Integrating Geographical Information Systems and ... - Springer Link

Annals of Operations Research 116, 243–269, 2002  2002 Kluwer Academic Publishers. Manufactured in The Netherlands.

Integrating Geographical Information Systems and Multi-Criteria Methods: A Case Study ELIANE GONÇALVES GOMES and MARCOS PEREIRA ESTELLITA LINS ∗

{eggomes;lins}@pep.ufrj.br Production Engineering Program – Federal University Rio de Janeiro, Centro de Tecnologia, Bloco F, Sala F-105, Cidade Universitária, Ilha do Fundão, CEP: 21945-970, Rio de Janeiro, RJ, Brazil

Abstract. This paper presents an application of the integration between Geographical Information Systems (GIS) and Multi-Criteria Decision Analysis (MCDA) to aid spatial decisions. We present a hypothetical case study to illustrate the GIS–MCDA integration: the selection of the best municipal district of Rio de Janeiro State, Brazil, in relation to the quality of urban life. The best municipal district is the one that presents the closest characteristics to those considered ideal by the decision-maker. The approach adopted is the Multi-Objective Linear Programming (MOLP) and the chosen method is the Pareto Race. Keywords: Geographical Information Systems (GIS), Multi-Objective Linear Programming (MOLP), GIS–MCDA integration, quality of urban life

Introduction Decision-making can be defined as the process of choices among alternatives. MultiCriteria Decision Analysis (MCDA), developed in the environment of Operational Research, aids analysts and decision-makers in situations in which there is a need for identification of priorities according to multiple criteria. This usually happens in situations where conflictive interests coexist. The Geographical Information Systems (GIS) support the solution of complex spatial problems, providing the decision-maker with a flexible environment in the process of the decision research and in the solution of the problem [7]. The visualization of the context, structure of the problem and its alternative solutions is one of the most powerful components of a decision support system [15]. Thus, the integration GIS–MCDA has the objective of favouring decision-makers, providing them with ways to evaluate several alternatives, based on multiple, conflictive criteria. Pereira and Duckstein [35] stress that the most important components of a multicriteria technique are those that involve interaction with the decision-maker, to develop a value function or to elucidate a group of weights for the evaluation criteria. These steps usually require interactive questions. The current GIS packages do not possess these capacities of interactive questions, and the interaction between the analyst and the decision-maker takes place outside the GIS environment. An important advantage in ∗ Corresponding author.

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using GIS to accomplish spatial multi-criteria analysis is the ease in developing criteria based on neighbourhood analysis operations. The role of GIS in spatial decision-making is to aid the decision-maker in designating priority weights to the criteria, to evaluate the feasible alternatives and to visualize the results of his choice [20]. The search usually results in the selection of a certain number of alternatives that satisfy minimum threshold values. The reduction of the set of alternatives and the selection of the best one usually require the use of multi-criteria techniques. Thus, improvement of the capacities of GIS in the decision-making can be achieved by the introduction of multi-criteria techniques in the GIS environment. Some examples can be found in [8,21–24,29,33–35]. Eastman [12], Malczewski [32], Gomes and Lins [15], Gomes [14] and Malczewski [31] present the state of the art in GIS– Multi-criteria integration. The objective of this paper is to show how the integration between GIS and the multi-criteria methods can support spatial decisions. We present a case study that is aimed at selecting the best municipal district of Rio de Janeiro State, Brazil, in relation to the quality of urban life. Five families of criteria are analysed: infrastructure, education, security, health and work. Selection of criteria in each of these families was based on the existence, collecting periodicity and reliability of the information. We used the vector GIS spatial data structure and the smallest geographical unit is the municipal district. Thus, each alternative is visualized as a municipality, or a polygon of the vector database. Furthermore, this paper contributes to the literature on GIS–MCDA integration, by making use of the Multi-Objective Linear Programming (MOLP) approach as the multi-criteria method (almost all of the case studies found in the literature employ the Multi-attribute approach), besides briefly describing the state of the art on this subject. Besides, as the problem presented has a multi-attribute nature, we present a method to convert it into a multi-objective one, aiming to make use of the latter’s facilities (no requirement of precisely specifying an a priori problem, interactive use of computational graphical interface, making the interaction analyst–decision-maker–modelling easier). 1.

GIS–Multi-criteria integration

Jankowski [20] proposed two methods, or architectures, for integrating GIS and Multicriteria techniques: the loose coupling strategy and the tight coupling strategy. The main idea of the loose coupling strategy is to facilitate the integration using a file exchange mechanism. The assumption behind this strategy is that multi-criteria techniques already exist in the form of stand-alone computer programs. The results of the decision analysis may be sent to GIS for display and spatial visualization. The loose coupling architecture is based on linking three modules (GIS module, Multi-criteria technique module and file exchange module), as seen in figure 1. The tight coupling strategy uses multiple criteria evaluation functions fully integrated into GIS, a shared database and a common user interface (figure 2). Differently from the previous architecture, the data manipulations and transferences between the boxes “Data input management functions” – “Spatial analysis functions” –

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Figure 1. The loose coupling architecture for linking GIS and Multi-criteria techniques (after [20]).

Figure 2. The tight coupling architecture for linking GIS and Multi-criteria techniques (after [20]).

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Figure 3. GIS–Multi-criteria integration model for the proposed case study.

