Stochastic Optimization Algorithm based on Deterministic

Approximations

Mohan Krishnamoorthy

1 a

, Alexander Brodsky

2 b

and Daniel A. Menasc

2 c

Mathematics and Computer Science Division, Argonne National Laboratory, Lemont, IL 60439, U.S.A.

Department of Computer Science, George Mason University, Fairfax, VA 22030, U.S.A.

Keywords:

Decision Support, Decision Guidance, Deterministic Approximations, Stochastic Simulation Optimization,

Heuristic Algorithm.

Abstract:

We consider steady-state production processes that have feasibility constraints and metrics of cost and through-

put that are stochastic functions of process controls. We propose an efﬁcient stochastic optimization algorithm

for the problem of ﬁnding process controls that minimize the expectation of cost while satisfying deterministic

feasibility constraints and stochastic steady state demand for the output product with a given high probability.

The proposed algorithm is based on (1) a series of deterministic approximations to produce a candidate set

of near-optimal control settings for the production process, and (2) stochastic simulations on the candidate

set using optimal simulation budget allocation methods. We demonstrate the proposed algorithm on a use

case of a real-world heat-sink production process that involves contract suppliers and manufacturers as well as

unit manufacturing processes of shearing, milling, drilling, and machining, and conduct an experimental study

that shows that the proposed algorithm signiﬁcantly outperforms four popular simulation-based stochastic

optimization algorithms.

1 INTRODUCTION

This paper considers steady-state processes that pro-

duce a discrete product and have feasibility con-

straints and metrics of cost and throughput that are

stochastic functions of process controls. We are con-

cerned with the development of a one-stage stochas-

tic optimization algorithm for the problem of ﬁnding

process controls that minimize the expectation of cost

while satisfying deterministic feasibility constraints

and stochastic steady state demand for the output

product with a given high probability. These prob-

lems are prevalent in manufacturing processes, such

as machining, assembly lines, and supply chain man-

agement. The problem tries to ﬁnd process controls

such as speed of the machines that not only satisfy the

feasibility constraints of being within a constant ca-

pacity, but also satisfy the production demand. In the

stochastic case, we want to ﬁnd these process controls

at the minimum expected cost and for demand that is

satisﬁed with a high probability of 95%.

https://orcid.org/0000-0002-0828-4066

https://orcid.org/0000-0002-0312-2105

https://orcid.org/0000-0002-4085-6212

Stochastic optimization have typically been per-

formed using simulation-based optimization tech-

niques (see (Amaran et al., 2016) for a review of

such techniques). Tools like SIMULINK (Dabney

and Harman, 2001) and Modelica (Provan and Ven-

turini, 2012) allow users to run stochastic simula-

tions on models of complex systems in mechani-

cal, hydraulic, thermal, control, and electrical power.

Tools like OMOptim (Thieriota et al., 2011), Efﬁ-

cient Traceable Model-Based Dynamic Optimization

(EDOp) (OpenModelica, 2009), and jMetal (Durillo

and Nebro, 2011) use simulation models to heuris-

tically guide a trial and error search for the optimal

answer. The above approaches are limited because

they use simulation as a black box instead of using

the problem’s underlying mathematical structure.

From the work on Mathematical Programming

(MP), we know that, for deterministic problems, uti-

lizing the mathematical structure can lead to sig-

niﬁcantly better results in terms of optimality of

results and computational complexity compared to

simulation-based approaches (e.g., (Amaran et al.,

2016) and (Klemmt et al., 2009)). For this reason, a

number of approaches have been developed to bridge

the gap between stochastic simulation and MP. For

Krishnamoorthy, M., Brodsky, A. and Menascé, D.

Stochastic Optimization Algorithm based on Deterministic Approximations.

DOI: 10.5220/0010343802870294

In Proceedings of the 10th International Conference on Operations Research and Enterprise Systems (ICORES 2021), pages 287-294

ISBN: 978-989-758-485-5

287

instance, (Thompson and Davis, 1990) propose an in-

tegrated approach that combines simulation with MP

where the MP problem is constructed from the orig-

inal stochastic problem with uncertainties being re-

solved to their mean values by using a sample of

black-box simulations. This strategy of extracting

an MP from the original problem is also used by

(Paraskevopoulos et al., 1991) to solve the optimal ca-

pacity planning problem by incorporating the original

objective function augmented with a penalty on the

sensitivity of the objective function to various types of

uncertainty. The authors of (Xu et al., 2016) propose

an ordinal transformation framework, consisting of a

two-stage optimization framework that ﬁrst extracts a

low ﬁdelity model using simulation or a queuing net-

work model using assumptions that simplify the orig-

inal problem and then uses this model to reduce the

search space over which high ﬁdelity simulations are

run to ﬁnd the optimal solution to the original prob-

lem. However, extraction of the mathematical struc-

ture through sampling using a black-box simulation is

computationally expensive, especially for real-world

production processes composed of complex process

networks.

