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Optimization of a maintenance strategy with considering the influence of the production plan on the manufacturing system degradation
Proceedings of the 13th IFAC Symposium on Information Control Problems in Manufacturing Moscow, Russia, June 3-5, 2009
Optimization of a maintenance strategy with considering the influence of the production plan on the manufacturing system degradation Hajjej Zied*, Dellagi Sofiene**, Rezg Nidhal*** INRIA Nancy Grand Est /LGIPM – Université de Metz, Ile du Saulcy, 57045 Metz Cedex - FRANCE (e-mail: *[email protected] - **[email protected] ***[email protected]) Abstract: This paper deals the degradation of the machine according to the production rate. An optimal maintenance strategy is developed in combination with the production plan for a manufacturing system satisfying a random demand. Firstly, for a given randomly demand, we established ,with a constrained stochastic production-planning problem under hypotheses of inventory variables, an optimal production plan which minimizes the average total holding and production costs. Secondly, using the optimal production plan obtained and its influence on the manufacturing system failure rate, by analytical study we established an optimal maintenance scheduling which minimizes the maintenance total expected cost. Finally, a numerical example is studied in order to apply the developed approach. Keywords: Failure rate, random demand, linear quadratic model, average cost, Minimal repair 1.
INTRODUCTION
Recently the problems treating the combination between maintenance and production plans represents one interesting frame work and attracts several researches . Actually the development of an aggregated optimal production/maintenance plan which minimizes the total cost including production, inventory and maintenance is one of the first actions of a hierarchical decision making process. The basic idea of this plan is to find the optimal production plan and maintenance strategy required by the company to manufacture their products which satisfy a random demand over future periods. It’s easy to note the complexity of this task since the various uncertainties associated with this decision process. These uncertainties are usually due to externs and interns factors. The randomly demand, which induce the incapacity of knowing exactly the demand behaviour through future periods, can be considered as an extern factor. The material availability variation can be considered as an intern factor. More then it’s interested to develop an optimal maintenance strategy with considering the manufacturing system degradation according to the production rate. In this context Silva,F. and Wagner, C.(2004) deals with a chance-constrained stochastic productionplanning problem under hypotheses of imperfect information of inventory variables. The optimal production plan obtained by the minimizing of the expected cost. The stochastic nature of the system is due to machines that are subject to break-downs and repairs or maintenance actions. The maintenance policies proposed in the literature are mainly policies of the critical age of a machine or a set of machines. These policies are based on models describing the degrading law of the equipment. Valdez-Flores and Feldman (1989) distinguish four classes of models: inspection models, models of minimal repair, shock models and replacement models. The cost/time of maintenance/repair is supposed to
978-3-902661-43-2/09/$20.00 © 2009 IFAC
be known and consequently the impact of a maintenance/failure cannot be analyzed. Under these conditions, it can be shown that the optimal policy is the one of critical age. If consists in carrying out a preventive maintenance action on a machine at its critical age. In this context Pham and Wang (1996) suggest that maintenance can be classified according to the degree to which the operating conditions of an item are restored by maintenance. They proposed five categories: Perfect repair or perfect maintenance, minimal repair or minimal maintenance, imperfect repair or imperfect maintenance, worse repair or maintenance and worst repair or maintenance. On the other hand, maintenance becomes even more significant with the production control policies implementation like Just-in-Time, which require the availability of machines at the right time. The traditional approach which dissociates maintenance and production is not satisfactory any more. An integrated approach of maintenance and production control becomes necessary. In this context Buzacott and Shanthikumar (1993) proved the importance of the choice of the maintenance policy for the minimization of the total cost. Rezg et Al. (2004) presented a common optimization of the preventive maintenance and stock control in a production line made up of N machines. In the same context of integrating maintenance and production Rezg and al. (2008) presented a mathematical model and a numerical procedure which allows determining a joint optimal inventory control and age based preventive maintenance policy for a randomly failing production system. New maintenance/production strategies by taking into account the context of subcontractor are studied by (Dellagi, S. et al., 2007). Dellagi, S. et al.(2007) developed and optimize a new maintenance policy with taking into account a machine subcontractor constraints.
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10.3182/20090603-3-RU-2001.0497
13th IFAC INCOM (INCOM'09) Moscow, Russia, June 3-5, 2009
This paper presents an optimal production and maintenance plan. It is strongly motivated by the lack of tools to assess the production system in the presence of maintenance activities. Indeed, the interaction between maintenance policies and production control is rarely studied in the literature. There are few pieces of work on the simultaneously optimization of maintenance and production policies for manufacturing systems. In particular, Cheung and Hausmann (1997) consider the simultaneous optimization of the strategic stock and the maintenance policy of the critical age type. In the literature the consideration of the materiel degradation according to the production rate is infrequently studied. That’s why, in order to establish an optimal maintenance strategy, we need to take into account the influence of the production plan on the material failure rate. In fact, in this work we established an economical production plan and we used it in the determination of the optimal maintenance strategy. The paper is organized as follows. Section 2 describes the problem and the working assumptions. Section 3 considers the single machine case and presents the problem formulation. In section 4 we show the influence of manufacturing system degradation on the optimal production plan determined. In section 5, considering the influence of the production plan in the manufacturing system degradation, we developed an analytical model in order to determine the optimal maintenance plan. We presented a numerical example, in order to apply the analytical results. Finally we conclude in section 6. 2. PROBLEM STATEMENT We are concerned with the problem of the optimal production planning problem formulation of a manufacturing system composed of machine M which produces a single product in order to meet the random demand d law characterized by a Normal distribution facing the system at a minimum cost. The Normal mean and standard deviation parameters are respectively and σd2. The problem is illustrated in Figure.1. The machine M is subject to a random failure. The probability degradation law of machine M is described by the probability density function of time to failure f(t) and for which the failure rate λ(t) increases with time and according to the production rate U(t) . Failures of machine M can be prevented by a preventive maintenance action which is scheduled according to its history. Our first objective is to establish an economical production plan satisfying the randomly demand with an. Secondly, using the optimal production plan obtained, we determined the optimal preventive maintenance period. The use of the optimal production plan in maintenance study is justified by the fact of taking into account the influence of the production rate on the evolution of the failure rate of the machine.
d
Machine
S Fig. 1. Problem description
3.
