A novel algorithm for fuzzy soft set based decision making from multiobserver input parameter data set

Accepted Manuscript

A novel algorithm for fuzzy soft set based decision making from multiobserver input parameter data set Jose´ Carlos R. Alcantud PII: DOI: Reference:

S1566-2535(15)00080-9 10.1016/j.inffus.2015.08.007 INFFUS 733

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Information Fusion

Received date: Revised date: Accepted date:

12 January 2015 30 July 2015 25 August 2015

Please cite this article as: Jose´ Carlos R. Alcantud, A novel algorithm for fuzzy soft set based decision making from multiobserver input parameter data set, Information Fusion (2015), doi: 10.1016/j.inffus.2015.08.007

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Highlights

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• Two innovations produce novel algorithm for fuzzy soft set based decision making • With multi-source information, new aggregation procedure for resultant fuzzy soft set

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• New algorithm produces unique score-based solution with high power of discrimination

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Jos´e Carlos R. Alcantud

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A novel algorithm for fuzzy soft set based decision making from multiobserver input parameter data set

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Facultad de Econom´ıa y Empresa and Multidisciplinary Institute of Enterprise (IME), University of Salamanca, 37007 Salamanca, Spain [email protected] URL: http://diarium.usal.es/jcr

Abstract

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We present two innovations that produce a novel approach to the problem of fuzzy soft set based decision making in the presence of multiobserver input parameter data sets. The first novelty consists of a new process of information fusion that furnishes a more reliable resultant fuzzy soft set from such input data set. The second one concerns the mechanism that decides among the alternatives in this resultant fuzzy soft set. It relies on scores computed from a relative comparison matrix. The advantages of our novel procedure are a higher power of discrimination and a well-determined final solution.

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Keywords: Fuzzy soft set; Resultant fuzzy soft set; Comparison table; Decision making. 1. Introduction

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Many real life problems require to use imprecise, uncertain or subjective data. Hence their solutions involve the application of mathematical principles that potentially capture these features. Fuzzy set theory caused a profound change in Mathematics by allowing partial membership. Since Zadeh [1] introduced fuzzy sets, a vast literature on their properties and applications to decision making has been produced. For example, Mardani et al. [2] is an extensive analysis of papers about fuzzy multi-criteria decision making published in the period 1994-2014. Tanino [3] or Fodor and Roubens [4] are a short sample of classical references. Furthermore, one of the most used preference structures in group decision making problems under uncertainty Preprint submitted to Information Fusion

August 31, 2015

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is the fuzzy preference relation (cf., Castro et al. [5], which provides an application to consensus-driven group recommender systems). However in some practical problems, imprecise individual or collective knowledge cannot be faithfully represented by fuzzy sets. This constraint calls for generalizations of this notion and related variations which may supply more suitable models. In this regard, Atanassov [6, 7] proposes the concept of intuitionistic fuzzy sets. New intuitionistic fuzzy multi-attribute group decision making methods are developed e.g., in Chen et al. [8] or Wei [9] among other recent references. The use of interval-valued Atanassov intuitionistic fuzzy sets in multi-expert decision making is exemplified in De Miguel et al. [10]. Bustince and Burillo [11] prove that the concept of vague set coincides with the notion of intuitionistic fuzzy set. Xu and Cai [12] provides a systematic introduction to intuitionistic fuzzy aggregation methods and their many application to decision making. A direct extension of fuzziness is the general field of Hesitancy in Fuzzy Sets. Hesitant fuzzy sets are introduced by Torra [13]. A reference work is Xu [14] (see also the special issue introduced by Herrera et al. [15], especially the survey Rodr´ıguez et al. [16]). Multiexpert multicriteria decision making under this requirement is explored by Xia et al. [17] or Tan et al. [18] among others. Given the proliferation of extending notions, it is also important to explore their relationships. Hesitant fuzzy sets can be represented as fuzzy multisets ([13, Lemma 14]) and as type-2 fuzzy sets ([13, Lemma 16]). Bustince et al. [19] is an updated account of types of fuzzy sets and their connections. These authors prove that the original mathematical formulation of an interval type-2 fuzzy set (resp., Atanassov intuitionistic fuzzy set, vague set, grey set, interval-valued fuzzy set, interval-valued Atanassov intuitionistic fuzzy set) corresponds to a set-valued fuzzy set or a hesitant fuzzy set. Set-valued fuzzy sets can be regarded as type-2 fuzzy sets: [19, Section V.A]. Bustince et al. [20] show that fuzzy sets and interval-valued fuzzy sets are particular cases of interval type-2 fuzzy sets. An overview of the mathematical relationships between intuitionistic fuzzy sets and other theories modelling imprecision is given by Deschrijver and Kerre [21]. 1.1. Soft sets and extensions In a different vein, Molodtsov [22] initiates the theory of soft sets. Quoting from Feng and Zhou [23, Section 1], soft set theory “is considered as a 3

