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Application of multicriterional analysis on the selection of the location for disposal of communal waste
European Journal of Operational Research 55 (1991) 211-217 North-Holland
211
Theory and Methodology
Application of multicriterional analysis on the selection of the location for disposal of communal waste * Drago Vuk, Bogomir Ko~elj Highschool for the Organization Science, Kranj, University of Maribor, Pregernova ll, 64000 Kranj, Yugoslavia
Nenad Mladineo GI. Faculty of Civil Engineering, University of Split, F. Maslege bb, 58000 Split, Yugoslavia Received September 1989; revised February 1990
Abstract: This paper presents the application of the PROMETHEEmethods and geometrical representation of multicriterional analysis (GAIA) o n an actual example of the selection of the location for the disposal of communal waste in the region Gorenjska in Slovenia. In addition to the model presentation the paper deals with the application of the method in the function of decision making on investments which is necessary to make the actual existing locations environmentally acceptable.
Keywords: Location, ecology, multiple criteria
1. Introduction
The improvement of the living standard is the result of the development of science and industry in general; however this has led to the presence of some accompanying phenomena which endanger the human environment, i.e. the eco system. It is a well-known fact that life and waste cannot be separated and that the quantities and type of waste are directly related to the standard of living and the development of industry. The starting point of our considerations is the structural and functional nature of the eco system. * In this paper the authors used material from the Ph.D. Thesis of D. Vuk.
If the eco system is considered as the sum of the influences of all the system elements, then both the influence exerted by the elements on the system and of the system on each element has to be taken into account. In the analyzed case, the disposal of communal waste was taken as the element of the eco system, since previous analyses have shown that the given structure and quantity of waste in one region is the optimal solution for programming the regional disposals. Consequently, as the influences on the system element are different and dependent on many factors, containing both the environmental and economic components, the PROMETHEEmethod was chosen for the decision making support to the optimal disposal locations.
0377-2217/91/$03.50 © 1991 - Elsevier Science Publishers B.V. All rights reserved
212
D. Vuk et al. / Application of multicriterional analysis
Although it is a well-known fact that the influences of the system on the element and vice versa are vector valued quantities which characterize those relations, the ordinal scale was used in our evaluations. All the values for that scale were obtained empirically and therefore the factors were given empirically. Since there are many factors, the accuracy for the multicriterional analysis, i.e. for the PRO~THEE method is considered to be satisfactory. There is another possibility for re-calculating each factor separately. In that case this example could be treated deterministically and thus the decisions could be made. Already in simple evaluations of the functionsstability, costs, environmental pollution, reliability and the stability, a minimum number of coordinates was established i.e. between 6 and 7 coordinates for defining the ecological examples of the system. The conditions were considered as 'loads' and they represent the product between the value of the factor and the importance of the problem. Thus it is possible to define a great number of alternatives for different conditions (financing, personnel, etc.). According to the presented facts, the model was developed for the selection of the optimal disposal location, and it will be briefly presented.
2. Definition of the model criteria In order to present each criterion efficiently by mathematical operations and tables, each criterion was given the algebraic notation ' k ' and two indices which designate the group and the current number of criteria. The criteria were classified into the following two groups: - the first group is represented by four location criteria and - the second group is represented by eleven technological-economic criteria. The criteria are presented in this order k~ = dispersion in space with regard to the central location (C1); h a = safety of the location (C2); k~= size of the location, exploitation time (Ca); k~= distance of the location from the settlement (64); time of the waste mineralization (neutralization) (65);
= specific land use (function of the location)
(66); = possibility for treating the waste from the city treatment plants (67); = energy budget (68); = usefulness of waste (exploitation of wastes)
(69); quantity of waste (610);
/672=influence of the environment (CH); k~= influence on the groundwater and
surface water (612); k92= influence on the atmosphere (613); k~0= investment costs (C~4); k?l= operation costs of the disposal plant, collection and transportation of communal wastes
(615). The given criteria represent a multi-parameter frame for estimating the potential location based on the data available in the phase of investment planning. For some criteria, it was possible to establish the exact parameter values (e.g. investment costs, waste quantity etc); however, for a great number of criteria, the expert estimation was given according to the attribute (descriptive) evaluation. Since all criteria do not have the same importance for the final location selection, the weight evaluation for each criterion also influences the result. The special advantage of the PROMETHEE method, i.e. of the software package PROMCALC is that the output includes 'weight stability intervals', so that it is possible to check the real influence of the given weights on the final solution. 3. Multicriteria analysis using the PROMETHEE methods We briefly outline in this section the PROMETHEE I and II methods which are used to rank the alternative locations. A complete description of these methods and their extensions is given in [1]. Let us call A the set of all the possible locations we want to rank, and ]'1, f2 . . . . . fk the k criteria that have been selected. The PROMETHEE methods first build a valued outranking relation on A. This relation is then used to obtain a partial (PROMETHEE I) or a complete preorder (PROMETHEEII) on ,,4. More general interval orders can also be obtained, using the PROMETHEE II) algorithm described in [1].
