Coordination in Closed-Loop Supply Chain with Price-Dependent Returns

Abstract

This paper proposes two Closed-loop Supply Chain (CLSC) games in which a manufacturer sets some green activity programs efforts and a retailer sets the selling price. Both strategies influence the return rate, which is a state variable. The pricing strategy plays a key role in the identification of the best contract to achieve coordination as well as in achieving environmental objectives. The pricing strategy influences the return rate negatively, as consumers delay the return of their goods when the purchasing (and repurchasing) price is high. We then compare a wholesale price contract (WPC) and a revenue sharing contract (RSC) mechanism as both have interesting pricing policy implications. Our result shows that firms coordinate the CLSC through a (WPC) when the sharing parameter is too low while the negative effect of pricing on returns is too severe. In that case, the low sharing parameter deters the manufacturer to accept any sharing agreements. Further, firms coordinate the CLSC when the sharing parameter is medium independent of the negative impact of pricing on returns. When the sharing parameter is too high the retailer never opts for an RSC. We find that the magnitude of pricing effect on returns determines the contract to be adopted: For certain sharing parameter, firms prefer an RSC when the price effect on return is low and a WPC when this effect is high. In all other cases, firms do not have a consensus on the contract to be adopted and coordination is then not achieved.

Appendix

Proof of Proposition 1

We search for a pair of bounded and continuously differentiable value functions $V_{M}^{\mathcal {W}}(r^{\mathcal {W}})$ and $V_{R}^{\mathcal {W}}(r^{\mathcal {W}})$ for which a unique solution for $r^{ \mathcal {W}}(t)$ does exist, and the HJB equations:

$$\displaystyle \begin{aligned} \begin{array}{rcl} \rho V_{M}^{\mathcal{W}} &\displaystyle =&\displaystyle \left( \alpha \sqrt{r^{\mathcal{W}}}-\beta p^{ \mathcal{W}}\right) \left( \omega ^{\mathcal{W}}+\Delta \sqrt{r^{\mathcal{W}} }\right) -\frac{\mu \left( A^{\mathcal{W}}\right) ^{2}}{2}\\&\displaystyle &\displaystyle +V_{M}^{{}^{\mathcal{ W}}\prime }\left( kA^{\mathcal{W}}-\lambda p^{\mathcal{W}}\sqrt{r^{\mathcal{W} }}-\delta r^{\mathcal{W}}\right) {} \end{array} \end{aligned} $$

(28)

$$\displaystyle \begin{aligned} \begin{array}{rcl} \rho V_{R}^{\mathcal{W}} &\displaystyle =&\displaystyle \left( \alpha \sqrt{r^{\mathcal{W}}}-\beta p^{ \mathcal{W}}\right) \left( p^{\mathcal{W}}-\omega ^{\mathcal{W}}\right) +V_{R}^{{}^{\mathcal{W}}\prime }\left( kA^{\mathcal{W}}-\lambda p^{\mathcal{W}} \sqrt{r^{\mathcal{W}}}-\delta r^{\mathcal{W}}\right)\\{} \end{array} \end{aligned} $$

(29)

are satisfied for any value of $r^{\mathcal {W}}\left ( t\right ) \in (0,1].$ We solve the game á la Stakelberg, where M is the leader. Therefore, we start by solving the R’s optimization problem. The optimization of R’s HJB with respect to the pricing strategy leads to:

$$\displaystyle \begin{aligned} p^{\mathcal{W}}=\frac{\omega ^{\mathcal{W}}\beta -\sqrt{r^{\mathcal{W}}} \left( \lambda V_{R}^{{}^{\mathcal{W}}\prime }-\alpha \right) }{2\beta } \end{aligned} $$

(30)

