Forward and reverse flows pricing decisions for two competing supply chains with common collection centers in an intuitionistic fuzzy environment

Abstract

This paper studies the selling and the acquisition price decisions for the products of two partially overlapping competing supply chains in an intuitionistic fuzzy environment. In the forward flows, the supply chains are assumed to produce substitutable products and sell them in a duopoly marketplace. The reverse flows are also concentrated on collecting and recycling the used products. While regularly it is assumed that the structures of the competing supply chains are completely separated, this paper assumes that the supply chains benefit from the services of the same collection centers. Considering the collection centers and the two supply chains to act as three integrated entities having enough bargaining powers to affect the decisions of each other, this paper concentrates on investigating the effects of different power structure scenarios on the optimal pricing decisions made by all of the parties involved in the problem. Another important feature of this study is that it employs the concept of intuitionistic fuzzy sets to integrate uncertainties and impressions inevitable in real-world situations into the decision-making process. To achieve this purpose, first, a new credibility-based definition for the concept of intuitionistic fuzzy variables is provided. This concept is then employed in the process of developing the intuitionistic fuzzy pricing models. A new expected value operator is also defined which is employed to transform the credibility-based intuitionistic fuzzy programming problems into their crisp equivalents. A numerical example is employed to discuss the properties of the introduced concepts, providing some interesting managerial implications regarding the defined problem.

Appendices

Appendix A: Proofs of propositions

Proof of proposition 1

Using Lemmas 2 and 3, the expected value of the profit function $\tilde{\pi }_{\text{cc}} \left( {p_{1} ,p_{2} ,r_{1} ,r_{2} } \right)$ is expressed as follows:

$$ \begin{aligned} E\left[ {\tilde{\pi }_{{{\text{cc}}}} } \right] & = \sum\limits_{{i = 1}}^{2} {\left( {E\left[ {\tilde{e}_{i} } \right]k_{i} + E\left[ {\tilde{\gamma }_{1} } \right]r_{i} k_{i} + E\left[ {\tilde{\gamma }_{2} } \right]p_{i} k_{i} } \right.} \\ & \quad \left. { - E\left[ {\tilde{e}_{i} } \right]r_{i} - E\left[ {\tilde{\gamma }_{1} } \right]r_{i}^{2} - E\left[ {\tilde{\gamma }_{2} } \right]p_{i} r_{i} } \right). \\ \end{aligned} $$

(53)

Considering (53), the first- and second-order derivatives of $E\left[ {\tilde{\pi }_{\text{cc}} } \right]$ with respect to $r_{1}$ and $r_{2}$ are determined as follows:

$$ \frac{{\partial E\left[ {\tilde{\pi }_{\text{cc}} } \right]}}{{\partial r_{i} }} = - E\left[ {\tilde{e}_{i} } \right] + E\left[ {\tilde{\gamma }_{1} } \right]k_{i} - E\left[ {\tilde{\gamma }_{2} } \right]p_{i} - 2E\left[ {\tilde{\gamma }_{1} } \right]r_{i} ,\quad i = 1,2, $$

(54)

$$ \frac{{\partial^{2} E\left[ {\tilde{\pi }_{\text{cc}} } \right]}}{{\partial^{2} r_{1} }} = \frac{{\partial^{2} E\left[ {\tilde{\pi }_{cc} } \right]}}{{\partial^{2} r_{2} }} = - 2E\left[ {\tilde{\gamma }_{1} } \right], $$

(55)

$$ \frac{{\partial^{2} E\left[ {\tilde{\pi }_{\text{cc}} } \right]}}{{\partial^{2} r_{1} r_{2} }} = \frac{{\partial^{2} E\left[ {\tilde{\pi }_{\text{cc}} } \right]}}{{\partial^{2} r_{2} r_{1} }} = 0. $$

(56)

Using (55) and (56), the Hessian matrix of the expected profit $E\left[ {\tilde{\pi }_{\text{cc}} } \right]$ with respect to $r_{1}$ and $r_{2}$ and its first- and second-order determinants are defined as follows:

$$ H = \left[ {\begin{array}{*{20}c} { - 2E\left[ {\tilde{\gamma }_{1} } \right]} & 0 \\ 0 & { - 2E\left[ {\tilde{\gamma }_{1} } \right]} \\ \end{array} } \right], $$

(57)

$$ d_{1} \left( H \right) = - 2E\left[ {\tilde{\gamma }_{1} } \right], $$

(58)

$$ d_{2} \left( H \right) = 4E^{2} \left[ {\tilde{\gamma }_{1} } \right]. $$

(59)

The Hessian matrix is negative definite. Therefore, it is concluded that $E\left[ {\tilde{\pi }_{\text{cc}} } \right]$ is jointly concave with respect to $r_{1}$ and $r_{2}$. Using Eq. (54), the optimal pricing decisions $r_{1}^{*} \left( {p_{1} } \right)$ and $r_{2}^{*} \left( {p_{2} } \right)$ are determined as provided in (33).

Proof of proposition 2

The expected value of the profit function $\tilde{\pi }_{{{\text{sc}}_{i} }}$ is expressed as follows:

$$ \begin{aligned} E\left[ {\tilde{\pi }_{{{\text{sc}}_{i} }} } \right] & = \left( {E\left[ {\tilde{d}_{i} } \right] - E\left[ {\tilde{\beta }_{1} } \right]p_{i} + E\left[ {\tilde{\beta }_{2} } \right]p_{j} + E\left[ {\tilde{\beta }_{3} } \right]r_{i}^{*} } \right)p_{i} \\ & \quad - I_{{i1}} + E\left[ {\tilde{c}_{i} \tilde{\beta }_{1} } \right]p_{i} - I_{{i2}} p_{j} - I_{{i3}} r_{i}^{*} \\ & \quad - \left( {k_{i} - s_{i} } \right)\left( {E\left[ {\tilde{e}_{i} } \right] + E\left[ {\tilde{\gamma }_{1} } \right]r_{i}^{*} + E\left[ {\tilde{\gamma }_{2} } \right]p_{i} } \right),\quad i = 1,2, \\ \end{aligned} $$

(60)

where $I_{i1}$, $I_{i2}$ and $I_{i3}$ are defined in “Appendix B.”

