Coordinating a decentralized hybrid push–pull assembly system with unreliable supply and uncertain demand

Abstract

We consider a decentralized assembly system in which $m+n$ unreliable suppliers sell complementary components to an assembler, who faces a random demand. We assume that $m$ of these suppliers are more powerful than the assembler and sell components via push contracts, while the remaining $n$ suppliers are less powerful than the assembler and sell components via pull contracts. The supply chain operations are modelled by a three-stage game. We first characterize the equilibrium decisions of all chain members, and find that supply chain efficiency depends heavily on system parameters. We then develop a mechanism to coordinate the supply chain. The mechanism consists of two policies, including a buyback policy between push suppliers and the assembler, and a subsidy policy between the assembler and pull suppliers. We show that the mechanism can eliminate the two sources of “double marginalization” that exist in the chain with unreliable supply and uncertain demand, and achieve Pareto improvements.

Appendix: Mathematical proofs

1.1 Appendix 1: Proof of Lemma 2

(i) We first prove that for any given $q_{I}$, the assembler should set $\{w_{j}\}$ such that $F^{-1}\left( 1-\frac{c_{m+1}}{\beta _{I}\beta _{m+1}w_{m+1}}\right) =\cdots =F^{-1} \left( 1-\frac{c_{m+n}}{\beta _{I}\beta _{m+n}w_{m+n}}\right)$. If there exists a $w_{j^{\prime }}$, such that $F^{-1}\left( 1-\frac{c_{j^{\prime }}}{\beta _{I}\beta _{j^{\prime }}w_{j^{\prime }}}\right) <\min \big [F^{-1}\left( 1-\frac{c_{j}}{\beta _{I}\beta _{j}w_{j}}\right) , j=m+1,\ldots ,j^{\prime }-1,j^{\prime }+1,\ldots ,m+n\big ]$, we know from Eq. (10) that the assembler’s expected profit function becomes

$$\begin{aligned}&\pi _{a}(q_{I}, w_{m+1},\ldots ,w_{m+n})\\&\quad =\beta _{I}\left( \beta _{J}p -\sum _{j=m+1}^{m+n}\beta _{j}w_{j}\right) E\left\{ \min \left[ X, q_{I},F^{-1} \left( 1-\frac{c_{j^{\prime }}}{\beta _{I}\beta _{j^{\prime }}w_{j^{\prime }}}\right) \right] \right\} -\sum _{i=1}^{m}\beta _{i}w_{i}q_{I}. \end{aligned}$$

Then, by reducing $w_{j}$, $j\ne j^{\prime }$, to a value such that $F^{-1}\left( 1-\frac{c_{j^{\prime }}}{\beta _{I}\beta _{j^{\prime }}w_{j^{\prime }}}\right) = \min \big [F^{-1}\left( 1-\frac{c_{j}}{\beta _{I}\beta _{j}w_{j}}\right) , j=m+1,\ldots ,j^{\prime }-1,j^{\prime }+1,\ldots ,m+n\big ]$, the assembler can increase its expected profit. Thus, the assembler should set $\{w_{j}\}$ such that $F^{-1}\left( 1-\frac{c_{m+1}}{\beta _{I}\beta _{m+1}w_{m+1}} \right) =\cdots =F^{-1}\left( 1-\frac{c_{m+n}}{\beta _{I}\beta _{m+n}w_{m+n}} \right)$.

Based upon the above analysis, we know that the assembler can set $w_{m+1}$ first, and then determine $(w_{m+2},\ldots ,w_{m+n})$ according to $\frac{c_{m+1}}{\beta _{m+1}w_{m+1}}\ldots = \frac{c_{m+n}}{\beta _{m+n}w_{m+n}}$.

