Available online at www.sciencediect.com ScienceDiect Pocedia Engineeing 122 (2015 ) 151 157 Opeational Reseach in Sustainable Development and Civil Engineeing - meeting of EURO woking goup and 15th Geman-Lithuanian-Polish colloquium (ORSDCE 2015) Influence of data tansfomation on multiciteia evaluation esult Askoldas Podviezko *, Valentinas Podvezko Vilnius Gediminas technical univesity, Sauletekis av. 11, Vilnius 10223, Lithuania Abstact The main idea of quantitative multiple citeia decision-making methods (MCDM) is compising values of a chosen set of citeia into a single cumulative citeion of evaluation. Units of measuement can be diffeent: pe cent, anks, gades, money units, physical units, etc. Consequently, thei incopoation into a single evaluation citeion is possible if values of citeia ae independent of units of measuement. Such dimensionless values ae obtained by nomalizing the values. Citeia can be both minimizing and maximizing. Some MCDA methods imply tansfomation of minimizing citeia into maximizing ones. Moeove, values of citeia can me negative (pofit, gowth ate, etc.), but some MCDA methods can use only positive citeia. Theefoe, maoity of MCDA methods use both nomalization and tansfomation of citeia with negative values. Thee ae diffeent fomulae available. Even in the same method diffeent tansfomation and nomalization fomulae can be used. Nevetheless, using diffeent tansfomation and nomalization fomulae can lead to diffeences in esults of evaluation. In this pape it is shown that diffeent types of tansfomation and nomalization of data applied to popula MCDA methods, such as SAW o TOPSIS may poduce consideable diffeences in evaluation. Consequently, attention has to be paid to making a choice of the type of nomalization, which eveals pefeences of decision-make. Dependence of evaluation esults on the chosen type of tansfomation o nomalization is demonstated. A case-study is povided. 2015 The The Authos. Published Published by Elsevie by Elsevie Ltd. Ltd. This is an open access aticle unde the CC BY-NC-ND license (http://ceativecommons.og/licenses/by-nc-nd/4.0/). Pee-eview Pee-eview unde unde esponsibility esponsibility of the of oganizing the oganizing committee committee of the of Opeational the Opeational Reseach Reseach in Sustainable in Sustainable Development Development and Civil and Civil Engineeing - meeting - meeting of of EURO EURO woking woking goup goup and 15th and 15th Geman-Lithuanian-Polish colloquium colloquium. Keywods: Nomalization, tansfomation, MCDA, multiple citeia, SAW, TOPSIS * Coesponding autho. Tel.: +370 60423820. E-mail addess: askoldas@gmail.com 1877-7058 2015 The Authos. Published by Elsevie Ltd. This is an open access aticle unde the CC BY-NC-ND license (http://ceativecommons.og/licenses/by-nc-nd/4.0/). Pee-eview unde esponsibility of the oganizing committee of the Opeational Reseach in Sustainable Development and Civil Engineeing - meeting of EURO woking goup and 15th Geman-Lithuanian-Polish colloquium doi:10.1016/.poeng.2015.10.019
152 Askoldas Podviezko and Valentinas Podvezko / Pocedia Engineeing 122 ( 2015 ) 151 157 1. Intoduction In decision-aid systems vaious quantitative multiple citeia decision-aid (MCDA) methods ae widely used. The pupose of the methods is to help a decision-make to evaluate seveal altenatives o pocesses in espect to an obective of evaluation by indicating the best altenative/pocess o by poviding a anking of the altenatives by the level of thei attactiveness. The whole plethoa of the MCDA methods exist with vaying degee of populaity. It is logical to conclude that neithe single univesal method, which most accuately would eflect chaactes of decisionmakes and pocess given data, is not yet identified. A coe featue of any decision-aid method is to eveal and embed pefeences of decision-makes with an acceptable level of fidelity into a decision-aid methodology fo the pupose of eflecting opinions of paticipating expets. Maoity of the methods use only