AN OPTIMIZATION METHOD TO SOLVE A FULLY INTUITIONISTIC FUZZY NON-LINEAR SEPARABLE PROGRAMMING PROBLEM

. This paper presents an optimization method to solve a non-linear separable programming problem with coefficients and variables as generalized trapezoidal intuitionistic fuzzy numbers. Such optimization problems are known as fully intuitionistic fuzzy non-linear separable programming problems. The optimization method is based on the linear approximation of fully intuitionistic fuzzy non-linear separable functions. The concept of an intuitionistic fuzzy line segment between two intuitionistic fuzzy points is introduced to find the required linear approximation. In this way, a fully intuitionistic fuzzy non-linear programming problem is converted into an intuitionistic fuzzy linear programming problem. The defuzzification and component-wise comparison techniques are then used to convert the fully intuitionistic fuzzy linear programming problem to a linear programming problem with crisp coefficients which can then be solved by using traditional optimization techniques. The application of the proposed approach in an investment problem faced by a businessman has been presented.


Introduction
Optimizing a linear objective subject to certain linear constraints is one of the most pioneering problem in the field of operations research.The inability to express objective function and constraints as a linear combination of variables give rise to non-linear programming problem.The introduction of an optimization problem where objective function and constraints are defined by non-linear functions, known as non-linear optimization problems, made it easy to model various real-life situations.Non-linear programming problems are used in various aspects of real life such as portfolio management problem and revenue management problem in finance, traffic equilibrium and traffic management problems in transportation, resource allocation problem and production planning problem in manufacturing processes and many more.

Literature review
There are many techniques to solve non-linear programming problem [11].Quadratic programming techniques [15] can be used when the objective function is quadratic and constraints are linear.Various fractional programming techniques such as Charnes Cooper transformation [7], denominator objective restriction method [28] and many more can be used when objective function is a ratio of concave and convex function and constraints are given by convex functions.Under differentiability and constraint qualifications, the Karush Kuhn Tucker (KKT) [39] conditions provides necessary conditions for a solution to be optimal.An efficient linear programming (LP) test algorithm is defined by using rectangles and parallelograms for solving a system of non linear equations in [41].A variable transformation method explains the use of surrounding parallelograms as a component when the non-linearity of the function becomes weak.
A separable programming problem [34] is a special case of non-linear programming problem where both objective function and constraints are given by separable functions.Separable programming problem was introduced by Miller for the very first time in 1963 [26] and he used simplex method to solve local separable programming problem.A method with quadratic order of convergence based on approximating convex function by means of first degree splines was given in 1971 by Cox [8].A solving method for a class of separable convex programming problem is presented in [38].A piece-wise linear approximation method is defined in [23] to solve a mixed integer quadratic constrained optimization problem with non-uniform partitioning of domain for various variables.The partitioning of variables is done in such a way that the density of breakpoints is higher in the neighbourhood of local optimal solution.In [20], the alternating direction method of multipliers is adopted to solve an optimization problem with linear constraints and objective function, a sum of two separable functions which are non-convex in nature.In [3], an approach that combines the idea of alternating direction method of multipliers and stochastic gradient method is developed for solving structured separable convex optimization problem.A sequential updating scheme of Lagrangian multiplier with relaxed step size is presented in [31] to solve a multi-block convex optimization problem with separable objective function and linear constraints.An ordinary differential equation solver approach for linearly constrained separable convex optimization problem is developed in [24].The characterization theorem for the classical knapsack problem with single objective function and a single equality constraint are formulated and proved in [35].In [21], Keha et al. presented two models for approximating non-linear function by piecewise linear functions with and without binary variables.The models presented are preferable for efficient computation as both the models have same LP bounds.Fasano and Pinter [14] developed an efficient piecewise linearization method for dealing with non-convex optimization problem in space engineering.Stefanov and Stefanov [37] approximated a separable problem by using a linear program and gave a restricted basis entry rule for solving the resultant linear programming problem by using the simplex method.A lattice piecewise linear model has been used in [42] to approximate non linear objective functions.The method defined in [42] is applicable for any continuous objective functions rather than only separable functions.In [9] three mixed integer linear programming models have been compared for solving separable programming problem by a piecewise linear approximation of cost function.An effective linear approximation method where only log 2 (+1) binary variables are used to linearise a concave function with  break points has been presented in [18].Resultantly, the method proposed is faster than the conventional one, especially when the number of break points become large.Chukwudi et al. [29] proposed an algorithm to solve optimization problem with non linear separable quadratic objective functions with linear constraints by approximating the given objective function to a piecewise linear function.Ambrosio et al. [10] proposed a fast heuristic algorithm based on constructive techniques for solving non-linear knapsack problem with separable non-convex function.
Investment problem is one basic optimization problem faced by businesses where decisions about investment projects, ventures and amount to be invested are to be made with an objective of achieving maximum returns in future.Numerous researchers have given various methods to solve the problem optimally.In [19], type-2 fuzzy logic is applied in stock market problems to generate simple and understandable models which are helpful in representing numerical and linguistic uncertainties.In [4], a fuzzy logic managerial decision tool for assets acquisition is proposed where objective is to optimize the ratio between acquisition cost and economic performance.Dynamic programming models have also been used by various authors to solve investment problems in businesses.In [22], a dynamic programming model is applied to obtain the optimal policy in an investment problem where chaos numbers are used to represent uncertainty of return data.In [33], a dynamic programming approach using multi-stage decision making has been given to solve an investment problem in an imprecise environment.The concept of possibility index is introduced which gives the possibility of making an imprecise investment to attain maximum profit.In [32], a dynamic programming method is used to solve the investment problem in continuous time.The consumption and investment problem of an agent is considered who faces a subsistence consumption constraint and has a quadratic utility function.An inventory control optimization problem is considered in [36] with stochastic demand and a stochastic quasi gradient method is developed to solve the problem.A piecewise linear approximation method for solving separable quadratic programming problem by using fuzzy goal programming methodology has been discussed in [30].In later years, various researchers also solved the problem in an imprecise environment.A fuzzy non-linear separable programming problem has been discussed by Chakraborty et al. [5] where they formulated a piecewise fuzzy linear approximation of fuzzy non-linear function.In [17], the separability of different functions is used to define an evaluation function; which is then used for attribute reduction in an imprecise environment.A descent algorithm for solving a non-convex multi-objective optimization problem under uncertainty is presented in [25].

