OCBA_UGM_121214

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OPTIMAL COMPUTING

BUDGET ALLOCATIONFOR CONSTRAINED

OPTIMIZATION

Department of Industrial and Systems

Engineering

Nugroho Artadi PUJOWIDIANTO

Supervisors:Dr. YAP Chee Meng and A/Prof. LEE Loo

Hay

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Outline2

1. Introduction

2. Literature Review

3. Asymptotic Simulation Budget Allocation

4. Explicit Consideration of Correlation between

Performance Measures in Simulation budget

Allocation

5. Bed Allocation Problem

6. Conclusion and Future Research

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Outline3

Our goals:

Enhancesimulation

efficiency whenthere arestochasticconstraints

Explicitly consider the correlationbetween theperformancemeasures

Develop easy-to-

Introduction

Literature Review

Computing Budget

Allocation Problem

1st

procedure 2nd

procedure

Bed Allocation Problem

Conclusions and FutureResearch

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Problem Context4

Constrained simulation optimization

One main objective

E.g. : minimize idle time / maximize utilization

Constrained measures

E.g. : waiting time should be less than certain limit

Both estimated via stochastic simulation

Simulation A design

(a set of

decision

variables)

Parameters of

uncertaintiesObjective function

value for the

design Constraint Measure(s)

value(s) for the design

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Motivation5

Simulation budget needs to be efficiently allocated

Computing

budget isoftenlimited

Eachsimulation

is time-consuming

Multiple

simulationreplicationsare needed

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How to run simulation

efficiently?

x1 x2 x3 x4 x5

95% Confidence Intervals

Intelligent

Equal Simulation

with the same number of total runs

x1 x2 x3 x4 x5

Option 3 is

better isolated

x1 x2 x3 x4 x5

6

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Literature Review7

Ranking and Selection

(R&S)Bechhofer et al. (1995), Swisher et al. (2003), Kim and Nelson (2006, 2007)

Unconstrained

Optimization

Feasibility

Determinati

on

Constrained

Optimization

Multi-

Objective

Optimization

Guaranteein

g the

desired level

of

confidence

Kim and Nelson

(2001), Nelson et

al. (2001)

Batur and

Kim (2010)

Andradóttir and

Kim (2010),

Healey et al.

(2010), Morrice

and Butler (2006)

Butler et al.

(2001)

Maximizing

Simulation

Efficiency

Chen et al.

(2000), Chick and

Inoue (2001),

Glynn and Juneja2004

Szechtman

and Yücesan

(2008)

? Lee et al.

(2010),

Teng et al.(2010)

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Computing Budget Allocation

Problem

Objective:

maximize the probability of correct selection

(P{CS})

Decision variable: The proportion of simulation budget allocated to

each design () max,…, * subject to 1

=

, ≥ 0

8

arg min=,,…, ℎ

subject to for all

1 2

Context: Selection of the best feasible

design ℎ

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9

=

=∩ >

=

Design 1 (the

best) is

estimated

feasible

Design is

estimated

feasible

Design appears to be

better than the

best design 1

≡1

=1 ≡

1

=1 where

The best feasible

design appears to

be feasible

The non-best design

appears to be better

than the best in

terms of main

objective

The non-best design

appears to be

feasible

The non-best design

appears to be

feasible

Correct

Selection (CS)

The non-best design

appears to be better

than the best in

terms of main

objective

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Proposed procedures

Maximize lower

bound of P{CS} 2 main categories of

non-best designs

Use only the most

critical performancemeasure

Allocation isindependent of the

correlation

Maximize decay rate of

P{False Selection}=1-P{CS}

3 main categories of

non-best designs

May consider more than

one performance

measure

Allocation depends on

10

1st Procedure 2nd Procedure

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Approximate term of P{CS}

( APCS)

P{CS} can be lower bounded by APCS as

follows

The revised computing budget allocation

problem becomes

≥

=

− min min , > ≠ 1 −

max,…, subject to 1= , ≥ 0

11

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Category of Non-best Designs

Simplifying the term

2 collectively exhaustive and mutually exclusive

sets

1,

≥

> ,

1, < > ,

min min , > ≠

where argmin .

hi

gi1

Constraint limit

Objective Function,

feasible The best

design (b)

hbest

Constraint

measure, SO

SF

minimization

12

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proce ure o or Constrained Optimization (OCBA-

CO) The computing budget allocation problem

becomes

Based on the KKT conditions, the optimal

simulation budget allocation rules for the bestfeasible design: max ,

∈,

∀ 1 where

13

max,…,

= − ∈− > ∈

1 − subject to 1= , ≥ 0.

