- Email: [email protected]
Applying voting theory in natural resource management: a case of multiple-criteria group decision support
Journal of Environmental Management (2002) 64, 127–137 doi:10.1006/jema.2001.0511, available online at http://www.idealibrary.com on
Applying voting theory in natural resource management: a case of multiple-criteria group decision support Sanna Laukkanen† , Annika Kangas*‡ and Jyrki Kangas‡ † University ‡ Finnish
of Joensuu, P.O. Box 111, FIN-80101 Joensuu, Finland Forest Research Institute, P.O. Box 44, FIN-69101 Kannus, Finland
Received 22 January 2001; accepted 7 September 2001
Voting theory has a lot in common with utility theory, and especially with group decision-making. An expected-utilitymaximising strategy exists in voting situations, as well as in decision-making situations. Therefore, it is natural to utilise the achievements of voting theory also in group decision-making. Most voting systems are based on a single criterion or holistic preference information on decision alternatives. However, a voting scheme called multicriteria approval is specially developed for decision-making situations with multiple criteria. This study considers the voting theory from the group decision support point of view and compares it with some other methods applied to similar purposes in natural resource management. A case study is presented, where the approval voting approach is introduced to natural resources planning and tested in a forestry group decision-making process. Applying multicriteria approval method was found to be a potential approach for handling some challenges typical for forestry group decision support. These challenges include (i) utilising ordinal information in the evaluation of decision alternatives, (ii) being readily understandable for and treating equally all the stakeholders in possession of different levels of knowledge on the subject considered, (iii) fast and cheap acquisition of preference information from several stakeholders, and (iv) dealing with multiple criteria. 2002 Elsevier Science Ltd.
Keywords: forest planning, group decision support, multicriteria approval, natural resource management, voting theory.
Introduction Nowadays, forest planning and forest decisionmaking are facing new requirements: planning solely for timber production is not enough any longer. Forest owners may have many other objectives such as those related to recreation, picking wild berries, beautiful scenery and nature conservation. Consequently, there is a need for multi-objective forest planning and decision support in forestry. This applies to all owner groups, and to natural resource management in a wider scope. Often, there are numerous decision-makers Ł
Corresponding author. Present address: University of Helsinki, P.O. Box 27, FIN-00014 Helsinki, Finland. Email: [email protected]
0301–4797/02/020127C11 $35.00/0
involved in planning as well. Each of them can have different aims and objectives concerning the use of forests or other natural resources. The aims are often contradictory and they should somehow be made commensurable to find the best or at least a good compromise solution. This is why effective methods are needed for group-decision support in forestry. In Finland the number of forest holdings owned by various consortia as well as heirs is increasing. At least 50 000 estates of the deceased and other consortia own forest. This means that 27% of the private non-industrial forests in Finland are owned by consortia and heirs at the moment (Simonen, 2000). These groups are problematic from the point of view of practical forest planning and forest counselling. Each action to be taken in forest needs to be approved by every shareholder. There has 2002 Elsevier Science Ltd.
128
S. Laukkanen et al.
been no method or application especially developed for this kind of group decision-making situations. Methods based on Multi-Attribute Utility Theory (MAUT) have been developed for purposes of natural resources planning. These methods have been applied to forest planning too. For example, the Analytic Hierarchy Process (AHP) has been used in supporting multi-objective group decisionmaking (e.g. Kangas, 1999). Methods generally applied to forest planning involve some problems when applied to multiobjective planning of a forest holding owned by a consortium. They may not be able to deal with information that is expressed on a scale other than the ratio scale. Consequently, there is a need for a method that can make use of ordinal information. (In order to utilise qualitative information in decision-making in general, a (possibly subjective) preference order is always needed.) Another problem is that in many decision support methods answering the necessary inquiries are difficult for decision-makers. The decision-makers may include experts familiar with the decision analysis methods. Therefore, they may have better capabilities to answer these inquiries than those who do not have any prior information about the methods. Especially in group decision-making, the methods the application of which necessitates many complicated questions to be made also demand more time and guidance. In group decision support, the planning system also needs to be difficult to manipulate. One possible solution for these problems are the outranking methods (e.g. Vincke, 1992). These include decision support methods that are able to exploit ordinal and imprecise data (Pirlot, 1997). They are rather easy for decision-makers in the sense that less questions are asked of them than in many other decision support methods. Also, ratio scale weights for the criteria are not required. On the other hand, the thresholds used in the analysis may be difficult to understand. What is more important, the method itself is far from being transparent to the decision-makers (e.g. Hokkanen and Salminen, 1997a,b; Kangas et al., 2001). The family of methods based on stochastic multicriteria acceptance analysis (SMAA), provides another possible solution. It is possible to carry out an analysis without any preference information, and imprecise information can also be handled. (Lahdelma et al., 1998; Hokkanen et al., 2000; Lahdelma and Salminen, 2001). Another possibility is to apply the voting theory (e.g. Cranor, 1996). Throughout the history of democracy, voting has proved to be an efficient tool for making choices among decision alternatives,
e.g. in different kind of elections. It is also, in one form or another, known to practically everyone. In the voting theory, special attention has been paid to systems that are neutral and difficult to manipulate (Nurmi, 1987). Therefore, voting theory can be seen as a credible alternative in group decision-making and in supporting it. Most voting systems are based on a single criterion only or holistic preference information on decision alternatives. Nevertheless, there is also a voting scheme specially developed for multicriteria decision-making situations, namely multicriteria approval. In this article, different voting systems are presented and their qualities are discussed from the point of view of natural resources management. The voting theory is considered especially in the context of group decision support, and methods based on voting theory are also compared with some other MCDM methods applied to similar purposes in the natural resource management. A case study is presented, showing an application of the approval voting approach to natural resources planning and testing of its applicability in a forestry group decision-making process.
Voting theory in decision support Voting procedures as decision support methods Many voting systems have a lot to do with the utility theory. The most common of these, plurality voting, for instance, takes into account the preference ordering of the voters, albeit only with respect to the best candidate. A major drawback of this method is that it is generally considered to be very easy to manipulate. However, it can be shown that any non-dictatorial voting scheme is subject to manipulation (Gibbard, 1973; Sattherthwaite, 1975). However, some methods require more information to manipulate an election than others (Nurmi, 1987). The approval voting method was independently proposed by several people in the 1970s (Brams and Fishburn, 1983). It is a voting procedure in which each voter votes for as many candidates as she/he wishes. In other words, the voter votes for all candidates of whom she/he ‘approves’ (McBurnett http://bcn.boulder). Each candidate approved by the voter receives one vote. The candidate receiving the greatest total number of votes is declared to be the winner. This method is harder to manipulate by any one voter than plurality voting. This
Multicriteria approval
is because it requires information about the distribution of approvals of the alternatives in order to be manipulated (Nurmi, 1987). However, it tends to promote moderate candidates. The Borda count (e.g. Saari, 1994) takes into account the whole preference ordering: given n candidates, each voter casts n votes for the most preferred candidate, n 1 votes for the second most preferred one and finally 1 vote for the least preferred candidate. The winner is the candidate getting the most votes. The problem with this method is that the result may be the function of the number of candidates (Riker, 1982). Also this method has been criticised because of its susceptibility to manipulation, but according to Nurmi (1987) it is more difficult to manipulate than the aforementioned methods. The aforementioned voting systems only use information of the preference ordering of the alternatives (ordinal information). So called utilitarian voting systems make use also of information on the intensity of the preferences (cardinal information). These systems include, for example, direct aggregation of cardinal utilities and the demand-based method (e.g. Riker, 1982; Mueller, 1989). In the former case, voters are required to assign utility values to the candidates, and the candidate receiving the largest sum of utility values wins. In the latter, voters offer to pay some money in order to obtain a preferred alternative, and the one getting the highest sum of bids wins. Both of these methods are easy to manipulate by inflating the utility value given to the most preferred alternative. Yet another method is the cumulative voting system, where each voter is given a number of votes they can distribute to the candidates in any way: all votes can, for instance, be given to one candidate or many candidates can be given an equal number of votes (e.g. Lakeman, 1974). A recently presented voting system is a so-called declared-strategy voting (Cranor, 1996). It has been proved that it is often in the best interest to vote for a candidate other than the one most preferred by the voter – i.e. to vote strategically or to manipulate the election. However, such manipulation needs information about the other voters’ preferences, which is not available to each voter. Therefore, the voters with more information can have more power. In declared-strategy voting, however, this problem has been turned to a virtue: in this system all voters can vote strategically. Voters are first asked to give cardinal utility values to the candidates, but the utilities are not summed up like in the systems mentioned above. Then, voters submit their strategy to a computer. Such
129
strategies may be, for example, to vote for the most probable winner in order to show support (so-called bandwagon strategy), or to vote against the least preferred alternative. These strategies are then aggregated to determine the election results. This system has been seen as being especially suitable for group decision-making. However, it requires special software for forming the voting strategies and aggregating them. What is more, the result may depend somewhat on the order that the strategies of different voters are handled.
