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DECISION ANALYSIS
Weighting procedures
Carlos Bana e CostaJoão LourençoMónica Oliveira
Academic year 2012/2013
Structuring Evaluation
Points of view
PlausibleDescriptors of Performance
Options’Performance profile
Partialvalues
Weights
Sensitivity andRobustness
analyses
Options
MULTICRITERIA STEPS:Structuring vs. evaluation
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Additive value model
)a(v.k)a(V j
n
1jj
V(a) global value of option ‘a’
vj(a) partial value of option ‘a’in FPV/criterion j
kj weighting coefficients(relative weights)of FPV/criterion j
n
1jj 1k and kj > 0 ( j = 1,…,n )
With:
0)(
100)(
,0)(
,100)(
everythinginworstV
everythinginbestV
jworstv
jbestv
jj
jj
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What do the weighting coefficients represent?
• They are scaling factors, scaling constants, or simply weights
• They represent the contribution for the global score (value) of each scoring unit in the PV j
• They allow to convert each partial value unit vj(.) into partial value unit V(.)
• They take into account the nature of substitution rates between FPV/criteria Operationalize the notion of compensation
• They cannot be seen as direct indicators of importance!
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The ranges of the performance levels of all the points of view are critical to obtain the weighting
coefficients
Building the weighting coefficients always requires the comparison between reference alternatives:
▫ Worst vs. Best▫ Neutral (neither attractive nor unattractive) vs. Good▫ Base vs. Target
Most common critical mistake: assigning weights that reflect the “importance” of the criteria, without
considering the range of the performance descriptor scales and the importance of those ranges!
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A more complex case: HierarchicalAdditive Model
(Lourenço, 2002)
The globalvalue isobtainedthrough abottom-upapproach
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Weighting procedures
• Swing weighting
• Trade-off procedure
• MACBETH
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Swing Weighting procedure
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Swing Weighting Procedure: 3 steps
1. Ordering of the weighting coefficients of the points of views
2. Their quantification
3. Normalization of the obtained values so that their sum is equal to 1
Note: In the examples, we use as reference the worst and best levels in each FPV, which correspond to the values of 0
and 100, respectively.Other two sufficiently different reference levels can be
adopted (such as “neutral” and “good”).
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First step: Ordering of the weighting coefficients
The facilitator asks the decision-maker (DM) to
consider all the fundamental points of
view of the decision problem (represented by PV1, PV2, PV3 and PV4) in
the worst levels.PV1
best1
PV2 PV3 PV4
best2
worst1
best3 best4
worst2 worst3 worst4
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Then, the DM answers to the following question: "If it was possible to go from the worst to the best level in a single point of view, which
point of view would you select for this change?”
Answer of the DM: “I would select the PV2”.
PV1
best1
PV2 PV3 PV4
best2
worst1
best3 best4
worst2 worst3 worst4
First step: Ordering of the weighting coefficients
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Next question: “Excluding PV2 that has already been selected by you, which point of view would you choose now to go from the lowest level to the highest level?”Answer: “I would select PV1”.
The DM would be successively asked about the point of view that would be chosen to go from the worst to the best level, excluding all the previous selected points of view, until only one point of view of the results has to be selected.Result: an ordering…
First step: Ordering of the weighting coefficients
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PV1
best 1
PV2
worst 2
PV3
PV4
best 2
worst1 worst 3
best 3
worst 4
best 4
jj3j34j41j12j2 ,worstworst,estbworst,bestworst,bestworst,best
Where: “X Y” indicates that X is preferable to Y.
The DM indicates that it is preferable to go from the worst level to the best level in PV2 than in PV1, which in turn is preferable to make this change in this PV (PV1) than in PV4, which in turn…
This preference relation can be symbolically represented by:
First step: Ordering of the weighting coefficients
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If you assign 100 points to the best levels and 0 points to the worst levels, being kj the weighting coefficient of the point of view j, we will get:
which is equivalent to:
0k0-100k0-100k0-100k0-100 3412
0kkkk 3412
First step: Ordering of the weighting coefficients
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Second step: quantification of the values to assign to the weighting coefficients of all the points of view
• Quantification step of the scaling factors of all PV. • Comparison of the increments (swings) from the
worst to the best level in each PV, with the swing from the worst to the best level in a reference point of view (which values 100 points).