In (Krishnamoorthy et al., 2015), instead of ex-

tracting the mathematical structure using black-box

simulation, the mathematical structure is extracted

from a white-box simulation code analysis as part of

a heuristic algorithm to solve a stochastic optimiza-

tion problem of ﬁnding controls for temporal produc-

tion processes with inventories as to minimize the to-

tal cost while satisfying the stochastic demand with

a predeﬁned probability. Similar to the previous ap-

proaches, the mathematical structure is used for ap-

proximating a candidate set of solutions by solving a

series of deterministic MP problems that approximate

the stochastic simulation. However, the class of prob-

lems considered in (Krishnamoorthy et al., 2015) is

limited to processes described using piece-wise linear

arithmetic. In (Krishnamoorthy et al., 2018), the ap-

proach from (Krishnamoorthy et al., 2015) is gener-

alized to solve stochastic optimization problems that

may involve non-linear arithmetic with stochastic ob-

jective and multiple demand constraints, like those in

temporal production processes. However, many pro-

duction processes, especially in manufacturing, have

models that are in steady-state and have only a sin-

gle demand constraint that can be solved more easily.

To close this gap, this paper specializes the approach

from (Krishnamoorthy et al., 2018) for a single de-

mand constraint to solve the stochastic optimization

problems for steady-state production processes de-

scribed using non-linear arithmetic.

More speciﬁcally, the contributions of this paper

are three-fold: (A) a specialized heuristic algorithm

called Stochastic Optimization Algorithm Based on

Deterministic Approximations (SODA) to solve the

problem of ﬁnding production process controls that

minimize the expectation of cost while satisfying

the deterministic process feasibility constraints and

stochastic steady state demand for the output prod-

uct with a given high probability. The proposed

algorithm is based on (1) a series of deterministic

approximations to produce a candidate set of near-

optimal control settings for the production process,

and (2) stochastic simulations on the candidate set

using optimal simulation budget allocation methods

(e.g., (Chen et al., 2000), (Chen and Lee, 2011),

(Lee et al., 2012)). (B) a demonstration of the pro-

posed algorithm on a use case of a real-world heat-

sink production process that involves 10 processes in-

cluding contract suppliers and manufacturers as well

as unit manufacturing processes of shearing, milling,

drilling, and machining with models from the litera-

ture that use non-linear physics-based equations. (C)

an initial experimental study using the heat-sink pro-

duction process to compare the proposed algorithm

with four popular simulation-based stochastic opti-

mization algorithms viz., Nondominated Sorting Ge-

netic Algorithm 2 (NGSA2) (Deb et al., 2002), Indi-

cator Based Evolutionary Algorithm (IBEA) (Zitzler

and K

unzli, 2004), Strength Pareto Evolutionary Al-

gorithm 2 (SPEA2) (Zitzler et al., 2001), and Speed-

constrained Multi-objective Particle swarm optimiza-

tion (SMPSO) (Nebro et al., 2009). The experimental

study demonstrates that SODA signiﬁcantly outper-

forms the other algorithms in terms of optimality of

results and computation time. In particular, running

over a 12-process problem using a 8-core server with

16GB RAM, in 40 minutes, SODA achieves a pro-

duction cost lower than that of competing algorithms

by 61%; in 16 hours SODA achieves 29% better cost;

and, in 3 days it achieves 7% better cost.

The rest of this paper is organized as follows.

Section 2 formally describes the stochastic optimiza-

tion problem over steady-state production processes.

SODA, including deterministic approximations, is

presented in section 3. Experimental results are pre-

sented in section 4. Finally, section 5 concludes with

some future research directions.

ICORES 2021 - 10th International Conference on Operations Research and Enterprise Systems

288

2 OPTIMIZATION OF

PRODUCTION PROCESSES

WITH SCFA

We now specialize the stochastic optimization prob-

lem from (Krishnamoorthy et al., 2018) for steady-

state processes with a single feasibility constraint over

a stochastic metric. The stochastic optimization prob-

lem for such processes assumes a stochastic closed-

form arithmetic (SCFA) simulation of the following

form. A SCFA simulation on input variable

X is a se-

quence y

= expr

,... ,y

= expr

where expr

,1 ≤ i ≤ n is either

(a) An arithmetic or boolean expression in terms of

a subset of the elements of

X and/or y

,. .. ,y

i−1

We say that y

is arithmetic or boolean if the expr

is arithmetic or boolean correspondingly.

