Supply chain coordination with risk-averse retailer and option contract: Supplier-led vs. Retailer-led

Journal Pre-proof Supply Chain Coordination with Risk-averse Retailer and Option Contract: Supplier-led vs. Retailer-led

Zhongyi Liu, Shengya Hua, Xin Zhai PII:

S0925-5273(19)30339-1

DOI:

https://doi.org/10.1016/j.ijpe.2019.107518

Reference:

PROECO 107518

To appear in:

International Journal of Production Economics

Received Date:

20 February 2019

Accepted Date:

08 October 2019

Please cite this article as: Zhongyi Liu, Shengya Hua, Xin Zhai, Supply Chain Coordination with Risk-averse Retailer and Option Contract: Supplier-led vs. Retailer-led, International Journal of Production Economics (2019), https://doi.org/10.1016/j.ijpe.2019.107518

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Journal Pre-proof

Supply Chain Coordination with Risk-averse Retailer and Option Contract: Supplier-led vs. Retailer-led Zhongyi Liu School of Management People’s Public Security University of China [email protected] Shengya Hua School of Economics and Management South China Normal University [email protected] Xin Zhai Guanghua School of Management Peking University [email protected]

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Journal Pre-proof

Supply Chain Coordination with Risk-averse Retailer and Option Contract: Supplier-led vs. Retailer-led Abstract We investigate the coordination of both the supplier-led and the retailer-led supply chains under option contract. Specifically, we consider the option pricing, ordering, and producing problems in a supply chain consisting of a risk-neutral supplier and a riskaverse retailer behaving under the conditional value-at-risk (CVaR) criteria. The results show that, in a supplier-led supply chain, the supplier’s production quantity equals the retailer’s order quantity only if the penalty cost is high enough. In a retailer-led supply chain, the retailer will set the option price as low as possible in order to shift more risks to the supplier. By comparing results under both supply chain structures, we find that the retailer purchases more options at a lower price in the retailer-led supply chain, but the supplier’s production quantities remain the same under both supply chain structures. Finally, we prove that both the supplier-led and the retailer-led supply chains can be coordinated under the same conditions, which is different from the conclusions obtained by previous studies. Key words: option contract; risk aversion; supply chain coordination, supply chain structure

1. Introduction Due to the long lead time and high uncertainty in consumer demand, the mismatch between supply and demand is one of the top concerns in many industries (Fisher, 1997; Hendricks and Singhal, 2014). For example, according to a 2010 Accenture report* on the energy sector, ethanol and biodiesel production in the European Union was highly inconsistent with consumption. The European Union produced four billion litres of ethanol and 11 billion litres of biodiesel in 2010; yet, there was still a considerably higher market demand of around five billion liters of the former and 13 billion litres of the latter. Similar disparities can be found in industries such as apparel, electricity, semiconductors etc. Hendricks and Singhal (2014) summarized three sources of

*Blending Biofuels in the European Union, Accenture Report, 2010.

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Journal Pre-proof demand-supply mismatch announced by publicly traded firms, including production disruption, excess inventory, and product introduction delay, all of which result in increased volatility in equity valuations. Over a two-year period around the announcement date, a firm’s mean abnormal equity volatility increases 5.62% for production disruption, 11.19% for excess inventory, and 6.28% for product introduction delay. To reconcile this mismatch between supply and demand, firms have increasingly sought ways to improve the flexibility of their supply policies. One strategy widely used in industries is the option contract. Options were originally used in the financial market to specify the terms for a future transaction on an asset between two parties. It was later introduced into the supply chain area as a means of hedging risks induced by uncertain demand. For example, Hewlett-Packard invested 35% of its procurement cost in option contracts and designed a customized option contract for the purchasing of memory chips (Martinez-de-Albeniz, 2005; Fu et al., 2010); Suning Commerce Group, one of the largest retailers in China, manages its inventory by using option contracts in order to avoid overstock (Wang and Liu, 2007). Under an option contract, at the cost of an option fee, the buyer gains the right but not the obligation to engage in some future transaction, while the seller gains the fee and incurs a corresponding obligation to fulfill the transaction. The option contract helps retailers ensure supply at a specified price to meet uncertain future demand when the future supply and/or price are uncertain, and provides flexibility in the quantity of products to be purchased after further demand information is available. In academic research, option contracts have been shown to enhance flexibility and be able to coordinate the decentralized supply chain in the risk-neutral setting (Zhao et al., 2010). However, it is well-known that decision makers may have preferences (or decision bias) beyond profit maximization (Schweitzer and Cachon, 2000). A survey conducted by McKinsey Company on 1500 executives from 90 countries indicates that those executives demonstrate extreme levels of risk aversion regardless of the investment size (Koller et al., 2012). Thus, the agents in supply chains are not necessarily risk neutral. Various risk measurements have been introduced in supply chain management to depict the decision behaviour of decision makers that are risk averse, such as downside risk (Gan et al., 2005), value at risk (Hsieh and Lu, 2010), loss aversion (Liu et al., 2014), and mean variance (Kouvelis et al., 2015). Although the wholesale price, buyback, and revenue sharing contracts have all been explored in 2

Journal Pre-proof the risk-averse setting, option contract, however, are rarely studied in such a setting. Research is needed to establish whether or not an option contract can improve supply chain performance under the risk-averse setting. In this research, we explore the interaction between a risk-neutral supplier and a risk-averse retailer in the context of a two-echelon supply chain. Most of the previous researches focus on the supplier-led supply chain in which supplier plays the leader role and has the right to design the contract. However, the downstream retailer may have a large market share and behaves as the core company in a supply chain, such as Amazon, Apple, JD, and Wal-Mart. In this scenario, the retailer gains more power in designing the contract. For example, Apple forced its suppliers to reduce fees for iPhone 7 by 10.0%-20.0% in 2017 (Tanner, 2017); Wal-Mart requires its suppliers to reduce the costs they charged by 15% in a summit so that it can regain its spot as the low-price leader (Sit, 2017). In this paper we make a deep probing into the coordination of supplier-led and retailer-led supply chain structures in a risk-averse setting. More precisely, we address three questions: (1) what are the supplier and retailer’s optimal decisions in both supply chain structures? (2) can both the supplier-led and the retailerled supply chains be coordinated in our settings? (3) which of the supply chain structures performs better? Our work makes three contributions as follows: (1) We explore the risk-neutral supplier’s and risk-averse retailer’s decisions under both supplier-led and retailer-led supply chains; (2) We derive the conditions under which supplier-led and retailer-led supply chain coordination is achieved by an option contract; (3) we make a comparison between the two supply chain structures from the perspective of channel performance and coordination conditions. Previous studies show that the retailer-led supply chain has a better performance than the supplier-led supply chain under wholesale price contract (Cachon 2004, Davis et al. 2014, Yang et al. 2018). In contrast, our results show that the channel performance (or the supplier’s production quantity) are the same under both supply chain structures, and the retailer purchases more options in the retailer-led supply chain under option contract. Moreover, the conditions that lead to coordination are the same for both channel structures. The rest of the paper is organized as follows: In Section 2 we review the related literature to identify the research gap and position our study. In Section 3 we introduce the model and assumptions. In Sections 4 and 5 we discuss the retailer’s and supplier’s decisions on ordering, option pricing, and production in a supplier-led and retailer-led supply chain, respectively. In Sections 6, we analyze the influence of supply chain 3

Journal Pre-proof structure. In Section 7, we derive the conditions under which supply chain coordination is achieved. In Section 8, we conclude this paper and provide suggestions for further research.

