KR20160011776A - Method for obtaining solutions based on weighting analytic hierarchy process, grey number and entropy for multiple-criteria group decision making problems - Google Patents

Abstract

According to an embodiment of the present invention, a method for deriving a solution of a multiple decision making problem comprises the following steps: generating, by the K number of decision makers participating in decision making, a decision making matrix consisting of linguistic variables which evaluate the n number of alternatives in accordance with the m number of references by each decision maker; obtaining normalized decision making matrixes; operating weight values for each reference based on an entropy in each reference; generating relative weight values which are operated based on each column value of a determination matrix which is generated by calculating relative importance for other reference of any one reference; operating normalized comprehensive weight values which multiply the weight values of each reference and the relative weight values of each reference, and generating a weighting normalized grey decision making matrix by using the operated comprehensive weight values and a normalized grey decision making matrix; generating a grey correlation coefficient and a grey correlation index, and operating a group separation measure by using a positive ideal solution, a negative ideal solution, and the weighting normalized decision making matrix; and determining rankings of the alternatives in accordance with a size order of relative proximities determined based on the group separation measure.

Description

METHOD FOR OBTAINING SOLUTIONS BASED ON WEIGHTING ANALYTIC HIERARCHY PROCESS, GRAY NUMBER AND ENTROPY FOR MULTIPLE-CRITERIA GROUP DECISION MAKING PROBLEMS BACKGROUND OF THE INVENTION 1. Field of the Invention < RTI ID =

The present invention relates to decision methodologies, and more particularly, to multiple decision methodologies.

In the real world, a large number of people with diverse opinions and preferences often make decisions about specific issues in groups. As a simple example, in a plan to have a group of several friends, decision makers can present opinions with different tastes and criteria on different criteria and scales such as size, location, cost, transportation, schedule, and so on. In any case, an alternative should ultimately be drawn, and the resulting alternative should be an alternative that satisfies more participants than any other alternative, although it may not satisfy all participants.

In order to systematically derive a solution to the conventional multiple-criterion decision making (MCDM) problem, a SAW (Simple Additive Weighting) technique, an ELEMENT (Elimination and Choice Translating Reality) ), And TOPSIS (Technique Ordered Preference by Similarity to the Ideal Solution).

Among the classical MCDM methodologies, TOPSIS is a method that proposes a solution with the shortest distance to the positive ideal solution and the farthest distance to the negative ideal solution. This is a methodology that imitates the decision logic of the solution.

However, these classical MCDM problem solving methodologies have been able to derive an appropriate solution in certain situations that are generally suited to simplified modeling, but are limited in dealing with incomplete and ambiguous problems in the real world.

In response, fuzzy theory and gray theory have been proposed, both of which can deal with uncertain information mathematically. Gray theory is more superior to fuzzy theory in that it can handle fuzzy situations more flexibly and can rank all alternatives by calculating the gray possibility degree between the alternative set of alternatives and the reference alternative .

On the other hand, as the high-performance mobile information communication devices become popular, the ubiquitous computing paradigm of the social network environment becomes real and many participants can interact more easily and more positively. Therefore, there is a need for a methodology to solve multiple decision problems, that is, many criteria to be considered, which are involved in many people at the same time, and decision-making problems of many decision-makers in such ubiquitous social environments.

"A Novel Interval Gray Number and Entropy-based Solution for Multiple-Criteria Group Decision Making Problem," Ubiquitous Intelligence & Computing and Autonomic & Trusted Computing (UIC / ATC), HK Kang, DG Kim, HW Jeong, GY Park, ), 2012 9th International Conference on, pp. 349-356, 2012. G. Li, D. Yamaguchi, and M. Nagai, "A gray-based decision-making approach to the supplier selection problem," Mathematical and Computer Modeling, vol. 46, no. 3-4, pp. 573-581, Aug. 2007. Z. Cao, L. Ma, N. Wang, and Y. Wang, "An entropy-based evaluation method of maintenance support system," Reliability, Maintainability and Safety (ICRMS), 2011 9th International Conference on pp. 842-848, 2011.

The present invention provides a method of deriving a solution of multiple decision problems by considering ambiguous and uncertain criteria of multiple participants in the real world based on hierarchical analysis technique (AHP), gray theory and entropy There is.

The problem to be solved by the present invention is to provide a method of deriving a solution of multiple decision problems by considering the subjectivity of multiple participants in the real world based on AHP, gray theory and entropy.

The solution to the problem of the present invention is not limited to those mentioned above, and other solutions not mentioned can be clearly understood by those skilled in the art from the following description.

A method for deriving a multiple decision problem solution performed by a computer according to an aspect of the present invention includes:

(a) an N × m decision matrix (l) consisting of linguistic variables evaluating n alternatives according to m criteria, and K decision makers participating in the decision (gray) -based decision matrices for each decision-maker;

(b) obtaining normalized decision matrices for each decision-maker's decision matrix by normalizing the interval gray numbers corresponding to the linguistic variables;

(c) computing weights for each criterion based on entropy in each criterion, for each decision-maker's normalization decision matrix;

(d) relative to each decision maker, relative weights calculated on the basis of the relative importance values of each row of the decision matrix generated by calculating the relative importance of one of the m criteria to the other criterion Generating a configured relative weight vector for each criterion;

(e) computing, for each decision maker, a comprehensive weight normalized by multiplying the weight of each criterion by the relative weight of each criterion, and using the computed global weights and the normalized gray decision matrix Generating a weighted normalized gray decision matrix;

(f) determining a positive ideal solution and a negative ideal solution from each weighted normalized gray decision matrix of each decision maker;

(g) computing a gray relation coefficient by each decision maker using the positive anomaly solution, the negative anomaly solution, and the weighted normalization decision matrix;

(h) computing a degree of gray relation from the gray correlation coefficient with respect to each of the alternatives;

(i) computing a group separation measure for each alternative for gray decision indices of all decision makers;

(j) determining a relative closeness for each alternative based on the group separation metric;

(k) ranking the alternatives according to the magnitude order of the relative proximities.

