EPL models with fuzzy imperfect production system including carbon emission: a fuzzy differential equation approach

Abstract

The paper outlines the production policies for maximum profit of a firm producing imperfect economic lot size with time-dependent fuzzy defective rate under the respective country’s carbon emission rules. Generally in economic production lot-size models, defective production starts after the passage of some time from production commencement. So the starting time of producing defective units is normally uncertain and imprecise. Thus, produced defective units are fuzzy, partially reworked instantly and sold as fresh units. As a result, the inventory level at any time becomes fuzzy and the relation between the production, demand and inventory level becomes a fuzzy differential equation (FDE). Nowadays, different governments have made environmental regulations following the United Nations Framework Convention on Climate Change to reduce carbon emission. Some governments use cape and trade policy on emission. Due to this, firms are in fix how to optimize the production. If the firms produce more, the profit increases along with more emission and corresponding tax. Here, models are formulated as profit maximization problems using FDE, and the corresponding inventory and environmental costs are calculated using fuzzy Riemann integration. An $\alpha$-cut of average profits is obtained and the reduced multi-objective crisp problems are solved using intuitionistic fuzzy optimization technique. The models are illustrated numerically and results are presented graphically. Considering different carbon regulations, an algorithm for a firm management is presented to achieve the maximum profit. Real-life production problems for the firms in Annex I and developing countries are solved.

Appendices

Appendix A

1.1 Mathematical prerequisites

1.1.1 Possibility/necessity measure in fuzzy environment

Let $\mathfrak {R}$ represent the set of real numbers and ${\tilde{A}}$ and ${\tilde{B}}$ be two fuzzy numbers with membership functions $\mu _{{\tilde{A}}}$ and $\mu _{{\tilde{B}}}$, respectively. Then, taking degree of uncertainty as the semantics of fuzzy number, according to Zadeh (1999), Dubois and Prade (1988) and Liu and Iwamura (1998):

$$\begin{aligned} \text{ Pos } ({\tilde{A}}\star {\tilde{B}}) = \hbox {sup}\{\hbox {min} (\mu _{{\tilde{A}}}(x), \mu _{{\tilde{B}}}(y)), x,y \in \mathfrak {R}, x\star y\} \end{aligned}$$

where the abbreviation Pos represents possibility and $\star$ is any one of the relations $>, <, =, \le , \ge$. Analogously, if ${\tilde{B}}$ is a crisp number, say b, then $\text{ Pos } ({\tilde{A}}\star b) = \hbox {sup}\{\mu _{{\tilde{A}}}(x),x \in R, x\star b\}$

On the other hand, necessity measure of an event ${\tilde{A}}\star {\tilde{B}}$ is a dual of possibility measure and defined as: $\text{ Nes } ({\tilde{A}}\star {\tilde{B}}) = 1- \text{ Pos } (\overline{{\tilde{A}}\star {\tilde{B}}})\nonumber$, where the abbreviation Nes represents necessity measure and $\overline{{\tilde{A}}\star {\tilde{B}}}$ represents complement of the event ${\tilde{A}}\star {\tilde{B}}$.

According to Liu and Iwamura (1998), Lemmas 1 and 2 can be easily derived.

Lemma 1

If ${\tilde{a}}=(a_1,a_2,a_3)$ be a TFN (triangular fuzzy number) with $0<a_1$ and b is a crisp number, $pos({\tilde{a}} \le b)\ge \alpha$ iff $\displaystyle \frac{b-a_1}{a_2-a_1} \ge \alpha$.

Lemma 2

If ${\tilde{a}}=(a_1,a_2,a_3)$ be a TFN with $0<a_1$ and b is a crisp number, $nes({\tilde{a}} \le b)\ge \alpha$ iff $\displaystyle \frac{a_3-b}{a_3-a_2} \le 1- \alpha$.

