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ISSN : 1598-7248 (Print)
ISSN : 2234-6473 (Online)
Industrial Engineering & Management Systems Vol.16 No.1 pp.22-43
DOI : https://doi.org/10.7232/iems.2017.16.1.022

# Mean-Variance Analysis for Optimal Operation and Supply Chain Coordination in a Green Supply Chain

Yamaguchi Shin, Hirofumi Goto, Etsuko Kusukawa*
Graduate School of Electrical and Electronic Systems, College of Engineering, Osaka Prefecture University, Sakai, Osaka, Japan
Course of Electrical and Electronic Systems, College of Engineering Osaka Prefecture University, Sakai, Osaka, Japan
Graduate School of Electrical and Electronic Systems, College of Engineering Osaka Prefecture University, Sakai, Osaka, Japan
Corresponding author : kusukawa@eis.osakafu-u.ac.jp
March 22, 2016 October 20, 2016 January 6, 2017

## ABSTRACT

It is urgently-needed to construct a green supply chain (GSC) from collection of used products through recycling of them to sales of products using the recycled parts. Besides, it is necessary to consider the uncertainty in product demand as a risk in a GSC. This study proposes the optimal operations for a GSC with a retailer and a manufacturer. A retailer pays an incentive for collection of used products from customers and sells a single type of products in a market. A manufacturer produces the products ordered by the retailer, using recyclable parts with acceptable quality and compensates the collection cost of used products as to the recycled parts. This paper discusses the following risk attitudes: risk-neutral attitude, risk-averse attitude, and risk-prone attitude. Using mean-variance analysis, the optimal decisions for product order quantity, collection incentive, and lower limit of quality level, in the decentralized GSC (DGSC) and the integrated GSC (IGSC) are made. DGSC optimizes the utility function of each member. IGSC does that of the whole system. The analysis numerically investigates how (i) risk attitude and (ii) quality of recyclable parts affect the optimal operations. Supply chain coordination between GSC members to shift IGSC from DGSC is discussed.

## 1.INTRODUCTION

From social concerns about 3R (Reduce-Reuse- Recycle) activity worldwide, it is urgently-needed to construct a new supply chain management which incorporates reverse chains/logistics into traditional forward chains/logistics (Guide and Wassenhove, 2009; Jayant et al., 2012; Souza, 2013; Govindan et al., 2014; Stindt and Sahamie, 2014; Schenkel et al., 2015; Radhi and Zhang, 2016, Cannella et al., 2016). The traditional forward chains/logistics consist of the flows from procurement of new materials through production of new products to selling them. The reverse chains/logistics are composed of the flows from collection of used products through recycling parts from the used products to reuse the recycled parts. Also, a supply chain which organizes the forward chains and the reverse chains has been called a closed-loop supply chain, a reverse supply chain, or a green supply chain (GSC) (Guide and Wassenhove, 2009; Souza, 2013; Govindan et al., 2014; Stindt and Sahamie, 2014; Bhattacharya and Kaur, 2015; Gurtu et al., 2015). A supply chain which is considered in this paper is called GSC in terms of greening effort/initiatives (Swami and Shah, 2013; Ghosh and Shah, 2015) in promoting collection activity of used products and remanufacturing activity. The manufacturing to reuse recycled parts is called the remanufacturing (Bhattacharya and Kaur, 2015; Radhi and Zhang, 2016).

Remanufacturing is an approach used by many companies from different industries such as Dell, Hewlett- Packard(HP), IBM, Kodak, Xerox. An example of the different remanufactured products includes the following: photocopies, cellular telephones, single-use cameras, car’s engines and transmissions and retreaded tires (Radhi and Zhang, 2016). It is necessary to take some measures and policies in order to promote 3R activities in the GSC.

In order to conduct and promote 3R activities, in general, it is considerable for the system operation in a GSC to face the uncertainties in demand of a single type of products and quality of a single type of used products collected from a market. For the above topics, many previous studies regarding a GSC determined the optimal operations in the GSC so as to maximize the expected profit or minimize the expected cost under the uncertainties (Schenkel et al., 2015). There are some overviews of above topics (Fleischmann et al., 1997; Guide and Wassenhove, 2009; Souza, 2013; Govindan et al., 2013; Stindt and Sahamie, 2014; Bazan et al., 2016: Thiripura Sundari and Vijayalakshmi, 2016). However, only considering the expectation for decision-making is insufficient because it ignores the risk attitude of the decision makers (Choi et al., 2008a; Chiu and Choi, 2016). For example, sometimes the decision makers want to stabilize their profits, and sometimes they want to earn enormous profits even if there is a possibility to decrease the profits significantly.

Therefore, this paper clarifies decision-making in a GSC with considering the risk attitude. To discuss the risk attitude, this paper applies the classical mean-variance (MV) theory (Markowitz, 1959; Choi et al., 2008b) to determining the optimal operations of GSC members and the whole system in the GSC under the uncertainty in product demand.

This paper is closely relevant to Watanabe et al. (2013) and Watanabe and Kusukawa (2014). The system framework of a GSC which is considered in this paper is the same as that in Watanabe et al. (2013) and Watanabe and Kusukawa (2014). The GSC consists of a retailer and a manufacturer, and it is assumed that a contract for cooperation regarding collection of used products is concluded between both members. The retailer collects used products from consumers by paying an incentive, and then delivers the collected used products to a manufacturer. Also, the retailer orders a single type of products from the manufacturer and sells them in a market. The manufacturer produces the same quantity of products ordered by the retailer, using recyclable parts with acceptable quality and compensates the collection cost of used products as to the recycled parts.

Here, this study discusses three types of risk attitude regarding the uncertainty in product demand: risk-neutral attitude without consideration of variance of profit in a GSC, risk-averse attitude with negative consideration of variance of profit in a GSC, where those who are with the negative consideration hope to stabilize their profit, and risk-prone attitude with positive consideration of variance of profit in a GSC, where those who are with the positive consideration weigh heavily improvement in chances to generate large profit rather than stability of their profit. Using the mean-variance analysis for each risk attitude, the theoretically optimal decisions for product order quantity, collection incentive, and lower limit of quality level, in the decentralized GSC (DGSC) and the integrated GSC (IGSC) are made. DGSC optimizes the utility function of each member, based on the Stackelberg game (Nagarajan and Sosic, 2008; Yan and Sun, 2012, Watanabe et al., 2013, Watanabe and Kusukawa, 2014, Hong et al., 2015). IGSC optimizes that of the whole system. In general, the optimal operation in IGSC is better than that in DGSC from the aspect of the total optimization. Therefore, supply chain coordination is introduced into IGSC to encourage both members to shift the optimal operations as to risk attitude of GSC members from those in DGSC, by guaranteeing the both members’ expected profits improvement in IGSC. Concretely, the unit wholesale price and compensation for collection incentive of used products are coordinated between members as to risk attitudes in IGSC as Nash bargaining solution (Nash, 1950, 1953; Kohli Park, 1989, Nagarajan and Sosic, 2008; Hong et al., 2013; Watanabe et al., 2013; Watanabe and Kusukawa, 2014; Ghosh and Shah, 2015).

The numerical analyses in this paper investigate how (i) three types of risk attitude regarding product demand and (ii) quality of recyclable parts affect the optimal operations for DGSC and IGSC. Also, the optimal operation in DGSC is compared with that in IGSC as to risk attitudes (risk neutral, risk averse, and risk prone).

The outcomes obtained from the theoretical analyses and the numerical analyses regarding a GSC will answer the following questions for practitioners, academic researchers, and real-world policymakers, regarding operations in a GSC:

• How the optimal decisions for product order quantity, collection incentive, and lower limit of quality level, in DGSC and IGSC, are made when risk attitudes are discussed by using mean-variance theory,

• How the risk attitude affects the optimal operations in DGSC and IGSC,

• How the supply chain coordination to shift from DGSC to IGSC is made when risk attitudes are discussed by using mean-variance theory.

