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Ordering quantity decisions considering uncertainty in supply-chain logistics operations
Int. J. Production Economics 134 (2011) 16–27
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Int. J. Production Economics journal homepage: www.elsevier.com/locate/ijpe
Ordering quantity decisions considering uncertainty in supply-chain logistics operations Hyoungtae Kim, Jye-Chyi Lu , Paul H. Kvam 1, Yu-Chung Tsao
2
School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA 30332, United States
a r t i c l e i n f o
a b s t r a c t
Article history: Received 29 December 2009 Accepted 13 February 2011 Available online 22 February 2011
This research seeks to determine the optimal order amount for a retailer given uncertainty in a supplychain’s logistics network due to unforeseeable disruption or various types of defects (e.g., shipping damage, missing parts and misplacing products). Mixture distribution models characterize problems from solitary failures and contingent events causing network to function ineffectively. The uncertainty in the number of good products successfully reaching distribution centers (DCs) and retail stores poses a challenge in deciding product-order amounts. Because the commonly used ordering plan developed for maximizing expected profits does not allow retailers to address concerns about contingencies; this research proposes two improved procedures with risk-averse characteristics towards low probability and high impact events. Several examples illustrate the impact of a DC’s operation policies and model assumptions on a retailer’s product-ordering plan and resulting sales profit. & 2011 Published by Elsevier B.V.
Keywords: Contingency Logistics network Mixture distribution Stochastic optimization
1. Introduction In this era of global sourcing, to reduce purchase costs and attract a larger base of customers, retailers such as Wal-Mart, Home Depot and Dollar General are constantly seeking suppliers with lower prices and finding them at greater and greater distances from their distribution centers (DCs) and stores. Consequently, a significant proportion of shipped products from overseas suppliers is susceptible to defects. Reasons for defects include missing parts, misplaced products (at DCs, stores) or mistakes in orders and shipments. Sometimes, products are damaged due to mishandling in transportation or are affected by low probability and high impact contingencies such as extreme weather, labor disputes or even terrorist attacks. When there are traffic problems or logistics delays due to security inspections at a country’s borders and seaports, orders, packagings and shipments fail to arrive at DCs or stores on time. Regardless of problems resulting from the supply sources or logistics operations, this article considers all of them as supply and logistics defects. Our past project studies with a major retailing chain indicate that the proportion of such defects could reach as high as 20%. This creates a significant challenge in productordering and shelf-space management. Corresponding author. Tel.: + 1 404 894 2318.
E-mail addresses: [email protected] (J.-C. Lu), [email protected] (P.H. Kvam), [email protected] (Y.-C. Tsao). 1 Tel.: +1 404 894 6515. 2 Tel.: +1 626 203 5835. 0925-5273/$ - see front matter & 2011 Published by Elsevier B.V. doi:10.1016/j.ijpe.2011.02.017
If the defect rate is not accounted for in the purchase order, the resulting product shortages serve as precursors to consequences such as inconveniencing their customers, compromising the retailer’s reputation for service quality, and then having to trace, sell, repair or return the defective products. For a large retail chain like Home Depot, the stock-out problem can cost more than billion dollars. On the other hand, the use of excessive inventory to handle uncertain supply and logistics defects is also costly. This article models the defect process in a three-level supply chain network that consists of suppliers, one DC and one store, and links it to the DC’s operation policy for developing an optimal productordering scheme. In recent years, supply chain or enterprise risk management have attracted a great deal of attention from both researchers and practitioners. Wu and Olson (2008) provided three types of risk evaluation models to evaluate and improve supplier selection decisions in an uncertain supply chain environment. Wu and Olson (2010a) developed DEA VaR model as a new tool to conduct risk management in vendor selection. Wu and Olson (2010b) and Wu and Olson (2009) demonstrates support to risk management through validation of predictive scorecards for a large bank and a small bank, respectively. Wu et al. (2010) developed a risk model in concurrent engineering product development. Complexity and uncertainty in many practical problems require new methods and tools (Wu and Olson, 2009). The literature on supply-chain contract decisions feature methods that utilize a high-level general model to describe supply uncertainties without getting into any degree of logistics details. See examples in Sculli and Wu (1981), Ramasech et al.
H. Kim et al. / Int. J. Production Economics 134 (2011) 16–27
(1991), Lau and Lau (1994), Parlar and Perry (1995), Parlar (1997), Weng and McClurg (2003), and Mohebbi (2003) for diverse implications of random production lead-time on inventory policies. Gulyani (2001) studied the effects of poor transportation on the supply chain (i.e., highly ineffective freight transportation systems) and showed how it increases the probability of incurring damage in transit and total inventories, while also increasing overhead costs. Silver (1976) used the economic order quantity (EOQ) formulation to model the situation, where the order quantity received from the supplier does not necessarily match the quantity requisitioned. He showed that the optimal order quantity depends on the mean and the standard deviation of the amount received. Shih (1980) studied the optimal ordering schemes in the case, where the proportion of defective products (PDP) in the accepted lots has a known probability distribution. Noori and Keller (1984) extended Silver’s model to obtain an optimal production quantity when the amount of products received at stores assumes probability distributions such as uniform, normal and gamma. Papers concerned with logistics or probabilistic networks are more common in the area of transportation, especially in the hazardous material routing problem. In the transportation of a hazardous material, the event of an accident with a truck fully loaded with a hazardous material can be treated as a contingency. However, in most past models, the main objective has been to find the optimal route to minimize the expected total system cost, without concern for supply-chain contract decisions such as the optimal ordering quantity. In general, these models do not link the supply chain and logistics decision processes together. Without understanding the details of the logistics networks (e.g., how defects occurred in the supply network, how different operation policies in the DC affect the defect process), the supply-chain contract decisions are prone to inaccuracies, especially in dealing with stochastic optimizations due to supply uncertainties. For example, suppose a typical defect rate is yn and a contingent event occurs with probability p (e.g., 0.00001), and upon occurrence, yc 100% of the total shipment is damaged. Then, the overall defect rate is ð1IÞ yn þ I yc ¼ yn þI ðyc yn Þ, where I¼ 1 under the contingency and I ¼0 otherwise. Even though yc 100% (e.g., 90%) product damage under the contingent situation can cause enormous stock-out costs to the retailer, the average defect rate yn þ p ðyc yn Þ is nearly the same as yn without the contingency due to the very small probability p. Consequently, orders based on the average defect rate (as seen in most supply-chain contracts) do not prepare the retailer for potentially severe losses that accompany contingencies. Thus, it is important to know how defects will impact uncertainty in the amount of good products arriving at stores and to develop optimization strategies then to compensate for these situations. This article describes two procedures with which retailers can generate reasonable solutions that exhibit risk-averse characteristics toward extreme events. Section 2 models processes for product defect rates between any two points in the network, which are directly linked to logistics operations. Next, we construct a random variable representing the total proportion of defective products (TPDP) by integrating models of defects at various stages of the supply-chain network. The TPDP is based on a series of mixture distributions and characterizes overall service levels (defect rates) of logistics operations in contingent and noncontingent circumstances. Moreover, we investigate the impact of two different DC operation policies on the resulting distributions of TPDP. Section 3.1 shows the ineffectiveness of using the expected profit in locating the optimal ordering quantity. In Section 3.2, a probability constrained optimization procedure is developed to handle the low probability and high impact events
17
that lead to logistics uncertainties. Numerical examples are provided in Section 4. Section 5 summarizes the results of the article and outlines future research opportunities.
2. Models for product defect and retail profit 2.1. A model for total proportion of defective products Consider the problem of a retailer who is buying products from k identical suppliers. Each supplier provides the retailer with identical products at the same price. Products from the k suppliers are transported and stored in a single DC before being sent out to the retail outlet (see Fig. 1). A contingent event on products shipped from supplier j to the DC is assumed to have a high impact and low probability of occurrence and affects logistics operations (e.g., product defects or delivery delays) between the suppliers and the DC. We assume that given a contingency, the damaged proportion, denoted by XjC, j¼1,y,n are independent of the size of actual shipment, with distribution function Gc. More generally, define XjC
Proportion of defective products (PDP) due to contingency between supplier j and DC XjC GjC , where EðXjC Þ ¼ mjC , VarðXjC Þ ¼ s2jC ,
XjN
PDP due to non-contingency between supplier j and DC XjN GjN , where EðXjN Þ ¼ mjN , VarðXjN Þ ¼ s2jN ,
Ij
PjDC
1 if a contingent event occurs between supplier and DC; 0 otherwise, where {I1 ; . . . ; Ik g are independent with PðIj ¼ 1Þ ¼ pj , and ð1Ij ÞXjN þ Ij XjC , PDP from supplier j to DC.
Note that PjDC has a mixture distribution, where Ij serves as the Bernoulli mixing distribution. In real-life operations some companies allow their products to be shipped with other companies’ products in the same truck, but some do not. For keeping the presentation brief this article focuses on the first case and provides comments on the other case. Details are provided on the web site http://crier.isye. gatech.edu/apps/research-papers/papers/jclu-2004-07.pdf. Next, we model the total proportion of defects (TPDP) from products shipped from suppliers, through the DC and on to the retail store. Note that DC serves as an inspection station to screen out defective products not shipped to the store. Define similar notations XC , XN and I0 as the PDPs and contingency indicator between DC and store. Note that there is no index j in these notations due to the products mixing process in the DC. Let PR be the PDP of the products that are shipped from DC to store. Then, similar to PjDC, PR ¼ ð1I0 ÞXN þI0 XC , and PjDCR ¼ PjDC þ PR ð1PjDC Þ is the total PDP for products shipped the supplier
Fig. 1. Supply-chain network with k suppliers and one DC.
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H. Kim et al. / Int. J. Production Economics 134 (2011) 16–27
j, passed to the DC and reaching the retail store. Finally, P Y ¼ kj ¼ 1 PjDCR =k is the average TPDP from all suppliers. The mean and variance of the average TPDP (Y) are needed for product order-quantity decision. In particular, they are used in later sections to investigate the effect of contingency on the optimal order quantity and retailer’s profit function. Note that the total PDPs ðP1DCR , . . . ,PkDCR Þ are not clearly independent due to the shared risks. In their derivations different distributions for XjC, XjN and Ij, and also for XC , XN and I0, made the expressions of the moments of Y very complicated and difficult in numerical evaluations. To employ the first-pass analysis for supporting product order-quantity decisions, this article assumes that all XjC and XC have the same distribution with mean mC and variance s2C ; all XjN and XN have the same mean mN and variance s2N ; all Ij and I0 have the same expectation equal to p. Note that we assume the distributions of these random variables are all mutually independent. This assumption is reasonable in real-life due to separate operations in each supply route. Since the mean and variance derivations are from standard expectation operations of random variables, the following shows the results without detail: E½Y ¼
k 1X E½PjDC þ PR ð1PjDC Þ ¼ ½ð1pÞmN þ pmC ð2½ð1pÞmN þ pmC Þ, kj¼1
Var½Y ¼ ¼
k 1 X 1 XX Var½PjDCR þ 2 CovðPiDCR ,PjDCR Þ 2 k j¼1 k iaj
1 f½ð1pÞs2N þ ps2C ð2½1ðð1pÞmN þ pmC Þ þð1pÞs2N þ ps2C Þg k 1 k þ 2 ð2:1Þ ðð1pÞs2N þ ps2C Þð1½ð1pÞmN þ pmC Þ2 : k 2
White (1970) considers the problem of deciding the optimum batch production quantity Q when the probability of producing a good-for-sale item is p; thus, the total number of good items is distributed binomial(Q,p). White’s expected profit function is strictly concave and leads to the optimum batch production quantity. The next section addresses the concavity of retailer’s expected profit function and derives the optimal order quantity Q that maximizes the expected profit. Then, the impact of changing defect-model’s parameters to the optimal solution is explored.
