Joint optimal lot sizing and preventive maintenance policy for a production facility subject to condition monitoring

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Joint optimal lot sizing and preventive maintenance policy for a production facility subject to condition monitoring L. Jafari, V. Makis

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S0925-5273(15)00283-2 http://dx.doi.org/10.1016/j.ijpe.2015.07.034 PROECO6171

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Int. J. Production Economics

Received date: 21 April 2015 Revised date: 16 July 2015 Accepted date: 29 July 2015 Cite this article as: L. Jafari, V. Makis, Joint optimal lot sizing and preventive maintenance policy for a production facility subject to condition monitoring, Int. J. Production Economics, http://dx.doi.org/10.1016/j.ijpe.2015.07.034 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Joint optimal lot sizing and preventive maintenance policy for a production facility subject to condition monitoring L. Jafari, V. Makis∗ Mechanical and Industrial Engineering Department, University of Toronto, Toronto, Ontario, Canada

Abstract In this paper, we consider the joint optimization of economic manufacturing quantity (EMQ) and preventive maintenance (PM) policy for a production facility subject to deterioration and condition monitoring (CM). Unlike the previous joint models of EMQ and maintenance policy which used traditional maintenance approaches, we propose the proportional hazards model (PHM) to consider CM information as well as the age of the production facility. The deterioration process is determined by the age and covariate values and the covariate process is modeled as a continuous-time Markov process. The condition information is available at each inspection epoch, which is the end of each production run. The hazard rate is estimated after obtaining the new information through CM. The problem is formulated and solved in the semi-Markov decision process (SMDP) framework. The objective is to minimize the long-run expected average cost per unit time. Also, the mean residual life (MRL) of the production facility is calculated as an important statistic for practical applications. A numerical example is provided and a comparison with the age-based policy shows an outstanding performance of the new model and the control policy proposed in this paper. Keywords: Condition-based maintenance (CBM), Semi-Markov decision process (SMDP), Proportional hazards model (PHM), Economic manufacturing quantity (EMQ).

∗ Corresponding author. Tel: +1 416 9784184 E-mail addresses: {ljafari, makis}@mie.utoronto.ca

Preprint submitted to Elsevier

July 30, 2015

ACRONYMS AND ABBREVIATIONS PM CM CBM PHM EMQ MRL SMDP

Preventive Maintenance Condition Monitoring Condition-Based Maintenance Proportional Hazards Model Economic Manufacturing Quantity Mean Residual Life Semi-Markov Decision Process NOTATION

Q p d T ξ f (t) qij Pi,j (t) νi ∆ z h(t) W CI CL CH CP CF K τ

Production lot size Production rate Demand rate Predetermined age to perform preventive maintenance of the production facility Failure time of the production facility Density function of ξ Instantaneous transition rate of the covariate process from state i to j Transition probability from state i to j in t time units Exponential distribution parameter of the covariate process in state i, i = 0, 1 Length of the inspection interval Value of the covariate process Hazard rate at time t Preventive maintenance level Inspection cost Lost production cost rate Inventory holding cost rate Preventive maintenance cost Corrective maintenance cost Set-up cost Expected sojourn time

1. Introduction There has been a growing interest in determining the economic manufacturing quantity (EMQ) of products in the production and inventory control. The traditional EMQ models are rarely applied in practice, because their assumptions are not completely satisfied. To make these models more realistic, considerable research has been done to relax some of these assumptions, which can be summarized as follows: • Multi-product systems (Ben-Daya & Hariga (2000); Pasandideh et al. (2010); Taleizadeh et al. (2014)), • Dynamic demand rate (Bouslah et al. (2013); Larson (1997); Pal et al. (2013); Shah et al. (2014); Sicilia et al. (2014)), • Imperfect quality of the manufactured product (Makis (1998); Moon et al. (2002); Avinadav & Raz (2003); Khan et al. (2011); Tai (2013); Sarkar et al. (2014)), 2

• Deterioration of the production facility and the age-based PM policy (Groenevelt et al. (1992); Tse & Makis (1994); Fung & Makis (1998); Ben-Daya (2002)), • CM of the deteriorating production facility and utilization of this information for production and maintenance decision making. This is a new practical extension, which is the main focus in this paper. Recently, Glock (2012) and Glock et al. (2014) provided a literature review on lot sizing. They mentioned that deterioration will affect the productivity of an inventory system. However, most of the models have considered the joint optimization of production quantity and quality control of the product deterioration which can be classified as imperfect quality. Another kind of deterioration in the system can be the production facility deterioration with usage and age which may result in its failure causing substantial downtime and failure cost. In one of the early works, Groenevelt et al. (1992) developed a model considering a random machine breakdowns and analyzed the optimal lot size and the associated re-ordering policy. To incorporate PM in the context of lot-sizing problems, Tse & Makis (1994) presented a model where they considered both PM and failure replacement to minimize the long-run expected average cost per unit time. They introduced two types of failure: major and minor failures. When the major failure happens, the machine is replaced by a new one. However, minor failure is corrected by minimal repair and production can be resumed immediately at lower cost. They found the optimal lot size and preventive replacement time. Also, Ben-Daya (2002) considered imperfect PM on EMQ model. The author assumed that the age of the system is reduced proportional to the PM level. The system is restored to as good as new condition, when it is out of control or after m inspections. The decision variable is the number of inspections (m) to minimize the expected total cost per unit time. Then, Suliman & Jawad (2012) extended Ben-Daya (2002) by assuming that the failure may occur at any time after system shifts to the out-of-control state. They also incorporated shortage cost and non-negligible PM time. Similarly, Liao et al. (2011) proposed a model considering perfect and imperfect PM in a deteriorating production system with increasing hazard rate. They assumed that the probability of performing perfect PM depends on the number of imperfect PM since the last renewal cycle. When failure occurs, the delayed repair is triggered and the production resumes with lower production rate until perfect PM is performed. They minimized the total cost to find the optimum run time in EMQ model. Recently, Gomez et al. (2013) developed a model to determine the optimal production plan in a system subject to reliability and quality deterioration. They considered different maintenance actions (minimal repair, major repair and preventive maintenance) in their model to minimize the related cost including the inventory holding, backlog, overhaul, preventive maintenance, and defective costs. Then, Sarkar & Sarkar (2013) proposed an interesting model in the joint optimization of EMQ, maintenance and quality framework. They assumed exponential demand, time-varying production rate under the effect of inflation and time value of money. Costs are dependent on the system reliability, because a more reliable system can produce more products. The reliability parameter is a decision variable to maximize the profit in a production system. However, the above-mentioned joint optimization of EMQ and maintenance policies have been applied only to traditional maintenance policies and CBM has not been considered yet for the EMQ models. CBM analysis recommends maintenance actions based on the information collected through CM. Due to the low cost and wide availability of highly effective hardware and software for installing data collection and information systems, CM and CBM have become the main elements of modern maintenance management programs with an increased focus on predictive maintenance. 3

