problem.

Minimum Cost Flow (MCF) problems have many applications in almost all industries, such as agriculture, communications, education, energy, manufacturing, medicine, and transportation [1]. Generally, the MCF problem minimizes the cost of transporting products that are available at some sources and required at some destinations. However, there are few MCF problems with only a single objective in the real world. Therefore, in recent years, many authors have considered multiple objective minimum cost flow problems [2]. Another complexity in the real issues is the impreciseness of values of coefficients of the variables in the objective functions, availability, and demand of the products. The fuzzy set theory introduced by Zadeh [3] is a good alternative for this impreciseness. To our knowledge, Zimmermann [4] proposed the first formulation of fuzzy multiobjective linear programming. Also, for the first time, Shih and Lee [5] considered a fuzzy MCF problem. After that, this problem has been studied by many researchers from several viewpoints; see [6]–[8] and the references therein.

Our motivation for this paper is the recent works of Kaur and Kumar[8], [9]. In [9], the authors consider fuzzy multiobjective transportation problems where there exist some nodes, called intermediate nodes, at which the product may be stored in case of the excess of the available product, and later on, the product may be supplied from these intermediate nodes to the destinations. They assumed that there are different types of conveyances, such as trucks, cargo flights, trains, ships, etc., for transporting the products from sources to destinations. Such multiobjective transportation problems in which the conveyances and intermediate nodes are used simultaneously are known as multiobjective Solid Minimal Cost Flow (SMCF) problems [9]. The SMCF problem with fuzzy data was studied by several authors [9]–[13]. Sometimes, we must transport more than one commodity from sources to destinations. These problems are called multicommodity flow problems. Ghatee and Hashemi [14] studied fuzzy multicommodity flow problem, and Chakraborty et al. [15], Dalman et al. [16], Kundu et al. [17], and Rani et al. [18] considered multiobjective multi-item solid transportation problem under uncertainty. This paper considers a Fuzzy Multiobjective Multicommodity Minimal Cost Flow (FMMMCF) problem when conveyance limitations exist. In our model, a conveyance may be allowed to transport a certain amount of a commodity. To our knowledge, this model has no research, even for deterministic data.

This paper is organized into 6 sections. In the next section, some preliminaries of fuzzy numbers are reviewed. In Section 3, we describe our model, and a formulation of the FMMMCF problem is introduced. In Section 4, the new method is proposed, and we illustrate this method with some numerical examples in Section 5. The conclusion and some suggestions are given in Section 6.

In this section, we provide some preliminaries.

Definition 1 ([19]). A function L: [0, ∞) → [0,1] (or R: [0, ∞) → [0,1]) is said to be a reference function of fuzzy numbers if and only if I. L(0) = 1 (or R(0) = 1).

II. L (R) is non-increasing on [0, ∞).

Definition 2 ([20]). A fuzzy number a ̃ = (m, n, α, β)LR is said to be an LR flat fuzzy number if its membership function μ (x) is given by m − x L ( ) , for x ≤ m, α > 0 μ = x − n a ̃ R ( ), for x ≥ n, β > 0

{1, otherwise Definition 3 ([20]). Two LR flat fuzzy numbers a ̃1 = (m1, n1, α1, β1)LR and a ̃2 = (m2, n2, α2, β2)LR are said to be equal, i.e., a ̃1 = a ̃2 if and only if m1 = m2, n1 = n2, α1 = α2, and β1 = β2.

Definition 4 ([21]). An LR flat fuzzy number a ̃ = (m, n, α, β)LR is said to be a non-negative LR flat fuzzy number if and only if m − α ≥ 0.

Let a ̃1 = (m1, n1, α1, β1)LR and a ̃2 = (m2, n2, α2, β2)LR be two LR flat fuzzy numbers. Then

I. a ̃1 ⊕ a ̃2 = (m1 + m2, n1 + n2, α1 + α2, β1 + β2)LR.

II. Let a ̃1 and a ̃2 be non-negative LR flat fuzzy numbers. Then a ̃1 ⊗ a ̃2 ≃ (m1 m2, n1 n2, (m1 − α1)(m2 − α2) − m1 m2, (n1 + β1)(n2 + β2) − n1 n2)LR.

III. Let λ be a real number. Then

(λ m1, λ n1, λ α1, λ β1)LR λ ≥ 0, λa ̃1 = { (λ n1, λ m1, −λ β1, −λ α1)LR λ < 0.

(λ m1, λ n1, λ α1, λ β1)LR λ ≥ 0, λa ̃1 = { (λ n1, λ m1, −λ β1, −λ α1)LR λ < 0 .

In this paper, we use modified Liou and Wang's [22] ranking for the comparison of fuzzy numbers.

