A simple utility function with the rules-verified weights for analyzing
the top competitiveness of WCY 2012
Yu-Chien Ko
a, Hamido Fujita
b,⇑, Gwo-Hshiung Tzeng
c,d aDepartment of Information Management, Chung Hua University, 707, Sec.2 Wufu Road, Hsinchu 30012, Taiwan b
Software and Information Science, Iwate Prefectural University, Takizawa, Japan c
Graduate Institute of Urban Planning, College of Public Affairs, National Taipei University, 151, University Rd., San Shia District, New Taipei City 23741, Taiwan d
Institute of Management of Technology, National Chiao Tung University, Ta-Hsuch Road, Hsinchu 300, Taiwan
a r t i c l e
i n f o
Article history: Received 29 May 2013
Received in revised form 10 October 2013 Accepted 13 October 2013
Available online 24 October 2013
Keywords:
Uncertainty reduction
Evidential weight based on preferences (EWP) Utility
Induction
World Competitiveness Yearbook (WCY)
a b s t r a c t
Uncertainty always causes hesitations during decision-making. The uncertainty reduction however is not available through simple operations and easy interpretation. This research solves this problem by propos-ing an evidential weight based on preferences (EWP). The key technique of EWP is the integration of the roughness measures of an induction rule to reduce noises in doubts. The utilities composed of the derived weight from EWP are empirically used on World Competitiveness Yearbook 2012 to analyze the bench-marking nations. This case study shows European welfare nations (Denmark, Finland, Norway, and Swe-den) focus on the long term strategic competitiveness while Asian tiger nations (Singapore, Hong Kong, Korea, and Taiwan) are energetic on short term surviving competitiveness.
Ó 2013 Elsevier B.V. All rights reserved.
1. Introduction
Evidence and inference are the fundamentals of decision-mak-ing. Uncertainty is always a challenge to both of these elements. In the theoretical frameworks likeFig. 1, the evidential weight, as proposed by Keynes, is based on the probability relations to ex-press the rational belief about the importance and relevance between a primary proposition (premise) and a secondary proposi-tion (conclusion)[1,2]. However, the uncertainty concerns such as the incomplete information for the probability judgment [3], probability unreliability [4–6], and suspicious conduct over all probabilities[7], can cause hesitation in decision-making. These of-ten happen when there is difficulty of consisof-tent interpretation be-tween the relevance and importance of the propositions. Moreover, this inconsistency is normally an effect of the increase or decrease of evidential probabilities. For instance, an increased number of evidences may subsequently increase noise, which cannot raise the importance of the premises. Or, a high relevance might trade off the importance, and vice versa, thus making the interpretation ineffective. These inconsistency problems should be resolved through uncertainty reduction. Recently, the roughness [8], fuzziness[9], statistical reasoning[10], extended from the theory
of evidence [6,7], are used to reduce various uncertainties. However, they have not been able to get rid of the uncertainty in the evidential weight, thus cannot provide a good consistency be-tween importance and relevance to explain implications.
This research aims to propose an evidential weight based on preferences (EWP) with a reduced uncertainty. The technique of the roughness theory is used to approximate a weight having a consistent relevance and importance. The derived weights based on preferences are further designed to formulate a simple utility function (SUF) for analyzing the top competitiveness. The utility of SUF is the product of the derived weight and an observation va-lue, thus different from Keeney and Raiffa’s[11]. To achieve this goal, a methodology is designed by the roughness theory which can induce rules indifferently thus making EWP indifferent from each other. The resultant EWP is then standardized as the rules-verified weight for utility. Empirically, a competitiveness study about Asian Tigers nations (Singapore, Hong Kong, Korea, and Tai-wan) and European welfare nations (Denmark, Finland, Norway, and Sweden) is used for illustration. The difference between these two groups is interpreted relevantly and importantly by our pro-posed utilities.
Roughness is the key concept to solve the uncertainty of eviden-tial weight. Rough sets theory (RST) extended the theory of evi-dence[6,7]to present the vagueness of approximations with the rough membership function (the accuracy rate) in 1995–1997
[12–14]. Later, RST proposed a certainty measure and a coverage
0950-7051/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved.
http://dx.doi.org/10.1016/j.knosys.2013.10.017 ⇑Corresponding author. Tel.: +81 6646316.
E-mail addresses: eugene@chu.edu.tw (Y.-C. Ko), issam@iwate-pu.ac.jp
(H. Fujita),ghtzeng@mail.ntpu.edu.tw,ghtzeng@cc.nctu.edu.tw(G.-H. Tzeng).
