# Pricing and Warranty Decisions of Substitutable Products for a Fuzzy Two-Echelon Supply Chain

Authors: Yongzhao Wang
Journal: Discrete Dynamics in Nature and Society
 (2017)

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## Abstract

This paper considers the optimal decisions of pricing and warranty level for substitutable products in a fuzzy supply chain environment, where the consumer demands, manufacturing costs, and warranty costs are characterized as fuzzy variables. Expected value manufacturer-leader Stackelberg models and two chance-constrained programming models, that is, theλ-optimistic and λ-pessimistic manufacturer-leader Stackelberg models, are established. The corresponding closed-form solutions are obtained by using the game-theoretical approach and fuzzy theory. Finally, numerical examples are presented to illustrate the effectiveness of the models and to provide managerial insights for decision makers. The results show that it would be beneficial for the whole supply chain when the warranty services are introduced into the market, and the firm which provides the warranty service has the advantage in holding more dominant position.


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## Body

## 1. Introduction

Price is the most important and direct criterion that consumers use to evaluate a product at the market. With the technological advancements accelerated, which have shortened product life cycles and intensified business competition. More and more firms have improved their market competitiveness by improving services and/or product quality, so as to build brand loyalty to avoid traditional competition which focuses solely on price. For instance, IBM and Dell have a good reputation with dependence on their customer services and supports [1]. Therefore, warranty has become an effective measure for encouraging market demand by reducing risks for consumers and has also associated with the firms’ corporate social responsibility which is a key characteristic of corporate sustainability and business sustainability (Kimble and Lesher [2], Van Marrewijk [3], Ahi and Searcy [4], and Jeyakumar et al. [5]). A warranty represents the commitment for producers to restore the functionalities of faulty product within a certain period after it is sold. The majority of consumers prefer to purchase a product from a manufacturer with a warranty ensuring the replacement or repairing of the product during the warranty period, with little or no charge [6], which can also ensure an increasing of sales leading to more profits for the supply-chain members.Warranty has received extensive attention in literature. Chung and Wee [7] considered the pricing policy, imperfect production, inspection planning, warranty period, and the stock-level-dependent demand with the Weibull deterioration, partial backorder, and inflation and investigated an integrated production-inventory-deteriorating model. Wu et al. [8] developed a decision model for producers in the static demand market to determine the optimal price, warranty length, and production rate of a product to maximize profit based on the predetermined life cycle. Chen et al. [9] discussed the issues associated with a manufacturer’s pricing strategies in a supply chain that comprises one manufacturer and two competing retailers, with warranty period-dependent demands. Wei et al. [1] explored the optimal pricing and warranty period decisions of two complementary products in a supply chain from a two-stage game theoretic perspective, where there are two manufacturers and one common retailer, such as Lin and Shue [10], Wu et al. [11], Tsao et al. [12], and Huang et al. [13].Recently, several powerful retail chains have emerged around the world (Walmart, Carrefour, etc.). These powerful retailers have a dominant position relative to most manufacturers on the market, and multiple substitutable products of an item sourced from two different manufacturers will appear on Walmart’s shelves at the same time. For instance, Walmart retails substitute brands of detergents like Unilever brands (Surf, Wisk) and P&G brands (Tide, Gain, and Cheer) [14]. The influence of these powerful retailers continues to expand on the market, and their perspective of channel problems attracts more and more attention of scholars [15, 16].In order to make effective supply chain strategies, we cannot ignore the uncertainties that happen in the real world. And those uncertainties are usually associated with the product supply, the customer demand, the manufacturing cost, and so on [17–19]. More recently, predecessors provided a new foundation for optimization problems in the fuzzy environment, in which various models such as the expected value and the chance-constrained programming were proposed to deal with optimization problems (Dubois and Prade [20]; Liu [21, 22]). In recent supply chain studies, Zhou et al. [17] established the expected value models as well as chance-constrained programming models to determine the pricing strategies for the retailer and the manufacturer. Song et al. [23] discussed the optimal decisions of pricing and selling effort for a two-echelon supply chain with uncertain consumer demands, manufacturing costs, and selling costs. Zhao and Wang [24] studied the pricing and retail service decisions of three different game structure models in a supply chain which was assumed in a fuzzy uncertainty environment. Zhao and Wei [25], Chakraborty et al. [26], and so on. Table 1 illustrates some recent papers on pricing strategies within supply chain in the fuzzy environment as follows [14, 17, 23, 24, 27–31].Table 1
Pricing strategies in the fuzzy environment.
Paper Decisions Product status Supply chain structure Gu and Ji [32] Pricing policy; Single One manufacturer, one retailer recycling Zhou et al. [17] Pricing policy Single One manufacturer, one retailer Guo and Liu [27] Pricing decision; Single One manufacturer, one retailer collaboration Wei and Zhao [28] Pricing policy; Single One manufacturer, two retailers remanufacturing rate Zhao et al. [29] Pricing policy Substitutable One manufacturer, two retailers Zhao et al. [14] Pricing policy; Substitutable Two manufacturers, one retailer service level Wei et al. [30] Pricing policy Single One manufacturer, duopolistic retailers Song et al. [23] Pricing policy; Single One manufacturer, one retailer selling effort Wang et al. [31] Pricing policy Complementary Two manufacturers Zhao and Wang [24] Pricing policy; Single One manufacturer, two retailers service levelDiffering from those of prior studies, by considering a fuzzy supply chain where two manufacturers produce two substitutable products and sell them to a retailer who sells them to the end consumers, this paper studies the optimal wholesale prices, warranty levels, and retail prices’ decisions. The main contributions of this paper are as follows: first, we establish one expected value MS model and two chance-constrained programming models (i.e.,λ-optimistic and λ-pessimistic). Second, by using the game-theoretical approach and fuzzy theory, the optimal wholesale prices, warranty levels, and retail prices are obtained in the three decision models. Third, through numerical studies, we get some managerial insights: (1) the firm’s optimal pricing and warranty strategy and the corresponding profits vary with the ranking measure. The strategy in the λ-optimistic Stackelberg model is always the highest, and the strategy in the λ-pessimistic Stackelberg model is always the lowest. (2) It would be beneficial for the whole supply chain when the warranty services are introduced into the market. The firm which provides the warranty service has the advantage in holding more dominant position and so on.The rest of the paper is organized as follows. The problem description and notations are presented in Section2. In Section 3, we formulate the expected value model and two chance-constrained programming MS Stackelberg models with fuzzy consumer’s demands, fuzzy manufacturer costs, and fuzzy warranty costs. Section 4 gives some numerical examples to illustrate the effectiveness of the models. A conclusion is presented in Section 5.


