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Weekly Updates1. Introduction
The problem of optimally controlling the production rates of manufacturing systems has been widely discussed in the scientific literature. This is particularly true for one-product flow lines. This type of system is of special interest because it is usually used in mass production, and is composed of costly specialized equipment that is dedicated to the production of this single product. The owner of such a system would thus want to reach the profit threshold as rapidly as possible, especially considering that product life-cycle durations are continuously decreasing. It is a complex problem that can be solved only for very simple manufacturing systems. The solution for the two-machine transfer line is presented in this paper in order to define the structure of the control policy, which is necessary for the approach proposed here, for a general transfer line. The solution obtained for the two-machine transfer line points to an extended version of the hedging point policy, which consists in building up a fixed level of inventory whenever possible, and then attempting to maintain this level. Because of the complexity of this problem, it is still not possible to solve it for general cases such as the one we examine in this paper. Heuristic methods have to be developed in order to obtain satisfactory performances from the system.
We extend the hedging point policy to the transfer line problem, and obtain a class of suboptimal policies similar to the well-known kanban policy. We are then confronted with a one-parameter-per-machine optimization problem consisting in finding the hedging levels, or buffer capacities, of the in-process and finished goods buffers. Although knowing the structure of the policy is of great value for simplifying the problem, the complexity of the problem grows with the length of the transfer line. Since there is no analytical method to find the optimal solution, we need to use a combination of a performance-estimating tool and an optimization technique. Some very fast tools that have been developed to estimate certain performance measures, usually throughput, are decomposition techniques. Dallery et al. (1989) introduced the DDX (for Dallery-David-Xie) algorithm that used a continuous flow of material. Burman (1995) extended the algorithm to consider the case of non-homogenous lines. Dallery and Le Bihan (1999) extended the algorithm to more general failure and repair distributions with the use of generalized exponential distributions. Schor (1995) and Gershwin and Schor (2000) developed algorithms for two problems: (i) the allocation of a fixed buffer space in order to achieve maximum throughput; and (ii) the minimization of buffer space while attaining a fixed throughput. All the contributions mentioned above used the average throughput of the line as a main performance measure. They assumed a saturated demand i.e., finished goods are never stocked. Bonvik (1996) observed that most decomposition techniques tend to slightly over estimate the throughput. This might be a problem when estimating backlog in a high-utilization system, where a small overestimation of a throughput could lead to a large underestimation of backlog. Such an underestimation of backlog for high-utilization ratios was observed in Bonvik et al. (2000). This can be problematic in some cases.
Discrete-event simulation is a very effective way of estimating almost any system performance given that the input data is accurate. However, it can be very time-consuming, especially for long lines. We have observed that performing an inventory/backlog cost minimization using design of experiments and a response surface methodology can take weeks for a single experimental design for a six machine line on a 1.8 MHz Pentium[R] IV processor. This task demands a tremendous effort for a 20-machine line. We have successfully used a combined discrete-event/continuous-simulation model that can be used to reduce the number of events generated during a run and have thus reduced the computational time. However, even using this faster model, the optimization of long lines remains too time-consuming to perform for long lines using one parameter per machine.
