Modeling and analysis of an unreliable flow line composed of parallel-machine stages.

ALAIN PATCHONG [1][*]

DIDIER WILLAEYS [2]

This paper focuses on flow lines composed of multiple parallel-machine stages. The system is similar to a classical flow line, the only difference being that a given stage may consist of parallel machines. A method for modeling and

analyzing this type of flow lines is presented. The technique used amounts to replacing each parallel-machine stage by a single equivalent machine in order to obtain a classically-structured flow line with machines in series. Equations for computing the parameters of equivalent machines are submitted. These parameters can then be used to carry out performance diagnosis of the corresponding stage, or as input data for existing analysis methods to evaluate the production performance of flow lines. The method presented in this paper has been successfully implemented and tested in the industry over the past few years.

1. Introduction

A transfer line (also called manufacturing flow line or production line) is made of material, work areas and finite storage areas. The material flows from a work area to the following storage area. Work areas may consist of machines or operators who perform a value-added operation on the material (Strosser, 1991). The lime spent by the material in a work area may be considered as deterministic when it does not vary from one part to the next; stochastic when it varies randomly from part-to-part. This randomness may be due to random processing times, random failures and repair events, or both. When a failure occurs at a machine, it becomes unavailable for a certain amount of time, the machine is said to be down. A machine is said to be operational if it is not down, it can then either be working if it is processing a part or otherwise idle. An idle machine is starved or blocked. A machine is said to be blocked if, after the completion of its service on a part, the downstream buffer becomes full. The blocked mac hine is stopped from processing until a place becomes available in the downstream buffer. On the other hand, we assume that a machine is starved if, after completion of its service on a part, the upstream buffer becomes empty. The starved machine is prevented from processing until a part arrives in the upstream buffer. The blocking mechanism described here is known as blocking after service (David et al., 1990). In this article, we deal with transfer lines comprised of multiple parallel-machine stages. Such systems are similar to a classical transfer line except for the fact that a given stage may consist of parallel machines. Burman (1995) refers to this kind of systems as series-parallel flow lines.

In general, the manufacturing engineer has at his disposal three means of increasing the throughput of a manufacturing system: (i) increase intermediate storage capacity; (ii) improve the availability of machines; and (iii) increase the working capacity of machines (Marris, 1994). All these solutions are limited. In fact, failures may not be completely eliminated in a manufacturing system. Increasing intermediate storage capacity requires that space be available and that possible resulting storage costs be taken into account. In such a situation, multiple-machine stages may be considered as a solution if, for technological reasons, machines are prevented from processing, beyond a specific speed.

We now present the contents of this paper. Section 2 gives a succinct review of the main existing papers familiar to us that deal with the subject of parallel machines. A new analysis method based on replacing multiple-machine stages by an equivalent machine is presented in Section 3. In Section 4, the method is tested on some examples and results are compared with those yielded by simulation. Finally, Section 5 compares on one hand the proposed method and assorted results, and on the other hand results obtained from major existing methods.

It should be mentioned that the method presented in this paper has been successfully implemented in industry over the past few years. It was initially developed to compute input data necessary to carry out the diagnosis technique described in Cauffriez et al. (1996).

2. Brief literature review

The literature abounds with papers describing methods for analyzing manufacturing systems. However, the area of series--parallel manufacturing systems has remained somewhat an unexplored field (Dallery and Gershwin, 1992; Patchong et al., 1997). In this section, we present a succinct review of our quest for the main existing papers relating to parallel machines.

Ignall and Silver (1977) propose an approximate method for systems made of stages of identical machines. The method evolves from the observation by Buzacott which gives the following general formulation for the production rate of a two-machine flow line:

P(N) = [P.sub.0] + ([P.sub.[infinity]] - [P.sub.0]) x m(N),

where [P.sub.0] and [P.sub.[infinity]] are the production rates of the line with no buffer and with infinite buffer respectively, and m(N) is a monotonically increasing function of the buffer N. The above formula can be extended to multiple-machine systems by adjusting [P.sub.0] and [P.sub.[infinity]].

Mitra (1988) presents a numerical study showing that multiple-machine stages are preferable to a single-machine stage when the isolated production rates are equal.

Forestier (1980) studies an automated production system comprising of two-stage machines separated by an intermediate storage buffer. As in Dubois and Forestier (1982), which is a particular case, machines are modeled by a Markov chain with three-variable states. The solution is applied to the specific case of single-machine stages.

Ancelin and Semery (1987) present a method that uses for the first time the technique of replacing a parallel-machine stage by a single equivalent machine.

The equivalent processing rate of the set of parallel machines ([U.sub.eq]) is defined as the sum of the processing rates of all machines in the set:

[U.sub.eq] = [[[sigma].sup.n].sub.j=1] [U.sub.j].

The equivalent failure rate ([[lambda].sub.eq]) is given by

[[lambda].sub.eq] = [[[sigma].sup.n].sub.i=1] ([[lambda].sub.i] [[[pi].sup.n].sub.j=I j[not equal to]i] [[micro].sub.i]/[[micro].sub.i] + [[lambda].sub.i]).

The equivalent repair rate ([[micro].sub.eq]) is derived from the following relations

[P.sub.eq] = [U.sub.eq]/(1 + [[lambda].sub.eq]/[[micro].sub.eq]) with [P.sub.eq] = [[[sigma].sup.n].sub.j=1] [P.sub.j],

where n is the number of machines of the stage, [P.sub.j], [[micro].sub.j] and [[lambda].sub.j] are respectively the production rate in isolation, repair rate and failure rate of machine j.

