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cost ratios in analytic hierarchy process
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COMPUTER MODELLING
Mathematical
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Computer
Modelling
39 (2004)
279-286 www.elsevier.com/locate/mcm
Note on Incremental Benefit/Cost Ratios in Analytic Hierarchy Process KUO-LUNG
YANG
Department of Management Science Chinese Military Academy, Taiwan, R.O.C. yangklungOyahoo.com.tw
PETER Central
Department Police
CHU
of Traffic Science University, Taiwan,
R.O.C.
una211Qsun4.cpu.edu.tw
WEN-TAI Comptroller
Bureau,
CHOUHUANG
Ministry of National chchQms3.url.com.tw
(Received
Aprzl
2003;
Defense,
accepted
May
Taiwan,
R.0.C
2003)
Abstract-Scholars have frequently applied benefit-cost analysis to policy issue decisions. They hope to measure the relative importance among alternatives and provide systematical decision-making information using the benefit and cost factors. In this paper, the work of Bernhard and Canada [l] is studied based on incremental benefit/cost ratios with a cutoff ratio to analyze and extend the benefit/cost problem. This note has three purposes. First, we derive some theoretical properties for the incremental benefit/cost ratios with the cutoff ratio and a corrected method to show that the previous results are incomplete. Second, we provide an efficient method for solving the incremental benefit/cost ratios with cutoff ratio problems, Third, Bernhard and Canada [l] assumed that the desired incremental benefit/cost ratios could be derived using a comparison matrix under the perfectly consistent condition. We offer a counter-example to highlight the difference between the normalized benefit from the benefit and the normalized benefit vector from the AHP. We hope these findings will be useful for decision makers in applying benefit/cost analysis. @ 2004 Elsevier Ltd. All rights reserved.
Keywords-Analytic
hierarchy
process
(AHP),
Benefit/cost
analysis
1. INTRODUCTION discussed the AHP in 1970. It is a systematized decision model that provides a popular technique for decision problems. Utilizing deduction to divide a complex and nonstructure objective into hierarchical attributes, it considers both certain and uncertain factors. The decision-makers use the numerals provided using subjective judgments of each attribute’s relative importance. Induction is then applied to synthesize those judgments for determining the alternative priority. This method is useful for decision-maker reference in choosing the optimum project. AHP
Saaty
0895-7177/04/s - see front doi: 10.1016/S0895-7177(04)00013-5
matter
@ 2004
Elsevier
Ltd.
All
rights
reserved.
Typeset
by
A,+@-'QX
K.-L.
280
YANG et al.
is applied popularly in multiple criteria decision problems in marketing management, resource allocation, and behavioral science. Benefit/cost analysis is another AHP applied category. AHP can determine the order ranking for making choices. This procedure is directed at different alternative attributes for constructing two separate hierarchies affecting the related cost and benefit factors. A decision-maker utilizes the benefit priorities to compare the cost priorities when seeking which option has the highest benefit priority/cost priority ratio. The decision process is an uncertain situation [2] that may not be able to calculate an accurate cost and benefit valuation. Decisions made according to the AHP characteristic express a numeral type and calculate the effects that provide the necessary information for a sound decision. Saaty and Kearns [3] pointed out that AHP is similar to regular benefit/cost analysis using money as the common currency. Both benefit and cost priorities are measured in the same commensurable units. Therefore, in the calculation process, decisionmakers must obey the assumption of more objectivity. In recent decades, scholars have discussed AHP research and application using benefit-cost methods. For example, Azis [4] evaluated and determined the Tans-Sumatra highway project. Clayton and Wright [5] considered whether riverboat gambling should be authorized on Pennsylvania’s rivers and lakes and analyzed the potential benefits and possible costs. Bennett and Saaty [6] questioned the knapsack multiple resource allocation in benefit/cost analysis using the AHP method. They formulated different linear programming types for knapsack allocation problems. They expected to obtain the benefit/cost ratios, or net benefits for a package of projects that could be maximized subject to different types of resource constraints. Alternatively, the objective could be to minimize costs subject to resource constraints. Bernhard and Canada [l] showed that Saaty’s procedure was sensitive to arbitrary semantic changes in benefit and cost labeling. They thought that the benefits and costs should be known with certainty and measured in dollars. Their suggestion for AHP benefit/cost ratios use was to calculate a cutoff rate that produced a benefit priority to the cost priority ratio. Wedley et al. [7] referred to Bernhard and Canada’s criticism, which under different proportions makes magnitude adjustments for AHP benefit/cost ratios. They tried to put benefit and cost into commensurate terms such that the benefit/cost ratios would lead the decision-maker to make a judgment. Recently, Saaty and Cho [8] have extended the benefit/cost analysis to include four hierarchies: benefits, costs, opportunities, and risks for the decision by the U.S. Congress on China’s trade status. Their results helped leaders and the members of Congress to decide the best possible outcome. In the next section, the results from Bernhard and Canada [l] are reviewed. In the third section, we will analyze and improve their method. Some theoretical properties for the incremental benefit/cost ratios with cutoff ratio and a correct method to show the. incompleteness of the previous results are derived. In the fourth section, Bernhard and Canada assumed that any predetermined incremental benefit/cost ratios could be derived using a comparison matrix with a perfectly consistent assumption. We offer a counter-example to show the difference between the normalized benefit from the benefit and the normalized benefit vector from the AHP. The conclusion follows.
2. REVIEW
OF BERNHARD
AND
CANADA’S
RESULTS
They mentioned that two major features are missing from Saaty’s standard benefit/cost ratios analysis procedure. The first is the incremental benefit/cost ratio consideration. The second is an additional requirement for further information with’ regard to the decision-maker’s relative preferences for benefit increments vs. cost increments. They claimed that the cutoff ratio should provide further information. We review their results. Bernhard and Canada considered Bi E benefit, in dollars, procurable
from project i, and
Ci E cost, in dollars, which would be incurred by project i.
t
They used the net profit from the profit, subtracted the cost of Bernhard [9] as the criterion to determine the best project. In the following, we review two modifications of Bernhard and Canada .[lj. For the first modification, they mentioned that a project i will be attractive to the decision-maker if and only if the net gain from it is nonnegative, that is, if and only if B, > C,. Hence, they only considcrccl those projects that satisfy the constraint B, ) C,. For the second modification, without loss of generality, they relabeled the indices such that Ci < Cz < < C,,, They then cancelled out project i when it satisfied the following constraint: an index j:
with j < i and B, 2 B,.
