Type-2 Fuzzy Stochastic Transportation Problem with Gamma Distribution

Babita Chaini; Narmada Ranarahu

doi:10.5391/IJFIS.2023.23.1.91

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International Journal of Fuzzy Logic and Intelligent Systems 2023; 23(1): 91-105

Published online March 25, 2023

https://doi.org/10.5391/IJFIS.2023.23.1.91

Type-2 Fuzzy Stochastic Transportation Problem with Gamma Distribution

Babita Chaini and Narmada Ranarahu

Department of Mathematics, SOA deemed to be University, Bhubaneswar, India

Correspondence to :
Babita Chaini (babitachaini@gmail.com)

Received: December 6, 2021; Revised: January 9, 2023; Accepted: March 6, 2023

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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Abstract
1. Introduction
2. Basic Preliminary
3. Mathematical Models
4. Transformation Techniques
5. Case Study
6. Result and Discussion
7. Conclusion
Conflicts of Interest
Tables
References
Biographies

The transportation problem in real-life is an uncertain problem. Particularly when goods are transported from the source to destinations with the best transportation setup that satisfies the decision maker’s preferences by taking into account the competing objectives/criteria such as maintaining exact relationships between a few linear parameters, such as actual transportation fee/total transportation cost, delivery fee/desired path, total return/total investment, etc. Due to the uncertainty of nature, such a relationship is not deterministic. In this stochastic transportation problem supplies are considered as fuzzy random variables, which follow fuzzy gamma distribution, with shape parameter α and scale parameter β. Here β is a perfectly normal interval type-2 fuzzy random variable. This paper proposes a solution methodology for solving the fuzzy stochastic transportation problem, where fuzziness and randomness occur under one roof. Therefore, we converted it to an equivalent deterministic mathematical programming problem by applying the following two steps. In the first step of the solution procedure, fuzziness is removed by using alpha-cut technique to obtain stochastic transportation problem. In the second step, the stochastic transportation problem is converted to an equivalent crisp transportation problem using the chance constrained technique. This mathematical model is solved by existing methodology or software. In order to illustrate the methodology a case study is provided.

Keywords: Stochastic programming, Fuzzy stochastic transportation problem, Fuzzy gamma random variables, Optimization techniques, Type 2 fuzzy set

1. Introduction

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Abstract
1. Introduction
2. Basic Preliminary
3. Mathematical Models
4. Transformation Techniques
5. Case Study
6. Result and Discussion
7. Conclusion
Conflicts of Interest
Tables
References
Biographies

Transportation problem is a special type of linear programming problem (LPP). The objective of this transportation problem is to minimize the cost of distributing a product from a number of sources to a number of destination. In many real-life problem we used transportation problem such as production, investment, scheduling, deciding plant location and inventory control, etc. In real-life we face many diverse situations due to uncertainty in judgement, lack of evidence, etc. Fuzzy transportation problem is more appropriate to model and solve the real-world problems. Being a particular type of LPP, fuzzy linear programming approach can be used in solving fuzzy transportation problem.

Type-2 fuzzy sets (T2FS) are the extension of type-1 fuzzy sets (T1FS), which can convey more uncertainty information in solving decision-making problems. Interval type-2 fuzzy numbers are a special kind of type-2 fuzzy numbers. These numbers can be described by triangular, trapezoidal, etc. There are different type of membership function in type-2 fuzzy such as trapizoidal membership function, triangular membership function, Gaussian membership function, power membership function, etc. The popular fuzzy numbers, i.e., used in type-2 fuzzy are triangular fuzzy number, trapizoidal fuzzy number, Gaussian fuzzy number, etc, and this fuzzy numbers are all particular type of LR flat fuzzy numbers.

In the real-world decision-making circumstances, we regularly need to settle on a choice under questionable information or data. In many concrete situations, it is difficult to present the mathematical models using random parameters. So, true issues have been demonstrated by thinking about probabilistic vulnerability parameters. Stochastic programming (SP) is a standout amongst the most critical methodologies to handle the uncertainty. SP problem is one of the mathematical programming problems that involves randomness. At the point when uncertainty occurs on the market demands for a commodity, the issue of booking shipments from supply points to demand points is called a stochastic transportation problem.

In general, the coefficient of the stochastic transportation problem (STP) are described by unverifiable parameters, for example, random, fuzzy, and multi-choice parameters. Different researchers have been considering different indicators containing random variables, and fuzzy variables such as normal, log-normal, exponential, Cauchy, Weibull, and others for the source and destination parameters of STP in the SP model under type-1 fuzzy environment. In our work, attention has been given to solving the fuzzy STP (FSTP) under type-2 fuzzy environment, where the supplies are fuzzy random variables that follows the fuzzy gamma distribution, in which the scale parameter β is perfectly normal interval type-2 fuzzy number.

The concept of fuzzy random variable, complementary relationship between randomness and fuzziness, fuzzy number, fuzzy probability are first introduced by Zadeh [1]. Chance-constrained mathematical programs aim at finding optimal solutions to problems where the probability of an undesirable outcome is limited by a given threshold which is developed by Charnes et al. [2]. The concept of fuzzy set theory was first introduced by Zadeh [3]. Solution methodology of different types of multi-objective fuzzy probabilistic programming problems have developed by several researchers [4–9]. After this motivation, Mendel et al. [10] discussed how type-2 fuzzy is different from type-1 fuzzy. They developed the unnecessary to take the route from general T2FS to interval type-2 fuzzy set (IT2FS). The authors of [11] discussed perfectly normal interval type-2 trapezoidal fuzzy numbers with their left, right-hand spreads and their core. Ma et al. [12] developed a chance-constraint programming model under a fuzzy environment, where waste generation amount are supposed to be type-2 fuzzy variable and treated capacities of facilities are assumed to be type-1 fuzzy variables. Agrawal and Ganesh [13] developed developed a method to solve fuzzy fractional transportation problem in which the parameters of the transportation problem, supply, and demand, are stochastic in nature and considered as a fuzzy random variable that follows the exponential distribution with fuzzy mean and fuzzy variance. Suo et al. [14] developed a type-2 fuzzy chance-constraint programming method for supporting energy systems planning under uncertainty. The results are helpful for managers to adjust the city’s current energy structure. Verdegay [15] developed the application of Zimmermann’s fuzzy programming approach and presented the additive fuzy programming model. Kundu et al. [16] considered two fixed charge transportation problems with type-2 fuzzy variables. The first one with transportation and fixed costs as type-2 fuzzy variables and the second one with transportation costs, fixed costs, supplies and demands all as type-2 fuzzy variables. Par and Kar [17] proposed a hybridized forecasting method on weight adjustment of neural networks with back-propagation learning using general T2FSs. Kundu et al. [18] developed a method to solve linear programming network problems with constraints using interval type-2 fuzzy variables. Das et al. [19] formulated a model as profit maximization problems in stochastic and fuzzy-stochastic environments, where the idle time of the machine leads to an additional cost for the loss of man-hours. They have taken the time between successive breakdowns of the machines is random and the maintenance time is also considered as random. Kundu et al. [20] introduced nearest interval approximation method of continuous type-2 fuzzy variable so that decision-making problem with type-2 fuzzy parameters can be solved easily. Dan et al. [21] investigated the notion of a minimum spanning tree (MST) of an undirected type-2 fuzzy weighted connected graph (UT2FWCG), where the edge weights are represented as discrete type-2 fuzzy variables.

Also, from the literature survey, we two studies [5,13]. In [5], the authors considered multi-objective STP, where parameters are independent normally distributed fuzzy random variables. Agrawal and Ganesh [13] developed a method to solve the fuzzy fractional transportation problem in which the parameters of the transportation problem, supply, and demand, are stochastic in nature and considered as a fuzzy random variable that follows the exponential distribution with fuzzy mean and fuzzy variance. In this paper, a novel strategy is developed for STP involving fuzzy gamma distribution. The main difference between this paper and the above two papers are follows. Firstly, as we know T2FSs have more degrees of freedom to describe uncertainty and are capable of handling inexact data in a logically correct manner. Many researchers have used interval type-2 fuzzy logic systems (T2FLSs) in the practical field even though generalized T2FLSs are computationally demanding. In comparison to the generalized T2FLS, computations in the IT2FS are more manageable. The main difference between the generalized and interval type-2 fuzzy membership functions is that the interval type-2 membership function’s secondary membership value is, in general, equal to 1. Here this paper is solved under interval type-2 fuzzy environment not type-1. Secondly, only the supply parameter is fuzzy random variables and the scale parameter β˜˜ is perfectly normal interval type-2 fuzzy number.

In this paper, we have considered a fuzzy STP with fuzzy gamma distribution involving interval type-2 fuzzy parameters. The proposed mathematical model for fuzzy STP cannot be solved directly by mathematical approaches. so here we proposed a solution procedure for solving the model by transforming it to a crisp programming problem. Here, defuzzification and de-randomization have been perform in two steps.

(i) The first step, fuzziness is removed by using alpha-cut technique to obtain STP.
(ii) The second step, randomness is removed by using the chance constrained technique, to obtain crisp transportation problem.

Finally, the solution procedure for solving the model is illustrated with the help of a case study.

The rest of the paper is structured as follows. In Section 2, we recall the basic concept and discuss the interval type-2 trapezoidal fuzzy number. In Section 3, we discuss the mathematical model of STP and its deterministic equivalent form is derived in Section 4. Section 5 illustrates a case study. The results of the example are given in Section 6, and Section 7 provides the concluding remarks.

2. Basic Preliminary

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Abstract
1. Introduction
2. Basic Preliminary
3. Mathematical Models
4. Transformation Techniques
5. Case Study
6. Result and Discussion
7. Conclusion
Conflicts of Interest
Tables
References
Biographies

Definition 2.1 (Crisp set, [22])

A set is a well-defined collection of distinct elements or objects. Let X be the universal set and A is a subset of X. Then characteristic function (membership function or discrimination function) of the set is denoted by X_A and is defined by the mapping X_A : X → {0, 1}, such that for every element x ∈ X

(1)XA(x)={1iff x∈X,0iff x∉X,

where 1 indicates membership and 0 indicates non-membership.

Definition 2.2 (Fuzzy set, [23])

A fuzzy set is a class of objects in which there is no sharp boundary between those objects that belong to the class and those do not. Fuzzy sets generalize the idea of crisp set by extending the range of characteristic function from the bounded pair (0, 1) to the unit interval [0, 1].

Let X be a collection of distinct objects and x be an element of X. Then a fuzzy set Ã in X is a set of order pairs

(2)A˜={(x,μA˜(x))∣x∈X},

where μ_Ã(x) is called the membership function or membership value or grade function (generalized characteristic function) or degree of belongingness of x ∈ X in the fuzzy set Ã : X → M. M is known as membership space, which is considered as the closed interval [0, 1]. If supremum of μ_Ã(x) = 1 then fuzzy set Ã is known as normalized fuzzy set. Support of Ã = {(x ∈ X|μ_Ã(x)) > 0}. The maximum value of the membership is called height.

Definition 2.3 (Fuzzy number, [24])

A fuzzy number Ã is a convex normalized fuzzy set Ã of the real line R, with membership function

μA˜:R→[0;1],

satisfying the following conditions.

There exists exactly one interval I ∈ R such that μ_Ã(x) = 1; x ∈ I.
The membership function μ_Ã is piecewise continuous.

Definition 2.4 (α-cut set, [25])

α-cut of the fuzzy number Ã is the set {x|μ_A(x) ≥ α} for 0 < α < 1 and denoted Ã[α].

Definition 2.5 (Convex fuzzy set, [26])

If all the α-cut sets are convex, then the fuzzy set with these α-cut sets is convex. Mathematically, if a relation μ_Ã(t) ≥ min{μ_Ã(r), μ_Ã(s)} holds, the fuzzy set Ã is convex. where t = λr + (1 − λ)s, λ ∈ [0, 1]

Definition 2.6 (Type-2 fuzzy set, [10])

A T2FS denoted as A˜˜ is a fuzzy set with membership function whose membership value is itself a fuzzy set. It is defined as

A˜˜={((x,u) μA˜˜(x,u))∣x∈X,∀u∈Jx⊆[0,1]}

in which 0≤μA˜˜(x,u)≤1.A˜˜ can also be expressed as

A˜˜=∫x∈X∫u∈JxμA˜˜ (x,u)/(x,u), Jx⊆[0,1],

where ∫ ∫ denotes union over all admissible x and u. For discrete universe of discourse, ∫ s replaced by ∑.

Definition 2.7 (Interval type-2 fuzzy set, [27])

An IT2FS is fully characterized by primary membership function and lower membership function i.e., when all μA˜˜(x,u)=1. And it is defined as

A˜˜={((x,u),1)∣x∈X,∀u∈Jx⊆[0,1]}.

Definition 2.8 (Normal IT2FS and perfectly normal IT2FS, [27])

An IT2FS A˜˜ is said to be normal if its upper membership function is normal. It is said to be perfectly normal if both of its upper and lower membership functions are normal.

Definition 2.9 (Type-2 fuzzy number, [27])

A T2FS denoted as A˜˜ in the universe R is said to be Type-2 fuzzy number if the following conditions are satisfied:

A˜˜ is normal.
A˜˜ is convex.
The support of A˜˜ is closed and bounded.

Definition 2.10 (Primary α-cut of perfectly normal IT2FS, [11])

The primary α-cut of PnIT2FS is A˜˜α={(x,u)∣Jx≥α,u∈[0,1]} which is bounded by two regions

μ_A˜˜α(x)={(x,μ_A˜˜(x))∣μ_A˜˜(x)≥α,∀α∈[0,1]},μ¯A˜˜α(x)={(x,μ¯A˜˜(x))∣μ¯A˜˜(x)≥α,∀α∈[0,1]}.