Multicriteria approval The ideas of approval voting can be utilised also in multicriteria decision support. Fraser and Hauge (1998) presented an application of approval voting, called multicriteria approval, which can be used in multicriteria decision-making. In multicriteria approval, the voters of social choice theory are substituted with multiple criteria. The criteria are ranked with respect to importance, unlike voters in social theory. The standard version of multicriteria approval has been developed for one decisionmaker, but if the decision-makers can agree upon the importance order of the criteria and the border for approval for each criterion, the standard version suits also for group decision-making. Multicriteria approval method begins with the determination of the alternatives and the criteria under which the alternatives will be compared. Secondly, decision-makers rank the criteria by their importance. (There is also the possibility of using cardinal weights of the criteria if a solution cannot be found using ordinal information and if cardinal weighting information is available.) The next step is to determine which of the alternatives will be approved for each criterion. This is done by defining where the border between approval and disapproval regarding each criterion lies. Fraser and Hauge (1998) defined the border as the average evaluation of the alternatives with respect to the criterion considered. In other words, each alternative i, with respect to each criterion j, is approved if the criterion value is above average, and disapproved if otherwise. With numerical criterion values cj .ai /, the border is the mean P cj .ai / of the m alternatives, criterion value cj D m iD1 m with respect to each criterion j. The average value has been chosen as the borderline because, according to approval voting theory, it can be proved that the optimal strategy for a voter is to vote for all the candidates that are above
130
S. Laukkanen et al.
average (e.g. Kim and Roush, 1980). The border between approval and disapproval could, however, be defined also in other ways. For example, if the utility achieved is a non-linear function of a criterion variable, the optimal borderline should be the average utility value, not the average criterion value (Fraser and Hauge, 1998). The borderlines could also be defined as, for example, some proportion of the maximum value for each criterion or as some percentile of the range of variation. The borderline could also be defined in a different way for each criterion. Such an approach may not be reasonable in voting theory, where all voters are defined equally important. In decision support, however, some criteria usually are more important than others. Then, increasing the borderline value would emphasise the importance of a criterion; decreasing it would do the opposite. Ordinal information, with a verbal scale such as Excellent, Good, Average, Passable, Poor, can be readily used for approval and disapproval. If there are qualitative criteria, the preferences of decisionmakers need to be separately questioned. In the group decision-making situation, each decisionmaker can form his/her own judgement for each alternative. The final judgement of approval or disapproval of the alternatives will then be formed based on combinations of these judgements. For example, an alternative which is at least average for all (or majority of) decision-makers is approved and the others are disapproved. The same method can be used if the values of some criterion were evaluated by expert opinion. The determination of the voting result begins after defining the dichotomous preferences. There are five possible classes of voting result: Unanimous, Majority, Ordinally Dominant, Deadlocked and Indeterminate. In this article the classification is presented the way Fraser and Hauge (1998) have described it. The voting result is Unanimous if one alternative has been approved with respect to all criteria and no other alternative has been approved with respect to all criteria. Majority result occurs when one alternative has been approved with respect to the majority of criteria which have been defined to be the most important. If one alternative has been defined to be superior on the grounds of the order of criteria, and the dichotomous preferences, the result is Ordinally Dominant. The class Unanimous is a subclass of Majority and both of these are subclasses of Ordinally Dominant. The result is Deadlocked if there are two or more alternatives that are defined to belong to the subsets Unanimous, Majority or Ordinally Dominant.
These alternatives are approved with respect to the same criteria. They are also disapproved with respect to the same criteria, and so it is not possible to determine one single superior alternative. The result is Indeterminate if there is not enough information to determine one superior alternative. There are two or more alternatives between which it is impossible to name the best. More information is needed in addition to the criteria order. The first phase in determining the voting class is to find out the number of existing Ordinally Dominant alternatives. According to Fraser and Hauge (1998) this is done by using a variation of the ordinal deductive selection system (1) (Meister and Fraser, 1995). Alternative k is classified to be Ordinally Dominant if f .nŁ /ki ½0 8nŁ , 1nŁ n, 8i6D k where
n X
.1/
Ł
Ł
f .n /ki D
gijk
jD1
and
1, if cj .ak />cj ^cj .ai /cj 0, if cj .ak />cj ^cj .ai />cj gijk D 0, if cj .ak /cj ^cj .ai /cj 1, if cj .ak /cj ^cj .ai />cj
Fraser and Hauge (1998) describe the principle of this formula as follows: ‘Alternative k is classified Ordinally Dominant if for every possible value of nŁ , and all other alternatives i6D k, f .nŁ /ki remains greater than or equal to zero. If at anytime, for any i6D k it becomes negative, alternative k is labelled as Indeterminate’. The process is based on the idea that approval with respect to a more important criterion can completely compensate disapproval with respect to a less important criterion (Fraser and Hauge, 1998). After this process each alternative is labelled as either Ordinally Dominant or Indeterminate. If there is only one single alternative that is Ordinally Dominant next step is to examine if it belongs to subclasses of Ordinally Dominant. If it does not belong to either Unanimous or Majority then it is classified as Ordinally Dominant.
Case study Alternative plans and their evaluation The multicriteria approval method was used in a real forest planning situation. The problem was to
Multicriteria approval
choose a forest plan for a forest holding owned by a consortium consisting of three forest owners. All of them have a 1/3 share in the forest holding. The forest holding is 30 hectares in size and consists of three separate blocks. The holding has been divided into a total of 13 forest stands, i.e. compartments, each being relatively homogeneous as regards the soil and stand characteristics. As is usual in the management of boreal coniferous-dominated forests, these compartments were regarded as basic decision units. In forest planning one treatment schedule is chosen for each compartment, to be implemented throughout the compartment. The planning period consisted of two 10-year periods. Twenty alternative forest plans were produced, each consisting of a combination of compartment-wise treatment schedules for the 20year planning horizon. The forest owners had different objectives as regards forest utilisation. One of them aimed at timber production, the beauty of the forest landscape and wild berry yield. The second aimed at timber production, biodiversity of the forest, the beauty of the forest landscape, and conserving habitats for capercaillie (Tetrao urogallus), a valued game bird. The third mainly preferred timber production. After discussions held with the forest owners the following decision criteria were settled upon. The choices regarding the criteria were made by the forest owners, guided by a consultant. In the case of the timber production, the criteria were (i) the present value of net incomes over the 20years planning horizon and (ii) the estimated value of future timber production at the end of the planning period. Both of these were calculated utilising simulation of forest development, and by applying a discount rate of 3%. The wild berry yield was measured with the bilberry yield index. The criteria values of timber production and bilberry yield were obtained directly via the Monsu forest planning software (Pukkala, 1998). Monsu includes a simulation part, predicting the criteria values in each compartment, under different treatment schedules, using several interrelated statistical models. Monsu also includes a mathematical optimisation part, which was utilised in producing plans that were efficient in regard to the criteria that could be taken into consideration in Monsu. Judgements of the other criteria were partly subjective, and therefore these judgements were made after alternative plans were prepared. In all, twenty forest plans were produced. The quality of the forest with respect to the habitat needs of capercaillie was the only criterion
131
that could not to be taken into consideration in the calculations of Monsu at all. However, as both the viability of the capercaillies and biodiversity were supposed to depend on the amount of the old forest and felling, those criteria were combined and this criterion was called biodiversity. Concerning this criterion the alternative plans were evaluated with the help of an expert on the ecology of capercaillie and on forest biodiversity, by means of pairwise comparisons as in the AHP (Saaty, 1977; see Kangas et al., 1993). These values were linearly scaled for the purposes of multicriteria approval. The evaluation of the scenic beauty of forest landscape was also partly subjective. However, the pairwise comparisons of the twenty plans would have been too laborious and time-consuming for the decision-makers. Consequently, the evaluations were carried out so that each decision-maker individually formed the weights for the different blocks of the forest holding by means of pairwise comparisons, as different blocks had different importance with respect to scenic values. The final landscape beauty evaluations were made with the help of these weights and the scenic beauty indices given by Monsu. The criterion values in alternative forest plans are presented in Table 1. The next phase in the case study was to rank the criteria by their importance. A mutual analysis of the decision-makers was agreed upon. The mutual order was obtained so that, using values 1–7, all decision-makers defined the weights that describe the importance of the criteria. The final order of the criteria was formed on the basis of the average values of the decision-makers’ weightings. The average values were used for importance ranking to enable the utilisation of the same data also in PROMETHEE analysis (see Laukkanen et al., 2001). This method was approved of by all decisionmakers. Another possibility, perhaps more in accordance with the ideas of multicriteria approval, would have been to use some voting system (e.g. Borda count, which combines individual rankings). The order of the criteria from the most important to the least important was: Net incomes, Biodiversity, Monetary value of the future timber production, Scenic beauty of the forest landscape, and Wild berry yield. Next, the limits between approval and disapproval for each criterion were defined (Table 2). The limits were calculated directly from the average of the criterion values of the alternatives. For the sake of sensitivity analysis, also the midpoint of the range of variation was used as a border of approval.
