# Performance Indicators for Spare Parts and Maintenance Management: An Analytical Study

Authors: Oumaima Bounou; Abdellah El Barkany; Ahmed El Biyaali
Journal: Journal of Engineering
 (2020)

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## Abstract

A properly implemented maintenance management system has an impact at different levels. Maintenance is defined as the set of actions to maintain a property in a specified state. The unavailability of the spare parts required, to carry out the maintenance intervention, causes an extension of the inactivity time of the installation. On the contrary, an excessive stock of spare parts confines enormous capital and entails an enormous cost of ownership. According to the literature already made, we have directed in our work to propose a model of joint management of maintenance and spare parts based on stochastic-deterministic batch Petri networks. We studied this model by simulation using a graphical interface dedicated to the graphical tool used. So, we present, in this paper, the analytical study of the model by defining the performance indicators and viewing the influence of system parameters on these indicators. The main stages of the analytical study are developing theμ-marking graph, the associated Markov process which gives the associated transition matrix, and the definition of performance indicators using the probability distribution of the states. We deal with an application of the analytical evaluation of the proposed model. We end this article with an analysis and synthesis.


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## Body

## 1. Introduction

A properly implemented maintenance management system has an impact at different levels: infrastructure, resources (human and material), management (spare parts, inventory, etc.) and security, as well as competitiveness criteria, quality, price, time, flexibility, service, and reputation [1]. Maintenance is defined as all of the actions enabling a property to be maintained or restored to a specified state or able to provide a specific service [1]. Corrective maintenance is very costly because it causes unnecessary machine time [2]. The unavailability of the desired spare parts for the maintenance intervention causes an extension of the inactivity time of the installation. On the contrary, an excessive stock of spare parts confines enormous capital and leads to an enormous cost of ownership [2].According to the review paper [3] on the joint management of spare parts and maintenance, the parameters, considered in some models, are demand forecasts, time of maintenance intervention, failure number, maintenance policy, maintenance frequency, spare parts’ inventories, repair, and inventory levels. Also, it is found that the preventive maintenance needs are considered in most cases separately of inventory management or integrated into the application history or the total cost considered. For the performance evaluation of the maintenance service, there are models based on Petri nets or using simulation software such as Arena. The criteria, which took into account the performance evaluation, are maintenance quality, maintenance costs, and time control. So, Table 1 presents a view of the parameters’ frequency in the literature review.Table 1
The parameters’ frequency in the literature review.
ReferencesDemand forecastsTime of maintenance interventionFailure numberMaintenance policyMaintenance frequencySpare parts’ inventory managementDecisionsRepairInventory levelsMaintenance agent numberPerformance evaluationCorrective maintenancePreventive maintenanceOrder quantityOrder intervalProductionReliability[4]xxx[5]xx[6]xx[7]xxx[8]x[9]xx[10]x[11]xx[12]x[13]xx[14]xxx[15]x[16]xxx[17]xxx[18]x[2]xxx[19]xx[20]xxx[21]xxx[22]xxxxxA spare part stock is necessary in the performance of maintenance interventions on production facilities. In the literature review of the joint management of spare parts and maintenance, the performance evaluation is rarely addressed. In this context, we have developed a model, proposed in [22], which allows us to evaluate the performance of the interventions of the maintenance and the supply of the spare parts.In this paper, we first present the model in Section2, and then in Section 3, the analytical study of the model is presented by defining the performance indicators and viewing the influence of system parameters on these indicators. We deal with an application of the analytical evaluation of the proposed model in Section 4. We end this article with an analysis and synthesis presented in Section 5.


## 2. Performance Evaluation Model for the Joint Management of Spare Parts and Maintenance

The maintenance and spare parts’ management are always linked because we cannot do maintenance interventions on production facilities without a necessary stock of spare parts. With regard to this joint management, most of the work is directed towards the selection of intervention policies and the determination of the number of parts required. On the contrary, joint management performance evaluation is rarely addressed. In this context, we present a model, proposed in [22], which allows us to evaluate the performance of the interventions of the maintenance and the supply of the spare parts. This model is based on the model proposed in [21] for the performance evaluation of the management of spare parts, which is inspired from the models presented in [5, 16] and the models presented in [4, 9] that deal with the performance evaluation of the maintenance. The model is presented in Figure 1, and its parameters are presented in Table 2. Thus, we add additional indicators to the indicators already presented in [21] for the section of the analytical study of the management of spare parts.Figure 1
Spare parts and maintenance management model.Table 2
Designation of model parameters.
DesignationSignificancePlacesSpare parts’ supplyP1StockP2Stock level and orders in progressP3Supply orders in progressP4Generation of a new review and inspection periodP5Inspection of the state of the stock level of spare partsP6The obsolescence inspection of the stockMaintenance managementP7Broken machineP8Machine waiting for corrective interventionP9Machine repairP10Test the machine operation after repairP11Operational machine before preventive interventionP12Machine waiting for preventive interventionP13Transfer to preventive interventionP14Put the machine back into working orderTechTechnicians’ availabilityTransitionsSpare parts’ supplyt1Output of stock and delivery of maintenance service requestt2The order procurement operationt3The ordering processt4The state of the stock level sufficientt5Deterministic timing for inspectionMaintenance managementt6Failure or unexpected breakdownt7_1 and t7_2Deterministic transitions allow to go to the waiting statet8Initiate the repair phaset9The end of repair and trigger the test phaset10Restarting in the case of good operationt11Repeat the repairt12Preventive maintenance ordert13Order of the intervention commencementt14The intervention end and start-upAccording to Bounou et al. [21], the characteristics of the spare parts’ stock supply system are recalled: the stock management policy considered is (T, s, S), the demand is stochastic, and following the fish process, the lead time is stochastic and follows the exponential law. Thus, the maintenance service request is lost or postponed when there is an out of stock due to a dead stock or supply delay too long and exceeds what is expected. This reported request will influence the maintenance action and later on the production system.(1)
MC stands for corrective maintenance(2)
MP means preventive maintenanceThe model’s operation is summarized from [21, 22] by the following:(1)
The stock level is modeled byM-marking M (P1).(2)
The stock level and the current supply orders are photographed using itsM-marking M (P2).(3)
The inhibitory arc connecting transition t3 and place P2 monitored the level of the stock.(4)
The policy we considered in our system is a policy with periodic review, and at each period, we have crossing of transition t5, and two tokens are placed in places P5 and P6.(5)
During the stock status inspection, there are two cases either to cross transition t3 or transition t4. In both cases of inspections, transition t3 has priority over transition t4. After the crossing in all cases, a token is deposited in place P4 to generate a new inventory revision period.(6)
When t6 is crossed (an unexpected failure), the machine is put on hold for corrective maintenance (P8).(7)
When the technician is available, transition t8 is crossed to begin the repair of the machine (P9). The completed repair task gives place crossing transition t9 in order to test the operation of the machine before restarting it.(8)
In the case of preventive maintenance, the fixed revision period gives place the crossing of transition t12. When the machine is operational, the token is in one of the places P11 and P12 which crosses one of the two transitions t7_1 and t7_2 and puts the machine in the waiting state of a corrective maintenance.According to the literature review, it is necessary to master the spare parts’ management and the maintenance management at the same time since these two elements are interrelated and essential for the industry in order to ensure continuity of service. Thus, we noticed that the probabilistic graphical methods are the most efficient techniques for the joint management for these two elements. As it is defined in Petri nets, the techniques of performance evaluation are divided into two main approaches: analytical and simulation. The proposed model is developed from the models proposed in the literature. We will apply the analytic approach for joint management performance evaluation with performance indicators presented in the next section. The new integrated indicators, in relation to our work, are linked to the management of maintenance in order to control it in parallel with the management of spare parts according to the parameters of supply and repair (maintenance).


## 3. Analytical Study of Performance Evaluation

Regarding the indicators that we can consider in the performance evaluation in the joint management of spare parts’ management and maintenance in addition to the criteria already used in [21], we will add the following indicators related to the maintenance: the frequency of unforeseen breakdowns, the frequency of planned interventions, and the duration, quality, and costs of interventions. We start the stages of the analytical study of our model proposed by developing the μ-marking graph, the associated Markov process, and the definition of performance indicators. At the end of the analytical study, we move on to graphically visualize the influence of the system rates on the performance indicators.


