- Email: [email protected]
An ordinal ranking model for the highway corridor selection problem
Cmpur. Envirou. L’rhm Syarms Vol. 9, No. 4. pp. 271-276. Printed in Great Britain. All rights reserved
1984 Copyright
AN ORDINAL RANKING MODEL CORRIDOR SELECTION WADE Faculty
of Administrative
0198-9715184 $3.00~0.00 .(‘: 1984 Pergamon Press Ltd
FOR THE HIGHWAY PROBLEMJ-
D. COOK
Studies, York University,
Toronto,
Canada
M31 2R6
and LAWRENCE Department Abstract -This paper present a model which desirabilityof various is given. An approach
of General
Business,
M. SEIFORD
University
of Texas, Austin, TX 78712, U.S.A.
examines the problem of corridor selection in highway design and routing. We is designed specifically to deal with ordinal information pertaining to the relative potential corridors. A technique for combining ordinal rankings into a consensus to the problem of determining criteria weights is suggested.
INTRODUCTION
A familiar problem in highway design and planning is one of route selection. Specifically, when a highway is to be constructed in a particular geographical region, the first major task facing the transport agency is that of selecting a corridor-a 30&500 wide strip of land along which the highway is to pass. In assessing the relative desirability of any set of potential corridors it is necessary to investigate various impact parameters. These include user and regional development models, both benefits, environmental and social benefits and costs,. . . etc. While numerous sophisticated and ad hoc, have been developed to deal with this problem, all appear to meet with difficulties at the implementation stage. This is partially due to the multidimensional nature of the problem. Generally, existing models attempt to quantify the impact of each dimension. However, since a large number of impact parameters, particularly those relating to environmental and social benefits/disbenefits, cannot easily be quantified (if at all), these models tend in many instances to misrepresent prevailing conditions. Since available information is often of the ordinal rather than cardinal type, these quantitative (cardinal) approaches often fall short of their goal. In this paper we present a model designed to take account of the ordinal nature of the problem. First we show how to combine ordinal information such as to obtain a ranking of the corridors on the basis of each dimension. We then describe a procedure for combining rankings across dimensions. While this new model is not intended as a direct replacement for “cardinal” approaches currently in use, it does have tremendous potential as a complementary tool of analysis. In the following section we describe the corridor selection problem. The third section presents a brief review of current approaches to this problem, along with associated literature. The fourth section then details an ordinal ranking model for the corridor selection problem. The final section discusses conclusions and future directions.
THE
CORRIDOR
SELECTION
PROBLEM
A problem which has attracted considerable interest during the past several decades is that of design and selection of highway corridors. The overwhelming impact of highways on the economy as well as on the social and environmental structure of a region or country has captured the attention of economists, operations researchers, engineers and government agencies. Over the past 40 years highway planning has been among the top two or three priority areas of study in developed and developing nations alike.
tSupported
in part under NSERC
Grant
No. A8966. 271
212
WADE D. COOKand LAWRENCEM. SEIFORLI
The dimensions of the problem can be represented via the following categories of benefits/ disbenefits: (1) regional development; (2) social; (3) environmental; (4) user; (5) right of way. Regional development parameters, for example, would include employment and business opportunities, net changes in housing costs due to highway improvements,. . . etc. Social benefits/ costs would include community cohesion and growth, relocation hardship, public reaction, transit mobility,. etc. We will not elaborate here on the other classes of benefits and costs. Benefits as outlined above, fall into two basic groups-“hard” and “soft”. In most highway planning problems there are an overwhelming number of soft or qualitative aspects of the problem, and while traditional cost/benefit and related techniques have attempted to capture such factors, they have, in most instances, been less than successful. Various information gathering devices (e.g. public opinion polls) have been utilized to assess the relative impacts of proposed corridors along each selected dimension. In a 1975 study [I] for example involving the design and routing of a highway bypassing the city of Peterborough, Ontario, an opinion poll was conducted in each municipality. For each poll a sample of residents was selected, and each was asked to assess the impact which various routings of corridors would have in terms of water pollution, noise, loss of natural land areas, damage to fish and game,. . . etc. The opinions of residents generally appeared in an ordinal format along several dimensions. Even along quantitative dimensions voters are often not able to supply hard numbers, but rather can only express preferences on an ordinal basis. Various modelling techniques have been proposed and used in the economic evaluation of highway corridors. Some of these are reviewed in the section to follow. In the fourth section we present an alternative to these models-one which specifically takes account of the ordinal nature of the problem.