Substituting Eq. (30) in the M’s HJB gives:

$$\displaystyle \begin{aligned} \begin{array}{rcl} \rho V_{M}^{\mathcal{W}} &\displaystyle =&\displaystyle \left( \alpha \sqrt{r^{\mathcal{W}}}-\beta \frac{ \omega ^{\mathcal{W}}\beta -\sqrt{r^{\mathcal{W}}}\left( \lambda V_{R}^{{}^{ \mathcal{W}}\prime }-\alpha \right) }{2\beta }\right)\\ &\displaystyle &\displaystyle \left( \frac{\omega ^{ \mathcal{W}}\beta -\sqrt{r^{\mathcal{W}}}\left( \lambda V_{R}^{{}^{\mathcal{W} }\prime }-\alpha \right) }{2\beta }+\Delta \sqrt{r^{\mathcal{W}}}\right) - \frac{\mu \left( A^{\mathcal{W}}\right) ^{2}}{2} \notag \\ &\displaystyle &\displaystyle +V_{M}^{{}^{\mathcal{W}}\prime }\left( kA^{\mathcal{W} }-\lambda \frac{\omega ^{\mathcal{W}}\beta -\sqrt{r^{\mathcal{W}}}\left( \lambda V_{R}^{{}^{\mathcal{W}}\prime }-\alpha \right) }{2\beta }\sqrt{r^{ \mathcal{W}}}-\delta r^{\mathcal{W}}\right){} \end{array} \end{aligned} $$

(31)

Maximizing with respect to green efforts, $A^{\mathcal {W}}$, and wholesale price, $\omega ^{\mathcal {W}}$, gives:

$$\displaystyle \begin{aligned} \begin{array}{rcl} A^{\mathcal{W}} &\displaystyle =&\displaystyle \frac{kV_{M}^{{}^{\mathcal{W}}\prime }}{\mu } {} \end{array} \end{aligned} $$

(32)

$$\displaystyle \begin{aligned} \begin{array}{rcl} \omega ^{\mathcal{W}} &\displaystyle =&\displaystyle \frac{\left( \lambda V_{R}^{{}^{\mathcal{W}}\prime }-\left( \Delta \beta +\lambda V_{M}^{{}^{\mathcal{W}}\prime }\right) \right) }{\beta }\sqrt{r^{\mathcal{W}}}{} \end{array} \end{aligned} $$

(33)

Plugging Eq. (33) in Eq. (30), we obtain the optimal price:

$$\displaystyle \begin{aligned} p^{\mathcal{W}}=\frac{\left( \alpha -\Delta \beta -\lambda V_{M}^{{}^{\mathcal{ W}}\prime }\right) }{2\beta }\sqrt{r^{\mathcal{W}}} \end{aligned} $$

(34)

Substituting the optimal strategies inside Eqs. (29) and (31) and simplifying we obtain:

$$\displaystyle \begin{aligned} \begin{array}{rcl} 4\beta \mu \rho V_{M}^{\mathcal{W}} &\displaystyle =&\displaystyle \mu \left( \alpha +\Delta \beta -\lambda V_{M}^{{}^{\mathcal{W}}\prime }\right) \left( \alpha +\Delta \beta +\lambda V_{M}^{{}^{\mathcal{W}}\prime }\right) r^{\mathcal{W}}+2\beta V_{M}^{{}^{\mathcal{W}}\prime 2}k^{2} \notag \\ &\displaystyle &\displaystyle +2\mu V_{M}^{{}^{\mathcal{W}}\prime }\left( \lambda \left( \Delta \beta -\alpha +\lambda V_{M}^{{}^{\mathcal{W}}\prime }\right) -2\beta \delta \right) r^{\mathcal{W}} {} \end{array} \end{aligned} $$

(35)

(36)

To solve the previous pair of equations, we can conjecture linear value functions $V_{M}^{\mathcal {W}}=B_{1}^{\mathcal {W}}r^{\mathcal {W}}+B_{2}^{ \mathcal {W}}$ and $V_{R}^{\mathcal {W}}=L_{1}^{\mathcal {W}}r^{\mathcal {W} }+L_{2}^{\mathcal {W}}.$ Substituting these conjectures and their derivatives inside Eqs. (35) and (36) gives:

$$\displaystyle \begin{aligned} \begin{array}{rcl} 4\beta \mu \rho \left( B_{1}^{\mathcal{W}}r^{\mathcal{W}}{+}B_{2}^{\mathcal{W} }\right) &\displaystyle {=}&\displaystyle \mu \left( \alpha {+}\Delta \beta {-}\lambda B_{1}^{\mathcal{W} }\right) \left( \alpha {+}\Delta \beta {+}\lambda B_{1}^{\mathcal{W}}\right) r^{ \mathcal{W}} \notag \\ &\displaystyle &\displaystyle {+}2\beta B_{1}^{{}^{\mathcal{W}}2}k^{2}{+}2\mu B_{1}^{\mathcal{W}}\left( \lambda \left( \Delta \beta {-}\alpha {+}\lambda B_{1}^{\mathcal{W}}\right){-}2\beta \delta \right) ^{\mathcal{W}}r\\ {} \end{array} \end{aligned} $$