Definition 2.11 (Totally symmetric type-2 trapezoidal fuzzy number, [27])

A type-2 fuzzy number represented as a triplet of type-1 fuzzy numbers is called TST2TFN if it satisfies the following condition:

Upper membership function and lower membership functions are symmetric trapezoidal fuzzy number and generalized symmetric trapezoidal fuzzy number, respectively.
For each reference point in the primary universe of discourse, secondary membership functions are symmetric trapezoidal fuzzy numbers.

Definition 2.12 (Footprint of uncertainty, [11])

Let A˜˜ be IT2FS; uncertainty in the primary membership of T2FS consists of a bounded region called the footprint of uncertainty, which is the union of all primary membership. Footprint of uncertainty is characterized by upper membership function and lower membership function. Both of the membership functions are T1FSs. Upper membership function is denoted by μ¯A˜˜ and lower membership function is denoted by μ_A˜˜, respectively.

Definition 2.13 (Perfectly normal interval type 2 trapezoidal fuzzy number, [11])

An interval type 2 fuzzy number A˜˜ is called perfectly normal interval type 2 trapezoidal fuzzy number (IT2TFN) when the upper membership function μ_A˜˜(x) and lower membership function μ¯A˜˜(x) are both trapezoidal fuzzy numbers; that is, A˜˜=[A_,A¯]=[<a_,b_,c_,d_>,<a¯,b¯,c¯,d¯>]. Where A and Ā are known as lower and upper interval valued bounds of A˜˜. Ã=<a, b, c, d> is the lower trapezoidal fuzzy number. A¯˜=<a¯,b¯,c¯,d¯> is the upper trapezoidal fuzzy number. μA˜˜(b_)=μA˜˜(b¯) is the core and μA˜˜(b_)=μA˜˜(b¯)=1.

b–a is called lower spread of Ã and b̄ – ā is called lower spread of A¯˜. c–b is called upper spread of Ã and c̄–b̄ is called upper spread of A¯˜.

The membership functions of x in Ã and A¯˜ are expressed as follows:

μ_A˜˜(x)={0,if x≤a_,x-a_b_-a_,if a_≤x≤b_,1,if b_≤x≤c_,d_-xd_-c_,if c_≤x≤d_,0,if x≥d_,μ¯A˜˜(x)={0,if x≤a¯,x-a¯b¯-a¯,if a¯≤x≤b¯,1,if b_≤x≤c_,d¯-xd¯-c¯,if c¯≤x≤d¯,0,if x≥d¯.

Definition 2.14 (α-cut of perfectly normal,[11])

α-cut of perfectly normal IT2TFN A˜˜=[A_,A¯]=[<a_,b_,c_,d_>,<a¯,b¯,c¯,d¯>] is A˜˜[α]=[A_[α], Ā[α]] = [[a + (b–a)[α], d–(d–c)[α]], [ā+(b̄–ā)[α], d̄–(d̄–c̄)[α]]][[a+(b–a)[α], ā + (b̄ – ā)[α]][d– (d –c)[α], d̄ – (d̄ – c̄)[α]]].

Definition 2.15 (Stochastic transportation problem, [28])

A stochastic problem is an optimization problem in which some or all problem parameters are uncertain or imprecise in stochastic sense and described by random variables with known probability distribution. When the market demands for a commodity are not known with certainty, the problem of scheduling shipments to a number of demand points from several supply points is a STP.

Definition 2.16 (Incomplete gamma function, Borwein(2009)

Before discussing the incomplete gamma function, first we have discussed about the complete gamma function. The gamma function is defined as

Γ(α1)=∫0∞tα1-1e-tdt.

The integral is complete because the bounds of integration is the complete positive real line (0,∞). An incomplete gamma function replaces the upper or lower limit of integration in the integral that defines the complete gamma function. If you replace the upper limit of integration (∞) by x, you get the lower incomplete gamma function.

p(α1,x)=∫0xtα1-1e-tdt

Here lower refers to integrating only the left side, which means values of t < x. If you replace the lower limit of integration (0) by x, you get the upper incomplete gamma function.

q(α1,x)=∫x∞tα1-1e-tdt.

Here upper refers to integrating only the right side, which means values of t > x.

Definition 2.17 (Fuzzy gamma distribution, [25])

A random variable X has a FGD whose probability density function (PDF) is denoted by f(x, α₁, β̃), where β̃ is the fuzzy scale parameter. Now the fuzzy probability of the FRV X̃ on the interval [u, v] is a FN whose α_1₁-cut is

P˜(u≤X˜≤v)[α]={∫uv1βα1Γ(α1)xα1-1e-x/βdx∣β∈β˜˜[α]}.

3. Mathematical Models

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Abstract
1. Introduction
2. Basic Preliminary
3. Mathematical Models
4. Transformation Techniques
5. Case Study
6. Result and Discussion
7. Conclusion
Conflicts of Interest
Tables
References
Biographies

General mathematical model of STP is represented as

M in Z=∑i=1m∑j=1nCij1xij,

subject to

P (∑j=1nxij≤ai)≥pi, i=1, 2, …, mP (∑i=1mxij≤bj)≥qj, j=1, 2, …, n,xij≥0, ∀i, j,

where 0 < β_i < 1, ∀i and 0 < γ_j < 1, ∀j and a_i and b_j are random variable.

Depending on the randomness of different parameters, the following cases are considered here. The mathematical programming model of different cases for fuzzy STP are presented as

Model 1

Let a_i, i = 1, 2, ..., m are gamma distributed independent FRVS. Let the FRV a_i be denoted as a˜˜i. In this model the mathematical programming model for FSTP is presented as

min Z=∑i=1m∑j=1nCij1xij,

subject to

P˜˜ (∑j=1nxij≤ai˜)≥pi˜, i=1, 2, …, m,∑i=1mxij≥bj, j=1, 2, …, n,xij≥0 ∀i, j,

where p˜˜, i = 1, 2, ..., m are fuzzy numbers and Cij1, b_j ∈ R ∀i, j.

Model 2

Let b_j, j = 1, 2, ..., n are gamma distributed independent FRVS. Let the FRV b_j be denoted as bj˜˜. In this model the mathematical programming model for FSTP is presented as

min Z=∑i=1m∑j=1nCij1xij,

subject to

∑j=1nxij≤ai, i=1, 2, …, m,P˜˜(∑i=1mxij≤bj˜˜)≥qj˜˜, j=1, 2, …, n,xij≥0 ∀i, j,

where q˜˜j, j = 1, 2, ..., n are fuzzy numbers and Cij1, b_j ∈ R ∀i, j.

Model 3

Let a_i (i = 1, 2, ...,m) and b_j (j = 1, 2, ..., n) are gamma distributed independent FRVS. Let the FRV a_i be denoted as ai˜˜, b_j be denoted as bj˜˜. In this model the mathematical programming model for FSTP is presented as

min Z=∑i=1m∑j=1nCij1xij,

subject to

P˜˜ (∑j=1nxij≤ai˜˜)≥pi˜˜, i=1, 2, …, m,P˜˜ (∑i=1mxij≥bj˜˜)≥qj˜˜, j=1, 2, …, n,xij≥0 ∀i, j,

where pi˜˜, i = 1, 2, ..., m and qj˜˜, j = 1, 2, ..., n are fuzzy numbers and Cij1∈R∀i, j.

4. Transformation Techniques

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Abstract
1. Introduction
2. Basic Preliminary
3. Mathematical Models
4. Transformation Techniques
5. Case Study
6. Result and Discussion
7. Conclusion
Conflicts of Interest
Tables
References
Biographies

The mathematical models presented in Section 3 are difficult to solve due to randomness and fuzziness. Therefore an equivalent crisp models are necessary to obtain solution of the proposed models. So in this section an attempt has been made in order to find crisp equivalent model of the above FSTP.

Theorem 4.1

If ai˜˜, i = 1, 2, ..., m are gamma distributed independent fuzzy random variable, whose scale parameter βi˜˜ is PNIT2FN, then

P˜˜ (∑j=1nxij≤ai˜˜)≽pi˜˜

is equivalent to

∑i=1nxij≤βi¯*γα1-1(1-p_i*)Γ(α11),∑i=1nxij≤βi_*γα1-1(1-pi¯*)Γ(α1),

where x_ij is real number. βi˜˜ are PNIT2FN.

Proof

It is assumed that ai˜˜, i = 1, 2, ..., m are gamma distributed independent fuzzy random variable whose mean, variance are α₁β, and α₁β², where the scale parameter βi˜˜ is PNIT2FN.

So α-cuts of ai˜˜,pi˜˜,βi˜˜ are presented as

α-cuts of ai˜˜ is

ai˜˜[α]=[ai_[α],ai¯[α]]=[[a_i*[α],a_i*[α]],[a¯i*[α],b¯i*[α]]],

where

ai_[α]=[a_i*[α],a_i*[α]],ai¯[α]=[a¯i*[α],a¯i*[α]].

α-cuts of pi˜ is

pi˜˜[α]=[pi_[α],pi¯[α]]=[[p_i*[α],p_i*[α]],[p¯i*[α],p¯i*[α]]],

where

pi_[α]=[p_i*[α],p_i*[α]],pi¯[α]=[p¯i*[α],p¯i*[α]].

α-cuts of β̃ is

βi˜˜[α]=[β_i[α],β¯i[α]]=[[β_i*[α],β_i*[α]],[β¯i*[α],β¯i*[α]]],

where

(3)β_[α]=[β_i*[α],β_i*[α]],β¯[α]=[β¯i*[α],β¯i*[α]],P˜ (∑j=1nxij≤ai˜˜)[α]≽pi˜˜.

Now the α-cut of the constraint (3) is expressed as

P˜˜(∑j=1nxij≤ai˜˜) [α]=P˜˜(Ai≤ai˜˜)[α], where Ai=∑j=1nxij={P(Ai≤ai)∣ai∈ai˜˜[α],βi˜˜[α]=[β_i[α],β¯i[α]]}={1-P(ai≤Ai)∣ai∈ai˜˜[α],βi˜˜[α]=[β_i[α],β¯i[α]]}={1-∫0Ai1βα1Γ(α1)aiα1-1e-ai/βdai∣ai∈ai˜˜[α],βi˜˜[α]=[β_i[α],β¯i[α]]}where aiβi=y⇒dai=βidy={1-∫0∑j=1nxij/βi1βiα11Γ(α11)(βiy)α1-1e-yβidy∣ai∈ai˜˜[α],βi˜˜[α]=[β_i[α],β¯i[α1]]}={1-1Γ(α11)∫0∑j=1nxij/βiyα111-1e-ydy}={1-1Γ(α1)γα1[∑j=1nxijβi]∣ai∈ai˜˜[α],βi˜˜[α]=[β_i[α],β¯i[α]]},

where

γα1(x)=∫0xyα1-1e-ydy

is an lower incomplete gamma function, which is an increasing function, so lower minimum value of 1-1Γ(α1)γα1 [∑j=1nxijβi] is

Min{1-1Γ(α1)γα1[∑j=1nxijβi]∣βi˜˜[α]=[β_i[α],β¯i[α]]}=1-1Γ(α1)γα1[∑j=1nxijβi*].

Lower maximum value of 1-1Γ(α1)γα1 [∑j=1nxijβi] is

Max{1-1Γ(α1)γα1[∑j=1nxijβi]∣βi˜˜[α]=[β_i[α],β¯i[α]]}=1-1Γ(α1)γα1[∑j=1nxijβ_i*].

Upper minimum value of 1-1Γ(α1)γα1 [∑j=1nxijβi] is

Min{1-1Γ(α1)γα1[∑j=1nxijβi]∣βi˜˜[α]=[β_i[α],β¯i[α]]}=1-1Γ(α1)γα1[∑j=1nxijβ_i*].

Upper maximum value of 1-1Γ(α1)γα1 [∑j=1nxijβi] is

Max{1-1Γ(α1)γα1[∑j=1nxijβi]∣βi˜˜[α]=[β_i[α],β¯i[α]]}=1-1Γ(α1)γα1[∑j=1nxijβ¯i*].

(4)P˜˜(∑j=1nxij≤ai˜˜) [α]=[1-1Γ(α1)γα1[∑j=1nxijβ¯i*],1-1Γ(α1)γα1[∑j=1nxijβ_i*],1-1Γ(α1)γα1[∑j=1nxijβ_i*],1-1Γ(α1)γα1[∑j=1nxijβ¯i*]]≥[p¯i*,p_i*],[p_i*p¯i*],1-1Γ(α1)γα1[∑j=1nxijβ¯i*],1-1Γ(α1)γα1[∑j=1nxijβ_i*]≥[p_i*p¯i*],1-1Γ(α1)γα1[∑j=1nxijβ¯i*]≥p_i*,

(5)1-1Γ(α1)γα1[∑j=1nxijβ_i*]≥p¯i*.

Using fuzzy inequality,the α − cut of the fuzzy constraint is expressed as

P˜˜(∑j=1nxij≤a˜˜i) [α]≽pi˜˜[α],

from (4), we have

1-1Γ(α1)γα11[∑i=1nxijβ¯i*]≥p_i*,1Γ(α1)γα1[∑i=1nxijβ¯i*]≤1-p_i*,1Γ(α1)(∑i=1nxijβ¯i*)≤γα1-1(1-p_i*),(∑i=1nxijβ¯i*)≤γα1-1(1-p_i*)Γ(α1),∑i=1nxij≤βi¯*γα1-1(1-p_i*)Γ(α1),

and

P˜˜ (∑j=1nxij≤a˜˜i) [α]≽pi˜˜[α],

from (5), we have

1-1Γ(α1)γα1[∑i=1nxijβ_i*]≥pi¯*,1Γ(α1)γα1[∑i=1nxijβ_i*]≤1-pi¯*,1Γ(α1)(∑i=1nxijβ_i*)≤γα1-1(1-pi¯*),(∑i=1nxijβ_i*)≤γα1-1(1-p_i*)Γ(α1),∑i=1nxij≤βi_*γα1-1(1-pi¯*)Γ(α1),

from the result of (4) and (5), we have

∑i=1nxij≤βi¯*γα1-1(1-p_i*)Γ(α1),∑i=1nxij≤βi_*γα1-1(1-pi¯*)Γ(α1).

