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An application of the TODIM method to the multicriteria rental evaluation of residential properties
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European Journal of Operational Research 193 (2009) 204–211 www.elsevier.com/locate/ejor
Decision Support
An application of the TODIM method to the multicriteria rental evaluation of residential properties Luiz Fla´vio Autran Monteiro Gomes b
a,*
, Luı´s Alberto Duncan Rangel
b,1
a Ibmec/RJ, Business Administration, Av. Presidente Wilson 118, 20030-020 Rio de Janeiro, RJ, Brazil School of Industrial and Metallurgical Engineering, Fluminense Federal University, Av. dos Trabalhadores 420, 27255-125 Volta Redonda, RJ, Brazil
Received 2 April 2007; accepted 26 October 2007 Available online 5 November 2007
Abstract This article presents an evaluation study of residential properties carried out together with real estate agents in the city of Volta Redonda, Brazil. The study aimed to define a reference value for the rents of these properties using the TODIM method of Multicriteria Decision Aiding. By applying this method to the ordering of properties with different characteristics, a ranking of all the properties was obtained and, as a result of this, diverse ranges of rental values for the properties under analysis. The study was complemented by an analysis of the sensitivity of the numerical results obtained. 2007 Elsevier B.V. All rights reserved. Keywords: Multiple criteria analysis; Real estate market; Prospect theory
1. Introduction Brazil is the fifth largest country in the world. It has a population of around 190 million people and 4500 miles of coastline. Due to the scale of the Brazilian territory, the real estate market presents a wide variety of economic conditions as well as differing supply and demand characteristics. The percentage of housing deficit varies according to the region of the country. It has historically been highest in the Northeast and lowest in the Southern region of the country. Real estate activity has been intense even in the most developed Brazilian states such as Sa˜o Paulo and Rio de Janeiro. The rental evaluation of a property is one of the most important tasks for those who work in the Brazilian real estate market. This evaluation is generally based on quantitative and qualitative criteria employing various simple methods (Guimara˜es Neto, 1992, Pessoa, 1992). Among *
Corresponding author. Tel.: +55 21 45034053; fax: +55 21 25126173. E-mail addresses: [email protected] (L.F. Autran Monteiro Gomes), [email protected]ff.br (L.A. Duncan Rangel). 1 Tel.: +55 24 33424896. 0377-2217/$ - see front matter 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2007.10.046
these methods, the most commonly used are the Comparative Method and the Income Method. In the Comparative Method, similar properties at locations close to the property in question are checked, and based on comparisons with the data obtained, the value of the rent is established. This method is the most practiced in the real estate market and the one which normally presents the value closest to the market value. The Income Method defines the value of the rent as a percentage of the sale value of the property. It is simple to apply, but depending on the supply and demand in the region in question, may not generate competitive and practicable values in the market. This article describes the use of a Multicriteria Decision Aiding method in evaluating residential properties available for rent in the city of Volta Redonda, in the State of Rio de Janeiro, Brazil. In the majority of cases, the criteria used to make the evaluation of the alternatives are conflicting. For example, what would be most valued: a small, old house in an excellent residential neighborhood, close to the center or a large, new property with a swimming pool and leisure area in a neighborhood far from the downtown area? In this study, the TODIM method (Gomes and Lima, 1992a,b; Trotta et al., 1999) was used to order the alternatives of
L.F. Autran Monteiro Gomes, L.A. Duncan Rangel / European Journal of Operational Research 193 (2009) 204–211
the residential properties for rent. After the ordering, it becomes easier to define the rental values, once previously evaluated properties, that is, those with previously defined rental values, have been included in the set of alternatives. Other examples of discrete multicriteria valuation can be seen in Stagl (2004, 2007) and Gamper et al. (2006). 2. The TODIM method The TODIM method (an acronym in Portuguese of Interactive and Multicriteria Decision Making), conceived in its current form at the beginning of the nineties, is a discrete multicriteria method based on Prospect Theory (Kahneman and Tversky, 1979). This means that underlying the method is a psychological theory, which was the subject of the Nobel Prize for Economics awarded in 2002 (Roux, 2002). Thus, while practically all other multicriteria methods start from the premise that the decision maker always looks for the solution corresponding to the maximum of some global measure of value – for example, the highest possible value of a multiattribute utility function, in the case of MAUT (Keeney and Raiffa, 1993) – the TODIM method makes use of a global measurement of value calculable by the application of the paradigm of Prospect Theory. In this way, the method is based on a description, proved by empirical evidence, of how people effectively make decisions in the face of risk. Although not all multicriteria problems deal with risk, the shape of the value function of the TODIM method is the same as the gain/loss function of Prospect Theory. The use of TODIM relies on a global multiattribute value function. This function is built in parts, with their mathematical descriptions reproducing the gain/loss function of Prospect Theory. The global multiattribute value function of TODIM then aggregates all measures of gains and losses over all criteria. In its calculations the TODIM method must test specific forms of the losses and gains functions. Each one of the forms depends on the value of one single parameter. The forms, once validated empirically, serve to construct the additive difference function of the method. This notion of an additive utility function is taken from Tversky (1969). The additive difference function is indeed a global multiattribute value function and reflects the dominance measurements of each alternative over each other alternative. In this sense, TODIM maintains a similarity with outranking ´ THE´E ([Brans and Mareschal, methods, such as PROME 1990), because the global value of each alternative is relative to its dominance over other alternatives in the set. Although it appears complicated to have to test the validity of the application of the paradigm to the database, which may on occasions oblige the decision analyst to use other forms of the losses and gains functions, in fact it is not so. Since the first practical uses of the TODIM method, back in the nineties, the same two mathematical forms have been used successfully, and have been validated empirically in different applications (Gomes and Lima, 1992a,b; Trotta et al., 1999).
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From the construction of the aforementioned TODIM additive difference function, which functions as a multiattribute value function and, as such, must also have its use validated by the verification of the condition of mutual preferential independence (Keeney and Raiffa, 1993; Clemen and Reilly, 2001), the method leads to a global ordering of the alternatives. It can be observed that the construction of the multiattribute value function, or additive difference function, of the TODIM method is based on a projection of the differences between the values of any two alternatives (perceived in relation to each criterion) to a referential criterion or reference criterion. The TODIM method makes use of pair comparisons between the decision criteria, using technically simple resources to eliminate occasional inconsistencies arising from these comparisons. It also allows value judgments to be carried out in a verbal scale, using a criteria hierarchy, fuzzy value judgments and making use of interdependence relationships among the alternatives. It is a noncompensatory method in the sense that tradeoffs do not occur (Bouyssou, 1986). Roy and Bouyssou (1993), talking about the TODIM method, state that it is: ‘‘...a method based on the French School and the American School. It combines aspects of the Multiattribute Utility Theory, of the AHP method and the ELECTRE methods’’ (p. 638). The concept of introducing expressions of losses and gains in the same multiattribute function, present in the formulation of the TODIM method, gives this method ´ THE´E methods, which some similarity to the PROME make use of the notion of net outranking flow. BarbaRomero and Pomerol (2000) have stated the following in respect of the TODIM method: ‘‘it is based on a notion ´ THE´E extremely similar to a net flow, in the PROME sense’’ (p. 229). Consider a set of n alternatives to be ordered in the presence of m quantitative or qualitative criteria, and assume that one of these criteria can be considered as the reference criterion. After the definition of these elements, experts are asked to estimate, for each one of the qualitative criteria c, the contribution of each alternative i to the objective associated with the criterion. This method requires the values of the evaluation, of the alternatives in relation to the criteria, to be numerical and to be normalized; consequently the qualitative criteria evaluated in a verbal scale are transformed into a cardinal scale. The evaluations of the quantitative criteria are obtained from the performance of the alternatives in relation to the criteria, such as, for example, the level of noise measured in decibels, the power of an engine measured in horsepower or a student’s mark in a subject, etc. TODIM can therefore be used for qualitative as well as quantitative criteria. Verbal scales of qualitative criteria are converted to cardinal ones and both types of scales are normalized. The relative measure of dominance of one alternative over another is found for each pair of alternatives. This measure is computed as the sum over all criteria