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Knowledge-Based Systems
measure for the induction rules in 2002[15]. The uncertainty def-inition for an induction rule was almost complete then. However, these three separated measures cannot identify a unique weight to consistently explain the evidential relevance and importance. Consequently, the dominance-based rough set approach (DRSA) was developed after RST to consider classification, sorting, choice, and ranking problems, and to specify noise as the imprecise rele-vance[16]. The noise is something like a sample (any objects; or in this paper, nations), which has an inconsistency between its pre-mise and conclusion. It usually cannot be avoided and hard to con-trol in the real world. The consistency level which is best to explain the importance and relevance of the premise is still non-determin-istic[17]. The uncertainty reduction in evidential weight becomes more difficult when the problems of measures integration and con-sistency level influence each other.
The characteristics of the benchmarking nations (the top ten or the upper half in competitiveness) can reveal the competitiveness strategies that stakeholders are interested in. With the aforemen-tioned uncertainty, applying the evidential weight to analyze the
benchmarking nations has challenges summarized as the
followings:
A simple utility function composed of evidential weights has not been successfully derived for competitiveness analysis. Generally, the utility knowledge can sufficiently help determine the key competitiveness characteristics. Especially, the eviden-tial weight and the observation can make utilities more illustra-tive. World Competitiveness Yearbook (WCY), however, assumes that every criterion performs equally and operates in a simple linear formula as Eq.(1) [18].
f ðxÞ ¼X
m
j¼1
wjrx;j ð1Þ
where wjis a weight of criterion qj, m is the number of criteria, x
represents a nation, rx,jrepresents a value of criterion qjwith
re-spect to nation x. Finally, f(x) is the competitiveness score of na-tion x. In the academic researches[19,20], the equal weights are criticized. After empirical testing, they claimed that the weights of WCY cannot be equal. A modified function becomes necessary and important.
The evidential weights are still vague or unreliable for utilities
[3,5,12–17]. Especially that they cannot clearly interpret the rel-evance and importance consistently.
To overcome the above challenges, a methodology for classifica-tion of the benchmarking naclassifica-tions named I-EWP (the inducclassifica-tion of EWP), is designed inFig. 2. I-EWP extends RST and DRSA to reduce uncertainty in the relevance and importance by integrating the roughness measures. I-EWP can be processed by Lingo 12 empirically.
I-EWP has two stages. Stage I solves EWP by considering the certainty, coverage, and accuracy rates of RST and DRSA. These are presented in Section2.3, and redesigned for the top competi-tiveness in Section 3.2. Stage II proposes SUF with the derived weight from I-EWP to classify the benchmarking nations. This SUF is verified with the classification results. Empirically, all na-tions and criteria of WCY 2012 are included to avoid subjective bias.
This paper has two main parts. The first part presents the design and implementation of I-EWP, the details of which are described in Section 3. The second part applies the rules-verified weight to a case study about European welfare nations and Asian tiger nations. Most of these nations belong to the top level but have different styles of competitiveness. Their difference is hard to distinguish because their competitiveness is close to one another. Therefore, the weighted utilities are aggregated to distinguish their competi-tiveness. This issue will be discussed in Section5.
The remainder of this paper is organized as follows: Section2
reviews the evidential weights. Section3presents the design and implementation of I-EWP. Section4addresses application results of I-EWP, and Section5presents discussions on the proposed util-ities and the case study. Finally, concluding remarks are stated to close the paper.
2. Literature review
The related theories of evidential weights are presented in this section; Section2.2is about the evidential weight; Section2.3is about DRSA and RST. The dataset of this research is described next. 2.1. Research dataset
International Institute for Management Development (IMD) annually publishes WCY, a well-known report which ranks and analyzes how a nation’s environment can create and develop sus-tainable enterprises [21,22]. WCY is a product cooperating with
Probability- based uncertainty (1921) Theory of evidence (1967~1976) Statistical Reasoning (1991) Roughness (1988) Fuzziness (1965)
Fig. 1. Probability-based uncertainty.
WCY with 59 nations Inducing EWP Deucing rules
Fig. 2. I-EWP for the benchmarking nations.
Table 1
4 factors and 20 criteria of WCY-IMD 2012.
Economic performance Business efficiency
q1 Domestic economy q11 Productivity and efficiency q2 International trade q12 Labor market
q3 International investment q13 Finance
q4 Employment q14 Management practices
q5 Prices q15 Attitudes and values
Government Efficiency Infrastructure
q6 Public finance q16 Basic infrastructure q7 Fiscal policy q17 Technological infrastructure q8 Institutional framework q18 Scientific infrastructure q9 Business legislation q19 Health and environment q10 Societal framework q20 Education
fifty-four partner institutes worldwide. Its ranking considers broad perspectives by gathering the latest and most relevant data on the subject and by analyzing the policy consequence. The dataset in-clude 59 nations, 4 consolidated factors, and 20 criteria inTable 1 [18].