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new mathematical tool for dealing with uncertainties which is free from the inadequacy of parameter tools. In soft set theory, the problem of setting the membership function simply does not arise as in fuzzy set theory, which makes the theory convenient and easy to use in practice.” Indeed, Molodtsov [22] shows its applicability to several fields and establishes some fundamental results subsequently complemented by works like Maji et al. [24] and Akta¸s and C ¸ a˘gman [25] among others. Interestingly, Molodtsov [22] shows that the models by fuzzy sets and soft sets are not independent. Other important works regarding the connection among soft sets, fuzzy sets and other soft computing models include Ali [26], Feng et al. [27] and Feng et al. [28]. Soft sets have been extended in various ways starting with Maji, Biswas and Roy [29] who introduce fuzzy soft sets. Wang, Li and Chen [30] introduce hesitant fuzzy soft sets, which combine the ideas of hesitancy (cf., Torra [13]) with the latter concept. Han et al. [31] and Zou and Xiao [32] are concerned with incomplete soft sets, which may arise from errors in data measurement, errors of data understanding or restrictions in data collection. Feng et al. [33] introduce choice value soft sets in order to improve and further extend C ¸ a˘gman and Engino˘glu’s [34] uni-int decision making method.

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1.2. Fuzzy soft sets and decision making Let us focus on the context of fuzzy soft set based decision making. Then the researcher must face the fact that there is no universally accepted criterion for evaluating the alternatives. This is the cost to pay for tackling problems whose nature is subjective or humanistic. The pioneering Roy and Maji [35] formulate a solution for an object recognition problem where the recognition strategy relies on multiobserver input parameter data set. Obviously this formulation can be adapted to other choice situations with inputs having the same structure. We intend to improve the performance of their algorithm at the two stages of their proposal, that we proceed to describe. Stage 1. Roy and Maji [35] propose to begin with an aggregation procedure that yields a single resultant fuzzy soft set from preliminary multi-source information. We show that their original approach, which is universally accepted in this context henceforth, may result into a heavy loss of information that ultimately generates uncertainty. Consequently we argue that it is convenient to use an alternative proposal. Stage 2. Here we address the pure decision making problem: How do we evaluate the alternatives from the information in the resultant fuzzy soft set?

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Roy and Maji [35] propose to construct a Comparison matrix that permits to compute scores for the alternatives. In order to solve the same problem, a different procedure at Stage 2 is given by Kong et al. [36]. These authors claim that the Roy and Maji’s algorithm is incorrect on the grounds of a single naive example. However there is little doubt that such “counterexample” does not support such an extreme view (cf., Feng et al. [37, Subsection 3.2]). In addition, Feng et al. [37] point out that the disparity of opinions between [36] and [35] is whether the criterion for making a decision should use scores or fuzzy choice values (understood as the sum of all membership values accross attributes). In this controversy we concur with Feng et al.’s argument that the completely redesigned approach by scores in Roy and Maji [35] is more suitable for making decisions in an imprecise environment. As to their own proposal for solving the problem at Stage 2, Feng et al. [37] fully incorporate subjectivity by proposing an adjustable method based on level soft sets. Therefore in their approach the optimal choice is dependent upon the selected level soft sets. Their model introduces potential sources of uncertainty, in the form of threshold fuzzy sets, threshold values, or a choice among decision rules (mid-level, top-level). The practitioner has neither assistance to decide among these possibilities, nor a proper comparative study which supports the idea that their method is more reliable than earlier proposals.

