Marouane Najib Mohamed Najid
email: najib.najid@univ-nantes.fr

Selsouli Alaoui
email: malaou07@emn.fr

Abdelmoula Mouhafid
email: abdelmoula.mohafid@univ-nantes.fr

Alaoui

Marouane Alaoui-Selsouli

An Integrated Production and Maintenance Planning Model with time windows and shortage cost

Keywords: time windows, shortage cost Production and maintenance planning, Time windows, Shortage cost

In this paper, we tackle the problem of integrating production and maintenance. Production problem addresses the issue of determining the production lot sizes of various items. Preventive maintenance is carried out in time windows to restore the production line to an 'as-good-as-new' status, and when a production line fails, a minimal repair is carried out to restore it to an 'as-bad-as-old' status. The resulting problem is modelled as a linear mixedinteger program. It takes into account demand shortage and the reliability of the production line. Computational experiments are carried out to show the effectiveness of the integrated model compared to classical separate model for different instances, and the obtained results are analyzed in detail.

Introduction

In the industrial environment, the relationship between production and maintenance has been conflictual in nature. This attitude is perpetuated by the lack of communication regarding the scheduling requirements of each function. Maintenance is generally a secondary process in companies that have production as their core business. Production management often views maintenance in the context of hours or days out of service and fails to realize the strategic importance of incorporating maintenance in the manufacturing planning. On the other hand, management for the maintenance function attempts to impose constraints on production that it deems necessary to achieve complete equipment reliability. The result is that the implementation of an optimal maintenance policy is constrained by the demands of production.

Currently, production planning and maintenance are performed separately. For instance, common ERP modules include production planning module that optimizes the utilization of manufacturing capacity, parts, components and material resources. Maintenance module is a peripheral module little known in fact, it is not part of core ERP application as production planning, finance, HR (Human Resources) Modules. it is considered as computerized maintenance management system (CMMS), and it allows managing preventive and corrective maintenance of all material assets of the company. We notice that each of these modules is used independently with specific data from each service, and no coordination is implemented to reduce production and maintenance costs. Therefore, maintenance planning should be an integral part of the overall business strategy and should be coordinated and scheduled with manufacturing activities. So, we should consider maintenance as integral parts of the production plan rather than as interruptions to that plan. Any violation of the maintenance schedule is treated as a violation of the production plan integrity. To solve this problem, we develop an integrated production planning and maintenance activities. This integration hedge against often avoidable failures and re-planning occurrences, and minimize loss of demand which is not without consequence for companies. In this area, we propose an integrated model of production and maintenance planning that minimizes production, inventory, set up, demand shortage, preventive and corrective maintenance costs.

The particularity of our model is that preventive maintenance actions are planned in time windows, and demand shortage is allowed when the resource capacity is not sufficient to meet all demand.

The remainder of the paper is organized as follows. In the second section a brief literature review is presented. In the third section, we describe separate production planning and maintenance problem, and we state its mathematical formulation. We illustrate our integrated problem in the fourth section. Computational results are showed in the fifth section, and we end up with conclusion and outlooks in the last section.

Brief literature review

The problem of joint planning of production and maintenance is a recent problem tackled in research due to its importance in the current highly competitive environment. Several relations exist in different ways between maintenance and production according to [START_REF] Budai | A review of planning models for maintenance & production[END_REF]. First of all, when planning maintenance takes into account production (e.g. [START_REF] Dekker | Prompt a decision support system for opportunity based preventive maintenance[END_REF]; [START_REF] Bäackert | The determination of maintenance strategies for plants subject to breakdown[END_REF]; [START_REF] Cassady | Selective maintenance modelling for industrial systems[END_REF], [START_REF] Marquez | Maintenance policies for a production system with constrained production rate and buffer capacity[END_REF]. Secondly , maintenance can also be seen as a production process which needs to be planned, see e.g. [START_REF] Dijkstra | Planning the size and organization of KLM's aircraft maintenance personnel[END_REF]. [START_REF] Shenoy | MRSRP-a tool for manpower resources and spares requirements planning[END_REF], Rosheil and Christy (1996), [START_REF] Yan | Airline short-term maintenance manpower supply planning[END_REF]. Finally, the integrated models of production and maintenance which we are interested in our research work.

Several studies of integrated production planning and maintenance problem were proposed with different approaches, resolution methods, and modeling tools. In the literature, we find models developed with stochastic aspect by using such Markov chains to oversee the state of machine or to determine an optimal buffer level according to the production rate in order to establish an interdependent policy of maintenance and production. In the last decade, large models were stated in this context, some of them treat the control problem of a stochastic manufacturing system with random breakdowns, repairs and preventive maintenance. The objective of the control models is to find the production and the preventive maintenance rates of the machines so as to minimize the total cost of inventory/backlog, repair and preventive maintenance see e.g. [START_REF] Kenné | Hierarchical control of production and maintenance rates in manufacturing systems[END_REF], [START_REF] Kenné | Stochastic optimal production control problem with corrective maintenance[END_REF], [START_REF] Gharbi | Maintenance scheduling and production control of multiple-machine manufacturing systems[END_REF], and [START_REF] Rezg | Modeling and optimizing a joint inventory control and preventive maintenance strategy for a randomly failing production unit: analytical and simulationapproaches[END_REF]. An extension was proposed by [START_REF] Kenné | Optimal production, maintenance and lockouttagout control policies in manufacturing systems[END_REF] considering preventive maintenance activities and differentiates two types of repairs: repairs Other research works deal with models of production systems with buffer inventory. The role of this buffer inventory is to satisfy demand when corrective or preventive maintenance is carried out. One of the earliest and basis work on this subject is [START_REF] Van Der | Maintenance optimization of a production system with buffer capacity[END_REF]. In their model the demand rate is constant and equal to d (units/time) and as long as the fixed buffer capacity (K) is not reached the installation operates at a constant rate of p units/time (p > d) and the excess output is stored in the buffer. When the buffer is full, the installation reduces its speed from p to d. Upon failure corrective maintenance starts and the installation becomes as good as new. The decision to start a preventive maintenance action is not only based on the condition of the installation, but also on the level of the buffer.