2.2. The evidential weight
The evidential weight was proposed by Keynes in 1921 to ex-press the degree of relevance in terms of probability. The main idea claims that the doubtful arguments relevant to decision should be considered quantitatively instead of Logic only. Its application re-quires considering not only the knowledge of decision makers but also circumstances for induction[23], thus can estimate the evidential relevance like goodness and risk [3,24]. However, the evolution of the evidential weight is criticized as below.
The numerical indeterminateness of probability. Mathematical expectation of probability cannot always possibly determine which alternative ought to be chosen[3].
The uncertainty from Bayesian measure. Bayesian’s assumption that a single probability measure over states can represent belief is challenged with unreliability. The reason is that only partial information is available, thus Bayesian measure can be epistemic and cannot be fit for decision[5].
The incomplete information for decision. This is explained by a stopping problem, wherein there is difficulty in finding a rational principle to decide when or where to stop the process of acquiring information in forming a probability judgment
[25].
The above weaknesses, which illustrate the uncertainty of evi-dential weights, have not been solved. In the literatures, there are three paradigms intending to solve this uncertainty and give judgment information. The first is the roughness, presented in Sec-tion2.3. The second is the fuzziness proposed in 1965 which is not included in this paper due to incomplete definition about the evi-dential weights. The third is the statistical reasoning which has the techniques of analyzing randomness, imprecision, and vagueness
[26,27]. Roughness concept in decision making is suggested as the most promising candidate for a unified theory to solve the uncertainty[28]. Therefore, the roughness theory is chosen in this paper to analyze the evidential weight.
2.3. DRSA and RST
DRSA is a powerful technique of relational structure and can in-duce conditional preferences for classification, sorting, choice, and ranking[16]. The induced preferences for the ranking can imply the evidences to achieve the dominance class. There are four parts to illustrate this concept. First is the ranking unions: ClPt (the
up-ward union of classes which includes objects ranked at least tth) and Cl<
t (the downward union of classes which includes objects
ranked less than tth), where Cl is a cluster set containing prefer-ence-ordered classes Clt, t 2 T and T = {1, 2, . . . , n}. The formulations
for the above statement can be expressed as Cl = {Cl1, . . . , Clt,
. . ., Cln}, Cl1= {y 2 U:y is ranked in the top position}, Cl2= {y 2 U:y
is ranked in the second position}, . . . , and Cln= {y 2 U:y is ranked in
the bottom position} where U is a set with decision- makers’ pref-erence orders and n is the number of prefpref-erence-ordered classes. For all s, t 2 T and s P t (rank of s P rank of t), every object in Cls
is preferred to be at least as good as any of object in Clt. The upward
union is constructed as ClP
t ¼ [sPtClsfor s P t; inversely, the
down-ward union as Cl<
t ¼ [s<tClsfor s < t. A representation of the upward
union, called the dominating set, can rely on a set of criteria, P. It follows the dominance principle of requiring each chosen object
at least as good as object x in all considered criteria of P. The gran-ules of a dominating set based on P can be viewed as the granular cones in the criteria value space. Vice versa, the dominated set for the downward union follows the dominance principle and has the granules in the opposite direction. These cones are named as P-dominating and P-dominated sets[25], respectively.
Second is about the dominance sets. For instance, object y dom-inates object x with respect to a criteria set P (denotation yDPx). A
dominance set means an important set. Given x, y 2 U and P, the dominance sets are formulated as:
P dominating set : DþPðxÞ ¼ fy 2 U; yDPxg P dominated set : DPðxÞ ¼ fy 2 U; xDPyg where x, y 2 Cl, y%qx for Dþ PðxÞ; x%qy for D PðxÞ, and all q 2 P.
Third is about the use of relevant evidences to explain the rank-ing unions with conditional preferences. For instance of assignrank-ing objects into P-dominating set, evidences have two types. One is called consistent evidence, i.e., objects can be properly assigned into Dþ
PðxÞ and Cl P
t . The other is inconsistent evidence, i.e., objects
assigned in ClP
t possibly violate the dominance principle of D þ PðxÞ.
In other word, this inconsistent evidence is not a member of a dominating set but assigned to the upward union. Therefore, inconsistent evidence is the major part making induction degener-ate. According to the dominance consistency, there are three approximations defined for relevant evidences.
