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Application of fuzzy AHP method based on eigenvalues for decision making in construction industry

Application of fuzzy AHP method based on eigenvalues for decision making in construction industry N. Praščević, Ž. Praščević Primjena neizrazite AHP metode utemeljene na vlastitim vrijednostima za donošenje odluka u građevinarstvu ISSN 1330-3651 (Print), ISSN 1848-6339 (Online) DOI: 10.17559/TV-20140212113942 APPLICATION OF FUZZY AHP METHOD BASED ON EIGENVALUES FOR DECISION MAKING IN CONSTRUCTION INDUSTRY Nataša Praščević, Živojin Praščević Original scientific paper The Analytic Hierarchy Process (AHP) has found wide application for multicriteria decision making (MCDM) in various areas. In the paper is given a short survey of the AHP application for decision making in the construction industry and realisation of construction projects. Authors have proposed a method based on the eigenvalue and eigenvector of the fuzzy comparison matrices for determination of local and global priorities of decision alternatives. These eigenvalues are determined using expected values of fuzzy numbers and their products. These fuzzy eigenvalues and eigenvectors are in further procedure applied for ranking the alternatives. Comparing the final ranks of alternatives, obtained according to proposed method and some other methods, the authors found good agreement. The fuzzy AHP, in comparison with AHP with crisp numbers, gives more complete, flexible and realistic results. A case study of the optimal selection of the structural system for a large industrial hall is presented in the paper. Keywords: construction industry; fuzzy AHP; fuzzy number; multicriteria decision making Primjena neizrazite AHP metode utemeljene na vlastitim vrijednostima za donošenje odluka u građevinarstvu Izvorni znanstveni članak Analitički hijerarhijski proces (AHP) našao je široku primjenu za višekriterijsko donošenje odluka (MCDM) u različitim područjima. U radu je da n kratak osvrt na primjenu AHP metode za donošenje odluka u građevinarstvu i realizaciji građevinskih projekata. Autori su predložili metodu za određivanje vlastitih vrijednosti i vlastitih vektora neizrazitih (fuzzy) matrica komparacije za nalaženje lokalnih i globalnih prioriteta alternat iva odlučivanja. Vlastite vrijednosti se određuju uporabom očekivane vrijednosti neizrazitih (fuzzy) brojeva i njihovih produkata. Dobivene vlastite vrijednosti i vlastiti vektori se u daljnjoj proceduri primjenjuju za rangiranje alternativa. Uspoređujući konačne rangove alternativa koji su dobiveni prema predloženoj metodi i nekim drugim metodama, autori su našli dobru suglasnost. AHP sa neizrazitim brojevima, u uspoređenju sa AHP sa izrazitim brojevima, daje kompletnije, fleksibilnije i realističnije rezultate. U radu je priložena studija slučaja optimalnog izbora konstruktivnog sustava jedne industrijske hale. Ključne riječi: građevinarstvo; neizraziti AHP; neizraziti broj; višekriterijsko odlučivanje 1 Introduction value of comparisons. This is important for application in the construction industry, where in the first phase of the The Analytic Hierarchy Process (AHP) for choosing construction project realization and preparation of factors that are important for decision making (DM) was preliminary feasibility studies, many important data proposed by Saaty [1]. This is one of the useful methods concerning costs, time of works execution and others, are in multi criteria decision making (MCDM), which has not precisely known, but the values of comparison of found wide application in many areas of science and important factors could be better assessed. Since these practice, so there are a big number of references about values cannot be expressed precisely by the crisp AHP. In this process factors are selected and formulated numbers, it is necessary to use the fuzzy numbers. The in a hierarchy structure descending from one overall goal usage of verbal judgements ("equal", "equal/moderate", to criteria and alternatives, as it is shown in Fig. 1. "moderate" to "extreme") for mutual comparison of Each level may represent different factors criteria, sub criteria and alternatives is more accurate than (economical, technical, social, etc.) that are evaluated by by integer or crisp numbers. experts. It provides an overall view of the complex At each level, the comparisons may be expressed relationships inherent in a considered situation. It helps numerically or linguistically in words. These non the decision maker to assess whether the issues in each numerical values are transformed to numerical ones level have the same order of magnitude, so he can according to the corresponding scale. According to compare such homogeneous elements accurately. As Ishizaka [3] “the absence of units in comparison values is Saaty [2] emphasises, "the most effective way to an important advantage, since these values are quotients concentrate judgments is to take a pair of elements and of two quantities of the same kind”. Application of fuzzy compare them on a single property without concern for numbers instead of crisp numbers gives more realistic other properties or other elements". results and better ranking of alternatives. Elements that have a global character are represented The first solution of fuzzy AHP was proposed by Van at the higher levels of the hierarchy. "The fundamental Laarhoven and Pedrycz [4]. They used triangular fuzzy approach of AHP is to decompose a big problem into numbers and employed the logarithmic least squares several smaller problems that are solved separately to method (LLSM) to generate elements of the priority determine their priority vectors. According to these values vector (fuzzy weights). Buckley [5] used trapezoidal of the separate priority vectors, the final priority vector of fuzzy numbers to express pair-wise comparison values. the alternatives is calculated taking into account Csutora and Buckley [6] proposed "Lambda-Max relationships between hierarchy levels" [2]. method", which is direct fuzzification of the eigenvector Unlike other methods of MCDM, here it is not method. Buckley et al. [7] revisited fuzzy hierarchical necessary to know the exact numerical values of the analysis and presented a new method of finding fuzzy factors being considered, so, it is enough to assess a good weights by fuzzifying an equivalent method. They Tehnički vjesnik 23, 1(2016), 57-64 57 Application of fuzzy AHP method based on eigenvalues for decision making in construction industry N. Praščević, Ž. Praščević designed an evolutionary algorithm to estimate fuzzy most appropriate alternative. Level 1 encompasses weights. Wang and Chin [8] proposed an eigenvector prescribed criteria and level 2 contains alternatives that method to generate interval or fuzzy weight estimate from are related to these criteria. an interval of fuzzy comparison matrix with trapezoidal and triangular fuzzy numbers, which differs from Orevall goal Level 0 mentioned Lambda-Max method, proposed by Csutora Overall goal and Buckley [6]. Mikhailov [9] developed a fuzzy programming method based on geometric representation Level 1 Criterion C Criterion C Criterion C 1 2 n Criteria of the prioritization process. The problem of finding elements of the priority vector is transformed into fuzzy programming problem. Level 2 Alternat. A Alternat. A Alternat. A 1 2 m The extent analysis, proposed by Chang [10], is used Alernatives in many papers for handling the fuzzy AHP and ranking Figure 1 Hierarchical levels alternatives. He applied this approach with triangular fuzzy numbers [11] for calculation of the synthetic extent Unlike other methods of multicriteria decision values S of the pair-wise comparison matrix and using the making, relative weights w of factors F (i = 1,2,...,k), i i principle of comparison of fuzzy numbers that was which in this case are criteria or alternatives, are proposed in [10], found the requested weight vector of the compared in dependence on corresponding level. These comparison matrix. weights are assessed usually by the decision making team. In a number of papers for solving multicriteria According to these values is determined the priority decision problems, the AHP and fuzzy AHP are combined matrix F = [f ] , with elements ij k×k with the TOPSIS and Fuzzy TOPSIS method. In this paper a method is proposed, which is different f = , i , j = 1,2,...k , from methods proposed by other authors, for (1) ij determination of fuzzy eigenvalues and fuzzy priority vectors based on expected values of fuzzy numbers. These fuzzy values are in further procedure used for where w and w are weights of corresponding criteria C i j i ranking of alternatives. and C . This matrix is known as a reciprocal matrix, since it has positive entries everywhere and satisfies the 2 Application of AHP in construction reciprocal property Like in other industries, AHP as one of the methods f = , f > 0, f = 1, i , j = 1,2,...,k. (2) of multicriteria decision making is used in the ij ji ii ji construction industry to solve many different problems, and here are emphasized some of them: This matrix is consistent, because the following - Selection of construction projects for realization, conditions are satisfied - Selection of a contractor for the project realization, - Selection of temporary facilities and machinery in f = f f , i , j = 1,2,...k ; p = 1,2,...k construction sites, ij ip pj - Selection of construction methods, - Choice of the maintenance