The criterion to minimize is the average inventory level and the average number of backorders. [START_REF] Iravani | Integrated maintenance and production control of a deteriorating production system[END_REF] extended the above model by assuming a stochastic demand and production process. [START_REF] Kyriakidis | Optimal preventive maintenance of production system with an intermediate buffer[END_REF] studied an infinite-state generalization of [START_REF] Van Der | Maintenance optimization of a production system with buffer capacity[END_REF]. Preventive maintenance actions can be regularly (after each T time periods) performed and the duration of corrective and preventive maintenance actions is random see e.g Chelbi and Ait [START_REF] Ait-Kadi | Analysis of a production /inventory system with random failing production unit submitted to regular preventive maintenance[END_REF], and the possibility of imperfect production can be included in model for example Zequeria and al. (2004). [START_REF] Dhouib | Availability and throughput of unreliable, unbuffered production lines with non-homogeneous deterministic processing times[END_REF] worked on production lines composed of several serial machines which are subject to random operation-dependent failures when it has no intermediate buffers between adjacent machines. Models of preventive maintenance policy with control quality of products for a randomly failing production system producing conforming and non-conforming units can be developed e.g, [START_REF] Radhoui | Integrated model of preventive maintenance, quality control and buffer sizing for unreliable and imperfect production systems[END_REF]. The considered system consists of one machine designed to fulfill a constant demand. According to the proportion l of non-conforming units observed on each lot and compared to a threshold value lm, one decides to undertake or not maintenance actions on the system.

A continuous time models with an infinite time horizon were presented such economic manufacturing quantity (EMQ) models with failure aspects. The study of those models is based on failures of production system and their impacts on the decisions of the lot sizing, see e.g. [START_REF] Groenevelt | Production batching with machine breakdowns and safety stocks[END_REF] who consider the effects of stochastic machine breakdowns and corrective maintenance on economic lot sizing decisions, and propose to use the EMQ as an approximation to the optimal production lot size. [START_REF]Approximations to production lot sizing with machine breakdowns[END_REF] provides a better approximation to the optimal production lot size, [START_REF] Makis | Optimal preventive replacement, lot sizing and inspection policy for a deteriorating production system[END_REF] proposed a model for joint determination of the lot size, inspection interval and preventive replacement time for a production facility that is subject to random failure. [START_REF] Srinivasan | Production-inventory systems with preventive maintenance[END_REF] considered an (S, s) policy, i.e. as soon as the inventory level reaches S, a preventive maintenance operation is initiated and the machine becomes as good as new. After the preventive Unlike the above mentioned continuous time models, the discrete time models assume that the planning horizon is finite and divided into discrete periods for which demand is given and may vary between periods. A few works exist in this area, they concern aggregate production planning problems known as multi items capacitated lot sizing problems at which, we integrate decision variables modeling preventive and corrective maintenance. To our knowledge, the only works developed within this framework are proposed by [START_REF] Weinstein | Integrating maintenance and production decisions in a hierarchical production planning environment[END_REF], [START_REF] Aghezzaf | An integrated production and preventive maintenance planning model[END_REF], and [START_REF] Aghezzaf | Integrated production and preventive maintenance in production systems subject to random failures[END_REF]. [START_REF] Weinstein | Integrating maintenance and production decisions in a hierarchical production planning environment[END_REF] 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. [START_REF] Aghezzaf | An integrated production and preventive maintenance planning model[END_REF] studied a production system that subjects to random failures. They assume that any maintenance action carried out on the system, in a period, reduces the system's available production capacity during that period.

The objective is to find an integrated lot-sizing and preventive maintenance strategy of the system that satisfies the demand for all items over the entire horizon without backlogging, and which minimizes the expected sum of production and maintenance costs. An extension of the above work is treated by [START_REF] Aghezzaf | Integrated production and preventive maintenance in production systems subject to random failures[END_REF]. They developed models which take into consideration the parameters of reliability of the system and consider parallel production lines. Our motivation by this paper is to provide an extension of these models to a more complete and more flexible model in relation to the date of preventive maintenance tasks. The aim of our study is to establish a joint production and maintenance planning problem at the tactical level to minimize the total costs associated with production and maintenance. Before describing our problem, we will define both of aggregate production and maintenance planning.

Statement of the separate problem

Maintenance problem

Throughout their operational life, complex industrial systems are subject to corrective operations and preventive maintenances to keep them in working order. Preventive maintenance is a schedule of planned maintenance actions aimed at the prevention of Maintenance problem is to answer the question: when do we perform preventive maintenance task, ie, what is the periodicity of preventive maintenance?

Several policies have been proposed to address this issue, the readers can refer to [START_REF] Wang | A survey of maintenance policies of deteriorating systems[END_REF] for a detailed review of these policies. Initially, in our policy the preventive maintenance actions are performed at fixed time intervals k (k=1, 2 …), and corrective maintenance actions are carried out at failures occurs. Then, we have to determine the optimal length of preventive maintenance period which minimizes the expected maintenance cost per unit time (example, see figure 1).

Parameters:

: Preventive maintenance cost.