D. Vuk et al. / Application of rnulticriterional analysis
3.1. PROMETHEEvalued outranking relation
213
giving the 'outranked character' of a; - a n d its net flow.
- Unicriterion preference function For each criterion fg and for every two locations a, b ~ A the preference of the decision-maker for location a over location b is expressed through a preference function Pj such that 0
Va, b ~ A .
b)~
pj(a, b) may be considered as the intensity of preference of a over b, it is equal to 0 in case of no preference of a over b or indifference between a and b, it is equal to 1 in case of strict p r e f e r e n c e of a over b, for the particular criterion ~. In practice, Pj(a, b) is often a function of d =~(a) - ~(b): Pj(a, b) = Pj(d). Pj is a non-decreasing function of d and is equal to 0 for negative values of d. Six useful shapes have been employed (see [1]). - Multicriteria preference index Let us suppose that a preference function Pj and a weight wj have been specified for each criterion ~ ( j = 1 . . . . . k). The multicriteria preference i n d e x / 7 is defined as k
~., w, Pg( a, b) H(a, b ) = j=l
k
, Va, b ~ A .
Ewj j=l lI(a, b) represents the intensity of preference of the decision-maker of a over b, when considering simultaneously all the criteria. This index defines a valued outranking relation on A. 3.2. Flows and rankings PROMETHEE defines for each a ~ A.
-its leaving flow. ++(a)= E n ( a , b), b~A
giving the 'outranking character' of a;
-its entering flow. +-(a)=
E II(b, a), b~A
i f ( a ) = q~+ ( a ) - 4~-(a). The higher the leaving flow the lower the entering flow, the better the action. The net flow is given as a complete preorder defined as: - a outranks b iff dp(a)> th(b), - a is indifferent to b iff O ( a ) = ¢(b). Every outranking method involves the determination of some parameters and weights. It is interesting to know the influence they have on the rankings when small deviations in their values are introduced. As the decision-maker usually cannot fix correctly their exact values, this stability problem is of major importance. It seems a necessary condition for a good outranking method. This was one of the principal purposes of PROMETHEE.
4. D e s c r i p t i o n
and
numerical
treatment
of
the
problem
The evaluation of the model for the optimal locations selection was carried out according to the actual demands for establishing the suitability of the locations in the region Gorenjska, Kranj, Yugoslavia. This region has ca. 500000 inhabitants with the annual gross social (national) income of $4000. The preliminary analysis served to identify the following 5 locations: - Teneti~a (A1); - Kovor (A 2)' - Me~aklja (A3); - Crnivec (An) ; - Draga (As); for which according to the defined criteria the data were gathered as presented in Table 1. The table also gives data on the defined types of criteria using the respective parameters and weights (see PROMETHEEmethod [1,2]). A superficial input data analysis can show two facts. First, the data are given in 'original' values units. It is not necessary to reduce to a common unit which is often wrong. Then, one part of criteria can be described as qualitative which is exceptionally favourable in preliminary investments project processing certain parameters can
D. Vuk et al. / Application of multicriterional analysis
214 Table 1 Criterion
Problem
Locations
type
A1
A2
A3
A4
A5
C1 C2 C3 Ca C5 C6 C7 C8 C9 Cto C11
min max max min min min max min max min min
3 3 10.25 7 50 1.5 5 1 3 46000 4
4 4 14.10 6 10 1.0 5 5 3 15000 4
4 3 6.4 6 50 2.0 5 4 3 31000 3
2 2 2.5 8 10 1.5 5 2 3 34000 2
C12 C13 C14 C15
min min min min
3 4 150 400
4 4 150 200
4 3 100 300
2 2 40 500
3 2 5.8 7 50 2.0 3 1 3 30000 3 2 2 90 300
be only descriptively, i.e. attributively, estimated. Second, certain criteria are conflicting so that standard ranking methods cannot be applied. Generally speaking, the PROMCALC is very favourable to the users and input data defining is very simple. Results obtained by the PROMETHEE method (Figure 1) show that the location A1, Tenet~e, is the lowest in ranking (the worst), while the location A2, Kovar, and A5, Draga, are the best locations, i.e. the preferred ones. The fact that location A2, Kovar, is the first in ranking consid-
PROMETHEE
I
Criterion type
Parameters
Weights
III I III III II III I III I V IV IV IV III II
2 12 3 10 2 4 100 0.5 0.5 0.5 100 250
10 10 10 5 2 2 1 10 5 10 5 5 5 10 10
25000 2.5 2.5 2.5 -