This proves the theorem.

Now using Theorem 4.1, the deterministic equivalent of mathematical Model 1 becomes DModel 1

Mni Z=(∑j=1nxij≤ai˜)≽pi˜,

subject to

∑i=1nxij≤βi¯*γα1-1(1-p_i*)Γ(α1),∑i=1nxij≤βi_*γα1-1(1-pi¯*)Γ(α1),

where x_ij is real number. ai˜˜ are FRVs βi˜˜ are PNIT2FNs.

Theorem 4.2

If bj˜˜, j = 1, 2, ..., n are gamma distributed independent fuzzy random variable, whose scale parameter βi˜˜ is PNIT2FN, then

P˜˜(∑i=1mxij≥bj˜˜)≽pi˜˜

is equivalent to

∑i=1nxij≤βi_*γα1-1(p_i*Γ(α1)),∑i=1nxij≤βi¯*γα1-1(pi¯*Γ(α1)),

where x_ij is real number. bj˜˜ are FRVs. βi˜˜ are PNIT2FNs.

Proof

It is assumed that bj˜˜, j = 1, 2, ..., n are gamma distributed independent fuzzy random variable whose mean, variance are α₁β, and α₁β², where the scale parameter βi˜˜

is PNIT2FN.

So α-cuts of bi˜˜,pi˜˜,βi˜˜ are presented as

α-cuts of bi˜˜ is

b˜˜i[α]=[bi_[α],bi¯[α]]=[[b_i*[α],b_i*[α]],[b¯i*[α],b¯i*[α]]],

where

bi_[α]=[b_i*[α],b_i*[α]],bi¯[α]=[b¯i*[α],b¯i*[α]].

α-cuts of pi˜˜ is

p˜˜i[α]=[pi_[α],pi¯[α]]=[[p_i*[α],p_i*[α]],[p¯i*[α],p¯i*[α]]],

where

pi_[α]=[p_i*[α],p_i*[α]],pi¯[α]=[p¯i*[α],p¯i*[α]].

α-cuts of β̃ is

β˜˜[α]=[β_[α],β¯[α]]=[β_*[α],β_*[α]],[β¯*[α],β¯*[α]],

where

(6)β_[α]=[β_*[α],β_*[α]],β¯[α]=[β¯*[α],β¯*[α]],P˜˜ (∑j=1nxij≤b˜˜j)[α]≽p˜˜i.

Now the α-cut of the constraint (6) is expressed as

P˜˜(∑j=1nxij≥b˜˜j) [α]=P˜˜(Ai≥b˜˜j)[α], where Ai=∑j=1nxij={P(Ai≥bj)∣bj∈bj˜˜[α],βi˜˜[α]=[β_i[α],β¯i[α]]}={P(bj≤Ai)∣bj∈bj˜˜[α],βi˜˜[α]=[β_i[α],β¯i[α]]}={∫0Ai1βiα1Γ(α1)aiα1-1e-ai/βidai∣bj∈bj˜˜[α],βi˜˜[α]=[β_i[α],β¯i[α]]}where aiβi=y⇒dai=βidy={∫0∑j=1nxij/βi1βiα1Γ(α11)(βiy)α1-1e-yβidy∣bj∈bj˜˜[α],βi˜˜[α]=[β_i[α],β¯i[α]]}={1Γ(α1)∫0∑j=1nxij/βiyα1-1e-ydy∣bj∈bj˜˜[α],βi˜˜[α]=[β_i[α],β¯i[α]]}={1Γ(α1)γα1[∑j=1nxijβi]∣bj∈bj˜˜[α],βi˜˜[α]=[β_i[α],β¯i[α]]},

where

γα1(x)=∫0xyα1-1e-ydy

is an lower incomplete gamma function. which is an increasing function, so lower minimum value of 1Γ(α1)γα1 [∑j=1nxijβi] is

Min {1Γ(α1)γα1 [∑j=1nxijβi]∣β˜˜i[α]=[β_i[α],β¯i[α]]}=1Γ(α1)γα1 [∑j=1nxijβ¯i *].

Lower maximum value of 1Γ(α1)γα1 [∑j=1nxijβi] is

Max {1Γ(α1)γα1 [∑j=1nxijβi]∣β˜˜i[α]=[β_i[α],β¯i[α]]}=1Γ(α1)γα1 [∑j=1nxijβ_i *].

Upper minimum value of 1Γ(α1)γα1 [∑j=1nxijβi] is

Min {1-1Γ(α1)γα1 [∑j=1nxijβ]∣β˜˜i[α]=[β_i[α],β¯i[α]]}=1Γ(α1)γα1 [∑j=1nxijβ_i*].

Upper maximum value of 1-1Γ(α1)γα1 [∑j=1nxijβi] is

Max {1-1Γ(α1)γα1 [∑j=1nxijβi]∣β˜˜i[α]=[β_i[α],β¯i[α]]}=1Γ(α1)γα1 [∑j=1nxijβ¯i*].

P˜˜ (∑j=1nxij≤bj) [α1]=[1Γ(α1)γα1 [∑j=1nxijβ¯i *],1Γ(α1)γα1 [∑j=1nxijβ_i *],1Γ(α1)γα1 [∑j=1nxijβ_i*],1Γ(α1)γα1 [∑j=1nxijβ¯i*]]≥[p¯i*,p_i*],[p_i*p¯i*],1Γ(α1)γα1 [∑j=1nxijβ¯i*],1Γ(α1)γα1 [∑j=1nxijβ_i*]≥[p_i*,p¯i*],1Γ(α1)γα1 [∑j=1nxijβ¯i*]≥p_i*,1Γ(α1)γα1 [∑j=1nxijβ_i*]≥p¯i*,

from (4), we have

1Γ(α1)γα1 [∑j=1nxijβ¯i*]≥p_i*,1Γ(α1)(∑i=1nxijβ¯i*)≥γα1 -1(p_i*),(∑i=1nxijβ¯i*)≤γα1 -1(p_i*)Γ(α1),∑i=1nxij≤β¯i*γα1 -1(p_i*)Γ(α1),

and from (5), we have

1Γ(α1)γα1 (∑i=1nxijβ_i*)≥pi¯*,1Γ(α1)γα1 (∑i=1nxijβ_*)≤pi¯*,1Γ(α1)(∑i=1nxijβ_i*)≤γα1 -1(pi¯*)(∑i=1nxijβ_i*)≤γα1-1((p¯i*)Γ(α1)),∑i=1nxij≤β_i*γα1-1((pi¯*)Γ(α1)).

Now from the result of (4) and (5), we have

∑i=1nxij≤β¯i*γα1 -1(p_i*)Γ(α1),∑i=1nxij≤β_i*γα1 -1(pi¯*)Γ(α1).

This proves the theorem.

Now using Theorem (4.2), the deterministic equivalent of mathematical Model 2 becomes DModel 2

Min Z=(∑i=1mxij≥b˜˜j)⊳p˜˜i,

subject to

∑i=1nxij≤β¯i*γα1 -1(p_i*)Γ(α1),∑i=1nxij≤β_i*γα1 -1(pi¯*)Γ(α1),

where x_ij is real number. b˜˜j are FRVs, β˜˜i is PNIT2FN.

5. Case Study

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Abstract
1. Introduction
2. Basic Preliminary
3. Mathematical Models
4. Transformation Techniques
5. Case Study
6. Result and Discussion
7. Conclusion
Conflicts of Interest
Tables
References
Biographies

A Potato and Banana chips making company “VF tango” (name changed) supplied chips by collecting chips from its three factories situated at three different places, Khurda, Rourkela, and Jharsuguda of Odisha, India by lorries to three destinations situated at Jamshedpur, Kolkata and Hyderabad of India through nine different routes. The main purpose is to minimize the transportation cost in order to maximize the profit. However, due to different unpredictable factors such as weather (flood, cyclone), season, different social activities the supply is not known. In the recent years as lockdown (due to COVID-19) may introduce latency and restrictions in terms of movement, the supply is totally unpredictable for decision makers. After discussing the manager of the company and analyzing the past data, it is finalized that the supply parameter follow gamma distribution. To manage the issue emerging due to previously mentioned cases, a fuzzy STP approach has been considered.

The unit transportation cost of one unit (in 500 packets) through a particular route during transportation are given in the Table 1. Now, the transportation problem is represented by

(7)min Z=18x11+25x12+13x13+15x21+12x22+13x23+16x31+12x32+11x33,

subject to

p˜˜ (∑j=13x1j≤a˜˜¯1)⊳p˜˜¯1,p˜˜ (∑j=13x1j≤a˜˜_1)⊳p˜˜_1,p˜˜ (∑j=13x2j≤a˜˜¯2)⊳p˜˜¯2,p˜˜ (∑j=13x2j≤a˜˜_2)⊳p˜˜_2,p˜˜ (∑j=13x3j≤a˜˜¯3)⊳p˜˜¯3,p˜˜ (∑j=13x1j≤a˜˜_3)⊳p˜˜_3,∑i=13xi1≥b1,∑i=13xi2≥b2,∑i=13xi3≥b3,xij≥0, i=1, 2, 3 and j=1, 2, 3,

where p˜˜i (i=1, 2, 3) are fuzzy numbers. a˜˜i (i=1, 2, 3) (i = 1, 2, 3) are gamma distributed fuzzy random variables. Here the scale parameter β˜˜ai are perfectly normal interval type 2 trapezoidal fuzzy numbers, whose lower membership function (β˜˜_a1) and upper membership function (β˜˜¯a1) are trapezoidal fuzzy numbers. Here the shape parameter α= 0.8 for all a˜˜i.

They are represented by

β˜˜¯a1=upper membership function of β˜˜a1=〈52,54,57,59〉,β˜˜_a1=lower membership function of β˜˜a1=〈53,55,56,58〉,β˜˜¯a2=upper membership function of β˜˜a2=〈62,64,67,69〉,β˜˜_a2=lower membership function of β˜˜a2=〈63,65,66,68〉,β˜˜¯a3=upper membership function of β˜˜a3=〈72,74,77,79〉,β˜˜_a3=lower membership function of β˜˜a3=〈73,75,76,78〉.

Specified probability levels:

p1˜˜¯=0.04˜˜¯=upper membership function of 0.04˜˜=〈0.02,0.04,0.07,0.09〉,p1˜˜_=0.04˜˜_=lower membership function of 0.04˜˜=〈0.03,0.05,0.06,0.08〉,p2˜˜¯=0.06˜˜¯=upper membership function of 0.06˜˜=〈0.01,0.03,0.06,0.08〉,p2˜˜_=0.06˜˜_=lower membership function of 0.06˜˜=〈0.02,0.04,0.05,0.07〉,p3˜˜¯=0.08˜˜¯=upper membership function of 0.08˜˜=〈0.02,0,04,0.07,0.09〉,p3˜˜_=0.08˜˜_=lower membership function of 0.08˜˜=〈0.03,0.05,0.06,0.08〉.

The α-cuts are

β˜˜¯a1[α]=[52+2α,59-2α],β˜˜_a1[α]=[53+2α,58-2α],β˜˜¯a2[α]=[62+2α,69-2α],β˜˜_a2[α]=[63+2α,68-2α],β˜˜¯a3[α]=[72+2α,79-2α],β˜˜_a3[α]=[73+2α,78-2α],0.04˜˜¯[α]=[0.02+0.02α,0.09-0.02α],0.04˜˜_[α]=[0.03+0.02α,0.08-0.02α],0.06˜˜¯[α]=[0.01+0.02α,0.08-0.02α],0.06˜˜_[α]=[0.02+0.02α,0.07-0.02α],0.08˜˜¯[α]=[0.02+0.02α,0.09-0.02α],0.08˜˜_[α]=[0.03+0.02α,0.08-0.02α].

Now using Theorem (4.1) the deterministic equivalent model of (7) becomes

Minimize 18x11+25x12+13x13+15x21+12x22+13x23+16x31+12x32+11x33,∑j=13x1j≤γα1 -1((0.92+0.02α)Γ(α1))(52+2α),∑j=13x1j≤γα1 -1((0.91+0.02α)Γ(α1))(53+2α),∑j=13x2j≤γα-1((0.93+0.02α)Γ(α1))(62+2α),∑j=13x2j≤γα1 -1((0.92+0.02α)Γ(α1))(63+2α),∑j=13x3j≤γα1 -1((0.92+0.02α)Γ(α1))(72+2α),∑j=13x3j≤γα1 -1(0.91+0.02α)Γ(α1))(73+2α),∑i=13xi1≥12,∑i=13xi2≥13,∑i=13xi3≥16,x11≥0, x12≥0, x13≥0, x21≥0, x22≥0,x23≥0, x31≥0, x32≥0, x33≥0.

Using R software and for α = 0.5, the deterministic equivalent becomes

Minimize 18x11+25x12+13x13+15x21+12x22+13x23+16x31+12x32+11x33,

Subject to

∑j=13x1j≤13.9303398,∑j=13x1j≤18.1516419,∑j=13x2j≤14.9077023,∑j=13x2j≤19.5031491,∑j=13x3j≤14.3020416,∑j=13x3j≤21.8529261,∑i=13x1j≥12,∑i=13x2j≥13,∑i=13x3j≥16,x11≥0, x12≥0, x13≥0, x21≥0, x22≥0,x23≥0, x31≥0, x32≥0, x33≥0.

Using LINGO software, the solution of the above problem

6. Result and Discussion

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Abstract
1. Introduction
2. Basic Preliminary
3. Mathematical Models
4. Transformation Techniques
5. Case Study
6. Result and Discussion
7. Conclusion
Conflicts of Interest
Tables
References
Biographies

The above problem is solved using the LINGO software and optimal compromise solutions are obtained for different values of α. From the tabular results shown above, it is found that for α = 0.5 the objective value is 535.5805, for α = 0.6 the objective value is 523.6332 and for α = 0.7 the objective value is 512.0000. For each value of α we can get a solution. x_ij (i = 1, 2, 3 and j = 1, 2, 3) represents the quantity transportated from i-th origin to j-th destination.

