به حداقل رساندن هزینه - زمان در یک مسئله حمل و نقل با پارامترهای فازی: مطالعه موردی

In real world applications the supply, the demand and the transportation cost per unit of the quantities in a transportation problem are hardly specified precisely because of the changing economic and environmental conditions. It is also important that the time required for transportation should be minimum. In this paper a method has been proposed for the minimization of transportation cost as well as time of transportation when the demand, supply and transportation cost per unit of the quantities are fuzzy. The problem is modeled as multi objective linear programming problem with imprecise parameters. Fuzzy parametric programming has been used to handle impreciseness and the resulting multi objective problem has been solved by prioritized goal programming approach. A case study has been made using the proposed approach.

The problem of minimizing the total cost of transportation has been studied since long and is well known. In a time minimizing transportation problem, the time of transporting goods is minimized to satisfy certain conditions in respect of availabilities at sources and requirements at destinations. The basic difference between cost minimizing and time minimizing transportation problem is that the cost of transportation changes with variations in the quantity but the time involved remains unchanged and irrespective of the quantities. The time minimizing transportation problem has been studied by Hammer[1,2], Garfinkel and Rao[3], Szware[4], Bhatia, Swarup and Puri[5], Ramakrishnan[6], Sharma and Swarup[7], Seshan and Tikekar[8] and by several other authors. Time-cost trade off means the problem of minimizing the transportation cost in addition to minimizing the time of the transportation. Time-cost trade off analysis has been discussed by Satya Prakash[9], Bhatia, Swarup and Puri[10] and several other authors. Satya Prakash[9] has used goal programming approach to solve the problem. Liu[11] discussed a method for solving the cost minimization transportation problem with varying demand and supply.Most of the models developed for solving the transportation problem are with the assumption that the supply, demand and the cost per unit values are exactly known. But in real world applications, the supply, the demand and the cost per unit of the quantities are generally not specified precisely i.e. the parameters are fuzzy in nature. Impreciseness in the parameters means the information for these parameters are not complete. But even with incomplete information, the model user is normally able to give a realistic interval for the parameters. Carlsson and Korhonen[12] and Chanas[13] discussed parametric approach to deal with the fuzzy parameters

In this paper an effort has been made to develop a model to minimize the transportation cost as well as to minimize the transportation time. As the parameters in real world problems generally are not found in precise form, we incorporated imprecise parameters in the model. In the proposed model we find that the objective function Transportation cost (11) and the resulting goal programming model (30) with (25)-(29) is a function of the membership function. Both the objectives, minimization of the transportation cost and minimization of transportation time are reasonably achieved. Also in Fig. 6 we see how the transportation cost changes with the change in membership function. Using fuzzy parametric programming and preemptive goal programming, the solution has been obtained which is a good compromise solution. The model takes care to minimize the total transportation duration. As the time minimization in transportation problem is a special type of problem to handle, we have utilized the concept of partitioned set (of time) and the well-known preemptive goal programming approach. Although in this model, linear and exponential membership functions have been considered, different types of membership functions may also be considered as per suitability. The proposed technique has been illustrated with a case study. The model may also be used to solve time-cost trade off transportation problems with several objectives with real field data.