Machine Life Cycle Cost Estimation via Monte-Carlo ...

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Machine Life Cycle Cost Estimation via Monte-Carlo Simulation 1

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Jürgen Fleischer , Marc Wawerla , Stephan Niggeschmidt 1

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Institute of Production Science, University of Karlsruhe, Germany

Abstract Recently an increasing number of customers demands more extensive warranties from the machine building industry. In order to control and maintain the arising costs from the seller’s point of view, the paper in hand presents a generic and comprehensive approach to estimate the distribution function of machine warranty cost. Based on the estimation of the failure rate distribution certain life cycle cost elements are quantified either deterministically or stochastically depending on their characteristic. The Monte-Carlo simulation is used for the flexible consideration of the entire system and the estimation of risk figures such as the Value-at-Risk. Keywords: Production Management, Life Cycle Cost, Monte-Carlo Method

1 MOTIVATION

Cost [€]

During the last several years more and more companies, especially in the automobile industry, detected the high relevance of the life cycle cost of their production equipment. Although life cycle costs are normally referred to as all of the costs generated during the life cycle of an item [1], the article in hand focuses on the maintenance costs because of two reasons. On the one hand the maintenance costs are mostly driven through the failure rate of an item which is normally modeled through the Weibull probability function [2, 3]. Ergo it is more complex to consider them than deterministic costs that can easily be added and varied. On the other hand analyses show that the maintenance costs are a key differentiator between offers of different suppliers. Initiated through this perception an increasing number of companies demand a warranty for the maintenance costs today [4]. These costs annually reach up to 10% of the machine acquision costs, in some cases even higher as shown in figure 1 [5].

like the experience based Failure Modes Effects and Criticality Analysis (FMECA) [7] and the data based Weibull analysis [3]. The calculation error generally depends on the existing experience, the available data and its quality. Thus precision depends on the efforts done regarding data quality. Apart from forecast accuracy, the second risk arises from the statistical variance of the failure rate. Assumed constant, the reciprocal value of the failure rate is the mathematical expectation of the time between failures; the Mean Time Between Failures (MTBF) [1]. A field data analysis of 10 identical turning machines in similar operations over a period of 12372 operating hours conducted by the Institute of Production Science shows a standard deviation of the MTBF of 580.1h at an average of 1030h (figure 2).

Average

Maintenance cost per year 193200.72800.-

122800.-

1000000.-

714000.-

827000.-

Investment

Investment

Investment

Machine 1

Machine 2

Machine 3

Figure 1: Comparison of the annual maintenance cost and investment [6]. For the machine manufacturer the following risks arise from this framework which is in the following referred to as life cycle cost contracts. First the machine reliability calculation can result in inaccurate values. State of the art are methods

14th CIRP Conference on Life Cycle Engineering

Figure 2: MTBF values of identical machines in similar operations. Obviously the maintenance costs of the machines also vary in a wide range. Thus the risk to the machine manufacturer in case of a life cycle cost contract is immense without precise life cycle cost estimation. The height of the penalties can even reach up to 30% of the machine price in current contracts [4]. Different scientific methods are known to calculate the mathematical expectation of the failure rate for

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technical systems such as the Bool model, the Markov model, the Fault Tree Analysis and the Monte-Carlo method [8]. Because of its flexibility the Monte-Carlo method is most promising in this context. It is defined as a methodology for obtaining estimates of the solution of mathematical problems by the means of random numbers [9]. In literature it was first mentioned in 1949 in context with the Manhattan project [10]. The Monte-Carlo method uses statistical distribution functions on component level as input parameters but results in average values [9, 11, 12]. Against, methods to combine deterministic and stochastic life cycle costs are a deficit in research today. Furthermore no approach exists to calculate the life cycle cost distribution function of technical systems. 2

AIMS

This article provides a general approach for the estimation of statistical life cycle cost distributions for machines and components to provide the basis for the risk calculation of life cycle cost contracts. In most cases only some data from different sources are available for the calculation. The proposed method is required to handle these circumstances. Beside the prognosis of stochastic cost elements, the calculation of deterministic cost elements is subject matter. The core aim is the combination of stochastic and deterministic cost elements to identify the statistical life cycle cost distribution. The result of the method enables the machine manufacturer to calculate the Value-at-risk of a certain offer. 3

PROCEDURE

The presented life cycle cost estimation procedure consists of three stages. The first step is the calculation of the failure rate distributions on component level to consider the stochastic cost influences. In the second step, the deterministic life cycle cost elements need to be calculated while existing life cycle cost standards must be considered. The third step consists of the Monte-Carlo simulation to combine the stochastic and deterministic life cycle cost elements. Flexibility regarding both, time intervals to cover different contractual periods and the machines part list to take different assemblies into account, is necessary. 3.1

Estimation of failure rate distributions

The two main methods to estimate the failure rate distribution of components, used in the proposed procedure, have been mentioned already. On the one hand there is the Weibull analysis which is based on data from machine service, spare parts, maintenance and fatigue tests [3]. A condition for the use of the Weibull analyses is the assignability of the failure times to root cause components. On the other hand there is the experience based FMECA which supports the user through a detailed structure but still feature uncertainties because extensive experience is required [7]. Especially in the offer preparation stage machine manufacturers face the following problems: •

Field data comes from several sources with variable quality levels. Particularly service and spare part data quality is arguable.