The rest of this paper is organized as follows. In Section 2, a review of the literature is presented with some topics. Section 3 provides model descriptions and formulates a utility function which consists of the expectation and the variance of profit of each member. In Section 4, the decision procedures for the optimal operations in a decentralized GSC and an integrated GSC are respectively described, and the introduction of profit sharing approach adopting Nash bargaining solution is discussed. Section 5 conducts numerical analyses, shows the results of the optimal operation, and describes managerial insights. Section 6 summarizes conclusions and future researches for this paper.

## 2.LITERATURE REVIEW

As issues related to this paper, the following four topics are often discussed: a variety of qualities of used products, effects of incentive for collection of used products, shift from DGSC to IGSC and supply chain coordination, and risk analysis using mean-variance theory.

First, some authors have discussed the optimal tactical production planning by incorporating uncertainty in the quality of used products into the GSC. Aras et al. (2004) investigated the issue of the stochastic nature of product returns and found conditions under which qualitybased categorization was most cost effective. Zikopoulos and Tagaras (2007) investigated how the profitability of reuse activities was affected by uncertainty regarding the quality of returned products in two collection sites and determined the unique optimal solution (procurement and production quantities). Guide et al. (2003) and Ferguson et al. (2009) assumed that returned products were classified into discrete quality category from 1 to integer n, and the procurement prices and the remanufacturing costs were different based on the corresponding quality level. Mukhopadhyay and Ma (2009) discussed a GSC consisting of a retailer who sold a single product and a manufacturer who collected used products from the market, remanufactured parts from the used products and then produced products. They assumed two situations for the remanufacturing ratio between reuse parts and used products: a constant situation and an uncertain situation. Under each situation, they proposed the optimal production strategy for the procurement quantity of used products, the remanufacturing quantity of parts from used products and the production quantity of new parts from new materials. Nenes et al. (2010) observed that both quality and quantity of returns (used products) were unfortunately highly stochastic, and investigated the optimal policies for ordering of new products and remanufacturing of products so as to maximize the companies’ performance, such as minimizing their expected cost or maximizing their expected profit. Watanabe et al. (2013) and Watanabe and Kusukawa (2014) discussed the optimal decision for lower limit of quality level of recyclable parts with a variety of quality level after disassembly of used products. Radhi and Zhang (2016) discussed the optimal minimum quality of used products with uncertain quality to accept into each operating factory in a remanufacturing supply network. Zikopoulos and Tagaras (2015) discussed a reverse supply chain with multiple collection sites and the possibility of return sorting and derived the optimal decisions for acquisition and remanufacturing lot-sizing under locations of the unreliable classification/sorting operation.

Some previous papers have discussed the second topic, effects of incentive for collection of end-of-used products on the optimal operation in a GSC. The effect of incentive for collection of end-of-used products which is paid from a manufacturer to consumers on the optimal tactical strategy in a GSC was discussed in Kaya (2010). A manufacturer who produced original products using virgin materials and remanufactured products by using returns from consumers was considered. The amount of returns depended on the incentive offered by the manufacturer. The optimal value of this incentive and the optimal production quantities of both remanufacturing parts and new parts in a stochastic demand were determined. The effect of incentive for the collection of end-of-used products which is paid from a manufacturer to a retailer on the optimal tactical production planning in a GSC was discussed in Lee et al. (2011). A model that integrated pricing, production, and inventory decisions in a reverse production system with retailer collection was discussed. The returned products were assumed valuable to the manufacturer for creating as-new products, but the retailer had to divide effort between selling the new product and collecting the returns. The manufacturer offered incentives to the retailer to participate in the overall system. The effects of a collection incentive contract on promotion of the collection and recycling activities in a GSC were discussed in Watanabe et al. (2013) and Watanabe and Kusukawa (2014). One is the incentive which is paid from a retailer to customers, the other is the incentive which is paid a manufacturer from a retailer. Two types of incentive enabled to promote the collection and the recy cling of used products. The optimal operations were made for a product order quantity, the unit collection incentive of used products and the lower limit of quality level for recycling used products. De Giovanni et al. (2016) showed that incentive strategies lead to the implementation of the total optimal solution in a dynamic closed-loop supply chain which consists of a manufacturer and a retailer.

Regarding the third topic, in a DGSC, all members in the GSC determine the optimal operation so as to maximize their profits. As one of the optimal decision-making approaches under a DGSC, the Stackelberg game has been adopted in several previous papers: Yan and Sun (2012), Watanabe et al. (2013), Watanabe and Kuskawa (2014), Hong et al. (2015), and Esmaeili et al. (2016). In the Stackelberg game, there is a single leader of the decision- making and a single (multiple) follower(s) of the decision-making of the leader. The leader of the decisionmaking determines the optimal strategy so as to maximize the leader’s (expected) profit. The follower(s) of the decision- making determine(s) the optimal strategy so as to maximize the follower(s)’s (expected) profit under the optimal strategy determined by the leader of the decisionmaking (Cachon and Netessine, 2004; Nagarajan and Sosic, 2008; Esmaeili and Zeephongsekul, 2010; Berr, 2011; Hu et al., 2011; Mukhopadhyay et al., 2011; Aust and Buscher, 2012). However, considering the aspect of the total optimization of a GSC, it is desirable to shift to the optimal operation in IGSC from that in DGSC. To achieve the shift, a variety of supply chain coordination approaches between all members which increase profits of all members in the integrated supply chain, have been discussed in the following papers and some overviews: Tsay et al. (1999), Cachon and Netessine (2004), Yano and Gilbert (2004), Nagarajan and Sosic (2008), Hu et al. (2011), Yan and Sun (2012) and Govindan et al. (2013).

The incorporation of the game theory into supply chain coordination in a GSC has been discussed by Govindan et al. (2013) as overview.

Kohli Park (1989), Hong et al. (2013), Watanabe et al. (2013), Watanabe and Kusukawa (2014), and Ghosh and Shah (2015) incorporated Nash bargaining solution (Nash, 1950; Nash, 1953; Nagarajan and Sosic, 2008) into supply chain coordination of GSC. Kohli and Park (1989) discussed a bargaining problem in which a buyer and a seller negotiate over the order quantity and the average unit price. Hong et al. (2013) discussed the Nash bargaining model which was utilized to implement profit sharing for the manufacturer and the retailer in the channel coordination of strategic alliance. Watanabe et al. (2013) and Watanabe and Kusukawa (2014) coordinated the degree of compensation for a retailer’s collection incentive of used products and the margin for wholesale per product as Nash bargaining solution between a retailer and a manufacturer. Ghosh and Shah (2015) coordinated cost sharing rate of greening cost as Nash bargaining solution between a retailer and a manufacturer.

Those previous papers which were related from the first topic to the third topic, mentioned above determined the optimal operations in a GSC under the assumption that GSC members and the decision-maker of the whole system are risk neural, that is, they determined the optimal operations so as to maximize the individual expected profit in the GSC. This implies that the above papers don’t consider the effect of variance of the individual profit on the optimal operation in a GSC. In contrast, this paper clarifies the optimal operation with considering the risk attitude in a GSC.

Risk analysis using the mean-variance theory, the last topic, is first incorporated into inventory control problem especially newsvendor problem that is the optimal product order quantity for a single type of products to sell in a single period (Lau, 1980; Chiu and Choi., 2016; Wu et al., 2009). Lau (1980) applied first the mean-variance theory in finance presented Markowitz (1959) into inventory control problem. Instead of maximizing the expected profit, this author proposed an alternative objective which aimed at maximizing an objective function of the expected profit and the standard deviation of profit. According to Chiu and Choi (2016), in 2000, Chen and Federgruen applied a mean-variance analysis to the single period newsvendor problem as well as the multi-period stock inventory problem. Wu et al. (2009) studied the risk-averse newsvendor model with a mean-variance objective function and showed that stockout cost had a significant impact on the newsvendor’s optimal ordering decisions.