: Corrective maintenance cost.

: Preventive maintenance period (periodicity).

: Optimal preventive maintenance period.

: Failure rate function.

: Number of preventive maintenance during the horizon.

: Expected number of failures during .

: Expected corrective maintenance cost between [t1, t2] : Expected maintenance cost for a period t.

: Expected cost per unit time of maintenance Gertsbakh, (2000).

: Expected total cost of maintenance throughout a finite time horizon.

[Insert figure 1 [Insert figure 2 here]

The mathematical formulation of the expected total maintenance cost throughout the horizon is given below:

We implement an algorithm to compute preventive maintenance period and all the costs described above including the calculation of the expected total maintenance cost CT in a finite horizon. This cost is important to compare a separate study of the problem with our integrated one in computational results in the following section. The algorithm is stated as follows:

Algorithm

Initialization:

Parameters of the used failure distribution;

Preventive maintenance cost;

Corrective maintenance cost;

Main program:

for t = 1: to Number_period do
Computation of expected number of failures in each period t (NB(t)).

Computation of expected maintenance (preventive and corrective) cost in each period t ( ).

Computation of expected maintenance cost per unit time in each period t (equation ( 1)).

End for

Determination of the optimal result .

Computation of expected maintenance cost per preventive maintenance period

Aggregate production planning

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.

The basis of our work is the multi items capacitated lot sizing problems which are the main 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. In our work, we will consider shortage cost. To our knowledge, there are few works that deal with lots sizing problems with demand shortage. [START_REF] Sandbothe | A forward algorithm for the capacitated lot size model with stockouts[END_REF] and [START_REF] Aksen | The single-item lot-sizing problem with immediate lost sales[END_REF] addressed a single item lot sizing problem with constant capacity and shortage cost. [START_REF] Absi | The multi capacitated lot sizing problem with set up time and shortage costs[END_REF] dealt with a multi item capacitated lot sizing problem with shortage cost, and an extension to a multi-item capacitated lot-sizing problem with setup times , safety stock and demand shortage [START_REF] Absi | The multi-item capacitated lot-sizing problem with setup times, safety stock and demand shortages[END_REF]. For more information about multi items capacitated lot sizing problems, the reader can refer to a detailed review presented by [START_REF] Karimi | The capacitated lot sizing problem: a review of models and algorithms[END_REF].

Let define the parameters and the mathematical formulation of the multi items capacitated lot sizing problem with demand shortage denoted (MCLSP-SC) below:

Model parameters:

: Demand of item i to satisfy during period t.

: Set-up cost of producing one unit of item i in period t.

: Fixed cost of producing one unit of item i period t. : Variable cost of holding one unit of item i by the end of period t.

: Unit cost for requirements not met regarding demand for item i in period t.

: Processing time for each item i.

: Capacity consumed by preventive maintenance action in period t.

(t) : Capacity consumed by corrective maintenance action in period t.

: Vector of N (Number of period) elements contains the expected number of failures NB(t) in each period t of horizon if we don't perform any preventive maintenance action (figure 5).

K (t) :

Available capacity in period t.

=

Decision variables:

: 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.

: Demand shortage for item i in period t.

: Binary preventive maintenance variable.

MCLSP-SC:

Subject to: The objective function (4) minimizes the sum of the set up, holding, production, demand shortage costs over the whole N-period horizon. Constraint ( 5) is the inventory balance equation. Constraint ( 6) is the capacity constraint where preventive maintenance actions and expected number of breakdowns in each period are known. Constraint (7) relates the continuous production variables to the binary setup variables. Constraint (8) expresses that quantity lost of item i in period t must be less than demand of item in period t. The constraints ( 9) and ( 10) express non-negativity and integrality constraints.

The multi items capacitated lot sizing problem often has been studied in literature. Solution methods of the problem can be classified into three main categories. The first is exact methods e.g [START_REF] Barany | Strong formulations for multi-item capacitated lot sizing[END_REF], [START_REF] Eppen | Solving multi-item capacitated lot sizing problems using variable redefinition[END_REF], the second category is commonsense or specialized heuristics e.g [START_REF] Maes | Multi-item single-level capacitated dynamic lotsizing heuristics: a general review[END_REF], [START_REF] Kirca | A new heuristic approach for the multi-item dynamic lot sizing problem[END_REF] [START_REF] Karni | A heuristic algorithm for the multi-item lot sizing problem with cacapacity constraints[END_REF] and the third category belong to mathematical programmingbased heuristics e.g. [START_REF] Thizy | Lagrangean relaxation for the multi-item capacitated lot-sizing problem: a heuristic implementation[END_REF], Trigeiro(1987), [START_REF] Diaby | Capacitated lot-sizing and scheduling by Lagrangean relaxation[END_REF].

There are few references dealing with lot-sizing problems with shortage costs. . [START_REF] Absi | The multi capacitated lot sizing problem with set up time and shortage costs[END_REF] used inequalities within a branch-and-cut framework to find near optimal solutions, and Absi and kedad ( 2009) developed a Lagrangian relaxation of the capacity constraints to obtain lower and upper bounds for their problem. To the best of our knowledge, there are the first papers that deal with shortage costs for the multi-item capacitated lot-sizing problem.

Let the total cost of maintenance and production in the separate study, it will be the sum of expected total cost of maintenance throughout a finite time horizon CT (determined in section 3.1) and the optimal cost obtained from MCLSP-SC model .