Burman (1995) also presents a method based on replacing of multiple-machine stages by a single equivalent machine. The flow line is represented by the continuous model (i.e., the discrete flow of parts is approximated by a continuous flow). The maximum processing rate of the equivalent machine is the sum of the processing rates of parallel machines. The production rate of the equivalent machine equals that of the set of parallel machines. The assumption that each machine in the set of parallel machines operates independently and the theorem due to Drake (which states that the expected value of the sum of a set of random variables is the sum of the expected values of the random variables in the set) are used to calculate the remaining parameters.

Ancelin and Semery (1987) and Burman (1995) are the only contribution in our literature review that can actually be used to analyze series-parallel flow lines. Unfortunately these methods did not give satisfaction in their application to performance diagnosis. The reasons are as follows:

* Performance diagnosis described in Cauffriez et al. (1996) includes analysis of the equivalent parameters of each parallel-machine stage, especially: failure rate and repair rate. We noticed that equivalent parameters yielded by the methods of Ancelin and Semery (1987) were often unrealistic and then meaningless. It is important to emphasize that equivalent failure rate and equivalent repair rate of parallel-machine stages must be physically meaningful to carry on the above-mentioned diagnosis.

* Burman (1995) is not easy to implement. In fact, use of the theorem of Drake leads to a time-dependent equation, resolution of which requires an appropriate choice of the time value. The technique proposed by the author for choosing this appropriate time value appears to be rather complicated. It should be noted that we had no prior knowledge of Burman's method at the time we developed the method presented in this paper.

Some of the above reasons prompted us to develop the method presented in Section 3. In fact, there was a need for a method that is both easy to implement and gives realistic equivalent repair rate and failure rate values.

3. Description of the proposed method

3.1. Notation

3.1.1. The basic parameters of an isolated machine

Many of the notations used here are consistent with those used by Dallery and Gershwin (1992) and Patchong (1997). We use lowercase letters for these basic parameters of isolated machines.

[t.sub.i] = average processing time of machine [M.sub.i];

[u.sub.i] = average processing rate of machine [M.sub.i];

[mttf.sub.i] = average time to failure of machine [M.sub.i];

[[lambda].sub.i] = average failure rate of machine [M.sub.i];

[mttr.sub.i] = average time to repair of machine [M.sub.i];

[[micro].sub.i] = average repair rate of machine [M.sub.i];

[e.sub.i] = isolated efficiency of machine [M.sub.i], that is the average proportion of time during which machine [M.sub.i] would be operational if it were operated without being starved or blocked;

[p.sub.i] = isolated production rate of machine [M.sub.i], that is the average number of parts that machine [M.sub.i] would have produced per unit of time if it were operated without being starved or blocked.

The previous parameters are related by the five equations below, in fact, there are only three independent parameters. The most common independent parameters are: [[lambda].sub.i], [[micro].sub.i] and [u.sub.i].

[u.sub.i] = 1/[t.sub.i]. (1)

[[lambda].sub.i] = 1/[mttf.sub.i]. (2)

[[micro].sub.i] = 1/[mttr.sub.i]. (3)

[e.sub.i] = [mttf.sub.i]/[mttf.sub.i] + [mttr.sub.i] = [[micro].sub.i]/[[micro].sub.i] + [[lambda].sub.i]. (4)

[p.sub.i] = [u.sub.i][e.sub.i] = [u.sub.i][[micro].sub.i]/[[micro].sub.i] + [[lambda].sub.i]. (5)

Should the isolated machine, whose parameters are defined above, be put in a flow line, additional parameters would be needed to characterize it. These parameters are given below.

3.1.2. The parameters of a machine performing in a flow line

That is parameters mostly used in performance evaluation. Their first letter is uppercase.

[Pw.sub.i] = probability that machine [M.sub.i] is working (also called utilization rate or efficiency);

[P.sub.i] = production rate of machine [M.sub.i] that is the average number of parts produced by [M.sub.i] in the flow line per unit of time;

[PS.sub.i] = probability of machine [M.sub.i] being starved;

[Pb.sub.i] = probability of machine [M.sub.i] being blocked;

[Pi.sub.i] = probability of machine [M.sub.i] being idle (starved or blocked);

[Pd.sub.i] = probability of machine [M.sub.i] being down.

The definition of an idle machine can be given by the following equation.

[Pi.sub.i] = [PS.sub.i] + [Pb.sub.i]. (6)

It is important to note that, (6) is correct if and only if a machine [M.sub.i] is prevented from being simultaneously starved and blocked. Hence, (6) is correct for continuous fluid model and approximate for discrete material flow (Gershwin, 1994).

3.2. Basic equations

In this section, we present basic relations that can be applied to manufacturing systems in general, regardless of processing time, uptime, and downtime distribution (Dallery and Gershwin, 1992).