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of both relative gain/loss values for these alternatives. The parts in this sum will be either gains, losses, or zeros, depending on the performance of each alternative with respect to every criterion. The evaluation of the alternatives in relation to all the criteria produces the matrix of evaluation, where the values are all numerical. Their normalization is then performed, using, for each criterion, the division of the value of one alternative by the sum of all the alternatives. This normalization is carried out for each criterion, thus obtaining a matrix, where all the values are between zero and one. It is called the matrix of normalized alternatives’ scores against criteria. P = [Pnm], with n indicating the number of alternatives and m the number of criteria, as shown in Table 1. After the attribution of the weights of the criteria and their normalization, the partial matrices of dominance and the final matrix of dominance must be calculated. The decision makers must indicate which criterion r is to be chosen as the reference criterion for the calculations according to the relative importance assigned to each criterion. In this way, the criterion with the highest value accorded to its importance will usually be chosen as the reference criterion. The weight of each criterion is determined by the decision makers on a numerical scale (e.g., from 1 to 5) and is then normalized. Thus, wrc is the weight of criterion c divided by the weight of the reference criterion r. Using wrc allows all pairs of differences between performance measurements to be translated into the same dimension, i.e. that of the reference criterion. The measurement of dominance of each alternative Ai over each alternative Aj, now incorporated to Prospect Theory, is given by the mathematical expression dðAi ; Aj Þ ¼
m X
Uc ðAi ; Aj Þ
8ði; jÞ:
ð1Þ
c¼1
Thus d(Ai, Aj) represents the measurement of dominance of alternative Ai over alternative Aj; m is the number of criteria; c is any criterion, for c = 1, . . . , m; wrc is equal to wc divided by wr, where r is the reference criterion; Pic and Pjc are, respectively, the performances of the alternatives Ai and Aj in relation to c; h is the attenuation factor of the losses; different choices of h lead to different shapes of the prospect theoretical value function in the negative quadrant. The expression Uc(Ai, Aj) represents the parcel of the contribution of criterion c to function d(Ai, Aj), when comparing alternative i with alternative j. If the value of Pic Pjc is positive, it will represent a gain for the function d(Ai, Aj) and, therefore the expression Uc(Ai, Aj) will be used, corresponding, that is, to Eq. (2). If Pic Pjc is nil, the value zero will be assigned to Uc(Ai, Aj), by applying Eq. (3). If Pic Pjc is negative, Uc(Ai, Aj) will be represented by Eq. (4). The construction of function Uc(Ai, Aj) in fact permits an adjustment of the data of the problem to the value function of Prospect Theory, thus explaining the aversion and the propensity to risk. This function has the shape of an ‘‘S’’, represented in Fig. 1. Above the horizontal axis, considered as a reference for this analysis, there is a concave curve representing the gains, and, below the horizontal axis, there is a convex curve representing the losses. The concave part reflects the aversion to risk in the face of gains and the convex part, in turn, symbolizes the propensity to risk when dealing with losses. After the diverse partial matrices of dominance have been calculated, one for each criterion, the final dominance matrix of the general element d(Ai, Aj) is obtained, through the sum of the elements of the diverse matrices. Expression (2) is used to determine the overall value of alternative i through normalization of the corresponding dominance measurements. The rank of every alternative originates from the ordering of their respective values.
When: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi wrc ðP ic P jc Þ if ðP ic P jc Þ > 0; m P wrc c¼1 0 Uc ðAi ; Aj Þ ¼ vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi if ðP ic P jc Þ ¼ 0; u m u P t wrc ðP jc P ic Þ 1 c¼1 h if ðP ic P jc Þ < 0; wrc
Value
ð2Þ ð3Þ
ð4Þ Losses
Gains
Table 1 Matrix of normalized alternatives’ scores against criteria Alternatives
A1 A2 ... Ai ... An
Criteria C1
C2
...
Cj
...
Cm
P11 P21 ... Pi1 ... Pn1
P12 P22 ... Pi2 ... Pn2
... ... ... ... ... ...
P1j P2j ... Pij
... ... ... ... ... ...
P1 P2
Pnj
m m
Pim ... Pnm
Fig. 1. Value function of the TODIM method (Gomes and Lima, 1992a).
L.F. Autran Monteiro Gomes, L.A. Duncan Rangel / European Journal of Operational Research 193 (2009) 204–211
Pn Pn j¼1 dðAi ; Aj Þ min j¼1 dðAi ; Aj Þ P P : ni ¼ n n max j¼1 dðAi ; Aj Þ min j¼1 dðAi ; Aj Þ
ð2Þ
Therefore, the global measures obtained computed by (2) permit the complete rank ordering of all alternatives. A sensitivity analysis should then be applied to verify the stability of the results based on the decision makers’ preferences. The sensitivity analysis should therefore be carried out on h as well as on the criteria weights, the choice of the reference criterion, and performance evaluations. 3. Case study The city of Volta Redonda is situated in the South of the State of Rio de Janeiro, Brazil. It has approximately 256,000 inhabitants and there are currently a large number of properties, residential and commercial, rented or available for rent. The study sought to determine a reference value for the rent of residential properties, serving in this way as an important source of information for landlords and real estate agents. A set of 6 alternatives with known rental values was used as a guideline. Computations in this case study were performed using an Excel-based spreadsheet. Commercial software for the calculations of the TODIM method is currently under development, though. In order to evaluate the properties, it was necessary to identify which criteria should be considered in the study. For this purpose, research was carried out together with several real estate companies in the city of Volta Redonda and interviews conducted with specialists in the field of property evaluations. 3.1. Definition of the criteria Based on the points of view expressed in the interviews with realtors, it was concluded that eight criteria should be selected for this particular application. All other possible criteria would already be reflected in these eight criteria, given the characteristics of the study area. The eight most important evaluation criteria for this analysis were identified, and are described as follows: C1 – Location: This is the most important evaluation criterion for rental properties. It is a qualitative criterion which seeks to define whether the property is in a valued location. The scale presented in Table 2 as Associated Score was defined to evaluate the performance of the alternatives in relation to this criterion. In this way, a property evaluated as being in a ‘‘good location’’, receives a score equal to 4. C2 – Constructed area: This is a quantitative criterion which describes the size of the property constructed. In this study, the unit of measurement was square meters (m2). C3 – Construction quality: This is a qualitative criterion which defines the standard of finishing of the property. Three levels of finishing and their respective scores were
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Table 2 Valuation of criterion C1 – location Location
Associated score
Periphery Periphery/Average location Average location Good location Excellent location
1 2 3 4 5
defined as presented in Table 3. Considering a property which has been constructed to a ‘‘high standard of finishing’’, the property receives a score equal to 3, the maximum value for this evaluation criterion. C4 – State of conservation: This is a qualitative criterion used to evaluate the general state of the property. This criterion considers whether any work has been carried out on the property, as well as habitation conditions. Table 4, as follows, presents the associated scores for the state of conservation of the property. For example, an apartment fit for habitation, needing minor repairs, in other words in an average ‘‘state of conservation’’, receives a score equal to 2. This criterion is not directly related to criterion C3, as the state of conservation is independent of the standard of finishing of the property. A property might have a high standard of finishing, using high quality material, but be in need of repairs because of its age. C5 – Number of Garage spaces: This is a quantitative criterion much valued by people who need to rent a property and have one or more cars. C6 – Number of rooms: This is a quantitative criterion which considers for valuation the number of rooms in the property: living rooms, bedrooms, kitchen/breakfast area and bathrooms. The difference between this criterion and criterion C2, which deals with the area constructed, should be highlighted. It is important to evaluate the property by the number of rooms, as two properties of the same total size may be in question, but one of them may have large rooms and the other, a larger number of smaller rooms. C7 – Attractions: This is a qualitative criterion which considers the existence of leisure areas such as swimming