According to one embodiment, for the linguistic variable, an interval gray number may be allocated by considering a benefit criterion type and a cost criterion type for each criterion, respectively.

According to one embodiment, the step (c)

의 구간 회색수 요소들

의 전체 합에 대한 구간 회색수 요소들

각각의 비율

을 연산하는 단계;Normalized Gray Decision Matrix

The gray-scale elements

Gt; < RTI ID = 0.0 >

Each ratio

;

로부터 기준

에 관한 엔트로피

를 다음 수학식ratio

From standard

Entropy on

To the following equation

;

에 관한 엔트로피

를 정규화한

를 다음 수학식standard

Entropy on

Normalized

To the following equation

;

에 기초하여 기준

에 관한 가중치

을 다음 수학식The calculated entropy

Based on

Weight

To the following equation

As shown in FIG.

According to one embodiment, in step (d)

은 다음 수학식The determination matrix

Is expressed by the following equation

(1≤i, j≤m)는 m 개의 기준들 중 i 번째 기준의 j 번째 기준에 대한 상대적 주관적 중요도이며,Lt; RTI ID = 0.0 >

(1 ≤ i, j ≤ m) is the relative subjective importance of the jth criterion of the ith criterion among the m criteria,

The step (d)

에 판단 행렬

를 곱한 보정 판단 행렬

을 획득하는 단계;The determination matrix

The judgment matrix

≪ / RTI >

;

의 각 행마다 요소들

의 합산 값들

을 산출하는 단계; 및The correction judgment matrix

For each row of elements

Lt; / RTI >

; And

의 전체 합 S에 대한 각 합산 값들

의 비율로써 상대적 가중치들

을 산출하고, 상대적 가중치들

로 상대적 가중치 벡터

를 각 기준

마다 생성하는 단계를 포함할 수 있다.The sum values

Lt; RTI ID = 0.0 > S < / RTI &

The relative weights < RTI ID = 0.0 >

And the relative weights < RTI ID = 0.0 >

The relative weight vector

To each criterion

.

According to one embodiment, the method for deriving a multi-

Between the step (d) and the step (e)

(d-1) verifying, for each decision maker, the consistency of the decision matrix based on the relative significance values of each column of the decision matrix; And

(d-2) If the consistency check fails, the process returns to step (d) to obtain the determination matrix again, and if consistency verification passes, proceeding to step (e).

According to one embodiment, step (d-1)

은 다음 수학식The determination matrix

Is expressed by the following equation

(1≤i, j≤m)는 m 개의 기준들 중 i 번째 기준의 j 번째 기준에 대한 상대적 주관적 중요도인 경우에,Lt; RTI ID = 0.0 >

(1? I, j? M) is a relative subjective importance for the jth reference of the ith reference among the m criteria,

의 각 열(column)마다 상대적 중요도 요소들

의 합산 값들

을 각각 산출하는 단계;The determination matrix

For each column of < RTI ID = 0.0 >

Lt; / RTI >

Respectively;

을 다음 수학식The first normalization evaluation factor matrix

To the following equation

;

의 각 행마다 요소들

의 평균값들

을 산출함로써 제2차 평가 인자 행렬

을 다음 수학식The first normalization evaluation factor matrix

For each row of elements

Average values

The second evaluation factor matrix < RTI ID = 0.0 >

To the following equation

;

를 다음 수학식The second evaluation factor matrix

To the following equation

에 곱하여 제3차 평가 인자 행렬

를 산출하는 단계;According to the judgment matrix

The third evaluation factor matrix < RTI ID = 0.0 >

;

의 각 열 내의 요소들의 합산값들

로 이루어진 고유벡터(eigenvector)

를 산출하는 단계;The third evaluation factor matrix

Lt; RTI ID = 0.0 >

(Eigenvector)

;

를 다음 수학식The largest eigenvalue

To the following equation

A step supported according to;

The consistency index (CI) is calculated using the following equation

(CR = CI / RI) obtained by dividing the calculated consistency index (CI) by a random index (RI) given according to the number of criteria, and comparing the consistency ratio with a predetermined threshold value, And verifying that it is consistent if it is lower.

과 부정적 이상 해법

을 이용하여 대안들의 회색 상관 계수들(grey relation coefficients)

,

을 다음 수학식According to one embodiment, in step (g), with respect to the k < th > decision maker,

And Negative Ideal Solutions

The gray correlation coefficients of the alternatives are used.

,

To the following equation

는 긍정적 이상 해법

과 가중 정규화된 대안

을 각각 의미하는 두 구간 회색수들의 편차이고,

는 부정적 이상 해법

과 가중 정규화된 대안

을 각각 의미하며,

는

인 차등 계수(differential coefficient)일 수 있다.here

Positive anomaly solution

And Weighted Normalized Alternatives

, Which is the deviation of the gray numbers of the two sections,

Negative adverse solution

And Weighted Normalized Alternatives

Respectively,

The

Which may be a differential coefficient.

,

는, 각 대안에 관하여 상기 회색 상관 계수들

,

을 기초로 다음 수학식According to one embodiment, in step (g), the gray correlation index

,

For each alternative, the gray correlation coefficients < RTI ID = 0.0 >

,

On the basis of the following equation

Lt; / RTI >

,

는 상기 회색 상관 지수

,

에 기초하여 다음 수학식According to one embodiment, in step (i), the group separation measure

,

The gray correlation index

,

On the basis of the following equation

Lt; / RTI >

는 각 대안에 관하여 상기 그룹 분리 척도

,

에 기초하여, 다음 수학식According to one embodiment, in step (h), the relative proximity

For each alternative,

,

On the basis of the following equation

Lt; / RTI >

According to another aspect of the present invention, there is provided a computer-readable recording medium storing a program capable of implementing a method for deriving a multiple decision problem solution according to an exemplary embodiment of the present invention.

