Fractional Transportation Problem with Non-Linear Discount Cost

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.


Introduction
Fractional Programming Problem (FPP) is a mathematical language used for describing the optimization problems. It is based on parameters, decision variables, objective function (rational form) subject to various types of constraints. FPP is an extension or generalization of linear programming problem where the objective function is in the form of ratio with linear constraints. The fractional programming problem has received the interest of many researchers due to its application in numerous essential fields such as production planning, financial and corporate planning, health care and hospital planning. It can be seen in many practical optimization problems where the objective functions are quotients of two functions which are used to achieve the highest ratio of outcome to cost, profit to time, profit to cost, output to employees, return to cost, actual cost to standard cost and minimizing inventory to sales, cost to time, student to teacher ratios etc., where the ratio representing the highest efficiency of a system. Several methods were suggested for solving this problem such as the variable transformation method Charnes and Cooper (1962) and the updated objective function method by Bitran and Novaes (1973).
Transportation problem is one of the most widely used applications of mathematical programming. Mainly, transportation problem is concerned with shipping methods or selecting routes in a product distribution network among the manufacturing plants and distribution warehouses situated in different regions or local outlets. In literature, it has been used in number of ways. The problem commonly faced in the distribution of goods from various supply points to different destination points is known as the transportation problem, first studied by Hitchcock (1941). Dantzig (1951) adapted the simplex method to solve the transportation problem formulated earlier by Hitchcock. Charnes et al. (1954) derived an intuitive presentation of Dantzig's procedure called the stepping-stone method which follows the basic logic of the simplex method but avoids the use of the tableau and the pivot operations required to get the inverse of the basis.
Originally, Swarup (1966) studied a fractional transportation problem under the assumption that the denominator is always positive and it has an important role in logistics and supply chain management for reducing cost and improving service. Transportation problem with discount cost has been studied by many researchers viz., Shetty (1959) formulated an algorithm to solve transportation problems taking non-linear costs. He considered the case when a convex production cost is included at each supply centre besides the linear transportation cost. Dhingra, et al.(2012) have proposed an algorithm to solve multi-objective fractional bottleneck transportation problem with restrictions. In this paper, we have proposed a method for fractional transportation problem to find out the per unit profit associated with each shipping along with some non-linear discount cost. We have also used KKT condition to get the optimal solution. The methodology has been verified through a numerical example.
This paper is organized in the following way. In Section 1, we have presented a background study of fractional programming problem and transportation problem. In Section 2, preliminaries used for solving fractional transportation problems and the KKT optimality condition have been discussed. In Section 3, algorithms for solving the fractional transportation problems are presented. The effectiveness of the algorithm is proven by the practical application in section 4. Finally, in section 5, concluding remarks are outlined.

Preliminaries
This research seeks to apply the existing general non-linear programming algorithms to solve the problem. Before formulating the problem of interest, it is necessary to have a IASSL ISSN-2424-6271 190 good understanding of the background of non-linear programming which is reproduced here from Mubashiru (2014):

Polyhedral Sets
A set P in an n dimensional normed vector space is called polyhedral set if it is the intersection of a finite number of closed-half spaces, i.e.

matrix of rank m, and b is an m vector. A point x is an extreme point of P if and only if A can be decomposed into [B, N] such that,
Any such solution is called a basic feasible solution for P. The number of extreme points of P is finite.

is an m x n matrix of rank m and b is an m vector. Then
P has at least one extreme point.

2.2Extreme Direction
Let P be a non-empty polyhedral set in where, j e is an n-m vector of zero except for in position j which is 1.

Programming Problem (NPP)
Given the non-linear programming problem: In the following subsections, we have discussed about the necessary and sufficient optimality conditions of KKT.

2.3.1KKT Necessary Optimality Conditions
Further, iff and i g are convex, each j h as affine, then the above necessary optimality condition will also be sufficient.