## 2. Problem Description

In our symmetric-information two-echelon supply chain structure in a fuzzy environment, there are two competitive manufactures (indexed bym1, m2) and one common retailer. The manufacturer mi produces a substitutable product with manufacturing cost ci and wholesales it with wholesale price wi to the common retailer, who in turn sells it to the consumer with retail price pi (i=1,2), respectively. The two manufacturers provide warranty t1 and t2 to the retailer.We assume all activity occurs within a single period and ignore the logistic cost components of each firm for analytical convenience. Further, the two manufacturers hold more market power than the common retailer in the market, and the manufacturers act as the Stackelberg leader. Under such a scenario, the two manufacturers and common retailer make their pricing and warranty decisions in order to achieve maximal expected profits.The consumer’s demand is sensitive to the retail prices and the manufacturer warranty levels. Similar to Wang and Zhao [33] and Wei et al. [1], we define the consumer’s demand function as a linear form in this paper, which is decreasing in its own price and increasing in the opponent’s price, moreover, increasing in its own warranty level and decreasing in the opponent’s warranty level.(1)Dip1,p2,t1,t2=αi-β1pi+β2pj+γ1ti-γ2tj,i=1,2,j=3-i.The parameter αi represents the primary market base of the product i. The parameters βi and γi are price and warranty elastic coefficients, which measure the effectiveness of the retail prices and manufacturer’s warranties in stimulating or restraining product’s consumer demand, respectively. Moreover, we characterize the total cost of providing the warranty by g(t), which is a strictly convex function and has been property of ∂g(t)/∂t>0 and ∂2g(t)/∂t2>0. Consistent with the extant literature (Xiao and Yang [34], etc.), we assume that the total cost of providing the warranty is gi(t)=liti2, where li denotes the warranty cost coefficient of manufacturer mi, i=1,2.Since the decision makers have little or no historical data on consumer’s demand, it is more appropriate to estimate the manufacturing cost and warranty cost coefficient based on decision makes judgements, intuitions, and experience. In this paper, we assume the parametersci, αi, βi, γi, and li are mutually independent nonnegative fuzzy parameters. The parameters βi and γi satisfy E[β1]>E[β2] and E[γ1]>E[γ2], which signify that the expected demand should be more sensitive to the changes in its own retail price and warranty than to the changes in the competitor’s. The unit marginal profit pi-wi and demand Di(p1,p2,t1,t2) are nonnegative in the real world; thus Pos{pi-ci<0}=0 and Pos{Di(p1,p2,t1,t2)<0}=0  (i=1,2).According to the problem descriptions and assumptions, the manufacturermi and the retailer’s profit functions can be expressed as follows:(2)Πmiwi,ti=wi-ciDip1,p2,t1,t2-liti2,i=1,2,(3)Πrp1,p2=∑i=12pi-wiDip1,p2,t1,t2.We impose the following conditions on the parameters to ensure that the profit expressions will be well behaved and possess a unique optimum:8β1λLliλL-(γ1λU)2>0 and (β1λL)2-(β2λU)2>0.


## 3. Fuzzy Models and Main Results

In this section, we consider a scenario that the sizes of the two manufacturers are larger compared to the common retailer’s in the market. The two manufacturers act as the Stackelberg leaders and the common retailer acts as the Stackelberg follower. Using the game-theoretical approach to analyze the model, we can get the timing of the game as follows: the manufacturers first choose the wholesale prices and the warranty levels of the product, simultaneously; after observing the wholesale prices and the warranty levels, the retailer then determines his retail prices of two products.


