The following definitions are used to compare any two TrFNs.

Definition 3.1

A trapezoidal fuzzy number à = [a₁, a₂, a₃, a₄] is said to be non negative trapezoidal fuzzy number if and only if ℜ(Ã) ≥ 0.

Definition 3.2

A trapezoidal fuzzy number à = [a₁, a₂, a₃, a₄] is said to be zero trapezoidal fuzzy number if and only if ℜ(Ã) = 0.

Definition 3.3

Two trapezoidal fuzzy numbers à = [a₁, a₂, a₃, a₄] and B̃ = [b₁, b₂, b₃, b₄] are said to be equal trapezoidal fuzzy number if and only if ℜ(Ã) = ℜ(B̃).

Definition 3.4 (Liou and Wang [32])

A ranking function is a function ℜ : F(R) → R, where F(R) is a set of fuzzy numbers defined on set of real numbers, which maps each fuzzy number into the real line.

Let à = [a₁, a₂, a₃, a₄] be a trapezoidal fuzzy number then R(A˜)=(a1+a2+a3+a44).

Let à = [a₁, a₂, a₃, a₄] and B̃ = [b₁, b₂, b₃, b₄] be two TrFNs. Then,

• ℜ(Ã) > ℜ(B̃) iff à ≻ B̃

• ℜ(Ã) < ℜ(B̃) iff à ≺ B̃

• ℜ(Ã) = ℜ(B̃) iff à ≈ B̃,

where R(A˜)=(a1+a2+a3+a44), R(B˜)=(b1+b2+b3+b44).

Theorem 1

The ranking function ℜ : F(R) → R is a linear function.

The following basic terminologies used in this article are defined in this section.

Definition 4.1

If the transportation problem has at least one of the parameter (cost) or two of the parameters (supply and demand) or all of the parameters (supply, demand and cost) are in fuzzy numbers then the problem is called FTP.

Further, FTP can be classified into four categories. They are

• Type-1 FTP

• Type-2 FTP

• Type-3 FTP (mixed FTP)

• Type-4 FTP (fully FTP)

Definition 4.2

A transportation problem having fuzzy availabilities and fuzzy demands but crisp costs is termed as fuzzy transportation problem of type-1.

Definition 4.3

A transportation problem having crisp availabilities and crisp demands but fuzzy costs is termed as fuzzy transportation problem of type-2.

Definition 4.4

The transportation problem is said to be the type-3 fuzzy transportation problem or mixed fuzzy transportation problem if all the parameters of the transportation problem (such as supplies, demands and costs) must be in the mixture of crisp numbers, triangular fuzzy numbers and trapezoidal fuzzy numbers.

Definition 4.5

The transportation problem is said to be the type-4 fuzzy transportation problem or fully fuzzy transportation problem if all the parameters of the transportation problem (such as supplies, demands and costs) must be in fuzzy numbers.

Definition 4.6

The transportation problem is said to be balanced fuzzy transportation problem if total fuzzy supply is equal to total fuzzy demand.

That is,

Definition 4.7

The transportation problem is said to be an unbalanced fuzzy transportation problem if total fuzzy supply is not equal to total fuzzy demand.

That is,

Definition 4.8

A set of fuzzy non negative allocations x̃_ij > 0̃ satisfies the row and column restriction is known as fuzzy feasible solution.

Definition 4.9

Any feasible solution is a fuzzy basic feasible solution if the number of non negative allocations is at most (m+n−1) where m is the number of rows and n is the number of columns in the m × n transportation table.

Definition 4.10

If the fuzzy basic feasible solution contains less than (m + n − 1) non negative allocations in m×n transportation table, it is said to be degenerate.

Definition 4.11

Any fuzzy feasible solution to a transportation problem containing m origins and n destinations is said to be fuzzy non degenerate, if it contains exactly (m+n−1) occupied cells.

Definition 4.12

The fuzzy basic feasible solution is said to be fuzzy optimal solution if it minimizes the total fuzzy transportation cost (or) it maximizes the total fuzzy transportation profit.