“Display and data output functions” and the box “Multicriteria evaluation functions” are performed endogenously, with no need of a file exchange module. Under this approach, the GIS evaluation functions facilitate spatial decision-making with multiple criteria. The multiple criteria evaluation functions can be seen as a part of the GIS toolbox, that is, one can select a function from the common GIS user interface. This design facilitates the map views of alternatives and their criteria. The IDRISI (http://www.clarklabs.org/03PROD/03prod.htm) and SPRING (Georeferenced Information Processing System – http://www.dpi.inpe.br/english/index.html) GIS software have multi-criteria evaluation functions that use the Analytical Hierarchic Process (AHP) method [38], and can be seen as examples of the tight coupling strategy implementation. In this paper the methodology adopted, presented in figure 3, is based on the integration method proposed by Jankowski and Richard [21], similar to the loose coupling strategy. On the whole, the integration involves three main stages. In the first stage, conducted in a GIS environment, there is a reduction in the number of alternatives, through physical and/or qualitative constraints imposed by the criteria. These constraints, in most cases, are related to topological operations and/or to search operations (known as spatial


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queries, which yield details or parameters about the features themselves, where the data is stored in a GIS database; the information processing is through database manipulation and mathematical analysis functions, using logic operators AND, OR, NOT), easily carried out in GIS. With this reduced set of alternatives, we proceed with the multi-criteria analysis to select the best alternative among these. The Multi-Objective Linear Programming (MOLP) problem was solved through the Pareto Race method [27,28], implemented by VIG software [25]. We chose this method on account of its interactivity, good graphical interface, permitting the use of a great number of objective functions, availability and compatibility with the operational systems handled by the authors. In the third stage, the MOLP results are introduced into GIS, for the final visualization of the choice of the decision-maker, so as to guarantee that “the most correct decision is that which best represents the interests of the decision-maker” [16]. 2.

Case study

The objective of this case study is to present a hypothetical example of the integration between GIS and multi-criteria methods in support of the spatial decision. We wish to select the best municipal district of Rio de Janeiro State, Brazil, in relation to the quality of urban life of the local population. Some characteristics of the public administrations were selected, and can be regarded as evaluators of each municipal government’s performance. These characteristics were gathered in criteria of infrastructure, education, security, health and work. These families of criteria are seen by Graeml and Erdmann [16] as indicators of quality of urban life, and could even be used as a type of performance measurement of the local administrations. Each municipal district is seen as an alternative (represented by polygons of the vector GIS database), and the best municipal district is the one that exhibits the characteristics of urban life quality closest to those desired by the decision-maker. Rio de Janeiro State is situated in the Southeast Region of Brazil (also composed of São Paulo, Minas Gerais and Espírito Santo States), which concentrates about 2/3 of the Brazilian GDP (Gross National Product). This region has communication and product flow routes with the markets of the Mercosul economic block (whose members, besides Brazil, include Argentina, Uruguay and Paraguay). Rio de Janeiro State has about 14.4 million inhabitants, 40% of whom (5.8 million) live in the capital, Rio de Janeiro City. The State is divided into 91 municipal districts (the vector GIS database used has 69 municipal districts – 1991 municipal configuration) (figure 4), grouped in eight Government Areas. The population of the State strengthened its trend towards low growth rates in the first half of the last decade, showing in 1996, a rise to an average rate of only 0.92% a year, representing an emigration of more than 32,000 people a year. The infant mortality rate presents decline, registering 29 deaths among those under 1 year old for each group of 1,000 live births, representing a reduction of more than 25% in 10 years. The literacy rate of the population in the age group 5 or more years, showed a significant rise from 1991 to 1996. In 1991, 87% of the population of Rio de Janeiro State

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Figure 4. Geographical location of Rio de Janeiro State, highlighting its municipal districts.

were literate and, in 1996, this percentage rose to 92%. In 1996, in Rio de Janeiro State, the school attendance rate of those from 7 to 14 years was 93.9%, and those from 4 to 6 years, 71%. Between 1995 and 1997, the number of homicides presented a reduction of approximately 8.1%. 2.1. Problem structuring 2.1.1. Defining the criteria The definition of which criteria should be adopted for the study of the quality of urban life of the population of a certain area depends on deep studies of the ease and cost of data collection, reliability of the collected data, decision-making model and its structure. For the decisions to be based on concrete data, it is necessary to think about what can be measured and how these data will be used [16]. In this case study, the criteria were chosen on the basis of the existence and ease of obtaining the data for each municipal district of Rio de Janeiro State, considering some basic areas in the definition of the indicators (criteria), such as dwelling/infrastructure, education, health, security and work. Figure 5 displays the decision criteria and their role in problem structuring. Each one of these criteria has an important role in the decision-making process, for instance, to measure the level of the public services, to assist the planning processes, administration and evaluation of policies, to infer the measure of the population’s socio-economic situation, etc. [14]. Figure 5 shows that some criteria are used for classification, through characteristics that must be maximized or minimized, while others are exclusion criteria that provide early elimination of alternatives whose bad performance cannot legitimately be compensated by good performance in some other criteria. As stated by Barba-Romero and Pomerol [3], the exclusion criteria can be used in a disjunctive manner (the condition sufficient for an alternative to be qualified is that it must comply at least with one of the criteria) or in a conjunctive one (the non-compliance with one of the criteria implies in the elimination of the alternative). The same authors


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Figure 5. Decision criteria.

stress that, in many cases, the conjunctive method is extremely rigorous and the disjunctive method is excessively permissive. This affirmation justifies a hybrid approach that considers as admissible alternatives the ones that comply with a certain number of exclusion criteria (this number must be between one and the total number of exclusion criteria). All the data come from official sources of information, namely, the Brazilian Institute of Geography and Statistics, the Ministry of Health and Rio de Janeiro State Health Secretariat, the Ministry of Social Welfare, the Ministry of Education, the National Institute of Studies and Educational Research, the United Nations Development Program (UNDP), and from the Brazilian Human Development Atlas. Next we define the classification and exclusion criteria: • Regular Domestic Waste Collection: this criterion is aimed at measuring the standard of basic sanitation in relation to waste collection. The values refer to the percentage