3. Solution strategies Section 3.1 shows how the optimization with the standard expected-value approach fails in the case of a low-probabilityhigh-consequence contingency event. Two risk-averse solutions are proposed in Section 3.2.
3.1. Risk neutral solutions Define the following expectation with respect to the density functions of demand x and TPDP (Y) and denote the expected sales profit (ESP) as the difference between the expected sales revenue and procurement cost. Let g(y) be the p.d.f. of Y and G be the c.d.f. of Y. ESP ¼ rE½Minðx,ð1YÞQ Þcð1E½YÞQ Z 1Z 1 ½xð1yÞQ f ðxÞgðyÞ dx dy, ¼ rE½xcð1E½YÞQ r EIC ¼ h
Z
1 0
þp
2.2. Formulation of k-supplier product-ordering problems
0
Z
½ð1yÞQ xf ðxÞgðyÞ dx dy
0 1Z 1
Z 0
In the k-supplier model, we assume that the total order quantity Q is split equally between k suppliers. The retail price is fixed and strictly greater than wholesale cost ðr 4 cÞ regardless of the terms of trade. The holding cost per period at the retail store level is h for each unsold product. In the event of a stock-out, unmet demand is lost, resulting in the margin being lost (to the retailer). The related stock-out penalty cost is p. All cost parameters are assumed to be known. For a given total order quantity Q from the retailer and the TPDP (Y) among Q in logistics transit, the retailer’s profit is
PðQ ,YÞ rMin½x,ð1YÞQ cð1YÞQ h½ð1YÞQ x þ p½xð1YÞQ þ , ð2:2Þ where ðxyÞ þ represents max½ðxyÞ,0. The first term, sales revenue (SR), is the unit retail price r times the amount of products sold. If the customer demand x is less than the available products for sale, ð1YÞQ , x is the amount of products sold. On the other hand, if the demand is more than the available products, the amount of products sold is (1 Y)Q. This article considers x as a random variable with a distribution function FðxÞ. The second term, procurement cost (PC), is the unit wholesale price c times the amount of products arrived and available at the store for sale. Note that the defective products that do not arrive at the store are not included in this procurement cost for retailer; it is typical that such cost is charged to suppliers or logistics companies. The third term is inventory costs (IC). The fourth term, overstock inventory cost, is the unit holding cost for unsold products h times the amount of unsold products. The last term, stock-out penalty cost, is the unit shortage cost p times the amount of product shortage at the store.
ð1yÞQ
ð1yÞQ
½xð1yÞQ f ðxÞgðyÞ dx dy:
ð1yÞQ
Then, retailer’s expected profit is the difference between the above two expectations, i.e., E½PðQ ,YÞ ¼ ESPEIC. Shih (1980) proved the convexity of EIC; to prove the concavity of E½PðQ ,YÞ it suffices to show the concavity of ESP from the following second derivative: Z 1 @2 ESR ¼ r ð1yÞ2 f ðð1yÞQ ÞgðyÞ dy o0 for all Q : ð3:1Þ 2 @Q 0 See Appendix for the proof of Eq. (3.1). Consider the case in which demand is uniformly distributed with boundary parameters a and b. Then, Z
ð1yÞQ
f ðxÞ dx ¼
0
ð1yÞQ a ba
and
Z
ð1yÞQ
xf ðxÞ dx ¼
0
ð1yÞ2 Q 2 : 2ðbaÞ
Thus, the expected profit is aþb 1 cð1mÞQ ½ðhþ r þ pÞðs2 þ ð1mÞ2 ÞQ 2 2 2ðbaÞ 2ð1mÞðah þ bðr þ pÞÞQ þ a2 hþ b2 ðr þ pÞ
E½PðQ ,YÞ ¼ r
ðh þr þ pÞðs2 þð1mÞ2 Þ 2ðbaÞ !2 ð1mÞ bðr þ pcÞ þ aðh þ cÞ Q r þpþh ð1mÞ2 þ s2
¼
ða þbÞ þ a2 hþ b2 ðr þ pÞ 2 ð1mÞ2 ðbðr þ pcÞ þ aðh þcÞÞ2 þ , ð1mÞ2 þ s2 2ðbaÞðh þr þ pÞÞ þ
where m ¼ E½Y and s2 ¼ Var½Y represent the effect of TPDP.
ð3:2Þ
H. Kim et al. / Int. J. Production Economics 134 (2011) 16–27
The following proposition links the optimal order quantity maximizing the expected retailer profit to the optimal order quantity in traditional studies that consider product defects. Proposition 3.1. Let Q0 ¼ fbðr þ pcÞ þ aðh þ cÞg=fr þ p þhg, which is the optimal order quantity in the conventional ‘‘news vendor’’ problem assuming no product defects in the order process (i.e., m ¼ s ¼ 0Þ. In terms of Q0, the order quantity which maximizes E½PðQ Þ is ð1mÞ bðr þ pcÞ þ aðh þcÞ ð1mÞ ¼ Q0 : Q ¼ r þ p þh ð1mÞ2 þ s2 ð1mÞ2 þ s2 This result coincides with results of Noori and Keller (1984), where Q is proportional to the mean of Y and is reduced by an increase in the variability of Y. The next proposition shows that the optimal order quantity is a weighted average of the optimal order quantities that maximize ESP and EIC separately. To understand the optimal order quantity Q , let us look into the following proposition. Proposition 3.2. The order quantities which maximize ESP and EIC are, respectively, ð1mÞ c QSP ¼ bðbaÞ , r ð1mÞ2 þ s2 QIC ¼
p : a þ ðbaÞ pþh ð1mÞ þ s2
" # ðbðr þ pcÞ þ aðh þ cÞÞ2 2ðbaÞðhþ r þ pÞ determines the maximum expected profit: Note that A and C are decreasing functions of m while B is an increasing function of m. When the mean increases, the expected profit curve broadens ð@A=@m o 0Þ, shifts to the right (e.g., the optimal order quantity increases, @B=@m 40), and the corresponding maximum expected profit decreases ð@C=@m o 0Þ. On the other hand, A is increasing in s2 , while B and C decrease in s2 . When the variance increases, the curve shrinks ð@A=@s2 40Þ, shifts to the left (e.g., the optimal order quantity decreases, @B=@s2 o0), and the maximum expected profit decreases ð@C=@s2 o 0Þ. When the contingency probability is small (e.g., p r 0:001), the mean and variance of Y given in Eq. (2.1) do not involve the proportion of defective products in the contingency situation; that is, mC and s2C are not involved. This implies that the optimal product-order quantity Q does not depend on mC and s2C in this case. Thus, the possible impact from a contingency cannot be realized in the above optimization approach based on expected profit. The following section introduces two alternative procedures allowing the retailer to generate reasonable solutions reflecting natural risk-averse characteristics toward extreme events that have low probability. 3.2. Risk-averse solutions
ð1mÞ 2
Furthermore, Q is a convex combination of QnSP and QnIC: þð1lÞQIC Q ¼ lQSP
where l ¼
r : r þ p þh
If r 4 p þ h, i.e., the unit retail price is larger than the inventory related cost (via the sum of holding cost and shortage cost), more weight is assigned to the quantity which maximizes ESP, the expected sales profit. The following result (see Appendix for proof) is useful to understand the variability of the retailer’s profit function. Proposition 3.3. Whenever ðc þ hÞ 4 ðr þ pcÞ, the retailer’s variability profit is increasing in order quantity Q. Note that c+ h is the unit wholesale price plus holding cost and ðrcÞ þ p is the unit profit margin plus shortage cost. Proposition 3.3 states that whenever the profit margin loss from the unit surplus in the inventory is greater than that from the unit shortage in the stock-out, the retailer’s profit variance is an increasing function of Q. The expected profit function in (3.2) can be reorganized into the following three components that demonstrate how the mean and variance of Y are linked to the expected retail profit. E½PðQ ,YÞ ¼ Aðm, s2 Þ½Q Bðm, s2 Þ2 þCðm, s2 Þ,
ð3:3Þ
The first method limits the solution space to the set of order quantities which guarantees an expected profit level under contingency. The second method features a constraint based on the percentiles of the profit distribution. While both methods restrict the solution space to control the consequence of the contingency, they differ in important ways; the first method considers only the measured contingency and not its probability, while the second method is based directly on the contingency distribution. 3.2.1. Constrained optimization I—profit constraint Given a contingent event, this method considers only solutions that lead to (conditionally) expected profit of at least P0 . That is max
ðh þ r þ pÞðs2 þ ð1mÞ2 Þ 2ðbaÞ
determines the spread of the profit function, Bðm, s2 Þ ¼
bðr þ pcÞ þaðhþ cÞ r þ p þh ð1mÞ2 þ s2 ð1mÞ
determines the optimal order quantity; and Cðm, s2 Þ ¼
ða þbÞ ð1mÞ2 þ a2 hþ b2 ðr þ pÞ þ 2 ð1mÞ2 þ s2
EG ½PðQ ,YÞ
Q Z0
s:t:
EGC ½PðQ ,YÞ EG ½PðQ ,YÞjI ¼ 1 Z P0 ,
ð3:4Þ
where I is the indicator function for a contingency. The retailer’s strong risk-aversion can be reflected by increasing the value of P0 . The restricted solution space is based on the following sets of order quantities: SP0
SP0 ,C
where Aðm, s2 Þ ¼
19
S1P0
Set of possible order quantities which lead to unconditional expected profit Z P0 , fQ jEG ½PðQ ,YÞ Z P0 g. Set of possible order quantities which lead to contingency expected profit Z P0 , fQ jEGC ½PðQ ,YÞZ P0 g. SP0 \ SP0 ,C .