Next, noticing the importance of agents’ risk preference on the supply chain decisions including supply chain coordination, studies on not only risk averse agents, but also risk prone agents become increasingly popular in recent years (Choi et al., 2008a; Choi et al., 2008c). Choi et al. (2008a) considered the newsvendor problem with different risk preferences (risk averse, risk neutral and risk prone). Choi et al. (2008c) discussed supply chain agents, a retailer and a manufacturer, with the above three risk preferences. They verified the impacts on the optimal order quantity of a single type of a product in a single period and setting the supply chain coordinating contract regarding a wholesale pricing policy when the supply chain agents have different risk preferences under an mean-variance objective which consists of both the expected profit and the variance of profit of supply chain agents.

Some previous studies applied risk analysis using mean-variance theory to reverse supply chains (Li and Cai, 2009; Zeballos et al., 2016). Li and Cai (2009) discussed a remanufacturing system considering random yield and random demand, the enterprise paid for the used product to end users to control the collection quantity. The optimization model was proposed to derive the analytical solution of the collection price before the random yield is realized, and the selling price per the remanufactured product before the demand randomness is realized so as to maximize the enterprise’s utilization, which is the mean variance function of the remanufacturer’s profit. However, they did not discuss not only the optimal decision for a remanufacturing ratio of used products in consideration of quality of used products, but also the promotion of the recycling activity of the used products. Sun and Da (2013) studied the effects of channel power structure and participants’ risk-averse attitude on differentiated pricing policies of closed-loop supply chain. Differently from the paper, this paper assumes the three types of risk attitude (risk-neutral, risk-averse, and risk-prone). Moreover, based on Watanabe et al. (2013) and Watanabe and Kusukawa (2014) with risk neutral, this paper coordinates the degree of compensation for a retailer’s collection incentive of used products and the margin for wholesale per product as discussed Nash bargaining solution between a retailer and a manufacturer as to GSC members’ risk attitudes in IGSC.

## 3.MODEL FORMULATION

### 3.1.Notation

#### 3.1.1.Indices and Sets

• E[⋅] :  expectation

• V[⋅] :  variance

• A :  a risk-averse attitude

• N :  a risk-neutral attitude

• P :  a risk-prone attitude

• i∈{A,N,P} :  a set of risk attitudes

• R :  a retailer

• M:  a manufacturer

• S :  the whole system

• j∈{R, M, S} :  a set of members in the GSC

• D :  a decentralized GSC

• I :  a integrated GSC

• k ∈{D, I} :  a set of methods of decision-making for the GSC

#### 3.1.2.Decision Variables

• Q :  product order quantity

• t :  collection incentive (collection cost) per unit of used products, referred to simply as collection incentive

• u :  lower limit of quality level to remanufacture recyclable parts after disassembly of used products, referred to simply as lower limit of quality level (0 ≤ u ≤ 1)

#### 3.1.3.Parameters

• tU :  upper limit of collection incentive t

• A(t) :  collection quantity of used products for collection incentive t

• R(t) :  compensation per unit of remanufactured recyclable parts paid to a retailer from a manufacturer for collection incentive t

• ca :  disassembly and inspection cost per unit of used products

• ct :  delivery cost per unit of used products collected from a market, to a manufacturer

• :  quality level of recyclable parts (0 ≤ ≤ 1)

• g () :  probability density function (pdf) of quality level

• cr () :  remanufacturing cost per unit of recyclable parts with quality level

• cd :  disposal cost per unit of un-recycled parts

• cn :  procurement cost per unit of new parts

• cm :  production cost per unit of products

• ma :  margin obtained from wholesale per unit of products

• w :  wholesale price per unit of products, referred to simply as the unit wholesale price

• p :  sales price per unit of products

• s :  shortage penalty cost per unit of products whose demand is unsatisfied

• hr :  inventory holding cost per unit of unsold products

• x :  demand of product in a market

• f (x) :  probability density function of demand x

• F (x) :  cumulative distribution function of demand x

• γ :  a degree of risk attitude

#### 3.1.4.Other Variables

• πj (Q, t, u):  profit of a member j

• $U j i ( Q , t , u )$ :  utility function of a member j under a risk attitude i

• $Q k i$ :  the optimal order quantity in a method of decisionmaking k under a risk attitude i

• tk :  the optimal collection incentive in a method of decision- making k

• uk :  the optimal lower limit of quality level in a method of decision-making k

### 3.2.Model Descriptions

A GSC discussed in this paper is the same as that in the following previous papers: Watanabe et al. (2013) and Watanabe and Kusukawa (2014).

#### 3.2.1.An Operational Flow of a GSC

Details of the GSC are provided as follows.

• (1) The GSC with a retailer and a manufacturer is considered. The GSC has an operational flow, from collection of a single type of used products, through remanufacturing a single type of recycled parts from the used products, to sales of a single type of products from either the recycled parts or new parts in a single period. A single type of product such as consumer electronics (mobile phone, personal computer), semiconductor, and electronic component is sold in a market.

• (2) The retailer pays collection incentive t per unit of used products (collection incentive) to collect them from a market and delivers the collected quantity A(t) of used products for collection incentive t to the manufacturer at the unit cost ct.

• (3) The manufacturer disassembles the used products to a single type of recyclable parts and inspects all the recyclable parts at the unit cost a ca. After the disassembly, the manufacturer classifies the recyclable parts into the quality level (0 ≤ ≤ 1) . The manufacturer optimally determines the lower limit of quality level u (0 ≤ u ≤ 1) for recyclable parts and remanufactures recycled parts from all the recyclable parts with quality level more than the lower limit of quality level u at the unit remanufacturing cost cr (). The manufacturer disposes all the recyclable parts with lower quality level than u at the unit cost cd.

• (4) The manufacturer pays the compensation to the retailer for the cooperation to collection of the used products. Concretely, the manufacturer pays the compensation R(t) per unit of the recycled parts to the retailer, who paid the collection incentive t to collect the quantity A(t) of the used products.

• (5) The retailer optimally determines the collection incentive t and the product order quantity Q. The retailer places an order of the products Q with the manufacturer.

• (6) The manufacturer produces the products at the unit cost cm to satisfy the order quantity Q from the retailer. If the required quantity of parts to produce Q is unsatisfied with the quantity of the recycled parts, the manufacturer procures the required quantity of new parts at the unit cost cn from external supplier.

• (7) The manufacturer sells the quantity Q of the products to the retailer at the unit wholesale price w.

• (8) The retailer sells the products in a market at the unit sales price p during a single period. The unit inventory holding cost hr of excess products for the product demand x per unit time is incurred, while the unit shortage penalty cost s for the unsatisfied product demand is incurred.

#### 3.2.2.Model Assumptions

Assumptions of the model described in this paper are as follows.

• (1) The collection quantity of used products, A(t) varies according to the collection incentive t . In general, the higher the collection incentive t is, the more a retailer can collect used products from a market. From the aspect of a retailer’s profit, the feasible range of t is 0 ≤ ttU < p . This paper assumes that the collection quantity A(t) is not enough to satisfy an order quantity of the products even if the retailer pays the upper limit tU of t.

• (2) A single type of recyclable parts is extracted from the unit of used products. The manufacturer remanufactures a single type of parts, using the single type of recyclable parts with higher quality level than u .

• (3) The variability of quality level of the recyclable parts is modeled as a probabilistic distribution with the probability density function g () .

• (4) The unit remanufacturing cost cr() to a recycled part from a recyclable part with varies as to the quality level (0 ≤ ≤1) . The lower quality level is, the higher the unit remanufacturing cost cr() is. Here, = 0 indicates the worst quality level of the recyclable parts, while =1 indicated the best quality level of recyclable parts. Therefore, cr() is a monotone decreasing function with respect to quality level .

• (5) The quality of each recycled part produced from recyclable parts is as good as that of products new parts.

• (6) The unit wholesale price w is calculated from the unit procurement cost cn of new parts, the unit production cost cm of products, and the unit margin ma obtained from wholesales per product.

• (7) The variability of product demand x is modeled as a probabilistic distribution with the probability density function f (x) .

### 3.3.Expectation and Variance of Individual Profit in a GSC

Based on Subsection 3.2, profits of the retailer, the manufacturer, the whole system, are formulated (Watanabe et al., 2013; Watanabe and Kusukawa, 2014). From the profits, expectations and variances of them are derived in this subsection to apply mean-variance theory to the determining the optimal operations in the GSC.