PROBLEM FORMULATION
3.1 Notation Cpr : unit production cost Cs : holding cost of a product unit during the period k Cλ : penality degradation cost CRM: minimal repair cost fi(t) : probability density function of time to failure for the machine H : the finite production horizon c2 : preventive maintenance action cost c1 : corrective maintenance action cost mu : monitory unit Ri(t) : Reliability function S(k) : The stock level at the end of the period k Umax : the maximal production rate U(k) : The production rate for period k Z : the total expected cost including production and inventory over a finite horizon H C : the maintenance total cost expected per time unit α : Probabilistic index; related to the customer satisfaction ∆t : period length En (T/∆t): floor function of a (T/∆t): greatest integer 3.2 Production policy This paper considers a single stochastic inventory balance system. The idea is to minimize the expected production and holding costs over a finite time horizon [0, H]. It’s assumed that the horizon H is portioned equally into H periods. The demand is satisfied at the end of each period. Thus, this kind of problem can be formulated as a linearstochastic optimal control problem under threshold stock level constraint. This subsection derives an analytical model for evaluating the total expected cost including production and inventory over a finite horizon H. In order to formulate the analytical model we assume that: • •
Storage and production costs Cpr , Cs are known and constant the demand Standard deviation σd(k) and the demand mean (k) for each period k are known and constant
We recall that our objective is to determine the optimal production plan over a time horizon H. Formally our problem is presented as following: min
1 , 2 , … . … .
The system model is described by a hybrid state with continuous component, the dynamic of the stock. The state equation of the stock level is given by:
1 ! 0 #
Where S0 is a given initial stock.
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(1)
13th IFAC INCOM (INCOM'09) Moscow, Russia, June 3-5, 2009
Respecting the machine capacity constraint, the production rate of every period k cannot exceed a given maximal production rate Umax. The expected cost including production and holding costs for the period k is given by: $%
% , &%
'( · & * '+, · *
(2)
Remark: The use of quadratic cost that allows penalizing both excess and shortage of inventory level Since that the total expected cost including production and inventory over a finite H periods is expressed by: ∑./0 %1#
%$* 2 '( 3 * ∑./0 %1# 4'( 3 '+, 3 * 5 (3)
Remark:
C(U(H))2 is not included in the cost formulation because we don’t consider the production command at the end of the horizon H. Since that our problem becomes: * 678 4'( 3 * ∑./0 %1# 4'( 3 '+, 3 * 55 (4)
Such that,
1 !
undisturbed by any repair of failure. It is assumed that the repair and replacement times are negligible. If we assume that λ(T) represents the machine failure rate I function and E F G# H .Then, the expected cost of the one interval [0,T] is: . 'J 1 K0 . E F K* K0 · E L N K* M Thus, the total expected cost until time H is: ' O O. 'J 1 O · 4K0 · E PO K* 5 · Q
•
The following figure describes dynamic system evolution in discrete time
(U(1),d(1)) (U(2),d(2))
(N=1, 2….)
(5)
Maintenance policy optimization
4. INFLUENCE OF MANUFACTURING SYSTEM DEGRADATION ON THE OPTIMAL PRODUCTION PLAN
0 A A BCD
K=2
[
In this subsection, we want to optimize the maintenance strategy adopted which is preventive maintenance with minimal repair. According to the production plan established during H periods, the objective is to determine the optimal partition number N* for which the preventive maintenance must be done. We noted that N* can exceed H, it means that no preventive maintenance to do in this case. It’s noted that we use in the maintenance study the optimal production plan obtained from last subsection. It’s clear that the maintenance policy is tightly related to the system degradation. That’s why we adopted the optimal production plan obtained in last subsection in order to take into account the influence of the production rate on the failure rate λ(t).
9:;<= > 0? >@
K=1
I
It has proven the existence of an optimal preventive maintenance period N*T in the case of increased failure rate (Patrick,L.2000)
Under the following constraints:
0
W
RS ·GX T U VU YRZ
In this section, a constrained state-space stochastic optimal control problem is formulated .It is used to represent a constrained aggregate production planning problem under the stock level constraint. At the same time, this section deals the influence of maintenance cost on the variation of the optimal production plan.
K=H (U(H),d(H))
4.1 Analytical study
t
Fig. 2. Discrete time
In order to justify this influence two cases are studied: 3.3 Maintenance policy
case1: we don’t consider the penalizing of the degradation
• The Maintenance cost model in the case of minimal reparation policy The maintenance strategy under consideration is the well known preventive maintenance with minimal repair (Faulkner, L.L.,). More precisely, the machine will operate over a given on a horizon H. the maintenance policy adopted is as follows: To maintain a unit, an interval H [0, H],is portioned equally into N parts T in which it is replaced at times k.T (k=1,2,….N) and NT=H. Then, we consider the replacement with minimal repair at failure we can consider that replacement. The unit is replaced or preventive maintenance actions are practiced at periodic time’s k.T and the unit is as good as new in each replacement or preventive maintenance action. When a unit fails between preventive maintenance actions, only minimal repair is made, and hence failure rate remains
case2: we considered the penalizing of the degradation In practice, the model provides a linear decision rule that allows the determination of an aggregate inventory and production. This rule is derived from the minimization of a quadratic production cost subject to linear equations that represent the balance among inventory and production components. We recall that our problem formulated in subsection 3.2 is:
678 * 678 4'( 3 * ∑./0 %1# 4'( 3 '+, 3 * 55 (6) Under the constraints:
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13th IFAC INCOM (INCOM'09) Moscow, Russia, June 3-5, 2009
1 ! 9:;<= > 0? > \
0 A A BCD Thus we can simplify our problem as: d/0 min] Z min] _'( 3 Sa H * ∑e1# 4'( 3 Sa k * '+, 3
u k * 5 '( 3 σh * 3
d d/0 *
i
(7)
With = mean stock level at the end of the period k under the
same constraints
Adding the penalizing of the degradation and the maintenance costs to problem. Our problem formulation becomes: d/0 min] Z min] _Ck · Sa H * ∑e1# 4Ck · Sa k * Clm 3 u k * 5 Ck 3 σh * 3
d d/0 *
* 'T 3 ∑. %1# H% i
{d1=431, d2= 444, d3= 442, d4= 340, d5=392, d6=400, d7= 350, d8=400, d9=350, d10= 370, d11= 395, d12= 415, d13= 431, d14= 444, d15= 442, d16= 340, d17= 392, d18= 375} (ii) the following data are used for the other parameters: Cpr =3 mu, Cs =2 mu/k, Umax=500 pieces (iii) Additional information: the standard deviation of the demand is σd=14 and the customer satisfaction degree, associated with the stock constraint, is equal to 95% (α=0.95). Applying the numerical procedure we obtained the optimal production plan for the two cases: Case 1: we don’t considered the penalizing of the degradation (Cn =0) Case 2: we considered the penalizing of the degradation (Cn =3000) The result is exhibited in figure 3. WƌŽĚƵĐƚŝŽŶƉůĂŶʄсϬ WƌŽĚƵĐƚŝŽŶƉůĂŶʄсϯϬϬϬ
ϲϬϬ
(8)
ƵƉƉĞƌďŽƵŶĚ
ϱϬϬ
Under the constraints:
ϰϬϬ
1 !
