Service quality guarantee design: obedience behavior, demand updating and information asymmetry

Abstract

With the increasingly fierce market competition and the rapid development of advanced logistics technology, service quality guarantee and demand updating become effective ways to promote procurement decisions in logistics supply chain. However, information asymmetry and obedience behavior have made it more complicated. In this paper, we considered the above factors, and studied the capacity procurement issue in a logistics service supply chain consisting of a logistics service integrator (LSI) and a functional logistics service provider (FLSP) in two periods. First, we find the optimal purchase quantities increase with the FLSP’s obedience factor, in specific conditions, the LSI’s guaranteed service quality and FLSP’s obedience behavior can reach the upper limit (or lower limit). Second, the information symmetry creates a win–win situation iff the penalty cost to the FLSP is moderate and the demand is incompletely revealed. Third, demand updating relaxes the condition for the LSI’s service quality guarantee reaching to the upper limit. For FLSP, when the penalty cost is moderate, the demand updating makes FLSP less obedient in case that the demand is completely revealed and more obedient when the demand is incompletely revealed.

Appendices

Appendix A: The proof of Proposition 1 and Lemma 1

1.1 Proof of the Proposition 1

• 1. Substitute $z_{1} = Q_{1} - l\left( {s_{1} } \right)$ into the LSI’s profit function

  $$ \begin{aligned} \Pi_{I 1} & = \left[ {p - k_{c} \left( {1 - \varphi_{1} } \right)} \right]\left[ {z_{1} + l\left( {s_{1} } \right) - \int\limits_{0}^{{z_{1} }} {F\left( x \right)dx} } \right] - g\left[ {\mu_{1} - z_{1} + \int\limits_{0}^{{z_{1} }} {F\left( x \right)dx} } \right] + v\int\limits_{0}^{{z_{1} }} {F\left( x \right)dx} \\ & \quad + \left[ {k_{i} \left( {1 - \varphi_{1} } \right) - w} \right]\left[ {z_{1} + l\left( {s_{1} } \right)} \right] \\ \end{aligned} $$

  First, we develop the Hessian matrix of $\Pi_{{_{I1} }} (\gamma_{1} ,z_{1} )$

  $$ H_{{\Pi_{I1} (\gamma_{1} ,z_{1} )}} = \left[ {\begin{array}{*{20}l} {\frac{{\partial^{2} \Pi_{I1} \left( {\gamma_{1} ,z_{1} } \right)}}{{\partial^{2} z_{1} }}} \hfill & {\frac{{\partial^{2} \Pi_{I1} \left( {\gamma_{1} ,z_{1} } \right)}}{{\partial z_{1} \partial \gamma_{1} }}} \hfill \\ {\frac{{\partial^{2} \Pi_{I1} \left( {\gamma_{1} ,z_{1} } \right)}}{{\partial \gamma_{1} \partial z_{1} }}} \hfill & {\frac{{\partial^{2} \Pi_{I1} \left( {\gamma_{1} ,z_{1} } \right)}}{{\partial^{2} \gamma_{1} }}} \hfill \\ \end{array} } \right] = \left[ {\begin{array}{*{20}l} { - \left[ {p - k_{c} \left( {1 - \varphi_{1} } \right) + g - v} \right]f\left( {z_{1} } \right)} \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill \\ \end{array} } \right] $$

In which the principle minors are $- \left[ {p - k_{c} \left( {1 - \varphi_{1} } \right) + g - v} \right]f\left( {z_{1} } \right) < 0$, and then zero. Thus the Hessian matrix is negative semidefinite, and $\Pi_{{_{I1} }} (\gamma_{1} ,z_{1} )$ is a joint concave function of $\gamma_{{_{1} }}$ and $z_{1}$.

Then we take the first order derivative of $\Pi_{I 1}$ with respect to $z_{1}$ and we have:

$$ \frac{{\partial \Pi_{I1} }}{{\partial z_{1} }} = \left[ {p + g - k_{c} \left( {1 - \varphi_{1} } \right) + k_{i} \left( {1 - \varphi_{1} } \right) - w} \right] - \left[ {p - k_{c} \left( {1 - \varphi_{1} } \right) + g - v} \right]F\left( {z_{1} } \right) $$

We further take the second order derivative of $\Pi_{I 1}$ with respect to $z_{1}$:

$$ \frac{{\partial^{2} \Pi_{I1} }}{{\partial^{2} z_{1} }} = - \left[ {p - k_{c} \left( {1 - \varphi_{1} } \right) + g - v} \right]f\left( {z_{1} } \right) $$

When $p - k_{c} \left( {1 - \varphi_{1} } \right) + g - v > 0$, $\frac{{\partial^{2} \Pi_{I1} }}{{\partial^{2} z_{1} }} < 0$. There is an optimal procurement factor $z_{1}$ in the first period. With $\frac{{\partial \Pi_{I1} }}{{\partial z_{1} }} = 0$, we obtain the optimal value:

$$ z_{1}^{*} = F^{ - 1} \left( {\frac{{p + g - k_{c} \left( {1 - \varphi_{1} } \right) + k_{i} \left( {1 - \varphi_{1} } \right) - w}}{{p + g - k_{c} \left( {1 - \varphi_{1} } \right) - v}}} \right). $$

According to Assumption 4, we have $w - k_{i} \left( {1 - \varphi_{1} } \right) \le p - k_{c} \left( {1 - \varphi_{1} } \right)$, then we have $0 \le \frac{{w - k_{i} \left( {1 - \varphi_{1} } \right) - v}}{{p + g - k_{c} \left( {1 - \varphi_{1} } \right) - v}} \le 1$, so it could get $0 \le F\left( {z_{1}^{*} } \right) \le 1$.

