1. INTRODUCTION

Making complex decisions must beviewed as a complex process. It takes placeat several levels and generally has a variableschedule depending on the overall scenario,that is decision-making environment. For

example, at the creative level, professionalskills and psychological background ofdecision makers are very important. At theexecutive level essential are willingness toreflect, ability of consistent reasoning andlevel of operational capability of a person tomake decisions based on their own intuition.

MODIFICATION OF THE ANALYTIC HIERARCHY PROCESS (AHP)

METHOD USING FUZZY LOGIC: FUZZY AHP APPROACH AS A

SUPPORT TO THE DECISION MAKING PROCESS CONCERNING

ENGAGEMENT OF THE GROUP FOR ADDITIONAL HINDERING

Darko Božanić, Dragan Pamučar and Dragan Bojanić*

Military Academy, University of Defence in Belgrade,Generala Pavla Jurišića Šturma 33, 11000 Belgrade, Serbia

(Received 30 November 2014; accepted 30 March 2015)

Abstract

This paper presents the modification of the AHP method, which takes into account the degree ofsuspense of decision maker, that is it allows that decision maker, with a certain degree of conviction(which is usually less than 100%), defines which linguistic expression corresponds to optimalitycriteria comparison. To determine the criteria weights and alternative values, fuzzy numbers are usedsince they are very suitable for the expression of vagueness and uncertainty. In this way, afterapplying the AHP method, we obtained values of criterion functions for each of the examinedalternatives, which corresponds to the value determined by the degree of conviction. This providesthat for different values of the degree of conviction can be made generation of different sets ofcriterion functions values. The set model was tested on choosing directions of action of the Groupfor additional hindering, as a procedure wich is often accompanied by greater or lesser degree ofuncertainty of criteria that are necessary in relevant decision making.

Keywords: Fuzzy algebras, fuzzy AHP, AHP method, the additional hindering group, the direction ofaction

* Corresponding author: draganbojanic@yahoo.com

S e r b i a n

J o u r n a l

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M a n a g e m e n t

Serbian Journal of Management 10 (2) (2015) 151 - 171

www.sjm06.com

DOI:10.5937/sjm10-7223

At the level of responsibility it is veryimportant the level of awareness of decisionmakers that decision making process is aresponsible act, as well as that the decisionmaking process and decisions rising from itcontain morality and understanding ofconsequences of decisions implementation.

In addition, it should be emphasized thatthe decision making is psychologicallycomplex and laborious process. It usuallycontains a multitude of interrelated andmutually dependent factors whose impact isnot simple to identify accurately(consistently) and link in the whole of theoutcome (decision). In this regard, manyauthors of scientific papers, and practitioners(the actual decision makers), indicate thatoptimal decisions are not usually made basedon personal reflection or intuition. Thisincreasingly points the necessity of usingscientifically based methods in decisionmaking process.

Since the theory of fuzzy sets (Zadeh,1965) was proposed in 1965, it has been usedfor handling fuzzy decision-makingproblems (Chen, 2000; Hong & Choi, 2014;Jae & Moon, 2002; Fan & Liu, 2015).Kickert (1978) has discussed the field offuzzy multi-criteria decision-making.Zimmermann (1987) illustrated a fuzzy setapproach to multi-objective decision-making, and he has compared someapproaches to solve multi-attribute decision-making problems based on fuzzy set theory.Yager (1978) presented a fuzzy multi-attribute decision-making method that usescrisp weights. Yager (1988) introduced anordered weighted aggregation operator andinvestigated its properties. Laarhoven andPedrycz (2003) presented a method formulti-attribute decision making using fuzzynumbers as weights.

Vague sets, which Gau and Buehrer

(2013) presented, are a generalized form offuzzy sets. These sets were used by Chen andTan (2014). They presented some newtechniques for handling multi-criteria fuzzydecision-making problems based on vagueset theory, where the characteristics of thealternatives are represented by vague sets.The proposed techniques used a scorefunction, S, to evaluate the degree ofsuitability to which an alternative satisfiesthe decision-maker’s requirement. Recently,Hong and Choi (2014) proposed an accuracyfunction, H, to measure the degree ofaccuracy in the grades of membership ofeach alternative, with respect to a set ofcriteria represented by vague values.However, in some cases, these functions donot give sufficient information aboutalternatives.

Chen and Tan (2014), Hong and Choi(2014), Liu and Wang (2007) and Ye (2010,2014) presented some new techniques forhandling fuzzy multi-criteria decision-making problems based on vague set theoryor intuitionistic fuzzy sets, where thecharacteristics of the alternatives arerepresented by vague sets or intuitionisticfuzzy sets and the criteria weights are givenby fuzzy numbers. However, intuitionisticfuzzy sets is the same as fuzzy sets, thedomains of which are discrete sets,intuitionistic fuzzy sets are used to indicatethe extent to which the criterion does or doesnot belong to some fuzzy concepts. Thenotion of a fuzzy number and the operationon fuzzy numbers were introduced byDubois and Prade (1978, 1987). Nehi andMaleki (2005) proposed Intuitionistictrapezoidal fuzzy numbers and someoperators for them, which are the extendingof intuitionistic triangular fuzzy numbers.Intuitionistic triangular fuzzy numbers andintuitionistic trapezoidal fuzzy numbers are

152 D.Božanić / SJM 10 (2) (2015) 151 - 171

the extending of intuitionistic fuzzy sets inanother way, which extends discrete set tocontinuous set, and they are the extending offuzzy numbers.

