Joint Condition-Based Production and Maintenance Optimization:
Balancing Maintenance and Inventory Costs by Adjusting
Production Rates
Emiel Krol
Master’s Thesis Econometrics, Operations Research and Actuarial Studies Supervisor: Dr. B. de Jonge
Joint Condition-Based Production and Maintenance Optimization:
Balancing Maintenance and Inventory Costs by Adjusting
Production Rates
Emiel Krol
10th August 2020
Abstract
We consider a production facility that processes orders of different sizes. Due to deterioration during production this facility requires maintenance. We consider a sys-tem that uses condition information, combines make-to-stock and make-to-order, and has an adjustable production speed. We compare this system with systems that solely consider to-stock or to-order. We find that the system that combines make-to-stock and make-to-order outperforms the other systems. For a higher stock index, maintenance is performed at a higher deterioration level and a lower production speed setting is preferred. We perform a sensitivity analysis which demonstrates the robust-ness of our results. Furthermore, we find that adding even a small amount of stocking capacity to a make-to-order model can greatly reduce costs.
1
Introduction
Maintenance planning and production planning are often studied separately. Maintenance planning can lead to decreased maintenance costs, whereas production planning can lead to a higher profit by producing the optimal number of items. When scheduling maintenance and production separately they yield conflicting or suboptimal schedules. In studies such as Cassady and Kutanoglu (2005) it has been shown that substantial gains can be achieved by combining the planning of maintenance and production.
orders. We consider both production after an order has arrived and preventive stocking. As such this system is a combination of make-to-order and make-to-stock systems.
Due to advances made in sensor technology, the condition of equipment can often be measured more accurately and frequently (Choi et al. 2018). This condition information can be used to find more accurate maintenance and production strategies. Condition-based main-tenance strategies that use this condition information lead to reduced cost when compared to other strategies such as time-based maintenance or block-based maintenance (De Jonge et al. 2017, Cherkaoui et al. 2018).
We consider a system that is capable of producing at several production speeds. How-ever, producing at a higher speed results in deterioration at a higher rate. As the system deteriorates preventive maintenance can be performed, with the aim to avoid failure and cor-rective maintenance. Corcor-rective maintenance is typically more expensive and requires more time than preventive maintenance. During maintenance no production is possible, which could lead to undesirably low stock levels or high numbers of back-orders. No maintenance is possible during production, and in the case of a failure during production, production on the current item being produced has to be started over. Over time dynamic decisions must be made about whether to perform preventive maintenance or to produce at a certain speed. When there are many back-orders, it might be worthwhile to produce at a higher speed at the cost of a higher deterioration rate. If there are sufficient items on stock it might be prefer-able to produce at a slower rate to avoid excess deterioration. Typically, it is worthwhile to perform preventive maintenance at different deterioration levels for different back-order or stock levels.
The remainder of this study is structured as follows. In Section 2 we review studies that consider make-to-stock systems, make-to-order systems, and production dependent deterior-ation. In Section 3 the problem we consider is described formally. In Section 4 we model this problem by formulating a Markov decision process in order to jointly optimize production and maintenance decisions by minimising costs. In Section 5 we provide a numerical analysis of our model for several instances of our problem. In Section 6 we give our conclusions and provide suggestions for future research.
2
Literature review
stock-ing items preemptively. These studies can be further divided into studies that consider a fixed set of jobs, a known demand for each period, a constant demand rate, and a random demand each time period. Finally these studies can be divided into studies that only consider corrective maintenance, preventive maintenance, and studies that combine both. Further-more we consider studies that allow for multiple production rates. These can be production dependent failure rates or production dependent deterioration rates. Some of these studies are combined with full-filling a certain type of demand.
This section is structured as follows. In 2.1 we consider studies that do not allow for stocking. In 2.2 we consider studies that do allow for stocking. In 2.3 we consider production dependent deterioration. Finally, in 2.4 we discuss the contribution of this paper to the literature.
2.1
No stocking
First we consider studies that do not allow for stocking products. We start with studies that consider a set number of jobs to be processed. Cassady and Kutanoglu (2005) consider a single machine that is repaired minimally upon failure, preventive maintenance renews the system. Their objective is to minimize the total expected weighted completion time. Yulan et al. (2008) extend this by assuming both types of maintenance require time. Fitouhi and Nourelfath (2012) integrate preventive maintenance and production planning for a single machine with only two states. Minimal repair is required upon failure, preventive mainten-ance is possible at the beginning of each production planning period. Their objective is to minimize the sum of preventive and corrective maintenance costs, setup costs, holding costs, back-order costs, and production costs, while satisfying the demand for all products over the entire horizon. They formulate a mixed integer problem in order to determine optimal policies for this system.
