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Article Excerpt
1. Introduction
1.1. Closed-loop systems
A loop is a material flow system that consists of work centers or machines separated by storage areas (buffers) in which material travels from machine to buffer to machine in a fixed sequence and returns to the first machine. An example is a manufacturing system, illustrated in Fig. 1, in which raw parts enter the system from outside and are loaded onto pallets or fixtures at a loading station (machine [M.sub.1]). The pallets and the associated parts then visit buffer [B.sub.1], machine [M.sub.2], ..., [M.sub.k-1], [B.sub.k-1]. Once all the operations have been performed, the part-pallet assembly goes to the unloading station ([M.sub.k]) where the part is unloaded from its pallet. The finished part leaves the system, while the empty pallet goes to the empty pallet buffer ([B.sub.k]) to wait for a new raw part. The total number of pallets in the system--the population--is constant since pallets are not added or removed from the production line. For the pallets, the production line is closed. The production rate of parts is the same as the rate at which pallets travel through each workstation, and the distribution of parts in the system is the same as the distribution of pallets, except for the empty pallet buffer. Examples of closed-loop production lines with pallets or fixtures can be observed in automotive production, electronic component assembly, food packaging and consumer manufacturing industries. Such loops are common where work pieces are loaded onto a support in order to ensure accuracy and stability during operations. In addition, loops occur in production systems controlled by Constant Work In Process (CONWIP) (Hopp and Spearman, 1996; Hopp and Roof, 1998; Spearman et at, 1990); basestock; production authorization cards (Buzacott and Shanthiku-mar, 1992, 1993); the control point policy (Gershwin, 2000); and other policies (Bonvik et al., 1997; Dallery and Liberopoulos, 2000). There is a single loop like that of Fig. 1 in a line controlled by CONWIP; in other cases, there are multiple loops. In systems controlled with such policies tokens or production authorization cards behave similarly to the pallets as described above, and the number of tokens (or another quantity) is constant either within the whole system or within a specific portion of the system.
[FIGURE 1 OMITTED]
1.2. Related literature
Although several analytical methods have been developed for the analysis of open production lines (Dallery and Gershwin, 1992), little work has been done on closed lines with unreliable machines and finite buffers. See the references in Tolio and Gershwin (1998). Frein et at. (1996) proposed an analytical method for the performance evaluation of closed lines that is an extension of the decomposition method developed for open lines. However, this method does not fully capture the correlation that exists among the parts held in each buffer of the system. The effects of this correlation increase when the number of machines decreases. As a result, the method is more accurate for larger systems than for smaller ones. In contrast, the method described here is well suited to small systems. Papers that have treated loops by other approaches include Lim and Meerkov (1993), Kim et al. (2002), Bozer and Hsieh (2005). See, also, Akyildiz (1988) and Perros (1990).
1.3. Contribution
This paper is a summary of the major results of the work in Maggio (2000). It presents an approximate analytical method for predicting the average throughput and average work in process in each buffer of a loop system. New decomposition equations are proposed for analyzing the behavior of a generalization of the Buzacott model, i.e., a discrete-time discrete-state system which is synchronous, with unreliable machines and finite buffers. These new equations take into account the relationship between the maximum number of parts in the buffers and the propagation of interruptions of flow in the system. This relationship is due to the constant population. To do this we extend the multi-failure-mode approach (Tolio and Matta, 1998; Tolio et al., 2002).
In principle, the method of this paper can be applied to loop systems of any size. In practice, it becomes unwieldy for loops with more than three machines. We therefore restrict the presentation to three-machine, three-buffer loops. The complete development for larger systems appears in Maggio (2000). An approach that circumvents the increased complexity of larger systems is described in Levantesi (2001), Werner (2001) and Gershwin and Werner (2007).
1.4. Outline
The class of systems that we model and analyze, and the important features of their behavior that require a new approach, are described in detail in Section 2. The decomposition technique is derived in Section 3. Section 4 contains numerical results from the method, including comparisons with simulation, and we conclude with Section 5.
2. Description of the system
The model described here is an extension of the deterministic processing time model of Gershwin (1994), Tolio and Matta (1998) and Tolio et al, (2002). We make all the assumptions and approximations of those models, follow all their conventions and use their notation. We review some of the main features below, and we also describe the additional features that are special to the three-machine loop.
2.1. Assumptions and notation
2.1.1. Machines
We denote machine i by M.sub.i] and its downstream buffer by [B.sub.i] (i = 1, ..., 3). We do not distinguish among parts, pallets and tokens. We model the system as though the items neither enter nor leave, and we refer to the items as parts. We use modulo 3 arithmetic in treating the indices that refer to machines and buffers, so [M.sub.1] is the machine downstream of [M.sub.3] and [M.sub.m+3] is the same as [M.sub.m].
The model we are considering is the same as the transfer line model of Tolio and Matta (1998) except for the loop structure. Processing times of each machine are equal, deterministic and constant. Time is scaled so that operations take one time unit. We assume, also, that all the machines start their operations at the same instant. Transportation time is negligible compared to the operation time.
Machines are unreliable and may have more than one failure mode; [F.sub.i] represents the number of failure modes of machine [M.sub.i]. If [M.sub.i] is operational and neither starved nor blocked (Section 2.1.2) at the beginning of a time step, it has a probability [p.sub.ij] of failing in mode j (j = 1, ..., [F.sub.i]) during that time step. On the other hand, if machine [M.sub.i] is down in mode j at the beginning of a time step, it has a probability [r.sub.ij] of being repaired. As a consequence, the times to failure and to repair are geometrically distributed. By convention, repairs and failures occur at the beginning of time steps and changes in the buffer levels (the amounts of material in the buffers) take place at the end of time steps. Models with deterministic processing times and geometrically distributed times to failure and times to repair are referred to as synchronous models (Dallery and Gershwin, 1992) or Buzacott models. We deal with unreliable machines that have operation-dependent failures (Buzacott and Hanrfin, 1978; Dallery and Gershwin, 1992). That is, a machine can fail only while it is working.
2.1.2. Buffers
Buffers are finite; buffer [B.sub.i] can hold a maximum of [N.sub.i] parts. If [B.sub.i] is full, [M.sub.i] is not allowed to operate and is said to be blocked; if [B.sub.i] is empty, [M.sub.i+1] is starved and cannot operate. If [M.sub.i] and [M.sub.i+1] both operate during a time step, the level of [B.sub.i] is unchanged. If neither [M.sub.i] nor [M.sub.i+1] operate, the level of [B.sub.i] is also unchanged. If [M.sub.i] operates and [M.sub.i+1] does not during a time step, the level of the buffer increases by one at the end of the time step, i.e., after the machine repair/failure states are changed. Similarly if [M.sub.i] does not operate and...
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