(37)

$$\displaystyle \begin{aligned} \begin{array}{rcl} 4\beta \mu \rho \left( L_{1}^{\mathcal{W}}r^{\mathcal{W}}+L_{2}^{\mathcal{W} }\right) &\displaystyle =&\displaystyle \mu \left( \left( \alpha +\Delta \beta +\lambda B_{1}^{\mathcal{W} }\right) ^{2}-4\alpha \lambda L_{1}^{\mathcal{W}}\right) r^{\mathcal{W} }\\&\displaystyle &\displaystyle +4\beta L_{1}^{\mathcal{W}}\left( k^{2}B_{1}^{\mathcal{W}}-\delta \mu r^{ \mathcal{W}}\right){} \end{array} \end{aligned} $$

(38)

By identification, we obtain the following system of equations:

$$\displaystyle \begin{aligned} \begin{array}{rcl} &\displaystyle -&\displaystyle 4\beta \mu \rho B_{1}^{\mathcal{W}}+\mu \left( \alpha +\Delta \beta -\lambda B_{1}^{\mathcal{W}}\right) \left( \alpha +\Delta \beta +\lambda B_{1}^{\mathcal{W}}\right)\\ &\displaystyle +&\displaystyle 2\mu B_{1}^{\mathcal{W}}\left( \lambda \left( \Delta \beta -\alpha +\lambda B_{1}^{\mathcal{W}}\right) -2\beta \delta \right) =0 {} \end{array} \end{aligned} $$

(39)

$$\displaystyle \begin{aligned} \begin{array}{rcl} &\displaystyle -&\displaystyle 4\beta \mu \rho B_{2}^{\mathcal{W}}+2\beta \left( B_{1}^{{}^{\mathcal{W} }}\right) ^{2}k^{2} =0 {} \end{array} \end{aligned} $$

(40)

$$\displaystyle \begin{aligned} \begin{array}{rcl} &\displaystyle -&\displaystyle 4\beta \mu \rho L_{1}^{\mathcal{W}}+\mu \left( \left( \alpha +\Delta \beta +\lambda B_{1}^{\mathcal{W}}\right) ^{2}-4\alpha \lambda L_{1}^{\mathcal{W} }\right) +4\beta L_{1}^{\mathcal{W}}\delta \mu =0 {} \end{array} \end{aligned} $$

(41)

$$\displaystyle \begin{aligned} \begin{array}{rcl} &\displaystyle -&\displaystyle 4\beta \mu \rho L_{2}^{\mathcal{W}}+4\beta L_{1}^{\mathcal{W}}k^{2}B_{1}^{ \mathcal{W}} =0{} \end{array} \end{aligned} $$

(42)

We can select the negative root of $B_{1}^{\mathcal {W}}$, which is given by

$$\displaystyle \begin{aligned} &B_{1}^{\mathcal{W}}=\\&\quad \frac{\left( \left( \alpha -\Delta \beta \right) \lambda +2\beta \left( \delta +\rho \right) -2\sqrt{\beta \left( \delta +\rho \right) \left( \left( \alpha -\Delta \beta \right) \lambda +\beta \left( \delta +\rho \right) \right) -\Delta \alpha \beta \lambda ^{2}}\right) }{ \lambda ^{2}} \end{aligned} $$

(43)

Then, the remaining parameters are given by:

$$\displaystyle \begin{aligned} \begin{array}{rcl} B_{2}^{\mathcal{W}} &\displaystyle =&\displaystyle \frac{\left( B_{1}^{{}^{\mathcal{W}}}\right) ^{2}k^{2}}{ 2\mu \rho } {} \end{array} \end{aligned} $$