Substituting the pricing decisions $r_{1}^{*} \left( {p_{1} } \right)$ and $r_{2}^{*} \left( {p_{2} } \right)$ into (60), the first- and second-order derivatives of the expected profit function $E\left[ {\tilde{\pi }_{{{\text{sc}}_{i} }} } \right]$ with respect to $p_{i}$ are determined as follows:

$$ \begin{aligned} \frac{{\partial E\left[ {\tilde{\pi }_{{{\text{sc}}_{i} }} } \right]}}{{\partial p_{i} }} & = \left( {E\left[ {\tilde{\gamma }_{1} } \right]} \right.\left( {2E\left[ {\tilde{d}_{i} } \right] + 2E\left[ {\tilde{c}_{i} \tilde{\beta }_{1} } \right] - E\left[ {\tilde{\gamma }_{2} } \right]\left( {k_{i} - s_{i} } \right)} \right. \\ & \quad \left. { - \,4E\left[ {\tilde{\beta }_{1} } \right]p_{i} + 2E\left[ {\tilde{\beta }_{2} } \right]p_{j} } \right) + E\left[ {\tilde{\gamma }_{2} } \right]I_{{i3}} \\ & \quad \left. { + \,E\left[ {\tilde{\beta }_{3} } \right]\left( {E\left[ {\tilde{\gamma }_{1} } \right]k_{i} - E\left[ {\tilde{e}_{i} } \right] - 2E\left[ {\tilde{\gamma }_{2} } \right]p_{i} } \right)} \right)/2E\left[ {\tilde{\gamma }_{1} } \right], \\ \end{aligned} $$

(61)

$$ \frac{{\partial^{2} E\left[ {\tilde{\pi }_{{{\text{sc}}_{i} }} } \right]}}{{\partial^{2} p_{i} }} = - \frac{{2E\left[ {\tilde{\beta }_{1} } \right]E\left[ {\tilde{\gamma }_{1} } \right] + E\left[ {\tilde{\beta }_{3} } \right]E\left[ {\tilde{\gamma }_{2} } \right]}}{{E\left[ {\tilde{\gamma }_{1} } \right]}}. $$

(62)

It follows from (62) that $E\left[ {\tilde{\pi }_{{{\text{sc}}_{i} }} } \right]$ is a concave function of $p_{i}$. Using Eq. (61), the optimal pricing decisions $p_{1}^{*}$ and $p_{2}^{*}$ are determined as provided in (34).

Proof of proposition 3

The concavity of the expected profit function $E\left[ {\tilde{\pi }_{{{\text{sc}}_{2} }} } \right]$ with respect to $p_{2}$ is investigated in the proof of proposition 2. Using Eq. (61), the optimal pricing decision $p_{2}^{*} \left( {p_{1} } \right)$ can be easily obtained.

Proof of proposition 4

The expected profit function of the leading supply chain is expressed as follows:

$$ \begin{aligned} E\left[ {\tilde{\pi }_{{{\text{sc}}_{1} }} } \right] & = \left( {E\left[ {\tilde{d}_{1} } \right] - E\left[ {\tilde{\beta }_{1} } \right]p_{1} + E\left[ {\tilde{\beta }_{2} } \right]p_{2}^{*} + E\left[ {\tilde{\beta }_{3} } \right]r_{1}^{*} } \right)p_{1} \\ & \quad - I_{{11}} + E\left[ {\tilde{c}_{1} \tilde{\beta }_{1} } \right]p_{1} - I_{{12}} p_{2}^{*} - I_{{13}} r_{1}^{*} \\ & \quad - \left( {k_{1} - s_{1} } \right)\left( {E\left[ {\tilde{e}_{1} } \right] + E\left[ {\tilde{\gamma }_{1} } \right]r_{1}^{*} + E\left[ {\tilde{\gamma }_{2} } \right]p_{1} } \right), \\ \end{aligned} $$

(63)

where $I_{11}$, $I_{12}$ and $I_{13}$ are defined in “Appendix B.”

Substituting the optimal pricing decisions $p_{2}^{*} \left( {p_{1} } \right)$ and $r_{1}^{*} \left( {p_{1} } \right)$ into (63), the first- and second-order derivatives of the expected profit $E\left[ {\tilde{\pi }_{{{\text{sc}}_{1} }} } \right]$ with respect to $p_{1}$ are determined as follows:

$$ \begin{aligned} \frac{{\partial E\left[ {\tilde{\pi }_{{{\text{sc}}_{1} }} } \right]}}{{\partial p_{1} }} & = \left( {2E\left[ {\tilde{\beta }_{1} } \right]E\left[ {\tilde{\gamma }_{1} } \right]\left( {E\left[ {\tilde{\gamma }_{1} } \right]2E\left[ {\tilde{d}_{1} } \right] + 2E\left[ {\tilde{c}_{1} \tilde{\beta }_{1} } \right]} \right.} \right. \\ & \quad \left. { - E\left[ {\tilde{\gamma }_{2} } \right]\left( {k_{1} - s_{1} } \right) - 4E\left[ {\tilde{\beta }_{1} } \right]p_{1} } \right) + E\left[ {\tilde{\gamma }_{2} } \right]I_{{13}} \\ & \quad \left. { + E\left[ {\tilde{\beta }_{3} } \right]\left( {E\left[ {\tilde{\gamma }_{1} } \right]k_{1} - E\left[ {\tilde{e}_{1} } \right]} \right)} \right) \\ & \quad + E\left[ {\tilde{\beta }_{2} } \right]E\left[ {\tilde{\gamma }_{1} } \right]\left( {E\left[ {\tilde{\gamma }_{1} } \right]\left( {2E\left[ {\tilde{d}_{2} } \right] + 2E\left[ {\tilde{c}_{2} \tilde{\beta }_{1} } \right]} \right.} \right. \\ & \quad \left. { - E\left[ {\tilde{\gamma }_{2} } \right]\left( {k_{2} - s_{2} } \right) - 2I_{{12}} + 4E\left[ {\tilde{\beta }_{2} } \right]p_{1} } \right) \\ & \quad \left. { + E\left[ {\tilde{\gamma }_{2} } \right]I_{{23}} - E\left[ {\tilde{\beta }_{3} } \right]\left( {E\left[ {\tilde{\gamma }_{1} } \right]k_{2} - E\left[ {\tilde{e}_{2} } \right]} \right)} \right) \\ & \quad + E\left[ {\tilde{\beta }_{3} } \right]E\left[ {\tilde{\gamma }_{2} } \right]\left( {E\left[ {\tilde{\gamma }_{1} } \right]\left( {2E\left[ {\tilde{d}_{1} } \right] + 2E\left[ {\tilde{c}_{1} \tilde{\beta }_{1} } \right]} \right.} \right. \\ & \quad \left. { - E\left[ {\tilde{\gamma }_{2} } \right]\left( {k_{1} - s_{1} } \right) - 8E\left[ {\tilde{\beta }_{1} } \right]p_{1} } \right) + E\left[ {\tilde{\gamma }_{2} } \right]I_{{13}} \\ & \quad \left. { + E\left[ {\tilde{\beta }_{3} } \right]\left( {E\left[ {\tilde{\gamma }_{1} } \right]k_{1} - E\left[ {\tilde{e}_{1} } \right] - 2E\left[ {\tilde{\gamma }_{2} } \right]p_{1} } \right)} \right) \\ & \quad /\left( {4E\left[ {\tilde{\beta }_{1} } \right]E^{2} \left[ {\tilde{\gamma }_{1} } \right] + 2E\left[ {\tilde{\beta }_{3} } \right]E\left[ {\tilde{\gamma }_{1} } \right]E\left[ {\tilde{\gamma }_{2} } \right]} \right), \\ \end{aligned} $$

(64)

$$ \begin{aligned} \frac{{\partial ^{2} E\left[ {{\tilde{\pi }}_{{{\text{sc}}_{1} }} } \right]}}{{\partial ^{2} p_{1} }} & = - \frac{{2E\left[ {{\tilde{\gamma }}_{1} } \right]\left( {2E^{2} \left[ {{\tilde{\beta }}_{1} } \right] - E^{2} \left[ {{\tilde{\beta }}_{2} } \right]} \right) + 2E\left[ {{\tilde{\beta }}_{1} } \right]E\left[ {{\tilde{\beta }}_{3} } \right]E\left[ {{\tilde{\gamma }}_{2} } \right]}}{{2E\left[ {{\tilde{\beta }}_{1} } \right]E\left[ {{\tilde{\gamma }}_{1} } \right] + E\left[ {{\tilde{\beta }}_{3} } \right]E\left[ {{\tilde{\gamma }}_{2} } \right]}} \\ & \quad -\, \frac{{E\left[ {{\tilde{\beta }}_{3} } \right]E\left[ {{\tilde{\gamma }}_{2} } \right]}}{{E\left[ {{\tilde{\gamma }}_{1} } \right]}}. \\ \end{aligned} $$

(65)

It follows from (65) that $E\left[ {\tilde{\pi }_{{{\text{sc}}_{1} }} } \right]$ is concave with respect to $p_{1}$. By setting (64) equal to 0 and solving the resulting equation with respect to $p_{1}$, the optimal pricing decision of supply chain ${\text{sc}}_{1}$ is determined as provided in (38).

Proof of proposition 5

The expected profits of the competing supply chains are expressed as follows:

$$ \begin{aligned} E\left[ {\tilde{\pi }_{{{\text{sc}}_{i} }} } \right] & = \left( {E\left[ {\tilde{d}_{i} } \right] - E\left[ {\tilde{\beta }_{1} } \right]p_{i} + E\left[ {\tilde{\beta }_{2} } \right]p_{j} + E\left[ {\tilde{\beta }_{3} } \right]r_{i} } \right)p_{i} \\ & \quad - I_{{i1}} + E\left[ {\tilde{c}_{i} \tilde{\beta }_{1} } \right]p_{i} - I_{{i2}} p_{j} - I_{{i3}} r_{i} \\ & \quad - \left( {k_{i} - s_{i} } \right)\left( {E\left[ {\tilde{e}_{i} } \right] + E\left[ {\tilde{\gamma }_{1} } \right]r_{i} + E\left[ {\tilde{\gamma }_{2} } \right]p_{i} } \right), \\ & \quad i = 1,2, \\ \end{aligned} $$

(66)

where $I_{i1}$, $I_{i2}$ and $I_{i3}$ are defined in “Appendix B.”

The first- and second-order derivatives of the expected profit function $E\left[ {\tilde{\pi }_{{{\text{sc}}_{i} }} } \right]$ with respect to $p_{i}$ are determined as follows:

$$ \begin{aligned} \frac{{\partial E\left[ {\tilde{\pi }_{{sc_{i} }} } \right]}}{{\partial p_{i} }} & = E\left[ {\tilde{d}_{i} } \right] - 2E\left[ {\tilde{\beta }_{1} } \right]p_{i} + E\left[ {\tilde{\beta }_{2} } \right]p_{j} \\ & \quad + E\left[ {\tilde{\beta }_{3} } \right]r_{i} + E\left[ {\tilde{c}_{i} \tilde{\beta }_{1} } \right] - E\left[ {\tilde{\gamma }_{2} } \right]\left( {k_{i} - s_{i} } \right), \\ \end{aligned} $$

(67)

$$ \frac{{\partial^{2} E\left[ {\tilde{\pi }_{{{\text{sc}}_{i} }} } \right]}}{{\partial^{2} p_{i} }} = - 2E\left[ {\tilde{\beta }_{1} } \right]. $$

(68)

It follows from (68) that the expected profit function $E\left[ {\tilde{\pi }_{{{\text{sc}}_{i} }} } \right]$ is concave with respect to $p_{i}$. Using (67), the optimal pricing decisions of the competing supply chains are determined as provided in (41).

