Multicriteria Choice Based on Fuzzy Information

Abstract

This paper proposes a new method for solving the problem of multicriteria optimization of a numerical vector function on a fuzzy set. The membership function of a fuzzy feasible set is joined to the original set of criteria that allows the original problem of multi-criteria optimization to be treated as the task of finding a suitable compromise (Pareto-optimal) solution with respect to an extended set of criteria. It is assumed that in a search for the “best” compromise solution there is only fuzzy information about the preferences of decision maker in the form of information quanta. At the first stage of the proposed method, the search for a compromise is made on the basis of an axiomatic approach, with which the Pareto set is reduced. The result of the reduction is a fuzzy set with a membership function, which is determined on the basis of the used fuzzy information. At the second stage, the obtained membership function is added to the extended set of criteria, after which the scalarization procedure based on the idea of goal programming is used to solve the formed multicriteria problem.

APPENDIX

Proof of Theorem 1. We choose an arbitrary interior point ${{z}^{0}} \in \text{int}Z$. There exists $\varepsilon > 0$ at which ${{U}_{\varepsilon }}({{z}^{0}}) \subset Z$ where ${{U}_{\varepsilon }}({{z}^{0}})$ – is the ε neighborhood of the point ${{z}^{0}}$. Due to invariance property without loss of generality, we can put ${{z}^{0}} = {{0}_{{m + 1}}}$.

Let us denote the membership function of a fuzzy binary relation $\succ$ by ${{\mu }_{Z}}$. We introduce a fuzzy set $\hat {Z}$ with membership function ${{\lambda }_{{\hat {Z}}}}:\,{{R}^{{m + 1}}} \to [0,1]$ defined by the equality

$${{\lambda }_{{\hat {Z}}}}(\hat {z}) = \left\{ \begin{gathered} {{\mu }_{Z}}{\text{(}}\hat {z}{{,0}_{{m + 1}}}{\text{)}}{\text{,}}\,\,\,{\text{if}}\,\,\,\hat {z} \in {{U}_{\varepsilon }}{\text{(}}{{0}_{{m + 1}}}{\text{)}} \hfill \\ {\text{and}}\,\,{{\mu }_{{\text{Z}}}}(\hat {z}{{,0}_{{m + 1}}}) > 0 \hfill \\ {\text{0}}{\text{,}}\,\,\,{\text{otherwise}} \hfill \\ \end{gathered} \right\}$$

as well as the fuzzy set generated by it K with membership function:

$${{\lambda }_{K}}(z) = \left\{ \begin{gathered} {{\lambda }_{{\hat {Z}}}}(\hat {z}),\,\,\,{\text{if}}\,\,z = \alpha \cdot \hat {z}\,\,{\text{for some}} \hfill \\ \alpha > 0\,\,\,{\text{and}}\,\,\hat {z} \in {\text{Supp(}}\hat {Z}) \hfill \\ 0,\,\,\,{\text{otherwise}} \hfill \\ \end{gathered} \right\}.$$

Due to the homogeneity of the function ${{\mu }_{Z}}$, the set K is a fuzzy cone. This cone does not contain the origin due to the irreflexivity of the relation $\succ$. We establish that this cone is acute and convex. In fact, if the given cone is not acute, then a nonzero vector $z = \alpha \cdot \hat {z}$ must be found for some $\alpha > 0$ and $\hat {z} \in {\text{Supp(}}\hat {Z})$ such that ${{\lambda }_{K}}(\alpha \cdot \hat {z}) > 0$ and ${{\lambda }_{K}}( - \alpha \cdot \hat {z}) > 0$. In this case, the number $\alpha$ can be chosen to be so small that both inequalities ${{\lambda }_{K}}(\alpha \cdot \hat {z})$ = ${{\lambda }_{K}}(z) = {{\mu }_{Z}}(z{{,0}_{{m + 1}}}) > 0$ and ${{\lambda }_{K}}( - \alpha \cdot \hat {z}) = {{\lambda }_{K}}( - z)$ = $\mu ({{0}_{{m + 1}}},z) > 0$ hold with some $z \in Z$. Using transitivity $\succ$, from here we arrive at the inequality $\mu (z,z) > 0$, which contradicts the irreflexivity of the relation $\succ$ on Z.