We again take the first order derivative of $z_{1}^{*}$ with respect to $\varphi_{1}$ and we have:

$$ \frac{{\partial z_{1}^{*} }}{{\partial \varphi_{ 1} }} = \frac{ 1}{{f\left( {z_{1}^{*} } \right)}} \cdot \frac{{k_{c} \left( {w - v} \right) - k_{i} \left( {p + g - v} \right)}}{{\left[ {p - k_{c} \left( {1 - \varphi_{ 1} } \right) + g - v} \right]^{2} }} $$

As $f\left( {z_{1}^{*} } \right) > 0$, and according to the Assumption 1 in Sect. 3, $k_{c} \left( {w - v} \right) - k_{i} \left( {p + g - v} \right) > 0$ is satisfied, we have $\frac{{\partial z_{1}^{*} }}{{\partial \varphi_{ 1} }} > 0$.

Thus $\frac{{\partial Q_{ 1}^{ *} }}{{\partial \varphi_{ 1} }} = \frac{{\partial \left( {z_{1}^{*} { + }l\left( {s_{1} } \right)} \right)}}{{\partial \varphi_{ 1} }} = \frac{{\partial z_{1}^{*} }}{{\partial \varphi_{ 1} }} > 0$.

• 2. We substitute $\hat{z}_{ 2} = \hat{Q}_{ 2} - l\left( {s_{ 2} } \right)$ into the LSI’s profit and take the second order derivative of $\hat{\Pi }_{I 2}$ with respect to $\hat{z}_{ 2}$, and we get:

  $$ \frac{{\partial^{2} \hat{\Pi }_{I 2} }}{{\partial^{2} \hat{z}_{ 2} }} = - \left[ { 1- F\left( {z_{1}^{*} } \right)} \right] \cdot \left[ {p - k_{c} \left( {1 - \varphi_{2} } \right) + g - v} \right]g\left( {\hat{z}_{2} } \right) $$

When $p - k_{c} \left( {1 - \varphi_{ 2} } \right) + g - v > 0$, we have $\frac{{\partial^{2} \hat{\Pi }_{I 2} }}{{\partial^{2} \hat{z}_{ 2} }} < 0$. There is an optimal procurement factor $\hat{z}_{ 2}^{*}$ in the second period, with $\frac{{\partial \hat{\Pi }_{I2} }}{{\partial z_{2} }} = 0$, we get:

$$ \hat{z}_{ 2}^{*} = G^{ - 1} \left( {\frac{{p + g - k_{c} \left( {1 - \varphi_{ 2} } \right) + k_{i} \left( {1 - \varphi_{ 2} } \right) - w}}{{p + g - k_{c} \left( {1 - \varphi_{ 2} } \right) - v}}} \right) $$

As $0 \le G\left( {\hat{z}_{ 2}^{*} } \right) \le 1$, $p + g - k_{c} \left( {1 - \varphi_{ 2} } \right) + k_{i} \left( {1 - \varphi_{ 2} } \right) - w \ge 0$ needs to be satisfied. We have $w - k_{i} \left( {1 - \varphi_{ 2} } \right) \ge v$.

We again take the first order derivative of $\hat{z}_{2}^{*}$ with respect to $\varphi_{2}$ and we have:

$$ \frac{{\partial \hat{z}_{2}^{*} }}{{\partial \varphi_{2} }} = \frac{ 1}{{g\left( {\hat{z}_{2}^{*} } \right)}} \cdot \frac{{k_{c} \left( {w - v} \right) - k_{i} \left( {p + g - v} \right)}}{{\left[ {p - k_{c} \left( {1 - \varphi_{ 1} } \right) + g - v} \right]^{2} }} $$

As $g\left( {\hat{z}_{2}^{*} } \right) > 0$, and according to the assumption in Sect. 3, $k_{c} \left( {w - v} \right) - k_{i} \left( {p + g - v} \right) > 0$ is satisfied, we have $\frac{{\partial \hat{z}_{2}^{*} }}{{\partial \varphi_{2} }} > 0$.

Thus $\frac{{\partial \hat{Q}_{ 2}^{ *} }}{{\partial \varphi_{ 2} }} = \frac{{\partial \left( {\hat{z}_{ 2}^{*} { + }l\left( {s_{ 2} } \right)} \right)}}{{\partial \varphi_{ 2} }} = \frac{{\partial \hat{z}_{ 2}^{*} }}{{\partial \varphi_{ 2} }} > 0$.

1.2 The Proof of Lemma 1

$$ F\left( {\hat{z}_{ 2}^{*} } \right) - F\left( {z_{1}^{*} } \right) = \frac{{p + g - k_{c} \left( {1 - \varphi_{ 2} } \right) + k_{i} \left( {1 - \varphi_{ 2} } \right) - w}}{{p + g - k_{c} \left( {1 - \varphi_{ 2} } \right) - v}}\left[ {1 - F\left( {z_{1}^{*} } \right)} \right] \ge 0 $$

That is $F\left( {\hat{z}_{ 2}^{*} } \right) > F\left( {z_{1}^{*} } \right)$. As $F\left( x \right)$ is a monotone increasing function, we have $\hat{z}_{ 2}^{*} \ge z_{1}^{*}$.