P ClP t ¼ x 2 U : DþPðxÞ # Cl P t ;P ClP t ¼ [ x2ClP t Dþ PðxÞ Bnp ClPt ¼ P ClPt P ClPt P Cl< t ¼ x 2 U; DPðxÞ # Cl < t ; P Cl< t ¼ [ x2Cl< t D PðxÞ; Bnp Cl<t ¼ P Cl<t P Cl<t where t ¼ 1; 2; . . . ; n; Bnp ClP t and Bnp Cl< t
are P-doubtful re-gions. Objects in P-doubtful regions are inconsistent. In a simple word, P ClPt
requires the largest union of P-dominating sets to be properly included in ClP
t. P Cl P t
requires the smallest union of P-dominating sets to contain all elements of ClP
t while allowing some
inconsistent objects.
Finally, the following is about the three measures related to the evidential weight.
Accuracy rate (AR)[14,29]
The accuracy rate presents the ratio of the proper assignment to the possible assignment. Two typical accuracy rates (
a
) are listed as:a
ClP t ¼jP Cl P t j jP ClPt j¼ jP ClPt j jUj jP Cl< t ja
Cl< t ¼jP Cl < t j jP Cl<t j¼ jP Cl<t j jUj jP ClPt jThe symbol
a
is used to present ‘a ratio of the cardinalities of P-lower approximation to those of P-upper approximation, i.e., the degree of the properly classifying approximation relative to the possibly classified approximation’.Coverage rates (CR)[15,29]
The coverage rate expresses ‘the probability of objects in the P-lower approximation relatively belonging to the corresponding union of decision classes’, defined by Pawlak and Greco. There are two typical coverage rates (CR) for the upward unions ClPt
and the downward union Cl<
t, which are formulated as follows:
CR ClPt ¼jP Cl P t j jClPt j ;CR Cl<t ¼jP Cl < t j jCl<tj
Certainty rate (Cer)[15]
A certainty rate of RST is formulated as:
Cerð/; wÞ ¼Cardk/ \ wk
Cardk/k for / ! w
where / and
w
are sets for condition and conclusion. Card jj means the number of elements in a set. In a reverse way to explain the cer-tainty rate, the ratio can be used to express the degree of the noise within the condition for implication.Saaty (2001) proposed that pair-wise comparisons and induc-tions can be formulated as ratios, and then transformed the com-parisons into the priority of criteria, or the criteria weights[30]. He also mentioned that the ratios represent how much more or less a criterion is as compared to another, and that its application can determine how close the criteria are. Furthermore, he empha-sized that ratio operations are independent from irrelevant alter-natives. Thus the ratio scales derived from different scales (criteria) can be implemented mathematically to generate a char-acteristic ratio with invariance. Based on these theories, a multipli-cation of ratios can express the quality of induction. These ratio operations can be further used to solve the evidential uncertainty, as mentioned next section.
3. I-EWP and The proposed utility
I-EWP is designed to reduce the uncertainty in EWP through the induction process for each criterion. Because the induction pro-cesses are independent from each other, the roughness measures are also independent between any two criteria. Thus, the product of the roughness measures can exist in an indifference curve dis-tinguished by preference orders. There are four parts in this sec-tion. Firstly, the data set for this research is presented next. Section3.2presents the uncertainty reduction of I-EWP and the proposed utilities for classifying the benchmarking nations. The validation of driving EWP is presented in Section 3.3. A simple example of I-EWP is illustrated in Section3.4.
3.1. Dataset
The dataset of this research is collected from WCY 2012, which adopts all criteria and nations, i.e., 20 criteria and 59 nations (ob-jects shown by x or z). The ranking union for the dominating com-petitiveness includes the top ten or the upper half nations: Canada, Germany, Hong Kong, Norway, Qatar, Singapore, Sweden, Switzer-land, Taiwan, and USA. Alternatively, the upper half has 29 nations. 3.2. I-EWP and the induced utilities
The roughness measures for an induction rule are defined with the coverage, accuracy, and certainty rates which are related to the evidential weight. RST has techniques such as a deterministic ap-proach to express the vagueness with approximations (1982)
[31], the probabilistic rough set model dealing with the informa-tion uncertainty (1988)[8], linking the belief function and plausi-bility of evidence theory to the lower and upper approximations (1996)[13,14], and the complete definition of the roughness mea-sures for an induction rule (2002) [15]. Alternatively, DRSA ex-tended the roughness theory to ranking relevance (1995) [32]. Later, DRSA provided evidential measures for the ranking union (2001)[29]and for the variable-consistency (2007)[17]. However, the variable-consistency measure does not specify the best level for uncertainty reduction. Up to now, all these measures individu-ally perform well while only one measure cannot clearly specify
uncertainty nor reduce uncertainty well. Therefore, an integration design is proposed in the followings.
The design of I-EWP for an induction rule has three parts. The first part is based on DRSA to approximate the optimal classifica-tion for the benchmarking naclassifica-tions. The evidential relevance and importance for the ranking union will be solved here. The second is based on RST to approximate the minimum uncertainty (or max-imum certainty) for the evidential relevance and importance. The third integrates the first and the second parts to approximate an optimal classification with uncertainty reduction. The resulting va-lue is assigned as EWP inFig. 3. This EWP will be standardized to replace the weight in Eq.(1)for presenting a simple utility function different from Keeney’s (1976).
