Interval-Valued Fuzzy Numbersand Concordance Analysisapproach Tothe Operational Organization

Interval-valued Fuzzy Numbersand Concordance AnalysisApproach tothe Operational Organization Type Selection

MING-MIIN YU1* and JING-SHING YAO2

1Department of Business Administration, Fu Jen Catholic University

No.510, Chung Cheng Rd, ShinChuang, Taipei, Taiwan 242, R.O.C

2Department of Mathematics, National Taiwan University

No.1, Sec. 4, Roosevelt Rd, Taipei, Taiwan 106, R.O.C

Abstract: - In this paper the concordance analysis method is applied to evaluate the candidate alternatives of operational organization types. To avoid the complex process in estimating the weight of sensitivity analysis and considering decision-makers’ opinions in the fuzzy sense, this research develops a fuzzy concordance analysis method to choose the most suitable operational organization type. Tao-Yuan’s rapid transit system is an example for studying the outranking relationships among operational organization types. The results are of primary importance in providing decisionmakers with a good reference under complex and uncertain circumstances.

Key-words:- Interval-valued Fuzzy Number; Concordance Analysis; Multicriteria Decision Model; Organizational Type; Rapid Transit.

Introduction

Decision making itself, is broadly defined to include any choice or selection of alternatives, and is undoubtedly one of the most fundamental activities of human beings. The subject of decision making is the study of how decisions are actually made and how they can be made better or more successfully. Classical decision making generally deals with a set of alternative actions that are available to the decision maker, a relation indicating the outcome to be expected from each alternative action, and finally, an objective function that orders the outcomes according to their desirability. When the outcomes for each action are characterized only approximately, we say that decisions are made under uncertainty. Decision makers are normally faced with a lack of precise information to assess a set of alternatives in an uncertain environment. To resolve this problem, the fuzzy set theory pioneered by Zadeh[1] is extensively used. Applications of fuzzy sets within the field of decision making have consisted of fuzzifications of the classical theories of decision making.

Amongst the practical applications reported in the literature (Saaty[2], Buckley[3], Cholewa[4], Barrett and Pattanaik[5], Ramakrishnan and Rao[6], Maeda and Murakamin[7], Chang and Chen[8], Teodorovic[9], and Yeh et al.[10]), fuzzy multicriteria analysis models have shown advantages in modeling different decision problems. Despite their applicability to decision problems, typical fuzzy multicriteria analysis models using ordinary fuzzy sets as their membership functions are often overly precise. However, for some concepts and contexts in which they are applied, we may be able to identify appropriate membership functions only approximately, and the confidence of subjective judgment can be achieved to a level less than 100%. Membership functions of interval-valued fuzzy sets are not as specific as their counter parts of ordinary fuzzy sets, but this lack of specificity makes them more realistic in some applications.

It does seem meaninglessly to obtain the best operational organization type from all possible alternatives, since differences between the best and the second best alternatives are quite limited in practice. In order to rank alternatives and understand the relationship among alternatives, a valued outranking relation should be constructed. The purpose of the concordance analysis is derived to identify the outranking relationship of non-dominated alternatives. Hence, in dealing with this type of decision making problem, a method developed in this paper by applying fuzzy linguistic terms and interval-valued fuzzy numbers into concordance analysis based on statistical data can be used to obtain a set of non-dominated alternatives, which can provide a better reference for making decisions under complex and uncertain circumstances.

Problem Formulation

Supposing we want to evaluate a set of n alternatives under a set of m criteria, which are independent of each other. Subjective assessments are to be given in linguistic terms to determine how important each criterion is for the problem evaluated, represented by a fuzzy weighting vector, and incorporating a statistical method with interval-valued fuzzy numbers to form the effectiveness matrix. In the following sections the description of the determination of the effectiveness matrix and weighting vectors by statistical method and fuzzy theory will be discussedin detail. Now, for the sake of convenience, assume that the effectiveness score of the th alternative and the ith criterion is represented by, and the elements are integrated into an effectiveness matrix of order. To define the weights, let denote a set of m criteria and the weights associated with the m criteria. For brevity, let be a weighting vector of order.

