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 DOI: 10.1177/0037549714548095

published online 12 September 2014SIMULATIONHui Xiao and Loo Hay Lee

Simulation optimization using genetic algorithms with optimal computing budget allocation  

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Simulation

Simulation: Transactions of the Society for

Modeling and Simulation International

1–12

� 2014 The Author(s)

DOI: 10.1177/0037549714548095

sim.sagepub.com

Simulation optimization using geneticalgorithms with optimal computingbudget allocation

Hui Xiao1 and Loo Hay Lee2

AbstractA method is proposed to improve the efficiency of simulation optimization by integrating the notion of optimal comput-ing budget allocation into the genetic algorithm, which is a global optimization search method that iteratively generatesnew solutions using elite candidate solutions. When applying genetic algorithms in a stochastic setting, each solutionmust be simulated a large number of times. Hence, the computing budget allocation can make a significant difference tothe performance of the genetic algorithm. An easily implementable closed-form computing budget allocation rule ofranking the best m solutions out of total k solutions is proposed. The proposed budget allocation rule can perform bet-ter than the existing asymptotically optimal allocation rule for ranking the best m solutions. By integrating the proposedbudget allocation rule, the search efficiency of genetic algorithms has significantly improved, as shown in the numericalexamples.

KeywordsRanking and selection, simulation, genetic algorithms, optimal computing budget allocation, stochastic optimization

1. Introduction

We consider the simulation optimization problem with

continuous search space, where the objective function can

only be estimated with noise via simulation. Traditional

approaches to tackle this type of problems include stochas-

tic approximation,1,2 simultaneous perturbation stochastic

approximation,3 sample path method,4 and response sur-

face methodology.5 With the fast development of comput-

ing technology, many metaheuristics have been adopted to

solve simulation optimization problems. These metaheur-

istics include simulated annealing,6 Tabu search,7 genetic

algorithms (GAs),8 the nested partition search,9 and the

locally convergent adaptive random search.10,11 The main

challenge of applying these methods is the evaluation of

the objective function because of its stochastic nature.

This paper aims to improve the search efficiency of

GAs in a stochastic environment. First introduced by

Holland,8 a GA is a heuristic adaptive search method that

mimics the process of natural evolution. While being suc-

cessfully applied to deterministic optimization prob-

lems,12–14 a GA becomes computationally expensive when

the evaluation of candidate solutions is subject to noise.

There is a variety of approaches in the literature to deal

with noisy fitness values. The first approach simply applies

the conventional GA to the noisy fitness function. The

underlying theory supporting this idea is that the GA has a

self-averaging nature, i.e., the solution with good fitness in

average survives as population.15 The second method is to

sample the noisy fitness value several times and use the

sample mean as an estimation.16–18 The third approach is

to evaluate an individual not only by its sampled value but

also by those near the individual. The underlying assump-

tion of this approach is that the fitness values of nearby

solutions give some information to the fitness value esti-

mation of the point of interest.19,20 In addition, using a

grouping procedure rather than an individual solution has

been suggested in recent years.21

The sample mean is commonly used as the estimation

of the true objective value. Therefore, each selected solu-

tion must be repeatedly evaluated in order to obtain a

1School of Statistics, Southwestern University of Finance and Economics,

Chengdu, China2Department of Industrial and Systems Engineering, National University

of Singapore, Singapore

Corresponding author:

Hui Xiao, School of Statistics, Southwestern University of Finance and

Economics, 555 Liutai Avenue, Wenjiang, Chengdu 611130, P. R. China.

Email: msxh@swufe.edu.cn

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statistically significant value of its sample mean. In this

paper, we aim to improve the search efficiency of a GA

by reducing the number of samples when the sample mean

is used as an estimation. In a GA, higher probability of

being selected to reproduce will be given to better candi-

date solutions so that the solutions in the next generation

will be on average better than that of the previous genera-

tion. Therefore, identifying the elite subset without rank-

ing is not enough even though the search efficiency of a

GA can be improved by using the optimal computing bud-

get allocating rule of selecting the optimal subset.22 The

relative ranking of the elite solutions needs to be identified

so that the probability of being selected to reproduce can

be assigned accurately to the corresponding solutions.

Therefore, the overall efficiency of a GA in simulation

optimization depends on how efficiently we simulate the

candidate solutions and how correctly we rank the elite

solutions. By intelligently allocating the computing effort

to the current solutions, the optimal subset can be ranked

more accurately, and thus the newly generated solutions

can be more promising. Therefore, the search efficiency of

a GA can be improved by optimally allocating the com-

puting effort within each iteration of the GA.

From the ranking and selection perspective, identifying

the relative ranking of the elite solutions is equivalent to

determining the ranking of the best m solutions out of total

k alternatives if the population size for a GA is k and the

size of the elite subset is m. In the literature, an asymptoti-

cally optimal allocation (AOA) rule was proposed to

determine the ranking of the best m solutions.23 However,

obtaining the AOA rule requires solving a system of non-

linear equations. Besides, the AOA rule itself is imple-

mented by a heuristic sequential allocation algorithm.