I-EWP is implemented from the induction rule, qP j;t0! Cl
P t , as
presented inDefinition 2. There are six definitions for completing the implementation. Definition 1 presents the competitiveness DRSA.Definition 2explains the induction rule of competitiveness.
Definition 3 talks about the induction evidences.Definition 4 is about the measures based on the induction evidences.Definition 5discusses the quality classification. And finally,Definition 6 dis-cusses the calculation of the rules-verified weights by standardized EWP.
Definition 1 (The competitiveness DRSA). DRSA ¼ ðU; Q ; f ; R; ClP t Þ,
where U = {yjy = 1, . . . ,n}, Q = {q1,q2, . . . ,qm}, f:U Q ? R, R is a
ranking set, R 2 f1th;2th; . . . ; nthg; ClPt is a ranking union having
nations at least t, and t is a rank place like 10th. Definition 2. An induction rule of competitiveness qP
j;t0! Cl P t
rep-resents how a criterion qjsupports nations to achieve the top t
posi-tions where qP j;t0;ðqPj;t0¼
S
sPt0qj;sÞ, is also a ranking union containing
the top t0positions with respect to q
j. This rule associates the
rank-ing evidences to a rankrank-ing union, which is independent to addition or removal of other criteria. Our design can be conceptualized as in
Fig. 4.
Definition 3 (The induction evidences (objects)). Under the induc-tion rule qP
j;t0! Cl P
t , there are two approximations defined with
boundaries x and x where x 2 ClP t ; x 2 Cl
P
t , and the rank of x is
always higher than or equal to that of x. x is assumed as the bound-ary of the important evidences and x as the boundary of the rele-vant evidences. These two types of evidences are defined as:
Important evidences: Dþ PðxÞ. Relevant evidences:DþPðxÞ. Approximating optimal classification by DRSA Approximating minimum uncertainty by RST Approximating the quality classification with uncertainty reduction CR AR Certainty EWP I-EWP
The important evidences belong to the upper part of the relevant evidences in Fig. 4. The approximations based on the induction evidences are defined as:
Important approximation: P0 ClP t ¼ DþPðxÞ \ ClPt . Relevant approximation: P0 ClP t ¼ DþPðxÞ. Doubtful region: Dþ PðxÞ DþPðxÞ.
Important approximation contains the important evidences belonging to the ranking union. It is same as the lower approxi-mation of DRSA. Relevant approxiapproxi-mation contains the evidences above the boundary x and requires that x belongs to the ranking union. Doubtful region contains the evidences that are relevant but not important. The noise in this area is dissimilar to the important evidence, and is called distinguished noise. Therefore, the noise within the approximations is defined as:
Undistinguished noises: Dþ PðxÞ P0 Cl P t . Distinguished noises: Dþ PðxÞ DþPðxÞ P Cl P t .
The distinguished noises are objects away from the important evidences, and normally located in the doubtful region. The undistinguished noises are together with the important evidences and cannot be separated by objective methods.
Obviously, the more evidences in P0 ClP t
the more important P is; the more noise in P0 ClP
t
the less relevant P is. Due to the impact of noises, x and x are non-deterministic priori. Therefore, x and x are presented as slash lines inFig. 4. They can be specified by approximating the optimal classification with the minimum distinguished noises.
Definition 4 (Measures of the induction evidences). Three measures related to the evidential weight ofFig. 4are defined below.
Evidence-accuracy rate (
a
0)[14,29]An accuracy rate presents the ratio of ‘Important approxima-tion’ to ‘Relevant approximaapproxima-tion’, i.e., the degree of the properly classified evidence relative to the possibly relevant evidences, and is defined as:
a
0¼jP 0 ClP t j jP0 ClP t ja
0for a logical implication represents the degree of necessarycondition of ‘Important approximation’ in the relevant
evidences.
Evidence-coverage rates (CR0)[15,29]
A coverage rate expresses the ratio of ‘Important approxima-tion’ relatively belonging to the ranking union, and is defined as: CR0 ¼jP 0 ClP t j jClPt j
CR0for a logical implication represents the degree of sufficient
con-dition that ‘Important approximation’ influences the ranking union.
Evidence-certainty rate (Cer0)[15]
A certainty rate expresses the ratio of objects in ‘Important approximation’ relatively belonging to the important evidences: Cer0 ¼jP 0ClP t ð Þj jDþ PðxÞj
where jj means the number of evidences in a set. Cer0represents the degree of reliability of P0 ClP
t
.