(ii) Now, together with $\frac{c_{m+1}}{\beta _{m+1}w_{m+1}}\ldots = \frac{c_{m+n}}{\beta _{m+n}w_{m+n}}$, we prove that the assembler should set $w_{m+1}$ and $q_{I}$ such that $q_{I}=F^{-1}\left( 1-\frac{c_{m+1}}{\beta _{I}\beta _{m+1}w_{m+1}}\right)$. If $q_{I}>F^{-1}\left( 1-\frac{c_{m+1}}{\beta _{I}\beta _{m+1}w_{m+1}}\right)$, the assembler’s expected profit function of Eq. (10) becomes

$$\begin{aligned} \pi _{a}(q_{I}, w_{m+1})= & {} \beta _{I}\left( \beta _{J}p-\frac{\beta _{m+1}w_{m+1}C_{J}}{c_{m+1}} \right) E\left\{ \min \left[ X, F^{-1}\left( 1-\frac{c_{m+1}}{\beta _{I}\beta _{m+1}w_{m+1}}\right) \right] \right\} \\&-\sum _{i=1}^{m}\beta _{i}w_{i}q_{I}. \end{aligned}$$

Then, by reducing $q_{I}$ to a value such that $q_{I}=F^{-1}\left( 1-\frac{c_{m+1}}{\beta _{I}\beta _{m+1}w_{m+1}}\right)$, the assembler can improve its expected profit.

If $q_{I}<F^{-1}\left( 1-\frac{c_{m+1}}{\beta _{I}\beta _{m+1}w_{m+1}}\right)$, the assembler’s expected profit function of Eq. (10) becomes

$$\begin{aligned} \pi _{a}(q_{I}, w_{m+1})=\beta _{I}\left( \beta _{J}p-\frac{\beta _{m+1}w_{m+1}C_{J}}{c_{m+1}} \right) E\left[ \min \left( X,q_{I}\right) \right] -\sum _{i=1}^{m}\beta _{i}w_{i}q_{I}. \end{aligned}$$

Then, by reducing $w_{m+1}$ to a value such that $F^{-1}\left( 1-\frac{c_{m+1}}{\beta _{I}\beta _{m+1}w_{m+1}}\right) =q_{I}$, the assembler can again improve its expected profit. Combining (i) and (ii), we complete the Proof of Lemma 2.

1.2 Appendix 2: Proof of Lemma 3

Assume that $h(q_{I})$ increases in $q_{I}$, then $\frac{d \pi _{a}(q_{I})}{d q_{I}}$ of Eq. (13) decreases in $q_{I}$. Note that $\lim \limits _{q_{I}\rightarrow 0}\frac{d \pi _{a}(q_{I})}{d q_{I}}=\beta _{I}\beta _{J}p-C_{J}-\sum _{i=1}^{m}\beta _{i}w_{i}>0$, and $\lim \limits _{q_{I}\rightarrow +\infty }\frac{d \pi _{a}(q_{I})}{d q_{I}}<0$. Consequently, there exists a unique $q_{I}^{*}\in (0,+\infty )$ such that $\frac{d \pi _{a}(q_{I})}{d q_{I}}=0$ at $q_{I}=q_{I}^{*}$, $\frac{d \pi _{a}(q_{I})}{d q_{I}}>0$ for $q_{I}<q_{I}^{*}$, and $\frac{d \pi _{a}(q_{I})}{d q_{I}}<0$ for $q_{I}>q_{I}^{*}$. So, $\pi _{a}(q_{I})$ is unimodal in $q_{I}$ and reaches its maximizer at $q_{I}^{*}$.

1.3 Appendix 3: Proof of Lemma 4

Assume that $h(q_{I}^{*})$, $g(q_{I}^{*})$, and $\frac{q_{I}^{*}f^{\prime }(q_{I}^{*})}{f(q_{I}^{*})}+2g(q_{I}^{*})$ all increase in $q_{I}^{*}$.

(i) If there exists a $q^{\prime }$ on $(0,+\infty )$ that satisfies $g(q^{\prime })=1$, then $\frac{d \pi _{i}(q_{I}^{*})}{d q_{I}^{*}}$ of Eq. (17) decreases in $q_{I}^{*}$ for $q_{I}^{*}\in (0, q^{\prime }]$ and is negative for $q_{I}^{*}\in (q^{\prime }, +\infty )$. In conjunction with the fact that $\lim \limits _{q_{I}^{*}\rightarrow 0}\frac{d \pi _{i}(q_{I}^{*})}{d q_{I}^{*}}= \beta _{I}\beta _{J}p-C_{J}-\sum _{l=1,l\ne i}^{m}\beta _{l}w_{l}-c_{i}>0$, we know there exists a unique $q^{\prime \prime }\in (0, q^{\prime }]$ such that $\frac{d \pi _{i}(q_{I}^{*})}{d q_{I}^{*}}=0$ at $q_{I}^{*}=q^{\prime \prime }$, $\frac{d \pi _{i}(q_{I}^{*})}{d q_{I}^{*}}>0$ for $q_{I}^{*}<q^{\prime \prime }$, and $\frac{d \pi _{i}(q_{I}^{*})}{d q_{I}^{*}}<0$ for $q_{I}^{*}>q^{\prime \prime }$. Thus, $\pi _{i}(q_{I}^{*})$ is unimodal in $q_{I}^{*}$.

