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A Simulation Budget Allocation Procedure for Enhancing the Efficiency of Optimal Subset Selection
Si Zhang
Dept. of Management Science & Engineering
School of Management
Shanghai University
2. Literature Review
1. Introduction
3. Optimal Computing Budget Allocation for optimal subset
selection
4. Convergence Rate Analysis
5. Conclusions and future research
Outline
2
Manufacturing industry Financial investment Service industry
Simulation Optimization
Simulation & Optimization
electronic circuit design Portfolio selection Spare parts inventory planning for airlines
Difficulties:
3
➢Optimization problems for complex system
minX
f X
Feasible region
• How to evaluate? simulation
• How to find the best? optimization
Challenges
1
1( ) [ ( , )] ( , )
N
iX
i
Min f X E f X W f X WN
Many Alternatives in Design Space
Multiple Simulation Runs (replications)
OCBA: Optimal Computing Budget Allocation
Goal: maximize the overall efficiency
1. Introduction1.2 Computing cost for simulation optimization
SimulatorOptimization
Engine
The performance of solutions
New solutions to evaluate
• n∞, 100% correct 100% CorrectTime consuming
Good enough
• n is finite, for each solution, make sure the correctness of selection maximal or at a high level (e.g. 95%)
5
minX
f X
1
1 ˆ( ) ( , )n
iX
i
Best f X f Xn
Noise
1. Introduction
How to efficiently allocate simulation replications budget?
1.3 How to run simulation efficiently?
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x1 x2 x3 x4 x5 x1 x2 x3 x4 x5 x1 x2 x3 x4 x5
Equal simulation Intelligent wayAllocate 100 replications to this 5 solutions. How? Each 20 replications More important, more
Optimal Computing Budget Allocation (OCBA)
90% Confidence Interval
2. Literature Review2.1 Literature review for Ranking and Selection (R&S)
Max P{CS}
Subject to Computing Budget
Get Computing replications
Subject to P{CS}>=P*
➢ Two-stage procedureDudewicz and Dalal (1975),
Rinott (1978)
➢ Indifference-zone procedure (IZ)Kim and Nelson (2001),
Nelson et al. (2001)
➢ Optimal Computing Budget Allocation (OCBA)Chen et al. (1996), Chen et al. (1997),
Chen et al. (2000), …
Ranking and Selection
determine the number of simulation replications in selecting the best solution(s) from a finite number of alternative solutions.
Bechhofer et al. (1995), Swisher et al. (2003), Kim and Nelson (2006, 2007)
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2. Literature Review2.2 Literature review for OCBA
➢ Extension work• Constraints: Pujowidianto et al. (2009)
• Multiple objectives: Chen and Lee (2009); Lee et al. (2010)
• Optimal subset selection: Chen et al. (2008)
• Correlation among solutions: Fu et al. (2004, 2007)
• Performance not normally distributed: Glynn and Juneja (2004)
➢ Basic framework for OCBAChen et al. (1996), Chen et al. (1997), Chen et al. (2000)
➢ Application of OCBA • Problems given a fixed set of alternatives
semiconductor wafer fab scheduling (Hsieh et al., 2001; Hsieh et al. 2007)• Problems with enormous size or continuous solution space
Nested Partition (Shi et al. 1999); Cross Entropy (He et al. 2010).
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1. Introduction of the Optimal Computing Budget Allocation
C. H. Chen and L. H. Lee, (2010). Stochastic Simulation Optimization: An Optimal Computing Budget Allocation. World Scientific Publishing.
2. Optimal Computing Budget Allocation for Optimal Subset Selection
S. Zhang, J. Xu, L.H. Lee, E.P. Chew and C.H. Chen (2016). A Simulation BudgetAllocation Procedure for Enhancing the Efficiency of Optimal Subset Selection, IEEETransactions on Automatic Control,61(1):62 ~ 75
In This Talk…
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Confidence Interval (C.I.)
f(X, wj) z
– z is the critical value for the standard normal distribution
NjN 1
1
N
Increase the number of
simulation runs (N)
N
99% Confidence Interval
Precision of Stochastic Simulation Estimator
Probability of Correct Selection: P{CS}
x1 x2 x3 x4 x5
99% Confidence Intervals for f(X)
x1 x2 x3 x4 x5
As N increases
• Simulation precision enhances
• Confidence intervals become narrower
• P{CS} increases
Increase N
Smarter Simulation Allocation
x1 x2 x3 x4 x5
99% Confidence
Intervals for J(X)
after some simulations
• Which designs should we simulate more?
