On relationships among different types of solutions of fuzzy optimization problems

Abstract

Since fuzzy optimization problems are extensively studied by considering the endpoint functions of level sets of fuzzy-valued objective function with respect to the fuzzy-max order, in this paper, by defining two related partial order relations we establish the complete relationships between the optimal solutions with respect to the fuzzy-max order and the ones with respect to the related partial order relations. These relationships reflect the intrinsic properties of fuzzy optimizations.

Keywords

Fuzzy numbers fuzzy-max order fuzzy optimization problems Pareto optimal

1 Introduction

Using the theory of fuzzy sets [9, 23], fuzzy optimization problems have been extensively studied since the seventies. The approaches in [19, 24] are the first approaches to solve the fuzzy linear programming problem. In [19], the main concern is with the application of the theory of fuzzy sets to decision problems involving fuzzy goals and strategies, etc.. However, the emphasis is on mathematical programming and the use of the concept of a level set to extend some of the classical results to problems involving fuzzy constraints and objective functions. In [24], fuzzy set theory is then applied to fuzzy linear programming problems and it is shown how fuzzy linear programming problems can be solved without increasing the computational effort.

Fuzzy optimization problems are extensively studied by considering the endpoint functions of level sets of fuzzy-valued objective function with respect to the fuzzy-max order [4 , 22]. By using strongly generalized derivative, the Karush-Kuhn-Tucker optimality conditions for a class of fuzzy optimization problems are presented in [4, 22]. In order to consider the differentiation of fuzzy-valued function, Wu invoked the Hausdorff metric to define the distance between two fuzzy numbers and the Hukuhara difference to define the difference of two fuzzy numbers in [21]. Under these settings, the optimality conditions for obtaining the non-dominated solutions are elicited naturally by introducing the Lagrange multipliers. In [15], Phu and Tri considered the fuzzy dynamic programming problems by using the generalized Hukuhara differentiability for fuzzy functions. In [7], Dey and Roy made an approach to solve multi-objective structural model using parameterized t-norms based on fuzzy optimization programming technique.

Since for the fuzzy-max order the fuzzy numbers is ranking by their left-hand endpoints and right-hand endpoints of level sets, to reveal the important role of this order in fuzzy optimizations, we will define two related partial order relations and will establish the complete relationships between the optimal solutions with respect to the fuzzy-max order and the ones with respect to the related partial order relations.

This paper is arranged as follows. After the Introduction, some necessary fundamental knowledge on fuzzy numbers is reviewed and three types of partial order relations are defined in Section 2. These partial orders allow us to define some concepts for optimal solutions in Section 3. We present a set of theorems and examples to give detailed insight in the relationships among these different types of solutions.

2 Preliminaries

Any mapping $μ : ℝ \to [0, 1]$ will be called a fuzzy set μ on $ℝ$ . Its α-level set of μ is $[μ]^{α} = {x \in ℝ : μ (x) \geq α}$ for each α ∈ (0, 1]. Specifically, for α = 0, the set [μ] ⁰ is defined by ${[μ]}^{0} = cl {x \in ℝ : μ (x) > 0}$ , where clA denotes the closure of a crisp set A. A fuzzy set μ is said to be a fuzzy number if it is normal, convex, upper semi-continuous and the set [μ] ⁰ is compact [8]. Equivalently, a fuzzy number μ is a fuzzy set with non-empty bounded level sets [μ] ^α = [μ_L (α) , μ_R (α)] for all α ∈ [0, 1], where [μ_L (α) , μ_R (α)] denotes a closed interval with the left-hand endpoint μ_L (α) and the right-hand endpoint μ_R (α). We denote the class of fuzzy numbers by ℱ.

Ordering of fuzzy numbers is an important and prerequisite procedure for fuzzy optimizations. In general, the fuzzy numbers are ranked by considering their left-hand endpoints and right-hand endpoints of level sets.

Definition 2.1. Let A = [a, b] and B = [c, d] be two bounded closed intervals. We define the following partial order relations:

A ≦ B, if and only if a ≤ c and b ≤ d;

$A ≨ B$ , if and only if A ≦ B and A ≠ B, i.e., a ≤ c and b < d, or a < c and b ≤ d;

A < B, if and only if a < c and b < d.

Definition 2.2. Given α ∈ [0, 1]. For any fuzzy numbers μ, ν ∈ ℱ, we say that

μ ≼ _αν, if and only if [μ] ^α ≦ [ν] ^α;

$μ ⋨_{α} ν$ , if and only if $[μ]^{α} ≨ [ν]^{α}$ ;

μ ≺ _αν, if and only if [μ] ^α < [ν] ^α.

The fuzzy-max order [13 , 20] for fuzzy numbers is defined as follows.

Definition 2.3. [13, 14] For any fuzzy numbers μ, ν ∈ ℱ, we say that

μ ≼ ν, if and only if [μ] ^α ≦ [ν] ^α for all α ∈ [0, 1];

$μ ⋨ ν$ , if and only if [μ] ^α ≦ [ν] ^α for all α ∈ [0, 1] and μ ≠ ν;

μ ≺ ν, if and only if $[μ]^{α} ≨ [ν]^{α}$ for all α ∈ [0, 1];

μ ⪡ ν, if and only if [μ] ^α < [ν] ^α for all α ∈ [0, 1].

