Static and Dynamic Pricing Strategies in a Closed-Loop Supply Chain with Reference Quality Effects

Remanufacturing of returned products has been increasingly recognized in industries as an effective approach to face environmental responsibility, government regulations, and increased awareness of consumers. In this paper, we address a closed-loop supply chain (CLSC) in which the manufacturer produces the brand-new products, as well as the remanufactured goods while the retailer sells these products to customers. We consider several different scenarios: the manufacturer and the retailer adopt a steady-state price or a dynamic price with reference quality effects in a centralized case; either, neither or both the manufacturer and the retailer price dynamically with reference quality effects, respectively, in a decentralized model. We solve the problem with the retailer recycling the sold copies and deduce the optimal pricing strategies while the manufacturer in charge of recovering the used items in such a CLSC. The result shows that dynamic pricing strategies are much more profitable for the supply chain and its members when compared with pricing statically; the dynamic pricing strategies with time-varying quality characterized by reference quality are more suited to a long-term and cooperative closed-loop supply chain. Moreover, the optimal recycling fraction relies on the recovery cost coefficient and proves to be uniform despite adopting a dynamic price and quality in all distinct cases.


Introduction
As the global economy prospered and advanced technology widely used, so many diversified products move towards consumers like an unstoppable wave that the life of products becomes shorter and shorter, the speed of renewing goods gets faster and faster, which make the total amount and the growth ratio of the waste rise. Approximately 1.3 billion tons of solid waste was generated in cities worldwide in 2017. It is estimated that by 2025, the annual amount of solid waste will be about two times greater than the amount in 2017 [1,2]. Meanwhile, the circular economy, replacing the "take, make, consume, and dispose" patterns with closed-loops of material flows by combining a number of different processes including maintenance, repair, reusing, refurbishing, remanufacturing, and recycling, has created much more concern in the academic world and in practice [2,3]. As such, remanufacturing, as a crucial component of the circular economy, has been regarded as a potential method to deal with economic, social, and environmental problems [4][5][6][7]. Many manufacturers have For the sake of comparing the effectiveness of the various distinct pricing strategies with dynamic quality characterized by reference quality, we attempt to solve the model in several scenarios: (1) The whole supply chain adopts static and dynamic pricing strategies in a centralized scenario respectively; and, (2) In a decentralized case, either, neither or both the manufacturer and the retailer price dynamically, respectively.
The remainder of this paper is organized as follows. A literature review is represented in Section 2. In Section 3, the model setting is formulated, and the notation is defined. The analytic solution is presented by solving the Stackelberg game between the manufacturer and the retailer in both the centralized and decentralized cases in Sections 4 and 5, respectively. Section 6 gives numerical examples and assesses the sensitivity of some key parameters to demonstrate the feasibility of the main results. Eventually, we conclude the paper with discussions and further research directions in Section 7.

Literature Review
The main purpose of our work is to investigate different pricing strategies in the presence of reference quality in a closed-loop supply chain. Hence, literature referenced in this paper are formed from several topics: closed-loop supply chain, pricing dynamically or statically, and the reference quality effects.

Closed-Loop Supply Chain
As for the CLSC, the quality of the brand-new and remanufactured product plays a significant role in the remanufacturing process. Atasu and Souza [5] proposed three typical product recovery policies and analyzed the impacts that they had on quality decisions. The results suggested that the way of recycling vigorously affected the quality choice of the closed-loop supply chain system. Product recovery form (product recycling reuses the components or product recycling is costly) generally promoted the quality decision of the firm, while the form (product recycling remains profitable without using the material or components) inversely increased the product quality choice. Maiti and Giri [15] considered a closed-loop supply chain in which the manufacturer provided a product with a decent quality available to consumers in the context of a third party collecting the used items. The best decisions were acquired for such a closed-loop supply chain with different cases. It was found that the centralized policy was always optimal, and among the four decentralized cases, the retailer-led one was much more profitable. Ahmed et al. [21] assumed that the recovery fraction of returned items was a variable controlled by two related decision coefficients, which were the buying price for returned goods corresponding to an acceptable quality level.
However, the demand is not only sensitive to a measurable quality of product, but also is affected by the consumer's perceived quality [22]. Hazen et al. [23] studied the perceptual quality of remanufactured goods from construct and measure development perspectives and found that perceived quality of remanufacturing products was sensitive to performance, features, lifetime, and service level. Wang and Hazen [6] revealed that purchase intention of remanufactured products was positively affected by perceived value, which was influenced by quality knowledge, cost knowledge, and green knowledge, respectively, via structural equation modeling.
Since dynamics is a prominent problem in the operation of modern enterprises, the decision variables have dynamic characteristics, particularly for a closed-loop supply chain in the market environment. Xu and Zhu [24] designed a non-linear dynamic pricing model to research the demand and the collection rate of used items affected by price and reference price, and pointed out that in the long term, the price of the ultimate product and the recycling price of used items was fixed to a certain value ignoring reference to the initial price of new goods and the recycling price of the used items. Huang and Nie [25] considered a closed-loop supply chain with a dynamic used-product collection fraction by building the differential equation about product return fraction. It was found that the recovery fraction and the collection effort were growing while the retail price and wholesale price were declining with time. Li et al. [26] proposed the concept of product reliability that relied on the quality and the time-utility value of the product and found that the pursuit of this strategy, especially the restructure and operation in a closed-loop supply chain, played a critical role in product quality.
Due to both the quality of the product and the reference quality perceived by the customer are sensitive to the profits of a supply chain and its members. In the market environment, quality has dynamic characteristics, as does the reference quality. Few researchers focus on the dynamic product quality and the reference quality for such a closed-loop supply chain in extant literatures. Furthermore, the decision variables are closely interrelated, for example, price is always a response to the quality of product. How does the dynamic quality affect the pricing strategies in a closed-loop supply chain? Consequently, we employed a dynamic pricing strategy with time-varying quality on the basis of reference quality effects in a closed-loop supply chain.