Let Q^ be the optimal ordering quantity of the original unconditional problem, Q^ C be the optimal ordering quantity considering only the contingency situation, and Qn be the optimal solution to the constrained optimization problem in (3.4). As the lower bound P0 in Eq. (3.4) increases, the solution Qn increases toward Q^ C ; as P0 decreases, the constraint eventually disappears. Fig. 2 illustrates the approach with the conditional and unconditional profit functions. This problem formulation provides flexibility to the decision maker, with P0 serving as an utility function that shrinks the unconstrained solution towards Q^ C in the case of contingency.
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H. Kim et al. / Int. J. Production Economics 134 (2011) 16–27
Fig. 2. Expected profits based on increased mean in defect distribution after contingency.
Fig. 4. Increased variance shifts expected profits to the left.
P0 ) reduces the order quantity due to the increased variance. The section that follows illustrates these solution procedures with a numerical example. 3.2.2. Constrained optimization II—probability constraint As an alternative to controlling the profit function by conditioning on the occurrence of a contingency, here we restrict the solution space by restricting the probability space of the profit distribution. That is, we bound from below (with g) the probability that the profit is less than an amount P1 . If we assume, for simplicity, that the demand level x is fixed, the problem becomes max Fig. 3. Increased mean causes infeasible solution.
s:t: To further explore the properties of the proposed constrained optimization method, following the discussion in Eq. (3.3), this section considers two ways contingency affects the expected profit through the distribution of Y: (1) contingency increases the mean of Y, and (2) contingency increases the variance of Y. We first examine the case where the contingency causes a location shift (to the right) in the defect distribution, so GC is increased from G by a constant. The location shift (to the right) in G causes both a location shift (to the right) and an increased variability in the expected profit (see Fig. 2). In the case where S1 ¼SP \ SP ,C a |, if Q^ A S1 , then Q ¼ Q^ ; otherwise, P0
0
P0
0
Q ¼ minQ A S1 fQ g. If S1P0 ¼ |, it is not possible to keep the P0
conditional expected profit above P0 (in case of a contingency) without allowing overall expected profit to go below P0 (see Fig. 3). To rectify this problem, P0 must be reduced until there is an overlap between SP0 and SP0 ,C . In general, the order quantity increases in the level of risk-aversion (e.g., when P0 increases, Q increases) under the location shift case. From Proposition 3.1, the optimal order quantity which maximizes expected profit decreases as the variance increases. Accordingly, increased variability in Y shifts the expected profit to the left as illustrated in Fig. 4. Furthermore, the increase in variance results in a decrease in expected profit. The vertical distance between the two local maxima in the expected profit curves represents the decrease in maximum possible profits due to the increase in variance. When the variance under contingency is s2 þ d, this distance is
D ¼ Cðm, s2 ÞCðm, s2 þ dÞ "
¼
ð1mÞ2 ð1mÞ2 þ s2
ð1mÞ2 ð1mÞ2 þ s2 þ d
#"
# ðbðr þ pcÞ þ aðh þcÞÞ2 : 2ðbaÞðh þr þ pÞÞ
If S1P0 ¼ SP0 \ SP0 ,C a |, then a unique solution exists; if Q^ A S1P0 , then Q ¼ Q^ ; otherwise, Q ¼ max 1 fQ g. Again, if no overlap Q A SP
Pg ðPðQ ,YÞ r P1 Þ r g,
ð3:5Þ
where (
PðQ ,YÞ ¼
r xcð1YÞQ hðð1YÞQ xÞ,
ð1YÞQ Z x,
r xcð1YÞQ ðr þ pÞðxð1YÞQ Þ,
ð1YÞQ r x:
In this case, the retailer’s strong risk-aversion can be reflected by either increasing the value of P1 or decreasing the value of g. Denote by Q^ the optimal order quantity for the unconstrained maximization problem that satisfies the following expression (see Shih, 1980): Z 1x=Q^ ð1mÞðrc þ pÞ ð1yÞgðyÞ dy ¼ : ð3:6Þ r þ hþ p 0 If we tacitly assume Q Z x, the probability constraint becomes px þ P1 x x P Y Z1 P½PðQ ,YÞ r P1 ¼ P Y Z1 Y Z1 ðr þ pcÞQ Q Q r x þ hxP1 x x P Y r1 : þ P Y r 1 Y r1 ðhþ cÞQ Q Q The following propositions characterize the solution to the stochastic constraint placed on the profit distribution. Proposition 3.4. For any fixed target profit level P1 , there exists a critical order size Q1 ¼ P1 =ðrcÞ such that for any demand Q r Q1 -P½PðQ ,YÞ r P1 ¼ 1, px þ P1 Q 4 Q1 -P½PðQ ,YÞ r P1 ¼ 1G 1 ðr þ pcÞQ r x þhxP1 þG 1 : ðh þcÞQ Proposition 3.5. For any given target profit level P1 and probability g, there exists a feasible set for (3.5) of the form SðA,BÞ ¼ fQ jQL rQ rQU g with
0
between SP0 and SP0 ,C exists, the constraint P0 must be reduced. Unlike the location shift case, strong risk-aversion (bigger value of
Eg ½PðQ ,YÞ
Q Z0
QL ¼
px þ P1 , ðr þ pcÞð1G1 ð1gÞÞ
H. Kim et al. / Int. J. Production Economics 134 (2011) 16–27
px þ P1 r x þ hxP1 þG 1 , QU ¼ Q : g ¼ 1G 1 ðr þ pcÞQ ðhþ cÞQ where G1 represents the inverse cumulative distribution function of Y. As P1 increases, the feasible set S shrinks; the lower boundary QL decreases in g while the upper boundary QU increases. Thus, S widens as g increases. Proposition 3.6. If g o g1 ¼ 1Gð1zðP1 ÞÞ, where zðtÞ ¼ ððpx þtÞ= ðr þ pcÞÞððr x þhxtÞ=ðhþ cÞÞ1 , then SðP1 , gÞ ¼ |, e.g., there is no feasible solution in (3.5). Proof. We need to show that P½PðQ ,YÞ r P1 has only one minimum point at ðr x þ hxP1 Þ=ðh þcÞ. Because 1Gð1ðpx þ P1 Þ= ððr þ pcÞQ ÞÞ is a decreasing function of Q and Gð1ðr x þ hxP1 Þ=ððh þcÞQU ÞÞ is increasing in Q, P½PðQ ,YÞ r P1 has its minimum at ððr x þ hxP1 Þ=ðhþ cÞ. & Proposition 3.7. If Q ZQ1 , then g1 is increasing in P1 and g1 ¼ 1 when P1 is set at the maximum profit level, e.g., P1 ¼ ðrcÞx, which can be achieved only when there are no product shortages or unsold products. Proof. It is easy to see that zðtÞ is an increasing function of t. If Q ZQ1 , it can be shown that zðP1 Þ r 1. When P1 ¼ ðrcÞx, then zðP1 Þ ¼ 1, hence g1 ¼ 1. & Proposition 3.8. For any given profit level P1 and probability g, g Z g1 , the optimal order quantity for the constrained optimization problem (3.5) is 8 ^ > if Q^ A SðP1 , gÞ, >
Contents lists available at ScienceDirect
Int. J. Production Economics journal homepage: www.elsevier.com/locate/ijpe
Ordering quantity decisions considering uncertainty in supply-chain logistics operations Hyoungtae Kim, Jye-Chyi Lu , Paul H. Kvam 1, Yu-Chung Tsao
2
School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA 30332, United States
a r t i c l e i n f o
a b s t r a c t
Article history: Received 29 December 2009 Accepted 13 February 2011 Available online 22 February 2011
This research seeks to determine the optimal order amount for a retailer given uncertainty in a supplychain’s logistics network due to unforeseeable disruption or various types of defects (e.g., shipping damage, missing parts and misplacing products). Mixture distribution models characterize problems from solitary failures and contingent events causing network to function ineffectively. The uncertainty in the number of good products successfully reaching distribution centers (DCs) and retail stores poses a challenge in deciding product-order amounts. Because the commonly used ordering plan developed for maximizing expected profits does not allow retailers to address concerns about contingencies; this research proposes two improved procedures with risk-averse characteristics towards low probability and high impact events. Several examples illustrate the impact of a DC’s operation policies and model assumptions on a retailer’s product-ordering plan and resulting sales profit. & 2011 Published by Elsevier B.V.
Keywords: Contingency Logistics network Mixture distribution Stochastic optimization
1. Introduction In this era of global sourcing, to reduce purchase costs and attract a larger base of customers, retailers such as Wal-Mart, Home Depot and Dollar General are constantly seeking suppliers with lower prices and finding them at greater and greater distances from their distribution centers (DCs) and stores. Consequently, a significant proportion of shipped products from overseas suppliers is susceptible to defects. Reasons for defects include missing parts, misplaced products (at DCs, stores) or mistakes in orders and shipments. Sometimes, products are damaged due to mishandling in transportation or are affected by low probability and high impact contingencies such as extreme weather, labor disputes or even terrorist attacks. When there are traffic problems or logistics delays due to security inspections at a country’s borders and seaports, orders, packagings and shipments fail to arrive at DCs or stores on time. Regardless of problems resulting from the supply sources or logistics operations, this article considers all of them as supply and logistics defects. Our past project studies with a major retailing chain indicate that the proportion of such defects could reach as high as 20%. This creates a significant challenge in productordering and shelf-space management. Corresponding author. Tel.: + 1 404 894 2318.
E-mail addresses: [email protected] (J.-C. Lu), [email protected] (P.H. Kvam), [email protected] (Y.-C. Tsao). 1 Tel.: +1 404 894 6515. 2 Tel.: +1 626 203 5835. 0925-5273/$ - see front matter & 2011 Published by Elsevier B.V. doi:10.1016/j.ijpe.2011.02.017
If the defect rate is not accounted for in the purchase order, the resulting product shortages serve as precursors to consequences such as inconveniencing their customers, compromising the retailer’s reputation for service quality, and then having to trace, sell, repair or return the defective products. For a large retail chain like Home Depot, the stock-out problem can cost more than billion dollars. On the other hand, the use of excessive inventory to handle uncertain supply and logistics defects is also costly. This article models the defect process in a three-level supply chain network that consists of suppliers, one DC and one store, and links it to the DC’s operation policy for developing an optimal productordering scheme. In recent years, supply chain or enterprise risk management have attracted a great deal of attention from both researchers and practitioners. Wu and Olson (2008) provided three types of risk evaluation models to evaluate and improve supplier selection decisions in an uncertain supply chain environment. Wu and Olson (2010a) developed DEA VaR model as a new tool to conduct risk management in vendor selection. Wu and Olson (2010b) and Wu and Olson (2009) demonstrates support to risk management through validation of predictive scorecards for a large bank and a small bank, respectively. Wu et al. (2010) developed a risk model in concurrent engineering product development. Complexity and uncertainty in many practical problems require new methods and tools (Wu and Olson, 2009). The literature on supply-chain contract decisions feature methods that utilize a high-level general model to describe supply uncertainties without getting into any degree of logistics details. See examples in Sculli and Wu (1981), Ramasech et al.
