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Generating consensus priority point vectors: a logarithmic goal programming approach
Computers & Operations Research 26 (1999) 637—643
Technical Note
Generating consensus priority point vectors: a logarithmic goal programming approach Noel (Kweku-Muata) Bryson , Anito Joseph * School of Business, Virginia Commonwealth University, Richmond, VA 23284, USA School of Business Administration, Department of Management Science, University of Miami, 417 Jenkins Building, Coral Gables, FL 33124, USA Received February 1997; received in revised form July 1998
Abstract The analytic hierarchy process (Saaty, The Analytic Hierarchy Process: Planning, Priority Setting, Resource Allocation, NewYork: McGraw-Hill 1980) is a popular technique for addressing multiple-criteria decision-making problems (MCDMs). Various techniques have been proposed for using the AHP in group situations. Fundamental to the AHP is the generation of priority point vectors from matrices of pairwise comparison data. In this paper, we present a logarithmic goal programming model for generating the ‘consensus’ priority point vector from the set of individual priority point vectors. 1999 Elsevier Science Ltd. All rights reserved. Scope and purpose Within modern organizations, multiple-criteria decision-making problems (MCDMs) often occur within a group context, and individual priorities for decision alternatives must be synthesized into a single set of priorities which represents the consensus opinion for the group. This requires a process for aggregating individual priorities into a set of group priorities. In this paper, we examine the use of the analytic hierarchy process (AHP) MCDM technique for the group situation, and present an approach for aggregating individual priorities into a set of group ‘consensus’ priorities. Keywords: AHP; Group decision making; MCDM; Goal programming
*Corresponding author. Tel.: 001 305 284 6595; fax: 001 305 284 2321. 0305-0548/99/$ — see front matter 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 3 0 5 - 0 5 4 8 ( 9 8 ) 0 0 0 8 3 - 5
638
N.(K.-M.) Bryson, A. Joseph / Computers & Operations Research 26 (1999) 637—643
1. Introduction The analytic hierarchy process [1] is a popular technique for addressing multiple-criteria decision-making (MCDM) problems. Given the increasing importance of group work in modern organizations, techniques have been proposed for using the AHP in group situations (e.g. Saaty [2] and Bryson [3]. Under the AHP model for an individual decision-maker, a decision involving a set of N objects (e.g. criteria, alternatives) is represented using a priority point vector (or weight vector) w"(w , 2 , w ), where w "1; and w '0 for j"1, 2 , N. , H H H In this representation the ratio (w/w ) reflects the decision-maker’s belief in the relative G H importance of object ‘‘i’’ compared to object ‘‘j’’. Similarly, a decision involving a group of decisionmakers is represented using a consensus priority point vector w"(w , 2 , w ), where again , each ratio (w /w ) reflects the group’s belief in the relative importance of object ‘‘i’’ compared to G H object ‘‘j’’. Techniques for determining the priority vector (e.g. w) involve the specification of a pairwise comparison matrix A"+a ,, where a is a numeric point estimate of the relative importance of GH GH object i compared to object j. If a given evaluator is perfectly consistent in his/her estimates then for each i, j, k we would have the relation a "a a . However, it is common that perfect consistency GH GI IH does not exist in the pairwise comparison data of the A matrix, and so an estimate of the relative consistency of the A matrix is provided. Various consistency indicators have been proposed by other researchers (e.g. Saaty, [1] and Golden and Wang [4]) but these indicators lack meaningful interpretation. In this paper we present an integrated logarithmic goal programming-based model (LGPM) for generating the ‘consensus’ priority point vector, w"(w , 2 , w ). Previous approaches have been , based on either the eigenvector method (EM) or the logarithmic least-squares method (LLSM). Unlike EM, LGPM does not require that the pairwise comparison matrix be reciprocal; and unlike LLSM, LGPM does not require any statistical assumptions. Both the EM and LLSM are overly sensitive to the presence of outlier opinions, while LGPM is ‘resistant’ to the presence of outliers. Our LGPM-based procedure for the group process is an integrated one that does not require the explicit computation or specification of a group value for each pairwise comparison entry. The method also has the additional advantage of providing an interpretable consistency indicator for the group data.