2. Literature review Our work brings together streams of research on option contracts, risk preference, and supply chain coordination. Although much work has been done in each of these areas, studies that combine the three aspects are relatively rare. In what follows, we review the closely related literature in the three areas, and then summarize the differences between our work and current literature, and highlight our contributions. It has been shown that an option contract can provide suppliers and retailers with flexibility to share risks caused by uncertain demand, and then improve supply chain performance. Barnes-Schuster et al. (2002) are among the first to study the option contract and supply chain coordination. Considering a two-period model with one supplier and one retailer, when demands in the two periods are correlated, they find that supply chain coordination can be achieved under an option contract if the exercise price is piecewise linear. Erkoc and Wu (2005) and Jin and Wu (2007) study the capacity option contract, which is used to share supply chain risk and encourage high-tech manufacturers to expand capacity. Zhao et al. (2010) address the coordination issues in a manufacturer-retailer supply chain under an option contract based on a cooperative game approach. Compared with the wholesale price contract, they find that an option contract can coordinate the supply chain and achieve Pareto-improvement for both the supplier and retailer. Zhao et al. (2013) develop a bidirectional option contract, which may be exercised as either a call option or a put option, to coordinate a manufactureretailer supply chain. They figure out the retailer’s optimal order strategies and the conditions under which supply chain coordination is achieved. The aforementioned studies propose an option or option-like contract to coordinate the supply chain under the assumption that the retailer is risk neutral. In practice, however, as retailers are closer to the end market and face more risks, they tend to be risk averse. Therefore, the problem of option contracts under risk aversion setting, is worth in-depth exploration. Among studies on options procurement by risk-averse retailers, Wu et al. (2010) show that the retailer’s order decision is influenced by both risk preference parameter and the option price. Buzacott et al. (2011) study a class of commitment-option

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Journal Pre-proof contracts with demand information updating in the mean-variance framework, and demonstrat that if the quality of information revision goes from low to medium, the mean-variance analysis is efficient and the risk reduction is significant. Studying the portfolio procurement problem under an option contract under both risk-neutral and risk-averse settings, Fu (2015) find that the procurement decision under risk-averse setting is more prudent than that under risk-neutral setting, risk-averse solution has more consistent performance and can mitigate the risk of high losses. Zhou et al. (2018) consider a supply chain and explore the coordination problem under the mean-variance framework. Contrasting with the results obtained by previous research, they find that supply chain coordination is not always obtained with an option contract. Similar to our research, these studies consider the options purchasing problem for a risk-averse buyer. However, they mainly focus on a single agent’s purchasing decision, ignore the upstream agent’s problem and the influence of supply chain structure. Our work fills this gap by considering the coordination problem in a supply chain with a risk-averse retailer. Researchers began exploring supply chain coordination in the risk-averse environment more than a decade ago (Agrawal and Seshadri, 2000) but from very different perspectives. Gan et al. (2004) investigate supply chain coordination in the risk-averse setting, and explore a coordinating contract that results in a Pareto-optimal solution. By considering three cases where agents have different risk preference, their work complements the traditional concept of supply chain coordination in the riskneutral setting. Gan et al. (2005) investigate a supply chain consisting of a risk-neutral supplier and a risk-averse retailer subject to downside risk, and find that the revenue sharing and buyback contracts cannot coordinate the supply chain when the retailer’s downside risk is too high. However, supplier-led supply chain coordination problem is a prevailing topic in the existing literatures. For example, Chen et al. (2014) consider a supplier-led supply chain with a loss averse retailer and option contracts, and find that there always exists a Pareto contract as compared to the non-coordinating contracts. Wei et al. (2017) investigate a supplier-led supply chain with two competing retailers. By considering fairness concerns, they find that when the retailers are more concerned with fairness, the coordination of the supply chain will be easier to achieve. Mallic et al. (2018) study a supplier-led supply chain with stochastic lead time demand and derive the optimal production rate, ordering quantity that maximizing joint total profit. The coordination problem in both supplier-led and retailer-led supply chains is considered 5

Journal Pre-proof in Yang et al. (2018). By assuming both the supplier and retailer are risk averse, they find that three-part tariff revenue sharing and buy-back contracts can coordinate both supplier-led and retailer-led supply chains. Similar to Yang et al. (2018), we study the coordination problem under different supply chain structures but consider a different purchasing contract, an option contract. As for the measures of risk preference, the three widely used measures in finance and insurance industry are mean-variance (MV), value-at-risk (VaR), and conditional value-at-risk (CVaR) (Chen et al., 2009). Each measure has its own merits and limitations, but as CVaR is convenient for optimization analysis, we use the CVaR criterion in this paper to model the retailer’s risk aversion for tractability.

3. Model description We consider a two-echelon supply chain consisting of one supplier and one retailer that are engaged in a single selling season. The retailer faces stochastic market demand D with probability density function (pdf) f ( x) and cumulative distribution function (cdf) F ( x) . The supplier sells a product to the retailer under the option contract. We define

the period before the market demand is realized as t1 and the period after demand realization as t2 . In period t1 , under the option contract, the retailer can purchase a number of options at the unit price o (option price). Each option gives the retailer the right but not the obligation to purchase one unit of the product at the pre-determined price e (exercise price) in period t2 . We denote the number of options that the retailer purchases as M . After receiving the retailer’s order, the supplier makes a capacity decision and then begins to produce the product at unit cost c . Define the number of products that the supplier produces as Q . In period t2 , the selling season starts and the retailer observes the market demand D and chooses to exercise some or all of the purchased option at unit price e . The number of options that the retailer exercises is min{M , D} . If the number of options that the retailer decides to exercise is greater than

the supplier’s production quantity Q , the supplier has to take some measures like urgent production or purchasing from other sources to cover the shortage at a higher cost. For simplicity, we denote the unit cost of obtaining the extra number of the products in period t2 as the penalty cost z with c  z  p . Then, the supplier ships the corresponding number of products to the retailer. After receiving these products,

6

Journal Pre-proof the retailer sells them at unit retail price p . In this study we consider a short lifecycle product or similar products that have little salvage value. Without loss of generality, we assume that the salvage value is zero and there is no goodwill loss to the retailer if a stock-out occurs. To avoid the trivialities, we assume c  o  e  p to ensure that the supply chain partners’ reservation payoffs are non-negative. In this study we consider the two cases where the supplier and the retailer is the supply chain leader, respectively, wherein the supply chain leader has the right to determine the option price. We consider the option exercise price as an exogenous parameter. This assumption enables the model to be tractable and is consistent with the existing literature on the option contract in supply chain management such as Burnetas and Ritchken (2005), Li et al. (2009), Chen and Shen (2012), and Liu et al. (2014). In some high-tech or capital-intensive industries, the option price is transferred to the supplier to build capacity. On the one hand, by paying the option price, the retailer obtains flexibility in ordering and reduces the risk of uncertain market demand. On the other hand, the supplier benefits by selling the option and obtaining market demand information from the retailer’s order (Hu et al., 2018). Against this background, we regard the option price as the price of flexibility charged by the supplier to the retailer, and the exercise price as the wholesale price determined by other factors, such as market competition or government regulations. Another assumption that we make is that the general failure rate of the market demand is increasing, i.e., Dh( D )  Df ( D) F ( D) increases in D . This assumption is commonly made in the related literature, such as Lariviere and Porteus (2001), which is satisfied by most common distributions such as the normal and uniform distributions. Previous studies have primarily focused on the risk-neutral setting in which the supply chain partners target profit maximization. However, there is evidence that the retailer (newsvendor) behaves as an independent decision maker and may have preferences (or decision bias) other than profit maximization (Schweitzer and Cachon, 2000). Risk preference is one important factor that influences a retailer’s decisionmaking. Therefore, in this research, we are interested in exploring the behaviour of the risk-averse retailer. A popular risk measurement in finance and the insurance industry is the CVaR criterion (Rockafellar and Uryasev, 2000), which measures the average profit falling below the  -quantile level and has better computational characteristics. Denoting  ( M ) as the retailer’s random profit, we define its  -CVaR value under

