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Let us also suppose that only the objective function coefficients are known ex- actly and that, by hypothesis, all the constraint matrix lines correspond to quantities that are expressed in the same type of unit (e.g., physical, monetary). The values of the constraint matrix coefficients are uncertain. A finite set S' of variable settings allows this imperfect knowledge to be modelled. In the general case, it is possible for the intersection of the feasible domains of all the variable settings to be empty. In addition, even if this intersection is not empty, the price of robustness, as Soyster refers to it, can be high. The decision maker might accept an unfeasible robust so- lution in a small subset of S,, but only if the cost is relatively low. In fact, it is often acceptable in practice to not respect equality; however, in this case, it is important for the non-zero deviations between the right and left members to be “small” and few in number. Thus, an unfeasible mathematical solution may be preferable to a much more costly solution that perfectly satisfies all the equalities. For a solution x, the deviations that must be taken into account are defined as follows:

Figure 9 Let us also suppose that only the objective function coefficients are known ex- actly and that, by hypothesis, all the constraint matrix lines correspond to quantities that are expressed in the same type of unit (e.g., physical, monetary). The values of the constraint matrix coefficients are uncertain. A finite set S' of variable settings allows this imperfect knowledge to be modelled. In the general case, it is possible for the intersection of the feasible domains of all the variable settings to be empty. In addition, even if this intersection is not empty, the price of robustness, as Soyster refers to it, can be high. The decision maker might accept an unfeasible robust so- lution in a small subset of S,, but only if the cost is relatively low. In fact, it is often acceptable in practice to not respect equality; however, in this case, it is important for the non-zero deviations between the right and left members to be “small” and few in number. Thus, an unfeasible mathematical solution may be preferable to a much more costly solution that perfectly satisfies all the equalities. For a solution x, the deviations that must be taken into account are defined as follows:

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Fig. 1.3. Two domains of competence set analysis
Fig. 1.3. Two domains of competence set analysis
Table 2.1 Data for an example on international business management (Empl. = employees)
Table 2.1 Data for an example on international business management (Empl. = employees)
Table 2.2, An example of objective ranking and classification for the data from Table 2.1 attachment (JAIST consists of three schools), nationality (Japanese or foreign — the latter constitute over 10% of young researchers at JAIST), research position (master students, doctoral students, research associates, etc.). In total, the data base was not very large, but large enough to create computational problems.
Table 2.2, An example of objective ranking and classification for the data from Table 2.1 attachment (JAIST consists of three schools), nationality (Japanese or foreign — the latter constitute over 10% of young researchers at JAIST), research position (master students, doctoral students, research associates, etc.). In total, the data base was not very large, but large enough to create computational problems.
Table 2.3 Four different types of reference profile distributions best ranked questions of the second (importance) type were more changeable, only three of them consistently were ranked among the best ones in diverse ranking pro- files. Moreover, a rank reversal phenomenon was observed: if the average reference distribution was used, best ranked were questions of rather obvious type, more in- teresting results were obtained when using more demanding reference profile. This rank reversal, however, influenced more the best ranked questions than worst ranked questions, more significant fc for university management. Tr meq 2te. en: (ec ee ee |
Table 2.3 Four different types of reference profile distributions best ranked questions of the second (importance) type were more changeable, only three of them consistently were ranked among the best ones in diverse ranking pro- files. Moreover, a rank reversal phenomenon was observed: if the average reference distribution was used, best ranked were questions of rather obvious type, more in- teresting results were obtained when using more demanding reference profile. This rank reversal, however, influenced more the best ranked questions than worst ranked questions, more significant fc for university management. Tr meq 2te. en: (ec ee ee |