According to the method of deriving a multiple decision problem solution of the present invention, when there are a plurality of choices of various aspects in a certain decision problem, an alternative which maximally satisfies a maximum number of participants by adjusting the preferences of a plurality of participants more efficiently can do.

According to the method of deriving the multiple decision problem solution of the present invention, based on AHP, gray theory and entropy, it is possible to overcome subjectiveity and to derive an optimal alternative more accurately than the TOPSIS technique.

The effects of the present invention are not limited to those mentioned above, and other effects not mentioned can be clearly understood by those skilled in the art from the following description.

FIG. 1 is a flowchart illustrating a method of deriving a multiple decision problem solution according to an exemplary embodiment of the present invention. Referring to FIG.
FIG. 2 is a flowchart illustrating a procedure for calculating a weight based on entropy for generating a weighted normalized gray decision matrix for a specific decision criterion of a specific decision-maker, from among multiple decision problem solution derivation methods according to an embodiment of the present invention. Fig.
FIG. 3 is a flowchart illustrating a procedure for calculating a relative weight vector for each evaluation factor of the hierarchical analysis technique among the multiple decision problem solution derivation methods according to an exemplary embodiment of the present invention.
FIG. 4 is a flowchart illustrating a method of deriving a multiple decision problem solution according to an exemplary embodiment of the present invention, using a relative weight vector for each evaluation factor of a hierarchical analysis technique to verify the consistency of a decision matrix assumed in a hierarchical analysis technique A flow chart illustrating the procedure in detail.
FIG. 5 is a graph comparing the proximity indexes derived using the multiple decision problem solution derivation methodology and the conventional TOPSIS methodology, respectively, according to an embodiment of the present invention.

For the embodiments of the invention disclosed herein, specific structural and functional descriptions are set forth for the purpose of describing an embodiment of the invention only, and it is to be understood that the embodiments of the invention may be practiced in various forms, The present invention should not be construed as limited to the embodiments described in Figs.

Hereinafter, preferred embodiments of the present invention will be described in detail with reference to the accompanying drawings. The same reference numerals are used for the same constituent elements in the drawings and redundant explanations for the same constituent elements are omitted.

Throughout this specification, variables such as i, j, k, etc. are natural numbers unless expressly defined otherwise or described in other contexts.

Gray theory

The gray theory is J.L. Deng "The Introduction of Gray System", The Journal of Gray System 1, 1989.

For the sake of clarity, the gray theory is one of the new mathematical theories based on the concept of gray set, which is effectively used to solve the problem in the indeterminate problem with discrete data and incomplete information. .

The interval gray number, the relative operator, and the gray relational coefficient of the sequences can be defined as follows.

1. Section gray number

과 상한

을 가지는 회색수를 가리키며, 상하한이 고정된 값을 가질 경우에

으로 표시한다. 만약

이면,

는 "결정론적 숫자(deterministic number)" 또는 "백색수(white number)"라고 한다. 반면에

인 경우에는,

는 "흑색수(black number)"라고 부른다.
Gray gray area is lower limit

And upper limit

, And when the upper and lower limits have a fixed value

. if

If so,

Is referred to as a " deterministic number "or a" white number ". On the other hand

In this case,

Is called a "black number ".

2. Operation of interval gray number

The arithmetic rule of the gray level of the interval is expressed by the following equation (1).

3. Gray correlation coefficient

,

이라 할 때, 거리 측정 함수

는 다음과 같이 정의된다.

,

, The distance measurement function

Is defined as follows.

In this case, if a sequence consisting of weight interval gray numbers standardizing the index values of the respective alternatives is expressed by the following Equation 3,

The reference sequence is expressed by the following equation (4)

At this time, the gray correlation coefficients of the interval gray numbers are defined as follows.

이고, ζ는 0≤ζ≤1인 구별 계수(distinguishing coefficient)로서 통상적으로 ζ=0.5로 사용한다.
here,

And ζ is the distinguishing coefficient of 0 ζ ≤ 1, which is usually used as ζ = 0.5.

AHP (Analytic Hierarchy Process)

AHP is effective in determining the weight coefficients by partitioning complex factors into several correlated hierarchies. Problems can be organized more systematically and more regularly by AHP.

1. Construction of judgment matrix

및

를 각각 평가 인자들(evaluation factors)이라고 하고,

를

에 대비한

의 중요도(importance value)라고 하면, 예를 들어 다음 표 1과 같이 중요도의 값과 그 언어적 의미를 정의할 수 있다. 여기서 판단 행렬의 평가 인자들은 의사 결정 행렬의 기준 인자들과 실질적으로 동등하다.

And

Are referred to as evaluation factors, respectively,

To

Against

The significance value and its linguistic meaning can be defined, for example, as shown in Table 1 below. Where the evaluation factors of the decision matrix are substantially equivalent to the reference factors of the decision matrix.

Importance value meaning One equal 3 Slightly more important 5 Fairly more important 7 Very important 9 Extremely important

의 관계가 있다.Meanwhile

.

In this case, the determination matrix P can be expressed by the following equation (6).

2. Normalization of decision matrix

The values of the respective elements of the column of the judgment matrix P are summed and normalized by dividing each of the values of the elements by the sum value.

3. Calculating Relative Weight Vector

A relative weight vector is calculated by the average of each horizontal row.

Entropy

Entropy is the numerical value of the amount of information needed to distinguish the derived value of a discrete random variable. With entropy, subjectivity and vagueness can be treated mathematically according to a person's subjective preference.