### 3.1. Evolution of the System and Resolution of the Associated Stochastic Process

Theμ-marking graph represents the evolution of the system modeled by the batch deterministic and stochastic Petri nets. It describes each state of the system, and each crossing represents the execution of an operation. It is constructed from the initial μ-marking and by considering all the possible crossings, by the repetition of its principle. Unstable states are eliminated by merging the tangible states in order to lighten the graph and the resolution method. So, this graph is isomorphic to a continuous-time Markov chain which will make it possible to determine the associated stochastic process and its characterization.After the precision of the Markovian stochastic process related to the studied system, we pass to the precision of the characteristics of the process: the probability distribution of the network’s states and the transition matrix constructed from the associated Markov chain graph, presented in Figure2. The parameters of the stock management policy considered (T, s, S), in the model presented in Figure 1, are (5, 2, 6). The initial μ-marking is μ0  = (6, 6, Ø, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, {1, 2}). To simplify the complexity of the study of this system and the resolution of the associated system, we consider that the lot size does not influence the demand and the time of supply. In this case, the associated transition matrix is simplified. The probability distribution of the states is a function of several parameters presented in Table 3. It is represented by a line vector ∏1 and obtained by solving the following system:(1)Π1.Q=0,∑iπi=1,with Q being a simplified transition matrix, Π1 being the probability distribution of the states, and πi representing the element of Π1.Figure 2
The graph of the associated Markov chain.Table 3
The parameters of the graph of the associated Markov chain.
ParametersDesignationλ1Demand rateλ2Delay rateλ6Unexpected failure rateλ11Rate to repair againλ12Rate of preventive maintenance interventionsμ9Repair rate for corrective maintenanceμ10Test success rateμ14Repair rate for preventive maintenance


### 3.2. Evaluation of the System Performance by Indicators

Using the performance indicators formalized by [5, 9, 16, 21] and the state distribution, the performance evaluation of the modeled system is done by the analysis of the parameters’ effect on the variation of the performance which depends on the parameters cited in Table 2.(1)
Average stockSmoy: the average stock function is given by the average M-marking of place P1:(2)Smoy=Mp1moy=∑i=028πi∗μp1.(2)
Average storage cost: the average storage cost function is given by multiplying the average stock function by the storage cost per unit CS:(3)CSmoy=Cs∗Smoy=Cs∗Mp1moy,CSmoy=Cs∗∑i=028πi∗μp1.(3)
Probability of the empty stock: the probability of the stock break is based on theμ-marking of place p1 when it is equal to zero:(4)ProbS=0=Probμp1=0,ProbS=0=π1+π26+π27+π28.(4)
Frequency of supply: based on theμ-marking graph, the average supply frequency corresponds to the average crossing frequency of transition t2:(5)St25=μ24,μ25,μ26,St26=μ1,μ27,μ28,FAmoy=∑iFt2i=∑x∑i|μiεSt2xλ2x∗πi,FAmoy=λ2∗π1+π24+π25+π26+π27+π28.(5)
Average ordering cost: the ordering cost is in the form of a product of the average supply order frequency and the unit price of placing an order Cc:(6)CCmoy=Cc∗Ft3=Cc∗FAmoy,CCmoy=Cc∗λ2∗π1+π24+π25+π26+π27+π28.(6)
Average purchase cost: the average purchase frequency is the average supply frequency (Ca: unit purchase cost of a product unit):(7)CAmoy=Ca∗FAmoy,CAmoy=Ca∗λ2∗π1+π24+π25+π26+π27+π28.(7)
Coverage rate: so, the average demand is equal to1/λ1. Regarding the average stock, it is already calculated based on the marking of place p1:(8)TC=SmoyDmoy=λ1∗Smoy.(8)
Average cost of inventory outage: out of stock is a nil stock and can be caused by a delay in the supply of orders (withCru: cost of breaking the unit stock):(9)CRmoy=Cru∗λ2∗ProbS=0,CRmoy=Cru∗λ2∗π1+π26+π27+π28.(9)
Probability of having a dead stock: in our system, there is an inspection over the life of the stored spare parts. Based on theμ-marking of place p6, the dead stock is represented by μ-marking equal to 1:(10)ProbDV=0=Probμp6=1=π1+π28.(10)
Cost of obsolescence: the dead stock generates an additional cost to the costs already considered. In the case of obsolescence, an order is passed (withCob: obsolescence cost per unit):(11)COmoy=Cob∗λ2∗ProbDV=0,COmoy=Cob∗λ2∗π1+π28.(11)
The frequency of unplanned failure is calculated based on the crossing of transition t6 which is related to the failure rate:(12)FPImoy=λ6∗π0.(12)
The frequency of planned interventions consists of calculating the frequency of crossing transition t12 and transitions t7_1 and t7_2 through preventive maintenance:(13)FIPmoy=λ12∗π0.(13)
The repair frequency consists of calculating the repair frequency given by the averageM-marking of place P9 in the case of corrective intervention or place P13 in the case of preventive intervention P13.Corrective intervention:(14)FRcmoy=λ6∗π3+λ11∗π8+π17+π20+π25.Preventive intervention:(15)FRpmoy=λ12∗π2.(14)
The intervention duration is measured by the token residence time either in place P9 for the corrective maintenance or place P13 for the preventive maintenance:(16)Dimoy=Diu∗FRcmoy+FRpmoy.(15)
The intervention quality is measured by the frequencies of crossing transitions t11 (bad repair) and t10 (good repair).Good repair:(17)QBRmoy=μ10∗π7+π11+π15+π19+π23.Bad repair:(18)QMRmoy=λ11∗π7+π11+π15+π19+π23.(16)
The costs of the interventions contain the costs of repair (Cr), technician work (Cth labor cost per hour), and parts (Cp unit cost), as well as production downtime presented by a shortfall (Cvpu unit cost of product sales and Qh quantity produced per hour). They are mainly based on the intervention duration:(19)CImoy=Cr+Cth∗Dimoy+Cp∗FPImoy+FIPmoy+Cvpu∗Qh∗Dimoy.


### 3.3. The Indicators Graphic Visualization

We now move on to graphically visualize the influence of the system rates on the performance indicators. The indicators are based on 8 parameters, which lead us to discretize the evaluation of the indicators in the form of cases by varying just two parameters, and the others are fixed. The parameters considered, in different cases of the analytical study, are presented in Table4.Table 4
The parameters considered in the analytical evaluation.
CasesParametersCase 1: variation of demand and delay ratesλ1Variableλ2Variableλ60.075λ110.01λ122μ90.34μ100.9μ140.13Case 2: rate variation of unexpected failure and maintenance interventionsλ14λ22λ6Variableλ110.01λ12Variableμ90.34μ100.9μ140.13Case 3: repair rate variation for corrective and preventive maintenanceλ14λ22λ60.075λ110.01λ122μ9Variableμ100.9μ14VariableCase 4: test rate variation (repair again or not)λ14λ22λ60.075λ11Variableλ122μ90.34μ10Variableμ140.13


#### 3.3.1. Average Stock

The different curves in Figure3 show the effects of the model rates on the average stock. Curve (1) in Figure 3 shows the effects of delay and demand rates on average inventory. This indictor has known large values when the demand rate is small. On the contrary, the average stock decreases with the increase in the demand rate. Regarding the delay rate, it does not have a big influence on the average stock like the demand rate.Figure 3
Average stock. (a) Variation of demand and delay rates. (b) Rate variation of unexpected failure and maintenance interventions. (c) Repair rate variation for corrective and preventive maintenance. (d) Test rate variation (repair again or not).
(a)(b)(c)(d)Then, curve (2) represents the effects of the rate variation of the unexpected failures and planned interventions. When these rates have small values, the average stock has peak values. For the large values of the preventive intervention rate and the small values of the failure rate, it is noted in this case that the average stock has larger values than in the case of the large values of the two rates. Thus, the average stock, in the latter case, is greater compared to the average stock, the large values of failure rate, and the small values of preventive intervention rate.On the contrary, curve (3) represents the effects of the variation of repair rates for corrective and preventive maintenance. If one of the repair rates (corrective or preventive) has small values, the average stock has experienced large values. On the contrary, the average stock becomes stable when the two rates increase. Finally, from curve (4), we observe the effects of the variation in repair test rates. We note that the average stock increases when the repair again rate increases. The increase in the test success rate (i.e., the test success is poor) causes the average stock level to decrease. The stock has maximum values at the zero value of the test success rate regardless of the repair again rate.


#### 3.3.2. Probability of Empty Stock

The different curves in Figure4 show the effects of the model rates on the probability of the empty stock. Curve (1) in Figure 4 shows the effects of delay and demand rates on the probability of empty inventory. We note that the stock shortage probability is high when the delay rate is small (i.e., the delay is long) which is logical since the risk of stock shortage is linked to the deadline. When the latter is long or exceeds what is expected, the risk of rupture becomes very likely.Figure 4
Probability of empty stock. (a) Variation of demand and delay rates. (b) Rate variation of unexpected failure and maintenance interventions. (c) Repair rate variation for corrective and preventive maintenance. (d) Test rate variation (repair again or not).
(a)(b)(c)(d)Then, curve (2) represents the effects of the variation of the unexpected failure and planned intervention rates on the probability. We notice that the probability has known a minimum value just at the level of the small values of the rates. So, the two rates influence in the same way; recently, when one of the rates increases, the probability increases. We note that the unexpected failure rate has a significant influence on the probability of stock shortage (empty stock).On the contrary, curve (3) represents the effects of the variation in repair rates for corrective and preventive maintenance. The remark retained is the probability of empty stock is small when one of the rates has small values. Finally, from curve (4), we observe the effects of the variation in repair test rates. In this case, we see that the repair again rate has a great influence on the probability of empty stock; on the contrary, the test success rate is logical. The probability increases if the repair again rate increases.


#### 3.3.3. Supply Frequency

The different curves in Figure5 show the effects of the model rates on the supply frequency. Curve (1) in Figure 5 shows the effects of delay and demand rates on this frequency. The conclusion adopted is that the demand rate does not have a significant influence on the frequency, in contrast to the delay rate, which influences randomness. These variations have large values during large delay rate values and small demand rate values (i.e., increases in demand).Figure 5
Supply frequency. (a) Variation of demand and delay rates. (b) Rate variation of unexpected failure and maintenance interventions. (c) Repair rate variation for corrective and preventive maintenance. (d) Test rate variation (repair again or not).
(a)(b)(c)(d)Then, curve (2) represents the effects of the variation in the unexpected failure and planned intervention rates. The minimum of frequency is at values close to zero for the two rates. The frequency increases suddenly to have a slow variation. A large frequency occurs at large values of the unexpected failure rate and small values of the preventive intervention rate. The second point is at an average value of the supply frequency when the two rates are large.On the contrary, curve (3) represents the effects of the variation in repair rates for corrective and preventive maintenance. The supply frequency is small when one of the rates is small, and the maximum value takes place when the two rates have large values. Note that the two rates influence the frequency in the same way.Finally, from curve (4), we observe the effects of the variation in repair test rates. Test rates have a major influence on the frequency of supply. The large values of this indicator are at the levels of small values of the test success rate (good repair) and large values of the repair again rate. So, the repair again rate has a big influence on the frequency so that the latter increases when the rate increases. On the contrary, the test success rate influences the frequency inversely compared to the repair again rate, i.e., increasing the test success rate decreases the frequency.


