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COST OPTIMIZATION FOR REALISATION OF MAINTENANCE COST
Alexej Chovanec •
Faculty of Special Technology / Alexander Dubcek University in Trencin Studentska 1, 911 50 Trencin, Slovak republic e-mail: chovanec@tnuni.sk
Anton Ambrozy •
Faculty of Special Technology / Alexander Dubcek University in Trencin Studentska 1, 911 50 Trencin, Slovak republic e-mail: ambrozy@tnuni.sk
Abstract: This paper deals with optimal control interval determination using minimization the financial costs. It clears conceptual, mathematical and simulation model of the problem solution. It enumerates and evaluates results of the simulation.
Keywords: sampling size, fault time, interval between failure, normal distribution, financial costs, simulation experiment, optimisation process, probability density, optimal maintenance interval
1. INTRODUCTION
In the stage of the technique usage some precautions of dependability support are realised at the technique user, expressed in the Complex care program, which determines:
- strategy of approaches to the realisation of the complex care control,
- methods of the activities in the realisation precautions,
- organisation work forms of executive workers,
- types and technological limitations of complex care tasks.
The non-operating state and maintenance costs minimizing, but also optimising of:
> dependability,
> safety,
> production quality,
> profit.
The optimal determination of time periods for maintenance making is important task of the maintenance systems control. The maintenance interval represents the maintenance period between single maintenance levels (the period for realisation of the maintenance of respective level).
At the optimisation of maintenance intervals we need the verified and usable data, which can be acquired from one or more sources, for example:
• Previous knowledge from similar systems, if the maintenance actions can be suitable also for the new designed product. The maintenance programs of similar products can provide after the critical analysis suitable base.
• The data from all kinds of tests at the producer can provide information about the efficiency, usefulness and the effectiveness of designed maintenance programs of the new product and its components.
• If no previous verified knowledge about the fault rate of other systems exist or if the previous and new systems are not similar enough, the maintenance interval can be determined by estimation (expert method). The experiences of producer, user specialists are utilised and it is progressing in accordance with the knowledge in the dependability, operation, operation conditions etc.
• If verified data about the failure rates of single system components exist, it is possible to use calculations in accordance with the restoration theory and rules of complicate systems dependability for determination of the maintenance intervals.
2. COST OPTIMISING PRINCIPLE
The optimal determination of time periods for maintenance making is important task of the maintenance systems control. The maintenance interval represents the maintenance period between single maintenance levels (the period for realisation of the maintenance of respective level).
At the optimisation of maintenance intervals we need the verified and usable data, which can be acquired from one or more sources, for example:
• Previous knowledge from similar systems, if the maintenance actions can be suitable also for the new designed product. The maintenance programs of similar products can provide after the critical analysis suitable base.
• The data from all kinds of tests at the producer can provide information about the efficiency, usefulness and the effectiveness of designed maintenance programs of the new product and its components.
• If no previous verified knowledge about the fault rate of other systems exist or if the previous and new systems are not similar enough, the maintenance interval can be determined by estimation (expert method). The experiences of producer, user specialists are utilised and it is progressing in accordance with the knowledge in the dependability, operation, operation conditions etc.
• If verified data about the failure rates of single system components exist, it is possible to use calculations in accordance with the restoration theory and rules of complicate systems dependability for determination of the maintenance intervals.
Determination of maintenance action interval thus so that the total unit costs of the object's life cycle should be minimized is the aim of the optimisation.
The total unit costs Nc at vanishing the costs components, which are not suggestible by the maintenance actions, consist of three components:
1. Supplying costs of the object (No), which are constant (independent on the maintenance interval time),
2. Costs of preventive maintenance (Np),
3. Costs of the corrective maintenance (Nn) of the object (their amount depend on the maintenance interval)
The expression of total unit costs:
Nc(t) = 1/t (N0 + Np(t) + Nn(t)) ( 1 )
where: t - is the maintenance interval, which we are looking for.
The expression is rearranged into the form, where each term on the right side expresses the relative
costs:
Nc(t) = No /t+ Np(t) /t+ Nn(t /t) ( 2 )
The relative costs for the preventive maintenance are constant in time and that's why we directly replace them by symbol Np :
Ncft) = No /t+ Np+ N„(t/t) ( 3 )
The optimising task lies in finding such a maintenance interval t, which ensures that the total unit costs expressed by the previous equation will be minimal.
