Two-objective method for crisp and fuzzy interval comparison in optimization

Abstract

In real optimization we always meet two main groups of criteria: requirements of useful outcomes increasing or expenses decreasing and demands of lower uncertainty or, in other words, risk minimization. Therefore, it seems advisable to formulate optimization problem under conditions of uncertainty, at least, two-objective on the basis of local criteria of outcomes increasing or expenses reduction and risk minimization. Generally, risk may be treated as the uncertainty of obtained result. In the considered situation, the degree of risk (uncertainty) may be defined in a natural way through the width of final interval objective function at the point of optimum achieved. To solve the given problem, the two-objective interval comparison technique has been developed taking into account the probability of supremacy of one interval over the other one and relation of compared widths of intervals. To illustrate the efficiency of the proposed method, the simple examples of minimization of interval double-extreme discontinuous cost function and fuzzy extension of Rosenbrock's test function are presented.

Introduction

In practice, there are many optimization problems formulated using imprecise parameters. Frequently, such parameters may be considered as intervals or fuzzy number. As the consequence, the optimization tasks with interval or fuzzy interval cost function of real arguments are obtained. It must be emphasized that mentioned problems differ essentially from so called global interval optimization when real value objective functions are used [1], [2]. In addition, the cost function often cannot be performed by any set of analytical expressions, because it is given by an algorithm. Of course, in such cases, it is impossible to use gradient methods, but different kinds of numerical direct search methods may be successively applied. The procedure for direct numerical solving of optimization task can generally be presented as a sequence of searching steps, on each of which we try to receive the smaller/greater objective function value than on previous one. It is clear that in such cases we are faced with a problem of consecutive interval or fuzzy internal reduction. Unfortunately, even in case of successful resolving of the problem there are no any guarantees that the result will be obtained with minimal uncertainty. It is easy to see that in our case such uncertainty can be presented in natural way through the interval width of minimized/maximized objective function. In a nutshell, in the considered case we deal with two local criteria. In general, simultaneous satisfaction the such criteria in a maximal degree is usually impossible. Thereby, the problem is to construct a compromise criterion, which may be performed as the aggregation of local criteria. To permit it, at first we must build the local criterion for quantitative assessment of degree in which one interval—crisp or fuzzy—is greater than another one. Of course, it must be presented in a form applicable for all cases of intervals placement. The problem of crisp and fuzzy intervals (numbers) ordering is of perennial interest, because of its direct relevance in practical modelling and optimization of real world processes. Theoretically, fuzzy numbers can only be partially ordered and hence cannot be compared. However, when fuzzy numbers are used in practical applications or when a choice has to be made among alternatives, the comparison of fuzzy numbers becomes necessary. There are numbers of definitions of the ordering relation over fuzzy quantities (as well as crisp intervals) [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15]. In most cases, the authors use some quantitative indices. The values of such indices present degree to which one interval (fuzzy or crisp) is greater/less than another interval. In some cases, even several indices are used simultaneously. For example, in [9] four indices of inequality and three of equality are proposed. It was noted in reviews of the most known approaches [14], [15], [16], [29] that, although some of these methods have shown more consistency and better performance in difficult cases, no singlemethod of fuzzy interval comparison may be put forward as the best. The existing approaches to fuzzy interval comparison may be clustered into three groups: methods of only qualitative fuzzy and crisp intervals ordering [3], [4], [5], [6], [7], methods permitting quantitative ordering by means of some indices obtained from the base definitions of fuzzy sets theory [3], [8], [9] and methods based on representation of fuzzy number as $\alpha$-level sets [10], [11], [12], [13]. It must be emphasized, that the last group of methods has some advantages. At first, they permit to operate with all the types of membership functions without any restrictions. This feature is of large practical importance, but can be actually used in the case of numerical computation. Secondly, $\alpha$-levels are in essence the set of usual crisp intervals. Therefore, the powerful tools of interval arithmetic can also be used to solve the problem of fuzzy interval ordering. In this paper we present further development of such methods. The proposed approach based on $\alpha$-level representation of fuzzy intervals and probability estimation states that a certain interval is greater/equal than another interval. It is necessary to note that probabilistic approach was used only to infer the set of formulae for deterministic quantitative estimation of intervals inequality/equality. The method allows to compare interval and real number and to take into account (implicitly) the widths of ordered intervals. The idea is not in principle novel. But now we can cite only few works [23], [24], [25], [26], [27], [28] which are based on it. Using the probability approach in [23] the set of expressions for estimation of probability $P(A<B)$ have been obtained. The expressions for evaluating $P(A<B)$ in the cases of overlapping and inclusion (see Fig. 1) presented in [23] are the same as those presented in our papers [25], [26], [27], [28]. But probability $P(A=B)$ in [23] has been presented (in our notation) as $P(A=B)={\epsilon }^2/(W(A)W(B))$, where $W(A),W(B)$ are the widths of the intervals to be compared and $\epsilon$ is an arbitrary small number. In[23] $\epsilon \approx 0$ is proposed. In other words, in [23] implicitly stated that in any case $P(A=B)=0$. Since in practice there may be situations when, even intuitively, we feel that $P(A=B)>P(A<B)$, assuming $P(A=B)=0$ for all cases seems to be rather useless. Furthermore, there are no any considerations of real number and interval comparison in [23] as well as the fuzzy interval relations. Only so-called double interval relations are presented. In [24] the expression for $P(A⩽B)$ for the case of overlapping (see Fig. 1) has been obtained in the form, which is an exact equivalent to the equality equation for $P(A<B)$ that we have proposed in [25], [26], [27], [28]. It has been shown [25], [26], [27], [28] that different expressions for the calculation of $P(A<B)$ and $P(A=B)$ in both overlapping and inclusion cases must be used. For the inclusion case, in [24] proposed (in our notation) $P(A⩽B)=(2b_2-a_2-a_1)/(2(b_2-b_1))$. It is easy to see that in the asymptotic limit, when $a_1\to a_2\to b_2$ we get $P(A⩽B)\to 1$. However, it is impossible to explain earnestly such a result. There are no any considerations of real number and interval comparison in [24]. Some expressions for fuzzy interval relations are proposed in [24], but only for the Gaussian form of fuzzy intervals. In this paper, we propose the complete set of interval relations involving separated equality and inequality relations and comparisons of real numbers and intervals. The method for fuzzy interval comparison based on their $\alpha$-level representation and probability approach is presented too. However, our main purpose is to present the general method for optimization in the case of crisp or fuzzy interval objective function using local criteria which express explicitly the widths of compared interval as the measure of uncertainty of the result. They may also be considered as the risk minimizing criteria. In order to do so, the rest of this paper is set out as follows. InSections 2 and 3 we briefly describe our version of a probabilistic approach to crisp and fuzzy interval comparison, which doesn’t suffer from the drawback mentioned above. Section 4 describes the main features of a method for building a generalized criterion on the base of an interval cost function and on the uncertainty of the obtained result. In Section 5, the simple examples of minimization of interval-valued double-extreme discontinuous cost function with a real argument and a fuzzy extension of Rosenbrock's test function are presented.