Transportation of Materials Under Fuzzy Environment Using Expected Monetary Value Strategy

The task of moving goods from a set of origins to a set of destinations while lessening the overall cost of transportation is commonly referred to as a transportation problem. Moving goods from one source to another while considering with regard to their respective supply and demand is the main objective of this particular type of linear programming problem (LPP). In order to deal with the increasing demands of the customers, different ideas and methods are to be implemented to fulfill their demands with minimum cost and within the stipulated time period.

Though there are different methods available to deal with the transportation problems, in real life situations, the demands, supply and availabilities are not known exactly. Fuzzy set theory proposed by Zadeh [1] is the main tool to tackle these kinds of situations.

Many authors have put in their efforts to solve single objective fuzzy transportation problems. However, we do not deal with a single objective at all circumstances, it is necessary to satisfy the other objectives like time and profit which play a major role in fuzzy transportation problems. To save time and money, the majority of the authors provided their solutions for numerous fuzzy transportation model problems. Some authors focused on the crucial aspect of the transportation problem with multiple objectives: the limitations on product blending while transferring raw materials with various levels of purity. In a realistic sense, the model's parameters are fuzzy because of various uncontrollable variables that exist.

Bit [2] was one of the mathematicians who worked on multi-objective fuzzy transportation problems. Abd El-Wahed, and Lee [3] presented a dynamic goal programming method for multi-objective transportation problem. An innovative approach for resolving linear multi-objective transportation problems utilizing fuzzy parameters was reported by Gupta and Kumar [4]. To deal with the objectives which are conflict in nature, Khan and Das [5] examined multi-objective transportation problems and presented a review to connect between the shortcomings of the problem and the fuzzy multiobjective optimization techniques. The optimal compromise approach to a multi-objective transportation problem (MOTP) was discovered by Abd El-Wahed [6]. Khoshnava and Mozaffarib [7] dealt with fully fuzzy transportation problems with multi objectives using weighted average method.

Uddin et al. [8] have used fuzzy goal programming approach to solve multi objective transportation problems. Bhageri et al. [9] is one of the authors who dealt with fuzzy multi-objective transportation problem using a system of weights. Ammar and Khalifa [10] solved multi-objective transportation problems using parametric analysis. Krishnaveni and Ganesan [11] proposed an effective approach for solving fuzzy transportation problem. Midya et al. [12] recommended multiobjective fractional fixed charge transportation problem using the fuzzy chance-constrained rough approximation method, which led to the most preferable optimal solution. Goal programming was used by Anukokila and Radhakrishnan [13] to analyse a fully fuzzy fractional multi-objective transportation problem. An example was also given to demonstrate the effectiveness of the multi-objective suggested strategy.

Fathy and Hassanien’s work [14] shows how multilevel multiobjective fuzzy linear programming problems can be effectively solved using the harmonic mean technique. They have applied the crisp linear approach which is then split into three crisp multiobjective linear programming problems at each level. Then, each crisp problem's multiobjective is combined into a single objective using the fuzzy harmonic mean technique. Secondly, the harmonic mean for each level is used to generate the resulting final, single-objective problem. A fuzzy harmonic mean approach was used by Kache and Singh [15] to address the fuzzy multi-objective transportation problem (FMOTP). Initially, the problem was mathematically formulated followed by the division into three levels of multi-objective LPPs using fuzzy arithmetic. The multi-objective LPPs are then reduced to single-objective linear programming problems utilizing the fuzzy harmonic mean as a technique. Such single objective linear programming problems for any of the three levels are subsequently addressed in order to yield the combined fuzzy optimal solution. To address the multi-objective transportation problem, an innovative approach based on nearest interval approximation was suggested by Niksirat [16] in 2022.

Taking all these methods into consideration, our focus has been on a FTP that has two goals: fuzzy time and fuzzy cost. The EMV analysis is performed to transform it to a single objective transportation problem. The ranking function and the arithmetic operations play the major role to preserve the fuzziness of the problem.

Using the location index and fuzziness index concepts to describe the decision parameters (fuzzy numbers) in their parametric form, we minimize the transportation cost and time in this work in the simplest possible manner.