Integrated production planning and maintenance problem

Consider a planning horizon H of length covering N periods of fixed length , and a set of item to be produced on a capacitated production line. During each period , a demand of item should be satisfied. Items are produced on a production line with known capacities given in unit time, and processing time expressed in unit time per item. Furthermore, we allow demand shortage unfulfilled due to insufficient capacity, and we will assign a high unit cost for each item lost in each period.

If in separate study the optimal length of preventive maintenance period is . It means that preventive maintenance tasks are performed periodically in the beginning of period's t =1, +1, 2 +1, 3 +1, +1…Our aim is not to plan these preventive maintenance actions at the beginning of those periods, but to plan them in time windows . In other words, a preventive maintenance task will be performed at the earliest in the beginning of the period or at the latest in the beginning of the period , and will complete within the period in which it started, with , and k is chosen to avoid the overlaps between time windows, so:

Moreover, we assume that each preventive or corrective maintenance action carried out on the production line consumes capacity units, at the beginning of the planning horizon the production line is in «as good as new (AGAN), the production line is considered here as a complex system and the failure rate is an overall rate of the whole line. When a preventive maintenance is carried out, the production line is restored to AGAN status and when a production line fails, a minimal repair is carried out to restore it to "as bad as old" (ABAO)

status. Finally we assume that the expected failures increase with elapsed time since the last preventive maintenance. Figure 3 shows an example of integrated production planning and maintenance.

[Insert figure 3 here]

The proposal mathematical program models the problem of determining optimal integrated production and maintenance plan in a production system. Indeed, the model determines the optimal production plan and optimal date of preventive maintenance actions taking into account the reliability of the production line, knowing that the failure rate increases from the last preventive maintenance task. The mathematical formulation of our problem denoted (MCLSP-TW-SC) is below. We keep the same parameters as in Section 3.2, and we add a new binary decision variable .

The variable 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 as:

This is a nonlinear expression, but can be solved by introducing the following three linear constraints below: that considers preventive and corrective maintenance. Constraint ( 14) relates the continuous production variables to the binary setup variables. Constraint (15) expresses that quantity lost of item i in period t must be less than demand of item in period t. The constraint ( 16)

ensures that one maintenance is carried out in the interval . The constraint (17) ensures that two preventive maintenance actions cannot be carried out successively. Constraints ( 18)-( 20) force the decision variable to 0 or 1. The constraints ( 21) and ( 22) express non-negativity and integrality constraints.

[Insert figure 4, 5 here]

It is time now to define how we calculate cost and capacity consumed when a preventive maintenance action is planned. From Figure 4, we observe that if a preventive maintenance is performed at a given period t, the maintenance cost in this period t is the sum of preventive and corrective maintenance costs. The corrective maintenance cost in period t is the expected number of failures in that period t multiplied by the cost of one corrective maintenance action. Notice that if no preventive maintenance action was performed, the maintenance cost in period t is the corrective maintenance cost. The same reasoning can be applied for the capacity consumed by maintenance actions in a given period t when preventive maintenance or not was carried out. Let be the expression of maintenance cost and capacity consumed by maintenance action in a given period t:

If preventive maintenance is planned at any period t, the failure rate is renewed. So the expected number of failures in this period t is the first element of vector NB which is NB(1).

This justifies equations (23), and ( 24).

If we eliminate constraints ( 16)-( 20), our problem is reduced to a separate study of multi capacitated lot sizing problem with shortage cost (MCLSP-SC) where preventive maintenance periodicity and expected number of breakdowns in each period are known.

Then, we can assimilate the (MCLSP-TW-SC) to one with set-up time (MCLSP-SC-ST) which is NP hard [START_REF] Absi | The multi capacitated lot sizing problem with set up time and shortage costs[END_REF].

Computational results

In this section, we discuss the results of some computational experiments carried out to test our integrated model. The tests aim to compare our integrated model to separate one. Remind that the separate model is the study done in section 3.1 and 3.2. We created five sets of randomly generated instances. We consider a following planning horizon composed of T [Insert table 1 here] We got a periodicity n = 3 periods ( ), and k=1. So, we can say that the time windows are [3p, 3p+2] with p = 1... . We will show computational results in the tables 3-5 obtained for each set of instances. All tests were run on a Pentium 4 (3 GHz) with 1 Gb of RAM. We used standard MIP software (XpressMP) with the solver default settings.

[Insert table 3 To assess the quality of our model solved by XpressMP, we considered four parameters: number of products, number of periods, shortage cost, and capacity tightness. For both series of results, we provide:

• #BestSol: it can be instances that could be solved to optimality (*value) and the computational times vary from 0.15 seconds to 5 minutes, or instances which the best integer solution (value) is got after 20 min of computations.

• Gab (%): for the instances that could not be solved to optimality, the average relative gap value obtained between the best integer solution and the best lower bound.

• C1: total cost of separate problem.

• C2: total cost of integrated problem. Now, we compare the results obtained with separated and integrated models. The results

show, when capacity is moderately loose, the separate problem gives better result for some instances and same results for others, the reason is that the problem is considered as uncapacitated lot sizing problem. Notice that all instances are solved to optimality, the computational time varies from 0.15 seconds to 10 seconds for all instances. When capacity is tight, the integrated problem provides better or same results. The computational times vary from 2 minutes to 5 minutes to solve to optimality for all instances with (N, T) = (3, 12), see tables 3, 4 and 5, and instances with (N, T) = (6, 18) in separate case when shortage cost equals to 65 and 75, see table 3 and4. When the capacity is too tight, for all instances the integrated problem gives better results. The computational times vary from 1 to 3 minutes to solve to optimality for all instances with (N, T) = (3, 12) (tables 3, 4 and 5). As a conclusion, the integrated model gives better solution than the separate model for different shortage cost and when the capacity is too tight.