#### 3.3.4. Coverage Rate

The different curves in Figure6 show the effects of the model rates on the coverage rate. Curve (1) in Figure 6 shows the effects of delay and demand rates on the coverage rate. It has increased at the small values of the demand rate and then decreased with the increase in the demand rate. Thus, we note that the delay rate does not have a great influence on the coverage rate.Figure 6
Coverage rate. (a) Variation of demand and delay rates. (b) Rate variation of unexpected failure and maintenance interventions. (c) Repair rate variation for corrective and preventive maintenance. (d) Test rate variation (repair again or not).
(a)(b)(c)(d)Then, curve (2) represents the effects of the variation in the unexpected failure and planned intervention rates. The coverage rate peaked at the small values for both rates. Thus, the small values of the unexpected failure rate give large values of the coverage rate, i.e., this later decreases with the increase in the unexpected failure rate. Regarding the preventive intervention rate, it does not influence in a remarkable way and slightly increases the coverage rate.On the contrary, curve (3) represents the effects of the variation in repair rates for corrective and preventive maintenance. The coverage rate has large values when one of the preventive and corrective repair rates has small values. For the rest of the rate values, the coverage rate remains stable after the decrease.Finally, from curve (4), we observe the effects of the variation in repair test rates. The repair again rate has a big influence on the coverage rate. At small values of the repair again rate, the test success rate does not have a large influence on the coverage rate. The increase in the two repair test rates influences having a high coverage rate, but the latter is lower than the value in the case of the test success rate is very small. Finally, we conclude that increasing the repair again rate increases the coverage rate; on the contrary, the increase in test success decreases it, which is logical.


#### 3.3.5. Obsolescence Probability

The different curves in Figure7 show the effects of the model rates on the probability of obsolescence. Curve (1) in Figure 7 shows the effects of delay rates and maintenance service demand on the obsolescence probability. This probability decreases with a large delay rate (i.e., the delay is small); however, small delay rate values give large probability values. The demand rate has a slight influence on the probability, and this influence is noticed with the small values of the delay rate (large delay). With large demand rate values (low demand) and small delay rate values (large delay), the obsolescence probability is equal to 1.Figure 7
Obsolescence probability. (a) Variation of demand and delay rates. (b) Rate variation of unexpected failure and maintenance interventions. (c) Repair rate variation for corrective and preventive maintenance. (d) Test rate variation (repair again or not).
(a)(b)(c)(d)Then, curve (2) represents the effects of the variation in the unexpected failure and planned intervention rates. We note, on the one hand, that the minimum probability of obsolescence is obtained in the area of the large preventive intervention rate and the low rate of unexpected failure, although the maximum value is obtained in the area of the low preventive intervention rate and high unexpected failure rate (i.e., low failures). On the other hand, the remark retained is that the unexpected failure rate influences more than the preventive intervention rate on the obsolescence probability.On the contrary, curve (3) represents the effects of the variation in repair rates for corrective and preventive maintenance. In this case, we note that the corrective and preventive repair rates influence in the same way on the probability. The maximum value of the probability is when both rates have large values. Finally, from curve (4), we observe the effects of the variation in repair test rates. We notice that(1)
Increasing the test success rate helps to decrease the probability, against increasing the repair again rate(2)
The maximum value takes place with large values of the repair again rate and small values of the test success rate (i.e., the success of the test is great, and repeating the test is very low), which is logical


#### 3.3.6. Interventions’ Frequency

Interventions for Unexpected Failures. The different curves in Figure 8 show the effects of model rates on the frequency of inventions due to unexpected failures. Curve (1) in Figure 8 shows the effects of delay and demand rates on the unexpected failure frequency. Firstly, we note that the delay rate does not influence the unexpected failure frequency. This frequency has a maximum value when the demand rate is low (i.e., the demand is high) which makes sense. Thus, it decreases with an increase in the demand rate (i.e., a decrease in the service demand).Figure 8
Interventions for unexpected failures. (a) Variation of demand and delay rates. (b) Rate variation of unexpected failure and maintenance interventions. (c) Repair rate variation for corrective and preventive maintenance. (d) Test rate variation (repair again or not).
(a)(b)(c)(d)Then, curve (2) represents the effects of the variation in the unexpected failure and planned intervention rates. The unexpected failure frequency is random according to the unexpected failure and preventive maintenance rates. It is similar to the sinusoidal shape with variable amplitude. The unexpected failure rate has a remarkable influence on the frequency.On the contrary, curve (3) represents the effects of the variation in repair rates for corrective and preventive maintenance. The conclusion reached is that repair rates influence the frequency in the same way. Finally, from curve (4), we observe the effects of the variation in repair test rates. We notice that(1)
The increase in the test success rate helps to decrease the unexpected failure frequency; on the contrary, the increase in the repair again rate increases the probability(2)
The maximum value takes place with large values of the repair again rate and small values of the test success rate (i.e., the test success is great, and repair again is very low), which is logicalPreventive Interventions. The different curves in Figure 9 show the effects of the model rates on the preventive intervention frequency. Curve (1) in Figure 9 shows the effects of delay and demand rates on this frequency. We note the same remark of the unexpected failure frequency: the demand rate also influences the preventive intervention frequency against the delay. This influence is focused on the area of small values of the demand rate.Figure 9
Preventive interventions. (a) Variation of demand and delay rates. (b) Rate variation of unexpected failure and maintenance interventions. (c) Repair rate variation for corrective and preventive maintenance. (d) Test rate variation (repair again or not).
(a)(b)(c)(d)Then, curve (2) represents the effects of the variation in the unexpected failure and planned intervention rates. The preventive maintenance rate influences in a way that it increases the frequency. However, the unexpected failure rate increases the frequency decreases.On the contrary, curve (3) represents the effects of the repair rate variation for corrective and preventive maintenance. Both rates increase the frequency in the small value area. The frequency stabilizes at the levels of the large rate values.Finally, from curve (4), we observe the effects of the repair test rate variation. The repair again rate increases the preventive intervention frequency. The latter also increases if the test success rate increases (the test success is low) which is logical.


#### 3.3.7. Repair Frequency

Corrective Repair. The different curves in Figure 10 show the effects of model rates on the corrective repair frequency. Curve (1) in Figure 10 shows the effects of delay and demand rates on this frequency. The curve of this frequency is similar to the unexpected failure frequency, which is logical. The difference is in the demand rate interval which is restricted in this case compared to the unexpected failure frequency. At this interval, the frequency values are large and stable.Figure 10
Corrective repair frequency. (a) Variation of demand and delay rates. (b) Rate variation of unexpected failure and maintenance interventions. (c) Repair rate variation for corrective and preventive maintenance. (d) Test rate variation (repair again or not).
(a)(b)(c)(d)Then, curve (2) represents the effects of the variation in the unexpected failure and planned intervention rates. The unexpected failure rate has a remarkable influence on the corrective repair frequency. The preventive intervention rate also influences the frequency but in a way less than the unexpected failure rate. On the contrary, curve (3) represents the effects of the repair rate variation for corrective and preventive maintenance. We notice that the rates increase the corrective repair frequency in the small value area of one of the rates. On the contrary, the frequency stabilizes for large values. The conclusion reached is that these rates increase the frequency corrective repair.Finally, from curve (4), we observe the effects of the repair test rate variation. The first remark retained is that the corrective repair frequency knew significant values in the area of large values of the test success rate (i.e., test success is low) and small values of the repair again rate (i.e., the possibility of redoing the repair is important) which is logical. The frequency decreases with increasing the repair again rate (i.e., decreasing the possibility of redoing the repair). In the area of small values of the test success rate (i.e., the test success is high), we notice that the frequency has small values and is not influenced by the repair again rate.Preventive Repair. The different curves in Figure 11 show the effects of the model rates on the preventive repair frequency. Curve (1) in Figure 11 shows the effects of delay and demand rates on this frequency. We note that the demand rate has a big influence on the frequency. Small demand rate values (i.e., large demand values) give large repair frequencies, which makes sense. Thus, we note that the demand rate interval, which gives these values, is narrower compared to the case of the corrective repair frequency.Figure 11
Preventive repair frequency. (a) Variation of demand and delay rates. (b) Rate variation of unexpected failure and maintenance interventions. (c) Repair rate variation for corrective and preventive maintenance. (d) Test rate variation (repair again or not).
(a)(b)(c)(d)Then, curve (2) represents the effects of the unexpected failure and planned intervention rates’ variation. The conclusion reached is that the preventive intervention rate has a remarkable influence on the frequency, which is logical.On the contrary, curve (3) represents the effects of the repair rate variation for corrective and preventive maintenance. The preventive repair frequency has the same curve as the corrective repair frequency. The preventive repair frequency increases at the level of small rate values; on the contrary, it stabilizes at the level of large rate values.Finally, from curve (4), we observe the effects of the repair test rate variation. We note that the repair again rate influences remarkably the preventive repair frequency. The rates influence this frequency the same way as the corrective repair frequency.


