1 Introduction
2 Literature review
2.1 Overview of ranking and selection
2.1.1 Performance measures
Objectives | ||||
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PCS | EOC | PGS | ||
Indifference zone | Frequentist | – |
Chick and Wu (2005) | |
Bayesian |
Frazier (2014) | – | – | |
OCBA | Frequentist |
Chen and Lee (2010) | – | – |
Bayesian |
Chen and Lee (2010) |
He et al. (2007) |
Branke et al. (2005) | |
EVI | Bayesian |
Chick and Inoue (2001) |
Chick and Inoue (2001) | – |
Small EVI | Bayesian |
Chick et al. (2010) | - | |
Racing | Bayesian |
Birattari et al. (2010) | – | – |
2.1.2 Major R&S methods
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The indifference-zone methods such as KN\(^{++}\) (Kim and Nelson 2006) which aim at identifying an alternative that is not worse by more than \(\delta \) compared to the true best. KN\(^{++}\) maintains a set of possibly best solutions and drops solutions from this set when it detects clear evidence that an alternative is unlikely to be best. The procedure iterates until only one solution remains.
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The expected value of information (EVI) procedure (Chick and Inoue 2001) which maximises the expected value of information in the next samples.
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The racing method such as F-race that is based on the nonparametric Friedman’s two-way analysis of variance by ranks (Birattari et al. 2010). Similar to KN\(^{++}\), racing methods drop alternatives from sampling that are unlikely to be the best based on the observations so far, until only one alternative remains. However, racing methods have no performance guarantee.
2.2 Overview of multi-objective ranking and selection
2.2.1 MORS performance measures
2.2.2 MORS methods
3 Assumptions and problem formulation
4 M-MOBA procedure
4.1 M-MOBA PCS procedure
4.2 M-MOBA indifference-zone procedure
4.2.1 New definition of indifference zone and good selection
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indifference-zone dominated if there is another solution that is at least \(\delta _h\) better in each objective h,
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borderline dominated, if it would become non-dominated by improving each objective h by \(\delta _{h}\),
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borderline non-dominated, if it is non-dominated, but would become dominated by worsening each objective h by \(\delta _{h}\),
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indifference-zone non-dominated if it remains non-dominated even if each objective h is worsened by \(\delta _h\).
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solution i indifference zone dominates solution j, denoted by \(i \prec _{IZ} j\), if \(\mu _{i,h} < \mu _{j,h}-\delta _h, \forall h= 1, 2, \ldots , H\),
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solution i borderline dominates solution j, denoted by \(i \precsim _{IZ} j\), if \(\mu _{i,h} < \mu _{j,h}\), \(\forall h= 1, 2, \ldots , H\) and \(\exists h \in \{ 1, 2, \ldots , H\}\), \(|\delta _{ijh}| \leqslant \delta _h\).
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indifference-zone dominated if \(\exists i \in \{ 1, 2, \ldots , m\}\), \(i \prec _{IZ} j\),
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borderline dominated if \(\not \exists i \in \{ 1, 2, \ldots , m\}\), \(i \prec _{IZ} j\) and \(\exists i \in \{ 1, 2, \ldots , m\}\), \(i \precsim _{IZ} j\),
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borderline non-dominated if \(\not \exists i \in \{ 1, 2, \ldots , m\}\), \(i \prec _{IZ} j\), \(\not \exists i \in \{ 1, 2, \ldots , n\}\), \(i \precsim _{IZ} j\) and \(\exists i \in \{ 1, 2, \ldots , m\}, h \in \{1, 2, \ldots , H\} \mu _{i,h} > \mu _{j,h} - \delta _h\),
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indifference-zone non-dominated if \(\not \exists i \in \{ 1, 2, \ldots , m\}\), \(i \prec _{IZ} j\) or \(i \precsim _{IZ} j\) and \( \not \exists i \in \{ 1, 2, \ldots , m\}, \not \exists h \in \{1, 2, \ldots , H\} \mu _{i,h} > \mu _{j,h} - \delta _h\).
Observed | True | |||
---|---|---|---|---|
Indifference-zone dominated | Borderline dominated | Borderline non-dominated | Indifference-zone non-dominated | |
Indifference-zone dominated | ✓ | ✓ | ✗ | ✗ |
Borderline dominated | ✓ | ✓ | ✓ | ✗ |
Borderline non-dominated | ✗ | ✓ | ✓ | ✓ |
Indifference-zone non-dominated | ✗ | ✗ | ✓ | ✓ |
4.2.2 M-MOBA IZ procedure
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For a solution that is borderline dominated, if all other solutions on the observed Pareto front are indifference-zone non-dominated, an example for the area that \(a_c\) needs to move out is shown in Fig. 8, i.e. the original area plus the striped area that allows \(a_c\) to become borderline non-dominated.
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For a solution that is borderline dominated, if a solution on the observed Pareto front is borderline non-dominated, the area that \(a_c\) needs to leave is the area discussed above plus the small rectangle around the borderline non-dominated solution. For example, if solution \(a_2\) shown in Fig. 9 is borderline non-dominated (with respect to \(a_c\)), the area with indifference zone for \(a_c\) is the shaded part.