(1)
After these two modifications, they assumed that the profits satisfy B1 < Bz < < B,, and the costs satisfy Cr < C2 < < C,,, respectively. They defined the following terms: k=B1+BZ+..,+B,: k’=
Cl +C2 +...+C,,
b, f
c, s
= normalized benefit for project i: and hence! bl + bz + CZ 77 ( >=
normalized
cost for project i. and hence, ci + c2 +
+ b,, = 1.
+ c,, = 1.
They claimed that if one further assumes that the decision-maker is perfectly consistent, the vectors [bl, ba.. 1b,] and [cl: ~2,. . . , c,,] are; respectively, those which would be yielded here by his or her consideration of the benefits and costs under the AHP. This assumption seems questionable. It will be stated in more detail in Section 4. For two projects, 3 and i, by considering the incremental benefit/cost ratios, the decision-maker will prefer the former to the latter if and only if B, - C, > Bi - C, or, equivalently, if and only if [(bj - bi)/(c, - cz)] > k’/k. Th ey used F/k as the cutoff ratio to provide the necessary “further information” cited above. Bernhard and Canada considered the river-crossing example in Saaty [lO,ll] with the known net gain to provide a transportation artery across a river. 1. 2. 3. 4.
Establish ferry service. Build a bridge. Build a tunnel, and implicitly, Do nothing.
there is a fourth alternative.
They assumed that the monetary benefit and cost for Project 3 (do nothing), Project 1 (the ferry), Project 2 (the bridge), and Project 3 (the tunnel) as Bo: B1, B2, B3, Co, Cl, Cz, and C:s. respectively, such that the known net gain is denoted as B 1 - Cl = 5, B2 - Cz = 20, and B3 - C’s = 25, where implicitly Bo = 0 and Cu = 0. They constructed two examples such that two possible benefits and costs systems may satisfy the predetermined net gain. We list their results in Table 1.
Example Range
1
1. The
bl/Cl
=
of k’/k
Optimal Alternative Example
Table
2
1.340
results
of Bernhard
b2/c2
= 1.110
< 0.763
(0.763,1.064)
Project 3 the tunnel
Project
bl/q
= 1.021
the
b2/c2
and
Canada
b3/c3
= 0.923
(1.064, 2
bridge
= 1.129
[l].
b3/c3
1
ferry
= 0.939
= 0.667 > 1.340
1.340)
Project the
k’/k
Project 4 do nothing
k’/k
= 0.677
282
K.-L.
YANG
et al.
For the first example, they have the following data: B1 = 10, Bz = 50, BS = 90, Cl = 5, Cz = 30, and C’s = 65, such that k = 150, k’ = 100, k’/k = 0.667. For the second example, they change the values of B1 and Cl to B1 = 15 and Cl = 10. For Example 1, from bi = &/k and ci = Ci/k’, they have two hierarchies with corresponding normalized weights for profit such that bo < bl < b2 < b3 and CO < cl < c2 < ~3 with (bo, bl, b2, b3) and cost (co,c~,c~,c~), bo = 0, bl = 0.067, b2 = 0.333, b3 = 0.600, ~0 = 0, cl = 0.050, c2 = 0.300, and cg = 0.650. They computed the incremental benefit/cost ratios for projects i and i - 1, with i = 1,2,3, to find that bl/q = 1.340, (b2 - bl)/(cz - ~1) = 1.064, and (b3 - b2)/(c3 - ~2) = 0.763. They also computed the benefit/cost ratios for projects i with i = 1,2,3. Similarly, for Example 2, they computed that k = 155, k’ = 105, bo = 0, bl = 0.097, b2 = 0.323, b3 = 0.581, Q = 0, cl = 0.095, c2 = 0.286, and cg = 0.619. They found bl/q = 1.021, (b2 - b,)/(cz - cl) = 1.183, (bg 1 b2)/(c3 - ~2) = 0.775, and the benefit/cost ratios for projects i with i = 1,2,3. We quote their results in Table 1. Bernhard and Canada [l] compared the cutoff ratio, k’/k with (bi - bi-l)/(ci - ci-.1) to find that [(bi - bi-l)/(ci - ci-I)] 2 k’/k with i = 1,2,3 for Examples 1 and 2. Hence, they concluded that the incremental benefit/cost ratios with the cutoff ratio would imply that Project 3 (the tunnel) is the best for both examples. This is consistent with the results for the net gain. They also observed Saaty’s procedure, the benefit/cost ratios. For Example 1, bl /cl is greater than b2/c2 and b3/c3, then Project 1 (the ferry) is the best project. For Example 2, bz/cz is greater than bl/q and bs/c3, then Project 2 (the bridge) is the best project. They criticized Saaty’s procedure [lO,ll] choosing from the uncorrected choice Project 1, the ferry, for the first example to another wrong choice Project 2, the bridge, for the second example.
3. OUR
ANALYSIS
AND
IMPROVEMENT
Before we explain the paradox in Bernhard and Canada [l], we developed some properties for the incremental benefit/cost ratios with the cutoff ratio. 3.1. Analysis
of Incremental
Benefit/Cost
Ratios
with
the Cutoff
Ratio
To simplify the expression and mnemonics, we denote “project j” better than “project i” by p(j) > p(i). Under the condition Cl < C2 < . . . < C, and B1 < B2 < . . . < B,, for two projects i and j, the following are equivalent. (4 p(j) > p(i). (b) Bj - Cj > Bi - Ci, using the net profit. (C) (bj - bi)k > (Cj - ci)k’ since Bj - Bi = (bj - bi)k and Cj - Ci = (~j - ci) k’. (d) If j > i, then cj > ci and (b, - bi)/(cj - c,) > k//k; and if j < i, then (bj - bi)/(q
Consequently,
ci
< ci and
- ci) < k’/k.
we obtain the following theorem.
THEOREM 1. Under the conditions bo < bl < b2 < b3 and co < cl < c2 < cg, for two projects i and j with i < j, if 0 < k’/k < (bj - bi)/(c.j - ci), then p(j) > p(i) and if (bj - bi)/(cj - ci) < k//k,
then p(i)
> p(j).
Bernhard and Canada [l] only compared the values of (bi - bi-l)/(cz - ci-1) for i = 1,2,3, with the cutoff ratio, k’/k such that their comparison was incomplete. We now state the improved version for the incremental benefit/cost ratios with the cutoff ratio. THEOREM 2. Under the conditions bo < bl < b2 < b3 and ~0 < cl < c2 < cg, with the ratio k//k, we need to use the incremental benefit/cost ratios (bj - bi)/(c3 - ci) for each pair (i,j) with i < j to partition the positive real number into subintervals. In each subinterval, using Theorem 1, we know the preference for every pair of projects. We will show that the preference for projects holds the transitive property, that is, if p(l) > p(m) and p(m) > p(n), cutoff
then p(l)
> p(n).