The condition of the machine is monitored at regular intervals. Once the condition is critical, the machine is preventively maintained. This approach has been applied widely, since it can increase the production efficiency by maximizing the equipment availability or/and minimizing the related costs. For an extensive review on CBM models readers are referred to Jardine et al. (2006); Rehorn et al. (2005); Si et al. (2011). One of the powerful and popular statistical model that is suitable for modeling degrading systems in the CBM framework is PHM. The PHM considers both age and condition variables called covariates using a hazard rate. This model was introduced by Cox (1972). Later, Kumar & Klefsjo (1994) reviewed the literature on the PHM. Increasingly, PHM is gaining popularity as a useful model for different applications such as Ding & He (2011); Feng et al. (2010); Tian & Liao (2011). Makis & Jardine (1992) developed the optimal maintenance policy for a PH model minimizing the long-run expected average cost per unit time. They considered a PHM with Markov covariate process and periodic monitoring. Then, Wu and Ryan (2010) extended their work by considering possible state transitions between inspection times. However, the joint optimization of manufacturing quantity and maintenance for a deteriorating production facility using PHM has not yet been considered, which is a better representation of real systems subject to CM. In this paper, we propose a joint optimization of EMQ and a preventive maintenance policy for a production system subject to CM at the opportunistic epochs when a production run is completed using PHM to minimize the long-run expected average cost per unit time. The main contributions of this paper can be summarized as follows: i. Development of the EMQ model considering production system deterioration and CBM. The production facility is subject to deterioration and the joint optimization of EMQ and maintenance policy using PHM is addressed. ii. Consideration of both maintenance and production costs which include the set-up, inventory holding, and shortage costs as well as inspection, preventive, and corrective maintenance costs. iii. Development of the semi-Markov decision process (SMDP) framework and the algorithm to obtain the optimal lot size and maintenance policy for the proposed model. iv. Numerical studies and a comparison with the traditional age-based policy. The remainder of the paper is organized as follows. Section 2 summarizes the assumptions and the details of the proposed model and presents problem formulation. In Section 3, a computational algorithm in the SMDP framework based on the policy iteration algorithm is developed. Section 4 deals with the computation of the MRL. In Section 5, we review the joint optimization of EMQ and age-based policy. The effectiveness of the proposed model is demonstrated in Section 6 by comparing the optimal policy with the traditional age-based policy. In Section 7, we discuss possible extensions of our model, and provide concluding remarks. 2. Model Assumptions and Formulation We consider a manufacturing system which produces the lot size Q in each production run, with constant production and demand rates denoted by p, and d, respectively. The production facility is subject to failure and its condition is inspected at equidistant time epochs (∆, 2∆, ...), where ∆ is the inspection interval and we suppose that inspection is performed at the end of each production run. Thus, the production lot size Q can be denoted as Q = p · ∆. To describe the behavior of the production facility deterioration process properly and obtain the optimal lot size and maintenance policy, we assume that the value of the covariate process (such as the level of metal particles in an engine oil, or a vibration monitoring level) indicate the machine 4

deterioration and is determined through inspection. To incorporate these values into the model, we apply PHM. The maintenance policy is to replace the production facility (machine) upon its failure or at a preventive maintenance time. The machine will be preventively replaced when its hazard rate exceeds the preventive maintenance level (W ) or its age reaches the pre-determined time T . We also assume that the machine failure occurs only in the production phase and it is observable at any time. Both corrective and preventive actions restore the production facility to the ”as good as new” condition. Considering the PH model, the hazard rate of the machine depends on its age and also on the value of a covariate process which indicates its deterioration. Therefore, it is assumed that the hazard rate is the product of a baseline hazard rate h0 (t) dependent on the age of the production facility and a positive function ψ(Zt ) dependent only on the values of the covariate process (see also Cox (1972); Makis & Jardine (1992)). Let Z be a continuous time Markov chain with the state space Ω = {0, 1, ..., J}. State 0 indicates the best possible state, which means that there is no deterioration and the facility is new or as good as new. We assume that there is no deterioration at time zero, i.e. Z0 = 0, and the last state J is an absorbing state. Thus, the hazard rate at time t can be expressed as follows: h(t, Zt ) = h0 (t).ψ(Zt ),

(1)

and the survival function is given by  Z t  P (ξ > t | Zs , 0 ≤ s ≤ t) = exp − h0 (s).ψ(Zs )ds ,

(2)

0

where ξ is the failure time of the production facility. The values of the covariate process (Z) are observed only at discrete points of time (∆, 2∆, ...). Inspections are scheduled periodically to estimate the condition of the production facility and the related maintenance actions are chosen based on the inspection results. The objective is to jointly optimize the lot size (Q) and the preventive maintenance level (W ) for the hazard rate to minimize the long-run expected average cost per unit time. We consider the following costs in the model: • CP : Preventive maintenance cost when PM is performed, which takes TP time units. • CF : Failure (replacement) cost incurred when corrective maintenance is performed, which takes TF time units. • CI : Inspection cost incurred when inspection is performed. • CH : Inventory holding cost incurred for one item per unit time. • CL : Cost rate of the lost production. • K: Set-up cost. From Eq. (1), the hazard rate of the machine at the nth inspection epoch is given by: h(n∆, Zn∆ ) = h0 (n∆) · ψ(Zn∆ ). 5

(3)

The Z process is a continuous-time Markov chain and its instantaneous transition rates qij , i, j ∈ Ω, are defined by qij = lim

u→0+

P ( Zt+u = j| Zt = i) < +∞, i 6= j ∈ Ω, u

and qii = −

X

qij .