Definition 2.4. [17] For a fuzzy number μ, we define a function $μ_{M} : [0, 1] \to ℝ$ by assigning the midpoint of each α-level set to μ_M (α) for all α ∈ [0, 1], i.e., $μ_{M} (α) = \frac{μ_{L} (α) + μ_{R} (α)}{2} .$

Then the function $μ_{M} : [0, 1] \to ℝ$ will be called the midpoint function of the fuzzy number μ.

Recently, Chalco-Cano and his coworkers obtained some interesting results on fuzzy optimizations by using the sum of endpoints of level sets of the fuzzy-valued objective function [1 , 4–6]. Inspired by their work, we introduce with geometric intuition the following partial order relations on fuzzy numbers by using the midpoints of level sets of fuzzy numbers.

Definition 2.5. For any fuzzy numbers μ, ν ∈ ℱ, we say that

μ ≼ _mν, if and only if their midpoint functions μ_M and ν_M satisfy μ_M (α) ≤ ν_M (α) for all α ∈ [0, 1];

$μ ⋨_{m} ν$ , if and only if μ ≼ _mν and there exists α₀ ∈ [0, 1] such that μ_M (α₀) < ν_M (α₀);

μ ≺ _mν, if and only if μ_M (α) < ν_M (α) for all α ∈ [0, 1].

3 Fuzzy optimizations based on different types of partial orders

Now we consider the following optimization problem with fuzzy-valued objective function:

$\begin{matrix} (FO) & minimize f (x) \\ subject to x \in Ω, \end{matrix}$ (1) where $Ω \subseteq ℝ^{n}$ is said to be the feasible set, and f : Ω → ℱ is a fuzzy function. It would be natural to decompose (FO) as a series of optimization problems with interval-valued objective function [12] as follows:

$\begin{matrix} (O_{α}) & minimize [f (x)]^{α} \\ subject to x \in Ω, \end{matrix}$ (2) for each α ∈ [0, 1]. In addition, one can also consider the following optimization problem with midpoint-function-valued objective function [12]:

$\begin{matrix} (O_{M}) & minimize f (x)_{M} \\ subject to x \in Ω . \end{matrix}$ (3)

However, for convenience we unify the above three types of optimization problems as (FO) with respect to the three different types of partial order relations on ℱ in the previous section (i.e., Definitions 2.2, 2.3 and 2.5). These partial order relations allow us to define different concepts for optimal solutions as an extension of those optimal solutions used in multi-objective programming problems [12].

Definition 3.1. [12] Let x^* be a feasible solution of (FO), i.e., x^* ∈ Ω. We say that

x^* is an optimal solution of (FO) with respect to ≼, if f (x^*) ≼ f (x) for all x ∈ Ω; the set of optimal solutions of (FO) is denoted by OS(f);

x^* is an ideal optimal solution of (FO) with respect to ≼, if $f (x^{*}) ⋨ f (x)$ for all x (≠ x^*) ∈ Ω; the set of ideal optimal solutions of (FO) is denoted by IOS(f);

x^* is a strongly optimal solution of (FO) with respect to ≼, if f (x^*) ≺ f (x) for all x (≠ x^*) ∈ Ω; the set of strongly optimal solutions of (FO) is denoted by SOS(f);

x^* is an absolutely optimal solution of (FO) with respect to ≼, if f (x^*) ⪡ f (x) for all x (≠ x^*) ∈ Ω; the set of absolutely optimal solutions of (FO) is denoted by AOS(f);

x^* is a Pareto optimal solution of (FO) with respect to ≼, if there exists no x ∈ Ω such that $f (x) ⋨ f (x^{*})$ ; the set of Pareto optimal solutions of (FO) is denoted by PS(f);

x^* is a strongly Pareto optimal solution of (FO) with respect to ≼, if there exists no x (≠ x^*) ∈ Ω such that f (x) ≼ f (x^*); the set of strongly Pareto optimal solutions of (FO) is denoted by SPS(f);

x^* is a weakly Pareto optimal solution of (FO) with respect to ≼, if there exists no x (≠ x^*) ∈ Ω such that f (x) ≺ f (x^*); the set of weakly Pareto optimal solutions of (FO) is denoted by WPS(f);

x^* is an absolutely Pareto optimal solution of (FO) with respect to ≼, if there exists no x (≠ x^*) ∈ Ω such that f (x) ⪡ f (x^*); the set of absolutely Pareto optimal solutions of (FO) is denoted by APS(f).

