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Opportunistic policy for optimal preventive maintenance of a multi-component system in continuous operating units
Computers and Chemical Engineering 33 (2009) 1499–1510
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Computers and Chemical Engineering journal homepage: www.elsevier.com/locate/compchemeng
Opportunistic policy for optimal preventive maintenance of a multi-component system in continuous operating units Radouane Laggoune a,∗ , Alaa Chateauneuf b , Djamil Aissani a a b
Laboratory of Modelisation and Optimisation of Systems (LAMOS), University of Bejaia, Targa Ouzemour, 06000 Bejaia, Algeria Laboratory of Mechanics and Engineering (LaMI), Université, Blaise Pascal, Campus des Cézeaux, BP 206, 63174 Aubière Cedex, France
a r t i c l e
i n f o
Article history: Received 31 May 2008 Received in revised form 19 December 2008 Accepted 10 March 2009 Available online 21 March 2009 Keywords: Opportunistic maintenance Multi-component system Economic dependence Production loss Monte Carlo
a b s t r a c t For continuous operating units such as petrochemical plants, the production loss due to downtime is high, and the economic profitability is conditioned by the implementation of suitable maintenance policy that could increase the availability and reduce the operating costs. In this paper, a preventive maintenance plan approach is proposed for a multi-component series system subjected to random failures, where the cost rate is minimized under general lifetime distribution. The expected total cost is given by corrective and preventive costs, which are related to the components, and by the common costs related to the whole system, especially the production loss during system shutdown. When the system is down, either correctively or preventively, the opportunity to replace preventively non-failed components is considered. A solution procedure based on Monte Carlo simulations with informative search method is proposed and applied to the optimisation of the component replacement for the hydrogen compressor in an oil refinery. © 2009 Elsevier Ltd. All rights reserved.
1. Introduction For continuous operating units, the production loss is often very large when unexpected shutdown occurs. Their economic profitability implies the implementation of suitable maintenance policy which could increase the availability and reduce the operating costs. To underline the consequences of unavailability, it can be mentioned that production losses in chemical plants can range from $5000 to $100,000 per hour (Tan & Kramer, 1997). For refineries, the total production losses soar to millions of dollars (Nahara, 1993). In addition, the safety requirements enforce to decrease the failure probability, on a very low level in order to avoid disastrous consequences. The preventive maintenance (PM) is often carried out to prevent or to slow down the deterioration processes. PM is a scheduled downtime, usually periodical, in which a well-defined set of tasks (e.g., inspection, replacement, cleaning, lubrication, adjustment and alignment) are performed. In oil refining facilities, the problems associated with part replacement are more concerned than other routine maintenance activities such as cleaning and lubricating, from the PM scheduling point of view. This is because the direct costs due to part failure and replacement are usually very high, and the impact of different replacement intervals on the overall main-
∗ Corresponding author. Tel.: +213 34 21 51 88; fax: +213 34 21 51 88. E-mail address: r [email protected] (R. Laggoune). 0098-1354/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.compchemeng.2009.03.003
tenance cost is often very sensitive and significant, in addition to the safety requirements. In a series system, the one-by-one preventive replacements of components improve the global system reliability on the account of its availability, which would be largely penalized, because of frequent shutdowns for component replacements. For multi-component systems, an optimal maintenance policy must take into account the interactions between the various components of the system. These interactions are of three types (Thomas, 1986): economic dependence, structural dependence and stochastic dependence. We are mainly concerned by the economic dependence reflecting the influence of component operation/maintenance costs on the overall system costs; in this case, saving in costs or downtime can be achieved when several components are jointly maintained. The objective of this paper is to develop a preventive/ corrective/opportunistic maintenance plan for a multi-component system subjected to high production losses and economic dependence. In the next section, we review the relevant literature, particularly that dealing with multi-component systems. In Section 3, we provide the cost formulation and the maintenance models in several cases; an algorithm allowing for combined preventive/corrective/opportunistic replacement of the system components is also presented. In Section 4, the proposed approach is illustrated by a simple example with two components, allowing to explain the formulation interest and to verify the convergence of the solution procedure. An industrial application is provided in
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Nomenclature
Section 5, where the results show the effectiveness of the proposed approach for practical systems.
C0c
2. Literature review
corrective common cost related to the system, to be paid at each repair upon failure p C0 preventive common cost related to the system, to be paid at each time the system undergoes a preventive maintenance. Cic specific corrective cost, to be paid at each replacement upon failure of component i p specific corrective cost, to be paid at each preventive Ci replacement of component i c Csyst,j expected total corrective cost of the whole system due to failure of component j p Csys expected total preventive maintenance cost of the whole system Ci ( i ) cost rate for component i (objective function of the mono-component policy) Cmono () cost rate for the mono-group policy (objective function) C(,k1 ,. . .,kq ) cost rate for the opportunistic multi-grouping policy (objective function) basic preventive maintenance interval ( = min i ) i=1,...,q
i ki i
subscripts indicating components, i = 1, 2, . . ., q integer multiplier of component i time interval (age) between preventive replacements of component i ( i = ki ) q number of system components K the least common multiple of all ki k kth scheduled time instant where preventive replacements take place, k = 1, 2, . . ., K i0 initial optimal replacement time interval of component i ki0 initial Integer multiplying factor of component i ti simulated lifetime of component i tj time instant of failure of component i simulated time instant of system failure due to the ti weakest component N total number of simulations s subscript indicating the simulation number IFk,s , IRk,s ∈ [0, 1] binary variables indicating the states of failure or operation, respectively Gp group (set) of components to be preventively replaced at a scheduled time instant group of non-failed components to be opportunistiGh cally replaced during a failure group of components to be preventively replaced at Gpk the kth scheduled time instant Gph group of components to be opportunistically replaced at the kth interval [(k − 1),k] cumulative distribution failure (CDF) of component Fi (·) i cumulative distribution failure of the whole system Fsys (·) Fsys,j (·) cumulative distribution failure of the system due to component j reliability function of the whole system Rsys (·) MTTF mean time to failure ˇ Weibull shape parameter Weibull scale parameter H2 hydrogen HC hydrocarbon LPG liquefied petrol gas
For multi-component systems, when no strong dependence exists between the different components, the traditional singleunit model developed by Barlow and Hunter (1960) can be independently applied to each unit, in order to provide optimal replacement schedule. However, the general case of multicomponent systems implies to take account for the interactions between various components. The economic dependence is common in most continuous operating systems, such as oil refineries, chemical processing facilities, mass-production manufacturing lines and power generators (Das, Lashkari, & Sengupta, 2007; Vassiliadis & Pistikopoulos, 2001). For this type of systems, the single shutdown cost is often much higher than the cost of the components to be replaced. Therefore, there is a great potential for cost savings by implementing suitable maintenance policy. A number of studies have reviewed the various maintenance policies for multi-component systems (Cho & Parlar, 1991; Dekker, Wildeman, & Van Der Duyn Schouten, 1997; Thomas, 1986; Wang, 2002). These reviews show that most of the authors use simplified assumptions, or deal with particular systems (special structure is assumed), in order to formulate the decision problem with less mathematical difficulty. From another point of view, most of the developed decision models are based on dynamic programming or Markovian approaches (Bâckert & Rippin, 1985; Lam & Yeh, 1994), which approximate continuous decision variables by finite discrete state decision variables. These restrictions in both maintenance policies and model formulations could affect the optimality of the solutions because of the reduction of the solution space. In addition, discrete state decision models are often difficult to apply to systems with large number of components and different failure distributions. The common planning approaches used for multi-component manufacturing systems include the group/block replacement models and the opportunistic maintenance models. In the block maintenance policy, an entire group of units or components are replaced at periodic intervals. The interval is decided based on time, cost or both (Dekker et al., 1997; Kardon & Fredendall, 2002; Tam, Chan, & Price, 2006). Performing maintenance periodically can be a costly option because maintenance activities are carried out even though components are in good condition. The concept of opportunistic maintenance comes from the fact that there is the possibility of dependence between the various components of the system. For example, the cost of simultaneous maintenance actions on various components would be less than the sum of the total cost of individual maintenance actions (Huang & Okogbaa, 1996); this is due to the potential savings in the common cost usually termed “set-up cost” in the literature. This is particularly true in the case of series system, where the failure of any component results in stopping the whole system. Therefore, providing the opportunity to carry out preventive maintenance on some components along with replacement of failed components, leads to very small additional cost, compared to separate replacements. Under these conditions, the maintenance decisions for one component depend on the states (ages) of the other components in the system (Rao & Bhadhury, 2000; Van Der Duyn Schouten & Vanneste, 1990). Several techniques and approaches dealing with multicomponent systems are provided in the specialized literature. Dekker and Roelvink (1995) presented a heuristic replacement criterion which is only useful when a fixed group of components is replaced. Huang and Okogbaa (1996) presented a replacement scheduling approach based on the cost boundary condition; they claim that it is more effective than some of the very sophisticated but improperly formulated “optimal” models. Van Der Duyn
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Fig. 1. Scheduled preventive maintenance plan.
