1. INTRODUCTION
From social concerns about 3R (ReduceReuse Recycle) activity worldwide, it is urgentlyneeded to construct new supply chain managements which incorporate reverse chains into forward chains (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 forward chains consist of the flows from procurement of new materials through production of new products to sales of the products. The reverse chains are composed of the flows from collection of used products through recycling parts from the used products to reuse of the recycled parts. Also, supply chains which organize the forward chains and the reverse chains have been called closedloop supply chains (CLSCs) or reverse supply chains (Guide and Wassenhove, 2009; Souza, 2013; Govindan et al., 2014; Stindt and Sahamie, 2014; Bhattacharya and Kaur, 2015; Gurtu et al., 2015).
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, singleuse cameras, and car’s engines, 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 CLSC.
In general, unlike forward chains, CLSCs face uncertainties in not only product demand but also collection quantity of used products and quality of used products (Nenes et al., 2010). These two uncertainties lead to the uncertainty in the quantity of recycled parts. Therefore, manufacturers in CLSCs need to procure new parts from a supplier to supply a shortage or dispose the excess recycled parts according to the quantity of the recycled parts in addition to product demand.
Here, as one of countermeasures against the uncertainties in production yield and product demand, a flexible ordering policy has been proposed by Hu et al. (2013). In FOP, the minimum order quantity and the maximum order quantity are determined and the practical order quantity is adjusted as to the production yield. In contrast, this paper calls an ordering policy that a single product order quantity is determined as a traditional ordering policy (TOP). Hu et al. (2013) found that FOP can reduce the influences of the uncertainties in production yield and product demand on the expected profit of a manufacturer, by comparing FOP with TOP.
Therefore, this paper discusses incorporation of a flexible ordering policy (FOP) (Hu et al., 2013) into a CLSC which consists of a buyer, a manufacturer, and a recycler, to deal with the uncertainties in product demand and collection quantity of used products. The recycler collects used products from a market, disassembles them to a single type of parts, classifies with quality level of the parts, and remanufactures a single type of parts. The manufacturer pays the recycling incentive to the recycler along with the quantity of the remanufactured parts in addition to expense for the parts and produces a single type of products from the remanufactured parts. The buyer sells the products in the market. In the CLSC with FOP, the wholesale quantity of the products varies along with the quantity of the remanufactured parts between the minimum product order quantity and the maximum product order quantity. Contrastively, in the CLSC with TOP, a single product order quantity is determined, and it always equals the wholesale quantity as a traditional ordering policy (TOP). This study focuses on the way to determining the minimum and the maximum product order quantities in the CLSC with FOP.
In addition to these order quantities, decisionmaking on recycling incentive for remanufactured parts and lower limit of quality level to remanufacture parts are discussed. Those four variables are optimally determined for each of the decentralized CLSC (DCLSC) and the integrated CLSC (ICLSC) with FOP. DCLSC maximizes the expected profit of each member, based on the Stackelberg game (Nagarajan and Sošić, 2008; Yan and Sun, 2012; Watanabe et al., 2013; Watanabe and Kusukawa, 2014; Hong et al., 2015). This study regards the buyer, the manufacturer, and the recycler, as the first leader, the second leader, and the follower, of the decisionmaking for DCLSC, respectively. Then, the buyer determines the optimal minimum and maximum product order quantities. The manufacturer determines the optimal recycling incentive under the optimal order quantities for the buyer. The recycler determines the optimal lower limit of quality level under the optimal recycling incentive for the manufacturer. In ICLSC with FOP, the four variables are optimally determined so as to maximize the expected profit of the whole system. Shift from DCLSC to ICLSC enhances the CLSC because the optimal operation in ICLSC is generally better than that in DCLSC from the aspect of the total optimization. However, the shift has possibilities to make the expected profits of some members lower. Therefore, profit sharing as supply chain coordination (SCC) is introduces into ICLSC with FOP to guarantee the improvement in the expected profits of all the members. As profit sharing approaches, this paper adopts (I) Nash bargaining solution (Nash, 1950, 1953; Kohli and Park, 1989; Nagarajan and Sošić, 2008; Hong et al., 2013; Watanabe et al., 2013; Watanabe and Kusukawa, 2014; Ghosh and Shah, 2015) where the unit wholesale price of the products and the unit sales price of the recycled parts are coordinated among the members in the CLSC, and (II) return on investments (Jaber et al., 2006) of the members in the CLSC.
Numerical analyses investigate how (i) the uncertainty in product demand, and (ii) the uncertainty in collection quantity of used products, and (iii) a quality distribution of the parts extracted from used products, affect the optimal operation and the expected profits in CLSC with FOP by providing numerical examples. In order to confirm the effect of FOP, the expected profit of the whole system under the optimal operation for ICLSC with FOP is compared with that for ICLSC with TOP. Moreover, the effect of SCC adopting profit sharing based on each of Nash bargaining solution and return on investments is confirmed.
The contributions of this paper for practitioners, academic researchers, and policymakers regarding a CLSC, are as follows:

Formulation of the expected profits of the members in the CLSC with FOP,

Proposal on methods, with use of mathematical analyses and numerical searches, to make decisions for the DCLSC and ICLSC with FOP,

Showing of influences of quality distribution of parts extracted from used products on the optimal operations for the DCLSC and ICLSC with FOP,

Verification of effects of FOP against the uncertainties in product demand and collection quantity of used products,