Definition 3.2. [12] Let x^* be a feasible solution of (FO), i.e., x^* ∈ Ω. For a given α ∈ [0, 1], we say that

x^* is an optimal solution of (FO) with respect to ≼_α, if f (x^*) ≼ _αf (x) for all x ∈ Ω; the set of optimal solutions of (FO) with respect to ≼_α is denoted by OS_α (f);

x^* is an ideal optimal solution of (FO) with respect to ≼_α, if $f (x^{*}) ⋨_{α} f (x)$ for all x (≠ x^*) ∈ Ω; the set of ideal optimal solutions of (FO) with respect to ≼_α is denoted by IOS_α (f);

x^* is a strongly optimal solution of (FO) with respect to ≼_α, if f (x^*) ≺ _αf (x) for all x (≠ x^*) ∈ Ω; the set of strongly optimal solutions of (FO) with respect to ≼_α is denoted by SOS_α (f);

x^* is a Pareto optimal solution of (FO) with respect to ≼_α, if there exists no x ∈ Ω such that $f (x) ⋨_{α} f (x^{*})$ ; the set of Pareto optimal solutions of (FO) with respect to ≼_α is denoted by PS_α (f);

x^* is a strongly Pareto optimal solution of (FO) with respect to ≼_α, if there exists no x (≠ x^*) ∈ Ω such that f (x) ≼ _αf (x^*); the set of strongly Pareto optimal solutions of (FO) with respect to ≼_α is denoted by SPS_α (f);

x^* is a weakly Pareto optimal solution of (FO) with respect to ≼_α, if there exists no x (≠ x^*) ∈ Ω such that f (x) ≺ _αf (x^*); the set of weakly Pareto optimal solutions of (FO) with respect to ≼_α is denoted by WPS_α (f).

Definition 3.3. [12] Let x^* be a feasible solution of (FO), i.e., x^* ∈ Ω. We say that

x^* is an optimal solution of (FO) with respect to ≼_m, if f (x^*) ≼ _mf (x) for all x ∈ Ω; the set of optimal solutions of (FO) with respect to ≼_m is denoted by OS_m (f);

x^* is an ideal optimal solution of (FO) with respect to ≼_m, if $f (x^{*}) ⋨_{m} f (x)$ for all x (≠ x^*) ∈ Ω; the set of ideal optimal solutions of (FO) with respect to ≼_m is denoted by IOS_m (f);

x^* is a strongly optimal solution of (FO) with respect to ≼_m, if f (x^*) ≺ _mf (x) for all x (≠ x^*) ∈ Ω; the set of strongly optimal solutions of (FO) with respect to ≼_m is denoted by SOS_m (f);

x^* is a Pareto optimal solution of (FO) with respect to ≼_m, if there exists no x ∈ Ω such that $f (x) ⋨_{m} f (x^{*})$ ; the set of Pareto optimal solutions of (FO) with respect to ≼_m is denoted by PS_m (f);

x^* is a strongly Pareto optimal solution of (FO) with respect to ≼_m, if there exists no x (≠ x^*) ∈ Ω such that f (x) ≼ _mf (x^*); the set of strongly Pareto optimal solutions of (FO) with respect to ≼_m is denoted by SPS_m (f);

x^* is a weakly Pareto optimal solution of (FO) with respect to ≼_m, if there exists no x (≠ x^*) ∈ Ω such that f (x) ≺ _mf (x^*); the set of weakly Pareto optimal solutions of (FO) with respect to ≼_m is denoted by WPS_m (f).

Now we present some basic properties of these different types of solutions and discuss the relationships among them.

Theorem 3.4. For the solution sets of (FO) with respect to ≼, $\begin{matrix} (1) & AOS (f) \subseteq SOS (f) \subseteq IOS (f) \subseteq OS (f) \\ \subseteq PS (f) \subseteq WPS (f) \subseteq APS (f); \\ (2) & IOS (f) \subseteq SPS (f) \subseteq PS (f) . \end{matrix}$

Moreover, if the set AOS (f) is nonempty, then

\begin{matrix} AOS (f) & = & SOS (f) = IOS (f) = OS (f) \\ = & SPS (f) = PS (f) = WPS (f) = APS (f), \end{matrix}

and it is a singleton set. The same conclusion also holds true for the sets SOS (f) and IOS (f).

Proof. It is easy to see from Definitions 2.3 and 3.1.

The following example shows there is no necessary relationship between OS (f) and SPS (f) in general.

Example 3.5. Let Ω = {1, 2, 3}. Define a fuzzy function f : Ω → ℱ by $f (1) = f (2) = f (3) = μ,$ where μ is a fuzzy number with level sets [μ] ^α = [α - 1, 1 - α] for all α ∈ [0, 1]. Then we get that OS (f) = Ω and SPS (f) =∅. Consequently, SPS (f) ⊂ OS (f).

However, we define another fuzzy function g : Ω → ℱ as g (1) = g (2) = μ but g (3) = ν, where ν is a fuzzy number with level sets [ν] ^α = [3α - 2, 1] for all α ∈ [0, 1]. Because the fuzzy numbers μ and ν are not comparable with respect to ≼, we obtain that OS (g) =∅ and SPS (g) = {3}, i.e., OS (g) ⊂ SPS (g).