weights of citeia fo this pupose. Citeia descibing chaacteistics of evaluated obects in espect of the goal of evaluation must be chosen in the initial step of the analysis. Values of citeia ae used in mentioned methodologies togethe with weights, which measue pefeences of expet decisionmakes in tems of impotance of chosen citeia fo evaluation. Values of citeia descibing chaacteistics of the altenatives can be deived fom existing statists o be estimated by expets. An MCDA method compises both components, namely values of citeia, which chaacteize the evaluated obects o pocesses, and weights elicited fom expets. Finally, both weights and values of citeia ae compised into a single citeion of the MCDA method. Such an aggegation is possible only if citeia ae measued in the same dimension. This is achieved by making values of citeia dimensionless. This is achieved by nomalization of values of each citeia, which is a mapping of values of citeia into the set of dimensionless eal numbes. Usually, seveal types of nomalization ae simultaneously available fo cetain methods, choices of ways of nomalization can be made. Contay, a few methods, such as TOPSIS o COPRAS, use only one popietay nomalization [1]. Diffeent types of nomalization natually may influence the esult of evaluation as they map values of citeia into the set of eal numbes diffeent way. Seveal types of citeia exist. Maximizing and minimizing citeia ae widely used in vast maoity of cases. Maximizing citeia ae such that the highe thei value is associated with eflection of an undelying bette appeciation of the altenative by a decision-make. Contay, smalle values of minimizing citeia mean that the coesponding altenative should be eflected as moe attactive. Some methods use exclusively maximizing citeia. This implies that an additional tansfomation of minimizing values to maximizing values is equied in ode to use such a method. Vitually, a nomalization type should attempt to adequately account opinions of paticipating expets on each citeion. Theefoe, using diffeent types of nomalization fo diffeent citeia is a natual solution in such cases, when peception of values of diffeent citeia is diffeent by paticipating expets. Thee could be cases, when neithe above-mentioned type of nomalization fits. In such cases, when an expet decision-make would be inclined to opt fo a paticula value of a citeion and has cetain pefeences, say, fo a cetain size of a house, age of an employee, etc., using neithe maximizing no minimizing nomalization would be plausible. Using a nomalization suitable fo a maximizing citeion in such a case would poduce a distotion. Fo example, in a case of choosing a house of a paticula size, say, of 100 squae metes, which will be discussed below in moe detail, such a nomalization designed fo a maximizing citeion would poduce a highe nomalized value fo 200 o 250 squae mete house, even if a buye would not pefe a bigge house because of highe heating bills to pay, moe cleaning effot equied, less efficient accessibility of all locations of the house, moe floos, etc. In the case, when an option fo a good available in the maket is consideed by a buye, it is plausible to conside the nomalization, which maps the wost value to zeo, and the best value to 1. This nomalization often can epesent peception of the situation in the maket by a buye decision-make. Even in case if a value of cetain chaacteistic of the wost commodity in the maket is athe good, this chaacteistic would often be assessed by the lowest gade (in ou case it is 0), while the chaacteistic of the best commodity by the best gade (in ou case 1). Unfotunately, availability of diffeent methods of nomalization yet do help to achieve a bette matching of decision-make s pefeences. Usually, MCDA methods use the same type of nomalization fo the entie set of citeia. And citeia can be peceived in a diffeent way. Also, eseach on pope matching of type of nomalization in accodance with pefeences of the decision-make is scace.
