Simultaneous determination of production lot size and preventive maintenance schedule for unreliable production system

The Authors

Anis Chelbi, Centre de Recherche en Productique (CEREP), Tunis, Tunisia

Nidhal Rezg, Laboratoire de Génie Industriel et de Production Mécanique (LGIPM), Metz, France

Mehdi Radhoui, Laboratoire de Génie Industriel et de Production Mécanique (LGIPM), Metz, France

Abstract

Purpose – The purpose of this study is to propose and model an integrated production-maintenance strategy for unreliable production systems producing conforming and non-conforming items.

Design/methodology/approach – The proposed integrated policy is defined and modeled mathematically.

Findings – The paper focuses on finding simultaneously the optimal values of the lot size Q and the age T at which preventive maintenance must be performed. These values minimize the total average cost per time unit over an infinite horizon.

Practical implications – The paper attempts to integrate in a single model the three main aspects of any manufacturing system: production, maintenance, and quality. It deals with the lot-sizing problem for a production system which may randomly shift to an out-of-control state and produce non-conforming units. The system is submitted to an age-based preventive maintenance policy. The effect of performing preventive maintenance on quality- and inventory-related costs is shown through a numerical example.

Originality/value – The paper proposes an integrated model that links EMQ, quality and an age-based preventive maintenance policy. It is shown that performing preventive maintenance yields reduction in inventory- and quality-related costs.

Article Type:

Research paper

Keyword(s):

Lot size; Preventive maintenance; Quality; Inventory.

Journal:

Journal of Quality in Maintenance Engineering

Volume:

14

Number:

2

Year:

2008

pp:

161-176

Copyright ©

Emerald Group Publishing Limited

ISSN:

1355-2511

I. Introduction

Considerable research has been undertaken to enrich the classical Economic Manufacturing Quantity (EMQ) model (see Silver et al. (1998)). A great number of these research efforts aimed at extending the EMQ model to single unit production systems subjected to random failures and submitted to different maintenance strategies (Hau et al., 1987; Groenevelt et al., 1992, Cheung and Hausman, 1997; Tadashi et al., 2001). Moueen et al. (1996) considered this lot sizing problem in presence of regular maintenance interruptions with no failure permitted within a production cycle. Chelbi et al. (2004) proposed a more general model considering regular maintenance with possibility of failure, still for a single unit production system. The case of production systems composed of n machines has also been dealt with in the literature; see for example (Dallery et al., 1992; Xie, 1993; Rezg et al., 2004).

The literature also provides models that deal with the effects of defective items produced by an imperfect production process on EMQ. Rosenblatt et al. (1986) analyzed the situation where the system is subject to a random process deterioration that shifts to an out-of-control state (and produces some proportion of defective items), they show that it is better to produce smaller lot sizes than those of the classical EMQ model. Porteus et al. (1985) reached the same conclusion and explored different options for investing in quality improvement and setup cost reduction.

More recently, Ben Daya (2002) proposed an integrated model that links EMQ, quality and maintenance requirements for an imperfect production process and different levels of preventive maintenance. He proved that performing preventive maintenance yields reductions in quality related costs.

Preventive maintenance policies adopted in the above mentioned papers consist in periodical inspections and/or regular preventive replacements. Age based preventive maintenance have not been considered.

In our present work, we consider a single unit production system which must satisfy a constant and continuous demand D. After a random period of operation, the system shifts to an out-of-control state and produces non-conforming units at a given rate. The system is submitted to an age-based preventive maintenance policy. A preventive maintenance action is performed as soon as the system reaches age T without having shifted to the out-of-control state. If the shift to the out-of-control state is detected, a restoration action of the system is planned for L time units later.

A mathematical model and a numerical procedure are developed in order to simultaneously determine nearly optimal values of the lot size Q and the age T for preventive maintenance. The total average cost per time unit is considered as the performance criterion of the proposed policy.

The next section defines the proposed strategy and states the working assumptions and the used notation. In section III, the mathematical model is presented. The fourth section is dedicated to a numerical procedure developed for finding the best combination of Q and T minimizing the total unitary average cost. This procedure is tested in section V through a numerical example. Finally, a summary of the work together with indications about extensions currently under consideration is provided in the last section of the paper.

