The Two-Stage Assembly Scheduling Problem: Complexity and Approximation

This paper introduces a new two-stage assembly scheduling problem. There are m machines at the first stage each of which produces a component of a job. When all m components are available, a single assembly machine at the second stage completes the job. The objective is to schedule jobs on the machines so that the makespan is minimized. It is shown that the search for an optimal solution may be restricted to permutation schedules. The problem is proved to be NP-hard in the strong sense even when m = 2. A schedule associated with an arbitrary permutation of jobs is shown to provide a worst-case ratio bound of 2, and a heuristic with a worst-case ratio bound of 2 lim is presented. The compact vector summation technique is applied for finding approximate solutions with worst-case absolute performance guarantees. Introduction The two-stage assembly problem is a generalization of the two-machine flow shop problem. Informally, it can be described as follows. There are n jobs to be processed. In the first stage, each of the machines M2 , i 1, m, m > 2, processes a component of a job; these machines work independently of each other. In the second stage, the assembly machine M4 assembles the m prepared components of each job. Each job Jj , j = 1, n, consists of a chain of sets of operations ({01,j, 0„,0 }, 0A0 ). An operation 0, 0 is to be processed on machine M„ this requires p.i,j time. Machine Mi can process at most one job at operation 0A,j is to be performed on MA and takes pA,) time. For 2,..., az, k = 1, m, i k, operations 0,0 and Ok j , j = 1, to be processed simultaneously. An assembly operation Oki may i = 1, m, and a time. An assembly any i and k, i = 1, n, are allowed start only after all operations 01,j , 0„,,j have been completed. The assembly machine MA can assemble the components of at most one job at a time. The criterion for optimality is the makespan Cmax , i.e., we need to minimize the time that all machines have completed all n jobs. The problem is frequently encountered in practice. Picture, for instance, the production of personal computers. Orders are assembled to customer specification at a packaging station. A customer typically requires a specific set of modules; a central processing unit, a hard disc, a video display unit, a printer, an appropriate keyboard, a set of manuals in the right language, etc. Although there may be only a few options for each module (e.g., there may only be five types of hard discs), a large variety of end products can still be offered to the customer by using different combinations at the packaging station. The modules are produced on independent feeder lines, say one line for the keyboards, one for the display units, etc. It is clear that this situation fits our assembly scheduling model. Of course, there are many other situations where a set of modules are produced on independent feeder lines, followed by an assembly or a packaging step. As many industries move closer to Just-In-Time systems, this type of layout will increasingly be found. Moreover, the market pressure for larger variety combined with the need to control costs in a global competitive environment forces companies to re-design products with flat bills of materials and modular structures. It follows that the problem discussed in this paper becomes increasingly relevant.