Jafari and Makis (2015) jointly optimize production lot size and maintenance planning to minimize costs of a single machine producing a single product type. At the end of each period the system is inspected such that the system condition is known. The condition is modelled using a discrete state space. They formulate a semi-Markov Decision Process (SMDP) in order to find the optimal decision structure.
maintenance. Corrective maintenance requires more time than preventive maintenance. A cost is incurred for each period a job is in the system. A Markov Decision Process (MDP) if formulated in order to find the optimal decision structure.
2.2
Stocking
Next we consider studies that allow for stocking products. The following studies consider a constant demand rate. Martinelli (2005) and Martinelli (2007) consider a system that produces a single type of product on a single machine subject to a non-homogeneous Markov failure/repair process with the failure rate depending on the production rate. Two production rates are considered. This is generalized in Martinelli (2010) by considering N failure rates dependent on production rates through an increasing piecewise linear function. This setting is extended by Francie et al. (2014) by considering two machines. Cheng et al. (2018) consider a system that produces a single type of product at a constant rate. They consider condition based maintenance, where the condition of the system is determined by inspections at the end of a production run. The quality of the the product is dependent on the deterioration level of the system. They use a simulation-based approach in order to to jointly optimize quality, production and maintenance. The decision variables are the inventory threshold which indicates when to start a new production run and the deterioration threshold which indicates when to start imperfect maintenance.
condition based maintenance system where products must be produced in lots. The system degradation is modelled using a stationary gamma process, the degradation level is inspected after in between production lots. They consider dynamic demand for each product which has to be satisfied at the end of each period. Both back-order and inventory are allowed at different costs. Imperfect preventative maintenance is required when the risk of failure exceeds a certain threshold. A fix-iterative model heuristic is proposed.
2.3
Production dependent deterioration
The following studies consider production dependent deterioration that is not subject to a form of demand. De Jonge and Jakobsons (2018) consider block based maintenance for a machine subject to random usage. The machine can be either turned on or off, in which case there is no deterioration. Uit het Broek et al. (2019) jointly optimize production profits and maintenance costs. Condition based maintenance is performed at pre-specified moments. For all time periods between these moments, a production rate needs to be selected. The deterioration rate is considered to be stochastic. They propose an MDP in order to find the optimal decision structure.
2.4
Contribution by the present study
From the above we conclude that ample research has been done combining condition based maintenance planning and production planning. Some of this research considers make-to-stock systems rather than make-to-order systems and some of this research considers produc-tion dependent failure rates. But very little research has been done considering producproduc-tion dependent deterioration. The aim of this study is to contribute to the current literature by combining a make-to-stock and make-to-order system, while considering production depend-ent deterioration.
3
Problem Description
model provides the possibility of avoiding back-order costs, thereby reducing the total costs. The machine can produce a single product at a time, and it only deteriorates while producing. We consider different production speeds that differ in their deterioration rate.
Let w denote the number of production speeds. If a higher production speed produces a product at x times the speed of another production speed, the deterioration rate of this higher production speed is at least x times as high as the deterioration rate of the lower production speed. Thus the expected amount of deterioration for producing a single product is higher for a higher production speed. The deterioration process is modelled using a discrete-time Markov chain with m`1 states. The transition probability matrix depends on the production speed and equals Pi for each time period when production speed i is used. These production
speeds are ordered such that speed i ` 1 produces at a faster rate than speed i.
Deterioration state 1 represents the as-good-as-new state, state m the most deteriorated state, and state m ` 1 the failed state. We assume that the condition of the machine cannot improve without maintenance and therefore that each Pi is upper triangular. Since
produc-tion can last longer than a single time period, we have that the transiproduc-tion matrices during the entire production length are given by Qi “ Piki, where ki is the production duration
corresponding to speed i. Since a higher production speed coincides with a higher deteri-oration rate, for any two transition probability matrices Qv and Qw, with v ă w, we have
řm`1 n1“jQi,n 1 v ď řm`1 n1“jQi,n 1
w for each i and j.
We consider two types of maintenance: Preventive maintenance, which can be carried out at any time as long as the machine is not in the failed state, and corrective maintenance, which needs to be carried out when the machine is in the failed state. Corrective maintenance requires Tcm time periods and has a cost ccm. Preventive maintenance requires Tpm time
periods and has cost cpm. We assume that corrective maintenance is at least as expensive
as preventive maintenance i.e. cpmď ccm, and that corrective maintenance takes at least as
long as preventive maintenance, Tpmď Tcm. Production is not possible during maintenance.