(4)

i6=j

To model monotonic system deterioration, we assume that the state process is non-decreasing with probability 1, i.e. qij = 0 for all j < i. The transition probability matrix P(t)=(Pi,j (t))i,j∈Ω , is obtained by solving the Kolmogorov backward differential equations, where Pi,j (t) = P (Zs+t = j | Zs = i).

(5)

The hazard rate of the production facility is calculated using its age as well as transition probabilities of the covariate process (Eq.(5)) which depend on the observable state of the process at each inspection epoch. These inspections are performed at the end of each production run which depends on EMQ, and provide the opportunity to determine the deterioration state (condition) of the production facility. When the hazard rate exceeds the preventive maintenance level W , then PM action is performed, otherwise the system is left operational until the next inspection epoch. The objective is to find the optimal values of the production lot size and the preventive maintenance level (Q∗ , W ∗ ) such that the long-run expected average cost per unit time is minimized. We now develop an efficient computational algorithm in the semi-Markov decision process (SMDP) framework to determine the optimal decision variables (Q∗ , W ∗ ). 3. Computational Algorithm in the SMDP Framework In this section, we develop a computational algorithm in the SMDP framework. We monitor the covariate process at each inspection epoch. Suppose that the machine is operational at the nth inspection epoch (n∆), then we compute the hazard rate using Eq. (3). We partition the interval of the hazard rate [0, H] into a fixed large L sub-intervals, where H is suitably selected upper bound for the hazard rate. The number of subintervals (L) should be selected properly. The more accurate results will be obtained by choosing the larger number of subintervals (L), however the computational time increases. Therefore, the number of subintervals should be selected big enough to get the precise results in a reasonable time. Before calculating the transition probabilities, the exact definitions of the state space for the SMDP is required: • State (0,0): machine is in ”as good as new” condition, • State (z, n): the first component represents the value of the covariate at time n∆ and the second component is the operating age of the machine i.e. (n∆), • State P M : the value of hazard rate crosses the preventive maintenance level, so PM action is performed. Note that PM action is also performed when the age of the machine exceeds the specified maximum age T . Thus, the state space of SMDP can be denoted by S = S1 ∪ S2 ∪ S3 , where S1 = {(z, n) | z ∈ Ω, n ∈ N }, S2 = {P M }, and S3 = {(0, 0)}. For the cost minimization problem, the SMDP can be determined by the following quantities: 6

1. Pm,k = the probability that the system will be in state k ∈ S at the next decision epoch given the current state is m ∈ S, 2. τm = the expected sojourn time until the next decision epoch given the current state is m ∈ S, 3. Cm = the expected cost incurred until the next decision epoch given the current state is m ∈ S. Once all these quantities are defined, for a fixed preventive maintenance level W and production lot size Q, the long-run expected average cost g(Q, W ) can be obtained by solving the following system of linear equations Tijms (1994): X Vm = Cm − g(Q, W ) · τm + Pm,k · Vk , k∈S

Vj

= 0 ,

for an arbitrary selected single state j ∈ S,

(6)

where {Vm } values are the unknown values together with g(Q, W ) which are determined by solving Eq.(6). {Vm } values are related to the so-called relative values (see Tijms (1994), pp. 168-169 for more details). To proceed with the SMDP approach, the calculation of conditional reliability function is required. Since the degradation state process (covariate process) is only observable at inspection epochs, then, it may transit at any time between two inspections (Wu and Ryan (2010)). Therefore, we consider this assumption to derive the conditional reliability function. However, we suppose that covariate process can make transition from healthy state to other states (warning or absorbing), whereas Wu and Ryan (2010) considered only the sequential degradation, i.e. in their paper the covariate process can make transition from state 0 to 1 and it cannot jump from state 0 to state 2. To illustrate the computational algorithm, we assume that Z process has three states, Zt ∈ {0, 1, 2} for all t, however it can be generalized to more states, which can be a suitable topic for future research. Then, the conditional reliability function is given by: R(n, z, t) = P (ξ > n∆ + t | ξ > n∆, Z1∆ , ..., Zn∆ , Zn∆ = z) h  Z n∆+t  i = E exp − h0 (s).ψ(Zs )ds | Zn∆ = z .

(7)

n∆

The above equation can be evaluated based on different values of z and conditioning on sojourn times in the healthy and unhealthy states (see Appendix A). 3.1. Transition Probabilities The SMDP transition probabilities for the states defined in the previous section are calculated as follows: 1. Assume that the system is in the state (z, (n−1)), where h((n−1)∆, z) < W , and n∆ < T . Then the transition probability to the next state (z 0 , n), where h(n∆, z 0 ) < W is given by: P(z,(n−1)),(z 0 ,n) = P (Zn∆ = z 0 , ξ > n∆ | Z(n−1)∆ = z, ξ> (n−1)∆).

