Uncertainty Assessment in the Decision Analysis Process
4.4 Local Sensitivity Analysis via Iterative Binary Search Algo- Algo-rithm
Sensitivity analysis addresses the question how the variation of input variables influences model output [55]. There are two categories: local and global sensitivity analysis [110].
Local sensitivity analysis varies input variables one at a time to determine which variables have the greatest effect on model output, while holding the others fixed at nominal values.
Local sensitivity analysis can provide an initial understanding of the sensitivity of an individual variable on model output over a small region around the nominal values of input variables, with efficient computation. However, local sensitivity analysis may not provide meaningful results when the model under consideration is nonlinear, or when input variables are perturbed simultaneously and by different amounts, and the effects of interactions among input variables on model output cannot be captured [46], [88].
Global sensitivity analysis varies all input variables simultaneously over the full range and investigates the influence of each variable averaged over all possible values of other input vari-ables [110], [46]. Global sensitivity analysis can provide insights into model behavior over the full range of model output, taking into account the variable interactions [88]. However, com-putational cost of global sensitivity analysis is higher than local sensitivity analysis and may become prohibitive for large complex models.
In this research, we take the perspectives that different types of sensitivity analysis reveal model behaviors in different domains of the variables [138], and global sensitivity analysis should not precede local sensitivity analysis [50]. This section focuses on local sensitivity analysis when solving evaluation decision making problems, and global sensitivity analysis is investigated in the next section.
Local Sensitivity Analysis in the Decision Analysis Process
When the MCDA methods are utilized in evaluation decision making problems, local sensitivity analysis can be conducted to determine the sensitivity of alternatives’ rankings to changes in input variables. A unified local sensitivity analysis approach for three MCDA methods including SAW, multiplicative weighting method, and AHP, was proposed [126], where two questions were addressed: (1) How sensitive the ranking of the best alternative or any alternative is to variations in the current weights or performance measures of decision criteria? (2) What is the smallest change in the current weights or performance measures of decision criteria which can alter the current ranking of two alternatives?
However, this sensitivity analysis approach is specific for these three MCDA methods and is not applicable to other MCDA methods. In addition, this approach was obtained through the analytical inferences of these three specified MCDA methods, which only involve simple mathematical calculation steps. For instance, SAW just has two calculation steps: multiplication and addition, multiplicative weighting method only involves multiplication, and AHP also merely involves multiplication and addition. Nonetheless, for other MCDA methods with complicated mathematical calculations, such as TOPSIS or ELECTRE, it is difficult to infer the sensitivity
4.4 Local Sensitivity Analysis via Iterative Binary Search Algorithm
coefficient for each input variable analytically. Thus, this sensitivity analysis approach cannot be extended for general MCDA methods.
In this study, an iterative binary search algorithm is developed to investigate the sensitivity of alternatives’ ranking to the variations of weighting factors or criteria values. The iterative binary search algorithm can overcome these drawbacks mentioned above, since it is a sampling-based method which will not be affected by the analytical calculation steps of MCDA methods.
Additionally, it can be generalized to other MCDA methods.
4.4.1 Iterative Binary Search Algorithm
The binary search technique has been widely used to find a target value in a sorted (usually ascending) sequence efficiently [131], [82]. This technique compares the middle element of the sorted sequence to the target value, if the middle element is equal to the target value, then the search terminates. If the target value is less than middle element, then the algorithm eliminates the right half of the sorted sequence and conducts the same search for the left side. If the target value is bigger than the middle element, then the algorithm ignores the left half of the sorted sequence and performs the same search for the right side. Otherwise, we can conclude that the target value is not in the sorted sequence.
For example, given a sorted sequence [0 5 12 17 23 25 50 60 80], assume that we want to find the target value 25. The binary search technique works as follows.
• First iteration: [0 5 12 172325 50 60 80]. The target value 25 is bigger than the middle element 23, ignore the left half of the sorted sequence, and perform the same search for the right side.
• Second iteration: [255060 80]. The target value 25 is smaller than the middle element 50, ignore the right side of the sorted sequence, and perform the same search for the left side.
• Third iteration: [25]. The target value 25 equals the element 25, the target value is found.
When using the MCDA methods to solve a given problem, input parameters are decision criteria, weighting factors, the original ranking of the alternatives, and the number of iterations.
The outputs of the iterative binary search algorithm are the minimum changes in decision criteria and weighting factors to alter the rankings of two alternatives. The iterative binary search algorithm varies one input variable at a time in order to find the minimum change in this input variable, which can alter the ranking of two alternatives.
The initialization of the iterative binary search algorithm is illustrated in Figure 4.4. The first step is to initialize input parameters: left lower boundll bound, left upper boundlu bound, right lower bound rl bound, and right upper bound ru bound. In the next step, the left trial
Figure 4.4: Initialization for the Iterative Binary Search Algorithm
valuel trialis calculated by the middle element in the left search space(ll bound+lu bound)/2, and the right trial value r trial is calculated by the middle element in the right search space (rl bound+ru bound)/2.