ering the input flow, and location A5, Draga, the highest by output flow ranking, practically does not allow the preference of only one location; thus it is necessary to use the PROMETHEEII method for determining the complete preorder. The results of complete ranking are presented in Figure 2, which shows final ranking of locations obtained by the PROMETHEEII method. Taking into account the fact that three locations have the positive resulting inflow and that the preference of the first locations in ranking is rather weak, more detailed studies should be car-
PfiI~IfiL
~nNXING
ESC : PROIE~HEEII - Any other key : detailed PRONE~HEEI,
Figure 1
D. Vuk et al. / Application of multicriterional analysis
PROMETHEE
II COMPLETE RANKING D~aga
govor
1, [I ~. Kov0r
~., i~ 5
~ afa
3, A4 rnlvee 4,~13 Mezaklda L A I Tenetise
215
Mezaklda
¢~nivee
Tenetise
(t,112)
(g,Q25) (0,010) (-0,064) (-0,083)
Figure 2
ried out on all three locations, and the final decision should follow a more detailed study of the preferred locations. Table 2 presents data on 'Weights Stability Intervals' (WSI) for each criterion with respect to the weights given by the experts. According to WSI none of the criteria are stressed as particularly 'non-stable' whereas a wide stability interval can be noted for criteria C9, C10 and C~3. Table 2 Weights Stability Intervals Criterion
Weight
Interval
%
% Interval
C, C2 Ca C4 C5 C6 C7 C~
10.00 10.00 10.00 5.00 2.00 2.00 1.00 10.00
[4.06, 12.301 [5.03, 17.42] [5.82, 14.92] [1.55, 13.91] [0.00, 3.15] [0.00, 6.60] [0.00, 2.15] [5.40, 12.10]
10.00 10.00 10.00 5.00 2.00 2.00 1.00 10.00
[4.32,12.02] [5.29,16.22] [6.07,14.22] [1.60,12.77] [0.00,3.11] [0.00,6,31] [0.00,2.13] [5.66,11.85]
k~l k~ kI k~ k~ k~ k~ k42
c~
k~
5.00
Clo CH Ca2 C~3 C~4 Cl~
k2 kv: k2 k2 k20 k~l
10.00 5.00 5.00 5.00 10.00 10.00
[0.00, ~]
5.00 I0.00,10o.0ol
[7.03, 70.67] 10.00 [7.25,43.98] [1.06, 8.84] 5.00 [1.10,8.51] [0.00, 10.26] 5.00 [0.00,9.75] [0.00, 16.60] 5.00 [0.00,14.87] [6.71, 12.50] 10.00 [6.94,12.20] [8.49, 17.81] 10.00 [8.62,16.52]
5. The GAlA method applied on problem of the location selection for disposal of communal waste Since the results obtained by the PROMETHEE method have aroused a great deal of discussion, the latest software GAIA was used, to ensure a more detailed insight into the problem, and it made it possible to study the criteria 'behaviour', using the geometric representation of the multicriterional analysis. Considering the k criteria, each action a i of A ( i = 1. . . . , n) can be represented by a point ~(ai) . . . . . Ck(ai)) in R k. To visualize this information, a two-dimensional representation of these points is built using Principal Components Analysis. More details on the procedure can be found in [5]. A plane is determined which is called the GAlA (U, v)-plane, using the two first principal components u and v. It is also possible to relate the GAIA method to PROMETHEE II: the PROMETHEE II method requires that weights be associated to the criteria and builds a complete preorder on A. The weight can be presented in the (u, v)-plane by PROMETHEE II decision axis H which points in the direction of the best ranked actions. It is possible to interac-
216
D. Vuk et al. / Application of multicriterional analysis
¢3 4
~5 C6 C7 C8 C9 C18 CII C12
k13
k14 k21 k22 k23 k24 k25 k26 k27 k28
C13 614
C15
k29 k218 k211
Cl',
kll
u: -B,23 v:
438
¢ow~nd : ¢~,ite;,ia ) Al,~vs : wove, 'A': glo],al view, '2': zoow, ESC to stop, Figure 3
alized criteria are taking into account the scalings of the evaluations. GAIA is providing interactively the decision PROMETHEE decision axis. The best possible geometrical representation on a plane is given. It is obtained by considering the two best eigenvalues of a particular preference matrix. GAIA is extremely useful for analysing the conflictual character of the criteria. Figures 3 and 4 present the geometric representation of the problem related to the locations selection for disposal of communal wastes apply-
tively modify the weights and observe the consequences on the PROMETHEEII ranking. The quality of the representation is measured by the percentage 8 of inertia explained by the two principal components u and o in many practical applications; this percentage is rather high (often more than 80%), thus giving a good representation of the multicriteria problem. GAIA (Geometrical Analysis for Interactive Aid) is based on the PROMETHEEn method for multicriteria decision making. Easy to define gener-
)A I A2 R3 A4 {l 5
Tenetise ~vo~ ~zaklja Cmivec Ih,aga
A1 :
Tenetise
2 ...................................................