II. Problem definition, working assumptions and notation

We consider a single unit production system producing a single item having a constant and continuous demand of D units per time unit. A production cycle starts with a new system which is assumed to be in an in-control state, producing items of acceptable quality at the maximum production rate P which is greater than the demand D. Whenever the process reaches a maximum inventory level Z, the production rate is made equal to D. Nevertheless, after a random period of time in production, the system may shift to an out-of-control state, producing non-conforming units at a rate α. The elapsed time τ for the production process to be in the in-control state, before the shift occurs, is a random variable which is assumed to follow a general distribution with an increasing hazard rate.

While being in the in-control state and in order to reduce the probability to shift to the out-of-control state, the production unit is submitted to preventive maintenance of negligible duration at age T. The shift to the out-of-control state is detected instantaneously. When this shift occurs, a restoration of the system to the in-control state and to the as good as new condition is planned for L time units later. During this period L of preparation of the different resources needed for restoration, the non-conforming units produced at rate α are immediately rejected. The restoration action has a random duration.

The production cycle ends at the end of period L following a shift to the out-of-control state. We suppose in this paper that failure may never occur before restoration takes place (Figure 1).

To ensure the continuous satisfaction of demand D, the lot size Q must be set so as to be able to compensate the average consumption during the restoration period, as well as the non-conforming units rejected during period L following the shift to the out-of control state.

Hence, the decision variables are the lot size Q and the age T at which preventive maintenance must be performed. Here are the assumptions and notations adopted to develop an integrated model for the joint determination of the optimal values of Q and T which will minimize the total average cost per time unit over an infinite horizon:

• The time τ which elapses until the production process shifts to the out-of-control state is a random variable and follows a general distribution with increasing hazard rate.
• The shift to the out-of-control state is detected instantaneously.
• As soon as the shift to the out-of-control state is detected, a restoration action of the unit is planned for L time units later.
• The restoration time is a random variable t.
• The production unit never fails before the restoration takes place
• The probability distribution associated to the time to restore the production unit to the as good as new state, is known.
• Each preventive maintenance action takes a negligible time and restores the system to the as good as new state.
• While in the out-of-control state, the system produces non-conforming items at a constant rate α.
• All costs are supposed to be constant and known.
• Shortage is possible. In this situation a shortage cost is associated to the supplementary effort deployed to ensure the satisfaction of demand.
• Unused buffer inventory is depleted to zero only after a restoration cycle. The depletion period is considered to be negligible.

II.1 Notation

• f (τ): probability density function associated to the time to shift to the out-of control state.
• F (τ): probability distribution function associated to the time to shift to the out-of control state:F(τ)=∫₀ ^τ f(x)dx
• R (τ)=1 – F(τ)
• r(.): Hazard rate function associated to the time to shift to the out-of control state:r(τ)=f(τ)/R(τ)
• h (t): probability density function associated to the production unit restoration time.
• H (t): probability distribution function associated to the production unit restoration time:H(t)=∫_{\kern4 0} ^t h(x)dx
• μ: average restoration time:μ=∫_{\kern6 0} ^∞ tdH(t)
• Q: lot size.
• Z: the maximum inventory level reached during a production cycle.
• P: maximum production rate (units/unit time).
• D: The demand (units/unit time).
• T: age at which the system is submitted to a preventive maintenance action.
• K: average setup cost.
• C _cm : average cost of a corrective maintenance action.
• C _pm : average cost of a preventive maintenance action.
• C _p : shortage cost per item.
• C _s : holding cost per item per unit time.
• C _nq : cost incurred by producing a non-conforming item per unit time.
• α: production rate of non-conforming items.
• X _T : time to shift to the out-of-control state under preventive maintenance at age T.
• E(X _T ): the average time to shift to the out-of-control state under preventive maintenance at age T.
• L: period of preparation of the different resources needed to restore the system to the state “as good as new”.

III. Model development

Our objective is to establish the expression of the the total average cost per time unit. The total expected cost corresponds to the sum of the setpup cost (K), the maintenance cost (CM), the inventory holding cost (CS), the shortage cost (CP) and the cost of non-conforming items (CNQ). This total cost will be divided by the average duration of a restoration cycle.

The model presented in this section is a stationary model that considers average of all possible realizations.

Figures 2 and 3 illustrate the two possible scenarios of the inventory evolution over a restoration cycle. The maximum inventory level is denoted by Z.

The condition to be in Scenario 1 is: Equation 1 and the one to be in scenario 2 is: Equation 2 These conditions represent an approximation since the probability density function f(τ) is not a pulse at: Equation 3

III.1 Average restoration cycle length

The average restoration cycle duration, RCD, is given by: Equation 4 As shown in Figure 1, the system is submitted N times to preventive maintenance at age T and the shift to the out-of-control state occurs between NT and (N+1)T for N=1,2,3, … .

