The quadratic programming problem with fuzzy relation inequality constraints
Highlights
► A fuzzy relation quadratic programming model is introduced. ► Some sufficient conditions are presented for its optimal solutions. ► The components having no effect on the solution process are determined and removed. ► The problem is converted into a traditional quadratic programming problem. ► An algorithm is proposed to solve it.
Introduction
The quadratic programming is a designed mathematical modeling technique to optimize the usage of limited resources. It has led to a number of interesting applications and the development of numerous useful results. The inventory management (Abdel-Malek & Areeratchakul, 2007), portfolio selection (Zhang & Nie, 2005), engineering design (Petersen & Bodson, 2006), molecular study (Pavlovi & Divni, 2007), and economics (Schwarz, 2006) are the examples. The quadratic programming is one of the most important optimization techniques in operations research.
Fuzzy set theory has been extensively employed in linear and non-linear optimization problems (Ammar, 2000, Chen, 2004, Liu, 2004, Liu, 2006, Mitsuo et al., 1997, Sakawa, 1993). Another completely different kind of fuzzy programming problem is the optimization of objective functions on the region whose set of feasible solutions have been defined as FRE or FRI constraints (Abbasi Molai, 2010, Fang and Li, 1999, Ghodousian and Khorram, 2008, Guo and Xia, 2006, Guu and Wu, 2010, Guu and Wu, 2002, Li and Fang, 2008, Lin et al., 2009, Loetamonphong and Fang, 2001, Lu and Fang, 2001, Pandey, 2004, Peeva, 1994, Shivanian and Khorram, 2009, Vasantha Kandasamy and Smarandache, 2004, Wang et al., 1991, Wu et al., 2002, Wu et al., 2008, Wu and Guu, 2005, Wu and Guu, 2004, Yang and Cao, 2005). The fuzzy relation equations (FRE) concept and description of its structure were firstly proposed by Sanchez (1976). FRE is a special case of the FRI. In recent years, the FRE and FRI have received a considerable development in applications and theory. They have already become an important element of fuzzy mathematics (Abbasi Molai and Khorram, 2008, Czogala and Predrycz, 1981, Di Nola et al., 1989, Klir and Folger, 1988, Li and Fang, 2008, Loia and Sessa, 2005, Pedrycz, 1985, Peeva and Kyosev, 2004, Vasantha Kandasamy and Smarandache, 2004).
The optimization model with a linear objective function subject to FRE with the max–min composition has been investigated by Fang and Li (1999). They converted the model into a 0–1 integer programming problem and solved this by the branch-and-bound method. The method was improved by Wu et al., 2002, Wu and Guu, 2005 by providing the upper bounds for the branch-and-bound procedure. Loetamonphong and Fang (2001) considered the problem with the max-product composition. They applied the similar idea to Fang and Li (1999) to solve the problem. For the recent problem, Guu and Wu (2002) provided a necessary condition for its optimal solution in terms of the derived maximum solution from FRE. Some other generalizations on this issue can be found in (Abbasi Molai, 2010, Di Nola et al., 1989, Ghodousian and Khorram, 2008, Guo and Xia, 2006, Guu and Wu, 2010, Li and Fang, 2008, Lu and Fang, 2001, Pandey, 2004, Peeva, 1994, Shivanian and Khorram, 2009, Vasantha Kandasamy and Smarandache, 2004, Wang et al., 1991, Wu and Guu, 2004, Wu et al., 2008, Yang and Cao, 2005, Zhang et al., 2003).
The linear objective function optimization problem with FRI was investigated by Zhang et al. (2003), where the fuzzy operator is considered as the max–min composition. Also, Guo and Xia (2006) presented an algorithm to accelerate the resolution of this problem. Ghodousian and Khorram (2008) studied the recent problem with the defined FRI by the max–min composition in which the fuzzy inequality replaces the ordinary inequality in the constraints and then proposed an algorithm to solve it.