The convexity of the fuzzy cone K follows from the convexity of a fuzzy set $\hat {Z}$, which, in turn, occurs due to the invariance and transitivity of the fuzzy relation $\succ$ on the Z. To prove the convexity of the set $\hat {Z}$ we choose two points arbitrarily ${{\hat {z}}^{1}},{{\hat {z}}^{2}} \in \hat {Z}$ and number $\theta \in [0,\,1]$. We have ${{\hat {z}}^{1}},\,{{\hat {z}}^{2}} \in {{U}_{\varepsilon }}({{0}_{{m + 1}}})$, Moreover,

$$\begin{gathered} \theta \cdot {{{\hat {z}}}^{1}} \in {{U}_{\varepsilon }}({{0}_{{m + 1}}}),\,\,\,\,(1 - \theta ) \cdot {{{\hat {z}}}^{2}} \in {{U}_{\varepsilon }}({{0}_{{m + 1}}}), \\ \theta \cdot {{{\hat {z}}}^{1}} + (1 - \theta ) \cdot {{{\hat {z}}}^{2}} \in {{U}_{\varepsilon }}({{0}_{{m + 1}}}). \\ \end{gathered} $$

The last inclusion is true due to ${{\hat {z}}^{1}},\,{{\hat {z}}^{2}} \in {{U}_{\varepsilon }}({{0}_{{m + 1}}})$ and the triangle inequality:

$$\left\| {\theta \cdot {{{\hat {z}}}^{1}} + (1 - \theta ) \cdot {{{\hat {z}}}^{2}}} \right\| \leqslant \theta \cdot \left\| {{{{\hat {z}}}^{1}}} \right\| + (1 - \theta ) \cdot \left\| {{{{\hat {z}}}^{2}}} \right\| < \varepsilon {\kern 1pt} .$$

If at least one of the equalities $\mu ({{\hat {z}}^{1}}{{,0}_{{m + 1}}}) = 0$ or $\mu ({{\hat {z}}^{2}}{{,0}_{{m + 1}}}) = 0$ holds then there is nothing to prove. Therefore, we will further assume that $\mu ({{\hat {z}}^{1}}{{,0}_{{m + 1}}}) > 0$ and $\mu ({{\hat {z}}^{2}}{{,0}_{{m + 1}}}) > 0$.

Using $(\theta - 1) \cdot {{\hat {z}}^{2}} \in {{U}_{\varepsilon }}({{0}_{{m + 1}}})$ and the transitivity of relation $\succ$ on the set ${{U}_{\varepsilon }}({{0}_{{m + 1}}})$ we obtain:

$$\begin{gathered} {{\mu }_{{\hat {Z}}}}(\theta \cdot {{{\hat {z}}}^{1}},(\theta - 1) \cdot {{{\hat {z}}}^{2}}) \geqslant \min\{ {{\mu }_{{\hat {Z}}}}(\theta \cdot {{{\hat {z}}}^{1}}{{,0}_{{m + 1}}}); \\ {{\mu }_{{\hat {Z}}}}({{0}_{{m + 1}}},(\theta - 1) \cdot {{{\hat {z}}}^{2}})\} . \\ \end{gathered} $$

Hence, based on the invariance of the same relation, it follows that

$$\begin{gathered} {{\mu }_{{\hat {Z}}}}(\theta \cdot {{{\hat {z}}}^{1}} + (1 - \theta ) \cdot {{{\hat {z}}}^{2}}{{,0}_{{m + 1}}}) \geqslant \min\{ {{\mu }_{{\hat {Z}}}}({{{\hat {z}}}^{1}}{{,0}_{{m + 1}}}); \\ {{\mu }_{{\hat {Z}}}}({{0}_{{m + 1}}}, - {{{\hat {z}}}^{2}})\} , \\ \end{gathered} $$

or

$${{\lambda }_{{\hat {Z}}}}(\theta \cdot {{\hat {z}}^{1}} + (1 - \theta ) \cdot {{\hat {z}}^{2}}) \geqslant \min\{ {{\lambda }_{{\hat {Z}}}}({{\hat {z}}^{1}});{{\lambda }_{{\hat {Z}}}}({{\hat {z}}^{2}})\} ,$$

which establishes the convexity of the set $\hat {Z}$.

