Simulation-Based Uncertainty Analysis
3.3 Parameter Uncertainty
3.3.1 Multi-Dimensional Inter-Criteria Sensitivity Analysis
A multi-dimensional approach is introduced to allow the consideration of simultaneous variations of the weights within a decision model and to facilitate the weight elicitation process by allowing the assignment of intervals instead of discrete weights. It should be emphasised that this approach is not aimed at substituting but at complementing the one-dimensional sensitivity analysis. Assuming, again, that n attributes are considered (n ∈ N), this means that instead of assigning one exact weight wi to each attribute i (i∈ {1, ..., n}), it is sufficient to assign an interval I(wi) by determining a lower (wli) and an upper (wiu) bound:
I(wi) =
wil, wui
. (3.8)
In groups, the intervals could for instance be obtained by permitting each group member to define his or her weights individually and then defining the intervals as the superset of the individual weights. Alternatively, the group members could be permitted to use intervals, too, and the group interval could be obtained by using the superset of the individual intervals. However, the uncertainties of the inter-criteria preferences for all n attributes can then be described by the n-dimensional interval
Cw =
n i=1
I(wi) =
w1l, wu1
× ... ×
wnl, wnu
, (3.9)
which can also be regarded as a generalised cuboid. However, not all points within Cw represent valid weight combinations since they do not necessarily fulfill the constraint that the sum of the weights of all attributes is equal to one. This constraint is represented by the hypersurface
H =
w∈ Rn : wi ≥ 0,
n i=1
wi = 1
. (3.10)
1
1 w3
w1
w2 1
H Cw∩ Cw
H
( )w1
I
( )w3
I
( )w2
I 1
1 w3
w1
w2 1
H Cw∩ Cw
H
( )w1
I
( )w3
I
( )w2
I
Figure 3.10: Intersection of the Weight Intervals and the (Hyper-)Surface Representing the Valid Weights in 3D
Hence, the intersection Cw ∩ H is the set of valid weight combinations within the n-dimensional weighting interval. Thus, this is the basic set for the Monte Carlo Simulation (see below). For n = 3 the set Cw∩H can be illustrated as the hatched area in Figure 3.10.
For higher dimensions, such a graphic illustration is not directly possible. However, the method can nevertheless be applied for any dimension. The set Cw ∩ H of valid weight
Chapter 3.3. Parameter Uncertainty 67
combinations is also called weight space in literature [cf. e.g. Lahdelma and Salminen, 2001; Hodgkin et al., 2005; Mavrotas and Trifillis, 2006].
It is important to choose the weight intervals and thus Cw in such a way that Cw∩H = ∅.
It is of course possible to construct intervals that result in Cw ∩ H = ∅. A procedure providing usable results for such parameter input would necessitate scaling up or down the input and would thus result in weights lying outside the afore assigned intervals (at least for some weights). Rather than considering such procedures, it is argued that, in practice, there should always be a facilitator who guides the decision making group in order to avoid the occurrence of such problems, to ensure that Cw∩ H = ∅ and to explain the difficulty to the members of the group of decision makers if necessary. For instance, a comparison of the ranges of the assigned weight intervals and the ranges of the actually drawn weights can provide the basis for a consistency check. The latter will be picked up again in more detail in Chapter 4.
Monte Carlo simulation can then be used to draw multiple samples of valid weight com-binations, where “valid” means that the samples are drawn under the constraint that the sum of the weights of all attributes must be equal to 1. Uniform distributions of the weights within the assigned intervals are presumed for the analyses carried out, if no other information is available. On the one hand, this assumption was made for reasons of simplicity. But on the other hand, the most important (and most understandable) part in a practical application, is to illustrate the spread of the results (the ranges in which the results can vary) and not the probabilistic structure inside the spread [Bertsch et al., 2007c]. Thus, when drawing the samples, all elements of the set Cw ∩ H are considered to be equiprobable.
While the restriction of the simulation space to the set Cw ∩ H, as illustrated in Figure 3.10, is conceptually simple, the question arises how this problem can be tackled com-putationally. Because of its straightforward implementability and its computation rate in practical applications, an approximate procedure as illustrated in Figure 3.11 is pro-posed. This procedure allows the sum of the attributes’ weights to vary within the interval [1− , 1 + ] whose size is determined by the accuracy factor ( > 0).
The accuracy factor certainly influences the computation rate. However, it could be ob-served in practical case studies that setting = 10−3 or = 10−4, for instance, gives good results in combination with an acceptable computation rate. Concerning the influence on the calculation speed, the same holds for the desired sample size τ . Here, practical experiences have shown that setting τ = 1000 or τ = 10000 usually leads to sufficiently good results, i.e. a graphical comparison has shown that the drawn samples provide a
6. Stop
4. Store sample as valid weight combination (counter = counter + 1)
5. Check if counter = τ
2. Draw sample from Cw, i.e. draw an n-tuple of weights wi
Yes 3. Check if ∑i=n1wi∈[1−ε +,1 ε]
1. Determine desired sample size τas well as accuracy factor εand set counter = 0
Yes
No No
6. Stop
4. Store sample as valid weight combination (counter = counter + 1)
5. Check if counter = τ
2. Draw sample from Cw, i.e. draw an n-tuple of weights wi
2. Draw sample from Cw, i.e. draw an n-tuple of weights wi
Yes 3. Check if ∑i=n1wi∈[1−ε +,1 ε]
3. Check if ∑i=n1wi∈[1−ε +,1 ε]
1. Determine desired sample size τas well as accuracy factor εand set counter = 0
Yes
No No
Figure 3.11: Proposed Procedure to Ensure that the Sum of the Weights is Equal to 1
good representation of the full range of the theoretical distribution. Once the samples are drawn, the approach is in principal straightforward and furthermore computationally easily feasible, i.e. the analysis is carried out in parallel for all samples.
As mentioned above, uniform distributions are used for the sampling of the weights wi within the respective intervals I(wi). It should be noted that it can thus be concluded that the resulting weight combinations, sampled according to the proposed procedure, are also uniformly distributed (in Cw∩ H). However, such a conclusion cannot be drawn for arbitrary distributions.