Diversification strategy with random yield suppliers for a mean–variance risk-sensitive manufacturer

Transportation Research Part E xxx (2016) xxx–xxx

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Transportation Research Part E journal homepage: www.elsevier.com/locate/tre

Diversification strategy with random yield suppliers for a mean–variance risk-sensitive manufacturer Weili Xue a, Tsan-Ming Choi b, Lijun Ma c,⇑ a

School of Economics and Management, Southeast University, Nanjing 210096, China Business Division, Institute of Textiles and Clothing, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong c Department of Management Science, College of Management, Shenzhen University, Shenzhen 518060, China b

a r t i c l e

i n f o

Article history: Received 5 September 2015 Received in revised form 18 December 2015 Accepted 25 January 2016 Available online xxxx Keywords: Risk analysis Inventory control Supplier selection Random yield Mean-variance analysis Newsvendor problem

a b s t r a c t We consider the diversification strategy for a mean–variance risk-sensitive manufacturer with unreliable suppliers. We first analyze the linear model and find that the suppliers are selected according to the descending order of their contributed marginal expected profit, and increasing the manufacturer’s risk-averseness leads to a more even allocation of demand across the suppliers. Then, we study the general newsvendor model. By approximating the leftover inventory with a normal distribution, we establish the general properties of the active supplier set and show that the supplier selection rule is similar to that under the risk-neutral setting when the demand uncertainty is large. Moreover, we conjecture that the selection rule also applies when the demand uncertainty is low, which we verify with an extensive numerical study. Our paper makes two contributions: First, we establish the properties of the optimal diversification strategy and develop corresponding insights into the trade off between cost and reliability under the mean–variance framework. Second, we perform comparative statics on the optimal solution, with a particular emphasis on investigating how changes in the supplier’s cost or reliability affect the risk-averse manufacturer’s ordering decisions and customer service level. Ó 2016 Elsevier Ltd. All rights reserved.

1. Introduction With the globalization of supply chain operations, many firms face risks from both the supply side (i.e., supply uncertainty) and the demand side (i.e., demand uncertainty). Supply uncertainty arises due to factors such as major machine malfunctions, damage during transportation, human-centered issues such as fraud and strike, natural hazards, terrorism, and political instability (see Kleindorfer and Saad, 2005). With supply uncertainty, the quantity of goods received by the firm may be less than the quantity ordered. For example, Starbucks may find that, in addition to the agreed price, the quality and quantity of the coffee beans supplied by farmers may vary as the weather varies. Diversification of suppliers is a fundamental strategy for effectively managing the supply risk. In operations management, an extensive body of literature studies the problems of supplier diversification. These studies focus on different aspects of the problem, varying from the allocation of demand among suppliers to the trade-off between reliability and procurement cost and the interplay between supplier diversification and flexible resources. For instance, Bernstein et al. (2015) argue that a manufacturer cannot mitigate the supply risk simply by diversifying its suppliers, which is modeled by a Bernoulli distribution, when the profit function is ‘‘linear” ⇑ Corresponding author. Tel.: +86 755 26535181; fax: +86 755 26534451. E-mail addresses: [email protected] (W. Xue), [email protected] (T.-M. Choi), [email protected] (L. Ma). http://dx.doi.org/10.1016/j.tre.2016.01.013 1366-5545/Ó 2016 Elsevier Ltd. All rights reserved.

Please cite this article in press as: Xue, W., et al. Diversification strategy with random yield suppliers for a mean–variance risk-sensitive manufacturer. Transport. Res. Part E (2016), http://dx.doi.org/10.1016/j.tre.2016.01.013

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W. Xue et al. / Transportation Research Part E xxx (2016) xxx–xxx

in the expected delivery quantity. They thus argue that the shape of the revenue and cost functions leads to the respective supplier diversification strategies. However, they consider the problem in a risk-neutral setting only. It is well established in the literature that decision makers have preferences other than risk neutrality. Risk aversion is a common decision preference. Pioneered by Markotwiz (1959) in the 1950s, the mean–variance framework has become the building block of modern financial theories. In fact, variance is a natural measure of risk. When variance is the measure of risk, the diversification strategy can reduce the supply risk even when the profit function is ‘‘linear” in the expected delivery quantity. Consider the following example. A manufacturer uses two independent suppliers to produce two units of a particular component. Suppose that the unit production cost for both suppliers is identical and denoted as c and the unit selling price for the product is p. All other expenses are negligible. The two suppliers face the risk of a natural disaster, described by a Bernoulli random variable with a probability of M in the normal state and thus a probability of 1
M in the disruptive state. If the manufacturer diversifies by sourcing one component from each supplier, its expected profit is 2Mðp
cÞ. Using a single source, its expected profit is again 2Mðp
cÞ. However, the profit variance when sourcing from both suppliers is 2Mð1
MÞðp
cÞ2 and the profit variance when sourcing from one supplier is 4Mð1
MÞðp
cÞ2 . Thus, even if the expected profit is the same for both strategies, supplier diversification can mitigate the risk by reducing the profit variance. In this paper we consider a monopoly manufacturer who acts as the buyer, he sources components from a portfolio of suppliers and assembles the components, then sells the final product. Suppliers are unreliable and supply uncertainty is modelled with respect to the proportional yield. Different suppliers have different level of reliability. The more reliable the supplier, the more expensive the procurement cost for the manufacturer is. The manufacturer makes component ordering decisions to balance the risk of uncertain supply, and the associated profit generated from sales to maximize his utility, which is measured by mean–variance. For the linear revenue model, we characterize the diversification strategy and find that the suppliers are selected in the ascending order of their contributed marginal expected profit. Moreover, we find that the order quantities of those suppliers increase even when the manufacturer is more risk averse. For the general newsvendor model, we approximate the leftover inventory by the normal distribution and establish the properties of the active supplier set, then develop the optimal diversification strategy with the corresponding algorithm when the demand uncertainty is high; that is, active suppliers are chosen in an ascending order of their effective marginal cost. Based on an extensive numerical study, we further propose a conjecture, which implies that the selection rule also applies when the demand uncertainty is low. Finally, we perform comparative statics on the optimal solution, with a particular emphasis on investigating how changes in a supplier’s cost or reliability affect the risk-averse manufacturer’s ordering decisions and customer service level. The rest of this paper is organized as follows. In Section 2, we review the related literature. In Section 3, we formulate the problem and analyze the linear revenue model and the newsvendor model. In Section 4, we carry out numerical studies. We conclude our study and discuss future research directions in Section 5.

2. Literature review This paper is related to several streams of research in inventory management. The first stream examines how the supply risk affects the performance of inventory management. Natural disasters such as hurricanes, earthquakes, tsunamis, human-centered issues such as fraud and strike, and machine malfunctions can all lead to supply uncertainty. The management of supply risk has attracted considerable attention from both practitioners and researchers in the inventory management field. Related research considering yield risk in inventory management has been carried out in various ways, including with single- or multi-period, single- or multi-stage, single- or multi-products, and single or multiple suppliers. We focus on the multiple supplier, one product procurement problem, where suppliers’ uncertainties are modeled with random yield. For other settings, we refer readers to the excellent reviews by Yano and Lee (1995), Grosfeld-Nir and Gerchak (2004), Tang (2006), Vakharia and Yenipazarli (2008), and Sadghiani et al. (2015) and the references therein. Almost all studies that consider random yield assume a single supplier. Parlar and Wang (1993) are among the first to demonstrate the benefits of dual sourcing in the presence of supply uncertainty. Anupindi and Akella (1993) and Swaminathan and Shanthikumar (1999) consider the total cost model with two suppliers. These authors also generalize some of their results to allow for multiple periods. Yano and Lee (1995) explain that the complexity of dealing with a general set of suppliers is extreme and hence it is difficult to obtain structural results. Recently, Tang and Kouvelis (2011) study supplier diversification strategies in the presence of yield uncertainty and buyer competition. Chaturvedi and Martínez de Albéniz (2011) analyze the optimal procurement auction mechanism with unreliable supply. They design optimal mechanisms that depend on the buyer’s level of information regarding the supplier’s cost of production and reliability. Giri (2011) studies the sourcing problem from two unreliable suppliers, who differ in their reliability and selling price. They investigate the optimal procurement policy in a risk-neutral setting and carry out the numerical study in a risk-averse setting, where an exponential utility function is used to measure the decision maker’s risk attitude. Cheong and Song (2013) study the value of information on supply risk under random yields within a newsvendor framework. In addition to the literature dealing with no more than two suppliers with supply risk, Agrawal and Nahmias (1997) seem to be the first work to consider the procurement problem with multiple unreliable suppliers, where supply uncertainty is modeled with random yield. The yield rates are assumed to be independent and normally distributed. Demand is deterministic and they also consider fixed order costs in their model. When the suppliers are identical, they find the optimal number of suppliers and order quantities. However, when the suppliers are not identical, they show the complexity of the optimization problem and focus on the dual sourcing problem. Chen et al. (2001) investigate the inventory control problem with Please cite this article in press as: Xue, W., et al. Diversification strategy with random yield suppliers for a mean–variance risk-sensitive manufacturer. Transport. Res. Part E (2016), http://dx.doi.org/10.1016/j.tre.2016.01.013

