Analysis of strategies for substitutable and complementary products in a two-levels fuzzy supply chain system

Abstract

In this paper, the pricing competition models have been studied between two substitutable and one complementary products in a two-echelon fuzzy supply chain system. Here, three manufacturers separately (one each) produce and sell their products through a common retailer in the cooperative and non-cooperative markets. The demand for each product depends linearly on the products’ prices. Here, manufacturing costs, base demands and price elasticity are characterized as fuzzy parameters. The closed-form expressions of optimal wholesale and retail pricing decisions have been derived for four different market situations under game theory to maximize the expected profit function of each participant of the supply chain. Sharing of profits under co-operation mechanism amongst the manufacturers and retailer in different proportions are presented numerically for the fuzzy model. As particular cases, the optimal pricing decisions have been derived for two substitutable products or two complementary products separately and the results of previous investigators are presented. In addition, the traditional deterministic version of the fuzzy problem is numerically solved. The results of fuzzy and crisp models are presented and compared. Finally, the problem is illustrated with some numerical data for four market situations, sensitivity analyses are performed and management decisions are explored. The products’ demands and supply chain profits are graphically presented for different price elasticity of the substitutable and complementary products.

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Acknowledgements

The first author is gratefully to University Grants Commission (UGC, India) for partially financial support to continue this research work by Innovative Research Project Grants (Ref. No. VU/Innovative/Sc/03/2015).

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Correspondence to Raghu Nandan Giri.