The mathematical formulation of Multi-Objective Fuzzy Transportation Problem (MOFTP) is defined as follows:

Let $\tilde{Z}_k=\left\{\tilde{Z}_1, \tilde{Z}_2, \ldots \ldots, \tilde{Z}_k\right\}$ be a vector of $\mathrm{k}$ objective functions, $\tilde{c}_{i j}^k$ denotes the fuzzy cost (fuzzy time); $\tilde{a}_i$ be the amount available at the source; $\tilde{b}_j$ be the amount available at the destination and $\tilde{x}_{i j}$ denotes the decision variable, then the mathematical formulation of the transportation problem is given below:

$\begin{aligned} & \text { Minimize } \tilde{\mathrm{Z}}_{\mathrm{k}}=\sum_{\mathrm{i}=1}^{\mathrm{m}} \sum_{\mathrm{j}=1}^{\mathrm{n}} \tilde{\mathrm{c}}_{\mathrm{ij}} \tilde{\mathrm{x}}_{\mathrm{ij}} \\ & \text { subject to } \sum_{\mathrm{i}=1}^{\mathrm{n}} \tilde{\mathrm{x}}_{\mathrm{ij}}=\tilde{\mathrm{a}}_{\mathrm{i}}, \mathrm{i}=1,2, \ldots \ldots \mathrm{m} \\ & \qquad \sum_{\mathrm{i}=1}^{\mathrm{m}} \tilde{\mathrm{x}}_{\mathrm{ij}}=\tilde{\mathrm{b}}_{\mathrm{j}}, \mathrm{j}=1,2, \ldots \ldots \mathrm{n} \\ & \text { and } \tilde{\mathrm{x}}_{\mathrm{ij}} \succeq \tilde{0}\end{aligned}$              (1)

This multi-objective fuzzy transportation problem can also be represented as an (m×n) cost matrix given in Table 1.

Table 1. Generalized multi-objective fuzzy transportation table

Definition 3.1

A feasible solution $\widetilde{\boldsymbol{x}}=\left\{\tilde{x}_{i j}\right\}$ is said to be non-dominated solution (an efficient solution or fuzzy Pareto efficiency or fuzzy Pareto optimality) of the problem (1) if there exists no other solution $\widetilde{\boldsymbol{x}}=\left\{\tilde{x}_{i j}\right\} \in X$ such that:

$\begin{array}{r}\sum_{i=1}^m \sum_{j=1}^n \tilde{\mathrm{c}}_{\mathrm{ij}}^{-\mathrm{t}} \tilde{\mathrm{x}}_{\mathrm{ij}} \preceq \sum_{\mathrm{i}=1}^{\mathrm{m}} \sum_{\mathrm{j}=1}^{\mathrm{n}} \tilde{\mathrm{c}}_{\mathrm{ij}} \tilde{\mathrm{x}}_{\mathrm{ij}}^* \text { for all } \mathrm{r} \text { and } \\ \sum_{\mathrm{i}=1}^{\mathrm{m}} \sum_{\mathrm{j}=1}^{\mathrm{n}} \tilde{\mathrm{c}}_{\mathrm{ij}}^{-\mathrm{t}} \tilde{\mathrm{x}}_{\mathrm{ij}} \neq \sum_{\mathrm{i}=1}^{\mathrm{m}} \sum_{\mathrm{j}=1}^{\mathrm{n}} \tilde{\mathrm{c}}_{\mathrm{ij}} \tilde{\mathrm{x}}_{\mathrm{ij}}^* \text { for some } \mathrm{r} .\end{array}$

3.1 Expected Monetary Value (EMV) analysis

When there are circumstances that may or may not occur, EMV analysis aids in computing the average results. It assists in calculating the sum required to control the identified risks and also enables us to select the task that will cost less money.

Here the risk identified is the fuzzy time and this EMV analysis helps us to select the optimal allocation that will reduce the fuzzy cost as well.

In the fuzzy MOTP (with fuzzy cost $\tilde{c}_{i j}$ and fuzzy time $\tilde{t}_{i j}$) which we have considered, the Expected Monetary Value for each source S_i and destination D_j are given by:

$\begin{gathered}\operatorname{EMV}\left(S_i\right)=\sum_{j=1}^{n \sum}\left(\tilde{c}_{i j} \times \text { memebershipvalue }\left(\tilde{t}_{i j}\right)\right), \\ i=1,2,3, \ldots, m \\ \operatorname{EMV}\left(D_j\right)=\sum_{i=1}^m\left(\tilde{c}_{i j} \times \text { memebershipvalue }\left(\tilde{t}_{i j}\right)\right), \\ j=1,2,3, \ldots, n\end{gathered}$

where, the membership value of $\tilde{t}_{i j}$ is $\frac{U-\tilde{t}_{i j}}{U-L}, U$ is the maximum of all $\tilde{t}_{i j}$ 's and $L$ is the minimum of all $\tilde{t}_{i j}$ 's.