Table 3 Valuation of criterion C3 – construction quality Construction quality
Associated score
Low standard of finishing Average standard of finishing High standard of finishing
1 2 3
Table 4 Valuation of criterion C4 – state of conservation State of conservation
Associated score
Bad Average Good Very good
1 2 3 4
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pools and barbecues among others. It is evaluated according to Table 5. A house with a swimming pool and barbecue receives 4 points. C8 – Security: This is a qualitative criterion used to describe the security conditions of the property. Table 6 presents the evaluation of the property according to this criterion. For example, an apartment in a building with a doorman and security cameras achieved a score of 1. All the criteria defined above are maximization criteria, in other words, the higher the score obtained in the evaluation of the alternatives in relation to each criterion, the better the performance. 3.2. Weighting of the criteria In accordance with the importance given to the criteria used to evaluate the properties in the study, their respective weights were defined by the decision makers through direct valuation and later normalized. The direct valuation consisted of assigning a number between 1 and 5 to each criterion, where 1 would mean ‘least important’ and 5 would mean ‘most important’. The assignment of weights was performed by the decision makers – in this example, the realtors. The information is presented in Table 7. 3.3. The alternatives of the decision process In this case study, fifteen properties in different neighborhoods in the city of Volta Redonda were evaluated. The properties and their respective locations have not been identified, as this was not deemed necessary for the purposes of the article. Only the characteristics of the properties have been shown, designated A1, A2, A3, . . . , A15. The following describes the 15 properties used in the evaluation:
Table 5 Valuation of criterion C7 – attractions Attractions
Associated score
Without attractions Backyard or terrace Barbecue Swimming pool Swimming pool, barbecue or others
0 1 2 3 4
Table 6 Valuation of Criterion C8 – Security Security
Associated score
No additional security Doorman and security cameras in the apartment building. Houses with manned security boxes in the streets
0 1
Table 7 Criteria ranks Criterion
Description
Assigned weights
Criteria weights
C1 C2 C3 C4 C5 C6 C7 C8
Localization Constructed area Quality of construction State of conservation Number of garage spaces Number of rooms Attractions Security
5 3 2 4 1 2 1 2
0.25 0.15 0.10 0.20 0.05 0.10 0.05 0.10
A1 – A house in an average location, with 290 m2 of constructed area, a high standard of finishing, in a good state of conservation, with one garage space, 6 rooms, a swimming pool, barbecue and other attractions, without a security system. A2 – A house in a good location, with 180 m2 of constructed area, an average standard of finishing, in an average state of conservation, with one garage space, 4 rooms, a backyard and terrace without a security system. A3 – A house in an average location, with 347 m2 of constructed area, a low standard of finishing, in an average state of conservation, two garage spaces, 5 rooms, a large backyard, without a security system. A4 – A house in an average location, with 124 m2 of constructed area, an average standard of finishing, in a good state of conservation, two garage spaces, 5 rooms, a fruit orchard, a swimming pool and barbecue, without security system. A5 – A house in an excellent location, with 360 m2 of constructed area, a high standard of finishing, in a very good state of conservation, four garage spaces, 9 rooms, a backyard and manned security boxes in the neighborhood streets. A6 – A house located between the periphery and the city center (periphery/average location) with 89 m2 of constructed area, an average standard of finishing, in a good state of conservation, with one garage space, 5 rooms, a backyard, without a security system. A7 – An apartment located in the periphery, with 85 m2 of constructed area, a low standard of finishing, in a bad state of conservation, one garage space, 4 rooms, a manned entrance hall with security. A8 – An apartment in an excellent location, with 80 m2 of constructed area, average standard of finishing, good state of conservation, with one garage space, 6 rooms, manned entrance hall with security. A9 – An apartment located between the periphery and the city center (periphery/average location), with 121 m2 of constructed area, an average standard of finishing, in a good state of conservation, no garage space, 6 rooms, without a security system. A10 – A house located between the periphery and the city center (periphery/average location), with 120 m2 of constructed area, a low standard of finishing, in a good
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state of conservation, with one garage space, 5 rooms, a large backyard, without a security system. A11 – A house in a good location, with 280 m2 of constructed area, an average standard of finishing, in an average state of conservation, with two garage spaces, 7 rooms, with an additional security system. A12 – An apartment located in the periphery, with 90 m2 of constructed area, a low standard of finishing, in a bad state of conservation, one garage space, 5 rooms, without additional security. A13 – An apartment located in the periphery in an average location, with 160 m2 of constructed area, a high standard of finishing, in a good state of conservation, two garage spaces, 6 rooms, with additional security features. A14 – An apartment in a good location, with 320 m2 of constructed area, high standard of finishing, in a good state of conservation, 2 garage spaces, 8 rooms, with in addition a security system. A15 – A house in a good location, with 180 m2 of constructed area, an average standard of finishing, in a very good state of conservation, one garage space, 6 rooms, with in addition a security system. Table 8, containing the Evaluation of Alternatives against Criteria, presents the complete evaluation of the properties studied in the analysis in relation to the criteria selected by the decision makers. In the case study, the attenuation factor of losses h has a value equal to 1, which means that the losses will contribute with their real value to the global value. In order to implement the method, it is necessary for these performances to be normalized. The matrix of normalized performances is then called the Matrix of Normalized Alternatives’ Scores against Criteria, presented in Table 9 below. In order to demonstrate how to determine the dominance measurements, the computations to obtain d(A2, Table 8 Evaluation of alternatives against Criteria
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Table 9 Matrix of normalized alternatives’ scores against criteria Alternatives
A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A12 A13 A14 A15
Criteria C1
C2
C3
C4
C5
C6
C7
C8
0.068 0.091 0.068 0.068 0.114 0.045 0.023 0.114 0.045 0.045 0.091 0.023 0.045 0.068 0.091
0.103 0.064 0.123 0.044 0.127 0.031 0.03 0.028 0.043 0.042 0.099 0.032 0.057 0.113 0.064
0.1 0.067 0.033 0.067 0.1 0.067 0.033 0.067 0.067 0.033 0.067 0.033 0.1 0.1 0.067
0.075 0.05 0.05 0.075 0.1 0.075 0.025 0.075 0.075 0.075 0.05 0.025 0.075 0.075 0.1
0.045 0.045 0.091 0.091 0.182 0.045 0.045 0.045 0 0.045 0.091 0.045 0.091 0.091 0.045
0.069 0.046 0.057 0.057 0.103 0.057 0.046 0.069 0.069 0.057 0.08 0.057 0.069 0.092 0.069
0.174 0.087 0.043 0.174 0.043 0.043 0 0 0 0.043 0.13 0.087 0.043 0.087 0.043
0 0 0 0 0.143 0 0.143 0.143 0 0 0.143 0 0.143 0.143 0.143
A4) are presented in the Appendix as an example. Similar computations would lead to a 15 · 15 matrix, where the values in the cells would be measurements of dominance. After the implementation of the mathematical formulation of the TODIM method, the overall values of the alternatives obtained through normalization of the corresponding dominance measurements are presented in Table 10. This table also presents the ordering of each alternative. 3.4. Results analysis Among the properties evaluated, some were inserted as references, simply to assist in the analysis, as their rental values were already known. These properties are presented in Table 11. In this way, by inserting these properties with known values, it is possible to establish a range of values for the other properties under analysis. For example, in this study, it can be seen that in the final ordering by the TODIM Table 10 Final values and ordering
Alternatives
Criteria C1
C2
C3
C4
C5
C6
C7
C8
Alternative
Normalized global value
Ordering
A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A12 A13 A14 A15
3 4 3 3 5 2 1 5 2 2 4 1 2 3 4
290 180 347 124 360 89 85 80 121 120 280 90 160 320 180
3 2 1 2 3 2 1 2 2 1 2 1 3 3 2
3 2 2 3 4 3 1 3 3 3 2 1 3 3 4
1 1 2 2 4 1 1 1 0 1 2 1 2 2 1
6 4 5 5 9 5 4 6 6 5 7 5 6 8 6
4 2 1 4 1 1 0 0 0 1 3 2 1 2 1
0 0 0 0 1 0 1 1 0 0 1 0 1 1 1
A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A12 A13 A14 A15
0.6916 0.3862 0.3992 0.6210 1.0000 0.2860 0.0000 0.4407 0.0202 0.2127 0.8576 0.1073 0.7188 0.9372 0.6733
5 10 9 7 1 11 15 8 14 12 3 13 4 2 6
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Table 11 Properties with a previously defined rental value Property
Monthly rental value
A3 A4 A5 A9 A10 A11
US$ US$ US$ US$ US$ US$
214 309 712 133 166 414