### 3.1. Theλ-Optimistic Manufacturer-Leader Stackelberg (MSU) Model

With a given condition levelλ∈(0,1], we formulate the maximum chance-constrained programming model for the manufacturer-leader supply chain as follows:(4)maxw1,t1Π¯m1w1maxw2,t2Π¯m2w2s.t.Poswi-ciαi-β1pi∗wi,ti,tj+β2pj∗wj,tj,ti+γ1ti-γ2tj≥Π¯mi≥λ,Poswi-ci<0=0,i=1,2,j=3-i.p1∗,p2∗arederivedfromsolvingtheproblemmaxp1,p2Π¯rp1,p2s.t.Pospi-wiαi-β1pi+β2pj+γ1ti-γ2tj≥Π¯r≥λ,Posαi-β1pi+β2pj+γ1ti-γ2tj<0=0,pi≥wi,i=1,2,j=3-i, where Π¯mi and Π¯r are the threshold values of the manufacturers and the retailer; namely, Π¯mi and Π¯r should be the maximum values that supply chain members’ profit functions achieve with at least possibility λ, which is a predetermined confidence level for the profit of the manufacturer and the retailer in model (4).Similar to [17], it is clear that model (4) can be transformed into model (5) in which the manufacturer and the retailer try to maximize their optimal -optimistic profits [Π¯m1]λU,  [Π¯m2]λU, and [Π¯r(p1,p2)]λU. The other conditions satisfy that the marginal profit and consumer’s demand are nonnegative. Model (4) can be transformed into the following, which is named optimistic manufacturer-leader Stackelberg (MSU) model.(5)maxw1,t1Π¯m1λU=w1-c1D1-l1t12λUmaxw2,t2Π¯m2λU=w2-c2D2-l2t22λUs.t.Poswi-ci<0=0,i=1,2,j=3-i.p1∗,p2∗arederivedfromsolvingtheproblemmaxp1,p2Π¯rp1,p2λU=∑i=12pi-wiDiλUs.t.Posαi-β1pi+β2pj+γ1ti-γ2tj<0=0,pi≥wi,i=1,2,j=3-i.Proposition 1.
In the MSU model, given the wholesale prices and warranty levels made earlier by manufacturers, the retailer’s optimal response functions, denoted byp1∗(w1,t1,t2) and p2∗(w2,t1,t2), are(6)p1∗w1,t1,t2=12w1+12β1λL2-β2λU2β1λLγ1λU-β2λUγ2λLt1+β2λUγ1λU-β1λLγ2λLt2+β1λLα1λU+β2λUα2λU,p2∗w2,t1,t2=12w2+12β1λL2-β2λU2β2λUγ1λU-β1λLγ2λLt1+β1λLγ1λU-β2λUγ2λLt2+β1λLα2λU+β2λUα1λUunder the conditions that Pos{α1-β1p1∗(w1,t1,t2)+β2p2∗(w2,t1,t2)+γ1t1-γ2t2<0}=0 and Pos{α2-β1p2∗(w2,t1,t2)+β2p1∗(w1,t1,t2)+γ1t2-γ2t1<0}=0 hold.Proof.
Theλ-optimistic value of the retailer’s profit is(7)Πrp1,p2λU=p1-w1α1λU-β1λLp1+β2λUp2+γ1λUt1-γ2λLt2+p2-w2α2λU-β1λLp2+β2λUp1+γ1λUt2-γ2λLt1.The following first and second-order derivatives of [Πr(p1,p2)]λU with respect to (p1,p2) can be obtained:(8)∂Πrp1,p2λU∂p1=α1λU-2β1λLp1+2β2λUp2+γ1λUt1-γ2λLt2+β1λLw1-β2λUw2,(9)∂Πrp1,p2λU∂p2=α2λU-2β1λLp2+2β2λUp1+γ1λUt2-γ2λLt1+β1λLw2-β2λUw1,(10)∂2Πrp1,p2λU∂p12=∂2Πrp1,p2λU∂p22=-2β1λL,(11)∂2Πrp1,p2λU∂p1∂p2=∂2Πrp1,p2λU∂p2∂p1=2β2λU. Using the assumptions (β1λL)2-(β2λU)2>0 (since E[β1]>E[β2]), the Hessian matrix is negative definite. Therefore, [Πr(p1,p2)]λU is a concave function. Letting (8) and (9) be equal to zero, we can obtain (6).Knowing the information of the retailer’s reaction, the two manufacturers would use them to maximize theirλ-optimistic profits by choosing the best wholesale prices and warranty levels, respectively.Proposition 2.
In the MSU model, the manufacturers’ optimal wholesale prices (denoted byw1U∗ and w2U∗) and warranty levels (denoted by t1U∗ and t2U∗) are(12)t1U∗=A12A23-A22A13A11A22-A12A21,t2U∗=A21A13-A11A23A11A22-A12A21,w1U∗=4l1λLγ1λUt1∗+c1λL,w2U∗=4l2λLγ1λUt2∗+c2λL,where(13)
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,under the conditions that Pos{c1>w1U∗}=0, Pos{c2>w2U∗}=0, Pos{α1-β1p1∗(w1U∗,t1U∗,t2U∗)+β2p2∗(w2U∗,t1U∗,t2U∗)+γ1t1U∗-γ2t2U∗<0}=0, and Pos{α2-β1p2∗(w2U∗,t1U∗,t2U∗)+β2p1∗(w1U∗,t1U∗,t2U∗)+γ1t2U∗-γ2t1U∗<0}=0 hold.Proof.
Substituting (6) into (3) results in(14)ΠmiλU=wi-ciλL-β1λLwi2+β2λUwj2+γ1λUti2-γ2λLtj2+αiλU2-liλLti2,i=1,2,j=3-i.The first- and second-order partial derivative with respect to (wi,ti) can be shown as(15)∂ΠmiλU∂wi=-β1λLwi+β2λUwj2+γ1λUti2-γ2λLtj2+αiλU2+ciλLβ1λL2,(16)∂ΠmiλU∂ti=γ1λU2wi-ciλL-2liλLti.(17)∂2ΠmiλU∂wi2=-β1λL<0,(18)∂2ΠmiλU∂ti2=-2liλL,(19)∂2ΠmiλU∂wi∂ti=∂2ΠmiλU∂ti∂wi=γ1λU2.Since 8β1λLliλL-(γ1λU)2>0, [Πmi]λU are concave functions. Setting (15) and (16) to be equal to zero and solving them, we can attain (12).Substituting (12) into (6), then the retailer’s optimal retail prices are(20)p⁡1U∗=12w1U∗+12β1λL2-β2λU2β1λLγ1λU-β2λUγ2λLt1U∗+β2λUγ1λU-β1λLγ2λLt2U∗+β1λLα1λU+β2λUα2λU,p2U∗=12w2U∗+12β1λL2-β2λU2β2λUγ1λU-β1λLγ2λLt1U∗+β1λLγ1λU-β2λUγ2λLt2U∗+β1λLα2λU+β2λUα1λU.Therefore, we can get a uniqueλ-optimistic Stackelberg equilibrium wiU∗,  tiU∗, and piU∗  (i=1,2) of model (4).


