European Journal of Operational Research 228 (2013) 536–544

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European Journal of Operational Research

journal homepage: www.elsevier .com/locate /e jor

Decision Support

Weighted Influence Non-linear Gauge System (WINGS) – An analysis method forthe systems of interrelated components q

Jerzy Michnik ⇑University of Economics in Katowice, ul. 1 Maja 50, 40-287 Katowice, Poland

a r t i c l e i n f o

Article history:Received 27 February 2012Accepted 2 February 2013Available online 13 February 2013

Keywords:Composite ImportanceDEMATELInterrelationsMultiple criteria decision analysis (MCDA)Structural modelingWINGS

0377-2217/$ - see front matter � 2013 Elsevier B.V. Ahttp://dx.doi.org/10.1016/j.ejor.2013.02.007

q Research partly supported by Polish Ministry of SResearch Grant No. NN111 438637.⇑ Tel.: +48 322577470; fax: +48 322577471.

E-mail address: jerzy.michnik@ue.katowice.pl

a b s t r a c t

The WINGS method has been derived from DEMATEL and can be widely used as a structural model foranalysis of intertwined factors and causal relations between them. Its novelty comes from an idea ofincluding in one mathematical mechanism both strength (importance) and influence of the system com-ponents. In particular, WINGS can be applied as the MCDA method for evaluating alternatives when inter-relations between criteria cannot be neglected. For the problem with independent criteria, WINGSreproduces the additive aggregation of preferences, a classical method in MCDA.

� 2013 Elsevier B.V. All rights reserved.

1. Introduction

There are plenty of approaches and methods that emerge fromtwo various research fields: system (structural) analysis and mod-eling and operational research (OR). In many cases, they met to-gether to develop methods for solving complex problems whichled to Soft OR. The problem structuring methods (PSMs) emergedin response to some of the constraints and limitations experiencedby managers and researchers using the existing quantitative ORmethods (Ackermann, 2012). To the popular approaches in PSMbelong Soft Systems Methodology (SSM) and multimethodology(Mingers and White, 2010). When making the right decision isone of the key problems, the support from systems thinking ap-proach integrated with more formal modeling can be invaluable.This paper presents an attempt to build a method that is generalenough to be helpful in the analysis of complex situations, whilealso including the quantitative tool for more precise assessments.

Quite a long time ago, in the seventies, DEMATEL appeared as aresult of the project conducted in Geneva Research Center of theBatelle Memorial Institute (Gabus and Fontela, 1973; Fontela andGabus, 1976). Originally, DEMATEL was aimed at the fragmentedand antagonistic phenomena of world societies and as a searchfor integrated solutions. Its main idea was to build and analyze astructural model. This model was to mirror the causal interrela-

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cience and Higher Education,

tions between its elements. The tabular and graphical form of theoutput was designed to illuminate the complex relationships in asystem. The universality and simplicity of DEMATEL allows it tobe applied in a wide range of various problems in social sciences.

In recent years, thanks to its universality, DEMATEL has been re-vived in Asia, especially in Japan and Taiwan. A growing number ofapplications have been observed since the beginning of 21st cen-tury. While numerous articles utilizing DEMATEL and its variousvariants to a wide range of problems were published during thelast 15 years, only a limited number are mentioned below.

DEMATEL has been found to be helpful in designing humaninterface for supervisory control systems (Hori and Shimizu,1999). The Composite Importance, a revised version of DEMATELhas been used to find the effective factors to resolve issues in orderto create safe, secure and reliable future society (Tamura andAkazawa, 2005b). The similar problem (Tamura and Akazawa,2005a) and modeling of uneasy factors over foods (Tamura et al.,2006) has been analyzed with the stochastic versions of DEMATELand Composite Importance. Fuzzy variant of DEMATEL has beenproposed for developing the global managers competencies (Wuand Lee, 2007). DEMATEL has served as a tool for identificationof building repair policy choice criteria roles (Dytczak and Ginda,2009). It has also been used in an interesting and atypical situationof identifying affective factors in visual arts, including government,technology, arts sponsors and the social conditions (Jasbi andFrmanfarmaee, 2010).

The numerous group of articles apply DEMATEL or its variants,very often combining it with other methods, to solving problems inmultiple criteria decision analysis (MCDA). There are a few

J. Michnik / European Journal of Operational Research 228 (2013) 536–544 537

examples from the last few years: combined ANP and DEMATELapproach used for the best vendor selection (Yang and Tzeng,2011); causal modeling of web-advertising effects using SEM mod-ified by DEMATEL technique (Wei et al., 2010); fuzzy DEMATELwith ANP for evaluation a firm environmental knowledge manage-ment in uncertainty (Tseng, 2011); fuzzy Delphi + DEMATEL + ANPemployed to construct a technology selection model regarding theeconomic and industrial prospects (Shen et al., 2011).

This article introduces Weighted Influence Non-linear GaugeSystem (WINGS) – a kind of structural model that extends the abil-ity of DEMATEL and similarly can be used as an aid in an analysis ofvarious systems of interrelated components.

The acronym of WINGS reflects its salient features. ‘Weighted’means that the measures of internal strength (importance) of thecomponents modify (weigh) the intensity of influence. ‘Influence’stresses the crucial role of interrelations between the components.The mathematical processing of input data brings the non-linearityinto the model. ‘Gauge System’ is self-explanatory.

