- Email: [email protected]
An Integrated Production and Maintenance Planning Model with time windows and shortage cost
Proceedings of the 13th IFAC Symposium on Information Control Problems in Manufacturing Moscow, Russia, June 3-5, 2009
An Integrated Production and Maintenance Planning Model with time windows and shortage cost M. Alaoui selsouli* N.M. Najib** A. Mohafid** E.H. Aghezzaf*** * Ecole des Mines de Nantes, 4 rue Alfred Kastler, BP 20722, F- 44307 Nantes cedex 3 France (Tel: +33(0)2-51-85-83-18; e-mail: [email protected] ). ** IUT de Nantes, avenue Pr Jean Rouxel, BP 539, 44475 Carquefou, France (e-mail: najib.najid, abdelmoula.mohafid @univ-nantes.fr) *** Department of industrial Management, Technologiepark 903, University of Ghent,9052, Zwijnaarde, Belgium (e-mail: [email protected] ) Abstract: We present in this paper a problem combining two planning problems studied in the literature: the problem of capacitated lot sizing with shortage cost and the problem of determining optimal maintenance cycle. The aim of our study is to build a new model of joint planning of production and maintenance where preventive maintenance tasks are carried out in time windows to better meet customer demand. We provide an illustrative example that show the effectiveness of integrated model if we compare it to separate model. Keywords: production, shortage, preventive maintenance, corrective maintenance, integrated planning, time windows. Another reason for considering joint production and maintenance is that in practice, the resource capacity is not sufficient to meet demand which generate to delay delivery or even a loss of demand. Such a situation is not without consequence for companies because it can cause loss of customer, deterioration in the company's image and ultimately a decrease in sales.
1. INTRODUCTION Usually, production systems are subjected to breakdowns which increase with age of machines. These frequent downtimes, caused by a production line failure, are considered to be a source of disturbance of production plan and lead to additional costs, loss of production, decrease productivity and quality, and inefficient use of personnel, equipments and facilities. Indeed, the availability of equipments, at required time, is a necessary condition for the good sequence of production and in the respect for lead time. However, the deployment of maintenance activities requires stopping production system. Thereby, the current production plan becomes obsolete. Revising the production plan in an emergency situation is often very expensive and deteriorates service quality.
Our work in this area is to propose an integrated model of production planning and maintenance which will allow us to draw up a production plan taking into account shortage cost, and plan preventive maintenance actions in time windows. The remainder of the paper is organized as follows. The second section treats a brief literature review about production planning and maintenance problems. The third section addresses the separate production planning and maintenance. The fourth section covers the integrated model, and its formulation. A series of tests is presented in the last section that will highlight the effectiveness and benefits of our model and then we end up with conclusion and outlooks.
We observe then that neither the production plan takes into account the actions of maintenance nor the maintenance activities consider the production plan, because usually there is not cooperation between both managers of these activities, what creates conflicting relationship between production and maintenance. In truth, maintenance is a secondary process in companies that have production as their core business. The result is that maintenance does not receive enough management attention, and management often looks at maintenance as a necessary evil, not as means to reduce costs.
2. BRIEF LITERATURE REVIEW The problem of joint planning of production and maintenance is a recent problem and so far little work has been published in this issue. In literature, we find papers that consider the relation between maintenance and production. The relation exists in several ways. Principally, the integrated models of production and maintenance which we are interested in our research work. These models can split into four main categories:
To solve this problem, we develop an integrated production planning and maintenance activities. This integration will hedge against often avoidable failures and re-planning occurrences.
978-3-902661-43-2/09/$20.00 © 2009 IFAC
822
10.3182/20090603-3-RU-2001.0458
13th IFAC INCOM (INCOM'09) Moscow, Russia, June 3-5, 2009
The first category develops production and maintenance rate optimization models. It was treated by Kenné and Boukas (2003), Gharbi and Kenné (2000), Kenné and al (2003), Kenné and Gharbi (2004), Gharbi and Kenné (2005), Charlot and al (2007), Kenné and Nkeungoue (2008). Those models treat the problem of optimal flow control where the decision variables are the production and maintenance rates. The objective is to minimize the cost of surplus and maintenance.
Parameters: : Preventive maintenance cost.
: Corrective maintenance cost.
: Failure rate.
: Length of preventive maintenance cycle.
We assume that the system lifetime is distributed according to weibull distribution with shape and scale parameters
The second category concerns models of production systems with buffer inventory (e.g. Schouten and Vanneste (1995); Iravani and Duenyas (2002); Kyriakidis and Dimitrakos (2006); Chelbi and Ait-Kadi (2004); Gharbi et al (2007); Kenné and al (2007)). In order to reduce effect of breakdowns on production process, a buffer inventory may be built up during the production up time. The role of this buffer inventory is to satisfy demand when corrective or preventive maintenance is carried out.
function.
!
"
: Failure probability density
: Failure rate function.
: Expected number of failures for cycle.
#$ %& : Expected cost per time unit of maintenance.
The third one is economic manufacturing quantity (EMQ) models with failure aspects, see e.g. Groenevelt and al (1992), Makis and Fung (1995), Chung (2003), Rahim and Ben-Daya (1998), Wang and Sheu (2003). The study of those models is based on failures of production system and their impact on the decisions of the lot sizing and the quality products. The reader can refer to a detailed review presented by Budai et al (2006) for more information about these categories of models.