### 3.2. Theλ-Pessimistic Manufacturer-Leader Stackelberg (MSL) Model

The minimax chance-constrained programming model for the manufacturer-leader Stackelberg supply chain can be formulated as follows, whereλ is a predetermined confidence level for the channel members’ profits.(21)maxw1,t1minΠ¯m1⁡Π¯m1w1maxw2,t2minΠ¯m2⁡Π¯m2w2s.t.Poswi-ciαi-β1pi∗wi,ti,tj+β2pj∗wj,tj,ti+γ1ti-γ2tj≤Π¯mi≥λ,Poswi-ci<0=0,i=1,2,j=3-i.p1∗,p2∗arederivedfromsolvingtheproblemmaxp1,p2minΠ¯r⁡Π¯rp1,p2s.t.Pospi-wiαi-β1pi+β2pj+γ1ti-γ2tj≤Π¯r≥λ,Posαi-β1pi+β2pj+γ1ti-γ2tj<0=0,pi≥wi,i=1,2,j=3-i.When the two manufacturers and retailer’s profits function to achieve with at least possibilityλ, Π¯mi and Π¯r should be the minimum value. The channel members try to maximize their optimal λ-pessimistic profits and model (21) can be transformed into the following.(22)maxw1,t1Π¯m1λL=w1-c1D1-l1t12λLmaxw2,t2Π¯m2λL=w2-c2D2-l2t22λLs.t.Poswi-ci<0=0,i=1,2,j=3-i.p1∗,p2∗arederivedfromsolvingtheproblemmaxp1,p2Π¯rp1,p2λL=∑i=12pi-wiDiλLs.t.Posαi-β1pi+β2pj+γ1ti-γ2tj<0=0,pi≥wi,i=1,2,j=3-i.Similar to the proof of Section3.1, we can easily obtain Proposition 3 as follows.Proposition 3.
In the MSL model, the uniqueλ-pessimistic Stackelberg equilibrium solutions are(23)t1L∗=B12B23-B22B13B11B22-B12B21,t2L∗=B21B13-B11B23B11B22-B12B21,w1L∗=4l1λUγ1λLt1∗+c1λU,w2L∗=4l2λUγ1λLt2∗+c2λU,p1L∗=12w1L∗+12β1λU2-β2λL2β1λUγ1λL-β2λLγ2λUt1L∗+β2λLγ1λL-β1λUγ2λUt2L∗+β1λUα1λL+β2λLα2λL,p2L∗=12w2L∗+12β1λU2-β2λL2β2λLγ1λL-β1λUγ2λUt1L∗+β1λUγ1λL-β2λLγ2λUt2L∗+β1λUα2λL+β2λLα1λL,where(24)B11=γ1λL-8β1λUl1λUγ1λL,B12=4β2λLl2λUγ1λL-γ2λU,B13=α1λL-β1λUc1λU+β2λLc2λU,B21=4β2λLl1λUγ1λL-γ2λU,B22=γ1λL-8β1λUl2λUγ1λL,B23=α2λL-β1λUc2λU+β2λLc1λU,under the conditions that Pos{c1>w1L∗}=0, Pos{c2>w2L∗}=0, Pos{α1-β1p1∗(w1L∗,t1L∗,t2L∗)+β2p2∗(w2L∗,t1L∗,t2L∗)+γ1t1L∗-γ2t2L∗<0}=0, and Pos{α2-β1p2∗(w2L∗,t1L∗,t2L∗)+β2p1∗(w1L∗,t1L∗,t2L∗)+γ1t2L∗-γ2t1L∗<0}=0 hold.


### 3.3. The Expected Value Manufacturer-Leader Stackelberg (MSE) Model

In this subsection, we formulate the following manufacturer-leader Stackelberg expected value model.(25)maxw1,t1EΠm1=Ew1-c1D1-l1t12maxw2,t2EΠm2=Ew2-c2D1-l2t22s.t.Poswi-ci<0=0,i=1,2,j=3-i.p1∗,p2∗arederivedfromsolvingtheproblemmaxp1,p2EΠrp1,p2=E∑i=12pi-wiDis.t.Posαi-β1pi+β2pj+γ1ti-γ2tj<0=0,pi≥wi,i=1,2,j=3-i.Similar to the proof of Section3.1, we can get the following Proposition 4.Proposition 4.
The unique expected Stackelberg equilibrium solutions of the MSE model (25) are(26)t1E∗=F12F23-F22F13F11F22-F12F21,t2E∗=F21F13-F11F23F11F22-F12F21,w1E∗=4El1Eγ1t1E∗+F10,w2E∗=4El2Eγ1t2E∗+F20,p1E∗=12w1E∗+12Eβ12-Eβ22Eβ1Eγ1-Eβ2Eγ2t1E∗+Eβ2Eγ1-Eβ1Eγ2t2E∗+Eβ1Eα1+Eβ2Eα2,p2E∗=12w2+12Eβ12-Eβ22Eβ2Eγ1-Eβ1Eγ2t1E∗+Eβ1Eγ1-Eβ2Eγ2t2E∗+Eβ1Eα2+Eβ2Eα1,
where(27)
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.Remark 5.
If two manufacturers’ warranty services are not considered, the optimal pricing decisions, where the manufacturers and the retailer try to maximize theirλ-optimistic profits, can be chosen as(28)w1U0∗=2β1λLα1λU+β2λUα2λU+2β1λLβ1λLc1λL+β1λLβ2λUc2λL4β1λL2-β2λU2,w2U0∗=2β1λLα2λU+β2λUα1λU+2β1λLβ1λLc2λL+β1λLβ2λUc1λL4β1λL2-β2λU2,p1U0∗=12w1U0∗+β1λUα1λL+β2λLα2λL2β1λU2-2β2λL2,p2U0∗=12w2U0∗+β1λUα2λL+β2λLα1λL2β1λU2-2β2λL2.Remark 6.
Considering only one manufacturer’s warranty level, without loss of generality, we assume thatm1 provides the warranty for the retailer, and the unique λ-optimistic Stackelberg equilibrium solutions of this model are as follows:(29)w1U1∗=2β1λLG12+β2λUG222β1λLG11+β2λUG21,w2U1∗=G11G22-G12G212β1λLG11+β2λUG21,t1U1∗=γ1λUw1U1∗4l1λL-γ1λUc1λL4l1λL,p1U1∗=12w1U1∗+β1λLγ1λU-β2λUγ1λL2β1λU2-2β2λL2t1U1∗+β1λUα1λL+β2λLα2λL2β1λU2-2β2λL2,p2U1∗=12w2U1∗+β2λUγ1λU-β1λLγ1λL2β1λU2-2β2λL2t2U1∗+β1λUα2λL+β2λLα1λL2β1λU2-2β2λL2,where(30)G11=2β1λL-γ1λUγ1λU4l1λL,G21=γ1λUγ1λL4l1λL-2β2λU,G12=α1λU-c1λLγ1λUγ1λU4l1λL+c1λLβ1λL,G22=α2λU+c1λLγ1λUγ1λL4l1λL+c2λLβ1λL.