Definition 5 (The quality classification rate (EWP)). The classifica-tion rate for qP
j;t0! Cl P
t needs to consider both sufficient and
neces-sary conditions. The product of CR0 and
a
0will be a unique valueon an indifference curve, which originates from the product of sufficient and necessary ratios for the indifferent induction rules. The induction measures are independent to addition or removal of other criteria. The product values thus can be used for prefer-ence orders. Further, the quality of classification needs have the reliability concern. According to the logical implication, a quality classification can be formulated as:
Quality classification , Minimum uncertainty.
‘Quality classification if and only if minimum uncertainty’ can be processed by mathematics to get a unique value on an indifference curve. Therefore, the quality classification rate based on evidential weight can be formulated below.
Model I: Solving EWPj
Max EWPj¼ Cer0 CR0
a
0s:t: P ¼ fqjg Cer0¼jP 0 ClP t j jDþPðxÞj ; CR0¼jP 0 ClP t j jClPt j ;
a
0¼jP0 Cl P t j jP0 ClP t j;Model I will approximate a unique value, EWPj, to consistently
enlighten the relevance and importance of criterion qjsupporting
nations to achieve the benchmarking positions. EWPjcan be used as
a weight like a slope inFig. 5.Fig. 5also illustrates how noise in the doubtful region is reduced by Model I. This process cuts nations into yes or no supporting evidences when approximating the quality classification. The vagueness in the doubtful region will diminish due to the optimal solution. The noise in ‘Important evidences’ will be counted as imprecision to the classification. The ranking position of x will becomes higher as much as possible to reduce the noise of ‘Important evidences’. The ranking position of x also becomes highest to reduce distinguished noises. When approximating the optimal solution, x and x will be adjusted to the same position, and EWPjis solved as the slope ofFig. 5.
Definition 6 (The rules-verified weights). EWPjcan be standardized
by Eq.(2)to get the rules-verified weights.
wj¼ EWPj Pm j¼1EWPj ð2Þ Important approximation
(
t)
P Cl
′
≥ The downward union of classes Beyond Relevant approximation Doubtful region , t j t q≥′→Cl≥ x x The upward union of classes (Clt≥) tFig. 4. Approximations based on the induction evidences.
Important appr.
No
Beyond Relevant appr. Doubtful regionYes
( )
f x
: j EWP The slope of dash line yes no espo timat io n uncertainty reduction xjr
where wjis within a range 0–1 thus each criterion can show its
rel-evance and importance consistently relative to others. The rules-verified weights have two merits. First, the weighs get rid of distin-guished noise in the doubtful region. Second, each weight explains the relevance and importance consistently. The rules-verified weight will further function in the utility discussed in the following. 3.3. Validation of the proposed utilities
SUF is designed by substituting wjof Eq.(1)with wjof Eq.(2).
Then, the deduction rules constructed with the utilities become able to classify the benchmarking nations. For instance,
‘if f(x) P h then x2 the top ten nations’ claims a boundary utility h to separate the benchmarking and non-benchmarking nations. h can be solved with constrains: min{f(x)jx 2 the top ten nations} = h and max{f(x)jx R the top ten nations} < h. SUF can be proved as below.
Proof. Let
P1: The benchmarking and non-benchmarking nations; P2: (w1, . . . , wm) is a tuple of the rules-verified weights and
SUFðxÞ ¼Pm j¼1wjrx;j;
P3: The utility classification,
SUFðxÞ j¼1;x2ClP t PSUFðzÞ j¼1;zRClP t where ClP
t is the set of the benchmarking nations.
According to the syllogism,
*P1 ! P2 can be proved by Model I
*P2 ! P3 can be proved by the classification results )P1 ! P3 is proved to be true
The validation of P2 ? P3 will be fulfilled in R1 and R2 based on
Table 2of Section4.1. Therefore, SUF is proved true for the bench-marking nations. h
3.4. An example of the rules-verified weight
An example here illustrates how a criterion classifies objects with an induction rule. By approximating the maximum ‘Important approximation’ and minimum ‘distinguished noise’, the slope of
Fig. 5can be solved as EWPjfor a criterion qj. The dataset of this
example has U = {x1, x2, x3, x4}, Q = {q1, q2}, Rq1= Rq2= {1, 2, 3, 4},
these values, 1, . . . , 4, are ranks, ClP
t ¼ fCl1;Cl2;Cl3;Cl4g, and the
ranking union is {Cl1, Cl2}.
As seen in the experiment results, the cells with bold values are the ‘Important approximations’ when EWP1 are optimally solved
for qP 1;t0! Cl
P
2 and EWP2 for qP2;t0! Cl P 2. Obviously, both qP 1;t0! Cl P 2 and q P 2;t0! Cl P
2 have quality classification. I-EWP
dimin-ishes vagueness and explains the relevance and importance of cri-teria consistently by a weight value only. The inconsistency problem between relevance and importance is solved here.
