Characterizations of the Solution Sets of Generalized Convex Fuzzy Optimization Problem

Definition 2.1

[21] A fuzzy number ũ is a fuzzy set with the following properties:

1. ũ is normal, i.e., there exists x₀ ∈ ℝⁿ such that ũ(x₀) = 1;

2. ũ is upper semi-continuous (u.s.c);

3. ũ(λx + (1 − λ)y) ⩾ min{ũ(x), ũ(y)} for all x, y ∈ ℝⁿ, λ ∈ [0, 1];

4. [ũ]⁰ is compact.

The family of fuzzy numbers is denoted by 𝔼. Clearly, ũ ∈ 𝔼 is a fuzzy number if and only if [ũ]^r can be represented by [ũ_*(r), ũ^*(r)] which is a nonempty compact convex subset for any r ∈ [0, 1], where ũ_*(r) denotes the left-hand endpoint of [ũ]^r and ũ^*(r) denotes the right-hand endpoint of [ũ]^r. Thus, a fuzzy number is determined by the endpoints of the interval [ũ_*(r), ũ^*(r)]. A precise number a ∈ ℝ is a special case of fuzzy number encoded as

In particular, the fuzzy number 0̃ is defined as 0̃(t) = 1 if t = 0, and 0̃(t) = 0 if t ≠ 0. Thus, a fuzzy number ũ can be identified by a parameterized triple {(ũ_*(r), ũ^*(r), r)∣r ∈ [0, 1]}.

The following lemma makes the connection between fuzzy numbers and their endpoint functions.

Lemma 2.1

[27] Assume that I = [0, 1], ũ_* : I → ℝ and ũ^* : I → ℝ satisfy the following conditions:

1. ũ_* : I → ℝ is a bounded increasing function;

2. ũ^* : I → ℝ is a bounded decreasing function;

3. ũ_*(1) ⩽ ũ^*(1);

4. lim_r→k⁻ ũ_*(r) = ũ_*(k) and lim_r→k⁻ ũ^*(r) = ũ^*(k) for 0 < k ⩽ 1;

5. lim_r→0⁺ ũ_*(r) = ũ_*(0) and lim_r→0⁺ ũ^*(r) = ũ^*(0).

Then, ũ : ℝ → I defined by ũ(x) = sup{r∣ũ_*(r) ⩽ x ⩽ ũ^*(r)} is a fuzzy number with parameterization given by {(ũ_*(r), ũ^*(r), r)∣r ∈ [0, 1]}. Moreover, if ũ : ℝ → I is a fuzzy number with parametrization given by {(ũ_*(r), ũ^*(r), r)∣r ∈ [0, 1]}, then the functions ũ_*(r) and ũ^*(r) satisfy the above conditions (i)-(v).

Triangular fuzzy numbers are a special type of fuzzy numbers which are defined by three real numbers a, b and c with a ⩽ b ⩽ c and we write ũ := 〈a, b, c〉 and [ũ]^r = [a + (b − a)r, c − (c − b)r], ∀ r ∈ [0, 1]. For given fuzzy numbers ũ, ṽ ∈ 𝔼 denoted by intervals [ũ_*(r), ũ^*(r)] and [ṽ_*(r), ṽ^*(r)] for any r ∈ [0, 1], respectively, and for any real number λ ∈ ℝ, we define the fuzzy addition ũ +̃ ṽ and scalar multiplication λũ as follows:

For ũ, ṽ ∈ 𝔼, λ ∈ ℝ, r ∈ [0, 1], we have [ũ+̃ṽ]^r = [ũ]^r+̃[ṽ]^r and [λũ]^r = λ [ũ]^r, i.e.,

Definition 2.2

[45] Let ũ, ṽ ∈ 𝔼. If there exists ω̃ ∈ 𝔼 such that ũ = ṽ+̃ω̃, then ω̃ is called the Hukuhara difference (H-difference, for short and denoted by ũ ⊖_Hṽ) of ũ and ṽ.