Therefore, we need to solve the system of nonlinear equa-

tions a large number of times if the AOA rule is directly

integrated with a GA. In this paper, we aim to derive an

easily implementable closed-form allocation rule to deter-

mine the ranking of the best m solutions out of total k

alternatives so that the allocation rule can be easily inte-

grated with a GA.

In the ranking and selection literature, many efficient

computing budget allocation procedures have been pro-

posed to compare different solutions. Three popular proce-

dures for selecting the single best solution include optimal

computing budget allocation (OCBA),24,25 the indifference

zone procedure (IZP),26 and the value information proce-

dure (VIP).27,28 OCBA focuses on the efficiency of simula-

tion by intelligently allocating further replications based

on both the mean and variance. OCBA has been extended

in various ways and applied in different areas.29–32 The

IZP aims to find a feasible way to guarantee the pre-

specified probability of correct selection can be achieved.

The VIP uses the Bayesian posterior distribution to

describe the evidence of correct selection and allocates fur-

ther replications based on maximizing the value

information. Other selection criteria include selecting the

optimal subset,22 selecting the Pareto set for multi-objec-

tive optimization problems,33,34 ranking all solutions com-

pletely,35 and selecting the single best solution subjected to

stochastic constraints.36,37 These computing budget alloca-

tion procedures have been applied to different search algo-

rithms in stochastic environment such as particle swarm

optimization,38 cross entropy method,39 population-based

incremental learning and neighborhood random

search.10,40

To reiterate, the objective of this paper is to derive an

easily implementable closed-form computing budget allo-

cation rule for ranking the best m solutions out of total k

alternatives so that this computing budget allocation rule

can be easily integrated with a GA. The contribution of

this paper is threefold. From simulation optimization per-

spective, we illustrate a better way of efficiently allocating

computing budget for a GA in a stochastic environment.

From ranking and selection perspective, we offer an easily

implementable heuristic for ranking the best m solutions

out of total k alternatives. From the computing budget allo-

cation perspective, the proposed heuristic demonstrates

how the OCBA framework can be extended to rank the

best m solutions. The rest of the paper is organized as fol-

lows: Section 2 formulates the best m ranking problem for

a GA. Section 3 derives the computing budget allocation

rule. Numerical comparisons for ranking the best m solu-

tions are conducted in Section 4. Numerical examples of

using a GA with the computing budget allocation rule to

solve continuous simulation optimization problems are

conducted in Section 5. Finally, we conclude this paper in

Section 6.

2. Problem formulation

In this section, we formally define the computing budget

allocation problem for a GA. An optimization model is

formulated based on optimal computing budget allocation

framework.

2.1. Genetic algorithms

A GA begins by randomly generating an initial population

of strings with size k. These strings represent possible

solutions to the problem. Each solution is evaluated by a

measurable objective function. In stochastic setting, each

solution must be evaluated repeatedly in order to use sam-

ple mean value as the estimation of the mean objective

value. A second population is generated from the elite

solutions of the first population. The better the objective

value is, the more likely the corresponding solution will

be chosen to reproduce. Searches are usually terminated

when reaching the predetermined number of generations.

The GA algorithm in a stochastic setting can be summar-

ized in Algorithm 1.

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In this algorithm, pi is the probability that the ith ranked

solution will be chosen to reproduce. The main challenge

of applying a GA in a stochastic setting is how to effi-

ciently evaluate and rank the candidate solutions. Suppose

that there is a total of n computing budget (or simulation

replications) available at each iteration of a GA, the prob-

lem to solve is how to efficiently allocate the n simulation

replications to the k solutions such that we can rank the

best m solutions as correctly as possible.

In this paper, computing budget and simulation replica-

tions are used interchangeably, as the solution is evaluated

via simulation. The computing budget refers to the number

of simulation replications. Likewise, designs and solutions

are used interchangeably. The performance of the design

in ranking and selection refers to the objective value of the

solution in the context of a GA.

2.2. Assumptions2.2.1. Assumption 1. The designs are simulated indepen-

dently of each other, i.e., the samples (Xi1, � � � ,Xi,ain) foreach i = 1, � � �, k are independent.

Assumption 1 is a common practice used in ranking and

selection area. We also require that no design has exactly

the same performance value as that of other designs. This

is a standard assumption in the literature that seeks an opti-

mal computing budget allocation, since it ensures that two

designs can be distinguished with a finite number of simu-

lation replications.

The second assumption is a standard practice in using

large deviation theory. Since the objective of this paper is

to derive the optimal computing budget allocation for the

designs, we replicate the assumption for completeness.

