Reliability centered maintenance using System Dynamics Approach Hadi A. Khorshidi
Indra Gunawan, M. Yousef Ibrahim
School of Applied Science and Engineering Faculty of Science, Monash University Australia
[email protected]
School of Engineering and Information Technology Federation University, Gippsland campus Churchill, Australia
[email protected] [email protected]
Abstract—This paper presents a system dynamic modeling technique for reliability centered maintenance (RCM). It provides a simulation model to analyze the impact of maintenance strategies such as preventive and corrective maintenance on system availability. Also, maintenance cost is associated with the model to have an optimization view for a manufacturing system. As a result, the optimal decisions can be made based on the results of the model. In addition, sensitivity analysis has been conducted for various parameters in the simulation model. The results were compared and discussed for various conditions. Keywords— Reliability centered maintenance (RCM), System Dynamics; Maintenance strategies; Reliability value; Optimization; System evaluation
I.
INTRODUCTION
Reliability centered maintenance (RCM) is defined in IEC standard [1] as “method to identify and select failure management policies to efficiently and effectively achieve the required safety, availability and economy of operation”. RCM is a systematic decision making process to keep balance between preventive and corrective maintenance actions and select cost-effective maintenance plans in order to improve the reliability [2]. Also, reliability has been widely known as a critical criterion for system design, operation, and maintenance [3]. System dynamics approach is used in different areas such as social science, economics, politics, engineering, manufacturing, and management to simulate the system interactions. Also, its capability in capturing dynamic behavior of complex operations has been verified [4]. In this paper, RCM is modelled by system dynamics’ tools to employ its ability for reliability evaluation through using various maintenance strategies. In fact, the dynamic behavior of the system reliability is simulated through system dynamics’ package. The system that is considered is a binary system in which there are functioning and failure states. An example is suggested to investigate the simulated model. Software iThink 9.0.3 is used to model the system. II.
SYSTEM DYNAMICS
System dynamics is first introduced by Prof. Forrester [5] in 1950s in Massachusetts Institute of Technology (MIT). This methodology is based on the system thinking concept to find
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the causal relationship between the effective elements. It models a system over time which is the main variable to define the dynamical behavior of the system. The system dynamics’ art is to represent feedback process by Causal Loop Diagrams (CLDs) and stock and flow structures to predict the future behavior of the system [6]. A. Causal loop diagram CLD is a useful way to capture the system structure, and feedback processes [7]. In a causal diagram, elements are connected by arrows to denote the impacts on each other. A positive arrow shows element A influences positively on element B where B is increased or decreased by increasing or decreasing in A respectively. Likewise, a negative arrow shows a negative relationship in which element B is increased or decreased inversely by decreasing and increasing in element A respectively [8]. These arrows can construct a CLD to represent feedback process among elements through causal loops. The causal loops also can be positive (reinforcing) or negative (balancing). There is an even number of negative arrows in a positive loop which is associated with an exponential growth. On the other hand, there is an odd number of negative arrows in a negative loop which tends to reach a balance point [9]. After developing a CLD, causal relationship should be converted into stock and flow structure. B. Stock and flow structure Stock and flow structures are another tool to represent system structure and feedback process. In these structures, mathematical equations are used for causal relationships to simulate the system behavior [6, 8]. Stock (level) is a variable that represents the quantity at one specific time which has been cumulated since the past. However, flow is a rate variable that shows change rate over a time interval [5]. In the software, stock is shown by a rectangle and flow is represented by a valve that might go in or out of the stock. Also, there are some auxiliary variables which are effective on the stock and flow variables. They are shown by circles in the simulation model.
III.