W. Xue et al. / Transportation Research Part E xxx (2016) xxx–xxx

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multiple unreliable suppliers and stochastic demand over an infinite horizon. Any unsatisfied demand is lost. They characterize the optimal replenishment policy by minimizing the long-run average cost. Dada et al. (2007) study the differences in sourcing strategies with completely reliable and unreliable suppliers. They show that the total order quantity is greater in the unreliable suppliers setting. Babich et al. (2012) extend the identical supplier case of Agrawal and Nahmias (1997) to include the analysis of financing decisions and constraints. Federgruen and Yang (2008) study the single period, single product sourcing problem with multiple unreliable suppliers by satisfying the customer service level constraint. They characterize the optimal policy for the identical supplier case and develop two approximations for the shortfall probability, one based on the large-deviation technique and one based on the central limit theorem. Federgruen and Yang (2009) consider the same setting as Federgruen and Yang (2008), but the suppliers differ in their yield distributions, procurement costs, and capacity levels. They consider both the service constraint model and the total cost model. Burke et al. (2009) study the implications of uncertain supplier reliability on a firm’s sourcing decisions with stochastic demand. They characterize the conditions under which a single supplier sourcing strategy is optimal. Li et al. (2013) also study a single period, multiple supplier procurement problem with both supply and demand uncertainties. Moreover, demand is price dependent and the random supply capacities are correlated. Federgruen and Yang (2011) extend their single period models (see Federgruen and Yang, 2008, 2009) to a general finite horizon model, in which unsatisfied demand is backlogged, and characterize the optimal policy. Federgruen and Yang (2014) extend Federgruen and Yang (2011) to an infinite horizon and characterize the optimal policy. Merzifonluoglu and Feng (2014) investigate the optimal procurement problem for a newsvendor with multiple unreliable suppliers, where both fixed ordering cost and capacity limits exist. Their focus is on developing efficient algorithms to solve the problem. However, none of these studies considers the risk-averse behavior of the decision maker except Giri (2011), who only studies the dual sourcing problem under the utility function with a numerical study. The second stream focuses on how risk attitude affects decision making. Since the 1950s, a rich line of research has considered risk measures, including utility function, Value at Risk (VaR), Conditional Value at Risk (CVaR), and Mean–Variance (MV). The original MV approach was first introduced by Markotwiz (1959) to study the portfolio management problem with stochastic yield. It usually minimizes the variance of the portfolio with a given mean, or maximizes the mean of the portfolio with a given variance. Pioneered by Lau (1980), the MV approach has also attracted extensive studies using various methods, including single- and multi-period models. Lau (1980) studies the mean-standard-deviation tradeoff in a newsvendor framework. Berman and Schnabel (1986) study the mean–variance tradeoff in a newsvendor model with risk-averse and risk-seeking settings. They find that the risk-averse newsvendor orders less than the risk-neutral newsvendor, while the risk-seeking newsvendor orders more than the risk-neutral newsvendor. Chen and Federgruen (2001) study the newsvendor problem with a quadratic utility function. They construct an efficient frontier for the noninferior solution points via a numerical study. They show that the stockout cost plays an important role in determining the optimal order quantity under the mean–variance framework. Choi et al. (2008a) extend Lau (1980) by considering three different kinds of risk attitudes in a newsvendor setting and study the problem with and without stock-out cost. Wu et al. (2009) study the newsvendor problem under the MV objective when the demand follows a power distribution, and prove that the optimal order quantity in this case can be greater than the one in risk-neutral setting. Choi et al. (2011) study the multi-period inventory control problem under the mean–variance framework. As the variance of profit is not separable for the original problem, they develop a primal–dual solution approach and show that the optimal policy is of a base-stock type. Liu et al. (2012) study the mass customization problem under the mean–variance framework with both demand and return uncertainties. They jointly investigate the retail pricing, consumer return, and level of modularity. They reveal the structural properties of the model and derive the closed-form solutions for the optimal decisions. Liu and Nagurney (2011) analyze the offshore outsourcing problem under the exchange-rate risk and competition between multiple suppliers within the mean–variance framework. Choi and Chiu (2012a) study the newsvendor problem with both the mean–variance and mean-downside-risk frameworks. The above literature does not consider demand information updating. Yet, in practice, there are usually opportunities to update the related demand information. Choi et al. (2003) is the first to study the two-stage two-ordering dynamic optimization model within the mean–variance framework under Bayesian information updating. Choi et al. (2004) extend Choi et al. (2003) to N stages with only one order opportunity (where N > 2). They formulate the problem as an optimal-stopping time model and determine the optimal order policy. Choi and Chow (2008) study the quick response program of a two-echelon supply chain under the mean–variance framework. They illustrate the conditions to achieve a win–win situation for the supply chain agents. Buzacott et al. (2011) study the procurement problem with a class of commitment-option supply contracts under the mean–variance framework where demand information can be updated. They show how the information updating quality and risk-averse coefficient affect the optimal order quantities. In addition to the single echelon mean–variance problems reviewed above, several works consider supply-chain coordination problems (see, for example, Lau and Lau, 1999; Agrawal and Seshadri, 2000; Tsay, 2002; Gan et al., 2004; de Albeniz and Simchi-Levi, 2006; Choi et al., 2008b; Wei and Choi, 2010). We refer readers to Choi and Chiu (2012b) for a complete review of this line of research. However, the majority of these works do not consider supply uncertainties.

3. Model formulation We consider a two-stage supply chain in which a monopoly manufacturer procures identical components (or closely substitutable components) from N suppliers and produces enough to satisfy the stochastic demand D. The suppliers are Please cite this article in press as: Xue, W., et al. Diversification strategy with random yield suppliers for a mean–variance risk-sensitive manufacturer. Transport. Res. Part E (2016), http://dx.doi.org/10.1016/j.tre.2016.01.013

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unreliable in the sense that each supplier’s delivered quantity may be strictly lower than the quantity ordered by the manufacturer, as defined in Dada et al. (2007). In line with the traditional procurement models, we assume that the procurement decision should be made and the components will arrive before the demand is realized. The assembly is then carried out. For simplicity, we do not consider the bill-of-material structure and only focus on the main component, which is a one-to-one mapping to the final product. Thus, the assembly lead time can be assumed to be zero (or a fixed time, which does not affect our conclusions). The manufacturer needs to decide the order quantity from each supplier to maximize his utility. In this paper, we use the proportional yield model to measure the supplier’s unreliable risk. Specifically, let yi be the order quantity from supplier i, and Ri 2 ½0; 1 be the corresponding yield rate of supplier i; i ¼ 1; 2; . . . ; N. Then, the quantity the P manufacturer receives from supplier i is Ri yi (see Xu and Lu, 2013), and the total quantity received is Ni¼1 Ri yi . Moreover, the suppliers are different in their supply risk and we assume the yield rate Ri is independent of Rj ; j – i and has a cumulative density function (CDF) Mi ðxÞ and a probability density function (PDF) mi ðxÞ; i ¼ 1; 2; . . . ; N. Denote r i ¼ ERi ; f2i ¼ VARðRi Þ as the mean and the variance of Ri ; i ¼ 1; 2; . . . ; N, respectively. The uncertain demand D is also independent of the supplier’s yield, and has a CDF FðxÞ and a PDF f ðxÞ. Denote l ¼ E½D and r2 ¼ VAR½D. Let ci be the unit order cost of supplier i; i ¼ 1; 2; . . . ; N. Denote the order vector as Y ¼ ½y1 ; . . . ; yN  and the random yield rate vector as R ¼ ½R1 ; R2 ; . . . ; RN . Then, with the order quantity Y, the manufacturer’s total order cost CðYÞ is given by P P CðYÞ ¼ Ni¼1 ci yi or CðYÞ ¼ Ni¼1 ci Ri yi , depending on the payment scheme, i.e., the production model or the procurement model (Xu and Lu, 2013). In our paper, we assume that the manufacturer needs to pay for the quantity ordered rather than the quantity received, i.e., our model considers the production model. This is true when the yield uncertainty is caused by the raw material quality. Different raw material quality leads to different yields during the process of manufacturing. Thus, the manufacturer needs to pay for the material ordered. Let p be the unit selling price of the end product, and h be the unit holding cost when h P 0 or the unit salvage cost when h < 0, for any excess components inventory after the demand realization. We call pri
ci the contributed marginal expected profit for supplier i and assume pr i
ci P 0 for all i 6 N. The unsatisfied demand is lost. Denote BðYjR; DÞ, depending on the cost of supply and demand risks, as the holding cost or salvage cost of the manufacturer given order vector Y, thus we have P þ BðYjR; DÞ ¼ hð Ni¼1 Ri yi
DÞ , where ðÞþ ¼ maxð; 0Þ. Similar to Bernstein et al. (2015), we assume that the manufacturer has P a revenue function VðYjR; DÞ ¼ p minfD; Ni¼1 Ri yi g, which depends on the total order quantity of the components from the suppliers. Then, the profit function with uncertain supply and demand for the manufacturer, PðYjR; DÞ, is given by

PðYjR; DÞ ¼ VðYjR; DÞ
CðYÞ
BðYjR; DÞ

ð1Þ

The manufacturer’s utility, denoted by UðYÞ, is measured by the mean–variance of his profit. Then, we have

UðYÞ ¼ E½PðYjR; DÞ
kVAR½PðYjR; DÞ

ð2Þ

where k P 0 is the risk-averse coefficient. The greater the value of k, the more risk averse the manufacturer is. Specifically, when k ¼ 0, the utility function becomes a risk-neutral utility function. We assume the feasible decision set as P Y 2 S ¼ fY : Ni¼1 yi 6 Q g. Here, Q can be treated as the budget constraint of the manufacturer or the predetermined production schedule (see Chaturvedi and Martínez de Albéniz, 2011; Bernstein et al., 2015). The manufacturer solves the following optimization problem.