– 2 & 3 are clearly superior
– 1, 4 & 5 have larger variances
• When is the optimal screening point?
Optimal Computing Budget Allocation (OCBA)
(P1) Minimize the total number of simulation runs in order to achieve a desired simulation quality:
[ N1 + N2 + .. + Nk ]
s.t. P{CS} > Psat (a satisfactory level)kNN ,..,1
min
(P2) Maximize the simulation quality with a given simulation budget:
P{CS}
s.t. N1 + N2 + .. + Nk = T (total comp. budget)kNN ,..,1
max
Selecting the best
1. Introduction of the Optimal Computing Budget Allocation
C. H. Chen and L. H. Lee, (2010). Stochastic Simulation Optimization: An Optimal Computing Budget Allocation. World Scientific Publishing.
2. Optimal Computing Budget Allocation for optimal subset selection
S. Zhang, J. Xu, L.H. Lee, E.P. Chew and C.H. Chen (2016). A Simulation BudgetAllocation Procedure for Enhancing the Efficiency of Optimal Subset Selection, IEEETransactions on Automatic Control,61(1):62 ~ 75
In This Talk…
14
Problem Motivation
How about qualitative criteria and political feasibility ?
Alternatives
Simulator
not
enough!A, B, C, D, E, …
Best
C
a more flexible and
people oriented way
Rough Experiment
D1, D2, D3,…, D100
Good designs:D4, D20, D46, D64, D82
Accurate Experiment
Best design: D64
screen procedure
Get certain solution(s)
Evaluate the performance of these solution(s)
Select an elite subset and determine the search direction
Move to the new generated solution(s)
population based
searching algorithms
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Problem Formulation for optimal subset selection
k alternatives
Total computing budget: T
i iN T
2,i iN
Replications allocated 1 2, , ,
1
max
. .
kN N N
k
i
i
P CS
s t N T
How to formulate it?
How to get the optimal allocation rule?
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Simulation
1, 2, …, k Top-m
7/28
▪P{CS} does not have close form expression▪Use lower bounds to approximate its true value, which are called the Approximated Probability of Correct Selection (APCS).
Expression of P{CS}
, for 1,2, , and 1, 2, ,i j
i j
P CS P X X i m j m m k
1X 2X 2mX mX kX1mX
The ordering of means: The optimal subset:
1,2, ,m1 2 ... k
9/28
P CS
1iX
2iX
1miX
2miX
miX
3miX
kiX
1kiX
1miX
2miX
1iX
2iX
1miX
2miX
miX
3miX
kiX
1kiX
1miX
2miX
Use as a thresholdmi
X
1
1
1 1
1p m m q
m k
i i i i
p q m
APCSm P X X P X X
One lower bound of
Boundaries
Expression of P{CS}
10/28
P CS
Use as a threshold1mi
X
Another lower bound of
1iX
2iX
1miX
2miX
miX
3miX
kiX
1kiX
1miX
2miX
1 12
1 2
1p m m q
m k
i i i i
p q m
APCSm P X X P X X
1 2{ } max ,P CS APCSm APCSm APCSm
Lemma 1.
Expression of P{CS}
Lemma 2. There exists a large enough such that both sub-problems 1 and 2 are convex with respect to the vector >0when .
Sub-problem 1
Sub-problem 2
1 2
1
1, , ,
1 1
1
max 1
. . 1
0, for 1,2, , .
k
m k
i m m j
i j m
k
i
i
i
APCSm P X X P X X
s t
i k
1 2
2 1 1, , ,
1 2
1
max 1
. . 1
0, for 1,2, , .
k
m k
i m m j
i j m
k
i
i
i
APCSm P X X P X X
s t
i k
1 2, , ,
1
max
. .
