)
Return ($/year)
Repair ($/year)
Return ($/year)
Repair ($/year)
1
0.01 0.1 1
2436.09
2436.57 2441.07 2486.07
1722.61
1723.56 1732.56 1822.56
10
0.01 0.1 1
2868.77
2869.02 2873.52 2918.52
2028.85
2029.35 2038.35 2128.35
100
0.01 0.1 1
4238.57
4236.57 4241.07 4286.07
3000.35
2996.35 3005.35 3095.35
policies. We make the following observations:
The return policy outperforms the repair policy unless the unit
repair cost cR is small enough (e.g., see cases with cR ¼ 0:01 in Table 1). This is in agreement with our previous finding: in order for the repair policy to dominate the return policy, cR must be lower than h=ð2DÞ. Increasing ymax from 0.05 to 0.1 reduces the average annual cost in both the return and the repair models. This implies that if we are willing to accept higher risk in the supply process, we may indeed achieve cost savings.
6. Conclusion In this paper, we have studied an EOQ model with Binomial yield. This approach to modeling supply unreliability is suitable for both the imperfect quality and the uncertain quantity contexts. Assuming that the fixed order cost and the unit variable cost are functions of unreliability level, we develop single supplier models in which the optimal order quantity and the optimal unreliability level are determined. The return and repair models, though seemingly similar, behave very differently. In the return model, when only the fixed order cost or the unit variable cost is dependent on the unreliability level, we find that unreliable suppliers are appropriate for higher demand rates while reliable suppliers are needed for higher holding costs. In the repair model, however, when the fixed order cost is strictly decreasing in the unreliability level, we find that reliable suppliers are needed for higher demand rates while unreliable suppliers are appropriate for higher holding costs. Interestingly, we observe that the average unit holding cost, h=ð2DÞ, in the return model plays the same role as the unit repair cost, cR , in the repair model. Similar to Porteus (1986), we can interpret ymax as the original unreliability level and consider the option of investing in
reliability improvement. That is, to reduce the unreliability level, one has to not only pay higher fixed order and/or unit variable costs but also invest directly in the (supplier’s) production process. Using our models, one might address this issue by adding the investment cost of changing the unreliability level to kðyÞ, the average total cost optimized over the order quantity. In this case, it would be interesting to see how the results presented in this paper would change.
References Barad, M., Braha, D., 1996. Control limits for multi-stage manufacturing processes with binomial yield (single and multiple production runs). The Journal of the Operational Research Society 47 (1), 98–112. Chen, J., Yao, D.D., Zheng, S., 2001. Optimal replenishment and rework with multiple unreliable supply sources. Operations Research 49 (3), 430–443. Cheng, T.C.E., 1991. EPQ with process capability and quality assurance considerations. The Journal of the Operational Research Society 42 (8), 713–720. Gerchak, Y., Parlar, M., 1990. Yield randomness, cost tradeoffs, and diversification in the EOQ model. Naval Research Logistics 37 (3), 341–354. Grosfeld-Nir, A., 2005. A two-bottleneck system with binomial yields and rigid demand. European Journal of Operational Research 165 (1), 231–250. Grosfeld-Nir, A., Ronen, B., 1993. A single bottleneck system with binomial yields and rigid demand. Management Science 39 (5), 650–653. Micheli, G.J.L., 2008. A decision-maker-centred supplier selection approach for critical supplies. Management Decision 46 (6), 918–932. Porteus, E.L., 1985. Investing in reduced setups in the EOQ model. Management Science 31 (8), 998–1010. Porteus, E.L., 1986. Optimal lot sizing, process quality improvement and setup cost reduction. Operations Research 34 (1), 137–144. Silver, E.A., 1976. Establishing the order quantity when the amount received is uncertain. INFOR 14 (1), 32–39. Treleven, M., Schweikhart, S.B., 1988. A risk/benefit analysis of sourcing strategies: single vs. multiple sourcing. Journal of Operations Management 7 (4), 93–114. Tripathy, P.K., Wee, W.-M., Majhi, P.R., 2003. An EOQ model with process reliability considerations. The Journal of the Operational Research Society 54 (5), 549–554. Yano, C.A., Lee, H.L., 1995. Lot sizing with random yields: a review. Operations Research 43 (2), 311–334. Zipkin, P.H., 2000. Foundations of Inventory Management. McGraw-Hill/Irwin, Boston.
Contents lists available at ScienceDirect
Journal of Purchasing & Supply Management journal homepage: www.elsevier.com/locate/pursup
Sole sourcing in EOQ models with Binomial yield M. Mahdi Tajbakhsh a,, Chi-Guhn Lee b, Saeed Zolfaghari c a b c
Department of Industrial Engineering, Dalhousie University, P.O. Box 1000, Halifax, Nova Scotia, Canada B3J 2X4 Department of Mechanical and Industrial Engineering, University of Toronto, 5 King’s College Road, Toronto, Ontario, Canada M5S 3G8 Department of Mechanical and Industrial Engineering, Ryerson University, 350 Victoria Street, Toronto, Ontario, Canada M5B 2K3
a r t i c l e in fo
abstract
Article history: Received 16 April 2009 Received in revised form 28 November 2009 Accepted 8 December 2009
We develop two EOQ-based inventory models in which an inventory manager must determine the order size as well as the supply reliability level in the presence of uncertainty in the quality and/or quantity of supply. The number of acceptable units in the order is captured by the Binomial yield model, and reliability is increased both by increasing the order setup cost and by increasing the unit price. For each developed model, we present an equation of which a solution is the optimal reliability level and a closed form solution for the optimal order size given the optimal reliability level. We then provide a comparison of the two models. & 2009 Elsevier Ltd. All rights reserved.