2. Generating group priority point vectors The process of generating the group priority point vector involves the following three stages: (1) Individual specification and review (a) For each pair of objects, pairwise comparison data is specified by each individual group member. (b) For each group member, the corresponding priority point vectors and consistency indicators are generated. Each group member examines this data, and if the degree of consistency is acceptable, the process goes forward to the next stage. Otherwise the group member repeats this stage.
N.(K.-M.) Bryson, A. Joseph / Computers & Operations Research 26 (1999) 637—643
639
(2) Group Synthesis This stage involves application of a procedure for synthesizing the individual pairwise comparison data into group pairwise comparison data. Two procedures have been suggested: consensus vote, and geometric averaging. In the consensus vote option, the entire group provides a single numeric value for each pairwise comparison resulting in a ‘consensus’ pairwise comparison matrix A"+a ,. In the geometric averaging option [5], each individual (say ‘‘t’’ in ¹, ¹ being the index set GH of the group members) provides a numeric value a that reflects her/his view of the relative GH importance of object ‘‘i’’ compared to object ‘‘j’’. The ‘consensus’ matrix A"+a , is computed by GH the formula: a "(% 3 a ) +, where ‘‘M’’ is the number of group members in ¹. GH R 2 GH (3) Prioritization: In this stage, the priority point vector generation technique is applied to obtain the ‘consensus’ priority point vector, w"(w , 2 , w ). Traditionally, either the right eigenvector method (see , Saaty [1]) or the logarithmic least squares (see Fichtner [6]) have been used for this stage.
3. A logarithmic goal programming model (LGPM) In this section we present a logarithmic goal programming model (LGPM) for generating the consensus priority vector. This model integrates the synthesis and prioritization stages. A variation of LGPM has also been used for individual priority vectors [7, 8]. Goal programming has been previously suggested by Arbel [9] for generating priority point vectors. In Arbel’s model the pairwise comparison matrix contained interval estimates, thus A"+(l , u ),. In order to generate the priority point vector, the following linear goal programming GH GH problem is solved: Min +w "l w !w #w "0;!u w !w #w "0; w "1, , G GH H G GH H G G 3' where I"+i: 1)i)N,, and w '0 ∀i3I. G Although this model has the advantage of being able to accommodate both precise (l "u ) and GH GH imprecise (i.e. l (u ) pairwise comparison information, sometimes the problem may be infeasible GH GH because of inconsistencies in the values of the A matrix. We therefore, propose an alternate goal programming model in which our objective is to generate a priority point vector w"(w , 2 , w ) , such that for the comparison between each pair of objects ‘i’ and ‘j’, the difference between the ratio (w /w ) and the decision-maker’s specified a is minimized. Now let there be real numbers p *1, G H GH GH q *1 such that (w /w )*(p /q )"a , where p and q cannot both be greater than 1. Then p " GH G H GH GH GH GH GH GH q "1 implies that (w /w )"a ; q '1 implies that (w /w )'a ; and p '1 implies that (w /w ) GH G H GH GH G H GH GH G H (a . Therefore, if p "q "1 for each pair of objects ‘i’ and ‘j’, then the set of point estimates GH GH GH provided by the decision-maker ‘t’ is consistent; otherwise, the data are inconsistent and our problem then is to minimize the product % 3 % 3 p q . Aczel and Saaty [5] suggested that the G ' H ' GH GH group ‘consensus’ pairwise comparison values should be the geometric average of the individual pairwise comparison values. Rather than explicitly focusing on each pairwise comparison, we will focus on the entire set of pairwise comparison values. Therefore if ¹ is the index set of the
640
N.(K.-M.) Bryson, A. Joseph / Computers & Operations Research 26 (1999) 637—643