To facilitate the making of qualitative assessments in ranking the alternatives, we introduce a general representation of k linguistic terms for some preference rating. Suppose the fuzzy numbers of each linguistic term are, and their membership function graph is illustrated in Fig. 1.

0 1

Fig. 1. Fuzzy numbers

Here,, .

If , then .

If , then.

Here,, are the fuzzy numbers of m criteria which have their different levels of importance, such as ‘Very low,’ ‘Low,’ ‘Medium,’ ‘High,’ ‘Very high,’ etc., representing fuzzy description. Decisionmaker finds its fuzzy number for the judgment of the importance of criterion as, where and it can be demonstrated as.

Let and be fuzzy addition and multiplication, respectively. The average value of the fuzzy number of criterion importance is and it is denoted as

(1)

In this paper we choose the center-of-area method to determine the best crisp value of the triangular membership function. Let denote the centroid of the fuzzy number, and we then have

(2)

where denotes the estimated weighting value of the ith criterion in the fuzzy sense. The weighting vector for the criteria is represented as.Given the effectiveness matrix and the weighting vector, the objective of the decision making problem is to outrank all the alternatives by giving each of them an overall effectiveness score with respect to all the criteria.

The Fuzzy Concordance Analysis

3.1 Interval-valuedfuzzynumbers

Let the membership function of fuzzy set be; then the membership grade of x is a point value,. For instance, the membership grade of a 30 year-old fuzzy linguistic term =”young man” is a point value. Alternatively, the membership grade of a 30 year-old can be noted as interval. In practice, it is easier to determine a membership grade within an interval than to determine an exact value point. In this regard, the interval-valued fuzzy numbers will be utilized in this paper.

Definition: Let be a fuzzy set on R. It is referred as an interval-valued fuzzy set, if its membership function is as given below.

The membership grade of x of interval-valued fuzzy set is located in the interval.

0 p a b x c q

Fig. 2. Level interval-valued fuzzy number

Let

(3)

and

(4)

Here, and denotes, where,is referred as a level interval-valued fuzzy number(see Fig.2). Let, and we can use the method of integration as an explanatory tool to obtain the centroid of as follows:

So, the centroid or weighted average of is

We define the crisp value of the level interval-valued fuzzy number as

. (8)

If , then Eq.(3) becomes , and then level interval-valued fuzzy number becomes level fuzzy number . From Eq. (5), we have, which is the centroid of level fuzzy number. The result of using Eq. (5) to defuzzy level interval-valued fuzzy number is an extension of the centroid of level fuzzy number. Therefore, we can use Eq. (5) to defuzzy level interval-valued fuzzy number.

3.2Constructing the level interval-valued fuzzy numbers based on statistical data

Suppose that there are r decision-makers. An evaluation is based on a three-dimension of order with elements, which represents the scores of the jth alternative with respect to the ith criterion judged by the sth decisionmaker. Once we have obtained the values of, we usually use the sample values in order to make some inferences about the population represented by the sample.

Let (unknown) be the population effectiveness score of the alternative with respect to criterion.We use theto refer to the value of. In other words, is a point estimate of, and the probability of the point estimate’s error is unknown. It is true that our accuracy increases with large samples, but there is still no reason that we should expect a point estimate from a given sample to be exactly equal to the population effectiveness score. The interval we evaluate is the parameter and is referred as an interval estimate.

If and are the mean and standard deviation of a random sample of size <30 from an approximate normal population with unknown variance, then a confidence interval for is given by

where is the t value with degrees of freedom, leaving an area of to the right and to the left, , , , . To obtain a point on the confidence interval, let level fuzzy number corresponding to the interval be

Again, let, and we can thenobtain level fuzzy number, where . We can define the as the effectiveness score of alternative and criterion ,which is obtained by using Eq. (5) to defuzzy in the fuzzy sense. Thus, the effectiveness matrixPin the fuzzy sense,, can be obtained. If one incorporatesthe weighting vectorEq. (2) with the defuzzied effectiveness matrix, then the concordance index Eq.(6) and discordance index Eq.(8) can be obtained.

3.3 Concordanceand discordance analysis

Let be a set corresponding to m criteria, and is a set of weights associated with the m criteria. Partition the set M into three subsets. Let

where, denotes the effectiveness score of the jth alternative and the ith criterion, denotes that alternative is preferable to alternative , and denotes that alternative and are equivalent. Alternative are arbitrary pairwise alternatives,.

