New Mean and Median Techniques to Solve Multiobjective Linear Fractional Programming Problem and Comparison with Other Techniques

In the feld of operation research, both linear and fractional programming problems have been more encountered in recent years because they are more realistic in expressing real-life problems. Fractional programming problem is used when several rates need to be optimized simultaneously such as resource allocation planning, fnancial and corporate planning, healthcare, and hospital planning. Tere are several techniques to solve the multiobjective linear fractional programming problem. However, because of the use of scalarization, these techniques have some limitations. Tis paper proposed two new mean and median techniques to solve the multiobjective linear fractional programming problem by overcoming the limitations. After utilizing mean and median techniques, the problem is converted into an equivalent linear fractional programming problem; then, the linear fractional programming problem is transformed into linear programming problem and solved by the conventional simplex method or mathematical software. Some numerical examples have been illustrated to show the efciency of the proposed techniques and algorithm. Te performance of these solutions was evaluated by comparing their results with other existing methods. Te numerical results have shown that the proposed techniques are better than other techniques. Furthermore, the proposed techniques solve a pure multiobjective maximization problem, which is even impossible with some existing techniques. Te present investigation can be improved further, which is left for future research.


Introduction
Optimization is the process of fnding the maxima or minima of a single or several objective functions subject to constraints.Many real-world problems are modeled as optimization problems.A linear programming approach has been frequently used to optimize a single linear objective function subject to fnite linear constraints [1].Tis approach is used to solve many real-world problems such as manufacturing, marketing, fnance, advertising, and agriculture.However, it can be difcult to optimize two or more objectives at the same time, and it is even more difcult if the objectives are conficting in nature.Te goal was to generate a compromise solution that achieves all the objective functions simultaneously.Multiobjective optimization techniques are helpful in improving the decision-making process in such situations.Several methods have been proposed to solve such problems by various scholars [2].However, due to the limitations of the methods, choosing a proper technique remains a subject of active research.In the optimization problem, linear fractional programming concerns the optimization problem of a ratio of two linear functions subject to some constraints.Recently, linear fractional programming problems have attracted the interest of many researchers due to their application in different disciplines such as production planning, fnancial and corporative planning, economics and management science, healthcare, and hospital planning [3].
In this paper, we focus our interest on a multiobjective linear fractional programming problem where more than one linear fractional objective function is optimized simultaneously subject to fnite linear constraints.Diferent researchers have proposed diferent scalarizing techniques to solve multiobjective optimization problems.Most of these techniques have been tested with nonconficting objectives that were not appropriate.A multiobjective optimization problem with multiple conficting objectives is the prerequisite for the application of every multiobjective optimization technique.Besides this, the existing scalarizing techniques have been evaluated when the denominator (scalarizing) quantity of the combined multiobjective function is nonzero.After analyzing the limitations of the previous scalarization techniques, improved scalarization techniques are proposed in this study.Tese proposed scalarizing quantities are very rare to be zero when compared to other existing techniques.
Tis study was motivated by the fact that diferent researchers solved nonconficting multiobjective programming problems by converting them into singleobjective programming problems, such as Chandra Sen, advanced transformation, new arithmetic average, new geometric average, advanced harmonic average, advanced mean deviation, and Pearson 2 skewness coefcient.However, solving the conficting multiobjective fractional programming problem using the proposed techniques gave a better result and overcame the limitations of other techniques.Tis study tried to fll this gap.
Te rest of the research is organized as follows.An overview of the literature is given in Section 2. Section 3 presented the methodology of the proposed methods.Section 4 described the problem formulation and the solution concept.Section 5 presented existing and proposed solving approaches.Section 6 discussed our algorithm.Section 7 provides illustrative examples, and the result is compared with the other existing approaches.Section 8 discussed the comparative study.Lastly, Section 9 presented conclusion and suggestions for future work.