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1.3. Contribution and organization of this paper We formulate another information fusion procedure that overcomes the handicap found at Stage 1. Put shortly, it consists of replacing the ‘AND’ operator used in [35] (namely, the minimum) by the other prominent example of t-norm in multi-valued logic: namely, the product. We also propose a novel decision method at Stage 2 of the problem. In line with Roy and Maji’s acclaimed proposal it appeals to scores and produces a unique well-determined outcome. But we produce a different comparison table that eschews the use of incongruous “crisp” values at the core of the definition of Roy and Maji’s Comparison matrix. To achieve this goal we evaluate the relative differences in membership values across the alternatives. Due to these innovations, our procedure is considerably less inconclusive than the aforementioned solutions in fuzzy soft set decision making. As is shown by examples from the literature, these procedures tend to produce many ties that are avoided under our position. 5

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2. Definitions: soft sets and fuzzy soft sets

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This paper is organized as follows. Section 2 recalls some terminology and definitions. Section 3 contains our main contributions. Firstly we present the problem. We discuss the aggregation issue when the input is a set of multiobserver data. Then we propose a novel solution for the problem, and we compare it with previous solutions by exploring examples that have provided arguments in the literature. We conclude in Section 4.

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We adopt the usual description and terminology for soft sets and their extensions: U denotes a universe of objects and E denotes a universal set of parameters. Definition 1 (Molodtsov [22]). A pair (F, A) is a soft set over U when A ⊆ E and F : A −→ P(U ), where P(U ) denotes the set of all subsets of U .

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A soft set over U is regarded as a parameterized family of subsets of the universe U , the set A being the parameters. For each parameter e ∈ A, F (e) is the subset of U approximated by e or the set of e-approximate elements of the soft set. To put an example, if U = {c1 , c2 , c3 , c4 } is a universe of cars and A contains the parameter e that describes “white color” and the parameter e0 that describes “diesel engine” then F (e) = {c1 } means that the only car with white color is c1 and F (e0 ) = {c1 , c3 } means that the only cars with diesel engine are c1 and c3 . Many papers conduct formal investigations of this and related concepts. For example, Maji, Bismas and Roy [24] develop this notion and define among other concepts: soft subsets and supersets, soft equalities, intersections and unions of soft sets, et cetera. Furthermore, Feng and Li [38] give a systematic study on several types of soft subsets and various soft equal relations induced by them. Concerning (pure) soft set based decision making, we refer the reader to Maji, Biswas and Roy [39], C ¸ a˘gman and Engino˘glu’s [34] and Feng and Zhou [23]. In order to model more general situations, the following notion is subsequently proposed and investigated in [29]: Definition 2 (Maji, Biswas and Roy [29]). A pair (F, A) is a fuzzy soft set over U when A ⊆ E and F : A −→ FS(U ), where FS(U ) denotes the set of all fuzzy sets on U . 6

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Obviously, every soft set can be considered as a fuzzy soft set. Following with our car example above, fuzzy soft sets permit to deal with other properties like “expensive” or “modernly designed” for which partial memberships are natural. As is well known, when both U and A are finite (as in the application references mentioned above) soft sets and fuzzy soft sets can be represented either by matrices or in tabular form. Rows are attached with objects in U , and columns are attached with parameters in A. In the case of a soft set, these representations are binary (i.e., all cells are either 0 or 1). Concerning fuzzy soft set based decision making, the most successful approaches are probably Roy and Maji [35], Kong et al. [36] and Feng et al. [37] as explained above. 3. Fuzzy soft set based decision making: a new proposal of solution In this section we approach the problem of applying fuzzy soft sets in decision making practice, in the terms expressed by the pioneering Roy and Maji [35]. In order to motivate the analysis we use the following practical situation that concerns an object recognition problem.