ϯϬϬ
9:;<= > 0? > \
ϮϬϬ
0 A A BCD
ϭ
For each period k we adopted the production rate U(k) established from the optimal production plan. Since that the failure rate evolves in each interval according to the production rate adopted in the interval and the failure rate cumulated at the beginning of the interval (Fig. 4). Formally the failure rate in the interval k is expressed as following: s t u0, ∆w
5.
ϵ
ϭϭ ϭϯ ϭϱ ϭϳ ϭϵ
Ŭ;ŵŽŶƚŚͿ
OPTIMISATION OF THE MAINTENANCE POLICY
5.1 Analytical Study
H 0 H#
xyz
ϳ
We noted that the optimal production plan obtained in case 2, is weak compared to the production plan obtained in the case 1.The result is scheduled. When we consider the penalizing of the degradation with the influence of the production rate on the degradation degree we have to decrease the production rate.
(9)
With tinf(k) represents the lower bound of interval k % · H ∆H% BCD nk(t) is expressed as following : 8 % · H H% H%/0 ∆
ϱ
Fig. 3. Optimal production plan
with nk(t) represents the failure rate at time t .
H% H%/0 opq % rH%
ϯ
{u0, ∆w
* The term 'T · ∑. %1# H% represent the penalizing degradation during horizon H.
4.2 Numerical Results A company, whose sales are subject to the effects of seasonal fluctuating demand, tries to develop an aggregated production plan which minimizes total costs over a finite planning horizon: H=18 months. The strategy discussed in previous section has been employed here to provide a decision policy for managers. A simple example of a multi-period, single product aggregated production planning problem is formulated by the generalized deterministic model. (i) the monthly mean demand % follows bellow
In last section, we have concluded that it exists a tightly influence of the production plan on the degradation degree of the machine. That why we focused on the joint optimization strategy in which we consider the maintenance strategy based the optimal production plan obtained. In fact, in this study, firstly we determine the optimal production plan minimizing the average production cost. Secondly using this optimal production plan in the maintenance cost formulation we will determine the optimal preventive maintenance strategy characterized by the optimal period N*T. ∆t k=1
k=2
H
0 T
2T
(N-1)T
Production periods
H Maintenance periods NT
En(T/∆t) Fig.4. Failure rate during H
The analytical model, in which we calculate the total cost expected of maintenance action, is developed under the following assumptions:
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o Probability density functions of time to failure, time to maintenance activities is known
13th IFAC INCOM (INCOM'09) Moscow, Russia, June 3-5, 2009
o Costs C2 and C1 incurred by the actions of preventive and corrective maintenance are known and constant, with C1>>C2 o Preventive maintenance is performed as soon as the age of the machine reaches T o Failures are detected instantaneously The failure rate at period T is expressed as following: H I H|pL W N ∆}
With
~L
W NS ∆}
xyz
(10)
U
With N=1,2,…..,
(11)
A M/0
~L
4
5
6
7
8
9
431
444
442
340
392
375
392
400
350
U
439
399
399
390
320
320
320
380
300
k
10
11
12
13
14
15
16
17
18
370
395
415
431
444
442
340
392
375
320
390
410
440
480
444
490
495
400
*
T=H/N
.
MY0
3
D U
}
S
2
*
I
Z
1
k D
∆U
}
Table 1. Optimal production plan
;
I
3 G|pL W N∆U H
We take the same data as previous section 4 i.e. the following data (Cpr =3 mu, Cs =2 mu/k, Umax=500 pieces). Applying the numerical procedure in the formulation problem established in subsection 3.2 we obtained the optimal production plan and the optimal maintenance period which are exhibited respectively in table1, and figure 5.
The analytic expression of the total cost expected of maintenance actions is defined by:
W
W NS }
xyz
O minimizing ' O if M A
Since that we obtain, if it exists, the optimal period N*.T, for which we practiced the preventive maintenance action.
|p
~L
1 We find an optimal partition number N* which minimizes C(N).
with: ' O O · K0 · G# H K*
∑o10∆} G# Ho G|pL W N∆U
5.2 Numerical example
Consequently, the problem is expressed as:
With I E F G# H
xyz
3 H
M
' O O · uK0 · E F K* w
.
Since that we can established the total cost expected of maintenance action
M
W
|pL N
∑o10 ∆} G#
With M O · E L N ! O 1 · E
7 FP∆ t 1,2, … … , , . . O
' O O · K0 · G# H K*
U
Lemma 1:
· H
En (T/∆t): integer part of T/∆t
O min ' O
W
|pL N
I
2 E F 7 L∆UN 3 ∆ 3 H# ∑o10 ∆} G# o
W NS }
xyz
3 H (12)
The failure rate at time t: Ho Ho/0 Δ · H xyz
We use the optimal production plan obtained in table 1 in the total cost expected of maintenance actions C(N) formulated in subsection 5.1. Concerning numerical example, we suppose that the failure time of machine M has a degradation law characterized by a Weibull distribution. The Weibull scale and shape parameters are respectively β=100 and α=2. The cost associated with a corrective and preventive maintenance action are respectively C1=3000 mu C2=500 mu monetary.
Since that we can express λ t as following: Ho H# # o · H
We recall that we have formulated λi(t) expression as following :
With: o ∑o/0 10
Ho H# # ∑o/0 10
xyz
· H ∆
xyz
@ U /0
0 0 and H# # H# Since that we obtained: W ∆}
|pL N
E F ∑o10
∆U G# Ho
W ~L NS I G|pL W N∆U } xyz }
3 H
Replacing the expression an λi(t) in L(T) expression W
|pL N
U
2E F ∑o10 ∆} G# H# # o
~L
W NS }
xyz
3
I G|pL W N∆U H }
xyz
3 H
xyz
·
· r/ /0
xyz
·
We recall that in the case of Weibull distribution W(α,β) we have: H
\ /0 ·
We recall that We have formulated L(T) as : I
E F G# H W
|p
∆U
I
∑o10∆} G# Ho G|pL W N∆U }
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~L
W NS }
xyz
3 H
13th IFAC INCOM (INCOM'09) Moscow, Russia, June 3-5, 2009
Since that, with replacing n(t) and ni(t) by its expression we obtained L(T) expression as following I
E F ∆ Q7 L N . H# #
∆U
xyz
W
|pL N
U S
xyz
W ∆}
|pL N
∑o10
/0 ∑o10 ∆} ∑o/0 10 · Δ
o
~L
W NS ∆}
xyz
·
0
I
·
_F ! 7 L N i
∆U
With ∆t=1 The last L(T) form obtained takes easy the numerical procedure problem resolution. Figure 5 presents the histogram of the total cost expected of maintenance actions, C(N), according to N. We conclude that the optimal maintenance period obtained is N*T=14T . Since that applying the preventive maintenance action at this optimal period N*T=3.5*∆t, we obtained a minimal total cost expected of maintenance action C*=1612.2944mu . ϮϳϬϬϬ ϮϰϬϬϬ ϮϭϬϬϬ ϭϴϬϬϬ ϭϱϬϬϬ
ĐŽƐƚŵĂŝŶƚĞŶĂŶĐĞ ŵŝŶŝŵƵŶĐŽƐƚ
ϭϮϬϬϬ ϵϬϬϬ ϲϬϬϬ
C*=1612.2944
ϯϬϬϬ
N*T=3.5 Ϭ Ϭ͘Ϯϱ Ϭ͘ϱ Ϭ͘ϳϱ ϭ ϭ͘Ϯϱ ϭ͘ϱ ϭ͘ϳϱ Ϯ Ϯ͘Ϯϱ Ϯ͘ϱ Ϯ͘ϳϱ ϯ ϯ͘Ϯϱ ϯ͘ϱ ϯ͘ϳϱ ϰ ϰ͘Ϯϱ
Ϭ
d;ƉĞƌŝŽĚŵĂŝŶƚĞŶĂŶĐĞͿ Fig. 5. Curve of total cost expected of maintenance according to N 6. CONCLUSION This paper described a sequential constrained linear-quadratic stochastic production planning problem in order to satisfy a random demand d. The machine is subjected to randomly failure. A minimal repair is practiced at every failure. In order to reduce the failure frequency, preventive maintenance actions are scheduled according to the manufacturing system history. Firstly, given a randomly demand and a satisfaction customer rate, we have formulated and solved a linear-quadratic stochastic production problem in order to obtain an optimal production plan and we show the influence of manufacturing system degradation on the optimal production plan. Secondly, using the optimal production plan in the maintenance problem formulation, we established an economical maintenance scheduling in which we take into account the influence of the production rate on the machine failure rate.
REFERENCES O.S.Silva F and Cezarino W,(2004) “An Optimal Production Policy Applied to a Flow-shop Manufacturing System “; Brazilian Journal of Operations & Production Management ,Brazil ,Vol.1,N.1,73-92. Valdez-Flores, C. and Feldman, R.M., (1989) ”A survey of preventive maintenance models for stochastically deteriorating single-unit systems”, Naval Research Logistics Quarterly 36, pp. 419–446. Wang, H. and Pham, H. (1996) “Optimal maintenance policies for several imperfect maintenance models”, International Journal of Systems Science . Buzacott, J.A., and Shanthikumar, J.G.,(1993) “Stochastic Models of manufacturing systems”;Prentice Hall,Englewood Cliffs,NJ. Rezg, N., Xie, X., Mati, Y.,(2004), “Joint optimization of preventive maintenance and inventory control in a production line using simulation”, International Journal of Production Research. 15 May, Vol. 42, No. 10, 20292046 Cheung, K. L, Hausmann, W. H.,(1997),”Joint optimization of preventive maintenance and safety stock in an unreliable production environment”; Naval Research Logistics, 44, 257-272. N. Rezg, S. Dellagi., A.Chelbi,(2008) "Optimal Strategy of Inventory Control and Preventive Maintenance ", International Journal of Production Research, Vol. 46, No. 19, p5349–5365. S. Dellagi, N. Rezg, X. Xie,(2007) "Preventive Maintenance of Manufacturing Systems Under Environmental Constraints", International Journal of Production Research, vol. 45, Issue 5, p1233-1254. L.L.Faulkner,(2005)“Maintenance,Replacement,and Reliability Theory and Applications”,Chapter 2 Component Replacement Decisions,Taylor & Francis Group,USA Patrick Lyounet,(2000)” la maintenance mathématiques et methods”,page 138,Editions TEC & DOC,Paris
The goal of this study is to establish an economical production plan and to use it in order to optimize the maintenance cost. It’s noted that the use of the optimal production plan in the maintenance cost formulation is justified by the significant influence of the production rate on the manufacturing system degradation.