The retailer’s profit consists of the collection cost of used products from a market, the delivery cost of used products to a manufacturer, the procurement cost of products, the compensation revenue for the retailer’s cooperation of the collection of used products, the product sales, the inventory holding cost of unsold products, and the shortage penalty cost for unsatisfied product demand in a market.

The retailer’s profit for the order quantity Q, the collection incentive t, and the lower limit of quality level u , πR (Q, t, u) is formulated as

$π R ( Q , t , u ) = − t A ( t ) − c t A ( t ) − w Q + { p x − h r ( Q − x ) ( 0 ≤ x ≤ Q ) p Q − s ( x − Q ) ( Q ≤ x ) + R ( t ) ∫ u 1 A ( t ) g ( ℓ ) d ℓ$
(1)

Taking expectation with respect to the product demand x in Eq. (1), the retailer’s expected profit for the order quantity Q, the collection incentive t, and the lower limit of quality level $E [ π R ( Q , t , u ) ]$ can be derived as

$E [ π R ( Q , t , u ) ] = − t A ( t ) − c t A ( t ) − w Q + R ( t ) ∫ u 1 A ( t ) g ( ℓ ) d ℓ + p { ∫ 0 Q x f ( x ) d x + ∫ Q ∞ Q f ( x ) d x } − h r ∫ 0 Q ( Q − x ) f ( x ) d x − s ∫ Q ∞ ( x − Q ) f ( x ) d x$
(2)

In Eq. (2), the first term is the collection cost of the used products from customers, the second term is the delivery cost of the used products to a manufacturer, the third term is the procurement cost of products, the fourth term is the compensation revenue from a manufacturer to a retailer, the fifth term is the expected product sales, the sixth term is the expected inventory holding cost of the unsold products, and the final term is the expected shortage penalty cost for unsatisfied product demand in a market.

From Eqs. (1) and (2), variance of the retailer’s profit for the order quantity Q, the collection incentive t, and the lower limit of quality level u , V [$π R ( Q , t , u )$] can be derived as

$V [ π R ( Q , t , u ) ] = E [ { π R ( Q , t , u ) } 2 ] { E [ π R ( Q , t , u ) ] } 2 = s 2 V [ x ] + 2 ( p + h r + s ) ( − p − h r + s ) ∫ 0 Q x F ( x ) d x − ( p + h r + s ) 2 { ∫ 0 Q F ( x ) d x } 2 + 2 ( p + h r + s ) { Q ( p + h r ) − s E [ x ] } ∫ 0 Q F ( x ) d x$
(3)

The manufacturer’s profit consists of the product wholesales, the disassembly and the inspection costs of the used products, the remanufacturing cost of recyclable parts, the compensation cost to the retailer, the disposal cost of the un-recycled parts, the procurement cost of new parts, and the production cost of products.

The manufacturer’s profit for the order quantity Q, the collection incentive t, and the lower limit of quality level u, πM (Q, t, u) is formulated as

$π M ( Q , t , u ) = w Q − R ( t ) ∫ u 1 A ( t ) g ( ℓ ) d ℓ − c a A ( t ) − ∫ u 1 A ( t ) c r ( ℓ ) g ( ℓ ) d ℓ − c d ∫ 0 u A ( t ) ( ℓ ) d ℓ − c n { Q − ∫ u 1 A ( t ) g ( ℓ ) d ℓ } − c m Q$
(4)

In Eq. (4), the first term is the product wholesales, the second term is the compensation cost to a retailer, the third term is the disassembly and the inspection cost of the used products, the fourth term is the remanufacturing cost of recyclable parts, the fifth term is the disposal cost of the un-recycled parts, the sixth term is the procurement cost of new parts, and the final term is the production cost of products.

Here, from Eq. (4), it can be seen that the manufacturer’s profit is unaffected by product demand x . Therefore, expectation and variance of the manufacturer’s profit for the order quantity Q , the collection incentive t , and the lower limit of quality level u , E[$π M ( Q , t , u )$] and V[$π M ( Q , t , u )$] can be respectively derived as

$E [ π M ( Q , t , u ) ] = π M ( Q , t , u )$
(5)

$V [ π M ( Q , t , u ) ] = 0$
(6)

The whole system’s profit for the order quantity Q , the collection incentive t , and the lower limit of quality level u , πS (Q, t, u) is calculated from the sum of the retailer’s profit and the manufacturer’s profit as

$π S ( Q , t , u ) = π R ( Q , t , u ) + π M ( Q , t , u ) .$
(7)

Therefore, the whole system’s expected profit E[πS (Q, t, u)] is obtained as the sum of both member’s expected profits, corresponding to

$E [ π S ( Q , t , u ) ] = E [ π R ( Q , t , u ) ] + E [ π M ( Q , t , u ) ] = − t A ( t ) − c t A ( t ) − c a A ( t ) − ∫ u 1 A ( t ) c r ( ℓ ) g ( ℓ ) − c d ∫ 0 u A ( t ) g ( ℓ ) d ℓ − c n { Q − ∫ u 1 A ( t ) g ( ℓ ) d ℓ } − c m Q + p { ∫ 0 Q x f ( x ) d x + ∫ Q ∞ Q f ( x ) d x } − h r ∫ 0 Q ( Q − x ) f ( x ) d x − s ∫ Q ∞ ( x − Q ) f ( x ) d x$
(8)

From Eq. (8), it can be seen that the terms regarding the wholesales of products and the compensation for the collection incentive accruing between the retailer and the manufacturer are canceled out.

From Eq. (6), accordingly, variance of the whole system’s profit is obtained as

$V [ π S ( Q , t , u ) ] = V [ π R ( Q , t , u ) ] .$
(9)

### 3.4.Mean-Variance Analysis of Profits in a GSC for Uncertainty in Product Demand

This subsection conducts the mean-variance analysis (Choi et al., 2008c; Wu et al., 2009) of individual profits in a GSC for the uncertainty in the product demand. Concretely, by using utility functions of a retailer, a manufac turer, and the whole system in a GSC, the risk analysis is conducted for three types of risk attitudes regarding the uncertainty in the product demand: a risk-neutral attitude (N), a risk-averse attitude (A), and a risk-prone attitude (P). The attitude N makes a decision without consideration of variance of profit in a GSC. The attitude A, with negative consideration of variance of profit in a GSC, hopes to stabilize the profit, while the attitude P, with positive consideration of variance of profit in a GSC, weighs heavily improvement in chances to generate large profit rather than stability of the profit.

The utility functions of a retailer, a manufacturer, and the whole system, in a GSC can be obtained based on the expectation and the variance of the individual profit from Eqs. (2), (3), (5), (6), (8) and (9).

Therefore, the utility functions of member j∈{R, M, S} under risk attitude i∈{N, A, P} for the order quantity Q, the collection incentive t, and the lower limit of quality level u are defined as

$U j i ( Q , t , u ) = E [ π j ( Q , t , u ) ] + γ V [ π j ( Q , t , u ) ]$
(10)

Here, γ denotes a degree of risk attitude. Accordingly, γ = 0 indicates the risk-neutral attitude (i = N) , γ < 0 indicates the risk-averse attitude (i = A) , and γ > 0 indicates the risk-prone attitude (i = P) .

## 4.DECISION-MAKING IN A GSC

Using the utility functions defined in Subsection 3.4, the optimal decisions for product order quantity, collection incentive, and lower limit of quality level, in DGSC and IGSC, are made.

### 4.1.Optimal Operation in a Decentralized GSC

For the optimal operations in a decentralized GSC (DGSC), the optimal decision approach of the Stackelberg game is adopted.

In the GSC discussed in this paper, the retailer’s optimal decision for collection incentive affects not only the maximization of the retailer’s utility function, but also the manufacturer’s optimal decision for the lower limit of quality level so as to maximize the manufacturer’s utility function, and vice versa. Accordingly, this paper determines the optimal operation in DGSC by adopting the Stackelberg game. In this paper, the retailer is the leader of the decision-making in DGSC and the manufacturer is the follower of the decision-making. This is because the following situation is considered in this paper: the retailer not only pays the collection incentive t to collect used products from a market so as to cooperate the encouragement of the manufacturer’s recycling activity of used products, but also faces stochastic demand of products in a market, sells the product in the market, and earns the most profit of the whole system in a GSC.

