1 Introduction

According to Vineyard and Meredith (1999) flexible manufacturing cell operate at 70–80 % utilization as compared to traditional machines which operates at as low as 20 % utilization. Thus, the components in FMC are more susceptible to wear and tear than the traditional machines operating over the same period of time. Over the last few decades, various researchers have investigated the reliability and maintenance aspects related to FMC’s. Lin et al. (2001) have presented sequential imperfect PM models incorporating adjustment/improvement factors in a hazard rate and effective age. Savsar (2000) developed mathematical models to study and compare the operations of a fully reliable and an unreliable flexible manufacturing cell (FMC), each with a flexible machine, a loading/unloading robot and a pallet handling system wherein the operation times, loading/unloading times and material handling times were assumed to be random. Sheu and Griffith (2001) have considered a generalized age replacement policy with age dependent minimal repair and random lead time. Bloch-Mercier (2002) has considered a repairable system subject to continuous time Markovian deterioration while running, that leads to failure. Gurler and Kaya (2002) have proposed a maintenance and replacement policy for a multi component, multi-state system, in which both the system and the components can be described through a range of performance levels varying from perfect functioning to complete failure. Wang (2002) surveyed, summarized and compared the various existing maintenance policies for both single and multi-unit systems. Chen et al. (2003) have proposed a combined state and time-dependent maintenance policy for a Markovian deteriorating multi-state system with inspection equipment connected. A PM policy for a degradation system with an acceptable reliability level has been proposed by Zhao (2003). Rupe and Kuo (2003) used Markov models to study the effect of equipment failures to the effectiveness of the FMS from the view point of the customer. Tsai et al. (2004) have applied PM in simultaneously considering three actions, viz. mechanical service, repair and replacement for a multi-component system based on availability. Optimal preventive maintenance policies are developed by Wu and Croome (2005) for a critical system that operates periodically, and is maintained with higher cost at the up state than at the down state. Wu and Croome (2005) have investigated the optimization problem of PM policies for situations where the quality of PM is a random variable with a certain probability distribution. Sheu et al. (2005) have described a general PM model that incorporates perfect and imperfect types of PM. Sheu et al. (2006) have proposed periodic PM policies, which maximize the availability of a repairable system with major repair at failure. The three types of PM considered are, imperfect PM, perfect PM and failed PM. Savsar (2005) has analyzed the performance measures of an FMS with different maintenance policies and different mean time to failure. Savsar and Aldaihani (2008) developed a stochastic model to analyze the performance of a two machine FMC served by a robot. Sharma and Kumar (2008) used Markovian approach to model failure and repair rates for critical engineering systems. Boschian et al. (2009) have presented a maintenance strategy for two machines working in parallel. Halim and Tang (2009) proposed a graphical method for determining the confidence interval for the optimal replacement interval of a deteriorating system, under age replacement and block replacement models, in which the inter-failure times follow a two-parameter Weibull distribution. Maheshwari et al. (2010) used birth death process to derive the differential difference equations governing the underlying Markov model for an unreliable FMC and the equations were solved by using Runge–Kutta method to find the probabilities for different system states. Philip and Sharma (2013) had used a stochastic reward net approach for reliability analysis of a flexible manufacturing module. Based upon the extent review of literature authors found that much effort has been made by various researchers to develop system models to solve the reliability and maintainability issues related with FMC’s. Being analytical and complex these models are not only difficult to understand but also difficult to practice. To this effect, in the present paper authors propose a contemporaneous adoption of two different methodologies such as Failure Mode and Effect Analysis (FMEA) and Petri Nets (PNs) to build an integrated and helpful framework that could prove beneficial to analyze the FMC in both qualitative and quantitative manner. In the qualitative approach detailed FMEA analysis of Robot is carried out by listing the potential failure modes w.r.t various components and their effect on system performance. The numerical values of parameters failure occurrence probability (Of), severity (S) and detectability (Od) are obtained from expert elicitation and data book (MIL-HDK 217F and RADC TR-85-194) to compute risk priority number (RPN) score. In quantitative analysis, use of Generalized Stochastic Petri Nets (GSPN) is made to model the FMC. Further simulation experiments were carried out to study the performance of FMC. A case of fully reliable cell and unreliable cell with five maintenance policies is investigated. The organization of the remaining part of the paper is as follows. In Sect. 2, the illustrative cases are discussed that are to be investigated during the development and simulation of the throughput of an FMC.