Let '()* the expected cost of maintenance for a cycle as : '()* '* + ', - .
2 34
"
/010
Mathematical formulation of expected cost per time unit is as follows (see e.g. Gertsbakh (2000)): 5
The last category presents aggregate production planning problems at which, we integrate decision variables modelling preventive and corrective maintenance. To our knowledge, the only works developed within this framework are proposed by Weinstein and Chung (1999). The authors presented a three part-model to resolve the conflicting objectives of system reliability and profit maximization. An aggregate production plan is first generated, and then a master production schedules is developed to minimize the weighted deviations from the specified aggregate production goals. Finally, work-center loading requirements, determined through rough cut capacity planning, are used to simulate equipment failures during the aggregate planning horizon. Also, we found work of Aghezzaf and al (2007) and Aghezzaf and Najid (2008). They developed models which take into consideration the parameters of reliability of the system at the beginning of the planning.
6789 2 34
(1)
Our aim is to determine the length of preventive maintenance cycle from (1) that minimizes expected cost per time unit. Once it is determined, we can get the expected total cost during a finite horizon ((2) or (3)) of length : - ; where; is the fixed length of production period and N is the number of period. ' : Expected total cost of maintenance during the horizon. 2
<= > 34? : Number of preventive maintenance during 2 the horizon. If
2
2 34
is integer then
' <= - '()* @ Else ' <= - '()* + ', .
3. SEPARATE PRODUCTION PLANNING AND MAINTENANCE
2AB 2 34
"
/010 C
3.2 Modelling production planning problem
As a first step, we choose to determine separately an optimal production plan and preventive maintenance cycle that minimizes the expected total cost of maintenance for the planning horizon.
A production plan establishes resource requirements to perform processing from raw material to finished products, in order to satisfy customers in the most efficient and most economical possible way. In other words, decisions to produce are made in the best report between the financial objective and the one of customer satisfaction.
3.1 Mathematical model The average cost per time unit for an infinite time horizon is defined by the renewal reward theorem as the ratio of the expected cost per cycle and the average length of the cycle.
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13th IFAC INCOM (INCOM'09) Moscow, Russia, June 3-5, 2009
lj
Lot sizing problem Aggregate lot sizing problems are production planning problems with setups between production lots. Because of these setups, it is often too costly to produce a given product in every period. On the other hand, generating fewer setups by producing large quantities to satisfy future demands results in high inventory holding costs. Thus, the objective is to determine the periods where production should take place, and the quantities to be produced, in order to satisfy demand while minimizing production, setup and inventory holding costs. Other costs might also be considered. Examples are backorder cost, backlogging cost, shortage cost etc. The reader can refer to a detailed review presented by Karimi and al (2003) for more information on capacitated lot sizing problem which is the basis of our work.
: Processing time for each item i.
Z*
: Capacity consumed by preventive maintenance.
Z,
: Capacity consumed by corrective maintenance.
j 1km =∑m opk 1jo Big number
Decision variables:
qjk
rjk
sjk
: Binary set up variable of item i in period t. : Quantity of item i produced in period t. : Inventory of item i at the end of period t.
/jk : Demand shortage for item i in period t.
(MCLSP-SC) Minimize : t t ]jk qjk + ajk rjk + ejk sjk + ijk /jk v
Mathematical formulation We consider a planning horizon H of length $ D - E covering N periods of fixed length E, and a set of item F G H to be produced on a production line of maximal capacity I JKL . During each period M G N, a demand O of item i should be satisfied. Also, the processing time of each item i is known. Period of preventive maintenance actions are known and each action consume respectively P Q - I JKL capacity units. Expected number of failures for each period between preventive maintenance actions is also known, and each corrective maintenance action consume P R - I JKL capacity units. Furthermore, we allow demand shortage unfulfilled due to insufficient capacity. We will assign a high unit cost for each item lost in each period. The aggregate lot sizing model addressed aims to minimize set up, production, holding, and shortage costs.
jG kGu
Subject to: rjk + /jk + sj k sjk 1jk M\ G ^ M 0 G w x t lj rjk y V0 M0 G w z jG
j rjk 1km qjk y { M\ G ^ M 0 G w | /jk y 1jk M\ G ^ M 0 G w } rjk sjk /jk ~ { M\ G ^ M 0 G w qjk G { M\ G ^ M 0 G w { s{ { The objective function (4) minimizes the sum of the set up, holding, production, and demand shortage costs over the whole N-period horizon. Constraint (5) is the inventory balance equation. Constraint (6) is the capacity constraint that considers preventive and corrective maintenance. Constraint (7) relates the continuous production variables to the binary setup variables. Constraint (8) express that quantity lost of item F G H in period t must be less than demand of item F G H in period t. constraints (9) and (10) characterize domain’s variables: production quantity, inventory level, demand shortage variables are nonnegative, and set-up variable is binary.
To initialize some of the variables, we introduce a dummy period 0, where inventory was zero. Index: i: Items. t: Periods. Model parameters: O : Demand of item i to satisfy in period t.
The expected result in this section is to calculate the total production and maintenance cost. This cost will be the sum of cost obtained by (2) or (3) and the cost given by the model (MCLSP-SC). This will allow us to compare this result with the result of our integrated model that we will describe in the next section.
I JKL : Maximal capacity of production line.
DST U Expected number of failures in each period t. I U Available capacity in period t.
