Selection of Supply Chain Contracts: The Case of Option Contracts

Abstract

Chapter 7 focuses on exploring issues concerning supply chain contract selection/implementation with the option contracts under consideration. From existing research it is known that various contracts have been developed to attain supply chain coordination and ensure arbitrary allocation of the resulting coordinating profit. However, since the extents to which the individuals involved improve their profits are different with different coordinating contracts, an important issue that remains to be resolved is how to select a coordinating contract that is acceptable for all the contracting partners. In Chap. 7 an effort to address this issue is made with the consideration of option contracts. In this research, the cooperative game approach is taken to consider the supply chain coordination issue with option contracts and to develop the contract negotiation model, taking into account supply chain members’ risk preferences and negotiating powers. The negotiation models developed with the option contract can be easily extended to other types of contracts such as the buyback contract, the revenue-sharing contract, the sales-rebate contract, etc. In this sense the research of this chapter presents a theoretical modeling framework for the selection/implementation issue of supply chain contracts.

The research of this chapter is based on Zhao et al. (2010).

Appendix: Proofs of the Main Results

Proof of Proposition 7.4.1.

From (7.4.1), the retailer’s problem with the wholesale price mechanism is to solve

$$\displaystyle{ \begin{array}{lll} \mathrm{P_{7A.1}:}\ \ \ \max \limits _{Q_{wr}\geq 0}E\Pi _{wr}(Q_{wr}) = E[p\min \{Q_{wr},X\} - wQ_{wr} + v\max \{Q_{wr} - X,0\}].\end{array} }$$

(7.7.1)

With some algebra, it is obtained from (7.7.1) that

$$\displaystyle{ \begin{array}{lll} E\Pi _{wr}(Q_{wr})& =&E[p\min \{Q_{wr},X\} - wQ_{wr} + v\max \{Q_{wr} - X,0\}] \\ & =&p\int _{0}^{Q_{wr}}xdF(x) + p\int _{Q_{ wr}}^{+\infty }Q_{ wr}dF(x) - wQ_{wr} + v\int _{0}^{Q_{wr}}(Q_{ wr} - x)dF(x) \\ & =&(p - w)Q_{wr} - (p - v)\int _{0}^{Q_{wr}}F(x)dx.\end{array} }$$

(7.7.2)

Further, from (7.7.2)

$$\displaystyle{ \begin{array}{lll} \frac{dE\Pi _{wr}(Q_{wr})} {dQ_{wr}} & =&(p - w) - (p - v)F(Q_{wr}), \end{array} }$$

(7.7.3)

$$\displaystyle{ \begin{array}{lll} \frac{d^{2}E\Pi _{ wr}(Q_{wr})} {dQ_{wr}^{2}} & =& - (p - v)f(Q_{wr}) <0.\end{array} }$$

(7.7.4)

It follows from (7.7.4) that $E\Pi _{wr}(Q_{wr})$ is strictly concave in Q _wr , so the first-order optimality condition works. We therefore obtain from (7.7.3) the retailer’s optimal order quantity as $\overline{Q}_{wr} = F^{-1}(\frac{p-w} {p-v} )$. Substituting $\overline{Q}_{wr}$ into (7.7.2), we see that the retailer’s maximum expected profit under the wholesale price mechanism, denoted by π _wr , is given by

$$\displaystyle{ \begin{array}{lll} \pi _{wr} = (p - w)\overline{Q}_{wr} - (p - v)\int _{0}^{\overline{Q}_{wr}}F(x)dx. \end{array} }$$

(7.7.5)

Based on the assumptions in our model, since the manufacturer uses the “make-to-order” production policy, which means that it is a quantity-taker, the manufacturer’s expected profit under the wholesale price mechanism, denoted by π _wm , is given by $\pi _{wm} = (w - c)\overline{Q}_{wr}$ with $\overline{Q}_{wr} = F^{-1}(\frac{p-w} {p-v} )$. Thus, the proof is completed. □ 

Proof of Proposition 7.4.2.