Appendix B: The proof of Proposition 2 and Lemma 2

2.1 Proof of Proposition 2

• 1. The optimal service quality guarantee in the first period

We take the first order derivative of $\Pi_{I 1}$ with respect to $\gamma_{1}$:

$$ \frac{{\partial \Pi_{I1} }}{{\partial \gamma_{1} }} = \beta \cdot \left( {1 - \lambda } \right)\left[ {p - w + \left( {k_{i} - k_{c} } \right)\left( {1 - \varphi_{1} } \right)} \right] $$

As $\beta \left( {1 - \lambda } \right) > 0$, and from Assumption 1, we know that $k_{c} > k_{i}$ holds. So when $\varphi_{1} = 1 - \frac{p - w}{{k_{c} - k_{i} }}$, $\frac{{\partial \Pi_{I1} }}{{\partial \gamma_{1} }} = 0$. Because of $\underline{\varphi } \le \varphi_{1} \le 1$, there are two cases to consider, one is the case of $\underline{\varphi } \ge 1 - \frac{p - w}{{k_{c} - k_{i} }}$, and another is the case of $\underline{\varphi } < 1 - \frac{p - w}{{k_{c} - k_{i} }}$.

• 1. The case of $\underline{\varphi } \ge 1 - \frac{p - w}{{k_{c} - k_{i} }}$

Because of $\underline{\varphi } \ge 1 - \frac{p - w}{{k_{c} - k_{i} }}$, we can get $\varphi_{1} > \underline{\varphi } > 1 - \frac{p - w}{{k_{c} - k_{i} }}$, and we have $\frac{{\partial \Pi_{I1} }}{{\partial \gamma_{1} }} > 0$.

Therefore, the LSI’s profit in the first period increases with the service quality guarantee, then $\gamma_{1}^{*} = 1$.

• 2. The case of $\underline{\varphi } < 1 - \frac{p - w}{{k_{c} - k_{i} }}$

When $\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\varphi } < \varphi_{1} < 1 - \frac{p - w}{{k_{c} - k_{i} }}$, we have $\frac{{\partial \Pi_{I1} }}{{\partial \gamma_{1} }} < 0$ and the LSI’s profit in the first period decreases with the service quality guarantee, then $\gamma_{1}^{*} = \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\gamma }$.

When $1 - \frac{p - w}{{k_{c} - k_{i} }} < \varphi_{1} < 1$, we have $\frac{{\partial \Pi_{I1} }}{{\partial \gamma_{1} }} > 0$ and the LSI’s profit in the first period increases with the service quality guarantee, then $\gamma_{1}^{*} = 1$.

Overall, if $\varphi_{1} > 1 - \frac{p - w}{{k_{c} - k_{i} }}$, then $\gamma_{1}^{*} = 1$; otherwise, if $\varphi_{1} \le 1 - \frac{p - w}{{k_{c} - k_{i} }}$, then $\gamma_{1}^{*} = \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\gamma }$.

• 2. The optimal service quality guarantee in the second period when market demand uncertainty is completely revealed.

We have the first order derivative of $\tilde{\Pi }_{I2}$ with respect to $\gamma_{2}$ as:

$$ \frac{{\partial \tilde{\Pi }_{I2} }}{{\partial \gamma_{2} }} = F\left( {z_{1}^{*} } \right)\left[ {p - w + \left( {k_{i} - k_{c} } \right)\left( {1 - \varphi_{2} } \right)} \right]\beta \left( {1 - \lambda } \right) + \eta $$

As $\beta \left( {1 - \lambda } \right) > 0$, and from Proposition 1, we know that $k_{c} > k_{i}$ holds. So when $\varphi_{2} = 1 - \frac{p - w}{{k_{c} - k_{i} }} - \frac{\eta }{{\beta \left( {1 - \lambda } \right) \cdot F\left( {z_{1}^{*} } \right)\left( {k_{c} - k_{i} } \right)}}$, $\frac{{\partial \Pi_{I2} }}{{\partial \gamma_{2} }} = 0$. Because of $\underline{\varphi } \le \varphi_{2} \le 1$, there are two cases to consider, one is the case of $\underline{\varphi } \ge 1 - \frac{p - w}{{k_{c} - k_{i} }} - \frac{\eta }{{\beta \left( {1 - \lambda } \right) \cdot F\left( {z_{1}^{*} } \right)\left( {k_{c} - k_{i} } \right)}}$, and another is the case of $\underline{\varphi } < 1 - \frac{p - w}{{k_{c} - k_{i} }} - \frac{\eta }{{\beta \left( {1 - \lambda } \right) \cdot F\left( {z_{1}^{*} } \right)\left( {k_{c} - k_{i} } \right)}}$.

• 1. The case of $\underline{\varphi } \ge 1 - \frac{p - w}{{k_{c} - k_{i} }} - \frac{\eta }{{\beta \left( {1 - \lambda } \right) \cdot F\left( {z_{1}^{*} } \right)\left( {k_{c} - k_{i} } \right)}}$

Because of $\underline{\varphi } \ge 1 - \frac{p - w}{{k_{c} - k_{i} }} - \frac{\eta }{{\beta \left( {1 - \lambda } \right) \cdot F\left( {z_{1}^{*} } \right)\left( {k_{c} - k_{i} } \right)}}$, we can get $\varphi_{2} > \underline{\varphi } > 1 - \frac{p - w}{{k_{c} - k_{i} }} - \frac{\eta }{{\beta \left( {1 - \lambda } \right) \cdot F\left( {z_{1}^{*} } \right)\left( {k_{c} - k_{i} } \right)}}$, and we have $\frac{{\partial \Pi_{I2} }}{{\partial \gamma_{2} }} > 0$.

Therefore, the LSI’s profit in the first period increases with the service quality guarantee, then $\gamma_{2}^{*} = 1$.