'""13 ...........................................
,~
i {ft
u-" Con,and : Actions > AI,PoMs : wove, ESC to stop, Figure 4
2,95 v=-19,6e
D. Vuk et al. / Application of multicriterional analysis
ing the GAIA programme and the input data given in Table 1. Figure 3 presents the interaction of the criteria, with a relatively low percentage of information obtained by the projection onto the two-dimensional coordinate system, which points to the fact that the problem is poorly structured. Furthermore, a complete divergence of the criteria directions can be easily noted, which results from their conflicts, since the summary vector resulting from such scattered vectors cannot be great. Figure 4 presents the position of the actions in the projection plane (mapping plane), as well as the size and direction of the summary vector. The 'grouping' of similar actions can be seen best on the example of action A1 and A3. However, the only firm conclusion which can be drawn is the exceptionally bad structure of the problem, which is illustrated by the size of the summary vector in Figure 4; thus, it is evident that small change in the input data would influence the sequence of the actions. Consequently (particularly bearing in mind the importance of the decision made), before making a decision it is necessary to carry out investigations of the locations under study, in order to achieve a better structure of the problem.
217
(with the certain losses in reducing the problem dimensions), it is possible to keep the 'interactions' between criteria and the significance of each criterion in relation to the others. In other words, the conflicting criteria will, in this representation, have a significantly different direction (low covariance among criteria leads to a low value of the scalar product of the vectors which represent them); the convergent criteria are represented by vectors of similar directions. Furthermore, the importance of criteria in decision making is geometrically represented by the length of the vectors, so that vectors with higher absolute values correspond to the dominating criteria. By summing up the vectors which represent criteria the summary vector is obtained, lts direction and size describe the resulting action of all criteria. If the summary vector has low absolute value, it means that the criteria are in conflict. It can be generally concluded that the geometrical representation of the multicriterional analysis represents powerful tools in the hands of the system analysis and a valuable help in solving problems which are characterized by partly or completely conflicting criteria, which unfortunately occurs more frequently in decision-making processes.
6. Conclusion References From the presented facts it can be concluded that the PROMETHEE method has a decisive positive contribution both to the process of decision making for the locations selection and to the process of decision making on investments with the objective of raising the disposal place class and thus to improve the quality of the given eco system. This fact represents an additional reason why, in our actual example, the PROMETHEE method was used, as it primarily yielded some results applicable to the decision making process and to the continuation of the research study. The described selection of the plane for the geometrical representation of a multi-criterional problem makes it possible to achieve the minim u m loss of information (with regard to the method of the least squares), which means that
[1] Brans, J.P., Mareschal B., and Vincke, Ph., '"PROMETHEE': A new family of autranking methods in multicriteria analysis", in: J.P. Brans (ed.), Operational Research "84, NorthHolland, Amsterdam, 1984, 447-490. [2] Mladineo, N., Margeta, J., Brans, J.P., Mareschal, B., "Multicriteria ranking of alternative locations for small scale hydro plants", European Journal of Operational Research 31 (1987) 215-222. [3] Mareschal, B., "Weight stability intervals in multicriteria decision aid", European Journal of Operational Research 33 (1984) 54-64. [4] Ribarovir, Z., Mladineo, N., "Application of multicriterional analysis to the ranking and evaluation of the investment programmes in the ready mixed concrete industry", Engineering Costs and Production Economics 12 (1987) 367374. [5] Mareschal, B., Brans, J.P., "Geometrical representation for MCDA", European Journal of Operational Research 34 (1988) 59-77.