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Optimization of a maintenance strategy with considering the influence of the production plan on the manufacturing system degradation Hajjej Zied*, Dellagi Sofiene**, Rezg Nidhal*** INRIA Nancy Grand Est /LGIPM – Université de Metz, Ile du Saulcy, 57045 Metz Cedex - FRANCE (e-mail: *[email protected] - **[email protected] ***[email protected]) Abstract: This paper deals the degradation of the machine according to the production rate. An optimal maintenance strategy is developed in combination with the production plan for a manufacturing system satisfying a random demand. Firstly, for a given randomly demand, we established ,with a constrained stochastic production-planning problem under hypotheses of inventory variables, an optimal production plan which minimizes the average total holding and production costs. Secondly, using the optimal production plan obtained and its influence on the manufacturing system failure rate, by analytical study we established an optimal maintenance scheduling which minimizes the maintenance total expected cost. Finally, a numerical example is studied in order to apply the developed approach. Keywords: Failure rate, random demand, linear quadratic model, average cost, Minimal repair 1.
INTRODUCTION
Recently the problems treating the combination between maintenance and production plans represents one interesting frame work and attracts several researches . Actually the development of an aggregated optimal production/maintenance plan which minimizes the total cost including production, inventory and maintenance is one of the first actions of a hierarchical decision making process. The basic idea of this plan is to find the optimal production plan and maintenance strategy required by the company to manufacture their products which satisfy a random demand over future periods. It’s easy to note the complexity of this task since the various uncertainties associated with this decision process. These uncertainties are usually due to externs and interns factors. The randomly demand, which induce the incapacity of knowing exactly the demand behaviour through future periods, can be considered as an extern factor. The material availability variation can be considered as an intern factor. More then it’s interested to develop an optimal maintenance strategy with considering the manufacturing system degradation according to the production rate. In this context Silva,F. and Wagner, C.(2004) deals with a chance-constrained stochastic productionplanning problem under hypotheses of imperfect information of inventory variables. The optimal production plan obtained by the minimizing of the expected cost. The stochastic nature of the system is due to machines that are subject to break-downs and repairs or maintenance actions. The maintenance policies proposed in the literature are mainly policies of the critical age of a machine or a set of machines. These policies are based on models describing the degrading law of the equipment. Valdez-Flores and Feldman (1989) distinguish four classes of models: inspection models, models of minimal repair, shock models and replacement models. The cost/time of maintenance/repair is supposed to
978-3-902661-43-2/09/$20.00 © 2009 IFAC
be known and consequently the impact of a maintenance/failure cannot be analyzed. Under these conditions, it can be shown that the optimal policy is the one of critical age. If consists in carrying out a preventive maintenance action on a machine at its critical age. In this context Pham and Wang (1996) suggest that maintenance can be classified according to the degree to which the operating conditions of an item are restored by maintenance. They proposed five categories: Perfect repair or perfect maintenance, minimal repair or minimal maintenance, imperfect repair or imperfect maintenance, worse repair or maintenance and worst repair or maintenance. On the other hand, maintenance becomes even more significant with the production control policies implementation like Just-in-Time, which require the availability of machines at the right time. The traditional approach which dissociates maintenance and production is not satisfactory any more. An integrated approach of maintenance and production control becomes necessary. In this context Buzacott and Shanthikumar (1993) proved the importance of the choice of the maintenance policy for the minimization of the total cost. Rezg et Al. (2004) presented a common optimization of the preventive maintenance and stock control in a production line made up of N machines. In the same context of integrating maintenance and production Rezg and al. (2008) presented a mathematical model and a numerical procedure which allows determining a joint optimal inventory control and age based preventive maintenance policy for a randomly failing production system. New maintenance/production strategies by taking into account the context of subcontractor are studied by (Dellagi, S. et al., 2007). Dellagi, S. et al.(2007) developed and optimize a new maintenance policy with taking into account a machine subcontractor constraints.
1609
10.3182/20090603-3-RU-2001.0497
13th IFAC INCOM (INCOM'09) Moscow, Russia, June 3-5, 2009
This paper presents an optimal production and maintenance plan. It is strongly motivated by the lack of tools to assess the production system in the presence of maintenance activities. Indeed, the interaction between maintenance policies and production control is rarely studied in the literature. There are few pieces of work on the simultaneously optimization of maintenance and production policies for manufacturing systems. In particular, Cheung and Hausmann (1997) consider the simultaneous optimization of the strategic stock and the maintenance policy of the critical age type. In the literature the consideration of the materiel degradation according to the production rate is infrequently studied. That’s why, in order to establish an optimal maintenance strategy, we need to take into account the influence of the production plan on the material failure rate. In fact, in this work we established an economical production plan and we used it in the determination of the optimal maintenance strategy. The paper is organized as follows. Section 2 describes the problem and the working assumptions. Section 3 considers the single machine case and presents the problem formulation. In section 4 we show the influence of manufacturing system degradation on the optimal production plan determined. In section 5, considering the influence of the production plan in the manufacturing system degradation, we developed an analytical model in order to determine the optimal maintenance plan. We presented a numerical example, in order to apply the analytical results. Finally we conclude in section 6. 2. PROBLEM STATEMENT We are concerned with the problem of the optimal production planning problem formulation of a manufacturing system composed of machine M which produces a single product in order to meet the random demand d law characterized by a Normal distribution facing the system at a minimum cost. The Normal mean and standard deviation parameters are respectively and σd2. The problem is illustrated in Figure.1. The machine M is subject to a random failure. The probability degradation law of machine M is described by the probability density function of time to failure f(t) and for which the failure rate λ(t) increases with time and according to the production rate U(t) . Failures of machine M can be prevented by a preventive maintenance action which is scheduled according to its history. Our first objective is to establish an economical production plan satisfying the randomly demand with an. Secondly, using the optimal production plan obtained, we determined the optimal preventive maintenance period. The use of the optimal production plan in maintenance study is justified by the fact of taking into account the influence of the production rate on the evolution of the failure rate of the machine.
d
Machine
S Fig. 1. Problem description
3.
