chapter 08 lecture 12 to 19 w08 431 scheduling

8/20/2019 Chapter 08 Lecture 12 to 19 w08 431 Scheduling

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LESSON 12: INTRODUCTION TO

OPERATIONS SCHEDULING

Outline

•Hierarchy of Production Decisions• Operations Scheduling

• Production Systems


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Hierarchy o Pro!uction Deci"ion"

• After a production facility and processes are set up, a seriesof production planning decisions are required. The entiredecision making process may e !ie"ed as a hierarchical

process. A conceptual !ie" of the hierarchical process isgi!en in Te#t $igure %&' (see the ne#t slide).

• $irst, one "ould like to kno" ho" much demand may ee#pected and "hen the demand may e e#pected. Thisquestion is addressed y forecasting.

• An aggregate plan determines the aggregate productionle!els and resource capacities o!er the planning hori*on. Theproduction le!els are often different from the


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A g g r e g a t e P l a n

$ o r e c a s t o f d e m a n d

+ a t e r i a l e q u i r e m e n t s P l a n n i n g S y s t e m

+ a s t e r P r o d u c t i o n S c h e d u l e

D e t a i l e d - o S h o p S c h e d u l e

#i$ure %&1: Hierarchy o Pro!uction Deci"ion"


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forecasted demand le!els ecause of seasonalityassociated "ith the demand, constraints on the a!ailaility ofproduction resources, etc. The resource capacities such as

"orkforce le!els are determined assuming one aggregateunit of production.

• +aster production schedule (+PS) translates the aggregateplan in terms of specific units of production and time period.

$or e#ample, an aggregate plan may require / units ofchairs in April. +PS then translates the requirement into 0//units of desk chair in 1eek ', '/ units of ladder&ack chairin 1eek 0 and 0// units of 2itchen chair in 1eek 3.


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• The +PS generates the production plan in terms of finishedproducts "hich often require components or suassemlies.The materials requirements planning (+P) computes the

changes in the in!entory le!els of components andsuassemlies o!er the planning hori*on and determines thesi*e and timing of ordering the components andsuassemlies.

• 4e#t comes operations scheduling. After generating demandforecast, "orkforce capacities, production plan, andpurchasing plan, it5s logical to ask "hich "orker or machine"ill e used to produce "hich product and "hen the products"ill e produced. This is ans"ered y operations scheduling.


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• An operations scheduling question is gi!en in the pre!iousslide. 6n more general terms, operations scheduling is theallocation of resources o!er time to perform a collection oftasks.

• 7#amples of resources8

9 1orkers, +achines, Tools

• 7#amples of tasks8

9 Operations that ring some physical changes to material inorder to e!entually manufacture products

9 Setups such as "alking to reach the "orkplace, otainingand returning tools, setting the required :igs and fi#tures,

positioning and inspecting material, cleaning etc.

O'eration" Sche!ulin$


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• Operations and -os

9 The ao!e definition of scheduling uses the more generalterm of task that includes oth operations and setup.

9 1e shall most often use the terms operation and :o. 9 A collection of operations on a single product is a :o.

• The planning hori*on

9 The planning hori*on of a scheduling decision is !ery short

say days, "eeks or months.• Schedule

9 A schedule is the final outcome of operations schedulingand gi!es a detail chart of "hat acti!ities "ill e done using

!arious resources o!er the planning hori*on.



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• Sequencing

9 Sequencing and scheduling are similar terms. ;utsequencing does not refer to time. $or e#ample, if a ankteller processes customers, the ank teller may :ustprocess the customers on a first come first ser!ed asis"ithout any planning aout e#act start and end times foreach customer. That5s sequencing. Scheduling, in contrast,produces a detail plan of !arious acti!ities o!er time.

• A


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< r i n d i n g m a c h i n e

0 = > % ' / ' 0

D a y s

' =

? a t h e m a c h i n e - o A ' 0

- o A ' 0

- o ; 0 3

- o ; 0 3


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9 The schedule sho"n on the


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9 As it is in the e#ample


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•


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Pro!uction Sy"te("

• Product characteristics differ.

• Some products are standard and require minimal or no!ariation from one item to another. These products are

produced in large !olume on the asis of demand forecast.7#amples of standard products include staple products,economy cars, etc.