If the machine breaks down, the product in production is lost. Both types of maintenance are considered perfect; they bring the machine back to the as-good-as-new state. Preventive maintenance is not possible during production.
Every time period an order arrives with probability p, implying that the discrete-time arrival process of orders is memoryless and that the resulting interarrival times are geomet-rically distributed. Let n denote the number of possible order sizes. Each order has size i1
“ 1, 2, . . . , n with probability qi1, such thatřn
i“1qi1 “ 1.
there is no limit on the amount of back-orders. Back-ordering an order places it in a first in first out (FIFO) queue. Stocking a single product unit has costs h for every time period. Costs for orders in the system are incurred by the total flow time. The cost for back-orders is c per production per period. In the model that only allows stocking a missed order cost µ is introduced. Without this cost it would be optimal to not produce at all and thus have zero stocking and maintenance costs.
4
Markov Decision Process
In order to minimize the joint cost of maintenance and production we formulate a Markov decision process (MDP) representation of the system, as described by Puterman (1994). The solution of the MDP will yield the optimal maintenance and production decisions to be made at each deterioration state and back-order/storage level in order to minimise total costs. These costs consist of storage costs and back-order costs.
The deterioration level of the system will be denoted by X, here X “ 1 denotes the as-good-as-new state and X “ m ` 1 the failed state. The production speed setting by Y which is 0 when idle, L denotes the production time remaining for the current product, T denotes the remaining maintenance time and Z denotes the number of items that are back-ordered, when negative, and the number of items stocked when positive. The state is of the system is thus given by s “ pX, Y, L, T, Zq. The following formulation is inspired by De Jonge (2020).
The state space of the MDP is given by
S “ tpX, Y, L, T, Zq :X P t1, . . . , m, m ` 1u, Y P t0, 1, 2, . . . , Ωu, L P t0, 1, 2u, T P t0, 1, . . . , Tcmu, Z P t. . . , ´1, 0, 1, . . . , zuu. T ą 0 ñ Y “ 0, L “ 0 (1) Y ą 0, L ą 0 ñ T “ 0 (2) Z “ z ñ Y “ 0, L “ 0, (3)
Our goal is to minimize the long-run cost rate. We apply the value iteration algorithm described in (Puterman, 1994, Chapter 8) in order to determine the optimal policies for all states s P S. The values v0, v1, . . . , vn are real-valued functions on the state space S. We set
the final reward v0
psq “ 0 for all s P S such that vnpsq can be interpreted as the minimum total expected cost as a function of the current state s when there are n periods remaining. We suggest a multi-step MDP, similar to the one used by De Jonge (2020). The first step is the arrival of a new order in the system. In the second step we will determine whether to start performing maintenance. In the third step we will determine whether to start production. Finally, in the fourth step the system moves to the next time period.
The first step is the arrival of a new order in the system, such that the only possible change is the position in the back-order/stocks. The transition probabilities are given by
p1ps1|sq “ $ ’ ’ ’ ’ ’ ’ & ’ ’ ’ ’ ’ ’ % 1, if X1 “ X, Y1 “ Y, L1 “ L, T1 “ T , Z1 “ Z “ Ξ, i “ 1, . . . , n, pqi, if X1 “ X, Y1 “ Y, L1 “ L, T1 “ T , Z1 “ Z ´ i, i “ 1, . . . , n, 1 ´ p, if X1 “ X, Y1 “ Y, L1 “ L, T1 “ T , Z1 “ Z, 0, elsewhere.
Here, if an order arrives the size of the order is subtracted from the inventory position, unless the number of back-orders is at its maximum capacity, in which case the excess orders are lost.
In the second step we must decide whether to start performing maintenance in this period. Maintenance must always be started when the system is in the failed state and maintenance has not yet started. If maintenance has already started in a previous period this is continued. We assume that preventive maintenance is not possible during a job. If the system is not in the failed state and the remaining maintenance time is zero either action is allowed. The possible actions are performing maintenance M and not performing maintenance N . Therefore,
A2psq “ $ ’ ’ ’ & ’ ’ ’ % tM u, if X “ m ` 1 and T “ 0, tN u, if T ą 0 or pX ‰ m ` 1 and L ą 0q, tM, N u, if otherwise.