Proof of proposition 6

The expected profit of the common collectors is expressed as follows:

$$ \begin{aligned} E\left[ {\tilde{\pi }_{{{\text{cc}}}} } \right] & = \sum\limits_{{i = 1}}^{2} {\left( {E\left[ {\tilde{e}_{i} } \right]k_{i} + E\left[ {\tilde{\gamma }_{1} } \right]r_{i} k_{i} + E\left[ {\tilde{\gamma }_{2} } \right]p_{i}^{*} k_{i} } \right.} \\ & \quad \left. { - E\left[ {\tilde{e}_{i} } \right]r_{i} - E\left[ {\tilde{\gamma }_{1} } \right]r_{i}^{2} - E\left[ {\tilde{\gamma }_{2} } \right]p_{i}^{*} r_{i} } \right). \\ \end{aligned} $$

(69)

Substituting $p_{1}^{*} \left( {r_{1} ,r_{2} } \right)$ and $p_{2}^{*} \left( {r_{1} ,r_{2} } \right)$ into (69), the first- and second-order derivatives of the function $E\left[ {\tilde{\pi }_{\text{cc}} } \right]$ with respect to $r_{1}$ and $r_{2}$ are determined as follows:

$$ \begin{aligned} \frac{{\partial E\left[ {\tilde{\pi }_{{{\text{cc}}}} } \right]}}{{\partial r_{i} }} & = \left( { - 2E\left[ {\tilde{\beta }_{1} } \right]\left( {E\left[ {\tilde{\gamma }_{2} } \right]\left( {E\left[ {\tilde{d}_{i} } \right] + E\left[ {\tilde{c}_{i} \tilde{\beta }_{1} } \right]} \right.} \right.} \right. \\ & \left. {\quad - E\left[ {\tilde{\gamma }_{2} } \right]\left( {k_{i} - s_{i} } \right)} \right) \\ & \quad - 2E\left[ {\tilde{\beta }_{1} } \right]\left( {E\left[ {\tilde{\gamma }_{1} } \right]k_{i} - E\left[ {\tilde{e}_{i} } \right] - 2E\left[ {\tilde{\gamma }_{1} } \right]r_{i} } \right) \\ & \quad \left. { - E\left[ {\tilde{\beta }_{3} } \right]E\left[ {\tilde{\gamma }_{2} } \right]\left( {k_{i} - 2r_{i} } \right)} \right) \\ & \quad - E\left[ {\tilde{\beta }_{2} } \right]\left( {E\left[ {\tilde{\gamma }_{2} } \right]\left( {E\left[ {\tilde{d}_{j} } \right] + E\left[ {\tilde{c}_{j} \tilde{\beta }_{1} } \right]} \right.} \right. \\ & \quad \left. { - E\left[ {\tilde{\gamma }_{2} } \right]\left( {k_{j} - s_{j} } \right)} \right) \\ & \quad + E\left[ {\tilde{\beta }_{2} } \right]\left( {E\left[ {\tilde{\gamma }_{1} } \right]k_{i} - E\left[ {\tilde{e}_{i} } \right] - 2E\left[ {\tilde{\gamma }_{1} } \right]r_{i} } \right) \\ & \quad \left. {\left. { - E\left[ {\tilde{\beta }_{3} } \right]E\left[ {\tilde{\gamma }_{2} } \right]\left( {k_{j} - 2r_{j} } \right)} \right)} \right) \\ & \quad /\left( {4E^{2} \left[ {\tilde{\beta }_{1} } \right] - E^{2} \left[ {\tilde{\beta }_{2} } \right]} \right) \\ \end{aligned} $$

(70)

$$ \frac{{\partial^{2} E\left[ {\tilde{\pi }_{\text{cc}} } \right]}}{{\partial^{2} r_{1} }} = \frac{{\partial^{2} E\left[ {\tilde{\pi }_{\text{cs}} } \right]}}{{\partial^{2} r_{2} }} = - 2E\left[ {\tilde{\gamma }_{1} } \right] - \frac{{4E\left[ {\tilde{\beta }_{1} } \right]E\left[ {\tilde{\beta }_{3} } \right]E\left[ {\tilde{\gamma }_{2} } \right]}}{{4E^{2} \left[ {\tilde{\beta }_{1} } \right] - E^{2} \left[ {\tilde{\beta }_{2} } \right]}}, $$

(71)

$$ \frac{{\partial^{2} E\left[ {\tilde{\pi }_{\text{cc}} } \right]}}{{\partial^{2} r_{1} r_{2} }} = \frac{{\partial^{2} E\left[ {\tilde{\pi }_{\text{cs}} } \right]}}{{\partial^{2} r_{2} r_{1} }} = - \frac{{2E\left[ {\tilde{\beta }_{2} } \right]E\left[ {\tilde{\beta }_{3} } \right]E\left[ {\tilde{\gamma }_{2} } \right]}}{{4E^{2} \left[ {\tilde{\beta }_{1} } \right] - E^{2} \left[ {\tilde{\beta }_{2} } \right]}}. $$

(72)

