Impacts of decision sequences on a random yield supply chain with a service level requirement

Abstract

The last decade has witnessed a rising trend of social network structures where decision making may be influenced by members’ channel power. This paper considers a decentralized supply chain that comprises a random yield supplier and a manufacturer with a service level requirement. The supplier determines the production quantity while the manufacturer decides on the replenishment quantity. We investigate several game-theoretic models with respect to decision sequences of the supplier and manufacturer with a service level requirement. We compare firms’ equilibrium production/order quantity and profits under these models. Both firms achieve a higher payoff in the manufacturer-leader game than in the supplier-leader and simultaneous move games, which indicates that the manufacturer-leader game is Pareto efficient from the channel structure perspective. A critical value is observed at the service level, beyond which the supplier’s profit decreases, while the manufacturer’s profit increases in the service level requirement. Interestingly, the critical value is independent of the models and it just depends on the wholesale price and production cost. We also examine the decentralized random yield supply chain under the vendor-managed inventory arrangement. Finally, numerical computations are conducted to evaluate supply chain performance with respect to various parameters.

Appendix

Proof of Proposition 4

Following the same logic in Sect. 3.1, we obtain $Q_m =D$. When $Q_m =D$ is substituted into the optimization problem (11), the constraint $P\left\{ {YQ_s \ge D} \right\} \ge \alpha$ can be rewritten as $1-F\left( {\frac{D}{Q_s }} \right) \ge \alpha$, which is equal to $Q_s \ge \frac{D}{F^{-1}\left( {1-\alpha } \right) }$. When $K_s^1 =\frac{1}{F^{-1}\left( {1-\alpha } \right) }$ is defined, we derive $Q_s =K_s D$ from the objective function, where $K_s =1/{\Lambda ^{-1}\left( {\frac{c}{w}} \right) }$. By combining these results, we obtain $Q_s^*=\left\{ {\begin{array}{lll} K_s^1 D&{} \textit{if} &{} \alpha \hbox {>}\overline{F} \left( {1/{K_{s}}}\right) \\ K_s D&{} \textit{if} &{} \alpha \le \overline{F} \left( {1/{K_s }} \right) \\ \end{array}} \right.$.

Proof of Corollary 1

With Eqs. (12) and (13), we observe that the optimal profits of the manufacturer and the supplier are independent of service level $\alpha$ if $\alpha \le \overline{F} \left( {1/{K_s }} \right)$. If $\alpha >\overline{F} \left( {1/{K_s }} \right)$, $\overline{\Pi }_s^{SS} =wDK_s^1 \left[ {\Lambda \left( {\frac{1}{K_s^1 }} \right) +\frac{\alpha }{K_s^1 }-\frac{c}{w}} \right]$ can be written as $\overline{\Pi }_s^{SS} =D\left[ {\frac{w\Lambda \left( {F^{-1}\left( {1-\alpha } \right) } \right) -c}{F^{-1}\left( {1-\alpha } \right) }+w\alpha } \right]$ based on Eq. (13). By differentiating this equation with respect to $\alpha$, we obtain the following:

$$\begin{aligned} \frac{d\overline{\Pi }_s^{SS} }{d\alpha }= & {} \left( {\frac{w\Lambda \left( {F^{-1}\left( {1-\alpha } \right) } \right) -c}{F^{-1}\left( {1-\alpha } \right) }} \right) ^{{\prime }}D+wD \nonumber \\= & {} \left\{ {\frac{\left( {w\left( {F^{-1}\left( {1-\alpha } \right) } \right) ^{2}f\left( {F^{-1}\left( {1-\alpha } \right) } \right) -w\Lambda \left( {F^{-1}\left( {1-\alpha } \right) } \right) +c} \right) \left( {\frac{1}{-f\left( {F^{-1}\left( {1-\alpha } \right) } \right) }} \right) }{\left( {F^{-1}\left( {1-\alpha } \right) } \right) ^{2}}+w} \right\} D\nonumber \\= & {} \frac{\left( {w\Lambda \left( {F^{-1}\left( {1-\alpha } \right) } \right) -c} \right) D}{\left( {F^{-1}\left( {1-\alpha } \right) } \right) ^{2}f\left( {F^{-1}\left( {1-\alpha } \right) } \right) }. \end{aligned}$$