strategy for the According to Saaty [1, 2] necessary and sufficient construction equipment, condition for consistency is that the principal eigenvalue - Choice of structural systems for design of bridges, of matrix F, for the eigenvalue problem buildings and other civil engineering projects, - Choice of strategy for the maintenance of structural Fw = kw, (3) systems, - Determination of weighting factors affecting safety has value λ = k. max on construction sites, For further analysis it is necessary to normalize - Choice of a supplier of resources for construction, vector w by dividing each of its elements by their sum - Ranking of real estates, etc. 3 Non fuzzy AHP w = , i = 1,2,...,k. (4) w + w + ....w 1 2 k In the first Saaty’s works is proposed and developed AHP with non fuzzy (crisp) data on several levels and The values f , according to Saaty [1], [2], represent ij many other authors have used this procedure to solve the pair-wise comparison or importance of the factor F different problems of decision making. In this paper is compared to the factor F at a certain level of the considered the problem of multicriteria decision making hierarchy. Hence, matrix F is called pair-wise comparison in which given alternatives A , A ,..., A are ranked for 1 2 m matrix. As Saaty [1] emphasizes, in a general decision prescribed criteria C , C ,...,C . One model with three 1 2 n making it is impossible to give precise values of elements levels for solving these problems is shown in Fig. 1. f according to Eq. (1), but only estimate them. For ij Level 0 is related to the overall goal, which includes elicitation of pair-wise comparison judgments of criteria, ranking of alternatives and determination of the best or he proposed fundamental scale of measurements. The 58 Technical Gazette 23, 1(2016), 57-64 N. Praščević, Ž. Praščević Primjena neizrazite AHP metode utemeljene na vlastitim vrijednostima za donošenje odluka u građevinarstvu differences Δij = f – w /w cause inconsistency of the ij i j μ (x) matrix F, and its principal eigenvalue is 1.00 A (α) A (α) λ ≥ k. (5) max μ (x) To every eigenvalue λ corresponds eigenvector w i i a x a a l m u that represents one solution of the system of k homogeneous linear Eq. (3). Maximal positive real Figure 2 Triangular fuzzy number A eigenvalue λ and the corresponding eigenvector w are max Parametric presentation of a triangular fuzzy number accepted for further calculation. Since the estimated matrix F is not a consistent one, Saaty [1] introduced the A at level α is consistency index CI and the consistency ratio CR for this matrix, that should be calculated by the following A = [ A (α ), A (α )], (8) α l u formulas where λ − k max CI = , (6) k − 1 A (α ) = a + (a − a )α , l l m l CI A (α ) = a − (a − a )α , (9) u u u m CR = . (7) 0 < α ≤ 1 a ≤ a ≤ a . RI l m u RI is called random consistency, which depends on Triangular fuzzy number is usually described by the size of matrix k, and its values proposed by Saaty [1] three characteristic values a , a and a which are crisp l m u are given in Tab. 1. numbers Table 1 Average random consistency RI A = (a , a , a ) . (10) K 1 2 3 4 5 6 7 8 9 10 l m u RI 0 0 0,58 0,90 1,12 1,24 1,32 1,41 1,45 1,49 ~ ~ −1 Reciprocal fuzzy number A to A is for a > 0 If CR ≤ 0,10, the estimates of the elements of the vector w are acceptable. Otherwise, the consistency of ~ 1  1 1  −1 the matrix F should be improved by changing values of A = = , , ~   some of its elements, taking into account that this matrix A (α ) A (α )  u l  must be reciprocal. Saaty’s method is based on calculation of the maximal eigenvalue and corresponding or approximatelly [4, 5, 6] eigenvectors, and hence is known as the eigenvector method.   ~ 1 1 1 −1   A = , , . (11)   a a a 4 Fuzzy AHP and determination of the priority vectors for u m l   ranking of alternatives since this fuzzy number is not exactly triangular one. Some of decision criteria are subjective and In this paper, the pair-wise comparison judgments, qualitative by nature, so the decision maker cannot easily that express relative importance between factors F and F i j express strengths of his preferences or provide exact pair- in the hierarchy, are expressed by the triangular fuzzy wise comparison. Hence, the crisp numbers are not so numbers f ij suitable to express these pair-wise comparison values due to their vagueness. Since judgments of the decision maker ~ ~ or his team are uncertain and imprecise, it is much better f = ( f , f , f ), f = (1, 1, 1) f > 0 ij ij ,l ij ,m ij ,u ii ij ,l (12) to give pair-wise comparisons as fuzzy values than as i , j = 1, 2,..., k , i ≠ j, crisp ones. As Wang et al. [8] emphasize, due to complexity and uncertainty involved in real world, which constitute a fuzzy comparison matrix F with the sometimes, it is unrealistic or impossible to acquire exact following elements judgments for these decision problems. To overcome these shortcomings due to crisp   numbers, the fuzzy AHP was developed for solving these 1 1 1 1   (13) f = = , , , i, j = 1, 2,..., k , i ≠ j. ji   problems of multicriteria decision making. The triangular f f f ij ,u ij ,m ij ,l ij   and trapezoidal fuzzy numbers are commonly adoptive due to their simplicity in mathematical modelling of many This matrix is consistent if and only if [5] problems in practice. In this paper are used triangular fuzzy numbers that are most frequently used by many ~ ~ ~ f ≈ f ⊗ f , i, j = 1,2,...k , p = 1,2,...k (14) ip ip pj authors. The triangular fuzzy number, as a special type of a fuzzy set over the set of real numbers (real line) R, is while the sign ⊗ denotes a fuzzy product. shown in Fig. 2. Tehnički vjesnik 23, 1(2016), 57-64 59 Application of fuzzy AHP method based on eigenvalues for decision making in construction industry N. Praščević, Ž. Praščević This method is based on the calculation of expected According to Eqs. (10) ÷ (13) the fuzzy matrix F can values of fuzzy numbers and their products. Expected be expressed by three characteristic crisp matrices ~ ~ value EV ( A) of the fuzzy number A = (a , a , a ) , l m u F = (F , F , F ). (15) written in the parametric forms (9), is [13] l m u , F , F are obtained according to (12) a + 2a + a Crisp matrices F ~ l m u l m u (20) EV ( A) = . and (13) as follows ~ ~  1 f ... f  12 ,l 1k ,l The product of two positive fuzzy numbers A ⊗ B ,   1 f 1 ... f 12 ,u 2k ,l with α cuts A = [ A (α ), A (α )] and F = , α l u   . . ... .   B = [B (α ), B (α )] 0 < α ≤ 1 can be written in the α l u 1 f 1 f ... 1   1k ,u 2k ,u   following form 1 f ... f   12 ,m 1k ,m   1 f 1 ... f 12 ,m 2k ,m A ⊗ B = [ A (α )B (α ), A (α )B (α )], (21) F = , (16)   α α l l u u . . ... .   1 f 1 f ... 1   1k ,m 2k ,m   Expressing A (α), A (α) in the form (9) by a , a , a l u l m u  1 f ... f  12 ,u 1k ,u and B (α), B (α) by b , b , b for a > 0, b > 0, in a similar l u l m u l l   1 f 1 ... f way, after integration, the expected value of the product 12 ,l 2k ,u F = .   . . ... . of two fuzzy numbers is obtained   1 f 1 f ... 1   1k ,l 2k ,l   ~ ~ 1 EV (A ⊗ B ) = [(2a + a )b + l m l (22) Buckley [5] has proved that fuzzy matrix is + (a + 4a + a )b + (a + 2a )b ] l m u m m u u consistent according to (15) if and only if its crisp matrix F is consistent. Some authors have proposed triangular fuzzy A system of the fuzzy linear Eq. (18) may be written in the following form numbers for expression of the intensity of importance on Saaty’s absolute scale. In this paper the following fuzzy ~ ~ ~ ~ ~ ~ ~ ~ numbers are used f ⊗ w ⊕ f ⊗ w ⊕ ... ⊕ f ⊗ w = λ ⊗ w , i = 1,2,....n, i1 1 i 2 2 in n i 1 = (1, 1, g ), x = ( x − g , x , x + g ) for where the sign ⊕ denotes the fuzzy addition. g i i d i g (17) The expected values of fuzzy products due to (22) are 1 < x < 9, 9 = (9 − g , 9, 9). i d where γ and γ are chosen real numbers. d g EV ( f ⊗ w ) = [(2 f + f )w + ij ij ij ,l ij ,m j ,l 12 (23) The new technique for calibration of membership functions in the fuzzy AHP proposed by Ishizaka and + ( f + 4 f + f )w + ( f + 2 f )w ] ij ,l ij ,m ij ,u j ,m ij ,m ij ,u j ,u Nguen [12] is very acceptable. The formula for definition ~ 1 of membership functions, given and used in this paper, is EV (λ ⊗ w ) = [(2λ + λ )w + i l m i,l (24) in accordance with that proposal. + (λ + 4λ +λ )w + (λ + 2λ )w ] l m u i,m m u i,u 4.2 Approximate method for determining fuzzy weights i, j = 1,2,..., k. Since Saaty’s AHP method is based on finding The expected value of the sum of fuzzy numbers is eigenvalue and eigenvectors of the fuzzy matrix F at the equal to the sum of the expected values of fuzzy numbers corresponding hierarchical level, here is proposed one method to solve the fuzzy eigenvalue and eigenvector ~ ~ problem and find solutions of the system of homogenous EV ( f ⊗ w ) = λ ⊗ w , i = 1,2,..., n. (25) ∑ ij j i fuzzy linear equations j =1 ~ ~ ~ ~ F ⊗ w = λ ⊗ w, (18) By introducing formulas for expected values of the fuzzy products (23) and (24) in these fuzzy equations, is obtained a system of fuzzy linear homogenous equations where is the fuzzy reciprocal comparison matrix of type [k×k]. ~ F w + F w + F w − λ w − λ w − λ w = 0, (26) l l m m u u l l m m u u Elements of the fuzzy matrix F , fuzzy vector and fuzzy eigenvalue λ are assumed as triangular fuzzy where are numbers, that may be denoted according to Eq. (10) as ~ F = 2F + F , F = F + 4F + F , F = F + 2F , (27) l l m m l m u u m u w = (w , w , w ), λ = (λ ,λ ,λ ) (19) l m u l m u 60 Technical Gazette 23, 1(2016), 57-64 