H. Kim et al. / Int. J. Production Economics 134 (2011) 16–27
(1991), Lau and Lau (1994), Parlar and Perry (1995), Parlar (1997), Weng and McClurg (2003), and Mohebbi (2003) for diverse implications of random production lead-time on inventory policies. Gulyani (2001) studied the effects of poor transportation on the supply chain (i.e., highly ineffective freight transportation systems) and showed how it increases the probability of incurring damage in transit and total inventories, while also increasing overhead costs. Silver (1976) used the economic order quantity (EOQ) formulation to model the situation, where the order quantity received from the supplier does not necessarily match the quantity requisitioned. He showed that the optimal order quantity depends on the mean and the standard deviation of the amount received. Shih (1980) studied the optimal ordering schemes in the case, where the proportion of defective products (PDP) in the accepted lots has a known probability distribution. Noori and Keller (1984) extended Silver’s model to obtain an optimal production quantity when the amount of products received at stores assumes probability distributions such as uniform, normal and gamma. Papers concerned with logistics or probabilistic networks are more common in the area of transportation, especially in the hazardous material routing problem. In the transportation of a hazardous material, the event of an accident with a truck fully loaded with a hazardous material can be treated as a contingency. However, in most past models, the main objective has been to find the optimal route to minimize the expected total system cost, without concern for supply-chain contract decisions such as the optimal ordering quantity. In general, these models do not link the supply chain and logistics decision processes together. Without understanding the details of the logistics networks (e.g., how defects occurred in the supply network, how different operation policies in the DC affect the defect process), the supply-chain contract decisions are prone to inaccuracies, especially in dealing with stochastic optimizations due to supply uncertainties. For example, suppose a typical defect rate is yn and a contingent event occurs with probability p (e.g., 0.00001), and upon occurrence, yc 100% of the total shipment is damaged. Then, the overall defect rate is ð1IÞ yn þ I yc ¼ yn þI ðyc yn Þ, where I¼ 1 under the contingency and I ¼0 otherwise. Even though yc 100% (e.g., 90%) product damage under the contingent situation can cause enormous stock-out costs to the retailer, the average defect rate yn þ p ðyc yn Þ is nearly the same as yn without the contingency due to the very small probability p. Consequently, orders based on the average defect rate (as seen in most supply-chain contracts) do not prepare the retailer for potentially severe losses that accompany contingencies. Thus, it is important to know how defects will impact uncertainty in the amount of good products arriving at stores and to develop optimization strategies then to compensate for these situations. This article describes two procedures with which retailers can generate reasonable solutions that exhibit risk-averse characteristics toward extreme events. Section 2 models processes for product defect rates between any two points in the network, which are directly linked to logistics operations. Next, we construct a random variable representing the total proportion of defective products (TPDP) by integrating models of defects at various stages of the supply-chain network. The TPDP is based on a series of mixture distributions and characterizes overall service levels (defect rates) of logistics operations in contingent and noncontingent circumstances. Moreover, we investigate the impact of two different DC operation policies on the resulting distributions of TPDP. Section 3.1 shows the ineffectiveness of using the expected profit in locating the optimal ordering quantity. In Section 3.2, a probability constrained optimization procedure is developed to handle the low probability and high impact events
17
that lead to logistics uncertainties. Numerical examples are provided in Section 4. Section 5 summarizes the results of the article and outlines future research opportunities.
2. Models for product defect and retail profit 2.1. A model for total proportion of defective products Consider the problem of a retailer who is buying products from k identical suppliers. Each supplier provides the retailer with identical products at the same price. Products from the k suppliers are transported and stored in a single DC before being sent out to the retail outlet (see Fig. 1). A contingent event on products shipped from supplier j to the DC is assumed to have a high impact and low probability of occurrence and affects logistics operations (e.g., product defects or delivery delays) between the suppliers and the DC. We assume that given a contingency, the damaged proportion, denoted by XjC, j¼1,y,n are independent of the size of actual shipment, with distribution function Gc. More generally, define XjC
Proportion of defective products (PDP) due to contingency between supplier j and DC XjC GjC , where EðXjC Þ ¼ mjC , VarðXjC Þ ¼ s2jC ,
XjN
PDP due to non-contingency between supplier j and DC XjN GjN , where EðXjN Þ ¼ mjN , VarðXjN Þ ¼ s2jN ,
Ij
PjDC
1 if a contingent event occurs between supplier and DC; 0 otherwise, where {I1 ; . . . ; Ik g are independent with PðIj ¼ 1Þ ¼ pj , and ð1Ij ÞXjN þ Ij XjC , PDP from supplier j to DC.
Note that PjDC has a mixture distribution, where Ij serves as the Bernoulli mixing distribution. In real-life operations some companies allow their products to be shipped with other companies’ products in the same truck, but some do not. For keeping the presentation brief this article focuses on the first case and provides comments on the other case. Details are provided on the web site http://crier.isye. gatech.edu/apps/research-papers/papers/jclu-2004-07.pdf. Next, we model the total proportion of defects (TPDP) from products shipped from suppliers, through the DC and on to the retail store. Note that DC serves as an inspection station to screen out defective products not shipped to the store. Define similar notations XC , XN and I0 as the PDPs and contingency indicator between DC and store. Note that there is no index j in these notations due to the products mixing process in the DC. Let PR be the PDP of the products that are shipped from DC to store. Then, similar to PjDC, PR ¼ ð1I0 ÞXN þI0 XC , and PjDCR ¼ PjDC þ PR ð1PjDC Þ is the total PDP for products shipped the supplier
Fig. 1. Supply-chain network with k suppliers and one DC.
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H. Kim et al. / Int. J. Production Economics 134 (2011) 16–27
j, passed to the DC and reaching the retail store. Finally, P Y ¼ kj ¼ 1 PjDCR =k is the average TPDP from all suppliers. The mean and variance of the average TPDP (Y) are needed for product order-quantity decision. In particular, they are used in later sections to investigate the effect of contingency on the optimal order quantity and retailer’s profit function. Note that the total PDPs ðP1DCR , . . . ,PkDCR Þ are not clearly independent due to the shared risks. In their derivations different distributions for XjC, XjN and Ij, and also for XC , XN and I0, made the expressions of the moments of Y very complicated and difficult in numerical evaluations. To employ the first-pass analysis for supporting product order-quantity decisions, this article assumes that all XjC and XC have the same distribution with mean mC and variance s2C ; all XjN and XN have the same mean mN and variance s2N ; all Ij and I0 have the same expectation equal to p. Note that we assume the distributions of these random variables are all mutually independent. This assumption is reasonable in real-life due to separate operations in each supply route. Since the mean and variance derivations are from standard expectation operations of random variables, the following shows the results without detail: E½Y ¼
k 1X E½PjDC þ PR ð1PjDC Þ ¼ ½ð1pÞmN þ pmC ð2½ð1pÞmN þ pmC Þ, kj¼1
Var½Y ¼ ¼
k 1 X 1 XX Var½PjDCR þ 2 CovðPiDCR ,PjDCR Þ 2 k j¼1 k iaj
1 f½ð1pÞs2N þ ps2C ð2½1ðð1pÞmN þ pmC Þ þð1pÞs2N þ ps2C Þg k 1 k þ 2 ð2:1Þ ðð1pÞs2N þ ps2C Þð1½ð1pÞmN þ pmC Þ2 : k 2
White (1970) considers the problem of deciding the optimum batch production quantity Q when the probability of producing a good-for-sale item is p; thus, the total number of good items is distributed binomial(Q,p). White’s expected profit function is strictly concave and leads to the optimum batch production quantity. The next section addresses the concavity of retailer’s expected profit function and derives the optimal order quantity Q that maximizes the expected profit. Then, the impact of changing defect-model’s parameters to the optimal solution is explored.
3. Solution strategies Section 3.1 shows how the optimization with the standard expected-value approach fails in the case of a low-probabilityhigh-consequence contingency event. Two risk-averse solutions are proposed in Section 3.2.
3.1. Risk neutral solutions Define the following expectation with respect to the density functions of demand x and TPDP (Y) and denote the expected sales profit (ESP) as the difference between the expected sales revenue and procurement cost. Let g(y) be the p.d.f. of Y and G be the c.d.f. of Y. ESP ¼ rE½Minðx,ð1YÞQ Þcð1E½YÞQ Z 1Z 1 ½xð1yÞQ f ðxÞgðyÞ dx dy, ¼ rE½xcð1E½YÞQ r EIC ¼ h
Z
1 0
þp
2.2. Formulation of k-supplier product-ordering problems
0
Z
½ð1yÞQ xf ðxÞgðyÞ dx dy
0 1Z 1
Z 0
In the k-supplier model, we assume that the total order quantity Q is split equally between k suppliers. The retail price is fixed and strictly greater than wholesale cost ðr 4 cÞ regardless of the terms of trade. The holding cost per period at the retail store level is h for each unsold product. In the event of a stock-out, unmet demand is lost, resulting in the margin being lost (to the retailer). The related stock-out penalty cost is p. All cost parameters are assumed to be known. For a given total order quantity Q from the retailer and the TPDP (Y) among Q in logistics transit, the retailer’s profit is
PðQ ,YÞ rMin½x,ð1YÞQ cð1YÞQ h½ð1YÞQ x þ p½xð1YÞQ þ , ð2:2Þ where ðxyÞ þ represents max½ðxyÞ,0. The first term, sales revenue (SR), is the unit retail price r times the amount of products sold. If the customer demand x is less than the available products for sale, ð1YÞQ , x is the amount of products sold. On the other hand, if the demand is more than the available products, the amount of products sold is (1 Y)Q. This article considers x as a random variable with a distribution function FðxÞ. The second term, procurement cost (PC), is the unit wholesale price c times the amount of products arrived and available at the store for sale. Note that the defective products that do not arrive at the store are not included in this procurement cost for retailer; it is typical that such cost is charged to suppliers or logistics companies. The third term is inventory costs (IC). The fourth term, overstock inventory cost, is the unit holding cost for unsold products h times the amount of unsold products. The last term, stock-out penalty cost, is the unit shortage cost p times the amount of product shortage at the store.