Furthermore, the expected value methodis also applied to ranking. Heilpern (1992)proposed the expected value of a fuzzynumber. Then Grzegrorzewski (2013)proposed the expected value and orderingmethod for intuitionistic fuzzy numbers byusing the expected interval of intuitionisticfuzzy numbers. Also Wang and Zhang(2014) defined some aggregation operators,including intuitionistic trapezoidal fuzzyweighted arithmetic averaging operator andweighted geometric averaging operator, andproposed an intuitionistic trapezoidal fuzzymulti-criteria decision-making method withknown weights based on expected values,score function, and accuracy function ofintuitionistic trapezoidal fuzzy numbers.

Fuzzy multi-criteria evaluation methodsare used widely in fields such as informationproject selection, material selection andmany other areas of management decisionproblems (Chen, 2000; Chen & Tzeng, 2014;Yeh & Deng, 2014) and strategy selectionproblems (Chiadamrong, 2013; Ding &Liang, 2015).

Li (1999) proposed a simple and efficientfuzzy model for dealing multi-judges andmulti-criteria decision making problems infuzzy environment and suggested a levelweighted fuzzy relation for comparing orranking sets. This method can avoid animmediately defuzzified process when it canprovide a precise solution. In addition, thetechnique of ideal and anti-ideal points isused easily to find the best alternative,considering that the chosen alternativeshould simultaneously have the shortestdistance from the positive ideal point and thelongest distance from the negative ideal

point (Anagnostopoulos et al., 2008; Chen &Tzeng, 2014; Kuo et al., 2007). The idealpoint is composed of all best criteriaavailable and the anti-ideal point iscomposed of all worst criteria attainable.Several extensions of TOPSIS have beenmade to incorporate fuzzy numbers in theprocess. Chen (2000), in order to solve agroup decision-making problem, measuredthe distance between two trapezoidal fuzzynumbers by a vertex method resulting in acrisp distance value and used the ideal andanti-ideal solutions to define a crisp overallscore for each alternative.

Chang and Yeh (2012) as well as Xu et al.(2010) proposed a multi-criteria methodbased on the similarity of each alternative tothe ideal and anti-ideal solutions and theyused fuzzy similarity measure instead ofdistances. Kuo et al. (2007) proposed amethod combining the efficient fuzzy model(Li, 1999) and the principles of TOPSIS tosolve multi-criteria in a fuzzy environment.This method results in fuzzy distance valueswhich are compared by a fuzzy rankingmethod. Further, grey relations and pairwisecomparison, to obtain preference relationsand ranking order of the alternatives, arecombined to create a fuzzy method by Kuo etal. (2007).

The technique often used in the field ofmulti-criteria decision making is the AnalyticHierarchy Process (Saaty, 1980). It is basedon the decomposition of a complex probleminto a hierarchy, with the goal at the top, andcriteria, sub criteria and alternatives at thelevels and sub-levels of hierarchy. Thedecision maker makes comparison ofelements in pairs at each level of hierarchy inrelation to the elements at a higher level byusing some of the preferred scales, usuallyso-called Saaty's scales. The end result arethe vectors of relative importance (priorities)

153D.Božanić / SJM 10 (2) (2015) 151 - 171

of alternatives in relation to the goal.Analytic Hierarchy Process (AHP) is a

method of scientific analysis of scenario anddecision making using consistent valuationof hierarchies. AHP has been applied invarious fields of strategic management andresource allocation, where decisions havefar-reaching importance and where thedecision makers need quality and reliableadvice during alternatives consideration anddetermination of their effects in relation tothe set objectives. The scientific capacity ofAHP is proved by a number of dissertationsat prestigious international universities andmany research papers at scientificconferences and journals (Saaty, 1996; Ray& Triantaphyllou, 1999; Raju & Pillai,1999; Arslan & Khisty, 2006; Boender etal.,1989; Chang, 1981; Chen, 1997; Zhu etal., 1999; Devetak & Terzić, 2011).

AHP method has been intensively used indecision-making processes in a militaryenvironment, especially in decision-makingin combat operations. The application ofAHP methods in support of decision makingfor choosing courses of action of the Groupfor additional hindering (GAH), will beshown through this work.

GAH is a temporary engineering structurethat is generally organized in defenseoperations with the basic task to hiderdirections of sudden invasion of opponents(especially their armored and mechanizedunits). They act independently or with otherelements of the antiarmored combat in thedirections (usually additional) where thereare not organized positions for antiarmorcombat, in the interstices, on the exposedsides, in the direction of stronger expansionof air-raids and similar (Military Lexicon,1981). It is equipped with a number of anti-tank mines, minelayer, and depending onneed it is assigned other light antitank means,

means of transport and mines and explosivemeans (Military Lexicon, 1981). One GAHis generaly determined with the startingpoint, one of two lines of action, and in everydirection of action two or three lines ofhidering (Military Lexicon, 1981). TheStarting point and lines of hindering areprimarily conditioned by the defineddirections of action of this group. In essence,defining lines of action of the GAH is thebasis for assigning specific tasks for it.

In order to have GAH properly used andto achieve maximum performance it isnecessary that the person who decides,makes the selection of GAH direction ofaction on which it will be most effectivelyused. Usually, in practice, a number ofoffered directions are imposed, and theperson who decides chooses one or two.