7. Conclusion

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Abstract
1. Introduction
2. Basic Preliminary
3. Mathematical Models
4. Transformation Techniques
5. Case Study
6. Result and Discussion
7. Conclusion
Conflicts of Interest
Tables
References
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In this paper, a fuzzy STP has been proposed. Where gamma distributed FRV is present on the right-hand side of the constraint. The scale parameter β˜˜ is a PNIT2FN. Here the fuzzy stochastic models have been converted into equivalent deterministic models in two steps. In the first step using the concept of α-cut the fuzziness has been removed and in the second step using the chance constraint method randomness has been removed. Here we have considered only the first model and its equivalent deterministic models are derived. Also, a case study is solved according to the first model. Here the mathematical model is solved by R and Lingo software. Similarly, other cases can be established, that is where demand parameters or both supply, and demand parameters are FRVs. Since the whole paper has been done with a single objective stochastic transportation problem, also we can solve multi-objective, multi-choice STP under a type-2 fuzzy environment considering other probability distributions. This work can also be applied to other mathematical programming problems namely: assignment problems, quadratic programming problems, bi-level programming problem, two-stage programming prolem. In future we will work on fractional transportation problem under interval type-2 fuzzy environment.

Conflicts of Interest

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Abstract
1. Introduction
2. Basic Preliminary
3. Mathematical Models
4. Transformation Techniques
5. Case Study
6. Result and Discussion
7. Conclusion
Conflicts of Interest
Tables
References
Biographies

No potential conflict of interest relevant to this article was reported.

Tables

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Abstract
1. Introduction
2. Basic Preliminary
3. Mathematical Models
4. Transformation Techniques
5. Case Study
6. Result and Discussion
7. Conclusion
Conflicts of Interest
Tables
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Table. 1.

Table 1. Transportation cost per unit (in rupees).

Source destination	Jamsedpur	Kolkata	Hyderabad
Khurda	18	25	13
Rourkela	15	12	13
Jharsuguda	16	12	11

Table. 2.

Table 2. Solution for different values of α.

α	0.5	0.6	0.7
x₁₁	0.00	0.00	0.00
x₁₂	0.00	0.00	0.00
x₁₃	11.79026	5.816579	0.00
x₂₁	12.00	12.00	12.00
x₂₂	2.907702	5.702769	8.827397
x₂₃	0.00	0.00	0.00
x₃₁	0.00	0.00	0.00
x₃₂	10.09230	7.297231	4.172603
x₃₃	4.209744	10.18342	16.0000
Z	535.5805	523.6332	512.0000

References

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Abstract
1. Introduction
2. Basic Preliminary
3. Mathematical Models
4. Transformation Techniques
5. Case Study
6. Result and Discussion
7. Conclusion
Conflicts of Interest
Tables
References
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Zadeh, LA (1968). Probability measures of fuzzy events. Journal of Mathematical Analysis and Applications. 23, 421-427.
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Zadeh, LA (1965). Fuzzy sets. Information and Control. 8, 338-353. https://doi.org/10.1016/S0019-9958(65)90241-X
Acharya, S, Ranarahu, N, Dash, JK, and Acharya, MM (2014). Solving multi-objective fuzzy probabilistic programming problem. Journal of Intelligent & Fuzzy Systems. 26, 935-948. https://doi.org/10.3233/IFS-130784
Acharya, S, Ranarahu, N, Dash, JK, and Acharya, MM (2014). Computation of a multi-objective fuzzy stochastic transportation problem. International Journal of Fuzzy Computation and Modelling. 1, 212-233. https://doi.org/10.1504/IJFCM.2014.067129
Ranarahu, N, Dash, JK, and Acharya, S (2017). Multiobjective bilevel fuzzy probabilistic programming problem. OPSEARCH. 54, 475-504. https://doi.org/10.1007/s12597-016-0290-5
Ranarahu, N, Dash, JK, and Acharya, S (2018). Multi-objective fuzzy quadratic probabilistic programming problem involving fuzzy Cauchy random variable. International Journal of Operational Research. 32, 495-513. https://doi.org/10.1504/IJOR.2018.093517
Ranarahu, N, Dash, JK, and Acharya, S (2019). Computation of multi-choice multi-objective fuzzy probabilistic transportation problem. Operations Research in Development Sector. Singapore: Springer, pp. 81-95 https://doi.org/10.1007/978-981-13-1954-96
Ranarahu, N, and Dash, JK (2020). A new type of fuzzy transportation problem. TEST Engineering and Management. 83, 6108-6111.
Mendel, JM, John, RI, and Liu, F (2006). Interval type-2 fuzzy logic systems made simple. IEEE Transactions on Fuzzy Systems. 14, 808-821. https://doi.org/10.1109/TFUZZ.2006.879986
Srinivasan, A, and Geetharamani, G (2016). Linear programming problem with interval type 2 fuzzy coefficients and an interpretation for ITS constraints. Journal of Applied Mathematics. 2016. article no 8496812
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Agrawal, P, and Ganesh, T (2020). Fuzzy fractional stochastic transportation problem involving exponential distribution. OPSEARCH. 57, 1093-1114. https://doi.org/10.1007/s12597-020-00458-5
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Kundu, P, Kar, S, and Maiti, M (2014). Fixed charge transportation problem with type-2 fuzzy variables. Information Sciences. 255, 170-186. https://doi.org/10.1016/j.ins.2013.08.005
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Kundu, P, Majumder, S, Kar, S, and Maiti, M (2019). A method to solve linear programming problem with interval type-2 fuzzy parameters. Fuzzy Optimization and Decision Making. 18, 103-130. https://doi.org/10.1007/s10700-018-9287-2
Das, D, Roy, A, and Kar, S (2011). A volume flexible economic production lot-sizing problem with imperfect quality and random machine failure in fuzzy-stochastic environment. Computers & Mathematics with Applications. 61, 2388-2400. https://doi.org/10.1016/j.camwa.2011.02.015
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Biographies

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Abstract
1. Introduction
2. Basic Preliminary
3. Mathematical Models
4. Transformation Techniques
5. Case Study
6. Result and Discussion
7. Conclusion
Conflicts of Interest
Tables
References
Biographies

Babita Chaini is a mathematician. He works as a researcher at the Department of Mathematics in SOA University, located in Odisha, India. Her research interests mainly include fuzzy optimization, and operation research and their practical applications.

E-mail: babitachaini@gmail.com

Narmada Ranarahu holds a Ph.D. in Operation Research from SOA University in Odisha, India. Currently, she works as an assistant professor of Mathematics at SOA university. Her research interests involve mathematical modelling, multiobjective optimization, fuzzy optimization, and operation research.

Email-narmada.ranarahu@gmail.com

Article

Original Article

International Journal of Fuzzy Logic and Intelligent Systems 2023; 23(1): 91-105

Published online March 25, 2023 https://doi.org/10.5391/IJFIS.2023.23.1.91

Type-2 Fuzzy Stochastic Transportation Problem with Gamma Distribution

Babita Chaini and Narmada Ranarahu

Department of Mathematics, SOA deemed to be University, Bhubaneswar, India

Correspondence to:Babita Chaini (babitachaini@gmail.com)

Received: December 6, 2021; Revised: January 9, 2023; Accepted: March 6, 2023

Abstract

Keywords: Stochastic programming, Fuzzy stochastic transportation problem, Fuzzy gamma random variables, Optimization techniques, Type 2 fuzzy set

1. Introduction

Also, from the literature survey, we two studies [5,13]. In [5], the authors considered multi-objective STP, where parameters are independent normally distributed fuzzy random variables. Agrawal and Ganesh [13] developed a method to solve the fuzzy fractional transportation problem in which the parameters of the transportation problem, supply, and demand, are stochastic in nature and considered as a fuzzy random variable that follows the exponential distribution with fuzzy mean and fuzzy variance. In this paper, a novel strategy is developed for STP involving fuzzy gamma distribution. The main difference between this paper and the above two papers are follows. Firstly, as we know T2FSs have more degrees of freedom to describe uncertainty and are capable of handling inexact data in a logically correct manner. Many researchers have used interval type-2 fuzzy logic systems (T2FLSs) in the practical field even though generalized T2FLSs are computationally demanding. In comparison to the generalized T2FLS, computations in the IT2FS are more manageable. The main difference between the generalized and interval type-2 fuzzy membership functions is that the interval type-2 membership function’s secondary membership value is, in general, equal to 1. Here this paper is solved under interval type-2 fuzzy environment not type-1. Secondly, only the supply parameter is fuzzy random variables and the scale parameter $\tilde{\tilde{β}}$ is perfectly normal interval type-2 fuzzy number.

(i) The first step, fuzziness is removed by using alpha-cut technique to obtain STP.
(ii) The second step, randomness is removed by using the chance constrained technique, to obtain crisp transportation problem.

Finally, the solution procedure for solving the model is illustrated with the help of a case study.

2. Basic Preliminary

Definition 2.1 (Crisp set, [22])

(1)

X_{A} (x) = {\begin{array}{l} 1 & iff x \in X, \\ 0 & iff x \notin X, \end{array}

where 1 indicates membership and 0 indicates non-membership.

Definition 2.2 (Fuzzy set, [23])

Let X be a collection of distinct objects and x be an element of X. Then a fuzzy set Ã in X is a set of order pairs

(2)

\tilde{A} = {(x, μ_{\tilde{A}} (x)) ∣ x \in X},

Definition 2.3 (Fuzzy number, [24])

A fuzzy number Ã is a convex normalized fuzzy set Ã of the real line R, with membership function

μ_{\tilde{A}} : R \to [0; 1],

satisfying the following conditions.

There exists exactly one interval I ∈ R such that μ_Ã(x) = 1; x ∈ I.
The membership function μ_Ã is piecewise continuous.

Definition 2.4 (α-cut set, [25])

α-cut of the fuzzy number Ã is the set {x|μ_A(x) ≥ α} for 0 < α < 1 and denoted Ã[α].

Definition 2.5 (Convex fuzzy set, [26])

Definition 2.6 (Type-2 fuzzy set, [10])

A T2FS denoted as $\tilde{\tilde{A}}$ is a fuzzy set with membership function whose membership value is itself a fuzzy set. It is defined as

\tilde{\tilde{A}} = {((x, u) μ_{\tilde{\tilde{A}}} (x, u)) ∣ x \in X, \forall u \in J_{x} \subseteq [0, 1]}

in which $0 \leq μ_{\tilde{\tilde{A}}} (x, u) \leq 1. \tilde{\tilde{A}}$ can also be expressed as

\tilde{\tilde{A}} = \int_{x \in X} \int_{u \in J_{x}} μ_{\tilde{\tilde{A}}} (x, u) / (x, u), J_{x} \subseteq [0, 1],

where ∫ ∫ denotes union over all admissible x and u. For discrete universe of discourse, ∫ s replaced by ∑.

Definition 2.7 (Interval type-2 fuzzy set, [27])

An IT2FS is fully characterized by primary membership function and lower membership function i.e., when all $μ_{\tilde{\tilde{A}}} (x, u) = 1$ . And it is defined as

\tilde{\tilde{A}} = {((x, u), 1) ∣ x \in X, \forall u \in J_{x} \subseteq [0, 1]} .

Definition 2.8 (Normal IT2FS and perfectly normal IT2FS, [27])

An IT2FS $\tilde{\tilde{A}}$ is said to be normal if its upper membership function is normal. It is said to be perfectly normal if both of its upper and lower membership functions are normal.

Definition 2.9 (Type-2 fuzzy number, [27])

A T2FS denoted as $\tilde{\tilde{A}}$ in the universe R is said to be Type-2 fuzzy number if the following conditions are satisfied:

$\tilde{\tilde{A}}$ is normal.
$\tilde{\tilde{A}}$ is convex.
The support of $\tilde{\tilde{A}}$ is closed and bounded.

Definition 2.10 (Primary α-cut of perfectly normal IT2FS, [11])

The primary α-cut of PnIT2FS is ${\tilde{\tilde{A}}}_{α} = {(x, u) ∣ J_{x} \geq α, u \in [0, 1]}$ which is bounded by two regions

\begin{array}{l} {\underline{μ}}_{{\tilde{\tilde{A}}}_{α}} (x) = {(x, {\underline{μ}}_{\tilde{\tilde{A}}} (x)) ∣ {\underline{μ}}_{\tilde{\tilde{A}}} (x) \geq α, \forall α \in [0, 1]}, \\ {\bar{μ}}_{{\tilde{\tilde{A}}}_{α}} (x) = {(x, {\bar{μ}}_{\tilde{\tilde{A}}} (x)) ∣ {\bar{μ}}_{\tilde{\tilde{A}}} (x) \geq α, \forall α \in [0, 1]} . \end{array}

Definition 2.11 (Totally symmetric type-2 trapezoidal fuzzy number, [27])

A type-2 fuzzy number represented as a triplet of type-1 fuzzy numbers is called TST2TFN if it satisfies the following condition:

Upper membership function and lower membership functions are symmetric trapezoidal fuzzy number and generalized symmetric trapezoidal fuzzy number, respectively.
For each reference point in the primary universe of discourse, secondary membership functions are symmetric trapezoidal fuzzy numbers.

Definition 2.12 (Footprint of uncertainty, [11])

Let $\tilde{\tilde{A}}$ be IT2FS; uncertainty in the primary membership of T2FS consists of a bounded region called the footprint of uncertainty, which is the union of all primary membership. Footprint of uncertainty is characterized by upper membership function and lower membership function. Both of the membership functions are T1FSs. Upper membership function is denoted by ${\bar{μ}}_{\tilde{\tilde{A}}}$ and lower membership function is denoted by ${\underline{μ}}_{\tilde{\tilde{A}}}$ , respectively.