(44)

$$\displaystyle \begin{aligned} \begin{array}{rcl} L_{1}^{\mathcal{W}} &\displaystyle =&\displaystyle \frac{\mu \left( \alpha +\Delta \beta +\lambda B_{1}^{ \mathcal{W}}\right) ^{2}}{4\mu \left( \alpha \lambda -\beta \left( \delta +\rho \right) \right) } {} \end{array} \end{aligned} $$

(45)

$$\displaystyle \begin{aligned} \begin{array}{rcl} L_{2}^{\mathcal{W}} &\displaystyle =&\displaystyle \frac{L_{1}^{\mathcal{W}}B_{1}^{\mathcal{W}}k^{2}}{ \mu \rho }{} \end{array} \end{aligned} $$

(46)

□

Proof of Corollary 1

The results in Corollary 1 follow the following derivatives:

• $\frac {\partial A_{SS}^{\mathcal {R}}}{\partial \phi }=\frac { k}{\mu }\frac {\partial B_{1}^{\mathcal {R}}}{\partial \phi }>0$

• $\frac {\partial r_{SS}^{\mathcal {R}}}{\partial \phi } =\left \{ 2\beta k^{2}\left ( -B_{1}^{\mathcal {R}}+\frac {\partial B_{1}^{ \mathcal {R}}}{\partial \phi }\left ( 1-\phi \right ) \right ) DEN[r_{SS}^{ \mathcal {R}}]+\mu NUM[r_{SS}^{\mathcal {R}}]\right .$ $\left .\left ( \left ( \alpha \lambda +2\beta \delta \right ) +\lambda ^{2}\frac {\partial L_{1}^{\mathcal {R}}}{ \partial \phi }\right ) \right \} /DEN[r_{SS}^{\mathcal {R}}]^{2}>0$

• $\frac {\partial p_{SS}^{\mathcal {R}}}{\partial \phi }=\frac { \left [ \left ( -\alpha -\frac {\partial L_{1}^{\mathcal {R}}}{\partial \phi } \lambda \right ) \sqrt {r}+\left ( \alpha \left ( 1-\phi \right ) -\lambda L_{1}^{ \mathcal {R}}\right ) \sqrt {\frac {\partial r_{SS}^{\mathcal {R}}}{\partial \phi }}\right ] DEN[p_{SS}^{\mathcal {R}}]+2\beta NUM[p_{SS}^{\mathcal {R}}]}{\left [ 2\beta \left ( 1-\phi \right ) \right ] ^{2}}>0$;

• $\frac {\partial D_{SS}^{\mathcal {R}}}{\partial \phi } =\alpha \sqrt {\frac {\partial r_{SS}^{\mathcal {R}}}{\partial \phi }}-\beta \frac {\partial p_{SS}^{\mathcal {R}}}{\partial \phi }>0;$

• $\frac {\partial V_{M_{SS}}^{\mathcal {R}}}{\partial \phi }= \frac {\partial B_{1}^{\mathcal {R}}}{\partial \phi }r_{SS}^{\mathcal {R}}+ \frac {\partial r_{SS}^{\mathcal {R}}}{\partial \phi }B_{1}^{\mathcal {R}}+ \frac {\partial B_{2}^{\mathcal {R}}}{\partial \phi }>0$

• $\frac {\partial V_{R_{SS}}^{\mathcal {R}}}{\partial \phi }= \frac {\partial L_{1}^{\mathcal {R}}}{\partial \phi }r_{SS}^{\mathcal {R}}+ \frac {\partial r_{SS}^{\mathcal {R}}}{\partial \phi }L_{1}^{\mathcal {R}}+ \frac {\partial L_{2}^{\mathcal {R}}}{\partial \phi }\{ \begin {array}{l} \geqslant 0, \forall \phi \in (0,\overline {\phi }] \\ <0, \text{otherwise} \end {array}$.

