Simulation-Based Discrete Optimization of Stochastic Discrete Event Systems Subject to Non Closed-Form Constraints

Abstract

This technical note addresses the discrete optimization of stochastic discrete event systems for which both the performance function and the constraint function are not known but can be evaluated by simulation and the solution space is either finite or unbounded. Our method is based on random search in a neighborhood structure called the most promising area proposed in and a moving observation area. The simulation budget is allocated dynamically to promising solutions. Simulation-based constraints are taken into account in an augmented performance function via an increasing penalty factor. We prove that under some assumptions, the algorithm converges with probability 1 to a set of true local optimal solutions. These assumptions are restrictive and difficult to verify but we hope that the encouraging numerical results would motivate future research exploiting ideas of this technical note.

Full text

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IEEE Transactions on Automatic Control, To appear
1

Abstract—This paper addresses the discrete optimization of
stochastic discrete event systems for which both the performance
function and the constraint function are not known but can be
evaluated by simulation and the solution space is either finite or
unbounded. Our method is based on random search in a
neighborhood structure called the most promising area proposed
in [7] and a moving observation area. The simulation budget is
allocated dynamically to promising solutions. Simulation-based
constraints are taken into account in an augmented performance
function via an increasing penalty factor. We prove that under
some assumptions, the algorithm converges with probability 1 to
a set of true local optimal solutions. These assumptions are
restrictive and difficult to verify but we hope that the
encouraging numerical results would motivate future research
exploiting ideas of this paper.

Index Terms— Simulation based optimization, discrete event
systems, most promising area, observation area, non closed-form
constraints

I. INTRODUCTION
ISCRETE optimization frequently arises in the design and
operation of real-life stochastic discrete event systems.
These problems are often difficult due to the complexity of
real-life discrete event systems, their intricate dynamics and
omnipresence of random disturbances. For some discrete
event systems, analytical solutions can be derived under some
simplification and often restrictive assumptions and, in this
case, the related discrete optimization problem becomes a
combinatorial optimization problem that can be solved with
usual discrete optimization techniques. Unfortunately, most
real-life stochastic discrete event systems do not have closed-
form solutions and discrete event simulation is usually the
only resort for performance evaluation.
Recently, Discrete Optimization via Simulation (DOvS)
techniques have been developed to solve stochastic
optimization problems for which the optimality is measured
based on simulation results and the decision variables are

Manuscript received September 18, 2007.
Jie LI was with LGIPM/ COSTEAM team – INRIA Nancy Grand Est,
Ecole Nationale d’Ingénieurs de Metz, Ile du Saulcy, 57045, Metz France. He
is now senior engineer with UFIDA Software CO. LTD, China, (e-mail:
lijie@ufida.com.cn).
Alexandre Sava is with LGIPM/ COSTEAM team – INRIA Nancy Grand
Est, Ecole Nationale d’Ingénieurs de Metz, Ile du Saulcy, 57045, Metz France
(e-mail: sava@enim.fr).
Xiaolan Xie is with Ecole des Mines de Saint-Etienne, 158 cours Fauriel,
42023 Saint-Etienne cedex 2, France (Tel: 33 4 77 42 66 95, Fax: 33 4 77 42
02 49, e-mail: xie@emse.fr). Corresponding author.
discrete (see [3, 13, 14] for surveys). Discrete optimization
problems with small size solutions space can be efficiently
solved using ranking and selection methods [4]. If discrete
event simulation is taken as a black box for performance
evaluation, some general meta-heuristics for deterministic
combinatorial optimization such as tabu search and genetic
algorithms can be applied directly. However such
straightforward applications neglect major barriers of discrete
optimization via simulation including the long computation
time for simulation and the noise of performance evaluation
using simulation. This has motivated development of
simulation-based optimization
unbounded solution space that takes into account issues
related to simulation budget and estimation errors. These
methods include but are not limited to ordinal optimization [5,
6], simulated annealing [1], nested partitions [12] and random
search in a most promising area [7, 8].
The results presented in this paper are closely related to the
COMPASS method (Convergent Optimization via Most-
Promising-Area Stochastic Search) proposed in [7] for
optimization problems for which the performance measures
are estimated via a stochastic discrete-event simulation and the
decision variables are integer ordered. It is based on random
search in a neighborhood structure called the most promising
area. It was proved that COMPASS algorithm converges to a
set of local optimal solutions with probability 1 under the
assumption that unlimited number of simulation observations
is performed for each visited solution. It was extended in [8],
which sets up a general framework of locally convergent
random-search algorithms that ensures the convergence of the
algorithms under some conditions. Convergence rates and a
central limit theorem were also established.
In this paper we consider discrete optimization problems for
which both the performance function and some constraint
functions need to be evaluated by simulation. Such constraints
will be called non closed-form constraints. Both finite and
unbounded solution space cases will be considered. Our
optimization approach combines the most promising area
concept of [7] with two new concepts: the moving observation
area and an augmented performance function. A finite moving
observation area around the current sample optimum is used to
identify the exploration area. It allows us to concentrate the
simulation budget on solutions near the current sample
optimum and avoid spending simulation budget on visited but
uninteresting solutions that are far away. We believe that this
could help improve the convergence to local optimum
solutions and quicker convergence than COMPASS algorithm
methods for large or
Simulation-based discrete optimization of stochastic discrete event
systems subject to non closed form constraints
Jie Li, Alexandre Sava. Xiaolan Xie
D