2. METHODOLOGY

2.1. Fuzzy logic

Fuzzy sets were introduced with the basicgoal that the uncertainty in linguistics isrepresented and modeled in a mathematicallyformalized way, and sets defined this waycan be perceived as a generalization ofclassical set theory. The basic idea of fuzzysets is very simple. In the classical (non-fuzzy) sets an element (a member of theuniversal set), either belongs or does notbelong to defined set. In this sense, fuzzy setis a generalization of classical set as themembership of an element can becharacterized by a number from interval[0,1]. In other words, the membershipfunction of fuzzy set maps each element ofthe universal set in this interval of realnumbers. One of the biggest differencesbetween classical and fuzzy sets is in the fact

154 D.Božanić / SJM 10 (2) (2015) 151 - 171

that classical sets always have a uniquemembership function, while for the fuzzy setthere is an infinite number of differentmembership functions which can describe it.This fact allows fuzzy systems to adaptappropriately to situations where applicable.This fact was pointed by Lotfi Zadeh (1965)who defined fuzzy sets, with a special notethat each area can be fuzzicated andtherefore it can generalize until thenconventional classical approach to the setstheory.

The concept of fuzzy number

In determining time required to performtasks in the Serbian Armed Forces' units veryoften it can be heard estimation that a taskcan be done for "about a couple of minutes."This means that, for example, "about threeminutes" is the nearest integer which in bestway expresses the approximate value of thetime necessary to perform the task.

The statement that the time required toperform the task is three minutes we willinterpret equally in any situation. However,when we say that the time required toperform an activity is nearly three minuteswe can ask ourselves: "How close?", "Whatis the maximum error?", and sometimes it isa sufficient information to us. If we say thatthe time required to perform activity is"about three", on the one hand it might besufficient information to us, and on the otherhand it may only increase the confusion.

Similar descriptions of situations onesuccessfully uses in decision makingprocess, and fuzzy logic allows us to usesuch, seemingly imprecise, information inmany scientific fields. Figure 1 illustratesthis idea, that is to apply, instead of preciseand rigorous descriptions of complexphenomena, the opposite approach and allowthem to be inaccurate (Pamučar, 2010).

The term of fuzzy set

The classical discrete set is a set ofelements with the same properties. Eachelement of a discrete set belongs to it 100%,or on a scale of zero to one, we can say thateach element of a discrete set belongs to thatset with the degree of 1. Of course, thediscrete element may not be the part of theset, then we can say that it belongs to the setwith the degree of 0.

Fuzzy set is an extension andgeneralisation of the classical discrete set(Jantzen, 1998). It represents a set ofelements with similar properties. The degreeof element membership in a fuzzy set can beany real number from the interval.

Definition (Teodorović & Kikuchi, 1994):Fuzzy set A on the non-empty set U is calledthe ordered pair (μA(x),x), where μA(x) is thedegree of membership of element x ∈ U infuzzy set A. The degree of membership is anumber from the interval [0,1]. Themembership degree bigger, the element ofthe universal set U correspond to a greater

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3 4 5 6 7210-13 4 5 6 7210-1

Three Close to three

3 4 5 6 7210-1

About three

Figure 1. Fuzzy number

extent to the characteristics of a fuzzy set.Formally, the fuzzy set A is defined as a

set of ordered pairs

(1)

If we define the reference set V = {o, p, r,s, t}, a fuzzy set could look like this B ={(0.3, o), (0.1, p), (0, r), (0, s), (0.9, t)}. Thismeans that the element o belongs to the set Bwith degree 0.3, p with degree 0.1, t withdegree 0.9, and r and s do not belong to theset B (Pamučar, 2010).

Membership functions

Each fuzzy set can be represented by itsmembership function. If the reference set isdiscrete, membership function is a set ofdiscrete values from the interval [0,1], as inthe previous example. If the reference set iscontinuous, it can be expressed analyticallyby using membership function.

The most commonly used forms ofmembership functions are (Pamučar et al.,2011a):

- Triangular functions, Figure 2c,- Trapezoidal functions, Figure 2a,- Gaussian curve, Figure 2d, and- Bell curve Figure 2b.

In the figure 2. the ordinate represents thedegree of membership. The fuzzy variable xis shown on the abscissa.

Mathematical expressions describing themembership functions shown in Figure 2.have the following form:

(2)

(3)

(4)

156 D.Božanić / SJM 10 (2) (2015) 151 - 171

, ,0 1A AA x x x X x

x

Figure 2. The most commonly used forms of membership functions

0, 0

( ) / ( ),

( ) / ( ),

0,

c

x ax a c a a x c

xe x e c c x e

x e

0, 0

( ) / ( ),

1,

( ) / ( ),

0,

a

x ax a b a a x b

x b x de x e c d x e

x e

21( )

2

x ce

d x e

(5)

Most tools for designing fuzzy systemsallow the user to define different randommembership functions (Pamučar, 2010).

Universe of discourse

The elements of fuzzy sets are taken fromthe universe of discourse. The universe ofdiscourse contains all the elements that canbe taken into consideration. It means thatfuzzy variable can take values only from theuniverse of discourse.

