Interactive Fuzzy Goal Programming Based on Taylor Series to Solve Multiobjective Nonlinear Programming Problems with Interval Type 2 Fuzzy Numbers
Abstract
This paper presents an interactive fuzzy goal programming (FGP) approach for solving multiobjective nonlinear programming problems (MONLPP) with interval type 2 fuzzy numbers (IT2 FNs). The cost and time of the objective functions, the resources, and the requirements of each kind of resources are taken to be trapezoidal IT2 FNs. Here, the considered problem is first transformed into an equivalent crisp MONLPP, and then the transformed MONLPP is converted into an equivalent Multiobjective Linear Programming Problem (MOLPP). By using a procedure based on Taylor series, this problem is reduced into a single objective linear programming problem (LPP) which can be easily solved by Maple 18.02 optimization toolbox. Finally, the proposed solution procedure is illustrated by two numerical examples.
Keywords: Fuzzy goal programming, Nonlinear Programming, Taylor series, Interval Type 2 fuzzy sets, Multiobjective nonlinear programming, Membership function.
1 Introduction
Most of the reallife problems are frequently characterized by multiple and conflicting criteria. Such conditions are normally estimated by optimizing multiple objective functions. Besides, when modeling realworld problems, often the parameters are included inexact quantities due to varied unmanageable factors. In practical mathematical programming problems, a decisionmaker generally encounters a situation of uncertainty as well as complexity, due to various unknown factors. Usually, it is required to optimize several nonlinear and conflicting objectives simultaneously. To address the uncertain parameters which result in such situations, different fuzzy numbers are employed. Fuzzy quantities are very suitable for modeling these type conditions.
The fuzzy set theory first developed by Zadeh [1] has been used to decisionmaking problems with imprecise information. Bellman and Zadeh [2] introduce that a fuzzy decision making is defined as the fuzzy set of options, obtaining from the intersection of the goals or objectives and constraints. The concept of fuzzy programming was first introduced by Tanaka et al. [3] in the structure of the fuzzy decision of Bellman and Zadeh. Later, fuzzy programming approach to linear programming with many objectives was investigated by Zimmermann [4].
The simplest approach for solving the fuzzy linear programming problem is converted it into the corresponding crisp programming problem. Zimmermann [4] has developed a fuzzy programming approach to solve the crisp multiobjective linear programming problem. Some authors have transformed the fuzzy programming problem into the crisp problem by using the ranking function [5, 6] and then solved it by conventional methods.
In many practical problems such as in industrial planning, financial and corporate planning, marketing and media selection, etc., there exist many fuzzy and nonlinear production, planning and scheduling problems. These problems cannot be expressed and solved by conventional techniques due to uncertain information. So, the investigation on modeling and optimization for nonlinear programming with interval type 2 fuzzy numbers (IT2 FNs) are not only significant in the fuzzy programming theory but also have a great and wide advantage in the application of the realworld practical problems of conflicting nature.
Type2 fuzzy sets are introduced by Zadeh et al. [7] as the extension of type1 fuzzy sets. Type2 fuzzy sets are characterized by two memberships to determine more degrees. Since type2 fuzzy sets have the advantage of modeling uncertain systems more correctly compared with type1 fuzzy sets. However, when the type2 fuzzy sets are employed to solve the problems, the computational procedures are very complicated [8]. So interval type 2 (IT2) fuzzy sets are widely employed with some relative illustrations to decrease dimensions, which are highly useful for computation and theoretical studies [9]. IT2 fuzzy sets can be observed as a particular illustration of common type2 fuzzy sets that all the values of secondary membership are equal to 1. Hence, it not only represents the uncertainty better than type1 fuzzy sets but also reduces the complexity compared to type2 fuzzy sets.