Remark 2.1

Clearly, if the H-difference ω̃ = ũ ⊖_Hṽ exists, then for any r ∈ [0, 1], ω̃_*(r) = (ũ ⊖_Hṽ)_*(r) = ũ_*(r) − ṽ_*(r) and ω̃^*(r) = (ũ ⊖_Hṽ)^*(r) = ũ^*(r) − ṽ^*(r). Moreover, we also have k(ũ ⊖_Hṽ) = kũ ⊖_H kṽ for any k ∈ ℝ₊.

Definition 2.3

[21] Let ũ, ṽ ∈ 𝔼.

1. ũ ≼ ṽ iff ũ_*(r) ⩽ ṽ_*(r) and ũ^*(r) ⩽ ṽ^*(r) for each r ∈ [0, 1];

2. If ũ ≼ ṽ and ṽ ≼ ũ, then ũ = ṽ;

3. ũ ≺ v iff ũ ≼ ṽ and there exists r₀ ∈ [0, 1] such that ũ_*(r₀) < ṽ_*(r₀) or ũ^*(r₀) < ṽ^*(r₀);

4. ũ and ṽ are comparable iff either ũ ≼ ṽ or ṽ ≼ ũ; otherwise they are non-comparable.

Note that ≼ is a partial order relation on 𝔼. It is sometimes convenient to write ṽ ≽ ũ (respectively, ṽ ≻ ũ) in place of ũ ≼ ṽ (respectively, ũ ≺ ṽ).

Remark 2.2

(i) Let [ũ]^r = [ũ_*(r), ũ^*(r)] for any r ∈ [0, 1]. Then, k[ũ]^r ≽ 0̃ iff kũ_*(r) ⩾ 0 and kũ^*(r) ⩾ 0 for any r ∈ [0, 1], where k ∈ ℝ₊ and 0̃ = [0, 0]. (ii) ũ ≼ ṽ iff ṽ_*(r) − ũ_*(r) ⩾ 0 and ṽ^*(r) − ũ^*(r) ⩾ 0 for any r ∈ [0, 1]. (iii) Let ũ ≼ ṽ. If λ ⩾ 0, then λũ ≼ λṽ. Accordingly, if λ < 0, then λũ ≽ λṽ.

Let f̃ : K(⊆ ℝⁿ) → 𝔼 be a fuzzy mapping. The r-cut of f̃ at x ∈ K, which is a closed and bounded interval, can be denoted by [f̃(x)]^r = f̃(x, r) := [f̃_*(x, r), f̃^*(x, r)] for any r ∈ [0, 1], where f̃_*(x, r) and f̃^*(x, r) are functions from K × [0, 1] to the set of real numbers ℝ. f̃_*(x, r) is a bounded increasing function of r and f̃^*(x, r) is a bounded decreasing function of r. Moreover, f̃_*(x, r) ⩽ f̃^*(x, r).

Definition 2.4

[36] Let f̃ : K → 𝔼 be a fuzzy mapping.

1. f̃ is called u.s.c at x₀ ∈ K iff both f̃_*(x, r) and f̃^*(x, r) are u.s.c at x₀ uniformly in r ∈ [0, 1]. f̃ is called u.s.c. on K iff it is u.s.c. at each point of K.

2. f̃ is called lower semicontinuous (l.s.c) at x₀ ∈ K iff both f̃_*(x, r) and f̃^*(x, r) are l.s.c at x₀ uniformly in r ∈ [0, 1]. f̃ is called l.s.c on K iff it is l.s.c at each point of K.

Lemma 2.2

[46] Let K be a nonempty closed and bounded subset of ℝⁿ. A real-valued function f : ℝⁿ → ℝ which is u.s.c (respectively, l.s.c) on K attains its maximum (respectively, minimum) on K.

Definition 3.1

[47] Let y ∈ K. Then the set K is called an α-invex at y with respect to (w.r.t., shortly) α and η iff for each x ∈ K, λ ∈ [0, 1], y + λα(x, y)η(x, y) ∈ K. K is called an α-invex set w.r.t. α and η iff K is α-invex at each y ∈ K.

Remark 3.1

Obviously, K is a convex set with α(x, y) = 1, η(x, y) = x − y and is an invex set in [48] with α(x, y) = 1 for all x, y ∈ K. However, the following example shows that the converse is not true.