132
S. Laukkanen et al. Table 1. Alternative plans
Criterion values in alternative plans Net incomes, present value 2001–2020, FIM
Biodiversity 2020
Value of the future timber production 2020, FIM
Scenic beauty 2020
Bilberry yield 2020, kg/ha
39 740 15 011 96 900 151 503 101 652 45 092 73 225 134 780 66 079 154 163 86 666 133 135 36 664 90 931 81 864 114 550 122 782 62 228 105 142 95 177
1Ð143 1Ð436 0Ð419 0Ð121 0Ð407 1Ð114 0Ð441 0Ð139 0Ð704 0Ð116 0Ð471 0Ð168 1Ð107 0Ð275 0Ð543 0Ð289 0Ð160 0Ð444 0Ð232 0Ð271
876 516 923 254 739 278 543 862 752 080 862 810 777 945 626 711 826 339 532 995 796 026 650 071 883 552 724 857 787 786 682 128 577 407 810 837 684 919 714 019
6Ð2356 6Ð2982 6Ð0517 5Ð7230 5Ð8641 6Ð2196 6Ð1927 5Ð3174 6Ð1106 5Ð3639 5Ð9070 5Ð7391 6Ð2356 6Ð1302 6Ð0517 5Ð9892 6Ð0356 6Ð2356 6Ð0517 6Ð1142
10Ð7 11Ð6 8Ð2 5Ð2 8Ð0 10Ð6 8Ð7 6Ð0 9Ð6 5Ð1 8Ð7 6Ð0 10Ð8 7Ð3 8Ð9 6Ð8 5Ð9 9Ð1 6Ð7 7Ð3
P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 P11 P12 P13 P14 P15 P16 P17 P18 P19 P20
Table 2. The limit between approval and disapproval for each criterion using the average criterion value, or the midpoint of the range of variation Criterion
cj D
m X cj .ai / iD1
Net incomes Biodiversity Value of the future timber production Scenic beauty Bilberry yield
m
90 364Ð2 0Ð5 738 669Ð6 5Ð99 8Ð06
cQ j D
max.cj /Cmin.cj / 2 84 587Ð0 0Ð8 728 124Ð5 5Ð81 8Ð35
Results of the case study The next step was to examine which of the alternatives were above and which were below the average for each criterion; in other words which alternatives were approved and which alternatives were disapproved for each criterion (Table 3). For identifying a class of voting result, a test was conducted to see how many Ordinally Dominant alternatives existed. Each alternative was compared with other alternatives in accordance with P the formula f .nŁ /ki D njD1 gijk . The alternatives P1, P2, P6, P9, P13, P15 were found to be acceptable for all other criteria except Net income. However, these alternatives could not be Ordinally Dominant because f .nŁ /ki took a negative value immediately in the beginning of the
comparison. The table of approval and disapproval shows that only alternatives P3 and P5 could be Ordinally Dominant. After comparison calculations it turned out that only alternative P3 was Ordinally Dominant, because in the pairwise comparisons alternative P3 did not lose to any other alternative. In other words, f .nŁ /3i never took a negative value. The alternatives P5 and P3 were even when the three most important criteria were examined but alternative P3 dominated alternative P5 as regards the two least important criteria. After this, alternative P3 was examined to find out whether or not it belonged to the subclasses of Ordinally Dominant. Since this forest plan alternative was not approved of by all five criteria it could not be an unanimous winner. Neither was it a majority winner since it was not approved by the second-ranked criterion. The conclusion was that plan alternative P3 did not belong to either of the subsets and therefore it was classified as Ordinally Dominant. Using the midpoint of the range of variation as the border decreased the border for Net incomes, Future value of timber production and Scenic beauty and increased the border for the other criteria (Table 2). Therefore, more alternatives were approved with respect to Net incomes, and less with respect to Wild berry yield, for example (Table 4). Then, alternative P3, which was Ordinally Dominant in the previous case, was no longer approved by the Wild berry yield criterion.
Multicriteria approval Table 3. border Alternative P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 P11 P12 P13 P14 P15 P16 P17 P18 P19 P20
133
Approvals (C) and disapprovals ( ) of the alternatives using average criterion value as Net incomes
Biodiversity
Value of the future timber production
Scenic beauty
Bilberry yield
C C
C C C
C C C
C C C
C
C C C
C C
C C
C
C
C
C
C C C C C
C
C
C C
C
C
C
C C C C C
C C C C C C C
C C C
Table 4. Approvals (C) and disapprovals ( ) of the alternatives using the midpoint of range of variation as border Alternative P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 P11 P12 P13 P14 P15 P16 P17 P18 P19 P20
Net incomes
Biodiversity
Value of the future timber production
Scenic beauty
Bilberry yield
C C
C C C
C C C
C C
C C C
C C C
C C
C
C
C
C
C
C
C
C C C C C C C C
C
C C C C C C C C C C
C C C C C C
Instead, alternative P11 became approved by the Net incomes and Scenic beauty as well, and was found Ordinally Dominant. Therefore, even small changes in the approval border may change the results of multicriteria analysis. Also, adding a new decision alternative to the analysis may cause a rank reversal among the ‘old’ alternatives, if approval borders are calculated as a function of the criterion values of the alternatives considered.
C C
Discussion Characteristics of multicriteria approval In this study, an application of voting theory called multicriteria approval was introduced into the field of natural resource management and tested in forestry group decision support. Multicriteria
134
S. Laukkanen et al.
approval has not been applied to forest planning before. It seems to satisfy many requirements of multicriteria forestry decision support, because its main aim is to find an alternative that is widely satisfactory among the multiple criteria. It is highly suitable for situations where information that other methods demand is difficult or expensive to obtain, or where the available information is of low quality. The method is easy to explain to the decision-makers and it is readily understandable. There can be different kind of decision-makers, experts and non-experts, and the method is still fair for everyone. The only preference information that the decision-makers need to be asked is the importance order of the criteria. This seems easy enough, but such ranking may be more problematic than it seems. For example, the importance order of the criteria may depend on the current values of the criteria, and on the range of values for the criteria. However, the situation is similar for most practical decision support systems (von Winterfeldt and Edwards, 1986). Due to its minimal information requirements multicriteria approval does not consume much time; thus, the costs of using this method remain low. Since multicriteria approval is so easy for decision-makers, queries can be carried out using a post enquiry or, for example, via the Internet. Therefore, it could also be used for consortia with plenty of owners. In Finland, for instance, some common forests have hundreds, or even thousands of owners. Approaches based on voting theory would be worth applying also in participatory planning of natural resources governed by public institutions (e.g. State-owned forests), where their ¨ ainen ¨ merits would be very useful (see Pykal et al., 1999). However, evaluation of the alternatives on the grounds of qualitative criteria would demand more time and involvement of decision-makers than quantitative ones. This is because decisionmakers have to judge every alternative separately with respect to the criterion in question. All the criteria of importance are usually not included in common forest planning software applications. In this kind of a situation, forest planning software (typically based on mathematical optimisation) alone cannot solve the problem in a satisfactory manner, but other approaches are also needed. By using the multicriteria approval method all the criteria could be included in the case study, e.g. the biodiversity criterion with satisfactory habitats being provided for capercaillie. The benefits of both a quantitative biodiversity index produced by forest planning software and experts’
qualitative evaluations of alternatives with respect to biodiversity could be utilised in the multicriteria approval context. One disadvantage of the multicriteria approval method is that it does not always offer a solution for a problem of multicriteria decision-making. It does not put all the alternatives in order but instead chooses one or more best alternatives. If one Ordinally Dominant alternative can be found this method will choose it, but if the voting result is Deadlocked it chooses more than one alternative. In this study, an Ordinally Dominant alternative was found. In the case of a Deadlocked result Fraser and Hauge (1998) suggest that it would be useful to try to find the solution by using the run-off procedure. This procedure tries to solve the situation by taking into account only those alternatives that were among the deadlocked set and reanalysing the case. By ignoring some of the worst alternatives the mean criterion values and, correspondingly, the result of the analysis may change. Another possibility is to directly heighten the approval border, in order to reduce the number of Deadlocked alternatives. In the case of an Indeterminate voting result the method does not give any information about the decision at all. Then there is a need for more preference information to enable the best alternative to be found. In the case study, the border of approval and disapproval of each alternative with respect to each criterion was calculated directly with average criterion value for the quantitative criteria, based on the approval voting theory. As another possibility, the midpoint of the range of variation was used. It was noted, that the method is quite sensitive to the approval border. Therefore, it is also subject to rank reversal, when new alternatives are introduced or some alternatives are excluded. However, if the approval borders were fixed (i.e. independent of the criterion values of alternatives considered), rank reversal does not occur even if the number of alternatives were changed. From the practical point of view, however, this need not be a serious problem.
Comparison with other decision support techniques Comparison of multicriteria approval with other methods applied to similar decision support purposes, such as MAUT and outranking, is interesting. Outranking methods used for ranking alternatives typically produce an ordinal preference
Multicriteria approval
order (e.g. Pirlot, 1997; Lootsma and Shujit, 1997). In some cases, e.g. PROMETHEE II, the rankings also have cardinal properties. MAUT methods, for instance the AHP and SMART, give a measure of relative priorities to the alternatives (Saaty, 1980; von Winterfeldt and Edwards, 1986). Preference information is also expressed in different ways in these approaches. MAUT methods are typically based on cardinal preferences. When using outranking methods, the weights of criteria need to be defined but the method can use either ordinal or cardinal information. The most frequently used MAUT methods are based on an additive linear utility model, and thus they are fully compensatory (von Winterfeldt and Edwards, 1986). Outranking methods have both compensatory and non-compensatory elements (e.g. Pirlot, 1997). Multicriteria approval, on the other hand, is clearly non-compensatory. When the AHP and PROMETHEE II were tested for the same case study as was used here (see Laukkanen et al., 2001), the results obtained clearly differed from those obtained with multicriteria approval (Table 5). The AHP, using most sophisticated information and being fully compensatory, put the alternatives with the biggest incomes above the others (incomes being the most important criteria). The PROMETHEE II analysis chose the alternatives with the lowest incomes but better values for other criteria. Multicriteria approval chose a compromise solution. These results reflect the different Table 5. The rankings of the alternative plans of the case study based on AHP and PROMETHEE II analysis (see Laukkanen et al., 2001) Alternative P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 P11 P12 P13 P14 P15 P16 P17 P18 P19 P20
AHP
PROMETHEE II
5 4 11 1 8 3 19 7 13 2 14 6 9 18 12 10 15 20 16 17
2 1 7 18 14 4 8 19 5 20 10 17 3 12 6 13 16 9 15 11
135
properties of the methods (e.g. compensatory – noncompensatory), the different amount of information (ratio scale – dichotomous), and different criteria weights (ratio scale – rank). When multicriteria approval is applied to group decision support, a mutual rank order of the criteria is determined. The mutual approach is used also in outranking methods. In the AHP (or other cardinal utility-based methods) it is possible to assess the criteria weights separately for each stakeholder and then computationally combine the preference information obtained. When combining the separate analyses, each decision-maker is assigned a definite weight. Using common weights or even a common rank of criteria may seem theoretically problematic. In fact, the famous theorem of Arrow (1951) states that there is no method of aggregating individual preferences over three or more alternatives that would satisfy several conditions for fairness and always produce a logical result. However, in any practical group decision support the preferences of the stakeholders need to be combined one way or other – if not, the only decision support left would be negotiation. However, negotiations may be even more susceptible to manipulation than any decision support technique. In addition to weights, outranking techniques require preference and indifference thresholds. Below the indifference threshold q a decisionmaker is indifferent between two alternatives and above the preference threshold p a decision-maker expresses strict preference of one alternative over another (Brans et al., 1986). There is a hesitation zone between these thresholds. The approval border of multicriteria approval resembles the threshold values of outranking methods. However, it is probably easier to understand for ordinary decision-makers, since it is based on the consideration of only one alternative at a time, whereas in outranking methods the thresholds are based on differences between two alternatives. Although the preference judgements in outranking methods might be rather easy to express, the handling of preference information in calculations is difficult to explain to non-specialists. Problems in the comprehensibility and interpretability of the results are the drawbacks of outranking methods in general (Vincke, 1992). Outranking methods require guiding and initiation to threshold values. The AHP, in turn, requires even more guiding and time because the weights of the criteria are established by using pairwise comparisons. The number of the comparisons increases rapidly with the number of alternatives and criteria. However, using statistical regression approaches instead of Saaty’s