Assume, a ̃ = (m, n, α, β)LR .

1 1 −1 1

ℜ(a ̃) = γ [ ∫ (mλ + n(1 − λ))dλ ] + (1 − γ) [λ ∫ (m − α L (ρ))dρ + (1 − λ) ∫ (n +

β R (ρ))dρ ],

where γ ∈ [0,1] and λ ∈ [0,1].

Let a ̃ and b ̃ be two LR flat fuzzy numbers. Then a ̃ ≥ ̃ b ̃ (a ̃ ≤ ̃ b ̃) if ℜ(a ̃) ≥ ℜ(b ̃) (ℜ(a ̃) ≤ ℜ(b ̃)).

In this section, we introduce a fully fuzzy multicommodity multiobjective model for solid minimal cost flow problems in which there are limitations on conveyances for the transport of the products. For example, assume we want to transport coal and petroleum by train from one city to another. We need a special tank for each commodity (coal and petroleum). Therefore, we cannot allocate all the train's capacity to a commodity. For another example, assume that we want to send grease and petroleum from one country to another country by ship. Assume that some rules for importing grease or petroleum by ship in the destination country do not allow you to send more than a particular value of these materials. Therefore, you cannot allocate all the ship's capacity to a commodity. These examples show we must design a new model to cover these problems.

Assume that G = (N, A) is a given network where N is the set of nodes and A is the set of links. We describe our problem with a simple example. Consider a network with two nodes, shown in Fig. 1. We want to send two commodity t1 and t2 from node 1 to node 2. There exist three conveyances between these two nodes, and each conveyance has a total capacity e and furthermore, each conveyance has a capacity for each commodity as et and et . Note that we cannot send commodities more of the total capacity e, while we can have et + et ≥ e.

Similar to [8], [9] we categorize the nodes as follows.

Purely source node: those nodes S in which there exists some node S′ such that the product may be supplied from S to S′ while there does not exist any node S′′ to transport product from S′′ to S. The set of all such nodes is denoted by NPS.

Purely destination node: those nodes D in which there does not exist any node D′ such that the product may be supplied from D to D′ while there exists some node D′′ to transport product from D′′ to D. The set of all such nodes is denoted by NPD.

Intermediate node: the intermediate nodes are another part of a network. In the following, we sort the types of these nodes.

I. Those nodes S have some quantity of the product for supplying to other nodes, and also, there exist some nodes such that some amount of the product is transported from those nodes to node S. Such nodes are called source nodes, and the set of all such nodes is dented by NS.

II. Those nodes D which require some quantity of the product, and also there exist some nodes such that the product is supplied from node D to those nodes. Such nodes are called destination nodes, and the set of all such nodes is dented by ND.

III. Those nodes T which neither any quantity of the product is available at them to transship, nor any quantity of the products is required, and all quantity of the product transferred from some nodes to node T is supplied from T to some other nodes. Such nodes are called transition nodes, and the set of all such nodes is dented by NT.

In the following, we list notations we use to represent our model.

I. a ̃i : the fuzzy availability of the product t at ith purely source node.

II. a ̃i′: the fuzzy availability of the product t at ith source node.

III. bj : the fuzzy demand of the product t at jth purely destination node.

IV. bj ′: the fuzzy demand of the product t at jth destination node.

V. e ̃k: the fuzzy capacity of the kth conveyance for transfer of the product t.

VI. e ̃k: the total fuzzy capacity of the kth conveyance.

tl

VII. c ̃ijk: the fuzzy penalty per unit of flow t from ith (purely) source to jth (purely) destination by means of the kth conveyance in the lth objective function.

VIII. x ̃ijk: the fuzzy quantity of the product t that should be transported from ith node to jth node by means of the kth conveyance to minimize all objective functions.

IX. SC: the set of all available conveyances.

t t ̃t ̃t t

We assume that a ̃i, a ̃i′ ,bj , bj′, e ̃k, e ̃k are non-negative LR flat fuzzy numbers. We also assume that there are L objective functions and T commodities. With these notations, a FMMMCF problem can be formulated into the following fuzzy multiobjective linear programming problem:

tl t

Minimum ∑ ∑ ∑ (c ̃ijk ⊗ x ̃ijk) l = 1, … , L, (1)