(8)

It is the probability that the value of the hazard rate will not exceed the preventive maintenance level W and the system will not fail in the next inspection interval. Then, this

7

probability can be calculated as: P(z,(n−1)),(z 0 ,n)=P (Zn∆ = z 0 , ξ > n∆ | Z(n−1)∆ = z, ξ> (n−1)∆) =P (Zn∆=z 0 | ξ >n∆, Z(n−1)∆=z, ξ>(n−1)∆)·P (ξ>n∆ | Z(n−1)∆=z, ξ>(n−1)∆) =Pz,z 0 (∆)·R((n − 1), z, ∆).

(9)

2. If the hazard rate crosses the preventive maintenance level (i.e., h(n∆, z 0 ) ≥ W ) or the age of the machine exceeds the pre-determined age T , where h((n−1)∆, z) < W , then the system goes to the P M state and the transition probability is given by: P  z 0 P (Zn∆ = z 0 | Z(n−1)∆ = z, ξ> (n−1)∆)·R((n−1), z, ∆) ; n∆
n ≥ 1.

(11)

4. When the machine is in the P M state, then mandatory replacement is performed and the system goes back to state (0, 0). We have: P(P M ),(0,0) = 1.

(12)

In the next two sections, the expected sojourn times and the expected cost in each state are determined. 3.2. Expected Sojourn Times The expected sojourn times should be calculated for each SMDP states. We have different scenarios depending on the depletion time and replacement or preventive maintenance time. If the production facility has failed before the next inspection epoch (∆ = Q/p), then the mean 0 sojourn time for the initial state depends on the time that inventory is depleted (t ) and also on the time to replace the machine (TF ) (see Figure 1). Since, the production will resume when the inventory is depleted, then t0 can be calculated as follows: 0

t =

p−d t. d

8

(13)

So, the sojourn time for the initial state can be written as: τ(0,0)= E(sojourn time | Z0 = 0, ξ > 0) = E(sojourn time | Z0 = 0, ξ > 0, ξ > ∆) · R(0, 0, ∆) Z ∆ E(sojourn time | Z0 = 0, ξ > 0, ξ ≤ ∆, ξ = t) · f (t)dt + 0 Z ∆h i Q p−d = ·R(0, 0, ∆)+ )t)·I{TF < p−d t}+(t+TF )·I{TF ≥ p−d t} · f (t)dt, (14) (t+( d d p d 0

Inventory Level

where term E(.) denotes the expectation operator, and f (t) is the failure time density of the production facility. The expected sojourn time incurred until the next decision epoch for state (z, n), where the hazard rate does not cross the preventive maintenance level, depends on the time that inventory is depleted 0 (t ) and the time to replace the machine (TF ). We have two scenarios: the first possibility is to perform corrective maintenance without any shortage while in the second scenario, when the shortage occurs (see Figure 1).

(p-d)Q/p Corrective maintenance p-d

-d

Q/p Q/d

t

t'

Time

TF

Figure 1: Corrective maintenance is performed with a shortage

Figure 1: Corrective maintenance is performed with a shortage

So, the sojourn time in state (z, n) is given by: τ(z,n) = E(sojourn time | Zn∆=z, ξ>n∆) = E(sojourn time | Zn∆=z, ξ>n∆, ξ>(n+1)∆)·R(n, z, ∆) Z ∆ + E(sojourn time | Zn∆ = z, ξ > n∆, ξ ≤ (n + 1)∆, (ξ − n∆) = t) · f (t | n, z)dt = 0 Z ∆h i Q Q Q p−d Q Q · R(n, z, ∆)+ ( − + t+( )t)·I{TF < p−d t}+( − +t+TF )·I{TF ≥ p−d t} ·f (t|n, z)dt,(15) d d d p d d p |d {z } |0 {z } 1

2

where f (t | n, z) =

d P (ξ < n∆ + t | ξ > n∆, Zn∆ = z). dt

(16)

The first term in Eq.(15) is the expected time that the system will spend in the state (z, n) until the next inspection epoch, i.e. the system will be reliable during the next production run. However, the second term represents the expected sojourn time when the production facility failure occurs with the possibility of shortage depending on depletion time (t0 ) and replacement time (TF ). 9 1

If the hazard rate crosses the preventive maintenance level, or the age of the machine exceeds the pre-determined age (T ), then the sojourn time depends on the time that inventory is depleted and the preventive maintenance time. Then, the preventive maintenance is performed without any shortage, or the shortage occurs. So, the sojourn time in P M state is given by:  Q Q Q  TP < Q  d − p; d − p τP M = (17)  Q Q  T ; TP ≥ d − p . P 3.3. Expected Costs The expected cost incurred at each SMDP state is computed in this section. Depending on the depletion time and corrective or preventive maintenance time, different costs will be incorporated. For the initial state, the expected cost can be obtained as follows: C(0,0) = E(cost | Z0 = 0, ξ > 0) = E(cost | Z0 = 0, ξ > 0, ξ > ∆)·R(0, 0, ∆) Z ∆ + E(cost | Z0 = 0, ξ > 0, ξ ≤ ∆, ξ = t) · f (t)dt

(18)

0

Z Z ∆  = K+ CH (p − d)tdt · R(0, 0, ∆)+ 0

0

∆h

Z t Z CF + CH (p − d)u du + 0

(p−d) t d

CH ((p − d)t − u · d)du

0

i (p − d) t) · d · I{TF ≥ p−d t} · f (t)dt = d d Z ∆h i p(p−d)
2 (p − d) CH (p−d)Q2 C +C ·R(0, + t +C ·(T − t)· d·I ·f (t)dt. K+ 0, ∆) p−d F H L F {TF ≥ d t} 2p2 2d d 0 | {z } | {z }

+CL · (TF −

1

2

The first term in Eq.(18) is the holding cost for the first production run, while the second term represents the corresponding costs when the failure occurs and lost production cost will be incurred when the replacement time is greater than the depletion time. Also, the set-up cost (K) will be incurred at the beginning of each production run. The average cost incurred until the next decision epoch for state (z, n) such that the hazard rate does not cross the preventive maintenance level, is given by: C(z,n) = E(cost | Zn = z, ξ > n∆) = E(cost | Zn∆ = z, ξ > n∆, ξ > (n + 1)∆) · R(n, z, ∆) Z ∆ + E(cost | Zn∆=z, ξ>n∆, ξ≤(n+1)∆, (ξ −n∆)=t) · f (t | n, z)dt 0 Q −Q d p

