A Visual Comparative Study of Multi-Objective Transportation Problem in Several Approaches

The main objective of this study was to present in one applet java program the five approaches for solving the MultiObjective Transportation Problem (MOTP). The program was built using the java programming language, to solve the MOTP and to visualize the solution steps. In addition, the study made a comparisons between the four different approaches used to solve the MOTP. For better understanding, the solution procedure is illustrated with a numerical example.


INTRODUCTION
The Transportation Problem (TP) is a special type of linear programming problem. It is considered as a minimum cost flow problem, which deals with the transportation of commodities from m sources to n destinations. This means to transfer a quantity of products from plants to warehouses such that the transportation cost is minimum. TP usually involves a single objective function. However the transportation problem which involves multiple objective functions is called Multi-Objective Transportation Problem (MOTP).
A number of studies has been available to obtain an optimal solution for balanced transportation problems. That is when the total supply is equal to the total demand then the transportation model is said to be balanced [23]. But in real life, the decision maker faces an unbalanced transportation problem in which the total supply is less than the total demand. This can be solved through compromised solutions. M. Zangiabadi and H. R. Maleki used a special type of nonlinear (hyperbolic and exponential) membership functions to solve MOTP, it gives an optimal compromise solution [17]. Kishore and Jayswal [10] introduced a method, called fuzzy approach, to solve unbalanced transportation problems with budgetary constraints. Charnsethikul Peerayuth and Sverasreni Saeree [3] discussed a method for solving the constrained bottleneck transportation problem under budgetary condition. Lin and Cheng [14] gave a genetic algorithm for solving a transportation network under a budget Constraint. Senapati and Tapan Kumar [22] investigated fuzzy multi-index transportation problems with budgetary restriction. Khanna, Bak hshiand Puri [9] introduced an algorithm for solving transportation flow under budgetary constraints. Tiwari, Dharmar and Rao [24] investigated how the preemptive priority structure can be used in fuzzy goal programming problems. Weighted goal programming for unbalanced single objective transportation problem with budgetary constraint has been discussed by Kishore and Jayswal [11]. Pandian and Natarajan [18] introduced the zero point method for finding an optimal solution to a classical transportation problem. Subsequently, a number of transportation problems have been appeared in the literature [15,7,6,21,8].
The main objectives of this study were to present in one applet java program the four approaches:
The study solved the MOTP and made a comparison between the four approaches . The program was built using the java programming language, to solve the MOTP and to visualize the solution steps. The study addressed the following steps: The first was to solve MOTP using the five approaches. The second was to combine the genetic algorithm and the five approaches. The third was to provide a visual solution which compare between the five approaches.
The remainder of this paper is organized as follows. In section (2) some basic concepts and definitions about the MOTP and the genetic algorithm were introduced. In section (3) the study described five methods which have been used to solve the MOTP. Section (4) showed the implementation of the problem and results. Section (5) analysed the results and compared between the the different solutions of MOTP. Section (6) is the conclusion of this research.

PRELIMINARIES
Given a transportation model with m sources and n destinations, the amount of supply available at source i is ai, the demand required at destination j is bj, the cost of transporting one unit between source i and destination j is cij and xij denotes the quantity transported from source i to destination j so the cost associated with this movement is cost  quantity ij ij c x  . The cost of transporting the commodity from source i to all destinations is given by Thus, the total cost * ( ) F X of transporting the commodity from all the sources to all the destinations is The sources may be factories, warehouses, etc. and they are characterized by available quantities denoted The destination may be warehouses, sales outlets, etc. and they are characterized by available N o v e m b e r , 2 0 1 3 quantities denoted 1 , , .
The transportation cost between a given source i to a given destination j pair is the penalty ij c . The unknown quantity to be transported between the source-destination pair ( , ) x . In the simplest case, the unit transportation cost is constant. The transportation problem is to find the optimal distribution plan for shipments from sources to destinations that minimizes the total transportation cost, in the same time, it seeks to find an optimal distribution plan for a single commodity. The mathematical form of MOTP can be stated as follows: Subject to 1 1, 2,..., X is the multiple objective functions, it is a vector of k objective functions and * X is a feasible vector solution of 1 P . The above MOTP form is knewn as the canonical form, by changing the pair of equality in (4) and (5) to a pair of inequality, it results the equivalent problem which is in standard form. Every minimization problem can be appeared as a maximization problem and vice versa.
Genetic algorithm has shown a great potential to work out several real-world problems in the point of optimization.
One of these problems is MOTP. That algorithmic model begins with the creation of a set of solutions which are referred to as a population of individuals. Each individual in a population consists of a set of parameter values which completely describe a solution. A solution is encoded in a string called a chromosome, which consists of genes that can take a number of values. Initially, the collection of solutions (population) is generated randomly and at each iteration a new generation of solutions is formed by applying genetic operators (crossover, mutation, selection). Each solution is evaluated using an objective function called a fitness function and this process is repeated until some form of convergence in fitness is achieved. The goal of the optimization process is to minimize or maximize the fitness.