#### 3.3.8. Repair Quality

Good Repair. The different curves in Figure 12 show the effects of the model rates on the good quality of repair. Curve (1) in Figure 12 shows the effects of delay and demand rates on this quality. We notice the demand rate influences the good repair; on the contrary, the delay rate has no effect on this quality. Then, curve (2) represents the effects of the unexpected failure and planned intervention rates’ variation. The maximum values for good repair are in the area of low preventive intervention rate. The increase in the preventive intervention rate (decrease in preventive interventions) decreases the amount of good repair. On the contrary, the increase in the unexpected failure rate increases the amount of good repair.Figure 12
Good repair quality. (a) Variation of demand and delay rates. (b) Rate variation of unexpected failure and maintenance interventions. (c) Repair rate variation for corrective and preventive maintenance. (d) Test rate variation (repair again or not).
(a)(b)(c)(d)On the contrary, curve (3) represents the effects of the repair rate variation for corrective and preventive maintenance. The curve looks like a few indicators based on repair rates. There is an increase in the good repair at the small values of one of the rates, and the stability of the indicator for the rest of the rates changes. Finally, from curve (4), we observe the effects of the repair test rate variation. The small values of the success test rate (success of the test is very low) give a large amount of good repair, which is close to reality. So, increasing the success test rate decreases the amount of good repair; however, when the repair again rate increases (i.e., redoing the repair is low), the good repair increases.Bad Repair. The different curves in Figure 13 show the effects of model rates on bad repair quality. Curve (1) in Figure 13 shows the effects of delay and demand rates on this quality. This appearance is similar to the case of the good repair amount. So, the demand rate influences the bad repair amount.Figure 13
Bad repair quality. (a) Variation of demand and delay rates. (b) Rate variation of unexpected failure and maintenance interventions. (c) Repair rate variation for corrective and preventive maintenance. (d) Test rate variation (repair again or not).
(a)(b)(c)(d)Then, curve (2) represents the effects of the unexpected failure and planned intervention rates’ variation. Increasing the preventive intervention rate decreases the poor repair amount. However, the increase in the unexpected failure rate increases the quality. Therefore, preventive maintenance improves the quality of repair.On the contrary, curve (3) represents the effects of the repair rate variation for corrective and preventive maintenance. The curve looks like a few indicators based on repair rates. There is an increase in poor repair at the small values of one of the rates, and the indicator stability for the rest of the rates changes.Finally, from curve (4), we observe the effects of the repair test rate variation. The first remark is that the appearance of poor repair varies inversely with that of good repair. The small values of the repair again rate make a large amount of poor repair. So, increasing the test success rate increases the bad repair amount, but at the repair again rate, when it increases (i.e., repair again is low), the bad repair decreases.


## 3.1. Evolution of the System and Resolution of the Associated Stochastic Process

Theμ-marking graph represents the evolution of the system modeled by the batch deterministic and stochastic Petri nets. It describes each state of the system, and each crossing represents the execution of an operation. It is constructed from the initial μ-marking and by considering all the possible crossings, by the repetition of its principle. Unstable states are eliminated by merging the tangible states in order to lighten the graph and the resolution method. So, this graph is isomorphic to a continuous-time Markov chain which will make it possible to determine the associated stochastic process and its characterization.After the precision of the Markovian stochastic process related to the studied system, we pass to the precision of the characteristics of the process: the probability distribution of the network’s states and the transition matrix constructed from the associated Markov chain graph, presented in Figure2. The parameters of the stock management policy considered (T, s, S), in the model presented in Figure 1, are (5, 2, 6). The initial μ-marking is μ0  = (6, 6, Ø, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, {1, 2}). To simplify the complexity of the study of this system and the resolution of the associated system, we consider that the lot size does not influence the demand and the time of supply. In this case, the associated transition matrix is simplified. The probability distribution of the states is a function of several parameters presented in Table 3. It is represented by a line vector ∏1 and obtained by solving the following system:(1)Π1.Q=0,∑iπi=1,with Q being a simplified transition matrix, Π1 being the probability distribution of the states, and πi representing the element of Π1.Figure 2
The graph of the associated Markov chain.Table 3
The parameters of the graph of the associated Markov chain.
ParametersDesignationλ1Demand rateλ2Delay rateλ6Unexpected failure rateλ11Rate to repair againλ12Rate of preventive maintenance interventionsμ9Repair rate for corrective maintenanceμ10Test success rateμ14Repair rate for preventive maintenance


## 3.2. Evaluation of the System Performance by Indicators

Using the performance indicators formalized by [5, 9, 16, 21] and the state distribution, the performance evaluation of the modeled system is done by the analysis of the parameters’ effect on the variation of the performance which depends on the parameters cited in Table 2.(1)
Average stockSmoy: the average stock function is given by the average M-marking of place P1:(2)Smoy=Mp1moy=∑i=028πi∗μp1.(2)
Average storage cost: the average storage cost function is given by multiplying the average stock function by the storage cost per unit CS:(3)CSmoy=Cs∗Smoy=Cs∗Mp1moy,CSmoy=Cs∗∑i=028πi∗μp1.(3)
Probability of the empty stock: the probability of the stock break is based on theμ-marking of place p1 when it is equal to zero:(4)ProbS=0=Probμp1=0,ProbS=0=π1+π26+π27+π28.(4)
Frequency of supply: based on theμ-marking graph, the average supply frequency corresponds to the average crossing frequency of transition t2:(5)St25=μ24,μ25,μ26,St26=μ1,μ27,μ28,FAmoy=∑iFt2i=∑x∑i|μiεSt2xλ2x∗πi,FAmoy=λ2∗π1+π24+π25+π26+π27+π28.(5)
Average ordering cost: the ordering cost is in the form of a product of the average supply order frequency and the unit price of placing an order Cc:(6)CCmoy=Cc∗Ft3=Cc∗FAmoy,CCmoy=Cc∗λ2∗π1+π24+π25+π26+π27+π28.(6)
Average purchase cost: the average purchase frequency is the average supply frequency (Ca: unit purchase cost of a product unit):(7)CAmoy=Ca∗FAmoy,CAmoy=Ca∗λ2∗π1+π24+π25+π26+π27+π28.(7)
Coverage rate: so, the average demand is equal to1/λ1. Regarding the average stock, it is already calculated based on the marking of place p1:(8)TC=SmoyDmoy=λ1∗Smoy.(8)
Average cost of inventory outage: out of stock is a nil stock and can be caused by a delay in the supply of orders (withCru: cost of breaking the unit stock):(9)CRmoy=Cru∗λ2∗ProbS=0,CRmoy=Cru∗λ2∗π1+π26+π27+π28.(9)
Probability of having a dead stock: in our system, there is an inspection over the life of the stored spare parts. Based on theμ-marking of place p6, the dead stock is represented by μ-marking equal to 1:(10)ProbDV=0=Probμp6=1=π1+π28.(10)
Cost of obsolescence: the dead stock generates an additional cost to the costs already considered. In the case of obsolescence, an order is passed (withCob: obsolescence cost per unit):(11)COmoy=Cob∗λ2∗ProbDV=0,COmoy=Cob∗λ2∗π1+π28.(11)
The frequency of unplanned failure is calculated based on the crossing of transition t6 which is related to the failure rate:(12)FPImoy=λ6∗π0.(12)
The frequency of planned interventions consists of calculating the frequency of crossing transition t12 and transitions t7_1 and t7_2 through preventive maintenance:(13)FIPmoy=λ12∗π0.(13)
The repair frequency consists of calculating the repair frequency given by the averageM-marking of place P9 in the case of corrective intervention or place P13 in the case of preventive intervention P13.Corrective intervention:(14)FRcmoy=λ6∗π3+λ11∗π8+π17+π20+π25.Preventive intervention:(15)FRpmoy=λ12∗π2.(14)
The intervention duration is measured by the token residence time either in place P9 for the corrective maintenance or place P13 for the preventive maintenance:(16)Dimoy=Diu∗FRcmoy+FRpmoy.(15)
The intervention quality is measured by the frequencies of crossing transitions t11 (bad repair) and t10 (good repair).Good repair:(17)QBRmoy=μ10∗π7+π11+π15+π19+π23.Bad repair:(18)QMRmoy=λ11∗π7+π11+π15+π19+π23.(16)
The costs of the interventions contain the costs of repair (Cr), technician work (Cth labor cost per hour), and parts (Cp unit cost), as well as production downtime presented by a shortfall (Cvpu unit cost of product sales and Qh quantity produced per hour). They are mainly based on the intervention duration:(19)CImoy=Cr+Cth∗Dimoy+Cp∗FPImoy+FIPmoy+Cvpu∗Qh∗Dimoy.


## 3.3. The Indicators Graphic Visualization

We now move on to graphically visualize the influence of the system rates on the performance indicators. The indicators are based on 8 parameters, which lead us to discretize the evaluation of the indicators in the form of cases by varying just two parameters, and the others are fixed. The parameters considered, in different cases of the analytical study, are presented in Table4.Table 4
The parameters considered in the analytical evaluation.
CasesParametersCase 1: variation of demand and delay ratesλ1Variableλ2Variableλ60.075λ110.01λ122μ90.34μ100.9μ140.13Case 2: rate variation of unexpected failure and maintenance interventionsλ14λ22λ6Variableλ110.01λ12Variableμ90.34μ100.9μ140.13Case 3: repair rate variation for corrective and preventive maintenanceλ14λ22λ60.075λ110.01λ122μ9Variableμ100.9μ14VariableCase 4: test rate variation (repair again or not)λ14λ22λ60.075λ11Variableλ122μ90.34μ10Variableμ140.13


### 3.3.1. Average Stock

The different curves in Figure3 show the effects of the model rates on the average stock. Curve (1) in Figure 3 shows the effects of delay and demand rates on average inventory. This indictor has known large values when the demand rate is small. On the contrary, the average stock decreases with the increase in the demand rate. Regarding the delay rate, it does not have a big influence on the average stock like the demand rate.Figure 3
Average stock. (a) Variation of demand and delay rates. (b) Rate variation of unexpected failure and maintenance interventions. (c) Repair rate variation for corrective and preventive maintenance. (d) Test rate variation (repair again or not).
(a)(b)(c)(d)Then, curve (2) represents the effects of the rate variation of the unexpected failures and planned interventions. When these rates have small values, the average stock has peak values. For the large values of the preventive intervention rate and the small values of the failure rate, it is noted in this case that the average stock has larger values than in the case of the large values of the two rates. Thus, the average stock, in the latter case, is greater compared to the average stock, the large values of failure rate, and the small values of preventive intervention rate.On the contrary, curve (3) represents the effects of the variation of repair rates for corrective and preventive maintenance. If one of the repair rates (corrective or preventive) has small values, the average stock has experienced large values. On the contrary, the average stock becomes stable when the two rates increase. Finally, from curve (4), we observe the effects of the variation in repair test rates. We note that the average stock increases when the repair again rate increases. The increase in the test success rate (i.e., the test success is poor) causes the average stock level to decrease. The stock has maximum values at the zero value of the test success rate regardless of the repair again rate.