(19)

Given $\alpha >\overline{F} \left( {1/{K_s }} \right)$, we obtain $F^{-1}\left( {1-\alpha } \right) <\Lambda ^{-1}\left( {\frac{c}{w}} \right)$, which is equal to $w\Lambda \left( {F^{-1}\left( {1-\alpha } \right) } \right) -c<0$. Therefore, the numerator of the right-hand side of (19) is negative and we derive $\frac{d\overline{\Pi }_s^{SS} }{d\alpha }<0$. Similarly, we can verify $\frac{d\overline{\Pi }_m^{SS} }{d\alpha }>0$ if $\alpha >\overline{F} \left( {1/{K_s }} \right)$.

Proof of Proposition 5

The proof of the first part of this proposition can be derived by following the same approach as that employed in Proposition 2. We prove the second part of this proposition as follows: if $\alpha >\overline{F} \left( {1/{K_s }} \right)$, substituting $Q_s =\frac{Q_m }{F^{-1}\left( {1-\alpha } \right) }$ into Eq. (16) will yield the following:

$$\begin{aligned} \Pi _{m} =pE\left[ {\min \left( {Y\frac{Q_m }{F^{-1}\left( {1-\alpha } \right) },D,Q_m } \right) } \right] -wE\left[ {\min \left( {Y\frac{Q_m }{F^{-1}\left( {1-\alpha } \right) },Q_m } \right) } \right] .\quad \quad \end{aligned}$$

(20)

We analyze Eq. (20) case by case.

• (1) $Q_m \le D$

In this case, Eq. (20) can be simplified as follows:

$$\begin{aligned} \Pi _{m} =\left( {p-w} \right) Q_m \left\{ {\int \limits _{0}^{F^{-1}({1-\alpha })}} \frac{yf\left( y \right) }{F^{-1}\left( {1-\alpha } \right) }dy+{\int \limits _{F^{-1}\left( {1-\alpha } \right) }^{1}} {f\left( y \right) dy} \right\} . \end{aligned}$$

(21)

By taking the derivative of Eq. (21) with respect to $Q_m$, we obtain $\frac{d\Pi _{m} }{dQ_m }>0$. Therefore, the optimal value $Q_m^*$ is obtained at the boundary point D.

• (2) $Q_m \ge D$

In this case, Eq. (20) can be written as follows:

$$\begin{aligned} \Pi _{m}= & {} p\left\{ {{\int \limits _{0}^{\frac{DF^{-1}\left( {1-\alpha } \right) }{Q_m }}} {\frac{yQ_m }{F^{-1}\left( {1-\alpha } \right) }f\left( y \right) dy+{\int \limits _{\frac{DF^{-1}\left( {1-\alpha } \right) }{Q_m }}^{1}} {Df\left( y \right) dy} } } \right\} \nonumber \\&-\,wQ_m \left\{ {{\int \limits _0^{F^{-1}\left( {1-\alpha } \right) }} {\frac{y}{F^{-1}\left( {1-\alpha } \right) }f\left( y \right) dy+{\int \limits _{F^{-1}\left( {1-\alpha } \right) }^{1}} {f\left( y \right) dy} } } \right\} . \end{aligned}$$

(22)

By taking the first and second derivatives of Eq. (22) with respect to $Q_m$, we obtain the following:

$$\begin{aligned} \frac{d\Pi _{m} }{dQ_m }= & {} p{\int \limits _{0}^{\frac{DF^{-1}\left( {1-\alpha } \right) }{Q_m }}s} {\frac{y}{F^{-1}\left( {1-\alpha } \right) }f\left( y \right) dy} -w\left\{ {\int \limits _0^{F^{-1}\left( {1-\alpha } \right) }} \frac{y}{F^{-1}\left( {1-\alpha } \right) }f\left( y \right) dy\right. \nonumber \\&\left. +\,{\int \limits _{F^{-1}\left( {1-\alpha } \right) }^{1}} {f\left( y \right) dy} \right\} , \end{aligned}$$