136
S. Laukkanen et al.
eigenvalue technique in the analysis of pairwise comparisons enables the use of incomplete comparison matrices (Alho et al., 1996). Ultimately, the basic differences between methods based on voting theory and other MCDM methods are not so clear as they might first seem. Several voting methods have been presented that make it possible to apply also cardinal preference information in the evaluation of alternative decisions. For example, the way the votes are cast in cumulative voting greatly resembles the evaluation of weights in some SMART methods (von Winterfeldt and Edwards, 1986). The Copeland score calculated from pairwise votes resembles the concordance index in outranking methods (see Nurmi, 1995). Indeed, according to Vincke (1992), the voting theory can be seen as an ancestor of outranking methods. From the point of view of practical natural resources management, perhaps the greatest advantage of multicriteria approval is in its ability to rank alternative decisions in a multiple-criteria planning situation on the grounds of much less and more rudimentary preference information than is required by other MCDM methods. Defects and low quality in the preference and evaluation information are problems frequently faced in forestry decision-making, especially in group decision support (e.g. cases of consortia-owned forests, heirs, and common forests) and in participatory planning processes. Besides, stakeholders normally have different abilities to participate in the decision support process and to give information on their preferences in modes required by MCDM methods. In voting approaches, the idea is to avoid giving an advantage to those with more expertise on the subject and on the methodology over other stakeholders. However, even though multicriteria approval can make use of low-quality information as it often persists in the reality, applying dichotomous preferences means loss of information in cases where the cardinal weighting of criteria and cardinal preference orderings of the alternatives with respect to single criteria, for example, are available (e.g. Niemi, 1984). Although the votingbased methods can be developed to handle also cardinal information, in such development the very benefits of voting approaches might get lost. In such cases, outranking methods, and the AHP and other MAUT techniques provide readily available solutions for handling more developed preference data. Inevitably, applications for different MCDM methods with different qualities can be found in
the field of natural resources decision support. No method is the best one in all the decision support processes. Choice of the decision support method should be based on the available information: in cases where cardinal information is available, the AHP or generally MAUT methods should be used. In cases of low-quality information, multicriteria approval may be a good solution.
References Alho, J., Kangas, J. and Kolehmainen, O. (1996). Uncertainty in the expert predictions of the ecological consequences of forest plans. Applied Statistics 45, 1–14. Arrow, K. (1951). Social Choice and Individual Values. New York: Wiley. Brams, S. T. and Fishburn, P. (1983). Approval Voting. Boston: Birkhauser. Brans, J. P., Vincke P. and Mareschal, B. (1986). How to select and how to rank projects: the PROMETHEE method. European Journal of Operational Research 24, 228–238. Cranor, L. F. (1996). Declared-strategy voting: an instrument for group decision-making. Academic dissertation. http://www.research.att.com/¾lorrie/pubs/diss/diss.html Fraser, N. M. and Hauge, J. W. (1998). Multicriteria approval: application of approval voting concepts to MCDM problems. Journal of Multi-Criteria Decision Analysis 7, 263–272. Gibbard, A. (1973). Manipulation of voting schemes: a general result. Econometrica 41, 587–601. Hokkanen, J. and Salminen, P. (1997a). Locating a waste treatment facility by multicriteria analysis. Journal of Multi-Criteria Decision Analysis 6, 175–184. Hokkanen, J. and Salminen, P. (1997b). ELECTRE III and IV decision aids in an environmental problem. Journal of Multi-Criteria Decision Analysis 6, 215–226. Hokkanen, J., Lahdelma, R. and Salminen, P. (2000). Multicriteria decision support in a technology competition for cleaning polluted soil in Helsinki. Journal of Environmental Management 60, 339–348. ¨ ainen, ¨ Kangas, A. S., Kangas, J. and Pykal J. (2001). Outranking methods as tools in strategic natural resources planning. Silva Fennica 35, 215–217. Kangas, J. (1999). The analytic hierarchy process (AHP): standard version, forestry application and advances. In Multiple Use of Forests and Other Natural Resources (F. Helles, P. Holten-Andersen and L. Wichmann, eds), pp. 96–105. Kluwer Academic Publishers, Forestry Sciences 61. Kangas, J., Karsikko, J., Laasonen, L. and Pukkala, T. (1993). A method for estimating the suitability function of wildlife habitat for forest planning on the basis of expertise. Silva Fennica 27, 259–268. Kim, K. H. and Roush, R. W. (1980). Introduction to Mathematical Consensus Theory. New York: Marcel Dekker.
Multicriteria approval Lahdelma, R., Hokkanen, J. and Salminen, P. (1998). SMAA – stochastic multiobjective acceptability analysis. European Journal of Operational Research 106, 137–143. Lahdelma, R. and Salminen, P. (2001). SMAA-2: stochastic multicriteria acceptability analysis for group decision-making. Operations Research 49(3). Lakeman, E. (1974). How Democracies Vote: a Study of Electoral Systems. 4th edition. London: Faber & Faber. Laukkanen, S., Kangas, A. and Kangas, J. (2001). Moni¨ a¨ at¨ ¨ ostuen tekniikoita metsal¨ ¨ on tavoitteisen ryhmap ¨ on suunnitteluun. [Multiple criteria hoidon ja kayt¨ group decision support techniques for forest management planning.] Research Notes of Faculty of Forestry 124, University of Joensuu. 61. Lootsma, F. A. and Schuijt, H. (1997). The multiplicative AHP, SMART, and ELECTRE in a common context. Journal of Multi-Criteria Decision Analysis 6, 185–196. McBurnett, N. Approval voting homepage. http://bcn.boulder.co.us/government/ approvalvote/center.html Meister, D. and Fraser, N. M. (1995). The Ordinal Deductive Selection System for Multicriteria DecisionMaking. Unpublished paper. University of Waterloo. Waterloo, Ontario, Canada. Mueller, D. C. (1989). Public Choice II. New York: Cambridge University Press. Niemi, R. G. (1984). Strategic behavior under approval voting. American Political Science Review 78, 952–958. Nurmi, H. (1987). Comparing Voting Systems. Dordrecht: D. Reidel Publishing Company. Nurmi, H. (1995). On the difficulty of making social choices. Theory and Decision 38, 99–119.
137
Pirlot, M. (1997). A common framework for describing some outranking methods. Journal of Multi-Criteria Decision Analysis 6, 86–92. Pukkala, T. (1998). Monikayt¨ ¨ on Suunnitteluohjelmisto Monsu. Ohjelmiston Toiminta ja kaytt¨ ¨ o. Finland. Mimeograph: University of Joensuu. ¨ ainen, ¨ Pykal J., Kangas, J. and Loikkanen, T. (1999). Interactive decision analysis in participatory strategic forest planning: experiences from state owned boreal forests. Journal of Forest Economics 5, 341–364. Riker, W. H. (1982). Liberalism Against Populism. Prospect Heights: Waveland Press, Inc. Saaty, T. L. (1977). A scaling method for priorities in hierarchical structures. Journal of Mathematical Psychology 15, 234–281. Saaty, T. L. (1980). The Analytic Hierarchy Process. p. 287. New York: McGraw-Hill. Saari, D. G. (1994). Geometry of voting, volume 3 of Studies in Economic Theory. New York: SpringerVerlag. Satterthwaite, M. A. (1975). Strategy-proofness and Arrow’s conditions: existence and correspondence theorems for voting procedures and social welfare functions. Journal of Economic Theory 10, 187–217. ¨ Simonen, R. (2000). Metsaammattilaiset kouluttamaan perikuntia. [In Finnish]. Metsalehti ¨ Makasiini 8, 15–17. Vincke, P. (1992). Multicriteria Decision-aid. Chichester: Wiley. von Winterfeldt, D. and Edwards, W. (1986). Decision Analysis and Behavioral Research. p. 604s. Cambridge: Cambridge University Press.
