Ordered Weighted Sum in infinite sequences environment with applications

Abstract

We study the Yager OWS aggregation and the Irregular normalized Yager OWA aggregation under infinite environment. Two evaluation problems are proposed to motivate their applications. Few distinguishing properties are discussed. We show that such aggregations use the same type of weights, infinite weights (sequence), to embody the involved preferences. Their respective applications in the fields of Scientometrics and Gradual Confidence Aggregation are proposed and analyzed.

Keywords

Aggregation function Decision making evaluation Infinite Ordered Weighted Sum OWA operators

1. Introduction

As a well known aggregation function [1 , 12], Yager OWA operators [17] and related aggregation techniques have been widely studied and used in numerous areas, see for example [3–10 , 18–21]. However, very few literatures discuss the possible applications of ordered aggregation that is suitable also in infinite environment. Recursive OWA aggregation [16] as a special type of infinite OWA aggregation has been studied and applied in long time evaluation problems [4]. Mesiar and Pap [11] discussed Aggregation of infinite sequences in a general sense. However, more general form of infinite ordered aggregations needs to be studied because their usages and applications are varied in nature.

This study will discuss two types of related aggregations in infinite environment: (i) The Infinite Ordered Weighted Sum (IOWS), and (ii) The Infinite Irregular normalized Yager OWA aggregations.

We first present a couple of evaluation cases as motivators to study the above mentioned two types of infinite ordered aggregations.

Case 1. (Ordered aggregation in Scientometrics): In Scientometrics, it is a common practice to consider both the number of published papers and their respective citations with regard to a particular scholar. In this consideration, an infinite non-increasing citation sequence ${x_{j}}_{j = 1}^{\infty}$ , ( $x_{j} \in ℕ)$ can represent a type of academic status of a scholar, where x_j represents the citations of her jth highest cited paper. It follows that for any such citation sequence $x = {x_{j}}_{j = 1}^{\infty}$ , there exists a “better" citation sequence, e.g., $x^{+} = {x_{j}^{+}}_{j = 1}^{\infty}$ , such that 1) $x_{j}^{+} = x_{j} + 1$ when j < q, 2) $x_{q}^{+} = 1$ , and 3) $x_{j}^{+} = 0$ when j > q, where q = inf {i|x_i = 0}. In this case, if F (x) represents the scientometric index of x to measure the academic impact of a scholar, then it should be monotonic with x. That is, for any two citation sequences $x = {x_{j}}_{j = 1}^{\infty}$ and $y = {y_{j}}_{j = 1}^{\infty}$ , if x_j ≥ y_j for any $j \in ℕ$ , then it is reasonable that we should have F (x) ≥ F (y). As a simple example to illustrate, we can choose $F (x) = \sum_{j = 1}^{\infty} x_{j}$ . In Section 2, we will discuss this case again with more definitions and properties introduced.

Case 2. (Gradual Confidence Aggregation): Suppose a manager needs to know whether a proposed project is feasible or not. She collects opinions gradually from the experts until she believes that she can make a decision according to some predetermined conditions. Take for example, her aggregated confidence from the opinion sequence must attain some predetermined threshold value V ∈ [0, 1] for the project to work, and the total number of experts whom she can consult, can exceed a given number $M \in ℕ$ . That is, if the manager does not have enough confidence, she may invite more opinions from additional experts. These opinions can be either positive or negative.

Suppose every distinct opinion from the experts can be expressed by a real number x_i in [0, 1], representing the extent to which Expert i thinks that the project is feasible. The experts can express their opinions in numerical values if they feel comfortable with that. Even if they are unwilling and can provide only linguistic feedbacks the same can be easily translated to their numerical counterparts. For example, let L = {notfeasible; conditionallyfeasible; generallyfeasible; fullyfeasible} be a set of linguistic feedbacks and S = {0, 0.3, 0.7, 1} be the corresponding set of numerical values. We then make a one to one correspondence between these two sets. Thus, when Expert i selects “conditionally feasible”, we may take x_i = 0.3. Our aim is the following. Given opinion sequences $x = {x_{i}}_{i = 1}^{n} (n \in ℕ$ ) (where i can be an index variable representing the ordering of the experts being consulted with), we need to obtain a final aggregated opinion F (x) which is obtained by aggregating opinion sequences $x = {x_{i}}_{i = 1}^{n}$ ( $n \in ℕ$ ). We note that in such case, given two opinion sequences $x = {x_{i}}_{i = 1}^{n}$ and $y = {y_{i}}_{i = 1}^{n + 1}$ with x_i = y_i for any i = 1, …, n, then both A (x) ≥ A (y) and A (x) ≤ A (y) are possible and reasonable. In Section 3, we will discuss this case again with more definitions and properties introduced.