Appendices

Appendix 1

The expected profit of the supply chain is given by

$$\begin{aligned} E\big [{\widetilde{\pi }}_c\big ]=\, & {} E\big [\big (s_1-{\tilde{c}}_1\big )\big (\tilde{a_1}-{\tilde{\beta }} s_1+\tau {\tilde{\beta }} s_2-\theta _1 {\tilde{\beta }} s_3\big )+\big (s_2-{\tilde{c}}_2\big )\big (\tilde{a_2}+\tau {\tilde{\beta }} s_1-{\tilde{\beta }} s_2-\theta _2 {\tilde{\beta }} s_3\big )\nonumber \\&+\big (s_3-{\tilde{c}}_3\big )\big (\tilde{a_3}-\theta _1 {\tilde{\beta }} s_1-\theta _2 {\tilde{\beta }} s_2-{\tilde{\beta }} s_3\big )\big ]\nonumber \\=\, & {} \frac{1}{2}\int _{0}^{1}\big \{\big [\big (s_1-{\tilde{c}}_1\big )\big (\tilde{a_1}-{\tilde{\beta }} s_1+\tau {\tilde{\beta }} s_2-\theta _1 {\tilde{\beta }} s_3\big )\big ]_{\alpha }^{R}+\big [\big (s_1-\tilde{c_1}\big )\big (\tilde{a_1}-{\tilde{\beta }} s_1+\tau {\tilde{\beta }} s_2-\theta _1 {\tilde{\beta }} s_3\big )\big ]_{\alpha }^{L}\nonumber \\&+\big [\big (s_2-{\tilde{c}}_2\big )\big (\tilde{a_2}+\tau {\tilde{\beta }} s_1-{\tilde{\beta }} s_2-\theta _2 {\tilde{\beta }} s_3\big )\big ]_{\alpha }^{R}+\big [\big (s_2-{\tilde{c}}_2\big )\big (\tilde{a_2}+\tau {\tilde{\beta }} s_1-{\tilde{\beta }} s_2-\theta _2 {\tilde{\beta }} s_3\big )\big ]_{\alpha }^{L}\nonumber \\&+\big [\big (s_3-{\tilde{c}}_3\big )\big (\tilde{a_3}-\theta _1 {\tilde{\beta }} s_1-\theta _2 {\tilde{\beta }} s_2-{\tilde{\beta }} s_3\big )\big ]_{\alpha }^{R}+\big [\big (s_3-{\tilde{c}}_3\big )\big (\tilde{a_3}-\theta _1 {\tilde{\beta }} s_1-\theta _2 {\tilde{\beta }} s_2-{\tilde{\beta }} s_3\big )\big ]_{\alpha }^{L}\big \}d\alpha \nonumber \\=\, & {} \frac{1}{2}\int _{0}^{1}\big \{\big (s_1-{\tilde{c}}_{1\alpha }^{L}\big )\big ({\tilde{a}}_{1\alpha }^{R}-{\tilde{\beta }}_{\alpha }^{L} s_1+\tau {\tilde{\beta }}_{\alpha }^{R} s_2-\theta _1 {\tilde{\beta }}_{\alpha }^{L} s_3\big )+\big [\big (s_1-{\tilde{c}}_{1\alpha }^{R}\big )\big ({\tilde{a}}_{1\alpha }^{L}-{\tilde{\beta }}_{\alpha }^{R} s_1+\tau {\tilde{\beta }}_{\alpha }^{L} s_2\nonumber \\&-\theta _1 {\tilde{\beta }}_{\alpha }^{R} s_3\big )\big ]+\big (s_2-{\tilde{c}}_{2\alpha }^{L}\big )\big ({\tilde{a}}_{2\alpha }^{R}+\tau {\tilde{\beta }}_{\alpha }^{R} s_1-{\tilde{\beta }}_{\alpha }^{L} s_2-\theta _2 {\tilde{\beta }}_{\alpha }^{L} s_3\big )+\big (s_2-{\tilde{c}}_{2\alpha }^{R}\big )\big ({\tilde{a}}_{2\alpha }^{L}+\tau {\tilde{\beta }}_{\alpha }^{L} s_1\nonumber \\&-{\tilde{\beta }}_{\alpha }^{R} s_2-\theta _2 {\tilde{\beta }}_{\alpha }^{R} s_3\big )+\big (s_3-{\tilde{c}}_{3\alpha }^{L}\big )\big ({\tilde{a}}_{3\alpha }^{R}-\theta _1 {\tilde{\beta }}_{\alpha }^{L} s_1-\theta _2 {\tilde{\beta }}_{\alpha }^{L} s_2-{\tilde{\beta }}_{\alpha }^{L} s_3\big )\nonumber \\&+\big (s_3-{\tilde{c}}_{3\alpha }^{R}\big )\big ({\tilde{a}}_{3\alpha }^{L}-\theta _1 {\tilde{\beta }}_{\alpha }^{R} s_1-\theta _2 {\tilde{\beta }}_{\alpha }^{R} s_2-{\tilde{\beta }}_{\alpha }^{R} s_3\big )\big \}d\alpha \nonumber \\=\, & {} \frac{1}{2}\int _{0}^{1}\big \{-\big ({\tilde{\beta }}_{\alpha }^{L}+{\tilde{\beta }}_{\alpha }^{R}\big )s_1^2-\big ({\tilde{\beta }}_{\alpha }^{L}+{\tilde{\beta }}_{\alpha }^{R}\big )s_2^2-\big ({\tilde{\beta }}_{\alpha }^{L}+{\tilde{\beta }}_{\alpha }^{R}\big )s_3^2+2\tau \big ({\tilde{\beta }}_{\alpha }^{L}+{\tilde{\beta }}_{\alpha }^{R}\big )s_1s_2\nonumber \\&-2\theta _1 \big ({\tilde{\beta }}_{\alpha }^{L}+{\tilde{\beta }}_{\alpha }^{R}\big )s_1s_3-2\theta _2\big ({\tilde{\beta }}_{\alpha }^{L}+{\tilde{\beta }}_{\alpha }^{R}\big )s_2s_3+ \big [\big ({\tilde{a}}_{1\alpha }^{R}+{\tilde{a}}_{1\alpha }^{L}\big )+\big ({\tilde{c}}_{1\alpha }^{L}{\tilde{\beta }}_{\alpha }^{L}+{\tilde{c}}_{1\alpha }^{R}{\tilde{\beta }}_{\alpha }^{R}\big )\nonumber \\&-\tau \big ({\tilde{c}}_{2\alpha }^{L}{\tilde{\beta }}_{\alpha }^{R}+{\tilde{c}}_{2\alpha }^{R}{\tilde{\beta }}_{\alpha }^{L}\big )+ \theta _1 \big ({\tilde{c}}_{3\alpha }^{L}{\tilde{\beta }}_{\alpha }^{L}+{\tilde{c}}_{3\alpha }^{R}{\tilde{\beta }}_{\alpha }^{R}\big )\big ]s_1 +\big [\big ({\tilde{a}}_{2\alpha }^{R}+{\tilde{a}}_{2\alpha }^{L}\big )-\tau \big ({\tilde{c}}_{1\alpha }^{L}{\tilde{\beta }}_{\alpha }^{R}+{\tilde{c}}_{1\alpha }^{R}{\tilde{\beta }}_{\alpha }^{L}\big )\nonumber \\&+\big ({\tilde{c}}_{2\alpha }^{L}{\tilde{\beta }}_{\alpha }^{L}+{\tilde{c}}_{2\alpha }^{R}{\tilde{\beta }}_{\alpha }^{R}\big )+ \theta _2 \big ({\tilde{c}}_{3\alpha }^{L}{\tilde{\beta }}_{\alpha }^{L}+{\tilde{c}}_{3\alpha }^{R}{\tilde{\beta }}_{\alpha }^{R}\big )\big ]s_2+\big [\big ({\tilde{a}}_{3\alpha }^{R}+{\tilde{a}}_{3\alpha }^{L}\big )+\theta _1 \big ({\tilde{c}}_{1\alpha }^{L}{\tilde{\beta }}_{\alpha }^{L}+{\tilde{c}}_{1\alpha }^{R}{\tilde{\beta }}_{\alpha }^{R}\big )\nonumber \\&+\theta _2 \big ({\tilde{c}}_{2\alpha }^{L}{\tilde{\beta }}_{\alpha }^{L}+{\tilde{c}}_{2\alpha }^{R}{\tilde{\beta }}_{\alpha }^{R}\big )+\big ({\tilde{c}}_{3\alpha }^{L}{\tilde{\beta }}_{\alpha }^{L}+{\tilde{c}}_{3\alpha }^{R}{\tilde{\beta }}_{\alpha }^{R}\big )\big ]s_3-\big [\big ({\tilde{c}}_{1\alpha }^{L}{\tilde{a}}_{1\alpha }^{R}+{\tilde{c}}_{1\alpha }^{R}{\tilde{a}}_{1\alpha }^{L}\big )\nonumber \\&+ \big ({\tilde{c}}_{2\alpha }^{L}{\tilde{a}}_{2\alpha }^{R}+{\tilde{c}}_{2\alpha }^{R}{\tilde{a}}_{2\alpha }^{L}\big )+\big ({\tilde{c}}_{3\alpha }^{L}{\tilde{a}}_{3\alpha }^{R}+{\tilde{c}}_{3\alpha }^{R}{\tilde{a}}_{3\alpha }^{L}\big )\big ]\big \}d\alpha \nonumber \\=\, & {} -E\big [{\tilde{\beta }}\big ]s_1^2-E\big [{\tilde{\beta }}\big ]s_2^2-E\big [{\tilde{\beta }}\big ]s_3^2+2\tau E\big [{\tilde{\beta }}\big ]s_1s_2-2\theta _1 E\big [{\tilde{\beta }}\big ]s_1s_3-2\theta _2 E\big [{\tilde{\beta }}\big ]s_2s_3\nonumber \\&+\big (E\big [{\tilde{a}}_{1}\big ]+E\big [{\tilde{c}}_{1}{\tilde{\beta }}\big ]+\theta _1 E\big [{\tilde{c}}_{3}{\tilde{\beta }}\big ]\big )s_1+\big (E\big [{\tilde{a}}_{2}\big ]+E\big [{\tilde{c}}_{2}{\tilde{\beta }}\big ]+\theta _2 E\big [{\tilde{c}}_{3}{\tilde{\beta }}\big ]\big )s_2+\big (E\big [{\tilde{a}}_{3}\big ]\nonumber \\&+\theta _1 E\big [{\tilde{c}}_{1}{\tilde{\beta }}\big ]+\theta _2 E\big [{\tilde{c}}_{2}{\tilde{\beta }}\big ]+E\big [{\tilde{c}}_{3}{\tilde{\beta }}\big ]\big )s_3-\frac{\tau }{2}\sum _{i=1}^{2}\left[ \int _0^1\left( {\tilde{c}}_{i \alpha }^{L}{\tilde{\beta }}_{\alpha }^{R}+{\tilde{c}}_{i \alpha }^{R}{\tilde{\beta }}_{\alpha }^{L}\right) d\alpha \right] s_{(3-i)}\nonumber \\&-\frac{1}{2}\sum _{j=1}^3\int _0^1\left( {\tilde{c}}_{j \alpha }^{L}{\tilde{a}}_{j\alpha }^{R}+{\tilde{c}}_{j \alpha }^{R}{\tilde{a}}_{j \alpha }^{L}\right) d\alpha \end{aligned}$$
(20)