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of the urban population that is attended at their domiciles, direct or indirectly, by the regular systems of waste collection, at a certain place and in a particular period of time. Sanitary Facilities: measures the coverage of the sewage service, through a collection network or septic tanks. It is represented by the percentage of the population that has drainage for their “waste” through domestic connection to the sewage network or septic tank, in a particular place and in a certain period. Road Network: allows the selection of the municipal districts that have organized road networks with access to the industrial complexes, market places, services and leisure areas. An inadequate transport infrastructure is responsible for the associated logistical costs, hindering compliance with the population’s needs and preventing the establishment of new enterprises in these regions. Population from 7 to 14 years old that does not attend schools: this criterion aims at evaluation of the living conditions in childhood, particularly with regard to access to education at the fundamental level. School Evasion: aims to portray the problem of fundamental schooling in the municipal districts, enabling monitoring of the educational sector and supporting the Brazilian government in educational planning and administration. Education Index: this index attempts to demonstrate the population’s access to knowledge [6]. It is measured by the combination between the illiteracy rate and the weighted admission rate at the three levels (fundamental, high school and further education). Homicides: seeks to appraise the risk of death by homicide, and has the purpose of aiding the planning, administration and evaluation process of the security policies. Infant Mortality: estimates the risk of child death in the first year of life. It is expressed by the number of deaths of children under one year old per thousand live births, at a particular place and in a certain period of time. In general, high infant mortality rates mean that the population has low health, socio-economic development and living condition levels. Maternal Mortality: this refers to the number of female deaths due to maternal causes, expressed in relation to 100 thousand live births, in a certain place and in a particular period of time, reflecting the quality of the health care for women. Immunization Cover: is expressed by the percentage of vaccinated by a type of vaccine, according to place. In our paper, we chose vaccination for measles and the target population was children under one year old. We chose measles because it represents a routine procedure of immunization cover for a contagious disease. Thus, it is possible to monitor the immunization figures in the medical centres. Longevity Index: portrays the health conditions of the population and it is measured by the life expectancy (average number of years that a newborn would expect to live if he/she were exposed to a mortality profile – [6]).


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• Employment: percentage of the employed population in relation to the Economically Active Population (EAP), aimed at measuring the EAP employment level. By economically active population we mean the part of the population that was employed in the whole or part of the reference period. The EAP comprises people from 10 to 65 years old classified as employed or unemployed in the reference period. • Occupational Accidents: this is represented by the ratio between the number of benefits granted due to occupational accidents and the resident population in the age range 10 to 65. • Income Index: this is defined by the population’s purchasing power, based on the average family income of the municipal district, adjusted to the local cost of living, through the methodology known as Purchasing Power Parity (PPP), aimed at describing income levels and distribution [6]. 2.1.2. Defining the alternatives Besides the definition of the criteria, the problem structuring covers identification of the set of alternatives. Since we are employing a multi-objective method, the set of alternatives must be continuous and characterized by a set of constraints. However, the natural approach to this case study is with discrete alternatives (the municipal districts), represented by polygons in the GIS database. To convert these discrete alternatives into a continuous set, we can use the artifice of using the characteristics of these discrete alternatives to formulate the constraints that this set must respect. We can consider the solution space as a set of points in n (where n is the number of classification criteria), which are convex linear combination of the real municipalities’ attributes. Nevertheless, not all the municipal districts are considered valid alternatives (in accordance with the exclusion criteria). Thus, the convex linear combination is accomplished by using just the pre-selected municipalities’ attributes, as explained below. 2.2. Multi-criteria method implementation Each municipal district is seen as an alternative, represented by polygons of the GIS vector database. The problem constraints serve as narrowing factors of the number of alternatives. In other words, in possession of the constraints, it is possible to carry out a pre-selection of the alternatives, attaining the set of feasible alternatives. This stage is performed in a GIS environment. After obtaining this reduced set of alternatives, the multi-criteria analysis is accomplished, more specifically the solution of an optimization problem (multi-objective linear programming problem), which should indicate the ideal municipal district. Afterwards, the results of the choice of the decision-maker are visualized in GIS. 2.2.1. Pre-selection of the alternatives: Multi-attribute class allocation problematic The physical and/or qualitative constraints of our problem act as restricting factors on the number of alternatives. A set of feasible alternatives is produced, characterizing the multi-attribute class allocation problematic [37], in which each municipal district

