THE USE OF COMBINED MULTICRITERIA METHOD FOR THE VALUATION OF REAL ESTATE

A considered problem is a valuation of real estate. It is important to specify their exact market value, which is the result of several factors. Valuation of property is made on the basis of information and transactions on the local market. Moreover, the valuation always is based on the data of the similar properties. A comprehensive set of data is needed for these reasons. It is quite confusing because the number of trans-actions on the local market often is not sufficient. The purpose of this paper is to present a method for multicriteria valuation of real estate. This procedure is based on the Analytic Hierarchy Process (AHP) and the Goal Programming (GP). It was designed especially for valuation in situation in which information are limited. The proposed method was used for the valuation of the real estate located on Warsaw.


Introduction
The considered problem is valuation of real estate as a subject of ownership. The study is limited only to the valuation of the market value of the property. In the paper are used definitions of a valuation and the market value of the real estate which are regulated by Real Estate Management Act [Act of 21.08.1997] and Council of Ministers On the Valuation of Property and Preparing the Appraisal [Act of 21.08.2004]. These definitions are based on recommendations of The International Valuation Standards, European Valuation Standards and EU directives [Mączyńska 2009;Trojanek, 2010, p. 66-75]. The valuation is defined as the process of estimating e.g. market value of property. The market value is defined as the price of a property that is available for purchase. This price is most likely to be concluded by buyers and sellers of the property [Mączyńska 2009;Trojanek, 2010, p. 66-75]. The correct valuation of real estate requires the fulfillment of certain conditions. Firstly, the correct valuation is based on comprehensive set of information and transactions from local real estate market. It means, that it is based on information of properties which were traded on the local market in the last two years. Therefore, all characteristics of real estate have a local nature. Secondly, the correct valuation is based on information of properties which are similar to the valued one. The similar property means, that it is comparable due to the type, location, legal status, purpose, method of use and other characteristics that affect its value. Also, in the valuation process is required the determining weights for the characteristics of properties which are compared [Mączyńska 2009;Trojanek, 2010, p. 66-75].
Unfortunately, on the local market often is not sufficient number of transactions that can be used for valuation. Moreover, values of characteristics of property mostly are intangible. Solution of this problem can be a method of valuation of real estate which does not need an extensive set of information to obtain reliable results.
The purpose of this paper is to present a combined multicriteria valuation method (CMVM). This procedure is based on a combination of two multicriteria decision-making methods i.e. the Analytic Hierarchy Process (AHP) and Goal Programming (GP). The mixed AHP and GP procedure for multicriteria real estate valuation has been designed especially for valuation in situation in which information are limited [Mixed valuation methods…, 2011, p. 221-238;Kozioł-Kaczorek, 2012, p. 236-239;Kozioł-Kaczorek, 2013, p. 87-93]. Originators of CMVM are J. Aznar, F. Guijarro and J.M. Moreno-Jimenez. They created this procedure during a research project of the Spanish Ministry of Education and Science. The application of the proposed methodology was illustrated by the example of the valuation of a peach plantation in the La Riberta district in Valencia in Spain.
The main aim of the CMVM is to extract the knowledge from specific characteristics of real estate during the valuation process. This technique can be used with both intangible and scarce information. The Analytic Hierarchy Process allows for the incorporate tangible and intangible aspects by means of using paired comparisons in the valuation procedure. The Goal Programming (GP) allows for the incorporate both the scarce information available (objective) and the individual appraiser's attitude with regards to the valuation process (subjective) [Mixed valuation methods…, 2011, p. 221-238;Kozioł-Kaczorek, 2012, p. 236-239;Kozioł-Kaczorek, 2013, p. 87-93]. The proposed method will be used for the valuation of the real estate located in Warsaw.