Conclusion

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 reliability parameters of the system and its capacity in the development of the optimal production and maintenance planning. Time windows are added to reduce demand shortage in period of high demand and to give more flexibility to preventive maintenance actions. The computational results show the effectiveness of our model in a more practical and realistic case when the capacity is too tight, which means that the capacity is not sufficient to meet all demands. The Optimization solver Xpress MP can solve to optimality problem with small instances, and for larger ones we observe that it is enable to solve them to optimality, in particular when capacity is too tight. Therefore, we think about solving problem with large instances using relaxation method, and meta-heuristic in future.

F

  tagout, associated with accident prevention, and repairs without lockout/tagout.

  production resumes as soon as the inventory level drops down to or below a prespecified value s, and the facility continues to produce items until the inventory level is raised back to S. If the facility breaks down during operation, it is minimally repaired and put back into commission.

  . The primary goal of preventive maintenance is to prevent equipment failure before it actually occurs. Corrective maintenance is defined as maintenance work which involves the repair or replacement of components which have failed or broken down.

  objective function (11) minimizes the sum of the set up, holding, production, demand shortage, and maintenance (preventive and corrective) costs over the whole N-period horizon.Constraint (12) is the inventory balance equation. Constraint (13) is the capacity constraint

  fixed length to produce a set of items on production line each with an available capacity. The production, set-up, and holding costs are respectively 10, 25, and 5. Four parameters are considered for the analysis:Problem dimension: The problem dimensions represented by the number of products N : The capacity required, in each period, is then computed as if lot-for-lot solutions were implemented. The capacity is then obtained by dividing the later result by the target average utilization of capacity . The factor is set to 0.85, 0.95 and 1.1 corresponding respectively to situations with moderately loose, tight and too tight capacity constraints.Demand pattern: the demand of each product at each period is generated randomly from the interval[20,100].Shortage cost: the shortage cost is fixed for all products for all periods. It is generated randomly from the interval[50,100].All problem tests are generated with Weibull distribution of production line. Let be and respectively its probability density and failure rate functions: The shape and the scale parameters are respectively and .

Figure 4 :Figure 5 :Figure 7 :

 457 Figure 4: Expected number of failures according to preventive maintenance actions

  We consider in this paper a finite time horizon of length T. Once, the length of preventive maintenance period is known, we compute the expected total maintenance cost throughout the horizon which is based on the expected maintenance cost per preventive

	maintenance period	and	(	) (example, see figure 2) 1 .
	F o r
	P
	e
		e r
		R
		e
			v i e
				w
				O n l
				y
				here]

Table 1

 1

		shows the expected number of failures as a function of system's age. The cost of
	preventive maintenance action is set to	, and the cost of minimal repair action is
	given by	. The capacity lost, when a preventive maintenance task and minimal repair
	action are carried out, is respectively	, and	.
		[Insert figure 6 here]
		[Insert figure 7 here]
		[Insert figure 8 here]
	Figures 6, 7, and 8 show the expected total cost of maintenance strategy as a function of the
	preventive replacement cycle T. We illustrate optimal preventive maintenance cycle and the

Table 1 :

 1 Expected number of failures of the system for T= 12, 24.

	Cost per
	unit time

Table 2 :

 2 Expected total cost of maintenance of the system for T= 12, 18, 24.

	Item	Period	Total				Shortage cost = 65
			cost	Capacity tightness
				Moderately loose		Tight			Too tight
				#BestSol Gap (%)	#BestSol	Gap (%)	#BestSol	Gap (%)
	3	12	C1	*22527	0		*24265		0	*40822	0
			C2	*22621	0		*22436		0	*39071	0
	6	18	C1	*62367	0		*79652		0	129797	0.017
			C2	*62367	0		79652		0.15	129687	0.09
	12	24	C1	*174927	0		220042		0.04	346727	0.04
			r o F C2 *175012	0		219977		0.12	346697	0.1
	Item	Period	Total cost	P Capacity tightness		Shortage cost = 75
				e Moderately loose		Tight			Too tight
	3	12	C1	#BestSol Gap (%) e r *23447 0	#BestSol Gap (%) *24982 0	#BestSol Gap (%) *47802 0
			C2	*23541	0		*23046		0	*46784	0
	6 12	18 24	C1 C2 C1	*67297 *67297 *173227	0 0 0	*84652 84652 e R 217387		0 0.11 0.02	136222 135911 371422	0.035 0.21 0.03
			C2	*173312	0		e i v 217387	0	370986	0.1
							w
									O n l
										y

Table 5 :

 5 Summary solution with shortage cost equals to 95.

	Item	Period	Total			Shortage cost = 95
			cost	Capacity tightness
				Moderately loose	Tight			Too tight
				#BestSol Gap (%)	#BestSol Gap (%)	#BestSol Gap (%)
	3	12	C1	*22947	0	*28092	0	*51647	0
			C2	*23041	0	*26145	0	*48231	0
	6	18	C1	* 64097	0	86132	0.03	160427	0.025
			C2	*64097	0	86132	0.1	160034	0.23
	12	24	C1	*172737	0	223212	0.025	428437	0.05
			C2	*172822	0	223212	0.1	427724	0.16

Table 3 :

 3 Summary solution with shortage cost equals to 65.

Table 4 :

 4 Summary solution with shortage cost equals to 75.

http://mc.manuscriptcentral.com/tprs Email: ijpr@lboro.ac.ukInternational Journal of Production Research

Acknowledgements

The authors would like to thank anonymous referees for their careful reading, thorough reviews and valuable comments that helped us improving the quality of the paper.


