Abstract
We study the optimal decisions of a supplier in terms of channel structure selection and marketing effort strategy when facing a competitor offering substitutable products. By employing game theoretic models for different retail competition scenarios, we show that equilibrium channel structures are primarily determined by the intensity of retail competition along with the marketing effort level. Channel structure selection and marketing effort strategy are interdependent and interactive. Our results show that the supplier has an incentive to make marketing effort and sell products directly to consumers in most cases. When the retail competition becomes fiercer, the supplier can benefit from selecting a distribution strategy based on customers’ sensitivity to the supplier’s marketing effort. In contrast, the supplier selling products through the downstream retailer will not make marketing effort under certain market conditions (e.g., low competition and high customer sensitivity to the supplier’s effort performance), while the competitor adopts the direct-sales strategy. Generally, the supplier will prefer making marketing effort relative to undertaking product distribution. Finally, we also examine the strategic effects of channel structure selection and marketing effort decisions on the competitor’s profit in the retail competition market.
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Acknowledgements
This work was supported by National Natural Science Foundation of China [Grant Number 71971113].
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All authors contributed to the study conception and design. Material preparation and model analysis were performed by [K.C.]. The first draft of the manuscript was written by [M.L.], and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.
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Appendices
Appendix A Symbols and functions
A.1. The following symbols are used to simplify the results, where \(i=1,2\) and \(i\ne j\).
A.2. The solutions \({\beta _i}\) and \({\gamma _j}\) presented in all propositions and corollaries are determined by \({Z_i}({\beta _i}\mid {{\gamma _j}}) = 0\), where the functions \({Z_i}({\beta _i}\mid {{\gamma _j}} )\) are given as
Appendix B Proofs
Proof of Theorem 1
For the four channel structures under Model I, we give the proof as follows.
(i) Channel structure CC
The demands of \(S_{1}\) and \(S_{2}\) are \({q_1} = a - {p_1} + \gamma {p_2}\) and \({q_2} = a - {p_2} + \gamma {p_1} + \beta \sqrt{{e_2}} \), respectively. The profit functions of \(S_{1}\) and \(S_{2}\) can be presented by \(\Pi _{S1}^{CC} = ({p_1} - c){q_1}\) and \(\Pi _{S2}^{CC} = ({p_2} - c){q_2} - {e_2}\), respectively.
The optimization problem of chains i is \({\mathop {\max }\limits _{{p_i}}}\ \Pi _{Si}^{CC}({p_i},{p_j})\). Because \({{\partial \Pi _{S1}^{CC}} /{\partial {p_1}}} = a - 2{p_1} + \gamma {p_2} + c\) and \({{{{\partial ^2}\prod _{S1}^{CC}} / {\partial {p_1}}}^2} = - 2 < 0\) , \(\Pi _{S1}^{CC}\) is concave over \({p_1}\). Similarly, we can show that \(\Pi _{S1}^{CC}\) is concave over \({p_2}\). From the first-order conditions, we have
Because \({{{{\partial ^2}\Pi _{S2}^{CC}} / {\partial {e_2}}}^2} < 0\), \(\Pi _{S2}^{CC}\) is concave over \({e_2}\) and we have
From Eqs. (B1) and (B2), the equilibrium solutions of decision and profit are given in Table 2.
(ii) Channel structure CD
The demand of \(S_{1}\) is \({q_1} = a - {p_1} + \gamma {p_2}\). The profit function of \(S_{1}\) is \(\Pi _{S1}^{CD} = ({p_1} - c){q_1}\). The optimization problem is \({\mathop {\max }\limits _{{p_1}}} \ \Pi _{S1}^{CD}({p_1},{p_2})\). Similarly to channel structure CC, we can prove that the function \(\Pi _{S1}^{CD}\) is concave over the price \({p_1}\). We have
The demand of \(S_{2}\) is \({q_2} = a - {p_2} + \gamma {p_1} + \beta \sqrt{{e_2}} \). The profit function of \(R_{2}\) is \(\Pi _{R2}^{CD} = ({p_2} - {w_2}){q_2}\). Given the unit wholesale price \({w_2}\) offered by its upstream \(S_{2}\), \(R_{2}\)’s optimal problem is written as \({\mathop {\max }\limits _{{p_2}}}\ \Pi _{R2}^{CD}({p_2}\mid {{w_2},{p_1}})\). Obviously, \({{\partial \Pi _{R2}^{CD}} / \partial }{p_2} = a - 2{p_2} + \gamma {p_1} + \beta \sqrt{{e_2}} + {w_2}\) and \({{{\partial ^2}\Pi _{R2}^{CD}} / \partial }{p_2}^2 = - 2 < 0\); therefore, we have \(\Pi _{R2}^{CD}\) is concave over \(p_{2}\). We have
From Eqs. (B3) and (B4), we can derive
The profit function of \(S_{2}\) is \(\Pi _{S2}^{CD} = ({w_2} - c){q_2} - {e_2}\). According to Eqs. (B5) and (B6), the optimization problem of \(S_{2}\) is \({\mathop {\max }\limits _{w_2}} \ \Pi _{S2}^{CD}({w_2},{p_2},{p_1})\).