Proposal on SCC to realize the shift from DCLSC to ICLSC and verification of effects of the SCC.
Utilizing CLSC with FOP, that is, ICLSC with FOP can reduce effects of uncertainties in demands of product and collection quantity of used products when the optimal order quantity of products are determined from the viewpoint of maximizing the expected profit. So, it is expected that utilizing CLSC with FOP, that is, ICLSC with FOP will be able to helpful when practitioners in companies which use remanufacturing approach (Radhi and Zhang, 2016), academic researchers, and policymakers make the optimal operation including raw materials/parts production in the manufacturing industry which produces products such as photocopies, cellular telephones, singleuse cameras, PCs, car’s engines, transmissions, retreaded tires and electric appliance and so on, where there are uncertainties in demand of product and collection quantity of used products.
The rest of this paper is organized as follows. In Section 2, a review of literature is presented with some topics. Section 3 provides model descriptions of a CLSC discussed in this paper and formulates the expected profits of the members in the CLSC. In Section 4, the optimal operations for DCLSC and ICLSC are derived and SCC is discussed. Section 5 shows results of numerical analyses and describes managerial insights. Finally, 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, effect of incentive for promotion of recycling activity, shift from DCLSC to ICLSC and supply chain coordination.
First, many authors have discussed the optimal operation with consideration of uncertainties about supply chains. Aras et al. (2004) investigated the issue of the stochastic nature of product returns and found conditions under which qualitybased categorization was the most cost effective. Zikopoulos and Tagaras (2007) investigated how the profitability of reuse activities was affected by uncertainty in the quality of returned products in two collection sites and determined the optimal procurement and production quantities as the unique solution. Guide et al. (2003) and Ferguson et al. (2009) assumed that returned products were classified into discrete quality rank from 1 to integer n, and the procurement prices and the remanufacturing costs were different based on the rank corresponding to quality level. Mukhopadhyay and Ma (2009) discussed a CLSC consisting of a retailer who sold a single of products and a manufacturer who collected used products from the market, remanufactured parts from the used products, and produced products. They assumed two situations for the remanufacturing ratio between reused 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 quantity of remanufactured parts, and the production quantity of new parts from new materials. Nenes et al. (2010) observed that both quality and quantity of used products were 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 operation for a CLSC with consideration of the uncertainties in product demand and quality of used products. Concretely, the optimal decision for lower limit of quality level of recyclable parts with a variety of quality level after disassembly of used products are conducted. Also, when the product demand follows uncertain distribution with known mean and known variance, the expected profit is formulated and the optimal operation is derived under the worst situation by applying distribution free approach (DFA). Zikopoulos and Tagaras (2015) considered a reverse supply chain with multiple collection sites and the possibility of return sorting, and derived the optimal decisions for acquisition and remanufacturing lotsizing under locations of the unreliable classification. Radhi and Zhang (2016) discussed the optimal lower quality of used products with uncertain quality to accept into each operating factory in a remanufacturing supply network. In order to restrain the uncertainties in production yield and product demand in a forward supply chain, Hu et al. (2013) presented a flexible ordering policy (FOP) with the minimum order quantity and the maximum order quantity, and found that FOP can reduce the influences of the two uncertainties and enhance the profit of the supply chain.
A presumption that incorporating FOP into CLSCs can work through the uncertainties in product demand and collection quantity of used products, is the biggest motivation of this study.
Many previous studies have discussed the second topic, effects of incentive for collection of used products on the optimal operations in CLSCs. Kaya (2010) discussed the effect of incentive for collection of used products which is paid from a manufacturer to consumers on the optimal tactical strategy in a CLSC. In the paper, 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. Lee et al. (2011) discussed a model that integrated pricing, production, and inventory decisions, in a reverse production system with retailer collection. The returned products were assumed valuable to the manufacturer for creating asnew 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. Effects of a collection incentive contract on promotion of the collection and recycling activities in a CLSC were discussed in Watanabe et al. (2013) and Watanabe and Kusukawa (2014). The papers considered the incentive which is paid from a retailer to customers, and the incentive which is paid from a manufacturer to a retailer as compensation. It was clarified that the two types of incentive enabled to promote the collection and the recycling of used products. The optimal operations for a product order quantity, the unit collection incentive of used products, and the lower limit of quality level for recycling used products, were determined. De Giovanni et al. (2016) clarified that incentive strategies lead to the implementation of the total optimal solution in a dynamic CLSC which consists of a manufacturer and a retailer. In the above papers which considered incentive contracts to promote recycling activity in CLSCs, FOP hadn’t been incorporated into CLSCs.
Regarding the third topic, in DCLSCs, each member determines the optimal operation so as to maximize its own profit. As one of the decisionmaking approaches in DCLSCs, the Stackelberg game has been adopted in several previous papers (Yan and Sun, 2012; Watanabe and Kuskawa, 2014; Hong et al., 2015; Esmaeili et al., 2016). In the Stackelberg game, there is a single (multiple) relationship( s) of a leader of the decisionmaking and a follower of the decisionmaking of the leader. The leader determines the optimal strategy so as to maximize profit of the leader. The follower determines the optimal strategy so as to maximize profit of the follower under the strategy determined by the leader (Cachon and Netessine, 2004; Nagarajan and Sošić, 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 CLSCs, it is desirable to shift to the optimal operation in ICLSC from that in DCLSC. To realize the shift, a variety of supply chain coordination (SCC) approaches which guarantee to increase profits of all the members in ICLSCs, have been discussed in the following papers and some overviews: Tsay et al. (1999), Cachon and Netessine (2004), Yano and Gilbert (2004), Nagarajan and Sošić (2008), Hu et al. (2011), Yan and Sun (2012), and Govindan et al. (2013). Kohli and Park (1989), Hong et al. (2013), Watanabe et al. (2013), Watanabe and Kusukawa (2014), and Ghosh and Shah (2015), adopted profit sharing based on Nash bargaining solution (Nash, 1950; Nash, 1953; Nagarajan and Sošić, 2008) as SCC. Kohli and Park (1989) dealt with 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 share profit for a manufacturer and a retailer in the channel coordination of strategic alliance. Watanabe et al. (2013) and Watanabe and Kusukawa (2014) coordinated the margin for wholesale per product and the degree of compensation for a retailer’s collection incentive of used products as Nash bargaining solution between a retailer and a manufacturer. Ghosh and Shah (2015) coordinated sharing rate of greening cost as Nash bargaining solution between a retailer and a manufacturer. Additionally, Jaber et al. (2006) introduced profit sharing based on return on investments as supply chain coordination (SCC) into a forward supply chain which consists of a supplier, a manufacturer, and a retailer. In comparison with profit sharing based on Nash bargaining solution, profit sharing based on return on investments can consider the performance of earning profit of the members. This paper applies profit sharing based on each of (I) Nash bargaining solution where the unit wholesale price of the products and the unit sales price of the remanufactured parts are coordinated, and (II) return on investments, as SCC, to ICLSC with FOP.
3. MODEL FORMULATIONS
3.1. Notation
3.1.1. Indices

B : a buyer

M : a manufacturer

R : a recycler

S : the whole system

i∈{B, M, R, S} : a set of the members in a CLSC and the whole system

D : a decentralized closedloop supply chain

I : a integrated closedloop supply chain
3.1.2. Decision Variables

q : the minimum product order quantity in a CLSC with FOP and the product order quantity in a CLSC with TOP (0 ≤ q)

Q : the maximum product order quantity in a CLSC with FOP (q ≤ Q)

t : recycling incentive per unit of remanufactured parts, referred to simply as recycling incentive $(0\le t\le {w}_{m}{c}_{m}{w}_{r})$

u : lower limit of quality level to remanufacture parts which used products are disassembled to (0 ≤ u ≤ 1)
3.1.3. Other Variables and Functions

E[․] : expectation

x_{c} : collection quantity of used products (0 ≤ x_{c})

A(t) : the expectation of x_{c}

ε : a stochastic additive variation in x_{c}

h(ε) : probability density function of ε

H(ε) : cumulative distribution function of ε

c_{c} : collection cost per unit of used products

c_{a} : disassembly cost per unit of used products and inspection cost per unit of parts extracted from the used products