Definition 2.13 (Perfectly normal interval type 2 trapezoidal fuzzy number, [11])

An interval type 2 fuzzy number $\tilde{\tilde{A}}$ is called perfectly normal interval type 2 trapezoidal fuzzy number (IT2TFN) when the upper membership function ${\underline{μ}}_{\tilde{\tilde{A}}} (x)$ and lower membership function ${\bar{μ}}_{\tilde{\tilde{A}}} (x)$ are both trapezoidal fuzzy numbers; that is, $\tilde{\tilde{A}} = [\underline{A}, \bar{A}] = [< \underline{a}, \underline{b}, \underline{c}, \underline{d} >, < \bar{a}, \bar{b}, \bar{c}, \bar{d} >]$ . Where A and Ā are known as lower and upper interval valued bounds of $\tilde{\tilde{A}}$ . Ã=<a, b, c, d> is the lower trapezoidal fuzzy number. $\tilde{\bar{A}} = < \bar{a}, \bar{b}, \bar{c}, \bar{d} >$ is the upper trapezoidal fuzzy number. $μ_{\tilde{\tilde{A}}} (\underline{b}) = μ_{\tilde{\tilde{A}}} (\bar{b})$ is the core and $μ_{\tilde{\tilde{A}}} (\underline{b}) = μ_{\tilde{\tilde{A}}} (\bar{b}) = 1$ .

b–a is called lower spread of Ã and b̄ – ā is called lower spread of $\tilde{\bar{A}}$ . c–b is called upper spread of Ã and c̄–b̄ is called upper spread of $\tilde{\bar{A}}$ .

The membership functions of x in Ã and $\tilde{\bar{A}}$ are expressed as follows:

\begin{array}{l} {\underline{μ}}_{\tilde{\tilde{A}}} (x) = {\begin{array}{l} 0, & if x \leq \underline{a}, \\ \frac{x - \underline{a}}{\underline{b} - \underline{a}}, & if \underline{a} \leq x \leq \underline{b}, \\ 1, & if \underline{b} \leq x \leq \underline{c}, \\ \frac{\underline{d} - x}{\underline{d} - \underline{c}}, & if \underline{c} \leq x \leq \underline{d}, \\ 0, & if x \geq \underline{d}, \end{array} \\ {\bar{μ}}_{\tilde{\tilde{A}}} (x) = {\begin{array}{l} 0, & if x \leq \bar{a}, \\ \frac{x - \bar{a}}{\bar{b} - \bar{a}}, & if \bar{a} \leq x \leq \bar{b}, \\ 1, & if \underline{b} \leq x \leq \underline{c}, \\ \frac{\bar{d} - x}{\bar{d} - \bar{c}}, & if \bar{c} \leq x \leq \bar{d}, \\ 0, & if x \geq \bar{d} . \end{array} \end{array}

Definition 2.14 (α-cut of perfectly normal,[11])

α-cut of perfectly normal IT2TFN $\tilde{\tilde{A}} = [\underline{A}, \bar{A}] = [< \underline{a}, \underline{b}, \underline{c}, \underline{d} >, < \bar{a}, \bar{b}, \bar{c}, \bar{d} >]$ is $\tilde{\tilde{A}} [α] = [\underline{A} [α]$ , Ā[α]] = [[a + (b–a)[α], d–(d–c)[α]], [ā+(b̄–ā)[α], d̄–(d̄–c̄)[α]]][[a+(b–a)[α], ā + (b̄ – ā)[α]][d– (d –c)[α], d̄ – (d̄ – c̄)[α]]].

Definition 2.15 (Stochastic transportation problem, [28])

Definition 2.16 (Incomplete gamma function, Borwein(2009)

Before discussing the incomplete gamma function, first we have discussed about the complete gamma function. The gamma function is defined as

Γ (α_{1}) = \int_{0}^{\infty} t^{α_{1} - 1} e^{- t} d t .

p (α_{1}, x) = \int_{0}^{x} t^{α_{1} - 1} e^{- t} d t

Here lower refers to integrating only the left side, which means values of t < x. If you replace the lower limit of integration (0) by x, you get the upper incomplete gamma function.

q (α_{1}, x) = \int_{x}^{\infty} t^{α_{1} - 1} e^{- t} d t .

Here upper refers to integrating only the right side, which means values of t > x.

Definition 2.17 (Fuzzy gamma distribution, [25])

\tilde{P} (u \leq \tilde{X} \leq v) [α] = {\int_{u}^{v} \frac{1}{β^{α_{1}} Γ (α_{1})} x^{α_{1} - 1} e^{- x / β} d x ∣ β \in \tilde{\tilde{β}} [α]} .

3. Mathematical Models

General mathematical model of STP is represented as

M i n Z = \sum_{i = 1}^{m} \sum_{j = 1}^{n} C_{i j}^{1} x_{i j},

subject to

\begin{array}{l} P (\sum_{j = 1}^{n} x_{i j} \leq a_{i}) \geq p_{i}, i = 1, 2, \dots, m \\ P (\sum_{i = 1}^{m} x_{i j} \leq b_{j}) \geq q_{j}, j = 1, 2, \dots, n, \\ x_{i j} \geq 0, \forall i, j, \end{array}

where 0 < β_i < 1, ∀i and 0 < γ_j < 1, ∀j and a_i and b_j are random variable.

Depending on the randomness of different parameters, the following cases are considered here. The mathematical programming model of different cases for fuzzy STP are presented as

Model 1

Let a_i, i = 1, 2, ..., m are gamma distributed independent FRVS. Let the FRV a_i be denoted as ${\tilde{\tilde{a}}}_{i}$ . In this model the mathematical programming model for FSTP is presented as

min Z = \sum_{i = 1}^{m} \sum_{j = 1}^{n} C_{i j}^{1} x_{i j},

subject to

\begin{array}{l} \tilde{\tilde{P}} (\sum_{j = 1}^{n} x_{i j} \leq \tilde{a_{i}}) \geq \tilde{p_{i}}, i = 1, 2, \dots, m, \\ \sum_{i = 1}^{m} x_{i j} \geq b_{j}, j = 1, 2, \dots, n, \\ x_{i j} \geq 0 \forall i, j, \end{array}

where $\tilde{\tilde{p}}$ , i = 1, 2, ..., m are fuzzy numbers and $C_{i j}^{1}$ , b_j ∈ R ∀i, j.

Model 2

Let b_j, j = 1, 2, ..., n are gamma distributed independent FRVS. Let the FRV b_j be denoted as $\tilde{\tilde{b_{j}}}$ . In this model the mathematical programming model for FSTP is presented as

min Z = \sum_{i = 1}^{m} \sum_{j = 1}^{n} C_{i j}^{1} x_{i j},

subject to

\begin{array}{l} \sum_{j = 1}^{n} x_{i j} \leq a_{i}, i = 1, 2, \dots, m, \\ \tilde{\tilde{P}} (\sum_{i = 1}^{m} x_{i j} \leq \tilde{\tilde{b_{j}}}) \geq \tilde{\tilde{q_{j}}}, j = 1, 2, \dots, n, \\ x_{i j} \geq 0 \forall i, j, \end{array}

where ${\tilde{\tilde{q}}}_{j}$ , j = 1, 2, ..., n are fuzzy numbers and $C_{i j}^{1}$ , b_j ∈ R ∀i, j.

Model 3

Let a_i (i = 1, 2, ...,m) and b_j (j = 1, 2, ..., n) are gamma distributed independent FRVS. Let the FRV a_i be denoted as $\tilde{\tilde{a_{i}}}$ , b_j be denoted as $\tilde{\tilde{b_{j}}}$ . In this model the mathematical programming model for FSTP is presented as

min Z = \sum_{i = 1}^{m} \sum_{j = 1}^{n} C_{i j}^{1} x_{i} j,

subject to

\begin{array}{l} \tilde{\tilde{P}} (\sum_{j = 1}^{n} x_{i j} \leq \tilde{\tilde{a_{i}}}) \geq \tilde{\tilde{p_{i}}}, i = 1, 2, \dots, m, \\ \tilde{\tilde{P}} (\sum_{i = 1}^{m} x_{i j} \geq \tilde{\tilde{b_{j}}}) \geq \tilde{\tilde{q_{j}}}, j = 1, 2, \dots, n, \\ x_{i j} \geq 0 \forall i, j, \end{array}

where $\tilde{\tilde{p_{i}}}$ , i = 1, 2, ..., m and $\tilde{\tilde{q_{j}}}$ , j = 1, 2, ..., n are fuzzy numbers and $C_{i j}^{1} \in R \forall i$ , j.

4. Transformation Techniques

Theorem 4.1

If $\tilde{\tilde{a_{i}}}$ , i = 1, 2, ..., m are gamma distributed independent fuzzy random variable, whose scale parameter $\tilde{\tilde{β_{i}}}$ is PNIT2FN, then

\tilde{\tilde{P}} (\sum_{j = 1}^{n} x_{i j} \leq \tilde{\tilde{a_{i}}}) ≽ \tilde{\tilde{p_{i}}}

is equivalent to

\begin{array}{l} \sum_{i = 1}^{n} x_{i j} \leq {\bar{β_{i}}}_{*} {γ_{α_{1}}}^{- 1} (1 - {\underline{p}}_{i}^{*}) Γ (α_{1_{1}}), \\ \sum_{i = 1}^{n} x_{i j} \leq {\underline{β_{i}}}_{*} {γ_{α_{1}}}^{- 1} (1 - {\bar{p_{i}}}^{*}) Γ (α_{1}), \end{array}

where x_ij is real number. $\tilde{\tilde{β_{i}}}$ are PNIT2FN.

Proof

It is assumed that $\tilde{\tilde{a_{i}}}$ , i = 1, 2, ..., m are gamma distributed independent fuzzy random variable whose mean, variance are α₁β, and α₁β², where the scale parameter $\tilde{\tilde{β_{i}}}$ is PNIT2FN.

So α-cuts of $\tilde{\tilde{a_{i}}}, \tilde{\tilde{p_{i}}}, \tilde{\tilde{β_{i}}}$ are presented as

α-cuts of $\tilde{\tilde{a_{i}}}$ is

\begin{array}{l} \tilde{\tilde{a_{i}}} [α] = [\underline{a_{i}} [α], \bar{a_{i}} [α]] \\ = [[{\underline{a}}_{i_{*}} [α], {\underline{a}}_{i}^{*} [α]], [{\bar{a}}_{i_{*}} [α], {\bar{b}}_{i}^{*} [α]]], \end{array}

where

\begin{array}{l} \underline{a_{i}} [α] = [{\underline{a}}_{i_{*}} [α], {\underline{a}}_{i}^{*} [α]], \\ \bar{a_{i}} [α] = [{\bar{a}}_{i_{*}} [α], {\bar{a}}_{i}^{*} [α]] . \end{array}

α-cuts of $\tilde{p_{i}}$ is

\begin{array}{l} \tilde{\tilde{p_{i}}} [α] = [\underline{p_{i}} [α], \bar{p_{i}} [α]] \\ = [[{\underline{p}}_{i_{*}} [α], {\underline{p}}_{i}^{*} [α]], [{\bar{p}}_{i_{*}} [α], {\bar{p}}_{i}^{*} [α]]], \end{array}

where

\begin{array}{l} \underline{p_{i}} [α] = [{\underline{p}}_{i_{*}} [α], {\underline{p}}_{i}^{*} [α]], \\ \bar{p_{i}} [α] = [{\bar{p}}_{i_{*}} [α], {\bar{p}}_{i}^{*} [α]] . \end{array}

α-cuts of β̃ is

\begin{array}{l} \tilde{\tilde{β_{i}}} [α] = [{\underline{β}}_{i} [α], {\bar{β}}_{i} [α]] \\ = [[{\underline{β}}_{i_{*}} [α], {\underline{β}}_{i}^{*} [α]], [{\bar{β}}_{i_{*}} [α], {\bar{β}}_{i}^{*} [α]]], \end{array}

where

(3)

\begin{array}{l} \underline{β} [α] = [{\underline{β}}_{i_{*}} [α], {\underline{β}}_{i}^{*} [α]], \\ \bar{β} [α] = [{\bar{β}}_{i_{*}} [α], {\bar{β}}_{i}^{*} [α]], \\ \tilde{P} (\sum_{j = 1}^{n} x_{i j} \leq \tilde{\tilde{a_{i}}}) [α] ≽ \tilde{\tilde{p_{i}}} . \end{array}

