Approaches for Solving Multi-Objective Transportation Problem

Transportation is important in the sense that it allows people to take part in human activities. The classical transportation problem can be described in a special case of linear programming problem and its models are applied to determine an optimal solution of the transportation problem required for deterministic of how many units of commodity to be shipped from each origin to various destinations where the objectives have to optimize (minimize or maximize) cost or time. The basic transportation problem is developed by F.L. Hitchcock in 1941. Transportation problem further developed by T.C. Koopmans in 1949 and G.B. Dantzing in 1951. Single objective i.e minimization of cost or time focused by Hitchcock, Koopmans and Dantzing in their studies. Before the seventieth century study of transportation problem by researchers were focuses upon optimizing the single objective. In the techniques of optimizing single objective transportation problem are not suitable when optimizing two or more than two objectives are given in transportation problem. When there is the situation of two or more than two objectives in transportation problem, then such types of problems are called as multi-objective transportation problem. Multi-objective transportation problem is the special extension of the transportation problem.

Optimization is a kind of the decision making, in which decision has to be taken to optimize one or more objective under some prescribed set of circumstances. These problems may be a single or multi-objective and are to be optimized (maximized or minimized) under a specified set of constraints. The constraints usually are in the form of inequalities or equalities. Such problems, which often arise as a result of mathematical modeling of many real-life situations, are called optimization problems. Multi-objective optimization or multi-objective programming is the process of simultaneously optimizing more than one objective subject to certain constraints. Applications of multi-objective optimization problems were found in the fields: product and process design, finance, aircraft design, the oil and gas industry, automobile design and many more.

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Goal Programming Problem Technique

Various techniques have been developed to solve MOTP by various researchers one of these is goal programming problem approach. Goal programming technique is used for solving multi-objective optimization problem that balance exchange as a compromise in conflict objective. In goal programming, there is a need to establish a hierarchy of importance among goals so that the lower order goals are considered only after the higher order goal are satisfied. Goal programming technique helps in complete the satisfactory level of all objectives. Goal programming methods have been frequently used to solve multi-objective transportation problem.

Sang Moon Lee is a pioneer of the solution technique of multi-objective transportation problem which is solved by using goal programming techniques in 1972. Almost all techniques for transportation problem have focused upon the optimization of a single objective condition, namely the minimization of total transportation costs used before seventeen century. They have generally neglected the multiple objectives, i.e various environmental constraints, unique organizational values of the firm, and bureaucratic decision structures involved in the problem. But in reality, these are important factors which greatly control the decision in organization. They studied these entire situations, and then developed new technique to solve MOTP by using goal programming.

A. Baidya et al. are used goal programming to solve an interval valued multi-item solid transportation problem with safety measures. They are introducing a new concept ‘safety factor’ in transportation problem. When items are transported from origins to destinations through different conveyances, there are some risks to transport the items due to bad road or some routes especially in developing countries. Due to this reason total safety factor is important in transportation and depending upon the nature of safety factor. They also formulate five models without and with safety factor, where this factor may be crisp, fuzzy, interval, stochastic in nature and solve all mathematical problems by using LINGO 12.0 Software. Multi item solid transportation problem with safety measure gives an idea about this new factor safety measure to transport commodities from some sources to some destinations by the means of different conveyances. The corresponding multi-objective transportation problem is formulated using ‘mean and width’ technique. Then the problem is converted to a single objective transportation problem taking convex combination of the objectives according to their weights. They are suggested as models can be extended to include the additional amount to be spent to increase the safety measures along different routes keeping extra securities, using very fast vehicle etc.

Wuttinum Nunkaew and Busaba Phruksaphantrat are developed relationship between customer to customer in a conventional transportation problem. Lexicographic goal programming is used to solve the MOTP with a minimization of the total transportation cost and the overall independence value. They also obtain the efficient reasonable solution that satisfied both consideration of depot to customer and customer to customer relationship that means the lowest total transportation cost and nearest locality of customer are determined. Also, each customer can be served by only one depot if the capacity of the depot is sufficient, these advantages are more compatible to the reality than the conventional transportation model.

Waiel F. El-Wahed and Sang M. Lee developed iterative fuzzy goal programming problem (IFGP) to solve MOTP. The approach controls the search direction via updating both upper bounds and aspiration level of each objective function. The solution results provide a preferred compromise solution which is more realistic from the decision maker (DM) view. The approach is a powerful method to determine appropriate aspiration levels of the objective functions. The performance of the suggested approach was evaluated by using a set of metric distance functions with respect to the two previously developed methods. They, also suggested combination of goal programming, fuzzy programming, and interactive programming in one methodology is a powerful tool for solving MOTP and other multi-objective optimization problems.

H. R. Maleki and S. Khodaparasti are developed to solve a special mathematical model of non-linear multi-objective transportation problem by using a fuzzy goal programming approach. In goal programming models, absolute deviations are obtained in the optimal solution. Another advantage of the models based on goal programming is the minimum changes required for sensitively analysis. Fuzzy goal programming is used when a decision maker is unable to specify accurate objective levels and hence changes of acceptable violations may frequently occur. The other methods can be affected by these changes more than goal programming methods.

M. Zangiabadi & H.R. Maleki proposed three special types of membership functions have been used to solve the multi-objective transportation problem. The optimal compromise solution does not change, when compared with the solution obtained by the linear membership function. But they used the exponential membership function, with different values of parameters, and then the optimal compromise solution does not change significantly. They also compare with the solution obtained by the linear membership function. Further, they conclude that for a multi-objective probabilistic transportation problem if the demand parameters are gamma random variables, then the deterministic problem becomes non-linear.