• Some other products are produced only on the asis of

customer order and there e#ist significant differencesamong the items. These products are produced in lo"!olumes. 7#amples of custom products include lu#ury cars,fashion clothes, etc.


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Pro!uction Sy"te("

• Different product characteristic requires a differentproduction system.

• Standard products require a

9 make&to&stock or assemle&to&stock production system• Custom products require a

9 make&to&order or assemle&to&order production system


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Pro!uction Sy"te(": #lo) Sho'

• A make&to&stock@assemle&to&stock production system isassociated "ith

9 High !olume of production8 so, it5s feasile to make a

high capital in!estment 9 ?ess !ariation8 so, there is less uncertainty

9 Standard products8 so, there is a predictale pattern offlo" of :os through the machines.

9 The ao!e characteristics of products encourages theuse of a flo" shop8 the machines are arranged in theorder in "hich e!ery :o !isits the machines. Theproduction system is not fle#ile it cannot produce items

if design changes significantly.


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64

OET

Pro!uction Sy"te(": #lo) Sho'

The ao!e picture is a conceptual !ie" of a flo" shop. Theo#es represent machines and the arro"s sho" the :o flo".

7!ery :o !isits the machines in the same order.


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Pro!uction Sy"te(": *o+ Sho'

• A make&to&order@assemle&to&order production system isassociated "ith

9 ?o" !olume of production8 so, it5s not feasile to make a

high capital in!estment 9 +ore !ariation8 so, there is more uncertainty

9 Custom products8 so, there is no predictale pattern offlo" of :os through the machines.

9 The ao!e characteristics of products encourages theuse of a :o shop8 the machines are arrangedfunctionally. So, the production system is fle#ile it canproduce items of a "ide !ariety of designs.


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? +

+

+

+

D

D

D

D

D

D

D

D

<

<

<

<

<

<

P

P

A A Aecei!ing and

Shipping Assemly

Painting Department

?athe Department +illing Department Drilling Department


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Pro!uction Sy"te(": ,atch Pro!uction

• 6n et"een flo" shop and :o shop, there is atchproduction that gi!es est of oth shops (flo" shop and :oshop) to some e#tent.

• A atch production system is fle#ile like a :o shop, utcapale of producing a moderately high !olume like a flo"shop.

• Similar ut different products are produced using the same

facility, machine or "orkforce. Ho"e!er, the system isassociated "ith setups. 1hene!er the facility, machine or"orker s"itches from producing one product to another, asetup time or cost is needed.


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• ;et"een t"o successi!e setups a atch of the sameproduct is produced.

• 4o", look at the prolem8

9 6f the atch si*e is too high, all the other productsrequiring the facility, "orker or machine "ill ha!e to "aitfor a !ery long time.

9 6f the atch si*e is too small, most of the time "ill e

spent on setups and there "ill e nothing to sell.• So, the atch si*e must e neither too large, nor too small.

Similarly, the !olume of production must e moderate.


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• 7#amples8

9 ;ooks

9 Pastry

9 Painting

9 Automoile gears (thepicture sho"s aComputer 4umerical

Control (C4C) machineused to produce gears)


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Pro!uction Sy"te("

Pro-ect an! Continuou" Pro!uction

• $lo" shop is not the e#treme case "ith respect to the high!olume of production. The continuous production system isused in process industry e.g., oil refinery is set up to

produce products "ith little or no !ariety 0= hours@day and Fdays@"eek.

• -o shop is not the e#treme case "ith respect to fle#iility orcustomi*ation. Huge pro:ects are managed to produce a

ridge, a sky&scraper, an aeroplane, a sumarine, etc.


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*o+

Sho'

,atch

'ro!uction

Sho'#lo) C

u " t o (

i . a t i o n

Hi$h

Lo)Hi$hLo)

/olu(e

Pro-ect

Pro!uction

Sy"te("

Continuou"

Pro!uction


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*o+

Sho'

,atch

'ro!uction

Sho'#lo) C

u " t o (

i . a t i o n

Hi$h

Lo)Hi$hLo)

/olu(e

Pro-ect

Continuou"

Pro!uction

Aircrat

Cu"to(&(a!e

0achine an! Part"

,oo"

Auto(o+ile

Oil

reinery

Pro!uction

Sy"te("


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*o+

Sho'

,atch

'ro!uction

Sho'#lo) C

u " t o (

i . a t i o n

Hi$h

Lo)Hi$hLo)

/olu(e

Pro-ect

Continuou"

Pro!uction

La+or

Inten"ie

Ca'ital

Inten"ie

Pro!uction

Sy"te("


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READING AND E3ERCISES

?esson '0

eading8

Section %.', pp. ='3&='> (=th 7d.), pp. =/=&=/ (th 7d.)

