Defuzzification of trapezoidal type-2 fuzzy variables and its application to solid transportation problem

Abstract

The main proposal of this paper is to derive two different reduction process for a trapezoidal type-2 fuzzy number. The first reduction method is based on critical values and the second method is based on α-cut of fuzzy number. As an application a multi-objective solid transportation problem, minimizing the cost and time has been developed using trapezoidal type-2 fuzzy number as system parameters and hereby solved. Finally after solving the proposed multi-objective problem by intuitionistic fuzzy programming technique a comparison between the two proposed reduction methods are discussed briefly. The proposed models and techniques are finally illustrated by providing numerical examples at the end. Also this paper present a comparative study between the proposed method to the KM algorithm and NT method for type reduction.

Keywords

Critical values defuzzification trapezoidal type-2 fuzzy set intuitionistic fuzzy programming technique solid transportation problem

1 Introduction

As an extension of the well known traditional transportation problem (TP) Shell [8] introduced the solid transportation problem (STP) and later on in the year 1962 Haley [15] developed the solution procedure for STP. Basically a TP deals with two set of constraints i.e. source and demand constraints, but in addition to it a STP consider another constraints viz. conveyance constraint. This constraint has been introduced depending upon the mode of transportation. A literature survey [1–3 , 26–30] divulge that, research work based on modelling and solution techniques of TP or STP in different environments become an interesting topic to many researchers. In their research they have proposed so many new methods to solve a TP or STP.

For a TP, the system parameters always plays an important role. The value of these system parameters are generally consider as a crisp number. But it has been observed that, all the time these values are not possible to consider in it’s exact form. Because the values of these system parameters are changing over time to time. For example let us consider the demand, it is the requirement for a particular destination. Normally it depends upon the unit price, per unit requirement, the road conditions between the supply and demand point, time etc. and all these are changeable. Consequently it bring an effect on the demand. In this way the other system parameters are also changes. This change directly effect the total transportation cost and time respectively. Also in many cases it become essential to make a future transportation plan in advance, where the transportation parameters are not predetermined. The crisp environment here fails to model this situation. But the fuzzy concept have the capability to do the same.

Fuzzy number first introduced by Zadeh [16] in 1975. The fuzzy number comes with a pair of member and it’s degree of membership. TP in fuzzy environment are available in literature. But there is problem with this fuzzy number. It become fail when there is a uncertainty exist in the membership function. It has been observed that, when the decision is made upon the different expert’s opinions then to define the type-1 membership function or the degree of membership for a particular member is become difficult. In such case using the type-2 fuzzy set is become very fruitful as type-2 fuzzy set is the second order approximation of uncertainty whereas the fuzzy number is the first order approximation. Type-2 fuzzy set was first proposed by Zadeh [16]. There are mainly two fuzzy numbers viz. triangular and trapezoidal fuzzy number, which are frequently used in literature to express the uncertainties [6 , 43]. However recently the researchers [42] have paid their attention to the use of Gaussian type-2 fuzzy set also. Compare to a type-1 fuzzy set a type-2 fuzzy set is more sophisticated, which allow us to incorporate uncertainty about the membership function of type-1 fuzzy sets. So many research works [12 , 44] related to a type-2 fuzzy sets are there in literature. T2FS are successfully applied in many important fields such as multi criteria group decision making process [39 –41], TP [5 , 28–30], neural network [33], DEA modelling [32] etc.

But due to the three dimensional structure, computational complexity is very high with a T2FS and hence to make it easier the defuzzification process has been introduced by the researchers. Defuzzification is the process by which we get sometimes direct crisp value from a type-2 fuzzy set or some time a type-1 fuzzy set, which is called a type reduced set (TRS). Applying the crisp conversation rules related to a type-1 fuzzy set the TRS convert to a crisp output. There are so many defuzzification process exist in literature. Interested readers are refer to [9 , 32]. Greenfield et al. [36, 37] developed two different defuzzufication method for T2FS, Coupland et al. [35] proposed a fast geometric defuzzification process where the defuzzification is based on converting a type-2 fuzzy sets into a geometric type-2 fuzzy sets. Recently in the year 2013 Chiclana et al. [7] introduced a new type-reduction of general type-2 fuzzy sets based on the type-1 OWA method. Also Qin et al. [32] introduced an effective defuzzification method based on three critical values (CV) viz. optimistic CV, pessimistic CV and CV of regular fuzzy variable (RFV). In [32] Qin et al. discussed the reductions for type-2 triangular, type-2 normal and type-2 gamma fuzzy variables.

This present study address some new developments within the field of type-2 fuzzy sets. Here in the following we have list out these.

Trapezoidal type-2 fuzzy variable with type-2 membership function is presented.

Defuzzification of a trapezoidal type-2 fuzzy variable using critical value based reduction method is proposed.

Also the nearest interval approximation for defuzzification of trapezoidal type-2 fuzzy variable is derived.

The intuitionistic fuzzy optimization technique applied for solving the multi-objective solid transportation problem.

Also as an application of the proposed methodology this paper present a multi-objective solid transportation problem where the system parameters are trapezoidal type-2 fuzzy by nature. We have defuzzified these parameters using the proposed defuzzification methods and hereby solved the multi-objective solid transportation problem (MOSTP). The intuitionistic fuzzy optimization technique has been applied to solve the multi-objective problem i.e. MOSTP. The related soft computations were done using the LINGO 13.0 software.

The organization of the paper as follows, Section 2 discuss some fundamental concept of type-2 fuzzy sets with some important definitions and examples, in Section 3 the proposed defuzzification methods for a trapezoidal type-2 fuzzy variables are given. Section 4 discuss the problem and it’s formulation. The solution of the problem and numerical experiment is in Section 5. In this section a comparative study of the proposed method to KM algorithm and Nie-Tan method are presented. In the last the conclusion and future scope of this present investigation are given in Section 6. A reference list is attached at the end.

2 Some fundamental concepts

In this section we present some fundamental concepts about type-2 fuzzy sets. Some important definitions with example are also given.

2.1 Type-2 fuzzy set (T2 FS)

Type-2 fuzzy set $\tilde{A}$ defined on a universe of discourse X, which is denoted as $\tilde{A} \subseteq X$ , is a set of pairs ${x, μ_{\tilde{A}} (x)}$ , where x an element of a fuzzy set is, and its grade of membership ${μ_{\tilde{A}} (x)}$ in the fuzzy set $\tilde{A}$ is a type-1 fuzzy set defined in the interval J_x ⊂ [0, 1], i.e. A T2 FS $\tilde{A}$ defined by Mendel and John [12] is $\tilde{A} = {((x, u), μ_{\tilde{A}} (x, u)) : \forall x \in X, J_{x} \subset [0, 1]},$

where $0 \leq μ_{\tilde{A}} (x, u) \leq 1$ is the type-2 membership function.

Example 1. Let us take $\tilde{A} {(x, μ_{\tilde{A}} (x)) : x \in X}$ whereX = 3, 6, 9 and the primary memberships of the points 3, 6, 9 are given by J₃ = 0.4, 0.8, 0.9, J₆ = 0.3, 0.7, 0.8, 0.9 and J₉ = 0.2, 0.7, 1.0 respectively. Then the secondary grade of the point 3 is $μ_{\tilde{A}} (3) = μ_{\tilde{A}} (3, u) = (0.5 / 0.4) + (0.7 / 0.8) + (0.3 / 0.9) \sim$ $(\begin{matrix} 0.4 & 0.8 & 0.9 \\ 0.5 & 0.7 & 0.3 \end{matrix})$ . i.e. $μ_{\tilde{A}} (3, 0.4) = 0.5, μ_{\tilde{A}} (3, 0.8) = 0.7, μ_{\tilde{A}} (3, 0.9) = 0.3 .$

More specifically $μ_{\tilde{A}} (3, 0.4) = 0.5$ means that the membership grade which is named as secondary membership grade that the point 3 has the primary membership 0.4 is 0.5. So $\tilde{A}$ considers on the value 3 with membership grade $(\begin{matrix} 0.4 & 0.8 & 0.9 \\ 0.5 & 0.7 & 0.3 \end{matrix})$ , which is a RFV.

Same as, $μ_{\tilde{A}} (6) = μ_{\tilde{A}} (6, u) = (0.4 / 0.3) + (0.6 / 0.7)$ $+ (0.5 / 0.8) + (0.8 / 0.9) \sim (\begin{matrix} 0.4 & 0.6 & 0.5 & 0.8 \\ 0.3 & 0.7 & 0.8 & 0.9 \end{matrix})$ and $μ_{\tilde{A}} (9) = μ_{\tilde{A}} (9, u) = (0.9 / 0.2) + (0.5 / 0.7) + (0.1 /$ $1.0) \sim (\begin{matrix} 0.9 & 0.5 & 0.1 \\ 0.2 & 0.7 & 1.0 \end{matrix})$ .