(23)

$$\begin{aligned} \frac{d^{2}\Pi _{m} }{dQ_m^2 }= & {} p\frac{-D^{2}F^{-1}\left( {1-\alpha } \right) }{Q_m^3 }f\left( {\frac{DF^{-1}\left( {1-\alpha } \right) }{Q_m }} \right) <0. \end{aligned}$$

(24)

Given that $\frac{d^{2}\Pi _{m}}{dQ_m^2 }<0$, the expected profit in Eq. (22) is concave on $Q_m$. By defining $\frac{DF^{-1}\left( {1-\alpha } \right) }{Q_m }\hbox {=}\frac{1}{K_m^1 }$ and $\frac{1}{F^{-1}\left( {1-\alpha } \right) }=K_s^1$, setting Eq. (23) to zero will yield the following:

$$\begin{aligned} {\int \limits _{0}^{\frac{1}{K_m^1 }}} {yf\left( y \right) dy} =\frac{w}{p}\left\{ {{\int \limits _0^{\frac{1}{K_s^{1}}}} {yf\left( y \right) dy+\frac{\alpha }{K_s^1 }} } \right\} . \end{aligned}$$

(25)

From Eq. (25), we derive the optimal order quantity $Q_m^*=\frac{K_m^1 D}{K_s^1 }$. The result holds if and only if $K_m^1 \ge K_s^1$ because the condition $Q_m \ge D$ must be satisfied.

By combining these two cases, we obtain the results presented in Proposition 5.

Proof of Corollary  2

By setting Eq. (23) to zero, the first order optimal condition can be rewritten as follows:

$$\begin{aligned} p{\int \limits _{0}^{\frac{DF^{-1}\left( {1-\alpha } \right) }{Q_m^*}}} {yf\left( y \right) dy} -w\left\{ {\int \limits _0^{F^{-1}\left( {1-\alpha } \right) }} {yf\left( y \right) dy+\alpha F^{-1}\left( {1-\alpha } \right) } \right\} =0. \end{aligned}$$

(26)

For notational simplicity, we define the following:

$$\begin{aligned} T\left( \alpha \right)= & {} F^{-1}\left( {1-\alpha } \right) \hbox { and } G\left( {Q_m ,\alpha } \right) =p{\int \limits _0^{\frac{DT\left( \alpha \right) }{Q_m }}} {yf\left( y \right) dy}\nonumber \\&-\,w\left\{ {{\int \limits _0^{T\left( \alpha \right) }} {yf\left( y \right) dy+\alpha T\left( \alpha \right) } } \right\} . \end{aligned}$$

By taking the derivative of $G\left( {Q_m ,\alpha } \right)$ with respect to $\alpha$, we obtain the following:

$$\begin{aligned} \frac{\partial G}{\partial \alpha }=\left\{ {p\frac{D^{2}T\left( \alpha \right) }{Q_m^2 }f\left( {\frac{DT\left( \alpha \right) }{Q_m }} \right) -w\alpha } \right\} T^{{\prime }}\left( \alpha \right) , \end{aligned}$$

(27)

where $T^{{\prime }}\left( \alpha \right) =\frac{dF^{-1}\left( {1-\alpha } \right) }{d\alpha }=-\frac{1}{f\left( {F^{-1}\left( {1-\alpha } \right) } \right) }<0$. From Eq. (26), we derive the following:

$$\begin{aligned} w\alpha \hbox {=}\frac{1}{T\left( \alpha \right) }\left\{ {p{\int \limits _0^{\frac{DT\left( \alpha \right) }{Q_m }}} {yf\left( y \right) dy} -w{\int \limits _{0}^{T\left( \alpha \right) }} {yf\left( y \right) dy} } \right\} . \end{aligned}$$

(28)