1.2 Fuzzy extension principle (Zadeh 1965)

If ${\tilde{a}}$ , ${\tilde{b}}$$\subseteq \mathfrak {R}$ and ${\tilde{c}}=f({\tilde{a}},{\tilde{b}})$, where $f:\mathfrak {R}\times \mathfrak {R}\rightarrow \mathfrak {R}$ is a binary operation, membership function $\mu _{{\tilde{c}}}$ of ${\tilde{c}}$ is defined as

$$\begin{aligned}&\text{ For } \text{ each }\,\,z\in \mathfrak {R},\,\,\, \mu _{{\tilde{c}}}(z) = \mathrm{sup}\{\mathrm{min}(\mu _{{\tilde{a}}}(x),\mu _{{\tilde{b}}}(y)), \nonumber \\&\quad x,y \in \mathfrak {R} \text{ and } z=f(x,y)\} \end{aligned}$$

(35)

1.3 $\alpha$-Cut of a fuzzy number

$\alpha$-Cut of a fuzzy number ${\tilde{A}}$ in X is denoted by $A[\alpha ]$ and is defined as the following crisp set

$$\begin{aligned} A[\alpha ]=\{x:\mu _{{\tilde{A}}}(x)\ge \alpha , x\in X\} \text{ where } \alpha \in [0,1] \end{aligned}$$

$A[\alpha ]$ is a non-empty bounded closed interval contained in X and it can be denoted by $A[\alpha ] = [A_L(\alpha ),A_R(\alpha )]$. $A_L(\alpha )$ and $A_R(\alpha )$ are the lower and upper bounds of the closed interval, respectively, and defined as

$$\begin{aligned}&A_L(\alpha )=\mathrm{inf}\{x\in X:\mu _{{\tilde{A}}}(x)\ge \alpha \}\nonumber \\&A_R(\alpha )=\mathrm{sup}\{x\in X:\mu _{{\tilde{A}}}(x)\ge \alpha \} \end{aligned}$$

(36)

1.3.1 Intuitionistic fuzzy set (IFS) (Atanassov 1986, 2010)

Let $X = {x_1, x_2, \ldots , x_n}$ be a finite universal set. An Atanassov’s IFS A is a set of ordered triples,

$$\begin{aligned} A = \{\langle x_i, \mu _A(x_i), \nu _A(x_i)\rangle : x_i \in X\} \end{aligned}$$

where $\mu _A(x_i)$ and $\nu _A(x_i)$ are functions mapping from X into [0, 1]. For each $x_i$$\in$X, $\mu _A(x_i)$ represents the degree of membership and $\nu _A(x_i)$ represents the degree of non-membership of the element $x_i$ to the subset A of X. For the functions $\mu _A(x_i)$ and $\nu _A(x_i)$ mapping into [0, 1], the condition $0 \le \mu _A(x_i) + \nu _A(x_i) \le 1$ holds .

1.3.2 Fuzzy differential equation (FDE) (Buckley and Feuring 1999)

Consider the first-order ordinary differential equation

$$\begin{aligned} \frac{\mathrm{d}Y}{\mathrm{d}t}=f(t,Y,k),\quad Y(0)=C, \end{aligned}$$

(37)

where $k =(k_1,k_2,\dots k_n)$ is a vector of constants and t is in some interval I (closed and bounded) which contains zero. Let Eq. (37) have a unique solution

$$\begin{aligned} Y=g(t, k, C),\quad \text{ for } t\in I ,\,\, k\in K \subset \mathfrak {R}^n,\,\, C \in \mathfrak {R} \end{aligned}$$

(38)

When ${\widetilde{k}} =({\widetilde{k}}_1,{\widetilde{k}}_2,\dots {\widetilde{k}}_n)$ is a vector of fuzzy numbers and ${\widetilde{C}}$ be another fuzzy number, Eq. (37) reduces to the following fuzzy differential equation (FDE),

$$\begin{aligned} \frac{\mathrm{d}{\widetilde{Y}}}{\mathrm{d}t}=f(t,{\widetilde{Y}},{\widetilde{k}}),\,\,\,\,\,\, {\widetilde{Y}}(0)={\widetilde{C}} \end{aligned}$$

(39)

Assuming that derivative of the unknown fuzzy function ${\widetilde{Y}}(t)$ exists, then according to Buckley and Feuring (1999),

$$\begin{aligned} {\widetilde{Y}}(t)=g(t, {\widetilde{k}}, {\widetilde{C}}) \end{aligned}$$