Dynamic Pricing Strategy
Being regarded as a superior pricing policy, many firms utilize time-varying pricing to gain more profits. Zhang et al. [27] compared the optimal strategies and profits for the supply chain with a time-varying reference price effect in distinct distribution channels. Shah et al. [28] constructed a Stackelberg game model with price markdown choice in a two-echelon supply chain. Wu et al. [19] employed a dynamic pricing and inventory strategy with reference effects being considered in both full retail price and price discount cases. Jia and Hu [29] investigated the time-varying ordering policy and dynamic price for deterioration items in a supply chain. Popescu and Wu [30] presented a dynamic pricing problem, in which demand was partly dependent on the firms' previous pricing. Xu and Liu [31] discussed the reference price effect on the effectiveness across three decentralized reverse channels in a closed-loop supply chain. Zhang et al. [32] found that a two-part tariff contract was meaningful to coordinate the channel in a green supply chain within dynamic pricing in different cases. Ferrara et al. [33] constructed a dynamic pricing model of supply chain for corporate social responsibility.
The above researchers investigated a dynamic pricing strategy with a reference price or other decision variables, such as green innovation and corporate social responsibility. Moreover, product quality directly determines the price of the product and is always closely linked to the price. Gavious and Lowengart [16] considered a case in which the reference quality was a state variable to investigate the players' pricing strategies. Zhang et al. [20] proposed dynamic and static pricing strategies in both centralized and decentralized scenarios with advertising taking reference quality effects into consideration. Kopalle and Winer [34] derived a case where the monopolist makes dynamic decisions considering price and expected product quality.
However, in spite of the obvious merits of time-varying pricing, some companies are still unwilling to modify pricing policies partly due to the potential adjustment costs and partly due to the consideration of customer psychology [35]. As a result, given that a comparison between dynamic and static pricing policies is vitally significant, which has already been seen in related researches, such as [20,35], it is clear that time-varying pricing policies apparently exceed steady-state pricing strategies with neglecting the adjustment cost, for the uniform pricing policy could be considered as a specific case of the time-varying one. By pricing statically, Ghosh and Shah [36] considered game theoretical models and presented how greening levels, pricing strategies, and profits were disrupted by channel structures. Krishnamurthi [37] investigated differences between the reference price, the purchase price, and ordering quantity, where consumers exhibited reference price. Liu et al. [38] focused on pricing strategies with three different supply chain network structures in two-stage Stackelberg game models. Karray and Sigue [39] gave a comparison about the pricing policy of different companies, in which one firm sold an independent product, while the other two sold complementary goods.
Additionally, the difference and the comparison between time-varying and steady-state pricing have rarely derived attention among researches. Cachon and Feldman [35] suggested that pricing statically provides more benefits compared with pricing dynamically when taking strategic consumers Sustainability 2018, 10, 157 5 of 31 into consideration. Zhang et al. [20] compared the time-varying and steady-state pricing policies in both centralized and decentralized scenarios with advertising. Gayon and Dallery [40] combined pricing strategies with the optimum production decision, and then presented that time-varying pricing could be a better choice to increase benefits when production is partly constrained in a make-to-stock queue.
Few studies researched a dynamic pricing strategy in a closed-loop supply chain, particularly with pricing dynamically with reference quality effects. What is the best pricing strategy, pricing dynamically or pricing statically, if the dynamic quality and dynamic reference quality are considered in a closed-loop supply chain? Hence, this work tries to compare the dynamic pricing and uniform pricing taking dynamic product quality and reference quality into consideration in a closed-loop supply chain.

Reference Quality Effect
Product quality improvement has derived much more attention among academia. Singer et al. [17] constructed models for different strategic behavior considering product quality in a supplier-retailer partnership for disposable items. Li et al. [41] developed a grocery retail supply chain to avoid food spoilage waste and gained the optimal food retailer's revenue via a pricing channel that is determined by dynamically identifying food shelf life. Chao et al. [42] investigated two contract mechanisms for product recycling cost sharing between a manufacturer and a supplier to stimulate quality improvement efforts. Xie et al. [43] researched a risk-averse supply chain combined with quality investment efforts and price decision under uncertain demand. Most of above studies think of quality as a uniform value, however, the firm always modifies the quality policy of product according to the market and consumers in a given period. The enterprise will make more (less) efforts to improve the product quality if customers are exposed to much more (less) qualified goods provided by its competitors. It signifies that relative quality among products does work when consumers make buying decisions, in other words, customers would set a benchmark of quality (perceived by previous product, observable information of the product and the product of its competitors) before they prepare to purchase this product.
Chenavaz [44] studied the dynamic quality strategy of a company whose customers utilized a reference quality when decision-making according to the principles of behavioral economics. The paper regards reference quality as an optimal control setting and determines solutions using Pontryagin's maximum principle without considering the fact that price and quality are closely related. Accordingly, in accordance with firms' practice in management, the relationship between the price and the dynamic quality strategy characterized by reference quality effects could not be neglected. Gavious and Lowengart [16] proposed an asymmetric and contractual agreement and investigated the impacts that the price and reference quality had on profits. Liu et al. [45] studied the reference quality effects combined with a revenue sharing contract in myopic and far-sighted scenarios. Kopalle and Winer [34] derived a case where the monopolists made dynamic decisions considering price and expected product quality. Zhang et al. [20] compared dynamic and static pricing strategies in a supply chain with advertising in the context of reference quality effect.
The aforementioned researches investigated the reference quality and dynamic quality in a traditional supply chain and few of them compared the pricing policies that are affected by dynamic quality and reference quality. However, in a closed-loop supply chain, the product system including brand-new products and remanufactured goods as well as the collection policy should be considered. Does this product system influence the optimal pricing strategy with dynamic quality characterized by reference quality? What is the best collection policy in such a pricing and quality strategy for a closed-loop supply chain? Hence, based on reference quality effect, we investigate dynamic quality and dynamic pricing strategies in a closed-loop supply chain.
In general, there is a large quantity of research literature on closed-loop supply chains, dynamic pricing, reference and dynamic quality. Nevertheless, there are few literatures researching the time-varying pricing strategies in the presence of dynamic quality characterized by reference quality Sustainability 2018, 10, 157 6 of 31 effects in a closed-loop supply chain. We provide a specific comparison between the key literatures and our model shown in Table 1. Above all, we develop a Stackelberg game between a manufacturer and a retailer in a closed-loop supply chain. The manufacturer is a leader who settles the quality level and the wholesale price, while the retailer is a follower who determines the retail price in the Stackelberg game. Additionally, the purpose of this paper is to solve the problem using differential game with dynamic and static pricing strategies taking reference quality into consideration and explore the best recovery fraction for the closed-loop supply chain simultaneously.