Schouten and Vanneste (1990) investigated (n, N)-strategies for the maintenance of a two component series system. They presented a numerical procedure to compute the optimal values of n and N. In this policy, a single unit undergoes a maintenance action whenever it fails or reaches the age N and the other unit can be included into the maintenance operation if and only if its age exceeds n. The generalization to systems with more than two units of the proposed algorithm is very difficult; this is due to the necessity to have detailed information on the age distribution of all units at each opportunity. Zheng and Fard (1992) proposed also an opportunistic maintenance policy allowing for partial joint replacement. However, the initiation of a joint replacement of a unit is restricted to preventive replacement only; the failures are not included as opportunities to carry out preventive maintenance on the non-failed units. Goyal and Gunasekaran (1992) presented a model for multicomponents with a very special deterioration structure, describing a transport fleet consisting of groups of vehicles; the failed components (vehicles) can be left in the failed condition for some period of time until an appropriate moment for repair arises. This is not possible in the case of a series system such as process units, where the failure of any item leads to the failure of the whole system. Samrout, Yalaoui, Châtelet, and Chebbo (2005) presented a method to minimize the preventive maintenance cost of series-parallel systems using Ant colony optimisation, but their method deals with only the case of constant failure rates. Duarte, Craveiro, and Trigo (2006) proposed an algorithm for maintenance management optimisation of a series system based on preventive maintenance, where all the components are assumed to exhibit linearly increasing hazard rate and constant repair rate. For many real systems, the assumptions mentioned above must be relaxed to provide more realistic models. This leads to more complexity in the mathematical formulation and/or more efforts in the solution procedure. To deal with these problems, especially for systems with a large number of components, the simulation-oriented approaches can be interesting and may perform well (Charles, Floru, Pantel, Pibouleau, & Domenech, 2003; Tan & Kramer, 1997). In the present work, a decision model is developed on the basis of multi-grouping optimisation that can assist a maintenance manager in making the preventive maintenance plan for series systems. It allows us to determine the replacement time of the system components exhibiting general and different failure rates, in such a way to minimize the total maintenance cost of the whole system. The methodology considers simultaneously corrective, preventive and opportunistic maintenance and allows us to deal with systems constituted by large number of components. The cost objective function for maintenance tasks (preventive, opportunistic and corrective) is defined for the whole system, the formulation takes into account both component and system-related costs. On the basis of informative method for defining the optimal maintenance groups, based on Monte Carlo simulations, the algorithm searches for the preventive
replacement times that minimize the total cost rate, including the total unplanned downtime of the system. 3. Formulation of the maintenance models 3.1. System description Let us consider a system composed of a set of q components arranged in series, the failure of any component leads to the failure of the whole system. Let 1 , 2 , . . ., q be the time intervals between preventive replacements of components 1, 2, . . ., q, respectively (Fig. 1). It is assumed that each replacement restores the component to the “as good as new” state because of its replacement by a new one from the same population (with the same reliability function). During the system downtime, either for preventive or for corrective maintenance, it is to decide if we can take the opportunity to replace preventively some of the non-failed components; this decision should be based on component degradation and risk undertaken if these components fail before reaching the following scheduled preventive time. 3.2. Costs structure The replacement costs can be divided into two parts: • The first part, considered as constant, is related to common system costs, such as mobilizing repair crew, safety provisions, disassembling machine, transportation, tools and the production loss related to the time lost in these tasks. The common system cost can be termed “set-up-cost”; it is noted C0c for corrective p replacement and C0 for preventive replacement. We consider that p C0c is greater than C0 because the emergency character of the corrective intervention does not allow for optimising tools and procedures, the staff is constrained to use available tools and fast but costly procedures. On the other hand, the preventive replacement can be prepared in advance and executed during a period where the production loss penalty is reduced. • The second part, considered as variable, is related to the specific characteristics of the component to be replaced, such as spare part costs, manpower costs, specific tools, and repair procedures. It is also related to production loss during the necessary time to extract and replace the component itself. For the ith component, p the specific corrective and preventive costs are noted Cic and Ci , p c respectively. We consider that Ci is greater than Ci , because the preventive replacement cost includes only the spare part and the manpower costs. Whereas for corrective replacement, in addition to the replacement cost, we must also add the failure cost which includes component-dependent production loss and other fees, such as compensatory solutions regarding the production.
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When a replacement is carried out, the impact on the total system cost is given by the sum of the above two parts of costs. For scheduled preventive maintenance, the system maintenance cost is given by: p
p
Csys = C0 +
p
Ci
as good as new, the replacement is instantaneous and the components are stochastically independent. The system renewal cycle is given by:
q
(1 − Fi (t))dt,
(1)
0
i ∈ Gp
where Gp is the group of components to be preventively replaced at the scheduled maintenance time (i.e. all components that reached the optimal age replacement as it will be detailed below). For corrective maintenance, when the jth component fails, the system is shut down and the failed component is replaced; the opportunity is taken to replace other critical components according to a predefined decision rule as it will be detailed later. In this case, assuming that the preventive replacement cost and the opportunistic one p are the same and noted Ci , the corrective system cost is given by: c Csys,j = C0c + Cjc +
p
Ci
then the system total cost per time unit is given by:
Cmono () =
(3)
i 0
(4)
where i is the time (age) for the preventive replacement of component i and Fi (·) its failure probability distribution, it represents the cumulative distribution function (CDF) of the random variable “time to failure”, and can be obtained from historical failure data.
K q C(, k1 , k2 , . . . , kq ) =
k=1
j=1
p
q
i=1
q
p C i=1 i
[1 − Fsys ()]
[1 − Fi (t)]dt (5)
i=1,...,q
p
(1 − Fi (t))dt
Fsys () + C0 +
components are defined by: i = ki , where ki are integer multipliers satisfying ki ≥ 1 for i = 1, 2, . . ., q (Fig. 1). In this context, the decision variables are (, k1 , k2 , . . ., kq ) where is continuous and ki are integer variables. The expression of the total cost per unit time requires the consideration of the involved costs along a cycle. In this case, the representative cycle is given by the expected time for replacing all the components simultaneously K (Gertsbakh, 2000), with K = lcm{k1 ,k2 ,. . .kq }, where lcm is the least common multiple. At the beginning of a new cycle, the system is completely renewed and becomes as good as new. As illustrated in Fig. 1, the life cycle of the system involves several replacements of the different components, and therefore, decisions concerning opportunistic and preventive replacements should be included in the cost model. At each replacement time k = k, the expected cost is calculated by including corrective, preventive and opportunistic terms. This formulation allows us to write the total expected cost per unit time as following:
3.3.1. Age-based policy for single component Having the corrective and preventive maintenance costs for a component i and its failure distribution, the expected cost per time unit is written by the sum of the expected corrective and preventive costs divided by the expected cycle length (Barlow & Hunter, 1960): Cic Fi (i ) + Ci (1 − Fi (i ))
3.3.3. Opportunistic multi-grouping approach For general industrial systems, the failure rates are very different from one component to another, and hence, a more realistic cost model has to be considered. The idea proposed in this work lies in the optimal definition of the preventive replacement times (ages) and grouping. In other words, the replacement time is selected, such as: (1) at each scheduled replacement, a decision has to be made for each component to define if it should be preventively replaced or not; and (2) at each system failure, a decision has to be taken for each component to see if it should be opportunistically replaced or left as it is until the next scheduled replacement. The final goal is to plan regular preventive replacements where optimal component grouping is defined (Fig. 1). In this way, the basic maintenance time is defined as the minimum replacement time: = min i . In addition, the maintenance times for the different
3.3. Models formulation for several maintenance policies
Ci (i ) =
Cc i=1 i
where Fsys () = 1 − Rsys () = 1 − i=1 [1 − Fi ()] is the failure probability of the overall system and Rsys () is the system reliability. This strategy seems to be suitable only when the systems are composed by identical components (with similar lifetime distributions). However, when the component lifetimes are different, a waste of money is observed when replacing reliable components under the pressure of other less reliable ones.
(2)
expected cost on one cycle C(t) = t expected length of a cycle
q
q
c where Csys,j is the corrective system cost due to the failure of component j, Gh is the group of components to be replaced preventively during this opportunity (Gh is defined according to a deterioration-based rule, analyzing the cost/benefit balance as it will be detailed later). It can be clearly seen in the Eqs. (1) and (2) that the set-up cost is paid only once independently of the number of components replaced at an opportunity. This reflects the opportunistic nature of costs and the economic dependence between components. According to the renewal theory and assuming infinite horizon, the expected cost per unit time is given by (Barlow & Proschan, 1965):
t→∞
C0c +
0
i ∈ Gh i= / j
C(T ) = lim
i=1
C0c + Cjc +
3.3.2. Equivalent mono-component approach Here we assume that all the system components are jointly replaced, at the system failure or after a certain time , whichever occurs first. Knowing that any component failure leads to system failure, assuming that after each replacement the system becomes
p
i ∈ Gh
k
Ci
p
Fsys,j (k) + C0 + K
p
i ∈ Gp
k
Ci
(1 − Fsys (k))
(6)
where Fsys,j (·) is the CDF of system failure due to the jth component. For a series system, the failure of any component leads to the system failure, then Fsys,j (·) = Fj (·). Ghk is a group of components to be replaced opportunistically when a failure occurs in the interval [(k − 1), k], Gpk is the group of components to be replaced preventively at instant k: all i|k/ki = Integer, i = 1, q; k = 1, K.
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Fig. 2. Decision cost basis for opportunistic replacement.
The minimization of the expected cost per time unit aims at finding the best set of the maintenance time i : Find : , k1 , k2 , . . . and kq that minimize : C(, k1 , k2 , . . . kq ) subject to : ≥ 0; ki ≥ 1 and ki are integers
(7)
3.4. Solution procedure As the cost function in problem (7) contains continuous and discrete variables, a solution strategy has to be developed for efficient computation of the optimal replacement plan, especially for large number of components. The numerical solution can be carried out by Monte Carlo simulations, allowing to deal with both integer and continuous variables in optimisation (Marquez, Gupta, & Ignizio, 2006). While the replacement time can be easily optimised by classical algorithms, the implication of the discrete variables ki leads to a very large number of possible combinations. It is thus
necessary to reduce the number of considered combinations for practical systems, without discarding the potentially optimal combinations. The proposed solution is a search method based on conditional information concerning the reliability levels of the components. As the search algorithm requires a starting point, the initial solution can be defined by optimal times for the individual components (minimum of costs in Eq. (4)). This solution gives reasonable initial values for ki , defined by the ratio between the component optimal 0 , which is replacement time i0 and the minimum optimal time min 0 ). Due to system effect and economic written: ki0 = Integer(i0 /min dependence, the search range for optimal groups, defined by ki , can vary from the initial groups, defined by ki0 , in the range ±1 defined by: ki0 − 1 ≤ ki ≤ ki0 + 1 with ki ≥ 1. This gives convenient bounds for optimal search and reduces strongly the number of combinations to only three times the number of components (instead of the factorial).
Fig. 3. Flowchart of the solution algorithm.
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Fig. 4. Example of corrective/opportunistic/preventive maintenance simulation.
The large number of random simulations ensures the stability of the estimates and guarantees the solution convergence to the optimal plan. As all significant combinations are considered, the optimal solution cannot be missed in the proposed procedure. It is also to note that the stability and the convergence of Monte Carlo simulations have been approved, in solving nonlinear and complex optimisation problems. In addition, the deterioration-based decision can be included by analyzing the cost/benefit balance of the component to be preventively replaced. Let us consider the case where the jth component fails at the time tj between two scheduled maintenance times k and (k + 1), as illustrated in Fig. 2. The opportunity of replacing the p component i leads to an expected cost Ci Ri (tj ). If the ith component is left without replacement, two cases are possible: either it remains operating until the following scheduled replacement, which will p cost Ci Ri ((k + 1)|tj ), or it fails before, leading to system breakdown, the corresponding cost is (C0c + Cic )Fi (t) with a maximum value at (k + 1). The decision-making criterion for opportunistic replacement can thus be defined by comparing the two costs. If the opportunistic replacement cost is less than the corrective one, it is better to change the component; otherwise it can be left till the following planned replacement: p
If : Ci (Ri (tj ) − Ri ((k + 1)|tj )) ≺ (C0c + Cic )Fi ((k + 1)|tj ) Otherwise
initial grouping configuration ki0 ; the solution procedure is given as following: 1. Generate random samples of the lifetimes of the components ti , according to the failure probability distributions. Then, the system failure time is defined by: tsys = min ti and the corresponding i=1,q
failed component producing the system shutdown is identified. The replacements are scheduled at the times k, where k is an integer varying from 1 to K. 2. At the kth replacement, the simulated system failure time tsys is compared with the scheduled time for preventive maintenance k. Two possibilities exist: a. If no failure is observed before k, the preventive maintenance can be carried out at k, according to the current grouping rule, as defined in the updated plan, and a move to the next scheduled time (k + 1) is done. b. If failure is observed, the system is down and the failed component is correctively replaced. On the basis of the conditional strategy described above, the opportunity of replacing other
⇒ then, make opportunistic replacement of component i at the time tj ⇒ leave it till the next scheduled replacement
The flowchart of the developed algorithm is depicted in Fig. 3. After introducing the component failure data (probability distribup tions and parameters), the cost parameters (C0c , C0 , Cic , Cic ) and the
components is considered and the related preventive costs are computed.