### 3.3.2. Probability of Empty Stock

The different curves in Figure4 show the effects of the model rates on the probability of the empty stock. Curve (1) in Figure 4 shows the effects of delay and demand rates on the probability of empty inventory. We note that the stock shortage probability is high when the delay rate is small (i.e., the delay is long) which is logical since the risk of stock shortage is linked to the deadline. When the latter is long or exceeds what is expected, the risk of rupture becomes very likely.Figure 4
Probability of empty stock. (a) Variation of demand and delay rates. (b) Rate variation of unexpected failure and maintenance interventions. (c) Repair rate variation for corrective and preventive maintenance. (d) Test rate variation (repair again or not).
(a)(b)(c)(d)Then, curve (2) represents the effects of the variation of the unexpected failure and planned intervention rates on the probability. We notice that the probability has known a minimum value just at the level of the small values of the rates. So, the two rates influence in the same way; recently, when one of the rates increases, the probability increases. We note that the unexpected failure rate has a significant influence on the probability of stock shortage (empty stock).On the contrary, curve (3) represents the effects of the variation in repair rates for corrective and preventive maintenance. The remark retained is the probability of empty stock is small when one of the rates has small values. Finally, from curve (4), we observe the effects of the variation in repair test rates. In this case, we see that the repair again rate has a great influence on the probability of empty stock; on the contrary, the test success rate is logical. The probability increases if the repair again rate increases.


### 3.3.3. Supply Frequency

The different curves in Figure5 show the effects of the model rates on the supply frequency. Curve (1) in Figure 5 shows the effects of delay and demand rates on this frequency. The conclusion adopted is that the demand rate does not have a significant influence on the frequency, in contrast to the delay rate, which influences randomness. These variations have large values during large delay rate values and small demand rate values (i.e., increases in demand).Figure 5
Supply frequency. (a) Variation of demand and delay rates. (b) Rate variation of unexpected failure and maintenance interventions. (c) Repair rate variation for corrective and preventive maintenance. (d) Test rate variation (repair again or not).
(a)(b)(c)(d)Then, curve (2) represents the effects of the variation in the unexpected failure and planned intervention rates. The minimum of frequency is at values close to zero for the two rates. The frequency increases suddenly to have a slow variation. A large frequency occurs at large values of the unexpected failure rate and small values of the preventive intervention rate. The second point is at an average value of the supply frequency when the two rates are large.On the contrary, curve (3) represents the effects of the variation in repair rates for corrective and preventive maintenance. The supply frequency is small when one of the rates is small, and the maximum value takes place when the two rates have large values. Note that the two rates influence the frequency in the same way.Finally, from curve (4), we observe the effects of the variation in repair test rates. Test rates have a major influence on the frequency of supply. The large values of this indicator are at the levels of small values of the test success rate (good repair) and large values of the repair again rate. So, the repair again rate has a big influence on the frequency so that the latter increases when the rate increases. On the contrary, the test success rate influences the frequency inversely compared to the repair again rate, i.e., increasing the test success rate decreases the frequency.


### 3.3.4. Coverage Rate

The different curves in Figure6 show the effects of the model rates on the coverage rate. Curve (1) in Figure 6 shows the effects of delay and demand rates on the coverage rate. It has increased at the small values of the demand rate and then decreased with the increase in the demand rate. Thus, we note that the delay rate does not have a great influence on the coverage rate.Figure 6
Coverage rate. (a) Variation of demand and delay rates. (b) Rate variation of unexpected failure and maintenance interventions. (c) Repair rate variation for corrective and preventive maintenance. (d) Test rate variation (repair again or not).
(a)(b)(c)(d)Then, curve (2) represents the effects of the variation in the unexpected failure and planned intervention rates. The coverage rate peaked at the small values for both rates. Thus, the small values of the unexpected failure rate give large values of the coverage rate, i.e., this later decreases with the increase in the unexpected failure rate. Regarding the preventive intervention rate, it does not influence in a remarkable way and slightly increases the coverage rate.On the contrary, curve (3) represents the effects of the variation in repair rates for corrective and preventive maintenance. The coverage rate has large values when one of the preventive and corrective repair rates has small values. For the rest of the rate values, the coverage rate remains stable after the decrease.Finally, from curve (4), we observe the effects of the variation in repair test rates. The repair again rate has a big influence on the coverage rate. At small values of the repair again rate, the test success rate does not have a large influence on the coverage rate. The increase in the two repair test rates influences having a high coverage rate, but the latter is lower than the value in the case of the test success rate is very small. Finally, we conclude that increasing the repair again rate increases the coverage rate; on the contrary, the increase in test success decreases it, which is logical.


### 3.3.5. Obsolescence Probability

The different curves in Figure7 show the effects of the model rates on the probability of obsolescence. Curve (1) in Figure 7 shows the effects of delay rates and maintenance service demand on the obsolescence probability. This probability decreases with a large delay rate (i.e., the delay is small); however, small delay rate values give large probability values. The demand rate has a slight influence on the probability, and this influence is noticed with the small values of the delay rate (large delay). With large demand rate values (low demand) and small delay rate values (large delay), the obsolescence probability is equal to 1.Figure 7
Obsolescence probability. (a) Variation of demand and delay rates. (b) Rate variation of unexpected failure and maintenance interventions. (c) Repair rate variation for corrective and preventive maintenance. (d) Test rate variation (repair again or not).
(a)(b)(c)(d)Then, curve (2) represents the effects of the variation in the unexpected failure and planned intervention rates. We note, on the one hand, that the minimum probability of obsolescence is obtained in the area of the large preventive intervention rate and the low rate of unexpected failure, although the maximum value is obtained in the area of the low preventive intervention rate and high unexpected failure rate (i.e., low failures). On the other hand, the remark retained is that the unexpected failure rate influences more than the preventive intervention rate on the obsolescence probability.On the contrary, curve (3) represents the effects of the variation in repair rates for corrective and preventive maintenance. In this case, we note that the corrective and preventive repair rates influence in the same way on the probability. The maximum value of the probability is when both rates have large values. Finally, from curve (4), we observe the effects of the variation in repair test rates. We notice that(1)
Increasing the test success rate helps to decrease the probability, against increasing the repair again rate(2)
The maximum value takes place with large values of the repair again rate and small values of the test success rate (i.e., the success of the test is great, and repeating the test is very low), which is logical


### 3.3.6. Interventions’ Frequency

Interventions for Unexpected Failures. The different curves in Figure 8 show the effects of model rates on the frequency of inventions due to unexpected failures. Curve (1) in Figure 8 shows the effects of delay and demand rates on the unexpected failure frequency. Firstly, we note that the delay rate does not influence the unexpected failure frequency. This frequency has a maximum value when the demand rate is low (i.e., the demand is high) which makes sense. Thus, it decreases with an increase in the demand rate (i.e., a decrease in the service demand).Figure 8
Interventions for unexpected failures. (a) Variation of demand and delay rates. (b) Rate variation of unexpected failure and maintenance interventions. (c) Repair rate variation for corrective and preventive maintenance. (d) Test rate variation (repair again or not).
(a)(b)(c)(d)Then, curve (2) represents the effects of the variation in the unexpected failure and planned intervention rates. The unexpected failure frequency is random according to the unexpected failure and preventive maintenance rates. It is similar to the sinusoidal shape with variable amplitude. The unexpected failure rate has a remarkable influence on the frequency.On the contrary, curve (3) represents the effects of the variation in repair rates for corrective and preventive maintenance. The conclusion reached is that repair rates influence the frequency in the same way. Finally, from curve (4), we observe the effects of the variation in repair test rates. We notice that(1)
The increase in the test success rate helps to decrease the unexpected failure frequency; on the contrary, the increase in the repair again rate increases the probability(2)
The maximum value takes place with large values of the repair again rate and small values of the test success rate (i.e., the test success is great, and repair again is very low), which is logicalPreventive Interventions. The different curves in Figure 9 show the effects of the model rates on the preventive intervention frequency. Curve (1) in Figure 9 shows the effects of delay and demand rates on this frequency. We note the same remark of the unexpected failure frequency: the demand rate also influences the preventive intervention frequency against the delay. This influence is focused on the area of small values of the demand rate.Figure 9
Preventive interventions. (a) Variation of demand and delay rates. (b) Rate variation of unexpected failure and maintenance interventions. (c) Repair rate variation for corrective and preventive maintenance. (d) Test rate variation (repair again or not).
(a)(b)(c)(d)Then, curve (2) represents the effects of the variation in the unexpected failure and planned intervention rates. The preventive maintenance rate influences in a way that it increases the frequency. However, the unexpected failure rate increases the frequency decreases.On the contrary, curve (3) represents the effects of the repair rate variation for corrective and preventive maintenance. Both rates increase the frequency in the small value area. The frequency stabilizes at the levels of the large rate values.Finally, from curve (4), we observe the effects of the repair test rate variation. The repair again rate increases the preventive intervention frequency. The latter also increases if the test success rate increases (the test success is low) which is logical.


