Startseite A New Method for Solving Single- and Multi-Objective Capacitated Solid Minimum Cost Flow Problems under Uncertainty
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A New Method for Solving Single- and Multi-Objective Capacitated Solid Minimum Cost Flow Problems under Uncertainty

  • Manjot Kaur und Rehan Sadiq EMAIL logo
Veröffentlicht/Copyright: 12. Dezember 2015
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Journal of Intelligent Systems
Aus der Zeitschrift Journal of Intelligent Systems Band 25 Heft 2

Abstract

In real life, a person may assume that an object belongs to a set, but it is possible that he (she) is not sure about it. In other words, there may be hesitation or confusion whether an object belongs to a set or not. In fuzzy set theory, there is no means to incorporate such type of hesitation or confusion. A possible solution is to use intuitionistic fuzzy set [K. T. Atanassov, Intutionistic fuzzy sets, Fuzzy Sets Syst.20 (1986), 87–96]. In this article, the concept of unbalanced fully fuzzy multi-objective capacitated solid minimum cost flow (SMCF) problems is generalized by unbalanced intuitionistic fully fuzzy multi-objective capacitated SMCF (CSMCF) problems and new methods are proposed for solving these problems. The main advantage of the proposed methods over the existing methods is that all the unbalanced fully fuzzy single- and multi-objective CSMCF problems that can be solved by the existing methods can also be solved by the proposed method.

Keywords: Multi-objective linear programming; intuitionistic fuzzy numbers; solid minimum cost flow

1 Introduction

Minimum cost flow (MCF) problem is an important problem in combinatorial optimization and network flows. Its applications can be found in practical problems such as transportation, communication, urban design, and job scheduling models [1, 3].

The aim of the classical MCF problem is to minimize the cost of transporting some products that are available at some sources and required at some destinations. In many actual problems, the cost, capacities, supplies, and demand parameters may be imprecise. In the literature, to deal quantitatively with imprecise information, the concept of classical MCF problems is extended by fuzzy MCF problems.

Ghatee and Hashemi [5, 6] proposed a method to find the fuzzy optimal solution of balanced fully fuzzy MCF (FFMCF) problems by converting its fuzzy linear programming formulation into an equivalent crisp linear programming formulation. Kaur and Kumar [11] pointed out that it is not genuine to use the Hukuhara’s difference in the fuzzy linear programming formulation of FFMCF problems. Also, Kaur and Kumar [11] pointed out that in the existing methods [7, 8], unbalanced FFMCF real-life problems are solved using the existing methods [5, 6], which is applicable only for finding the fuzzy optimal solution of balanced FFMCF problems. To overcome these shortcomings of the existing methods [5–8], Kaur and Kumar [11] proposed a new formulation without using Hukuhara’s difference by modifying the existing formulation [5, 6] and proposed a new method for finding the fuzzy optimal solution of unbalanced FFMCF problems.

Kaur and Kumar [14] pointed out that the existing method [9] can be used only to solve such multi-objective MCF problems in which only the cost coefficients are represented by fuzzy numbers. To overcome this limitation of the existing method [9], Kaur and Kumar [14] proposed a method for solving such multi-objective MCF problems in which all the parameters are represented by fuzzy numbers, which is the generalization of the existing method [11].

In crisp MCF problems as well as in fuzzy MCF problems, it is assumed that only the same type of conveyances (e.g. trucks, trains, or ships) are used for transporting the product. Kaur and Kumar [12] pointed out that in real-life problems, there is need to use more than one type of conveyances (e.g. trucks and trains simultaneously) for transporting the product and extended the concept of fuzzy MCF problems into fuzzy solid MCF (SMCF) problems and proposed a method for solving fully fuzzy single-objective uncapacitated SMCF problems. Kaur and Kumar [13] pointed out that the existing method [12] can neither be used for solving single-objective fully fuzzy single-objective unbalanced capacitated SMCF (CSMCF) problems nor fully fuzzy multi-objective unbalanced capacitated/uncapacitated SMCF problems. To overcome the limitations of the existing method [12], Kaur and Kumar [13] proposed a method for solving fully fuzzy multi-objective unbalanced CSMCF problems.

A membership function of a classical fuzzy set assigns to each element of the universe of discourse a number from the unit interval to indicate the degree of belongingness to the set under consideration. The degree of non-belongingness is just automatically the complement to one of the membership degree. However, a human being who expresses the degree of membership of given element in a fuzzy set very often does not express corresponding degree of non-membership as the complement to 1. This reflects a well-known psychological fact that the linguistic negation not always identifies with logical negation. Thus, Atanassov [2] introduced the concept of an intuitionistic fuzzy set that is characterized by two functions expressing the degree of belongingness and the degree of non-belongingness, respectively. This idea, which is a natural generalization of usual fuzzy set [24], seems to be useful in modeling many real-life situations.

Kumar and Kaur [15] pointed out the shortcomings and limitations of the existing ranking approaches [4, 10, 16–23] for intuitionistic fuzzy numbers and proposed an approach for comparing intuitionistic fuzzy numbers. Also, with the help of proposed ranking approach, Kumar and Kaur [15] proposed a method for solving unbalanced intuitionistic fully fuzzy single-objective capacitated MCF problems. After reviewing the literature, it can be concluded that there is no method in the literature that can be used for solving unbalanced intuitionistic fully fuzzy multi-objective CSMCF (IFFMOCSMCF) problems; thus, in this article, a new method is proposed for the same. Also, the unbalanced fully fuzzy single-objective capacitated MCF problems, unbalanced fully fuzzy multi-objective capacitated MCF problems, unbalanced fully fuzzy multi-objective CSMCF problems, unbalanced intuitionistic fully fuzzy single-objective capacitated MCF problems, and unbalanced intuitionistic fully fuzzy multi-objective capacitated MCF problems are the particular cases of the unbalanced IFFMOCSMCF problems; thus, all these problems can be solved by the proposed method.

This article is organized as follows: in Section 2, some basic definitions and arithmetic operations of LR flat intuitionistic fuzzy numbers are presented. In Section 3, the limitations of the existing methods [12, 13] are pointed out. In Section 4, comparison of intuitionistic fuzzy numbers is presented. In Section 5, linear programming formulation of balanced intuitionistic fully fuzzy single- and multi-objective CSMCF problems is proposed. In Section 6, new methods for solving unbalanced intuitionistic fully fuzzy single- and multi-objective CSMCF problems are proposed, and to illustrate the proposed method, numerical examples are solved. In Section 7, particular cases of the proposed method are discussed. The advantages of proposed methods over existing methods are discussed in Section 8. Results are compared in Section 9. The conclusions are discussed in Section 10.

2 Preliminaries

In this section, some basic definitions and arithmetic operations are presented [15].

2.1 Basic Definitions

In this section, some basic definitions are presented [15].

Definition 1: An intuitionistic fuzzy set A˜ = {(x, μA˜(x), νA˜(x))|x ∈ X} on the universal set X is characterized by a truth membership function μA˜, μA˜:X → [0, 1] and a false membership function νA˜, νA˜:X → [0, 1]. The values μA˜(x) and νA˜(x) represent the degrees of membership and non-membership for x∈X and always satisfy the condition μA˜(x) + νA˜(x) ≤ 1∀x ∈ X. The value (1 − μA˜(x) − νA(x)) represents the degree of hesitation for x∈X.

Definition 2: Let A˜ be an intuitionistic fuzzy set. Then, A˜α = {x ∈ X|μA˜(x) ≥ α, νA˜(x) ≤ (1 − α)} is said to be an α-cut of A˜.

Definition 3: An intuitionistic fuzzy set A˜ = {(x, μA˜(x), νA˜(x)|x ∈ X} is called intuitionistic fuzzy-normal if there exist at least two points x0, x1∈X such that μA˜(x0) = 1, νA˜(x1) = 1.

Definition 4: An intuitionistic fuzzy set A˜ = {(x, μA˜(x), νA˜(x)|x ∈ X} is called intuitionistic fuzzy-convex if ∀x1, x2∈X, λ∈[0, 1]:

μA˜(λx1 + (1 − λ)x2 ≥ min(μA˜(x1), μA˜(x2))νA˜(λx1 + (1 − λ)x2) ≤ max(νA˜(x1), νA˜(x2)

Definition 5: An intuitionistic fuzzy set A˜ = {x, μA˜(x), νA˜(x))|x ∈ X} defined on the universal set X is called intuitionistic fuzzy number if

  1. A˜ is intuitionistic fuzzy-normal,

  2. A˜ is intuitionistic fuzzy-convex,

  3. μA˜ is upper semicontinuous and νA˜ is lower semicontinuous,

  4. A˜ = {x ∈ X|νA˜(x) < 1} is bounded.

Definition 6: An intuitionistic fuzzy number A˜, defined on the universal set of real numbers ℝ, denoted as A˜ = {(a_, a¯, aL,aR)LR; (a′_, a′¯, a′L, a′R)LR}, where a′_ − a′L ≤ a_ − aL ≤ a′_ ≤ a_ ≤ a¯ ≤ a′¯ ≤ a¯ + aR ≤ a′¯ + a′R is said to be an LR flat intuitionistic fuzzy number if the degrees of membership μA˜(x) and non-membership νA˜(x) are given by

μA˜(x) = {L(a_ − xaL),x ≤ a_, aL > 0R(x − a¯aR),x ≥ a¯, aR > 01,a_ ≤ x ≤ a¯

νA˜(x) = {1 − L(a′_ − xa′L),x ≤ a′_, a′L > 01 − R(x − a′a′R),x ≥ a′, a′R > 00,a′_ ≤ x ≤ a′¯

Definition 7: Two LR flat intuitionistic fuzzy numbers A˜ = {(a_, a¯, aL, aR)LR; (a′_, a′¯, a′L, a′R)LR} and B˜ = {(b_, b¯, bL, bR)LR; (b′_, b′¯, b′L, b′R)LR} are said to be equal, i.e. A˜ = B˜ if and only if a_ = b_, a¯ = b¯, aL = bL, aR = bR, a′_ = b′_, a′¯ = b′¯, a′L = b′L and a′R = b′R.

Definition 8: An LR flat intuitionistic fuzzy number A˜ = {(a_, a¯, aL, aR)LR; (a′_, a′¯, a′L, a′R)LR} is said to be a zero LR flat intuitionistic fuzzy number if and only if a_ = 0, a¯ = 0, aL = 0, aR = 0, a′_ = 0, a′¯ = 0, a′L = 0 and a′R = 0.

Definition 9: An LR flat intuitionistic fuzzy number A˜ = {(a_, a¯, aL, aR)LR; (a′, a′¯, a′L, a′R)LR} is said to be a non-negative LR flat intuitionistic fuzzy number if and only if a′_ − a′L ≥ 0.

Remark 1: If a¯ = a¯ = a′_ = a′¯ = a (say) then an LR flat intuitionistic fuzzy number {(a_, a¯, aL, aR)LR; (a′_, a′¯, a′L, a′R)LR} is said to be an LR intuitionistic fuzzy number and is denoted as {(a, aL, aR)LR; (a, a′L, a′R)LR}.

Remark 2: If a_ = a¯ = a′_ = a′¯ = a (say) and L(x)=R(x)=maximum{0, 1−x}, then an LR flat intuitionistic fuzzy number {(a_, a¯, aL, aR)LR; (a′_, a′¯, a′L, a′R)LR} is said to be a triangular intuitionistic fuzzy number and is denoted as {(a1, a, a2);  (a′1, a, a′2)} or (a′1, a1, a, a2, a′2). where a′1 = a − a′L,  a1 = a − aL,  a2 = a + aR,  a′2 = a + a′R.

Remark 3: If a_ ≠ a¯, a′_ ≠ a′¯ and L(x)=R(x)=maximum{0, 1−x}, then an LR flat intuitionistic fuzzy number {(a_, a¯, aL, aR)LR; (a′_, a′¯, a′L, a′R)LR} is said to be a trapezoidal intuitionistic fuzzy number and is denoted as {(a1, a2, a3, a4); (a′1, a′2, a′3, a′4)} or (a′1, a1, a′2, a2, a3, a′3, a4, a′4). where a1 = a_ − aL, a2 = a_, a3 = a¯, a4 = a¯ + aR, a′1 = a′_ − a′L, a′2 = a′_, a′3 = a′¯, and a′4 = a′¯ + a′R.

2.2 Arithmetic Operations

In this section, some arithmetic operations between LR flat intuitionistic fuzzy numbers, defined on universal set of real numbers ℝ, are presented.

  1. Let A˜ = {(a_, a¯, aL, aR)LR; (a′_, a′¯, a′L, a′R)LR} and B˜ = {(b_, b¯, bL, bR)LR; (b′_,b′¯, b′L, b′R)LR} be two LR flat intuitionistic fuzzy numbers. Then,

    A˜ ⊕ B˜ = {(a_ + b_, a¯ + b¯, aL + bL, aR + bR)LR; (a′_ + b′_, a′¯ + b′¯, a′L + b′L, a′R + b′R)LR}.