Askoldas Podviezko and Valentinas Podvezko / Pocedia Engineeing 122 ( 2015 ) 151 157 153 In this pape it is shown that thee could be cases, when impope choice of type of nomalization is consideably alteing the esult of evaluation. A simple case of choosing a house by a buye is ceated and studied. A case, whee diffeent methods of nomalization simultaneously ae used, is poposed. The pape suggests an appoach of putting moe emphasis on making a choice of a method of nomalization and on ceation of moe appopiate types of nomalization to cove the whole vaiety of the ways of how decision-makes may peceive values of a citeion in accodance to his/he utility. Such an appoach may impove mapping of citeia to the set of nomalized citeia making it moe adequate. 2. Some popula MCDA methods, which use nomalization 2.1. The SAW (Simple Additive Weighing) method The method is one of the most popula [1, 2]. It exposes coe ideas of the MCDA methods to compise nomalized values of citeia and thei weights of impotance to a single citeion of the method by the following fomula (1) [4, 5]: S m i1 i (1) i whee S is the cumulative citeion; i ae weights of citeia; m is the numbe of chosen citeia; ae nomalized values of citeia; i index fo citeia; index fo altenatives. Nomalization of data is a mapping of function between the set of values of citeia, which have a cetain dimension o measuement, to the set of eal numbes. The initial idea of such a mapping is to make values of citeia dimensionless, nevetheless a mapping fo applied to a cetain citeion does not necessaily eflect opinion of paticipating expets The citeion S is calculated fo each altenative and shows the level of thei attactiveness in the quantitative way. The lage is the cumulative citeion S, the moe attactive is the altenative. The SAW method uses only maximizing citeia and only positive values. Theefoe, in the case if minimizing citeia ae pesent, they have to be tansfomed into the maximizing ones. Thee ae seveal ways to do this. The nomalization mapping values of citeia into the inteval [0,1] could be used, which allows to tansfom minimizing citeia into maximizing in one step. Nomalization of maximizing citeia is caied out by the fomula (2): min max min (2) While nomalization of minimizing citeia is caied out by the fomula (3): max. (3) max min Othe popula type of nomalization as shown by the fomula (4): max (4)
154 Askoldas Podviezko and Valentinas Podvezko / Pocedia Engineeing 122 ( 2015 ) 151 157 does not equie additional two tansfomation of minimizing citeia into maximizing ones as it is caied out by using the invese fomula (5) fo minimizing citeia instead of the fomula (4): min (5) The nomalization, which maps values of citeia to such nomalized values, which make up one in sum as is shown by the fomula (6) equies additional tansfomation of minimizing values into maximizing ones. Such a nomalization can be caied out by the following fomula (5), nevetheless it intoduces distotions [6]. i m 1 (6) In case thee ae negative values pesent, both above-mentioned types of nomalization equie an a-pioi tansfomation of negative values to positive ones. It could be caied out by the fomula (7) [7]: ˆ 1 min i (7) We note that such a tansfomation intoduces distotions as the esult depends on the magnitude of the shift of the set of values of citeia. 2.2. The TOPSIS (Technique fo Ode Pefeence by Similaity to an Ideal Solution) method The TOPSIS method uses a popietay vecto nomalization. It nomalizes values of citeia by fomula (8) in a way that the esulting nomalized vecto is of the unitay length [8, 9]. The nomalized vecto is constucted fo each citeion. Its co-odinates consist of values of the citeia fo each altenative, divided by the nomalizing constant: i n 2 1 (8) In the oiginal method the nomalization is unifom fo all the citeia. The oiginal method does not allow to alte any nomalization and to use an altenative nomalization in the case if pefeences of an expet decision-make diffe. 3. Case study: a poblem of choosing the best house Suppose a decision-make encountes a poblem of choosing the best house. Let him choose a set of citeia, which descibe a house as follows: 1) The size of the house, sq. m.; 2) Distance to public tanspot, km; 3) Adacent land aea, aes; 4) Distance to city cente 5) Ai pollution, gade; 6) Distance to a gocey stoe, km.