Substituting Eq. (28) into Eq. (27) will yield the following:

$$\begin{aligned}&p\frac{D^{2}T\left( \alpha \right) }{Q_m^2 }f\left( {\frac{DT\left( \alpha \right) }{Q_m }} \right) -\frac{1}{T\left( \alpha \right) }p{\int \limits _0^{\frac{DT\left( \alpha \right) }{Q_m }}} {yf\left( y \right) dy} +\frac{1}{T\left( \alpha \right) }w{\int \limits _0^{T\left( \alpha \right) }} {yf\left( y \right) dy} \\&\quad \ge \, p\frac{D^{2}T\left( \alpha \right) }{Q_m^2 }f\left( {\frac{DT\left( \alpha \right) }{Q_m }} \right) -\frac{1}{T\left( \alpha \right) }p\frac{DT\left( \alpha \right) }{Q_m }\frac{DT\left( \alpha \right) }{Q_m }f\left( {\frac{DT\left( \alpha \right) }{Q_m }} \right) \\&\qquad +\,\frac{1}{T\left( \alpha \right) }w{\int \limits _0^{T\left( \alpha \right) }} {yf\left( y \right) dy} \\&\quad =\frac{1}{T\left( \alpha \right) }w{\int \limits _{0}^{T\left( \alpha \right) }} {yf\left( y \right) dy} >0 \end{aligned}$$

where the first inequality holds because ${\int \nolimits _0^{\frac{DT\left( \alpha \right) }{Q_m }}} {yf\left( y \right) dy} \le \frac{DT\left( \alpha \right) }{Q_m }\frac{DT\left( \alpha \right) }{Q_m }f\left( {\frac{DT\left( \alpha \right) }{Q_m }} \right)$. Therefore, we obtain $\frac{\partial G}{\partial \alpha }<0$. Taking the derivative of $G\left( {Q_m ,\alpha } \right)$ with respect to $Q_m$ will yield the following:

$$\begin{aligned} \frac{\partial G}{\partial Q_m }=-p\frac{DT\left( \alpha \right) }{Q_m^2 }\frac{DT\left( \alpha \right) }{Q_m }f\left( {\frac{DT\left( \alpha \right) }{Q_m }} \right) <0. \end{aligned}$$

According to implicit function theory, we obtain $\frac{dQ_m^*}{d\alpha }=-\frac{{\partial G}/{\partial \alpha }}{{\partial G}/{\partial Q_m^*}}<0$, which indicates that the optimal order quantity $Q_m^*$ decreases in service level $\alpha$.

Proof of Corollary  3

It is easily to show the first part of this proposition. We examine the case when $\alpha >\overline{F} \left( {1/{K_s }} \right)$.

Case 1 $K_m^1 \ge K_s^1$. From Eq. (17), we derive $\overline{\Pi }_m^{MS} =pD\left( {1-F\left( {\frac{1}{K_m^1 }} \right) } \right)$. By substituting $Q_m^*=K_m^1 \frac{D}{K_s^1 }$ into this equation, we obtain the following:

$$\begin{aligned} \overline{\Pi }_m^{MS} =pD\left( {1-F\left( {\frac{D}{Q_m^*}F^{-1}\left( {1-\alpha } \right) } \right) } \right) . \end{aligned}$$

(29)

For notational simplicity, we define $A=\frac{D}{Q_m^*}F^{-1}\left( {1-\alpha } \right)$ and then take the derivative of Eq. (29) with respect to $\alpha$ to obtain $\frac{d\overline{\Pi }_m^{MS} }{d\alpha }=-\frac{dF}{dA}\frac{dA}{d\alpha }$. From Eq. (26), we obtain $\int _0^{\frac{DF^{-1}\left( {1-\alpha } \right) }{Q_m^*}} {yf\left( y \right) dy} =\frac{w}{p}\left\{ {\int _0^{F^{-1}\left( {1-\alpha } \right) } {yf\left( y \right) dy+\alpha F^{-1}\left( {1-\alpha } \right) } } \right\}$. By defining $B=\int _0^{F^{-1}\left( {1-\alpha } \right) } {yf\left( y \right) dy+\alpha F^{-1}\left( {1-\alpha } \right) }$ and noting $\Lambda \left( x \right) =\int _0^x {yf\left( y \right) } dy$, we obtain $\int _0^{\frac{DF^{-1}\left( {1-\alpha } \right) }{Q_m^*}} {yf\left( y \right) dy} =\Lambda \left( A \right) =\frac{wB}{p}$, which is equal to $A=\Lambda ^{-1}\left( {\frac{wB}{p}} \right)$. From $\frac{dB}{d\alpha }=-w\alpha \frac{1}{f\left( {F^{-1}\left( {1-\alpha } \right) } \right) }<0$, we derive $\frac{dA}{d\alpha }=\frac{w\frac{dB}{d\alpha }}{pf\left( {{wB}/p} \right) }<0$. Therefore, $\frac{d\overline{\Pi }_m^{MS} }{d\alpha }=-\frac{dF}{dA}\frac{dA}{d\alpha }>0$.

