Solving the problem of industry by formulating it as a fractional capacitated transportation problem with bounds on rim conditions (original) (raw)
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2017
This paper discusses a paradox in a capacitated transportation problem where the objective function is the sum of two fractional functions consisting of variable costs only. A paradoxical situation arises in a transportation problem when value of the objective function falls below the optimal value and this lower value is attainable by shipping larger quantities of goods over the same routes that were previously designated as optimal. Firstly, optimality condition at which a feasible solution of fractional plus fractional capacitated transportation problem will be an optimal solution is established. Then a sufficient condition for the existence of paradox is found. If paradox exists, then the procedure for finding the best paradoxical pair is proposed which ultimately gives a paradoxical range of flows. Moreover, a method is proposed to find the paradoxical solution for a specified flow . Developed algorithm is applied on the real data taken from the account keeping books of the fir...
Linear Plus Linear Fractional Capacitated Transportation Problem with Restricted Flow
American Journal of Operations Research, 2013
In this paper, a transportation problem with an objective function as the sum of a linear and fractional function is considered. The linear function represents the total transportation cost incurred when the goods are shipped from various sources to the destinations and the fractional function gives the ratio of sales tax to the total public expenditure. Our objective is to determine the transportation schedule which minimizes the sum of total transportation cost and ratio of total sales tax paid to the total public expenditure. Sometimes, situations arise where either reserve stocks have to be kept at the supply points, for emergencies or there may be extra demand in the markets. In such situations, the total flow needs to be controlled or enhanced. In this paper, a special class of transportation problems is studied where in the total transportation flow is restricted to a known specified level. A related transportation problem is formulated and it is shown that to each basic feasible solution which is called corner feasible solution to related transportation problem, there is a corresponding feasible solution to this restricted flow problem. The optimal solution to restricted flow problem may be obtained from the optimal solution to related transportation problem. An algorithm is presented to solve a capacitated linear plus linear fractional transportation problem with restricted flow. The algorithm is supported by a real life example of a manufacturing company.
The present paper presents an algorithm to solve a linear plus linear fractional transportation problem with enhanced flow. Sometimes, situations arise where either reserve stocks have to be kept at supply points say, for emergencies, or there may be extra demand in the markets. In such situations, the total flow needs to be controlled or enhanced. In this paper ,a special class of transportation problem is studied where total transportation flow is enhanced to a known specified level.A related transportation problem is formulated and it is established that a special type of feasible solution called corner feasible solution of related transportation problembear one to one correspondence with the feasible solutionof the givenenhanced flow problem. The optimal solution to enhanced flow problem may be obtained from the optimal solution to related transportation problem. An algorithm is presented to solve a capacitated linear plus linear fractional transportation problem with enhanced flow. A real life example of a company is included in support of the theory.
This paper presents an algorithm to find optimum cost-time trade off pairs in a fractional capacitated transportation problem with bounds on total availabilities at sources and total destination requirements. The objective function is a ratio of two linear functions consisting of variable costs and profits respectively. Sometimes, situations arise where either reserve stocks have to be kept at the supply points say, for emergencies or there is a shortfall in the production level. In such situations, the total flow needs to be curtailed. In this paper, a special class of transportation problems is studied where in the total transportation flow is restricted to a known specified level. A related transportation problem is formulated and the efficient cost-time trade off pairs to the given problem are shown to be derivable from this related transportation problem. Moreover , it is established that special type of feasible solution called corner feasible solution of related transportation problem bear one to one correspondence with the feasible solution of the given restricted flow problem. The optimal solution to restricted flow problem may be obtained from the optimal solution to related transportation problem. Numerical illustration is included in support of theory.
2015
In this paper, a comparison of optimum solution between transportation and transhipment problem is discussed whose objective function is fractional and the objective is to minimize the total actual transportation cost to total standard transportation cost. here, the fractional transportation problem is converted to an equivalent fractional transhipment problem and then solved by using the method of fractional transportation problem and concluded that in some situations, fractional transhipment will be less expensive than fractional transportation by means of numerical example.