The first and second order partial diff. of Eq. (20) w. r. to \(s_1\), \(s_2\) and \(s_3\) are as follows

$$\begin{aligned} \frac{\partial E\left[ {\widetilde{\pi }}_c\right] }{\partial s_1}=\, & {} -2E\left[ {\tilde{\beta }}\right] s_1+2\tau E\left[ {\tilde{\beta }}\right] s_2-2\theta _1E\left[ {\tilde{\beta }}\right] s_3+E\left[ {\tilde{a}}_{1}\right] +E\left[ {\tilde{c}}_{1}{\tilde{\beta }}\right] +\theta _1 E\left[ {\tilde{c}}_{3}{\tilde{\beta }}\right] -\frac{\tau }{2}\int _0^1\left( {\tilde{c}}_{2 \alpha }^{L}{\tilde{\beta }}_{\alpha }^{R}+{\tilde{c}}_{2 \alpha }^{R}{\tilde{\beta }}_{\alpha }^{L}\right) d\alpha ,\\ \frac{\partial E\left[ {\widetilde{\pi }}_c\right] }{\partial s_2}=\, & {} 2\tau E\left[ {\tilde{\beta }}\right] s_1-2E\left[ {\tilde{\beta }}\right] s_2-2\theta _2 E\left[ {\tilde{\beta }}\right] s_3+E\left[ {\tilde{a}}_{2}\right] +E\left[ {\tilde{c}}_{2}{\tilde{\beta }}\right] +\theta _2 E\left[ {\tilde{c}}_{3}{\tilde{\beta }}\right] -\frac{\tau }{2}\int _0^1\left( {\tilde{c}}_{1 \alpha }^{L}{\tilde{\beta }}_{\alpha }^{R}+{\tilde{c}}_{1 \alpha }^{R}{\tilde{\beta }}_{\alpha }^{L}\right) d\alpha ,\\ \frac{\partial E\left[ {\widetilde{\pi }}_c\right] }{\partial s_3}= \,& {} -2\theta _1E\left[ {\tilde{\beta }}\right] s_1-2\theta _2E\left[ {\tilde{\beta }}\right] s_2-2E\left[ {\tilde{\beta }}\right] s_3+E\left[ {\tilde{a}}_{3}\right] +\theta _1 E\left[ {\tilde{c}}_{1}{\tilde{\beta }}\right] +\theta _2 E\left[ {\tilde{c}}_{2}{\tilde{\beta }}\right] +E\left[ {\tilde{c}}_{3}{\tilde{\beta }}\right] ,\\ \frac{\partial ^2 E\left[ {\widetilde{\pi }}_c\right] }{\partial s^2_1}=\, & {} -2E\left[ {\tilde{\beta }}\right] ,\frac{\partial ^2 E\left[ {\widetilde{\pi }}_c\right] }{\partial s_2 \partial s_1}=2\tau E\left[ {\tilde{\beta }}\right] ,\frac{\partial ^2 E\left[ {\widetilde{\pi }}_c\right] }{\partial s_3 \partial s_1}=-2\theta _1E\left[ {\tilde{\beta }}\right] ,\frac{\partial ^2 E\left[ {\widetilde{\pi }}_c\right] }{\partial s_1 \partial s_2}=2\tau E\left[ {\tilde{\beta }}\right] ,\frac{\partial ^2 E\left[ {\widetilde{\pi }}_c\right] }{\partial s^2_2}=-2E\left[ {\tilde{\beta }}\right] ,\\ \frac{\partial ^2 E\left[ {\widetilde{\pi }}_c\right] }{\partial s_3 \partial s_2}=\, & {} -2\theta _2E\left[ {\tilde{\beta }}\right] ,\frac{\partial ^2 E\left[ {\widetilde{\pi }}_c\right] }{\partial s_1 \partial s_3}=-2\theta _1E\left[ {\tilde{\beta }}\right] ,\frac{\partial ^2 E\left[ {\widetilde{\pi }}_c\right] }{\partial s_2 \partial s_3}=-2\theta _2E\left[ {\tilde{\beta }}\right] \;{\mathrm{and}}\;\frac{\partial ^2 E\left[ {\widetilde{\pi }}_c\right] }{\partial s^2_3}=-2E\left[ {\tilde{\beta }}\right] . \end{aligned}$$

Determinant value of the Hessian matrix |H| is given by

$$\begin{aligned} \left| \begin{array}{lll} -2E[{\tilde{\beta }}] &{} 2\tau E[{\tilde{\beta }}] &{} -2\theta _1 E[{\tilde{\beta }}]\\ 2\tau E[{\tilde{\beta }}] &{} -2E[{\tilde{\beta }}] &{} -2\theta _2 E[{\tilde{\beta }}] \\ -2\theta _1 E[{\tilde{\beta }}] &{} -2\theta _2 E[{\tilde{\beta }}] &{} -2E[{\tilde{\beta }}]\end{array}\right| =8\{E[{\tilde{\beta }}]\}^3(\tau ^2+2\tau \theta _1\theta _2+\theta _1^2+\theta _2^2-1). \end{aligned}$$

Therefore, the profit expression given in Eq. (20) is concave if \(\tau ^2+2\tau \theta _1\theta _2+\theta _1^2+\theta _2^2<1\) and \(\tau ^2<1\).

At the extreme point, we have

$$\begin{aligned} \begin{aligned} -2E[{\tilde{\beta }}]s^*_1+2\tau E[{\tilde{\beta }}] s^*_2-2\theta _1 E[{\tilde{\beta }}] s^*_3&=-K_1,\\ \,\,\,\,2\tau E[{\tilde{\beta }}] s^*_1-2E[{\tilde{\beta }}]s^*_2-2\theta _2 E[{\tilde{\beta }}] s^*_3&=-K_2,\\ -2\theta _1 E[{\tilde{\beta }}] s^*_1-2\theta _2 E[{\tilde{\beta }}] s^*_2-2E[{\tilde{\beta }}]s^*_3&=-K_3, \end{aligned} \end{aligned}$$
(21)

where \(K_1=E[{\tilde{a}}_{1}]+E[{\tilde{c}}_{1}{\tilde{\beta }}]+\theta _1 E[{\tilde{c}}_{3}{\tilde{\beta }}]-\frac{\tau }{2}\int _0^1({\tilde{c}}_{2 \alpha }^{L}{\tilde{\beta }}_{\alpha }^{R}+{\tilde{c}}_{2 \alpha }^{R}{\tilde{\beta }}_{\alpha }^{L})d\alpha\), \(K_2=E[{\tilde{a}}_{2}]+E[{\tilde{c}}_{2}{\tilde{\beta }}]+\theta _2 E[{\tilde{c}}_{3}{\tilde{\beta }}]-\frac{\tau }{2}\int _0^1({\tilde{c}}_{1 \alpha }^{L}{\tilde{\beta }}_{\alpha }^{R}+{\tilde{c}}_{1 \alpha }^{R}{\tilde{\beta }}_{\alpha }^{L})d\alpha\) and \(K_3=E[{\tilde{a}}_{3}]+\theta _1 E[{\tilde{c}}_{1}{\tilde{\beta }}]+\theta _2 E[{\tilde{c}}_{2}{\tilde{\beta }}]+E[{\tilde{c}}_{3}{\tilde{\beta }}].\)