t ̃ t ∑j:(i,j)∈ A ∑k∈ S x ̃ ≤ a ̃ , i ∈ NPS, t = 1, … , T,

C ijk i t t t′

∑ ∑ x ̃ ≤ ̃ ∑ ∑ x ̃ ⊕ a ̃ , i ∈ N , t = 1, … , T,

j:(i,j)∈ A k∈ SC ijk j:(j,i)∈ A k∈ SC jik i S

t ̃ ̃t ∑i:(i,j)∈ A ∑k∈ S x ̃ ≥ b , j ∈ NPD, t = 1, … , T,

C ijk j t t t′

∑ ∑ x ̃ ≥ ̃ ∑ ∑ x ̃ ⊕ b ̃ , j ∈ N , t = 1, … , T,

i:(i,j)∈ A k∈ SC ijk i:(j,i)∈ A k∈ SC jik j D

∑j:(i,j)∈ A ∑k∈ S x ̃ = ∑j:(j,i)∈ A ∑k∈ S x ̃ , i ∈ NT, t = 1, … , T,

C ijk C jik

T t ̃ ∑t=1 ∑j:(i,j)∈ A x ̃ ≤ e ̃k , k ∈ SC,

ijk t ̃ t x ̃ijk ≤ e ̃k , k ∈ SC, (i, j) ∈ A, t = 1, … , T.

where x ̃ijk is a non-negative LR flat fuzzy number for all (i, j) ∈ A and k ∈ SC.

Almost in all available algorithms for MCF problems, we must examine whether the problem is balanced or unbalanced and convert an unbalanced one to a balanced one with some modifications. This process may be expensive, so solving our model without this assumption is better. In our model, there are some equalities and inequalities; therefore, with a generalization of the proposed algorithm in Subsection 5.4.2 of [23] and using existing methods [24] to solve multiobjective linear programming, we can obtain the optimal compromise solution for our model. We recall the definition of a fuzzy efficient solution from the literature. Definition 5. A fuzzy, feasible solution x ̃ = {x ̃ijk} is said to be a fuzzy efficient solution of the fully fuzzy multiobjective SMMMCF problem if there is no other fuzzy feasible solution x ̃′ = {x ̃ijk} such that

tl t′ tl t

∑ ∑ ∑ ℜ(c ̃ijk ⊗ x ̃ijk) ≤ ∑ ∑ ∑ ℜ(c ̃ijk ⊗ x ̃ijk),

t=1 (i,j)∈ A k∈ SC t=1 (i,j)∈ A k∈ SC

for all l ∈ {1, … , L}, and

tl t′ tl t

∑ ∑ ∑ ℜ(c ̃ijk ⊗ x ̃ijk) < ∑ ∑ ∑ ℜ(c ̃ijk ⊗ x ̃ijk),

t=1 (i,j)∈ A k∈ SC t=1 (i,j)∈ A k∈ SC

for at least one l ∈ {1, … , L}.

Note that, for real-world problems, we do not need to obtain the set of all fuzzy efficient solutions. It is sufficient to compute a fuzzy optimal compromise solution. A fuzzy optimal compromise solution of the FMMMCF problem is a feasible solution that the decision maker prefers to all other solutions. We accept that a fuzzy optimal compromise solution must be a fuzzy efficient solution.

tl tl tl tl tl t t t t t t t t t t t′

Step 1. Assume that c ̃ijk = (pijk, qijk, αijk, βijk)LR, x ̃ijk = (yijk, zijk, γijk, δijk)LR, a ̃i = (ri , si , εi , ζi)LR, a ̃i =

t′ t′ t′ t′ ̃t t t t t ̃t′ t′ t′ t′ t′ t (ri , si , εi , ζi )LR, bj = (vj , wj , ηj, θj )LR, bj = (vj , wj , ηj , θj )LR, e ̃k = (gk, hk, λk, μk)LR, and e ̃k =

t t t t

(gk, hk, λk, μk)LR .Therefore, Problem (1) can be written as

tl tl tl tl t t t t

Minimum ∑ ∑ ∑ (pijk, qijk, αijk, βijk)LR ⊗ (yijk, zijk, γijk, δijk)LR = 1, … , L, (2)

t t t t ̃ t t t t ∑j:(i,j)∈ A ∑k∈ S (y , z , γ , δ )LR ≤ (r , s , ε , ζ )LR, i ∈ NPS, t = 1, … , T,

C ijk ijk ijk ijk i i i i

t t t t ̃ t t t t ∑J:(I,j)∈ A ∑k∈ S (y , z , γ , δ )LR ≤ ∑j:(j,i)∈ A ∑k∈ S (y , z , γ , δ )LR ⊕

