Simulation of an Assembly Line Life Time using Weibull distribution Luis Neto SYSTEC-FoF, Faculdade de Engenharia, Universidade do Porto, Rua Dr. Roberto Frias, 4200-465 Porto, Portugal
[email protected] http://systec-fof.fe.up.pt
Abstract. This work consists in the study of two fault models and the applications of Weibull distribution. The fault models are applied to a simulation of a real case industrial scenario, then, Weibull distribution is used to make predictions about life-time statistics. Results are shown in form of plots and answers to life-time questions about industrial machines survivor probabilities. Keywords: Assembly Line Simulation ·Failure Modeling ·Cost Assessment ·FlexSim
1
Introduction
Machines reliability and life-time condition represent a major decision concern in order to decide when to design a supply chain. The present work pretends to give answers to re-use questions: – Will old machine be reliable to the next scheduling? – How much life is left in device? – It is a good decision to buy a new machine taking in account reliability of the old machine? This work scopes in the Reborn European Project 1 , innovative reuse of industrial equipment is the research motivation behind this project. The project consortium, constituted by academic and corporate partners, aims to develop cyber-physical components that can retrieve statistics of assembly lines which, further, can be used for multiple purposes. One of the challenging activities for this project is the development of simulations, based on concrete industrial cases from the consortium corporate partners, for estimation of machinery cost. Simulations must obey to functional parameters given by the companies, also, must embed stochastic methods for simulation of machinery fault and repair activities. The results of the simulations must show all failure historic, as statistics about production itself eg: number of pieces produced, idle time and processing time. 1
URL: http://www.reborn-eu-project.org/
To the results of the simulations, life-time statistics must be applied to determine reliability of the shop floor equipment. Due to the academic partners of the project consortium choice and a strong life-time analysis recommendation in the literature, the well known Weibull distribution was chosen to model the life-time of machines involved in the simulation. This statistical distribution allows to answer the above questions about the products life-time. By fitting this distribution to dataset of failures resulting from the simulation, and regarding the assessment objective, the distribution is able to retrieve: (1) mean life time, commonly known as mean time to failure (MTTF), which is the average time before a failure occurs; (2) failure rate, the number of failures per unit of time; (3) probability of failure or reliability given the time, chance to the machine fail or be reliable after t years of operation; (4) plots for reliability vs time, which can also be related with costs vs time. In this work, using a project partner operation scenario, a simulation was built using the FlexSim 2 simulator, the resulting data from the simulation was analyzed by fitting data to the Weibull distribution and the results were exposed and discussed. Along the document, pitfalls, limitations and assumptions are also exposed to the readers. In section 2, problem’s formal definition is presented. Section 3, shows literature review, the studied Failure models and details about Weibull distribution are shown. Section 4, the solving process is detailed along with assumptions, constraints and technology used in this work. In sections 4 and 5, respectively, results and conclusions are presented along with analysis.
2
Problem Definition
In the scenario we modeled (Fig 1), the shop floor is composed by 10 workstations, each workstations (WS) carries a product at time and that product passes through an assembly process in each WS. In total, 6 different products are assembled by this assembly line. Each type of product takes a different amount of time to be assembled in each WS. The products are transported by human operators between workstations, operators drop the product in the next WS, sequentially, trough the assembly line. When a products enters the simulation it is passed to a queue that holds it until the first WS is able to process it; when product reach the last WS of the line is discarded from the simulation. If a fault occurs during a specific product assembly process, that product is considered faulty and is accounted as defected; if no fault occurs, the product is accounted as a good product. Workstations are physical industrial machines that perform discrete assembly steps in a product, each machine is composed by: (1) a carrier roll, which transports the product within the machine operation area; (2) a set of other unspecified physical sub-components, performs the concrete assembly in the product. For application of fault models, because of the unspecified assembly components 2
URL: https://www.flexsim.com/
Fig. 1. Simulation model layout in FlexSim.
of the machine, we will focus just on the carrier roll operation. We will consider for fault modeling, the strength of the roll and the system that pushes the roll (engine and controllers). For future extensibility of the work, we considered the possibility of incorporating new fault models to enhance the simulations accuracy. More details about the fault models, life-time modeling and simulation parameters are given in the next chapters.
3
Background Work
As we are not concerned with the simulator choice, the focus of literature study for this work were the failure models to adopt and, understatement of Weibull distribution. The main concerns regarding failure models were the lack of information about machines to model and the choice of applicable models to this situation. Regarding the Weibull distribution, the concerns were: (1) how to extract the probabilistic information to answer the life-time condition questions; (2) in what manner should failure data be extracted from the simulation; (3) how to fit the extracted data to the distribution. 3.1
Failure Models
The supplied data about machines was very limited. The adoption of failure models was restricted by the lack of modeling data, such as: real failure data, details about subcomponents (eg: motors, semiconductors, bearings), consumption data (eg: current, voltage), operation data (eg: temperature, humidity, vibration). Table in Fig.4, shows the information used in the adopted models. In mechanical engineering and micro electrical mechanical systems (MEMS) fields, failure models are common topic. Also in the semi-conductors field a variety of models are used to modulate chip wear-out. In [3] the author presents different techniques to deal with the different components of microprocessors.