Notations
We define the variables with a set a concise and inerratic symbols. For ease of writing, we make some general rules for the symbols, such as R(t) means the reference quality, etc. (as shown in Table 2).

Model Description
The model considers a closed-loop supply chain consisting of one manufacturer and one retailer. The manufacturer is responsible for the production of new products and remanufactured goods, while the retailer is in charge of recycling the used products. When considering the dynamic strategies, the decision variables are the functions of time t. We develop a Stackelberg game between a manufacturer and a retailer in a closed-loop supply chain. The manufacturer is a leader who settles the quality level Sustainability 2018, 10, 157 7 of 31 q(t) and the wholesale price w(t), while the retailer is a follower who determines the retail price p(t) in the Stackelberg game.
Since the reference quality is closely related with the product quality, the reference quality R(t) is always time-varying. Decentralized and centralized decision-making models are two predominant mainstreams in supply chain management [15,16,20]. Although the product quality and the reference quality invariably change with time, alternative pricing strategies can be selected for the manufacturer and the retailer-one is a time-varying pricing strategy and another is pricing statically. The static pricing strategy can be seen as a special case of the dynamic one in the decentralized model. Hence, four different patterns of the pricing strategies for the closed-loop supply chain are theoretically feasible. However, the retail price would also vary with the wholesale prices. Accordingly, it rarely happens in reality that the manufacturer takes a time-varying wholesale price, whereas the retailer adopts a uniform price. Hence, the model in which the manufacturer adopts a time-varying wholesale price, whereas the retailer applies a steady-state retail price is absolutely eliminated in our work. We obtain three different pricing strategies in the decentralized scenario and two distinct ones in the centralized scenario for the closed-loop supply chain shown in Figure 1.

Model Description
The model considers a closed-loop supply chain consisting of one manufacturer and one retailer. The manufacturer is responsible for the production of new products and remanufactured goods, while the retailer is in charge of recycling the used products. When considering the dynamic strategies, the decision variables are the functions of time t. We develop a Stackelberg game between a manufacturer and a retailer in a closed-loop supply chain. The manufacturer is a leader who settles the quality level q(t) and the wholesale price w(t), while the retailer is a follower who determines the retail price p(t) in the Stackelberg game.
Since the reference quality is closely related with the product quality, the reference quality R(t) is always time-varying. Decentralized and centralized decision-making models are two predominant mainstreams in supply chain management [15,16,20]. Although the product quality and the reference quality invariably change with time, alternative pricing strategies can be selected for the manufacturer and the retailer-one is a time-varying pricing strategy and another is pricing statically. The static pricing strategy can be seen as a special case of the dynamic one in the decentralized model. Hence, four different patterns of the pricing strategies for the closed-loop supply chain are theoretically feasible. However, the retail price would also vary with the wholesale prices. Accordingly, it rarely happens in reality that the manufacturer takes a time-varying wholesale price, whereas the retailer adopts a uniform price. Hence, the model in which the manufacturer adopts a time-varying wholesale price, whereas the retailer applies a steady-state retail price is absolutely eliminated in our work. We obtain three different pricing strategies in the decentralized scenario and two distinct ones in the centralized scenario for the closed-loop supply chain shown in Figure 1. In Figure 1, five models of the manufacturer and the retailer are listed, with the red and dotted downward line as the dynamic variables (q(t) represent product quality the retailer got from the manufacturer, R(t) represents reference quality the customer obtained from the retailer, and w(t) and p(t) represent the dynamic wholesale price and retail price, respectively), with the black and solid line as the static variables (C(τ) means the costs the manufacturer spent to collect the returned products from customer, b means the costs the manufacturer spent to retrieve the sold items from the retailer, and w and p represent the static wholesale price and retail price, respectively).
We derive a closed-loop supply chain with the retailer collecting the sold items, as the retailer is frequently exposed to the market as well as consumers, the used product was recovered by retailer and then transformed to the manufacturer with the price b. The production cost of the new product is cm, and cr denotes the cost of remanufacturing, producing a new product with a recycled item costs less than manufacturing a new one, we assume cr < cm and b + cr < cm [10]. Proposing that a fraction τ is controlled by the retailer and a higher τ means more difficult to recover, namely, it will cost more for the retailer to reach a greater proportion of recycled products, 0 < τ <1, and τ = 0 denotes that the retailer will not take recycling policy. In Figure 1, five models of the manufacturer and the retailer are listed, with the red and dotted downward line as the dynamic variables (q(t) represent product quality the retailer got from the manufacturer, R(t) represents reference quality the customer obtained from the retailer, and w(t) and p(t) represent the dynamic wholesale price and retail price, respectively), with the black and solid line as the static variables (C(τ) means the costs the manufacturer spent to collect the returned products from customer, b means the costs the manufacturer spent to retrieve the sold items from the retailer, and w and p represent the static wholesale price and retail price, respectively).
We derive a closed-loop supply chain with the retailer collecting the sold items, as the retailer is frequently exposed to the market as well as consumers, the used product was recovered by retailer and then transformed to the manufacturer with the price b. The production cost of the new product is c m , and c r denotes the cost of remanufacturing, producing a new product with a recycled item costs less than manufacturing a new one, we assume c r < c m and b + c r < c m [10]. Proposing that a fraction τ is controlled by the retailer and a higher τ means more difficult to recover, namely, it will cost more for the retailer to reach a greater proportion of recycled products, 0 < τ < 1, and τ = 0 denotes that the retailer will not take recycling policy.
Product quality is determined by some core components (e.g., camera pixels, memory size, battery capacity, CPU for mobile phone) for the products. These components have a series of quality levels and the qualified manufacturer applies different key ingredients to decide the product quality that makes product quality become a possible time-varying variable [5]. The performance of these critical components gradually declines so that they eventually stop working during product usage. The remanufacturing process is replacing the used components with the corresponding new ones. Hence, we assume that the remanufactured goods are as good as the brand-new products in quality [35].