In view of the above description, a method of deriving a multi-decision problem solution method of the present invention which can overcome subjectivity and derive an optimal alternative more precisely than the TOPSIS technique based on AHP, gray theory and entropy will be described below.

FIG. 1 is a flowchart illustrating a method of deriving a multiple decision problem solution according to an exemplary embodiment of the present invention. Referring to FIG.

Referring to FIG. 1, a method for deriving a multiple decision problem solution can be expressed by the following 13 steps.

First, in step S11, a decision maker (DM) participating in a decision making process is defined as n x m (k) consisting of linguistic variables that evaluate n alternatives according to m criteria Gray-based decision matrices are generated.

Next, in step S12, normalized gray decision matrices are generated by normalizing the gray values of each section with respect to each decision maker.

In step S13, for each decision maker, the weights for each criterion are computed based on the entropy computed from the normalized gray number estimates of the alternatives in each criterion of the normalized gray decision matrix.

In step S14, for each decision maker, a judgment matrix is generated by calculating the relative importance of one of the m criteria to another criterion.

In step S15, for each decision maker, a relative weight vector consisting of relative weights calculated based on the relative importance values of each row of the determination matrix is generated for each criterion.

In step S16, for each decision maker, the consistency of the decision matrix is verified based on the relative importance values of each column of the decision matrix. Since the decision maker compares relatively criterionally with some criterion, it can be contradictory when the decision maker makes a mistake. Therefore, it is possible to verify the consistency by quantifying the degree of contradiction and comparing the degree of the numerical contradiction to a predetermined threshold value.

If the consistency check fails in step S16, the process returns to step S14 to obtain the determination matrix again.

If the consistency verification passes in step S16, the process can proceed to step S17.

In step S17, for each decision maker, comprehensive weights are obtained by normalizing the values obtained by multiplying the weight of each criterion by the relative weight of each criterion, and the computed comprehensive weights and the normalized gray decision matrix To generate a weighted normalized gray decision matrix.

In step S18, for each decision maker, a positive ideal solution and a negative ideal solution are determined from the weighted normalized gray decision matrix, respectively.

In step S19, for each decision maker, the gray relation coefficient of the alternatives is calculated using the positive and negative solutions.

In step S1A, for each decision maker, the degree of gray relation for the positive ideal solution and the negative ideal solution is calculated from the average of the gray correlation coefficients for each alternative.

In step S1B, all decision makers must be considered, so that for each alternative, group separation measures are calculated based on the gray correlation index values of all decision makers.

In step S1C, for each alternative, a relative closeness to the ideal solution is determined, according to each alternative group separation measures.

In step S1D, the order of the alternatives is determined according to the magnitude order of the relative proximity.

In the following, specific calculation procedures are described for each step.

을 생성한다.First, in step S11, the decision makers of K decision-making groups are divided into n × m gray-number-based decision matrices consisting of linguistic variables evaluating n alternatives according to m criteria

.

As shown in Table 2, linguistic variables represent linguistic expressions in which decision makers evaluate the value of each alternative, for example, very low (VL), low (L), medium (M), high H), and very high (VH).

The criterion can be classified into two kinds. One is a criterion that is a benefit criterion, and the other is a cost criterion. The smaller the value, the better the alternative. It is a meaning standard.

Therefore, for the linguistic variables, the interval gray numbers can be allocated by considering the benefit criterion type and the cost criterion type, respectively.

Linguistic variable Benefits Cost Basis Gray number Very low [1, 2] [9, 10] lowness [3, 4] [7, 8] usually [5, 6] [5, 6] height [7, 8] [3, 4] Very high [9, 10] [1, 2]

은 다음 수학식 7과 같이 주어질 수 있다.Using these linguistic variables and gray numbers, the n × m decision matrix

Can be given by Equation (7).

는 n 개의 대안들(alternatives)의 집합이고,

은 m 개의 기준들(criteria)의 집합이다.here

Is a set of n alternatives,

Is a set of m criteria.

는 k 번째 의사결정자가 i 번째 대안에 관하여 j 번째 기준으로 가치 평가를 내린 결과를 가리키는 언어적 변수이다. 언어적 변수 L에 구간 회색수가 할당되어 있기 때문에

를 이용하여 많은 수의 기준, 의사 결정자 집단 및 대안들을 수학적으로 다룰 수 있다.

Is a linguistic variable that indicates the result of evaluating the kth decision-maker at the j-th measure with respect to the i-th alternative. Since the linguistic variable L is assigned an interval gray number

Can be used to mathematically handle a large number of criteria, decision maker groups, and alternatives.

내의 구간 회색수들을 정규화함으로써, 정규화된 회색 의사 결정 행렬

를 구한다.Next, in step S12, with respect to the k-th decision maker, a decision matrix

By normalizing the gray numbers in the interval, a normalized gray decision matrix < RTI ID = 0.0 >

.

Typically, in a multi-criteria decision problem, each measured value must be normalized to be able to cross-operate with each other. Therefore, the decision matrix using the interval gray number is also normalized. In contrast, the conventional decision matrix does not consider that each criterion is classified into the benefit criterion type and the cost criterion type, The normalization can be performed considering the benefit criterion type and the cost criterion type.

는 다음 수학식 8과 같이 표현된다.Accordingly, the gray decision matrix

Is expressed by the following equation (8).

는 i 번째 대안을 j 번째 기준에 의해 평가한 언어적 표현 평가치에 상응하는 구간 회색수이다.here,

Is the interval gray number corresponding to the linguistic expression evaluation value evaluated by the j-th criterion of the i-th alternative.

는 다음 수학식 9와 같이 표현된다.Normalized Gray Decision Matrix

Is expressed by the following equation (9).