7

Journal Pre-proof the order quantity M as CVaR ( ( M ))  E[ ( M )   ( M )  q ( ( M ))] ,

where q ( ( M )) is the  -quantile of  ( M ) given as

q ( ( M ))  inf{  P( ( M )   )  } . For the convenience of calculation, Rockafellar and Uryasev (2000, 2002) provide an equivalent definition of  -CVaR as follows, which we adopt in this research:

1 CVaR ( ( M ))  max{  E (   ( M ))  }  R ,  where 

is a real number and  (0    1)

(1)

is the risk-aversion parameter,

Specifically, the smaller the value of  is, the more risk averse is the retailer. When

  1,

the

CVaR

model

degenerates

into

the

risk-neutral

model

and

CVaR ( ( M ))  E[ ( M )] . Table 1 summarizes the notation used throughout the paper. To simplify presentation, we provide all the proofs in the Appendix. Table 1. List of notation. c:

the supplier’s production cost per unit

p : the retailer’s selling price per unit os , or : the option prices in a supplier-led and a retailer-led supply chain, respectively e:

the option’s exercising price per unit

z : the supplier’s cost of obtaining an extra unit of the product in period t2 Ms, Mr :

the retailer’s purchased option quantities in the supplier-led and retailer-led supply chains, respectively M sc : The retailer’s purchased option quantity in the centralized supply chain

Qs , Qr :

the supplier’s production quantities in the supplier-led and retailer-led supply chains, respectively Qsc : the supplier’s production quantity in the centralized supply chain

D : the stochastic market demand with support [0, ) F (  ) : the cumulative distribution function of D f (  ) : the probability density function of D h(  ) : the failure rate of D , i.e., h(  )  f (  ) (1  F (  ))

 s ,  r : the retailer’s profits in the supplier-led and retailer-led supply chains, respectively s , r :

the supplier’s profits in the supplier-led and retailer-led supply chains, respectively

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Journal Pre-proof  sc : the centralized supply chain’s profit

4. Supplier-led supply chain In this section we study the case where the supplier is the supply chain’s leader, i.e., the supplier is the core of the supply chain and has the right to decide the option price os . Following Yang et al. (2018), this supply chain structure can also be called as push supply chain. Figure 1 summarize the sequence of events in the corresponding game model. To distinguish the supplier-led model from the retailer-led model in notation, we use the subscripts ‘ s ’ and ‘ r ’ to denote the supplier-led supply chain and retailerled supply chain, respectively. Supplier determines option price os and production policy

Supplier ships products to the retailer

Supplier begins to produce

t2

t1

Retailer purchases M s units of options

Retailer chooses the number of options to be exercised

Figure 1. Sequence of events when the supplier is the leader. 4.1 Retailer’s problem Using backward induction, we first derive the risk-averse retailer’s optimal order policy under the option contract. The risk-neutral retailer’s profit is  s ( M s )  p min( M s  D )  os M s  e min( M s  D ) .

(2)

The first term is the retailer’s sales revenue while the last two terms are the costs of buying and exercising options, respectively. If the retailer is risk averse, its utility function is formed by plugging Equation (2) into Equation (1), and the retailer’s CVaR is

1 CVaR ( s ( M s ))  max{  E[  (os M s  ( p  e) min( M s  D)] } .  R  As min( M s  D)  M s  ( M s  D)  , we can rewrite Equation (3) as

1 CVaR ( s ( M s ))  max{  E[  ( p  os  e) M s  ( p  e)( M s  D)  ] } .  R 

9

(3)

Journal Pre-proof Solving the above maximization problem, we obtain the retailer’s optimal ordering decision as follows: 1 Proposition 1. The risk-averse retailer’s optimal order quantity is M s  F (

and its optimal CVaR value is CVaR ( s ( M s ))  1 ( p  ce ) 

F 1 (

p os e ) p e

0

p  os  e p e

)

DdF ( D) .

Since os  0 , e  0 , and os  e  c , there is an upper bound on M s . When e  c , the

maximum 1 p  c p e

Ms  F (

M s  F 1 ( )

as

os  0 ; when

e c,

the

maximum

 ) as os  c  e . Thus, we define an upper bound on the retailer’s order

quantity as

 F 1 ( )   , if e  c, M max   1 p c  F ( p e  ), if e  c, where  is a very small number close to zero. For the convenience of expression, we omit term  and claim M max  F 1 ( ) when e  c . By Proposition 1, if   1 , the retailer is risk neutral and the classical risk-neutral 1 order quantity under the option contract is M s | 1  F (

p  os  e p e

) , which is obviously

greater than that of the risk-averse retailer. Similarly, when the retailer is loss averse and orders from a risk-neutral supplier via option contract, Chen et al. (2014) find that the loss-averse retailer orders less than a risk-neutral retailer. However, this research complements Chen et al. (2014) by considering a risk averse retailer, and further explores the case where the retailer acts as the leader and determine the option price. Corollary 1.

M s 

 0,

M s os

 0 , and

M s e

 0.

The first inequality in Corollary 1 implies that the more risk averse the retailer is, the fewer the number of options it orders. This makes intuitive sense as the risk-averse retailer would sacrifice part of its profit rather than take on more risk. The retailer can reduce the risk it faces by decreasing the order size and pay less for the option. The last two inequalities indicate that the higher the option price and exercise price are, the fewer the number of options the retailer orders. In the loss averse setting, Chen et al. (2014) derive similar results that the loss-averse retailer’s optimal order quantity is decreasing in the degree of loss aversion, option price and exercise price. 4.2 Supplier’s problem In the supplier-led supply chain, before the selling season starts, the supplier decides

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Journal Pre-proof the option price and its production policy to maximize its utility. Based on the production policy, once receiving the retailer’s order of the option, the supplier determines the number of products to produce. As it is probable that the retailer will not exercise all of the purchased options, a rational supplier would produce the quantity Qs that is no greater than M s , i.e., Qs  M s . We denote  s (os , Qs ) as the supplier’s

profit and z as the penalty cost incurred in period t2 for each exercised option that cannot be satisfied by the supplier’s regular production capacity. Hence, the supplier’s problem is

max  s (os , Qs )  os M s  e min( D M s )  cQs  z (min( D M s )  Qs )  , Qs  M s 

s.t.

M s  M max . The first two terms in the objective function are the revenues realized by the supplier from the number of options sold and the number of options exercised, respectively. As explained in Section 3, when the supplier’s production quantity cannot satisfy all the retailer’s exercised option requirement, it takes urgent measures to obtain extra units of the product at a higher cost to cover the unsatisfied number of the option. Therefore, in period t2 , the supplier can always ship up to M s units of the product to the retailer and the corresponding revenue is e min( D M s ) . The last two terms of the objective function are the production cost and penalty cost incurred in period t2 . As we assume that the supplier is a risk-neutral decision-maker, we can re-write the supplier’s objective function as max E ( s (os , Qs ))  os M s  eE (min( D M s ))  cQs  zE[(min( D M s )  Qs )  ] ,

which is equivalent to max E ( s (os , Qs ))  os M s  e 

Ms

0

F ( D)dD  cQs  z 

Ms

Qs

F ( D)dD ,

(4)

where F ( D)  1  F ( D) . Note that, by Proposition 1, there is a one-to-one mapping between os and M s . So we can express os as



os  ( p  e) 1 

F (Ms )



.