So discrete type-2 fuzzy variable $\tilde{A}$ is given by $\tilde{A} = (0.5 / 0.4) / 3 + (0.7 / 0.8) / 3 + (0.3 / 0.9) / 3 + (0.3 / 0.4) / 6 + (0.7 / 0.6) / 6 + (0.8 / 0.5) / 6 + (0.9 / 0.8) / 0.6 + (0.2 / 0.9) / 9 + (0.7 / 0.5) / 9 + (1.0 / 0.1) 9 .$

Type-2 fuzzy set $\tilde{A}$ can be also express as,

$\tilde{A} = {\begin{matrix} 3, with possibilities (0.4, 0.8, 0.9); \\ 6, with possibilities (0.3, 0.7, 0.8, 0.9); \\ 9, with possibilities (0.2, 0.7, 0.1); \end{matrix}$

2.2 Regular fuzzy variable (RFV) and their Critical values (CVs)

For a possibility space (φ, p, Pos), a regular fuzzy variable $\tilde{ξ}$ is defined as a measurable map from φ to [0, 1] in the sense that for every t ∈ [0, 1], one has ${γ \in φ | \tilde{ξ} (γ) \leq t} \in p$ .

A discrete RFV is represented as $\tilde{ξ} \sim (\begin{matrix} r_{1}, r_{2} \dots r_{n} \\ μ_{1}, μ_{2} \dots μ_{n} \end{matrix})$ where r_i ∈ [0, 1] and μ_i > 0, ∀ i and $\max_{i} (μ_{i}) = 1$ .

If $\tilde{ξ} = (r_{1}, r_{2}, r_{3})$ with 0 ≤ r₁ < r₂ < r₃ ≤ 1, then $\tilde{ξ}$ is called a triangular RFV.

Qin et al. [32] introduced three kinds of critical values (CVs). Let $\tilde{ξ}$ be a RFV. Then,

The optimistic CV of $\tilde{ξ}$ , denoted by ${CV}^{*} [\tilde{ξ}]$ is given by,

${CV}^{*} [\tilde{ξ}] = sup_{α \in [0, 1]} [α \land Pos (\tilde{ξ} \geq α)]$ (1)

The pessimistic CV of $\tilde{ξ}$ , denoted by ${CV}_{*} [\tilde{ξ}]$ is given by,

${CV}_{*} [\tilde{ξ}] = sup_{α \in [0, 1]} [α \land Nec (\tilde{ξ} \geq α)]$ (2)

The CV of $\tilde{ξ}$ , denoted by $CV [\tilde{ξ}]$ is given by,

$CV [\tilde{ξ}] = sup_{α \in [0, 1]} [α \land Cr (\tilde{ξ} \geq α)]$ (3)

Example 2. Let $\tilde{ξ}$ be a discrete RFV define as $ξ \sim (\begin{matrix} 0.3 & 0.5 & 0.8 & 0.9 \\ 0.1 & 0.9 & 0.6 & 0.3 \end{matrix})$

Then we can find out that,

$Pos (\tilde{ξ} \geq α) = {\begin{matrix} 0, & if α \leq 0.2 \\ 0.9, & if 0.2 < α \leq 0.5 \\ 0.6, & if 0.5 < α \leq 0.8 \\ 0.3, & if 0.8 < α \leq 1.0 \end{matrix}$

$Nec (\tilde{ξ} \geq α) = {\begin{matrix} 0.9, & if α \leq 0.3 \\ 0.1, & if 0.3 < α \leq 0.5 \\ 0.4, & if 0.5 < α \leq 0.8 \\ 0.7, & if 0.8 < α \leq 1.0 \end{matrix}$

and

$Cr (\tilde{ξ} \geq α) = {\begin{matrix} 0.9, & if α \leq 0.3 \\ 0.5, & if 0.3 < α \leq 0.5 \\ 0.5, & if 0.5 < α \leq 0.8 \\ 0.5, & if 0.8 < α \leq 1.0 \end{matrix}$

Then by the definitions of CVs, from (1)-(3), we have

$\begin{matrix} {CV}^{*} [\tilde{ξ}] = sup_{α \in [0, 1]} [α \land Pos (\tilde{ξ} \geq α)] \\ = sup_{α \in [0, 0.2]} [α \land 0] \lor sup_{α \in (0.2, 0.5]} [α \land 0.9] \\ \lor sup_{α \in (0.5, 0.8]} [α \land 0.6] \lor sup_{α \in [0.8, 1]} (α \land 0.3] \\ = 0 \lor 0.5 \lor 0.6 \lor 0.3 \\ = 0.6 \end{matrix}$

$\begin{matrix} {CV}_{*} [\tilde{ξ}] = sup_{α \in [0, 1]} [α \land Nec (\tilde{ξ} \geq α)] \\ = sup_{α \in [0, 0.3]} [α \land 0.9] \lor sup_{α \in (0.3, 0.5]} [α \land 0.1] \\ \lor sup_{α \in (0.5, 0.8]} [α \land 0.4] \lor sup_{α \in (0.8, 1]} [α \land 0.7] \\ = 0 \lor 0.1 \lor 0.4 \lor 0.7 = 0.7 \end{matrix}$

and

$\begin{matrix} CV [\tilde{ξ}] = sup_{α \in [0, 1]} [α \land Cr (\tilde{ξ} \geq α)] \\ = sup_{α \in [0, 0.3]} [α \land 0.9] \lor sup_{α \in (0.3, 0.5]} [α \land 0.5] \\ \lor sup_{α \in (0.5, 0.8]} [α \land 0.5] \lor sup_{α \in (0.8, 1]} [α \land 0.5] \\ = 0 \lor 0.5 \lor 0.5 \lor 0.5 = 0.5 \end{matrix}$

2.3 Critical values (CVs) of trapezoidal and triangular RFVs

The following theorems introduced the critical values (CVs) of trapezoidal and triangular RFVs.

Theorem 1.(Qin et al. [32]) Let $\tilde{ξ} = (r_{1}, r_{2}, r_{3}, r_{4})$ be a trapezoidal RFV. Then we have,

The optimistic CV of $\tilde{ξ}$ is ${CV}^{*} [\tilde{ξ}] = \frac{r_{4}}{(1 + r_{4} - r_{3})} .$

The pessimistic CV of $\tilde{ξ}$ is ${CV}_{*} [\tilde{ξ}] = \frac{r_{2}}{(1 + r_{2} - r_{1})} .$

The CV of $\tilde{ξ}$ is

$CV [\tilde{ξ}] = {\begin{matrix} \frac{2 r_{2} - r_{1}}{1 + 2 (r_{2} - r_{1})}, & if r_{2} \geq \frac{1}{2} \\ \frac{1}{2}, & if r_{2} \leq \frac{1}{2} < r_{3} \\ \frac{r_{4}}{1 + 2 (r_{4} - r_{3})}, & if r_{3} \leq \frac{1}{2} \end{matrix}$ .

Example 3. Let $\tilde{ξ} = (0.3, 0.4, 0.8, 0.9)$ be a trapezoidal RFV. Then according to the Theorem 1 we have, ${CV}^{*} [\tilde{ξ}] = \frac{9}{11}, {CV}_{*} [\tilde{ξ}] = \frac{4}{11}, CV [\tilde{ξ}] = \frac{1}{2} .$

Theorem 2.(Qin et al. [32]) Let $\tilde{ξ} = (r_{1}, r_{2}, r_{3})$ be a triangular RFV. Then we have,

The optimistic CV of $\tilde{ξ}$ is ${CV}^{*} [\tilde{ξ}] = \frac{r_{3}}{(1 + r_{3} - r_{2})} .$

The pessimistic CV of $\tilde{ξ}$ is ${CV}_{*} [\tilde{ξ}] = \frac{r_{2}}{(1 + r_{2} - r_{1})} .$

The CV of $\tilde{ξ}$ is $CV [\tilde{ξ}] = {\begin{matrix} \frac{2 r_{2} - r_{1}}{1 + 2 (r_{2} - r_{1})}, & if r_{2} \geq \frac{1}{2} \\ \frac{r_{3}}{1 + 2 (r_{3} - r_{2})}, & if r_{2} \leq \frac{1}{2} \end{matrix}$ .

Example 4. Let $\tilde{ξ} = (0.6, 0.7, 0.9)$ be a triangular RFV. Then according to the Theorem 2 we have, ${CV}^{*} [\tilde{ξ}] = \frac{3}{4}, {CV}_{*} [\tilde{ξ}] = \frac{7}{11}, CV [\tilde{ξ}] = \frac{2}{3} .$

3 Proposed reduction method

Reduction for a type-2 fuzzy set means reducing the fuzziness i.e. to convert a type-2 fuzzy into a type-1 fuzzy set. Here we have proposed two different techniques for the reduction of a trapezoidal type-2 fuzzy variable. The first method derived with the help of critical values (CV) of fuzzy number and the second is based on the concept of α-cut of fuzzy numbers. This section gives the details of these two proposed methods.