This paper regards the retailer and the manufacturer as the leader and the follower of the decision-making in DGSC, respectively. Then, the retailer determines the optimal product order quantity $Q D i$ and the optimal collection incentive tD so as to maximize the retailer’s utility function in Eq. (10) considering lower limit of quality level u under the degree γ of risk attitude i. The manufacturer determines the optimal lower limit of quality level uD so as to maximize the manufacturer’s utility function in Eq. (10) considering collection incentive t under the degree γ of risk attitude i .

#### 4.1.1.Optimal Product Order Quantity in DGSC

The optimal product order quantity in DGSC under the degree γ of risk attitude i∈{N, A, P} , $Q D i$ , is determined so as to maximize the retailer’s utility function $U R i ( Q , t , u )$.

• Proposition 1: $Q D i$ is determined regardless of t and u under any risk attitude i∈{N,A,P} .

• Proof: The first order partial derivative of $U R i ( Q , t , u )$ with respect to Q , $∂ U R i ( Q , t , ) / ∂ Q$, is obtained as

$∂ U R i ( Q , t , u ) ∂ Q = − w + p + s − ( p + h r + s ) F ( Q ) + 2 γ ( p + h r + s ) { s ( Q − E [ x ] ) F ( Q ) + ( p + h r ) ∫ 0 Q F ( x ) d x − ( p + h r + s ) F ( Q ) ∫ 0 Q F ( x ) d x }$
(11)

(See Appendix B). From Eq. (11), it can be seen that $∂ U R i ( Q , t , ) / ∂ Q$ doesn’t depend on t and u in DGSC. This results in Proposition 1.

• Proposition 2: The retailer’s utility function under attitude N, $U R N ( Q , t , u )$, is the concave function with respect to Q under t and u.

• Proof: From Eq. (11), the second order partial derivative of $U R N ( Q , t , u )$ under attitude N with respect to Q is obtained as

$∂ 2 U R N ( Q , t , u ) ∂ Q 2 = − ( p + h r + s ) f ( Q ) .$
(12)

Also, Eq. (12) is negative from general conditions p > 0 , 0, hr > and s > 0 . Therefore, $U R N ( Q , t , u )$ is the concave function with respect to Q under t and u.

• Proposition 3:  The optimal product order quantity in DGSC under attitude N, $Q D N$, can be determined as the following unique solution to maximize $U R N ( Q , t , u )$:(13)

$Q D N = F − 1 ( − w + p + s p + h r + s ) .$
(13)

• Proof. The solution of $∂ U R N ( Q , t , ) / ∂ Q$ results in Proposition 3.

However, it is impossible to prove that the utility function $U R i ( Q , t , u )$ under attitude A or attitude P is the concave function with respect to Q under t and u . Hence, $Q D i$ under attitude A or attitude P is determined by the numerical search so as to maximize the utility function $U R i ( Q , t , u )$ under attitude A or attitude P.

#### 4.1.2.Optimal Decisions for Collection Incentive and Lower Limit of Quality Level in DGSC

Under the optimal order quantity $Q D i$ determined in Subsection 4.1.1, the collection incentive t and the lower limit of quality level u are optimized. The variance of the retailer’s profit, V[$π R ( Q , t , u )$] in Eq. (3) is unaffected by t and u. Therefore, the optimal decisions for t and u in DGSC under any risk attitude i∈{N,A,P} result in same ones.

• Proposition 4: The optimal decision for u under DGSC is unaffected by $Q D i$ but depends on t .

• Proof: The first order partial derivative of the manufacturer’s utility function under $Q D i$ , $U M i ( Q D i , t , u )$, with respect to u, $∂ U M i ( Q D i , t , u ) / ∂ u$, is obtained as

$∂ U M i ( Q D i , t , u ) ∂ u = A ( t ) g ( u ) { c r ( u ) + R ( t ) − c d − c n }$
(14)

(See Appendix C). From Eq. (14), it can be seen that $∂ U M i ( Q D i , t , u ) / ∂ u$ doesn’t depend on $Q D i$ but depend on t in DGSC. This results in Proposition 4.

Here, Eq. (14) is zero if the following condition:(15)

$c r ( u ) + R ( t ) − c d − c n = 0$
(15)

is satisfied.

• Proposition 5: A process to determine the provisional optimal lower limit of quality level in DGSC, uD(t), is shown as follows:

$i . I f c r ( 1 ) + R ( t ) − c d − c n > 0 , u D ( t ) = 1. i i . I f c r ( 0 ) + R ( t ) − c d − c n < 0 , u D ( t ) = 0. i i i . O t h e r w i s e , u D ( t ) = c r − 1 { c d + c n − R ( t ) } .$

• Proof: See Appendix D.

• Proposition 6: The optimal decision for t in DGSC is unaffected by Q .

• Proof: From Eqs. (2) and (3), E[πR(Q, t, u)] has no term depending on both of t and Q , and V[$π R ( Q , t , u )$] doesn’t depend on t. Therefore, the first order partial derivative of the retailer’s utility function $U R i ( Q , t , u )$ with respect to , $∂ U R i ( Q , t , ) / ∂ t$, doesn’t depend on Q.

The decision procedure for the optimal collection incentive tD and the optimal lower limit of quality level uD is shown as follows:

• [Step 1]  Substitute t and uD (t) into the utility function $U R i ( Q D i , t , u )$.

• [Step 2]  By varying t within 0 ≤ ttU, determine the optimal combination of collection incentive t and the provisional optimal lower limit of quality level uD (t), which maximizes $U R i ( Q D i , t , u D ( t ) )$, as $( t D , u D )$ in DGSC.

Thus, the optimal collection incentive tD and the optimal lower limit of quality level uD are determined mutually between a retailer and a manufacturer in DGSC.

### 4.2.Optimal Operation in an Integrated GSC

Under an Integrated GSC (IGSC), the optimal product order quantity $Q I i$, the optimal collection incentive , tI and the lower limit of quality level uI are determined so as to maximize the whole system’s utility function $U S i ( Q , t , u )$ under the degree γ of risk attitude i∈{N, A, P} .

#### 4.2.1.Optimal Product Order Quantity in IGSC

The optimal product order quantity $Q I i$ in IGSC under the degree γ of risk attitude i∈{N, A, P} is determined so as to maximize the whole system’s utility function $U S i ( Q , t , u )$ .

• Proposition 7: $Q I i$ is determined regardless of t and u under any risk attitude.

• Proof: The first order partial derivative of $U S i ( Q , t , u )$ with respect to Q , $∂ U S i ( Q , t , ) / ∂ Q$ is obtained as

$∂ U S i ( Q , t , u ) ∂ Q = − c m − c n + p + s − ( p + h r + s ) F ( Q ) + 2 γ ( p + h r + s ) { s ( Q − E [ x ] ) F ( Q ) + ( p + h r ) ∫ 0 Q F ( x ) d x − ( p + h r + s ) F ( Q ) ∫ 0 Q F ( x ) d x }$
(16)

(See Appendix E). From Eq. (16), it can be seen that $∂ U S i ( Q , t , ) / ∂ Q$ doesn’t depend on t and u in IGSC.

This results in Proposition 7.

• Proposition 8: The whole system’s utility function under attitude N, $U S N ( Q , t , u )$, is the concave function with respect to Q under t and u.

• Proof: From Eq. (16), the second order partial derivative of $U S N ( Q , t , u )$ under attitude N with respect to Q is obtained as

$∂ 2 U S N ( Q , t , u ) ∂ Q 2 = ( p + h r + s ) f ( Q )$
(17)

Also, Eq. (17) is negative from general conditions p > 0, 0, hr > and s > 0. Therefore, $U S N ( Q , t , u )$ is the concave function with respect to Q under t and u .