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proce ure o or Constrained Optimization (OCBA-

CO) Based on the KKT conditions, the optimal

simulation budget allocation rules for non-best

designs:

∀ 1,

+

,if

∈

,

or if ∈

14

hi

gi1

Constraint limit

Objective Function,

feasible The best

design (b)

hbest

Constraint

measure, SO

SF

minimization

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Closed-form solution

For implementation, the rules can be simplified

by assuming that

In this case,

In addition,

≈ , ∀ ∈

≈ 1, , ,and ≈ 1, > , .

≫ ∈

15

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OCBA-CO Sequential Algorithm

Step 0 – Initialize and perform initial number of runs for each design:

Step 1 – Determine the best feasible design based on the available samples

Step 2 – If , stop

Step 3 – Increase the total computing budget by the increment and computethe number of simulation replications for each design based on the OCBA-COrule

Step 4 – Scale down the budget allocated for each alternative so that the totaladditional replications of is equal to the increment

Step 5 - Determine the best feasible alternative based on the available samplesup to the current stage and go to step 2

T N k

i

l

i

1

0 ⋯

+ + 1 Δ

max(0,+ − )

16

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Numerical Example17

Structure of the Problem Scenario

feasible

minimization

Constraint limit

1

2

3

4

5

6

7

8

9

10

Constraint

measure, J 1

Objective

Function, J 0

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Numerical Results18

0 2,000 4,000 6,000 8,000 10,000 12,000 14,00050%

60%

70%

80%

90%

95%

100%

Total Computing Budget, T

P r o b a b i l i t

y o f C o r r e c t S e l e c t i o n , P

C S

EA (k=10)

OCBA-CO (k=10)

EA (k=100)

OCBA-CO (k=100)

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Large Deviations Theory19

The tail probability approaches zero

exponentially fast

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Probability of False Selection

(*) 20

False selection (a suboptimal solution is

selected as the best), the opposite of correct

selection, happens when

The best feasible design appears to be infeasible A non-best design appears to be feasible and

better than the best design in terms of the

objective function

*+ = 1 > =1 ∪ ≤ =1 ∩1 > ≠1

Design 1 is

estimated

infeasible

Design is

estimated

feasible

Design appears to be

better thanDesign 1

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Rate of Decay of * 21

The rate function of * approaching to zero

Based on Large Deviations Theory (see Dembo and

Zeitouni, 1998)

− lim→∞1 log*+

=

−lim

→∞

1

log max

max1

≤≤ 1

>

, max

≠1

≤

=1 ∩1 >

− lim

→∞

1

log*+

min min1≤≤ 11(1 = ) ,min≠1

inf ≤1 ,≤ 11(1)+ ,∈{,…,} =

The rate function for the

probability that system 1

is classified infeasibledue to constraint

The rate function for the probability that

system

appears to be feasible and better

than system 1 in terms of the objectivefunction

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Simulation Budget Allocation

Problem22

What vector proportions of maximizes the

rate of decay of *? Problem Q

max1 ,…, min min1≤≤ 11(1 = ) ,min≠1

inf ≤1 ,≤ 11(1)+ ,∈{,…,}

= 1, ≥ 0 subject

to

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Properties of the Optimal Allocation23

Based on Karush-Kuhn-Tucker (KKT)

conditions for the simulation budget allocation

problem Q,

Based on KKT conditions for the simulationbudget allocation problem ,

min∈{1,…,}1

∗1(1 = ) ≥ (1∗,∗) = 1

∗, ∗ ∀ , ≠ 1

(1∗,∗) 1(1∗,∗) = 1≠1

(, ) (, ) ∗ ∗, ∗

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Classification of constraints and

designs24

Let 0and 0, be the Lagrange

multipliers for problem . The constraint measures can be divided into

two:

∗ *: < 0 and ∗ ;When the performance

measures are independent:

violated constraints

∗ : 0 and ∗ . feasible constraints

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Classification of constraints and

designs25

The non-best designs can be divided into

three main groups

∗ : < 0, ∗ ∗ and ∗ nonempty, 1 .

Γ∗ : < 0, ∗ ∗ and ∗ empty, 1 ;

∗

: 0, ∗

∗

and ∗

nonempty, 1 ;

When the performancemeasures areindependent:feasible designs

Infeasible designs

that are better than

design 1 in main

objective functionvalue

Infeasible designs

that are worse than

design 1 in main

objective functionvalue

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Correlation between Performance

Measures26

Positive

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Correlation between Performance

Measures27

Negative

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Illustration of the three sets