Solving the system of linear Eqs. given in (21), we have

$$\begin{aligned} s^*_1=-\frac{(1-\theta _2^2)K_1+(\tau +\theta _1\theta _2)K_2-(\theta _1+\tau \theta _2)K_3}{2E[{\tilde{\beta }}](\tau ^2+2\tau \theta _1\theta _2+\theta _1^2+\theta _2^2-1)} \end{aligned}$$
(22)
$$\begin{aligned} s^*_2=-\frac{(\tau +\theta _1\theta _2)K_1+(1-\theta _1^2)K_2-(\theta _2+\tau \theta _1)K_3}{2E[{\tilde{\beta }}](\tau ^2+2\tau \theta _1\theta _2+\theta _1^2+\theta _2^2-1)} \end{aligned}$$
(23)
$$\begin{aligned} s^*_3=\frac{(\theta _1+\tau \theta _2)K_1+(\theta _2+\tau \theta _1)K_2-(1-\tau ^2)K_3}{2E[{\tilde{\beta }}](\tau ^2+2\tau \theta _1\theta _2+\theta _1^2+\theta _2^2-1)} \end{aligned}$$
(24)

From the constraint \({{\textit{Pos}}}(\{\tilde{a_1}\ge {\tilde{\beta }} s^*_1-\tau {\tilde{\beta }} s^*_2+\theta _1{\tilde{\beta }} s^*_3\})>\eta _1\) and using Definitions 3 and 4, we have

$$\begin{aligned} {\tilde{a}}^{R}_{1\eta _1}>{\tilde{\beta }}^{L}_{\eta _1} s^*_1-\tau {\tilde{\beta }}^{R}_{\eta _1} s^*_2+\theta _1{\tilde{\beta }}^{L}_{\eta _1} s^*_3. \end{aligned}$$

Similarly, from the constraints \({{\textit{Pos}}}(\{\tilde{a_2}\ge -\tau {\tilde{\beta }} s^*_1 +{\tilde{\beta }} s^*_2+\theta _2{\tilde{\beta }} s^*_3\})>\eta _1\), \({{\textit{Pos}}}(\{\tilde{a_3}\ge \theta _1{\tilde{\beta }} s^*_1+\theta _2{\tilde{\beta }} s^*_2+{\tilde{\beta }} s^*_3\})>\eta _1\) and \({{\textit{Pos}}}(\{\tilde{c_i} \le s^*_i\})>\eta _2,\,i=1,2,3\), we have

$$\begin{aligned} {\tilde{a}}^{R}_{2\eta _1}>-\tau {\tilde{\beta }}^{R}_{\eta _1} s^*_1+{\tilde{\beta }}^{L}_{\eta _1} s^*_2 +\theta _2{\tilde{\beta }}^{L}_{\eta _1} s^*_3,\,\,{\tilde{a}}^{R}_{3\eta _1}>\theta _1{\tilde{\beta }}^{L}_{\eta _1} s^*_1+\theta _2{\tilde{\beta }}^{L}_{\eta _1} s^*_2+{\tilde{\beta }}^{L}_{\eta _1} s^*_3 \text{ and } {\tilde{c}}^{L}_{i\eta _2}<s^*_i,\,i=1,2,3. \end{aligned}$$

Appendix 2

The expected profit of the retailer in the MS approach is

$$\begin{aligned} E[{\widetilde{\pi }}_r]=\, & {} (s_1-p_1)(E[\tilde{a_1}]-E[{\tilde{\beta }}] s_1+\tau E[{\tilde{\beta }}] s_2-\theta _1 E[{\tilde{\beta }}] s_3)+(s_2-p_2)(E[\tilde{a_2}]+\tau E[{\tilde{\beta }}] s_1\nonumber \\&-E[{\tilde{\beta }}] s_2-\theta _2 E[{\tilde{\beta }}] s_3)+(s_3-p_3)(E[\tilde{a_3}]-\theta _1 E[{\tilde{\beta }}] s_1-\theta _2 E[{\tilde{\beta }}] s_2-E[{\tilde{\beta }}] s_3) \end{aligned}$$
(25)

Proceeding as “Appendix 1”, retailer’s expected profit expression in the MS approach is concave if

$$\begin{aligned} \tau ^2+2\tau \theta _1\theta _2+\theta _1^2+\theta _2^2<1 \text{ and } \tau ^2<1 \end{aligned}$$

and the retailer’s optimal pricing decisions become

$$\begin{aligned} s^*_1=\frac{p_1}{2}-\frac{(1-\theta _2^2)E[{\tilde{a}}_1]+(\tau +\theta _1\theta _2)E[{\tilde{a}}_2]-(\theta _1+\tau \theta _2)E[{\tilde{a}}_3]}{2E[{\tilde{\beta }}](\tau ^2+2\tau \theta _1\theta _2+\theta _1^2+\theta _2^2-1)} \end{aligned}$$
(26)
$$\begin{aligned} s^*_2=\frac{p_2}{2}-\frac{(\tau +\theta _1\theta _2)E[{\tilde{a}}_1]+(1-\theta _1^2)E[{\tilde{a}}_2]-(\theta _2+\tau \theta _1)E[{\tilde{a}}_3]}{2E[{\tilde{\beta }}](\tau ^2+2\tau \theta _1\theta _2+\theta _1^2+\theta _2^2-1)} \end{aligned}$$
(27)
$$\begin{aligned} s^*_3=\frac{p_3}{2}+\frac{(\theta _1+\tau \theta _2)E[{\tilde{a}}_1]+(\theta _2+\tau \theta _1)E[{\tilde{a}}_2]-(1-\tau ^2)E[{\tilde{a}}_3]}{2E[{\tilde{\beta }}](\tau ^2+2\tau \theta _1\theta _2+\theta _1^2+\theta _2^2-1)} \end{aligned}$$
(28)

From the constraints \({{\textit{Pos}}}(\{\tilde{a_1}\ge {\tilde{\beta }} s^*_1-\tau {\tilde{\beta }} s^*_2+\theta _1{\tilde{\beta }} s^*_3\})>\eta _1\), \({{\textit{Pos}}}(\{\tilde{a_2}\ge -\tau {\tilde{\beta }} s^*_1 +{\tilde{\beta }} s^*_2+\theta _2{\tilde{\beta }} s^*_3\})>\eta _1\), \({{\textit{Pos}}}(\{\tilde{a_3}\ge \theta _1{\tilde{\beta }} s^*_1+\theta _2{\tilde{\beta }} s^*_2+{\tilde{\beta }} s^*_3\})>\eta _1\) and \(s^*_i > p_i\) (i=1,2,3) as similar as “Appendix 1”, we have