Theorem 3.6. Let α be a fixed number in [0, 1]. Then for the solution sets of (FO) with respect to ≼_α, $\begin{matrix} (1) & {SOS}_{α} (f) \subseteq {IOS}_{α} (f) \subseteq {OS}_{α} (f) \subseteq {PS}_{α} (f) \\ \subseteq {WPS}_{α} (f); \\ (2) & {IOS}_{α} (f) \subseteq {SPS}_{α} (f) \subseteq {PS}_{α} (f) . \end{matrix}$ Moreover, if the set SOS_α (f) is nonempty, then $\begin{matrix} {SOS}_{α} (f) = {IOS}_{α} (f) = {OS}_{α} (f) = {SPS}_{α} (f) \\ = {PS}_{α} (f) = {WPS}_{α} (f), \end{matrix}$ and it is a singleton set. The same conclusion also holds true for the set IOS_α (f).

Proof. It follows immediately from Definitions 2.2 and 3.2.

The following example shows there is no necessary relationship between OS_α (f) and SPS_α (f) in general.

Example 3.7. Let Ω = {1, 2} and f : Ω → ℱ be a fuzzy function such that [f (1)] ^α = [α - 1, 2 - 2α] for all α ∈ [0, 1] and ${[f (2)]}^{α} = {\begin{matrix} [3 α - 2, 1], & if α \in [0.5, 1], \\ [3 α - 2, 3 - 4 α], & if α \in [0, 0.5) . \end{matrix}$

Since [f (1)] ^0.5 = [f (2)] ^0.5 = [-0.5, 1], we get that OS_0.5 (f) = Ω and SPS_0.5 (f) =∅. Consequently, SPS_0.5 (f) ⊂ OS_0.5 (f).

However, since [f (1)] ^0.3 = [-0.7, 1.4] and [f (2)] ^0.3 = [-1.1, 1.8] are not comparable with respect to ≦, we obtain that OS (f) _0.3 =∅ and SPS_0.3 (f) = Ω, i.e., OS (f) _0.3 ⊂ SPS (f) _0.3.

Theorem 3.8. For the solution sets of (FO) with respect to ≼_m, we have the inclusions: $\begin{matrix} (1) & {SOS}_{m} (f) \subseteq {IOS}_{m} (f) \subseteq {OS}_{m} (f) \subseteq {PS}_{m} (f) \\ \subseteq {WPS}_{m} (f); \\ (2) & {IOS}_{m} (f) \subseteq {SPS}_{m} (f) \subseteq {PS}_{m} (f) . \end{matrix}$

Moreover, if the set SOS_m (f) is nonempty, then

\begin{matrix} {SOS}_{m} (f) = {IOS}_{m} (f) = {OS}_{m} (f) = {SPS}_{m} (f) \\ = {PS}_{m} (f) = {WPS}_{m} (f), \end{matrix}

and it is a singleton set. The same conclusion also holds true for the set IOS_m (f).

Proof. It is easy to see from Definitions 2.5 and 3.3.

The following example shows there is no necessary relationship between OS_m (f) and SPS_m (f) in general.

Example 3.9. Let Ω = {1, 2, 3} and f : Ω → ℱ be a fuzzy function such that [f (1)] ^α = [0, 2 - 2α] and [f (2)] ^α = [f (3)] ^α = [α - 1, 3 - 3α] for all α ∈ [0, 1]. Since f (1) _M (α) = f (2) _M (α) = f (3) _M (α) =1 - α, we get that OS_m (f) = Ω and SPS_m (f) =∅. Consequently, SPS_m (f) ⊂ OS_m (f).

However, we define another fuzzy function g : Ω → ℱ as g (1) = f (1), g (2) = f (2) but [g (3)] ^α = [α, 1] for all α ∈ [0, 1]. Because g (3) are not comparable with g (1) and g (2) with respect to ≼_m, we obtain that OS (g) =∅ and SPS (g) = {3}, i.e., OS (g) ⊂ SPS (g).

Theorem 3.10. For the solution sets of (FO),

OS (f) =⋂ _α∈[0,1]OS_α (f) ;

SOS (f) =⋂ _α∈[0,1]IOS_α (f) ;

AOS (f) = ⋂ _α∈[0,1]SOS_α (f) .

Proof. It is easy to see from Definitions 2.2, 2.3, 3.1 and 3.2.

Corollary 3.11.

If SOS (f) is nonempty, then SOS (f) = IOS_α (f) for any α ∈ [0, 1];

If AOS (f) is nonempty, then AOS (f) = SOS_α (f) for any α ∈ [0, 1].

Proof. We only prove the first assertion. The second assertion can be proved similarly. Suppose SOS (f) is nonempty. From Theorem 3.1 it follows that SOS (f) is a singleton set. By Theorem 3.4, we have SOS_α (f)≠ ∅ for any α ∈ [0, 1]. Thus for all α ∈ [0, 1], by Theorem 3.2 we obtain that the sets SOS_α (f) are the same singleton set as SOS (f).

Corollary 3.12.