  2. Let A˜ = {(a_, a¯, aL, aR)LR; (a′_, a′¯, a′L, a′R)LR} be an LR flat intuitionistic fuzzy number. Then,

    λA˜ = {{(λa_, λa¯, λaL, λaR)LR; (λa′_, λa′¯, λa′L, λa′R)LR},λ ≥ 0.{(λa¯, λa_, −λaR, −λaL,)RL; (λa′¯, λa′_, −λa′R, −λa′L)RL},λ ≤ 0.

  3. Let A˜ = {(a_, a¯, aL, aR)LR; (a′_, a′¯, a′L, a′R)LR} and B˜ = {(b_, b¯, bL, bR)LR; (b′_, b′¯, b′L, b′R)LR} be two non-negative LR flat intuitionistic fuzzy numbers. Then,

    A˜ ⊗ B˜ = {(a_ b_, a¯ b¯, a_bL + b_aL − aLbL, a¯bR + b¯aR + aRbR)LR; (a′_ b′_, a′¯ b′¯, a′_b′L +b′_a′L − a′Lb′L, a′¯b′R + b′¯a′R + a′Rb′R)LR}.

3 Limitations of the Existing Methods

To the best of our knowledge, only the existing method [12] can be used for solving fully fuzzy single-objective uncapacitated SMCF problems, and only the existing method [13] can be used for solving fully fuzzy multi-objective CSMCF problems. However, neither of these two existing methods nor any other existing method can be used for solving the following problems:

(1) Fully fuzzy single and multi-objective CSMCF problems:

Example 1: Find the fuzzy optimal compromise solution of the fully fuzzy multi-objective CSMCF problem depicted in Figure 1. The data are listed in Tables 1 and 2.

Figure 1: Network Representing Fully Fuzzy CSMCF Problem.
Figure 1:

Network Representing Fully Fuzzy CSMCF Problem.

Table 1

Fuzzy Penalties (c˜ijk1), (c˜ijk2), (c˜ijk3), Minimum Fuzzy Amount (l˜ijk), and Maximum Fuzzy Amount (u˜ijk).

ijkc˜ijk1c˜ijk2c˜ijk3l˜ijku˜ijk
131(8, 10, 2, 2)LR(5, 8, 3, 3)LR(6, 9, 2, 3)LR(2, 3, 0, 5)LR(10, 20, 10, 0)LR
132(4, 8, 3, 2)LR(8, 10, 2, 2)LR(3, 6, 2, 3)LR(7, 10, 5, 3)LR(30, 40, 10, 20)LR
211(8, 10, 4, 4)LR(9, 12, 6, 3)LR(5, 8, 3, 3)LR(0, 0, 0, 0)LR(30, 40, 10, 10)LR
212(6, 8, 4, 4)LR(8, 10, 4, 4)LR(9, 12, 6, 3)LR(0, 0, 0, 0)LR(40, 50, 20, 10)LR
231(9, 12, 6, 3)LR(6, 8, 4, 4)LR,(2, 4, 2, 2)LR(0, 0, 0, 0)LR(40, 60, 20, 10)LR
232(3, 6, 2, 3)LR(6, 9, 2, 3)LR(4, 8, 3, 2)LR(2, 3, 1, 2)LR(40, 60, 20, 20)LR

Where L(x)=maximum{0, 1−x4} R(x)=maximum{0, 1−x}.

Table 2

Fuzzy Supply (a˜i/e˜i), Fuzzy Demand (b˜j/d˜j), and Fuzzy Capacities (f˜k).

Nodesa˜i/e˜ib˜j/d˜jConveyancesf˜k
1(60, 80, 20, 20)LR–1(60, 80, 20, 10)LR
2(50, 70, 20, 20)LR–2(50, 70, 20, 40)LR
3–(50, 80, 30, 50)LR––

Where L(x)=maximum{0, 1−x4} R(x)=maximum{0, 1−x}.

(2) Intuitionistic fully fuzzy single- and multi-objective CSMCF problems:

Example 2: Find the intuitionistic fuzzy optimal solution of the intuitionistic fully fuzzy single-objective CSMCF (IFFSOCSMCF) problem depicted in Figure 2. The data are listed in Tables 3–6.

Figure 2: Network Representing Intuitionistic Fully Fuzzy CSMCF Problem.
Figure 2:

Network Representing Intuitionistic Fully Fuzzy CSMCF Problem.

Table 3

Intuitionistic Fuzzy Cost (c˜ijk).

ijkc˜ijk
131{(400, 4000, 360, 36000)LR ; (200, 22000, 180, 38000)LR }
132{(200, 2000, 180, 18000)LR ; (100, 11000, 90, 19000)LR }
121{(100, 1000, 90, 9000)LR ; (50, 5500, 45, 9500)LR }
122{(50, 200, 48, 1800)LR ; (10, 1100, 9, 1900)LR }
231{(300, 3000, 270, 27000)LR ; (150, 16500, 135, 28500)LR }
232{(500, 5000, 450, 45000)LR ; (250, 27500, 225, 47500)LR }

Where L(x)=maximum{0, 1−x}, R(x)=maximum{0, 1−x}.

Table 4

Minimum Intuitionistic Fuzzy Amount (l˜ijk) and Maximum Intuitionistic Fuzzy Amount (u˜ijk).

ijkl˜ijku˜ijk
131{(0, 0, 0, 0)LR ; (0, 0, 0, 0)LR }{(40, 50, 20, 20)LR ; (30, 60, 20, 40)LR }
132{(6, 7, 3, 2)LR ; (4, 8, 3, 3)LR }{(150, 220, 100, 40)LR ; (100, 240, 80, 60)LR }
121{(0, 0, 0, 0)LR ; (0, 0, 0, 0)LR }{(65, 70, 35, 10)LR ; (60, 75, 45, 10)LR }
122{(2, 3, 1, 1)LR ; (1, 3, 1, 2)LR }{(100, 150, 80, 150)LR ; (60, 170, 60, 180)LR }
231{(0, 0, 0, 0)LR ; (0, 0, 0, 0)LR }{(150, 220, 100, 40)LR ; (100, 240, 80, 60)LR }
232{(0, 0, 0, 0)LR ; (0, 0, 0, 0)LR }{(50, 55, 42, 10)LR ; (13, 60, 9, 11)LR }

Where L(x)=maximum{0, 1−x}, R(x)=maximum{0, 1−x}.

Table 5

Intuitionistic Fuzzy Supply (a˜i/e˜i) and Intuitionistic Fuzzy Demand (b˜j/d˜j).

Nodea˜i/e˜ib˜j/d˜j
1{(200, 250, 100, 50)LR ; (150, 270, 100, 80)LR }–
2–{(100, 150, 80, 50)LR ; (60, 170, 60, 80)LR }
3–{(50, 100, 20, 0)LR ; (40, 100, 40, 50)LR }

Where L(x)=maximum{0, 1−x}, R(x)=maximum{0, 1−x}.

Table 6

Intuitionistic Fuzzy Capacity (f˜k).

Conveyancef˜k
1{(0, 50, 0, 0)LR ; (0, 50, 0, 50)LR }
2{(200, 250, 100, 50)LR ; (150, 270, 100, 80)LR }

Where L(x)=maximum{0, 1−x}, R(x)=maximum{0, 1−x}.

Example 3: Consider an IFFMOCSMCF problem with the same network flow structure and data as in Example 2. The first objective is same as in Example 2, and the second objective is to minimize the total intuitionistic fuzzy passing time. The intuitionistic fuzzy passing time required for transporting one unit quantity of product from ith source to jth destination by means of kth conveyance is listed in Table 7.

Table 7

Intuitionistic Fuzzy Passing Time (c˜ijk2).

ijkc˜ijk2
131{(500, 5000, 450, 45,000)LR ; (250, 27,500, 225, 47,500)LR }
132{(0, 0, 0, 0)LR ; (0, 0, 0, 0)LR }
121{(150, 950, 141, 8050)LR ; (45, 5000, 41, 11,000)LR }
122{(45, 150, 42, 1950)LR ; (11, 1000, 9, 1900)LR }
231{(400, 4000, 360, 36,000)LR ; (200, 22,000, 180, 38,000)LR }
232{(200, 2000, 180, 18,000)LR ; (100, 11,000, 90, 19,000)LR }

Where L(x)=maximum{0, 1−x}, R(x)=maximum{0, 1−x}.

Remark 4: The intuitionistic fully fuzzy single- and multi-objective uncapacitated SMCF problems can neither be solved by any of existing methods [12, 13] nor by any of the other existing method.

4 Comparison of Intuitionistic Fuzzy Numbers

Kumar and Kaur [15] pointed out the limitations of the existing methods [4, 10, 16, 17, 19, 20, 22, 23] as well as shortcomings of the existing methods [16, 18, 21] for comparing intuitionistic fuzzy numbers and proposed a method to overcome the limitations of the existing methods [4, 10, 16, 17, 19, 20, 22, 23] as well as to resolve the shortcomings of the existing methods [16, 18, 21]. If A˜ = {(a_, a¯, aL, aR)LR; (a′_, a′¯, a′L, a′R)LR)} and B˜ = {(b_, b¯, bL, bR)LR; (b′_, b′¯, b′L, b′R)LR)} are two LR flat intuitionistic fuzzy numbers, then according to the method proposed by Kumar and Kaur [15], these numbers can be compared as follows:

Step 1: Calculate Mμβ,k′(A˜) = β[k′ + 12∫01rk′(a_ + a¯)dr] + (1 − β)[k′ + 12∫01rk′[a_ − aLL−1(r) + a¯ + aRR−1(r)]dr], and Mμβ,k′(B˜) = β[k′ + 12∫01rk′(b_ + b¯)dr] + (1 − β)[k′ + 12∫01rk′[b_ − bLL−1(r) + b¯ + bRR−1(r)]dr], and check if Mμβ,k′(A˜) > Mμβ,k′(B˜),Mμβ,k′(A˜) < Mμβ,k′(B˜), or Mμβ,k′(A˜) = Mμβ,k′(B˜).

Case (i): If Mμβ,k′(A˜) > Mμβ,k′(B˜), then A˜ ≻ B˜, i.e. minimum (A˜, B˜) = B˜.

Case (ii): If Mμβ,k′(A˜) < Mμβ,k′(B˜), then A˜ ≺ B˜, i.e. minimum (A˜, B˜) = A˜.

Case (iii): If Mμβ,k′(A˜) = Mμβ,k′(B˜), then go to Step 2.

Step 2: Calculate

Mνβ,k′(A˜) = β[k′ + 12∫01rk′(a′_ + a′¯)dr] + (1 − β)[k′ + 12∫01rk′[a′_ − a′LL−1(r) + a′¯ + a′RR−1(r)]dr] and Mνβ,k′(B˜) = β[k′ + 12∫01rk′(b′_ + b′¯)dr] + (1 − β)[k′ + 12∫01rk′[b′_ − b′LL−1(r) + b′¯ + b′RR−1(r)]dr], and check if −Mνβ,k′(A˜) > −Mνβ,k′(B˜), −Mνβ,k′(A˜) < −Mνβ,k′(B˜), or Mνβ,k′(A˜) = Mνβ,k′(B˜).

Case (i): If −Mνβ,k′(A˜) > −Mνβ,k′(B˜), then A˜ ≻ B˜, i.e. minimum (A˜, B˜) = B˜.

Case (ii): If −Mνβ,k′(A˜) < −Mνβ,k′(B˜), then A˜ ≺ B˜, i.e. minimum (A˜, B˜) = A˜.

Case (iii): If Mνβ,k′(A˜) = Mνβ,k′(B˜), then A˜ ∼ B˜.

Remark 5: For β = 13 and k′=0, the indices for membership and non-membership functions Mμ1/3,0(A˜i) and Mν1/3,0(A˜i) are as follows:

Mμ1/3,0(A˜i) = 13(3a_ + 3a¯ − aL + aR2) and Mν1/3,0(A˜i) = 13(3a′_ + 3a′¯ − a′L + a′R2).

5 Proposed Linear Programming Formulation of Balanced Intuitionistic Fully Fuzzy Single- and Multi-Objective CMCF Problems

In this section, by generalizing the existing linear programming formulation of balanced fully fuzzy single- and multi-objective uncapacitated SMCF problems [12, 13], the linear programming formulation of balanced intuitionistic fully fuzzy single- and multi-objective CSMCF problems is proposed.