In order to more generally and strictly model above mentioned two cases, in this study we will discuss and present two corresponding types of ordered aggregations. In addition, concluding remarks will be found in Section 4.

2. Infinite Ordered Weighted Sum with applications

Yager OWA aggregation techniques as well-known and powerful information fusion tools can efficiently and flexibly model optimism/pessimism preferences of decision makers [17, 21]. A traditional and often used form of OWA aggregation is rephrased as follows according to the standard expressions used in this study.

Let $x = {x_{j}}_{j = 1}^{n} \in ℝ^{n}$ be an input finite sequence, $x_{σ} = (x_{σ (j)})_{j = 1}^{n}$ denote a rearranged form of x with non-increasing order where σ : {1, …, n} → {1, …, n} is any allowed permutation such that x_σ(a) ≥ x_σ(b) whenever a < b. Let $w = (w_{j})_{j = 1}^{n} \in [0, 1]^{n} (\sum_{j = 1}^{n} w_{j} = 1)$ be a normalized weights sequence called Yager OWA weights (sequence). Then Yager OWA aggregation (of dimension n) to input x with w is defined by the function $F_{w} : ℝ^{n} \to ℝ$ such that [17]: $F_{w} (x) = \sum_{j = 1}^{n} w_{j} x_{σ (j)}$ (1) The orness of Yager OWA weights (sequence) w is defined by [17]: $orness (w) = \sum_{j = 1}^{n} \frac{n - j}{n - 1} w_{j}$ (2)

Generally, bigger orness of Yager OWA weights w often shows more optimistic attitude of the decision maker, and vice-versa. And in general Yager OWA aggregation returns larger final aggregation result with bigger orness of Yager OWA weights (but it is not always the case unless we put some further restrictions, e.g., the restriction of the considered Yager OWA weights w to some special families such as family of Recursive OWA weights [16]).

When $w = (w_{j})_{j = 1}^{n} \in [0, 1]^{n}$ is not necessarily a normalized weights sequence, then we call F_w in Equation (1) the Ordered Weighted Sum (OWS) of input x with w.

In some special applications, Ordered Weighted Sum with extended weights (sequence) are more suitable. Return to Case 1 as introduced in Section 1. Since the citation sequence $x = {x_{j}}_{j = 1}^{\infty}$ can always be made better, therefore the Yager OWA weights (sequence) in finite dimension is not suitable to aggregate x. Therefore, we need to extend it and introduce some new extended ordered weights (sequence). Note that in the infinite sequence environment, the decision to normalize the weight sequences generally will not affect the final alternative orderings and decision results.

Definition 1. (Infinite Ordered Weighted Sum) Let χ_∞ be the set of all real non-negative sequences such that for any $x = {x_{j}}_{j = 1}^{\infty} \in χ_{\infty}$ , $\sum_{j = 1}^{\infty} x_{j} < + \infty$ (+∞ is the extended real number that is the supremum of all real numbers). An infinite weights (sequence) is defined to be an infinite non-negative bounded sequence $w = {w_{j}}_{j = 1}^{\infty}$ (i.e., there exists A > 0 such that for any $j \in ℕ, w_{j} \leq A$ ). We define $W$ to be the set of all infinite weights (sequence). Then Infinite Ordered Weighted Sum (IOWS) to input x with w is defined to be the function $F_{w} : ℝ^{\infty} \to ℝ$ such that:

$F_{w} (x) = \sum_{j = 1}^{\infty} w_{j} x_{σ (j)}$ (3) whenever $σ : ℕ \to ℕ$ is any possible bijective mapping (i.e., a permutation of $ℕ$ ) satisfying x_σ(a) ≥ x_σ(b) whenever a < b. When such a σ does not exist, we do not define Infinite Ordered Weighted Sum, and in the remainder of this study, we always assume that such a σ exists in the related infinite environment.