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IEEE Transactions on Automatic Control, To appear
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in [7] was observed in the extended numerical experiments
conducted in [10, 11]. Discrete optimization with non closed-
form constraints poses additional technical barriers due the
need for feasibility assessment. We introduce a penalty cost
function to combine the simulation-based constraints into an
augmented objective function. The optimization algorithms
are proved to converge to true local optimum under some
conditions.
Although most simulation-based optimization techniques
can be extended to address problems considered in this paper,
to the best of our knowledge, non closed-form constraints are
hardly explicitly addressed in discrete simulation-based
optimization methods. One exception is the constraint ordinal
optimization approach proposed in [9] that replaces non
closed-form constraints by some order constraints. As shown
in this paper, non closed-form constraints pose additional
difficulties in the investigation of the convergence properties
of the simulation-based optimization algorithms. Our
theoretical results are obtained under some assumptions that
are restrictive and difficult to verify. However we hope that
the encouraging numerical results given in [10, 11] would
motivate future research exploiting ideas of this paper.
The remainder of the paper is organized as follows. Section
2 introduces DOvS problems with non closed-form constraints
and presents the optimization algorithms. Section 3 establishes
formally the convergence properties.

II. DISCRETE OPTIMIZATION WITH NON-CLOSED FORM
CONSTRAINTS
A. Problem setting
The problems considered in this paper are discrete
optimization problems of the following form:

min
x Θ∈
subject to

([
xFE
where Θ = Φ∩Zd is the solutions space, Φ is a closed set in
d
ℜ , and Zd is the set of d-dimensional integer vectors, w
represents all random uncontrollable factors. For the sake of
simplicity, the constraint function is assumed to be a scalar
function. Extension to multiple non-closed form constraints is
immediate and all results of this paper hold for the general
case.
We assume that closed-form expressions for both the cost
function
[] ),(
wxGE
and the constraint function
are not always available. However both
[] ),(
wxFE
can be estimated by sample mean of repetitive
simulation observations of G(x,w) and F(x,w). Let
g(x)=E[G(x,w)] and f(x)=E[F(x,w)].
We call the problem defined above a Discrete Optimization
via Simulation (DOvS) problem with non closed form
constraints. The problem is said to be fully constrained if Φ
)],([
wxGE
(1)
α>
)],
w
(2)
[] ),(
wx
and
F
w
E
x