The flow chart of the iterative binary search algorithm is shown in Figure 4.5, wherelstands for left and uupper, llstands for left lower, lu left upper,rl right lower, andru right upper.
deltais the minimum change in weights or decision criteria when two rankings are altered. The default setting is that it is non-feasible to change the current parameter to alter the ranking of two alternatives. The number of iterationruns determines the precision of the calculation [82].
For instance, when the iteration runs is set as runs = 30, the precision of the calculation is log(2runs) =log(230)≈9.
The new trial values of the parameter under consideration are calculated and new rankings of alternatives are computed. The rankings in the left search space will be evaluated first.
If the rankings using left new trial value change, then we will assign true to the judgment variable isFeasible, and calculate the relative quantity of the parameter under consideration delta decrement, and the left new trial value l trial is assigned to the left lower bound ll bound. If the ranking using left new trial value does not change, then, the left new trial valuel trialis given to the left upper boundlu bound. After the evaluation of the left search space, the similar procedure is performed to the right search space. The algorithm is terminated when the number of iteration is finished. Finally, if the judgment variable isFeasibleistrue, the absolute magnitude of the relative quantities delta decrement and delta increment is compared. The smaller quantity delta is the minimum change which can alter the ranking of two alternatives. Otherwise, we can conclude that it is not feasible to change the current parameter so that the ranking of two alternatives is altered.
4.4.2 Interactive Sensitivity Analysis for Weighting Factors
It is observed that weighting factors are often highly subjective considering that they are elicited based on DM’s experience or intuition. The inherent uncertainties and subjectivities of weighting factors have significant impacts on the final result of a decision making problem. In this study,
4.4 Local Sensitivity Analysis via Iterative Binary Search Algorithm
Figure 4.5: Flow Chart of the Iterative Binary Search Algorithm
an interactive sensitivity analysis for weighting factors is developed. The basic idea is to vary the weighting factor of one criterion from 0 to 100%, while keeping the weighting factors of other criteria the same proportion as in the original setting.
Local Sensitivity Analysis for an Aircraft Selection Example
Local sensitivity analysis for an aircraft selection example, as described in Subsection 2.3.4, is conducted in this subsection. The decision matrix is shown in Table 4.4.
When SAW is used to solve the aircraft selection example, the ranking of the three
alterna-Table 4.4: Decision Matrix of an Aircraft Selection Example for Local Sensitivity Analysis Criteria
C1: Comfort C2: Cost C3: Environmental friendliness Alternatives w1: 0.3 w2: 0.4 w3: 0.3
Aircraft A 8 7 10
Aircraft B 9 6 5
Aircraft C 6 7 8
tives is [Aircraft A Aircraft B Aircraft C]. The developed iterative binary search algorithm can answer the question: What is the smallest change in the weighting factors so that the ranking of the most preferred alternative or any alternative will be altered?
The absolute minimum changes in the weighting factors which can alter the ranking of the alternatives are summarized in Table 4.5. For the convenience of comparison, the relative minimum changes are also presented in Table 4.6. The relative minimum changes are the absolute minimum changes scaled against the original values of the weighting factors. In these two tables, N/F (Non-Feasible) means that it is not mathematically feasible to alter the ranking of the alternatives through the change of the current parameter.
The first two rows in Table 4.6 show that when the weighting factor ofC3decreases−39.69%, Aircraft B becomes the most preferred alternative, and it is not possible to change the weighting factors so that Aircraft C ranks first. Moreover, it can be seen from the whole table that the weighting factor ofC3 is most sensitive to the ranking of the three alternatives.
Furthermore, following the proposed idea of varying the weighting factor of one criterion Table 4.5: Absolute Minimum Changes in Weighting Factors to Alter the Rankings of Alternatives in an Aircraft Selection Example
Pairs of rankings C1 C2 C3 A1:A2 0.54 0.42 -0.12 A1:A3 N/F N/F N/F A2:A3 -0.21 N/F 0.23
Table 4.6: Relative Minimum Changes in Weighting Factors to Alter the Rankings of Alternatives in an Aircraft Selection Example
Pairs of rankings C1 C2 C3
A1:A2 178.58% 104.17% -39.69%
A1:A3 N/F N/F N/F
A2:A3 -67.15% N/F 74.61%
4.5 Global Sensitivity Analysis using Partial Rank Correlation Coefficients
Figure 4.6: Interactive Sensitivity Analysis for the Weighting Factor ofC1 in an Aircraft Selection Example
from 0 to 100%, while keeping the weighting factors of other criteria the same proportion as in the original setting, the interactive sensitivity analysis for the weighting factor ofC1is illustrated as an example in Figure 4.6, where an intersection of two lines indicates that there is a ranking change between two alternatives.