THE
LITERATURE
Among the earliest applications of cost/benefit analysis were those in the transport field. Projects ranged from highway development studies in U.S.A., Canada, Australia, Papua New Guinea and Dominican Republic (see [2-41) to canal and airport construction in Kenya and elsewhere. Adler [.5] documents numerous studies from roads and railways to ports and pipelines in literally dozens of countries throughout the world. Hoyle [4] discusses the use of cost/benefit methods in the selection of highway routes in Uganda and Kenya. Within the past 25 years Liberia for example has been transformed from a series of disconnected ports to a politically unified country through road and rail development. Stanley [6] reports on the use of cost/benefit analysis in this context. Cost/benefit methodology does however possess certain shortcomings as described by Hill [3] : “When costs and benefits are not available in market prices, the cost/benefit model imputes them as if they were subject to market transactions. When imputing market prices the analyst should distinguish between those items which are ordinarily evaluated in the market.. .and the intangibles which.. .cannot be priced in monetary terms. In practice these intangibles fall outside the central criterion and beyond the scope of the tools of measurement of cost/benefit analysis.”
Attempts to rectify apparent flaws in cost/benefit procedures have given rise to a number of adjuncts to the conventional approach. The most important class of these adjuncts is the set of multiattribute ranking procedures. While there are many different ranking models in existence one of the more popular approaches vis-a-vis problems such as the one under discussion is that developed by Roy [7,8]. Roy’s method begins with a set of alternatives (e.g. corridors) and a set of criteria/dimensions. It is then assumed that the planner is able to assign a numerical (cardinal) rating to each alternative with respect to
Ordinal
ranking
model for highway
corridor
selection
213
each criterion. Weights are then applied to each criterion. Roy then defines a concordance index for each pair of alternatives and uses this set of indicies to arrive at a “best” alternative. Massam [9, lo] has extended Roy’s method to produce a complete ranking (not just the first ranked) of the alternatives. Singapore transit data is used to test the model. Massam [lo] provides an extensive coverage of these concepts. On the operational side Guigow [I l] and Van Delft and Nijkamp [12] have applied Roy’s method to particular spatial location problems, Similarly, ranking techniques such as those discussed above have been used in subway alignment problems. See Massam and Miller [ 131. A number of other similar ranking procedures have been used to examine the route/corridor selection problem. These will not be discussed here. It is sufficient to say that in general all current approaches subscribe to the same “cardinal number” rules. Either it is specifically required that cardinal, not ordinal, data be submitted to the model or the model or decision maker imputes cardinal numbers wherever ordinal information is the only data available. We now present a model which provides an alternative approach for dealing with those selection problems in which the information available is primarily ordinal in nature. It should be pointed out, however, that the following section is not an in-depth study of corridor selection, but rather is intended to illustrate how ordinal techniques could be applied to such a problem.