[] ),(
GE
is a compact set, i.e. Θ is a finite set. Otherwise, the problem
is said to be partially constrained or unconstrained.
When the closed-form expression for ( ) x
equivalent DOvS formulation without constraint (2) can be
obtained by introducing the constraint (2), i.e. ( )
the definition of Φ. For this reason, the DOvS of the following
form:
()
[]
wxGE
x
Θ∈
constraints. A general framework to solve this class of DOvS
problems was proposed in [7, 8].
f
is available, an
α>
xf
, in
,min
is called DOvS with closed-form
B. Fully constrained discrete optimization with non closed-
form constraints
The COMPASS algorithm proposed in [7] is adapted to
take into account non closed-form constraints. The new
ingredient is the introduction of a penalty cost function to
combine the simulation-based constraints into an augmented
objective function. The COMPASS algorithm of [7] is
summarized hereafter by taking into account the non closed-
form constraints.
More precisely, the algorithm does not evaluate all
solutions x∈Θ. Solutions to consider are generated
progressively. Let Vk be the set of all solutions visited through
iteration k. Further, the simulation budget, i.e. the number of
simulation replications also called observations of each
solution x, is allocated dynamically. Let
number of observations allocated to solution x up to iteration
k. Let
)(xGk
and
)(xFk
denote the sample mean of all
observations ),(
wxGi
and,(
xFi
),(
wxF
until iteration k (i.e. x∈Vk):
∑=
)(
k
xN
∑=
)(
k
xN
In order to take into account the non-closed form
constraints, a penalty of estimated constraint function is added
to the cost function and the sample optimum is determined
according to the following augmented cost function:
+=
)()(
SxGxH
kk
where Sk(x) is the penalty factor for constraint violation, (x)+ =
max(0, x).
The algorithm focuses
)(minarg
xHx
k
Vkx
∈
solutions to consider in a so-called most promising area:
{
:
kk
xxxxC
≤−Θ∈=
where
yx−
denotes the Euclidean distance between x and y.
At each iteration, a Simulation Allocation Rule (SAR) is
used to determine the additional number ak(x) of observations
to allocate to solution x at iteration k. As a result,
∑=
i
0
)(xNk
be the total
)
w
, i=1,…,k of ),(
wxG
and
=
)(
1
),(
1
)(
xN
i
i
k
k
wxGxG

=
)(
1
),(
1
)(
xN
i
i
k
k
wxFxF

+
−
))( )((
xFx
k
k
α

on sample optimum
*
k
=
and samples randomly additional
}
*
k
*
,
k
xyandVyy
≠∈∀−

=
k
ik
xaxN
)()(
is the total number of observations on

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IEEE Transactions on Automatic Control, To appear
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solution x until iteration k.

Algorithm 1 ( Fully constrained DOvS with non closed-form
constraints)
Step 1: Initialization
1.1 Set iteration count
0
=
k
.
1.2 Fix an initial solution
∈
Θ=
0
C
.
1.3 Determine
()
0
a
according to a Simulation Allocation
Rule (SAR).
1.4 Perform
()
0
a
simulation replications for x0, set
()()
000
xaxN
=
, calculate
Step 2: Let
1
+= kk. Sample new solutions and allocate
additional simulation budget
2.1. Samplemsolutions
kk
xx
,
1
2.2. Let
{
kkkk
xxVV
,,
211
K
−
2.3. Determine
( ) xak
according to the SAR for all
2.4. For all
k
Vx∈
, perform
update
( ) xNk
,
)(xGk
and
Fk
Step 3: Determine the new sample optimum
most promising area
k
C , and go to Step 2. ?

Θ
0 x
, set
{ }
0
x
0
*
00
,
xxV
==
and
0x
0x
)(0
0 xG
and
)(0
0 xF
.
km
.
x
}
,,
,
2
K
x
from
1
−
k
C
.
km
∪=
k
Vx∈
.
( ) xak
)
observations, then
(x
.
*
k x , update the
C. Partially constrained discrete optimization with non
closed-form constraints
The direct application of Algorithm 1 to this case is
impossible as the most promising area is unbounded. In this
subsection, Algorithm 1 is extended by introducing the
concept of moving observation area from which new solutions
are sampled.
At each iteration of the optimization process, the
observation area D is defined as the cube around the current
sample optimum x*. Thus,
[] qxqxD
ii
d
i
+−=Π =
) *() *(
1
,
where q >0 is the width of the observation area. Note that the
observation area D is a moving but finite observation window
during the optimization process. This is different from the
COMPASS algorithms of [7, 8] that use an observation
hyperbox around some given point and covering all visited
solutions. The observation hyperbox grows as the algorithm
progresses. With the moving observation area, the simulation
budget is devoted to the best sampled optimum x* and to
solutions that are in the nearby area of x* instead of those that
are far away from x*. We hope to be able to speedup the
convergence in terms of the total number of simulation
observations to true local optimal solutions of DOvS.