Remark 1. The absolute convergence of ${\sum_{k = 1}^{j} x_{k}}_{j = 1}^{\infty}$ ensures that $\sum_{j = 1}^{\infty} w_{j} x_{σ (j)} \leq sup {w_{j}}_{j = 1}^{\infty} . \sum_{j = 1}^{\infty} x_{σ (j)}$ = $sup {w_{j}}_{j = 1}^{\infty} . \sum_{j = 1}^{\infty} x_{j} < \infty$ , thus Equation (3) is meaningful only when such a σ exists. In addition, even though σ is non-unique in some of the cases, the aggregation results are not affected by this fact.

Remark 2. If there are only finitely many positive members in the input sequence $x = {x_{j}}_{j = 1}^{\infty}$ , then the permutation σ clearly exists. If there are infinitely many positive members, we can omit all zero members, thus obtaining a new sequence $x^{'} = {x^{'}}_{j = 1}^{\infty}$ consisting of only positive members, and then again σ the permutation exists. For example, consider $x = {x_{j}}_{j = 1}^{\infty}$ such that x_j = 0 whenever j is odd, and x_j = 2^-j/2 whenever j is even. Clearly there is no suitable permutation for x. Therefore, we may omit all zero members in x and obtain a new sequence $x^{'} = {x_{j}^{'}}_{j = 1}^{\infty} = {2^{- j}}_{j = 1}^{\infty}$ . Thus there exists a suitable permutation σ with σ (k) = k for any $k \in ℕ$ .

Remark 3. The infinite weights (sequence) $w = {w_{j}}_{j = 1}^{\infty}$ , defined in Definition 1 only needs to be bounded, and since $\sum_{j = 1}^{\infty} w_{j}$ is allowed to be unbounded (i.e., series ${\sum_{k = 1}^{j} w_{k}}_{j = 1}^{\infty}$ diverges), then we do not consider the normalization of $w = {w_{j}}_{j = 1}^{\infty}$ .

Remark 4. Since a mapping $σ : ℕ \to ℕ$ satisfying x_σ(a) ≤ x_σ(b) whenever a < b generally does not exist unless x_j = 0 for all $j \in ℕ$ we do not consider the reversed OWA aggregation. In addition, here we do not discuss the orness of any infinite weights (sequence) also.

Proposition 1. (Monotonicity) For any x, y ∈ χ_∞ with x ≤ y (i.e., x_j ≤ y_j for any $j \in ℕ$ ), let $w \in W$ be any infinite weights (sequence), then we have F_w (x) ≤ F_w (y).

Proof. Let $σ_{1} : ℕ \to ℕ$ be any bijective mapping satisfying x_σ₁(a) ≥ x_σ₁(b) whenever a < b; and $σ_{2} : ℕ \to ℕ$ be any bijective mapping satisfying y_σ₂(a) ≥ y_σ₂(b) whenever a < b.

We next prove that for any $k \in ℕ$ , y_σ₂(k) ≥ x_σ₁(k). Suppose there exists some $k \in ℕ$ such that x_σ₁(r+1) > y_σ₂(r+1), then we necessarily have max {t | y_σ₂(t) ≥ x_σ₁(r+1)} ≤ r while max {t|x_σ₁(t) ≥ x_σ₁(r+1)} = r + 1. Therefore, there must exist h ∈ {1, …, r + 1} such that x_σ₁(h) ≥ x_σ₁(r+1) > y_σ₁(h), which leads to contradiction to the given condition “ x_j ≤ y_j for any $j \in ℕ$ ”. Consequently, $F_{w} (x) = \sum_{j = 1}^{\infty} w_{j} x_{σ_{1} (j)} \leq \sum_{j = 1}^{\infty} w_{j} y_{σ_{2} (j)} = F_{w} (y) .$ □

Definition 2. For any infinite weights (sequence) $w = {w_{j}}_{j = 1}^{\infty}$ ,

if w_a ≥ w_b whenever a < b, then we call w monotonic non-increasing; and we denote the set of all monotonic non-increasing infinite weights (sequences) by $W_{D}$ , with $W_{D} \subset W$ .

if the series ${\sum_{k = 1}^{j} w_{k}}_{j = 1}^{\infty}$ converges, then we call w convergent; and we denote the set of all convergent infinite weights (sequences) by $W_{C}$ , with $W_{C} \subset W$ .

