1. IOE/MFG 543
Chapter 14: General purpose procedures for scheduling in practice
Sections : Dispatching rules and filtered beam search

2. 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

3. Overview Dispatching (or priority) rules Composite dispatching 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

4. 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 …

5. 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

6. 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?)

7. 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?

8. 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
K is the scaling parameter
p is the average of the processing time of the remaining jobs
Ij(t)= wj
exp(-
max(dj-pj-t, 0) ) pj
K p

9. Apparant Tardiness Cost (ATC) rule for 1||S wjTj (2)
How do we determine the value of K?
Due date tightness factor
Due date range factor
R = (dmax-dmin)/Cmax
Empirical studies
K = R if R≤0.5
K = 6 - 2R if R>0.5
t = 1-
(Sdj)/n
Cmax

10. 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

11. Apparant Tardiness Cost with Setups (ATCS) (2)
Ranking index for job j when job l has just completed its processing
s is the average of the setup times of the jobs remaining to be scheduled
Ij(t, l)= wj
exp(-
max(dj-pj-t, 0)
) exp(-
slj ) pj
K1 p
K2 s

12. 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)

13. ATCS Example 14.2.1 Job data Setup times k 1 2 3 4 s0k s1k - s2k s3k
s3k
s4k
job j 1 2 3 4 pj 13 9 10 dj 12 37 21 22 wj 5

14. Implementing a general composite rule
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

15. 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:

16. 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

17. 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

18. 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

19. Crude prediction and thorough evaluation
Example of a crude prediction
Combine
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

20. Example 14.3.1 Solve 1 || SwjTj using the following data
Use beam width = 2 and no filter
job j 1 2 3 4 pj 10 13 dj 12 wj 14