ð3Þ

maxUðYÞ Y2S

For a general newsvendor function, solving the above optimization problem is very challenging (as the objective function is not concave in order quantities). Thus, in the following analysis, we mainly study the linear revenue model first, which is actually a special case of the newsvendor model when the salvage cost h is equal to
p. By studying this model, we can see how the demand risk affects the supplier’s diversification strategy and the supplier’s selection rule. As we show in the following, the supplier’s selection strategy under the mean–variance framework is different from that in the general newsvendor model. Then, we analyze the general newsvendor model by approximation in which the demand and the random yields follow a normal distribution. We establish the suppliers’ selection rules with some relationships between the order quantities from those suppliers. We use the subscripts l; n to denote the linear revenue model and the general newsvendor model for the utility, profit, revenue, and cost functions, respectively. 3.1. Deterministic linear revenue model In the deterministic linear revenue model, the salvage cost function is given by

!þ N X BðYjR; DÞ ¼
p Ri yi
D i¼1

Note that here, h ¼
p in the general holding cost function represents the situation where there is no inventory left over and inventory can be depleted to satisfy the demand. Therefore, the manufacturer’s profit function becomes

Please cite this article in press as: Xue, W., et al. Diversification strategy with random yield suppliers for a mean–variance risk-sensitive manufacturer. Transport. Res. Part E (2016), http://dx.doi.org/10.1016/j.tre.2016.01.013

W. Xue et al. / Transportation Research Part E xxx (2016) xxx–xxx N N X X Ri y i
c i yi

Pl ðYjR; DÞ ¼ p

i¼1

5

ð4Þ

i¼1

This model is also considered as the fixed-order quantity model in Chaturvedi and Martínez de Albéniz (2011) and Bernstein et al. (2015), which represents a production model that follows pre-determined orders (e.g., a production schedule). In this model, the manufacturer’s profit is linear in the supply yield rate. Suppose that suppliers are indexed by decreasing order in pri
ci ; i.e., the smaller the value of i, the greater the value of pri
ci is. Then, with a linear revenue model, we have N N X X E½Pl ðYjR; DÞ ¼ p r i yi
ci yi i¼1

i¼1

N X VAR½Pl ðYjR; DÞ ¼ p2 f2i y2i i¼1

and the optimization problem (3) can be simplified as N N N X X X max p r i yi
ci yi
kp2 f2i y2i Y2S

i¼1

i¼1

ð5Þ

i¼1

We now present the result for the optimal order quantity from each supplier obtained by the following proposition. Proposition 3.1. (a) When k ¼ 0, it is (weakly) optimal for the manufacturer to order from a single supplier with the largest r i p
ci , i.e., the supplier 1. (b) When k > 0, suppliers will be selected in a decreasing order of ri p
ci , and the optimal number of P pr j
cj suppliers to be selected, n , will be N when N j¼1 2kp2 f2 6 Q , or the largest value n satisfying j

n
1 1 X ðpr j
cj Þ
ðprn
cn Þ 6Q 2k j¼1 p2 f2j

when

PN

pr j
cj j¼1 2kp2 f2 j

yi ¼

ð6Þ

> Q . The corresponding optimal order quantity from each supplier satisfies

8 > pr i
ci > > ; > 2 2 > < 2kp fi

if

1 > f2 > pr i
ci > i > P
 2 n > 1 : 2kp2 fi j¼1 2 f

j

1 2k

n X ðpr
c Þ j

j¼1

p2 f2j

j

!
Q ; if

N X pr
c j

j¼1

N X pr
c j

j¼1

j

2kp2 f2j

j

2kp2 f2j

6 Q; > Q:

Proof. The proof is a standard constrained quadratic problem and thus the proof is omitted. h Part (a) of Proposition 3.1 shows that when there is no risk aversion, i.e., k ¼ 0, single sourcing from the supplier with the highest expected per unit profit r i p
ci is optimal for the manufacturer. However, as part (b) of this proposition shows, once risk aversion is considered, it is optimal for the manufacturer to diversify his order quantities among suppliers in a decreasing order of ri p
ci when Q is large enough. Thus, this proposition suggests that supplier diversification in the presence of linear risk is the result of the interplay between the inherent sourcing risk and the risk attitude of the manufacturer. Our results are complementary to Bernstein et al. (2015), in that diversification cannot mitigate the yield risk for a riskneutral manufacturer in a linear revenue model. However, when the decision maker has a mean–variance preference, a diversification strategy can indeed mitigate the yield risk. pr i
ci As for the order quantities, this proposition shows that the manufacturer should order an unconstrained quantity, 2kp , 2 f2 i

from each supplier when the total order quantities are lower than the capacity constraint Q. However, he should choose a small number of suppliers, i.e., n 6 N, and adjust his order quantities from each supplier by deducing the risk-adjusted   1 Pn ðprj
cj Þ p2 f2 1 quantity, i.e., Pn i 1 2k
Q , if the total unconstrained order quantity exceeds his capacity, i.e., 2 j¼1 p2 f 1 2k

PN j¼1

j¼1 p2 f2 j

ðpr j
cj Þ p2 f2j

j


Q P 0.

We can further see how the number of qualified suppliers and the corresponding order quantities depend both on the value of risk aversion k and the supply risk f2i . From Eq. (6), we know that the left-hand side is decreasing in k and f2i given 



r 2
c2 Þ , we have n ¼ 1. Therefore, single sourcing n. Thus, the optimal n is increasing in k and f2i . Moreover, given kf21 6 pr1
c12p
ðp 2Q

is optimal when the decision maker is less risk averse or supplier 1’s supply variance is small enough. This is consistent with the traditional wisdom as Bernstein et al. (2015). Please cite this article in press as: Xue, W., et al. Diversification strategy with random yield suppliers for a mean–variance risk-sensitive manufacturer. Transport. Res. Part E (2016), http://dx.doi.org/10.1016/j.tre.2016.01.013

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W. Xue et al. / Transportation Research Part E xxx (2016) xxx–xxx

By taking the partial derivative of yi with respect to k, we have

8 >  > >
pr2i
c2 i2 ; > > < 2k p fi

n X pr
c j

if

@yi ¼ 1 @k > f2 > pr i
ci > i > P
þ  2 2 n > 1 : 2k p2 fi j¼1 2

j¼1 1 2k2

n X ðpr
c Þ j¼1

f j

j j p2 f2j

n X pr
c

; if

j

j¼1

Then, define l as the first index i that satisfies Pn1

6 Q;

j

2kp2 f2j

> Q:

j

2kp2 f2j

Pn

1 j¼1 f2 j

ðpr j
cj Þ

j¼1

f2j

P pr i
ci when

Pn

pr j
cj j¼1 2kp2 f2 j

> Q , and l ¼ N when

Pn

pr j
cj j¼1 2kp2 f2 j

6Q

i is positive when i P l. As the manufacturer chooses the suppliers according to their contributed marginal expected then, @y @k profit, when the manufacturer becomes more risk averse, he decreases the order quantities from the suppliers with a higher contributed marginal expected profit and increases the order quantities from the suppliers with a lower contributed marginal expected profit. This shows that the manufacturer will allocate his order quantities more evenly among the suppliers when he is more risk averse, i.e, when k becomes large. To the best of our knowledge, this is a novel finding in the literature on supplier selection with risk consideration. By summarizing the above argument, we have the following result.

Lemma 3.1. (a) Given n , there exists a value of l, such that the order quantity from each supplier i is decreasing in k for i 6 l, and the order quantity is increasing in k for i P l. 