Case 2 $K_m^1 \le K_s^1$. From Eq. (17), we derive $\overline{\Pi }_m^{MS} =\left( {p-w} \right) K_s^1 D\left( {\Lambda \left( {\frac{1}{K_s^1 }} \right) +\frac{\alpha }{K_s^1 }} \right)$. After simplification, we obtain $\overline{\Pi }_m^{MS} =\left( {p-w} \right) D\left( {K_s^1 \Lambda \left( {\frac{1}{K_s^1 }} \right) +\alpha } \right)$. Differentiating this equation with respect to $\alpha$ will yield $\frac{d\overline{\Pi }_m^{MS} }{d\alpha }=\left( {K_s^1 } \right) ^{2}\Lambda \left( {\frac{1}{K_s^1 }} \right) f\left( {\frac{1}{K_s^1 }} \right) >0$.

By combining these cases, we derive that the expected profit of the manufacturer increases with $\alpha$. Similarly, we verify that $\frac{d\overline{\Pi }_s^{MS} }{d\alpha }<0$; that is, the expected profit of the supplier decreases with $\alpha$.

Proof of Proposition 7

$\Pi _s^{SS} =\Pi _s^{VN} \le \Pi _s^{MS}$ can easily be verified by comparing the profits of the supplier in Table 1. With respect to the manufacturer, if $K_m \le K_s$, then $\Pi _m^{MS} =\left( {p-w} \right) D\left[ {1+K_s \frac{c}{w}-F\left( {\frac{1}{K_s }} \right) } \right]$ and we obtain $\Pi _m^{SS} =\Pi _m^{VN} =\Pi _m^{MS}$. If $K_m =K_s$, we derive $\frac{c}{p}+\frac{w}{p}\cdot \frac{1-F\left( {1/{K_s }} \right) }{K_s }\hbox {=}\frac{c}{w}$, which is equal to $-pK_s \frac{c}{w}-wF\left( {\frac{1}{K_s }} \right) +w+K_s c\hbox {=}0$, from $\int _0^{\frac{1}{K_m }} {yf\left( y \right) dy} =\frac{c}{p}+\frac{w}{p}\cdot \frac{1-F\left( {1/{K_s }} \right) }{K_s }$ and $\int _0^{\frac{1}{K_s }} {yf\left( y \right) dy} =\frac{c}{w}$. Therefore, we obtain the following:

$$\begin{aligned}&pD\left[ {1-F\left( {\frac{1}{K_m }} \right) } \right] -\left( {p-w} \right) D\left[ {1+K_s^ \frac{c}{w}-F\left( {\frac{1}{K_s }} \right) } \right] \nonumber \\&\quad =\left\{ {p\left( {F\left( {\frac{1}{K_s }} \right) -F\left( {\frac{1}{K_m }} \right) } \right) -pK_s^ \frac{c}{w}-wF\left( {\frac{1}{K_s }} \right) +w+K_s^ c} \right\} \hbox {=0} , \end{aligned}$$

(30)

which indicates that $\Pi _m^{MS}$ is continuous at $K_m = K_s$. According to the proof of the manufacturer’s optimal order quantity, $\Pi _m^{MS}$ is a linear function of the order quantity Q when $Q\le D$ (i.e., $K_m \le K_s)$ and is a concave function of order quantity Q when $Q\ge D$ (i.e., $K_m \ge K_s)$, Therefore, we obtain $\Pi _m^{MS} \ge \Pi _m^{SS} =\Pi _m^{VN}$ when $K_m \ge K_s$.