### 3.3.7. Repair Frequency

Corrective Repair. The different curves in Figure 10 show the effects of model rates on the corrective repair frequency. Curve (1) in Figure 10 shows the effects of delay and demand rates on this frequency. The curve of this frequency is similar to the unexpected failure frequency, which is logical. The difference is in the demand rate interval which is restricted in this case compared to the unexpected failure frequency. At this interval, the frequency values are large and stable.Figure 10
Corrective repair frequency. (a) Variation of demand and delay rates. (b) Rate variation of unexpected failure and maintenance interventions. (c) Repair rate variation for corrective and preventive maintenance. (d) Test rate variation (repair again or not).
(a)(b)(c)(d)Then, curve (2) represents the effects of the variation in the unexpected failure and planned intervention rates. The unexpected failure rate has a remarkable influence on the corrective repair frequency. The preventive intervention rate also influences the frequency but in a way less than the unexpected failure rate. On the contrary, curve (3) represents the effects of the repair rate variation for corrective and preventive maintenance. We notice that the rates increase the corrective repair frequency in the small value area of one of the rates. On the contrary, the frequency stabilizes for large values. The conclusion reached is that these rates increase the frequency corrective repair.Finally, from curve (4), we observe the effects of the repair test rate variation. The first remark retained is that the corrective repair frequency knew significant values in the area of large values of the test success rate (i.e., test success is low) and small values of the repair again rate (i.e., the possibility of redoing the repair is important) which is logical. The frequency decreases with increasing the repair again rate (i.e., decreasing the possibility of redoing the repair). In the area of small values of the test success rate (i.e., the test success is high), we notice that the frequency has small values and is not influenced by the repair again rate.Preventive Repair. The different curves in Figure 11 show the effects of the model rates on the preventive repair frequency. Curve (1) in Figure 11 shows the effects of delay and demand rates on this frequency. We note that the demand rate has a big influence on the frequency. Small demand rate values (i.e., large demand values) give large repair frequencies, which makes sense. Thus, we note that the demand rate interval, which gives these values, is narrower compared to the case of the corrective repair frequency.Figure 11
Preventive repair frequency. (a) Variation of demand and delay rates. (b) Rate variation of unexpected failure and maintenance interventions. (c) Repair rate variation for corrective and preventive maintenance. (d) Test rate variation (repair again or not).
(a)(b)(c)(d)Then, curve (2) represents the effects of the unexpected failure and planned intervention rates’ variation. The conclusion reached is that the preventive intervention rate has a remarkable influence on the frequency, which is logical.On the contrary, curve (3) represents the effects of the repair rate variation for corrective and preventive maintenance. The preventive repair frequency has the same curve as the corrective repair frequency. The preventive repair frequency increases at the level of small rate values; on the contrary, it stabilizes at the level of large rate values.Finally, from curve (4), we observe the effects of the repair test rate variation. We note that the repair again rate influences remarkably the preventive repair frequency. The rates influence this frequency the same way as the corrective repair frequency.


### 3.3.8. Repair Quality

Good Repair. The different curves in Figure 12 show the effects of the model rates on the good quality of repair. Curve (1) in Figure 12 shows the effects of delay and demand rates on this quality. We notice the demand rate influences the good repair; on the contrary, the delay rate has no effect on this quality. Then, curve (2) represents the effects of the unexpected failure and planned intervention rates’ variation. The maximum values for good repair are in the area of low preventive intervention rate. The increase in the preventive intervention rate (decrease in preventive interventions) decreases the amount of good repair. On the contrary, the increase in the unexpected failure rate increases the amount of good repair.Figure 12
Good repair quality. (a) Variation of demand and delay rates. (b) Rate variation of unexpected failure and maintenance interventions. (c) Repair rate variation for corrective and preventive maintenance. (d) Test rate variation (repair again or not).
(a)(b)(c)(d)On the contrary, curve (3) represents the effects of the repair rate variation for corrective and preventive maintenance. The curve looks like a few indicators based on repair rates. There is an increase in the good repair at the small values of one of the rates, and the stability of the indicator for the rest of the rates changes. Finally, from curve (4), we observe the effects of the repair test rate variation. The small values of the success test rate (success of the test is very low) give a large amount of good repair, which is close to reality. So, increasing the success test rate decreases the amount of good repair; however, when the repair again rate increases (i.e., redoing the repair is low), the good repair increases.Bad Repair. The different curves in Figure 13 show the effects of model rates on bad repair quality. Curve (1) in Figure 13 shows the effects of delay and demand rates on this quality. This appearance is similar to the case of the good repair amount. So, the demand rate influences the bad repair amount.Figure 13
Bad repair quality. (a) Variation of demand and delay rates. (b) Rate variation of unexpected failure and maintenance interventions. (c) Repair rate variation for corrective and preventive maintenance. (d) Test rate variation (repair again or not).
(a)(b)(c)(d)Then, curve (2) represents the effects of the unexpected failure and planned intervention rates’ variation. Increasing the preventive intervention rate decreases the poor repair amount. However, the increase in the unexpected failure rate increases the quality. Therefore, preventive maintenance improves the quality of repair.On the contrary, curve (3) represents the effects of the repair rate variation for corrective and preventive maintenance. The curve looks like a few indicators based on repair rates. There is an increase in poor repair at the small values of one of the rates, and the indicator stability for the rest of the rates changes.Finally, from curve (4), we observe the effects of the repair test rate variation. The first remark is that the appearance of poor repair varies inversely with that of good repair. The small values of the repair again rate make a large amount of poor repair. So, increasing the test success rate increases the bad repair amount, but at the repair again rate, when it increases (i.e., repair again is low), the bad repair decreases.


## 3.3.1. Average Stock

The different curves in Figure3 show the effects of the model rates on the average stock. Curve (1) in Figure 3 shows the effects of delay and demand rates on average inventory. This indictor has known large values when the demand rate is small. On the contrary, the average stock decreases with the increase in the demand rate. Regarding the delay rate, it does not have a big influence on the average stock like the demand rate.Figure 3
Average stock. (a) Variation of demand and delay rates. (b) Rate variation of unexpected failure and maintenance interventions. (c) Repair rate variation for corrective and preventive maintenance. (d) Test rate variation (repair again or not).
(a)(b)(c)(d)Then, curve (2) represents the effects of the rate variation of the unexpected failures and planned interventions. When these rates have small values, the average stock has peak values. For the large values of the preventive intervention rate and the small values of the failure rate, it is noted in this case that the average stock has larger values than in the case of the large values of the two rates. Thus, the average stock, in the latter case, is greater compared to the average stock, the large values of failure rate, and the small values of preventive intervention rate.On the contrary, curve (3) represents the effects of the variation of repair rates for corrective and preventive maintenance. If one of the repair rates (corrective or preventive) has small values, the average stock has experienced large values. On the contrary, the average stock becomes stable when the two rates increase. Finally, from curve (4), we observe the effects of the variation in repair test rates. We note that the average stock increases when the repair again rate increases. The increase in the test success rate (i.e., the test success is poor) causes the average stock level to decrease. The stock has maximum values at the zero value of the test success rate regardless of the repair again rate.


## 3.3.2. Probability of Empty Stock

The different curves in Figure4 show the effects of the model rates on the probability of the empty stock. Curve (1) in Figure 4 shows the effects of delay and demand rates on the probability of empty inventory. We note that the stock shortage probability is high when the delay rate is small (i.e., the delay is long) which is logical since the risk of stock shortage is linked to the deadline. When the latter is long or exceeds what is expected, the risk of rupture becomes very likely.Figure 4
Probability of empty stock. (a) Variation of demand and delay rates. (b) Rate variation of unexpected failure and maintenance interventions. (c) Repair rate variation for corrective and preventive maintenance. (d) Test rate variation (repair again or not).
(a)(b)(c)(d)Then, curve (2) represents the effects of the variation of the unexpected failure and planned intervention rates on the probability. We notice that the probability has known a minimum value just at the level of the small values of the rates. So, the two rates influence in the same way; recently, when one of the rates increases, the probability increases. We note that the unexpected failure rate has a significant influence on the probability of stock shortage (empty stock).On the contrary, curve (3) represents the effects of the variation in repair rates for corrective and preventive maintenance. The remark retained is the probability of empty stock is small when one of the rates has small values. Finally, from curve (4), we observe the effects of the variation in repair test rates. In this case, we see that the repair again rate has a great influence on the probability of empty stock; on the contrary, the test success rate is logical. The probability increases if the repair again rate increases.


## 3.3.3. Supply Frequency

The different curves in Figure5 show the effects of the model rates on the supply frequency. Curve (1) in Figure 5 shows the effects of delay and demand rates on this frequency. The conclusion adopted is that the demand rate does not have a significant influence on the frequency, in contrast to the delay rate, which influences randomness. These variations have large values during large delay rate values and small demand rate values (i.e., increases in demand).Figure 5
Supply frequency. (a) Variation of demand and delay rates. (b) Rate variation of unexpected failure and maintenance interventions. (c) Repair rate variation for corrective and preventive maintenance. (d) Test rate variation (repair again or not).
(a)(b)(c)(d)Then, curve (2) represents the effects of the variation in the unexpected failure and planned intervention rates. The minimum of frequency is at values close to zero for the two rates. The frequency increases suddenly to have a slow variation. A large frequency occurs at large values of the unexpected failure rate and small values of the preventive intervention rate. The second point is at an average value of the supply frequency when the two rates are large.On the contrary, curve (3) represents the effects of the variation in repair rates for corrective and preventive maintenance. The supply frequency is small when one of the rates is small, and the maximum value takes place when the two rates have large values. Note that the two rates influence the frequency in the same way.Finally, from curve (4), we observe the effects of the variation in repair test rates. Test rates have a major influence on the frequency of supply. The large values of this indicator are at the levels of small values of the test success rate (good repair) and large values of the repair again rate. So, the repair again rate has a big influence on the frequency so that the latter increases when the rate increases. On the contrary, the test success rate influences the frequency inversely compared to the repair again rate, i.e., increasing the test success rate decreases the frequency.