Due to a major similarity with our case, the models analyzed with more detail are based in MEMS and mechanical systems summarized in [7][2]. The examples analyzed, which lead to obtain acceleration factors (AF), used in the calculation of failure rates, were then: – Coffin-Manson Relationship - Used to modulate the materials fatigue under cyclic wear-out. Uses the temperature, a material dependent constant and an empirically determined constant to take in account the elastic regime of materials. – Arrhenius Equation - Used for chemical reaction rate modeling, the parameter activation energy can be varied to test a wide spectrum of stresses. Requires an activation energy constant and temperature. – Eyring Model - Based on Arrhenius model allows to extend the stress factors beyond the temperature (eg: use of humidity). – Power law (Peck) model - Used to modulate the corrosion of circuitry encapsulated in plastics. Requires a material dependent constant, temperature and humidity. Given lack of parameters for the simulation, the utilization of the studied models was compromised by the assumptions that would need to be done. Material dependent constants cannot be assumed because we do not have knowledge about the materials we are dealing. Ambient dependent variables, such as temperature and humidity are easier to assume, based on that, the steps presented in [8][10] to determine MTTF, based on Arrhenius model, are a suitable option to this case. The constants to use in calculations, namely, activation energy and temperature, are standard for a variety of test cases. [10] The only variable left was the time used in the Arrhenius model. This time variable corresponds to the time that takes under tests to a failure to occur. In Fig. 3 table , the expected fault time for an year of machine processing is given. We can assume that the total time of operation for a given machine in a year, minus the expected fault time, gives the limit of time in which a machine will have no failure and we can use this time in the equation. Due to the long time that takes to a fail, considering machine processing time, we assumed the simulation time to generate failures in this model, as the working hours in a year (table in Fig. 3) - simulation time - is larger than machines working time in a year (table in Fig. 4), and failures can happen in simulation time. The process of failure generation for this model is illustrated in the digram of Fig. 6. Another model for generating failures was used based in [6]. The authors present a simplistic failure model based on load and strength of components given by normal distribution. We adapted this model to our simulation. The WSs in simulation have a random strength and the items are dropped on them with a given load force. The values of load and strength are given by normal distributions and if the load is superior to the force, a failure is triggered. The process is illustrated in the digram of Fig. 5.
3.2
Weibull Distribution
Weibull was chosen to modulate the life-time of the machines by the project consortium. In addiction, the literature shows that is flexible and widely used in life-time modeling. [4][11][1] These authors contribute and briefly summarize the works done in modifying the distribution to better adapt a bathtub shape of life-time. This bathtub shape accurately describes the evolution of life in tested components; it starts with infant mortality cases, then components pass through a constant failure probability, and lastly, the probability increases exponentially representing the end of life. These works also describe the way to treat the failure data information, applying a sequence of steps to extract the parameters of the distribution, η, the scale parameter and β, the shape parameter. Some equations related with reliability, and relevant to address the questions we want to give answer, are derived from the distribution and shown in Fig. 2.
Fig. 2. Internal Gateway configuration.
Another reenforcement to the usage of this distribution was its adoption to treat life-time data from real components testing [1][12] and, with an high degree of similarity with our case, the use of Monte Carlo [6], Markov Chain and Bayes inference [5][9] methods to generate and treat life-time data.
4
Problem Modulation
Tables used in simulation behavior parameterization are in Fig. 3 and Fig. 4, these parameters are used to program times each WS spend in each item type. SWS1 and SWS2 perform two sequential tasks in each item. Normal WSs perform a single task per item. One table not shown is the factory time table, this table establishes the schedule for the company. The simulation runs in week days, from 8:00 am. to 17:00 pm., with a break of 15 min. between 12:00 pm. and 12:15 pm.. This time table, excluding holidays, totals the number of working hours in a year given by the table in Fig. 3.
Fig. 3. Time information table.
Items in the simulation start being processed after leaving the Queue to WS1, sequentially the items follow the assembly line shown in the layout of Fig. 1. Times spent in each WS are shown in the table in Fig. 4.
Fig. 4. WS time information table.