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3. For the replaced components in step 2, new lifetimes are generated (as new components are installed); a move to the next replacement time (k + 1) is performed, and so on, until the end of the system cycle (until the replacement of all components simultaneously). 4. For the simulated scenario, the life cycle length and the corresponding total cost are computed. 5. Repeat steps 1–4, to generate new scenarios by random sampling, until the prescribed number of simulations is reached. 6. The expected total cost per unit time is estimated in terms of the mathematical average of the computed costs and the cycle span of the simulated scenarios. For all the sampled scenarios, the total cost per unit time in terms of the statistical expectations is given by:
⎛
⎛⎛
⎞
p 1 ⎝ ⎝⎝C0c + Cic + Ci ⎠ IFk,s E[C(t)] = N N
s=1
K
k=1
⎛ + ⎝C0 + p
⎞ p
Ci
⎞⎞
⎠ IRk,s ⎠⎠
(8)
where N is the total number of simulations, IFk,s and IRk,s are binary indicators for the states of failure and operation, respectively. For the kth replacement interval of the sth simulation, these indicators are defined by: IFk,s = 1; IRk,s = 0 IFk,s = 0; IRk,s = 1
Table 1 Data and optimisation results for the illustrative example. p
Component
ˇ
cic (D )
ci (D )
Topt (days)
CT (D /day)
1 2
500 200
2.5 2.7
5000.00 400.00
600.00 200.00
177 74
11.57 17.59
in Fig. 4b. New lifetimes are then generated for the new components (i.e. components 1 and 2). The scheduled replacement at is now examined; in our example, components 1, 2 and 3 are preventively replaced (Fig. 4b). In the next replacement interval ( ≺ t ≤ 2), components 1 and 3 appear to have high deterioration rate and are then replaced at 2 (Fig. 4c). Then, component 5 shows a failure before reaching the scheduled replacement at 3, and so on. This illustration shows how a component can be dynamically considered in function of the possible opportunities. 4. Illustrative example with two components
i ∈ Gh
i ∈ Gp
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if failure if operation
7. The procedure is repeated for different replacement intervals and grouping configurations ki . The search for the optimum solution allows us to update the scheduled maintenance plan, by changing and ki , in order to define a better combination of component grouping. The iterative scheme is stopped when the optimum solution cannot be improved. Fig. 4 shows an example of a sample of five components, where the scheduled preventive replacement has a span equal to . The above described algorithm is applied to generate random scenarios. The component lifetimes are firstly generated, as shown in Fig. 4a; it is observed that component 2 fails before the scheduled replacement and should be correctively replaced. Given the deterioration of component 1, basing on the decision rule provided above, the opportunity of replacing it preventively is carried out, as illustrated
In order to illustrate the proposed policy, let us consider the case of two components with the failure and cost data in Table 1. The corrective and preventive common costs are c0c = 8000 D and p c0 = 600 D , respectively. Table 1 gives also the optimal times and costs for separate replacement of the two components (Eq. (4)). 4.1. Mono-group policy If the two components have to be changed simultaneously (mono-group), the optimal time is 71 days with a cost per unit time of 31.59 D /day (Fig. 5). It can be seen that mono-grouping is less interesting than independent replacement of each one of the two components, as the sum of the costs in Table 1 leads to 29.18 D /day. 4.2. Multi-group policy Even for this simple problem, the choice of the optimal policy is not that obvious. For independent replacements, the optimal time ratio is given by: 177/74 = 2.39; it is not an evidence to say that the policy {2,} is better than {3,}. Table 2 shows the results for different combinations. It is clearly seen that policy 2-1 (i.e. k1 = 2; k2 = 1) leads to the lowest cost per unit time (23.822 D /day), followed by the policy 3-1 (i.e. k1 = 3; k2 = 1) with 24.625 D /day. The policies 1-2 and 3-2 are far from being optimal, which could be expected by looking at the single component results. In this example, the policy 2-1 leads to a cost saving of 8% with respect to simultaneous replacement of the two components (i.e. policy 1-1).
Fig. 5. Expected maintenance cost rate for single components and mono-group policies.
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Table 2 Optimal solutions for different replacement groups. Replacement groups k1 -k2
System life cycle (days)
(days)
Optimal replacement times, 1 / 2 (days)
Minimum expected cost (D /day)
1-1 1-2 2-1 3-1 3-2
68 158 130 189 372
68 79 65 63 62
68/68 79/158 130/65 189/63 186/124
25.797 33.259 23.822 24.625 32.285
Fig. 6. Maintenance cycle of the two-component system.
In order to validate the solution procedure, let us now consider the policy 2-1 (i.e. component 1 is replaced each two replacements of component 2). The maintenance cycle for this policy is depicted in Fig. 6. At a given time, each component may have one of the three possibilities: (1) surviving till the next scheduled replacement, (2) failing before reaching the scheduled time, or (3) replacement by opportunity when other components fail or replaced preventively. Table 3 shows the convergence of the expected cost estimates in terms of the number of random samples. Convergence is clearly shown by the monotonic decrease of the coefficient of variation (C.O.V.) with the increase of the sample size. It is also shown that the cost estimate becomes stable for more than 500 Monte Carlo samples. High precision is reached for cost estimate with 10000 samples, as the coefficient of variation is less than 0.1%. For this simple case, it is possible to write analytically Eq. (6) without opportunistic terms, the result leads to the same replacement time as Monte Carlo (i.e. 65 days) with the cost 25.49 D /day. As it could be expected, Monte Carlo method converges to lower cost estimate due to the consideration of opportunistic replacement and real failure times between the scheduled replacements. Table 3 Convergence of Monte Carlo simulations. Number of samples
Mean cost estimate
C.O.V. of the estimate
10 50 100 500 1,000 5,000 10,000 50,000
14.902 19.100 22.990 23.938 23.872 23.801 23.817 23.804
5.30% 2.27% 2.11% 0.42% 0.37% 0.12% 0.09% 0.01%
5. Industrial application 5.1. Refinery centrifugal compressor The proposed methodology is applied to the centrifugal compressor of the catalytic reforming unit of Skikda refinery, which is the most important oil refinery in Algeria; the first units started production in 1980, its capacity is about 15 millions tons/year. The multiple staging compressor is driven by a steam turbine; it is essentially constituted by the stator (diaphragms, landings, tightness subsystem) and the rotor (shaft, wheels, equilibrium piston. . .). The case study is chosen according to an earlier study (Laggoune & Aissani, 2000), which has revealed that the hydrogen compressor was the leading cause of Skikda catalytic reformer breakdowns. This confirms the fact, stated in (Tan & Kramer, 1997), that pumps and compressors are the leading equipment failures in refineries. The catalytic reforming objective is to convert low-octane naphtha, obtained from the topping unit, into high-octane reformate for gasoline blending and/or to provide aromatics (benzene, toluene, and xylene) for petrochemical plants. It also produces highly purified hydrogen for hydro-treating processes. The major catalytic reforming reactions are: dehydrogenation of naphthenes to aromatics (C6 ); isomerisation of naphthenes to aromatics (C5 ); dehydrocyclization of paraffins to aromatics; isomerisation of normal paraffins to iso-paraffins; and hydrocracking of paraffins into smaller molecules. The catalytic reforming unit of Skikda refinery is a semiregenerative type, with successive reactors, each one with a fixed bed of catalyst (Fig. 7). The catalyst used contains platinum and rhenium on a chlorinated alumina support base. The noble metals (platinum and rhenium) are catalytic sites for the dehydrogenation reactions and the chlorinated alumina provides the acid sites needed for isomerisation, cyclization and hydrocrack-
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Fig. 7. Schematic diagram of a typical semi-regenerative catalytic reforming unit.
