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Email: international@du.ac.ir

Damghan University

University Blvd, Damghan, IR

Ali Abbasi-Molai

Associate Professor of Applied Mathematics

- Linear Optimization
- Nonlinear Optimization
- Advanced Operations Research I
- Advanced Linear Programming
- Integer Programming and Network Flows
- Optimization of Nonlinear Models
- General Mathematics I, II, III
- Operations Research I, II

DOI: 10.1016/j.ins.2018.10.061

The conditions (36) for existence of the solution in the decomposed constraints (35) in [2] are not necessary and sufficient. These conditions are only necessary conditions. Also, the authors in [2] claimed that the maximal elements of the system of bipolar max-min equality constraints or set D are either g+ or belong to the set G={Supi=1,…,mgi|(∀i)(gi∈Gi−)}. This expression is not correct. The set G is revised such that it contains the maximal elements of D. © 2018 Elsevier Inc.

AUTHOR KEYWORDS: Bipolar fuzzy relation equations; Linear optimization; Max-min composition operator; Maximal elements

INDEX KEYWORDS: Artificial intelligence; Software engineering, Bipolar fuzzy relations; Equality constraints; Linear optimization; Max-min; Max-min composition; Maximal elements, Linear programming

PUBLISHER: Elsevier Inc.

DOI: 10.1007/s40995-016-0108-6

In this paper, the linear fractional programming problem subject to a system of fuzzy relation inequalities (FRI) with the max-Hamacher composition operator is studied. First, the structure of its feasible domain is investigated, and its feasible solution set determined. Then, some sufficient conditions are given that under them, some of the optimal components of the problem are directly determined. The optimal solution of the linear fractional programming problem might not be any of the minimal solutions. However, in the process of obtaining the optimal solution, we need to compute the minimal solutions. Therefore, some reductions are presented that under them, we can compute the minimal solutions fast. The original problem can be transformed into some traditional linear fractional programming subproblems and eventually optimized in a small search space. Finally, an algorithm is designed to solve the problem based on the above reductions. An application and some numerical examples are provided to illustrate the procedure. © 2016, Shiraz University.

AUTHOR KEYWORDS: Fuzzy optimization; Fuzzy relation inequality; Linear fractional programming problem; Max-Hamacher composition

PUBLISHER: Springer International Publishing

DOI: 10.22111/ijfs.2018.3762

This paper studies the nonlinear optimization problems subject to bipolar max-min fuzzy relation equation constraints. The feasible solution set of the problems is non-convex, in a general case. Therefore, conventional nonlinear optimization methods cannot be ideal for resolution of such problems. Hence, a Genetic Algorithm (GA) is proposed to find their optimal solution. This algorithm uses the structure of the feasible domain of the problems and lower and upper bound of the feasible solution set to choose the initial population. The GA employs two different crossover operations: 1-N-points crossover and 2-Arithmetic crossover. We run the GA with two crossover operations for some test problems and compare their results and performance to each other. Also, their results are compared with the results of other authors’ works. © 2018, University of Sistan and Baluchestan. All rights reserved.

AUTHOR KEYWORDS: Bipolar fuzzy relation equations; Genetic algorithm; Max-min composition; Nonlinear optimization

PUBLISHER: University of Sistan and Baluchestan

DOI: 10.1007/s00500-014-1464-9

This paper studies the minimization problem of a linear objective function subject to mixed fuzzy relation inequalities (MFRIs) over finite support with regard to max-$$T_1$$T1 and max-$$T_2$$T2 composition operators, where $$T_1$$T1 and $$T_2$$T2 are two pseudo-t-norms. We first determine the structure of its feasible domain and then show that the solution set of a MFRI system is determined by a maximum solution and a finite number of minimal solutions. Moreover, sufficient and necessary conditions are proposed to check whether the feasible domain of the problem is empty or not. The MFRI path is defined to determine the minimal solutions of its feasible domain. The resolution process of the optimization problem is also designed based on the structure of its feasible domain. Procedures are proposed to reduce the size of the problem. With regard to the above points and the procedures, an algorithm is designed to solve the problem. Its application is expressed in the area of investing and covering. Finally, the algorithm is compared with other approaches. © 2014, Springer-Verlag Berlin Heidelberg.

