**4. Aggregation: proposal of a ranking index**

If the Pareto optimality principle cannot rank two fuzzy numbers, trade-off consideration is required to finalize the ranking decision. That is, a fuzzy number of a higher rank must have some "weaker" aspect in terms of the three attributes but its "stronger" aspect is sufficient to bring it to a higher rank overall. This ranking process should involve an aggregation that combines all aspects into an overall evaluation and then determines the ranking result. This section will propose a ranking index for aggregation along with numerical examples.

#### **4.1 Discount factors and ranking index**

As discussed in Section 3, representative x-values are used as the primary attribute to rank fuzzy numbers. Then, we view the information of range and membership ratio as secondary attributes that will "discount" the representative x-values. To illustrate, consider a crisp number, 5, which has the representative x-value of 5, range of 0 and membership ratio of 1. If a fuzzy number with the representative x-value of 5 has a range larger than 0 and a membership ratio less than 1, this fuzzy number should be ranked lower than the crisp number 5. The discount factors are intended to capture this idea. Let *Irank*(*FA*) be the index as the discounted representative x-value of *FA* for ranking, and it can be formulated as follows.

$$I\_{rank}(F\_A) = d\_{mg}(F\_A) \cdot d\_{mem}(F\_A) \cdot rep(F\_A, w) \tag{12}$$

where *drng*(*FA*) and *dmem*(*FA*) are the discount factors associated with range and membership ratio, respectively. To quantify these discount factors, we consider the following conditions:


Apparently, many forms of formulations can be used for the discount factors and satisfy these conditions. In this chapter, we use a simple ratio with respect to some reference (or extreme) values. Let *rngmin* be the minimum reference for range, and *memmax* be the maximum reference for membership ratio. We also set that *rngmin* > 0 and 0 < *memmax* ≤ 1. Then, the discount factors for *FA* can be formulated as follows.

$$d\_{\rm rg}(F\_A) = \frac{\rm rg\_{\rm min}}{\rm rg\_{\rm g}(F\_A, \alpha)}\tag{13}$$

$$d\_{mem}(F\_A) = \frac{mem(F\_A, a)}{m \varepsilon m\_{max}} \tag{14}$$

With these discount factors, if *FA* has a range equal to *rngmin*, its discount factor, *drng*(*FA*), is equal to 1 (i.e., no discount). A similar effect is also set for *dmem*(*FA*). The selection of the values for *rngmin* and *memmax* depends on how decision makers interpret the discount ratio for their ranking problems. One suggestion is to identify the minimum range and the maximum membership ratio from the set of fuzzy numbers to be ranked. That is, suppose that *FR* = {*FA*, *FB*, *FC*, … } be the set of fuzzy numbers that need to be ranked in a problem. We can select *rngmin* and *memmax* according to the following equations. Then, we can interpret the discount ratio with respect to the "best values" among the set of fuzzy numbers in the problem.

$$\text{rng}\_{\text{min}} = \min\left\{ \text{rng}(\text{F}\_{\text{A}}, a), \text{rng}(\text{F}\_{\text{B}}, a), \text{rng}(\text{F}\_{\text{C}}, a) \dots \right\} \tag{15}$$

$$mem\_{\max} = \max\left\{mem(F\_A, a), mem(F\_B, a), mem(F\_C, a) \dots \right\} \tag{16}$$

### **4.2 Overview of the ranking method**

After defining the attributes in Section 3 and the ranking index in Section 4.1, this sub-section will overview our proposed approach to rank fuzzy numbers. The procedure to determine the ranking index is illustrated in **Figure 5**. Given a fuzzy number *FA*, we first determine the values of three attributes: representative x-value, x-value range and overall membership ratio. Then, we can evaluate the discount factors for xvalue range and overall membership ratio. In the end, we can determine the ranking index for the given fuzzy number.

Suppose that we are tasked to rank a set of fuzzy numbers. We first determine the ranking index for each fuzzy number. Then, we can use the index, *Irank*, for this ranking task. That is, if *Irank*ð Þ *FA* ≥*Irank*ð Þ *FB* , we rank *FA* higher than *FB*, symbolically, *FA*≽*FB*.

#### **4.3 Numerical examples**

As a recall from Section 3, we set *w* = 1/12 and *α* = 0 to evaluate representative xvalue, range and membership ratio. We use Eqs. (15) and (16) to obtain *rngmin* and *memmax* and then calculate the values of the discounts and the ranking index. We reuse the numerical examples from Section 3.5 with the cases where Pareto optimality cannot finalize the ranking. The results are presented in **Table 6**.

Case e comes from Group 1 (see **Table 1**), where *FE*<sup>3</sup> is ranked on the top per Pareto optimality (same result from the ranking index). Between *FE*<sup>1</sup> and *FE*2, though *FE*<sup>2</sup> should be ranked higher intuitively, trade-off is involved logically because *FE*<sup>2</sup> has a large range (reflected in its range discount of 0.5 as well). Per the ranking index, we still have *FE*2≽*FE*1, which matches the general intuition.

Cases g and h come from Group 3 (see **Table 3**), where wide trapezoidal fuzzy numbers are compared with narrow triangular fuzzy numbers. Per the ranking index, the triangular fuzzy numbers are ranked higher mainly because of the large difference of the range discount (1 vs. 0.2). In contrast, the difference of the membership ratio discount is less substantial.

Cases j and k come from Group 4 (see **Table 4**), where two triangular fuzzy numbers are nested in a trapezoidal fuzzy number. In both cases, the wider triangular fuzzy numbers (i.e., *FJ*<sup>2</sup> and *FK*2) are ranked lowest as they receive both discounts

**Figure 5.** *Procedure to determine the ranking index.*


*Decoupling of Attributes and Aggregation for Fuzzy Number Ranking DOI: http://dx.doi.org/10.5772/intechopen.109992*

#### **Table 6.**

*Ranking index results for cases with trade-off consideration.*

(i.e., *drng* = *dmem* = 0.6667). In case j, the narrower triangular fuzzy number (*FJ*1) is ranked first because its representative x-value and range can "win" over its weaker membership ratio as compared to the trapezoidal fuzzy number (*FJ*3). In contrast, in case k, the trapezoidal fuzzy number (*FK*1) "wins" because it has better representative x-value and membership ratio as compared to *FK*3.

Cases n, o and r come from Group 5 (see **Table 5**). In case n, we have *FN*2≽*FN*1, as *FN*<sup>2</sup> has higher representative x-value despite lower membership ratio (associated with the discount *dmem*(*FN*2) = 0.8). In cases o, we have *FO*1≽*FO*<sup>2</sup> because the membership ratio of *FO*<sup>2</sup> is substantially lower despite its higher representative x-value. In case r, the trade-off between range and membership ratio is relatively close. In the end, we have *FR*1≽*FR*2, as *FR*<sup>2</sup> has a lower value of the discount from membership ratio (i.e., *dmem*(*FR*2) = 0.2 vs. *drng*(*FR*1) = 0.25).

Notably, the judgment for ranking with trade-off can become difficult when the trade-off among the three attributes is getting close. We argue that such difficulty is fundamentally embedded into the problem structure of FNR, which involves multiple dimensions of considerations. Thus, our solution strategy is not about providing the best ranking procedure. Instead, we emphasize the importance of defining attributes to quantify the "intuition". Then, decision makers can explicitly explain their tradeoff considerations in the ranking process.