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is allocated to one of the two classes, namely, acceptable for the subsequent analysis (feasible) or rejected a priori. This phase is conducted in GIS environment. This phase can be compared with the suitability analysis phase described by Jankowski [20] and Pereira and Duckstein [35], which aims at selecting feasible alternatives. One possible strategy to reduce feasible alternatives suggests satisfying technical or physical suitability factors. Another strategy suggests introducing threshold values that must be passed by an alternative in order to qualify. Our approach uses these two strategies together in order to pre-select the alternatives. The first condition a municipal district must fulfill to be selected is to be crossed by paved federal highways with good/regular conservation conditions, which is a basic requirement for trading expansion and commodities attainment. About 65% of Brazilian freight transportation is made by road modal [13]. This constraint is easily analysed in GIS, because it represents a physical constraint, typically a topological operation. Superimposing these two thematic layers, “municipal_districts” and “road_network”, a process known as overlay (process of comparing spatial features in two or more map layers [30], we select the alternatives that fully satisfy the pre-defined conditions. After overlaying these two thematic layers, we want to select the objects of the layer “municipal_districts” that are intercepted by objects of the layer “road_network”. Besides this, the descriptive attributes of the latter layer (type of highway, physical aspects, etc.) should fulfill the following conditions: “highway_type” = federal paved AND “physical_aspect” = good OR regular. This procedure combines typical GIS functions (overlay, spatial query and search), which, according to Maguire and Dangermong [30], undertake complex analysis. This first constraint reduces the set of alternatives from 69 to 40, the procedure for which may be visualized in figure 6. Apart from the constraint of the conservation aspect of the highways, the municipal districts should also meet the condition that the values of the Indexes of Education, Longevity and Income must be larger than the average values for the State, 0.652, 0.641 and 0.705, respectively. These indexes are the basic components of the Human Development Index (HDI) supplied by the UNDP. They refer, respectively, to the access to knowledge (as measured by the adult literacy rate and the combined primary, secondary and tertiary gross enrolment ratio), long healthy life (as measured by life expectancy at birth) and a decent standard of living (as measured by GDP per capita – PPP US$) [6]. The average values for the State are lower than the values computed for Brazil as a whole, 0.83 (education), 0.71 (longevity) and 0.71 (income), which by its turn does not have high values considering the international context [18]. So, it seems unacceptable to consider as candidates the municipalities that present indexes lower than the average for the Sate. This constraint presents a Boolean algebra equation: the municipal districts must meet the 1st AND 2nd AND 3rd criteria. Similar to the previous step, this one is easily visualized in GIS. This condition was extremely restrictive, and the query resulted in an extremely reduced set of alternatives. Making use of the hybrid approach already cited (reducing the strictness of the conjunctive method and not allowing the excessive

Figure 6. Thematic map of the physical aspect of the highways and Rio de Janeiro State municipalities selected by this exclusion criterion.

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permissiveness of the disjunctive method), this constraint was relaxed: the municipal district should simultaneously fulfill at least two of these conditions. The query conducted in GIS presents the following structure: to select the objects of the layer, “municipal_districts” that fulfill the condition expressed by [(Longevity 0.641 AND Education 0.652) OR (Income 0.705 AND Education 0.652) OR (Longevity 0.641 AND Income 0.705)], where the numbers represent the average values of these indexes for Rio de Janeiro State, acquired directly from [6], which has its own methodology to obtain these normalized values. The closer the value of the indicator is to 1, the greater is the human development level in the municipality or region in the considered dimension [6]. This search resulted in the selection of 34 of the 69 municipal districts, as may be visualized in figure 7. The following stage is to overlay the two layers of information generated, creating a third layer that contains the municipal districts that fulfill both constraints: to be crossed by paved federal highways, in a good or regular state of conservation, AND to present at least two of the constituent indicators of HDI greater than the average indexes for the State. The result of this overlay operation (displayed in figure 8) produced a set of 26 alternatives that, in the 2nd stage, are appraised by multi-criteria analysis. It can seem that we loose information in the next steps, not considering the municipalities excluded by the exclusion criteria. One should notice that these informations are used in different phases that have their usefulness in the global solution. The information about infrastructure (highways), health (longevity), education and work (income) are considered in Model (2) (section 2.2.3) through other variables. If the information used to pre-select some alternatives were considered in these posterior phases we would not respect the non-redundancy axiom [37], since we would consider criteria that would evaluate characteristics already evaluated by other criteria. This would be the case of the longevity index (that evaluates the life expectance), used to preselect some alternatives, and the infant mortality rate in Model (2). 2.2.2. Choice of multi-criteria evaluation method The Multi-criteria Decision Aid can be divided into multi-attribute and multi-objective problems. The former deal with discrete alternatives and the latter with a continuous space of alternatives. Among the multi-attribute problems there is commonly a classification of the methods used as either belonging to the American or to the French Schools of Decision Aid. The French School is based on outranking relations and we can mention the methods of the ELECTRE [36] and PROMETHEE [5] families. The American School reduces the multiple criteria to a synthesis criterion, in most cases applying a weighted sum. Due to their apparent mathematical simplicity, these methods have great popularity. From this class of methods, we can mention the AHP [38], MACBETH [2] and UTA [19] methods. In spite of its supposed simplicity, these methods meet some reluctance on the part of the decision-makers, particularly the need for weight assignment. The multi-objective problems are, as a rule, mathematically more difficult, although they demand the decision-maker’s constant presence. In a multi-objective context, the notion of optimal solution gives way to the concept of efficient or Pareto optimal

Figure 7. Result of the search for municipal districts that fulfill the condition imposed by the longevity, education and income indexes of HDI.

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Figure 8. Indication of the 26 municipal districts pre-selected for the multi-objective optimization.