는 하한은

이고 상한은

으로서

에 할당된 구간 회색수이며, 구간 회색수

를 정규화한 결과는 구간 회색수

라고 칭한다.In other words,

The lower limit is

And the upper limit is

As

Is the gray number assigned to the segment,

The result of normalizing

Quot;

에 대한 가중치

를 정규화 회색 의사 결정 행렬

의 각 기준에서의 대안들의 정규화된 회색수 평가치들

로부터 산출되는 엔트로피(entropy)에 기초하여 연산한다.In step S13, with respect to the k-th decision maker,

Weight for

Normalized Gray Decision Matrix

The normalized gray number estimates of alternatives at each criterion

Based on the entropy computed from the input data.

In common sense, when valuing each criterion by each decision maker, it is conceivable that each criterion will have a subjective weight that varies from one criterion to another. However, it is not an appropriate solution for many decision makers to set their own weights each time reflecting their subjective intentions.

In this regard, in order to objectively estimate the weights to reflect subjective intentions, the more scattered the evaluation scores given by decision makers on each criterion, the more important that the criterion plays a more important role than the other criteria It is possible to deduce that

For example, if you are a decision maker who thinks that travel standards are important in your travel plans, then the criteria for travel alternatives will vary widely. On the other hand, if the decision maker thinks that the price is relatively insignificant, the evaluation of the alternatives will not be much different according to the price criterion. If so, this decision maker can consider that the travel criteria rather than the price criteria are of relative importance for the final decision of the travel plan.

의 분산(diversity)에 의해 k 번째 의사 결정자의 기준들

에 관한 주관적 의중을 객관적으로 수치화할 수 있도록, k 번째 의사 결정자의 정규화 회색 의사 결정 행렬

에서 각 기준

에 관하여, 엔트로피

의 크기에 상응하는 가중치

를 연산할 수 있다.Accordingly, in the present invention, the normalized gray decision matrix < RTI ID = 0.0 >

The decision of the k-th decision maker

, The k-th decision maker's normalized gray decision matrix

In each criterion

, Entropy

A weight corresponding to the size of

Can be calculated.

에 관한 엔트로피

를 이용하여 가중치

를 연산하는 절차를 예시하기 위해 도 2를 참조하면, 도 2는 본 발명의 일 실시예에 따른 다중 의사 결정 문제 해법 도출 방법 중에서, 특정 의사결정자의 특정 판단 기준에 대한 가중 정규화 회색 의사 결정 행렬을 생성하기 위해 엔트로피에 기초한 가중치를 연산하는 절차를 구체적으로 예시한 순서도이다.Specifically,

Entropy on

Lt; / RTI >

FIG. 2 is a flow chart illustrating a method of deriving a weighted normalized gray decision matrix for a specific decision criterion of a specific decision maker from among multiple decision problem solution derivation methods according to an embodiment of the present invention. FIG. 4 is a flow chart specifically illustrating a procedure for calculating an entropy-based weight to generate an entropy-based weight.

의 j번째 열(column), 즉 j 번째 기준

의 엔트로피

는 단계(S131)에서, 정규화 회색 의사 결정 행렬

의 구간 회색수 요소들

의 전체 합에 대한 구간 회색수 요소들

각각의 비율

을 다음 수학식 10과 같이 연산한다.2, a normalized gray decision matrix

(J) th column

Entropy of

In step S131, the normalized gray decision matrix < RTI ID = 0.0 >

The gray-scale elements

Gt; < RTI ID = 0.0 >

Each ratio

Is calculated according to the following equation (10).

로부터 기준

에 관한 엔트로피

를 수학식 11과 같이 연산한다.In step S132,

From standard

Entropy on

Is calculated as shown in Equation (11).

에 관한 엔트로피

를 정규화한

를 수학식 12과 같이 얻는다.In step S133,

Entropy on

Normalized

Is obtained as shown in Expression (12).

는 항상 0보다 크거나 같고 1보다 작거나 같은 어떤 실수이다.Where m is the number of criteria,

Is any real number that is always greater than or equal to 0 and less than or equal to 1.

에 기초하여 기준

에 관한 가중치

을 수학식 13과 같이 얻는다. In step S134, the normalized entropy

Based on

Weight

As shown in Equation (13).

는 기준

에 관한 분산을 의미한다. 기준

에 기초하여 대안들에 대해 k 번째 의사 결정자가 부여한 가치 평가 점수들이 넓게 분포되어 있을수록 분산은 더 클 것이고, 이는 기준

가 의사결정의 결과를 좌우할 가능성이 커짐을 의미하며, 따라서 상대적으로 중요한 기준임을 의미한다.here,

Standard

. ≪ / RTI > standard

, The variance will be larger as the valuation scores given by the k-th decision maker are broadly distributed to the alternatives,

Means that there is a greater likelihood that the outcome of the decision will influence the outcome of the decision, and thus is a relatively important criterion.

를 산출하여 다음 수학식 14와 같은 판단 행렬

을 생성한다. 여기서, 1≤i, j≤m임에 주의한다.Referring again to Fig. 1, at step S14, with respect to the k-th decision maker, the relative importance of the j-th reference of the i-th reference among the m

And a decision matrix < RTI ID = 0.0 >

. Note that 1? I, j? M.

의 각 행(row)의 상대적 중요도 값들

에 기초하여 연산되는 상대적 가중치들

로 구성된 상대적 가중치 벡터

를 각 기준

마다 생성한다.In step S15, with respect to the k-th decision maker,

The relative importance values of each row of < RTI ID = 0.0 >

Lt; RTI ID = 0.0 >

The relative weight vector

To each criterion

.