3.2.1. Between production rate and efficiency

This equation results from the definition of the production rate:

[P.sub.i] = [u.sub.i][Pw.sub.i]. (7)

3.2.2. Between probability of failures and efficiency

If we assume operation-dependent failures (i.e., failures that occur only while the machine is processing a part), the behavior of the machine is characterized by a three-state Markov chain (Fig. 1). Using balance equations, we can write:

[[lambda].sub.i][Pw.sub.i] = [[micro].sub.i][Pd.sub.i] (8)

3.2.3. Between probability of failure, probability of idleness and efficiency

As shown in Fig. 1, at any time, machine [M.sub.i] is in one of the three following states: down, working, or idle. That can be equated as follows:

[Pw.sub.i] + [Pd.sub.i] + [Pi.sub.i] = 1. (9)

3.2.4. Between probability of idleness and efficiency

This equation is derived from (4), (8) and (9):

[Pw.sub.i] = [e.sub.i](1 - [Pi.sub.i]). (10)

Note that for reliable machines, [e.sub.i] = 1 and [Pd.sub.i] = 0, hence (9) and (10) become the same equation and reduce to the following:

[Pw.sub.i] + [Pi.sub.i] = 1. (11)

3.3. Additional notations and equations

Similarly to (7), we define the following production losses:

[I.sub.i] = [u.sub.i][Pi.sub.i]. (12)

[D.sub.i] = [u.sub.i][Pd.sub.i]. (13)

That is production losses due to idleness and failures respectively.

Equations (7), (9), (12) and (13) yield:

[P.sub.i]/[u.sub.i] + [D.sub.i]/[u.sub.i] + [I.sub.i]/[u.sub.i] = 1 (14)

In the remainder of this paper, we use (14) as follows:

[P.sub.i] + [D.sub.i] + [I.sub.i] = [u.sub.i] (15)

3.4. Solution technique

Figure 2 depicts a series-parallel flow line made of k work centers (set of parallel machines). A work center i is composed of [J.sub.i] parallel machines and is separated from the downstream by intermediate buffer [B.sub.i] (except the last). A machine [M.sub.ij] represents the jth machine of the ith set of parallel machines. The method presented here was initially developed for performance diagnosis. Therefore, its original goal was to provide both the isolated parameters of the equivalent machine [M.sub.i] (i.e., [[micro].sub.i], [[lambda].sub.i], [u.sub.i] and a means to assess the effects of idleness ([Pi.sub.i]), using data measured on each machine [M.sub.ij]. The technique is based on replacing sets of parallel machines by an equivalent machine so that the flow line in Fig. 2 is replaced by the one in Fig. 3. The parameters of the equivalent machine must be defined so that its behavior in the line (in-line behavior) matches the one of the set of parallel machines. To apply the method presented here we should know or be able to calculate at least four independent parameters for each machine [M.sub.ij]: the three classical independent parameters related to the machine and one parameter related to the presence of the machine in the flow line. Stated this way, the problem amounts to formulating four equations needed to calculate the four unknown quantities (for example: [[micro].sub.i], [[lambda].sub.i], [u.sub.i], [Pi.sub.i]). We choose to develop the proposed method using the four independent parameters: [[lambda].sub.ij], [u.sub.ij], [Pw.sub.ij], [Pd.sub.ij]. So, in the remainder of this paper, we assume that these parameters are known or can be computed. Note that this choice does not restrict the applicability of the method. There is just a need for four independent parameters. Cauffriez et al. (1998) have written a whole paper to justify the need for these parameters. Should more details be needed, see this paper. The above quantities ([[lambda].sub.ij], [u.sub.ij], [Pw.sub.ij], [Pd.sub.ij]) can be calculated using Equations (A1)-(A7) of Appendix A.

3.4.1. Equation 1: processing rate of the equivalent machine

We define the processing rate [u.sub.i] of the equivalent machine ([M.sub.i]) corresponding to stage i as the sum of the processing rates of all parallel machines when the set of machines in stage i work normally (without failure):

[u.sub.i] = [[[sigma].sup.Ji].sub.j=1] [u.sub.ij]. (16)

3.4.2. Equation 2. production rate of the equivalent machine

The production rate [P.sub.i] of the equivalent machine ([M.sub.i] is the sum of production rates of all machines in stage i:

[P.sub.i] = [[[sigma].sup.Ji].sub.j=1] [P.sub.ij]. (17)

3.4.3. Equation 3: failure rate of the equivalent machine

We state that the equivalent machine ([M.sub.i]) is considered to be down if at least one of the [J.sub.i] machines in stage i is down. In other words, it suffices that one of parallel machines in stage i fails (while the others [J.sub.i] -- 1 machines remain operational) for [M.sub.i] to be considered down. If a subsequent failure of a parallel machine in stage i occurs, it is not counted as a failure of [M.sub.i] (since [M.sub.i] is already down), however, the resulting slowing down of the processing rate is taken into account. The above can be summarized as follows: there is already a failure in the system (stage i) that will not be considered as a failure but will slow down the processing rate of the equivalent machine. That can be summarized as follows:

[[lambda].sub.i][Pw.sub.i] = [[[sigma].sup.Ji].sub.j=1] [[lambda].sub.ij][Pw.sub.ij] [[pi].sub.k[not equal to]j] (1 - [Pd.sub.ik]). (18)

From (18) we can derive the failure rate of the equivalent machine as follows:

[[lambda].sub.i] = 1/[Pw.sub.i] [[[sigma].sup.[J.sub.i]].sub.j=1] [[lambda].sub.ij][Pw.sub.ij] [[pi].sub.k[not equal to]j] (1 - [Pd.sub.jk] (19)

[Pw.sub.i] is derived from (7), (16) and (17)

[Pw.sub.i] = [[[sigma].sup.[J.sub.i]].sub.j=1] [u.sub.ij][Pw.sub.ij]/[[[sigma].sup.[J.sub.i]].sub.j=1] [u.sub.ij]. (20)