PROBLEM FORMULATION
3.1 Notation Cpr : unit production cost Cs : holding cost of a product unit during the period k Cλ : penality degradation cost CRM: minimal repair cost fi(t) : probability density function of time to failure for the machine H : the finite production horizon c2 : preventive maintenance action cost c1 : corrective maintenance action cost mu : monitory unit Ri(t) : Reliability function S(k) : The stock level at the end of the period k Umax : the maximal production rate U(k) : The production rate for period k Z : the total expected cost including production and inventory over a finite horizon H C : the maintenance total cost expected per time unit α : Probabilistic index; related to the customer satisfaction ∆t : period length En (T/∆t): floor function of a (T/∆t): greatest integer 3.2 Production policy This paper considers a single stochastic inventory balance system. The idea is to minimize the expected production and holding costs over a finite time horizon [0, H]. It’s assumed that the horizon H is portioned equally into H periods. The demand is satisfied at the end of each period. Thus, this kind of problem can be formulated as a linearstochastic optimal control problem under threshold stock level constraint. This subsection derives an analytical model for evaluating the total expected cost including production and inventory over a finite horizon H. In order to formulate the analytical model we assume that: • •
Storage and production costs Cpr , Cs are known and constant the demand Standard deviation σd(k) and the demand mean (k) for each period k are known and constant
We recall that our objective is to determine the optimal production plan over a time horizon H. Formally our problem is presented as following: min
1 , 2 , … . … .
The system model is described by a hybrid state with continuous component, the dynamic of the stock. The state equation of the stock level is given by:
1 ! 0 #
Where S0 is a given initial stock.
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(1)
13th IFAC INCOM (INCOM'09) Moscow, Russia, June 3-5, 2009
Respecting the machine capacity constraint, the production rate of every period k cannot exceed a given maximal production rate Umax. The expected cost including production and holding costs for the period k is given by: $%
% , &%
'( · & * '+, · *
(2)
Remark: The use of quadratic cost that allows penalizing both excess and shortage of inventory level Since that the total expected cost including production and inventory over a finite H periods is expressed by: ∑./0 %1#
%$* 2 '( 3 * ∑./0 %1# 4'( 3 '+, 3 * 5 (3)
Remark:
C(U(H))2 is not included in the cost formulation because we don’t consider the production command at the end of the horizon H. Since that our problem becomes: * 678 4'( 3 * ∑./0 %1# 4'( 3 '+, 3 * 55 (4)
Such that,
1 !
undisturbed by any repair of failure. It is assumed that the repair and replacement times are negligible. If we assume that λ(T) represents the machine failure rate I function and E F G# H .Then, the expected cost of the one interval [0,T] is: . 'J 1 K0 . E F K* K0 · E L N K* M Thus, the total expected cost until time H is: ' O O. 'J 1 O · 4K0 · E PO K* 5 · Q
•
The following figure describes dynamic system evolution in discrete time
(U(1),d(1)) (U(2),d(2))
(N=1, 2….)
(5)
Maintenance policy optimization
4. INFLUENCE OF MANUFACTURING SYSTEM DEGRADATION ON THE OPTIMAL PRODUCTION PLAN
0 A A BCD
K=2
[
In this subsection, we want to optimize the maintenance strategy adopted which is preventive maintenance with minimal repair. According to the production plan established during H periods, the objective is to determine the optimal partition number N* for which the preventive maintenance must be done. We noted that N* can exceed H, it means that no preventive maintenance to do in this case. It’s noted that we use in the maintenance study the optimal production plan obtained from last subsection. It’s clear that the maintenance policy is tightly related to the system degradation. That’s why we adopted the optimal production plan obtained in last subsection in order to take into account the influence of the production rate on the failure rate λ(t).
9:;<= > 0? >@
K=1
I
It has proven the existence of an optimal preventive maintenance period N*T in the case of increased failure rate (Patrick,L.2000)
Under the following constraints:
0
W
RS ·GX T U VU YRZ
In this section, a constrained state-space stochastic optimal control problem is formulated .It is used to represent a constrained aggregate production planning problem under the stock level constraint. At the same time, this section deals the influence of maintenance cost on the variation of the optimal production plan.
K=H (U(H),d(H))
4.1 Analytical study
t
Fig. 2. Discrete time
In order to justify this influence two cases are studied: 3.3 Maintenance policy
case1: we don’t consider the penalizing of the degradation
• The Maintenance cost model in the case of minimal reparation policy The maintenance strategy under consideration is the well known preventive maintenance with minimal repair (Faulkner, L.L.,). More precisely, the machine will operate over a given on a horizon H. the maintenance policy adopted is as follows: To maintain a unit, an interval H [0, H],is portioned equally into N parts T in which it is replaced at times k.T (k=1,2,….N) and NT=H. Then, we consider the replacement with minimal repair at failure we can consider that replacement. The unit is replaced or preventive maintenance actions are practiced at periodic time’s k.T and the unit is as good as new in each replacement or preventive maintenance action. When a unit fails between preventive maintenance actions, only minimal repair is made, and hence failure rate remains
case2: we considered the penalizing of the degradation In practice, the model provides a linear decision rule that allows the determination of an aggregate inventory and production. This rule is derived from the minimization of a quadratic production cost subject to linear equations that represent the balance among inventory and production components. We recall that our problem formulated in subsection 3.2 is:
678 * 678 4'( 3 * ∑./0 %1# 4'( 3 '+, 3 * 55 (6) Under the constraints:
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13th IFAC INCOM (INCOM'09) Moscow, Russia, June 3-5, 2009
1 ! 9:;<= > 0? > \
0 A A BCD Thus we can simplify our problem as: d/0 min] Z min] _'( 3 Sa H * ∑e1# 4'( 3 Sa k * '+, 3
u k * 5 '( 3 σh * 3
d d/0 *
i
(7)
With = mean stock level at the end of the period k under the
same constraints
Adding the penalizing of the degradation and the maintenance costs to problem. Our problem formulation becomes: d/0 min] Z min] _Ck · Sa H * ∑e1# 4Ck · Sa k * Clm 3 u k * 5 Ck 3 σh * 3
d d/0 *
* 'T 3 ∑. %1# H% i
{d1=431, d2= 444, d3= 442, d4= 340, d5=392, d6=400, d7= 350, d8=400, d9=350, d10= 370, d11= 395, d12= 415, d13= 431, d14= 444, d15= 442, d16= 340, d17= 392, d18= 375} (ii) the following data are used for the other parameters: Cpr =3 mu, Cs =2 mu/k, Umax=500 pieces (iii) Additional information: the standard deviation of the demand is σd=14 and the customer satisfaction degree, associated with the stock constraint, is equal to 95% (α=0.95). Applying the numerical procedure we obtained the optimal production plan for the two cases: Case 1: we don’t considered the penalizing of the degradation (Cn =0) Case 2: we considered the penalizing of the degradation (Cn =3000) The result is exhibited in figure 3. WƌŽĚƵĐƚŝŽŶƉůĂŶʄсϬ WƌŽĚƵĐƚŝŽŶƉůĂŶʄсϯϬϬϬ
ϲϬϬ
(8)
ƵƉƉĞƌďŽƵŶĚ
ϱϬϬ
Under the constraints:
ϰϬϬ
1 !