(40)

is solution of (39) if its $\alpha$-cut ${\widetilde{Y}}(t)[\alpha ]=[Y_L(t,\alpha ),Y_R(t,\alpha )]$ satisfies the following conditions.

$$\begin{aligned} \frac{\mathrm{d}Y_L(t,\alpha )}{\mathrm{d}t}= & {} f_L(t,\alpha ),\, \frac{\mathrm{d}Y_R(t,\alpha )}{\mathrm{d}t}=f_R(t,\alpha ),\,\forall \alpha \in [0,1].\nonumber \\ \frac{\mathrm{d}Y_L(0,\alpha )}{\mathrm{d}t}= & {} C_L(\alpha ),\, \frac{\mathrm{d}Y_R(0,\alpha )}{\mathrm{d}t}=C_R(\alpha ),\,\forall \alpha \in [0,1].\nonumber \\ \end{aligned}$$

(41)

$$\begin{aligned}&\frac{\mathrm{d}Y_L(t,\alpha )}{\mathrm{d}t}\,\,\text{ and } \,\,\frac{\mathrm{d}Y_R(t,\alpha )}{\mathrm{d}t}\,\, \text{ are } \text{ continuous } \text{ on } \,\,I \times [0,1].\nonumber \\&\frac{\mathrm{d}Y_L(t,\alpha )}{\mathrm{d}t} \,\,\text{ is } \text{ an } \text{ increasing } \text{ function } \text{ of }\,\, \alpha \,\,\text{ for } \text{ each }\,\, t\in I.\nonumber \\&\frac{\mathrm{d}Y_R(t,\alpha )}{\mathrm{d}t} \,\,\text{ is } \text{ a } \text{ decreasing } \text{ function } \text{ of }\,\, \alpha \,\,\text{ for } \text{ each }\,\, t\in I.\nonumber \\&\frac{\mathrm{d}Y_L(t,1)}{\mathrm{d}t} \le \frac{\mathrm{d}Y_R(t,1)}{\mathrm{d}t}, \forall \, t\in I. \end{aligned}$$

(42)

where ${\widetilde{f}}(t)[\alpha ]=[f_L(t,\alpha ),f_R(t,\alpha )]$, ${\widetilde{C}}[\alpha ]=[C_L(\alpha ),C_R(\alpha )]$ are obtained following Eq. (36) and membership function of ${\widetilde{Y}}(t)$ is obtained using fuzzy extension principle (35).

1.3.3 Fuzzy Riemann integration (FRI)

Wu (2000) defined integration of fuzzy mapping over crisp and fuzzy intervals. Depending upon the limits of integral, two types of FRIs are as follows:

Fuzzy Riemann integral of type-I (Wu 2000): let ${\tilde{f}}(x)$ be a closed and bounded fuzzy-valued function on [a, b] and $[f_L(\alpha ,x),f_R(\alpha ,x)]$ be $\alpha$-cut of ${\tilde{f}}(x)$$\forall x \in [a,b]$. If $f_L(\alpha ,x)$ and $f_R(\alpha ,x)$ are Riemann integrable on [a, b], $\forall \alpha$, then the fuzzy Riemann integral $\int \limits _{a}^{b}{\tilde{f}}(x)\mathrm{d}x$ is a closed fuzzy number and its $\alpha$-level set is given by

$$\begin{aligned} \left( \int \limits _{a}^{b}{\tilde{f}}(x)\mathrm{d}x\right) [\alpha ]= \left[ \int \limits _{a}^{b}f_L(\alpha ,x)\mathrm{d}x,\int \limits _{a}^{b}f_R(\alpha ,x)\mathrm{d}x \right] \end{aligned}$$

Fuzzy Riemann integral of type-II (Wu 2000): let ${\tilde{f}}({\tilde{x}})$ be a bounded and closed fuzzy-valued function defined on the closed fuzzy interval $[{\tilde{a}},{\tilde{b}}]$ and ${\tilde{f}}(x)$ be induced by ${\tilde{f}}({\tilde{x}})$. $[f_L(\alpha ,x),f_R(\alpha ,x)]$ be $\alpha$-cut of ${\tilde{f}}(x)$ and ${\tilde{f}}(x)$ is either non-negative or non-positive.

