Selasa, 14 Februari 2012

Liberatore



AHP method provides a fundamental scale to assign pairwise comparison judgment, as shown on figure 3. Figure 4 shows the way proposed by Saaty [Saaty & al 94]. The meaning of the table is that criterion A is strongly more important than criterion B. For the vendor selection problem, the fundamental scale is not sharp enough to assign relevant pairwise judgements. AHP method proposes to create as many refinements as needed for the specific problem, and to estimate verbally the value of each new point of the scale. This work has to be done by the evaluation team, in order to obtain a consensus about the evaluation scale. The evaluation team had created a scale divided in 1/10, from 1 to 9 (1.1, 1.2, …, 8.8, 8.9, 9). Also evaluators use their own comparison ruler, but using the same principle than the basic one.






Level
Definition
Explanation
1
Equal importance
Two factors contribute equally to the objective
3
Moderate importance
Experience and judgment slightly favor one factor over another
5
Strong importance
Experience and judgment strongly favor one criterion over another
7
Very strong or demonstrated importance
A factor is favored very strongly over another; its dominance demonstrated in practice
9
Extreme importance
The evidence favoring one factor over another is of the highest possible order of affirmation
Figure 3: Fundamental scale for AHP pairwise comparison


Figure 4: Basic pairwise comparison ruler
Each evaluator has to compare elements of the same hierarchy level (see fig.2). First, cost is compared to quality. Then, on one hand, capital expenditure compared to operating expenditure, and on the other hand three couples have to be compared: (Technical / Operational), (Technical / Vendor) and (Operational / Vendor). At last, each element of the third level is considered as a root and so, all the sub-criteria under the same root have to be compared with this same pairwise method. Results of these judgments are summarized in pairwise comparison judgement matrices (PCJM) as shown on figure 5.

Operating expenditure
OC
MC
CSS
Priority vector
Operating cost (OC)
1
3.0
1.6
0.518
Maintenance cost (MC)
0.333
1
0.77
0.195
Cost of support services (CSS)
0.625
1.3
CR = 0.01
 
1
0.287

Figure 5: Pairwise comparison judgment matrix
The meaning of this matrix is that, according to the evaluation team, the most important sub-criterion, under the Operating Expenditure root, is “Operating cost” (priority: 0.518) followed by “Cost of support services” (priority: 0.287) and finally “Maintenance cost” (priority: 0.195). Priorities are computed by this way (example for Operating Cost factor):
The number of criteria divides result of the sum, in our case there are three. So priority for the first sub-criterion is:
The last cell of the PCJM is related to consistency ratio (CR). According to Saaty [Saaty & al 94], AHP method can introduce some inconsistency in judgement because experts not only use rational ways to compare criteria but also their feelings, so results can be not totally consistent. For example, with three criteria (A, B and C), if the first line of the matrix is  and , the logical following should be: . But it is possible to find  and it means there is some inconsistency. So evaluators have to compute a consistency ratio to determine if their judgements are good or too inconsistent.
First, each column of PCJM is multiplied by the corresponding priority and lines are summed as show below in figure 6
Operating expenditure
OC (0.518)
MC (0.195)
CSS (0.287)
Line sum
Operating cost (OC)
0.518
0.585
0.459
1.562
Maintenance cost (MC)
0.172
0.195
0.221
0.588
Cost of support services (CSS)
0.324
0.254
0.287
0.865
Figure 6: Inconsistency matrix
After that line some vector is divided by priority vector:
And so evaluators can compute lmax =
The next step is to compute the consistency index: CI, where n is the number of criteria.
AHP method provides a random index (RI), which depends of the matrix size. This index is used to compute the consistency ratio (CR). For a 3´3 matrix RI= 0.58. So finally:
. According to Saaty [Saaty & al 94], if , it means that judgment is consistent enough and can be approved. In our example, each PCJM has a CR equal to or less than 0.03 so we can consider that experts’ judgements are consistent. Results of the pairwise comparisons are summarized in figure 7. Local weights (L.W.) are directly from PCJM and global weights are computed by multiplying local weights from the root to the sub-criteria. For example global weight for Unit Cost factor is:

When numbers of alternatives and sub-criteria increase, using PCJM from usual AHP method  becomes  computationally  difficult   and  sometimes   infeasible. In  our  case,  the



Strategic issues
L. W.
Criteria
L.W.
Sub-criteria
L. W.
G. W.
Cost





Quality
0.565





0.435
Capital expenditure


Operating expenditure


Technical








Operational




Vendor specific
0.524


0.476


0.519








0.313




0.169
Capital investment
Unit cost
Cost of NMS
Operating Cost
Maintenance cost
Cost of support services
Technical features/characteristics
System capacity
System reliability/availability
System performance
Comply to standards
Interoperability with other systems
Future technology development
System redundancy
Upgradability on H/W and S/W
Ease of operations
Performance monitoring capability
Fault diagnosis capabilities
Billing flexibility
System security features
Delivery lead time
Quality of support services
Experience in related products
Problem solving capabilities
Technical expertise
Vendor’s reputation
0.268
0.521
0.211
0.518
0.195
0.287
0.096
0.054
0.210
0.148
0.077
0.093
0.113
0.094
0.115
0.098
0.214
0.249
0.181
0.258
0.084
0.275
0.092
0.290
0.175
0.084
Total:
0.079
0.154
0.062
0.139
0.052
0.077
0.022
0.012
0.047
0.033
0.017
0.021
0.026
0.021
0.026
0.013
0.029
0.034
0.025
0.035
0.006
0.020
0.007
0.021
0.013
0.006
1.000
Figure 7: Composite priority weights for critical sub-criteria

number of alternatives is only three, but the evaluation of vendor proposals of a particular telecommunications system sometimes involves a large number of technical details consisting of several sub-criteria. It may be practically too difficult to make pairwise comparison among the vendor system with respect to every sub-criteria, and it’s also a time-consuming process. Liberatore [Liberatore 87] proposes the rating scale that is the fifth level of the hierarchy (see figure 2). Instead of assign pairwise comparison for each sub-criterion, related to each alternative, Liberatore suggests affecting a mark, which can be outstanding (O), good (G), average (A), fair (F) and poor (P). By the way you can reduce the time affected to the evaluation of alternatives. Liberatore uses a PCJM to convert the rating scale in a score scale, as shown in figure 8.




O
G
A
F
P
Priorities
O
G
A
F
P
1
1/3
1/5
1/7
1/9
3
1
1/3
1/5
1/7
5
3
1
1/3
1/5
7
5
3
1
1/3
9
7
5
3
CR=0.053
 
1
0.513
0.261
0.129
0.063
0.024
Figure 8: PCJM for rating scale
The evaluation team can now use Liberatore’s scale to affect a score to each sub-criterion and for each alternative, and then compute the total score of each system in order to determine which is the best system for the company. For each system scores of sub-criteria are added and the highest score shows the best system for the company. When several evaluators are involved in selecting a vendor system, we can use different method to obtain consensus about evaluation. Saaty [Saaty & al 94] proposes geometric approach; it means for n evaluators, each one affecting one score s, the consensus score will be . Also we can use the Delphi method or as suggested by Liberatore [Liberatore 87], we can find the mean and the median of the global priority weights of vendor systems of team members and use them to select the best vendor. Figure 9 is a part of final evaluation, if evaluation of alternatives only concerns criteria of figure 9, results would be: best alternative is system C (highest score) followed by system B and the last one would be system A.







Figure 9: Example of evaluation results

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