Mini-course 1 Decision Analysis (Dr. Mariya Sodenkamp) / Class 5 / ITB_L5_ 2015_04_27
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How to measure consistency?
(approximate method)
1) |
Find Principal Eigen value |
λmax : add up the products of each |
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element of priority vector |
and the summarized pairwise comparison |
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values in columns. |
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Note: λmax |
must be equal to or larger than the number of compared |
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elements (n): λmax ≥ n |
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2) |
Find Consistency Index (C.I.): |
C.I. = |
λmax − n |
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n −1 |
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3) |
Find Consistency Ratio (C.R.): |
C.R. = |
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C.I. |
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R.I. |
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n |
1 |
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2 |
Random |
Consistency |
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7 |
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9 |
10 |
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3 |
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6 |
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R.I. |
0 |
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0 |
0.58 |
0.9 |
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1.12 |
1.24 |
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1.32 |
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1.41 |
1.45 |
1.49 |
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n |
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10 |
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R.I. |
0 |
0 |
0.58 |
0.9 |
1.12 |
1.24 |
1.32 |
1.41 |
1.45 |
1.49 |
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1) |
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Find |
λmax |
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Burj |
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Petronas |
Empire |
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Priority |
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Height |
Dubai |
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Towers |
State |
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Vector |
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Burj Dubai |
1 |
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2 |
2 |
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0.47 |
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Petronas |
1/2 |
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1 |
4 |
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0.38 |
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Towers |
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Number of compared elements: |
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Empire State |
1/2 |
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1/4 |
1 |
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0.15 |
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n = 3 |
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Sum of |
2 |
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13/4 |
7 |
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Column |
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λ = 2 0.47 + |
13 |
0.38 + 7 0.15 = 0.94 +1.235 +1.05 = 3.225 |
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max |
4 |
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λmax >= n ? |
yes |
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2) |
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Calculate consistency index: |
C.I. = |
3.325 − 3 |
= 0.1125 |
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3) |
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Find final consistency: |
C.R. = |
0.1125 |
= 0.19 > 0.1 |
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inconsistent |
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0.58 |
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Fuel |
Car A |
Car B |
Car C |
Priority |
cons. |
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Vector |
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Car A |
1 |
1/4 |
1/7 |
0.08 |
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Car B |
4 |
1 |
1/3 |
0.26 |
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Car C |
7 |
3 |
1 |
0.66 |
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1) |
Find |
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λmax |
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GOAL |
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Price |
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Maintenance |
Horse- |
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Fuel consump3on |
Priority Vector |
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power |
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Price |
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1 |
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5 |
7 |
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3 |
0.58 |
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Maintenance |
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1/5 |
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1 |
2 |
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1/2 |
0.12 |
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Horse-power |
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1/7 |
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1/2 |
1 |
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1/4 |
0.07 |
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Fuel consump3on |
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1/3 |
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2 |
4 |
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1 |
0.23 |
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n = 4 |
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Sum of Column |
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176/105 |
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17/2 |
14 |
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19/4 |
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λ |
= |
176 |
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0.58 + |
17 |
0.12 +14 0.07 + |
19 |
0.23 = 0.972 +1.02 + 0.98 +1.093 = 4.065 |
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max |
105 |
2 |
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4 |
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λmax >= n ? yes
2) |
Calculate consistency index: |
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C.I. = |
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4.065 − 4 |
= 0.0217 |
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3) |
Find final consistency: |
C.R. = |
0.0217 |
= 0.02 < 0.1 |
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0.9 |
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consistent |
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Fuel |
Car A |
Car B |
Car C |
Priority |
cons. |
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Vector |
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Car A |
1 |
1/4 |
1/7 |
0.08 |
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Car B |
4 |
1 |
1/3 |
0.26 |
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Car C |
7 |
3 |
1 |
0.66 |
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12 |
4.25 |
1.48 |
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λmax =14 0.07 + 5.2 0.24 +1.375 0.69 = 0.98 +1.248 + 0.948 = 3.176
C.I. = 3.176 − 3 = 0.088 2
C.R. = 00.089.58 = 0.15 > 0.1- acceptable?
λmax =1.31 0.75 + 6.33 0.18 +13 0.07 = 0.983+1.139 + 0.91 = 3.032
C.I. = 3.032 − 3 = 0.016 2
C.R. = 00.016.58 = 0.03 - acceptable
λmax =1.375 0.71+ 5.33 0.21+12 0.08 = 0.976 +1.119 + 0.96 = 3.055
C.I. = 3.055 − 3 = 0.0275 2
C.R. = 0.02750.58 = 0.05 - acceptable
λmax =12 0.08 + 4.25 0.26 +1.48 0.66 = 0.96 +1.105 + 0.977 = 3.042
C.I. = 3.042 − 3 = 0.021 2
C.R. = 00..02158 = 0.04 - acceptable
Causes of Inconsistency
Clerical Error
The most common cause. When entering one or more judgments into a computer, the wrong value, or perhaps the inverse of what was intended is entered. Clerical errors can be very detrimental and often go undetected in many analyses.
When using professional software (e.g., Expert Choice/Decision Lens), one can find and correct such errors.
Lack of Information
If one has little or no information about the factors being compared, then judgments will appear to be random and a high inconsistency ratio will result. Sometimes we fool ourselves into thinking that we know more than we really do. It is useful to find out that a lack of information exists, although sometimes we might be willing to proceed without immediately spending time and money gathering additional information in order to ascertain if the additional information is likely to have a significant impact on the decision.
Causes of Inconsistency
Lack of Concentration
This can happen if the people making judgments become fatigued or are not really interested in the decision.
Real World Is Not Always Consistent
The real world is rarely perfectly consistent and is sometimes fairly inconsistent. Professional sports is a good example. It is not too uncommon for Team A to defeat Team B, after which Team B defeats Team C, after which Team C defeats Team A!
Inconsistencies such as this may be explained as being due to random fluctuations, or to underlying causes (such as match-ups of personnel), or to a combination. Regardless of the reasons, real world inconsistencies do exist and thus will appear in our judgments.
Consistency: Conclusions
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§ Consistency itself is a necessary condition for a better understanding of relations in the world but it is not sufficient.
§ It is fortunate that the mind is not programmed to be always consistent. Otherwise, it could not integrate new information by changing old relations.
§ Inconsistency of 10 % (0.1) can be tolerated
Exercise: Outsourcing Partner Selection
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Vendor A |
Vendor B |
Vendor C |
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Supplier's |
130 |
150 |
110 |
suggested price, |
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euro |
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Distance, km |
12,000 |
3,500 |
10,000 |
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Organizational |
good |
excellent |
satisfactory |
behavior |
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Adaption with |
complete |
partial |
complete |
the purchaser‘s |
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procedures |
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Based on the information above, which vendor is the most preferred (using the AHP)?
Step 4: Compute judgment consistency…