5.1 Linear Programming Formulation of Balanced IFFSOCSMCF Problems

Let a˜i and e˜i be the intuitionistic fuzzy supply of the product at ith purely source node and at ith source node, b˜j and d˜j be the intuitionistic fuzzy demand of the product at jth purely destination node and at the jth destination node, f˜k be the intuitionistic fuzzy capacity of the kth conveyance, i.e. the amount of product that can be carried by the kth conveyance, c˜ijk be the intuitionistic fuzzy cost per unit of flow from ith source to jth destination by means of the kth conveyance, x˜ijk be the intuitionistic fuzzy quantity of the product that should be transported from ith node to jth node by means of the kth conveyance, l˜ijk be the minimum intuitionistic fuzzy amount that can flow from ith node to jth node by means of the kth conveyance, and u˜ijk be the maximum intuitionistic fuzzy amount that can flow from ith node to jth node by means of the kth conveyance. Then, any balanced intuitionistic fully fuzzy CSMCF problem, i.e. ∑i ∈ NPSa˜i ⊕ ∑i ∈ NSe˜i = ∑j ∈ NPDb˜j ⊕ ∑j ∈ NDd˜j = ∑k ∈ SCf˜k, can be formulated into the following intuitionistic fuzzy single-objective linear programming problem:

(P1)Minimize ∑(i,j) ∈ A ∑k ∈ SC(c˜ijk ⊗ x˜ijk)Subject to∑j:(i,j) ∈ A ∑k ∈ SCx˜ijk = a˜ii ∈ NPS∑j:(i,j) ∈ A ∑k ∈ SCx˜ijk = ∑j:(j,i) ∈ A ∑k ∈ SCx˜jik ⊕ e˜ii ∈ NS∑i:(i,j) ∈ A ∑k ∈ SCx˜ijk = b˜jj ∈ NPD∑i:(i,j) ∈ A ∑k ∈ Scx˜ijk = ∑i:(j,i) ∈ A ∑k ∈ SCx˜jik ⊕ d˜jj ∈ ND∑j:(i,j) ∈ A ∑k ∈ SCx˜ijk = ∑j:(j,i) ∈ A ∑k ∈ SCx˜jiki ∈ NT∑(i,j) ∈ Ax˜ijk = f˜kk ∈ SCl˜ijk ≺_ x˜ijk ≺_ u˜ijk   ∀(i, j) ∈ A,k ∈ SC  (P1)

x˜ijk is a non-negative LR flat intuitionistic fuzzy number ∀(i, j)∈A, k∈SC , where A is set of arcs joining different nodes and SC is the set of all available conveyances.

5.2 Linear Programming Formulation of Balanced IFFMOCSMCF Problems

If there are more than one objective (say P), then any balanced IFFMOCSMCF problem can be formulated into intuitionistic fully fuzzy multi-objective linear programming problem (P2).

(P2)Minimize ∑(i,j) ∈ A ∑k ∈ SC(c˜ijkη ⊗ x˜ijk); η = 1, 2, …, PSubject to∑j:(i,j) ∈ A ∑k ∈ SCx˜ijk = a˜ii ∈ NPS∑j:(i,j) ∈ A ∑k ∈ SCx˜ijk = ∑j:(j,i) ∈ A ∑k ∈ SCx˜jik ⊕ e˜ii ∈ NS∑i:(i,j) ∈ A ∑k ∈ SCx˜ijk = b˜jj ∈ NPD∑i:(i,j) ∈ A ∑k ∈ SCx˜ijk = ∑i:(j,i) ∈ A ∑k ∈ SCx˜jik ⊕ d˜jj ∈ ND∑j:(i,j) ∈ A ∑k ∈ SCx˜ijk = ∑j:(j,i) ∈ A ∑k ∈ SCx˜jiki ∈ NT∑(i,j) ∈ Ax˜ijk = f˜kk ∈ SCl˜ijk ≺_ x˜ijk ≺_ u˜ijk   ∀(i, j) ∈ A, k ∈ SC  (P2)

x˜ijk is a non-negative LR flat intuitionistic fuzzy number ∀(i, j)∈A, k∈SC , c˜ijkη is the intuitionistic fuzzy penalty per unit of flow from ith source to jth destination by means of the kth conveyance in the ηth objective function, and P is the total number of objectives.

Remark 6: If ∑i ∈ NPSa˜i ⊕ ∑i ∈ NSe˜i = ∑j ∈ NPDb˜j ⊕ ∑j ∈ NDd˜j = ∑k ∈ SCf˜k, then an intuitionistic fully fuzzy CSMCF problem is said to be a balanced intuitionistic fully fuzzy CSMCF problem; otherwise, it is said to be an unbalanced intuitionistic fully fuzzy CSMCF problem.

6 Proposed Methods

In this section, to overcome the limitations of the methods discussed in Section 3, a new method for solving unbalanced IFFSOCSMCF problems as well as a new method for solving unbalanced IFFMOCSMCF problems is proposed by generalizing the existing methods [12, 13].

6.1 Proposed Method for Solving Unbalanced IFFSOCSMCF Problems

In this section, a new method is proposed for finding the intuitionistic fuzzy optimal solution of unbalanced IFFSOCSMCF problems by generalizing the existing method [12].

The steps of the proposed method are as follows:

Step 1: Find ∑i ∈ NPSa˜i ⊕ ∑i ∈ NSe˜i, ∑j ∈ NPDb˜j ⊕ ∑j ∈ NDd˜j, and ∑k ∈ SCf˜k. Let ∑i ∈ NPSa˜i ⊕ ∑i ∈ NSe˜i = {(m_, m¯, mL, mR)LR; (m′_, m′¯, m′L, m′R)LR},∑j ∈ NPDb˜j ⊕ ∑j ∈ NDd˜j = {(n_, n¯, nL, nR)LR; (n′_, n′¯, n′L, n′R)LR}, and ∑k ∈ SCf˜k = {(f_, f¯, fL, fR)LR; (f′_, f′¯, f′L, f′R)LR}. Examine if the problem is balanced or unbalanced.

Case 1: If the problem is balanced, i.e. ∑i ∈ NPSa˜i ⊕ ∑i ∈ NSe˜i = ∑j ∈ NPDb˜j ⊕ ∑j ∈ NDd˜j = ∑k ∈ SCf˜k, then go to Step 4.

Case 2: If the problem is unbalanced, i.e. ∑i ∈ NPSa˜i ⊕ ∑i ∈ NSe˜i = ∑j ∈ NPDb˜j ⊕ ∑j ∈ NDd˜j ≠ ∑k ∈ SCf˜k,∑i ∈ NPSa˜i ⊕ ∑i ∈ NSe˜i ≠ ∑j ∈ NPDb˜j ⊕ ∑j ∈ NDd˜j = ∑k ∈ SCf˜k, ∑i ∈ NPSa˜i ⊕ ∑i ∈ NSe˜i = ∑k ∈ SCf˜k ≠ ∑j ∈ NPDb˜j ⊕ ∑j ∈ NDd˜j, or ∑i ∈ NPSa˜i ⊕ ∑i ∈ NSe˜i ≠ ∑j ∈ NPDb˜j ⊕ ∑j ∈ NDd˜j ≠ ∑k ∈ SCf˜k, then go to Step 2.

Step 2: Check if ∑i ∈ NPSa˜i ⊕ ∑i ∈ NSe˜i = ∑j ∈ NPDb˜j ⊕ ∑j ∈ NDd˜j or ∑i ∈ NPSa˜i ⊕ ∑i ∈ NSe˜i ≠ ∑j ∈ NPDb˜j ⊕ ∑j ∈ NDd˜j.

Case 1: If ∑i ∈ NPSa˜i ⊕ ∑i ∈ NSe˜i = ∑j ∈ NPDb˜j ⊕ ∑j ∈ NDd˜j, then go to Step 3.

Case 2: If ∑i ∈ NPSa˜i ⊕ ∑i ∈ NSe˜i ≠ ∑j ∈ NPDb˜j ⊕ ∑j ∈ NDd˜j, then convert ∑i ∈ NPSa˜i ⊕ ∑i ∈ NSe˜i ≠ ∑j ∈ NPDb˜j ⊕ ∑j ∈ NDd˜j into ∑i ∈ NPSa˜i ⊕ ∑i ∈ NSe˜i = ∑j ∈ NPDb˜j ⊕ ∑j ∈ NDd˜j as follows:

Case 2a: If m′_ − m′L ≤ n′_ − n′L, (m_ − mL) − (m′_ − m′L) ≤ (n_ − nL) − (n′_ − n′L), m′_ − (m_ − mL) ≤ n′_ −(n_ − nL), m_ − m′_ ≤ n_ − n′_,m¯ − m_ ≤ n¯ − n_, m′¯ − m¯ ≤ n′¯ − n¯, (m¯ + mR) − m′¯ ≤ (n¯ + nR) − n′¯, and (m′¯ + m′R) − (m¯ + mR) ≤ (n′¯ + n′R) − (n¯ + nR), then introduce a dummy purely source node with intuitionistic fuzzy supply {(n_ − m_, n¯ − m¯, nL − mL, nR − mR)LR; (n′_ − m′_, n′¯ − m′¯, n′L − m′L, n′R − m′R)LR}, so that ∑i ∈ NPSa˜i ⊕ ∑i ∈ NSe˜i = ∑j ∈ NPDb˜j ⊕ ∑j ∈ NDd˜j. Go to Step 3.

Case 2b: If m′_ − m′L ≥ n′_ − n′L, (m_ − mL) − (m′_ − m′L) ≥ (n_ − nL) − (n′_ − n′L),m′_ − (m_ − mL) ≥ n′_ − (n_ − nL), m_ − m′_ ≥ n_ − n′_, m¯ − m_ ≥ n¯ − n_,m′¯ − m¯ ≥ n′¯ − n¯, (m¯ + mR) − m′¯ ≥ (n¯ + nR) − n′¯, and (m′¯ + m′R) −(m¯ + mR) ≥ (n′¯ + n′R) − (n¯ + nR), then introduce a dummy purely destination node with intuitionistic fuzzy demand {(m_ − n_, m¯ − n¯, mL − nL, mR − nR)LR; (m′_ − n′_, m′¯ − n′¯, m′L − n′L, m′R − n′R)LR}, so that ∑i ∈ NPSa˜i ⊕ ∑i ∈ NSe˜i = ∑j ∈ NPDb˜j ⊕ ∑j ∈ NDd˜j. Go to Step 3.

Case 2c: If neither Case 2a nor Case 2b is satisfied then introduce a dummy purely source node with intuitionistic fuzzy supply {(C_,C¯, CL, CR)LR; (C′_, C′¯, C′L, C′R)LR} and also introduce a dummy destination node with intuitionistic fuzzy demand {(D_,D¯, DL, DR)LR; (D′_, D′¯, D′L, D′R)LR}, so that ∑i ∈ NPSa˜i ⊕ ∑i ∈ NSe˜i = ∑j ∈ NPDb˜j ⊕ ∑j ∈ NDd˜j. Go to Step 3, where

C′_ = maximum(0, [(n′_ − n′L) − (m′_ − m′L)]) + maximum(0, {[(n_ − nL) − (n′_ − n′L)] − [(m_ − mL) − (m′_ − m′L)]}) + maximum(0, [(n′_ − (n_ − nL)) − (m′_ − (m_ − mL)]),

C_ = C′_ + maximum(0, [(n_ − n′_) − (m_ − m′_)])

C¯ = C_ + maximum(0, [(n¯ − n_) − (m¯ − m_)])

C′¯ = C¯ + maximum(0, [(n′¯ − n¯) − (m′¯ − m¯)])

C′L = maximum(0, [(n_ − nL − n′_ + n′L) − (m_ − mL − m′_ + m′L)]) + maximum(0, [(n′_ − n_ + nL) − (m′_ − m_ + mL)])

CL = maximum(0, [(n′_ − n_ + nL) − (m′_ − m_ + mL)]) + maximum(0, [(n_ − n′_) − (m_ − m′_)])

CR = maximum(0, [(n′¯ − n¯) − (m′¯ − m¯)]) + maximum(0, [(n¯ + nR − n′¯) − (m¯ + mR − m′¯)])

C′R = maximum(0, [(n¯ + nR − n′¯) − (m¯ + mR − m′¯)]) + maximum(0, [(n′¯ + n′R − n¯ − nR) − (m′¯ + m′R − m¯ − mR)]),

D′_ = maximum(0, [(m′_ − m′L) − (n′_ − n′L)] + maximum(0, [(m_ − mL) − (m′_ − m′L)] − [(n_ − nL) − (n′_ − n′L)]) +maximum(0, [(m′_ − (m_ − mL)) − (n′_ − (n_ − nL)]), and

D_ = D′_ + maximum(0, [(m_ − m′_) − (n_ − n′_)])

D¯ = D_ + maximum(0, [(m¯ − m_) − (n¯ − n_)])

D′¯ = D¯ + maximum(0, [(m′¯ − m¯) − (n′¯ − n¯)])

D′L = maximum(0, [(m_ − mL − m′_ + m′L) − (n_ − nL − n′_ + n′L)]) + maximum(0, [(m′_ − m_ + mL)− (n′_ − n_ + nL)])

DL = maximum (0, [(m′_ − m_ + mL) − (n′_ − n_ + nL)]) + maximum(0, [(m_ − m′_) − (n_ − n′_)])

DR =  maximum (0, [(m′¯ − m¯) − (n′¯ − n¯)]) + maximum(0, [(m¯ + mR − m′¯) − (n¯ + nR − n′¯)])

D′R = maximum(0, [(m¯ + mR − m′¯) − (n¯ + nR − n′¯)]) + maximum(0, [(m′¯ + m′R − m¯ − mR)− (n′¯ + n′R − n¯ − nR)]).