decision-makers and M""¹", it follows that, for the group, our problem is to minimize the product % 3 % 3 % 3 p q . This translates to solving the following linear goal programming problem: R 2 G ' H ' GH GH LGPP: lg#"Min(1/M) lg# R s.t. lgv !lgv # lgp !lgq "lga ∀t3¹; i, j3I G H GH GH GH (1/K) (lgp #lgq )!lg#"0 ∀t3¹ GH GH G 3' H 3' where K"N*(N!1); I"+1, 2, 2 , N,; lgv "log(v ); lgp "log(p ); lgq "log(q ); lg#" G G GH GH GH GH log(#); lg#"log(#); and all variables are nonnegative. If A is a reciprocal matrix, then some of the constraints are redundant and can be ignored. The solution of this problem results in the unnormalized vector v"(v , 2 , v ), which can be normalized to give our normalized consensus , priority point vector w"(w , 2 , w ) where (v /v )"(w /w ) for each (i, j). It should be noted that , G H G H unlike Arbel’s goal programming model, our logarithmic goal programming model is never infeasible. The reader might have also observed that LGPP is mathematically similar to the minimum sum of absolute errors (MSAE) regression model which is known to be resistant to the presence of outliers [10]. Likewise, it can be shown that LGPM is resistant to outliers (e.g. Bryson [8]). Hence the presence of outliers in the pairwise comparison preference data should not have an adverse effect. Another interesting property of LGPP is that it provides a set of potentially meaningful consistency indicators. Given that lg#"log(#) is the objective function value, then # is the minimum average value that each entry in the comparison matrix would have to be multiplied or divided by in order to make the set of pairwise comparison values consistent. Thus if the decision-makers’ estimates were consistent we would have log(#)"0 and #"1. Otherwise we would have log (#)'0 and #'1. Associated with #, is its dual indicator p"1/# which takes on values in the interval (0, 1], and is an average of the fractions a /w (when p (q ) and w /a (when GH GH GH GH GH GH p 'q ), where w "w /w . p is thus a measure of the average consistency of the pairwise GH GH GH H comparisons. Thus, p"1 indicates a situation of total consistency, and p between 0 and 1 indicate the relative consistency of the pairwise comparison matrix. Of course, similar to other proposed consistency indicators (e.g. Saaty’s C.R., and Golden and Wang’s G), there are no objective cut-off values for # and p. However, since users can associate meaning with # and p values, they could adopt an action learning process that would result in the development of personal cut-off rules. It should be noted that corresponding consistency indicator p("1/#) are associated with each decision-maker ‘‘t’’.
4. Illustrative example In our illustrative example, a three-person selection committee has to determine the criteria and corresponding weights that are to be used in the selection of a database management system. After analysis they decided that the most appropriate evaluation criteria are Cost, Functionality,
N.(K.-M.) Bryson, A. Joseph / Computers & Operations Research 26 (1999) 637—643
641
Usability, Reliability, and Vendor support. The decision makers on the committee decide to use pairwise comparison techniques to generate the corresponding group weights with p"0.80 as the cut-off value for an acceptable degree of consistency. The reader may recall that p is the minimum average value that each entry in the comparison matrix would have to be multiplied or divided by in order to make the set of pairwise comparison values consistent. In Step 1, each decision-maker entered his/her pairwise comparison data as shown in Table 1. The LGPP was formulated and solved for each decision-maker, i.e., ¹"+1,, resulting in the generation of the corresponding individual priority point (or weight) vector and consistency indicator. Table 2 displays the resulting individual priority point vectors and consistency indicators. Since each of the individual consistency indicators implied an acceptable degree of consistency (i.e. p*0.80), then the group LGPP, i.e., ¹"+1, 2, 3, was formulated and solved. The resulting group priority point vector and consistency indicator were generated. These results are displayed in Table 2. From this we can see that there is again an acceptable level of consistency for Table 1 Functionality
(a) Pairwise comparison data for decision-maker 1 Cost 1.05 Functionality Usability Reliability (b) Pairwise comparison data for decision-maker 2 Cost 1.19 Functionality Usability Reliability (c) Pairwise comparison data for decision-maker 3 Cost 1.10 Functionality Usability Reliability
Usability
Reliability
Vendor support
1.33 1.38
1.74 1.25 0.95
1.70 1.54 1.73 1.05
1.57 1.21
1.38 1.54 1.22
2.48 2.19 1.78 1.67
1.30 1.35
1.70 1.35 1.20
2.20 1.65 1.75 1.45
Table 2 Decision maker
Priority point vector
1 2 3 Group mean
(0.247, (0.280, (0.265, (0.250,
0.235, 0.243, 0.241, 0.238,
0.185, 0.200, 0.199, 0.188,
0.180, 0.164, 0.175, 0.176,
0.153) 0.113) 0.120) 0.147)
#
p
1.10 1.05 1.05 1.15
0.91 0.95 0.95 0.87
642
N.(K.-M.) Bryson, A. Joseph / Computers & Operations Research 26 (1999) 637—643
the group pairwise comparison data. It should be noted that even though the individual pairwise comparison data could each have an acceptable degree of consistency, that the corresponding group pairwise comparison data could fail to have an acceptable degree of consistency.