• Proposition 9: The optimal product order quantity in IGSC under attitude N, $Q I N$, can be determined as the following unique solution to maximize $U S N ( Q , t , u )$:(18)

$Q I N = F − 1 ( − c m − c n + p + s p + h r + s ) .$
(18)

• Proof: The solution of $∂ U S N ( Q , t , u ) / ∂ Q = 0$ results in Proposition 9.

However, it is impossible to prove that the utility function $U S i ( Q , t , u )$ under attitude A or attitude P is the concave function with respect to Q under t and u . Hence, $Q I i$ under attitude A or attitude P is determined by the numerical search so as to maximize the utility function $U S i ( Q , t , u )$ under attitude A or attitude P.

#### 4.2.2.Optimal Decisions for Collection Incentive and Lower Limit of Quality Level in IGSC

Under the optimal order quantity $Q I i$ determined in Subsection 4.2.1, the collection incentive t and the lower limit of quality level u are optimized. As with DGSC, the optimal decisions for t and u in IGSC under any risk attitude i∈{N, A, P} result in same ones.

• Proposition 10: The optimal decision for u under IGSC is unaffected by $Q I i$ and t .

• Proof: The first order partial derivative of the whole system’s utility function under $Q I i$, $U S i ( Q I i , t , u )$ , with respect to u, $∂ U S i ( Q I i , t , u ) / ∂ u$ , is obtained as

$∂ U S i ( Q I i , t , u ) ∂ u = A ( t ) g ( u ) { c r ( u ) − c d − c n }$
(19)

(See Appendix F). From Eq. (19), it can be seen that $∂ U S i ( Q I i , t , u ) / ∂ u$ doesn’t depend on $Q I i$ and t. This results in Proposition 10.

Here, Eq. (19) is zero if the following condition:(20)

$c r ( u ) − c d − c n = 0$
(20)

is satisfied.

• Proposition 11: A process to determine the optimal lower limit of quality level in IGSC, uI, is shown as follows:

$i . I f c r ( 1 ) − c d − c n > 0 , u I = 1. i i . I f c r ( 0 ) − c d − c n < 0 , u I = 0. i i i . O t h e r w i s e , u I = c r − 1 ( c d + c n ) .$

• Proof:  See Appendix G.

• Proposition 12: The optimal decision for t in IGSC is unaffected by Q .

• Proof: From Eqs. (3) and (8), E[πS (Q, t, u)] has no term depending on both of t and Q , and V[πS (Q, t, u)] doesn’t depend on t. Therefore, the first order partial derivative of the whole system’s utility function $U S i ( Q , t , u )$ with respect to , $∂ U S i ( Q , t , ) / ∂ t$ doesn’t depend on Q.

The decision procedure for the optimal collection incentive tI in IGSC is shown as follows:

• [Step 1]  Substitute t into the utility function $U S i ( Q I i , t , u I )$.

• [Step 2]  By varying t within 0 ≤ ttU, determine the optimal collection incentive, which maximizes $U S i ( Q I i , t , u I )$, as tI in IGSC.

### 4.3.Profit Sharing as Supply Chain Coordination

When the optimal operation in a GSC shifts from DGSC to IGSC, it is more likely to increase the whole system’s expected profit. However, note that there is a probability that the whole system’s expected profit under IGSC is lower than that in DGSC because the objective function is the utility function which isn’t equal to the expected profit under attitudes A and P. Also, it is not always guaranteed that the expected profit of each member in IGSC can be improved even if the whole system’s expected profit increases in IGSC. Nevertheless, it is desirable to shift to the optimal operation in IGSC from that in DGSC, considering the aspect of the total optimization maximizing the expected profit of the whole system in a GSC.

This subsection discusses supply chain coordination (SCC) between a retailer and a manufacturer to guarantee the more expected profit for both members under the optimal decision in IGSC when the optimal operation in a GSC shifts from DGSC to IGSC and the whole system’s expected profit in IGSC is higher than that in DGSC.

As SCC in IGSC, this paper adopts the profit sharing approach using the Nash bargaining solution (Watanabe et al., 2013; Watanabe and Kusukawa, 2014).

Concretely, this paper coordinates the unit wholesale price w and the unit compensation R(t) of used products collected by a retailer at the collection incentive t between both members. Here, w and R(t) are respectively set as

$w = c n + c m + m a$
(21)

$R ( t ) = ( 1 + α ) t$
(22)

Eqs. (21), (22) are similar to Watanabe et al. (2013), Watanabe and Kusukawa (2014). Then, as to the optimal operation for the degree γ of risk attitude, the degree α of compensation for the retailer’s collection incentive and the margin ma for wholesale per product are coordinated as the Nash bargaining solution $( α i N a s h , m a i N a s h )$. The wholesale price wi Nash and compensation RiNash (t) coordinated under IGSC are calculated by substituting the coordinated parameters in IGSC αi Nash and mai Nash into Eqs. (21) and (22).

From the aspect of the retailer’s profit, 0 ≤ wi Nashp is satisfied. Also, it is natural to consider that a retailer can cooperate to collect aggressively used products only if the following condition is satisfied:

$r < R ( t ) ∫ u 1 g ( ℓ ) d ℓ .$
(23)

Eq. (23) indicates the condition that the compensation which a manufacturer pays to a retailer is higher than the unit collection inventive t which a retailer incurs.

From the aspect of the manufacturer’s profit, it is natural to consider that a manufacturer can pay the compensation to a retailer only if the following condition is satisfied:

$R ( t ) ∫ u 1 g ( ℓ ) d ℓ + c a + ∫ u 1 c r ( ℓ ) g ( ℓ ) d ℓ + c d ∫ 0 u g ( ℓ ) d ℓ ≤ c n .$
(24)

Eq. (24) indicates the condition that the total unit cost associated with the recycling of used products, which is the sum of the unit cost regarding the compensation of collection of used products, the disassembly and inspection of used products, the remanufacturing, and disposal of recyclable parts, is lower than the unit procurement cost of new parts.

Here, the profits of the retailer and the manufacturer are redefined as(25)

$π j , k i N a s h ( α , m a ) = π j ( Q k i , t k , u k )$
(25)

so that the degree α of compensation and the margin ma are variables.

The coordinated parameters (αi Nash, mai Nash) are determined so as to maximize Eq. (26) satisfying the constrained conditions in Eqs. (27) and (28):

$T ( α i N a s h , m a i N a s h ) = { E [ π R , I i N a s h ( α i N a s h , m a i N a s h ) ] − E [ π R , I i N a s h ( α , m a ) ] } × { E [ π M , I i N a s h ( α i N a s h , m a i N a s h ) ] − E [ π M , D i N a s h ( α , m a ) ] }$
(26)

subject to

${ E [ π R , I i N a s h ( α i N a s h , m a i N a s h ) ] − E [ π R , D i N a s h ( α , m a ) ] } > 0 ,$
(27)

${ E [ π M , I i N a s h ( α i N a s h , m a i N a s h ) ] − E [ π M , D i N a s h ( α , m a ) ] } > 0.$
(28)

The details of Eqs. (26)-(28) are provided as follows. Eq. (26) coordinates the unit wholesale price w and the unit compensation R(t) of used products as αi Nash and mai Nash under the optimal operation in IGSC for the degree γ of risk attitude. αi Nash and mai Nash are determined so as to maximize the multiplication of (the difference of the expected profit of the retailer for αi Nash and mai Nash coordinated under the optimal decision in IGSC as to the optimal operation for the degree γ of risk attitude and that for R(t) and w not coordinated under the optimal decision in DGSC as to the optimal operation for the degree γ of risk attitude) and (the difference of the expected profit of the manufacturer for αi Nash and mai Nash coordinated under the optimal decision in IGSC as to the optimal operation for the degree γ of risk attitude and that for R(t) and w not coordinated under the optimal decision of DGSC as to the optimal operation for the degree γ of risk attitude).

Eq. (27) is the constrained condition to guarantee the situation where the expected profit of the retailer for αi Nash and mai Nash coordinated under the optimal decision in IGSC as to the optimal operation for the degree γ of risk attitude is always higher than that for R(t) and w not coordinated under the optimal decision in DGSC as to the optimal operation for the degree γ .