Novelty, contribution and motivation
Many methods have been defined in the literature to solve separable programming problems when the coefficients and variables are deterministic.But, in real life situations, we often deal with uncertainty.In practical decision-making situations, technological coefficients are usually imprecise because of various uncertain factors and hence are represented by using fuzzy numbers.For example, the profit associated with a certain commodity depends on market demand, smoothness of supply chain, availability of raw material, position of the company doing business in the market; and these factors can not be always defined precisely.So, the profit associated is usually given in the form of "around x dollars" and such information can be easily represented by using fuzzy sets.But the moment a logical decision maker represents their point of view, they also give their confidence level; this usually represent their depth of knowledge in the context of information.For example, a decision maker having expertise in nuclear weapons can give coefficients related to his/her field with more accuracy and confidence as compared to other field such as environment, wireless communication, logistics, production planning and many more.So, it is better that the information about various coefficients is usually revealed by a logical decision maker.This way, the logical information revealed by decision maker will be always represented in an imprecise form with some specified confidence level.The semantic representation of these coefficients is very well done when impreciseness can be clubbed with hesitation.The inclusion of hesitation along with impreciseness bridges the gap between mathematical representation and linguistic expression.Intuitionistic fuzzy numbers are one such tools which can be used for representing this hesitation by giving a non-membership function along with the membership function; which is not the exact complement of membership function.
We know that objectives of various real life optimization problems such as portfolio management problem, business investment problem, stock management problems and various other problems can be very well represented by using non-linear separable functions.So, there is a need to introduce methods for solving non-linear separable programming problem in an environment similar to real life, i.e. an environment where imprecise and indeterminant information can be well represented.So, in this work, we present an optimization method for solving a fully intuitionistic fuzzy non-linear separable programming problem.The coefficients of the problem are represented by using intuitionistic fuzzy numbers which are considered as one of the best tools for representing imprecise information which also has some scope of hesitation in it.The basic idea to solve the problem is to convert it into linear optimization problem.For this, we extended the idea of fuzzy line segment between two fuzzy points and introduced the idea of an intuitionistic fuzzy line segment between two intuitionistic fuzzy points.By doing so, we reduced an intuitionistic fuzzy non linear separable programming problem to an intuitionistic fuzzy linear programming problem without causing much loss of important information.The reduced problem is then solved by using ranking function on coefficients of objective functions and reducing them to crisp number and using component-wise comparison of right and left side of the constraints.
The paper is organized as follows: In Section 2, preliminaries and concepts regarding intuitionistic fuzzy set theory and separable programming problem have been reviewed.In Section 3, a mathematical model representing non-linear separable programming problem under intuitionistic fuzzy environment has been formulated and its solution methodology has been discussed.A real world application of the proposed problem in an investment problem is illustrated with the help of a numerical example in Section 4. Section 5 comprises of the concluding remarks.