4. Application results
The application results have two parts. The first includes the rules-verified weights and the deduction rules, which are con-structed from SUF to illustrate the classification of multiple criteria. The second is about the aggregated utilities for illustrating eco-nomic performance, government efficiency, business efficiency, and infrastructure. Therefore, stakeholders can catch points for policy making.
4.1. The resulted weights and induction rules
The rules-verified weights for the top ten and the upper half levels in 2012 are solved and presented inTable 3where the bold figures represent the highest weights.
The deduction rules based on SUF for the top ten and the upper half nations are obtained as:
R1:
if f ðxÞ P 67:85 then x 2 the top ten nations Cer0
¼ 1; CR0¼ 1;
a
0¼ 1SUFðxÞ ¼ 0:08 rx;1þ 0:05 rx;2þ þ 0:03 rx;20
R2:
if f ðxÞ P 60:64 then x 2 the upper half nations Cer0¼ 1; CR0¼ 1;
a
0¼ 1SUFðxÞ ¼ 0:04 rx;1þ 0:03 rx;2þ þ 0:06 rx;20
Obviously, R1 and R2 successfully classify the benchmarking nations and prove that SUF really exists for the benchmarking na-tions. The proposed SUF utilities are thus proved true by our rules-verified weights. Another example of the aggregated utilities is illustrated below.
4.2. The aggregated utilities
The aggregated utilities of SUF can consistently compare the rel-evance and importance of economics, government, business, and infrastructure for competitiveness. This can be formulated by using Eq.(3) and by setting criteria value as one. The results are pre-sented inTable 4. As shown in the table, the infrastructure plays as more important for the upper half nations and the business effi-ciency as a more important factor for the top ten nations.
fPðxÞ ¼
XjPj j2P
wj ð1Þ ð3Þ
Another application of SUF gives a utilities pattern of the top ten na-tions for the competitiveness factors as seen inFig. 6. Results show that the business efficiency has highest relevance and importance
Table 2
An example of EWP.
q1 q2 ClPt Induction rules and EWPj
x1 1 1 Cl1 qP 1;t0! Cl P 2 x2 2 3 Cl2 EWP1= 1 1 1 = 1 for q1 x3 3 2 Cl3 qP 2;t0! Cl P 2 x4 4 4 Cl4 EWP2= 1 0.5 1 = 0.5 for q2 EWPj= Cer0 CR0a0 Table 3
The rules-verified weights (wj) for WCY 2012.
For the top level For the upper half level
w1 0.08 w11 0.08 w1 0.04 w11 0.06 w2 0.05 w12 0.02 w2 0.03 w12 0.04 w3 0.05 w13 0.08 w3 0.05 w13 0.06 w4 0.02 w14 0.08 w4 0.04 w14 0.06 w5 0.02 w15 0.05 w5 0.02 w15 0.05 w6 0.07 w16 0.03 w6 0.03 w16 0.06 w7 0.03 w17 0.07 w7 0.02 w17 0.06 w8 0.06 w18 0.04 w8 0.08 w18 0.06 w9 0.05 w19 0.05 w9 0.06 w19 0.06 w10 0.04 w20 0.03 w10 0.06 w20 0.06
than the other factors. Further, eight nations of the top ten, Canada, Germany, Hong Kong, Norway, Singapore, Switzerland, and Taiwan had their government efficiency as their second important factor. Exceptions are USA and Qatar. Obviously, the aggregated utilities of government are the second important and relevant for the top ten nations. Details ofFig. 6below thus verify
Table 4.
4.3. Achievements of I-EWP methodology
This research has accomplished three achievements. First, the evidential weight which can keep relevance and importance con-sistent is solved by Model I. Second, SUF with the rules-verified weights is effective for the benchmarking nations. Third, the com-petitiveness patterns can be formed by the aggregated utilities. These findings are further discussed in Section5.
5. Discussions on EWP and the case study
This section has two parts. One is about the technique discus-sion. The other is a case study about implications on European wel-fare nations and Asian tiger nations.
5.1. Technique discussion
The technique discussion has four stages, the goal, methodol-ogy, applications, and comparison. This research aims to improve the uncertainty of evidential weights by integrating the roughness measures and proposing the use of SUF with our derived weights.
I-EWP methodology identifies the induction evidences, defines the measures of the induction evidences, integrates the inductions measures, reduces the distinguished noise to the minimum by approximating the maximum ‘Important approximation’, and ap-plies the utility function with the derived weights for classification. These derived weights are used as priori knowledge for deduction thus making them different from the regression weights which are initiated with unknown or non-deterministic variables [33–35]. SUF, which has a known priori, obviously is easier and simpler than the general regression.