ing reactions. It is worth to note that the catalyst effectiveness in a semi-regenerative catalytic reformer is reduced over time during operation by carbonaceous coke deposition and chloride loss. Therefore the catalyst is periodically regenerated in situ by high temperature oxidation of the coke followed by chlorination. To underline the vital role of the compressor in the reforming unit, we summarize its function in the process. Indeed, the necessary hydrogen for the various reactions of catalytic reforming is recycled by the compressor; it is the presence of hydrogen which limits the formation of coke. The speed of coke formation is very influenced by the hydrogen pressure; a compressor dysfunction can involve a pressure loss and consequently an acceleration of coking. Reduction in the hydrogen flow implies the reduction in hydrocarbons to maintain a preset H2 /HC rate, in other words a reduction in the load rate which would generate production losses. In addition, the compressor is necessary for the unit pre-heating during operation starting after shut-down; it also provides the necessary air for the coke combustion during catalyst regeneration. 5.2. Data collection and analysis 5.2.1. Failure data for reliability analysis The data available at the refinery site are the failure dates of the compressor, the component inducing failure, the preventive maintenance dates and the maintenance durations. The first analysis consists in extracting the times to failure for each component in order to constitute the samples, the times to preventive replace-
ment are considered as censored data; so the data sample includes failures and censored data (Table 4). The samples are then fitted by two-parameter Weibull model, this model was chosen because it adjusts well to mechanical components, it allows describing alternately the three phases of the component life (early-life, mid-life, and wear-out), and it is efficient with a minimum of only five failures (Summers-Smith, 1989). The Weibull reliability function is given by:
ˇ t
R(t) = 1 − F(t) = exp −
,
where t is the time to failure of individual components, is the scale parameter and ˇ is the shape parameter. The most important parameter is ˇ, it takes different values depending on the component life phase (failure mode): When ˇ < 1, the failure rate is decreasing (infantile mortality); when ˇ = 1, the failure rate is constant (hazardous failure); and when ˇ > 1, the failure rate is increasing (wear-out). The Weibull parameter estimation is obtained by the “Statistica” software (Table 4), which uses the maximum likelihood method, and allows us to deal with censored samples. 5.2.2. Cost data The maintenance cost assessment is a very hard and complex task, as it depends on many parameters. Therefore, we have to use some simplifying assumptions and criteria to provide approx-
Table 4 Failure data and Weibull parameters of the system components. Component
Code
Sample size
Observed failures
Shape parameter ˇ
Scale parameter
MTBF days
Sheathing Sheathing Tightness Stub bearing Tightness ring Carrying bearing Stub bearing Labyrinth support
C286 C285 C275 C230 C460 C419 C401 C780
24 23 21 21 34 15 17 14
14 15 15 8 14 8 8 7
1.73 1.88 2.43 2.53 2.14 3.55 2.68 2.09
486 507 286 898 905 736 1094 1388
483 475 240 787 844 636 888 1047
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Table 5 Common costs and maintenance costs. Component
Code
Corrective cost (D )
Common costs Sheathing Sheathing Tightness Stub bearing Tightness ring Carrying bearing Stub bearing Labyrinth support
C286 C285 C275 C230 C460 C419 C401 C780
36,000 14,868 39,204 44,880 57,876 73,860 46,752 48,568 74,232
Preventive cost (D )
Cost ratio corrective/preventive
1,200 3,639 5,438 7,398 8,277 13,554 14,130 21,356 24,348
imations close to the real costs, that is why the mean costs are often used. The common costs include the set-up cost and the production loss. The set-up cost is obtained by the mean cost of the previous maintenance operations multiplied by an actualisation factor. The production loss is obtained by multiplying an hourly cost by a mean time making the equipment busy when subjected to the maintenance operation; it includes the entire time span since the production stopping and system disassembly to reach the component concerned by the replacement. During a preventive maintenance, most provisions are usually taken before stopping the production, such as scaffolding installation, safety provisions, getting available maintenance crew and tools. Therefore a preventive replacement takes only the necessary time to open the system; consequently the corrective common cost is larger than the preventive one. The component preventive cost includes the spare part and labour costs; the corrective one includes the spare part, the labour, the production loss, and other fees. The production loss added to the component corrective cost concerns only the necessary time to extract the component and to replace it by a new one. Table 5 gives the specific component costs, as well as the system common cost. The large ratio of corrective to preventive costs can be easily observed for these data. The maintenance optimisation is carried out according to different assumptions in order to compare the optimal solutions and to show the benefits of the proposed maintenance plan.
30 4.1 7.2 6.1 7.0 5.4 3.3 2.3 3.1
Fig. 8. Expected cost rates for the separate components.
5.3. Maintenance optimisation 5.3.1. Single component maintenance optimisation As a first step, the policy based on separate components is considered to obtain optimal replacement times for individual components. The expected cost is computed for each component, independently, and a one-by-one optimisation is applied. Fig. 8 shows the expected total costs for individual components (Eq. (4)). Table 6 gives the optimal solutions for individual components. As discussed above, the ratios of individual optimal times can be used as a starting point for group maintenance plan. From Table 6, these 0 , leading to: k ˜ 1 = 1.82, k˜ 2 = 1.63, ratios are given by: k˜ i = i0 /min ˜k3 = 1.0, k˜ 4 = 3.09, k˜ 5 = 3.73, k˜ 6 = 3.56, k˜ 7 = 6.82 and k˜ 8 = 7.89.
Fig. 9. Expected cost rate for the mono-group policy.
5.3.2. Mono-group policy In order to underline, the system effect, the whole system is now considered as a macro-component, where simultaneous replacement is performed for all the components (Eq. (5)). The expected cost, depicted in Fig. 9, shows a minimum at 99 days, with an optimal cost of 1864.92 D for the whole system. Naturally, this solution is not optimal as it does not take into account the specific costs related to different components.
0 )). Later, the nearest integer is used for ki (ki0 = Integer(i0 /min
Table 6 Optimal solutions for individual components. Component
Code
MTBF (days)
Optimum time, i0 (days)
Cost (D /day)
0 Rate, k˜ i = i0 /min
Sheathing Sheathing Tightness stub bearing Tightness ring Carrying bearing Stub bearing Labyrinth support
C286 C285 C275 C230 C460 C419 C401 C780
483 475 240 787 844 636 888 1047
197 176 108 334 403 385 737 852
68.7588 87.5325 139.541 48.2451 74.9058 56.5187 55.2709 70.7366
1.82 1.63 1.00 3.09 3.73 3.56 6.82 7.89
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Table 7 Optimal solutions for different replacement groups. No.
Replacement groups, k1 -k2 -k3 -k4 -k5 -k6 -k7 -k8
1 2 3 4 5 6 7 8 9 10
1-1-1-1-1-1-1-1 1-1-1-4-4-2-7-8 1-1-1-3-4-4-6-7 1-1-1-3-4-4-7-7 2-2-1-4-3-2-6-6 2-2-1-4-4-4-7-7 2-2-1-3-4-4-7-8 2-2-2-2-3-3-6-6 2-2-2-2-3-3-6-7 2-2-2-4-4-4-6-8
System life cycle (days) 99 6,944 18,648 9,240 1,620 2,940 20,832 744 5,208 2,520
5.3.3. Opportunistic multi-grouping optimisation The proposed solution is based on optimal grouping of components replacement (Eq. (6)) including opportunistic replacements. Regarding the results obtained when considering individual components, the natural and evident grouping is 2-2-1-3-4-4-7-8 with = 124 days, but this configuration is not optimal as it is shown in Table 7, indicating the optimal solutions for significant grouping configurations. From this table, it can be shown that the minimum cost is achieved for the configuration 2-2-2-2-3-3-6-6 with = 124 days where three groups are considered, according to k1 = k2 = k3 = k4 = 2, k5 = k6 = 3 and k7 = k8 = 6: (1) a group of compressor components (C286, C285, C275 and C230) with replacement at the age of 248 days, (2) a group of turbine components (C460 and C419) with replacement at the age of 372 days and (3) another group of turbine components (C401 and C780). The minimal cost is only 534.871 D , which represents 71% of reduction with respect to the case of only one group (1-1-1-1-1-1-1-1) and 5% with respect to the evident grouping. This reduction shows clearly the importance of the optimisation over groups and the interest of the proposed model. It shows also the capacity and the stability of the procedure in dealing with practical systems where no prior solution can be provided. 6. Conclusions The proposed maintenance plan is based on opportunistic multigrouping replacement optimisation for multi-component systems. As in continuous operating units such as chemical plants, the production losses are very high. Therefore, the maintenance optimisation has the potential for substantially reducing the operating costs and for increasing corporate profit by increasing availability and production. The proposed approach is based on the analysis of individual components, according to the well-known age-based model. The optimisation algorithm allows rearranging the optimal individual replacement times in such a way that all component times become multiple of the smallest one to allow for joint replacements. In this way, the times obtained by the multi-grouping approach do not give individual optimality conditions, but satisfies the optimal cost regarding the whole system. The proposed algorithm is a numerical procedure, where the life cycles are simulated and the optimal solution is numerically searched. The large number of random simulations by Monte Carlo method, which is very useful in solving nonlinear and complex optimisation problems, ensures the stability of the estimates and guarantees the solution convergence to the optimal one. As all significant combinations are considered, the optimal solution cannot be missed. The most important facts revealed in this study are: (1) the effectiveness of the opportunistic policy in cost saving for multi-component systems and the capacity of the Monte Carlo simulations in solving these complex problems, and (2) the evident grouping configuration, which is sometimes adopted by
(days)
Optimal replacement times, 1 / 2 / 3 / 4 / 5 / 6 / 7 / 8 (days)
Minimum expected cost (D /day)
99 124 111 110 135 105 124 124 124 105
99/99/99/99/99/99/99/99 124/124/124/496/496/248/868/992 111/111/111/333/444/444/666/777 110/110/110/330/440/440/770/770 270/270/135/540/405/270/810/810 210/210/105/420/420/420/735/735 248/248/124/372/496/496/868/992 248/248/248/248/372/372/744/744 248/248/248/248/372/372/744/868 210/210/210/420/420/420/630/850
1864.92 590.894 576.362 576.039 561.913 564.305 560.669 534.871 535.111 540.032
maintenance managers, is not necessary cost-effective; therefore, an optimisation procedure must be considered. It is well known that in process units such as oil refineries and chemical plants, the production loss is proportional to the downtime and the costs induced are very high. It seems to be more suitable to relax the assumption of the instantaneous maintenance actions to highlight the effect of this variable on the grouping configuration; this topic is currently under consideration. Acknowledgement The authors would like to acknowledge the anonymous reviewers for their fruitful comments and recommendations, which have made it possible to improve upon the value and the clarity of the original paper. References Bâckert, W., & Rippin, D. W. T. (1985). The determination of maintenance strategies for plants subject to breakdown. Computers and Chemical Engineering, 9, 113–126. Barlow, R. E., & Hunter, L. C. (1960). Optimum preventive maintenance policies. Operation Research, 8, 90–100. Barlow, R. E., & Proschan, F. (1965). Mathematical theory of reliability. New York: John Wiley & Sons. Charles, A. S., Floru, I. R., Pantel, C., Pibouleau, L., & Domenech, S. (2003). Optimization of preventive maintenance strategies in a multipurpose batch plant: Application to semiconductor manufacturing. Computers and Chemical Engineering, 27, 449–467. Cho, I. D., & Parlar, M. (1991). A survey of maintenance models for multi-unit systems. European Journal of Operational Research, 51, 1–23. Das, K., Lashkari, R. S., & Sengupta, S. (2007). Machine reliability and preventive maintenance planning for cellular manufacturing systems. European Journal of Operational Research, 183, 162–180. Dekker, R., & Roelvink, I. F. K. (1995). Marginal cost criteria for preventive replacement of a group of components. European Journal of Operational Research, 84, 467–480. Dekker, R., Wildeman, R. E., & Van Der Duyn Schouten, F. A. (1997). A review of multi-component maintenance models with economic dependence. Mathematical Methods of Operational Research, 45, 411–435. Duarte, J. A. C., Craveiro, J. C. T. A., & Trigo, T. P. (2006). Optimisation of the preventive maintenance plan of a series components system. International Journal of Pressure Vessels and Piping, 83, 244–248. Gertsbakh, I. (2000). Reliability Theory: With applications to preventive maintenance. Berlin: Springer. Goyal, S. K., & Gunasekaran, A. (1992). Determining economic maintenance frequency for a family of machines. International Journal of Systems Science, 4, 655–659. Huang, J., & Okogbaa, O. G. (1996). A heuristic replacement scheduling approach for multi-unit systems with economic dependency. International Journal of Reliability, Quality and Safety Engineering, 3, 1–10. Kardon, B., & Fredendall, L. D. (2002). Incorporating overall probability of system failure into a preventive maintenance model for serial system. Journal of Quality in Maintenance Engineering, 8(4), 331–345. Laggoune, R., & Aissani, D. (2000). Repeat Failure Analysis for Oil Refinery Maintenance Optimization: Case Study of Skikda Refinery Compressing Magnaforming. In Proceedings of the Second International conference on Mathematical Methods in Reliability (MMR’2000), vol. 2 Bordeaux, France, (pp. 671–674). Lam, T. C., & Yeh, R. H. (1994). Optimal maintenance policies for deteriorating systems under various maintenance strategies. IEEE Transactions on Reliability, 43, 423–430.