A multi-objective fuzzy transportation problem in which the information for the fuzzy cost, time, supply, and demand are displayed in Table 2.

Table 2. Multi-objective fuzzy transportation problem (MOFTP)

Each of the fuzzy number in Table 2 is observed to be a symmetric fuzzy number. The Table 3 is obtained by converting all fuzzy numbers into parametric form defined in Definition 2.2.

Table 3. Parametric form of MOFTP

Table 4. Fuzzy transportation table with membership values of fuzzy time

Table 5. Fuzzy transportation problem with single objective

After shifting that problem to its parametric form, we cannot solve it directly. The multi-objective problem must be reduced to a single objective problem before any strategy can be used to attain the desired answer. The membership value of the fuzzy quantities excluding the fuzzy cost is calculated. Later the EMV is obtained for each cell. The next step is to find the EMV for each row first and then for each column. The Table 4 depicts a single objective transportation problem that is ready to be dealt with.

The row or the column which has the least Expected Monetary Value is identified according to Step 6 of the algorithm. The column or the row which is so selected will be given priority and then from that row (column), the cell with largest EMV is selected. The cell which is chosen is given the maximum allocation. The process from step 4 till step 7 goes on and the Table 5 is obtained.

Table 6. Fuzzy transportation problem after giving the allocation

According to Table 6, the optimum fuzzy transportation cost is:

=(9, 1-δ, 1-δ)(10, 5-5δ, 5-5δ)+(12, 2-2δ, 2-2δ)(4, 2-2δ, 2-2δ)+(10, 2-2δ, 2-2δ)(15, 3-3δ, 3-3δ)+(14, 4-4δ, 4-4δ)(1, 2-2δ, 2-2δ)+(6, 1-δ, 1-δ)(12, 3-3δ, 3-3δ)=(90, 5-5δ, 5-5δ)+(48, 2-2δ, 2-2δ)+(150, 3-3δ, 3-3δ)+(14, 4-4δ, 4-4δ)+(72, 3-3δ, 3-3δ)=(374, 5-5δ, 5-5δ)=(513+5δ, 518, 523-5δ)

Optimum fuzzy transportation time is:

=(16, 2-2δ, 2-2δ)(10, 5-5δ, 5-5δ)+(12, 4-4δ, 4-4δ)(4, 2-2δ, 2-2δ)+(13, 4-4δ, 44δ) (15, 3-3δ, 3-3δ)+(19, 1-δ, 1-δ) (1, 2-2δ, 2-2δ)+(8, 2-2δ, 2-2δ)(12, 3-3δ, 33δ)=(160, 5-5δ, 5-5δ)+(48, 4-4δ, 4-4δ)+(195, 4-4δ, 4-4δ)+(19, 22δ, 2-2δ)+(96, 3-3δ, 3-3δ)=(518, 5-5δ, 5-5δ)=(369+5δ, 374,379-5δ).

The following table (Table 7) gives the fuzzy optimum solution to multi-objective fuzzy transportation problem for different values of δ.

Table 7. Fuzzy optimum solution to multi-objective fuzzy transportation problem

On the other hand, we convert this multi-objective fuzzy transportation problem in to an equivalent crisp transportation problem using the proposed ranking method.

Table 8. Multi-objective crisp transportation problem

Converting this multi-objective crisp transportation problem (Table 8) in to a single objective crisp transportation problem (Table 9) using weighted mean, we have:

Table 10 gives the optimum allocation for the given multi-objective fuzzy transportation problem by converting to an equivalent crisp multi-objective transportation problem and which is given by x₁₁=9, x₁₃=5, x₂₁=1, x₂₂=15, x₃₃=12.

The corresponding Transportation cost is 518 cost units and the transportation time is 374-time units. Here we got only the crisp solution for the fuzzy problem. We are not sure that how far this crisp solution (outputs) is suitable when the inputs are fuzzy parameters.

Table 9. Single objective crisp transportation problem

Table 10. Optimum allocation for the single objective crisp transportation problem