C ijk ijk ijk ijk C jik jik jik jik

′ ′ ′ ′

t t t t

(ri , si , εi , ζi ) LR, i ∈ NS, t = 1, … , T, t t t t ̃ t t t t ∑ ∑ (yijk, zijk, γijk, δijk)LR ≥ (vj , wj , ηj, θj), j ∈ NPD, t = 1, … , T,

i:(i,j)∈ A k∈ SC

t t t t ̃ t t t t ∑ ∑ (yijk, zijk, γijk, δijk)LR ≥ ∑ ∑ (yjik, zjik, γjik, δjik)LR

i:(i,j)∈ A k∈ SC i:(j,i)∈ A k∈ SC

′ ′ ′ ′

t t t t

⊕ (vj , wj , ηj , θj ) LR, j ∈ ND, t = 1, … , T,

t t t t t t t t

∑ ∑ (yijk, zijk, γijk, δijk)LR = ∑ ∑ (yjik, zjik, γjik, δjik)LR, i ∈ NT, t

j:(i,j)∈ A k∈ SC j:(j,i)∈ A k∈ SC

= 1, … , T, t t t t ̃ ∑ ∑ (yijk, zijk, γijk, δijk)LR ≤ (gk, hk, λk, μk)LR, k ∈ SC,

t=1 j:(i,j)∈ A

t t t t ̃ t t t t (yijk, zijk, γijk, δijk)LR ≤ (gk, hk, λk, μk)LR, k ∈ SC, (i, j) ∈ A, t = 1, … , T.

t t t t

where (y , z , γ , δ )LR is a non-negative LR flat fuzzy number for all k ∈ SC, (i, j) ∈ A, t = 1, … , T.

ijk ijk ijk ijk

Step 2. Assume that

tl tl tl tl t t t t tl tl tl tl

(pijk, qijk, αijk, βijk)LR ⊗ (yijk, zijk, γijk, δijk)LR = (oijk, uijk, φijk, τijk)LR,

t t t t t′ t′ t′ t′ t t t t

(yjik, zjik, γjik, δjik)LR ⊕ (ri , si , εi , ζi )LR = (mjik, njik, πjik, σjik) ,

LR

and

t t t t t′ t′ t′ t′ t′ t′ t′ t′

(yjik, zjik, γjik, δjik)LR ⊕ (vj , wj , ηj , θj )LR = (mjik, njik, πjik, σjik) .

LR

With these notations, Problem (2) can be written as

tl tl tl tl

Minimum ∑ ∑ ∑ (oijk, uijk, φijk, τijk)LR , l = 1, … , L, (3)

t t t t ̃ t t t t ∑ ∑ (yijk, zijk, γijk, δijk)LR ≤ (ri , si , εi , ζi)LR, i ∈ NPS, t = 1, … , T,

j:(i,j)∈ A k∈ SC

t t t t ̃ t t t t ∑j:(i,j)∈ A ∑k∈ S (y , z , γ , δ )LR ≤ (m , n , π , σ ) , i ∈ NS, t = 1, … , T,

C ijk ijk ijk ijk jik jik jik jik LR

t t t t ̃ t t t t ∑ ∑ (yijk, zijk, γijk, δijk)LR ≥ (vj , wj , ηj, θj) , j ∈ NPD, t = 1, … , T,

i:(i,j)∈ A k∈ SC

t t t t ̃ t′ t′ t′ t′ ∑ ∑ (yijk, zijk, γijk, δijk)LR ≥ ∑ ∑ (mjik, njik, πjik, σjik) , j ∈ ND, t = 1, … , T,

LR i:(i,j)∈ A k∈ SC i:(j,i)∈ A k∈ SC t t t t t t t t

∑ ∑ (yijk, zijk, γijk, δijk)LR = ∑ ∑ (yjik, zjik, γjik, δjik)LR ,

j:(i,j)∈ A k∈ SC j:(j,i)∈ A k∈ SC

i ∈ NT, t = 1, … , T,

t t t t ̃ ∑ ∑ (yijk, zijk, γijk, δijk)LR ≤ (gk, hk, λk, μk)LR, k ∈ SC,

t=1 j:(i,j)∈ A

t t t t ̃ t t t t (yijk, zijk, γijk, δijk)LR ≤ (gk, hk, λk, μk)LR, k ∈ SC, (i, j) ∈ A, t = 1, … , T.

t t t t

where (yijk, zijk, γijk, δijk)LR is a non-negative LR flat fuzzy number for all k ∈ SC, (i, j) ∈ A, t = 1, … , T.

