Reliability analysis of a flexible manufacturing cell

Reliability Engineering and System Safety 67 (2000) 147–152 www.elsevier.com/locate/ress

Reliability analysis of a flexible manufacturing cell M. Savsar* Department of Mechanical and Industrial Engineering, College of Engineering and Petroleum, Kuwait University, P.O. Box 5969, Safat 13060, Kuwait Received 4 June 1998; accepted 6 September 1999

Abstract In this study, mathematical models are developed 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. The operation times, loading/unloading times, and material handling times by the pallet are assumed to be random. The operation of the reliable cell is compared to that of an unreliable cell with respect to utilization of the cell components, including the machine, robot, and pallet handling system. The unreliable cell is assumed to operate under random (machine and robot) failures with constant failure rates for the machine and the robot. The pallet handling system is assumed to be completely reliable. q 2000 Elsevier Science Ltd. All rights reserved. Keywords: Flexible manufacturing cell; Flexible manufacturing systems; Reliability analysis

1. Introduction In recent years a great deal of attention has been given to the automation of manufacturing systems. In order to meet increased demand for customized products and to reduce production lot sizes, the industry has adapted new techniques and production concepts by introducing flexibility into the production machines so that a variety of products can be manufactured on the same equipment. As indicated by Snader [1] and Chan and Bedworth [2], at present, the most feasible approach for automating the job shop process seems to be through flexible manufacturing cells (FMCs), which require lower investment, less risk, and also satisfy many of the benefits gained through flexible manufacturing systems (FMSs). While FMSs are very expensive and generally require investments in millions of dollars, FMCs are less costly, smaller and less complex systems. Therefore, for smaller companies with restricted capital resources, a gradual integration is initiated with limited investment in a small FMC, which facilitate subsequent integration into a larger system, a FMS. An FMC consists of a robot, one or more flexible machines including inspection, and an external material handling system such as an automated pallet for moving blanks and finished parts into and out of the cell. The robot is utilized for internal material handling which includes machine loading and unloading. The FMC is capable of doing different operations * Tel.: 1965-4811188; fax: 1965-484-7131. E-mail address: [email protected] (M. Savsar).

on a variety of parts, which usually form a part family with selection by a group technology approach. The cell performance depends on several operational and system characteristics, which include, part scheduling, robot, machine and pallet characteristics. Smith et al. [3] have presented a survey of the characteristics of US FMSs. Most of the research related to operational characteristics of FMCs are directed to the scheduling aspects. Scheduling algorithms are used to determine the sequence of parts, which are continuously introduced to the cell. Chan and Bedworth [2], Hitomi and Yoshimura [4], Seidman [5], and Hutchinson et al. [6] have developed models for static and dynamic scheduling in FMCs. However, system characteristics, such as configuration, design, and reliability of FMCs, have significant effects on its performance. Machining rate, pallet capacity, robot speed and pallet speed, are important system characteristics affecting FMC performance. Several models have been developed for FMSs and FMCs in relation to the effects of different parameters on system performance. Buzacott and Yao [7], Henneke and Choi [8], Sabuncuoglu and Hommertzheim [9], and Savsar and Cogun [10] have presented stochastic and simulation models for evaluating the performance of FMCs and FMSs with respect to system configuration and component speeds, such as machining rate, robot and pallet speeds. Koulamas [11] looked into the reliability and maintenance aspects and presented a stochastic model for an FMC, which operates under a stochastic environment with tool failure and replacement consideration. He developed a semi-Markov model to study the effects of tool failures on system performance.

0951-8320/00/$ - see front matter q 2000 Elsevier Science Ltd. All rights reserved. PII: S0951-832 0(99)00056-3

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M. Savsar / Reliability Engineering and System Safety 67 (2000) 147–152

Fig. 1. A flexible manufacturing cell with a robot and a pallet.

Thus, with the exception of the study by Koulamas [11], previous researches have been mostly directed to the scheduling aspects of reliable FMC systems. Not much research has been done on the reliability aspects of such systems. Vineyard and Meredith [12] have looked into the effects of maintenance policies on FMS failures by studying some real systems using simulation. Their conclusion suggests that the FMSs experience four times more wear and tear during their useful life than the traditional machine tools. This is mainly because of the accelerated usage of the FMSs during their useful life. Past studies, such as Smith [3], indicate that while the FMS and FMC systems typically operate 70–80% of the time during their useful life, the traditional machine tools are only utilized 20% of the time during their useful life. Therefore, reliability analysis of flexible systems is extremely important in understanding and increasing the utilization and productivity of such systems. In this study, we present a stochastic model to determine the performance of a FMC under random operational conditions, including random failures of cell components (machine tool and robot) in addition to random processing times, random machine loading and unloading times, and random pallet transfer times. Stochastic modeling is necessary due to different operational requirements of a variety of parts dynamically scheduled to enter into the cell. Further, production machine and robot failures are very important consideration in system performance. Stochastic models are presented to study and compare the utilization of the components of a reliable cell to that of an unreliable cell. In particular, utilization rates of the machine tool, the robot and the pallet handling system are formulated and compared under the fully reliable system and the unreliable system with specified component hazard rates and operational conditions.

