Inventory of a deteriorating green product with preservation technology cost using a hybrid algorithm

Abstract

Nowadays, in a competitive market circumstance, a distributor/wholesaler permits a settled trade credit to the retailer(s) for more deal, and in turn, retailer offers a fraction of that credit period to the customers for the quick-moving of the business, i.e. to increase the demand. In reality, all the customers are not fully trustworthy. There are some customers who do not pay the dues after enjoying the trade credit. The number of default customers increases with the increase in trade credit period given to them. Thus, these two contradict each other. There is a decent market with increasing demand for greener (contamination free) item(s), especially for deteriorating item(s). Here, in the present investigation, customers’ demand increases with both the trade credit given to customers and the greenness of the product. Obviously, the price tag will be higher for green items. Increased price always negates the demand. Hence, greenness and the increased price of the green products are contradictory to each other. Once more, to reduce the deterioration rate, the retailer incurs an expenditure termed as preservation technology cost. Due to preservation, deterioration of the items decreases but cost increases. Here, control of deterioration and preservation act contradictory to each other. The base demand, effects (coefficients) of the trade credit, and greenness are taken as fuzzy. Incorporating the above facts, a model is developed with a fuzzy differential equation and solved by Chalco-Cano $\alpha$-cut method. The profit is calculated by assuming two-level trade credit and default customers. In this investigation, a new concept of clearing the supplier’s dues is introduced. Instead of payment of all dues at the end of business cycle period, the retailer clears his dues as and when he has sufficient money for this purpose. Numerical experiments for different cases are performed along with their physical interpretations. Imprecise nature of the profit is shown through its membership function and depicted graphically. Interestingly, it is demonstrated that positive effect of deterioration due to the increase in preservation technology and increase in profit due to greenness are up to certain extent. After that, these have the reverse effects.

Appendices

Appendix-A

Here, we present some well-known definitions of fuzzy numbers which are required to develop the model.

Triangular fuzzy number (TFN) A TFN ${\widetilde{P}}= (p_1, p_2, p_3)$ has three parameters $p_1, p_2$ and $p_3$, where $p_1<p_2< p_3$ and its membership function $\mu _{{\widetilde{P}}}(x)$ is of the form:

$$\begin{aligned} \mu _{{\widetilde{P}}}(x)=\left\{ \begin{array}{ll} \frac{x-p_1}{p_2-p_1} &{} \text{ for } p_1\le x\le p_2\\ \frac{p_3-x}{p_3-p_2} &{} \text{ for } p_2\le x\le p_3\\ 0 &{} \text{ otherwise. } \end{array}\right. \end{aligned}$$

(27)

$\mathbf {\alpha }$-cut of a fuzzy number: $\alpha$-cut of a fuzzy number ${\widetilde{Z}}$ in $\mathfrak {R}$ is denoted by ${\tilde{Z}}(\alpha )$ and is defined as,

$$\begin{aligned} {\tilde{Z}}(\alpha ) = \{ z \in \mathfrak {R}/ \mu _{{\widetilde{Z}}}(z)\ge \alpha \}. \end{aligned}$$

Fuzzy extension principle (Zadeh 1978): If ${\widetilde{Z}}_1$ , ${\widetilde{Z}}_2$ $\subseteq \mathfrak {R}$ and ${\widetilde{Z}}=f({\widetilde{Z}}_1, {\widetilde{Z}}_2)$, where $f :\mathfrak {R}\times \mathfrak {R}\rightarrow \mathfrak {R}$ is a binary operation, membership function $\mu _{{\widetilde{Z}}}(z)$ of ${\widetilde{Z}}$ is defined as

$$\begin{aligned}&\text{ For } \text{ each }\, \, z\in \mathfrak {R}, \nonumber \\&\mu _{{\widetilde{Z}}}(z)=sup\{min(\mu _{{\widetilde{Z}}_1} (z_1),\mu _{{\widetilde{Z}}_2}(z_2)),\nonumber \\&z_1,z_2 \in \mathfrak {R} \text{ and } z=f(z_1,z_2)\}. \end{aligned}$$