MODELING
First of all, the probability that a system is in functioning state which is known as availability should be investigated. As it is mentioned before, the system is in binary state. Figure 1
stable situation. This trend is like as beginning periods of life in bathtub curve [12], and goal seeking pattern of behavior in system dynamics [6]. 1: Av ailability 1:
1
λ10 1
1
0 1:
µ01
1
1 1
Figure 1. State transition in the system
shows how a system transits from working state (state 1) to failed state (state 0) and vice versa where λ10 is the failure rate, and µ01 is the repair rate. The probability of being in each state (j) at time t (Pj(t)) follows the differential equations (1) and (2) [10, 11]. dP0(t)/dt = -μ01× P0(t)+λ10× P1(t)
(1)
dP1(t)/dt = - λ10× P1(t)+μ01× P0(t)
(2)
where P1(t) shows the availability, and P1(t)=1- P0(t) because the system is binary. Also, initial conditions of the system are P1(t)=1, P0(t)=0. This process can be demonstrated by CLD as figure 2. Failure rate
Repair rate + Function
1
0.00
3.00
1
6.00 Time
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Av ailability trend
Figure 4. Availability trend Time
TABLE 1. AVAILABILITY MEASURE IN TIME PERIODS 0 1 2 3 4 5 6 7 8 9
A
1
.91
.86
.83
.81
.8
.79
.79
.78
where A is availability, and Initial A means initial value of the availability. Preventive Maintenance
Failure
Failure rate
Repair rate
Figure 2. Causal loop for availability
As can be seen, failure and function have negative and positive effects on availability respectively. However, availability positively influences on these two elements according to Eq. 2. Also, there are two causal loops. The loop between function and availability is positive, and the loop between failure and availability is a negative one. In addition, increasing the failure and repair rates leads to increase in failure and functioning states respectively. The stock and flow structure can be developed based on CLD as figure 3.
.78
A(t) = (μ01× (1-A(t-1))- λ10× A(t-1))×dt; Initial A=1 (3)
Corrective Maintenance
-
.78
10
Model equation is used in iThink package for availability is as Eq. 3.
+
Availability
+
1:
+ Function
+
Availability
+
Failure
-
Figure 5. Causal loop for improvement strategies
There are some improvement strategies which influence on availability. These strategies, that are also known as reliability/availability allocation, can reduce failure rate or
Correctiv e Maintenance Action
Failure rate
Repair rate
Prev entiv e Maintenance Action
Av ailability ~
ef f ect of PM
ef f ect of CM
Function
Failure
Initial repair rate
Initial f ailure rate
Figure 3. Stock and flow structure for availability
The model is run with failure rate equal to 0.1 and repair rate 0.35. Figure 4 shows availability variation over time. Also, TABLE 1 provides availability measures for 10 time periods. Accordingly, the availability of the system decreases to reach a
Av ailability
Function
1933
Failure rate
Repaire rate
Failure
Figure 6. Stock and flow structure for improvement strategies
increase repair rate [13]. In figure 5, the impact of corrective and preventive maintenance as improvement strategies on repair and failure rates respectively is shown as CLD. In corrective maintenance, the mean time to repair (MTTR) can be reduced by allocating more resource and manpower in repair division. Since repair rate and MTTR have an inverse relationship (µ=1/MTTR), repair rates will be influenced. For example, if the allocated resource and manpower for repair is doubled, it leads to shorten MTTR by half, and the repair rate is multiplied by 2. Consequently, there is a linear relationship between corrective maintenance action and repair rate. Also in preventive maintenance, there are replacement policies and technical actions that increase the mean time between failures (MTBF). Since failure rate and MTBF have an inverse relationship (λ=1/MTBF), repair rates will be reduced. Figure 6 shows these relationships in stock and flow structure. The elements as effect of CM and PM quantify the impact of changing on corrective and preventive maintenance respectively which is shown by equations 4 and 5. Effect_of_CM = Corrective_Maintenance_action
level. TABLE 2 gives availability measures for the scenarios through 10 time periods. It shows that the stable point for the scenarios is different. Time
TABLE 2. AVAILABILITY MEASURE FOR SCENARIOS 0 1 2 3 4 5 6 7 8 9
(2)
1
.93
.9
.88
.88
.88
.88
.88
.88
.88
.88
(3)
1
.94
.9
.87
.85
.84
.84
.83
.83
.83
.83
(4)
1
.94
.92
.91
.91
.9
.9
.9
.9
.9
.9
According to figure 7, using more improvement actions leads to the higher availability value. However, employing maintenance strategies impose cost on the system. As a result, an optimization viewpoint is useful to make decision on applying improvement strategies. System availability and system cost should be considered concurrently in evaluating effect of maintenance strategies.