A fuzzy acute convex cone K that contains no origin uniquely generates a fuzzy conical binary relation with this cone on ${{R}^{{m + 1}}}$. According to Lemma 7.2 [14], this relation has the properties of irreflexivity, transitivity, and invariance. Clearly, this relationship is an extension of the relationship $\succ$. The uniqueness of the extension follows from the uniqueness of the cone K generated by the set $\hat {Z}$.

Proof of Theorem 3. This theorem for Condition 1 was proved in [15]. Therefore, we continue the proof, assuming that Condition 2 is satisfied.

Necessity: let $x{\text{*}}$ be an arbitrary weakly efficient point. It is easy to understand that there is a positive number $\alpha$, where the vector $w = (\alpha ,\alpha ,...,\alpha ) \in {{R}^{{p + 1}}}$ satisfies inclusion $u = \hat {g}(x^*) + w \in U$. As proof, we suppose the contrary, i.e., there is a point $x \in {\text{Supp}}(X)$ for which $\mathop {\max}\limits_{i = 1,2,...,p + 1} ({{u}_{i}} - {{g}_{i}}(x^*))$ > $\mathop {\max}\limits_{i = 1,2,...,p + 1} ({{u}_{i}} - {{g}_{i}}(x))$. From here for every i it follows:

$$\mathop {\max}\limits_{i = 1,2,...,p + 1} ({{u}_{i}} - {{g}_{i}}(x^*)) > {{u}_{i}} - {{g}_{i}}(x).$$

Substituting to the both sides of the last inequality ${{u}_{i}} = {{g}_{i}}(x^*) + \alpha$ we obtain ${{g}_{i}}(x) < {{g}_{i}}(x^*),$ $i = 1,2,...,p + 1$, which contradicts the weakly efficiency of $x{\text{*}}$.

Sufficiency: we suppose that for vector u from Condition 2 of this theorem the inequality (4) holds. We use the opposite reasoning: point $x{\text{*}}$ is not weakly efficient, i.e., there is a point $x \in {\text{Supp}}(X)$ such that ${{g}_{i}}(x) > {{g}_{i}}(x^*),$ $i = 1,2,...,p + 1$, or ${{u}_{i}} - {{g}_{i}}(x^*) > {{u}_{i}} - {{g}_{i}}(x),$ $i = 1,2,...,p + 1$. This implies the inequality $\mathop {max}\limits_{i = 1,2,...,p + 1} ({{u}_{i}} - {{g}_{i}}(x^*))$ > ${{u}_{i}} - {{g}_{i}}(x)$ for any $i$, which means that $\mathop {\max }\limits_{i = 1,2,...,p + 1} ({{u}_{i}} - {{g}_{i}}(x^*))$ > $\mathop {\max }\limits_{i = 1,2,...,p + 1} ({{u}_{i}} - {{g}_{i}}(x))$contradicting (4).

Proof of Theorem 4. According to Theorem 4, the right-hand side of inclusion (5) is a set of vectors that correspond to weakly efficient points with respect to the vector function $\hat {g}$ on the set ${\text{Supp}}(X)$. Theorem 2 (see the Remark on this theorem) guarantees that $Max(\hat {Y}) \subset \hat {f}({{P}_{g}}({\text{Supp}}(X)))$. Because $\hat {g} = (g,{{g}_{{p + 1}}})$, the inclusion ${{P}_{g}}({\text{Supp}}(X)) \subset {{P}_{{\hat {g}}}}({\text{Supp}}(X))$ is true, and we have $\hat {f}({{P}_{g}}({\text{Supp}}(X))) \subset \hat {f}({{P}_{{\hat {g}}}}({\text{Supp}}(X)))$. Therefore $Max(\hat {Y}) \subset \hat {f}({{P}_{{\hat {g}}}}({\text{Supp}}(X)))$ is satisfied. However, the right-hand side of inclusion (5) includes $\hat {f}({{P}_{{\hat {g}}}}({\text{Supp}}(X)))$, since any Pareto-optimal point is weakly efficient with respect to the same vector function $\hat {g}$. Thus, inclusion (5) is proved.