(ii) If such a $q^{\prime }$ does not exist, it means $g(q_{I}^{*})<1$ for $q_{I}^{*}>0$. Thus, $\frac{d \pi _{i}(q_{I}^{*})}{d q_{I}^{*}}$ decreases in $q_{I}^{*}$ for $q_{I}^{*}>0$. Consequently, $\pi _{i}(q_{I}^{*})$ is again unimodal in $q_{I}^{*}$. Combining (i) and (ii), we complete the proof of Lemma 4.

1.4 Appendix 4: Proof of Corollary 1

If $q^{*}\ge q^{*}_{c}$, we have

$$\begin{aligned} \beta _{I}\beta _{J}p[1-F(q^{*}_{c})]\ge & {} \beta _{I}\beta _{J}p[1-F(q^{*})]>\beta _{I}\beta _{J}p[1-F(q^{*})] -\left( \sum _{i=1}^{m}\beta _{i}w_{i}^{*}-C_{I}\right) \\&-\,C_{J}h(q^{*}). \end{aligned}$$

Comparing Eq. (5) with Eqs. (14) and (19), we can get

$$\begin{aligned} \beta _{I}\beta _{J}p[1-F(q^{*}_{c})]=C= \beta _{I}\beta _{J}p[1-F(q^{*})]-\left( \sum _{i=1}^{m}\beta _{i}w_{i}^{*} -C_{I}\right) -C_{J}h(q^{*}). \end{aligned}$$

The above two expressions are contradictory. Thus, $q^{*}<q^{*}_{c}$.

1.5 Appendix 5: Proof of Corollary 2

(i) Because $h(q)$, $g(q)$ and $\frac{qf^{\prime }(q)}{f(q)}+2g(q)$ all increase in $q$, we know from Eq. (19) that: (a) $\frac{dq^{*}}{dc_{k}}<0$, $k=1,\ldots ,m+n$. (b) $\frac{dq^{*}}{dC_{J}}<0$ for any given $C$. Consequently, $\frac{d q^{*}}{d C_{I}}>0$, i.e., $\frac{d q^{*}}{d \alpha }>0$.

(ii) From Eq. (4), we have

$$\begin{aligned} \frac{d\pi _{c}(q^{*})}{dc_{k}}= & {} \beta _{I}\beta _{J}p[1-F(q^{*})]\frac{dq^{*}}{dc_{k}}- \left( q^{*} +C\frac{dq^{*}}{dc_{k}}\right) \\= & {} \left\{ \beta _{I}\beta _{J}p[1-F(q^{*})] -C\right\} \frac{dq^{*}}{dc_{k}}-q^{*}, \end{aligned}$$

and

$$\begin{aligned} \frac{d\pi _{c}(q^{*})}{d\alpha }= & {} \frac{d\pi _{c}(q^{*})}{dq^{*}} \frac{dq^{*}}{d\alpha }\\= & {} \{\beta _{I}\beta _{J}p[1-F(q^{*})]-C\}\frac{dq^{*}}{d\alpha }. \end{aligned}$$

Because $q^{*}<q_{c}^{*}$, then $\beta _{I}\beta _{J}p[1-F(q^{*})]-C>0$. In conjunction with $\frac{dq^{*}}{dc_{k}}<0$ and $\frac{dq^{*}}{d\alpha }>0$, we have $\frac{d\pi _{c}(q^{*})}{dc_{k}}<0$ and $\frac{d\pi _{c}(q^{*})}{d\alpha }>0$. Combining (i) and (ii), we complete the Proof of Corollary 2.