From \({{\partial \Pi _{S2}^{CD}} / \partial }{w_2} = [(2 + \gamma )a + 2\beta \sqrt{{e_2}} + (2{\gamma ^2} - 4){w_2}{{ + (\gamma - {\gamma ^2} + 2)c]} / {(4 - {\gamma ^2})}}\) and \({{{\partial ^2}\Pi _{S2}^{CD}} / \partial }{w_2}^2 = 2{\gamma ^2} - 4 < 0\), the function \(\Pi _{S2}^{CD}\) is concave over \(w_{2}\). Then, from the first-order condition, we have
Since \({{{{\partial ^2}\prod _{S2}^{CD}}/ {\partial {e_2}}}^2} < 0\), the optimal marketing effort level of \(S_{2}\) should satisfy the first-order condition for the given prices, which is written as follows.
From Eqs. (B7) and (B8), we can derive the equilibrium solutions as shown in Table 2.
(iii) Channel structure DC
The demand of \(S_{1}\) is \({q_1} = a - {p_1} + \gamma {p_2}\). The profit function of \(R_{1}\) is \(\Pi _{R1}^{DC} = ({p_1} - {w_1}){q_1}\). Given the unit wholesale price determined by the upstream \(S_{1}\), the optimal problem of \(R_{1}\) is \({\mathop {\max }\limits _{{p_1}}}\ \Pi _{R1}^{DC}( {{p_1}} \mid {w_1},{p_2})\). Since \({{\partial \prod _{R1}^{DC}} / \partial }{p_1} = a - 2{p_1} + \gamma {p_2} + {w_1}\) and \({{{\partial ^2}\Pi _{R1}^{DC}} / \partial }{p_1}^2 = - 2 < 0\), we have that the function \(\Pi _{R1}^{DC}\) is concave over \(p_{1}\). From the first-order condition of \({{\partial \Pi _{R1}^{DC}} / {\partial {p_1}}}\), we have
The demand of \(S_{2}\) is \({q_2} = a - {p_2} + \gamma {p_1} + \beta \sqrt{{e_2}} \). The profit of \(S_{2}\) is written as \({\mathop {\max }\limits _{{p_2}}} \ \Pi _{S2}^{DC} = ({p_2} - c){q_2}\). Similarly, we can show that \(\Pi _{S2}^{DC}\) is concave over \(p_{2}\). We have
From Eqs. (B9) and (B10), we can derive the following results.
According to Eqs. (B11) and (B12), the optimal problem of \(S_{1}\) is written as \({\mathop {\max }\limits _{{w_1}}} \ \Pi _{S1}^{DC}({w_1},{p_1},{p_2})\). Since \(\displaystyle \frac{{\partial \Pi _{S1}^{DC}}}{{\partial {w_1}}}=\) \( \displaystyle \frac{{(2 + \gamma )a + \gamma \beta \sqrt{{e_2}} + 2({\gamma ^2} - 2){w_1} + (\gamma - {\gamma ^2} + 2)c}}{{4 - {\gamma ^2}}}\) and \(\displaystyle \frac{{{\partial ^2}\Pi _{S1}^{DC}}}{{\partial {w_1}^2}} = 2({\gamma ^2} - 2) < 0\), the function \(\Pi _{S1}^{DC}\) is concave over \(w_{1}\). From the first-order condition, we have
Similarly, the optimal problem of \(S_{2}\) is \({\mathop {\max }\limits _{e_2}} \ \Pi _{S2}^{DC}({e_2},{p_1},{p_2})\). Since \({{{{\partial ^2}\Pi _{S2}^{DC}} / {\partial {e_2}}}^2} < 0\), the optimal marketing effort level of \(S_{2}\) should satisfy the first-order condition \({{\partial \Pi _{S2}^{DC}} / {\partial {e_2}}} = 0\) for the given prices and we have
According to Eqs. (B12), (B13) and (B14), we can derive the equilibrium solutions as shown in Table 2.