Step 3: Using Step 2, ∑i ∈ NPSa˜i ⊕ ∑i ∈ NSe˜i = ∑j ∈ NPDb˜j ⊕ ∑j ∈ NDd˜j. Let ∑i ∈ NPSa˜i ⊕ ∑i ∈ NSe˜i = ∑j ∈ NPDb˜j ⊕ ∑j ∈ NDd˜j = {(g_, g¯, gL, gR)LR; (g′_, g′¯, g′L, g′R)LR} and ∑k ∈ SCf˜k = {(f_, f¯, fL, fR)LR; (f′_, f′¯, f′L, f′R)LR}.

Now check ∑i ∈ NPSa˜i ⊕∑i ∈ NSe˜i = ∑j ∈ NPDb˜j ⊕ ∑j ∈ NDd˜j = ∑k ∈ SCf˜k or ∑i ∈ NPSa˜i ⊕ ∑i ∈ NSe˜i = ∑j ∈ NPDb˜j ⊕ ∑j ∈ NDd˜j ≠ ∑k ∈ SCf˜k.

Case 1: If ∑i ∈ NPSa˜i ⊕ ∑i ∈ NSe˜i = ∑j ∈ NPDb˜j ⊕ ∑j ∈ NDd˜j = ∑k ∈ SCf˜k, then go to Step 4.

Case 2: If ∑i ∈ NPSa˜i ⊕ ∑i ∈ NSe˜i = ∑j ∈ NPDb˜j ⊕ ∑j ∈ NDd˜j ≠ ∑k ∈ SCf˜k, then convert ∑i ∈ NPSa˜i ⊕ ∑i ∈ NSe˜i = ∑j ∈ NPDb˜j ⊕ ∑j ∈ NDd˜j ≠ ∑k ∈ SCf˜k into ∑i ∈ NPSa˜i ⊕ ∑i ∈ NSe˜i = ∑j ∈ NPDb˜j ⊕ ∑j ∈ NDd˜j = ∑k ∈ SCf˜k as follows:

Case 2a: If g′_ − g′L ≤ f′_ − f′L, (g_ − gL) − (g′_ − g′L) ≤ (f_ − fL) − (f′_ − f′L),g′_ −(g_ − gL) ≤ f′_ − (f_ − fL),g_ − g′_ ≤ f_ − f′_, g¯ − g_ ≤ f¯ − f_, g′¯ − g¯ ≤ f′¯ − f¯, (g¯ + gR) − g′ ≤ (f + fR) − f′ and (g′¯ + g′R) − (g¯ + gR) ≤ (f′¯ + f′R) − (f¯ + fR), then check in Step 2 if a dummy purely source node and/or a dummy purely destination node is introduced or not.

Case (i): If both the dummy purely source node and dummy purely destination node are introduced, then increase both the intuitionistic fuzzy supply of the already introduced dummy purely source node and the intuitionistic fuzzy demand of the already introduced dummy purely destination node by the same intuitionistic fuzzy quantity {(f_ − g_, f¯ − g¯, fL − gL, fR − gR)LR; (f′_ − g′_, f′¯ − g′¯, f′L − g′L, f′R − g′R)LR}, so that ∑i ∈ NPSa˜i ⊕ ∑i ∈ NSe˜i = ∑j ∈ NPDb˜j ⊕ ∑j ∈ NDd˜j = ∑k ∈ SCf˜k. Go to Step 4.

Case (ii): If a dummy purely source node is introduced but no dummy purely destination node is introduced, then increase the intuitionistic fuzzy supply of the already introduced dummy purely source node by the intuitionistic fuzzy quantity {(f_ − g_, f¯ − g¯, fL − gL, fR − gR)LR; (f′_ − g′_, f′¯ − g′¯, f′L − g′L, f′R − g′R)LR} and also introduce a dummy purely destination node with intuitionistic fuzzy demand {(f_ − g_, f¯ − g¯, fL − gL, fR − gR)LR; (f′_ − g′_, f′¯ − g′¯, f′L − g′L, f′R − g′R)LR}, so that ∑i ∈ NPSa˜i ⊕ ∑i ∈ NSe˜i = ∑j ∈ NPDb˜j ⊕ ∑j ∈ NDd˜j = ∑k ∈ SCf˜k. Go to Step 4.

Case (iii): If a dummy purely destination node is introduced but no dummy purely source node is introduced, then increase the intuitionistic fuzzy demand of the already introduced dummy purely destination node by the intuitionistic fuzzy quantity {(f_ − g_, f¯ − g¯, fL − gL, fR − gR)LR; (f′_ − g′_, f′¯ − g′¯, f′L − g′L, f′R − g′R)LR} and also introduce a dummy purely source node with intuitionistic fuzzy supply {(f_ − g_, f¯ − g¯, fL − gL, fR − gR)LR; (f′_ − g′_, f′¯ − g′¯, f′L − g′L, f′R − g′R)LR}, so that ∑i ∈ NPSa˜i ⊕ ∑i ∈ NSe˜i = ∑j ∈ NPDb˜j ⊕ ∑j ∈ NDd˜j = ∑k ∈ SCf˜k. Go to Step 4.

Case 2b: If g′_ − g′L ≥ f′_ − f′L, (g_ − gL) − (g′_ − g′L) ≥ (f_ − fL) − (f′_ − f′L),g′_ − (g_ − gL) ≥ f′_ − (f_ − fL), g_ − g′_ ≥ f_ − f′_, g¯ − g_ ≥ f¯ − f_, g′¯ − g¯ ≥ f′¯ − f¯,(g¯ + gR) − g′¯ ≥ (f¯ + fR) − f′¯, and (g′¯ + g′R) − (g¯ + mR) ≥ (f′¯ + f′R) − (f¯ + fR), then introduce a dummy conveyance with intuitionistic fuzzy capacity {(g_ − f_, g¯ − f¯, gL − fL, gR − fR)LR; (g′_ − f′_, g′¯ − f′¯, g′L − f′L, g′R − f′R)LR}, so that ∑i ∈ NPSa˜i ⊕ ∑i ∈ NSe˜i = ∑j ∈ NPDb˜j ⊕ ∑j ∈ NDd˜j = ∑k ∈ SCf˜k. Go to Step 4.

Case 2c: If neither Case 2a nor Case 2b is satisfied, then check in Step 2 if a dummy purely source node and/or a dummy purely destination node is introduced or not.

Case (i): If both the dummy purely source node and dummy purely destination node are introduced, then increase both the intuitionistic fuzzy supply of the already introduced dummy purely source node and the intuitionistic fuzzy demand of the already introduced dummy purely destination node by the same intuitionistic fuzzy quantity {(I_, I¯, IL, IR)LR; (I′_, I′¯, I′L, I′R)LR} and also introduce a dummy purely conveyance with intuitionistic fuzzy capacity {(J_, J¯, JL, JR)LR; (J′_, J′¯, J′L, J′R)LR}, so that ∑i ∈ NPSa˜i ⊕ ∑i ∈ NSa˜′i = ∑j ∈ NPDb˜j ⊕ ∑j ∈ NDb˜′j = ∑k ∈ SCe˜k. Go to Step 4, where

I′_ = maximum(0, [(f′_ − f′L) − (g′_ − g′L)] + maximum(0, [(f_ − fL) − (f′_ − f′L)] − [(g_ − gL) − (g′_ − g′L)]) + maximum(0, [(f′_ − (f_ − fL)) − (g′_ − (g_ − gL)]),

I_ = I′_ + maximum(0, [(f_ − f′_) − (g_ − g′_)])

I¯ = I_ + maximum(0, [(f¯ − f_) − (g¯ − g_)])

I′¯ = I¯ + maximum(0, [(f′¯ − f¯) − (g′¯ − g¯)])

I′L = maximum(0, [(f_ − fL − f′_ + f′L) − (g_ − gL − g′_ + g′L)]) + maximum(0, [(f′_ − f_ + fL) − (g′_ − g_ + gL)])IL = maximum(0, [(f′_ − f_ + fL) − (g′_ − g_ + gL)]) + maximum(0, [(f_ − f′_) − (g_ − g′_)])IR = maximum(0, [(f′¯ − f¯) − (g′¯ − g¯)]) + maximum(0, [(f¯ + fR − f′¯) − (g¯ + gR − g′¯)])I′R = maximum(0, [(f¯ + fR − f′¯) − (g¯ + gR − g′¯)]) + maximum(0, [(f′¯ + f′R − f¯ − fR)− (g′¯ + g′R − g¯ − gR)]), andJ′_ = maximum(0, [(g′_ − g′L) − (f′_ − f′L)] + maximum(0, [(g_ − gL) − (g′_ − g′L)] − [(f_ − fL) − (f′_ − f′L)]) +maximum(0, [(g′_ − (g_ − gL)) − (f′_ − (f_ − fL)]),

J_ = J′_ +maximum(0, [(g_ − g′_) − (f_ − f′_)])

J¯ = J_ + maximum(0, [(g¯ − g_) − (f¯ − f_)])

J′¯ = J¯ + maximum(0, [(g′¯ − g¯) − (f′¯ − f¯)])

J′L = maximum(0, [(g_ − gL − g′_ + g′L) − (f_ − fL − f′_ + f′L)]) + maximum(0, [(g′_ − g_ + gL) − (f′_ − f_ + fL)])

JL = maximum(0, [(g′_ − g_ + gL) − (f′_ − f_ + fL)]) + maximum(0, [(g_ − g′_) − (f_ − f′_)])

JR = maximum(0, [(g′¯ − g¯) − (f′¯ − f¯)]) + maximum(0, [(g¯ + gR − g′¯) − (f¯ + fR − f′¯)])

J′R = maximum(0, [(g¯ + gR − g′¯) − (f¯ + fR − f′¯)]) + maximum(0, [(g′¯ + g′R − g¯ − gR) − (f′¯ + f′R − f¯ − fR)]).

Case (ii): If a dummy purely source node is introduced but no dummy purely destination node is introduced, then increase the intuitionistic fuzzy supply of the already introduced dummy purely source node by the intuitionistic fuzzy quantity {(I_, I¯, IL, IR)LR; (I′_, I′¯, I′L, I′R)LR} and introduce a dummy purely destination node with intuitionistic fuzzy demand {(I_, I¯, IL, IR)LR; (I′_, I′¯, I′L, I′R)LR}. Also, introduce a dummy conveyance with intuitionistic fuzzy capacity {(J_, J¯, JL, JR)LR; (J′_, J′¯, J′L, J′R)LR}, so that ∑i ∈ NPSa˜i ⊕ ∑i ∈ NSa˜′i = ∑j ∈ NPDb˜j ⊕ ∑j ∈ NDb˜′j = ∑k ∈ SCe˜k. Go to Step 4.

Case (iii): If a dummy purely destination node is introduced but no dummy purely source node is introduced, then increase the intuitionistic fuzzy demand of the already introduced dummy purely destination node by the intuitionistic fuzzy quantity {(I_, I¯, IL, IR)LR; (I′_, I′¯, I′L, I′R)LR} and introduce a dummy purely source node with intuitionistic fuzzy supply {(I_, I¯, IL, IR)LR; (I′_, I′¯, I′L, I′R)LR}. Also, introduce a dummy conveyance with intuitionistic fuzzy capacity {(J_, J¯, JL, JR)LR; (J′_, J′¯, J′L, J′R)LR}, so that ∑i ∈ NPSa˜i ⊕ ∑i ∈ NSa˜′i = ∑j ∈ NPDb˜j ⊕ ∑j ∈ NDb˜′j = ∑k ∈ SCe˜k. Go to Step 4.

Step 4: The balanced IFFSOCSMCF problem, obtained using Steps 1–3, can be formulated into the intuitionistic fuzzy linear programming problem (P1) by assuming the following intuitionistic fuzzy transportation cost as zero LR flat intuitionistic fuzzy numbers:

  1. If any dummy purely source node is introduced, then assume the intuitionistic fuzzy transportation cost for transporting one unit quantity of the product from the introduced dummy purely source node to all purely destination nodes and all intermediate nodes by all conveyance as zero LR flat intuitionistic fuzzy number.

  2. If any dummy purely destination node is introduced, then assume the intuitionistic fuzzy transportation cost for transporting one unit quantity of the product from all purely source nodes and all intermediate nodes to the introduced dummy purely destination node by all conveyance as zero LR flat intuitionistic fuzzy number.

  3. If any dummy conveyance is introduced, then assume the intuitionistic fuzzy transportation cost for transporting one unit quantity of the product from all purely source nodes and intermediate nodes to all intermediate nodes and all purely destination nodes by introduced dummy conveyance as zero LR flat intuitionistic fuzzy number.