7#ercises8

%.3a, %.3, p. =0 (=th 7d.), p. ='3 (th 7d.)

%.3' (identify the :o and machine) p. =>= (=th 7d.),p. =/ (th 7d.)


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LESSON 14: SCHEDULING O,*ECTI/ES

Outline

• -o Characteristics

• Comparison of Schedules

• Scheduling Terms

• Scheduling O:ecti!es


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*o+ Characteri"tic"

• ?esson ' pro!ides a rief discussion on production systems.1e ha!e discussed some alternati!e "ays of arrangingmachines. 6n this lesson, "e shall first discuss some :ocharacteristics and scheduling o:ecti!es.

•


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*o+ Characteri"tic"

• -o characteristics are important inputs to :o shop, atchproduction and flo" shop scheduling. 7!ery :o has a

9 ready time8 the time "hen the :o arri!es at the shop floor

9 processing time8 the time required to process the :o

9 due date8 the time "hen the :o must e completed• 4otation8

jd

jt

jr

j

j

j

:ofor dateDue

:ofor timeProcessing

:ofor timeeady

=

=

=


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*o+ Characteri"tic"

• Assumptions8 9 A machine can process one :o at a time

9 A :o can e processed y one machine at a time

9 1e usually assume an equal importance and the same

arri!al time for all :os (7#ample ' is an e#ception, "here :os arri!e at different times). $urther, "e assume thatpreemption is not allo"ed. So, once a :o is started on amachine, the :o must e completed efore another :o

can e processed y that machine.


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Co('ari"on o Sche!ule"

• $irst, "e shall see an e#ample "here no schedule is estin e!ery criteria.

• E5a('le 1: Suppose that a production facility starts at%83/ am. T"o :os "ill e processed. -o ' can e started

right a"ay, "ill require 0 hours to process and thecustomer "ants the :o done y '083/ pm. -o 0 can estarted not until 83/ am, "ill require ' hour to processand the customer "ants the :o done y '/83/ am.

• 6t5s customary to set start time, and e#press allother times "ith a suitale unit.

• So, set at %83/ am and e#press all other times inhours.

0=t

0=t


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• So, -o ' has a ready time of / hour (can e started righta"ay), processing time of 0 hours and due date of = hours(due at '083/ pm, = hours after the start of the process)

• Similarly, -o 0 has a ready time of ' hour (can e started

not until 83/ am), processing time of ' hour and due dateof 0 hours (due at '/83/ am, 0 hours after the start of theprocess)

• The :o characteristics so otained are sho"n elo"8

- o e a d yT i m e ( h r )

P r o c e s s i n gT i m e ( h r )

D u eD a t e ( h r )

' / 0 =

0 ' ' 0

j jr jt jd


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• There are t"o alternati!e schedules (sequences)Schedule '8 process -o ' first, then -o 0.

Schedule 08 process -o 0 first, then -o '.

• Schedule ' (see the ne#t slide)8

9 Only 3 hours "ill e needed to complete the :os. 9 Ho"e!er, -o 0 can e completed at time 3 "hich is

late y ' hour.

• Schedule 0 (see the ne#t slide)8

9 ;oth the :os are completed right "hen they areneeded.

9 Ho"e!er, a total of = hours "ill e needed


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• Schedule 'requires thefacility to eopen for fe"erhours (3 hoursonly in contrastto = hoursrequired ySchedule 0)

• Schedule 0meets the duedates(Schedule '

does not)

T i m e ,

- o '

t ' 0 3 =/

S c h e d u l e '

- o 0

2d 1d

T i m e , t

- o '

' 0 3 =/

S c h e d u l e 0

- o 0

2d 1d


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• 4e#t, "e shall see a similar e#ample "ith a different pairof criteria.