(28)

Let F(A) be the space of all compact and convex fuzzy sets on A. If $f:\mathfrak {R}^n\rightarrow \mathfrak {R}^n$ is a continuous function, then ${\tilde{f}}:F(\mathfrak {R}^n)\rightarrow F(\mathfrak {R}^n)$ is well defined function and its $\alpha$-cut say, ${\tilde{f}}(x)(\alpha )$ is given by (Roman-Flores et al. 2001)

$$\begin{aligned} {\tilde{f}}(x)(\alpha )=f([x]^\alpha ),\forall \alpha \in [0,1],\forall x\in F(\mathfrak {R}^n) \end{aligned}$$

(29)

where $f(A)=\{f(a)/a\in A\}$.

Interval and interval comparison $\alpha$-cut of the fuzzy number (membership function should be continuous) is termed as interval. An interval $I_N$ in $\mathfrak {R}$ has two components $[I_\mathrm{NL}, I_\mathrm{NR}]$ and defined as $I=\{x\in \mathfrak {R}| I_\mathrm{NL}\le x \le I_\mathrm{NR}\}$. In other words, the number is represented by its mid-point $m_d(I_N)$ and the average width $w_h(I_N)$ as $I_N=<m_d(I_N),w_h(I_N)>$. The mid-point of the number $m_d(I)$ is defined by $m_d(I)=(I_\mathrm{NL}+I_\mathrm{NR})/2$ and half width $w_h(I)$ is defined by $w_h(I)=(I_\mathrm{NR}-I_\mathrm{NL})/2$.

As the profit is obtained as an interval, its comparison is necessary to reveal the best optimum value of the profit. In the literature of fuzzy mathematics, several researchers have presented their approaches to compare intervals. In 2009, Sengupta and Pal discussed the available methods of ranking of intervals and proved that their approach is more acceptable to compare interval. For this reason, Sengupta and Pal’s approach is used.

Definition

(Seikkla derivative) Seikkla, presented a method to differentiate a fuzzy number. Let, ${\tilde{Z}}(x)$ be a fuzzy number and its derivative be $SD{\tilde{Z}}(t)$. Also let, $[Z_1(x,\alpha ),Z_2(x,\alpha )]$ be the $\alpha$-cuts of the fuzzy number ${\tilde{Z}}(x)$ for all $x\in I$. Then, the derivative $SD{\tilde{Z}}(x)$ exists and $SD{\tilde{Z}}(x)(\alpha )=[Z_1^{'}(x,\alpha ),Z_2^{'}(x,\alpha )]$.

Fuzzy differential equation (FDE) (Chalco-Cano and Roman-Flores 2009): Suppose, an initial value problem in fuzzy form is given by

$$\begin{aligned} {\tilde{Z}}'(t)={\tilde{u}}(t,{\tilde{Z}}(t)),\,\,{\tilde{Z}}(0)={\tilde{Z}}_0 \end{aligned}$$

(30)

where $u:[0,T]\times F(V)\rightarrow F(\mathfrak {R}^n)$ is obtained from extension principle (28) (Zadeh) on a continuous function $v:[0,T]\times V\rightarrow \mathfrak {R}^n$, $V\subset \mathfrak {R}^n$. Here, u is continuous as v is continuous (Roman-Flores, 2001) and following the rule given by Eq. (29), we may write,

$$\begin{aligned}{}[u(t,Z)]^\alpha =v(t,[Z]^\alpha )\,\,\,\,\,\, \text{ and }\,\,\,\, v(t,A)=\{v(t,a)/a\in A\} \end{aligned}$$

Now, consider the differential equation (Deterministic ) associated with FDE (30) and is of the form

$$\begin{aligned} x'(t)=v(t,x(t)),\,\,x(0)=x_0 \end{aligned}$$

(31)

Here, $x'(t)$ is the derivative of $x:[0,T]\rightarrow \mathfrak {R}^n$ in crisp form. Then, a solution of (30) in fuzzy form can be derived from Eq. (31) following the rule given below

• At first, solve Eq. (31) and let $x(t,x_0)$ be its solution.