Marie-Bernadette Lepetit

Bernard Mercey

Charles Simon

Understanding interface effects in perovskite thin films

Understanding interface effects in perovskite thin films

The control of matter properties (transport, magnetic, dielectric,. . . ) using synthesis as thin films is strongly hindered by the lack of reliable theories, able to guide the design of new systems, through the understanding of the interface effects and of the way the substrate constraints are imposed to the material. The present paper analyses the energetic contributions at the interfaces, and proposes a model describing the microscopic mechanisms governing the interactions at an epitaxial interface between a manganite and another transition metal oxide in perovskite structure (as for instance SrTiO3). The model is checked against experimental results and literature analysis.

The technological importance of spin valves or spin injectors as potential applications of manganese oxides induced a large number of works on manganite thin films 2,3 . For this reason the lost of magnetization of La 2/3 Sr 1/3 MnO 3 (LSMO) or La 2/3 Ca 1/3 MnO 3 (LCMO) near an SrTiO 3 (STO) interface has been the subject of many interpretations. Let us cite (i) homogeneous substrate strain 4 (ii) electronic and/or chemical phase separation 5 related to structural inhomogeneities at the interface 6 , (iii) manganese e g orbital reconstruction inducing C-type antiferromagnetism 7,8 . None of these interpretations however provide a good understanding of the observed phenomena. For instance, it was shown that an homogeneous substrate strain of the in-plane parameters does not relax for film thickness smaller than 1000 Å5 , while a drastic change in the transport properties is observed for films thinner than a few unit cells (∼ 3 -4 on STO substrate 7,9 , ∼ 30 on LaAlO 3 substrate 7 ,. . . ). In the second hypothesis (ii), there is no clear proposition of the nature of the inhomogeneities, their origin, the way they may act in order to induce the observed properties. Finally, ferromagnetic hysteresis loops were found in very thin films up to only three unit cells 9 (u.c.), in contradiction with the proposed C-type AFM ordering resulting from orbital ordering (iii). In any case, whatever the reasons put forward, the existence of a so-called "dead layer" at the interface between the manganite film and most perovskite substrates seems to be established 9,10 . This "dead layer" is of a few unit cells width and exhibits a large decrease of the conductivity ; however its origin is not at all understood.

We believe that a careful analysis allow us to infer a model for the interface effects between a manganite and an oxide substrate with a perovskite structure. The main concepts of our model can be summarized as an energy balance at the interface.

• It is well known that the strongest effect of the substrate is to constrain the film inplane cell parameters to fit the substrate ones. a film = a substrate b film = b substrate This constraint is quite strong since it is associated with bond elongation, i.e. the most energetic vibrational modes 11 . It thus relaxes slowly (not before 250 u.c., 1000 Å, on a STO substrate 5 ). In the literature, it is associated with an u.c. volume constraint V film ≃ V bulk . While there is indeed, in the film free energy, an elastic term favoring

V film = V bulk : V 2κ ∆V V
2 , this term cannot not be treated as a constraint imposed by the substrate. It should rather be evaluated against the other terms of the constrained film free energy.

• The substrate imposes to the film its in-plane symmetry operations. This constraint is usually weak since, acting on bond angles and dihedral angles, it is related to low energy vibrational modes. One can thus expects that, after constraining the few first cells at the interface, these constraints start to relax according to the energy of the associated vibrational mode.

• The electronic structure of the film and substrate interact at the interface. In particular the possible delocalization effects at the interface should be taken into account.

As a matter of example let us see how these constraints apply to a LSMO thin film on an STO (001) substrate. The in-plane parameters of the LSMO are imposed by the STO substrate and the film in under tensile strain at the interface. As already mentioned, this constraint holds over a large number of monolayers (ML).The STO in-plane symmetry operations impose to the first layers of LSMO to present a 4-fold symmetry axis, perpendicular to the film, and untilted octahedra. It results, that in these layers, the crystal-field split 3d orbitals of the Mn atom share their ( x, y, z) orthogonal axes with the ( a, b, c) substrate lattice vectors ( c being the out-of-plane direction) ; that is the Mn orbitals are d ab , d ac , d bc for the low energy ones and d 3c 2 -r 2 , d a 2 -b 2 for the high energy ones. Any further distortion (such as Jahn-Teller) of the MnO 6 octahedra should thus respect those constraints and are therefore restricted to atomic movements along the c direction. Let us now analyze the energy minimization of the LSMO layers at the interface under the above constraints. On one hand the film is under tensile strain and the minimization of the elastic energy favors a contraction of the LSMO monolayer u.c. along the c axis. On the other hand the film Fermi level orbitals are the Mn partially filled d z 2 and d x 2 -y 2 ones while the lowest empty orbitals of the STO are the Ti empty 3d orbitals. It is thus quite natural that the e g manganese orbitals delocalize to some extend into the titanium 3d empty ones. This interaction is particularly favored in the case of the 3d z 2 orbitals of the Mn and Ti atoms (see figure 1). Indeed, the through-bridge bonding mechanism

t d Mn z 2 ,d Ti z 2 =-d Mn z 2 | Ĥ|p O z Op O z | Ĥ|d Ti z 2 / (ε d -εp).
acting via the 2p O z orbitals of the oxygens is energetically favored, due to the large d Mn z 2 -2p O z -d Ti z 2 overlaps (fig. 1). This delocalization energy tends to favor a Jahn-Teller distortion increasing the occupation of the d Mn z 2 orbitals, that is an elongation of the LSMO monolayer u.c. along the c axis. The physics at the interface thus results from the competition between