Example 3.1

Let K = [−3, −2] ∪ [2, 5],

where x, y ∈ K. Obviously, K is an α-invex set w.r.t. α and η for each x, y ∈ K and λ ∈ [0, 1]. Indeed,

However, let x̄ = −2, ȳ = 5, λ̄ = 12 . We have ȳ + λ̄η(x̄, ȳ) = ȳ + λ̄(ȳ−2) = 132 ∉ K, which implies that K is not an invex set w.r.t. the same η.

Definition 3.2

[26] Let K be a nonempty convex subset of ℝⁿ. A fuzzy mapping f̃ : K → 𝔼 is called convex on K iff

Lemma 3.1

[31] Let K be a nonempty convex subset of ℝⁿ. f̃ is convex on K iff the endpoint functions f̃_*(x, r) and f̃^*(x, r) are convex on K, i.e.,

Based on Definition 3.2, Noor [49] introduced the preinvex fuzzy mapping which is a proper generalization of the convex fuzzy mapping.

Definition 3.3

[49] Let K be a nonempty invex set of ℝⁿ w.r.t. η. A fuzzy mapping f̃ : K → 𝔼 is called preinvex on K w.r.t. η iff

Lemma 3.2

[33] Let K be a nonempty invex set of ℝⁿ w.r.t. η. f̃ is preinvex on K w.r.t. η iff the endpoint functions f̃_*(x, r) and f̃^*(x, r) are preinvex on K w.r.t. η, i.e.,

Remark 3.2

When η(x, y) = x − y, Lemma 3.2 reduces to Lemma 3.1. It is clear that if the invex set K is not a convex set, then the fuzzy preinvexity of f̃ does not imply the fuzzy convexity of f̃. The following example shows that, even if K is convex, the fuzzy preinvexity of f̃ cannot imply the fuzzy convexity of f̃.

Example 3.2

Let K = ℝ and f̃ : K → 𝔼 be a fuzzy mapping defined by f̃(x) = 〈0, 1, 2〉 (−∣x∣). Let

It is easy to check that f̃ is preinvex on K w.r.t. η. However, there exist x̄ = 2, ȳ = −1, λ̄ = 12 and r₀ ∈ (0, 1) such that f̃_*(λ̄x̄ + (1 − λ̄)ȳ), r₀) = −12r0⪇−32 r₀ = λf̃_*(x, r₀) + (1 − λ)f̃^*(y, r₀). It follows from Lemma 3.1 that f̃ is not convex on K.

Definition 3.4

Let K be a nonempty α-invex set of ℝⁿ w.r.t. α and η. A fuzzy mapping f̃ : K → 𝔼 is called α-preinvex on K w.r.t. α and η iff

Similar to Lemma 3.1 and 3.2, we have the following property.

Proposition 3.1

Let K be a nonempty α-invex set of ℝⁿ w.r.t. α and η. f̃ is α-preinvex on K w.r.t. α and η iff the endpoint functions f̃_*(x, r) and f̃^*(x, r) are α-preinvex on K w.r.t. α and η, i.e.,

Remark 3.3

When α(x, y) = 1, Proposition 3.1 becomes Lemma 3.2. Furthermore, Proposition 3.1 becomes Lemma 3.1 when α(x, y) = 1 and η(x, y) = x − y. However, The following example shows that α-preinvexity of f̃ does not imply preinvexity of f̃. Thus, Definition 3.4 is a proper generalization of Definition 3.3.

Example 3.3

Let f̃ : K → 𝔼 be a fuzzy mapping defined by f̃(x) = 〈0, 1, 2〉 x, x ∈ K = [0, +∞). For any x, y ∈ K, let

1. Clearly, K is an α-invex set w.r.t. α and η. Moreover, K is an invex set w.r.t. the same η since y + λη(x, y) = 13λx+(1−12λ)y∈K.

2. It is easy to verify that f̃ is α-preinvex on K w.r.t. α and η. However, there exist r₀ ∈ (0, 1), x̄ = 3, ȳ = 5 and λ̄ = 12 such that f̃_*(ȳ + λ̄η̄(x̄, ȳ), r₀) = 174 r₀ > 4r₀ = λ̄f̃_*(x̄, r₀) + (1 − λ̄)f̃_*(ȳ, r₀). It follows from Lemma 3.2 that f̃ is not a preinvex fuzzy function on K w.r.t. the same η.