Substituting os into Equation (4), we see that the problem of optimizing (os , Qs ) is equivalent to one of optimizing ( M s , Qs ) , i.e., max E ( s ( M s , Qs )) . For the convenience of expression, let ( M s ) 

p e



11

F ( M s ) 1  ( epze)  M s h( M s )  

p e



(1   )

Journal Pre-proof represent the first-order derivative of E ( s ( M s , Qs )) with respect to M s . Then, we summarize the supplier’s optimal production and order quantity decisions in Proposition 2. Proposition 2. When cooperating with a risk-averse retailer under the option contract, the risk-neutral supplier’s optimal decisions on production quantity are (1) if (Q )  0 , Qs*  M s*  min{Q, M max } ;

Qs*  Q, M s*  M , if M  M max ,  * (2) if (Q )  0 , Qs  Q, M s*  M max , if Q  M max  M , Q*  M *  M , if M max  Q, s max  s z c where Q  F 1 ( z ) and M satisfies ( M )  0 .

Given

M s* , the corresponding option price is

os*  ( p  e)(1 

F ( M s* )



) . In

Proposition 2, Q can be viewed as the supplier’s optimal production quantity without constraints. When we ignore the constraints, it is interesting to note that Qs*  Q is independent of the option price and exercise price. This result stems from our assumption that the supplier can always satisfy all the retailer’s exercised option requirement by fulfilling the unsatisfied number of options from other channels in period t2 . That means that the expected revenue from selling options and satisfying the exercised option requirement, i.e., the sum of the first and second terms in Equation (4), is fixed given the retailer’s order quantity. Thus, when the supplier makes the production decision, it needs to consider only the tradeoff between the unit underage cost z  c and the unit overage cost c . However, as there is an upper bound M max on the order quantity, the supplier has to set the production quantity as Qs*  min{Q, M max } .

From Proposition 2, we also observe that the equilibrium production quantity and order quantity are impacted by the sign of (Q ) . From the proof of Proposition 2 in the appendix, we know () is a decreasing function. In the first part of Proposition 2, when (Q )  0 , the production quantity without constraints Q is larger than or equal to the equilibrium order quantity, which may result from a low overage cost c . But as the constraint Qs  M s should always be satisfied, the equilibrium production quantity and order quantity are equal. In the second part of Proposition 2, when

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Journal Pre-proof (Q )  0 , which means Q is less than M , the equilibrium production quantity may be smaller than the equilibrium order quantity. Nevertheless, if M max  Q , i.e., the retailer is very risk-averse, the retailer orders a very low quantity and consequently,

Qs*  M s*  M max holds. Next, we study the effects of the penalty cost z and the retailer’s risk attitude

 on

(Q ) . By definition, Q is a function of

z . Then, by substituting

Q ( z ) into

 and taking the first-order derivative of ( z ) with respect to z , we obtain Proposition 3. Proposition 3. Given the other parameters, if z  zˆ , the supplier’s production quantity Qs*  M s* ;

z  zˆ , the supplier’s production quantity

if

zˆ  min{z , 1 F ( Mc

max )

Qs*  M s* ,

where

} and z uniquely solves ( z )  0 .

The conclusion in Proposition 3 is intuitive. When the penalty cost z is small, the supplier evaluates the tradeoff between overstocking and shortage but does not produce all the number of sold options; when z is large enough, the shortage risk dominates the overstocking risk, so the supplier produces all the number of sold options to avoid shortage. In addition, from the definition of Q , we observe that Q increases in z , i.e., the higher the penalty cost is, the more number of options the supplier will produce. To investigate how the retailer’s risk attitude  impacts the relationship between M s* and Qs* , we take the firs-order derivative of (Q ) with respect to

 and

summarize the results in Proposition 4. For the convenience of expression, we define

  p  c  e z z c , 1 as the unique solution of (Q ) |   0 , and 1

z c , if e  c,   z 2    ( z  c)( p  e) , if e  c.  z ( p  c) 

Proposition 4. Given the other parameters, (1) when   0 , if   max{1 ,2 } , Qs*  M s* ;

if

  max{1 , 2 } ,

Qs*  M s* ;

(2)

when

 0,

if

  (0, min{1 , 2 })  [1 ,1] , Qs*  M s* ; if   [min{1 , 2 },1 ) , Qs*  M s* ; (3) when

  0,

 has no effect on the relationship between

13

Qs* and M s* .

Journal Pre-proof In Proposition 4, it is intuitive that the retailer’s risk attitude may have an impact on the relationship between Qs* and M s* . However, the direction of the impact is not consistent and determined by the value of  . The reason behind this is that, the retailer’s risk attitude  has a direct impact on the order quantity M s* and, in turn, M s* impacts the supplier’s expected profit. For the supplier, it needs to keep the order

quantity at a certain level to balance the overstocking risk and shortage risk. Moreover, since M s  M max and  impacts M max as well, the impacts of  on the supplier’s profit, and the relationship between Qs* and M s* become quite complicated.

5. Retailer-led supply chain With the development of the customer-oriented service economy, the structure of the supply chain is changing. Giant retailers, such as Wal-Mart and Alibaba, started to lead the supply chain, representing a pull supply chain. In this case, the retailer is able to impact its upstream supplier’s production and delivery decisions (Wang and Liu, 2007). Such business practices motivate us to study the retailer-led supply chain under the option contract. To the best of our knowledge, Wang and Liu (2007) pioneered the study of the retailer-led supply chain structure under option contract. Unlike their research, we focus on a more generalized risk-averse retailer setting. In a retailer-led supply chain, as characterized by Wang and Liu (2007), the retailer holds the right of pricing with the coordinating variable being the production quantity rather than the order quantity. In this section we study the case where the retailer is the supply chain leader that decides the option price or and number of options M r it will purchase. Given the retailer’s offer (or , M r ) , the supplier will accept the contract as long as it can obtain a non-negative expected profit and decides the production quantity Qr . Figure 2 depicts the sequence of events in the retailer-led supply chain game model.

14

Journal Pre-proof Supplier chooses the production policy and begins to produce

Supplier ships products to the retailer

t2

t1

Retailer determines option price or and purchases M r units of options

Retailer chooses the number of options to be exercised

Figure 2. Sequence of events when the retailer is the leader. 5.1 Supplier’s problem Using backward induction, we first analyze the supplier’s production decision. As the supplier is a risk-neutral decision-maker, our goal is to maximize the supplier’s expected profit. Given the retailer’s offer (or , M r ) , the supplier’s problem is

max E ( r (Qr ))  or M r  eE[min( D M r )]  cQr  zE[(min( D M r )  Qr )  ], s.t.

Qr  M r .

The first and second terms are the revenues from selling options and satisfying the retailer’s exercised option requirement. The third term is the production cost and the last term is the expected penalty cost incurred in period t2 . By analyzing the first-order condition, we obtain the supplier’s optimal production policy in Proposition 5. Proposition 5. The risk-neutral supplier’s optimal production quantity is

 Q, if Q  M r , Qr    M r , if Q  M r , z c where Q  F 1 ( z ) .