## 3.1. Theλ-Optimistic Manufacturer-Leader Stackelberg (MSU) Model

With a given condition levelλ∈(0,1], we formulate the maximum chance-constrained programming model for the manufacturer-leader supply chain as follows:(4)maxw1,t1Π¯m1w1maxw2,t2Π¯m2w2s.t.Poswi-ciαi-β1pi∗wi,ti,tj+β2pj∗wj,tj,ti+γ1ti-γ2tj≥Π¯mi≥λ,Poswi-ci<0=0,i=1,2,j=3-i.p1∗,p2∗arederivedfromsolvingtheproblemmaxp1,p2Π¯rp1,p2s.t.Pospi-wiαi-β1pi+β2pj+γ1ti-γ2tj≥Π¯r≥λ,Posαi-β1pi+β2pj+γ1ti-γ2tj<0=0,pi≥wi,i=1,2,j=3-i, where Π¯mi and Π¯r are the threshold values of the manufacturers and the retailer; namely, Π¯mi and Π¯r should be the maximum values that supply chain members’ profit functions achieve with at least possibility λ, which is a predetermined confidence level for the profit of the manufacturer and the retailer in model (4).Similar to [17], it is clear that model (4) can be transformed into model (5) in which the manufacturer and the retailer try to maximize their optimal -optimistic profits [Π¯m1]λU,  [Π¯m2]λU, and [Π¯r(p1,p2)]λU. The other conditions satisfy that the marginal profit and consumer’s demand are nonnegative. Model (4) can be transformed into the following, which is named optimistic manufacturer-leader Stackelberg (MSU) model.(5)maxw1,t1Π¯m1λU=w1-c1D1-l1t12λUmaxw2,t2Π¯m2λU=w2-c2D2-l2t22λUs.t.Poswi-ci<0=0,i=1,2,j=3-i.p1∗,p2∗arederivedfromsolvingtheproblemmaxp1,p2Π¯rp1,p2λU=∑i=12pi-wiDiλUs.t.Posαi-β1pi+β2pj+γ1ti-γ2tj<0=0,pi≥wi,i=1,2,j=3-i.Proposition 1.
In the MSU model, given the wholesale prices and warranty levels made earlier by manufacturers, the retailer’s optimal response functions, denoted byp1∗(w1,t1,t2) and p2∗(w2,t1,t2), are(6)p1∗w1,t1,t2=12w1+12β1λL2-β2λU2β1λLγ1λU-β2λUγ2λLt1+β2λUγ1λU-β1λLγ2λLt2+β1λLα1λU+β2λUα2λU,p2∗w2,t1,t2=12w2+12β1λL2-β2λU2β2λUγ1λU-β1λLγ2λLt1+β1λLγ1λU-β2λUγ2λLt2+β1λLα2λU+β2λUα1λUunder the conditions that Pos{α1-β1p1∗(w1,t1,t2)+β2p2∗(w2,t1,t2)+γ1t1-γ2t2<0}=0 and Pos{α2-β1p2∗(w2,t1,t2)+β2p1∗(w1,t1,t2)+γ1t2-γ2t1<0}=0 hold.Proof.
Theλ-optimistic value of the retailer’s profit is(7)Πrp1,p2λU=p1-w1α1λU-β1λLp1+β2λUp2+γ1λUt1-γ2λLt2+p2-w2α2λU-β1λLp2+β2λUp1+γ1λUt2-γ2λLt1.The following first and second-order derivatives of [Πr(p1,p2)]λU with respect to (p1,p2) can be obtained:(8)∂Πrp1,p2λU∂p1=α1λU-2β1λLp1+2β2λUp2+γ1λUt1-γ2λLt2+β1λLw1-β2λUw2,(9)∂Πrp1,p2λU∂p2=α2λU-2β1λLp2+2β2λUp1+γ1λUt2-γ2λLt1+β1λLw2-β2λUw1,(10)∂2Πrp1,p2λU∂p12=∂2Πrp1,p2λU∂p22=-2β1λL,(11)∂2Πrp1,p2λU∂p1∂p2=∂2Πrp1,p2λU∂p2∂p1=2β2λU. Using the assumptions (β1λL)2-(β2λU)2>0 (since E[β1]>E[β2]), the Hessian matrix is negative definite. Therefore, [Πr(p1,p2)]λU is a concave function. Letting (8) and (9) be equal to zero, we can obtain (6).Knowing the information of the retailer’s reaction, the two manufacturers would use them to maximize theirλ-optimistic profits by choosing the best wholesale prices and warranty levels, respectively.Proposition 2.
In the MSU model, the manufacturers’ optimal wholesale prices (denoted byw1U∗ and w2U∗) and warranty levels (denoted by t1U∗ and t2U∗) are(12)t1U∗=A12A23-A22A13A11A22-A12A21,t2U∗=A21A13-A11A23A11A22-A12A21,w1U∗=4l1λLγ1λUt1∗+c1λL,w2U∗=4l2λLγ1λUt2∗+c2λL,where(13)
A
11
=
γ
1
λ
U
-
8
β
1
λ
L
l
1
λ
L
γ
1
λ
U
,
A
12
=
4
β
2
λ
U
l
2
λ
L
γ
1
λ
U
-
γ
2
λ
L
,
A
13
=
α
1
λ
U
-
β
1
λ
L
c
1
λ
L
+
β
2
λ
U
c
2
λ
L
,
A
21
=
4
β
2
λ
U
l
1
λ
L
γ
1
λ
U
-
γ
2
λ
L
,
A
22
=
γ
1
λ
U
-
8
β
1
λ
L
l
2
λ
L
γ
1
λ
U
,
A
23
=
α
2
λ
U
-
β
1
λ
L
c
2
λ
L
+
β
2
λ
U
c
1
λ
L
,under the conditions that Pos{c1>w1U∗}=0, Pos{c2>w2U∗}=0, Pos{α1-β1p1∗(w1U∗,t1U∗,t2U∗)+β2p2∗(w2U∗,t1U∗,t2U∗)+γ1t1U∗-γ2t2U∗<0}=0, and Pos{α2-β1p2∗(w2U∗,t1U∗,t2U∗)+β2p1∗(w1U∗,t1U∗,t2U∗)+γ1t2U∗-γ2t1U∗<0}=0 hold.Proof.
Substituting (6) into (3) results in(14)ΠmiλU=wi-ciλL-β1λLwi2+β2λUwj2+γ1λUti2-γ2λLtj2+αiλU2-liλLti2,i=1,2,j=3-i.The first- and second-order partial derivative with respect to (wi,ti) can be shown as(15)∂ΠmiλU∂wi=-β1λLwi+β2λUwj2+γ1λUti2-γ2λLtj2+αiλU2+ciλLβ1λL2,(16)∂ΠmiλU∂ti=γ1λU2wi-ciλL-2liλLti.(17)∂2ΠmiλU∂wi2=-β1λL<0,(18)∂2ΠmiλU∂ti2=-2liλL,(19)∂2ΠmiλU∂wi∂ti=∂2ΠmiλU∂ti∂wi=γ1λU2.Since 8β1λLliλL-(γ1λU)2>0, [Πmi]λU are concave functions. Setting (15) and (16) to be equal to zero and solving them, we can attain (12).Substituting (12) into (6), then the retailer’s optimal retail prices are(20)p⁡1U∗=12w1U∗+12β1λL2-β2λU2β1λLγ1λU-β2λUγ2λLt1U∗+β2λUγ1λU-β1λLγ2λLt2U∗+β1λLα1λU+β2λUα2λU,p2U∗=12w2U∗+12β1λL2-β2λU2β2λUγ1λU-β1λLγ2λLt1U∗+β1λLγ1λU-β2λUγ2λLt2U∗+β1λLα2λU+β2λUα1λU.Therefore, we can get a uniqueλ-optimistic Stackelberg equilibrium wiU∗,  tiU∗, and piU∗  (i=1,2) of model (4).