Clarifying the concept of universe ofdiscourse will be observed through thevariable time required to perform the task. Interms of time required to perform the taskthere is a high degree of uncertainty, but weare assured that this time will not be longerthan t2 or less than t1. In other words, we areassured that time belongs to the closedinterval [t1, t2]. This closed interval is calledthe universe of discourse and it issimbolicaly marked as T = [t1, t2], Figure 3.

Determination of the universe ofdiscourse of each fuzzy variable is the task ofthe designer and the most natural solution isto adopt the universe of discourse so that itmatches the physical boundaries of thevariable. If the variable is not of a physicalorigin, one of the standard universes ofdiscourse is adopted or an abstract universeof discourse is defined (Božanić & Pamučar,2010; Pamučar et al., 2011b).

In addition to the universe of discourse,the triangular fuzzy number, in our casefuzzy time, is characterized by the degree ofconviction. The concept based on which thefuzzy number is expressed using universe ofdiscourse and corresponding degrees ofconviction is suggested by Kaufmann andGupta (1985). Figure 3 shows the fuzzynumber Ã. The universe of discourse whichcorresponds to the degree of conviction α ismarked as .

2.2. The methodological basis of AHP

The analytic hierarchy process belongs tothe class of methods for soft optimization.Basically, this is a specific tool for theanalysis of hierarchically arranged elementsof decision making. AHP method performsthe previous analysis and decomposition ofthe evaluation problem in pairs of hierarchyelements (goals, criteria and alternatives),mostly in the "top-down" direction, andalthough it may be reversed or combined. Atthe end, all evaluations are sintetized andaccording to the established mathematicalmodel they determine weighted values of thehierarchy elements. Since the sum of theweighed values of the elements at eachhierarchy level is 1, the decision maker canrank all elements of decision making in thehorizontal and vertical sense.

AHP allows interactive analysis of the

157D.Božanić / SJM 10 (2) (2015) 151 - 171

2

1

1b x

x c

Figure 3. Fuzzy à number with thecorresponding universe of discourse and degreeof conviction

Ã

1 2,a a

sensitivity of the evaluation process at thefinal ranks of the hierarchy of elements.Additional feature of the method is thatduring the evaluation of elements of thehierarchy, until the end of the synthesisprocedure and the results, the consistency ofdecision maker reasoning can be verified inorder to monitor the correctness of theobtained weight values and rankings ofalternatives and criteria.

Methodologically speaking, AHP is atechnique based on the decomposition of acomplex problem into a hierarchy where thegoal is at the top of the hierarchy, and thecriteria, subcriteria and alternatives are atlower levels. As an illustration, figure 4shows the hierarchy consisting of the goal,eight criteria and four alternatives. Thehierarchy does not have to be complete, eg.an element at some level does not have to bea criterion for all elements in the sub-level,so that the hierarchy can be divided intosubhierarchies, whose only common elementis at the top of the hierarchy.

AHP is a flexible method that allowscomplex problems with many criteria andalternatives to relatively easily find arelationship between the influencing factors,recognize their relative influence andimportance in practical applications anddetermine the dominance of one factor overanother. The method anticipates the fact thateven the most complex problem can be

broken down into a hierarchy. AHP holds allparts of the hierarchy in an order, so that it iseasy to see how changes of one factor affectthe other factors. In essence, the method doesnot insist on the differences betweenquantitative and / or qualitative criteria, orpossible differences in the matrix ofquantitative criteria.

In addition to the hierarchical structuringof the problem, the AHP methodology differsfrom other multicriteria methods because itperforms comparison in pairs of elementsE1, E2, ..., En at a given level of hierarchy inrelation to the elements at а higher level.Each element at the higher level needs n(n-1)/2 comparisons of semantic or numerictype as defined by Saaty's scale in table 1.

Looking from the top of the hierarchy, thegoal is at the top and it does not compare

158 D.Božanić / SJM 10 (2) (2015) 151 - 171

Figure 4. An example of the hierarchical structure of the problem in the AHP method

Table 1. Saaty's scale for comparison inpairs

Standard values Definition

1 Equal significance

3 Low dominance

5 High dominance

7 Highly dominance

9 Apsolut dominance

2, 4, 6, 8 Average values

1 1 1 1 1 1 1 1, , , , , , , ,1, 2,3, 4,5,6,7,8,9

9 8 7 6 5 4 3 2S

with any of the other elements. At the level 1there are the criteria which are compared inpairs, each to each, with respect to theimmediate parent element at a higher level(the goal here is at the zero level). Theprocedure is applied by going down throughthe hierarchy, until at the last level k there arenot performed comparisons of allalternatives with respect to the parent sub-sub-... sub-criteria at the penultimate (k-1)level, figure 5.

The main problem with the comparisonsin pairs is how to quantify the linguisticalyformulated selections - the phrases. In mostmethods that use comparisons it is achievedby using the appropriate numerical values,usually expressed by fractions with wholenumbers. When by comparision one wants toexpress the similarity, rather than the relativeratio, instead of fractions there can be useddifferences of integers (Triantaphyllou &Lin, 1996). In developing the scale ofevaluation, there are two approaches: linear(Saaty, 1990) and exponential (Lootsma,1988; Boender et al., 1989; Lootsma et al.,1990). Both approaches are based on certaintheories from the field of psychology, andhere only the first one is of interest which byfar dominates in the application.