(iv) Channel structure DD
The demand of \(S_{1}\) is \({q_1} = a - {p_1} + \gamma {p_2}\) and that of \(S_{2}\) is \({q_2} = a - {p_2} + \gamma {p_1} + \beta \sqrt{{e_2}} \). The profit functions of \(S_{1}\) and \(R_{1}\) are \(\Pi _{S1}^{DD} = ({p_1} - {w_1}){q_1}\) and \(\Pi _{R1}^{DD} = ({w_1} - c){q_1}\), respectively. Similarly, it is easy to show that \(\Pi _{S1}^{DD}\) is concave over \(p_{1}\) and \(\Pi _{R1}^{DD}\) is concave over \(w_{1}\). Given \(w_{1}\) offered by the upstream \(S_{1}\), \(R_{1}\)’s optimal problem is \({\mathop {\max } \limits _{{p_1}}} \ \Pi _{R1}^{DD}({p_1},{w_1},{p_2})\). We have
For supply chain 2, the profit functions of \(S_{2}\) and \(R_{2}\) are \(\Pi _{S2}^{DD} = ({p_2} - {w_2}){q_2} - {e_2}\) and \(\Pi _{R2}^{DD} = ({w_2} - c){q_2}\), respectively. Similarly, it is easy to show \(\Pi _{S2}^{DD}\) and \(\Pi _{R2}^{DD}\) are concave over \(p_{2}\) and \(w_{2}\), respectively. Given the unit wholesale price (\(w_{2}\)) determined by the upstream \(S_{2}\), the optimal problem of \(R_{2}\) is \({\mathop {\max } \limits _{{p_2}}} \ \Pi _{R2}^{DD}({p_2},{w_2},{p_1})\). We have
From Eqs. (B15) and (B16), we can obtain the best responses as follows.
Based on Eqs. (B17) and (B18), the profit of \(S_{i}\) is written as \({\mathop {\max } \limits _{w_i}} \ \Pi _{Si}^{DD}({w_i}\mid {{p_i},{p_j}} )\), where \(i,j = 1,2\) and \(i \ne j\). We have \({{\partial \Pi _{S1}^{DD}} / \partial }{w_1} = {{[a - {p_1} + \gamma {p_2} + ({\gamma ^2} - 2)({w_1} - c)]} }{/ {(4 - {\gamma ^2})}} = 0\) and \({{\partial \Pi _{S2}^{DD}} / \partial }{w_2} = [a - {p_2} + \gamma {p_1} + \beta \sqrt{{e_2}} {{ + ({\gamma ^2} - 2)({w_2} - c)]} / {(4 - {\gamma ^2})}} = 0\). Therefore, we obtain
Similar to (ii), we decide the optimal marketing effort level of \(S_{2}\) according to the first-order condition (i.e., \({{\partial \Pi _{S2}^{DD}} / {\partial {e_2}}} = {{({w_2} - c)\beta } / {(2\sqrt{{e_2}} }}) - 1 = 0\)) and have
From Eqs. (B19–B21), we can derive the equilibrium solutions as shown in Table 2. \(\square \)
Proof of Corollary 1
(i) From Theorem 1, we have
Here, we have \(2({\gamma ^2} + {\beta ^2} - 4) < 0\), and \(({\beta ^2} - 4 - 2\gamma )a + ({\beta ^2} - 4 - 2\gamma + \gamma {\beta ^2})c + 2[(2 + \gamma )a + (2 + \gamma - {\beta ^2})c] = {\beta ^2}[a + (\gamma - 1)c]\). Meanwhile, since \(\beta ,\gamma \in [0,1]\), we obtain \({\beta ^2}[a + (\gamma - 1)c] \ge 0\) and \({\beta ^2}[a + (\gamma - 1)c] = 0\) if and only if \(\beta = 0\). Therefore, we have \(p_1^{*CC} \le p_2^{*CC}\).
