Using DEA and AHP for Multiplicative Aggregation of Hierarchical Indicators

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1 Ameican Jounal of Opeation Reeach, 205, 5, Publihed Online Septembe 205 in SciRe. Uing DEA and AHP fo Multiplicative Aggegation of Hieachical Indicato Mohammad Sadegh Paa Faculty of Management, Lauentian Univeity, Sudbuy, Canada Received 23 June 205; accepted 2 July 205; publihed 24 July 205 Copyight 205 by autho and Scientific Reeach Publihing Inc. Thi wo i licened unde the Ceative Common Attibution Intenational Licene (CC BY). Abtact The autho [Paa, M.S. (204) Uing Data Envelopment Analyi and Analytic Hieachy Poce to Contuct Compoite Indicato. Jounal of Applied Opeational Reeach, 6(3), ] ecently popoed a multiplicative appoach uing Data Envelopment Analyi (DEA) and Analytic Hieachy Poce (AHP) to eflect the pioity weight of indicato in contucting compoite indicato (CI). Nonethele, thi appoach i limited to the ituation with a ingle level hieachy which might not atify the need of a multiple level hieachy. Theefoe, the cuent pape extend thi appoach to the ituation in which the indicato of imila chaacteitic can be gouped into ub-categoie and futhe lined into categoie to fom a thee-level hieachical tuctue. An illutative example of oad afety pefomance fo a et of Euopean countie highlight the uefulne of the popoed extended appoach. Keywod Data Envelopment Analyi, Analytic Hieachy Poce, Compoite Indicato, Multiplicative Aggegation, Hieachical Stuctue. Intoduction A compoite indicato (CI) i a mathematical tool to aggegate a et of multidimenional indicato in ode to poduce a ingle meaue of pefomance. In a ecent pape, Paa [] popoed a multiplicative appoach uing DEA and AHP to eflect the pioity weight of indicato in contucting CI. Thi appoach can be oganized into the following tep: ) Uing a multiplicative DEA baed-ci model to compute the compoite value of each Deciion Maing Unit (DMU). The computed compoite value ae ued in the next tep. 2) Uing a i ditance model to obtain the optimal weight of indicato fo each DMU (imum compoite lo). How to cite thi pape: Paa, M.S. (205) Uing DEA and AHP fo Multiplicative Aggegation of Hieachical Indicato. Ameican Jounal of Opeation Reeach, 5,

2 M. S. Paa 3) Uing the i ditance model bounded by AHP to obtain the pioity weight of indicato fo each DMU (imum compoite lo). 4) Uing a paamete goal pogamg model to ae the pefomance of each DMU in tem of it elative cloene to the pioity weight of indicato. In the baic multiplication DEA baed-ci model, all indicato ae imply teated to be at the ame level of hieachy [2]. Nonethele, thee indicato might alo belong to diffeent ub-categoie and futhe be lined to one anothe contituting a thee-level hieachical tuctue. To ovecome thi limitation, we integate AHP to a thee-level DEA-baed CI model in a multiplicative context. A thee-level DEA baed-ci model can eflect the chaacteitic of the genealized multi-level DEA baed-ci model developed by [3]. 2. Methodology 2.. DEA-Baed CI Model A DEA-baed CI model can be fomulated imila to a multiplicative DEA model without explicit input [2]. In the following, and in line with the moe common CI teology, we will often efe to output a indicato. In ode to eliate the cale diffeence between all (output) indicato, and moeove, to enue that all of them ae in the ame diection of change the nomalized countepat of indicato, uing a - method, ae computed a follow [4]: y j yˆ ˆ j y ( ) = + x yˆ ˆ ( ) y ( ), yˆ ( ) { yˆ, yˆ,, yˆ } = fo deiable indicato, () 2 n yˆ ( ) yˆ j yj = + x, yˆ yˆ ˆ ( ) y ( ) = { yˆ ˆ ˆ, y 2,, yn} fo undeiable indicato. (2) ( ) Hee, y j i the nomalized value of (output) indicato ( =, 2,, ) fo DMU j( j =, 2,, n). Since in a multiplicative aggegation, the value of each indicato mut alway be lage than, we add a poitive contant x to the nomalized value of each indicato. We chooe x o that ( y ( ) + x) tun to.0 while y ( ) i the imum nomalized value of indicato fo all DMU. Although the model ued in thi pape doe not atify the deiable unit invaiant popety, it i vey obut to change in the meauement unit [2]. Theefoe, thi would only lightly change the compoite value without maing a ignificant change in DMU aning. Then a multiplicative optimization model in the contuction of a compoite indicato can be fomulated a u u CI = y, ubject to yj e with u 0, whee CI i the compoite value of DMU = = ( =, 2,, n) o the DMU unde aement, u i the weight of indicato (, 2,, ) = and e i the bae of the natual logaithm. Taing logaithm with bae e, the multiplicative model can be conveted to the following log-linea pogamg model: Max.t. = CI = u y (3) = uy j j, (4) u 0, (5) whee the tilde ymbol (~) denote natual logaithm. The combination of (3)-(5) fom a ingle level DEAbaed CI model in a log-linea context that loo lie an output-oiented DEA model without explicit input. Thi model i theoetically imila to the log-linea DEA model fo efficiency analyi intoduced in [5] Thee-Level DEA-Baed CI Model We develop ou fomulation baed on a genealized ditance model (fo example, ee [6]) in uch a way that the 328