Hence, E(X _T ) is given by the following expression: Equation 5 Hence: Equation 6

III.2 Scenario 1: case where the maximum inventory level Z is reached after τ (see Figure 2)

III.2.1 The average maintenance cost: CM. The number of preventive maintenance actions before the shift to the out-of-control state is given by: Equation 7 where (.) stands for the integer part function.

The expected number of preventive maintenance actions can be expressed as follows: Equation 8 Hence, the maintenance cost over a restoration cycle is: Equation 9

III.2.2 The average inventory holding cost: CS. Referring to Figure 2, we distinguish five different inventory zones. Let A_i be the average inventory held in Zone 1.

Zone 1: Equation 10

Zone 2: Equation 11

Zone 3: Equation 12

Zone 4 (see Figure 4): Equation 13

Equation 14

A ₄ is then expressed as follows: Equation 15

Zone 5: Equation 16 Thus, considering together zones 1 to 5, the expected total inventory holding cost is given by the following expression: Equation 17

III.2.3 The average cost of non-conforming units: CNQ. Referring to Figure 5, the average number of non-conforming units is given for each zone: Equation 18 Equation 19 The average cost of units lost is then given by: Equation 20 III.2.4 The average shortage cost: CP. Shortage occurs in case the restoration action lasts longer than the period necessary for the consumption of the remaining inventory at the end the preparation period L. The average shortage cost can be expressed as follows: Equation 21 Where t _c is the period necessary for the consumption of the remaining stock available at the beginning of the restoration action: Equation 22 Thus, the expected total cost for scenario 1 is expressed by the sum of the different costs detailed so far: Equation 23 The total expected cost per unit of time is consequently given by: Equation 24 It is important to point out that for this first scenario, the lot size Q (the quantity produced during the actual production time for each restoration cycle) is given by the following expression as a function of the maximum inventory level Z: Equation 25 with: Equation 26 (see Figure 2). This expression will be used later in next sections to derive the optimal lot size Q _opt from any given optimal value of the maximum inventory level Z _opt.

III.3 Scenario 2: case where the maximum inventory level Z is built before τ (see Figure 3)

III.3.1 The average maintenance cost: CM. Like for scenario 1, this cost is given by: Equation 27

III.3.2 The average inventory holding cost: CS. Referring to Figure 3, we distinguish four different inventory zones. Let A _i be the average inventory held in Zone 1.

Zone 1: Equation 28 Zone 2: Equation 29

Zone 3: Equation 30

Zone 4: Equation 31 Hence, considering together zones 1 to 4, the expected total inventory holding cost is given by the following expression: Equation 32

III.3.3 The average cost of non-conforming units: CNQ. The average number of non-conforming units is given by: Equation 33 The average cost of units lost is then: Equation 34 III.3.4 The average shortage cost: CP. As in the situation of scenario 1, shortage occurs in case the restoration action lasts longer than the period necessary for the consumption of the remaining inventory at the end the preparation period L. The difference with scenario 1 is the remaining inventory level at the end of period L, which is equal in this case to Z-αL. The average shortage cost can then be expressed as follows: Equation 35 Thus, the expected total cost of scenario 2 is expressed by the following sum: Equation 36 The total expected cost per time unit is therefore given by: Equation 37 Like for the first scenario, it is possible to express, for this second scenario, the lot size Q as a function of the maximum inventory level Z: Equation 38 This expression will also be used later to derive the optimal lot size Q _opt from any given optimal value of the maximum inventory level Z _opt.

Finally, we obtain the following non-linear optimisation problem: Equation 39 In fact, according to the proposed strategy in this paper, the maximum inventory level Z can not be larger than Z _max which is the sum of the quantity built before the shift to the out-of-control state and the quantity built during period L.

Z _max is given by: Equation 40 It should be noted that considering the average of all possible realizations on a stationary basis, the cost model is an approximation. What controls how close is the approximation is the time to shift out of control distribution. If the standard deviation of X _T is small (the distribution of X _T is concentrated around its average) the approximation will be close to the exact cost value. The wider is the distribution the more important is the error.

IV. Numerical procedure

Owing to the complexity of the model, an iterative numerical procedure has been developed to obtain the best solution (Z _opt,T _opt) and the corresponding (Q _opt,T _opt), for any given system and costs configuration.