로부터 상대적 가중치 벡터

를 연산하는 절차를 예시하기 위해 도 3을 참조하면, 도 3은 본 발명의 일 실시예에 따른 다중 의사 결정 문제 해법 도출 방법 중에서, 계층적 분석 기법의 각 평가 인자에 대한 상대적 가중치 벡터를 연산하는 절차를 구체적으로 예시한 순서도이다.Specifically,

Lt; RTI ID = 0.0 >

FIG. 3 is a flow chart illustrating a method of calculating a relative weight vector for each evaluation factor of a hierarchical analysis technique among multiple decision problem solution derivation methods according to an embodiment of the present invention A flow chart illustrating the procedure in detail.

를 위한 상대적 가중치 벡터

를 획득하기 위해, 먼저 단계(S151)에서, 판단 행렬

에 판단 행렬

를 곱하여 다음 수학식 15와 같이 보정 판단 행렬

을 획득한다.In Fig. 3,

Lt; RTI ID = 0.0 >

(S151), the determination matrix < RTI ID = 0.0 >

The judgment matrix

The correction judgment matrix < RTI ID = 0.0 >

.

의 각 행(row)마다 요소들

의 합산 값들

을 다음 수학식 16과 같이 각각 산출한다.In step S152, the correction determination matrix

For each row of elements < RTI ID = 0.0 >

Lt; / RTI >

Respectively, as shown in the following Equation (16).

의 전체 합 S에 대한 각 합산 값들

의 비율로써 다음 수학식 17과 같이 상대적 가중치들

을 산출한다.In step S153, the summed values

Lt; RTI ID = 0.0 > S < / RTI &

The relative weights < RTI ID = 0.0 >

.

로 다음 수학식 18과 같이 구성된 상대적 가중치 벡터

를 각 기준

마다 생성한다.In step S154, the relative weights

A relative weight vector < RTI ID = 0.0 >

To each criterion

.

의 각 열의 상대적 중요도 값들

에 기초하여 판단 행렬

의 일관성을 검증한다.Referring back to Fig. 1, in step S16, with respect to the k-th decision maker, this time,

The relative importance values of the respective columns

Lt; RTI ID = 0.0 >

.

의 제3차 정규화 평가 인자 행렬

의 최대 고유값(largest eigenvalue)로부터 얻은 일관성 인덱스(CI, Consistency Index)와 랜덤 인덱스(RI, Random Index)의 일관성 비율(CR)을 소정의 문턱값, 예를 들어 0.1에 비교하여 만약 비율(CR)이 문턱값보다 적으면, 일관성이 있다고 판정하고 그렇지 않으면 일관성이 없다고 판정할 수 있다. 여기서 랜덤 인덱스(RI)는 기준 요소들의 개수에 따라 결정되어 주어지며, 평균 일관성 인덱스라 할 수 있다.In accordance with an embodiment,

Of the third normalization evaluation factor matrix

(CR) of a consistency index (CI) and a random index (RI) obtained from a largest eigenvalue of a CR (RI) to a predetermined threshold value, for example, 0.1, ) Is less than the threshold, it is determined that there is consistency, and otherwise, it can be determined that there is inconsistency. Here, the random index (RI) is determined according to the number of reference elements and can be referred to as an average consistency index.

로부터 일관성을 검증하는 절차를 예시하기 위해 도 4를 참조하면, 도 4는 본 발명의 일 실시예에 따른 다중 의사 결정 문제 해법 도출 방법 중에서, 계층적 분석 기법의 각 평가 인자에 대한 상대적 가중치 벡터를 이용하여 계층적 분석 기법에서 가정한 판단 행렬의 일관성을 검증하는 절차를 구체적으로 예시한 순서도이다.More specifically,

4 is a block diagram illustrating a method of deriving a multiple decision problem solution according to an exemplary embodiment of the present invention. Referring to FIG. 4, a relative weight vector for each evaluation factor of a hierarchical analysis technique A procedure for verifying the consistency of a judgment matrix assumed in the hierarchical analysis technique is illustrated in detail.

의 각 열(column)마다 상대적 중요도 요소들

의 합산 값들

을 다음 수학식 19와 같이 각각 산출한다.In Fig. 4, in step S161,

For each column of < RTI ID = 0.0 >

Lt; / RTI >

As shown in the following equation (19).

을 다음 수학식 20과 같이 산출할 수 있다.In step S162, the first order normalization evaluation factor matrix

Can be calculated by the following equation (20).

의 각 행마다 요소들

의 평균값들

을 산출함로써 제2차 평가 인자 행렬

을 다음 수학식 21과 같이 산출할 수 있다.In step S163, the first-order normalization evaluation factor matrix

For each row of elements

Average values

The second evaluation factor matrix < RTI ID = 0.0 >

Can be calculated by the following equation (21).

를 다음 수학식 22와 같이 판단 행렬

에 곱하여 제3차 평가 인자 행렬

를 산출한다.In step S164, the second evaluation factor matrix

As shown in the following equation (22)

The third evaluation factor matrix < RTI ID = 0.0 >

.

의 각 열 내의 요소들의 합산값들

로 이루어진 고유벡터(eigenvector)

를 다음 수학식 23과 같이 산출한다.In step S165, the third evaluation factor matrix

Lt; RTI ID = 0.0 >

(Eigenvector)

As shown in the following equation (23).

를 다음 수학식 24와 같이 산출한다.In step S166, the largest eigenvalue is calculated,

Is calculated by the following equation (24).

Here, 1? I? M.

In step S167, the coherence index CI is calculated according to the following equation (25), and the consistency ratio (CR = CI / RI) obtained by dividing the calculated coherence index CI by a random index ) To a predetermined threshold value, for example, 0.1, to verify consistency.

On the other hand, the random index can be given as shown in Table 3 below.