2 Illustrative case

The Robot failures are analyzed by performing Failure Mode and Effect Analysis as shown in Table 1. The numerical values of parameters failure occurrence probability (Of), Severity (S), detectability (Od). It is observed from the table that for Actuators RPN Score is highest i.e. 216.

Table 1 FMEA analysis of different components of robot in an FMC
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As the Robot is used to transport the parts among the machines and is shared by all the parts of the system. Thus investigating the failure aspects of robot becomes important. So GSPN model is developed and simulation experiments are designed to study the failure aspects. The key assumptions made are as follows:

  1. 1.

    The failure considered is due to robot and not by machines.

  2. 2.

    The failure and repair distributions are deterministic in nature.

  3. 3.

    Weibull, Gamma and Uniform failure rate distribution are considered.

Considered in the study is flexible manufacturing cell (FMC) shown in Fig. 1. It comprises of Load/Unload (LU) station (used to mount raw parts on pallets and unload finished parts from them) and three machines (M1, M2, M3) that are used to process two different kinds of parts i.e. Part A and Part B. there are various alternative working schedules (i.e. machine processing sequences) for each part type, that are summarized in Table 2. Parts of each type have to be loaded onto pallets of corresponding type and there is a limited number of pallets of each type.

Fig. 1
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Schematization of FMC with robot as transportation system

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Table 2 Working schedules
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Type “A” parts have two possible working schedules that involve the movements from LU to M1, from M1 to either M2 or M3 and from M2/M3 to LU.

Type “B” parts have two possible working schedules involving the movements from LU to either M2 or M3and from M2/M3 to LU.

Since, the parts have alternative working schedules (both types of parts can choose between M2 and M3), a policy is needed to select the machine to be used. The policy modelled is simple: if both machines are free, the part chooses the fastest one, while if two types of parts are waiting for the same machine, the part with shortest processing time is selected to be processed on that machine. This is represented in the Petri Net model (Figs. 2, 3) by means of the four inhibitor arcs connected to transitions t5 and t6. The transition/weight rates of transition is shown in Table 3.

Fig. 2
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GSPN model without failure

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Fig. 3
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GSPN model with failure

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Table 3 Transition rate/weight for GSPN model
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In the GSPN model, the circles represent places while transitions are represented by squares. The Robot is connected by arcs with the transitions and machines (circles). The arc towards the transition represent the loading of part to the machine while the arc away from the transition represent the unloading of the finished part. Only one component at a time is considered to be under maintenance. The pallet contains 5 parts that are to be finished. The finished products are to be processed by two schedules as shown in Table 2. The arc represents the movement of parts from one place to another.

3 Experiment 1

Several simulation experiments are carried out to study the performance of FMC operations under different maintenance policies i.e. (a) a fully reliable cell (FRC); (b) a cell with Corrective Maintenance policy; (c) a cell with block based preventive policy: (d) a cell with Age based preventive policy; (e) a cell with Opportunity triggered policy. Opportunity triggered maintenance policy triggers when the failure occurs and along with corrective maintenance of the Robot component, preventive maintenance triggers which leads to lesser time for maintenance as the Robot is in idle state i.e. in the corrective maintenance mode. The performance measure considered was the throughput during the simulation period.

Each simulation experiment was carried out for the operation of the production cell over a period of 1 month (20 working days and 8 h. per day or a period of 160 h). In the case of PM introduction, it was assumed that PM time of 30 min (or 15 min when combined with CM) is added to 8 h at the end of each shift. Ten simulation replicates are made and the performance measure, the throughput during the month, was obtained for each case. Other simulation related parameters are given for each experiment. The repair rate is considered to be uniform for all the reliable and unreliable FMC.

Table 4 shows the comparison between throughput of reliable FMC and FMC with failure and Fig. 4 can easily shows the comparison between reliable and unreliable FMC considering failure only due to robot.

Table 4 Throughput of FMC of different time between failure for reliable and unreliable cell
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Fig. 4
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Comparison of reliable and unreliable FMC when the failure rate distribution is uniform

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Different failure maintenance policies were used during the simulation which shows the variation of production rates with different MTBFs and different maintenance policies as shown in Table 5 and can be seen in Fig. 5.