It is believed that people are generally notable to conduct selections if they have the

infinite set of possibilities, for example, onefinds it difficult to distinguish values such as5.00 and 5.09. Since the psychologicalexperiments have shown that one can notsimultaneously compare more than sevenobjects (plus or minus 2) (Miller, 1956),Saaty defined the scale that has the highestvalue 9, the lowest value 1 and the differencebetween the notches 1. Saaty scale isgenerally considered a standard for AHP, andit is used for comparison in pairs. According

to this scale, which will be called for easyidentification Scale 1, the available valuesfor the comparison in pairs are the elementsof a discrete set of 17 values.

Scale 1 = Saaty's scale = {9, 8, 7, 6, 5, 4,3, 2, 1, 1/2, 1/3, 1/4, 1/5, 1/6, 1/7, 1/8, 1/9}.

Elements symmetric with respect to 1 arereciprocal. Values from Scale 1 can begrouped into two intervals: [1, 9] and [1/9,1]. As shown above, the values in the interval[1, 9] are uniformly distributed, while thevalues in the interval [1/9, 1] are grouped onthe right side of the interval. There is nogood reason that for the defined scale in theinterval [1, 9] the values are properlydistributed. An alternative could be that thevalues in the interval [1/9, 1] are properlydistributed, and that other values are their

159D.Božanić / SJM 10 (2) (2015) 151 - 171

Figure 5. The general hierarchical model in AHP

reciprocal values. Ma and Zeng (1991) havesuggested the scale of this kind (Scale 2).

Scale 2 = Ma and Zeng’s scale = {9, 9/2,9/3, 9/4, 9/5, 9/6, 9/7, 9/8,1, 8/9, 7/9, 6/9, 5/9,4/9, 3/9, 2/9, 1/9}.

In the interval [1/9, 1], the intervalbetween successive values is (1-1/9) / 8 =1/9, so that the values are properlydistributed. The values in the interval [1, 9]are the reciprocal values from the interval[1/9, 1].

Similar to Scale 2 the other, for exampleweighing values from the previous scalesmay be defined. For the interval [1/9, 1]values can be computed using the formula:

NV = V (Scale1) + [V (Scale 2) – V (Scale1)]*(α/100) (6)

where labels NV and V mean the new valueand value, respectively, a parameter α mayvary from 0 to 100. The values in the interval[1, 9] are reciprocal values for the valuescalculated by the top form. For α = 0 isobtained Scale 1, and for α = 100 is obtainedthe Scale 2.

Previous analyzes have shown that doesnot exist a scale that is the most applicablefor all situations, nor it can be said that thereis the non-applicable scale (Triantaphyllou etal., 1998).

The Saaty's scale from table 1 is dominantin applications even though it has certaindisadvantages. One of the disadvantages isthat the half of the scale is linear, and theother half is non-linear. This means thatwhen a decision maker or analyst performscomparisons in pairs, based on semanticpreferences from the right column (table 1)or by direct association, a number of valuesin the left column are put in the square

matrix of comparisons.

(7)

As is true aji = 1/aij and aii = 1 for each i,j = 1, 2,..., n, the matrix A is positive,balanced and reciprocal. The essentialinformation about preferences of elementsE1, E2,..., En can be found only in the uppertriangle of the matrix, but all the proceduresfor its further analysis use the reciprocalvalues of the lower triangle.

The main disadvantage of suchapproaches is the "non-flexible" definition ofthe intensity of the importance of linguisticexpressions, which can be solved by thefuzzification of linguistic expressions. Thefuzzification of Saaty's scale is described inmany papers (Chang, 1981; Boender etal.,1989; Chen, 1997; Zhu et al., 1999;Arslan & Khisty, 2006; Devetak & Terzić,2011; Pamučar et al., 2011c, 2012, 2015;Božanić et al., 2013).

Common to all these approaches is"sharp" fuzzification of linguisticexpressions in the Saaty's scale which arepresented whit triangular fuzzy numbers. Bythe "sharp" fuzzification we mean that forthe fuzzy number T = (t1, t2, t3) the universeof discourse is determined ahead i.e. it isdefined ahead that the value of the fuzzynumber will not be greater than t3 or less thant1. In other words, we are confident that thevalue of linguistic expressions belongs to aclosed interval [t1, t3].

In the papers (Pamučar et al., 2011c,2012, 2015; Božanić et al., 2011, 2013), themodification-Saaty's scale was used, and itwas also used in this study. DuringFuzzification of the Saaty's scale, triangularfuzzy numbers were used. Unlike thepreviously mentioned papers in which the

160 D.Božanić / SJM 10 (2) (2015) 151 - 171

{ }ij nxnA a R

process of the Fuzzification of the AHP/ANPmethods was described, the degree ofuncertainty of the decision-makers has beentaken into account in the model applied inthis paper. In this way, after the applicationof AHP / ANP method, the values of criterionfunctions for each of the consideredalternatives which corresponds to the valuedetermined by the degree of certainty areobtained. Advantages of the scalefuzzyficated in this way are abilities togenerate different criteria functions fordifferent values of the degree of certainty.