Further explanation can be found in the book by Dembo

and Zeitouni.41

Let the cumulant generating functions of �Xi(ain) be

L�Xi(ain)i (u)= In E(eu�Xi(ain)) for each i = 1, � � �, k, where

u 2 R. Let the effective domain of a function f(x) be

denoted by Df = {x: f(x) \ N} and its interior is denoted

by D0f . The extended real number is defined as

R�=R [ f+‘g. Let DLi= fu 2 R : Li(u)\ ‘g and

Fi = fL0i(u) : u 2 D0Lig.

2.2.2. Assumption 2. For every i = 1, � � �, k, the following

is true:

(1) The limit Li(u)= limn!‘

(1=ain)L�Xi(ain)i (ainu)

exists as an extended real number for all u 2 R.

(2) The origin belongs to the interior of DLi, i.e.,

0 2 D0Li.

(3) Li( � ) is strictly convex and differentiable on D0Li.

(4) Li( � ) is steep, i.e., limn!‘jL0i(un)j=‘, where {un}

is a sequence converging to the boundary point

of DLi.

Assumption 2 ensures the large deviation principle holds

for the estimators �Xi(ain) with good, strictly convex rate

function Ii(x)= supu2R

ux� Li(u)ð Þ.

2.2.3. Assumption 3. The closure of the convex hull of all

points mi 2 R is a subset of the intersection of the interiors

of the effective domains of the rate function Ii(x) for each

i = 1, � � �, k, i.e., ½m1,mk � � \ki= 1F

0i .

Assumption 3 ensures that the sample mean of every

design can take any value in the interval [m1, mk] and

P �Xi(ain)5 �Xi+ 1(ai+ 1n)). 0ð .42

2.3. Computing budget allocation problem

From the ranking and selection perspective, the problem

discussed above can be described as follows. Consider a

finite number of designs, each with an unknown perfor-

mance value mi 2 R, i = 1, � � �, k. The goal is to find the

computing budget allocation rule that maximizes the prob-

ability of correctly ranking the best m (m \ k) designs.

Let a = (a1, � � �, ak) be the proportion of total computing

budget n allocated to each design such thatPki= 1 ai = 1, i= 1, � � � , k: Let �Xi(ain)= (ain)

�1Painj= 1

Xij denote the sample mean performance of design i,

where (Xi1, � � � ,Xi, ai, n) denotes the samples from popula-

tion i. The objective is to find the optimal allocation rule

a�=(a�1, � � � ,a�k) such that the probability of correctly

Algorithm 1. The genetic algorithm in a stochastic environment:

INITIALIZATION: Generate a starting population with size k.LOOP: While the termination criterion is not met,

EVALUATION: Determine the computing budget allocation of the k solutions and simulate the k solutions.SELECTION: Select the best m solutions to reproduce according a predetermined probability vector

p = (p1, � � �, pm) such thatPm

i= 1 pi = 1.REPRODUCTION: Crossover the parental strings at a random point in the gene string so that offspring

consists of a portion of each parent.MUTATION: Randomly alter the genetic makeup to avoid local optima.

END OF LOOP

Xiao and Lee 3

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ranking the best m designs can be maximized with a fixed

limited computing budget n.

Denote the mean performance of each design by

m1, � � �, mk, m1 \ � � � \ mi \ � � � \ mk. The best m

designs are correctly ranked if and only if�Xi(ain)4 �Xi+ 1(ai+ 1n) for all i = 1, � � �, m– 1 and�Xm(amn)4 �Xj(ajn) for all j = m + 1, � � �, k. The prob-

ability of correctly ranking the best m designs can be

expressed as follows.

P(CRm)=P\m�1

i= 1

�Xi(ain)4 �Xi+ 1(ai+ 1n)ð Þ( )

\ \kj=m+ 1

�Xm(amn)4 �Xj(ajn)� �( )!

ð1Þ

The optimal computing budget allocation problem can

be formulated by maximizing the probability of correctly

ranking the best m designs.

maxa1, ���,ak

P(CRm)

s:t:Xk

i= 1

ai = 1,ai 5 0, i= 1, � � � , k ð2Þ

3. Approximated closed-form allocationrule

Given the optimization model (2), we aim to derive an eas-

ily implementable closed-form allocation rule in this sec-

tion. The derivation is based on the large deviation theory.

3.1. Approximated allocation rule

Define a strictly increasing sequence {ci, i = 0, 1, � � �, m},

i.e., c0 \ � � �\ ci \ � � �\ cm with ci = mi, i = 1, � � �,m, c0 = –N, where mi is the mean performance value of

design i. P(CRm) can be approximated by the following

expression.

P(CRm)=P\m�1

i= 1

(�Xi(ain)4 �Xi+ 1(ai+ 1n))

( )

\ \kj=m+ 1

�Xm(amn)4 �Xj(ajn)� �( )!