Mendel et al. presented some definitions and concepts of IT2 fuzzy sets in [8]. Mitchell [10] and Zeng and Li [11] suggested methods to describe the connection between IT2 fuzzy sets. To accomplish limitations in these methods, Wu and Mendel [12] suggested a method called vector similarity method to convert IT2 fuzzy sets into the word more effectively. Ondrej and Milos [13] used IT2 fuzzy sets to generate a fuzzy voter design for faulttolerant systems. Shu and Liang [14] proposed a new method based on IT2 Fuzzy Logic Systems (FLSs) to investigate and evaluate the network lifetime for wireless sensor networks. Wu and Mendel [15] defined linguistic weighted average and used it to handle hierarchical multicriteria decisionmaking problems. Han and Mendel [16] IT2 FNs in deciding the logistic location, and the result has been demonstrated to be more comforting. Chen and Lee [17] proposed the definition of possibility degree of trapezoidal IT2 FNs and some arithmetic operations. Sinha et. al. [18] used IT2 FNs for modeling a multiobjective solid transportation problem. Li et. al. [19] investigated the problem of filter design for IT2 FLSs with D stability constraints based on a new performance index. Up to now, IT2 FNs were used by many authors for decisionmaking problems [20, 21, 22, 23, 24, 25]. Due to their facility to handle with the high level of uncertainty, IT2 FLSs further performed in various realworld applications, containing intelligent control [26, 27], time series predictions [28, 29, 30], pattern recognition [31], image processing [32] and many others.
In this paper, an interactive fuzzy goal programming (FGP) approach based on Taylor series is presented to achieve the highest degree of membership function for multiobjective nonlinear programming (MONLPP) with trapezoidal IT2 FNs. The FGP approach first introduced by Narasimhan [33] and then Hannan [34] presented different membership functions, i.e., piecewise linear membership functions into FGP model. Tiwari et al. [35] introduced a weighted additive model that incorporates each goal’s weight into the objective function, where the weights reveal the relative importance of the fuzzy goals. Mohamed [36] discussed the relationship between goal programming and fuzzy programming where the highest degree of each of the membership goals is achieved by minimizing over deviation variables. Several methods are suggested to linearize the fractional and/or nonlinear functions in literature. In the case of a nonlinear programming, the most common methods are based on linearization procedures [37, 38, 39, 40].
Here, all the parameters of MONLPP are considered trapezoidal IT2 FNs. To the best of my knowledge, no work has been studied on MONLPP with trapezoidal IT2 FNs under the nonlinear constraints. In order to convert the considered fuzzy model into its crisp equivalent, the expected value of trapezoidal IT2 FNs is first employed and then aspiration levels and the tolerance limits of the objective functions are determined by getting individual optimal solutions and thereby the feasible region for the problem is reconstructed by using the upper of lower limits of decision variables. After these operations, the nonlinear membership functions, which are associated with each nonlinear objective of the problem are constructed and then with the use of Taylor series approach around its maximal solution, each nonlinear membership function is converted into linear functions. In this way, this problem is reduced into a single objective linear programming problem and then an interactive solution procedure is presented to determine the optimal solution for MNLOPP with IT2 FNs. Finally, numerical examples are presented to demonstrate the feasibility of the suggested procedures.
The paper is constructed as follows: Sect. 2 deals with some definitions and arithmetic operations on IT2 FNs. Section 3 deals with problem formulation and its solution procedure. In Sect. 4, numerical examples are given to illustrate the methodology. Finally, we concluded in Sect. 5.
2 Preliminaries
2.1 Interval Type2 Fuzzy Set
Definition 1 (Mendel et. al. [8]) Let be a type2 fuzzy set, then is defined as
where is the universe of discourse and denotes the membership function of can be defined as
Definition 2 (Mendel et. al. [8]) If all then called an IT2 fuzzy set i.e.
Uncertainty in the first memberships of a type2 fuzzy set consists of a bounded region that we call the footprint of uncertainty. It is the union of all first memberships.
The footprint of uncertainty is characterized by the upper membership function and the lower membership function, and are denoted by and (Mendel et. al. [8]).
Definition 3 An IT2 FN is called a trapezoidal IT2 FN where the upper membership function and the lower membership function are both trapezoidal fuzzy numbers, i.e.,
(1) 
where and denote membership values of the corresponding elements and respectively.