In Proposition 5, Q is the supplier’s unconstrained optimal production quantity. When M r is large enough so that Qr  M r is naturally guaranteed, the supplier just needs to solve the unconstrained problem, which yields Qr  Q . When M r is small so that M r  Q , since the supplier will never produce more units of products than the number of options that the retailer has purchased, the optimal production quantity is Qr  M r .

15

Journal Pre-proof

5.2 Retailer’s problem As discussed above, in the retailer-led supply chain, the retailer determines the order size and option price. In this subsection, we study the retailer’s problem. Figure 2 shows that the retailer decides or and M r simultaneously. Following a similar analysis in Subsection 4.1, we see that the retailer’s objective function is

CVaR ( r (or , M r ))  max{   R

1



E[  ( p  or  e) M r  ( p  e)( M r  D)  ] } .

Proposition 6. The risk-averse retailer’s optimal option pricing and ordering decisions are (1) if e  c , or*  0 and M r* |or* 0  F 1 ( ) ; (2) if

e  c , or*  c  e

and M r*  F 1 ( pp ce  ) .

Combined with the definition of M max , we can also express the optimal order quantity in Proposition 6 as M r*  M max . Proposition 6 concludes that when or  e  c and or  0 in the retailer-led supply chain, the retailer intends to set the option price as low as possible, even to the extent that or*  e  c . Although, when the retailer is not that risk-averse so that the order quantity is larger than the supplier’s production quantity, the supplier may benefit from unexercised options. In practice, the phenomenon that the retailer bullies the supplier to such an extent ( or*  e  c ) is not rare. For example, Walmart often requires its suppliers to keep the wholesale prices at very low levels to keep the promise of “Everyday Low Price”. But to meet Walmart’s demand, some suppliers have to lose money on each sale (Rey, 2017). In China market, KFC suppresses the purchasing price of chicken meat at a very low level such that its big suppliers only obtain slender profits and most small suppliers are losing money (Finance.ifeng.com). When e  c , it is obvious that M r* increases in  but is independent of the other parameters. The reason behind this is that as the option price approaches zero, the risk faced by the retailer also approaches zero. Therefore, given any other parameters, the retailer will purchase as many of the option as it can. Specifically, when   1 , the number of options purchased by the retailer equals the upper bound of the demand. When

e c,

we summarize the sensitivity of M r* relative to the other parameters in 16

Journal Pre-proof Corollary 2. Corollary 2. When

e c,

M r* 

 0 and

M r* e

0.

The impact of  on the retailer’s order size in Corollary 2 is the same as that in Corollary 1, i.e., the more risk-averse the retailer is, the fewer number of options the retailer orders. However, the impacts of the exercise price

e

on M r* are different in

Corollary 1 and Corollary 2. The reason behind this is that in the retailer-led supply chain, the retailer determines the option price and prefers to set the option price as low as possible so that it takes less risk and orders more of the product. When larger

e

e c,

the

is, the smaller is or* . But in the supplier-led supply chain, the supplier has

the right to determine the option price, so the retailer prefers a lower exercise price to decrease its cost.

6. Supplier-led supply chain vs. Retailer-led supply chain We examine the impacts of the supply chain structure on the retailer’s and supplier’s decisions in this section. We start with the retailer’s order decisions in different cases. Proposition 7. M s*  M r* , where the equality holds when (1) e  c and os  0 or (2) ec

and os  e  c . Proposition 7 is straightforward. Combined with Proposition 6, in the retailer-led

supply chain, the retailer sets the option price as low as possible. By doing this, the retailer can take full advantage of the option contract for risk management by purchasing a larger number of options at a lower cost (even zero cost) and obtaining more profit consequently. In the supplier-led supply chain, however, the supplier determines the option price, which, in turn, affects the retailer’s order quantity. If the option price os* is higher than or* , the order quantity becomes smaller and M s*  M r* . Furthermore, given Proposition 1 and 7, it is obvious that os*  or* , i.e., the option price in the supplier-led supply chain is higher than that in the retailer-led supply chain. Please note that in Proposition 7, when “=” holds, M s*  M r*  M max , which means the retailer is highly risk averse, so it restricts the order quantity at a very low level in both the supplier-led and retailer-led supply chains. Proposition 8. Qs*  Qr  min{Q, M max } .

17

Journal Pre-proof Proposition 8 implies that the supply chain structure has no impact on the supplier’s production quantity. However, much of the literature focusing on the wholesale price contract concludes that the order quantity in the retailer-led supply chain is higher than that in the supplier-led supply chain (e.g., Cachon 2004, Davis et al. 2014, Yang et al. 2018), which means the retailer-led supply chain ends up with higher total supply chain profit than the supplier-led supply chain does. This conclusion does not hold in this research, i.e., the retailer is risk averse and purchases from a riskneutral supplier under the option contract. The reason is that, compared with the wholesale price contract, under the option contract, both the retailer and supplier prefer a higher quantity of the production to satisfy more market demand.

7. Supply chain coordination In this section we derive the conditions under which the option contract coordinates the supply chain consisting of a risk-neutral supplier and a risk-averse retailer under the CVaR criterion. Specifically, since the supplier and retailer follow different performance measures and we do not know the profit of the supply chain as a whole, we cannot use the traditional coordination concept for risk-neutral agents (Cachon, 2003). Instead, we follow the definition of supply chain coordination introduced by Gan et al. (2004, 2005) for the risk-averse retailer. By their definition, for a supply chain consisting of one risk-neutral supplier and one risk-averse retailer, the option contract agreed upon by the two parties is said to coordinate the supply chain if (1) each agent’s individual rationality or participation conditions are satisfied, (2) the retailer’s CVaR criterion is maximized, and (3) the integrated supply chain’s expected profit is maximized. The first condition is straightforward as long as the supplier’s and retailer’s individual profits are greater than zero, and the second condition is characterized in Sections 4 and 5. Hence, we consider only the third condition in the following. From the supply chain’s perspective, the integrated supply chain’s profit, denoted as  sc (Qsc ) , is  sc (Qsc )  p min( D Qsc )  cQsc . Hence, the integrated supply chain’s expected profit is E ( sc (Q))  p 

Qsc

0

F ( D)dD  cQsc .

(6)

p c Lemma 1. The integrated supply chain’s optimal order quantity is Qsc*  F 1 ( p ) .

According to the three conditions in Gan et al. (2004, 2005), a sufficient condition 18

Journal Pre-proof that coordinates the supplier-led/retailer-led supply chain is

M s  Qs  Qsc* / *

M r  Qr  Qsc* . Thus, the integrated supply chain’s optimal order quantity Qsc is a

benchmark. 7.1 Supplier-led supply chain coordination In the supplier-led supply chain, the supplier produces its optimal quantity Qs and makes sure that Qs achieves the system-wide optimum, i.e., Qs  Qsc* . Meanwhile, since the supplier’s production quantity is not greater than the number of options that the retailer has purchased, the supplier needs to make sure that the retailer’s order size is not smaller than the system-wide optimum, i.e., M s*  Qsc* . By investigating the conditions under which the equality Qs  Qsc* and inequality M s*  Qsc* hold, we derive the following proposition. Proposition 9. In the supplier-led supply chain, channel coordination is achieved in the following two cases (1) when e  c , z  p and   1  cp ; (2) when

e c,

z  p , and   1  ep .