The Hessian matrix of the expected profit $E\left[ {\tilde{\pi }_{\text{cc}} } \right]$ with respect to $r_{1}$ and $r_{2}$ and its first- and second-order determinants are defined as follows:

$$ H = \left[ {\begin{array}{*{20}c} { - 2E\left[ {\tilde{\gamma }_{1} } \right] - \frac{{4E\left[ {\tilde{\beta }_{1} } \right]E\left[ {\tilde{\beta }_{3} } \right]E\left[ {\tilde{\gamma }_{2} } \right]}}{{4E^{2} \left[ {\tilde{\beta }_{1} } \right] - E^{2} \left[ {\tilde{\beta }_{2} } \right]}}} & { - \frac{{2E\left[ {\tilde{\beta }_{2} } \right]E\left[ {\tilde{\beta }_{3} } \right]E\left[ {\tilde{\gamma }_{2} } \right]}}{{4E^{2} \left[ {\tilde{\beta }_{1} } \right] - E^{2} \left[ {\tilde{\beta }_{2} } \right]}}} \\ { - \frac{{2E\left[ {\tilde{\beta }_{2} } \right]E\left[ {\tilde{\beta }_{3} } \right]E\left[ {\tilde{\gamma }_{2} } \right]}}{{4E^{2} \left[ {\tilde{\beta }_{1} } \right] - E^{2} \left[ {\tilde{\beta }_{2} } \right]}}} & { - 2E\left[ {\tilde{\gamma }_{1} } \right] - \frac{{4E\left[ {\tilde{\beta }_{1} } \right]E\left[ {\tilde{\beta }_{3} } \right]E\left[ {\tilde{\gamma }_{2} } \right]}}{{4E^{2} \left[ {\tilde{\beta }_{1} } \right] - E^{2} \left[ {\tilde{\beta }_{2} } \right]}}} \\ \end{array} } \right], $$

(73)

$$ d_{1} \left( H \right) = - 2E\left[ {\tilde{\gamma }_{1} } \right] - \frac{{4E\left[ {\tilde{\beta }_{1} } \right]E\left[ {\tilde{\beta }_{3} } \right]E\left[ {\tilde{\gamma }_{2} } \right]}}{{4E^{2} \left[ {\tilde{\beta }_{1} } \right] - E^{2} \left[ {\tilde{\beta }_{2} } \right]}}, $$

(74)

$$ d_{2} \left( H \right) = 4E^{2} \left[ {\tilde{\gamma }_{1} } \right] + \frac{{4E\left[ {\tilde{\beta }_{3} } \right]E\left[ {\tilde{\gamma }_{2} } \right]\left( {4E\left[ {\tilde{\beta }_{1} } \right]E\left[ {\tilde{\gamma }_{1} } \right] + E\left[ {\tilde{\beta }_{3} } \right]E\left[ {\tilde{\gamma }_{2} } \right]} \right)}}{{\left( {4E^{2} \left[ {\tilde{\beta }_{1} } \right] - E^{2} \left[ {\tilde{\beta }_{2} } \right]} \right)}}. $$

(75)

It follows from (74) and (75) that the expected profit function $E\left[ {\tilde{\pi }_{\text{cc}} } \right]$ is jointly concave with respect to $r_{1}$ and $r_{2}$. Using Eq. (70), the optimal pricing decisions of the common collection centers are determined as provided in (42).

Proof of proposition 7

The concavity of the expected profit function $E\left[ {\tilde{\pi }_{{sc_{2} }} } \right]$ with respect to $p_{2}$ is investigated in the proof of proposition 5. Using Eq. (67), the optimal pricing decision of the second supply chain is obtained as provided in (45).

Proof of proposition 8

The expected value of the profit function $\tilde{\pi }_{{{\text{sc}}_{1} }} \left( {p_{1} ,p_{2}^{*} \left( {p_{1} ,r_{2} } \right),r_{1} } \right)$ is expressed as follows:

$$ \begin{aligned} E\left[ {\tilde{\pi }_{{{\text{sc}}_{1} }} } \right] & = \left( {E\left[ {\tilde{d}_{1} } \right] - E\left[ {\tilde{\beta }_{1} } \right]p_{1} + E\left[ {\tilde{\beta }_{2} } \right]p_{2}^{*} + E\left[ {\tilde{\beta }_{3} } \right]r_{1} } \right)p_{1} \\ & \quad - I_{{11}} + E\left[ {\tilde{c}_{1} \tilde{\beta }_{1} } \right]p_{1} - I_{{12}} p_{2}^{*} - I_{{13}} r_{1} \\ & \quad - \left( {k_{1} - s_{1} } \right)\left( {E\left[ {\tilde{e}_{1} } \right] + E\left[ {\tilde{\gamma }_{1} } \right]r_{1} + E\left[ {\tilde{\gamma }_{2} } \right]p_{1} } \right), \\ \end{aligned} $$

(76)

where $I_{11}$, $I_{12}$ and $I_{13}$ are defined in “Appendix B.”

Substituting the optimal pricing decision $p_{2}^{*} \left( {p_{1} ,r_{2} } \right)$ into (76), the first- and second-order derivatives of expected profit function $E\left[ {\tilde{\pi }_{{{\text{sc}}_{1} }} } \right]$ with respect to $p_{1}$ are determined as follows:

$$ \begin{aligned} \frac{{\partial E\left[ {\tilde{\pi }_{{{\text{sc}}_{1} }} } \right]}}{{\partial p_{1} }} & = \left( {2E\left[ {\tilde{\beta }_{1} } \right]\left( {E\left[ {\tilde{d}_{1} } \right] + E\left[ {\tilde{c}_{1} \tilde{\beta }_{1} } \right]} \right.} \right. \\ & \quad \left. { + E\left[ {\tilde{\beta }_{3} } \right]r_{1} - 2E\left[ {\tilde{\beta }_{1} } \right]p_{1} } \right) \\ & \quad + E\left[ {\tilde{\beta }_{2} } \right]\left( {E\left[ {\tilde{d}_{2} } \right] + E\left[ {\tilde{c}_{2} \tilde{\beta }_{1} } \right] + E\left[ {\tilde{\beta }_{3} } \right]r_{2} } \right. \\ & \quad \left. { - I_{{12}} + 2E\left[ {\tilde{\beta }_{2} } \right]p_{1} } \right)/\left( {2E\left[ {\tilde{\beta }_{1} } \right]} \right), \\ \end{aligned} $$

(77)

$$ \frac{{\partial^{2} E\left[ {\tilde{\pi }_{{sc_{1} }} } \right]}}{{\partial^{2} p_{1} }} = - \frac{{2E^{2} \left[ {\tilde{\beta }_{1} } \right] - E^{2} \left[ {\tilde{\beta }_{2} } \right]}}{{E\left[ {\tilde{\beta }_{1} } \right]}}. $$

(78)

It follows from (78) that $E\left[ {\tilde{\pi }_{{{\text{sc}}_{1} }} } \right]$ is concave with respect to $p_{1}$. By setting (77) equal to 0 and solving the resulting equation with respect to $p_{1}$, the optimal pricing decision $p_{1}^{*} \left( {r_{1} ,r_{2} } \right)$ is determined as provided in (46).