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solution (it is a possible solution if and only if there is not one other solution that improves the value of one objective function without worsening the value of at least one other objective function). Due to the mathematical complexity of this approach, it is common to develop methods that restrict the number of alternatives, making it discrete, meaning that we are changing a multi-objective problem into a multi-attribute one [42]. Although we gain mathematical simplicity with this transformation, we lose interactivity and we oblige the decision-maker to explicitly state his/her preferences (and even he/she may not know what they are). If the analyst is not concerned about mathematical complexity but the impossibility of the decision-maker to supply coherent information, he/she must seek the interactivity of the multi-objective problems, particularly if they are allied to a software with a great visual appeal, which enables the decision-maker to implicitly express preferences and to learn throughout the process. It is also worth pointing out that the multi-objective approach enables global visualization of the feasible solutions space, as well as the efficient frontier (set of all efficient solutions), making it easier to understand the problem. If the initial problem has a multi-attribute structure, it is necessary to convert the alternatives’ space into a continuous set that contains it, so that this problem can be solved as a multi-objective one. There are many manners to carry out this action, and in this paper we favoured, for its simplicity, to consider that the alternatives’ space is the set of vectors whose co-ordinates are a convex linear combination of the original alternatives’ co-ordinates. So, the problem is solved as though it had a multi-objective structure, having as a result a solution that belongs to the new space generated, but with a great probability of not belonging to the original alternatives’ space. It is thus necessary to have one other phase that consists of choosing from among the real alternatives the one that most resembles this virtual alternative. Considering that the space produced is metric, it is enough to find which real alternative has the smallest distance to the virtual one. The metric can be any of the existing ones, the usual ones being the Euclidean, also called L2 norm, which has a compensatory characteristic (the low performance of an alternative in one criterion is compensated by a high performance of the same alternative in another criterion; see equation (3)), and the Tchebycheff metric, also called L∞ norm, non-compensatory. For each alternative, the Tchebycheff metric considers only the criterion that yields the greatest distance to the reference point, being indifferent to the other criteria, not allowing compensations. At present, there are many multi-objective methods, and we can mention the STEM method [4], the Zionts and Wallenius [44] procedure, the TRIMAP method [10], and the Pareto Race method [27,28]. We chose the Pareto Race multi-objective method, implemented in the VIG software [25]. The Pareto Race method and the VIG software are described later (section 2.2.4). 2.2.3. MOLP problem formulation The MOLP problem to be formulated presents 10 objective functions: to maximize % population with sanitary facilities (PSF); to minimize homicide rate (HRT); to minimize

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% population from 7 to 14 years old that do not attend schools (PNS); to minimize infant mortality rate (IMR); to maximize % population with regular domestic waste collection (PDW); to minimize the school evasion rate (SER); to maximize immunization cover for measles (ICM); to minimize the maternal mortality coefficient (MMC); to maximize the occupation rate (ORT); to minimize the occupational accident rate (OAR). The formulation of the problem is based on the Halme et al. [17] model, which incorporates the decision-maker preference information. It was originally conceived to incorporate the decision-maker’s preferences in the Decision-Making Units’ efficiency evaluation in Data Envelopment Analysis models. The decision-maker is introduced in the search to the best combination of efficient alternatives (understood as the decisionmaker’s best combination, or favourite). The general MOLP model for this case may be seen in (1), where Xi , Yi , . . . , represent the value of the criterion X, Y, . . . , for the alternative i; λi are the decision variables that represent the decision-maker’s preferences for the alternative i, i = 1, . . . , n. max max .. . s.t.

n i=1 n

Xi λi Y i λi (1)

i=1 n i=1

λi = 1,

λi 0,

i = 1, . . . , n.

In the literature [40,43], arguments are found against the use of importance weights for the criteria to deduce and to represent the decision-maker’s preference information. Many decision-makers are not able, neither technically nor psychologically, to assign a weight scale. Technically, they cannot understand what the weight assigned means [9], and psychologically, they can face ethical opposition. One example is the weight assignment to the criteria cost of construction and probability of fatal accidents when dealing with the design of a highway. Assigning these weights would mean specifying the worth of a human life [11]. On the other hand, there is the opposite question: once the weights are assigned, the decision-maker can feel apart from the decision process, presented with a final result in which he/she may judge that he/she has not taken part. Therefore, in order to bypass these difficulties, great interactivity is necessary, a characteristic of some of the MOLP methods. The Halme et al. [17] approach introduces the decision-maker’s preference in the efficiency analysis, by explicitly locating his most preferred solution vector on the efficient frontier. The same authors highlight that when systematically exploring the neighbourhoods of the Most Preferred Solution (MPS), one does not know explicitly the decision-maker’s value function, but its form becomes known when the end of the search for MPS is reached. MOLP interactive methods are the most appropriate in the


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MPS search, because they are interactive procedures and the decision-maker can learn through the process. For this case study, model (1) is employed in the search for the ideal municipal district, alternatively to its original conception, which was to support the Value Efficiency Analysis in DEA. The MOLP model to be optimized is that presented in (2), where PSF = % pop. with sanitary facilities; HRT = homicides rate; PNS = % pop. 7 to 14 years old not attending schools; IMR = infant mortality rate; PDW = % pop. with assessment of domestic waste; SER = rate of school evasion; ICM = immunization cover for measles; MMC = coefficient of maternal mortality; ORT = occupation rate; OAR = occupational accident rate; i = 1, . . . , n alternatives (municipal districts); λ vector representing the decision maker preferences.

max

26

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i=1

min min min max min max min max

26 i=1 26 i=1 26

HRT i λi PNSi λi IMRi λi

i=1 26 i=1 26

SERi λi

i=1 26 i=1 26

PDW i λi

(2)

ICM i λi

MMCi λi

i=1 26

ORT i λi

i=1

min s.t.

26

OARi λi

i=1 26 i=1

λi = 1,

λi 0,

i = 1, . . . , n.