Proposition 9 implies that in the supplier-led supply chain, coordination is achieved only when the retailer is not too risk averse, i.e.,   1  cp or   1  ep , which ensures that the retailer purchases enough number of options that is no less than the supply chain coordination order size Qsc* . Proposition 2 establishes that the supplier always chooses the production quantity as condition

z p

ensures that

Q  Qsc* ;

Qs  min Q, M max  . Therefore, the

  1  cp

or   1  ep

M max  Qsc* . Especially, when the retailer is risk neutral, i.e.,

ensures that

  1 , supply chain

coordination is achieved as long as z  p . When the retailer is risk neutral, Zhao et al. (2010) proved that coordination of the supply chain is feasible when (os  e)  {(os  e)  cp os  e  p} . However, in their work, the supplier adopts the “make-to-order” policy and has no right to decide the option price. Compared with the result in Zhao et al. (2010), we find from Proposition 9 that when the retailer is risk neutral, supply chain coordination in our study is independent of the option price and exercise price. The reason behind this is that the supplier in the

19

Journal Pre-proof supplier-led supply chain determines the option price and can produce fewer units of products than the number of options that the retailer has purchased. By adjusting the option price, the supplier can always induce the retailer to purchase a greater number of options than the channel coordination order size. Moreover, as explained in Subsection 4.2, the supplier’s production quantity is independent of the option price and exercise price. Then it is intuitive that supply chain coordination in this case is independent of os and

e.

7.2 Retailer-led supply chain coordination We derive the conditions under which the retailer-led supply chain coordination is achieved. In the retailer-led supply chain, the retailer needs to make sure that its order size is

* . Moreover, the supplier’s production quantity should satisfy M r*  Qsc

* . Qr  Qsc

Proposition 10. In the retailer-led supply chain, channel coordination is achieved in the following two cases (1) when e  c , z  p , and   1  cp ; (2) when

e c,

z  p , and   1  ep .

By comparing Proposition 10 and Proposition 9, we note that the conditions under which the supply chain is coordinated are the same in the supplier-led supply chain and the retailer-led supply chain. The key reason behind this is the retailer’s risk attitude and the supplier’s production flexibility, i.e., the supplier can produce fewer units of the product than the order quantity. In both the supplier-led and retailer-led supply chains, the condition Qs*,r  Qsc* , leading to

z  p , ensures that the supplier’s

production quantity can coordinate the supply chain; the condition Qsc*  M s*,r , leading to   1 

c p

or   1  ep , ensures that the retailer’s order quantity is not too small

because of its risk-averse attitude.

8. Conclusion We investigate supply chain coordination in a two-echelon supply chain consisting of a risk-neutral supplier and a risk-averse retailer. The retailer purchases from the supplier via an option contract to satisfy uncertain market demand. We study two supply chain structures, i.e., the supplier-led supply chain and the retailer-led supply chain. For both

20

Journal Pre-proof structures, we study the supply chain leader’s option pricing problem, the retailer’s optimal option order quantity under CVaR criterion and the supplier’s optimal production quantity. The results show that risk-averse attitude has negative influence on the retailer’s order quantity, and the supplier’s production quantity equals the retailer’s order quantity only if the penalty cost is high enough. In a retailer-led supply chain, the retailer will set the option price as low as possible to transfer more risks to the supplier. By comparing results under both supply chain structures, we find that the retailer purchases more options at a lower price in the retailer-led supply chain, but the supplier’s production quantities remain the same under both supply chain structures. Finally, we prove that both the supplier-led and the retailer-led supply chains can be coordinated under the same conditions, which is different from the conclusions obtained by previous studies. This research can be extended in several ways. First, we consider only a single supplier and a single retailer. When multiple retailers are introduced, following Wei et al. (2017) and taking two competing retailers as an example, the two retailers play the Cournot game to determine their sale price and their corresponding order quantity. The demand faced by one retailer is a linear function of its sale price and its rival’s sale price, subjecting to random perturbation of the market simultaneously. The competition between the retailers may affect the supplier’s production decision. Therefore, whether or not supply chain coordination can be achieved should be re-considered in the presence of retailers’ competition. Second, we assume that the information (e.g., demand information, retailer’s risk attitude) is symmetric between the supply chain partners. However, information asymmetry can be considered to make the problem more realistic. For example, the retailer may have private information about the market demand and may choose to disclose the market information to the manufacturer at various degrees of confidentiality, which captures the confidentiality nature of vertical information sharing. In this scenario, the retailer reveals a signal to the manufacturer, and the manufacturer uses this signal in determining its wholesale price and production quantity. For an information sharing structure like this, Li et al. (2008) show that a higher degree of confidentiality makes the manufacturer worse off, the retailers better off, and the whole supply chain better off. When all retailers share their information confidentially, the chain-optimal equilibrium wholesale price is lower with a higher degree of confidentiality. Therefore, in the setting of our research with information asymmetry, we may infer that achieving supply chain coordination is dependent on the 21

Journal Pre-proof degree of information confidentiality. Finally, our study approaches the problem from a theoretical perspective, leaving empirical validation of our results for future research.

Acknowledgements This research is supported by grants from National Natural Science Foundation of China (No. 71772006, 71673011, 71804185).

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25

Journal Pre-proof Appendix Proof of Proposition 1. We denote

g (M s  )     

1



Ms

0

1



E[  ( p  os  e) M s  ( p  e)( M s  D)  ]

[  os M s  ( p  e) D] dF ( D) 

1





Ms

[  ( p  os  e) M s ] dF ( D)

Case 1: When   os M s ,

g (M s  )   . g ( M s   ) 1 0 

Case 2: When os M s    ( p  os  e) M s , g (M s  )     

1



 os M s p e

0

1



[  os M s  ( p  e) D]dF ( D)

(  os M s ) F (

  os M s pe

)

1



( p  e) 

 os M s

0

p e

DdF ( D) .

g ( M s   ) 1   os M s  1 F ( )   pe

It is the case that

g ( M s  ) 

is decreasing in  . Substituting the two endpoints

  os M s and   ( p  os  e) M s into

g ( M s  ) 

, we obtain

g ( M s   ) 1    os M s g ( M s   ) 1  1  F (M s )     ( p  os  e ) M s

.

Case 3: When   ( p  os  e) M s ,

g (M s  )    

1

1

Ms

 0





Ms

[  os M s  ( p  e) D]dF ( D)

[  ( p  os  e) M s ]dF ( D)

Ms 1 1    (  os M s ) F ( M s )  ( p  e)  DdF ( D) .





1  [  ( p  os  e) M s ](1  F ( M s ))



g ( M s   ) 1  1  0  

26

0

Journal Pre-proof Based on the above analysis, g ( M s   ) is increasing in  in Case 1 and decreasing in  for Case 3. Also, as g ( M s   ) is a continuous function, we know  that the optimal value 

M s  F 1 ( ) , then value

can only be achieved in Case 2. Specifically, if

g ( M s  )    ( p  os  e ) M s

 1  1 F ( M s )  0 . We can obtain the optimal

  from the first-order condition, i.e.,    ( p  e) F 1 ( )  os M s . If

M s  F 1 ( ) , then

g ( M s  )    ( p  os  e ) M s

 1  1 F ( M s )  0 . In this case, the optimal

 value  can be obtained at the right endpoint, i.e.,    ( p  os  e) M s .