2.2 The Arithmetic Operations of Interval Type2 Fuzzy Set
Suppose and are two trapezoidal IT2 FNs, then the following procedures are satisfied (Li et. al [19]):
(2) 
(3) 
(4) 
(5) 
(6) 
(7) 
(8) 
2.3 Defuzzification of Trapezoidal Interval Type2 Fuzzy Numbers
Let us consider a trapezoidal IT2 FNcharacterized by Equation (1). The expected value of is determined as follows (Hu et. al [20]):
(9) 
Assuming that and are two trapezoidal IT2 FNs, then we get if and only if
Whenand the trapezoidal IT2 FN reduces to trapezoidal fuzzy number, just as The expected value of is
3 Problem Formulation
A conventional MONLPP is formulated as:
(10) 
where “” denotes minimization and maximization; are multiple and nonlinear objectives to be optimized; are realvalued nonlinear constraints.is dimensionel decision vector.
In realworld decisionmaking problems such as in production, planning, scheduling, etc. the present quantity of resources as well as the production quantity or the demand quantity or the target over a period might be imprecise and possess various types of fuzziness due to many factors such as market price, existence of men power, perception with the operators, weather, rain, transportation, traffic, etc. Also, the objectives characterized by the decisionmaker may be illdefined due to estimated parameters. Thus, IT2 FNs appears to be more practical in such conditions. In formulating such problems, the detailed concepts and notations are given in papers [41, 42].
Assuming that the objective functions and the resource constraint functions are nonlinear with estimated coefficient parameters which are in terms of trapezoidal IT2 FNs. The MONLPP with trapezoidal IT2 FNs can be formulated as:
(11) 
where
Therefore, MONLPP with trapezoidal IT2 FNs (11) can be genarally formulated as follows:
(12) 
By using the expected value function as defined in (9), problem (12) is transformed into an equivalent crisp MONLPP as:
(13) 
where the expected values of , and are
3.1 Construction of Fuzzy Multiobjective Nonlinear Goal Programming
In a multiobjective programming, if an imprecise aspiration level is injected to each of the objectives, then these fuzzy objectives are expressed as fuzzy goals. Let be the aspiration level assigned to the objective Then the fuzzy goals are for the maximization objective in (13) and for the minimization objective of (13) where and represent the fuzzified inequalities.
Therefore, the fuzzy multiobjective goal programming problem can be formulated as follows:
(14) 
Now, consider the fuzzy goal Its membership function can be defined as follows:
(15) 
where is the lower tolerance limit for the fuzzy goal and is the tolerant interval which is subjectively selected, respectively. They are determined as follows:
and (LABEL:GrindEQ__16_)
Similarly, consider the fuzzy goal of . Its membership function can be defined as follows:
(16) 
where is the upper tolerance limit for the fuzzy goal and the tolerant interval which is subjectively selected, respectively. They are determined as follows:
and (LABEL:GrindEQ__18_)
3.2 Linearization Nonlinear Membership and Constraint Functions Using the Taylor Series
Several methods are implemented to linearize the fractional and/or nonlinear functions in literature [37, 38, 39, 40]. In this section, problem (13) will transform into an equivalent multiobjective linear programming problem (MOLPP).
Note that the feasible region for a programming problem is the whole set of alternatives for the decision variables over which the objective function is to be optimized. Therefore, it can be illustrated with the limits of decision variables, and thereby the nonlinear constrained region can be easily converted to the linear inequalities.
The suggested solution procedure can be continued as follows:

Construct problem (13)

Solve problem (13) as a single objective nonlinear programming problem, taking each time only one objective as objective function and ignoring all others.

Compute the value of each objective function at each solution and then define the feasible region by the limits of decision variables.

Determine which is the solution that is employed to maximize the nonlinear membership function associated with nonlinear objective

Then, transform nonlinear membership functions by using Taylor series approach around the solution as follows:
(19)
Here, functions approximate the nonlinear functions around the maximal solution So, Taylor series approach generally provides a relatively good approximation to a differentiable function but only around a given point, and not over the entire domain.