Step 3. Using rank function ℜ, we solve the following problem:

tl tl tl tl

Minimum ℜ (∑ ∑ ∑ (oijk, uijk, φijk, τijk)LR) l = 1, … , L, (4)

t t t t t t t t

ℜ(∑j:(i,j)∈ A ∑k∈ S (y , z , γ , δ )LR) ≤ ℜ(r , s , ε , ζ )LR, i ∈ NPS, t = 1, … , T,

C ijk ijk ijk ijk i i i i t t t t t t t t

ℜ(∑j:(i,j)∈ A ∑k∈ S (y , z , γ , δ )LR) ≤ ℜ(m , n , π , σ ) , i ∈ NS, t = 1, … , T,

C ijk ijk ijk ijk jik jik jik jik LR

t t t t t t t t

ℜ ( ∑ ∑ (yijk, zijk, γijk, δijk)LR) ≥ ℜ(vj , wj , ηj, θj), j ∈ NPD, t = 1, … , T,

i:(i,j)∈ A k∈ SC ′ ′ ′ ′

t t t t t t t t

ℜ ( ∑ ∑ (yijk, zijk, γijk, δijk)LR) ≥ ℜ ( ∑ ∑ (mjik, njik, πjik, σjik) ),

LR i:(i,j)∈ A k∈ SC i:(j,i)∈ A k∈ SC

j ∈ ND, t = 1, … , T,

∑ ∑ yijk = ∑ ∑ yjik , i ∈ NT,

j:(i,j)∈ A k∈ SC j:(j,i)∈ A k∈ SC

∑ ∑ zijk = ∑ ∑ zjik , i ∈ NT,

j:(i,j)∈ A k∈ SC j:(j,i)∈ A k∈ SC

∑ ∑ γijk = ∑ ∑ γjik , i ∈ NT,

j:(i,j)∈ A k∈ SC j:(j,i)∈ A k∈ SC

∑ ∑ δijk = ∑ ∑ δjik , i ∈ NT,

j:(i,j)∈ A k∈ SC j:(j,i)∈ A k∈ SC

t t t t

ℜ (∑ ∑ (yijk, zijk, γijk, δijk)LR) ≤ ℜ(gk, hk, λk, μk)LR k ∈ SC,

t=1 j:(i,j)∈ A t t t t t t t t

ℜ(yijk, zijk, γijk, δijk)LR ≤ ℜ(gk, hk, λk, μk)LR, k ∈ SC, (i, j) ∈ A, t = 1, … , T,

t t t t t t

yijk − γijk, zijk − yijk, γijk, δijk ≥ 0, k ∈ SC, (i, j) ∈ A, t = 1, … , T.

Step 4. Concerning the linear property of the rank Function (4) can be written as:

tl tl tl tl

Minimum ∑ ∑ ∑ ℜ(oijk, uijk, φijk, τijk)LR l = 1, … , L, (5)

t t t t t t t t

∑j:(i,j)∈ A ∑k∈ S ℜ(y , z , γ , δ )LR ≤ ℜ(r , s , ε , ζ )LR, i ∈ NPS, t = 1, … , T,

C ijk ijk ijk ijk i i i i t t t t t t t t

∑j:(i,j)∈ A ∑k∈ S ℜ(y , z , γ , δ )LR ≤ ℜ(m , n , π , σ ) , i ∈ NS, t = 1, … , T,

C ijk ijk ijk ijk jik jik jik jik LR t t t t t t t t

∑ ∑ ℜ(yijk, zijk, γijk, δijk)LR ≥ ℜ(vj , wj , ηj, θj) , j ∈ NPD, t = 1, … , T,

i:(i,j)∈ A k∈ SC

′ ′ ′ ′

t t t t t t t t

∑ ∑ ℜ(yijk, zijk, γijk, δijk)LR ≥ ∑ ∑ ℜ (mjik, njik, πjik, σjik) ,

LR i:(i,j)∈ A k∈ SC i:(j,i)∈ A k∈ SC

j ∈ ND, t = 1, … , T,

∑ ∑ yijk = ∑ ∑ yjik , i ∈ NT,

j:(i,j)∈ A k∈ SC j:(j,i)∈ A k∈ SC

∑ ∑ zijk = ∑ ∑ zjik , i ∈ NT,

j:(i,j)∈ A k∈ SC j:(j,i)∈ A k∈ SC

∑ ∑ γijk = ∑ ∑ γjik , i ∈ NT,

j:(i,j)∈ A k∈ SC j:(j,i)∈ A k∈ SC

∑ ∑ δijk = ∑ ∑ δjik , i ∈ NT,

j:(i,j)∈ A k∈ SC j:(j,i)∈ A k∈ SC

t t t t

∑ ∑ ℜ(yijk, zijk, γijk, δijk)LR ≤ ℜ(gk, hk, λk, μk)LR, k ∈ SC,

t=1 j:(i,j)∈ A t t t t t t t t

ℜ(yijk, zijk, γijk, δijk)LR ≤ ℜ(gk, hk, λk, μk)LR, k ∈ SC, (i, j) ∈ A, t = 1, … , T, t t t t t t

yijk − γijk, zijk − yijk, γijk, δijk ≥ 0, k ∈ SC, (i, j) ∈ A, t = 1, … , T.