3 M. S. Paa hieachical tuctue of indicato, uing a weighted-aveage appoach, ae taen into conideation [3]. Let y = y, 2,, l l =, 2,, S of categoy j be the value of indicato ( = ) of ub-categoy ( ) l( l =, 2,, S) fo DMU j( j, 2,, n) ln ( ll j ) of indicato of ub-categoy l' of categoy l while fo the DMU j i defined a while S l = p = afte nomalizing the oiginal data. Let u ll be the intenal weight j j = u ll = =. Then the value of ub-categoy l' of categoy l y = u y. Let p ll be the intenal weight of ub-categoy l' of categoy l =. Then the value of categoy l i defined a S y = p y. Let p l be the weight of categoy l. lj j l = To develop a linea model, the new multiplie of indicato of ub-categoy l' of categoy l i defined a: u = pp l u. Similaly, the new multiplie of ub-categoy l' of categoy l i defined a: p = pp l. Let CI ( =, 2,, n) be the bet attainable compoite value fo the DMU unde aement, calculated fom the one-level DEA-baed CI model. We want the compoite value CI ( u ll ), calculated fom the et of weight u ll to be cloet to CI. Ou definition of cloet i that the laget ditance i at it imum. Hence we chooe the fom of the i model: u { ( )} ll CI CI u to imize a ingle deviation which i equivalent to the following linea model: Min θ (6).t. S S l= l = = CI u y θ, (7) S S j j l= l = = S l = = u y CI j, (8) u = pl l, (9) u = p ll,, (0) = u, p, p> 0 ll,,, () l θ 0. (2) The combination of (6)-(2) fom a thee-level DEA baed-ci model in the log-linea context that identifie the imum compoite lo θ needed to aive at an optimal et of weight. Containt (7) enue that each DMU loe no moe than θ of it bet attainable compoite value, CI. The econd et of containt (8) atifie that the compoite value of all DMU ae le than o equal to thei uppe bound of CI j. Two et of containt (9) and (0) ae added to the model. Thi implie that the um of weight unde each (ub-)ub-categoy equal to the weight of that (ub-)ub-categoy. It hould be noted that the oiginal (o intenal) weight ued fo calculating the weighted aveage ae obtained a u = u p and p = p pl Pioitizing Indicato Weight Uing AHP The thee-level DEA baed-ci model identifie the imum compoite lo θ needed to aive at a et of weight of indicato by the intenal mechanim of DEA. On the othe hand, the pioity weight of indicato, and the coeponding (ub-)categoie ae defined out of the intenal mechanim of DEA by AHP. In ode to moe clealy demontate how AHP i integated into the thee-level DEA-baed CI model, thi eeach peent an analytical poce in which indicato weight ae bounded by the AHP method. The AHP pocedue fo impoing weight bound may be boen down into the following tep: Step : A deciion mae mae a paiwie compaion matix of diffeent citeia, denoted by A, with the entie of alq ( l = q =, 2,, S). The compaative impotance of citeia i povided by the deciion mae uing a ating cale. Saaty [7] ecommend uing a -9 cale. 329

