KARYA TULIS
JOURNAL REPORT
ON APLICATION OF MULTI OBJECTIVE LINEAR PROGRAMMING
(MOLP)
INTEGRATING GEOGRAPHICAL INFORMATION SYSTEMS AND
MULTI-CRITERIA METHODS:
A CASE STUDY
BY:
RAHMAWATY
DEPARTEMEN KEHUTANAN
FAKULTAS PERTANIAN
KATA PENGANTAR
Puji syukur penulis panjatkan kepada Tuhan Yang Maha Esa, yang telah
memberikan segala rahmat dan karunia-Nya sehingga KARYA TULIS ini dapat
diselesaikan. Judul yang dipilih adalah
“
INTEGRATING GEOGRAPHICAL INFORMATIONSYSTEMS AND MULTI-CRITERIA METHODS: A CASE STUDY”. Tulisan ini merupakan suatu
kritik mengenai JOURNAL REPORT ON APLICATION OF MULTI OBJECTIVE LINEAR
PROGRAMMING (MOLP), Penulisnya adalah Eliane Goncalves Gomes, Marcos Pereira Estellita Lins.
dan bersumber dari Journal Annals of Operations Research
:Oct 2002.Vol.116, Iss. 1; pg. 243.
Kami menyadari bahwa karya tulis ini masih jauh dari sempurna, oleh karena itu
kami mengharapkan saran dan kritik yang bersifat membangun untuk lebih
menyempurnakan karya tulis ini. Akhir kata kami ucapkan semoga karya tulis ini dapat
bermanfaat.
Medan, April 2010
DAFTAR ISI
I.
Title of the Study
1
II. Introduction
1
A. Background
2
B. Objectives of the Study
III. Multi Criteria Decision Analysis ( MCDA) and Multi Objective
Linear Programming (MOLP)
2
IV. GIS-Multi-criteria Integration
4
V. Case Study
4
A. Problem Structuring
4
B. Multi-Criteria Method Implementation
6
C. Results
11
D. Results Analysis
12
VI. Conclusion
13
VII. General Comments on the Paper
14
A Critique on
INTEGRATING GEOGRAPHICAL INFORMATION SYSTEMS (GIS)
AND MULTI-CRITERIA METHODS: A CASE STUDY
Author:Eliane Goncalves Gomes, Marcos Pereira Estellita Lins.
Journal: Annals of Operations Research: Oct 2002.Vol.116, Iss. 1; pg. 243
By: Rahmawaty
I. Title of the Study
As a researcher, the title provided by the authors has the element of simplicity, brevity, specificity and location and subject matter focused. The reader can easily determine what the study is all about and what it tries to investigate (Integrating GIS and multi-criteria methods: A case study) and what mathematical model? A Multi-Criteria Decision Analysis (MCDA)/multi objective Linear Programming (MOLP). Brief title but very informative. To more make informative, it would be better if the author mention the location of the study on the title, so the reader know the location of the study.
Just going through the title, one can easily understand that the concern of the research study is related to subject optimization model in forestry (FRM 294), because using the keywords such as “multi-criteria methods” clearly indicate that the subject matter is related.
II. Introduction
A. Background of the Study:
The authors looked into the 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, using Integrating GIS and multi-criteria methods.
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 (Gomes and Lins, 1999). Thus, the integration GIS-MCDA has the objective of favoring decision-makers, providing them with ways to evaluate several alternatives, based on multiple, conflictive criteria.
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.
B. Objectives of the Study:
The objective of this paper is (to present a hypothetical example of the integration and to show how the integration between GIS and the multi-criteria methods can support spatial decisions.
The authors 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. 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.
III. Multi-Criteria Decision Analysis (MCDA) and Multiple Objective Linear Programming (MOLP) Models
A. Multi-Criteria Decision Analysis (MCDA)
According to Gomes and Lins (2002), Decision-making can be defined as the process of choices among alternatives. Multi-Criteria 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.
Multi-Criteria Decision Analysis (MCDA), or Multi Criteria Decision Making (MCDM), is a procedure aimed at supporting decision maker(s) whose problem involves numerous and conflicting evaluations.
MCDA aims at highlighting these conflicts and deriving a way to come to a compromise in a transparent process. For example, European Parliament may apply MCDA to arrive on a number of conclusions on whether introducing software patents in Europe would help or destroy European software industry.