Case-1: If ${\tilde{f}}(x)$ is non-negative and $f_L(\alpha ,x)$ and $f_R(\alpha ,x)$ are Riemann integrable on $[a_R(\alpha ),b_L(\alpha )]$ and $[a_L(\alpha ),b_R(\alpha )]$, respectively, $\forall \alpha$, then the fuzzy Riemann integral $\int \nolimits _{{\tilde{a}}}^{{\tilde{b}}}{\tilde{f}}({\tilde{x}})\mathrm{d}{\tilde{x}}$ is a closed fuzzy number and its $\alpha$-level set is given by

$$\begin{aligned}&\left( \int \limits _{{\tilde{a}}}^{{\tilde{b}}} {\tilde{f}}({\tilde{x}})\mathrm{d}{\tilde{x}}\right) [\alpha ]\nonumber \\&\quad = \left\{ \!\begin{array}{ll} \bigg [\int \nolimits _{a_R(\alpha )}^{b_L(\alpha )}f_L(\alpha ,x)\mathrm{d}x, \int \nolimits _{a_L(\alpha )}^{b_R(\alpha )}f_R(\alpha ,x)\mathrm{d}x \bigg ] &{}\quad \text{ if } b_L(\alpha )>a_R(\alpha )\\ \bigg [0,\int \nolimits _{a_L(\alpha )}^{b_R(\alpha )}f_R(\alpha ,x)\mathrm{d}x \bigg ]&{} \quad \text{ if } b_L(\alpha )\le a_R(\alpha ) \end{array}\right. \end{aligned}$$

Case-2: If ${\tilde{f}}(x)$ is non-positive and $f_L(\alpha ,x)$ and $f_R(\alpha ,x)$ are Riemann integrable on $[a_L(\alpha ),b_R(\alpha )]$ and $[a_R(\alpha ),b_L(\alpha )]$, respectively, $\forall \alpha$, then the fuzzy Riemann integral $\int \nolimits _{{\tilde{a}}}^{{\tilde{b}}}{\tilde{f}}({\tilde{x}})\mathrm{d}{\tilde{x}}$ is a closed fuzzy number and its $\alpha$-level set is given by

$$\begin{aligned}&\left( \int \limits _{{\tilde{a}}}^{{\tilde{b}}} {\tilde{f}}({\tilde{x}})\mathrm{d}{\tilde{x}}\right) [\alpha ]\\&\quad = \left\{ \begin{array}{ll} \left[ \int \nolimits _{a_L(\alpha )}^{b_R(\alpha )}f_L(\alpha ,x)\mathrm{d}x, \int \nolimits _{a_R(\alpha )}^{b_L(\alpha )}f_R(\alpha ,x)\mathrm{d}x \right] &{}\quad \text{ if } \,b_L(\alpha )>a_R(\alpha )\\ \left[ \int \nolimits _{a_L(\alpha )}^{b_R(\alpha )}f_L(\alpha ,x)\mathrm{d}x,0 \right] &{}\quad \text{ if } \,b_L(\alpha )\le a_R(\alpha ) \end{array}\right. \end{aligned}$$

Appendix B

In the interval$[0,{\tilde{\tau }}]$:

$\frac{\mathrm{d}I(t)}{\mathrm{d}t}=P - D$ with the initial condition I(0)=0. Solving this equation in crisp environment, we get, $I(t)= (P - D)t$.

Table 9 Individual minimum and maximum of objective functions

In the interval$[{\tilde{\tau }},t_1]$:

The fuzzy differential equation is $\frac{\mathrm{d}{\widetilde{I}}(t)}{\mathrm{d}t}=P - D -(1-\theta )\gamma P(t-{\widetilde{\tau }})$ with fuzzy initial condition ${\widetilde{I}}({\tilde{\tau }})= (P - D){\tilde{\tau }}$. Solving this fuzzy differential equation, we get,

$$\begin{aligned} \left\{ \begin{array}{ll} I_{L}(\alpha ,t)=(P-D)t-\frac{(1-\theta )\gamma P}{2}(t-\tau _L)^2\\ I_{R}(\alpha ,t)=(P-D)t-\frac{(1-\theta )\gamma P}{2}(t-\tau _R)^2 \end{array} \right. \end{aligned}$$