Failures follow the two models presented in Failure Models subsection. For the load/strength failure mode, the logic that generates the failures is represented in the flux diagram of the Fig. 5. Flux diagram in Fig. 6 represents the logic used to modulate the Arrhenius fault model. This model generates far less failures than the previous one, in a simulation of 10 years, a maximum of one failure per WS occurs within this model. To explain a complete simulation process the illustrative flux diagram is shown in Fig. 7. After simulation run for the desired number of years, all failure data is exported to a spreadsheet. Then, data is sorted by failure time, a linear regression for extraction of Weibull parameters is applied and the distribution is used for calculations. Each machine has an independent equivalent time for each failure mode, when a failure occurs, the timer is reset. This operation happens because we assume the faulty component is repaired and its state after repair is new (machine time is reset).
Fig. 5. Load model failure generation.
Fig. 6. Arrhenius model failure generation.
Fig. 7. Internal Gateway configuration.
5
Results
Independently for the two models, the following results were obtained by ordering the failure data set by time of failure. Failure data from the different WSs was fused in a larger data set to proceed to life-time analysis. The fit to the Weibull distribution was done using a Matlab fitting module. The selected correlation level was 95%. The following two subsections illustrate obtained results for lifetime prediction in device operation hours. As shown in the table of Fig. 3, an year of active work has 1884.85 hours, this simulation was run for 10 years, which gives a total of 18848.5 hours of simulated work. Using the distribution parameters determined we will answer the question stated in the this paper introduction, as that was our motivation. 5.1
Load Failure Model Results
The results for the parameters η (scale parameter) and β (shape parameter) were, respectively, 156.93 and 1.33. For Weibull distribution values of β > 1 show a clear wear-out behavior [6], this failure model shows an high accumulation of failures during time, reflecting in a accentuated degradation over time. A series of plots for this fault model can be seen in Fig. 5.1. Regarding the equations in Fig. 2, for the shape parameters η = 156.93 and β = 1.33, probability of the roll mechanism of a machine to survive an year of processing (1884.85 hours) is 1.4222E-12. Which clearly means that in one year of processing state, machines will for sure have a roll fail. For a machine to have a 50% chance of not having a roll fault, it needs to work less than 117.80 hours. If we regard on the WS time information table in Fig. 4, and take the
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WS1 as reference, we see this machine works precisely 131 hours/year, which means that the machine will survive with a probability of 45% for one year of operation. Taking the equation in Fig. 2 to calculate the mean time to failure, this gives M T T F = 144.296 hours due to load failures, for every WS. 5.2
Arrhenius Failure Model Results
Regarding faults due to thermal effects, the results for the parameters η (scale parameter) and β (shape parameter) were, respectively, 96626 and 3.47; once again, this values for the parameters reveal a clear wear-out behavior. Contrary to the previous model, the faults predicted by the Arrhenius model happen later in life curve of machines as the aging process is slow. A series of plots for this fault model can be seen in Fig. 5.2. Regarding the equations in Fig. 2, for the shape parameters η = 156.93 and β = 1.33. The probability a circuitry component of a machine to survive an year of processing (1884.85 hours) is 99.99%. Which clearly means that in one year of processing state, machines will for sure survive to thermal effects in circuit components. For a machine to have a 50% chance of having a fault, it needs to work at least 46 years (not referring to simulation time), which is 86703 hours of machine processing. These results can be explained because of the machine that first failed (WS7), due to Arrhenius model, failed after 52952 hours of simulation time, sixth year in simulation. The last failure did not happened in simulation time, it was predicted to happen after 131474 hours of simulation, 15 years of simulation. If we regard on the simulation time for 7 years, which gives 13193 hours, and take the WS7 as reference, we see that this machine will survive with a probability of 99%. Then we can conclude that this failure happened mostly because of the interferece caused by the normal distribution applied to the MTTF (Arrhenius model failure generation flow in Fig. 6). Taking the equation in Fig. 2 to calculate the mean time to failure for all WSs, this gives M T T F = 86899.4 hours to failure using the Arrhenius model.
6
Conclusion and Future Work
Due to the lack of real data, the models and life-time prediction are penalized. In the results section, a big discrepancy in failure times can be seen due to the assumptions that need to be done in lack of concrete data. Nevertheless, for the assumptions made, the studied life distribution has proven to be accurate comparing the results with simulation behavior. Models more approximated of reality, in combination with the Weibull distribution, can be applied in simulations like these to accurately make prediction and extract statistics about factory plant. As future work, we need to extract and present statistics regarding plant operation. Such as: number of correct and bad produced pieces, costs of machine maintenance, times for each machine in different states (producing, idle, break down) and throughput given the failures. A second step would consist in creating
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fault models more approximated to the reality, further research needs to be done in order to find new models that allow us to make more realistic assumptions, and perhaps, get life-time data from real machines. Adding information of the costs to repair can value the assessment task by estimating money quantities involved easily using the predictions.
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