robot reaches to the machine, grips the part, moves to the pallet and releases the part in its spot. Then, it picks up another blank, moves to the machine and loads it to the machine. This cycle of operations is continued until all n blanks are completed. The pallet then moves completed parts out of the cell and a new pallet with a set of n blanks is delivered to the cell automatically. Due to the introduction of different parts into the FMC and the characteristics of the system operation, processing times as well as the loading/unloading times are random, which present a complication in studying and modeling the cell performance. The problem is further complicated if random failures of the machine tool as well as the robot are incorporated into the model. In the following section, stochastic mathematical models are presented to study cell performance under the mentioned characteristics and operational conditions. 3. Stochastic models Stochastic models are developed for the FMC discussed above and illustrated in Fig. 1. Processing times on the machine, robot loading and unloading times, pallet transfer times and the equipment up and down times are all assumed as random quantities that follow exponential distribution. Models are presented for the unreliable cell and the reliable cell with no failures. In order to model the FMC operation, the following system states are defined: Sijk …t† Pijk …t† i j

k

state of the FMC at time t probability that the system will be in state Sijk …t† number of blanks in the FMC (on the pallet and on the machine or the robot gripper) state of the production machine (j ˆ 0 if the M/C is idle; j ˆ 1 if the machine is operating on a part; and j ˆ d if the machine is down under repair) state of the robot (k ˆ 1 if the robot is loading/ unloading the machine; k ˆ 0 if the robot is not engaged in loading/unloading the machine; and k ˆ d if the robot is down under repair)

The following notation is used for the system parameters in the model.

i u z v l

2. Operation of the cell

m

The FMC considered in this study is illustrated in Fig. 1. An automated pallet handling system delivers n blanks consisting of different parts into the cell. The robot reaches to the pallet, grips a blank, moves to the machine and loads the blank. After the machining operation is completed, the

a b n n

loading rate of the robot (parts/unit time) unloading rate of the robot (parts/unit time) combined loading/unloading rate of the robot (parts/unit time) pallet transfer rate (pallets/unit time) failure rate of the production machine (1/l ˆ mean time between machine failures) repair rate of the production machine (1/m ˆ mean machine repair time) failure rate of the robot repair rate of the robot machining rate (or production rate) of the machine (parts/unit time) pallet capacity (number of parts/pallet)

M. Savsar / Reliability Engineering and System Safety 67 (2000) 147–152

Qc

149

production output rate of the cell in terms of parts/ unit time

Using the state probability definitions and the notations above, the stochastic transition diagram of the unreliable FMC operation, with machine tool and robot failures, is shown in Fig. 2. Using the fact that the net flow rate at each state is equal to the difference between the rates of flow in and flow out, the following system of differential equations are constructed for the unreliable FMC with machine and robot failures. dPn;0;0 …t†=dt ˆ vP0;0;0 2 iPn;0;0 dPn21;d;0 …t†=dt ˆ lPn21;1;0 2 mPn21;d;0 dPn21;1;0 …t†=dt ˆ iPn;0;0 1 mPn21;d;0 2 …l 1 n†Pn21;1;0 dPn21;0;1 …t†=dt ˆ nPn21;1;0 1 bPn21;0;d 2 …a 1 z†Pn21;0;1 dPn21;0;d …t†=dt ˆ aPn21;0;1 2 bPn21;0;d

dPn2x;d;0 …t†=dt ˆ lPn2x;1;0 2 mPn2x;d;0 dPn2x;1;0 …t†=dt ˆ zPn2x11;0;1 1 mPn2x;d;0 2 …l 1 n†Pn2x;1;0 dPn2x;0;1 …t†=dt ˆ nPn2x;1;0 1 bPn2x;0;d 2 …a 1 z†Pn2x;0;1 dPn2x;0;d …t†=dt ˆ aPn2x;0;1 2 bPn2x;0;d

where P0;0;0 ˆ 1={nvn21 …1 1 s† 1 v…1 1 r†…nz21 1 u21 2 z21 †

dP0;d;0 …t†=dt ˆ lP0;1;0 2 mP0;d;0

1vi21 1 1}

dP0;1;0 …t†=dt ˆ zP1;0;1 1 mP0;d;0 2 …l 1 n†P0;1;0 dP0;0;1 …t†=dt ˆ nP0;1;0 1 bP0:0:d 2 …a 1 u†P0;0;1

…1†

dP0;0;d …t†=dt ˆ aP0;0;1 2 bP0;0;d dP0;0;0 …t†=dt ˆ uP0;0;1 2 vP0;0;0 For the steady state solution, we let t ! ∞ and thus dP…t†=dt ! 0 in the Equation set (1) above. The resulting set of difference equations are solved by using the fact that the sum of all probabilities is 1, i.e. n X d X d X

Pijk ˆ 1:

Fig. 2. Stochastic transition diagram of the FMC with machine tool and robot failures.