method, A8 is in the 8th place, A3 is in the 9th place, with a rental value of US$ 214, and A4 is in the 7th place, and should be rented for US$ 309. It can then be said that A8 must be rented for a value between US$ 214 and US$ 309. In the same way it can be seen that property A14 is in the 2nd place. A5 is in the 1st place, with a rental value of US$ 712, and A11 is in the 3rd place with a rental value of US$ 414. Therefore property A14 must be rented for a value of between US$ 712 and US$ 414. If the ranking order obtained by TODIM did not coincide with the ranking based on dollar values this would mean that the properties’ dollar values would have to be reconsidered and then changed accordingly. These value intervals are defined for each alternative under analysis, serving as a reference to obtain values for the rent. In this way, the decision process becomes less complex. Table 12 suggests the appropriate intervals for the properties with non-defined rental values. This shows that realtors would considerably benefit from using a more comprehensive, multicriteria approach through the use of TODIM when they assign dollar values to the properties. 3.5. Sensitivity analysis After obtaining the results through the implementation of the TODIM method, a sensitivity analysis was carried out varying the weighting given to the criterion of greatest importance to the decision makers, which is criterion C1. When its weighting was reduced from 5 to 3 there was no variation in the ranking of the 15 alternatives. Another analysis was carried out varying the value of h, the attenuation factor of losses. In the first implementation carried
Table 12 Suggested rental values of the alternatives analyzed Property
Ordering
Monthly rental value
A1 A2 A6 A7 A8 A12 A13 A14 A15
5 10 11 15 8 13 4 2 6
Between US$ 414 and Between US$ 214 and Between US$ 214 and Less than US$ 133 Between US$ 309 and Between US$ 166 and Between US$ 414 and Between US$ 712 and Between US$ 414 and
US$ 309 US$ 166 US$ 166 US$ US$ US$ US$ US$
214 133 309 414 309
out, the value used for h was 1. In the sensitivity analysis the value of h was altered from 1 to 5. When this alteration was carried out, the only change in the ordering of the values occurred in the values of A13 with A15, indicating that, in spite of the alteration in the value of the attenuation index of losses, the ordering obtained was consistent. In this particular application the decision makers felt that sensitivity analyses should be conducted on the weight of the reference criterion and the value of h only. 4. Conclusion The analysis of the alternatives using the TODIM method led to an ordering which showed it to be satisfactory and in agreement with the expectations of the specialists. Through its formulation it became easier to resolve conflicts between criteria, as, sometimes, in order to achieve a good performance in a determined criterion of the analysis, it is necessary not to be concerned about performance in another (Belton and Stewart, 2002). In the case of property evaluation, the method is capable of assisting professionals in the real estate market to evaluate the alternatives more clearly in relation to the criteria defined by the specialists. In this study it was also possible to identify that three properties, A1, A13 and A15, fell into the same range established by the alternatives A4 and A11. From Table 10, it can be seen that the order of these three properties is: A13, A1 and A15. In this way, the rental values of these three properties could be the same or, alternatively, giving a reference for the rental value of the property according to the ordering supplied by the method, the greatest value attributed to A13 and the lowest value to A15. Thus, the analysis and the solution of the problem presented here, by means of the TODIM multicriteria method, reflected in their results the preferences of the decision agents, experts in the multiple dimensions of the problem analyzed. Consequently, it can be concluded that the method constitutes efficient support for the evaluation of property. As new properties are included in the portfolio of a realtor, the TODIM method must be run again taking into account the characteristics of these new properties. After the rank ordering is obtained for these new properties their suggested rental values will be determined with the help of Table 11. Realtors concluded that applying the TODIM method to the rental evaluation of residential properties can provide a considerable help to them, given the extreme difficulties in assigning dollar values to all evaluation criteria. Given new evaluation scenarios, with a new set of evaluation criteria, however, new applications of the multicriteria analysis would have to be performed. Changes in the scenarios can lead to changes in estimated rental values even for properties whose rental values have previously been defined. For future studies efforts should also be focused on trying to quantify the monetary consequences associated to every criterion.
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By applying Eq. (1) one determines the dominance measurement d(A2, A4) as equal to 2.833.
Acknowledgements The authors are grateful to the referees for their insightful comments on the first version of this paper. They are also grateful to the National Council for Scientific and Technological Development (CNPq) of Brazil for supporting the research. Appendix Eq. (1) is used to determine the values of the dominance measurements. For instance, when determining d(A2, A4) one must first obtain the different values of Uc(A2, A4) for all criteria c. This is accomplished by making use of Eqs. (2)–(4). The computations below are performed for: h = 1; r = C1 (the reference criterion); data in Tables 9 and 7. • For c = C1 one gets P21 P41 > 0 (a gain). By using Eq. (2), with wrc = 1.0, U1(A2, A4) = 0.075, i.e. vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u w11 ðP 21 P 41 Þ uwrc ðP ic P jc Þ u ¼ m P t 4 wrc c¼1
211
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð0:25=0:25Þð0:091 0:068Þ ¼ 4 ¼ 0:075:
• For c = C2 one gets P22 P42 > 0 (a gain). By using Eq. (2), with wrc = 0.6, U2 (A2, A4) = 0.054. • For c = C3 one gets P23 P43 = 0 (neither a gain nor a loss). By using Eq. (3), with wrc = 0.4, U3(A2, A4) = 0.0. • For c = C4 one gets P24 P44 < 0 (a loss). By using Eq. (4), with wrc = 0.8, U4(A2, A4) = 0.353, i.e. vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u P u m sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uð wrc ÞðP jc P ic Þ 1 t c¼1 1 4ðP 44 P 24 Þ ¼ h h w14 wrc sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 4ð0:075 0:050Þ ¼ 1 ð0:20=0:25Þ ¼ 0:353: • For c = C5 one gets P25 P45 < 0 (a loss). By using Eq. (4), with wrc = 0.2, U5(A2, A4) = 0.959. • For c = C6 one gets P26 P46 < 0 (a loss). By using Eq. (4), with wrc = 0.4, U6(A2 ,A4) = 0.331. • For c = C7 one gets P27 P47 < 0 (a loss). By using Eq. (4), with wrc = 0.2, U7(A2, A4) = 1.319. • For c = C8 one gets P28 P48 = 0 (neither a gain nor a loss). By using Eq. (3), with wrc = 0.4, U8(A2, A4) = 0.0.
References Barba-Romero, S., Pomerol, J.C., 2000. Multicriterion Decision in Management: Principles And Practice. Kluwer Academic Publishers, Boston. Belton, V., Stewart, T.J., 2002. Multiple Criteria Decision Analysis: An Integrated Approach. Kluwer Academic Publishers, Massachusetts. Bouyssou, D., 1986. Some remarks on the notion of compensation in MCDM. European Journal of Operational Research 26 (1), 150– 160. ´ THE ´ E methods for Brans, J.P., Mareschal, B., 1990. The PROME MCDM, the PROMCALC GAIA and BANDADVISER software. In: Bana e Costa, C.A. (Ed.), Readings in Multiple Criteria Decision Aid. Springer Verlag, Berlin. Clemen, R.T., Reilly, T., 2001. Making Hard Decisions with Decision Tools. Pacific Grove, Duxbury. Gamper, C.D., Tho¨rni, M., Weck-Hannemann, H., 2006. A conceptual approach to the use of Cost Benefit and Multi Criteria Analysis in natural hazard management. Natural Hazard and Earth System Sciences 6, 293–302. Gomes, L.F.A.M., Lima, M.M.P.P., 1992a. TODIM: Basics and application to multicriteria ranking of projects with environmental impacts. Foundations of Computing and Decision Sciences 16 (4), 113– 127. Gomes, L.F.A.M., Lima, M.M.P.P., 1992b. From modeling individual preferences to multicriteria ranking of discrete alternatives: A look at Prospect Theory and the additive difference model. Foundations of Computing and Decision Sciences 17 (3), 171– 184. Guimara˜es Neto, C.A., 1992. Crite´rios de Avaliac¸a˜o e NB502-89. In: Introduc¸a˜o a` Engenharia de Avaliac¸o˜es e Perı´cias Judiciais. Instituto de Engenharia Legal, Sa˜o Paulo, pp. 45–60. Kahneman, D., Tversky, A., 1979. Prospect theory: An analysis of decision under risk. Econometrica 47, 263–292. Keeney, R.L., Raiffa, H., 1993. Decisions with Multiple Objectives: Preferences and Value Tradeoffs. Cambridge University Press, Cambridge. Pessoa, M., 1992. Laudos te´cnicos, conceitos ba´sicos e definic¸o˜es. In: Introduc¸a˜o a Engenharia de Avaliac¸o˜es e Perı´cias Judiciais. Instituto de Engenharia Legal, Sa˜o Paulo, pp. 1–26. ´ conomie, 2nd ed. Economica, Paris. Roux, D., 2002. Nobel en E Roy, B., Bouyssou, D., 1993. Aide multicrite`re a` la de´csion: Me´thodes et cas. Economica, Paris. Stagl, S., 2004. Valuation for sustainable development – The role of multicriteria evaluation. Vierteljahrshefte zur Wirtschaftsforschung 73 (1), 53–62. Stagl, S., 2007. SDRN Rapid Research and Evidence Review on Emerging Methods for Sustainability Valuation and Appraisal. University of Sussex, Brighton. Trotta, L.T.F., Nobre, F.F., Gomes, L.F.A.M., 1999. Multi-criteria decision making – An approach to setting priorities in health care. Statistics in Medicine 18, 3345–3354. Tversky, A., 1969. Intransitivity of preferences. Psychological Review 76 (1), 31–48.