The model presented in this study takesinto account the degree of uncertaintymarked with the parameter β. In doing so, thevalue of β = 0 describes the maximumpossible uncertainty, while the value of β = 1corresponds to a situation where we knowwith absolute certainty what linguisticexpression corresponds to a givencomparison of the optimality criterion. Thevalue of the parameter β can be any numberin the interval [0,1]. Using the describedprocedure the fuzzification of Saaty's scale

was performed as shown in table 1.In this way, for the set value of parameter

β, we choose lower and upper limit of theuniverse of discourse fuzzy numberrandomly, so that they are within the limitsdefined by the following expression:

(8)

where the value t2 is the value of linguisticexpression where membership function hasthe maximum value, i.e. t2 = 1.

After fuzzification of the Saaty's scale,table 2., the process of implementation ofAHP method is the same as in the classicalAHP method.

A detailed description of the phases ofAHP method is shown in the literature(Lootsma, 1988; Kujačić, 2001).Defuzzification of fuzzy numbers is doneusing the method of Centre of gravityaccording to the expression:

161D.Božanić / SJM 10 (2) (2015) 151 - 171

1 2 1 2 1 2

1 2 3 2 2 2

3 2 3 2 2 3

, , , 1/9,9

, , , 1/9,9

2 , , , 1/9,9

t t t t t tT t t t t t t

t t t t t t

Table 2. Fuzzy Saaty's scale for comparison in pairsFazzy values

Definition Fuzzy number Inverse fuzzy number

Equal significance

(1, 1, 1) If the comparison is with itself,

1 2, ,t x t in other cases

1 / (2 ),1 / ,1/x

in other cases

Low dominance 1 2,3,t t 2 11 ,1/ 3,1t t

High dominance 1 2,5,t t 2 11 ,1/ 5,1t t

Highly dominance 1 2,7,t t 2 11 ,1/ 7,1t t

Absolute dominance 1 2,9,t t 2 11 ,1/ 9,1t t

Medium values 1 2, , ,t x t

2, 4, 6, 8x2 11 ,1/ ,1t x t

2, 4, 6, 8x

(9)

In this way, after the application of AHPmethod, the values of criteria functions foreach of the studied alternatives are given.The obtained values of criteria functionscorrespond to a certain parameter value β.For different values of parameter β it ispossible to generate different sets of valuesof criteria functions. In the paper there willbe generated five sets of values of criteriafunctions for three different values of theparameter β , β = 0, β = 0.5 and β = 1. Thus,for the different degrees of conviction of adecision maker we get different appearanceof the scale.

2.3.Consistency of matrices in fuzzy

AHP method

Fuzzy AHP makes it possible to identifyand analyze inconsistency of decision makerin the process of reasoning and evaluation ofthe hierarchy elements. One is rarelyconsistent in estimating the values or therelationship of quantitative and qualitativeelements in the hierarchy. Fuzzy AHP in acertain way recognizes this fact in the way ithas an approximate mechanism formeasuring the consistency which is based oncertain premises and simple matrixoperations.

When one had the ability to accuratelydetermine the values of weight coefficientsof all elements which are mutuallycomparable at a given level of hierarchy, thevalues of the matrix itself would becompletely consistent. However, if it isclaimed that A is much more important thanB, B a little bit more important than C, C alittle bit more important than A, that resultsin inconsistency in solving problems and in

reducing the reliability of results. Thegeneral view is that redundancy ofcomparisons in pairs makes AHP the methodthat is not too sensitive to errors in reasoning.It also provides the ability to measure errorsin reasoning by calculating an index ofconsistency for the resulting matrix ofcomparisons, and then calculates the degreeof consistency.

In order to calculate the degree ofconsistency , we should first calculate theindex of consistency according to therelation:

(10)

where the λmax is a maximum net value of thecomparison matrix. The λmax closer to thenumber n, the inconsistency is smaller.

In order to calculate λmax , first the matrixof comparisons should be multiplied with thevector of weight coefficients to determinethe vector b.

(11)

Dividing the corresponding elements ofvector b and w if is obtained λmax.

(12)

The degree of consistency (CR) is the

162 D.Božanić / SJM 10 (2) (2015) 151 - 171

13 1 2 1 1 = 3defuzzy T t t t t t

max

1

nCIn

11 12 1 1 1

21 22 2 2 2

1 2

. .

. .

. . . .

. . . .

. .

n

n

n n nn n n

a a a w ba a a w b

a a a w b

1

11

22

2

max1

1. ...

n

ii

nn

n

bw

bw

n

bw

ratio of consistency index (CI) and randomindex (RI).

(13)

Random index (RI) depends on the orderof the matrix, and it is taken from table 3where the first row is the row of matrix ofcomparison, and the second row is the row ofrandom indexes (Saaty, 1980; Peneva &Popchev, 1998).

If the degree of consistency (CR) is lessthan 0.10, the result is sufficiently accurateand there is no need for adjustments in thecomparisons and repeating the calculation. Ifthe degree of consistency is greater than thisvalue, one should determine the reasons forinconsistency, remove them by partialrepetition of the comparison in pairs, and ifrepeated procedure in several steps doesen'tlead to a reduction of the degree ofconsistency to tolerable limits, all resultsshould be discarded and the whole processshould be repeated from the very start.

3. RESULTS AND DISCUSSIONS

3.1. Defining criteria for selection of

action directions of GAH

Those who make a decision aresometimes in a situation to consider only onelocation, and then decision-making isreduced to accepting or rejecting thelocation. However, they are often in a

situation to rank a number of proposedlocations and conclude which one is the bestto be chosen. Ranking locations is performedby evaluation of each location with the aimof selecting the best out of the set ofproposed options, in relation to theimportance of the chosen criteria. If there isa possibility of changes, the number ofoptions is larger, thus the optimization ofselection is more complex (Pamučar et al.,2011a, 2011b).