• Note:▫ Any point of view may serve as a reference PV.
Usually, it is chosen the first point of view that was selected in the first step.
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Question: “In how much would you quantify the swing from the worst to the best level in the PV1, if it was known that it
was assigned 100 points to the swing from the worst to the
best level in PV2?”
Answer: “In my opinion, the swing from the worst to the best level in PV1 is equal to 80% of the swing in PV2”.
Second step: quantification of the values to assign to the weighting coefficients of all the points of view
2 1 4 3
100
0
80
PV2
best1
PV1 PV4 PV3
best2
worst1
best3 best4
worst2 worst3 worst4
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The process is repeated: the DMcompares the swing from theworst to the best level in PV4with the increment of the worstlevel to the best level in PV2;and the question is repeatedusing PV3 in the comparison.
Assuming that the DM answers 60 and 20, respectively, this means that these values represent the scaling constants of the points of view 4 and 3, being thus completed the
second step of this procedure.
Second step: quantification of the values to assign to the weighting coefficients of all the points of view
100
0
80
60
20
PV2
best1
PV1 PV4 PV3
best2
worst1
best3 best4
worst2 worst3 worst4
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Third step: normalization of the weighting coefficients
Normalization of the weighting coefficients obtained in the previous step such that their sum is equal to 1. The following expression is used:
n
1j
'j
'j
j
k
kk , with (j =1,…, n)
'jk
jk
where: - is the non-normalized weighting coefficient of the point of view j (obtained in the second step);
- is the normalized weighting coefficient of the point of view j.
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• In our example, the normalized weighting coefficients would be:
0,31602010080
80k1
0,38602010080
100k 2
0,08602010080
20k 3
0,23602010080
60k 4
Third step: normalization of the weighting coefficients
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Swing Weighting in the hierarchical additive model• It was described the simplest situation!▫ The application of this weighting procedure to a tree of points of
view with more than one hierarchical level is significantly more complicated
• It is necessary to repeat the weighting procedure as many times as the number of sub-trees in the tree of points of view.
• The questions posed to the DM will have to be adapted to the existing hierarchy of points of views.
• In order to determine the weighting coefficients of all the points of view, it should be used a bottom-up approach:A. Weighting the FPV present in the lower hierarchical levelsB. Determination of the weighting coefficients of the PV
hierarchically superior to these FPV and so successively until reaching the top of the hierarchy .
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Global
PV1
PV1.1.1
PV2
PV1.1 PV1.2
PV3
PV1.1.2 PV1.1.3 PV1.2.1 PV1.2.2 PV1.2.3 PV1.2.4
PV3.1 PV3.2
100 4060 100 75 50 25
100 25
Swing Weighting in the hierarchical additive model - Example
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There are two options (or combination of both options) to determine the weighting coefficients of PV1.1 and PV1.2:
1st Option: Comparison of the swing from the lowest reference levels to the highest reference levels in all the fundamental points of view of PV1.1 (in simultaneous)with the equivalent swing in all the fundamental points of view of PV1.2 (simultaneous);
2nd Option: Comparison of the swing from the lowest reference level to the highest reference level in onefundamental point of view of PV1.1 with the equivalent swing in one fundamental point of view of PV1.2.
Swing Weighting in the hierarchical additive model - Example
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1st option: Swing Weighting in the hierarchical additive model
• Initial question: "If all the fundamental points of view (PVFS) from PV1.1 and PV1.2 were in the lower reference levels, which of these two sets of points of views would you select first to go to the upper reference levels?”
• Answer: “The set of points of view from PV1.1”.
• Next question: “In how much would you quantify the swing from the lowest reference levels to the highest reference levels in the FPV of PV1.2, if it was known that it was assigned 100 points to the equivalent swing in the FPV of PV1.1?”.