European Journal of Operational Research 193 (2009) 204–211 www.elsevier.com/locate/ejor
Decision Support
An application of the TODIM method to the multicriteria rental evaluation of residential properties Luiz Fla´vio Autran Monteiro Gomes b
a,*
, Luı´s Alberto Duncan Rangel
b,1
a Ibmec/RJ, Business Administration, Av. Presidente Wilson 118, 20030-020 Rio de Janeiro, RJ, Brazil School of Industrial and Metallurgical Engineering, Fluminense Federal University, Av. dos Trabalhadores 420, 27255-125 Volta Redonda, RJ, Brazil
Received 2 April 2007; accepted 26 October 2007 Available online 5 November 2007
Abstract This article presents an evaluation study of residential properties carried out together with real estate agents in the city of Volta Redonda, Brazil. The study aimed to define a reference value for the rents of these properties using the TODIM method of Multicriteria Decision Aiding. By applying this method to the ordering of properties with different characteristics, a ranking of all the properties was obtained and, as a result of this, diverse ranges of rental values for the properties under analysis. The study was complemented by an analysis of the sensitivity of the numerical results obtained. 2007 Elsevier B.V. All rights reserved. Keywords: Multiple criteria analysis; Real estate market; Prospect theory
1. Introduction Brazil is the fifth largest country in the world. It has a population of around 190 million people and 4500 miles of coastline. Due to the scale of the Brazilian territory, the real estate market presents a wide variety of economic conditions as well as differing supply and demand characteristics. The percentage of housing deficit varies according to the region of the country. It has historically been highest in the Northeast and lowest in the Southern region of the country. Real estate activity has been intense even in the most developed Brazilian states such as Sa˜o Paulo and Rio de Janeiro. The rental evaluation of a property is one of the most important tasks for those who work in the Brazilian real estate market. This evaluation is generally based on quantitative and qualitative criteria employing various simple methods (Guimara˜es Neto, 1992, Pessoa, 1992). Among *
Corresponding author. Tel.: +55 21 45034053; fax: +55 21 25126173. E-mail addresses: [email protected] (L.F. Autran Monteiro Gomes), [email protected]ff.br (L.A. Duncan Rangel). 1 Tel.: +55 24 33424896. 0377-2217/$ - see front matter 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2007.10.046
these methods, the most commonly used are the Comparative Method and the Income Method. In the Comparative Method, similar properties at locations close to the property in question are checked, and based on comparisons with the data obtained, the value of the rent is established. This method is the most practiced in the real estate market and the one which normally presents the value closest to the market value. The Income Method defines the value of the rent as a percentage of the sale value of the property. It is simple to apply, but depending on the supply and demand in the region in question, may not generate competitive and practicable values in the market. This article describes the use of a Multicriteria Decision Aiding method in evaluating residential properties available for rent in the city of Volta Redonda, in the State of Rio de Janeiro, Brazil. In the majority of cases, the criteria used to make the evaluation of the alternatives are conflicting. For example, what would be most valued: a small, old house in an excellent residential neighborhood, close to the center or a large, new property with a swimming pool and leisure area in a neighborhood far from the downtown area? In this study, the TODIM method (Gomes and Lima, 1992a,b; Trotta et al., 1999) was used to order the alternatives of
L.F. Autran Monteiro Gomes, L.A. Duncan Rangel / European Journal of Operational Research 193 (2009) 204–211
the residential properties for rent. After the ordering, it becomes easier to define the rental values, once previously evaluated properties, that is, those with previously defined rental values, have been included in the set of alternatives. Other examples of discrete multicriteria valuation can be seen in Stagl (2004, 2007) and Gamper et al. (2006). 2. The TODIM method The TODIM method (an acronym in Portuguese of Interactive and Multicriteria Decision Making), conceived in its current form at the beginning of the nineties, is a discrete multicriteria method based on Prospect Theory (Kahneman and Tversky, 1979). This means that underlying the method is a psychological theory, which was the subject of the Nobel Prize for Economics awarded in 2002 (Roux, 2002). Thus, while practically all other multicriteria methods start from the premise that the decision maker always looks for the solution corresponding to the maximum of some global measure of value – for example, the highest possible value of a multiattribute utility function, in the case of MAUT (Keeney and Raiffa, 1993) – the TODIM method makes use of a global measurement of value calculable by the application of the paradigm of Prospect Theory. In this way, the method is based on a description, proved by empirical evidence, of how people effectively make decisions in the face of risk. Although not all multicriteria problems deal with risk, the shape of the value function of the TODIM method is the same as the gain/loss function of Prospect Theory. The use of TODIM relies on a global multiattribute value function. This function is built in parts, with their mathematical descriptions reproducing the gain/loss function of Prospect Theory. The global multiattribute value function of TODIM then aggregates all measures of gains and losses over all criteria. In its calculations the TODIM method must test specific forms of the losses and gains functions. Each one of the forms depends on the value of one single parameter. The forms, once validated empirically, serve to construct the additive difference function of the method. This notion of an additive utility function is taken from Tversky (1969). The additive difference function is indeed a global multiattribute value function and reflects the dominance measurements of each alternative over each other alternative. In this sense, TODIM maintains a similarity with outranking ´ THE´E ([Brans and Mareschal, methods, such as PROME 1990), because the global value of each alternative is relative to its dominance over other alternatives in the set. Although it appears complicated to have to test the validity of the application of the paradigm to the database, which may on occasions oblige the decision analyst to use other forms of the losses and gains functions, in fact it is not so. Since the first practical uses of the TODIM method, back in the nineties, the same two mathematical forms have been used successfully, and have been validated empirically in different applications (Gomes and Lima, 1992a,b; Trotta et al., 1999).