$$\begin{aligned}&{\tilde{a}}^{R}_{1\eta _1}>{\tilde{\beta }}^{L}_{\eta _1} s^*_1-\tau {\tilde{\beta }}^{R}_{\eta _1} s^*_2+\theta _1{\tilde{\beta }}^{L}_{\eta _1} s^*_3,\,\, {\tilde{a}}^{R}_{2\eta _1}>-\tau {\tilde{\beta }}^{R}_{\eta _1} s^*_1+{\tilde{\beta }}^{L}_{\eta _1} s^*_2 +\theta _2{\tilde{\beta }}^{L}_{\eta _1} s^*_3,\,\,{\tilde{a}}^{R}_{3\eta _1}>\theta _1{\tilde{\beta }}^{L}_{\eta _1} s^*_1+\theta _2{\tilde{\beta }}^{L}_{\eta _1} s^*_2+{\tilde{\beta }}^{L}_{\eta _1} s^*_3,\\&E[{\tilde{\beta }}](\tau ^2+2\tau \theta _1\theta _2+\theta _1^2+\theta _2^2-1)p_1+(1-\theta _2^2)E[{\tilde{a}}_1]+(\tau +\theta _1\theta _2)E[{\tilde{a}}_2]>(\theta _1+\tau \theta _2)E[{\tilde{a}}_3],\\&E[{\tilde{\beta }}](\tau ^2+2\tau \theta _1\theta _2+\theta _1^2+\theta _2^2-1)p_2+(\tau +\theta _1\theta _2)E[{\tilde{a}}_1]+(1-\theta _1^2)E[{\tilde{a}}_2]>(\theta _2+\tau \theta _1)E[{\tilde{a}}_3],\\ \text{ and }&E[{\tilde{\beta }}](\tau ^2+2\tau \theta _1\theta _2+\theta _1^2+\theta _2^2-1)p_3+(1-\tau ^2)E[{\tilde{a}}_3]>(\theta _1+\tau \theta _2)E[{\tilde{a}}_1]+(\theta _2+\tau \theta _1)E[{\tilde{a}}_2]. \end{aligned}$$

By substituting the Eqs. (26), (27) and (28) into the Eqs. (4),  (5) and (6), the manufacturers’ expected profits can be expressed as

$$\begin{aligned} E\left[ {\widetilde{\pi }}_{M_1}\right]=\, & {} -\frac{1}{2}E\left[ {\tilde{\beta }}\right] p_1^2+\frac{\tau }{2} E\left[ {\tilde{\beta }}\right] p_1p_2-\frac{\theta _1}{2} E\left[ {\tilde{\beta }}\right] p_1p_3+ \frac{1}{2}\left\{ E\left[ {\tilde{a}}_{1}\right] +E\left[ {\tilde{c}}_{1}{\tilde{\beta }}\right] \right\} p_1\nonumber \\&-\frac{\tau }{4}\left[ \int _0^1\left( {\tilde{c}}_{1 \alpha }^{L}{\tilde{\beta }}_{\alpha }^{R}+{\tilde{c}}_{1 \alpha }^{R}{\tilde{\beta }}_{\alpha }^{L}\right) d\alpha \right] p_2+\frac{\theta _1}{2} E\left[ {\tilde{c}}_{1}{\tilde{\beta }}\right] p_3+\frac{E\left[ {\tilde{c}}_{1}{\tilde{\beta }}\right] E\left[ {\tilde{a}}_{1}\right] }{2E\left[ {\tilde{\beta }}\right] }-\frac{1}{2}\int _0^1\left( {\tilde{c}}_{1 \alpha }^{L}{\tilde{a}}_{1\alpha }^{R}+{\tilde{c}}_{1 \alpha }^{R}{\tilde{a}}_{1 \alpha }^{L}\right) d\alpha \end{aligned}$$
(29)
$$\begin{aligned} E\left[ {\widetilde{\pi }}_{M_2}\right]& {} =\, \frac{\tau }{2} E\left[ {\tilde{\beta }}\right] p_1p_2-\frac{1}{2}E\left[ {\tilde{\beta }}\right] p_2^2-\frac{\theta _2}{2} E\left[ {\tilde{\beta }}\right] p_2p_3-\frac{\tau }{4}\left[ \int _0^1\left( {\tilde{c}}_{2 \alpha }^{L}{\tilde{\beta }}_{\alpha }^{R}+{\tilde{c}}_{2 \alpha }^{R}{\tilde{\beta }}_{\alpha }^{L}\right) d\alpha \right] p_1\nonumber \\&\quad +\frac{1}{2}\left\{ E\left[ {\tilde{a}}_{2}\right] +E\left[ {\tilde{c}}_{2}{\tilde{\beta }}\right] \right\} p_2+\frac{\theta _2}{2} E\left[ {\tilde{c}}_{2}{\tilde{\beta }}\right] p_3+\frac{E\left[ {\tilde{c}}_{2}{\tilde{\beta }}\right] E\left[ {\tilde{a}}_{2}\right] }{2E\left[ {\tilde{\beta }}\right] }-\frac{1}{2}\int _0^1\left( {\tilde{c}}_{2 \alpha }^{L}{\tilde{a}}_{2\alpha }^{R}+{\tilde{c}}_{2 \alpha }^{R}{\tilde{a}}_{2 \alpha }^{L}\right) d\alpha \end{aligned}$$
(30)
$$\begin{aligned} E\left[ {\widetilde{\pi }}_{M_3}\right]=\, & {} -\frac{1}{2}\theta _1 E\left[ {\tilde{\beta }}\right] p_1p_3-\frac{1}{2}\theta _2 E\left[ {\tilde{\beta }}\right] p_2p_3- \frac{1}{2}E\left[ {\tilde{\beta }}\right] p_3^2+\frac{\theta _1}{2} E\left[ {\tilde{c}}_{3}{\tilde{\beta }}\right] p_1+\frac{\theta _2}{2} E\left[ {\tilde{c}}_{3}{\tilde{\beta }}\right] p_2\nonumber \\&+\frac{1}{2}\left\{ E\left[ {\tilde{a}}_{3}\right] +E\left[ {\tilde{c}}_{3}{\tilde{\beta }}\right] \right\} p_3+\frac{E\left[ {\tilde{c}}_{3}{\tilde{\beta }}\right] E\left[ {\tilde{a}}_{3}\right] }{2E\left[ {\tilde{\beta }}\right] }-\frac{1}{2}\int _0^1\left( {\tilde{c}}_{3 \alpha }^{L}{\tilde{a}}_{3\alpha }^{R}+{\tilde{c}}_{3 \alpha }^{R}{\tilde{a}}_{3\alpha }^{L}\right) d\alpha \end{aligned}$$
(31)

The first and second order partial diff. of Eq. (29) w. r. to \(p_1\) are as follows

$$\begin{aligned} \frac{\partial E[{\widetilde{\pi }}_{M_1}]}{\partial p_1}=-E[{\tilde{\beta }}] p_1+\frac{\tau }{2}E[{\tilde{\beta }}]p_2-\frac{\theta _1}{2}E[{\tilde{\beta }}]p_3+\frac{1}{2}(E[{\tilde{a}}_1]+E[{\tilde{\beta }}{\tilde{c}}_1]),\,\,and\,\,\frac{ \partial ^2 E[{\widetilde{\pi }}_{M_1}]}{\partial p^2_1}=-E[{\tilde{\beta }}]<0 \end{aligned}$$

