Linear optimization with fuzzy variable over fuzzy polytope

Abstract

In the present paper a fuzzy optimization problem with fuzzy variable and fuzzy constraints is under discussion. The purpose of the paper consists in finding a fuzzy optimal solution for this problem for the linear case. A sensible attempt to solve a fuzzy optimization problem through reformulation to a crisp optimization problem using level-cuts of the fuzzy polytope is considered.

Keywords

Fuzzy optimization fuzzy polytope fuzzy goal programming fuzzy optimal solution

1 Introduction

The mathematical area of optimization plays a very important role in our modern life. In our age of scarce natural resources such as oil and gas, it is particulary important not to waste. This requires an optimal use of these supplies. The same applies to other limited resources as well a time. This strive for optimal solutions is not limited to industrial applications but has long reached our daily life.

Our quest is complicated by the fact that we more often than not have to make decisions without being able to rely on precise information. We have to deal with amount of uncertainty every single day.

In crisp optimization problems it is assumed that the decision-maker has exact and full information on the data entering the problem. Even when this is the case, the decision-maker usually finds it more convenient to express his / her knowledge in linguistic terms, i.e. through conventional linguistic variables (see e.g. [27 –29]), rather than by using high precision numerical data.

Moreover, our attitude to the linguistic variables often depends on e.g. mood or age. Let us consider the phrase “The shop is near the house”. In terms of fuzzy logic we understand word “near” differently. It strongly depends on the age of the decision making person (or decision-maker for short). Thus, the young decision-maker can say that the phrase is true, even if the distance from the house to the shop exceed 2 km. For the old person, the phrase holds true only if the distance is less than 800 m. Of course, in this example we have to define what words “young” and “old” mean. However, it becomes now clear that different persons define usual things different. Thus, it makes perfect sense to talk about fuzzy optimization problems from a vague predicate approach, as it is understood that this vagueness arises from the way we use to express the decision-makers’ knowledge and not from any random event. In short, it is supposed that the nature of the data defining the problem is fuzzy.

One commonly used approach to deal with these problems is to model them as fuzzy optimization problems, see e.g. [26]. This approach proved to be very useful in many applied sciences, such as engineering, economics, applied mathematics, physics, as well as in other disciplines: [4 , 30].

The problems usually considered in optimization are mathematical models and, thus, idealizations of real world problems. Therefore, the classical “achieve the best value of the objective function” approach may be too restrictive. Often a set of alternative solutions is more valuable to the decision-maker.

Many authors (see e.g. [15, 23]) try to find a single best solution of the fuzzy optimization problem in the linear case. These approaches are based on the extension principle of Bellman and Zadeh [1]. We suggest to reflect the uncertainty in fuzzy optimization problems through the existence of a set of optimal solutions [11]. Under the assumption that this set consists of more than one element, the decision-maker can improve the choice relying on some criteria that are not a priory considered in the optimization problem.

Therefore, it is natural to consider the solution of a fuzzy optimization problem as fuzzy. Hence, a criterion for comparing the elements of the fuzzy set of optimal solutions (i.e. fuzzy solution) is required. As soon as this fuzzy set has membership function, the possible criteria for comparison the elements of the fuzzy solution can be the values of the membership function. For this it is necessary to compute the values of the membership function exactly. An approach to calculate such membership function values is suggested in [7 –9] and further developed in [11] for fuzzy linear optimization problem with fuzzy objective and crisp constraints. The same idea we use in the present paper to find the membership function value of fuzzy linear optimization problem with both fuzzy objective and constraints.

With this innovative approach the decision-maker obtains a collection of some basic solutions, each accompanied by a measure of the extent to which it is the optimal solution of the initial fuzzy optimization problem. It is up to the decision-maker to make the final choice - the decision-maker can restrict himself / herself to the solutions which are equal to a degree greater than a fixed value, or the solutions which have membership function values greater than a fixed value, or equal to one, etc. In any case our fuzzy solution will constitute an important support and source of information for the decision-maker.

In the previous discussions we have considered optimization problems only with fuzzy objective functions [10 , 22]. In many cases, however, the constraints can also be fuzzy. A fuzzy optimization problem with fuzzy constraints have been examined for the linear case e.g. in [6 , 25] using the min-max approach and [3] using the possibilistic approach. The main target of this paper is to present a completely new idea of solution for fuzzy optimization problem with fuzzy objectives and fuzzy constraints in the linear case. A main difficulty in formulation of this problem consist in definition of a fuzzy polytope.