3.1 Defuzzification using CV

According to Chen et al. [45] for a trapezoidal type-2 fuzzy fuzzy variable $\tilde{ξ} = (r_{1}, r_{2}, r_{3}, r_{4}; θ_{l}, θ_{r})$ , where r_i ∈ R, ∀ i and θ_l, θ_r ∈ [0, 1] are the two parameters that characterize the degree of uncertainty that $\tilde{ξ}$ takes a value say x and the corresponding secondary possibility distribution function ${\tilde{μ}}_{\tilde{ξ}} (x)$ is given by,

${\tilde{μ}}_{\tilde{ξ}} (x) = (\frac{x - r_{1}}{r_{2} - r_{1}} - θ_{l} \min {\frac{x - r_{1}}{r_{2} - r_{1}}, \frac{r_{2} - x}{r_{2} - r_{1}}}, \frac{x - r_{1}}{r_{2} - r_{1}}, \frac{x - r_{1}}{r_{2} - r_{1}}$ $+ θ_{r} \min {\frac{x - r_{1}}{r_{2} - r_{1}}, \frac{r_{2} - x}{r_{2} - r_{1}}})$

For any x ∈ [r₁, r₂], ${\tilde{μ}}_{\tilde{ξ}} (x) = \tilde{1}$ for x ∈ (r₂, r₃) and ${\tilde{μ}}_{\tilde{ξ}} (x) = (\frac{r_{4} - x}{r_{4} - r_{3}} - θ_{l} \min {\frac{r_{4} - x}{r_{4} - r_{3}}, \frac{x - r_{3}}{r_{4} - r_{3}}}, \frac{r_{4} - x}{r_{4} - r_{3}},$ $\frac{r_{4} - x}{r_{4} - r_{3}} + θ_{r} \min {\frac{r_{4} - x}{r_{4} - r_{3}}, \frac{x - r_{3}}{r_{4} - r_{3}}})$

For any x ∈ [r₃, r₄].

Theorem 3.(Qin et al. [32]) Let $\tilde{ξ} = (r_{1}, r_{2}, r_{3}, r_{4}; θ_{l}, θ_{r})$ be a type-2 trapezoidal fuzzy variable. Then we have,

Using the optimistic CV reduction method, the reduction ξ₁ of $\tilde{ξ}$ has the following possibility distribution,

$μ_{ξ_{1}} (x) = {\begin{matrix} \frac{(1 + θ_{r}) (x - r_{1})}{r_{2} - r_{1} + θ_{r} (x - r_{1})}, & if x \in [r_{1}, \frac{r_{1} + r_{2}}{2}] \\ \frac{(1 - θ_{r}) x - r_{1} + θ_{r} r_{2}}{r_{2} - r_{1} + θ_{r} (r_{2} - x)}, & if x \in (\frac{r_{1} + r_{2}}{2}, r_{2}] \\ \tilde{1}, & if x \in (r_{3}, r_{4}) \\ \frac{(1 + θ_{r}) (r_{4} - x)}{r_{4} - r_{3} + θ_{r} (r_{4} - x)}, & if x \in (r_{3}, \frac{r_{3} + r_{4}}{2}] \\ \frac{(- 1 + θ_{r}) x - θ_{r} r_{3} + r_{4}}{r_{4} - r_{3} + θ_{r} (x - r_{4})}, & if x \in (\frac{r_{3} + r_{4}}{2}, r_{4}] . \end{matrix}$

Using the pessimistic CV reduction method, the reduction ξ₂ of $\tilde{ξ}$ has the following possibility distribution, $μ_{ξ_{2}} (x) = {\begin{matrix} \frac{(x - r_{1})}{r_{2} - r_{1} + θ_{l} (x - r_{1})}, & if x \in [r_{1}, \frac{r_{1} + r_{2}}{2}] \\ \frac{(x - r_{1})}{r_{2} - r_{1} + θ_{l} (r_{1} - x)}, & if x \in (\frac{r_{1} + r_{2}}{2}, r_{2}] \\ \tilde{1}, & if x \in (r_{3}, r_{4}) \\ \frac{(r_{4} - x)}{r_{4} - r_{3} + θ_{l} (x - r_{3})}, & if x \in (r_{3}, \frac{r_{3} + r_{4}}{2}] \\ \frac{(r_{4} - x)}{r_{4} - r_{3} + θ_{l} (r_{4} - x)}, & if x \in (\frac{r_{3} + r_{4}}{2}, r_{4}] . \end{matrix}$

Using the CV reduction method, the reduction ξ₃ of $\tilde{ξ}$ has the following possibility distribution, $μ_{ξ_{3}} (x) = {\begin{matrix} \frac{(1 + θ_{r}) (x - r_{1})}{r_{2} - r_{1} + 2 θ_{r} (x - r_{1})}, & if x \in [r_{1}, \frac{r_{1} + r_{2}}{2}] \\ \frac{(1 - θ_{l}) x + θ_{l} r_{2} - r_{1}}{r_{2} - r_{1} + 2 θ_{l} (r_{2} - x)}, & if x \in (\frac{r_{1} + r_{2}}{2}, r_{2}] \\ \tilde{1}, & if x \in (r_{3}, r_{4}) \\ \frac{(- 1 + θ_{l}) x - θ_{l} r_{3} + r_{4}}{r_{4} - r_{3} + 2 θ_{l} (x - r_{3})}, & if x \in (r_{3}, \frac{r_{3} + r_{4}}{2}] \\ \frac{(1 + θ_{r}) (r_{4} - x)}{r_{4} - r_{3} + 2 θ_{r} (r_{4} - x)}, & if x \in (\frac{r_{3} + r_{4}}{2}, r_{4}] . \end{matrix}$

Theorem 4.(Qin. et al. [32]) Supposeξ_ibe the reduction of type-2 fuzzy variable $\tilde{ξ} = ({\tilde{r}}_{1}^{i}, {\tilde{r}}_{2}^{i}, {\tilde{r}}_{3}^{i}, {\tilde{r}}_{4}^{i}; θ_{l, i}, θ_{r, i})$ obtained by the CV reduction method for i = 1, 2, 3, ⋯ , n and ξ₁, ξ₂, ⋯ , ξ_n are mutually independent, and k_i ≥ 0 for i = 1, 2, 3, ⋯ , n .

Given the generalized credibility level α ∈ (0, 0.5], if α ∈ (0, 0.25], then $\tilde{Cr} {\sum_{i = 1}^{n} k_{i} ξ_{i} \leq t} \geq α$ is equivalent to, $\sum_{i = 1}^{n} \frac{(1 - 2 α + (1 - 4 α) θ_{r, i}) k_{i} r_{1}^{i} + 2 α k_{i} r_{2}^{i}}{1 + (1 - 4 α) θ_{r, i}} \leq t,$ and if α ∈ (0.25, 0.5], then $\tilde{Cr} {\sum_{i = 1}^{n} k_{i} ξ_{i} \leq t} \geq α$ is equivalent to $\sum_{i = 1}^{n} \frac{(1 - 2 α) k_{i} r_{1}^{i} + (2 α + (4 α - 1) θ_{l, i}) k_{i} r_{2}^{i}}{1 + (4 α - 1) θ_{l, i}} \leq t .$

Given the generalized credibility level α ∈ (0.5, 1], if α ∈ (0.5, 0.75], then $\tilde{Cr} {\sum_{i = 1}^{n} k_{i} ξ_{i} \leq t} \geq α$ is equivalent to $\sum_{i = 1}^{n} \frac{(2 α - 1) k_{i} r_{4}^{i} + (2 (1 - α) + (3 - 4 α) θ_{l, i}) k_{i} r_{3}^{i}}{1 + (3 - 4 α) θ_{l, i}} \leq t,$ and if α ∈ (0.75, 1], then $\tilde{Cr} {\sum_{i = 1}^{n} k_{i} ξ_{i} \leq t} \geq α$ is equivalent to $\sum_{i = 1}^{n} \frac{(2 α - 1 + (4 α - 3) θ_{r, i}) k_{i} r_{4}^{i} + 2 (1 - α) k_{i} r_{3}^{i}}{1 + (4 α - 3) θ_{r, i}} \leq t .$

Proof: We are proof here only case 2, the other can be proof in similar way.

For a trapezoidal type-2 fuzzy variable the possibility distribution obtained by CV reduction method is given by, $μ_{ξ_{3}} (x) = {\begin{matrix} \frac{(1 + θ_{r}) (x - r_{1})}{r_{2} - r_{1} + 2 θ_{r} (x - r_{1})}, & if x \in [r_{1}, \frac{r_{1} + r_{2}}{2}] \\ \frac{(1 - θ_{l}) x + θ_{l} r_{2} - r_{1}}{r_{2} - r_{1} + 2 θ_{l} (r_{2} - x)}, & if x \in (\frac{r_{1} + r_{2}}{2}, r_{2}] \\ \tilde{1}, & if x \in (r_{3}, r_{4}) \\ \frac{(- 1 + θ_{l}) x - θ_{l} r_{3} + r_{4}}{r_{4} - r_{3} + 2 θ_{l} (x - r_{3})}, & if x \in (r_{3}, \frac{r_{3} + r_{4}}{2}] \\ \frac{(1 + θ_{r}) (r_{4} - x)}{r_{4} - r_{3} + 2 θ_{r} (r_{4} - x)}, & if x \in (\frac{r_{3} + r_{4}}{2}, r_{4}] . \end{matrix}$

Let ξ^′ = ∑kξ

Now if α > 0.5 then we have,

$\tilde{Cr} (ξ^{'} \leq t) = \frac{1}{2} (1 + sup_{x \leq t} μ_{ξ^{'}} (x) - sup_{x > t} μ_{ξ^{'}} (x))$

$= \frac{1}{2} (1 + 1 - sup_{x > t} μ_{ξ^{'}} (x))$

Therefore $\tilde{Cr} (ξ \leq t) \geq α$ is equivalent to

If we define $ξ_{s u p}^{'} (α) = s u p {r | \sup_{x > t} μ_{ξ^{'}} (x) \geq α}$ for α ∈ (0, 1] then we have, $ξ_{s u p}^{'} (2 - 2 α) \leq t$

So we can write, using the above, $\begin{array}{l} ξ_{s u p}^{'} (2 - 2 α) \leq t = {(\sum k ξ)}_{s u p} (2 - 2 α) \\ = \sum k ξ_{s u p} (2 - 2 α) \leq t \end{array}$

Now if we put $x = \frac{r_{3} + r_{4}}{2}$ in the possibility distribution function then it become 0.5 i.e. $μ_{ξ} (\frac{r_{3} + r_{4}}{2}) = 0.5$

If (2 -2α) ≥0.5 i.e. α ∈ (0.5, 0.75] then ξ_sup (2 -2α) is the solution of the following equation.