AUTHOR KEYWORDS: Max-pseudo-t-norm composition; Maximum solution; Minimal solution; Mixed fuzzy relation inequality; Non-convex optimization

INDEX KEYWORDS: Constraint theory; Convex optimization; Linear programming; Mathematical operators; Optimization, Composition operators; Fuzzy relations; Inequality constraint; Linear objective functions; Minimal solutions; Nonconvex optimization; Pseudo-t-norm; Sufficient and necessary condition, Problem solving

PUBLISHER: Springer Verlag

DOI: 10.1016/j.cie.2014.03.024

The minimization problem of a quadratic objective function with the max-product fuzzy relation inequality constraints is studied in this paper. In this problem, its objective function is not necessarily convex. Hence, its Hessian matrix is not necessarily positive semi-definite. Therefore, we cannot apply the modified simplex method to solve this problem, in a general case. In this paper, we firstly study the structure of its feasible domain. We then use some properties of n × n real symmetric indefinite matrices, Cholesky's decomposition, and the least square technique, and convert the problem to a separable programming problem. Furthermore, a relation in terms of a closed form is presented to solve it. Finally, an algorithm is proposed to solve the original problem. An application example in the economic area is given to illustrate the problem. Of course, there are other application examples in the area of digital data service and reliability engineering. © 2014 Elsevier Ltd. All rights reserved.

AUTHOR KEYWORDS: Fuzzy relation inequality; Least square technique; Max-product composition; Minimal solution; Quadratic programming; Separable programming

INDEX KEYWORDS: Algorithms; Constraint theory; Quadratic programming, Fuzzy relations; Least-square techniques; Max-product composition; Minimal solutions; Modified simplex methods; Quadratic objective functions; Quadratic programming problems; Reliability engineering, Problem solving

PUBLISHER: Elsevier Ltd

DOI: 10.1007/s00500-014-1249-1

In this paper, a system of fuzzy polynomial equations is studied. Two solution types are defined for this system, called solution and (Formula presented.)-cut solution. Then sufficient and necessary conditions are proposed for existence of solution and (Formula presented.)-cut solution of the system, respectively. The solution set of the system is also determined. Moreover, a new algorithm is designed to find all the solutions and all the (Formula presented.)-cut solutions of the system based on the eigenvalue method. Finally, some examples are given to illustrate the concepts and the algorithm. © 2014, Springer-Verlag Berlin Heidelberg.

AUTHOR KEYWORDS: Eigenvalue method; Fuzzy numbers; Gröbner basis; Real solution; System of fuzzy polynomial equations

INDEX KEYWORDS: Eigenvalues and eigenfunctions; Fuzzy sets, Eigenvalue methods; Existence of Solutions; Fuzzy numbers; Polynomial equation; Real solutions; Solution set; Solution types; Sufficient and necessary condition, Polynomials

PUBLISHER: Springer Verlag

DOI: 10.1016/j.ins.2011.04.012

In this paper, the LU-factorization is extended to the fuzzy square matrix with respect to the max-product composition operator called L°U- factorization. Equivalently, we will find two fuzzy (lower and upper) triangular matrices L and U for a fuzzy square matrix A such that A = L°U, where "°" is the max-product composition. An algorithm is presented to find the matrices L and U. Furthermore, some necessary and sufficient conditions are proposed for the existence and uniqueness of the L°U-factorization for a given fuzzy square matrix A. An algorithm is also proposed to find the solution set of a square system of Fuzzy Relation Equations (FRE) using the L°U-factorization. The algorithm finds the solution set without finding its minimal solutions and maximum solution. It is shown that the two algorithms have a polynomial-time complexity as O(n3). Since the determination of the minimal solutions is an NP-hard problem, the algorithm can be very important from the practical point of view. © 2011 Elsevier Inc. All rights reserved.

AUTHOR KEYWORDS: Fuzzy relation equation; L°U-factorization; Max-product operator

INDEX KEYWORDS: Existence and uniqueness; Fuzzy relation equations; Max-product; Max-product composition; Minimal solutions; Polynomial time complexity; Sufficient conditions; Triangular matrices, Algorithms; Computational complexity; Factorization; Fuzzy systems; Product development, Matrix algebra