Algorithm 2 (Partially constrained DOvS with non closed-
form constraints)
Identical to Algorithm 1 with Step 2.1 replaced by :
2.1’. Sample m solutions
kmkk
x ,, x ,
1
x
K
2
from Ck−1∩Dk−1
where the observation area is
[] qxqxD
i
k
i
k
d
i
k
+−=Π =
) *() *(
1
,
.
III. CONVERGENCE OF DOVS ALGORITHMS
In this section, we prove that under some assumptions the
above algorithms converge to a set of local optimums defined
as follows:
( )(
( )
⎩
∈∀
xNHy
where
:
≤−Θ∈=
yxyxNH
of x. By definition, NH(x) includes x. Further, a solution x
which satisfies the constraint and whose neighbors y∈ NH(x)
do not satisfy the constraint, is a local optimum regardless the
value of the objective function.
The following assumptions are needed for the convergence
proofs of the DOvS algorithms.
⎡
∞→
N
∀
Θ∈
x
.
⎡
∞→
N
∀
Θ∈
x
.
Assumption 3:
⇒∞→
)(
xN
k
Assumption 4: The SAR guarantees that
/
−
∈
k
Vx
. In addition, with probability β > 0,
for each
kk
xNHxx
∩∪∈
−−
)(
11
Assumption 5: x 0 is a feasible solution.
Assumption 6:
=
∞→
k
Assumption 7: There is no solution x∈Φ such that f(x) = α.
Assumption 8: In Step 2.1 or Step 2.1’, (i) the solution
sampling is made independently for different iterations k and
(ii) for each iteration k, any solution x in the sampling area,
i.e. x ∈ Ck−1 in algorithm 1 and x ∈ Ck−1∩Dk−1 in algorithm 2,
is sampled with a probability of at least γ for some γ > 0.
Remark 1: Assumptions 1-2 are standard assumptions for
simulated-based optimization and are met by most simulation
observations. Assumption 3 is needed to ensure the
convergence to feasible solutions. The penalty factor can be
chosen either deterministically or stochastically. Assumption 8
can be easily satisfied as our sampling area is always bounded
whether the solution space is bounded or not. The uniform
sampling scheme proposed in [7] can be used if the DOvS
problem is defined on a polyhedron Φ.
Remark 2: Assumption 4 ensures that every new solution is
simulated at least once and that, with some positive
probability, additional simulation budget is allocated to the
solutions which are in the neighborhood of the last optimum
sample
11
optimization algorithms defined in this paper. It is
significantly weaker than related assumption made in [7]
( )
x
( )
y
( )
y
)
⎭
⎬
⎫
⎨
⎧
≤≤>Θ∈
=
forggandxfx
M
,:
αα

( ) {} 1
is the local neighborhood
Assumption 1:
1)(),(
1
N
lim
0
=
⎥⎦
⎤
⎢⎣
=
∑=
N
i
i
xgwxGP
,
Assumption 2:
1)(),(
1
N
lim
0
=
⎥⎦
⎤
⎢⎣
=
∑=
N
i
i
xfwxFP
,
∞→
)(
xS
k
, ∀
ak
Θ
1
ak
∈
)
x
x
.
, for all
)(
x
(
≥
1
{
k V
1
≥

}
k
V
**
.
+∞
)( lim
0 xNk
.
*
kx− including
*
kx−. This assumption is very weak for

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IEEE Transactions on Automatic Control, To appear
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which requires additional simulation runs for all solutions in
Vk. However, it is similar to related assumption made in [8]
brought to our attention during the review of this paper.
Remark 3: Assumption 5 is restrictive and difficult to
check. In practice, it seems reasonable for most experimental
systems for which a feasible benchmark solution, often the
current solution, does exist. If such a solution is not available,
the problem is solved in two phases by starting with the
maximization of f(x).
Remark 4: Assumption 7 is impossible to check a priori.
We describe here one practical approach to apply the
optimization algorithms and results of this paper. If the DOvS
algorithms ends with a solution x* such that the estimation
)(
xFk
of f(x*) is very close to α, then we then replace α
*
with α+ε with a small positive number ε such that
α+ε and re-run the DOvS algorithms.
Lemma 1. Under
∞=
∞→
k
and
)()( lim
xgxHk
k
∞→
Proof. Let ε = |f(x) – α|/2. According to Assumption 7, ε >
0. Furthermore, by Assumption 2, w.p. 1, there exists K > 0
such that
ε
<−
)()(
xFxf
k
for all k> K (For the sake of
0
()
k
Fx
>
Assumptions
then
∞→
k
if f(x)>α.
1-3
∞
and 7,
f(x)<α
if
)(lim
xNk
,
=
)(lim
xHk
if
=
simplicity, the expression “w.p. 1” will be omitted in the
following proofs). Hence,
Fk
)(xFk
– α<-ε if f(x)<α. As a result, if f(x)>α, by Assumption
1,
[
[
)()( lim
xgxG
k
k
∞→
If f(x) < α, Assumption 3 leads to

>
∞→∞→
kk
Lemma 2: Under Assumptions 1-7, w.p. 1, any solution
x∈Θ such that
solution, i.e. f(x) < α.
Proof. Assume that there exists x such that
f(x) < α. From Assumption 4, Nk(x) → ∞ as k increases. From
Lemma 1, w.p. 1,
=
∞→
k
5-6 and Lemma 1, w.p. 1,
lim
k
∞→
two relations imply that, w.p. 1, x can only be sample
optimum a finite number of times which contradicts
i.o. and concludes the proof. ?.
Theorem 1: If Assumptions 1-8 are satisfied and if Θ is
finite, then the sequence [ ]
k
converges w.p. 1 to the set M in the sense that w.p. 1, there
exists K > 0 such that
Mxk∈
, for all k> K.
)(x
– α > ε if f(x) > α and
]
]
))( )(()( lim
k
=
)( lim
k
=
xFxSxGxH
k
k
k
k
=−+=
+
∞→∞→
α

[] ∞=+
ε )
x
()(lim)( lim
SxGxH
k
k
k
.
?
*
kx = x i.o. (infinitely often) is a feasible
xxk=
*
i.o. and
∞
)(lim
xHk
. Similarly, by Assumptions
)()(
00
xgxHk
=
. The above
xxk=
*

∞
=0
*
k x
generated by Algorithm 1
*
Theorem 2: Under Assumptions 1-7 and if Θ is finite, w.p.
1,
)( min
Ux
∈
)(lim
k
→
*
k
xgxg
∞
∞
=

where U∞ is the set of feasible solutions for which an
unlimited number of simulations is performed in Algorithm 1.
Proof of Theorem 1: From the finiteness of Θ, the theorem
holds if {
..
=∉
oiMxP
k
that we prove by contradiction. If
Mxk∉
i.o., there exists an x∈Θ and x∉M such that
i.o. From Lemma 2, x is a feasible solution. Since x∉M, there
exists
Θ∩∈
)(xNH y
such that g(y) < g(x) and f(y) > α. Since
} 0
*
**
k xx =

zyxy
−≤=−
1
for all z≠y, y∈ Ck if
*
k x x =
and y∉Vk.
according to Assumption 8,
{
P
}
0
*
k
1
>≥∉=∈
+
γ
kk
Vy andxxVy

Since
*
k xx =
i.o., there exists K1> 0 such that y∈ Vk, ∀ k≥ K1.
x =
i.o. implies
By Assumption 4,
*
k x
∞→
∞→
)(lim
k
yNk
. Since
both x and y are feasible and g(y) < g(x), from Lemma 1, there
exists an integer K2 such that Hk(x) > Hk(y) for all k ≥ K2. As a
result, x can be sample optimum (i.e.
number of times. This is a contradiction. Therefore,
{
..
=∉
oiMxP
k
which concludes the proof. ?
Proof of Theorem 2: Since the set Θ of solutions is finite,
there exists a constant K1> 0 such that for all k≥ K1,
belongs to the set of solutions that are sample optimum
infinitely often. By Lemma 2,
⎩⎨⎧
∞
Uy
1
⎜⎝
∞∞
to infinity for all
∞
∈Ux
, by Lemma 1, there exists a constant
K2> K1 such that for all k ≥ K2,
*
kxx =
) only a finite
} 0
*
*
kx
∞
∈Uxk
*
, ∀ k≥ K1.
Let
⎭⎬⎫
=∈=
∈
∞
)(min)(:
ygxgUxS
. As
∞
U is nonempty and
finite,
0)(min
∈
y
)( min
∈
uy
2
>
⎟⎠
⎞⎛
−=
−
ygyg
uS
ε
. Since Nk(x) tends
ε+<
∞
∈
)( min
Uy
)(
ygxH
k
, ∀x∈S
and
ε+>
∞
∈
)(min
Uy
)(
ygxH
k
, ∀x∈ U∞-S. This implies that
Sxk∈
Remark 5: For DOvS with closed-form constraints, i.e.
DOvS without constraint (2), Theorems 1 and 2 hold under
only Assumptions 1, 4 and 8. Lemma 1 is just Assumption 1,
Lemma 2 trivially holds. The proofs of Theorems 1 and 2 are
the same. For this special case, although Theorems 1-2 are
similar to related results of [7, 8], our proofs are significantly
simpler than those of [7, 8].
We now consider the convergence of Algorithm 2 for
optimization of partially constrained DOvS problems with non
closed-form constraints.
The following additional assumption is needed.
Assumption 9. For the given starting point x0, there exists a
compact finite set Π (not necessarily known) and a positive
*
, ∀k≥K2 and concludes the proof. ?

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IEEE Transactions on Automatic Control, To appear
5
constant
δ > 0 such that
0x ∈Π
Θ
where Πc is the
Θ
I
and
δ+≥
)(),(
0 xgwxG
complementary set of Π.
Remark 6: This assumption ensures the existence of finite
optimal solution. It is more restrictive than related assumption
of [7, 8] that only requires
for any
∩Π∈
c
x
δ+≥
)()(
0 x
w
gxg
for
any
holds for most real-life applications. Assumption 9 directly
holds if the cost function ,(
wxG
that go to infinity as x goes to infinity. Many applications
have this property as x typically corresponds to resources
capacities and
),(
wxG
contains resource capacity related
cost. Even if Assumption 9 does not hold directly, under the
assumption of the existence of a finite optimal solution, we
can approximate
),(
wxG
by
where ε > 0 is small enough such that ( )
good approximation of g(x) in the area of interest. The new
sample cost function
,(
xU
≥
wxG
. More generally, if the error term
the cost function is too large for the solution determined by
Algorithm 2, we can reduce ε accordingly and re-run
Algorithm 2.
Θ∩Π∈
c
x
. Focus now on the case
0),(
≥
xG
which
) contains deterministic terms
0
),
ε
(
+
),(
xxwxGwxU
−+=ε
−

0
g xxx
is a
)
w
satisfies Assumption 9
as
0),(
0 xx−
ε
of
Let
⎥⎦
⎤
⎢⎣
⎡
Π=Π
=
)(
)(
1
,
i
i
d
i
bb
be the compact finite set
considered in Assumption 9, where
the (not necessarily known) lower and upper bounds.
Theorem 3 Under Assumptions 1-9, the sequence [ ]
generated by Algorithm 2 converges w.p. 1 to the set M in the
sense that w.p. 1, there exists K > 0 such that
for all k > K.
Theorem 4: Under assumptions 1-7 and 9, w.p. 1,
)( lim
xg
k
k
∞→
where
∞
U is the set of feasible solutions for which an
unlimited number of simulations is performed in algorithm 2.
Proof of Theorem 3: Note that Lemma 1 and 2 still hold. For
any sequence {V0, V1, . . .} generated by Algorithm 2, since
Θ⊆⊆
+1
kk
VV
, ∀k≥ 0,
V
)(i
b
and
)(i
b, i=1,…, d are
∞
k
=0
*
k x