5. Conclusion In this paper we have explored features of a logarithmic goal programming method for generating consensus priority point vectors. This method possesses some attractive properties that make it an appealing alternative to other proposed methods. It is resistant to the presence of ‘outliers’ in pairwise comparison data that result in distorted priority vectors being produced by other methods; it does not require any statistical assumptions; and does not require that the pairwise comparison matrix be reciprocal. It also provides a set of potentially useful indicators for interpreting the consistency of the pairwise comparison matrix.
References [1] Saaty T. The analytic hierarchy process: planning, priority setting. Resource allocation, New York: McGraw-Hill, 1980. [2] Saaty T. Group decision making and the AHP. In: Golden B, Wasil E, Harker P, editors. The Analytic Hierarchy Process: Application and Studies 1989;59—67. [3] Bryson N. Group decision-making and the analytic hierarchy process: exploring the consensus-relevant information content. Computers & Operations Research 1996;23:27—35 [4] Golden B, Wang Q. An alternate measure of consistency. In: Golden B, Wasil E, Harker P, editors. The Analytic Hierarchy Process: Application and Studies 1989;68—81. [5] Aczel J, Saaty T. Procedures for synthesizing ratio judgments. Journal of Mathematical Psychology 1983;27: 93—102. [6] Fichtner J. On deriving priority vectors from matrices of pairwise comparisons. Socio-Economic Planning Sciences 1986;20(6):399—405. [7] Bryson N, Mobolurin Q, Ngwenyama O. Modeling pairwise comparisons on ratio scales. European Journal of Operational Research 1995;83:639—54. [8] Bryson N. A goal programming for generating priority vectors. Journal of the Operational Research Society 1995;46:641—8. [9] Arbel A. A linear programming approach for processing approximate articulation of preference. In: Korhonen P, Lewandowski A, Wallenius J, editors. Multiple Criteria Decision Support 1989:81—6. [10] Narula S. The minimum sum of absolute errors regression, Journal of Quality Technology 1987;19:37—45. [11] Basak I, Saaty T. Group decision making using the analytic hierarchy process. Mathematical and Computer Modelling 1993;17(4/5):101—109.
Kweku-Muata (Noel) Bryson is Professor of Information Systems in the School of Business at Virginia Commonwealth University in Richmond, Virginia. Previously he was Professor of Information Systems and Decision Sciences in the School of Business at Howard University in Washington, DC. He holds a B.Sc. in Natural Sciences from the University of the West Indies at Mona; an M.S. in Systems Engineering from Howard University, and a Ph.D. in Applied Mathematics (Management Science & Information Systems) from The University of Maryland at College Park. Dr. Bryson’s research areas include: group support systems, multicriteria decision analysis, expert systems, spatial and distributed database systems, and integer programming. He has published articles in a variety of journals including:
N.(K.-M.) Bryson, A. Joseph / Computers & Operations Research 26 (1999) 637—643
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Computers & Operations Research, IEEE ¹ransactions on Knowledge & Data Engineering, Decision Support Systems, Information Systems, Information Processing & Management, Data & Knowledge Engineering, Journal of the Operational Research Society, European Journal of Operational Research, and Journal of Multiple Criteria Decision Analysis. Anito Joseph is an Assistant Professor in the Department of Management Science, School of Business Administration, University of Miami. She holds a B.Sc. in Natural Science from the University of the West Indies at St. Augustine, and M.S. and Ph.D, in Management Science/Statistics from the University of Maryland at College Park. She has published in Computers & Operations Research, ¹ransportation Science, European Journal of Operational Research, and Mathematical and Computer Modelling. Her research interests include integer programming, cluster analysis, multicriteria decision analysis and applications of operations research in information systems.