The input data are: f(.), h(.), C _cm , C _pm , C _p , C _s , C _nq , P, D, L, K, α, ΔZ, Ti, ΔT and T _max

where:

Ti: Initial value of T.

T _max: Maximum value of T.

ΔZ Increment of the maximum inventory level.

ΔT Increment of time.

T _max must be chosen large enough to cover a wide range of values of T (see Figure 6).

V. Numerical example and discussion

To illustrate our approach, we consider a situation with the following input data which have been arbitrarily chosen considering nevertheless realistic settings:

• Distribution associated to the time to shift to the out-of-control state: Weibull law with shape parameter 1.25 and scale parameter 1, giving a mean value of 0.93 unit time. (one unit time=1 month).
• Restoration time distribution: Gamma law with shape parameter 2 and scale parameter 40 (mean value=0.05 month).
• C _cm =1000 $, C _pm =75 $, K=500 $.
• Cs=1 $/unit/month, C _p =30 $/unit short.
• C _nq =10 $/unit lost/month.
• P=32,400 units/month.
• D=20,160 units/month.
• α=3,600 units/mont.
• Tmax=2.5 months, ΔZ=5 units, ΔT=0.1 month, L=0.03 month

The obtained results are shown in Table I and Figure 7.

Hence, the best strategy consists in performing a preventive maintenance action whenever a restored system reaches age T _opt=0.2 month. The lot size to be produced between consecutive restorations is equal to Q _opt (33,524 units). According to these settings, the maximum inventory level that would be reached during a production cycle is equal to Z* (2,540 units).

If this best strategy is adopted (see Table I), the period of time between consecutive restorations of the system is expected to be equal to 1.58 month (RCD=1.58). During this period, an average of seven preventive maintenance actions would be carried out (E(NPM)=6.98), and the shift to the out-of-control state would occur in average after 1.50 month (E(X _T )=1.50). The total corresponding unitary operating cost would be equal to 4203,54 $/month. Also notice, that in this case, we would be in a configuration corresponding to scenario 2: Equation 41 The expected time to shift to the out-of-control state without preventive maintenance being equal to 0.93 (mean of the used Weibull distribution), it is interesting to point out that the adopted age-based preventive maintenance policy (with T=0.2 month) delays the start of non-conforming units production (the shift to the out-of-control state) by (1.50-0.93)=0.57 month in average.

More generally, for very small values of T (important preventive maintenance effort), the average time to shift to the out-of-control state increases, which means that the start of production of non-conforming units is delayed. On the other hand, for high values of T (negligible preventive maintenance effort), compared to the optimal solution, even if maintenance unitary costs are lower, the quality and inventory related unitary costs are higher. Hence, performing preventive maintenance yields reduction in inventory and quality related costs.

VI. Conclusion

This paper presented a joint operating-preventive maintenance and quality control strategy for a randomly failing single unit production system, which must continuously satisfy a demand of D units per time unit, and may randomly shift to an out-of-control state and start producing non-conforming units. When this shift occurs a full restoration of the system is planned for L time units later.

This strategy is characterized by two decision variables: T that represents the age at which the system must undergo preventive maintenance, and the production lot-size Q to be set so as to satisfy the demand compensating the average consumption during the restoration period, as well as the non-conforming units rejected during period L following the shift to the out-of control state.

A mathematical model and a numerical procedure have been developed to generate the optimal strategy (Q _opt,T _opt) which minimizes the total average unitary cost.

Some extensions to this work are currently under consideration. Particularly, the possibility that the system fails during the preparation period L, and the consideration of preventive maintenance actions with non-negligible duration.

Equation 1

Equation 2

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Equation 15

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Equation 24

Equation 25

Equation 26

Equation 27

Equation 28

Equation 29

Equation 30

Equation 31

Equation 32

Equation 33

Equation 34

Equation 35

Equation 36

Equation 37

Equation 38

Equation 39

Equation 40

Equation 41

Figure 1The operating strategy: Scenario 1

Figure 2Inventory evolution over a restoration cycle: case where Z is reached after the shift to the out-of-control state: Scenario 2

Figure 3Inventory evolution over a restoration cycle: case where Z is reached before the shift to the out-of-control state

Figure 4Detailed view of zone 4

Figure 5Detailed view of the non-conforming units zones

Figure 6Numerical procedure

Figure 7Evolution of the average total unitary cost C(Z*, T)

Table INumerical results

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Corresponding author

Anis Chelbi can be contacted at: anis.chelbi@planet.tn