Z ∆ Z ∆h Z t i Q CH ((p − d) −td)dt + CH (p−d)tdt · R(n, z,∆)+ CF + CH (p−d)udu p 0 0 0 0 Z (p−d) t Z Q−Q i d d p Q (p−d) + CH ((p−d)t− u·d)du+ CH ((p−d) −td)dt+CL ·(TF − t)·d·I{TF ≥ p−d t} ) f (t|n,z)dt d p d 0 0

Z = K + CI + h

10

Z ∆h h (p − d)Q2 i p(p − d)
2 CH (p − d)2 Q2 = K + CI +CH CF + CH · R(n, z, ∆)+ t + + 2pd 2d 2p2 d | {z } |0 {z } 1

2

i (p − d) CL · (TF − t)· d · I{TF ≥ p−d t} ) · f (t | n, z)dt . d d | {z }

(19)

2

The first term in Eq.(19) is the holding cost of the depleting inventory in the previous production run, and holding cost of the current production run, plus the inspection cost. The second term represents the replacement cost, holding cost, and lost production cost if shortage occurs. Finally, the average cost incurred in P M state is: CP M

Z Q−Q d p Q Q Q = CP + CL · I{TP ≥( Q − Q )} · (TP − ( − )) · d + CH ((p − d) − td)dt d p d p p 0 2 2 Q Q CH (p − d) Q = CP + CL · I{TP ≥( Q − Q )} · (TP − ( − )) · d + . d p d p 2p2 d

(20)

By substituting the above quantities in Eq. (6), the optimal production quantity and maintenance level (Q∗ , W ∗ ) and the corresponding optimal long-run expected average cost per unit time can be obtained. 4. Mean Residual Life In this section, we derive the formula for the mean residual life (MRL). The MRL statistic is important for practical applications, since it describes the average remaining life of the system. For any state (z, n), the MRL is given by: M RL(z,n) = E{ξ | ξ > n∆, Zn∆ = z} Z ∞ Z = P (ξ > n∆ + t | ξ > n∆, z)dt = 0

∞

R(n, z, t)dt,

(21)

0

The conditional reliability function of the production facility is provided in Appendix A. 5. Model Development for the Joint EMQ and Age-based Maintenance Policy The joint optimization of EMQ and age-based maintenance policy is developed in this section to investigate the value of CM in the proposed model. In the age-based policy, the production facility will be replaced whenever it fails or N production runs are completed to perform preventive maintenance. Let F (y) denotes the failure time distribution function of the production facility. We have: F (y) = 1 − R(0, 0, y),

(22)

where R(0, 0, y) is obtained using Appendix A. From renewal theory, the long-run expected average cost per unit time for this policy C(N, Q) is

11

given by: E(CC) Expected Cycle Cost = , Expected Cycle Length E(CL)

C(N, Q) =

(23)

where E(CC) and E(CL) represent the expected cycle cost and expected cycle length, respectively. Similarly, Fung & Makis (1998) derived the expected average cost for EMQ model considering the inspection and random machine failure. We make an extension by considering preventive maintenance action and shortage cost in the joint optimization of EMQ and age-based policy. In order to find E(CC) and E(CL), we will derive the expected cost and the expected cycle length in two cases: (i) one completed production cycle, (ii) incomplete production cycle. The cost components of the first case are given by (see also Figure 2): E[Set up cost] = K. Q p

Z E[Holding cost] = |0





Z

CH (p − d)u du + {z } |0 1

2p2

Q −Q d p

  Q CH (p − d) − u · d du p {z } 2

(p − d)2 Q2 i (p − d)Q2 + = C . H 2p2 d 2pd

Inventory Level

= CH ·

h (p − d)Q2

(24)

(25)

Preventive maintenance

(p-d)Q/p p-d

-d

Q/d

Q/p Q/d

Time

N

Figure 2: Preventive maintenance is performed after N completed production runs

Figure 2: Preventive maintenance is performed after N completed production runs

The first term in Eq. (25) is the holding cost of the production run, and the second term corresponds to the holding cost of depleting inventory in the current production run. The expected cycle length for the first case will be: E[Cycle length] =

Q . d

(26)

If preventive action is performed at the end of production run, then we should also add the following cost components: Q Q E[Lost production cost] = CL ·(TP −( − ))·d · I{TP ≥( Q−Q )} . d p d p 12

(27)

E[Preventive maintenance cost] = CP ,

(28)

and the cycle length is increased by adding the following term: E[Cycle length] = (

Q Q − ) · I{TP <( Q − Q )} + TP · I{TP ≥( Q − Q )} . d p d p d p

(29)

When failure occurs, then we have incomplete production cycle, i.e., the second case (see also Figure 3). Let ξ = y. Then t = y − n · Q p and the cost components are given by: E[Set up cost] = K. Z E[Holding cost] =

t

CH 0

|



Z t0  Z Q−Q    d p Q CH (p − d) − u · d du (p − d)u du + CH (p − d)t − u · d du + p {z } |0 {z } |0 {z } 1

= CH

(30)

2

d)2 Q2

p(p − d) 2 (p − t + CH 2d 2p2 d

3

(31)

,

Inventory Level

and the first term in Eq. (31) represents the holding cost in a production run until failure occurs, the second term corresponds to the holding cost of depleting inventory in the current production run, and the third term is the holding cost of depleting inventory in the previous production run.

