1. Introduction

Transportation problem is one of the most useful and applied branches of linear programming. It deals with finding the optimal distribution of goods from several sources to several destinations. The optimal distribution means that this distribution costs the lowest transportation cost, and there is no other distribution than the optimal one that costs less. Transportation problems can be solved to minimize transportation cost, distance, time, deterioration of goods, vehicle fuel consumption, etc... The transportation problem can also be solved to maximize the companies’ earnings, profits, etc.

The transportation problem was first formalized by the French mathematician Gaspard Monge (1871) [1]. Tolstoi, A.N. was one of the first to study the transportation problem mathematically in the 1920s. In 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. Tolstoi (1939) illuminated his approach by applications to the transportation of salt, cement, and other cargo between sources and destinations along the railway network of the Soviet Union. In particular, for that large-scale, instance of the transportation problem was solved to optimality. F.L. Hitchcock (1941) worked on the distribution of 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].

Usually, the solution to the transportation problem passes through three main steps. These steps include finding the initial basic feasible solution, the test of optimality, and moving towards optimality. Enormous researches were prepared to find the initial basic feasible solution as well as the other two steps [3-5]. But different ways give the optimal solution directly. Out of these methods are the Lingo codes and excel solver.

More advanced researches were carried out to solve the fuzzy transportation problem, multi-objective transportation problem, multiple criteria transportation problem, multi-index, multi-level transportation problem, and transshipment [2,6-12].

In this paper, we prepared a lingo code to solve the transportation problem while applying this code to solve a real-world problem with actual data. The prepared lingo code based on the transportation problem technique is used to solve a real transportation problem. The actual problem deals with re-distributing flour, which is produced in middle Egypt milestones to warehouses. The company milestones exist in 16 different cities. The flour was required to be distributed to 5 governorate cities, which are 26 major cities.

2. Transportation Problem

The main parts of the transportation problem are the number of sources “m”, the number of destinations “n” and the unit costs “c_ij” of transporting goods from each source “i=1,2,…,m” to each destination “j=1,2,…,n”. The amounts of goods that should be transported from source “i” to destination “j” is x_ij. The total transportation cost “Z” can be calculated using the following transportation model:

Subject To:

By solving the above model of the transportation problem, we’ll get the optimal solution to the problem. This model is used to build a lingo code to solve an actual transportation problem, as in the following sections.

3. Lingo

LINGO is a well-known and powerful Operational Research software tool available today designed to make building and solving problems easier and more efficient. It is designed to build and solve Linear, Nonlinear, Quadratic, Quadratically Constrained, Semi-Definite, Second Order Cone, Integer and Stochastic optimization models faster, easier, and more efficient.

LINGO includes integrated packages that include a powerful language for the aim of expressing optimization models, a full-featured environment for editing and building problems, and a set of fast built-in solvers.

Lingo lets us formulate our linear, nonlinear and integer problems quickly in a highly readable format. LINGO modeling language allows us to express models in a straight-forward intuitive manner using summations and subscripted variables, much like we would with pencil and paper. The key benefits of LINGO are easy model expression, convenient data options, create turn-key applications and extensive documentation and help [13].

Models become more comfortable to build, easier to understand, and, therefore, much easier to maintain. LINGO can also exploit multiple CPU cores for faster model generation. Lingo codes were prepared to solve transportation problem examples as in [14-16].

4. Case Study

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.

The collected data of this problem include the distances in kilometers between all cities where mills exist, to all cities where they want to send flour. Also, all mills availabilities and all major cities requirements in tons through the studied month. Table (1) represents the collected data.

Table 1. Transportation problem collected data sources capacity and destination demands and distances between them

A lingo code is prepared to solve the transportation problem case study that is studied in this research. Figure (1) illustrates the prepared lingo code for the case study. The prepared lingo code based on the transportation problem formulation is applied to solve the problem and redistribute flour from the company mills to the major cities. It is found that the total ton-kilometers was 973,060 for the actual distribution for the studied month. But it is reduced to become 883,828 ton-kilometers for the optimal distribution for the same month. The total saving was found to be 89,232 ton-kilometers. That is 9.1702% in savings.

Figure 1. Lingo code for the studied transportation problem

To compare the results of the LINGO code, i.e. the optimal solution with the actual distribution of the problem, we add the actual distribution of the studied transportation problem case study in Table (2). Total ton-kilometers of the actual distribution is 973,060.

Table 2. Transportation problem actual distribution between sources and destination in (ton)

The optimal distribution of flour is illustrated in Table (3), while Table (4) summarizes the comparison between actual and optimal distributions. Total ton-kilometers of the optimal distribution is 883,828. This reduces the total ton-kilometers by a value of 89,232, which is a 9.1702% in savings.

Table 3. The optimal distribution of M.E.M.C. flour from sources to each destination

Table 4. The optimal versus actual distribution of M.E.M.C.

5. Conclusions

A LINGO code is prepared and applied to solve an actual transportation problem. Lingo is a potent, fast, and easy tool to solve such transportation problems. The case study is a mills company called M.E.M.C. that exists in middle Egypt. We could achieve the optimal distribution of flour from the company mills which exist in 16 cities to 26 major cities. The total ton-kilometers of the actual distribution was 973,060. Total ton-kilometers of the optimal distribution is 883,828. The generated optimal solution could reduce a total of 89,232 ton-kilometers, which represents 9.1702 % of the actual ton-kilometers. Both lingo code, actual and optimal distribution of flour are included in this paper.