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Example 1 ([35], Section 4). We consider a set of objects U = {o1 , ..., o6 }. They have different colours, sizes and surface textures. The colour space is represented by A = {blackish (a1 ), dark brown (a2 ), yellowish (a3 ), reddish (a4 )}. The size space is represented by B = {large (b1 ), very large (b2 ), small (b3 ), very small (b4 ), average (b5 )}. The surface texture space is represented by C = {course (c1 ), moderately course (c2 ), fine (c3 ), extra fine (c4 )}. The universal set of parameters is E = A ∪ B ∪ C. A fuzzy soft set (F1 , A) describes the ‘objects having colour space’. A fuzzy soft set (F2 , B) describes the ‘objects having size’. A fuzzy soft set (F3 , C) describes the ‘objects having surface texture granularity’. They are defined by the tabular representations in Figure 1. We assume that the set of choice parameters of an observer is

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P = {p1 = a1 ∧ b1 ∧ c1 , p2 = a1 ∧ b5 ∧ c3 , p3 = a2 ∧ b1 ∧ c2 , p4 = a2 ∧ b4 ∧ c4 , p5 = a3 ∧ b3 ∧ c3 , p6 = a4 ∧ b4 ∧ c3 , p7 = a4 ∧ b5 ∧ c4 }.

This archetypal example shows that there are two possible stages in a fuzzy soft set based decision making problem. In the first place, we may need an aggregation procedure that yields a single resultant fuzzy soft set 7

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(F1 , A)

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Figure 1: Tabular representations of the fuzzy soft sets (F1 , A), (F2 , B) and (F3 , C) in Example 1.

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from the original information (e.g., because we have multiobserver data in terms of various sets of parameters in the problem of object recognition). In the second place, one always wants to make the final decision from the overall information, independently of whether it is produced as a resultant fuzzy soft set or not.

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Remark 1. Observe that the decision maker may proceed with only an observed subset of all aggregate sets of parameters. For example, in Example 1 the decision maker proceeds with combined parameters like “blackish colour, large size and course surface texture” and “reddish colour, average size and extra fine surface texture” but not with “reddish colour, large size and course surface texture”. This restriction reduces the size of the problem but does not require any further methodological change. In order to solve fuzzy soft set based decision making problems like Example 1, the application of Roy and Maji’s pioneering algorithm is as follows: 8

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Algorithm 1 ([35]).

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1. Input the fuzzy soft sets (F, A), (G, B), (H, C), . . . 2. Compute the resultant fuzzy soft set on k objects o1 , . . . , ok using the minimum as an “AND” operator, and place it in the form of a table whose cell (i, j) is denoted tij . 3. Construct a k × k matrix C = (cij )k×k where cij is the number of parameters for which the membership value of oi is greater or equal than the membership value of oj . This is Roy and Maji’s Comparison matrix. Observe that we can describe this step as follows: cij is the number of parameters m for which tim − tjm > 0, or the number of non-negative values in the finite sequence ti1 − tj1 , ti2 − tj2 , . . . 4. For each i = 1, ..., k, compute ri as the sum of the elements in row i of C, and ti as the sum of the elements in column i of C. Then for each i = 1, ..., k, compute the score si = ri − ti of object i. 5. The decision is any object ok that maximizes the score, i.e., any ok such that sk = maxi=1,...,k si .

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Thus if we use Algorithm 1 to give a solution to the problem in Example 1, we proceed as follows:

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Example 2 (Example 1 continued). According to Algorithm 1, in order to solve Example 1 the corresponding resultant fuzzy soft set in Step 2 is (S, P ) given by Table 1 (cf., [35, Section 4]). As a result, Roy and Maji [35] show that option o5 should be selected because s5 > s3 > s6 > s2 > s1 > s4 .

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Feng et al. [37, Subsection 3.3] give an interesting discussion on the validity and limitations of Algorithm 1. We are especially concerned by their argument that there exist some fuzzy soft set based decision problems in which Algorithm 1 may not be successfully used to find the optimal decision. To overcome this criticism we devise new approaches for Steps 2 and 3 in Algorithm 1. In subsection 3.1 we explore the problem at Step 2, namely the aggregation of multi-source data about different sets of parameters. We propose our new procedure for the computation of a Comparison matrix in subsection 3.2 below, where we formulate a newly designed algorithm that incorporates both innovations. Then Subsection 3.3 contains further discussion about the advantages of our algorithm. 9

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Table 1: Tabular representation of the fuzzy soft set (S, P ) in Roy and Maji [35].