52 180 145 130 120 90 125 125 140 125 130 135 140 150 125 130 140 130 135 130 125 53 170 120 175 190 210 135 175 190 190 175 110 190 150 155 145 190 155 145 135 125 S4 175 155 170 195 205 150 175 195 165 195 120 190 165 160 150 195 135 130 155, 135 S5 165 165 175 190 200 145 175 190 165 160 195 170 160 155 165 200 130 125 150 155 160 195 175 195 205 150 190 195 155 155 150 175 150 145 170 195 120 145 145 150 175 175 160 165 125 140 1 175 200 170 S9 160 165 170 190 190 165 175 190 165 170 150 165 140 155 195 190 145 125 150 155 S10 175 170 175 195 195 195 175 195 150 155 130 160 150 160 170 190 125 130 145 150 Si] 170 155 190 190 195 155 190 180 150 145 145 190 165, 155 155 195 195 135 135 145 S12 165, 160 190 195 200 145 195 185 165, 140 155 165 170 160 150 160 135 195 140 135 S13 155 165 195 190 205 170 190 180 170 155 145 170 145 165 165 175 125 160 195 155 S14 160 170 190 195 210 155 195 185 165 160 160 190 160 170 160 160 155 155 150 195 S15 170 155 195 190 215 145 190 180 160 155 155 190 170 155 155 175 135 150 145 140 S16 165 155 190 200 210 140 195 185 155 160 150 190 155 145 170 160 120 145 155 145 S17 170 170 195 190 210 145 190 180 165 155 135, 190 170 165 165 150 110 140 S18 165 165 190 195 205 135 180 165 170 160 170 175 135 Sig S 180 195 195 200 220 160 200 125 150 145 150 190 160 165 180
52 180 145 130 120 90 125 125 140 125 130 135 140 150 125 130 140 130 135 130 125 53 170 120 175 190 210 135 175 190 190 175 110 190 150 155 145 190 155 145 135 125 S4 175 155 170 195 205 150 175 195 165 195 120 190 165 160 150 195 135 130 155, 135 S5 165 165 175 190 200 145 175 190 165 160 195 170 160 155 165 200 130 125 150 155 160 195 175 195 205 150 190 195 155 155 150 175 150 145 170 195 120 145 145 150 175 175 160 165 125 140 1 175 200 170 S9 160 165 170 190 190 165 175 190 165 170 150 165 140 155 195 190 145 125 150 155 S10 175 170 175 195 195 195 175 195 150 155 130 160 150 160 170 190 125 130 145 150 Si] 170 155 190 190 195 155 190 180 150 145 145 190 165, 155 155 195 195 135 135 145 S12 165, 160 190 195 200 145 195 185 165, 140 155 165 170 160 150 160 135 195 140 135 S13 155 165 195 190 205 170 190 180 170 155 145 170 145 165 165 175 125 160 195 155 S14 160 170 190 195 210 155 195 185 165 160 160 190 160 170 160 160 155 155 150 195 S15 170 155 195 190 215 145 190 180 160 155 155 190 170 155 155 175 135 150 145 140 S16 165 155 190 200 210 140 195 185 155 160 150 190 155 145 170 160 120 145 155 145 S17 170 170 195 190 210 145 190 180 165 155 135, 190 170 165 165 150 110 140 S18 165 165 190 195 205 135 180 165 170 160 170 175 135 Sig S 180 195 195 200 220 160 200 125 150 145 150 190 160 165 180
Table 5.1 Parametric families of continuous distributions Table 5.2. Cycle-transitivity for the continuous distributions in Table 5.1
Table 5.1 Parametric families of continuous distributions Table 5.2. Cycle-transitivity for the continuous distributions in Table 5.1
In the cases of the unimodal uniform, Gumbel, Laplace and normal distribu- ions we have fixed one of the two parameters in order to restrict the family to a yne-parameter subfamily, mainly because with two free parameters, the formulae yecome utmost cumbersome. The one exception is the two-dimensional family of 1ormal distributions. In [18], we have shown that the corresponding reciprocal rela- ion is in that case moderately stochastic transitive.
In the cases of the unimodal uniform, Gumbel, Laplace and normal distribu- ions we have fixed one of the two parameters in order to restrict the family to a yne-parameter subfamily, mainly because with two free parameters, the formulae yecome utmost cumbersome. The one exception is the two-dimensional family of 1ormal distributions. In [18], we have shown that the corresponding reciprocal rela- ion is in that case moderately stochastic transitive.
to MCDA, together with System Dynamics and Viable Systems Modelling are also briefly outlined, as “near neighbours” of PSMs [83, Chapter 12]. Other approaches which it is felt might lay claim to share the same characteristics [83, p xv] are Ack- off’s Idealized Planning, Mason and Mitroff’s Strategic Assumption Surfacing and Testing and Saaty’s Analytic Hierarchy Process. The key features of the five prin- cipal approaches and the potential for integration with MCDA are summarised in Table 8.1.
to MCDA, together with System Dynamics and Viable Systems Modelling are also briefly outlined, as “near neighbours” of PSMs [83, Chapter 12]. Other approaches which it is felt might lay claim to share the same characteristics [83, p xv] are Ack- off’s Idealized Planning, Mason and Mitroff’s Strategic Assumption Surfacing and Testing and Saaty’s Analytic Hierarchy Process. The key features of the five prin- cipal approaches and the potential for integration with MCDA are summarised in Table 8.1.