Let , which is referred to as a concordance set. Term ‘’ denotes a weak outranking relationship, which represents that alternative is preferable to or no different to alternative. A discordance set of alternative is defined as, where ‘’ denotes ‘not preferable to’.

The concordance index is hence defined as

(6)

where

(7)

This index can be viewed as a measurement on the satisfaction degree of the decisionmaker if the alternative is chosen over alternative.It is convenient to present the concordance indices in a concord matrix C, where is the element of the jth row and column.

The discordance index can be defined as , (8)

where .The discordance index can be viewed as a measurement on the dissatisfaction degree of selecting alternative over alternative. Again, a discordance matrix D can be constructed in which is the element of the jth row andth column.

To outrank the alternatives based on the concordance index Eq.(6) and discordance index Eq.(8), the concept of choosing alternative by the maximum concord index and minimum discord index is used. A well-known approach [11] concerns the use of graphs by which the various relationships between the pairs of alternatives can be structured. We categorize alternatives into two classes: accepted and rejected. Accepted alternatives have to fulfill the following conditions:

, (9)

where the symbols and are the thresholds which must be determined exogenously.

There are two methods to determine the threshold. The first one is determined by decisionmakers; the second one is the average of the concordance index and discordance index; that is

. (10)

As such, the accepted alternatives must satisfy the concordance and discordance conditions as follows:

. (11)

The concordance condition and discordance condition are used to define the outranking relation. The outranking relation is then used to form a composite graph. Composite graphs are defined by controlling the concordance index and discordance index of the arc that is allowed to belong to a graph. Specifically, alternative is preferable to alternative (i.e., an arc () will appear in the composite graph if and only if the concordance and discordance conditions are satisfied). The comprehensive outranking relationship is used to form a graph in which each node in the graph represents a non-dominated alternative.

Case Study

The following case discusses the selection process of operational organization types to operate Tao-Yuan’s mass rapid transit system in Taiwan. There are five operational organization types (Government-owned-and-operated,; Government-owned-and-operated-contracted-out,; Privately-built-and-government-operated,; Government-owned-and-privately-operated,; Incorporated,).

There are nine evaluation criteria(Government finance, ; Government policy implementation,; Establishment of company,; Level of service,; Financial condition,; Subsidiary business,; Self-governed operation right,; Professional employee,; Operation performance,). The evaluation results are provided by sixteen experts. The whole process is prepared to select the most appropriate operational organization type among these five alternatives. From nine selected criteria, sixteen experts will choose the most appropriate type among five operational organization types.

All nine selected criteria with scales are in five levels of degree: ‘Very low,’ ‘Low,’ ‘Medium,’ ‘High,’ ‘Very high,’ which are of linguistic terms whose linguistic terms are represented by triangular fuzzy numbers that represent their approximate value range between 0 and 1 and are used to reflect the fuzziness of the term. These expressed by the linguistic terms are defined in Table 1.

Table 1. Fuzzy numbers for linguistic values used by the weighting vectors

Linguistic values / Fuzzy numbers
Very low /
Low /
Medium /
High /
Very high /

The weighting vectors for the nine criteria are given by the sixteen experts involved using the linguistic terms in Table 1. After the process of aggregating and averaging, we obtain the fuzzy number of weight to be as shown in Table 2, and the centroid of weighting fuzzy numbers can also be found. The weighting vector W in Table 2 represents a compromised assessment result.

Table 2. The fuzzy numbers and crisp value (centroid) of weights in each criterion

Criteria / Fuzzy number / Crisp value
/ （0.456, 0.687, 0.821） / 0.6547
/ （0.612, 0.756, 0.834） / 0.7340
/ （0.352, 0.631, 0.710） / 0.5643
/ （0.453, 0.685, 0.873） / 0.6703
/ （0.501, 0.750, 0.895） / 0.7153
/ （0.315, 0.489, 0.675） / 0.4930
/ （0.612, 0.834, 0.923） / 0.7900
/ （0.465, 0.687, 0.845） / 0.6657
/ （0.813, 0.857, 0.938） / 0.7793

To facilitate the making of qualitative assessments in evaluating the effectiveness scores of each alternative, linguistic terms defined in Table 3 are used. These linguistic terms are characterized by triangular fuzzy numbers for representing their approximate value ranges between 0 to 5.