Nanda and Kar, [26], introduced the notion of a quasiconvex fuzzy mapping. Panigrahi in [28] pointed out that, however, the notion for finding the maximum of two fuzzy numbers has not been discussed in their paper. It may happen that two fuzzy numbers are not comparable. Therefore, Panigrahi defined a class of comparable and non-comparable fuzzy functions (see the following Definition 3.6) and furnished some reasonable examples (see Examples 3.9 and 3.10 in [28]). Furthermore, Panigrahi modified the definition of quasiconvex fuzzy mappings (see Definition 4.8 in [28]). Motivated by Panigrahi, [28], and Noor, [38], we now define the α-prequasiinvex fuzzy mapping.

Definition 3.5

[28] A fuzzy mapping f̃ : K → 𝔼 is called comparable iff f̃(x₁) and f̃(x₂) are comparable for every pair x₁ ≠ x₂ ∈ K. Otherwise, f̃ is called non-comparable. Let 𝓕 denote the set of all comparable fuzzy functions.

Definition 3.6

Let K be a nonempty α-invex set of ℝⁿ w.r.t. α and η. A fuzzy mapping f̃ : K → 𝔼 is called α-prequasiinvex on K w.r.t. α and η iff

where f̃(x) and f̃(y) are comparable.

Remark 3.4

If α(x, y) = 1, then Definition 3.6 reduces to the prequasiinvex fuzzy mapping introduced by Wu and Xu in [30]. Obviously, if the α-invex set K is not an invex set, then the fuzzy α-prequasiinvexity of f̃ cannot imply the fuzzy prequasiinvexity of f̃. The following example shows that, even if K is an invex set w.r.t. the same η, the α-prequasiinvexity of f̃ does not imply the prequasiinvexity of f̃, either. Therefore, the fuzzy α-prequasiinvexity is a proper generalization of the fuzzy prequasiinvexity.

Example 3.4

Let K = [0, +∞). For any x, y ∈ K, let

and f̃(x) = 〈0, 1, 2〉 x. Indeed, it follows from Definition 3.5 that f̃ is a comparable fuzzy function. Moreover, it is easy to verify that K is an α-invex set and f̃ is α-prequasiinvex fuzzy function on K w.r.t. α and η. On the other hand, K is also an invex set w.r.t. the same η. But there exist r₀ ∈ (0, 1), x̄ = 2, ȳ = 1 and λ̄ = 23 such that f̃_*(ȳ + λ̄η(x̄, ȳ), r₀) = 73 r₀ > 2r₀ = xr₀ = max{f̃_*(x̄, r₀), f̃_*(ȳ, r₀)}. Hence, f̃ is not prequasiinvex on K w.r.t. the same η.

Remark 3.5

It is not hard to see that the fuzzy α-preinvexity of f̃ implies that the fuzzy α-prequasiinvexity of f̃. However, the example shows that the fuzzy α-prequasiinvexity of f̃ does not imply the fuzzy α-preinvexity of f̃. Hence, the fuzzy α-prequasiinvexity is also a proper generalization of the fuzzy α-preinvexity.

Example 3.5

Let K = ℝ. For any x, y ∈ K, let

and η(x, y) = x − y. Then, we obtain, for all r ∈ [0, 1],

Clearly, K is an α-invex set w.r.t. α and η. Moreover, it is easy to check that f̃ is α-prequasiinvex on K w.r.t. α and η. However, there exist r₀ ∈ (0, 1), x̄ = 2, ȳ = −1 and λ̄ = 12 such that

It follows from Proposition 3.1 that f̃ is not α-prequasiinvex on K w.r.t. the same α and η.

Motivated by the fuzzy directional derivative in [30], we introduce the concept of the fuzzy αη-directional derivative by virtue of H-difference.