methods which define the different approaches to problem structuring and MCDA can be effectively combined. The diagrammatic representation of the MCDA pro- cess in Fig. 8.1 suggests a natural way to do so, with the problem structuring phase supported by one of the more general PSMs and providing a rich description of the problem from which an appropriate multicriteria model may be derived. This way of working, illustrated in Fig. 8.3A, provides the foundation for an Honours year class, entitled “Problem Structuring to Evaluation”, which one of the authors has taught, together with Fran Ackermann since 2003. Whilst there is rich potential for interaction and iteration between models, in reality this is dependent on practical circumstances, in particular constraints on time and resource. Figure 8.3B illustrates a way of working which is perhaps more likely if using MCDA with one of the more focused PSMs (see, for example [61]); however, such an intervention might be fur- ther enhanced if initiated and supported by the use of a general PSM, as shown in Fig. 8.3C.
methods which define the different approaches to problem structuring and MCDA can be effectively combined. The diagrammatic representation of the MCDA pro- cess in Fig. 8.1 suggests a natural way to do so, with the problem structuring phase supported by one of the more general PSMs and providing a rich description of the problem from which an appropriate multicriteria model may be derived. This way of working, illustrated in Fig. 8.3A, provides the foundation for an Honours year class, entitled “Problem Structuring to Evaluation”, which one of the authors has taught, together with Fran Ackermann since 2003. Whilst there is rich potential for interaction and iteration between models, in reality this is dependent on practical circumstances, in particular constraints on time and resource. Figure 8.3B illustrates a way of working which is perhaps more likely if using MCDA with one of the more focused PSMs (see, for example [61]); however, such an intervention might be fur- ther enhanced if initiated and supported by the use of a general PSM, as shown in Fig. 8.3C.
Fig. 8.4 Causal map from Hawston workshops As illustration, the causal map extracted from the Hawston/Hermanus commu- nity is shown in Fig. 8.4. A central theme is probably that of concept 103, namely that the “wrong” people were receiving rights, rather than the traditional fishing communities. A number of reasons for this feature are identified, suggesting pos- sible courses of action and external forces. More importantly to the interests of MCDA, a small number of concepts appear near the top of the map, and which are at the end of chains of links. These concepts can then be interpreted (at least tentatively) as the primary objectives of the rights allocation process as per- ceived by the Hawston/Hermanus community, namely (a) reduction of poverty and unemployment in the community, (b) countering of gangsterism and crime, and (c) preservation of stocks. Although, different structures emerged from the three community groups, it was
Fig. 8.4 Causal map from Hawston workshops As illustration, the causal map extracted from the Hawston/Hermanus commu- nity is shown in Fig. 8.4. A central theme is probably that of concept 103, namely that the “wrong” people were receiving rights, rather than the traditional fishing communities. A number of reasons for this feature are identified, suggesting pos- sible courses of action and external forces. More importantly to the interests of MCDA, a small number of concepts appear near the top of the map, and which are at the end of chains of links. These concepts can then be interpreted (at least tentatively) as the primary objectives of the rights allocation process as per- ceived by the Hawston/Hermanus community, namely (a) reduction of poverty and unemployment in the community, (b) countering of gangsterism and crime, and (c) preservation of stocks. Although, different structures emerged from the three community groups, it was
Fig. 8.5 Overall value tree for fisheries rights allocation ee ee Interaction with the relevant government department, the Marine and Coastal Management (MCM) directorate of the Department of Environmental Affairs and Tourism, was carried out separately from that with community representatives. The process was similar, but also included a deconstruction of effective criteria that were in operation in previous rights allocations (even if not explicitly stated at the time). Similar structures evolved, and the objectives identified with MCM did in- clude those identified by the communities. However, additional criteria emerged. notably those related to concerns for financial stability of the industry and regional economic development. Finally, an aggregate value tree incorporating all concerns could be tabled as a basis for future rights allocation decisions. This tree is presented in Fig. 8.5.
Fig. 8.5 Overall value tree for fisheries rights allocation ee ee Interaction with the relevant government department, the Marine and Coastal Management (MCM) directorate of the Department of Environmental Affairs and Tourism, was carried out separately from that with community representatives. The process was similar, but also included a deconstruction of effective criteria that were in operation in previous rights allocations (even if not explicitly stated at the time). Similar structures evolved, and the objectives identified with MCM did in- clude those identified by the communities. However, additional criteria emerged. notably those related to concerns for financial stability of the industry and regional economic development. Finally, an aggregate value tree incorporating all concerns could be tabled as a basis for future rights allocation decisions. This tree is presented in Fig. 8.5.