□

Proof of Proposition 2

We search for a pair of bounded and continuously differentiable value functions $V_{M}^{\mathcal {R}}(r^{\mathcal {R}})$ and $V_{R}^{\mathcal {R}}(r^{\mathcal {R}})$ for which a unique solution for $r^{ \mathcal {R}}(t)$ does exist, and the HJB equations:

$$\displaystyle \begin{aligned} \begin{array}{rcl} \rho V_{M}^{\mathcal{R}} &\displaystyle =&\displaystyle \left( \alpha \sqrt{r^{\mathcal{R}}}-\beta p^{ \mathcal{R}}\right) \left( p^{\mathcal{R}}\phi +\Delta \sqrt{r^{\mathcal{R}}} \right)\\ &\displaystyle &\displaystyle -\frac{\mu \left( A^{\mathcal{R}}\right) ^{2}}{2}+V_{M}^{{}^{\mathcal{R} }\prime }\left( kA^{\mathcal{R}}-\lambda p^{\mathcal{R}}\sqrt{r^{\mathcal{R} }}-\delta r^{\mathcal{R}}\right) {} \end{array} \end{aligned} $$

(47)

$$\displaystyle \begin{aligned} \begin{array}{rcl} \rho V_{R}^{\mathcal{R}} &\displaystyle =&\displaystyle \left( \alpha \sqrt{r^{\mathcal{R}}}-\beta p^{ \mathcal{R}}\right) p^{\mathcal{R}}\left( 1-\phi \right) +V_{R}^{{}^{\mathcal{R} }\prime }\left( kA^{\mathcal{R}}-\lambda p^{\mathcal{R}}\sqrt{r^{\mathcal{R} }}-\delta r^{\mathcal{R}}\right)\quad {} \end{array} \end{aligned} $$

(48)

are always satisfied for any value of $r^{\mathcal {R}}\left ( t\right ) \in (0,1]$. We solve the game á la Stakelberg, where M is the leader. Nevertheless, the pricing and green efforts strategies are independent; therefore, solving the Stakelberg game corresponds to solving the Nash game. In fact, the firms reaction functions are given by

$$\displaystyle \begin{aligned} \begin{array}{rcl} \frac{\partial V_{M}^{\mathcal{R}}}{\partial A^{\mathcal{R}}} &\displaystyle =&\displaystyle kV_{M}^{{}^{ \mathcal{R}}\prime }-A^{\mathcal{R}}\mu {} \end{array} \end{aligned} $$

(49)

$$\displaystyle \begin{aligned} \begin{array}{rcl} \frac{\partial V_{R}^{\mathcal{R}}}{\partial p^{\mathcal{R}}} &\displaystyle =&\displaystyle p^{\mathcal{ R}}\beta \phi -p^{\mathcal{R}}\beta -\sqrt{r^{\mathcal{R}}}\lambda V_{R}^{{}^{ \mathcal{R}}\prime }+\left( 1-\phi \right) \left( \sqrt{r^{\mathcal{R}}} \alpha -p^{\mathcal{R}}\beta \right){} \end{array} \end{aligned} $$

(50)

Therefore, the optimal strategies result as follows:

$$\displaystyle \begin{aligned} \begin{array}{rcl} A^{\mathcal{R}} &\displaystyle =&\displaystyle \frac{kV_{M}^{{}^{\mathcal{R}}\prime }}{\mu } {} \end{array} \end{aligned} $$

(51)

$$\displaystyle \begin{aligned} \begin{array}{rcl} p^{\mathcal{R}} &\displaystyle =&\displaystyle \frac{\alpha \left( 1-\phi \right) -\lambda V_{R}^{{}^{ \mathcal{R}}\prime }}{2\beta \left( 1-\phi \right) }\sqrt{r^{\mathcal{R}}}{} \end{array} \end{aligned} $$

(52)