• the elastic energy that favors c < a, V 2κ ∆V V 2
• and the delocalization energy that favors c > a

- (t d Mn z 2 ,d z 2 Ti ) 2 ε d Ti z 2 -ε d Mn z 2
where

t d Mn z 2 ,d Ti z 2 ≃ - d Mn z 2 |p O z p O z |d Ti z 2 ε d -ε p
On one hand, the relatively short metal-oxygen distances (≃ 1.95 Å), the strong directionality of the d z 2 and p z orbitals inducing a very large d Mn z 2 |p O z p O z |d Ti z 2 overlap term, and the relatively weak orbital energy difference between the oxygen 2p and the metal 3d orbitals results in a very large effective transfer integral between the d z 2 orbitals of the Mn and Ti atoms (of the order of ≃ 1eV 12 ) and thus of the delocalization energy. Indeed, one do not expect the ε d Ti z 2 -ε d Mn z 2 energy difference to be larger than a few electron-Volts (∼1,2). On the other hand, the elastic energy can be evaluated for different c values. For a cubic structure (c = c ST O = 3.905 Å), the LSMO volume increase is of 2%, and the associated elastic energy can be estimated to ≃ 0.015 eV (the LSMO compressibility being taken from ref. 13), that is much weaker (an order of magnitude) than the expected delocalization energy. Even if these energetic evaluations are only qualitative, one can expect with quite confidence, that for LSMO films (LCMO and similar films) on a STO substrate the first few ML at the interface are elongated in the c direction, despite their already in-plane tensile strain.

After a few unit cells, there is no more delocalization energy to gain from a larger occupation of the d z 2 orbitals. At the same time, the in-plane symmetry operations imposed by the substrate should start to relax, and in particular the ones associated with the LSMO vibrational modes with the lowest frequencies : the octahedra tilt. The consequence of this constraint relaxation should be to adjust the e g orbitals occupations toward a bulklike value (compared to the first interface ML it means an increase of the d x 2 -y 2 occupation and a decrease of the d z 2 one). The c ML parameter should thus start to decrease toward the value expected from the elastic energy minimization (monolayer u.c. volume conservation), that is a contraction of the monolayers c parameter and a tendency to a larger occupation of the d x 2 -y 2 orbitals compared to the d z 2 ones.

What are the consequences of the above structural considerations, in term of magnetic and transport properties? As far as the first few layers at the interface are concerned, the tendency to occupy the d z 2 orbitals at the expense of the d x 2 -y 2 ones should result in a decrease of the (in-plane) double-exchange and thus in a strong reduction of both the Curie temperature and the conductivity 19 . Let us note that even the d z 2 orbitals are subject to the (in-plane) double-exchange, however the d Mn z 2 -p O x/y -d Mn z 2 delocalization process is much less effective than the d Mn x 2 -y 2 -p O x/y -d Mn x 2 -y 2 one, with the consequence that the effective (in-plane) exchange integral is much weaker and thus the Curie temperature. After a few layers, the relaxation of the e g orbitals occupation should result in an increase of the double exchange and thus of both the Curie temperature and the film conductivity. The temporary limit being set by the in-plane a, b parameters constraints.

Finally, for very thick films this later constraint is relaxed and one should retrieve the bulk properties.

Let us now check the predictions issued from our model against the experimental data. LSMO thin films were grown by Laser-MBE using a setting detailed in ref. 17. The temperature during the deposition is 620 • C and the pressure is 4 × 10 -4 mbar. The gas is a mixture of O 2 and O 3 . The intensity oscillations of the specular beam (RHEED) are used to measure the number of deposited layers. After the deposition, the pressure is increased to 5 × 10 -3 mbar associated with a higher O 3 concentration, to insure a good oxidation. The films are then cooled to room temperature in this atmosphere. Such deposition conditions allow us to grow films with a T c higher than 325 K for thicknesses larger than 100 Å. After the growth, the films were studied by X-ray diffraction (θ-2θ mode) Seifert XP3000 system. From these data, the c lattice parameter and the films thicknesses were determined. These measured thicknesses agree well with the number of RHEED oscillations. Magnetization measurements were carried out in a Quantum Design SQUID system, under a 500 Oersteds magnetic field parallel to the substrate. Transport measurements, four probes method, were carried out in a PPMS Quantum Design system.

LSMO ultrathin films with thicknesses ranging from 31 Å (8 ML) to 92 Å (24 ML) were grown onto 001oriented SrTiO 3 substrates, previously etched to ensure a TiO 2 terminating layer 14 . The out-of-plane average ML lattices parameters, c , of the different films were determined by X-ray diffraction and are reported in table I. One sees immediately that the variation of c as a func- tion of the number of ML does not correspond to the value expected from the conservation of the manganite volume. Indeed, the LSMO tensile strain at the interface let us expect a reduction of the ML average c parameter compared to the bulk value. Table I exhibits large c values associated with ML volumes larger than V bulk . Such a behavior was also observed by Herger & al 15 on even thinner films with only 1,2,3,4,6 and 9 ML of LSMO. Our films show an increase of c when reducing the thickness (in agreement with films of ref. 15). Further examining the data of reference 15, one sees that while the first few layers at the interface exhibit c values larger than the STO one (cumulative displacement compared to c STO : ∆z positive and increasing), after about 3 monolayers c retrieves a value smaller than a = a STO (decreasing ∆z). At this point, let us notice that all these structural results are in full agreement with our theoretical predictions.