AN
Ranking
ORDINAL
MODEL
FOR
CORRIDOR
SELECTION
by dimension
In the model to be developed in this section let us assume that each member of a group of individuals (municipal committee, strategic task force,. . . etc.) is asked to rank the corridors, with a ranking being supplied corresponding to each dimension. While some criteria may be such that the voter can provide a cardinal ranking, we will concentrate here on ordinal preferences. In the cardinal case techniques described above can be applied. For each qualitative dimension i assume that each committee member j supplies an ordinal ranking (LEE, a?, . . . d,. . ., a! where d is th e rank assigned to the Jth corridor along the ith dimension. For example, if i = 1 = recreational benefits, and if there are 5 corridors, this ordinal ranking might take the form (2, 1, 4, 3, 5) i.e. ai = 2, u? = 1, . . . etc. Thus corridor 1 is given a rank of 2, corridor 2 a rank of 1, etc. In assigning ranks to corridors, it is expected that each member would give consideration to both the benefits as well as the cost of doing the project. In order to combine the ordinal rankings [(ni’j, . . ., &)]jE1 of the 171committee members into a compromise or consensus ranking, many different possibilities exist. Kemeny and Snell [14], Kendall [lS], Bogart [16, 173 and Cook and Seiford [18] all suggest different procedures for doing such. Since the Cook/Seiford method is designed to handle large problems, it is particularly appropriate in the problem under discussion. It, like a number of the other approaches possesses certain desirable properties which any consensus method should exhibit. These properties are similar in some respects to the social choice axioms of Arrow [19]. A full discussion is given in [18]. The Cook/Seiford consensus method for a particular criterion proceeds as follows. Given a set of rankings A 1, AZ,. . ., A, [where Aj = (uj, a:,. . ., a:)] of the L corridors, the consensus is a ranking fi = (6’, 6’, . . ., 6’) which minimizes the function,
Cook and Seiford show that the optimal ranking can be determined by formulating linear assignment problem.
tThe Index i for criteria
has been suppressed.
M(B) as a
WADE D. COOK and LAWRENCEM. SEIFORD
214
Example. Consider
the following
set of rankings:
(m = 10 committee
members, 1 4 3 2 5
Let us now construct
the distance
matrix
n = 5 objects) 4 1 5 3 2
1 3 4 2 5
(d/k). Note that:
k = 1, 2,. . ., 5 ranks; m = 1, 2,. . ., 10 members ; /=1,2 d/k is the sum of the deviations aj’ of object /. For example: dll= dzl=
between
,...,
5objects.
a selected
rank
k and the 10 given member
ll-11+14-ll+13-1(+...+13-11 ~4-1~+)3-1~+~2-1l+...+~4-1~
ranks
=13; =21.
Thus, we have: Distance 13 21 24 22 20 Using
the assignment
11 15 16 16 12
algorithm
we get:
Object
1
2
110000 2 0 3 0 4 0 5 0
0 0 0 1
Matrix
(d/k)
11 11 12 14 12
17 11 12 14 16
Priority 3
1 0 0 0
27 19 16 18 20
4
5
0 1 0 0
0 0 1 0
That is B = [l, 3, 4, 5, 21 and M(g) = 66 is the minimum distance. it is clear that alternate medians exist (e.g. B = [l, 4, 3, 5, 23).
By viewing
the cost matrix,
Combining dimensions While ordinal rankings associated with any given dimension can be used to arrive at a consensus or compromise ranking relative to that dimension, any technique used to aggregate dimensions in order to get an overall consensus must rely on cardinal dimension weights. Unless the relative importance of the various dimensions can be quantified no operational method exists for getting the consensus. Let us assume initially that a set of fixed known weights (r/t;}{= 1 are available. If Bi denotes the consensus associated with criterion i, a logical overall consensus is, therefore, B=
i i=l
WBi
(1)
Ordinal
ranking
model for highway
corridor
selection
275
Specifically, if 6/i ~6, for all /#/I, then corridor /r is the preferred route. Typically the I% values will possess some degree of error, hence there is some concern as to the stability of the first place alternative kr in B. However, because of the simplistic nature of equation (1) it is easy to perform a sensitivity analysis on each dimension weight. For example, if 6 represents the possible error factor in w1 say (i.e. the weight on the iiih criterion is II$, +6), then B is changed to B+6&,. It is easily shown that corridor /i is still optimal provided,
Obtaining weights through sampling
Weights expressing the relative importance of qualitative criteria such as public reaction, relocation hardship and transit mobility must be regarded as purely subjective. Each individual opinion as to what the weight should be can simply be viewed, in a statistical sense, as an estimate of the true weight. Perhaps, then, the correct weight l% on criterion i is the population average of the weights-that is, the average of the values suggested by all individuals in the population affected by the corridor(s). If one accepts this rationalization as to the interpretation of the IX, then an estimate of this population average can be obtained through sampling. In fact, a sample estimate (sample mean) can be made arbitrarily close to the population mean by selecting a large enough sample. Thus if a 95% level of confidence, say, is desired when a given corridor is selected as being “best”, 6 in equation (2) can be made as small as is necessary if a proper sample is taken when the I% are being estimated.