As with Eq. (27), Eq. (28) is the constrained condition to guarantee the situation where the expected profit of the manufacturer for αi Nash and mai Nash coordinated under the optimal decision in IGSC as to the optimal operation for the degree γ of risk attitude is always higher than that for R(t) and w not coordinated under the optimal decision in DGSC as to the optimal operation for the degree γ .

The decision procedure for the optimal combination of (αi Nash, mai Nash) is shown as follows:

• [Step 1]  Substitute (αi Nash, mai Nash) into T(αi Nash, mai Nash) in Eq. (26).

• [Step 2]  By varying (αi Nash, mai Nash) to satisfy 0 ≤ wi Nashp and Eqs. (23)-(24), determine the optimal combination (αi Nash, mai Nash), which maximizes T(αi Nash, mai Nash) with satisfying Eqs. (27) and (28).

## 5.NUMERICAL ANALYSES

This section numerically investigates how (i) three types of risk attitude regarding product demand and (ii) quality of recyclable parts affect the optimal operations for DGSC and IGSC. Concretely, the optimal operation regarding the optimal product order quantity, the optimal collection incentive, and the optimal lower limit of quality level, and the expected profits in DGSC are compared with those in IGSC by changing (i) the degree γ of risk attitude and (ii) the quality distribution of recyclable parts. Moreover, it shows that SCC can bring more profits to a retailer and a manufacturer in IGSC and enable to shift to IGSC from DGSC. Concretely, the unit wholesale price and compensation for collection incentive of used products are coordinated between both members in IGSC as Nash bargaining solutions.

### 5.1.Numerical Examples

The following numerical examples are used as p = 150, s = 175, hr = 15, ca = 1, cd = 1, ct = 1 ct = 1, cn = 35, cm = 2, ma = 15. A(t) and cr () are respectively set as A(t) =100 + 50t and cr () = 40(1-0.9), satisfying the properties of functions in Subsection 3.2.2 (1), (4), and (6) (Watanabe et al., 2013; Watanabe and Kusukawa, 2014).

The degree of compensation α for the retailer’s unit collection incentive t is set as α = 0.7 within the feasible range to satisfy Eqs. (23) and (24).

The product demand x follows the normal distribution with the mean μ =1,000 and the variance σ2 = 3002.

In this paper, the variation in the quality level (0 ≤ ≤1) of recyclable parts in used products is modeled by using the beta distribution. This is because the beta distribution is widely used to measure relative parameters like level (0 ≤ ≤1) , or anything that is between 0-1. Also, the beta distribution can express various shapes of distribution of recyclable parts in used products such as the uniform distribution-type shape, the normal distribution-type shape, the exponential distribution-type shape, the left-biased distribution shape, the right-biased distribution shape by using the following probability density function g () of the beta distribution with parameters (a,b) :(29)

$g ( ℓ ) = Γ ( a + b ) Γ ( a ) + Γ ( b ) ℓ a − 1 ( 1 − ℓ ) b − 1$
(29)

where Γ(⋅) denotes the gamma function. This paper provides the following four cases of quality distribution B(a, b) of recyclable parts (Watanabe et al., 2013; Watanabe and Kusukawa, 2014):

• Case 1 B(1, 1) :  a situation where quality of each recyclable parts is distributed uniformly,

• Case 2 B(2, 2) :  a situation where there are many recyclable parts with middle quality level,

• Case 3 B(3, 2) :  a situation where there are many recyclable parts with the relatively high quality level,

• Case 4 B(2, 3) :  a situation where there are many recyclable parts with the relatively low quality level.

Figure 1 shows four cases of the quality distribution of recyclable parts in used products modeled as the beta distribution B( | a, b) in Cases 1-4.

### 5.2.Results of Numerical Analyses

#### 5.2.1.Effect of Risk Attitude on the Optimal Product Order Quantities, the Expected Profits, and the Variance of the Profits, in DGSC and IGSC

Here, effect of the degree γ of risk attitude on the optimal product order quantity and the expected profits in DGSC and IGSC is discussed.

Table 1 shows the optimal product order quantities in DGSC and IGSC, $Q k i$ (k ∈{D, I}) , as to the degree γ of risk attitude. Table 2 shows the effect of the degree γ of risk attitude on the expected profits of a retailer, a manufacturer, and the whole system in DGSC and IGSC in Case 2 of quality level of recyclable parts. Figures 2 shows the effect of the degree γ of risk attitude on the optimal product order quantities $Q k i$. Figure 3 shows the effect of the product order quantity Q on the variance of the retailer’s profit, V[$π R ( Q , t , u )$].

From Table 1, it can be seen that, except for γ = 1.0 × 10-3, the higher γ is, the more $Q k i$ is in DGSC and IGSC. Accordingly, γ represents the characteristics that a decision-maker with attitude A tends to decrease the optimal order quantities, while that with attitude P tends to increase them. Note that the quality distribution doesn’t affect the optimal order quantities.

Next, from Figure 2, it can be seen that the higher the degree γ of risk attitude is, the larger the optimal product order quantities $Q k i$ are, but $Q k i$ happen to drop sharply, as with Table 1. This is because γ is so high that the influence of V[$π R ( Q , t , u )$] on the utility functions of a retailer and the whole system becomes very large. It leads to disregarding the expected profits of them. In addition, from Figure 3, it can be seen that V[$π R ( Q , t , u )$] for smaller product order quantity than the product order quantity which minimizes V[$π R ( Q , t , u )$] is higher than V[$π R ( Q , t , u )$] for larger product order quantity than the product order quantity. As a result, very small product order quantity is determined as the optimal product order quantity. From Table 2，it is confirmed that the expected profits of a retailer and the whole system are quite low under γ =1.0×10−3 . Therefore, it is necessary for a retail- er in DGSC and the decision-maker in IGSC with attitude P to estimate carefully the degree γ of risk attitude when the optimal product order quantity is decided.

Table 3 shows effect of the degree γ of risk attitude on the variances of profits and the utility functions in addition to the expected profits in DGSC and IGSC. From Table 3, it can be seen that the variance of the retailer’s profit and the retailer’s utility function in DGSC and the variance of the whole system’s profit and the whole system’s utility function in IGSC increase as the degree γ . Also, when the absolute of the degree γ is high (γ = −1.0×103, 1.0×103 ), the utility functions are almost composed of the only profit variances. It is one of the causes of the sharp drop in the product order quantities in Table 1 and the expected profits in Table 2 and Table 3.

Figure 4 shows the behaviors of the utility function $U R A ( Q , t D , u D )$, the expected profit E[$π R ( Q , t D , u D )$] , and the variance multiplied by γ , γV[$π R ( Q , t D , u D )$], of a retailer with attitude A for product order quantity Q under tD , uD , and γ = −1.0×10−5 in Case 2 of the quality distribution. Figure 5 shows the behaviors of the utility function $U R P ( Q , t D , u D )$, the expected profit E[$π R ( Q , t D , u D )$], and the variance multiplied by γ , γV[$π R ( Q , t D , u D )$], of a retailer with attitude P for product order quantity Q under tD , uD , and γ =1.0×10−5 in Case 2 of the quality distribution. Note that the utility function is the sum of the expected profit and the variance of the profit multiplied by γ .

From Figure 4, it can be seen that the optimal product order quantity under γ = −1.0×10−5 (attitude A) to maximize the utility function of a retailer is smaller than that under attitude N. In contrast, from Figure 5, it can be seen that the optimal product order quantity under γ =1.0×10−5 (attitude P) to maximize the utility function of a retailer is larger than that under attitude N. The reason is that the product order quantity which minimize the variance of the retailer’s expected profit is smaller than the optimal order quantity under attitude N.

#### 5.2.2.Comparison of Optimal Operations and Expected Profits in DGSC and IGSC

In this subsection, the optimal decisions for the product order quantity, the collection incentive, and the lower limit of quality level in DGSC are compared with those in IGSC.

##### i.The optimal product order quantity

Table 1 indicates that the optimal product order quantity in IGSC tend to be larger than the optimal product order quantity in DGSC. Under γ = −1.0×10−3 , the optimal product order quantity in IGSC is equal to that in DGSC. This is because the expected profits are disregarded and the variance of a retailer’s profit and that of the whole system’s one are same as Eq. (9).

