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Research Article | Open Access
Optimality Conditions for Fuzzy Number Quadratic Programming with Fuzzy Coefficients
The purpose of the present paper is to investigate optimality conditions and duality theory in fuzzy number quadratic programming (FNQP) in which the objective function is fuzzy quadratic function with fuzzy number coefficients and the constraint set is fuzzy linear functions with fuzzy number coefficients. Firstly, the equivalent quadratic programming of FNQP is presented by utilizing a linear ranking function and the dual of fuzzy number quadratic programming primal problems is introduced. Secondly, we present optimality conditions for fuzzy number quadratic programming. We then prove several duality results for fuzzy number quadratic programming problems with fuzzy coefficients.
Quadratic programming is a mathematical modeling technique designed to optimize the usage of limited resources. It has led to a number of interesting applications and the development of numerous useful results (see [1–7]). Quadratic programming is one of the most important optimization techniques in operations research. In real-world applications, quadratic programming models usually are formulated to find some future course of action. The parameter values used would be based on a prediction of future conditions which inevitably involves some degree of uncertainty. If some parameters are imprecise or uncertain, then some crisp values are usually assigned to those uncertain parameters to make the conventional quadratic program workable. The occurrences of randomness and imprecision in the real world are inevitable owing to some unexpected situations. The research on fuzzy mathematical programming has been an active area since Bellman and Zadeh proposed the definition of fuzzy decision [8–12]. As the definition of Bellman and Zadeh, fuzzy decision may be described as the best balance degree of fuzzy objective and resource constraints. In light of the above view, Zimmermann developed a tolerance approach [10, 11] for a symmetric model of fuzzy linear programming. Intuitively, when the cost and constraint coefficients and the right-hand sides are fuzzy numbers, the derived objective value is fuzzy as well.
Recently, Liu  proposes an effective solution method to solve a class of fuzzy quadratic programming problems. Based on Zadeh’s extension principle [11, 14], the fuzzy quadratic programming problem is transformed into a pair of two-level mathematical programs to calculate the upper and lower bounds of the objective value at possibility level . The membership function of the fuzzy objective value is derived numerically by enumerating different values of. Mahdavi-Amiria and Nasseria [15, 16], using a linear ranking function, establish the dual problem of the linear programming problem with trapezoidal fuzzy variables and hence deduce some duality results.
The organization and content of this paper can be summarized as follows. In Section 2, we provide some properties of fuzzy numbers and introduce the concept of fuzzy scalar product. Then we provide a discussion of fuzzy numbers and linear ranking functions for ordering them. The definition of the FNQP problem is given and the equivalent quadratic programming of FNQP is presented by utilizing a linear ranking function in Section 3. Section 4 optimality conditions for fuzzy number quadratic programming are presented. We establish duality for the FNQP problem in Section 5 and deduce the duality results. We conclude in Section 6.
2.1. Fuzzy Numbers
We review the fundamental notions of fuzzy set theory, initiated by Bellman and Zadeh . Let be a universal set. The fuzzy subset of is defined by its membership function . The -level set of fuzzy set is given by
Definition 1. Let be a fuzzy number, that is, a convex normalized fuzzy subset of the real line in the sense that(a) and , where is the membership function specifying to what degree belongs to .(b) is a piecewise continuous function.
The lower and upper bounds of any -level set are represented by and .
A function, usually denoted by “" or “", is a reference function of a fuzzy number if and only if (1), (2), (3) is nonincreasing on .
Definition 2. A convenient representation of fuzzy numbers is in the form of an - fuzzy number which is defined as
where is the core of , are the lower and upper modal values of , and are the left-hand and right-hand spreads .
A flat fuzzy number is denoted by . Among the various types of fuzzy numbers, trapezoidal fuzzy numbers, denoted by , are of the greatest importance. We denote the set of all trapezoidal fuzzy numbers by . If , then we obtain a triangular fuzzy number, and we show it with .
Let and , both being trapezoidal fuzzy numbers. We next define arithmetic on the fuzzy numbers and as follows:
2.2. Ranking Functions
There are different methods for comparison of fuzzy numbers [17–22]. One of the most convenient methods is comparison by use of ranking functions [23, 24]. In fact, an efficient approach for ordering the elements of is to define a ranking function which maps each fuzzy number into the real line, where a natural order exists. We define orders on by where . Also we write if and only if .
We restrict our attention to linear ranking functions, that is, a ranking function such that for all and . The following lemma is now immediate.
Lemma 3 (see ). Let be any linear ranking function and, without loss of generality, let ; then (i) if and only if if and only if . (ii)If and , then . (iii)Let ; if , then ; otherwise,.
