Theory and MethodologyOptimization of printed circuit board manufacturing: Integrated modeling and algorithms
Introduction
There has been an increasing interest regarding the modeling and analysis of manufacturing systems in the electronics and semiconductor industry. This interest has been influenced by the advancements made in the design of computerized numerical control (CNC) machines. The sophistication of these machines varies tremendously and, in this paper, we study the surface mount technology with a CNC machine that has a rotary head to perform the operations. A CNC functions as follows: a PCB is placed on a table and adjacent to it are multiple feeder locations with a different component type assigned to each. A head is used to grasp these components from their feeder locations and mount them on the PCB with the help of an arm (the head is usually located on top of the arm). Some heads are capable of picking up more than one component simultaneously, and these are called “rotary heads”. These heads are widely used in the Surface Mount Technology. Our particular focus is on a CNC with a rotary single head that can pick up a certain number of components of the same type at a single time and mount them on the board (e.g. Quad 400 series). Fig. 1 is a generalized illustration of a CNC that is used for manufacturing a PCB. The components that are mounted on a PCB are discrete devices such as resistors, diodes, transistors, transformers, or connectors.
The manufacturing of a PCB consists of three basic steps or subproblems (see, e.g., Ball and Magazine, 1988):
- 1.
allocation of component types to machines,
- 2.
allocation of component types to feeders at each machine,
- 3.
pick-and-placement sequencing.
Earlier studies have formulated the third subproblem, pick-and-placement sequencing, independent of the allocation decisions. However, the sequencing of components on a PCB is dependent on the feeder location of the component types. When the assignment of component types to feeders is not done carefully, even if pick-and-placement sequencing is solved for optimality, it can result in an extremely poor performance. The efficiency/inefficiency of such independent approaches can best be understood by presenting an error guarantee. The error guarantee can be defined as the ratio of the difference between the feasible solution and the lower bound values to the value of the lower bound.
In general, there is no guarantee for the quality of optimal solutions in the PCB manufacturing problem, and this study aims to enlighten the quality of solutions. As a result of the need for an integrated approach, we integrate the above-mentioned three subproblems into a single problem. We use Lagrangian Relaxation to decompose the integrated mathematical model into two subproblems and then propose an algorithm that attempts to solve both of these subproblems. Given the optimal costs of the vehicle routing problem, we propose a method which finds an optimal solution for the integrated PCB problem. We also show that our integrated algorithm has an error guarantee of ε if the error bound of each vehicle routing problem is ε.
Section snippets
Literature review
Among the many studies which examine the PCB manufacturing problem are Ball and Magazine (1988), Christopher et al. (1991), Drezner and Nof (1984), Gavish and Seidmann (1988), Leipala and Nevalainen (1989), McGinnis et al. (1992), McGinnis et al. (1993), and Or and Demirkol (1996). Ball and Magazine (1988) describe the three subproblems defined in the previous section and consider only a subset of our problem, sequencing the placement for insertion technology machines. These machines are
The integrated model and an algorithm
In this section, we provide a mathematical model that simultaneously considers the three subproblems defined earlier. We discuss the implications of two different technologies; one with the capability of moving the feeder locator, and the other with no feeder locator movement.
The first subproblem, allocation of component types to machines, can be reduced to the second subproblem, allocation of components to feeders at each machine, by indexing the feeder locations. For each component type, the
Computational experiments
In this section, we present the computational experiments that show the benefits of using the integrated approach. We start with describing a typical PCB used in these experiments. The PCB is rectangular with dimensions of 500 units of length and 100 units of width. In order to represent the real characteristics of a PCB, some of the components are allocated on a line (vertical, horizontal, diagonal, and circular), some of them are randomly distributed on the free space, and the rest are placed
Conclusions and future extensions
This paper describes an optimization problem that has significant implications for the productivity of printed circuit board manufacturing using Surface Mount Technology with a rotary head. The problem presents a series of optimization subproblems which need to be addressed simultaneously in order to realize system-wide improvements. Since the overall problem is NP-hard, it is unlikely that optimal solutions can be obtained in polynomial time. Our proposed approach simultaneously accounts for
Acknowledgements
During the progression of this research, Kemal Altinkemer and Burak Kazaz were partially supported by the Center for Management of Manufacturing Enterprises (CMME), Purdue University. Murat Köksalan was partially supported by the Turkish Scientific and Technical Research Council, TÜBİTAK.
Authors gracefully acknowledge the referees and the editor whose comments immensely improved the presentation of the paper.
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