211
Theory and Methodology
Application of multicriterional analysis on the selection of the location for disposal of communal waste * Drago Vuk, Bogomir Ko~elj Highschool for the Organization Science, Kranj, University of Maribor, Pregernova ll, 64000 Kranj, Yugoslavia
Nenad Mladineo GI. Faculty of Civil Engineering, University of Split, F. Maslege bb, 58000 Split, Yugoslavia Received September 1989; revised February 1990
Abstract: This paper presents the application of the PROMETHEEmethods and geometrical representation of multicriterional analysis (GAIA) o n an actual example of the selection of the location for the disposal of communal waste in the region Gorenjska in Slovenia. In addition to the model presentation the paper deals with the application of the method in the function of decision making on investments which is necessary to make the actual existing locations environmentally acceptable.
Keywords: Location, ecology, multiple criteria
1. Introduction
The improvement of the living standard is the result of the development of science and industry in general; however this has led to the presence of some accompanying phenomena which endanger the human environment, i.e. the eco system. It is a well-known fact that life and waste cannot be separated and that the quantities and type of waste are directly related to the standard of living and the development of industry. The starting point of our considerations is the structural and functional nature of the eco system. * In this paper the authors used material from the Ph.D. Thesis of D. Vuk.
If the eco system is considered as the sum of the influences of all the system elements, then both the influence exerted by the elements on the system and of the system on each element has to be taken into account. In the analyzed case, the disposal of communal waste was taken as the element of the eco system, since previous analyses have shown that the given structure and quantity of waste in one region is the optimal solution for programming the regional disposals. Consequently, as the influences on the system element are different and dependent on many factors, containing both the environmental and economic components, the PROMETHEEmethod was chosen for the decision making support to the optimal disposal locations.
0377-2217/91/$03.50 © 1991 - Elsevier Science Publishers B.V. All rights reserved
212
D. Vuk et al. / Application of multicriterional analysis
Although it is a well-known fact that the influences of the system on the element and vice versa are vector valued quantities which characterize those relations, the ordinal scale was used in our evaluations. All the values for that scale were obtained empirically and therefore the factors were given empirically. Since there are many factors, the accuracy for the multicriterional analysis, i.e. for the PRO~THEE method is considered to be satisfactory. There is another possibility for re-calculating each factor separately. In that case this example could be treated deterministically and thus the decisions could be made. Already in simple evaluations of the functionsstability, costs, environmental pollution, reliability and the stability, a minimum number of coordinates was established i.e. between 6 and 7 coordinates for defining the ecological examples of the system. The conditions were considered as 'loads' and they represent the product between the value of the factor and the importance of the problem. Thus it is possible to define a great number of alternatives for different conditions (financing, personnel, etc.). According to the presented facts, the model was developed for the selection of the optimal disposal location, and it will be briefly presented.
2. Definition of the model criteria In order to present each criterion efficiently by mathematical operations and tables, each criterion was given the algebraic notation ' k ' and two indices which designate the group and the current number of criteria. The criteria were classified into the following two groups: - the first group is represented by four location criteria and - the second group is represented by eleven technological-economic criteria. The criteria are presented in this order k~ = dispersion in space with regard to the central location (C1); h a = safety of the location (C2); k~= size of the location, exploitation time (Ca); k~= distance of the location from the settlement (64); time of the waste mineralization (neutralization) (65);
= specific land use (function of the location)
(66); = possibility for treating the waste from the city treatment plants (67); = energy budget (68); = usefulness of waste (exploitation of wastes)
(69); quantity of waste (610);
/672=influence of the environment (CH); k~= influence on the groundwater and
surface water (612); k92= influence on the atmosphere (613); k~0= investment costs (C~4); k?l= operation costs of the disposal plant, collection and transportation of communal wastes
(615). The given criteria represent a multi-parameter frame for estimating the potential location based on the data available in the phase of investment planning. For some criteria, it was possible to establish the exact parameter values (e.g. investment costs, waste quantity etc); however, for a great number of criteria, the expert estimation was given according to the attribute (descriptive) evaluation. Since all criteria do not have the same importance for the final location selection, the weight evaluation for each criterion also influences the result. The special advantage of the PROMETHEE method, i.e. of the software package PROMCALC is that the output includes 'weight stability intervals', so that it is possible to check the real influence of the given weights on the final solution. 3. Multicriteria analysis using the PROMETHEE methods We briefly outline in this section the PROMETHEE I and II methods which are used to rank the alternative locations. A complete description of these methods and their extensions is given in [1]. Let us call A the set of all the possible locations we want to rank, and ]'1, f2 . . . . . fk the k criteria that have been selected. The PROMETHEE methods first build a valued outranking relation on A. This relation is then used to obtain a partial (PROMETHEE I) or a complete preorder (PROMETHEEII) on ,,4. More general interval orders can also be obtained, using the PROMETHEE II) algorithm described in [1].