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Marquez, A. C., Gupta, J. N. D., & Ignizio, J. P. (2006). Improving preventive maintenance scheduling in semiconductor fabrication facilities. Production Planning & Control, 17(7), 742–754. Nahara, K. (1993). Total productive management in the refinery of the 21st century. In Proceedings of the conference on foundation of computer aided operations, FOCAP-O Park City, U.K., (pp. 111–132). Rao, A. N., & Bhadhury, B. (2000). Opportunistic maintenance of multi-equipment systems: A case study. Quality and Reliability Engineering International, 16(6), 487–500. Samrout, M., Yalaoui, F., Châtelet, E., & Chebbo, N. (2005). New method to minimize the preventive maintenance cost of series-parallel systems using ant colony optimisation. Reliability Engineering and System Safety, 89, 346–354. Summers-Smith, J. D. (1989). Fault diagnosis as an aid to process machine reliability. Quality and Reliability Engineering International, 5, 203–205. Tam, A. S. B., Chan, W. M., & Price, J. W. H. (2006). Optimal maintenance intervals for multi-component system. Production Planning & Control, 17(8), 769–779.
Tan, J. S., & Kramer, M. A. (1997). A general framework for preventive maintenance optimization in chemical process operations. Computers and Chemical Engineering, 21(12), 1451–1469. Thomas, L. C. (1986). A survey of maintenance and replacement models of multi-item systems. Reliability Engineering, 16, 297–309. Van Der Duyn Schouten, F. A., & Vanneste, S. G. (1990). Analysis and computation of (n,N)-strategies for maintenance of a two-component system. European Journal of Operational Research, 48, 260–274. Vassiliadis, C. G., & Pistikopoulos, E. N. (2001). Maintenance scheduling and process optimization under uncertainty. Computers & Chemical Engineering, 25(2–3), 217–236. Wang, H. (2002). A survey of maintenance policies of deteriorating systems. European Journal of Operational Research, 139, 469–489. Zheng, X., & Fard, N. (1992). Hazard-rate tolerance method for an opportunisticreplacement policy. IEEE Transactions on Reliability, 41(1), 13–20.
Contents lists available at ScienceDirect
Computers and Chemical Engineering journal homepage: www.elsevier.com/locate/compchemeng
Opportunistic policy for optimal preventive maintenance of a multi-component system in continuous operating units Radouane Laggoune a,∗ , Alaa Chateauneuf b , Djamil Aissani a a b
Laboratory of Modelisation and Optimisation of Systems (LAMOS), University of Bejaia, Targa Ouzemour, 06000 Bejaia, Algeria Laboratory of Mechanics and Engineering (LaMI), Université, Blaise Pascal, Campus des Cézeaux, BP 206, 63174 Aubière Cedex, France
a r t i c l e
i n f o
Article history: Received 31 May 2008 Received in revised form 19 December 2008 Accepted 10 March 2009 Available online 21 March 2009 Keywords: Opportunistic maintenance Multi-component system Economic dependence Production loss Monte Carlo
a b s t r a c t For continuous operating units such as petrochemical plants, the production loss due to downtime is high, and the economic profitability is conditioned by the implementation of suitable maintenance policy that could increase the availability and reduce the operating costs. In this paper, a preventive maintenance plan approach is proposed for a multi-component series system subjected to random failures, where the cost rate is minimized under general lifetime distribution. The expected total cost is given by corrective and preventive costs, which are related to the components, and by the common costs related to the whole system, especially the production loss during system shutdown. When the system is down, either correctively or preventively, the opportunity to replace preventively non-failed components is considered. A solution procedure based on Monte Carlo simulations with informative search method is proposed and applied to the optimisation of the component replacement for the hydrogen compressor in an oil refinery. © 2009 Elsevier Ltd. All rights reserved.
1. Introduction For continuous operating units, the production loss is often very large when unexpected shutdown occurs. Their economic profitability implies the implementation of suitable maintenance policy which could increase the availability and reduce the operating costs. To underline the consequences of unavailability, it can be mentioned that production losses in chemical plants can range from $5000 to $100,000 per hour (Tan & Kramer, 1997). For refineries, the total production losses soar to millions of dollars (Nahara, 1993). In addition, the safety requirements enforce to decrease the failure probability, on a very low level in order to avoid disastrous consequences. The preventive maintenance (PM) is often carried out to prevent or to slow down the deterioration processes. PM is a scheduled downtime, usually periodical, in which a well-defined set of tasks (e.g., inspection, replacement, cleaning, lubrication, adjustment and alignment) are performed. In oil refining facilities, the problems associated with part replacement are more concerned than other routine maintenance activities such as cleaning and lubricating, from the PM scheduling point of view. This is because the direct costs due to part failure and replacement are usually very high, and the impact of different replacement intervals on the overall main-
∗ Corresponding author. Tel.: +213 34 21 51 88; fax: +213 34 21 51 88. E-mail address: r [email protected] (R. Laggoune). 0098-1354/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.compchemeng.2009.03.003
tenance cost is often very sensitive and significant, in addition to the safety requirements. In a series system, the one-by-one preventive replacements of components improve the global system reliability on the account of its availability, which would be largely penalized, because of frequent shutdowns for component replacements. For multi-component systems, an optimal maintenance policy must take into account the interactions between the various components of the system. These interactions are of three types (Thomas, 1986): economic dependence, structural dependence and stochastic dependence. We are mainly concerned by the economic dependence reflecting the influence of component operation/maintenance costs on the overall system costs; in this case, saving in costs or downtime can be achieved when several components are jointly maintained. The objective of this paper is to develop a preventive/ corrective/opportunistic maintenance plan for a multi-component system subjected to high production losses and economic dependence. In the next section, we review the relevant literature, particularly that dealing with multi-component systems. In Section 3, we provide the cost formulation and the maintenance models in several cases; an algorithm allowing for combined preventive/corrective/opportunistic replacement of the system components is also presented. In Section 4, the proposed approach is illustrated by a simple example with two components, allowing to explain the formulation interest and to verify the convergence of the solution procedure. An industrial application is provided in
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Nomenclature
Section 5, where the results show the effectiveness of the proposed approach for practical systems.
C0c
2. Literature review
corrective common cost related to the system, to be paid at each repair upon failure p C0 preventive common cost related to the system, to be paid at each time the system undergoes a preventive maintenance. Cic specific corrective cost, to be paid at each replacement upon failure of component i p specific corrective cost, to be paid at each preventive Ci replacement of component i c Csyst,j expected total corrective cost of the whole system due to failure of component j p Csys expected total preventive maintenance cost of the whole system Ci ( i ) cost rate for component i (objective function of the mono-component policy) Cmono () cost rate for the mono-group policy (objective function) C(,k1 ,. . .,kq ) cost rate for the opportunistic multi-grouping policy (objective function) basic preventive maintenance interval ( = min i ) i=1,...,q
i ki i
subscripts indicating components, i = 1, 2, . . ., q integer multiplier of component i time interval (age) between preventive replacements of component i ( i = ki ) q number of system components K the least common multiple of all ki k kth scheduled time instant where preventive replacements take place, k = 1, 2, . . ., K i0 initial optimal replacement time interval of component i ki0 initial Integer multiplying factor of component i ti simulated lifetime of component i tj time instant of failure of component i simulated time instant of system failure due to the ti weakest component N total number of simulations s subscript indicating the simulation number IFk,s , IRk,s ∈ [0, 1] binary variables indicating the states of failure or operation, respectively Gp group (set) of components to be preventively replaced at a scheduled time instant group of non-failed components to be opportunistiGh cally replaced during a failure group of components to be preventively replaced at Gpk the kth scheduled time instant Gph group of components to be opportunistically replaced at the kth interval [(k − 1),k] cumulative distribution failure (CDF) of component Fi (·) i cumulative distribution failure of the whole system Fsys (·) Fsys,j (·) cumulative distribution failure of the system due to component j reliability function of the whole system Rsys (·) MTTF mean time to failure ˇ Weibull shape parameter Weibull scale parameter H2 hydrogen HC hydrocarbon LPG liquefied petrol gas
For multi-component systems, when no strong dependence exists between the different components, the traditional singleunit model developed by Barlow and Hunter (1960) can be independently applied to each unit, in order to provide optimal replacement schedule. However, the general case of multicomponent systems implies to take account for the interactions between various components. The economic dependence is common in most continuous operating systems, such as oil refineries, chemical processing facilities, mass-production manufacturing lines and power generators (Das, Lashkari, & Sengupta, 2007; Vassiliadis & Pistikopoulos, 2001). For this type of systems, the single shutdown cost is often much higher than the cost of the components to be replaced. Therefore, there is a great potential for cost savings by implementing suitable maintenance policy. A number of studies have reviewed the various maintenance policies for multi-component systems (Cho & Parlar, 1991; Dekker, Wildeman, & Van Der Duyn Schouten, 1997; Thomas, 1986; Wang, 2002). These reviews show that most of the authors use simplified assumptions, or deal with particular systems (special structure is assumed), in order to formulate the decision problem with less mathematical difficulty. From another point of view, most of the developed decision models are based on dynamic programming or Markovian approaches (Bâckert & Rippin, 1985; Lam & Yeh, 1994), which approximate continuous decision variables by finite discrete state decision variables. These restrictions in both maintenance policies and model formulations could affect the optimality of the solutions because of the reduction of the solution space. In addition, discrete state decision models are often difficult to apply to systems with large number of components and different failure distributions. The common planning approaches used for multi-component manufacturing systems include the group/block replacement models and the opportunistic maintenance models. In the block maintenance policy, an entire group of units or components are replaced at periodic intervals. The interval is decided based on time, cost or both (Dekker et al., 1997; Kardon & Fredendall, 2002; Tam, Chan, & Price, 2006). Performing maintenance periodically can be a costly option because maintenance activities are carried out even though components are in good condition. The concept of opportunistic maintenance comes from the fact that there is the possibility of dependence between the various components of the system. For example, the cost of simultaneous maintenance actions on various components would be less than the sum of the total cost of individual maintenance actions (Huang & Okogbaa, 1996); this is due to the potential savings in the common cost usually termed “set-up cost” in the literature. This is particularly true in the case of series system, where the failure of any component results in stopping the whole system. Therefore, providing the opportunity to carry out preventive maintenance on some components along with replacement of failed components, leads to very small additional cost, compared to separate replacements. Under these conditions, the maintenance decisions for one component depend on the states (ages) of the other components in the system (Rao & Bhadhury, 2000; Van Der Duyn Schouten & Vanneste, 1990). Several techniques and approaches dealing with multicomponent systems are provided in the specialized literature. Dekker and Roelvink (1995) presented a heuristic replacement criterion which is only useful when a fixed group of components is replaced. Huang and Okogbaa (1996) presented a replacement scheduling approach based on the cost boundary condition; they claim that it is more effective than some of the very sophisticated but improperly formulated “optimal” models. Van Der Duyn
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Fig. 1. Scheduled preventive maintenance plan.