## 3.2. Theλ-Pessimistic Manufacturer-Leader Stackelberg (MSL) Model

The minimax chance-constrained programming model for the manufacturer-leader Stackelberg supply chain can be formulated as follows, whereλ is a predetermined confidence level for the channel members’ profits.(21)maxw1,t1minΠ¯m1⁡Π¯m1w1maxw2,t2minΠ¯m2⁡Π¯m2w2s.t.Poswi-ciαi-β1pi∗wi,ti,tj+β2pj∗wj,tj,ti+γ1ti-γ2tj≤Π¯mi≥λ,Poswi-ci<0=0,i=1,2,j=3-i.p1∗,p2∗arederivedfromsolvingtheproblemmaxp1,p2minΠ¯r⁡Π¯rp1,p2s.t.Pospi-wiαi-β1pi+β2pj+γ1ti-γ2tj≤Π¯r≥λ,Posαi-β1pi+β2pj+γ1ti-γ2tj<0=0,pi≥wi,i=1,2,j=3-i.When the two manufacturers and retailer’s profits function to achieve with at least possibilityλ, Π¯mi and Π¯r should be the minimum value. The channel members try to maximize their optimal λ-pessimistic profits and model (21) can be transformed into the following.(22)maxw1,t1Π¯m1λL=w1-c1D1-l1t12λLmaxw2,t2Π¯m2λL=w2-c2D2-l2t22λLs.t.Poswi-ci<0=0,i=1,2,j=3-i.p1∗,p2∗arederivedfromsolvingtheproblemmaxp1,p2Π¯rp1,p2λL=∑i=12pi-wiDiλLs.t.Posαi-β1pi+β2pj+γ1ti-γ2tj<0=0,pi≥wi,i=1,2,j=3-i.Similar to the proof of Section3.1, we can easily obtain Proposition 3 as follows.Proposition 3.
In the MSL model, the uniqueλ-pessimistic Stackelberg equilibrium solutions are(23)t1L∗=B12B23-B22B13B11B22-B12B21,t2L∗=B21B13-B11B23B11B22-B12B21,w1L∗=4l1λUγ1λLt1∗+c1λU,w2L∗=4l2λUγ1λLt2∗+c2λU,p1L∗=12w1L∗+12β1λU2-β2λL2β1λUγ1λL-β2λLγ2λUt1L∗+β2λLγ1λL-β1λUγ2λUt2L∗+β1λUα1λL+β2λLα2λL,p2L∗=12w2L∗+12β1λU2-β2λL2β2λLγ1λL-β1λUγ2λUt1L∗+β1λUγ1λL-β2λLγ2λUt2L∗+β1λUα2λL+β2λLα1λL,where(24)B11=γ1λL-8β1λUl1λUγ1λL,B12=4β2λLl2λUγ1λL-γ2λU,B13=α1λL-β1λUc1λU+β2λLc2λU,B21=4β2λLl1λUγ1λL-γ2λU,B22=γ1λL-8β1λUl2λUγ1λL,B23=α2λL-β1λUc2λU+β2λLc1λU,under the conditions that Pos{c1>w1L∗}=0, Pos{c2>w2L∗}=0, Pos{α1-β1p1∗(w1L∗,t1L∗,t2L∗)+β2p2∗(w2L∗,t1L∗,t2L∗)+γ1t1L∗-γ2t2L∗<0}=0, and Pos{α2-β1p2∗(w2L∗,t1L∗,t2L∗)+β2p1∗(w1L∗,t1L∗,t2L∗)+γ1t2L∗-γ2t1L∗<0}=0 hold.


