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Choice criteria in conditional preventive maintenance
Mechanical Systems and Signal Processing (1999) 13(1), 163–168 Article No. mssp.1998.0176, available online at http://www.idealibrary.com on
SHORT PAPER CHOICE CRITERIA IN CONDITIONAL PREVENTIVE MAINTENANCE S. L Groupe de Me´canique Applique´e, Universite´ de Reims Champagne, Ardenne, BP1053 51687 , Reims Cedex 2 , France
(Received October 1997 , accepted after revisions June 1998) The purpose of this study was to improve the availability of production equipment by selecting the best maintenance management method. Corrective maintenance, systematic preventive maintenance and conditional maintenance costs are compared and the Weibull law is used to represent reliability and evaluate maintenance costs. Finally, an example is presented from a flow show factory, and the maintenance servise is shown to be more efficient through better organisation and fewer costs.
7 1999 Academic Press
1. GOAL
The reduction of production costs, by reducing stocks, has made production systems increasingly vulnerable to risks, and so breakdowns need to be reduced as much as possible. Thus the maintenance service has become a key sector of production. After decades of corrective maintenance, systematic preventive maintenance has emerged, but its costs remain high, since parts are changed before the end of their service life. Nowadays, many companies have changed to a conditional preventive maintenance. However, its installation is often costly while results are difficult to predict. This article offers a procedure that allows the company to choose the most suitable maintenance method best, or to calculate the maximum investment allowed, so that the installation of a preventive conditional maintenance system will remain the most economical. The choice criteria is always the production cost. The reliability of parts is calculated using the Weibull law, in order to calculate their probability of malfunction and to compare them with maintenance costs. The maintenance methods evaluated here are corrective maintenance, systematic preventive maintenance, and conditional preventive maintenance. The procedure concerns linear and non-linear production systems. Systems that include risks, for which maintenance management is security management for personnel and goods, are excluded. This method has been adapted to an industrial situation, in a mass-production factory.
2. CALCULATION OF MAINTENANCE COSTS
For the calculations, two types of costs are considered. The expenses of the maintenance service have been put together under the name of intervention costs. This includes the costs 0888–3270/99/010163 + 06 $30.00/0
7 1999 Academic Press
.
164
of parts, material, and maintenance personnel needed for the repairs of equipment, or for the replacement of parts in a poor condition. The consequences of a breakdown are considered as a penalty, which includes the shutting down of the machinery, delays in, and disorganisation of, production. Although the exact value of the different parameters can be difficult to obtain, analysing the data using a computer-aided maintenance system allows a sufficiently accurate evaluation [1]. 2.1. If I is the cost of the intervention of maintenance, and P is the penalty for breakdown, then Boucly [2] offers a method of comparison between the costs of corrective maintenance and systematic preventive maintenance. If the law of service life is placed in a model by b the Weibull law, R(t) = e−((t − g)/h) , then the ratio of systematic preventive maintenance costs and corrective maintenance costs is: R=
1 + (1 − R'(Xs ))r G[1 + 1/b] × Xs 1+r f0 R'(x) dx
(1) b
where r is P/I, the ratio of the penalty and intervention costs; R'(x) = e−(x) ; x = (t − g)/h, t is the variable of time; b, g and h are the parameters of the Weibull law; G[1 + 1/b] is a law showing the MTBF; and Xs = (Ts − g)/h. This ratio is then a function of the systematic exchange period, Xs = (Ts − g)/h, and two parameters: the ratio P/I and the coefficient b. Study of this function shows that, in some cases, a minimum below 1 is found. In such cases, the value of the ratio R (possible gains of systematic maintenance in relation to corrective maintenance) and the optimal period of systematic exchange is found to be Ts = hXs + g. An example is presented in Fig. 1. 2.2. Conditional preventive maintenance is intended to prevent sudden breakdowns. If a surveillance capable of preventing all unexpected breakdowns is assumed, there no longer exists a penalty for breakdown. Only intervention costs remain, to which one must add the cost of surveillance. The cost of surveillance is caused by a sole investment, or by continuing expenses. The investment corresponds to the cost of acquiring surveillance material, or to the cost of training personnel in analysis. Thus, a time period for write-off is fixed. The permanent expenses correspond to a use of products, and, most of all, services,
5
1.5
2 1.0 20 10 0.5 100 0.0 0.0
0.4
0.8
1.2
Xs Figure 1. Ratio of systematic preventive maintenance cost on corrective maintenance cost with b = 3 and r ranging from 1 to 100.