Proof of proposition 9

The expected value of the profit function $\tilde{\pi }_{\text{cc}} (r_{1} ,r_{2} ,p_{1}^{*} \left( {r_{1} ,r_{2} } \right),p_{2}^{*} \left( {r_{1} ,r_{2} } \right)$ is expressed as follows:

$$ \begin{aligned} E\left[ {\tilde{\pi }_{{{\text{cc}}}} } \right] & = \sum\limits_{{i = 1}}^{2} {\left( {E\left[ {\tilde{e}_{i} } \right]k_{i} + E\left[ {\tilde{\gamma }_{1} } \right]r_{i} k_{i} + E\left[ {\tilde{\gamma }_{2} } \right]p_{i}^{*} k_{i} } \right.} \\ & \quad \left. { - E\left[ {\tilde{e}_{i} } \right]r_{i} - E\left[ {\tilde{\gamma }_{1} } \right]r_{i}^{2} - E\left[ {\tilde{\gamma }_{2} } \right]p_{i}^{*} r} \right)_{i} . \\ \end{aligned} $$

(79)

Substituting the optimal pricing decisions $p_{1}^{*} \left( {r_{1} ,r_{2} } \right)$ and $p_{2}^{*} \left( {r_{1} ,r_{2} } \right)$ into (79), the first- and second-order derivatives of $E\left[ {\tilde{\pi }_{cc} } \right]$ with respect to $r_{1}$ and $r_{2}$ are determined as follows:

$$ \begin{aligned} \frac{{\partial E\left[ {\tilde{\pi }_{{{\text{cc}}}} } \right]}}{{\partial r_{1} }} & = \left( { - 2E\left[ {\tilde{\beta }_{1} } \right]\left( {E\left[ {\tilde{\gamma }_{2} } \right]\left( {E\left[ {\tilde{d}_{1} } \right] + E\left[ {\tilde{c}_{1} \tilde{\beta }_{1} } \right]} \right.} \right.} \right. \\ & \quad \left. { - E\left[ {\tilde{\gamma }_{2} } \right]\left( {k_{1} - s_{1} } \right) - E\left[ {\tilde{\beta }_{3} } \right]\left( {k_{1} - 2r_{1} } \right)} \right) \\ & \quad \left. { - 2E\left[ {\tilde{\beta }_{1} } \right]\left( {E\left[ {\tilde{\gamma }_{1} } \right]k_{1} - E\left[ {\tilde{e}_{1} } \right] - 2E\left[ {\tilde{\gamma }_{1} } \right]r_{1} } \right)} \right) \\ & \quad - E\left[ {\tilde{\beta }_{2} } \right]\left( {E\left[ {\tilde{\gamma }_{2} } \right]\left( {E\left[ {\tilde{d}_{2} } \right] + E\left[ {\tilde{c}_{2} \tilde{\beta }_{1} } \right]} \right.} \right. \\ & \quad \left. { - E\left[ {\tilde{\gamma }_{2} } \right]\left( {k_{2} - s_{2} } \right) - E\left[ {\tilde{\beta }_{3} } \right]\left( {k_{1} - 2r_{1} } \right) - I_{{12}} } \right) \\ & \left. {\quad + 2E\left[ {\tilde{\beta }_{2} } \right]\left( {E\left[ {\tilde{\gamma }_{1} } \right]k_{1} - E\left[ {\tilde{e}_{1} } \right] - 2E\left[ {\tilde{\gamma }_{1} } \right]r_{1} } \right)} \right) \\ & \quad /\left( {4E^{2} \left[ {\tilde{\beta }_{1} } \right] - 2E^{2} \left[ {\tilde{\beta }_{2} } \right]} \right), \\ \end{aligned} $$

(80)

$$ \begin{aligned} \frac{{\partial E\left[ {\tilde{\pi }_{{{\text{cc}}}} } \right]}}{{\partial r_{2} }} & = \left( { - 4E^{2} \left[ {\tilde{\beta }_{1} } \right]\left( {E\left[ {\tilde{\gamma }_{2} } \right]\left( {E\left[ {\tilde{d}_{2} } \right] + E\left[ {\tilde{c}_{2} \tilde{\beta }_{1} } \right]} \right.} \right.} \right. \\ & \quad \left. { - E\left[ {\tilde{\gamma }_{2} } \right]\left( {k_{2} - s_{2} } \right) - E\left[ {\tilde{\beta }_{3} } \right]\left( {k_{2} - 2r_{2} } \right)} \right) \\ & \quad \left. { - 2E\left[ {\tilde{\beta }_{1} } \right]\left( {E\left[ {\tilde{\gamma }_{1} } \right]k_{2} - E\left[ {\tilde{e}_{2} } \right] - 2E\left[ {\tilde{\gamma }_{1} } \right]r_{2} } \right)} \right) \\ & \quad + E^{2} [\tilde{\beta }_{2} ]\left( {E[\tilde{\gamma }_{2} ]\left( {E[\tilde{d}_{2} ] + E[\tilde{c}_{2} \tilde{\beta }_{1} ]} \right.} \right.E[\tilde{\gamma }_{2} ] \\ & \quad (k_{2} - s_{2} ) - E[\tilde{\beta }_{3} ](k_{2} - 2r_{2} ) + I_{{12}} ) \\ & \quad \left. { - 4E[\tilde{\beta }_{1} ](E[\tilde{\gamma }_{1} ]k_{2} - E[\tilde{e}_{2} ] - 2E[\tilde{\gamma }_{1} ]r_{2} )} \right) \\ & \quad - 2E\left[ {\tilde{\beta }_{1} } \right]E\left[ {\tilde{\beta }_{2} } \right]E\left[ {\tilde{\gamma }_{2} } \right]\left( {E\left[ {\tilde{d}_{1} } \right] + E\left[ {\tilde{c}_{1} \tilde{\beta }_{1} } \right]} \right. \\ & \quad \left. { - E\left[ {\tilde{\gamma }_{2} } \right]\left( {k_{1} - s_{1} } \right) - E\left[ {\tilde{\beta }_{3} } \right]\left( {k_{1} - 2r_{1} } \right)} \right) \\ & \quad /\left( {8E^{3} \left[ {\tilde{\beta }_{1} } \right] - 4E\left[ {\tilde{\beta }_{1} } \right]E^{2} \left[ {\tilde{\beta }_{2} } \right]} \right), \\ \end{aligned} $$