Lohgaonkar M.H. et al. introduced fuzzy goal programming approach to unbalanced transportation problem with additive multiple fuzzy goals, when the goals are considered to be of equal importance. But, in reality, all goals may not be of equal importance. They also discussed two different ways of assigning weights to additional model described in the paper. The direct weights are used in fuzzy goal programming model for unbalanced multi-objective transportation problem.

Surapati Pramanik et al. proposed an alternative solution approach for multi-objective quadratic programming problem (MOQPP). They first transform MOQPP in to equivalent multi-objective linear programming problem by first order Taylor series approximation. Then fuzzy goal programming approach is used to solve the problem by minimizing negative deviational variables. Also, proposed concept to collection problems, decentralized bi-level and multi-level quadratic programming problem.

Genetic Algorithm Approach

Genetic algorithm (GA) is a method for solving both constrained and unconstrained optimization problems based on a natural selection process that mimics biological evolution. The algorithm has repeatedly modified a population of individual solutions. Genetic algorithms belong to the larger class of evolutionary algorithms (EA), which generate solutions to optimization problems using techniques inspired by natural evolution. Genetic algorithm method is another frequently used to solving multi-objective transportation problem.

Mistuo Gen et al. proposed new approach by using spanning tree based genetic algorithm (GA) to solved MOTP. Spanning tree-based encoding was implemented with decoding from which an infeasible chromosome (i.e. Prufer number) and adopted to represent of balanced transportation solution. In case of small-scale problem, there is no great difference on the computing result, this type of problem solved by genetic algorithm approach. Also, when large scale problem, spanning tree-based GA approach can get the Pareto solution with less time than the matrix GA approach and most of the results are not dominated by those obtained in the matrix-based GA approach. Therefore, in the sense of Pareto optimality, this spanning tree-based GA approach is more effective than the matrix-based GA. They also say that spanning tree-based GA approach much more efficient than the matrix-based GA on the transportation problem.

A.A Mousa et al. presented an efficient evolutionary algorithm for solving multi objective transportation problem. Also they proposed some approaches such as, effectively applied to solve the MOTP with no limitation in handling higher dimensional problems, conclude that integration of GA and local search technique has improved the quality of founded solution, where the computational time grows with the number of achieved solution and simulation results verify and advantage of the proposed approach.

Sayed A. Zaki et al. presented improved algorithm for solving MOTP was presented. They Firstly, the algorithm is an iterative multi-objective genetic algorithm with an external population of Pareto optimal solutions that best conform a Pareto front. Secondly the algorithm implements GA to provide the initial set (close to the Pareto set as possible) followed by local search method to improve the quality of the solutions. They also concluded that integration of GA and local search technique has improved the solution’s quality and avoid an awesome number of solutions clustering algorithm saves the most representative solutions, which gets iteratively updated in the presence of new solutions.

H.C.W. Lau et al. presented multi-objective vehicle routing problem with multiple depots, multiple customers, and multiple products has been studied. The objective has been to simultaneously minimize both the total traveling distance and the total traveling time. A multi-objective evolutionary algorithms (MOEA) called fuzzy logic guided non dominated sorting genetic algorithm 2 (FL-NSGA2) was proposed to solve this multi-objective optimization problem. The role of fuzzy logic is to dynamically adjust the crossover rate and mutation rate after ten consecutive generations.

Anthony Chen et al. presented two mean-variance models for determining the optimal toll and capacity of a build-operate-transfer (BOT) project under demand uncertainty. These two models were formulated as a stochastic bi-level mathematical program with multiple objectives, which is difficult to solve using traditional optimization methods. A simulation-based multi-objective genetic algorithm (SMOGA) procedure that integrates stochastic simulation, a traffic assignment algorithm, a distance-based method, and a genetic algorithm was developed to solve. They verified the feasibility of using the SMOGA procedure by solving a BOT project in China as a case study and found that the proposed procedure is robust in generating good non-dominated solutions with respect to different GA’s parameters, and performs better than the weighted-sum method.

Zhang Hong-Wei et al. presented a new genetic algorithm based on the theory of Lamarckian evolution (Lam-GA) to solve multi-objective transportation optimization problem. The algorithm carries through some local transformation according to certain rules after distributing transportation counts on the fuzzy rule basis, which can increase the strength for searching better solution. Experimental data shows that after strengthening the mutation locally, the new algorithm can get better Pareto front and Pareto optimal solutions in solving large-scale transport problems, so that Lam-GA is more effective than Fuzzy-GA, st-GA and m-GA.

Conclusion

Transportation problem (TP) is a special case of linear programming problem (LPP) in which cost optimization has been made on the base of demand and resources. Combining of two or more than two objectives in T.P. then this type of problem is called as multi-objective transportation problem (MOTP). Different approached methods for solving MOTP by various authors. In the present paper, we are comparing between goal programming technique and genetic algorithm technique used to solve MOTP. We made an attempt to collect possible work on goal programming (GP) technique and genetic algorithm (GA) technique used to solve MOTP in various situations.

GP model have been applied to solve large-scale multi-criteria decision-making problems, analytical structure that a decision maker can use to provide optimal solutions to multiple and conflicting objectives, many objectives while the decision making is looking for the best solution from among a set of feasible solutions etc.

Genetic algorithm (GA) is a method for solving both constrained and unconstrained optimization problems based on a natural selection process that mimics biological evolution. The algorithm has repeatedly modified a population of individual solutions. Genetic algorithms belong to the larger class of evolutionary algorithms (EA), which generate solutions to optimization problems using techniques inspired by natural evolution. Genetic algorithm method is another frequently used to solving multi-objective transportation problem. This will be helpful to new researchers for their initial level studies in goal programming and genetic algorithm used in MOTP.

References

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