Model Setting
Assuming that the product quality is observable by its features in the selling process, when deciding to purchase this product, consumers tend to evaluate product quality based on a benchmark, called reference quality. This product quality consumers perceived is affected through using some information, such as, consumers' previous experience about this product, past product quality levels, and the quality of its competitive counterparts, etc. To obtain the reference quality, customers calculate the continuous weighted average of all the previous product quality. Similar to extant research [34,44], reference quality writes where θ > 0 represents the continuous memory coefficient (or forgetting parameter) and R 0 denotes the initial reference quality at time t = 0. As such, solving the derivatives of R(t), that is dR(t)/dt or • R(t), we employ an exponential smoothing process of this historical quality to characterize the dynamics of reference quality, as follows, which is seen in research [16,27,45].
Similar to previous literature [20,46], we now describe market demand, which is influenced by market capacity, retail price, product quality, and reference quality. The function of reference quality on market demand relies on the difference between product quality and reference quality, and the market demand function is given by: where α ≥ 0 denotes market capacity, β ≥ 0 and γ 1 ≥ 0 represents the effects of retail price, and quality on current demand, respectively, γ 2 ≥ 0 captures the effect of the difference between product quality and reference quality on current sales. The total costs for the whole supply chain composed of manufacturing/remanufacturing cost, quality cost undertaken by the manufacturer, and recovery cost owned by retailer. c m and b + c γ represent the unit cost of manufacturing and remanufacturing, respectively. In accordance with former literatures [45,47], a quadratic cost function about quality level is proposed below: The recovery cost mainly consists of fixed expenses in accord with returning fraction τ caused by early investment, such as establishing the recycling channel and enhancing customers' awareness of recycling. Analogous to the extant literature [6,12,48], we obtain the following recovery cost function: Follow [44], the objective functions of profits for the stakeholders, the manufacturer, and the retailer of a closed-loop supply chain are denoted, respectively, as follows:

Centralized Decision System Scenario
In the centralized case, the manufacturer and the retailer adopt various strategies to seek the maximal profit for the whole supply chain. The profit maximization problem in the centralized scenario can be given by: We develop two cases in this section: one in which the closed-loop supply system applies a steady-state retail price, the other where the closed-loop supply system is pricing dynamically. Next, the optimal solutions about price, quality, and reference quality in both cases, are revealed, respectively.

The Static Price Model
When considering a uniform retail price, the centralized closed-loop supply chain system will maximize its profit function (7). Now, the following proposition depicts the optimum strategies on price, quality, and reference quality in the presence of pricing uniformly in such a closed-loop supply chain system. All the proofs of propositions and corollaries are disclosed in the Appendix A for the sake of the readability of this paper. Proposition 1. By pricing statically, the equilibrium strategies on retail price, quality, and reference quality for the centralized closed-loop supply chain are revealed below: (11) where M denotes in the Appendix A1.

The Dynamic Price Model
When considering a time-varying retail price, the centralized closed-loop supply chain system will maximize its profit function (7). Now, the following proposition describes the optimal strategies on price, quality, and reference quality in the context of pricing dynamically in such a closed-loop supply chain system. Proposition 2. By pricing dynamically, the equilibrium strategies on retail price, quality, and reference quality for the centralized closed-loop supply chain are displayed below: where

Decentralized Decision System Scenario
In the decentralized case, the manufacturer and the retailer regard maximizing their own interests as their decision criteria. Hence, the profit maximization problem in the decentralized case is shown in (14): The equilibrium strategies on price, quality, and reference quality are presented, respectively, in triple scenarios according to distinct pricing strategies, viz., dynamically or uniformly. In the first scenario, both of the members of the closed-loop chain adopt a uniform price, the second is the manufacturer applies the steady-state wholesale price, whereas the time-varying retail price is adopted by the retailer and the third is both the stakeholders of the closed-loop chain employ the time-varying price in such a closed-loop supply chain.