To make our discussions clear, it can be reasonable to outline it once more, that in early approaches it was sufficient to find a single solution with maximal membership function value. In our approach we are searching for the whole set of optimal solutions with its membership function. Thus, these approaches are poorly comparable. However, as a fuzzy optimization problem can be never solved directly, we propose in the present manuscript a proper reformulation of the problem saving the initial properties of fuzziness in the optimal set of optimal solutions. This approach is also used for solution linear fuzzy bilevel optimization problem with adopted k-th best algorithm [5].

As soon as we consider a fuzzy linear optimization problem, we suppose, that a solution can be found in one of the vertices of the fuzzy polytope. However, the membership function value of the polytope in a certain vertex can differ from the membership function value of the solution.

The above described ideas are presented as follows. To clarify a notion about a fuzzy polytope, significant definitions as respects to a fuzzy line and intersections of fuzzy lines are given in Section 2.

The solution method of the fuzzy optimization problem is based on taking level-cuts of the fuzzy polytope. Thus, for a fixed α-cut a pair of crisp polytopes is obtained and the fuzzy optimization problem can be splitted into two crisp optimization problems.

In previous discussions we have already derived that a solution of the problem, that has amount of uncertainty, cannot be exact [11]. Thus, a solution of the fuzzy optimization problem considered to be fuzzy. We take into account the inherently uncertain nature of the fuzzy optimization problem and consider all the solutions of the corresponding crisp problems simultaneously. Finally, we describe a fuzzy solution as a union of each crisp optimization problem for all level-cuts. Of course, a fuzzy solution has to be enriched with its membership function. Again, we are not interested in computation of the membership function itself, but only its values on some crucial points are a matter of interest. Taking into consideration fuzzy nature of the feasible set, they can easily be computed. When the membership function values of the elements of the set of fuzzy optimal solutions are known, it enables the decision-maker to make an educated choice. Our approach equips the decision-maker with a correlation among all significant solutions and quantitatively measure the advantage of his / her choice over other.

The formulation of the problem and the solution method are presented in Section 3.

In Section 4 an illustrative example is given.

2 Preliminaries

To identify points that belong to the same object (e.g. a fuzzy line), it is also necessary to define the concept of fuzzy connectedness.

Definition 1. [20] Given a fuzzy set of points $F^{2}$ (over $ℝ^{2}$ ). The degree of fuzzy connectedness of two points p and q within $F^{2}$ is defined as $C_{F^{2}} (p, q) = max [min μ_{F^{2}} (r)],$ where the maximum is taken over all paths connecting these points and the minimum is taken over all points r on each path.

Definition 2.[19] Two points p and q are said to be fuzzily connected in $F^{2}$ if $C_{F^{2}} (p, q) \geq min [μ_{F^{2}} (p), μ_{F^{2}} (q)] .$ In other words, two points are connected in a fuzzy set of points if there exists a path between them which is composed of only points which also belong to this fuzzy set.

This definition is also consistent with the concept of connectedness of two points within a crisp set whose membership values are all equal to 1.

Let us consider two fuzzy points $\tilde{p}$ and $\tilde{q}$ in $F^{2}$ with membership functions $μ_{\tilde{p}} (p)$ and $μ_{\tilde{q}} (q)$ , respectively.

Definition 3. A fuzzy line $\tilde{pq}$ which connects two points $\tilde{p} = (p, μ_{\tilde{p}} (p))$ and $\tilde{q} = (q, μ_{\tilde{q}} (q))$ is defined as a fuzzy set each of whose members is a linear combination of a pair of points p and q with the membership function defined as $μ_{\tilde{pq}} (r) = min {μ_{\tilde{p}} (p), μ_{\tilde{q}} (q)},$ (1) where $\tilde{pq} = (r, μ_{\tilde{pq}} (r))$ and r = λp + (1 - λ) q (λ ∈ [0, 1]). Moreover, for the elements p ₁ and p ₂ of the fuzzy point $\tilde{p}$ and q ₁ and q ₂ of the fuzzy point $\tilde{q}$ $p_{1} q_{1} ∣ ∣ p_{2} q_{2} and ∣ p_{1} q_{1} ∣ = ∣ p_{2} q_{2} ∣$ hold true.

A fuzzy line may be visualised as a centre line with a variable thickness (see Figs. 1 and 2). This thin area of space (or thin volume of space in $F^{3}$ (over $ℝ^{3}$ )) bounds a family of crisp lines which are formed by pairs of endpoints belonging to the two fuzzy sets of endpoints.