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Remark 2. We emphasize that the only source of divergence among the foremost references [37, 36, 35] appears after Step 2 (the problem that we explore in subsection 3.1). Observe that Feng et al. [37] need not discuss the aggregation issue, although they explicitly recognize that the inputted fuzzy soft set in the first step of their Algorithm 2 could be the resultant fuzzy soft set obtained from an aggregation process.

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3.1. A new aggregation procedure for deriving resultant fuzzy soft sets When the practitioner has an array of fuzzy soft sets on a common universe of objects, as in the motivating situation of Example 1, then she first needs to aggregate all this information in order to decide among the resulting outcomes. We explain above that Roy and Maji advocate for using the minimum as an “AND” operator in order to produce their resultant fuzzy soft set from the primitives in Step 2 of their Algorithm 1. This procedure is universally adopted in this literature without further discussion. Example 2 recalls its use. However we believe that such procedure needs improvement. The fact that the value of a combination of parameters is measured by the minimum of the respective values of the parameters is a naive approach that produces undesirable effects. The following brief example clarifies this argument: Example 3. Consider the fuzzy soft sets (F0 , A0 ), (F1 , A1 ) and (F2 , A1 ) whose tabular representations appear in Table 2. For clarification, we let A0 = {dark, bright}, A1 = {large, small}. Then we are interested in combining these parameters for the two objects o1 , o2 that constitute our universe U . Suppose that we use the minimum as an “AND” operator. In this case 10

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the resultant fuzzy soft set that arises from (F0 , A0 ) and (F1 , A1 ) coincides with the one that arises from (F0 , A0 ) and (F2 , A1 ). This is the fuzzy soft set whose tabular representation appears in Table 3. As a result, we conclude that there is a loss of information when we apply Roy and Maji’s aggregation procedure. The reason is that although (F1 , A1 ) and (F2 , A1 ) define rather different fuzzy soft sets, the application of such procedure ends in the same resultant fuzzy soft set. e2

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Table 2: Tabular representation of the fuzzy soft sets (F0 , A0 ), (F1 , A1 ) and (F2 , A1 ) respectively.

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Table 3: Tabular representation of the resultant fuzzy soft set that arises from performing either “(F0 , A0 ) AND (F1 , A1 )” or “(F0 , A0 ) AND (F2 , A1 )” when “AND” is the min operator (Roy and Maji’s [35] procedure).

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In multi-valued logic, t-norms generalize conjunction (“AND”). The most prominent examples of t-norms are the minimum and the product t-norms. Inspired by this fact we propose a new way to aggregate multisource information in situations like Examples 1 and 2. In our procedure we use the product instead of the minimum as the “AND” operator. The objective is to avoid potentially dramatic losses of information (cf., Example 3). Let us clarify this procedure and its advantages with the following two illustrations. Example 4. Let us assume that we have the input data of Example 3. Suppose that we now use our proposed procedure, i.e., the product as the “AND” operator. 11

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Then the resultant fuzzy soft set that arises from combining (F0 , A0 ) and (F1 , A1 ) is the fuzzy soft set whose tabular representation appears in Table 4. However when we combine (F0 , A0 ) and (F2 , A1 ) we obtain the resultant fuzzy soft set whose tabular representation appears in Table 5. It is apparent that now they are different. As a result, with this example we conclude that the introduction of a more precise “AND” operator permits to better discriminate among resultant fuzzy soft sets.

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Table 4: Tabular representation of the resultant fuzzy soft set that arises from performing “(F0 , A0 ) AND (F1 , A1 )” when “AND” is the product operator.

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Table 5: Tabular representation of the resultant fuzzy soft set that arises from performing “(F0 , A0 ) AND (F2 , A1 )” when “AND” is the product operator.

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Example 5. Let us assume that we have the input data of Examples 1 and 2. Suppose that we now apply our proposed procedure, i.e., the product as the “AND” operator. Then the resultant fuzzy soft set that arises is (S 0 , P 0 ) in Table 6. If we continue with Steps 3 to 5 in Algorithm 1 then one concludes that o6 (instead of o5 ) should be selected. For brevity we skip the details of such wellknown analysis. But we emphasize the fact that by changing the aggregation procedure in order to better capture the real features of the observations, the modified Algorithm 1 detects that the optimal decision should be different. 12

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Table 6: Tabular representation of the resultant fuzzy soft set (S 0 , P 0 ) derived from the inputs in Example 1, when “AND” is the product operator.