CONCLUSIONS
AND DISCUSSION
The principal difference between existing corridor selection models and that proposed herein is that the latter is designed specifically to deal with those situations where ordinal rather than cardinal data is available. The Cook/Seiford [18] and other consensus methods provide a mechanism for collapsing a set of ordinal preferences into a compromise ranking. The problem of combining dimensions has always been an item of controversy. The concept of assigning a weight to each of several criteria where, in practical terms, those criteria are not comparable seems undesirable. This is, however, unavoidable. Ultimately one must always come to terms with the question “Is corridor A which ranks 1st on two dimensions and 6th on a third dimension preferable to corridor B which scores 2nd, 2nd and 4th respectively?’ Hence criteria weights must be assigned in some manner. In this paper we have suggested a sampling procedure for obtaining such. While the ordinal approach proposed in this paper is not intended as a direct replacement for existing methodologies, it does possess certain interesting attributes which would make it an important complementary tool. It appears that further work should be directed toward modifying existing tools to take more direct account of the ordinal elements of such selection problems. Models which can deal with both cardinal and ordinal data are highly desirable. REFERENCES 1. De Leut Cather. Application of evaluation techniques to multidiciplinary feasibility studies. Roads and Transportation Association of Canada (1975). 2. Cohen Benjamin I. Capital Budgeting for Transportation in Underdeveloped Countries. Mimeographed, Cambridge, MA. Harvard Transportation and Economic Development Seminar, Discussion Paper No. 39 (1966). 3. Hill Morris. Planning,for Multiple Objectives. Regional Science Research Institute, Pennsylvania (1973). 4. Hoyle B. S. (Ed.) Transport and Development, Macmillan, London (1973). 5. Adler H. Economic Appraisal c$ Transport Projects. Indiana Univ. Press, Bloomington (197 I ). 6. Stanley William R. Transport expansion in Liberia. In B. S. Hoyle (Ed.), Transport and Decelopmenf. Macmillan, London (1973). 7. Roy B., Sussman B. and Bouayoun R. A decision method in presence of multiple view-points: ELECTRE. Paper presented at the Study SesGons on Methods of Calculation in the Social Sciences, Rome (July, 1966). 8. Roy B. Classement et choix en presence de points de vue multiples. Revue fr. Reck opkr. 8, 57775 (1968).
276
WADE D. COOK and LAWRENCE M. SEIF~RD
9. Massam Bryan H. The search for the best alternative using multiple criteria; Singapore transit study. Econ. Geogr. 54(3), 245-253 (1978). Applications to Phnning Problems in the Public Sector. Pergamon Press, Oxford (1980). 10. Massam Bryan H. SpatialSearch: units. Reg. Urban Econ. 1, 107-138 (1971). 11. Guigou J.-L. On french location models for production R. A multi-objective decision model for regional development, environmental quality 12. Van Delft A. and Niikamp control and industrial land use. Paper prepared for the European Meeting qf the Regional Science Associution, Budapest (August, 1974). to the solution of complex location problems: Spadina subway 13. Massam Bryan H. and Miller Paul. An approach alignment, Toronto. Discussion Paper No. 23, Dept of Geography, York Univ., Toronto (1980). 14. Kemeny J. G. and Snell L. J. Preference ranking: an axiomatic approach. In Muthematicul Models ;,I the Sociul Scier~es, pp. 9-23. Ginn, New York (1962). 15. Kendall M. Rank Correlation Methods. 3rd edn. Hafner, New York (1962). 16. Bogart Kenneth P. Preference structures I: distances between transitive preference relations. J. math.Social. 3, 49967 (1973). II: distances between asymmetric relations. SIAM J. appl. Muth. Vol. 29(2), 17. Bogart Kenneth P. Preference structures 254-262 (1975). 18. Cook Wade D. and Seiford Lawrence M. Priority ranking and consensus formation. Mgrnr Sci. 24(16), 1721-1732 (1978). 19. Arrow K. J. Social Choice and Individual Values. Wiley, New York (195 1).