$\begin{matrix} \frac{(- 1 + θ_{l}) x - θ_{l} r_{3} + r_{4}}{r_{4} - r_{3} + 2 θ_{l} (x - r_{3})} = 2 - 2 α \end{matrix}$ (4) Solving the equation (4) we get, $x = \frac{(2 α - 1) k_{i} r_{4}^{i} + (2 (1 - α) + (3 - 4 α) θ_{l, i}) k_{i} r_{3}^{i}}{1 + (3 - 4 α) θ_{l, i}},$

3.2 Defuzzification using nearest interval approximation of fuzzy number

A trapezoidal type-2 fuzzy fuzzy variable is defined as $\tilde{ξ} = (r_{1}, r_{2}, r_{3}, r_{4}; θ_{l}, θ_{r})$ , then by optimistic CV, pessimistic CV and CV reduction method we have

Using the optimistic CV reduction method, the reduction ξ₁ of $\tilde{ξ}$ has the following possibility distribution, $μ_{ξ_{1}} (x) = {\begin{matrix} \frac{(1 + θ_{r}) (x - r_{1})}{r_{2} - r_{1} + θ_{r} (x - r_{1})}, & if x \in [r_{1}, \frac{r_{1} + r_{2}}{2}] \\ \frac{(1 - θ_{r}) x - r_{1} + θ_{r} r_{2}}{r_{2} - r_{1} + θ_{r} (r_{2} - x)}, & if x \in (\frac{r_{1} + r_{2}}{2}, r_{2}] \\ \tilde{1}, & if x \in (r_{3}, r_{4}) \\ \frac{(1 + θ_{r}) (r_{4} - x)}{r_{4} - r_{3} + θ_{r} (r_{4} - x)}, & if x \in (r_{3}, \frac{r_{3} + r_{4}}{2}] \\ \frac{(- 1 + θ_{r}) x - θ_{r} r_{3} + r_{4}}{r_{4} - r_{3} + θ_{r} (x - r_{4})}, & if x \in (\frac{r_{3} + r_{4}}{2}, r_{4}] . \end{matrix}$

Now according to the definition of α-cut (Wu [10]), we have the following α-cuts of the reductions of $\tilde{ξ}$

Using the optimistic CV reduction method, $ξ_{1 L} (α) = {\begin{matrix} \frac{(1 + θ_{r}) r_{1} + (r_{2} - r_{1} - θ_{r} r_{1}) α}{(1 + θ_{r}) - θ_{r} α}, & if 0 \leq α \leq 0.5 \\ \frac{(r_{1} - θ_{r} r_{1}) + (r_{2} - r_{1} - θ_{r} r_{2}) α}{(1 - θ_{r}) + θ_{r} α}, & if 0.5 < α \leq 1 \end{matrix}$

$ξ_{1 R} (α) = {\begin{matrix} \frac{(r_{4} - θ_{r} r_{3}) - (r_{4} - r_{3} - θ_{r} r_{3}) α}{(1 - θ_{r}) - θ_{r} α}, & if 0.5 \leq α \leq 1 \\ \frac{(1 + θ_{r}) r_{4} - (r_{4} - r_{3} + θ_{r} r_{4}) α}{(1 + θ_{r}) - θ_{r} α}, & if 0 \leq α < 0.5 \end{matrix}$

Using the pessimistic CV reduction method,

$ξ_{2 L} (α) = {\begin{matrix} \frac{r_{1} + (r_{2} - r_{1} - θ_{r} r_{1}) α}{1 - θ_{l} α}, & if 0 \leq α \leq 0.5 \\ \frac{r_{1} + (r_{2} - r_{1} + θ_{r} r_{2}) α}{1 + θ_{l} α}, & if 0.5 < α \leq 1 \end{matrix}$

$ξ_{2 R} (α) = {\begin{matrix} \frac{r_{4} - (r_{4} - r_{3} - θ_{l} r_{2}) α}{1 + θ_{l} α}, & if 0.5 \leq α \leq 1 \\ \frac{r_{4} - (r_{4} - r_{3} + θ_{l} r_{4}) α}{1 - θ_{l} α}, & if 0 \leq α < 0.5 \end{matrix}$

Using the CV reduction method,

$ξ_{3 L} (α) = {\begin{matrix} \frac{(1 + θ_{r}) r_{1} + (r_{2} - r_{1} - 2 θ_{r} r_{1}) α}{(1 + θ_{r}) - 2 θ_{r} α}, & if 0 \leq α \leq 0.5 \\ \frac{(r_{1} - θ_{l} r_{2}) + (r_{2} - r_{1} - θ_{l} r_{2}) α}{(1 - θ_{l}) + 2 θ_{l} α}, & if 0.5 < α \leq 1 \end{matrix}$

$ξ_{3 R} (α) = {\begin{matrix} \frac{(r_{4} - θ_{l} r_{3}) + (r_{4} - r_{3} - 2 θ_{l} r_{3}) α}{(1 - θ_{l}) + 2 θ_{l} α}, & if 0.5 \leq α \leq 1 \\ \frac{(1 + θ_{r}) r_{4} - (r_{4} - r_{3} + 2 θ_{r} r_{4}) α}{(1 + θ_{r}) - 2 θ_{r} α}, & if 0.5 \leq α < 1 \end{matrix}$

According to Grzegorzewski [25] the nearest interval approximation of a fuzzy number

\tilde{A}

with distance metric d is given by

C_{d} (\tilde{A}) = [C_{L}, C_{R}]

, where

C_{L} = \int_{0}^{1} A_{L} (α) d α

and

C_{R} = \int_{0}^{1} A_{R} (α) d α

. Here the distance metric d measure the distance of fuzzy number

\tilde{A}

from

C_{d} (\tilde{A})

by,

d (\tilde{A}, C_{d} (\tilde{A}))

= \sqrt{\int_{0}^{1} {A_{L} (α) - C_{L}}^{2} d α + \int_{0}^{1} {A_{R} (α) - C_{R}}^{2} d α}

With this method for the α-cuts of optimistic CV, pessimistic CV or CV reduction of the trapezoidal type-2 fuzzy number $\tilde{ξ}$ we can find the nearest interval approximation of $\tilde{ξ}$ .

3.2.1 Nearest interval approximation of $\tilde{ξ}$ (using optimistic CV reduction ξ₁) based on α-cut

The nearest interval approximation of the trapezoidal type-2 fuzzy variable $\tilde{ξ}$ here obtained as a closed interval [C_L, C_R] where C_L and C_R are define as follows,

$\begin{matrix} C_{L} = \int_{0}^{1} ξ_{1 L} (α) d α \\ = \int_{0}^{0.5} \frac{(1 + θ_{r}) r_{1} + (r_{2} - r_{1} - θ_{r} r_{1}) α}{(1 + θ_{r}) - θ_{r} α} d α \\ + \int_{0.5}^{1} \frac{(r_{1} - θ_{r} r_{2}) + (r_{2} - r_{1} + θ_{r} r_{2}) α}{(1 - θ_{r}) + θ_{r} α} d α \\ = C_{L 1} + C_{L 2} \end{matrix}$

where

$\begin{matrix} C_{L 1} = \frac{(1 + θ_{r}) r_{1}}{θ_{r}} \ln (\frac{1 + θ_{r}}{1 + 0.5 θ_{r}}) \\ - \frac{r_{2} - r_{1} - θ_{r} r_{1}}{θ_{r}^{2}} [0.5 θ_{r} - (1 + θ_{r}) \ln (\frac{1 + θ_{r}}{1 + 0.5 θ_{r}})] \\ C_{L 2} = - \frac{r_{1} - θ_{r} r_{2}}{θ_{r}} \ln (1 - 0.5 θ_{r}) \\ + \frac{r_{2} - r_{1} + θ_{r} r_{2}}{θ_{r}^{2}} [0.5 θ_{r} + (1 - θ_{r}) \ln (1 - 0.5 θ_{r})] \end{matrix}$