WINGS, as a descendant of DEMATEL, inherits all merits of itspredecessor: it can handle complex problems of reacting factors,and its mathematical operations are clear and simple. Yet, it alsohas its own unique features. First of all, WINGS evaluates boththe strength of the acting factor and the intensity of its influence,while DEMATEL takes into consideration only the latter. In addi-tion, a special form of WINGS can serve as the MCDA method forevaluating alternatives when interrelations between criteria can-not be neglected. It has been shown that, when the criteria areindependent, WINGS reduces to the additive aggregation, com-monly used in MCDA.

The remainder of the article is organized as following. Section2 contains a short presentation of DEMATEL and CompositeImportance. The method WINGS is introduced in Section 3. Thisis a main part of the article and comprises also a series of exam-ples illustrating the main features of WINGS. A comparison be-tween WINGS and the similar methods from structuralmodeling and MCDA is presented at the end of Section 3. Sum-mary and remarks on future directions of study are placed inSection 4 (Conlusions).

1 It may happen that all sums of rows and columns are equal and the total-influencematrix will not exist. However it is very unlikely in practice.

2. DEMATEL and Composite Importance

This section shortly presents the essence of the DEMATEL meth-od. It is followed by the description of the Composite Importance, aconcept that sprang up from DEMATEL.

2.1. Outline of DEMATEL

We consider the system of n elements. The verbal scores for aninfluence assessment is translated into the non-negative integersfrom 0 to 4, according to the following mapping: ‘no influence’ ? 0,‘low influence’ ? 1, ‘medium influence’ ? 2, ‘high influence’ ? 3,‘very high influence’ ? 4. The value representing the influence ofelement i on element j is denoted as dij and becomes the elementof the initial direct-relation matrix A = [aij], i, j = 1, . . . ,n. By assump-tion, the principal diagonal elements are all equal zero (aii = 0,i = 1, . . . ,n).

The normalized matrix B is

B ¼ 1v A; ð1Þ

where the normalizing factor v is given by

v ¼ maxi;j¼1;...;n

Xn

j¼1

aij;Xn

i¼1

aij

( ): ð2Þ

If 9kPn

j¼1akj < 1, the power series of normalized matrix con-verges to zero matrix and the total-influence matrix X is welldefined.1 It comprises the direct influence between elements (B)and all indirect influences (B2,B3, . . .) as follows:

X ¼ Bþ B2 þ B3 þ � � � ¼ BI� B

; ð3Þ

where I is n � n unit matrix.ri ¼

Pnj¼1xij – the sum of all elements of the i row of the total

influence matrix X is interpreted as the total influence exerted byelement i on all other elements in the system. Similarly, the sumof all elements of the j column of total influence matrixcj ¼

Pni¼1xij is interpreted as the total influence exerted by all other

elements on the element j.Additionally, the two-dimensional chart, called an impact-rela-

tions map, is used to illustrate the causal relations in the system.In this graph, each element is represented as a point with twoco-ordinates: ri + ci on the horizontal axis, ri � ci on the verticalaxis. The value of ri + ci combines interrelations of both directionsof the element i and therefore is interpreted as an overall influencestrength of that element. ri � ci shows the difference between ex-erted and received influence and is a basis for classification of ele-ments. Those elements for which ri � ci is positive are consideredas ‘causal’ components of the system, those for which ri � ci is neg-ative are considered as ‘affected’.

2.2. Concept of Composite Importance

In the series of two articles Tamura and Akazawa notice that‘‘. . . the original DEMATEL is not taking into account the impor-tance of each factor itself. Hence, it is not possible to evaluatethe priority among the factors (Tamura and Akazawa, 2005b). Alsothey argue: ‘‘We need to take into account both the strength ofrelationships among factors and the importance of each factor’’(Tamura and Akazawa, 2005a). To overcome this problem theypropose to use the n-dimensional vector y whose componentsmeasure the importance of each element itself. The vector y is nor-malized by division of each component of y by the largest one. Thenormalized vector y is denoted as yr. Then, this normalized vectoris used to define the Composite Importance z:

z ¼ yr þ Xyr ¼ ðIþ XÞyr: ð4Þ

The ith element of the Composite Importance vector measureshow much the ith factor can improve overall structure, that canbe thought as a kind of priority ranking.

3. Weighted Influence Non-linear Gauge System – WINGS

When one considers the interactions between two elements,the common sense suggests that the effect that interaction de-pends not only on ‘‘the intensity of affecting’’ but also on ‘‘thestrength of factor’’ that acts. This general observation is supportedby many specific cases.

In classical physics there are two analogical laws: law of univer-sal gravitation and Coulomb’s law. In both magnitude of the forceon each of two elements (masses or electric charges) depends ontheir masses (charges) and the distance between them. We canthink about distance as a measure of ‘‘the intensity of influence’’and about mass as a measure of ’’the strength of element’’. Simi-larly, in the elastic collision, an effect of the collision depends onboth mass and velocity of the colliding bodies.

When we translocate to the field of social sciences we can alsofind the analogical examples. In management one has to analyze

Fig. 1. (A) An initial directed graph of the system with five components and arrowsrepresenting the non-zero influences. (B) The same system with numericalassignments for strengths and influences (the internal strength of component C4

is zero).