D. Vuk et al. / Application of rnulticriterional analysis
3.1. PROMETHEEvalued outranking relation
213
giving the 'outranked character' of a; - a n d its net flow.
- Unicriterion preference function For each criterion fg and for every two locations a, b ~ A the preference of the decision-maker for location a over location b is expressed through a preference function Pj such that 0
Va, b ~ A .
b)~
pj(a, b) may be considered as the intensity of preference of a over b, it is equal to 0 in case of no preference of a over b or indifference between a and b, it is equal to 1 in case of strict p r e f e r e n c e of a over b, for the particular criterion ~. In practice, Pj(a, b) is often a function of d =~(a) - ~(b): Pj(a, b) = Pj(d). Pj is a non-decreasing function of d and is equal to 0 for negative values of d. Six useful shapes have been employed (see [1]). - Multicriteria preference index Let us suppose that a preference function Pj and a weight wj have been specified for each criterion ~ ( j = 1 . . . . . k). The multicriteria preference i n d e x / 7 is defined as k
~., w, Pg( a, b) H(a, b ) = j=l
k
, Va, b ~ A .
Ewj j=l lI(a, b) represents the intensity of preference of the decision-maker of a over b, when considering simultaneously all the criteria. This index defines a valued outranking relation on A. 3.2. Flows and rankings PROMETHEE defines for each a ~ A.
-its leaving flow. ++(a)= E n ( a , b), b~A
giving the 'outranking character' of a;
-its entering flow. +-(a)=
E II(b, a), b~A
i f ( a ) = q~+ ( a ) - 4~-(a). The higher the leaving flow the lower the entering flow, the better the action. The net flow is given as a complete preorder defined as: - a outranks b iff dp(a)> th(b), - a is indifferent to b iff O ( a ) = ¢(b). Every outranking method involves the determination of some parameters and weights. It is interesting to know the influence they have on the rankings when small deviations in their values are introduced. As the decision-maker usually cannot fix correctly their exact values, this stability problem is of major importance. It seems a necessary condition for a good outranking method. This was one of the principal purposes of PROMETHEE.
4. D e s c r i p t i o n
and
numerical
treatment
of
the
problem
The evaluation of the model for the optimal locations selection was carried out according to the actual demands for establishing the suitability of the locations in the region Gorenjska, Kranj, Yugoslavia. This region has ca. 500000 inhabitants with the annual gross social (national) income of $4000. The preliminary analysis served to identify the following 5 locations: - Teneti~a (A1); - Kovor (A 2)' - Me~aklja (A3); - Crnivec (An) ; - Draga (As); for which according to the defined criteria the data were gathered as presented in Table 1. The table also gives data on the defined types of criteria using the respective parameters and weights (see PROMETHEEmethod [1,2]). A superficial input data analysis can show two facts. First, the data are given in 'original' values units. It is not necessary to reduce to a common unit which is often wrong. Then, one part of criteria can be described as qualitative which is exceptionally favourable in preliminary investments project processing certain parameters can
D. Vuk et al. / Application of multicriterional analysis
214 Table 1 Criterion
Problem
Locations
type
A1
A2
A3
A4
A5
C1 C2 C3 Ca C5 C6 C7 C8 C9 Cto C11
min max max min min min max min max min min
3 3 10.25 7 50 1.5 5 1 3 46000 4
4 4 14.10 6 10 1.0 5 5 3 15000 4
4 3 6.4 6 50 2.0 5 4 3 31000 3
2 2 2.5 8 10 1.5 5 2 3 34000 2
C12 C13 C14 C15
min min min min
3 4 150 400
4 4 150 200
4 3 100 300
2 2 40 500
3 2 5.8 7 50 2.0 3 1 3 30000 3 2 2 90 300
be only descriptively, i.e. attributively, estimated. Second, certain criteria are conflicting so that standard ranking methods cannot be applied. Generally speaking, the PROMCALC is very favourable to the users and input data defining is very simple. Results obtained by the PROMETHEE method (Figure 1) show that the location A1, Tenet~e, is the lowest in ranking (the worst), while the location A2, Kovar, and A5, Draga, are the best locations, i.e. the preferred ones. The fact that location A2, Kovar, is the first in ranking consid-
PROMETHEE
I
Criterion type
Parameters
Weights
III I III III II III I III I V IV IV IV III II
2 12 3 10 2 4 100 0.5 0.5 0.5 100 250
10 10 10 5 2 2 1 10 5 10 5 5 5 10 10
25000 2.5 2.5 2.5 -