$\begin{matrix} C_{R} = \int_{0}^{1} ξ_{1 R} (α) d α \\ = \int_{0}^{0.5} \frac{(1 + θ_{r}) r_{4} - (r_{4} - r_{3} + θ_{r} r_{4}) α}{(1 + θ_{r}) - θ_{r} α} d α \\ + \int_{0.5}^{1} \frac{(r_{1} - θ_{r} r_{2}) + (r_{2} - r_{1} + θ_{r} r_{2}) α}{(1 - θ_{r}) + θ_{r} α} d α \\ = C_{R 1} + C_{R 2} \end{matrix}$

where

$\begin{matrix} C_{R 1} = \frac{(1 + θ_{r}) r_{4}}{θ_{r}} \ln (\frac{1 + θ_{r}}{1 + 0.5 θ_{r}}) \\ + \frac{r_{4} - r_{3} - θ_{r} r_{4}}{θ_{r}^{2}} [0.5 θ_{r} - (1 + θ_{r}) \ln (\frac{1 + θ_{r}}{1 + 0.5 θ_{r}})] \\ C_{R 2} = - \frac{r_{4} - θ_{r} r_{3}}{θ_{r}} \ln (1 - 0.5 θ_{r}) \\ - \frac{r_{4} - r_{3} + θ_{r} r_{3}}{θ_{r}^{2}} [0.5 θ_{r} + (1 - θ_{r}) \ln (1 - 0.5 θ_{r})] \end{matrix}$

4 Statement of the problem and model formulation

In this paper a multi-objective solid transportation problem with type-2 fuzzy variable is considered, where the objectives are minimizing the cost and time of transportations. Let there are m (i = 1, 2, ⋯ , m) number of source point exist, from where the transportable goods to be deliver to the n (j = 1, 2, ⋯ , n) destinations using the l (k = 1, 2, ⋯ , l) available vehicles. Also let ${\tilde{c}}_{ijk}$ is the fuzzy unit transportation cost, where index i indicate the particular source, j indicate the particular demand point and k indicate the particular vehicles that used. Similarly let ${\tilde{t}}_{ijk}$ is the fuzzy expected unit transportation time. x_ijk is amount of transported goods from i-th source to j-th destination using k-th conveyance. So the multi-objective fuzzy solid transportation problem can be state as follows: $Min Z_{1} = \sum_{i = 1}^{m} \sum_{j = 1}^{n} \sum_{k = 1}^{l} {\tilde{c}}_{ijk} x_{ijk}$ (5) $Min Z_{2} = \sum_{i = 1}^{m} \sum_{j = 1}^{n} \sum_{k = 1}^{l} {\tilde{t}}_{ijk} y_{ijk}$ (6) $\begin{matrix} s . t . \\ \sum_{j = 1}^{n} \sum_{k = 1}^{l} x_{ijk} \leq {\tilde{a}}_{i}, i = 1, 2, \dots, m . \end{matrix}$ (7) $\sum_{i = 1}^{m} \sum_{k = 1}^{l} x_{ijk} \geq {\tilde{b}}_{j}, j = 1, 2, \dots, n .$ (8) $\sum_{i = 1}^{m} \sum_{j = 1}^{n} x_{ijk} \leq {\tilde{e}}_{k}, k = 1, 2, \dots, l .$ (9)

$x_{ijk} \geq 0, \forall i, j, k \forall i, j .$ (10)

Here (5) indicate the cost minimization objective and (6) is time minimizing objective function. Constraint (7) is the source constraint, which implies that total amount of transported goods $(\sum_{j = 1}^{n} \sum_{k = 1}^{l} x_{ijk}, i = 1, 2, \dots, m)$ is less or equal to the available source. Similarly (8) and (9) are the demand and conveyance constraint respectively. The last constraint (10) is the non negativity restriction for the transportation system. In this problem we consider some important assumption and these assumptions are as follows:

Assumption:

y_ijk is a binary operation define as

$y_{ijk} = {\begin{matrix} 1 & if x_{ijk} > 0 \\ 0 & otherwise \end{matrix} .$

The fuzzy membership functions for the system parameters are defined based on the previous statistical information and following the expert’s opinion about the parameters.

The problem is an unbalanced problem i.e. $\sum_{i = 1}^{m} {\tilde{a}}_{i} \neq \sum_{j = 1}^{n} {\tilde{b}}_{j} \neq \sum_{k = 1}^{l} {\tilde{e}}_{k}$ .

5 Solution and numerical experimentation

The proposed fuzzy problem has been solved by applying two different approach. One is based on chance-constrained programming technique and another approach is nearest interval approximation of fuzzy number.

5.1 Chance-constrained programming using generalized credibility

Chance-constrained programming with fuzzy number was developed by Liu et al. [4], Yang et al. [17], Kundu et al. [28, 29] using credibility measure. Since the parameters of the proposed problem are taken as trapezoidal type-2 fuzzy variable, so we defuzzify them using the proposed defuzzification methods (given in Section 3).

Let ${\tilde{c}}_{ijk}^{'}, {\tilde{t}}_{ijk}^{'}, {\tilde{a}}_{i}^{'}, {\tilde{b}}_{j}^{'}, {\tilde{e}}_{k}^{'}$ are type reduced form for the type-2 fuzzy variables ${\tilde{c}}_{ijk}, {\tilde{t}}_{ijk}, {\tilde{a}}_{i}, {\tilde{b}}_{j}, {\tilde{e}}_{k}$ respectively based on critical value reduction method [32]. With this reduced form, we formulate a chance constrained programming model to find solution of the proposed problem. Basically in chance constrained programming, the uncertain constraints are allowed to be violated such that constraints constraints must be satisfied at some chance or confidence level specified by the DM. Also we can not use the usual credibility measure to the type reduced variables as they may not be normalized and hence we use the generalized credibility. With this, we formulate the following chance constrained programming model for the proposed problem given by (5)–(10). $\begin{matrix} \underset{x}{\underset{︸}{Min}} \underset{\bar{f_{1}}}{\underset{︸}{Min}} \bar{f_{1}} \\ \underset{x}{\underset{︸}{Min}} \underset{\bar{f_{2}}}{\underset{︸}{Min}} \bar{f_{2}} \\ s . t . \end{matrix}$ (11) $\tilde{Cr} {\sum_{i = 1}^{m} \sum_{j = 1}^{n} \sum_{k = 1}^{l} {\tilde{c}}_{ijk}^{'} x_{ijk} \leq \bar{f_{1}}} \geq α^{c}$ $\begin{matrix} \tilde{Cr} {\sum_{i = 1}^{m} \sum_{j = 1}^{n} \sum_{k = 1}^{l} {\tilde{t}}_{ijk}^{'} y_{ijk} \leq \bar{f_{2}}} \geq α^{t} \\ \tilde{Cr} {\sum_{j = 1}^{n} \sum_{k = 1}^{l} x_{ijk} \leq {\tilde{a}}_{i}^{'}} \geq α_{i}, i = 1, 2, \dots, m \\ \tilde{Cr} {\sum_{i = 1}^{m} \sum_{k = 1}^{l} x_{ijk} \geq {\tilde{b}}_{j}^{'}} \geq α_{j}, j = 1, 2, \dots, n \\ \tilde{Cr} {\sum_{i = 1}^{m} \sum_{j = 1}^{n} x_{ijk} \leq e_{k}^{'}} \geq α_{k}, k = 1, 2, \dots, l \end{matrix}$ with non negativity restriction given by constraint (10). Here α^c, α^t, α_i, α_j, and α_k, ∀ i, j, k are the predetermined different generalized credibility levels for the chance constraints respectively.

5.1.1 Equivalent crisp form of chance constrained programming model (11)

Let all the parameters i.e. ${\tilde{c}}_{ijk}, {\tilde{t}}_{ijk}, {\tilde{a}}_{i}, {\tilde{b}}_{j}, {\tilde{e}}_{k}$ are mutually independent trapezoidal type-2 fuzzy variables and for all i, j, k represent as follows:

${\tilde{c}}_{ijk} = (c_{ijk}^{1}, c_{ijk}^{2}, c_{ijk}^{3}, c_{ijk}^{4}; θ_{l, ijk}^{c}, θ_{r, ijk}^{c}),$

${\tilde{t}}_{ijk} = (t_{ijk}^{1}, t_{ijk}^{2}, t_{ijk}^{3}, t_{ijk}^{4}; θ_{l, ijk}^{t}, θ_{r, ijk}^{t}),$

${\tilde{a}}_{i} = (a_{i}^{1}, a_{i}^{2}, a_{i}^{3}, a_{i}^{4}; θ_{l, i}^{a}, θ_{r, i}^{a}),$

${\tilde{b}}_{j} = (b_{j}^{1}, b_{j}^{2}, b_{j}^{3}, b_{j}^{4}; θ_{l, j}^{b}, θ_{r, j}^{b})$

and ${\tilde{e}}_{i} = (e_{k}^{1}, e_{k}^{2}, e_{k}^{3}, e_{k}^{4}; θ_{l, k}^{e}, θ_{r, k}^{e}) .$