The immediate costs associated with this action are
Since the second step does not contain any stochasticity, we let f2ps, aq denote the
determ-inistic function that returns the new state after action a P A2psq is chosen in state s:
f2ps, aq “ $ & % s, if a “ N, p1, 0, 0, Tcm1X“m`1` Tpm1Xďm`1, Zq, if a “ M,
here the state remains unchanged if no maintenance is started. If maintenance is performed the system is once again as good as new, current jobs being processed are reset so the remaining process time is set to 0, the remaining maintenance time is set to the appropriate maintenance length.
In the third step we choose whether to start producing a new product. If maintenance is currently being performed, if a product is currently being produced, or if stock is at its maximum capacity no new production can be started. Here N denotes not starting a new production cycle, so either idle or continuing a production cycle started in an earlier period, and Si denotes starting a production cycle on production setting i “ 1, 2. The possible
actions are A3psq “ $ & % tN u, if T ą 0 or L ą 0 or Z “ z, tS1, S2u, if T “ 0.
Since the third step does not contain any stochasticity either, we let f3ps, aq denote the
deterministic function that returns the new state after action a P A3psq is chosen in state s:
f3ps, aq “ $ ’ ’ ’ & ’ ’ ’ % s, if a “ N, pX, 1, 2, 0, Zq, if a “ S1, pX, 2, 1, 0, Zq, if a “ S2.
The fourth step is the step to the next time period. Let p4ps1|sq denote the probability of
moving from state s to state s1 in this step. The transition probabilities are given by
Here the state of the system will not change if the system is idle and if maintenance is currently being performed then the only change is the maintenance time remaining. If the system does not reach the failed state the production time remaining is reduced by one for both production speeds. If a product is finished in this production cycle it is added to the stock. If the system does reach the failed state the production time remaining is reset to 0. The flow time costs are given by
CZpsq “ $ & % hZ, if Z ě 0, cZ, if Z ă 0.
If there are items in stock, Z ě 0, the stocking costs are incurred for each unit stocked. If there are ordered items Z ă 0, flow time costs are incurred for the number of back-ordered items. These costs can be added during either the second step or the third step since Z does not change here.
By using the four steps described above in reverse order any value vn can be determined
based on the value vn´1 in the following way:
un3psq “ ÿ s1PS p4ps1|sqvn´1ps1q, un2psq “ min aPA3psqtu n 3pf3ps, aqqu ` CZpsq, un1psq “ min aPA2psq tCmps, aq ` un2pf2ps, aqqu, vnpsq “ ÿ s1PS p1ps1|squn1ps 1 q. Here un
3psq are the values before the transition to vn´1psq. un2psq are the values after the cost
minimizing decision of whether to to start a new producing is taken. un
1psq are the values
after the cost minimizing decision of whether to perform maintenance is taken. And finally vn
psq denotes the expected value in state s in iteration n before a new order might arrive. The value algorithm determines the span for each iteration (Puterman, 1994, Chapter 8), where the span is defined as follows:
sppvnpsq ´ vn´1psqq “ max sPS tv n psq ´ vn´1psqu ´ min sPStv n psq ´ vn´1psqu,
which terminates when sppvnpsq ´ vn´1psqq ă for some ą 0. The values maxsPStvnpsq ´
vn´1psqu and minsPStvnpsq ´ vn´1psqu respectively provide upper and lower bounds for the
optimal gain. Once the algorithm terminates an optimal policy has been determined for all states s P S by choosing actions a2 P A2 and a3 P A3 in order to minimize un2 and un3
5
Numerical Analysis
In this section we will discuss numerical results of the Markov Decision Process formulated in Section 4. For all problem instances discussed in this section we set the stopping criterion to “ 10´6. We consider two production settings, the deterioration process of both are
modeled by a stationary gamma process. We discretize these gamma processes by using the approach described by De Jonge (2019). The parameters are selected such that the mean deterioration rate of the rate that produces twice as fast is at least twice as high with respect to the slower production speed. For a gamma deterioration process with shape parameter a and scale parameter b the mean deterioration increment per time period with length 1 is a ¨ b and the coefficient of variation is 1{?a. The first production setting has parameters a1 “ 2.5, b1 “ 0.2 and the second setting has parameters a2 “ 2.5, b2 “ 0.5. Thus the mean
deterioration increments per time unit are 0.5 and 1.25 for the first and second deterioration processes respectively. The coefficient of variation is 1{?2.5 « 0.63 for both processes. We consider m “ 10 deterioration states before failure and time steps of length ∆t “ 0.1. The resulting transition probability matrices of the Markov chains that model the deterioration are displayed in Appendix A. Note that the deterioration probabilities displayed in P2 of
deteriorating past a certain deterioration level is at least as high as the probabilities displayed in P1.