Keywords: Inventory management Supply management Binomial yield
1. Introduction Uncertainty in supply quantity and/or quality has been one of the major concerns in procurement decisions (Treleven and Schweikhart, 1988). This type of uncertainty, also known as random yield, affects lot sizing decisions in production control and inventory management. Applications are numerous (Yano and Lee, 1995) and include electronic fabrication and assembly, chemical processes, and any procurement system with suppliers that may produce defective products. In inventory systems, random yield usually refers to one of two cases:
The quantity delivered may contain defective units (quality uncertainty).
The quantity supplied may not be equal to the quantity ordered (quantity uncertainty). Random yield has been studied extensively in the literature on production system control and inventory management. Yano and Lee (1995) present five main approaches to modeling yield uncertainty in production and inventory systems, and discuss their advantages and disadvantages. Only three models are relevant to pure inventory management: Binomial yield, stochastically proportional yield, and random capacity. In the Binomial yield model, the acceptable quantity is Binomially distributed Corresponding author. Tel.: + 1 902 494 6173.
E-mail addresses: [email protected] (M.M. Tajbakhsh), [email protected] (C.-G. Lee), [email protected] (S. Zolfaghari). 1478-4092/$ - see front matter & 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.pursup.2009.12.005
with two parameters: the order quantity and the probability of an individual item being non-defective. On the other hand, in the stochastic proportional yield model the acceptable quantity is a multiplication of the yield rate, which is a random variable, and the order quantity. In the random capacity model, the received quantity is the minimum of the order quantity and the random capacity. The Binomial yield model has been used in discrete production systems to describe random yield by some researchers (GrosfeldNir and Ronen, 1993; Barad and Braha, 1996; Grosfeld-Nir, 2005). As Yano and Lee (1995) explain, the Binomial yield model assumes that the supplier’s production process is stationary and that the generation of one unit (at the supplier’s facility) is independent of the generation of other units. This representation is suitable when the (supplier’s) production system is in control for long durations. Examples include high-tech products (Grosfeld-Nir and Ronen, 1993), and electronic or mechanical products (Barad and Braha, 1996). Although random yield has been studied extensively, most researchers assume that the pattern of yield randomness, or the yield distribution, is given and focus on determining optimal lot sizes. However, as argued by Gerchak and Parlar (1990), a firm may be able to improve yield by implementing a tighter quality assurance program in its supply base at the expense of increased unit purchasing cost and/or order setup cost. Gerchak and Parlar (1990) consider an EOQ model with stochastically proportional yield, where the expected number of acceptable units in an order is the order size times the mean yield rate (i.e., expected fraction acceptable) and the variance of the acceptable units is the square of order size times the variance of the yield rate. Assuming that
164
M.M. Tajbakhsh et al. / Journal of Purchasing & Supply Management 16 (2010) 163–170
the yield rate variance is a control variable but not the mean yield rate, they analyzed two cases: the Cost-Per-Order model and the Unit-Cost model. In the former, when the order setup cost is a decreasing, convex function of the standard deviation of the yield rate and the unit cost is fixed, Gerchak and Parlar (1990) show that the optimal order size is decreasing in the standard deviation of the yield rate (this relationship has also been observed by Silver, 1976 in a simpler model). They also show that the optimal order size is decreasing in the mean yield rate for a fixed standard deviation, and that the optimal value for the standard deviation does not depend on the demand rate and unit holding cost. In the latter, it is the cost per unit ordered that depends on yield variability. Using a logarithmic functional form for the cost per unit, they show that the optimal value of the standard deviation is increasing in the demand rate and decreasing in the unit holding cost. Cheng (1991) considers an EOQ model in which the expected fraction acceptable (mean yield rate) can be changed through quality assurance efforts. While the yield variance (variability) is a decision variable in Gerchak and Parlar (1990), it is implicitly ignored by Cheng (1991). Assuming that the unit production cost is an increasing power function of the acceptable fraction, Cheng (1991) obtains closed-form expressions for the optimal production run and the optimal fraction acceptable. Moreover, he shows that the higher the quality, the smaller the lot size. Tripathy et al. (2003) generalize the model and the results of Cheng (1991). In their model, the unit production cost is a power function of both the acceptable fraction and the demand rate, increasing in the former and decreasing in the latter. Thus, in addition to the lot size and acceptable fraction, the demand rate is a decision variable. Tripathy et al. (2003) obtain closed-form expressions for the optimal production run, the optimal fraction acceptable, and the optimal demand rate. Moreover, they show that higher quality may lead to both a smaller lot size and a lower demand rate. Porteus (1986) has also shown a significant relationship between lot sizing and quality, where he modifies the EOQ model as follows. The production process begins in the ‘‘in-control’’ state after each setup. While producing any single unit in the lot, the production process (machine) can go ‘‘out-of-control’’ with a given probability. Once out of control, the process produces defective units throughout the current lot and until the next setup (which returns the process to the in-control state). Here, the probability of going out of control while producing each unit is constant, and thus, the time to failure is geometrically distributed. Moreover, it is assumed that each defective unit can be repaired (reworked) instantaneously at a cost. Porteus (1986) provides three options for investing in quality improvements: (a) reducing the probability that the process goes out of control; (b) reducing the setup cost; and (c) using the two previous options simultaneously. He shows that the optimal lot size is smaller than the classical EOQ. By assuming a logarithmic form of the investment cost function, he also shows that the optimal process quality level (i.e., probability that the process goes out of control) is decreasing in the demand rate, increasing in the unit holding cost, and decreasing in the unit repair cost. One should note that there are two main differences between the work of Porteus (1986) and the other papers reviewed thus far. First, Porteus (1986) assumes that defective items are repaired while the previous papers do not consider repair or rework. Second, Porteus (1986) assumes a constant setup cost and unit production cost to obtain the optimal lot size, and then, introduces an investment cost to change (and thereby, optimize) the probability of producing a defective unit or the setup cost. The previous papers, however, define the setup cost or unit cost as a function of random yield parameters, and optimize both the order quantity and the yield distribution simultaneously.