Proof of Theorem 2 is in the Appendix.
Analytic
Hierarchy
Process
283
EXAMPLE 3. Here, we reconsider the same Example 2 from Bernhard and Canada [l] to find the range for V/k. We compute the incremental benefit/cost ratios (b, - bi)/(cj - cZ) for every pair (i,j) with i < j, according to their size. This shows that (b3 - ba)/(cg - ~2) = 0.77, (b3 - bl)/(c3 -cl) = 0.92, b3/c3 = 0.94, bl/cl = 1.02, b2/ c2 = 1.13, (62 - bl)/(c:! - cl) = 1.18, an d use them to partition the positive real numbers into subintervals. The results are listed jn Table 2. The range for V/lc is listed in the first row in Table 2 Table Range
of k’/k
2. The
incremental
(0,0.77)
benefit/cost
ratios
and
the preferences
for two
(0.77,0.92)
(0.92,0.94)
(0.94,1.02)
(1.02,1.13)
P(3)
> P(2)
P(2)
> P(3)
P(2)
> P(3)
P(2)
’ P(3)
P(2)
> P(3)
P(3)
> P(l)
P(3)
> P(l)
P(l)
> P(3)
P(l)
> P(3)
P(l)
> P(3)
Compare
P(3)
> P(O)
P(3)
> P(O)
P(3)
> P(0)
P(O)
> P(3)
P(0)
> P(3)
Projects
P(l)
> P(O)
P(l)
> P(0)
P(l)
> P(O)
P(l)
> P(0)
P(0)
> P(l)
P(O)
P(2)
> r-40)
P(2)
> P(0)
P(l)
P(2)
> P(l)
P(2) >
P(2) > 742) >
P(O) P(l)
P(2) > P(2) >
P(O) P(l)
P(2) > PC‘4>
projects
P(l)
A subinterval was chosen as (0.94,1.02) to illustrate the transitivity of Theorem 2. We have the preferences for two projects as p(2) > p(3), ~(1) > p(3), p(O) > p(3), p(1) > p(O), p(2) > p(O), and p(2) > p(L). W e can then rearrange them as p(2) > p(l) > p(O) > p(3) demonstrating the transitivity of Theorem 2. From Table 2, the best project for each subinterval is implied and the results are listed in Table 3. Bernhard and Canada [l] used only the values for bl/cl = 1.02, (b2 - b,)/(c2 - cl) = 1.18, and (03 - b2)/( cg - cp) = 0.77 to partition the positive real number. We offer a counterexample to illustrate that their method is incomplete. Table The Range
of V/k
The
results
from
our
results
from
Undecided
imoroved
Project 2 the bridge
the method
< 0.94
for the
(0.77,1.13)
Project 3 the tunnel
of k’/k
Optimal Alternative
comparison
< 0.77
Optimal Alternative
Range
3. The
of Bernhard
best
policy
method. > 1.13 Project 4 the ferry and
Canada
(0.94,1.02)
(1.02,1.13)
Project 2 the bridge
Project 2 the bridge
[l]. > 1.13 Project 4 the ferry
Hence, in the subinterval (0,0.94), they could not decide on the best policy. This indicates that their procedure has problems. In brief, the incremental benefit/cost analysis with cutoff ratio by Bernhard and Canada [I] is incomplete as demonstrated through the second example in their paper. 3.2. A Simple
Improved
Approach
Back to the benefit/cost analysis in AHP. Under the same condition as in Bernhard and Canada [l], we reordered the project and deleted the negative net gain project. We have two hierarchies with benefit vector g(bl, . , b,,), with Cy=“=,b, = 1 and cost vector (cl,. , cn), with Crf, ci = 1, such that bl < . < b, and cl < . . . < c,. In general, we did not know the benefit Bi and cost Ci. Therefore, in the AHP environment, we cannot find the net gain Bi - Ci. That means, without knowing the cutoff ratio, k//k, we cannot determine the magnitude of B, - C, = (b, - cik’/k)k. C onversely, if we know the cutoff ratio k//k, we can directly compute bi - cik’/k for i = 1,. , 7%to select the maximum. We derived a simple approach that obtained the same result as the net gain. We obtained the same results as the incremental benefit/cost ratios with cutoff ratio suggested by Bernhard and Canada [l]. The key factor is how to derive the cutoff ratio k’/k.
284
3.3.
K.-L.
Criticism
of Bernhard
and
Canada’s
YANG et al. Method
In Bernhard and Canada’s procedure [I] on Saaty’s benefit/cost ratios, they assumed that they knew the net gain for each project. If the decision-makers know the net gain for each project and they accept the net gain as the criterion for selecting the best choice, without any work on the incremental benefit/cost ratios or cutoff ratio, they could determine the best choice, if the best project, j, satisfies Bj - Cj = max {& - Ci, for all project i} . However, under the AHP situation, we only have cl < c2 < . .. < c,, bi < b2 < . .. < b,, CG, C, = 1, and Cy’i bi = 1. W e d onothavethedatafor{&-Ci:i=l,...,n}. Uptonow, there has been no well-constructed method that can determine the total benefit values, Ic, total cost, Ic’, and the cutoff ratio, V//c. Therefore, different decision-makers might choose different cutoff ratios. According to Theorem 2, they might have different conclusions. This indicates that the incremental benefit/cost analysis by Bernhard and Canada [I] (after the revision of Theorem 2) or our simplified method for incremental benefit/cost analysis is dependent on the cutoff ratio value choice, Ic’/lc. We claim that the incremental benefit/cost analysis is not suitable for the AHP situation. Their criticism for the first example in Project 1 (the ferry) is an incorrect choice and for the second example in Project 2 (the bridge), it is another incorrect choice based on their criterion “to maximize the net gain”. However, using Saaty’s procedure, the criterion is “to maximize the benefit/cost ratio”. The results from different criterion can be compared, but the project priority cannot be discriminated.