Applying voting theory in natural resource management: a case of multiple-criteria group decision support Sanna Laukkanen† , Annika Kangas*‡ and Jyrki Kangas‡ † University ‡ Finnish
of Joensuu, P.O. Box 111, FIN-80101 Joensuu, Finland Forest Research Institute, P.O. Box 44, FIN-69101 Kannus, Finland
Received 22 January 2001; accepted 7 September 2001
Voting theory has a lot in common with utility theory, and especially with group decision-making. An expected-utilitymaximising strategy exists in voting situations, as well as in decision-making situations. Therefore, it is natural to utilise the achievements of voting theory also in group decision-making. Most voting systems are based on a single criterion or holistic preference information on decision alternatives. However, a voting scheme called multicriteria approval is specially developed for decision-making situations with multiple criteria. This study considers the voting theory from the group decision support point of view and compares it with some other methods applied to similar purposes in natural resource management. A case study is presented, where the approval voting approach is introduced to natural resources planning and tested in a forestry group decision-making process. Applying multicriteria approval method was found to be a potential approach for handling some challenges typical for forestry group decision support. These challenges include (i) utilising ordinal information in the evaluation of decision alternatives, (ii) being readily understandable for and treating equally all the stakeholders in possession of different levels of knowledge on the subject considered, (iii) fast and cheap acquisition of preference information from several stakeholders, and (iv) dealing with multiple criteria. 2002 Elsevier Science Ltd.
Keywords: forest planning, group decision support, multicriteria approval, natural resource management, voting theory.
Introduction Nowadays, forest planning and forest decisionmaking are facing new requirements: planning solely for timber production is not enough any longer. Forest owners may have many other objectives such as those related to recreation, picking wild berries, beautiful scenery and nature conservation. Consequently, there is a need for multi-objective forest planning and decision support in forestry. This applies to all owner groups, and to natural resource management in a wider scope. Often, there are numerous decision-makers Ł
Corresponding author. Present address: University of Helsinki, P.O. Box 27, FIN-00014 Helsinki, Finland. Email: [email protected]
0301–4797/02/020127C11 $35.00/0
involved in planning as well. Each of them can have different aims and objectives concerning the use of forests or other natural resources. The aims are often contradictory and they should somehow be made commensurable to find the best or at least a good compromise solution. This is why effective methods are needed for group-decision support in forestry. In Finland the number of forest holdings owned by various consortia as well as heirs is increasing. At least 50 000 estates of the deceased and other consortia own forest. This means that 27% of the private non-industrial forests in Finland are owned by consortia and heirs at the moment (Simonen, 2000). These groups are problematic from the point of view of practical forest planning and forest counselling. Each action to be taken in forest needs to be approved by every shareholder. There has 2002 Elsevier Science Ltd.
128
S. Laukkanen et al.
been no method or application especially developed for this kind of group decision-making situations. Methods based on Multi-Attribute Utility Theory (MAUT) have been developed for purposes of natural resources planning. These methods have been applied to forest planning too. For example, the Analytic Hierarchy Process (AHP) has been used in supporting multi-objective group decisionmaking (e.g. Kangas, 1999). Methods generally applied to forest planning involve some problems when applied to multiobjective planning of a forest holding owned by a consortium. They may not be able to deal with information that is expressed on a scale other than the ratio scale. Consequently, there is a need for a method that can make use of ordinal information. (In order to utilise qualitative information in decision-making in general, a (possibly subjective) preference order is always needed.) Another problem is that in many decision support methods answering the necessary inquiries are difficult for decision-makers. The decision-makers may include experts familiar with the decision analysis methods. Therefore, they may have better capabilities to answer these inquiries than those who do not have any prior information about the methods. Especially in group decision-making, the methods the application of which necessitates many complicated questions to be made also demand more time and guidance. In group decision support, the planning system also needs to be difficult to manipulate. One possible solution for these problems are the outranking methods (e.g. Vincke, 1992). These include decision support methods that are able to exploit ordinal and imprecise data (Pirlot, 1997). They are rather easy for decision-makers in the sense that less questions are asked of them than in many other decision support methods. Also, ratio scale weights for the criteria are not required. On the other hand, the thresholds used in the analysis may be difficult to understand. What is more important, the method itself is far from being transparent to the decision-makers (e.g. Hokkanen and Salminen, 1997a,b; Kangas et al., 2001). The family of methods based on stochastic multicriteria acceptance analysis (SMAA), provides another possible solution. It is possible to carry out an analysis without any preference information, and imprecise information can also be handled. (Lahdelma et al., 1998; Hokkanen et al., 2000; Lahdelma and Salminen, 2001). Another possibility is to apply the voting theory (e.g. Cranor, 1996). Throughout the history of democracy, voting has proved to be an efficient tool for making choices among decision alternatives,
e.g. in different kind of elections. It is also, in one form or another, known to practically everyone. In the voting theory, special attention has been paid to systems that are neutral and difficult to manipulate (Nurmi, 1987). Therefore, voting theory can be seen as a credible alternative in group decision-making and in supporting it. Most voting systems are based on a single criterion only or holistic preference information on decision alternatives. Nevertheless, there is also a voting scheme specially developed for multicriteria decision-making situations, namely multicriteria approval. In this article, different voting systems are presented and their qualities are discussed from the point of view of natural resources management. The voting theory is considered especially in the context of group decision support, and methods based on voting theory are also compared with some other MCDM methods applied to similar purposes in the natural resource management. A case study is presented, showing an application of the approval voting approach to natural resources planning and testing of its applicability in a forestry group decision-making process.
Voting theory in decision support Voting procedures as decision support methods Many voting systems have a lot to do with the utility theory. The most common of these, plurality voting, for instance, takes into account the preference ordering of the voters, albeit only with respect to the best candidate. A major drawback of this method is that it is generally considered to be very easy to manipulate. However, it can be shown that any non-dictatorial voting scheme is subject to manipulation (Gibbard, 1973; Sattherthwaite, 1975). However, some methods require more information to manipulate an election than others (Nurmi, 1987). The approval voting method was independently proposed by several people in the 1970s (Brams and Fishburn, 1983). It is a voting procedure in which each voter votes for as many candidates as she/he wishes. In other words, the voter votes for all candidates of whom she/he ‘approves’ (McBurnett http://bcn.boulder). Each candidate approved by the voter receives one vote. The candidate receiving the greatest total number of votes is declared to be the winner. This method is harder to manipulate by any one voter than plurality voting. This
Multicriteria approval
is because it requires information about the distribution of approvals of the alternatives in order to be manipulated (Nurmi, 1987). However, it tends to promote moderate candidates. The Borda count (e.g. Saari, 1994) takes into account the whole preference ordering: given n candidates, each voter casts n votes for the most preferred candidate, n 1 votes for the second most preferred one and finally 1 vote for the least preferred candidate. The winner is the candidate getting the most votes. The problem with this method is that the result may be the function of the number of candidates (Riker, 1982). Also this method has been criticised because of its susceptibility to manipulation, but according to Nurmi (1987) it is more difficult to manipulate than the aforementioned methods. The aforementioned voting systems only use information of the preference ordering of the alternatives (ordinal information). So called utilitarian voting systems make use also of information on the intensity of the preferences (cardinal information). These systems include, for example, direct aggregation of cardinal utilities and the demand-based method (e.g. Riker, 1982; Mueller, 1989). In the former case, voters are required to assign utility values to the candidates, and the candidate receiving the largest sum of utility values wins. In the latter, voters offer to pay some money in order to obtain a preferred alternative, and the one getting the highest sum of bids wins. Both of these methods are easy to manipulate by inflating the utility value given to the most preferred alternative. Yet another method is the cumulative voting system, where each voter is given a number of votes they can distribute to the candidates in any way: all votes can, for instance, be given to one candidate or many candidates can be given an equal number of votes (e.g. Lakeman, 1974). A recently presented voting system is a so-called declared-strategy voting (Cranor, 1996). It has been proved that it is often in the best interest to vote for a candidate other than the one most preferred by the voter – i.e. to vote strategically or to manipulate the election. However, such manipulation needs information about the other voters’ preferences, which is not available to each voter. Therefore, the voters with more information can have more power. In declared-strategy voting, however, this problem has been turned to a virtue: in this system all voters can vote strategically. Voters are first asked to give cardinal utility values to the candidates, but the utilities are not summed up like in the systems mentioned above. Then, voters submit their strategy to a computer. Such
129
strategies may be, for example, to vote for the most probable winner in order to show support (so-called bandwagon strategy), or to vote against the least preferred alternative. These strategies are then aggregated to determine the election results. This system has been seen as being especially suitable for group decision-making. However, it requires special software for forming the voting strategies and aggregating them. What is more, the result may depend somewhat on the order that the strategies of different voters are handled.
