We consider a multiobjective linear fractional transportation problem (MLFTP) with several fractional criteria, such as, the maximization of the transport profitability like profit/cost or profit/time, and its two properties are source and destination. Our aim is to introduce MLFTP which has not been studied in literature before and to provide a fuzzy approach which obtain a compromise Paretooptimal solution for this problem. To do this, first, we present a theorem which shows that MLFTP is always solvable. And then, reducing MLFTP to the Zimmermann’s “min” operator model which is the maxmin problem, we construct Generalized Dinkelbach’s Algorithm for solving the obtained problem. Furthermore, we provide an illustrative numerical example to explain this fuzzy approach.
A lot of research work has been conducted on transportation problems. These problems and their solution techniques play an important role in logistics and supply chain management for reducing cost, improving service quality, and so forth. One of the hot research topics of transportation problems is the use of fuzzy set theory. In 1965, the concept of the fuzzy set was first introduced by Zadeh. Several authors gave the most significant papers on fuzzy sets and their applications. In practical applications, the required data of the reallife problems may be imprecise. Thus, adaptation of fuzzy sets theory in the solution process increases the flexibility and effectiveness of the proposed approaches. There are recent papers by Zadeh published in 2005 and 2008 [
Fractional programming (FP) for singleobjective optimization problems was studied extensively from the viewpoint of its application to several reallife problems. For instance, cost benefit analysis in agricultural production planning, faculty and other staff allocation problems for minimizing certain ratios of students’ enrolments and stuff structure within academic units of educational institutions, and other optimization problems frequently involve the fractional objectives in a decision situation [
Now, since most of the decision making problems in practice are multiobjective in nature, FP with multiplicity of objectives has also been studied by the pioneer researches in the field [
In reallife situations, multipleobjective linear transportation problem (MLTP) is more common. A lot of fuzzy research work has been conducted on MLTP. In [
Transportation problem with fractional objective function is widely used as performance measure in many reallife situations, for example, in the analysis of financial aspects of transportation enterprises and undertaking and in transportation management situations, where an individual or a group of a community is faced with the problem of maintaining good ratios between some crucial parameters concerned with the transportation of commodities from certain sources to various destinations. In transportation problems, examples of fractional objectives include optimization of total actual transportation cost/total standard transportation cost, total return/total investment, risk assets/capital, and total tax/total public expenditure on commodity [
To deal with LFTP, it can be observed from literature that few works have been carried out except Bajalinov’s work [
In this paper, our aim is to obtain a compromise Paretooptimal solution for MLFTP by means of Zimmermann’s “min” operator. Using Generalized Dinkelbach’s Algorithm, MLFTP that has nonlinear structure is reduced to a sequence of linear programming problems. However, the solution generated by “min” operator does not guarantee Paretooptimality. For this reason, we offer a Paretooptimality test to obtain a compromise Paretooptimal solution.
This paper is organized as follows. Section
Assume that there are
MLFTP is solvable if and only if the above inequality (
We will show that
From (
Finally, since
Hence, the MLFTP is solvable.
If total demand equals total supply, that is,
If a transportation problem has a total supply that is strictly less than total demand, the problem has no feasible solution. In this situation, it is sometimes desirable to allow the possibility of leaving some demand unmet. In such a case, we can balance a transportation problem by creating a dummy supply point that has a supply equal to the amount of unmet demand and associating a penalty with it.
By converting inequalities (
MLFTP in a canonical form is solvable if and only if the above balanced condition (
This theorem can be proved by a procedure similar to Theorem
In the context of multiple criteria, definitions of efficient and nondominated or Paretooptimal solutions are used instead of the optimal solution concept in singleobjective programming. In the multipleobjective linear fractional programming, the concept of the strongly efficient solution which corresponds to the standard definition of the efficient solution in multiobjective linear programming is insufficient, and therefore the weakly efficient concept is considered. In theory, finding the strongly efficient solutions is desired. However, in practice, solution algorithms tend to find weakly efficient solutions. This is because, vertexes of the weakly Paretooptimal solution set (
The point
The point
According to these definitions,
A feasible point
According to Definition
In general, an optimal solution which simultaneously maximizes all the objective functions in MLFTP does not always exist when the objective functions conflict with each other. When a certain Paretooptimal solution is selected, any improvement of one objective function can be achieved only at the expense of at least one of the other objective functions. Thus, the above Definition
In this section, using Zimmermann’s “min” operator, we will give a fuzzy approach to obtain a compromise Paretooptimal solution for MLFTP.
There are several membership functions in the literature, for example, linear, hyperbolic, and piecewiselinear, and so forth [
First of all, by using the “min” fuzzy operator model proposed by Zimmermann [
Here, since the membership function
Problem (
The transportation problem in (
One of the most popular and general strategies for fractional programming (not necessary linear) is the parametric approach described by W. Dinkelbach. In the case of linear fractional programming, this algorithm reduces the solution of a sequence of linear programming problems [
In this section, we consider the maxmin problem (
Let
parametric function
if problem (
if
Assume that
Equations (
Parametric function
The sequence
This lemma and theorem provide the necessary theoretical basis for a generalization of Dinkelbach’s algorithm. Algorithm is suggested that finds the constrained maximum of the minimum of finitely many ratios. The method involves a sequence of linear subproblems if the ratios are linear.
Now we can give Generalized Dinkelbach’s Algorithm for MLFTP.
The convergence of sequence
for all
the sequence
A current solution
Let us apply the fuzzy approach presented above to the following numerical example. Consider
Three linear fractional transportation problems are solved as nonlinear programming problems directly by using any available nonlinear programming package, for example, GAMS [
The individual maxima and minima and corresponding solutions.