## 3.3. The Expected Value Manufacturer-Leader Stackelberg (MSE) Model

In this subsection, we formulate the following manufacturer-leader Stackelberg expected value model.(25)maxw1,t1EΠm1=Ew1-c1D1-l1t12maxw2,t2EΠm2=Ew2-c2D1-l2t22s.t.Poswi-ci<0=0,i=1,2,j=3-i.p1∗,p2∗arederivedfromsolvingtheproblemmaxp1,p2EΠrp1,p2=E∑i=12pi-wiDis.t.Posαi-β1pi+β2pj+γ1ti-γ2tj<0=0,pi≥wi,i=1,2,j=3-i.Similar to the proof of Section3.1, we can get the following Proposition 4.Proposition 4.
The unique expected Stackelberg equilibrium solutions of the MSE model (25) are(26)t1E∗=F12F23-F22F13F11F22-F12F21,t2E∗=F21F13-F11F23F11F22-F12F21,w1E∗=4El1Eγ1t1E∗+F10,w2E∗=4El2Eγ1t2E∗+F20,p1E∗=12w1E∗+12Eβ12-Eβ22Eβ1Eγ1-Eβ2Eγ2t1E∗+Eβ2Eγ1-Eβ1Eγ2t2E∗+Eβ1Eα1+Eβ2Eα2,p2E∗=12w2+12Eβ12-Eβ22Eβ2Eγ1-Eβ1Eγ2t1E∗+Eβ1Eγ1-Eβ2Eγ2t2E∗+Eβ1Eα2+Eβ2Eα1,
where(27)
F
10
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β
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0
1
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+
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0
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4
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22
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1
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13
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1
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2
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1
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20
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2
+
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β
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F
10
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2
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20
+
E
c
2
β
1
.Remark 5.
If two manufacturers’ warranty services are not considered, the optimal pricing decisions, where the manufacturers and the retailer try to maximize theirλ-optimistic profits, can be chosen as(28)w1U0∗=2β1λLα1λU+β2λUα2λU+2β1λLβ1λLc1λL+β1λLβ2λUc2λL4β1λL2-β2λU2,w2U0∗=2β1λLα2λU+β2λUα1λU+2β1λLβ1λLc2λL+β1λLβ2λUc1λL4β1λL2-β2λU2,p1U0∗=12w1U0∗+β1λUα1λL+β2λLα2λL2β1λU2-2β2λL2,p2U0∗=12w2U0∗+β1λUα2λL+β2λLα1λL2β1λU2-2β2λL2.Remark 6.
Considering only one manufacturer’s warranty level, without loss of generality, we assume thatm1 provides the warranty for the retailer, and the unique λ-optimistic Stackelberg equilibrium solutions of this model are as follows:(29)w1U1∗=2β1λLG12+β2λUG222β1λLG11+β2λUG21,w2U1∗=G11G22-G12G212β1λLG11+β2λUG21,t1U1∗=γ1λUw1U1∗4l1λL-γ1λUc1λL4l1λL,p1U1∗=12w1U1∗+β1λLγ1λU-β2λUγ1λL2β1λU2-2β2λL2t1U1∗+β1λUα1λL+β2λLα2λL2β1λU2-2β2λL2,p2U1∗=12w2U1∗+β2λUγ1λU-β1λLγ1λL2β1λU2-2β2λL2t2U1∗+β1λUα2λL+β2λLα1λL2β1λU2-2β2λL2,where(30)G11=2β1λL-γ1λUγ1λU4l1λL,G21=γ1λUγ1λL4l1λL-2β2λU,G12=α1λU-c1λLγ1λUγ1λU4l1λL+c1λLβ1λL,G22=α2λU+c1λLγ1λUγ1λL4l1λL+c2λLβ1λL.