VXY) Z* Z, - :S[0 \] ^_ \` ab/]c/db1 g VXY) Z, - :S[0 c0eb/f\`b O U Set-up cost of producing one unit of item i in period t. V0 W
4. INTEGRATED PRODUCTION PLANNING AND MAINTENANCE 4.1 Problem description
O U Fixed cost of producing one unit of item i period t.
Considering the same description of lot sizing problem with shortage cost (MCLSP-SC) described above. We integrate a binary decision variables associated to preventive maintenance, and additional constraints. The model assigns the appropriate expected number of breakdowns to each period between preventive maintenance actions. Moreover, if < - ; is the
hO : Variable cost of holding one unit of item i by the end of period t.
ijk : Unit cost for requirements not met regarding demand for item i in period t.
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13th IFAC INCOM (INCOM'09) Moscow, Russia, June 3-5, 2009
: k a @ z < kp*Ao k + k y 0 @ : | k y k M0 G w { 0 + 0 } k y k M0 G w { 0
length of preventive maintenance cycle in separate model. It means that preventive maintenance tasks are performed periodically in the beginning of period’s t =1, < +1, 2< +1, 3< +1…Our aim is not to plan these preventive maintenance actions at the beginning of those periods, but to plan them in time windows a< - ; a< + - ;. In other words, a preventive maintenance will be performed at the earliest at the beginning of the period 0 a< + or at the latest at the beginning of the m m period 0 a< + + with a @ > ?, where > ? is A A the number of preventive maintenance actions during planning horizon, and k is chosen to avoid the overlaps between time windows, so: > number,
A
A
t
*Ao
k
k ~ t o + op
(20) 0 M0 G w { 0 rjk sjk /jk ~ { M\ G ^ M 0 G w @ qjk k k G { M\ G ^ M 0 G w { 0 @@
? if n is an even
We introduce a dummy period 0, where the preventive maintenance was performed, and where inventory was zero.
otherwise.
Finally, we assume that when a production line fails, a minimal repair is carried out to restore it to “as bad as old” status. When a preventive maintenance is carried out, the production line is restored to “as good as new “status. We also assume that the expected failures increase with elapsed time since the last preventive maintenance.
s{ { _{ The variable jk must equal one when the last preventive maintenance was occurred in period j, and we are now in period t. To ensure this, we define jk as:
4.2 mathematical formulation
This is a nonlinear expression, but can be solved by introducing the following three constraints (18), (19), and (20).
jk k k @C
Parameter: DT : Vector of N elements contains the expected number of failures in each period t if we don’t consider preventive maintenance
The constraint (13) is the new capacity constraint. The constraint (16) ensures that one maintenance is carried out in the interval a< + a< + + . The constraint (17) ensures that two preventive maintenance actions cannot be carried out successively. The constraint (21) and (22) express non-negativity and integrality constraints.
DT .
Decision variables: U Binary preventive maintenance variable (1 if preventive maintenance is carried out in the beginning of period t, 0 otherwise).
Our problem can be assimilated to a capacitated lot sizing problem with set-up time, and shortage cost (CLSP-STSC). It is well known that the capacitated lot sizing problem with set-up time is an NP-hard problem (Trigeiro and al (1989)). For the (CLSP-ST-SC), it is also NP-hard; see Absi and Kedad-Sidhoum (2008). Then, we can say that our problem (MCLSP-SC-M-TW) is NP hard.
U Binary variable (1 if in period t the last preventive maintenance ended in period j, 0 otherwise). (MCLSP-SC-M-TW): Minimize
5. AN ILLUSTRATIVE EXAMPLE
t t ]jk qjk + ajk rjk + ejk sjk +ijk /jk jG kGu
Subject to:
+ t '* k + ', t kGu
Let consider the following planning horizon composed of 20 production periods of fixed length ; , each with a maximal capacity selected from [16 30]. The number of product is selected from [2 4]. The production, set-up, and holding costs are respectively 5, 25, and 2. The shortage cost for item i in period t is selected from [50 110]. The processing time of item i is lj C. The demand to be satisfied of each item is selected from [3 6].
(11) k p"
:[0 k
rjk + /jk + sj k sjk 1jk M\ G ^ M 0 G w @ t lj rjk + Z* k y VXY) jG
Z, t
k
Now, if we consider the preventive maintenance model with minimal repair at failure as the selected maintenance strategy of the production system with the following parameters: the cost of preventive maintenance action is set to '* @{, and the cost of minimal repair action is given by ', Cx. The capacity lost, when a preventive maintenance task and minimal repair action are carried out, is respectively Z* {VXY) , and Z, {CCVXY) .
(13) :[0 k M0 G w
p"
j rjk 1km qjk y { M\ G ^ M 0 G w v /jk y 1jk M\ G ^ M 0 G w x
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13th IFAC INCOM (INCOM'09) Moscow, Russia, June 3-5, 2009
The shape and scale parameters are respectively C, and v.
We will show some tests in the table 1. This is a comparison of the total cost of production and maintenance in both integrated and separate models. All tests were run on a Pentium 4 (3 GHz) with 1 Gb of RAM, running under Windows XP. We used a standard MIP software (XpressMP) with the solver default settings. The computational times vary from 0.15 seconds to 300 seconds for loose capacity (K_max=30) and 2 items to too tight capacity (K_max=16) and 4 items.