In a study of the optimal buffer space allocation in transfer lines under saturated demands, Hillier et al. (1993) observed what they called the storage bowl phenomenon. Using an exhaustive simulation of all possible combinations of buffers, they observed that the buffers in the middle portion of the line should be given more storage than the extremity buffers when allocating a fixed amount of storage among machines. An approximate relation for the distribution of buffer space in balanced transfer lines with variable processing times is given by Hillier (2000). The proposed relation gives the increase in buffer space for the middle buffers in relation to the end buffers (first and last). This study was conducted on transfer lines of up to six machines. Schor (1995) presented an optimization algorithm for decomposition methods. Results for some homogenous lines composed of ten unreliable machines showed that the decrease in buffer space should be applied to more than just the extremity buffers (three buffers) for longer lines. On a case presented on the MIT website (Gershwin, 1996), the solution of a 20-machine homogenous line obtained with the algorithm from Gershwin and Goldis (1995) and Schor (1995) showed a transitional portion of approximately four buffers at both ends. The largest step in capacity, however, is always between the extremity and their adjacent buffers. All of these studies have been conducted under the saturated demand assumption, meaning that the system operates at maximum capacity all the time and does not respond to a fixed demand rate. However, the observed recurring profile lets us believe that saturated demand may not be the only context in which a distinct pattern arises in the distribution of buffer space/hedging levels. Based on these observations, we focus our study on the profile of the optimal stock allocation for the inventory/backlog cost-minimization problem. We believe that finding a recurring profile in the results would enable us to propose a parameterized profile that would greatly reduce the number of parameters for reasonable size transfer lines. This reduction of parameters would result in a significant reduction in the number of runs necessary for optimization of long lines, limiting the optimization problem to only a few parameters. The subsequent reduction in complexity would allow the optimization of large systems that would otherwise be too time-consuming to tackle. Profile-based heuristics have been proposed in the past for the kanban allocation problem. The most well known is the inverted bowl profile from Hillier et al. (1993) and Hillier (2000). However, the inverted bowl profile is applicable only when the objective is to minimize the storage space while attaining a target system throughput or maximize the throughput with a maximum allowable storage space. The actual utilization of this storage space is not considered as an optimization variable and production is not triggered by customer demand. The backlog level is therefore also excluded from the equation. The current problem statement considers the average inventory level, the backlog level and the satisfaction of customer demand from the finished product buffer.
The rest of this paper is organized as follows: the problem statement is presented in Section 2, the experimental approach in Section 3, the simulation model is briefly presented in Section 4. Section 5 presents the design of experiments and response surface methodology. Section 6 gives numerical results and analysis for the m-parameter optimized lines and Section 7 presents the proposed profile-based distribution of buffer space and its validation. Concluding remarks are presented in Section 8.
2. Problem statement
The studied system is a tandem line composed of m machines separated by in-process buffers ([B.sub.i]: i = 1,..., m-1.). The maximum production rate of each machine is [u.sub.max]. Every part goes through every stage of the production and every buffer before reaching the finished goods buffer. The constant rate demand (d) is satisfied through the finished goods buffer ([B.sub.m]). The raw material supply for machine [M.sub.1] is considered infinite and all transportation times are considered null. This system is illustrated in Fig. 1 with machines represented by circles and buffers by triangles.
Demands that are not satisfied directly from the finished goods buffer are backlogged. The backlog cost is a function of the backlog level.
Let [u.sub.i] (t) and [x.sub.i](t) be the production rate of machine [M.sub.i], and the stock level associated with machine [M.sub.i]. Denoting the state of machine [M.sub.i] by[[xi].sub.i] (t), we can describe the model of the system by a hybrid state with continuous (dynamics of the stocks) and discrete components (states of the machines) as follows.
Stock dynamics: We denote the number of parts in the Work In Process (WIP) as [x.sub.j], j = 1,...., m - 1 and the difference between cumulative production and demand as [x.sub.m]. The surplus [x.sub.m] can be positive (finished good inventories) or negative (backlogs). The state constraint is then 0 [less than or equal to] [x.sub.j](t) [less than or equal to] [B.sub.j] where [B.sub.j] is the size of WIP j. Let S = [0, [B.sub.1] x ... x [0, [B.sub.m-1]] x R [subset] [R.sup.m] denote the state constraint domain. The state equations can be written as follows:
[FIGURE 1 OMITTED]
[d/[dt]]([x.xub.i](t)) = [u.sub.i](t)[u.sub.i + 1](t), [x.sub.i](0) = [x.sub.i0], i = 1,...,m - 1, (1)
[d/[dt]]([x.sub.m](t) = [u.sub.m](t) - d, [x.sub.m](0) = [x.sub.m0], (2)
where [x.sub.i0] and [x.sub.m0] are the given initial WIP and surplus.
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