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We propose the solution concepts for the fuzzy optimization problems in the quotient space of fuzzy numbers. The Karush-Kuhn-Tucker (KKT) optimality conditions are elicited naturally by introducing the Lagrange function multipliers. The effectiveness is illustrated by examples.

The fuzzy set theory was introduced initially in 1965 by Zadeh [

Zimmermann and Rödder initially applied fuzzy sets theory to the linear programing problems and linear multiobjective programing problems by using the aspiration level approach [

These above results of fuzzy optimization are based on well-known and widely used algebraic structures of fuzzy numbers and the differentiability of fuzzy mappings was based on the concept of Hukuhara difference. However these operations can have some disadvantages for both theory and practical application. In [

We start this section by recalling some pertinent concepts and key lemmas from the function of bounded variation, fuzzy numbers, and fuzzy number equivalence classes which will be used later.

Let

Let

Let

for any contents

Every monotonic function

Any mapping

Let

Let

For any

Let

It is easy to verify that the equivalence relation defined above is reflexive, symmetric, and transitive [

Let

Let

Let

The addition operation defined by Definition

Let

For any

Let

In this paper, we always suppose that the range of fuzzy mappings is the set of all fuzzy number equivalence classes.

Let

the mappings

for all

Let

Let

Let

Let

We say that

We say that

If

Sometimes we may write

Let

Let

Let

We say that

We say that

Let

Let

The result follows from Definitions

Let

Let

Let

Now we are in a position to present the KKT optimality conditions for nondominated solutions of problem (

Let

Suppose that conditions (1) and (2) are satisfied and

Let

Since conditions (1) and (2) are satisfied, taking the midpoint function of (1) and (2), we obtain the following new conditions:

Let

Let

System I:

System II:

Let

Suppose that conditions (1) and (2) are satisfied and

System I:

System II:

Define a fuzzy mapping

In this present investigation, the KKT optimality conditions are elicited naturally by introducing the Lagrange function multipliers, and we also provided some examples to illustrate the main results. The research on the quotient space of fuzzy numbers can be traced back to the works of Mareš [

The authors declare that there are no conflicts of interest regarding the publication of this paper.

This work was supported by The National Natural Science Foundations of China (Grants nos. 11671001 and 61472056), The Natural Science Foundation Project of CQ CSTC (cstc2015jcyjA00034, cstc2014jcyjA00054), and The Graduate Teaching Reform Research Program of Chongqing Municipal Education Commission (no. YJG143010).