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Not all possible types of data are available in any case. Especially data from customers and suppliers is often unavailable.

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Field data refers to different machine types which only share some identical parts and assemblies.

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Fatigue tests are limited by development times and expenses.

shrinking

product

Nevertheless, in order to calculate failure rate distributions with sufficient precision for life cycle cost contracts, the calculation procedure needs to be planned in detail. Machines normally consist of a huge number of components, but the statistic broadness of the field data for a precise analysis is only given for some components. So it is important to break down the machines bill of material to a manageable detail scale by defining life cycle relevant machine components. For the appraisal of the relevant machine components, the analysis of the required spare parts during the two year free replacement warranty period is used. Although long term effects of changing failure rates during the life cycle are underestimated, this procedure provides a reasonable approximation. After the identification of the relevant parts of the machine, the available data sources must be assigned to every machine component. The most precise field data for reliability examinations is normally gained from the production data acquisition and the maintenance department of the machine operating company (see customer data in figure 4). Maintenance data consist of a manual description of maintenance actions divided into preventive and reactive maintenance. This information is mostly linked to the required spare parts which are ordered from the machine manufacturer or a third party. The advantage of spare part data is the possibility to link it directly to the bill of material through the identification number. Field data from production data acquisition also include short term breakdowns that are mostly not caused by the production equipment. The differentiation of equipment breakdowns, the assignment of the root causes for breakdowns and the identification of broken machine components is normally not possible in retrospect. In most cases machine manufacturers have only limited access to in-service field data. Then only data from spare part business and warranty services is available. Because Weibull analyzed field data provides most valid results, this method is preferred dependent on data quality. In order to minimize statistical variance influenced by different stress requirements, the machine pool needs to be divided into stress classes according to the in-service utilization at the customers. By an increasing number of stress classes the prediction quality gains accuracy. Against the data availability limits the number of stress classes to a maximum of three. The Weibull analyses then needs to be conducted for every stress class. If no data is available at all, statements about the MTBF need to be requested from the machine manufacturer suppliers. If even from the supplier no statement is available, knowledge based approaches offer the possibility to come to average values for the MTBF and the Mean Time To Repair (MTTR). Due to requirements from most automotive customers, the FMEA is commonly used in the machine development process [13]. In order to implement it systematically for reliability forecasts, the standard FMEA sheet needs to be adapted by a quantification of the failure rate, the MTTR and the spare part cost according to figure 3. Not needed for calculation is the criticality assessment as it is for example required in a FMECA procedure. Once the mean values of the repair time and the time between failures in the adapted FMECA have been estimated or were given from the

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supplier, the mathematical distribution must be assigned assuming an exponential distribution. In this case the reciprocal value of the MTBF is equal to the failure rate which is constant and fully describes the exponential distribution.

Rating

Service order

Av. of failure rate class

Rating

Failure Occurrence Severity Detection RPN probability Effect Conse- Cost MTTR Effectiv. of quence [€] prevention in measure service −5 Voltage 3 5 ∗ 10 Mach6 590 3.5 h 9 No 180 regulat. ine prevention untight breakmeasure in down place

methodology works with any kind of deterministic cost schedules. Examples are cost elements like power supply cost, investment cost or tool cost. Because most of them can only be modeled with discontinuous functions or number sequences, the addition to one overall number sequence reduces complexity. This task is graphically shown in figure 5. The overall cost number sequence is needed in the following to implement the deterministic cost elements into the machine cost calculation. The stochastic cost elements are not finally calculated until the third step – the Monte-Carlo-Simulation – starts. Accumulated [€] power supply costs Accumulated [€] capital costs

Figure 3: Adapted FMECA sheet. Once analyzed, the different results of the calculation must be assessed considering the given constraints. In the first instance the number of input failures is a knock out criteria. Although the Weibull analyses using linear regression works already with more than one failure record, a reasonable calculation of the Weibull distribution parameters starts with about 10 failure records [3]. The matching of the distribution function and the input data is described by the correlation coefficient which is used to prioritize the results from the different data sources. Comprehensive examinations show that data from the described source normally fit with a correlation coefficient of 0.9