Using fuzzy numbers in knapsack problems

This paper investigates knapsack problems in which all of the weight coefficients are fuzzy numbers. This work is based on the assumption that each weight coefficient is imprecise due to the use of decimal truncation or rough estimation of the coefficients by the decision-maker. To deal with this kind of imprecise data, fuzzy sets provide a powerful tool to model and solve this problem. Our work intends to extend the original knapsack problem into a more generalized problem that would be useful in practical situations. As a result, our study shows that the fuzzy knapsack problem is an extension of the crisp knapsack problem, and that the crisp knapsack problem is a special case of the fuzzy knapsack problem.     

Some new results in linear programs with trapezoidal fuzzy numbers: Finite convergence of the Ganesan and Veeramani’s method and a fuzzy revised simplex method

In a recent paper, Ganesan and Veermani [K. Ganesan, P. Veeramani, Fuzzy linear programs with trapezoidal fuzzy numbers, Ann. Oper. Res. 143 (2006) 305–315] considered a kind of linear programming involving symmetric trapezoidal fuzzy numbers without converting them to the crisp linear programming problems and then proved fuzzy analogues of some important theorems of linear programming that lead to a new method for solving fuzzy linear programming (FLP) problems. In this paper, we obtain some another new results for FLP problems. In fact, we show that if an FLP problem has a fuzzy feasible solution, it also has a fuzzy basic feasible solution and if an FLP problem has an optimal fuzzy solution, it has an optimal fuzzy basic solution too. We also prove that in the absence of degeneracy, the method proposed by Ganesan and Veermani stops in a finite number of iterations. Then, we propose a revised kind of their method that is more efficient and robust in practice. Finally, we give a new method to obtain an initial fuzzy basic feasible solution for solving FLP problems.     

Solving a full fuzzy linear programming using lexicography method and fuzzy approximate solution

This paper discusses full fuzzy linear programming (FFLP) problems of which all parameters and variable are triangular fuzzy numbers. We use the concept of the symmetric triangular fuzzy number and introduce an approach to defuzzify a general fuzzy quantity. For such a problem, first, the fuzzy triangular number is approximated to its nearest symmetric triangular number, with the assumption that all decision variables are symmetric triangular. An optimal solution to the above-mentioned problem is a symmetric fuzzy solution. Every FLP models turned into two crisp complex linear problems; first a problem is designed in which the center objective value will be calculated and since the center of a fuzzy number is preferred to (its) margin. With a special ranking on fuzzy numbers, the FFLP transform to multi objective linear programming (MOLP) where all variables and parameters are crisp.     

Some duality results on linear programming problems with symmetric fuzzy numbers

Recently, linear programming problems with symmetric fuzzy numbers (LPSFN) have considered by some authors and have proposed a new method for solving these problems without converting to the classical linear programming problem, where the cost coefficients are symmetric fuzzy numbers (see in [4]). Here we extend their results and first prove the optimality theorem and then define the dual problem of LPSFN problem. Furthermore, we give some duality results as a natural extensions of duality results for linear programming problems with crisp data.     

Solving the knapsack problem with imprecise weight coefficients using genetic algorithms

This paper investigates solving the knapsack problem with imprecise weight coefficients using genetic algorithms. This work is based on the assumption that each weight coefficient is imprecise due to decimal truncation or coefficient rough estimation by the decision-maker. To deal with this kind of imprecise data, fuzzy sets provide a powerful tool to model and solve this problem. We investigate the possibility of using genetic algorithms in solving the fuzzy knapsack problem without defining membership functions for each imprecise weight coefficient. The proposed approach simulates a fuzzy number by distributing it into some partition points. We use genetic algorithms to evolve the values in each partition point so that the final values represent the membership grade of a fuzzy number. The empirical results show that the proposed approach can obtain very good solutions within the given bound of each imprecise weight coefficient than the fuzzy knapsack approach. The fuzzy genetic algorithm concept approach is different, but gives better results than the traditional fuzzy approach.     

Solving fuzzy multi-objective linear programming problems using deviation degree measures and weighted max–min method

This paper proposes a method for solving fuzzy multi-objective linear programming (FMOLP) problems where all the coefficients are triangular fuzzy numbers and all the constraints are fuzzy equality or inequality. Using the deviation degree measures and weighted max–min method, the FMOLP problem is transformed into crisp linear programming (CLP) problem. If decision makers fix the values of deviation degrees of two side fuzzy numbers in each constraint, then the δ-pareto-optimal solution of the FMOLP problems can be obtained by solving the CLP problem. The bigger the values of the deviation degrees are, the better the objectives function values will be. So we also propose an algorithm to find a balance-pareto-optimal solution between two goals in conflict: to improve the objectives function values and to decrease the values of the deviation degrees. Finally, to illustrate our method, we solve a numerical example.     