538 J. Michnik / European Journal of Operational Research 228 (2013) 536–544

the interrelations between many intertwined elements. Let’s as-sume that we are going to modernize the production line and,among others, we consider two important criteria: ‘uncertaintyof project results’ and ‘technological competencies’. They are notindependent – the latter has some influence on the former. Theinfluence of ‘technological competencies’ on ‘uncertainty of projectresults’ should combine the importance of criterion ‘technologicalcompetencies’ in the studied system and how strong it acts on‘uncertainty of project results’. The other example refers to intro-ducing a new product. It is obvious that a financial risk is influ-enced by the competitors’ reaction. But an aggregated effect ofweak reaction of big (strong, important, more influential) compet-itor can be more important than strong reaction of less influentialone. One more example comes from market behavior. Let’s con-sider a particular change of a price of some product which is pur-chased by two different groups of customers. One group isnumerous (strong, important), the other comparably small (weak).If members of both groups react to the price change similarly, thechange in a total demand of the numerous group will be much big-ger than that of the weaker one. In general, we can say that the fi-nal effect of the interactions in the system depends on acombination of ‘strength’ of an acting factor and ‘intensity’ of anaction.

Tamura and Akazawa (2005a,b) introduced in their model theimportance of the element itself, but they neglected the role of thatimportance in the interactions between elements. Similarly to ori-ginal DEMATEL, they separately calculate the total influence matrixand then use it to modify the initial importance vector.

Above considerations lead to the idea that both strength (inter-nal power or importance of the factor) and influence (intensity ofaffecting) are intertwined together and need to ‘cooperate’ in themodel to adequately reflect the interactions of elements in a com-pound system. The procedure WINGS – introduced in this article –was designed to fulfill this requirement.

The basic assumptions of WINGS grow from the philosophy ofstructural modeling in social sciences and are settled on the para-digm that the system behavior and its important features can bestudied with the model of interrelations between system’s compo-nents. We assume that:

� Two basic features of the system components are responsiblefor the interrelations: internal strength and influence.

� The objective mechanism of interactions should includedirect and also all possible indirect relations between com-ponents which is a result of the transitivity of interactions.

� The more complex interactions, involving more than twocomponents, can be characterized with enough approxima-tion by two-component interactions.

� However the objective measurement is not possible, theexperienced specialist can make rational assessments (alsoexpressed in numbers) of the strength (importance) of thecomponents and influences between them.

Concerning the second assumption, transitivity seems obvious.Also the indirect influence should weaken with the number ofintermediary components. It means that we need the mechanismthat will be able to express the total evaluation of the infinite seriesof indirect influences. Such a mechanism has been proposed inDEMATEL and makes the foundation of WINGS, too.

3.1. Procedure of WINGS

We assume that the problem can be solved by the analysis of amodel consisting of finite number, n P 2, components (they alsomay be called factors or simply elements). They are selected bythorough analysis and/or during a discussion if a group user is in-

volved in a process (for convenience we will use from now theterm ‘user’ for any kind of subject or the group of subjects inter-ested in application of the method: a decision maker, a researcher,an analyst, an expert, etc.). Then, for each component, its strengthand interrelations in a system are assessed.

The strength (importance), introduced by WINGS, can be alsocalled the initial or internal strength, as it enters the model asthe input value assessed by the user. The strength of the compo-nents may be of various nature. Especially, some componentsmay have no strength or negligible strength which is modeled bythe numerical value of zero. However (similarly to DEMATEL) thecomponent with no internal strength acquires the non-zero valuein a system via process of interactions with other components.

The WINGS procedure is divided into seven steps as follows:

Step 1:1. The user selects the n P 2 components that constitute

the system. The directed graph representing the systemcan be very helpful during the beginning phase of theWINGS procedure. In the graph: (1) Nodes representthe components of the system. (2) Arrow from influenc-ing node to influenced node represents the non-zeroinfluence. An example of the directed graph for the sys-tem of five components is presented in Fig. 1A.

2. Verbal scale of strength. The user evaluates the strength ofall system components using the following 5-point ver-bal scale: ‘low strength’, ‘medium strength’, ‘highstrength’, ‘very high strength’. ‘No strength’ is used inthe following cases: (1) The user feels that the internalstrength of the component is negligible; (2) The user isnot able to assign any other verbal term to the compo-nent; and (3) Some components of the system, by theirvery nature, should not be assigned the strength inadvance.

3. Verbal scale of influence. The user evaluates the levels ofinfluence between all system components using the fol-lowing 5-point verbal scale: ‘no influence’, ‘low influ-ence’, ‘medium influence’, ‘high influence’, ‘very highinfluence’. If the user feels that the 5-point scale is toonarrow to handle his evaluations, the scale can be easilyenlarged.

Step 2: The user assigns the numerical values to the verbal eval-uations. The values and their relations depend on the userassessment, however to keep balance between strengthand influence, we suggest to use the same mapping forboth measures.The two generic aspects of the method determine thecharacter of numerical scales:

1. The natural zero appears as an equivalent of both verbalassessments: ‘no strength’ and ‘no influence’.

J. Michnik / European Journal of Operational Research 228 (2013) 536–544 539

2. The final evaluation of the components depends on sumsof products of initial evaluations.

The above conditions imply that if the method is to be meaning-ful, the scales for strength and influence should be the ratio scales(see e.g. Roberts, 1985; Bouyssou et al., 2006, chap. 3). It can bealso convenient to choose the lowest non-zero level as the unit le-vel (as it is chosen in examples below).