The main objective of our paper is to develop inventory models to simultaneously optimize the order quantity and the reliability level of an unreliable supplier, when the unreliability is captured by the Binomial yield model. To this end, we study two EOQ models when the acceptable (usable) quantity is Binomially distributed with two parameters: the order quantity and the reliability level. In the first model called the return model, the supplier assumes responsibility for the supply unreliability, whereas in the second model called the repair model, the buyer absorbs the unreliability risks at a cost. The major contribution of this study is twofold. Firstly, we allow both the order setup cost and the unit cost to simultaneously change with unreliability level, whereas Gerchak and Parlar (1990) allow both but not simultaneously, Cheng (1991) allows only the latter, and Porteus (1986) allows neither. Secondly, and more importantly, by investigating the return and repair models in a unified framework, we can compare these two modeling approaches and highlight their implications on the obtained results, depending on whether the supplier or the buyer takes responsibility for supply unreliability. The previous papers in the literature, however, assume that either the supplier (similar to our return model) or the buyer (similar to our repair model) absorbs the unreliability risks, and thus, have not compared the results of these two different assumptions. The assumption that the order setup cost and unit variable cost are functions of unreliability level deserves more explanation. Consider a company that uses a production process with a certain level of capability to manufacture a product, and suppose that a quality assurance program is in place to monitor the quality of the items. The capability of the process and the effectiveness of the quality assurance program depend upon various factors such as production technology, machine capability, work methods, use of online monitoring systems, skill level of workers, and inspection policies (Cheng, 1991). Enforcing a high level of process capability and stringent quality assurance will evidently result in highquality products with lower defectives (i.e., higher reliability). This, however, requires substantial investment in plant, machinery, equipment, and employee training, thereby increasing the unit production cost. Furthermore, adopting advanced production technology necessitates the use of more skilled labor and more expensive tools to initiate production runs. This in turn implies higher setup costs. It is worth mentioning that Zipkin (2000) has also studied the EOQ setting with imperfect quality. Specifically, the return and repair models in our paper are similar to, respectively, the ‘‘immediate detection, reimbursement’’ and ‘‘immediate detection, no reimbursement’’ cases in Section 3.6 of Zipkin (2000). However, there are three main points that distinguish our work from his. First, the unreliability level in our models is a decision variable while it is a given parameter in his models, i.e., he only focuses on determining optimal order quantities. Second, the order setup cost and the unit cost are functions of the unreliability level in our models whereas they are fixed in his models. Third, the way he has modeled unreliability differs from ours. Zipkin assumes that each batch contains a fixed fraction of defective items, irrespective of the batch size, and thereby, he does not take yield variability into account (similar to Cheng, 1991). Using the Binomial yield model, however, we are able to incorporate the yield variance into our models. The organization of the paper is as follows. In the next section, we explain the main assumptions and notation of our models. Then, we analyze the return and repair models where determination of supply reliability is taken into consideration. Next, we compare these two models. The last section provides concluding remarks.
M.M. Tajbakhsh et al. / Journal of Purchasing & Supply Management 16 (2010) 163–170
2. Preliminaries In this paper, we consider inventory problems with a single unreliable supplier, in which both the optimal order quantity and the optimal unreliability level must be determined simultaneously. That is, in addition to determining the optimal order quantity, the inventory manager can choose the best unreliability level, noting that as supply reliability increases, the item becomes more expensive and the fixed order cost may increase. The fact that unreliability level is a decision variable affecting the unit cost and fixed order cost may have one of the following two interpretations: 1. When dealing with a predetermined supplier, the inventory manager may ask for a higher level of supply reliability and may be willing to pay more. 2. When selecting a supplier from a supplier pool, the inventory manager can choose a more reliable supplier with a possibly higher fixed order cost and acquire the product at a higher price. Notice that supplier selection is generally based on minimizing total costs (e.g., supplier management, transportation, and inventory costs). Thus, one might reduce the total costs by procuring the product from an unreliable supplier and saving on the purchase price and order cost. This is obviously applicable to items that are not critical. For more discussion, the reader is referred to Micheli (2008) and the references therein.
165
They are general enough to capture common functional forms in the literature, yet they allow us to obtain analytical results for our models. One should note that the log-convexity assumption for AðyÞ is not too strong. For example, the power b function AðyÞ ¼ a y ; a; b4 0, and the logarithmic function AðyÞ ¼ ab ln y; a Z b 40, are log-convex. These two functional forms have been used extensively in the literature to model the order setup cost as a function of uncertainty or variability measures (e.g., see Gerchak and Parlar, 1990; Porteus, 1986). 2.2. Binomial yield In this paper, we use the Binomial yield model: When an order of size Q is placed with a supplier, the number of usable items supplied or the quantity delivered, YQ , is a Binomial random variable with parameters Q and 1y, i.e., ! Q Q y PrfYQ ¼ yg ¼ ; y ¼ 0; 1; . . . ; Q : ð1yÞy y y Thus, we can write the mean and variance of random yield, respectively, as follows: E½YQ ¼ Q ð1yÞ;
ð1Þ
VarðYQ Þ ¼ Q yð1yÞ:
ð2Þ
As observed, Binomial yield can be used to model either of the following two cases:
Quality uncertainty: A proportion of any quantity supplied 2.1. Notation and assumptions We modify the classical EOQ model and use the following notation:
D ¼ Constant demand rate (units/unit time). h ¼ Holding cost per unit per unit time ($/unit/unit time). Q ¼ Order quantity. y ¼ Unreliability level: Probability that a unit is defective or is not delivered. AðyÞ ¼ Fixed order cost ($/replenishment). cðyÞ ¼ Unit variable cost of the item ($/unit). YQ ¼ Random yield: usable items supplied or quantity delivered by a supplier (random variable). ETðQ ; yÞ ¼ ðExpectedÞ length of a cycle (a cycle is defined as the time between two consecutive replenishments). ECðQ ; yÞ ¼ Expected total cost per cycle. GðQ ; yÞ ¼ Long run average cost per unit time ($/unit time).