If there exist two numbers α₁, α₂ ∈ [0, 1] such that OS_α
₁ (f)∩ OS_α
₂ (f) = ∅, then OS (f) is empty;

If there exist two numbers α₁, α₂ ∈ [0, 1] such that IOS_α
₁ (f) ≠ IOS_α
₂ (f), then SOS (f) is empty;

If there exist two numbers α₁, α₂ ∈ [0, 1] such that SOS_α
₁ (f) ≠ SOS_α
₂ (f), then AOS (f) is empty.

Proof. It is easy to see from Theorem 3.4 and Corollary 3.1.

Example 3.13. Let Ω = {1, 2} and f : Ω → ℱ be a fuzzy function such that [f (1)] ^α = [0, 2 - 2α], [f (2)] ^α = [-1 + 2α, 1] for all α ∈ [0, 1]. Since OS_0.1 (f) = {2} and OS_0.6 (f) = {1}, we get that OS (f) =∅.

Theorem 3.14. For the solution sets of (FO),

⋃_α∈[0,1]OS_α (f)⊆ WPS (f) ;

⋃_α∈[0,1]IOS_α (f)⊆ SPS (f) ;

⋃_α∈[0,1]SOS_α (f) ⊆ SPS (f) .

Proof. For any fixed number α₀ ∈ [0, 1], suppose x^* ∈ OS_{α
₀} (f). If there exists an x ∈ Ω such that f (x) ≺ f (x^*), i.e., $[f (x)]^{α} ≨ [f (x^{*})]^{α}$ for all α ∈ [0, 1], then we have $f (x) ⋨_{α_{0}} f (x^{*})$ . This contradicts with x^* ∈ OS_{α
₀} (f). Consequently, we get x^* ∈ WPS (f). We have proved assertion (1). Similarly, we can prove the assertions (2) and (3).

Now we show that the inclusions in Theorem 3.14 may be strict and can not be improved further by the following examples.

Example 3.15. Let Ω = {1, 2, 3} and f : Ω → ℱ be a fuzzy function such that [f (1)] ^α = [2α - 1, 1], [f (2)] ^α = [0, 2 - α] for all α ∈ [0, 1], and ${[f (3)]}^{α} = {\begin{matrix} [2 α - 1, 1], & if α \in [0, 0.5], \\ [α - 0.5, 1.5 - α], & if α \in (0.5, 1] . \end{matrix}$

Then we have ∪_α∈[0,1]OS_α (f) = {1, 3}, WPS (f) = Ω and PS (f) = {2, 3} which implies that $⋃_{α \in [0, 1]} O S_{α} (f) \subset WPS (f),$ but $⋃_{α \in [0, 1]} O S_{α} (f) ⊈ PS (f) .$

Example 3.16. Let Ω = {1, 2, 3} and f : Ω → ℱ be a fuzzy function such that [f (1)] ^α = [1, 4] for all α ∈ [0, 1], ${[f (2)]}^{α} = {\begin{matrix} [0, 4], & if α \in [0, 0.5], \\ [2, 4], & if α \in (0.5, 1], \end{matrix}$ and ${[f (3)]}^{α} = {\begin{matrix} [0, 4], & if α \in [0, 0.5], \\ [3, 4], & if α \in (0.5, 1] . \end{matrix}$

Then we get that $⋃_{α \in [0, 1]} IO S_{α} (f) = {1} \subset {1, 2} = SPS (f) .$

Example 3.17. Let Ω = {1, 2, 3} and f : Ω → ℱ be a fuzzy function such that ${[f (1)]}^{α} = {\begin{matrix} [0, 7], & if α \in [0, 0.5], \\ [4, 7], & if α \in (0.5, 1], \end{matrix}$ ${[f (2)]}^{α} = {\begin{matrix} [1, 8], & if α \in [0, 0.5], \\ [3, 6], & if α \in (0.5, 1], \end{matrix}$ and ${[f (3)]}^{α} = {\begin{matrix} [2, 9], & if α \in [0, 0.5], \\ [2, 5], & if α \in (0.5, 1] . \end{matrix}$

Then we get that $⋃_{α \in [0, 1]} SO S_{α} (f) = {1, 3} \subset {1, 2, 3} = SPS (f) .$

Theorem 3.18. For the solution sets of (FO), $⋂_{α \in [0, 1]} P S_{α} (f) \subseteq PS (f) .$

Proof. Suppose x^* ∈ ∩ _α∈[0,1]PS_α (f). If there exists an x ∈ Ω such that $f (x) ⋨ f (x^{*})$ , then there exists an α₀ ∈ [0, 1] such that $[f (x)]^{α_{0}} ≨ [f (x^{*})]^{α_{0}}$ . Thus we have $f (x) ⋨_{α_{0}} f (x^{*})$ . This contradicts with x^* ∈ PS_{α
₀} (f). Consequently, we have x^* ∈ PS (f).

Here we give an example for which the inclusion may be strict in Theorem 3.18.

