A novel iterative method to solve a linear fractional transportation problem

The linear fractional transportation problem (LFTP) is widely encountered as a particular type of transportation problem (TP) in real-life. In this paper, a novel algorithm, based on the traditional definition of continuity, is presented to solve the LFTP. An iterative constraint is constructed by combining the objective function of the LFTP and the supply-demand condition since the fractional objective function is continuous at every point of the feasible region. By this constraint obtained, LFTP is converted into an iterative linear programming (LP) problem to reach the optimum solution. In this study, the case of asymptotic solution for LFTP is discussed for the first time in the literature. The numerical examples are performed for the linear and asymptotic cases to illustrate the method, and the approach proposed is compared with the other existing methods to demonstrate the efficiency of the algorithm. Also, an application had environmentalist objective is solved by proposed mathematical method using the software general algebraic modeling system (GAMS) with data set of the real case. Finally, some computational results from tests performed on randomly generated large-scale transportation problems are provided.

The TP includes decisions having great importance in logistics and supply chain management in terms of reducing costs and providing the best service. The TP with a fractional objective is a widely used LFP problem having ratios such as profit/cost, income/capital, profit/labor, total actual transportation cost/total standard transportation cost, and risk assets/capital, etc. Firstly, Swarup (1966) dealt with a fractional transportation problem (FTP) and supposed that the denominator is always positive. Moanta (2007) studied to obtain an optimum solution with the simplex method for the three-dimensional TP having the objective function as the ratio of two positive linear functions. Sheikhi et al.  The rest of the paper is arranged as follows. The definition of the LFTP and some preliminaries are mentioned in Section 2. In the next section, we describe the solution methodology. We introduce our algorithm and present its flow chart in Section 4. There are experiments in order to illustrate our algorithm in Section 5. The last section emphasizes our conclusions.

Problem definition and preliminaries
LFTP is a problem optimizing the rate of profitability expressed as cost/ profit, time/profit or the cost/amount of materials to be transported under supply and demand constraints. The mathematical model of a general LFTP can be stated as follows: is decision variable which refers to product amount transported from ℎ supply source to ℎ demand region. We assume that ( ) 0 P  x (Since it is an objective for the TP, it will always be positive.), ( ) 0 , where the feasible region S is a convex and non-empty feasible set defined by constraints (1b)-(1d), and which is a necessary and sufficient condition for the existence of a feasible solution to the problem (1a)-(1d). Because the total supply is not less than the total demand.
The continuity can be stated in terms of traditional neighborhoods as follows: x is continuous at the given k mn  xR .

Our methodology
Considering the algorithm in (Ozkok, 2020), the convergence ensures for every is obtained. From the fractional objective function (1a) of LFTP, we get Combining the convergence condition (3) and the other expression of the fractional objective function (4), we have: Following the algebraic operations, A novel iterative method to solve a linear fractional transportation problem 155 is obtained.
Z and Z are used to explain the objective function for the LFTP and the LP problem, respectively. The following iterative LP problem denominated LFTP-LP(k) is constructed to solve the LFTP (1): Here, the superscript   0,1, 2, k  points out the iteration counter and (6.2) denotes our iterative constraint. Our algorithm starts with an initial point ( , , , , ,

Determining an initial solution
Firstly, the initial point needs to be determined to start the algorithm. Choosing an initial solution ( ) , the only condition to be considered is that the fractional objective function must not be undefined at that point. In this study, the initial feasible point is chosen by solving the following LP problem: Since we are solving LFTP, it is possible to obtain the initial feasible solution by using The Northwest Corner

Execution of the proposed algorithm
Considering the previous statements, our algorithm is as follows: A novel iterative method to solve a linear fractional transportation problem 157 Step 1: Choose the initial solution ( ) 0 0 , Z x for the LFTP (1) and M which is a very big number, set 0 k = .
Step 2: Obtain the optimum solution * x of LFTP-LP(k).
Step 3: Find the product value of * 1.x .
Step : If Step : is the optimum value of LFTP. STOP.
Step : If 1 k k ZZ +  , then put 1 kk =+ , and return to Step 2.
The flow chart of the proposed method is illustrated in Figure 1. The algorithm is presented with the direction of the minimum objective. On the other hand, we notify that the procedure can be updated to the direction of the maximum objective with respect to the structure of the LFTP thanks to the flexibility of the algorithm.   Step : Since 0 1 ZZ  , then put 1 k = , and return to Step 2.
Step 2: The LFTP-LP(1) is constituted as: A novel iterative method to solve a linear fractional transportation problem 160 The optimum solution of (9) is We state that our algorithm finds the same optimum solution with the transportation simplex method (Bajalinov, 2003). However, our algorithm generates the optimum solution by applying two iterations containing one linear programming problem instead of many computations.