A fuzzy plane, which is an extension of a fuzzy line, is a thin planar shell with variable thickness. This shell encloses a family of crisp planes which is an extension of the family of crisp lines representing the fuzzy line. These concepts of fuzzy lines and planes encapsulate exact lines and exact planes as special cases.

Definition 4. [19] The intersection of two fuzzy lines $\tilde{pq}$ and $\tilde{rs}$ is a fuzzy point $\tilde{t}$ which is represented by a fuzzy set $\tilde{t} = (t, μ_{\tilde{t}} (t))$ , where $μ_{\tilde{t}} (t) = min {μ_{\tilde{pq}} (t), μ_{\tilde{rs}} (t)} .$

This Definition is illustrated on Fig. 3.

We can extend all aforesaid concepts to cover the intersection of a fuzzy line and a crisp plane, or of a fuzzy line and a fuzzy plane, or of two fuzzy planes, or of two fuzzy surfaces. Thus, the intersection of these geometry entities can be performed as two separate tasks:

Compute the intersection of pairs of crisp geometry entities (which belongs to the two families of fuzzy entities) in the same way as in crisp geometry.

Compute the membership value for each resulting entity.

Remark 1. There is no reason why all the definitions have to be formulated especially in $F^{2}$ . Thus, we extend them to $F^{n}$ .

A fuzzy traverse is composed of fuzzy vertices and fuzzy edges. Thus, it may be visualised as having edges of variable thickness, as described for fuzzy lines.

Definition 5.[16] Given a point set, its convex hull is the smallest convex set containing the given set.

Definition 6. Given a finite set of fuzzy points ${{\tilde{p}}^{1}, \dots$ $\dots, {\tilde{p}}^{n}}$ . A fuzzy traverse is a closed shape $\tilde{P} = conv {{\tilde{p}}^{1}, \dots, {\tilde{p}}^{n}} \ int (conv {{\tilde{p}}^{1}, \dots, {\tilde{p}}^{n}})$ composed of a finite sequence of fuzzy line / hyperplane segments (faces).

An illustration of the fuzzy traverse is presented on Fig. 4.

Remark 2. Membership functions of fuzzy lines $\tilde{p^{1} p^{2}}$ , $\tilde{p^{2} p^{3}}, \dots, \tilde{p^{n - 1} p^{n}}$ , $\tilde{p^{n} p^{1}}$ can be calculated using formula (1). In high dimensional space, membership functions of fuzzy hyperplanes are composed using convex linear combinations of the vertices belonging to this face.

Obviously, the fuzzy traverse can be given by a finite number of linear equations.

This concept is readily extended to that of a fuzzy polytope. The difference between the fuzzy traverse and the fuzzy polytope is that the fuzzy polytope includes its interior.

Definition 7. For any given finite set of fuzzy points ${{\tilde{p}}^{1}, \dots, {\tilde{p}}^{n}}$ , a fuzzy polytope is defined as $P = conv {{\tilde{p}}^{1}, \dots, {\tilde{p}}^{n}}$ . It can be composed into a fuzzy traverse $\tilde{P} = P \ int (P)$ and its interior $int (P)$ . The membership function $μ_{int (P)}$ of every point p of $int (P)$ is equal to $μ_{int (P)} (p) = sup_{p = \sum_{i = 1}^{n} ω_{i} p_{i}} min {μ_{{\tilde{p}}^{1}} (p_{1}), \dots, μ_{{\tilde{p}}^{n}} (p_{n})},$ where ω _i > 0 for i = 1, …, n, $\sum_{i = 1}^{n} ω_{i} = 1$ and n is a number of the vertices of the fuzzy polytope.

The illustration is given on Fig. 5. The fuzzy polytope can be given by a finite number of linear inequalities.

Definition 8. Extreme points of the fuzzy polytope are the fuzzy points on its boundary.

Since the fuzzy traverse $\tilde{P}$ can be considered as a fuzzy set, we can take its level-cut P (α) for some fixed α ∈ [0, 1]. That means, that we take the same α-cut of all fuzzy vertices ${\tilde{p}}^{1}, \dots, {\tilde{p}}^{n}$ , namely, $[p_{L}^{1} (α), p_{R}^{1} (α)]$ , …, $[p_{L}^{n} (α), p_{R}^{n} (α)]$ .

Remark 3. The number of vertices in the level-cuts of P (α) can be different for different α, i.e. we denote through $p_{L}^{k}$ and $p_{R}^{k}$ (k = 1, …, n) the sets of the vertices that belong to one fuzzy vertex ${\tilde{p}}^{k}$ . This is illustrated in Section 4.