The individual interviews revealed a strong consistency in employee values and desire to see the company grow in order to realise, eventually, a “pot of gold”. How- ever, different perspectives on how this might be achieved were apparent and the interview process led one employee to realise that they had more substantial con- cerns about developments that would ensue from accepting the venture capital than they had previously thought. The key concepts from the individual maps were con- solidated into one map (Fig. 8.6) by the team of facilitators before the workshop and, as this was of a manageable size, it was manually recreated step-by-step in front of the group of employees. As the map unfolded, participants were able to comment on the concepts and links and to further develop these. Se ee ee ee er i . . , . i fe
The individual interviews revealed a strong consistency in employee values and desire to see the company grow in order to realise, eventually, a “pot of gold”. How- ever, different perspectives on how this might be achieved were apparent and the interview process led one employee to realise that they had more substantial con- cerns about developments that would ensue from accepting the venture capital than they had previously thought. The key concepts from the individual maps were con- solidated into one map (Fig. 8.6) by the team of facilitators before the workshop and, as this was of a manageable size, it was manually recreated step-by-step in front of the group of employees. As the map unfolded, participants were able to comment on the concepts and links and to further develop these. Se ee ee ee er i . . , . i fe
some of the issues and uncertainties faced. This was complemented by a stakeholder analysis, which sought to identify and map all stakeholders against the two dimen- sions of interest in the issue and their power to influence outcomes (positively or negatively). See Eden and Ackermann [35, p121—125] for a more detailed discus- sion of this type of analysis. This power-interest grid groups stakeholders in four categories: players (high interest and power) need to be managed closely; subjects (high interest, low power) should be kept informed; context setters (high power but low interest) need to be kept satisfied; and the crowd (low interest and power) who should be monitored.
some of the issues and uncertainties faced. This was complemented by a stakeholder analysis, which sought to identify and map all stakeholders against the two dimen- sions of interest in the issue and their power to influence outcomes (positively or negatively). See Eden and Ackermann [35, p121—125] for a more detailed discus- sion of this type of analysis. This power-interest grid groups stakeholders in four categories: players (high interest and power) need to be managed closely; subjects (high interest, low power) should be kept informed; context setters (high power but low interest) need to be kept satisfied; and the crowd (low interest and power) who should be monitored.
Fig. 8.9 Multicriteria model for King Communications and Security Ltd
Fig. 8.9 Multicriteria model for King Communications and Security Ltd
concepts. There did not seem to be a need for causal mapping, as the “alterna- tives” were defined by the applicants, and the stakeholders and environment were self-evident. The MCDA perspective led to using emergent criteria as the rationale for clustering. In between the first and second workshops a value tree could thus be structured and circulated to participants, and was broadly accepted in the form displayed in Fig. 8.10.
concepts. There did not seem to be a need for causal mapping, as the “alterna- tives” were defined by the applicants, and the stakeholders and environment were self-evident. The MCDA perspective led to using emergent criteria as the rationale for clustering. In between the first and second workshops a value tree could thus be structured and circulated to participants, and was broadly accepted in the form displayed in Fig. 8.10.
such a comparison is suggested in Table 8.2. Of course, MCDA does not end with structuring. On occasions, a good struc- uring may make the solution self-evident, but in most cases the analytical or onvergent phase of MCDA will follow the structuring. The advantage of using the VICDA framework as a template for structuring is that the transition from divergent tructuring to convergent analysis is essentially seamless. The general feature of the ‘ase studies above was that the analytical models (in these cases multiattribute value unctions, but the method is not central) flowed out of the structuring as a natural ‘onsequence.
such a comparison is suggested in Table 8.2. Of course, MCDA does not end with structuring. On occasions, a good struc- uring may make the solution self-evident, but in most cases the analytical or onvergent phase of MCDA will follow the structuring. The advantage of using the VICDA framework as a template for structuring is that the transition from divergent tructuring to convergent analysis is essentially seamless. The general feature of the ‘ase studies above was that the analytical models (in these cases multiattribute value unctions, but the method is not central) flowed out of the structuring as a natural ‘onsequence.
