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Decentralization versus coordination in competing supply chains under retailers’ extended warranties

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Abstract

This paper studies a two-tier duopoly competing supply chain system consisting of two manufacturers and two exclusive retailers. Both manufacturers produce differentiated products and both retailers provide extended warranties for the products they sell. Two types of channel-structure strategy options are considered: a decentralized structure with a wholesale price contract and a coordinated structure with a sophisticated contract. We first derive the equilibrium outcomes under three possible chain-to-chain competition scenarios. Subsequently, we reveal how manufacturers control their retail channels to gain more supply chain system profit under an interactive environment with supply chain competition and retailers’ extended warranties. We find that pure coordinated channel competition and pure decentralized channel competition may both reach equilibrium. Furthermore, the interaction forces of supply chain competition and extended warranty service significantly impact the characteristics of the equilibria. Finally, we analyze the competing supply chain’s coordination contract design by using the example of a two-part tariff contract, and determine the feasible contract parameter range that results in a win-win solution for supply chain members.

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Acknowledgements

The authors gratefully acknowledge the editor and two anonymous referees for their helpful comments and suggestions which significantly improve the quality of the paper. This research is supported by the National Natural Science Foundation of China (Grant Nos. 71402101, 71372140, 71402102, 71472126, 71432003, 71572114, and 71531003).

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Correspondence to Wen Yang.

Appendix

Appendix

Proof of equilibrium outcomes under DD case

At the competition stage of the DD case, given manufacturer i’s wholesale price \(w_i\), retailer i makes decisions on the product retail price and the extended warranty price to maximize its own profit. Noting that the contracts are unobservable to the firms of the rival supply chain, retailer i’s best response in terms of retail price and extended warranty price \((p_i, x_i)\) depends on its own contract parameter \(w_i\) and on the influence of its competitor’s contract parameter \(w_j\), which is implicitly captured through \((p_j, x_j)\), where \(i=1,2\); \(j=3-i\). Equation (A.1) gives the expression of retailer i’s best response functions, \(i=1,2; j=3-i\).

$$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} p_{i}(p_j, x_j |w_i)=\frac{(2d-2dr^2-t)[(1-r)a+rp_j]+d(1-r^2)(2w_i+ct^2)}{4d-4dr^2-t}\\ x_{i}(p_j, x_j |w_i)=\frac{t[(1-r)a-w_i+rp_j+ct(2d-2dr^2-t)]}{4d-4dr^2-t} \end{array} \right. . \end{aligned}$$
(A.1)

Manufacturers take the retailers’ retail price decisions into account when setting their wholesale prices, which yield the manufacturers’ wholesale price decisions as the functions of \(p_i\) and \(p_j\),

$$\begin{aligned} w_i(p_i, p_j)=\frac{ \begin{array}{cc} (2+r)(1-r)(2d+2dr-2dr^2-2dr^3-rt)a+r(2d \\ -2dr^2-t)[rp_i+(r^2-2)p_j]-cdt^2(2+r)(1+r)(1-r)^2 \end{array}}{2d(4-r^2)(1-r^2)}. \end{aligned}$$
(A.2)

Substituting the wholesale price decisions given above into the response functions of the retail prices and extended warranty prices given in (A.1), we obtain the equilibria of the retail prices \(p_i^{i(D)j(D)}\) and the extended warranty prices \(x_i^{i(D)j(D)}\) as given in Table 2. The equilibrium product quantity \(q_i^{i(D)j(D)}\) and extended warranty quantity \(Q_i^{i(D)j(D)}\) can be determined based on the equilibrium retail prices and equilibrium extended warranty prices. Again, substituting equilibrium retail prices into the wholesale prices response functions as given in (A.2) yields the equilibrium wholesale price \({w_i}^{i(D)j(D)}\) as given in Table 2. Using the equilibrium expressions \(p_i^{i(D)j(D)}\), \(x_i^{i(D)j(D)}\) and \(w_i^{i(D)j(D)}\), we find the supply chain players’ profits and total chain profit, which are shown in Table 2. \(\square \)