Askoldas Podviezko and Valentinas Podvezko / Pocedia Engineeing 122 ( 2015 ) 151 157 155 Suppose, the decision-make has a popietay pefeence fo the size of the house 100 squae metes because of the following easons aleady mentioned above: heating bills to pay, cleaning effot equied, efficient accessibility of all locations of the house, pevailing single floo fo this size. Neithe of descibed above methods of nomalization would coectly map values of coesponding citeion epesenting the size of the house in case the decision-make is consideing the following altenatives of houses A, B, C, and D with the following values of citeia. Fo the pupose of the case-study we outline the following weights to the citeia, as is shown in Table 1. Table 1. Values of citeia, which epesent altenative houses consideed. Altenatives: No. of citeion Citeia type A B C D Weights 1. Size, sq. m. max o 110 60 200 250 0.32 popietay 2. Distance to public tanspot, km min 5 2 1.5 7 0.27 3. Adacent land aea, aes max 5.5 5 6.5 6 0.13 4. Distance to city cente, km min 12 7 20 18 0.11 5. Ai pollution, gade min 3 7 2 1 0.12 6. Distance to a gocey stoe, km min 2 0.5 3 10 0.05 Tansfomed values fo the SAW method by fomulae (5) (6) ae pesented in Table 2. Table 2. Values of nomalized citeia fo the SAW method. Altenatives: No. of citeion Citeia A B C D 1. Size, sq. m. 0.177 0.097 0.323 0.403 2. Distance to public tanspot, km 0.132 0.331 0.442 0.095 3. Adacent land aea, aes 0.239 0.217 0.283 0.261 4. Distance to city cente, km 0.251 0.431 0.151 0.167 5. Ai pollution, gade 0.169 0.072 0.253 0.506 6. Distance to a gocey stoe, km 0.170 0.682 0.114 0.034 Tansfomed values fo the TOPSIS method by fomula (8) ae pesented in Table 3. Table 3. Values of nomalized citeia fo the TOPSIS method. Altenatives: No. of citeion Citeia A B C D 1. Size, sq. m. 0.320 0.175 0.582 0.727 2. Distance to public tanspot, km 0.558 0.223 0.167 0.781 3. Adacent land aea, aes 0.476 0.433 0.563 0.519 4. Distance to city cente, km 0.396 0.231 0.660 0.594 5. Ai pollution, gade 0.378 0.882 0.252 0.126 6. Distance to a gocey stoe, km 0.188 0.047 0.282 0.940 As the citeion 1 should eflect pefeence of the decision-make fo the house of 100 squae metes, popietay tansfomation (9) should be used, values of which is pesented in Table 4.
156 Askoldas Podviezko and Valentinas Podvezko / Pocedia Engineeing 122 ( 2015 ) 151 157 0 2 ( ) 2 i ep z exp, whee z, (9) i 0 The mean i is the most desiable value of the citeion, and i is peceived standad deviation of the mapping of values of citeia into the set of eal numbes. We used 100 squae metes as the mean, and athe lage numbe 50 fo standad deviation to eplicate a possibly quite high level of indiffeence of the buye to the size of the house. No. of citeion Table 4. Values of nomalized citeion 1 by fomula (9). Citeia Altenatives: A B C D 1. Size, sq. m. 0.980 0.726 0.135 0.011 This case of tansfomation will povide altenative values of the citeion 2, which will be used to calculate values of SAW and TOPSIS altenative cumulative citeia to compae them with the cumulative citeia, obtained fom nomalized values fom Table 2 and 3. Results ae shown in Table 5 and 6. Table 5. Values of cumulative citeion of the SAW method. Altenatives: A B C D Cumulative citeion obtained fom classic nomalization 0.180 0.239 0.312 0.269 obtained fom popietay nomalization 0.293 0.333 0.232 0.142 Table 6. Values of cumulative citeion of the TOPSIS method. Altenatives: A B C D Cumulative citeion obtained fom classic nomalization 0.387 0.451 0.767 0.530 obtained fom popietay nomalization 0.707 0.684 0.455 0.229 Compaison of esults obtained using both SAW and TOPSIS methods shows huge discepancies between obtained ankings of the altenatives consideed (Table 7). This clealy suggests necessity to pay much highe attention to the choice of types of nomalization. Table 7. Compaison of ankings of altenatives obtained using SAW and TOPSIS methods. Altenatives: A B C D Cumulative citeion obtained using classic nomalization, SAW 4 3 1 2 obtained using classic nomalization, TOPSIS 4 3 1 2 obtained using popietay nomalization, SAW 2 1 3 4 obtained using popietay nomalization, TOPSIS 1 2 3 4 The nomalization (9), which sustains the pefeence of the decision-make fo a smalle house, consideably adusts esults in favo of the altenatives 1 and 2. Adustment is visible because of the athe high weight assigned to