• E5a('le 2: An auto repair shop has a space prolem andrequires parking fees for all cars "aiting for ser!ice. The

shop starts at %83/ am and t"o cars are "aiting to erepaired. Car ' "ill require ' hour and the customer "antsthe :o done y '083/ pm. -o 0 "ill require 3 hours andthe customer "ants the :o done y ''83/ am.

C a r e a d y

T i m e ( h r )P r o c e s s i n g

T i m e ( h r )D u e

D a t e ( h r )

' / ' =

0 / 3 3

j jr jt jd


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• Schedule 3requires lessparking fees ('hour only incontrast to 3hours requiredy Schedule =)

• Schedule =meets the due

dates(Schedule 3does not)

T i m e ,

C a r 0

t

' 0 3 =/

S c h e d u l e 3

C a r '

2d 1d

T i m e ,

C a r 0

t

' 0 3 =/

S c h e d u l e =

C a r '

2d 1d


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Sche!ulin$ Ter("

• Static !ersus dynamic scheduling 9 6n static scheduling, :os are all a!ailale for

processing right at time "hen the processingstarts. An e#ample is a grocery store that collects

orders online. 1hen the manager comes to the storeat %83/ am, the manager finds orders collectedthroughout the night and must e deli!ered on that day.

9 6n dynamic scheduling, :os are not a!ailale all at time

. An e#ample is a ank "here customers arri!ethroughout its ser!ice hours.0=t

0=t


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Sche!ulin$ Ter("

• Deterministic !ersus stochastic scheduling 9 6n deterministic scheduling, the :o characteristics such

as ready time, processing time, and due date,are all kno"n "ith certainty. Processing times required

y machines may e predicted precisely. 9 6n stochastic scheduling, the :o characteristics are

uncertain. Some parameters such as mean and!ariance are assumed to e kno"n from historical

oser!ation. The ser!ice times required y "orkers areoften assumed to e uncertain.

jr jd jt


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Sche!ulin$ Ter("

• $easile !ersus 6nfeasile schedule 9 A feasile schedule meets all the constraints and an

infeasile schedule does not. Precedence constraintsoften associate "ith the scheduling process. A

precedence constraint puts restriction on the sequenceof operations. $or e#ample, a painting process mayha!e three stages8 cleaning the product, surfaceacti!ation for paint adhesion, and select andapplication of paint. These stages must e performedin the stated sequence. A schedule is not feasile if the

:os are not processed in that sequence.


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Sche!ulin$ Ter("

M 1 M 2 M 3

E n t e r

E x i t

J o b s

M a c h i n e s

• Parallel !ersus sequentialprocessing

9 6n parallel processing, thereare multiple identical

machines and a :o can eprocessed y any machine.

An e#ample is a computer"ith multiple processors.

Another e#ample is a ank"ith multiple tellers. On theright, see a conceptual !ie""ith 3 parallel machines, +',+0 and +3.


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Sche!ulin$ Ter("

• Parallel !ersus sequential processing 9 6n a sequential processing, :os are sequentially

processed through the machines. An e#ample is+cDonald5s dri!e through that has three stages8 place

order, pay and pick&up. ;elo", see a conceptual !ie""ith 3 sequential machines +', +0 and +3.

M 1 M 2 M 3E n t e r E x i t

J o b s

M a c h i n e s


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Sche!ulin$ Ter("

• Completion time !ersus flo" time 9 Completion time of -o is the epoch "hen a :o

is completed. The length of time is computed from thestart of operations.

9 $lo" time is a similar term. Ho"e!er, the length of timeis computed from the arri!al of the :o. $lo" time of-o is the length of time et"een arri!al andcompletion of the :o. So,

9 ;et"een arri!al and completion, a :o is either inqueue for a machine or processed y a machine.So, "here is the "aiting time of -o

jC j ,

j F j ,

j j j r C F −=

j j j t W F += jW j


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Sche!ulin$ Ter("

• Completion time !ersus flo" timeE5a('le 4: Consider 7#ample ', Schedule ' on slides

&%. The facility starts at %83/ am, -o 0 arri!es at 83/am, "aits for an hour efore its processing starts at

'/83/ am, and the :o is completed at ''83/ am. Thecompletion time and flo" time are sho"n elo"8

• Set at %83/ am and the unit of time G hour

• -o 0 is completed at ''83/ am, 3 hours after the

start of operation, so completion time, G 3 hours.• -o 0 arri!es at 83/ am, ' hour after the start of

operation, so

0=t

2C

hours

andhour

213

1

222

2

=−=−=

=

r C F

r


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Sche!ulin$ Ter("

• Completion time !ersus flo" time

9 Important note: if all the jobs are ready for processingat , as they are in static scheduling, then

completion time is just the same as flow time.