Return to Case 1 (ordered aggregation in scientometrics) proposed in Section 1. For any citation sequence x ∈ χ_∞ (which is already in non-increasing order), though we have not been equipped with orness degree, we can still select a family of infinite weights (sequences)

{w^{< α >}}_{α \in [0, + \infty)} = {{j^{- α}}_{j = 1}^{\infty}}_{α \in [0, + \infty)} (\subset W_{D} \subset W

) to well model the preference in this special case. Notice that we have

{{j^{- α}}_{j = 1}^{\infty}}_{α \in (1, + \infty)} (\subset W_{C} \subset W

Given $x = {x_{j}}_{j = 1}^{\infty}$ ( $x_{j} \in ℕ$ ), note that once we select α =∞, then $w^{< \infty >} = {j^{- \infty}}_{j = 1}^{\infty} = {1, 0, 0, . . .}$ . Thus, Infinite OWS of input x with w such that $F_{w^{< \infty >}} (x) = \sum_{j = 1}^{\infty} j^{- \infty} x_{σ (j)} = x_{σ (1)} = x_{1}$ returns the citation numbers of the highest cited paper of one scholar. As another extreme situation, consider α = 0, then $w^{< 0 >} = {j^{- 0}}_{j = 1}^{\infty} = {1}_{j = 1}^{\infty} = {1, 1, 1, . . .}$ . Thus, Infinite Yager OWA aggregation to input x with w such that $F_{w^{< 0 >}} (x) = \sum_{j = 1}^{\infty} j^{- 0} x_{σ (j)} = \sum_{j = 1}^{\infty} x_{σ (j)} = \sum_{j = 1}^{\infty} x_{j}$ returns the totally summed citation numbers of one scholar’s published papers.

Choice of larger or smaller α can result in two different preferences: one is to take preference only on the highest citation numbers, showing the “representative works” preferred attitude; the other is to take preference on the total citation numbers, showing the “comprehensive quality” preferred attitude. As a simple example, consider two citations sequences x = (3, 0, 0, …) and y = (1, 1, 1, 1, 0, 0, …) showing scholar x only publishes one paper with 3 citations, while scholar y publishes 4 papers each with only 1 citation. Clearly, when having “representative works” preferred attitude and choosing α =∞, we have F_{w
^<∞>} (x) =3 > 1 = F_{w
^<∞>} (y); when having “comprehensive quality” preferred attitude and choosing α = 0, we have F_{w
^<0>} (x) =3 < 4 = F_{w
^<0>} (y). Different preferences for infinite weights (sequences) may lead to different evaluation results.

3. Irregular normalized Yager OWA aggregation with applications

In this section, we consider another type of input sequences. Recall in Case 2 as introduced in Section 1 that, for each n ( $n \in ℕ$ ) the decision maker may seek one more new opinion from an expert distinct from the earlier ones. Thus she obtains opinion sequences with increasing dimensions, ${x_{i}}_{i = 1}^{n} (n \in ℕ$ ). However, we can still use the one infinite weights (sequence) $w \in W$ to perform the Yager OWA aggregations of dimension n ( $n \in ℕ$ ). But as we mentioned in Section 1 that there is no monotonicity with respect to opinion sequences is needed, then we should use a normalized type of this Infinite Yager OWA weights (sequence) $w \in W$ ; that is, to normalize the first n ( $n \in ℕ$ ) entries in $w = {w_{j}}_{j = 1}^{\infty}$ and then to do Yager OWA aggregations (of dimension n ( $n \in ℕ$ ), respectively). In the following we present the definition of the Irregular normalized Yager OWA aggregation.

Definition 3. (Irregular normalized Yager OWA aggregation) Let $χ = {x = {x_{j}}_{j = 1}^{\infty} : n \in ℕ, x_{j} \geq 0}$ be the set of all non-negative real sequences with finite dimensions. Let $w = {w_{j}}_{j = 1}^{\infty} \in W_{D}$ be a monotonic non-increasing infinite weights (sequences). Then the Irregular normalized Yager OWA aggregation to input $x = {x_{j}}_{j = 1}^{\infty} \in χ$ with w is defined to be the function $F_{w}^{(n)} : χ \to ℝ$ such that: $F_{w}^{(n)} (x) = \frac{\sum_{j = 1}^{n} w_{j} x_{σ (j)}}{\sum_{k = 1}^{n} w_{k}}$ (4) where σ : {1, 2, …, n} → {1, 2, …, n} is any bijective mapping as mentioned earlier. If σ satisfies x_σ(a) ≥ x_σ(b) whenever a < b the Irregular normalized Yager OWA aggregation is said to be optimism oriented; If σ satisfies x_σ(a) ≤ x_σ(b) whenever a < b, then the Irregular normalized Yager OWA aggregation is said to be pessimism oriented.