## 3.3.4. Coverage Rate

The different curves in Figure6 show the effects of the model rates on the coverage rate. Curve (1) in Figure 6 shows the effects of delay and demand rates on the coverage rate. It has increased at the small values of the demand rate and then decreased with the increase in the demand rate. Thus, we note that the delay rate does not have a great influence on the coverage rate.Figure 6
Coverage rate. (a) Variation of demand and delay rates. (b) Rate variation of unexpected failure and maintenance interventions. (c) Repair rate variation for corrective and preventive maintenance. (d) Test rate variation (repair again or not).
(a)(b)(c)(d)Then, curve (2) represents the effects of the variation in the unexpected failure and planned intervention rates. The coverage rate peaked at the small values for both rates. Thus, the small values of the unexpected failure rate give large values of the coverage rate, i.e., this later decreases with the increase in the unexpected failure rate. Regarding the preventive intervention rate, it does not influence in a remarkable way and slightly increases the coverage rate.On the contrary, curve (3) represents the effects of the variation in repair rates for corrective and preventive maintenance. The coverage rate has large values when one of the preventive and corrective repair rates has small values. For the rest of the rate values, the coverage rate remains stable after the decrease.Finally, from curve (4), we observe the effects of the variation in repair test rates. The repair again rate has a big influence on the coverage rate. At small values of the repair again rate, the test success rate does not have a large influence on the coverage rate. The increase in the two repair test rates influences having a high coverage rate, but the latter is lower than the value in the case of the test success rate is very small. Finally, we conclude that increasing the repair again rate increases the coverage rate; on the contrary, the increase in test success decreases it, which is logical.


## 3.3.5. Obsolescence Probability

The different curves in Figure7 show the effects of the model rates on the probability of obsolescence. Curve (1) in Figure 7 shows the effects of delay rates and maintenance service demand on the obsolescence probability. This probability decreases with a large delay rate (i.e., the delay is small); however, small delay rate values give large probability values. The demand rate has a slight influence on the probability, and this influence is noticed with the small values of the delay rate (large delay). With large demand rate values (low demand) and small delay rate values (large delay), the obsolescence probability is equal to 1.Figure 7
Obsolescence probability. (a) Variation of demand and delay rates. (b) Rate variation of unexpected failure and maintenance interventions. (c) Repair rate variation for corrective and preventive maintenance. (d) Test rate variation (repair again or not).
(a)(b)(c)(d)Then, curve (2) represents the effects of the variation in the unexpected failure and planned intervention rates. We note, on the one hand, that the minimum probability of obsolescence is obtained in the area of the large preventive intervention rate and the low rate of unexpected failure, although the maximum value is obtained in the area of the low preventive intervention rate and high unexpected failure rate (i.e., low failures). On the other hand, the remark retained is that the unexpected failure rate influences more than the preventive intervention rate on the obsolescence probability.On the contrary, curve (3) represents the effects of the variation in repair rates for corrective and preventive maintenance. In this case, we note that the corrective and preventive repair rates influence in the same way on the probability. The maximum value of the probability is when both rates have large values. Finally, from curve (4), we observe the effects of the variation in repair test rates. We notice that(1)
Increasing the test success rate helps to decrease the probability, against increasing the repair again rate(2)
The maximum value takes place with large values of the repair again rate and small values of the test success rate (i.e., the success of the test is great, and repeating the test is very low), which is logical


## 3.3.6. Interventions’ Frequency

Interventions for Unexpected Failures. The different curves in Figure 8 show the effects of model rates on the frequency of inventions due to unexpected failures. Curve (1) in Figure 8 shows the effects of delay and demand rates on the unexpected failure frequency. Firstly, we note that the delay rate does not influence the unexpected failure frequency. This frequency has a maximum value when the demand rate is low (i.e., the demand is high) which makes sense. Thus, it decreases with an increase in the demand rate (i.e., a decrease in the service demand).Figure 8
Interventions for unexpected failures. (a) Variation of demand and delay rates. (b) Rate variation of unexpected failure and maintenance interventions. (c) Repair rate variation for corrective and preventive maintenance. (d) Test rate variation (repair again or not).
(a)(b)(c)(d)Then, curve (2) represents the effects of the variation in the unexpected failure and planned intervention rates. The unexpected failure frequency is random according to the unexpected failure and preventive maintenance rates. It is similar to the sinusoidal shape with variable amplitude. The unexpected failure rate has a remarkable influence on the frequency.On the contrary, curve (3) represents the effects of the variation in repair rates for corrective and preventive maintenance. The conclusion reached is that repair rates influence the frequency in the same way. Finally, from curve (4), we observe the effects of the variation in repair test rates. We notice that(1)
The increase in the test success rate helps to decrease the unexpected failure frequency; on the contrary, the increase in the repair again rate increases the probability(2)
The maximum value takes place with large values of the repair again rate and small values of the test success rate (i.e., the test success is great, and repair again is very low), which is logicalPreventive Interventions. The different curves in Figure 9 show the effects of the model rates on the preventive intervention frequency. Curve (1) in Figure 9 shows the effects of delay and demand rates on this frequency. We note the same remark of the unexpected failure frequency: the demand rate also influences the preventive intervention frequency against the delay. This influence is focused on the area of small values of the demand rate.Figure 9
Preventive interventions. (a) Variation of demand and delay rates. (b) Rate variation of unexpected failure and maintenance interventions. (c) Repair rate variation for corrective and preventive maintenance. (d) Test rate variation (repair again or not).
(a)(b)(c)(d)Then, curve (2) represents the effects of the variation in the unexpected failure and planned intervention rates. The preventive maintenance rate influences in a way that it increases the frequency. However, the unexpected failure rate increases the frequency decreases.On the contrary, curve (3) represents the effects of the repair rate variation for corrective and preventive maintenance. Both rates increase the frequency in the small value area. The frequency stabilizes at the levels of the large rate values.Finally, from curve (4), we observe the effects of the repair test rate variation. The repair again rate increases the preventive intervention frequency. The latter also increases if the test success rate increases (the test success is low) which is logical.


## 3.3.7. Repair Frequency

Corrective Repair. The different curves in Figure 10 show the effects of model rates on the corrective repair frequency. Curve (1) in Figure 10 shows the effects of delay and demand rates on this frequency. The curve of this frequency is similar to the unexpected failure frequency, which is logical. The difference is in the demand rate interval which is restricted in this case compared to the unexpected failure frequency. At this interval, the frequency values are large and stable.Figure 10
Corrective repair frequency. (a) Variation of demand and delay rates. (b) Rate variation of unexpected failure and maintenance interventions. (c) Repair rate variation for corrective and preventive maintenance. (d) Test rate variation (repair again or not).
(a)(b)(c)(d)Then, curve (2) represents the effects of the variation in the unexpected failure and planned intervention rates. The unexpected failure rate has a remarkable influence on the corrective repair frequency. The preventive intervention rate also influences the frequency but in a way less than the unexpected failure rate. On the contrary, curve (3) represents the effects of the repair rate variation for corrective and preventive maintenance. We notice that the rates increase the corrective repair frequency in the small value area of one of the rates. On the contrary, the frequency stabilizes for large values. The conclusion reached is that these rates increase the frequency corrective repair.Finally, from curve (4), we observe the effects of the repair test rate variation. The first remark retained is that the corrective repair frequency knew significant values in the area of large values of the test success rate (i.e., test success is low) and small values of the repair again rate (i.e., the possibility of redoing the repair is important) which is logical. The frequency decreases with increasing the repair again rate (i.e., decreasing the possibility of redoing the repair). In the area of small values of the test success rate (i.e., the test success is high), we notice that the frequency has small values and is not influenced by the repair again rate.Preventive Repair. The different curves in Figure 11 show the effects of the model rates on the preventive repair frequency. Curve (1) in Figure 11 shows the effects of delay and demand rates on this frequency. We note that the demand rate has a big influence on the frequency. Small demand rate values (i.e., large demand values) give large repair frequencies, which makes sense. Thus, we note that the demand rate interval, which gives these values, is narrower compared to the case of the corrective repair frequency.Figure 11
Preventive repair frequency. (a) Variation of demand and delay rates. (b) Rate variation of unexpected failure and maintenance interventions. (c) Repair rate variation for corrective and preventive maintenance. (d) Test rate variation (repair again or not).
(a)(b)(c)(d)Then, curve (2) represents the effects of the unexpected failure and planned intervention rates’ variation. The conclusion reached is that the preventive intervention rate has a remarkable influence on the frequency, which is logical.On the contrary, curve (3) represents the effects of the repair rate variation for corrective and preventive maintenance. The preventive repair frequency has the same curve as the corrective repair frequency. The preventive repair frequency increases at the level of small rate values; on the contrary, it stabilizes at the level of large rate values.Finally, from curve (4), we observe the effects of the repair test rate variation. We note that the repair again rate influences remarkably the preventive repair frequency. The rates influence this frequency the same way as the corrective repair frequency.