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As is apparent from these examples, when we use the product in order to derive resultant fuzzy soft sets from preliminary fuzzy soft sets we obtain several benefits. The resultant fuzzy soft set gives a finer assessment of the combined parameters that incorporates all the original constituents. Hence we avoid undesirable situations like the aggregation conflict in Example 3. It is more difficult to get ties when scores are applied in order to rank the resultant fuzzy soft set. And therefore the final ranking better discriminates among the objects.

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3.2. A new algorithm for ranking fuzzy soft sets Now we proceed to formulate a newly redesigned algorithm, that overcomes the drawbacks of earlier approaches that we have brought to light. The discussion in subsection 3.1 suggests that Algorithm 1 can be modified so as to incorporate a more sophisticated construction of resultant fuzzy soft sets at Step 2. Another aspect that we find controversial is the definition of C in Algorithm 1. Its cells are integer numbers between 0 and the number of parameters, thus ri , ti , and the crucial si are integers too. This is a source of indeterminacy because it makes more likely that the decision ends in a draw. The same criticism applies to Kong et al.’s algorithm, which is reexamined and clarified in Feng et al. [37, Subsection 3.2]. And it is more pressing in the algorithms in the adjustable approach by Feng et al. [37]. It is clear that these algorithms tend to produce many ties, especially when there are few parameters. In view of all these shortcomings, we advocate for using a new construction that avoids the use of “crisp” values at the core of the definition of Roy 13

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and Maji’s Comparison matrix. Instead of establishing whether an object beats another one or not in an absolute manner, we measure this notion in terms of the difference of their membership values relative to the maximum membership among objects. The idea behind this modification is that a difference of say, 0.1, is much more significant when the objects’ memberships are bound by 0.25 than when there are objects with large memberships like 0.95 or 1. Therefore our proposal incorporates this correction as well as the novelty explained in subsection 3.1 as follows: Algorithm 2.

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1. Input the fuzzy soft sets (F, A), (G, B), (H, C), . . . 2. Compute the resultant fuzzy soft set on k objects o1 , . . . , ok using the product as an “AND” operator, and place it in the form of a table whose cell (i, j) is denoted tij . 3. For each parameter j, let Mj be the maximum membership value of any object, i.e., Mj = maxi=1,...,k tij for each j = 1, ..., q. Now construct a k × k comparison matrix A = (aij )k×k where for each i, j, we let aij be the sum of the non-negative values in the following finite sequence: tiq − tjq ti1 − tj1 ti2 − tj2 , , ......, . M1 M2 Mq

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This matrix can be equivalently displayed as a comparison table. 4. For each i = 1, ..., k, compute Ri as the sum of the elements in row i of A, and Ti as the sum of the elements in column i of A. Then for each i = 1, ..., k, compute the score Si = Ri − Ti of object i. 5. The decision is any object ok that maximizes the score, i.e., any ok such that Sk = maxi=1,...,k Si . The following example illustrates the application of Algorithm 2 to a concrete situation.

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Example 6. Let us assume that we have the input data of Example 1. Example 5 gives the resultant fuzzy soft set obtained in Step 2 of Algorithm 2, namely, (S 0 , P 0 ) in Table 6. Now it is easy to compute its Comparison table by Algorithm 2, which is given in Table 7. Then Table 8 shows its associated scores. As a result, one concludes that o6 should be selected when we consider the input data of Example 1 (instead of o5 , the decision by Algorithm 1). 14

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Table 7: Comparison table of the fuzzy soft set (S 0 , P 0 ) in Example 1 using Algorithm 2.

7.16 5.05 4.02 5.87 10.08 9.55

8.46 8.23 7.80 8.11 4.84 4.29

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Table 8: Score table of the fuzzy soft set (S 0 , P 0 ), derived from its Comparison table by Algorithm 2.