4. PERFECTLY
CONSISTENT
CONDITION
In Bernhard and Canada [6, p. 591, they asserted that if the decision-makers were perfectly consistent, given a normalized benefit vector [bi, b2, . . , b,], derived by themselves, the benefits under the AHP could be determined. We know that the relative judgment scale is restricted in AHP from l/9, l/8, . . . , 1,2,. . . to 9. Not in every situation can it produce a perfectly consistent comparison matrix. For completeness, we offer a counterexample to show that their assertion is invalid. Suppose that bl = lO/ll, b2 = l/11, and the comparison matrix constructed by the decision-maker as ( l/i& “f’ ). From ( iiilZ “i’)( “i’) = 2( “i’) and any two by two reciprocal matrix, we know that the normalized eigenvector of the comparison matrix must have the expression (ai2/1 +ui2, l/l + ai2) with ~12 E {l/9,1/8,. . . ,1,2,. . ,9}. Th erefore, we cannot select ai2 = 10 to satisfy the equation (uiz/l +uiz, l/l +uiz) = (lO/ll, l/11). C onsequently, when decision-makers study the benefit-cost problem, we advise them to be aware of the difference between (i) the normalized (ii) the normalized
benefit (bi) from the benefit Bi with bi = Bilk, benefit vector from the AHP.
and
5. CONCLUSION If decision-makers know the monetary benefit and cost for each project, Bernhard and Canada [l] suggest using the net gain as the criterion for selecting the best project. In this paper, we derive some theoretical properties that are correct for using the incremental benefit/cost ratios and provide a more simple and efficient way for judgment. From Theorem 2, different cutoff ratios might imply different choices for the best project. Hence, Bernhard and Canada’s method (with our Theorem 2 improvement) or our simplified method for the incremental benefit/cost ratios both suffer with a debatable cutoff ratio V/k. Unless decision-makers understand the subtle benefit and cost, then the cutoff ratio V/k cannot be determined. In general, the benefit and cost are difficult to decide. Consequently, we suggest that decision-makers adopt Saaty’s AHP
Analytic
Hierarchy
285
Process
benefits, costs, opportunities, and risks method IS] and benefit/cost procedures [la, p. itil- 1661 for selecting the best project. On the basis of many reports, Saaty’s procedures are popular and flexible and can effectively deal with this kind of problem.
APPENDIX PROOF OF THEOREM 2. We consider every i and j with i < j project pair to compute the in cremental benefit/cost ratios (b, - bl)/(cJ - cz). We tl len use them to partition the positive real numbers. First, we consider those subintervals not on the boundary, for a subinterval ((bt - b,)/(ct - c,), (6, - b)l(c3 -G)) with s < t and i < j. In this subinterval, supposing that p(l) > p(m) and p(m) > p(n). we will prove that p(l) > p(lr,). We divide the problem into four cases: (a) (b) (c) (d)
1 > m and na 1 > m and m 1 < m and m 1 < 7n and m
> n, < 7~. > n, and < n.
For Case (a), using Theorem 1, p(l) > p(m) and 1 > nt, we have
h - b,,,2 (ci - cm) (4 - h) (5 - co Similarly, we obtain bm - 6, 2 (cm - cn)
(4 - h) cc, -4’
Combining equations (2) and (3) with 1 > n, we imply p(2) > For Case (b), we divide the problem into two cases:
p(n)
(bl) 1 > n, and (b2) 1 < 71. For Case (bl), using Theorem 1, p(l) > p(m) and
bn- b,, 5 (G, - cm)
m
<
71, we
have
(bt - 6s) < (G, - cm) (b, - bt) (Ct- cs) (Cj - CT)
([I)
Combining equations (2) and (4) with 1 > 71, we imply p(l) > p(n) For Case (b2), using Theorem 1, p(l) > p(m) and 1 > m, we have
(bt.- bs) - bl) br - b,, 2 (cl - cm) (b, (cJ_ ci) 2 (Cl- cm,)(ct - e,)’ Again, using Theorem 1 and
m
<
n,
we
(51
know
b,, - b,, F (cn - cm)
(bt - b,s) (ct - cs)‘
Combining equations (5) and (6) with 1 < 7z, we imply p(l) > p(n). Similarly, we can solve Cases (c) and (d), respectively. Next, we consider the intervals on the boundary as (0, (b, - bg)/ (cn - cp)) and ((b, - ho)/ (c, - co), oo) where (b, - bo)/(ca - cP)=min((b, - b,j/(c, - c,) : ? < J’} and (by - b~)/(q - co) = max{(b, - bi)/(cj - ci) : i < j}. For the interval (0, (b, - bp)/(ca - CD)), we only need to consider Case (a) 1 > rn. and 711 > II, since 1 < m and m < n will not occur, owing to Theorem 1. Similarly, for the interval ((by - be)l(c, - co)>m), we only need to consider Case (d) 1 < m and m < n. The remaiuing proofs are similar to our previous discussion, so they are omitted.
K.-L.
YANG et al.
REFERENCES 1. R.H. Bernhard and J.R. Canada, Some problems in using benefit/cost ratios with the Analytic Hierarchy Process, The Engineering Economist 36 (l), 56-65, (1990). 2. T.L. Saaty, Multi-Criteria Decision Making-The Analytic Hierarchy Process, pp. 113-114, RWS, Pittsburgh, PA, (1990). 3. T.L. Saaty and K.P. Kearns, Analytic Planning, RWS, Pittsburgh, PA, (1994). 4. I.J. Azis, AHP in the benefit-cost framework: A post-evaluation of the Trans-Sumatra highway project, European Journal of Operational Research 48, 38-48, (1990). 5. W.A. Clayton and M. Wright, Benefit cost analysis of riverboat gambling, Mathl. Comput. Modelling 17 (4/5), 187-194, (1993). 6. J.P. Bennett and T.L. Saaty, Knapsack allocation of multiple resources in benefit-cost analysis by way of the analytic hierarchy process, Mathl. Comput. Modelling 17 (4/5), 55-72, (1993). 7. W.C. Wedley, E.U. Choo and B. Schoner, Magnitude adjustment for AHP benefit/cost ratios, European journal of Operational Research 133, 342-351, (2001). 8. T.L. Saaty and Y. Cho, The decision by the US Congress on China’s trade status: A multicriteria analysis, Socio-Economic Planning Sciences 35, 243-252, (2001). 9. R.H. Bernhard, A comprehensive comparison and critique of discounting indices proposed for capital investment evaluation, The Engineering Economist 16 (3), 157-186, (1971). 10. T.L. Saaty, The Analytic Hierarchy Process, McGraw-Hill, New York, NY, (1980). 11. T.L. Saaty, Decision Making for Leaders, RWS, Pittsburgh, PA, (1995). 12. T.L. Saaty, Fundamentals of Decision Making and Priority Theory, RWS, Pittsburgh, PA, (1994).