• 2. The case of $\underline{\varphi } < 1 - \frac{p - w}{{k_{c} - k_{i} }} - \frac{\eta }{{\beta \left( {1 - \lambda } \right) \cdot F\left( {z_{1}^{*} } \right)\left( {k_{c} - k_{i} } \right)}}$

When $\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\varphi } < \tilde{\varphi }_{2} < 1 - \frac{p - w}{{k_{c} - k_{i} }} - \frac{\eta }{{\beta \left( {1 - \lambda } \right) \cdot F\left( {z_{1}^{*} } \right)\left( {k_{c} - k_{i} } \right)}}$, we have $\frac{{\partial \Pi_{I2} }}{{\partial \tilde{\gamma }_{2} }} < 0$, and the LSI’s profit in the first period decreases with the service quality guarantee, then $\tilde{\gamma }_{2}^{*} = \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\gamma }$.

When $1 - \frac{p - w}{{k_{c} - k_{i} }} - \frac{\eta }{{\beta \left( {1 - \lambda } \right) \cdot F\left( {z_{1}^{*} } \right)\left( {k_{c} - k_{i} } \right)}} < \tilde{\varphi }_{2} < 1$, we have $\frac{{\partial \Pi_{I2} }}{{\partial \tilde{\gamma }_{2} }} > 0$ and the LSI’s profit in the first period increases with the service quality guarantee, then $\tilde{\gamma }_{2}^{*} = 1$.

Overall, if $\tilde{\varphi }_{2} > 1 - \frac{p - w}{{k_{c} - k_{i} }} - \frac{\eta }{{\beta \left( {1 - \lambda } \right) \cdot F\left( {z_{1}^{*} } \right)\left( {k_{c} - k_{i} } \right)}}$, then $\tilde{\gamma }_{2}^{*} = 1$; otherwise, if $\tilde{\varphi }_{2} \le 1 - \frac{p - w}{{k_{c} - k_{i} }} - \frac{\eta }{{\beta \left( {1 - \lambda } \right) \cdot F\left( {z_{1}^{*} } \right)\left( {k_{c} - k_{i} } \right)}}$, then $\tilde{\gamma }_{2}^{*} = \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\gamma }$.

• 3. When market demand uncertainty is incompletely revealed in the second period, the proof of the optimal service quality guarantee’s solution is similar to the situation that market demand uncertainty is completely revealed.

2.2 Proof of the Lemma 2

To make the optimal service quality guarantee reach its upper limit, the obedience factor should satisfy $\varphi_{1} > 1 - \frac{p - w}{{k_{c} - k_{i} }}$ in the first period, and $\tilde{\varphi }_{2} > 1 - \frac{p - w}{{k_{c} - k_{i} }} - \frac{\eta }{{\beta \left( {1 - \lambda } \right) \cdot F\left( {z_{1}^{*} } \right)\left( {k_{c} - k_{i} } \right)}}$, $\hat{\varphi }_{2} > 1 - \frac{p - w}{{\left( {k_{c} - k_{i} } \right)}} - \frac{\eta }{{\left[ { 1- F\left( {z_{1}^{*} } \right)} \right] \cdot \beta \left( {1 - \lambda } \right) \cdot \left( {k_{c} - k_{i} } \right)}}$ in the second period, respectively.

As $\frac{\eta }{{\beta \left( {1 - \lambda } \right) \cdot F\left( {z_{1}^{*} } \right)\left( {k_{c} - k_{i} } \right)}} > 0$ and $\frac{\eta }{{\left[ { 1- F\left( {z_{1}^{*} } \right)} \right] \cdot \beta \left( {1 - \lambda } \right) \cdot \left( {k_{c} - k_{i} } \right)}} > 0$

Therefore, we can get that $1 - \frac{p - w}{{k_{c} - k_{i} }} > 1 - \frac{p - w}{{k_{c} - k_{i} }} - \frac{\eta }{{\beta \left( {1 - \lambda } \right) \cdot F\left( {z_{1}^{*} } \right)\left( {k_{c} - k_{i} } \right)}}$, and

$$ 1 - \frac{p - w}{{k_{c} - k_{i} }} > 1 - \frac{p - w}{{\left( {k_{c} - k_{i} } \right)}} - \frac{\eta }{{\left[ { 1- F\left( {z_{1}^{*} } \right)} \right] \cdot \beta \left( {1 - \lambda } \right) \cdot \left( {k_{c} - k_{i} } \right)}}. $$

Appendix C: The proof of Proposition 3

• 1. When LSI does not share the information to the FLSP

Take the first order derivative of $\tilde{\Pi }_{F 2}$ with respect to $\tilde{\varphi }_{2}^{IA}$, we have

$$ \frac{{\partial \tilde{\Pi }_{F 2} }}{{\partial \tilde{\varphi }_{2}^{IA} }} = F\left( {z_{1}^{*} } \right)Q_{1}^{*} \left( {k_{i} - \rho \tilde{\gamma }_{2}^{*} } \right) $$

If $k_{i} - \rho \tilde{\gamma }_{2}^{*} > 0$, then $\frac{{\partial \tilde{\Pi }_{F 2} }}{{\partial \tilde{\varphi }_{2}^{IA} }} > 0$ and $\tilde{\Pi }_{F 2}$ increases with the obedience factor $\tilde{\varphi }_{2}^{IA}$ in the second period. If $k_{i} - \rho \tilde{\gamma }_{2}^{*} < 0$, then $\frac{{\partial \tilde{\Pi }_{F 2} }}{{\partial \tilde{\varphi }_{2}^{IA} }} < 0$ and $\tilde{\Pi }_{F 2}$ decreases with the obedience factor $\tilde{\varphi }_{2}^{IA}$ in the second period.