Example 2 (Gupta et al., 1993) (Asymptotic case):
The following LFTP is considered ( )  Let S denotes the feasible region.
Step : Since Step : Since 0 1 ZZ  , then put 1 k = , and return to Step 2.
Step 2: Using 2 x and 2 Z , the problem LFTP-LP(2) is constituted as:  (12) The optimum solution of (12) is Step : Since 32 ZZ  , then put 3 k = , and return to Step 2.
Step 2: Using 3 x and 3 Z , the problem LFTP-LP(3) is constituted as: The optimum solution of (13) is Step : Since Step : Since 43 ZZ  , then put 4 k = , and return to Step 2.
Step 3: Since Step : 54 ZZ  , so the limit value of the objective function at the asymptotic solution of the LFTP is 40292.395 0.83367 48331.474  . Namely, Thus, the solution generated by our algorithm yields a better objective function value than Gupta et al.'s (1993) optimum solution. A comparison of the solutions is presented in Table 1.

Practical applications
Let us consider a part of supply chain network of a company which produces and markets textile products. It is desired to implement logistic processes most efficiently by focusing on customer satisfaction. In the part of network, the products are transported from six distribution centers to meet ten customer zones demand. By minimizing the carbon footprint by taking into account the 2 CO emission that occurs during transportation, an environmentalist perspective will be gained to the supply chain management. The requirements of the customer zones are respectively 4830, 2900, 4910, 2720, 4800, 2760,3740, 4520, 7460,3500 units of product and the supply of distribution centers are respectively 6600,9040,7800,9600,7200,1900 units of product. Moreover, the data of profit and transportation cost for one unit of product to be transported from ℎ distribution center to ℎ customer zone are given in Table 2-A novel iterative method to solve a linear fractional transportation problem 163 3. How many units should be transported from each distribution centers to each customer zones in order that the profitability ratio of the company, expressed as profit/total 2 CO emission, is maximum with the fixed amount of 2 CO emission of processing at distribution centers being 165000 units?  The fractional programming of the above transportation problem is as follows: A novel iterative method to solve a linear fractional transportation problem 164 6 10 ij ij i=1 j=1 6 10 ij ij i=1 j=1 px max Z( ) c x +165000 =   x (15) subject to

Conclusion
In this study, we expanded the iterative method developed by Ozkok (2020) for the LFP problem to LFTP. Firstly, it is considered that the objective function is continuous when linearization is performed for the proposed solution procedure based on the traditional definition of continuity. Then, a linear iterative structure is constructed with the linearization. This iterative method plays an important role in solving LFTP in real-life situations due to the effectiveness and easiness of computation. Our approach overcomes successfully the shortcoming in terms of computation by reason of increasing process as the problem size raises since it is an iterative procedure based on LP unlike existing methods. The proposed algorithm can be easily applied to even large-scale LFTPs since it solves LFTP by a series of LP problems. Another advantage of our algorithm is that it has the ability to solve the LFTP with mixed constraints since the method does not depend on constraints. Moreover, the case of asymptotic solution for LFTP was handled for the first time.
The numerical examples and a case study were executed to demonstrate the proposed mathematical solution method, and the solutions of the examples were also compared with existing methods. Finally, the proposed method was performed on randomly generated large-scale problems, and computational results were presented. We coded the proposed algorithm in the software GAMS 35.1.0. For further study, the expanded iterative solution procedure can be adapted to different kinds of LFTP (i.e interval or solid LFTP) thanks to its flexibility of algorithm. As another future research, the algorithm can be improved to solve fuzzy LFTPs since introducing the parameters with fuzzy numbers would give more realistic result in modelling of real situations.