The Static Wholesale Price and Retail Price Model
When considering the uniform wholesale and retail price, the stakeholders of the supply chain, the manufacturer and the retailer, will maximize their own profit function (14). Now, the following proposition characterizes the optimum strategies on price, quality, and reference quality in the presence of pricing uniformly in such a closed-loop supply chain system. Proposition 3. By pricing statically, the equilibrium strategies on wholesale price, retail price, quality, and reference quality for the manufacturer and the retailer are revealed below: And the corresponding reference quality is: where C 1 denotes in Appendix A3.

The Static Wholesale Price and Dynamic Retail Price Model
This model considers a uniform wholesale price and a dynamic retail price, the stakeholders of the supply chain, the manufacturer and the retailer, will maximize their own profit function (14). Now, the following proposition characterizes the optimum strategies on price, quality, and reference quality in the presence of pricing dynamically in such a closed-loop supply chain system. Proposition 4. Given a steady-state wholesale price and time-varying retail price, the equilibrium strategies on wholesale price, retail price, quality and reference quality for the manufacturer and the retailer are revealed below: Corollary 1. The optimum wholesale price, quality, reference quality and the benefits of the manufacturer in the "ss" model are equivalent to those in the "sd" model, that is, q ss (t) = q sd (t) (25) R ss (t) = R sd (t) Based on Corollary 1, the optimal wholesale price, quality, reference quality, and the benefits of the manufacturer will not be affected by the pricing strategy of the retailer, dynamically or statically, if the manufacturer applies a steady-state wholesale price.

The Dynamic Wholesale Price and Retail Price Model
When considering the dynamic wholesale and retail price, the stakeholders of the supply chain, the manufacturer and the retailer, will maximize their own profit function (14). Now, the following proposition characterizes the optimum strategies on price, quality, and reference quality in the presence of pricing dynamically in such a closed-loop supply chain system. Proposition 5. By pricing dynamically, the equilibrium strategies on wholesale price, retail price, quality, and reference quality for the manufacturer and the retailer are disclosed below: w dd = 2k 1 (α+β(c m +2bτ−∆τ))+(γ 1 +γ 2 ) 2 (∆τ−bτ−c m ) where N 2 , A 2 , B 2 , h i , r i , i = 3, 4 are characterized in Appendix A6.
Proposition 6. According to Propositions 1-5, the demand of the product in which the system adopts the static and dynamic pricing strategies in centralized and decentralized case are presented as follows respectively:

Numerical Examples
In this section, we firstly investigate the flexibility analysis of several of the significance parameters to find out how they affect the equilibrium strategies and then derive numerical examples to examine the feasibility and effectiveness of our work. Finally, we explore the optimal recycling policy for the CLSC.
We evaluate the flexibility analysis of several of the importance parameters, for instance, β, θ, γ 1 and γ 2 to study their influences on revenues for supply chain and its members in each case, which are shown in Table 3. When comparing time-varying and steady-state pricing strategies, an apparent result suggests that the benefits of a centralized closed-loop supply chain are invariably much better than those of a decentralized one. The supply chain achieves more profitable revenue with dynamic pricing than with pricing statically both in the centralized and decentralized form, that is, J cd > J cs > J dd > J sd > J ss . It means that the centralized form and the dynamic pricing benefit the whole supply chain. In the centralized form, the retailer and the manufacturer obtain more profitable revenue in dynamic pricing than in uniform pricing, namely, J dd M > J ss M = J sd M , J dd R > J sd R > J ss R . From Table 2, we can find that the revenues of the whole supply chain, the manufacturer and the retailer constantly grow with γ 1 and θ increasing with whatever pricing strategy is adopted. Nevertheless, the higher β and γ 2 create less benefits for the stakeholders of the closed-loop supply chain. It signifies that the whole closed-loop supply chain and its members will obtain the most benefits if quality effect γ 1 and the impact of continuous memory coefficient of reference quality θ are higher, while the marginal contribution of the difference between quality and reference quality γ 2 and the price elasticity coefficient β are lower.