(43)

In the interval$[t_1,{\tilde{T}}]$:

The differential equation is $\frac{\mathrm{d}I(t)}{\mathrm{d}t}= - D$ with boundary condition $I(T)= 0$. Solving the above equation, we get $I(t)=D(T-t)$. If ${\tilde{T}}$ is fuzzy in nature, then using the Jadeh’s extension principle we have,

$$\begin{aligned} I_{L}(\alpha ,t)= & {} D(T_L-t)\nonumber \\ I_{R}(\alpha ,t)= & {} D(T_R-t) \end{aligned}$$

(44)

Checking of Buckley–Feuring conditions:

In the interval$[{\tilde{\tau }},t_1]$:

Here, ${\tilde{\tau }}=(\tau _1,\tau _2,\tau _3)$ is a triangular fuzzy number having $\alpha$-cut ${\tilde{\tau }}[\alpha ]=[\tau _L,\tau _R]$, where $\tau _L=\tau _1+\alpha (\tau _2-\tau _1)$ and $\tau _R=\tau _3-\alpha (\tau _3-\tau _2)$. Also, ${\tilde{I}}(t)[\alpha ]=[I_L(\alpha ,t),I_R(\alpha ,t)]$, where $I_{L}(\alpha ,t)=(P-D)t-\frac{(1-\theta )\gamma P}{2}(t-\tau _L)^2$ and $I_{R}(\alpha ,t)=(P-D)t-\frac{(1-\theta )\gamma P}{2}(t-\tau _R)^2$.

Therefore, $\frac{\mathrm{d}I_{L}(\alpha ,t)}{\mathrm{d}t}=(P-D)-(1-\theta )\gamma P(t-\tau _L)$ and $\frac{\mathrm{d}I_{R}(\alpha ,t)}{\mathrm{d}t}=(P-D)-(1-\theta )\gamma P(t-\tau _R)$.

Differentiating the above equations with respect to $\alpha$, we get

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}\alpha }\bigg (\frac{\mathrm{d}I_{L}(\alpha ,t)}{\mathrm{d}t}\bigg )= & {} (1-\theta )\gamma P\frac{\mathrm{d}}{\mathrm{d}\alpha }(\tau _L) \\= & {} (1-\theta )\gamma P\frac{\mathrm{d}}{\mathrm{d}\alpha }\{\tau _1+\alpha (\tau _2-\tau _1)\} \\= & {} (1-\theta )\gamma P(\tau _2-\tau _1)>0. \end{aligned}$$

and

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}\alpha }\bigg (\frac{\mathrm{d}I_{R}(\alpha ,t)}{\mathrm{d}t}\bigg )= & {} (1-\theta )\gamma P\frac{\mathrm{d}}{\mathrm{d}\alpha }(\tau _R) \\= & {} (1-\theta )\gamma P\frac{\mathrm{d}}{\mathrm{d}\alpha }\{\tau _3-\alpha (\tau _3-\tau _2)\} \\= & {} -(1-\theta )\gamma P(\tau _3-\tau _2)<0. \end{aligned}$$

Also, $\frac{\mathrm{d}I_{L}(1,t)}{\mathrm{d}t}=\frac{\mathrm{d}I_{R}(1,t)}{\mathrm{d}t}=(P-D)-(1-\theta )\gamma P(t-\tau _2)$.

Hence, all the equations and conditions defined by (41) and (42), respectively, are satisfied.

Appendix C

Individual minimum of the objective functions $ACEC_k$ for all $k=L,C,R$ is obtained and given in Table 9. Now we calculate $L_L=4862.07$, $L_C=5057.22$, $L_R=5243.88$, $U_L=4876.38$, $U_C=5063.69$, $U_R=5265.32$. we formulate the following problem as:

(45)

Table 10 Optimum results of Eq. (45) for $w=0.10$

The solutions obtained for Eq. (45) are given in Table 10. Now we perform the Pareto-optimal solution test for strong or weak solutions. The Pareto-optimal results are presented in Table 11. In Table 11, the value of $V^*$ is quite small, and hence, the optimal results in Table 11 are strong Pareto-optimal solution and can be accepted.

Table 11 Pareto-optimal results