^{1,2}

^{1}

^{3}

^{1}

^{2}

^{3}

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 [

Recently, Liu [

The organization and content of this paper can be summarized as follows. In Section

We review the fundamental notions of fuzzy set theory, initiated by Bellman and Zadeh [

Let

The lower and upper bounds of any

A function, usually denoted by “

A convenient representation of fuzzy numbers is in the form of an

A flat fuzzy number is denoted by

Let

There are different methods for comparison of fuzzy numbers [

We restrict our attention to linear ranking functions, that is, a ranking function

Let

If

Let

Here, we introduce a linear ranking function that is similar to the ranking function adopted by Maleki et al. [

We shall say that the real number

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.

Let

Any

The following theorem shows that any FNQP can be reduced to a quadratic programming problem.

The following quadratic programming problem (QPP) and the FNQP in (

The method of proof is the same as Lemma

Then, we have

So,

Now, let

From the ranking function (

From Theorem

If QPP does not have a solution, then FNQP does not have a solution either.

Let

Let

If

If

If the fuzzy number matrix

From Theorem

(i) If

(ii) If

Consider the following FNQP:

We apply the ranking function (

Then we see that coefficient matrix of (

The optimal solution of (

Let us now state the necessary optimality conditions of problem (

Let

Since

From the ranking function (

Turning to sufficient conditions, we first define

Let us now state the sufficient optimality conditions for problem (

Let

From the ranking function (

Then, from Theorem 7.2 in Avriel [

Next, we give a sufficient and necessary optimality condition for FNQP (

Let

Using the ranking function (

By Theorem 3.4 in Lee et al. [

Similar to the duality theory in quadratic programming (see, e.g., Mangasarian [

For the FNQP

We see that (

Consider the given FNQP in Example

Now if we apply the ranking function (

The optimal solution is

We shall discuss here the relationships between the fuzzy number quadratic programming problem and its corresponding dual. Let

Dual of DFNP is FNQP.

Use Lemma

Lemma

Let

By the ranking function (

The following corollaries are immediate consequences of Theorem

Let

We say that FNLPP (or DFNLPP) is unbounded if feasible solutions exist with arbitrary small (or large) ranking values for the fuzzy objective function.

If either problem is unbounded, then the other problem has no feasible solution.

Let

Since

For an illustration of the above theorem, consider the FNQP and DFNQP given in Examples

Let

Since

Let

Since

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.

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.