Technical Note
Generating consensus priority point vectors: a logarithmic goal programming approach Noel (Kweku-Muata) Bryson , Anito Joseph * School of Business, Virginia Commonwealth University, Richmond, VA 23284, USA School of Business Administration, Department of Management Science, University of Miami, 417 Jenkins Building, Coral Gables, FL 33124, USA Received February 1997; received in revised form July 1998
Abstract The analytic hierarchy process (Saaty, The Analytic Hierarchy Process: Planning, Priority Setting, Resource Allocation, NewYork: McGraw-Hill 1980) is a popular technique for addressing multiple-criteria decision-making problems (MCDMs). Various techniques have been proposed for using the AHP in group situations. Fundamental to the AHP is the generation of priority point vectors from matrices of pairwise comparison data. In this paper, we present a logarithmic goal programming model for generating the ‘consensus’ priority point vector from the set of individual priority point vectors. 1999 Elsevier Science Ltd. All rights reserved. Scope and purpose Within modern organizations, multiple-criteria decision-making problems (MCDMs) often occur within a group context, and individual priorities for decision alternatives must be synthesized into a single set of priorities which represents the consensus opinion for the group. This requires a process for aggregating individual priorities into a set of group priorities. In this paper, we examine the use of the analytic hierarchy process (AHP) MCDM technique for the group situation, and present an approach for aggregating individual priorities into a set of group ‘consensus’ priorities. Keywords: AHP; Group decision making; MCDM; Goal programming
*Corresponding author. Tel.: 001 305 284 6595; fax: 001 305 284 2321. 0305-0548/99/$ — see front matter 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 3 0 5 - 0 5 4 8 ( 9 8 ) 0 0 0 8 3 - 5
638
N.(K.-M.) Bryson, A. Joseph / Computers & Operations Research 26 (1999) 637—643
1. Introduction The analytic hierarchy process [1] is a popular technique for addressing multiple-criteria decision-making (MCDM) problems. Given the increasing importance of group work in modern organizations, techniques have been proposed for using the AHP in group situations (e.g. Saaty [2] and Bryson [3]. Under the AHP model for an individual decision-maker, a decision involving a set of N objects (e.g. criteria, alternatives) is represented using a priority point vector (or weight vector) w"(w , 2 , w ), where w "1; and w '0 for j"1, 2 , N. , H H H In this representation the ratio (w/w ) reflects the decision-maker’s belief in the relative G H importance of object ‘‘i’’ compared to object ‘‘j’’. Similarly, a decision involving a group of decisionmakers is represented using a consensus priority point vector w"(w , 2 , w ), where again , each ratio (w /w ) reflects the group’s belief in the relative importance of object ‘‘i’’ compared to G H object ‘‘j’’. Techniques for determining the priority vector (e.g. w) involve the specification of a pairwise comparison matrix A"+a ,, where a is a numeric point estimate of the relative importance of GH GH object i compared to object j. If a given evaluator is perfectly consistent in his/her estimates then for each i, j, k we would have the relation a "a a . However, it is common that perfect consistency GH GI IH does not exist in the pairwise comparison data of the A matrix, and so an estimate of the relative consistency of the A matrix is provided. Various consistency indicators have been proposed by other researchers (e.g. Saaty, [1] and Golden and Wang [4]) but these indicators lack meaningful interpretation. In this paper we present an integrated logarithmic goal programming-based model (LGPM) for generating the ‘consensus’ priority point vector, w"(w , 2 , w ). Previous approaches have been , based on either the eigenvector method (EM) or the logarithmic least-squares method (LLSM). Unlike EM, LGPM does not require that the pairwise comparison matrix be reciprocal; and unlike LLSM, LGPM does not require any statistical assumptions. Both the EM and LLSM are overly sensitive to the presence of outlier opinions, while LGPM is ‘resistant’ to the presence of outliers. Our LGPM-based procedure for the group process is an integrated one that does not require the explicit computation or specification of a group value for each pairwise comparison entry. The method also has the additional advantage of providing an interpretable consistency indicator for the group data.