(81)

$$ \frac{{\partial^{2} E\left[ {\tilde{\pi }_{\text{cc}} } \right]}}{{\partial^{2} r_{1} }} = - 2\left[ {\tilde{\gamma }_{1} } \right] - \frac{{2E\left[ {\tilde{\beta }_{1} } \right]E\left[ {\tilde{\beta }_{3} } \right]E\left[ {\tilde{\gamma }_{2} } \right]}}{{2E^{2} \left[ {\tilde{\beta }_{1} } \right] - E^{2} \left[ {\tilde{\beta }_{2} } \right]}}, $$

(82)

$$ \begin{aligned} \frac{{\partial ^{2} E\left[ {\tilde{\pi }_{{cc}} } \right]}}{{\partial ^{2} r_{2} }} & = - 2\left[ {\tilde{\gamma }_{1} } \right] - \frac{{E\left[ {\tilde{\beta }_{3} } \right]E\left[ {\tilde{\gamma }_{2} } \right]}}{{E\left[ {\tilde{\beta }_{1} } \right]}} \\ & \quad \left( {1 + \frac{{E^{2} \left[ {\tilde{\beta }_{2} } \right]}}{{4E^{2} \left[ {\tilde{\beta }_{1} } \right] - 2E^{2} \left[ {\tilde{\beta }_{2} } \right]}}} \right), \\ \end{aligned} $$

(83)

$$ \frac{{\partial^{2} E\left[ {\tilde{\pi }_{cc} } \right]}}{{\partial^{2} r_{1} r_{2} }} = \frac{{\partial^{2} E\left[ {\tilde{\pi }_{cs} } \right]}}{{\partial^{2} r_{2} r_{1} }} = - \frac{{E\left[ {\tilde{\beta }_{2} } \right]E\left[ {\tilde{\beta }_{3} } \right]E\left[ {\tilde{\gamma }_{2} } \right]}}{{2E^{2} \left[ {\tilde{\beta }_{1} } \right] - E^{2} \left[ {\tilde{\beta }_{2} } \right]}}, $$

(84)

where $I_{12}$ is defined in “Appendix B.”

The hessian matrix of $E\left[ {\tilde{\pi }_{\text{cs}} } \right]$ with respect to $r_{1}$ and $r_{2}$ and its first- and second-order determinants are defined as follows:

$$ H = \left[ {\begin{array}{*{20}l} { - 2\left[ {\tilde{\gamma }_{1} } \right] - \frac{{2E\left[ {\tilde{\beta }_{1} } \right]E\left[ {\tilde{\beta }_{3} } \right]E\left[ {\tilde{\gamma }_{2} } \right]}}{{2E^{2} \left[ {\tilde{\beta }_{1} } \right] - E^{2} \left[ {\tilde{\beta }_{2} } \right]}}} \hfill & { - \frac{{E\left[ {\tilde{\beta }_{2} } \right]E\left[ {\tilde{\beta }_{3} } \right]E\left[ {\tilde{\gamma }_{2} } \right]}}{{2E^{2} \left[ {\tilde{\beta }_{1} } \right] - E^{2} \left[ {\tilde{\beta }_{2} } \right]}}} \hfill \\ { - \frac{{E\left[ {\tilde{\beta }_{2} } \right]E\left[ {\tilde{\beta }_{3} } \right]E\left[ {\tilde{\gamma }_{2} } \right]}}{{2E^{2} \left[ {\tilde{\beta }_{1} } \right] - E^{2} \left[ {\tilde{\beta }_{2} } \right]}}} \hfill & { - 2\left[ {\tilde{\gamma }_{1} } \right] - \frac{{E\left[ {\tilde{\beta }_{3} } \right]E\left[ {\tilde{\gamma }_{2} } \right] }}{{E\left[ {\tilde{\beta }_{1} } \right]}}\left( {1 + \frac{{E^{2} \left[ {\tilde{\beta }_{2} } \right]}}{{4E^{2} \left[ {\tilde{\beta }_{1} } \right] - 2E^{2} \left[ {\tilde{\beta }_{2} } \right]}}} \right)} \hfill \\ \end{array} } \right], $$

(85)

$$ d_{1} \left( H \right) = - 2\left[ {\tilde{\gamma }_{1} } \right] - \frac{{2E\left[ {\tilde{\beta }_{1} } \right]E\left[ {\tilde{\beta }_{3} } \right]E\left[ {\tilde{\gamma }_{2} } \right]}}{{2E^{2} \left[ {\tilde{\beta }_{1} } \right] - E^{2} \left[ {\tilde{\beta }_{2} } \right]}}, $$