0
kN N N
k
i
i
i
APCSm
s t N T
N
➢ OCBA model
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Problem Formulation for optimal subset selection
*T T
*T
OCBA for optimal subset selection
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Noise to signal ratio Square root rule
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OCBA for optimal subset selection
Theorem 3.1. The asymptotically optimal allocation rule, named as OCBAm+,
to maximize APCSm+ is
*1 *1 *2
1 2* * * *
1 2*2 *1 *2
1 2
if , , ,
if k
APCSm APCSm
APCSm APCSm
Corollary 1. If m equals to one, the allocation rule OCBAm+, will be the
OCBA1 rule expressed as follows.
2
i i i i
j j j j
N
N
, 1i j for
2
1 1 22
ki
i i
NN
,
1i i
22
1
16/28
Asymptotic Convergence Rate Analysis
• How does an allocation rule perform?• How do we compare the performance of different allocation
rules?
Numerical experiments Theoretical framework?❖When T goes to infinity, P{CS} convergences to 1 and P{IS} convergences to 0.
❖Given an allocation rule, we can calculate the convergence rate of P{CS} (or
P{IS}) as a measure to characterize the performance of this allocation rule.
❖The allocation rule with higher convergence rate is better than rules with
lower convergence rates.
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• Suppose for k designs, we have
• If we have
1 2 1 1m m k k
1 1,2, ,S m 2 1, 2, ,S m m k
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max ( ) min ( )i i j jj Si S
P IS P X T X T
1 2 1 2
1 2, ,
max ( ) ( ) max ( ) ( )i i j j i i j ji S j S i S j S
P X T X T P IS S S P X T X T
1 2
1lim log ( ) ( ) , , ,i i j j ij i jT
P X T X T G i S j ST
• Then
1 2,
1lim log min ,ij i jT i S j S
P IS GT
Convergence rate
• A lower bound and an upper bound of P IS
Asymptotic Convergence Rate Analysis
19/28
Dembo, A. and Zeitouni, O. (1992); Peter Glynn and Sandeep Juneja (2004)
• The convergence rate for equal allocation rule (EA) under
equal variance case
• Large deviation theory
2
2 2,
2
i j
ij i j
i i j j
G
1i k
2 2 2 2
1 2 k
2
2,
4
i j
ij i jGk
1 2
2
1
11 2,min , ,
4
m m
ij i j m mm mi S j S
G Gk
Asymptotic Convergence Rate Analysis
EA rule:
20/28
Lemma 5(a) The asymptotic convergence rate obtained byOCBAm in equal variance case
Lemma 5(b) The asymptotic convergence rate obtained by OCBAm+ in equal variance case
2 2
1 1
11 1 1 2 2
1 1
min , , , min ,2 1 1 2 1 1
m k m
mk m k m m
m k m
G G
2
1
11 2
1
,2 1 1
m mL L
m mm m L L
m m
G
Asymptotic Convergence Rate Analysis
21/28
If the means of all designs form an arithmeticalprogression, and the variances of all populations areequal, that is and
Theorem 2. The asymptotic convergence rates gained byOCBAm+ and OCBAm are greater than the rate gainedby equal allocation rule.
Theorem 3. The asymptotic convergence rate gained byOCBAm+ is always no less than OCBAm when m=1 or
1 , for 1,2, , 1i i d i k 2 2 2 2
1 2 k
Conclusions: 1.OCBAm+ and OCBAm are better than the equal allocation rule.2.OCBAm+ is better than OCBAm in most situations.
Asymptotic Convergence Rate Analysis
2 and 5m k m
Numerical Experiment for OCBAm+➢ Base Experiment: k=50; m=5; Solution i ~ N(i , 102)
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0.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
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P{C
S}
T
EA
OCBAm+
Rule EA OCBAm+
Convergence Rate 0.50×10-4 6.59×10-4<
88003750
Conclusions
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We derive OCBAm+, an allocation rule for the optimal subset selection problem
1. When m=1, OCBAm+ goes to OCBA1;2. OCBAm+ performs much better than EA.
We propose a framework of asymptotic convergence rate analysis on the probability of correct selection given allocation rules for optimal subset selection problems.
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Thank you for your kind attention!
Any questions or suggestions are welcomed!