Example 1: The ‘low’ level is represented by 1. Then user deter-mines ‘medium’ level as two times higher than ‘low’, the level‘high’ as three times higher than ‘low’, and level ‘very high’ as fourtimes higher than ‘low’. This gives the assignment similar to DEM-ATEL: ‘no’ = 0, ‘low’ = 1, ‘medium’ = 2, ‘high’ = 3, ‘very high’ = 4.Example 2: 1 for level ‘low’, ‘medium’ = 2 � ‘low’, ‘high’ = 3 � ‘low’,‘very high’ = 2 � ‘high’. This gives the following assignment:‘no’ = 0, ‘low’ = 1, ‘medium’ = 2, ‘high’ = 3, ‘very high’ = 6.

Fig. 1B shows an example of the numerical assignments addedto the graph of the system from Fig. 1A.

Step 3: The numbers assessed in Step 2 are inserted into the directstrength–influence matrix D. This is n � n matrix with ele-ments dij.Values representing strength of components are insertedinto principal diagonal, i.e. dii = strength of component i.Values representing influences are inserted in such away that for i – j, dij = influence of component i on compo-nent j.

Step 4: Matrix D is calibrated according to the following formula:

2 The cthe sub(Grinste

C ¼ 1s

D; ð5Þ

where calibrating factor is defined as a sum af all elements ofmatrix D, i.e.

s ¼Xn

i¼1

Xn

j¼1

dij: ð6Þ

Remarks:1. This way of a calibration ensures the existence of the total

strength–influence matrix T defined in Eq. (7) if there are atleast two positive elements in matrix D and both are not inthe same row. An opposite situation may be excluded fromthe analysis, as it actually does not represent any system.

2. This calibration, alike that used in DEMATEL, ensures thatthe results are invariant under the positive homothetictransformation dij ! d0ij ¼ adij; a > 0, for i, j = 1, . . . ,n. Thisis in an agreement with the remark about meaningfulnessmade in Step 2.

Step 5: Calculate the total strength–influence matrix T from theformula:

T ¼ Cþ C2 þ C3 þ � � � ¼ CI� C

: ð7Þ

Remark: The series in above equation converge, and consequentlythe total strength–influence matrix T exists, if at least one rowsum of matrix C elements is less than 1. This is ensured by the cal-ibration defined in Step 3.2

Step 6:1. For each element in the system the row sum ri and column

sum cj of the matrix T are calculated:

alibrated matrix, with at least one row sum of its elements less than 1, is like-matrix of transient states of the matrix representing absorbing Markov chainad and Snell, 2006, chap. 11).

ri ¼Xn

j¼1

tij; cj ¼Xn

i¼1

tij; ð8Þ

where tij are the elements of matrix T.

2. For each element in the system ri + ci and ri � ci are

calculated.Step 7: The ri and ci represent the total impact and the total recep-

tivity of component. ri + ci shows the total engagement ofthe component in the system; the sign of ri � ci indicatesthe role (position) of the component in the system: posi-tive means the component belongs to the influencing(cause) group, negative means that the component belongsto the influenced (result) group. Following the DEMATEL,we propose to draw the auxiliary chart (r � c vs. r + c)which can be called engagement-position map, whichtogether with the numerical output will facilitate the finalanalysis and discussion.

3.2. Examples

In the first example the small system consisted of three ele-ments is considered. It follows the WINGS procedure step by stepand shows the mechanism of the method. Similar calculations forthe original DEMATEL and Composite Importance have been doneto reveal the differences between the three methods.

In Examples 2a and 2b we test how WINGS works with theMCDA problem. When applied to problem with the independenceprinciple, WINGS reduces to the weighted sum aggregation meth-od (Example 2a). Then, in Example 2b we show how WINGS dealswith the case with dependencies. Example 3 shows that, whenapplied to hierarchical MCDA problem, WINGS and the weightedsum method lead to the different formulas for final score ofalternative.

3.2.1. Example 1

Step 1:1. The user selected three components that constitute the

system.2. Verbal scale of strength. The user evaluated the strength of

all system components: Component C1 – very highstrength, Component C2 – medium strength, ComponentC3 – medium strength.

3. Verbal scale of influence. The user evaluated the levels ofinfluence between the system components as follows:C1 on C2 – low, C1 on C3 – very high, C2 on C1 – high, C2

on C3 – medium, C3 on C1 – medium, C3 on C2 – high.Step 2: The user preceives that the following scale is appropriate:

‘no’ = 0, ‘low’ = 1, ‘medium’ = 2, ‘high’ = 3, ‘very high’ = 4.The assignments translated into numbers are presentedin Fig. 2.

Step 3: These numbers are inserted into direct strength–influencematrix:

D1 ¼4 1 43 2 22 3 2

264

375; ð9Þ

where the input data for three components: C1, C2, and C3

are placed consecutively into matrix rows.