• 2. When LSI shares the information to the FLSP

Take the first order derivative of $\tilde{\Pi }_{F 2}$ with respect to $\tilde{\varphi }_{2}^{IS}$, we have

$$ \frac{{\partial \tilde{\Pi }_{F 2} }}{{\partial \tilde{\varphi }_{2}^{IS} }} = F\left( {z_{1}^{*} } \right)\tilde{Q}_{2}^{*} \left( {k_{i} - \rho \tilde{\gamma }_{2}^{*} } \right) $$

And the analysis process is similar to the above.

Appendix D: The proof of Propositions 4–6 and Lemma 3

• 1. We take the first order derivative of $\Pi_{F1}$ with respect to $\varphi_{1}$

  $$ \frac{{\partial \Pi_{F1} }}{{\partial \varphi_{1} }} = \frac{{\partial z_{1}^{*} }}{{\partial \varphi_{ 1} }}\left[ {w - c - \rho \varphi_{1} \gamma_{1}^{*} - k_{i} \left( {1 - \varphi_{1} } \right)} \right] + \left[ {z_{1}^{*} + l\left( {s_{1} } \right)} \right]\left[ { - \rho \gamma_{1}^{*} + k_{i} } \right] $$

If $- \rho \gamma_{1}^{*} + k_{i} > 0$, that is $k_{i} > \rho \gamma_{1}^{*}$, then $\frac{{\partial \Pi_{F1} \left( {\varphi_{1} } \right)}}{{\partial \varphi_{1} }} > 0$, the FLSP’s profit increases with obedience factor $\varphi_{1}$. Thus the FLSP’s optimal obedience factor in the first period is $\varphi_{1}^{*} = 1$.

If $- \rho \gamma_{1}^{*} + k_{i} < 0$, with the second order derivative of $\Pi_{F1}$ with respect to $\varphi_{1}$, we have

$$ \frac{{\partial^{2} \Pi_{F 1} \left( {\varphi_{1} } \right)}}{{\partial^{2} \varphi_{ 1} }} = \frac{{\partial^{2} z_{ 1}^{*} }}{{\partial^{2} \varphi_{ 1} }}\left[ {w - c - \rho \varphi_{ 1} \gamma_{ 1}^{*} - k_{i} \left( {1 - \varphi_{ 1} } \right)} \right] + 2\left( { - \rho \gamma_{ 1}^{*} + k_{i} } \right)\frac{{\partial z_{ 1}^{*} }}{{\partial \varphi_{ 1} }} $$

where

$$ \begin{aligned} \frac{{\partial^{2} z_{1}^{*} }}{{\partial^{2} \varphi_{1} }} & = \frac{ 1}{{f\left( {z_{1}^{*} } \right)}} \cdot \frac{{2k_{c} \left[ {k_{c} \left( {w - v} \right) - k_{i} \left( {p + g - v} \right)} \right]}}{{\left[ {p - k_{c} \left( {1 - \varphi_{ 1} } \right) + g - v} \right]^{3} }} - \frac{1}{{f^{2} \left( {z_{1}^{*} } \right)}}\frac{{\partial z_{1}^{*} }}{{\partial \varphi_{1} }}\frac{{k_{c} \left( {w - v} \right) - k_{i} \left( {p + g - v} \right)}}{{\left[ {p - k_{c} \left( {1 - \varphi_{ 1} } \right) + g - v} \right]^{2} }} \\ & = \frac{ 1}{{f\left( {z_{1}^{*} } \right)}} \cdot \frac{{k_{c} \left( {w - v} \right) - k_{i} \left( {p + g - v} \right)}}{{\left[ {p - k_{c} \left( {1 - \varphi_{ 1} } \right) + g - v} \right]^{3} }}\left[ { - 2k_{c} - \frac{1}{{f^{2} \left( {z_{1}^{*} } \right)}}\frac{{k_{c} \left( {w - v} \right) - k_{i} \left( {p + g - v} \right)}}{{p - k_{c} \left( {1 - \varphi_{ 1} } \right) + g - v}}} \right] < 0 \\ \end{aligned} $$

As $w - c - \rho \varphi_{ 1} \gamma_{ 1}^{*} - k_{i} \left( {1 - \varphi_{ 1} } \right) > 0$ and $\frac{{\partial z_{ 1}^{*} }}{{\partial \varphi_{ 1} }} > 0$, we have $\frac{{\partial^{2} \Pi_{F 1} \left( {\varphi_{1} } \right)}}{{\partial^{2} \varphi_{ 1} }} < 0$.

Let $\frac{{\partial \Pi_{F1} \left( {\varphi_{1} } \right)}}{{\partial \varphi_{1} }} = 0$ and we obtain $\varphi_{1}^{*} = \frac{{w - c - k_{i} }}{{\rho \gamma_{1}^{*} - k_{i} }} - \frac{{z_{1}^{*} + l\left( {s_{1} } \right)}}{{{{\partial z_{1}^{*} } \mathord{\left/ {\vphantom {{\partial z_{1}^{*} } {\partial \varphi_{1}^{*} }}} \right. \kern-0pt} {\partial \varphi_{1}^{*} }}}}$.

Since the range of $\varphi$ is $\varphi \in \left[ {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\varphi } ,1} \right]$, so if $\frac{{w - c - k_{i} }}{{\rho \gamma_{1}^{*} - k_{i} }} - \frac{{z_{1}^{*} + l\left( {s_{1} } \right)}}{{{{\partial z_{1}^{*} } \mathord{\left/ {\vphantom {{\partial z_{1}^{*} } {\partial \varphi_{1}^{*} }}} \right. \kern-0pt} {\partial \varphi_{1}^{*} }}}} < \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\varphi }$.