ℓ : quality level of each part extracted from the used products (0 ≤ ℓ ≤1)

g(ℓ) : probability density function of ℓ

G(ℓ) : cumulative distribution function of ℓ

c_{r} (ℓ) : remanufacturing cost per unit of the parts with quality level ℓ

c_{d} : disposal cost per unit of the unremanufactured parts

x_{r} : quantity of remanufactured parts

w_{r} : sales price per unit of remanufactured parts

d : wholesale quantity of products

c_{m} : production cost per unit of products

c_{n} : procurement cost per unit of new parts

p_{s} : salvage value in a disposal market per unit of remanufactured parts

w_{m} : wholesale price of products

x : product demand (0 ≤ x)

f (x) : probability density function of x

F(x) : cumulative distribution function of x

p_{m} : sales price per unit of products

h_{m} : inventory holding cost per unit of products

s_{m} : shortage penalty cost per unit of products

π_{i} (q, Q, t, u) : profit of member i for q, Q, t, and u, in a CLSC with FOP

${\pi}_{i}^{T}\left(q,t,u\right)$ : profit of member i for q, t, and u, in a CLSC with TOP

q_{D} : the optimal minimum product order quantity for DCLSC with FOP

Q_{D} : the optimal maximum product order quantity for DCLSC with FOP

t_{D} : the optimal recycling incentive for DCLSC with FOP

u_{D} : the optimal lower limit of quality level for DCLSC with FOP

u_{D}(t) : the provisional lower limit of quality level for DCLSC with FOP for t

q_{I} : the optimal minimum product order quantity for ICLSC with FOP

Q_{I} : the optimal maximum product order quantity for ICLSC with FOP

t_{I} : the optimal recycling incentive for ICLSC with FOP

u_{I} : the optimal lower limit of quality level for ICLSC with FOP

${q}_{I}^{T}$ : the optimal product order quantity for ICLSC with TOP

${t}_{I}^{T}$ : the optimal recycling incentive for ICLSC with TOP

${u}_{I}^{T}$ : the optimal lower limit of quality level for ICLSC with TOP

${q}_{I}^{T}(t,\hspace{0.17em}u)$ : the provisional product order quantity for ICLSC with TOP for t and u

${w}_{r}^{Nash}$ : the unit sales price of remanufactured parts which is coordinated as Nash bargaining solution

${w}_{m}^{Nash}$ : the unit wholesale price of products which is coordinated as Nash bargaining solution

$T({w}_{r}^{Nash},\hspace{0.17em}{w}_{m}^{Nash})$ : objective function of Nash bargaining solution

$T{C}_{i}(q,\hspace{0.17em}Q,\hspace{0.17em}t,\hspace{0.17em}u)$ : total cost of the member $i\in \{B,\hspace{0.17em}M,\hspace{0.17em}R\}$ for q, Q, t, and u, in a CLSC with FOP

${r}_{i}$ : return on investment of the member i∈{B, M, R}

${r}_{i}^{N}$ : the standardized return on investment of the member i∈{B, M, R}

$\Delta {\pi}_{S}^{E}$ : improvement in the expected profit of the whole system by the shift from DCLSC to ICLSC

$\Delta {\pi}_{i}^{E}$ : improvement in the expected profit of the member i∈{B, M, R} by the shift from DCLSC to ICLSC after profit sharing based on return on investments

${\pi}_{i}^{E}$ : the expected profit of the member i∈{B, M, R} in ICLSC after profit sharing based on return on investments
3.2. Model Descriptions
3.2.1. Operational Flow of a CLSC
In this study, a CLSC with a retailer, a manufacturer, and a recycler is considered. It is supposed that a single type of products such as consumer electronics (mobile phone, personal computer), semiconductor, and electronic component is sold in a market.
The operational flow of the CLSC is as follows.

(1) A recycler collects used products, whose quantity is x_{c}, at the unit cost c_{c} from a market.

(2) The recycler disassembles the used products to a single type of parts and inspects the parts at the unit cost c_{a}. After that, the parts are classified into quality level ℓ(0 ≤ ℓ ≤1). The recycler remanufactures the parts with quality level which is greater than or equal to the lower limit of quality level $u(0\le u\le 1),u\le \ell \le 1,$, at the unit cost c_{r}(ℓ).

(3) The parts with quality level which is less than u, 0 ≤ ℓ < u, are disposed at the unit cost c_{d}.

(4) A manufacturer buys all of the remanufactured parts, whose quantity is x_{r}, from the recycler at the unit price w_{r}.

(5) The manufacturer pays recycling incentive t per unit of remanufactured parts to the recycler.

(6) The manufacturer produces the products, whose quantity is d, at the unit cost c_{m} and sells them to a buyer at the unit wholesale price . w_{m} In the CLSC with TOP, if the quantity of remanufactured parts x_{r} is less than the product order quantity q, the manufacturer procures new parts, whose quantity is q − x_{r}, from an external supplier at the unit cost c_{n}. Accordingly, wholesale quantity d is always equal to the product order quantity q. Meanwhile, in the CLSC with FOP, d has three cases. If x_{r} is less than the minimum product order quantity q, the manufacturer procures new parts q − x_{r} from the external supplier at the unit cost c_{n}, and d is equal to q, as with TOP. If x_{r} is greater than or equal to q and less than or equal to the maximum product order quantity Q, d is equal to x_{r}. If x_{r} is greater than Q, d is equal to Q.

(7) In both of the CLSC with TOP and that with FOP, the excess remanufactured parts are sold at the unit salvage value p_{s} in a disposal market.

(8) The buyer sells the products in a market at the unit sales price p_{m} during a single period. Also, the buyer incurs the inventory holding cost h_{m} per unit of unsold products for product demand x and the shortage penalty cost s_{m} per unsatisfied product demand.
3.2.2. Model Assumptions
The model assumptions in this study are as follows.

(1) The expected collection quantity of used products, A(t), varies as to the recycling incentive t. In general, the higher t is, the more used products the recycler can collect from a market. Therefore, A(t) has the following characteristic: $\partial A(t)/\partial t>0.$ Here, from the aspect of the manufacturer’s profit, the feasible range of t is $0\le t\le {w}_{m}{c}_{m}{w}_{r}.$

(2) Collection quantity of used products, x_{c}, is modeled as(1)
where ε is a stochastic variation. ε follows a probabilistic distribution with the probability density function h(ε) and the cumulative distribution function H (ε ). Here, the expectation E[ε ] = 0, and $A(t)\le \epsilon $ due to 0 .c ≤ x

(3) The variability of quality level ℓ of the parts extracted from used products is modeled as a probabilistic distribution with probability density function g(ℓ) and cumulative distribution function G(ℓ).

(4) The remanufacturing cost per unit of the parts, c_{r}(ℓ), varies as to the quality level ℓ of the parts. As the quality level ℓ is lower, c_{r}(ℓ) is higher. Here, ℓ = 0 indicates the worst quality level, and ℓ =1 indicates the best quality level. Therefore, ( ) r c ℓ has the following characteristic: ( ) / 0.

(5) The quality of remanufactured parts is the same as that of new parts.

(6) The salvage value in a disposal market per unit of remanufactured parts p_{s} is lower than the unit procurement cost of new parts, c_{n}, and the unit procurement cost of remanufactured parts, w_{r}.