Step 4: Calibrated matrix for this example is given by

C1 ¼0:174 0:043 0:1740:130 0:087 0:0870:087 0:130 0:087

264

375: ð10Þ

Fig. 2. The graph of the system discussed in Example 1.

540 J. Michnik / European Journal of Operational Research 228 (2013) 536–544

Step 5: The total importance–influence matrix T is as follows:

Table 1Exampl

C1

C2

C3

W – W

T1 ¼0:252 0:095 0:2470:193 0:125 0:1440:147 0:170 0:139

264

375: ð11Þ

Step 6: The comparison of the results of WINGS, DEMATEL andComposite Importance calculations are presented in Table1.

Step 7: When we compare the columns r + c and z (CompositeImportance), we can see that:

e 1

IN

� WINGS results in the following ranking of the totalimpact: C1, C3, C2.

� DEMATEL underestimates the role of the component C1,because it does not take into account its strength whichis very high (d11 = 4). The ranking is: C3, C1, C2.

� Composite Importance method includes importance inits calculations and therefore it places C1 at the samefirst place as WINGS (the ranking is: C1, C2, C3). How-ever it underestimates C3, because it does not take intoaccount the very strong influence of very strong com-ponent C1 on C3 (d13 = 4).

� Both, WINGS and DEMATEL recognize the C1 and C2 asthe cause components, while C3 as the influencedcomponent.

3.2.2. Example 2a – multiple criteria decision problem withindependent criteria

A model with such a problem should comprise of two compo-nents of different characters: criteria and alternatives. In the caseof criteria, the internal strength is a measure of relative impor-tance. In contrast to the criteria, the alternatives have no internalstrength (importance). Their position in the system is acquiredby the evaluation of their influence on the criteria. This influenceis interpreted as the ability to fulfill the objectives representedby the criteria. Finally, the total engagement (which, in this case,is equal to the total impact) defines the ranking (weak order) ofthe alternatives.

To show how WINGS deals with the multiple criteria decisionproblem, we have chosen the minimal possible model that consistsof only two criteria (C1 and C2) and two decision alternatives (A1

and A2). Its graph is shown in Fig. 3A. In direct strength–influencematrix the criteria take placed of the first two components whilealternatives take place of the next two components.

– comparison of the results of WINGS, DEMATEL and Composite Importance.

r c r

W D W D W

0.594 5.000 0.591 4.919 10.462 5.000 0.390 4.351 00.456 5.000 0.531 5.730 0

GS, D – DEMATEL, CI – Composite Importance.

The verbal values for importance for both criteria, after transla-tion into numbers, are inserted into strength–influence matrix (d11

and d22). Then, the user estimates verbally the influence of eachalternative on each criterion. Equivalently, it means answering tothe question: how far given alternative fulfills the objective repre-sented by given criterion? Again, the numerical estimates are in-serted into strength–influence matrix (d31 and d32 for firstalternative, d41 and d42 for the second).

After calibration (Step 4) the matrix C will have the followingform:

C2a ¼

w1 0 0 00 w2 0 0

a11 a12 0 0a21 a22 0 0

26664

37775; ð12Þ

where w1 and w2 represent the relative importance of first and sec-ond criterion, respectively. a11 and a12 (a21 and a22) represent theinfluence of the first (second) alternative on first and second crite-rion, respectively. This notation facilitate the distinction betweencriteria and alternatives. As a result of calibration, all non-zero ele-ments of matrix C2a are less than 1 (in particular w1 + w2 6 1).

For this example the total importance–influence matrix is

T2a ¼

w11�w1

0 0 0

0 w21�w2

0 0a11

1�w1

a121�w2

0 0a21

1�w1

a221�w2

0 0

2666664

3777775: ð13Þ

The total engagements for the first and second alternatives areðr þ cÞai

¼ rai¼ ai1=ð1�w1Þ þ ai2=ð1�w2Þ, where i = 1,2. It is seen

that for the decision problem with independent criteria, the totalengagement will be always equal to the total impact. This effectis caused by the special structure of the initial (and calibrated) ma-trix (the column of zeros for each alternative).

The above result can be easily generalized to the arbitrary num-bers of criteria and alternatives. With nc – the number of criteria,WINGS will lead to the following formula for the total engagementof ith alternative:

ðr þ cÞai¼Xnc

j¼1

aij

1�wj: ð14Þ

The above result shows that in the case of independent criteriaWINGS reduces to the weighted sum aggregation method. It is notessential that the initial criteria importances have been changedinto w0j ¼ 1=ð1�wjÞ by the increasing transformation. It is alwayspossible to change the procedure and choose the proper strategyfor setting the weights w0j directly (see discussion in Bouyssouet al., 2006, chap. 5) and then adjust the scale for influence. In prac-tice, it is no sense to use WINGS, but sooner the weighted sumaggregation method. This example has rather theoretical thenpractical meaning. It supports the statement that in the case ofMCDA problem, WINGS can be the considered as the extension ofweighted sum method, because, when the dependencies are ne-glected, it reduces to that method.

+ c r � c z

D W D CI

.185 9.919 0.003 0.081 4.216

.851 9.351 0.072 0.649 3.892

.986 10.730 �0.075 �0.730 3.851

Fig. 3. (A) The graph of the multiple criteria decision problem discussed in Example2a (independent criteria). (B) The graph of the same problem as in (A) but withdependent criteria (Example 2b).