The Numerical Example on Equilibrium Strategies
Next, due to Propositions 1 and 2, the equilibrium strategies about quality, reference quality, and price for the whole supply chain are acquired in the centralized model. By applying Propositions 3-5, we derive the optimum policies on quality, reference quality, and wholesale and retail price for the manufacturer and the retailer in the decentralized case, respectively, which are denoted in Figures 2-4 below.
An obvious result shown in Figure 2 is that regardless of case, the quality increases in the entire planning horizon. It indicates that the manufacturer would continuously enlarge the quality input regardless of what pricing policy the manufacturer and the retailer will adopt. The product quality in which the manufacturer takes a time-varying price is a little bit higher when compared with that in which the manufacturer adopts a steady-state price in both the centralized and decentralized scenarios. The manufacturer will put much more investment in product quality in the centralized case than in the decentralized scenario, where the manufacturer and the retailer are pricing dynamically and in the case where they are pricing uniformly. An obvious result shown in Figure 2 is that regardless of case, the quality increases in the entire planning horizon. It indicates that the manufacturer would continuously enlarge the quality input regardless of what pricing policy the manufacturer and the retailer will adopt. The product quality in which the manufacturer takes a time-varying price is a little bit higher when compared with that in which the manufacturer adopts a steady-state price in both the centralized and decentralized scenarios. The manufacturer will put much more investment in product quality in the centralized case than in the decentralized scenario, where the manufacturer and the retailer are pricing dynamically and in the case where they are pricing uniformly.
It visibly signifies that time-varying pricing and the centralized decision form are propitious to product quality. Dynamically pricing strategies require a high level of input to continuously improve product quality. However, the improvement of product quality is not accomplished overnight, so that enterprises preparing for adopting dynamic pricing strategies should do some basic research and development work regarding product quality in advance.
Clearly, the Figure 3 shows that the retail price and the wholesale price always grow throughout the cycle in dynamic pricing cases. In the centralized case, the retail price in which the supply chain adopts dynamic pricing strategies outclasses that with the supply chain pricing statically. The peak retail and wholesale price belongs to the case where the supply chain is pricing statically in the first semi-period, and in which both the members of the supply chain adopt a time-varying price in the last part in decentralized case. We can infer that dynamic prices exceed static ones and these differences are increasingly large within a longer planning horizon, which firmly means dynamic pricing is more suitable for a long-term cooperative supply chain.
When comparing quality and price strategies presented in Figures 2 and 3, quality level and retail price continue to grow in a period when adopting dynamic pricing strategy both in the decentralized and centralized cases. It suggests that dynamic pricing strategies, that is, a policy with gradually increased high-quality product and higher and higher price, will lead to an upscale brand famous for top-level quality and peak price throughout its life-cycle among like products. Dynamic pricing strategies would be more applicable for those companies identified as "high-end firms". As it obviously depicted in Figure 4, the reference quality where the system adopts a uniform price is slightly smaller than that with the system pricing dynamically. The reference quality presents an overall tendency to decline as time goes by in all distinct pricing strategies in the entire planning horizon. The reason can be found in the first beginning, q(t) < R(t), which means ( ) 0 R t • < and derivative of R(t) less than zero, that is, the reference quality decreased. An interesting inference can be obtained that R(t) will increase in the long term for q(t) growing and R(t) decreasing, once q(t) > R(t) that means ( ) 0 R t • > and derivative of R(t) more than zero, which lead the reference  Since the product quality is less than the reference quality and the gap decreases as time goes on, the effect the gap has on current demand increases and the total demand of the product constantly grows with a static pricing strategy, according to Equation (2). In dynamic pricing, the product quality and the difference between the product quality and the reference quality have a positive effect on current demand, whereas the gradually increasing price has a negative effect on demand. The relationship between the general demand and time is unclear, however, we can infer that total demand is declining in a long enough planning horizon due to a substantially increased price from Figures 3 and 4.
The above strategies about retail price, wholesale price, and product quality are the best solutions where the retailer takes the recycling policy. Note that the recycling cost is fixed with the rate of recycling and this fraction does not work on the equilibrium strategies mentioned before; we can obtain the optimal strategies where the manufacturer takes recycling policy represented in Corollary 2.

Corollary 2.
Let b = 0, the Propositions 1-5, are the optimal pricing strategies in which the manufacturer takes recycling policy.
According to Corollary 2, retail price, wholesale price, and product quality strategies in corresponding scenarios with the manufacturer taking the recycling policy can be regarded as a special case of that with the retailer adopting the recycling policy. For the manufacturer, it will be responsible for the recycling activities where the recycling cost is lower than expense spending on the transaction cost to the retailer, that is, k2τ 2 /2 < bτD(t).

Analysis of the Optimal Recycling Fraction
Especially, we discuss the influence the recovery fraction τ has on the whole supply chain and its members. It should be considered that the profits for all of the participants increase gradually with τ increases. Actually, this summary only works with specific parameter k2. A higher τ means saving more manufacturing costs, but leading to additional recovery costs simultaneously. The retailer benefits more with a high level of recovery where the rate of increasing sales is larger than that of the growing recovery cost. Consequently, the impact the recovery fraction has on the profit is mainly dependent on the recovery cost coefficient k2.
The relation between τ and profits for members of the supply chain vary considerably in precise value of k2. Applying Proposition 3, taking τ ss as an example, we drive the profit of the It visibly signifies that time-varying pricing and the centralized decision form are propitious to product quality. Dynamically pricing strategies require a high level of input to continuously improve product quality. However, the improvement of product quality is not accomplished overnight, so that enterprises preparing for adopting dynamic pricing strategies should do some basic research and development work regarding product quality in advance.
Clearly, the Figure 3 shows that the retail price and the wholesale price always grow throughout the cycle in dynamic pricing cases. In the centralized case, the retail price in which the supply chain adopts dynamic pricing strategies outclasses that with the supply chain pricing statically. The peak retail and wholesale price belongs to the case where the supply chain is pricing statically in the first semi-period, and in which both the members of the supply chain adopt a time-varying price in the last part in decentralized case. We can infer that dynamic prices exceed static ones and these differences are increasingly large within a longer planning horizon, which firmly means dynamic pricing is more suitable for a long-term cooperative supply chain.
When comparing quality and price strategies presented in Figures 2 and 3, quality level and retail price continue to grow in a period when adopting dynamic pricing strategy both in the decentralized and centralized cases. It suggests that dynamic pricing strategies, that is, a policy with gradually increased high-quality product and higher and higher price, will lead to an upscale brand famous for top-level quality and peak price throughout its life-cycle among like products. Dynamic pricing strategies would be more applicable for those companies identified as "high-end firms".
As it obviously depicted in Figure 4, the reference quality where the system adopts a uniform price is slightly smaller than that with the system pricing dynamically. The reference quality presents an overall tendency to decline as time goes by in all distinct pricing strategies in the entire planning horizon. The reason can be found in the first beginning, q(t) < R(t), which means • R(t) < 0 and derivative of R(t) less than zero, that is, the reference quality decreased. An interesting inference can be obtained that R(t) will increase in the long term for q(t) growing and R(t) decreasing, once q(t) > R(t) that means • R(t) > 0 and derivative of R(t) more than zero, which lead the reference quality to a constant growth.
It suggests that the consumers' reference quality is in accord with product quality. Due to reference quality effects, though the manufacturer has been improving the product quality, the product quality consumers really apperceived decreased firstly and then increased in the long term. In other words, once the manufacturer decides to adopt dynamic pricing strategies and produce a continuously high-quality product, the reference quality perceived by customers decreases first and eventually grows with the manufacturer persistently improving product quality, which results in a virtuous cycle where the quality and the reference quality promote and rise mutually.
Since the product quality is less than the reference quality and the gap decreases as time goes on, the effect the gap has on current demand increases and the total demand of the product constantly grows with a static pricing strategy, according to Equation (2). In dynamic pricing, the product quality and the difference between the product quality and the reference quality have a positive effect on current demand, whereas the gradually increasing price has a negative effect on demand. The relationship between the general demand and time is unclear, however, we can infer that total demand is declining in a long enough planning horizon due to a substantially increased price from Figures 3 and 4.
The above strategies about retail price, wholesale price, and product quality are the best solutions where the retailer takes the recycling policy. Note that the recycling cost is fixed with the rate of recycling and this fraction does not work on the equilibrium strategies mentioned before; we can obtain the optimal strategies where the manufacturer takes recycling policy represented in Corollary 2.