## 4. Numerical Example

The pricing and warranty strategies for different Stackelberg, that is, theλ-optimistic, the λ-pessimistic, and the expected value Stackelberg, of the supply chain have been discussed. All the expressions are in a very complicated form; therefore, we use numerical examples to compare the optimal results in the different decision scenarios and to analyze the effect of some key parameters on the equilibrium solutions. Assume that the relation between linguistic expressions and triangular fuzzy variables for the manufacturing costs, primary market bases, price elastic coefficients, warranty elastic coefficients, and warranty cost coefficients are determined by experts’ experiences as shown in Table 2. On the basis of comparison and analysis, some managerial insights are derived.Table 2
Relation between linguistic expression and triangular fuzzy variable.
Linguistic expression Triangular fuzzy variable c1 Low (about 2) (1, 2, 3) High (about 6) (5, 6, 7) c2 Low (about 6) (5, 6, 7) High (about 10) (8, 10, 12) α1 Small (about 1500) (1300, 1500, 1700) Large (about 2000) (1900, 2000, 2100) α2 Small (about 600) (400, 600, 800) Large (about 1000) (950, 1000, 1050) β1 Sensitive (about 150) (120, 150, 180) Very sensitive (about 200) (180, 200, 220) β2 Sensitive (about 60) (50, 60, 70) Very sensitive (about 100) (90, 100, 110) γ1 Sensitive (about 20) (10, 20, 30) Very sensitive (about 50) (40, 50, 60) γ2 Sensitive (about 10) (5, 10, 15) Very sensitive (about 20) (10, 20, 30) l1 Sensitive (about 15) (12, 15, 18) Very sensitive (about 25) (20, 25, 30) l2 Sensitive (about 10) (8, 10, 12) Very sensitive (about 15) (10, 15, 20)Consider the case that the manufacturing costs are in low levels, the primary market bases are in large levels, and all the elastic coefficients are in very sensitive levels. By using Table2, the expected values of all triangular fuzzy variables are E[c1]=(1+2×2+3)/4=2, E[c2]=(5+2×6+7)/4=6, E[α1]=2000, E[α2]=1000, E[β1]=200, E[β2]=100, E[γ1]=50, E[γ2]=20, E[l1]=25, and E[l2]=15. The λ-optimistic and λ-pessimistic values are, respectively, c1λU=1+λ, c1λL=3-λ, c2λU=5+λ, c2λL=7-λ, α1λU=1900+100λ, α1λL=2100-100λ, α2λU=950+50λ, α2λL=1050-50λ, β1λU=180+20λ, β1λL=220-20λ, β2λU=90+10λ, β2λL=110-10λ, γ1λU=40+10λ, γ1λL=60-10λ, γ2λU=10+10λ, γ2λL=30-10λ, l1λU=20+5λ, l1λL=30-5λ, l2λU=10+5λ, and l2λL=20-5λ. We can also calculate E[c1β1]=406.67, E[c2β1]=1206.7, (1/2)∫01‍(c1λLβ2λU+c1λUβ2λL)dλ=196.67, (1/2)∫01‍(c1λLγ1λU+c1λUγ1λL)dλ=96.67, (1/2)∫01‍(c2λLβ2λU+c2λUβ2λL)dλ=596.67, and (1/2)∫01‍(c2λLγ1λU+c2λUγ1λL)dλ=296.67.Discussion 1 (comparison of the optimal decisions in different models).
From the results obtained in the above scenarios, the optimal wholesale prices, warranty levels, retail prices, and the maximal profits of each channel member can be calculated as shown in Tables3 and 4 with a confidence value λ=0.8.
Tables3 and 4 indicate that two substitutable products have different wholesale prices, different retail prices, and different warranty levels in the MSU, MSE, and MSL models. Therefore, the following insights are obtained.
(1-1) From Table2, we can see that wiU∗>wiE∗>wiL∗, tiU∗>tiE∗>tiL∗, piU∗>piE∗>piL∗, ΠmiU∗>ΠmiE∗>ΠmiL∗, ΠrU∗>ΠrE∗>ΠrL∗, and ΠtU∗>ΠtE∗>ΠtL∗  (i=1,2); that is, the firm’s optimal pricing and warranty strategy and the corresponding profits vary with the ranking measure. The strategy in the λ-optimistic Stackelberg model is always the highest, and the strategy in the λ-pessimistic Stackelberg model is always the lowest.
(1-2) The total profits of the two manufacturers are higher than that of the retailer; that is because the manufacturers play the leader’s role which hold advantage in acquiring more profits in this paper.Table 3
The optimal decisions of prices and warranty levels.
Scenario w 1 ∗ w 2 ∗ t 1 ∗ t 2 ∗ p 1 ∗ p2∗ MSU 8.467 7.755 3.611 1.815 13.735 11.474 MSE 8.244 7.602 3.208 1.478 12.908 10.718 MSL 7.999 7.410 2.676 0.908 12.146 10.008Table 4
The maximal profits in different models.
Scenario Π m 1 Π m 2 Π r Πt MSU 4042.4 328.2 4154.5 8525.1 MSE 3621.0 218.5 3385.3 7224.8 MSL 3243.6 136.2 2773.5 6153.3Discussion 2 (the influence of the manufacturers’ warranty levels on equilibrium solutions).
Considering three different scenarios (a) there is no warranty service to be considered in the supply chain, (b) just manufacturerm1 provides the warranty for the retailer, and (c) two manufacturers’ warranty services are considered in the supply chain, we compare the λ-optimistic Stackelberg equilibrium solutions in this subsection. The corresponding results are shown in Tables 5 and 6.
(2-1) The optimal prices and warranty levels under the condition of considering two manufacturers’ warranty services are larger than the other scenarios. The firms’ maximal profits under the condition of considering two manufacturers’ warranty services are higher than the other scenarios. That is because the warranty provided by the manufacturers will stimulate consumer’s purchase desire and facilitate final purchase behavior. It would be beneficial for the whole supply chain when the warranty services are introduced into the market.
(2-2) If just manufacturerm1 provides the warranty in the market, product 1’s optimal pricing decisions increase obviously than that when there is no warranty. On the contrary, product 1’s optimal wholesale prices and retail prices decrease more clearly than that when there is no warranty. The firm which provides the warranty service has the advantage in holding more dominant position than without the warranty.
(2-3) In comparison of no warranty and one warranty level in Table5, it can be known that the wholesale price of product 1, on account of provision of warranty services, increased by 0.377, which raised the retailer’s purchasing cost. However, the improvement of the retail price ensures the increase in profits. In this case, product 2 had to reduce the wholesale price to ensure the market share because of no warranty service, and the decline in market competitiveness has led to lower retail prices and lower profits. As is shown in Table 6, it is not difficult to find that the market share of product 1 increased from 92.6% in the case of no warranty level to 95.2% in the case of one warranty level, and the profits of retailer and the entire supply chain have been significantly improved. In conclusion, the firm which provides the warranty service has the advantage in holding more dominant position, and it would be beneficial for the whole supply chain when the warranty services are introduced into the market.Table 5
Comparison of the optimal decisions with and without warranty level.
Scenario w 1 ∗ w 2 ∗ t 1 ∗ t 2 ∗ p 1 ∗ p2∗ No warranty level 8.021 7.564 — — 12.916 10.993 One warranty level 8.408 7.226 3.579 — 13.448 10.562 Two warranty levels 8.467 7.755 3.611 1.815 13.735 11.474Table 6
Comparison of the maximal profits with and without warranty level.
Scenario Πm1 Πm2 Πr Π t No warranty level 3792.9 304.8 3577.2 7674.9 One warranty level 3971.9 199.3 3730.2 7901.4 Two warranty levels 4042.4 328.2 4154.5 8525.1Discussion 3 (sensitivity analysis of parameterλ).
Numerical studies will be used to illustrate the effects of parameterλ on the optimal decisions in MSU and MSL models. By varying the value of λ (from 0.1 to 0.9), the changes of the optimal decisions can be obtained, which is shown in Figures 1 and 2.
(3-1) It follows from Figures1 and 2 that the optimal wholesale prices, the optimal warranty levels, the optimal retail prices, the maximal profits of the manufacturer m1 and the retailer decrease obviously as λ increases in the MSU model and, conversely, increase clearly as λ increases in the MSL model. This is consistent with our intuition that the larger the risk resulting from the given confidence level, the more the benefit to the firm.Figure 1
Changes of prices and warranty levels withλ in MSU and MSL model.Figure 2
Changes of firms’ maximal profits withλ in MSU and MSL model.


## 5. Conclusion

Based on game theory and fuzzy theory, we analyze the price and warranty decisions for two substitutable products in a fuzzy supply chain, where the consumer demand, manufacturing costs, and warranty costs for each product are characterized as fuzzy variables. Considering the manufacturers behaving as the Stackelberg leader, we establish expected value model and two chance-constrained programming models. The closed form solutions in each scenario have been proved. Some important analyses about the results are obtained which present managerial insights into the economic behavior of firms and can be afforded as the basis for empirical study in the future, such as the following:(1) the total profits of the two manufacturers are higher than that of the retailer’s; this is because the manufacturers play the leader’s role which holds the advantage in acquiring more profits in this paper. (2) The warranty provided by the manufacturers will stimulate consumer’s purchase desire and facilitate final purchase behavior. It would be beneficial for the whole supply chain when the warranty services are introduced into the market.Several extensions to this article are possible. First, this paper just considers the scenario where the manufacturers behave as the Stackelberg leader. Further research can extend the models to include Retailer-leader Stackelberg and Nash game models. Second, we establish the models under assuming the symmetry information. Information asymmetry, such as cost and demand information, will be considered in the future. Otherwise, more general forms of the demand function can be used to analyze the problem except the linear form.



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