Multicriteria approval The ideas of approval voting can be utilised also in multicriteria decision support. Fraser and Hauge (1998) presented an application of approval voting, called multicriteria approval, which can be used in multicriteria decision-making. In multicriteria approval, the voters of social choice theory are substituted with multiple criteria. The criteria are ranked with respect to importance, unlike voters in social theory. The standard version of multicriteria approval has been developed for one decisionmaker, but if the decision-makers can agree upon the importance order of the criteria and the border for approval for each criterion, the standard version suits also for group decision-making. Multicriteria approval method begins with the determination of the alternatives and the criteria under which the alternatives will be compared. Secondly, decision-makers rank the criteria by their importance. (There is also the possibility of using cardinal weights of the criteria if a solution cannot be found using ordinal information and if cardinal weighting information is available.) The next step is to determine which of the alternatives will be approved for each criterion. This is done by defining where the border between approval and disapproval regarding each criterion lies. Fraser and Hauge (1998) defined the border as the average evaluation of the alternatives with respect to the criterion considered. In other words, each alternative i, with respect to each criterion j, is approved if the criterion value is above average, and disapproved if otherwise. With numerical criterion values cj .ai /, the border is the mean P cj .ai / of the m alternatives, criterion value cj D m iD1 m with respect to each criterion j. The average value has been chosen as the borderline because, according to approval voting theory, it can be proved that the optimal strategy for a voter is to vote for all the candidates that are above
130
S. Laukkanen et al.
average (e.g. Kim and Roush, 1980). The border between approval and disapproval could, however, be defined also in other ways. For example, if the utility achieved is a non-linear function of a criterion variable, the optimal borderline should be the average utility value, not the average criterion value (Fraser and Hauge, 1998). The borderlines could also be defined as, for example, some proportion of the maximum value for each criterion or as some percentile of the range of variation. The borderline could also be defined in a different way for each criterion. Such an approach may not be reasonable in voting theory, where all voters are defined equally important. In decision support, however, some criteria usually are more important than others. Then, increasing the borderline value would emphasise the importance of a criterion; decreasing it would do the opposite. Ordinal information, with a verbal scale such as Excellent, Good, Average, Passable, Poor, can be readily used for approval and disapproval. If there are qualitative criteria, the preferences of decisionmakers need to be separately questioned. In the group decision-making situation, each decisionmaker can form his/her own judgement for each alternative. The final judgement of approval or disapproval of the alternatives will then be formed based on combinations of these judgements. For example, an alternative which is at least average for all (or majority of) decision-makers is approved and the others are disapproved. The same method can be used if the values of some criterion were evaluated by expert opinion. The determination of the voting result begins after defining the dichotomous preferences. There are five possible classes of voting result: Unanimous, Majority, Ordinally Dominant, Deadlocked and Indeterminate. In this article the classification is presented the way Fraser and Hauge (1998) have described it. The voting result is Unanimous if one alternative has been approved with respect to all criteria and no other alternative has been approved with respect to all criteria. Majority result occurs when one alternative has been approved with respect to the majority of criteria which have been defined to be the most important. If one alternative has been defined to be superior on the grounds of the order of criteria, and the dichotomous preferences, the result is Ordinally Dominant. The class Unanimous is a subclass of Majority and both of these are subclasses of Ordinally Dominant. The result is Deadlocked if there are two or more alternatives that are defined to belong to the subsets Unanimous, Majority or Ordinally Dominant.
These alternatives are approved with respect to the same criteria. They are also disapproved with respect to the same criteria, and so it is not possible to determine one single superior alternative. The result is Indeterminate if there is not enough information to determine one superior alternative. There are two or more alternatives between which it is impossible to name the best. More information is needed in addition to the criteria order. The first phase in determining the voting class is to find out the number of existing Ordinally Dominant alternatives. According to Fraser and Hauge (1998) this is done by using a variation of the ordinal deductive selection system (1) (Meister and Fraser, 1995). Alternative k is classified to be Ordinally Dominant if f .nŁ /ki ½0 8nŁ , 1nŁ n, 8i6D k where
n X
.1/
Ł
Ł
f .n /ki D
gijk
jD1
and
1, if cj .ak />cj ^cj .ai /cj 0, if cj .ak />cj ^cj .ai />cj gijk D 0, if cj .ak /cj ^cj .ai /cj 1, if cj .ak /cj ^cj .ai />cj
Fraser and Hauge (1998) describe the principle of this formula as follows: ‘Alternative k is classified Ordinally Dominant if for every possible value of nŁ , and all other alternatives i6D k, f .nŁ /ki remains greater than or equal to zero. If at anytime, for any i6D k it becomes negative, alternative k is labelled as Indeterminate’. The process is based on the idea that approval with respect to a more important criterion can completely compensate disapproval with respect to a less important criterion (Fraser and Hauge, 1998). After this process each alternative is labelled as either Ordinally Dominant or Indeterminate. If there is only one single alternative that is Ordinally Dominant next step is to examine if it belongs to subclasses of Ordinally Dominant. If it does not belong to either Unanimous or Majority then it is classified as Ordinally Dominant.
Case study Alternative plans and their evaluation The multicriteria approval method was used in a real forest planning situation. The problem was to
Multicriteria approval
choose a forest plan for a forest holding owned by a consortium consisting of three forest owners. All of them have a 1/3 share in the forest holding. The forest holding is 30 hectares in size and consists of three separate blocks. The holding has been divided into a total of 13 forest stands, i.e. compartments, each being relatively homogeneous as regards the soil and stand characteristics. As is usual in the management of boreal coniferous-dominated forests, these compartments were regarded as basic decision units. In forest planning one treatment schedule is chosen for each compartment, to be implemented throughout the compartment. The planning period consisted of two 10-year periods. Twenty alternative forest plans were produced, each consisting of a combination of compartment-wise treatment schedules for the 20year planning horizon. The forest owners had different objectives as regards forest utilisation. One of them aimed at timber production, the beauty of the forest landscape and wild berry yield. The second aimed at timber production, biodiversity of the forest, the beauty of the forest landscape, and conserving habitats for capercaillie (Tetrao urogallus), a valued game bird. The third mainly preferred timber production. After discussions held with the forest owners the following decision criteria were settled upon. The choices regarding the criteria were made by the forest owners, guided by a consultant. In the case of the timber production, the criteria were (i) the present value of net incomes over the 20years planning horizon and (ii) the estimated value of future timber production at the end of the planning period. Both of these were calculated utilising simulation of forest development, and by applying a discount rate of 3%. The wild berry yield was measured with the bilberry yield index. The criteria values of timber production and bilberry yield were obtained directly via the Monsu forest planning software (Pukkala, 1998). Monsu includes a simulation part, predicting the criteria values in each compartment, under different treatment schedules, using several interrelated statistical models. Monsu also includes a mathematical optimisation part, which was utilised in producing plans that were efficient in regard to the criteria that could be taken into consideration in Monsu. Judgements of the other criteria were partly subjective, and therefore these judgements were made after alternative plans were prepared. In all, twenty forest plans were produced. The quality of the forest with respect to the habitat needs of capercaillie was the only criterion
131
that could not to be taken into consideration in the calculations of Monsu at all. However, as both the viability of the capercaillies and biodiversity were supposed to depend on the amount of the old forest and felling, those criteria were combined and this criterion was called biodiversity. Concerning this criterion the alternative plans were evaluated with the help of an expert on the ecology of capercaillie and on forest biodiversity, by means of pairwise comparisons as in the AHP (Saaty, 1977; see Kangas et al., 1993). These values were linearly scaled for the purposes of multicriteria approval. The evaluation of the scenic beauty of forest landscape was also partly subjective. However, the pairwise comparisons of the twenty plans would have been too laborious and time-consuming for the decision-makers. Consequently, the evaluations were carried out so that each decision-maker individually formed the weights for the different blocks of the forest holding by means of pairwise comparisons, as different blocks had different importance with respect to scenic values. The final landscape beauty evaluations were made with the help of these weights and the scenic beauty indices given by Monsu. The criterion values in alternative forest plans are presented in Table 1. The next phase in the case study was to rank the criteria by their importance. A mutual analysis of the decision-makers was agreed upon. The mutual order was obtained so that, using values 1–7, all decision-makers defined the weights that describe the importance of the criteria. The final order of the criteria was formed on the basis of the average values of the decision-makers’ weightings. The average values were used for importance ranking to enable the utilisation of the same data also in PROMETHEE analysis (see Laukkanen et al., 2001). This method was approved of by all decisionmakers. Another possibility, perhaps more in accordance with the ideas of multicriteria approval, would have been to use some voting system (e.g. Borda count, which combines individual rankings). The order of the criteria from the most important to the least important was: Net incomes, Biodiversity, Monetary value of the future timber production, Scenic beauty of the forest landscape, and Wild berry yield. Next, the limits between approval and disapproval for each criterion were defined (Table 2). The limits were calculated directly from the average of the criterion values of the alternatives. For the sake of sensitivity analysis, also the midpoint of the range of variation was used as a border of approval.