Step 5. With solving the crisp programming Problem (5), find the optimal compromise solution xijk =

t∗ t∗ t∗ t∗

(y , zijk, γijk, δijk) LR.

ijk

Step 6. Find the fuzzy optimal value of each objective function by putting the values of

t∗ t∗ t∗ t∗ t∗ T tl t

x = (y , z , γ , δ )LR in ∑t=1 ∑(i,j)∈ A ∑k∈ S (c ̃ ⊗ x ̃ ).

ijk ijk ijk ijk ijk C ijk ijk

In this section, we illustrate our method with an example.

Example 1. Consider the network Fig. 2 with the following data. Fuzzy penalty for 1st objective function to transport commodity 1st

c131 = (3, 4, 2, 2)LR, c132 = (2, 3, 1, 2)LR.

c211 = (4, 5, 3, 3)LR, c212 = (5, 6, 3, 3)LR.

c231 = (5, 7, 4, 3)LR, c232 = (3, 4, 2, 3)LR.

Fuzzy penalty for 2nd objective function to transport commodity 1st

c131 = (4, 6, 3, 3)LR, c132 = (3, 4, 2, 2)LR.

c211 = (5, 6, 4, 3)LR, c212 = (6, 7, 4, 4)LR.

c231 = (6, 8, 4, 4)LR, c232 = (4, 5, 2, 3)LR.

Fuzzy penalty for 1st objective function to transport commodity 2nd

c131 = (2, 3, 1, 1)LR, c132 = (2, 4, 1, 2)LR.

c211 = (4, 5, 3, 2)LR, c212 = (2, 3, 2, 2)LR.

c231 = (4, 6, 3, 3)LR, c232 = (2, 3, 1, 2)LR.

Fuzzy penalty for 2nd objective function to transport commodity 2nd

c131 = (3, 4, 2, 1)LR, c132 = (3, 4, 2, 2)LR.

c211 = (5, 7, 4, 3)LR, c212 = (5, 6, 3, 3)LR.

c231 = (5, 6, 4, 3)LR, c232 = (2, 3, 1, 1)LR.

Fuzzy availability of the commodity 1st at source node 1 and purely source node 2

1′ 1

a1 = (30, 40, 20, 10)LR, a2 = (30, 40, 20, 20)LR.

Fuzzy availability of the commodity 2nd at source node 1 and purely source node 2

2′ 2

a1 = (40, 60, 20, 30)LR, a2 = (40, 50, 30, 30)LR.

Fuzzy demand of the commodity 1st at purely destination node 3 b3 = (30, 50, 20, 30)LR.

Fuzzy demand of the commodity 2nd at purely destination node 3 b3 = (40, 50, 30, 30)LR.

Fuzzy capacity of the 1st conveyance for transfer of the commodities 1st and 2nd e1 = (60, 70, 40, 30)LR, e1 = (60, 70, 30, 30)LR.

Fuzzy capacity of the 2nd conveyance for transfer of the commodities 1st and 2nd e2 = (60, 70, 30, 30)LR, e2 = (60, 70, 20, 20)LR.

The total fuzzy capacity of the 1st and 2nd conveyances e1 = (70, 70, 30, 30)LR, e2 = (70, 80, 20, 20)LR.

We assume that L(x) = R(x) = max {0, 1 − x }. Therefore, for a fuzzy number a ̃ = (m, n, α, β), ℜ(a ̃) = (m + n) + (β − α) (see Remark 1 in [9]).

The model will be as follows:

Minimum (3, 4, 2, 2)LR ⊗ x131 ⊕ (2, 3, 1, 2)LR ⊗ x132 ⊕ (4, 5, 3, 3)LR ⊗ x211

⊕ (5, 6, 3, 3)LR ⊗ x212 ⊕ (5, 7, 4, 3)LR ⊗ x231 ⊕ (3, 4, 2, 3)LR ⊗ x232

⊕ (2, 3, 1, 1)LR ⊗ x131 ⊕ (2,4, 1, 2)LR ⊗ x132 ⊕ (4, 5, 3, 2)LR ⊗ x211

⊕ (2, 3, 2, 2)LR ⊗ x212 ⊕ (4, 6, 3, 3)LR ⊗ x231 ⊕ (2, 3, 1, 2)LR ⊗ x232, Minimum (4, 6, 3, 3)LR ⊗ x131 ⊕ (3, 4, 2, 2)LR ⊗ x132 ⊕ (5, 6, 4, 3)LR ⊗ x211