2. Generating group priority point vectors The process of generating the group priority point vector involves the following three stages: (1) Individual specification and review (a) For each pair of objects, pairwise comparison data is specified by each individual group member. (b) For each group member, the corresponding priority point vectors and consistency indicators are generated. Each group member examines this data, and if the degree of consistency is acceptable, the process goes forward to the next stage. Otherwise the group member repeats this stage.
N.(K.-M.) Bryson, A. Joseph / Computers & Operations Research 26 (1999) 637—643
639
(2) Group Synthesis This stage involves application of a procedure for synthesizing the individual pairwise comparison data into group pairwise comparison data. Two procedures have been suggested: consensus vote, and geometric averaging. In the consensus vote option, the entire group provides a single numeric value for each pairwise comparison resulting in a ‘consensus’ pairwise comparison matrix A"+a ,. In the geometric averaging option [5], each individual (say ‘‘t’’ in ¹, ¹ being the index set GH of the group members) provides a numeric value a that reflects her/his view of the relative GH importance of object ‘‘i’’ compared to object ‘‘j’’. The ‘consensus’ matrix A"+a , is computed by GH the formula: a "(% 3 a ) +, where ‘‘M’’ is the number of group members in ¹. GH R 2 GH (3) Prioritization: In this stage, the priority point vector generation technique is applied to obtain the ‘consensus’ priority point vector, w"(w , 2 , w ). Traditionally, either the right eigenvector method (see , Saaty [1]) or the logarithmic least squares (see Fichtner [6]) have been used for this stage.
3. A logarithmic goal programming model (LGPM) In this section we present a logarithmic goal programming model (LGPM) for generating the consensus priority vector. This model integrates the synthesis and prioritization stages. A variation of LGPM has also been used for individual priority vectors [7, 8]. Goal programming has been previously suggested by Arbel [9] for generating priority point vectors. In Arbel’s model the pairwise comparison matrix contained interval estimates, thus A"+(l , u ),. In order to generate the priority point vector, the following linear goal programming GH GH problem is solved: Min +w "l w !w #w "0;!u w !w #w "0; w "1, , G GH H G GH H G G 3' where I"+i: 1)i)N,, and w '0 ∀i3I. G Although this model has the advantage of being able to accommodate both precise (l "u ) and GH GH imprecise (i.e. l (u ) pairwise comparison information, sometimes the problem may be infeasible GH GH because of inconsistencies in the values of the A matrix. We therefore, propose an alternate goal programming model in which our objective is to generate a priority point vector w"(w , 2 , w ) , such that for the comparison between each pair of objects ‘i’ and ‘j’, the difference between the ratio (w /w ) and the decision-maker’s specified a is minimized. Now let there be real numbers p *1, G H GH GH q *1 such that (w /w )*(p /q )"a , where p and q cannot both be greater than 1. Then p " GH G H GH GH GH GH GH GH q "1 implies that (w /w )"a ; q '1 implies that (w /w )'a ; and p '1 implies that (w /w ) GH G H GH GH G H GH GH G H (a . Therefore, if p "q "1 for each pair of objects ‘i’ and ‘j’, then the set of point estimates GH GH GH provided by the decision-maker ‘t’ is consistent; otherwise, the data are inconsistent and our problem then is to minimize the product % 3 % 3 p q . Aczel and Saaty [5] suggested that the G ' H ' GH GH group ‘consensus’ pairwise comparison values should be the geometric average of the individual pairwise comparison values. Rather than explicitly focusing on each pairwise comparison, we will focus on the entire set of pairwise comparison values. Therefore if ¹ is the index set of the
640
N.(K.-M.) Bryson, A. Joseph / Computers & Operations Research 26 (1999) 637—643