Similarly, from Theorem 1, we have \(q_1^{*CC} - q_2^{*CC} = {\beta ^2}[a + (\gamma - 1)c]/[2({\gamma ^2} + {\beta ^2} - 4)]\). It is easy to find \(q_1^{*CC} - q_2^{*CC} \le 0\). Obviously, \(q_1^{*CC} - q_2^{*CC} = 0\) if and only if \(\beta = 0\).
Meanwhile, from Theorem 1, we have
Because \(\beta ,\gamma \in [0,1]\), we have \(\Pi _{S2}^{*CC} \ge \Pi _{S1}^{*CC}\). Obviously, \(\Pi _{S2}^{*CC} = \Pi _{S1}^{*CC}\) if and only if \(\beta = 0\).
From Theorem 1, we have
Because \(\beta ,\gamma \in [0,1]\), we have \(p_1^{*DD} \le p_2^{*DD}\). Obviously, \(p_1^{*DD} = p_2^{*DD}\) if and only if \(\beta = 0\).
Similarly, from Theorem 1, we have
Because \(\beta ,\gamma \in [0,1]\) and \(ST > 0\), we have \(q_1^{*DD} \le q_2^{*DD}\). Obviously, \(q_1^{*DD} = q_2^{*DD}\) if and only if \(\beta = 0\).
Moreover, from Theorem 1, we have
Because \(\beta ,\gamma \in [0,1]\) and \(ST > 0\), we have \(w_1^{*DD} \le w_2^{*DD}\). Obviously, \(w_1^{*DD} = w_2^{*DD}\) if and only if \(\beta = 0\).
Meanwhile, we can prove \(\Pi _{S1}^{*DD} \le \Pi _{S2}^{*DD}\) by the similar method.
(ii) From Theorem 1, we have
Because \(\beta ,\gamma \in [0,1]\), we have \(p_1^{*CD} \le p_2^{*CD}\). Obviously, \(p_1^{*CD} = p_2^{*CD}\) if and only if \(\beta = 0\).
Meanwhile, from Theorem 1, we have
Because \(\beta ,\gamma \in [0,1]\) and \(A > 0\), then we have \(q_1^{*CD} \ge q_2^{*CD}\).
From Theorem 1, we have
It is easy to find that \(p_1^{*DC} \ge p_2^{*DC}\) is satisfied if and only if \(w_1^{*DC} - c \ge \beta \sqrt{e_2^{*DC}} \) is satisfied. Then, we have \(w_1^{*DC} - c - \beta \sqrt{e_2^{*DC}} = {{[A + (\gamma - 4 + 2{\gamma ^2})\beta \sqrt{e_2^{*DC}} ]} /{(4 - 2{\gamma ^2})}}\).
Since \(4 - 2{\gamma ^2} > 0\), we find \(w_1^{*DC} - c \ge \beta \sqrt{e_2^{*DC}} \) is equivalent to \(A + (\gamma - 4 + 2{\gamma ^2})\beta \sqrt{e_2^{*DC}} \ge 0\).
Then, we have \(A + (\gamma - 4 + 2{\gamma ^2})\beta \sqrt{e_2^{*DC}} = [4(2 - {\gamma ^2})(4 - {\gamma ^2}) + (20{\gamma ^2} - 24 - 4{\gamma ^4}){{{\beta ^2}]A} /M}\).
Because \(\beta ,\gamma \in [0,1]\), we have \(p_1^{*DC} \ge p_2^{*DC}\).
Similarly, from Theorem 1, we can also show that \(q_1^{*DC} \le q_2^{*DC}\).
Meanwhile, we can prove \(\Pi _{S1}^{*CD} > \Pi _{S2}^{*CD}\) and \(\Pi _{S1}^{*DC} < \Pi _{S2}^{*DC}\) by the similar method. \(\square \)
Proof of Proposition 1
(i) From Theorem 1, we have
It is easy to find \({{{A^2}} /{\{ 4(4 - {\gamma ^2}){{({\gamma ^2} + {\beta ^2} - 4)}^2}}}{{{{[4(2 - {\gamma ^2}) - 2{\beta ^2}]}^2}\} }} > 0\). We assume
Then, there is \(\Pi _{S2}^{*CC} - \Pi _{S2}^{*CD} \ge 0\) satisfied, if and only if the condition \({f_1}(\beta ,\gamma ) \ge 0\) is established. Here, we can regard \({f_1}(\beta ,\gamma )\) as a quadratic function of \({\beta ^2}\). Based on the properties of quadratic functions, we calculate
Finally, we can obtain \(\Pi _{S2}^{*CC} \ge \Pi _{S2}^{*CD}\).