Available at
MATHEMATICAL
www.ElsevierMathematics.com COWERED .v 8CIENCE (if4DIRECT.
COMPUTER MODELLING
Mathematical
and
Computer
Modelling
39 (2004)
279-286 www.elsevier.com/locate/mcm
Note on Incremental Benefit/Cost Ratios in Analytic Hierarchy Process KUO-LUNG
YANG
Department of Management Science Chinese Military Academy, Taiwan, R.O.C. yangklungOyahoo.com.tw
PETER Central
Department Police
CHU
of Traffic Science University, Taiwan,
R.O.C.
una211Qsun4.cpu.edu.tw
WEN-TAI Comptroller
Bureau,
CHOUHUANG
Ministry of National chchQms3.url.com.tw
(Received
Aprzl
2003;
Defense,
accepted
May
Taiwan,
R.0.C
2003)
Abstract-Scholars have frequently applied benefit-cost analysis to policy issue decisions. They hope to measure the relative importance among alternatives and provide systematical decision-making information using the benefit and cost factors. In this paper, the work of Bernhard and Canada [l] is studied based on incremental benefit/cost ratios with a cutoff ratio to analyze and extend the benefit/cost problem. This note has three purposes. First, we derive some theoretical properties for the incremental benefit/cost ratios with the cutoff ratio and a corrected method to show that the previous results are incomplete. Second, we provide an efficient method for solving the incremental benefit/cost ratios with cutoff ratio problems, Third, Bernhard and Canada [l] assumed that the desired incremental benefit/cost ratios could be derived using a comparison matrix under the perfectly consistent condition. We offer a counter-example to highlight the difference between the normalized benefit from the benefit and the normalized benefit vector from the AHP. We hope these findings will be useful for decision makers in applying benefit/cost analysis. @ 2004 Elsevier Ltd. All rights reserved.
Keywords-Analytic
hierarchy
process
(AHP),
Benefit/cost
analysis
1. INTRODUCTION discussed the AHP in 1970. It is a systematized decision model that provides a popular technique for decision problems. Utilizing deduction to divide a complex and nonstructure objective into hierarchical attributes, it considers both certain and uncertain factors. The decision-makers use the numerals provided using subjective judgments of each attribute’s relative importance. Induction is then applied to synthesize those judgments for determining the alternative priority. This method is useful for decision-maker reference in choosing the optimum project. AHP
Saaty
0895-7177/04/s - see front doi: 10.1016/S0895-7177(04)00013-5
matter
@ 2004
Elsevier
Ltd.
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rights
reserved.
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A,+@-'QX
K.-L.
280
YANG et al.
is applied popularly in multiple criteria decision problems in marketing management, resource allocation, and behavioral science. Benefit/cost analysis is another AHP applied category. AHP can determine the order ranking for making choices. This procedure is directed at different alternative attributes for constructing two separate hierarchies affecting the related cost and benefit factors. A decision-maker utilizes the benefit priorities to compare the cost priorities when seeking which option has the highest benefit priority/cost priority ratio. The decision process is an uncertain situation [2] that may not be able to calculate an accurate cost and benefit valuation. Decisions made according to the AHP characteristic express a numeral type and calculate the effects that provide the necessary information for a sound decision. Saaty and Kearns [3] pointed out that AHP is similar to regular benefit/cost analysis using money as the common currency. Both benefit and cost priorities are measured in the same commensurable units. Therefore, in the calculation process, decisionmakers must obey the assumption of more objectivity. In recent decades, scholars have discussed AHP research and application using benefit-cost methods. For example, Azis [4] evaluated and determined the Tans-Sumatra highway project. Clayton and Wright [5] considered whether riverboat gambling should be authorized on Pennsylvania’s rivers and lakes and analyzed the potential benefits and possible costs. Bennett and Saaty [6] questioned the knapsack multiple resource allocation in benefit/cost analysis using the AHP method. They formulated different linear programming types for knapsack allocation problems. They expected to obtain the benefit/cost ratios, or net benefits for a package of projects that could be maximized subject to different types of resource constraints. Alternatively, the objective could be to minimize costs subject to resource constraints. Bernhard and Canada [l] showed that Saaty’s procedure was sensitive to arbitrary semantic changes in benefit and cost labeling. They thought that the benefits and costs should be known with certainty and measured in dollars. Their suggestion for AHP benefit/cost ratios use was to calculate a cutoff rate that produced a benefit priority to the cost priority ratio. Wedley et al. [7] referred to Bernhard and Canada’s criticism, which under different proportions makes magnitude adjustments for AHP benefit/cost ratios. They tried to put benefit and cost into commensurate terms such that the benefit/cost ratios would lead the decision-maker to make a judgment. Recently, Saaty and Cho [8] have extended the benefit/cost analysis to include four hierarchies: benefits, costs, opportunities, and risks for the decision by the U.S. Congress on China’s trade status. Their results helped leaders and the members of Congress to decide the best possible outcome. In the next section, the results from Bernhard and Canada [l] are reviewed. In the third section, we will analyze and improve their method. Some theoretical properties for the incremental benefit/cost ratios with cutoff ratio and a correct method to show the. incompleteness of the previous results are derived. In the fourth section, Bernhard and Canada assumed that any predetermined incremental benefit/cost ratios could be derived using a comparison matrix with a perfectly consistent assumption. We offer a counter-example to show the difference between the normalized benefit from the benefit and the normalized benefit vector from the AHP. The conclusion follows.
2. REVIEW
OF BERNHARD
AND
CANADA’S
RESULTS
They mentioned that two major features are missing from Saaty’s standard benefit/cost ratios analysis procedure. The first is the incremental benefit/cost ratio consideration. The second is an additional requirement for further information with’ regard to the decision-maker’s relative preferences for benefit increments vs. cost increments. They claimed that the cutoff ratio should provide further information. We review their results. Bernhard and Canada considered Bi E benefit, in dollars, procurable
from project i, and
Ci E cost, in dollars, which would be incurred by project i.
t
They used the net profit from the profit, subtracted the cost of Bernhard [9] as the criterion to determine the best project. In the following, we review two modifications of Bernhard and Canada .[lj. For the first modification, they mentioned that a project i will be attractive to the decision-maker if and only if the net gain from it is nonnegative, that is, if and only if B, > C,. Hence, they only considcrccl those projects that satisfy the constraint B, ) C,. For the second modification, without loss of generality, they relabeled the indices such that Ci < Cz < < C,,, They then cancelled out project i when it satisfied the following constraint: an index j:
with j < i and B, 2 B,.