3.4.4. Equation 4: repair rate of the equivalent machine

Assuming inter-independence of events (especially failures), let us observe machine j of stage i at the level of the corresponding equivalent machine. If machine j fails, we can reasonably estimate that the resulting production shortfall probability should match the proportion of losses due to the inactivity of machine j multiplied by the probability of failure of machine j. Hence, the probability of failure of the equivalent machine can be written as follows:

[Pd.sub.i] = [[[sigma].sup.[J.sub.i]].sub.j=1] = [Pd.sub.ij] X [u.sub.ij]/[u.sub.i]. (21a)

Another way to obtain the previous equation is to observe that the loss due to a failing machine leads to the shortfall of an equivalent machine i, in proportion with its speed. Hence Equation (21b):

[Pd.sub.i] X [u.sub.i] = [[[sigma].sup.[J.sub.i]].sub.j=1] [Pd.sub.ij] X [u.sub.ij]. (21b)

From (8), (20) and (21a) we can derive the repair rate of the equivalent machine as follows:

[[micro].sub.i] = [[[sigma].sup.[J.sub.i]].sub.j=1] [[lambda].sub.ij][Pw.sub.ij] [[pi].sub.k[not equal to]j](1 - [Pd.sub.ik])/([[[sigma].sup.[J.sub.i]].sub.j=1] [Pd.sub.ij] X [u.sub.ij]/[u.sub.i]), (22)

where [u.sub.i] is computed using (16).

3.4.5. Idleness of the equivalent machine

The probability of idleness of the equivalent machine is derived from the four previous equations. The yielded result is sometimes needed to perform performance diagnosis of work centers.

Equation (15) is valid for each machine j of stage i:

[P.sub.ij] + [D.sub.ij] + [I.sub.ij] = [u.sub.ij]. (23)

Equation (15) is also valid for the equivalent machine corresponding to stage i:

[P.sub.i] + [D.sub.i] + [I.sub.i] = [u.sub.i]. (24)

Using (23) for every machine in stage i and summing up yields:

[[[sigma].sup.[J.sub.i]].sub.j=1] [P.sub.ij] + [[[sigma].sup.[J.sub.i]].sub.j=1] [D.sub.ij] + [[[sigma].sup.[J.sub.i]].sub.j=1] [I.sub.ij] = [[[sigma].sup.[J.sub.i]].sub.j=1] [u.sub.ij]. (25)

Using (16) and (17), (25) becomes:

[P.sub.i] + [[[sigma].sup.[J.sub.i]].sub.j=1] [D.sub.ij] + [[[sigma].sup.[J.sup.i]].sub.j=1] [I.sub.ij] = [u.sub.i]. (26)

Furthermore, (21a) can be written as follows:

[D.sub.i] = [[[sigma].sup.[J.sub.i]].sub.j=1] [D.sub.ij]. (27)

From (24), (26) and (27) we obtain

[I.sub.i] [[[sigma].sup.[J.sub.i]].sub.j=1] = [I.sub.ij]. (28)

Using (12) and (10), we obtain

[Pi.sub.i] = 1/[u.sub.i] [[[sigma].sup.[J.sub.i]].sub.j=1] [u.sub.ij] (1 - [Pw.sub.ij]/[e.sub.ij]), (29)

where [u.sub.i] is computed using (16).

3.5. Case of machines that are not subject to idleness

In this section, we focus on the specific case of machines that are not subject to idleness (neither starved, nor blocked). This is equivalent to considering machines as performing in isolation (out-line evaluation). This variant of the proposed method can be used to compute optimal parameters when designing manufacturing systems.

If we consider that machines are performing in isolation, we have:

[Pi.sub.ij] = 0, [forall] i [epsilon] [1,...,k] and [forall] j [epsilon] [1,...,[J.sub.i]]. (30)

Equation 1, Equation 2, and Equation 4 translated by (16), (17), and (21) respectively, are linked. That is, this set of three equations is equivalent to a set of two independent equations. Therefore, the original set of four equations (Equation 1, Equation 2, Equation 3 and Equation 4) can be reduced to a set of three independent equations with three independent variables. The unique set of solutions to these equations is as follows:

[u.sub.i] = [[[sigma].sup.[J.sub.i]].sub.j=1] [u.sub.ij],

[[lambda].sub.i] = 1/[Pw.sub.i] [[[sigma].sup.[J.sub.i]].sub.j=1] [[lambda].sub.ij] [[[pi].sup.[J.sub.i]].sub.k=1] [Pw.sub.ik],

[u.sub.i] = [[[sigma].sup.[J.sub.i]].sub.j=1] [[lambda].sub.ij] [[[pi].sup.[J.sub.i]].sub.k=1] [Pw.sub.ik]/([[[sigma].sup.[J.sub.i]].sub.j=1] [Pd.sub.ij] X [u.sub.ij]/[u.sub.i]). (31)

These solutions are derived from (16), (19) and (22) respectively.

