Solving Fuzzy Multi-Objective Linear Programming Problem by Applying Statistical Method

In this paper, the statistical averaging method and the new statistical averaging methods have been used to solve the fuzzy multi-objective linear programming problems. These methods have been applied to form a single objective function from the fuzzy multi-objective linear programming problems. At first, a numerical example of solving fuzzy multi-objective linear programming problem has been provided to validate the maximum risk reduction by the proposed method. The proposed method has been applied to assess the risk of damage due to natural calamities like flood, cyclone, sidor, and storms at the coastal areas in Bangladesh. The proposed method of solving the fuzzy multi-objective linear programming problems by the statistical method has been compared with the Chandra Sen’s method. The numerical results show that the proposed method maximizes the risk reduction capacity better than Chandra Sen’s method.


Introduction
Real world circumstances are not always deterministic. There exist different kinds of uncertainties in social, industrial and economic systems. Different kinds of uncertainties are defined as stochastic uncertainty and fuzziness [1]. A system with a stochastic uncertainty is solved by the stochastic optimization technique using the probability theory. Also, fuzzy programming technique is widely used to solve problem with uncertainty. A system with uncertainty can be optimized to reduce the risk factors in the system with fuzzy conditions. An optimization of a system with uncertainty using fuzzy conditions is called a fuzzy Optimization. Modelling under a fuzzy environment is called fuzzy modelling. Fuzzy linear programming is one of the most frequently applied fuzzy decision making techniques. We get fuzzy linear programming problem (FLPP) by interchanging the parameters of linear programming problem by fuzzy numbers. Several methods have been proposed in the literature to solve FLPP.
Firstly, Bellman and Zadeh in [2] proposed decision making in fuzzy condition. They explained the use of decision making in fuzzy condition by using a controlled system which is either stochastic or deterministic with multistage decision processes, Zimmermann proposed a formulation of FLPP using theory of fuzzy sets. He used FLPP to the linear vector maximum problem and found that the solutions obtained using FLPP are always efficient. Considering parameters ambiguity, Tanaka and Asai also proposed a formulation of FLPP to find a logical solution [3]. They highlighted that the FLPP with fuzzy numbers may be considered as a model of decision problems in a system with influential human estimations. Not only linear programming problems (LPP), but also multi-objective LPP (MOLPP) and multi-objective nonlinear programming problems were solved using fuzzy approaches which are available in the literature.
Laganathan and Lalitha [4] solved a multi-objective nonlinear programming problem using α-cut method in fuzzy approach and compared the solution with the solution obtained by Zimmermann [5] who used membership function. Beher et al. [6] also used α-cut method to solve multi-objective linear programming problem (MOLPP) in fuzzy approach and compared the solutions to the solution obtained by Zimmermann [5] who used membership function.
Thakre et al. [7] solved an objective function with constraint matrix and cost coefficients which are fuzzy in nature. They used MOLPP with the constraints to solve FLPP and proved that the solutions are independent of weights. Nahar and Alim [8] discussed Chandra Sen's approach, statistical averaging method and new statistical averaging method to solve MOLPP. For MOLFPP, a new geometric average technique was proposed by Nahar and Alim [9]. Veeramani and Sumathi [10] solved fuzzy linear fractional programming problem (FLFPP) where the cost, resources and technological coefficients of the objective function were triangular fuzzy numbers. In the solution procedure, they converted the FLFPP into a multi-objective linear fractional programming problem.
There are a lot of methods for solving fully fuzzy linear programming problems in the literature. Ebrahim nejad and Tavana [11] converted the FLPP into an equivalent crisp linear programming problem and solved by simplex method.
Here Lotfi et al. [13] proposed a method to obtain the approximate solution of fully fuzzy linear programming problems. A method to solve fully fuzzy linear programming problems is proposed by Amit Kumar et al. [14] using idea of crisp linear programming and ranking function. FLPP is solved by S. Nahar et al. [15].
To solve FLPP, Nahar, S. et al. [16] used weighted sum method. Here both equal and unequal weight has been used. In this paper there is a discussion for ranking function. Here they used triangular and trapezoidal fuzzy number. Sen, C. [17] developed averaging technique of multi-objective optimization for rural development planning. Akter, M. et al. [18] proposed fuzzy synthetic evaluation method for risk assessment on the natural hazard. Nishad, A. [19] used alpha cut fuzzy number for solving fractional programming problem in fuzzy field. Pitam, S., [20] developed goal programming approach for fuzzy multi-objective linear and fuzzy multi-objective linear programming problem.
A good averaging represents maximum characteristics of the data. There are five averages, among them mean, median, mode are called simple averages. Geometric mean and harmonic mean are called special averages. In this work we consider arithmetic mean, geometric mean and harmonic mean. In this paper, we solve fuzzy multi-objective linear programming problem by applying Chandra Sen's method, statistical averaging method and new statistical averaging method.
These methods are applied for making single objective from multi-objective linear programming problem. A real life example is given to reduce risk in coastal region. From that data list a FLPP is structured. From that FLPP we convert into MOLPP.