Since MCDA involves a certain element of subjectiveness, morals and ethics of the researcher implementing MCDA plays a significant part on accuracy and fairness of MCDA's conclusion. The ethical point is very important when one is making a decision that seriously impact on other people as opposed to a personal decision.
Some of the MCDA models are: Analytic Hierarchy Process (AHP), Multi-Attribute Global Inference of Quality (MAGIQ), Goal Programming, ELECTRE (Outranking), and Data Envelopment Analysis. Which model is most appropriate depends on the problem at hand and may be to some extent which model decision maker is most comfortable with (http://en.wikipedia.org/wiki/Multi-criteria_decision_analysis).
B. Multiple Objectives Linear Programming (MOLP) Models
According to Briassoulis (2007),there are two main groups of LP models, the single and the multiple objective (or, multiobjectives). The first deal with problems in which there is one objective to optimize and the second address the more realistic situation of finding solutions which satisfy more than one objective. In both cases, the structure of the optimization problem includes one or more (in the case of multiple objectives) objective functions and a set of constraints.
maximization of population income, minimization of the cost of development (or maximization of the benefits of development), etc.
The constraints which can be taken into account depend on the case but representative objectives include: lower and upper limits on land use (reflecting, for example, zoning or natural constraints such as land suitability), other constraints on development, availability of labour, and so on.
According to Briassoulis (2007), multiple objective linear programming models (MOLP)
address the question of land use solutions which meet more than one objective. Of particular importance in this context are environmental objectives and constraints. The role of environmental factors in determining the optimal allocation of land uses in a region has always been of high importance in the context of planning in agricultural regions.
In addition, the need for detailed information on spatial data as well as for the spatial representation of the optimal land configurations always figured high on the researchers’ wish lists.
Progress on and diffusion of GIS techniques and technology since the 1980s mostly has made possible the use of information of better spatial detail and specificity. Linear programming models for agricultural regions appeared which are sensitive to the distribution of environmental conditions in the study areas and which are linked to GIS to provide for mappings of the optimal solutions produced by the models.
According to Briassoulis (2007), the objective function of the (multiple objective) LP model seeks to minimize the cost of meeting these demands and includes two components: (a) the cost of local production and (b) the cost of imports to complement local production to meet local demand. The assumption is that the economic costs of production determine whether local demand will be met by local production or by imports subject, among others, to the natural resources constraints facing the study region.
The MOLP problem can be generalized as follows (Sakawa, 1993 cited by Tomas, 2006):
Minimize k linear objective functions Z1(x)=c1x
Z2(x)=c2x .
. .
Zk(x)=ckx
Subject to the linear inequality constrains
AX b,
And the non-negativity conditions
X 0
Where : z1,…,zk are the measures of effectiveness (profits or costs).
i = 1,…,k
Ci = (ci1,…,cin) represent the cost/profit coefficients. X = (x1,…,xn)T refer to the decision variables. A = a11, …, a1n
. . .
Am1,…,amn
B = (b1,…,bm)T are the available resources.
If the notion of optimality for single-objective LP is directly applied to this MOLP, we arrive at the following notion of a complete optimal solution.
IV. GIS-Multi-criteria integration
There are two methods proposed by Jankowski (1995), 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" - "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 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 (1994), 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 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, the authors 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 (Korhonen and Laakso, 1986; Korhonen and Wallenius, 1988), implemented by VIG software (Korhonen, 1987). The authors 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" (Graeml and Erdmann, 1998).
V. Case study
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.
A. Problem structuring
1. Defining the criteria
The decision criteria and their role in problem structuring displays in Figure 5. 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. (Gomes, 1999).
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.
The authors 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 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 (UNDP, 1998). 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).
• 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 centers.
• Longevity Index: portrays the health conditions of the population and it are 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 - (UNDP, 1998).
• 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 (UNDP, 1998).
2. Defining the alternatives
According to the authors, 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 mn (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.
B. 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.
problem), which should indicate the ideal municipal district. Afterwards, the results of the choice of the decision-maker are visualized in GIS.
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 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 constraint is easily analyzed 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 (Manguire and Dangermond, 1991), 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 Dangermond (1991), 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 (see appendix).