It can be proved that Equation 1, Equation 2 and Equation 4 are linked by showing that combining two of them may yield the third. We will show that combining

Equation 1 and Equation 2 yields Equation 4. We know in a system with no idleness, (9) reduces to:

[Pw.sub.i] + [Pd.sub.i] = 1. (32)

Recall that this equation is both applicable to parallel machines [M.sub.ij] and equivalent machine [M.sub.i]. Hence Equation (20), which evolves from Equation 1 and Equation 2, can be written:

1 - [Pd.sub.i] = [[[sigma].sup.[J.sub.i]].sub.j=1] [u.sub.ij](1 - [Pd.sub.ij])/ [[[sigma].sup.[J.sub.i]].sub.j=1] [u.sub.ij],

or:

[Pd.sub.i] = [[[sigma].sup.[J.sub.i]].sub.j=1] [u.sub.ij][Pd.sub.ij])/ [[[sigma].sup.[J.sub.i]].sub.j=1] [u.sub.ij]

The above equation is equivalent to Equation 4 translated by (21a).

When using the method presented here in performance diagnosis of series-parallel lines, metrology should be performed first. Metrology is the measurement of times necessary to computing needed data (see Appendix A).

4. Calculations

4.1. Numerical results

Analytic methods give approximate estimates of performance parameters. In order to avoid possible inaccuracies due to imperfections of analytic methods, we simulate both the original series-parallel flow lines and the equivalent series-structured flow lines wherein the sets of parallel machines are replaced by corresponding equivalent machines. We perform discrete flow simulation using a simulator software called ARENA[TM]. The duration of the simulation is 350 000 time units. During the first 50 000 time units, statistics are not collected. This warm-up period (the first 50 000 time units) is used to avoid transient effects on the final results. There exist no universal examples that we can use to evaluate the accuracy of the proposed method. So, we first use examples similar to those given by Burman (1995). Next, we examine examples of flow lines containing stages with five parallel machines. Testing examples are three-stage series-parallel flow lines. The first and the third stages are composed of sin gle identical machines ([[micro].sub.1] = [[micro].sub.3] = 0.1; [[lambda].sub.1] = [[lambda].sub.3] = 0.01; [u.sub.1] = [u.sub.3] = 1). The second stage is composed of two (see Fig. 4) or five parallel machines. Each system is run twice. In the first run, capacities of the intermediate buffers are set to 10. Then, they are dropped to two. This allows us to observe the effect of intermediate storage on the accuracy of estimates yielded by the proposed method. Parameters for testing flow lines are given in Appendix B. We perform two types of estimation. The first simulation termed In-Line Estimation (ILE) is based on (16), (19) and (22). Here, machines are assumed to be processing within a flow line, which implies existence of blocking and starvation. The second simulation termed Out-Line Estimation (OLE) is based on the set of Equations (31). Here, machines are assumed to processing without idleness (or out-of-line). Results for the simulation of the original flow line are referred to as Original Line Estimat ion (ORLE). The quantities of interest are, the production rate (P), the equivalent failure rate of the second stage ([[lambda].sub.2]), the equivalent repair rate of the second stage ([[lambda].sub.2]), the average level of the first intermediate buffer [B.sub.1] (n.sub.1] and the average level of the second intermediate buffer [B.sub.2] ([n.sub.2]). Numerical results presented in this section are those of the analysis of the three following sets of examples.

4.1.1. Redundant parallel machines

All parallel machines of the second stage are identical to the first and third machines. In Appendix B, this type of machines corresponds to Case 1 and Case 2 (for two-machine second stage flow lines), Case 7 and Case 8 (for five-machine second stage flow lines). Corresponding results are given in Table 1 and Table 2.

4.1.2. Slow parallel machines

The parallel machines of the second stage are identical to the first and third machines except that their speeds are slowed down in order to have the same isolated production rate as the first and third machines. Hence, all these flow lines are balanced (their isolated production rates are equal). In Appendix B, this type of machines corresponds to Case 3 and Case 4, (for two-machine second stage flow lines), Case 9 and Case 10 (for five-machine second stage flow lines). In the case of two-machine second stage flow lines, each of the parallel machines operates at half of the speed of the first and the third machines. In the case of five-machine second stage flow lines, each of the parallel machines operates at one-fifth of the speed of the first and the third machines. Results for estimations are given in Table 3 and Table 4.

4.1.3. Unreliable parallel machines

The parallel machines of the second stage are identical to the first and third machines except that their reliabilities are degraded in order to have the same isolated production rate as the first and third machines. As a result, all these flow lines are balanced and each of the parallel machines of the second stage is half as reliable as those in the first and third stages. In Appendix B, these types of machines correspond to Case 5 and Case 6, for two-machine stage flow lines. Corresponding results are given in Table 5. We do not examine the extreme case of a five-machine stage which we found unrealistic and therefore uninteresting.

4.1.4. Concluding remarks

It appears that the accuracy of results are not directly related to the capacities of intermediate storage buffers, especially when the isolated production rate of the set of parallel machines is greater than those of the first and third machines (Tables 1 and 4). We notice that Results for ILE and OLE are very close. The average difference is 0.1%, for production rates. This closeness is logical since it is an empirical confirmation of the proof given in Section 3.5. That also proves the consistency of equations used, particularly, the correctness of Equation 4. The ILE should be used when the computation of the probability of idleness of the set of parallel machines is needed. In this case, the probability of idleness can be calculated using (29). This estimation also allows us to perform performance evaluation using data collected directly on the actual flow line. The OLE should be used when designing flow lines or when idleness data are not available. The less accurate results are due to flow lines whose second stage machines are less reliable than the first and third machines (see Table 5). The number of parallel machines of a stage does not significantly affect the accuracy of the estimations (see Tables 1, 2, 3 and 4). Finally, estimations of the average buffer levels does not present any pertinent singularity to be commented upon. Finally, the machines we tested closely match the field of industrial (realistic) cases as defined by Dallery and Gershwin (1992). That may justify the success of industrial utilization of the method presented.