However, the case with a non-linear objective function has been developing very slowly. It is difficult to solve this problem by using traditional non-linear optimization methods. The initial research on the non-linear objective function optimization problem can be found in Mitsuo et al. (1997). Wang et al. (1991) presented the latticized linear programming problem subject to FRI constraints with the max–min composition. Their method can be utilized to solve such problems based on the obtained results from a set of all conservative paths. Peeva (1994) found all the minimal solutions to the same problem, and compared their corresponding objective function values to obtain optimal solutions. Yang and Cao (2005) devised another variant of the model, called a fuzzy relational geometric programming subject to FRE with the max–min composition. They disassembled the problem into two sub-problems depending on the negative and non-negative exponents in the objective function, and indicated that the formed sub-problem by negative exponents is easily solved by the maximum solution. Conversely, a minimal solution can be employed to optimize the other formed sub-problem by non-negative exponents. Wu et al. (2008) considered a linear fractional programming problem provided to FRE constraints with the max-Archimedean t-norm composition. They firstly developed some theoretical results based on the properties of the max-Archimedean t-norm composition and reduced its feasible domain. Then the linear fractional programming problem was converted into a traditional linear programming problem and solved in a small search space. Shivanian and Khorram (2009) studied the monomial geometric programming problem with fuzzy relation inequality constraints. The objective function of the problem is a monomial in x1, …, xn, i.e., it is a product of form , where αj ∈ R and c > 0. With regard to the structure of the problem, the minimal solutions and the maximum solution of its feasible domain guarantee the resolution of the problem. Their proposed algorithm is based on the structure and the pairwise comparison of quasi-minimal solutions. In this paper, we study the optimization of a quadratic function as subject to the fuzzy relation inequality constraints.
In view of the importance of quadratic programming and the fuzzy relation inequality in theory and applications, we are motivated to propose a fuzzy relation quadratic programming. So it is of interest to solve the quadratic programming with fuzzy relation inequality constraints with the max-product composition. Since the objective function is non-linear, the minimal solutions and the maximum solution of its feasible domain cannot guarantee the resolution of the problem. Some sufficient conditions are presented to determine some of its optimal components without solving the problem. Using the conditions, some procedures are proposed to simplify the problem. Then the simplified problem is converted into a traditional quadratic programming problem. Finally, the modified simplex method in Hilier and Liberman (1981) or the numerical algorithms in Bazaraa, Sherali, and Shetty (1993) are applied to solve the simplified problem.
This paper is organized as follows. In Section 2, the quadratic programming problem with FRI constraints is formulated and presented some concepts about FRI and studied some properties of its feasible solution set. In Section 3, some sufficient conditions are proposed to determine the optimal solution of the problem. Some results are also given to reduce the size of the problem. In Section 4, some procedures are suggested to simplify the reduced problem and an algorithm is designed to solve the simplified problem. Some numerical examples are presented to illustrate the algorithm. Furthermore, we show that one of the minimal solutions or the maximum solution of the feasible domain of the problem or their combinational may not be an optimal solution for the problem. Finally, conclusions are presented in Section 5.
Section snippets
Formulation of the quadratic programming problem with FRI constraints and some concepts
Let A = [aij] and B = [bij] are m × n and l × n fuzzy relation matrices with 0 ⩽ aij, bij ⩽ 1, respectively. Also, assume that and . Furthermore, c = [c1, … , cn] is a vector of cost coefficients and Q = [qij] is the n × n matrix of the quadratic form. In this section, we shall consider the following problem.where x = [x1, … , xn]T is the vector of decision variables to be determined. The operator of “•” denotes the max-product
Some sufficient conditions and some results for the reduction of the size of problem (1)
This section provides some theoretical results with respect to the objective function in problem (1), and then develops a procedure to compute the optimal solution to the problem based on the results. The procedure firstly reduces the size of the problem by determining the decision variables and then converts the FRI into the constraint in the form of the general linear programming problem. The optimal solution of problem (1) can eventually be obtained in a small search space. We suppose that X(
A procedure for the resolution of problem (1)
Similar to Fang and Li (1999), one of the minimal solutions or the maximum solution of system (2) or their combinational may not be an optimal solution for problem (1), in a general case. The following example explains this point. Example 1 Consider the following quadratic programming problem subject to FRI with the max-product composition.The maximum solution of its feasible domain is computed by formula (3) as
Conclusion
In this paper, the quadratic programming problem with fuzzy relation inequality constraints was introduced. Some sufficient conditions were proposed to determine the optimal solution of the problem without solving it, directly. Some sufficient conditions were also presented to find some components of the optimal solutions of the problem. Based on these conditions, some procedures were given to simplify the problem. Hence, the problem can be reduced to a problem with a smaller feasible domain.
Acknowledgment
The author thanks the anonymous reviewers for their valuable comments that improved the quality of this paper.
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2016, Computers and Industrial EngineeringCitation Excerpt :(b) It can be easily verified by the definition of FRI path. Now, we compute Example 3 in Molai (2012). After Step 7 of Example 3, it can be rewritten the rest of the steps until Step 11 in details as follows.