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2.2.4. Solution to the MOLP problem In order to search for the MPS, MOLP interactive methods are the most appropriate, mainly due to their interactive feature and possibility of learning through the process. The VIG software was chosen, which implements the Pareto Race method. This choice was made based upon its interactivity, good graphical interface, permitting the use of a great number of objective functions. The ADBASE solver [39], for instance, does not enable free search of the efficient frontier, providing the decision-maker just the extreme efficient points; the TRIMAP software [10], although interactive, does not allow more than three objective functions, and, therefore, was not suitable for this case study. The Pareto Race is an interactive, visual, dynamic, search procedure for exploring the efficient frontier of a multiple objective linear programming program. In Pareto Race, the user sees the objective function values (called flexible goals) on a display in numeric form and as bar graphs, as he/she travels along the efficient frontier, which allows a dynamic and visual search [26]. This procedure allows the decision-maker to look freely at any part of the efficient frontier, controlling the speed and the direction of motion [25]. By pressing number keys corresponding to ordinal numbers of the objectives, the decision-maker expresses which objectives he/she would like to improve and how greatly. In other words, the decision-maker specifies which criterion he/she believes to be distant from the acceptable level and changes the direction of the search. In this way, he/she implicitly specifies a reference direction [17]. The theoretical basis of Pareto Race is in the reference direction approach to MOLP, developed by Korhonen and Laakso [27]. In this approach, any direction specified by the decision-maker is projected onto the efficient frontier. Using a reference direction, a subset of efficient solutions (an efficient curve) is generated and presented for the decision-maker’s evaluation. The interface is based on a graphical representation. One picture is produced for each interaction. Pareto Race improves the former procedure (reference direction approach) making it dynamic [26]. The decision-maker can move in any direction (on the efficient frontier) he likes, and no specific assumptions concerning his/her underlying utility function are needed during the search process. The only assumption made about the decision-maker’s value function is that it is pseudo-concave at the moment when the search for the MPS is terminated [17]. This is implied by pseudo-concavity with respect to the objective function, provided all objective functions are concave and differentiable. As stated by Korhonen and Laakso [27], the visual representation gives the decision-maker a holistic perception of changes in objective function values as he/she moves to a given direction on the efficient frontier. The reference direction is built from the specification of the aspiration levels for each objective function (example, table 1). Projected over the efficient solutions set, this direction produces a path over the efficient frontier, and this is presented to the decision-maker. The search ends when the decision-maker believes that the values of the objective functions are his/her most preferred values, that is, his/her MPS (example, table 2). This MOLP model’s solution supplies the best or “ideal” alternative that presents the objective functions’ ideal values. This “ideal” alternative is the linear combination


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Table 1 Multi-objective optimization initial result, without conducting the Pareto Race. Criteria

Initial goalsa

Solutions obtained

%_with_sanit_facil rt_homicides %_not_school infant_mort_rt %_with_waste_collect rt_sch_evasion measles_immun rt_mater_mort occup_rt rt_occup_accid

85 25 15 20 80 50 100 100 65 50

97.9956 28.6478 14.8884 22.9182 75.5992 5.7296 97.4708 114.591 55.5158 5.7296

Decision variables different from zerob λ4 λ24 λ27 λ44 λ50 λ64 λ66

0.1949 0.0311 10−5 0.1580 0.2865 0.1230 0.2064

a Supplied by the decision-maker, just serve as delimitation for the Pareto Race search

area. b λ , λ , etc. refer to the values of the λ decision variables; the number refers to the 1 2 i

alphabetical position of the municipal district in the original database.

of the other alternatives. The value of each λi can be interpreted as the contribution of each alternative i to the composition of the ideal alternative. In the case in which the optimization result is λi = 1, and all the others λj = 0, j = i, the ideal municipal district is thus represented exactly by the alternative i. 2.3. Results Tables 1 and 2 show some results. In table 1 we found the results obtained by the software VIG before the Pareto Race. This is the “optimum” (default) solution given by the program, without the intervention of the decision-maker. Table 2 displays some results obtained after the Pareto Race (#I#, #II#, #III#), i.e., they are some of the decisionmaker’s MPS. In these cases, there is interference on the part of the decision-maker in the search for the best decision, which is guided in the direction of his preference. These races were obtained, in most cases, with the intention of improving the values of the critical objectives, the ones that prevented the decision-maker’s reference direction from changing according to his/her convenience. The municipal districts that have λ different from zero for all the solutions found are: Angra dos Reis, Arraial do Cabo, Barra do Piraí, Itaperuna, Itatiaia, Macaé, Mangaratiba, Petrópolis, Piraí, Rio de Janeiro, Teresópolis, Três Rios, Vassouras and Volta Redonda. The Pareto Race solutions are efficient and feasible, resulting from an optimization procedure. The choice of one of the solutions reflects the decision-maker’s preference for a particular configuration of values, to the detriment of others. However, the plain choice of one of the solutions does not dictate the final solution, given that the main

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Table 2 Some results after the Pareto Race. Criteria

Initial goals

%_with_sanit_facil rt_homicides %_not_school infant_mort_rt %_with_waste_collect rt_sch_evasion measles_immun rt_mater_mort occup_rt rt_occup_accid

85 25 15 20 80 50 100 100 65 50

Solutions obtained #I#

#II#

#III#

97.9559 33.4201 14.539 23.0988 81.1036 6.7136 119.4069 80.5677 61.6684 6.7136

96.9804 40.0747 14.9869 22.819 84.9479 8.0284 135.045 67.6736 58.1019 4.7309

98.2774 30.4195 14.9469 213.9808 79.0655 6.1134 110.4016 51.6402 63.3737 6.1134

Decision variables different from zero #I#

#II#

#III#

λ1 = 0.2137 λ3 = λ24 = λ25 = λ27 = λ44 = λ68 = 10−5 λ50 = 0.3871 λ64 = 0.0549 λ66 = 0.3443