We use the software Matlab to determine the expected number of breakdowns for each period, and the optimal periodicity of preventive maintenance that will minimize the total cost of maintenance. We got a periodicity < = 3 periods ( C;), and k=1. So, we can say that the time windows are [3p, 3p+2] with p = 1...6.
Table 1. Integrated and separate cost for different shortage costs and maximal capacities Shortage cost = 110 2Items 3Items 4Items
Cost Cost Cost Cost Cost Cost
1 2 1 2 1 2
Kmax = 16 14370 13553
Kmax = 18 13007 12924
Kmax = 20 11662 10844
Kmax = 22 10231 9427.67
Kmax = 25 8175.04 7372
Kmax = 30 4803.04 4720.67
25810 24993
24447 24365
23100 22282
21667 20863
19607 18803.7
17545 16741.7
37140 36322.7
35777 35695
34428 33610.7
32995 NR
30933 30130
28871 28067.7
Shortage cost = 75 2Items 3Items 4Items
Cost Cost Cost Cost Cost Cost
1 2 1 2 1 2
Kmax = 16 10170 9597.67
Kmax = 18 9262.04 9179.67
Kmax = 20 8372.04 7799.67
Kmax = 22 7431.04 6872.67
Kmax = 25 6075.04 5516.67
Kmax = 30 4313.04 3775.67
17970 17397.7
17062 NR
16170 15597.7
15227 14668.7
13867 13309
12505 11946.7
25695 25122.7
24787 24705
23893 23321
21588 21029.7
20226 19668
22950 22391.7
Shortage cost = 50 2Items 3Items 4Items
Cost Cost Cost Cost Cost Cost
1 2 1 2 1 2
Kmax = 16 7170.0 6772.67
Kmax = 18 6587.0 6504.67
Kmax = 20 6022 5624.67
Kmax = 22 5431 5047.67
Kmax = 25 4575 4191.67
Kmax = 30 3463 3100.67
12370 11972.7
11787 NR
11220 10822.7
10627 10243.7
9767.04 9383.67
8905.04 8521.67
17520 17123
16937 16854.7
16368 15970.7
15775 15392
14913 14529.7
14051 13668
1
: Separated case : Integrated case NR: No result for this instance
2
reliability parameters of the system and its capacity in the development of the optimal production and maintenance planning. Also, we allow demand shortage; we add the concept of time windows where preventive maintenance will be carried out. Those time windows will help to reduce demand shortage in period of high demand, and will give more flexibility to preventive maintenance actions. The illustrative example shows the effectiveness of our model. Solve problem with a relaxation method, and an extension to several production lines would be interesting to investigate in future work.
To assess the quality of our model solved by XpressMP, we considered three parameters: number of products, shortage cost, and capacity tightness. The capacity is chosen to be tight, and too tight to create shortage. For some instances, we didn’t get solution because of a bad lower bound given by the solver. All results are illustrated in table above. We see from table 1 that integrated model gives better solution than separate model for different capacities, and shortage costs. 6. CONCLUSION
REFERENCES
A joint production and maintenance planning model for a production system subject to random failures has been proposed. This model takes, explicitly, into account the
Absi, N. and Kedad-Sidhoum, S. (2008). The multi capacitated lot sizing problem with set up time and
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13th IFAC INCOM (INCOM'09) Moscow, Russia, June 3-5, 2009
maintenance, Computers & industrial engineering, 46, 865-875.
shortage costs. European journal of operational research, 185, 1351-1374 Aghezzaf, E.H. Jamali, M.A. and Ait-Kadi, D. (2007). An integrated production and preventive maintenance planning model. European journal of operational research, 181, 676-685. Aghezzaf, E.H. and Najid, M.N. (2008). Integrated production and preventive maintenance in production systems subject to random failures. Information science, 178, 3382-3392. Budai, G. Dekker, R. and Nicolai, R. (2006). A review of planning models for maintenance & production. http://en.scientificcommons.org/17697690 Charlot, E. Kenné, J.P. and Nadeau, S. (2007). Optimal production, maintenance and lockouttagout control policies in manufacturing systems, International journal of production economics, 107, 435-450. Chelbi, A. and Ait-Kadi, D. (2004). Analysis of a production /inventory system with random failing production unit submitted to regular preventive maintenance. European journal of operational research, 156, 712-718. Chung, K. (2003). Approximations to production lot sizing with machine breakdowns. Computers & operations research, 30, 1499-1507. Gertsbakh. I (2000), reliability: Theory with Applications to Preventive Maintenance,2nd ed, Springer. Gharbi, A. and Kenné, J.P. (2000). Production and preventive maintenance rates control for a manufacturing system: an experimental design approach. International Journal of Production Economics, 65, 275-287. Gharbi, A. and Kenné, J.P. (2005). Maintenance scheduling and production control of multiplemachine manufacturing systems. Computers & industrial engineering, 48, 693-707. Gharbi, A. Kenné, J.P. and Beit, M. (2007). Optimal safety stocks and preventive maintenance periods in unreliable manufacturing systems. International Journal of Production Economics, 107, 422-434. Groenevelt, H. Pintelon, L. and Seidmann, A. (1992). Production batching with machine breakdowns and safety stocks. Operations research, 40, 959-971. Iravani, S. and Duenyas, I. (2002). Integrated maintenance and production control of a deteriorating production system. IIE Transactions, 34, 423-435, 2002. Karimi, B. Fatemi Ghomi, S. M. T. and Wilson, J. M. (2003). The capacitated lot sizing problem: a review of models and algorithms. Omega, 31,365-378 Kenné, J.P. and Boukas, E. (2003). Hierarchical control of production and maintenance rates in manufacturing systems. Journal of quality in maintenance engineering, 9, 66-82. Kenné, J.P. Boukas, E. and Gharbi, A. (2003). Control of production and corrective maintenance rates in a multiple-machine multiple-product manufacturing system. Mathematical and Computer Modelling, 38, 351-365. Kenné, J.P. and Gharbi, A. (2004). Stochastic optimal production control problem with corrective
