Uncertain pricing decision problem in closedloop supply chain with riskaverse retailer
 Hua Ke^{1},
 Yanzhun Li^{1} and
 Hu Huang^{1}Email authorView ORCID ID profile
https://doi.org/10.1186/s4046701500393
© Ke et al. 2015
Received: 12 August 2015
Accepted: 7 October 2015
Published: 26 October 2015
Abstract
This paper considers a pricing and remanufacturing problem in closedloop supply chain which includes one dominant manufacturer and one retailer under uncertain environment. The remanufacturing cost, market demand, and collection cost are characterized as uncertain variables. The firms’ optimal strategies are obtained by uncertaintytheorybased and gametheorybased models. Besides, the impacts of the risk sensitivity on the performances of the closedloop supply chain members are given by the comparison of different degrees of the retailer’s risk aversion and numerical studies. It is found that only the manufacturer makes more profits when the retailer is more risk sensitive but this condition in the closedloop supply chain still leads to better performance of the total system.
Keywords
Introduction
With the worldwide promotion of lowcarbon economy and sustainable development, closedloop supply chain management has played an increasingly important role in both the environment and economic activities throughout this decade [1–6]. Reverse collection and remanufacturing are becoming more and more important under this background. Increasing researches about reverse supply chain management have been conducted in this decade [7–11]. For instance, Wei and Zhao [8] addressed the decisions of reverse channel with three different used products collection channels from customers and investigated how the implications of these models affect the decisions of the manufacturer, the retailer, and the third party. Govindan [10] reviewed recently published papers in reverse logistic and closedloop supply chain in scientific journals and suggested future research opportunities according to these papers. Wei and Zhao [11] considered the pricing and remanufacturing decisions in a duopoly market with two competing supply chains, each of which includes one manufacturer and one retailer. Five game decision models were established to explore the chain members’ optimal strategies on price and/or remanufacturing and examined the influences of some key parameters and chain members’ maximum profits through numerical studies.
According to the lack of referable historical data and uncertainties in collection of used products, we adopt uncertainty theory instead of probability theory to solve problems with such variables. For now, uncertainty theory has been successfully applied to many fields, such as option pricing problem [12], stock problem [13], production control problem [14], supply chain pricing problem [15], uncertain random process [16], project scheduling problem [17], and so on. In this paper, we consider the remanufacturing cost, market demand, and collection cost as uncertain variables.
Besides, many researches also take the risk in supply chain into consideration. Agrawal and Seshadri [18] considered a singleperiod inventory model in which a riskaverse retailer faces random customer demand and decides the selling price with the objective of maximizing expected utility. This research also provided a better understanding of retailers’ pricing behavior that could lead to improved price contracts and channel management policies. Barry [19] addressed that the world is at risk and the supply chain is not exempt and raised some essential supply chain questions that have impacts on the field which is from outside of the supply chain. Xiao and Yang [20] developed a priceservice competition model of two supply chains, each including one riskneutral supplier and one riskaverse retailer to investigate the optimal decisions of players under demand uncertainty and analyzed the effects of the retailers’ risk sensitivity on the players’ optimal strategies. Ke et al. [21] studied a pricing decision problem in fuzzy supply chain with one manufacturer and two risksensitive retailers.
The goal of this paper is to analyze how risk sensitivity affects the performances of the channel members. In consideration of different degrees of the risk aversion of the retailer, two different models are built to derive the optimal management strategies of the supply chain members under different decision scenarios. Moreover, the impacts of the risk sensitivity on the performances of the closedloop supply chain are also given by numerical studies. It is found that only the manufacturer makes more profits when the retailer is more risk sensitive. Meanwhile, this choice in the closedloop supply chain leads to better performance of the total system. The results also show how the uncertainty in the supply chain influences the pricing and remanufacturing decisions.
The rest of the paper is organized as follows: Preliminaries are presented in “Preliminaries” section. Some useful notations and necessary assumptions are discussed in “Problem description” section and two uncertain models are detailed in “Models and solution approaches” section. In the “Analysis of the strategy decision” section, numerical experiments are applied to analyze the effects of the uncertainty and power structures on the optimal decisions and maximal profits. Some conclusions are given in the “Conclusions” section.
Problem description
In this paper, we consider a closedloop supply chain pricing problem which includes one dominant manufacturer and one retailer and analyze how the retailer’s risk sensitivity affects the performances of the channel members. In the forward supply chain, the manufacturer produces new products by using the original component at unit cost c _{ m }, remanufactures products from used products at unit cost \(\tilde {c_{r}}\), which is an uncertain variable, and wholesales the products to the retailer with unit wholesale price w _{ r }, which is a decision variable. Then, the retailer sells products to the costumer at unit sales price p _{ r }, which is a decision variable. In the reverse supply chain, the used products are recycled from the consumers by different channels.

c _{ m }: unit manufacturing cost of the product

w _{ r }: unit wholesale price of the product, a decision variable

p _{ r }: unit sales price of the product, a decision variable

\(\tilde {c_{r}}\): unit remanufacturing cost of the product, an uncertain variable