To summarize, ( p  e) F 1 ( )  os M s  M s  F 1 ( )   M s  F 1 ( ) ( p  os  e) M s  

Thus, we can express CVaR ( s ( M s )) as

CVaR ( s ( M s ))  max( g ( M s   ))  g ( M s    ) .  R

Then, we further optimize CVaR ( s ( M s )) with respect to M s . 1  Case 1: If M s  F ( ) , then    ( p  e) F 1 ( )  os M s . Plugging  into

g ( M s    ) , we have g (M s   )    

1



  os M s p e

0

[   os M s  ( p  e) D]dF ( D)

F 1 ( ) 1  os M s  ( p  e)  DdF ( D) .



Taking the first derivative of

0

g (M s   )

with respect to

M s , we obtain

g ( M s    )  1  c0  0 . Thus, g ( M s   ) is decreasing in M s . As M s  F ( ) , the M s optimal value can be reached at M s*  F 1 ( ) . The retailer’s optimal CVaR utility is F 1 ( ) 1 CVaR1 ( s ( M s* ))  os F 1 ( )  ( p  e)  DdF ( D) .



Case 2: If

0

M s  F 1 ( ) , then    ( p  os  e) M s . Plugging  

g ( M s    ) , we obtain

27

into

Journal Pre-proof g (M s  )    



1



  os M s

[   os M s  ( p  e) D]dF ( D)

p e

0

Ms 1 1  ( p  os  e) M s  ( p  e) M s F ( M s )  ( p  e)  DdF ( D).





0

 Taking the first and second derivatives of g ( M s   ) with respect to M s , we have

g ( M s    ) 1  ( p  os  e)  ( p  e) F ( M s ) M s   2 g (M s   ) 1   ( p  e) f ( M s )  0 2 M s 

.

 Hence, g ( M s   ) is a concave function with respect to M s . We have p  os  e

M s*  F 1 (

p e



) . The retailer’s optimal CVaR utility is

CVaR2 ( s ( M s* ))  ( p  os  e) M s*  

1



( p  e) 

F 1 (

1



p os e ) p e

0

( p  e) M s* F ( M s* ) 

1



( p  e) 

M s*

0

DdF ( D)

DdF ( D).

Subtracting CVaR1 ( s ( M s )) from CVaR2 ( s ( M s )) , we obtain

CVaR2 ( s ( M s* ))  CVaR1 ( s ( M s* )) 

1



( p  e) 

F 1 (

p os e ) p e

DdF ( D)  [os F 1 ( ) 

0

1



( p  ce ) 

F 1 ( )

0

DdF ( D)]

F 1 ( ) 1  os F 1 ( )  ( p  e)  1 pos e DdF ( D)





F 1 ( )

p o e F 1 ( pse  )

As

1



F

(

p e

)

1 [ ( p  e) F ( D)  ( p  os  e)]dD.



( p  e) F ( D) is increasing in the integral support [ F 1 (

1



( p  e) F ( D ) 

1



( p  e) F ( F 1 (

p  os  e p e

 ) F 1 ( )] ,

p  os  e  ))  p  os  e . pe

Then,



F 1 ( )

p o e F 1 ( pse  )

1 [ ( p  e) F ( D)  ( p  os  e)]dD  0 .



Hence, CVaR2 ( s ( M s ))  CVaR1 ( s ( M s )) . Thus, the risk-averse retailer’s optimal order

quantity

is

M s  F 1 ( p posee  )

CVaR ( s ( M s ))  1 ( p  e) 

F 1 (

0

p os e ) p e

and

DdF ( D) .

28

its

optimal

CVaR

value

is

Journal Pre-proof

Proof of Corollary 1. According to the implicit function derivation rule and taking M s* ’s derivative with respect to  , os , and

e,

respectively, we have

M s p  os  e M s p  os  e 1  f (M s )    0,   pe  f (M s ) p  e M s M s 1 1 * f (M s )     0, os pe os ( p  e) f ( M s* ) M s os M s os 1 f ( M s )     0. 2  e ( p  e) e f ( M s ) ( p  e) 2



Proof of Proposition 2. Substituting os  ( p  e) 1 

F (Ms )





into Equation (4), we

have Ms Ms  F (M s )  E ( s ( M s , Qs ))  ( p  e) 1   M s  e 0 F ( D)dD  cQs  z Qs F ( D)dD .   

Case 1. If the supplier sets Qs  M s , then the objective function is equivalent to Qs  F (Qs )  E ( s (Qs ))  ( p  e) 1   Qs  e 0 F ( D)dD  cQs .   

Analyzing the first-order condition, we find that the optimal production quantity Qs*  Qˆ , where Qˆ satisfies

 e ˆ (Qˆ )   c  1   , F (Qˆ ) 1   Qh  pe   pe

(A1)

and the optimal order quantity is M s*  Qs* . However, the constraint M s  M max should hold. Hence, if Qˆ  M max , M s*  Qs*  M max ; otherwise, if Qˆ  M max , M s*  Qs*  Qˆ . Case 2. If the supplier sets Qs  M s , taking the first-order derivatives of E ( s ( M s , Qs )) with respect to

M s and Qs , we obtain

Case 2. If the supplier sets Qs  M s , taking the first- and second-order derivatives of E ( s ( M s , Qs )) with respect to M s and Qs , we obtain

 (e  z )  pe E ( s ( M s , Qs )) p  e  F ( M s ) 1   M s h( M s )   (1   ) , (A2) M s  pe   

29

Journal Pre-proof E ( s ( M s , Qs ))  zF (Qs )  c , Qs

(A3)

  2 E ( s ( M s , Qs )) p  e  (e  z )  1  M s h( M s )  F ( M s )  h( M s )  M s h( M s )  ,  2 M s   pe   2 E ( s ( M s , Qs ))   zf (Qs )  0 . Qs 2 From Equation (A2), it is obvious that the supplier will constraint M s lower than a

M s h( M s )  1  ( epze)

certain level so that  2 E (  s ( M s ,Qs )) M s 2

 2 E (  s ( M s ,Qs )) M s Qs

is satisfied. Therefore, we know

 0 . Moreover, combined with the second-order mixed partial derivatives 

 2 E (  s ( M s ,Qs )) Qs M s

 0 , the Hessian matrix is

  2 E ( s ( M s , Qs ))  M s 2    0  

Since   0 and

 2 E (  s ( M s ,Qs )) M s 2

 0

  .  2 E ( s ( M s , Qs ))   Qs 2  0

, we know the Hessian matrix is negative definite.

Thus, the optimal Qs*  Q and M s*  M , where Q  F 1

  z c z

and M solves

( M )  F ( M ) 1  ( epze)  Mh( M )     1  0 . Taking the first-order derivative of ( M s ) with respect to

M s , we have

    (e  z ) ( M s )   F ( M s )  h( M s )  2   M s h( M s )   M s h( M s )  . pe    





It is evident that  decreases in M s for M s  M : Mh( M )  2  ( epze) . To ensure that the constraint Qs  M s holds, we need (Q )  0 . In addition, combined with the constraint M s  M max , we know if (Q )  0 and ( M max )  0 , i.e., Q  M and M  M max , then Qs*  Q and M s*  M . If (Q )  ( M max )  0 , i.e., Q  M max  M ,

then Qs*  Q

and

M s*  M max . If

( M max )  (Q )  0 , i.e.,

M max  Q  M , no

feasible solution exists because of the constraint Qs  M s . In this case, the supplier has to set Qs*  M s*  M max . Moreover, when (Q )  0 , i.e., M  Q , we can re-write (Q ) as 30

Journal Pre-proof   e c (Q )  F (Q ) 1   Qh(Q )    1  0 ,  pe  p  ce

(A4)

which means that Equation (A4) is equivalent to (A1). Therefore, we know Qˆ  Q . When (Q )  0 , the constraint Qs  M s cannot be satisfied and the supplier will set Qs*  M s* .