Preliminaries: concepts and definitions
In this section, some basic preliminaries of intuitionistic fuzzy set theory have been discussed.
Definition 1 (Intuitionistic Fuzzy Set [2]).Let  be a universe of discourse.Then an intuitionistic fuzzy set Ã in  is defined by equation (1).
where  ; ) be a trapezoidal intuitionistic fuzzy number.Then the -cut and -cut of Ã is given by equations (A.4) and (A.5) respectively.The ranking function of the intuitionistic fuzzy number can be given by equation (4).
) is said to be separable if it can be expressed as sum of  single variable functions; i.e.
where  1 ( 1 ),  2 ( 2 ), . . .,   (  ) are single variable functions of  1 ,  2 , . . .,   respectively.Definition 7 (Separable Programming Problem).A non-linear programming problem in which the objective function and(or) constraints can be expressed as a linear combination of several different single variable functions, of which some or all are non-linear, is called a separable programming problem.

Problem formulation and model development
In order to find an optimal solution for a non-linear separable programming problem, its formulation is required.In this section, a methodology for solving non-linear separable programming problem in an intuitionistic fuzzy environment is presented.The proposed model can be used to solve optimization problem when the objective function depends upon the combination of various factors; e.g. the profit function of a company producing numerous independent goods is dependent on the profit functions (may be linear or non-linear) of individual goods.In the same way, the constraints of such an optimization problem, which may represent factors such as capital investment required, machine hours requirements, power supply requirements etc., can be very well represented by using linear or non-linear separable functions as constraints.In this work, we have used intuitionistic fuzzy numbers to represent technological coefficients, variables, resources available and coefficients of objective function.Intuitionistic fuzzy numbers are considered as one of the best tools to represent real life situations that are imprecise in nature and also includes the hesitation factor of the decision maker.While representing the intuitionistic fuzzy numbers, the degree of non-belongingness is considered along with the degree of belongingness; two of which are not the exact complement of each other and this helps in representing the indeterminacy of the information revealed by the decision maker.So, the model explained in this work is beneficial to solve real-world optimization problem when the objective function and constraints are written as the sum of functions of single variables.The advantage of semantic and logical representation of information, as revealed by decision maker, adds up because of use of intuitionistic fuzzy numbers.
A standard form of a fully intuitionistic fuzzy non-linear separable programming problem is given by equation (6).
In order to find the linear approximation of a non-linear separable functions, f  (x  ) and g  (x  ), the partitioning of variables into various intervals is required and then the separable functions are approximated by a line segment in those intervals.For that, we need the equation of a line segment passing through two intuitionistic fuzzy points.The parametric equation of an intuitionistic fuzzy line segment can be obtained by using  − cut set of coordinates of intuitionistic fuzzy points.Suppose P ]︂}︂ .