Fractional transportation problem with non-linear discount cost
Sri Lankan Journal of Applied Statistics
The generalization of linear programming is a fractional programming where the objective function is a proportion of two linear functions. Likewise, in fractional transportation problem the aim is to optimize or improve the ratio of two cost functions or damage functions or demand functions. Since the ratio of two functions is considered, the fractional programming models become more appropriate for dealing with real life problems. The fractional transportation problem (FTP) plays a very important role in supply management for reducing cost and amending service. In real life, the parameters in the models are rarely known exactly and have to be evaluated. This paper investigates the fractional transportation problem (FTP) with some discount cost that avails during the shipment time. The transportation problem, which is one of integer programming problems, deals with distributing any commodity from any group of 'sources' to any group of destinations or 'sinks' in the most effective way with a given 'supply' and 'demand' constraints. The volume of goods to be transported from one place to another incurs some discount cost that could effectively reduce the shipment cost which is directly related to the profit associated with the shipment. This paper is aimed at studying the optimal solution for the problem has been achieved by using Karush-Kuhn-Tucker (KKT) optimality algorithm. Finally, a numerical example is illustrated to support the algorithm.
A Linear Fractional Interval Transportation Problem with and Without Budgetary Constraints
2015
In this paper, we consider a linear fractional inte rval transportation problem with and without budgetary constraints. The decisio n makers can make the correct decisions according to their budget by finding the number of units transported between the intervals of the total supply / total demand. H ere, a solution procedure is provided and it is verified by means of an example.
A Solution Procedure to Solve Multi objective Fractional Transportation Problem
In decision making process if the objective function is ratio of two linear functions and objective function is to be optimized. For example one may be interested to know the ratio of total cost to total time required for transportation. This ratio is an objective function which is fractional objective function. When there are several such fractional objectives to be optimized simultaneously then the problem becomes multi objective fractional programming problem (MOFLPP). Initially Hungarian mathematician BelaMartos constructed such type of problem and named it as hyperbolic programming problem. Same problem in general referred as Linear Fractional Programming Problem. Fractional programming problem can be converted into linear programming problem (LPP) by using variable transformation given by Charnes and Cooper. Then it can be solved by Simplex Method for Linear Programming Problem.. In this paper we propose to solve multi objective fractional transportation problem. Initially will solve each of the transportation problem as single objective and then using Taylor series approach expand each of the problem about its optimal solution and ignoring second and higher order error terms each of the objective is converted into linear one. Then the problem reduces to MOLTPP. Evaluate each of the objectives at every optimal solution and obtain evaluation matrix. Define hyperbolic membership function using best and worst values of objective function with reference to evaluation matrix. These membership functions are fuzzy functions Compromise solution is obtained using weighted a.m. of hyperbolic membership functions and also weights quadratic mean of hyperbolic membership functions. Propose o solve problem at the end to explain the procedure.
Fractional Plus Fractional Capacitated Transportation Problem with Enhanced Flow
2018
This paper presents an algorithm to solve a fractional plus fractional capacitated transportation problem with enhanced flow (EP ). A related transportation problem (RTP ) is formed and it is shown that to each corner feasible solution to (RTP ),there is a corresponding feasible solution to enhanced flow problem (EP ). An optimal solution to (EP ) is shown to be determined from an optimal solution to (RTP ). A numerical example is given in support of the theory and is verified by using a computing software Excel Solver.
A Case Study on Solutions of Linear Fractional Programming Problems
International Journal of Mechanical and Industrial Engineering, 2022
In some decision making problems, objective function can be defined as the ratio of two linear functional subjects to given constraints. These types of problems are known as linear fractional programming problems. The importance of linear fractional programming problems comes from the fact that many real life problems can be expressed as the ratio of physical or economical values represented by linear functions, for example traffic planning, game theory and production planning etc. In this article, correspond to a production planning problem the mathematical model developed, is a linear fractional programming and in order to solve it, various fractional programming techniques has been used. Finally result is compared with the solution obtained by graphical method. To illustrate the efficiency of stated method a numerical example has given.