Example 3.19. Let Ω = {1, 2, 3} and f : Ω → ℱ be a fuzzy function such that [f (1)] ^α = [0, 6] for all α ∈ [0, 1], ${[f (2)]}^{α} = {\begin{matrix} [1, 6], & if α \in [0, 0.5], \\ [3, 5], & if α \in (0.5, 1], \end{matrix}$ and ${[f (3)]}^{α} = {\begin{matrix} [2, 6], & if α \in [0, 0.5], \\ [2, 4], & if α \in (0.5, 1] . \end{matrix}$

Then we get that $⋂_{α \in [0, 1]} P S_{α} (f) = {1} \subset {1, 2, 3} = PS (f) .$

Theorem 3.20. For the solution sets of (FO),

⋃_α∈[0,1]WPS_α (f)⊆ APS (f) ;

⋃_α∈[0,1]SPS_α (f)⊆ SPS (f) ;

⋃_α∈[0,1]PS_α (f) ⊆ WPS (f) .

Proof. We only prove the first assertion. The other assertions can be proved similarly. For any fixed number α₀ ∈ [0, 1], suppose x^* ∈ WPS_{α
₀} (f). If there exists an x (≠ x^*) ∈ Ω such that f (x) ⪡ f (x^*), then by Definitions 2.2 and 2.3 we have f (x) ≺ _{α
₀}f (x^*). This contradicts with x^* ∈ WPS_{α
₀} (f). Consequently, we have x^* ∈ APS (f).

The following examples show that the inclusions may be strict and can not be improved further in Theorem 3.20.

Example 3.21. Let Ω = {1, 2, 3} and f : Ω → ℱ be a fuzzy function such that ${[f (1)]}^{α} = {\begin{matrix} [3, 10], & if α \in [0, 0.5], \\ [5, 7], & if α \in (0.5, 1], \end{matrix}$ ${[f (2)]}^{α} = {\begin{matrix} [2, 10], & if α \in [0, 0.5], \\ [4, 6], & if α \in (0.5, 1], \end{matrix}$ and ${[f (3)]}^{α} = {\begin{matrix} [1, 10], & if α \in [0, 0.5], \\ [4, 6], & if α \in (0.5, 1] . \end{matrix}$

Then we have $⋃_{α \in [0, 1]} WP S_{α} (f) = {1, 2, 3} = APS (f),$ but PS (f) = {3} and WPS (f) = {2, 3}.

However, if let Ω = {1, 2, 3, 4} and define another fuzzy function g : Ω → ℱ as g (1) = f (1), g (2) = f (2), g (3) = f (3), and ${[g (4)]}^{α} = {\begin{matrix} [0, 9], & if α \in [0, 0.5], \\ [5, 8], & if α \in (0.5, 1] . \end{matrix}$

Now we obtain that $⋃_{α \in [0, 1]} WP S_{α} (g) = {2, 3, 4} \subset Ω = APS (g) .$

Example 3.22. Let Ω = {1, 2, 3} and f : Ω → ℱ be a fuzzy function such that [f (1)] ^α = [3α - 2, 1], [f (2)] ^α = [-1, 2 - 3α] for all α ∈ [0, 1], and ${[f (3)]}^{α} = {\begin{matrix} [3 α - 2, 1], & if α \in [0, 0.5], \\ [2 α - 2, 2 - 2 α], & if α \in (0.5, 1] . \end{matrix}$

Then we get that $⋃_{α \in [0, 1]} SP S_{α} (f) = {2} \subset {2, 3} = SPS (f) .$

Example 3.23. Let Ω = {1, 2} and f : Ω → ℱ be a fuzzy function such that ${[f (1)]}^{α} = {\begin{matrix} [0, 3], & if α \in [0, 0.5], \\ [2, 3], & if α \in (0.5, 1], \end{matrix}$ and ${[f (2)]}^{α} = {\begin{matrix} [0, 3], & if α \in [0, 0.5], \\ [0, 1], & if α \in (0.5, 1] . \end{matrix}$

Then we get that $⋃_{α \in [0, 1]} P S_{α} (f) = {1, 2} = WPS (f),$ but PS (f) = {2} .

However, if let Ω = {1, 2, 3} and define another fuzzy function g : Ω → ℱ as [g (1)] ^α = [1, 3] for all α ∈ [0, 1], ${[g (2)]}^{α} = {\begin{matrix} [1, 3], & if α \in [0, 0.5], \\ [2, 3], & if α \in (0.5, 1], \end{matrix}$ and ${[g (3)]}^{α} = {\begin{matrix} [0, 3], & if α \in [0, 0.5], \\ [2, 3], & if α \in (0.5, 1], \end{matrix}$

Now we obtain that $⋃_{α \in [0, 1]} P S_{α} (g) = {1, 3} \subset Ω = WPS (g) .$

Corollary 3.24. Let f : Ω → ℱ be a fuzzy function and x^* ∈ Ω.