Substituting the optimal strategies inside the firms’ HJBs gives:

$$\displaystyle \begin{aligned} \begin{array}{rcl} 4\mu \beta \left( 1-\phi \right) ^{2}\rho V_{M}^{\mathcal{R}} &\displaystyle =&\displaystyle \mu \left( \left( 1-\phi \right) \left( 2\Delta \beta +\alpha \phi \right) -\lambda \phi V_{R}^{{}^{\mathcal{R}}\prime }\right) \allowbreak \left( \alpha \left( 1-\phi \right) +\lambda V_{R}^{{}^{\mathcal{R}}\prime }\right) r^{\mathcal{R} }\\&\displaystyle &\displaystyle +2\mu \left( 1-\phi \right) V_{M}^{{}^{\mathcal{R}}\prime }\left( -\lambda \left( \alpha \left( 1-\phi \right) -\lambda V_{R}^{{}^{ \mathcal{R}}\prime }\right) -2\beta \delta \left( 1-\phi \right) \right) r^{ \mathcal{R}} \notag \\ &\displaystyle &\displaystyle +2\beta \left( 1-\phi \right) ^{2}k^{2}V_{M}^{{}^{\mathcal{R} }\prime 2} {} \end{array} \end{aligned} $$

(53)

$$\displaystyle \begin{aligned} \begin{array}{rcl} 4\left( 1-\phi \right) \beta \mu \rho V_{R}^{\mathcal{R}} &\displaystyle =&\displaystyle \mu \left( \alpha \phi -\alpha +\lambda V_{R}^{{}^{\mathcal{R}}\prime }\right) ^{2}r^{ \mathcal{R}}\allowbreak +4\left( 1-\phi \right) \beta V_{R}^{{}^{\mathcal{R} }\prime }\left( k^{2}V_{M}^{{}^{\mathcal{R}}\prime }-\delta \mu r^{\mathcal{R} }\right){} \notag \\ &\displaystyle &\displaystyle \end{array} \end{aligned} $$

(54)

To solve the previous pair of equations, we can conjecture linear value functions $V_{M}^{\mathcal {R}}=B_{1}^{\mathcal {R}}r^{\mathcal {R}}+B_{2}^{ \mathcal {R}}$ and $V_{R}^{\mathcal {R}}=L_{1}^{\mathcal {R}}r^{\mathcal {R} }+L_{2}^{\mathcal {R}}.$ Substituting these conjectures and their derivatives inside Eqs. (53) and (54) gives:

(55)

$$\displaystyle \begin{aligned} \begin{array}{rcl} &\displaystyle &\displaystyle \left( -\lambda \left( \alpha \left( 1-\phi \right) -\lambda L_{1}^{ \mathcal{R}}\right) -2\beta \delta \left( 1-\phi \right) \right) r\\&\displaystyle &\displaystyle +2\beta \left( 1-\phi \right) ^{2}k^{2}\left( B_{1}^{{}^{\mathcal{R}}}\right) ^{2} {} \end{array} \end{aligned} $$

(56)

$$\displaystyle \begin{aligned} \begin{array}{rcl} 4\left( 1-\phi \right) \beta \mu \rho \left( L_{1}^{\mathcal{R}}r^{\mathcal{R} }+L_{2}^{\mathcal{R}}\right) &\displaystyle =&\displaystyle \mu \left( \alpha \phi -\alpha +\lambda L_{1}^{\mathcal{R}}\right) ^{2}r^{\mathcal{R}}\\&\displaystyle &\displaystyle +4\left( 1-\phi \right) \beta L_{1}^{\mathcal{R}}\left( k^{2}B_{1}^{\mathcal{R}}-\delta \mu r^{\mathcal{R}}\right){} \end{array} \end{aligned} $$

(57)

By identification, the model parameters are:

$$\displaystyle \begin{aligned} \begin{array}{rcl} &\displaystyle &\displaystyle -4\mu \beta \left( 1-\phi \right) ^{2}\rho B_{1}^{\mathcal{R}}+\mu \left( \left( 1-\phi \right) \left( 2\Delta \beta +\alpha \phi \right) -\lambda \phi L_{1}^{\mathcal{R}}\right) \allowbreak \left( \alpha \left( 1-\phi \right) +\lambda L_{1}^{\mathcal{R}}\right)\\ &\displaystyle &\displaystyle \quad +2\mu \left( 1-\phi \right) B_{1}^{\mathcal{R}}\left( -\lambda \left( \alpha \left( 1-\phi \right) -\lambda B_{1}^{\mathcal{R}}\right) -2\beta \delta \left( 1-\phi \right) \right) =0 {} \end{array} \end{aligned} $$