Kenné, J.P. Gharbi, A. and Beit, M. (2007). Agedependent production planning and maintenance strategies in unreliable manufacturing systems with lost sale. European journal of operational Research, 107, 422-434. Kenné, J.P. and Nkeungoue, L.J.(2008). Simultaneous control of production, preventive and corrective maintenance rates of failure prone manufacturing system. Applied numerical mathematics, 58, 180-194. Kyriakidis, E.G. and Dimitrakos, T.D. (2006). Optimal preventive maintenance of production system with an intermediate buffer. European journal of operational research, 168, 86-99. Makis, V. and Fung, J. (1995). Optimal preventive replacement, lot sizing and inspection policy for a deteriorating production system. Journal of quality in maintenance engineering, 1, 41-55. Rahim, M. and Ben-Daya, M. (1998). A generalized economic model for joint determination of production run, inspection schedule and control chart design. International journal of production research, 36, 277289. Trigeiro, W. Thomas, L.J. and McLain, J.O., (1989). Capacitated lot sizing with setup time. Management science, 35, 353-366. Van der Duyn Schouten, F. and Vanneste, S. (1995). Maintenance optimization of a production system with buffer capacity. European journal of operational research 82, 323-338. Wang, C. and Sheu, S. (2003). Determining the optimal production-maintenance policy with inspection errors: using a markov chain. Computers & operations research, 30, 1-17. Weinstein, L. and Chung, C. (1999). Integrating maintenance and production decisions in a hierarchical production planning environment. Computers & operations research, 26, 1059-1074.
827
An Integrated Production and Maintenance Planning Model with time windows and shortage cost M. Alaoui selsouli* N.M. Najib** A. Mohafid** E.H. Aghezzaf*** * Ecole des Mines de Nantes, 4 rue Alfred Kastler, BP 20722, F- 44307 Nantes cedex 3 France (Tel: +33(0)2-51-85-83-18; e-mail: [email protected] ). ** IUT de Nantes, avenue Pr Jean Rouxel, BP 539, 44475 Carquefou, France (e-mail: najib.najid, abdelmoula.mohafid @univ-nantes.fr) *** Department of industrial Management, Technologiepark 903, University of Ghent,9052, Zwijnaarde, Belgium (e-mail: [email protected] ) Abstract: We present in this paper a problem combining two planning problems studied in the literature: the problem of capacitated lot sizing with shortage cost and the problem of determining optimal maintenance cycle. The aim of our study is to build a new model of joint planning of production and maintenance where preventive maintenance tasks are carried out in time windows to better meet customer demand. We provide an illustrative example that show the effectiveness of integrated model if we compare it to separate model. Keywords: production, shortage, preventive maintenance, corrective maintenance, integrated planning, time windows. Another reason for considering joint production and maintenance is that in practice, the resource capacity is not sufficient to meet demand which generate to delay delivery or even a loss of demand. Such a situation is not without consequence for companies because it can cause loss of customer, deterioration in the company's image and ultimately a decrease in sales.
1. INTRODUCTION Usually, production systems are subjected to breakdowns which increase with age of machines. These frequent downtimes, caused by a production line failure, are considered to be a source of disturbance of production plan and lead to additional costs, loss of production, decrease productivity and quality, and inefficient use of personnel, equipments and facilities. Indeed, the availability of equipments, at required time, is a necessary condition for the good sequence of production and in the respect for lead time. However, the deployment of maintenance activities requires stopping production system. Thereby, the current production plan becomes obsolete. Revising the production plan in an emergency situation is often very expensive and deteriorates service quality.
Our work in this area is to propose an integrated model of production planning and maintenance which will allow us to draw up a production plan taking into account shortage cost, and plan preventive maintenance actions in time windows. The remainder of the paper is organized as follows. The second section treats a brief literature review about production planning and maintenance problems. The third section addresses the separate production planning and maintenance. The fourth section covers the integrated model, and its formulation. A series of tests is presented in the last section that will highlight the effectiveness and benefits of our model and then we end up with conclusion and outlooks.
We observe then that neither the production plan takes into account the actions of maintenance nor the maintenance activities consider the production plan, because usually there is not cooperation between both managers of these activities, what creates conflicting relationship between production and maintenance. In truth, maintenance is a secondary process in companies that have production as their core business. The result is that maintenance does not receive enough management attention, and management often looks at maintenance as a necessary evil, not as means to reduce costs.
2. BRIEF LITERATURE REVIEW The problem of joint planning of production and maintenance is a recent problem and so far little work has been published in this issue. In literature, we find papers that consider the relation between maintenance and production. The relation exists in several ways. Principally, the integrated models of production and maintenance which we are interested in our research work. These models can split into four main categories:
To solve this problem, we develop an integrated production planning and maintenance activities. This integration will hedge against often avoidable failures and re-planning occurrences.