This proposed method is used for finding the optimal basic feasible solution in fuzzy environment and the following step by step procedure is utilized to find out the same.

Step 1

Consider the TP having all the parameters such as supply, demand and costs must be in fuzzy numbers (This situation is known as FFTP).

Step 2

Examine whether the total fuzzy supply equals total fuzzy demand. If not, introduce a dummy row/column having all its cost elements as fuzzy zero and fuzzy supply/fuzzy demand as the positive difference of fuzzy supply and fuzzy demand.

Step 3

After using Step 2, transform the FFTP into its equivalent crisp TP using the ranking procedure as mentioned in Section 3.

Step 4

Now, the crisp TP having all the entries of supply, demand and costs are integers then kept as it is. Otherwise at least one or all of the supply, demand and costs are not in integers then rewrite its nearest integer value.

Step 5

After using Step 4 of the proposed method, now solve the crisp TP by using any one of the existing methods (MODI, Zero Point Method) or software packages such as TORA, LINGO and so on. This step yields the optimal allocation and optimal objective value of the crisp TP (The optimal allotted cell in crisp transportation table is referred as occupied cells. The remaining cells are called unoccupied cells. The numbers of occupied cells in crisp TP which are exactly m + n − 1 and all have zero cost. Similarly in FFTP also have the same m + n − 1 number of occupied cells but its corresponding costs are fuzzy zeros). Now, construct the new fully fuzzy transportation table (FFTT) whose occupied cells costs are fuzzy zeros and the remaining cells costs are its original cost. Subtract each row entries for the current table from the row minimum. Next subtract each column entries of the reduced table from the column minimum. Clearly, each row and each column of the resulting table has at least one fuzzy zero. The current resulting table is the allotment table.

Step 6

After using Step 5 of the proposed method, now we check the allotment table if one or more rows/columns having exactly one occupied cell (fuzzy zero) then allot the maximum possible value to that cell and adjust the corresponding supply or demand with a positive difference of supply and demand. Otherwise, if all the rows/columns having more than one occupied cells then select a cell in the α-row and β-column of the transportation table whose cost is maximum (If the maximum cost is more than one i.e., a tie occurs then select arbitrarily) and examine which one of the cells is minimum cost (If the minimum cost is more than one i.e., a tie occurs then select arbitrarily) among all the occupied cells in that row and column then allot the maximum possible value to that cell. In this manner proceeds selected row and column entirely. If the entire row and column of the occupied cells having fully allotted then select the next maximum cost of the transportation table and examine which one of the cells is minimum cost among all the occupied cells in that row and column then allot the maximum possible value to that cell. Repeat this process until all the fuzzy supply points are fully used and all the fuzzy demand points are fully received. This allotment yields the fully fuzzy solution to the given fully fuzzy transportation problem.

Remark 5.1

Allot the maximum possible value to the occupied cells in type-2 and type-4 FTP which is the most preferable row/column having exactly one occupied cell.

Remark 5.2

From the MODI method, we conclude that the TP have exactly m + n − 1 number of non-negative independent allocations.

Remark 5.3

From the zero point method, we can make exactly m+n−1 number of zeros (zero referred to as zero cost) in the cost matrix. All these zeros (costs) are in independent positions.

Remark 5.4

From Remark 5.2 and Remark 5.3, we can directly replace the fuzzy zeros instead of original costs in the occupied cells in the original FTP. This modification does not affect the originality of the problem

Now, we prove the following theorems which are used to derive the solution to a fuzzy transportation problem obtained by the PSK method is a fuzzy optimal solution to the FFTP.

Theorem 5.1

Any optimal solution to the fully fuzzy transportation (P₁) where

(P₁) Minimize Z˜*=∑i=1m∑j=1n(c˜ij⊖u˜i⊖v˜j)x˜ij

Subject to (1) to (3) are satisfied, where ũ_i (minimum of ith row of the newly constructed transportation table c̃_ij) and ṽ_j (minimum of jth column of the resulting transportation table [c̃_ij ⊖ ũ_i]) are some real TrFNs, is an optimal solution to the problem (P) where

(P) Minimize Z˜=∑i=1m∑j=1nc˜ij⊗x˜ij

Subject to (1) to (3) are satisfied.