…2†

iˆ0 jˆ0 kˆ0

…4†

and s ˆ l=m; r ˆ a=b: System performance is measured by the utilization rate of the production machine (Lm), utilization rate of the robot (Lr) and utilization rate of the pallet handling system (Lh). These measures are calculated by using the system state probabilities determined above. P0,0,0 represents the utilization rate of the pallet handling system. It is fraction of the time that handling system is loading and unloading a pallet at a rate of v pallets/unit time or nv parts/unit time. Thus, Lh ˆ P0;0;0

…5†

Similarly, utilization rate of the machine is the fraction of time that the machine is operational, and is given by nX 21

The following general solutions are obtained for the state probabilities:

Lm ˆ

Pi;1;0 ˆ …v=n†P0;0;0 ; i ˆ 0; …n 2 1;

and utilization rate of the robot is the fraction of time that the robot is operational and is given by

Pi;d;0 ˆ …lv=mn†P0;0;0 ; i ˆ 0; …n 2 1;

Pi;1;0 ˆ …nv=n†P0;0;0

Lr ˆ Pn;0;0 1

Pi;0;1 ˆ …v=z†P0;0;0 ; i ˆ 1; …n 2 1;

…6†

iˆ0

nX 21

Pi;0;1 1 P0;0;1

iˆ1

P0;0;1 ˆ …v=u†P0;0;0 ; Pi;0;d ˆ …av=bz†P0;0;0 ; i ˆ 1; …n 2 1;

ˆ ‰v=i 1 …n 2 1†v=z 1 v=uŠP0;0;0

P0;0;d ˆ …av=bu†P0;0;0 ;

The above model is for the unreliable cell with machine

Pn;0;0 ˆ …v=i†P0;0;0 ;

(3)

…7†

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M. Savsar / Reliability Engineering and System Safety 67 (2000) 147–152

number of parts processed by the machine per unit time. It is obtained for both, reliable and unreliable cells as follows: Qc ˆ Lm n ˆ …nv=n†P0;0;0 n ˆ nvP0;0;0

…11†

Eqs. (5)–(11) are easily used to determine the utilization rates of the cell components, as well as the production output rate of the cell, for both, the reliable and unreliable cell systems. In order to illustrate the applications of these formulas, the solution results of an FMC case problem are shown in the next section. It is interesting to note that the ratios Lm =Lh and Lr =Lh are the same for reliable and unreliable cells. This can be easily verified by substituting the corresponding values and determining the ratios as follows: Lm =Lh ˆ nv=n is the same for both reliable and unreliable cells. Similarly, Lr =Lh ˆ …n 2 1†v=z 1 v=i 1 v=u is also the same for reliable and unreliable cells.

Fig. 3. Effects of pallet capacity on machine utilization rate.

tool and robot failures. For the reliable FMC without machine and robot failures, system states corresponding to Si;d;0 and Si;0;d ; where i ˆ 0; 1; …; n 2 1; are not applicable. A procedure similar to the above could be applied to the rest of the transition diagram and the utilization rates of the reliable FMC components could be obtained. However, an easier way is to use the fact that a reliable FMC is a system with no failures, i.e.l ˆ 0 and a ˆ 0: Thus, setting s ˆ l=m ˆ 0 and r ˆ a=b ˆ 0 in Eqs. (5)–(7), the following set of Eqs. (8)–(10) are easily obtained for the reliable FMC. Lh ˆ P0;0;0 ˆ 1=‰1 1 nv=n 1 …n 2 1†w=z 1 …u 1 i†v=uiŠ …8† nv P Lm ˆ n 0;0;0 Lr ˆ ‰…n 2 1†v=z 1 v=i 1 v=uŠP0;0;0

…9† …10†

Production output rate of the cell, Qc, is defined as the

The implications of these results are that failures of system components have no effects on the two ratios given above. The relative impact of failures is the same for both subsystems due to the fact that machine and robot utilization rates are directly proportional to the pallet handling system utilization rate. The functional relationships or the proportionality rates are the same regardless of the cell reliability. In other words, relative utilization rates of the machine and the robot remain constant regardless of the degree of reliability introduced.