205
From the construction of the aforementioned TODIM additive difference function, which functions as a multiattribute value function and, as such, must also have its use validated by the verification of the condition of mutual preferential independence (Keeney and Raiffa, 1993; Clemen and Reilly, 2001), the method leads to a global ordering of the alternatives. It can be observed that the construction of the multiattribute value function, or additive difference function, of the TODIM method is based on a projection of the differences between the values of any two alternatives (perceived in relation to each criterion) to a referential criterion or reference criterion. The TODIM method makes use of pair comparisons between the decision criteria, using technically simple resources to eliminate occasional inconsistencies arising from these comparisons. It also allows value judgments to be carried out in a verbal scale, using a criteria hierarchy, fuzzy value judgments and making use of interdependence relationships among the alternatives. It is a noncompensatory method in the sense that tradeoffs do not occur (Bouyssou, 1986). Roy and Bouyssou (1993), talking about the TODIM method, state that it is: ‘‘...a method based on the French School and the American School. It combines aspects of the Multiattribute Utility Theory, of the AHP method and the ELECTRE methods’’ (p. 638). The concept of introducing expressions of losses and gains in the same multiattribute function, present in the formulation of the TODIM method, gives this method ´ THE´E methods, which some similarity to the PROME make use of the notion of net outranking flow. BarbaRomero and Pomerol (2000) have stated the following in respect of the TODIM method: ‘‘it is based on a notion ´ THE´E extremely similar to a net flow, in the PROME sense’’ (p. 229). Consider a set of n alternatives to be ordered in the presence of m quantitative or qualitative criteria, and assume that one of these criteria can be considered as the reference criterion. After the definition of these elements, experts are asked to estimate, for each one of the qualitative criteria c, the contribution of each alternative i to the objective associated with the criterion. This method requires the values of the evaluation, of the alternatives in relation to the criteria, to be numerical and to be normalized; consequently the qualitative criteria evaluated in a verbal scale are transformed into a cardinal scale. The evaluations of the quantitative criteria are obtained from the performance of the alternatives in relation to the criteria, such as, for example, the level of noise measured in decibels, the power of an engine measured in horsepower or a student’s mark in a subject, etc. TODIM can therefore be used for qualitative as well as quantitative criteria. Verbal scales of qualitative criteria are converted to cardinal ones and both types of scales are normalized. The relative measure of dominance of one alternative over another is found for each pair of alternatives. This measure is computed as the sum over all criteria
206
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of both relative gain/loss values for these alternatives. The parts in this sum will be either gains, losses, or zeros, depending on the performance of each alternative with respect to every criterion. The evaluation of the alternatives in relation to all the criteria produces the matrix of evaluation, where the values are all numerical. Their normalization is then performed, using, for each criterion, the division of the value of one alternative by the sum of all the alternatives. This normalization is carried out for each criterion, thus obtaining a matrix, where all the values are between zero and one. It is called the matrix of normalized alternatives’ scores against criteria. P = [Pnm], with n indicating the number of alternatives and m the number of criteria, as shown in Table 1. After the attribution of the weights of the criteria and their normalization, the partial matrices of dominance and the final matrix of dominance must be calculated. The decision makers must indicate which criterion r is to be chosen as the reference criterion for the calculations according to the relative importance assigned to each criterion. In this way, the criterion with the highest value accorded to its importance will usually be chosen as the reference criterion. The weight of each criterion is determined by the decision makers on a numerical scale (e.g., from 1 to 5) and is then normalized. Thus, wrc is the weight of criterion c divided by the weight of the reference criterion r. Using wrc allows all pairs of differences between performance measurements to be translated into the same dimension, i.e. that of the reference criterion. The measurement of dominance of each alternative Ai over each alternative Aj, now incorporated to Prospect Theory, is given by the mathematical expression dðAi ; Aj Þ ¼
m X
Uc ðAi ; Aj Þ
8ði; jÞ:
ð1Þ
c¼1
Thus d(Ai, Aj) represents the measurement of dominance of alternative Ai over alternative Aj; m is the number of criteria; c is any criterion, for c = 1, . . . , m; wrc is equal to wc divided by wr, where r is the reference criterion; Pic and Pjc are, respectively, the performances of the alternatives Ai and Aj in relation to c; h is the attenuation factor of the losses; different choices of h lead to different shapes of the prospect theoretical value function in the negative quadrant. The expression Uc(Ai, Aj) represents the parcel of the contribution of criterion c to function d(Ai, Aj), when comparing alternative i with alternative j. If the value of Pic Pjc is positive, it will represent a gain for the function d(Ai, Aj) and, therefore the expression Uc(Ai, Aj) will be used, corresponding, that is, to Eq. (2). If Pic Pjc is nil, the value zero will be assigned to Uc(Ai, Aj), by applying Eq. (3). If Pic Pjc is negative, Uc(Ai, Aj) will be represented by Eq. (4). The construction of function Uc(Ai, Aj) in fact permits an adjustment of the data of the problem to the value function of Prospect Theory, thus explaining the aversion and the propensity to risk. This function has the shape of an ‘‘S’’, represented in Fig. 1. Above the horizontal axis, considered as a reference for this analysis, there is a concave curve representing the gains, and, below the horizontal axis, there is a convex curve representing the losses. The concave part reflects the aversion to risk in the face of gains and the convex part, in turn, symbolizes the propensity to risk when dealing with losses. After the diverse partial matrices of dominance have been calculated, one for each criterion, the final dominance matrix of the general element d(Ai, Aj) is obtained, through the sum of the elements of the diverse matrices. Expression (2) is used to determine the overall value of alternative i through normalization of the corresponding dominance measurements. The rank of every alternative originates from the ordering of their respective values.
When: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi wrc ðP ic P jc Þ if ðP ic P jc Þ > 0; m P wrc c¼1 0 Uc ðAi ; Aj Þ ¼ vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi if ðP ic P jc Þ ¼ 0; u m u P t wrc ðP jc P ic Þ 1 c¼1 h if ðP ic P jc Þ < 0; wrc
Value
ð2Þ ð3Þ
ð4Þ Losses
Gains
Table 1 Matrix of normalized alternatives’ scores against criteria Alternatives
A1 A2 ... Ai ... An
Criteria C1
C2
...
Cj
...
Cm
P11 P21 ... Pi1 ... Pn1
P12 P22 ... Pi2 ... Pn2
... ... ... ... ... ...
P1j P2j ... Pij
... ... ... ... ... ...
P1 P2
Pnj
m m
Pim ... Pnm
Fig. 1. Value function of the TODIM method (Gomes and Lima, 1992a).
L.F. Autran Monteiro Gomes, L.A. Duncan Rangel / European Journal of Operational Research 193 (2009) 204–211
Pn Pn j¼1 dðAi ; Aj Þ min j¼1 dðAi ; Aj Þ P P : ni ¼ n n max j¼1 dðAi ; Aj Þ min j¼1 dðAi ; Aj Þ
ð2Þ
Therefore, the global measures obtained computed by (2) permit the complete rank ordering of all alternatives. A sensitivity analysis should then be applied to verify the stability of the results based on the decision makers’ preferences. The sensitivity analysis should therefore be carried out on h as well as on the criteria weights, the choice of the reference criterion, and performance evaluations. 3. Case study The city of Volta Redonda is situated in the South of the State of Rio de Janeiro, Brazil. It has approximately 256,000 inhabitants and there are currently a large number of properties, residential and commercial, rented or available for rent. The study sought to determine a reference value for the rent of residential properties, serving in this way as an important source of information for landlords and real estate agents. A set of 6 alternatives with known rental values was used as a guideline. Computations in this case study were performed using an Excel-based spreadsheet. Commercial software for the calculations of the TODIM method is currently under development, though. In order to evaluate the properties, it was necessary to identify which criteria should be considered in the study. For this purpose, research was carried out together with several real estate companies in the city of Volta Redonda and interviews conducted with specialists in the field of property evaluations. 3.1. Definition of the criteria Based on the points of view expressed in the interviews with realtors, it was concluded that eight criteria should be selected for this particular application. All other possible criteria would already be reflected in these eight criteria, given the characteristics of the study area. The eight most important evaluation criteria for this analysis were identified, and are described as follows: C1 – Location: This is the most important evaluation criterion for rental properties. It is a qualitative criterion which seeks to define whether the property is in a valued location. The scale presented in Table 2 as Associated Score was defined to evaluate the performance of the alternatives in relation to this criterion. In this way, a property evaluated as being in a ‘‘good location’’, receives a score equal to 4. C2 – Constructed area: This is a quantitative criterion which describes the size of the property constructed. In this study, the unit of measurement was square meters (m2). C3 – Construction quality: This is a qualitative criterion which defines the standard of finishing of the property. Three levels of finishing and their respective scores were