decision-makers and M""¹", it follows that, for the group, our problem is to minimize the product % 3 % 3 % 3 p q . This translates to solving the following linear goal programming problem: R 2 G ' H ' GH GH LGPP: lg#"Min(1/M) lg# R s.t. lgv !lgv # lgp !lgq "lga ∀t3¹; i, j3I G H GH GH GH (1/K) (lgp #lgq )!lg#"0 ∀t3¹ GH GH G 3' H 3' where K"N*(N!1); I"+1, 2, 2 , N,; lgv "log(v ); lgp "log(p ); lgq "log(q ); lg#" G G GH GH GH GH log(#); lg#"log(#); and all variables are nonnegative. If A is a reciprocal matrix, then some of the constraints are redundant and can be ignored. The solution of this problem results in the unnormalized vector v"(v , 2 , v ), which can be normalized to give our normalized consensus , priority point vector w"(w , 2 , w ) where (v /v )"(w /w ) for each (i, j). It should be noted that , G H G H unlike Arbel’s goal programming model, our logarithmic goal programming model is never infeasible. The reader might have also observed that LGPP is mathematically similar to the minimum sum of absolute errors (MSAE) regression model which is known to be resistant to the presence of outliers [10]. Likewise, it can be shown that LGPM is resistant to outliers (e.g. Bryson [8]). Hence the presence of outliers in the pairwise comparison preference data should not have an adverse effect. Another interesting property of LGPP is that it provides a set of potentially meaningful consistency indicators. Given that lg#"log(#) is the objective function value, then # is the minimum average value that each entry in the comparison matrix would have to be multiplied or divided by in order to make the set of pairwise comparison values consistent. Thus if the decision-makers’ estimates were consistent we would have log(#)"0 and #"1. Otherwise we would have log (#)'0 and #'1. Associated with #, is its dual indicator p"1/# which takes on values in the interval (0, 1], and is an average of the fractions a /w (when p (q ) and w /a (when GH GH GH GH GH GH p 'q ), where w "w /w . p is thus a measure of the average consistency of the pairwise GH GH GH H comparisons. Thus, p"1 indicates a situation of total consistency, and p between 0 and 1 indicate the relative consistency of the pairwise comparison matrix. Of course, similar to other proposed consistency indicators (e.g. Saaty’s C.R., and Golden and Wang’s G), there are no objective cut-off values for # and p. However, since users can associate meaning with # and p values, they could adopt an action learning process that would result in the development of personal cut-off rules. It should be noted that corresponding consistency indicator p("1/#) are associated with each decision-maker ‘‘t’’.
4. Illustrative example In our illustrative example, a three-person selection committee has to determine the criteria and corresponding weights that are to be used in the selection of a database management system. After analysis they decided that the most appropriate evaluation criteria are Cost, Functionality,
N.(K.-M.) Bryson, A. Joseph / Computers & Operations Research 26 (1999) 637—643
641
Usability, Reliability, and Vendor support. The decision makers on the committee decide to use pairwise comparison techniques to generate the corresponding group weights with p"0.80 as the cut-off value for an acceptable degree of consistency. The reader may recall that p is the minimum average value that each entry in the comparison matrix would have to be multiplied or divided by in order to make the set of pairwise comparison values consistent. In Step 1, each decision-maker entered his/her pairwise comparison data as shown in Table 1. The LGPP was formulated and solved for each decision-maker, i.e., ¹"+1,, resulting in the generation of the corresponding individual priority point (or weight) vector and consistency indicator. Table 2 displays the resulting individual priority point vectors and consistency indicators. Since each of the individual consistency indicators implied an acceptable degree of consistency (i.e. p*0.80), then the group LGPP, i.e., ¹"+1, 2, 3, was formulated and solved. The resulting group priority point vector and consistency indicator were generated. These results are displayed in Table 2. From this we can see that there is again an acceptable level of consistency for Table 1 Functionality
(a) Pairwise comparison data for decision-maker 1 Cost 1.05 Functionality Usability Reliability (b) Pairwise comparison data for decision-maker 2 Cost 1.19 Functionality Usability Reliability (c) Pairwise comparison data for decision-maker 3 Cost 1.10 Functionality Usability Reliability
Usability
Reliability
Vendor support
1.33 1.38
1.74 1.25 0.95
1.70 1.54 1.73 1.05
1.57 1.21
1.38 1.54 1.22
2.48 2.19 1.78 1.67
1.30 1.35
1.70 1.35 1.20
2.20 1.65 1.75 1.45
Table 2 Decision maker
Priority point vector
1 2 3 Group mean
(0.247, (0.280, (0.265, (0.250,
0.235, 0.243, 0.241, 0.238,
0.185, 0.200, 0.199, 0.188,
0.180, 0.164, 0.175, 0.176,
0.153) 0.113) 0.120) 0.147)
#
p
1.10 1.05 1.05 1.15
0.91 0.95 0.95 0.87
642
N.(K.-M.) Bryson, A. Joseph / Computers & Operations Research 26 (1999) 637—643
the group pairwise comparison data. It should be noted that even though the individual pairwise comparison data could each have an acceptable degree of consistency, that the corresponding group pairwise comparison data could fail to have an acceptable degree of consistency.