Thus, we obtain two crisp traverses P _L (α) and P _R (α). The left-hand side traverse P _L (α) is a closed figure, composed with the use of left-hand side bounds of all fuzzy vertices in the same order as fuzzy traverse $\tilde{P}$ :

$\begin{matrix} P_{L} (α) & = & conv {p_{L}^{1} (α), \dots, p_{L}^{n} (α)} \ \\ \ int (conv {p_{L}^{1} (α), \dots, p_{L}^{n} (α)}) . \end{matrix}$ (2)

Analogously, the right-hand side traverse is also a closed figure. It can be written as

$\begin{matrix} P_{R} (α) & = & conv {p_{R}^{1} (α), \dots, p_{R}^{n} (α)} \ \\ \ int (conv {p_{R}^{1} (α), \dots, p_{R}^{n} (α)}) . \end{matrix}$ (3)

Definition 9. A level-cut of the fuzzy traverse $\tilde{P}$ for some fixed α ∈ [0, 1] is defined as $P (α) = conv {P_{R} (α)} \ int (conv {P_{L} (α)}) .$

Unlike a fuzzy traverse, a fuzzy polytope includes its interior. Thus, under the assumption that taking an α-cut we obtain a left- and a right-hand side polytope, denoted as $P_{L} (α)$ and $P_{R} (α)$ , respectively, we can write the following:

Definition 10. A level-cut of the fuzzy polytope $P$ for some fixed α ∈ [0, 1] is given by $P (α) = P_{R} (α) \ int (P_{L} (α)),$ where, by analogy to formulas (2) and (3), $P_{L} (α) = conv {p_{L}^{k}}$ and $P_{R} (α) = conv {p_{R}^{k}}$ (k = 1, …, n).

The α-cut of a fuzzy polytope is illustrated in Fig. 6.

Definition 11. A normalized fuzzy number $\tilde{c}$ is a fuzzy number with the membership function, that reaches value equal to 1: $max μ_{\tilde{c}} (x) = 1 \forall x \in ℝ .$

Remark 4. Under the assumption that all fuzzy vectors are defined by normalized fuzzy components with triangular [3] (or at least bell-shaped [12]) membership function, $P (1)$ is a crisp traverse.

Remark 5. $P_{L} (α) \subset P_{R} (α)$ for all α ∈ [0, 1).

Proposition 1. Let 0 ≤ α ₁ < α ₂ ≤ 1. Then $P (α_{2}) \subset P (α_{1})$ .

For further discussion let us make

Assumption 1. Assume that $p_{L}^{k} (α) \to p^{k} (1)$ and $p_{R}^{k} (α) \to p^{k} (1)$ for α ↑ 1 (see Fig. 7).

That means that we assume that all the vertices that belong to one fuzzy vertex ${\tilde{p}}^{k}$ converge to the corresponding vertex on the α-cut for α = 1 p ^k (1).

3 Formulation and solution method

We present a solution algorithm for the linear case of fuzzy optimization problem defined as

$\begin{matrix} F (\tilde{c}) & = & d^{⊤} \tilde{c} \to min \\ s . t . \tilde{c} \in P, \end{matrix}$ (4) where

$P \subset F^{n}$ is a fuzzy polytope;

d ≠ 0 is a known crisp vector in $ℝ^{n}$ ;

$\tilde{c}$ is a fuzzy variable in $F^{n}$ .

The fuzzy polytope $P$ is defined according to Definition 7 by a given set of fuzzy points ${{\tilde{p}}^{1}, \dots, {\tilde{p}}^{n}}$ .

For the solution of linear optimization problems with fuzzy objective function and fuzzy constraints we use the notions introduced in Section 2. Let us consider problem (4) on some α ∈ [0, 1]:

$\begin{matrix} F (c_{α}) & = & d^{⊤} c_{α} \to min \\ s . t . c_{α} \in P (α) \end{matrix}$ (5) Let us denote the set of optimal solutions of problem (5) through $Ψ (α) = \arg min {d^{⊤} c_{α}^{*} : c_{α}^{*} \in P_{α}} .$ Ψ (α) is a point-to-set mapping which maps α ∈ [0, 1] to the set of global optimal solutions of problem (5).