(b) An expression invoking PD(

P), a function that

draws from a probability distribution using pa-

rameters

P that are a subset of the elements of

and/or y

,. .. ,y

i−1

We say that y

,1 ≤ i ≤ n is stochastic if, recursively,

(a) expr

invokes PD(

P), or

(b) expr

uses at least one stochastic variable y

,1 ≤

j < i

If y

is not stochastic, we say that it is deterministic.

Also, we say that a SCFA simulation S computes a

variable v if v = y

, for some 1 ≤ i ≤ n.

This paper considers the stochastic optimization

problem of ﬁnding process controls that minimize the

cost expectation while satisfying deterministic pro-

cess constraints and steady state demand for the out-

put product with a given probability. More formally,

the stochastic optimization problem is of the form:

minimize

X∈

E(cost(

X))

subject to C(

X)∧

P(thru(

X) ≥ θ) ≥ α

(1)

where

D = D

× ··· × D

is the domain for deci-

sion variables

X is a vector of decision variables

over

D, cost(

X) is a random variable deﬁned in terms

X, thru(

X) is a random variable deﬁned in terms

X, C(

X) is a deterministic constraint in

X i.e., a

function C :

D → {true, f alse}, θ ∈ R is a through-

put threshold, α ∈ [0,1] is a probability threshold,

and P(thru(

X) ≥ θ) is the probability that thru(

X) is

greater than or equal to θ

Note in this problem that upon increasing θ to

some θ

, the size of the space of alternatives that sat-

isfy the stochastic demand constraint in (1) increases

and hence it can be said that the best solution, i.e.,

the minimum expected cost is monotonically improv-

ing in θ

. We assume that the random variables,

cost(

X) and thru(

X) as well as the deterministic con-

straint C(

X) are expressed by an SCFA simulation S

that computes the stochastic arithmetic variable (cost,

thru) ∈

as well as the deterministic boolean vari-

able C ∈ {true, f alse}. Many complex real-world

processes can be formulated as SCFA simulations as

described in (Krishnamoorthy et al., 2017).

3 STOCHASTIC OPTIMIZATION

ALGORITHM BASED ON

DETERMINISTIC

APPROXIMATIONS

This section presents the Stochastic Optimization

Algorithm Based on Deterministic Approximations

(SODA). The problem of optimizing stochastic pro-

duction processes can be solved using simulation-

based optimization approaches by initializing the con-

trol settings and performing simulations to check

whether the throughput satisﬁes the demand with suf-

ﬁcient probability. But that approach is inefﬁcient be-

cause the stochastic space of this problem is very large

and hence this approach will typically converge very

slowly to the optimum solution. So, the key idea of

SODA is that instead of working with a large number

of choices in the stochastic space, we use determin-

istic approximations to generate a small set of candi-

date control settings and then validate these control

settings in the stochastic space using simulations.

An overview of SODA is shown in Fig. 1. To gen-

erate a small set of candidate control settings, SODA

performs deterministic approximations of the original

stochastic problem by deﬁning a deterministic com-

putation S

from the SCFA simulation S described

in section 2 by replacing every expression that uses

a probability distribution PD(

P) with the expectation

of that distribution. The deterministic approximations

cost

(

X) and thru

(

X) of cost(

X) and thru(

X), re-

spectively, can be expressed using S

. To optimize

this reduced problem, a deterministic optimization

problem (see (2)) approximates the stochastic opti-

mization problem of (1) is used as a heuristic.

minimize

X∈

cost

(

subject to C(

X)∧

thru

(

X) ≥ θ

(2)

where θ

≥ θ is a conservative approximation of θ.

Stochastic Optimization Algorithm based on Deterministic Approximations

289

Generate candidates with deterministic optimization

Run stochastic simulation

Heuristically

(exponentially)

inﬂate the demand

bound as a function

of the demand

satisfaction difference

Calculate exp. cost and demand

satisfaction probabilities

Is avg. cost min

until now?

Store run

Store candidate as

the best so far

likelihood

of demand

sat. suff.

high?

yes

can't say

Calculate discrete demand between when likelihood of

demand sat. was suff. high and when it was not / can't say

Store run

For all discrete demands

Generate candidates with deterministic optimization

Run stochastic simulation

Calculate exp. cost and demand

satisfaction probabilities

Is avg. cost min

until now?