132
S. Laukkanen et al. Table 1. Alternative plans
Criterion values in alternative plans Net incomes, present value 2001–2020, FIM
Biodiversity 2020
Value of the future timber production 2020, FIM
Scenic beauty 2020
Bilberry yield 2020, kg/ha
39 740 15 011 96 900 151 503 101 652 45 092 73 225 134 780 66 079 154 163 86 666 133 135 36 664 90 931 81 864 114 550 122 782 62 228 105 142 95 177
1Ð143 1Ð436 0Ð419 0Ð121 0Ð407 1Ð114 0Ð441 0Ð139 0Ð704 0Ð116 0Ð471 0Ð168 1Ð107 0Ð275 0Ð543 0Ð289 0Ð160 0Ð444 0Ð232 0Ð271
876 516 923 254 739 278 543 862 752 080 862 810 777 945 626 711 826 339 532 995 796 026 650 071 883 552 724 857 787 786 682 128 577 407 810 837 684 919 714 019
6Ð2356 6Ð2982 6Ð0517 5Ð7230 5Ð8641 6Ð2196 6Ð1927 5Ð3174 6Ð1106 5Ð3639 5Ð9070 5Ð7391 6Ð2356 6Ð1302 6Ð0517 5Ð9892 6Ð0356 6Ð2356 6Ð0517 6Ð1142
10Ð7 11Ð6 8Ð2 5Ð2 8Ð0 10Ð6 8Ð7 6Ð0 9Ð6 5Ð1 8Ð7 6Ð0 10Ð8 7Ð3 8Ð9 6Ð8 5Ð9 9Ð1 6Ð7 7Ð3
P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 P11 P12 P13 P14 P15 P16 P17 P18 P19 P20
Table 2. The limit between approval and disapproval for each criterion using the average criterion value, or the midpoint of the range of variation Criterion
cj D
m X cj .ai / iD1
Net incomes Biodiversity Value of the future timber production Scenic beauty Bilberry yield
m
90 364Ð2 0Ð5 738 669Ð6 5Ð99 8Ð06
cQ j D
max.cj /Cmin.cj / 2 84 587Ð0 0Ð8 728 124Ð5 5Ð81 8Ð35
Results of the case study The next step was to examine which of the alternatives were above and which were below the average for each criterion; in other words which alternatives were approved and which alternatives were disapproved for each criterion (Table 3). For identifying a class of voting result, a test was conducted to see how many Ordinally Dominant alternatives existed. Each alternative was compared with other alternatives in accordance with P the formula f .nŁ /ki D njD1 gijk . The alternatives P1, P2, P6, P9, P13, P15 were found to be acceptable for all other criteria except Net income. However, these alternatives could not be Ordinally Dominant because f .nŁ /ki took a negative value immediately in the beginning of the
comparison. The table of approval and disapproval shows that only alternatives P3 and P5 could be Ordinally Dominant. After comparison calculations it turned out that only alternative P3 was Ordinally Dominant, because in the pairwise comparisons alternative P3 did not lose to any other alternative. In other words, f .nŁ /3i never took a negative value. The alternatives P5 and P3 were even when the three most important criteria were examined but alternative P3 dominated alternative P5 as regards the two least important criteria. After this, alternative P3 was examined to find out whether or not it belonged to the subclasses of Ordinally Dominant. Since this forest plan alternative was not approved of by all five criteria it could not be an unanimous winner. Neither was it a majority winner since it was not approved by the second-ranked criterion. The conclusion was that plan alternative P3 did not belong to either of the subsets and therefore it was classified as Ordinally Dominant. Using the midpoint of the range of variation as the border decreased the border for Net incomes, Future value of timber production and Scenic beauty and increased the border for the other criteria (Table 2). Therefore, more alternatives were approved with respect to Net incomes, and less with respect to Wild berry yield, for example (Table 4). Then, alternative P3, which was Ordinally Dominant in the previous case, was no longer approved by the Wild berry yield criterion.
Multicriteria approval Table 3. border Alternative P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 P11 P12 P13 P14 P15 P16 P17 P18 P19 P20
133
Approvals (C) and disapprovals ( ) of the alternatives using average criterion value as Net incomes
Biodiversity
Value of the future timber production
Scenic beauty
Bilberry yield
C C
C C C
C C C
C C C
C
C C C
C C
C C
C
C
C
C
C C C C C
C
C
C C
C
C
C
C C C C C
C C C C C C C
C C C
Table 4. Approvals (C) and disapprovals ( ) of the alternatives using the midpoint of range of variation as border Alternative P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 P11 P12 P13 P14 P15 P16 P17 P18 P19 P20
Net incomes
Biodiversity
Value of the future timber production
Scenic beauty
Bilberry yield
C C
C C C
C C C
C C
C C C
C C C
C C
C
C
C
C
C
C
C
C C C C C C C C
C
C C C C C C C C C C
C C C C C C
Instead, alternative P11 became approved by the Net incomes and Scenic beauty as well, and was found Ordinally Dominant. Therefore, even small changes in the approval border may change the results of multicriteria analysis. Also, adding a new decision alternative to the analysis may cause a rank reversal among the ‘old’ alternatives, if approval borders are calculated as a function of the criterion values of the alternatives considered.
C C
Discussion Characteristics of multicriteria approval In this study, an application of voting theory called multicriteria approval was introduced into the field of natural resource management and tested in forestry group decision support. Multicriteria
134
S. Laukkanen et al.
approval has not been applied to forest planning before. It seems to satisfy many requirements of multicriteria forestry decision support, because its main aim is to find an alternative that is widely satisfactory among the multiple criteria. It is highly suitable for situations where information that other methods demand is difficult or expensive to obtain, or where the available information is of low quality. The method is easy to explain to the decision-makers and it is readily understandable. There can be different kind of decision-makers, experts and non-experts, and the method is still fair for everyone. The only preference information that the decision-makers need to be asked is the importance order of the criteria. This seems easy enough, but such ranking may be more problematic than it seems. For example, the importance order of the criteria may depend on the current values of the criteria, and on the range of values for the criteria. However, the situation is similar for most practical decision support systems (von Winterfeldt and Edwards, 1986). Due to its minimal information requirements multicriteria approval does not consume much time; thus, the costs of using this method remain low. Since multicriteria approval is so easy for decision-makers, queries can be carried out using a post enquiry or, for example, via the Internet. Therefore, it could also be used for consortia with plenty of owners. In Finland, for instance, some common forests have hundreds, or even thousands of owners. Approaches based on voting theory would be worth applying also in participatory planning of natural resources governed by public institutions (e.g. State-owned forests), where their ¨ ainen ¨ merits would be very useful (see Pykal et al., 1999). However, evaluation of the alternatives on the grounds of qualitative criteria would demand more time and involvement of decision-makers than quantitative ones. This is because decisionmakers have to judge every alternative separately with respect to the criterion in question. All the criteria of importance are usually not included in common forest planning software applications. In this kind of a situation, forest planning software (typically based on mathematical optimisation) alone cannot solve the problem in a satisfactory manner, but other approaches are also needed. By using the multicriteria approval method all the criteria could be included in the case study, e.g. the biodiversity criterion with satisfactory habitats being provided for capercaillie. The benefits of both a quantitative biodiversity index produced by forest planning software and experts’
qualitative evaluations of alternatives with respect to biodiversity could be utilised in the multicriteria approval context. One disadvantage of the multicriteria approval method is that it does not always offer a solution for a problem of multicriteria decision-making. It does not put all the alternatives in order but instead chooses one or more best alternatives. If one Ordinally Dominant alternative can be found this method will choose it, but if the voting result is Deadlocked it chooses more than one alternative. In this study, an Ordinally Dominant alternative was found. In the case of a Deadlocked result Fraser and Hauge (1998) suggest that it would be useful to try to find the solution by using the run-off procedure. This procedure tries to solve the situation by taking into account only those alternatives that were among the deadlocked set and reanalysing the case. By ignoring some of the worst alternatives the mean criterion values and, correspondingly, the result of the analysis may change. Another possibility is to directly heighten the approval border, in order to reduce the number of Deadlocked alternatives. In the case of an Indeterminate voting result the method does not give any information about the decision at all. Then there is a need for more preference information to enable the best alternative to be found. In the case study, the border of approval and disapproval of each alternative with respect to each criterion was calculated directly with average criterion value for the quantitative criteria, based on the approval voting theory. As another possibility, the midpoint of the range of variation was used. It was noted, that the method is quite sensitive to the approval border. Therefore, it is also subject to rank reversal, when new alternatives are introduced or some alternatives are excluded. However, if the approval borders were fixed (i.e. independent of the criterion values of alternatives considered), rank reversal does not occur even if the number of alternatives were changed. From the practical point of view, however, this need not be a serious problem.
Comparison with other decision support techniques Comparison of multicriteria approval with other methods applied to similar decision support purposes, such as MAUT and outranking, is interesting. Outranking methods used for ranking alternatives typically produce an ordinal preference
Multicriteria approval
order (e.g. Pirlot, 1997; Lootsma and Shujit, 1997). In some cases, e.g. PROMETHEE II, the rankings also have cardinal properties. MAUT methods, for instance the AHP and SMART, give a measure of relative priorities to the alternatives (Saaty, 1980; von Winterfeldt and Edwards, 1986). Preference information is also expressed in different ways in these approaches. MAUT methods are typically based on cardinal preferences. When using outranking methods, the weights of criteria need to be defined but the method can use either ordinal or cardinal information. The most frequently used MAUT methods are based on an additive linear utility model, and thus they are fully compensatory (von Winterfeldt and Edwards, 1986). Outranking methods have both compensatory and non-compensatory elements (e.g. Pirlot, 1997). Multicriteria approval, on the other hand, is clearly non-compensatory. When the AHP and PROMETHEE II were tested for the same case study as was used here (see Laukkanen et al., 2001), the results obtained clearly differed from those obtained with multicriteria approval (Table 5). The AHP, using most sophisticated information and being fully compensatory, put the alternatives with the biggest incomes above the others (incomes being the most important criteria). The PROMETHEE II analysis chose the alternatives with the lowest incomes but better values for other criteria. Multicriteria approval chose a compromise solution. These results reflect the different Table 5. The rankings of the alternative plans of the case study based on AHP and PROMETHEE II analysis (see Laukkanen et al., 2001) Alternative P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 P11 P12 P13 P14 P15 P16 P17 P18 P19 P20
AHP
PROMETHEE II
5 4 11 1 8 3 19 7 13 2 14 6 9 18 12 10 15 20 16 17
2 1 7 18 14 4 8 19 5 20 10 17 3 12 6 13 16 9 15 11
135
properties of the methods (e.g. compensatory – noncompensatory), the different amount of information (ratio scale – dichotomous), and different criteria weights (ratio scale – rank). When multicriteria approval is applied to group decision support, a mutual rank order of the criteria is determined. The mutual approach is used also in outranking methods. In the AHP (or other cardinal utility-based methods) it is possible to assess the criteria weights separately for each stakeholder and then computationally combine the preference information obtained. When combining the separate analyses, each decision-maker is assigned a definite weight. Using common weights or even a common rank of criteria may seem theoretically problematic. In fact, the famous theorem of Arrow (1951) states that there is no method of aggregating individual preferences over three or more alternatives that would satisfy several conditions for fairness and always produce a logical result. However, in any practical group decision support the preferences of the stakeholders need to be combined one way or other – if not, the only decision support left would be negotiation. However, negotiations may be even more susceptible to manipulation than any decision support technique. In addition to weights, outranking techniques require preference and indifference thresholds. Below the indifference threshold q a decisionmaker is indifferent between two alternatives and above the preference threshold p a decision-maker expresses strict preference of one alternative over another (Brans et al., 1986). There is a hesitation zone between these thresholds. The approval border of multicriteria approval resembles the threshold values of outranking methods. However, it is probably easier to understand for ordinary decision-makers, since it is based on the consideration of only one alternative at a time, whereas in outranking methods the thresholds are based on differences between two alternatives. Although the preference judgements in outranking methods might be rather easy to express, the handling of preference information in calculations is difficult to explain to non-specialists. Problems in the comprehensibility and interpretability of the results are the drawbacks of outranking methods in general (Vincke, 1992). Outranking methods require guiding and initiation to threshold values. The AHP, in turn, requires even more guiding and time because the weights of the criteria are established by using pairwise comparisons. The number of the comparisons increases rapidly with the number of alternatives and criteria. However, using statistical regression approaches instead of Saaty’s