Theorem 5.2

If { x˜ijo, i = 1, 2, …, m and j = 1, 2,…, n} is a feasible solution to the problem (P) and (c̃_ij ⊖ ũ_i ⊖ ṽ_j) ≽ 0̃, for all i and j where ũ_i and ṽ_j are some real TrFNs, such that the minimum ∑i=1m∑j=1n(c˜ij⊖u˜i⊖v˜j)x˜ij subject to (1) to (3) are satisfied, is fuzzy zero, then { x˜ijo, i = 1, 2, …, m and j = 1, 2, …, n} is a fuzzy optimal solution to the problem (P).

Theorem 5.3

A solution obtained by the PSK method for a FFTP with equality constraints (P) is a fully fuzzy optimal solution for the fully fuzzy transportation problem (P).

6.1 Example 1

A firm has three factories S₁, S₂, and S₃ that manufacture the same product of air coolers in three different places. The firm manager would like to transport air coolers from three different factories to four different warehouses D₁, D₂, D₃ and D₄. All the factories are connected to all the warehouses by roads and air coolers are transported by lorries. The availability (availability of air coolers are depends on its production but production depends on men, machine, etc.) of air coolers are not known exactly due to long power cut, labour’s over time work, unexpected failures in machine etc. The demand of air coolers is not known exactly due to seasonal changes (In sunny days the sale of air coolers are more when compared to rainy days). Similarly the transportation cost is not known exactly due to variations in rates of petrol, traffic jams, weather in hilly areas etc. So, all the parameters of the TP are in uncertain quantities which are given in terms of TrFN. The transportation costs (Rupees in hundreds) for an air cooler from different factories to different warehouses are given below from the past experience.

Find the optimal allocation which minimizes total fuzzy transportation cost.

Solution by proposed method:

Now using Step 2 we get, ∑i=1ma˜i=∑j=1nb˜j=60, the given problem is balanced FFTP.

Now, using Step 3 of proposed method, in conformation to Model (2) FFTP can be transformed into its equivalent crisp transportation problem by using the ranking method ℜ(Ã) = (a₁ + a₂ + a₃ + a₄)/4.

Since in transportation problems supplies, demands and costs are not in decimal values because no physical meaning of fractional value. Therefore whenever the rank values occur in the form of decimal then we convert it into its nearest integer value. Otherwise, all the rank values of the FFTP is in integers then kept as it is.

After using Step 5 of the proposed method, the optimal allotment of the above problem is

The optimum solution is x₁₁ = 3, x₂₃ = 3, x₂₄ = 4, x₃₁ = 1, x₃₂ = 3, x₃₃ = 1.

The minimum objective value Z = (2×3)+(5×3)+(4× 4) + (7 × 1) + (6 × 3) + (6 × 1) = 68 (Rupees in hundreds).

After using Step 5 of the proposed method, now using Step 6, we get the optimal allotment directly for the FFTP is shown in the following table.

The fuzzy optimal solution in terms of trapezoidal fuzzy number is

Hence, the total fuzzy transportation cost is

6.2 Example 2

There are three sources (rows) and four destinations (columns), all the sources are connected to all the destinations by roads and the goods are transported by trucks. The supply and demand of goods are well known crisp quantities but the transportation cost is not known exactly (due to variations in rates of petrol, traffic jams, weather in hilly areas etc). Hence crisp supply, crisp demand and unit transportation cost (given in terms of TrFNs) are given in the following table. Find the optimal allocation which minimizes total fuzzy transportation cost.

Solution by proposed method:

Now using Step 2, we get, ∑i=1mai=∑j=1nbj=20, the given problem is balanced.

Now, using Step 3 and Step 4 of the proposed method, we get the following crisp TP (P).