Review of the Literature
Tis section presents a review of diferent pieces of the literature related to mean and median scalarizing techniques for multiobjective linear fractional programming problems.
Life is about making decisions, and the choice of optimal solutions is not an exclusive subject of scientists, engineers, and economists.Decision-making is present in daily life; when looking for an enjoyable vacancy, everyone will formulate an optimization problem like with a minimum amount of money, visiting a maximum number of places in a minimum amount of time and with the maximum comfort [4].Multiobjective programming is a part of mathematical programming dealing with decision problems characterized by multiple conficting objective functions that must be simultaneously optimized on a feasible set of decisions [5].Fractional programming is used in several practical applications, such as cutting stock problems, shipping schedule problems, and diferent felds such as education, hospital administration, court systems, air force maintenance units, resource allocation, transportation, and bank branches.Diferent methods were suggested to solve linear fractional programming problem such as the variable transformation approach [6] and updated solution procedures [3].
Many researchers and scholars have studied how to convert multiobjective optimization problems into singleobjective optimization problems and solve them using several techniques.Some of them are listed in Ref. [7].Improved scalarization techniques using mean, harmonic mean, and geometric mean have been applied for solving multioptimization linear programming problems.Reference [8] solved the multiobjective linear fractional programming problem by using mean and median.New average techniques (geometric and arithmetic) had been proposed by Refs.[9,10], which suggested harmonic average and advanced harmonic average techniques to solve multiobjective linear fractional programming problems.Advanced transformation and new statistical averaging techniques have been applied to solve a multiobjective linear programming problem [11][12][13].Also, new mean deviation and advanced mean deviation to solve a multiobjective fractional programming problem have been proposed.Moreover, regarding mean and median, Ref. [14] ofered the Pearson 2 skewness coefcient technique to turn a multiobjective linear fractional programming problem into a singleobjective linear fractional programming problem.Te main goal of this research is to solve a multiobjective linear fractional programming problem with less computational difculties using the new proposed techniques.Furthermore, the proposed techniques address the limitations of existing approaches.

Methodology
In this methodology, we have used two new mean and median techniques to solve a multiobjective linear fractional programming problem.First, the multiobjective linear fractional programming problem was converted into an equivalent linear fractional programming problem by the proposed techniques.Second, the linear fractional programming problem was transformed into a linear programming problem by variable transformation and solved by the conventional simplex method or mathematical software, which compared their results with the other existing methods.Lastly, to illustrate the proposed methods, numerical examples are given.

Mathematical Formulation and Solution Concept
4.1.Linear Fractional Programming Problem.Te general linear fractional programming problem is formulated as follows: According to the method introduced by Charnes and Cooper [6], problem (1) is changed into the following linear problem by the use of variable transformations. Let Teorem [15]: let (y * , λ * ) be the optimal solution of (1), then x * � (y * /λ * ) is the optimum solution for (1).

Multiobjective Linear Fractional Programming Problem (MOLFPP).
In this section, the techniques for transforming a multiobjective linear fractional programming problem into a linear fractional linear programming problem will be explained.
Te general problem of linear fractional programming with multiple objectives may be expressed as [16] MaxZ where r denotes the quantity of objective functions that must be maximized, n − r is the number of objective functions that need to be minimized, and n is the total number of objective functions that must be maximized plus minimized, Defnition 1.A point x * ∈ R n is said to be an efcient solution of a multiobjective linear fractional programming problem (3) if there is no x∈ R n such that Z i (x) ≥ Z i (x * ) for all i � 1, 2, . . .r, and Z i (x) > Z i (x * ) for at least one i, and Z r+1 (x) ≤ Z i (x * ) for all i � r + 1, r + 2, . . .., n, and Z i (x) < Z i (x * ) for at least one i � r + 1.

Applied Techniques of Converting Multiobjective Linear Fractional Programming Problem into Single Linear Fractional Programming Problem
Tere are many techniques, such as Chandra Sen, advanced transformation, new arithmetic average, new geometric average, advanced harmonic average, advanced mean deviation, and Pearson 2 skewness coefcient, presented in the literature to solve multiobjective programming problems.Tese techniques are concisely described in the following.
After individually optimizing all fractional objective functions subject to the given constraints using the above variable transformation (2), we get where c i (i � 1, 2, . . ., n) are the aspiration values of the objective functions.[17].According to Chandra Sen's technique, the multiobjective linear fractional programming problem given in (3) can be converted into a single-objective function as

Chandra Sen Technique
Tis can be solved using the simplex method with the same constraints (3).[11].According to advanced transformation technique, the multiobjective linear fractional programming problem given in ( 3) is converted into a single-objective function as

Advanced Transformation Technique
where subject to the same constraints as in (3).
Journal of Optimization [9].According to a new arithmetic average technique, the combined objective function under the same constraints (3) can be formulated as [9].Te combined objective function by a new geometric average technique under the same constraints (3) is expressed as follows:

New Geometric Average Technique
where , where (10) [10].Te combined objective function by the advanced harmonic average technique under the same constraints ( 3) is expressed as follows:

Advanced Harmonic Average Technique
where AH av � ((2|m 5.6.Advanced Mean Deviation Technique [13].According to the advanced mean deviation technique, the combined objective function under the same constraints (3) can be formulated as where is the total number of objective functions.[14].According to Pearson 2 skewness coefcient technique, the combined objective function under the same constraints (3) can be formulated as

