a simulation budget allocation procedure for enhancing the ... · how to efficiently allocate...

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?

6

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)

7

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).

8

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…

9

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

15

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?

16

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

20

Problem Formulation for optimal subset selection

*T T

*T

OCBA for optimal subset selection

21

Noise to signal ratio Square root rule

3

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.

17/28

• 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

21

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


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


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


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.


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!

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