Number of criteria One 2 3 4 5 6 7 8 RI - - 0.58 0.92 1.12 1.24 1.32 1.14

에 관하여 산출된 일관성 인덱스(CI)를 무작위적으로 설정된 판단 행렬들에서 산출된 평균 일관성 인덱스인 랜덤 인덱스(RI)에 비교한 값이다. 따라서, 일관성 비율(CR)이 문턱값보다 작으면 인덱스(CI)가 랜덤 인덱스(RI)에 비해 훨씬 작다는 의미이고, 이는 의사 결정자가 판단 행렬

을 모순없이 일관적으로 설정하였음을 의미한다.The consistency ratios (CR) are determined by the decision maker

To the random index (RI), which is the average consistency index calculated from randomly set decision matrices. Therefore, if the coherence ratio CR is less than the threshold value, it means that the index CI is much smaller than the random index RI,

Is consistently set without inconsistencies.

을 다시 구해야 한다.If the consistency verification fails in step S167, the process returns to step S14,

You need to find out again.

If the consistency verification passes in step S167, the process can proceed to step S17.

의 가중치

와 각 기준

의 상대적 가중치

를 곱한 값들을 정규화한 포괄적 가중치들(comprehensive weight)

을 수학식 26과 같이 연산하고, 연산된 포괄적 가중치들

과 정규화 회색 의사 결정 행렬

을 이용하여 가중 정규화 회색 의사 결정 행렬

을 생성한다.Referring back to Fig. 1, in step S17, with respect to the k-th decision maker,

Weight of

And each criterion

Relative weight

Gt; < tb >< sep > Comprehensive weights <

As shown in Equation (26), and the calculated global weights

And Normalized Gray Decision Matrix

A weighted normalized gray decision matrix < RTI ID = 0.0 >

.

로부터 가중 정규화된 가중 정규화 회색 의사 결정 행렬

의 각각의 정규화된 요소들

은 다음 수학식 27과 같이 표현될 수 있다.Normalized Gray Decision Matrix

Normalized weighted normalized gray decision matrix < RTI ID = 0.0 >

Each normalized element < RTI ID = 0.0 >

Can be expressed by the following equation (27).

의 상한은

이고 하한은

이다.here,

The upper limit of

And the lower limit is

to be.

로부터 긍정적 이상 해법

과 부정적 이상 해법

을 각각 다음 수학식 28, 29와 같이 결정한다.In step S18, for the kth decision maker, a weighted normalized gray decision matrix

Positive anomaly solution from

And Negative Ideal Solutions

Respectively, as shown in the following equations (28) and (29).

In Equations (28) and (29), J is a larger-the-better criterion that the larger the value as a benefit criterion, the smaller the value as a cost criterion, the smaller the criterion -the-better criterion.

과 부정적 이상 해법

을 이용하여 대안들의 회색 상관 계수들(grey relation coefficients)

,

를 다음 수학식 30 및 수학식 31과 같이 연산한다.In step S19, with respect to the k-th decision maker,

And Negative Ideal Solutions

The gray correlation coefficients of the alternatives are used.

,

(30) and (31).

는 긍정적 이상 해법

과 가중 정규화된 대안

을 각각 의미하는 두 구간 회색수들의 편차이고,

는 통상적으로 0.5의 값을 가지는

인 차등 계수(differential coefficient)이다.here

Positive anomaly solution

And Weighted Normalized Alternatives

, Which is the deviation of the gray numbers of the two sections,

Lt; RTI ID = 0.0 > 0.5 < / RTI >

Which is a differential coefficient.

는 부정적 이상 해법

과 가중 정규화된 대안

을 각각 의미하는 두 구간 회색수들의 편차이고,

는 통상적으로 0.5의 값을 가지는

인 차등 계수이다.here

Negative adverse solution

And Weighted Normalized Alternatives

, Which is the deviation of the gray numbers of the two sections,

Lt; RTI ID = 0.0 > 0.5 < / RTI >

Lt; / RTI >

,

의 평균으로부터, 각 i 번째 대안마다, 긍정적 이상 해법과 부정적 이상 해법에 대한 회색 상관 지수들(degree of grey relation)

,

을 연산한다.In step S1A, for the k-th decision maker, gray correlation coefficients < RTI ID = 0.0 >

,

For each ith alternative, the degree of gray relation for the positive ideal solution and the negative ideal solution,

,

.

,

에 기초하여, 그룹 분리 척도들(group separation measures)

,

을 연산한다.In step S1B, all K decision makers must be considered, so that for each alternative, the gray correlation index values of all decision makers

,

Group separation measures < RTI ID = 0.0 >

,

.

,

에 따라, 이상 해법에 대한 상대적 근접도(relative closeness)

를 결정한다.In step S1C, for each alternative, each alternative grouping measures

,

The relative closeness to the ideal solution,

.

는 0부터 1 사이의 실수이고, 값이 높을수록 더 좋은 대안임을 의미한다.

Is a real number between 0 and 1, the higher the value, the better the alternative.

의 크기 순서에 따라 대안들의 순위를 결정한다.In step < RTI ID = 0.0 > S1D,

Order of the alternatives according to their size order.

FIG. 5 is a graph comparing the proximity indexes derived using the multiple decision problem solution derivation methodology and the conventional TOPSIS methodology, respectively, according to an embodiment of the present invention.

Referring to FIG. 5, both the conventional TOPSIS methodology and the method of deriving the multiple decision problem solution of the present invention are closer to the positive ideal solution method, and the closer to the negative ideal solution method, the higher the relative proximity.

In FIG. 5, the multiple decision problem solution derivation methodology of the present invention, compared to the conventional TOPSIS methodology, shows that the proximity index of the alternatives is generally larger, thereby increasing the spacing between the relative proximity values of the alternatives, Can be more easily judged.

It is to be understood that both the foregoing general description and the following detailed description of the present invention are exemplary and explanatory and are intended to provide further explanation of the invention as claimed. It will be understood that variations and specific embodiments which may occur to those skilled in the art are included within the scope of the present invention.