1.6 Appendix 6: Proof of Corollary 3

(i) Because $h(q)$, $g(q)$ and $\frac{qf^{\prime }(q)}{f(q)}+2g(q)$ all increase in $q$, we know from Eq. (19) that $\frac{dq^{*}}{d\beta _{k}}>0$, $k=1,\ldots ,m+n$. Second, we know from Eq. (4) that

$$\begin{aligned} \frac{d\pi _{c}(q^{*})}{d\beta _{k}}= & {} \Pi _{l=1,l\ne k}^{m+n}\beta _{l}p\int _0^{q^{*}} [1-F(x)]dx +\beta _{I}\beta _{J}p[1-F(q^{*})] \frac{dq^{*}}{d\beta _{k}}-C\frac{dq^{*}}{d\beta _{k}}\\= & {} \{\beta _{I}\beta _{J}p[1-F(q^{*})]-C\}\frac{dq^{*}}{d\beta _{k}}+ \Pi _{l=1,l\ne k}^{m+n}\beta _{l}p\int _0^{q^{*}} [1-F(x)]dx. \end{aligned}$$

Due to $q^{*}<q_{c}^{*}$, we have $\beta _{I}\beta _{J}p[1-F(q^{*})]-C>0$, and hence $\frac{d\pi _{c}(q^{*})}{d\beta _{k}}>0$, we complete the proof of Corollary 3 (i).

(ii) Corollary 3 (ii) can be easily verified from Eqs. (4) and (19).

1.7 Appendix 7: Proof of Corollary 4

(i) Because $h(q)$, $g(q)$ and $\frac{qf^{\prime }(q)}{f(q)}+2g(q)$ all increase in $q$, we know from Eq. (19) that $\frac{d q^{*}}{d m}<0$. Consequently, the marginal procurement cost $\sum _{j=m+1}^{m+n}\beta _{I}\beta _{j}w_{j}^{*}= \frac{C_{J}}{1-F(q^{*})}$ increases in $q^{*}$ and decreases in $m$. Then, from Eqs. (18) and (19), we can write the marginal procurement cost of components $1,\ldots ,m$ as $\sum _{i=1}^{m}\beta _{i}w_{i}^{*}= \beta _{I}\beta _{J}p[1-F(q^{*})]-C_{J}[1+h(q^{*})]$. Clearly, $\sum _{i=1}^{m}\beta _{i}w_{i}^{*}$ decreases in $q^{*}$ and increases in $m$.

(ii) It can be seen from Eq. (19) that the production quantity is not related to the number of pull suppliers. Consequently, we know from Eqs. (18) and (20) that the marginal procurement costs $\sum _{i=1}^{m}\beta _{i}w_{i}^{*}$ and $\sum _{j=m+1}^{m+n}\beta _{I}\beta _{j}w_{j}^{*}$ are also not related to the number of pull suppliers.

(iii) From Eq. (4), we have

$$\begin{aligned} \frac{d\pi _{c}(q^{*})}{dm}= & {} \frac{d\pi _{c}(q^{*})}{dq^{*}} \frac{dq^{*}}{dm}\\= & {} \{\beta _{I}\beta _{J}p[1-F(q^{*})]-C\}\frac{dq^{*}}{dm}. \end{aligned}$$

Due to $q^{*}<q_{c}^{*}$, we have $\beta _{I}\beta _{J}p[1-F(q^{*})]-C>0$, and hence $\frac{d\pi _{c}(q^{*})}{dm}<0$. Using the same analysis would verify $\frac{d\pi _{c}(q^{*})}{dn}=0$. Combining (i), (i) and (iii), we complete the Proof of Corollary 4.

1.8 Appendix 8: Proof of Theorem 2

Using backward induction, we first investigate the production decisions of pull suppliers, and then analyze the assembler’s decision problem.