Our assumptions are as follows: Assumption S1: Q and y are assumed to be continuous decision variables. Assumption S2: cðyÞ is assumed to be a non-increasing and convex function of y. Assumption S3: AðyÞ is assumed to be a non-increasing and logconvex function of y. Assumption S4: We assume that 0 r y r ymax o1, where ymax is the highest unreliability level acceptable to management. In fact, ymax shows the maximum risk that management is willing to accept in the supply process, and is expected to be moderate in practice. That cðyÞ and AðyÞ are non-increasing in y is to establish that higher reliability comes at a price; that is, as y decreases (reliability improves), the unit variable cost and the fixed order cost will go up. The convexity assumption on cðyÞ and the log-convexity assumption on AðyÞ are technical assumptions.
may be defective, i.e., each unit supplied is defective with probability y and non-defective with probability 1y. Here, y is called the defective rate. Quantity uncertainty: A proportion of any quantity ordered may not be delivered, i.e., any item ordered may not be delivered with probability y. Here, 1y is called the yield ratio.
In order to capture both the quality uncertainty and the quantity uncertainty cases, we refer to y as unreliability level. Clearly, as y increases, supply reliability decreases.
3. The return model In this section, we consider a pure inventory model (procurement environment) where each unit produced at the supplier’s facility is defective with probability y and non-defective with probability 1y. The model is called the return model and is interpreted in two ways:
The buyer performs 100% inspection after receiving
the quantity ordered, and returns defective units to the supplier for refund. Here, the buyer encounters quality uncertainty. The supplier performs 100% inspection before dispatching the buyer’s order, and removes imperfect units from the batch. Here, the buyer experiences quantity uncertainty. For instance, this is the case when the supplier is not able to replace the defectives due to a lack of enough inventory for immediate replacement. Outsourcing the item, the supplier requires some time to receive new orders; manufacturing the item, the supplier needs time to produce new items.
In the return model, implicit assumptions are that the quality imperfection is the supplier’s responsibility, and that the defective items identified in a batch cannot be immediately reworked or
166
M.M. Tajbakhsh et al. / Journal of Purchasing & Supply Management 16 (2010) 163–170
replaced by non-defective items. Therefore, the buyer does not pay for the units defective or not delivered, i.e., cðyÞ is the cost per (non-defective) unit received.
00
The function uðyÞ is convex in y (i.e., u ðyÞ Z0) if and only if 00
2AðyÞA ðyÞ½A0 ðyÞ2 Z 0()
½AðyÞ2 A ðyÞ AðyÞA ðyÞ½A ðyÞ2 þ Z 0() Z0: AðyÞ ½AðyÞ2 00
3.1. Model formulation In the return model, the expected total cost per cycle and the expected cycle length, respectively, are ECðQ ; yÞ ¼ AðyÞ þcðyÞ E½YQ þ h ETðQ ; yÞ ¼
E½YQ2 2D
AðyÞA ðyÞ½A0 ðyÞ2 00
¼
d A0 ðyÞ d d d2 ¼ ln AðyÞ ¼ ln AðyÞ: 2 dy AðyÞ dy dy dy
Thus, uðyÞ is convex in y if and only if 00
A ðyÞ d2 þ 2 ln AðyÞ Z 0: AðyÞ dy
In accordance with the Renewal Reward Theorem, we can write the long-run average cost per unit time as 2 ECðQ ; yÞ AðyÞD h E½YQ GðQ ; yÞ ¼ ¼ cðyÞD þ þ ; ETðQ ; yÞ E½YQ 2 E½YQ
AðyÞ D h þ ½Q ð1yÞ þ y: Q ð1yÞ 2
d2 2
ln AðyÞ Z 0; 00
moreover, A ðyÞ Z 0, because a log-convex function is also a convex function. Therefore, Inequality (4) is satisfied, implying that uðyÞ is a convex function of y. & ð3Þ
The optimization problem is represented as fminGðQ ; yÞjQ 4 0; 0 r y r ymax g: 3.2. Model analysis
In the proof of Lemma 1, we notice that the log-convexity of AðyÞ is a sufficient (not necessary) condition for the convexity of uðyÞ. Using Lemma 1, we observe that kðyÞ is the sum of three convex functions, and hence, is a convex function of y. Now, we have A0 ðyÞD
Examining the second partial derivatives of GðQ ; yÞ with respect to Q and y, one can easily verify that GðQ ; yÞ is strictly convex in Q but is not convex in y. Thus, GðQ ; yÞ is not jointly convex in ðQ ; yÞ. A reasonable approach to the optimization problem is to fix y, optimize over Q to obtain Q ðyÞ, and then optimize over y. Therefore, to obtain the optimal values of Q and y, we proceed as follows. First, for a fixed y, we obtain the optimal order quantity, Q ðyÞ, by setting @GðQ ; yÞ=@Q ¼ 0, i.e., qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi AðyÞD hð1yÞ 1 ¼ 0 ) Q ðyÞ ¼ 2AðyÞD=h: þ 2 1y Q 2 ð1yÞ If AðyÞ ¼ A (i.e., the fixed order cost is a constant independent of y), then Q ðyÞ is increasing in y. In this case, improved quality (low value of y) corresponds to small order quantities. Gerchak and Parlar (1990) and Cheng (1991) have also pointed out this behavior. Next, we substitute Q ðyÞ in GðQ ; yÞ to obtain kðyÞ as follows: AðyÞD h þ ½Q ðyÞð1yÞ þ y ¼ cðyÞD kðyÞ :¼ GðQ ðyÞ; yÞ ¼ cðyÞD þ Q ðyÞð1yÞ 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi AðyÞD h hy pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ 2AðyÞDh: þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ ½ 2AðyÞD=h þ y ¼ cðyÞD þ 2 2AðyÞD=h 2