ering the input flow, and location A5, Draga, the highest by output flow ranking, practically does not allow the preference of only one location; thus it is necessary to use the PROMETHEEII method for determining the complete preorder. The results of complete ranking are presented in Figure 2, which shows final ranking of locations obtained by the PROMETHEEII method. Taking into account the fact that three locations have the positive resulting inflow and that the preference of the first locations in ranking is rather weak, more detailed studies should be car-
PfiI~IfiL
~nNXING
ESC : PROIE~HEEII - Any other key : detailed PRONE~HEEI,
Figure 1
D. Vuk et al. / Application of multicriterional analysis
PROMETHEE
II COMPLETE RANKING D~aga
govor
1, [I ~. Kov0r
~., i~ 5
~ afa
3, A4 rnlvee 4,~13 Mezaklda L A I Tenetise
215
Mezaklda
¢~nivee
Tenetise
(t,112)
(g,Q25) (0,010) (-0,064) (-0,083)
Figure 2
ried out on all three locations, and the final decision should follow a more detailed study of the preferred locations. Table 2 presents data on 'Weights Stability Intervals' (WSI) for each criterion with respect to the weights given by the experts. According to WSI none of the criteria are stressed as particularly 'non-stable' whereas a wide stability interval can be noted for criteria C9, C10 and C~3. Table 2 Weights Stability Intervals Criterion
Weight
Interval
%
% Interval
C, C2 Ca C4 C5 C6 C7 C~
10.00 10.00 10.00 5.00 2.00 2.00 1.00 10.00
[4.06, 12.301 [5.03, 17.42] [5.82, 14.92] [1.55, 13.91] [0.00, 3.15] [0.00, 6.60] [0.00, 2.15] [5.40, 12.10]
10.00 10.00 10.00 5.00 2.00 2.00 1.00 10.00
[4.32,12.02] [5.29,16.22] [6.07,14.22] [1.60,12.77] [0.00,3.11] [0.00,6,31] [0.00,2.13] [5.66,11.85]
k~l k~ kI k~ k~ k~ k~ k42
c~
k~
5.00
Clo CH Ca2 C~3 C~4 Cl~
k2 kv: k2 k2 k20 k~l
10.00 5.00 5.00 5.00 10.00 10.00
[0.00, ~]
5.00 I0.00,10o.0ol
[7.03, 70.67] 10.00 [7.25,43.98] [1.06, 8.84] 5.00 [1.10,8.51] [0.00, 10.26] 5.00 [0.00,9.75] [0.00, 16.60] 5.00 [0.00,14.87] [6.71, 12.50] 10.00 [6.94,12.20] [8.49, 17.81] 10.00 [8.62,16.52]
5. The GAlA method applied on problem of the location selection for disposal of communal waste Since the results obtained by the PROMETHEE method have aroused a great deal of discussion, the latest software GAIA was used, to ensure a more detailed insight into the problem, and it made it possible to study the criteria 'behaviour', using the geometric representation of the multicriterional analysis. Considering the k criteria, each action a i of A ( i = 1. . . . , n) can be represented by a point ~(ai) . . . . . Ck(ai)) in R k. To visualize this information, a two-dimensional representation of these points is built using Principal Components Analysis. More details on the procedure can be found in [5]. A plane is determined which is called the GAlA (U, v)-plane, using the two first principal components u and v. It is also possible to relate the GAIA method to PROMETHEE II: the PROMETHEE II method requires that weights be associated to the criteria and builds a complete preorder on A. The weight can be presented in the (u, v)-plane by PROMETHEE II decision axis H which points in the direction of the best ranked actions. It is possible to interac-
216
D. Vuk et al. / Application of multicriterional analysis
¢3 4
~5 C6 C7 C8 C9 C18 CII C12
k13
k14 k21 k22 k23 k24 k25 k26 k27 k28
C13 614
C15
k29 k218 k211
Cl',
kll
u: -B,23 v:
438
¢ow~nd : ¢~,ite;,ia ) Al,~vs : wove, 'A': glo],al view, '2': zoow, ESC to stop, Figure 3
alized criteria are taking into account the scalings of the evaluations. GAIA is providing interactively the decision PROMETHEE decision axis. The best possible geometrical representation on a plane is given. It is obtained by considering the two best eigenvalues of a particular preference matrix. GAIA is extremely useful for analysing the conflictual character of the criteria. Figures 3 and 4 present the geometric representation of the problem related to the locations selection for disposal of communal wastes apply-
tively modify the weights and observe the consequences on the PROMETHEEII ranking. The quality of the representation is measured by the percentage 8 of inertia explained by the two principal components u and o in many practical applications; this percentage is rather high (often more than 80%), thus giving a good representation of the multicriteria problem. GAIA (Geometrical Analysis for Interactive Aid) is based on the PROMETHEEn method for multicriteria decision making. Easy to define gener-
)A I A2 R3 A4 {l 5
Tenetise ~vo~ ~zaklja Cmivec Ih,aga
A1 :
Tenetise
2 ...................................................