There is no capacity limit on the number of back-orders, however our model requires a finite state space. Therefore in the models that consider back-orders there is a maximum capacity of 25 back-orders, however only the optimal decisions up to 15 are considered since this imposed capacity causes incoherent behaviour as the back-order level approaches this capacity. This behaviour has very little to no effect on the rest of the model since the probability of reaching these states nears zero. In the models that consider stocking there is a maximum stock z “ 15. The costs for each unit in storage and each back-ordered unit are c “ 0.003 and h “ 0.01 respectively. For the model that only considers stocking a missed order cost is required, otherwise there would be no reason to ever produce any products. An order is missed when there is no product in stock when an order arrives. The cost per missed order per time period is µ “ 1.2. Preventive maintenance costs are cpm “ 0.8 and
corrective maintenance costs are ccm“ 1. Preventive maintenance takes Tpm“ 1 time period
and corrective maintenance takes Tcm “ 3 time periods. A new job arrives with probability
p “ 0.3. The probabilities of this job having sizes 1, 2 and 3 are q1 “ 1{2, q2 “ 1{3, and
q3 “ 1{6 respectively.
decision structures of the instance described above for all three models. In 5.2 we provide a sensitivity analysis. Finally, in 5.3 we provide results for additional instances.
5.1
Optimal decision structures
5.1.1 Back-order only
First we consider the model with only back-orders, resulting in the optimal decision structure shown in Figure 1. Implementing this decision structure would yield a minimized cost rate of 0.67636. In this model it is optimal to produce at the higher speed for most back-order levels. It is very costly to be at these levels therefore we aim to fulfill outstanding orders as quickly as possible. Once there are only a few back-orders the slow production speed is preferred. These levels of back-orders are not that expensive and the slower deterioration of the slow production speed outweighs their cost. At zero back-orders there is no production since it is not allowed here; maintenance is scheduled at a lower deterioration level as a consequence. For other back-order levels, maintenance is performed at a point such that the probability of failure is relatively low. At a high number of back-orders, maintenance is scheduled at a lower deterioration level. A breakdown in this state would keep the system in a state of a high number of back-orders due to the extra time requirement of corrective maintenance. The per period costs during a breakdown decreases as the number of back-orders decreases, as a result PM is scheduled at a higher deterioration level.
5.1.2 Stock only
Next, we consider the scenario where there is only stocking. This results in the optimal decision structure as shown in Figure 2. Implementing this decision structure would yield a minimized cost rate of 0.60184.
-15 -14 -13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0Backorder index / Stock index 1 2 3 4 5 6 7 8 9 10 State
Figure 1: Optimal decision structure, back-order only: White, light grey, darker grey, and black indicate do nothing, low production speed, high production speed, and perform preventive maintenance respectively.
high. As the stock level increases, maintenance is postponed until a higher deterioration level is reached. This is due to the system producing on a lower production speed on the preceding deterioration levels. Furthermore, a failure at this stock level is less risky since the probability of paying a missed order cost during maintenance is lower here. The exception is when there is zero stock, here maintenance is postponed due to the very high risk of paying a missed order cost when performing maintenance in this state.
5.1.3 Back-orders and stock
0 1 2 3 Backorder index / Stock index4 5 6 7 8 9 10 11 12 13 1 2 3 4 5 6 7 8 9 10 State
Figure 2: Optimal decision structure, stock only: White, light grey, darker grey, and black indicate do nothing, low production speed, high production speed, and perform preventive maintenance respectively.
-15-14-13-12-11-10-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9Backorder index / Stock index
1 2 3 4 5 6 7 8 9 10 State
-15-14-13-12-11-10-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8Backorder index / Stock index 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92 96 100 State
Figure 4: Optimal decision structure for X “ 100, both back-orders and stocking: White, light grey, darker grey, and black indicate do nothing, low production speed, high production speed, and perform preventive maintenance respectively.
the system is closer to a state where there is stock rather than back-orders, which has lower costs, furthermore the costs of a failure are relatively low here due to the lower the per period costs. The structure of when to produce at high speed rather than low speed and when to produce at low speed rather than not producing at all is very similar to the stocking only model. When there are a number of items in stock the optimal decision is to produce at the low production speed. It is optimal to have some level of stock but not too much, and at this production speed this level can be maintained. As the deterioration level increases it becomes optimal to produce at high speed for a higher stock level in order to have a small stock buffer during maintenance. At a lower deterioration level, a lower stock level is preferred since it is unlikely that maintenance will be required soon. As a consequence the low production speed is preferable to the high production speed.