ð1yÞQ
ð1yÞQ
½xð1yÞQ f ðxÞgðyÞ dx dy:
ð1yÞQ
Then, retailer’s expected profit is the difference between the above two expectations, i.e., E½PðQ ,YÞ ¼ ESPEIC. Shih (1980) proved the convexity of EIC; to prove the concavity of E½PðQ ,YÞ it suffices to show the concavity of ESP from the following second derivative: Z 1 @2 ESR ¼ r ð1yÞ2 f ðð1yÞQ ÞgðyÞ dy o0 for all Q : ð3:1Þ 2 @Q 0 See Appendix for the proof of Eq. (3.1). Consider the case in which demand is uniformly distributed with boundary parameters a and b. Then, Z
ð1yÞQ
f ðxÞ dx ¼
0
ð1yÞQ a ba
and
Z
ð1yÞQ
xf ðxÞ dx ¼
0
ð1yÞ2 Q 2 : 2ðbaÞ
Thus, the expected profit is aþb 1 cð1mÞQ ½ðhþ r þ pÞðs2 þ ð1mÞ2 ÞQ 2 2 2ðbaÞ 2ð1mÞðah þ bðr þ pÞÞQ þ a2 hþ b2 ðr þ pÞ
E½PðQ ,YÞ ¼ r
ðh þr þ pÞðs2 þð1mÞ2 Þ 2ðbaÞ !2 ð1mÞ bðr þ pcÞ þ aðh þ cÞ Q r þpþh ð1mÞ2 þ s2
¼
ða þbÞ þ a2 hþ b2 ðr þ pÞ 2 ð1mÞ2 ðbðr þ pcÞ þ aðh þcÞÞ2 þ , ð1mÞ2 þ s2 2ðbaÞðh þr þ pÞÞ þ
where m ¼ E½Y and s2 ¼ Var½Y represent the effect of TPDP.
ð3:2Þ
H. Kim et al. / Int. J. Production Economics 134 (2011) 16–27
The following proposition links the optimal order quantity maximizing the expected retailer profit to the optimal order quantity in traditional studies that consider product defects. Proposition 3.1. Let Q0 ¼ fbðr þ pcÞ þ aðh þ cÞg=fr þ p þhg, which is the optimal order quantity in the conventional ‘‘news vendor’’ problem assuming no product defects in the order process (i.e., m ¼ s ¼ 0Þ. In terms of Q0, the order quantity which maximizes E½PðQ Þ is ð1mÞ bðr þ pcÞ þ aðh þcÞ ð1mÞ ¼ Q0 : Q ¼ r þ p þh ð1mÞ2 þ s2 ð1mÞ2 þ s2 This result coincides with results of Noori and Keller (1984), where Q is proportional to the mean of Y and is reduced by an increase in the variability of Y. The next proposition shows that the optimal order quantity is a weighted average of the optimal order quantities that maximize ESP and EIC separately. To understand the optimal order quantity Q , let us look into the following proposition. Proposition 3.2. The order quantities which maximize ESP and EIC are, respectively, ð1mÞ c QSP ¼ bðbaÞ , r ð1mÞ2 þ s2 QIC ¼
p : a þ ðbaÞ pþh ð1mÞ þ s2
" # ðbðr þ pcÞ þ aðh þ cÞÞ2 2ðbaÞðhþ r þ pÞ determines the maximum expected profit: Note that A and C are decreasing functions of m while B is an increasing function of m. When the mean increases, the expected profit curve broadens ð@A=@m o 0Þ, shifts to the right (e.g., the optimal order quantity increases, @B=@m 40), and the corresponding maximum expected profit decreases ð@C=@m o 0Þ. On the other hand, A is increasing in s2 , while B and C decrease in s2 . When the variance increases, the curve shrinks ð@A=@s2 40Þ, shifts to the left (e.g., the optimal order quantity decreases, @B=@s2 o0), and the maximum expected profit decreases ð@C=@s2 o 0Þ. When the contingency probability is small (e.g., p r 0:001), the mean and variance of Y given in Eq. (2.1) do not involve the proportion of defective products in the contingency situation; that is, mC and s2C are not involved. This implies that the optimal product-order quantity Q does not depend on mC and s2C in this case. Thus, the possible impact from a contingency cannot be realized in the above optimization approach based on expected profit. The following section introduces two alternative procedures allowing the retailer to generate reasonable solutions reflecting natural risk-averse characteristics toward extreme events that have low probability. 3.2. Risk-averse solutions
ð1mÞ 2
Furthermore, Q is a convex combination of QnSP and QnIC: þð1lÞQIC Q ¼ lQSP
where l ¼
r : r þ p þh
If r 4 p þ h, i.e., the unit retail price is larger than the inventory related cost (via the sum of holding cost and shortage cost), more weight is assigned to the quantity which maximizes ESP, the expected sales profit. The following result (see Appendix for proof) is useful to understand the variability of the retailer’s profit function. Proposition 3.3. Whenever ðc þ hÞ 4 ðr þ pcÞ, the retailer’s variability profit is increasing in order quantity Q. Note that c+ h is the unit wholesale price plus holding cost and ðrcÞ þ p is the unit profit margin plus shortage cost. Proposition 3.3 states that whenever the profit margin loss from the unit surplus in the inventory is greater than that from the unit shortage in the stock-out, the retailer’s profit variance is an increasing function of Q. The expected profit function in (3.2) can be reorganized into the following three components that demonstrate how the mean and variance of Y are linked to the expected retail profit. E½PðQ ,YÞ ¼ Aðm, s2 Þ½Q Bðm, s2 Þ2 þCðm, s2 Þ,
ð3:3Þ
The first method limits the solution space to the set of order quantities which guarantees an expected profit level under contingency. The second method features a constraint based on the percentiles of the profit distribution. While both methods restrict the solution space to control the consequence of the contingency, they differ in important ways; the first method considers only the measured contingency and not its probability, while the second method is based directly on the contingency distribution. 3.2.1. Constrained optimization I—profit constraint Given a contingent event, this method considers only solutions that lead to (conditionally) expected profit of at least P0 . That is max
ðh þ r þ pÞðs2 þ ð1mÞ2 Þ 2ðbaÞ
determines the spread of the profit function, Bðm, s2 Þ ¼
bðr þ pcÞ þaðhþ cÞ r þ p þh ð1mÞ2 þ s2 ð1mÞ
determines the optimal order quantity; and Cðm, s2 Þ ¼
ða þbÞ ð1mÞ2 þ a2 hþ b2 ðr þ pÞ þ 2 ð1mÞ2 þ s2
EG ½PðQ ,YÞ
Q Z0
s:t:
EGC ½PðQ ,YÞ EG ½PðQ ,YÞjI ¼ 1 Z P0 ,
ð3:4Þ
where I is the indicator function for a contingency. The retailer’s strong risk-aversion can be reflected by increasing the value of P0 . The restricted solution space is based on the following sets of order quantities: SP0
SP0 ,C
where Aðm, s2 Þ ¼
19
S1P0
Set of possible order quantities which lead to unconditional expected profit Z P0 , fQ jEG ½PðQ ,YÞ Z P0 g. Set of possible order quantities which lead to contingency expected profit Z P0 , fQ jEGC ½PðQ ,YÞZ P0 g. SP0 \ SP0 ,C .
Let Q^ be the optimal ordering quantity of the original unconditional problem, Q^ C be the optimal ordering quantity considering only the contingency situation, and Qn be the optimal solution to the constrained optimization problem in (3.4). As the lower bound P0 in Eq. (3.4) increases, the solution Qn increases toward Q^ C ; as P0 decreases, the constraint eventually disappears. Fig. 2 illustrates the approach with the conditional and unconditional profit functions. This problem formulation provides flexibility to the decision maker, with P0 serving as an utility function that shrinks the unconstrained solution towards Q^ C in the case of contingency.
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H. Kim et al. / Int. J. Production Economics 134 (2011) 16–27
Fig. 2. Expected profits based on increased mean in defect distribution after contingency.
Fig. 4. Increased variance shifts expected profits to the left.
P0 ) reduces the order quantity due to the increased variance. The section that follows illustrates these solution procedures with a numerical example. 3.2.2. Constrained optimization II—probability constraint As an alternative to controlling the profit function by conditioning on the occurrence of a contingency, here we restrict the solution space by restricting the probability space of the profit distribution. That is, we bound from below (with g) the probability that the profit is less than an amount P1 . If we assume, for simplicity, that the demand level x is fixed, the problem becomes max Fig. 3. Increased mean causes infeasible solution.