978-3-902661-43-2/09/$20.00 © 2009 IFAC
822
10.3182/20090603-3-RU-2001.0458
13th IFAC INCOM (INCOM'09) Moscow, Russia, June 3-5, 2009
The first category develops production and maintenance rate optimization models. It was treated by Kenné and Boukas (2003), Gharbi and Kenné (2000), Kenné and al (2003), Kenné and Gharbi (2004), Gharbi and Kenné (2005), Charlot and al (2007), Kenné and Nkeungoue (2008). Those models treat the problem of optimal flow control where the decision variables are the production and maintenance rates. The objective is to minimize the cost of surplus and maintenance.
Parameters: : Preventive maintenance cost.
: Corrective maintenance cost.
: Failure rate.
: Length of preventive maintenance cycle.
We assume that the system lifetime is distributed according to weibull distribution with shape and scale parameters
The second category concerns models of production systems with buffer inventory (e.g. Schouten and Vanneste (1995); Iravani and Duenyas (2002); Kyriakidis and Dimitrakos (2006); Chelbi and Ait-Kadi (2004); Gharbi et al (2007); Kenné and al (2007)). In order to reduce effect of breakdowns on production process, a buffer inventory may be built up during the production up time. The role of this buffer inventory is to satisfy demand when corrective or preventive maintenance is carried out.
function.
!
"
: Failure probability density
: Failure rate function.
: Expected number of failures for cycle.
#$ %& : Expected cost per time unit of maintenance.
The third one is economic manufacturing quantity (EMQ) models with failure aspects, see e.g. Groenevelt and al (1992), Makis and Fung (1995), Chung (2003), Rahim and Ben-Daya (1998), Wang and Sheu (2003). The study of those models is based on failures of production system and their impact on the decisions of the lot sizing and the quality products. The reader can refer to a detailed review presented by Budai et al (2006) for more information about these categories of models.
Let '()* the expected cost of maintenance for a cycle as : '()* '* + ', - .
2 34
"
/010
Mathematical formulation of expected cost per time unit is as follows (see e.g. Gertsbakh (2000)): 5
The last category presents aggregate production planning problems at which, we integrate decision variables modelling preventive and corrective maintenance. To our knowledge, the only works developed within this framework are proposed by Weinstein and Chung (1999). The authors presented a three part-model to resolve the conflicting objectives of system reliability and profit maximization. An aggregate production plan is first generated, and then a master production schedules is developed to minimize the weighted deviations from the specified aggregate production goals. Finally, work-center loading requirements, determined through rough cut capacity planning, are used to simulate equipment failures during the aggregate planning horizon. Also, we found work of Aghezzaf and al (2007) and Aghezzaf and Najid (2008). They developed models which take into consideration the parameters of reliability of the system at the beginning of the planning.
6789 2 34
(1)
Our aim is to determine the length of preventive maintenance cycle from (1) that minimizes expected cost per time unit. Once it is determined, we can get the expected total cost during a finite horizon ((2) or (3)) of length : - ; where; is the fixed length of production period and N is the number of period. ' : Expected total cost of maintenance during the horizon. 2
<= > 34? : Number of preventive maintenance during 2 the horizon. If
2
2 34
is integer then
' <= - '()* @ Else ' <= - '()* + ', .
3. SEPARATE PRODUCTION PLANNING AND MAINTENANCE
2AB 2 34
"
/010 C
3.2 Modelling production planning problem
As a first step, we choose to determine separately an optimal production plan and preventive maintenance cycle that minimizes the expected total cost of maintenance for the planning horizon.
A production plan establishes resource requirements to perform processing from raw material to finished products, in order to satisfy customers in the most efficient and most economical possible way. In other words, decisions to produce are made in the best report between the financial objective and the one of customer satisfaction.
3.1 Mathematical model The average cost per time unit for an infinite time horizon is defined by the renewal reward theorem as the ratio of the expected cost per cycle and the average length of the cycle.
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13th IFAC INCOM (INCOM'09) Moscow, Russia, June 3-5, 2009
lj
Lot sizing problem Aggregate lot sizing problems are production planning problems with setups between production lots. Because of these setups, it is often too costly to produce a given product in every period. On the other hand, generating fewer setups by producing large quantities to satisfy future demands results in high inventory holding costs. Thus, the objective is to determine the periods where production should take place, and the quantities to be produced, in order to satisfy demand while minimizing production, setup and inventory holding costs. Other costs might also be considered. Examples are backorder cost, backlogging cost, shortage cost etc. The reader can refer to a detailed review presented by Karimi and al (2003) for more information on capacitated lot sizing problem which is the basis of our work.
: Processing time for each item i.
Z*
: Capacity consumed by preventive maintenance.
Z,
: Capacity consumed by corrective maintenance.
j 1km =∑m opk 1jo Big number
Decision variables:
qjk
rjk
sjk
: Binary set up variable of item i in period t. : Quantity of item i produced in period t. : Inventory of item i at the end of period t.
/jk : Demand shortage for item i in period t.