(1)
After these two modifications, they assumed that the profits satisfy B1 < Bz < < B,, and the costs satisfy Cr < C2 < < C,,, respectively. They defined the following terms: k=B1+BZ+..,+B,: k’=
Cl +C2 +...+C,,
b, f
c, s
= normalized benefit for project i: and hence! bl + bz + CZ 77 ( >=
normalized
cost for project i. and hence, ci + c2 +
+ b,, = 1.
+ c,, = 1.
They claimed that if one further assumes that the decision-maker is perfectly consistent, the vectors [bl, ba.. 1b,] and [cl: ~2,. . . , c,,] are; respectively, those which would be yielded here by his or her consideration of the benefits and costs under the AHP. This assumption seems questionable. It will be stated in more detail in Section 4. For two projects, 3 and i, by considering the incremental benefit/cost ratios, the decision-maker will prefer the former to the latter if and only if B, - C, > Bi - C, or, equivalently, if and only if [(bj - bi)/(c, - cz)] > k’/k. Th ey used F/k as the cutoff ratio to provide the necessary “further information” cited above. Bernhard and Canada considered the river-crossing example in Saaty [lO,ll] with the known net gain to provide a transportation artery across a river. 1. 2. 3. 4.
Establish ferry service. Build a bridge. Build a tunnel, and implicitly, Do nothing.
there is a fourth alternative.
They assumed that the monetary benefit and cost for Project 3 (do nothing), Project 1 (the ferry), Project 2 (the bridge), and Project 3 (the tunnel) as Bo: B1, B2, B3, Co, Cl, Cz, and C:s. respectively, such that the known net gain is denoted as B 1 - Cl = 5, B2 - Cz = 20, and B3 - C’s = 25, where implicitly Bo = 0 and Cu = 0. They constructed two examples such that two possible benefits and costs systems may satisfy the predetermined net gain. We list their results in Table 1.
Example Range
1
1. The
bl/Cl
=
of k’/k
Optimal Alternative Example
Table
2
1.340
results
of Bernhard
b2/c2
= 1.110
< 0.763
(0.763,1.064)
Project 3 the tunnel
Project
bl/q
= 1.021
the
b2/c2
and
Canada
b3/c3
= 0.923
(1.064, 2
bridge
= 1.129
[l].
b3/c3
1
ferry
= 0.939
= 0.667 > 1.340
1.340)
Project the
k’/k
Project 4 do nothing
k’/k
= 0.677
282
K.-L.
YANG
et al.
For the first example, they have the following data: B1 = 10, Bz = 50, BS = 90, Cl = 5, Cz = 30, and C’s = 65, such that k = 150, k’ = 100, k’/k = 0.667. For the second example, they change the values of B1 and Cl to B1 = 15 and Cl = 10. For Example 1, from bi = &/k and ci = Ci/k’, they have two hierarchies with corresponding normalized weights for profit such that bo < bl < b2 < b3 and CO < cl < c2 < ~3 with (bo, bl, b2, b3) and cost (co,c~,c~,c~), bo = 0, bl = 0.067, b2 = 0.333, b3 = 0.600, ~0 = 0, cl = 0.050, c2 = 0.300, and cg = 0.650. They computed the incremental benefit/cost ratios for projects i and i - 1, with i = 1,2,3, to find that bl/q = 1.340, (b2 - bl)/(cz - ~1) = 1.064, and (b3 - b2)/(c3 - ~2) = 0.763. They also computed the benefit/cost ratios for projects i with i = 1,2,3. Similarly, for Example 2, they computed that k = 155, k’ = 105, bo = 0, bl = 0.097, b2 = 0.323, b3 = 0.581, Q = 0, cl = 0.095, c2 = 0.286, and cg = 0.619. They found bl/q = 1.021, (b2 - b,)/(cz - cl) = 1.183, (bg 1 b2)/(c3 - ~2) = 0.775, and the benefit/cost ratios for projects i with i = 1,2,3. We quote their results in Table 1. Bernhard and Canada [l] compared the cutoff ratio, k’/k with (bi - bi-l)/(ci - ci-.1) to find that [(bi - bi-l)/(ci - ci-I)] 2 k’/k with i = 1,2,3 for Examples 1 and 2. Hence, they concluded that the incremental benefit/cost ratios with the cutoff ratio would imply that Project 3 (the tunnel) is the best for both examples. This is consistent with the results for the net gain. They also observed Saaty’s procedure, the benefit/cost ratios. For Example 1, bl /cl is greater than b2/c2 and b3/c3, then Project 1 (the ferry) is the best project. For Example 2, bz/cz is greater than bl/q and bs/c3, then Project 2 (the bridge) is the best project. They criticized Saaty’s procedure [lO,ll] choosing from the uncorrected choice Project 1, the ferry, for the first example to another wrong choice Project 2, the bridge, for the second example.
3. OUR
ANALYSIS
AND
IMPROVEMENT
Before we explain the paradox in Bernhard and Canada [l], we developed some properties for the incremental benefit/cost ratios with the cutoff ratio. 3.1. Analysis
of Incremental
Benefit/Cost
Ratios
with
the Cutoff
Ratio
To simplify the expression and mnemonics, we denote “project j” better than “project i” by p(j) > p(i). Under the condition Cl < C2 < . . . < C, and B1 < B2 < . . . < B,, for two projects i and j, the following are equivalent. (4 p(j) > p(i). (b) Bj - Cj > Bi - Ci, using the net profit. (C) (bj - bi)k > (Cj - ci)k’ since Bj - Bi = (bj - bi)k and Cj - Ci = (~j - ci) k’. (d) If j > i, then cj > ci and (b, - bi)/(cj - c,) > k//k; and if j < i, then (bj - bi)/(q
Consequently,
ci
< ci and
- ci) < k’/k.
we obtain the following theorem.
THEOREM 1. Under the conditions bo < bl < b2 < b3 and co < cl < c2 < cg, for two projects i and j with i < j, if 0 < k’/k < (bj - bi)/(c.j - ci), then p(j) > p(i) and if (bj - bi)/(cj - ci) < k//k,
then p(i)
> p(j).
Bernhard and Canada [l] only compared the values of (bi - bi-l)/(cz - ci-1) for i = 1,2,3, with the cutoff ratio, k’/k such that their comparison was incomplete. We now state the improved version for the incremental benefit/cost ratios with the cutoff ratio. THEOREM 2. Under the conditions bo < bl < b2 < b3 and ~0 < cl < c2 < cg, with the ratio k//k, we need to use the incremental benefit/cost ratios (bj - bi)/(c3 - ci) for each pair (i,j) with i < j to partition the positive real number into subintervals. In each subinterval, using Theorem 1, we know the preference for every pair of projects. We will show that the preference for projects holds the transitive property, that is, if p(l) > p(m) and p(m) > p(n), cutoff
then p(l)
> p(n).
