辅导 MATH6119、Python/c++编程语言讲解
Case Study - Information Sheet
Printed Circuit Board Assembly
1 Background
Philips Electronics produces printed circuit boards (PCBs) for use in consumer electronics (audio and video equipment, personal computers) and in professional industries
(telecommunication systems, aircraft navigation, medical equipment). Special machines
are used to mount components onto the boards. The problem to be considered is one
of production preparation: for each type of PCB, it is necessary to specify the way in
which the components should be mounted.
This case study is based on work carried out for one of the major product divisions
of Philips. Although some simplifications have been made to the original problem, this
does not affect the main characteristics of the decisions that have to be taken.
2 Assembly of PCBs
PCBs are assembled by automated machines. A conveyor feeds each board into the first
machine, and transports the boards between machines. Thus, a board passes into the
first machine where a selected subset of components are mounted, then passes to the
second machine where further component mountings are performed, and so on, until the
last machine completes the PCB. The conveyor moves all partially assembled boards to
the next machine simultaneously. Thus, the conveyor only moves when every machine
has completed its work on a board. For example, if there are eleven components to be
mounted by three machines, A, B and C, where machine A mounts components 1, 2,
3 and 4 and takes 8 seconds, machine B mounts components 5, 6 and 7 and takes 12
seconds, and machine C mounts components 8, 9, 10 and 11 and takes 9 seconds, then the
conveyor moves the boards every 12 seconds. Consequently, the first PCB is completed
after 36 seconds, and a further PCB is completed at the end of each subsequent 12-second
interval.
The components to be mounted on the board are contained in feeders on one side of
the conveyor. The components are classified into different types, and each feeder only
contains components of a single type.
Each machine has a robot arm with three heads. The heads are each fitted with a
piece of equipment which can pick components from the feeders and subsequently place
them on the board. At most one component at a time can be carried by a head. Note that
each component type can only be handled by a subset of the set of head equipments. In
other words, a head with a given piece of equipment can only pick and place components
of a limited set of component types.
The mounting process consists of a sequence of pick-and-place moves. In the picking
phase, the heads pick a component from the relevant feeders in turn. The order is
fixed: the first head picks first, then the second, and finally the third head. During
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the placing phase, the robot arm moves to appropriate points on the board so that
the components can be sequentially mounted. In contrast to the picking, the order in
which the components are mounted can be chosen. Note that it is possible to choose
pick-and-place moves in which only one or two of the three heads are used.
To illustrate the process, consider the following example in which a single machine is
used to mount five components on a board. There are two components of type α, one of
type β, and two of type γ. The feeders for these component types have coordinates (5,
0), (10, 0), and (15, 0), respectively. Further, the two α components have to be mounted
at locations with coordinates a1 = (10, 2) and a2 = (10, 12), the β component has to
be mounted at a location with coordinates b1 = (6, 5), and the two γ components have
to be mounted at locations with coordinates c1 = (15, 8) and c2 = (14, 16). Suppose
that, on the first pick-and-place move, the second α component is assigned to head 1,
the β component is assigned to head 2, and the first γ component is assigned to head
3, and that these components are mounted in the order γ, α and β. Then, the robot
arm moves successively between locations (5, 0), (10, 0), (15, 0), (15, 8), (10, 12) and
(6, 5): a total distance of 5 + 5 + 8 + √
41 + √
65 = 32.49. Suppose that on the second
pick-and-place move, the first α component is assigned to head 1, and the second γ
component is assigned to head 2, and that these components are mounted in the order
α and γ. Starting at the location (6, 5) where the last component was mounted on the
previous move, the robot arm moves successively between locations (5, 0), (15, 0), (10, 2)
and (14, 16): a total distance of √
26 + 10 + √
29 + √
212 = 35.04. These pick-and-place
moves are illustrated in Figures 1a and 1b, respectively.
Figure 1: (a) First pick and place move; (b) Second pick and place move (the solid line
denotes the route travelled by the arm during this move)
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3 Throughput
It is required to maximize the throughput of the assembly line, which is determined by
the machine with the heaviest workload. The workload of a machine is proportional to
the total distance travelled by the robot arm. In computing this distance, you should
assume that, when the conveyor moves, the robot arm remains at the position where the
last component was placed until the new board arrives. Increased throughput is achieved
by:
• avoiding large movements of the robot arm;
• balancing the workload between machines.
The scheme that is selected should minimize the time required for the busiest machine
to complete all of its mounting operations.
4 Data
Relevant data for the problem are as follows. There are three machines, and 102 components to be mounted on the board. The components are of 10 different types, which are
labelled as type A, type B, etc., up to type J. There are nine pieces of equipment which
are to be fitted on the heads, and no piece of equipment is duplicated. The following
table lists the equipment, giving the component types that each piece of equipment can
handle.
Piece of equipment Types of components that can be mounted
Figure 2 displays the feeder positions, and shows the positions of the component
types on the board. The feeders for components of types A, . . . , J have coordinates (2,
0), . . ., (11, 0), respectively. The board is depicted below the feeders. The components
to be mounted at the top of the board have coordinates (1, 2), . . . , (12, 2), and the
components to be mounted at the bottom of the board have coordinates (1, 13), . . . ,
(12, 13). There are some locations on the board where no component is to be fitted.
For example, no component is required at the position with coordinated (9, 5), while a
component of type J is to be mounted at the position with coordinates (10, 5).
Note that the times required to pick a component from a feeder, to mount a component on a board, and to move the board on the conveyor are neglibible.
eeders
Figure 2: Layout of the PCB board
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