\(\tilde {p_{c}}\): unit collection cost of used product, an uncertain variable

τ: the used products collecting rate, 0 ≤τ≤1
In order to attain the closedform solutions, some assumptions of this work are made as follows:
Assumption 1.
where \(\tilde {d}\), \(\tilde {\beta }\) are uncertain variables, \(\tilde {d}\) denotes the primary demand of the new products, while \(\tilde {\beta }\) denotes the measure of the responsiveness of the new product’s demand to its own price.
Assumption 2.
All the uncertain coefficients are assumed nonnegative and mutually independent.
Assumption 3.
(Dominant manufacturer) The manufacturer is the Stackelberg leader, and the retailer is the follower under both the two collection models.
Assumption 4.
(Riskaverse retailer) The retailer is assumed to be risk averse.
Throughout this paper, we adopt the convention of pronoun “he” to refer to the manufacturer and pronoun “she” to refer to the retailer. The manufacturer in each decision scenario makes his decision to maximize his own expected profit, subjected to the retailer’s response. We solve the retailer’s response function firstly, given that she has observed the manufacturer’s decision.
Models and solution approaches
In this section, Stackelberg game models are employed to derive the equilibrium prices in different scenarios.
EV model
Proposition 1.
Proposition 2.
According to Propositions 1 and 2, Proposition 3 can be easily obtained.
Proposition 3.
CC model
The CC model is to maximize the objective function under a certain confidence level and has been widely used to deal with problems with risksensitive decision makers.
In this paper, we assume that the retailer is risk averse, which means that the confidence level α>0.5. To solve the model, we should transform the uncertain model into an equivalent model first.
Proposition 4.
Proposition 5.
According to Propositions 4 and 5, Proposition 6 can be easily obtained.
Proposition 6.
Analysis of the strategy decision
Due to the complicated forms of the equilibrium prices, numerical examples rather than analytical comparisons are conducted to explore the effects of the risk sensitivity of the retailer on equilibrium prices.
Similarly, we can attain the expected values of \(E\left [\tilde {c}_{r}^{\alpha }\tilde {d}^{1\alpha }\right ]\), \(E\left [\tilde {p}_{c}\tilde {\beta }\right ]\), and \(E\left [\tilde {c}_{r}\tilde {\beta }\right ]\).
Optimal decisions and maximal expected profits
Decision scenario  α  \({w}_{r}^{*}\)  τ ^{∗}  π _{ m }  p ^{∗}  π _{ r }  π _{ m }+π _{ r } 

EV    185.9999  0.2807  4119.13  221.5714  996.45  5115.58 
0.6  152.8141  0.2797  5303.43  198.6293  2111.14  7414.57  
0.7  155.4727  0.2865  5556.37  197.2485  1970.43  7526.80  
CC  0.8  158.0673  0.2932  5808.88  195.9011  1824.65  7633.53 
0.9  160.6000  0.2997  6060.80  194.5857  1674.26  7735.06  
1  163.0732  0.3060  6311.95  193.3013  1519.72  7831.67 

It is clear that the optimal values of the wholesale price and the collecting rate of used products are becoming higher along with the increase of α in the CC model, which means that the manufacturer can decide higher wholesale price when the retailer is more sensitive to the risk.

The sales price is becoming lower with the increase of α in the CC model.

Consumer may prefer that the retailer is risk sensitive because the sales price is highest in the EV model.

Both the maximal profits of the whole system and the manufacturer are becoming higher when α increases in the CC model.

The maximal profit of the retailer is becoming lower when α increases in the CC model.

The maximal profits of the whole system and individual firms are lowest in the EV model which means that the manufacturer prefers that the retailer is more risk sensitive.
Conclusions
In this paper, we considered an uncertain pricing decision problem in closedloop supply chain with riskaverse retailer. The remanufacturing cost, consumer demand, and used products collecting cost were defined as uncertain variables. Uncertaintytheorybased and gametheorybased models were applied to obtain the optimal strategies. The equilibrium behaviors of the participants in the operational level and the optimal decisions on wholesale price, sales price, and used products collecting rate were derived from these models. Numerical experiments were also given to supervise the effects of the retailer’s risk sensitivity in strategy level.
Our works mainly focus on the impacts of the retailer’s risk sensitivity on the performances of the closedloop supply chain members with the dominant manufacturer. The different dominant participants and the selections of the recycling channels are the important directions for the future research.
Appendix
Preliminaries
In this section, we will introduce some important concepts and theorems of uncertainty theory for modeling the pricing decision problem with human belief degree.
Let Γ be a nonempty set and a σalgebra over Γ. Each element Λ in is called an event. The set function , initiated by Liu [22] and refined by Liu [23], is called an uncertain measure if it satisfies:
Besides, the product uncertain measure on the product σalgebra was defined by Liu [24] as follows:
Axiom 4.
where A _{ k } are arbitrarily chosen events from for k=1,2,⋯, respectively.
Definition 1.
Definition 2.
Definition 3.
An uncertainty distribution Φ is referred to be regular if its inverse function Φ ^{−1}(α) exists and is unique for each α∈[0,1].
Lemma 1.
Definition 4.
Lemma 2.
Lemma 3.
Example 1.
Example 2.
Lemma 4.
provided that the expected value E[ξ] exists.
Example 3.
Definition 5.
Proof of Proposition 1
By solving Eq. (30), Eq. (6) can be obtained. Proposition 1 is proved.
Proof of Proposition 2
Solving Eq. (34), we obtain Eq. (9). Proposition 2 is proved.
Proof of Proposition4
By solving Eq. (37), Eq. (15) can be obtained. Proposition 4 is proved.
Proof of Proposition5
Solving Eq. (41), we obtain Eq. (18). Proposition 5 is proved.
Declarations
Acknowledgements
This work was supported by National Natural Science Foundation of China (Nos. 71371141,71001080).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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