(58)

$$\displaystyle \begin{aligned} \begin{array}{rcl} &\displaystyle &\displaystyle \quad -4\mu \beta \left( 1-\phi \right) ^{2}\rho B_{2}^{\mathcal{R}}+2\beta \left( 1-\phi \right) ^{2}k^{2}\left( B_{1}^{\mathcal{R}}\right) ^{2} =0 {} \end{array} \end{aligned} $$

(59)

$$\displaystyle \begin{aligned} \begin{array}{rcl} &\displaystyle &\displaystyle \quad -4\left( 1-\phi \right) \beta \mu \rho L_{1}^{\mathcal{R}}+\mu \left( \alpha \phi -\alpha +\lambda L_{1}^{\mathcal{R}}\right) ^{2}-4\delta \mu \left( 1-\phi \right) \beta L_{1}^{\mathcal{R}} =0 {} \end{array} \end{aligned} $$

(60)

$$\displaystyle \begin{aligned} \begin{array}{rcl} &\displaystyle &\displaystyle \quad -4\left( 1-\phi \right) \beta \mu \rho B_{2}^{\mathcal{R}}+4\left( 1-\phi \right) \beta L_{1}^{\mathcal{R}}k^{2}B_{1}^{\mathcal{R}} =0{} \end{array} \end{aligned} $$

(61)

We can see that there exists one solution only for $B_{1}^{\mathcal {R}}$ while we take the negative root for $R_{1}^{\mathcal {R}}$. The solution is given as follows:

$$\displaystyle \begin{aligned} \begin{array}{rcl} B_{1}^{\mathcal{R}} &\displaystyle =&\displaystyle \frac{\mu \left( \lambda L_{1}^{\mathcal{R}}+\alpha \left( 1-\phi \right) \right) \left( \left( 2\Delta \beta +\alpha \phi \right) \left( 1-\phi \right) -\lambda \phi L_{1}^{\mathcal{R}}\right) }{ 2\mu \left( 1-\phi \right) \left( \left( \lambda \alpha +2\beta \left( \delta +\rho \right) \right) \left( 1-\phi \right) -\lambda ^{2}L_{1}^{ \mathcal{R}}\right) \allowbreak } {} \end{array} \end{aligned} $$

(62)

$$\displaystyle \begin{aligned} \begin{array}{rcl} B_{2}^{\mathcal{R}} &\displaystyle =&\displaystyle \frac{k^{2}\left( B_{1}^{\mathcal{R}}\right) ^{2}}{ 2\mu \rho } {} \end{array} \end{aligned} $$

(63)

$$\displaystyle \begin{aligned} \begin{array}{rcl} L_{1}^{\mathcal{R}} &\displaystyle =&\displaystyle \frac{\left( \alpha \lambda +2\beta \left( \delta +\rho \right) \right) \left( 1-\phi \right) -2\sqrt{\beta \left( 1-\phi \right) ^{2}\left( \delta +\rho \right) \left( \alpha \lambda +\beta \left( \delta +\rho \right) \right) }}{\lambda ^{2}} {} \end{array} \end{aligned} $$

(64)

$$\displaystyle \begin{aligned} \begin{array}{rcl} L_{2}^{\mathcal{R}} &\displaystyle =&\displaystyle \frac{k^{2}B_{1}^{\mathcal{R}}L_{1}^{\mathcal{R}}}{ \mu \rho }{} \end{array} \end{aligned} $$

(65)

□

1.1 Pareto Analysis on Different Sets

Hereby, we carry out the Pareto analysis on two different parameter sets to demonstrate the robustness of our findings. Specifically, we use the following two parameter sets:

• high parameter values, by fixing the parameters as follows: α = 3, β = 0.8, k = 0.15, Δ = 0.3, δ = 0.3, ρ = 0.15 and μ = 1.5;

• low parameter values, by fixing the parameters as follows: α = 1, β = 0.4, k = 0.05, Δ = 0.1, δ = 0.1, ρ = 0.05 and μ = 0.5.

As we display in Figs. 11 and 12, the findings that we obtain in Fig. 10 are confirmed when taking different parameter sets.

Fig. 11

Pareto improving region with high parameter values

Fig. 12

Pareto improving region with low parameter values