Dominated Solution:
A solution X (1) is said to dominate the other solution X (2), if both conditions 1 and 2 are true: The solution X(1) is no worse than X(2) in all objectives, or no value of F j ( X (1) ) is greater than any value of F j ( X(2)) for all j= 1,2…M.
The solution X(1) is strictly better than X(2) in at least one objective, or F j (X(1))  F j(X (2)) for at least one j{ 1,2…M}.
denotes an optimal value for the i-th function is the vector of variables which optimizes Where the power p represents a distance parameter 1≤ p ≤ ∞, d k is the degree of closeness of the preferred compromise solution vector * X to the optimal solution vector with respect to the k-th objective function, and in minimum problem d k equals: The optimal solution of The preferred compromise solution of The distance functions with 1,2, and p  are and

EXPERIMENTAL, RESULTS AND DISCUSSIONS
The four solution's approaches for solving MOTP and the genetic algorithm in java language were implemented to obtain the optimal solution for each approach. Also, practical comparisons between the four approaches were presented. Figure 1( A) gives the user interface of the JAVA-APPLET program. The input to the frame shows transport products from three sources to four destinations center. The user started to input the general data of the problem that determines the size of the problem, the number of the multi-objective functions of the optimization problem and population size (the population size is the number of chromosomes in the population). The data that describe the details of the problem are entered through two buttons on this frame; data from frame and data from frame and file. "Minimum or Maximum" buttons determine the type of the problem. This frame is revealing in the four solution approaches for the MOTP.  The structure of the chromosome corresponding to this solution concatenated the twelve variable x i in a binary numbers; 0101, 11, 0001, 1011, 0101, 11, 0111, 01111, 0101, 01, 0110, and 01011. These give one chromosome as; C:01011100011011010111011101111010101011001011.

Fig 2 : Binary encoded chromosome
The selection operator was intended to improve the average quality of the population by giving the high-quality chromosomes a better chance to get copied into the next generation. Selection chromosomes are selected as parents to produce children and the chromosome with the best fitness values get selected to become the parent. The Tournament selection technique was used to pick a small subset of chromosomes (two or three) from the mating pool randomly. Crossover exchanges information between two parent chromosomes in order to produce two new offsprings for the next population. A modified uniform crossover was presented, where one offspring was constructed by choosing every sub-chromosome with a probability P (= 0.64) from either parent, as shown in Figure 3.

EXPERIMENTAL RESULTS
Interactive algorithms (I & II): Interactive methods may be characterized by the following steps: First, one or more non-dominated solutions are generated. Second, the decision maker is required for tradeoff information concerning these solutions, and the problem is modified based on the decision maker's responses. These two steps are then repeated iteratively until the decision maker is satisfied with the current solution. interactive programming in one methodology. Where, the basic idea of Goal Programming (GP) is a simple and easy to use. In GP, the decision maker should define the goal that he wants to accomplish. The GP model converts all the objective functions to constraints with the input of deviation variables from the goals. It can be expressed as the following:     . The good performance of the Pareto-based approach could be conveyed to the following reasons: Pareto-based approach aims to maintain diversity in the population instead of converging to a single solution and it uses non-domination to assess the fitness of individuals. This approach is dividing generation into a number of fronts. The best solution can be selected from the best front (the front that minimizes or maximizes the fittness function) [23].

STRATEGY OF REANALYSIS
The Interactive Fuzzy goal programming approach was relatively better than the Interactive (I) approach and Fuzzy approach. This could be explained by the lower value of the distance functions 1 2 and D D obtained using the Interactive Fuzzy goal approach compared to the Interactive (I) approach. Also the value of the distance functions were lower for the Interactive Fuzzy goal approach than the Fuzzy approach. The good performance of the Interactive Fuzzy goal programming approach could be due to the fact that the Interactive Fuzzy goal programming approach is a compilation of goal programming approach and the fuzzy approach. Therefore, the Interactive Fuzzy goal programming approach is an improvement for Fuzzy approach. Moreover, the Interactive (I) approach was better than the Fuzzy approach based on the value of the distance functions 1 2 and D D .

CONCLUSIONS
This paper introduced a visual emplementation for the four approaches; Pareto-based approach, Interactive (I) approach, Fuzzy approach and Interactive Fuzzy goal approach for solving MOTP by object-oriented programming (Java). After running the java program for the four methods it could be concluded that the Pareto-based approach gave the best