(p-d)Q/p

Corrective maintenance p-d

-d

Q/p Q/d n

t

t'

Time

Figure 3: Corrective maintenance is performed upon failure

Figure 3: Corrective maintenance is performed upon failure

E[Lost production cost] = CL ·(TF −

p−d t)·d·I{TF ≥ p−d t} . d d

E[Corrective maintenance cost] = CF ,

(32)

(33)

and the corresponding expected cycle length is given by: E[Cycle length]= t+

p−d t·I{TF < p−d t}+TF ·I{TF ≥ p−d t} , d d d 13

(34)

Now, the expressions for E(CC) and E(CL) can be derived by combining the results in case (i) and case (ii), as follows: Z E(CC) =

(n+1)Q p

N −1Z X

∞

E(CC | ξ=y)f (y)dy+

NQ p

E(CC | ξ=y)f (y)dy

nQ p

n=0

h  i (p − d)Q2  NQ  Q Q = N · K +CH · +CP +CL ·(TP −( − ))·d·I{TP ≥( Q−Q )} · 1−F ( ) d p 2pd d p p | {z } 1

Z (n+1)Qh  p (p−d)Q2 nQ p(p − d) nQ 2 + n· K+CH · · (1−F ( )) + K +CH · ·(y − ) nQ 2pd p 2d p n=0 p | {z } N −1 X

2

 i (p − d)2 Q2 p−d nQ  +CH · + C · T − ·(y− ) ·d·I +C p − d nQ L F F ·f (y)dy {TF ≥ d (y− p )} 2p2 d d p | {z }

(35)

2

The first term in Eq. (35) represents the expected PM cost for N completed cycles, while the second term shows the expected failure cost for n completed cycles and one incomplete cycle. Z E(CL) =

(n+1)Q p

N −1Z X

∞

E(CL | ξ=y)f (y)dy+

NQ p

n=0

nQ p

E(CL | ξ=y)f (y)dy

i  h Q Q NQ  Q Q ) = (N − 1) · + + ( − ) · ITp <( Q − Q ) + TP ·ITp ≥( Q − Q ) · 1−F ( d p d p d p d p p | {z } 1

Z (n+1)Q h p nQ nQ p−d nQ nQ (y− · (1 − F ( )) + )+( )·(y− )·I{TF < p−d (y−nQ )} + nQ d p d p p d p n=0 p {z } | 2 i +TF · I{TF ≥ p−d (y− nQ )} ·f (y)dy, d p | {z } N −1 X

(36)

2

where the first term is the cycle length when preventive maintenance occurs, while the second term corresponds to the expected cycle length when failure of the production facility occurs. 6. Experimental Results We assume that the production facility is subject to stochastic deterioration described by a PH model. The covariate values are determined through CM at the end of production, which depends on EMQ, which is also a control parameter. The covariate process is described by a continuous time Markov process with the set of possible values Ω = {0, 1, 2}. The sojourn times in state 0 and 1 are exponentially distributed with parameter ν0 = q01 + q02 , and ν1 = q12 , respectively. The

14

transition rates of the state process are given by: q01 = 0.16, q02 = 0.09 and q12 = 0.39. Then, the corresponding transition probability matrix is obtained by solving Kolmogorov backward differential equations and it is given by:   −ν1 t −ν0 t ) −ν t −ν t −ν0 t − q01 (e 1 −e 0 ) e−ν0 t q01 (e ν0 −ν−e 1 − e ν0 −ν1 1   P = [Pi,j (t)] = 0 , e−ν1 t 1 − e−ν1 t 0 0 1 which is used to compute the SMDP transition probabilities. To describe the hazard rate of the production facility, PHM is applied. For the model parameters, we were motivated by two published real case studies which focused on PM policies (see Makis et al. (2006); Kim et al. (2011)). The baseline hazard rate is assumed to follow Weibull distribution β−1 h0 (t) = βtαβ , where α = 1.755e + 004, β = 1.723, and ψ(Zt ) = e0.5Zt . Also, the predetermined age of the production facility to perform preventive maintenance is T = 12, 000 working hours. The maximum value of the hazard rate in this experiment is 20.2 × 10−5 . Production and demand rates are constant ( see e.g. Suliman & Jawad (2012); Taleizadeh et al. (2014)) and they will be considered as p = 20 and d = 10, respectively. The system preventive and replacement time parameters are given by: TP = 100, and TF = 100 hours, where TP is the time to perform preventive maintenance, and TF is the time to replace the system upon its failure. The following costs will be considered in the experiment: CI = 10, CL = 20, CH = 0.5, CP = 1560, CF = 6780, K = 600 The computational algorithm for the optimal decision variables (Q∗ , W ∗ ) requires an appropriate discretization (L) to get the sufficiently precise result in a reasonable time. All computations were coded in Matlab 2012 on an Intel Core (TM) i5 CPU with 2.27 GHz. Using the proposed model in Section 3, the long-run expected average cost per unit time and the computational time of each run for different level of discretization (L) are shown in Table 1. Table 1: The optimal long-run expected average cost per unit time and computational time for different L.

Discretization level (L)

10

20

30

40

Average cost per unit time

2.0068

1.9065

1.8575

1.8502

Computational time (sec)

18.10

24.48

35.25

51.16

We have found that when L ≥ 30, the partition leads to a sufficient degree of precision, so L does not need to be chosen very large.

15

Table 2: The optimal parameters and the long-run expected average cost per unit time.

Optimal Policy

Lot size (Q )

Preventive maintenance level (W ∗ × 10−5 )

Proposed Approach

11000

5.33

—

1.7112

Age-based

12000

—

15

2.3171

∗

Completed production runs (N ∗ )

Average cost per unit time

Figure 4a shows the hazard rate plot for the simulated data with the optimal decision variables The stars show the hazard rate at each inspection epoch. Once the hazard rate exceeds the preventive maintenance level (W = 5.33 × 10−5 ), PM will be performed. For the data history in Figure 4a, this occurred at the 14th inspection epoch. (Q∗ , W ∗ ).