s:t: To further explore the properties of the proposed constrained optimization method, following the discussion in Eq. (3.3), this section considers two ways contingency affects the expected profit through the distribution of Y: (1) contingency increases the mean of Y, and (2) contingency increases the variance of Y. We first examine the case where the contingency causes a location shift (to the right) in the defect distribution, so GC is increased from G by a constant. The location shift (to the right) in G causes both a location shift (to the right) and an increased variability in the expected profit (see Fig. 2). In the case where S1 ¼SP \ SP ,C a |, if Q^ A S1 , then Q ¼ Q^ ; otherwise, P0
0
P0
0
Q ¼ minQ A S1 fQ g. If S1P0 ¼ |, it is not possible to keep the P0
conditional expected profit above P0 (in case of a contingency) without allowing overall expected profit to go below P0 (see Fig. 3). To rectify this problem, P0 must be reduced until there is an overlap between SP0 and SP0 ,C . In general, the order quantity increases in the level of risk-aversion (e.g., when P0 increases, Q increases) under the location shift case. From Proposition 3.1, the optimal order quantity which maximizes expected profit decreases as the variance increases. Accordingly, increased variability in Y shifts the expected profit to the left as illustrated in Fig. 4. Furthermore, the increase in variance results in a decrease in expected profit. The vertical distance between the two local maxima in the expected profit curves represents the decrease in maximum possible profits due to the increase in variance. When the variance under contingency is s2 þ d, this distance is
D ¼ Cðm, s2 ÞCðm, s2 þ dÞ "
¼
ð1mÞ2 ð1mÞ2 þ s2
ð1mÞ2 ð1mÞ2 þ s2 þ d
#"
# ðbðr þ pcÞ þ aðh þcÞÞ2 : 2ðbaÞðh þr þ pÞÞ
If S1P0 ¼ SP0 \ SP0 ,C a |, then a unique solution exists; if Q^ A S1P0 , then Q ¼ Q^ ; otherwise, Q ¼ max 1 fQ g. Again, if no overlap Q A SP
Pg ðPðQ ,YÞ r P1 Þ r g,
ð3:5Þ
where (
PðQ ,YÞ ¼
r xcð1YÞQ hðð1YÞQ xÞ,
ð1YÞQ Z x,
r xcð1YÞQ ðr þ pÞðxð1YÞQ Þ,
ð1YÞQ r x:
In this case, the retailer’s strong risk-aversion can be reflected by either increasing the value of P1 or decreasing the value of g. Denote by Q^ the optimal order quantity for the unconstrained maximization problem that satisfies the following expression (see Shih, 1980): Z 1x=Q^ ð1mÞðrc þ pÞ ð1yÞgðyÞ dy ¼ : ð3:6Þ r þ hþ p 0 If we tacitly assume Q Z x, the probability constraint becomes px þ P1 x x P Y Z1 P½PðQ ,YÞ r P1 ¼ P Y Z1 Y Z1 ðr þ pcÞQ Q Q r x þ hxP1 x x P Y r1 : þ P Y r 1 Y r1 ðhþ cÞQ Q Q The following propositions characterize the solution to the stochastic constraint placed on the profit distribution. Proposition 3.4. For any fixed target profit level P1 , there exists a critical order size Q1 ¼ P1 =ðrcÞ such that for any demand Q r Q1 -P½PðQ ,YÞ r P1 ¼ 1, px þ P1 Q 4 Q1 -P½PðQ ,YÞ r P1 ¼ 1G 1 ðr þ pcÞQ r x þhxP1 þG 1 : ðh þcÞQ Proposition 3.5. For any given target profit level P1 and probability g, there exists a feasible set for (3.5) of the form SðA,BÞ ¼ fQ jQL rQ rQU g with
0
between SP0 and SP0 ,C exists, the constraint P0 must be reduced. Unlike the location shift case, strong risk-aversion (bigger value of
Eg ½PðQ ,YÞ
Q Z0
QL ¼
px þ P1 , ðr þ pcÞð1G1 ð1gÞÞ
H. Kim et al. / Int. J. Production Economics 134 (2011) 16–27
px þ P1 r x þ hxP1 þG 1 , QU ¼ Q : g ¼ 1G 1 ðr þ pcÞQ ðhþ cÞQ where G1 represents the inverse cumulative distribution function of Y. As P1 increases, the feasible set S shrinks; the lower boundary QL decreases in g while the upper boundary QU increases. Thus, S widens as g increases. Proposition 3.6. If g o g1 ¼ 1Gð1zðP1 ÞÞ, where zðtÞ ¼ ððpx þtÞ= ðr þ pcÞÞððr x þhxtÞ=ðhþ cÞÞ1 , then SðP1 , gÞ ¼ |, e.g., there is no feasible solution in (3.5). Proof. We need to show that P½PðQ ,YÞ r P1 has only one minimum point at ðr x þ hxP1 Þ=ðh þcÞ. Because 1Gð1ðpx þ P1 Þ= ððr þ pcÞQ ÞÞ is a decreasing function of Q and Gð1ðr x þ hxP1 Þ=ððh þcÞQU ÞÞ is increasing in Q, P½PðQ ,YÞ r P1 has its minimum at ððr x þ hxP1 Þ=ðhþ cÞ. & Proposition 3.7. If Q ZQ1 , then g1 is increasing in P1 and g1 ¼ 1 when P1 is set at the maximum profit level, e.g., P1 ¼ ðrcÞx, which can be achieved only when there are no product shortages or unsold products. Proof. It is easy to see that zðtÞ is an increasing function of t. If Q ZQ1 , it can be shown that zðP1 Þ r 1. When P1 ¼ ðrcÞx, then zðP1 Þ ¼ 1, hence g1 ¼ 1. & Proposition 3.8. For any given profit level P1 and probability g, g Z g1 , the optimal order quantity for the constrained optimization problem (3.5) is 8 ^ > if Q^ A SðP1 , gÞ, >
> :Q if Q^ ZQU , U where Q^ is determined by (3.6). Whenever the order quantity Q^ satisfies the probability constraint (e.g., Q^ A SðP1 , gÞ), then the retailer orders Q^ . Otherwise, the retailer should order either QL or QU according to whether Q^ r QL or not. Fig. 5 shows the result in Proposition 3.8. In the previous section, only the mean and the variance of Y are needed to derive the optimal solution, but in this case, the distribution of Y must be known to apply probability constraints and derive an optimal solution. If the distribution is known along with appropriate values of P1 and g, profit loss can be avoided in
21
the case of contingency. However, because of the small probability of contingency, the value of g must be selected carefully. Remarks. It is interesting to note that traditional Mean–Variance and Max–Min procedures are not appropriate to generate riskaverse solutions under a possible contingency. First, we consider the Mean–Variance procedure. The objective function can be written as max EG ½PðQ ,YÞaVar G ½PðQ ,YÞ Q Z0
or max ðEGN ½PðQ ,YÞaVar GN ½PðQ ,YÞÞP½I ¼ 0 Q Z0
þ ðEGC ½PðQ ,YÞaVar GC ½PðQ ,YÞÞP½I ¼ 1: Assuming p ¼ P½I ¼ 1 is very small, the solution to the above maximization problem will maximize the objective function under the non-contingency case. Due to its small probability, the contingency does not affect the derivation of risk-averse solutions. Now, the formal objective function under the Max–Min procedure is maxmin EG ½PðQ ,YÞ: Q
G
The Max–Min procedure provides a solution which maximizes the expected profit under the worst case scenario. Figures in Section 3.2.1 can be used to illustrate Max–Min solutions. Max–Min solutions of Figs. 2 and 3 correspond to ordering quantities where two curves intersect. In Fig. 4, the Max–Min solution coincides with the ordering quantity which maximizes the expected profit under contingency. In this way, the Max–Min solution does not provide any flexibility in terms of the resulting ordering quantity. Contrary to these two traditional methods for risk-averse solutions, the two proposed methods in this section provide not only a way to handle the low-probability events, but also they offer great flexibility in deriving ordering quantity decisions in accordance with the decision maker’s various degrees of risk preferences by adjusting parameter values.
4. Numerical examples To further understand the implication of results from the previous section, we consider numerical examples employing a variety of model parameters. 4.1. Constrained optimization I—profit constraint For the following example, we assign parameter values: ðr,c, p,hÞ ¼ ð$50,$10,$30,$2Þ with the demand distribution of Uða ¼ 100, b ¼ 150Þ. We assume the (unconditional) mean and the variance of Y are ðm, s2 Þ ¼ ð0:01,0:01Þ. In the case of contingency, we use the parameter values listed in Table 1. We let mc and s2c , respectively, represent the mean and the variance of Y under contingency. Table 1 Parameter values in the case of contingency.
Fig. 5. Shape of the probability constraint.
ðmc , s2 Þ
(0.05, 0.01) (0.4, 0.01)
(0.1, 0.01) (0.5, 0.01)
(0.2, 0.01) (0.6, 0.01)
(0.3, 0.01) (0.7, 0.01)
ðm, s2c Þ
(0.01, 0.05)
(0.01, 0.1)
(0.01, 0.2)
(0.01, 0.3)
(0.01, 0.4)
(0.01, 0.5)
(0.01, 0.6)
(0.01, 0.7)
22
H. Kim et al. / Int. J. Production Economics 134 (2011) 16–27
Table 2 Example: case 1 versus case 2. Mean
Variance
Q^
E (profit)
%
Mean
Variance
Q^
E (profit)
%
0.01 0.05 0.1 0.2 0.3 0.4 0.5 0.6 0.7
0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01
143 149 157 176 200 231 274 336 428
$4575 $4561 $4540 $4487 $4410 $4293 $4102 $3762 $3075
100.0 0.3 0.8 1.9 3.6 6.2 10.3 17.8 32.8
0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01
0.01 0.05 0.1 0.2 0.3 0.4 0.5 0.6 0.7
143 137 131 120 110 102 95 89 84
$4575 $3934 $3198 $1915 $831 $95 $896 $1595 $2211
100.0 14.0 30.1 58.1 81.8 102.1 119.6 134.9 148.3
Table 3 Solutions when P0 ¼ $4000 and s2 ¼ 0:01. S4000,C
S4000
0.05 0.1 0.2 0.3 0.4 0.5 0.6 0.7
122, 129, 146, 169, 201, 253, | |
117, 117, 117, 117, 117, 117, 117, 117,
175 184 205 231 262 296
S 4000 169 169 169 169 169 169 169 169
122, 129, 146, 169, | | | |
169 169 169 169
Qn
E½PðQ Þ
143 143 146 169 201 253 Infeasible Infeasible
4575 4575 4566 4012 1813 5308 NA NA
0 Expected Profits
mc
S3000
S 3000
0.05 0.1 0.2 0.3 0.4 0.5 0.6 0.7
103, 109, 123, 142, 167, 203, 262, 398,
99, 99, 99, 99, 99, 99, 99, 99,
103, 109, 123, 142, 167, | | |
mu=0.2
mu=0.6 mu=0.7
E½PðQ Þ
143 143 143 143 167 203 262 398
4575 4575 4575 4575 4095 1620 6986 48 355
Q 186 186 186 186 186
100
150
250
300
350
Fig. 6. Profit functions under fixed variance.
Table 5 Solutions when P0 ¼ $3000 and m ¼ 0:01.
s2c 0.05 0.1 0.2 0.3 0.4 0.5 0.6 0.7
Eg ½PðQ ,YÞ
S3000,C
S3000
S 3000
Qn
E½PðQ Þ
104, 170 116, 145 | | | | | |
99, 99, 99, 99, 99, 99, 99, 99,
104, 170 116, 145 | | | | | |
143 143 Infeasible Infeasible Infeasible Infeasible Infeasible Infeasible
4575 4575 NA NA NA NA NA NA
x 104
Case2 – Mean is fixed at 0.01
186 186 186 186 186 186 186 186
1
EgC ½PðQ ,YÞ Z4000:
When contingency increases the mean to mc ¼ 0:05, we have S14000 ¼ S4000 \ S4000,C ¼ f117 rQ r169g \ f122 rQ r 175g ¼ f122 rQ r169g. The solution Q ¼ Q^ ¼ 143 because 143 A S14000 . Table 3 summarizes results using different values of mc . When mc Z0:4, S14000 is the empty set, so there is no solution which guarantees a minimum expected profit of $4000 regardless of contingency. If we choose a constraint value of P0 smaller than $4000, say $3000, we have a nonempty set S13000 when mc ¼ 0:4 as shown in Table 4. Fig. 6 shows plots of expected profits versus ordering quantity at different levels of m given s2 ¼ 0:01. If a contingency leads to an increase in the variance of Y, the optimal ordering quantity decreases. For example, at P0 ¼ $3000, Table 5 shows solutions for eight different values of s2c between 0.05 and 0.70. In Fig. 7, we fix the mean at a constant value, m ¼ 0:01, and change the variance to see the effect on expected profit. As expected, the optimal order quantity increases
200
Q (Ordering Quantity)
var = 0.01
0 var = 0.05
–1 Expected Profits
s:t:
mu=0.3 mu=0.4
–15000
Table 2 lists optimal ordering quantities which maximize expected profits under various combinations of the mean and the variance of Y. As stated previously, Q^ increases as the mean of Y increases and decreases as the variance of Y increases. For the unconditional case, the optimal ordering quantity is Q^ ¼ 143 with E½PðQ^ Þ ¼ $4575. If the expected profit under contingency is restricted to be at or above P0 ¼ $4000, the solution satisfies Q Z0
mu=0.1
–10000
50
S3000,C
max
mu=0.01 mu=0.05
–20000
mc
186 186 186 186 186 186 186 186
–5000
mu=0.5
Table 4 Solutions when P0 ¼ $3000 and s2 ¼ 0:01.