Table 3. Fuzzy numbers for linguistic values used by the effectiveness matrix

Linguistic values / Fuzzy numbers
Very poor /
Poor /
Fair /
Good /
Very good /

The linguistic assessment results are obtained by guiding the decision maker through a subjective assessment process of comparing the alternative’s characteristic data with the linguistic terms. The fuzzy effective score for the evaluation criteria with each alternative can be given directly by the decision maker. Let the average effectiveness score of alternative with criterion be, the standard deviation. The results are shown in Table 4.

Table 4. The mean value and standard deviation of the effectiveness scores of the five operational organization types with nine criteria

Operational organization types
Criteria / / / / /
/ 1.1（0.612） / 1.9（0.815） / 3.6（0.892） / 2.9（0.827） / 1.8（0.913）
/ 4.7（1.235） / 3.2（1.036） / 4.8（1.032） / 2.7（0.737） / 3.7（1.024）
/ 4.8（1.285） / 4.3（1.183） / 1.2（0.843） / 2.1（0.946） / 3.2（0.677）
/ 3.2（0.816） / 3.8（0.432） / 3.3（0.683） / 4.6（1.345） / 4.1（1.072）
/ 4.9（0.905） / 4.8（1.535） / 4.7（1.337） / 3.3（1.042） / 4.3（0.731）
/ 3.8（1.013） / 2.9（0.464） / 2.9（0.842） / 4.8（1.178） / 4.1（0.641）
/ 3.1（0.878） / 4.1（0.843） / 3.2（1.022） / 4.9（0.864） / 4.3（0.923）
/ 2.9（0.537） / 3.6（0.647） / 3.3（0.936） / 4.9（1.512） / 4.7（1.164）
/ 3.3（0.624） / 3.7（0.519） / 3.1（0.691） / 4.7（0.833） / 4.2（0.523）

*The number in the parenthesis denotes standard deviation.

In reference to section 3.1 and using the statistical method mentioned in section 3.2, defuzzification of the fuzzy effectiveness matrix (using Table 4) and weighting vectors (using Table 2) are employed in the ordinary concordance and discordance analysis model to find outranking relationships among alternatives, and the results are shown as follows.

By using level fuzzy numbers to obtain an effectiveness matrix, the confidence level should be determined first. Let , for each , then , ; and let , for each , then , . From the table of t distributions the t value with degrees of freedom can be obtained and results are,,, .By applying level (0.85,0.965) interval-valued fuzzy numbers, the effectiveness matrix can be obtained as follows:

The concordance matrix and discordance matrix are:

As we adopt the average method to determinethe threshold, the threshold is,. The concordance, discordance, and comprehensive outranking relationship matrices can now be written:

From, the composite graph can be constructed as:

５

１２３

Fig.3. Composite graph

The resulting graph is shown above in Fig.3. Alternatives are accepted and alternative is rejected. The ranking orders for alternatives are also shown in Fig. 3. The best selection of operational organization type is alternative.

Conclusions

In this paperwe have proposed a new method for the construction of a fuzzy outranking relation in view of its exploitation in the decision making problem. Fuzzy concordance analysis has been outlined to evaluate a discrete number of alternatives by means of statistical methods, fuzzy linguistic terms, level fuzzy numbers, and ordinary concordance analysis. Despite the explicit mathematical presentation in this paper, it must be emphasized that the analytical approach is, in essence, simple and straightforward. The method has been illustrated by an exemplary application to Tao-Yuan’s mass rapid transit system for studying the outranking relationship among operational organization types. The fuzzy concordance analysis method can reflect different value judgments under multisociety circumstances. It can also simplify the computation problem by imposing fuzzy concepts. Therefore, it is very appropriate to the evaluation problem of operational organization type selection.The primary purpose is to assist decisionmakers to make decision under complex and uncertain circumstances.

Acknowledgement

This research was supported in part by the National Science Council of Taiwan, Republic of China, under Grant No. NSC92-2416-H-030-016. The author is grateful to express appreciation to the National Science Council of Taiwan for financial support.

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