Similarly, we can prove \(\Pi _{S1}^{*CC} \ge \Pi _{S1}^{*DC}\) and omit the details here.
(ii) From Theorem 1, we have
Let \({Z_1}({\beta _1}\mid \gamma ) = 4{(4 - {\gamma ^2})^{1/2}}{(2 - {\gamma ^2})^{3/2}} - 2(4{\gamma ^4} - 17{\gamma ^2} + 16) + [8 - 3{\gamma ^2} - 2{(4 - {\gamma ^2})^{1/2}}{(2 - {\gamma ^2})^{1/2}}]\beta _1^2\).
Then, we find that \(\Pi _{S1}^{*CD} - \Pi _{S1}^{*DD} \ge 0\) is equivalent to the condition of \({Z_1}({\beta _1}\mid \gamma ) \ge 0\).
-
(i)
When \({\beta _1} = 0\), we have \({{\partial {Z_1}(0\mid \gamma )}/ {\partial \gamma }} > 0\), \({Z_1}(0\mid 0) < 0\) and \({Z_1}(0\mid 1) > 0\);
-
(ii)
When \({\beta _1} = 1\), we have \({\partial {Z_1}(1\mid \gamma )} / {\partial \gamma } > 0\), \({Z_1}(1\mid 0) < 0\) and \({Z_1}(1\mid 1) > 0\). Finally, we can obtain
-
if \((\gamma ,\beta ) \in \{ 0 \le \gamma \le {\gamma _1}, 0 \le \beta \le 1\}\)
\(\cup \{ {\gamma _1} \le \gamma \le {\gamma _2}, 0 \le \beta \le {\beta _1}\} \), \(\Pi _{S1}^{*DD} \le \Pi _{S1}^{*CD}\);
-
if \((\gamma ,\beta ) \in \{ {\gamma _1} \le \gamma \le {\gamma _2}, {\beta _1} < \beta \le 1\} \)
\(\cup \{ {\gamma _2} < \gamma \le 1, 0 \le \beta \le 1\} \), \(\Pi _{S1}^{*DD} > \Pi _{S1}^{*CD}\), where the thresholds \(\gamma _{1}\) (\(\approx 0.8062\)) and \(\gamma _{2}\) (\(\approx 0.9309\)) are uniquely determined by the equations as \({Z_1}(0\mid {\gamma _1}) = 0\) and \({Z_1}(1\mid {\gamma _2}) = 0\).
-
-
(iii)
Similar to the proof of part (ii), we can obtain the results in part (iii).
\(\square \)
Proof of Corollary 2
(i) From Theorem 1, we have \({{\partial \Pi _{S1}^{*CC}} /{\partial {\beta ^2}}} = {{A\gamma } / {[2{{(4 - {\gamma ^2} - {\beta ^2})}^2}]}}\). Because \(\beta ,\gamma \in [0,1]\), we have \({{\partial \Pi _{S1}^{*CC}} / {\partial {\beta ^2}}} \ge 0\).
Similarly, we can derive
(ii) From Theorem 1, we have
It is easy to find \({{2(2 - {\gamma ^2}){A^2}\varphi } /{\{ (4 - {\gamma ^2})}}{[2ST - (8 - 3{\gamma ^2})}{{\beta ^2}]^2}\} > 0\). Let \({f_2}(\beta ,\gamma ) = 2(2{\gamma ^2} - 4 - \gamma ) + (2 - \gamma ){\beta ^2}\).
Then, we find that \({{\partial \Pi _{S1}^{*DD}} / {\partial {\beta ^2}}} > 0\) is equivalent to the condition of \({f_2}(\beta ,\gamma ) < 0\). We can regard \({f_2}(\beta ,\gamma )\) as a quadratic function of \(\beta \).