• Following Eq. (28) to $x(t,x_0)$ (in relation to $x_0$), the extension can be written as ${\tilde{Z}}(t)={\tilde{x}}(t,{\tilde{Z}}_0)$ (according to Chalco-Cano and Roman-Flores 2009) for each fixed t. This is called the fuzzy solution of (30) provided that the following theorem holds.

Theorem 1

(Chalco-Cano and Roman-Flores 2009): Let, V be an open set in $\mathfrak {R}^n$ and $Z_0(\alpha ) \subset V$. If v is continuous for each $c\in V$, then the solution x(., c) of the deterministic problem (31) is unique and x(t, .) must be continuous on V for each $t\in [0,T]$ fixed. If this condition holds then, there exists a solution ${\widetilde{Z}}(t)={\tilde{x}}(t,Z_0)$ of the FDE (30) and which is the unique fuzzy solution.

Fuzzy Riemann integration (Wu 2000): Let, a closed and bounded fuzzy-valued function ${\tilde{u}}(x)$ is defined on a closed interval [a,b]. Also, let the $\alpha$-cut of ${\tilde{u}}(x)$ is defined as ${\tilde{u}}(x)(\alpha )=[u_L(\alpha ,x),u_R(\alpha ,x)]$, where the left-cut $u_L(\alpha ,x)$ and right-cut $u_R(\alpha ,x)$ are Riemann integrable on [a, b], $\forall$ $\alpha$.

Then, $\int \limits _a^b{\tilde{u}}(x)dx$ is fuzzy Riemann integrable and its $\alpha$-cut is given by

$$\begin{aligned} \bigg (\int \limits _a^b{\tilde{u}}(x)\mathrm{d}x\bigg )(\alpha )= \bigg [\int \limits _a^b u_L(\alpha ,x)\mathrm{d}x,\int \limits _a^b u_R(\alpha ,x)dx\bigg ]. \end{aligned}$$

Appendix-B

List of test functions (TF)

TF-1: :

(Taken from Michalewicz, 1992): Minimize $F(x_1, x_2)=(x_1-2)^2+(x_2-1)^2$,

such that $-x_1^2+x_2\ge 0, x_1+x_2\le 2$, $-5\le x_1, x_2\le 5$,

It has one global minima at $(x_1,x_2)$ =(1, 1), and $F(1,1)=1$.

TF-2: :

(Taken from Michalewicz, 1992) Minimize $F(x_1, x_2)=100(x_2-x_1^2)^2+(x_1-1)^2$,

such that $x_1+x_2^2\ge 0, x_1^2+x_2\ge 0$, $-0.5\le x_1\le 0.5$, $-1.0\le x_2\le 1.0$

It has one global minima at $(x_1,x_2)=(0.5,0.25)$, and $F(0.5,0.25)=0.25$.

TF-3: :

(Taken from Bessaou and Siarry, 2001):

$DJ(x_1,x_2,x_3)=x_1^2+x_2^2+x_3^2$, $-5.12\le x_1,x_2,x_3 \le 5.12$.

It has one global minima at $(x_1,x_2,x_3)=(0,0,0)$ and $DJ(0,0,0)=0$.

TF-4: :

(Taken from Bessaou and Siarry, 2001):

$F2(x_1,x_2)=100\times (x_2^2-x_1)+(1-x_1)$, $-2.048\le x_1, x_2 \le 2.048$.

It has one minima at $(x_1,x_2)=(2.048,0)$ and $F2(2.048,0)=-205.8480$.

TF-5: :

(Taken from Bessaou and Siarry, 2001):

$ES(x_1,x_2)=-cos(x_1)\times cos(x_2)\times exp\{-[(x_1-\pi )^2+(x_2-\pi )^2]\}$, $-10\le x_1, x_2 \le 10$.