If there exists an α₀ such that f (x^*) _L (α₀) < f (x) _L (α₀) for all x (≠ x^*) ∈ Ω, then x^* ∈ SPS (f);

If there exists an α₀ such that f (x^*) _L (α₀) ≤ f (x) _L (α₀) for all x (≠ x^*) ∈ Ω, then x^* ∈ APS (f).

Proof. (1) By the hypothesis, we know that there exists an α₀ ∈ [0, 1] such that f (x^*) _L (α₀) < f (x) _L (α₀) for all x (≠ x^*) ∈ Ω. For any x (≠ x^*) ∈ Ω, either f (x^*) _R (α₀) ≤ f (x) _R (α₀), we have $f (x^{*}) ⋨_{α_{0}} f (x)$ ; or f (x^*) _R (α₀) > f (x) _R (α₀), we have the two fuzzy numbers f (x^*) and f (x) are not comparable with respect to ≼_{α
₀}. Thus there exists no x (≠ x^*) ∈ Ω such that f (x) ≼ _{α
₀}f (x^*) which implies x^* ∈ SPS_{α
₀} (f). Consequently, from Theorem 3.20 it follows that x^* ∈ SPS (f).

(2) Similarly, by the hypothesis we know that there exists an α₀ ∈ [0, 1] such that f (x^*) _L (α₀) ≤ f (x) _L (α₀) for all x (≠ x^*) ∈ Ω. For any x (≠ x^*) ∈ Ω, if f (x^*) _R (α₀) ≤ f (x) _R (α₀), we have f (x^*) ≼ _{α
₀}f (x); if f (x^*) _L (α₀) = f (x) _L (α₀) and f (x^*) _R (α₀) > f (x) _R (α₀), we have $f (x) ⋨_{α_{0}} f (x^{*})$ but $f (x) ⋨_{α_{0}} f (x^{*})$ ; if f (x^*) _L (α₀) < f (x) _L (α₀) and f (x^*) _R (α₀) > f (x) _R (α₀), we have the two fuzzy numbers f (x^*) and f (x) are not comparable with respect to ≼_{α
₀}. Thus there exists no x (≠ x^*) ∈ Ω such that f (x) ≺ _{α
₀}f (x^*) which implies x^* ∈ WPS_{α
₀} (f). Consequently, from Theorem 3.20 it follows that x^* ∈ APS (f).

Corollary 3.25. Let f : Ω → ℱ be a fuzzy function and x^* ∈ Ω.

If there exists an α₀ such that f (x^*) _R (α₀) < f (x) _R (α₀) for all x (≠ x^*) ∈ Ω, then x^* ∈ SPS (f);

If there exists an α₀ such that f (x^*) _R (α₀) ≤ f (x) _R (α₀) for all x (≠ x^*) ∈ Ω, then x^* ∈ APS (f).

Theorem 3.26. For the solution sets of (FO),

AOS (f)⊆ SOS (f) ⊆ SOS_m (f) ;

OS (f)⊆ OS_m (f) ;

IOS (f) ⊆ IOS_m (f) .

Moreover, if the set AOS (f) is nonempty, then

AOS (f) = SOS (f) = {SOS}_{m} (f),

and it is a singleton set. The same conclusion also holds true for the sets SOS (f) and IOS (f).

Proof. Since for any x ∈ Ω and α ∈ [0, 1], we have $f (x)_{M} (α) = \frac{{f (x)}_{L} (α) + {f (x)}_{R} (α)}{2} .$

Consequently, for any x, x^* ∈ Ω and α ∈ [0, 1] we obtain that

if ${\begin{matrix} f (x^{*})_{L} (α) \leq f (x)_{L} (α), \\ f (x^{*})_{R} (α) < f (x)_{R} (α), \end{matrix}$ or ${\begin{matrix} f (x^{*})_{L} (α) < f (x)_{L} (α), \\ f (x^{*})_{R} (α) \leq f (x)_{R} (α), \end{matrix}$ then f (x^*) _M (α)< f (x) _M (α) ;

if ${\begin{matrix} f (x^{*})_{L} (α) \leq f (x)_{L} (α), \\ f (x^{*})_{R} (α) \leq f (x)_{R} (α), \end{matrix}$ then f (x^*) _M (α) ≤ f (x) _M (α) . Therefore, the inclusions follow immediately from Definitions 3.1 and 3.3.

We omit the proof of the following result because it is actually equivalent to Theorem 4 in [6].

Theorem 3.27. [6] For the solution sets of (FO), $O S_{m} (f) \subseteq PS (f) .$

An example for which the inclusion will become strict in Theorem 3.27 is shown as follows.