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$) (UNDP, 1998). 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 (Human Development Report, 2001). 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 permissiveness of the disjunctive method), this constraint was relaxed: the municipal district should simultaneously fulfill at least two of these conditions.
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(see original paper)) 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 in formations 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. 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. 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 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 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). According the authors, 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 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 favored, 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.
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 % 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. (1999) 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 decision maker’s best combination, or favorite). 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
∑
= n i
i
Xi
1λ
Max
∑
= n i
i
Yi
1λ
. . . s.t..
,...,
1
,
0
1
1n
i
i
i
n i=
≥
=
∑
=λ
λ
(1)
The Halme et al. (1999) 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 neighborhoods 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 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 1 i
i
PSFi
λ
Min
∑
Min
∑
= 26 1 ii
PNSi
λ
Min
∑
= 26 1 i
i
IMRi
λ
Max
∑
= 26 1 i
i
PDWi
λ
Min
∑
= 26 1 i
i
SERi
λ
Max
∑
= 26 1 i
i
ICMi
λ
Min
∑
= 26 1 i
i
MMCi
λ
Max
∑
= 26 1 i
i
ORTi
λ
Min
∑
= 26 1 i
i
OARi
λ
s.t..
,...,
1
,
0
1
26 1n
i
i
i
i=
≥
=
∑
=λ
λ
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 theoretical basis of Pareto Race is in the reference direction approach to MOLP, developed by Korhonen and Laakso (1986). 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. 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.
Projected over the efficient solutions set, this direction produces a path over the efficient frontier, and this is presented to the 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. some results after the Pareto Race, see appendix)).
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 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.
C. Results
According to the authors, 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 decision maker’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 Pirai, Itaperuna, Itatiaia, Macae, Mangaratiba, Petropolis, Pirai, Rio de Janeiro, Teresopolis, Tres 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 objective, the choice of the best municipal district of Rio de Janeiro State, in terms of the quality of urban life, was not achieved.
[image:14.612.201.388.585.695.2]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 value of the alternative i in the objective k and a*k is the normalized ideal value in the objective k. (4) is the normalization equation.
L2k =
∑
=
−
n
k
ak
aik
1
2
*
Aik =
MinjXjk
MaxjXjk
MinjXjk
Xik
−
−
1/2
(3)
The result of the deviation estimate is in Table 3 (see appendix). This result is then exported to the GIS software for visualization, as displayed in Figure 9 (see appendix). 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 (Arrow and Raynaud, 1986) 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 hold.
According to this classification (Table 3, see appendix), the best municipal district of Rio de Janeiro State, in terms of the quality of urban life, is Petropolis (followed directly by Teresopolis).
D. 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 analyzed; 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 analyzed 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 analyzed, 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 Petropolis 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 analyzed, 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, favoring alternatives with high values in these criteria.
classification), as the 4th Brazilian City in terms of the quality of life. This classification uses the longevity, education and income indexes, in the construction 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.
VI. Conclusions
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 computational effort. Besides this, some existing multi-criteria software has 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 does not fit into the traditional multi-objective linear programming models. In this study, the authoe 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, the authors 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 centers. 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 other exclusion criteria to be analyzed 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. The authors 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.
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.
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.
VII. General Comments on the Paper
On the whole, this paper is very good, because of giving information to us related to the subject optimization model in forestry (FRM 294) about the Integrating GIS and multi-criteria methods: A case study.
More appropriate using the Integrating GIS and multi-criteria methods approach than other mathematical model (like: integer programming, dynamic programming because this case has more than one objective and according to Burrough (1986), 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. According to Gomes (1999), an integrated GIS-Multi-criteria system, it can be affirmed that :
• 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.
This paper used a model, so this study more effective because no need go to the field to all the measurement. Using the Integrating GIS and multi-criteria methods approach, case study in Rio de Janeiro State, Brazil, in relation to the quality of urban life can be analyzed and assessed.
This paper appropriate using the integration of multi-criteria within the tight coupling strategy would extend the usefulness of GIS as support for spatial decision. According to Korhonen and Laakso(1986), the system generated by this perfect integration GIS- MCDA can be inserted in the context of the Spatial Decision Support Systems (SDSS) 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 analyze conflicts in a spatial context.
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