1. Introduction

The French mathematician Gaspard Monge prepared the first transportation problem (T.P.) formula, Abdelwali et al [1]. Then on in the 1920s, Tolstoi, A.N. was one of the first to study the transportation problem mathematically. On 1930, on the collection Transportation Planning Volume number 1, for the National Commissariat of Transportation for the Soviet Union, he published a paper Methods of Finding the Minimal Kilometers in Cargo-transportation in space. Once again, Tolstoi (1939) illuminated his approach by applications to the transportation of cement, salt, and other cargo between different sources and destinations along the railway network of the Soviet Union. On 1941, F.L. Hitchcock worked on the distribution of some products from several sources to numerous localities. Koopman also worked on the optimum utilization of the transportation system and used a model of transportation, in activity analysis of production and allocation. It is known as the Hitchcock Koopman transportation problem [2]. More other papers were published on this topic with more features and methods of solution.

T.P. is a special nature of linear programming. It can be solved by the linear programming simplex method. But due to its special nature, T.P. can be solved easily through its table. To solve any T.P., three steps are needed. These steps are: Finding the initial basic feasible solution, test of optimality and moving towards optimality. There are some packages and software were prepared to find the optimal solution of any T.P. directly, like Tora by Hamdy Taha [3], Manager by Sang. M. Lee [4], and more other packages. Excel solver [5] and Lingo [1] can be used, too, to solve T.P. According to the total availabilities of problem sources and total requirements of destinations, the T.P. could be balanced or unbalanced. Unbalanced problems need to be changed into balanced T.P. by adding a dummy source or a dummy destination. The T.P. data should include more than one source with known availabilities of each source, more than one destination with known requirements of each destination, and the unit cost between each source and each destination. The optimal solution of a classical T.P. generates the distribution of a single product from all sources to all destinations, while this distribution gives the minimum transportation cost. More advanced researches on transportation problem had been introduced to study multi-objective T.P., multilevel T.P., multi-stage, fuzzy T.P., fuzzy multi-objective T.P., interactive fuzzy multi-objective T.P. and more other related advanced researches.

The classical T.P.s are solved to achieve just a single objective function. This objective could be minimizing the total transportation distances, time, or cost. In real world applications, more objectives should be considered while solving a T.P. In general, the T.P. objectives are conflicting in nature, as a result the simultaneous optimization of objectives is impossible. Multi-objective programming deals with trying to obtain a set of efficient or Pareto optimal solutions. This leads the decision makers (DMs) to seek a most preferred compromise solution rather than optimal one [6]. There are many different methods that are used to solve Multi-objective Transportation Problems (MOTPs). From these methods, goal programming, the weighting method, multiple criteria decision-making procedures, the decomposition approach, the interactive method, the minimize distance method, and many other different methods. Some of these methods are illustrated and implemented in these research papers [2,7,8,9,10,11,12,13,14]. In this paper, the minimize distance method strategy is applied to find the efficient solutions of a real multi-objective transportation problem.

2. The Minimize Distance Method

Kamal et al [6] introduced a distance-based method for solving multi-objective optimization problems. It is a new model which depends on the goal programming weighted method. The method is proposed based on minimizing the distances between the ideal objectives to the feasible objective space. This method provides the best compromised solution for Multi Objective Linear Programming Problems (MOLPP). The proposed model tackles MOLPP by solving a series of single objective subproblems, where the objectives are transformed into constraints. The generated compromise solution may be improved by defining priorities in terms of the weight. A criterion is also proposed for deciding the best compromise solution. The main advantage of the proposed approach over other approaches is that it can obtain the compromise solution without any preference and for different preferences.

3. Problem Formulation

The multi-objective linear programming (MOLP) formulation based on the minimize distance method is presented in [6]. Based on this method and its formula, the multi-objective transportation problem, with two objective functions, can be derived as follows [6]:

Subject to:

Where:

f^*₁, f^*₂: the obtained ideal objective values by solving single objective T.P.s.

w₁, w₂: weights for objective 1 and objective 2 respectively.

f1, f2: the objective values for the other efficient solutions.

d: the general deviational variable for all objectives.

the unit cost for objectives 1 and 2 from source i to destination j.

the amount to be shipped when optimizing for objectives 1 and 2 from source i to destination j.