Askoldas Podviezko and Valentinas Podvezko / Pocedia Engineeing 122 ( 2015 ) 151 157 157 the citeion 2. Values of the cumulative citeion of the TOPSIS method fo the two altenatives diffe by only 3%, which make the two altenatives of simila attactiveness. 4. Conclusions A coe featue of any decision-aid method is to eveal and embed pefeences of decision-makes with an acceptable level of fidelity into a decision-aid methodology fo the pupose of eflecting opinions of paticipating expets. Maoity of the methods use only weights of citeia fo this pupose. Diffeent types of tansfomation and nomalization of data ae available fo popula MCDA methods, such as SAW o TOPSIS. If diffeent methods ae applied, it may poduce consideable diffeences in evaluation. Consequently, much moe attention has to be paid to making a choice of the type of nomalization, which should eveal pefeences of decision-make in the best possible way. Diffeent types of nomalization natually may influence the esult of evaluation as they map values of citeia into the set of eal numbes diffeent way. Using diffeent types of nomalization fo diffeent citeia is a natual solution in such cases, when peception of values of diffeent citeia is diffeent by paticipating expets. Thee could be cases, when neithe maximizing, no minimizing type of nomalization fits. In such cases, when an expet decision-make would be inclined to opt fo a paticula value of a citeion and has cetain pefeences, say, fo a cetain size of a house, age of an employee, etc., a special popietay nomalization should be used. In this pape it is shown that thee could be cases, when impope choice of type of nomalization is consideably alteing the esult of evaluation. A case study of choosing a house by a buye is ceated and studied. A case, whee diffeent methods of nomalization simultaneously ae used, is poposed. The pape suggests an appoach of putting moe emphasis on making a choice of a method of nomalization and on ceation of moe appopiate types of nomalization to cove the whole vaiety of the ways of how decision-makes may peceive values of a citeion in accodance to his/he utility. Such an appoach may impove mapping of citeia to the set of nomalized citeia making it moe adequate and could povide an additional epoting tool fo MCDA methods [10]. Refeences [1] O.Kapliski, L. Tupenaite, Review of the multiple citeia decision making methods, intelligent and biometic systems applied in moden constuction economics, Tansfomations in Business & Economics 10 (2011) 166 181. [2] O. Kapliski, F. Peldschus, The poblems of quantitative evaluation of socio-economic systems development, Engineeing Economics 22 (2011) 345 355 [3] R. Ginevicius, A. Podviezko, Spendim paamos metod taikymo ypatumai komecini bank finansinio stabilumo vetinime, Veslas teo. i pakt. 13 (2012) 314 323. [4] A. Podviezko, V. Podvezko, Absolute and Relative Evaluation of Socio-Economic Obects Based on Multiple Citeia Decision Making Methods, Eng. Econ. 25 (2014) 522 529. doi:10.5755/01.ee.25.5.6624. [5] R. Ginevicius, A. Podviezko, The evaluation of financial stability and soundness of Lithuanian banks, Ekon. Istaz.-Econ. Res. 26 (2013) 191 208. [6] A. Podviezko, Distotions Intoduced by Nomalisation of Values of Citeia in Multiple Citeia Methods of Evaluation, LMD dab. 55 A (2014) 51 56. [7] R. Ginevicius, V. Podvezko, Some poblems of evaluating multiciteia decision methods, Int. J. Manag. Decis. Mak. 8 (2007) 527. doi:10.1504/ijmdm.2007.013415. [8] K. Yoon, C.L. Hwang, Multiple attibute decision making: an intoduction, Sage Publications, Thousand Oaks, CA, 1995. [9] W. Baues, R. Ginevicius, A. Podviezko, Development of a methodology of evaluation of financial stability of commecial banks, Panoeconomicus. 61 (2014) 349 367. doi:10.2298/pan1403349b. [10] A. Podviezko, Augmenting Multiciteia Decision Aid Methods by Gaphical and Analytical Repoting Tools, in: L. Niedite, R. Stazdina, B. Wangle (Eds.), Wokshop Bus. Infom. Res., Spinge Belin Heidelbeg, 2012: pp. 236 251. http://dx.doi.og/10.1007/978-3-642-29231-6_19.