ϯϬϬ
9:;<= > 0? > \
ϮϬϬ
0 A A BCD
ϭ
For each period k we adopted the production rate U(k) established from the optimal production plan. Since that the failure rate evolves in each interval according to the production rate adopted in the interval and the failure rate cumulated at the beginning of the interval (Fig. 4). Formally the failure rate in the interval k is expressed as following: s t u0, ∆w
5.
ϵ
ϭϭ ϭϯ ϭϱ ϭϳ ϭϵ
Ŭ;ŵŽŶƚŚͿ
OPTIMISATION OF THE MAINTENANCE POLICY
5.1 Analytical Study
H 0 H#
xyz
ϳ
We noted that the optimal production plan obtained in case 2, is weak compared to the production plan obtained in the case 1.The result is scheduled. When we consider the penalizing of the degradation with the influence of the production rate on the degradation degree we have to decrease the production rate.
(9)
With tinf(k) represents the lower bound of interval k % · H ∆H% BCD nk(t) is expressed as following : 8 % · H H% H%/0 ∆
ϱ
Fig. 3. Optimal production plan
with nk(t) represents the failure rate at time t .
H% H%/0 opq % rH%
ϯ
{u0, ∆w
* The term 'T · ∑. %1# H% represent the penalizing degradation during horizon H.
4.2 Numerical Results A company, whose sales are subject to the effects of seasonal fluctuating demand, tries to develop an aggregated production plan which minimizes total costs over a finite planning horizon: H=18 months. The strategy discussed in previous section has been employed here to provide a decision policy for managers. A simple example of a multi-period, single product aggregated production planning problem is formulated by the generalized deterministic model. (i) the monthly mean demand % follows bellow
In last section, we have concluded that it exists a tightly influence of the production plan on the degradation degree of the machine. That why we focused on the joint optimization strategy in which we consider the maintenance strategy based the optimal production plan obtained. In fact, in this study, firstly we determine the optimal production plan minimizing the average production cost. Secondly using this optimal production plan in the maintenance cost formulation we will determine the optimal preventive maintenance strategy characterized by the optimal period N*T. ∆t k=1
k=2
H
0 T
2T
(N-1)T
Production periods
H Maintenance periods NT
En(T/∆t) Fig.4. Failure rate during H
The analytical model, in which we calculate the total cost expected of maintenance action, is developed under the following assumptions:
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o Probability density functions of time to failure, time to maintenance activities is known
13th IFAC INCOM (INCOM'09) Moscow, Russia, June 3-5, 2009
o Costs C2 and C1 incurred by the actions of preventive and corrective maintenance are known and constant, with C1>>C2 o Preventive maintenance is performed as soon as the age of the machine reaches T o Failures are detected instantaneously The failure rate at period T is expressed as following: H I H|pL W N ∆}
With
~L
W NS ∆}
xyz
(10)
U
With N=1,2,…..,
(11)
A M/0
~L
4
5
6
7
8
9
431
444
442
340
392
375
392
400
350
U
439
399
399
390
320
320
320
380
300
k
10
11
12
13
14
15
16
17
18
370
395
415
431
444
442
340
392
375
320
390
410
440
480
444
490
495
400
*
T=H/N
.
MY0
3
D U
}
S
2
*
I
Z
1
k D
∆U
}
Table 1. Optimal production plan
;
I
3 G|pL W N∆U H
We take the same data as previous section 4 i.e. the following data (Cpr =3 mu, Cs =2 mu/k, Umax=500 pieces). Applying the numerical procedure in the formulation problem established in subsection 3.2 we obtained the optimal production plan and the optimal maintenance period which are exhibited respectively in table1, and figure 5.
The analytic expression of the total cost expected of maintenance actions is defined by:
W
W NS }
xyz
O minimizing ' O if M A
Since that we obtain, if it exists, the optimal period N*.T, for which we practiced the preventive maintenance action.
|p
~L
1 We find an optimal partition number N* which minimizes C(N).
with: ' O O · K0 · G# H K*
∑o10∆} G# Ho G|pL W N∆U
5.2 Numerical example
Consequently, the problem is expressed as:
With I E F G# H
xyz
3 H
M
' O O · uK0 · E F K* w
.
Since that we can established the total cost expected of maintenance action
M
W
|pL N
∑o10 ∆} G#
With M O · E L N ! O 1 · E
7 FP∆ t 1,2, … … , , . . O
' O O · K0 · G# H K*
U
Lemma 1:
· H
En (T/∆t): integer part of T/∆t
O min ' O
W
|pL N
I
2 E F 7 L∆UN 3 ∆ 3 H# ∑o10 ∆} G# o
W NS }
xyz
3 H (12)
The failure rate at time t: Ho Ho/0 Δ · H xyz
We use the optimal production plan obtained in table 1 in the total cost expected of maintenance actions C(N) formulated in subsection 5.1. Concerning numerical example, we suppose that the failure time of machine M has a degradation law characterized by a Weibull distribution. The Weibull scale and shape parameters are respectively β=100 and α=2. The cost associated with a corrective and preventive maintenance action are respectively C1=3000 mu C2=500 mu monetary.