Now we can write the parametric equation of line segment passing through the points ({𝑐
and where , then above equations ( 7) and ( 8) can be written as, and where }︂ .
Now we can write the parametric equation of line segment passing through the points ({ and where , then above equations ( 11) and ( 12) can be written as, and where . Now, from equations ( 9), (13) and from equations (10), (14) we can write, and where Now, from equation ( 15) and from equation ( 16) we can write, Above, equations ( 17) and ( 18) are called parametric equation of intuitionistic fuzzy line-segment.Figure 2 represents the parametric equation of a line segment joining two trapezoidal intuitionistic fuzzy points ( ̃︁ . The trapezoids with their base on horizontal axis represents the -coordinates of two intuitionistic fuzzy point.The trapezoids with their base on the line marked as  represents the -coordinates of the two  (19). where (4) Non-zero   are associated with adjacent points.
In Figure 3, the blue dashed curves represent the linear intuitionistic fuzzy approximation of the given nonlinear intuitionistic fuzzy function given by the solid black curves by using the -cut method.
Suppose in equation ( 6), if each x  ∈ [ ã  , b  ] and partition of this interval is taken as equation (20).
(4) These non-zero  ′   are always associated with adjacent points.Then ranking function; as defined in equation ( 4) is used to convert the intuitionistic fuzzy coefficients of the objective function to crisp coefficients and comparison technique, as defined in Definition 5 is used to present constraints in a deterministic form.The use of Definition 5 ensures that no information revealed by decision maker is lost while solving the problem.The use of ranking function, i.e. equation ( 4) makes it easier to solve the problem in lesser dimension and reach a solution using reasonable time and space.
Computationally, the proposed methodology requires solving a linear programming problem with some additional constraints in the form of adjacency criteria.Thus, the problem can be solved in polynomial amount of time.The time complexity of simplex algorithm, one of the most effective algorithm to solve linear programming problem, is (( + )) where  is the number of variable and  is the number of inequality constraints.We solved the problem by using LINGO 13.0 solver.

Solution methodology
In this subsection, a step-by-step procedure for solving the concerned mathematical model is presented.
Step 1. Use the information about upper and lower bound of the decision variables to create the partitions of each decision variable.
Step 2. Approximate the non-linear intuitionistic fuzzy function present in objective as well as in constraints to a linear intuitionistic fuzzy function in each interval using the partitions of corresponding decision variable.
Step 3. Use the ranking function as defined in equation ( 4) to convert the intuitionistic fuzzy coefficients of the objective function to the corresponding crisp coefficients.
Step 4. Use the comparison technique as defined in Definition 5 to transform the intuitionistic fuzzy constraints to the corresponding deterministic problem.
Step 5. Solve the obtained linear programming problem by using some software package like LINGO, MATLAB etc.

Flowchart
The flowchart for the solution methodology is given by Figure 4.

An illustrative example: a business investment problem
The investment problem faced by a businessman is considered.The mathematical model for the problem is constructed which is then solved with the help of methodology proposed in this work.
First we approximate all the non-linear functions to the corresponding linear functions and then we use ranking function as given in equation ( 4) and comparison techniques as given in ( 5) to reduce the model to a linear programming problem.The corresponding linear programming model is given by equation (26).
Maximize  = 30.65875 and the constraints representing adjacency criteria are given by equation (27).

Sensitivity analysis
The sensitivity analysis of the linear programming model corresponding to the crisp version of equation ( 25) is given by Tables 1 and 2 where the corresponding linear programming model is given by equation (28).Maximize  = 30.65875 Table 1 represents the allowable increase and decrease in the various coefficients of objective functions which does not affect the current optimal value.An increase(decrease) in the coefficients greater than the allowable range will worsen the optimal value.Table 2 represents the allowable increase(decrease) in the right hand side coefficients of the constraints which does not alter the current optimal solution of the problem.

Comparison analysis
One of the common methods for solving optimization problems with imprecise coefficients is to transform the problem into a deterministic one by using various defuzzification techniques and then solving the deterministic version of the problem by using classical approach.It has also been found that no method for solving non-linear separable programming problem in an intuitionistic fuzzy environment has been defined in the literature till date.However, Chakraborty et al. [5] defined a solution method for solving non-linear separable programming problem in a fuzzy environment.In their work, the fuzzy constraints are reduced to crisp constraints by using Yager's ranking function [40] and then the fuzzy simplex method [13] is used to solve the problem.On extensive study of literature, we found that, as of now, there are various defuzzification methods for intuitionistic fuzzy numbers.3.500 ∞ 2.600 [5] 4.000 ∞ 2.980 [6] 4.20 ∞ 3.050 [7] 4.50 A method for ranking of intuitionistic fuzzy numbers based on centroid point method is given by Arun Prakash et al. [1].Dubey and Mehra [12] presented an approach based on value and ambiguity index for defuzzification of triangular intuitionistic fuzzy numbers.An approach based on the score function of membership and nonmembership functions is described by Nagoorgani and Ponnalagu [27] for the same.So, for comparison, we use various defuzzification methods as defined in [1,12,27] and formulate the problem in a deterministic environment.Table 3 comprises of the deterministic problem and its solution by using different ranking approaches.
During the solution process, the first step included the defuzzification of imprecise information which led to information loss.As one can clearly see, there is hardly any resemblance among the defuzzified values which may lead to a confusion as to which defuzzification method can be considered as a better approximation of the original problem.Also, the nature of the decision variables and that of solution obtained by defuzzification is not intuitionistic but crisp which is not much helpful as it does not have the factor of hesitation which was initially involved in our discussed problem.Thus, the proposed method in this work not only keeps the nature of the problem intact; as the solution obtained by using proposed method is intuitionistic in nature; but also corresponds to lesser information loss as compared to other methods.Thus, there are two fold benefits of the method proposed in this work.