Fig. 9.1 Necessary partial ranking at the first iteration Fig. 9.2 Necessary partial ranking at the second iteration
Fig. 9.1 Necessary partial ranking at the first iteration Fig. 9.2 Necessary partial ranking at the second iteration
Analyzing this first result, the DM observes that the necessary ranking is still very poor, which makes it difficult to discriminate among the solutions in A. He/she reacts by stating that s4 is preferred to ss. Considering this new piece of preference information, the necessary ranking is computed again and shown in Fig. 9.2. At this second iteration, it should be observed that the resulting necessary ranking has been enriched as compared to the first iteration (bold arrows), narrowing the set of “best choices,” i.e., solutions that are not preferred by any other solution in the necessary ranking: {s1, 53, 56, Sg, S10, 514, 515, 8517, S18, 519, S20 }, The DM believes that this necessary ranking is still insufficiently decisive and adds a new nairwise comparison: se is preferred to sin. Once again. the necessarv
Analyzing this first result, the DM observes that the necessary ranking is still very poor, which makes it difficult to discriminate among the solutions in A. He/she reacts by stating that s4 is preferred to ss. Considering this new piece of preference information, the necessary ranking is computed again and shown in Fig. 9.2. At this second iteration, it should be observed that the resulting necessary ranking has been enriched as compared to the first iteration (bold arrows), narrowing the set of “best choices,” i.e., solutions that are not preferred by any other solution in the necessary ranking: {s1, 53, 56, Sg, S10, 514, 515, 8517, S18, 519, S20 }, The DM believes that this necessary ranking is still insufficiently decisive and adds a new nairwise comparison: se is preferred to sin. Once again. the necessarv
Fig. 9.3. Necessary partial ranking at the third iteration
Fig. 9.3. Necessary partial ranking at the third iteration
Fig. 10.1 Traditional approach: decision model determines the “best” solution based on criteria measurements and DMs’ preferences
Fig. 10.1 Traditional approach: decision model determines the “best” solution based on criteria measurements and DMs’ preferences
Fig. 10.2 Inverse approach: identify preferences that are favourable for each alternative solution
Fig. 10.2 Inverse approach: identify preferences that are favourable for each alternative solution
be maximized Table 10.1 Problem with four alternatives and two criteria to be maximized
be maximized Table 10.1 Problem with four alternatives and two criteria to be maximized
Fig. 10.4 Favourable rank weights and rank acceptability indices in deterministic 2-criterion case with linear utility function tee a eee ee ee Ee Se Ee ee eee ee a Besides the weights that make an alternative x; most preferred, we can als identify the sets of favourable rank weights W,’ that give a particular rank r fc alternative x;. Figure 10.4 illustrates the favourable rank weights for the alternz tives. We can see that alternative x2 will always obtain either rank | or 2 and neve rank 3 or 4, while x4 can only receive either one of the two last ranks. The rele tive size of set W,” is the rank acceptability index, and it measures the variety c weights that give rank r to alternative x;. We can see that the second and third ran favourable weights are not necessarily connected sets. Therefore the midpoints c the favourable rank weights are meaningful only for the first rank. The weight space analysis is simple to perform graphically or analytically i
Fig. 10.4 Favourable rank weights and rank acceptability indices in deterministic 2-criterion case with linear utility function tee a eee ee ee Ee Se Ee ee eee ee a Besides the weights that make an alternative x; most preferred, we can als identify the sets of favourable rank weights W,’ that give a particular rank r fc alternative x;. Figure 10.4 illustrates the favourable rank weights for the alternz tives. We can see that alternative x2 will always obtain either rank | or 2 and neve rank 3 or 4, while x4 can only receive either one of the two last ranks. The rele tive size of set W,” is the rank acceptability index, and it measures the variety c weights that give rank r to alternative x;. We can see that the second and third ran favourable weights are not necessarily connected sets. Therefore the midpoints c the favourable rank weights are meaningful only for the first rank. The weight space analysis is simple to perform graphically or analytically i
Fig. 10.5 Rank acceptability indices in deterministic 2-criterion problem
Fig. 10.5 Rank acceptability indices in deterministic 2-criterion problem
Fig. 10.6 Examples of simulated cardinal values for four ranks is represented by a directed acyclic graph between the alternatives. In SMAA-O, ordinal criteria are treated by simulating corresponding (unknowr cardinal criteria values. The best alternative(s) corresponds to cardinal value and the worst alternative(s) corresponds to 0. In the case of a complete ranking the intermediate ranks correspond to values between 0 and 1. These values ar generated randomly from a uniform distribution, sorted into descending order, an assigned for the intermediate ranks. Equally good alternatives on the same ran level are assigned the same value. In the case of a partial ranking, sorting can b implemented efficiently using a variant of the topological sorting algorithm [22 Figure 10.6 illustrates random cardinal values generated for a complete rankin with four different ranks. — 1 1 1 7 a 1 ee er 7 1 le ae 1
Fig. 10.6 Examples of simulated cardinal values for four ranks is represented by a directed acyclic graph between the alternatives. In SMAA-O, ordinal criteria are treated by simulating corresponding (unknowr cardinal criteria values. The best alternative(s) corresponds to cardinal value and the worst alternative(s) corresponds to 0. In the case of a complete ranking the intermediate ranks correspond to values between 0 and 1. These values ar generated randomly from a uniform distribution, sorted into descending order, an assigned for the intermediate ranks. Equally good alternatives on the same ran level are assigned the same value. In the case of a partial ranking, sorting can b implemented efficiently using a variant of the topological sorting algorithm [22 Figure 10.6 illustrates random cardinal values generated for a complete rankin with four different ranks. — 1 1 1 7 a 1 ee er 7 1 le ae 1
In the extreme case, no weight information is available. However, even in that case, we can assume something about the weights. Normally the MCDA problem can be formulated so that all DMs want to maximize the outcome for each criterion. There- fore, we can assume that the weights should be non-negative. Also, practically all decision models are insensitive to scaling all weights by a positive factor. There- fore, we can, without loss of generality, assume that the weights are normalized somehow, e.g., so that their sum equals |. This means that the feasible weight space is an (n — 1)-dimensional simplex (see Eq. 10.3). Figure 10.7 illustrates the feasi- ble weight space in the 3-criterion case as a triangle with corners (1,0,0), (0,1,0) and (0,0,1). We therefore represent missing weight information by a uniform weight dis- tribution in the feasible weight space. Because the uniform distribution carries the least amount of information, it is well justified to use it to represent missing weight information.