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3.3. A comparison with earlier solutions The advancements in the topic of prioritizing fuzzy soft sets are based on discussions about the performance of existing solutions via examples. In this regard, we now analyze several additional situations that fed the controversy about this problem. They serve us to illustrate the application of our Algorithm 2 and to contrast it with existing proposals.

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Example 7. Table 9 puts forward the tabular representation of the fuzzy soft set (G, B) in Feng et al. [37]. With that example these authors argue that Algorithm 1 is more suitable than Kong et al.’s [36] algorithm. The gist of their argument is that Roy and Maji’s Algorithm 1 clearly determines that o2 should be chosen, while Kong et al.’s approach suggests the choice of o1 . However the choice of o1 seems rather counterintuitive. Let us show that our Algorithm 2 concurs with Feng et al.’s reasonable judgment. 15

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e2

e3

e4

e5

0.9 0.19

0.1 0.2 0.1 0.3 0.3 0.4 0.3 0.4

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o1 o2

e1

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Figure 2 shows how our Algorithm 2 is applied to (G, B). A first table captures the differences in membership values when any two different objects are compared accross parameters, as well as the maxima of the values of the input table for each parameter (i.e., for each column). Then the Comparison table is designed by adding up the non-negative values at each row, divided by the respective maxima at their column. Finally, the Si scores are computed in order to decide among the two objects, which yields the choice of o2 because S2 ≈ 1.29 and S1 ≈ −1.29. We abbreviate this outcome as o2  o1 .

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Table 9: Tabular representation of the fuzzy soft set (G, B) in Feng et al. [37, Subsection 3.3].

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e2

o1

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e4

e5

o2

0 0.2/0.3 + 0.2/0.4 + 0.2/0.3 + 0.1/0.4 ≈ 2.08

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e3

0.71 −0.2 −0.2 −0.2 −0.1 −0.71 0.2 0.2 0.2 0.1 0.9 0.3 0.4 0.3 0.4

Diff. memberships o1 vs. o2 Diff. memberships o2 vs. o1 Mj

o1 o2

e1

Row-sum (Ri )

Column-sum (Ti )

0.79 2.08

2.08 0.79

0.71 0.9

≈ 0.79 0

Score (Si ) −1.29 1.29

Figure 2: Computing the Comparison table and scores of the fuzzy soft set (G, B) through Algorithm 2.

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e1 o1 o2 o3

e2

e3

e4

0.9 0.7 0.3 0.2 0.8 0.6 0.4 0.3 0.7 0.5 0.5 0.4

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Example 8. Feng et al. [37] consider the fuzzy soft set (S1 , P1 ) whose tabular representation is given in Table 10. In order to apply Algorithm 2, its Comparison table is given in Table 11 and Table 12 gives its associated score table. Henceforth our procedure produces o3  o2  o1 , i.e., S3 > S2 > S1 .

o1

o2

o3

0 0.254 0.508 0.45 0 0.254 0.9 0.45 0

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o1 o2 o3

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Table 10: Tabular representation of the fuzzy soft set (S1 , P1 ) in Feng et al. [37, Subsection 3.3].

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Table 11: Comparison table of the fuzzy soft set (S1 , P1 ) in Table 10.

Column-sum (Ti )

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Row-sum (Ri ) 0.762 0.704 1.35

1.35 0.704 0.762

−0.588 0 0.588

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o1 o2 o3

Score (Si )

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Table 12: Score table of the fuzzy soft set (S1 , P1 ) in Table 10.

In Example 8 our outcome coincides with Feng et al.’s [37, Definition 4.1] solution for level 0.4 and is similar to their solution for level 0.5, namely, o3  o1 ∼ o2 : the same option is placed at the top. However it is only barely similar to their top-level solution, o3 ∼ o1  o2 (see [37, Table 10]). It must be emphasized that according to Feng et al. [37], the procedures by Roy and 17