⊕ (6, 7, 4, 4)LR ⊗ x212 ⊕ (6, 8, 4, 4)LR ⊗ x231 ⊕ (4, 5, 2, 3)LR ⊗ x232

⊕ (3, 4, 2, 1)LR ⊗ x131 ⊕ (3, 4, 2, 2)LR ⊗ x132 ⊕ (5, 7, 4, 3)LR ⊗ x211 ⊕ (5, 6, 3, 3)LR ⊗ x212 ⊕ (5, 6, 4, 3)LR ⊗ x231 ⊕ (2, 3, 1, 1)LR ⊗ x232,

1 1 1 1 ̃ x211 ⊕ x212 ⊕ x231 ⊕ x232 ≤ (30, 40, 20, 20)LR, 2 2 1 2 ̃ x211 ⊕ x212 ⊕ x231 ⊕ x232 ≤ (40, 50, 30, 30)LR, 1 1 ̃ 1 1 x131 ⊕ x132 ≤ x211 ⊕ x212 ⊕ (30, 40, 20, 10)LR, 2 2 ̃ 2 2 x131 ⊕ x132 ≤ x211 ⊕ x212 ⊕ (40, 60, 20, 30)LR, 1 1 1 1 ̃ x131 ⊕ x132 ⊕ x231 ⊕ x232 ≥ (30, 50, 20, 30)LR, 2 2 2 2 ̃ x131 ⊕ x132 ⊕ x231 ⊕ x232 ≥ (40, 50, 30, 30)LR, 1 1 1 2 2 2 ̃ x211 ⊕ x231 ⊕ x131 ⊕ x211 ⊕ x231 ⊕ x131 ≤ (70, 70, 30, 30)LR, x211 ≤ (60, 70, 40, 30)LR, x212 ≤ (60, 70, 30, 30)LR, x231 ≤ (60, 70, 40, 30)LR, x232 ≤ (60, 70, 30, 30)LR, x211 ≤ (60, 70, 30, 30)LR, x212 ≤ (60, 70, 20, 20)LR, x231 ≤ (60, 70, 30, 30)LR, x231 ≤ (60, 70, 20, 20)LR.

And xijk (i = j = 1, 2, 3, k, t = 1, 2) is a non-negative LR flat fuzzy number. Concerning Steps 3 and 4 in Section 4, the fuzzy optimal solution can be obtained by solving the following problem:

1 1 1 1 1 1 1 1

Minimum (61 y131 + 76 z131 + 8 γ131 + 48 δ131 + 38 y132 + 61 z132 + 8 γ132 +

1 1 1 1 1 1 1 1

40 δ132 + 84 y211 + 99 z211 + 8 γ211 + 64 δ211 + 99 y212 + 114 z212 + 16 γ212 +

1 1 1 1 1 1 1 1 1

72 δ212 + 107y231 + 129 z231 + 8 γ231 + 80 δ231 + 61y232 + 84 z232 + 8 γ232 + 56 δ232 +

2 2 2 2 2 2 2 2 2

38 y131 + 53 z131 + 8 γ131 + 32 δ131 + 38 y132 + 76 z132 + 8 γ132 + 48 δ132 + 84 y211 +

2 2 2 2 2 2 2 2 2

91 z211 + 8 γ211 + 56 δ211 + 46y212 + 61 z212 + 0 γ212 + 40 δ212 + 84y231 + 114 z231 +

2 2 2 2 2 2

8 γ231 + 72 δ231 + 38y232 + 61 z232 + 8 γ232 + 40 δ232),

1 1 1 1 1 1 1

Minimum (84 y131 + 114 z131 + 8 γ131 + 72 δ131 + 61 y132 + 76 z132 + 8 γ132

1 1 1 1 1 1

+ 48 δ132 + 107 y211 + 114 z211 + 8 γ211 + 72 δ211 + 122 y212

1 1 1 1 1 1

+ 137 z212 + 16 γ212 + 88 δ212 + 122y231 + 152 z231 + 16 γ231

1 1 1 1 1 2

+ 96 δ231 + 76y232 + 99 z232 + 16 γ232 + 64 δ232 + 61 y131

2 2 2 2 2 2 2

+ 68 z131 + 8 γ131 + 40 δ131 + 61 y132 + 76 z132 + 8 γ132 + 48 δ132

2 2 2 2 2 2

+ 107 y211 + 129 z211 + 8 γ211 + 80 δ211 + 99y212 + 114 z212

2 2 2 2 2 2

+ 16 γ212 + 72 δ212 + 107y231 + 114 z231 + 8 γ231 + 72 δ231

2 2 2 2

+ 38y232 + 53 z232 + 8 γ232 + 32 δ232),

1 1 1 1 1 1 1 1

(y211 + y212 + y231 + y232 + z211 + z212 + z231 + z232)