Step 5: Assuming c˜ijk = {(c_ ijk, c¯ijk, cijkL, cijkR)LR; (c′_ijk, c′¯ijk, c′ijkL, c′ijkR)LR}, x˜ijk = {(x_ijk, x¯ijk, xijkL, xijkR)LR;(x′_ijk, x′¯ijk, x′ijkL, x′ijkR)LR},a˜i = {(a_i, a¯i, aiL, aiR)LR; (a′_i, a′¯i, a′iL, a′iR)LR}, e˜i = {(e_i, e¯i, eiL, eiR)LR; (e′_i, e′¯i, e′iL, e′iR)LR},b˜j = {(b_j, b¯j, bjL, bjR)LR; (b′_j, b′¯j, b′jL, b′jR)LR}, d˜j = {(d_j, d¯j, djL, djR)LR; (d′_j, d′¯j, d′jL, d′jR)LR},f˜k = {(f_k, f¯k, fkL, fkR)LR; (f′_k, f′¯k, f′kL, f′kR)LR}, l˜ijk = {(l_ijk, l¯ijk, lijkL, lijkR)LR; (l′_ijk, l′¯ijk, l′ijkL, l′ijkR)LR}, and u˜ijk = {(u_ijk, u¯ijk, uijkL, uijkR)LR; (u′_ijk, u′¯ijk, u′ijkL, u′ijkR)LR}, the intuitionistic fuzzy linear programming problem (P1) can be written as

Minimize∑(i,j) ∈ A∑k ∈ SC({(c_ijk, c¯ijk, cijkL, cijkR)LR; (c′_ijk, c′¯ijk, c′ijkL, c′ijkR)LR} ⊗ {(x_ijk, x¯ijk, xijkL, xijkR)LR; (x′_ijk, x′¯ijk, x′ijkL, x′ijkR)LR})Subject to∑j:(i,j) ∈ A ∑k ∈ SC{(x_ijk, x¯ijk, xijkL, xijkR)LR; (x′_ijk, x′¯ijk, x′ijkL, x′ijkR)LR} = {(a_i, a¯i, aiL, aiR)LR; (a′_i, a′¯i, a′iL, a′iR)LR}   i∈NPS

∑j:(i,j) ∈ A ∑k ∈ SC{(x_ijk, x¯ijk, xijkL, xijkR)LR; (x′_ijk, x′¯ijk, x′ijkL, x′ijkR)LR} =∑j:(j,i) ∈ A ∑k ∈ SC{(x_jik,x¯jik, xjikL, xjikR)LR; (x′_jik, x′¯jik, x′jikL, x′jikR)LR} ⊕ {(e_i, e¯i, eiL, eiR)LR; (e′_i, e′¯i, e′iL, e′iR)LR}   i ∈ NS

(P3)∑i:(i,j) ∈ A ∑k ∈ SC{(x_ijk, x¯ijk, xijkL, xijkR)LR; (x′_ijk, x′¯ijk, x′ijkL, x′ijkR)LR} = {(b_j, b¯j, bjL, bjR)LR; (b′_j, b′¯j, b′jL, b′jR)LR}   j ∈ NPD∑i:(i,j) ∈ A ∑k ∈ SC{(x_ijk, x¯ijk, xijkL, xijkR)LR; (x′_ijk, x′¯ijk, x′ijkL, x′ijkR)LR}=∑i:(j,i) ∈ A ∑k ∈ SC{(x_jik,x¯jik, xjikL, xjikR)LR; (x′_jik, x′¯jik, x′jikL, x′jikR)LR} = ⊕ {(d_j, d¯j, djL, djR)LR; (d′_j, d′¯j, d′jL, d′jR)LR}   j∈ND  (P3)

∑j:(i,j) ∈ A ∑k ∈ SC{(x_ijk, x¯ijk, xijkL, xijkR)LR; (x′_ijk, x′¯ijk, x′ijkL, x′ijkR)LR} =∑j:(j,i) ∈ A ∑k ∈ SC{(x_jik, x¯jik, xjikL, xjikR)LR; (x′_jik, x′¯jik, x′jikL, x′jikR)LR}i ∈ NT∑(i,j) ∈ A{(x_ijk, x¯ijk, xijkL, xijkR)LR; (x′_ijk, x′¯ijk, x′ijkL, x′ijkR)LR}={(f_k, f¯k, fkL, fkR)LR; (f′_k, f′¯k, f′kL, f′kR)LR}  k ∈ SC{(l_ijk, l¯ijk, lijkL, lijkR)LR; (l′_ijk, l′¯ijk, l′ijkL, l′ijkR)LR} ≺_ {(x_ijk, x¯ijk, xijkL, xijkR)LR; (x′_ijk, x′¯ijkx′ijkL, x′ijkR)LR} ≺_ {(u_ijk, u¯ijk, uijkL, uijkR)LR;(u′_ijk, u′¯ijk, u′ijkL, u′ijkR)LR}∀(i, j) ∈ A,   k ∈ SC

{(x_ijk, x¯ijk, xijkL, xijkR)LR; (x′_ijk, x′¯ijk, x′ijkL, x′ijkR)LR} is non-negative LR flat intuitionistic fuzzy number ∀(i, j)∈A, k∈SC .

Step 6: Using the arithmetic operations of LR flat intuitionistic fuzzy numbers, defined in Section 2.2, the balanced IFFSOCSMCF problem (P3) can be written as

Minimize ∑(i,j) ∈ A∑k ∈ SC{(c_ijkx_ijk, c¯ijkx¯ijk, c_ijkxijkL + cijkLx_ijk − cijkLxijkL, c¯ijkxijkR + cijkRx¯ijk + cijkRxijkR)LR; (c′_ijkx′_ijk, c′¯ijkx′¯ijk, c′_ijkx′ijkL + c′ijkLx′_ijk − c′ijkLx′ijkL, c′¯ijkx′ijkR + c′ijkRx′¯ijk + c′ijkRx′ijkR)LR}

Subject to{(∑j:(i,j) ∈ A ∑k ∈ SCx_ijk, ∑j:(i,j) ∈ A ∑k ∈ SCx¯ijk, ∑j:(i,j) ∈ A ∑k ∈ SCxijkL, ∑j:(i,j) ∈ A ∑k ∈ SCxijkR)LR; (∑j:(i,j) ∈ A ∑k ∈ SCx′_ijk, ∑j:(i,j) ∈ A ∑k ∈ SCx′¯ijk, ∑j:(i,j) ∈ A ∑k ∈ SCx′ijkL,∑j:(i,j) ∈ A ∑k ∈ SCx′ijkR)LR} = {(a_i, a¯i, aiL, aiR)LR; (a′_i, a′¯i, a′iL, a′iR)LR}   i ∈NPS

{(∑j:(i,j) ∈ A ∑k ∈ SCx_ijk, ∑j:(i,j) ∈ A ∑k ∈ SCx¯ijk, ∑j:(i,j) ∈ A ∑k ∈ SCxijkL, ∑j:(i,j) ∈ A ∑k ∈ SCxijkR)LR; (∑j:(i,j) ∈ A ∑k ∈ SCx′_ijk, ∑j:(i,j) ∈ A ∑k ∈ SCx′¯ijk, ∑j:(i,j) ∈ A ∑k ∈ SCx′ijkL,∑j:(i,j) ∈ A ∑k ∈ SCx′ijkR)LR} = {(∑j:(j,i) ∈ A ∑k ∈ SCx_jik, ∑j:(j,i) ∈ A∑k ∈ SCx¯jik, ∑j:(j,i) ∈ A ∑k ∈ SCxjikL, ∑j:(j,i) ∈ A ∑k ∈ SCxjikR)LR; (∑j:(j,i) ∈ A  ∑k ∈ SCx′_jik,∑j:(j,i) ∈ A ∑k ∈ SCx′¯jik,∑j:(j,i) ∈ A∑k ∈ SCx′jikL,∑j:(j,i) ∈ A ∑k ∈ SCx′jikR)LR} ⊕ {(e_i, e¯i, eiL, eiR)LR; (e′_i, e′¯i, e′iL, e′iR)LR}   i ∈ NS

(P4){(∑i:(i,j) ∈ A ∑k ∈ SCx_ijk, ∑i:(i,j) ∈ A ∑k ∈ SCx¯ijk, ∑i:(i,j) ∈ A ∑k ∈ SCxijkL, ∑i:(i,j) ∈ A ∑k ∈ SCxijkR)LR; (∑i:(i,j) ∈ A ∑k ∈ SCx′_ijk, ∑i:(i,j) ∈ A ∑k ∈ SCx′¯ijk, ∑i:(i,j) ∈ A ∑k ∈ SCx′ijkL, ∑i:(i,j) ∈ A ∑k ∈ SCx′ijkR)LR} = {(b_j, b¯j, bjL, bjR)LR; (b′_j, b′¯j, b′jL, b′jR)LR}   j ∈ NPD  (P4)

{(∑i:(i,j) ∈ A ∑k ∈ SCx_ijk, ∑i:(i,j) ∈ A ∑k ∈ SCx¯ijk, ∑i:(i,j) ∈ A ∑k ∈ SCxijkL, ∑i:(i,j) ∈ A ∑k ∈ SCxijkR)LR; (∑i:(i,j) ∈ A ∑k ∈ SCx′_ijk, ∑i:(i,j) ∈ A ∑k ∈ SCx′¯ijk, ∑i:(i,j) ∈ A ∑k ∈ SCx′ijkL, ∑i:(i,j) ∈ A∑k ∈ SCx′ijkR)LR} = {(∑i:(j,i) ∈ A ∑k ∈ SCx_jik, ∑i:(j,i) ∈ A∑k ∈ SCx¯jik, ∑i:(j,i) ∈ A ∑k ∈ SCxjikL, ∑i:(j,i) ∈ A ∑k ∈ SCxjikR)LR; (∑i:(j,i) ∈ A ∑k ∈ SCx′_jik, ∑i:(j,i) ∈ A  ∑k ∈ SCx′¯jik,∑i:(j,i) ∈ A∑k ∈ SCx′jikL,∑i:(j,i) ∈ A ∑k ∈ SCx′jikR)LR} ⊕ {(d_j, d¯j, djL, djR)LR; (d′_j, d′¯j, d′jL, d′jR)LR}   j∈ND

{(∑j:(i,j) ∈ A ∑k ∈ SCx_ijk, ∑j:(i,j) ∈ A ∑k ∈ SCx¯ijk, ∑j:(i,j) ∈ A ∑k ∈ SCxijkL, ∑j:(i,j) ∈ A ∑k ∈ SCxijkR)LR; (∑j:(i,j) ∈ A ∑k ∈ SCx′_ijk, ∑j:(i,j) ∈ A ∑k ∈ SCx′¯ijk, ∑j:(i,j) ∈ A ∑k ∈ SCx′ijkL,∑j:(i,j) ∈ A∑k ∈ SCx′ijkR)LR} = {(∑j:(j,i) ∈ A ∑k ∈ SCx_jik, ∑j:(j,i) ∈ A∑k ∈ SCx¯jik, ∑j:(j,i) ∈ A ∑k ∈ SCxjikL, ∑j:(j,i) ∈ A ∑k ∈ SCxjikR)LR; (∑j:(j,i) ∈ A ∑k ∈ SCx′_jik, ∑j:(j,i) ∈ A ∑k ∈ SCx′¯jik,∑j:(j,i) ∈ A∑k ∈ SCx′jikL, ∑j:(j,i) ∈ A ∑k ∈ SCx′jikR)LR}   i ∈ NT{(∑(i,j) ∈ Ax_ijk, ∑(i,j) ∈ Ax¯ijk, ∑(i,j) ∈ AxijkL, ∑(i,j) ∈ AxijkR)LR; (∑(i,j) ∈ Ax′_ijk, ∑(i,j) ∈ Ax′¯ijk, ∑(i,j) ∈ Ax′ijkL, ∑(i,j) ∈ Ax′ijkR)LR} = {(f_k, f¯k, fkL, fkR)LR; f′_k, f′¯k, f′kL, f′kR)LR)}   k ∈ SC{(l_ijk, l¯ijk, lijkL, lijkR)LR; (l′_ijk, l′¯ijk, l′ijkL, l′ijkR)LR} ≺_ {(x_ijk, x¯ijk, xijkL, xijkR)LR; (x′_ijk, x′¯ijk, x′ijkL, x′ijkR)LR} ≺_ {(u_ijk, u¯ijk, uijkL, uijkR)LR;(u′_ijk, u′¯ijk,u′ijkL, u′ijkR)LR}   ∀(i, j) ∈ A, k ∈ SC{(x_ijk, x¯ijk, xijkL, xijkR)LR; (x′_ijk, x′¯ijk, x′ijkL, x′ijkR)LR} is non-negative LR flat intuitionistic fuzzy number ∀(i, j)∈A, k∈SC

Step 7: Using Definitions 7 and 9, the intuitionistic fuzzy single-objective linear programming problem (P4) can be written as

Minimize ∑(i,j) ∈ A ∑k ∈ SC{(c_ijkx_ijk, c¯ijkx¯ijk, c_ijkxijkL + cijkLx_ijk − cijkLxijkL, c¯ijkxijkR + cijkRx¯ijk + cijkRxijkR)LR; (c′_ijkx′_ijk, c′¯ijkx′¯ijk, c′_ijkx′ijkL + c′ijkLx′_ijk − c′ijkLx′ijkL, c′¯ijkx′ijkR+ c′ijkRx′¯ijk + c′ijkRx′ijkR)LR}