1 Þ
ðpr 2
c2 Þ , single sourcing is (b) The number of effective suppliers is increasing in k and fi ; i 6 n . Moreover, when Q k 6 ðpr1
c2f 2 2 p 1

optimal. 3.2. Newsvendor revenue model We now come back to the general newsvendor model. The newsvendor model is one of the fundamental tools to assist operational decisions under uncertainty (see Porteus, 2002). As shown in the following analysis, this model is too complex to be analytically tractable. Thus, we develop an approximate model using the technique proposed by Federgruen and Yang (2008) to gain some insight on the optimal diversification policy. Moreover, to ease the analysis, we assume that there is no capacity constraint; i.e., the capacity Q ! 1.1 P Let g ¼ Ni¼1 Ri yi be a random variable with a cumulative probability density function WðÞ, and a probability density function wðÞ, both of which are functions of yi ; i ¼ 1; 2; . . . ; N. Then, the inventory level at the end of the selling season, denoted by I, is I ¼ g
D, and the realized profit of the manufacturer, denoted by Pn ðYjR; DÞ, is given by

Pn ðYjR; DÞ ¼ p minfg; Dg


N X

þ

ci yi
hI ¼ pg


i¼1

N X ci yi
ðp þ hÞIþ i¼1

The mean of the random profit PðYjR; DÞ, i.e., E½PðYjR; DÞ, is given by N N X X E½Pn ðYjR; DÞ ¼ p r i yi
ci yi
ðp þ hÞE½Iþ  i¼1

ð7Þ

i¼1

and the variance of the profit, denoted by VAR½PðYjR; DÞ, is given by N

VAR½Pn ðYjR; DÞ ¼ ðp þ hÞ VAR½Iþ  þ p2 m f2i y2i
2pðp þ hÞCovðg; Iþ Þ 2

i¼1

in which

E½Iþ  ¼

Z Z

g

ð8Þ

ðg
DÞf ðDÞwðgÞdDdg

0 þ 2

2

VAR½Iþ  ¼ E½I 
ðE½Iþ Þ Z Z g 2 Z Z g ¼ ðg
DÞ2 f ðDÞwðgÞdDdg
ðg
DÞf ðDÞwðgÞdDdg 0

0

CovðI; gÞ ¼ E½gIþ 
EgEIþ Z Z g Z Z g ¼ g ðg
DÞf ðDÞwðgÞdDdg
g ðg
DÞf ðDÞwðgÞdDdg 0

0

1 Actually, as we show in the following analysis, the utility function is not concave in the order quantities even when Q ! 1. Thus, when Q is finite, the determination of whether the optimal total order quantity is within Q or not is analytically very complicated.

Please cite this article in press as: Xue, W., et al. Diversification strategy with random yield suppliers for a mean–variance risk-sensitive manufacturer. Transport. Res. Part E (2016), http://dx.doi.org/10.1016/j.tre.2016.01.013

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W. Xue et al. / Transportation Research Part E xxx (2016) xxx–xxx

As the distributions WðÞ and wðÞ are functions of decision variables, determining the optimal set of suppliers, the aggregate order, and its allocation among the suppliers is very difficult even when the objective is to maximize the expected revenue. Moreover, as Dada et al. (2007) note, their technique is not applicable in our production model. Thus, to focus on developing managerial insights, in this part, we employ the approximation of the leftover inventory I based on the central limit theorem, which is first proposed by Federgruen and Yang (2008). Specifically, we approximate I by the following normally distributed random variable

eI ¼ y
e in which ye ¼

! rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi XN 2 2 e þ D f y i¼1 i i



PN



i¼1 r i yi ;

 follows a standard normal distribution with a cumulative probability density function UðÞ and the

e follows a normal distribution with mean l and standard variance r, independent of probability density function /ðÞ, and D qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi PN 2 2 PN 2 2ffi e 2 e . Let u ¼  i¼1 fi yi þ D, which follows a normal distribution with mean l and variance i¼1 fi yi þ r , then, I ¼ ye
u. In PN 2 2 the following analysis, we let m ¼ i¼1 fi yi to simplify the notation. e P 0 and Here, we consider the parameters that ensure the demand and the random yields are well defined, that is, D Ri 2 ½0; 1. Actually, we can imagine that such an approximation is equivalent to the scenario in which we assume that each e also follows a normal distrirandom yield Ri is independently distributed under a normal distribution, and the demand D bution. Therefore, given the condition that fi 6 1=6; 1
3fi P r i P 3fi and l P 3r, almost all of the feasible values of those variables are well defined. Given the normal approximation of I, let g 1 ðye ; mÞ; g 2 ðye ; mÞ and g 3 ðye ; mÞ be the approximations of E½Iþ ; E½Iþ  and E½gIþ , respectively. Then, we have. 2

Lemma 3.2. The approximations of g 1 ðye ; mÞ; g 2 ðye ; mÞ and g 3 ðye ; mÞ are given by

  pffiffiffiffiffiffiffiffiffiffiffiffiffiffi   ye
l ye
l þ m þ r2 / pffiffiffiffiffiffiffiffiffiffiffiffiffiffi g 1 ðye ; mÞ ¼ ðye
lÞU pffiffiffiffiffiffiffiffiffiffiffiffiffiffi m þ r2  m þ r2  pffiffiffiffiffiffiffiffiffiffiffiffiffiffi  y
l  y
l e e þ ðye
lÞ m þ r2 / pffiffiffiffiffiffiffiffiffiffiffiffiffiffi g 2 ðye ; mÞ ¼ ððye
lÞ2 þ m þ r2 ÞU pffiffiffiffiffiffiffiffiffiffiffiffiffiffi m þ r2 m þ r2   pffiffiffiffiffiffiffiffiffiffiffiffiffiffi  y
l  ye
l e g 3 ðye ; mÞ ¼ ðye ðye
lÞ þ mÞU pffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ ye r2 þ m/ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2 þ m r2 þ m Proof. From the definition, we have

  ðye
uÞ u
l pffiffiffiffiffiffiffiffiffiffiffiffiffiffi / pffiffiffiffiffiffiffiffiffiffiffiffiffiffi du m þ r2 m þ r2
1     Z ye Z ye ðye
lÞ u
l ðu
lÞ u
l pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi / pffiffiffiffiffiffiffiffiffiffiffiffiffiffi du ¼ / pffiffiffiffiffiffiffiffiffiffiffiffiffiffi du
m þ r2 m þ r2 m þ r2 m þ r2
1
1   pffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 Z ye ðu
lÞ2 ye
l mþr ðu
lÞ
¼ ðye
lÞU pffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffiffiffiffiffi exp 2ðmþr2 Þ du 2 2 mþr 2p
1 m þ r   pffiffiffiffiffiffiffiffiffiffiffiffiffiffi   ye
l ye
l ¼ ðye
lÞU pffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ m þ r2 / pffiffiffiffiffiffiffiffiffiffiffiffiffiffi m þ r2 m þ r2   Z ye ðye
uÞ2 u
l pffiffiffiffiffiffiffiffiffiffiffiffiffiffi / pffiffiffiffiffiffiffiffiffiffiffiffiffiffi du g 2 ðye ; mÞ ¼ m þ r2 m þ r2
1   Z ye 2 ðye
lÞ þ ðu
lÞ2
2ðye
lÞðu
lÞ u
l pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ / pffiffiffiffiffiffiffiffiffiffiffiffiffiffi du m þ r2 m þ r2
1   Z ye   ye
l ðu
lÞ2
2ðye
lÞðu
lÞ u
l pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ðye
lÞ2 U pffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ / pffiffiffiffiffiffiffiffiffiffiffiffiffiffi du m þ r2 m þ r2 m þ r2
1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z y     pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðu
lÞ2 ye
l ye
l m þ r2 e
¼ ðye
lÞ2 U pffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ 2ðye
lÞ m þ r2 / pffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffiffiffiffiffi ðu
lÞdexp 2ðmþr2 Þ 2 2 mþr mþr 2p
1     p ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ye
l ye
l þ ðye
lÞ m þ r2 / pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ððye
lÞ2 þ m þ r2 ÞU pffiffiffiffiffiffiffiffiffiffiffiffiffiffi m þ r2 m þ r2

g 1 ðye ; mÞ ¼

Z

ye

respectively. Please cite this article in press as: Xue, W., et al. Diversification strategy with random yield suppliers for a mean–variance risk-sensitive manufacturer. Transport. Res. Part E (2016), http://dx.doi.org/10.1016/j.tre.2016.01.013

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W. Xue et al. / Transportation Research Part E xxx (2016) xxx–xxx

e then, E½gIþ  can be approximated by Now, we come to approximate E½gIþ . Note that g can be approximated by eI þ D, e eI þ . Moreover, by definition we have E½ðeI þ DÞ

EðeIeI þ Þ ¼

  Z 1   Z ye   2 u
l u
l u
l ðye
uÞðye
uÞdU pffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ ðye
uÞ  0dU pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ðye
uÞðye
uÞdU pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ EðeI þ Þ 2 2 2 mþr mþr mþr ye
1
1