4 M. S. Paa Step 2: The AHP method obtain the pioity weight of citeia by computing the eigenvecto of matix A w= w, w2,, ws, which i elated to the laget eigenvalue, λ. Aw = λ w. (3) (Equation (3)), ( ) T To detee whethe o not the inconitency in a compaion matix i eaonable the andom conitency atio, C.R., can be computed by the following equation: CR.. = λ N RI.. ( N ) whee R.I. i the aveage andom conitency index and N i the ize of a compaion matix. In a imila way, the pioity weight of (ub-)ub-citeia unde each (ub-)citeion can be computed. To obtain the weight bound fo indicato weight in the thee-level DEA-baed CI model, thi tudy aggegate the pioity weight of thee diffeent level in AHP a follow: u = we f, l S wl =, l= S e = and l = f ll = (4) = (5) whee l l l =,, S in AHP and e ll i the pioity weight of ub-citeion l ( l =, 2,, S ) unde citeion l and f ll i ub-ub-citeion ( =,, ) unde ub-citeion l. In ode to etimate the imum compoite lo θ neceay to achieve the pioity weight of indicato fo each DMU the following et of containt i added to the thee-level DEA-baed CI model: w i the pioity weight of citeion ( ) u = αu ll,,, while α > 0. (6) The et of containt (6) change the AHP computed weight to weight fo the new ytem by mean of a caling facto α. The caling facto α i added to avoid the poibility of contadicting containt leading to infeaibility o undeetimating the elative compoite coe of DMU [8] Paametic Goal Pogamg Model In thi tage we develop a paametic goal pogamg model that can be olved epeatedly to geneate the vaiou et of weight fo the dicete value of the paamete θ, uch that θ θ θ. The pupoe of the model i to imize the total deviation fom the pioity weight of indicato with a city bloc ditance meaue. Chooing uch a ditance meaue, each deviation i being equally weighted ubject to the following containt: S S + Min Z ( θ ) = ( d + d ).t. (7) l= l = = + u d + d = αu ll,,, (8) + d, d 0 ll,,, (9) and containt (7)-(2). Hee, d + and d ae the poitive and negative deviation fom the pioity weight of indicato unde ub-categoy l' of categoy l, fo DMU. The et of equation (8) indicate the goal equation whoe ight-hand ide ae the pioity weight of hieachical indicato adjuted by the obtained value of the caling vaiable in (6).Becaue the ange of deviation computed by the objective function i diffeent fo each DMU, it i neceay to nomalize it by uing elative deviation athe than abolute one. Hence, the nomalized deviation can be computed by: whee Z ( θ ) ( θ ) ( θ ) Z ( θ) Z ( θ ) Z =, (20) i the optimal value of objective function (7) fo θ θ θ θ a a meaue of cloene which epeent the elative cloene of each DMU to the weight obtained fom the thee-. We define ( ) 330