Therefore, the manufacturer’s (\(M_1\)) profit expression in MS approach is concave as \(E[{\tilde{\beta }}]>0\) and at the extreme point, we have

$$\begin{aligned} -E[{\tilde{\beta }}] p^*_1+\frac{\tau }{2}E[{\tilde{\beta }}]p^*_2-\frac{\theta _1}{2}E[{\tilde{\beta }}]p^*_3=-\frac{1}{2}(E[{\tilde{a}}_1]+E[{\tilde{\beta }}{\tilde{c}}_1]). \end{aligned}$$
(32)

Similarly, it can be shown that manufacturers’ (\(M_2\) and \(M_3\)) profit expressions for MS approach are also concave and at the extreme point they become

$$\begin{aligned} \frac{\tau }{2}E[{\tilde{\beta }}]p^*_1-E[{\tilde{\beta }}] p^*_2-\frac{\theta _2}{2}E[{\tilde{\beta }}]p^*_3=\, & {} -\frac{1}{2}(E[{\tilde{a}}_2]+E[{\tilde{\beta }}{\tilde{c}}_2]) \end{aligned}$$
(33)
$$\begin{aligned} -\frac{\theta _1}{2}E[{\tilde{\beta }}] p^*_1-\frac{\theta _2}{2}E[{\tilde{\beta }}]p^*_2-E[{\tilde{\beta }}] p^*_3=\, & {} -\frac{1}{2}(E[{\tilde{a}}_3]+E[{\tilde{\beta }}{\tilde{c}}_3]) \end{aligned}$$
(34)

Solving Eqs. (32), (33) and  (34), we have

$$\begin{aligned} p^*_1=\, & {} -\frac{(4-\theta _2^2)(E[{\tilde{a}}_1]+E[{\tilde{\beta }}{\tilde{c}}_1])+(2\tau +\theta _1\theta _2)(E[{\tilde{a}}_2]+E[{\tilde{\beta }}{\tilde{c}}_2])-(2\theta _1+\tau \theta _2)(E[{\tilde{a}}_3]+E[{\tilde{\beta }}{\tilde{c}}_3])}{2E[{\tilde{\beta }}](\tau ^2+\tau \theta _1\theta _2+\theta _1^2+\theta _2^2-4)} \end{aligned}$$
(35)
$$\begin{aligned} p^*_2=\, & {} -\frac{(2\tau +\theta _1\theta _2)(E[{\tilde{a}}_1]+E[{\tilde{\beta }}{\tilde{c}}_1])+(4-\theta _1^2)(E[{\tilde{a}}_2]+E[{\tilde{\beta }}{\tilde{c}}_2])-(2\theta _2+\tau \theta _1)(E[{\tilde{a}}_3]+E[{\tilde{\beta }}{\tilde{c}}_3])}{2E[{\tilde{\beta }}](\tau ^2+\tau \theta _1\theta _2+\theta _1^2+\theta _2^2-4)}\end{aligned}$$
(36)
$$\begin{aligned} p^*_3=\, & {} \frac{(2\theta _1+\tau \theta _2)(E[{\tilde{a}}_1]+E[{\tilde{\beta }}{\tilde{c}}_1])+(2\theta _2+\tau \theta _1)(E[{\tilde{a}}_2]+E[{\tilde{\beta }}{\tilde{c}}_2])-(4-\tau ^2)(E[{\tilde{a}}_3]+E[{\tilde{\beta }}{\tilde{c}}_3])}{2E[{\tilde{\beta }}](\tau ^2+\tau \theta _1\theta _2+\theta _1^2+\theta _2^2-4)} \end{aligned}$$
(37)

From the constraints \({{\textit{Pos}}}(\{\tilde{c_i} \le p^*_i\})>\eta _2,\,i=1,2,3\), we have \({\tilde{c}}^{L}_{i\eta _2}<p^*_i,\,i=1,2,3.\)

Appendix 3

The expected profit of the manufacturer \(M_1\) is

$$\begin{aligned} E\left[ {\widetilde{\pi }}_{M_1}\right]= & {} -E\left[ {\tilde{\beta }}\right] p_1^2+\tau E\left[ {\tilde{\beta }}\right] p_1p_2-\theta _1 E\left[ {\tilde{\beta }}\right] p_1p_3+ \left\{ E\left[ {\tilde{a}}_{1}\right] -\left( m_1-\tau m_2+ \theta _1 m_3\right) E\left[ {\tilde{\beta }}\right] \right. \nonumber \\&\left. +\,E\left[ {\tilde{c}}_{1}{\tilde{\beta }}\right] \right\} p_1-\frac{\tau }{2}\left[ \int _0^1\left( {\tilde{c}}_{1 \alpha }^{L}{\tilde{\beta }}_{\alpha }^{R}+{\tilde{c}}_{1 \alpha }^{R}{\tilde{\beta }}_{\alpha }^{L}\right) d\alpha \right] p_2+\theta _1 E\left[ {\tilde{c}}_{1}{\tilde{\beta }}\right] p_3+E\left[ {\tilde{c}}_{1}{\tilde{\beta }}\right] m_1\nonumber \\&+\theta _1 E\left[ {\tilde{c}}_{1}{\tilde{\beta }}\right] m_3-\frac{\tau m_2}{2}\left[ \int _0^1\left( {\tilde{c}}_{1 \alpha }^{L}{\tilde{\beta }}_{\alpha }^{R}+{\tilde{c}}_{1 \alpha }^{R}{\tilde{\beta }}_{\alpha }^{L}\right) d\alpha \right] -\frac{1}{2}\int _0^1\left( {\tilde{c}}_{1 \alpha }^{L}{\tilde{a}}_{1\alpha }^{R}+{\tilde{c}}_{1 \alpha }^{R}{\tilde{a}}_{1 \alpha }^{L}\right) d\alpha \end{aligned}$$
(38)

The first and second order partial diff. of Eq. (38) w. r. to \(p_1\) are as follows

$$\begin{aligned} \frac{\partial E\left[ {\widetilde{\pi }}_{M_1}\right] }{\partial p_1}=E\left[ {\tilde{a}}_1\right] -E\left[ {\tilde{\beta }}\right] (p_1+m_1)+\tau E\left[ {\tilde{\beta }}\right] (p_2+m_2)-\theta _1E\left[ {\tilde{\beta }}\right] (p_3+m_3)-E\left[ {\tilde{\beta }}\right] p_1+E\left[ {\tilde{\beta }}{\tilde{c}}_1\right] \end{aligned}$$

and \(\frac{ \partial ^2 E[{\widetilde{\pi }}_{M_1}]}{\partial p^2_1}=-2E[{\tilde{\beta }}]<0.\) Therefore, the manufacturer’s (\(M_1\)) expected profit expression in RS approach is concave as \(E[{\tilde{\beta }}]>0\) and at the extreme point, optimum pricing decisions of \(M_1\) is

$$\begin{aligned} p^*_1=\frac{E[{\tilde{a}}_1]+E[{\tilde{\beta }}{\tilde{c}}_1]}{E[{\tilde{\beta }}]}-s_1+\tau s_2-\theta _1 s_3. \end{aligned}$$
(39)