Proof of Proposition 2

Based on the calculations given in Sects. 3.1 and 3.3, it can be proved that

$$\begin{aligned} p_i^{i(D)j(D)}- {p_i}^{i(C)j(D)}= & {} \frac{d(4d-4dr^2-t)^2(2a-ct^2)}{2L_{DD}L_H}>0, \\ x_i^{i(D)j(D)}- {x_i}^{i(C)j(D)}= & {} -\frac{t(4d-4dr^2-t)(4d-3dr^2-t)(2a-ct^2)}{4L_{DD}L_H}<0, \\ q_i^{i(D)j(D)}- {q_i}^{i(C)j(D)}= & {} -\frac{d(4d-4dr^2-t)(4d-3dr^2-t)(2a-ct^2)}{2L_{DD}L_H}<0, \\ Q_i^{i(D)j(D)}- {Q_i}^{i(C)j(D)}= & {} -\frac{d(4d-4dr^2-t)(4d-3dr^2-t)(2a-ct^2)}{4L_{DD}L_H}<0, \end{aligned}$$

where \(i=1, 2\); \(j=3-i\). Based on the calculations given in Sects. 3.2 and 3.3, we can compute

$$\begin{aligned} p_i^{i(C)j(C)}- p_i^{i(D)j(C)}= & {} -\frac{d(4d-4dr^2-t)^2(2a-ct^2)}{2L_{CC}L_H}<0, \\ q_i^{i(C)j(C)}- q_i^{i(D)j(C)}= & {} \frac{d(4d-4dr^2-t)(4d-2dr^2-t)(2a-ct^2)}{2L_{CC}L_H}>0, \\ Q_i^{i(C)j(C)}- Q_i^{i(D)j(C)}= & {} \frac{d(4d-4dr^2-t)(4d-2dr^2-t)(2a-ct^2)}{4L_{CC}L_H}>0, \\ x_i^{i(C)j(C)}- x_i^{i(D)j(C)}= & {} \frac{t(4d-4dr^2-t)(4d-2dr^2-t)(2a-ct^2)}{4L_{CC}L_H}>0, \end{aligned}$$

where \(i=1, 2\); \(j=3-i\). \(\square \)

Proof of Theorem 1

Given that supply chain \(j (j=3-i)\) is decentralized, the difference in profit for the supply chain \(i(i=1,2)\) from choosing a coordinated channel or a decentralized channel is as follows:

$$\begin{aligned} \pi _i^{i(C)j(D)}-\pi _i^{i(D)j(D)}=\frac{d(2a-ct^2)^2(4d-4dr^2-t)G_1G_2}{16L_{DD}^2L_H^2}, \end{aligned}$$

where

$$\begin{aligned} G_1= & {} 6(3-\sqrt{3})d^2r^4+[(9-2\sqrt{3})t+(6\sqrt{3}-34)d]dr^2+(4d-t)^2>0,\\ G_2= & {} 6(3+\sqrt{3})d^2r^4+[(9+2\sqrt{3})t-(6\sqrt{3}+34)d]dr^2+(4d-t)^2. \end{aligned}$$

It can be calculated that if \(0<t<0.9282d\) and \(0\le r<r_{cd,dd}\), or \(0.9282d<t\le t_0\), then \(G_2>0\), which indicates \(\pi _i^{i(C)j(D)}>\pi _i^{i(D)j(D)}\). Otherwise, \(\pi _i^{i(C)j(D)}\le \pi _i^{i(D)j(D)}\). Moreover, iff \(0<t<0.9282d\) and \(r=r_{cd,dd}\), \(\pi _i^{i(D)j(D)}=\pi _i^{i(C)j(D)}\). Next, we can compute the difference in the total chain profit under a coordinated or decentralized channel structure given that the rival supply chain is coordinated. The difference between \(\pi _i^{i(C)j(C)}\) and \(\pi _i^{i(D)j(C)}\) is