5 M. S. Paa level DEA baed-ci model in the ange [0, ] afte adding the et of containt (6) to it. Inceaing the paamete ( θ ), we impove the deviation between the two ytem of weight obtained fom the thee-level DEA baed-ci model befoe and afte adding the et of containt (6). Thi may lead to diffeent aning poition fo each DMU in compaion to the othe DMU. It hould be noted that in a pecial cae whee the paamete θ = θ = 0, we aume ( θ ) =. 3. Numeical Example In thi ection we peent the application of the popoed appoach to ae the oad afety pefomance of a et of Euopean countie (o DMU). The data fo eight hieachical indicato that compoe oad afety pefomance indicato (SPI) fo Euopean countie ha been adopted fom [9]. Table peent the nomalized data, uing () and (2), fo SPI on a logaithmic cale. The notation in Table ae a follow: y = Road ue behavio, y 2 = Vehicle, y = Alcohol, y 2 = Seat belt, y = Roadide police alcohol tet pe 000 population in 2008, y 2 = The pecentage of dive above legal alcohol limit in oadide chec in 2008, y 2 = Daytime eat belt weaing ate on font eat aggegated of ca in 2009, y 22 = Daytime weaing ate of eat belt on ea eat of ca in 2009, y 2 = y 2 = The aveage pecentage of occupant potection coe fo new ca old in 2008, y 22 = y 22 = The aveage pecentageof pedetian potection coe fo new ca old in 2008, y 23 = y 23 = Renewal ate of paenge ca in 2007, y 24 = y 24 = Median age of paenge ca in AT = Autia, BE = Belgium, BG = Bulgaia, CY = Cypu, CZ = Czech Republic, DK = Denma, EE = Etonia, FI = Finland, FR = Fance, DE = Gemany, EL = Geece. The eult of the AHP model fo pioitizing hieachical SPI a contucted by the autho in Expet Choice oftwae ae peented in Table 2. One can ague that the pioity weight of SPI mut be judged by oad afety expet. Howeve, ince the aim of thi ection i jut to how the application of the popoed appoach on numeical data, we ee no poblem to ue ou judgment alone. Solving the thee-level DEA baed-ci model fo the county unde aement, we obtain an optimal et of weight with imum compoite lo ( θ ). It hould be noted that the compoite value of all countie calculated fom the thee-level DEA baed-ci model i identical to that calculated fom the one-level DEA baed-ci model. Theefoe, the imum compoite lo fo the county unde aement i θ = 0 (Table 3). Thi implie that the meaue of elative cloene to the AHP weight fo the county unde aement i ( θ ) = 0. On the othe hand, olving the thee-level DEA baed-ci model fo the county unde aement Table. Nomalized data fo hieachical SPI on a logaithmic cale. y y 2 County y y y y y y y y y y y y y y AT BE BG CY CZ DK EE FI FR DE EL

6 M. S. Paa Table 2. The AHP hieachical model fo SPI. Objective level Citeia level Sub-citeia level Sub-ub-citeia level Pioitizing oad afety pefomance indicato Road ue behavio w = 0.65 Vehicle w 2 = 0.35 Roadide police alcohol tet f = 0.40 Alcohol e = 0.60 Diving above legal alcohol limit f 2 = 0.60 Seat belt weaing in font eat f 2 = 0.70 Seat belt e 2 = 0.40 Seat belt weaing in ea eat f 22 = 0.30 Occupant potection fo ca e 2 = 0.30 Occupant potection fo ca f 2 =.00 Pedetian potection fo ca e 22 = 0.20 Pedetian potection fo ca f 22 =.00 Renewal ate of paenge ca e 23 = 0.40 Renewal ate of paenge ca f 23 =.00 Age of paengeca e 24 = 0.0 Age of paenge ca, f 24 =.00 Table 3. Minimum and imum loe in compoite value fo each county. Countie CI θ θ AT BE BG CY CZ DK EE FI FR DE EL afte adding the et of containt (6), we adjut the pioity weight of hieachical SPI obtained fom AHP in uch a way that they become compatible with the weight tuctue in the thee level DEA-baed CI model. Table 4 peent the optimal weight of hieachical SPI a well a it caling facto fo all countie. It hould be noted that the pioity weight of AHP ued fo incopoating weight bound on indicato weight afte addu ing (6) to the thee-level model ae obtained a u =. Similaly, the pioity weight of AHP at citeia α S pl level can be obtained a wl = while u = pl and u = p. α l = = = In addition, the pioity weight of AHP at ub-citeia and ub-ub-citeia level can be obtained a e = p pl and f = u p, epectively. The imum compoite lo fo each county to achieve the coeponding weight in the thee-level DEA-baed CI model afte adding (6) i equal to θ (Table 3). A a eult, the meaue of elative cloene to the pioity weight of SPI fo the county unde aement i ( θ ) =. Going one tep futhe to the olution poce of the paametic goal pogamg model, we poceed to the etimation of total deviation fom the AHP weight fo each county while the paamete θ i 0 θ θ. Table 5 epeent the aning poition of each county baed on the imum deviation fom the pioity weight of indicato fo θ = 0. It hould be noted that in a pecial cae whee the paamete θ = θ = 0 we 332