The comparison among the rules-verified weights, the regres-sion weights, and Bayesian weights are enlisted inTable 5below. Their comparison conditions are fairly constructed with the same dataset. Their priori and posterior information are the following:
The regression weights are posterior of unknown or non-deter-ministic variables.
Bayesian weights are posterior of data.
The rules-verified weights are posterior of preferences. The comparisons show Bayesian weight considers evidential relevance but does not consider the evidential importance and uncertainty reduction for propositions. In the case of regression weights, which are solved from the evaluation of expectations and observations, they are good at uncertainty reduction but do not directly consider the evidential relevance and importance. Only the rules-verified weights have all these three merits.
5.2. The case study
The case study about the relative advantages between European welfare nations and Asian tiger nations is based on the utilities of SUF. The aggregated utilities can provide an easy and simple vision to interpret the relative advantages by Eq.(4), which is formulated as: Fj¼ X4 x2welfare wjrxj X4 y2tigers wjryj ð4Þ Table 4
The aggregated utilities of WCY 2012.
For the top ten level For the upper half
wEconomic 0.23 wEconomic 0.18
wGovernment 0.25 wGovernment 0.25
wBusiness 0.30 wBusiness 0.26
wInfrastructure 0.22 wInfrastructure 0.30
Fig. 6. A competitiveness pattern of the top ten 2012.
Table 5
Comparison among the related techniques.
Weakness Bayesian weights Regression weights Rules-verified weights Evidential relevance 1 0 1 Evidential importance 0 0 1 Uncertainty reduction 0 1 1 Total advantages 1 1 3
Fig. 7. The relative advantages between European welfare nations and Asian tiger nations.
The application results of Eq. (4) on the case study are pre-sented inFig. 7wherein the positive values mean European na-tions have advantages over Asian tiger nana-tions and the negative values mean Asian tiger nations have the advantages. According toFig. 7, users can tell Asian tiger nations have more advantages (13 of 20) than those of European welfare nations. As a whole, they are more competitive. In details, European welfare nations are better in price (q5), societal framework (q10), productivity
and efficiency (q11), management practices (q14), basic
infrastruc-ture (q16), health and environment (q19), and education (q20).
Generally, developing these advantages requires a long time. Especially that societal framework (q10), management practices
(q14), health and environment (q19), and education (q20) all
influ-ence future generations. Alternatively, Asian tiger nations had thirteen criteria performing better than European welfare na-tions. The major advantages are international trade (q2),
interna-tional investment (q3), employment (q4), public finance (q6),
fiscal policy (q7), attitude (q15), and technology infrastructure
(q17). Asian tiger nations seem to have energetic and aggressive
attitudes. The benefits of these achievements can be seen in a shorter time. We would like to give a comment ‘European wel-fare nations focus more on a long term strategy in competitive-ness while Asian tiger nations are more energetic on short-term surviving competitiveness’. In the future there are two issues deserving of further exploration. First, criteria relationship can be discovered and visualized for competitiveness. Second, more methods can be available to interpret competitiveness. It is our great expectation that all these explorations can be achieved and collected in the knowledge-based systems in the near future.
6. Concluding remarks
This research supposes that the distinguished noise in the doubtful region can cause bigger uncertainty to the evidential weight. In this research an evidential weight based on preferences (EWP) is induced by reducing distinguished noise, and standard-ized to a rules-verified weight. The derived weights are used to formulate a simple utility function (SUF) for analyzing the top competitiveness. In our design, the rules-verified weight keeps evidential relevance and importance consistent thus capable of correct interpretation. There are fourth achievements in this paper. First, the epistemic uncertainty originated in 1921 is improved in probability relation. Second, the utility function proposed in 1973 and 1976 is improved with our derived weights and proved correct for classification. Third, the roughness measures for evidential weight from 1988–2007 are integrated. Fourth, the economic, gov-ernment, business, and infrastructure of WCY 2012 can be com-pared by the proposed utilities. The business efficiency plays the most important role in the top competitiveness level while the infrastructure is the most important one in the upper half level.
The case study between two competitiveness patterns shows that European welfare nations adopt a long term strategy for future generations. They gain advantages over Asian tiger nations in soci-etal framework (q10), management practices (q14), basic
infrastruc-ture (q16), health and environment (q19), and education (q20). On
the other hand, Asian tiger nations are energetic and focus on the short term surviving competitiveness. They are good at interna-tional trade (q2), international investment (q3), employment (q4),
public finance (q6), fiscal policy (q7), attitude (q15), and technology
infrastructure (q17).
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