Concept of a Fuzzy Set
The concept of a fuzzy set is an extension of the concept of a crisp set. A crisp set on a universal set U is defined by its characteristic function from U to {0, 1}. A fuzzy set on a domain U is defined by its membership function from U to [0, 1]. Let U be a nonempty set, to be called the universal set or the universe of discourse or simply a domain. Then by a fuzzy set on U is meant a function [ ]

Fuzzy Linear Programming Problem
A linear programming problem with fuzzy values is called fuzzy linear programming problem. In this paper, any fuzzy number is denoted by using ~ above fuzzy number, e.g., j c  , ij a  etc. Consider a fuzzy linear programming problem as in Equation (1).
The membership functions of ij a and i b have been expressed as in Equation (2) and (3), respectively.

( )
be fuzzy numbers. Therefore, the constraints in Equation (1) can be modified as in Equation (4).

Numerical Examples
Consider the fuzzy multi-objective linear programming problem (FMOLPP) as in Equation (8).
where the membership function of ; 25 elsewhere From Equation (8) we get, , 0 For the first objective function in Equation (13)

Statistical Averaging Method
There are three mean known as arithmetic mean, geometric mean and harmonic mean. For ungrouped raw data, mean is defined as the sum of the objectives divided by the number of observations. It is easy to understand and easy to calculate. If the number of items is sufficiently large, it is more accurate and more re- For this objective function in Equation (17) with constraints in Equation (14) For this objective function in Equation (18) with constraints in Equation (14), we get z = 4.3660 with (0.4706, 0.8529) Table 1 shows the comparison between the Chandra Sen's method and the statistical averaging methods. Here statistical averaging method consists of arithmetic mean, geometric mean and harmonic mean. The statistical averaging method gives better result than the Chandra Sen's method. These methods are used for obtaining a single objective function from multi-objective functions. We can see the comparison graphically "as shown in Figure 1".

New Statistical Averaging Method
Choosing minimum from the optimal values of maximum type in Chandra Sen's method we get ( ) ( ) For this objective function in Equation (19) with constraints in Equation (14), we get z = 6.3233 with (0.4706, 0.8529) We can find a single objective function from multi objective functions by using any method among Chandra Sen's method, statistical averaging method, and new statistical averaging method. The Table 2 shows that the statistical averaging method and the new statistical averaging method gives better optimization than the Chandra Sen's method. Table 3 shows secondary data of four parameters such as cropping intensity,  shelter, erosion, and population density at four coastal areas in Bangladesh. These data set have been collected from IWFM (BUET). The objective in this paper is to reduce the risk and hazard of the coastal area during natural disasters by solving FMOLPP. The targets set in this paper are to maximize cropping intensity and shelter and to minimize erosion and population density. For these purposes, those four parameters have been defined using four variables which are defined as decision variables. Our decision variables are 1 2 3 4 , , , x x x x which are risk and vulnerability indicators. 1 x for cropping intensity, 2 x for shelter, 3 x for erosion and 4 x for population density. Constant vector is terminated value for risk reduction capacity (on expert opinion). Decision variables are in different scales. For this reason, these variables have been normalized. Constraints are established based on the data of four coastal areas. The risk reduction coefficients of the objective functions are calculated using weighted method.

Finding Fuzzy Objective Function for Fuzzy Linear Programming Problem
Writing Equation (21) in matrix form, we get

Finding Constraints of Fuzzy Linear Programming Problem
Triangular fuzzy number for the coefficient of x 1 is found from Table 8. Triangular fuzzy number for the coefficient of x 2 is found from Table 9.
S. Nahar et al.
For the first objective function in Equation (27)

Chandra Sen's Method
Applying Chandra Sen's method for making single objective function from multi objective functions

Statistical Averaging Method
Applying arithmetic mean, geometric mean and harmonic mean among 1 2 3 , , φ φ φ   Table 13 shows that the statistical averaging method, the arithmetic averaging, and the harmonic averaging gives better result than the Chandra Sen's method.

Conclusion
In this paper, a fuzzy multi-objective linear programming problem has been solved using the Chandra Sen's method, the statistical averaging method, and the new statistical averaging method. The fuzzy multi-objective linear programming problem with constraints has been established from the data of cropping intensity, shelter, erosion, and population density at four coastal areas in Bangladesh. To maximize the risk reduction capacity, those data of cropping intensity, shelter, erosion, and population density have been set as fuzzy parameters which are triangular fuzzy numbers. The solutions obtained using the statistical averaging method and the new statistical averaging method are better than that of the Chandra Sen's method.