Since that we can express λ t as following: Ho H# # o · H
We recall that we have formulated λi(t) expression as following :
With: o ∑o/0 10
Ho H# # ∑o/0 10
xyz
· H ∆
xyz
@ U /0
0 0 and H# # H# Since that we obtained: W ∆}
|pL N
E F ∑o10
∆U G# Ho
W ~L NS I G|pL W N∆U } xyz }
3 H
Replacing the expression an λi(t) in L(T) expression W
|pL N
U
2E F ∑o10 ∆} G# H# # o
~L
W NS }
xyz
3
I G|pL W N∆U H }
xyz
3 H
xyz
·
· r/ /0
xyz
·
We recall that in the case of Weibull distribution W(α,β) we have: H
\ /0 ·
We recall that We have formulated L(T) as : I
E F G# H W
|p
∆U
I
∑o10∆} G# Ho G|pL W N∆U }
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~L
W NS }
xyz
3 H
13th IFAC INCOM (INCOM'09) Moscow, Russia, June 3-5, 2009
Since that, with replacing n(t) and ni(t) by its expression we obtained L(T) expression as following I
E F ∆ Q7 L N . H# #
∆U
xyz
W
|pL N
U S
xyz
W ∆}
|pL N
∑o10
/0 ∑o10 ∆} ∑o/0 10 · Δ
o
~L
W NS ∆}
xyz
·
0
I
·
_F ! 7 L N i
∆U
With ∆t=1 The last L(T) form obtained takes easy the numerical procedure problem resolution. Figure 5 presents the histogram of the total cost expected of maintenance actions, C(N), according to N. We conclude that the optimal maintenance period obtained is N*T=14T . Since that applying the preventive maintenance action at this optimal period N*T=3.5*∆t, we obtained a minimal total cost expected of maintenance action C*=1612.2944mu . ϮϳϬϬϬ ϮϰϬϬϬ ϮϭϬϬϬ ϭϴϬϬϬ ϭϱϬϬϬ
ĐŽƐƚŵĂŝŶƚĞŶĂŶĐĞ ŵŝŶŝŵƵŶĐŽƐƚ
ϭϮϬϬϬ ϵϬϬϬ ϲϬϬϬ
C*=1612.2944
ϯϬϬϬ
N*T=3.5 Ϭ Ϭ͘Ϯϱ Ϭ͘ϱ Ϭ͘ϳϱ ϭ ϭ͘Ϯϱ ϭ͘ϱ ϭ͘ϳϱ Ϯ Ϯ͘Ϯϱ Ϯ͘ϱ Ϯ͘ϳϱ ϯ ϯ͘Ϯϱ ϯ͘ϱ ϯ͘ϳϱ ϰ ϰ͘Ϯϱ
Ϭ
d;ƉĞƌŝŽĚŵĂŝŶƚĞŶĂŶĐĞͿ Fig. 5. Curve of total cost expected of maintenance according to N 6. CONCLUSION This paper described a sequential constrained linear-quadratic stochastic production planning problem in order to satisfy a random demand d. The machine is subjected to randomly failure. A minimal repair is practiced at every failure. In order to reduce the failure frequency, preventive maintenance actions are scheduled according to the manufacturing system history. Firstly, given a randomly demand and a satisfaction customer rate, we have formulated and solved a linear-quadratic stochastic production problem in order to obtain an optimal production plan and we show the influence of manufacturing system degradation on the optimal production plan. Secondly, using the optimal production plan in the maintenance problem formulation, we established an economical maintenance scheduling in which we take into account the influence of the production rate on the machine failure rate.
REFERENCES O.S.Silva F and Cezarino W,(2004) “An Optimal Production Policy Applied to a Flow-shop Manufacturing System “; Brazilian Journal of Operations & Production Management ,Brazil ,Vol.1,N.1,73-92. Valdez-Flores, C. and Feldman, R.M., (1989) ”A survey of preventive maintenance models for stochastically deteriorating single-unit systems”, Naval Research Logistics Quarterly 36, pp. 419–446. Wang, H. and Pham, H. (1996) “Optimal maintenance policies for several imperfect maintenance models”, International Journal of Systems Science . Buzacott, J.A., and Shanthikumar, J.G.,(1993) “Stochastic Models of manufacturing systems”;Prentice Hall,Englewood Cliffs,NJ. Rezg, N., Xie, X., Mati, Y.,(2004), “Joint optimization of preventive maintenance and inventory control in a production line using simulation”, International Journal of Production Research. 15 May, Vol. 42, No. 10, 20292046 Cheung, K. L, Hausmann, W. H.,(1997),”Joint optimization of preventive maintenance and safety stock in an unreliable production environment”; Naval Research Logistics, 44, 257-272. N. Rezg, S. Dellagi., A.Chelbi,(2008) "Optimal Strategy of Inventory Control and Preventive Maintenance ", International Journal of Production Research, Vol. 46, No. 19, p5349–5365. S. Dellagi, N. Rezg, X. Xie,(2007) "Preventive Maintenance of Manufacturing Systems Under Environmental Constraints", International Journal of Production Research, vol. 45, Issue 5, p1233-1254. L.L.Faulkner,(2005)“Maintenance,Replacement,and Reliability Theory and Applications”,Chapter 2 Component Replacement Decisions,Taylor & Francis Group,USA Patrick Lyounet,(2000)” la maintenance mathématiques et methods”,page 138,Editions TEC & DOC,Paris
The goal of this study is to establish an economical production plan and to use it in order to optimize the maintenance cost. It’s noted that the use of the optimal production plan in the maintenance cost formulation is justified by the significant influence of the production rate on the manufacturing system degradation.
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