1 1 1 1 1 1 1 1

+ (δ211 + δ212 + δ231 + δ232 − γ211 − γ212 − γ231 − γ232) ≤ 35,

2 2 2 2 2 2 2 2

(y211 + y212 + y231 + y232 + z211 + z212 + z231 + z232)

2 2 2 2 2 2 2 2

+ (δ211 + δ212 + δ231 + δ232 − γ211 − γ212 − γ231 − γ232) ≤ 45,

1 1 1 1 1 1 1 1

(y131 + y132 + z131 + z132) + (δ131 + δ132 − γ131 − γ132)

1 1 1 1

≤ (70 + y211 + y212 + z211 + z212)

1 1 1 1

+ (δ211 + δ212 − γ211 − γ212 − 10),

1 2 2 2 2 4 2 2 2 2 1 2 2

(y131 + y132 + z131 + z132) + (δ131 + δ132 − γ131 − γ132) ≤ (100 + y211 + y212 +

2 2 4 2 2 2 2

z211 + z212) + (δ211 + δ212 − γ211 − γ212 + 10),

1 1 1 1 1 1 1 1 1 4 1 1 1

(y131 + y132 + y231 + y232 + z131 + z132 + z231 + z232) + (δ131 + δ132 + δ231 +

1 1 1 1 1 128

δ232 − γ131 − γ132 − γ231 − γ232) ≥ , 2 2 2 2 2 2 2 2

(y131 + y132 + y231 + y232 + z131 + z132 + z231 + z232)

2 2 2 2 2 2 2 2

+ (δ131 + δ132 + δ231 + δ232 − γ131 − γ132 − γ231 − γ232) ≥ 45,

1 1 1 2 2 2 1 1 1 2 2 2

(y211 + y231 + y131 + y211 + y231 + y131 + z211 + z231 + z131 + z211 + z231 + z131)

1 1 1 2 2 2 1 1 1

+ (δ211 + δ231 + δ131 + δ211 + δ231 + δ131 − γ211 − γ231 − γ131 − γ211 − γ231 − γ131) ≤ 70,

1 1 1 4 1 1 187

(y211 + z211) + (δ211 − γ211) ≤ ,

1 1 1 1

(y212 + z212) + (δ212 − γ212) ≤ 65,

1 1 1 1

(y231 + z231) + (δ231 − γ231) ≤ ,

1 1 1 1

(y232 + z232) + (δ232 − γ232) ≤ 65,

2 2 2 2

(y211 + z211) + (δ211 − γ211) ≤ 65,

2 2 2 2

(y212 + z212) + (δ212 − γ212) ≤ 65,

2 2 2 2

(y231 + z231) + (δ231 − γ231) ≤ 65,

2 2 2 2

(y232 + z232) + (δ232 − γ232) ≤ 65,

t t t t t t

yijk − γijk, zijk − yijk, γijk, δijk ≥ 0, k ∈ SC, (i, j) ∈ A, t = 1, … , T.

By solving this problem using the weighted sum method [24], we have

y231 0.0415 y232 0.0103 y211 0.0627 z231 0.0198 z232 0.0103 z211 0.0979 γ231 0 γ232 0.6951 γ211 0 δ231 0 δ232 0 δ211 0 y212 0.1259 y131 0.0322 y132 0.0481 z212 0.1259 z331 0.0322 z132 0.2467 γ212 0 γ131 0 γ132 0 δ212 0 δ131 0.1873 δ132 0

y231 0.0208 y232 0.0437 y211 0.0474 z231 0.1824 z232 0.0437 z211 0.087 γ231 0 γ232 0 γ211 0.0215 δ231 0 δ232 0 δ211 0

y212 -0.1491 y131 -0.3316 y132 -0.4967 z212 -0.1491 z131 -0.3316 z132 -1.0268 γ212 0 γ131 0.3438 γ132 0 δ212 0 δ131 0 δ132 0

In this paper, we introduced a new model for fully fuzzy multiobjective multicommodity minimal cost flow problems in which there are several commodities to transport from sources to destinations, and there is more than one conveyance for this transporting. We also assume that each commodity has distinct capacities in each conveyance. We proposed a method for solving this problem without considering a balanced version of that. Our method can also be considered a generalization of some methods for solving fuzzy multiobjective methods in the presence of equalities and fuzzy inequalities.

Conceptualization, M. Gh.; Methodology, B. Kh., Software, Validation, formal analysis, and investigation, V. I..; writing-creating the initial design, M. Gh.; writing-reviewing and editing, B. kh. All authors have read and agreed to the published version of the manuscript.

This research received no external funding.

All the data are available in this paper.

The authors declare no conflict of interest.