Return to Case 2 in Section 1. As a representative example of infinite weights, here we consider again the special family of monotonic non-increasing infinite weights (sequences), ${w^{< α >}}_{α \in [0, + \infty)} = {{j^{- α}}_{j = 1}^{\infty}}_{α \in [0, + \infty)}$ ( $\subset W_{D} \subset W$ ). In detail, we will discuss two situations from which we will obtain very different characteristics of the related Irregular normalized Yager OWA aggregations when dimension n gradually approaches infinity. The two situations divide [0, + ∞) into corresponding two disjoint subsets, [0, 1] and (1, + ∞).

Recall that when α ∈ [0, 1], the series ${\sum_{j = 1}^{n} w_{j}^{< α >}}_{n = 1}^{\infty}$ $= {\sum_{j = 1}^{n} j^{- α}}_{n = 1}^{\infty}$ is unbounded, i.e., $w^{< a >} \in W \ W_{C}$ ; when α ∈ (1, + ∞), the series ${\sum_{j = 1}^{n} w_{j}^{< α >}}_{n = 1}^{\infty}$ $= {\sum_{j = 1}^{n} j^{- α}}_{n = 1}^{\infty}$ is convergent, i.e., $w^{< a >} \in W_{C}$ .

First we consider α ∈ [0, 1]. This would imply that ${\sum_{j = 1}^{n} w_{j}^{< α >}}_{n = 1}^{\infty}$ is unbounded. Then for any given $n \in ℕ$ and $x = {x_{j}}_{j = 1}^{\infty} \in χ$ with $F_{w^{< α >}}^{(n)} (x) = \frac{\sum_{j = 1}^{n} w_{j}^{< α >} x_{σ (j)}}{\sum_{k = 1}^{n} w_{k}^{< α >}}$ , let $y = {y_{j}}_{j = 1}^{m} \in χ$ (m > n) with y_j = x_j when j ≤ n and y_j = c when n < j ≤ m, then we have

$\begin{matrix} F_{w^{< α >}}^{(m)} (y) & = & \frac{\sum_{j = 1}^{m} w_{j}^{< α >} y_{η_{m} (j)}}{\sum_{k = 1}^{m} w_{k}^{< α >}} \\ = & \frac{\sum_{j = 1}^{n} w_{r (j)}^{< α >} x_{j}}{\sum_{k = 1}^{m} w_{k}^{< α >}} \\ + c . \frac{\sum_{j = 1}^{m} w_{j}^{< α >} - \sum_{j = 1}^{n} w_{r (j)}^{< α >}}{\sum_{k = 1}^{m} w_{k}^{< α >}} \end{matrix}$ (5) where η_m : {1, …, m} → {1, …, m} (m > n) is any bijective mapping satisfying y_{η_m(a)} ≥ y_{η_m(b)}(or y_{η_m(a)} ≤ y_{η_m(b)}) whenever a < b; and r : {1, …, n} → {1, …, m} is some mapping.

When m→ ∞, meaning there have infinitely many new opinions all with x_j = c (j > n), then we take limit form of Equation (5) and obtain $\begin{matrix} lim_{m \to \infty} F_{w^{< a >}}^{(m)} (y) = lim_{m \to \infty} \frac{\sum_{j = 1}^{m} w_{j}^{< α >} y_{η_{m} (j)}}{\sum_{k = 1}^{m} w_{k}^{< α >}} \\ = lim_{m \to \infty} \frac{\sum_{j = 1}^{n} w_{r (j)}^{< α >} x_{j}}{\sum_{k = 1}^{m} w_{k}^{< α >}} \\ + c . lim_{m \to \infty} \frac{\sum_{j = 1}^{m} w_{j}^{< α >} - \sum_{j = 1}^{n} w_{r (j)}^{< α >}}{\sum_{k = 1}^{m} w_{k}^{< α >}} \\ = 0 + c . 1 = c . \end{matrix}$ It shows that the earlier aggregation results can be completely replaced by new results; that is, no matter what aggregation result is employed at time n, it is possible to have any value c in [0, 1] as the new aggregated value under the limiting condition. It is not difficult to observe that a smaller α means that the earlier aggregation result is easier to “remove”, and vice-versa.

