session 10 university of southern california ise514 september 24, 2015 geza p. bottlik page 1...

Session 10University of Southern California

ISE514 September 24, 2015

Geza P. Bottlik

Outline• Questions? Comments?

• Quiz

• Introduction to scheduling

• Notation

• Measures

• Optimality

• Equivalency

• Formal routine for active and non-delay schedules

• Next quiz on 10/1

• First midterm on 10/8, review on 10/6, covers everything up to and including material covered on 10/1



Geza P. Bottlik

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



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Notations and definitionsdi - 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)

m

kiki WW

1



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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

m

kkijikii pWrC

1)( )( )( )(kijikii pWrC



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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

i



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Scheduling - the ordering of operations subject to restrictions and providing start and finishing times for each operation

Closed Shop - serves customers from inventory (make to stock)

Open shops - Jobs are made to order



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Wi m

Oi j(2)

Wi 2

Notations and definitions (continued)- Schematic

M1

M2

Mm

Processing Order - Mm-1, Mj, Mm …..M1, M2

Mj

Mm-1Oi m-1(1)

Oi2(m)

Oi1(m-1)

Ciri

Oi m(3)

pi m-1(1)



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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



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Measures (continued)

Based on completion Time

Maximum Flow time

Maximum Completion time

Average Flow time

Average Completion time

Weighted Average Completion time



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Based on due dates

Average Lateness

Maximum Lateness

Average Tardiness

Maximum Tardiness

Number of tardy jobs



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Based on Inventory or Utilization

Average number of waiting jobs

Average number of unfinished jobs (WIP)

Average number of completed jobs (finished goods)

Average Idle time

Maximum Idle time



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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



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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



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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.



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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



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Some further definitions

Jobs and ready times fixed = Static

Parameters known and fixed = Deterministic

Random arrival of jobs = Dynamic

Uncertain processing times = stochastic



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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:



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Optimality (continued)

Optimal

all possible

feasible

semi active

Active

non - delay



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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



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Schedule Generation

As a start, we will define a routine that will generate an active schedule

A semiactive schedule is one that starts every job as soon as it can, while obeying the technological and scheduling sequences. Also, the set of all semiactive schedules for a problem contains the optimal schedule

Fortunately, the set of active schedules also contains the optimum and is a smaller set.

We can forget about generating semiactive schedules



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Active scheduling

For a given problem there will be many active schedules

The routine we will use generates only one and we will have to make frequent choices. Were we to follow each of these decision paths, we would generate all the active schedules and find the optimum

However, our purpose here is to make those choices as intelligently as possible, even though it is difficult to foresee their eventual consequence

An active schedule is one in which no operation could be started earlier without delaying another operation or violating the technological constraints



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Definitions

First we will define some terminology useful for our routine:

Class of problems - n/m/G/B with no restrictions

Stage - step in the routine that places an operation into the schedule - there are therefore nm stages

t - counter for stages

Pt - partial schedule at stage t

Schedulable operation - an operation with all its predecessors in Pt

St - set of schedulable operations at stage t



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Definitions (continued)

sigmak - the earliest time an operation ok in St could be started

phik - the earliest time that ok in St could be finished

phik = sigmak + pk



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Routine by Giffler and Thompson1. t = 1, S1 is the set of first operations in all jobs

2. Find min{phik in St} and designate it phi*

Designate M on which phi* occurs as M* (could be arbitrary)

3. Choose oj in St such that it satisfies these conditions:

a. It uses M*

b. sigmaj < phi*

4. a. Add oj to Pt, which now becomes Pt+1

b. Delete oj from St which now becomes St+1

c. Add the operation that follows oj in the same job to St+1

d. Increment t by 1



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Routine by Giffler and Thompson (continued)

5. If there are operations left to schedule, go to step 2, else stop

Note well that at step 3b. sigmaj < phi*, we will often have several choices. We always have at least one, namely, phi*

These choices are an extensive topic that we will cover later

Follow the example I have taken from French

Generating these schedules is tedious work, so leave yourselves some extra time for that homework.



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Non-delay schedules

Non-delay schedules are a smaller set than the active schedules and therefore are a tempting set to explore

Unfortunately, they do not always contain the optimum

We will not let that deter us, because non-delay schedules have been found to be usually very good, if not optimal

A non-delay schedule is one where every operation is started as soon as it can be



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Non-delay schedules (continued)

We change two steps in the procedure for active schedules to obtain a non-delay procedure:

Step 2. instead of phi, we select sigma

Find min{sigmak in St} and designate it sigma*

Designate M on which sigma* occurs as M* (could be arbitrary)

Step 3 b. sigmaj = sigma*

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