Similarly, we can show that manufacturers’ (\(M_2\) and \(M_3\)) expected profit expressions in RS approach are concave as \(E[{\tilde{\beta }}]>0\) and optimum pricing decisions for \(M_2\) and \(M_3\) are respectively given by

$$\begin{aligned} p^*_2= & {} \frac{E[{\tilde{a}}_2]+E[{\tilde{\beta }}{\tilde{c}}_2]}{E[{\tilde{\beta }}]}+\tau s_1- s_2-\theta _2 s_3 \end{aligned}$$
(40)
$$\begin{aligned} p^*_3= & {} \frac{E[{\tilde{a}}_3]+E[{\tilde{\beta }}{\tilde{c}}_3]}{E[{\tilde{\beta }}]}-\theta _1 s_1-\theta _2 s_2- s_3 \end{aligned}$$
(41)

From the constraints \({{\textit{Pos}}}(\{\tilde{c_i} \le p^*_i\})>\eta _2,\,i=1,2,3\), we have \({\tilde{c}}^{L}_{i\eta _2}<p^*_i,\,i=1,2,3.\)

The retailer’s expected profit after manufacturers’ optimal decisions in RS approach is given by

$$\begin{aligned} E[{\widetilde{\pi }}_r]= & {} (s_1-p^*_1)(a_1-E[{\tilde{\beta }}] s_1+\tau E[{\tilde{\beta }}] s_2-\theta _1 E[{\tilde{\beta }}] s_3)+(s_2-p^*_2)(a_2 +\tau E[{\tilde{\beta }}] s_1-E[{\tilde{\beta }}] s_2\nonumber \\&-\theta _2 E[{\tilde{\beta }}] s_3) +(s_3-p^*_3)(a_3-\theta _1 E[{\tilde{\beta }}] s_1-\theta _2 E[{\tilde{\beta }}] s_2-E[{\tilde{\beta }}] s_3) \end{aligned}$$
(42)

The first and second order partial diff. of Eq. (42) w. r. to \(s_1\), \(s_2\) and \(s_3\) are as follows

$$\begin{aligned} \frac{\partial E[{\widetilde{\pi }}_{r}]}{\partial s_1} & {} = -P s_1+Q s_2-R s_3+B_1,\,\,\,\frac{\partial E[{\widetilde{\pi }}_{r}]}{\partial s_2}=Q s_1-S s_2-T s_3+B_2,\\ \frac{\partial E[{\widetilde{\pi }}_{r}]}{\partial s_3}& {} = -R s_1-T s_2-U s_3+B_3,\,\,\frac{\partial ^2 E[{\widetilde{\pi }}_{r}]}{\partial s^2_1}=-P,\,\,\frac{\partial ^2 E[{\widetilde{\pi }}_{r}]}{\partial s_2 \partial s_1}=Q,\,\,\frac{\partial ^2 E[{\widetilde{\pi }}_{r}]}{\partial s_3 \partial s_1}=-R,\\ \frac{\partial ^2 E[{\widetilde{\pi }}_{r}]}{\partial s_1 \partial s_2}& {} = Q,\,\,\frac{\partial ^2 E[{\widetilde{\pi }}_{r}]}{\partial s^2_2}=-S,\,\,\frac{\partial ^2 E[{\widetilde{\pi }}_{r}]}{\partial s_3 \partial s_2}=-T,\,\,\frac{\partial ^2 E[{\widetilde{\pi }}_{r}]}{\partial s_1 \partial s_3}=-R,\,\,\frac{\partial ^2 E[{\widetilde{\pi }}_{r}]}{\partial s_2 \partial s_3}=-T,\,\,\, \frac{\partial ^2 E[{\widetilde{\pi }}_{r}]}{\partial s^2_3}=-U,\\ \text{ where } P& {} = 2E[{\tilde{\beta }}](2+\tau ^2+\theta _1^2),\,\,Q= 2E[{\tilde{\beta }}](3\tau -\theta _1\theta _2),\,\, R=2E[{\tilde{\beta }}](3\theta _1-\tau \theta _2),\,\,\\ S& {}=\, 2E[{\tilde{\beta }}](2+\tau ^2+\theta _2^2),\,\,T=2E[{\tilde{\beta }}](3\theta _2-\tau \theta _1),\,\,U=2E[{\tilde{\beta }}](2+\theta _1^2+\theta _2^2),\\ B_1& {}= 3E[{\tilde{a}}_1]-2\tau E[{\tilde{a}}_2]+2\theta _1E[{\tilde{a}}_3]+E[{\tilde{c}}_1{\tilde{\beta }}]-\tau E[{\tilde{c}}_2{\tilde{\beta }}]+\theta _1E[{\tilde{c}}_3{\tilde{\beta }}],\,\,\\ B_2& {}=\, 3E[{\tilde{a}}_2]-2\tau E[{\tilde{a}}_1]+2\theta _2E[{\tilde{a}}_3]-\tau E[{\tilde{c}}_1{\tilde{\beta }}]+E[{\tilde{c}}_2{\tilde{\beta }}]+\theta _2E[{\tilde{c}}_3{\tilde{\beta }}],\,\,\\ B_3& {}=\, 3E[{\tilde{a}}_3]+2\theta _1E[{\tilde{a}}_1]+2\theta _2E[{\tilde{a}}_2]+\theta _1E[{\tilde{c}}_1{\tilde{\beta }}]+\theta _2E[{\tilde{c}}_2{\tilde{\beta }}]+E[{\tilde{c}}_3{\tilde{\beta }}]. \end{aligned}$$

The above retailer’s expected profit expression in RS approach is concave if

$$\begin{aligned} \tau ^2+2\tau \theta _1\theta _2+\theta _1^2+\theta _2^2<1\,and\,\tau ^2+\tau \theta _1\theta _2+\theta _1^2+\theta _2^2<4. \end{aligned}$$

At the extreme point, we have

$$\begin{aligned} P s^*_1-Q s^*_2+R s^*_3=B_1,\,\,\,-Q s^*_1+S s^*_2+T s^*_3=B_2\,\,\,{\mathrm{and}}\,\,\,R s^*_1+T s^*_2+U s^*_3=B_3 \end{aligned}$$
(43)