Z

ye

Then,

e eI þ ¼ EðeI þ Þ2 þ Eð D e eI þ Þ EðeI þ DÞ e follows a normal distribution with the probability density function As the random variable (vector) x ¼ ðu; DÞ 0
1 1 1 pffiffiffiffi exp
2ðx
DÞ R ðx
DÞ 2p R

in which D ¼ ðl; lÞ and

"

R¼

#

r2 þ m; r2 ; r2 ; r2

we have

e eI þ  ¼ E½ D

Z Z

ye


1

Z ¼

ye


1

Z ¼

ye


1

Z ¼

r m Z

ðye
uÞ

e

e

2 2 r2 ðu
D Þ þmð D
lÞ

e D pffiffiffi exp 2pr m

Z ðye
uÞ

r

m


2mr2

m

e

2 2 2 þmÞð D
r uþmlÞ r2 þm
u
l
2mr2 2ðr2 þmÞ

ðr e D pffiffiffi exp

2pr

e d Ddu e d Ddu

r2 uþmlÞ Z e pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðr2 þmÞðe D
r2 þm D r2 þ m e
2mr2 pffiffiffiffiffiffiffiffiffi exp d Ddu r 2pm 2

ye


1

Z

! ! e e
l e Dðy D u
D e
uÞ e pffiffiffi / pffiffiffi / dud D

u
l ðye
uÞ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp 2ðr2 þmÞ 2 2pðr þ mÞ

u
l ðye
uÞ r2 u þ ml
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp 2ðr2 þmÞ du r2 þ m 2pðr2 þ mÞ
1  2  Z ye u
l ððye
lÞ
ðu
lÞÞ r ðu
lÞ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp 2ðr2 þmÞ du þ l ¼ 2 þm 2 r 2pðr þ mÞ
1  2  ! Z 2 2 r ðye
lÞ ye
r u
l e ye
D r2 þm ðu
lÞ þ r2 þm
l ðu
lÞ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp 2ðr2 þmÞ du ¼ lðye
lÞU pffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ 2 þ mÞ r2 þ m 2 p ð r
1   pffiffiffiffiffiffiffiffiffiffiffiffiffiffi  y
l  ye
l e 2 ¼ ðlðye
lÞ
r ÞU pffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ l r2 þ m/ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2 þ m r2 þ m

¼

ye

Therefore, EðgIþ Þ can be approximated by g 3 ðye ; mÞ, which is defined as

  pffiffiffiffiffiffiffiffiffiffiffiffiffiffi  y
l  ye
l e þ ye r2 þ m/ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi g 3 ðye ; mÞ ¼ ðye ðye
lÞ þ mÞU pffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2 þ m r2 þ m

This completes the proof. h Based on this result, the approximations of the expected value and the variance of the manufacturer’s profit, denoted by E½PTn ðYjR; DÞ and VAR½PTn ðYjR; DÞ, are given by N N X X E½PTn ðYjR; DÞ ¼ p ri yi
ci yi
ðp þ hÞg 1 ðye ; mÞ i¼1

VAR½P

T n ðYjR; DÞ

i¼1

¼ ðp þ hÞ ðg 2 ðye ; mÞ
g 1 ðye ; mÞ2 Þ þ p2 m
2pðp þ hÞðg 3 ðye ; mÞ
ye g 1 ðye ; mÞÞ 2

and the manufacturer’s approximated ulility U T ðYÞ, is given by

U Tn ðYÞ ¼ E½PTn ðYjR; DÞ
kVAR½PTn ðYjR; DÞ Based on this approximation, we first analyze the properties of the profit variance and the utility function given the order quantities, to show some preliminary results. Then, we analyze the policy for choosing the suppliers and their order quantities using the corresponding algorithms. Please cite this article in press as: Xue, W., et al. Diversification strategy with random yield suppliers for a mean–variance risk-sensitive manufacturer. Transport. Res. Part E (2016), http://dx.doi.org/10.1016/j.tre.2016.01.013

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W. Xue et al. / Transportation Research Part E xxx (2016) xxx–xxx

3.2.1. Analysis under given order quantities Taking the first order derivative of VAR½PT ðYjR; DÞ with respect to

@VAR½P @m

T n ðYjR; DÞ

m and ye , we have

     2 ! ye
l ye
l ye
l ye
l 2 1
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi / pffiffiffiffiffiffiffiffiffiffiffiffiffiffi
/ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ðp þ hÞ U pffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ p2
2pðp þ hÞ r2 þ m r2 þ m r2 þ m r2 þ m    ! ye
l 1 ye
l ye
l p ffiffiffiffiffiffiffiffiffiffiffiffiffiffi p ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
m U / 2 ðm þ r2 Þ32 r2 þ m r2 þ m

   pffiffiffiffiffiffiffiffiffiffiffiffiffiffi     @VAR½PTn ðYjR; DÞ ye
l ye
l ye
l 2 þ m þ r2 / pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1
U pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 2ðp þ hÞ ðye
lÞU pffiffiffiffiffiffiffiffiffiffiffiffiffiffi @ye m þ r2  m þ r2 r2 þ m   m ye
l
2pðp þ hÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi / pffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2 þ m r2 þ m T

T

Given Y, the signs of @VAR½P@nmðYjR;DÞ and @VAR½P@yn ðYjR;DÞ indicate how the supply risk and the expected supply yield influence the e

variance of the manufacturer’s profit by the definition of m and ye . Fig. 1 shows that the supply risk and the expected yield can either increase or decrease the variance of the manufacturer’s profit for given order quantities, which implies that the signs may be positive or negative. The reason for the unclear overall effect is that, given the order quantities, an increase in r i and the yield variance fi could also increase the correlation between I and g. As for the manufacturer’s utility, we have

@U Tn ðYÞ @g ðy ; mÞ @VAR½PTn ðYjR; DÞ ¼
ðp þ hÞ 1 e
k @m @m @m @U Tn ðYÞ @g 1 ðye ; mÞ @VAR½PTn ðYjR; DÞ ¼ p
ðp þ hÞ
k @ye @ye @ye ðye ;mÞ in which p
ðp þ hÞ @g1@ðyme ;mÞ and p
ðp þ hÞ @g1@y are the first order derivatives of E½PTn ðYjR; DÞ with respect to

  @g 1 ðye ; mÞ 1 1 ye
l ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi / pffiffiffiffiffiffiffiffiffiffiffiffiffiffi @m 2 m þ r2 r2 þ m   @g 1 ðye ; mÞ ye
l ¼ U pffiffiffiffiffiffiffiffiffiffiffiffiffiffi @ye r2 þ m

Thus, from the expressions of

@g 1 ðye ;mÞ @m

and

e

@g 1 ðye ;mÞ , @ye

m and ye , and

we find that the expected profit of the manufacturer (i.e., k ¼ 0) is always

decreasing in fi . However, it is not always increasing in r i when h > 0. This is summarized in Lemma 3.3. Lemma 3.3. Given the order quantities Y and letting k ¼ 0, we have (a) The utility function U n ðYÞ is decreasing in f2i for each i. (b) The utility function U n ðYÞ is increasing in ri for each i when the holding cost h ¼ 0.

4

9.5

5

x 10

1.6

x 10

1.5

Profit Variance

Profit Variance

1.4 9

8.5

1.3 1.2 1.1 1 0.9 0.8

8

0

0.05

0.1

ζ

0.15

0.2

0.7 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

The expected yield

Fig. 1. Illustration of the effects f and r on the profit variance (instances: p ¼ 15; h ¼ 0; c ¼ 8; n ¼ 1; r ¼ 0:7; f ¼ 0:13; l ¼ 100; r ¼ 20; y ¼ 200).

Please cite this article in press as: Xue, W., et al. Diversification strategy with random yield suppliers for a mean–variance risk-sensitive manufacturer. Transport. Res. Part E (2016), http://dx.doi.org/10.1016/j.tre.2016.01.013

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W. Xue et al. / Transportation Research Part E xxx (2016) xxx–xxx

Federgruen and Yang (2009) also characterize part (a) of Lemma 3.3, and show that the results hold under the condition where ye þ I0 P l (I0 is the initial inventory, which is zero in our paper). Here, we extend this result to all possible ye . Under a general k > 0, as the signs of

@VAR½PTn ðYjR;DÞ @m

and

@VAR½PTn ðYjR;DÞ @ye

may be positive or negative, the expected utility of the

manufacturer is not always increasing or decreasing in the yield variance fi and the expected yield r i ; i.e., the supply risk and the expected yield do not always hurt or benefit the manufacturer, given his order quantity Y. When the order quantity and the demand variance go to infinity, we can still show some monotone properties of the profit variance for a general k, which are characterized by the following lemmas. Lemma 3.4. When yi ! 1, we have (a)

@VAR½PTn ðYjR;DÞ @m

P 0 and

@VAR½PTn ðYjR;DÞ @m

is constant; (b)

@VAR½PTn ðYjR;DÞ @yi

P 0.