## 3.3.8. Repair Quality

Good Repair. The different curves in Figure 12 show the effects of the model rates on the good quality of repair. Curve (1) in Figure 12 shows the effects of delay and demand rates on this quality. We notice the demand rate influences the good repair; on the contrary, the delay rate has no effect on this quality. Then, curve (2) represents the effects of the unexpected failure and planned intervention rates’ variation. The maximum values for good repair are in the area of low preventive intervention rate. The increase in the preventive intervention rate (decrease in preventive interventions) decreases the amount of good repair. On the contrary, the increase in the unexpected failure rate increases the amount of good repair.Figure 12
Good repair quality. (a) Variation of demand and delay rates. (b) Rate variation of unexpected failure and maintenance interventions. (c) Repair rate variation for corrective and preventive maintenance. (d) Test rate variation (repair again or not).
(a)(b)(c)(d)On the contrary, curve (3) represents the effects of the repair rate variation for corrective and preventive maintenance. The curve looks like a few indicators based on repair rates. There is an increase in the good repair at the small values of one of the rates, and the stability of the indicator for the rest of the rates changes. Finally, from curve (4), we observe the effects of the repair test rate variation. The small values of the success test rate (success of the test is very low) give a large amount of good repair, which is close to reality. So, increasing the success test rate decreases the amount of good repair; however, when the repair again rate increases (i.e., redoing the repair is low), the good repair increases.Bad Repair. The different curves in Figure 13 show the effects of model rates on bad repair quality. Curve (1) in Figure 13 shows the effects of delay and demand rates on this quality. This appearance is similar to the case of the good repair amount. So, the demand rate influences the bad repair amount.Figure 13
Bad repair quality. (a) Variation of demand and delay rates. (b) Rate variation of unexpected failure and maintenance interventions. (c) Repair rate variation for corrective and preventive maintenance. (d) Test rate variation (repair again or not).
(a)(b)(c)(d)Then, curve (2) represents the effects of the unexpected failure and planned intervention rates’ variation. Increasing the preventive intervention rate decreases the poor repair amount. However, the increase in the unexpected failure rate increases the quality. Therefore, preventive maintenance improves the quality of repair.On the contrary, curve (3) represents the effects of the repair rate variation for corrective and preventive maintenance. The curve looks like a few indicators based on repair rates. There is an increase in poor repair at the small values of one of the rates, and the indicator stability for the rest of the rates changes.Finally, from curve (4), we observe the effects of the repair test rate variation. The first remark is that the appearance of poor repair varies inversely with that of good repair. The small values of the repair again rate make a large amount of poor repair. So, increasing the test success rate increases the bad repair amount, but at the repair again rate, when it increases (i.e., repair again is low), the bad repair decreases.


## 4. Numerical Case Study

By finalizing this work with a case study in which all the parameters are fixed, we determine the state probability distribution and the indicators. The parameters considered in this study are presented in Table5, and the results obtained are summarized in Tables 6 (distribution vector) and 7 (indicators).Table 5
The case study parameters.
ParametersValueλ14λ22λ60.075λ110.01λ122μ90.34μ100.9μ140.13Table 6
Vector of the probability distribution of states: case study.
π00.022889338945166487873277817877829π10.00034413485289087115448365608128819π20.011444669472583243936638908938914π30.00048099402568446831790987020791476π40.35214367607948442881965873658198π50.27494509378502955924977399239793π60.045778677890332975746555635755658π70.10272673833726379136804742573109π80.00020545347667452758273609485146219π90.092454064503537412231242683157984π100.0024170997255826774439540570760257π110.00090309220516275860543338396247115π120.00081278298464648274489004556622404π130.0000018061844103255172108667679249423π140.00002124922835677079071607962264638π150.0000079392721332989767510627161535925π160.0000071453449199690790759564445382333π170.000000015878544266597953502125432307185π180.00000018680640313644651178971096831982π190.000000069795798974056938470881021130484π200.00000000013959159794811387694176204226097π210.000000062816219076651244623792919017435π220.000000001642254093507222081667788732482π230.0000000006135894415301708876560968890592π240.00000000027611524868857689944524360007664π250.00000000000087655634504310126808013841294171π260.0000000000017531126900862025361602768258834π270.022889338945166487873277817877829π280.069526366769827958226504472358662Table 7
Indicators: case study.
IndicatorsExpressionAverage stockSmoySmoy=4.5668073502421514796131061334359Average storage costCSmoyCSmoy=Cs∗4.5668073502421514796131061334359Probability of the empty stockProbS=0ProbS=0=0.092759840569638429944352148853939Frequency of supplyFAmoyFAmoy=0.18551968169326046995594429913453Average ordering costCCmoyCCmoy=Cc∗0.18551968169326046995594429913453Average purchase costCAmoyCAmoy=Ca∗0.18551968169326046995594429913453Coverage rateTCTC=18.267229400968605918452424533744Average cost of inventory outageCRmoyCRmoy=Cru∗0.092759840569638429944352148853939Probability of having a dead stockProbDV=0ProbDV=0=0.06987050162271882938098812843995Cost of obsolescenceCOmoyCOmoy=Cob∗0.06987050162271882938098812843995Unplanned failure frequencyFPImoyFPImoy=0.0017167004208874865904958363408372Planned intervention frequencyFIPmoyFIPmoy=0.045778677890332975746555635755658Repair frequencyCorrective interventionFRcmoyFRcmoy=0.0017187551158443560752343020927045Preventive interventionFRpmoyFRpmoy=0.022889338945166487873277817877829Intervention durationDimoyDimoy=Diu∗0.0246080940610108439485121199705335Intervention qualityGood repairQBRmoyQBR,moy=0.093274056201553438083607107852153Bad repairQMRmoyQMR,moy=0.013472919229113274389854360023089Intervention costsCImoy=Cr+Cth∗Diu∗0.0246080+Cp∗0.047495378+Cvpu∗Qh∗Diu∗0.0246080940According to Table8, some indicators in the form of costs are based on unit costs which can be variable or constant depending on the domains and parameters linked to the system. For the other indicators, we note the average stock indicator, the empty stock, and obsolescence probabilities have the low values. Also, the average coverage rate, the unexpected failures, and corrective repair frequencies have the low values; however, the preventive intervention frequency has high value and finally the repair quality indicators have high medium value.Table 8
The rates that influence the indicators.
IndicatorsRatesDemand rateDelay rateUnexpected failure rateRate of preventive maintenance interventionsRepair rate for corrective maintenanceRepair rate for preventive maintenanceRate to repair againTest success rateAverage stockSmoy∗∗∗∗∗Probability of the empty stockProbS=0∗∗∗∗∗Frequency of supplyFAmoy∗∗∗∗∗∗Coverage rateTC∗∗∗∗∗Probability of having a dead stockProbDV=0∗∗∗∗∗Unplanned failure frequencyFPImoy∗∗∗∗∗Planned intervention frequencyFIPmoy∗∗∗∗∗Repair frequencyCorrective interventionFRcmoy∗∗∗∗∗Preventive interventionFRpmoy∗∗∗∗∗Intervention qualityGood repairQBRmoy∗∗∗∗∗Bad repairQMRmoy∗∗∗∗∗


## 5. Analysis and Synthesis

We summarize and compare the rates that influence the indicators:(1)
The demand rate influences more the majority of the indicators compared to the delay rate which influences the probabilities. The difference between the majority of indicators in this case is in the demand rate interval in which the indicators have large values.(2)
The two repair rates influence all the indicators in the same way.(3)
The preventive intervention rate influences the average stock, the coverage rate, and the preventive intervention and repair frequencies. The large values of this rate increase these indicators. For the other indicators, they are influenced more by the unexpected failure rate which increases the values of the indicators.(4)
The test success rate influences, in order to increase the values of the indicators, on the corrective intervention and repair frequencies and the indicator of bad repair. For the others, they are influenced by the repair again rate.Table8 provides a summary of the rates that increase the indicator values. Of course, the rates, which decrease the indicators, are not mentioned by (∗) in the table. This summary makes it easier to determine the rates on which we must react to improve the performance.Finally, we noticed that preventive maintenance improves the performance of joint management of spare parts and maintenance. The risks of stock-out and obsolescence are influenced by the delay rate. To minimize them, it is necessary to have a large value of the delay rate (i.e., a small value of the delay), which is logical.The difference between the model proposed in [21] and this model of joint management is found at the level of the part added in the graph on maintenance which had to integrate its parameters (failure and repair rate) in the construction of performance indicators, i.e., its expressions are different between the two models. The indicators’ expressions, in joint management, are based on the supply and maintenance parameters; on the contrary, in the management of spare parts, their expressions are based only on the supply parameters. So, their behavior will also be different and depend, in this paper, on the parameters of supply and maintenance according to the type of maintenance. Thus, the evaluation of maintenance by indicators is not considered in [21]. By comparing the two models, we can conclude that the maintenance consideration has an influence on the management of the spare parts in a way to minimize the risks and the stock even if we will additional costs related to maintenance. These costs can be minimized along with the costs of spare parts by finding a combination of procurement and maintenance policies.


## 6. Conclusion

In this paper, we have treated the performance evaluation of joint management of spare parts and maintenance through indicators that are developed from our model presented in our previous works. The indicators are evaluated in the form of a presentation of the rate effects (model parameters) on the proposed indicators. This presentation is made in the form of cases by varying two rates that we want to see their effects on the indicators, and the other parameters are fixed. From this analytical study, we were able to determine the parameters that have a great influence on our considered model.Among these parameters, the demand rate influences all the indicators. Preventive maintenance improves joint management. To avoid and eliminate the risks, it is necessary to intervene on the delay rate. Of course, maintenance management is always linked to production in addition to the spare parts’ management. Corrective maintenance will take place when the asset is stopped (broken down). For preventive maintenance, interventions with production must be well planned.Certainly, the maintenance and spare parts are an important part of the production and necessary to continue the service. So, we want to assess in our future works the link between maintenance and production with spare parts’ management. On the other way, we will treat this axis by integrating the production plan with the joint management of maintenance and spare parts.



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*Source: 2950789-2020-10-28.xml*