5. Comparison with main existing methods

In this section, the proposed method is compared with major existing methods, namely Ancelin and Semery (1987) and Burman (1995).

5.1. Comparison with the method of Ancelin and Semery (1987)

When the parallel machines of the stage are identical as in the study cases we used in this work, the results of the proposed method (OLE) and the one of Ancelin and Semery (1987) are the same. So, it is not relevant to use those study cases to compare the two methods. On the other hand, when the machines are not identical, we noticed in some cases that unlike the proposed method, the repair rate and the failure rate of Ancelin and Semery (1987) did not match reality. The comparison we made focus on this weakness. In the example reported here, we have compared the results for both methods on a two-parallel-machine stage ([M.sub.1] and [M.sub.2]). The machines are initially set identical: [[micro].sub.1] = [[micro].sub.2] = 0.1; [[lambda].sub.1] = [[lambda].sub.2] = 0.01; [u.sub.1] = [u.sub.3] = 1. Then, we make the failure rate of machine [M.sub.1] vary from 0.01 to one. The evolution of the equivalent failure rate and repair rate are presented in Figs. 5 and 6 respectively. The results given by Ancelin and S emery (1987) are denoted A&S while those of the proposed method are denoted as P&W. As shown on Fig. 5, the manner of the equivalent failure rate denoted P&W is more stable than the one denoted A&S, since it increases much more slowly towards an asymptote, and reaches a steady value. In fact, the most important criticism that can be made about the method of Ancelin and Semery (1987) is related to the fact that Fig. 6 shows that, unlike P&W, the equivalent repair rate of the stage of A&S increases when [M.sub.1] becomes less reliable. So, should this method be used to carry on a performance diagnosis on a real production line this will be interpreted by: "degrading the reliability of one of the machines would improve the repair rate of the stage". This conclusion seems very difficult to admit and not easy to explain to field manufacturing engineers. More generally, Figs. 5 and 6 show that the parameters of P&W are more stable and more realistic than those of A&S.

5.2. Comparison with results for Burman (1995)

In Table 6, the results for the proposed method and those of Burman (1995) are compared for Cases 1, 3 and 5. Bur1 and Bur2 refer to the two types of methods proposed by Burman. Unlike the second, the first method does not take into account the variance of events in the flow line. For each Estimation E (OLE, Bur1 or Bur2), Error of Estimation E (Error OLE, Error Bur1 and Error Bur2) is computed as follows:

Error of Estimation E = Estimation E -- ORLE/ORLE

Table 6 shows that the proposed method is often better than either one or both of Burman's estimations (especially Bur1), but not substantially enough to conclude that it is superior. Similar conclusions can be drawn from the study of the cases with small buffers.

6. Conclusion

In this paper, we have presented an easily-applicable method to model and analyze series-parallel flow lines. Though initially dedicated to performance diagnosis, this method can be used to estimate equivalent parameters of sets of parallel machines, therefore allowing the utilization of classical algorithms of flow line analysis. The equivalent parameters estimated may also be used to perform simulation. The proposed method can be applied with any random distribution of uptime, downtime and processing time. Moreover, it is credited with successful implementation in industry. It also appears that the main benefits of the proposed method are its simplicity of implementation and its applicability to performance diagnosis.

Acknowledgements

We thank Jean-Paul Dessap for his useful suggestions. We are grateful to the anonymous referees for their valuable comments and suggestions. The research reported in this paper was supported by PSA Peugeot-Citroen.

(1.) PSA Peugeot Citroen, Route de Gisy, 78140 Velizy-Villacoublay, France E-mail: patchon1@mpsa.com

(2.) LAMIH URA 1775 CNRS, Universite de Valenciennes, 59304 Valenciennes, France

(*.) Corresponding author

Biographies

Alain Patchong is currently working with PSA Peugeot-Citroen, a carmaker group. He is in charge of the design of manufacturing flow lines of the body in white plants. He holds Engineering degrees in Electro-mechanics (1992) and in Robotics (1993) respectively from Ecole Polytechnique de Yaounde, Cameroon and Ecole des Mines de Douai, France. He received his Master (1994) and Ph.D. (1997) degrees in Industrial and Human Automatics from the Universite de Valenciennes, France. His primary research interests arc in the areas of diagnosis and performance evaluation of manufacturing systems.

Didier Willacys is a Professor of Automatic Control and Engineering of Manufacturing Systems at Universite de Valenciennes, France. He is also a senior research scientist at the Laboratoire d'Automatique et de Mecanique industrielles et Humaines, a laboratory associated with the French national center of scientific researches (CNRS). His research interests include diagnosis methods using Boolean logic, fuzzy set theory and Al. He holds patents in the field of diagnosis systems for the certification of PLC programs. Since 1992 he has worked on diagnosis methods to improve the performance of manufacturing systems and applies his results in many European factories.

Contributed by the Manufacturing Systems Modeling Department.

References

Ancelin, B. and Semery, A. (1987) Calcul de la productivite d'unc ligne integree do fabrication: CALIF, une methode analytique industrielle. RAIRO APII, 21(3), 209-238.

Burman, M.H, (1995) New results in flow line analysis. Ph.D. Dissertation, MIT, Cambridge, MA.