5P\m

i= 1

(ci�1 4 �Xi(ain)4 ci+ 1)

( )

\ \kj=m+ 1

�Xj(ajn)5 cm

� �( )!

Hence,

P(FRm)= 1� P(CRm)

4P[

i= 1, ���,m

�Xi(ain)4 ci�1ð Þ [ �Xi(ain)5 ci+ 1ð Þð Þ !(

[ [j=m+ 1, ���, k

�Xj(ajn)4 cm

� � !)

=Xm

i= 1P �Xi(ain)4 ci�1ð Þ [ �Xi(ain)5 ci+ 1ð Þf g

+Xk

j=m+ 1P �Xj(ain)4 cm

� �=P(AFRm) ð3Þ

where the equality follows from the assumption that every

design is simulated independently.

Note that P(AFRm) is bounded below by

max maxi= 1, ���,m

P �Xi(ain)4 ci�1ð Þ [ �Xi(ain)5 ci+ 1ð Þf g,�

maxj=m+ 1, ���, k

P �Xj(ajn)4 cm

� ��

and bounded above by

k*max maxi= 1, ���,m

P �Xi(ain)4 ci�1ð Þ [ �Xi(ain)5 ci+ 1ð Þf g,�

maxj=m+ 1, ���, k

P �Xj(ajn)4 cm

� ��

such that, assuming the limit exists,

limn!‘

1

nlnP(AFRm)

= limn!‘

1

nlnmax max

i= 1, ���,mP �Xi(ain)4 ci�1ð Þ [ �Xi(ain)ðf

�

5 ci+ 1Þg, maxj=m+ 1, ���, k

P �Xj(ajn)4 cm

� ��

Theorem 1 below states that the limit exists and the overall

convergence rate function is the minimum rate function of

each probability.

Theorem 1. The rate function of P(AFRm) is given by

� limn!‘

1

nlnP(AFRm)

= min mini= 1, ���,m

min aiIi(ci�1),aiIi(ci+ 1)ð Þf g,�

minj=m+ 1, ���, k

ajIj(cm)

)

Proof: If there exist functions R1(�) and R2(�) such that

4 Simulation: Transactions of the Society for Modeling and Simulation International

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limn!‘

1

nlnP �Xi(ain)4 ci�1f g= � R1(ai, ci�1)

limn!‘

1

nlnP �Xi(ain)5 ci+ 1f g= � R2(ai, ci+ 1)

for i = 1, � � �, m, and

limn!‘

1

nlnP �Xj(ajn)4 cm

� �= � R1(aj, cm)

for j = m + 1, � � �, k.

It can be concluded that

limn!‘

1

nlnP �Xi(ain)4 ci�1ð Þ [ �Xi(ain)5 ci+ 1ð Þf g

= limn!‘

1

nln P �Xi(ain)4 ci�1ð Þ+P �Xi(ain)5 ci+ 1ð Þf g

= �min R1(ai, ci�1),R2(ai, ci+ 1)f g

Therefore, the rate function of P(AFRm) can be denoted

as

limn!‘

1

nlnP(AFRm)

= limn!‘

1

nln

Xm

i= 1

P �Xi(ain)4 ci�1ð Þ [ �Xi(ain)5 ci+ 1ð Þf g,

Xk

j=m+ 1

P �Xj(ajn)4 cm

� �!

= �min mini= 1, ���,m

min R1(ai, ci�1),R2(ai, ci+ 1)ð Þf g,�

minj=m+ 1, ���, k

R1(aj, cm)

�

We are now in the position to derive the assumed function

R1(�) and R2(�). Under assumption 2, we will have

1

nlnE(eu�Xi(ain))=ai lnLi(u=ai)

and

supu2R

ci�1u� aiLi u=aið Þf g

=ai supu=ai

ci�1u=ai � Li u=aið Þf g=aiIi(ci�1)

supu2R

ci+ 1u� aiLi u=aið Þf g

=ai supu=ai

ci+ 1u=ai � Li u=aið Þf g=aiIi(ci+ 1)

By the Gartner–Ellis theorem,41 �Xi(ain) satisfies the

large deviation principle, therefore

limn!‘

1

nlnP �Xi(ain)4 ci�1f g= � aiIi(ci�1)

limn!‘

1

nlnP �Xi(ain)5 ci+ 1f g= � aiIi(ci+ 1)

Therefore, the rate function for approximated probability of

false ranking is

� limn!‘

1

nlnP(AFRm)= min

(min

i= 1, ���,mmin R1(ai, ci�1),ðf

R2(ai, ci+ 1)Þg, minj=m+ 1, ���, k

R1(aj, cm)

)

= min mini= 1, ���,m

min aiIi(ci�1),aiIi(ci+ 1)ð Þf g,�

minj=m+ 1, ���, k

ajIj(cm)

�:

� limn!‘

1

nlnP(AFRm)= min

�min

i= 1, ���,mmin R1(ai, ci�1),ðf

R2(ai, ci+ 1)Þg, minj=m+ 1, ���, k

R1(aj, cm)