Schouten and Vanneste (1990) investigated (n, N)-strategies for the maintenance of a two component series system. They presented a numerical procedure to compute the optimal values of n and N. In this policy, a single unit undergoes a maintenance action whenever it fails or reaches the age N and the other unit can be included into the maintenance operation if and only if its age exceeds n. The generalization to systems with more than two units of the proposed algorithm is very difficult; this is due to the necessity to have detailed information on the age distribution of all units at each opportunity. Zheng and Fard (1992) proposed also an opportunistic maintenance policy allowing for partial joint replacement. However, the initiation of a joint replacement of a unit is restricted to preventive replacement only; the failures are not included as opportunities to carry out preventive maintenance on the non-failed units. Goyal and Gunasekaran (1992) presented a model for multicomponents with a very special deterioration structure, describing a transport fleet consisting of groups of vehicles; the failed components (vehicles) can be left in the failed condition for some period of time until an appropriate moment for repair arises. This is not possible in the case of a series system such as process units, where the failure of any item leads to the failure of the whole system. Samrout, Yalaoui, Châtelet, and Chebbo (2005) presented a method to minimize the preventive maintenance cost of series-parallel systems using Ant colony optimisation, but their method deals with only the case of constant failure rates. Duarte, Craveiro, and Trigo (2006) proposed an algorithm for maintenance management optimisation of a series system based on preventive maintenance, where all the components are assumed to exhibit linearly increasing hazard rate and constant repair rate. For many real systems, the assumptions mentioned above must be relaxed to provide more realistic models. This leads to more complexity in the mathematical formulation and/or more efforts in the solution procedure. To deal with these problems, especially for systems with a large number of components, the simulation-oriented approaches can be interesting and may perform well (Charles, Floru, Pantel, Pibouleau, & Domenech, 2003; Tan & Kramer, 1997). In the present work, a decision model is developed on the basis of multi-grouping optimisation that can assist a maintenance manager in making the preventive maintenance plan for series systems. It allows us to determine the replacement time of the system components exhibiting general and different failure rates, in such a way to minimize the total maintenance cost of the whole system. The methodology considers simultaneously corrective, preventive and opportunistic maintenance and allows us to deal with systems constituted by large number of components. The cost objective function for maintenance tasks (preventive, opportunistic and corrective) is defined for the whole system, the formulation takes into account both component and system-related costs. On the basis of informative method for defining the optimal maintenance groups, based on Monte Carlo simulations, the algorithm searches for the preventive
replacement times that minimize the total cost rate, including the total unplanned downtime of the system. 3. Formulation of the maintenance models 3.1. System description Let us consider a system composed of a set of q components arranged in series, the failure of any component leads to the failure of the whole system. Let 1 , 2 , . . ., q be the time intervals between preventive replacements of components 1, 2, . . ., q, respectively (Fig. 1). It is assumed that each replacement restores the component to the “as good as new” state because of its replacement by a new one from the same population (with the same reliability function). During the system downtime, either for preventive or for corrective maintenance, it is to decide if we can take the opportunity to replace preventively some of the non-failed components; this decision should be based on component degradation and risk undertaken if these components fail before reaching the following scheduled preventive time. 3.2. Costs structure The replacement costs can be divided into two parts: • The first part, considered as constant, is related to common system costs, such as mobilizing repair crew, safety provisions, disassembling machine, transportation, tools and the production loss related to the time lost in these tasks. The common system cost can be termed “set-up-cost”; it is noted C0c for corrective p replacement and C0 for preventive replacement. We consider that p C0c is greater than C0 because the emergency character of the corrective intervention does not allow for optimising tools and procedures, the staff is constrained to use available tools and fast but costly procedures. On the other hand, the preventive replacement can be prepared in advance and executed during a period where the production loss penalty is reduced. • The second part, considered as variable, is related to the specific characteristics of the component to be replaced, such as spare part costs, manpower costs, specific tools, and repair procedures. It is also related to production loss during the necessary time to extract and replace the component itself. For the ith component, p the specific corrective and preventive costs are noted Cic and Ci , p c respectively. We consider that Ci is greater than Ci , because the preventive replacement cost includes only the spare part and the manpower costs. Whereas for corrective replacement, in addition to the replacement cost, we must also add the failure cost which includes component-dependent production loss and other fees, such as compensatory solutions regarding the production.
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When a replacement is carried out, the impact on the total system cost is given by the sum of the above two parts of costs. For scheduled preventive maintenance, the system maintenance cost is given by: p
p
Csys = C0 +
p
Ci
as good as new, the replacement is instantaneous and the components are stochastically independent. The system renewal cycle is given by:
q
(1 − Fi (t))dt,
(1)
0
i ∈ Gp
where Gp is the group of components to be preventively replaced at the scheduled maintenance time (i.e. all components that reached the optimal age replacement as it will be detailed below). For corrective maintenance, when the jth component fails, the system is shut down and the failed component is replaced; the opportunity is taken to replace other critical components according to a predefined decision rule as it will be detailed later. In this case, assuming that the preventive replacement cost and the opportunistic one p are the same and noted Ci , the corrective system cost is given by: c Csys,j = C0c + Cjc +
p
Ci
then the system total cost per time unit is given by:
Cmono () =
(3)
i 0
(4)
where i is the time (age) for the preventive replacement of component i and Fi (·) its failure probability distribution, it represents the cumulative distribution function (CDF) of the random variable “time to failure”, and can be obtained from historical failure data.
K q C(, k1 , k2 , . . . , kq ) =
k=1
j=1
p
q
i=1
q
p C i=1 i
[1 − Fsys ()]
[1 − Fi (t)]dt (5)
i=1,...,q
p
(1 − Fi (t))dt
Fsys () + C0 +
components are defined by: i = ki , where ki are integer multipliers satisfying ki ≥ 1 for i = 1, 2, . . ., q (Fig. 1). In this context, the decision variables are (, k1 , k2 , . . ., kq ) where is continuous and ki are integer variables. The expression of the total cost per unit time requires the consideration of the involved costs along a cycle. In this case, the representative cycle is given by the expected time for replacing all the components simultaneously K (Gertsbakh, 2000), with K = lcm{k1 ,k2 ,. . .kq }, where lcm is the least common multiple. At the beginning of a new cycle, the system is completely renewed and becomes as good as new. As illustrated in Fig. 1, the life cycle of the system involves several replacements of the different components, and therefore, decisions concerning opportunistic and preventive replacements should be included in the cost model. At each replacement time k = k, the expected cost is calculated by including corrective, preventive and opportunistic terms. This formulation allows us to write the total expected cost per unit time as following:
3.3.1. Age-based policy for single component Having the corrective and preventive maintenance costs for a component i and its failure distribution, the expected cost per time unit is written by the sum of the expected corrective and preventive costs divided by the expected cycle length (Barlow & Hunter, 1960): Cic Fi (i ) + Ci (1 − Fi (i ))
3.3.3. Opportunistic multi-grouping approach For general industrial systems, the failure rates are very different from one component to another, and hence, a more realistic cost model has to be considered. The idea proposed in this work lies in the optimal definition of the preventive replacement times (ages) and grouping. In other words, the replacement time is selected, such as: (1) at each scheduled replacement, a decision has to be made for each component to define if it should be preventively replaced or not; and (2) at each system failure, a decision has to be taken for each component to see if it should be opportunistically replaced or left as it is until the next scheduled replacement. The final goal is to plan regular preventive replacements where optimal component grouping is defined (Fig. 1). In this way, the basic maintenance time is defined as the minimum replacement time: = min i . In addition, the maintenance times for the different
3.3. Models formulation for several maintenance policies
Ci (i ) =
Cc i=1 i
where Fsys () = 1 − Rsys () = 1 − i=1 [1 − Fi ()] is the failure probability of the overall system and Rsys () is the system reliability. This strategy seems to be suitable only when the systems are composed by identical components (with similar lifetime distributions). However, when the component lifetimes are different, a waste of money is observed when replacing reliable components under the pressure of other less reliable ones.
(2)
expected cost on one cycle C(t) = t expected length of a cycle
q
q
c where Csys,j is the corrective system cost due to the failure of component j, Gh is the group of components to be replaced preventively during this opportunity (Gh is defined according to a deterioration-based rule, analyzing the cost/benefit balance as it will be detailed later). It can be clearly seen in the Eqs. (1) and (2) that the set-up cost is paid only once independently of the number of components replaced at an opportunity. This reflects the opportunistic nature of costs and the economic dependence between components. According to the renewal theory and assuming infinite horizon, the expected cost per unit time is given by (Barlow & Proschan, 1965):
t→∞
C0c +
0
i ∈ Gh i= / j
C(T ) = lim
i=1
C0c + Cjc +
3.3.2. Equivalent mono-component approach Here we assume that all the system components are jointly replaced, at the system failure or after a certain time , whichever occurs first. Knowing that any component failure leads to system failure, assuming that after each replacement the system becomes
p
i ∈ Gh
k
Ci
p
Fsys,j (k) + C0 + K
p
i ∈ Gp
k
Ci
(1 − Fsys (k))
(6)
where Fsys,j (·) is the CDF of system failure due to the jth component. For a series system, the failure of any component leads to the system failure, then Fsys,j (·) = Fj (·). Ghk is a group of components to be replaced opportunistically when a failure occurs in the interval [(k − 1), k], Gpk is the group of components to be replaced preventively at instant k: all i|k/ki = Integer, i = 1, q; k = 1, K.
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Fig. 2. Decision cost basis for opportunistic replacement.