Now the α-cut of the constraint (3) is expressed as

\begin{array}{l} \tilde{\tilde{P}} (\sum_{j = 1}^{n} x_{i j} \leq \tilde{\tilde{a_{i}}}) [α] \\ = \tilde{\tilde{P}} (A_{i} \leq \tilde{\tilde{a_{i}}}) [α], where A_{i} = \sum_{j = 1}^{n} x_{i j} \\ = {P (A_{i} \leq a_{i}) ∣ a_{i} \in \tilde{\tilde{a_{i}}} [α], \tilde{\tilde{β_{i}}} [α] = [{\underline{β}}_{i} [α], {\bar{β}}_{i} [α]]} \\ = {1 - P (a_{i} \leq A_{i}) ∣ a_{i} \in \tilde{\tilde{a_{i}}} [α], \tilde{\tilde{β_{i}}} [α] = [{\underline{β}}_{i} [α], {\bar{β}}_{i} [α]]} \\ = {1 - \int_{0}^{A_{i}} \frac{1}{β^{α_{1}} Γ (α_{1})} {a_{i}}^{α_{1} - 1} e^{- a_{i} / β} d a_{i} ∣ a_{i} \in \tilde{\tilde{a_{i}}} [α], \tilde{\tilde{β_{i}}} [α] = [{\underline{β}}_{i} [α], {\bar{β}}_{i} [α]]} \\ where \frac{a_{i}}{β_{i}} = y \Rightarrow d a_{i} = β_{i} d y \\ = {1 - \int_{0}^{\sum_{j = 1}^{n} x_{i j} / β_{i}} \frac{1}{β_{i}^{α_{1_{1}}} Γ (α_{1_{1}})} {(β_{i} y)}^{α_{1} - 1} e^{- y} β_{i} d y ∣ a_{i} \in \tilde{\tilde{a_{i}}} [α], \tilde{\tilde{β_{i}}} [α] = [{\underline{β}}_{i} [α], {\bar{β}}_{i} [α_{1}]]} \\ = {1 - \frac{1}{Γ (α_{11})} \int_{0}^{\sum_{j = 1}^{n} x_{i j} / β_{i}} y^{α_{1_{11}} - 1} e^{- y} d y} \\ = {1 - \frac{1}{Γ (α_{1})} γ_{α_{1}} [\frac{\sum_{j = 1}^{n} x_{i j}}{β_{i}}] ∣ a_{i} \in \tilde{\tilde{a_{i}}} [α], \tilde{\tilde{β_{i}}} [α] = [{\underline{β}}_{i} [α], {\bar{β}}_{i} [α]]}, \end{array}

where

γ_{α_{1}} (x) = \int_{0}^{x} y^{α_{1} - 1} e^{- y} d y

is an lower incomplete gamma function, which is an increasing function, so lower minimum value of $1 - \frac{1}{Γ (α_{1})} γ_{α_{1}} [\frac{\sum_{j = 1}^{n} x_{i j}}{β_{i}}]$ is

\begin{array}{l} M i n {1 - \frac{1}{Γ (α_{1})} γ_{α_{1}} [\frac{\sum_{j = 1}^{n} x_{i j}}{β_{i}}] ∣ \tilde{\tilde{β_{i}}} [α] = [{\underline{β}}_{i} [α], {\bar{β}}_{i} [α]]} \\ = 1 - \frac{1}{Γ (α_{1})} γ_{α_{1}} [\frac{\sum_{j = 1}^{n} x_{i j}}{β_{i_{*}}}] . \end{array}

Lower maximum value of $1 - \frac{1}{Γ (α_{1})} γ_{α_{1}} [\frac{\sum_{j = 1}^{n} x_{i j}}{β_{i}}]$ is

\begin{array}{l} M a x {1 - \frac{1}{Γ (α_{1})} γ_{α_{1}} [\frac{\sum_{j = 1}^{n} x_{i j}}{β_{i}}] ∣ \tilde{\tilde{β_{i}}} [α] = [{\underline{β}}_{i} [α], {\bar{β}}_{i} [α]]} \\ = 1 - \frac{1}{Γ (α_{1})} γ_{α_{1}} [\frac{\sum_{j = 1}^{n} x_{i j}}{{\underline{β}}_{i_{*}}}] . \end{array}

Upper minimum value of $1 - \frac{1}{Γ (α_{1})} γ_{α_{1}} [\frac{\sum_{j = 1}^{n} x_{i j}}{β_{i}}]$ is

\begin{array}{l} M i n {1 - \frac{1}{Γ (α_{1})} γ_{α_{1}} [\frac{\sum_{j = 1}^{n} x_{i j}}{β_{i}}] ∣ \tilde{\tilde{β_{i}}} [α] = [{\underline{β}}_{i} [α], {\bar{β}}_{i} [α]]} \\ = 1 - \frac{1}{Γ (α_{1})} γ_{α_{1}} [\frac{\sum_{j = 1}^{n} x_{i j}}{{\underline{β}}_{i}^{*}}] . \end{array}

Upper maximum value of $1 - \frac{1}{Γ (α_{1})} γ_{α_{1}} [\frac{\sum_{j = 1}^{n} x_{i j}}{β_{i}}]$ is

\begin{array}{l} M a x {1 - \frac{1}{Γ (α_{1})} γ_{α_{1}} [\frac{\sum_{j = 1}^{n} x_{i j}}{β_{i}}] ∣ \tilde{\tilde{β_{i}}} [α] = [{\underline{β}}_{i} [α], {\bar{β}}_{i} [α]]} \\ = 1 - \frac{1}{Γ (α_{1})} γ_{α_{1}} [\frac{\sum_{j = 1}^{n} x_{i j}}{{\bar{β}}_{i}^{*}}] . \end{array}

(4)

\begin{array}{l} \tilde{\tilde{P}} (\sum_{j = 1}^{n} x_{i j} \leq \tilde{\tilde{a_{i}}}) [α] \\ = [1 - \frac{1}{Γ (α_{1})} γ_{α_{1}} [\frac{\sum_{j = 1}^{n} x_{i j}}{{\bar{β}}_{i_{*}}}], 1 - \frac{1}{Γ (α_{1})} γ_{α_{1}} [\frac{\sum_{j = 1}^{n} x_{i j}}{{\underline{β}}_{i_{*}}}], \\ 1 - \frac{1}{Γ (α_{1})} γ_{α_{1}} [\frac{\sum_{j = 1}^{n} x_{i j}}{{\underline{β}}_{i_{*}}}], 1 - \frac{1}{Γ (α_{1})} γ_{α_{1}} [\frac{\sum_{j = 1}^{n} x_{i j}}{{\bar{β}}_{i}^{*}}]] \\ \geq [{\bar{p}}_{i_{*}}, {\underline{p}}_{i_{*}}], [{\underline{p}}_{i}^{*} {\bar{p}}_{i}^{*}], \\ 1 - \frac{1}{Γ (α_{1})} γ_{α_{1}} [\frac{\sum_{j = 1}^{n} x_{i j}}{{\bar{β}}_{i_{*}}}], \\ 1 - \frac{1}{Γ (α_{1})} γ_{α_{1}} [\frac{\sum_{j = 1}^{n} x_{i j}}{{\underline{β}}_{i_{*}}}] \geq [{\underline{p}}_{i}^{*} {\bar{p}}_{i}^{*}], \\ 1 - \frac{1}{Γ (α_{1})} γ_{α_{1}} [\frac{\sum_{j = 1}^{n} x_{i j}}{{\bar{β}}_{i_{*}}}] \geq {\underline{p}}_{i}^{*}, \end{array}

(5)

1 - \frac{1}{Γ (α_{1})} γ_{α_{1}} [\frac{\sum_{j = 1}^{n} x_{i j}}{{\underline{β}}_{i_{*}}}] \geq {\bar{p}}_{i}^{*} .

Using fuzzy inequality,the α − cut of the fuzzy constraint is expressed as

\tilde{\tilde{P}} (\sum_{j = 1}^{n} x_{i j} \leq {\tilde{\tilde{a}}}_{i}) [α] ≽ \tilde{\tilde{p_{i}}} [α],

from (4), we have

\begin{array}{l} 1 - \frac{1}{Γ (α_{1})} γ_{α_{1_{1}}} [\frac{\sum_{i = 1}^{n} x_{i j}}{{\bar{β}}_{i_{*}}}] \geq {\underline{p}}_{i}^{*}, \\ \frac{1}{Γ (α_{1})} γ_{α_{1}} [\frac{\sum_{i = 1}^{n} x_{i j}}{{\bar{β}}_{i_{*}}}] \leq 1 - {\underline{p}}_{i}^{*}, \\ \frac{1}{Γ (α_{1})} (\frac{\sum_{i = 1}^{n} x_{i j}}{{\bar{β}}_{i_{*}}}) \leq {γ_{α_{1}}}^{- 1} (1 - {\underline{p}}_{i}^{*}), \\ (\frac{\sum_{i = 1}^{n} x_{i j}}{{\bar{β}}_{i_{*}}}) \leq {γ_{α_{1}}}^{- 1} (1 - {\underline{p}}_{i}^{*}) Γ (α_{1}), \\ \sum_{i = 1}^{n} x_{i j} \leq {\bar{β_{i}}}_{*} {γ_{α_{1}}}^{- 1} (1 - {\underline{p}}_{i}^{*}) Γ (α_{1}), \end{array}

and

\tilde{\tilde{P}} (\sum_{j = 1}^{n} x_{i j} \leq {\tilde{\tilde{a}}}_{i}) [α] ≽ \tilde{\tilde{p_{i}}} [α],

from (5), we have

\begin{array}{l} 1 - \frac{1}{Γ (α_{1})} γ_{α_{1}} [\frac{\sum_{i = 1}^{n} x_{i j}}{{\underline{β}}_{i_{*}}}] \geq {\bar{p_{i}}}^{*}, \\ \frac{1}{Γ (α_{1})} γ_{α_{1}} [\frac{\sum_{i = 1}^{n} x_{i j}}{{\underline{β}}_{i_{*}}}] \leq 1 - {\bar{p_{i}}}^{*}, \\ \frac{1}{Γ (α_{1})} (\frac{\sum_{i = 1}^{n} x_{i j}}{{\underline{β}}_{i_{*}}}) \leq {γ_{α_{1}}}^{- 1} (1 - {\bar{p_{i}}}^{*}), \\ (\frac{\sum_{i = 1}^{n} x_{i j}}{{\underline{β}}_{i_{*}}}) \leq {γ_{α_{1}}}^{- 1} (1 - {\underline{p}}_{i}^{*}) Γ (α_{1}), \\ \sum_{i = 1}^{n} x_{i j} \leq {\underline{β_{i}}}_{*} {γ_{α_{1}}}^{- 1} (1 - {\bar{p_{i}}}^{*}) Γ (α_{1}), \end{array}

from the result of (4) and (5), we have

\begin{array}{l} \sum_{i = 1}^{n} x_{i j} \leq {\bar{β_{i}}}_{*} {γ_{α_{1}}}^{- 1} (1 - {\underline{p}}_{i}^{*}) Γ (α_{1}), \\ \sum_{i = 1}^{n} x_{i j} \leq {\underline{β_{i}}}_{*} {γ_{α_{1}}}^{- 1} (1 - {\bar{p_{i}}}^{*}) Γ (α_{1}) . \end{array}

This proves the theorem.

Now using Theorem 4.1, the deterministic equivalent of mathematical Model 1 becomes DModel 1

M n i Z = (\sum_{j = 1}^{n} x_{i j} \leq \tilde{a_{i}}) ≽ \tilde{p_{i}},

subject to

\begin{array}{l} \sum_{i = 1}^{n} x_{i j} \leq {\bar{β_{i}}}_{*} {γ_{α_{1}}}^{- 1} (1 - {\underline{p}}_{i}^{*}) Γ (α_{1}), \\ \sum_{i = 1}^{n} x_{i j} \leq {\underline{β_{i}}}_{*} {γ_{α_{1}}}^{- 1} (1 - {\bar{p_{i}}}^{*}) Γ (α_{1}), \end{array}

where x_ij is real number. $\tilde{\tilde{a_{i}}}$ are FRVs $\tilde{\tilde{β_{i}}}$ are PNIT2FNs.

Theorem 4.2

If $\tilde{\tilde{b_{j}}}$ , j = 1, 2, ..., n are gamma distributed independent fuzzy random variable, whose scale parameter $\tilde{\tilde{β_{i}}}$ is PNIT2FN, then

\tilde{\tilde{P}} (\sum_{i = 1}^{m} x_{i j} \geq \tilde{\tilde{b_{j}}}) ≽ \tilde{\tilde{p_{i}}}

is equivalent to

\begin{array}{l} \sum_{i = 1}^{n} x_{i j} \leq {\underline{β_{i}}}_{*} {γ_{α_{1}}}^{- 1} ({\underline{p}}_{i}^{*} Γ (α_{1})), \\ \sum_{i = 1}^{n} x_{i j} \leq {\bar{β_{i}}}_{*} {γ_{α_{1}}}^{- 1} ({\bar{p_{i}}}^{*} Γ (α_{1})), \end{array}

where x_ij is real number. $\tilde{\tilde{b_{j}}}$ are FRVs. $\tilde{\tilde{β_{i}}}$ are PNIT2FNs.

Proof

It is assumed that $\tilde{\tilde{b_{j}}}$ , j = 1, 2, ..., n are gamma distributed independent fuzzy random variable whose mean, variance are α₁β, and α₁β², where the scale parameter $\tilde{\tilde{β_{i}}}$

is PNIT2FN.

