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...is motivated by our recent work with several local electronics assembly companies. In typical security device manufacturing company, more than 200 different devices are produced. Although the products vary greatly, the major differentiation among these products is mainly due to the power supply, signal frequencies and sensor detection ranges. In order to compete successfully in the market, it is crucial for the company to improve its response time to market demands. However, unpunctual assembly material arrivals, uncertain demands, and the snowball production syndrome cause great concerns for the company's managers.
The company produces a family of micro-controller-based motion detectors, which detect an intrusion by using digital and analog signal processing technology. Based on dual microwave/PIR technology, the detector uses an on-board micro-controller for micro-controller advanced processing to enhance performance and to make the detectors immune to false alarms. Two microwave sensors are required for each module; one sends the microwave signal and the other receives the signal. About a dozen key components make one product different from another. The assembly process consists of some major operations, such as material debugging, board processing, SMT (Surface Mount Technology), stuffing, tin oven, test and final assembly. During the assembly process, various assembly materials, such as sensors and support circuits, are required. These assembly materials include raw materials/components purchased from outside suppliers as well as components manufactured by internal sub-assemblies. In contrast to materials and components manufactured internally or purchased locally, the leadtime of those components purchased from overseas is much more random compounded with the uncertainty of customs clearance. The assembly process (specifying the operation sequence and the starting time of the assembly process) is designed in advance and work orders are issued to the workers. At the same time, in order to ensure system quality, a testing plan, which may involve a number of testing devices, is also determined. With instructions in the form of work orders that are determined by the assembly process, the workers assemble the products by adding various materials and components to the semi-products until the end product is created. Once the assembly process is determined, the operations must exactly follow the assembly sequence.
The company faces many difficulties in the design of an assembly process. It is common for the assembly materials not to arrive punctually due to frequent late deliveries from both outside suppliers and the internal sub-assemblies. If these uncertainties are not taken into account, the result is a poor assembly design, which, in turn, results in frequently late deliveries to customers. In particular, if an assembly process is designed such that an operation involving assembly materials with a highly uncertain arrival time is processed at an early stage, it is highly likely that later operations may have to wait for the completion of that operation even if their corresponding materials are available. In the past, in order to avoid such uncertainties and improve the response time, manufacturers often built up large safety stocks of those materials with highly uncertain arrival times. However, with the growing variety of assembly materials and components, manufacturers find that this policy results in prohibitively high inventory costs. Moreover, since the life cycle of electronic products is now very short, holding large safety stocks potentially increases the risk of material obsolescence.
Generally, the assembly sequence is primarily determined by the product assembly-ability and the existing location of assembly facilities. However, the order of assembly operations can be arranged in more than one sequence. For example, during the process of assembling security devices, the SMT operation can be carried out either at the early or the late stage of the assembly process. Interested readers may refer to Lee (1996) and Lee and Tang (1997) for other examples. It is widely recognized that the design of the operation sequence in the assembly process offers great potential for improving total production performance.
In this paper, we investigate how to design the sequence of an assembly process taking into account the stochastic characteristics of material arrival times. Given the nature of stochastic material arrivals, the product finishing time is also uncertain. Therefore, our objective is to design an assembly sequence such that the probability of on-time delivery to customers is maximized. This study focuses on the design level of this problem, rather than on its operational level. As such, it is primarily concerned with the design of the sequence in the assembly process to maximize the possibility of on-time delivery. It is important to note that once the assembly process is designed, the operations must exactly follow the assembly sequence regardless of material arrival times.
There is literature related to assembly and manufacturing process re-engineering although most of it concentrates on the issue of how to reduce inventory costs by redesigning the operation sequence. Child et al. (1992) provide a comprehensive discussion on how a company reorganized its operations and strategies. Lee (1996) and Lee and Tang (1997) use a delayed differentiation concept to study product/process redesign to minimize inventory costs. In a recent paper, Lee and Tang (1998) investigate the problem of reducing the variability of production volume by interchanging adjacent operations, which is known as operations reversal. Swaminathan and Tayur (1999) utilize two-stage stochastic programming to study the problem of assembly sequence design and vanilla box configurations. Their objective function is to reduce the number of changes to the sequence design and to reduce inventory costs. Yan et al. (2002) investigate how to minimize investment in safety stocks by reengineering the manufacturing process through sequencing and merging operations. They consider one and two product cases and demonstrate that the methodology can be applied in a hierarchical manner to solve problems with more than two products.
The rest of this paper is organized as follows. In Section 2, the basic mathematical model and the optimality properties for the problem are presented. A few special cases are discussed in Section 3. Two heuristic algorithms are proposed in Section 4, both of which have computational time complexity O(n2). In Section 5, by developing an upper bound, a branch and bound algorithm to solve two general cases optimally is presented. Section 6 presents the computational experimental results. The paper concludes with Section 7, which suggests future research directions.
2. The basic model and optimality properties
We consider the simplest version of an assembly process in which one end product is assembled through a series of assembly operations. The assembly process is sequential and the operations can be done in any sequence without constraints. Without loss of generality, we assume that exactly one specific material is required to carry out the associated assembly operations. Different operations require different materials. Each material is supplied by an outside vendor and the delivery time of the material is an independent random variable with known cumulative and density distribution functions. To accommodate the stochastic material arrivals and to increase the performance of the assembly process, our objective is to design an assembly sequence such that the probability of on-time delivery to customers is maximized. This study focuses on the design level of this problem, rather than on its operational level. Nevertheless, in designing the sequence, we assume that, at the operational level, only after a specific assembly material arrives can an associated operation be carried out. At any time, no more than one operation can be occurring and no preemption is allowed. We use the following notation to signify the given data:
n = number of assembly operations;
N = {1, 2, ..., n}, set of assembly operations; d = the due date of the end product;
[pi] = the assembly process design that defines an operation sequence.
For all i [member of] N,
[M.sub.i] = material required by operation i;
[X.sub.i] = arrival time of [M.sub.i], which is a random variable;
[P.sub.i] = the processing time of assembly operation i;
[f.sub.i](t) = a probability density function for the arrival time of [M.sub.i];
[F.sub.i](t) = Prob{[X.sub.i] [less than or equal to] t}, a cumulative distribution function for the arrival time of [M.sub.i], assume that [F.sub.i](t) is continuous;
MS = [[ZIGMA].sup.n.sub.t=1] [P.sub.i], the sum of the processing times of all assembly operations. To avoid trivial cases, assume that MS [less than or equal to] d.
To accommodate the stochastic material arrivals, our objective is to find an assembly sequence design such that the probability of finishing the end product on time is maximized. Given [pi], denote g([pi]) as the probability of finishing the end product on time. Let [pi](i) be the operation of the ith position in [pi] and [t.sub.[pi](i)] be the planned starting time of operation [pi](i) in [pi]. Since g([pi]) is determined by the probability of being able to start each operation of [pi] as planned, we have
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
According to the definition of g([pi]), we have the following properties for the optimal assembly design. See the Appendix for the proofs.
Property 1. There is no idle time between operations in the optimal...
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