• (i) From (7.4.4), we see that the retailer’s problem with the option mechanism (o, e) is to solve

  $$\displaystyle{ \begin{array}{lll} \mathrm{P_{7A.2}}:\ \ \ \max \limits _{Q_{or}\geq 0}E\Pi _{or}(Q_{or}) = E[(p - e)\min \{Q_{or},X\} - oQ_{or}].\end{array} }$$

  (7.7.6)

  With some algebra, we obtain

  $$\displaystyle{ \begin{array}{lll} E\Pi _{or}(Q_{or})& =&(p - e)\int _{0}^{Q_{or}}xdF(x) + (p - e)\int _{Q_{ or}}^{+\infty }Q_{ or}dF(x) - oQ_{or} \\ & =&(p - o - e)Q_{or} - (p - e)\int _{0}^{Q_{or}}F(x)dx.\end{array} }$$

  (7.7.7)

  From (7.7.7), we obtain

  $$\displaystyle{ \begin{array}{lll} \frac{dE\Pi _{or}(Q_{or})} {dQ_{or}} & =&(p - o - e) - (p - e)F(Q_{or}),\end{array} }$$

  (7.7.8)
  $$\displaystyle{ \begin{array}{lll} \frac{d^{2}E\Pi _{ or}(Q_{or})} {dQ_{or}^{2}} & =& - (p - e)f(Q_{or}) <0.\end{array} }$$

  (7.7.9)

  It follows from (7.7.9) that $E\Pi _{or}(Q_{or})$ is strictly concave in Q _or , so the first-order optimality condition works. We therefore obtain from (7.7.8) that the retailer’s optimal reserve quantity is $\overline{Q}_{or} = F^{-1}(\frac{p-o-e} {p-e} )$. Similarly, from (7.4.5), we know the manufacturer’s optimal production quantity with the option contract mechanism is to solve

  $$\displaystyle{ \begin{array}{lll} \mathrm{P_{7A.3}}:\ \max \limits _{Q_{om}\geq 0}E\Pi _{om}(Q_{om}) = E[oQ_{om} + e\min \{Q_{om},X\} + v\max \{Q_{om} - X,0\} - cQ_{om}].\end{array} }$$

  (7.7.10)

  With some algebra, we obtain

  $$\displaystyle{ \begin{array}{lll} E\Pi _{om}(Q_{om}) = (e - c + o)Q_{om} - (e - v)\int _{0}^{Q_{om}}F(x)dx.\end{array} }$$

  (7.7.11)

  In a similar way, we can show the strict concavity of $E\Pi _{om}(Q_{om})$ in Q _om . Therefore, by the first-order optimality condition, the manufacturer’s optimal production quantity is obtained as $\overline{Q}_{om} = F^{-1}(\frac{e+o-c} {e-v} )$.

• (ii) Given o + e, from (7.7.7) and (7.7.11), it is clear that $E\Pi _{or}(Q_{or})$ is decreasing in o or increasing in e, and $E\Pi _{om}(Q_{om})$ is increasing in o or decreasing in e.

• (iii) Since the retailer’s optimal reserve quantity is $\overline{Q}_{or} = F^{-1}(\frac{p-o-e} {p-e} )$ and the manufacturer’s optimal production quantity is $\overline{Q}_{om} = F^{-1}(\frac{e+o-c} {e-v} )$, only if $F^{-1}(\frac{p-o-e} {p-e} ) = F^{-1}(\frac{e+o-c} {e-v} ),$ i.e., $e = p -\frac{p-v} {c-v}o$, will the retailer’s optimal reserve quantity be just consistent with the manufacturer’s optimal production quantity. Again, in view of the assumptions on the model parameters, we require that o < c − v. Thus, the proof is completed. □ 

Proof of Proposition 7.4.3.

