Coordination of fuzzy closed-loop supply chain with price dependent demand under symmetric and asymmetric information conditions

Abstract

This paper investigates the coordination issue of a two-echelon fuzzy closed-loop supply chain. Two coordinating models with symmetric and asymmetric information about retailer’s collecting scale parameter are established by using game theory, and the corresponding analytical solutions are obtained. Theoretical analysis and numerical example show that the maximal expected profits of the fuzzy closed-loop supply chain in two coordination situations are equal to that in the centralized decision case and greater than that in the decentralized decision scenario. Furthermore, under asymmetric information contract, the maximal expected profit obtained by the low-collecting-scale-level retailer is higher than that under symmetric information contract.

Appendix: Preliminaries

A possibility space is defined as a triplet $(\Theta ,\mathcal {P}(\Theta ), \mathrm{Pos})$, where $\Theta$ is a nonempty set, $\mathcal {P}(\Theta )$ the power set of $\Theta$, and Pos a possibility measure. Each element in $\mathcal {P}(\Theta )$ is called a fuzzy event. For each event A, Pos(A) indicates the possibility that A will occur. Nahmias (1978) and Liu (2002) gave the following four axioms

Axiom 1.:

$\mathrm{Pos}(\Theta )=1.$

Axiom 2.:

$\mathrm{Pos}(\phi )=0, \text{ where }~\phi ~\text{ denotes } \text{ the } \text{ empty } \text{ set. }$

Axiom 3.:

$\mathrm{Pos}\bigg (\displaystyle \bigcup ^m_{i=1}A_i\bigg )=\displaystyle \sup _{1\le i\le m}\mathrm{Pos}(A_i)~\text{ for } \text{ any } \text{ collection }~A_i~\text{ in }~\mathcal {P}(\Theta ).$

Axiom 4.:

$\text{ Let }~\Theta _i~\text{ be } \text{ nonempty } \text{ sets, } \text{ on } \text{ which }~\mathrm{Pos}_i~\text{ is } \text{ possibility } \text{ measure } \text{ satisfying } \text{ the }~~~$ first three axioms,$~i=1,2,\ldots ,n,~\text{ and }~\Theta =\prod ^n_{i=1}\Theta _i.$ Then

$$\begin{aligned} \mathrm{Pos}(A)=\mathop {\hbox {sup}}_{(\theta _1,\theta _2,\ldots ,\theta _n)\in A}\mathrm{Pos}_1(\theta _1)\wedge \mathrm{Pos}_2(\theta _2)\wedge \cdots \wedge \mathrm{Pos}_n(\theta _n), \end{aligned}$$

$\text{ for } \text{ each }~A\in \mathcal {P}(\Theta ).~\text{ In } \text{ that } \text{ case } \text{ we } \text{ write }~\mathrm{Pos}=\wedge _{i=1}^n\mathrm{Pos}_i.$

Lemma 1

(Liu 2002) Suppose that $(\Theta _i, \mathcal {P}(\Theta _i), \mathrm{Pos}_i)$ is a possibility space, $i=1,2,\ldots , n$. By Axiom 4, $(\prod ^n_{i=1}\Theta _i, \mathcal {P}(\prod ^n_{i=1}\Theta _i), \wedge ^n_{i=1}\mathrm{Pos}_i)$ is also a possibility space, which is called the product possibility space.

Definition 1

(Nahmias 1978) A fuzzy variable is defined as a function from the possibility space $(\Theta , \mathcal {P}(\Theta ), \mathrm{Pos})$ to the set of real numbers and its membership function is derived from the possibility by

$$\begin{aligned} \mu _\xi (x)=\mathrm{Pos}\left( \{\theta \in \Theta \mid \xi (\theta )=x\}\right) , \quad \forall x\in R . \end{aligned}$$

Definition 2

(Liu 2002) A fuzzy variable $\xi$ is said to be nonnegative (or positive) if  $\mathrm{Pos}(\{\xi <0\})=0$ (or $\mathrm{Pos}(\{\xi \le 0\})=0)$.

Definition 3

(Liu 2002) Let $f: R^n\rightarrow R$ be a function, and $\xi _i$ a fuzzy variable defined on the possibility space $(\Theta _i, \mathcal {P}(\Theta _i), \mathrm{Pos}_i), i=1,2,\ldots , n$, respectively. Then $\xi =f(\xi _1,\xi _2,\ldots ,\xi _n)$ is a fuzzy variable defined on the product possibility space $(\prod ^n_{i=1}\Theta _i, \mathcal {P}(\prod ^n_{i=1}\Theta _i), \wedge ^n_{i=1}\mathrm{Pos}_i)$.

The independence of fuzzy variables was discussed by several researchers, such as Liu (2002), Nahmias (1978) and Zadeh (1978).

Definition 4

(Liu 2002) The fuzzy variables $\xi _1,\xi _2,\ldots ,\xi _n$ are independent if for any sets $\mathcal {B}_1,\mathcal {B}_2,\ldots ,\mathcal {B}_n$ of R,

$$\begin{aligned} \mathrm{Pos}(\{\xi _i\in \mathcal {B}_i,i=1,2,\ldots ,n\})=\mathop {\min }_{1\le i\le n}\mathrm{Pos}(\{\xi _i\in \mathcal {B}_i\}). \end{aligned}$$

Lemma 2

(Liu 2004) Let $\xi _i$ be independent fuzzy variable, and $f_i: R\rightarrow R$ function, $i=1,2,\ldots ,m$. Then $f_1(\xi _1),f_2(\xi _2),\ldots ,f_m(\xi _m)$ are independent fuzzy variables.