N. Praščević, Ž. Praščević Primjena neizrazite AHP metode utemeljene na vlastitim vrijednostima za donošenje odluka u građevinarstvu c = (c , c , c ) , (i, j=1,2,…,n) using appropriate λ = 2λ + λ , λ = λ + 4λ + λ , λ = λ + 2λ . (28) ij ij ,l ij ,m ij ,u l l m m l m u u m u comparison scale adjusted to fuzzy values according to and vectors written in the transposed form (17). Express the fuzzy matrix C by three matrices C , C l m and C according to (15) and (16). Solve the fuzzy ~ ~ ~ ~ w = [w ,w ,...,w ] , 1,l 2 ,l n ,l l eigenvalue problem C ⊗ w = λ ⊗ w ,as described in the w = [w ,w ,...,w ] , (29) previous section, and determine the principal fuzzy m 1,m 2 ,m n ,m w = [w ,w ,...,w ] . eigenvalue λ = (λ , λ , λ ) and corresponding fuzzy u 1,u 2 ,u n ,u l m u eigenvectors w = (w ,w ,w ) . Normalize these vectors l m u Since all the values in these equations are positive using formulas (32) and (33) to obtain the fuzzy priority ones, the system of Eqs. (26) may be decomposed into ~ vectors of criteria w = (w ,w ,w ) . l m u three systems, which represent three crisp eigenvalue For the matrix C , calculate the consistency index CI problems and consistency ratio CR according to (6) and (7). If CR≤0,10, accept the assessed fuzzy elements of the pair- F w = λ w , F w = λ w , F w = λ w . (30) l l l l m m m m u u u u wise matrix C and obtained eigenvalues and eigenvectors. If CR > 0,10, improve the consistency of the By solving these three auxiliary eigenvalue problems, matrix C by changing some of its elements and repeat eigenvectors w , w and w and auxiliary eigenvalues l m u the procedure until this condition is satisfied. Fourth step. Formulate the pair-wise comparison λ , λ and λ are obtained. After that the requested m u ( j ) matrices for the alternatives related to the criterion eigenvalues λ , λ and λ are determined by solving the l m u C (j = 1,2,…, n) system of linear Eqs. (28). To meet the requirements for the principal C A A ...... A eigenvalues λ ≤ λ ≤ λ , normalised eigenvectors j 1 2 m l m u ( j ) ( j ) ~ ~   should satisfy the next condition A 1 a .... a 1 12 1m   ( j ) ( j ) ~ −1 ~ (34) ~ A ( j ) 2 (a ) 1 ... a  12 2m  A = , j = 1,.,n. w ≤ w ≤ w . (31) ... l m u  .... .... .... ....  ( j ) −1 ( j ) −1 ~ ~ A   m (a ) (a ) .... 1 1m 2m   The calculated eigenvectors w , w and w should be l m u normalized according to the following formulas Express these fuzzy matrices by the matrices ( j ) ( j ) ( j ) A , A , A according to (16). Solve the fuzzy l m u w λ l l w = , eigenvalue problems s λ l m ~ ~ m ( j ) ~( j ) ( j ) ~( j ) w = , A ⊗ p = λ ⊗ p , j = 1,2,...,n (35) (32) w λ u u to find the fuzzy principal eigenvalues w = , s λ ( j ) ( j ) ( j ) ( j ) u m and fuzzy eigenvectors λ = (λ , λ , λ ) max m u ~ ( j ) ( j ) ( j ) ( j ) (j) p = ( p , p , p ) , consistency indices CI and where max l m u (j) consistency ratios CR , according to (6) and (7) for n n n ( j ) matrices A , (j=1,2,…,n). If the consistency ratio is s = w , s = w , s = w . (33) l ∑ i,l m ∑ i,m u ∑ i,u (j) CR > 0,10, change some of the assessed values to i =1 i =1 i =1 ij ,m obtain the satisfactory consistency of this matrix. 4.2 Steps in the execution of the fuzzy AHP ~ ( j ) ( j ) ( j ) ( j ) Normalize vectors p = ( p , p , p ) by the formulas l m u (32) and (33) to obtain normalized local priority vectors Fuzzy AHP performs in several steps in a similar way ( j ) ( j ) ( j ) ( j ) as the procedure with crisp numbers, and is briefly p = ( p , p , p ) . This procedure is the same as in l m u explained here. the step 3. First step. Define the problem, overall goal that has Fifth step. Formulate local priority fuzzy to be attained, criteria, subcriteria, if necessary, and alternatives. matrix P = ( P , P , P ) , that contains normalized local l m u Second step. Define the hierarchy structure from the priority vectors, where top level through intermediate levels that contains the criteria and subcriteria to the lowest level, which is related (1) (2) (n) P = [ p p ... p ], l l l l to the alternatives, as shown in Fig. 1. (1) (2) (n) P = [ p p ... p ], (36) Third step. Formulate the pair-wise comparison m m m m (1) (2) (n) C for the criteria C ,C ,…,C by P = [ p p ... p ]. reciprocal fuzzy matrix 1 2 n u u u u assessing the priority values as fuzzy numbers Tehnički vjesnik 23, 1(2016), 57-64 61 Application of fuzzy AHP method based on eigenvalues for decision making in construction industry N. Praščević, Ž. Praščević Multiply these matrices from the right by the priority CV = , i = 1,2,...,m. (42) vectors of the criteria respectively, which are determined i ,e in the third step A fuzzy number or an alternative A with a smaller T T w = [w w ... w ] , * l 1,l 2,l n ,l CV is ranked better, and the best ranked alternative A is w = [w w ... w ] , (37) alternative A with minimal CV . i i m 1,m 2,m n ,m T According to this procedure, the authors have w = [w w ... w ] . u 1,u 2,u n ,u developed a corresponding computer program in MATLAB, which has been used to solve several and obtain vectors of global priorities g , g and g l m u. problems of ranking alternatives in the construction industry. The recently proposed method and computer g = P w = [g g ... g ] . program have been used for making the optimal choice l l l 1,l 2 ,l m ,l for the new railway trace in Montenegro, which is around g = P w = [g g ... g ] . (38) m m m 1,m 2,m m ,m 200 km long. g = P w = [g g ... g ] . u u u 1,u 2,u m ,u 5 Case study These vectors constitute fuzzy matrix of global priorities G = ( g , g , g ) of alternatives A , A ,…A . Th e pr o posed m e thod of the fuzzy AHP was applied 1 2 m l m u for the choice of the optimal structural reinforced concrete For every alternative A (i = 1, 2,...,m), elements of system of an industrial two-part hall with dimensions at these vectors are expressed by the corresponding the base of 2 × 24,50 m × 120,00 m. This hall under approximate triangular fuzzy numbers construction is shown in Fig. 3. The choice of appropriate ~ structural system and technology of construction, which g = ( g , g , g ), i = 1,2,..., m. (39) i i.l i,m i,u affects costs and speed of construction, is one of main tasks of the design and construction team. The period of Sixth step. Alternatives A (i = 1, 2,...,m) are ranked design and construction was limited on seven months, in this step according to their global priorities that are according to the contract between the investor and expressed by triangular fuzzy numbers g (i = 1,2,..., m). contractor. More proposals for ranking fuzzy numbers exist in the literature, and here is used Lee and Le’s[14] method improved by Cheng [15]. In this paper, comparison of the fuzzy numbers is based on the probability measure of fuzzy events, which was introduced by Zadeh [16]. The fuzzy numbers are ranked according to the generalized fuzzy mean (expected value) and generalized fuzzy spread (standard deviation). For the triangular probability distribution of the triangular fuzzy number as a fuzzy event, these values for the fuzzy number g are calculated by the following formulas [15]: - Generalized fuzzy mean (expected value) g + 2g + g i ,l i ,m i ,u g = , i = 1,2,...,m (40) i ,e Figure 3 Industrial hall under construction According to these requirements, the design team, in - Generalized spread (standard deviation) which the second author of this paper was included, established alternatives and criteria for the choice of the 2 2 2 σ = [ (3g + 4g + 3g − 4g g − most acceptable structural system. For the choice of this i i ,l i ,m i ,u i,l i,m 80 (41) system, the next three alternatives have been considered: 1/ 2 − 2g g − 4g g )] , i = 1, 2,..., m. i,l i,u i,m i,u Alternative A – Two chord reinforced concrete and steel girder supported by the reinforced concrete columns, According to Lee and Li [14], a fuzzy number with a shown in Fig. 1; Alternative A – Prestressed concrete girder supported higher mean value and, at the same time, a lower spread is ranked better. However, it is not easy to compare two by the reinforced concrete columns; Alternative A – Classical frame structure of fuzzy numbers when one of them has a higher mean value and, at the same time, a higher spread and the other has a reinforced concrete. Four main criteria have been used: lower mean and, at the same time, a lower spread. Therefore, Cheng [15] proposed to rank fuzzy numbers C – Summary costs of the design and the construction of the hall, according to the coefficient of variation CV C – Costs of annual maintenance of the hall, C – Time necessary for the construction works of the hall in weeks, 62 Technical Gazette 23, 1(2016), 57-64 N. Praščević, Ž. Praščević Primjena neizrazite AHP metode utemeljene na vlastitim vrijednostima za donošenje odluka u građevinarstvu C – Technological possibilities of the contracting 4 w = [0,3211 0,2007 0,1898 0,1283], firm to construct this industrial hall in the chosen system. w = [0,3845 0,2391 0,2222 0,1541], These criteria are usually used as the most important w = [0,4820 0,3135 0,2872 0,2015]. in the construction industry for all civil engineering u works, especially for this type of building. Besides these criteria, depending on the type and purpose of a building, For the matrix C are found consistency index CI = often are taken into account the other criteria such as 0,0025 and consistency ratio CR = 0,0028 < 0,10. functional, aesthetical, environmental and so on. Since the consistency index and consistency ratio are The authors of this paper have assessed pair-wise very small, the assessed matrix C is very consistent and comparison values on the basis of real technical, may be accepted. technological and financial data from this and similar ( j ) For matrices A of alternatives A (i=1,2,3) related to projects and investments. According to the data, pair-wise the criteria C , (j = 1,2,3,4) the following results have comparison fuzzy matrices have been formulated: been obtained: Fuzzy matrix C for the criteria Principal eigenvalues are given in Tab. 2. Table 2 Principal eigenvectors (1,1,1) (1,1.5 ,2) (1.5, 2,2.5) (2,2.5,3)   ~ ( j ) ( j ) ( j ) ( j )   1 1 λ λ Matrix A l m u ( , ,1) (1,1,1) (1,1,1.5) (1,1.5,2)   2 1.5 ~ (1) ~   2,6627 3,0015 3,4097 C = 1 1 1 1   ( , , ) ( ,1,1) (1,1,1) (1,1.5,2) ~ (2) 2,6880 3,0002 3,5207   A 2.5 2 1.5 1.5   1 1 1 1 1 1 1 (3) 2,5978 3,0015 3,5071 ( , , ) ( , ,1) ( , ,1) (1,1,1)    3 2.5 2 2 1.5 2 1.5  (4) 2,5604 3,0013 3,5654 ( j ) ( j ) Fuzzy matrices A related to the criterion C (j =1, Since the principal eigenvalue of matrices A , (j=1, 2, 3,4) are 2, 3, 4) are very close to the number of their columns n = 3, these matrices are consistent ones.   