Proof. Let ti ¼ fri . For part (a), given yi ! 1, we have i

@VAR½PTn ðYjR; DÞ 1 2 ¼ ðp þ hÞ ðUðti Þð1
ti /ðt i ÞÞ
/ðt i Þ2 Þ þ p2
2pðp þ hÞðUðt i Þ
t i /ðt i ÞÞ @m 2 We only need to show that the above equation has no positive roots. That is, we need to show that

  2 1 1 2 t i þ 1 /ðt i Þ2 6 0 ðUðt i Þ
ti /ðti ÞÞ
ðUðti Þð1
ti /ðti ÞÞ
/ðt i Þ2 Þ ¼ Uðt i Þ2
Uðt i Þ þ 2 4 Let wðt i Þ ¼ Uðt i Þ2
Uðt i Þ þ

1

t2 4 i

 þ 1 /ðti Þ2 , then we have

  @wðt i Þ 1 3 3 t i þ t i /ðt i ÞÞ ¼ /ðt i Þð2Uðti Þ
1
@t i 2 2   @2Uðt i Þ
1
12 t 3i þ 32 t i /ðt i Þ 3 3 ¼ 2/ðt i Þ þ ðt4i þ t2i
Þ/ðti Þ 2 2 @t i

Thus,

@ 2 wðt i Þ @t 2i

> 0 and

@wðt i Þ @t i

iÞ P @wðt jti ¼0 ¼ 0, which is equivalent to the case where wðti Þ is increasing in t i . As wðt i Þjti !1 ¼ 0, we @t i

have wðti Þ 6 0 for all possible t i P 0. For part (b), when yi ! 1, we have

@VAR½PTn ðYjR; DÞ 2 ¼ 2f2i yi ½ðp þ hÞ ðUðt i Þð1
t i /ðt i ÞÞ
/ðti Þ2 þ t i ðt i Uðti Þ þ /ðt i ÞÞð1
Uðt i ÞÞÞ þ p2
2pðp þ hÞUðt i Þ @yi We only need to show that the above equation has no positive roots. That is, we need to show that

w2 ðti Þ ¼ Uðt i Þ2
Uðt i Þ þ

/ðt i Þ2 t i ð2Uðt i Þ
1Þ/ðti ÞÞ þ 60 1 þ t2i 1 þ t 2i

Taking the first-order derivative, we have

@w2 ðti Þ ¼ /ðt i Þð1 þ t i Þ4 ð2Uðti Þ
1
ti /ðti ÞÞ @ti which is positive when ti P 0. Thus, w2 ðti Þ is increasing in t i , and as w2 ðti Þjti !1 ¼ 0, we have w2 ðt i Þ 6 0 for t i P 0. This completes the proof. h This result shows that when the order quantities are very large, the profit variance is increasing in the yield uncertainty fi ; i ¼ 1; 2; . . . ; n. However, the effect of the yield rate on the profit variance is still vague; i.e., the profit variance may still be increasing or decreasing in the yield rate ri . However, the overall effect is that the profit variance is increasing in the order quantities. @VAR½PT ðYjR;DÞ

n Lemma 3.5. When r ! 1, we have (a) @m @VAR½PTn ðYjR;DÞ is unbounded; (b) P 0. @y

@VAR½PTn ðYjR;DÞ @ye

P 0 and

@VAR½PTn ðYjR;DÞ @ye

P 0; moreover,

@VAR½PTn ðYjR;DÞ @m

is constant while

i

Proof. For part (a), given

@VAR½P @m

T n ðYjR; DÞ

r ! 1, we have 

¼ ðp þ hÞ

2

 1 1 1 þ p2
2pðp þ hÞ
2 2p 2

@VAR½PTn ðYjR; DÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ffi 2 1 ¼ v þ r ðp þ hÞ pffiffiffiffiffiffiffi @ye 2p T

The right-hand side of the above formulations are all positive, and thus @VAR½P@nmðYjR;DÞ P 0. Part (b) is straightforward following part (a). This completes the proof. h Please cite this article in press as: Xue, W., et al. Diversification strategy with random yield suppliers for a mean–variance risk-sensitive manufacturer. Transport. Res. Part E (2016), http://dx.doi.org/10.1016/j.tre.2016.01.013

11

W. Xue et al. / Transportation Research Part E xxx (2016) xxx–xxx

From this result, we can see that there is a threshold rH such that given the order quantities Y, when r P rH , the profit variance is increasing in both the yield uncertainty fi ; i ¼ 1; 2; . . . ; n and the yield rate ri , the overall effect is that the profit variance is increasing in the order quantities. Moreover, by the formulation of the utility, we can see that when r > rH , the manufacturer’s utility is decreasing in fi and r i , given the order quantities Y. 3.2.2. Analysis of the diversification strategy Next, we analyze the optimal suppliers’ diversification strategies with the corresponding order quantities. When there is no supply risk, Choi et al. (2008a) show that the variance of the profit is increasing in the order quantity. Moreover, they show that the optimal order quantity is less than the risk neutral order quantity. However, when the random supply risk is incorporated, Fig. 2(a) shows that the variance of profit is no longer monotone in the order quantity (even when holding/salvage cost is zero). As a result, Fig. 2(b) shows that the optimal order quantity may be greater than that under the risk-neutral setting. Thus, the mean–variance analysis under supply risk is more complex than that without supply risk. Nevertheless, we can still establish some relationships between the optimal order quantities for k > 0. Moreover, as we have shown that the profit variance is increasing in the order quantities when the demand variance is large enough (i.e., when r > rH ), we can develop the diversification strategy in this case to provide some managerial insights on the optimal portfolio with the corresponding order quantities. Define a supplier to be active if it is optimal for the newsvendor to place a positive order with that supplier. Then, the active set A is defined as follows. Definition 3.1. The active set A is the set of active suppliers, i.e., the optimal order quantities for those suppliers in A are greater than zero. Define

m1 ðye ; mÞ ¼ ðp þ hÞ

@g 1 ðye ; mÞ @VAR½PTn ðYjR; DÞ þk @ye @ye

ð9Þ

m2 ðye ; mÞ ¼ ðp þ hÞ

@g 1 ðye ; mÞ @VAR½PTn ðYjR; DÞ þk @m @m

ð10Þ

The following result characterizes the properties of the general expression for the manufacturer’s optimal order quantity from each supplier, given the active set A. Proposition 3.2. The optimal order quantity for supplier i 2 A in the approximated model, denoted by yi , is determined by

k2 f2i yi ¼ r i

  pr i
ci
k1 r i

ð11Þ



in which ðk1 ; k2 Þ 2 fk1 6 prir
ci and k2 P 0g i

k1 ¼ k2 ¼

S  fk1 P prir
ci and k2 6 0g are determined by i

m

m1 ðY IE ðk1 ; k2 Þ; I ðk1 ; k2 ÞÞ 2m2 ðY IE ðk1 ; k2 Þ; I ðk1 ; k2 ÞÞ

m

5

x 10

4

300

Profit Variance

4.5 200

4

h=0

3.5

h=5

100

3 2.5

0 Expected Utility

2

Expected Profit

−100

1.5 1

−200

0.5 0

0

20

40

60

80

100

120

Value of y

140

160

180

200

−300 0

20

40

60

80

100

120

140

160

180

200

Value of y

Fig. 2. Illustration of the effect of y (instances: p ¼ 15; h ¼ 0; c ¼ 8; n ¼ 1; k ¼ 0:0044; r ¼ 0:7; f ¼ 0:13; l ¼ 100; r ¼ 10).

Please cite this article in press as: Xue, W., et al. Diversification strategy with random yield suppliers for a mean–variance risk-sensitive manufacturer. Transport. Res. Part E (2016), http://dx.doi.org/10.1016/j.tre.2016.01.013

12

W. Xue et al. / Transportation Research Part E xxx (2016) xxx–xxx

P

where Y IE ðk1 ; k2 Þ ¼

  i2I r i yi

mI ðk1 ; k2 Þ ¼

and

P

2 2 i2I fi yi .

Proof. Given any supplier i 2 A, we have the following first order condition

@E½PTn ðYjR; DÞ ¼ pr i
ci
r i m1 ðye ; mÞ
2f2i yi m2 ðye ; mÞ ¼ 0 @yi Then, let k1 ¼ m1 ðye ; mÞ and k2 ¼ m2 ðye ; mÞ, we can have the desired result. h Although the values of k1 and k2 are difficult to obtain, this result provides an important tool to allocate the order quantities among those active suppliers when the active set A is given. Specifically, given the order quantities for any two active suppliers i and j, denoted by yi and yj , the other active supplier k’s order quantity, yk , satisfies 

i yj prri
c
yi f2 i i

pr j
cj r j f2 j

yj f12
yi f12 i





i k yk prri
c
yi prrk
c f2 f2

¼

i i

ð12Þ

k k

yk f12
yi f12

j

i

k

This result also gives a framework for providing some insights into the properties of those active suppliers. This is characterized by the following result. Lemma 3.6.

c

(a) If k2 – 0, for any suppliers i; j 2 A such that cr i ¼ r j , and they are unreliable in the sense that fi > 0 and fj > 0, then we have i

f2i r i

yi

¼

f2j r j

j

yj .