Store run

Store candidate as

the best so far

likelihood

of demand

sat. suff.

high?

yes

can't say

Store run

Inﬂate

Deﬂate

For all Stored Runs

Run stochastic simulation for the

number of simulations allocated

Calculate exp. cost and demand

satisfaction probabilities

Is avg. cost min

until now?

Store candidate as

the best so far

likelihood

of demand

sat. suff.

high?

yes

Remove candidate

from candidate set

Reﬁne Candidates using OCBA-CO

Simulation budget allocation

based on OCBA-CO

Figure 1: Overview of SODA.

This deterministic approximation is performed it-

eratively such that the control settings found in the

current iteration are more likely to generate through-

puts that satisfy demand with the desired probability

than in the previous iterations. This is possible be-

cause of the inﬂate phase (left box of Fig. 1) and the

deﬂate phase (middle box of Fig. 1) of SODA.

Intuitively, the inﬂate phase tries to increase the

throughput to satisfy the demand with the desired

probability. When the current candidate control set-

tings do not generate the throughput that satisﬁes the

original user-deﬁned demand with a desired probabil-

ity, the demand parameter itself is exponentially in-

ﬂated such that this inﬂation does not render future

iterations infeasible. This is ensured by precomput-

ing the feasible throughput range, which is the inter-

val between the requested demand and the through-

put bound where the production process is at capac-

ity. This bound is computed by solving the optimiza-

tion problem in (3). Then, the demand parameter is

inﬂated within the throughput range. This inﬂation

of the demand parameter yields higher controls for

the machines and thus increases the overall through-

put. However, this may result in the throughput over-

shooting the demand in the stochastic setting, which

degrades the objective cost.

maximize

X∈

thru(

subject to C(

(3)

To overcome this, in the deﬂate phase, SODA

scales back the demand by splitting it into a number

of points, separated by a small epsilon, in the interval

between the inﬂated demand where throughput over-

shot the original demand and the previous demand at

which the last inﬂation occurred. Deterministic ap-

proximations are run for this lower demand to check

whether the throughput still satisﬁes the demand with

desired probability while yielding a better objective

cost. In this way, the inﬂate and deﬂate phases ﬁnd a

better demand threshold for which it can get the right

balance between optimum cost and demand satisfac-

tion with desired probability.

After the iterative inﬂate and deﬂate procedure,

more simulations may be needed to check if a promis-

ing candidate that currently is not a top candidate

could be the optimal solution or to choose an optimal

solution from multiple candidates. To resolve this,

these candidates are further reﬁned in the reﬁneCan-

didates phase (right box in Fig. 1). In this phase,

SODA reﬁnes the candidates in the candidate set gen-

erated using deterministic approximation by running

Monte Carlo simulations on them. To maximize the

likelihood of selecting the best candidate, i.e., can-

didate with the least cost and sufﬁcient probability

of demand satisfaction, this phase uses the Optimal

Computing Budget Allocation method for Constraint

Optimization (OCBA-CO) (Lee et al., 2012) for the

ranking and selection problem with stochastic con-

straints. OCBA-CO allocates budget among the can-

didates so that the greatest fraction of the budget is

allocated to the most critical candidates in the candi-

date set. This apportions the computational budget in

a way that minimizes the effort of simulating candi-

dates that are not critical or those that have low vari-

ability. This phase is performed iteratively for some

delta budget that is allocated among the candidates

using OCBA-CO and, in each iteration, the additional

simulations on the candidates can yield a better ob-

jective cost or a higher conﬁdence of demand satis-

faction. In each subsequent iteration, OCBA-CO uses

the updated estimates to ﬁnd new promising candi-

dates and allocates a greater fraction of the delta bud-

get to them to check if these candidates can be further

reﬁned.

In this way, SODA uses the model knowledge in

inﬂate, deﬂate, and reﬁneCandidates phases to pro-

vide optimal control settings of the non-linear pro-

cesses that a process operator can use on the manu-

facturing ﬂoor in a stochastic environment.

ICORES 2021 - 10th International Conference on Operations Research and Enterprise Systems

290

4 EXPERIMENTAL RESULTS

This section discusses experimental results that evalu-

ate SODA by comparing the quality of objective cost

and rate of convergence obtained from SODA with

other metaheuristic simulation-based optimization al-

gorithms. We used the SCFA simulation of the Heat-

Sink Production Process (HSPP) described in (Kr-

ishnamoorthy et al., 2017) in the experiments. The

SCFA simulation of HSPP is written in JSONiq and

SODA is written in Java. The deterministic approxi-

mations for SODA are performed using a system that

automatically converts the SCFA simulation of HSPP

into a deterministic optimization problem, which is a

deterministic abstraction of the original SCFA simu-

lation.

