A two-echelon supply chain coordination with quantity discount incentive for fixed lifetime product in a fuzzy environment

Abstract

Efficient and effective management of a supply chain is of great importance for the success of the digital economy. Coordination among the members of a supply chain play a vital role for its effective management. The members of the supply chain may agree to cooperate initially, but owing to the competition prevailing in business environments, they may be tempted to maximize their profits and deviate from any agreement. So, effective mechanism is essential to enforce coordination among the members in a supply chain. In today’s competitive environment, the inventory managers are also interested in simple and easy procedures to apply them in their organizations. This paper investigates a single-manufacturer and a single-buyer two echelon supply chain model for a fixed lifetime product in a fuzzy cost environment with a quantity discount strategy as a coordination mechanism. Crisp models are developed under different scenarios (1) without coordination (2) with coordination and (3) system optimization. Fuzzy models are also formulated by representing the ordering cost and the holding cost of the manufacturer by trapezoidal fuzzy numbers. Signed distance method is adopted for defuzzification. Numerical results highlighting the sensitivity of various parameters are also elucidated.

Appendix

1.1 Preliminaries

The fuzzy set theory was introduced to deal with problems in which fuzzy phenomena exist. In a universe of discourse X, a fuzzy subset $\tilde{a}$ of X is defined by the membership function $\mu _{\tilde{a}}(x)$ which maps each element x in X to a real number in the interval [0, 1]. The function value of $\mu _{\tilde{a}}(x)$ denotes the grade of membership.

Definition 1

Fuzzy normal (Vijayan and Kumaran 2008) A fuzzy set is normal if the largest grade obtained by any element in that set is 1.

Definition 2

Fuzzy covex (Vijayan and Kumaran 2008) A fuzzy set $\tilde{a}$ on X is convex iff $\mu _{\tilde{a}}(\lambda x_{1}+ (1-\lambda )x_{2}) \ge min(\mu _{\tilde{a}}(x_{1}),\mu _{\tilde{a}}(x_{2}))$

Definition 3

Fuzzy point (Pu and Liu 1980) Let $\tilde{a}$ be a fuzzy set on R = $(-\infty ,\infty )$. It is called a fuzzy point if its membership function is

$$\begin{aligned} \mu _{\tilde{a}}(x)= \left\{ \begin{array}{ll} 1, &\quad x = a\\ 0, &\quad x\ne a \end{array}\right. \end{aligned}$$

(8.1)

Definition 4

Fuzzy number (Vijayan and Kumaran 2008) A fuzzy number is a fuzzy subset of the real line which is both normal and convex. For a fuzzy number $\tilde{A}$, its membership function is denoted by

$$\begin{aligned} \mu _{\tilde{a}}(x)= \left\{ \begin{array}{ll} l(x), &\quad x < m \\ 1, &\quad m \le x \le n\\ u(x), &\quad x > n \end{array}\right. \end{aligned}$$

(8.2)

where l(x) is upper semi continuous, strictly increasing for $x < m$ and there exist $m_{1} < m$ such that $l(x) = 0$ for $x \le m_{1}$, u(x) is continuous, strictly decreasing function for $x > n$ and there exist $n_{1} \ge n$ such that $u(x) = 0$ for $x \ge n_{1}$, l(x) and u(x) are called left and right reference functions respectively.

Definition 5

Trapezoidal fuzzy number (Zimmerman 1991) The fuzzy number $\tilde{A}$ is said to be a trapezoidal fuzzy number if it is fully determined by ( $a_{1}$, $a_{2}$, $a_{3}$, $a_{4}$ ) of crisp numbers such that $a_{1}< a_{2}< a_{3} < a_{4}$, whose membership function, representing a trapezoid, can be denoted by

$$\begin{aligned} \mu _{\tilde{A}}(x)= \left\{ \begin{array}{ll} \frac{x-a_{1}}{a_{2}-a_{1}}, &\quad a_{1} \le x \le a_{2}\\ 1, &\quad a_{2} \le x \le a_{3}\\ \frac{x-a_{4}}{a_{3}-a_{4}}, &\quad a_{3} \le x \le a_{4}\\ 0, &\quad otherwise \end{array}\right. \end{aligned}$$