(MCLSP-SC) Minimize : t t ]jk qjk + ajk rjk + ejk sjk + ijk /jk v
Mathematical formulation We consider a planning horizon H of length $ D - E covering N periods of fixed length E, and a set of item F G H to be produced on a production line of maximal capacity I JKL . During each period M G N, a demand O of item i should be satisfied. Also, the processing time of each item i is known. Period of preventive maintenance actions are known and each action consume respectively P Q - I JKL capacity units. Expected number of failures for each period between preventive maintenance actions is also known, and each corrective maintenance action consume P R - I JKL capacity units. Furthermore, we allow demand shortage unfulfilled due to insufficient capacity. We will assign a high unit cost for each item lost in each period. The aggregate lot sizing model addressed aims to minimize set up, production, holding, and shortage costs.
jG kGu
Subject to: rjk + /jk + sj k sjk 1jk M\ G ^ M 0 G w x t lj rjk y V0 M0 G w z jG
j rjk 1km qjk y { M\ G ^ M 0 G w | /jk y 1jk M\ G ^ M 0 G w } rjk sjk /jk ~ { M\ G ^ M 0 G w qjk G { M\ G ^ M 0 G w { s{ { The objective function (4) minimizes the sum of the set up, holding, production, and demand shortage costs over the whole N-period horizon. Constraint (5) is the inventory balance equation. Constraint (6) is the capacity constraint that considers preventive and corrective maintenance. Constraint (7) relates the continuous production variables to the binary setup variables. Constraint (8) express that quantity lost of item F G H in period t must be less than demand of item F G H in period t. constraints (9) and (10) characterize domain’s variables: production quantity, inventory level, demand shortage variables are nonnegative, and set-up variable is binary.
To initialize some of the variables, we introduce a dummy period 0, where inventory was zero. Index: i: Items. t: Periods. Model parameters: O : Demand of item i to satisfy in period t.
The expected result in this section is to calculate the total production and maintenance cost. This cost will be the sum of cost obtained by (2) or (3) and the cost given by the model (MCLSP-SC). This will allow us to compare this result with the result of our integrated model that we will describe in the next section.
I JKL : Maximal capacity of production line.
DST U Expected number of failures in each period t. I U Available capacity in period t.
VXY) Z* Z, - :S[0 \] ^_ \` ab/]c/db1 g VXY) Z, - :S[0 c0eb/f\`b O U Set-up cost of producing one unit of item i in period t. V0 W
4. INTEGRATED PRODUCTION PLANNING AND MAINTENANCE 4.1 Problem description
O U Fixed cost of producing one unit of item i period t.
Considering the same description of lot sizing problem with shortage cost (MCLSP-SC) described above. We integrate a binary decision variables associated to preventive maintenance, and additional constraints. The model assigns the appropriate expected number of breakdowns to each period between preventive maintenance actions. Moreover, if < - ; is the
hO : Variable cost of holding one unit of item i by the end of period t.
ijk : Unit cost for requirements not met regarding demand for item i in period t.
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13th IFAC INCOM (INCOM'09) Moscow, Russia, June 3-5, 2009
: k a @ z < kp*Ao k + k y 0 @ : | k y k M0 G w { 0 + 0 } k y k M0 G w { 0
length of preventive maintenance cycle in separate model. It means that preventive maintenance tasks are performed periodically in the beginning of period’s t =1, < +1, 2< +1, 3< +1…Our aim is not to plan these preventive maintenance actions at the beginning of those periods, but to plan them in time windows a< - ; a< + - ;. In other words, a preventive maintenance will be performed at the earliest at the beginning of the period 0 a< + or at the latest at the beginning of the m m period 0 a< + + with a @ > ?, where > ? is A A the number of preventive maintenance actions during planning horizon, and k is chosen to avoid the overlaps between time windows, so: > number,
A
A
t
*Ao
k
k ~ t o + op
(20) 0 M0 G w { 0 rjk sjk /jk ~ { M\ G ^ M 0 G w @ qjk k k G { M\ G ^ M 0 G w { 0 @@
? if n is an even
We introduce a dummy period 0, where the preventive maintenance was performed, and where inventory was zero.
otherwise.
Finally, we assume that when a production line fails, a minimal repair is carried out to restore it to “as bad as old” status. When a preventive maintenance is carried out, the production line is restored to “as good as new “status. We also assume that the expected failures increase with elapsed time since the last preventive maintenance.
s{ { _{ The variable jk must equal one when the last preventive maintenance was occurred in period j, and we are now in period t. To ensure this, we define jk as:
4.2 mathematical formulation
This is a nonlinear expression, but can be solved by introducing the following three constraints (18), (19), and (20).
jk k k @C
Parameter: DT : Vector of N elements contains the expected number of failures in each period t if we don’t consider preventive maintenance
The constraint (13) is the new capacity constraint. The constraint (16) ensures that one maintenance is carried out in the interval a< + a< + + . The constraint (17) ensures that two preventive maintenance actions cannot be carried out successively. The constraint (21) and (22) express non-negativity and integrality constraints.
DT .
Decision variables: U Binary preventive maintenance variable (1 if preventive maintenance is carried out in the beginning of period t, 0 otherwise).
Our problem can be assimilated to a capacitated lot sizing problem with set-up time, and shortage cost (CLSP-STSC). It is well known that the capacitated lot sizing problem with set-up time is an NP-hard problem (Trigeiro and al (1989)). For the (CLSP-ST-SC), it is also NP-hard; see Absi and Kedad-Sidhoum (2008). Then, we can say that our problem (MCLSP-SC-M-TW) is NP hard.