Proof of Proposition 8

If $\alpha \le \overline{F} \left( {1/{K_s }} \right)$, we derive $\overline{\Pi }_s^{SS} =\overline{\Pi }_s^{VN} =wD\left[ {1-F\left( {\frac{1}{K_s^1 }} \right) } \right]$ and $\overline{\Pi }_s^{MS} =\left\{ {\begin{array}{lll} w\frac{K_m }{K_s }D\left[ {1-F\left( {\frac{1}{K_s }} \right) } \right] &{} \textit{if} &{} K_m^ \ge K_s \\ wD\left[ {1-F\left( {\frac{1}{K_s }} \right) } \right] &{} \textit{if} &{} K_m \le K_s\\ \end{array}} \right.$ from Table  2. Therefore, we obtain $\overline{\Pi }_s^{SS} =\overline{\Pi }_s^{VN} \le \overline{\Pi }_s^{MS}$. If $\alpha >\overline{F} \left( {1/{K_s }} \right)$, then $\overline{\Pi }_s^{SS} =\overline{\Pi }_s^{VN} \le \overline{\Pi }_s^{MS}$ can be easily observed. Given that corollary 4 shows the upper bounds (i.e., ${\alpha }^{MS}={\alpha }^{SS}={\alpha }^{VN})$ of the service level requirement under different game theoretic models, the definition of $\overline{\alpha }={\alpha }^{MS}={\alpha }^{SS}={\alpha }^{VN}$ must satisfy the inequality $\alpha \le \overline{\alpha }$. With regard to the profit of the manufacturer, $\overline{\Pi }_m^{SS} =\overline{\Pi }_m^{VN} \le \overline{\Pi }_m^{MS}$ if $\alpha \le \overline{F} \left( {1/{K_s }} \right)$. If $\alpha >\overline{F} \left( {1/{K_s }} \right)$ and $K_m^1 \le K_s^1$, we observe that. $\overline{\Pi }_m^{MS} =\overline{\Pi }_m^{SS}$. If $\alpha >\overline{F} \left( {1/{K_s }} \right)$ and $K_m^1 \ge K_s^1$, $\overline{\Pi }_m^{MS} =pD\left( {1-F\left( {\frac{1}{K_m^1 }} \right) } \right)$, $\overline{\Pi }_m^{MS}$ becomes continuous at $K_m^1 \,{=}\,K_s^1$. According to the proof of Proposition 5, $\overline{\Pi }_m^{MS}$ is a linear function of the order quantity Q when $Q\le D$ (i.e., $K_m^1 \le K_s^1 )$ and is a concave function of order quantity Q when $Q\ge D$ (i.e., $K_m^1 \ge K_s^1 )$. Therefore, $\overline{\Pi }_m^{MS} \ge \overline{\Pi }_m^{SS}$ when $K_m^1 \ge K_s^1$. By combining these analyses, we obtain the results in Proposition 8.

Proof of Proposition 9

From Tables 1 and 2, we derive $\Pi _s^{SS} =\overline{\Pi }_s^{SS}$ if $\alpha \le \overline{F} \left( {1/{K_s }} \right)$. If $\alpha >\overline{F} \left( {1/{K_s }} \right)$, then $\Pi _s^{SS} =wD\overline{F} \left( {1/{K_s }} \right)$ and $\overline{\Pi }_s^{SS} =wDK_s^1 \left[ {\Lambda \left( {\frac{1}{K_s^1 }} \right) +\frac{\alpha }{K_s^1 }-\frac{c}{w}} \right]$. The continuity of $\overline{\Pi }_s^{SS}$ and the monotonicity of $\overline{\Pi }_s^{SS}$ in Proposition  4 indicate that $wD\overline{F} \left( {1/{K_s }} \right) \ge wDK_s^1 \left[ {\Lambda \left( {\frac{1}{K_s^1 }} \right) +\frac{\alpha }{K_s^1 }-\frac{c}{w}} \right]$, that is, $\Pi _s^{SS} \ge \overline{\Pi }_s^{SS}$. Similarly, we can obtain the other results according to the continuity and monotonicity properties of the expected profits of both firms.