IOE/MFG 543
Chapter 14: General purpose
procedures for scheduling in practice
Sections 14.1-14.3: Dispatching rules
and filtered beam search
1
Motivation



So far we have (mostly) discussed
algorithms for obtaining an optimal
solution for a specific problem
Many scheduling problems are
difficult to solve in practice
Important to have general methods
that can give good schedules in a
relatively short amount of time
2
Overview

Dispatching (or priority) rules
– E.g., WSPT

Composite dispatching rules
– Try to intelligently combine two or more
dispatching rules

Filtered beam search
– Heuristic implementation of branch and bound

Local search
– Try to find a better schedule in a neighborhood
of the current best schedule
– Methods: Simulated annealing, tabu search,
genetic algorithms
3
Section 14.1
Dispatching rules


There are several ways to classify the
dispatching rules
Static vs. Dynamic
– WSPT is …
– SRPT is …

Global vs. Local
– LAPT and Johnson’s rules are …
– WSPT is …
4
Applicability of elementary
dispatching rules

Dispatching rules can work well when
there is only a single objective
– See Table 14.1 page 337 for a list of rules
and in what environments they tend to
work well (sometimes optimal)

More sophisticated rules can often do
better
– Minimizing a combination of objectives
– The objective can change with time and
with the jobs waiting to be processed
5
Section 14.2 Composite
dispatching rules (CDR)



A CDR is a ranking expression that
combines two or more elementary
dispatching rules
An elementary rule is a function of
attributes of jobs and/or machines
Each elementary rule has a scaling
parameter
– Depends on the attributes
– E.g., compute statistics for the set of jobs (are
the due dates tight?)
6
Composite rule for 1||S wjTj

Recall: Problem 1||S wjTj is NP-hard
– We developed a pseudopolynomial DP
algorithm for 1||S Tj

What rules could produce good
schedules?
– If all due dates (and release dates) are
zero?
– If due dates are loose and spread out?
7
Apparant Tardiness Cost
(ATC) rule for 1||S wjTj

Combines WSPT and minimum slack (MS)
– Slack of job j is max(dj-pj-t, 0)

Ranking index for job j

wj
max(dj-pj-t, 0)
Ij(t)=
exp()
pj
Kp
K is the scaling parameter
p is the average of the processing time of
the remaining jobs

8
Apparant Tardiness Cost
(ATC) rule for 1||S wjTj (2)



How do we determine the value of K?
Due date tightness factor
(Sdj)/n
t = 1Cmax
Due date range factor
R = (dmax-dmin)/Cmax

Empirical studies
K = 4.5 + R
K = 6 - 2R
if R≤0.5
if R>0.5
9
Apparant Tardiness Cost
with setups (ATCS)

Generalization of the ATC rule to take
sequence dependent setup times into
account
– sjk is the setup time of job k if it comes after job
j on the machine
– s0k is the setup time if job k is scheduled first on
the machine

SST rule (Chapter 4)
– The job with the smallest setup time goes first

ATCS combines WSPT, MS and SST
10
Apparant Tardiness Cost
with Setups (ATCS) (2)

Ranking index for job j when job l has
just completed its processing
wj
max(dj-pj-t, 0)
Ij(t, l)=
exp(pj
K1 p

)
slj
exp(K2 s
s is the average of the setup times of
the jobs remaining to be scheduled
11
)
Apparant Tardiness Cost
with Setups (ATCS) (3)



Choosing the scaling parameters
Function of t, R and h = s/p
t is a function of Cmax
– Cmax is now schedule dependent
– Estimate Cmax as
Cmax = Spj + ns


K1 is computed the same way as K in ATC
K2 = t / (2√h)
12
ATCS
Example 14.2.1
Job data
job j 1 2 3 4
pj
13 9 13 10
dj
12 37 21 22
wj
2 4 2 5
Setup times
k
s0k
s1k
s2k
s3k
s4k
1
1
0
1
4
2
1
4
2
3
3
3
1
1
1
4
4
3
0
3
13
Implementing a general
composite rule
1.
2.
3.
4.
Choose the elementary rules
Compute the required job and/or
machine statistics
Use the statistics to compute the
values of the scaling values
Apply the dispatching rule to the set
of jobs
14
Section 14.3
Filtered beam search (FBS)

Enumerative branch and bound is one
of the most widely used procedures
for solving NP-hard problems
– It can be used to optimally solve any of
the scheduling problems that we have
considered so far

Problem:
15
Branch and bound (B&B)


In branch and bound we can eliminate all
nodes such that the lower bound is higher
than the cost of the best feasible solution
found so far
Consequences:
– If we can start off with a good solution then
many nodes can potentially be eliminated

Bad news:
– The number of nodes can still be large
16
Beam width



FBS is a B&B-based method in which
only some nodes at any given level are
evaluated
Nodes that are not evaluated are
discarded permanently
The number of nodes that are retained
is called the beam width
17
Filter width



Evaluating each node at a given level
can be computationally expensive
Instead, do a “crude prediction” of the
quality of all nodes at a given level
Evaluate a number of nodes
thoroughly
– This number is called the filter width
18
Crude prediction and
thorough evaluation

Example of a crude prediction
– Combine
1.
2.

The contribution to the objective of the jobs
already scheduled
Some job statistic, e.g., due date factor
Example of a thorough evaluation
– Schedule the remaining jobs according
to a composite dispatching rule
– This gives an upper bound on the value
at this node
19
Example 14.3.1

Solve 1 || SwjTj using the following
data
job j
pj
dj
wj

1
10
4
14
2
10
2
12
3
13
1
1
4
4
12
12
Use beam width = 2 and no filter
20