�

= min mini= 1, ���,m

min aiIi(ci�1),aiIi(ci+ 1)ð Þf g,�

minj=m+ 1, ���, k

ajIj(cm)

�:

Given the upper bound of false ranking probability

P(AFRm), minimizing the upper bound of false ranking

probability is equivalent to maximizing its rate function at

which P(AFRm) goes to zero, i.e., find a which solves the

following optimization model.

maxmin mini= 1, ���,m

min aiIi(ci�1),aiIi(ci+ 1)ð Þf g,�

minj=m+ 1, ���, k

ajIj(cm)

�

s:t:Xk

i= 1

ai = 1,ai 5 0, 8i= 1, � � � , k ð4Þ

This model can be re-expressed as follows.

max z

s:t: min

�min

i= 1, ���,mfminðaiIi(ci�1),aiIi(ci+ 1)Þg,

minj=m+ 1, ���, k

ajIj(cm)

�� z5 0

Xiao and Lee 5

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Xk

i= 1

ai = 1,ai 5 0, i= 1, � � � , k ð5Þ

Theorem 2. If the optimal allocation a�. 0,Pk

i= 1

a�i = 1 minimizes the approximated probability of false

ranking asymptotically, then,

a�p minðIp(cp�1), Ip(cp+ 1)Þ=a�q minðIq(cq�1),

Iq(cq+ 1)Þ=a�j Ij(cm)

where p, q 2 f1, � � � ,mg, j 2 fm+ 1, � � � , kg:

Proof: We first rewrite the optimization model (5) as

follows.

max z

s:t: min aiIi(ci�1),aiIi(ci+ 1)ð Þ � z5 0, i= 1, � � � ,m

ajIj(cm)� z5 0, j=m+ 1, � � � , k

Xk

i= 1

ai = 1,ai 5 0, i= 1, � � � , k

ð6Þ

Model (6) is a concave programming problem as shown in

the literature.42 Thus, the first order condition is also the

optimality condition. Therefore, under the Karush–Kuhn–

Tucker condition, there exist li and g . 0 such that

1�Xk

i= 1li = 0 ð7Þ

li min Ii(ci�1), Ii(ci+ 1)ð Þ= g, 8i= 1, � � � ,m ð8Þ

ljIj(cm)= g, 8j=m+ 1, � � � , k ð9Þ

li z� a�imin Ii(ci�1), Ii(ci+ 1)ð Þ

� = 0, 8i= 1, � � � ,m

ð10Þ

lj z� a�jIj(cm)

� = 0, 8j=m+ 1, � � � , k ð11Þ

li 5 0, 8i= 1, � � � , k ð12Þ

Equation (7) implies that there must exist some li, i = 1,

� � �, k, such that li is strictly positive. However, the rate

function Ii(ci), i = 1, � � �, k is strictly positive, therefore, any

li = 0 will lead all other li = 0. Therefore, it can be con-

cluded that li . 0 for every i. Therefore, by the com-

plementary slackness conditions in equations (10) and (11),

a�p minðIp(cp�1), (cp+ 1)Þ = a�j Ij(cm)=a�q min Iq(cq+ 1),Iq(cq+ 1)Þ, for any p, q 2 {1, � � �, m} and j 2 {m + 1,

� � �, k}.

3.2. Sequential allocation algorithm

To obtain the allocation rule, the rate functions must be

computed first. In order to obtain the rate function, the

population parameters must be known. However, no infor-

mation on the population parameters is known before

actual simulation experiments are conducted. Therefore,

we suggest a heuristic sequential allocation algorithm to

implement the allocation rule. At each step of this algo-

rithm, we update the population parameters estimation of

each design using the sample statistics so that the alloca-

tion rule can be determined using more accurate estimates.

In Algorithm 2, l is the iteration number. The ranking of

best m designs may change from iteration to iteration,

although it will converge to the correct ranking when total

computing budget goes to infinity. When the ranking of best

m designs changes, the budget allocation will be applied

immediately. Therefore, the actual proportion of the com-

puting budget for every design will converge to the optimal

proportion when the number of iterations is sufficient large.

Each design is initially simulated with n0 replications in

the first iteration. Additional D replications are increased in

each iteration until the total computing budget is exhausted.

The selection of n0 should keep good balance between

accuracy and efficiency. Previous research shows that a

good choice of n0 will be between 5 and 20. The value of

D should not be too large to allow the correction by the

next iteration. Empirically, D must be smaller than 100.

4. Numerical experiments

To illustrate the effectiveness of the proposed computing

budget allocation rule, we conduct several numerical

experiments in this section to compare the proposed allo-

cation rule with the asymptotically optimal allocation

rule,23 and equal allocation. In the numerical experiments,

the performance of each design is assumed to follow nor-

mal distribution. The assumption of normal distribution is

generally held in simulation experiments since the output

is obtained from an average performance or batch means,

so that central limit theorem effects hold. The empirical

probability of correctly ranking the best m designs is used

as the performance measurement.