1984 Copyright
AN ORDINAL RANKING MODEL CORRIDOR SELECTION WADE Faculty
of Administrative
0198-9715184 $3.00~0.00 .(‘: 1984 Pergamon Press Ltd
FOR THE HIGHWAY PROBLEMJ-
D. COOK
Studies, York University,
Toronto,
Canada
M31 2R6
and LAWRENCE Department Abstract -This paper present a model which desirabilityof various is given. An approach
of General
Business,
M. SEIFORD
University
of Texas, Austin, TX 78712, U.S.A.
examines the problem of corridor selection in highway design and routing. We is designed specifically to deal with ordinal information pertaining to the relative potential corridors. A technique for combining ordinal rankings into a consensus to the problem of determining criteria weights is suggested.
INTRODUCTION
A familiar problem in highway design and planning is one of route selection. Specifically, when a highway is to be constructed in a particular geographical region, the first major task facing the transport agency is that of selecting a corridor-a 30&500 wide strip of land along which the highway is to pass. In assessing the relative desirability of any set of potential corridors it is necessary to investigate various impact parameters. These include user and regional development models, both benefits, environmental and social benefits and costs,. . . etc. While numerous sophisticated and ad hoc, have been developed to deal with this problem, all appear to meet with difficulties at the implementation stage. This is partially due to the multidimensional nature of the problem. Generally, existing models attempt to quantify the impact of each dimension. However, since a large number of impact parameters, particularly those relating to environmental and social benefits/disbenefits, cannot easily be quantified (if at all), these models tend in many instances to misrepresent prevailing conditions. Since available information is often of the ordinal rather than cardinal type, these quantitative (cardinal) approaches often fall short of their goal. In this paper we present a model designed to take account of the ordinal nature of the problem. First we show how to combine ordinal information such as to obtain a ranking of the corridors on the basis of each dimension. We then describe a procedure for combining rankings across dimensions. While this new model is not intended as a direct replacement for “cardinal” approaches currently in use, it does have tremendous potential as a complementary tool of analysis. In the following section we describe the corridor selection problem. The third section presents a brief review of current approaches to this problem, along with associated literature. The fourth section then details an ordinal ranking model for the corridor selection problem. The final section discusses conclusions and future directions.
THE
CORRIDOR
SELECTION
PROBLEM
A problem which has attracted considerable interest during the past several decades is that of design and selection of highway corridors. The overwhelming impact of highways on the economy as well as on the social and environmental structure of a region or country has captured the attention of economists, operations researchers, engineers and government agencies. Over the past 40 years highway planning has been among the top two or three priority areas of study in developed and developing nations alike.
tSupported
in part under NSERC
Grant
No. A8966. 271
212
WADE D. COOKand LAWRENCEM. SEIFORLI
The dimensions of the problem can be represented via the following categories of benefits/ disbenefits: (1) regional development; (2) social; (3) environmental; (4) user; (5) right of way. Regional development parameters, for example, would include employment and business opportunities, net changes in housing costs due to highway improvements,. . . etc. Social benefits/ costs would include community cohesion and growth, relocation hardship, public reaction, transit mobility,. etc. We will not elaborate here on the other classes of benefits and costs. Benefits as outlined above, fall into two basic groups-“hard” and “soft”. In most highway planning problems there are an overwhelming number of soft or qualitative aspects of the problem, and while traditional cost/benefit and related techniques have attempted to capture such factors, they have, in most instances, been less than successful. Various information gathering devices (e.g. public opinion polls) have been utilized to assess the relative impacts of proposed corridors along each selected dimension. In a 1975 study [I] for example involving the design and routing of a highway bypassing the city of Peterborough, Ontario, an opinion poll was conducted in each municipality. For each poll a sample of residents was selected, and each was asked to assess the impact which various routings of corridors would have in terms of water pollution, noise, loss of natural land areas, damage to fish and game,. . . etc. The opinions of residents generally appeared in an ordinal format along several dimensions. Even along quantitative dimensions voters are often not able to supply hard numbers, but rather can only express preferences on an ordinal basis. Various modelling techniques have been proposed and used in the economic evaluation of highway corridors. Some of these are reviewed in the section to follow. In the fourth section we present an alternative to these models-one which specifically takes account of the ordinal nature of the problem.