The area of each set changes as the

correlation changes

28

µ0

µ1

Constraint limit

Objective Function,

minimization

feasible b

µ0b

Constraint

measure, S0

S1

S2

+ 1

µ0

µ1

Constraint limit

Objective Function,

minimization

feasible b

µ0b

Constraint

measure, S0

S1

S2

+ 0.9

µ0

µ1

Constraint limit

Objective Function,

minimization

feasible b

µ0b

Constraint

measure, S0

S1

S2

+ 0.6

µ0

µ1

Constraint limit

Objective Function,

minimization

feasible b

µ0b

Constraint

measure, S0

S1

S2

+ 0.3

µ0

µ1

Constraint limit

Objective Function,

minimization

feasible b

µ0b

Constraint

measure, S0

S1

S2

0

µ0

µ1

Constraint limit

Objective Function,

minimization

feasible b

µ0b

Constraint

measure, S0

S1

S2

– 0.3

µ0

µ1

Constraint limit

Objective Function,

minimization

feasible b

µ0b

Constraint

measure, S0

S1

S2

– 0.6

µ0

µ1

Constraint limit

Objective Function,

minimization

feasible b

µ0b

Constraint

measure, S0

S1

S2

– 0.9

µ0

µ1

Constraint limit

Objective Function,

minimization

feasible b

µ0b

Constraint

measure, S0

S1

S2

– 1

Note:

• The variance is

equal

• No feasible designs

have the same main

objective value as

the best (b)

Γ

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Key Assertions29

∗ ∗,∗ ∗ ∗, ∗ for all , ∈ Γ∗ ∪ ∗

Based on the 1st KKT condition of problem Q,

the sumands in the 2nd condition are within a

positive finite constant from each other

Therefore, as

Γ∗

∪

∗

→ ∞,

∗∗ → 0 for all , ∈ Γ∗ ∪ ∗ Optimality dictates that the bestsystem receive far more sample than

others to minimize the probability of

occurrence of the most likely of the

numerous false selection events

when

Γ∗ ∪

∗ → ∞

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Key Assertions30

Based on the property of the rate function(Bucklew, 1990)

Thus,

Based on one of the KKT conditions of problem ,

Therefore, as Γ∗ ∪ ∗ → ∞, the following holds:∗∗ → 0 for all , ∈ Γ∗ ∪ ∗ Optimality dictates that the best

system receive far more sample than

others to minimize the probability of

occurrence of the most likely of the

numerous false selection eventswhen Γ∗ ∪ ∗ → ∞

∗ 0 implies that ∗ 0

∗

∗

∗ ∗, ∗

0

∗

→ 0as

Γ∗ ∪ ∗ → ∞while

∗

,∗

0for all

, ∈ Γ∗ ∪ ∗

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Allocation to Non-best Designs31

From the 1st KKT condition of problem Q

As Γ∗ ∪ ∗ → ∞

(∗ ,∗)∗ (∗)∗ i n f ≤,≤ ( , ) for all 2, … ,

∗, ∗ ∗, ∗ for all , 1

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Allocation to Non-best Systems32

Thus, as Γ∗ ∪ ∗ → ∞, the allocation to non-best systems 2, … , is determined by the scorefunction

where the score for non-best system is givenas

When the number of designs is large, theallocation to the non-best system becomes

inversely proportional to its score.

∗

∗ →

inf ≤, ≤ ( , )

inf ≤,≤ ( , ) i n f ≤,≤ ( , )for all 2, … ,

Th S i M lti i t

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The Score in Multivariate

Normal33

When the performance measures areindependent∈ ℎ ℎ − ℎ

2

∈ ( ∈) − 2∈

∈ ( ℎ, ∈) ℎ − ℎ 2 − 2∈

Th S i M lti i t

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The Score in Multivariate

Normal34

When the performance measures arecorrelated∈ ℎ ℎ − ℎ

2

∈ ( ∈)

∈ ( ℎ, ∈)

=1

2′ − ′ ∈() Σ ∈(−1 ( − )∈()

=1

2ℎ1 − ℎ ′ − ′ ∈() Σ ,∈(−1 ℎ1 − ℎ( − )∈()

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Problem Scenarios35

We consider scenarios

where there are no

infeasible systems that are

better than the best 40% of non-best systems

are feasible

The constraint limits are

feasible

ObjectiveFunction, h

25

Constraint

measure, g

5

7

9

3

1

1 2 3 5 4 0

26

1

24

23

22

20

21

13

14

15

16

17

18

19

11

12

10

2

3

4

5

6

7

8

9

Constraint

limit, 4

minimization 2 × 0.4 × − 1 3

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Performance Comparison36

() = Rate of decay of given an

allocation

: Equal Allocation

: Proposed Closed-form allocation

∗ : Exact allocation via a solver

Computation time based on MATLAB profiler

(in second)

= min min1≤≤ 11(1 = ) ,min≠1

inf ≤1 ,≤ 11(1)+ ,∈{,…,}

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Effect of Number of Designs37

Although the rate of the closed-form is inferior tothat of the solver, the rate gap becomes smaller as

the number of systems increases This gap is relatively much smaller if compared to

the gap between EA and the optimal allocationobtained via solver

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Effect of Number of Designs

0

100

200

300

400

500

600

700

0 500 1,000 1,500 2,000 2,500 3,000

Time(Closed-Form) Time(Solver)

0.000

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0 500 1,000 1,500 2,000 2,500 3,000

z(EA) z(Closed-Form) z*(Solver)

38

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Bed Allocation Problem

The number

of beds is not

matched with

the increasing

population 5 millions

people vs.