For the comparison algorithms, we used the

jMetal package (Durillo and Nebro, 2011). The al-

gorithms chosen for comparison include Nondomi-

nated Sorting Genetic Algorithm 2 (NGSA2) ‘(Deb

et al., 2002), Indicator Based Evolutionary Algo-

rithm (IBEA) ‘(Zitzler and K

unzli, 2004), Strength

Pareto Evolutionary Algorithm 2 (SPEA2) ‘(Zitzler

et al., 2001), and Speed-constrained Multi-objective

Particle swarm optimization (SMPSO) ‘(Nebro et al.,

2009). These are popular multi-objective simulation-

based optimization algorithms and they were cho-

sen because they performed better than other single

and multi-objective simulation-based optimization al-

gorithms available in jMetal. The swarm/population

size for the selected algorithms was set to 100. Also,

we ran these algorithms with the following operators

(wherever applicable): (a) polynomial mutation op-

erator with probability of (noOfDecisionVariables)

−1

and distribution index of 20; (b) simulated binary

crossover operator with probability of 0.9 and distri-

bution index of 20; and (c) binary tournament selec-

tion operator. For each algorithm, the maximum num-

ber of evaluations was set to 500,000 and the maxi-

mum time to complete these evaluations was set to the

time that SODA ran for. All these algorithms use the

SCFA of HSPP to perform the cost, throughput and

feasibility constraint computations. The computation

of the demand satisfaction information from expected

throughput is also similar to that of SODA. The cost

and demand satisfaction information is then used by

the jMetal algorithms to further increase or decrease

the control settings using heuristics. The source code

for SCFA simulation of HSPP, SODA algorithm and

the comparison using the jMetal package is available

at (Krishnamoorthy et al., 2020)

Initially, SODA was run with two different set-

tings. In the ﬁrst setting, SODA was run for 16 hours

and the number of candidates collected from the In-

Figure 2: Estimated average cost for the elapsed time of 16

hours.

ﬂateDeﬂate phase was 2,000. In the second setting,

SODA was run for approximately 72 hours (3 days)

and the number of candidates collected was 2,000.

The inﬂateDeﬂate was run for a certain amount of

time and to accumulate the required candidates within

this time, the bound on the deterministic constraints

over each production process in HSPP was increased

so that the throughput range was wide. Additionally,

all production processes were stochastic due to 5%

of standard normal noise added to the controls. Fi-

nally, the demand from the HSPP was set to be 4. The

comparison algorithms were also run with the same

(relevant) parameters.

Figure 3: Estimated average cost for the elapsed time of 3

days.

The data collected from the experiments include

the estimated average costs achieved at different

elapsed time points for SODA in two settings and all

the comparison algorithms. Figure 2 shows the esti-

mated average costs achieved after 16 hours whereas

Fig. 3 shows the estimated average costs achieved

after about three days. Each experiment was run mul-

tiple times and 95% conﬁdence bars are included at

certain elapsed time points in both ﬁgures.

It can be seen that both settings of SODA per-

form better than the comparison algorithms initially.

This is because SODA uses deterministic approxima-

tion to reduce the search space of potential candidates

quickly whereas the competing algorithms start at a

Stochastic Optimization Algorithm based on Deterministic Approximations

291

random point in the search space and need a number

of additional iterations (and time) to ﬁnd candidates

closer to those found by SODA.

As time progresses, SODA in both settings is able

to achieve much better expected cost in the inﬂat-

eDeﬂate phase than the other algorithms. Hence,

SODA converges quicker toward the points close to

the (near) optimal solution found at the end of the al-

gorithm. This is because SODA uses heuristics in the

inﬂate and deﬂate phases that reduce the search space

quickly, which allows SODA to quickly converge to-

ward a more promising candidate.

Also, the solutions found by SODA at the end

of the experiment are much better than those found

by the competing algorithms. After 16 hours, the

expected cost found by SODA was 29% better than

the nearest comparison algorithm (SMPSO) and 49%

better than the second best comparison algorithm

(NSGA2) (see Fig. 2). After three days though, the

advantage of SODA over the nearest comparison al-

gorithm (SMPSO) reduces to 7% and that to the sec-

ond best algorithm (NSGA2) reduces to 17% (see Fig.