THE
LITERATURE
Among the earliest applications of cost/benefit analysis were those in the transport field. Projects ranged from highway development studies in U.S.A., Canada, Australia, Papua New Guinea and Dominican Republic (see [2-41) to canal and airport construction in Kenya and elsewhere. Adler [.5] documents numerous studies from roads and railways to ports and pipelines in literally dozens of countries throughout the world. Hoyle [4] discusses the use of cost/benefit methods in the selection of highway routes in Uganda and Kenya. Within the past 25 years Liberia for example has been transformed from a series of disconnected ports to a politically unified country through road and rail development. Stanley [6] reports on the use of cost/benefit analysis in this context. Cost/benefit methodology does however possess certain shortcomings as described by Hill [3] : “When costs and benefits are not available in market prices, the cost/benefit model imputes them as if they were subject to market transactions. When imputing market prices the analyst should distinguish between those items which are ordinarily evaluated in the market.. .and the intangibles which.. .cannot be priced in monetary terms. In practice these intangibles fall outside the central criterion and beyond the scope of the tools of measurement of cost/benefit analysis.”
Attempts to rectify apparent flaws in cost/benefit procedures have given rise to a number of adjuncts to the conventional approach. The most important class of these adjuncts is the set of multiattribute ranking procedures. While there are many different ranking models in existence one of the more popular approaches vis-a-vis problems such as the one under discussion is that developed by Roy [7,8]. Roy’s method begins with a set of alternatives (e.g. corridors) and a set of criteria/dimensions. It is then assumed that the planner is able to assign a numerical (cardinal) rating to each alternative with respect to
Ordinal
ranking
model for highway
corridor
selection
213
each criterion. Weights are then applied to each criterion. Roy then defines a concordance index for each pair of alternatives and uses this set of indicies to arrive at a “best” alternative. Massam [9, lo] has extended Roy’s method to produce a complete ranking (not just the first ranked) of the alternatives. Singapore transit data is used to test the model. Massam [lo] provides an extensive coverage of these concepts. On the operational side Guigow [I l] and Van Delft and Nijkamp [12] have applied Roy’s method to particular spatial location problems, Similarly, ranking techniques such as those discussed above have been used in subway alignment problems. See Massam and Miller [ 131. A number of other similar ranking procedures have been used to examine the route/corridor selection problem. These will not be discussed here. It is sufficient to say that in general all current approaches subscribe to the same “cardinal number” rules. Either it is specifically required that cardinal, not ordinal, data be submitted to the model or the model or decision maker imputes cardinal numbers wherever ordinal information is the only data available. We now present a model which provides an alternative approach for dealing with those selection problems in which the information available is primarily ordinal in nature. It should be pointed out, however, that the following section is not an in-depth study of corridor selection, but rather is intended to illustrate how ordinal techniques could be applied to such a problem.
AN
Ranking
ORDINAL
MODEL
FOR
CORRIDOR
SELECTION
by dimension
In the model to be developed in this section let us assume that each member of a group of individuals (municipal committee, strategic task force,. . . etc.) is asked to rank the corridors, with a ranking being supplied corresponding to each dimension. While some criteria may be such that the voter can provide a cardinal ranking, we will concentrate here on ordinal preferences. In the cardinal case techniques described above can be applied. For each qualitative dimension i assume that each committee member j supplies an ordinal ranking (LEE, a?, . . . d,. . ., a! where d is th e rank assigned to the Jth corridor along the ith dimension. For example, if i = 1 = recreational benefits, and if there are 5 corridors, this ordinal ranking might take the form (2, 1, 4, 3, 5) i.e. ai = 2, u? = 1, . . . etc. Thus corridor 1 is given a rank of 2, corridor 2 a rank of 1, etc. In assigning ranks to corridors, it is expected that each member would give consideration to both the benefits as well as the cost of doing the project. In order to combine the ordinal rankings [(ni’j, . . ., &)]jE1 of the 171committee members into a compromise or consensus ranking, many different possibilities exist. Kemeny and Snell [14], Kendall [lS], Bogart [16, 173 and Cook and Seiford [18] all suggest different procedures for doing such. Since the Cook/Seiford method is designed to handle large problems, it is particularly appropriate in the problem under discussion. It, like a number of the other approaches possesses certain desirable properties which any consensus method should exhibit. These properties are similar in some respects to the social choice axioms of Arrow [19]. A full discussion is given in [18]. The Cook/Seiford consensus method for a particular criterion proceeds as follows. Given a set of rankings A 1, AZ,. . ., A, [where Aj = (uj, a:,. . ., a:)] of the L corridors, the consensus is a ranking fi = (6’, 6’, . . ., 6’) which minimizes the function,
Cook and Seiford show that the optimal ranking can be determined by formulating linear assignment problem.