Mehar’s method for solving fully fuzzy linear programming problems with L-R fuzzy parameters

To the best of our knowledge, there is no method in literature for solving such fully fuzzy linear programming (FLP) problems in which some or all the parameters are represented by unrestricted L-R flat fuzzy numbers. Also, to propose such a method, there is need to find the product of unrestricted L-R flat fuzzy numbers. However, there is no method in the literature to find the product of unrestricted L-R flat fuzzy numbers.In this paper, firstly the product of unrestricted L-R flat fuzzy numbers is proposed and then with the help of proposed product, a new method (named as Mehar’s method) is proposed for solving fully FLP problems. It is also shown that the fully FLP problems which can be solved by the existing methods can also be solved by the Mehar’s method. However, such fully FLP problems in which some or all the parameters are represented by unrestricted L-R flat fuzzy numbers can be solved by Mehar’s method but can not be solved by any of the existing methods.     

Posynomial geometric programming with interval exponents and coefficients

Geometric programming provides a powerful tool for solving nonlinear problems where nonlinear relations can be well presented by an exponential or power function. In the real world, many applications of geometric programming are engineering design problems in which some of the problem parameters are estimates of actual values. This paper develops a solution method when the exponents in the objective function, the cost and the constraint coefficients, and the right-hand sides are imprecise and represented as interval data. Since the parameters of the problem are imprecise, the objective value should be imprecise as well. A pair of two-level mathematical programs is formulated to obtain the upper bound and lower bound of the objective values. Based on the duality theorem and by applying a variable separation technique, the pair of two-level mathematical programs is transformed into a pair of ordinary one-level geometric programs. Solving the pair of geometric programs produces the interval of the objective value. The ability of calculating the bounds of the objective value developed in this paper might help lead to more realistic modeling efforts in engineering optimization areas.     

A new approach for solving fully intuitionistic fuzzy transportation problems

In this paper, a well-known network-structured problem called the transportation problem (TP) is considered in an uncertain environment. The transportation costs, supply and demand are represented by trapezoidal intuitionistic fuzzy numbers (TrIFNs) which are the more generalized form of trapezoidal fuzzy numbers involving a degree of acceptance and a degree of rejection. We formulate the intuitionistic fuzzy TP (IFTP) and propose a solution approach to solve the problem. The IFTP is converted into a deterministic linear programming (LP) problem, which is solved using standard LP algorithms. The main contributions of this paper are fivefold: (1) we convert the formulated IFTP into a deterministic classical LP problem based on ordering of TrIFNs using accuracy function; (2) in contrast to most existing approaches, which provide a crisp solution, we propose a new approach that provides an intuitionistic fuzzy optimal solution; (3) in contrast to existing methods that include negative parts in the obtained intuitionistic fuzzy optimal solution and intuitionistic fuzzy optimal cost, we propose a new method that provides non-negative intuitionistic fuzzy optimal solution and optimal cost; (4) we discuss about the advantages of the proposed method over the existing methods for solving IFTPs; (5) we demonstrate the feasibility and richness of the obtained solutions in the context of two application examples.     

Duality in Fuzzy Linear Programming: Some New Concepts and Results

A class of fuzzy linear programming (FLP) problems based on fuzzy relations is introduced, the concepts of feasible and -efficient solutions are defined. The class of crisp (classical) LP problems and interval LP problems can be embedded into the class of FLP ones. Moreover, for FLP problems a new concept of duality is introduced and the weak and strong duality theorems are derived. The previous results are applied to the special case of interval LP and compared to the existing literature.     

The dual simplex method and sensitivity analysis for fuzzy linear programming with symmetric trapezoidal numbers

In this paper, we first extend the dual simplex method to a type of fuzzy linear programming problem involving symmetric trapezoidal fuzzy numbers. The results obtained lead to a solution for fuzzy linear programming problems that does not require their conversion into crisp linear programming problems. We then study the ranges of values we can achieve so that when changes to the data of the problem are introduced, the fuzzy optimal solution remains invariant. Finally, we obtain the optimal value function with fuzzy coefficients in each case, and the results are described by means of numerical examples.     

A revisit of numerical approach for solving linear fractional programming problem in a fuzzy environment

In the article, Veeramani and Sumathi [10] presented an interesting algorithm to solve a fully fuzzy linear fractional programming (FFLFP) problem with all parameters as well as decision variables as triangular fuzzy numbers. They transformed the FFLFP problem under consideration into a bi-objective linear programming (LP) problem, which is then converted into two crisp LP problems. In this paper, we show that they have used an inappropriate property for obtaining non-negative fuzzy optimal solution of the same problem which may lead to the erroneous results. Using a numerical example, we show that the optimal fuzzy solution derived from the existing model may not be non-negative. To overcome this shortcoming, a new constraint is added to the existing fuzzy model that ensures the fuzzy optimal solution of the same problem is a non-negative fuzzy number. Finally, the modified solution approach is extended for solving FFLFP problems with trapezoidal fuzzy parameters and illustrated with the help of a numerical example.     

Fuzzy linear programming problems as bi-criteria optimization problems

In this paper, we consider fuzzy linear programming (FLP) problems which involve fuzzy numbers only in coefficients of objective function. First, we shall give concepts of optimal solutions to (FLP) problems and investigate their properties. Next, in order to find all optimal solutions, we define three types of bi-criteria optimization problems.     