In this situation, the opinions aggregation is suitable for the decision maker who has optimistic or pessimistic attitude, but it is easy for him to consider any new opinions. In application, the decision maker can have a set of fixed conditions, say $M \in ℕ$ and V ∈ [0, 1]; then, if n ≥ M she may make decision according to if or not $F_{w^{< α >}}^{(n)} (x) \geq V$ , and if so, the project can be thought as feasible.

Secondly, we consider α ∈ (1, + α), and in this situation ${\sum_{j = 1}^{n} w_{j}^{< α >}}_{n = 1}^{\infty}$ is convergent. Then, it is possible that further different opinions will, only to some extent affect the previous aggregation results. Once n becomes larger, further opinions will have lesser effect. For example, when $x = {1}_{j = 1}^{n} \in χ$ , and we take optimism oriented Irregular normalized Yager OWA aggregation, then we have $F_{w^{< α >}}^{(n)} (x) = 1$ for any $w^{< α >} \in W_{D}$ . Now, let $y = {y_{j}}_{j = 1}^{m} \in χ$ with y_j = x_j = 1 when j ≤ n and y_j = c when n ≤ j ≤ m, then we have

$\begin{matrix} F_{w^{< α >}}^{(m)} (y) & = & \frac{\sum_{j = 1}^{m} w_{j}^{< α >} y_{j}}{\sum_{k = 1}^{m} w_{k}^{< α >}} = 1 . \frac{\sum_{j = 1}^{n} w_{j}^{< α >}}{\sum_{k = 1}^{m} w_{k}^{< α >}} \\ + c . \frac{\sum_{j = 1}^{m} w_{j}^{< α >} - \sum_{j = 1}^{n} w_{r (j)}^{< α >}}{\sum_{k = 1}^{m} w_{k}^{< α >}} \end{matrix}$ (6)

When m→ ∞, meaning that there have infinitely many new opinions all with x_j = c (j > n), then we take limit form of Equation (6) and obtain $\begin{matrix} lim_{m \to \infty} F_{w^{< α >}}^{(m)} (y) & = & lim_{m \to \infty} \frac{\sum_{j = 1}^{m} w_{j}^{< α >} y_{j}}{\sum_{k = 1}^{m} w_{k}^{< α >}} \\ = & 1 . lim_{m \to \infty} \frac{\sum_{j = 1}^{n} w_{j}^{< α >}}{\sum_{k = 1}^{m} w_{k}^{< α >}} \\ + c . lim_{m \to \infty} \frac{\sum_{j = 1}^{n} w_{j}^{< α >} - \sum_{j = 1}^{m} w_{r (j)}^{< α >}}{\sum_{k = 1}^{m} w_{k}^{< α >}} \\ = & λ . 1 + (1 - λ) . c \end{matrix}$ (7) $where λ = \frac{\sum_{j = 1}^{n} w_{j}^{< α >}}{\sum_{k = 1}^{m} w_{k}^{< α >}} \in (0, 1])$ (8) It shows that the earlier aggregation results can not be fully removed, and the larger n, the bigger is the λ. In addition, note that c < λ . 1 + (1 - λ) . c ≤ 1 implies that it is impossible to reduce the aggregation results below c. It is not difficult to observe that a larger α means that the earlier aggregation result has more effect in both the opinions to add and the corresponding aggregation result, and vice-versa. In this situation, the opinions aggregation is suitable for the decision maker who has optimistic or pessimistic attitude, and it is difficult to change her earlier impression and feeling if the new opinions are opposite to them. In application, the decision maker may have a set of fixed conditions, say d ∈ [0, 1] and V ∈ [0, 1]; then, once $\frac{\sum_{j = 1}^{n} w_{j}^{< α >}}{\sum_{j = 1}^{\infty} w_{j}^{< α >}} \geq d$ , she makes a decision based on whether or not $F_{w}^{(n)} (x) \geq V$ , and if so, the project can be thought as feasible.

4. Conclusions

Yager OWA operators and Yager OWA aggregation are powerful tools in decision making. However, apart from Recursive OWA operators there are few literatures and knowledge about ordered aggregation for infinite sequences. In this study, we listed two commonly faced real life problems which involve necessity to discuss ordered aggregation in infinite environment.

We defined and discussed two types of infinite ordered aggregations: (i) Infinite OWS which only has optimistic type, and (ii) Irregular normalized Yager OWA aggregation which has both optimism oriented type and pessimism oriented type. In distinct ways, both of these two types of infinite ordered aggregations used infinite weights (sequence) to express the involved preferences of decision makers. Correspondingly, two different evaluation problems are also analyzed throughout this study.

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