The minimization of the expected cost per time unit aims at finding the best set of the maintenance time i : Find : , k1 , k2 , . . . and kq that minimize : C(, k1 , k2 , . . . kq ) subject to : ≥ 0; ki ≥ 1 and ki are integers
(7)
3.4. Solution procedure As the cost function in problem (7) contains continuous and discrete variables, a solution strategy has to be developed for efficient computation of the optimal replacement plan, especially for large number of components. The numerical solution can be carried out by Monte Carlo simulations, allowing to deal with both integer and continuous variables in optimisation (Marquez, Gupta, & Ignizio, 2006). While the replacement time can be easily optimised by classical algorithms, the implication of the discrete variables ki leads to a very large number of possible combinations. It is thus
necessary to reduce the number of considered combinations for practical systems, without discarding the potentially optimal combinations. The proposed solution is a search method based on conditional information concerning the reliability levels of the components. As the search algorithm requires a starting point, the initial solution can be defined by optimal times for the individual components (minimum of costs in Eq. (4)). This solution gives reasonable initial values for ki , defined by the ratio between the component optimal 0 , which is replacement time i0 and the minimum optimal time min 0 ). Due to system effect and economic written: ki0 = Integer(i0 /min dependence, the search range for optimal groups, defined by ki , can vary from the initial groups, defined by ki0 , in the range ±1 defined by: ki0 − 1 ≤ ki ≤ ki0 + 1 with ki ≥ 1. This gives convenient bounds for optimal search and reduces strongly the number of combinations to only three times the number of components (instead of the factorial).
Fig. 3. Flowchart of the solution algorithm.
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Fig. 4. Example of corrective/opportunistic/preventive maintenance simulation.
The large number of random simulations ensures the stability of the estimates and guarantees the solution convergence to the optimal plan. As all significant combinations are considered, the optimal solution cannot be missed in the proposed procedure. It is also to note that the stability and the convergence of Monte Carlo simulations have been approved, in solving nonlinear and complex optimisation problems. In addition, the deterioration-based decision can be included by analyzing the cost/benefit balance of the component to be preventively replaced. Let us consider the case where the jth component fails at the time tj between two scheduled maintenance times k and (k + 1), as illustrated in Fig. 2. The opportunity of replacing the p component i leads to an expected cost Ci Ri (tj ). If the ith component is left without replacement, two cases are possible: either it remains operating until the following scheduled replacement, which will p cost Ci Ri ((k + 1)|tj ), or it fails before, leading to system breakdown, the corresponding cost is (C0c + Cic )Fi (t) with a maximum value at (k + 1). The decision-making criterion for opportunistic replacement can thus be defined by comparing the two costs. If the opportunistic replacement cost is less than the corrective one, it is better to change the component; otherwise it can be left till the following planned replacement: p
If : Ci (Ri (tj ) − Ri ((k + 1)|tj )) ≺ (C0c + Cic )Fi ((k + 1)|tj ) Otherwise
initial grouping configuration ki0 ; the solution procedure is given as following: 1. Generate random samples of the lifetimes of the components ti , according to the failure probability distributions. Then, the system failure time is defined by: tsys = min ti and the corresponding i=1,q
failed component producing the system shutdown is identified. The replacements are scheduled at the times k, where k is an integer varying from 1 to K. 2. At the kth replacement, the simulated system failure time tsys is compared with the scheduled time for preventive maintenance k. Two possibilities exist: a. If no failure is observed before k, the preventive maintenance can be carried out at k, according to the current grouping rule, as defined in the updated plan, and a move to the next scheduled time (k + 1) is done. b. If failure is observed, the system is down and the failed component is correctively replaced. On the basis of the conditional strategy described above, the opportunity of replacing other
⇒ then, make opportunistic replacement of component i at the time tj ⇒ leave it till the next scheduled replacement
The flowchart of the developed algorithm is depicted in Fig. 3. After introducing the component failure data (probability distribup tions and parameters), the cost parameters (C0c , C0 , Cic , Cic ) and the
components is considered and the related preventive costs are computed.
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3. For the replaced components in step 2, new lifetimes are generated (as new components are installed); a move to the next replacement time (k + 1) is performed, and so on, until the end of the system cycle (until the replacement of all components simultaneously). 4. For the simulated scenario, the life cycle length and the corresponding total cost are computed. 5. Repeat steps 1–4, to generate new scenarios by random sampling, until the prescribed number of simulations is reached. 6. The expected total cost per unit time is estimated in terms of the mathematical average of the computed costs and the cycle span of the simulated scenarios. For all the sampled scenarios, the total cost per unit time in terms of the statistical expectations is given by:
⎛
⎛⎛
⎞
p 1 ⎝ ⎝⎝C0c + Cic + Ci ⎠ IFk,s E[C(t)] = N N
s=1
K
k=1
⎛ + ⎝C0 + p
⎞ p
Ci
⎞⎞
⎠ IRk,s ⎠⎠
(8)
where N is the total number of simulations, IFk,s and IRk,s are binary indicators for the states of failure and operation, respectively. For the kth replacement interval of the sth simulation, these indicators are defined by: IFk,s = 1; IRk,s = 0 IFk,s = 0; IRk,s = 1
Table 1 Data and optimisation results for the illustrative example. p
Component
ˇ
cic (D )
ci (D )
Topt (days)
CT (D /day)
1 2
500 200
2.5 2.7
5000.00 400.00
600.00 200.00
177 74
11.57 17.59
in Fig. 4b. New lifetimes are then generated for the new components (i.e. components 1 and 2). The scheduled replacement at is now examined; in our example, components 1, 2 and 3 are preventively replaced (Fig. 4b). In the next replacement interval ( ≺ t ≤ 2), components 1 and 3 appear to have high deterioration rate and are then replaced at 2 (Fig. 4c). Then, component 5 shows a failure before reaching the scheduled replacement at 3, and so on. This illustration shows how a component can be dynamically considered in function of the possible opportunities. 4. Illustrative example with two components
i ∈ Gh
i ∈ Gp
1505
if failure if operation
7. The procedure is repeated for different replacement intervals and grouping configurations ki . The search for the optimum solution allows us to update the scheduled maintenance plan, by changing and ki , in order to define a better combination of component grouping. The iterative scheme is stopped when the optimum solution cannot be improved. Fig. 4 shows an example of a sample of five components, where the scheduled preventive replacement has a span equal to . The above described algorithm is applied to generate random scenarios. The component lifetimes are firstly generated, as shown in Fig. 4a; it is observed that component 2 fails before the scheduled replacement and should be correctively replaced. Given the deterioration of component 1, basing on the decision rule provided above, the opportunity of replacing it preventively is carried out, as illustrated
In order to illustrate the proposed policy, let us consider the case of two components with the failure and cost data in Table 1. The corrective and preventive common costs are c0c = 8000 D and p c0 = 600 D , respectively. Table 1 gives also the optimal times and costs for separate replacement of the two components (Eq. (4)). 4.1. Mono-group policy If the two components have to be changed simultaneously (mono-group), the optimal time is 71 days with a cost per unit time of 31.59 D /day (Fig. 5). It can be seen that mono-grouping is less interesting than independent replacement of each one of the two components, as the sum of the costs in Table 1 leads to 29.18 D /day. 4.2. Multi-group policy Even for this simple problem, the choice of the optimal policy is not that obvious. For independent replacements, the optimal time ratio is given by: 177/74 = 2.39; it is not an evidence to say that the policy {2,} is better than {3,}. Table 2 shows the results for different combinations. It is clearly seen that policy 2-1 (i.e. k1 = 2; k2 = 1) leads to the lowest cost per unit time (23.822 D /day), followed by the policy 3-1 (i.e. k1 = 3; k2 = 1) with 24.625 D /day. The policies 1-2 and 3-2 are far from being optimal, which could be expected by looking at the single component results. In this example, the policy 2-1 leads to a cost saving of 8% with respect to simultaneous replacement of the two components (i.e. policy 1-1).
Fig. 5. Expected maintenance cost rate for single components and mono-group policies.
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Table 2 Optimal solutions for different replacement groups. Replacement groups k1 -k2
System life cycle (days)
(days)
Optimal replacement times, 1 / 2 (days)
Minimum expected cost (D /day)
1-1 1-2 2-1 3-1 3-2
68 158 130 189 372
68 79 65 63 62
68/68 79/158 130/65 189/63 186/124
25.797 33.259 23.822 24.625 32.285
Fig. 6. Maintenance cycle of the two-component system.
In order to validate the solution procedure, let us now consider the policy 2-1 (i.e. component 1 is replaced each two replacements of component 2). The maintenance cycle for this policy is depicted in Fig. 6. At a given time, each component may have one of the three possibilities: (1) surviving till the next scheduled replacement, (2) failing before reaching the scheduled time, or (3) replacement by opportunity when other components fail or replaced preventively. Table 3 shows the convergence of the expected cost estimates in terms of the number of random samples. Convergence is clearly shown by the monotonic decrease of the coefficient of variation (C.O.V.) with the increase of the sample size. It is also shown that the cost estimate becomes stable for more than 500 Monte Carlo samples. High precision is reached for cost estimate with 10000 samples, as the coefficient of variation is less than 0.1%. For this simple case, it is possible to write analytically Eq. (6) without opportunistic terms, the result leads to the same replacement time as Monte Carlo (i.e. 65 days) with the cost 25.49 D /day. As it could be expected, Monte Carlo method converges to lower cost estimate due to the consideration of opportunistic replacement and real failure times between the scheduled replacements. Table 3 Convergence of Monte Carlo simulations. Number of samples
Mean cost estimate
C.O.V. of the estimate
10 50 100 500 1,000 5,000 10,000 50,000
14.902 19.100 22.990 23.938 23.872 23.801 23.817 23.804
5.30% 2.27% 2.11% 0.42% 0.37% 0.12% 0.09% 0.01%
5. Industrial application 5.1. Refinery centrifugal compressor The proposed methodology is applied to the centrifugal compressor of the catalytic reforming unit of Skikda refinery, which is the most important oil refinery in Algeria; the first units started production in 1980, its capacity is about 15 millions tons/year. The multiple staging compressor is driven by a steam turbine; it is essentially constituted by the stator (diaphragms, landings, tightness subsystem) and the rotor (shaft, wheels, equilibrium piston. . .). The case study is chosen according to an earlier study (Laggoune & Aissani, 2000), which has revealed that the hydrogen compressor was the leading cause of Skikda catalytic reformer breakdowns. This confirms the fact, stated in (Tan & Kramer, 1997), that pumps and compressors are the leading equipment failures in refineries. The catalytic reforming objective is to convert low-octane naphtha, obtained from the topping unit, into high-octane reformate for gasoline blending and/or to provide aromatics (benzene, toluene, and xylene) for petrochemical plants. It also produces highly purified hydrogen for hydro-treating processes. The major catalytic reforming reactions are: dehydrogenation of naphthenes to aromatics (C6 ); isomerisation of naphthenes to aromatics (C5 ); dehydrocyclization of paraffins to aromatics; isomerisation of normal paraffins to iso-paraffins; and hydrocracking of paraffins into smaller molecules. The catalytic reforming unit of Skikda refinery is a semiregenerative type, with successive reactors, each one with a fixed bed of catalyst (Fig. 7). The catalyst used contains platinum and rhenium on a chlorinated alumina support base. The noble metals (platinum and rhenium) are catalytic sites for the dehydrogenation reactions and the chlorinated alumina provides the acid sites needed for isomerisation, cyclization and hydrocrack-
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Fig. 7. Schematic diagram of a typical semi-regenerative catalytic reforming unit.