12000

20

11000

18 Hazards rate 16

10000 Mean Residual Life (hrs)

W*

Hazard rate × 10−5

14 12 10 8 6

9000 8000 7000 6000 5000

4

4000

2 0

3000 0

2

4

6

8 10 Inspection epoch

12

14

16

(a)

Figure 4 a: The illustration of the proposed approach to monitor the hazard rate at each inspection epoch.

0

2

4

6 8 Inspection epoch

10

12

14

Figure 4 b: The MRL of (b) the production facility.

Figure 4: (a) The illustration of the proposed approach to monitor the hazard rate at each inspection epoch. (b) The MRL of the production facility.

We also illustrate the MRL of the production facility at each inspection epoch in Figure 4b. It can be observed that when the covariate value switches to warning state, the MRL reduction is greater than in the healthy state. When the hazard rate exceeds the preventive maintenance level, the MRL is low and it is the time to preventively maintain the production facility. We also also verified the superiority of the proposed approach by comparing the results with the age-based policy, which does not take CM information into account. This policy was described in Section 5. The optimal decision variables (N, Q) are obtained by minimizing the long-run expected average cost using Eq. (23). Considering the same parameters as in the previous example, the minimum long-run expected average cost using the age-based maintenance policy increased to 2.3171, which is higher than the optimal average cost obtained using the proposed approach with PHM in EMQ model. Table 2 shows the long-run expected average cost per unit time and the 1 optimal decision variables for the age-based maintenance policy. 1 When the inspection cost is high, then the proposed model is not effective any more. As shown 16

in the sensitivity analysis provided in Table 3, the average cost increases with the increase of the inspection cost, which is intuitively clear. Table 3: Impact of increasing inspection cost CI on the optimal average cost of the proposed model.

Inspection Cost (CI )

5

10

50

100

150

200

250

Average Cost per unit time

1.7054

1.7112

1.8121

1.9008

2.0527

2.1263

2.2972

The cost savings are considerable for lower values of CI which is the case in many real situations due to the low cost of the current CM systems. Finally, the effect of the changes of the Weibull scale parameter is investigated and illustrated in Figure 5.

Average cost per unit time

1.8

1.75

1.7

1.65

1

1.5

2 Scale parameter

2.5 4

x 10

Figure 5: Average cost per unit time for different scale parameter values.

Figure 5: Average cost per unit time for different scale parameter values.

When the scale parameter increases, the hazard rate function decreases, which means that the system is more reliable. Therefore, we have slower degradation and the long-run expected average cost per unit time decreases. As expected, the average cost decreases with the increase of the scale parameter. 7. Conclusions and Future Research In this paper, we have developed a model and a computational algorithm that can be used to determine the optimal lot sizing and maintenance policy for a deteriorating production system subject to CM. Unlike the previous research in the joint optimization of lot-sizing and traditional maintenance policy, we have considered CM information which is utilized for making the optimal decisions. The opportunity to perform CM and determine the condition of the production facility is provided at the end of each production cycle. The deterioration of the production facility is described by a PH model, where the covariate process is modeled as a continuous-time Markov chain. Once the values of the covariate process are obtained, then the hazard rate is updated and a decision is made whether to start PM or continue with the next production run considering the 17 1

optimal preventive maintenance level for the hazard rate. Such a control limit policy has direct practical value as it can be implemented for on-line decision-making. A cost comparison with the traditional age-based maintenance policy has been done by considering the set-up cost, inventory holding, preventive, corrective maintenance, and shortage costs, which illustrates the benefit of the joint optimization of CBM and lot-sizing. The computational algorithm developed in the SMDP framework has been used to minimize the long-run expected average cost per unit time. We have demonstrated that the proposed model is substantially better than the traditional EMQ and age-based policy. The obtained numerical results also indicate which approach is more economical when the inspection cost increases. CBM leads to a higher cost compared to the age-based policy for the large values of the inspection cost. However, due to the wide availability of highly effective hardware and software for installing information systems, CM information is available in real systems at a reasonable cost, and as the results in Tables 2 and 3 indicate, the CBM policy proposed in this paper is considerably more cost effective than the traditional EMQ and age-based policy. Finally, we have investigated the effect of Weibull parameter on the average cost per unit time. We suggest several possible directions for future research. We have considered a covariate process that can be characterized by a continuous time Markov chain. In practice, this assumption can be relaxed by considering more general distributions such as Erlang or Weibull distributions of the covariate process sojourn times (Dong et al. (2006)). Such an extension may lead to both interesting theoretical and practical results. Another direction could be to consider joint optimization of production, maintenance and quality using CM information. Finally, we can consider the joint optimization of EMQ and maintenance policy using imperfect information for the covariate process in PH model, which has been introduced in the maintenance area by Ghasemi et al. (2007).

Acknowledgement The authors would like to thank the two anonymous referees for their constructive comments and to Ontario Centres of Excellence (OCE) for their financial support under grant No. 201461.

18

Appendix A Assume that the sojourn times of the covariate process Z in the healthy and unhealthy states i = 0, 1 are exponentially distributed with parameters νi . h  Z R(n, z, t) = P (ξ > n∆+t | ξ > n∆, Z1∆ , ..., Zn∆ , Zn∆=z)=E exp −

n∆+t

 i h0 (s)ψ(Zs )ds | Zn∆=z (A.1)

n∆

If z = 0, then Eq. (A.1) can be written as follows: ∞

 i h  Z n∆+t h0 (s)ψ(Zs )ds | t0 = u · p01 · ν0 e−ν0 u du E exp − n∆ 0 Z ∞ h  i  Z n∆+t h0 (s)ψ(Zs )ds | t0 = u · p02 · ν0 e−ν0 u du E exp − + Z