In the extreme case, no weight information is available. However, even in that case, we can assume something about the weights. Normally the MCDA problem can be formulated so that all DMs want to maximize the outcome for each criterion. There- fore, we can assume that the weights should be non-negative. Also, practically all decision models are insensitive to scaling all weights by a positive factor. There- fore, we can, without loss of generality, assume that the weights are normalized somehow, e.g., so that their sum equals |. This means that the feasible weight space is an (n — 1)-dimensional simplex (see Eq. 10.3). Figure 10.7 illustrates the feasi- ble weight space in the 3-criterion case as a triangle with corners (1,0,0), (0,1,0) and (0,0,1). We therefore represent missing weight information by a uniform weight dis- tribution in the feasible weight space. Because the uniform distribution carries the least amount of information, it is well justified to use it to represent missing weight information.
linear inequality constraints based on the intervals. Generation of weights from the restricted distributioncan be implemented easily by modifying the above procedure to reject weights that do not satisfy the interval constraints. Upper bounds can be treated efficiently using the rejection technique, but in high-dimensional cases lower bounds may cause a high rejection rate. A special scaling technique can be applied to treat lower bounds efficiently [69]. Figure 10.9 illustrates the resulting weight distribution. The interval constraints are parallel to the sides of the triangle.
linear inequality constraints based on the intervals. Generation of weights from the restricted distributioncan be implemented easily by modifying the above procedure to reject weights that do not satisfy the interval constraints. Upper bounds can be treated efficiently using the rejection technique, but in high-dimensional cases lower bounds may cause a high rejection rate. A special scaling technique can be applied to treat lower bounds efficiently [69]. Figure 10.9 illustrates the resulting weight distribution. The interval constraints are parallel to the sides of the triangle.
importance ranking for some criteria (w; ? w;) or equal importance (w; = wz ;). In the general case, ordinal preference information means that the DMs’ preference statements correspond to a partial importance ranking of the criteria. Multiple DMs may either agree on a common partial ranking, or they can provide their own rank- ings, which can then be combined into a partial ranking that is consistent with each DM’s preferences. Without any consensus, the combined order may result into ab- sent preference information. ru . _ y ae 1 cr aot 4, a ao ae
importance ranking for some criteria (w; ? w;) or equal importance (w; = wz ;). In the general case, ordinal preference information means that the DMs’ preference statements correspond to a partial importance ranking of the criteria. Multiple DMs may either agree on a common partial ranking, or they can provide their own rank- ings, which can then be combined into a partial ranking that is consistent with each DM’s preferences. Without any consensus, the combined order may result into ab- sent preference information. ru . _ y ae 1 cr aot 4, a ao ae
Implicit weight information can result, e.g., from DMs’ holistic preference state- ments “alternative x; is more preferred than x;”. This can be expressed as x; > Xx, and it means that the weights and other possible preference parameters must be such that the decision model ranks the alternatives consistent with the preference state- ments. In the case of an additive utility or value model, such holistic constraints result into general linear constraints in the weight space, as illustrated in Fig. 10.12. In the general case, with non-additive utility/value function model, outranking mod- els, etc., holistic constraints correspond to non-linear constraints in the weight space. The rejection technique can be used to generate the weights with arbitrary decision models.
Implicit weight information can result, e.g., from DMs’ holistic preference state- ments “alternative x; is more preferred than x;”. This can be expressed as x; > Xx, and it means that the weights and other possible preference parameters must be such that the decision model ranks the alternatives consistent with the preference state- ments. In the case of an additive utility or value model, such holistic constraints result into general linear constraints in the weight space, as illustrated in Fig. 10.12. In the general case, with non-additive utility/value function model, outranking mod- els, etc., holistic constraints correspond to non-linear constraints in the weight space. The rejection technique can be used to generate the weights with arbitrary decision models.