As we know, the level-cut of fuzzy polytope $P (α)$ can be decomposed into two crisp polytopes $P_{L} (α) = conv {p_{L}^{1} (α), \dots, p_{L}^{n} (α)}$ and $P_{R} (α) = conv {p_{R}^{1} (α), \dots, p_{R}^{n} (α)} .$ For these polytopes let us consider two optimization problems and then take the convex hull to obtain all solutions of problem (5) on the α-cut.

So, problem (5) can be split on

$\begin{matrix} F (c_{L} (α)) & = & d^{⊤} c_{L} (α) \to min \\ s . t . c_{L} (α) \in P_{L} (α) \end{matrix}$ (6) and

$\begin{matrix} F (c_{R} (α)) & = & d^{⊤} c_{R} (α) \to min \\ s . t . c_{R} (α) \in P_{R} (α), \end{matrix}$ (7) where c _L (α) and c _R (α) denotes the left- and right-hand side bounds of the α-cut of the fuzzy variable $\tilde{c}$ .

These problems, in turn, are crisp linear optimization problems, which can be solved using any of the standard methods (see e.g. [2]). Let the sets of optimal solutions of problems (6) and (7) be denoted as $c_{L}^{*} (α)$ and $c_{R}^{*} (α)$ , respectively. The optimal solutions of problems (6) and (7) $c_{L}^{*} (α)$ and $c_{R}^{*} (α)$ correspond to elements of the vertices $p_{L}^{k} (α)$ and $p_{R}^{k} (α)$ for some k = 1, …, n.

Let us consider problem (5) for α = 1:

$\begin{matrix} F (c_{1}) & = & d^{⊤} c_{1} \to min \\ s . t . c_{α} \in P (1), \end{matrix}$ (8) and let c ^* (1) be its optimal solution.

Theorem 1. $c_{L}^{*} (α) \to c^{*} (1)$ and $c_{R}^{*} (α) \to c^{*} (1)$ for α ↑ 1 (see Fig. 8).

Proof. The convergence in the formulation of thetheorem is caused by nothing but the following fact: $P_{L} (α) ⟶ P (1)$ and $P_{R} (α) ⟶ P (1)$ for α ↑ 1 (see Fig. 9) and Assumption 1 □

Within Definition 10 and Theorem 1 we obtain

Definition 12. The set of optimal solutions of optimization problem (5) is $c_{α}^{*} = conv {c_{L}^{*} (α), c^{*} (1)} \cup conv {c_{R}^{*} (α), c^{*} (1)} .$ (9)

Definition 13. A fuzzy optimal solution of problem (4) is a union of the optimal solutions of problem (5) for all α-cuts:∥ $\begin{matrix} {\tilde{c}}^{*} & = & ⋃_{α \in [0, 1]} (conv {c_{L}^{*} (α), c^{*} (1)} \cup \\ \cup conv {c_{R}^{*} (α), c^{*} (1)}) . \end{matrix}$ (10)

By analogy to linear crisp optimization, solution ${\tilde{c}}^{*}$ of the fuzzy optimization problem (4) is a subset of a vertex of the fuzzy polytope or of a fuzzy hyperplane defined by its vertices.

Hence, we write

Theorem 2. A fuzzy optimal solution of fuzzy linear optimization problem (4) is a subset of the set of extreme points of $P$ . In general, ${\tilde{c}}^{*} \subset conv {{\tilde{p}}^{k}}$ (k = 1, …, n) and the number of ${\tilde{p}}^{k}$ can not exceed n - 1, where n is a number of the vertices of the fuzzy polytope.

Proof. The proof is straightforward and follows directly from the convergence Theorem 1 and considerations for the crisp case. □

The fuzzy polytope $P$ has its own membership function $μ_{P}$ . Thus, for each component of the fuzzy solution we obtain its membership function value by straight forward calculation using

Definition 14. The membership function value of an element c ^* of fuzzy optimal solution ${\tilde{c}}^{*}$ of problem (4) can be calculated as∥ $\begin{matrix} μ_{{\tilde{c}}^{*}} (c^{*}) & = & ∣ {α ∣ c^{*} \in conv {c_{L}^{*} (α), c^{*} (1)} \cup \\ \cup conv {c_{R}^{*} (α), c^{*} (1)}, α \in [0, 1)} ∣ \end{matrix}$ (11) where |^:.:| stands for the geometric measure of the set.

Then, the final choice can be such an element c ^* of the fuzzy solution that has a maximal membership function value. However, other solutions can exist. This we can better explain in the next Section.