Table 5 Production rates of an FMC under different maintenance policies and different MTBFs
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Fig. 5
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Throughput for different TBF’s and different maintenance policies. CM corrective maintenance, BB block based PM, AB age based PM, OT opportunity triggered

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The Fig. 4 shows steep increase in the production rate between 500 and 1,000 h. Rapid change in the value of throughput at TBF 500–1,000 is observed because components in Robot may fail in the initial stage i.e. stage 1 of bath tub curve. The failures might be because of improper design, or selection of operating parameters etc. As the time progresses the failure rate becomes uniform. Thus, at the subsequent intervals i.e. 1,000, 1,500, 2,000, 2,500, 3,000, 3,500, 4,000, uniform change in throughput is observed. The robot failure in Fig. 4 consider CM policy only.

Figure 5 and Table 5 shows the comparison of different maintenance policies for the failure in Robot. In Fig. 5, OT found out to be the best maintenance policy for the robot failure. Irrespective of the maintenance policies the pattern of the increase in production rate w.r.t TBF is nearly same except CM in which no preventive maintenance is considered.

4 Experiment 2

The next experiment compares the effect of various time to failure distributions, including uniform, gamma, weibull distributions, on the FMC performance under the CM policy only. All of the FMC parameters related to operation times, repair times, and PM times are kept the same as given in the first experiment. Only time to failure distributions and the related parameters were changed such that TBF was varied between 500 and 4,000.

In the case of the gamma distribution, E (T) = αβ. Thus by changing the values of α and β, time between failures could be changed as required. For example, α = 250 and β = 2 resulted in a TBF of 500; α = 750 and β = 2 resulted in a TBF = 1,500; α = 1,250 and β = 2 resulted in a TBF = 2,500; and α = 2,000 and β = 2 resulted in a TBF = 4,000, which are the same values specified in the previous experiment for the uniform distribution.

For the Weibull distribution which has TBF = E(T) = β Г(1/α)/α, two parameters α(shape parameter) and β(scale parameter) have to be defined. For example, if TBF = 500 and α = 2, then on substituting the values β = 564.2. Similarly for TBF = 1,500, α = 2 and β = 1,692.2, for TBF = 2,500, α = 2 and β = 2,820.95, and for TBF = 4,000, α = 2 and β = 4,513.5 are used.

Comparisons of the three distributions, uniform, gamma and weibull, with respect to the CM policy are presented in Table 6 for FMC production rate. All the distributions show the same trend with respect to increasing production rates at increasing TBF values (Table 7).

Table 6 Parameters of the distributions used in simulation
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Table 7 Comparisons of FMC production rate under various distributions for the CM policy
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As it is seen in Fig. 6, uniformly distributed time between failures resulted in significantly different FMC production rate as compared to other distributions. This is because in a uniform distribution, which is structurally different from the other distributions, probability of failure is likely at all the possible values that the random variable can take, while in other distribution cases, probability concentration is around the central value. The FMC performance was almost the same under the other distributions investigated. This indicates that the type of distribution has no critical effects on FMC performance under CM policy if the distribution shapes are same. But of all three distributions Weibull distribution was found out to be the best for maximum throughput.

Fig. 6
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Throughput for different failure distributions

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5 Conclusion

In the present study, authors successfully presented simultaneous adoption of both qualitative and quantitative approach to model and analyze a FMC. The qualitative analysis using Failure mode effect analysis helps to list the potential failure modes w.r.t. various components and their effect on system performance. The numerical values of parameters failure occurrence probability (Of), severity (S) and detectability (Od) are obtained from expert elicitation and data book (MIL- HDK 217F and RADC) to compute RPN score. The effect of various maintenance strategies on the production rate of FMC under various time to failure distributions is investigated by performing simulation experiments.

Four maintenance policies i.e. (a) a fully reliable cell (FRC); (b) a cell with Corrective Maintenance policy (CM); (c) a cell with block based preventive policy (BB): (d) a cell with Age based preventive policy (AB); (e) a cell with Opportunity triggered policy (OT) are identified and their effects on FMC performance, are analyzed by using Generalized Stochastic Petri Net model. The results of the analysis of various cases show that maintenance of any form has significant effect on the throughput of the FMC. However, the type of maintenance applied is important and should be carefully studied before implementation. In the particular example studied, the best policy in all cases was the opportunity-triggered maintenance policy and worst policy was the corrective maintenance policy.

Future studies can be carried out on analyzing failure aspects of other FMC components such as machines, pallets with robot/AGVs. Apart from the maintenance policies considered in the study the effect of other maintenance policies can be investigated. Further, the cost of maintenance can be taken in consideration to investigate the impact of maintenance strategies in an unreliable FMC.