The suitable (convex) form of the curve Nc(t) and sufficiently sharp and expressive minimum in point D(topt; Nc(t) min), which enables relatively exact identification of the optimum point position, are the conditions of the application successfulness of the optimising method.
If the function minimum Nc(t) is flat and inexpressive, the determination of the optimal maintenance interval is more difficult, but it enables easier combination of maintenance actions without negative impact in economical field.
All variants of economical optimising of the technique life cycle or maintenance intervals are based on a similar principle.
The suitable (convex) form of the curve Nc(t) and sufficiently sharp and expressive minimum in point D(topt; Nc(t) min), which enables relatively exact identification of the optimum point position, are the conditions of the application successfulness of the optimising method.
3. OPTIMISING MODEL OF PERIODIC CONTROLS INTERVAL
The device Z is formed by elements Pi, P2,......PK, which are in operable or failure state.
The diagnostic operations with costs per hour C1 are made for the detection of failure state occurrence in periodical times tK1, tK2, ... tKN. During the diagnostic inspection all or several elements can be diagnosed.
In the case of failure occurrence until the closest control of the operability the device causes with its operation the loss per hour C2.
6tpi 6tpfr
5tp4 |
3tpi
ltp2
î Î «
tfvl fe fe
(6tpi, ltpj) ... Interval between failures
"1 r
3tpi ltp2 5tp+ 2tp5 6tp6 4tp? 2tpS
J I i I i ill.
tKi
Oki- ltp2) ... fault time
Fig. 3.1 Scheme of failures moments and diagnostic controls tKi - control time, I tpi - time of failure occurrence on the I - element
Operation times to the failure occurrence of single elements have the probability distribution of continuous type with distribution function:
F(t) = P ( T < t) ( 4 )
d
and with probability density ft) = — F (t).
dt
Due to the failures, which appear in times tPi, tP2, . . . . tPM, due to the work in unsatisfactory conditions or due to non operating state, losses rise up to the closest diagnostic control.
Np =ZC2 . (tKi - tPi) ( 5 )
If no failure occurs, losses rise from the unavailing usage of the diagnostic device C1.
The sum of costs of planned diagnostic controls and losses from the device operation in the fault state
Nc = ¿Ci + C2 . (tKl - tpj) ( 6 )
must be minimal and so also the mean costs of one cycle of periodic control.
4. SIMULATION MODEL OF MAINTENANCE INTERVALS OPTIMIZATION
In the case of knowledge of the mean time values of failures occurrence, deterministic calculation for various control intervals is made according to previous relations and the minimal value of test function is being searched.
If we have an elaborated statistical set of data and information about failures in the form of type and distribution parameters of times between failures in the operation units, we can utilise the simulation approach.
Flow diagram fig. 4.4. shows the process described below.
1. Insertion of input data (elements number, unit controlling costs C1, unit costs - losses C2, time of simulation end in operation units, distribution parameters of periods between failures of elements, set-up of initial values of variables).
2. We generate the size of intervals between failures. We determine the smallest and the biggest interval between failures and the step of tested cycle.
3. We realise the cycle of optimisation experiment and change the step (increase or decrease of the control interval).
4. We determine the number of controls. We calculate the prevention controlling activity costs Np. We calculate the operation times in fault state and the loss size Nn. We calculate the total costs Nc.
5. We process the output data.
6. We verify the conditions of experiment continuation.
7. We diagrammatise the size of costs depending on the size of control intervals.
Fig. 4.1 Diagram of optimal size of control interval for the simulation experiment 1 (inputparameters tab.5.1)
End of experiment?
Print of output documents
C END )
Fig. 4.2 Flow diagram
5. REALISATION AND EVALUATION OF RESULTS OF SIMULATION EXPERIMENTS
The simulation experiments enable to follow the changes of the output parameters values depending on the change of input parameters.
Which is the change of total costs and size of control interval in dependence on the change of unit costs values on control operations at constant costs of losses from the failure operation?
Tab. 4.1 represents the input and output parameters of simulations. Each experiment provides the determination visualisation of the control interval optimal size in the studied range of possible control interval. The optimisation process and its range for the simulation experiment 1 are shown in fig. 4.3.
Tab. 5.1 Input and output parameters of simulations
INPUT:
Generation of failure times: Normal distribution: MU=1200
SIGMA=200
Simulation: 1. 2. 3. 4. 5.