Pearson 2 Skewness Coefcient Technique
where S k2 � 3|mean(c i ) − median(c i )/s| ∀i � 1, 2, . . ., n, where s is a standard deviation for the value of all objective functions.5.8.Proposed Techniques.Te mean, median, and standard deviation have been used for scalarizing the multiobjective linear fractional programming problem.Te proposed techniques to solve the multiobjective fraction programming problem based on the mean and median idea are briefy described in the following.
Te process of translating a multiobjective fraction programming problem into a single linear fractional programming problem is as follows: where NMMT denotes new mean and median techniques and is calculated as where

The New Mean and Median Algorithm for Solving MOLFPP
We propose an algorithm for solving multiobjective linear fractional programming problem (3) as follows: Step 1: fnd the individual optimal value of each of the fractional objective functions subject to the constraints using variable transformation.
Step 3: construct the combined single fractional objective functions using formula (15).
Step 4: optimize the combined fractional objective function under the same constraints.

Numerical Examples
In this section, illustrative examples are presented to demonstrate how the techniques and the algorithm work.Some examples have been taken from previous studies to show the diferences with our proposed techniques.
Example 1. (see Ref. [18]): It is clear from the results of Table 1 that both objectives are individually achieved by diferent solutions (values of x 1 , x 2 , x 3, and x 4 ).Here, most of recently proposed methods such as advanced transformation, new arithmetic average, advanced harmonic average, and Pearson 2 skewness coefcient did not solve the problem.

New Mean and
Example 2 (see Ref. [14]): It is clear from the results of Table 2; all seven objectives are not individually achieved by the same solution.
Example 3. Consider the following multiobjective linear fractional programming problem: Using the value of mean, median, and standard deviation from Table 3, we fnd the following.

New Mean and Median Technique 2. MaxZ � (􏽐
After solving (30), we get Max Z � 151.35 at (40, 0, 0).Now, solve Example 4 using other techniques such as Example 5 (see Ref. [19]): Using the value of mean, median, and standard deviation from Table 5, we fnd the following.In Example 5, advanced transformation, advanced harmonic average, and Pearson 2 skewness coefcient did not also solve the problem.Tis is one of the limitations.

Comparative Study
Te following table summarizes the results of the MOLFPP using diferent techniques.It shows the comparison between the techniques studied in this paper and other techniques.Te solution to the problem obtained by the proposed techniques gave a better result than the other techniques that were previously studied, as shown in Table 6.In addition to this, the proposed techniques solve problems that could not be solved by some existing techniques.
From Table 6, "-" represents that the techniques did not solve the given problems.It is evident from this table that the results of Examples 1-5 show that when using the two proposed techniques called New Mean and Median Technique 1 and New Mean and Median Technique 2, the results are better than other techniques.

Conclusion and Future Work
In this paper, we have proposed two new techniques to convert a multiobjective linear fractional programming problem into a single linear fractional programming problem, and then the single linear fractional programming problem is solved by variable transformation in simple and easy ways.Tese techniques provided a more compressive and efective solution to the conficting multiobjective linear fractional programming problem.To illustrate the solution process and motivation, some numerical examples have been solved and the techniques were compared with the other existing techniques that were previously studied, as shown in Table 6.Te numerical results confrm that our proposed techniques optimize the problem better than those other techniques.Furthermore, the limitations of the existing methods have been pointed out.
Future studies would compare these new mean and median techniques with other techniques other than Chandra Sen, advanced transformation, new arithmetic average, new geometric average, advanced harmonic average, advanced mean deviation, and Pearson 2 skewness coefcient to reinforce the results and come up with better results.In addition to this, the proposed algorithm can be improved to solve multilevel multiobjective fractional programming problems.

2
Journal of Optimization where c t x + α, d t x + β are the real valued and continuous functions on S, d t x + β ≠ 0 for all x ∈ S, and c, d ∈ R n , b ∈ R m are the column vectors,α, β ∈ R are scalars, and A ∈ R m×n represent the m × n matrix.

Table 1 :
After all objective functions have been optimized individually, the results of Example 1 are given below.

Table 2 :
After all objective functions have been optimized individually, the results of Example 2 are given below.

Table 3 :
After all objective functions have been optimized individually, the results of Example 3 are given below.

Table 4 ,
we fnd the following.

Table 4 :
After all objective functions have been optimized individually, the results of Example 4 are given below. ) [14]By the Pearson 2 skewness coefcient[14], the result is Max Z � 1.29 at (3.6, 2.6 ).

Table 5 :
After all objective functions have been optimized individually, the results of Example 5 are given below.

Table 6 :
Te comparison of the fve numerical results that are obtained from previous examples is presented as follows.