Then according to the Theorem-4, the chance constrained programming model given by (11) convert the following equivalent crisp parametric programming problems. $\begin{matrix} Min Z_{1} = \sum_{i = 1}^{m} \sum_{j = 1}^{n} \sum_{k = 1}^{l} F_{ijk}^{c} x_{ijk} \\ Min Z_{2} = \sum_{i = 1}^{m} \sum_{j = 1}^{n} \sum_{k = 1}^{l} F_{ijk}^{t} y_{ijk} \\ s . t . \\ \sum_{j = 1}^{n} \sum_{k = 1}^{l} x_{ijk} \leq F_{i}^{a}, i = 1, 2, \dots, m \\ \sum_{i = 1}^{m} \sum_{k = 1}^{l} x_{ijk} \geq F_{j}^{b}, j = 1, 2, \dots, n \\ \sum_{i = 1}^{m} \sum_{j = 1}^{n} x_{ijk} \leq F_{k}^{e}, k = 1, 2, \dots, l \\ and the constraint (10) . \end{matrix}$ Where $F_{i}^{a}, F_{j}^{b}, F_{k}^{e}, F_{ijk}^{c}, F_{ijk}^{t}$ are equivalent crisp form of fuzzy parameters respectively and given by (5)–(10). $F_{i}^{a} = {\begin{matrix} \frac{(1 - 2 α_{i} + (1 - 4 α_{i}) θ_{r, i}^{a}) a_{i}^{1} + 2 α_{i} a_{i}^{2}}{1 + (1 - 4 α_{i}) θ_{r, i}^{a}} & if 0 < α_{i} \leq 0.25 \\ \frac{(1 - 2 α_{i}) a_{i}^{1} + (2 α_{i} + (4 α_{i} - 1) θ_{l, i}^{a}) a_{i}^{2}}{1 + (4 α_{i} - 1) θ_{l, i}^{a}} & if 0.25 < α_{i} \leq 0.5 \\ \frac{(2 α_{i} - 1) a_{i}^{4} + (2 (1 - α_{i}) + (3 - 4 α_{i}) θ_{l, i}^{a}) a_{i}^{3}}{1 + (3 - 4 α_{i}) θ_{l, i}^{a}} & if 0.5 < α_{i} \leq 0.75 \\ \frac{(2 α_{i} - 1 + (4 α_{i} - 3) θ_{r, i}^{a}) a_{i}^{4} + 2 (1 - α_{i}) a_{i}^{3}}{1 + (4 α_{i} - 3) θ_{r, i}^{a}} & if 0.75 < α_{i} \leq 1 \end{matrix}$ (12) $F_{j}^{b} = {\begin{matrix} \frac{(1 - 2 α_{j} + (1 - 4 α_{j}) θ_{r, j}^{b}) b_{j}^{1} + 2 α_{j} b_{j}^{2}}{1 + (1 - 4 α_{j}) θ_{r, j}^{a}} & if 0 < α_{j} \leq 0.25 \\ \frac{(1 - 2 α_{j}) b_{j}^{1} + (2 α_{j} + (4 α_{j} - 1) θ_{l, j}^{b}) b_{j}^{2}}{1 + (4 α_{j} - 1) θ_{l, j}^{b}} & if 0.25 < α_{j} \leq 0.5 \\ \frac{(2 α_{j} - 1) b_{j}^{4} + (2 (1 - α_{j}) + (3 - 4 α_{j}) θ_{l, j}^{b}) b_{j}^{3}}{1 + (3 - 4 α_{j}) θ_{l, j}^{b}} & if 0.5 < α_{j} \leq 0.75 \\ \frac{(2 α_{j} - 1 + (4 α_{j} - 3) θ_{r, j}^{b}) b_{j}^{4} + 2 (1 - α_{j}) b_{j}^{3}}{1 + (4 α_{j} - 3) θ_{r, j}^{b}} & if 0.75 < α_{j} \leq 1 \end{matrix}$ (13) $F_{k}^{e} = {\begin{matrix} \frac{(1 - 2 α_{k} + (1 - 4 α_{k}) θ_{r, k}^{e}) e_{k}^{1} + 2 α_{k} e_{k}^{2}}{1 + (1 - 4 α_{i} k) θ_{r, k}^{e}} & if 0 < α_{k} \leq 0.25 \\ \frac{(1 - 2 α_{k}) e_{k}^{1} + (2 α_{k} + (4 α_{k} - 1) θ_{l, k}^{e}) e_{k}^{2}}{1 + (4 α_{k} - 1) θ_{l, k}^{e}} & if 0.25 < α_{k} \leq 0.5 \\ \frac{(2 α_{k} - 1) e_{k}^{4} + (2 (1 - α_{k}) + (3 - 4 α_{k}) θ_{l, k}^{e}) e_{k}^{3}}{1 + (3 - 4 α_{k}) θ_{l, k}^{e}} & if 0.5 < α_{k} \leq 0.75 \\ \frac{(2 α_{k} - 1 + (4 α_{k} - 3) θ_{r, k}^{e}) e_{k}^{4} + 2 (1 - α_{k}) e_{k}^{3}}{1 + (4 α_{k} - 3) θ_{r, k}^{e}} & if 0.75 < α_{k} \leq 1 \end{matrix}$ (14) $\begin{matrix} F_{ijk}^{c} = {\begin{matrix} \frac{(1 - 2 α^{c} + (1 - 4 α^{c}) θ_{r, ijk}^{c}) c_{ijk}^{1} + 2 α^{c} c_{ijk}^{2}}{1 + (1 - 4 α^{c}) θ_{r, ijk}^{c}} & if 0 < α^{c} \leq 0.25 \\ \frac{(1 - 2 α^{c}) c_{ijk}^{1} + (2 α^{c} + (4 α^{c} - 1) θ_{l, ijk}^{c}) c_{ijk}^{2}}{1 + (4 α^{c} - 1) θ_{l, ijk}^{c}} & if 0.25 < α^{c} \leq 0.5 \\ \frac{(2 α^{c} - 1) c_{ijk}^{4} + (2 (1 - α^{c}) + (3 - 4 α^{c}) θ_{l, ijk}^{c}) c_{ijk}^{3}}{1 + (3 - 4 α^{c}) θ_{l, ijk}^{c}} & if 0.5 < α^{c} \leq 0.75 \\ \frac{(2 α^{c} - 1 + (4 α^{c} - 3) θ_{r, ijk}^{c}) c_{ijk}^{4} + 2 (1 - α^{c}) c_{ijk}^{3}}{1 + (4 α^{c} - 3) θ_{r, ijk}^{c}} & if 0.75 < α^{c} \leq 1 \end{matrix} \end{matrix}$ (15) $\begin{matrix} F_{ijk}^{t} = {\begin{matrix} \frac{(1 - 2 α^{t} + (1 - 4 α^{t}) θ_{r, ijk}^{t}) t_{ijk}^{1} + 2 α^{t} t_{ijk}^{2}}{1 + (1 - 4 α^{t}) θ_{r, ijk}^{t}} & if 0 < α^{t} \leq 0.25 \\ \frac{(1 - 2 α^{t}) t_{ijk}^{1} + (2 α^{t} + (4 α^{t} - 1) θ_{l, ijk}^{t}) t_{ijk}^{2}}{1 + (4 α^{t} - 1) θ_{l, ijk}^{t}} & if 0.25 < α^{t} \leq 0.5 \\ \frac{(2 α^{t} - 1) t_{ijk}^{4} + (2 (1 - α^{t}) + (3 - 4 α^{t}) θ_{l, ijk}^{t}) t_{ijk}^{3}}{1 + (3 - 4 α^{t}) θ_{l, ijk}^{t}} & if 0.5 < α^{t} \leq 0.75 \\ \frac{(2 α^{t} - 1 + (4 α^{t} - 3) θ_{r, ijk}^{t}) t_{ijk}^{4} + 2 (1 - α^{t}) t_{ijk}^{3}}{1 + (4 α^{t} - 3) θ_{r, ijk}^{t}} & if 0.75 < α^{t} \leq 1 \end{matrix} \end{matrix}$ (16) for all i, j, k.

5.2 Solution using nearest interval approximation

As given in Section 5.1.1, let ${\tilde{c}}_{ijk}, {\tilde{t}}_{ijk}, {\tilde{a}}_{i}, {\tilde{b}}_{j} and {\tilde{e}}_{k}$ are mutually independent trapezoidal type-2 fuzzy variables. Then using the nearest interval approximation method for trapezoidal type-2 fuzzy variables discussed in Section 3.2, we have the following deterministic approximated intervals for the parameters: ${\tilde{a}}_{i} \sim [a_{iL}, a_{iR}], {\tilde{b}}_{j} \sim [a_{jL}, c_{jR}], {\tilde{e}}_{k} \sim [e_{kL}, e_{kR}], {\tilde{c}}_{ijk} \sim [c_{ijkL}, c_{ijkR}], {\tilde{t}}_{ijk} \sim [t_{ijkL}, t_{ijkR}]$ respectively. Using these intervals in the model given by (5)–(10) we got the following problem. $\begin{matrix} Min Z_{1} = \sum_{i = 1}^{m} \sum_{j = 1}^{n} \sum_{k = 1}^{l} [c_{ijkL}, c_{ijkR}] x_{ijk} \\ Min Z_{2} = \sum_{i = 1}^{m} \sum_{j = 1}^{n} \sum_{k = 1}^{l} [t_{ijkL}, t_{ijkR}] y_{ijk} \\ s . t . \end{matrix}$ (17) $\begin{matrix} \sum_{j = 1}^{n} \sum_{k = 1}^{l} x_{ijk} \leq [a_{iL}, a_{iR}] i = 1, 2, \dots, m . \\ \sum_{i = 1}^{m} \sum_{k = 1}^{l} x_{ijk} \geq [b_{jL}, b_{jR}] j = 1, 2, \dots, n . \\ \sum_{i = 1}^{m} \sum_{j = 1}^{n} x_{ijk} \leq [e_{kL}, e_{kR}] k = 1, 2, \dots, l . \end{matrix}$ and the constraint (10).