The Analytic Hierarchy Process (AHP)
The Analytic Hierarchy Process (AHP) was developed by T. L. Saaty. It is discrete multicriteria method which combines a decisions theory and mathematics. AHP is used to making optimal choices for the multicriteria decisions problems by reducing them to a series of pairwise comparisons. These pairwise comparisons are carried out by experts and allow to designate a numerical measure of a validity of analysed variants. Other words, the optimal choice is made on the basis of weights of criteria and features determined by using AHP. These criteria and characteristics can be in the form of both metric or nonmetric (ordinal). AHP algorithm consists of few steps: definition of a problem, construction of a hierarchical model, pairwise comparison of the validity of analysed variants on each level of a hierarchy, construction of a vector of priorities for analysed decisions elements, verification of a consistency of comparisons at each level of the hierarchy, analysis of results [Saaty 1980, Saaty 2001, Wysocki 2010 Stage I. This stage contains general definition of the problem being solved. It contains a determination of an overarching objective, a determination of main and partial criteria and also a determination of analysed variants of these criteria [Saaty, 1980;Saaty, 2001;Wysocki, 2010].
In the case of property valuation, the overarching objective is prioritization of those characteristics of property that affect its market value. It consists in assigning weights for each of characteristic. The main criterion are real estate, which have been the subject of a transaction on an analysed local market, along with the prices for 1 m 2 . Partial criteria are characteristics of real estate and their values. The set of characteristics depends primarily on the type of property (undeveloped or developed property, premises realty etc.) and on the specificity of the local market. And so, the set of the characteristics of e.g. premises realty may consist position and location, the neighbourhood and the environment, surface, place, location of the premises in the building, the condition of the building, the standard finish of premises, etc. [Dydenko 2006]. Variants of partial criteria are values of the selected characteristics. Those values are determinants of both partial and the main criteria.
Stage II. This stage contains construction of a multicriteria hierarchical tree. The main criterion is always on the top of this structure. There are also below partial criteria. The rule indicates that in the tree structure cannot be less than two partial criteria but not more than seven. This rule is based on the psychological fact that human brain can compare, with no mistake, no more than 7 ± 2 objects [Saaty, 1980;Saaty, 2001;Wysocki, 2010].
In the case of property valuation, the form of the hierarchical model is depend on the type of real estate (undeveloped or developed property, premises realty etc.), as well as the peculiarities of the local market. The main criteria may include the following: the property with the lowest price (A), the property with the price somewhat below average (B), the property with the price slightly above average (C), the property with the highest price (D). Partial criteria are discussed above.  One item moderately more important than another 5 One item significantly more important than another 7 One item much more important than another 9 One item very much more important than another 2, 4, 6, 8 Intermediate situations Source: own study based on literature [Saaty 1980, Saaty 2001 In the case of real estate, this stage involve construction of five matrices of comparisons: the matrix of comparisons of real estate due to prices, Elements of hierarchical model are arranged according to the size of elements of the priorities vector , in order of their importance. A higher value means more the significance of the item [Saaty 1980, Wysocki 2010].
In the case of real estate, the vector of priorities includes weights of the individual characteristics of the property. This stage involve construction of five vectors of priorities: the vector of priorities for four chosen real estate; the vector of priorities for property with the lowest price; the vector of priorities for property with the price somewhat below average; the vector of priorities for property with the price slightly above average; = it means that evaluations are completely consistent [Saaty, 1980;Saaty, 2001;Trzaskalik, 2006;Wysocki, 2010].
A coefficient of inconsistencies can be also computed by the formula = , where is a random index, which is dependent on the matrix order. If the value of the coefficient of inconsistencies ≤ 0.1, then it can be concluded that comparisons are consistent. If the value of the coefficient of inconsistencies = 0, then it can be concluded that comparisons are completely consistent. When > 0.1 it is necessary to repeat (all or some) comparisons in pairs to remove the discrepancy [Saaty, 1980;Saaty, 2001;Trzaskalik, 2006;Wysocki, 2010].

The Goal Programming (GP)
The Goal Programming (GP) is an extension of linear programming. However, it is not focused on optimisation like linear programming. The Goal Programming is focused on satisfaction and searching of a number objectives at the same time. It is mathematical nonlinear programming but it can be easy linearized. Objectives are formulated by specifying the desired values of the analyzed criteria. The main task of goal programming is to find the best solution in which all the conditions are satisfied. A lot of variants of GP exist in literature. In this paper two of them are applied: Weighted Goal Programming and MinMax. The form of the basic model for WGP is: is the negative deviation and is positive deviations with respect to the j-th goal ( ̂ ). Moreover, denotes a normalisation factor. Note that, since the values are already normalised in the distribution mode, there is no need for normalisation factor, so = 1 is taken [Mixed valuation methods…, 2011 pp. 221-238]. In the model occurs strong constrains. Their determine feasible region i.e. ( ), = 1, … , . There are also weak constrains for the goals considered ( ̂ , = 1, … , ).
The goal vector ̂= ( ̂ , … , ̂ ) contains prices of each real estate comparable to the subject one during the valuation of real estate. The valuation function is on the form:

The combined multicriteria valuation method (CMVM)
As it was written before, the combined multicriteria method (CMVM) is a combination of the Analytic Hierarchy Process (AHP) and Goal Programming (GP). The algorithm of CMVM contains three stages. The first stage is the use of AHP to quantify the subjective information about the elements being compared. The second stage is the use of GP for estimating the parameters of regression model to obtain the market value of real estate. The GP is used in two different ways, according to a Manhattan norm and a Tchebychef norm ∞ . The Manhattan norm introduces scarce information available into the model. The Tchebychef norm ∞ captures the subjective attitude with respect to the valuation process (i.e. the greater distance between the compared elements contribute to the greater subjectivity of the assessment). Furthermore, this norm allows consideration of the proximity of the subject asset to one of the comparable sets of assets that does not follow common or majority behaviour. These norms are used to determine the objective function to be optimised [Mixed valuation methods…, 2011, p. 221-238;Kozioł-Kaczorek, 2013]. The third stage is the final valuation of the real estate. The final market value of real estate is a result of a convex combination of values obtained in stage two. The formula of the market value is: ( ) = (1 − ) ( ) + ∞ ( ), ∈ 0,1 where ( ) and ∞ ( ) are the and ∞ values. The level of depends on the expert knowledge. The level = 1 means that the valued property is very similar to comparable properties. On the other hand, level = 0 means that the valued properties is significant different from comparable properties [Mixed valuation methods…, 2011, p. 221-238;Kozioł--Kaczorek, 2013].

The valuation of real estate
The above-described methodology was applied to the valuation of premises realty located on Mokotów district in Warsaw. Therefore, the set of data also comes from the real estate market in the Mokotów district in Warsaw. It was collected by the property appraisers of Institute of Property Consultancy.
The first step of the CMVM is the use of AHP to determine the weights of characteristics of real estate. The overarching objective was the assigning weights for characteristics of valued premises realty. Main criteria were properties with the lowest price (A), the price somewhat below average (B), the price slightly above average (C), the highest price (D).
Partial criteria were characteristics of premises realty i.e.: the position and location, the neighbourhood and the environment, the surface, the location of the premises in the building, the condition of the building, the standard of finish of premises. Values of the selected characteristics of premises realty i.e.: very good, good, sufficient, bad were variants of partial criteria. The hierarchical model were then on the form presented in above table 2. The property A (the lowest price) is premises realty located on 11th floor in eleven storey building. The total area of the apartment is about 65 m 2 . Requires overhaul of apartment. The building also needs global renovation due to its technical condition. The price of 1 m 2 of the property was PLN 4 443.
The property B (the price somewhat below average) is premises realty located on 5th floor in ten storey building. The total area of the apartment is about 37 m 2 . Requires renovation of apartment. The building is in good technical condition. The price of 1 m 2 of the property was PLN 8 706.
The property C (the price slightly above average) is premises realty located on 5th floor in ten storey building. The total area of the apartment is about 31 m 2 . Requires renovation of apartment. The building is in good technical condition. The price of 1 m 2 of the property was PLN 8 710.
The property D (the highest price) is premises realty located on 3rd floor in fife storey building. The total area of the apartment is about 45 m 2 . The apartment was completely renovated, and the standard of it is very high. The building is new and in very good technical condition. The price of 1 m 2 of the property was PLN 13 283 .
Presented in table 2 hierarchical tree was established by analysis of the local market and the appraiser's knowledge. Then it was constructed five matrices of comparisons: -  Table 3.
Letter V denotes the valued property. It is premises realty located on 2nd floor in fife storey building. The total area of the apartment is about 39 m 2 . The apartment was completely renovated, and the standard of it is very high. The building is in very good technical condition. The second stage of the CMVM contains the GP application. The GP was used in two different ways (the Manhattan norm and the Tchebychef norm ∞ ) for estimating the regression parameter. In this paper the mentioned regression model was on the form = + + + + + + , = 1,2,3,4 where denotes weight of one of six characteristics of -th premises realty and ( ) denotes their prices. Obtained parameters were used to calculate the values of valued property by: -WGP: ( ) = 10 564 PLN/m 2 ; -MinMax: ( ) = 11 861PLN/m 2 . The obtained market value of the valued premises realty (V) is within the range (10 564; 11 862) PLN/m 2 . This market value is considerably in excess of the average price of 1 m 2 of premises realty on this local real estate market. This is due to the fact, that the valued premises realty (V) was more similar to the property with the highest price than to those with the average price.

Conclusions
The use of usual valuation method often is not possible because of the common problem with insufficient number of transactions on local real estate market. Described in the paper method of valuation (CMVM) can be solution of this problem. It is the combination of two multicriteria methodologies i.e. Analytic Hierarchy Process (AHP) and Goal Programming (GP). The AHP enables to quantify qualitative variables and include the weight of the importance of preferences. The GP captures the information from the limited information and the attitude of the appraiser in the valuation process. The calculated value range enables the expert to define the final market value.