Priority fuzzy matrices, P , P and P , that contain l m u   (1,1,1) (1.2,1.5,1.8) (1.7,2,2.3) normalized eigenvectors (local priority vectors) according   ~ 1 1 1 (1) to (36) are A = ( , , ) (1,1,1) (1.2,1.5,1.8) 1.8 1.5 1.2   1 1 1 1 1 1   ( , , ) ( , , ) (1,1,1) 0,4073 0,3525 0,3415 0,3242      2.3 2 1.7 1.8 1.5 1.2    P = 0,2813 0,2742 0,2985 0,2470 ,     0,1986 0,2359 0,2254 0,2818     (1,1,1) (1,1.3,1.6) (1.2,1.5,1.8)   0,4586 0,4082 0,3968 0,3814 ~ 1 1 (2) A =  ( , ,1) (1,1,1) (0.9,1.2,1.5)   P = 0,3194 0,3201 0,3459 0,2874 1.6 1.3     1 1 1 1 1 1 0,2220 0,2718 0,2573 0,3312     ( , , ) ( , , ) (1,1,1)   1.8 1.5 1.2 1.5 1.2 0.9   0,5178 0,4742 0,4617 0,4494     P = 0,3658 0,3781 0,4065 0,3437 .    (1,1,1) (0.9,1.2,1.5) (1.2,1.5,1.8) 0,2524 0,3212 0,3003 0,3949     1 1 1 (3) A = ( , , ) (1,1,1) (1.1,1.4,1.7)  1.5 1.2 0.9   Vectors of global priorities g , g and g are l m u 1 1 1 1 1 1   ( , , ) ( , , ) (1,1,1) calculated according to the expression (38), multiplying    1.8 1.5 1.2 1.7 1.4 1.1  matrices P , P and P by normalised vectors l m u   w ,w and w respectively, and results are shown in   (1,1,1) (1,1.3,1.6) (0.9,1.2,1.5) l m u   ~ 1 1 (4) Tab. 3. For each alternative components of these vectors A = ( , ,1) (1,1,1) (0.9,1.2,1.5)   1.6 1.3   form fuzzy numbers g = ( g , g , g ) (i= 1, 2, 3). i i,l i,m i,u 1 1 1 1 1 1   ( , , ) ( , , ) (1,1,1) For these fuzzy numbers, generalized fuzzy means    1.5 1.2 0.9 1.5 1.2 0.9  (expected values), standard deviations (spreads) and are determined for coefficients of variations V Elements of the comparison matrices are expressed alternatives A (i=1,2,3) using the expressions (40), (41) by the proposed formula (16) with γ = γ = 0,5 for d g and (42). The alternatives are ranked according to these elements of matrix C and γ = γ = 0,3 for matrices values and the results are shown in Tab. 4. d g ( j ) According to Chang's method based on the extent A , j = 1,2,3,4. analysis [11] priority normalized vector of alternatives Applying the mentioned computer program, the ~ d = [0,5165 0,3194 0,1641] is obtained. Alternative following principal eigenvalues for the matrix C and with higher component of this vector is better ranked. normalised eigenvectors are obtained Alternative A is the best ranked according to the λ = 3,3663, λ = 4,0075, λ = 5,1464, l m u expected value g = 0,443 and the coefficient of 1,e Tehnički vjesnik 23, 1(2016), 57-64 63 Application of fuzzy AHP method based on eigenvalues for decision making in construction industry N. Praščević, Ž. Praščević variation V = 11,41 %. This alternative has a discernible 7 References advantage over the other alternatives. This problem has [1] Saaty, T. L. The Analytic hierarchy Process. McGraw-Hill, been solved by the authors using the modified fuzzy New York, 1980. TOPSIS method [17] and the same order of alternatives is [2] Saaty, T. L. How to make a decision: Analytic Hierarchy obtained. Process. // European Journal of Operational Research. 48, (1990), pp. 9-26. DOI: 10.1016/0377-2217(90)90057-I Table 3 Vectors of global priorities, expected values and standard [3] Ishizaka, A. Clusters and Pivots for Evaluating a Large deviations Number of Alternatives in AHP, SOBRAPO, Brazilian Vector Vector Vector Exp. val. Stand. Alternative Operational Research Society, 2012. URL: g g g g dev. (%) l m u i,e www.scielo.br/pope. A 0,308 0,421 0,621 0,443 11,41 [4] Van Laarhoven, P. J. M.; Pedrycz, W. A fuzzy extension of A 0,106 0,195 0,442 0,235 11,71 Saaty‘s priority theory. // Fuzzy Sets and Systems. 11, A 0,120 0,245 0,575 0,297 11,78 (1983), pp. 229-241. DOI: 10.1016/S0165-0114(83)80082-7 [5] Buckley, J. J. Fuzzy hierarchical analysis. // Fuzzy Sets and Table 4 Ranks of alternatives Systems. 17, (1985), pp. 233-247. DOI: 10.1016/0165- Rank Exp. Coeffic. V Ranking of Alternative value Alternative (%) vector d 0114(85)90090-9 alter. g (Cheng) (Chang) i,e [6] Csutora, R.; Buckley, J. J. Fuzzy hierarchical analysis: the * * 1 A =A 0,443 A =A 11,41 0,5165 Lambda-Max method. // Fuzzy Sets and Systems. 120, 1 1 (2001) pp. 181-195. DOI: 10.1016/S0165-0114(99)00155-4 2 A 0,339 A 11,71 0,3194 2 3 [7] Buckley, J. J.; Feuring, T.; Hayashi, Y. Fuzzy hierarchical 3 A 0,274 A 11,78 0,1641 3 2 analysis revisited. // European Journal of Operational Research. 129, (2001), pp. 48-64. DOI: 10.1016/S0377- The alternative of the structural system A has been 2217(99)00405-1 chosen for realization and the hall has been successfully [8] Wang, J. M., Chin, K-S., An eigenvector method for finished in the planned and contracted time between the generalizing normalized interval and fuzzy weights. // investor and contractor. Applied Mathematics and Computation, 181, (2006), pp. 1257-1275. DOI: 10.1016/j.amc.2006.02.026 6 Conclusions [9] Mikhailov, L. A fuzzy programming method for deriving priorities in analytic hierarchy process. // Journal of the Operational Research Society. 51, (2000), pp. 341-349. DOI: The fuzzy AHP is one of very useful methods for 10.1057/palgrave.jors.2600899 multicriteria decision making (MCDM) in which the [10] Chang, D.-Y.; Extent Analysis and Synthetic Decision, factors that affect the decision making (criteria, sub- Optimization Techniques and Applications. Vol. 1, World criteria and alternatives) are hierarchically arranged scientific, Singapore, 1992. descending from one overall goal to the criteria, sub- [11] Chang, D.-Y. Application of the extent analysis method on criteria and alternatives in successive levels. In this fuzzy AHP. // European Journal of Operational Research, process, at all levels, these factors are mutually compared 95, 12(1996), pp. 649-655. DOI: 10.1016/0377-2217(95)00300- and these comparisons are expressed numerically or linguistically in words which are further transformed into [12] Ishizaka, A.; Nguyen, N. H. Calibrated fuzzy AHP for current bank account selection. // Expert systems with numbers according to the given comparison scale. applications. 40, (2013), pp. 3775-3783. DOI: This method, as other methods of MCDM, found 10.1016/j.eswa.2012.12.089 useful applications in the construction industry and [13] Chanas, S. On the interval approximation fuzzy numbers. // construction project management as a help for rational Fuzzy Sets and Systems. 122, (2001), pp. 353-356. DOI: decision making, so a large number of references in this 10.1016/S0165-0114(00)00080-4 field exists in the literature. [14] Lee, E. S.; Li, R. L. Comparison of fuzzy numbers based on The fuzzy AHP method gives more complete, flexible the probability measure of fuzzy events. // Comput. and realistic results, especially for the decision criteria Mathematic Application. 15, (1988), pp. 887-896. DOI: that have qualitative nature. The procedure for 10.1016/0898-1221(88)90124-1 [15] Cheng, C-H. A new approach for ranking fuzzy numbers by determination of approximate eigenvalues and distance method. // Fuzzy Sets and Systems, 95, (1992), pp. eigenvectors of the reciprocal matrices, proposed in this 307-317. DOI: 10.1016/S0165-0114(96)00272-2 paper, is used to find priority vectors for alternatives. This [16] Zadeh, L. A. Probability measures of fuzzy events. // procedure and corresponding computer program have Journal of Mathematic Analysis and Applications. 23, been used several times for choosing most appropriate (1968), pp. 421-427. DOI: 10.1016/0022-247X(68)90078-4 alternative for realisation of some construction projects. [17] Prascevic, Z.; Prascevic, N. One modification of fuzzy Comparisons that authors have made with application of TOPSIS method. // Journal of Modelling in Management. 8, some other methods of MCDM, have shown good (2013), pp. 81-102. DOI: 10.1108/17465661311311996 accordance with results obtained by the fuzzy AHP. In many situations, especially when number of alternatives is Authors’ addresses large, it is advisable to combine fuzzy AHP with other methods of MCDM, using AHP for determinations of the Doc. dr. sc. Nataša Praščević, dipl. ing .građ. criteria weights and after that chose the method for Prof. dr. sc. Živojin Praščević, dipl. ing. građ. ranking alternatives like TOPSIS, VICOR or some other University of Belgrade, Faculty of Civil Engineering method for ranking alternatives for multicriteria decision Bulevar kralja Aleksandra 73, 11120 Belgrade, Serbia E-mail: [email protected] making. 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Application of fuzzy AHP method based on eigenvalues for decision making in construction industry