207
Table 2 Valuation of criterion C1 – location Location
Associated score
Periphery Periphery/Average location Average location Good location Excellent location
1 2 3 4 5
defined as presented in Table 3. Considering a property which has been constructed to a ‘‘high standard of finishing’’, the property receives a score equal to 3, the maximum value for this evaluation criterion. C4 – State of conservation: This is a qualitative criterion used to evaluate the general state of the property. This criterion considers whether any work has been carried out on the property, as well as habitation conditions. Table 4, as follows, presents the associated scores for the state of conservation of the property. For example, an apartment fit for habitation, needing minor repairs, in other words in an average ‘‘state of conservation’’, receives a score equal to 2. This criterion is not directly related to criterion C3, as the state of conservation is independent of the standard of finishing of the property. A property might have a high standard of finishing, using high quality material, but be in need of repairs because of its age. C5 – Number of Garage spaces: This is a quantitative criterion much valued by people who need to rent a property and have one or more cars. C6 – Number of rooms: This is a quantitative criterion which considers for valuation the number of rooms in the property: living rooms, bedrooms, kitchen/breakfast area and bathrooms. The difference between this criterion and criterion C2, which deals with the area constructed, should be highlighted. It is important to evaluate the property by the number of rooms, as two properties of the same total size may be in question, but one of them may have large rooms and the other, a larger number of smaller rooms. C7 – Attractions: This is a qualitative criterion which considers the existence of leisure areas such as swimming
Table 3 Valuation of criterion C3 – construction quality Construction quality
Associated score
Low standard of finishing Average standard of finishing High standard of finishing
1 2 3
Table 4 Valuation of criterion C4 – state of conservation State of conservation
Associated score
Bad Average Good Very good
1 2 3 4
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pools and barbecues among others. It is evaluated according to Table 5. A house with a swimming pool and barbecue receives 4 points. C8 – Security: This is a qualitative criterion used to describe the security conditions of the property. Table 6 presents the evaluation of the property according to this criterion. For example, an apartment in a building with a doorman and security cameras achieved a score of 1. All the criteria defined above are maximization criteria, in other words, the higher the score obtained in the evaluation of the alternatives in relation to each criterion, the better the performance. 3.2. Weighting of the criteria In accordance with the importance given to the criteria used to evaluate the properties in the study, their respective weights were defined by the decision makers through direct valuation and later normalized. The direct valuation consisted of assigning a number between 1 and 5 to each criterion, where 1 would mean ‘least important’ and 5 would mean ‘most important’. The assignment of weights was performed by the decision makers – in this example, the realtors. The information is presented in Table 7. 3.3. The alternatives of the decision process In this case study, fifteen properties in different neighborhoods in the city of Volta Redonda were evaluated. The properties and their respective locations have not been identified, as this was not deemed necessary for the purposes of the article. Only the characteristics of the properties have been shown, designated A1, A2, A3, . . . , A15. The following describes the 15 properties used in the evaluation:
Table 5 Valuation of criterion C7 – attractions Attractions
Associated score
Without attractions Backyard or terrace Barbecue Swimming pool Swimming pool, barbecue or others
0 1 2 3 4
Table 6 Valuation of Criterion C8 – Security Security
Associated score
No additional security Doorman and security cameras in the apartment building. Houses with manned security boxes in the streets
0 1
Table 7 Criteria ranks Criterion
Description
Assigned weights
Criteria weights
C1 C2 C3 C4 C5 C6 C7 C8
Localization Constructed area Quality of construction State of conservation Number of garage spaces Number of rooms Attractions Security
5 3 2 4 1 2 1 2
0.25 0.15 0.10 0.20 0.05 0.10 0.05 0.10
A1 – A house in an average location, with 290 m2 of constructed area, a high standard of finishing, in a good state of conservation, with one garage space, 6 rooms, a swimming pool, barbecue and other attractions, without a security system. A2 – A house in a good location, with 180 m2 of constructed area, an average standard of finishing, in an average state of conservation, with one garage space, 4 rooms, a backyard and terrace without a security system. A3 – A house in an average location, with 347 m2 of constructed area, a low standard of finishing, in an average state of conservation, two garage spaces, 5 rooms, a large backyard, without a security system. A4 – A house in an average location, with 124 m2 of constructed area, an average standard of finishing, in a good state of conservation, two garage spaces, 5 rooms, a fruit orchard, a swimming pool and barbecue, without security system. A5 – A house in an excellent location, with 360 m2 of constructed area, a high standard of finishing, in a very good state of conservation, four garage spaces, 9 rooms, a backyard and manned security boxes in the neighborhood streets. A6 – A house located between the periphery and the city center (periphery/average location) with 89 m2 of constructed area, an average standard of finishing, in a good state of conservation, with one garage space, 5 rooms, a backyard, without a security system. A7 – An apartment located in the periphery, with 85 m2 of constructed area, a low standard of finishing, in a bad state of conservation, one garage space, 4 rooms, a manned entrance hall with security. A8 – An apartment in an excellent location, with 80 m2 of constructed area, average standard of finishing, good state of conservation, with one garage space, 6 rooms, manned entrance hall with security. A9 – An apartment located between the periphery and the city center (periphery/average location), with 121 m2 of constructed area, an average standard of finishing, in a good state of conservation, no garage space, 6 rooms, without a security system. A10 – A house located between the periphery and the city center (periphery/average location), with 120 m2 of constructed area, a low standard of finishing, in a good
L.F. Autran Monteiro Gomes, L.A. Duncan Rangel / European Journal of Operational Research 193 (2009) 204–211
state of conservation, with one garage space, 5 rooms, a large backyard, without a security system. A11 – A house in a good location, with 280 m2 of constructed area, an average standard of finishing, in an average state of conservation, with two garage spaces, 7 rooms, with an additional security system. A12 – An apartment located in the periphery, with 90 m2 of constructed area, a low standard of finishing, in a bad state of conservation, one garage space, 5 rooms, without additional security. A13 – An apartment located in the periphery in an average location, with 160 m2 of constructed area, a high standard of finishing, in a good state of conservation, two garage spaces, 6 rooms, with additional security features. A14 – An apartment in a good location, with 320 m2 of constructed area, high standard of finishing, in a good state of conservation, 2 garage spaces, 8 rooms, with in addition a security system. A15 – A house in a good location, with 180 m2 of constructed area, an average standard of finishing, in a very good state of conservation, one garage space, 6 rooms, with in addition a security system. Table 8, containing the Evaluation of Alternatives against Criteria, presents the complete evaluation of the properties studied in the analysis in relation to the criteria selected by the decision makers. In the case study, the attenuation factor of losses h has a value equal to 1, which means that the losses will contribute with their real value to the global value. In order to implement the method, it is necessary for these performances to be normalized. The matrix of normalized performances is then called the Matrix of Normalized Alternatives’ Scores against Criteria, presented in Table 9 below. In order to demonstrate how to determine the dominance measurements, the computations to obtain d(A2, Table 8 Evaluation of alternatives against Criteria
209
Table 9 Matrix of normalized alternatives’ scores against criteria Alternatives
A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A12 A13 A14 A15
Criteria C1
C2