It is clear that, the problem (17) is an interval valued multi-objective solid transportation problem. interval valued STP were solved by Das et al. [38], Kundu et al. [29], Safi et al. [21]. Recently Karmakaret al. [14] proposed an alternative optimization technique for interval-valued objective constrained optimization problems via multi-objective programming. Here we adopt the solution techniques from [14] to solve our problem given by (17).

So the objective functions of the problem (17) can be rewritten as follows: $\begin{array}{l} F_{M i n Z_{1}} = \sum_{i = 1}^{m} \sum_{j = 1}^{n} \sum_{k = 1}^{l} 〈c_{i j k}^{C}, c_{i j k}^{W}〉 x_{i j k} = \sum_{i = 1}^{m} \sum_{j = 1}^{n} \sum_{k = 1}^{l} 〈F_{i j k}^{C}, F_{i j k}^{W}〉 \\ E_{M i n Z_{2}} = \sum_{i = 1}^{m} \sum_{j = 1}^{n} \sum_{k = 1}^{l} 〈t_{i j k}^{C}, t_{i j k}^{W}〉 y_{i j k} = \sum_{i = 1}^{m} \sum_{j = 1}^{n} \sum_{k = 1}^{l} 〈E_{i j k}^{C}, E_{i j k}^{W}〉 \end{array}$ where $F_{ijk}^{C}, F_{ijk}^{W}, E_{ijk}^{C}$ and $E_{ijk}^{W}$ are given by $\begin{matrix} \sum_{i = 1}^{m} \sum_{j = 1}^{n} \sum_{k = 1}^{l} F_{ijk}^{C} = \sum_{i = 1}^{m} \sum_{j = 1}^{n} \sum_{k = 1}^{l} c_{ijk}^{C} x_{ijk} and \\ \sum_{i = 1}^{m} \sum_{j = 1}^{n} \sum_{k = 1}^{l} F_{ijk}^{W} = \sum_{i = 1}^{m} \sum_{j = 1}^{n} \sum_{k = 1}^{l} c_{ijk}^{W} x_{ijk} \end{matrix}$ (18) $\begin{matrix} \sum_{i = 1}^{m} \sum_{j = 1}^{n} \sum_{k = 1}^{l} E_{ijk}^{C} = \sum_{i = 1}^{m} \sum_{j = 1}^{n} \sum_{k = 1}^{l} t_{ijk}^{C} y_{ijk} and \\ \sum_{i = 1}^{m} \sum_{j = 1}^{n} \sum_{k = 1}^{l} E_{ijk}^{W} = \sum_{i = 1}^{m} \sum_{j = 1}^{n} \sum_{k = 1}^{l} t_{ijk}^{W} y_{ijk} \end{matrix}$ (19) With these, the interval valued multi-objective solid transportation problem (17) reduce into the corresponding non-interval deterministic multi-objective solid transportation problem as follows: $\begin{matrix} Min (\sum_{i = 1}^{m} \sum_{j = 1}^{n} \sum_{k = 1}^{l} F_{ijk}^{C}) = \sum_{i = 1}^{m} \sum_{j = 1}^{n} \sum_{k = 1}^{l} c_{ijk}^{C} x_{ijk} \\ Min (\sum_{i = 1}^{m} \sum_{j = 1}^{n} \sum_{k = 1}^{l} F_{ijk}^{W}) = \sum_{i = 1}^{m} \sum_{j = 1}^{n} \sum_{k = 1}^{l} c_{ijk}^{W} x_{ijk} \\ Min (\sum_{i = 1}^{m} \sum_{j = 1}^{n} \sum_{k = 1}^{l} E_{ijk}^{C}) = \sum_{i = 1}^{m} \sum_{j = 1}^{n} \sum_{k = 1}^{l} t_{ijk}^{C} y_{ijk} \\ Min (\sum_{i = 1}^{m} \sum_{j = 1}^{n} \sum_{k = 1}^{l} E_{ijk}^{W}) = \sum_{i = 1}^{m} \sum_{j = 1}^{n} \sum_{k = 1}^{l} t_{ijk}^{W} y_{ijk} \end{matrix}$ subject to, $\begin{matrix} \sum_{j = 1}^{n} \sum_{k = 1}^{l} x_{ijk} \leq a_{iR}, \sum_{j = 1}^{n} \sum_{k = 1}^{l} x_{ijk} \geq a_{iL} \\ i = 1, 2, \dots, m . \\ \sum_{i = 1}^{m} \sum_{k = 1}^{l} x_{ijk} \leq b_{jR}, \sum_{i = 1}^{m} \sum_{k = 1}^{l} x_{ijk} \geq b_{jL} \\ j = 1, 2, \dots, n . \\ \sum_{i = 1}^{m} \sum_{j = 1}^{n} x_{ijk} \leq e_{kR}, \sum_{i = 1}^{m} \sum_{j = 1}^{n} x_{ijk} \geq e_{kL} \\ k = 1, 2, \dots, l . \\ and the constraint (10) . \end{matrix}$

Now the problem become a multi-objective STP in deterministic form. We solve this MOSTP using the intuitionistic fuzzy programming technique [34].

5.2.1 Intuitionistic fuzzy programming technique for multi-objective solid transportation problem

The intuitionistic fuzzy programming technique is basically based on defining two different membership function, one is yes membership (μ_{f
_q} : q ∈ N, indicatethenumberofobjectivefunction) and another is non membership function (ν_{f
_q}) for each objective functions.

An exponential membership function is defined as $μ_{f_{q}} = {\begin{matrix} 1, & if f_{q} \leq L_{q} \\ \frac{e^{- w (\frac{f_{q} - L_{q}}{U_{q} - L_{q}}) - e^{- w}}}{1 - e^{- w}}, & if L_{q} \leq f_{q} \leq U_{k} \\ 0, & if f_{q} \geq U_{k} \end{matrix}$ (20) A quadratic non-membership function is defined as $ν_{f_{q}} = {\begin{matrix} 1, & if f_{q} \leq L_{q} \\ {(\frac{f_{q} - L_{q}}{U_{q} - L_{q}})}^{2}, & if L_{q} \leq f_{q} \leq U_{k} \\ 1, & if f_{q} \geq U_{k} \end{matrix}$ (21)

Here L_q and U_q are the lower bounds (i.e. lower optimal values) and upper bounds (i.e. upper optimal values) of the objective functions obtained by solving the multi-objective optimization problem as a single optimization problem.

Defining these two membership function given by (20) and (21) for each objective function following Chakraborty et al. [34] the optimization model converts to the following crisp model. $\begin{matrix} Max (α - β) \\ s . t . \end{matrix}$ (22) $\begin{matrix} α \leq \frac{e^{- w (\frac{f_{q} - L_{q}}{U_{q} - L_{q}}) - e^{- w}}}{1 - e^{- w}} \\ β \geq {(\frac{f_{q} - L_{q}}{U_{q} - L_{q}})}^{2} \\ α \geq β \\ α + β \leq 1 and α, β \geq 0 . \end{matrix}$ Now according to the importance of the objective function the DM solve the following problem to get the optimal objective value of the corresponding objective function. $\begin{matrix} Min / Max f_{imp} \\ s . t . \\ f_{imp} \leq L_{imp} - \frac{U_{imp} - L_{imp}}{w} \\ [\ln {α^{*} (1 - e^{- w})} + e^{- w}] \\ f_{imp} \geq L_{imp} + \sqrt{β^{*}} U_{imp} - L_{imp} \\ α \geq β \\ α + β \leq 1 and α, β \geq 0 . \end{matrix}$

Here α^* and β^* are optimal solution for the system given by (22).

Next with optimum decision variables the pareto-optimal solution test is performed according to Sakawa et al. [20] to define the best optimal solution.

5.3 Numerical experiment

For numerical experiment we consider two source, two demand and two conveyance and the corresponding data are given by the Table 1 represent the cost parameters, the type-2 fuzzy resources, demands and conveyance capacities are given by Table 2.

