1. The basics of scheduling
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The basics of scheduling

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3. Exam Results

4. Basics of scheduling
Definitions
Creating a schedule
Difference between optimum and good
Do optima exist?
Performance objectives
Basic rules
SPT
EDD
Moore
Lawler
Johnson
Heuristics

5. Introduction to Scheduling
The general job shop problem
All problems are subsets or relaxations of basic assumptions
Organized research and study of this area followed W.W.II
n jobs {J1, J2, ….Jn}
a job is a task or lot or batch that is to processed as a unit. If it is broken into several jobs, we restart the problem with each part a new job
m machines {M1, M2, ….Mm}
a machine is a processor or resource that performs a specific function required by a job

6. Introduction to Scheduling (continued)
Processing of a job on a machine is called an operation
nm operations oij is the ith job on the jth machine
Each job passes through each machine once and only once (Real jobs may repeat or skip)
Each job has a prescribed order in which it is processed by the machines - This is called its Technological Constraint (TC). Other names are processing order, routing, process plan.
General Job Shop - each job has its own processing order
Flow Shop - all jobs have the same processing order
Permutation Shop - all machines see the jobs in the same order

7. Introduction to Scheduling (continued)
A production line with all machines tied together is an example of a permutation shop
If all jobs have the same processing order, but the machines are not tied together, we have a Flow shop, because some machines may not see all jobs in the same order. (They may pass each other between machines – think of cars and successive toll booths)

8. Examples – Gantt chartsJ2 J1 J3
Flow Permutation J2 J1 J3 M1 M2 J2 J1 J3
Flow non-Permutation J1 J3 J2 M1 M2 J1 J2 J2 J1
General J2 J1 J2 J1

9. Assumptions
1. No two operations of a job may be processed simultaneously
2. Once an operation is started, it must be finished (no preemption)
3. Jobs may not be cancelled
4. Each job has m operations, one on each machine
5. Processing time pij is independent of the schedule (no set - up dependency)
6. In process inventory is allowed
7. There is only one of each machine

10. Assumptions (continued)
8. Machines may be idle
9. No machine may process more than one operation at a time
10. Machines are always available (no breakdowns)
11. All parameters are known and are deterministic - no randomness - no changes in the technological constraints
We relax these as required by our situation and solve a different problem

11. Notations and definitions
di - due date of Job i
ri - ready time of Job i
ai - allowance = di - ri
si - slack = di - remaining operations
pij - the time required to process oij
Wik - the waiting time of Job i preceding its kth operation (not the work on Mk):
J1’s time = W11+p11+W12 +p13+W13 +p12+W14+p1 17, if the TC is M1 to M3 to M2 to M17
We designate the kth operation as oij(k)

12. Notations and definitions (continued)
Ci is the completion time of Job i
Fi is the flow time of Job i = Ci - ri
Even though the English words have an identical meaning, we distinguish between Lateness and Tardiness
Lateness Li =Ci - di, therefore maybe positive or negative depending on whether we complete a job before or after its due date

13. Notations and definitions (continued)
Tardiness is non- zero only if the job is completed after its due date:
Ti = max{Li, 0}
We also define Earliness as Ei = Max{-Li, 0}
The weight or importance of a job is indicated either by wi or
Some of our definitions refer to instants in time
Completion, readiness
Others refer to elapsed time
Processing, Waiting, Flow

14. Notations and definitions (continued)
Scheduling - the ordering of operations subject to restrictions and providing start and finishing times for each operation

15. Notations and definitions (continued)- Schematic
Processing Order - Mm-1, Mj, Mm …..M1, M2 M1 M2
Oi1(m-1)
Oi2(m)
Wi m Mj
Mm-1
Oi j(2)
Wi 2
Oi m-1(1)
pi m-1(1)
Oi m(3) Mm ri Ci

16. Measures
Optimality or goodness of schedules only makes sense if we define the measure under which we are considering optimality or goodness.
There are three broad categories of measures:
Completion time
Due dates
Inventory or utilization
We also define a general class of measures called regular measures

17. Measures (continued)
Based on completion Time
Maximum Flow time
Maximum Completion time
Average Flow time
Average Completion time
Weighted Average Completion time

18. Measures (continued)
Based on due dates
Average Lateness
Maximum Lateness
Average Tardiness
Maximum Tardiness
Number of tardy jobs

19. Measures (continued)
Regular measures
Always are minimized
Are non decreasing in completion times
Examples
Average and maximum completion time
Average and maximum flow time
Average and maximum lateness
Average and maximum tardiness
Number of tardy jobs

20. Measures (continued)
Given two sets of completions times obtained under two schedules generated for the same problem:
C and C’
if Ci <= Ci’ implies that R(C)<=R(C’) then R is regular

21. Inventory
Inventory can exist in many places
Transit
Raw material
Work in Process (WIP)
Finished Goods (FG)
Distribution
Retail Location
Spares

22. Classification notation
All problems can be classified as n/m/A/B where
n - number of jobs
m - number of machines
A - pattern
F - Flow Shop
P - Permutation
G - General Job Shop
B - Measure
Cmax, Fmax etc.

23. Classification notation (continued)
for example n/3/F/Fbar means
any number of jobs on three machines in a flow shop being measured on the basis of average flow time

24. Some further definitions
Jobs and ready times fixed = Static
Parameters known and fixed = Deterministic
Random arrival of jobs = Dynamic
Uncertain processing times = stochastic

25. Kinds of scheduling
Taking sequences and placing them in a schedule is called time tabling (creating a Gantt chart)
Semiactive - process each job as soon as it can be (slide to the left on the chart)
Active - No operation can be started earlier without delaying some other operation
Non-delay - no machine is kept idle
Non-feasible - does not meet Technological Constraints
Number of possible schedules (including non-feasible ones) = (n!)m

26. Optimality
Since for any given problem there are a countable number of possible schedules (as long as we do not allow preemption or unnecessary delays) there must be an optimum (or optima) because we can (theoretically) compare all possible schedules and select the best one
If we look at the space that contains our schedules and attempt to locate the optimum we find that:

27. Optimality (continued)
all possible
Optimal
feasible
Active
semi active
non - delay

28. Equivalent measures
Two measures are equivalent if a schedule optimal with respect to one is also optimal with respect to the other and vice versa.
Cbar, Fbar, Wbar, Lbar are equivalent
Note that a schedule optimal to Lmax is optimal with Tmax, but not vice versa
Cmax, Nbarp and I bar are also equivalent
Cbar, Fbar, Lbar, Nbaru, Nbarw are equivalent for one machine
The choice of measure depends on the circumstances