4. Solution Algorithm

The solution algorithm as well as a flow-chart for the minimize distance method for solving MOLP problem are introduced in Kama et al [6]. Here is the derived solution algorithm steps for a multi-objective transportation problem.

Step 1. Consider the first objective function only. Solve the transportation problem as a single objective problem ignoring all other objectives subject to the constraints. Then consider the second objective function only and solve the transportation problem as a single objective problem ignoring all other objectives subject to the constraints. If there is more than two objectives, do the same for the other objective functions one by one.

Step 2. Based on the solutions of (Step 1), obtain the Ideal objective values (f^*₁, f^*₂). Then formulate the multi-objective optimization model as a single objective optimization model using the above model.

Step 3. Solve the prepared model (in Step 2) using any of the available solvers such as LINGO (a modelling language and optimizer) or any other solver.

Step 4. If the decision maker is satisfied with the solution so obtained then the process terminates, otherwise proceed to next step.

Step 5. Ask the decision maker to define weights (w₁, w₂) for each objective and repeat from Step 3 to Step 5 until the process terminates.

5. Case Study and Result Analysis

The minimize distance method formulation and solution algorithm are applied on the M.E.M.C. company. The M.E.M.C. is a big company that produces and distributes flour in 5 governorates in Egypt. Their mills exist in 16 cities while the company distributes flour to 26 major cities. The company produces several types of flour. We considered just one product in this paper. The data of the M.E.M.C. was introduced and solved as a single objective transportation problem by Abdelwali et al [1]. The data required for the multi-objective T.P. are prepared and illustrated in Table (1) below. Due to the road surface, number of lanes and the different speed limits from one road to the other in this transportation network, distance and time are independent of each other. So, these two objectives are conflicting. As a result, the ideal solution as well as the best compromise solution differ from each other.

Table 1. MOTP collected data source capacities, destination demands, and distances and time between them

A LINGO code is prepared to solve the M.E.M.C. transportation problem as a multi-objective problem. Table (2) below summarizes the output of the studied problem. This code could generate the best compromise solution of the studied problem which exists at the lowest distance from the ideal solution. The distances from each efficient solution from the ideal solution are calculated based on the distance formula that is prepared by El-Wahed et al [12]. Another LINGO code is prepared to validate the results of the first code and to be ready for the decision maker (DM) if s/he is not satisfied by the generated compromise solution. This second code is based on the weights of each objective (w₁, w₂) that the DM would like to choose or change.

Table 2. MOTP ideal and efficient distribution between sources and destinations

The best compromise solution is illustrated in Table (2), row (12). The ideal solution for objective 1 is included in Table (2), row (11), while the ideal solution for objective 2 is included in Table (2), row (1). Rows (2) to (10) includes more efficient solutions to be ready for the D.M. to choose from based on different weights of both objectives. Finally, row (13) presented the results of the second LINGO code to validate the results of code 1.

From the results of Table (1,2), it is clear that the best compromise solution exists at the minimum distance of the generated solution from the ideal solutions. This reflects the power of the minimize distance method to find the best compromise solution of multi-objective transportation problems.

6. Conclusions

The minimize distance method by Kamal et al [6] is applied on a real two objective transportation problem. This methos is applied on a mills company called M.E.M.C. that exists in middle Egypt. Two LINGO codes are prepared to find the best compromise solution and more other efficient solutions to be ready for the decision maker to choose from if he is not satisfied with the generated best compromise solution. All the output results are included in this paper. It is found that the gives the minimum distance from the ideal solution of both objectives. This reflects the power of the minimize distance method. When compared by the actual distribution, there are huge savings of both distance and time with the best compromise solution, and all other efficient solutions.