Ordering of Fuzzy Numbers

This chapter describes different methods for comparing and ordering fuzzy numbers. Theoretically, fuzzy numbers can only be partially ordered, and hence cannot be compared. However, in practical applications, such as decision making, scheduling, market analysis or optimisation with fuzzy uncertainties, the comparison of fuzzy numbers becomes crucial. Theoretically, fuzzy numbers can only be partially ordered, and hence cannot be compared. However, when they are used in practical applications, e.g., when a decision must be made among alternatives or an optimal value of an objective function must be found, the comparison of fuzzy numbers becomes crucial. There are numerous approaches to the ordering relation between fuzzy numbers [1–6] qualitative, quantitative and based on α–cuts. Jain [7] and Dubois and Prade [4] were the first who considered this problem. Somemethods to rank fuzzy numbers were reviewed by Bortolan and Degani [8]. Detyniecki and Yager [3] proposed the α-weighted valuations of fuzzy numbers. Hong and Kim [9] proposed an easy way to compute the min and max operation for fuzzy numbers. Asady and Zendehnam [10] proposed the ranking fuzzy numbers by distance minimisation method. Comparison of various rankingmethods for fuzzy numberswith the possibility of ranking the crisp numbers was described by Thorani et al. [11]. The problem of comparing of fuzzy numbers was also considered by Allahviranloo et al. [12]. They proposed a method based on the centroid point of a fuzzy number and its area. Sevastjanov and Róg [13] developed a probability-based comparison of fuzzy numbers. The probabilistic approach was also considered in [14]. The large number of fuzzy ordering methods can be justified by the fact that different methods can be useful for different purposes. For example, problems involving ranking, prioritising or choosing between large number of alternatives will benefit from methods that assign to fuzzy numbers crisp values thus reducing the fuzzy ordering problem to ordering of real numbers. An overview of selected approaches to ordering (ranking) of fuzzy numbers is presented below. The presented approaches can be generally divided into two groups. The first group consists of methods which enable two fuzzy numbers to be compared. Included in this group are such methods as probabilistic approach, centroid point approach or radius of gyration approach. To order a set of fuzzy numbers using © Springer International Publishing Switzerland 2015 I. Skalna et al., Advances in Fuzzy Decision Making, Studies in Fuzziness and Soft Computing 333, DOI 10.1007/978-3-319-26494-3_2 27 28 2 Ordering of Fuzzy Numbers these methods, some dedicated procedures are required. The second group consists of methods, which assign to a fuzzy number a crisp value. These are methods such as Yager ranking index based approach, defuzzification approach or weighted average. The methods from the second group can be directly used to order a set of fuzzy numbers, by employing one of the several methods for ordering (sorting) real numbers. All the above mentioned methods are compared using an example of ordering four triangular fuzzy numbers. 2.1 Probabilistic Approach The probabilistic (also known as probability degree-based or probability-based) approach to ordering fuzzy numbers is based on the α–cuts representation of fuzzy numbers. The α–cuts based orderings are so attractive, because they can be used regardless the type of themembership function.Moreover, eachα-level is an interval, so the powerful tools of interval arithmetic [15] can be employed to solve the problem of fuzzy ordering [13]. Let a = [a1, a2] and b = [b1, b2] be two closed and compact intervals. The possibility degree-based ranking method which is shown in Table2.1 was proposed by Jiang et al. [16]. The non-overlapping cases are omitted as they are obvious. A similar, but slightly extended approach to ordering of intervals was proposed in [13]. Let the real values a ∈ a and b ∈ b be given. They can be considered as two independent uniform random variables. If a and b overlaps, then some disjoint subintervals can be distinguished. The fall of random variables a and b in the subintervals [a1, b1], [b1, a2], [a2, b2] may be treated as a set of independent random events. Let the events Hk : a ∈ ai , b ∈ b j be defined for k = 1, . . . , n, where ai and b j are certain subintervals of intervals a and b in accordance with a = ⋃ i ai and b = ⋃ i bi (n = 4 for the case depicted in Fig. 2.1) [13]. Let P(Hk) be the probability of event Hk , and P(b > a|Hk) be the conditional probability of b > a given Hk . Hence, the composite probability may be expressed as follows: