where ;. and B are given maximal production and preventive maintenance rates of an individual ma-
chine respectively. Our decision variables are the production rate and preventive maintenance rates
over time and K a is the set of admissible decisions
at the mode a. By controlling u , one decreases
the machines failure frequencies and hence im- proves the system availability. Let G
a, a, x, u, u be
the cost rate de ned as follows:
G
a, a, x, u, uc
`x`c~x~ca, ∀a3M, 6 where c
` and c~ are respectively, cost incurred per unit produced part for positive inventory and back-
log, x `max0, xt, x~maxxt, 0. The
constant c a is used to penalize the repair and pre-
ventive maintenance activities and is de ned as
follows: c
aa 1
c3a2c. with a1a2ma 7
where c3 and c. are the costs of a machine repair and maintenance respectively.
Our objective is to control the production rate ut and the preventive maintenance rate
ut to minimize the expected discounted cost given
by
J
a, x, uE
GP
= e
~otGkt, x, udt D x0x, a
0a, k0 a
H
8 subject to constrains given by Eqs. 1
}6. The value function of such a problem is
v
a, x inf
u
|Ka J
a, a, u ∀a3M.
9 A su
cient condition for optimal control states that the value function given by 9 satis
es the following Hamilton
}Jacobi}Bellman HJB equa- tions:
ova, r min
u,u
o
Ka H
a, x, v , x, v
x
a, x, u, u ∀x 3 R1`m, a3M
10 where v
x
x, a is the partial derivative of v . The
optimality conditions given by 10 and elementary properties of the value function can be found in
Kenne [8] and Boukas and Kenne [13]. The optimal control policy uH , uH denotes
a minimizer over K a of the right-hand side of Eq.
10. This policy corresponds to the value function described previously. Then, when the value func-
tion is available, an optimal control policy can be obtained from Eq. 10. Some authors [4] used
numerical approaches, restricted to small size sys- tems for example a one or two machines manufac-
turing system producing one part type to show that the hedging point policy remains valid and it
depends on the age of the machine. It is now well known that the analytical solution of Eqs. 10 for
obtaining the value function and the related opti- mal control policy, is almost impossible. Instead of
solving Eq. 10, either analytically or numerically, we propose an alternative solution which is based
on the heuristic control policy presented in the next section.
3. Heuristic control policy The experimental design approach and response
surface methodology with simulation experiment has been proved to be useful tools in the control of
manufacturing systems [5]. The purpose of this paper is to
combine such simulation-based models with the modi
ed hedging point policy as described by Kenne and Gharbi [5] for the
case of one-machine, one-product manufacturing system. In Section 3.1 we
rst extend the modied hedging point policy, presented by Kenne and
Gharbi [5], to multiple and non-identical machines manufacturing systems. Machines are said to be
non-identical herein if the transition rates of their stochastic process are di
erent. The complex- ity of the proposed approach is well illustrated
in such a situation where the optimal production control problem remains an open question in the
literature [8]. Next Section 3.2, the modi ed
hedging point policy is extended to the case con- sidered in this paper multiple machines with the
same transition rates called herein identical ma- chines.
278 A. Gharbi, J.P. Kenne
Int. J. Production Economics 65 2000 275 }287
3.1. Multiple and non-identical machines systems The modi
ed hedging point policy, presented later, is described by the following parameters:
X The value of the threshold or number of parts
to maintain in stock to hedge against machines breakdowns.
Ai The age of the machine i, at which it is neces- sary to stock parts. Before this age, the ma-
chine i is assumed to be new, and a production at the demand rate is suggested if the other
machines are new.
Bi The age at which the machine i is sent to preventive maintenance when the threshold
value X is achieved. In such a situation, the machine will be sent to preventive maintenance
randomly. There is a random delay given by a random distribution with mean equal to 1B
from the machine age Bi to the maintenance
time. With these parameters, the proposed control policy
states the following: 1. Preventive maintenance policy: The maintenance
rate ui can be described by the following
machine-age-dependent policy: 1.
C if ait Bi for all i3M1,2, mN then ui 0;
1. C if ait
Bi ui , a
G
B if xtX,
otherwise. 11
2. Production control policy: The production rate u
can be described by the following machine- age-dependent policy:
1. C if ait
Ai for all i3M1,2, mN u
., a
G
a;. if xt0, d
if xt0, if xt0,
12
1. C if there exist a machine i such that ait
Ai u
., a
G
a;. if xtX, d
if xtX, if xtX.
13 With this parameterized control policy depending
on Ai, Bi and X, i1,2, m, the best approxima- tion of v
a, x, given by 9, is found for some values