Example 3.28. Let Ω = {1, 2, 3} and f : Ω → ℱ be a fuzzy function such that [f (1)] ^α = [α - 1, 1 - α], [f (2)] ^α = [α - 0.5, 1.5 - α] and [f (3)] ^α = [4α - 2, 2] for all α ∈ [0, 1]. Then we get that $O S_{m} (f) = {1} \subset {1, 3} = PS (f) .$

Theorem 3.29. For the solution sets of (FO), $SP S_{m} (f) \subseteq PS (f) .$

Proof. Suppose x^* ∈ SPS_m (f). If there exists an x ∈ Ω such that $f (x) ⋨ f (x^{*})$ , then for all α ∈ [0, 1] we have ${\begin{matrix} f (x)_{L} (α) \leq f (x^{*})_{L} (α), \\ f (x)_{R} (α) \leq f (x^{*})_{R} (α) . \end{matrix}$

Consequently, we get that f (x) _M (α) ≤ f (x^*) _M (α) for all α ∈ [0, 1], i.e., f (x) ≼ _mf (x^*), which contradicts with x^* ∈ SPS_m (f). Thus we have $SP S_{m} (f) \subseteq PS (f) .$

We give an example for which the inclusion in Theorem 3.29 becomes strict.

Example 3.30. Let Ω = {1, 2} and f : Ω → ℱ be a fuzzy function such that [f (1)] ^α = [α - 1, 2 - 2α] and [f (2)] ^α = [0, 1.5 - 1.5α] for all α ∈ [0, 1]. Then we get that $SP S_{m} (f) = {1} \subset {1, 2} = PS (f) .$

Theorem 3.31. For the solution sets of (FO), we have $P S_{m} (f) \subseteq PS (f) .$

Proof. Suppose x^* ∈ PS_m (f). If there exists an x ∈ Ω such that $f (x) ⋨ f (x^{*})$ , we have that for all α ∈ [0, 1], ${\begin{matrix} f (x)_{L} (α) \leq f (x^{*})_{L} (α), \\ f (x)_{R} (α) \leq f (x^{*})_{R} (α), \end{matrix}$ and there exists one α₀ ∈ [0, 1] such that ${\begin{matrix} f (x)_{L} (α_{0}) \leq f (x^{*})_{L} (α_{0}), \\ f (x)_{R} (α_{0}) < f (x^{*})_{R} (α), \end{matrix}$ or ${\begin{matrix} f (x)_{L} (α_{0}) < f (x^{*})_{L} (α_{0}), \\ f (x)_{R} (α_{0}) \leq f (x^{*})_{R} (α_{0}) . \end{matrix}$

Consequently, we get that f (x) _M (α) ≤ f (x^*) _M (α) for all α ∈ [0, 1] and f (x) _M (α₀) < f (x^*) _M (α₀), i.e., $f (x) ⋨_{m} f (x^{*})$ , which contradicts with x^* ∈ PS_m (f). Thus we have PS_m (f) ⊆ PS (f) .

The inclusion can become strict in Theorem 3.31.

Example 3.32. Let Ω = {1, 2, 3} and f : Ω → ℱ be a fuzzy function such that [f (1)] ^α = [4α, 4], [f (2)] ^α = [1, 5 - 4α] and [f (3)] ^α = [6α - 1, 9 - 4α] for all α ∈ [0, 1]. Then we get that $P S_{m} (f) = {1, 2} \subset {1, 2, 3} = PS (f) .$

Theorem 3.33. For the solution sets of (FO), we have $WP S_{m} (f) \subseteq WPS (f) .$

Proof. Suppose x^* ∈ WPS_m (f). If x^* ∉ WPS (f), there exists an x (≠ x^*) ∈ Ω such that f (x) ≺ f (x^*), i.e., ${\begin{matrix} f (x)_{L} (α) \leq f (x^{*})_{L} (α_{0}), \\ f (x)_{R} (α) < f (x^{*})_{R} (α), \end{matrix}$ or ${\begin{matrix} f (x)_{L} (α) < f (x^{*})_{L} (α_{0}), \\ f (x)_{R} (α) \leq f (x^{*})_{R} (α_{0}), \end{matrix}$ for all α ∈ [0, 1]. Thus we have f (x) ≺ _mf (x^*) which contradicts with x^* ∈ WPS_m (f). Therefore we have WPS_m (f) ⊆ WPS (f) .

The following example shows that the inclusion in Theorem 3.33 may become strict.

Example 3.34 Let Ω = {1, 2, 3} and f : Ω → ℱ be a fuzzy function such that [f (1)] ^α = [2α + 1, 3], [f (2)] ^α = [2 + α, 4 - α] and [f (3)] ^α = [4α, 8 - 4α] for all α ∈ [0, 1]. Then we get that

WP S_{m} (f) = {1, 2} \subset {1, 2, 3} = WPS (f),

but PS (f) = {1, 3}.

4 Conclusions

To reveal the intrinsic properties of fuzzy optimizations, we present some theorems to give detailed insight in the relationships among those different types of solutions with respect to the fuzzy-max order and two other related partial order relations. This study is complete since the converse implications are disproven by illustrative counterexamples whenever required. From these obtained results, we can see that though the fuzzy optimization problem can be decomposed as a series of optimization problems with interval-valued objective function or can derive some non-fuzzy optimization problems, they are not equivalent (see Theorems 3.14, 3.18, 3.20, 3.31 and 3.33). Thus there would be some interesting findings that appear when one considers fuzzy optimizations [2 , 20].

Footnotes

Acknowledgments

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

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