Definition 3.7

Let f̃ : K → 𝔼 be a fuzzy mapping, where K is an α-invex set w.r.t. α and η. If for any x, y ∈ K, there exists δ > 0 such that the H-difference f̃(y + λα(x, y)η(x, y))⊖_H{f̃(y)} exists for any real number λ ∈ (0, δ), and there exists h ∈ 𝔼 such that lim_λ→0⁺((f̃(y + λα(x, y)η(x, y))⊖_Hf̃(y))∖λ) = h, then f̃ is called fuzzy αη-directionally differentiable at y and h (denote f̃′(y; α(x, y)η(x, y))) is called fuzzy αη-directional derivative at y in the direction α(x, y)η(x, y).

Remark 3.6

It follows from Proposition 3.1 in [21] that if f̃ is fuzzy αη-directionally differentiable at y in the direction α(x, y)η(x, y), then for any fixed r ∈ [0, 1], f̃_*(x, r) and f̃^*(x, r) are αη-directionally differentiable at y in the direction α(x, y)η(x, y), and

where f~∗′ ((y; α(x, y)η(x, y)), r) and f̃^*′((y; α(x, y)η(x, y)), r) are respectively, the αη-directional derivatives of f̃_*(x, r) and f̃^*(x, r) at y in the direction α(x, y)η(x, y). That is,

The following example is given to illustrate Definition 3.7.

Example 3.6

Let K = {(x₁, x₂) ∈ ℝ²∣x₂ > x₁ > 0}. Let α(x, y) = p ∈ (0, 2), η(x, y) = x − 12 y for any x and y ∈ K. Define f~(x1,x2)=〈0,1,2〉x12+~〈0,1,2〉x22+~〈1,3,5〉 for any (x₁, x₂) ∈ K. Obviously, K is an α-invex set w.r.t. α and η. For any r ∈ [0, 1], we have

Thus, we obtain f~∗((x1,x2),r)=rx12+rx22+(1+2r) and f~∗((x1,x2),r)=(2−r)x12+(2−r)x22+(5−2r). Take x=(32,2) and y = (1, 2). By a direct calculation, we get f~∗′ ((y; α(x, y)η(x, y)), r) = 6pr and f̃^*′((y; α(x, y)η(x, y)), r) = −6pr + 12p. Hence, f̃′((y; α(x, y)η(x, y)), r) = [6pr, −6pr + 12p].

By applying the fuzzy αη-directional derivative, the concepts of the fuzzy α-pseudoinvexity and the fuzzy α-pseudomonotonicity are defined.

Definition 3.8

Let K be a nonempty α-invex set of ℝⁿ w.r.t. α and η. A fuzzy mapping f̃ : K → 𝔼 is called α-pseudoinvex on K w.r.t. α and η iff, for any x, y ∈ K,

The above implication is equivalent to the following implication:

The following example is used to illustrate Definition 3.8.

Example 3.7

Let K = {(x₁, x₂) ∈ ℝ²∣x₂ > x₁ > 0}. The fuzzy mapping f̃ : K → 𝔼 is defined by f̃(x₁, x₂) = 〈0, 1, 2〉 ⋅ x2x1 +̃ 〈1, 3, 5〉, ∀(x₁, x₂) ∈ K. Let α(x, y) = 2 and η(x, y) = x + y for any x, y ∈ K. Clearly, K is an α-invex set w.r.t. α and η. On the other hand, the endpoint functions of the fuzzy mapping f̃ are

It follows from a direct computation that

By Eqs. (3.1)-(3.4), f̃ is a fuzzy α-pseudoinvex mapping.

Definition 3.9

Let K be a nonempty α-invex set of ℝⁿ w.r.t. α and η. A fuzzy mapping f̃ : K → 𝔼 is called αη-pseudomonotone on K w.r.t. α and η iff, for any x, y ∈ K,

The above implication is equivalent to the following implication:

The following example will be used to illustrate Definition 3.9.

Example 3.8

In Example 3.7, we have

By Eqs. (3.1), (3.2), (3.5) and (3.6), f̃ is an αη-pseudomonotone mapping.

Definition 3.10

Let f̃ : ℝⁿ → 𝔼 be a fuzzy mapping.

1. f̃ is called positive homogeneous iff f̃(θ x) = θf̃(x) for any x ∈ ℝⁿ and θ > 0.