That is $k_{i} < \rho \gamma_{1}^{*} - \frac{{w - c - \rho \gamma_{1}^{*} }}{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\varphi } - 1 + {{\left( {z_{1}^{*} + l\left( {s_{1} } \right)} \right)} \mathord{\left/ {\vphantom {{\left( {z_{1}^{*} + l\left( {s_{1} } \right)} \right)} {\left( {{{\partial z_{1}^{*} } \mathord{\left/ {\vphantom {{\partial z_{1}^{*} } {\partial \varphi_{1}^{*} }}} \right. \kern-0pt} {\partial \varphi_{1}^{*} }}} \right)}}} \right. \kern-0pt} {\left( {{{\partial z_{1}^{*} } \mathord{\left/ {\vphantom {{\partial z_{1}^{*} } {\partial \varphi_{1}^{*} }}} \right. \kern-0pt} {\partial \varphi_{1}^{*} }}} \right)}}}}$, $\varphi_{1}^{*} = \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\varphi }$.

If $\frac{{w - c - k_{i} }}{{\rho \gamma_{1}^{*} - k_{i} }} - \frac{{z_{1}^{*} + l\left( {s_{1} } \right)}}{{{{\partial z_{1}^{*} } \mathord{\left/ {\vphantom {{\partial z_{1}^{*} } {\partial \varphi_{1}^{*} }}} \right. \kern-0pt} {\partial \varphi_{1}^{*} }}}} > 1$, that is $k_{i} > \rho \gamma_{1}^{*} - \frac{{w - c - \rho \gamma_{1}^{*} }}{{{{\left( {z_{1}^{*} + l\left( {s_{1} } \right)} \right)} \mathord{\left/ {\vphantom {{\left( {z_{1}^{*} + l\left( {s_{1} } \right)} \right)} {\left( {{{\partial z_{1}^{*} } \mathord{\left/ {\vphantom {{\partial z_{1}^{*} } {\partial \varphi_{1}^{*} }}} \right. \kern-0pt} {\partial \varphi_{1}^{*} }}} \right)}}} \right. \kern-0pt} {\left( {{{\partial z_{1}^{*} } \mathord{\left/ {\vphantom {{\partial z_{1}^{*} } {\partial \varphi_{1}^{*} }}} \right. \kern-0pt} {\partial \varphi_{1}^{*} }}} \right)}}}}$, $\varphi_{1}^{*} { = 1}$.

From the above, if $k_{i} < \rho \gamma_{1}^{*} - \frac{{w - c - \rho \gamma_{1}^{*} }}{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\varphi } - 1 + {{\left( {z_{1}^{*} + l\left( {s_{1} } \right)} \right)} \mathord{\left/ {\vphantom {{\left( {z_{1}^{*} + l\left( {s_{1} } \right)} \right)} {\left( {{{\partial z_{1}^{*} } \mathord{\left/ {\vphantom {{\partial z_{1}^{*} } {\partial \varphi_{1}^{*} }}} \right. \kern-0pt} {\partial \varphi_{1}^{*} }}} \right)}}} \right. \kern-0pt} {\left( {{{\partial z_{1}^{*} } \mathord{\left/ {\vphantom {{\partial z_{1}^{*} } {\partial \varphi_{1}^{*} }}} \right. \kern-0pt} {\partial \varphi_{1}^{*} }}} \right)}}}}$, $\varphi_{1}^{*} = \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\varphi }$.

If $\rho \gamma_{1}^{*} - \frac{{w - c - \rho \gamma_{1}^{*} }}{{{{\left( {z_{1}^{*} + l\left( {s_{1} } \right)} \right)} \mathord{\left/ {\vphantom {{\left( {z_{1}^{*} + l\left( {s_{1} } \right)} \right)} {\left( {{{\partial z_{1}^{*} } \mathord{\left/ {\vphantom {{\partial z_{1}^{*} } {\partial \varphi_{1}^{*} }}} \right. \kern-0pt} {\partial \varphi_{1}^{*} }}} \right)}}} \right. \kern-0pt} {\left( {{{\partial z_{1}^{*} } \mathord{\left/ {\vphantom {{\partial z_{1}^{*} } {\partial \varphi_{1}^{*} }}} \right. \kern-0pt} {\partial \varphi_{1}^{*} }}} \right)}}}} < k_{i} < \rho \gamma_{1}^{*} - \frac{{w - c - \rho \gamma_{1}^{*} }}{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\varphi } - 1 + {{\left( {z_{1}^{*} + l\left( {s_{1} } \right)} \right)} \mathord{\left/ {\vphantom {{\left( {z_{1}^{*} + l\left( {s_{1} } \right)} \right)} {\left( {{{\partial z_{1}^{*} } \mathord{\left/ {\vphantom {{\partial z_{1}^{*} } {\partial \varphi_{1}^{*} }}} \right. \kern-0pt} {\partial \varphi_{1}^{*} }}} \right)}}} \right. \kern-0pt} {\left( {{{\partial z_{1}^{*} } \mathord{\left/ {\vphantom {{\partial z_{1}^{*} } {\partial \varphi_{1}^{*} }}} \right. \kern-0pt} {\partial \varphi_{1}^{*} }}} \right)}}}}$,

$$ \varphi_{1}^{*} = \frac{{w - c - k_{i} }}{{\rho \gamma_{1}^{*} - k_{i} }} - \frac{{z_{1}^{*} + l\left( {s_{1} } \right)}}{{{{\partial z_{1}^{*} } \mathord{\left/ {\vphantom {{\partial z_{1}^{*} } {\partial \varphi_{1}^{*} }}} \right. \kern-0pt} {\partial \varphi_{1}^{*} }}}}. $$