fminkðyÞj0 r y r ymax g: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Letting uðyÞ :¼ 2AðyÞDh, we obtain the following result. Lemma 1. uðyÞ is a convex function of y. Proof. We have rffiffiffiffiffiffiffi! Dh A0 ðyÞ 0 pffiffiffiffiffiffiffiffiffi u ðyÞ ¼ 2 AðyÞ and ) rffiffiffiffiffiffiffi( 00 0 Dh A ðyÞ 1 ½A ðyÞ2 pffiffiffiffiffiffiffiffiffi : u ðyÞ ¼ 2 2 ½AðyÞ3=2 AðyÞ
h 2
k0 ðyÞ ¼ c0 ðyÞD þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ : 2AðyÞD=h
Now the optimization problem reduces to
ð4Þ
Since AðyÞ is assumed to be log-convex in y, we have dy
which, using (1) and (2), yields
00
0
Now, we can write ½AðyÞ
E½YQ : D
GðQ ; yÞ ¼ cðyÞD þ
00
2
;
2AðyÞA ðyÞ½A0 ðyÞ2 00
By setting k0 ðyÞ ¼ 0, we get f ðyÞ ¼
h ; 2D
ð5Þ
where A0 ðyÞ f ðyÞ :¼ c0 ðyÞ þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 2AðyÞD=h Assume that y^ is a solution to Eq. (5), i.e., f ðy^ Þ ¼ h=2D. Moreover, let y denote the optimal unreliability level. Due to convexity of kðyÞ in y, we must consider the following three cases:
If y^ o 0, then y ¼ 0. If 0 r y^ r ymax , then y ¼ y^ . If y^ 4 ymax , then y ¼ ymax . Moreover, because cðyÞ and uðyÞ are convex functions of y, then c0 ðyÞ and u0 ðyÞ=D are both non-decreasing in y. This implies that f ðyÞ is a non-decreasing function of y. Thus, using Eq. (5), we observe that, if either the fixed order cost or the unit variable cost is a constant independent of y (i.e., either A0 ðyÞ ¼ 0 or c0 ðyÞ ¼ 0), then y is non-decreasing in D and non-increasing in h. Using the above results, we now ascertain the following theorem. Theorem 2. If y^ is a solution to f ðyÞ ¼ h=2D, then the following hold:
(a) The optimal 8 > > <0 y ¼ y^ > > :y max
unreliability level, y , is if y^ o 0; if 0 r y^ r ymax ; if y^ 4 ymax :
(b) If either AðyÞ or cðyÞ is a constant independent of y, then y is a non-decreasing function of D and a non-increasing function of h.
M.M. Tajbakhsh et al. / Journal of Purchasing & Supply Management 16 (2010) 163–170
(c) The optimal order quantity, Q , is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 Q ¼ 2Aðy ÞD=h: 1y (d) The optimal average cost per unit time, GðQ ; y Þ, is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi GðQ ; y Þ ¼ cðy ÞD þ 2Aðy ÞDh þhy =2: Here, y is assumed to be continuous. Because kðyÞ is convex in y, Part (a) can be easily modified to address the case of discrete unreliability level. To do so, assume that y is to be selected from a set of discrete values represented by Y :¼ fy0 ; y1 ; y2 ; . . . ; ym j0 r yi o yi þ 1 ; i ¼ 0; 1; . . . ; m1g, where ym ¼ ymax . Then, Theorem 2(a) can be modified as follows: 8 > if y^ o y0 ; > < y0 y ¼ argminfkðyi Þ; kðyi þ 1 Þg if yi r y^ o yi þ 1 ; i ¼ 0; 1; . . . ; m1; > > :y if y^ Z ym : m The discrete case can be used in supplier selection where a supplier is to be chosen from a set of suppliers with different prices, order costs, and unreliability levels and the goal is to minimize the average cost per unit time. Here, yi refers to the unreliability level of supplier i. Part (b) shows that as the demand rate increases and/or the unit holding cost decreases, no improvement in supply reliability should be carried out (i.e., it may be economical to choose a less reliable supplier). In particular, assume that either AðyÞ is strictly log-convex in y or cðyÞ is strictly convex in y, so that f ðyÞ is strictly increasing in y. Our observations then suggest that as the market grows in medium ranges, one should rely on less reliable suppliers, and that as storage cost increases in medium ranges, more reliable suppliers should be chosen. As mentioned before, Gerchak and Parlar (1990) obtain similar results in their Unit-Cost model. Namely, with a logarithmic unit-cost function, they find that the optimal yield variability is increasing in D and decreasing in h. In general, Eq. (5) may not have a solution at all or may have more than one solution. Here, we present examples in which a unique, closed-form solution to Eq. (5) can be found. Example 1. Let AðyÞ ¼ A. We now consider two functional forms for the unit variable cost:
Porteus (1985) to denote investment costs of reducing the order setup cost. We can easily verify that AðyÞ is a decreasing, logconvex function of y. We now consider two cases for the unit variable cost: 1. Letting cðyÞ ¼ c, we can write Eq. (5) as rffiffiffiffiffiffiffi! ah h b=21 ; ¼ by 2D 2D which yields rffiffiffiffiffiffiffiffiffi!2=ðb þ 2Þ 2aD y^ ¼ b : h Thus, we get 9 8 rffiffiffiffiffiffiffiffiffi! 2=ðb þ 2Þ = < 2aD y ¼ min b ; ymax : ; : h b=2
2. Letting cðyÞ ¼ cy ; c 4 0, we can write Eq. (5) as rffiffiffiffiffiffiffi! bc b=21 ah h b=21 ; y ¼ by 2 2D 2D which gives " pffiffiffiffiffiffiffiffiffiffiffiffiffiffi #2=ðb þ 2Þ bDðc þ 2ah=DÞ y^ ¼ : h Hence, we get 8" 9 #2=ðb þ 2Þ < bDðc þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi = 2ah=DÞ y ¼ min ; ymax : : ; h
4. The repair model In this section, we consider a production-inventory model where each unit produced (at the supplier’s facility) is defective with probability y and non-defective with probability 1y. The model is called the repair model and is interpreted in two ways:
The buyer performs 100% inspection after receiving the
1. Logarithmic function. Let cðyÞ ¼ ab ln y;
a; b 40:
This functional form has been used by Gerchak and Parlar (1990) to describe the unit variable cost as a function of yield variability. One can easily verify that cðyÞ is strictly decreasing and convex in y. Now, using Eq. (5), we get b=y ¼ h=2D, which yields y^ ¼ 2bD=h. Finally, we have y ¼ minf2bD=h; ymax g. 2. Power function. Let b
cðyÞ ¼ ay
;
a; b 40:
This functional form has been used by Cheng (1991) to describe the unit production cost as a function of fraction acceptable. One can easily verify that cðyÞ is strictly decreasing b1 and convex in y. Now, using Eq. (5), we get aby ¼ h=2D, 1=ðb þ 1Þ ^ which yields y ¼ ð2abD=hÞ . Finally, we have y ¼ minfð2abD=hÞ1=ðb þ 1Þ ; ymax g. b
Example 2. Let AðyÞ ¼ ay ; a; b 40 (power function). This functional form has been used by Gerchak and Parlar (1990) to describe the order cost as a function of yield variability, and by
167
quantity ordered, and repairs (reworks) defective units at an additional cost. Alternatively, this can represent a manufacturing environment in which the production process is unreliable and imperfect units are repaired (reworked). The supplier performs 100% inspection before dispatching the buyer’s order, and replaces the defective units with nondefective units at an additional cost. As mentioned by Chen et al. (2001), this represents a new feature that has not been addressed in other random yield models: At a premium (in addition to the purchase price) per unit, the supplier guarantees to replace the defective units and delivers the entire order.
In the former, we let cR denote the repair (or rework) cost of each defective item ($/unit). In the latter, ycR denotes the additional premium per unit ordered, which is paid to guarantee the delivery. In the repair model, implicit assumptions are that the quality imperfection is the buyer’s responsibility, and that the defective items identified in a batch can be instantaneously reworked or replaced by non-defective items. Therefore, the buyer also pays for the defective units, i.e., cðyÞ is the cost per unit ordered. Gerchak and Parlar (1990), in their Unit-Cost model, also assume
168
M.M. Tajbakhsh et al. / Journal of Purchasing & Supply Management 16 (2010) 163–170
that the unit variable cost is the cost per unit ordered, which is more appropriate in manufacturing environments (see Yano and Lee, 1995 for more discussion).
Therefore, if either A0 ðyÞ ¼ 0 or A0 ðyÞ o 0, then y is nonincreasing in D and non-decreasing in h. Using the above results, we now ascertain the following theorem. Theorem 4. If y^ is a solution to f ðyÞ ¼ cR , then the following hold:
4.1. Model formulation
Using (1), the mean of random yield is Q ð1yÞ, and thus, the expected number of defective items in a lot of size Q is Q y. Therefore, the expected total cost per cycle is 2
ECðQ ; yÞ ¼ AðyÞ þcðyÞ Q þ
hQ þcR Q y: 2D
Since the cycle length, ETðQ Þ, is ETðQ Þ ¼
Q ; D
we can write the long-run average cost per unit time as GðQ ; yÞ ¼
ECðQ ; yÞ AðyÞD hQ ¼ cðyÞD þ þ þ cR Dy: ETðQ Þ Q 2
ð6Þ
(b) If AðyÞ is either a constant independent of y or strictly decreasing in y, then y is a non-increasing function of D and a nondecreasing function of h. (c) y is a non-increasing function of cR . (d) The optimal order quantity, Q , is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Q ¼ 2Aðy ÞD=h:
(e) The optimal average cost per unit time, GðQ ; y Þ, is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi GðQ ; y Þ ¼ cðy ÞD þ 2Aðy ÞDh þ cR Dy :
As before, the optimization problem is represented as fminGðQ ; yÞjQ 4 0; 0 r y r ymax g: 4.2. Model analysis As we observe, the cost function per unit time in the repair model is very similar to that in the return model. Indeed, one could obtain Expression (6) by making the following transformations in Expression (3): h=2 to cR D in the linear term hy=2, and Q ð1yÞ to Q. Using this relationship, we can shorten the analysis of the repair model. For a fixed y, we obtain the optimal order quantity Q ðyÞ as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Q ðyÞ ¼ 2AðyÞD=h: Thus, in the repair model, Q ðyÞ is a non-increasing function of y. That is, improved quality implies large order quantities, which has also been observed by Porteus (1986). Next, substituting Q ðyÞ in GðQ ; yÞ, we obtain kðyÞ as pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi kðyÞ ¼ GðQ ðyÞ; yÞ ¼ cðyÞD þ 2AðyÞDh þcR Dy: By setting k0 ðyÞ ¼ 0, we get f ðyÞ ¼ cR ;
(a) The optimal unreliability level, y , is 8 > if y^ o 0; > <0 y ¼ y^ if 0 r y^ r ymax ; > > :y if y^ 4 ymax : max
ð7Þ
where A0 ðyÞ f ðyÞ ¼ c0 ðyÞ þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 2AðyÞD=h Assuming that f ðy^ Þ ¼ cR , one can obtain y in the same way as in the return model (see Theorem 2(a)). We have shown that f ðyÞ is a non-decreasing function of y. Thus, using Eq. (7), we observe that y is non-increasing in cR . Obviously, if AðyÞ ¼ A, then y is independent of D and h.