(b) If k2 – 0 and supplier i 2 A is reliable, i.e., fi ¼ 0, then any other reliable supplier j 2 I satisfies (c) If k2 – 0 and supplier i 2 A is reliable, then any unreliable suppler j 2 A satisfies (d) If there is a reliable supplier i 2 A, then, any other supplier j with

ci r i

>

cj r j

cj r j

– crii .

cj r j

– cr i .

cj r j

¼ crii .

is active.

Proof. (a) From Eq. (11), we have

f2i yi f2j yj

r i



pr i
ci r i


k1



 ¼ pr
c rj jr j
k1 j

Thus, when

ci r i

c

¼ r j , we have j

(b) If fi ¼ fj ¼ 0, we have

pr i
ci r i

f2i r i

yi ¼

f2j r j

yj .

pr j
cj r j .

¼ k1 ¼

That is,

cj r j

¼ crii .

(c) If fi ¼ 0, we have

pr i
ci r i

¼ k1 . Then, if fj – 0, we have

(d) If fi ¼ 0, we have

pr i
ci r i

¼ k1 , thus,

pr i
ci r i



– k1 ¼ prir
ci . That is, i

i

@E½PTn ðYjR; DÞ jyj !0 ¼ pr j
cj
rj k1 > 0 @yj which implies that the optimal order quantity for supplier j will be greater than zero. h Note that cr i ; i 2 A, denotes the effective per unit cost of supplier i. Lemma 3.6 has some implications for the active set i

when k2 – 0. Let K 2 A be the set of those suppliers with the same effective per unit cost, i.e., crii ¼ m; i 2 K; then, by defining P r2 P r2 a virtual supplier e with re ¼ f2e ¼ i2K fi2 and ce ¼ m i2K fi2 , part (a) shows that we can treat those unreliable suppliers as a i

i



supplier e with an order quantity ye by Eq. (11), and the corresponding order quantity for each supplier i 2 K is fr2i ye . Part (b) i

shows that the reliable suppliers in the active set A, if they exist, have the same value of effective per unit cost. Part (c) implies that if there is a reliable supplier i in A, there is no active unreliable supplier j with the same effective per unit cost as supplier i. Part (d) shows that if there is a reliable supplier in the active set and there are other suppliers with lower effective unit costs, then the active sets are composed of all those suppliers. We now establish the optimal diversification strategy when r is large, i.e., r > rH , as follows.

Please cite this article in press as: Xue, W., et al. Diversification strategy with random yield suppliers for a mean–variance risk-sensitive manufacturer. Transport. Res. Part E (2016), http://dx.doi.org/10.1016/j.tre.2016.01.013

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W. Xue et al. / Transportation Research Part E xxx (2016) xxx–xxx

Lemma 3.7. 2 

r > rH , for any unreliable supplier i; j 2 A such that crii 6 crjj , we have firyi i P c (b) If r > rH , for any supplier i 2 A, any other supplier j with cr i > r j is active. i j (a) If

f2j yj r j

.

Proof. When r > rH , we know that @VAR½P@nmðYjR;DÞ P 0 (from Lemma 3.5). Thus, k2 P 0, which leads to result (a). Moreover, if supplier i 2 A, we have T

pr i
ci
r i k1
2f2i yi k2 ¼ 0 which leads to pr i
ci
ri k1 P 0. For any other supplier j, the first-order derivative of the utility with respect to yj , when yj ! 0, is pr j
cj
r j k1 . Therefore, this derivative is greater than zero when

ci r i

c

> rjj . That is, supplier j is also active. h

The proof of this result depends on the fact that k2 P 0 when the demand uncertainty is high, i.e., r > rH . Lemma 3.7 shows that in this case, the active suppliers are always selected in an order of their effective per unit costs. Thus, the traditional supplier selection rule by the ascending order of the effective unit costs, i.e., cr i , under the risk-neutral setting is still i

optimal under the mean–variance framework for large demand uncertainties. When r is small (and k > 0), we conduct extensive numerical studies and show that k2  0 for almost all settings. Fig. 3 illustrates 100 instances among the extensive numerical studies for N ¼ 1, with the price p randomly chosen from ½0; 50, the holding cost h randomly chosen from ð
p; 50, the unit cost c randomly chosen from ½0; p, the risk-aversion coefficient k randomly chosen from ½0; 1, the expected supply yield r randomly chosen from ½pc ; 1, the standard variance of supply yield

f randomly chosen from ½0; minf1
3 r ; 3r g, the mean demand l randomly chosen from ½0; 400, and the demand variance r randomly chosen from ½0; l3 . For a general N, the value of k2 is determined by the first-order condition of each supplier i as

pr i
ci
r i k1
2f2i yi k2 ¼ 0 which is equivalent to the case when N ¼ 1 from supplier i with the adjusted demand mean and variance. Thus, we conjecture that k2 P 0 even when r is small. However, the proof for analytically deriving this conjecture is rather challenging and is left for future investigation. As the objective function UðYÞ is not concave in Y, we also establish the following result on the bounds of the optimal order quantities. , in which Lemma 3.8. For each supplier i, the optimal order quantity yi 6 y

(  ¼ arg max yjpr min y
cy
ðp þ hÞg 1 y

f2 r min y; min y2 2

!

) ¼0

with cmin ¼ minfc1 ; . . . ; cN g rmin ¼ minfr1 ; . . . ; r N g; fmin ¼ minff1 ; . . . ; fN g.

50 45 40

Value of k 2

35 30 25 20 15 10 5 0 0

10

20

30

40

50

60

70

80

90

100

Instances Fig. 3. Illustration of the value of k2 when N ¼ 1. Parameter values are generated from uniform distributions, p 2 ½0; 50; h 2 ð
p; 50; c 2 ½0; p; k 2 ½0; 1; r 2 ½pc ; 1; f 2 ½0; l3; l 2 ½0; 400; r 2 ½0; l3.

Please cite this article in press as: Xue, W., et al. Diversification strategy with random yield suppliers for a mean–variance risk-sensitive manufacturer. Transport. Res. Part E (2016), http://dx.doi.org/10.1016/j.tre.2016.01.013

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W. Xue et al. / Transportation Research Part E xxx (2016) xxx–xxx

Proof. According to Lemma 3.3, E½PT ðYjR; DÞ is decreasing in fi and r i , thus, we have N N N N X X X X E½P ðYjR; DÞ 6 pr min yi
ci yi
ðp þ hÞg 1 r min yi ; f2min y2i

!

T

i¼1

i¼1

i¼1

i¼1

N N N N X X X X 6 pr min yi
cmin yi
ðp þ hÞg 1 r min yi ; f2min y2i i¼1

i¼1

0

i¼1

!

i¼1

N N N N X X X X f2 6 pr min yi
cmin yi
ðp þ hÞg 1 @r min yi ; min yi 2 i¼1 i¼1 i¼1 i¼1

PN in which the last inequality holds as

y i¼1 i 2

2 6

!2 1 A

PN

2 i¼1 yi .

P T , we have N  Thus, for each Y, whenever yi P y i¼1 yi P y; therefore, we have E½P ðYjR; DÞ 6 0, and the utility function T  UðyÞ 6 E½P ðYjR; DÞ 6 0. This implies that yi P y cannot be a candidate optimal solution as UðYÞjY¼0 ¼ 0. This completes the proof. h  is easy to calculate as the function in the brackets is concave. Then, by combining Eq. (12) with the The upper bound y selection rule proposed by Lemma 3.7 and the conjecture, the bounds on the order quantities proposed by Lemma 3.8, we can modify the algorithm proposed by Dada et al. (2007), as in Algorithm 1, for the mean–variance model. Algorithm 1. Finding the available suppliers with corresponding order quantities.  defined in Lemma 3.8. Step 1 Sort the suppliers in an increasing order of crii (i 2 ½1; N). Find the y   ½0; y  with a step size T. Initialize Y ¼ ½yð1Þ; . . . ; yðNÞ ¼ 0. Step 2 For each ½y1 ; y2  2 ½0; y Let yð1Þ ¼ y1 ; yð2Þ ¼ y2 . For j ¼ 3; . . . ; N, determine the value of yðjÞ by Eq. (12) (yðjÞ ¼ 0 if it produces a negative result). Evaluate the utility function UðYÞ. End Step 3 Output the optimal Y  among all Ys with the maximal utility UðYÞ.