Subject to∑j:(i,j) ∈ A ∑k ∈ SCx_ijk = a_ii ∈ NPS∑j:(i,j) ∈ A ∑k ∈ SCx¯ijk = a¯ii ∈ NPS∑j:(i,j) ∈ A ∑k ∈ SCxijkL = aiLi ∈ NPS∑j:(i,j) ∈ A ∑k ∈ SCxijkR = aiRi ∈ NPS∑j:(i,j) ∈ A ∑k ∈ SCx′_ijk = a′_ii ∈ NPS∑j:(i,j) ∈ A ∑k ∈ SCx′¯ijk = a′¯ii ∈ NPS∑j:(i,j) ∈ A ∑k ∈ SCx′ijkL = a′iLi ∈ NPS∑j:(i,j) ∈ A ∑k ∈ SCx′ijkR = a′iRi ∈ NPS∑j:(i,j) ∈ A ∑k ∈ SCx_ijk = ∑j:(j,i) ∈ A ∑k ∈ SCx_jik + e_ii ∈ NS∑j:(i,j) ∈ A ∑k ∈ SCx¯ijk = ∑j:(j,i) ∈ A ∑k ∈ SCx¯jik + e¯ii ∈ NS∑j:(i,j) ∈ A ∑k ∈ SCxijkL = ∑j:(j,i) ∈ A ∑k ∈ SCxjikL + eiLi ∈ NS∑j:(i,j) ∈ A ∑k ∈ SCxijkR = ∑j:(j,i) ∈ A ∑k ∈ SCxjikR + eiRi ∈ NS∑j:(i,j) ∈ A ∑k ∈ SCx′_ijk = ∑j:(j,i) ∈ A ∑k ∈ SCx′_jik + e′_ii ∈ NS∑j:(i,j) ∈ A ∑k ∈ SCx′¯ijk = ∑j:(j,i) ∈ A ∑k ∈ SCx′¯jik + e′¯ii ∈ NS∑j:(i,j) ∈ A ∑k ∈ SCx′ijkL = ∑j:(j,i) ∈ A ∑k ∈ SCx′jikL + e′iLi ∈ NS∑j:(i,j) ∈ A ∑k ∈ SCx′ijkR = ∑j:(j,i) ∈ A ∑k ∈ SCx′jikR + e′iRi ∈ NS∑i:(i,j) ∈ A ∑k ∈ SCx_ijk = b_jj ∈ NPD∑i:(i,j) ∈ A∑k ∈ SCx¯ijk = b¯jj∈NPD∑i:(i,j) ∈ A ∑k ∈ SC = xijkL = bjLj ∈ NPD∑i:(i,j) ∈ A ∑k ∈ SCxijkR = bjRj ∈ NPD∑i:(i,j) ∈ A ∑k ∈ SCx′_ijk = b′_jj ∈ NPD∑i:(i,j) ∈ A ∑k ∈ SCx′¯ijk = b′¯jj ∈ NPD∑i:(i,j) ∈ A ∑k ∈ SCx′ijkL = b′jLj ∈ NPD∑i:(i,j) ∈ A ∑k ∈ SCx′ijkR = b′jRj ∈ NPD∑i:(i,j) ∈ A ∑k ∈ SCx_ijk = ∑i:(j,i) ∈ A ∑k ∈ SCx_jik + d_jj ∈ ND∑i:(i,j) ∈ A ∑k ∈ SCx¯ijk = ∑i:(j,i) ∈ A ∑k ∈ SCx¯jik + d¯jj ∈ ND∑i:(i,j) ∈ A ∑k ∈ SCxijkL = ∑i:(j,i) ∈ A ∑k ∈ SCxjikL + djLj ∈ ND∑i:(i,j) ∈ A ∑k ∈ SCxijkR = ∑i:(j,i) ∈ A ∑k ∈ SCxjikR + djRj ∈ ND∑i:(i,j) ∈ A ∑k ∈ SCx′_ijk =∑i:(j,i) ∈ A ∑k ∈ SCx′_jik + d′_jj ∈ ND∑i:(i,j) ∈ A ∑k ∈ SCx′¯ijk = ∑i:(j,i) ∈ A ∑k ∈ SCx′¯jik + d′¯jj ∈ ND∑i:(i,j) ∈ A ∑k ∈ SCx′ijkL = ∑i:(j,i) ∈ A ∑k ∈ SCx′jikL + d′jLj ∈ ND

(P5)∑i:(i,j) ∈ A ∑k ∈ SCx′ijkR = ∑i:(j,i) ∈ A ∑k ∈ SCx′jikR + d′jRj ∈ ND∑j:(i,j) ∈ A ∑k ∈ SCx_ijk = ∑j:(j,i) ∈ A ∑k ∈ SCx_jiki ∈ NT∑j:(i,j) ∈ A ∑k ∈ SCx¯ijk = ∑j:(j,i) ∈ A ∑k ∈ SCx¯jiki ∈ NT∑j:(i,j) ∈ A ∑k ∈ SCxijkL =∑j:(j,i) ∈ A ∑k ∈ SCxjikLi ∈ NT∑j:(i,j) ∈ A ∑k ∈ SCxijkR = ∑j:(j,i) ∈ A ∑k ∈ SCxjikRi ∈ NT∑j:(i,j) ∈ A ∑k ∈ SCx′_ijk = ∑j:(j,i) ∈ A ∑k ∈ SCx′_jiki ∈ NT∑j:(i,j) ∈ A ∑k ∈ SCx′¯ijk = ∑j:(j,i) ∈ A ∑k ∈ SCx′¯jiki ∈ NT∑j:(i,j) ∈ A ∑k ∈ SCx′ijkL = ∑j:(j,i) ∈ A ∑k ∈ SCx′jikLi ∈ NT∑j:(i,j) ∈ A ∑k ∈ SCx′ijkR = ∑j:(j,i) ∈ A ∑k ∈ SCx′jikRi ∈ NT∑(i,j) ∈ Ax_ijk = f_kk ∈ SC∑(i,j) ∈ Ax¯ijk = f¯kk ∈ SC∑(i,j) ∈ AxijkL = fkLk ∈ SC∑(i,j) ∈ AxijkR = fkRk ∈ SC∑(i,j) ∈ Ax′_ijk = f′_kk ∈ SC∑(i,j) ∈ Ax′¯ijk = f′¯kk ∈ SC∑(i,j) ∈ Ax′ijkL = f′kLk ∈ SC∑(i,j) ∈ Ax′ijkR = f′kRk ∈ SC  (P4)

x′_ijk − x′ijkL ≥ 0, (x_ijk − xijkL) − (x′_ijk − x′ijkL) ≥ 0, x′_ijk − (x_ijk − xijkL) ≥ 0,x_ijk − x′_ijk ≥ 0, x¯ijk − x_ijk ≥ 0, x′¯ijk − x¯ijk ≥ 0, (x¯ijk + xijkR) − x′¯ijk ≥ 0,(x′¯ijk + x′ijkR) − (x¯ijk + xijkR) ≥ 0   ∀(i, j) ∈ A, k ∈ SC

(C1){(l_ijk, l¯ijk, lijkL, lijkR)LR; (l′_ijk, l′¯ijk, l′ijkL, l′ijkR)LR} ≺_ {(x_ijk, x¯ijk, xijkL, xijkR)LR;(x′_ijk, x′¯ijk, x′ijkL, x′ijkR)LR} ≺_ {(u_ijk, u¯ijk, uijkL, uijkR)LR; (u′_ijk, u′¯ijk, u′ijkL,u′ijkR)LR},   ∀(i, j) ∈ A,k ∈ SC}.  (C1)

Step 8: Suppose the intuitionistic fuzzy linear programming problem (P5) has f basic feasible solutions and {((x_ijk)w, (x¯ijk)w, (xijkL)w, (xijkR)w)LR; ((x′_ijk)w, (x′¯ijk)w, (x′ijkL)w, (x′ijkR)w)LR} be the wth basic feasible solution then the goal is to find such a basic feasible solution corresponding to which the value of the objective function is minimum, i.e.

minimum1 ≤ w ≤ f(∑(i,j) ∈ A∑k ∈ SC{(c_ijkx_ijk,c¯ijkx¯ijk,c_ijkxijkL+x_ijkcijkL−cijkLxijkL,c¯ijkxijkR+x¯ijkcijkR+cijkRxijkR)LR; (c′_ijkx′_ijk, c′¯ijkx′¯ijk, c′_ijkx′ijkL + x′_ijkc′ijkL − c′ijkLx′ijkL, c′¯ijkx′ijkR + x′¯ijkc′ijkR + c′ijkRx′ijkR)LR})

which can be obtained using the ranking approach described in Section 5, i.e. the intuitionistic fuzzy optimal solution of the intuitionistic fuzzy linear programming problem (P5), can be obtained by solving the following crisp single-objective linear programming problem:

(P6)Minimize∑(i,j) ∈ A∑k ∈ SCMμβ,k′({(c_ijkx_ijk, c¯ijkx¯ijk,c_ijkxijkL + x_ijkcijkL − cijkLxijkL, c¯ijkxijkR + x¯ijkcijkR + cijkRxijkR)LR; (c′_ijkx′_ijk, c′¯ijkx′¯ijk, c′_ijkx′ijkL + x′_ijkc′ijkL − c′ijkLx′ijkL, c′¯ijkx′ijkR+ x′¯ijkc′ijkR + c′ijkRx′ijkR)LR})Subject toMμβ,k′{(l_ijk, l¯ijk, lijkL, lijkR)LR; (l′_ijk, l′¯ijk, l′ijkL, l′ijkR)LR} ≤ Mμβ,k′{(x_ijk, x¯ijk, xijkL, xijkR)LR; (x′_ijk, x′¯ijk, x′ijkL, x′ijkR)LR} ≤ Mμβ,k′{(u_ijk, u¯ijk, uijkL,uijkR)LR; (u′_ijk, u′¯ijk, u′ijkL, u′ijkR)LR}  (P6)

as well as all the constraints of problem (P5) except (C1).

Case (i): If there does not exist any alternative optimal solution, then put the values of x_ijk, x¯ijk, xijkL, xijkR, x′_ijk, x′¯ijk, x′ijkL and x′ijkR in x˜ijk = {(x_ijk, x¯ijk, xijkL, xijkR)LR; (x′_ijk, x′¯ijk, x′ijkL, x′ijkR)LR} to find the intuitionistic fuzzy optimal solution {x˜ijk} and find the intuitionistic fuzzy optimal value ∑(i,j) ∈ A ∑k ∈ SC(c˜ijk ⊗ x˜ijk) by putting the values of x˜ijk.

Case (ii): If alternative solution exist, then go to Step 9.

Step 9: Solve the crisp linear programming problem (P7) to find the optimal solution {x_ijk,x¯ijk, xijkL, xijkR, x′_ijk, x′¯ijk, x′ijkL, x′ijkR}.

(P7)Minimize∑(i,j) ∈ A ∑k ∈ SCMνβ,k′({(c_ijkx_ijk, c¯ijkx¯ijk, c¯ijkxijkL + x_ijkcijkL − cijkLxijkL, c¯ijkxijkR + x¯ijkcijkR + cijkRxijkR)LR; (c′_ijkx′_ijk, c′¯ijkx′¯ijk, c′_ijkx′ijkL + x′_ijkc′ijkL − c′ijkLx′ijkL, c′¯ijkx′ijkR+ x′¯ijkc′ijkR + c′ijkRx′ijkR)LR})subject toMνβ,k′{(l_ijk, l¯ijk, lijkL, lijkR)LR; (l′_ijk, l′¯ijk, l′ijkL, l′ijkR)LR} ≤ Mνβ,k′{(x_ijk, x¯ijk, xijkL, xijkR)LR; (x′_ijk, x′¯ijk, x′ijkL, x′ijkR)LR} ≤ Mνβ,k′{(u_ijk, u¯ijk, uijkL, uijkR)LR; (u′_ijk, u′¯ijk, u′ijkL, u′ijkR)LR}∑(i,j) ∈ A∑k ∈ SCMμβ,k′({(c_ijkx_ijk, c¯ijkx¯ijk, c_ijkxijkL + x_ijkcijkL − cijkLxijkL, c¯ijkxijkR + x¯ijkcijkR + cijkRxijkR)LR; (c′_ijkx′_ijk, c′¯ijkx′¯ijk, c′_ijkx′ijkL + x′_ijkc′ijkL − c′ijkLx′ijkL, c′¯ijkx′ijkR + x′¯ijkc′ijkR + c′ijkRx′ijkR)LR}) = a (P7)

as well as all the constraints of problem (P5) except (C1), where a is the optimal value of the crisp linear programming problem (P6).

Step 10: Put the values of x_ijk, x¯ijk, xijkL, xijkR, x′_ijk, x′¯ijk, x′ijkL, and x′ijkR in x˜ijk = {(x_ijk, x¯ijk, xijkL, xijkR)LR; (x′_ijk, x′¯ijk, x′ijkL, x′ijkR)LR} to find the intuitionistic fuzzy optimal solution {x˜ijk}.

Step 11: Put the values of x˜ijk obtained from Step 10 in ∑(i,j) ∈ A ∑k ∈ SC(c˜ijk ⊗ x˜ijk), to find the minimum total intuitionistic fuzzy transportation cost.

6.2 Proposed Method for Solving Unbalanced IFFMOCSMCF Problems

In this section, a new method is proposed for solving unbalanced IFFMOCSMCF problems by generalizing the existing method [13].