Collecting data on the possible actionsdirections of GAH is done byreconnaissance. Based on the acquired data,the selection of direction on which GAH willact is performed. The first following step indeciding is to formulate alternatives, then theranking is performed - evaluation andrejection of those solutions that do not meetthe defined criteria.

Using Delphy method, data collection andselection of criteria by which the selection ofGAH routs of action is performed, arecarried out (Božanić et al., 2011).

The first criterion (K1) are "estimatesrelated to the penetration of an opponent in acertain direction." Consideration of thiscriterion is done through two sub-criteria.The first sub-criterion is "the probability ofpenetration of an opponent in a certaindirection" (K11). Through the assessment ofthis sub-criterion we take into accountwhether it is a less protected direction, thenthe direction where is organized less antitankdefence in the interstices, exposed side, the

163D.Božanić / SJM 10 (2) (2015) 151 - 171

CICRRI

Table 3. Random index

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

0.0

0.0

0.5

8

0.9

1.1

2

1.2

4

1.3

2

1.4

1

1.4

5

1.4

9

1.5

1

1.4

8

1.5

6

1.5

7

1.5

9

area adequate for performance of an airassault and the like. The second sub-criterionis the "impact that can be expressed over ourunits in the case of penetration of the enemyon the line" (K12). Defining this sub-criterionincludes consideration of the impact that anypenetration of the enemy in that directionwould have to the further outcome andcourse of operations (impact on theengagement of forces from the reserve, aswell as other elements of the fightingschedule, estimates of our losses due to thesuch actions of the enemy, etc.), as well as tothe action of GAH.

The second criterion (K2) is the "impactthat is achieved toward the opponents byclosing a given direction." The first of twosub-criteria defining this criterion is the"degree of possibility of slowing the pace ofopponents attack" (K21), where would beevaluated the extent to which the activities ofGAH could possibly slow the pace ofattacking of an opponent in a given direction.Through the second sub-criterion, "possiblelosses of opponents in personnel andmaterial technical means" (K22), it isestimated the possible losses of an opponentwhich would follow as a consequence of theactivities of GAH.

By the third criterion (K3) are estimated"characteristics of directions". They areassessed through four sub-criteria. The first(K31) is the "influence of soil properties onthe organization of hindering." Very often thelocation where the operation is performedhas a large impact on the manner of their

execution. In this case, it may play animportant role, and therefore it is estimatedthe way the land affects the operations ofGAH. On spatial dimensions it will dependhow many lines of hindering on a givendirection will be organized, then if there isany possibility to avoid minefields, howeffectively it is possible to set fire to protectthe mine fields and other soil parameters thatcan affect the organization of hindering. Thesecond sub-criterion (K32) "time needed toprepare the hindering line on the direction"means a self-assessment of number and sizeof the minefields that would effectively shutthe given direction, soil and other soilparameters that affect the speed of hinderingline preparation. Through the third sub-criterion (K33) "time needed for taking thedirection and executing the hindering" itwould be estimated how quickly the GAHcan get to the direction of hindering andhindering at particular direction in relation tothe location of the initial position. The lastsub-criterion (K34) is the estimation of"negative influences of the directionhindering on the subsequent actions of ourunits." Within this sub-criterion it isestimated whether and to what extent thehindering of a certain direction can influencethe subsequent actions of our units. Here weshould consider the wider aspects such as theconcept of the operation itself and possibleassignments of units on the direction.

Mutual comparison of the two elements ofthe hierarchy (models) was performed usingSaaty’s scale and it is presented in Tables 4 to7.

164 D.Božanić / SJM 10 (2) (2015) 151 - 171

K1 K2 K3

K1 - 3 4 K2

13 - 3

K3 1

4 13 -

I.R = 0.00

Table 4. The first level of criteria → first level of criteria

3.2.Analysis, synthesis and ranking of

alternatives

In order to define the relative importanceof criteria K1 - K3 their comparison is done inpairs, according to the linguistic expressionswhich are given in Saaty's fuzzy scale (Table2). Linguistic expressions that are used forcomparison of criteria form the matrix A.

All elements of the matrix are fuzzynumbers from Table 2. The elements that areon the main diagonal represent the fuzzynumber (1,1,1), while the matrix elementswhich are below the main diagonal are the

reciprocal values of elements below the maindiagonal.

In the next part of the paper a procedureof calculating the elements of AHP method(criteria and subcriteria and criteria function)will be presented for the degree of convictionof a decision maker β = 0.5. For other valuesof the degree of conviction of decisionmakers there will be presented the finalvalues of criteria functions for the givenalternatives.

The weight vector w of each of the criteriaof the matrix A is the sum of linguisticexpressions that describe the criteria in thesame row of the matrix A, which is dividedby the sum of all the linguistic expressionsthat describe the criteria of the matrix A.