Further, the apparatus according to the present invention can be implemented as a computer-readable code on a computer-readable recording medium. A computer-readable recording medium includes all kinds of recording apparatuses in which data that can be read by a computer system is stored. Examples of the recording medium include ROM, RAM, optical disk, magnetic tape, floppy disk, hard disk, nonvolatile memory and the like. The computer-readable recording medium may also be distributed over a networked computer system so that computer readable code can be stored and executed in a distributed manner.

Claims (11)

A method for deriving a multiple decision problem solution performed by a computer,
(a) an N × m decision matrix (l) consisting of linguistic variables evaluating n alternatives according to m criteria, and K decision makers participating in the decision (gray) -based decision matrices for each decision-maker;
(b) obtaining normalized decision matrices for each decision-maker's decision matrix by normalizing the interval gray numbers corresponding to the linguistic variables;
(c) computing weights for each criterion based on entropy in each criterion, for each decision-maker's normalization decision matrix;
(d) relative to each decision maker, relative weights calculated based on the relative importance values of each row of the decision matrix generated by calculating the relative importance of one of the m criteria to another criterion Generating a configured relative weight vector for each criterion;
(e) computing, for each decision maker, a comprehensive weight normalized by multiplying the weight of each criterion by the relative weight of each criterion, and using the computed global weights and the normalized gray decision matrix Generating a weighted normalized gray decision matrix;
(f) determining a positive ideal solution and a negative ideal solution from each weighted normalized gray decision matrix of each decision maker;
(g) computing a gray relation coefficient by each decision maker using the positive anomaly solution, the negative anomaly solution, and the weighted normalization decision matrix;
(h) computing a degree of gray relation from the gray correlation coefficient with respect to each of the alternatives;
(i) computing a group separation measure for each alternative for gray decision indices of all decision makers;
(j) determining a relative closeness for each alternative based on the group separation metric;
(k) ranking the alternatives according to the magnitude order of the relative proximities. The method of claim 1, wherein for the linguistic variable, a gray level is assigned to each of the criteria by considering a benefit criterion type and a cost criterion type for each criterion. The method of claim 1, wherein step (c)
Normalized Gray Decision Matrix

The gray-scale elements

Gt; < RTI ID = 0.0 >

Each ratio

;
ratio

From standard

Entropy on

To the following equation

;
standard

Entropy on

Normalized

To the following equation

;
The computed normalized entropy

Based on

Weight

To the following equation

And calculating a solution to the multiple decision problem solution. The method of claim 1, wherein in step (d)
The determination matrix

Is expressed by the following equation

Lt; RTI ID = 0.0 >

(1 ≤ i, j ≤ m) is the relative subjective importance of the jth criterion of the ith criterion among the m criteria,
The step (d)
The determination matrix

The judgment matrix

≪ / RTI >

;
The correction judgment matrix

For each row of elements

Lt; / RTI >

; And
The sum values

Lt; RTI ID = 0.0 > S < / RTI &

The relative weights < RTI ID = 0.0 >

And the relative weights < RTI ID = 0.0 >

The relative weight vector

To each criterion

Wherein the step of generating the multiple decision problem solution comprises: The method of claim 1, further comprising, between step (d) and step (e)
(d-1) verifying, for each decision maker, the consistency of the decision matrix based on the relative importance values of each column of the decision matrix; And
(d-2) if the consistency check fails, returning to step (d) to reclaim the decision matrix, and if consistency verification passes, proceeding to step (e) How to derive a solution to a decision problem. 6. The method of claim 5, wherein step (d-1)
The determination matrix

Is expressed by the following equation

Lt; RTI ID = 0.0 >

(1? I, j? M) is a relative subjective importance for the jth reference of the ith reference among the m criteria,
The determination matrix

For each column of < RTI ID = 0.0 >

Lt; / RTI >

Respectively;
The first normalization evaluation factor matrix

To the following equation

;
The first normalization evaluation factor matrix

For each row of elements

Average values

The second evaluation factor matrix < RTI ID = 0.0 >

To the following equation

;
The second evaluation factor matrix

To the following equation

According to the judgment matrix

The third evaluation factor matrix < RTI ID = 0.0 >

;
The third evaluation factor matrix

Lt; RTI ID = 0.0 >

(Eigenvector)

;
The largest eigenvalue

To the following equation

A step supported according to;
The consistency index (CI) is calculated using the following equation

(CR = CI / RI) obtained by dividing the calculated consistency index (CI) by a random index (RI) given according to the number of criteria, and comparing the consistency ratio with a predetermined threshold value, And if it is lower, verifying consistency. ≪ Desc / Clms Page number 20 > 2. The method of claim 1, wherein, in step (g), with respect to the kth decision maker,

And Negative Ideal Solutions

The gray correlation coefficients of the alternatives are used.

,

To the following equation

here

Positive anomaly solution

And Weighted Normalized Alternatives

, Which is the deviation of the gray numbers of the two sections,

Negative adverse solution

And Weighted Normalized Alternatives

Respectively,

The

Wherein the difference coefficient is a differential coefficient that is a difference between the first and second coefficients. 8. The method of claim 7, wherein in step (g)

,

For each alternative, the gray correlation coefficients < RTI ID = 0.0 >

,

On the basis of the following equation

Wherein the step of calculating a multi-decision problem solving method comprises: 9. The method according to claim 8, wherein, in step (i)

,

The gray correlation index

,

On the basis of the following equation

Wherein the step of calculating a multi-decision problem solving method comprises: The method according to claim 9, wherein, in step (h)

For each alternative,

,

On the basis of the following equation

Wherein the step of calculating a multi-decision problem solving method comprises: A computer-readable recording medium on which a program for implementing a method for deriving a multiple decision problem solution according to any one of claims 1 to 10 is recorded on a computer.

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