(i) From Eq. (28), we can rewrite supplier $j$’s expected profit function as

$$\begin{aligned}&\Pi _{j}(q_{j})\\&\quad =\left\{ \begin{array}{l@{\quad }l} \textstyle \beta _{I}\beta _{j}\left\{ w_{j} \int _0^{q_{j}} [1-F(x)]dx+b_{j}\int _0^{q_{j}} F(x)dx\right\} -c_{j}q_{j},&{}q_{j}\le \min (q_{I},q_{Jj}),\\ \textstyle \beta _{I}\beta _{j}\left\{ w_{j} \int _0^{\min (q_{I},q_{Jj})} [1-F(x)]dx+b_{j}q_{j}\right. \\ \textstyle \left. -\,b_{j}\int _0^{\min (q_{I},q_{Jj})} [1-F(x)]dx\right\} -c_{j}q_{j},&{}q_{j}>\min (q_{I},q_{Jj}). \end{array} \right. \end{aligned}$$

Substituting $b_{j}=\frac{\frac{\beta _{J}c_{j}}{\beta _{j}}p-w_{j}C}{\beta _{I}\beta _{J}p-C}$ of Eq. (30) into the above expression and taking the first derivative of $\Pi _{j}(q_{j})$ with respect to $q_{j}$, after some algebra, we have

$$\begin{aligned} \frac{d \Pi _{j}(q_{j})}{d q_{j}}=\left\{ \begin{array}{l@{\quad }l} \textstyle (\beta _{I}\beta _{j}w_{j}-c_{j}) \left[ 1-\frac{\beta _{I}\beta _{J}p}{\beta _{I}\beta _{J}p-C}F(q_{j})\right] , &{}q_{j}\le \min (q_{I},q_{Jj}),\\ \textstyle \frac{-(\beta _{I}\beta _{j}w_{j}-c_{j})C}{\beta _{I}\beta _{J}p-C}, &{}q_{j}>\min (q_{I},q_{Jj}). \end{array} \right. \end{aligned}$$

In conjunction with the fact that $(\beta _{I}\beta _{j}w_{j}-c_{j})\left[ 1-\frac{\beta _{I}\beta _{J}p}{\beta _{I}\beta _{J}p-C}F(q_{j})\right]$ decreases in $q_{j}$, we know $\Pi _{j}(q_{j})$ is concave in $q_{j}$ for $q_{j}\le \min (q_{I},q_{Jj})$ and decreases in $q_{j}$ for $q_{j}>\min (q_{I},q_{Jj})$. Second, $(\beta _{I}\beta _{j}w_{j}- c_{j})\left[ 1-\frac{\beta _{I}\beta _{J}p}{\beta _{I}\beta _{J}p-C}F(q_{j})\right] >0$ at $q_{j}=0$, and $(\beta _{I}\beta _{j}w_{j}-c_{j}) \left[ 1-\frac{\beta _{I}\beta _{J}p}{\beta _{I}\beta _{J}p-C}F(q_{j})\right] =0$ at $q_{j}=q_{c}^{*}$. Thus, if $q_{c}^{*}\le \min (q_{I},q_{Jj})$, $\Pi _{j}(q_{j})$ increases in $q_{j}$ for $q_{j}\le q_{c}^{*}$ and decreases in $q_{j}$ for $q_{j}>q_{c}^{*}$; otherwise, $\Pi _{j}(q_{j})$ increases in $q_{j}$ for $q_{j}\le \min (q_{I},q_{Jj})$ and decreases in $q_{j}$ for $q_{j}>\min (q_{I},q_{Jj})$. This means that $\Pi _{j}(q_{j})$ is unimodal in $q_{j}$, and increases in $q_{j}$ for $q_{j}\le \min (q_{I},q_{Jj}, q_{c}^{*})$ and decreases in $q_{j}$ for $q_{j}>\min (q_{I},q_{Jj}, q_{c}^{*})$. Consequently, for any given $q_{I}$ and $q_{Jj}$, the optimal production quantity of supplier $j$ is $\min (q_{c}^{*}, q_{I},q_{Jj})$.

We denote

$$\begin{aligned} q_{Jc}^{*}=\min (q_{I},q_{c}^{*}). \end{aligned}$$

Similar to the illustration of Eq. (9), we can conclude that $q_{Jc}^{*}=\min (q_{I},q_{c}^{*})$ is the reaction function of pull suppliers here.

