T h e flo w time o f a jo b through a simple shop comprises processing time and
waiting time. T h e waiting time w ill depend on the priority given to this jo b in
com petition with other jobs through the fa cility, and on the provision o f capacity
to cope with the workload.
C onw ay et al (1 1 ) approached the multi-stage scheduling problem by means o f
tw o extreme cases:
( i ) pure jo b shop - in w hich a jo b leavin g a machine is equ ally likely to
go to any other machine in the shop, and
( i i ) pure flo w shop - in w hich there is o n ly one path through the shop that
all jo b s w ill follow.
T o distinguish these cases from alternative routing, it should be noted that the
w h ole route is d ecid ed before the first operation in both cases. Thus, there is no
"ch o ice" o f route at the end o f any operation.
A special case in multiple machine problems concerns the handling o f parallel
processors, or multiprocessors.
T h e type o f scheduling problem most closely related to alternative routing is
concerned with 'parallel processors' o r 'multiprocessors’. In these problems, more
than one processor is available to carry out the same operation, by relaxing the
rule stated in 2.3.11, i.e. that there is only one machine available o f each machine
type.
T h e interest in multiprocessors is not confined to manufacturing facilities.
An alogies may be drawn w ith packet movement in data communications
networks, vehicle control in transportation networks and task assignment in
multiprocessor computer systems. These examples all require dynamic solutions,
where a controller observes the network and the route chosen depends on the state
o f the network at the time when a routing decision is required. Dynamic routing
is discussed in sections 3.7 and 3.10.
In general three classes o f multiprocessor problems have been studied - identical,
uniform or unrelated machines. W hen the machines are identical, the processing
t i n » is the same on all machines. When the machines are uniform, the processing
times vary in a uniform manner, but i f the machines are unrelated, the processing
times vary arbitrarily between machines.
T h e first point to consider when scheduling multiprocessors is whether a jo b may
be divided am ong 2 or more machines. I f not, (batch splitting w ill not be allowed
in this w ork), then n jobs must be d ivided into m distinct subsets (w h ere n is the
number o f jobs and m is the number o f machines).
C offm an (1 8 ) describes a schedule fo r multiprocessors as com prising m blocks or
subsets, where the tasks in each block are ordered by a permutation to yield the
order o f task executions fo r a processor.
C onw ay et al (1 1 ) show that to m inim ise the flo w time, the jobs should be divided
among the machines so as to balance the workload as far as possible, using S P T as
the dispatching rule, and also balancing the distribution o f lon g and short jobs
between machines. A lthough the simplest equations demand identical machines,
the principle o f balancing the workload among non-identical machines still holds
and processing times on different machines may be v iew ed as a m atrix o f n jobs
by m machines (fig .8 ) where p y gives the time to perform a single operation i on
machine j.
A regular measure o f performance m ay still be m inimised using the S P T rule. I f a
jo b can be divided between 2 o r m ore machines, better schedules are possible
(because o f the reduced processing tim e, reduced idle times, reduced waiting
tim es) but determination o f the schedules is more difficult. C on w ay et al prove
clear advantages fo r simultaneous processing ranging from a minimum o f 25%
reduction in average flo w time, fo r 2 machines, to a maxim um o f 50% , for many
machines, in the idealised, identical machine, identical jo b condition. Ignoring
the penalties o f multiple set-up and tooling, C onw ay e t al con sider that any
schedule can be im proved by taking advantage o f parallel operations on identical
machines.
In the next sections, some o f the analytical techniques w hich have been used to
tackle scheduling problems w ill be described.
It is worth restating that most practical production environments have the
fo llo w in g features:
( i ) T h e y are dynamic - jobs are continually arriving and m oving through
the network.
( i i ) T h e y are stochastic - although expected operation times are known
beforehand, operator or material or machine o r tool problems can
cause significant variations.
(ii i ) G o o d due date achievement and flo w time performance is required.
Generally a range o f performance measures w ou ld be used in a
production facility including W I P level, due date achievement, lead
tim e, unit cost, quality, machine utilisation.
In addition, investigation o f alternative routing demands that rules stated in 2.3.10
and 2.3.11 be relaxed, such that
( i v ) Job routing is not necessarily fix ed at the start
( v ) M ore than one machine o f the same type may be present