3). This is not surprising since SMPSO and NSGA2

use strong meta-heuristics and given enough time

such algorithms will eventually converge toward the

(near) optimal solution found by SODA. Although,

it should be noted from Fig. 3 that SODA reaches

close to the optimal solution found at the end much

more quickly than its competition. Also, since the

x-axis in Fig. 3 is in log scale, it should be noted

that out of the total experiment time of about 259,200

sec, the other algorithms stop improving at around

180,911 sec whereas SODA continues to improve

for a longer time (until about 210,562 sec) due to a

good collection of candidates from the InﬂateDeﬂate

phase and performing simulation reﬁnements using

an optimal budget allocation scheme of OCBA-CO

in ReﬁneCandidates phase. We also ran the Tukey-

Kramer procedure for pairwise comparisons of corre-

lated means on all six algorithm types. By doing so,

we conﬁrmed that SODA was indeed better than the

other algorithms when the experiment ended.

We believe that SODA is sensitive to three cat-

egories of parameters of the production process viz.

(1) the throughput range as determined by the inter-

val between the requested demand and the throughput

bound where the production process is at capacity, (2)

the level of noise added to the controls of each pro-

duction process, and (3) the coefﬁcient used to inﬂate

the demand parameter, where the higher value cor-

responds to inﬂating more aggressively. To evaluate

SODA against the comparison algorithms for these

categories, we ran the experiment for different set-

tings of each of these parameters. For the through-

Figure 4: Estimated average cost for the elapsed time of

3 days for a wide throughput range where the noise level

and inﬂation jump is A: high (10%) and very aggressive

(1.3x); B: high (10%) and aggressive (1x); C: high (10%)

and less aggressive (0.8x); D: low (5%) and very aggressive

(1.3x); E: low (5%) and aggressive (1x); F: low (5%) and

less aggressive (0.8x).

put range, we chose a wide range where the capacity

of each production process of HSPP was high and a

narrow range where the capacity was lower. For the

level of noise, we chose 5% of standard normal noise

as before as well as a higher noise of 10% of standard

normal noise. Finally, the inﬂation jump was (a) more

aggressive when a larger fraction of multiplicative ex-

ponential factor was added to the current demand, e.g,

1.3x, (b) aggressive when a smaller fraction of this

factor was added to the current demand, e.g, 1x, and

this factor was added to the current demand, e.g, 0.8x.

To understand the effects of these parameters com-

prehensively, SODA and the comparison algorithms

were run for 72 hours (3 days). The time for the in-

ﬂateDeﬂate phase and the number of candidates col-

lected during this phase was maintained the same as

before. Also, the demand of HSPP was maintained at

4 so that we could perform a direct comparison of the

estimated average costs over the elapsed time across

these experiments.

Figures 4 and 5 show the estimated average costs

achieved after three days when the throughput range

was wide and when the throughput range was narrow,

respectively. To increase or decrease the throughput

range, the capacity of the production process was in-

creased or decreased such that the throughput bound

obtained by solving the optimization model in equa-

tion 3 would change accordingly. In Figs. 4 and 5

the ﬁrst row shows results for the case when the noise

level is 10% of the standard normal noise and the sec-

ond row shows results for the case where the noise

level is 5% of that noise. The ﬁrst, second and third

column of both these ﬁgures show results for the case

when the level of inﬂation jump was more aggressive,

aggressive and less aggressive, respectively. Since

ICORES 2021 - 10th International Conference on Operations Research and Enterprise Systems

292

Figure 5: Estimated average cost for the elapsed time of 3

days for a narrow throughput range where the noise level

and inﬂation jump is A: high (10%) and very aggressive

(1.3x); B: high (10%) and aggressive (1x); C: high (10%)

and less aggressive (0.8x); D: low (5%) and very aggressive

(1.3x); E: low (5%) and aggressive (1x); F: low (5%) and

less aggressive (0.8x).

the inﬂation jump occurs in the inﬂateDeﬂate phase

of SODA, the results from the comparison algorithms

remained the same in any particular row of Figs. 4

and 5.

Table 1 shows the percentage difference of the es-

timated average costs between SODA and the most

competitive comparison algorithm, SMPSO, at the

end of the experiment. When the throughput range is

wide and the inﬂation jump is more aggressive (see

Figs. 4A and 4D), SODA decreases the cost more

rapidly than when the inﬂation jump is less aggres-

sive (see Figs. 4C and 4F). This is because SODA is

able to quickly ﬁnd candidates that satisfy the demand

with sufﬁcient probability because of the more ag-

gressive inﬂation jump. The initial candidates found

by the more aggressive case (not shown here) have

high expected cost due to the throughput range be-

ing wide. Additionally, SODA is affected by greater

amount of noise since greater inﬂation is required to

satisfy the demand with sufﬁcient probability. This

yields worse candidates collected during the inﬂat-

eDeﬂate phase when the inﬂation jump is less aggres-

sive and the noise is high (see Fig. 4C). It seems that

when the throughput range is wide, the most effec-

tive strategy for inﬂation jump is for it to be neither

less nor more aggressive because the cost decreases

rapidly in the inﬂateDeﬂate phase and the candidates

collected in this phase give the best gain in the ex-

pected cost over the comparison algorithms at the end

of the experiment (see Figs. 4B and 4E).