DOI: 10.1016/j.ins.2012.07.029

The occurrence of imprecision in the real world is inevitable due to some unexpected situations. The imprecision is often involved in any engineering design process. The imprecision and uncertainty are often interpreted as fuzziness. Fuzzy systems have an essential role in the uncertainty modelling, which can formulate the uncertainty in the actual environment. In this paper, a new approach is proposed to solve a system of fuzzy polynomial equations based on the Gröbner basis. In this approach, first, the h-cut of a system of fuzzy polynomial equations is computed, and a parametric form for the fuzzy system with respect to the parameter of h is obtained. Then, a Gröbner basis is computed for the ideal generated by the h-cuts of the system with respect to the lexicographical order using Faugère's algorithm, i.e., F 4 algorithm. The Gröbner basis of the system has an upper triangular structure. Therefore, the system can be solved using the forward substitution. Hence, all the solutions of the system of fuzzy polynomial equations can easily be obtained. Finally, the proposed approach is compared with the current numerical methods. Some theorems together with some numerical examples and applications are presented to show the efficiency of our method with respect to the other methods. © 2012 Elsevier Inc. All rights reserved.

AUTHOR KEYWORDS: F 4 algorithm; Faugère's algorithm; Fuzzy number; Gröbner basis; Lexicographical order; System of fuzzy polynomial equation

INDEX KEYWORDS: Actual environments; Engineering design process; Fuzzy numbers; Lexicographical order; Numerical example; Parametric forms; Polynomial equation; Triangular structures; Uncertainty modelling, Algorithms; Fuzzy sets; Fuzzy systems; Uncertainty analysis, Polynomials

DOI: 10.1016/j.cie.2011.09.012

The quadratic programming has been widely applied to solve real world problems. The quadratic functions are often applied in the inventory management, portfolio selection, engineering design, molecular study, and economics, etc. Fuzzy relation inequalities (FRI) are important elements of fuzzy mathematics, and they have recently been widely applied in the fuzzy comprehensive evaluation and cybernetics. In view of the importance of quadratic functions and FRI, we present a fuzzy relation quadratic programming model with a quadratic objective function subject to the max-product fuzzy relation inequality constraints. Some sufficient conditions are presented to determine its optimal solution in terms of the maximum solution or the minimal solutions of its feasible domain. Also, some simplification operations have been given to accelerate the resolution of the problem by removing the components having no effect on the solution process. The simplified problem can be converted into a traditional quadratic programming problem. An algorithm is also proposed to solve it. Finally, some numerical examples are given to illustrate the steps of the algorithm.

AUTHOR KEYWORDS: Fuzzy relation inequality; Max-product composition; Modified simplex method; Non-convex optimization; Quadratic programming

INDEX KEYWORDS: Engineering design; Fuzzy comprehensive evaluation; Fuzzy mathematics; Fuzzy relations; Inequality constraint; Inventory management; Max-product; Max-product composition; Minimal solutions; Modified simplex methods; Nonconvex optimization; Numerical example; Optimal solutions; Portfolio selection; Quadratic function; Quadratic objective functions; Quadratic programming model; Quadratic programming problems; Real-world problem; Solution process; Sufficient conditions, Algorithms; Computer programming; Constraint theory; Convex optimization; Economics; Functions; Inventory control; Optimization; Quadratic programming, Problem solving

DOI: 10.1016/j.mcm.2010.01.006

In this paper, we firstly consider an optimization problem with a linear objective function subject to a system of fuzzy relation inequalities using the max-product composition. Since its feasible domain is non-convex, traditional linear programming methods cannot be applied to solve it. An algorithm is proposed to solve this problem using fuzzy relation inequality paths. Then, a more general case of the problem, i.e., an optimization model with one fuzzy linear objective function subject to fuzzy-valued max-product fuzzy relation inequality constraints, is investigated in this paper. A new approach is proposed to solve this problem based on Zadeh's extension principle and the algorithm. This paper develops a procedure to derive the fuzzy objective value of the recent problem. A pair of mathematical program is formulated to compute the lower and upper bounds of the problem at the possibility level α. From different values of α, the membership function of the objective value is constructed. Since the objective value is expressed by a membership function rather than by a crisp value, more information is provided to make decisions. © 2010 Elsevier Ltd. All rights reserved.

AUTHOR KEYWORDS: Extension principle; Fuzzy relation inequality; Fuzzy set; Membership function; Non-convex optimization

INDEX KEYWORDS: Extension principles; Fuzzy relations; Fuzzy set membership; Inequality constraint; Linear objective functions; Lower and upper bounds; Mathematical program; Max-product; Max-product composition; New approaches; Nonconvex optimization; Optimization models; Optimization problems; Problem-based; Zadeh's extension principles, Constraint theory; Convex optimization; Fuzzy sets; Linear programming; Linearization; Optimization; Problem solving, Membership functions