Step 1: Use Steps 1–3 of the method proposed in Section 6.1 to obtain the balanced IFFMOCSMCF problem.

Step 2: Formulate the chosen IFFMOCSMCF problem into the intuitionistic fuzzy multi-objective linear programming problem (P2) by assuming the following intuitionistic fuzzy penalty as zero LR flat intuitionistic fuzzy numbers:

  1. If any dummy purely source node is introduced then assume the intuitionistic fuzzy penalty for transporting one unit quantity of the product from the introduced dummy purely source node to all purely destination nodes and all intermediate nodes by all conveyance as zero LR flat intuitionistic fuzzy number.

  2. If any dummy purely destination node is introduced, then assume the intuitionistic fuzzy penalty for transporting one unit quantity of the product from all purely source nodes and all intermediate nodes to the introduced dummy purely destination node by all conveyance as zero LR flat intuitionistic fuzzy number.

  3. If any dummy conveyance is introduced, then assume the intuitionistic fuzzy penalty for transporting one unit quantity of the product from all purely source nodes and intermediate nodes to all intermediate nodes and all purely destination nodes by introduced dummy conveyance as zero LR flat intuitionistic fuzzy number.

Step 3: Use the method proposed in Section 6.1 to find P intuitionistic fuzzy optimal solutions X˜1, X˜2, …, X˜P of the problems (P′1), (P′2), …, (P′P), respectively.

Minimize ∑(i,j) ∈ A ∑k ∈ SC(c˜ijk1 ⊗ x˜ijk)Subject to          (P′1)Constraints of problem (P1).Minimize ∑(i,j) ∈ A ∑k ∈ SC(c˜ijk2 ⊗ x˜ijk)Subject to           (P′2)Constraints of problem (P1).⋮Minimize ∑(i,j) ∈ A ∑k ∈ SC(c˜ijkP ⊗ x˜ijk)Subject to           (P′P)Constraints of problem (P1).

Step 4: Find the value of each objective function corresponding to the each intuitionistic fuzzy optimal solution obtained in Step 3. Let the value of ith objective function Z˜i corresponding to jth intuitionistic fuzzy optimal solution X˜j be denoted by Z˜i(X˜j).

Step 5: Find

Ui = maximum1 ≤ j ≤ P{ℜ(Z˜i(X˜j))}  ∀ i = 1, 2, …, PLi = minimum1 ≤ j ≤ P{ℜ(Z˜i(X˜j))}  ∀ i = 1, 2, …, P

where ℜ(Z˜i(X˜j)) = Mμβ,k′(Z˜i(X˜j)) if no alternative solution exists for the ith fuzzy linear programming problem (P′i); otherwise, ℜ(Z˜i(X˜j)) = Mνβ,k′(Z˜i(X˜j)).

Step 6: Define the linear membership function μi(ℜ(Z˜i))  ∀ i = 1, 2, …, P.

μi(ℜ(Z˜i)) = {1,ℜ(Z˜i) ≤ Li 1 − ℜ(Z˜i) − LiUi − Li,Li ≤ ℜ(Z˜i) ≤ Ui0,ℜ(Z˜i) ≥ Ui

Step 7: Using the linear membership function, obtained in Step 6, the intuitionistic fuzzy multi-objective linear programming problem (P2) can be converted into the following single-objective crisp linear programming problem:

(P8)Maximize λ′Subject toℜ(Z˜i) + λ′(Ui − Li) ≤ Ui,   i = 1, 2, …, P,λ′ ≥ 0. (P8)

as well as constraints of the problem (P6).

Step 8: Solve the crisp linear programming problem (P8) to find the optimal values of x_ijk, x¯ijk, xijkL, xijkR, x′_ijk, x′¯ijk, x′ijkL, and x′ijkR.

Step 9: Put the value of x_ijk, x¯ijk, xijkL, xijkR, x′_ijk, x′¯ijk, x′ijkL, and x′ijkR in x˜ijk = {(x_ijk, x¯ijk, xijkL, xijkR)LR; (x′_ijk, x′¯ijk, x′ijkL, x′ijkR)LR} to find the intuitionistic fuzzy optimal solution {x˜ijk}.

Step 10: Find the intuitionistic fuzzy optimal values of each function by putting the value of x˜ijk in ∑(i,j) ∈ A∑k ∈ SC(c˜ijkη ⊗ x˜ijk);  η = 1, 2, …, P.

7 Existing Methods as a Particular Case of the Proposed Method

In this section, it is depicted that the existing methods [11, 13–15] are the particular cases of the proposed method and can be obtained in the following manner:

  1. If in the linear programming formulation (P2) there is only one type of conveyance (SC is a singleton set) and η=1, then the proposed method will be converted into the existing method [15].

  2. Let A˜ = {(a_, a¯, aL, aR)LR; (a′_, a′¯, a′L, a′R)LR} be an LR flat intuitionistic fuzzy number. If a′_ − a′L = a_−aL, a′_ = a_, a¯ = a′¯, and a¯ + aR = a′¯ + a′R, then LR flat intuitionistic fuzzy number A˜ will be converted into LR flat fuzzy number. If in the linear programming formulation (P2) all the LR flat intuitionistic fuzzy number c˜ijkη, x˜ijk, a˜i, e˜i, b˜j, d˜j, f˜k, l˜ijk, and u˜ijk are converted into LR flat fuzzy numbers, then the proposed method will be converted into the existing method [13].

  3. If in the linear programming formulation (P2) there is only one type of conveyance (SC is a singleton set) and all the LR flat intuitionistic fuzzy number c˜ijkη, x˜ijk, a˜i, e˜i, b˜j, d˜j, f˜k, l˜ijk, and u˜ijk are converted into LR flat fuzzy numbers, then the proposed method will be converted into the existing method [14].

  4. If in the linear programming formulation (P2) there is only one type of conveyance (SC is a singleton set), η=1, and all the LR flat intuitionistic fuzzy number c˜ijkη, x˜ijk, a˜i, e˜i, b˜j, d˜j, f˜k, l˜ijk, and u˜ijk are converted into LR flat fuzzy numbers, then the proposed method will be converted into the existing method [11].

8 Advantages of Proposed Method over Existing Methods

The main advantage of the proposed methods over the existing methods [12, 13] is that all the problems that can be solved by the existing methods [12, 13] can also be solved by the methods proposed in this article. However, as discussed in Section 3, there exist several problems that can be solved by the methods proposed in this article but cannot be solved by the existing methods [12, 13]. To show the advantages of the proposed methods, the intuitionistic fully fuzzy single- and multi-objective CSMCF problems chosen in Examples 2 and 3 that cannot be solved using the existing methods [12, 13] are solved by proposed methods.

8.1 Intuitionistic Fuzzy Optimal Solution of the Chosen IFFSOCSMCF Problem

The intuitionistic fuzzy optimal solution of IFFSOCSMCF problem chosen in Example 2 can be obtained as follows:

Step 1: Total intuitionistic fuzzy supply ∑i ∈ NPSa˜i ⊕ ∑i ∈ NSe˜i = {(200, 250, 100, 50)LR; (150, 270, 100, 80)LR}, total intuitionistic fuzzy demand ∑j ∈ NDb˜j ⊕ ∑j ∈ NPDd˜j = {(150, 250, 100, 50)LR; (100, 270, 100, 130)LR}, and total fuzzy capacity ∑k ∈ SCf˜k = {(200, 300, 100, 50)LR; (150, 320, 100, 130)LR}. As ∑i ∈ NPSa˜i ⊕ ∑i ∈ NSe˜i ≠ ∑j ∈ NDb˜j ⊕ ∑j ∈ NPDd˜j ≠ ∑k ∈ SCf˜k, it is an unbalanced IFFSOCSMCF problem.

Step 2: As described in Case 2c of Step 2 of the method proposed in Section 6.1, there is need to introduce a dummy purely source node 4 with intuitionistic fuzzy supply a˜4 = {(0, 50, 0, 0)LR; (0, 50, 0, 50)LR} and a dummy purely destination node 5 with intuitionistic fuzzy demand b˜5 = {(50, 50, 0, 0)LR; (50, 50, 0, 0)LR}, so that ∑i ∈ NPSa˜i ⊕ ∑i ∈ NSe˜i = ∑j ∈ NDb˜j ⊕ ∑j ∈ NPDd˜j.

Step 3: As ∑i ∈ NPSa˜i ⊕ ∑i ∈ NSe˜i = ∑j ∈ NDb˜j ⊕ ∑j ∈ NPDd˜j = {(200, 300, 100, 50)LR; (150, 320, 100, 130)LR} = ∑k ∈ SCe˜k, the IFFSOCSMCF problem obtained in Step 2 is a balanced IFFSOCSMCF problem.

Step 4: As a dummy purely source node (4) and a dummy purely destination node (5) are introduced, as described in Step 2 of the method proposed in Section 6.1, assuming c˜4jk = c˜i5k = {(0, 0, 0, 0)LR; (0, 0, 0, 0)LR} ∀ i = 1, 2, 4;j=2, 3, 5; k=1, 2, the intuitionistic fuzzy linear programming formulation of the balanced IFFSOCSMCF problem obtained from Step 3 can be written as

Minimize ({(100, 1000, 90, 9000)LR; (50, 5500, 45, 9500)LR} ⊗ x˜121 ⊕ {(400, 4000, 360, 36,000)LR; (200, 22,000, 180, 38,000)LR} ⊗ x˜131 ⊕ {(300,3000, 270, 27,000)LR; (150, 16,500, 135, 28,500)LR} ⊗ x˜231 ⊕ {(50, 200, 48, 1800)LR; (10, 1100, 9, 1900)LR} ⊗ x˜122 ⊕ {(200, 2000, 180, 18,000)LR; (100, 11,000, 90, 19,000)LR} ⊗ x˜132 ⊕ {(500, 5000, 450, 45,000)LR; (250, 27,500, 225, 47,500)LR} ⊗ x˜232 ⊕ {(0, 0, 0, 0)LR; (0, 0, 0, 0)LR} ⊗ x˜421 ⊕ {(0, 0, 0, 0)LR; (0, 0, 0, 0)LR} ⊗ x˜431 ⊕ {(0, 0, 0, 0)LR; (0, 0, 0, 0)LR} ⊗ x˜451 ⊕ {(0, 0, 0, 0)LR; (0, 0, 0, 0)LR} ⊗x˜151 ⊕ {(0, 0, 0, 0)LR; (0, 0, 0, 0)LR} ⊗ x˜251 ⊕ {(0, 0, 0, 0)LR; (0, 0, 0, 0)LR} ⊗ x˜422 ⊕ {(0, 0, 0, 0)LR; (0, 0, 0, 0)LR} ⊗ x˜432 ⊕ {(0, 0, 0, 0)LR; (0, 0, 0, 0)LR} ⊗ x˜452 ⊕ {(0, 0, 0, 0)LR; (0, 0, 0, 0)LR} ⊗ x˜152 ⊕ {(0, 0, 0, 0)LR; (0, 0, 0, 0)LR} ⊗ x˜252)

Subject to∑k = 12(x˜12k ⊕ x˜13k ⊕ x˜15k) = {(200, 250, 100, 50)LR; (150, 270, 100, 80)LR}

∑k = 12(x˜12k ⊕ x˜42k) = ∑k = 12(x˜23k ⊕ x˜25k) ⊕ {(100, 150, 80, 50)LR; (60, 170, 60, 80)LR}

∑k = 12(x˜13k ⊕ x˜23k ⊕ x˜43k) = {(50, 100, 20, 0)LR; (40, 100, 40, 50)LR}

∑k = 12(x˜42k ⊕ x˜43k ⊕ x˜45k) = {(0, 50, 0, 0)LR; (0, 50, 0, 50)LR}

∑k = 12(x˜15k ⊕ x˜25k ⊕ x˜45k) = {(50, 50, 0, 0)LR; (50, 50, 0, 0)LR}

x˜121 ⊕ x˜131 ⊕ x˜151 ⊕ x˜231 ⊕ x˜251 ⊕ x˜421 ⊕ x˜431 ⊕ x˜451 = {(0, 50, 0, 0)LR; (0, 50, 0, 50)LR}

x˜122 ⊕ x˜132 ⊕ x˜152 ⊕ x˜232 ⊕ x˜252 ⊕ x˜422 ⊕ x˜432 ⊕ x˜452 = {(200, 250, 100, 50)LR; (150, 270, 100, 80)LR}

{(0, 0, 0, 0)LR; (0, 0, 0, 0)LR} ≺_ x˜121 ≺_ {(65, 70, 35, 10)LR; (60, 75, 45, 10)LR}

{(0, 0, 0, 0)LR; (0, 0, 0, 0)LR} ≺_ x˜131 ≺_ {(40, 50, 20, 20)LR; (30, 60, 20, 40)LR}

{(0, 0, 0, 0)LR; (0, 0, 0, 0)LR} ≺_ x˜231 ≺_ {(150, 220, 100, 40)LR; (100, 240, 80, 60)LR}

{(2, 3, 1, 1)LR; (1, 3, 1, 2)LR} ≺_ x˜122 ≺_ {(100, 150, 80, 150)LR; (60, 170, 60, 180)LR}

{(6, 7, 3, 2)LR; (4, 8, 3, 3)LR} ≺_ x˜132 ≺_ {(150, 220, 100, 40)LR; (100, 240, 80, 60)LR}

{(0, 0, 0, 0)LR; (0, 0, 0, 0)LR} ≺_ x˜232 ≺_ {(50, 55, 42, 10)LR; (13, 60, 9, 11)LR}

x˜ijk are non-negative LR flat intuitionistic fuzzy numbers ∀i=1, 2, 4; j=2, 3, 5; k=1, 2.