Corollary 2.
Let b = 0, the Propositions 1-5, are the optimal pricing strategies in which the manufacturer takes recycling policy.
According to Corollary 2, retail price, wholesale price, and product quality strategies in corresponding scenarios with the manufacturer taking the recycling policy can be regarded as a special case of that with the retailer adopting the recycling policy. For the manufacturer, it will be responsible for the recycling activities where the recycling cost is lower than expense spending on the transaction cost to the retailer, that is, k 2 τ 2 /2 < bτD(t).

Analysis of the Optimal Recycling Fraction
Especially, we discuss the influence the recovery fraction τ has on the whole supply chain and its members. It should be considered that the profits for all of the participants increase gradually with τ increases. Actually, this summary only works with specific parameter k 2 . A higher τ means saving more manufacturing costs, but leading to additional recovery costs simultaneously. The retailer benefits more with a high level of recovery where the rate of increasing sales is larger than that of the growing recovery cost. Consequently, the impact the recovery fraction has on the profit is mainly dependent on the recovery cost coefficient k 2 .
The relation between τ and profits for members of the supply chain vary considerably in precise value of k 2 . Applying Proposition 3, taking τ ss as an example, we drive the profit of the retailer with the recovery cost coefficient k 2 and the recycling fraction coefficient τ gradually increasing at the same time. Figure 5 depicts that the relationship between the profit for the retailer and the recycling ratio τ is not regular, namely, the retailer will get a continuously increasing profit with τ gradually increasing, in which k 2 is less than a specific range, and an increased then declining revenue with τ increasingly growing where k 2 is beyond that range. In dynamic pricing strategies, the recycling fraction should be determined by recovery cost, the retailer should take a higher recycling rate when the recovery cost is low, while adopting a lower recycling ratio when the recovery cost is high. The optimal recycling fraction is specific to a given recovery cost from Figure 5; to gain the concrete relationship between recycling fraction and the recycling cost, we get Corollary 3.
increasing, in which k2 is less than a specific range, and an increased then declining revenue with τ increasingly growing where k2 is beyond that range. In dynamic pricing strategies, the recycling fraction should be determined by recovery cost, the retailer should take a higher recycling rate when the recovery cost is low, while adopting a lower recycling ratio when the recovery cost is high. The optimal recycling fraction is specific to a given recovery cost from Figure 5; to gain the concrete relationship between recycling fraction and the recycling cost, we get Corollary 3.

Corollary 3.
No matter in which case, as a rational retailer, they would take the return contract only if that will make them more benefit than not accepting the recycling policy. Let ∆J denote the additional profit for the retailer generated by taking the recovery policy; the necessary condition for the retailer to adopt the recycling contract is ∆J > 0. Taking "ss" state as an example, hence, the most appropriate recycling fraction must meet the following statement: where H denotes in the Appendix A.8.
As a rational retailer, an appropriate return policy can be based on a given recycling cost coefficient where the closed-loop supply chain adopts dynamic pricing and quality strategies. The best recycling fraction and the recovery cost coefficient are inversely proportional from Corollary 3. To be economically profitable, the retailer will reject the recycling contract no matter which recovery fraction if the recycling cost is high enough, while the retailer should collect all of the sold products with a much lower recycling cost. It is impossible for the retailer to recycle one hundred percent of

Corollary 3.
No matter in which case, as a rational retailer, they would take the return contract only if that will make them more benefit than not accepting the recycling policy. Let ∆J denote the additional profit for the retailer generated by taking the recovery policy; the necessary condition for the retailer to adopt the recycling contract is ∆J > 0. Taking "ss" state as an example, hence, the most appropriate recycling fraction must meet the following statement: where H denotes in the Appendix A8.
As a rational retailer, an appropriate return policy can be based on a given recycling cost coefficient where the closed-loop supply chain adopts dynamic pricing and quality strategies. The best recycling fraction and the recovery cost coefficient are inversely proportional from Corollary 3. To be economically profitable, the retailer will reject the recycling contract no matter which recovery fraction if the recycling cost is high enough, while the retailer should collect all of the sold products with a much lower recycling cost. It is impossible for the retailer to recycle one hundred percent of the used items. To reach a higher fraction of recycling and a lower recovery cost coefficient, the simplicity and low cost of recycling is essential for the retailer in practice.
However, the profit of the manufacturer continuously increases with τ growth. To get higher revenues, the manufacturer wants the retailer to take a higher rate of recovering used goods which indicates a lower recycling cost coefficient is urgently needed for the retailer and the manufacturer. The manufacturer should make contributions, such as helping the retailer to set up and enlarge the recovery channel or advocate more in recycling among consumers, to make the retailer get a smaller k 2 , so that the retailer will invest more in returning the used goods, which would lead to a win-win situation for both of them.