Combining Cases 1 and 2, the supplier’s decisions on the production quantity and order quantity are (1) if (Q )  0 , Qs*  M s*  min{Q, M max } ; (2) if (Q )  0 ,

Qs*  Q, M s*  M , if M  M max ,  * Qs  Q, M s*  M max , if Q  M max  M , Q*  M *  M , if M max  Q. s max  s



Given the optimal order size M s* , the optimal option price is os*  ( p  e) 1 

Proof of Proposition 3. Q is a function of z and find that

(Q )  F (Q ) 1  ( epze)  Qh(Q )     1

dQ dz



c 2

z f (Q )

F ( M s* )



.

 0 . Then we can easily

decreases in

z . As

zc,

( z )  ( ppce)  0 and z   , ( z )    1  0 , there exists a unique z that solves

( z )  0 . From Proposition 2, we know when ( z )  0 or ( z )  0 and Qs*  M s* ; when



zˆ  min z , 1 F ( Mc

( z )  0 and

max )

 . If

Q  M max ,

Q  M max ,

Qs*  M s* . Therefore, we can define

z  zˆ , Qs*  M s* ; otherwise, Qs*  M s* .

Proof of Proposition 4. Substituting Q into  and taking the first-order derivative of  with respect to  , we have

 c c p  c  e 1   (e  z ) d z  z   .  1  (A5) d pe pe pe From Proposition 2, we know if (Q )  0 or (Q )  0 and Q  M max , Qs*  M s* ; if (Q )  0

and Q  M max , Qs*  M s* . Then combined with Equation (A5) and the

definitions of 1 and 2 , Proposition 4 can be easily proved. Proof of Proposition 5. We re-write the supplier’s expected profit as

31

Journal Pre-proof E ( r (Qr ))  or M r  e 

Mr

0

F ( D)dD  cQr  z 

Mr

Qr

F ( D)dD .

(A6)

Taking the first- and second-order derivatives of E ( r (Qr )) with respect to Qr , we have

 2 E ( r (Qr )) E ( r (Qr ))   zf (Qr )  0 .  zF (Qr )  c , Qr 2 Qr Therefore, the optimal production quantity without constriants is Qr  Q . When

Qr  M r is loose and the final Qr  Q . However, when

M r  Q , the constriant M r  Q , since

E (  r ( Qr )) Qr

|Qr  M r  0 and “=” is achieved when Qr  M r  Q , we have

Qr  M r .

Proof of Proposition 6. We re-write the retailer’s CVaR as g (or , M r ,  )   

  1



Mr

[  or M r  ( p  e) D] dF ( D)   [  ( p  or  e) M r ] dF ( D)

0

Mr

.

Following a similar procedure to the proof of Proposition 1, we have or M r   *  ( p  or  e) M r and

( p  e) F 1 ( )  or M r  M r  F 1 ( ),   M r  F 1 ( ).  ( p  or  e) M r  

Case 1. When

M r  F 1 ( ) , substituting

* g (or , M r ,  ) , we obtain g (or , M r ,  )  or M r 

p e



   ( p  e) F 1 ( )  or M r



F 1 ( )

into

DdF ( D) . Taking the first-

0

order derivatives of g (or , M r ,  * ) with respect to or and M r , we obtain

g (or , M r ,  * ) g (or , M r    )  M r  0 ,  or  0 . or M r That means the retailer will set the option price or and order size M r as low as possible. Combined with the constraints M r  F 1 ( ) and or  e  c , we know M r*  F 1 ( ) ; if e  c , or*  0 ; if

e  c , or*  c  e .

However, if or*  0 , the retailer

faces no risk and incurs zero cost when purchasing the option. Under this scenario, the retailer will maximize its expected profit instead of the CVaR value. Then it is obvious that the optimal order size for the retailer M r*  max{D}  F 1 ( ) . Hence, if e  c , the retailer’s objective function is E ( r (or* , M r* ))  ( p  e) 



0

32

DdF ( D) . If

e c,

the

Journal Pre-proof retailer’s CVaR is CVaR1 ( r (or* , M r ))  or* F 1 ( )  Case 2. If

p e





F 1 ( )

0

DdF ( D) .

M r  F 1 ( ) , then    ( p  or  e) M r . Substituting   into

g (or , M r    ) , g (or , M r    )  ( p  or  e) M r  1 ( p  e) M r F ( M r ) 

p e





Mr

0

DdF ( D) .

 Analyzing the first-order derivative of g (or , M r   ) with respect to or , we obtain

the same conclusion for the optimal option price or* as in Case 1. If e  c , or*  0 and

M r*  max{D}  F 1 ( ) .

E ( r (or* , M r* ))  ( p  e) 



0

The

DdF ( D) . If

retailer’s

objective

e  c , or*  c  e .

function

is

Taking the first and second

 derivatives of g (or , M r   ) with respect to M r , we obtain

g (or , M r    ) 1  ( p  or  e)  ( p  e) F ( M r ), M r   2 g (or , M r    ) 1   ( p  e) f ( M r )  0. 2 M r  Hence, when

e c,

the optimal

M r*  F 1

retailer’s CVaR is CVaR2 ( r (or* , M  )) 

p e







p c p e

e



  F 1 ( ) . Consequently, the

F 1 ( ppcc  )

0

.

DdF ( D) .

In Case 1 and Case 2, if e  c , we find that the retailer always takes the same decisions, i.e., or*  0 and M r*  max{D} . In the following analysis, we only focus on the condition where

e c.

Following a similar procedure to the proof of Proposition 1, we obtain

CVaR2 ( r (or* , M r ))  CVaR1 ( r (or* , M r )) . Therefore, the risk-averse retailer’s optimal p c order quantity is M r  F 1 ( p e  ) and the optimal option price is or*  c  e .

Proof of Lemma 1. Taking Equation (6)’s first- and second-order derivatives with respect to Qsc , we have

 2 E ( sc (Qsc )) E ( sc (Qsc ))   pf (Qsc )  0 .  pF (Qsc )  c , Qsc 2 Qsc Then  sc (Qsc ) is a concave function. We can obtain the integrated supply chain’s optimal order quantity from the first-order condition. Hence Qsc*  F 1 ( p pc ) .

33

Journal Pre-proof Proof of Proposition 8. In the supplier-led supply chain, by Proposition 2, we conclude that the supplier’s optimal production quantity Qs*  min Q, M max  . In the retailer-led supply chain, the retailer’s order quantity is M r*  M max . Then, combined with Proposition 5, we know Qr*  Qs*  min Q, M max  . Proof of Proposition 9. We explore the conditions that make

M s*  Qsc*

and

Qs  min Q, M max   Qsc* hold.

Case 1. When e  c , M max  F 1 ( ) . We first focus on the case Q  F 1 ( ) . In this case, to ensure that Q  Qsc* , we need z  p and   1  cp . Moreover, to ensure M s*  Qsc* , we also need

the constraint

p c  p (1 )

p c  p (1 )

or  e  p . However, since

p c  p (1 )

 1 and or  e  p ,

or  e  p is loose. Second, we consider the case where

Q  F 1 ( ) . In this case, to coordinate the supply chain, we need

  1  cp and

1  cz   . However, as z  p , these two conditions cannot be satisfied at the same time. Case 2. When e  c , the second part of Proposition 9 can be proved by taking a similar analysis to that for the case where e  c . Proof of Proposition 10. The proof is similar to that of Proposition 9.

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