This test function has one global minima at $(x_1,x_2)=(\pi ,\pi )$ and $ES(\pi ,\pi )=-1$.

TF-6: :

(Taken from Bessaou and Siarry, 2001)

Minimize $F_n(x_1,x_2,\ldots ,x_n)=\sum \limits _{i=1}^{n-1}\big [100\times (x_j^2-x_{j+1})^2+(1-x_j)^2\big ]$,

$-1\le x_1,x_2,\ldots ,x_n\le 5$,

This problem has one global minima at $(x_1,x_2,\ldots , x_n)=(1,1,\ldots ,1)$ and $F_n(1,1,\ldots ,n)=0$. Three functions $F_4$ are solved for this purpose.

TF-7: :

(Taken from Bessaou and Siarry, 2001):

$MZ(x_1,x_2,\ldots ,x_n)=-\sum \limits _{i=1}^{n}sin(x_i).$

$[sin(i.(x_i)^2/\pi )]^{2m}$, $-\pi \le x_1,x_2,\ldots ,x_n \le \pi$, where $m=10$.

For $n=2$, it has one global minima at $(x_1,x_2)=(2.25,1.57)$ and $MZ(2.25,1.57)=-1.80$.

TF-8: :

(Taken from Bessaou and Siarry, 2001):

$RC(x_1,x_2)=\{x_2-[5/(4\times \pi ^2)].x_1^2+(5/\pi )\times x_1-6\}^2+10\times \{1-[1/(8\pi )]\}\times cos(x_1)+10$, $-5\le x_1\le 10, 0\le x_2\le 15$.

This problem has three global minima at $(x_1,x_2)=(-\pi ,12.275),(\pi ,2.275)$, (9.42478, 2.475) and $RC(x_1,x_2)=0.397887$ at any one of these minima.

TF-9: :

(Taken from Bessaou and Siarry, 2001):

$Z_n(x_1,x_2,\ldots ,x_n)=\bigg (\sum \limits _{j=1}^{n}x_j^2\bigg ) +\bigg (\sum \limits _{j=1}^{n}0.5j\times x_j\bigg )^2 +\bigg (\sum \limits _{j=1}^{n}0.5j\times x_j\bigg )^4$,

$-5\le x_1,x_2,\ldots ,x_n \le 5$.

It has one global minima at $(x_1,x_2,\ldots ,x_n)=(0,0,\ldots ,0)$ and $Z_n(0,0,\ldots ,0)=0.$

TF-10: :

(Taken from Bessaou and Siarry, 2001):

$BH(x_1,x_2)=x_1^2+2\times x_2^2-0.3\times cos(3\pi \times x_1)\times cos(4\pi \times x_2)+0.3$, $-5\le x_1,x_2 \le 5$.

This problem has one global minima at $(x_1,x_2)=(0,0)$ and $BH(0,0)=0.$

	TF-1	TF-2	TF-3	TF-4	TF-5	TF-6	TF-7	TF-8	TF-9	TF-10	Mean
$X_1$ (say)	$-$ 1	1	0	$-$ 1	0	2	1	2	$-$ 1	1	${\overline{X}}_1=0.4$
$X_2$ (say)	2	1	2	3	0	0	2	3	$-$ 2	0	${\overline{X}}_2=1.1$
$X_3$ (say)	6	7	6	6	5	6	7	7	6	5	$\overline{X}_3=6.1$

The above test functions are solved using the algorithm GA, PSO, and GPSA, and number of success/wins of finding optimal solutions for each test function are tabulated in Table-ER1. It is observed that the number of wins is highest in the case of GPSA for all the test functions.

For calculation of different steps of ANOVA easily, we subtract 42 (without loss of generality) from each number and Table-ER1 reduces to

Here, total sample size of each algorithm is equal and say $I=10$ and number of algorithms is, say, $J=3$.

Appendix-C