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N. Praščević, Ž. Praščević Primjena neizrazite AHP metode utemeljene na vlastitim vrijednostima za donošenje odluka u građevinarstvu ISSN 1330-3651 (Print), ISSN 1848-6339 (Online) DOI: 10.17559/TV-20140212113942 APPLICATION OF FUZZY AHP METHOD BASED ON EIGENVALUES FOR DECISION MAKING IN CONSTRUCTION INDUSTRY Nataša Praščević, Živojin Praščević Original scientific paper The Analytic Hierarchy Process (AHP) has found wide application for multicriteria decision making (MCDM) in various areas. In the paper is given a short survey of the AHP application for decision making in the construction industry and realisation of construction projects. Authors have proposed a method based on the eigenvalue and eigenvector of the fuzzy comparison matrices for determination of local and global priorities of decision alternatives. These eigenvalues are determined using expected values of fuzzy numbers and their products. These fuzzy eigenvalues and eigenvectors are in further procedure applied for ranking the alternatives. Comparing the final ranks of alternatives, obtained according to proposed method and some other methods, the authors found good agreement. The fuzzy AHP, in comparison with AHP with crisp numbers, gives more complete, flexible and realistic results. A case study of the optimal selection of the structural system for a large industrial hall is presented in the paper. Keywords: construction industry; fuzzy AHP; fuzzy number; multicriteria decision making Primjena neizrazite AHP metode utemeljene na vlastitim vrijednostima za donošenje odluka u građevinarstvu Izvorni znanstveni članak Analitički hijerarhijski proces (AHP) našao je široku primjenu za višekriterijsko donošenje odluka (MCDM) u različitim područjima. U radu je da n kratak osvrt na primjenu AHP metode za donošenje odluka u građevinarstvu i realizaciji građevinskih projekata. Autori su predložili metodu za određivanje vlastitih vrijednosti i vlastitih vektora neizrazitih (fuzzy) matrica komparacije za nalaženje lokalnih i globalnih prioriteta alternat iva odlučivanja. Vlastite vrijednosti se određuju uporabom očekivane vrijednosti neizrazitih (fuzzy) brojeva i njihovih produkata. Dobivene vlastite vrijednosti i vlastiti vektori se u daljnjoj proceduri primjenjuju za rangiranje alternativa. Uspoređujući konačne rangove alternativa koji su dobiveni prema predloženoj metodi i nekim drugim metodama, autori su našli dobru suglasnost. AHP sa neizrazitim brojevima, u uspoređenju sa AHP sa izrazitim brojevima, daje kompletnije, fleksibilnije i realističnije rezultate. U radu je priložena studija slučaja optimalnog izbora konstruktivnog sustava jedne industrijske hale. Ključne riječi: građevinarstvo; neizraziti AHP; neizraziti broj; višekriterijsko odlučivanje 1 Introduction value of comparisons. This is important for application in the construction industry, where in the first phase of the The Analytic Hierarchy Process (AHP) for choosing construction project realization and preparation of factors that are important for decision making (DM) was preliminary feasibility studies, many important data proposed by Saaty [1]. This is one of the useful methods concerning costs, time of works execution and others, are in multi criteria decision making (MCDM), which has not precisely known, but the values of comparison of found wide application in many areas of science and important factors could be better assessed. Since these practice, so there are a big number of references about values cannot be expressed precisely by the crisp AHP. In this process factors are selected and formulated numbers, it is necessary to use the fuzzy numbers. The in a hierarchy structure descending from one overall goal usage of verbal judgements ("equal", "equal/moderate", to criteria and alternatives, as it is shown in Fig. 1. "moderate" to "extreme") for mutual comparison of Each level may represent different factors criteria, sub criteria and alternatives is more accurate than (economical, technical, social, etc.) that are evaluated by by integer or crisp numbers. experts. It provides an overall view of the complex At each level, the comparisons may be expressed relationships inherent in a considered situation. It helps numerically or linguistically in words. These non the decision maker to assess whether the issues in each numerical values are transformed to numerical ones level have the same order of magnitude, so he can according to the corresponding scale. According to compare such homogeneous elements accurately. As Ishizaka [3] “the absence of units in comparison values is Saaty [2] emphasises, "the most effective way to an important advantage, since these values are quotients concentrate judgments is to take a pair of elements and of two quantities of the same kind”. Application of fuzzy compare them on a single property without concern for numbers instead of crisp numbers gives more realistic other properties or other elements". results and better ranking of alternatives. Elements that have a global character are represented The first solution of fuzzy AHP was proposed by Van at the higher levels of the hierarchy. "The fundamental Laarhoven and Pedrycz [4]. They used triangular fuzzy approach of AHP is to decompose a big problem into numbers and employed the logarithmic least squares several smaller problems that are solved separately to method (LLSM) to generate elements of the priority determine their priority vectors. According to these values vector (fuzzy weights). Buckley [5] used trapezoidal of the separate priority vectors, the final priority vector of fuzzy numbers to express pair-wise comparison values. the alternatives is calculated taking into account Csutora and Buckley [6] proposed "Lambda-Max relationships between hierarchy levels" [2]. method", which is direct fuzzification of the eigenvector Unlike other methods of MCDM, here it is not method. Buckley et al. [7] revisited fuzzy hierarchical necessary to know the exact numerical values of the analysis and presented a new method of finding fuzzy factors being considered, so, it is enough to assess a good weights by fuzzifying an equivalent method. They Tehnički vjesnik 23, 1(2016), 57-64 57 Application of fuzzy AHP method based on eigenvalues for decision making in construction industry N. Praščević, Ž. Praščević designed an evolutionary algorithm to estimate fuzzy most appropriate alternative. Level 1 encompasses weights. Wang and Chin [8] proposed an eigenvector prescribed criteria and level 2 contains alternatives that method to generate interval or fuzzy weight estimate from are related to these criteria. an interval of fuzzy comparison matrix with trapezoidal and triangular fuzzy numbers, which differs from Orevall goal Level 0 mentioned Lambda-Max method, proposed by Csutora Overall goal and Buckley [6]. Mikhailov [9] developed a fuzzy programming method based on geometric representation Level 1 Criterion C Criterion C Criterion C 1 2 n Criteria of the prioritization process. The problem of finding elements of the priority vector is transformed into fuzzy programming problem. Level 2 Alternat. A Alternat. A Alternat. A 1 2 m The extent analysis, proposed by Chang [10], is used Alernatives in many papers for handling the fuzzy AHP and ranking Figure 1 Hierarchical levels alternatives. He applied this approach with triangular fuzzy numbers [11] for calculation of the synthetic extent Unlike other methods of multicriteria decision values S of the pair-wise comparison matrix and using the making, relative weights w of factors F (i = 1,2,...,k), i i principle of comparison of fuzzy numbers that was which in this case are criteria or alternatives, are proposed in [10], found the requested weight vector of the compared in dependence on corresponding level. These comparison matrix. weights are assessed usually by the decision making team. In a number of papers for solving multicriteria According to these values is determined the priority decision problems, the AHP and fuzzy AHP are combined matrix F = [f ] , with elements ij k×k with the TOPSIS and Fuzzy TOPSIS method. In this paper a method is proposed, which is different f = , i , j = 1,2,...k , from methods proposed by other authors, for (1) ij determination of fuzzy eigenvalues and fuzzy priority vectors based on expected values of fuzzy numbers. These fuzzy values are in further procedure used for where w and w are weights of corresponding criteria C i j i ranking of alternatives. and C . This matrix is known as a reciprocal matrix, since it has positive entries everywhere and satisfies the 2 Application of AHP in construction reciprocal property Like in other industries, AHP as one of the methods f = , f > 0, f = 1, i , j = 1,2,...,k. (2) of multicriteria decision making is used in the ij ji ii ji construction industry to solve many different problems, and here are emphasized some of them: This matrix is consistent, because the following - Selection of construction projects for realization, conditions are satisfied - Selection of a contractor for the project realization, - Selection of temporary facilities and machinery in f = f f , i , j = 1,2,...k ; p = 1,2,...k construction sites, ij ip pj - Selection of construction methods, - Choice of the maintenance strategy for the According to Saaty [1, 2] necessary and sufficient construction