4.1. Allocation rule for normal distribution

Suppose the performance of each design follows the

normal distribution, i.e., Xi;N (mi,s2i ), i= 1, � � � , k: The

rate function for normal distribution is Ii(x)=(x�mi)

2

2s2i

:

Therefore, the allocation rule in Theorem 2 can be repre-

sented as follows.

For any p, q 2 1, � � � ,mf g and j 2 m+ 1, � � � , kf g,Pki= 1 ai = 1 and

a�pa�q

=s2

p=minf(mp�mp+ 1)2, (mp�mp�1)

2gs2

q=minf(mq�mq+ 1)2, (mq�mq�1)

2ga�qa�

j

=s2

q=minf(mq�mq+ 1)2, (mq�mq�1)

2gs2

j=(mj�mm)

2

8><>: ð13Þ

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4.2. Allocation procedures

(1) Equal allocation (EA): the simulation replications

are allocated equally to each design, i.e.,

ai = 1=k, i= 1, � � � , k. This is the simplest allo-

cation rule and it can serve as a benchmark for all

other procedures.

(2) Asymptotically optimal allocation (AOA-m): this

allocation rule is proposed in the literature.23 The

allocation rule a�i , i= 1, � � � , k is such that

min(mi � mi+ 1)

2

2(s2i =a�i +s2

i+ 1=a�i+ 1),

(mi � mi�1)2

2(s2i =a�i +s2

i�1=a�i�1)

� �

=(m1 � m2)

2

2(s21=a�1 +s2

2=a�2)=

(mj � mm)2

2(s2j =a�j +s2

m=a�m),

i= 2, � � � ,m� 1; j=m+ 1, � � � , k

Xk

i= 1

ai = 1,ai . 0

8>>>>>>>>>>>><>>>>>>>>>>>>:

ð14Þ

(3) Approximated allocation (AA-m): this is the

closed-form allocation rule proposed in Theorem

2. The allocation rule a�i , i= 1, � � � , k for nor-

mally distributed design performance is deter-

mined by equation (13).

4.3. Numerical results

To compare the performance of the procedures, we carried

out numerical experiments for the different allocation pro-

cedures. Let the initial simulation replications n0 be 20.

The simulation replications are then gradually increased

by D = 40. The probability of correctly ranking the best

five designs is estimated as the number of times correct

ranking occurs out of the total 10,000 independent simula-

tion runs.

The mean and variance of each design are summarized

in Table 1. Equal spacing refers to the scenario where the

mean differences between consecutive designs are the

same but the variance of each design is different. Equal

variance refers to the scenario where the variance of each

design is the same but the mean differences between con-

secutive designs are different. Increasing spacing but

decreasing variance scenario refers to the situation where

both variance of each design and the mean differences

between consecutive designs are different.

4.3.1. Examples with expected mean and variance. The first

set of experiments is conducted assuming the mean and

variance for each of the 20 designs are known. Hence, the

allocation rule for AOA-m can be directly computed using

equation (14) and the AA-m rule can be computed using

equation (13). Denote the AOA-m rule as ~a and the AA-m

rule as a. Both allocation rules are implemented using the

sequential allocation algorithm suggested in Section 3.2.

After allocating n0 = 20 to each design, future computing

budget is always allocated according to ~a for AOA-m rule,

and a for the AA-m rule. The performances of the three

allocation rules are shown in Figures 1, 2, and 3, respec-

tively, for equal spacing, the equal variance scenario, and

an increasing spacing decreasing variance scenario.

We can see that the AA-m performs slightly better than

AOA-m within a finite computing budget. However,

AOA-m catches up with AA-m quickly when the comput-

ing budget becomes large. Either AOA-m or AA-m can

outperform EA significantly.

Algorithm 2. Sequential allocation algorithm

INPUT: k: total number of designs, m: number of designs needs ranking,n: total computing budget, n0: size of initial simulation replications,D: size of incremental budget in one iteration,

INITIALIZE: Perform n0 simulation replications for each design. l 0,

N l1 =N l

2 = � � � =N lk = n0.

LOOP: WhilePk

i= 1 Nli 4 n,

UPDATE: Calculate the rate function using the new simulation output.ALLOCATE: Increase the computing budget by D and determine the computing budget for each design, i.e.,

N l+ 1i =a�i

Pki= 1 N

li +D

� , such that

Pki= 1 a�i = 1 and

a�pa�q

=min Iq(cq�1), Iq(cq+ 1)� �

min Ip(cp�1), Ip(cp+ 1)� � , p, q 2 f1, � � � ,mg

a�pa�j

=Ij(cm)

min Ip(cp�1), Ip(cp+ 1)� � , p 2 f1, � � � ,mg, j 2 fm+ 1, � � � , kg

8>>><>>>:

SIMULATE: Perform additional max (0,N l+ 1i � N l

i) simulation runs for designi, i = 1, � � �, k, l l+ 1.