Proof of Theorem 2 is in the Appendix.
Analytic
Hierarchy
Process
283
EXAMPLE 3. Here, we reconsider the same Example 2 from Bernhard and Canada [l] to find the range for V/k. We compute the incremental benefit/cost ratios (b, - bi)/(cj - cZ) for every pair (i,j) with i < j, according to their size. This shows that (b3 - ba)/(cg - ~2) = 0.77, (b3 - bl)/(c3 -cl) = 0.92, b3/c3 = 0.94, bl/cl = 1.02, b2/ c2 = 1.13, (62 - bl)/(c:! - cl) = 1.18, an d use them to partition the positive real numbers into subintervals. The results are listed jn Table 2. The range for V/lc is listed in the first row in Table 2 Table Range
of k’/k
2. The
incremental
(0,0.77)
benefit/cost
ratios
and
the preferences
for two
(0.77,0.92)
(0.92,0.94)
(0.94,1.02)
(1.02,1.13)
P(3)
> P(2)
P(2)
> P(3)
P(2)
> P(3)
P(2)
’ P(3)
P(2)
> P(3)
P(3)
> P(l)
P(3)
> P(l)
P(l)
> P(3)
P(l)
> P(3)
P(l)
> P(3)
Compare
P(3)
> P(O)
P(3)
> P(O)
P(3)
> P(0)
P(O)
> P(3)
P(0)
> P(3)
Projects
P(l)
> P(O)
P(l)
> P(0)
P(l)
> P(O)
P(l)
> P(0)
P(0)
> P(l)
P(O)
P(2)
> r-40)
P(2)
> P(0)
P(l)
P(2)
> P(l)
P(2) >
P(2) > 742) >
P(O) P(l)
P(2) > P(2) >
P(O) P(l)
P(2) > PC‘4>
projects
P(l)
A subinterval was chosen as (0.94,1.02) to illustrate the transitivity of Theorem 2. We have the preferences for two projects as p(2) > p(3), ~(1) > p(3), p(O) > p(3), p(1) > p(O), p(2) > p(O), and p(2) > p(L). W e can then rearrange them as p(2) > p(l) > p(O) > p(3) demonstrating the transitivity of Theorem 2. From Table 2, the best project for each subinterval is implied and the results are listed in Table 3. Bernhard and Canada [l] used only the values for bl/cl = 1.02, (b2 - b,)/(c2 - cl) = 1.18, and (03 - b2)/( cg - cp) = 0.77 to partition the positive real number. We offer a counterexample to illustrate that their method is incomplete. Table The Range
of V/k
The
results
from
our
results
from
Undecided
imoroved
Project 2 the bridge
the method
< 0.94
for the
(0.77,1.13)
Project 3 the tunnel
of k’/k
Optimal Alternative
comparison
< 0.77
Optimal Alternative
Range
3. The
of Bernhard
best
policy
method. > 1.13 Project 4 the ferry and
Canada
(0.94,1.02)
(1.02,1.13)
Project 2 the bridge
Project 2 the bridge
[l]. > 1.13 Project 4 the ferry
Hence, in the subinterval (0,0.94), they could not decide on the best policy. This indicates that their procedure has problems. In brief, the incremental benefit/cost analysis with cutoff ratio by Bernhard and Canada [I] is incomplete as demonstrated through the second example in their paper. 3.2. A Simple
Improved
Approach
Back to the benefit/cost analysis in AHP. Under the same condition as in Bernhard and Canada [l], we reordered the project and deleted the negative net gain project. We have two hierarchies with benefit vector g(bl, . , b,,), with Cy=“=,b, = 1 and cost vector (cl,. , cn), with Crf, ci = 1, such that bl < . < b, and cl < . . . < c,. In general, we did not know the benefit Bi and cost Ci. Therefore, in the AHP environment, we cannot find the net gain Bi - Ci. That means, without knowing the cutoff ratio, k//k, we cannot determine the magnitude of B, - C, = (b, - cik’/k)k. C onversely, if we know the cutoff ratio k//k, we can directly compute bi - cik’/k for i = 1,. , 7%to select the maximum. We derived a simple approach that obtained the same result as the net gain. We obtained the same results as the incremental benefit/cost ratios with cutoff ratio suggested by Bernhard and Canada [l]. The key factor is how to derive the cutoff ratio k’/k.
284
3.3.
K.-L.
Criticism
of Bernhard
and
Canada’s
YANG et al. Method
In Bernhard and Canada’s procedure [I] on Saaty’s benefit/cost ratios, they assumed that they knew the net gain for each project. If the decision-makers know the net gain for each project and they accept the net gain as the criterion for selecting the best choice, without any work on the incremental benefit/cost ratios or cutoff ratio, they could determine the best choice, if the best project, j, satisfies Bj - Cj = max {& - Ci, for all project i} . However, under the AHP situation, we only have cl < c2 < . .. < c,, bi < b2 < . .. < b,, CG, C, = 1, and Cy’i bi = 1. W e d onothavethedatafor{&-Ci:i=l,...,n}. Uptonow, there has been no well-constructed method that can determine the total benefit values, Ic, total cost, Ic’, and the cutoff ratio, V//c. Therefore, different decision-makers might choose different cutoff ratios. According to Theorem 2, they might have different conclusions. This indicates that the incremental benefit/cost analysis by Bernhard and Canada [I] (after the revision of Theorem 2) or our simplified method for incremental benefit/cost analysis is dependent on the cutoff ratio value choice, Ic’/lc. We claim that the incremental benefit/cost analysis is not suitable for the AHP situation. Their criticism for the first example in Project 1 (the ferry) is an incorrect choice and for the second example in Project 2 (the bridge), it is another incorrect choice based on their criterion “to maximize the net gain”. However, using Saaty’s procedure, the criterion is “to maximize the benefit/cost ratio”. The results from different criterion can be compared, but the project priority cannot be discriminated.