So α-cuts of $\tilde{\tilde{b_{i}}}, \tilde{\tilde{p_{i}}}, \tilde{\tilde{β_{i}}}$ are presented as

α-cuts of $\tilde{\tilde{b_{i}}}$ is

\begin{array}{l} {\tilde{\tilde{b}}}_{i} [α] = [\underline{b_{i}} [α], \bar{b_{i}} [α]] \\ = [[{\underline{b}}_{i_{*}} [α], {\underline{b}}_{i}^{*} [α]], [{\bar{b}}_{i_{*}} [α], {\bar{b}}_{i}^{*} [α]]], \end{array}

where

\begin{array}{l} \underline{b_{i}} [α] = [{\underline{b}}_{i_{*}} [α], {\underline{b}}_{i}^{*} [α]], \\ \bar{b_{i}} [α] = [{\bar{b}}_{i_{*}} [α], {\bar{b}}_{i}^{*} [α]] . \end{array}

α-cuts of $\tilde{\tilde{p_{i}}}$ is

\begin{array}{l} {\tilde{\tilde{p}}}_{i} [α] = [\underline{p_{i}} [α], \bar{p_{i}} [α]] \\ = [[{\underline{p}}_{i_{*}} [α], {\underline{p}}_{i}^{*} [α]], [{\bar{p}}_{i_{*}} [α], {\bar{p}}_{i}^{*} [α]]], \end{array}

where

\begin{array}{l} \underline{p_{i}} [α] = [{\underline{p}}_{i_{*}} [α], {\underline{p}}_{i}^{*} [α]], \\ \bar{p_{i}} [α] = [{\bar{p}}_{i_{*}} [α], {\bar{p}}_{i}^{*} [α]] . \end{array}

α-cuts of β̃ is

\begin{array}{l} \tilde{\tilde{β}} [α] = [\underline{β} [α], \bar{β} [α]] \\ = [{\underline{β}}_{*} [α], {\underline{β}}^{*} [α]], [{\bar{β}}_{*} [α], {\bar{β}}^{*} [α]], \end{array}

where

(6)

\begin{array}{l} \underline{β} [α] = [{\underline{β}}_{*} [α], {\underline{β}}^{*} [α]], \\ \bar{β} [α] = [{\bar{β}}_{*} [α], {\bar{β}}^{*} [α]], \\ \tilde{\tilde{P}} (\sum_{j = 1}^{n} x_{i j} \leq {\tilde{\tilde{b}}}_{j}) [α] ≽ {\tilde{\tilde{p}}}_{i} . \end{array}

Now the α-cut of the constraint (6) is expressed as

\begin{array}{l} \tilde{\tilde{P}} (\sum_{j = 1}^{n} x_{i j} \geq {\tilde{\tilde{b}}}_{j}) [α] \\ = \tilde{\tilde{P}} (A_{i} \geq {\tilde{\tilde{b}}}_{j}) [α], where A_{i} = \sum_{j = 1}^{n} x_{i j} \\ = {P (A_{i} \geq b_{j}) ∣ b_{j} \in \tilde{\tilde{b_{j}}} [α], \tilde{\tilde{β_{i}}} [α] = [{\underline{β}}_{i} [α], {\bar{β}}_{i} [α]]} \\ = {P (b_{j} \leq A_{i}) ∣ b_{j} \in \tilde{\tilde{b_{j}}} [α], \tilde{\tilde{β_{i}}} [α] = [{\underline{β}}_{i} [α], {\bar{β}}_{i} [α]]} \\ = {\int_{0}^{A_{i}} \frac{1}{β_{i}^{α_{1}} Γ (α_{1})} {a_{i}}^{α_{1} - 1} e^{- a_{i} / β_{i}} d a_{i} ∣ b_{j} \in \tilde{\tilde{b_{j}}} [α], \tilde{\tilde{β_{i}}} [α] = [{\underline{β}}_{i} [α], {\bar{β}}_{i} [α]]} \\ where \frac{a_{i}}{β_{i}} = y \Rightarrow d a_{i} = β_{i} d y \\ = {\int_{0}^{\sum_{j = 1}^{n} x_{i j} / β_{i}} \frac{1}{β_{i}^{α_{1}} Γ (α_{1_{1}})} {(β_{i} y)}^{α_{1} - 1} e^{- y} β_{i} d y ∣ b_{j} \in \tilde{\tilde{b_{j}}} [α], \tilde{\tilde{β_{i}}} [α] = [{\underline{β}}_{i} [α], {\bar{β}}_{i} [α]]} \\ = {\frac{1}{Γ (α_{1})} \int_{0}^{\sum_{j = 1}^{n} x_{i j} / β_{i}} y^{α_{1} - 1} e^{- y} d y ∣ b_{j} \in \tilde{\tilde{b_{j}}} [α], \tilde{\tilde{β_{i}}} [α] = [{\underline{β}}_{i} [α], {\bar{β}}_{i} [α]]} \\ = {\frac{1}{Γ (α_{1})} γ_{α_{1}} [\frac{\sum_{j = 1}^{n} x_{i j}}{β_{i}}] ∣ b_{j} \in \tilde{\tilde{b_{j}}} [α], \tilde{\tilde{β_{i}}} [α] = [{\underline{β}}_{i} [α], {\bar{β}}_{i} [α]]}, \end{array}

where

γ_{α_{1}} (x) = \int_{0}^{x} y^{α_{1} - 1} e^{- y} d y

is an lower incomplete gamma function. which is an increasing function, so lower minimum value of $\frac{1}{Γ (α_{1})} γ_{α_{1}} [\frac{\sum_{j = 1}^{n} x_{i j}}{β_{i}}]$ is

\begin{array}{l} M i n {\frac{1}{Γ (α_{1})} γ_{α_{1}} [\frac{\sum_{j = 1}^{n} x_{i j}}{β_{i}}] ∣ {\tilde{\tilde{β}}}_{i} [α] = [{\underline{β}}_{i} [α], {\bar{β}}_{i} [α]]} \\ = \frac{1}{Γ (α_{1})} γ_{α_{1}} [\frac{\sum_{j = 1}^{n} x_{i j}}{{\bar{β}}_{i}^{*}}] . \end{array}

Lower maximum value of $\frac{1}{Γ (α_{1})} γ_{α_{1}} [\frac{\sum_{j = 1}^{n} x_{i j}}{β_{i}}]$ is

\begin{array}{l} M a x {\frac{1}{Γ (α_{1})} γ_{α_{1}} [\frac{\sum_{j = 1}^{n} x_{i j}}{β_{i}}] ∣ {\tilde{\tilde{β}}}_{i} [α] = [{\underline{β}}_{i} [α], {\bar{β}}_{i} [α]]} \\ = \frac{1}{Γ (α_{1})} γ_{α_{1}} [\frac{\sum_{j = 1}^{n} x_{i j}}{{\underline{β}}_{i}^{*}}] . \end{array}

Upper minimum value of $\frac{1}{Γ (α_{1})} γ_{α_{1}} [\frac{\sum_{j = 1}^{n} x_{i j}}{β_{i}}]$ is

\begin{array}{l} M i n {1 - \frac{1}{Γ (α_{1})} γ_{α_{1}} [\frac{\sum_{j = 1}^{n} x_{i j}}{β}] ∣ {\tilde{\tilde{β}}}_{i} [α] = [{\underline{β}}_{i} [α], {\bar{β}}_{i} [α]]} \\ = \frac{1}{Γ (α_{1})} γ_{α_{1}} [\frac{\sum_{j = 1}^{n} x_{i j}}{{\underline{β}}_{i_{*}}}] . \end{array}

Upper maximum value of $1 - \frac{1}{Γ (α_{1})} γ_{α_{1}} [\frac{\sum_{j = 1}^{n} x_{i j}}{β_{i}}]$ is

\begin{array}{l} M a x {1 - \frac{1}{Γ (α_{1})} γ_{α_{1}} [\frac{\sum_{j = 1}^{n} x_{i j}}{β_{i}}] ∣ {\tilde{\tilde{β}}}_{i} [α] = [{\underline{β}}_{i} [α], {\bar{β}}_{i} [α]]} \\ = \frac{1}{Γ (α_{1})} γ_{α_{1}} [\frac{\sum_{j = 1}^{n} x_{i j}}{{\bar{β}}_{i_{*}}}] . \end{array}

\begin{array}{l} \tilde{\tilde{P}} (\sum_{j = 1}^{n} x_{i j} \leq b_{j}) [α_{1}] \\ = [\frac{1}{Γ (α_{1})} γ_{α_{1}} [\frac{\sum_{j = 1}^{n} x_{i j}}{{\bar{β}}_{i}^{*}}], \frac{1}{Γ (α_{1})} γ_{α_{1}} [\frac{\sum_{j = 1}^{n} x_{i j}}{{\underline{β}}_{i}^{*}}], \\ \frac{1}{Γ (α_{1})} γ_{α_{1}} [\frac{\sum_{j = 1}^{n} x_{i j}}{{\underline{β}}_{i_{*}}}], \frac{1}{Γ (α_{1})} γ_{α_{1}} [\frac{\sum_{j = 1}^{n} x_{i j}}{{\bar{β}}_{i_{*}}}]] \\ \geq [{\bar{p}}_{i_{*}}, {\underline{p}}_{i_{*}}], [{\underline{p}}_{i}^{*} {\bar{p}}_{i}^{*}], \\ \frac{1}{Γ (α_{1})} γ_{α_{1}} [\frac{\sum_{j = 1}^{n} x_{i j}}{{\bar{β}}_{i}^{*}}], \\ \frac{1}{Γ (α_{1})} γ_{α_{1}} [\frac{\sum_{j = 1}^{n} x_{i j}}{{\underline{β}}_{i}^{*}}] \geq [{\underline{p}}_{i}^{*}, {\bar{p}}_{i}^{*}], \\ \frac{1}{Γ (α_{1})} γ_{α_{1}} [\frac{\sum_{j = 1}^{n} x_{i j}}{{\bar{β}}_{i}^{*}}] \geq {\underline{p}}_{i}^{*}, \\ \frac{1}{Γ (α_{1})} γ_{α_{1}} [\frac{\sum_{j = 1}^{n} x_{i j}}{{\underline{β}}_{i}^{*}}] \geq {\bar{p}}_{i}^{*}, \end{array}

from (4), we have

\begin{array}{l} \frac{1}{Γ (α_{1})} γ_{α_{1}} [\frac{\sum_{j = 1}^{n} x_{i j}}{{\bar{β}}_{i}^{*}}] \geq {\underline{p}}_{i}^{*}, \\ \frac{1}{Γ (α_{1})} (\frac{\sum_{i = 1}^{n} x_{i j}}{{\bar{β}}_{i}^{*}}) \geq γ_{α_{1}}^{- 1} ({\underline{p}}_{i}^{*}), \\ (\frac{\sum_{i = 1}^{n} x_{i j}}{{\bar{β}}_{i}^{*}}) \leq γ_{α_{1}}^{- 1} ({\underline{p}}_{i}^{*}) Γ (α_{1}), \\ \sum_{i = 1}^{n} x_{i j} \leq {\bar{β}}_{i}^{*} γ_{α_{1}}^{- 1} ({\underline{p}}_{i}^{*}) Γ (α_{1}), \end{array}

and from (5), we have

\begin{array}{l} \frac{1}{Γ (α_{1})} γ_{α_{1}} (\frac{\sum_{i = 1}^{n} x_{i j}}{{\underline{β}}_{i}^{*}}) \geq {\bar{p_{i}}}^{*}, \\ \frac{1}{Γ (α_{1})} γ_{α_{1}} (\frac{\sum_{i = 1}^{n} x_{i j}}{{\underline{β}}^{*}}) \leq {\bar{p_{i}}}^{*}, \\ \frac{1}{Γ (α_{1})} (\frac{\sum_{i = 1}^{n} x_{i j}}{{\underline{β}}_{i}^{*}}) \leq γ_{α_{1}}^{- 1} ({\bar{p_{i}}}^{*}) \\ (\frac{\sum_{i = 1}^{n} x_{i j}}{{\underline{β}}_{i}^{*}}) \leq {γ_{α_{1}}}^{- 1} (({\bar{p}}_{i}^{*}) Γ (α_{1})), \\ \sum_{i = 1}^{n} x_{i j} \leq {\underline{β}}_{i}^{*} {γ_{α_{1}}}^{- 1} (({\bar{p_{i}}}^{*}) Γ (α_{1})) . \end{array}

Now from the result of (4) and (5), we have

\begin{array}{l} \sum_{i = 1}^{n} x_{i j} \leq {\bar{β}}_{i}^{*} γ_{α_{1}}^{- 1} ({\underline{p}}_{i}^{*}) Γ (α_{1}), \\ \sum_{i = 1}^{n} x_{i j} \leq {\underline{β}}_{i}^{*} γ_{α_{1}}^{- 1} ({\bar{p_{i}}}^{*}) Γ (α_{1}) . \end{array}

This proves the theorem.

Now using Theorem (4.2), the deterministic equivalent of mathematical Model 2 becomes DModel 2

M i n Z = (\sum_{i = 1}^{m} x_{i j} \geq {\tilde{\tilde{b}}}_{j}) ⊳ {\tilde{\tilde{p}}}_{i},

subject to

\begin{array}{l} \sum_{i = 1}^{n} x_{i j} \leq {\bar{β}}_{i}^{*} γ_{α_{1}}^{- 1} ({\underline{p}}_{i}^{*}) Γ (α_{1}), \\ \sum_{i = 1}^{n} x_{i j} \leq {\underline{β}}_{i}^{*} γ_{α_{1}}^{- 1} ({\bar{p_{i}}}^{*}) Γ (α_{1}), \end{array}

where x_ij is real number. ${\tilde{\tilde{b}}}_{j}$ are FRVs, ${\tilde{\tilde{β}}}_{i}$ is PNIT2FN.

5. Case Study

The unit transportation cost of one unit (in 500 packets) through a particular route during transportation are given in the Table 1. Now, the transportation problem is represented by

(7)

min Z = 18 x_{11} + 25 x_{12} + 13 x_{13} + 15 x_{21} + 12 x_{22} + 13 x_{23} + 16 x_{31} + 12 x_{32} + 11 x_{33},

subject to

\begin{array}{l} \tilde{\tilde{p}} (\sum_{j = 1}^{3} x_{1 j} \leq {\bar{\tilde{\tilde{a}}}}_{1}) ⊳ {\bar{\tilde{\tilde{p}}}}_{1}, \\ \tilde{\tilde{p}} (\sum_{j = 1}^{3} x_{1 j} \leq {\underline{\tilde{\tilde{a}}}}_{1}) ⊳ {\underline{\tilde{\tilde{p}}}}_{1}, \\ \tilde{\tilde{p}} (\sum_{j = 1}^{3} x_{2 j} \leq {\bar{\tilde{\tilde{a}}}}_{2}) ⊳ {\bar{\tilde{\tilde{p}}}}_{2}, \\ \tilde{\tilde{p}} (\sum_{j = 1}^{3} x_{2 j} \leq {\underline{\tilde{\tilde{a}}}}_{2}) ⊳ {\underline{\tilde{\tilde{p}}}}_{2}, \\ \tilde{\tilde{p}} (\sum_{j = 1}^{3} x_{3 j} \leq {\bar{\tilde{\tilde{a}}}}_{3}) ⊳ {\bar{\tilde{\tilde{p}}}}_{3}, \\ \tilde{\tilde{p}} (\sum_{j = 1}^{3} x_{1 j} \leq {\underline{\tilde{\tilde{a}}}}_{3}) ⊳ {\underline{\tilde{\tilde{p}}}}_{3}, \\ \sum_{i = 1}^{3} x_{i 1} \geq b_{1}, \\ \sum_{i = 1}^{3} x_{i 2} \geq b_{2}, \\ \sum_{i = 1}^{3} x_{i 3} \geq b_{3}, \\ x_{i j} \geq 0, i = 1, 2, 3 and j = 1, 2, 3, \end{array}

where ${\tilde{\tilde{p}}}_{i} (i = 1, 2, 3)$ are fuzzy numbers. ${\tilde{\tilde{a}}}_{i} (i = 1, 2, 3)$ (i = 1, 2, 3) are gamma distributed fuzzy random variables. Here the scale parameter ${\tilde{\tilde{β}}}_{a_{i}}$ are perfectly normal interval type 2 trapezoidal fuzzy numbers, whose lower membership function ( ${\underline{\tilde{\tilde{β}}}}_{a_{1}}$ ) and upper membership function ( ${\bar{\tilde{\tilde{β}}}}_{a_{1}}$ ) are trapezoidal fuzzy numbers. Here the shape parameter α= 0.8 for all ${\tilde{\tilde{a}}}_{i}$ .