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For an indifference-zone non-dominated solution, if a solution on the observed Pareto front is borderline non-dominated, the area that \(a_c\) needs to move out is the area in Fig. 3 plus the stripe areas around the borderline non-dominated solution. Furthermore, if two borderline non-dominated solutions are neighbours on the Pareto front, the small square area between the two stripe areas also needs to be added. For example, in Fig. 10, \(a_1\) and \(a_2\) are both borderline non-dominated (due to \(a_4\) and \(a_5\), respectively), the area \(a_c\) that needs to leave in order to bring a change is the shaded part shown in Fig. 10.
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For a borderline non-dominated solution, if a solution on the observed Pareto front is borderline non-dominated, the shaded area that \(a_c\) needs to leave is the area discussed in Fig. 10 plus the stripe area on the upper right side. For example, similar to the situation in Fig. 10 where \(a_1\) and \(a_2\) are both borderline non-dominated, the area that \(a_c\) needs to leave in order to bring a change is the shaded part shown in Fig. 12.
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If the new Pareto-optimal solutions after the solution under consideration is removed are all indifference-zone non-dominated, we only need to check solutions that define the shaded area shown in Fig. 2. If some solutions that define the left and down sides of the shaded area are borderline non-dominated, the shaded area can be extended accordingly. For example, in Fig. 13, since \(a_2\) is borderline non-dominated, the area that \(a_c\) needs to leave is as the figure shows.
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If the new Pareto-optimal solution after the solution under consideration is removed is borderline non-dominated, the situation is so complex that we have not found a good method to summarise. For this situation, we use a brute-force method that divides the whole plane into different cells based on each solution’s objective values and the indifference zone in each objective accordingly, and checks for each cell whether it would change the current Pareto front in case the currently considered solution were to fall into this cell. For example, if we have four solutions in total as in Fig. 14, the number of cells that need to be considered is \((4*3)^2=144\). Please note that for the sake of clear demonstration, the domination relationship in this figure does not exactly conform to the situation that new Pareto-optimal solution after the solution under consideration is removed is borderline non-dominated.
4.3 M-MOBA hypervolume procedure
4.3.1 Mathematical calculation of the expected HV change
4.4 M-MOBA procedure for differential sampling between the objectives
5 Empirical results and analysis
5.1 M-MOBA PCS procedure
Index | Obj. 1 | Obj. 2 |
---|---|---|
0 | 1 | 2 |
1 | 3 | 1 |
2 | 5 | 5 |
Index | Obj. 1 | Obj. 2 | Index | Obj. 1 | Obj. 2 |
---|---|---|---|---|---|
1 | 0.5 | 5.5 | 9 | 4.8 | 5.5 |
2 | 1.9 | 4.2 | 10 | 5.2 | 5 |
3 | 2.8 | 3.3 | 11 | 5.9 | 4.1 |
4 | 3 | 3 | 12 | 6.3 | 3.8 |
5 | 3.9 | 2.1 | 13 | 6.7 | 7.2 |
6 | 4.3 | 1.8 | 14 | 7 | 7 |
7 | 4.6 | 1.5 | 15 | 7.9 | 6.1 |
8 | 3.8 | 6.3 | 16 | 9 | 9 |
Index | Obj. 1 | Obj. 2 | Index | Obj. 1 | Obj. 2 | Index | Obj. 1 | Obj. 2 |
---|---|---|---|---|---|---|---|---|
1 | 1 | 8 | 6 | 3 | 7 | 11 | 2.6 | 3.9 |
2 | 2 | 5 | 7 | 3.05 | 2.2 | 12 | 2 | 7 |
3 | 3.5 | 5.01 | 8 | 1.5 | 6 | 13 | 2.5 | 6 |
4 | 3 | 2 | 9 | 2.1 | 5.2 | |||
5 | 2.5 | 8 | 10 | 2.5 | 4 |
5.2 M-MOBA IZ procedure
5.3 M-MOBA HV procedure
Index | Obj. 1 | Obj. 2 | Index | Obj. 1 | Obj. 2 |
---|---|---|---|---|---|
1 | 1 | 5 | 6 | 4 | 2.1 |
2 | 5 | 1 | 7 | 2.1 | 4 |
3 | 3 | 3 | 8 | 5.5 | 5 |
4 | 3.1 | 2 | 9 | 3.5 | 5 |
5 | 2 | 3.1 | 10 | 6 | 6 |
Index | Obj. 1 | Obj. 2 |
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1 | 1 | 5 |
2 | 5 | 1 |
3 | 3.2 | 2.1 |
4 | 3 | 2 |
5 | 2 | 3.1 |
6 | 6 | 4 |
7 | 5 | 5 |
8 | 4 | 6 |
Number of alternatives | Best | Worst | Average |
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10 | 0.079110s | 0.093693s | 0.083943s |
20 | 0.099771s | 0.11613s | 0.106111s |
50 | 0.239546s | 0.399704s | 0.259820s |
100 | 0.497097s | 0.585236s | 0.515388s |
5.4 M-MOBA DS PCS procedure
Purpose | Method |
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Minimise PCS | M-MOBA PCS |
Minimise PCS but ignore small differences | M-MOBA IZ |
Any performance measure, but if objective functions are derived from different simulation models independently | M-MOBA DS |
Minimise hypervolume difference | M-MOBA HV |