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Maji [35] and by Kong et al. [36] produce a triple tie hence they provide no prioritization whatsoever (cf., [37, Subsection 3.3]). On this basis Feng et al. conclude that these two algorithms have limited power of discrimination. Incidentally, their top-level decision rule ends in a tie for first place between o1 and o3 : cf., [37, Table 10]. As a coda, we mention that these and other known examples insist on the proliferation of ties in the application of prior algorithms. Another indication is that Feng et al.’s 0.5-level decision rule applied to their Example 2.3 (with a universe of 5 houses and 3 parameters) ends in a tie for first place too: cf., [37, Table 8]. Although Feng et al. [37, Section 5] suggest that this difficulty may be overcome by the recourse to the concept of weighted soft sets, here we do not explore that route. Finally, in Table 13 we compare the focal criteria that we have discussed according to their main characteristics. Table 13: Features of the main fuzzy soft set based decision making procedures: a comparative study. Multi-source aggregation

Ranking Methodology

[35]

Min operator

[36]

Not discussed

Scores computed from Roy and Maji’s (crisp) comparison matrix Fuzzy choice values

[37]

Not discussed

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Procedure

Corrected objections

Yes

Ties proliferate Loss of information at aggregation stage Criterion is very controversial Ties proliferate Richness at the cost of indeterminacy Additional inputs needed (e.g., threshold fuzzy set)

Yes

Choice value of level soft set

No

Scores from new relative comparison matrix

Yes

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Algorithm 2 Product operator

Unique solution

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4. Final comments and conclusion In this paper we reexamine the application of fuzzy soft sets in decision making practice as first studied by Roy and Maji [35]. We concur with Feng et al. [37] when they argue that the choice value designed for crisp soft set based decision making is ill-suited for this case, and that the more appropriate method using scores has some limitations in the original formulation by Roy and Maji [35]. 18

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We show that the latter methodology can be further exploited in order to avoid many of its drawbacks. The advantages of our new algorithm are noteworthy. Our proposal admits multi-source data set but it aggregates these inputs into a resultant fuzzy soft set by a more accurate operator. Although it relies on newly constructed scores, it is still very easy to understand and produces a well-determined choice. However, it has a better power of discrimination than prior solutions. As to computational complexity, it can be easily and efficiently implemented in a computer. The main technical novelties of our Algorithm 2 are as follows.

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(1) When the original data originate in multi-source input parameter data sets our algorithm avoids the loss of information that the min operator imposes. The recourse to the product as the “AND” operator at the information fusion stage guarantees a more faithful assessment of the combined parameters than in earlier solutions, since it incorporates all the original constituents. This permits to avoid undesirable situations like the aggregation conflict in Example 3. (2) Constructing a Comparison table in a novel way permits to produce a compelling computation of new scores. It is based on relative rather than absolute differences. This produces a better assessment of intrinsic variations among alternatives, with respect to each attribute.

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In subsection 3.3 we prove by examples that Algorithm 2 is indeed different from prior proposals in the literature.

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As to future research, one can discuss whether these ideas can help to manage fuzzy soft set based decision making problems under incomplete information in the terms of Zou and Xiao [32] (see also Han et al. [31] and even Qin et al. [40] for other interesting approaches to this problem). It would also be interesting to extend our approach to inputs in the form of weighted fuzzy soft sets (cf., Feng et al. [37, Section 5], generalised fuzzy soft sets (cf., Majumdar and Samanta [41, 42]), or intuitionistic fuzzy soft sets (cf., Maji et al. [43, 44]). Finally, we should ponder to what extent decision making practice is limited by the use of indices with the aspiration for precise distinction of the alternatives. In the context of this paper it would be natural to presume that the uncertainty of the information creates decision uncertainty regions, which hampers direct applicability. Obviously this is not an original reflection. For example, Ekel and Schuffner Neto [45, Section 5] have expressed this concern very accurately in a context of comparing alternatives in a fuzzy 19

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environment on the basis of the ranking of fuzzy quantities. Their proposal is associated with transition to multicriteria choosing alternatives because the application of additional criteria can convincingly help to contract the decision uncertainty regions. Developing these deep ideas is far beyond the scope of this paper. Acknowledgements

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The author is grateful to three anonymous reviewers and Roc´ıo de Andr´es Calle for valuable comments and suggestions. Financial support by the Spanish Ministerio de Econom´ıa y Competitividad (Project ECO2012–31933) is gratefully acknowledged.

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