If $k_{i} > \rho \gamma_{1}^{*} - \frac{{w - c - \rho \gamma_{1}^{*} }}{{{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\varphi } - 1 + \left( {z_{1}^{*} + l\left( {s_{1} } \right)} \right)} \mathord{\left/ {\vphantom {{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\varphi } - 1 + \left( {z_{1}^{*} + l\left( {s_{1} } \right)} \right)} {\left( {{{\partial z_{1}^{*} } \mathord{\left/ {\vphantom {{\partial z_{1}^{*} } {\partial \varphi_{1}^{*} }}} \right. \kern-0pt} {\partial \varphi_{1}^{*} }}} \right)}}} \right. \kern-0pt} {\left( {{{\partial z_{1}^{*} } \mathord{\left/ {\vphantom {{\partial z_{1}^{*} } {\partial \varphi_{1}^{*} }}} \right. \kern-0pt} {\partial \varphi_{1}^{*} }}} \right)}}}}$, $\varphi_{1}^{*} { = 1}$.

• 2. When the LSI shares the information to the FLSP

We take the first order derivative of $\hat{\Pi }_{F2}$ with respect to $\hat{\varphi }_{2}^{IS}$

$$ \frac{{\partial \hat{\Pi }_{F2} }}{{\partial \hat{\varphi }_{2}^{IS} }} = \left[ { 1- F\left( {z_{1}^{*} } \right)} \right] \cdot \left\{ {\frac{{\partial \hat{z}_{2}^{*} }}{{\partial \hat{\varphi }_{2}^{IS} }}\left[ {w - c - \rho \hat{\varphi }_{2}^{IS} \hat{\gamma }_{2}^{*} - k_{i} \left( {1 - \hat{\varphi }_{2}^{IS} } \right)} \right] + \left[ {\hat{z}_{2}^{*} + l\left( {s_{2} } \right)} \right]\left[ { - \rho \hat{\gamma }_{2}^{*} + k_{i} } \right]} \right\} $$

$- \rho \hat{\gamma }_{2}^{*} + k_{i} > 0$, that is $k_{i} > \rho \hat{\gamma }_{2}^{*}$, because $\frac{{\partial \hat{z}_{ 2}^{*} }}{{\partial \hat{\varphi }_{2}^{IS} }}\left[ {w - c - \rho \hat{\varphi }_{2}^{IS} \hat{\gamma }_{2}^{*} - k_{i} \left( {1 - \hat{\varphi }_{2}^{IS} } \right)} \right] > 0$ and $\hat{z}_{ 2}^{*} + l\left( {s_{2} } \right) > 0$, we have $\frac{{\partial \hat{\Pi }_{F 2} }}{{\partial \hat{\varphi }_{2} }} > 0$ and $\hat{\Pi }_{F 2}$ increases with the obedience factor $\hat{\varphi }_{2}^{IS}$ in the second period. Thus $\hat{\varphi }_{2}^{IS*} { = 1}$.

If $- \rho \hat{\gamma }_{2}^{*} + k_{i} < 0$, we have the second order derivative of $\hat{\Pi }_{F 2}$ with respect to $\hat{\varphi }_{2}^{IS}$ as

$$ \frac{{\partial^{2} \hat{\Pi }_{F 2} }}{{\partial^{2} \hat{\varphi }_{2}^{IS} }} = \left[ { 1- F\left( {z_{1}^{*} } \right)} \right] \cdot \left\{ {\frac{{\partial^{2} \hat{z}_{ 2}^{*} }}{{\partial^{2} \hat{\varphi }_{2}^{IS} }}\left[ {w - c - \rho \hat{\varphi }_{2}^{IS} \hat{\gamma }_{2}^{*} - k_{i} \left( {1 - \hat{\varphi }_{2}^{IS} } \right)} \right] + 2\left( { - \rho \hat{\gamma }_{2}^{*} + k_{i} } \right)\frac{{\partial \hat{z}_{ 2}^{*} }}{{\partial \hat{\varphi }_{2}^{IS} }}} \right\} $$

where

$$ \begin{aligned} \frac{{\partial^{2} \hat{z}_{ 2}^{*} }}{{\partial^{2} \hat{\varphi }_{2}^{IS} }} & = \frac{ 1}{{g\left( {\hat{z}_{ 2}^{*} } \right)}} \cdot \frac{{2k_{c} \left[ {k_{c} \left( {w - v} \right) - k_{i} \left( {p + g - v} \right)} \right]}}{{\left[ {p - k_{c} \left( {1 - \hat{\varphi }_{2}^{IS} } \right) + g - v} \right]^{3} }} - \frac{1}{{f^{2} \left( {\hat{z}_{ 2}^{*} } \right)}}\frac{{\partial \hat{z}_{ 2}^{*} }}{{\partial \hat{\varphi }_{2}^{IS} }}\frac{{k_{c} \left( {w - v} \right) - k_{i} \left( {p + g - v} \right)}}{{\left[ {p - k_{c} \left( {1 - \hat{\varphi }_{2}^{IS} } \right) + g - v} \right]^{2} }} \\ & = \frac{ 1}{{g\left( {\hat{z}_{ 2}^{*} } \right)}} \cdot \frac{{k_{c} \left( {w - v} \right) - k_{i} \left( {p + g - v} \right)}}{{\left[ {p - k_{c} \left( {1 - \hat{\varphi }_{2}^{IS} } \right) + g - v} \right]^{3} }}\left[ { - 2k_{c} - f^{2} \left( {\hat{z}_{ 2}^{*} } \right)\frac{{k_{c} \left( {w - v} \right) - k_{i} \left( {p + g - v} \right)}}{{p - k_{c} \left( {1 - \hat{\varphi }_{2}^{IS} } \right) + g - v}}} \right] < 0 \\ \end{aligned} $$