 


2.111  4.972  1.736 

2.059  4.138  1.687 

2.111  4.138  1.687 

2.059  4.972  1.736 

2.059  4.972  1.736 

2.111  4.138  1.687 

2.059  4.972  1.736 

2.111  4.138  1.687 
By using (
Solving this problem, we obtain
The results corresponding to five iterations for the problem (
The results corresponding to five iterations for the problem (
Iteration 









1  0  0  0  — 


0.527  0.419  0.527  0.419  0.462 


0.48  0.465  0.574  0.465  0.053 


0.473  0.471  0.58  0.471  0.007 


0.473  0.472  0.58  0.472  0.001 


0.472  0.472  0.581  0.472  0.0004 
For
As known, unimodularity feature is observed in the coefficient matrix of the constraints of transportation problem where the determinant of all the square submatrices is either 0 or +1 or −1. If this feature is not satisfied, then the integer solution does not guarantee and the problem does not fit the standard form of the transportation problem [
In our fuzzy approach, model (
The results corresponding to two iterations for the problem (
Iteration 









1  0  0  0  — 


0.530  0.416  0.524  0.416  0.459 


0.470  0.475  0.583  0.470  0.047 
All solutions are obtained by using the GAMS computer package.
In this study, MLFTP has been introduced for the first time according to the best of our knowledge. First, we provided a theorem emphasizing the fact that MLFTP always has feasible solution; its set of feasible solutions is bounded, and, hence, MLFTP is always solvable. It is also proposed a fuzzy approach which generates a compromise Paretooptimal solution for MLFTP by reducing the problem to the Zimmermann’s “min” operator model and then constructing a solution algorithm based on Generalized Dinkelbach’s Algorithm.
By means of Generalized Dinkelbach’s Algorithm, MLFTP with nonlinear structure is reduced to a sequence of linear problems. Using LP is one of the most advantageous aspects of this method. We note that, although choosing linear membership function gives essential cause for this linearization property, using another different shape of membership functions such as hyperbolic and exponential and piecewiselinear, by means of Generalized Dinkelbach’s Algorithm, does not.
Our method generates a weakly Paretooptimal solution for MLFTP, if the problem has alternative solutions. In this case, by using the Paretooptimality test, a compromise Paretooptimal solution is found. Otherwise, our method can directly find a compromise Paretooptimal solution.
The sequence
An illustrative example is presented to show our fuzzy approach. As seen, at each iteration LP problems were solved rather than linear fractional ones.
This paper mainly provided an introduction to MLFTP. Many areas are still needed to be explored and developed in this direction. Some fuzzy approaches—using the goal programming, and/or using the nonlinear membership functions—can be developed for MLFTP, to find its compromise Paretooptimal solution. Computer codes must be written to be used to solve largescale applications of the method, such as applications to transportation system problems. Moreover, fuzzy approaches with fuzzy parameters for MLFTP together with different shapes of membership functions and/or their stochastic models, solid transportation problems, network problems, and so forth may become new topics in further research.
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors are very indebted to the anonymous referees for their critical suggestions for improvements.