Fig. 4. Graph for the multiple criteria problem with hierarchical structure (forclarity only a part of influence factors is shown).

J. Michnik / European Journal of Operational Research 228 (2013) 536–544 541

3.2.3. Example 2b – multiple criteria decision problem with dependentcriteria

Let’s now slightly modify the above problem by introducingsome influence between criteria, namely: second criterion influ-ence the first one, so we introduce w21 > 0 in the position: secondrow, first column. Now, the evaluations for the alternatives are asfollows:

ðr þ cÞai¼ ai1

1�w1þ ai2

1�w2þ ai1w21

ð1�w1Þð1�w2Þð15Þ

for i = 1, 2.Eq. (15) differs from Eq. (14) by the third term that represents

the indirect influence of alternative i on the first criterion throughthe second criterion.

We continue this example with a numerical illustration. Let’sassume that the 0–4 scale is used. The importance of first criterionis very high (w1 = 4) and that of the second medium (w2 = 2). Thevalues of influence on criteria for the first alternative are: a11 = 4,a12 = 1; for the second: a21 = 1, a22 = 4. At the moment, there arenot interdependencies between criteria. With this data the initialstrength–influence matrix has the form

D2b ¼

4 0 0 00 2 0 04 1 0 01 4 0 0

26664

37775; ð16Þ

The numerical values of total engagement for alternatives, cal-culated from final matrix, are: ðr þ cÞa1

¼ 0:405; ðr þ cÞa2¼ 0:369.

It means that first alternative is evaluated as better then thesecond. Now if the second criterion influences the first, it can beeasily calculated, that for medium influence (d21 = 2) bothalternatives will be evaluated equally ðra1 ¼ ra2 ¼ 0:357Þ. For highinfluence (d21 = 3) the second alternative will prevail the firstðra1 ¼ 0:337; ra2 ¼ 0:349Þ. This result is in agreement with theintuitive reasoning. Without interrelations between criteria, thesecond alternative having better score for second – less important– criterion, has placed as worse. But the influence exerted by thesecond criterion on the first raises the evaluation of the secondalternative.

At the end, let’s add the influence of first criterion on the secondwith d12 > 0. Obviously the formulas for tij become more compli-cated. The total engagement of alternative is given by

ðr þ cÞai¼ 1

W½ai1ð1�w2 þw12Þ þ ai2ð1�w1 þw21Þ�; ð17Þ

where W = (1 � w1)(1 �w2) �w12w21, i = 1, 2. It is clear that all ofthe elements cooperate to give the final result, particularly the rela-tions between criteria that also modify the denominator. Numeri-cally, the two combinations of influences: w12 = 1, w21 = 3 andw12 = 2, w21 = 4 lead to the equal positions of both alternatives.

3.2.4. Example 3 – multiple criteria decision problem with a hierarchyof criteria

We consider the hierarchy presented in Fig. 4. It embodies twocriteria C1 and C2 at the top level. Each of them is split into two sub-

criteria (C1 into C3 and C4, C2 into C5 and C6). Three alternativesmake the bottom level.

After the calibration, initial strengths of criteria and subcriteriabecome the weights (w1 � w6). Then, the user assesses all (non-zero) influences represented by arrows in Fig. 4. After the calibra-tion they also appear in matrix C3, as it is shown in the followingequation:

C3 ¼

w1 0 0 0 0 0 0 0 00 w2 0 0 0 0 0 0 0

c31 0 w3 0 0 0 0 0 0c41 0 0 w4 0 0 0 0 00 c52 0 0 w5 0 0 0 00 c62 0 0 0 w6 0 0 00 0 a13 a14 a15 a16 0 0 00 0 a23 a24 a25 a26 0 0 00 0 a33 a34 a35 a36 0 0 0

2666666666666664

3777777777777775

: ð18Þ

In the above matrix, similarly to Example 2a, the specific nota-tion is used to help the distinction between criteria and alterna-tives and make the final result more readable (see Fig. 4). cij

stands for the influence of subcriterion i on its ‘parent’ criterionj; aij stands for the influence of alternative i on subcriterion j.

The non-zero elements of the matrix T, relating to alternatives,appear in rows 7–9 and are given by the following formulas:

tiþ6;1 ¼ai;3c31

ð1�w1Þð1�w3Þþ ai;4c41

ð1�w1Þð1�w4Þ;

tiþ6;2 ¼ai;5c52

ð1�w2Þð1�w5Þþ ai;6c62

ð1�w2Þð1�w6Þ;

ð19Þ

tiþ6;3 ¼ai;3

1�w3; tiþ6;4 ¼

ai;4

1�w4; tiþ6;5 ¼

ai;5

1�w5; tiþ6;6 ¼

ai;6

1�w6;

ð20Þ

where i = 1, 2, 3.To assess the value of each alternative Ai, we calculate its index

ðr þ cÞai¼ rai

(similarly to the Example 2a, 2b, cai¼ 0 for each alter-

native). It is a sum of all elements in i + 6 row of matrix T, fromwhich the only non-zero elements are shown on the right handsides of Eqs. (19) and (20). The values from Eq. (20) represent thedirect impact of alternative Ai on sub-criteria C3–C6. In Eq. (19)there are the indirect impacts of alternative Ai on criteria C1 andC2. In both cases this indirect impact consists of two componentsthat represent the indirect impact via two subcriteria.