Table 10.2, SMAA applications
Table 10.2, SMAA applications
The existence of a collective preference therefore presents a dilemma: We must either accept that group preference can be intransitive, or we must make one person a dictator. The first option is to accept that even though A is preferred to B and B to C, it does not follow that A is preferred to C, as transitivity would require—in this case, in fact, the opposite preference holds. The second option is to accept that a collective preference is transitive only because one person is more important than the other two combined. If the collective ordering in the Condorcet example A to B to C, for instance, then person | is more important than 2 and 3 combined, in that 1’s preference of A to C outweighs 2’s and 3’s preferences of C to A. Simply put, person | a dictator, so the group preference is the same as 1’s preference, and 2 and 3 simply don’t matter. In any case, we must accept that even when individual preferences are well-behaved, a well-behaved group preference that reflects all in- dividual preferences may not exist.
The existence of a collective preference therefore presents a dilemma: We must either accept that group preference can be intransitive, or we must make one person a dictator. The first option is to accept that even though A is preferred to B and B to C, it does not follow that A is preferred to C, as transitivity would require—in this case, in fact, the opposite preference holds. The second option is to accept that a collective preference is transitive only because one person is more important than the other two combined. If the collective ordering in the Condorcet example A to B to C, for instance, then person | is more important than 2 and 3 combined, in that 1’s preference of A to C outweighs 2’s and 3’s preferences of C to A. Simply put, person | a dictator, so the group preference is the same as 1’s preference, and 2 and 3 simply don’t matter. In any case, we must accept that even when individual preferences are well-behaved, a well-behaved group preference that reflects all in- dividual preferences may not exist.
Further Investigation: In 2002, after several meetings and public consultations, Kitchener City Council adopted alternative A!. In a study of this project [16], the authors reanalyzed the decision process by applying MCDA tools to A!, A”, and A?, the three alternatives that survived the screening process. Criteria were selected for the evaluation of these alternatives, as follows: C,: protection of human health and the environment; C2: acceptability to the community; C3: operational and main- tenance costs; C4: expected property tax returns; Cs: effect on property values; C¢: demolition time; C7: amount of waste to be excavated. Tha analxoie wae pnnndirtead wocing tho Awnalsstinr Li ararvnhty Dennancc (A LID)\ FQ21
Further Investigation: In 2002, after several meetings and public consultations, Kitchener City Council adopted alternative A!. In a study of this project [16], the authors reanalyzed the decision process by applying MCDA tools to A!, A”, and A?, the three alternatives that survived the screening process. Criteria were selected for the evaluation of these alternatives, as follows: C,: protection of human health and the environment; C2: acceptability to the community; C3: operational and main- tenance costs; C4: expected property tax returns; Cs: effect on property values; C¢: demolition time; C7: amount of waste to be excavated. Tha analxoie wae pnnndirtead wocing tho Awnalsstinr Li ararvnhty Dennancc (A LID)\ FQ21
This rather typical environmental conflict was studied by Kilgour et al. [54], who assessed what negotiation support could have been provided by the Graph Model ‘or Conflict Resolution and the DSS GMCR II [34, 53]). The application of GMCR I takes into account not only multilateral negotiation, but also the possible benefits of coalition formation. Essentially, one asks how well each DM can do on his or 1er own, and then whether some group of DMs would benefit by cooperating in a -oalition.
This rather typical environmental conflict was studied by Kilgour et al. [54], who assessed what negotiation support could have been provided by the Graph Model ‘or Conflict Resolution and the DSS GMCR II [34, 53]). The application of GMCR I takes into account not only multilateral negotiation, but also the possible benefits of coalition formation. Essentially, one asks how well each DM can do on his or 1er own, and then whether some group of DMs would benefit by cooperating in a -oalition.