$\overline{Q}_{wr} <\overline{Q}_{or}$ iff

$$\displaystyle{ F^{-1}(\frac{p - w} {p - v} ) <F^{-1}(\frac{p - o - e} {p - e} ), }$$

(7.7.12)

which is equivalent to $o <\frac{(p-e)(w-v)} {p-v},$ thus establishing (i). So is (ii) established in a similar way. □ 

Proof of the relations $\overline{Q}_{s}> \overline{Q}_{wr}$ and $\overline{Q}_{s}> \overline{Q}_{wm}$. It is obvious that $\overline{Q}_{s}> \overline{Q}_{wr}$. Again, since

$$\displaystyle{ \frac{p - c} {p - v} - \frac{w - c} {w - v} = \frac{(p - w)(c - v)} {(p - v)(w - v)}> 0, }$$

(7.7.13)

and F(x) is strictly increasing on $[0,+\infty )$, it follows that

$$\displaystyle{ \overline{Q}_{s} = F^{-1}(\frac{p - c} {p - v})> F^{-1}(\frac{w - c} {w - v}) = \overline{Q}_{wm}. }$$

(7.7.14)

Hence, the proof is completed. □ 

Proof of Theorem 7.5.1.

By (7.7.7), we know that under the option contract (o, e), the retailer’s expected profit function is given by

$$\displaystyle{ \begin{array}{lll} E\Pi _{or}(Q_{or}) = (p - o - e)Q_{or} - (p - e)\int _{0}^{Q_{or}}F(x)dx,\end{array} }$$

(7.7.15)

where Q _or denotes the retailer’s reservation quantity with the option contract. By (7.5.1), the supply chain system’s profit function is given by

$$\displaystyle{ \begin{array}{lll} E\Pi _{s}(Q_{s}) = (p - c)Q_{s} - (p - v)\int _{0}^{Q_{s}}F(x)dx.\end{array} }$$

(7.7.16)

where Q _s denotes the production quantity of the supply chain system. Let

$$\displaystyle{ \left \{\!\!\begin{array}{llll} p - o - e =\lambda (p - c),\\ p - e =\lambda (p - v). \\ \end{array} \right. }$$

(7.7.17)

By the assumption that e > v, we require $\lambda \in [0,1)$. Substituting (7.7.17) into (7.7.15), we obtain

$$\displaystyle{ \begin{array}{lll} E\Pi _{or}(Q_{or}) =\lambda [(p - c)Q_{or} - (p - v)\int _{0}^{Q_{or}}F(x)dx].\end{array} }$$

(7.7.18)

Comparing (7.7.18) with (7.7.16), we know that any option contract (o, e) satisfying (7.7.17) will push the retailer to reserve as much as $\overline{Q}_{s}$, i.e., $\overline{Q}_{or} = \overline{Q}_{s},$ where $\overline{Q}_{s} = F^{-1}(\frac{p-c} {p-v})$ corresponds to the system-wide optimal production quantity for the supply chain. Again, since the manufacture adopts the “make-to-order” production policy, any option contract (o, e) satisfying (7.7.17) will make the supply chain system achieve the maximum expected profit for the channel. Taking o and e as variables and solving (7.7.17), we obtain the following option contract set, denoted as M,

$$\displaystyle{ M =\{ (o,e): o =\lambda (c - v),e = (1-\lambda )p +\lambda v,\lambda \in [0,1)\}. }$$

(7.7.19)

Substituting $o =\lambda (c - v)$ and $e = (1-\lambda )p +\lambda v$ into $\overline{Q}_{om} = F^{-1}(\frac{e+o-c} {e-v} )$ leads to $\overline{Q}_{om} = F^{-1}(\frac{p-c} {p-v}) = \overline{Q}_{s}$. Thus, with any option contract (o, e) in M, we have $\overline{Q}_{or} = \overline{Q}_{om} = \overline{Q}_{s}$. Again, substituting $o =\lambda (c - v),e = (1-\lambda )p +\lambda v$ into (7.7.11), together with (7.7.18), we see that under the option contract (o, e) in the set M associated with the parameter $\lambda$, the retailer’s expected profit, denoted as $\pi _{or}(\lambda )$, is given by

$$\displaystyle{ \begin{array}{lll} \pi _{or}(\lambda )& =&E\Pi _{or}(\overline{Q}_{or}) =\lambda [(p - c)\overline{Q}_{or} - (p - v)\int _{0}^{\overline{Q}_{or}}F(x)dx] \\ & =&\lambda [(p - c)\overline{Q}_{s} - (p - v)\int _{0}^{\overline{Q}_{s}}F(x)dx] \\ & =&\lambda E\Pi _{s}(\overline{Q}_{s}) =\lambda \pi _{c}, \end{array} }$$