equipment, condition for consistency is that the principal eigenvalue - Choice of structural systems for design of bridges, of matrix F, for the eigenvalue problem buildings and other civil engineering projects, - Choice of strategy for the maintenance of structural Fw = kw, (3) systems, - Determination of weighting factors affecting safety has value λ = k. max on construction sites, For further analysis it is necessary to normalize - Choice of a supplier of resources for construction, vector w by dividing each of its elements by their sum - Ranking of real estates, etc. 3 Non fuzzy AHP w = , i = 1,2,...,k. (4) w + w + ....w 1 2 k In the first Saaty’s works is proposed and developed AHP with non fuzzy (crisp) data on several levels and The values f , according to Saaty [1], [2], represent ij many other authors have used this procedure to solve the pair-wise comparison or importance of the factor F different problems of decision making. In this paper is compared to the factor F at a certain level of the considered the problem of multicriteria decision making hierarchy. Hence, matrix F is called pair-wise comparison in which given alternatives A , A ,..., A are ranked for 1 2 m matrix. As Saaty [1] emphasizes, in a general decision prescribed criteria C , C ,...,C . One model with three 1 2 n making it is impossible to give precise values of elements levels for solving these problems is shown in Fig. 1. f according to Eq. (1), but only estimate them. For ij Level 0 is related to the overall goal, which includes elicitation of pair-wise comparison judgments of criteria, ranking of alternatives and determination of the best or he proposed fundamental scale of measurements. The 58 Technical Gazette 23, 1(2016), 57-64 N. Praščević, Ž. Praščević Primjena neizrazite AHP metode utemeljene na vlastitim vrijednostima za donošenje odluka u građevinarstvu differences Δij = f – w /w cause inconsistency of the ij i j μ (x) matrix F, and its principal eigenvalue is 1.00 A (α) A (α) λ ≥ k. (5) max μ (x) To every eigenvalue λ corresponds eigenvector w i i a x a a l m u that represents one solution of the system of k homogeneous linear Eq. (3). Maximal positive real Figure 2 Triangular fuzzy number A eigenvalue λ and the corresponding eigenvector w are max Parametric presentation of a triangular fuzzy number accepted for further calculation. Since the estimated matrix F is not a consistent one, Saaty [1] introduced the A at level α is consistency index CI and the consistency ratio CR for this matrix, that should be calculated by the following A = [ A (α ), A (α )], (8) α l u formulas where λ − k max CI = , (6) k − 1 A (α ) = a + (a − a )α , l l m l CI A (α ) = a − (a − a )α , (9) u u u m CR = . (7) 0 < α ≤ 1 a ≤ a ≤ a . RI l m u RI is called random consistency, which depends on Triangular fuzzy number is usually described by the size of matrix k, and its values proposed by Saaty [1] three characteristic values a , a and a which are crisp l m u are given in Tab. 1. numbers Table 1 Average random consistency RI A = (a , a , a ) . (10) K 1 2 3 4 5 6 7 8 9 10 l m u RI 0 0 0,58 0,90 1,12 1,24 1,32 1,41 1,45 1,49 ~ ~ −1 Reciprocal fuzzy number A to A is for a > 0 If CR ≤ 0,10, the estimates of the elements of the vector w are acceptable. Otherwise, the consistency of ~ 1  1 1  −1 the matrix F should be improved by changing values of A = = , , ~   some of its elements, taking into account that this matrix A (α ) A (α )  u l  must be reciprocal. Saaty’s method is based on calculation of the maximal eigenvalue and corresponding or approximatelly [4, 5, 6] eigenvectors, and hence is known as the eigenvector method.   ~ 1 1 1 −1   A = , , . (11)   a a a 4 Fuzzy AHP and determination of the priority vectors for u m l   ranking of alternatives since this fuzzy number is not exactly triangular one. Some of decision criteria are subjective and In this paper, the pair-wise comparison judgments, qualitative by nature, so the decision maker cannot easily that express relative importance between factors F and F i j express strengths of his preferences or provide exact pair- in the hierarchy, are expressed by the triangular fuzzy wise comparison. Hence, the crisp numbers are not so numbers f ij suitable to express these pair-wise comparison values due to their vagueness. Since judgments of the decision maker ~ ~ or his team are uncertain and imprecise, it is much better f = ( f , f , f ), f = (1, 1, 1) f > 0 ij ij ,l ij ,m ij ,u ii ij ,l (12) to give pair-wise comparisons as fuzzy values than as i , j = 1, 2,..., k , i ≠ j, crisp ones. As Wang et al. [8] emphasize, due to complexity and uncertainty involved in real world, which constitute a fuzzy comparison matrix F with the sometimes, it is unrealistic or impossible to acquire exact following elements judgments for these decision problems. To overcome these shortcomings due to crisp   numbers, the fuzzy AHP was developed for solving these 1 1 1 1   (13) f = = , , , i, j = 1, 2,..., k , i ≠ j. ji   problems of multicriteria decision making. The triangular f f f ij ,u ij ,m ij ,l ij   and trapezoidal fuzzy numbers are commonly adoptive due to their simplicity in mathematical modelling of many This matrix is consistent if and only if [5] problems in practice. In this paper are used triangular fuzzy numbers that are most frequently used by many ~ ~ ~ f ≈ f ⊗ f , i, j = 1,2,...k , p = 1,2,...k (14) ip ip pj authors. The triangular fuzzy number, as a special type of a fuzzy set over the set of real numbers (real line) R, is while the sign ⊗ denotes a fuzzy product. shown in Fig. 2. Tehnički vjesnik 23, 1(2016), 57-64 59 Application of fuzzy AHP method based on eigenvalues for decision making in construction industry N. Praščević, Ž. Praščević This method is based on the calculation of expected According to Eqs. (10) ÷ (13) the fuzzy matrix F can values of fuzzy numbers and their products. Expected be expressed by three characteristic crisp matrices ~ ~ value EV ( A) of the fuzzy number A = (a , a , a ) , l m u F = (F , F , F ). (15) written in the parametric forms (9), is [13] l m u , F , F are obtained according to (12) a + 2a + a Crisp matrices F ~ l m u l m u (20) EV ( A) = . and (13) as follows ~ ~  1 f ... f  12 ,l 1k ,l The product of two positive fuzzy numbers A ⊗ B ,   1 f 1 ... f 12 ,u 2k ,l with α cuts A = [ A (α ), A (α )] and F = , α l u   . . ... .   B = [B (α ), B (α )] 0 < α ≤ 1 can be written in the α l u 1 f 1 f ... 1   1k ,u 2k ,u   following form 1 f ... f   12 ,m 1k ,m   1 f 1 ... f 12 ,m 2k ,m A ⊗ B = [ A (α )B (α ), A (α )B (α )], (21) F = , (16)   α α l l u u . . ... .   1 f 1 f ... 1   1k ,m 2k ,m   Expressing A (α), A (α) in the form (9) by a , a , a l u l m u  1 f ... f  12 ,u 1k ,u and B (α), B (α) by b , b , b for a > 0, b > 0, in a similar l u l m u l l   1 f 1 ... f way, after integration, the expected value of the product 12 ,l 2k ,u F = .   . . ... . of two fuzzy numbers is obtained   1 f 1 f ... 1   1k ,l 2k ,l   ~ ~ 1 EV (A ⊗ B ) = [(2a + a )b + l m l (22) Buckley [5] has proved that fuzzy matrix is + (a + 4a + a )b + (a + 2a )b ] l m u m m u u consistent according to (15) if and only if its crisp matrix F is consistent. Some authors have proposed triangular fuzzy A system of the fuzzy linear Eq. (18) may be written in the following form numbers for expression of the intensity of importance on Saaty’s absolute scale. In this paper the following fuzzy ~ ~ ~ ~ ~ ~ ~ ~ numbers are used f ⊗ w ⊕ f ⊗ w ⊕ ... ⊕ f ⊗ w = λ ⊗ w , i = 1,2,....n, i1 1 i 2 2 in n i 1 = (1, 1, g ), x = ( x − g , x , x + g ) for where the sign ⊕ denotes the fuzzy addition. g i i d i g (17) The expected values of fuzzy products due to (22) are 1 < x < 9, 9 = (9 − g , 9, 9). i d where γ and γ are chosen real numbers. d g EV ( f ⊗ w ) = [(2 f + f )w + ij ij ij ,l ij ,m j ,l 12 (23) The new technique for calibration of membership functions in the fuzzy AHP proposed by Ishizaka and + ( f + 4 f + f )w + ( f + 2 f )w ] ij ,l ij ,m ij ,u j ,m ij ,m ij ,u j ,u Nguen [12] is very acceptable. The formula for definition ~ 1 of membership functions, given and used in this paper, is EV (λ ⊗ w ) = [(2λ + λ )w + i l m i,l (24) in accordance with that proposal. + (λ + 4λ +λ )w + (λ + 2λ )w ] l m u i,m m u i,u 4.2 Approximate method for determining fuzzy weights i, j = 1,2,..., k. Since Saaty’s AHP method is based on finding The expected value of the sum of fuzzy numbers is eigenvalue and eigenvectors of the fuzzy matrix F at the equal to the sum of the expected values of fuzzy numbers corresponding hierarchical level, here is proposed one method to solve the fuzzy eigenvalue and eigenvector ~ ~ problem and find solutions of the system of homogenous EV ( f ⊗ w ) = λ ⊗ w , i = 1,2,..., n. (25) ∑ ij j i fuzzy linear equations j =1 ~ ~ ~ ~ F ⊗ w = λ ⊗ w, (18) By introducing formulas for expected values of the fuzzy products (23) and (24) in these fuzzy equations, is obtained a system of fuzzy linear homogenous equations where is the fuzzy reciprocal comparison matrix of type [k×k]. ~ F w + F w + F w − λ w − λ w − λ w = 0, (26) l l m m u u l l m m u u Elements of the fuzzy matrix F , fuzzy vector and fuzzy eigenvalue λ are assumed as triangular fuzzy where are numbers, that may be denoted according to Eq. (10) as ~ F = 2F + F , F = F + 4F + F , F = F + 2F , (27) l l m m l m u u m u w = (w , w , w ), λ = (λ ,λ ,λ ) (19) l m u l m u 60 Technical Gazette 23, 1(2016), 57-64 N. Praščević, Ž. Praščević Primjena neizrazite AHP metode utemeljene na vlastitim vrijednostima