To this effect, a recent study [32] has demonstrated that a particular EMO procedure, starting from random non-optimal solutions, can progress towards the theoretical Karush-Kuhn-Tucker (KKT) points with iterations in real-valued multi-objective optimization problems. The main difference and advantage of using an EMO com- pared to a posteriori MCDM procedures is that multiple trade-off solutions can be found in a single run of an EMO algorithm, whereas most a posteriori MCDM methodologies would require multiple independent runs. Tee QGaencee 1 Af thw CRAATL *\.. hee eee bee ils eee Ae BRA Hee Sie bcaw
To this effect, a recent study [32] has demonstrated that a particular EMO procedure, starting from random non-optimal solutions, can progress towards the theoretical Karush-Kuhn-Tucker (KKT) points with iterations in real-valued multi-objective optimization problems. The main difference and advantage of using an EMO com- pared to a posteriori MCDM procedures is that multiple trade-off solutions can be found in a single run of an EMO algorithm, whereas most a posteriori MCDM methodologies would require multiple independent runs. Tee QGaencee 1 Af thw CRAATL *\.. hee eee bee ils eee Ae BRA Hee Sie bcaw
Fig. 12.9 Obtained nondominated solutions using NSGA Fig. 12.10 Four trade-off trajectories
Fig. 12.9 Obtained nondominated solutions using NSGA Fig. 12.10 Four trade-off trajectories
Fig. 13.1 Vector and raster data models in GIS GIS utilizes two basic types of data: spatial data and attribute data [76]. The former describes the absolute and relative locations of spatial (geographic) entities (e.g., building, parcel of land, street, river, lake, state, country, etc.). The attributes re- fer to the properties of spatial entities. These properties can be quantitative and/or qualitative in nature. Attribute data are often referred to as tabular data. Spatial data are typically arranged in a GIS using one of two models: vector and raster (Fig. 13.1). Entities in vector format are represented by strings of coordinates.
Fig. 13.1 Vector and raster data models in GIS GIS utilizes two basic types of data: spatial data and attribute data [76]. The former describes the absolute and relative locations of spatial (geographic) entities (e.g., building, parcel of land, street, river, lake, state, country, etc.). The attributes re- fer to the properties of spatial entities. These properties can be quantitative and/or qualitative in nature. Attribute data are often referred to as tabular data. Spatial data are typically arranged in a GIS using one of two models: vector and raster (Fig. 13.1). Entities in vector format are represented by strings of coordinates.
Although the advent of desktop computing and cartographic modeling in the 1980s was instrumental in stimulating the development of GIS-MCDA, it was not until the 1990s that GIS-MCDA established itself as an identifiable subfield of research within the GIS literature [16, 29, 36, 66, 82, 104, 105, 115, 118, 145, 173]. The pro- liferation of GIS-MCDA can be illustrated by examining the increasing number of relevant publications in refereed journals. Figure 13.2 shows that there has been an exponential growth of the number of refereed publications on GIS-MCDA in the post-1990 period [18]. The rapid increase in the volume of GIS-MCDA research can be attributed to two main factors. First, during the 1990s, increasingly powerful personal computer-based GIS software was developed, refined, and utilized in ap- plications. GIS has gradually been regarded as a routine software application within
Although the advent of desktop computing and cartographic modeling in the 1980s was instrumental in stimulating the development of GIS-MCDA, it was not until the 1990s that GIS-MCDA established itself as an identifiable subfield of research within the GIS literature [16, 29, 36, 66, 82, 104, 105, 115, 118, 145, 173]. The pro- liferation of GIS-MCDA can be illustrated by examining the increasing number of relevant publications in refereed journals. Figure 13.2 shows that there has been an exponential growth of the number of refereed publications on GIS-MCDA in the post-1990 period [18]. The rapid increase in the volume of GIS-MCDA research can be attributed to two main factors. First, during the 1990s, increasingly powerful personal computer-based GIS software was developed, refined, and utilized in ap- plications. GIS has gradually been regarded as a routine software application within
Table 13.1 Classification of the GIS-MCDA articles according to the GIS data model, the spatial dimension of the evaluation criteria (EC), and the spatial definition of decision alternatives (DA)
Table 13.1 Classification of the GIS-MCDA articles according to the GIS data model, the spatial dimension of the evaluation criteria (EC), and the spatial definition of decision alternatives (DA)
Table 13.2 Classification of the GIS-MCDA articles according to the multicriteria decision rule multiattribute decision analysis (MADA) and multiobjective decision analysis (MODA). Some articles presented more than one combination rule 13.4.3 MCDA Components of GIS-MCDA
Table 13.2 Classification of the GIS-MCDA articles according to the multicriteria decision rule multiattribute decision analysis (MADA) and multiobjective decision analysis (MODA). Some articles presented more than one combination rule 13.4.3 MCDA Components of GIS-MCDA
Note: percentages of the total are given in brackets Table 13.3 Classification of GIS-MCDA papers according to the type of multicriteria decisior method for individual decision maker
Note: percentages of the total are given in brackets Table 13.3 Classification of GIS-MCDA papers according to the type of multicriteria decisior method for individual decision maker
Note: percentages of the total are given in brackets Table 13.4 Classification of GIS-MCDA papers according to the type of multicriteria decision methods for group decision making
Note: percentages of the total are given in brackets Table 13.4 Classification of GIS-MCDA papers according to the type of multicriteria decision methods for group decision making
Related topics:
Multiple Criteria Decision MakingMulticriteria Decision AnalysisProductivity and Efficiency Measurement
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