-
(i)
If \(\beta =0\), \({f_2}(0,\gamma ) = 2(2{\gamma ^2} - 4 - \gamma ) < 0\);
-
(ii)
If \(\beta =1\), \({f_2}(1,\gamma ) = 2(2{\gamma ^2} - 3 - \gamma ) - \gamma < 0\).
Therefore we have \({f_2}(\beta ,\gamma ) < 0,\ \ \forall \beta ,\gamma \in [0,1]\), namely \({{\partial \Pi _{S1}^{*DD}} / {\partial {\beta ^2}}} > 0\).
From Theorem 1, we have
It is easy to find \({{2{A^2}} / {\{ (4 - {\gamma ^2})}}{[2ST - (8 - 3{\gamma ^2}){\beta ^2}]^2}\} > 0\). Let \({Z_3}({\beta _3}\mid \gamma ) = 2({\gamma ^2} - 4)ST + 8(8 - 3{\gamma ^2})(2 - {\gamma ^2}) - (4 - {\gamma ^2})(8 - 3{\gamma ^2})\beta _3^2\), and \({{\partial \Pi _{S2}^{*DD}} / {\partial {\beta ^2}}} > 0\) is equivalent to the condition of \({Z_3}({\beta _3}\mid \gamma ) > 0\). Based on the properties of quadratic functions, we have
-
(i)
If \(\beta _{3}=0\), \({Z_3}(0\mid \gamma ) = 2{\gamma ^2}(4{\gamma ^4} - 21{\gamma ^2} + 28) \ge 0\);
-
(ii)
If \(\beta _{3}=1\), \({Z_3}(1\mid \gamma ) = 8{\gamma ^6} - 45{\gamma ^4} + 76{\gamma ^2} - 32\).
Next, we discuss \({Z_3}(1\mid \gamma )\). We have \({{\partial {Z_3}(1\mid \gamma )} / \partial }({\gamma ^2}) = 24{({\gamma ^2})^2} - 90{\gamma ^2} + 76 > 0\). Given \({Z_3}(1\mid 0) = - 32 < 0\) and \({Z_3}(1\mid 1) = 7 > 0\), \(\exists {\gamma _5} \in [0,1],\ \ s.t.{Z_3}(1\mid \gamma ) = 0\).
Finally, we have \(\left\{ \begin{array}{l} {Z_3}(1\mid \gamma ) \le 0\ \ \ \ 0 \le \gamma \le {\gamma _5} \\ {Z_3}(1\mid \gamma ) > 0\ \ \ \ {\gamma _5} < \gamma \le 1 \\ \end{array} \right. \) and \(\left\{ \begin{array}{l} {Z_3}({\beta _3}\mid \gamma ) \ge 0\ \ \ \ if\ 0 \le \gamma \le {\gamma _5}, 0 \le {\beta ^2} \le \beta _3^2 \\ {Z_3}({\beta _3}\mid \gamma ) \le 0\ \ \ \ if\ 0 \le \gamma \le {\gamma _5}, \beta _3^2< {\beta ^2} \le 1 \\ {Z_3}({\beta _3}\mid \gamma ) \ge 0\ \ \ \ if\ {\gamma _5} < \gamma \le 1, \forall {\beta ^2} \in [0,1] \\ \end{array} \right. \), where \(\gamma _{5}\) (\(\approx 0.7933\)) is determined by \({Z_3}(1\mid 0) = 0\) and \(\beta _{3}\) is determined by \({Z_3}({\beta _3}\mid \gamma ) = 0\). \(\square \)
Proof of Theorem 2
Because the proof method of Theorem 2 is similar to that of Theorem 1, we omit the details here. \(\square \)
Proof of Corollary 3
Similar to the proof method of Corollary 3. \(\square \)
Proof of Proposition 2
Similar to the proof method of Proposition 2. \(\square \)
Proof of Corollary 4
Similar to the proof method of Corollary 4. \(\square \)
Proof of Proposition 3
Similar to the proof method of Proposition 3. \(\square \)
Proof of Proposition 4
(i). From Theorems 1 and 2, we have
We find that the inequality of \({({\tilde{e}}_2^{*CC})^{1/2}} \ge {(e_2^{*CC}\mathrm{{)}}^{1/2}}\) is satisfied if and only if \(2(4 - {\beta ^2} - {\gamma ^2}) - (2 + \gamma )(4 - {\beta ^2} - 2\gamma ) \ge 0\) is satisfied. Assuming \({f_6}(\beta ,\gamma ) = 2(4 - {\beta ^2} - {\gamma ^2}) - (2 + \gamma )(4 - {\beta ^2} - 2\gamma )\), we find that \({f_6}(\beta ,\gamma ) \ge 0\) due to \(\gamma ,\beta \in [0,1]\), we have \({\tilde{e}}_2^{*CC} \ge e_2^{*CC}\).