4. A case problem and results A case problem has been selected with the following cell parameters in order to illustrate the application of the models. The results are presented in graphical form. The following are the assumed mean values for various cell parameters: Operation time per part: n21 ˆ 5 time units Robot loading time (for the first part): i 21 ˆ 0.06 time units Robot loading/unloading time for subsequent parts: z21 ˆ 1:0 time units Robot unloading time for the last part: u21 ˆ 0:6 time units Time between machine tool failures: l21 ˆ 100 time units Repair time (down time) of the machine tool: m 21 ˆ 10 time units Time between robot failures: a 21 ˆ 120 time units Repair time (down time) of the robot: b 21 ˆ 12 time units Pallet transfer time: v 21 ˆ 5 time units per pallet Pallet capacity, n, has been varied from 1 to 40 parts/pallet.

Fig. 4. Effects of pallet capacity on robot utilization rate.

Utilization rates of the production machine, the robot, and the pallet handling systems are compared for the reliable and unreliable FMC with component failures in order to visualize the effects of these failures on the utilization of different components for different pallet capacities. Fig. 3

M. Savsar / Reliability Engineering and System Safety 67 (2000) 147–152

Fig. 5. Effects of pallet capacity on pallet utilization rate.

illustrates the utilization rate of the production machine. As it can be seen from this figure, machine tool utilization is highly affected by the pallet capacity up to a certain level and stabilizes thereafter. However, there is a significant gap between the fully reliable cell and the slightly unreliable cell, with specified component hazard rates. Decrease in machine tool utilization is directly reflected in cell productivity. The mentioned gap increases with increasing pallet capacity. The production output rate of the cell, Qc, is obtained by multiplying the machine tool utilization with the average production output rate. For example, in case of the pallet capacity of 40 parts/pallet, production output rate of the fully reliable cell would be about …0:82†…1=5† ˆ 0:164 parts/time unit, while the production output rate of the unreliable cell would be about …0:74†…1=5† ˆ 0:148 parts/time unit (see the approximate utilization values in the graph in Fig. 3). Note that, since the average processing time is 5 time units, the average output rate is 1=5 ˆ 0:20 parts/time

Fig. 6. FMC production output rate at two different machine failure rates.

151

unit if the machine is fully utilized. Fig. 4 shows the trend in utilization of the robot under two cases at different pallet capacities. It is very similar to the case for the machine tool. Fig. 5 shows the utilization rate of the pallet handling system under the two conditions at different pallet capacities. As the pallet capacity is increased, the pallet system utilization decreases. However, the difference between the utilizations of the pallet systems for the fully reliable and unreliable cells is not as significant as it was for the robot and the machine tool. The utilization rate of the production machine is more important since it reflects the productivity of the system. These results show the significant effects of the failures on the utilizations of the FMC system components and thus its productivity. The effects of machine tool and robot failure rates on the production output rate of the FMC, Qc, are further illustrated in Figs. 6 and 7. In both figures all the parameters are as specified above for the case problem. However, in Fig. 6, two levels of machine tool failure rates are compared to the fully reliable case with respect to cell production output rate at different pallet capacities. The same illustration is done for the robot failure rates in Fig. 7. The effects of machine failures on production output rate is more pronounced as expected. A two-fold increase in robot failure rate has a slight reduction on production output rate.

5. Conclusions FMCs are gaining wide acceptance in today’s dynamic manufacturing environment. Thus, reliability analysis of FMC systems is also important. While reliability modeling of traditional machines and production systems has been subject of extensive research over the past several years, only few researches could be found on reliability studies of FMC systems. The stochastic models and the closed form solution formulas obtained in this paper could be used to analyze the productivity of a FMC under different failure, repair, and other operational characteristics. The formulas are relatively simple and can be used with simple calculations to obtain exact solutions. The models presented here could be extended to the case where two machine tools are served by a single robot and a pallet handling system. Production machines could work in series or in parallel. Reliability analysis of such systems is useful in determining the exact production output rates of the FMC or utilization of system components. The stochastic models developed in this paper are based on the assumption that time between failures, as well as the repair times, are exponentially distributed. In real life failure times are usually exponentially distributed. Repair times usually follow Erlang distribution, which includes exponential as a special form. Future extension of this research could be to study the reliability analysis of FMC systems under other types of time to failure and time to repair distributions.

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References

Fig. 7. FMC production output rate at two different robot features rates.

Simulation modeling could be a viable approach for such studies. Preventive maintenance schedules could also be included in such models to analyze the combined effects of corrective maintenance and preventive maintenance on system performance. Acknowledgements The author would like to thank the two referees whose valuable comments have substantially improved the quality and presentation of this paper.

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