194 204 228 258 296 346 409 411
Case 1– When Variance fixed at 0.01
5000
–2 –3
var = 0.7 var = 0.6
–4
var = 0.5
–5 var = 0.4
–6
var = 0.3 var = 0.2
–7
var = 0.1
–8 50
100
150
200
250
Q (Ordering Quantity) Fig. 7. Profit functions under fixed mean.
300
350
H. Kim et al. / Int. J. Production Economics 134 (2011) 16–27
as the mean of Y increases and decreases as the variance increases. It can be seen that the maximum expected profit is much more sensitive to changes in the variance compared to changes in the mean. The expected profit loss from an increase in the mean can be compensated by increasing the quantity of the order. The profit loss from an increase in variance can be much more dramatic, however, and the constrained maximization problem becomes infeasible quickly in this case. Profit loss becomes worse with larger order quantities because the larger order naturally creates more variability in the amount of total defects. The proposed approach provides a more robust solution to the optimization problem and is more appropriate when the contingency increases the mean of Y while variance remains stable. Otherwise, even with a reasonable constraint amount P0 , the feasible set SP0 can be empty. 4.2. Constrained optimization II—probability constraint
Table 6 Parameter values I. Parameter
Value
r
$50 120 units per period $2 $30 0.1, 0.01, 0.001 $3000,$4000 ($10, $10), ($5, $15), ($1, $19) i.i.d. Beta(10, 10) i.i.d. Beta(1, 99) i.i.d. Bernoulli with p ¼ 0.01
x h
p g P1 (c1,c2) XjC, XnC XjN, XnN Ij, Inj , I0
Table 7 Solutions under separated and integrated logistics operations.
p1 ¼ p2 ¼ $30
This example uses the same parameter values as before, ðr, p,hÞ ¼ ð$50,$30,$2Þ, with a fixed demand level at 120 units per period and k¼2 suppliers, and the following distributional assumptions:
400 350 300 250 200 150 100 50
Integrated operations
g
Qn
E½PðQ Þ
Qn
E½PðQ Þ
c1 ¼ $10 c2 ¼ $10
$3000 $3000 $3000 $4000 $4000 $4000
0.1 0.01 0.001 0.1 0.01 0.001
124 132 139 124 152 160
$4703 $4661 $4598 $4703 $4477 $4398
124 134 154 124 154 177
$4690 $4630 $4432 $4690 $4432 $4190
c1 ¼ $5 c2 ¼ $15
$3000 $3000 $3000 $4000 $4000 $4000
0.1 0.01 0.001 0.1 0.01 0.001
124 124 129 124 140 149
$4703 $4703 $4663 $4703 $4561 $4476
124 130 161 124 150 186
$4690 $4645 $4332 $4690 $4445 $4057
c1 ¼ $1 c2 ¼ $19
$3000 $3000 $3000 $4000 $4000 $4000
0.1 0.01 0.001 0.1 0.01 0.001
124 124 137 124 143 158
$4703 $4703 $4589 $4703 $4533 $4387
124 124 158 124 142 182
$4690 $4690 $4363 $4690 $4525 $4102
XjN ,XN i:i:d: Beta ð1,99Þ,
Parameters of the beta distributions are chosen, so that the damage proportion without contingency has a smaller mean and a smaller variance compared to the mean and the variance under contingency. Thus, in our example, we assume that the proportion of damaged products incurred during logistics operations under the normal operation without contingency follows a beta distribution with parameters a¼1 and b¼99, a mean of a/ (a+ b)¼ 0.01 and a variance of ab=ða þ b þ1Þða þ bÞ2 ¼ ð9:802Þ 105 . On the other hand, we assume Beta (10,10) for the damage proportion under contingency, which has a mean of a=ða þ bÞ ¼ 0:5 and a variance of ab=ða þ bþ 1Þða þbÞ2 ¼ 0:0119. Fig. 8 shows the shape of the simulated distribution of Y under the separated logistics operations assumption. Four different values of P1 and three different probabilities for g are considered: P1 A f1000,2000,3000,4000g and g A f0:1,0:01,0:001g. Table 6 summarizes values of considered parameters. The optimal order quantity ðQ^ Þ for the unconstrained maximization problem is 124 units independent of two different logistics operations. Resulting solutions under ‘‘separated’’ and
Separated operations
P1
XjC ,XC i:i:d: Beta ð10,10Þ,
Ij ,Ij ,I0 i:i:d: Bernoulli with p ¼ 0:01:
23
‘‘integrated’’ logistics operations (described in Section 2) are summarized in Table 7. Furthermore, Fig. 9 gives detailed comparisons between solutions from two cases graphically. For each fixed value of P1 , the optimal order quantity Q increases as g decreases, reflecting strong risk-aversion of the decision maker. In Fig. 9, expected profits at the optimal order quantities computed by adopting separated logistics operations are always higher compared to the optimal expected profits in case of integrated logistics operations. In this example, we can conclude that when products’ retail prices, holding costs, and shortage costs are identical, separated logistics channels for products from different suppliers always produce more profits than the integrated logistics channel as long as there are no additional costs for adopting separated logistics channels. This improved performance from the separated logistics channels is due to the smaller variability in the damage amount compared to that of the integrated logistics channel. However, despite the reduced variability of separated logistics channels, if shortage costs, which are due to product damages, are not the same for products from different suppliers, the integrated logistics channel may perform better under certain situations. In the following section, we illustrate this using numerical examples. 4.3. Constrained optimization II—risk-pooling using integrated operations
0 0
0.1
0.2
0.3
0.4
0.5
Fig. 8. Simulated pdf of YNM.
0.6
0.7
In this section, we show that the integrated logistics operation performs better than the separated one under different product
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H. Kim et al. / Int. J. Production Economics 134 (2011) 16–27
$4,705 $4,700 Separated Operaions Integrated Operation
$4,695 $4,690 $4,685 $4,680 1000
2000
3000
$4,750 $4,700 $4,650 $4,600 $4,550 $4,500 $4,450 $4,400 $4,350 $4,300 $4,250
4000
c1=5, c2=15, r=0.1
Integrated Operation
1000
Separated Operaions Integrated Operation
$4,695 $4,690 $4,685 $4,680
$4,600 Separated Operations Integrated Operation
$4,400
Separated Operations
$4,200
Integrated Operation
$4,000 $3,800 $3,600
2000
3000
4000
1000
Desired profit level
c1=1, c2=19, r=0.1
$4,695 $4,690
4000
c1=1, c2=19, r=0.001
$4,700 Separated Operations Integrated Operaion
3000
$4,800
$4,750
$4,700
2000
Desired profit level
c1=1, c2=19, r=0.01
$4,705
4000
$4,800
1000
Desired profit level
3000
c1=5, c2=15, r=0.001
$4,750 $4,700 $4,650 $4,600 $4,550 $4,500 $4,450 $4,400 $4,350 $4,300
4000
2000
Desired profit level
c1=5, c2=15, r=0.01
$4,700
3000
Separate Operations
Desired profit level
$4,705
2000
Separage Operations Integrated Operation
$4,800 $4,700 $4,600 $4,500 $4,400 $4,300 $4,200 $4,100 $4,000 $3,900
1000 2000 3000 4000
Desired profit level
1000
c1=c2=10, r=0.001
c1=c2=10, r=0.01
c1=c2=10, r=0.1
$4,600
$4,650
Separated Operations integrated Operation
$4,600 $4,550
$4,400
Separated Operations
$4,200
Integrated Operation
$4,500
$4,685
$4,000
$4,450
$4,680
$3,800
$4,400 1000 2000 3000 4000 Desired profit level
1000
2000
3000
1000
4000
2000
3000
4000
Desired profit level
Desired profit level
Fig. 9. Comparison between separated logistics operation and integrated logistics operation.
shortage costs. We first consider a simple example to convey the idea and consider a more general example later. Suppose that proportions of damaged products from two suppliers under separate logistics operations and integrated logistics operation are distributed as follows: ( P1 ðand P2 Þ ¼ ( P10 ðand P20 Þ ¼
pi_1=$1, pi_2=$29, r=0.5 $4,700 $4,600 $4,500 $4,400
0 0:2
with 0:5 probability, with 0:5 probability:
$4,300
0
with 0:5 probability,
$4,100
0:2
with 0:5 probability:
$4,000
Separated logistics operations (e.g., two trucks to separately ship products from two suppliers) result in independence of P1 and P2 while integrated logistics operation renders P10 and P20 dependent. We adopt a shock model to explain the dependency between proportions of product damages from two suppliers in case of integrated operation. When only one truck is used to transport products from two suppliers to the DC and then to the retailer, only two contingency cases are possible. If a contingency occurs, products from both suppliers will jointly experience an increased rate of product damages. If a contingency does not occur, proportions of damaged products will be smaller. Based on this, we construct the following distributions for the total proportion of damaged products: 8 with 0:25 probability, > <0 YNM ¼ 0:1 with 0:5 probability, > : 0:2 with 0:25 probability:
$4,200 Separated Integrated
$3,900 4200
4250
4300
Fig. 10. Risk-pool effects of integrated logistics operation.
( YM ¼
0
with 0:5 probability,
0:2
with 0:5 probability:
In our first example, we use different shortage costs for different suppliers p1 ¼ $1, p2 ¼ $29, g is fixed at 0.5 and P1 ¼ $4200, $4250,$4300. All other parameter values are kept the same as in Section 4.2. Fig. 10 shows how the integrated logistics operations perform better than the separated operations. This second example illustrates the risk-pooling effects of the integrated operation with more general distributional assumptions. Instead of using simple two-point mass discrete distributions as in the first example, we use the set of beta distributions shown
H. Kim et al. / Int. J. Production Economics 134 (2011) 16–27
25
pi_1=$5, pi_2=$25, Pi_1=$3800
Table 8 Parameter values II.