4. PERFECTLY
CONSISTENT
CONDITION
In Bernhard and Canada [6, p. 591, they asserted that if the decision-makers were perfectly consistent, given a normalized benefit vector [bi, b2, . . , b,], derived by themselves, the benefits under the AHP could be determined. We know that the relative judgment scale is restricted in AHP from l/9, l/8, . . . , 1,2,. . . to 9. Not in every situation can it produce a perfectly consistent comparison matrix. For completeness, we offer a counterexample to show that their assertion is invalid. Suppose that bl = lO/ll, b2 = l/11, and the comparison matrix constructed by the decision-maker as ( l/i& “f’ ). From ( iiilZ “i’)( “i’) = 2( “i’) and any two by two reciprocal matrix, we know that the normalized eigenvector of the comparison matrix must have the expression (ai2/1 +ui2, l/l + ai2) with ~12 E {l/9,1/8,. . . ,1,2,. . ,9}. Th erefore, we cannot select ai2 = 10 to satisfy the equation (uiz/l +uiz, l/l +uiz) = (lO/ll, l/11). C onsequently, when decision-makers study the benefit-cost problem, we advise them to be aware of the difference between (i) the normalized (ii) the normalized
benefit (bi) from the benefit Bi with bi = Bilk, benefit vector from the AHP.
and
5. CONCLUSION If decision-makers know the monetary benefit and cost for each project, Bernhard and Canada [l] suggest using the net gain as the criterion for selecting the best project. In this paper, we derive some theoretical properties that are correct for using the incremental benefit/cost ratios and provide a more simple and efficient way for judgment. From Theorem 2, different cutoff ratios might imply different choices for the best project. Hence, Bernhard and Canada’s method (with our Theorem 2 improvement) or our simplified method for the incremental benefit/cost ratios both suffer with a debatable cutoff ratio V/k. Unless decision-makers understand the subtle benefit and cost, then the cutoff ratio V/k cannot be determined. In general, the benefit and cost are difficult to decide. Consequently, we suggest that decision-makers adopt Saaty’s AHP
Analytic
Hierarchy
285
Process
benefits, costs, opportunities, and risks method IS] and benefit/cost procedures [la, p. itil- 1661 for selecting the best project. On the basis of many reports, Saaty’s procedures are popular and flexible and can effectively deal with this kind of problem.
APPENDIX PROOF OF THEOREM 2. We consider every i and j with i < j project pair to compute the in cremental benefit/cost ratios (b, - bl)/(cJ - cz). We tl len use them to partition the positive real numbers. First, we consider those subintervals not on the boundary, for a subinterval ((bt - b,)/(ct - c,), (6, - b)l(c3 -G)) with s < t and i < j. In this subinterval, supposing that p(l) > p(m) and p(m) > p(n). we will prove that p(l) > p(lr,). We divide the problem into four cases: (a) (b) (c) (d)
1 > m and na 1 > m and m 1 < m and m 1 < 7n and m
> n, < 7~. > n, and < n.
For Case (a), using Theorem 1, p(l) > p(m) and 1 > nt, we have
h - b,,,2 (ci - cm) (4 - h) (5 - co Similarly, we obtain bm - 6, 2 (cm - cn)
(4 - h) cc, -4’
Combining equations (2) and (3) with 1 > n, we imply p(2) > For Case (b), we divide the problem into two cases:
p(n)
(bl) 1 > n, and (b2) 1 < 71. For Case (bl), using Theorem 1, p(l) > p(m) and
bn- b,, 5 (G, - cm)
m
<
71, we
have
(bt - 6s) < (G, - cm) (b, - bt) (Ct- cs) (Cj - CT)
([I)
Combining equations (2) and (4) with 1 > 71, we imply p(l) > p(n) For Case (b2), using Theorem 1, p(l) > p(m) and 1 > m, we have
(bt.- bs) - bl) br - b,, 2 (cl - cm) (b, (cJ_ ci) 2 (Cl- cm,)(ct - e,)’ Again, using Theorem 1 and
m
<
n,
we
(51
know
b,, - b,, F (cn - cm)
(bt - b,s) (ct - cs)‘
Combining equations (5) and (6) with 1 < 7z, we imply p(l) > p(n). Similarly, we can solve Cases (c) and (d), respectively. Next, we consider the intervals on the boundary as (0, (b, - bg)/ (cn - cp)) and ((b, - ho)/ (c, - co), oo) where (b, - bo)/(ca - cP)=min((b, - b,j/(c, - c,) : ? < J’} and (by - b~)/(q - co) = max{(b, - bi)/(cj - ci) : i < j}. For the interval (0, (b, - bp)/(ca - CD)), we only need to consider Case (a) 1 > rn. and 711 > II, since 1 < m and m < n will not occur, owing to Theorem 1. Similarly, for the interval ((by - be)l(c, - co)>m), we only need to consider Case (d) 1 < m and m < n. The remaiuing proofs are similar to our previous discussion, so they are omitted.
K.-L.
YANG et al.
REFERENCES 1. R.H. Bernhard and J.R. Canada, Some problems in using benefit/cost ratios with the Analytic Hierarchy Process, The Engineering Economist 36 (l), 56-65, (1990). 2. T.L. Saaty, Multi-Criteria Decision Making-The Analytic Hierarchy Process, pp. 113-114, RWS, Pittsburgh, PA, (1990). 3. T.L. Saaty and K.P. Kearns, Analytic Planning, RWS, Pittsburgh, PA, (1994). 4. I.J. Azis, AHP in the benefit-cost framework: A post-evaluation of the Trans-Sumatra highway project, European Journal of Operational Research 48, 38-48, (1990). 5. W.A. Clayton and M. Wright, Benefit cost analysis of riverboat gambling, Mathl. Comput. Modelling 17 (4/5), 187-194, (1993). 6. J.P. Bennett and T.L. Saaty, Knapsack allocation of multiple resources in benefit-cost analysis by way of the analytic hierarchy process, Mathl. Comput. Modelling 17 (4/5), 55-72, (1993). 7. W.C. Wedley, E.U. Choo and B. Schoner, Magnitude adjustment for AHP benefit/cost ratios, European journal of Operational Research 133, 342-351, (2001). 8. T.L. Saaty and Y. Cho, The decision by the US Congress on China’s trade status: A multicriteria analysis, Socio-Economic Planning Sciences 35, 243-252, (2001). 9. R.H. Bernhard, A comprehensive comparison and critique of discounting indices proposed for capital investment evaluation, The Engineering Economist 16 (3), 157-186, (1971). 10. T.L. Saaty, The Analytic Hierarchy Process, McGraw-Hill, New York, NY, (1980). 11. T.L. Saaty, Decision Making for Leaders, RWS, Pittsburgh, PA, (1995). 12. T.L. Saaty, Fundamentals of Decision Making and Priority Theory, RWS, Pittsburgh, PA, (1994).