C3
C4
C5
C6
C7
C8
0.068 0.091 0.068 0.068 0.114 0.045 0.023 0.114 0.045 0.045 0.091 0.023 0.045 0.068 0.091
0.103 0.064 0.123 0.044 0.127 0.031 0.03 0.028 0.043 0.042 0.099 0.032 0.057 0.113 0.064
0.1 0.067 0.033 0.067 0.1 0.067 0.033 0.067 0.067 0.033 0.067 0.033 0.1 0.1 0.067
0.075 0.05 0.05 0.075 0.1 0.075 0.025 0.075 0.075 0.075 0.05 0.025 0.075 0.075 0.1
0.045 0.045 0.091 0.091 0.182 0.045 0.045 0.045 0 0.045 0.091 0.045 0.091 0.091 0.045
0.069 0.046 0.057 0.057 0.103 0.057 0.046 0.069 0.069 0.057 0.08 0.057 0.069 0.092 0.069
0.174 0.087 0.043 0.174 0.043 0.043 0 0 0 0.043 0.13 0.087 0.043 0.087 0.043
0 0 0 0 0.143 0 0.143 0.143 0 0 0.143 0 0.143 0.143 0.143
A4) are presented in the Appendix as an example. Similar computations would lead to a 15 · 15 matrix, where the values in the cells would be measurements of dominance. After the implementation of the mathematical formulation of the TODIM method, the overall values of the alternatives obtained through normalization of the corresponding dominance measurements are presented in Table 10. This table also presents the ordering of each alternative. 3.4. Results analysis Among the properties evaluated, some were inserted as references, simply to assist in the analysis, as their rental values were already known. These properties are presented in Table 11. In this way, by inserting these properties with known values, it is possible to establish a range of values for the other properties under analysis. For example, in this study, it can be seen that in the final ordering by the TODIM Table 10 Final values and ordering
Alternatives
Criteria C1
C2
C3
C4
C5
C6
C7
C8
Alternative
Normalized global value
Ordering
A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A12 A13 A14 A15
3 4 3 3 5 2 1 5 2 2 4 1 2 3 4
290 180 347 124 360 89 85 80 121 120 280 90 160 320 180
3 2 1 2 3 2 1 2 2 1 2 1 3 3 2
3 2 2 3 4 3 1 3 3 3 2 1 3 3 4
1 1 2 2 4 1 1 1 0 1 2 1 2 2 1
6 4 5 5 9 5 4 6 6 5 7 5 6 8 6
4 2 1 4 1 1 0 0 0 1 3 2 1 2 1
0 0 0 0 1 0 1 1 0 0 1 0 1 1 1
A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A12 A13 A14 A15
0.6916 0.3862 0.3992 0.6210 1.0000 0.2860 0.0000 0.4407 0.0202 0.2127 0.8576 0.1073 0.7188 0.9372 0.6733
5 10 9 7 1 11 15 8 14 12 3 13 4 2 6
210
L.F. Autran Monteiro Gomes, L.A. Duncan Rangel / European Journal of Operational Research 193 (2009) 204–211
Table 11 Properties with a previously defined rental value Property
Monthly rental value
A3 A4 A5 A9 A10 A11
US$ US$ US$ US$ US$ US$
214 309 712 133 166 414
method, A8 is in the 8th place, A3 is in the 9th place, with a rental value of US$ 214, and A4 is in the 7th place, and should be rented for US$ 309. It can then be said that A8 must be rented for a value between US$ 214 and US$ 309. In the same way it can be seen that property A14 is in the 2nd place. A5 is in the 1st place, with a rental value of US$ 712, and A11 is in the 3rd place with a rental value of US$ 414. Therefore property A14 must be rented for a value of between US$ 712 and US$ 414. If the ranking order obtained by TODIM did not coincide with the ranking based on dollar values this would mean that the properties’ dollar values would have to be reconsidered and then changed accordingly. These value intervals are defined for each alternative under analysis, serving as a reference to obtain values for the rent. In this way, the decision process becomes less complex. Table 12 suggests the appropriate intervals for the properties with non-defined rental values. This shows that realtors would considerably benefit from using a more comprehensive, multicriteria approach through the use of TODIM when they assign dollar values to the properties. 3.5. Sensitivity analysis After obtaining the results through the implementation of the TODIM method, a sensitivity analysis was carried out varying the weighting given to the criterion of greatest importance to the decision makers, which is criterion C1. When its weighting was reduced from 5 to 3 there was no variation in the ranking of the 15 alternatives. Another analysis was carried out varying the value of h, the attenuation factor of losses. In the first implementation carried
Table 12 Suggested rental values of the alternatives analyzed Property
Ordering
Monthly rental value
A1 A2 A6 A7 A8 A12 A13 A14 A15
5 10 11 15 8 13 4 2 6
Between US$ 414 and Between US$ 214 and Between US$ 214 and Less than US$ 133 Between US$ 309 and Between US$ 166 and Between US$ 414 and Between US$ 712 and Between US$ 414 and
US$ 309 US$ 166 US$ 166 US$ US$ US$ US$ US$
214 133 309 414 309
out, the value used for h was 1. In the sensitivity analysis the value of h was altered from 1 to 5. When this alteration was carried out, the only change in the ordering of the values occurred in the values of A13 with A15, indicating that, in spite of the alteration in the value of the attenuation index of losses, the ordering obtained was consistent. In this particular application the decision makers felt that sensitivity analyses should be conducted on the weight of the reference criterion and the value of h only. 4. Conclusion The analysis of the alternatives using the TODIM method led to an ordering which showed it to be satisfactory and in agreement with the expectations of the specialists. Through its formulation it became easier to resolve conflicts between criteria, as, sometimes, in order to achieve a good performance in a determined criterion of the analysis, it is necessary not to be concerned about performance in another (Belton and Stewart, 2002). In the case of property evaluation, the method is capable of assisting professionals in the real estate market to evaluate the alternatives more clearly in relation to the criteria defined by the specialists. In this study it was also possible to identify that three properties, A1, A13 and A15, fell into the same range established by the alternatives A4 and A11. From Table 10, it can be seen that the order of these three properties is: A13, A1 and A15. In this way, the rental values of these three properties could be the same or, alternatively, giving a reference for the rental value of the property according to the ordering supplied by the method, the greatest value attributed to A13 and the lowest value to A15. Thus, the analysis and the solution of the problem presented here, by means of the TODIM multicriteria method, reflected in their results the preferences of the decision agents, experts in the multiple dimensions of the problem analyzed. Consequently, it can be concluded that the method constitutes efficient support for the evaluation of property. As new properties are included in the portfolio of a realtor, the TODIM method must be run again taking into account the characteristics of these new properties. After the rank ordering is obtained for these new properties their suggested rental values will be determined with the help of Table 11. Realtors concluded that applying the TODIM method to the rental evaluation of residential properties can provide a considerable help to them, given the extreme difficulties in assigning dollar values to all evaluation criteria. Given new evaluation scenarios, with a new set of evaluation criteria, however, new applications of the multicriteria analysis would have to be performed. Changes in the scenarios can lead to changes in estimated rental values even for properties whose rental values have previously been defined. For future studies efforts should also be focused on trying to quantify the monetary consequences associated to every criterion.
L.F. Autran Monteiro Gomes, L.A. Duncan Rangel / European Journal of Operational Research 193 (2009) 204–211
By applying Eq. (1) one determines the dominance measurement d(A2, A4) as equal to 2.833.
Acknowledgements The authors are grateful to the referees for their insightful comments on the first version of this paper. They are also grateful to the National Council for Scientific and Technological Development (CNPq) of Brazil for supporting the research. Appendix Eq. (1) is used to determine the values of the dominance measurements. For instance, when determining d(A2, A4) one must first obtain the different values of Uc(A2, A4) for all criteria c. This is accomplished by making use of Eqs. (2)–(4). The computations below are performed for: h = 1; r = C1 (the reference criterion); data in Tables 9 and 7. • For c = C1 one gets P21 P41 > 0 (a gain). By using Eq. (2), with wrc = 1.0, U1(A2, A4) = 0.075, i.e. vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u w11 ðP 21 P 41 Þ uwrc ðP ic P jc Þ u ¼ m P t 4 wrc c¼1
211
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð0:25=0:25Þð0:091 0:068Þ ¼ 4 ¼ 0:075:
• For c = C2 one gets P22 P42 > 0 (a gain). By using Eq. (2), with wrc = 0.6, U2 (A2, A4) = 0.054. • For c = C3 one gets P23 P43 = 0 (neither a gain nor a loss). By using Eq. (3), with wrc = 0.4, U3(A2, A4) = 0.0. • For c = C4 one gets P24 P44 < 0 (a loss). By using Eq. (4), with wrc = 0.8, U4(A2, A4) = 0.353, i.e. vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u P u m sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uð wrc ÞðP jc P ic Þ 1 t c¼1 1 4ðP 44 P 24 Þ ¼ h h w14 wrc sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 4ð0:075 0:050Þ ¼ 1 ð0:20=0:25Þ ¼ 0:353: • For c = C5 one gets P25 P45 < 0 (a loss). By using Eq. (4), with wrc = 0.2, U5(A2, A4) = 0.959. • For c = C6 one gets P26 P46 < 0 (a loss). By using Eq. (4), with wrc = 0.4, U6(A2 ,A4) = 0.331. • For c = C7 one gets P27 P47 < 0 (a loss). By using Eq. (4), with wrc = 0.2, U7(A2, A4) = 1.319. • For c = C8 one gets P28 P48 = 0 (neither a gain nor a loss). By using Eq. (3), with wrc = 0.4, U8(A2, A4) = 0.0.
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