Inherited Fuzzy Interpolation Based on the Inherited Lower-upper Factorization

In this paper, a new scheme is proposed to find the fuzzy interpolation polynomial. In this case, the nodes are crisp data and the values are fuzzy numbers. In order to obtain the interpolation polynomial, a linear system is solved with crisp coefficients matrix and fuzzy right hand side. Then, the inherited lower-upper (LU) triangular factorization and inherited interpolation are applied to solve this system. The examples illustrate the applicability, simplicity and efficiency of the proposed method.     

Solving a multiobjective possibilistic problem through compromise programming

Real decision problems usually consider several objectives that have parameters which are often given by the decision maker in an imprecise way. It is possible to handle these kinds of problems through multiple criteria models in terms of possibility theory.Here we propose a method for solving these kinds of models through a fuzzy compromise programming approach.To formulate a fuzzy compromise programming problem from a possibilistic multiobjective linear programming problem the fuzzy ideal solution concept is introduced. This concept is based on soft preference and indifference relationships and on canonical representation of fuzzy numbers by means of their α-cuts. The accuracy between the ideal solution and the objective values is evaluated handling the fuzzy parameters through their expected intervals and a definition of discrepancy between intervals is introduced in our analysis.     

A simple approach to fuzzy critical path analysis in project networks

This paper develops a simple approach to critical path analysis in a project network with activity times being fuzzy numbers. The idea is based on the linear programming (LP) formulation and fuzzy number ranking method. The fuzzy critical path problem is formulated as an LP model with fuzzy coefficients of the objective function, and then on the basis of properties of linearity and additivity, the Yager’s ranking method is adopted to transform the fuzzy LP formulation to the crisp one which can be solved by using the conventional streamlined solution methods. Consequently, the critical path and total duration time can be obtained from the derived optimal solution. Moreover, in this paper we also define the most critical path and the relative path degree of criticality, which are theoretically sound and easy to use in practice. An example discussed in some previous studies illustrates that the proposed approach is able to find the most critical path, which is proved to be the same as that derived from an exhausted comparison of all possible paths. The proposed approach is very simple to apply, and it is not require knowing the explicit form of the membership functions of the fuzzy activity times.     

Towards a new strategy for solving fuzzy optimization problems

Fuzzy Optimization models and methods has been one of the most and well studied topics inside the broad area of Soft Computing. Particularly relevant is the field of fuzzy linear programming (FLP). Its applications as well as practical realizations can be found in all the real world areas. As FLP problems constitute the basis for solving fuzzy optimization problems, in this paper a basic introduction to the main models and methods in FLP is presented and, as a whole, Linear Programming problems with fuzzy costs, fuzzy constraints and fuzzy coefficients in the technological matrix are analyzed. But fuzzy sets and systems based optimization methods do not end with FLP, and hence in order to solve more complex optimization problems, fuzzy sets based Meta-heuristics are considered, and two main operative approaches described. Provided that these techniques obtain efficient and/or effective solutions, we present a fuzzy rule based methodology for coordinating Meta-heuristics and in addition, to provide intelligence, we propose a process of extraction of the knowledge to conduct the coordination of the system.     

An (α ,β)-Optimal solution concept in Fuzzy Optimization problems

We propose an (α,β)-optimal solution concept of fuzzy optimization problem based on the possibility and necessity measures. It is well known that the set of all fuzzy numbers can be embedded into a Banach space isometrically and isomorphically. Inspired by this embedding theorem, we can transform the fuzzy optimization problem into a biobjective programming problem by applying the embedding function to the original fuzzy optimization problem. Then the (α,β)-optimal solutions of fuzzy optimization problem can be obtained by solving its corresponding biobjective programming problem. We also consider the fuzzy optimization problem with fuzzy coefficients (i.e., the coefficients are assumed as fuzzy numbers). Under a setting of core value of fuzzy numbers, we provide the Karush–Kuhn–Tucker optimality conditions and show that the optimal solution of its corresponding crisp optimization problem (the usual optimization problem) is also a (1,1)-optimal solution of the original fuzzy optimization problem.     

Numerical Approach of Multi-Objective Optimal Control Problem in Imprecise Environment

In this paper, realistic production-inventory models without shortages for deteriorating items with imprecise holding and production costs for optimal production have been formulated. Here, the rate of production is assumed to be a function of time and considered as a control variable. Also the demand is time dependent and known. The imprecise holding and production costs are assumed to be represented by fuzzy numbers which are transformed to corresponding interval numbers. Following interval mathematics, the objective function is changed to respective multi-objective functions and thus the single-objective problem is reduced to a multi-objective decision making(MODM) problem. The MODM problem is then again transformed to a single objective function with the help of weighted sum method and then solved using global criteria method, calculus method, the Kuhn–Tucker conditions and generalized reduced gradient(GRG) technique. The models have been illustrated by numerical data. The optimum results for different objectives are obtained for different types of production function. Numerical values of demand, production function and stock level are presented in both tabular and graphical forms