##### ii.The optimal collection incentive and the optimal lower limit of quality level

Table 4 shows results of the optimal decisions for the collection incentive tk (k ∈{D, I}) and the lower limit of quality level uk (k ∈{D, I}) in the four cases of quality distribution of recyclable parts. Note that tk and uk are unaffected by the optimal product order quantity $Q k i$ from Eqs. (14) and (19). Also, as described in Subsection 4.1.1, the variance of the retailer’s profit, V[$π R ( Q , t , u )$], is unaffected by t and u. Therefore, k t and k u in Table 4 are the same as those in Watanabe and Kusukawa (2014), which determined them with no consideration of the variance.

From Table 4, the following results can be seen:

• When there are many recyclable parts with high quality level as Case 3, the optimal collection incentives in DGSC and IGSC become the highest among other cases in order to increase the number of collected used products, and vice versa,

• The optimal collection incentive in IGSC is higher than that in DGSC in any case. This is because the compensation calculated from the unit collection incentive impacted on only the optimal decision in DGSC. The compensation term is canceled out when the expected profit of the whole system is calculated in IGSC. Also, described in Subsection 3.2.2, the higher the collection incentive t is, the more a retailer can collect used products. Therefore, shift to IGSC promotes the collection activity of used products,

• The optimal lower limit of quality level in IGSC is lower than that in DGSC in any case. It means increment in remanufacturing ratio. Therefore, shift to IGSC promotes the remanufacturing activity. Moreover, the optimal lower limit of quality level in IGSC is unaffected by the quality distribution of recyclable parts.

##### iii.The expected profits of a retailer, a manufacturer, and the whole system

From Table 2, it can be seen that the whole system’s expected profit in IGSC is higher than that in DGSC except for γ = 1.0 × 10−5 and 1.0 × 10−4 .

However, there are situations where the whole system’s expected profit in IGSC is lower than that in DGSC because the objective function is not an expected profit but a utility function. Also, even if the whole system’s expected profit in IGSC is higher than that in DGSC, the manufacturer’s expected profit in DGSC may be lower than that in IGSC under attitude A and the retailer’s expected profit in DGSC may be lower than that in IGSC under attitude P.

Under the situation, in order to shift from the optimal decisions in DGSC to that in IGSC, some reasonable profit sharing is necessary to guarantee more profit of both members in IGSC than that in DGSC.

##### 5.2.3.Effect of supply chain coordination (SCC) on the expected profits in IGSC

Here, the unit wholesale price and the degree of compensation for the collection incentive are adjusted as supply chain coordination (SCC) using Nash bargaining solutions by Eqs. (26)- (28) in Subsection 4.3.

Table 5 shows effect of supply chain coordination (SCC) on the expected profits in IGSC in Case 2 of quality level of recyclable parts. In Table 5, * denotes that it is impossible to conduct SCC between a retailer and a manufacturer in IGSC. From Table 5, it can be seen that the SCC can guarantee that the expected profits of both members in IGSC are higher than those in DGSC when the whole system’s expected profit in IGSC is higher than that in DGSC. Meanwhile, when the whole system’s expected profit in IGSC is lower than that in DGSC, shift from the optimal decisions in DGSC to those in IGSC shouldn’t be conducted.

Therefore, it is verified that the incorporation of the SCC into the optimal operation in IGSC enables to shift from DGSC to IGSC, when the whole system’s expected profit in IGSC is higher than that in DGSC.

## 6.CONCLUSIONS

This paper incorporated mean-variance analysis on the uncertainty in product demand into the modeling and the theoretical analysis in a GSC with a retailer and a manufacturer. Concretely, the following three types of risk attitude regarding the uncertainty in product demand were discussed: risk-neutral attitude without consideration of variance of profit in a GSC, risk-averse attitude with negative consideration of variance of profit in a GSC, and risk-prone attitude with positive consideration of variance of profit in a GSC. In each risk attitude, the optimal decisions for product order quantity, collection incentive, and lower limit of quality level, were conducted in DGSC and IGSC by mathematical analyses and numerical searches.

Numerical analysis clarified how (i) three types of risk attitude regarding product demand and (ii) quality of recyclable parts affected the optimal operations for DGSC and IGSC.

In addition, when the whole system’s expected profit in IGSC is higher than that in DGSC, the profit sharing approach was introduced into IGSC as supply chain coordination to encourage the both members to make the shift from DGSC to IGSC.

Here, the profit sharing approach which coordinated the unit wholesale price and the compensation per unit of recycled parts as Nash bargaining solutions was discussed. It was verified that the shift of the optimal operation from DGSC to IGSC could guaranteed the improvement of the expected profits of a retailer and a manufacturer by conducting the profit sharing approach with Nash bargaining solutions when the whole system’s expected profit in IGSC is higher than that in DGSC.

Results of theoretical analysis and numerical analysis in this paper gave the following managerial insights:

• Using the mathematical analyses and the numerical searches, the optimal decisions for product order quantity, collection incentive, and lower limit of quality level, in DGSC and IGSC as to risk attitude, can be made,

• The optimal order quantity depends on the risk attitude, while the optimal collection incentive and the optimal lower limit of quality level are unaffected by the risk attitude,

• As a degree of risk attitude is high, the optimal product order quantity tends to be larger, but decision makers with risk-prone attitude should be careful for the optimal product order quantity too small and the expected profit too low,

• Under the decision-making with mean-variance theory, there is a probability that the expected profit of the whole system in IGSC is lower than that in DGSC,

• When the expected profit of the whole system in IGSC is higher than that in DGSC, the SCC with Nash bargaining solutions enables to shift from DGSC to IGSC.

As future researches, it will be necessary to incorporate the following topics into a GSC model in this paper:

• Risk analysis in a GSC and formulation of the utility functions of GSC members and the whole system considering not only the uncertainty in product demand but also the uncertainty in collectable quantity of used products from customers,

• Incorporation of new frameworks of into a GSC to encourage the collection of used products and the remanufacturing of products,

• The situation where the multiple types of the used products and the products are handled in the GSC.

## ACKNOWLEDGMENT

This research has been supported by the Grant-in- Aid for Scientific Research C No. 25350451 from the Japan Society for the Promotion of Science.

## Figure

Four cases of quality distribution of recyclable parts in used products modeled as the beta distribution B(ℓ | a,b) in Cases 1-4 (Watanabe et al., 2013; Watanabe and Kusukawa, 2014).

Effect of the degree γ of risk attitude on the optimal product order quantities in DGSC and IGSC, Qki.

Effect of the product order quantity Q on the variance of the retailer’s profit, V[πR(Q, t, u)].

Behaviors of the utility function URA(Q, tD, uD), the expected profit E[πR(Q, tD, uD)], and the variance multiplied by the degree γ of risk attitude, γV[πR(Q, tD, uD)] of a retailer with attitude A for product order quantity Q under tD , uD , and γ = −1.0 ×10−5 in Case 2 of the quality distribution.

Behaviors of the utility function URP(Q, tD, uD), the expected profit E[πR(Q, tD, uD)], and the variance multiplied by the degree γ of risk attitude, γV[πR(Q, tD, uD)] of a retailer with attitude P for product order quantity Q under tD , uD , and γ = 1.0 ×10−5 in Case 2 of the quality distribution.

## Table

The optimal product order quantities in DGSC and IGSC as to the degree γ of risk attitude

Effect of the degree γ of risk attitude on the expected profit of the retailer, the manufacturer, the whole system in DGSC and IGSC (Case 2 of quality level of recyclable parts)

Effect of the degree γ of risk attitude on the variances of the profits and the utility functions in DGSC and IGSC (Case 2 of quality level of recyclable parts)

Optimal decisions for collection incentive and lower limit of quality level in DGSC and IGSC

Effect of supply chain coordination (SCC) on the expected profits in IGSC (Case 2 of quality level of recyclable parts)

*denotes that it is impossible to conduct SCC between a retailer and a manufacturer in IGSC.

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