Cauffriez, L., Willaeys. D. and Defrenne, J. (1996) A method and a diagnosis system to value the production performances of manufacturing flow-line systems, in Proceedings of the CESA'96 IMACS IEEE-SMC Multiconference, 2(2), pp. 814-819.

Cauffriez, L., Willaeys, D. and Defrenne, J. (1998) Mesure des indicateurs de performance do lignes de production: presentation d'une methode et retour d'experience. RAIRO-APII-JESA. (in the press).

CNOMO (1987) Moyens do production, agrement-fiabilite-maintenabilite-disponibilite, temps d'etat d'un moyen, definition. Normalisation Equipement E41.50.520.N.

Dallery, Y. and Gershwin, S.B. (1992) Manufacturing flow lines systems: a review of models and analytic results. Queueing Systems: Theory and Applications, 12(1-2), 3-94.

David, R., Xie, XL. and Dallery, Y. (1990) Properties of continuous model of transfer lines with unreliable machines and finite buffers. IMA J. Math. Bus. Ind., 6, 281-308.

Dubois, D. and Forestior, J.P. (1982) Productivite et en-cours moyens d'un ensemble do deux machines separees par une zone do stockage. RAIRO Automatique, 16(2), 105-132.

Forestier, J.P. (1980) Modelisation stochastique et comportement asymptotique d'un systeme automatise do production. RAIRO Automatique. 14(2), 127-143.

Gershwin, S. B. (1994) Manufacturing Systems Engineering, Prentice Hall, Englewood Cliffs, NJ.

Ignall, E. and Silver, A. (1977) The output of a two-stage system with unreliable machines and limited storage. AIIE Transactions, 9, 183-188.

Marris, Ph. (1994) Management par les Contraintes en Gestion Industrielle: Trouver le bon Desequilibre, Editions d'Organisations, Paris.

Mitra, D. (1988) Stochastic theory of a fluid model of multiple failure-susceptible producers and consumers coupled by a buffer. Advances in Applied Probability, (in the press).

Patchong, A. (1997) Methodes dc modelisation, d'analyse et de diagnostic pour l'amelioration de l'efficacite d'un atelier de production. Ph.D. Dissertation, Universite de Valenciennes, France.

Patchong, A., Willaeys, D. and Defrenne, J. (1997) Analysis of a transfer line with automated and manual stations and, no intermediate storage, in Proceedings of the EDA 97 International Conference on Engineering Design and Automation.

Strosser, C.M. (1991) Diagnosis of production shortfall in a transfer line by means of analytic and simulation methods. Master's Thesis, MIT, Cambridge, MA.

                    Results for flow lines with two
                      redundant parallel machines
                  Case 1: [N.sub.i] = 10
                           ORLE            OLE    ILE
P                         0.872           0.863  0.861
[n.sub.1]                 3.717           3.567  3.433
[n.sub.2]                 6.382           6.647  6.520
[[micro].sub.2]            n/a            0.182  0.193
[[lambda].sub.2]           n/a            0.018  0.019
                  Case 2: [N.sub.i] = 2
                          ORLE            OLE    ILE
P                         0.834          0.826  0.827
[n.sub.1]                 0.786          0.784  0.758
[n.sub.2]                 1.250          1.293  1.249
[[micro].sub.2]            n/a           0.182  0.193
[[lambda].sub.2]           n/a           0.018  0.019
                    Results for flow lines with five
                      redundant parallel machines
                  Case 7: [N.sub.i] = 10
                           ORLE            OLE    ILE
P                         0.884           0.873  0.872
[n.sub.1]                 3.502           3.495  3.539
[n.sub.2]                 6.794           6.467  6.509
[[micro].sub.2]            n/a            0.342  0.472
[[lambda].sub.2]           n/a            0.034  0.047
                  Case 8: [N.sub.i] = 2
                          ORLE            OLE    ILE
P                         0.859          0.844  0.844
[n.sub.1]                 0.659          0.783  0.764
[n.sub.2]                 1.287          1.226  1.234
[[micro].sub.2]            n/a           0.342  0.473
[[lambda].sub.2]           n/a           0.034  0.047
                      Results for flow lines with
                       two slow parallel machines
                  Case 3: [N.sub.i] = 10
                           ORLE            OLE    ILE
P                         0.830           0.832  0.832
[n.sub.1]                 6.672           6.540  6.647
[n.sub.2]                 3.519           3.265  3.413
[[micro].sub.2]            n/a            0.182  0.183
[[lambda].sub.2]           n/a            0.018  0.018
                  Case 4: [N.sub.i] = 2
                          ORLE            OLE    ILE
P                         0.761          0.794  0.792
[n.sub.1]                 1.318          1.396  1.391
[n.sub.2]                 0.673          0.601  0.604
[[micro].sub.2]            n/a           0.182  0.185
[[lambda].sub.2]           n/a           0.018  0.018
                      Results for flow lines with
                      five slow parallel machines
                  Case 9: [N.sub.i] = 10
                           ORLE            OLE    ILE
P                         0.847           0.838  0.837
[n.sub.1]                 7.054           6.869  6.669
[n.sub.2]                 3.549           3.162  3.013
[[micro].sub.2]            n/a            0.342  0.353
[[lambda].sub.2]           n/a            0.034  0.035
                  Case 10: [N.sub.i] = 2
                           ORLE            OLE    ILE
P                         0.746           0.797  0.798
[n.sub.1]                 1.180           1.494  1.495
[n.sub.2]                 0.844           0.501  0.473
[[micro].sub.2]            n/a            0.342  0.369
[[lambda].sub.2]           n/a            0.034  0.036
                    Results for flow lines with two
                      unreliable parallel machines
                  Case 5: [N.sub.i] = 10
                           ORLE            OLE    ILE
P                         0.757           0.680  0.686
[n.sub.1]                 5.961           5.275  5.390
[n.sub.2]                 4.214           4.766  4.638
[[micro].sub.2]            n/a            0.091  0.013
[[lambda].sub.2]           n/a            0.109  0.107
                  Case 6: [N.sub.i] = 2
                          ORLE            OLE    ILE
P                         0.624          0.586  0.587
[n.sub.1]                 1.130          1.003  1.029
[n.sub.2]                 0.788          1.005  0.999
[[micro].sub.2]            n/a           0.091  0.123
[[lambda].sub.2]           n/a           0.109  0.155
                 Comparison with the results of Burman
                                 (1995)
            ORLE    OLE  Error  Bur 1  Error  Bur 2  Error
                           OLE         Bur 1         Bur 2
                           (%)           (%)           (%)
Case 1
P          0.872  0.863   -1.0  0.853   -2.2  0.861   -1.3
[n.sub.1]  3.717  3.567   -4.0  3.572   -3.9  3.437   -7.5
[n.sub.2]  6.382  6.647    4.2  6.541    2.5  6.543    2.5
Case 3
P          0.830  0.832    0.2  0.824   -0.7  0.830    0.0
[n.sub.1]  6.672  6.540   -2.0  6.124   -8.2  6.603   -1.0
[n.sub.2]  3.319  3.265   -1.6  3.397    2.4  3.397    2.4
Case 5
P          0.757  0.680  -10.2  0.682   -9.9  0.728   -3.8
[n.sub.1]  5.961  5.275  -11.5  5.179  -13.1  5.393   -9.5
[n.sub.2]  4.214  4.766   13.1  4.826   14.5  4.598    9.1