4 Example

Let us consider the fuzzy optimization problem

$\begin{matrix} F (\tilde{c}) & = & {\tilde{c}}_{1} + 2 {\tilde{c}}_{2} \to max \\ s . t . {\tilde{c}}_{1} + {\tilde{c}}_{2} ⪯ \tilde{7} \\ 0 ⪯ {\tilde{c}}_{1} ⪯ 3 \\ 0 ⪯ {\tilde{c}}_{2} ⪯ 5, \end{matrix}$ (12) where $\tilde{7} = (6, 7, 9)$ . These inequalities define the fuzzy polytope $P$ . The number of vertices of the crisp polytopes $P_{L} (α)$ and $P_{R} (α)$ obtained for different α-cuts depends on the value of α.

To discuss this situation seriatim, let us consider problem (12) for two different level-cuts: for α = 0.3 and α = 0.7. Solutions for these case are visualized in Figs. 10 and 11.

$\begin{matrix} F (c_{L} (0.3)) & = & c_{1 L} (0.3) + 2 c_{2 L} (0.3) \to max \\ s . t . c_{1 L} (0.3) + c_{2 L} (0.3) \leq 6.3 \\ 0 \leq c_{1 L} (0.3) \leq 3 \\ 0 \leq c_{2 L} (0.3) \leq 5 \end{matrix}$ (13) and

$\begin{matrix} F (c_{R} (0.3)) & = & c_{1 R} (0.3) + 2 c_{2 R} (0.3) \to max \\ s . t . c_{1 R} (0.3) + c_{2 R} (0.3) \leq 8.4 \\ 0 \leq c_{1 R} (0.3) \leq 3 \\ 0 \leq c_{2 R} (0.3) \leq 5 . \end{matrix}$ (14)

By analogy, for α = 0.7 we obtain

$\begin{matrix} F (c_{L} (0.7)) & = & c_{1 L} (0.7) + 2 c_{2 L} (0.7) \to max \\ s . t . c_{1 L} (0.7) + c_{2 L} (0.7) \leq 6.7 \\ 0 \leq c_{1 L} (0.7) \leq 3 \\ 0 \leq c_{2 L} (0.7) \leq 5 \end{matrix}$ (15) and

$\begin{matrix} F (c_{R} (0.7)) & = & c_{1 R} (0.7) + 2 c_{2 R} (0.7) \to max \\ s . t . c_{1 R} (0.7) + c_{2 R} (0.7) \leq 7.6 \\ 0 \leq c_{1 R} (0.7) \leq 3 \\ 0 \leq c_{2 R} (0.7) \leq 5 . \end{matrix}$ (16)

To compute the fuzzy optimal solution ${\tilde{c}}^{*}$ of problem (12) we have to take a convex hull for all α-cuts. And we obtain that ${\tilde{c}}^{*} = conv {(1, 5), (3, 5)}$ . And it is easy to see that for different α-cuts we have different Pareto optimal solutions. It is easy to see that solution $c_{1}^{*} = (2, 5)$ (that is obtained for α = 1) does not changes for all level-cuts and solution $c_{2}^{*} = (3, 5)$ stays optimal for all α ∈ [0, 0.5]. Thus, membership functions of the elements of the fuzzy solution $c_{1}^{*}$ and $c_{2}^{*}$ can be computed: $μ (c_{1}^{*}) = 1 and μ (c_{2}^{*}) = 0.5 .$

Remark 6. In this example it is easy to see that the maximal convex hull can be obtained for α = 0. However, it is also clear that it is not sufficient to consider α = 0 only.

5 Conclusions

In the paper the solution approach for fuzzy optimization problem (with both fuzzy objectives and constraints) is presented for the linear case. A solution approach is based on the definition of the fuzzy polytope. The approach is based on taking level-cuts. Thus, for each α-cut the initial fuzzy linear optimization problem is splitted into two crisp linear optimization problems. Then, considering all α-cuts a fuzzy optimal solution is found as a union of the convex hulls of corresponding optimal solutions.

A special attention is given to the computation of the membership function of the fuzzy solution of the fuzzy optimization problem in the linear case. Knowledge of the membership function values of the elements of the fuzzy optimal solution enables the decision-maker to make an educated choice. Moreover, using our approach, a decision-maker can see a correlation among solutions and quantitatively measure the advantage of his / her choice over other solutions.

To simplify a crisp choice of the decision-maker, for all crucial points of the fuzzy solution, corresponding membership function values can be found. This makes a choice of the decision-maker well-grounded.

This approach is quite innovative one and we try to extend it to nonlinear fuzzy optimization problems in our coming work.

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