136
S. Laukkanen et al.
eigenvalue technique in the analysis of pairwise comparisons enables the use of incomplete comparison matrices (Alho et al., 1996). Ultimately, the basic differences between methods based on voting theory and other MCDM methods are not so clear as they might first seem. Several voting methods have been presented that make it possible to apply also cardinal preference information in the evaluation of alternative decisions. For example, the way the votes are cast in cumulative voting greatly resembles the evaluation of weights in some SMART methods (von Winterfeldt and Edwards, 1986). The Copeland score calculated from pairwise votes resembles the concordance index in outranking methods (see Nurmi, 1995). Indeed, according to Vincke (1992), the voting theory can be seen as an ancestor of outranking methods. From the point of view of practical natural resources management, perhaps the greatest advantage of multicriteria approval is in its ability to rank alternative decisions in a multiple-criteria planning situation on the grounds of much less and more rudimentary preference information than is required by other MCDM methods. Defects and low quality in the preference and evaluation information are problems frequently faced in forestry decision-making, especially in group decision support (e.g. cases of consortia-owned forests, heirs, and common forests) and in participatory planning processes. Besides, stakeholders normally have different abilities to participate in the decision support process and to give information on their preferences in modes required by MCDM methods. In voting approaches, the idea is to avoid giving an advantage to those with more expertise on the subject and on the methodology over other stakeholders. However, even though multicriteria approval can make use of low-quality information as it often persists in the reality, applying dichotomous preferences means loss of information in cases where the cardinal weighting of criteria and cardinal preference orderings of the alternatives with respect to single criteria, for example, are available (e.g. Niemi, 1984). Although the votingbased methods can be developed to handle also cardinal information, in such development the very benefits of voting approaches might get lost. In such cases, outranking methods, and the AHP and other MAUT techniques provide readily available solutions for handling more developed preference data. Inevitably, applications for different MCDM methods with different qualities can be found in
the field of natural resources decision support. No method is the best one in all the decision support processes. Choice of the decision support method should be based on the available information: in cases where cardinal information is available, the AHP or generally MAUT methods should be used. In cases of low-quality information, multicriteria approval may be a good solution.
References Alho, J., Kangas, J. and Kolehmainen, O. (1996). Uncertainty in the expert predictions of the ecological consequences of forest plans. Applied Statistics 45, 1–14. Arrow, K. (1951). Social Choice and Individual Values. New York: Wiley. Brams, S. T. and Fishburn, P. (1983). Approval Voting. Boston: Birkhauser. Brans, J. P., Vincke P. and Mareschal, B. (1986). How to select and how to rank projects: the PROMETHEE method. European Journal of Operational Research 24, 228–238. Cranor, L. F. (1996). Declared-strategy voting: an instrument for group decision-making. Academic dissertation. http://www.research.att.com/¾lorrie/pubs/diss/diss.html Fraser, N. M. and Hauge, J. W. (1998). Multicriteria approval: application of approval voting concepts to MCDM problems. Journal of Multi-Criteria Decision Analysis 7, 263–272. Gibbard, A. (1973). Manipulation of voting schemes: a general result. Econometrica 41, 587–601. Hokkanen, J. and Salminen, P. (1997a). Locating a waste treatment facility by multicriteria analysis. Journal of Multi-Criteria Decision Analysis 6, 175–184. Hokkanen, J. and Salminen, P. (1997b). ELECTRE III and IV decision aids in an environmental problem. Journal of Multi-Criteria Decision Analysis 6, 215–226. Hokkanen, J., Lahdelma, R. and Salminen, P. (2000). Multicriteria decision support in a technology competition for cleaning polluted soil in Helsinki. Journal of Environmental Management 60, 339–348. ¨ ainen, ¨ Kangas, A. S., Kangas, J. and Pykal J. (2001). Outranking methods as tools in strategic natural resources planning. Silva Fennica 35, 215–217. Kangas, J. (1999). The analytic hierarchy process (AHP): standard version, forestry application and advances. In Multiple Use of Forests and Other Natural Resources (F. Helles, P. Holten-Andersen and L. Wichmann, eds), pp. 96–105. Kluwer Academic Publishers, Forestry Sciences 61. Kangas, J., Karsikko, J., Laasonen, L. and Pukkala, T. (1993). A method for estimating the suitability function of wildlife habitat for forest planning on the basis of expertise. Silva Fennica 27, 259–268. Kim, K. H. and Roush, R. W. (1980). Introduction to Mathematical Consensus Theory. New York: Marcel Dekker.
Multicriteria approval Lahdelma, R., Hokkanen, J. and Salminen, P. (1998). SMAA – stochastic multiobjective acceptability analysis. European Journal of Operational Research 106, 137–143. Lahdelma, R. and Salminen, P. (2001). SMAA-2: stochastic multicriteria acceptability analysis for group decision-making. Operations Research 49(3). Lakeman, E. (1974). How Democracies Vote: a Study of Electoral Systems. 4th edition. London: Faber & Faber. Laukkanen, S., Kangas, A. and Kangas, J. (2001). Moni¨ a¨ at¨ ¨ ostuen tekniikoita metsal¨ ¨ on tavoitteisen ryhmap ¨ on suunnitteluun. [Multiple criteria hoidon ja kayt¨ group decision support techniques for forest management planning.] Research Notes of Faculty of Forestry 124, University of Joensuu. 61. Lootsma, F. A. and Schuijt, H. (1997). The multiplicative AHP, SMART, and ELECTRE in a common context. Journal of Multi-Criteria Decision Analysis 6, 185–196. McBurnett, N. Approval voting homepage. http://bcn.boulder.co.us/government/ approvalvote/center.html Meister, D. and Fraser, N. M. (1995). The Ordinal Deductive Selection System for Multicriteria DecisionMaking. Unpublished paper. University of Waterloo. Waterloo, Ontario, Canada. Mueller, D. C. (1989). Public Choice II. New York: Cambridge University Press. Niemi, R. G. (1984). Strategic behavior under approval voting. American Political Science Review 78, 952–958. Nurmi, H. (1987). Comparing Voting Systems. Dordrecht: D. Reidel Publishing Company. Nurmi, H. (1995). On the difficulty of making social choices. Theory and Decision 38, 99–119.
137
Pirlot, M. (1997). A common framework for describing some outranking methods. Journal of Multi-Criteria Decision Analysis 6, 86–92. Pukkala, T. (1998). Monikayt¨ ¨ on Suunnitteluohjelmisto Monsu. Ohjelmiston Toiminta ja kaytt¨ ¨ o. Finland. Mimeograph: University of Joensuu. ¨ ainen, ¨ Pykal J., Kangas, J. and Loikkanen, T. (1999). Interactive decision analysis in participatory strategic forest planning: experiences from state owned boreal forests. Journal of Forest Economics 5, 341–364. Riker, W. H. (1982). Liberalism Against Populism. Prospect Heights: Waveland Press, Inc. Saaty, T. L. (1977). A scaling method for priorities in hierarchical structures. Journal of Mathematical Psychology 15, 234–281. Saaty, T. L. (1980). The Analytic Hierarchy Process. p. 287. New York: McGraw-Hill. Saari, D. G. (1994). Geometry of voting, volume 3 of Studies in Economic Theory. New York: SpringerVerlag. Satterthwaite, M. A. (1975). Strategy-proofness and Arrow’s conditions: existence and correspondence theorems for voting procedures and social welfare functions. Journal of Economic Theory 10, 187–217. ¨ Simonen, R. (2000). Metsaammattilaiset kouluttamaan perikuntia. [In Finnish]. Metsalehti ¨ Makasiini 8, 15–17. Vincke, P. (1992). Multicriteria Decision-aid. Chichester: Wiley. von Winterfeldt, D. and Edwards, W. (1986). Decision Analysis and Behavioral Research. p. 604s. Cambridge: Cambridge University Press.