[Graph omitted]

[Graph omitted]

Appendices

Appendix A: measurement of times

Notations and definitions of times given in this section are consistent with the CNOMO norm (CNOMO, 1987) used by most French companies.

[TR.sub.i] = required time of machine [M.sub.i];

[TF.sub.i] = uptime of machine [M.sub.i];

For each machine [M.sub.i], the above-defined times are related as follows:

[TAP.sub.i] = downtime of machine [M.sub.i];

[NAP.sub.i] = number of breaks (mainly failures) of machine [M.sub.i];

[TSAT.sub.i] = saturation time (mainly blocking) of machine [M.sub.i];

[TDES.sub.i] = removal time (mainly starvation) of machine [M.sub.i];

[TR.sub.i] = [TF.sub.i] + [TAP.sub.i] + [TSAT.sub.i] + [TDES.sub.i] (Al)

Main equations relating the machine parameters and the times measured

These relationships help calculate input data for the method developed using results from metrology.

[[micro].sub.i] = [NAP.sub.i]/[TAP.sub.i]. (A2)

[[lambda].sub.i] = [NAP.sub.i]/[TF.sub.i]. (A3)

[Pw.sub.i] = [TF.sub.i]/[TR.sub.i]. (A4)

[Pd.sub.i] = [TAP.sub.i]/[TR.sub.i]. (A5)

[Ps.sub.i] = [TDES.sub.i]/[TR.sub.i] (A6)

[Pb.sub.i] = [TSAT.sub.i]/[TR.sub.i] (A7)

[Pi.sub.i] is computed, using Equations (6), (A6) and (A7).

                      Appendix B: testing examples
           Parameters
         [[micro].sub.i]  [[lambda].sub.i]  [U.sub.i]  [N.sub.i]
Case I
1             0.100            0.010          1.000       10
2,1           0.100            0.010          1.000       n/a
2,2           0.100            0.010          1,000       10
3             0.100            0.010          1.000       n/a
Case 2
1             0.100            0.010          1.000        2
2,1           0.100            0.010          1.000       n/a
2,2           0.100            0.100          1.000        2
3             0.100            0.010          1.000       n/a
Case 3
1             0.100            0.010          1.000       10
2,1           0.100            0.010          0.500       n/a
2,2           0.100            0.010          0.500       10
3             0.100            0.010          1.000       n/a
Case 4
1             0.100            0.010          1.000        2
2,1           0.100            0.010          0.500       n/a
2,2           0.100            0.010          0.500        2
3             0.100            0.010          1.000       n/a
Case 5
1             0.100            0.010          1.000       10
2,1           0.100            0.120          1.000       n/a
2,2           0.100            0.120          1.000       10
3             0.100            0.010          1.000       n/a
Case 6
1             0.100            0.010          1.000        2
2,1           0.100            0.120          1.000       n/a
2,2           0.100            0.120          1.000        2
3             0.100            0.010          1.000       n/a
Case 7
1             0.100            0.010          1.000       10
2,1-5         0.100            0.010          1.000       10
3             0.100            0.010          1.000       n/a
Case 8
1             0.100            0.010          1.000        2
2,1-5         0.100            0.010          1.000        2
3             0.100            0.010          1.000       n/a
Case 9
1             0.100            0.010          1.000       10
2,1-5         0.100            0.010          0.020       10
3             0.100            0.010          1.000       n/a
Case 10
1             0.100            0.010          1.000        2
2,1-5         0.100            0.010          0.020        2
3             0.100            0.010          1.000       n/a