'""13 ...........................................
,~
i {ft
u-" Con,and : Actions > AI,PoMs : wove, ESC to stop, Figure 4
2,95 v=-19,6e
D. Vuk et al. / Application of multicriterional analysis
ing the GAIA programme and the input data given in Table 1. Figure 3 presents the interaction of the criteria, with a relatively low percentage of information obtained by the projection onto the two-dimensional coordinate system, which points to the fact that the problem is poorly structured. Furthermore, a complete divergence of the criteria directions can be easily noted, which results from their conflicts, since the summary vector resulting from such scattered vectors cannot be great. Figure 4 presents the position of the actions in the projection plane (mapping plane), as well as the size and direction of the summary vector. The 'grouping' of similar actions can be seen best on the example of action A1 and A3. However, the only firm conclusion which can be drawn is the exceptionally bad structure of the problem, which is illustrated by the size of the summary vector in Figure 4; thus, it is evident that small change in the input data would influence the sequence of the actions. Consequently (particularly bearing in mind the importance of the decision made), before making a decision it is necessary to carry out investigations of the locations under study, in order to achieve a better structure of the problem.
217
(with the certain losses in reducing the problem dimensions), it is possible to keep the 'interactions' between criteria and the significance of each criterion in relation to the others. In other words, the conflicting criteria will, in this representation, have a significantly different direction (low covariance among criteria leads to a low value of the scalar product of the vectors which represent them); the convergent criteria are represented by vectors of similar directions. Furthermore, the importance of criteria in decision making is geometrically represented by the length of the vectors, so that vectors with higher absolute values correspond to the dominating criteria. By summing up the vectors which represent criteria the summary vector is obtained, lts direction and size describe the resulting action of all criteria. If the summary vector has low absolute value, it means that the criteria are in conflict. It can be generally concluded that the geometrical representation of the multicriterional analysis represents powerful tools in the hands of the system analysis and a valuable help in solving problems which are characterized by partly or completely conflicting criteria, which unfortunately occurs more frequently in decision-making processes.
6. Conclusion References From the presented facts it can be concluded that the PROMETHEE method has a decisive positive contribution both to the process of decision making for the locations selection and to the process of decision making on investments with the objective of raising the disposal place class and thus to improve the quality of the given eco system. This fact represents an additional reason why, in our actual example, the PROMETHEE method was used, as it primarily yielded some results applicable to the decision making process and to the continuation of the research study. The described selection of the plane for the geometrical representation of a multi-criterional problem makes it possible to achieve the minim u m loss of information (with regard to the method of the least squares), which means that
[1] Brans, J.P., Mareschal B., and Vincke, Ph., '"PROMETHEE': A new family of autranking methods in multicriteria analysis", in: J.P. Brans (ed.), Operational Research "84, NorthHolland, Amsterdam, 1984, 447-490. [2] Mladineo, N., Margeta, J., Brans, J.P., Mareschal, B., "Multicriteria ranking of alternative locations for small scale hydro plants", European Journal of Operational Research 31 (1987) 215-222. [3] Mareschal, B., "Weight stability intervals in multicriteria decision aid", European Journal of Operational Research 33 (1984) 54-64. [4] Ribarovir, Z., Mladineo, N., "Application of multicriterional analysis to the ranking and evaluation of the investment programmes in the ready mixed concrete industry", Engineering Costs and Production Economics 12 (1987) 367374. [5] Mareschal, B., Brans, J.P., "Geometrical representation for MCDA", European Journal of Operational Research 34 (1988) 59-77.