(86)

$$ d_{2} \left( H \right) = \frac{{4E\left[ {\tilde{\beta }_{1} } \right]E^{2} \left[ {\tilde{\gamma }_{1} } \right]\left( {2E^{2} \left[ {\tilde{\beta }_{1} } \right] - E^{2} \left[ {\tilde{\beta }_{2} } \right]} \right) + E\left[ {\tilde{\beta }_{3} } \right]E\left[ {\tilde{\gamma }_{1} } \right]E\left[ {\tilde{\gamma }_{2} } \right]\left( {8E^{2} \left[ {\tilde{\beta }_{1} } \right] - E^{2} \left[ {\tilde{\beta }_{2} } \right]} \right) + 2E\left[ {\tilde{\beta }_{1} } \right]E^{2} \left[ {\tilde{\beta }_{3} } \right]E^{2} \left[ {\tilde{\gamma }_{2} } \right]}}{{2E^{3} \left[ {\tilde{\beta }_{1} } \right] - E\left[ {\tilde{\beta }_{1} } \right]E^{2} \left[ {\tilde{\beta }_{2} } \right]}}. $$

(87)

The Hessian matrix is negative definite. Therefore, it is concluded that $E\left[ {\tilde{\pi }_{\text{cc}} } \right]$ is jointly concave with respect to $r_{1}$ and $r_{2}$. By setting Eqs. (86) and (87) equal to 0 and simultaneously solving the resulting equations with respect to $r_{1}$ and $r_{2}$, the optimal pricing decisions $r_{1}^{*}$ and $r_{2}^{*}$ are determined as provided in (47) and (48).

Proof of proposition 10

The concavity of the expected profit functions $E\left[ {\tilde{\pi }_{{{\text{sc}}_{i} }} } \right]$ with respect to $p_{i}$ and $E\left[ {\tilde{\pi }_{\text{cc}} } \right]$ with respect to $r_{1}$ and $r_{2}$ is investigated in Proofs of Propositions 1 and 5, respectively. Using Eqs. (54) and (67), the optimal pricing decisions $p_{1}^{*}$, $p_{2}^{*}$, $r_{1}^{*}$ and $r_{2}^{*}$ are determined as provided in (51) and (52).

Appendix B: Notations for models

$$ \begin{aligned} I_{{i1}} & = \frac{1}{2}\int\limits_{0}^{1} {\left( {\lambda \left( {\tilde{c}_{{i\alpha P}}^{ + } \tilde{d}_{{i\alpha O}}^{ + } + \tilde{c}_{{i\alpha O}}^{ + } \tilde{d}_{{i\alpha P}}^{ + } } \right)} \right.} \\ & \quad \left. { + \left( {1 - \lambda } \right)\left( {\tilde{c}_{{i\alpha P}}^{ - } \tilde{d}_{{i\alpha O}}^{ - } + \tilde{c}_{{i\alpha O}}^{ - } \tilde{d}_{{i\alpha P}}^{ - } } \right)} \right){\text{d}}\alpha , \\ \end{aligned} $$

$$ \begin{aligned} I_{{i2}} & = \frac{1}{2}\int\limits_{0}^{1} {\left( {\lambda \left( {\tilde{c}_{{i\alpha P}}^{ + } \tilde{\beta }_{{2\alpha O}}^{ + } + \tilde{c}_{{i\alpha O}}^{ + } \tilde{\beta }_{{2\alpha P}}^{ + } } \right)} \right.} \\ & \quad \left. { + \left( {1 - \lambda } \right)\left( {\tilde{c}_{{i\alpha P}}^{ - } \tilde{\beta }_{{2\alpha O}}^{ - } + \tilde{c}_{{i\alpha O}}^{ - } \tilde{\beta }_{{2\alpha P}}^{ - } } \right)} \right){\text{d}}\alpha , \\ \end{aligned} $$

$$ \begin{aligned} I_{{i3}} & = \frac{1}{2}\int\limits_{0}^{1} {\left( {\lambda \left( {\tilde{c}_{{i\alpha P}}^{ + } \tilde{\beta }_{{3\alpha O}}^{ + } + \tilde{c}_{{i\alpha O}}^{ + } \tilde{\beta }_{{3\alpha P}}^{ + } } \right)} \right.} \\ & \quad \left. { + \left( {1 - \lambda } \right)\left( {\tilde{c}_{{i\alpha P}}^{ - } \tilde{\beta }_{{3\alpha O}}^{ - } + \tilde{c}_{{i\alpha O}}^{ - } \tilde{\beta }_{{3\alpha P}}^{ - } } \right)} \right){\text{d}}\alpha , \\ \end{aligned} $$

where $\tilde{c}_{i\alpha P}^{ + }$, $\tilde{c}_{i\alpha O}^{ + }$, $\tilde{c}_{i\alpha P}^{ - }$, $\tilde{c}_{i\alpha O}^{ - }$, $\tilde{d}_{i\alpha P}^{ + }$, $\tilde{d}_{i\alpha O}^{ + }$, $\tilde{d}_{i\alpha P}^{ - }$, $\tilde{d}_{i\alpha O}^{ - }$, $\tilde{\beta }_{2\alpha P}^{ + }$, $\tilde{\beta }_{2\alpha O}^{ + }$, $\tilde{\beta }_{2\alpha P}^{ - }$, $\tilde{\beta }_{2\alpha O}^{ - }$, $\tilde{\beta }_{3\alpha P}^{ + }$, $\tilde{\beta }_{3\alpha O}^{ + }$, $\tilde{\beta }_{3\alpha P}^{ - }$ and $\tilde{\beta }_{3\alpha O}^{ - }$ are the $\alpha$-pessimistic and $\alpha$-optimistic values corresponding to the intuitionistic fuzzy variables $\tilde{c}_{i}$, $\tilde{d}_{i}$, $\tilde{\beta }_{2}$ and $\tilde{\beta }_{3}$, respectively.