To compare the above result with other methods, we will re-place all 1/(1 � wj) by w0j. The total engagement of i alternative willbecome

ðr þ cÞai¼X6

k¼3

w0kai;k þw01w03ai;3c31 þw01w04ai;4c41

þw02w05ai;5c52 þw02w06ai;6c62: ð21Þ

3 The example with three criteria is much more illustrative, but this one coincideswith two criteria example of WINGS and is sufficient for a comparison.

542 J. Michnik / European Journal of Operational Research 228 (2013) 536–544

The weighted sum approach for analyzed hierarchy would looklike follows (Keeney and Raiffa, 1993, chap. 3) (the same formula isalso used in AHP (Saaty, 2005)):

vðaiÞ ¼ w1ðw3ai;3 þw4ai;4Þ þw2ðw5ai;5 þw6ai;6Þ; ð22Þ

where we assumed that w1 and w2 represent the conditionalweights in regards to the main objective, w3–w6 represent the con-ditional weights in regards to criteria from higher level.

The right hand side of Eq. (21) is a direct consequence ofWINGS’ structural approach. In particular:

� WINGS adds the terms of the form w0kai;k representingdirect influence of alternatives on subcriteria.

� In the terms that are similar to weighted sum approach, theadditional multiplier appears. It represents the direct influ-ence of subcriterion on criterion.

Although the ratios of w0 in Eq. (21) can be assessed similarly tothe weighted sum method, interpretation is slightly different andthey are also normalized (calibrated) differently. So, the signifi-cance of the difference between Eqs. (21) and (22) cannot bejudged without more theoretical and experimental research.

3.3. Comparison of WINGS with the other models of interactionsbetween components of the system

3.3.1. Structural modelingThere are plenty of various approaches and models in problem

structuring methods. WINGS shares some basic concepts and tech-nical aspects with a number of them. Here we limit ourselves tocomparison with small representative set of those methods thatseem to be closer to WINGS than the others.

Cognitive mappingCognitive mapping is often referred to as a problem structuring

method. In fact, the name of ‘‘cognitive map’’ covers a rich family ofvarious methods. The only thing that they share is the generalstatement: a cognitive map is a collection of nodes linked by somearcs (Marchant, 1999). It seems that WINGS (and obviously DEM-ATEL) is mostly related to the cognitive map with quantitativeassessment of strength developed by Roberts (1976). However,that version (like all types of cognitive maps) can work only withthe network that is an acyclic graph, while DEMATEL and WINGSallow graphs with cycles.

The authors of the Reasoning Map method tried to build abridge between structural modeling and decision making (Monti-beller et al., 2008; Montibeller and Belton, 2009). This method em-ploys qualitative assessment of preferences within ordinal scale,utilizes aggregation operators for qualitative data and providesalso qualitative outputs. It allows positive and as well as negativeinfluence, but it is also limited to acyclic graphs. In the ReasoningMap, the decision alternatives stay outside of the map and the bot-tom level is made by the attributes. The performance of a decisionalternative is evaluated in terms of its qualitative performance oneach attribute.

Interpretive Structural Modeling (ISM) (a clear presentation ofISM is given by Janes (1988), while the mathematical aspects arestudied by Warfield (1974)).

In ISM the elements of a system (named structure) and theirinterrelations are also presented in the digraph. The nodes repre-sent the elements of the issue or problem being studied, whilethe arcs denote a specific relation between the elements. Thismethod allows only two answers (‘yes’ or ‘no’) to the questionabout interrelation between components. These answers are repre-sented by ‘1’ or ‘0’ respectively and, in turn, are inserted into binarymatrix. Finally, with Boolean operations, the reachibility matrix isderived. WINGS and ISM have two aspects in common. One is

the transitivity of relations, the second is the information thatcan be derived from the final matrix. In ISM the sum of ‘1’ in rowand in column can be interpreted as the ‘driving power’ and the‘dependence power’, respectively. They are the parallels of totalimpact and total receptivity in WINGS.

3.3.2. Multiple criteria decision analysisIn the field of MCDA, WINGS shares with some other methods

its objective to consider interrelations between the criteria. Thereare not many such methods of aggregation. To the most popularmethods belong Choquet integral and the ANP.

Choquet integralThe Choquet integral is a generalization of the Lebesgue inte-

gral, defined with respect to a non-classical measure, often calledfuzzy measure. It is able to represent the interrelations betweencriteria, including redundancy (negative interaction) and synergy(positive interaction) (Grabisch, 1996). In the case of finite set ofcriteria, Choquet integral represents the aggregated score of analternative. It is a sum in which, besides the weights of individualcriteria, the weights of all ‘coalitions’ of criteria contribute to theaggregated evaluation. When there are no interactions, Choquetintegral reduces to the weighted sum aggregation. For two crite-ria,3 Choquet integral for the alternative Ai can be written as follows:

CIai¼ ai;1½lðC1;C2Þ � lðC2Þ� þ ai;2lðC2Þ; ð23Þ

where l(�) 2 [0,1] is the fuzzy measure and represents the weight ofa given subset of the set of criteria; the indices of criteria have to bepermuted so that ai,1 6 ai,2.