ing reactions. It is worth to note that the catalyst effectiveness in a semi-regenerative catalytic reformer is reduced over time during operation by carbonaceous coke deposition and chloride loss. Therefore the catalyst is periodically regenerated in situ by high temperature oxidation of the coke followed by chlorination. To underline the vital role of the compressor in the reforming unit, we summarize its function in the process. Indeed, the necessary hydrogen for the various reactions of catalytic reforming is recycled by the compressor; it is the presence of hydrogen which limits the formation of coke. The speed of coke formation is very influenced by the hydrogen pressure; a compressor dysfunction can involve a pressure loss and consequently an acceleration of coking. Reduction in the hydrogen flow implies the reduction in hydrocarbons to maintain a preset H2 /HC rate, in other words a reduction in the load rate which would generate production losses. In addition, the compressor is necessary for the unit pre-heating during operation starting after shut-down; it also provides the necessary air for the coke combustion during catalyst regeneration. 5.2. Data collection and analysis 5.2.1. Failure data for reliability analysis The data available at the refinery site are the failure dates of the compressor, the component inducing failure, the preventive maintenance dates and the maintenance durations. The first analysis consists in extracting the times to failure for each component in order to constitute the samples, the times to preventive replace-
ment are considered as censored data; so the data sample includes failures and censored data (Table 4). The samples are then fitted by two-parameter Weibull model, this model was chosen because it adjusts well to mechanical components, it allows describing alternately the three phases of the component life (early-life, mid-life, and wear-out), and it is efficient with a minimum of only five failures (Summers-Smith, 1989). The Weibull reliability function is given by:
ˇ t
R(t) = 1 − F(t) = exp −
,
where t is the time to failure of individual components, is the scale parameter and ˇ is the shape parameter. The most important parameter is ˇ, it takes different values depending on the component life phase (failure mode): When ˇ < 1, the failure rate is decreasing (infantile mortality); when ˇ = 1, the failure rate is constant (hazardous failure); and when ˇ > 1, the failure rate is increasing (wear-out). The Weibull parameter estimation is obtained by the “Statistica” software (Table 4), which uses the maximum likelihood method, and allows us to deal with censored samples. 5.2.2. Cost data The maintenance cost assessment is a very hard and complex task, as it depends on many parameters. Therefore, we have to use some simplifying assumptions and criteria to provide approx-
Table 4 Failure data and Weibull parameters of the system components. Component
Code
Sample size
Observed failures
Shape parameter ˇ
Scale parameter
MTBF days
Sheathing Sheathing Tightness Stub bearing Tightness ring Carrying bearing Stub bearing Labyrinth support
C286 C285 C275 C230 C460 C419 C401 C780
24 23 21 21 34 15 17 14
14 15 15 8 14 8 8 7
1.73 1.88 2.43 2.53 2.14 3.55 2.68 2.09
486 507 286 898 905 736 1094 1388
483 475 240 787 844 636 888 1047
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Table 5 Common costs and maintenance costs. Component
Code
Corrective cost (D )
Common costs Sheathing Sheathing Tightness Stub bearing Tightness ring Carrying bearing Stub bearing Labyrinth support
C286 C285 C275 C230 C460 C419 C401 C780
36,000 14,868 39,204 44,880 57,876 73,860 46,752 48,568 74,232
Preventive cost (D )
Cost ratio corrective/preventive
1,200 3,639 5,438 7,398 8,277 13,554 14,130 21,356 24,348
imations close to the real costs, that is why the mean costs are often used. The common costs include the set-up cost and the production loss. The set-up cost is obtained by the mean cost of the previous maintenance operations multiplied by an actualisation factor. The production loss is obtained by multiplying an hourly cost by a mean time making the equipment busy when subjected to the maintenance operation; it includes the entire time span since the production stopping and system disassembly to reach the component concerned by the replacement. During a preventive maintenance, most provisions are usually taken before stopping the production, such as scaffolding installation, safety provisions, getting available maintenance crew and tools. Therefore a preventive replacement takes only the necessary time to open the system; consequently the corrective common cost is larger than the preventive one. The component preventive cost includes the spare part and labour costs; the corrective one includes the spare part, the labour, the production loss, and other fees. The production loss added to the component corrective cost concerns only the necessary time to extract the component and to replace it by a new one. Table 5 gives the specific component costs, as well as the system common cost. The large ratio of corrective to preventive costs can be easily observed for these data. The maintenance optimisation is carried out according to different assumptions in order to compare the optimal solutions and to show the benefits of the proposed maintenance plan.
30 4.1 7.2 6.1 7.0 5.4 3.3 2.3 3.1
Fig. 8. Expected cost rates for the separate components.
5.3. Maintenance optimisation 5.3.1. Single component maintenance optimisation As a first step, the policy based on separate components is considered to obtain optimal replacement times for individual components. The expected cost is computed for each component, independently, and a one-by-one optimisation is applied. Fig. 8 shows the expected total costs for individual components (Eq. (4)). Table 6 gives the optimal solutions for individual components. As discussed above, the ratios of individual optimal times can be used as a starting point for group maintenance plan. From Table 6, these 0 , leading to: k ˜ 1 = 1.82, k˜ 2 = 1.63, ratios are given by: k˜ i = i0 /min ˜k3 = 1.0, k˜ 4 = 3.09, k˜ 5 = 3.73, k˜ 6 = 3.56, k˜ 7 = 6.82 and k˜ 8 = 7.89.
Fig. 9. Expected cost rate for the mono-group policy.
5.3.2. Mono-group policy In order to underline, the system effect, the whole system is now considered as a macro-component, where simultaneous replacement is performed for all the components (Eq. (5)). The expected cost, depicted in Fig. 9, shows a minimum at 99 days, with an optimal cost of 1864.92 D for the whole system. Naturally, this solution is not optimal as it does not take into account the specific costs related to different components.
0 )). Later, the nearest integer is used for ki (ki0 = Integer(i0 /min
Table 6 Optimal solutions for individual components. Component
Code
MTBF (days)
Optimum time, i0 (days)
Cost (D /day)
0 Rate, k˜ i = i0 /min
Sheathing Sheathing Tightness stub bearing Tightness ring Carrying bearing Stub bearing Labyrinth support
C286 C285 C275 C230 C460 C419 C401 C780
483 475 240 787 844 636 888 1047
197 176 108 334 403 385 737 852
68.7588 87.5325 139.541 48.2451 74.9058 56.5187 55.2709 70.7366
1.82 1.63 1.00 3.09 3.73 3.56 6.82 7.89
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Table 7 Optimal solutions for different replacement groups. No.
Replacement groups, k1 -k2 -k3 -k4 -k5 -k6 -k7 -k8
1 2 3 4 5 6 7 8 9 10
1-1-1-1-1-1-1-1 1-1-1-4-4-2-7-8 1-1-1-3-4-4-6-7 1-1-1-3-4-4-7-7 2-2-1-4-3-2-6-6 2-2-1-4-4-4-7-7 2-2-1-3-4-4-7-8 2-2-2-2-3-3-6-6 2-2-2-2-3-3-6-7 2-2-2-4-4-4-6-8
System life cycle (days) 99 6,944 18,648 9,240 1,620 2,940 20,832 744 5,208 2,520
5.3.3. Opportunistic multi-grouping optimisation The proposed solution is based on optimal grouping of components replacement (Eq. (6)) including opportunistic replacements. Regarding the results obtained when considering individual components, the natural and evident grouping is 2-2-1-3-4-4-7-8 with = 124 days, but this configuration is not optimal as it is shown in Table 7, indicating the optimal solutions for significant grouping configurations. From this table, it can be shown that the minimum cost is achieved for the configuration 2-2-2-2-3-3-6-6 with = 124 days where three groups are considered, according to k1 = k2 = k3 = k4 = 2, k5 = k6 = 3 and k7 = k8 = 6: (1) a group of compressor components (C286, C285, C275 and C230) with replacement at the age of 248 days, (2) a group of turbine components (C460 and C419) with replacement at the age of 372 days and (3) another group of turbine components (C401 and C780). The minimal cost is only 534.871 D , which represents 71% of reduction with respect to the case of only one group (1-1-1-1-1-1-1-1) and 5% with respect to the evident grouping. This reduction shows clearly the importance of the optimisation over groups and the interest of the proposed model. It shows also the capacity and the stability of the procedure in dealing with practical systems where no prior solution can be provided. 6. Conclusions The proposed maintenance plan is based on opportunistic multigrouping replacement optimisation for multi-component systems. As in continuous operating units such as chemical plants, the production losses are very high. Therefore, the maintenance optimisation has the potential for substantially reducing the operating costs and for increasing corporate profit by increasing availability and production. The proposed approach is based on the analysis of individual components, according to the well-known age-based model. The optimisation algorithm allows rearranging the optimal individual replacement times in such a way that all component times become multiple of the smallest one to allow for joint replacements. In this way, the times obtained by the multi-grouping approach do not give individual optimality conditions, but satisfies the optimal cost regarding the whole system. The proposed algorithm is a numerical procedure, where the life cycles are simulated and the optimal solution is numerically searched. The large number of random simulations by Monte Carlo method, which is very useful in solving nonlinear and complex optimisation problems, ensures the stability of the estimates and guarantees the solution convergence to the optimal one. As all significant combinations are considered, the optimal solution cannot be missed. The most important facts revealed in this study are: (1) the effectiveness of the opportunistic policy in cost saving for multi-component systems and the capacity of the Monte Carlo simulations in solving these complex problems, and (2) the evident grouping configuration, which is sometimes adopted by
(days)
Optimal replacement times, 1 / 2 / 3 / 4 / 5 / 6 / 7 / 8 (days)
Minimum expected cost (D /day)
99 124 111 110 135 105 124 124 124 105
99/99/99/99/99/99/99/99 124/124/124/496/496/248/868/992 111/111/111/333/444/444/666/777 110/110/110/330/440/440/770/770 270/270/135/540/405/270/810/810 210/210/105/420/420/420/735/735 248/248/124/372/496/496/868/992 248/248/248/248/372/372/744/744 248/248/248/248/372/372/744/868 210/210/210/420/420/420/630/850
1864.92 590.894 576.362 576.039 561.913 564.305 560.669 534.871 535.111 540.032
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