2. f̃ is called subodd iff f̃(x)+̃f̃(−x) ≽ 0̃ for any x ∈ ℝⁿ∖{0}.

Example 3.9

Let f̃ : X → 𝔼 be defined by f̃(x) = 〈0, 1, 2〉∣x∣. Then, f̃(x, r) = [rx,(2 − r)x] for any r ∈ [0, 1] and x ∈ X. If X := ℝⁿ, then f̃(θ x) = [r(θ∣x∣), (2 − r)(θ ∣x∣)] = θ[r∣x∣,(2 − r)∣x∣] = θf̃(x) for θ > 0. If X := ℝⁿ∖{0}, then f̃(x)+̃f̃(−x) = [2r∣x∣, 2(2 − r)∣x∣] ≽ 0̃ for any r ∈ [0, 1] and x ∈ ℝⁿ∖{0}.

Remark 3.7

The fuzzy αη-directional derivative is positive homogeneous w.r.t. the direction α(x, y)η(x, y).

Definition 3.11

Let f̃ : K → 𝔼 be a fuzzy mapping. f̃ is called fuzzy radially upper semicontinuous (r.u.s.c) iff the function φ̃(λ) := f̃(y + λα(x, y)η(x, y)) is u.s.c for any x, y ∈ K and λ ∈ [0, 1].

To obtain some results in Section 4 and 5, we will give the following assumption regarding the fuzzy function f̃. This assumption plays an important part in studying the properties of generalized convex fuzzy functions.

Assumption A

Let f̃ : K ⊆ ℝⁿ → 𝔼 satisfy the assumption

To understand the Assumption A, we consider the following example.

Example 3.10

In Example 3.3. Let x ⩾ y, in this case, we have f̃(y + α(x, y)η(x, y), r) = [r(16x+34y),(2−r)(16x+34y)] ≼ f̃(x, r), since

for any r ∈ [0, 1]. A similar result holds, if x < y.

We also need the following assumption regarding the functions α(⋅, ⋅) and η(⋅, ⋅).

Assumption B

[50] Let α(⋅, ⋅) : K × K → ℝ∖{0} and η(⋅, ⋅) : K × K → ℝⁿ satisfy the assumptions:

Remark 3.8

Clearly, η(y, y) = 0. If Assumption B holds and α(x, y) = α(y, y + λα(x, y)η(x, y)) for any x, y ∈ K, then we have η(y + λα(x, y)η(x, y), y) = λη(x, y). Indeed,

Lemma 3.3

[51] Let K be a nonempty α-invex set of ℝⁿ w.r.t. α and η. For any x, y ∈ K and λ ∈ [0, 1], if η(y, y + λα(x, y)η(x, y)) = −λη(x, y) and α(x, y) = α(y, y + λα(x, y)η(x, y)), then for any λ₁, λ₂ ∈ [0, 1] and λ₁ > λ₂, the following equalities hold,

Theorem 4.1

Let K be a nonempty α-invex set of ℝⁿ w.r.t. α and η. Suppose that the following conditions hold:

1. assumptions A and B are satisfied;

2. α satisfies

   α(x,y)=α(y,y+λα(x,y)η(x,y)),∀x,y∈K,λ∈[0,1].

   Then, the fuzzy mapping f̃ is α-preinvex on K w.r.t. α and η iff the fuzzy mapping φ̃(λ) := f̃(y + λα(x, y)η(x, y)) is convex on [0, 1].

Proof

Necessity. Suppose that the fuzzy mapping f̃ is α-preinvex on K w.r.t. α and η. In order to verify that the fuzzy mapping φ̃(λ) is convex on [0, 1], we need to show

From Eq. (4.1) and Lemma 3.1, we only need to prove that

By Lemma 3.3 and the α-preinvexity of f̃, we get

which means Eq. (4.2) holds. Similarly, Eq. (4.3) holds. Hence, φ̃(λ) is convex on [0, 1].

Sufficiency. Since Assumptions A and B hold, we have

Similarly, we have

It follows from Eqs. (4.4) and (4.5) that f̃ is α-preinvex on K w.r.t. α and η. □