(7) A probabilistic distribution with probability density function f(x) and cumulative distribution function F(x).
3.3. Formulations of Expected Profits
In this section, the expected profits of a buyer, a manufacturer, and a recycler, and the whole system, are formulated based on Subsection 3.2. From Subsection 3.2.1 (2) and Subsection 3.2.2 (2), the quantity of remanufactured parts, x_{r}, is calculated as
3.3.1. Expected Profits in a CLSC with FOP
From Subsection 3.2.1 (6), in the CLSC with FOP, the wholesale quantity of products from the manufacturer to the buyer, d, is given by
Note that if Q = q, then d = q, or the CLSC with FOP is corresponding to that with TOP. By solving , x_{r} < q, $q\le {x}_{r}\le Q,$ and Q < x_{r} with respect to the additive variation ε$A(t)\le \epsilon <L,\hspace{0.17em}L\le \epsilon \le U,$, and U < ε, where
are derived, respectively. However, L and U can’t be defined for u = 1. If u = 1, x_{r} = 0, or d = q regardless of ε from Eq. (3). Accordingly, the CLSC with FOP comes down to the CLSC with TOP for u = 1.
Profit of the buyer in the CLSC with FOP consists of the product sales, the inventory holding cost of unsold products, shortage penalty cost of unsatisfied product demand, and the procurement cost of products. The profit of the buyer for the minimum product order quantity q, the maximum product order quantity Q, the recycling incentive t, and the lower limit of quality level u, ${\pi}_{B}(q,\hspace{0.17em}Q,\hspace{0.17em}t,\hspace{0.17em}u),$ is formulated as(6)
By taking expectation with respect to the product demand x and the additive variation ε , the expected profit of the buyer in the CLSC with FOP for q, Q, t, and $E\left[{\pi}_{B}\left\{q,\hspace{0.17em}\hspace{0.17em}Q,\hspace{0.17em}\hspace{0.17em}t,\hspace{0.17em}\hspace{0.17em}\left(u\ne 1\right)\right\}\right]$, is derived as
(See Eq. (A8) in Appendix A). Here, E_{x}[․] denotes an expectation with respect to x.
Profit of the manufacturer in the CLSC with FOP consists of the product wholesales, the manufacturing cost of products, the procurement cost of remanufactured parts, the recycling incentive to the recycler, the procurement cost of new parts, and the salvage sales of remanufactured parts. The profit of the manufacturer for q, Q, t, and ${\pi}_{M}(q,\hspace{0.17em}\hspace{0.17em}Q,\hspace{0.17em}\hspace{0.17em}t,\hspace{0.17em}\hspace{0.17em}u)$ is formulated as(8)
By taking expectation with respect to ε , the expected profit of the manufacturer in the CLSC with FOP for q, Q, t, and $E[{\pi}_{M}\{q,\hspace{0.17em}Q,\hspace{0.17em}t,\hspace{0.17em}(u\ne 1)\}]$, is derived as
(See Eq. (A9) in Appendix A).
Profit of the recycler in the CLSC with FOP consists of the sales of remanufactured parts, the recycling incentive from the manufacturer, the remanufacturing cost of the parts, the disposal cost of the parts, the disassembly and inspection cost of the parts extracted from used products, and the collection cos2t of used products. The profit of the recycler for t and $u,\hspace{0.17em}\hspace{0.17em}{\pi}_{R}(t,\hspace{0.17em}\hspace{0.17em}u),$ is formulated as(10)
By taking expectation with respect to ε , the expected profit of the recycler in the CLSC with FOP for t and u, $E[{\pi}_{R}(t,\hspace{0.17em}u)]$ is derived as
(See Eq. (A10) in Appendix A). Here, from Eq. (11), note that the expected profit of the recycler doesn’t have to be divided whether u is equal to 1 because it is independent of L and U.
The expected profit of the whole system in the CLSC with FOP for q, Q, t, and $E[{\pi}_{S}\{q,\hspace{0.17em}\hspace{0.17em}Q,\hspace{0.17em}\hspace{0.17em}t,\hspace{0.17em}\hspace{0.17em}(u\ne 1)\}]$ is calculated as the sum of the expected profit of each member,
(See Eq. (A11) in Appendix A).
3.3.2. Expected Profit in a CLSC with TOP
From Subsection 3.2.1 (6), in the CLSC with TOP, the wholesale quantity of products from the manufacturer to the buyer, d, is always equal to the product order quantity q.
Therefore, the expected profit of the buyer in the CLSC with TOP can be derived as
(See Eq. (A12) in Appendix A). Here, note that the expected profit of the buyer in the CLSC with TOP is independent of the recycling incentive t and the lower limit of quality level u. In addition, the CLSC with FOP q = Q comes down to that with TOP.
The expected profit of the manufacturer in the CLSC with TOP can be derived as(14)
(See Eqs. (A13) and (A15) in Appendix A).
The profit of the recycler is independent of the product order quantity/ies. Therefore, the following equation(16)
satisfied.
The expected profit of the whole system in the CLSC with TOP is calculated as the sum of the expected profit of each member,
(See Eqs. (A14) and (A16) in Appendix A).
4. DECISIONMAKING FOR A CLSC
In this section, the optimal operations for the CLSC are derived. With FOP, the optimal operation for each of the decentralized CLSC (DCLSC) and the integrated CLSC (ICLSC) is discussed, respectively. Meanwhile, with TOP, it is reported that the optimal operations for ICLSCs are superior to those for DCLSCs (Hu et al., 2011; Yan and Sun, 2012; Watanabe et al., 2013; Watanabe and Kusukawa, 2014). Therefore, in this paper, the optimal operation only for ICLSC with FOP is discussed.
4.1. Optimal Operation for DCLSC
The optimal operation for DCLSC with FOP is discussed.
In DCLSC, the optimal decision approach for a Stackelberg game is adopted. This study regards a buyer, a manufacturer, and a recycler, as the first leader, the second leader, and the follower, of the decisionmaking for DCLSC, respectively. Then, the buyer determines the optimal minimum product order quantity q_{D} and the optimal maximum product order quantity Q_{D} so as to maximize its own expected profit $E[{\pi}_{B}(q,\hspace{0.17em}Q,\hspace{0.17em}t,\hspace{0.17em}u)].$. The manufacturer determines the optimal recycling incentive t_{D} so as to maximize its own expected profit under q_{D} and Q_{D}, $E[{\pi}_{M}({q}_{D},\hspace{0.17em}{Q}_{D},\hspace{0.17em}t,\hspace{0.17em}u)].$, The recycler determines the optimal lower limit of quality level u_{D} so as to maximize its own expected profit under t_{D}, $E[{\pi}_{R}({t}_{D},\hspace{0.17em}u)].$
4.1.1. Optimal Minimum Product Order Quantity and Optimal Maximum Product Order Quantity for DCLSC with FOP
In DCLSC, the optimal minimum product order quantity q_{D} and the optimal maximum product order quantity Q_{D} are determined so as to maximize the expected profit of the buyer, $E[{\pi}_{B}\{q,\hspace{0.17em}Q,\hspace{0.17em}t,\hspace{0.17em}(u\ne 1)\}].$
Proposition 1.$E[{\pi}_{B}\{q,\hspace{0.17em}Q,\hspace{0.17em}t,\hspace{0.17em}(u\ne 1)\}]$ in Eq. (7) is monomodal with respect to q and Q.
Proof. The first order partial derivatives of the expected profit of the buyer in the CLSC with FOP, $E[{\pi}_{B}\{q,\hspace{0.17em}Q,\hspace{0.17em}t,\hspace{0.17em}(u\ne 1)\}],$ with respect to q and Q are respectively derived as(20)