tThe Index i for criteria
has been suppressed.
M(B) as a
WADE D. COOK and LAWRENCEM. SEIFORD
214
Example. Consider
the following
set of rankings:
(m = 10 committee
members, 1 4 3 2 5
Let us now construct
the distance
matrix
n = 5 objects) 4 1 5 3 2
1 3 4 2 5
(d/k). Note that:
k = 1, 2,. . ., 5 ranks; m = 1, 2,. . ., 10 members ; /=1,2 d/k is the sum of the deviations aj’ of object /. For example: dll= dzl=
between
,...,
5objects.
a selected
rank
k and the 10 given member
ll-11+14-ll+13-1(+...+13-11 ~4-1~+)3-1~+~2-1l+...+~4-1~
ranks
=13; =21.
Thus, we have: Distance 13 21 24 22 20 Using
the assignment
11 15 16 16 12
algorithm
we get:
Object
1
2
110000 2 0 3 0 4 0 5 0
0 0 0 1
Matrix
(d/k)
11 11 12 14 12
17 11 12 14 16
Priority 3
1 0 0 0
27 19 16 18 20
4
5
0 1 0 0
0 0 1 0
That is B = [l, 3, 4, 5, 21 and M(g) = 66 is the minimum distance. it is clear that alternate medians exist (e.g. B = [l, 4, 3, 5, 23).
By viewing
the cost matrix,
Combining dimensions While ordinal rankings associated with any given dimension can be used to arrive at a consensus or compromise ranking relative to that dimension, any technique used to aggregate dimensions in order to get an overall consensus must rely on cardinal dimension weights. Unless the relative importance of the various dimensions can be quantified no operational method exists for getting the consensus. Let us assume initially that a set of fixed known weights (r/t;}{= 1 are available. If Bi denotes the consensus associated with criterion i, a logical overall consensus is, therefore, B=
i i=l
WBi
(1)
Ordinal
ranking
model for highway
corridor
selection
275
Specifically, if 6/i ~6, for all /#/I, then corridor /r is the preferred route. Typically the I% values will possess some degree of error, hence there is some concern as to the stability of the first place alternative kr in B. However, because of the simplistic nature of equation (1) it is easy to perform a sensitivity analysis on each dimension weight. For example, if 6 represents the possible error factor in w1 say (i.e. the weight on the iiih criterion is II$, +6), then B is changed to B+6&,. It is easily shown that corridor /i is still optimal provided,
Obtaining weights through sampling
Weights expressing the relative importance of qualitative criteria such as public reaction, relocation hardship and transit mobility must be regarded as purely subjective. Each individual opinion as to what the weight should be can simply be viewed, in a statistical sense, as an estimate of the true weight. Perhaps, then, the correct weight l% on criterion i is the population average of the weights-that is, the average of the values suggested by all individuals in the population affected by the corridor(s). If one accepts this rationalization as to the interpretation of the IX, then an estimate of this population average can be obtained through sampling. In fact, a sample estimate (sample mean) can be made arbitrarily close to the population mean by selecting a large enough sample. Thus if a 95% level of confidence, say, is desired when a given corridor is selected as being “best”, 6 in equation (2) can be made as small as is necessary if a proper sample is taken when the I% are being estimated.