END OF LOOP

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4.3.2. Examples with unknown mean and variance. The sec-

ond set of experiments is conducted assuming that no prior

information on the mean and variance of each design is

known. Both AA-m and AOA-m are implemented using

the sequential allocation algorithm. Each design is allo-

cated with n0 = 20 initial replications. At each iteration of

the sequential algorithm, sample mean and sample var-

iance are used as the estimation of the population mean

and variance for each design. They are substituted into

equation (14) for AOA-m rule or equation (13) for AA-m

rule to compute the computing budget allocation rule for

next iteration. Hence, the computing budget allocation rule

is changing for each iteration since the sample mean and

variance are changing when new simulation output is gen-

erated. In addition, it is worth noting that nonlinear equa-

tions (14) must be solved for each iteration in order to

obtain the AOA-m rule. The numerical results for the three

scenarios are shown in Figures 4, 5, and 6, respectively.

Given a fixed finite computing budget, the results indi-

cate that AA-m performs the best in all scenarios. AOA-m

performs much better than EA, however, the performance

difference between AA-m and AOA-m is significant.

Recall that the performances of AOA-m and AA-m are

similar if the mean and variance of each design are given.

Table 1. Parameters for the numerical experiments.

Equal spacing Equal variance Increasing spacing decreasing variance

Design Mean Variance Mean Variance Mean Variance

I 1 400 1 100 1 400II 2 361 2 100 2 361III 3 324 4 100 4 324IV 4 289 7 100 7 289V 5 256 11 100 11 256VI 6 225 16 100 16 225VII 7 196 22 100 22 196VIII 8 169 29 100 29 169IX 9 144 37 100 37 144X 10 121 46 100 46 121XI 11 100 56 100 56 100XII 12 81 67 100 67 81XIII 13 64 79 100 79 64XIV 14 49 92 100 92 49XV 15 36 106 100 106 36XVI 16 25 121 100 121 25XVII 17 16 137 100 137 16XVIII 18 9 154 100 154 9XIX 19 4 172 100 172 4XX 20 1 191 100 191 1

Figure 1. Probability of correctly ranking the best m solutionswith expected mean and variance for equal spacing scenario.

Figure 2. Probability of correctly ranking the best m solutionswith expected mean and variance for equal variance scenario.

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Hence, the reason why AOA-m performs much worse than

AA-m is because AOA-m rule needs to solve a system of

nonlinear equations at each iteration of the sequential allo-

cation algorithm. The solution of the nonlinear equations

is very sensitive toward the estimated mean and variance.

A small change in mean or variance could result a signifi-

cant change in the solution. However, the closed-form

allocation rule proposed in this paper is more stable from

iteration to iteration. In other words, the allocation rule aobtained from AOA-m rule fluctuates more than that from

the AA-m rule. Some designs are allocated more than

needed and some designs are allocated less than required.

Although the allocation can be corrected in subsequent

iterations when the computing budget becomes very large,

the performance in terms of correct ranking probability

will be affected within finite computing budget.

5. Simulation optimization using GA

In these numerical examples, we integrate the AA-m rule

with GA to solve the continuous simulation optimization

problems. The AA-m rule is used in the evaluation and

selection step of GA to identify the ranking of the best m

solutions. The resulting performance of GA with AA-m is

compared with the performance of GA with OCBA-m,22

and GA with EA. OCBA-m is the computing budget allo-

cation rule for selecting the best m solutions without identify-

ing their relative ranking. The AOA-m rule is not used for

comparison in these examples since it requires solving the

system of nonlinear equations lots of times, which increases

the computing burden significantly. Besides, we have shown

that AA-m is able to outperform AOA-m within finite com-

puting budget in Section 4. The purpose of these examples is

Figure 3. Probability of correctly ranking the best m solutionswith expected mean and variance for increasing spacing butdecreasing variance scenario.

Figure 4. Probability of correctly ranking the best m solutionswith unknown mean and variance for equal spacing scenario.

Figure 5. Probability of correctly ranking the best m solutionswith unknown mean and variance for equal variance scenario.

Figure 6. Probability of correctly ranking the best m solutionswith unknown mean and variance for increasing spacing butdecreasing variance scenario.

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not to find the best configuration of GA, but rather to explore

whether the AA-m rule can enhance the efficiency of simula-

tion optimization using GA.