When the throughput range is narrow and noise

is high, we can see that the more aggressive case

shown in Fig. 5A performs very poorly. A possi-

ble explanation is that the inﬂated demand keeps hit-

ting the throughput boundary due to more aggressive

inﬂation. This either yields higher expected cost or

for the interval explored by deﬂate not to be wide

enough for the expected cost to improve. This phe-

nomenon improves when the level of aggression is

reduced as shown in Fig. 5B and the candidates col-

lected in the inﬂateDeﬂate phase are the best for this

scenario when the inﬂation level is less aggressive as

shown in Fig. 5C. When the noise level is reduced, the

candidates collected in the inﬂateDeﬂate phase with

more aggressive inﬂation jump (see Fig. 5D) are bet-

ter as compared to when the noise level is higher (see

Fig. 5A). This happens because more candidates sat-

isfy the demand with sufﬁcient probability within the

narrow throughput range when the noise level is low.

Hence, when the throughput range is narrow, it seems

that the most effective strategy for inﬂation level is

for it to be between normal and less aggressive since

SODA achieves the best gain over the comparison al-

gorithms at this inﬂation level. This is because SODA

explores the narrow throughput range better for fea-

sible candidates and ﬁnds better optimal candidates

among them at this inﬂation level.

Table 1: Percentage difference of the estimated average cost

between SODA and the most competitive comparison algo-

rithm SMPSO found at the end of the experiment where

positive percentages mean SODA did better and negative

percentages mean SMPSO did better.

Inﬂation jump level

Wide throughput

range and

10% noise

Wide throughput

range and

5% noise

Narrow throughput

range and

10% noise

Narrow throughput

range and

5% noise

Very aggressive (1.3x) -16 2 -23 1

Aggressive (1x) 9 7 2 8

Less aggressive (0.8x) 5 2 -2 7

5 CONCLUSIONS

This paper presented an efﬁcient stochastic optimiza-

tion algorithm called SODA for the problem of ﬁnd-

ing process controls of a steady-state production pro-

cesses that minimize the expectation of cost while

satisfying the deterministic feasibility constraints and

stochastic steady state demand for the output product

with a given high probability. SODA is based on per-

forming (1) a series of deterministic approximations

to produce a candidate set of near-optimal control set-

tings for the production process, and (2) stochastic

simulations on the candidate set using optimal sim-

ulation budget allocation methods.

This paper also demonstrated SODA on a use case

of a real-world heat-sink production process that in-

volves contract suppliers and manufacturers as well

as unit manufacturing processes of shearing, milling,

drilling, and machining. Finally, an experimental

study showed that SODA signiﬁcantly outperforms

four popular simulation-based stochastic optimiza-

tion algorithms. In particular, the study showed that

SODA performs better than the best competing algo-

Stochastic Optimization Algorithm based on Deterministic Approximations

293

rithm by 29% after 16 hours and by 7% after three

days. The study also showed that SODA can solve

the optimization problem over a complex network of

production processes and SODA is able to scale well

for such a network. However, the efﬁciency of SODA

is limited by the total budget allocated to OCBA-CO,

which is an input parameter to the algorithm.

Future research directions include: (a) dynami-

cally executing the inﬂate deﬂate phase and candidate

reﬁnement phase of SODA to improve the exploration

of the search space; and (b) comparing SODA with an

existing Stochastic Optimization Algorithm Based on

Deterministic Approximations.

ACKNOWLEDGEMENTS

The authors are partly supported by the National Insti-

tute of Standards and Technology Cooperative Agree-

ment 70NANB12H277.

REFERENCES

Amaran, S., Sahinidis, N. V., Sharda, B., and Bury, S. J.

(2016). Simulation optimization: a review of algo-

rithms and applications. Annals of Operations Re-

search, 240(1):351–380.

Chen, C. H. and Lee, L. H. (2011). Stochastic Simulation

Optimization: An Optimal Computing Budget Alloca-

tion. World Scientiﬁc Publishing Company, Hacken-

sack, NJ, USA.

Chen, C.-H., Lin, J., Y

ucesan, E., and Chick, S. E. (2000).

