A simple utility function with the rules-verified weights for analyzing the top competitiveness of WCY 2012
Introduction
Evidence and inference are the fundamentals of decision-making. Uncertainty is always a challenge to both of these elements. In the theoretical frameworks like Fig. 1, the evidential weight, as proposed by Keynes, is based on the probability relations to express the rational belief about the importance and relevance between a primary proposition (premise) and a secondary proposition (conclusion) [1], [2]. However, the uncertainty concerns such as the incomplete information for the probability judgment [3], probability unreliability [4], [5], [6], and suspicious conduct over all probabilities [7], can cause hesitation in decision-making. These often happen when there is difficulty of consistent interpretation between 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 between 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 value, 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 Taiwan) 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 proposed utilities.
Roughness is the key concept to solve the uncertainty of evidential weight. Rough sets theory (RST) extended the theory of evidence [6], [7] to present the vagueness of approximations with the rough membership function (the accuracy rate) in 1995–1997 [12], [13], [14]. Later, RST proposed a certainty measure and a coverage measure for the induction rules in 2002 [15]. The uncertainty definition 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 relevance [16]. The noise is something like a sample (any objects; or in this paper, nations), which has an inconsistency between its premise and conclusion. It usually cannot be avoided and hard to control in the real world. The consistency level which is best to explain the importance and relevance of the premise is still non-deterministic [17]. The uncertainty reduction in evidential weight becomes more difficult when the problems of measures integration and consistency 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 aforementioned uncertainty, applying the evidential weight to analyze the benchmarking nations has challenges summarized as the followings:
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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 evidential weight and the observation can make utilities more illustrative. World Competitiveness Yearbook (WCY), however, assumes that every criterion performs equally and operates in a simple linear formula as Eq. (1) [18].where wj is a weight of criterion qj, m is the number of criteria, x represents a nation, rx,j represents a value of criterion qj with respect to nation x. Finally, f(x) is the competitiveness score of nation 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.
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The evidential weights are still vague or unreliable for utilities [3], [5], [12], [13], [14], [15], [16], [17]. Especially that they cannot clearly interpret the relevance and importance consistently.
To overcome the above challenges, a methodology for classification of the benchmarking nations named I-EWP (the induction of EWP), is designed in Fig. 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 Section 2.3, and redesigned for the top competitiveness 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 nations 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 competitiveness. This issue will be discussed in Section 5.
The remainder of this paper is organized as follows: Section 2 reviews the evidential weights. Section 3 presents the design and implementation of I-EWP. Section 4 addresses application results of I-EWP, and Section 5 presents discussions on the proposed utilities and the case study. Finally, concluding remarks are stated to close the paper.
Section snippets
Literature review
The related theories of evidential weights are presented in this section; Section 2.2 is about the evidential weight; Section 2.3 is about DRSA and RST. The dataset of this research is described next.
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 processes 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 distinguished by preference orders. There are four parts in this section. Firstly, the data set for this research is presented next. Section 3.2 presents the uncertainty reduction of
Application results
The application results have two parts. The first includes the rules-verified weights and the deduction rules, which are constructed from SUF to illustrate the classification of multiple criteria. The second is about the aggregated utilities for illustrating economic performance, government efficiency, business efficiency, and infrastructure. Therefore, stakeholders can catch points for policy making.
Discussions on EWP and the case study
This section has two parts. One is about the technique discussion. The other is a case study about implications on European welfare nations and Asian tiger nations.
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 standardized 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
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