λ1 = 0.5264 λ3 = λ24 = λ25 = λ27 = λ64 = 10−5 λ50 = 0.4082 λ66 = 0.0654

λ1 = 0.0178 λ3 = λ24 = λ25 = λ27 = λ43 = λ44 = λ68 = 10−5 λ50 = 0.3521 λ64 = 0.3823 λ66 = 0.2478

objective, the choice of the best municipal district of Rio de Janeiro State, in terms of the quality of urban life, was not achieved. To carry on the study, solution #III# (table 2) was chosen as the decision-maker’s Most Preferred Solution. From this solution, it can be interpreted that, for instance, municipal district 64 contributes 38.23% to the construction of the “ideal” municipal district. But is this the best municipal district in Rio de Janeiro State? To answer this question, it is necessary to have one other phase, in which we used the smallest distance method [3], which consists of choosing from among the feasible alternatives the one that most resembles the virtual alternative (MPS). Thus, we search the real alternative whose distance to the MPS is the smallest and the subsequent comparison of these distances, in a multi-variate space, where each criterion is an axis of the space formed, because none of the alternatives presented λi = 1. The best alternative has the best level of the attributes of the alternatives in each criterion, and in this case it is the vector formed by the elements of the column Solutions obtained in table 2, #III#. The smaller the distance, the closer to the ideal the alternative is. In case (improbable) the best alternative is a real one, the smallest distance value is zero, whatever the metric used. To represent the deviation of each alternative from the ideal point, the Euclidean distance was chosen, whose mathematical expression is (3), where aik is the normalized


263

value of the alternative i in the objective k and ak∗ is the normalized ideal value in the objective k. (4) is the normalization equation.

L2k

n aik − a ∗ 2 = k

1/2 ,

(3)

xik − Minj xj k . Maxj xj k − Minj xj k

(4)

k=1

aik =

The result of the deviation estimate is in table 3. This result is then exported to the GIS software for visualization, as displayed in figure 9. One point should be addressed: in equation (4), the addition of a district s with very low or high value of xsk modifies the results. This is a consequence of Arrow’s Theorem [1] that assures that there is not a multi-criteria method that satisfies simultaneously to the conditions of universality, Pareto’s unanimity, transitivity, totality, relevance of all criteria and independence in relation to irrelevant alternatives. In this case, the latter condition could not held. Table 3 Municipal districts in order of degree of proximity to the ideal point. Municipal district

Code

Petrópolis Teresópolis Três Rios Barra Mansa Macaé Rio de Janeiro Resende Volta Redonda Niterói Angra dos Reis Barra do Piraí Paraíba do Sul Cabo Frio Vassouras Itaperuna Casimiro de Abreu Campos dos Goytacazes Itaguaí São Gonçalo Duque de Caxias Mangaratiba Piraí Itatiaia Parati Arraial do Cabo Magé

i i i i i i i i i i i i i i i i i i i i i i i i i i

= 43 = 64 = 66 =5 = 27 = 50 = 46 = 69 = 36 =1 =4 = 40 =8 = 68 = 24 = 14 = 11 = 21 = 54 = 18 = 29 = 44 = 25 = 41 =3 = 28

Euclidean distance 0.3532 0.3758 0.4582 0.5635 0.6170 0.6375 0.6516 0.6538 0.6768 0.7268 0.7387 0.7565 0.7664 0.8776 0.8819 0.8989 0.9882 1.0342 1.0545 1.0738 1.0997 1.1076 1.1431 1.1807 1.2511 1.3509

Figure 9. Final classification of the municipal districts in relation to the ideal of quality of urban life (the values in the legend refer to the distances to the ideal point).

264 GOMES AND LINS


265

According to this classification (table 3), the best municipal district of Rio de Janeiro State, in terms of the quality of urban life, is Petrópolis (followed directly by Teresópolis). 2.4. Results analysis In the course of the Pareto Race we noticed that some optimization criteria (objective functions) restricted the analysis, that is, they prevented the reference direction specified by the decision-maker from being projected onto the efficient frontier. These critical criteria were the homicide rate, the coefficient of maternal mortality and the immunization cover for measles. When trying to minimize the first two and to maximize the third (seeking improvement of these objectives), other criteria were strongly influenced by the choice of this direction. For instance, when trying to minimize the homicides rate simultaneously with the immunization coverage maximization, the criteria % of population with sanitation facilities and the school evasion rate moved rapidly to a direction opposed to that desired by the decision-maker, obtaining extreme values in a negative path for these two former criteria, in the order of 10−6 and 104 , respectively. A criterion that showed little influence by any adopted direction was the rate of occupational accidents. Result #III# of table 2 was not the only MPS to be analysed; another 6 results were studied and were attained in the manner of the proceedings, terminating the search when the decision-maker believed that that solution portrayed the MPS, that is, the solution had the values of the objective functions that were in agreement with his/her preferences. A second result (also an MPS) considered acceptable was analysed so that the disposition of the alternatives would be checked. Comparing these two MPS’s, it was noticed that the first six municipal districts on the list remained in the same order, changing the hierarchy as of the 7th alternative. The hierarchy of other solutions was further analysed, including the case without the Pareto Race. Three of these cases presented some differences in the hierarchy of the alternatives. However, the first 4 alternatives remained unaltered, the alternative Petrópolis standing out as the alternative in first place in the hierarchy. We verified that in one other case (5th), the criteria homicide rate and immunization coverage for measles, were the ones with the highest distortion when compared to the values of the solution #III# (the first negatively and the second positively). This difference may be responsible for a great modification in the hierarchy. In the 6th case analysed, the criteria homicide rate and infant mortality rate, presented, respectively, values 44% and 34% higher than the preferred ones, in direction of raising values that should be minimized. This could significantly contribute to changing the disposition of the alternatives, favouring alternatives with high values in these criteria. Table 3 shows the municipal district, Petrópolis in 1st place in the hierarchy, followed by the municipal district, Teresópolis. The municipal district of Niterói is in 9th place in this hierarchy. This municipal district was considered by UNDP (making use of the HDI classification), as the 4th Brazilian City in terms of the quality of life. This classification uses the longevity, education and income indexes, in the con-