Conclusions
In this paper, we employ a Stackelberg game between a manufacturer and a retailer in a closed-loop supply chain. The manufacturer is a leader who settles the quality level and the wholesale price while the retailer is a follower who determines the retail price in the Stackelberg game. The main purpose of our work is to investigate different pricing strategies in the context of dynamic quality characterized by reference quality effects for such a closed-loop supply chain. Particularly, we determined the optimal strategies in several cases: one is where the whole supply chain adopts static and dynamic pricing strategies in a centralized scenario; the other one is either, neither, or both the manufacturer and the retailer take time-varying pricing strategies, respectively, in a decentralized case. In addition, the optimal recycling fractions for such a closed-loop supply chain in each scenario are also discussed throughout this paper.
Furthermore, the primary results of our work stated below are available by investigating numerical examples and flexibility analysis: (1) The profits where the supply chain take pricing decisions in a centralized case is better than profits in a decentralized scenario no matter whether pricing dynamically or statically. (2) The case in which the whole supply chain takes a time-varying price is better than the case in which either or neither of the manufacturer and the retailer price dynamically regardless of being in a centralized or decentralized model. (3) The reference quality move towards the product quality and dynamic pricing strategies are suitable more for the long term, according to comprehensive effects between price and demand. (4) The profits of the whole supply chain, the manufacturer, and the retailer always grow with γ 1 and θ increasing, while they decrease with β and γ 2 growing. (5) The best recycling fraction is obtained for the whole supply chain in all distinct cases. Some practical implications for industry according to the results are as follows. Firstly, the effects of different price-making models on the manufacturer's profit are consistent with that on the retailer's. The centralized model will make the closed-loop supply chain and its members achieve more revenues than a dynamic pricing strategy. It signifies that a dynamic pricing strategy with reference quality effects works more on a cooperative closed-loop supply chain.
Secondly, the manufacturer should continuously improve product quality no matter which pricing strategy the retailer and the manufacturer would make with reference quality effects. As for pricing statically, including the static wholesale price and the static retail price, the closed-loop system and its members could achieve less revenue than that with dynamic pricing; the gap of profit would enlarge over a longer term. It is far more likely that the manufacturer and the retailer will take the dynamic pricing strategies from a far-sighted perspective.
Dynamic quality and dynamic price are then promoted mutually, namely, dynamic pricing with reference quality effects is a policy with gradually increasing high-quality product with a higher price, will lead to an upscale brand throughout its life-cycle among its counterparts. Meanwhile, a dynamic pricing strategy with reference quality requires a high level of continuously improving product quality in the long term, which makes it more suitable for qualified companies to promote product quality increasingly or innovation-oriented firms.
Fourthly, these coefficients play an important role in the profits of the closed-loop system even in determining whether the manufacturer and the retailer choose to adopt the dynamic pricing strategy. Specifically, when customers are more sensitive to the product quality, when the memory coefficient of reference quality is higher, or when customers are less sensitive to the quality gap, when price elasticity coefficient is lower, the manufacturer and the retailer would be more willing to adopt these kinds of pricing strategies with reference quality. Therefore, it suggests that realizing some important parameters for the manufacturer and the retailer before producing products is useful in deciding the optimal pricing strategy.
Finally, the best recycling fraction is not affected by the dynamic decision-making model and proved to be uniform for the retailer in all of the distinct cases throughout the cycle. Since a higher recovery ratio is much more beneficial to the manufacturer, the manufacturer could make some efforts or pay added profit to the retailer to stimulate getting more used products returned.
Though this study has identified some important managerial insights, there are several directions worth investigating in further research. Initially, it would be meaningful to consider a distinct product system, including new product and remanufactured ones with different quality levels in which the whole supply chain and its members take a steady-state and time-varying pricing strategies, respectively. Then, the available extension of our study would be to investigate what impact the recovery policy has on the whole supply chain and its members in which the recycling fraction is a state variable. Lastly, some comprehensive contracts can be adopted, such as revenue sharing contracts and quantity discount contracts, to the closed-loop supply chain when making dynamic decisions.
Substitute Equations (A5) and (A6) into Equation (A7), and the optimum steady-state retail price can be obtained by optimizing Equation (A7), which is shown as follow Substitute Equation (A8) into Equations (A5) and (A6) respectively, and the best strategy on corresponding quality and the reference quality are obtained as follows The proof is completed.

Appendix A.2. Proof of Proposition 2
According to the maximum principle, we can gain the Hamitonian function of the centralized scenario as follow.
where λ cd denotes the adjoint variable which is correlated with R, on the basis of the optimization condition, ∂H cd ∂q = 0, ∂H cd ∂p = 0 and adjoint equation • λ cd = − ∂H cd ∂R , we can obtain Next, we can acquire  (2), respectively They can be rewritten as  The eigenvalues of A are given by Next, we get the corresponding matrix associated with eigenvectors of A P = Hence, we have According to the boundary conditions The analytical solutions of R cd (t), λ cd (t) can be obtained as Substitute Equations (A25) and (A26) into Equations (A12) and (A13) respectively, the optimal strategy on corresponding quality and the retail price are obtained as The proof is completed.