165
costs of measures, and of their analysis. This study accounts for the working hours of the maintenance service, used for collecting and analysing the data, as well as operations put out to contract. In both cases, these figures could be shown as an average surveillance cost, S, per operating hour of equipment. For conditional maintenance, the delay between two interventions is approximately equal to MTBT. If surveillance operations are done outside operating hours, its cost is: Cp =
I+S . MTBF
(2)
If surveillance operations of conditional maintenance require an interruption of production, Ps , the cost of the penalty for performing the surveillance, is added, which makes S' = S + Ps the new cost of surveillance. This allows this case to be treated identically: Cp =
I + S' . MTBF
(3)
If not all breakdowns are avoided, the risk of breakdown is evaluated as a chance of non-detection. This chance cannot be calculated, but, in principle, it can be found by experiment, by analogy to existing situations, in relation to the surveillance technique and the frequency of measures. Thus the mathematical chance of the cost of penalty for breakdown is calculated using: Esp(P) = chance(non-detection) × P. One only has to note S0 = S + Esp(P) to find a recognisable format: Cp =
I + S0 . MTBF
(4)
3. CHOICE OF A TYPE OF MAINTENANCE
In order to have any values, conditional preventive maintenance must have a cost inferior to those of the two other types of maintenance. They can be compared by calculating the ratio between conditional costs and corrective costs, which has to be inferior of R(1) or 1 if R is bigger than 1: Cp /Cc =
I+S MTBF I + S 1 + S/r × = = MTBF I+P I+P 1+r
(5)
as illustrated in Fig. 2. This graph can be used for two purposes: either the cost S of surveillance is known, and it can be determined immediately if the ratio is inferior to R, which has been calculated beforehand; or S is unknown, and one can find the maximum
1.50 1.00 0.50 0.00 0.2
2.6
8.0 Ratio S/I
16.0
55.0
Figure 2. Ratio of conditional preventive maintenance cost to corrective maintenance cost. r values: W, 1; Q, 2; E, 5; (, 10; w, 20; q, 100.
.
166
Data I and P
No
Is systematic maintenance possible? Yes Data β, γ, η Ratio R (Fig. 1)
No
Is systematic maintenance best?
Yes
Systematic maintenance Ts = η Xs + γ R = Cs/Cc
Corrective maintenance (R = 1)
Fig. 2
No
Yes
No
Ratio < 1
Corrective maintenance
Yes Ratio < R
Conditional maintenance
Systematic maintenance
Conditional maintenance
Figure 3. Choice of maintenance method.
admissible cost of surveillance by using R. Most often, this second situation is used to determine the acceptable investment for surveillance [3, 4]. Figure 3 shows a grow chart of the choice-making procedure. 4. INDUSTRIAL EXAMPLE
4.1. The factory examined cutting presses for iron, and the six biggest have a capacity of 200–450 t. Depending on the required production, these machines work either autonomously, or in a chain. These machines are critical to production; as any absence of the primary parts could stop the activity of other assembly stations further down the process. The reduction of stock of products ‘in process’, which is demanded of each station, increases the risk, since intermediary stocks are reduced year after year. 4.2. A classification of the duration of maintenance interventions shows that the clutch is the crucial part of the machines. The cost of maintenance—parts, work hours, and contracting out of activities—for the clutches represents over 90% of the total maintenance
167
3 1
2
5 2
10
1
20 100 0 0.0
0.4
1.2
0.8
1.6
2.0
Xs Figure 4. Ratio of systematic maintenance cost on corrective maintenance cost for b = 1.6
costs of these machines. One of these presses is particularly costly, with 910 h of intervention and at a cost of 590 000 francs in 5 years. The listed costs are the costs for delivery of parts, material and service. The historical study reveals six interventions for breakdowns, which allows the following Weibull model to be used: b = 1.67, g 3 0, h = 43 weeks, G[1 + 1/b] = 0.89. The Kolmogorov–Smirnov test is positive, so the model is valid. The cost of one intervention is high. The total duration of a repair is 130 h, to which 4000 francs worth of parts have to be added. This totals about 30 000 francs for each intervention to change the clutch disks. A breakdown requires an immediate, unscheduled intervention of maintenance personnel, often during overtime, or the repair is contracted out, depending on the importance of the work to be done. Furthermore, the breakdown disorganises surrounding production, and production following that particular station, which means operational delays of operators and a loss of production. Therefore, the cost for a breakdown is estimated to be twice the cost of the intervention, which is about 60 000 francs. 4.3. Taking into account the cost of the parts concerned, the systematic maintenance is managed individually, so a planned intervention can be executed without penalty for the production. The curve to use is therefore the ratio between systematic and corrective maintenance, Cs /Cc with b = 1.6 (Fig. 4). A ratio, r = P/I = 2, gives a minimum for Xs = 1.1 with R = Cs /Cc being 0.95. This means that preventive systematic maintenance is 5% less expensive than corrective maintenance, with an optimal intervention period of Ts = Xs h + g = 47 weeks. In that case, the average duration of operation between two interventions is: h
g
Xs
R(t) dt = 0.7445 × 43 = 32 weeks.
