Multi-objective optimization of cost-effective and customer-centric closed-loop supply chain management model in T-environment

Abstract

This article presents one real-life-based cost-effective and customer-centric closed-loop supply chain management model. The review of the existing literature identifies the classical performance indicators to any supply chain management model as the aggregate revenue, the customer satisfaction and the environmental concern. However, this review fails to find a single optimization-based supply chain management model that considers these three indicators, simultaneously. In this article, the proposed model maximizes the customer-satisfaction index and the aggregate revenue both under the environmental considerations via the reverse chain, whereas many existing studies took the reverse chain and the associated subsidies into account; this is the first mathematical model that optimizes the customer-satisfaction index, at the same time. This article employs the T-set that represents the inherent impreciseness to objective functions to the proposed model. The corresponding optimal values are superior than stipulated goals to both the objective functions in T-environment. The managerial insights extracted from sensitivity analysis of parameters suggest the managers to stabilize the environmental concern and the customer satisfaction, while ensuring the cost-effectiveness in real-life-based T-environment. Also, this analysis finds that the subsidy assists any supply chain to sustain, only if it is offered without any break and within the optimally determined bounds.

Appendices

Appendix A: Symbols and descriptions

See Table 11.

Table 11 Symbols with the descriptions used in the proposed CLSCM model

Appendix B: Drawbacks of classical fuzzy set

The drawbacks of the membership function to classical fuzzy set as well as the redefined membership function in Wu et al. (2015) are as follows (Garai et al. 2016, 2017)

• Firstly, the employments of redefined membership functions in Wu et al. (2015) were not as per the definitions presented in mathematical model as well as in numerical examples of that article.

• Secondly, the mathematical model in Wu et al. (2015) employed strictly monotonic redefined membership functions over the entire real line. However, the definitions in that article specified otherwise.

• In addition, the researchers could only employ the part of redefined membership functions, at which the corresponding objective values lied within the goals and goal plus tolerances, to the minimization type of objective functions. However, so far, researchers did not consider the above prerequisite (in form of constraints) within the mathematical models.

• Thirdly, Wu et al. (2015) discarded the upper bound at unity to membership functions of fuzzy objective functions. However, the corresponding lower bound to membership functions of fuzzy objective functions remained intact at zero. Here, the authors find this to be arbitrary and biased.

• Most importantly, the classical fuzzy membership function could provide satisfying solution, only when the extreme ends of imprecise information lied within zero and one. However, Garai et al. (2016, 2017) showed that the proper mathematical representation of impreciseness cannot be confined within any closed and bounded interval of real line, at all times.

Figure 5 illustrates the drawbacks of membership function to classical fuzzy set (Garai et al. 2017).

Fig. 5

Drawbacks of the membership function to classical fuzzy set

Appendix C: T-set and related definitions

The characteristic function of crisp subset assigns either 1 or 0 to elements of universal set and thereby discriminates between the members and non-members. Again, membership function to fuzzy subset generalizes the characteristic function and the values assigned to elements of universal set fall within the specified range of closed unit interval [0, 1]. However, Garai et al. (2016, 2017) observed that the membership function failed to discriminate between yes and certainly yes by assigning same value ‘unity’ to both cases and failed to discriminate between no and certainly no by assigning same value ‘zero’ to both cases (Garai et al. 2016, 2017).

In other words, elements to universal set must be or be or partly be or not be or never be lying in a subset. The concept of being, partly being or not being is well measured by membership function of fuzzy subset. However, this cannot explain the cases of must being (since membership value is unity, at all times) as well as never being (since membership value is zero, at all times) for elements of fuzzy subset. Garai et al. (2017) presented the following example to illustrate all these as follows

• The task is to pick up one tall policeman among all policemen for the guard of honour ceremony of the President of USA during the official visit to North Korea. To assign the job solely to some tall person in force is a bold step to North Korea. If one policeman with height of more than 6′ is considered to be tall, he can be assigned unity as the membership value in fuzzy subset of tall policemen.

Assume that Mr. Bansal is 6′4″ tall and Mr. Chowd is 6′6″ tall. In fuzzy set theory, both are given unity as the same membership value. However, if the DM were all machines with zero emotion, this would have been acceptable to choose any one of two persons (or in fact, any policeman taller than 6′). Nevertheless, as human beings, the mind plays key role. DM can have full happiness only after selecting the tallest person. Here, all the persons taller than 6′ are assigned the same membership value (unity). So, DM fail to find the most suitable person in force for the prestigious ceremony.

This marks that the upper bound of membership function of fuzzy subset at unity causes the dilemma.

Consequently, Garai et al. (2016, 2017) introduced the T-characteristic function and then T-set as follows

Definition 1

Let S be universal set and A is any subset of S. Then, T-characteristic function of A is denoted by $T_{A}$ and is defined as $T_{A} :S \to IR$. So, this assigns real number $T_{A} \left( x \right)$ to each element $x \in S$. Higher the value of $T_{A} \left( x \right)$, larger the value of membership of $x \in S$ in A. Here $IR$ denotes set of real numbers.

Definition 2

Let S be universal set. Then, T-subset A of S is defined as the ordered pair $A = \{ (x,T_{A} (x)):x \in S\}$, where T-characteristic function $T_{A} :S \to IR$ assigns real number $T_{A} \left( x \right)$ as membership value to each $x \in S.$

The universal set S is not necessarily an ordered set in imprecise environment. However, the function $T_{A} :S \to IR$ imposes an ordering over elements of S based on the value of $T_{A} \left( x \right)$$\forall x \in S$.

Definition 3

The union of two T-subsets A and B of S, which is denoted by $A \cup B$, is defined as $A \cup B = \{ (x,T_{A \cup B} (x)):x \in S\}$, where T-characteristic function $T_{A \cup B} (x)$ of $A \cup B$ is defined as $T_{A \cup B} (x) = \hbox{max} \{ T_{A} (x),T_{B} (x)\} ,\forall x \in S$.

Definition 4

The intersection of two T-subsets A and B of S, which is denoted by $A \cap B$, is defined as $A \cap B = \{ (x,T_{A \cap B} (x)):x \in S\}$, where T-characteristic function $T_{A \cap B} (x)$ of $A \cap B$ is defined as $T_{A \cap B} (x) = \hbox{min} \{ T_{A} (x),T_{B} (x)\} ,\forall x \in S$.

Definition 5

The complement of T-subset A of S, denoted by $\bar{A}{\text{ or }}A^{c}$, is defined as $\bar{A} = \{ (x,T_{{\bar{A}}} (x)):x \in S\}$, where T-characteristic function $T_{A} \left( x \right)$ of $\bar{A}{\text{ or }}A^{c}$ is defined as $T_{{\bar{A}}} (x) = 1 - T_{A} (x),\forall x \in S.$