To compare formulas of WINGS and Choquet integral, let’s takethe first three terms in the expansion for matrix T from Eq. (7):

ðr þ cÞai¼ ai;1 þ ai;2 þ ai;1ðw1 þw12Þ þ ai;2ðw2 þw21Þ

þ ai;1 w21 þw12w21 þw1w12 þw2w12

� �þ ai;2 w2

2 þw12w21 þw1w21 þw2w21� �

þ � � � ð24Þ

It is clearly seen, that owing to the different approach to theproblem, WINGS and Choquet integral use different input datafor evaluation of interrelations between criteria and differentlyprocess them in further steps. So, the direct comparison is at leastvery difficult, if not impossible.

The fuzzy measure is able to enlarge the score in the case of syn-ergy between criteria and decrease the score for redundant criteria.In WINGS, so far, there is no negative influence. Redundancy can bemodeled only by input of weak (or zero) direct influence betweenredundant criteria that results in weakening their relative position.

When the number of criteria is greater than three, the evalua-tion of fuzzy measure becomes a difficult task, since the user hasto consider the importance for sets containing 3 or more criteria.The number of parameters that have to be assessed grows expo-nentially and lead also to computational problems (Grabisch andRoubens, 2000)). In WINGS the number of parameters grows line-arly since the user needs to assess only the direct influences be-tween each pair of criteria. The interrelations between 3 or morecriteria appear in the higher order terms as the products of directinfluences.

ANP (the detailed description of the ANP procedure is presentedin Saaty (2005)).

Both methods use two kinds of input data: importances andinfluences, but they combine them differently.

In the ANP, the relative importance of influence is a central con-cept (this and the pairwise comparisons are taken over from theAHP). The input information comes out from answers to two kinds

J. Michnik / European Journal of Operational Research 228 (2013) 536–544 543

of questions (Saaty, 2005, chap. 2): (1) Which of the two elementsis more dominant with respect to a criterion? (2) Which of the twoelements influences the third element more with respect to a cri-terion? In WINGS, all the initial assessments can be assumed tobe made in respect to a whole system.

In the ANP, importance is represented by the weight of the clus-ter and serves to normalize the initial supermatrix into the sto-chastic supermatrix. When there is no influence betweenclusters, the corresponding weight is zero. That may be not clearto the user. In contrast, WINGS assigns importance directly to eachcomponent, independently of its connections.

Both methods apply the limiting process to their normalized(calibrated) matrix. However, in the case of the ANP, an analysisis much more complicated since the final result depends on reduc-ibility, primitivity and cyclicity of the stochastic matrix and, inmany cases, some additional manipulations are needed (Saaty,2005, chap. 2).

4 Both traditional operational research approach and problem structuring methodsdeveloped pragmatically and were only theorized and systematized at later stages(Mingers and Rosenhead, 2004).

4. Conclusions

The method WINGS (Weighted Influence Non-linear Gauge Sys-tem) has been designed as a quantitative tool to analyze and solvethe problems of compound systems with the interrelated compo-nents. It can serve as an aid in exploring various issues in the fieldof social sciences. The numerical outcome of WINGS helps to eluci-date the causal relationships between components and to ranktheir importance/position in the system.

In WINGS, two basic features of system components – strengthand influence make the foundations for system analysis. The com-ponents can be homogeneous or can have different nature and canplay a different role in a system. To reveal the overall strength andposition of the component, WINGS

� combines both the internal importance of the componentand its external influence on the other components,

� derives the indirect influences (higher order terms) fromtwo-component interactions,

� sums up the direct influence and indirect influences of allorders to obtain the total relations between components.

Information required for WINGS operations is qualitatively andquantitatively nondemanding and can be easily elicited from theuser. The method also gives high flexibility to the user allowingthe choice of verbal scale and its numerical representation. WINGSdoes not need any specialized software as it employs only elemen-tary matrix algebra. Thanks to the above features, WINGS can be-come a valuable alternative to other methods in the structuralmodeling and MCDA.

Though WINGS borrows a lot from DEMATEL, it brings one newimportant feature. In contrast to DEMATEL which counts only theinfluences of components, WINGS joins together internal strength(importance of the component) and its influence (intensity ofaffecting). Due to this aspect, WINGS can be considered as a com-plete method that can be used alone or can be a part of more com-plex models.

The special form of WINGS can be applied in the field of MCDAas a model of problems with interrelation between criteria. Whenthere are no influences between criteria, WINGS reduces toweighted sum aggregation. WINGS is able to deal also with hierar-chical problems, however its final formula for ranking of alterna-tives differs from that of weighted sum and AHP.

Several theoretical and practical questions arise and these sug-gest areas for further research. First of all, as this work is formu-lated more in terms of procedure to follow and illustrativeexamples, it should be followed by more precise analysis of formal

features and axioms.4 Then, the operational performance should beexamined in several practical applications including comparisonwith other competitive methods. An area, that merits more research,comprises the possible extensions of WINGS: i.e., problem of uncer-tain data, clustering of components, possibility to include negativeinfluences.

Acknowledgements

The author thanks the three anonymous referees and the Editorfor their insightful and valuable comments.

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