Optimization of printed circuit board manufacturing: Integrated modeling and algorithms

Abstract

This paper focuses on an integrated optimization problem that is designed to improve productivity in printed circuit board (PCB) manufacturing. We examine the problems of allocating the components to feeders and sequencing the placement of these components on the PCBs, populated by a rotary head machine with surface mount technology. While previous research focuses on sequencing the placement and only considers this subproblem as part of an interrelated set of problems, we provide an integrated approach which tackles all subproblems simultaneously as a single problem. Given an ε-approximation algorithm for the vehicle routing problem we present a solution with an ε-error gap for the PCB problem.

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):

• allocation of component types to machines,

• allocation of component types to feeders at each machine,

• 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.