As shown for Theorem 2(a), we can easily modify Part (a) to handle the case of discrete unreliability level. Part (b) shows that as the demand rate decreases and/or the unit holding cost increases, no improvement in supply reliability should be carried out (i.e., it may be economical to choose a less reliable supplier). In particular, assume that AðyÞ is strictly log-convex in y or cðyÞ is strictly convex in y, so that f ðyÞ is strictly increasing in y. Our observations then suggest that as the market grows in medium ranges, one should rely on more reliable suppliers, and that as storage cost increases in medium ranges, less reliable suppliers should be chosen. Part (c) implies that as the unit repair cost increases in medium ranges, more reliable suppliers should be selected. As mentioned before, Porteus (1986) obtains similar results. Namely, with a logarithmic investment cost function, he shows that the optimal process quality probability (i.e., probability that the production process becomes out-of-control) is decreasing in D, increasing in h, and decreasing in cR . Similar to Eq. (5), Eq. (7) may not have a solution at all or may have more than one solution. We now present examples in which a unique, closed-form solution to Eq. (7) can be found. Example 3. Once again, we let AðyÞ ¼ A, and consider two functional forms for the unit variable cost: the logarithmic and power forms: 1. Logarithmic function. Let cðyÞ ¼ ab ln y;
a; b4 0:
We use Eq. (7) to get y^ ¼ b=cR . Therefore, y ¼ minfb=cR ; ymax g. 2. Power function. Let
Lemma 3. If AðyÞ is strictly decreasing in y, then the following hold:
cðyÞ ¼ ay
(a) y^ is strictly decreasing in D. (b) y^ is strictly increasing in h.
We use Eq. (7) to get y ¼ minfðab=cR Þ1=ðb þ 1Þ ; ymax g.
Proof. (a) y^ is a solution to Eq. (7). Moreover, since AðyÞ is strictly decreasing in y (i.e., A0 ðyÞ o0), f ðyÞ is strictly increasing in D. Thus, as D increases, f ðy^ Þ increases. Since f ðyÞ is a non-decreasing function of y, a smaller value of y^ is needed to bring f ðy^ Þ back down to cR . This means that as D increases, y^ decreases. (b) The argument is very similar to that of Part (a). The only difference is that f ðyÞ is strictly decreasing in h. &
b
;
a; b 4 0:
b
y^ ¼ ðab=cR Þ1=ðb þ 1Þ .
Hence,
Example 4. Let AðyÞ ¼ ay ; a; b4 0 (power function). We now consider two cases for the unit variable cost: 1. Let cðyÞ ¼ c. Using Eq. (7) yields
y^ ¼
b cR
rffiffiffiffiffiffiffi!2=ðb þ 2Þ ah : 2D
M.M. Tajbakhsh et al. / Journal of Purchasing & Supply Management 16 (2010) 163–170
Thus,
8 9 ffi!2=ðb þ 2Þ < b rffiffiffiffiffiffi = ah y ¼ min ; ymax : : cR 2D ; b=2
; c 4 0. Using Eq. (7) yields 2. Let cðyÞ ¼ cy " pffiffiffiffiffiffiffiffiffiffiffiffiffiffi #2=ðb þ 2Þ bðc þ 2ah=DÞ y^ ¼ : 2cR Hence,
8" 9 #2=ðb þ 2Þ < bðc þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi = 2ah=DÞ y ¼ min ; ymax : : ; 2cR 5. Comparison of the return and repair models As mentioned before, the return model represents a procurement environment in which the buyer returns defective items to the supplier for refund, whereas the repair model represents a production-inventory environment in which the buyer repairs the defective items. The firm may own the production facility in the repair model. We also pointed out that as the demand rate (or the unit holding cost) changes, the optimal unreliability level in the return model and in the repair model may show opposite behavior. This is an interesting observation that implies, depending on whether the buyer or the supplier assumes the risk of quality imperfection, the buyer should adopt completely different strategies in supplier selection. Despite their different implications, the return and repair models turn out to be very similar in their costs. Comparing the results of the previous two sections, we find out that the average unit holding cost, h=ð2DÞ, in the return model plays the same role as the unit repair cost, cR , in the repair model. To provide an intuitive interpretation of this equivalence, we assume that y is given and then compare our models with the classic EOQ model. The return model requires a larger order quantity than the EOQ model does. This is intuitive because in the return model, the defectives cannot be repaired, and thereby, one has to order, on average, Dy extra units per unit time to receive the required number of non-defective items (D units per unit time). This additional inventory would then translate into an additional average holding cost of ½h=ð2DÞ ðDyÞ per unit time. Thus, as shown before, the minimum total cost per unit time for the return model is equal to that of the EOQ model plus hy=2, i.e., cðyÞD þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi hy : 2AðyÞDh þ 2
On the other hand, the repair model requires the same order quantity as the EOQ model. This is so because in the repair model,
169
the defective items are repaired and ultimately become usable. In this case, one has to repair, on average, Dy units per unit time out of the D units received, which results in an average repair cost of cR ðDyÞ per unit time. Hence, as seen before, the minimum total cost per unit time for the repair model becomes equal to that of the EOQ model plus cR Dy, i.e., cðyÞD þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2AðyÞDh þcR Dy:
Following the above argument and comparing the minimum cost expressions, we can easily see that h=ð2DÞ and cR have exactly the same role: They both represent the cost imposed on the inventory system by a defective unit. Before closing this section, we provide a numerical example in which we illustrate the practical relevance of our models and results. Illustration: Consider a firm that procures an item from a single supplier. After receiving each batch, the firm inspects the items to identify and remove the defectives. The number of usable items in a batch is believed to follow a Binomial distribution. The firm may ask the supplier for higher quality (i.e., a lower defective rate) but high-quality items cost more in terms of both the unit price and the order cost. Moreover, the firm may accept defective rates up to 5%, i.e., ymax ¼ 0:05. The unit price and fixed order cost are functions pof ffiffiffi the defective rate, respectively, as follows: cðyÞ ¼ 0:5= y and AðyÞ ¼ 1=y. These two functions are shown in Fig. 1. A defective rate of 5%, for instance, translates into a unit price of $2.24/unit and a fixed order cost of $20/order. To get a defective rate of 1%, the firm has to pay much more: $5/unit and $100/order. The annual demand for the item is D ¼ 1000 units, the holding cost is h ¼ $1=unit=year, and the repair cost is cR ¼ $1=unit. If the firm follows the return policy where defective units are returned to the supplier for refund, then the optimal defective rate is y ¼ 0:05 (using Example 2), the optimal order quantity is Q ¼ 211 (by Theorem 2(c)), and the average annual cost is $2436.09 (by Theorem 2(d)). If, on the other hand, the firm decides to apply the repair policy where defective units are repaired, then y ¼ 0:05 (using Example 4), Q ¼ 200 (by Theorem 4(d)), and the average annual cost is $2486.07 (by Theorem 4(e)). In this example, implementing either policy would require that the firm purchase low-quality items. However, the firm is better off returning the defectives to the supplier for refund rather than repairing the defectives itself. Unless the firm is able to cut the unit repair cost cR to below $0.0005 ½ ¼ h=ð2DÞ, the return policy will dominate the repair policy. To obtain more insights into the problem, we now perform sensitivity analysis on the parameters h, cR , and ymax . We change the parameter values as follows: h ¼ 1; 10; 100; cR ¼ 0:01; 0:1; 1; and ymax ¼ 0:05; 0:1. Table 1 presents the average annual cost for the return and repair
Fig. 1. Unit price and order cost as functions of defective rate.