8,064 beds

Source: http://www.straitstimes.com/BreakingNews/Singapore/Story/STIStory_767553.html

39

http://www.straitstimes.com/BreakingNews/Singapore/Story/STIStory_767553.html



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Problem Formulation

Objective:

Maximize the Average of Daily Bed Occupancy

Rate (BOR) given a certain demand rate

Constraints (Upper Limits) on: The 99th Percentile of Daily Turn-Around-Time

(TAT)

Average Daily Number of Overflow

Decision Variables:

Number of Bed for Each of the 5 specialties

The selection is based on the sample means

40

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EMD

Registration

, Triage,Consultatio

n

Elective

Patients

Patient

Arrival

Patient

Departure

Patient

Departur eBeds

Bed

Management Unit

System Description41

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Overflow Protocol

Specialty 1st Overflow 2nd Overflow 3rd Overflow

Medicine Oncology Cardiac Not Applicable

Cardiac Medicine Surgery Orthopedic

Oncology Medicine Surgery Orthopedic

Surgery Medicine Oncology Cardiac

Orthopedic Surgery Medicine Not Applicable

42

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Proposed Solution

Yes

Yes

Initialization

Stop

Terminate?

Generating the next set of

solutions by partitioning

and random sampling

Estimating thepromising index

Simulation

OCBA-CO

The best

feasible

alternative

No

Backtrack ?

Selecting the sub-

region containing

the best as the Most

Promising Region

Selecting the super-

region as the Most

Promising Region

No

43

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Input Parameters

Two types of Arrivals (Non-stationary Poisson

process)

64% of emergency patients are admitted

The length of stay (LOS) is exponentiallydistributed

0 2 4 6 8 10 12 14 16 18 20 22

Emergency

Patients 5.25 3.8 3 4.8 7 8.25 9 7.75 7.75 8 6.5 3.25

Elective

Patients 0 0 0 0 0.2 0.4 0.7 4.7 5.3 3.2 0.8 0.3

Arrival

Rates

Time

44

1 2 3 4 5

Medicine Cardiac Oncology Surgery Orthopedic

Length of Stay (days) 6.3 3.8 9.1 4.8 11.2

Proportion of Admitted

Emergency Patients 50% 14% 5% 18% 13%

Proportion of Elective

Patients 14% 22% 20% 28% 16%

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Constraint limits for

The 99th Percentile of Daily Turn-Around-Time

(TAT):

480 minutes Average Daily Number of Overflow

30%

Simulation settings

4 warm-up days

90 days are simulated

Other Settings45

Selection from a large number of

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alternatives

For each specialty:

Minimum number of bed = 5

Maximum number of bed = 500

Given 5 specialties: 3 x 1013 (30 trillions) bed configurations

46


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alternatives

NP+OCBA-CO is able to converge in terms of

the main objective value

0.729

0.808

0.858

0.8680.875

0.882 0.884 0.886 0.887 0.887

0.7

0.72

0.74

0.76

0.78

0.8

0.82

0.84

0.860.88

0.9

0 1 2 3 4 5 6 7 8 9 10

Iteration Number of NP

Average BOR

47


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alternatives

Effect of the Constraint on the Turn-Around-

Time

Note:

Number of beds

Limit on the

Turn-Around-Time

1 2 3 4 5

Medicine Cardiac Oncology Surgery Orthopedic

Length of Stay (days) 6.3 3.8 9.1 4.8 11.2

0 100 200 300 400 500 600 700 800 900

360

480

Medicine

Cardiac

Oncology

Surgery

OrthopedicLower BOR

48

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Contributions

Extend the OCBA approach to address the

constrained ranking and selection problem in the

presence of multiple stochastic constraints

Characterize the effect of correlation to thecomputing budget allocation and provide the

framework for extending the result for the general

distribution case

Generalize the OCBA for selecting the best design Provide the proof that the optimal allocation can

be approximated by closed-form expressions as

the number of designs approach infinity

49

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Future Research

Consider selection based on quantile instead

of sample mean

Explore the issue of correlation between

designs Develop a searching algorithm to suit the

proposed simulation budget allocation rule

50

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Any questions or suggestions are

welcomed

Thank you for your kind attention51

OCBA_UGM_121214

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