Solving the system of linear Eqs. given in (43), we have

$$\begin{aligned} s^*_1= \,& {} \frac{B_1(T^2-SU)-B_2(RT+QU)+B_3(QT+RS)}{(UQ^2+2QRT+SR^2+PT^2-PSU)}\nonumber \\ s^*_2=\, & {} \frac{-B_1(RT+QU)+B_2(R^2-PU)+B_3(QR+PT)}{(UQ^2+2QRT+SR^2+PT^2-PSU)}\nonumber \\ s^*_3= \,& {} \frac{B_1(QT+RS)+B_2(QR+PT)+B_3(Q^2-PS)}{(UQ^2+2QRT+SR^2+PT^2-PSU)} \end{aligned}$$
(44)

From the constraints \({{\textit{Pos}}}(\{\tilde{a_1}\ge {\tilde{\beta }} s^*_1-\tau {\tilde{\beta }} s^*_2+\theta _1{\tilde{\beta }} s^*_3\})>\eta _1\), \({{\textit{Pos}}}(\{\tilde{a_2}\ge -\tau {\tilde{\beta }} s^*_1 +{\tilde{\beta }} s^*_2+\theta _2{\tilde{\beta }} s^*_3\})>\eta _1\) and \({{\textit{Pos}}}(\{\tilde{a_3}\ge \theta _1{\tilde{\beta }} s^*_1+\theta _2{\tilde{\beta }} s^*_2+{\tilde{\beta }} s^*_3\})>\eta _1\) as similar as “Appendix 1”, we have

$$\begin{aligned} {\tilde{a}}^{R}_{1\eta _1}>{\tilde{\beta }}^{L}_{\eta _1} s^*_1-\tau {\tilde{\beta }}^{R}_{\eta _1} s^*_2+\theta _1{\tilde{\beta }}^{L}_{\eta _1} s^*_3,\,\, {\tilde{a}}^{R}_{2\eta _1}>-\tau {\tilde{\beta }}^{R}_{\eta _1} s^*_1+{\tilde{\beta }}^{L}_{\eta _1} s^*_2 +\theta _2{\tilde{\beta }}^{L}_{\eta _1} s^*_3,\,\,{\tilde{a}}^{R}_{3\eta _1}>\theta _1{\tilde{\beta }}^{L}_{\eta _1} s^*_1+\theta _2{\tilde{\beta }}^{L}_{\eta _1} s^*_2+{\tilde{\beta }}^{L}_{\eta _1} s^*_3. \end{aligned}$$

Appendix 4

The optimum decisions of three manufacturers and one retailer are derived in RS and MS approaches and given in Eqs. (39), (40),  (41), (26), (27) and (28). For Nash-equilibrium strategies, solving these six equations, we have

$$\begin{aligned} p^*_1=\, & {} -\frac{\left( 9-\theta _2^2\right) \left( E\left[ {\tilde{a}}_1\right] +E\left[ {\tilde{c}}_1{\tilde{\beta }}\right] \right) +\left( 3\tau +\theta _1\theta _2\right) \left( E\left[ {\tilde{a}}_2\right] +E\left[ {\tilde{c}}_2{\tilde{\beta }}\right] \right) -\left( 3\theta _1+\tau \theta _2\right) \left( E\left[ {\tilde{a}}_3\right] +E\left[ {\tilde{c}}_3{\tilde{\beta }}\right] \right) }{E\left[ {\tilde{\beta }}\right] \left( 3\tau ^2+2\tau \theta _1\theta _2+3\theta _1^2+3\theta _2^2-27\right) } \end{aligned}$$
(45)
$$\begin{aligned} p^*_2=\, & {} -\frac{\left( 3\tau +\theta _1\theta _2\right) \left( E\left[ {\tilde{a}}_1\right] +E\left[ {\tilde{c}}_1{\tilde{\beta }}\right] \right) +\left( 9-\theta _1^2\right) \left( E\left[ {\tilde{a}}_2\right] +E\left[ {\tilde{c}}_2{\tilde{\beta }}\right] \right) -\left( 3\theta _2+\tau \theta _1\right) \left( E[{\tilde{a}}_3]+E[{\tilde{c}}_3{\tilde{\beta }}]\right) }{E[{\tilde{\beta }}]\left( 3\tau ^2+2\tau \theta _1\theta _2+3\theta _1^2+3\theta _2^2-27\right) }\end{aligned}$$
(46)
$$\begin{aligned} p^*_3=\, & {} \frac{\left( 3\theta _1+\tau \theta _2\right) \left( E[{\tilde{a}}_1]+E[{\tilde{c}}_1{\tilde{\beta }}]\right) +\left( 3\theta _2+\tau \theta _1\right) \left( E[{\tilde{a}}_2]+E[{\tilde{c}}_2{\tilde{\beta }}]\right) -\left( 9-\tau ^2\right) \left( E[{\tilde{a}}_3]+E[{\tilde{c}}_3{\tilde{\beta }}]\right) }{E[{\tilde{\beta }}]\left( 3\tau ^2+2\tau \theta _1\theta _2+3\theta _1^2+3\theta _2^2-27\right) }\end{aligned}$$
(47)
$$\begin{aligned} s^*_1= \,& {} \frac{p^*_1}{2}-\frac{\left( 1-\theta _2^2\right) E[{\tilde{a}}_1]+\left( \tau +\theta _1\theta _2\right) E[{\tilde{a}}_2]-\left( \theta _1+\tau \theta _2\right) E[{\tilde{a}}_3]}{2E[{\tilde{\beta }}]\left( \tau ^2+2\tau \theta _1\theta _2+\theta _1^2+\theta _2^2-1\right) }\end{aligned}$$
(48)
$$\begin{aligned} s^*_2= \,& {} \frac{p^*_2}{2}-\frac{\left( \tau +\theta _1\theta _2\right) E[{\tilde{a}}_1]+\left( 1-\theta _1^2\right) E[{\tilde{a}}_2]-\left( \theta _2+\tau \theta _1\right) E[{\tilde{a}}_3]}{2E[{\tilde{\beta }}]\left( \tau ^2+2\tau \theta _1\theta _2+\theta _1^2+\theta _2^2-1\right) }\end{aligned}$$
(49)
$$\begin{aligned} s^*_3=\, & {} \frac{p^*_3}{2}+\frac{\left( \theta _1+\tau \theta _2\right) E[{\tilde{a}}_1]+\left( \theta _2+\tau \theta _1\right) E[{\tilde{a}}_2]-\left( 1-\tau ^2\right) E[{\tilde{a}}_3]}{2E[{\tilde{\beta }}]\left( \tau ^2+2\tau \theta _1\theta _2+\theta _1^2+\theta _2^2-1\right) } \end{aligned}$$
(50)

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Giri, R.N., Mondal, S.K. & Maiti, M. Analysis of strategies for substitutable and complementary products in a two-levels fuzzy supply chain system. Oper Res Int J 21, 485–524 (2021). https://doi.org/10.1007/s12351-018-0443-9

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Keywords

  • Supply chain
  • Fuzzy variable
  • Substitutable products
  • Complementary products
  • Stackelberg game