Here, we introduce a linear ranking function that is similar to the ranking function adopted by Maleki et al. . For any fuzzy number , we use ranking function as follows: where and and are inverse function of and , respectively. In the case of trapezoidal fuzzy numbers , we have where is the -level set of . Then, for trapezoidal fuzzy numbers and , we can get
Definition 4. We shall say that the real number corresponds to the fuzzy number , with respect to a given linear ranking function , if .
3. Fuzzy Number Quadratic Programming Problems
In this section, we first define fuzzy number quadratic programming problems with fuzzy coefficients. Then, using ranking functions for comparison of fuzzy numbers, we define a crisp model which is equivalent to the fuzzy quadratic programming problem with fuzzy coefficients and use optimal solution of this model as the optimal solution of fuzzy number quadratic programming problem with fuzzy number coefficients.
Definition 5. Let be the set of all trapezoidal fuzzy numbers. The model or where , , , and , for , is called a fuzzy number quadratic programming model.
Definition 6. Any which satisfies the set of constraints of FNQP is called a feasible solution. Let be the set of all feasible solutions of FNQP. We shall say that is a global optimal feasible solution for FNQP if for any . We say that is a local optimal feasible solution for FNQP if there exists a neighborhood of such that for all .
The following theorem shows that any FNQP can be reduced to a quadratic programming problem.
Theorem 7. The following quadratic programming problem (QPP) and the FNQP in (9) are equivalent: or where , , and are real numbers corresponding to the fuzzy numbers with respect to a given linear ranking function , respectively.
Proof. The method of proof is the same as Lemma in . However, to provide a self-contained presentation, and because this result is central to this paper, we provide a direct proof. Let and be the set of all feasible solutions of (9) and (11), respectively. We first prove that . We see
Then, we have . Let be an optimal feasible solution for (9); then, for all , we have
So, is an optimal feasible solution for (11) or (12).
Now, let be an optimal feasible solution for (11) or (12). We shall assume that is not an optimal feasible solution for (9) and exhibit a contradiction. If is not an optimal feasible solution for (9), then there exists a such that
From the ranking function (8), (15) can be written as which contradicts the facts that is an optimal feasible solution for (11), since .
From Theorem 7, we have that the sets of all feasible solutions of FNQP and QPP are the same. And if is an optimal feasible solution for FNQP, then is an optimal feasible solution for QPP. Then we can easily see the following result.
Corollary 8. If QPP does not have a solution, then FNQP does not have a solution either.
Definition 9. is denoted by a symmetric fuzzy number matrix if and for any .
Let be a real symmetric matrix corresponding to the symmetric fuzzy numbers matrix and . In the following, we only consider the symmetric fuzzy numbers matrix .
Definition 10. Let be a symmetric fuzzy numbers matrix and by the ranking function (8). (i) is called a fuzzy number positive definite matrix (resp., fuzzy number negative definite matrix) if (resp., ) for all nonzero in or is a positive definite matrix (resp., negative definite matrix). (ii)If (resp., ) for all in , or is a positive semidefinite matrix (resp., negative semidefinite matrix), then is said to be a fuzzy number positive semidefinite (resp., fuzzy number negative semidefinite). (iii)If is an indefinite matrix, then is said to be a fuzzy number indefinite matrix.
Definition 11. If the fuzzy number matrix is fuzzy number positive semidefinite matrix, then (9) is called a convex fuzzy number quadratic programming problem. If is fuzzy number positive definite matrix, (9) is a strict convex FNQP. Equation (9) is said to be a nonconvex FNQP if is fuzzy number indefinite matrix.
From Theorem 7 and definition above, the following result is now immediate.
Corollary 12. (i) If is fuzzy number positive semidefinite matrix, then the local solution of (9) is a global solution.
(ii) If is fuzzy number positive definite matrix, then the local solution of (9) is a unique global solution.
Example 13. Consider the following FNQP:
where coefficients are trapezoidal fuzzy numbers.
We apply the ranking function (8) to solve the above FNQP. The problem reduces to
Then we see that coefficient matrix of (18) is . Since is a positive definite matrix, (17) is a strict convex FNQP and the local solution of (18) is a global solution of (17).
The optimal solution of (18) is ; then the optimal solution of (17) is , and the optimal objective value is , , , . So, by the ranking function (8), .
4. Optimality Conditions for Fuzzy Number Quadratic Programming
Let us now state the necessary optimality conditions of problem (9).