0
Figure 2 allows the costs of conditional preventive maintenance and systematic preventive maintenance to be compared. Conditional preventive maintenance can be used until the cost reaches a value corresponding to about S/I = 2. A more detailed calculation
168
.
can be done quickly. The ratio between conditional cost and corrective cost has to be inferior to R, i.e. Cp 1 + S/I = QR Cc 1+r \ S/I Q [R(1 + r) − 1] = 0.95(1 + 2) − 1 = 1.85. With I = 30 000 francs, S is, at the most, 55 500 francs. This is the maximum budget that can be allocated to surveillance between the two service intervals, i.e. during the MTBF, with the condition that surveillance is efficient and avoids breakdowns. This corresponds to a yearly budget of 76 000 francs. An investment of 30 000–40 000 francs allows for the installation of a chain of vibration analysis measures, and thus improvements on the maintenance budget of this machine. 5. CONCLUSION
The described procedure allows to choose, as a function of the law of service life and maintenance costs, the most economical maintenance method. Later it would be possible to install a more efficient maintenance service, since it is better organised, and less costly. REFERENCES 1. S. L 1997 Association Universitaire de Me´canique 4, 545–548. Pour une G.M.A.O. plus efficace. 2. F. B 1989 Achats et entretien 336, 27–33. Condition optimale de remplacement pre´ventif. 3. S. L 1994 The`se de doctorat de l’Universite´ de Reims Champagne Ardennes, Reims, France. Ame´lioration de la disponibilite´ des e´quipements de production par l’optimisation de la gestion des stocks de maintenance. 4. S. L and A. P 1995 Revue d’Automatique et de Productique Applique´es 8, 285–290. Gestion de stock en maintenance corrective.
SHORT PAPER CHOICE CRITERIA IN CONDITIONAL PREVENTIVE MAINTENANCE S. L Groupe de Me´canique Applique´e, Universite´ de Reims Champagne, Ardenne, BP1053 51687 , Reims Cedex 2 , France
(Received October 1997 , accepted after revisions June 1998) The purpose of this study was to improve the availability of production equipment by selecting the best maintenance management method. Corrective maintenance, systematic preventive maintenance and conditional maintenance costs are compared and the Weibull law is used to represent reliability and evaluate maintenance costs. Finally, an example is presented from a flow show factory, and the maintenance servise is shown to be more efficient through better organisation and fewer costs.
7 1999 Academic Press
1. GOAL
The reduction of production costs, by reducing stocks, has made production systems increasingly vulnerable to risks, and so breakdowns need to be reduced as much as possible. Thus the maintenance service has become a key sector of production. After decades of corrective maintenance, systematic preventive maintenance has emerged, but its costs remain high, since parts are changed before the end of their service life. Nowadays, many companies have changed to a conditional preventive maintenance. However, its installation is often costly while results are difficult to predict. This article offers a procedure that allows the company to choose the most suitable maintenance method best, or to calculate the maximum investment allowed, so that the installation of a preventive conditional maintenance system will remain the most economical. The choice criteria is always the production cost. The reliability of parts is calculated using the Weibull law, in order to calculate their probability of malfunction and to compare them with maintenance costs. The maintenance methods evaluated here are corrective maintenance, systematic preventive maintenance, and conditional preventive maintenance. The procedure concerns linear and non-linear production systems. Systems that include risks, for which maintenance management is security management for personnel and goods, are excluded. This method has been adapted to an industrial situation, in a mass-production factory.
2. CALCULATION OF MAINTENANCE COSTS
For the calculations, two types of costs are considered. The expenses of the maintenance service have been put together under the name of intervention costs. This includes the costs 0888–3270/99/010163 + 06 $30.00/0
7 1999 Academic Press
.