165D.Božanić / SJM 10 (2) (2015) 151 - 171

Table 5. Estimates related to the penetration of an opponent in the direction → Estimatesrelated to the penetration of an opponent in the direction

K11 K12

K11 - 2 K12

12 -

Table 6. The impact on the opponent, which is achieved by closing direction → Impact onthe opponent, which is achieved by closing direction

K21 K22

K21 - 3 K22

13 -

I.R = 0.02

Table 7. Characteristics of directions → Characteristics of directions K31 K32 K33 K34

K31 - 1 12 2

K32 1 - 13 2

K33 2 3 - 3 K34

12 1

2 13 -

I.R = 0.01

1 2 3

11

2

1 13

1 3 4

3 1 3

4 3 1

K K K

KA

KK

)985.0,359.0,101.0(1~

3~

4~

3~

1~

3~

4~

3~

1~

4~

3~

1~

1111w

In the next step, by using fuzzycomparisons in pairs, there are formedmatrices of comparing the sub-criteria and

there are defined the weights coefficients foreach of the gives sub-criteria.

166 D.Božanić / SJM 10 (2) (2015) 151 - 171

1

2

3

0.101,0.359,0.985

(0.090,0.265,0.728)

0.025,0.052,0.214

ww w

w

1 1

11

112

1 2 0.073,0.226,0.662

0.030,0.104,0.2942 1KK A

wA w

w

2 2

21

122

1 3 0.153,0.474,1.165

0.085,0.261,0.7283 1KK A

wA w

w

3 3

1

311

32

33

1 1 1 34

1 1 2 2 0.130,0.408,1.037

0.032,0.061,0.2311 1 3 2

0.116,0.315,0.8072 3 1 30.099,0.216,0.576

2 2 3 1

KK A

ww

A www

1

,1 1 1

1,..., , 1,...,kj kjk

j il il j jl i l

w a a w j M p k (14)

11

1 1

12

1'1

1

0.101,0.359,0.985 0.073,0.226,0.662

0.101,0.359,0.985 0.030,0.104,0.294

0.007,0.081,0.652

(0.003,0.037,0.289)

KK K

K

www w w

w w

21

2 2

22

2'2

2

(0.090,0.265,0.728) 0.153,0.474,1.165

(0.090,0.265,0.728) 0.085,0.261,0.728

0.014,0.126,0.848

0.008,0.069,0.531

KK K

K

www w w

w w

Using fuzzy arithmetic and expressions:

where wj is an aggregation of weight vectorfor sub-criterion.

Using the above expression we get the

weighed values of sub-criteria according tothe following:

After completing the calculation ofweight coefficients of criteria and subcriteriawe get the final values of criteria functionsfor each of the studied alternatives.

To test the above model there were usedillustrative data that describe the fourpossible directions of action of the additionalhindering group. Characteristics of selectedsites are described through the previouslydefined criteria.

In Table 8 it is shown the final ranking ofthe alternatives obtained by using fuzzy AHPmethod. Calculation of the elements in thetable 8 was performed by using theexpression (15).

(15)

In (15), λ represents an optimism indexwhich expresses the decision maker’sattitude towards risk. A larger value of λindicates a higher degree of optimism. Inpractical applications, values 0, 0.5 and 1 areused respectively to represent thepessimistic, moderate and optimistic views

of the decision maker. For given fuzzynumbers A and B it is said that if ,then A<B; if then A=B; and if

, then A>B.

4. CONCLUSION

Fuzzification of AHP method leads to theconclusion that the values of the criteriafunctions change depending on the degree ofconviction of the decision maker. Thus, withincreasing the degree of conviction, thevalue of weighting coefficient of theobserved criterion increases, while withreducing of the degree of conviction thevalue of weighting coefficient decreases.

By applying the method modified thisway to the example of the choice of directionof GAH action it is led to the conclusion thatthe developed approach enables an optimal

167D.Božanić / SJM 10 (2) (2015) 151 - 171

31

32

3 3

33

34

3

3'3

3

3

0.025,0.052,0.214 0.130,0.408,1.037

0.025,0.052,0.214 0.032,0.061,0.231

0.025,0.052,0.214 0.116,0.315,0.807

0.025,0.052,0.214 0.099,0.216,0

K

KK K

K

K

wwww

w w ww ww w .576

0.0032,0.2121,0.2219

0,0008,0.0031,0.0494

0.0029,0.0163,0.1726

0.0024,0.0112,0.1232

1

2

3

4

0.103,0.206,0.309

0.131,0.263,0.396

(0.097,0.193,0.289)

0.116,0.231,0.347

FF

FFF

13 2 11 2 , 0,1TI A a a a

T TI A I B

T TI A I B

T TI A I B

Table 8. Final alternatives ranking

Index of optimism Alte-rnative =0.0

(pessimistic) =0.5

(moderate) =1.0

(optimistic)

Fina

l ra

nkin

g

A1 0.205 0.212 0.220 4 A2 0.285 0.295 0.305 1 A3 0.210 0.220 0.230 3 A4 0.245 0.255 0.265 2

choice from a set of offered alternatives, so

that the fuzzy AHP can be successfully used

in the formulation of strategy of decision

making process in this case. The presented

model greatly reduces the stress of decision

maker, but also allows people who have less

experience to make optimal decisions.

The method concepted this way could be

used in other examples of decision making

especially when it comes to decision making

where there are different degrees of

conviction of decision makers.

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МОДИФИКАЦИЈА МЕТОДЕ АНАЛИТИЧКОГ ХИЈЕРАРХИЈСКОГ

ПРОЦЕСА (AXП) ПРИМЕНОМ ФАЗИ ЛОГИКЕ: ФАЗИ АХП

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