Three well-known continuous deterministic optimiza-

tion problems are used in the experiments. However, we

assume that the objective function is subject to a nor-

mally distributed noise. The noise for all experiments is

assumed to be normally distributed with mean 0

and standard deviation of 50. The population size of the

GA is set to be 20, and the best 10 solutions will be

ranked as they will be selected to reproduce. Exponential

ranking selection scheme of the GA is used in the numer-

ical experiments. For ranked solution 1 to solution 10,

the probability of being selected to produce offspring is

set to be pi = (c– 1)c10 2 i / (c10 2 1), i = 1, � � �, 10.

The parameter c is set to be 0.7 in all experiments.

1000 simulation replications are available for each itera-

tion, and GA terminates when the total number of itera-

tions reaches 1000. It is worthy to note that the choice of

the parameter setting of GA is rather arbitrary since

the purpose of these numerical examples is not to find

the best parameter setting of GA. The goal is to compare

the performances of GA when it is integrated with

AA-m, OCBA-m and EA given the same experiment

setting.

Experiment 1: Goldstein–Price function

S(X )= 1+(x1+x2+1)2(19� 14x1+ 3x22 � 14x2 + 6x1x2+ 3x22)� �

� (18� 32x1 + 12x21 + 48x2 � 36x1x2 + 27x22) 30+(2x1 � 3x2)2

� �

where X = (x1, x2), –3 4 xi 4 3, i = 1, 2.

The function has a unique optimal solution at (0, –1)

with the objective value of 3. However, four local optima

exist in the given feasible region.

Experiment 2: Griewank function

S(X )=1

40(x21 + x22)� cos (x1) cos

x2ffiffiffi2p� �

+ 2

where X = (x1, x2), –10 4 xi 4 10, i = 1, 2.

The unique optimal solution is at (0, 0) with the objec-

tive value of 1. Many local optima exist in the given region.

Experiment 3: Spherical function

S(X )=X5i= 1

(X 2i � c)

where c = 5, –5 4 xi 4 15, i = 1, 2, 3, 4, 5.

Figure 7. Numerical results of simulation optimization usinggenetic algorithm integrated with simulation budget allocationrules: AA-m, OCBA-m, and EA for Goldstein–Price function.

Figure 8. Numerical results of simulation optimization usinggenetic algorithm integrated with simulation budget allocationrules: AA-m, OCBA-m, and EA for Giewank function.

Figure 9. Numerical results of simulation optimization usinggenetic algorithm integrated with simulation budget allocationrules: AA-m, OCBA-m, and EA for spherical function.

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The value of c can be arbitrary. We use c = 5 in the

experiment. This function has the optimal value of zero

with X = (5, 5, 5, 5, 5).

The numerical results for Goldstein–Price, Griewank,

and spherical functions are shown in Figures 7, 8, and 9,

respectively. We see that the optimality gap decreases for

all procedures as the available computing budget increases.

In all three examples, GA with AA-m is able to outperform

GA with OCBA-m and GA with EA. Based on the results,

it can be concluded that the efficiency of GA in simulation

optimization has been significantly improved by integrat-

ing the AA-m rule proposed in this paper.

6. Conclusion

Motivated by the idea of applying of the ranking and

selection procedure to genetic algorithms, we derive an

easily implementable closed-form allocation rule for rank-

ing the best m designs out of k alternatives. The proposed

closed-form allocation rule is integrated with GAs to solve

continuous simulation optimization problems. Numerical

experiments indicate that the closed-form allocation rule

can even perform better than the existing asymptotically

optimal allocation rule when the probability of correct

ranking is used as the performance measure within finite

computing budget. The numerical examples of using GAs

to solve simulation optimization problems show that the

proposed closed-form allocation can further enhance the

search efficiency of GAs compared with the OCBA-m rule

and EA. In general, the allocation rule proposed in this

paper can be applied and integrated with all other

population-based evolutionary algorithms, which require

knowing the ranking information of the best m solutions.

It provides a new way to increase the search efficiency for

population-based evolutionary algorithms by using the

available computing budget in the most efficient way.

Funding

This research received no specific grant from any funding agency

in the public, commercial, or not-for-profit sectors.

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Author biographies

Hui Xiao is currently an assistant professor at School of

Statistics, Southwestern University of Finance and

Economics. His research focuses on stochastic simulation

optimization, optimal computing budget allocation, system

reliability modeling and optimization.

Loo Hay Lee is an associate professor in the Department

of Industrial and Systems Engineering at National

University of Singapore. His research focuses on the

simulation-based optimization, maritime logistics that

includes port operations and the modeling and analysis for

the logistics and supply chain system. He has published

around 80 papers in international journals and has served

as the associate editor for IEEE Transactions on

Automatic Control, IIE Transactions, IEEE Transactions

on Automation Science and Engineering, Flexible Services

and Manufacturing Journal, Simulation: Transactions of

The Society for Modeling and Simulation International,

the Asia Pacific Journal of Operational Research, and the

International Journal of Industrial Engineer: Theory,

Applications and Practice. He is currently the co-editor

for Journal of Simulation and is a member of the advisory

board for OR Spectrum.

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