za donošenje odluka u građevinarstvu c = (c , c , c ) , (i, j=1,2,…,n) using appropriate λ = 2λ + λ , λ = λ + 4λ + λ , λ = λ + 2λ . (28) ij ij ,l ij ,m ij ,u l l m m l m u u m u comparison scale adjusted to fuzzy values according to and vectors written in the transposed form (17). Express the fuzzy matrix C by three matrices C , C l m and C according to (15) and (16). Solve the fuzzy ~ ~ ~ ~ w = [w ,w ,...,w ] , 1,l 2 ,l n ,l l eigenvalue problem C ⊗ w = λ ⊗ w ,as described in the w = [w ,w ,...,w ] , (29) previous section, and determine the principal fuzzy m 1,m 2 ,m n ,m w = [w ,w ,...,w ] . eigenvalue λ = (λ , λ , λ ) and corresponding fuzzy u 1,u 2 ,u n ,u l m u eigenvectors w = (w ,w ,w ) . Normalize these vectors l m u Since all the values in these equations are positive using formulas (32) and (33) to obtain the fuzzy priority ones, the system of Eqs. (26) may be decomposed into ~ vectors of criteria w = (w ,w ,w ) . l m u three systems, which represent three crisp eigenvalue For the matrix C , calculate the consistency index CI problems and consistency ratio CR according to (6) and (7). If CR≤0,10, accept the assessed fuzzy elements of the pair- F w = λ w , F w = λ w , F w = λ w . (30) l l l l m m m m u u u u wise matrix C and obtained eigenvalues and eigenvectors. If CR > 0,10, improve the consistency of the By solving these three auxiliary eigenvalue problems, matrix C by changing some of its elements and repeat eigenvectors w , w and w and auxiliary eigenvalues l m u the procedure until this condition is satisfied. Fourth step. Formulate the pair-wise comparison λ , λ and λ are obtained. After that the requested m u ( j ) matrices for the alternatives related to the criterion eigenvalues λ , λ and λ are determined by solving the l m u C (j = 1,2,…, n) system of linear Eqs. (28). To meet the requirements for the principal C A A ...... A eigenvalues λ ≤ λ ≤ λ , normalised eigenvectors j 1 2 m l m u ( j ) ( j ) ~ ~   should satisfy the next condition A 1 a .... a 1 12 1m   ( j ) ( j ) ~ −1 ~ (34) ~ A ( j ) 2 (a ) 1 ... a  12 2m  A = , j = 1,.,n. w ≤ w ≤ w . (31) ... l m u  .... .... .... ....  ( j ) −1 ( j ) −1 ~ ~ A   m (a ) (a ) .... 1 1m 2m   The calculated eigenvectors w , w and w should be l m u normalized according to the following formulas Express these fuzzy matrices by the matrices ( j ) ( j ) ( j ) A , A , A according to (16). Solve the fuzzy l m u w λ l l w = , eigenvalue problems s λ l m ~ ~ m ( j ) ~( j ) ( j ) ~( j ) w = , A ⊗ p = λ ⊗ p , j = 1,2,...,n (35) (32) w λ u u to find the fuzzy principal eigenvalues w = , s λ ( j ) ( j ) ( j ) ( j ) u m and fuzzy eigenvectors λ = (λ , λ , λ ) max m u ~ ( j ) ( j ) ( j ) ( j ) (j) p = ( p , p , p ) , consistency indices CI and where max l m u (j) consistency ratios CR , according to (6) and (7) for n n n ( j ) matrices A , (j=1,2,…,n). If the consistency ratio is s = w , s = w , s = w . (33) l ∑ i,l m ∑ i,m u ∑ i,u (j) CR > 0,10, change some of the assessed values to i =1 i =1 i =1 ij ,m obtain the satisfactory consistency of this matrix. 4.2 Steps in the execution of the fuzzy AHP ~ ( j ) ( j ) ( j ) ( j ) Normalize vectors p = ( p , p , p ) by the formulas l m u (32) and (33) to obtain normalized local priority vectors Fuzzy AHP performs in several steps in a similar way ( j ) ( j ) ( j ) ( j ) as the procedure with crisp numbers, and is briefly p = ( p , p , p ) . This procedure is the same as in l m u explained here. the step 3. First step. Define the problem, overall goal that has Fifth step. Formulate local priority fuzzy to be attained, criteria, subcriteria, if necessary, and alternatives. matrix P = ( P , P , P ) , that contains normalized local l m u Second step. Define the hierarchy structure from the priority vectors, where top level through intermediate levels that contains the criteria and subcriteria to the lowest level, which is related (1) (2) (n) P = [ p p ... p ], l l l l to the alternatives, as shown in Fig. 1. (1) (2) (n) P = [ p p ... p ], (36) Third step. Formulate the pair-wise comparison m m m m (1) (2) (n) C for the criteria C ,C ,…,C by P = [ p p ... p ]. reciprocal fuzzy matrix 1 2 n u u u u assessing the priority values as fuzzy numbers Tehnički vjesnik 23, 1(2016), 57-64 61 Application of fuzzy AHP method based on eigenvalues for decision making in construction industry N. Praščević, Ž. Praščević Multiply these matrices from the right by the priority CV = , i = 1,2,...,m. (42) vectors of the criteria respectively, which are determined i ,e in the third step A fuzzy number or an alternative A with a smaller T T w = [w w ... w ] , * l 1,l 2,l n ,l CV is ranked better, and the best ranked alternative A is w = [w w ... w ] , (37) alternative A with minimal CV . i i m 1,m 2,m n ,m T According to this procedure, the authors have w = [w w ... w ] . u 1,u 2,u n ,u developed a corresponding computer program in MATLAB, which has been used to solve several and obtain vectors of global priorities g , g and g l m u. problems of ranking alternatives in the construction industry. The recently proposed method and computer g = P w = [g g ... g ] . program have been used for making the optimal choice l l l 1,l 2 ,l m ,l for the new railway trace in Montenegro, which is around g = P w = [g g ... g ] . (38) m m m 1,m 2,m m ,m 200 km long. g = P w = [g g ... g ] . u u u 1,u 2,u m ,u 5 Case study These vectors constitute fuzzy matrix of global priorities G = ( g , g , g ) of alternatives A , A ,…A . Th e pr o posed m e thod of the fuzzy AHP was applied 1 2 m l m u for the choice of the optimal structural reinforced concrete For every alternative A (i = 1, 2,...,m), elements of system of an industrial two-part hall with dimensions at these vectors are expressed by the corresponding the base of 2 × 24,50 m × 120,00 m. This hall under approximate triangular fuzzy numbers construction is shown in Fig. 3. The choice of appropriate ~ structural system and technology of construction, which g = ( g , g , g ), i = 1,2,..., m. (39) i i.l i,m i,u affects costs and speed of construction, is one of main tasks of the design and construction team. The period of Sixth step. Alternatives A (i = 1, 2,...,m) are ranked design and construction was limited on seven months, in this step according to their global priorities that are according to the contract between the investor and expressed by triangular fuzzy numbers g (i = 1,2,..., m). contractor. More proposals for ranking fuzzy numbers exist in the literature, and here is used Lee and Le’s[14] method improved by Cheng [15]. In this paper, comparison of the fuzzy numbers is based on the probability measure of fuzzy events, which was introduced by Zadeh [16]. The fuzzy numbers are ranked according to the generalized fuzzy mean (expected value) and generalized fuzzy spread (standard deviation). For the triangular probability distribution of the triangular fuzzy number as a fuzzy event, these values for the fuzzy number g are calculated by the following formulas [15]: - Generalized fuzzy mean (expected value) g + 2g + g i ,l i ,m i ,u g = , i = 1,2,...,m (40) i ,e Figure 3 Industrial hall under construction According to these requirements, the design team, in - Generalized spread (standard deviation) which the second author of this paper was included, established alternatives and criteria for the choice of the 2 2 2 σ = [ (3g + 4g + 3g − 4g g − most acceptable structural system. For the choice of this i i ,l i ,m i ,u i,l i,m 80 (41) system, the next three alternatives have been considered: 1/ 2 − 2g g − 4g g )] , i = 1, 2,..., m. i,l i,u i,m i,u Alternative A – Two chord reinforced concrete and steel girder supported by the reinforced concrete columns, According to Lee and Li [14], a fuzzy number with a shown in Fig. 1; Alternative A – Prestressed concrete girder supported higher mean value and, at the same time, a lower spread is ranked better. However, it is not easy to compare two by the reinforced concrete columns; Alternative A – Classical frame structure of fuzzy numbers when one of them has a higher mean value and, at the same time, a higher spread and the other has a reinforced concrete. Four main criteria have been used: lower mean and, at the same time, a lower spread. Therefore, Cheng [15] proposed to rank fuzzy numbers C – Summary costs of the design and the construction of the hall, according to the coefficient of variation CV C – Costs of annual maintenance of the hall, C – Time necessary for the construction works of the hall in weeks, 62 Technical Gazette 23, 1(2016), 57-64 N. Praščević, Ž. Praščević Primjena neizrazite AHP metode utemeljene na vlastitim vrijednostima za donošenje odluka u građevinarstvu C – Technological possibilities of the contracting 4 w = [0,3211 0,2007 0,1898 0,1283], firm to construct this industrial hall in the chosen system. w = [0,3845 0,2391 0,2222 0,1541], These criteria are usually used as the most important w = [0,4820 0,3135 0,2872 0,2015]. in the construction industry for all civil engineering u works, especially for this type of building. Besides these criteria, depending on the type and purpose of a building, For the matrix C are found consistency index CI = often are taken into account the other criteria such as 0,0025 and consistency ratio CR = 0,0028 < 0,10. functional, aesthetical, environmental and so on. Since the consistency index and consistency ratio are The authors of this paper have assessed pair-wise very small, the assessed matrix C is very consistent and comparison values on the basis of real technical, may be accepted. technological and financial data from this and similar ( j ) For matrices A of alternatives A (i=1,2,3) related to projects and investments. According to the data, pair-wise the criteria C , (j = 1,2,3,4) the following results have comparison fuzzy matrices have been formulated: been obtained: Fuzzy matrix C for the criteria Principal eigenvalues are given in Tab. 2. Table 2 Principal eigenvectors (1,1,1) (1,1.5 ,2) (1.5, 2,2.5) (2,2.5,3)   ~ ( j ) ( j ) ( j ) ( j )   1 1 λ λ Matrix A l m u ( , ,1) (1,1,1) (1,1,1.5) (1,1.5,2)   2 1.5 ~ (1) ~   2,6627 3,0015 3,4097 C = 1 1 1 1   ( , , ) ( ,1,1) (1,1,1) (1,1.5,2) ~ (2) 2,6880 3,0002 3,5207   A 2.5 2 1.5 1.5   1 1 1 1 1 1 1 (3) 2,5978 3,0015 3,5071 ( , , ) ( , ,1) ( , ,1) (1,1,1)    3 2.5 2 2 1.5 2 1.5  (4) 2,5604 3,0013 3,5654 ( j ) ( j ) Fuzzy matrices A related to the criterion C (j =1, Since the principal eigenvalue of matrices A , (j=1, 2, 3,4) are 2, 3, 4) are very close to the number of their columns n = 3, these matrices are consistent ones.   