(8.3)

where $a_{1}, a_{2}, a_{3}$ and $a_{4}$ are the lowerlimit, lower mode, upper mode and upper limit respectively of the fuzzy number $\tilde{A}$. The interval [$a_{1}, a_{4}$] is called the support of the fuzzy number and it gives the range of all possible values of $\tilde{A}$ that are least marginally possible or plausible. The interval [$a_{2}, a_{3}$] corresponds to the core of fuzzy number and gives the range of most plausible values. The intervals [$a_{1}, a_{2}$] and [$a_{3}, a_{4}$] are called penumbra of the fuzzy number $\tilde{A}$.

Let $\tilde{A_{1}} = (a_{11}, a_{12},a_{13}, a_{14})$, $\tilde{A_{2}} = (a_{21}, a_{22},a_{23}, a_{24})$ be two trapezoidal fuzzy numbers, then $\tilde{A_{1}}+\tilde{A_{2}}= (a_{11}+a_{21},\, a_{12}+a_{22},\, a_{13}+a_{23},\, a_{14}+a_{24})$ and for all $b \ge 0$, $b\tilde{A_{1}} = (ba_{11}, ba_{12}, ba_{13}, ba_{14})$.

The set $\tilde{A}(\alpha )=\{x:\mu _{\tilde{A}}(x)\ge \alpha \}$, where $\alpha \varepsilon [0,1]$ is called the $\alpha$ cut of $\tilde{A}$. $\tilde{A}(\alpha )$ is a nonempty bounded closed interval contained in the set of real numbers and it can be denoted by $\tilde{A}(\alpha )=[\tilde{A}_{L}(\alpha ), \tilde{A}_{R}(\alpha )]$. $\tilde{A}_{L}(\alpha )$ and $\tilde{A}_{R}(\alpha )$ are respectively the left and right limits of $\tilde{A}(\alpha )$ and are usually known as the left and right $\alpha$ cuts of $\tilde{A}$. $\tilde{A}_{L}(\alpha ) = a_{1} +(a_{2}-a_{1})\alpha$ and $\tilde{A}_{R}(\alpha ) = a_{4} - (a_{4}-a_{3})\alpha$ for a trapezoidal number $\tilde{A} = ( a_{1}, a_{2}, a_{3}, a_{4})$.

Definition 6

Level $\alpha$ Fuzzy interval (Chiang et al. 2005) Let [a, b; $\alpha$] be a fuzzy set on $R = (-\infty , \infty )$. It is called a level $\alpha$ fuzzy interval, $0\le \alpha \le 1$, $a<b$, if its membership function is

$$\begin{aligned} \mu _{[\, a, \,b; \,\alpha \, ]}(x)= \left\{ \begin{array}{ll} \alpha , &\quad if \; a \le x \le b \\ 0,&\quad otherwise \end{array}\right. \end{aligned}$$

(8.4)

1.2 Signed distance method (Yao and Wu 2000)

The signed distance between the real numbers a and 0, denoted by $d_{0}(a,0)$ is given by $d_{0}(a,0) =a$. Hence the signed distance of $\tilde{A}_{L}(\alpha )$ and $\tilde{A}_{R}(\alpha )$ measured from 0 are $d_{0}(\tilde{A}_{L}(\alpha ),0) =\tilde{A}_{L}(\alpha )$ and $d_{0}(\tilde{A}_{R}(\alpha ),0) =\tilde{A}_{R}(\alpha )$.