5. Conclusion In this paper we have explored features of a logarithmic goal programming method for generating consensus priority point vectors. This method possesses some attractive properties that make it an appealing alternative to other proposed methods. It is resistant to the presence of ‘outliers’ in pairwise comparison data that result in distorted priority vectors being produced by other methods; it does not require any statistical assumptions; and does not require that the pairwise comparison matrix be reciprocal. It also provides a set of potentially useful indicators for interpreting the consistency of the pairwise comparison matrix.
References [1] Saaty T. The analytic hierarchy process: planning, priority setting. Resource allocation, New York: McGraw-Hill, 1980. [2] Saaty T. Group decision making and the AHP. In: Golden B, Wasil E, Harker P, editors. The Analytic Hierarchy Process: Application and Studies 1989;59—67. [3] Bryson N. Group decision-making and the analytic hierarchy process: exploring the consensus-relevant information content. Computers & Operations Research 1996;23:27—35 [4] Golden B, Wang Q. An alternate measure of consistency. In: Golden B, Wasil E, Harker P, editors. The Analytic Hierarchy Process: Application and Studies 1989;68—81. [5] Aczel J, Saaty T. Procedures for synthesizing ratio judgments. Journal of Mathematical Psychology 1983;27: 93—102. [6] Fichtner J. On deriving priority vectors from matrices of pairwise comparisons. Socio-Economic Planning Sciences 1986;20(6):399—405. [7] Bryson N, Mobolurin Q, Ngwenyama O. Modeling pairwise comparisons on ratio scales. European Journal of Operational Research 1995;83:639—54. [8] Bryson N. A goal programming for generating priority vectors. Journal of the Operational Research Society 1995;46:641—8. [9] Arbel A. A linear programming approach for processing approximate articulation of preference. In: Korhonen P, Lewandowski A, Wallenius J, editors. Multiple Criteria Decision Support 1989:81—6. [10] Narula S. The minimum sum of absolute errors regression, Journal of Quality Technology 1987;19:37—45. [11] Basak I, Saaty T. Group decision making using the analytic hierarchy process. Mathematical and Computer Modelling 1993;17(4/5):101—109.
Kweku-Muata (Noel) Bryson is Professor of Information Systems in the School of Business at Virginia Commonwealth University in Richmond, Virginia. Previously he was Professor of Information Systems and Decision Sciences in the School of Business at Howard University in Washington, DC. He holds a B.Sc. in Natural Sciences from the University of the West Indies at Mona; an M.S. in Systems Engineering from Howard University, and a Ph.D. in Applied Mathematics (Management Science & Information Systems) from The University of Maryland at College Park. Dr. Bryson’s research areas include: group support systems, multicriteria decision analysis, expert systems, spatial and distributed database systems, and integer programming. He has published articles in a variety of journals including:
N.(K.-M.) Bryson, A. Joseph / Computers & Operations Research 26 (1999) 637—643
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Computers & Operations Research, IEEE ¹ransactions on Knowledge & Data Engineering, Decision Support Systems, Information Systems, Information Processing & Management, Data & Knowledge Engineering, Journal of the Operational Research Society, European Journal of Operational Research, and Journal of Multiple Criteria Decision Analysis. Anito Joseph is an Assistant Professor in the Department of Management Science, School of Business Administration, University of Miami. She holds a B.Sc. in Natural Science from the University of the West Indies at St. Augustine, and M.S. and Ph.D, in Management Science/Statistics from the University of Maryland at College Park. She has published in Computers & Operations Research, ¹ransportation Science, European Journal of Operational Research, and Mathematical and Computer Modelling. Her research interests include integer programming, cluster analysis, multicriteria decision analysis and applications of operations research in information systems.