$4,800
Parameter
Value
r
$50 120 units per period $2 p1 ¼ $5, p1 ¼ $25 0.1, 0.05, 0.04, 0.03, 0.02, 0.01 $3800, $4000, $4200 $10 i.i.d. Beta(10, 10) i.i.d. Beta(1, 99) i.i.d. Bernoulli with p ¼0.01
x h
p g P1 c ¼ c1 ¼ c2 XjC, XnC XjN, XnN Ij, Inj , I0
$4,700 $4,600 $4,500 $4,400 $4,300
Separated Operations Integrated Operation
$4,200 $4,100 0.1
Table 9 Solutions under separated and integrated logistics operations with different shortage costs.
p1 ¼ 5, p2 ¼ 25
Separated operations
P1
g
Q
$3800 $3800 $3800 $3800 $3800 $3800 $4000 $4000 $4000 $4000 $4000 $4000 $4200 $4200 $4200 $4200 $4200 $4200
0.1 0.05 0.04 0.03 0.02 0.01 0.1 0.05 0.04 0.03 0.02 0.01 0.1 0.05 0.04 0.03 0.02 0.01
124 124 124 133 141 151 124 124 133 143 153 166 124 124 128 143 154 164
n
Integrated operations
E½PðQ Þ
Qn
E½PðQ Þ
$4709 $4709 $4709 $4627 $4548 $4447 $4701 $4701 $4621 $4524 $4425 $4290 $4709 $4709 $4677 $4529 $4417 $4311
124 124 124 124 138 161 124 124 124 139 151 171 124 124 124 133 153 NA
$4717 $4717 $4717 $4717 $4577 $4331 $4714 $4714 $4714 $4565 $4438 $4221 $4710 $4710 $4710 $4625 $4415 NA
0.05
0.04 0.03 gamma
0.02
0.01
pi_1=$5, pi_2=$25, Pi_1=$4000
$4,800 $4,700 $4,600 $4,500 $4,400 $4,300 $4,200 Separated Operations
$4,100
Integrated Operation
$4,000 $3,900 0.1
0.05
0.04 0.03 gamma
0.02
0.01
pi_1=$5, pi_2=$25, Pi_1=$4200
$4,800 $4,700
in Table 8. In this example, we assume shortage costs of p1 ¼ $5 and p2 ¼ $25. As explained in Section 4.2, the beta distribution parameters are chosen, so that the distribution of damage proportion without contingency has a smaller mean and variance compared to the mean and the variance under contingency. The results are summarized in Table 9, Figs. 11 and 12. Again, the integrated logistics operations perform better in the cases, where the parameters are P1 ¼ $3800,$4000,$4200 and g ¼ 0:02,0:03, 0:04,0:05,0:1. In both examples, by adopting an integrated logistics operation, the retailer will produce more profits whenever the logistics operations are exposed to possible contingencies and inventory costs such as unit shortage costs are different for products from different suppliers (e.g., p1 ¼ $5, p2 ¼ $25Þ. A contingency to the logistics operation for products with higher unit shortage cost may significantly affect the retailer profit. However, if the retailer employs an integrated operation, then the total resulting damages from contingency may be evenly spread among pooled products from different suppliers. In this way, product-pooling, using the integrated logistics operation, dampens the possible risk of having a large portion of damaged products, which have higher shortage costsby introducing the risk of having evenly distributed damaged products among all products.
5. Summary and conclusions In brief, the main goal of this research is to build a bridge between the quantitative uncertain-supply problem and the
$4,600 $4,500 $4,400 $4,300
Separated Operations Integrated Operation
$4,200 $4,100 0.1
0.05
0.04 0.03 gamma
0.02
0.01
Fig. 11. Risk-pooling effects of integrated logistics operation I.
problem of logistics network planning and vehicle routing, so that retailers can incorporate supply-chain logistics uncertainties in product-ordering decisions. We adopt statistical concepts to characterize the underlying uncertainty. We also investigate the impact of two different logistics operational policies on the resulting solutions. To achieve this goal, we examine the consequences of supply disruption on the retailer’s profits. The supply disruptions take the form of high-impact and low-probability contingencies which can threaten large sections of the supply chain. The traditional expected-value approaches for product-ordering decisions fail to provide the retailer with sufficient means of protection against the effects of contingency. To provide the decision-maker facing
26
H. Kim et al. / Int. J. Production Economics 134 (2011) 16–27
r=0.04
r=0.01 $4,750
$4,500 Separated
$4,400
Operations
$4,300
Integrated
$4,200
Operation
Separated
$4,700
Operations
$4,650
Integrated
$4,600
Operation
$4,550
$4,100 4000
3800
3800
4000
4200
r=0.05
r=0.02 $4,600
$4,720 Separated
$4,500
Operations Integrated
$4,400
Operation
$4,300
Separated
$4,710
Operations Integrated
$4,700
Operation
$4,690 3800
4000
4200
3800
4200
4000
r=0.03
r=0.1
$4,800
$4,720 Separated
$4,700
Operation
$4,600
Integrated
$4,500
Operation
Separated
$4,710
Operations Integrated
$4,700
Operation
$4,690
$4,400 3800
4000
4200
3800
4200
4000
Fig. 12. Risk-pooling effects of integrated logistics operation II.
possible contingency with a systematic way to handle this situation, we propose two procedures. In the first procedure, by constraining the maximization problem with respect to conditional expected profit, a more stable and risk-averse solution can be found. We consider two cases, where contingency either increases the mean of the proportion of damage distribution or increases the variance of the distribution. An increase in variance causes a rapid drop in expected profit, leaving no other alternatives to compensate the profit loss. On the other hand, the more robust methods introduced here compensate for a mean shift of the profit curve, resulting in an increased quantity of the order. In practice, it is recommended that the retailer investigate the characteristic of potential contingency to see how and if it affects the mean or the variance of Y. If the model implies that the contingency changes only the mean, then the retailer can benefit from the constrained optimization solution in Section 4.1. However, if the contingency adversely affects the variance, the decision maker should find a way to reduce the variance, perhaps by using multiple sourcing or purchasing options by which the damage distribution can be truncated. In the other procedure, we utilize the probability constraint to restrict possible solutions. With this procedure, the decisionmaker has more options to include risk preferences in the solution; one can change either the target profit level or the target probability level to derive risk-averse solutions. This procedure illustrates the risk-pooling effects of the integrated logistics operations under certain conditions. Whenever the inventory holding cost, the shortage cost and retail prices of products from different suppliers are identical, separated logistics operations between the distribution center and the retailer generate solutions with higher expected profits compared to those of the integrated logistics operation case. But in our examples, we show that the resulting expected profits may be higher under the integrated logistics operation strategy than the expected profits under separated logistics operations when shortage costs are significantly different.
The investigation of the effects of non-i.i.d. defect distributions associated with different routes can be considered for future research. To make our model more practical, we also need to introduce the logistics cost element in the retailer’s profit function. In that case, it will be an interesting problem to study the systematic trade-off methods between the cost saving effects due to the reduced variance from separated logistics operations and the additional logistics costs required for separated logistics operations. Extending our problem to a multi-period setting is another challenging task.
Appendix A A.1. Proof of Eq. (3.3) The first derivative is Z
@ESR @ ¼ cð1mÞr @Q @Q
0
Z
0
Z
@ @Q Z Z 1 ð1yÞQ
¼ cð1mÞr
1
1 0
!
1
½xð1yÞQ f ðxÞgðyÞ dx dy
ð1yÞQ
Z
1
xf ðxÞ dxgðyÞ dy
ð1yÞQ
!
1
f ðxÞ dxgðyÞ dy , ð1yÞQ
where ! Z 1Z 1 Z 1 @ xf ðxÞ dxgðyÞ dy ¼ ð1yÞ2 Qf ðð1yÞQ ÞgðyÞ dy, @Q 0 ð1yÞQ 0
@ @Q
Z
Z
1
f ðxÞ dxgðyÞ dy
ð1yÞQ 0
Z 0
1
!
1 ð1yÞQ
ð1yÞ2 Qf ðð1yÞQ ÞgðyÞ dy:
Z
1
ð1yÞF ðð1yÞQ ÞgðyÞ dy
¼ 0
H. Kim et al. / Int. J. Production Economics 134 (2011) 16–27
27
If we simplify the above expression we have Z 1 @ESR ¼ cð1mÞ þ r ð1yÞF ðð1yÞQ ÞgðyÞ dy: @Q 0
and similarly
Now it is easy to see the following result: Z 1 @2 ESR ¼ r ð1yÞ2 f ðð1yÞQ ÞgðyÞ dy, @Q 2 0
and fY r 1ðr x þ hxP1 Þ=ðh þ cÞQ g\ fY r1x=Q g ¼ fY r 1x=Q g. From these results, the conditional probabilities from the first and the second term in P½PðQ ,YÞ r P1 become equal to 1, so that P½PðQ ,YÞ r P1 ¼ PðY Z1x=Q Þ þ PðY r 1x=Q Þ ¼ 1. If Q 4Q1 , then fY Z 1ðpx þ P1 Þ=ðr þ pcÞQ g \fY Z 1x=Q g ¼ fY Z1ðpx þ P1 Þ=ðr þ pcÞQ g and fY r1ðr x þhxP1 Þ=ðhþ cÞQ g \ fY r 1 x=Q g ¼ fY r 1ðrx þ hxP1 Þ=ðh þ cÞQ g yield px þ P1 r x þ hxP1 þ G 1 : & P½PðQ ,YÞ r P1 ¼ 1G 1 ðr þ pcÞQ ðh þ cÞQ
which is negative for all possible values of Q. A.2. Proof of Proposition 3.3 The variance of the profit is x x P Y r 1 Var½PðQ ,YÞ ¼ Var PðQ ,YÞjY r 1 Q Q x x P Y 4 1 þVar PðQ ,YÞjY 4 1 Q Q x ¼ Var ðcð1YÞQ þ hðð1YÞQ xÞÞP Y r 1 Q x þVarðcð1YÞQ þ ðr þ pÞðxð1YÞQ ÞÞP Y 4 1 Q x 2 ¼ ðc þhÞ Var½ð1YÞQ G 1 Q x 2 þðr þ pcÞ Var½ð1YÞQ 1G 1 Q x x 2 2 2 2 2 2 þ ðr þ pcÞ Q s 1G 1 ¼ ðc þ hÞ Q s G 1 Q Q
x x þ ðr þ pcÞ2 1G 1 Q 2 s2 : ¼ ðc þ hÞ2 G 1 Q Q The first derivative of this variance with respect to the order quantity Q is
@Var½PðQ ,YÞ x x þ ðr þ pcÞ2 1G 1 ¼ 2s2 Q ðc þ hÞ2 G 1 Q Q @Q
x : þ s2 xððc þhÞ2 ðr þ pcÞ2 Þg 1 Q
The first term is always positive, and the second term is positive whenever ðc þhÞ 4ðr þ pcÞ.
Appendix B. Proof of Proposition 3.4 Proof. If Q rQ1 , using the previous assumption, Q Z x, we can show that
px þ P1 px þðrcÞQ px þ ðrcÞx Z Z ¼ x: ðr þ pcÞ ðr þ pcÞ ðr þ pcÞ Thus, fY Z 1ðpx þ P1 Þ=ðr þ pcÞQ g \ fY Z 1x=Q g ¼ fY Z1x=Q g
r x þ hxP1 r x þ hxðrcÞQ r x þ hxðrcÞx r r ¼ x, ðh þcÞ ðh þ cÞ ðhþ cÞ
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