164
of parts, material, and maintenance personnel needed for the repairs of equipment, or for the replacement of parts in a poor condition. The consequences of a breakdown are considered as a penalty, which includes the shutting down of the machinery, delays in, and disorganisation of, production. Although the exact value of the different parameters can be difficult to obtain, analysing the data using a computer-aided maintenance system allows a sufficiently accurate evaluation [1]. 2.1. If I is the cost of the intervention of maintenance, and P is the penalty for breakdown, then Boucly [2] offers a method of comparison between the costs of corrective maintenance and systematic preventive maintenance. If the law of service life is placed in a model by b the Weibull law, R(t) = e−((t − g)/h) , then the ratio of systematic preventive maintenance costs and corrective maintenance costs is: R=
1 + (1 − R'(Xs ))r G[1 + 1/b] × Xs 1+r f0 R'(x) dx
(1) b
where r is P/I, the ratio of the penalty and intervention costs; R'(x) = e−(x) ; x = (t − g)/h, t is the variable of time; b, g and h are the parameters of the Weibull law; G[1 + 1/b] is a law showing the MTBF; and Xs = (Ts − g)/h. This ratio is then a function of the systematic exchange period, Xs = (Ts − g)/h, and two parameters: the ratio P/I and the coefficient b. Study of this function shows that, in some cases, a minimum below 1 is found. In such cases, the value of the ratio R (possible gains of systematic maintenance in relation to corrective maintenance) and the optimal period of systematic exchange is found to be Ts = hXs + g. An example is presented in Fig. 1. 2.2. Conditional preventive maintenance is intended to prevent sudden breakdowns. If a surveillance capable of preventing all unexpected breakdowns is assumed, there no longer exists a penalty for breakdown. Only intervention costs remain, to which one must add the cost of surveillance. The cost of surveillance is caused by a sole investment, or by continuing expenses. The investment corresponds to the cost of acquiring surveillance material, or to the cost of training personnel in analysis. Thus, a time period for write-off is fixed. The permanent expenses correspond to a use of products, and, most of all, services,
5
1.5
2 1.0 20 10 0.5 100 0.0 0.0
0.4
0.8
1.2
Xs Figure 1. Ratio of systematic preventive maintenance cost on corrective maintenance cost with b = 3 and r ranging from 1 to 100.
165
costs of measures, and of their analysis. This study accounts for the working hours of the maintenance service, used for collecting and analysing the data, as well as operations put out to contract. In both cases, these figures could be shown as an average surveillance cost, S, per operating hour of equipment. For conditional maintenance, the delay between two interventions is approximately equal to MTBT. If surveillance operations are done outside operating hours, its cost is: Cp =
I+S . MTBF
(2)
If surveillance operations of conditional maintenance require an interruption of production, Ps , the cost of the penalty for performing the surveillance, is added, which makes S' = S + Ps the new cost of surveillance. This allows this case to be treated identically: Cp =
I + S' . MTBF
(3)
If not all breakdowns are avoided, the risk of breakdown is evaluated as a chance of non-detection. This chance cannot be calculated, but, in principle, it can be found by experiment, by analogy to existing situations, in relation to the surveillance technique and the frequency of measures. Thus the mathematical chance of the cost of penalty for breakdown is calculated using: Esp(P) = chance(non-detection) × P. One only has to note S0 = S + Esp(P) to find a recognisable format: Cp =
I + S0 . MTBF
(4)
3. CHOICE OF A TYPE OF MAINTENANCE
In order to have any values, conditional preventive maintenance must have a cost inferior to those of the two other types of maintenance. They can be compared by calculating the ratio between conditional costs and corrective costs, which has to be inferior of R(1) or 1 if R is bigger than 1: Cp /Cc =
I+S MTBF I + S 1 + S/r × = = MTBF I+P I+P 1+r
(5)
as illustrated in Fig. 2. This graph can be used for two purposes: either the cost S of surveillance is known, and it can be determined immediately if the ratio is inferior to R, which has been calculated beforehand; or S is unknown, and one can find the maximum
1.50 1.00 0.50 0.00 0.2
2.6
8.0 Ratio S/I
16.0
55.0
Figure 2. Ratio of conditional preventive maintenance cost to corrective maintenance cost. r values: W, 1; Q, 2; E, 5; (, 10; w, 20; q, 100.
.
166
Data I and P
No
Is systematic maintenance possible? Yes Data β, γ, η Ratio R (Fig. 1)
No
Is systematic maintenance best?