(4)
Effect_of_PM = 1-0.005×Preventive_Maintenance_action(5) where Corrective_Maintenance_action is represented by a number denotes how many times repair resources increase or decrease, and Preventive_Maintenance_action is represented by a percentage of replacement has been done. Also, repair and failure rates are updated by equation 6 and 7. Repair_rate = Initial_repair_rate × Effect_of_CM
(6)
Failure_rate = Initial_failure_rate × Effect_of_PM
(7)
Four scenarios have been considered. The first scenario shows no change on corrective and preventive maintenance policy. In the second one, the allocated resource and manpower for corrective maintenance becomes twice, it means Corrective_Maintenance_action is equal to 2. In the third scenario, there is fifty percent replacement rate that it means Preventive_Maintenance_action is 50 percent. The last scenario includes two changes simultaneously. As it can be seen in figure 7, improvement actions promote the availability
Preventive Maintenance
Corrective Maintenance
+
+ Function
+ Failure
Availability
+
Net value
Cost
Figure 8. Causal loop in optimization view
CM cost
Correctiv e Maintenance Action Prev entiv e Maintenance Action ~
PM cost
ef f ect of PM
ef f ect of CM Initial repair rate
Initial f ailure rate
Failure rate
Repaire rate Av ailability : 1 - 2 - 3 - 4 -
Maintenance cost
1
Transition to f ailure
Transition to working Av ailability
1 2 3 1:
+
Maintenance Cost
Failure rate
Repair rate
where initial amounts of repair and failure rates are 0.35 and 0.1 respectively as previous.
1:
4 4
1
4
2
Function
4
2
Failure
2
3 3
1 1 1: Page 1
1
0.00
10
3.00
6.00 Time
3
Reliability v alue
1
9.00 12.00 5:39 PM Tue, 11 Mar 2014
Net v alue
Perf ormance rate
Av ailability trend
Figure 9. Final stock and flow structure
Figure 7. Availability trend in four scenarios
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Cost
Figure 8 adds cost in CLD of the whole system. In this view, net value is a criterion which is influenced by availability and cost, and it can be considered to analyze improvement actions. As it is presented, corrective and preventive maintenance actions have both positive and negative impacts on net value via availability and cost respectively. This CLD can be transferred into stock and flow structure as shown in figure 9. For corrective maintenance, there is a linear relationship with slope 2 as Eq. 8. If the allocated resource and manpower for repair becomes twice, the CM cost becomes double to be 4, and similarly if repair resource be shorten by half, the CM cost becomes half to be 1. However, there is an exponential relationship between preventive maintenance and its cost, it means that the PM cost increases exponentially by increasing in the replacement percentage as Eq. 9. CM_cost = 2×Corrective_Maintenance_action
not reasonable. The cost of the scenarios 1-4 are 3, 5, 3.65, and 5.65 respectively. Net value amounts for the scenarios are brought in TABLE 3 through 10 time periods. Time
0
Maintenance cost is sum of CM and PM costs, and the cost is equal to the maintenance cost. On the other hand, reliability value which is employed in [14-17] is used to make the availability homogeneous with the cost. In manufacturing systems, the system can generate income as much as it is available. Therefore, there is a relationship between availability and income generation. It provides an opportunity to translate system availability into money. As a result, the performance rate is considered as the income that can be generated during a time period when the system is available. In this model, it is assumed 25. Also, the reliability value is calculated by Eq. 10. Reliability_value = Availability×Performance_rate (10)
22 19.95 18.59 17.68 17.09 16.69 16.43 16.25 16.13 16.06 16.01
(2)
20 18.10 17.11 16.59 16.32 16.18 16.11 16.07 16.05 16.04 16.03
(3) 21.35 19.78 18.71 17.98 17.49 17.15 16.92 16.77 16.66 16.59 16.54 (4) 19.35 17.90 17.12 16.70 16.47 16.35 16.29 16.25 16.24 16.23 16.22
After time period 2, scenario 3 can generate the most net value for the system. The area below the diagram of each scenario shows which scenario is the best one to select by decision makers as Eq. 12. The higher amount of area denotes the better scenario for the system. Area = ∫Net_value(t)dt =∑ Net_value(t)×Δt
(12)
According to this, the area value for the scenarios through 10 time periods is as follow: scenario (1)=192.88; scenario (2)=184.6; scenario (3)=196.39; scenario (4)=185.12. Therefore, the scenarios have been ranked based on the net value in the order of 3, 1, 4, and 2. Scenario (3) is the best one i.e. improving preventive maintenance is the optimum strategy in this case. Finally, two different conditions on system cost have been analyzed. These two are if the cost of both corrective and preventive maintenance actions have been increased in double or reduced in half. These variations might happen in system cost based on salary and component price changes. Therefore, Net Value: 1 - 2 - 3 - 4 -
As a result, reliability value is obtained as income which is money like cost. Therefore, reliability value can be differentiated from cost to have net value of the system as Eq. 11.