Step 5: Assuming β = 13, k′ = 0 and using Steps 6–9 of the method proposed in Section 6.1, the optimal values of x_122, x¯122, x122L, x122R, x′_122, x′¯122, x′122L, x′122R, x_132, x¯132, x132L, x′_132, x′¯132, x′132L, x_152, x¯152, x′_152, x′¯152, x¯431, x′¯431, and x′431R are 100, 150, 80, 50, 60, 170, 60, 80, 50, 50, 20, 40, 50, 40, 50, 50, 50, 50, 50, 50, and 50, respectively. Putting the values of x_ijk, x¯ijk, xijkL, xijkR, x′_ijk, x′¯ijk, x′ijkL, and x′ijkR in x˜ijk = {(x_ijk, x¯ijk, xijkL, xijkR)LR; (x′_ijk, x′¯ijk, x′ijkL, x′ijkR)LR}, the intuitionistic fuzzy optimal solution is

x˜122 = {(100, 150, 80, 50)LR; (60, 170, 60, 80)LR},x˜132 = {(50, 50, 20, 0)LR; (40, 50, 40, 0)LR},x˜152 = {(50, 50, 0, 0)LR; (50, 50, 0, 0)LR}, x˜431 = {(0, 50, 0, 0)LR; (0, 50, 0, 50)LR}, and the remaining x˜ijk are {(0, 0, 0, 0)LR ; (0, 0, 0, 0)LR }, and corresponding the minimum total intuitionistic fuzzy transportation cost is (15,000, 130,000, 14,360, 13,870,000)LR , (4600, 737,000, 4600, 1,513,000)LR }.

8.2 Intuitionistic Fuzzy Optimal Compromise Solution of the Chosen IFFMOCSMCF Problem

The intuitionistic fuzzy optimal compromise solution of IFFMOCSMCF problem chosen in Example 3 can be obtained as follows:

Step 1: Using Steps 1–3 of the method proposed in Section 6.2, the intuitionistic fuzzy optimal solutions X˜1 and X˜2, corresponding to objectives 1 and 2, respectively, are as follows:

X˜1 = {x˜122 = {(100, 150, 80, 50)LR; (60, 170, 60, 80)LR},x˜132 = {(50, 50, 20, 0)LR; (40, 50, 40, 0)LR},x˜431 = {(0, 50, 0, 0)LR; (0, 50, 0, 50)LR},x˜152 = {(50, 50, 0, 0)LR; (50, 50, 0, 0)LR}and the remaining x˜ijk are {(0, 0, 0, 0)LR; (0, 0, 0, 0)LR}

X˜2 = {x˜122 = {(100, 100, 80, 50)LR; (60, 120, 60, 30)LR},x˜132 = {(50, 100, 20, 0)LR; (40, 100, 40, 50)LR},x˜421 = {(0, 50, 0, 0)LR; (0, 50, 0, 50)LR},x˜152 = {(50, 50, 0, 0)LR; (50, 50, 0, 0)LR}and the remaining x˜ijk are {(0, 0, 0, 0)LR; (0, 0, 0, 0)LR}

Step 2: Using Step 4 of the method proposed in Section 6.2, the values of Z˜1(X˜1), Z˜1(X˜2), Z˜2(X˜1), and Z˜2(X˜2) are

{(15,000, 130,000, 14,360, 13,870,000)LR ; (4600, 737,000, 4600, 1,513,000)LR }, {(15,000, 220,000, 14,360, 2,080,000)LR ; (4600, 1,232,000, 4600, 3,718,000)LR , {(4500; 22,500; 4440; 397,500)LR ; (660; 170,000; 660; 555,000)LR }, and {(4500, 15,000, 4440, 300,000)LR ; (660, 120,000, 660, 315,000)LR }, respectively.

Step 3: Using Step 5 of the method proposed in Section 6.2, the values of U1, U2, L1, and L2 are 7,423,200, 1,066,320, 3,733,200, and 676,320, respectively.

Step 4: Using Step 6 of the methods proposed in Section 6.2, the linear membership function μ1(Mνβ,k′(Z˜1)) and μ2(Mνβ,k′(Z˜2)) are as follows:

μ1(Mνβ,k′(Z˜1)) = {1,Mνβ,k′(Z˜1) ≤ 3,733,2001 − Mνβ,k′(Z˜1) − 3,733,2003,690,000,3,733,200 ≤ Mνβ,k′(Z˜1) ≤ 7,423,2000, Mνβ,k′(Z˜1) ≥ 7,423,200

μ2(Mνβ,k′(Z˜2)) = {1,Mνβ,k′(Z2) ≤ 676,3201 − Mνβ,k′(Z˜2) − 676,320390,000,676,320 ≤ Mνβ,k′(Z˜2) ≤ 1,066,3200,Mνβ,k′(Z˜2) ≥ 1,066,320

Step 5: Using the linear membership functions obtained from Step 4 and applying Steps 7–10 of the method proposed in Section 6.2, the minimum total intuitionistic fuzzy transportation cost and minimum total intuitionistic fuzzy passing time are (15,000, 148,704.48, 14,360, 1,438,340.34)LR ; (4600, 839,874.65, 4600, 3,040,692.58)LR } and(4500, 20,941.29, 4440, 377,236.81)LR ; (660, 159,608.62, 660, 390,256.38)LR }, respectively.

9 Comparative Study

To show the advantage of the proposed method over the existing methods [12, 13], the results of some existing and chosen problems, obtained using the existing methods [12, 13] and the method proposed in this article, are compared in Table 8 and are explained as follows:

  1. The existing method [12] can be used for solving fully fuzzy single-objective uncapacitated SMCF problems. As the existing problem 6 [12, pp. 405] is fully fuzzy single-objective uncapacitated SMCF problem, it can be solved by the existing method [12]. However, the existing problem 3 [13, pp. 244] and the problem chosen in Examples 1, 2, and 3 are fully fuzzy multi-objective uncapacitated SMCF, fully fuzzy multi-objective CSMCF, IFFSOCSMCF, and IFFMOCSMCF problems, respectively. Thus, these problems cannot be solved using the existing method [12].

  2. The existing method [13] can be used for solving fully fuzzy single- and multi-objective uncapacitated SMCF problems. As the existing problems 6 [12, pp. 405] and 3 [13, pp. 244] are fully fuzzy single-objective uncapacitated SMCF problem and fully fuzzy multi-objective uncapacitated SMCF problem, respectively, these problems can be solved by the existing method [13]. However, the problems chosen in Examples 1, 2, and 3 are fully fuzzy multi-objective CSMCF problem, IFFSOCSMCF problem, and IFFMOCSMCF problem, respectively. Thus, these problems cannot be solved using the existing method [13].

  3. The methods proposed in this article can be used for solving intuitionistic fully fuzzy single- and multi-objective capacitated and uncapacitated SMCF problems. As the problems chosen in Examples 2 and 3 are IFFSOCSMCF and IFFMOCSMCF problems, respectively, these problems can be solved by the methods proposed in this article. Also, the method proposed in this article is the generalization of the existing methods [12, 13]. Thus, all the problems that can be solved by the existing methods [12, 13] can also be solved by the method proposed in this article.

Table 8

Comparison of Results Obtained Using Existing Methods and Method Proposed in This Article.

ExampleExisting method [14]Existing method [13]Proposed methods
6 [14, pp. 405](54.99, 97.40, 42.41, 42.41)LR(54.99, 97.40, 42.41, 42.41)LR(54.99, 97.40, 42.41, 42.41)LR
3 [13, pp. 244]Not applicable(320, 713.7007, 260, 619.5545)LR(320, 713.7007, 260, 619.5545)LR
(290, 674.2347, 250, 735.3198)LR(290, 674.2347, 250, 735.3198)LR
(180, 559.1662, 180, 590.4454)LR(180, 559.1662, 180, 590.4454)LR
1Not applicableNot applicable(304.45, 730.48, 251.63, 589.96)LR
(298.99, 706.57, 251.81, 726.40)LR
(172.16, 483.63, 168.57, 569.69)LR
2Not applicableNot applicable{(15,000, 130,000, 14,360, 13,870,000)LR , (4600, 737,000, 4600, 1,513,000)LR }
3Not applicableNot applicable{(15,000, 148,704.48, 14,360, 1,438,340.34)LR , (4600, 839,874.65, 4600, 3,040,692.58)LR }
{(4500, 20,941.29, 4440, 3,772,381)LR , (660, 159,608.62,660, 3,902,538)LR }

10 Conclusions

Based on the present study, it can be concluded that all the fuzzy MCF problems and their extensions that can be solved by the existing methods [5–9, 11, 13–15] can also be solved by the proposed method. However, the unbalanced IFFMOCSMCF problem cannot be solved using any of the existing methods [5–9, 11, 13–15] but can be solved by the proposed method.

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Received: 2015-9-22
Published Online: 2015-12-12
Published in Print: 2016-4-1

©2016 by De Gruyter

This article is distributed under the terms of the Creative Commons Attribution Non-Commercial License, which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

Artikel in diesem Heft

  1. Frontmatter
  2. Can Fuzzy Relational Calculus Bring Complex Issues in Selection of Examiners into Focus?
  3. Many-Valued Logic and Zadeh’s Fuzzy Sets: A Stone Representation Theorem for Interval-Valued Łukasiewicz–Moisil Algebras
  4. Fuzzy-EKF Controller for Intelligent Wheelchair Navigation
  5. Neuro-Fuzzy Modeling for Multi-Objective Test Suite Optimization
  6. Efficacy of Fuzzy-Stat Modelling in Classification of Gynaecologists and Patients
  7. A New Method for Solving Single- and Multi-Objective Capacitated Solid Minimum Cost Flow Problems under Uncertainty
  8. A Novel Approach to Construct Fuzzy Systems Based on Fireworks Optimization Algorithm
  9. A New Similarity Measure of Intuitionistic Fuzzy Sets Considering Abstention Group Influence and Its Applications
  10. Sliding Mode-Like Fuzzy Logic Control With Adaptive Boundary Layer for Multiple-Variable Discrete Nonlinear Systems
  11. Intuitionistic Fuzzy Similarity Measures and Their Role in Classification
  12. A Multicriteria Interval-Valued Intuitionistic Fuzzy Set TOPSIS Decision-Making Approach Based on the Improved Score Function
  13. The Rough Linear Approximate Space and Soft Linear Space
  14. A Genetic Algorithm Approach for Discovering Tuned Fuzzy Classification Rules with Intra- and Inter-Class Exceptions
  15. Fuzzy Logic-Based Formalisms for Gynecology Disease Diagnosis
  16. Classification of Ordered Semigroups in Terms of Generalized Interval-Valued Fuzzy Interior Ideals
https://doi.org/10.1515/jisys-2015-0108
Schlagwörter für diesen Artikel
Multi-objective linear programming; intuitionistic fuzzy numbers; solid minimum cost flow
Creative Commons
BY-NC-ND 3.0

Artikel in diesem Heft

  1. Frontmatter
  2. Can Fuzzy Relational Calculus Bring Complex Issues in Selection of Examiners into Focus?
  3. Many-Valued Logic and Zadeh’s Fuzzy Sets: A Stone Representation Theorem for Interval-Valued Łukasiewicz–Moisil Algebras
  4. Fuzzy-EKF Controller for Intelligent Wheelchair Navigation
  5. Neuro-Fuzzy Modeling for Multi-Objective Test Suite Optimization
  6. Efficacy of Fuzzy-Stat Modelling in Classification of Gynaecologists and Patients
  7. A New Method for Solving Single- and Multi-Objective Capacitated Solid Minimum Cost Flow Problems under Uncertainty
  8. A Novel Approach to Construct Fuzzy Systems Based on Fireworks Optimization Algorithm
  9. A New Similarity Measure of Intuitionistic Fuzzy Sets Considering Abstention Group Influence and Its Applications
  10. Sliding Mode-Like Fuzzy Logic Control With Adaptive Boundary Layer for Multiple-Variable Discrete Nonlinear Systems
  11. Intuitionistic Fuzzy Similarity Measures and Their Role in Classification
  12. A Multicriteria Interval-Valued Intuitionistic Fuzzy Set TOPSIS Decision-Making Approach Based on the Improved Score Function
  13. The Rough Linear Approximate Space and Soft Linear Space
  14. A Genetic Algorithm Approach for Discovering Tuned Fuzzy Classification Rules with Intra- and Inter-Class Exceptions
  15. Fuzzy Logic-Based Formalisms for Gynecology Disease Diagnosis
  16. Classification of Ordered Semigroups in Terms of Generalized Interval-Valued Fuzzy Interior Ideals
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