4. Numerical study In this section, we conduct some numerical studies to derive more insights into how the random yield and the mean–variance ratio k affect the manufacturer’s optimal order quantity, the service level, and the manufacturer’s utility. We define the service level as the probability in stock. To focus on the main insight, all results here are established for a single supplier, i.e., N ¼ 1, as this is also equivalent to the case with multiple identical suppliers. The system parameters are set as in Table 1. Fig. 4 illustrates how the risk-aversion coefficient influences the optimal order quantity and the service level. We can see that the optimal order quantity is first increasing, then suddenly decreasing in the risk-aversion coefficient. The underlying reason is the tradeoff between the expected profit and the profit variance when the manufacturer decides his order quantities. Moreover, in Fig. 2, we can see that the profit variance is decreasing in y when y is large enough. Therefore, when k is small, he focuses more on the expected profit than the profit variance, and sets a higher order quantity. As the variance is decreasing in the order quantity when y is large, the supplier will choose to order more as k becomes large. Conversely, when k is large enough, the supplier focuses more on the profit variance and sets a lower order quantity, and as the variance is increasing in the order quantity when y is small, the supplier will choose to order less when k becomes large. The service level can also illustrate this phenomenon. As shown in Fig. 4, when k becomes large, the service level first increases to almost 80%, then, as the supplier only focuses on his profit variance, the service level suddenly drops to almost zero. This also implies that, when considering supply risk, the risk-averse manufacturer will increase or decrease his service level, depending on the level of his risk aversion. Fig. 5 illustrates how the yield rate r and the yield variance f affect the optimal order quantity and the service level. As we can see, the order quantity is first increasing and then decreasing in r due to the tradeoff between the marginal utility brought by the increase in the expected order quantity and the marginal disutility brought by the increase in the supply risk.

Table 1 Parameter values in the base-case scenario. p

c

r

f

r

l

k

h

15

5

0.7

0.13

10

100

0.0044

0

Please cite this article in press as: Xue, W., et al. Diversification strategy with random yield suppliers for a mean–variance risk-sensitive manufacturer. Transport. Res. Part E (2016), http://dx.doi.org/10.1016/j.tre.2016.01.013

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W. Xue et al. / Transportation Research Part E xxx (2016) xxx–xxx 200

0.8

Order quantity Sevice level

0.6

100

0.4

50

0.2

0 0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

Service level

Order quantity

150

0 0.1

Valueof λ Fig. 4. The effect of the risk-aversion coefficient k on the order quantity and service level. (Instances: p ¼ 15; h ¼ 0; c ¼ 5; n ¼ 1; k ¼ 0:0044; r ¼ 0:7; f ¼ 0:13; l ¼ 100; r ¼ 10).

0.8

200

Order quantity Expected received quantity Service level

100

0.1

Order quantity for λ=0.04 Order quantity for λ=0.0044 Service level for λ=0.04

50

0.05

Service level

0.4

100

0.2

50

0 0.35

0.15

0.6

Service level Order quantity

Order quantity

150

150

0.4

0.45

0.5

0.55

0.6

Value of expected yield

0.65

0 0.7

0

0

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

Value of ζ

Fig. 5. The effect of the supply risk on the order quantity and service level. (Instances: p ¼ 15; h ¼ 0; c ¼ 5; n ¼ 1; k ¼ 0:0044; r ¼ 0:7; f ¼ 0:13; l ¼ 100; r ¼ 10).

That is, when r is low (large enough), the marginal utility brought by the additional order quantity is larger (lower) than the disutility it brings. We can also see that, consistent with intuition, the expected order quantity received by the manufacturer is always increasing in r, and the service level approaches zero when r is small and increases as r becomes large. As to the effect of the random yield f, extensive numerical studies show that the order quantity and the service level are not always monotone with f. Fig. 5 shows that when k ¼ 0:0044, the order quantity is always increasing in f, yet when k ¼ 0:04, the order quantity is first increasing and then decreasing in f. The reason lies in the non-monotone property of the variance with respect to f and the order quantity y; thus, it is possible for the manufacturer to order more when f becomes large. However, the overall effect is not clear. The service level, which depends on both the order quantity and the supply risk, is also unclear. For example, when k ¼ 0:04, Fig. 5 shows that the service level first increases to about 12% then suddenly drops to almost zero. Fig. 6 illustrates how the demand mean l and demand variance r affect the order quantity and the service level. As expected, the order quantity and the service level are increasing in the demand mean l and decreasing in the demand variance r because, when l becomes large (or r becomes small), the demand risk measured by the coefficient of variation (cv) is decreasing, thus, the manufacturer can order more to satisfy more demand by bearing some additional supply risk and the corresponding service level is also high. Fig. 7 illustrates how the supply risk parameters r and f, affect the manufacturer’s utility. Comparing it to Fig. 1, which depicts the effect of r and f given the order quantity Y, we find that when the manufacturer can choose the optimal order quantities, his utility is increasing in r and decreasing in f. This observation is intuitive in the sense that a high expected yield and a low supply risk always benefit the manufacturer.

Please cite this article in press as: Xue, W., et al. Diversification strategy with random yield suppliers for a mean–variance risk-sensitive manufacturer. Transport. Res. Part E (2016), http://dx.doi.org/10.1016/j.tre.2016.01.013

W. Xue et al. / Transportation Research Part E xxx (2016) xxx–xxx 1

400

0 20

40

60

80

100

120

140

160

180

Order quantity

0.5

200

Service level

Order quanity

Order quantity Service level

0 200

160

0.8

150

0.7

140

0.6 0.5

130 Order quantity Service level

120 110

0.3

100

0.2

90

0

5

risk

on

the

order

quantity

and

400

20

25

0.1 30

service

level.

(Instances:

0.02

0.04

p ¼ 15; h ¼ 0; c ¼ 5; n ¼ 1; k ¼ 0:0044;

700

300

680

Expected utility

Expected utility

15

720

350

250 200 150 100

660 640 620 600

50 0 0.35

10

Value of σ

Value of μ

Fig. 6. The effect of the demand r ¼ 0:7; f ¼ 0:13; l ¼ 100; r ¼ 10).

0.4

Service level

16

580 0.4

0.45

0.5

0.55

0.6

Value of expected yield

0.65

0.7

560

0

0.06

0.08

0.1

Value of ζ

Fig. 7. The effect of the supply risk on the manufacturer’s utility. (Instances: p ¼ 15; h ¼ 0; c ¼ 5; n ¼ 1; k ¼ 0:0044; r ¼ 0:7; f ¼ 0:13; l ¼ 100; r ¼ 10).

5. Conclusion As supply chains operate worldwide, firms face uncertainties from both the demand side and the supply side. Natural disasters such as hurricanes, earthquakes, tsunamis, machine malfunctions, and damages during transportation can all cause supply uncertainties. It is a natural strategy for a buyer to procure from multiple suppliers with supply uncertainty. For a risk-neutral buyer, Bernstein et al. (2015) show that the diversification strategy cannot mitigate the supply risk in the linear revenue model. Yet, for a risk-averse buyer, the diversification strategy indeed can mitigate the supply risk. In this paper, we consider a monopoly manufacturer with a mean–variance risk measure that buys identical (highly substitute) components from a portfolio of suppliers and assembles the components, then sells the final product to the market. Suppliers have different levels of reliability and higher reliability requires a higher cost of procurement. The manufacturer makes diversification and ordering decisions to balance the risk of uncertain supply, and the associated profit generated from sales. We first analyze the linear revenue model and characterize the diversification strategy, which shows that the optimal order quantity with mean–variance consideration may be greater than that in the risk-neutral setting and the manufacturer will allocate order quantities more evenly among the suppliers as he is more risk averse. For the general newsvendor model, we establish its approximation using the technique in Federgruen and Yang (2008). This approximation model allows us to obtain some properties of the profit variance, the utility, and the active supplier set. Moreover, we show that it is optimal to choose the suppliers according to the ascending order of their effective per unit costs when the demand uncertainty is large. Based on our extensive numerical studies, we further conjecture that this selection rule also applies when the demand uncertainty is low. The solution algorithm is developed, and finally, we perform some comparative statics on the optimal solution, with a particular emphasis on investigating the effect of the suppliers’ cost and reliability on the risk-averse manufacturer’s ordering decisions and customer service level.

Please cite this article in press as: Xue, W., et al. Diversification strategy with random yield suppliers for a mean–variance risk-sensitive manufacturer. Transport. Res. Part E (2016), http://dx.doi.org/10.1016/j.tre.2016.01.013

W. Xue et al. / Transportation Research Part E xxx (2016) xxx–xxx

17

Our models are not without limitations. First, we only study the linear revenue model and the normal approximation of the general model. A future research direction is to extend the current analysis to a general parameter setting. Second, even for the approximated newsvendor model, the objective function is no longer concave in the decision variables. Thus, a natural extension is to propose a novel algorithm to determine the good suppliers and their corresponding order quantities. Although we have shown that the selection rule is optimal when the demand uncertainty is high, we still need to prove whether the conjecture (which is based on an extensive numerical study in this paper) always holds in our future work. Eventually, we can consider other risk measures such as the utility function, and the Conditional Value-at-Risk (CVaR), to study the supplier selection problem with supply uncertainties. 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Please cite this article in press as: Xue, W., et al. Diversification strategy with random yield suppliers for a mean–variance risk-sensitive manufacturer. Transport. Res. Part E (2016), http://dx.doi.org/10.1016/j.tre.2016.01.013