Theorem 14 (necessary conditions). Let be a local minimizer of the fuzzy number quadratic programming problem (9). Then there exists a -vector such that where is the th row of fuzzy number matrix , and, for every , satisfying where we have
Proof. Since is a local minimizer of fuzzy number quadratic programming problem (9), is a local minimizer of quadratic programming problem (11) by Theorem 7. Then, from Theorem 7.1 in Avriel , there exists a -vector such that
where for any , and is the th row of matrix , and, for every , satisfying
From the ranking function (8), formulas (19)–(22) and (23)–(26) are equivalent and the proof is complete.
Turning to sufficient conditions, we first define
Let us now state the sufficient optimality conditions for problem (9).
Proof. From the ranking function (8), there exists a vector satisfying (23)–(26), and for every such that
it follows that
Then, from Theorem 7.2 in Avriel , is a strict local minimum of (11). So is a strict local minimum of FNQP by Theorem 7.
Next, we give a sufficient and necessary optimality condition for FNQP (9).
Proof. Using the ranking function (8), formulas (19)–(22) and formulas (23)–(26) are equivalent. Then (32) and (33) are respectively equivalent to the following formulas:
By Theorem 3.4 in Lee et al. , is a local minimizer of (11) if and only if there exists such that (23)-(26) and (34)-(35) hold. Then, based on Theorem 7, we complete the proof.
5. Duality for Fuzzy Number Quadratic Programming
Similar to the duality theory in quadratic programming (see, e.g., Mangasarian ), for every FNQP, there is an associated problem which satisfies some important properties. We shall call this related FNQP the dual fuzzy number quadratic programming (DFNQP).
5.1. Dual Problem
Example 17. Consider the given FNQP in Example 13. The dual to this problem follows:
Now if we apply the ranking function (8), we have
The optimal solution is , and the optimal objective value is . So, by the ranking function (8), .
5.2. The Relationships between FNQP and DFNQP
We shall discuss here the relationships between the fuzzy number quadratic programming problem and its corresponding dual. Let and be the feasible solution sets of FNQP and DFNQP, respectively.
Lemma 18. Dual of DFNP is FNQP.
Proof. Use Lemma 3 and the definition of DFNQP.
Lemma 18 indicates that the duality results can be applied to any of the primal or dual problems posed as the primal problem.
Theorem 19 (weak duality theorem). Let be fuzzy number positive semidefinite matrix, , ; then we have
Proof. By the ranking function (8), (41) holds if and only if where are real numbers corresponding to the fuzzy numbers . Equation (42) follows from Theorem 8.1.3 in  by observing that is convex on if the matrix is positive semidefinite.
The following corollaries are immediate consequences of Theorem 19.
Corollary 20. Let be fuzzy number positive semidefinite matrix. If and are feasible solutions to FNQP and DFNQP, respectively, and , then and are optimal solutions to their respective problems.
Definition 21. We say that FNLPP (or DFNLPP) is unbounded if feasible solutions exist with arbitrary small (or large) ranking values for the fuzzy objective function.
Corollary 22. If either problem is unbounded, then the other problem has no feasible solution.
Proof. Since solves FNQP (36), then is the optimal solutions to QPP (11). From Dorn’s duality theorem , there exist and some solves problem (38) since is positive semidefinite matrix, and . Then, solves DFNQP (37) and . For if , then, by the ranking function (8), , which contradicts .
For an illustration of the above theorem, consider the FNQP and DFNQP given in Examples 13 and 17, respectively. Symmetric fuzzy number matrix is fuzzy number positive definite, since matrix is positive definite. We see that the optimal solution for the FNQP is ; then there exists such that , , solves DFNQP and by the ranking function (8).
Proof. Since is fuzzy number positive definite and is an optimal solution to DFNQP (37), we have that is positive definite and is an optimal solution to problem (38). From Theorem 8.2.5 in , solves QPP (11) and . From Theorem 7 above, then solves FNQP (36), and − + .
Proof. Since is fuzzy number positive semidefinite and solves DFNQP (37), we have that is positive semidefinite and is an optimal solution to problem (38). From Theorem in , some (not necessarily equal to ), satisfying , solves FNQP (11), and − . From Theorem 7 above, solves FNQP (36) and .
We used a linear ranking function to define the dual of fuzzy number quadratic programming primal problems. We provide optimality conditions for fuzzy number quadratic programming. Similar to general quadratic programming, we presented several duality results.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors thank the PhD Start-up Fund of Natural Science Foundation of Guangdong Province, China (no. S2013040012506) and Project Science Foundation of Guangdong University of Finance (no. 2012RCYJ005) for their support.
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