7 M. S. Paa Table 4. Optimal weight of hieachical SPI obtained fom thee-level DEA baed-ci model bounded by AHP weight fo all countie. Weight of categoie p =.2322 Weight of ub-categoie p = p = Weight of ub-ub-categoie u = u = u = u = p =.99 u = p = p = u = p = u = p = u = α =.8957 Table 5. The aning poition of each county baed on the imum ditance to pioity weight of hieachical SPI. Countie Z ( θ ) Ran AT BE BG CY CZ DK EE FI FR DE EL.5300 aume ( θ ) =. Table 5 how that Fance (FR) i the bet pefome in tem of the CI value and the elative cloene to the pioity weight of indicato in compaion to the othe countie. Nevethele, inceaing the value of θ fom 0 to θ ha two main effect on the pefomance of the othe countie: impoving the degee of deviation and educing the value of compoite indicato. Thi, of coue, i a phenomenon, one expect to obeve fequently. The gaph of ( θ ) veu θ, a hown in Figue, i ued to decibe the elation between the elative cloene to the pioity weight of indicato and compoite lo fo each county. Thi may eult in diffeent aning poition fo each county in compaion to the othe countie (Table A). 4. Concluion We develop a multiplicative (o log-linea) aggegation appoach baed on DEA and AHP methodologie to contuct CI fo hieachical indicato. We define two et of weight of hieachical indicato in a theelevel DEA famewo. All indicato ae teated a benefit type which atify the popety of the lage the bette. The fit et epeent the weight of indicato with imum compoite lo. The econd et epeent 333

8 M. S. Paa Figue. The elative cloene to the pioity weight of hieachical indicato [ (θ)], veu compoite lo (θ) fo each county. the coeponding pioity weight of hieachical indicato, uing AHP, with imum compoite lo. We ae the pefomance of each DMU in compaion to the othe DMU baed on the elative cloene of the fit et of weight to the econd et of weight. Impoving the meaue of elative cloene in a defined ange of compoite lo, we exploe the vaiou aning poition fo the DMU unde aement in compaion to the othe DMU. To demontate the effectivene of the popoed appoach, we apply it to contuct a compoite oad afety pefomance index fo eight hieachical indicato that compoe SPI fo Euopean countie. Refeence [] Paa, M.S. (204) Uing Data Envelopment Analyi and Analytic Hieachy Poce to Contuct Compoite Indicato. Jounal of Applied Opeational Reeach, 6, [2] Zhou, P., Ang, B.W. and Zhou, D.Q. (200) Weighting and Aggegation in Compoite Indicato Contuction: A Multiplicative Optimization Appoach. Social Indicato Reeach, 96, [3] Shen, Y., Heman, E., Bij, T. and Wet, G. (203) Data Envelopment Analyi fo Compoite Indicato: A Multiple Laye Model. Social Indicato Reeach, 4, [4] OECD (2008) Handboo on Contucting Compoite Indicato: Methodology and Ue Guide. OECD Publihing. [5] Chane, A., Coope, W.W., Seifod, L. and Stutz, J. (982) A Multiplicative Model fo Efficiency Analyi. Socio- Economic Planning Science, 6, [6] Hahimoto, A. and Wu, D.A. (2004) A DEA-Compomie Pogamg Model fo Compehenive Raning. Jounal of the Opeation Reeach Society of Japan, 47, [7] Saaty, T.S. (980) The Analytic Hieachy Poce. McGaw-Hill, New Yo. [8] Podinovi, V.V. (2004) Suitability and Redundancy of Non-Homogeneou Weight Retiction fo Meauing the Relative Efficiency in DEA. Euopean Jounal of Opeational Reeach, 54, [9] Bax, C., Weemann, P., Gitelman, V., Shen, Y., Goldenbeld, C., Heman, E., Doveh, E., Haet, S., Wegman, F. and Aat, L. (202) Developing a Road Safety Index. Deliveable 4.9 of the EC FP7 Poject DaCoTA. 334

9 M. S. Paa Appendix Table A. The meaue of elative cloene to the pioity weight of hieachical SPI [Δ (θ)] v. compoite lo [θ] fo each county. θ AT BE BG CY CZ DK EE FI FR DE EL Ran N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A Ran Ran Ran Ran Ran Ran Ran Ran Ran Ran Ran Ran Ran Ran Ran Ran Ran

10 M. S. Paa Continued Ran Ran Ran Ran Ran Ran Ran Ran Ran Ran Ran Ran Ran Ran Ran Ran Ran Ran 336