(See Eqs. (B5) and (B6) in Appendix B). From Eq. (19), $\partial E\left[{\pi}_{B}\left\{q,\hspace{0.17em}Q,\hspace{0.17em}t,\hspace{0.17em}\left(u\ne 1\right)\right\}\right]/\partial q$ is positive when $0<F\left(q\right)<\left({p}_{m}{w}_{m}+{s}_{m}\right)/\left({p}_{m}+{h}_{m}+{s}_{m}\right),$ and it is negative when $\left({p}_{m}{w}_{m}+{s}_{m}\right)/\left({p}_{m}+{h}_{m}+{s}_{m}\right)<F\left(q\right)\le 1.$. Also, F(q) is a monotonically increasing function with respect to q. Therefore, $E[{\pi}_{B}\{q,\hspace{0.17em}Q,\hspace{0.17em}t,\hspace{0.17em}(u\ne 1)\}]$ is monomodal with respect to q. In a similar way, $E[{\pi}_{B}\{q,\hspace{0.17em}\hspace{0.17em}Q,\hspace{0.17em}\hspace{0.17em}t,\hspace{0.17em}\hspace{0.17em}(u\ne 1)\}]$ is monomodal with respect to Q, too.
Proposition 2.q_{D} and Q_{D} are determined regardless of t and u as the following unique solutions to maximize $E[{\pi}_{B}\{q,\hspace{0.17em}Q,\hspace{0.17em}t,\hspace{0.17em}(u\ne 1)\}]$ :
Proof. See Appendix C.
From the above analyses, q_{D} = Q_{D} can be elicited. Therefore, even if FOP is introduced into DCLSC, DCLSC with FOP comes down to that with TOP. This means that FOP doesn’t function in DCLSC. The reason is that the buyer isn’t rewarded by FOP in the considered model.
4.1.2. Optimal Recycling Incentive and Optimal Lower Limit of Quality Level for DCLSC with FOP
Under the optimal minimum product order quantity q_{D} and the optimal maximum product order quantity Q_{D} determined in Subsection 4.1.1, the optimal recycling incentive t_{D} and the optimal lower limit of quality level u_{D} are determined.
The first order partial derivative of the expected profit of the recycler in the CLSC with FOP, $E[{\pi}_{R}(t,\hspace{0.17em}u)],$ with respect to u is derived as
(See Eq. (B7) in Appendix B). From Eq. (23), it can be seen that $\partial E[{\pi}_{R}(t,\hspace{0.17em}u)]/\partial u$ depends on t. Eq. (23) is zero if the following equation:(24)
is satisfied.
Proposition 3.The provisional lower limit of quality level for t, u_{D}(t), is determined as follows:
$\begin{array}{l}i.\hspace{0.17em}\hspace{0.17em}If\hspace{0.17em}{c}_{r}\left(0\right)\left({w}_{r}+t+{c}_{d}\right)<0,\hspace{0.17em}\hspace{0.17em}{u}_{D}(t)=0,\\ ii.\hspace{0.17em}If\hspace{0.17em}{c}_{r}\left(1\right)\left({w}_{r}+t+{c}_{d}\right)>0,\hspace{0.17em}\hspace{0.17em}{u}_{D}(t)=1,\\ iii.\hspace{0.17em}Otherwise,\hspace{0.17em}{u}_{D}\left(t\right)={c}_{r}{}^{1}\left({w}_{r}+t+{c}_{d}\right).\end{array}$
Proof. See Appendix D.
It is impossible to derive the optimal recycling incentive t_{D} by mathematical analyses. In this study, t_{D} is determined by a numerical search. The decision procedure for t_{D} is shown as follows:
Step 1. Substitute t and u_{D}(t) into the expected profit of the manufacturer, $E[{\pi}_{M}({q}_{D},\hspace{0.17em}\hspace{0.17em}{Q}_{D},\hspace{0.17em}\hspace{0.17em}t,\hspace{0.17em}\hspace{0.17em}u)],$, with varying t within $0\le t\le {w}_{m}{c}_{m}{w}_{r}.$
Step 2. Determine the optimal t, which maximizes $E[{\pi}_{M}\{{q}_{D},\hspace{0.17em}{Q}_{D},\hspace{0.17em}t,\hspace{0.17em}{u}_{D}(t)\}],$ as t_{D}.
The optimal lower limit of quality level u_{D} is derived as u_{D}(t_{D}).
4.2. Optimal Operations for ICLSC
The optimal operation for ICLSC with each of FOP and TOP is discussed, respectively.
In ICLSC with FOP, the optimal minimum product order quantity q_{I}, the optimal maximum product order quantity, Q_{I} the optimal recycling incentive , t_{I} and the optimal lower limit of quality level, u_{I} are determined so as to maximize the expected profit of the whole system, $E[{\pi}_{S}(q,\hspace{0.17em}Q,\hspace{0.17em}t,\hspace{0.17em}u)].$
4.2.1. Optimal Minimum Product Order Quantity and Optimal Maximum Product Order Quantity for ICLSC with FOP
In ICLSC, the optimal minimum product order quantity q_{I} and the optimal maximum product order quantity Q_{I} are determined so as to maximize the expected profit of the buyer, $E[{\pi}_{S}\{q,\hspace{0.17em}Q,\hspace{0.17em}t,\hspace{0.17em}(u\ne 1)\}].$
Proposition 4.$E[{\pi}_{S}\{q,\hspace{0.17em}Q,\hspace{0.17em}t,\hspace{0.17em}(u\ne 1)\}]$ in Eq. (12) is monomodal with respect to q and Q.
Proof. The first order partial derivatives of the expected profit of the whole system in the CLSC with FOP, $E[{\pi}_{S}\{q,\hspace{0.17em}Q,\hspace{0.17em}t,\hspace{0.17em}(u\ne 1)\}],$ with respect to q and Q are respectively derived as(26)
(See Eqs. (B8) and (B9) in Appendix B). From Eq. (25), $\partial E\left[{\pi}_{S}\left\{q,\hspace{0.17em}Q,\hspace{0.17em}t,\left(u\ne 1\right)\right\}\right]/\partial q$ is positive when $0<F\left(q\right)<\left({p}_{m}+{s}_{m}{c}_{m}{c}_{n}\right)/\left({p}_{m}+{h}_{m}+{s}_{m}\right)$, and it is negative when $\left({p}_{m}+{s}_{m}{c}_{m}{c}_{n}\right)/\left({p}_{m}+{h}_{m}+{s}_{m}\right)<F(q)\le 1.$ Also, F(q) is a monotonically increasing function with respect to q. Therefore, $E[{\pi}_{S}\{q,\hspace{0.17em}\hspace{0.17em}Q,\hspace{0.17em}\hspace{0.17em}t,\hspace{0.17em}\hspace{0.17em}(u\ne 1)\}]$ is monomodal with respect to q. In a similar way, $E[{\pi}_{S}\{q,\hspace{0.17em}\hspace{0.17em}Q,\hspace{0.17em}\hspace{0.17em}t,\hspace{0.17em}\hspace{0.17em}(u\ne 1)\}]$ is monomodal with respect to Q, too.
Proposition 5.q_{I} and Q_{I} are determined regardless of t and u as the following unique solutions to max$E[{\pi}_{S}\{q,\hspace{0.17em}Q,\hspace{0.17em}t,\hspace{0.17em}(u\ne 1)\}]$:
Proof. See Appendix E.
Due to $\partial F\left(x\right)/\partial x>0$ and ${p}_{s}<{c}_{n}$ as described in Subsection 3.2.2 (6), ${q}_{I}<{Q}_{I}$ can be elicited. This means the introduction of FOP into ICLSC improves the expected profit of the whole system.
4.2.2. Optimal Recycling Incentive and Optimal Lower Limit of Quality Level for ICLSC with FOP
Under the optimal minimum product order quantity q_{I} and the optimal maximum product order quantity Q_{I} determined in Subsection 4.2.1, the optimal recycling incentive t_{I} and the optimal lower limit of quality level u_{I} are determined.
Unlike DCLSC, it is impossible to derive not only the optimal recycling incentive t_{I} but also the optimal lower limit of quality level u_{I} by mathematical analyses in ICLSC. In this study, t_{I} and u_{I} are determined by a numerical search. The decision procedure for t_{I} and u_{I} is shown as follows:
Step 1. Substitute t and u into the expected profit of the whole system, $E[{\pi}_{S}({q}_{I},\hspace{0.17em}{Q}_{I},\hspace{0.17em}t,\hspace{0.17em}u)]$ with varying t within $0\le t\le {w}_{m}{c}_{m}{w}_{r}$ and u within $0\le u\le 1.$
Step 2. Determine the optimal combination of t and u, which maximizes $E[{\pi}_{S}({q}_{I},\hspace{0.17em}{Q}_{I},\hspace{0.17em}t,\hspace{0.17em}u)],$ as $({t}_{I},\hspace{0.17em}\hspace{0.17em}{u}_{I}).$
4.2.3. Optimal Operation for ICLSC with TOP