Similarly, we can show other results. In the following, we give the comparisons on profits.
(ii) From Theorems 1 and 2, we have
It is easy to find that the inequality of \(\tilde{\Pi }_{S2}^{*CC} \ge \Pi _{S2}^{*CC}\) is equivalent to \(16 - 4{\gamma ^2} - 4{\beta ^2} - \gamma {\beta ^2} \ge 0\), which is always satisfied. Therefore, we obtain \(\tilde{\Pi }_{S2}^{*CC} \ge \Pi _{S2}^{*CC}\). Meanwhile, \(\tilde{\Pi }_{S2}^{*CC} = \Pi _{S2}^{*CC}\) if and only if \(\beta \) or \(\gamma \) is 0.
From Theorems 1 and 2, we find that the inequality of \(\tilde{\Pi }_{S2}^{*CD} \ge \Pi _{S2}^{*CD}\) is equivalent to the condition of \(8 - 4{\gamma ^2} + 2\gamma + (\gamma - 2){\beta ^2} \ge 0\). Let \({f_7}(\beta ,\gamma ) = 8 - 4{\gamma ^2} + 2\gamma + (\gamma - 2){\beta ^2}\). Obviously, the function \({f_7}(\beta ,\gamma )\) is a quadratic function of \(\beta \).
Because \({f_7}(0,\gamma ) > 0\) and \({f_7}(1,\gamma ) > 0\), we can conclude that \({f_7}(\beta ,\gamma ) > 0\), \(\forall \gamma ,\beta \in [0,1]\). Therefore, we have \(\tilde{\Pi }_{S2}^{*CD} \ge \Pi _{S2}^{*CD}\) and \(\tilde{\Pi }_{S2}^{*CD} = \Pi _{S2}^{*CD}\) if and only if \(\beta \) or \(\gamma \) is 0. The proof of \(\tilde{\Pi }_{S2}^{*DC} \ge \Pi _{S2}^{*DC}\) is similar to that of \(\tilde{\Pi }_{S2}^{*CD} \ge \Pi _{S2}^{*CD}\), and we omit the details here.
From Theorems 1 and 2, we find that the inequality of \(\tilde{\Pi }_{S2}^{*DD} \ge \Pi _{S2}^{*DD}\) is satisfied if and only if the condition of \(2(2{\gamma ^2} - 4 + \gamma )(2{\gamma ^2} - 4 - \gamma )(3 - {\gamma ^2}) + (\gamma - 3)( - {\gamma ^3} - 3{\gamma ^2} + 3\gamma + 8){\beta ^2} \ge 0\) is satisfied. We assume
-
(i)
When \(\beta =0\), we have \({Z_8}(0\mid \gamma ) > 0\);
-
(ii)
When \(\beta =1\), we have \({Z_8}(1\mid \gamma ) = - 8{\gamma ^6} + 57{\gamma ^4} - 122{\gamma ^2} - \gamma + 72\).
Then, we use MATLAB to calculate the solution of equation \( - 8{\gamma ^6} + 57{\gamma ^4} - 122{\gamma ^2} - \gamma + 72 = 0\) in \(\gamma \in [0,1]\). The result is given by \(\left\{ \begin{array}{l} {Z_8}(1\mid \gamma ) \ge 0\ \ \ 0 \le \gamma \le {\gamma _{12}} \\ {Z_8}(1\mid \gamma )< 0\ \ \ {\gamma _{12}} < \gamma \le 1 \\ \end{array} \right. \), where \({\gamma _{12}} \approx 0.9705\). Finally, we obtain \(\left\{ \begin{array}{l} {Z_8}({\beta _8}\mid \gamma ) \ge 0\ \ 0 \le \gamma \le {\gamma _{12}}, \forall \beta \in \mathrm{{[0,1]}} \\ {Z_8}({\beta _8}\mid \gamma ) \ge 0\ \ {\gamma _{12}}< \gamma \le 1, 0 \le \beta \le {\beta _8} \\ {Z_8}({\beta _8}\mid \gamma )< 0\ \ {\gamma _{12}}< \gamma \le 1, {\beta _8} < \beta \le 1 \\ \end{array} \right. \), where the threshold \(\beta _{8}\) is the unique solution determined by the equation as \({Z_8}({\beta _8}\mid \gamma ) = 0\). \(\square \)
Proof of Proposition 5
From Propositions 1 and 2, we can obtain the main results of Proposition 5. \(\square \)
Proof of Proposition 6
Here, comparisons of decision variables are obvious, and we omit them here. We only give the comparisons of profits.