(ii) Substituting $q_{j}=q_{J}=q_{Jc}^{*}=\min (q_{I},q_{c}^{*})$ into Eq. (27), we can rewrite the assembler’s expected profit function as

$$\begin{aligned}&\Pi _{a}(q_{I},w_{m+1},\ldots ,w_{m+n},b_{m+1},\ldots ,b_{m+n})\\&\quad =\left\{ \begin{array}{ll} \textstyle \beta _{I}\beta _{J}p\int _0^{q_{I}} [1-F(x)]dx-\sum \nolimits _{i=1}^{m}\beta _{i}w_{i}q_{I}+\sum \nolimits _{i=1}^{m}\beta _{i}b_{i}\int _0^{q_{I}}F(x)dx&{}\\ \textstyle - \sum \nolimits _{j=m+1}^{m+n}\beta _{I}\beta _{j}w_{j}\int _0^{q_{I}} [1-F(x)]dx-\sum \nolimits _{j=m+1}^{m+n}\beta _{I}\beta _{j}b_{j} \int _0^{q_{I}}F(x)dx,&{}q_{I}\le q_{c}^{*},\\ \textstyle \beta _{I}\beta _{J}p\int _0^{q_{c}} [1-F(x)]dx-\sum \nolimits _{i=1}^{m}\beta _{i} w_{i}q_{I}&{}\\ \textstyle - \sum \nolimits _{j=m+1}^{m+n}\beta _{I}\beta _{j}w_{j}\int _0^{q_{c}} [1-F(x)]dx-\sum \nolimits _{j=m+1}^{m+n}\beta _{I}\beta _{j}b_{j}\int _0^{q_{c}}F(x)dx,&{}q_{I}>q_{c}^{*}. \end{array} \right. \end{aligned}$$

Substituting $b_{i}=\frac{\left( w_{i}-\frac{c_{i}}{\beta _{i}} \right) \beta _{I}\beta _{J}p}{\beta _{I}\beta _{J}p-C}$ of Eq. (29) and $b_{j}=\frac{\frac{\beta _{J}c_{j}}{\beta _{j}}p -w_{j}C}{\beta _{I}\beta _{J}p-C}$ of Eq. (30) into the above expression and taking the first derivative of $\Pi _{a}(q_{I},w_{m+1},\ldots ,w_{m+n})$ with respect to $q_{I}$, after some algebra, we have

$$\begin{aligned}&\frac{d \Pi _{a}(q_{I},w_{m+1},\ldots ,w_{m+n})}{d q_{I}}\\&\quad =\left\{ \begin{array}{l@{\quad }l} \textstyle \left( \beta _{I}\beta _{J}p-\sum \nolimits _{i=1}^{m}\beta _{i} w_{i}-\sum \nolimits _{j=m+1}^{m+n}\beta _{I}\beta _{j}w_{j}\right) \left[ 1-\frac{\beta _{I}\beta _{J}p}{\beta _{I}\beta _{J}p-C}F(q_{I})\right] , &{}q_{I}\le q_{c}^{*},\\ \textstyle -\sum \nolimits _{i=1}^{m}\beta _{i} w_{i},&{}q_{I}>q_{c}^{*}. \end{array} \right. \end{aligned}$$

From the above expression, we are able to compute that $\frac{d \Pi _{a}(q_{I},w_{m+1},\ldots ,w_{m+n})}{d q_{I}}>0$ for $q_{I}<q_{c}^{*}$, $\frac{d\Pi _{a}(q_{I},w_{m+1},\ldots ,w_{m+n})}{d q_{I}}<0$ for $q_{I}>q_{c}^{*}$, and $\frac{d\Pi _{a}(q_{I},w_{m+1},\ldots ,w_{m+n})}{d q_{I}}=0$ for $q_{I}=q_{c}^{*}$. Thus, the unique maximizer of the assembler’s expected profit function is given by $q_{c}^{*}$. Combining (i) and (ii), we complete the Proof of Theorem 2.

1.9 Appendix 9: Proof of Corollary 6

In Eqs. (34), (35) and (36) can be obtained by solving $\Pi _{i}(w_{i})>\pi _{i}^{*}$, $\Pi _{j}>\pi _{j}^{*}$ and $\Pi _{a}(w_{m+1},\ldots ,w_{m+n})>\pi _{a}^{*}$, respectively.