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GOMES AND LINS

struction of HDI. The hierarchy built in this case study, uses criteria that go beyond those used in the HDI (such as the homicide rate and the occupation rate), which can be considered responsible for this absence of coincidence between these two methodologies. In relation to the estimation of deviations from the best solution using the Euclidean distance, the compensatory character of this methodology should be stressed, denoting that the low performance of an alternative in one criterion is compensated by a high performance of the same alternative in another criterion. For the purpose of comparison, we also used the Tchebycheff metric to calculate the deviations. The results proved to be quite similar to the former, keeping the three first municipal districts. 3.

Conclusions

When dealing with complex spatial problems, decisions made inside a spatial context prevent decision-makers from losing the perspective of the set of criteria and concentrating on a particular criterion [15]. The results analysis in a spatial context increases the quality of the decision. It can be affirmed that in an integrated GIS–Multi-criteria system [14]: • GIS help to clarify the decision process, providing structure to a non-structured decision process; • GIS make it possible to take into consideration a larger range of alternatives, eliminate those alternatives that are not feasible a priori, and offer the opportunity to include new subjects; • Possibilities of discussion and changes in the decision criteria; • Possibilities to explore conflictive decision criteria and the incorporation of methods for the solution of these spatial conflicts. Although the integrated GIS–Multi-criteria system represents an advance in spatial decision analysis, there is a clear need for continued research in this area. Topics for new research include the development of new methods for the generation of alternatives in GIS (the development of such methods would increase the usefulness of the system), and continued research in the so-called tight coupling [20] of these two tools. The integration of multi-criteria within the tight coupling strategy would extend the usefulness of GIS as support for spatial decision. The system generated by this perfect integration GIS– MCDA can be inserted in the context of the Spatial Decision Support Systems (SDSS) [27], which are designed to provide the user with a decision-making environment that allows analysis of the geographical information to be handled in a flexible way, making it possible to analyse conflicts in a spatial context. The first stage of the GIS–MCDA integration, the preliminary study of the alternatives, which leads to the reduction of the set of feasible alternatives, is an important initial stage, bringing reflections in the following stages, mainly in the reduction of the


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computational effort. Besides this, some existing multi-criteria software have constraints regarding the number of criteria and alternatives that can be used. When there exist physical and/or qualitative constraints, which can be implemented in GIS, the integration is shown to be quite effective. The case study presented and the problem proposed do not fit into the traditional multi-objective linear programming models. In this study, we wish to select the municipal district with the best quality of urban life. For such purpose, criteria were used that do not exhibit explicit constraints. The multi-objective model of incorporating the decision-maker’s preferences was shown to be viable and appropriate to the structure of the proposed case study, which dealt with the quality of public services. The use of MOLP interactive methods, like the Pareto Race that makes it possible to search for solutions on the efficient frontier that are in agreement with the objectives of the decision-maker, fits into the concept that the most correct decision is that which best represents the interests of the decision-makers. Comparing the results achieved with the Euclidean and the Tchebycheff metric, we verify that the three best municipal districts (considering the decision-maker preference information) belong to the so-called Serrana (“pertaining to the mountain”) Region of Rio de Janeiro State, showing that there is a quality of life breakdown in the big urban centres. On the other hand, our model just considers non-subjective indicators, that is, does not consider subjective preferences. Thus, someone wanting to live by the sea will not agree with the results of our analysis, and must include another exclusion criteria to be analysed in GIS (for instance, select the municipal districts that are in the boundary of the sea or which are X km away from it). The case study presented in this paper proposes a methodology that can be adjusted to support public decision-making. We surveyed the existence of the variables that could represent the quality of urban life (particularly the variables that are related to the quality of the public services) in the Brazilian agencies of information, and how to use them to subsidize the decision-making process. A former study [16] raised a considerable quantity of these variables, but the authors were not worried about whether these variables really exist, if they are periodically collected, if they are reliable, and which methodology would be proper to evaluate the quality of urban life. As for the selected criteria, we must stress that an urban society where the essential needs are almost fully satisfied, certainly criteria, such as leisure options, climate, ease of movement, and architectural/urbanistic factors, must be included to evaluate the quality of urban life. However, for an urban society where the essential needs are not already fulfilled, criteria, such as illiteracy, infant mortality and others, are of extreme importance. Certainly, subjects, such as security and road networks would be in common with both circumstances. An interesting development is the selection of the worst municipal district in terms of the quality of urban life. This situation can be compared to the decisions that should be taken by municipal planners when choosing areas for investment, with the objective of investing in precarious areas. Figure 4 displays a clear example: if regular or good conditions of road network were offered to the areas not shaded on the map, these municipal districts would be candidates (feasible alternatives) to the choice in the following

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stages, placing them in the same conditions as the other municipal districts selected by this constraint. In this case, the GIS–Multi-criteria integration is an appropriate tool for public decision-making, especially in municipal planning.

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