They are represented by

\begin{array}{l} {\bar{\tilde{\tilde{β}}}}_{a_{1}} = upper membership function of {\tilde{\tilde{β}}}_{a_{1}} \\ = 〈 52, 54, 57, 59 〉, \\ {\underline{\tilde{\tilde{β}}}}_{a_{1}} = lower membership function of {\tilde{\tilde{β}}}_{a_{1}} \\ = 〈 53, 55, 56, 58 〉, \\ {\bar{\tilde{\tilde{β}}}}_{a_{2}} = upper membership function of {\tilde{\tilde{β}}}_{a_{2}} \\ = 〈 62, 64, 67, 69 〉, \\ {\underline{\tilde{\tilde{β}}}}_{a_{2}} = lower membership function of {\tilde{\tilde{β}}}_{a_{2}} \\ = 〈 63, 65, 66, 68 〉, \\ {\bar{\tilde{\tilde{β}}}}_{a_{3}} = upper membership function of {\tilde{\tilde{β}}}_{a_{3}} \\ = 〈 72, 74, 77, 79 〉, \\ {\underline{\tilde{\tilde{β}}}}_{a_{3}} = lower membership function of {\tilde{\tilde{β}}}_{a_{3}} \\ = 〈 73, 75, 76, 78 〉 . \end{array}

Specified probability levels:

\begin{array}{l} \bar{\tilde{\tilde{p_{1}}}} = \bar{\tilde{\tilde{0.04}}} = upper membership function of \tilde{\tilde{0.04}} \\ = 〈 0.02, 0.04, 0.07, 0.09 〉, \\ \underline{\tilde{\tilde{p_{1}}}} = \underline{\tilde{\tilde{0.04}}} = lower membership function of \tilde{\tilde{0.04}} \\ = 〈 0.03, 0.05, 0.06, 0.08 〉, \\ \bar{\tilde{\tilde{p_{2}}}} = \bar{\tilde{\tilde{0.06}}} = upper membership function of \tilde{\tilde{0.06}} \\ = 〈 0.01, 0.03, 0.06, 0.08 〉, \\ \underline{\tilde{\tilde{p_{2}}}} = \underline{\tilde{\tilde{0.06}}} = lower membership function of \tilde{\tilde{0.06}} \\ = 〈 0.02, 0.04, 0.05, 0.07 〉, \\ \bar{\tilde{\tilde{p_{3}}}} = \bar{\tilde{\tilde{0.08}}} = upper membership function of \tilde{\tilde{0.08}} \\ = 〈 0.02, 0, 04, 0.07, 0.09 〉, \\ \underline{\tilde{\tilde{p_{3}}}} = \underline{\tilde{\tilde{0.08}}} = lower membership function of \tilde{\tilde{0.08}} \\ = 〈 0.03, 0.05, 0.06, 0.08 〉 . \end{array}

The α-cuts are

\begin{array}{l} {\bar{\tilde{\tilde{β}}}}_{a_{1}} [α] = [52 + 2 α, 59 - 2 α], \\ {\underline{\tilde{\tilde{β}}}}_{a_{1}} [α] = [53 + 2 α, 58 - 2 α], \\ {\bar{\tilde{\tilde{β}}}}_{a_{2}} [α] = [62 + 2 α, 69 - 2 α], \\ {\underline{\tilde{\tilde{β}}}}_{a_{2}} [α] = [63 + 2 α, 68 - 2 α], \\ {\bar{\tilde{\tilde{β}}}}_{a_{3}} [α] = [72 + 2 α, 79 - 2 α], \\ {\underline{\tilde{\tilde{β}}}}_{a_{3}} [α] = [73 + 2 α, 78 - 2 α], \\ \bar{\tilde{\tilde{0.04}}} [α] = [0.02 + 0.02 α, 0.09 - 0.02 α], \\ \underline{\tilde{\tilde{0.04}}} [α] = [0.03 + 0.02 α, 0.08 - 0.02 α], \\ \bar{\tilde{\tilde{0.06}}} [α] = [0.01 + 0.02 α, 0.08 - 0.02 α], \\ \underline{\tilde{\tilde{0.06}}} [α] = [0.02 + 0.02 α, 0.07 - 0.02 α], \\ \bar{\tilde{\tilde{0.08}}} [α] = [0.02 + 0.02 α, 0.09 - 0.02 α], \\ \underline{\tilde{\tilde{0.08}}} [α] = [0.03 + 0.02 α, 0.08 - 0.02 α] . \end{array}

Now using Theorem (4.1) the deterministic equivalent model of (7) becomes

\begin{array}{l} Minimize 18 x_{11} + 25 x_{12} + 13 x_{13} + 15 x_{21} + 12 x_{22} + 13 x_{23} + 16 x_{31} + 12 x_{32} + 11 x_{33}, \\ \sum_{j = 1}^{3} x_{1 j} \leq γ_{α_{1}}^{- 1} ((0.92 + 0.02 α) Γ (α_{1})) (52 + 2 α), \\ \sum_{j = 1}^{3} x_{1 j} \leq γ_{α_{1}}^{- 1} ((0.91 + 0.02 α) Γ (α_{1})) (53 + 2 α), \\ \sum_{j = 1}^{3} x_{2 j} \leq {γ_{α}}^{- 1} ((0.93 + 0.02 α) Γ (α_{1})) (62 + 2 α), \\ \sum_{j = 1}^{3} x_{2 j} \leq γ_{α_{1}}^{- 1} ((0.92 + 0.02 α) Γ (α_{1})) (63 + 2 α), \\ \sum_{j = 1}^{3} x_{3 j} \leq γ_{α_{1}}^{- 1} ((0.92 + 0.02 α) Γ (α_{1})) (72 + 2 α), \\ \sum_{j = 1}^{3} x_{3 j} \leq γ_{α_{1}}^{- 1} (0.91 + 0.02 α) Γ (α_{1})) (73 + 2 α), \\ \sum_{i = 1}^{3} x_{i 1} \geq 12, \\ \sum_{i = 1}^{3} x_{i 2} \geq 13, \\ \sum_{i = 1}^{3} x_{i 3} \geq 16, \\ x_{11} \geq 0, x_{12} \geq 0, x_{13} \geq 0, x_{21} \geq 0, x_{22} \geq 0, \\ x_{23} \geq 0, x_{31} \geq 0, x_{32} \geq 0, x_{33} \geq 0. \end{array}

Using R software and for α = 0.5, the deterministic equivalent becomes

Minimize 18 x_{11} + 25 x_{12} + 13 x_{13} + 15 x_{21} + 12 x_{22} + 13 x_{23} + 16 x_{31} + 12 x_{32} + 11 x_{33},

Subject to

\begin{array}{l} \sum_{j = 1}^{3} x_{1 j} \leq 13.9303398, \\ \sum_{j = 1}^{3} x_{1 j} \leq 18.1516419, \\ \sum_{j = 1}^{3} x_{2 j} \leq 14.9077023, \\ \sum_{j = 1}^{3} x_{2 j} \leq 19.5031491, \\ \sum_{j = 1}^{3} x_{3 j} \leq 14.3020416, \\ \sum_{j = 1}^{3} x_{3 j} \leq 21.8529261, \\ \sum_{i = 1}^{3} x_{1 j} \geq 12, \\ \sum_{i = 1}^{3} x_{2 j} \geq 13, \\ \sum_{i = 1}^{3} x_{3 j} \geq 16, \\ x_{11} \geq 0, x_{12} \geq 0, x_{13} \geq 0, x_{21} \geq 0, x_{22} \geq 0, \\ x_{23} \geq 0, x_{31} \geq 0, x_{32} \geq 0, x_{33} \geq 0. \end{array}

Using LINGO software, the solution of the above problem

6. Result and Discussion

7. Conclusion

In this paper, a fuzzy STP has been proposed. Where gamma distributed FRV is present on the right-hand side of the constraint. The scale parameter $\tilde{\tilde{β}}$ is a PNIT2FN. Here the fuzzy stochastic models have been converted into equivalent deterministic models in two steps. In the first step using the concept of α-cut the fuzziness has been removed and in the second step using the chance constraint method randomness has been removed. Here we have considered only the first model and its equivalent deterministic models are derived. Also, a case study is solved according to the first model. Here the mathematical model is solved by R and Lingo software. Similarly, other cases can be established, that is where demand parameters or both supply, and demand parameters are FRVs. Since the whole paper has been done with a single objective stochastic transportation problem, also we can solve multi-objective, multi-choice STP under a type-2 fuzzy environment considering other probability distributions. This work can also be applied to other mathematical programming problems namely: assignment problems, quadratic programming problems, bi-level programming problem, two-stage programming prolem. In future we will work on fractional transportation problem under interval type-2 fuzzy environment.

Abstract
1. Introduction
2. Basic Preliminary
3. Mathematical Models
4. Transformation Techniques
5. Case Study
6. Result and Discussion
7. Conclusion

Table 1 . Transportation cost per unit (in rupees).

Source destination	Jamsedpur	Kolkata	Hyderabad
Khurda	18	25	13
Rourkela	15	12	13
Jharsuguda	16	12	11

Table 2 . Solution for different values of α.

α	0.5	0.6	0.7
x₁₁	0.00	0.00	0.00
x₁₂	0.00	0.00	0.00
x₁₃	11.79026	5.816579	0.00
x₂₁	12.00	12.00	12.00
x₂₂	2.907702	5.702769	8.827397
x₂₃	0.00	0.00	0.00
x₃₁	0.00	0.00	0.00
x₃₂	10.09230	7.297231	4.172603
x₃₃	4.209744	10.18342	16.0000
Z	535.5805	523.6332	512.0000

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SCOPUS 2

International Journalof Fuzzy Logic andIntelligent Systems

Original Article

Type-2 Fuzzy Stochastic Transportation Problem with Gamma Distribution

Abstract

1. Introduction

2. Basic Preliminary

Definition 2.1 (Crisp set, [22])

Definition 2.2 (Fuzzy set, [23])

Definition 2.3 (Fuzzy number, [24])

Definition 2.4 (α-cut set, [25])

Definition 2.5 (Convex fuzzy set, [26])

Definition 2.6 (Type-2 fuzzy set, [10])

Definition 2.7 (Interval type-2 fuzzy set, [27])

Definition 2.8 (Normal IT2FS and perfectly normal IT2FS, [27])

Definition 2.9 (Type-2 fuzzy number, [27])

Definition 2.10 (Primary α-cut of perfectly normal IT2FS, [11])

Definition 2.11 (Totally symmetric type-2 trapezoidal fuzzy number, [27])

Definition 2.12 (Footprint of uncertainty, [11])

Definition 2.13 (Perfectly normal interval type 2 trapezoidal fuzzy number, [11])

Definition 2.14 (α-cut of perfectly normal,[11])

Definition 2.15 (Stochastic transportation problem, [28])

Definition 2.16 (Incomplete gamma function, Borwein(2009)

Definition 2.17 (Fuzzy gamma distribution, [25])

3. Mathematical Models

Model 1

Model 2

Model 3

4. Transformation Techniques

Theorem 4.1

Theorem 4.2

5. Case Study

6. Result and Discussion

7. Conclusion

Conflicts of Interest

Tables

References

Biographies

Article

Original Article

Type-2 Fuzzy Stochastic Transportation Problem with Gamma Distribution

Abstract

1. Introduction

2. Basic Preliminary

Definition 2.1 (Crisp set, [22])

Definition 2.2 (Fuzzy set, [23])

Definition 2.3 (Fuzzy number, [24])

Definition 2.4 (α-cut set, [25])

Definition 2.5 (Convex fuzzy set, [26])

Definition 2.6 (Type-2 fuzzy set, [10])

Definition 2.7 (Interval type-2 fuzzy set, [27])

Definition 2.8 (Normal IT2FS and perfectly normal IT2FS, [27])

Definition 2.9 (Type-2 fuzzy number, [27])

Definition 2.10 (Primary α-cut of perfectly normal IT2FS, [11])

Definition 2.11 (Totally symmetric type-2 trapezoidal fuzzy number, [27])

Definition 2.12 (Footprint of uncertainty, [11])

Definition 2.13 (Perfectly normal interval type 2 trapezoidal fuzzy number, [11])

Definition 2.14 (α-cut of perfectly normal,[11])

Definition 2.15 (Stochastic transportation problem, [28])

Definition 2.16 (Incomplete gamma function, Borwein(2009)

Definition 2.17 (Fuzzy gamma distribution, [25])

3. Mathematical Models

Model 1

Model 2

Model 3

4. Transformation Techniques

Theorem 4.1

Theorem 4.2

5. Case Study

6. Result and Discussion

7. Conclusion

References

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