R(n, 0, t) =

(A.2)

n∆

0

The first part of the above equation can be evaluated by splitting the integral, ∞

h  Z n∆+t  i E exp − h0 (s)ψ(Zs )ds | t0 = u · p01 · ν0 e−ν0 u du t n∆ Z t h  Z n∆+t  i + E exp − h0 (s)ψ(Zs )ds | t0 = u · p01 · ν0 e−ν0 u du 0 n∆ Z ∞  Z n∆+t  = exp − h0 (s)ψ(0)ds · p01 · ν0 e−ν0 u du t n∆ Z t h  Z n∆+t  i + E exp − h0 (s)ψ(Zs )ds | t0 = u, Zt0 = 1 · p01 · ν0 e−ν0 u du

Z

0

(A.3)

n∆

The second part of the above equation can be determined by conditioning on the sojourn time in the warning state, h  Z n∆+t  i E exp − h0 (s)ψ(Zs )ds | t0 = u, Zt0 = 1 · p01 · ν0 e−ν0 u du 0 n∆ Z tZ ∞ h  Z n∆+t  i = E exp − h0 (s)ψ(Zs )ds | t0 = u, Zt0 = 1, t1 = v · p01 · ν0 e−ν0 u · ν1 e−ν1 v dvdu 0 n∆ 0 Z t Z t−u h  Z n∆+t  i = E exp − h0 (s)ψ(Zs )ds | t0 = u, Zt0 = 1, t1 = v · p01 · ν0 e−ν0 u · ν1 e−ν1 v dvdu 0 0 n∆ Z tZ ∞ h  Z n∆+t  i + E exp − h0 (s)ψ(Zs )ds | t0 = u, Zt0 = 1, t1 = v · p01 · ν0 e−ν0 u · ν1 e−ν1 v dvdu Z

t

0

t−u

n∆

Z tZ t−u  Z n∆+u Z = exp − ( h0 (s)ψ(0)ds+ 0

0

n∆

Z tZ

∞

+ 0

t−u

Z h0 (s)ψ(1)ds+

n∆+u

Z exp − ( 

n∆+u+v

n∆+u

n∆

h0 (s)ψ(0)ds +

n∆+t

 h0 (s)ψ(2)ds) ·p01 ·ν0 e−ν0 u ·ν1 e−ν1 v dvdu

n∆+u+v

Z

n∆+t

 h0 (s)ψ(1)ds) · p01 · ν0 e−ν0 u · ν1 e−ν1 v dvdu.

n∆+u

19

(A.4)

The second part of Eq.(A.2) can be written by splitting the integral as follows: ∞

 Z n∆+t  i E exp − h0 (s)ψ(Zs )ds | t0 = u · p02 · ν0 e−ν0 u du t n∆ Z t h  i  Z n∆+t h0 (s)ψ(Zs )ds | t0 = u · p02 · ν0 e−ν0 u du E exp − + n∆ 0 Z ∞   Z n∆+t h0 (s)ψ(0)ds · p02 · ν0 e−ν0 u du exp − = n∆ t Z t h i  Z n∆+t  + E exp − h0 (s)ψ(Zs )ds | t0 = u, Zt0 = 2 · p02 · ν0 e−ν0 u du 0 n∆ Z ∞  Z n∆+t  = exp − h0 (s)ψ(0)ds · p02 · ν0 e−ν0 u du t n∆ Z n∆+t Z t  Z n∆+u  h0 (s)ψ(2)ds) · p02 · ν0 e−ν0 u du. h0 (s)ψ(0)ds + exp − ( + Z

h

(A.5)

n∆+u

n∆

0

So, the reliability function in Eq.(A.2) is given by: ∞

 Z n∆+t  exp − h0 (s)ψ(0)ds · p01 · ν0 e−ν0 u du R(n, 0, t)= n∆ t Z tZ t−u  Z n∆+u Z n∆+u+v Z n∆+t  + exp − ( h0 (s)ψ(0)ds+ h0 (s)ψ(1)ds+ h0 (s)ψ(2)ds) ·p01 ·ν0 e−ν0 u ·ν1 e−ν1 v dvdu Z

0

0

Z tZ

n∆ ∞

+ 0

n∆+u

Z exp − ( 

t−u ∞

n∆+u

n∆+u+v

Z

n∆+t

h0 (s)ψ(0)ds +

n∆

 h0 (s)ψ(1)ds) · p01 · ν0 e−ν0 u · ν1 e−ν1 v dvdu

n∆+u

 Z n∆+t  + exp − h0 (s)ψ(0)ds · p02 · ν0 e−ν0 u du t n∆ Z t Z n∆+t  Z n∆+u  + exp − ( h0 (s)ψ(0)ds + h0 (s)ψ(2)ds) · p02 · ν0 e−ν0 u du, Z

0

n∆

(A.6)

n∆+u

where t0 is the time during which the covariate process is in the healthy state, p01 = 02 p02 = q01q+q . 02 When z = 1, then the Eq. (A.1) is given by:

q01 q01 +q02 ,

Z ∞ h  Z n∆+t  i R(n, 1, t) = E exp h0 (s)ψ(Zs )ds | t1 = v · ν1 · e−ν1 v dv 0 n∆ Z ∞  Z n∆+t  = exp − h0 (s)ψ(1)ds · ν1 · e−ν1 v dv t n∆ Z t h  Z n∆+t  i + E exp h0 (s)ψ(Zs )ds | t1 = v, Zt1 = 2 · ν1 · e−ν1 v dv 0 n∆  Z n∆+t  = exp − h0 (s)ψ(1)ds · e−ν1 t n∆ Z t Z n∆+v Z n∆+t   + exp − ( h0 (s)ψ(1)ds + h0 (s)ψ(2)ds) · ν1 · e−ν1 v dv. 0

n∆

n∆+v

20

and

(A.7)

Finally, the conditional reliability when z = 2 is as follows: h

R(n, 2, t) = E exp

Z

n∆+t

 i  Z h0 (s)ψ(Zs )ds | Zs = 2 = exp −

n∆

n∆+t

n∆

21

 h0 (s)ψ(2)ds . (A.8)

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