The signed distance of the interval ($\tilde{A}_{L}(\alpha ),\tilde{A}_{R}(\alpha )$) measured from the origin 0 by

$$\begin{aligned} d_{0}((\tilde{A}_{L}(\alpha ),\tilde{A}_{R}(\alpha )),0) =\frac{1}{2}[d_{0}(\tilde{A}_{L}(\alpha ),0)+d_{0}(\tilde{A}_{R}(\alpha )]= \frac{1}{2}(\tilde{A}_{L}(\alpha )+\tilde{A}_{R}(\alpha )) \end{aligned}$$

(8.5)

where $\tilde{A}_{L}(\alpha )$ and $\tilde{A}_{L}(\alpha )$ exist and are integrable for $\alpha \varepsilon [0,1]$.

For each $\alpha \,\varepsilon \, [0,1]$, the crisp interval [$\tilde{A}_{L}(\alpha ),\tilde{A}_{R}(\alpha )$] and the level $\alpha$ fuzzy interval [[$\tilde{A}_{L}(\alpha ),\tilde{A}_{R}(\alpha )$]; $\alpha$] are in one to one correspondence. The signed distance from [$\tilde{A}_{L}(\alpha ),\tilde{A}_{R}(\alpha )$] to $\tilde{0}$ (where $\tilde{0}$ is the 1 level fuzzy point which maps to the origin) is

$$\begin{aligned} ([\tilde{A}_{L}(\alpha ),\tilde{A}_{R}(\alpha );\alpha ], \tilde{0})= d_{0}((\tilde{A}_{L}(\alpha ),\tilde{A}_{R}(\alpha )),0)=\frac{1}{2}(\tilde{A}_{L}(\alpha )+\tilde{A}_{R}(\alpha )) \end{aligned}$$

(8.6)

The signed distance of $\tilde{A}$ measured from 0 is defined as

$$\begin{aligned} d(\tilde{A},\tilde{0})=\frac{1}{2}\int ^ {1}_0(\tilde{A}_{L}(\alpha )+\tilde{A}_{R}(\alpha ))d\alpha \end{aligned}$$

(8.7)

Lemma 1

(Linearity property of the operator d (Vijayan and Kumaran 2008))

Let $\tilde{A}_{i},i=1,2,{\ldots }N$ be N fuzzy numbers and $b_{i},i = 1,2,{\ldots }N$ are real crisp constants. Then

$$\begin{aligned} d\left( \sum ^{N}_{i=1}b_{i}\tilde{A}_{i},\tilde{0}\right) = \sum ^{N}_{i=1}b_{i}d(\tilde{A}_{i},\tilde{0}) \end{aligned}$$

(8.8)

Proof

By definition,

$$\begin{aligned} d(\sum ^{N}_{i=1}b_{i}\tilde{A}_{i},\tilde{0})&= \frac{1}{2}\int ^ {1}_0\left( \left( \sum ^{N}_{i=1}b_{i}\tilde{A}_{i}\right) _{L}(\alpha )+\left( \sum ^{N}_{i=1}b_{i}\tilde{A}_{i}\right) _{R}(\alpha )\right) d\alpha \\&= \frac{1}{2}\int ^ {1}_0\left( \sum ^{N}_{i=1}b_{i}\tilde{A}_{iL}(\alpha )+\sum ^{N}_{i=1}b_{i}\tilde{A}_{iR}(\alpha )\right) d\alpha \\&= \frac{1}{2}\int ^ {1}_0\sum ^{N}_{i=1}b_{i}(\tilde{A}_{iL}(\alpha )+\tilde{A}_{iR}(\alpha ))d\alpha \\&= \sum ^{N}_{i=1}b_{i}\frac{1}{2}\int ^ {1}_0(\tilde{A}_{iL}(\alpha )+\tilde{A}_{iR}(\alpha ))d\alpha \\&= \sum ^{N}_{i=1}b_{i}d(\tilde{A}_{i},\tilde{0}) \end{aligned}$$

where $\tilde{A}_{iL}(\alpha )$ and $\tilde{A}_{iL}(\alpha )$ are respectively the left and right $\alpha$ cuts of the fuzzy number $\tilde{A}_{i}$. Hence the lemma is proved. $\square$