As $w - c - \rho \hat{\varphi }_{2} \hat{\gamma }_{2}^{*} - k_{i} \left( {1 - \hat{\varphi }_{2}^{IS} } \right) > 0$ and $\frac{{\partial \hat{z}_{ 2}^{*} }}{{\partial \hat{\varphi }_{2}^{IS} }} > 0$, thus, when $- \rho \hat{\gamma }_{2}^{*} + k_{i} < 0$, we have $\frac{{\partial^{2} \hat{\Pi }_{F 2} \left( {\varphi_{2} } \right)}}{{\partial^{2} \hat{\varphi }_{2}^{IS} }} < 0$. Accordingly, there is an optimal $\hat{\varphi }_{2}^{IS*}$ to maximize the FLSP’s profit. Let $\frac{{\partial \hat{\Pi }_{F 2} \left( {\hat{\varphi }_{2}^{IS} } \right)}}{{\partial \hat{\varphi }_{2}^{IS} }} = 0$, we have $\hat{\varphi }_{2}^{IS*} = \frac{{w - c - k_{i} }}{{\rho \hat{\gamma }_{2}^{*} - k_{i} }} - \frac{{\hat{z}_{2}^{*} + l\left( {s_{2} } \right)}}{{{{\partial \hat{z}_{2}^{*} } \mathord{\left/ {\vphantom {{\partial \hat{z}_{2}^{*} } {\partial \varphi_{2} }}} \right. \kern-0pt} {\partial \varphi_{2} }}}}$.

Similar to above proof process, the optimal $\hat{\varphi }_{2}^{IS*}$ is proved.

4.1 Proof of the Lemma 3

When LSI does not share the information to the FLSP

We take the first order derivative of $\hat{\Pi }_{F2}$ with respect to $\hat{\varphi }_{2}^{IA}$

$$ \frac{{\partial \hat{\Pi }_{F 2} }}{{\partial \hat{\varphi }_{2}^{IA} }} = F\left( {\hat{z}_{2}^{*} } \right)Q_{1}^{*} \left( {k_{i} - \rho \hat{\gamma }_{2}^{*} } \right) $$

If $k_{i} - \rho \hat{\gamma }_{2}^{*} > 0$, then $\frac{{\partial \hat{\Pi }_{F 2} }}{{\partial \hat{\varphi }_{2}^{IA} }} > 0$ and $\hat{\Pi }_{F 2}$ increases with the obedience factor $\hat{\varphi }_{2}^{IA}$ in the second period. If $k_{i} - \rho \hat{\gamma }_{2}^{*} < 0$, then $\frac{{\partial \hat{\Pi }_{F 2} }}{{\partial \hat{\varphi }_{2}^{IA} }} < 0$ and $\hat{\Pi }_{F 2}$ decreases with the obedience factor $\hat{\varphi }_{2}^{IA}$ in the second period.

4.2 Proof of Proposition 5

Because $\rho \hat{\gamma }_{2}^{*} > B_{2} = \rho \hat{\gamma }_{2}^{*} - \frac{{w - c - \rho \hat{\gamma }_{2}^{*} }}{{{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\varphi } - 1 + \left( {\hat{z}_{2}^{*} + l\left( {s_{2} } \right)} \right)} \mathord{\left/ {\vphantom {{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\varphi } - 1 + \left( {\hat{z}_{2}^{*} + l\left( {s_{2} } \right)} \right)} {\left( {{{\partial \hat{z}_{2}^{*} } \mathord{\left/ {\vphantom {{\partial \hat{z}_{2}^{*} } {\partial \hat{\varphi }_{2}^{{}} }}} \right. \kern-0pt} {\partial \hat{\varphi }_{2}^{{}} }}} \right)}}} \right. \kern-0pt} {\left( {{{\partial \hat{z}_{2}^{*} } \mathord{\left/ {\vphantom {{\partial \hat{z}_{2}^{*} } {\partial \hat{\varphi }_{2}^{{}} }}} \right. \kern-0pt} {\partial \hat{\varphi }_{2}^{{}} }}} \right)}}}}$, according to the “Appendix C”, and Propositions 3 and 4, we can derive the Proposition 5.

4.3 Proof of Proposition 6

Similar, because $\rho \hat{\gamma }_{2}^{*} > B_{1} = \rho \gamma_{1}^{*} - \frac{{w - c - \rho \gamma_{1}^{*} }}{{{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\varphi } - 1 + \left( {z_{1}^{*} + l\left( {s_{1} } \right)} \right)} \mathord{\left/ {\vphantom {{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\varphi } - 1 + \left( {z_{1}^{*} + l\left( {s_{1} } \right)} \right)} {\left( {{{\partial z_{1}^{*} } \mathord{\left/ {\vphantom {{\partial z_{1}^{*} } {\partial \varphi_{1}^{*} }}} \right. \kern-0pt} {\partial \varphi_{1}^{*} }}} \right)}}} \right. \kern-0pt} {\left( {{{\partial z_{1}^{*} } \mathord{\left/ {\vphantom {{\partial z_{1}^{*} } {\partial \varphi_{1}^{*} }}} \right. \kern-0pt} {\partial \varphi_{1}^{*} }}} \right)}}}}$, according to the “Appendix C”, and Propositions 3 and 4, we can derive the Proposition 6.