After using Step 4 of the proposed method, now using Step 5 we have

The optimal solution is x₁₃ = 3, x₂₄ = 5, x₃₁ = 5, x₃₂ = 4, x₃₃ = 0, x₃₄ = 3

The total transportation cost MinZ = 8× 3 + 6 × 5 + 5 × 5 + 8 × 4 + 15 × 0 + 10 × 3 = 141.

Now, construct the new transportation table (P*) whose occupied cells costs are zeros and the remaining cells costs are its original cost. Subtract each row entries for the current table from the row minimum. Next subtract each column entries of the reduced table from the column minimum. Clearly, each row and each column of the resulting table has at least one fuzzy zero. The current resulting table is the allotment table.

Now using Step 6 of the proposed method we get

The total transportation cost MinZ^* = 0× 3 + 0 × 5 + 0 × 5 + 0 × 4 + 0 × 0 + 0 × 3 = 0 (We could not minimize cost below zero).

Therefore, Any optimal solution of problem (P^*) (subject to all the constraints are satisfied) is also an optimal solution of problem (P) (subject to all the constraints are satisfied).

Hence the solution obtained by PSK method for solving FTP is always optimal. The optimal allotment is

(We could not minimize fuzzy cost below fuzzy zero).

Hence, the optimal solution is x₁₃ = 3, x₂₄ = 5, x₃₁ = 5, x₃₂ = 4, x₃₃ = 0, x₃₄ = 3.

The total fuzzy transportation cost is

Results and discussion

The minimum total fuzzy transportation cost (Rupees in hundreds) of problem 1 is

The result in (7) can be explained (Refer to Figure 1) as follows:

• (a) Transportation cost (Rupees in hundreds) lies in [12, 117].

• (b) 100% expect are in favour that the transportation cost (Rupees in hundreds) is [55, 88] as μ_Z̃(c) = 1, x = 55 to 88.

• (c) Assuming that μ_Z̃(c) is a membership value at c. Then 100μ_Z̃(c)% experts are in favour that the transportation cost (Rupees in hundreds) is c.

Values of μ_Z̃(c) at different values of c can be determined using equations given below.

Advantages of the proposed method

By using the proposed method a decision maker has the following advantages:

• The optimum objective value of the type-2 and type-4 FTP is non-negative TrFN i.e., there is no negative part in the obtained TrFN.

• The proposed method is computationally very simple and easy to understand. The solution obtained by PSK method for solving FTP is always optimal. Hence we need not check optimality criteria.

On the basis of the present study, it can be concluded that the FTP and FFTP which can be solved by the existing methods (Pandian and Natarajan [9], Dinagar and Palanivel [8], Rani et al. [13], Basirzadeh [16], Gani and Razak [11]) can also be solved by the proposed method. However, it is much easier to apply the proposed method as compared to all the existing methods. Also, new method and new multiplication operation on TrFN is proposed to compute the optimal objective values in terms of TrFN which are very simple and easy to understand and it can be easily applied by decision maker to solve type-2 and type-4 FTP. The solution obtained by this method the objective value of the FFTP remains always positive i.e., there is no negative part in the TrFN. Hence the proposed method is physically meaningful and computationally very simple when compared to all the existing methods.

The author would like to thank to the Editor-in-Chief “Professor Keon-Myung Lee” and the anonymous reviewers for their suggestions which have led to an improvement in both the quality and clarity of the paper.

Table 1. Tabular representation of FFTP.

Table 2. Tabular representation of crisp TP.

Table 3. Type-4 FTP.

Table 4. Crisp version of type-4 FTP.

Table 5. Crisp optimum table of type-4 FTP.

Table 6. Fuzzy optimum table of type-4 FTP.

Table 7. Type-2 FTP.

Table 8. Crisp version of type-2 FTP (P).

Table 9. Crisp optimum table of type-2 FTP (P*).

Table 10. Crisp optimum allocation table of type-2 FTP.

Table 11. Fuzzy optimum table of type-2 FTP.

Table 12. Comparative study.