3.3 A Fuzzy Goal Programming Model to Multiobjective Linear Programming Problem
The FGP approach was originally introduced by Zimmermann [4] in 1978. He employed the concept of membership functions. Tiwari et al. [35] suggested a weighted additive model that associates each goal’s weight into the objective function, where weights show the relative importance of the fuzzy goals. Afterward, Mohamed [36] suggested a kind of fuzzy goal, which is introduced in the general form of FGP model. In [43], Mohamed’s approach used to present a FGP approach for solving multiobjective programming problems and then Gupta and Bhattacharjee [40] formulated two FGP model for solving multiobjective programming problems.
According to paper [36], the highest degree of membership function is 1 and therefore, the nonlinear membership functions in (15) and (16) can be constructed as the following nonlinear membership goals;
(20) 
(21) 
where and represent the negative and positive deviations from the aspired levels, respectively. In addition, any positive deviation from 1 shows the full attainment of the membership value. Hence to reach the aspired levels of the fuzzy goal, it is sufficient to minimize its negative deviational variable from 1. At the same time, presentation of both deviation variables in the membership goal is unneeded, and the positive deviational variables are not necessary [40]. Thus the above membership goals can be written as follows:
(22) 
Here, the membership goals as defined in (22) are naturally nonlinear when the objective functions are nonlinear, and this may generate computational difficulties in the solution procedure of nonlinear problems. Therefore, by using Taylor series approach, the nonlinear membership goal in (22) can be written as the following linear function:
(23) 
Now let us consider the minmax form of fuzzy programming (18). If we put then we obtain the following equivalent fuzzy linear programming model:
(24) 
where denotes that the limits of decision variables derived from the individual optimal solutions of each objective.
To formulate the above fuzzy problem as a FGP model, the negative deviational variables in (24) can be defined as:
Thus, we obtain where
Then, an equivalent linear FGP model for problem (13) can be developed as follows:
(25) 
where represent the negative deviations from the aspired levels, respectively.
3.4 Interactive Fuzzy Goal Programming Approach Based on Taylor Series for MNLOPP with IT2 FNs
In this section, an interactive fuzzy goal programming algorithm is presented to achieve the highest degree for the membership functions.
The complete suggested solution procedures can be summarized as follows.
Step 1 Construct the mathematical model of MONLPP with IT2 FNs (12).
Step 2 By using the expected value function as defined in (9), obtain the corresponding crisp MONPP.
Step 3 Solve the MONPP as a single objective problem, considering each time only one objective as the objective function and ignoring all others.
Step 4 Compute the value of each objective function at each solution derived in Step 3. Then define the feasible region by the lower and upper limits of decision variables.
Step 5 From Step 4, determine the upper and lower tolerance limits of each objective function.
Step 7 Maximize each nonlinear membership functions under the feasible region derived in Step 4, individually and then determine the maximal solutions for each nonlinear membership function.
Step 8 Linearize each nonlinear membership function using Taylor series at the maximal solutions
Step 9 Construct the FGP model as formulated in (25), then solve it to obtain the optimal solution.
Step 10 If the decisionmaker is satisfied by the current solution, in Step 9, go to Step 11, else go to Step 12.
Step 11 The current solution is the optimal solution for MONPP with IT2 FNs.
Step 12 Compare the lower (upper) tolerance limit of each objective with the new value of the objective function. If the new value is higher (lower) than the lower tolerance limit, take this as a new lower (upper) tolerance limit. If else, hold the old one as is and then go to step (5).
4 Numerical Examples
Example 1
A manufacturing factory is going to produce 3 kinds of products A; B and C in a period (say one month). The production of A; B and C require three kinds of resources and . Here, to deal with uncertainties possessing doubt, let us consider that all parameters of the problem are IT2 FNs. Thus, the requirements of each kind of resource to produce each product A are
Assuming that the planned production quantities of A; B and C are respectively. Moreover, assuming that unit cost and sale’s price of product A, B and C are and
(Step 1): This problem can be formulated as follows:
(26) 
where