From Theorem 1, we find that \(\Pi _{S2}^{*CC} \ge \Pi _{S2}^{*DC}\) is satisfied if and only if the condition of \( - 2(2 + \gamma ) + {\beta ^2} \ge 0\) is satisfied. Because \(\beta ,\gamma \in [0,1]\), it is easy to prove \( - 2(2 + \gamma ) + {\beta ^2} < 0\) . Therefore, we have \(\Pi _{S2}^{*CC} \le \Pi _{S2}^{*DC}\) . Similarly, we can prove that \(\Pi _{S2}^{*DD} \ge \Pi _{S2}^{*CD}\), \(\tilde{\Pi }_{S2}^{*CC} \le \tilde{\Pi }_{S2}^{*DC}\) and \(\tilde{\Pi }_{S2}^{*DD} \ge \tilde{\Pi }_{S2}^{*CD}\). \(\square \)
Proof of Proposition 7
(i) Here, the comparisons of decision variables will not be described in detail. The following discussions are mainly about comparisons of profits.
From Theorems 1 and 2, we find that \(\tilde{\Pi }_{S1}^{*DC} \ge \Pi _{S1}^{*CC}\) is satisfied if and only if the following condition is satisfied
Because \({{{\partial ^2}{Z_9}({\beta _9}\mid \gamma )} / {{{(\beta _9^2)}^2}}} < 0\) and \({{\partial {Z_9}({\beta _9}\mid \gamma )} / {(\beta _9^2)}} > 0\), we find that \({{\partial {Z_9}({\beta _9}\mid \gamma )} / {(\beta _9^2)}}\) is monotonically decrease of \(\beta _9^2\) and \({Z_9}({\beta _9}\mid \gamma )\) is monotonically increasing of \(\beta _9^2\). In addition, since \({Z_9}(0\mid \gamma ) < 0\) and \({Z_9}(1\mid \gamma ) > 0\), we have \(\left\{ \begin{array}{ll} {\tilde{\Pi }_{S1}^{*DC} \le \Pi _{S1}^{*CC}} &{} {0 \le \beta \le {\beta _9},\ \forall \gamma \in [0,1]} \\ {\tilde{\Pi }_{S1}^{*DC} > \Pi _{S1}^{*CC}} &{} {{\beta _9} < \beta \le 1,\ \forall \gamma \in [0,1]} \\ \end{array} \right. \), where \(\beta _{9}\) is the solution of equation as \({Z_9}({\beta _9}\mid \gamma ) = 0\).
Based on Proposition 3 (i) and Proposition 5 (ii) (a), we can finally obtain Proposition 7 (i) (a).
In addition, we have \(\tilde{\Pi }_{S1}^{*CC} \ge \Pi _{S1}^{*CC}\) and \(\Pi _{S1}^{*DC} < \Pi _{S1}^{*CC}\) according to Propositions 3 (i) and Proposition 5 (i) (a). Therefore, we have \(\Pi _{S1}^{*DC} \le \tilde{\Pi }_{S1}^{*CC}\).
(ii) Similar to the proofs of part (i), we can obtain the results of part (ii). \(\square \)
Proof of Proposition 8
Similar to the proof method of Proposition 7. \(\square \)
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Li, M., Chen, K. Channel structure selection in a competitive supply chain under consideration of marketing effort strategy. Soft Comput 26, 12155–12177 (2022). https://doi.org/10.1007/s00500-022-07468-z
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DOI: https://doi.org/10.1007/s00500-022-07468-z