5.3.1 Solution obtained by chance-constrained programming

With these numerical data we first formulate chance constrained programming model following (11). Also we consider the predetermined generalized credibility levels for this chance constrained programming problem as α^c = α^t = α_i = α_j = α_k = 0.95 for i = j = k = 1, 2. Hence the equivalent crisp form for this problem can be represent as follows:

$\begin{matrix} Min \sum_{i = 1}^{2} \sum_{j = 1}^{2} \sum_{k = 1}^{2} [\frac{((0.9 + 0.8 θ_{r, ijk}^{c}) c_{ijk}^{4} + 0.1 c_{ijk}^{3}) x_{ijk}}{1 + 0.8 θ_{r, ijk}^{c}}] \\ Min \sum_{i = 1}^{2} \sum_{j = 1}^{2} \sum_{k = 1}^{2} [\frac{((0.9 + 0.8 θ_{r, ijk}^{t}) t_{ijk}^{4} + 0.1 t_{ijk}^{3}) y_{ijk}}{1 + 0.8 θ_{r, ijk}^{t}}] \\ s . t . \\ \sum_{j = 1}^{2} \sum_{k = 1}^{2} x_{ijk} \leq F_{i}^{a}, i = 1, 2 . \\ \sum_{i = 1}^{2} \sum_{k = 1}^{2} x_{ijk} \geq F_{j}^{b}, j = 1, 2 . \\ \sum_{i = 1}^{2} \sum_{j = 1}^{2} x_{ijk} \leq F_{k}^{e}, k = 1, 2 . \end{matrix}$ and the constraint (10).

Here $F_{i}^{a}, F_{j}^{b} and F_{k}^{e}$ are calculated using (12)–(14) as follows:

$F_{1}^{a} = 26.82558, F_{2}^{a} = 22.76744, F_{1}^{b} = 17.93902,$ $F_{2}^{b} = 16.93243, F_{1}^{e} = 19.93590, F_{2}^{e} = 18.93243 .$

Next using the LINGO 13.0 solver we solve the problem and the corresponding solutions are presented in Table 3.

5.3.2 Solution obtained by nearest interval approximation method

The nearest interval approximations (credibilistic) of the given trapezoidal type-2 fuzzy parameters are calculated and then the interval based multi-objective solid transportation problem so obtained solved using intuitionistic fuzzy optimization technique. The results are presented in the Table 3.

5.4 Experimental evaluation with different size numerical examples

This section provide some experiments to the proposed model with the help of numerical data given in Tables 2 and 4 respectively, to justify the proposed method of defuzzification for a trapezoidal type-2 fuzzy variables. Here we have choose some multi-objective solid transportation problems of different size with trapezoidal type-2 fuzzy parameters and solve them using the proposed method. The optimal results are given in Table 4 result.

5.5 The Karnik-Mendel (KM) [23] and Nie-Tan (NT) algorithms [19] for type reduction

The Karnik-Mendel (KM) algorithm [23] is a well-known type reduction method designed for dealing with interval type-2 fuzzy sets (IT2FS). This method initiates with a simple average value in order to find the initial switch point between the upper and the lower membership values. Then, two different algorithms are proposed for computing the left and right endpoints. The first step of both algorithms is the same, e.g., finding an initial solution through a simple averaging. Then, through an iterative procedure, endpoints of the resulting type-1 intervals are obtained. The KM algorithms are known to converge monotonically and superexponentially fast, however, several iterations are needed before convergence. The results of the KM type-reducer are obtained by first calculating the generalized centroid [c_l, c_r] of an IT2FS and then taking the average i.e. the centroid $x_{c} = \frac{c_{l} + c_{r}}{2}$ , where [c_l, c_r] are the lower and upper bound of the type-reduced set obtained using the KMalgorithm.

The Nie-Tan (NT) algorithm [19] has been developed using the vertical-slice representation. The main idea behind the NT algorithm is to reduce the order of an IT2FS by using vertical slices. As for an IT2FSs, each vertical slice is an embedded type-1 fuzzy set that can be easily type-reduced.

5.5.1 Comparative study with IT2FS representation of the proposed model

We know that any number can be represent as an interval number by introducing the lower and upper limit of the interval as same. Using this property of number we have defuzzified the inputs of our proposed model using the KM algorithm and also by the NT algorithm and then solve using LINGO 13.0 software to make a comparison between our proposed defuzzify technique and the two well known techniques KM and NT algorithms. The results of this experiment are given in Table 6.

Observing the optimal results from the Table 6, we conclude that our proposed method of defuzzification gives the best optimal solution compare to the KM and NT algorithm for this problem. Here it should be mentioned that the generalized credibility level (α) is 0.45. It is also true that for α = 0.95 the optimal cost and time obtained using the proposed method are greater than the cost and time obtained using NT and KM algorithms respectively. So here our conclusion is that for α ∈ [0, 0.5] our method gives the best solution and in other range it gives a compromise solution. From this one can draw a important managerial decision as follows.

From this analyses the decision maker (DM) have the opportunity to choose the generalized credibility level according to the situation of the objective function.

For the problem with minimization of objective, considering α ∈ [0, 0.5] will be profitable and for the maximization problem α ∈ (0.5, 1.0] gives the more profit.

5.6 Discussion and comparison between proposed methods

We solve here a multi-objective solid transportation problem with all of it’s parameters are as trapezoidal type-2 fuzzy variables using two different method. Observing the optimal solution from the Table 3, here it should be mentioned that although we solve the problem with same inputs but the format of results are different. So the comparison between this two optimal result are not likely. The reason for obtained different solutions is that the defuzzified values of the type-2 fuzzy parameters are different. One comes in an interval form and another in a single number.

Also the result shows that, when we use the interval approximation method for trapezoidal type-2 fuzzy variables, we find then the minimum cost and time for this particular minimization problem comparing with the use of generalized credibility method. However it can be noticed that the cost and time (i.e. 793.5189 and 276.6469) so obtained using generalized credibility method does belong to the interval [131.1083,643.8172] and [16.85627,78.00338], which are the solutions obtained using nearest interval approximation method. So from that, the decision maker can choose his/her best fit solution as well as the solution techniques.

5.7 Some advantages of the proposed method over existing methods

In this subsection we list out some advantages of the proposed defuzzification method by comparing with other existing methods.

In this paper, the proposed defuzzification method provide crisp output in two different form (one is in number and another in interval form) for a general type-2 (GT2FS) fuzzy set, this is one advantage of the proposed method.

Most of the defuzzification methods like KM algorithm, NT method, Collapsing method, EKM etc. were developed for IT2FS. A few number of defuzzification methods for GT2FS were exist in literature. So in this context our proposed methods have a advantage that it used a GT2FS.

Defuzzification of a GT2FS by centroid type reduction method most of the time fail to give the exact centroid or the exact crisp value, but our approach always gives the exact crisp value. Because in centroid type reduction method the primary domain become change due to the discretization of the continuous domain into finite number of points, whereas the proposed method i.e. CV based reduction method does not effect the domain of the original set.

Type reduction process through an iterative process (e.g. KM algorithm) are almost time consuming and most of the time it depend on soft computing based programming. But our proposed method is free from iteration process and hence it take less time for computation. Also in our proposed method without any soft computation one can convert a type-2 fuzzy number into a crisp value.

6 Conclusion and future scope

This investigation present the defuzzification process of type-2 trapezoidal fuzzy variables based on CV based reduction method and nearest interval approximation. Also a multi-objective minimizing the cost ans time of transportation are solved in this paper where the the parameters are taken as trapezoidal type-2 fuzzy variable. Intutionistic fuzzy optimization technique has been applied to solve the multi-objective problem. This optimization technique allows one to define a degree of rejection which may not simply the degree of acceptance. The major of contribution of this research in this filed can be summarized as follows:

Introduce some theoretical aspects of trapezoidal type-2 fuzzy sets.

Introduce the concept of defuzzification methods of trapezoidal type-2 fuzzy variables based on generalized credibility and nearest interval approximation method.

The concept intutionistic fuzzy optimization technique are applied to the multi-objective solid transportation problem.

As a future work the proposed defuzzification process can be use to solve the different optimization problems, DEA models involving trapezoidal type-2 fuzzy variable. Another thing is that the defuzzification process for trapezoidal type-2 fuzzy variable can be derive based on the mean reduction method, centroid method etc. All this work work are consider as future scope of this study.

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Defuzzification of trapezoidal type-2 fuzzy variables and its application to solid transportation problem

Abstract

Keywords

1 Introduction

2 Some fundamental concepts

2.1 Type-2 fuzzy set (T2 FS)

2.2 Regular fuzzy variable (RFV) and their Critical values (CVs)

3 Proposed reduction method

3.1 Defuzzification using CV

3.2.1 Nearest interval approximation of ξ ˜ (using optimistic CV reduction ξ1) based on α-cut

4 Statement of the problem and model formulation

5.1 Chance-constrained programming using generalized credibility

5.3.1 Solution obtained by chance-constrained programming

5.3.2 Solution obtained by nearest interval approximation method

5.4 Experimental evaluation with different size numerical examples

5.5 The Karnik-Mendel (KM) [23] and Nie-Tan (NT) algorithms [19] for type reduction

5.5.1 Comparative study with IT2FS representation of the proposed model

5.6 Discussion and comparison between proposed methods

5.7 Some advantages of the proposed method over existing methods

6 Conclusion and future scope

References

3.2.1 Nearest interval approximation of $\tilde{ξ}$ (using optimistic CV reduction ξ₁) based on α-cut