$$\begin{aligned} \pi _i^{i(C)j(C)}-\pi _i^{i(D)j(C)}=\frac{d(2a-ct^2)^2(4d-4dr^2-t)G_3G_4}{16L_{CC}^2L_H^2}, \end{aligned}$$

where

$$\begin{aligned} G_3= & {} t^2+2d[(4-\sqrt{3})r^2-4]t+4d^2(1-r^2)[4-(3-\sqrt{3})r^2]>0, \\ G_4= & {} t^2+2d[(4+\sqrt{3})r^2-4]t+4d^2(1-r^2)[4-(3+\sqrt{3})r^2]. \end{aligned}$$

It can be calculated that if \(0\le r<0.9194\) and \(0<t\le t_0\), or \(0.9194<r<1\) and \(t_{cc,cd}< t\le t_0\), then \(G_4>0\), which indicates that \(\pi _i^{i(C)j(C)}>\pi _i^{i(D)j(C)}\); otherwise, \(\pi _i^{i(C)j(C)}\le \pi _i^{i(D)j(C)}\). Moreover, iff \(0.9194<r<1\) and \(t=t_{cc,cd}\), \(\pi _i^{i(C)j(C)}=\pi _i^{i(D)j(C)}\). \(\square \)

Proof of Theorem 3

Based on the calculations given in Sects. 3.1 and 3.2, we can compute

$$\begin{aligned} p_i^{i(C)j(C)}- p_i^{i(D)j(D)}= & {} \frac{-d(1+r)(4d-4dr^2-t)(2a-ct^2)^2}{2L_{CC}L_{DD}}<0, \\ q_i^{i(C)j(C)}- q_i^{i(D)j(D)}= & {} \frac{d(4d-4dr^2-t)(2a-ct^2)^2}{2L_{CC}L_{DD}}>0, \\ x_i^{i(C)j(C)}- x_i^{i(D)j(D)}= & {} \frac{t(4d-4dr^2-t)(2a-ct^2)^2}{4L_{CC}L_{DD}}>0, \\ Q_i^{i(C)j(C)}- Q_i^{i(D)j(D)}= & {} \frac{d(4d-4dr^2-t)(2a-ct^2)^2}{4L_{CC}L_{DD}}>0. \end{aligned}$$

Again, we can compute

$$\begin{aligned} \pi _i^{i(C)j(C)}- \pi _i^{i(D)j(D)}=\frac{d(4d-4dr^2-t)(2a-ct^2)^2G_5G_6}{16L_{CC}^2L_{DD}^2}, \end{aligned}$$

where

$$\begin{aligned} G_5= & {} 2(3-\sqrt{3})dr^2+2(1-\sqrt{3})dr-4d+t<0, \\ G_6= & {} 2(3+\sqrt{3})dr^2+2(1+\sqrt{3})dr-4d+t. \end{aligned}$$

It can be concluded that if \(0\le r<\min \{r_{cc,dd}, \sqrt{1-\frac{t}{2d}}\}\) (i.e.,\(0\le r<r_{cc,dd}\) and \(0<t\le t_0\)), then \(G_6<0\), which indicates that \(\pi _i^{i(C)j(C)}>\pi _i^{i(D)j(D)}\); otherwise, \(\pi _i^{i(C)j(C)}\le \pi _i^{i(D)j(D)}\)( \(\pi _i^{i(C)j(C)}=\pi _i^{i(D)j(D)}\) only if \(0<t<1.3764\) and \(r=r_{cc,dd}\)). \(\square \)

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Ma, J., Ai, X., Yang, W. et al. Decentralization versus coordination in competing supply chains under retailers’ extended warranties. Ann Oper Res 275, 485–510 (2019). https://doi.org/10.1007/s10479-018-2871-6

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