In Figure 4 we observe the optimal decision structure for the same instance, but with more precise condition information X “ 100. We now observe a similar pattern with more detail. Implementing this decision structure would yield a cost rate of 0.4417, which is 96.3% of the cost rate for X “ 10. Since this percentage is very high, investing in technologies in order to obtain more precise condition information might not be worthwhile for this instance.
5.2
Sensitivity analysis
Figure 5 shows the cost rate of the optimal policy as a function of the deterioration scale parameter of the high production speed. Note that in the analysis above a scale parameter of b2 “ 0.5 is used. As the deterioration rate increases in value the stock only model and the
model with both options increase at roughly the same rate, however the back-order only model increases at a higher rate. The model that only allows back-orders has to account for this higher deterioration rate by either having a higher risk of failure or producing at the slower speed, which would result in a higher average time spend in a state with a higher number of back-orders. Both of these options result in extra costs. The model that includes stock can increase the stock level to account for a riskier higher production speed. Maintaining a higher stock level is cheaper, which is why the cost rate of the stock inclusive models increases at a slower pace. For all values of b2 the model where both options are allowed results in a lower
cost rate.
Figure 6 shows the cost rate of the optimal policy as a function of the deterioration shape parameter of both production settings. We change the shape parameter for both production settings simultaneously such that the coefficient of variation is equal for both settings. Similarly to the cost rate when changing only the b2 parameter, the cost rate for
the back-order only model increases at a faster rate than the stock only model as a, and thus the mean deterioration rate, increases. In all cases the cost rate of the model that allows both back-orders and stock outperforms the models that allow only one of them.
0.400 0.425 0.450 0.475 0.500 0.525 0.550 0.575 0.600 b_2 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 Cost rate both back-order only stock only
Figure 5: Cost rate of optimal policy as a function of the deterioration scale parameter b2 of the high speed production setting.
1.0 1.5 2.0 2.5 3.0 3.5 4.0 a 0.2 0.4 0.6 0.8 1.0 Cost rate both back-order only stock only
0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 c_pm 0.40 0.45 0.50 0.55 0.60 0.65 0.70 Cost rate both back-order only stock only
Figure 7: Cost rate of the optimal policy as a function of the preventive maintenance cost cpm.
immediate cost of preventive and corrective maintenance are now nearly equal, the only ma-jor downside of corrective maintenance is that it requires more time. The models that allow for stocking can decrease costs by increasing stock levels to account for this and perform maintenance less often.
Figure 8 shows the cost rate of the optimal policy as a function of the corrective ance cost. The cost rate for all three models considered increases as the corrective mainten-ance costs increases. However the cost rate of the back-order only model increases slightly faster than the other two models. The models that allow stocking can account for the cost of higher maintenance by having a higher stock level. However the back-order only model can only account for this by scheduling preventive maintenance earlier, so that there is less time to produce products and thus having a higher number of back-orders on average. Since it is more expensive to back-order an item than to stock it, this models cost rate increases at a faster rate as a function of the corrective maintenance cost.
0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.15 1.20 c_cm 0.45 0.50 0.55 0.60 0.65 0.70 Cost rate both back-order only stock only
Figure 8: Cost rate of the optimal policy as a function of the corrective maintenance cost ccm.
stocking capacity to a back-order only model.
5.3
Other instances
In this section we consider other problem instances. In 5.3.1 we consider an instance with a significantly higher corrective maintenance duration. In 5.3.2 we consider an instance where both types of maintenance have a longer duration.
5.3.1 High corrective maintenance duration
We now consider an instance with a lower PM cost, cpm “ 0.4, and with a much longer
corrective maintenance cost, Tcm“ 10. The remaining parameter values are unchanged from
0 2 4 6 8 10 Maximum stock level
0.45 0.50 0.55 0.60 0.65 Cost rate
Figure 9: Cost rate of the optimal policy as a function of maximum stocking capacity z. Back-orders are allowed for the values displayed in this graph.
occurs at a higher stock index for a higher deterioration level, whereas in this instance this threshold occurs at a lower stock index for a higher deterioration level. Due to the high TCM,
creating extra stock buffer for orders arriving during maintenance, is not worth the higher risk of failure of the high production speed. The cost rate of the high TCM instance is 0.62774
when only considering stocking and 0.75698 when only considering back-orders. The model that considers both has a cost rate of 0.47105, which is 75.0% of the stocking only model and 62.3% of the back-order only model.