Control operation unit costs 5000 4000 3000 2000 1000
Losses caused by failure operation 120
Ol UTPUT:
Simulation: 1. 2. 3. 4. 5.
Interval between controls 330 290 250 210 130
Total unit costs 3946376 3540940 3050004 2479416 1752340
4500000 4000000 „ 3500000 lis 3000000 £ 2500000 3 2000000 2 1500000 ° 1000000 500000 0
3946376
3540940
3050004
2479416
1752340
Simulation
3
4
5
Fig. 5.1 Reduction of total costs and size of optimal control interval at decreasing of control operations costs
according to tab. 5.1
Which is the change of total costs and control interval size depending on the change of unit costs values occurring due to the failure operation at constant costs of control operations?
Input and output parameters, optimisation process, increase of total costs and reduction of optimal control interval size at costs increase due to the failure operation is shown in tables and graphs below.
Tab. 5.2 Input and output parameters of simulations
INPUT:
Generation of failure times: Normal distribution: MU=1200
SIGMA=200
Simulation: 1. 2. 3. 4. 5.
Losses caused by failure operation 120 220 320 420 520
Control operation unit costs 5000
OUTPUT:
Simulation: 1. 2. 3. 4. 5.
Interval between controls 330 250 210 170 160
Total unit costs 3936012 5327617 6408536 7411640 8174268
Fig. 5.2 Diagram of optimal size of control interval for simulation experiment 5 (inputparameters tab. 5.2)
9000000 8000000 7000000 6000000 5000000 4000000 3000000 2000000 1000000 0
8174268 7411640
6408536 5327617 3936012
Simulation
350 300 250 200 150 100 50 0
330
250
210
170
160
Simulation
1
5
Fig. 5.3 Increase of total costs and the reduction of optimal control interval size at increase of costs due to
failure operation according to tab. 5.2
The following result from the results of simulation experiments:
1. The total costs decrease with the decreasing values of unit control costs.
2. The size of optimal control interval decreases with the decreasing value of the unit control costs and the number of control activities intervals increases.
3. The total costs will increase and the size of the optimal control interval will decrease with the increasing value of the unit costs from losses caused by the failure activity.
6. CONCLUSION
The simulation model optimises the interval of maintenance action so that the total costs (on control and costs caused by failure) should be minimised.
The processing of simulation model arises from deterministic process of resolution of the mentioned problem. The simulation model was verified by processes of classical computer methods.
The algorithm representing the simulation model is drawn so that it could be easily modifiable. It enables to remove the lacks of classical computer methods and to get the required results in short time.
The obtained results can be applied to the determination of the number of control activities intervals and the size of control activities interval. They enable to gain an idea about the number of expended financial resources for the control operations, about the costs of the control operations and total financial costs.
The simulation model creates the bases for further development of problems resolution possibilities being connected with the determination of the optimal control interval size and minimising of total costs. It expands the range of application utilisation of simulation modelling for solution of problems related to the operation of specific technical systems. Generally it can be the mobile technique, elaboration device or constructional groups of technical devices, which during their operation get into the failure state.
The simulation model provides general graphical outputs of costs changes in dependence on the size of control interval and the number of control intervals. It can be used for the graphical and didactical support of the explanation of optimisation problems of maintenance intervals.
REFERENCES
1. HOLUB, R. - VINTR, Z.: Aplikovne techniky spolehlivosti. Cast 1: Specifikace pozadavku na spolehlivost. [Skripta]: Vojenska akademie 2002.
2. FURCH, J.: Mozne pristupy pro stanoveni optimalniho intervalu doby zivotnosti vozidla. In: Sbornik mezinarodni konference" OPOTREBENI SPOLEHLIVOST DIAGNOSTIKA 2005, Univerzita obrany, Brno 2005, Ceska republika, str. 49 - 54, ISBN 80-7231-026-7
3. MENClK, J,: Optimalizace navrhu technickych objektu z hlediska nakladu. In: Sbornik mezinarodni konference" OPOTREBENI SPOLEHLIVOST DIAGNOSTIKA 2005, Univerzita obrany, Brno 2005, Ceska republika, str. 125 - 130, ISBN 80-7231-026-7
This article was elaborated at the tasks realisation of granted project VEGA 1/2094/05