170
M.M. Tajbakhsh et al. / Journal of Purchasing & Supply Management 16 (2010) 163–170
Table 1 Return vs. repair model. Parameters
ymax ¼ 0:05
ymax ¼ 0:1
h ($/unit/year)
cR ($/unit)
Return ($/year)
Repair ($/year)
Return ($/year)
Repair ($/year)
1
0.01 0.1 1
2436.09
2436.57 2441.07 2486.07
1722.61
1723.56 1732.56 1822.56
10
0.01 0.1 1
2868.77
2869.02 2873.52 2918.52
2028.85
2029.35 2038.35 2128.35
100
0.01 0.1 1
4238.57
4236.57 4241.07 4286.07
3000.35
2996.35 3005.35 3095.35
policies. We make the following observations:
The return policy outperforms the repair policy unless the unit
repair cost cR is small enough (e.g., see cases with cR ¼ 0:01 in Table 1). This is in agreement with our previous finding: in order for the repair policy to dominate the return policy, cR must be lower than h=ð2DÞ. Increasing ymax from 0.05 to 0.1 reduces the average annual cost in both the return and the repair models. This implies that if we are willing to accept higher risk in the supply process, we may indeed achieve cost savings.
6. Conclusion In this paper, we have studied an EOQ model with Binomial yield. This approach to modeling supply unreliability is suitable for both the imperfect quality and the uncertain quantity contexts. Assuming that the fixed order cost and the unit variable cost are functions of unreliability level, we develop single supplier models in which the optimal order quantity and the optimal unreliability level are determined. The return and repair models, though seemingly similar, behave very differently. In the return model, when only the fixed order cost or the unit variable cost is dependent on the unreliability level, we find that unreliable suppliers are appropriate for higher demand rates while reliable suppliers are needed for higher holding costs. In the repair model, however, when the fixed order cost is strictly decreasing in the unreliability level, we find that reliable suppliers are needed for higher demand rates while unreliable suppliers are appropriate for higher holding costs. Interestingly, we observe that the average unit holding cost, h=ð2DÞ, in the return model plays the same role as the unit repair cost, cR , in the repair model. Similar to Porteus (1986), we can interpret ymax as the original unreliability level and consider the option of investing in
reliability improvement. That is, to reduce the unreliability level, one has to not only pay higher fixed order and/or unit variable costs but also invest directly in the (supplier’s) production process. Using our models, one might address this issue by adding the investment cost of changing the unreliability level to kðyÞ, the average total cost optimized over the order quantity. In this case, it would be interesting to see how the results presented in this paper would change.
References Barad, M., Braha, D., 1996. Control limits for multi-stage manufacturing processes with binomial yield (single and multiple production runs). The Journal of the Operational Research Society 47 (1), 98–112. Chen, J., Yao, D.D., Zheng, S., 2001. Optimal replenishment and rework with multiple unreliable supply sources. Operations Research 49 (3), 430–443. Cheng, T.C.E., 1991. EPQ with process capability and quality assurance considerations. The Journal of the Operational Research Society 42 (8), 713–720. Gerchak, Y., Parlar, M., 1990. Yield randomness, cost tradeoffs, and diversification in the EOQ model. Naval Research Logistics 37 (3), 341–354. Grosfeld-Nir, A., 2005. A two-bottleneck system with binomial yields and rigid demand. European Journal of Operational Research 165 (1), 231–250. Grosfeld-Nir, A., Ronen, B., 1993. A single bottleneck system with binomial yields and rigid demand. Management Science 39 (5), 650–653. Micheli, G.J.L., 2008. A decision-maker-centred supplier selection approach for critical supplies. Management Decision 46 (6), 918–932. Porteus, E.L., 1985. Investing in reduced setups in the EOQ model. Management Science 31 (8), 998–1010. Porteus, E.L., 1986. Optimal lot sizing, process quality improvement and setup cost reduction. Operations Research 34 (1), 137–144. Silver, E.A., 1976. Establishing the order quantity when the amount received is uncertain. INFOR 14 (1), 32–39. Treleven, M., Schweikhart, S.B., 1988. A risk/benefit analysis of sourcing strategies: single vs. multiple sourcing. Journal of Operations Management 7 (4), 93–114. Tripathy, P.K., Wee, W.-M., Majhi, P.R., 2003. An EOQ model with process reliability considerations. The Journal of the Operational Research Society 54 (5), 549–554. Yano, C.A., Lee, H.L., 1995. Lot sizing with random yields: a review. Operations Research 43 (2), 311–334. Zipkin, P.H., 2000. Foundations of Inventory Management. McGraw-Hill/Irwin, Boston.