(7.7.20)

where $\pi _{c} = E\Pi _{s}(\overline{Q}_{s})$ denotes the system-wide optimal profit, and the manufacturer’s expected profit, denoted as $\pi _{om}(\lambda )$, is given by

$$\displaystyle{ \begin{array}{lll} \pi _{om}(\lambda )& =&E\Pi _{om}(\overline{Q}_{om}) = (e - c + o)\overline{Q}_{om} - (e - v)\int _{0}^{\overline{Q}_{om}}F(x)dx \\ & =&(1-\lambda )[(p - c)\overline{Q}_{s} - (p - v)\int _{0}^{\overline{Q}_{s}}F(x)dx] \\ & =&(1-\lambda )E\Pi _{s}(\overline{Q}_{s}) = (1-\lambda )\pi _{c}. \end{array} }$$

(7.7.21)

By Proposition 7.4.1, we obtain that with the wholesale price mechanism, the entire supply chain’s total profit is

$$\displaystyle{ \begin{array}{lll} \pi _{wr} +\pi _{wm} = (p - c)\overline{Q}_{wr} - (p - v)\int _{0}^{\overline{Q}_{wr}}F(x)dx.\end{array} }$$

(7.7.22)

Hence, with any option contract (o, e) in M, the increased profit of the supply chain system is

$$\displaystyle{ \begin{array}{lll} \Delta \pi & =&\pi _{c} - (\pi _{wr} +\pi _{wm}) \\ & =&(p - c)(\overline{Q}_{s} -\overline{Q}_{wr}) - (p - v)\int _{\overline{Q}_{ wr}}^{\overline{Q}_{s}}F(x)dx. \end{array} }$$

(7.7.23)

To summarize, Theorem 7.5.1 follows. □ 

Proof of the inequalities $\frac{(c-v)(w-e)} {w-v} <o <\frac{(p-e)(w-v)} {p-v}$. Let (o, e) be any option contract in the set M, i.e.,

$$\displaystyle{ \begin{array}{lll} o =\lambda (c - v)\mathrm{\ and\ }e = (1-\lambda )p +\lambda v\mathrm{\ for\ some\ }\lambda \in [0,1). \end{array} }$$

(7.7.24)

Substituting $e = (1-\lambda )p +\lambda v$ into $\frac{(p-e)(w-v)} {p-v}$ and $\frac{(c-v)(w-e)} {w-v}$ respectively, we obtain

$$\displaystyle{ \begin{array}{lll} \frac{(p-e)(w-v)} {p-v} =\lambda (w - v)>\lambda (c - v) = o, \end{array} }$$

(7.7.25)

$$\displaystyle{ \begin{array}{lll} \frac{(c-v)(w-e)} {w-v} = \frac{(c-v)[\lambda (p-v)+w-p]} {w-v},\end{array} }$$

(7.7.26)

where the inequality in (7.7.25) follows by w > c. Again, with some algebra, we obtain

$$\displaystyle{ \begin{array}{lll} &&\lambda (w - v) -\lambda (p - v) - (w - p) = (1-\lambda )(p - w)> 0. \end{array} }$$

(7.7.27)

Hence

$$\displaystyle{ \begin{array}{lll} \lambda (w - v)>\lambda (p - v) + (w - p), \end{array} }$$

(7.7.28)

which, together with (7.7.26), leads to

$$\displaystyle{ \begin{array}{lll} \frac{(c-v)(w-e)} {w-v} <\frac{\lambda (c-v)(w-v)} {w-v} =\lambda (c - v) = o.\end{array} }$$

(7.7.29)

Thus, the desired result follows. □