• Answer : “This swing is equal to 60% of the swing in the FPV of PV1.1”.
Thus, the weighting coefficients of PV1.1 and PV1.2 would be 0,625 e 0,375, respectively
(obtained respectively from 100/160 and 60/160).
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2ª opção: Swing Weighting in hierarchical additive model• Simplest option. • First, two FPV are selected, one from PV1.1 and
another one from PV1.2, being a basis of comparison. ▫ Assuming that the FPV with higher weighting
coefficients in each sub-tree are chosen, i.e., PV1.1.1 and PV1.2.1.
• In a first step, it is necessary to make a preference ordination between those FPV.
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• Question: “Being both FPV in the lowest reference levels, which one would you select first to move to the upper reference levels?”
• Answer: “PV1.1.1”
• Next question: “In how much would you quantify the swing from the lowest reference level to the highest reference level in PV1.2.1, if it was known that it was assigned 100 points to the equivalent swing in PV1.1.1?”
• Answer: “That swing is equal to 48% of the swing in PV1.1.1”
2nd option: Swing Weighting in the hirarchical additive model
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2nd option: Swing Weighting in the hirarchical additive model
The weighting coefficients of PV1.1 and PV1.2 would be obtained through the following
expressions:
625025048200100
200100
kkkk
kk
k4
1j
1.2.j'
121
3
1j
1.1.j'
111
3
1j
'1.1.j111
1.1 ,''
..''
..
''..
375025048200100
25048
kkkk
kk
k4
1j
1.2.j'
121
3
1j
1.1.j'
111
4
1j
'1.2.j121
1.2 ,''
..''
..
''..
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Two possible approaches for Swing Weighting in the hierarchical additive model
Comparison option by levelsComparison option between levels
Comparison of swings from the lowest reference levels to the highest reference levels in the points of view hierarchically inferior to PV1 (which correspond to the same swings in all the FPVs of PV1.1 and PV1.2) with the swings in PV2 and PV3 (which correspond to the swings in the respective FPV).
Comparison of swings of points of views on any hierarchical level, offering this choice to the decision maker, leaving the burden of computational complexity to the facilitator.
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Trade-Off Procedure
The key idea of this weighting procedure is to compare two fictitious alternatives at the time, with these alternatives with performance differing only in two Fundamental Points of View (consequently, the two alternatives have the same performance in the remaining Points of View (PV)).
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IntroductionC1
LOW1
UPP1
C2
A B
LOW2
UPP2
Alternative A has a performance at the upper reference level in PV C1 and a performance at the lower reference level in PV C2; alternative B has a performance at the lower reference level in PV C1 and a performance at the upper level in PV C2 .
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IntroductionThe choice of the preferred alternative for the decision-
maker indicates which PV, among the two considered, has the highest relative importance, i.e., which is the one with the highest weighting coefficient (in this case it was chosen alternative A, so C1 has a higher weight than C2). The PV with the higher weight is typically used as the reference PV, although other PV can be selected for that purpose.
The trade-off of this procedure consists in adjusting the level of impact of one of the alternatives of the reference PV, so as to obtain a relation of indifference between the two alternatives.
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IntroductionTo do so, or it is worsened the impact of the alternative that has the highest performance in the reference PV, or it is improved the impact of the alternative that has the lowest performance in the reference PV.
C1=Reference
LOW1
UPP1
C2
A B
LOW2
UPP2
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IntroductionThese adjustments must be repeated for n-1 pairs of the fictitious alternatives!
If the value functions associated with the PV are known, the weighting coefficients of PV can be determined by solving a system of equations that incorporates a constraint of normalization and the n-1 relations of indifference obtained previously.
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ExampleConsider a decision problem that involves the purchase of a new office where you will use the three following PV (where we also show the best and worst performances of the alternatives were analyzed):
PV/Criterion Description Best Worst
1 Price (M€) 1 2
2Centrality(distance to the centre in km)
0 50
3Parking (nº of places)
15 5
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Example
Consider these valuefunctions for the PV.