Yes
Systematic maintenance Ts = η Xs + γ R = Cs/Cc
Corrective maintenance (R = 1)
Fig. 2
No
Yes
No
Ratio < 1
Corrective maintenance
Yes Ratio < R
Conditional maintenance
Systematic maintenance
Conditional maintenance
Figure 3. Choice of maintenance method.
admissible cost of surveillance by using R. Most often, this second situation is used to determine the acceptable investment for surveillance [3, 4]. Figure 3 shows a grow chart of the choice-making procedure. 4. INDUSTRIAL EXAMPLE
4.1. The factory examined cutting presses for iron, and the six biggest have a capacity of 200–450 t. Depending on the required production, these machines work either autonomously, or in a chain. These machines are critical to production; as any absence of the primary parts could stop the activity of other assembly stations further down the process. The reduction of stock of products ‘in process’, which is demanded of each station, increases the risk, since intermediary stocks are reduced year after year. 4.2. A classification of the duration of maintenance interventions shows that the clutch is the crucial part of the machines. The cost of maintenance—parts, work hours, and contracting out of activities—for the clutches represents over 90% of the total maintenance
167
3 1
2
5 2
10
1
20 100 0 0.0
0.4
1.2
0.8
1.6
2.0
Xs Figure 4. Ratio of systematic maintenance cost on corrective maintenance cost for b = 1.6
costs of these machines. One of these presses is particularly costly, with 910 h of intervention and at a cost of 590 000 francs in 5 years. The listed costs are the costs for delivery of parts, material and service. The historical study reveals six interventions for breakdowns, which allows the following Weibull model to be used: b = 1.67, g 3 0, h = 43 weeks, G[1 + 1/b] = 0.89. The Kolmogorov–Smirnov test is positive, so the model is valid. The cost of one intervention is high. The total duration of a repair is 130 h, to which 4000 francs worth of parts have to be added. This totals about 30 000 francs for each intervention to change the clutch disks. A breakdown requires an immediate, unscheduled intervention of maintenance personnel, often during overtime, or the repair is contracted out, depending on the importance of the work to be done. Furthermore, the breakdown disorganises surrounding production, and production following that particular station, which means operational delays of operators and a loss of production. Therefore, the cost for a breakdown is estimated to be twice the cost of the intervention, which is about 60 000 francs. 4.3. Taking into account the cost of the parts concerned, the systematic maintenance is managed individually, so a planned intervention can be executed without penalty for the production. The curve to use is therefore the ratio between systematic and corrective maintenance, Cs /Cc with b = 1.6 (Fig. 4). A ratio, r = P/I = 2, gives a minimum for Xs = 1.1 with R = Cs /Cc being 0.95. This means that preventive systematic maintenance is 5% less expensive than corrective maintenance, with an optimal intervention period of Ts = Xs h + g = 47 weeks. In that case, the average duration of operation between two interventions is: h
g
Xs
R(t) dt = 0.7445 × 43 = 32 weeks.
0
Figure 2 allows the costs of conditional preventive maintenance and systematic preventive maintenance to be compared. Conditional preventive maintenance can be used until the cost reaches a value corresponding to about S/I = 2. A more detailed calculation
168
.
can be done quickly. The ratio between conditional cost and corrective cost has to be inferior to R, i.e. Cp 1 + S/I = QR Cc 1+r \ S/I Q [R(1 + r) − 1] = 0.95(1 + 2) − 1 = 1.85. With I = 30 000 francs, S is, at the most, 55 500 francs. This is the maximum budget that can be allocated to surveillance between the two service intervals, i.e. during the MTBF, with the condition that surveillance is efficient and avoids breakdowns. This corresponds to a yearly budget of 76 000 francs. An investment of 30 000–40 000 francs allows for the installation of a chain of vibration analysis measures, and thus improvements on the maintenance budget of this machine. 5. CONCLUSION
The described procedure allows to choose, as a function of the law of service life and maintenance costs, the most economical maintenance method. Later it would be possible to install a more efficient maintenance service, since it is better organised, and less costly. REFERENCES 1. S. L 1997 Association Universitaire de Me´canique 4, 545–548. Pour une G.M.A.O. plus efficace. 2. F. B 1989 Achats et entretien 336, 27–33. Condition optimale de remplacement pre´ventif. 3. S. L 1994 The`se de doctorat de l’Universite´ de Reims Champagne Ardennes, Reims, France. Ame´lioration de la disponibilite´ des e´quipements de production par l’optimisation de la gestion des stocks de maintenance. 4. S. L and A. P 1995 Revue d’Automatique et de Productique Applique´es 8, 285–290. Gestion de stock en maintenance corrective.