CONCLUSIONS
AND DISCUSSION
The principal difference between existing corridor selection models and that proposed herein is that the latter is designed specifically to deal with those situations where ordinal rather than cardinal data is available. The Cook/Seiford [18] and other consensus methods provide a mechanism for collapsing a set of ordinal preferences into a compromise ranking. The problem of combining dimensions has always been an item of controversy. The concept of assigning a weight to each of several criteria where, in practical terms, those criteria are not comparable seems undesirable. This is, however, unavoidable. Ultimately one must always come to terms with the question “Is corridor A which ranks 1st on two dimensions and 6th on a third dimension preferable to corridor B which scores 2nd, 2nd and 4th respectively?’ Hence criteria weights must be assigned in some manner. In this paper we have suggested a sampling procedure for obtaining such. While the ordinal approach proposed in this paper is not intended as a direct replacement for existing methodologies, it does possess certain interesting attributes which would make it an important complementary tool. It appears that further work should be directed toward modifying existing tools to take more direct account of the ordinal elements of such selection problems. Models which can deal with both cardinal and ordinal data are highly desirable. REFERENCES 1. De Leut Cather. Application of evaluation techniques to multidiciplinary feasibility studies. Roads and Transportation Association of Canada (1975). 2. Cohen Benjamin I. Capital Budgeting for Transportation in Underdeveloped Countries. Mimeographed, Cambridge, MA. Harvard Transportation and Economic Development Seminar, Discussion Paper No. 39 (1966). 3. Hill Morris. Planning,for Multiple Objectives. Regional Science Research Institute, Pennsylvania (1973). 4. Hoyle B. S. (Ed.) Transport and Development, Macmillan, London (1973). 5. Adler H. Economic Appraisal c$ Transport Projects. Indiana Univ. Press, Bloomington (197 I ). 6. Stanley William R. Transport expansion in Liberia. In B. S. Hoyle (Ed.), Transport and Decelopmenf. Macmillan, London (1973). 7. Roy B., Sussman B. and Bouayoun R. A decision method in presence of multiple view-points: ELECTRE. Paper presented at the Study SesGons on Methods of Calculation in the Social Sciences, Rome (July, 1966). 8. Roy B. Classement et choix en presence de points de vue multiples. Revue fr. Reck opkr. 8, 57775 (1968).
276
WADE D. COOK and LAWRENCE M. SEIF~RD
9. Massam Bryan H. The search for the best alternative using multiple criteria; Singapore transit study. Econ. Geogr. 54(3), 245-253 (1978). Applications to Phnning Problems in the Public Sector. Pergamon Press, Oxford (1980). 10. Massam Bryan H. SpatialSearch: units. Reg. Urban Econ. 1, 107-138 (1971). 11. Guigou J.-L. On french location models for production R. A multi-objective decision model for regional development, environmental quality 12. Van Delft A. and Niikamp control and industrial land use. Paper prepared for the European Meeting qf the Regional Science Associution, Budapest (August, 1974). to the solution of complex location problems: Spadina subway 13. Massam Bryan H. and Miller Paul. An approach alignment, Toronto. Discussion Paper No. 23, Dept of Geography, York Univ., Toronto (1980). 14. Kemeny J. G. and Snell L. J. Preference ranking: an axiomatic approach. In Muthematicul Models ;,I the Sociul Scier~es, pp. 9-23. Ginn, New York (1962). 15. Kendall M. Rank Correlation Methods. 3rd edn. Hafner, New York (1962). 16. Bogart Kenneth P. Preference structures I: distances between transitive preference relations. J. math.Social. 3, 49967 (1973). II: distances between asymmetric relations. SIAM J. appl. Muth. Vol. 29(2), 17. Bogart Kenneth P. Preference structures 254-262 (1975). 18. Cook Wade D. and Seiford Lawrence M. Priority ranking and consensus formation. Mgrnr Sci. 24(16), 1721-1732 (1978). 19. Arrow K. J. Social Choice and Individual Values. Wiley, New York (195 1).