5.3.2 High preventive and corrective maintenance duration
In this instance the preventive maintenance duration is set to Tpm “ 3 and the corrective
maintenance duration is set to Tcm “ 4. Furthermore, the probability of a job arriving now
-15-14-13-12-11-10-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 1011Backorder index / Stock index 1 2 3 4 5 6 7 8 9 10 State
Figure 10: Optimal decision structure of the instance with high CM duration, both back-orders and stock: White, light grey, darker grey, and black indicate do nothing, low production speed, high production speed, and perform preventive maintenance respect-ively.
however preventive maintenance is no longer as strongly preferred over corrective mainten-ance. At a higher stock level maintenance is postponed until a higher deterioration level since a failure at this level is less costly. At the maximum stock level there is no production since it is not allowed here, maintenance is scheduled at a lower deterioration level due to this. When only considering stocking this results in a cost rate of 0.9334 back-orders this results in a cost rate of 1.1964. When allowing both back-orders and stocking the cost rate is 0.8182, which is 87.7% of the stocking only model and 68.4% of the back-order only model.
6
Conclusion and suggestions for future work
In this study we have considered the joint optimization of production and maintenance of a deteriorating production facility. The production facility produces items in order to satisfy incoming orders. We considered a make-to-order system, a make-to-stock system and a com-bination of both. We considered also multiple production speeds with increasing deterioration rates.
-15-14-13-12-11-10-9-8-7-6-5-4-3-2-1 0 1 2 3 4 5 6 7 8 9 101112131415Backorder index / Stock index 1 2 3 4 5 6 7 8 9 10 State
Figure 11: Optimal decision structure of the high repair time instance, both back-orders and stock: White, light grey, darker grey, and black indicate do nothing, low production speed, high production speed, and perform preventive maintenance respectively.
problem from a cost perspective, with costs for orders in the system according to the total weighted flow-time, stocking costs and maintenance costs. We determined optimal decisions by formulating the system as a Markov decision process. The optimal decisions are dependent on the deterioration level and on the back-order/stock level of the system. We found that when there are back-orders it is worthwhile to produce at a higher speed and once there is a certain level of stock it is preferable to produce at a lower speed. Furthermore, maintenance is performed at a lower deterioration level for a higher number of back-orders and, as the number of back-orders decreases or as the stock increases, maintenance is performed at a higher deterioration level. We found that the make-to-order and make-to-stock systems are outperformed by the system that combines both. Our sensitivity analysis demonstrated the robustness of this result. Adding even a small stocking capacity to a model with only back-orders can greatly decrease the cost rate.
However, since the stock level is no longer dependent on a single machine, one could possibly avoid risking failure in order to achieve a more desirable stock level by balancing production speeds over the multiple machines.
In this paper we have considered only a single product type, it would be interesting to consider multiple product types with varying production time lengths. Products that need a short production time might be clustered together and produced at a high speed, whereas products that need a long production time might require a freshly maintained machine and a slow production speed in order to avoid failure before finishing the job.
In our numerical analysis we have only considered two production speeds. This could be further analyzed numerically by considering a higher number of production speeds. This would result in a more detailed decision structure and, in fields where there is a large number of production settings, a more realistic one.
Finally, we have assumed that the condition information is always available and perfect. An interesting extension to this paper would be to require inspections or to consider imper-fect condition information. The uncertainty that these additions would bring would likely result in more frequent maintenance and a higher preference to a lower production speed. Furthermore, we have assumed that maintenance can be started instantaneously. Adding a lead time to maintenance would yield an interesting interaction between the length of the lead time and the different production speeds.
Acknowledgements
Appendix A
P1“ » — — — — — — — — — — — — — — — — — — — — — — — — — — — – 0.7 0.19 0.06 0.02 0.01 0.01 0 0 0 0 0 0 0.7 0.19 0.06 0.02 0.01 0.01 0 0 0 0 0 0 0.7 0.19 0.06 0.02 0.01 0.01 0 0 0 0 0 0 0.7 0.19 0.06 0.02 0.01 0.01 0 0 0 0 0 0 0.7 0.19 0.06 0.02 0.01 0.01 0.01 0 0 0 0 0 0.7 0.19 0.06 0.02 0.01 0.01 0 0 0 0 0 0 0.7 0.19 0.06 0.02 0.02 0 0 0 0 0 0 0 0.7 0.19 0.06 0.05 0 0 0 0 0 0 0 0 0.7 0.19 0.1 0 0 0 0 0 0 0 0 0 0.7 0.3 0 0 0 0 0 0 0 0 0 0 1 fi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi flFigure 12: Transition probability matrix for the slow production speed.