KKT Sufficient Optimality Conditions
Let x be any feasible point different form x*. From the first KKT conditions, we obtain,

KKT Optimality Condition for Fractional Transportation Problem
The general fractional transportation problem is modeled as: We suppose that The fractional transportation table can be represented as: 11 ) ( where, x is the current basic solution.
IASSL ISSN-2424-6271 194 The Lagrange function for the non-linear fractional transportation is formulated in the form: where,  and w are Lagrange multipliers and   ℝ nm .
The optimal point x should satisfy the KKT conditions: Specifically, for each cell (i, j), we have, General solution procedure for the NTP:  Initialization: Find an initial basic feasible solution x .
 Iteration: Step 1-If x is KKT point, stop. Otherwise, go to the next step.
Step 2-Find the new feasible solution that improves the cost function and go to step 1.

Algorithm for Solving Non-linear Fractional Transportation Problem
Step ( Step (2): Find the initial basic feasible solution of the FTP by using anyone of the wellknown method for solving transportation problem.
Step ( x c x p Z 

Test for Optimality (or improvement):
After obtaining the initial feasible solution, the next step is to test whether it is optimal or not. If the solution is non-optimal, then we improve the solution by exchanging non-basic variable for a basic variable. In other words, we rearrange the allocation by transferring units from an occupied cell to an empty cell that has the largest net cost change or improvement index and then shift the units from other related cells so that all the rim (supply, demand) requirements are satisfied. This is done by tracing a closed path or closed loop.
Otherwise, go to step (7), where, ij NB x are the non-basic variables.
Step (7): Check the following given conditions: x is the basic variable comes to zero first while making the adjustment. Then, find the new basic variable and go to step 1.
The feasible set of our problem is a non-empty polyhedral set. And by definition, a polyhedral set P is a set bounded with a finite number of hyper planes from which it follows that it possesses finite number of extreme points. In each step of the algorithm, we jump from one extreme point to another looking for a better feasible solution implying that the algorithm will terminate after a finite iteration. In addition, since for all i and j, , P is bounded that guarantees the existence of minimum value.

Numerical Example
In this section, we consider a computational study of the above solution procedures. Emphasis will be given to a FTP where discounts are given to volume on quantity of goods transported which is concave in nature. Consider the following FTP with profit and shipping cost. We formulate the problem to maximize the per unit profit associated with shipments. Profit matrix  Table 1.  || || associated with each shipment and it is directly related to the unit of commodity purchased and transported and the discount (%) are shown in Table 3.

Table 3: Discount Cost (%) where A, B, C and D are sinks and E, F and G are sources
If we suppose that discount cost associated with each shipping from i to j, then the nonlinear fractional transportation problem can be formulated as follows: 11   150.
Therefore, we get the following terms of cost function: If we allow the discount on each transported product i from the source to each of the destination j as given in Table 3, the cost function becomes: Using any known method for solving transportation problem, we get the initial basic solution. The solution tableau is shown below:  Now, we use the KKT optimality conditions to improve our solution. The partial derivatives at x for the required function are given as: 11 1 Letting 1 u = 0 in the above equations, we obtain the following values: Next, we find the net evaluation factor or reduced cost for the non-basic variables using the following equation: The presence of negative value for the reduced cost signifies non-optimality. From the above equations, the minimum reduced cost for the non-basic variable is 12 x . Therefore, 12 x should enter the basis since it is the only negative reduced cost. We then move to the

Conclusion
In this paper, we have demonstrated that how fractional transportation problem can be solved efficiently for maximizing the per unit profit. This paper aimed at solving fractional transportation problem with volume discount on quantity of goods shipped which is a nonlinear fractional transportation problem. We applied KKT optimality algorithm to solve the fractional transportation problem. Maximization of profit is realized with discounts on large volumes, which means the determination of the best transportation route that would lead to low transportation cost and the effective transportation of these goods. This method can be used in all cases where discount cost, road tax, damage during the transportation is given. We then conclude that given discount on the cost of shipment could lead to increased productivity of producers. It provides the decision maker an alternative view to finding out the positive extent while transporting a product and hence this technique can contribute significantly to the literature of fractional programming.