Table 3. Comparison with various defuzzification techniques.
Method based on membership and non-membership functions [27] Method based on value and ambiguity index [12] Method based on centroid concept [

Conclusion
In this paper, a methodology to solve fully intuitionistic fuzzy non-linear separable programming problem has been introduced.One of the major advantage of the proposed method is its ability to represent imprecise information of the coefficients which also has components of hesitation involved in them.In this work, the concept of an intuitionistic fuzzy line segment between two intuitionistic fuzzy points is defined by using an , -cut method.The reduction of the problem to its intuitionistic fuzzy linear counterpart does not cost much loss of information as well and the mathematical model thus obtained is comparatively easier to solve.The linear approximation of non-linear functions of single variable is done by introducing grid points in the domain of the corresponding variable.The accuracy of the linear approximation increases as we increase the number of grid points.As the number of grid points increases, we realize that the size of the problem increases (i.e. the number of variables and constraints increases).Thus, in the proposed work, we are trading off accuracy of approximation (and thus in turn of that of solution) against computer time.This comes up as a major limitation of the proposed work.Further, this limitation of the work can be eliminated by extending the given approach to an iterative approach where new grid points are introduced in the grid corresponding to the optimal solution in the previous iteration.Then, a ranking function is used to defuzzify the intuitionistic fuzzy coefficients in the objective function and comparison techniques are used to represent the constraints deterministically.The problem is then reduced to a classical linear programming problem which can be solved easily by using classical optimization methods.
While solving optimization problems in an imprecise environment, various researchers use defuzzification of imprecise coefficients and then solve the resultant deterministic classical optimization problem.In comparison analysis section, we used three different defuzzification operators and solved the corresponding classical optimization problem.We observed that there is a wide difference in defuzzified coefficients and thus choosing one defuzzification operator which gives a better representation of the corresponding fuzzy number is a difficult task.Moreover, the defuzzification of imprecise coefficient leads to significant information loss.Also, the value of objective function and decision variable come up as deterministic numbers, thus changing the nature of decision variable and solution.In this work, a real life application of proposed problem has been explained in the investment planning problem faced by a businessman.The proposed model can be extended for solving hierarchical optimization problems as well.In future, the proposed work can also be extended for solving multiobjective non-linear separable programming problems and thus can prove to be beneficial for solving real world optimization problems.The approach presented in this work can also be implemented on real life problems ranging from industries such as transportation, communication networks, scheduling, manufacturing and many more.
where,  = min{ 1 ,  2 }. (3) Multiplication: If C = Ã ⨂︀ B then, )︂ , )︂ ,  From above -cut set and -cut set, we get generalized trapezoidal intuitionistic fuzzy number by taking standard approximation followed by (, )-cut method.Sometime it might be required to break -cut set or -cut set (for multiplication or division) into two parts.Here, we will take a simple example to understand the process.

𝛼
and ̃︁  IR  , ̃︁  IL  and ̃︁  IR  are not linear polynomials in  and  respectively.So, for making GTIFN closed under multiplication and division, we will take standard approximation.

Figure 1
represents the membership and nonmembership function of TIFN defined above.

Figure 4 .
Figure 4. Flowchart for the proposed methodology.

Figure 5 .
Figure 5. Solution of the investment problem.

Figure 6 .
Figure 6.Value of the objective function of investment problem.

Figure 7 .
Figure 7. Profit of the investment problem.

Figure 8 .
Figure 8. Risk of the investment problem.

Table 2 .
Right hand side ranges.