0
20
40
60
80
100
1 1,2 1,4 1,6 1,8 2
V(p
ric
e)
Price (106 €)
0
20
40
60
80
100
0 10 20 30 40 50
V(d
ista
nce
to th
e
ce
ntr
e)
distance to the centre in km
0
20
40
60
80
100
5 6 7 8 9 10 11 12 13 14 15N.º lugares de estacionamento
V(n
.º l
ug
ares
de
esta
cio
nam
ento
)
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Parking places
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Trade-Off Procedures: Example
1ª Step:Establish an ordering preferably between various
PV considered (step identical to the first step in Swing Weighting).
2ª Step :Select the PV which will serve as a reference. To this
end, it was selected PV “Price".
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ExampleQuestion the decision-maker:
Consider an office A located in the city center with a price of 2 Mio €. How much should be the price for an office B, situated 50 km from the city center, so that A and B were indifferent (knowing that both offices have the same number of parking places)?
This question might be represented by thefollowing expression (in which the symbol ~ meansindifference):
(2 M€, 0 km, x places) ~ (? €, 50 km, x places)
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ExampleAnswer:
The two offices would be indifferent if the price ofthe office B was equal to 1,55 M€.
The answer can be represented by:
(2 M€, 0 km, x car places) ~ (1,55 M€, 50 km, x carplaces)
This trade-off means that a variation in price between € 2 M and € 1.55 M equals the difference between an office being located 50 km from the city center and an office located in the center.
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Example
PV
F2
Dis
tan
ceto
th
ece
ntr
e (
km)
0
50
PVF1 Price (106 €)
12 1,55
Indifferent
In the additive model, thisratio can be equated asan indifference:
Which means:
(2 M€) (0 km) (x lugares)
(1,55 M€) (50 km) (x lugares)1 1 2 2 3 3
1 1 2 2 3 3
k v k v k v
k v k v k v
12 kk 53,75 100
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places
places
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ExampleThe decision-maker is then questioned:
Consider an office that has 15 parking places and a price equal to € 2 million. How much should be the price of an office B with 5 parking places, so that A and B were indifferent (knowing that both offices are at the same distance from the city center)?
This question can be represented by:
(2 M€, y km, 15 places) ~ (? €, y km, 5 places)
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ExampleAnswer:
Both offices would be indifferent if the price of office B was equal to 1.7 M €.
Then:
(2 M€, y km, 15 places) ~ (1,7 M€, y km, 5 places)
This trade-off means that, for the decision maker, a variation in price between € 2 M and € 1.7 M equals the difference between an office having 15 or 5 parking places.
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Example
Then:
And:
FP
V3
N.º
pla
ces
15
5
PV1 Price(106 €)
12 1,7
Indifferent
13 kk 27,5 100
(2 M€) (y km) (15 lugares)
(1,7 M€) (y km) (5 lugares)1 1 2 2 3 3
1 1 2 2 3 3
k v k v k v
k v k v k v
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places
places
ExampleSince the sum of the weights must equal one, a third
equation is added. Then solve the following system of equations:
0,15
0,30
0,55
1
27,5100
53,75 100
3
2
1
321
13
12
k
k
k
kkk
kk
kk
Note: as seen in the example of this method requires knowledge of the function value from the point of view used as the reference.
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What if the decision does not want to elicit hispreferences by providing numbers?
Use the MACBETH methodology!
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References• Keeney, R. L.; 1980. Sitting Energy Facilities,
Academic Press.• Keeney, R. L.; Nair, K.; 1977; Selecting Nuclear
Power Plant Sites in the Pacific Northwest UsingDecision Analysis, in D. E. Bell, R.L. Keeney, H. Raiffa (eds.), Conflicting Objectives in Decisions, John Wiley, 298-322.
• Lourenço, J. C. (2002) Modelo aditivo hierárquico: exemplos de métodos de ponderação e problemas associados, Working Paper 13-2002, CEG-IST.
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