Simulation budget allocation for further enhancing the

efﬁciency of ordinal optimization. Discrete Event Dy-

namic Systems, 10(3):251–270.

Dabney, J. B. and Harman, T. L. (2001). Mastering

SIMULINK 4. Prentice Hall, Upper Saddle River, NJ,

USA, 1st edition.

Deb, K., Pratap, A., Agarwal, S., and Meyarivan, T. (2002).

A fast and elitist multiobjective genetic algorithm:

NSGA-II. IEEE Transactions on Evolutionary Com-

putation, 6(2):182–197.

Durillo, J. J. and Nebro, A. J. (2011). jMetal: A Java frame-

work for multi-objective optimization. Advances in

Engineering Software, 42(10):760 – 771.

Klemmt, A., Horn, S., Weigert, G., and Wolter, K.-J.

(2009). Simulation-based optimization vs. mathemat-

ical programming: A hybrid approach for optimiz-

ing scheduling problems. Robotics and Computer-

Integrated Manufacturing, 25(6):917–925.

Krishnamoorthy, M., Brodsky, A., and Menasc, D. A.

(2018). Stochastic decision optimisation based on de-

terministic approximations of processes described as

closed-form arithmetic simulation. Journal of Deci-

sion Systems, 27(sup1):227–235.

Krishnamoorthy, M., Brodsky, A., and Menasc

e, D. (2015).

Optimizing stochastic temporal manufacturing pro-

cesses with inventories: An efﬁcient heuristic algo-

rithm based on deterministic approximations. In Pro-

ceedings of the 14th INFORMS Computing Society

Conference, pages 30–46.

Krishnamoorthy, M., Brodsky, A., and Menasc

e, D. A.

(2017). Stochastic optimization for steady state pro-

duction processes based on deterministic approxima-

tions. Technical Report GMU-CS-TR-2017-3, 4400

University Drive MSN 4A5, Fairfax, VA 22030-4444

USA.

Krishnamoorthy, M., Brodsky, A., and Menasc

e, D. A.

(2020). Source code for SODA algorithm.

Lee, L. H., Pujowidianto, N. A., Li, L. W., Chen, C. H., and

Yap, C. M. (2012). Approximate simulation budget

allocation for selecting the best design in the presence

of stochastic constraints. IEEE Transactions on Auto-

matic Control, 57(11):2940–2945.

Nebro, A., Durillo, J., Garc

ıa-Nieto, J., Coello Coello,

C., Luna, F., and Alba, E. (2009). Smpso: A new

pso-based metaheuristic for multi-objective optimiza-

tion. In 2009 IEEE Symposium on Computational In-

telligence in Multicriteria Decision-Making (MCDM

2009), pages 66–73. IEEE Press.

OpenModelica (2009). Efﬁcient Traceable Model-

Based Dynamic Optimization - EDOp, https://

openmodelica.org/research/omoptim/edop. Open-

Modelica.

Paraskevopoulos, D., Karakitsos, E., and Rustem, B.

(1991). Robust Capacity Planning under Uncertainty.

Management Science, 37(7):787–800.

Provan, G. and Venturini, A. (2012). Stochastic simula-

tion and inference using modelica. In Proceedings

of the 9th International Modelica Conference, pages

829–837.

Thieriota, H., Nemera, M., Torabzadeh-Tarib, M., Fritzson,

P., Singh, R., and Kocherry, J. J. (2011). Towards

design optimization with openmodelica emphasizing

parameter optimization with genetic algorithms. In

Modellica Conference. Modelica Association.

Thompson, S. D. and Davis, W. J. (1990). An integrated ap-

proach for modeling uncertainty in aggregate produc-

tion planning. IEEE Transactions on Systems, Man,

and Cybernetics, 20(5):1000–1012.

Xu, J., Zhang, S., Huang, E., Chen, C., Lee, L. H., and

Celik, N. (2016). MO

TOS: Multi-ﬁdelity optimiza-

tion with ordinal transformation and optimal sam-

pling. Asia-Paciﬁc Journal of Operational Research,

33(03):1650017.

Zitzler, E. and K

unzli, S. (2004). Indicator-based selection

in multiobjective search. In Proceedings of the 8th

International Conference on Parallel Problem Solving

from Nature, 2004, pages 832–842. Springer.

Zitzler, E., Laumanns, M., and Thiele, L. (2001). SPEA2:

Improving the strength pareto evolutionary algorithm.

Technical Report 103, Computer Engineering and

Networks Laboratory (TIK), Swiss Federal Institute

of Technology (ETH), Zurich, Switzerland.

ICORES 2021 - 10th International Conference on Operations Research and Enterprise Systems

294