P2“ » — — — — — — — — — — — — — — — — — — — — — — — — — — — – 0.58 0.19 0.08 0.05 0.03 0.02 0.01 0.01 0.01 0.01 0.02 0 0.58 0.19 0.08 0.05 0.03 0.02 0.01 0.01 0.01 0.03 0 0 0.58 0.19 0.08 0.05 0.03 0.02 0.01 0.01 0.03 0 0 0 0.58 0.19 0.08 0.05 0.03 0.02 0.01 0.04 0 0 0 0 0.58 0.19 0.08 0.05 0.03 0.02 0.06 0 0 0 0 0 0.58 0.19 0.08 0.05 0.03 0.08 0 0 0 0 0 0 0.58 0.19 0.08 0.05 0.11 0 0 0 0 0 0 0 0.58 0.19 0.08 0.15 0 0 0 0 0 0 0 0 0.58 0.19 0.23 0 0 0 0 0 0 0 0 0 0.58 0.42 0 0 0 0 0 0 0 0 0 0 1 fi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi fl
References
Ayed, S., D. Sofiene, and R. Nidhal (2012). Joint optimisation of maintenance and production policies considering random demand and variable production rate. International Journal of Production Research 50 (23), 6870–6885.
Uit het Broek, M. A. J., R. H. Teunter, B. De Jonge, J. Veldman, and N. D. Van Foreest (2019). Condition-based production planning: Adjusting production rates to balance out-put and failure risk. Manufacturing & Service Operations Management 22 (4), 792–811. Cassady, C. R. and E. Kutanoglu (2005). Integrating preventive maintenance planning and
production scheduling for a single machine. IEEE Transactions on Reliability 54 (2), 304– 309.
Cheng, G. Q., B. H. Zhou, and L. Li (2018). Integrated production, quality control and condition-based maintenance for imperfect production systems. Reliability Engineering and System Safety 175, 251–264.
Cherkaoui, H., K. T. Huynh, and A. Grall (2018). Quantitative assessments of performance and robustness of maintenance policies for stochastically deteriorating production systems. International Journal of Production Research 56 (3), 1089–1108.
Choi, T.-M., S. W. Wallace, and C. Wang (2018). Big data analytics in operations manage-ment. Production and Operations Management 27 (10), 1868–1883.
De Jonge, B. (2019). Discretizing continuous-time continuous-state deterioration processes, with an application to condition-based maintenance optimization. Reliability Engineering and System Safety 188, 1–5.
De Jonge, B. (2020). Joint condition-based production and maintenance optimization for a deteriorating production system. TBD TBD (TBD), TBD.
De Jonge, B. and E. Jakobsons (2018). Optimizing block-based maintenance under random machine usage. European Journal of Operational Research 265 (2), 703–709.
Fitouhi, M.-C. and M. Nourelfath (2012). Integrating noncyclical preventive maintenance scheduling and production planning for a single machine. International Journal of Pro-duction Economics 136 (2), 344–351.
Francie, Kouedeu Annie, Kenne Jean-Pierre, Dejax Pierre, Songmene Victor, and Polotski Vladimir (2014). Stochastic optimal control of manufacturing systems under production-dependent failure rates. International Journal of Production Economics 150, 174 – 187. Iravani, Seyed M.R. and Izak. Duenyas (2002). Integrated maintenance and production
control of a deteriorating production system. IIE Transactions 34 (5), 423–435.
Jafari, L. and V. Makis (2015). Joint optimal lot sizing and preventive maintenance policy for a production facility subject to condition monitoring. International Journal of Production Economics 169, 156–168.
Martinelli, Francesco (2005). Control of manufacturing systems with a two-value, production-dependent failure rate. Automatica 41 (11), 1943 – 1948.
Martinelli, F. (2007). Optimality of a two-threshold feedback control for a manufacturing system with a production dependent failure rate. IEEE Transactions on Automatic Con-trol 52 (10), 1937–1942.
Martinelli, F. (2010). Manufacturing systems with a production dependent failure rate: Structure of optimality. IEEE Transactions on Automatic Control 55 (10), 2401–2406. Puterman, ML (1994). Markov decision processes. 1994. Jhon Wiley & Sons, New Jersey. Wang, L., Z. Lu, and X. Han (2019). Joint optimal production planning and proactive
maintenance policy for a system subject to degradation. Journal of Quality in Maintenance Engineering 25 (2), 236–252.
Xiang, Yisha, C. Richard Cassady, Tongdan Jin, and Cai Wen Zhang (2014). Joint production and maintenance planning with machine deterioration and random yield. International Journal of Production Research 52 (6), 1644–1657.