Π∩∈Mxk
*
,
)( min
Ux
∈
*
xg
∞
=

U
∞
=
∞=
0
k
k
V
exists and
Θ⊆
∞
V
.
From lemma 1 and assumptions 5-6, there exists a constant
K1>0 such that for all k≥K1,
Hk
definition of
)(xHk
and Assumption 9,
c
Vx
∩Θ∩Π∈∀
. Thus,
(
xH
k
δ
≥
<−
)()(
00
xgx
. From the
δ
+
)()(
0 x
(
z
gxHk
)
≥
,
k
)min
Vz
∈
()
0
HxH
kk
k
>
, ∀ k
≥ K1 and
for all k ≥ K1, which implies
k
c
Vx
∩Θ∩Π∈∀
. Thus,
Π∈
*
k x
V
and
+
Π⊆
k
D

+
Π∪⊂
1
Kk
V
, where
⎥⎦
⎤
⎢⎣
⎡
Π⊆Π
=
)(
)(
1
,
i
i
d
i
bb
and
⎥⎦
⎤
⎢⎣
⎡
+−Π⊆Π
=
+
qbqb
i
i
d
i
)(
)(
1
,.
Since
∞<
1
K
V
and
∞<Π+
, the set V∞ is finite, i.e.
∞<
∞
V
.
Since
{
x
k
*
xk
∉
x =
Theorem 1. ?
Proof of Theorem 4: From the proof of Theorem 3, there
exists K1>0 such that
k
V
Π∈
*
k x
∩
for all k≥ K1, the theorem holds if
} 0.
=
o
that we prove by contradiction.
.
o
, there exists x∈Π and x∉M such that
.
*
Π∉
iMP
If
.
iM
Π∩
*
k x
i.o. The remaining proof is the same as that of
+
Π∪⊂
1
K
V
for all k≥K1
where
⎥⎦
⎤
⎢⎣
⎡
+−Π⊆Π
=
+
qbqb
i
i
d
i
)(
)(
1
,
. As a result,
∞
U is a
nonempty finite set and
similar to that of Theorem 2 with Θ replaced by the finite set
+
Π∪
1
K
V
. ?
∞
∈Ux0
. The remaining proof is
IV. CONCLUSION
In this paper we proposed a simulation based algorithm for
optimizing stochastic discrete event systems where both the
performance measure and some constraints are estimated by
simulation. Our approach combines most promising area
concept proposed in [7] with two new concepts: the moving
observation area and an augmented cost function. The non-
closed form constraints are taken into account in an
augmented performance function by means of an increasing
penalty factor. Moving observation area is introduced to
naturally concentrate the simulation budget on solutions in the
nearby area of the current sample optimum. Convergence
properties are established under reasonable conditions and
with very simple proofs.
The algorithms proposed in this paper have been applied to
the optimization of a multi-stage serial production-distribution
system controlled by base stock policies (see [11]). The
algorithms were first used to determine the optimum value of
base stock levels in order to minimize the inventory holding
costs and demand backlogging cost. The proposed algorithms
were shown to converge much faster than the COMPASS
algorithms of [7]. The algorithms were then applied to the
minimization of inventory holding costs subject to fill rate
constraints. The algorithms converge to fairly good feasible
solutions in a surprisingly small number of simulation
observations, in less than 1000 simulation observations for a
5-stage system. Our algorithms have also been applied in [10]
for optimization of various control strategies of serial supply
chains. Quick convergence was observed and the resulting
solutions follow properties of optimal solutions observed in
other supply chain studies.
The weak conditions on the SAR, the solution sampling
scheme and the selection of the observation areas
(Assumptions 4, 6, 8) allow non uniform sampling of the most