In ICLSC with TOP, the optimal product order quantity ${q}_{I}^{T},$ the optimal recycling incentive ${t}_{I}^{T},$ and the optimal lower limit of quality level ${u}_{I}^{T},$ are determined so as to maximize the expected profit of the whole system, E[ T ( , , )] S π
The first order partial derivative of the expected profit of the whole system in the CLSC with TOP, $E[{\pi}_{S}^{T}(q,\hspace{0.17em}t,\hspace{0.17em}u)],$ with respect to q is derived as(29)
(See Eqs. (B10) and (B11) in Appendix B). From Eqs. (4) and (30), it can be seen that $\partial E[{\pi}_{S}^{T}\{q,\hspace{0.17em}t,\hspace{0.17em}(u\ne 1)\}]/\partial q$ depends on t and u. Meanwhile, from Eq. (31), $\partial E[{\pi}_{S}^{T}\{q,\hspace{0.17em}\hspace{0.17em}t,\hspace{0.17em}\hspace{0.17em}(u=1)\}]/\partial q$ is independent of t and u.
Proposition 6.$E[{\pi}_{S}^{T}(q,\hspace{0.17em}t,\hspace{0.17em}u)]$ in Eqs. (17) and (18) are concave with respect to q.
Proof. See Appendix F.
The provisional product order quantity for t and u ≠ 1, ${q}_{I}^{T}\{t,\hspace{0.17em}(u\ne 1)\},$ is derived as q which satisfies the first order condition of $E[{\pi}_{S}^{T}\{q,\hspace{0.17em}\hspace{0.17em}t,\hspace{0.17em}\hspace{0.17em}(u\ne 1)\}]$ with respect to q, $\partial E[{\pi}_{S}^{T}\{q,\hspace{0.17em}\hspace{0.17em}t,\hspace{0.17em}\hspace{0.17em}(u\ne 1)\}]/\partial q=0.$ Also, by solving $\partial E\left[{\pi}_{S}^{T}\left\{q,\hspace{0.17em}t,\hspace{0.17em}\left(u=1\right)\right\}\right]/\partial q=0$ with respect to q, ${q}_{I}^{T}\{t,\hspace{0.17em}\hspace{0.17em}(u=1)\}$ is determined regardless of t as the following unique solution:
As with FOP, it is impossible to derive the optimal recycling incentive ${t}_{I}^{T}$ and the optimal lower limit of quality leve l ${u}_{I}^{T}$ by mathematical analyses in ICLSC with TOP. In this study, ${t}_{I}^{T}$ and ${u}_{I}^{T}$ are determined by a numerical search. The decision procedure for ${t}_{I}^{T}$, and ${u}_{I}^{T}$, is shown as follows:
Step 1. Substitute ${q}_{I}^{T}\left(t,\hspace{0.17em}u\right)$, t, and u, and u, into the expected profit of the whole system, $E[{\pi}_{S}^{T}(q,\hspace{0.17em}\hspace{0.17em}t,\hspace{0.17em}\hspace{0.17em}u)]$ with varying t within $0\le t\le {w}_{m}{c}_{m}{w}_{r}$ and u within 0 ≤ u ≤ 1 .
Step 2. Determine the optimal combination of t and u, which maximizes $E[{\pi}_{S}^{T}\{{q}_{I}^{T}(t,\hspace{0.17em}\hspace{0.17em}u),\hspace{0.17em}\hspace{0.17em}t,\hspace{0.17em}\hspace{0.17em}u\}],$ as $({t}_{I}^{T},\hspace{0.17em}\hspace{0.17em}{u}_{I}^{T})$.
From the above, the optimal product order quantity ${q}_{I}^{T}$ is derived as ${q}_{I}^{T}({t}_{I}^{T},\hspace{0.17em}\hspace{0.17em}{u}_{I}^{T})$.
4.3. Supply Chain Coordination
In general, by shifting from the optimal operation for DCLSC to that for ICLSC, the expected profit of the whole system increases. Furthermore, FOP can improve the expected profit of the whole system in ICLSC, as shown in Subsection 4.2.1. However, it is not guaranteed that the expected profit of each member improves. Nevertheless, it is desirable to shift from the optimal operation for DCLSC to that for ICLSC from the aspect of the total optimization of the expected profit.
This subsection discusses supply chain coordination (SCC) among the buyer, the manufacturer, and the recycler, to guarantee the improvement in the expected profit of each member. In this study, introduction of the following two profit sharing approaches as SCC are discussed: (I) profit sharing based on Nash bargaining solution, (II) profit sharing based on return on investments.
4.3.1. Profit Sharing based on Nash Bargaining Solution
In profit sharing based on Nash bargaining solution, the unit procurement cost of remanufactured parts , w_{r} , and the unit wholesale price of products, w_{m} , are coordinated as the Nash bargaining solution $\left({w}_{r}^{Nash},{w}_{m}^{Nash}\right)$
r m w w The coordinated parameters $\left({w}_{r}^{Nash},{w}_{m}^{Nash}\right)$ are determined so as to maximize $T\left({w}_{r}^{Nash},{w}_{m}^{Nash}\right)$ in Eq. (32) with satisfying the constrained conditions in Eqs. (33)(35), as follows:(34)
Eqs. (33)(35), are constraint conditions to guarantee that the expected profit of each member in ICLSC with FOP after the profit sharing (I) is higher than that in DCLSC.
4.3.2. Profit Sharing based on Return on Investments
In profit sharing based on Return on Investments, the improvement in the expected profit of the whole system attributed to the shift from DCLSC to ICLSC, $\Delta {\pi}_{S}^{E},$, is distributed based on return on investment of each member. $\Delta {\pi}_{S}^{E}$ is calculated as
Also, the return on investment of the member i∈{B, M, R}, r_{i} is calculated as
Here, $E[T{C}_{i}(q,\hspace{0.17em}\hspace{0.17em}Q,\hspace{0.17em}\hspace{0.17em}t,\hspace{0.17em}\hspace{0.17em}u)]$ denotes the total cost of the member i. From each of Eqs. (7), (13), (9), (15), and (11), it is formulated as(38)(39)(40)(41)(42)
So as to satisfy ${r}_{B}+{r}_{M}+{r}_{R}=1,$, r_{i} is standardized as
With the use of the standardized return on investment, ${r}_{i}^{N},$, the distributed expected profit to the member $i,\hspace{0.17em}\Delta {\pi}_{i}^{E},$ is calculated as(44)
As a result, the expected profit of the member i in ICLSC with FOP after the profit sharing (II), ${\pi}_{i}^{E},$ is derived as
5. NUMERICAL ANALYSES
This section investigates how (i) the uncertainty in product demand, and (ii) the uncertainty in collection quantity of used products, and (iii) a quality distribution of the parts extracted from used products, affect the optimal operation and the expected profits in the CLSC with FOP by providing numerical examples. Also, in order to clarify the effects of FOP, the expected profit of the whole system under the optimal operation for ICLSC with FOP is compared with that for ICLSC with TOP. Moreover, the benefits of SCC adopting profit sharing based on each of Nash bargaining solution and return on investments are shown.
The following numerical examples are used as system parameters in a CLSC: 150, p_{m} = 15, h_{m} = 175, s_{m} = 70, w_{m} = 10, c_{m} = 40, c_{n} = 10, s_{p} = 20, w_{r} = 5, c_{d} = 3, c_{a} = 1. c_{c} = A(t) and ${c}_{r}(\ell )$ are respectively set as A(t) = 500 + 50t and ${c}_{r}(\ell )=40(10.9\ell ),$ satisfying the characteristics in Subsection 3.2.2 (1) and (4). As a result, the upper limit of t is 40. ${w}_{m}{c}_{m}{w}_{r}=40.$ The product demand x follows the normal distribution with mean ${\mu}_{x}=1000$ and variance ${\sigma}_{x}^{2}={300}^{2}.$ The additive variation ε follows the normal distribution with mean ${\mu}_{\epsilon}=0$ and variance ${\sigma}_{\epsilon}^{2}={100}^{2}.$ The quality level of the parts extracted from used products, ℓ, follows the beta distribution with parameters (a, b).
The probability density function g (ℓ  a, b) of the beta distribution is provided as(46)
where $\Gamma (\xb7)$ denotes the gamma function. This study provides the following four cases of quality distribution $B\left(\ell a,\hspace{0.17em}b\right)$ of the parts:

Case 1 B(ℓ 1,1) : a case where quality of the parts are distributed uniformly,

Case 2 B(ℓ  2, 2) : a case where there are many parts with middle quality level,

Case 3 B(ℓ  3, 2) : a case where there are many parts with high quality level,

Case 4 B(ℓ  2, 3) : a case where there are many parts with low quality level.
5.1. Influence of the Quality Distribution on the Optimal Operation for each of DCLSC and ICLSC with FOP
Table 1 shows the optimal operation for each of DCLSC and ICLSC with FOP in the four cases, Cases 14, of quality distribution of the parts. The optimal operation under DCLSC $\left({q}_{D},\hspace{0.17em}\hspace{0.17em}{Q}_{D},\hspace{0.17em}\hspace{0.17em}{t}_{D},\hspace{0.17em}\hspace{0.17em}{u}_{D}\right)$ can be obtained from Eqs. (21) and (22) and Subsection 4.1.2 through the numerical calculation and the numerical search. The optimal operation under ICLSC with FOP $\left({q}_{I},\hspace{0.17em}\hspace{0.17em}{Q}_{I},\hspace{0.17em}\hspace{0.17em}{t}_{I},\hspace{0.17em}\hspace{0.17em}{u}_{I}\right)$ can be obtained from Eqs. (27) and (28) and Subsection 4.2.2 through the numerical calculation and the numerical search.
${E}_{D}[{x}_{r}]$ in Table 1 denotes the expected quantity of remanufactured parts under DCLSC. ${E}_{D}[{x}_{r}]$ is calculated through the numerical calculation as(47)
${E}_{I}[{x}_{r}]$ denotes the expected quantity of remanufactured parts under ICLSC with FOP. ${E}_{I}[{x}_{r}]$ is calculated through the numerical calculation as(49)
From Table 1, it can be seen that the optimal recycling incentive t_{I} and the optimal lower limit of quality level u_{I} in ICLSC are higher than t_{D} and u_{D} in DCLSC in any case. Here, from Eq. (2), as t is higher or u is lower, the quantity of remanufactured parts,x_{r}, is larger. As a result, ${E}_{I}[{x}_{r}]$ in ICLSC with FOP is larger than ${E}_{D}[{x}_{r}]$ in DCLSC. Therefore, it can be said that the remanufacturing activity can be promoted in ICLSC with FOP. Also, due to Propositions 2 and 5, the optimal minimum product order quantity and the optimal maximum product order quantity in Table 1 are independent of quality distribution of the parts.
5.2. Influence of the Uncertainty in Product Demand on the Optimal Operation for ICLSC with each of TOP and FOP
Table 2 shows the optimal operation for ICLSC and the expected profit of the whole system in ICLSC with each of TOP and FOP in Case 2, against the standard deviation of product demand x, σ_{x}. Here, ${\pi}_{S}^{T}\left({q}_{I}^{T},\hspace{0.17em}{t}_{I}^{T},\hspace{0.17em}{u}_{I}^{T}\right)$ denotes the expected profit of the whole system under the optimal operation for ICLSC with TOP, and ${\pi}_{S}^{}\left({q}_{I},\hspace{0.17em}{Q}_{I},\hspace{0.17em}{t}_{I},\hspace{0.17em}{u}_{I}\right)$ denotes that under the optimal operation for ICLSC with FOP. Also, the improvement rate of the expected profit of the whole system, P (%), appearing in Table 2 is calculated as
From Table 2, it can be seen that as σ_{x} increases, ${\pi}_{S}^{T}\left({q}_{I}^{T},\hspace{0.17em}\hspace{0.17em}{t}_{I}^{T},\hspace{0.17em}\hspace{0.17em}{u}_{I}^{T}\right)$ and ${\pi}_{S}^{}\left({q}_{I},\hspace{0.17em}\hspace{0.17em}{Q}_{I},\hspace{0.17em}\hspace{0.17em}{t}_{I},\hspace{0.17em}\hspace{0.17em}{u}_{I}\right)$ decrease, and P improves. Therefore, it is verified that, in ICLSC, FOP can relieve the decrement in the expected profit of the whole system caused by increment in the uncertainty in product demand. In addition, as the uncertainty in product demand increases, the optimal collection incentive becomes higher, or ICLSC tends to collect more used products and remanufacture higher quality parts selectively. Also, these results can be seen in any Case of the quality distribution.
5.3. Influence of the Uncertainty in Collection Quantity of Used Products on the Optimal Operation for ICLSC with Each of TOP and FOP
Table 3 shows the optimal operation for ICLSC and the expected profit of the whole system in ICLSC with each of TOP and FOP in Case 2, against the standard deviation of the additive variation ε in collection quantity of used products, σ_{ε}.
In a similar way to Subsection 5.2, from Table 3, it can be said that, in ICLSC, FOP relieves the decrement in the expected profit of the whole system caused by increment in the uncertainty in collection quantity of used products. Also, as σ_{ε} increases, not only the optimal recycling incentive but also the optimal lower limit of quality level become higher. Therefore, as the uncertainty in the collection quantity becomes higher, ICLSC gets to collect more used products and remanufacture higher quality parts selectively. These results are same as those in the other Cases of the quality distribution.
5.4. Benefits of Supply Chain Coordination (SCC)
First, Tables 47 shows the results of a SCC adopting (I) profit sharing based on Nash bargaining solution where the unit wholesale price of the products and sales price of the remanufactured parts are coordinated, in the four cases of the quality distribution. These results are calculated so as to satisfy Eqs. (32)(35) in Subsection 4.3.1. 5, 6
From Tables 47, the following things can be seen in any Case of the quality distribution:

The expected profits of the buyer and the manufacturer in ICLSC without the SCC are lower than those in DCLSC,

The SCC can improve the expected profits of all the members.
Therefore, the benefits of the SCC adopting profit sharing based on Nash bargaining solution is confirmed.
Second, Tables 811 shows the results of a SCC adopting (II) profit sharing based on return on investments in the four cases of the quality distribution. ${r}_{i},\hspace{0.17em}\hspace{0.17em}{r}_{i}^{N},\hspace{0.17em}\text{and}\hspace{0.17em}{\pi}_{i}^{E}\left(i\in \left\{B,M,R\right\}\right),$ in Tables 811 are calculated from Eqs. (37), (43), and 45, respectively. 9, 10
From Tables 811, it can be seen that the SCC can improve the expected profits of all the members in any Case. In addition, return on investment of the buyer, r_{B}, is less affected by the quality distribution of the parts in
comparison with return on investments of the other members. In Case 4 of the four cases, r_{M} is the highest and r_{R} is the lowest. This is because, in Case 4, the optimal recycling incentive is the lowest as showed in Table 1, due to the low quality of the parts.
From the above consideration, it is found that both of the profit sharing approaches (I) and (II) can guarantee the improvement in the expected profit of all the members in ICLSC and enable to show the benefits of the shift from the optimal operation for DCLSC to that for ICLSC.
Next, the benefits of the profit sharing approaches (I) and (II) are compared. In the profit sharing approach (I), the unit procurement cost of remanufactured parts, w_{r}, and the unit wholesale price of products, w_{m}, are coordinated as the Nash bargaining solution to share the expected profit among the buyer, the manufacturer, and the recycler. As a result, the improvement in the expected profit of the whole system tends to be equally distributed to each member in ICLSC with satisfying Eqs. (33)(35). Therefore, the profit sharing approach (I) doesn’t reflect the performance of earning profit of each member in the sharing. In the profit sharing approach (II), the return on investments, the ratio of the profit to the total cost, are reflected in the profit sharing among the buyer, the manufacturer, and the recycler. This means that the performance of earning profit of each member is considered. Accordingly, all the members in ICLSC can accept this profit sharing, and management effort of each member to earn profit will be promoted.
Therefore, the profit sharing approach (II) can be more acceptable profit sharing approach among the members in a CLSC than the profit sharing approach (I).
6. CONCLUSIONS
This paper discussed a CLSC with operational flows from collection of used products through remanufacturing of them to sales of products produced of the remanufactured parts. To deal with the uncertainties in product demand and collection quantity of used products, which the CLSC faced, incorporation of FOP into the CLSC is discussed, and the expected profits of the members in the CLSC were formulated with consideration of the uncertainties. This paper optimally determined the minimum product order quantity, the maximum product order quantity, the recycling incentive, and the lower limit of quality level, for each of DCLSC and ICLSC. In the result of the mathematical analyses, it was verified that the optimal operation in DCLSC with FOP came down to that with TOP, and FOP could improve the expected profit of the whole system in ICLSC. In order to realize the shift from DCLSC to ICLSC, this paper discussed introduction of the profit sharing based on each of (I) Nash bargaining solution, where the unit wholesale price of the products and the unit sales price of the recycled parts were coordinated among the members in the CLSC, and (II) return on investments of the members in the CLSC.
Numerical analyses investigated how (i) the uncertainty in product demand, and (ii) the uncertainty in collection quantity of used products, and (iii) a quality distribution of the parts extracted from used products, affect the optimal operation and the expected profits in CLSC with FOP. In addition, by comparing the expected profit of the whole system in ICLSC with FOP with that in ICLSC with TOP, the effects of FOP was showed. Moreover, the benefits of SCC adopting profit sharing based on each of Nash bargaining solution and return on investments were confirmed.
Results of the mathematical analyses and the numerical analyses gave the following managerial insights to practitioners, academic researchers, and policymakers:

By using the mathematical analyses and the numerical searches, the optimal decisions of the minimum product order quantity, the maximum product order quantity, the recycling incentive, and the lower limit of quality level, can be made for DCLSC and ICLSC with FOP,

The optimal operation for DCLSC with FOP comes down to that for DCLSC with TOP,

FOP adjusts the wholesale quantity of the products between the minimum product order quantity and the maximum product order quantity as to the quantity of the remanufactured parts, and it can improve the expected profit of the whole system in the CLSC which faces the uncertainties in product demand and collection quantity of used products,

In any Case of quality distribution of the parts, the remanufacturing activity can be promoted by the shift from DCLSC to ICLSC with FOP,

As the uncertainties in product demand and collection quantity of used products are higher, the improvement in the expected profit of the whole system is larger, or FOP can relieve the decrement in the expected profit of the whole system caused by increment in the two uncertainties,

The SCC adopting profit sharing based on each of (I) Nash bargaining solution and (II) return on investments can guarantee the improvement in the expected profits of all the members in the CLSC,

The profit sharing approach (II) can be more recommended than the profit sharing approach (I) because the approach (II) can reflect the performance of each member of earning profit in the sharing.
As future researches, it will be necessary to discuss the following topics:

Incorporation of a contract which causes FOP to function in DCLSC,

Way to the decisionmaking when the manufacturer, the buyer, and the recycler, are regarded as the first leader, the second leader, and the follower, respectively,

Development of a CLSC model in which a target value of collection quantity of used products is determined by a recycler.