Thickness ( Å) Monolayers c ( Å) V (
Let us now examine the magnetic and transport properties. The magnetization of the different films is reported on figure 2.For films thicker than 14 ML, the magnetization remains close to the values observed for the thickest films and the T c remains high. However, when the number of ML is decreased to 8, the magnetization at 10 K as well as T c are strongly decreased. These results agree well with the values reported in other works 7,9 for ultrathin LSMO films. The resistivity measurements FIG. 2: Magnetization under a 500 Oersteds field of LSMO films on a STO (001) substrate.

agree with the suppression of the double exchange suggested for 8 ML films on the magnetization curves (fig. 2). Indeed, the three thickest samples (14, 18 and 24 ML) exhibit a metal-insulator transition at T c while the thinner one (8 ML) presents a semiconductor-like behavior with a large magneto-resistive effect. These peculiar magnetic and transport properties of the first few (∼ 6-8) LSMO layers at the interface with STO are usually referred at as the dead layer 7,9 .

At this point let us notice that these experimental results do agree nicely with our theoretical predictions, and allow us to determine the number of layers over which each constraint extends. For instance the dominating d z 2 occupation can be associated with only the first 2-3 ML (see T c or ∆z variation in the 9 ML film of ref. 15). In the next layers the e Mn g orbitals occupations start to relax toward a more balanced one as supported by the increase of both T c and conductivity as a function of the number of layers in 3 to 8 ML films. The fact that the T c value reaches its saturation limit, that the metalinsulator transition is restored tells us that after about 8 ML, the double-exchange is fully restored and thus the manganese e g orbitals occupations.

The last point of our predictions one would like to see verified, is the actual dominating d Mn z 2 occupation in the first layers at the STO interface. LD-XAS experiments were conducted by different authors that reached opposite conclusions. Indeed, while Tebano et al 7 report the signature of a preferential d z 2 orbital occupation at the interface, Huijben et al argue that the experimental evidences are not significant. Very recently ARPES measurements 8 were done on 4,6 and 10 ML films and clearly exhibit an increase of the Γ point signal (attributed to the d Mn z 2 orbitals contribution) when the thickness of the film decreases. It thus seem that the experimental data confirm our findings from constraints and energetic considerations.

Another way to check the validity of our interface model is to test it against the properties of ultrathin films deposited on a buffered substrate. Indeed, the deposition of a buffer layer at the interface between the STO and the LSMO should strongly modify the electronic structure of the first LSMO layers. We thus grew either a (LaMnO 3 )(SrMnO 3 ) buffer (BUF1) or a [(LaMnO 3 )(SrMnO 3 )] 2 (BUF2) on STO, prior to the LSMO film. Let us point out that the (LaMnO 3 )(SrMnO 3 ) superlattices present an antiferromagnetic behavior 18 , when deposited in these conditions. The buffer layer will thus break both the symmetry constraint preventing the LSMO octahedra to be tilted in the first LSMO layers at the interface and the possibility for the LSMO manganese to delocalize its d z 2 population in the empty d Ti z 2 orbitals. Of course some delocalization of the LSMO d Mn z 2 orbitals can still occur since the buffers d Mn z 2 orbitals are most probably not totally filled. This effect is however expected to be much weaker than at the LSMO/STO interface both because of the buffer d Mn z 2 orbital occupation and because of the starting of the MnO 6 octahedra tilt in the LSMO. We thus expect that the buffered LSMO films will be more "bulklike" than the unbuffered ones, the thicker the buffer, the more "bulk-like" the film (the wider the distance between the film and the STO surface, the lesser the symmetry constraints will be imposed to the LSMO) . It means that for equal film thicknesses, the thicker the buffer, the highest the T c and the film conductivity should be. Table II experimental data show that the average c ML parameter decreases when the number of buffer layers goes from 0 to 2. This results in a decrease of the c/a ratio and induces a larger occupation of the d x 2 -y 2 orbitals with respect to the d z 2 ones. Such an electronic structure favors a more robust ferromagnetic behavior of the films as the thickness of the buffer layer increases. Indeed, both the magnetization (see fig. 3) and T c increase with increasing buffer thickness, as expected from the c/a ratio. The film buffered by two (LMO)(SMO) layers exhibits a T c close to 275 K, that is about 100 K higher than the T c observed for the film of same thickness deposited on a bare STO substrate. The resistivity of the buffered films behave as for a regular ferromagnetic materials with a maximum value at T c .

In conclusion, we presented in this paper a simple model able to explain and predict the interface effects between perovskites oxides. We applied our model to the difficult case of the nature and origin of the so-called dead layer at the LSMO interface with an STO substrate. Our model successfully predicted and explained all experimental features of this interface, up to now not under- stood.

  Marie-Bernadette Lepetit, Bernard Mercey, and Charles SimonCRISMAT, ENSICAEN-CNRS UMR6508, 6 bd. Maréchal Juin, 14050 Caen, FRANCE (Dated: May 4, 2011)

FIG. 1 :

 1 FIG. 1: (a) d Mn z 2 -2p O z -d Ti z 2 and d Mn x 2 -y 2 -(O)-d Ti x 2 -y 2 delocalization mechanism. (b) Through bridge delocalization mechanisms. The resulting effective Mn-Ti transfer integral is at the second order of perturbation

  Average c ML parameters for 8 ML LSMO films deposited either on bare or on buffered substrates.

FIG. 3 :

 3 FIG. 3:Magnetization under a 500 Oersteds field, of 8 ML LSMO films deposited on bare or buffered (BUF1 : (LMO)(SMO) × 1 ; BUF2 : (LMO)(SMO) × 2) SrTiO3 substrates. The magnetization of an unbuffered 24 ML film is given for comparison.

Acknowledgments

The authors thank ANR program SEMOME and STREP MACOMUFI for the support of this work.