Definition 5

(Liu 2002) Let $\xi$ be a fuzzy variable on the possibility space $(\Theta , \mathcal {P}(\Theta ), \mathrm{Pos})$, and $\alpha \in (0,1]$. Then

$$\begin{aligned} \xi ^L_\alpha =\inf \{r|\mathrm{Pos}(\{\xi \le r\})\ge \alpha \}~~\text{ and }~~\xi ^U_\alpha =\sup \{r|\mathrm{Pos}(\{\xi \ge r\})\ge \alpha \} \end{aligned}$$

are called the $\alpha$-pessimistic value and the $\alpha$-optimistic value of $~\xi$, respectively.

Example 1

The triangular fuzzy variable $\xi =(a_1,a_2,a_3)$ has its $\alpha$-pessimistic value and $\alpha$-optimistic value

$$\begin{aligned} \xi ^L_\alpha =a_2\alpha +a_1(1-\alpha ) ~~\text{ and }~~ \xi ^U_\alpha =a_2\alpha +a_3(1-\alpha ). \end{aligned}$$

Lemma 3

(Wang et al. 2007) Let $\xi _i$ be independent fuzzy variables defined on the possibility spaces $(\Theta _i, \mathcal {P}(\Theta _i), \mathrm{Pos}_i)$ with continuous membership function, $i=1,2,\ldots ,n$, and $f: X\subset \mathcal {R}^n\rightarrow \mathcal {R}$ a measurable function. If $f(x_1,x_2,\ldots ,x_n)$ is monotonic with respect to $x_i$, respectively, then

(a):

$f^U_{\alpha }(\xi )=f(\xi _{1\alpha }^V,\xi _{2\alpha }^V,\ldots ,\xi _{n\alpha }^V),$ where $~\xi _{i\alpha }^V=\xi _{i\alpha }^U$, if $f(x_1,x_2,\ldots ,x_n)$ is nondecreasing with respect to $x_i$; $\xi _{i\alpha }^V=\xi _{i\alpha }^L$, otherwise,

(b):

$f^L_\alpha (\mathbf {\xi })=f(\xi _{1\alpha }^{\overline{V}},\xi _{2\alpha }^{\overline{V}},\ldots ,\xi _{n\alpha }^{\overline{V}}),$ where $~\xi _{i\alpha }^{\overline{V}}=\xi _{i\alpha }^L$, if $f(x_1,x_2,\ldots ,x_n)$ is nondecreasing with respect to $x_i$; $\xi _{i\alpha }^{\overline{V}}=\xi _{i\alpha }^U$, otherwise,

where $f^U_\alpha (\mathbf {\xi })$ and $f^L_\alpha (\mathbf {\xi })$ denote the $\alpha$-optimistic value and the $\alpha$-pessimistic value of the fuzzy variable $f(\xi )$, respectively.

Definition 6

(Liu and Liu 2002) Let $(\Theta , \mathcal {P}(\Theta ), \mathrm{Pos})$ be a possibility space and A a set in $\mathcal {P}(\Theta )$. The credibility measure of A is defined as

$$\begin{aligned} \mathrm{Cr}(A)=\frac{1}{2}\left( 1+\mathrm{Pos}(A)-\mathrm{Pos}(A^c)\right) , \end{aligned}$$

where $A^c$ denotes the complement of A.

Definition 7

(Liu and Liu 2002) Let $\xi$ be a fuzzy variable. The expected value of $\xi$ is defined as

$$\begin{aligned} E[\xi ]=\int _0^{+\infty }\mathrm{Cr}(\{\xi \ge x\}){\mathrm{d}}x-\int ^0_{-\infty }\mathrm{Cr}(\{\xi \le x\}){\mathrm{d}} x \end{aligned}$$

provided that at least one of the two integrals is finite.

Example 2

The triangular fuzzy variable $\xi =(a_1,a_2,a_3)$ has an expected value

$$\begin{aligned} E[\xi ]=\frac{a_1+2a_2+a_3}{4}. \end{aligned}$$

Definition 8

(Liu and Liu 2002) Let f be a function on $R\rightarrow R$ and $\xi$ be a fuzzy variable. Then the expected value $E[f(\xi )]$ is defined as

$$\begin{aligned} E[f(\xi )]=\int _0^{+\infty }\mathrm{Cr}(\{f(\xi )\ge x\}) {\mathrm{d}}x-\int ^0_{-\infty }\mathrm{Cr}(\{f(\xi )\le x\}){\mathrm{d}}x \end{aligned}$$

provided that at least one of the two integrals is finite.

Lemma 4

rm (Liu and Liu 2003) Let $\xi$ be a fuzzy variable with finite expected value. Then

$$\begin{aligned} E[\xi ]=\frac{1}{2}\int ^1_0\left( \xi ^L_\alpha +\xi ^U_\alpha \right) {\mathrm{d}}\alpha . \end{aligned}$$

Lemma 5

(Liu and Liu 2003) Let $\xi$ and $\eta$ be independent fuzzy variables with finite expected values. Then for any numbers a and b,

$$\begin{aligned} E[a\xi +b\eta ]=aE[\xi ]+bE[\eta ]. \end{aligned}$$