1:
Net_value = Reliability_value - Cost
1:
(11)
10
(1)
(8)
PM_cost = EXP(0.01×Preventive_Maintenance_action) (9)
TABLE 3. NET VALUE MEASURE FOR SCENARIOS 1 2 3 4 5 6 7 8 9
19 1
3 15
1 3
1
2
1
3
3
Net Value: 1 - 2 - 3 - 4 1:
23
2
4 1:
1
11
0.00
2
3.00
Net Value: 1 - 2 - 3 - 4 -
2
1: 1
2
Page 1
16
0.00
3.00
24 1
3
4
1:
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Figure 11. Net value trend when cost is double
3
19
4 6.00 Time
Page 1
1:
2
4
4
1
3 2
6.00 Time
4
3 1
2
4
9.00 12.00 4:50 PM Wed, 19 Mar 2014
2 1:
Av ailability trend
21
3 4 4 2
Figure 10. Net value trend in four scenarios
1
This value is helpful to find out which scenario is appropriate for the system based on reliability and cost. Figure 10 provides a comparison view on net value for abovementioned scenarios. As it can be seen, the scenario 4 is not the best one when cost is added in evaluation. Because the value that scenario 4 generates in comparison with its cost is
1935
4
1:
18
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3.00
2 3
1
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4
2
3
6.00 Time
3 1 9.00 12.00 4:23 PM Mon, 24 Mar 2014
Figure 12. Net value trend when cost is half
the impact of these variations on four scenarios, and subsequently on optimal decision can be investigated. Net value trend of the scenarios for cost increase is shown in figure 11. In this particular case, doing nothing is preferred as optimal decision because improvement strategies impose much more cost on the system. Also, net value trend for cost reduction is presented in figure 12. In this condition, doing nothing is not reasonable. As you can see, the 4th scenario is the optimal solution after time period 3 because the improvement strategies are not too much costly anymore. These two conditions show the impact of cost and price on system operation. IV.
CONCLUSION
In this paper, a new methodology was introduced using system dynamics approach to model maintenance plans in terms of reliability. In this methodology, a CLD is drawn to present the causal relationship between the main elements. Also, a stock and flow structure is developed to simulate the proposed system. Both CLD and stock and flow structure are introduced in the context step by step. As it is mentioned in introduction, RCM aims to make decision on corrective and preventive maintenance actions in an economic way in order to improve system reliability. Accordingly in this study, various maintenance strategies have been applied in the model, and their impact is analyzed on the system’s availability and cost. As a result, the strategies have been ranked in an optimal view to identify the best strategy. This paper contributes to better understanding of the dynamic behavior of system reliability via system dynamics methodology. It allows all players, such as operation managers and maintenance managers, to find out how they influence on the dynamic behavior of the entire system [18]. Benefit of system dynamics models lies in using simple diagrams and their underlying mathematical expressions to model system behavior. This approach removes many of the ambiguities stem from the conventional decision analysis and system evaluation techniques. Therefore in this paper, a visual simulation model is developed that provides an opportunity to consider more effective parameters simply in system modelling. Since system dynamics approach has the capability to simulate complex systems, more elements which are effective on system reliability can be added to the model for further investigation. Also, through this methodology, redundancy allocation can be applied for system reliability improvement. REFERENCES [1] IEC. Dependability management - Part 3-11: Application guide Reliability centred maintenance 1999, IEC standards 60300-3-11. [2] Yssaad, B., Khiat, M. & Chaker, A., Reliability centered maintenance optimization for power distribution systems,
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