Priority fuzzy matrices, P , P and P , that contain l m u   (1,1,1) (1.2,1.5,1.8) (1.7,2,2.3) normalized eigenvectors (local priority vectors) according   ~ 1 1 1 (1) to (36) are A = ( , , ) (1,1,1) (1.2,1.5,1.8) 1.8 1.5 1.2   1 1 1 1 1 1   ( , , ) ( , , ) (1,1,1) 0,4073 0,3525 0,3415 0,3242      2.3 2 1.7 1.8 1.5 1.2    P = 0,2813 0,2742 0,2985 0,2470 ,     0,1986 0,2359 0,2254 0,2818     (1,1,1) (1,1.3,1.6) (1.2,1.5,1.8)   0,4586 0,4082 0,3968 0,3814 ~ 1 1 (2) A =  ( , ,1) (1,1,1) (0.9,1.2,1.5)   P = 0,3194 0,3201 0,3459 0,2874 1.6 1.3     1 1 1 1 1 1 0,2220 0,2718 0,2573 0,3312     ( , , ) ( , , ) (1,1,1)   1.8 1.5 1.2 1.5 1.2 0.9   0,5178 0,4742 0,4617 0,4494     P = 0,3658 0,3781 0,4065 0,3437 .    (1,1,1) (0.9,1.2,1.5) (1.2,1.5,1.8) 0,2524 0,3212 0,3003 0,3949     1 1 1 (3) A = ( , , ) (1,1,1) (1.1,1.4,1.7)  1.5 1.2 0.9   Vectors of global priorities g , g and g are l m u 1 1 1 1 1 1   ( , , ) ( , , ) (1,1,1) calculated according to the expression (38), multiplying    1.8 1.5 1.2 1.7 1.4 1.1  matrices P , P and P by normalised vectors l m u   w ,w and w respectively, and results are shown in   (1,1,1) (1,1.3,1.6) (0.9,1.2,1.5) l m u   ~ 1 1 (4) Tab. 3. For each alternative components of these vectors A = ( , ,1) (1,1,1) (0.9,1.2,1.5)   1.6 1.3   form fuzzy numbers g = ( g , g , g ) (i= 1, 2, 3). i i,l i,m i,u 1 1 1 1 1 1   ( , , ) ( , , ) (1,1,1) For these fuzzy numbers, generalized fuzzy means    1.5 1.2 0.9 1.5 1.2 0.9  (expected values), standard deviations (spreads) and are determined for coefficients of variations V Elements of the comparison matrices are expressed alternatives A (i=1,2,3) using the expressions (40), (41) by the proposed formula (16) with γ = γ = 0,5 for d g and (42). The alternatives are ranked according to these elements of matrix C and γ = γ = 0,3 for matrices values and the results are shown in Tab. 4. d g ( j ) According to Chang's method based on the extent A , j = 1,2,3,4. analysis [11] priority normalized vector of alternatives Applying the mentioned computer program, the ~ d = [0,5165 0,3194 0,1641] is obtained. Alternative following principal eigenvalues for the matrix C and with higher component of this vector is better ranked. normalised eigenvectors are obtained Alternative A is the best ranked according to the λ = 3,3663, λ = 4,0075, λ = 5,1464, l m u expected value g = 0,443 and the coefficient of 1,e Tehnički vjesnik 23, 1(2016), 57-64 63 Application of fuzzy AHP method based on eigenvalues for decision making in construction industry N. Praščević, Ž. Praščević variation V = 11,41 %. This alternative has a discernible 7 References advantage over the other alternatives. This problem has [1] Saaty, T. L. The Analytic hierarchy Process. McGraw-Hill, been solved by the authors using the modified fuzzy New York, 1980. TOPSIS method [17] and the same order of alternatives is [2] Saaty, T. L. How to make a decision: Analytic Hierarchy obtained. Process. // European Journal of Operational Research. 48, (1990), pp. 9-26. DOI: 10.1016/0377-2217(90)90057-I Table 3 Vectors of global priorities, expected values and standard [3] Ishizaka, A. Clusters and Pivots for Evaluating a Large deviations Number of Alternatives in AHP, SOBRAPO, Brazilian Vector Vector Vector Exp. val. Stand. Alternative Operational Research Society, 2012. URL: g g g g dev. (%) l m u i,e www.scielo.br/pope. A 0,308 0,421 0,621 0,443 11,41 [4] Van Laarhoven, P. J. M.; Pedrycz, W. A fuzzy extension of A 0,106 0,195 0,442 0,235 11,71 Saaty‘s priority theory. // Fuzzy Sets and Systems. 11, A 0,120 0,245 0,575 0,297 11,78 (1983), pp. 229-241. DOI: 10.1016/S0165-0114(83)80082-7 [5] Buckley, J. J. Fuzzy hierarchical analysis. // Fuzzy Sets and Table 4 Ranks of alternatives Systems. 17, (1985), pp. 233-247. DOI: 10.1016/0165- Rank Exp. Coeffic. V Ranking of Alternative value Alternative (%) vector d 0114(85)90090-9 alter. g (Cheng) (Chang) i,e [6] Csutora, R.; Buckley, J. J. Fuzzy hierarchical analysis: the * * 1 A =A 0,443 A =A 11,41 0,5165 Lambda-Max method. // Fuzzy Sets and Systems. 120, 1 1 (2001) pp. 181-195. DOI: 10.1016/S0165-0114(99)00155-4 2 A 0,339 A 11,71 0,3194 2 3 [7] Buckley, J. J.; Feuring, T.; Hayashi, Y. Fuzzy hierarchical 3 A 0,274 A 11,78 0,1641 3 2 analysis revisited. // European Journal of Operational Research. 129, (2001), pp. 48-64. DOI: 10.1016/S0377- The alternative of the structural system A has been 2217(99)00405-1 chosen for realization and the hall has been successfully [8] Wang, J. M., Chin, K-S., An eigenvector method for finished in the planned and contracted time between the generalizing normalized interval and fuzzy weights. // investor and contractor. Applied Mathematics and Computation, 181, (2006), pp. 1257-1275. DOI: 10.1016/j.amc.2006.02.026 6 Conclusions [9] Mikhailov, L. A fuzzy programming method for deriving priorities in analytic hierarchy process. // Journal of the Operational Research Society. 51, (2000), pp. 341-349. DOI: The fuzzy AHP is one of very useful methods for 10.1057/palgrave.jors.2600899 multicriteria decision making (MCDM) in which the [10] Chang, D.-Y.; Extent Analysis and Synthetic Decision, factors that affect the decision making (criteria, sub- Optimization Techniques and Applications. Vol. 1, World criteria and alternatives) are hierarchically arranged scientific, Singapore, 1992. descending from one overall goal to the criteria, sub- [11] Chang, D.-Y. Application of the extent analysis method on criteria and alternatives in successive levels. In this fuzzy AHP. // European Journal of Operational Research, process, at all levels, these factors are mutually compared 95, 12(1996), pp. 649-655. DOI: 10.1016/0377-2217(95)00300- and these comparisons are expressed numerically or linguistically in words which are further transformed into [12] Ishizaka, A.; Nguyen, N. H. Calibrated fuzzy AHP for current bank account selection. // Expert systems with numbers according to the given comparison scale. applications. 40, (2013), pp. 3775-3783. DOI: This method, as other methods of MCDM, found 10.1016/j.eswa.2012.12.089 useful applications in the construction industry and [13] Chanas, S. On the interval approximation fuzzy numbers. // construction project management as a help for rational Fuzzy Sets and Systems. 122, (2001), pp. 353-356. DOI: decision making, so a large number of references in this 10.1016/S0165-0114(00)00080-4 field exists in the literature. [14] Lee, E. S.; Li, R. L. Comparison of fuzzy numbers based on The fuzzy AHP method gives more complete, flexible the probability measure of fuzzy events. // Comput. and realistic results, especially for the decision criteria Mathematic Application. 15, (1988), pp. 887-896. DOI: that have qualitative nature. The procedure for 10.1016/0898-1221(88)90124-1 [15] Cheng, C-H. A new approach for ranking fuzzy numbers by determination of approximate eigenvalues and distance method. // Fuzzy Sets and Systems, 95, (1992), pp. eigenvectors of the reciprocal matrices, proposed in this 307-317. DOI: 10.1016/S0165-0114(96)00272-2 paper, is used to find priority vectors for alternatives. This [16] Zadeh, L. A. Probability measures of fuzzy events. // procedure and corresponding computer program have Journal of Mathematic Analysis and Applications. 23, been used several times for choosing most appropriate (1968), pp. 421-427. DOI: 10.1016/0022-247X(68)90078-4 alternative for realisation of some construction projects. [17] Prascevic, Z.; Prascevic, N. One modification of fuzzy Comparisons that authors have made with application of TOPSIS method. // Journal of Modelling in Management. 8, some other methods of MCDM, have shown good (2013), pp. 81-102. DOI: 10.1108/17465661311311996 accordance with results obtained by the fuzzy AHP. In many situations, especially when number of alternatives is Authors’ addresses large, it is advisable to combine fuzzy AHP with other methods of MCDM, using AHP for determinations of the Doc. dr. sc. Nataša Praščević, dipl. ing .građ. criteria weights and after that chose the method for Prof. dr. sc. Živojin Praščević, dipl. ing. građ. ranking alternatives like TOPSIS, VICOR or some other University of Belgrade, Faculty of Civil Engineering method for ranking alternatives for multicriteria decision Bulevar kralja Aleksandra 73, 11120 Belgrade, Serbia E-mail: [email protected] making. E-mail: [email protected] 64 Technical Gazette 23, 1(2016), 57-64

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