U Binary variable (1 if in period t the last preventive maintenance ended in period j, 0 otherwise). (MCLSP-SC-M-TW): Minimize
5. AN ILLUSTRATIVE EXAMPLE
t t ]jk qjk + ajk rjk + ejk sjk +ijk /jk jG kGu
Subject to:
+ t '* k + ', t kGu
Let consider the following planning horizon composed of 20 production periods of fixed length ; , each with a maximal capacity selected from [16 30]. The number of product is selected from [2 4]. The production, set-up, and holding costs are respectively 5, 25, and 2. The shortage cost for item i in period t is selected from [50 110]. The processing time of item i is lj C. The demand to be satisfied of each item is selected from [3 6].
(11) k p"
:[0 k
rjk + /jk + sj k sjk 1jk M\ G ^ M 0 G w @ t lj rjk + Z* k y VXY) jG
Z, t
k
Now, if we consider the preventive maintenance model with minimal repair at failure as the selected maintenance strategy of the production system with the following parameters: the cost of preventive maintenance action is set to '* @{, and the cost of minimal repair action is given by ', Cx. The capacity lost, when a preventive maintenance task and minimal repair action are carried out, is respectively Z* {VXY) , and Z, {CCVXY) .
(13) :[0 k M0 G w
p"
j rjk 1km qjk y { M\ G ^ M 0 G w v /jk y 1jk M\ G ^ M 0 G w x
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13th IFAC INCOM (INCOM'09) Moscow, Russia, June 3-5, 2009
The shape and scale parameters are respectively C, and v.
We will show some tests in the table 1. This is a comparison of the total cost of production and maintenance in both integrated and separate models. All tests were run on a Pentium 4 (3 GHz) with 1 Gb of RAM, running under Windows XP. We used a standard MIP software (XpressMP) with the solver default settings. The computational times vary from 0.15 seconds to 300 seconds for loose capacity (K_max=30) and 2 items to too tight capacity (K_max=16) and 4 items.
We use the software Matlab to determine the expected number of breakdowns for each period, and the optimal periodicity of preventive maintenance that will minimize the total cost of maintenance. We got a periodicity < = 3 periods ( C;), and k=1. So, we can say that the time windows are [3p, 3p+2] with p = 1...6.
Table 1. Integrated and separate cost for different shortage costs and maximal capacities Shortage cost = 110 2Items 3Items 4Items
Cost Cost Cost Cost Cost Cost
1 2 1 2 1 2
Kmax = 16 14370 13553
Kmax = 18 13007 12924
Kmax = 20 11662 10844
Kmax = 22 10231 9427.67
Kmax = 25 8175.04 7372
Kmax = 30 4803.04 4720.67
25810 24993
24447 24365
23100 22282
21667 20863
19607 18803.7
17545 16741.7
37140 36322.7
35777 35695
34428 33610.7
32995 NR
30933 30130
28871 28067.7
Shortage cost = 75 2Items 3Items 4Items
Cost Cost Cost Cost Cost Cost
1 2 1 2 1 2
Kmax = 16 10170 9597.67
Kmax = 18 9262.04 9179.67
Kmax = 20 8372.04 7799.67
Kmax = 22 7431.04 6872.67
Kmax = 25 6075.04 5516.67
Kmax = 30 4313.04 3775.67
17970 17397.7
17062 NR
16170 15597.7
15227 14668.7
13867 13309
12505 11946.7
25695 25122.7
24787 24705
23893 23321
21588 21029.7
20226 19668
22950 22391.7
Shortage cost = 50 2Items 3Items 4Items
Cost Cost Cost Cost Cost Cost
1 2 1 2 1 2
Kmax = 16 7170.0 6772.67
Kmax = 18 6587.0 6504.67
Kmax = 20 6022 5624.67
Kmax = 22 5431 5047.67
Kmax = 25 4575 4191.67
Kmax = 30 3463 3100.67
12370 11972.7
11787 NR
11220 10822.7
10627 10243.7
9767.04 9383.67
8905.04 8521.67
17520 17123
16937 16854.7
16368 15970.7
15775 15392
14913 14529.7
14051 13668
1
: Separated case : Integrated case NR: No result for this instance
2
reliability parameters of the system and its capacity in the development of the optimal production and maintenance planning. Also, we allow demand shortage; we add the concept of time windows where preventive maintenance will be carried out. Those time windows will help to reduce demand shortage in period of high demand, and will give more flexibility to preventive maintenance actions. The illustrative example shows the effectiveness of our model. Solve problem with a relaxation method, and an extension to several production lines would be interesting to investigate in future work.
To assess the quality of our model solved by XpressMP, we considered three parameters: number of products, shortage cost, and capacity tightness. The capacity is chosen to be tight, and too tight to create shortage. For some instances, we didn’t get solution because of a bad lower bound given by the solver. All results are illustrated in table above. We see from table 1 that integrated model gives better solution than separate model for different capacities, and shortage costs. 6. CONCLUSION
REFERENCES
A joint production and maintenance planning model for a production system subject to random failures has been proposed. This model takes, explicitly, into account the
Absi, N. and Kedad-Sidhoum, S. (2008). The multi capacitated lot sizing problem with set up time and
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maintenance, Computers & industrial engineering, 46, 865-875.
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