0=t

0== j j j r C F if

T i m e ,

- o '

t

' 0 3 =/

S c h e d u l e '

- o 0

2r

2C 2 F


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Sche!ulin$ Ter("

• ?ateness !ersus tardiness 9 ?ateness of :o

9 So, lateness of a :o may e

• negati!e, if the :o is early

• *ero, if the :o is completed right "hen it5s due

• positi!e, if the :o is late

9 Often, earliness is not re"arded ut lateness ispenali*ed. So, another term, tardiness is used "hich is

a non&negati!e quantity. Tardiness is *ero or the sameas lateness. 6f the :o is early or completed "hen it5sdue, tardiness is set to *ero. Other"ise, the :o is late,and tardiness is set to its lateness.

j j j d C L j −=,


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Sche!ulin$ Ter("

• ?ateness !ersus tardiness 9 Tardiness of :o

9 6n other "ords, tardiness of :o

• /, if it5s completed on or efore its due date,

• , if it5s completed after its due date,

E5a('le 6: Consider 7#ample ', Schedule ' on slides&%. . The lateness andtardiness of -os ' and 0 are computed on the ne#t

slide.

( ) j j LT j ,0max, =

0≤ j L

0> j L j L

2,4,3,2 2121 ==== d d C C

= jT j ,


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Sche!ulin$ Ter("

• ?ateness !ersus tardiness

9 -o ' is early. So, its lateness is negati!e and itstardiness is /.

9 -o 0 is late (or tardy). So, its lateness is positi!e andits tardiness is the same as its lateness.

( ) ( )( ) ( ) 11,0max,0max

02,0max,0max

123

242

22

11

222

111

===

=−==

=−=−=

−=−=−=

LT LT

d C L

d C L

22

1 0

LT

T

=

=


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Sche!ulin$ Ter("

• +akespan, ma#imum lateness, total completion time 9 These are three important criteria (scheduling

o:ecti!e) "hich are often used to choose a estschedule ecause of their important practical

implications. 9 +akespan is the completion time of the last :o

processed. Although makespan is defined as acompletion time of a :o, it actually measures ho" long

the production facility should remain open. 9 +akespan is denoted y ,

( )nC C C C C ,,,,max 321max =max

C


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Sche!ulin$ Ter("

• +akespan, ma#imum lateness, total completion time 9 6n 7#ample ', Schedule ', Slide %, -o 0 is the last

processed :o. So, makespan is the completion time of-o 0.

9 6n 7#ample ', Schedule 0, Slide %, -o ' is the lastprocessed :o. So, makespan is the completion time of-o '.

9 Oser!e that makespan is completion time of -o ' or-o 0 depending on "hich :o is processed last.

( ) ( ) 221max 33,2max,max C C C C ====

( ) ( ) 121max 42,4max,max C C C C ====


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Sche!ulin$ Ter("

• +akespan, ma#imum lateness, total completion time 9 $or a single machine static scheduling prolem (i.e.,

:os are all a!ailale for processing in the eginning),makespan is the sum of all processing times. Ho"e!er,

for a multi&machine prolem, makespan may edifferent from the sum of all processing times.

9 +a#imum lateness is denoted y

9 6n 7#ample ', Schedule ', Slide F,

( )n L L L L L ,,,,max 321max =max L

( ) ( ) 11,2max,max

123,242

21max

222111

=−==

=−=−=−=−=−=

L L L

d C Ld C L


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Sche!ulin$ Ter("

• +akespan, ma#imum lateness, total completion time 9 6n 7#ample ', Slides &%, Schedule ' minimi*es makespan and Schedule 0

minimi*es ma#imum lateness.

9 Total completion time is the sum of all completion times. 4otice that totalcompletion time is not the same as makespan. Total completion time is denotedy

9 6n 7#ample 0, Slides &'/, Schedule 3 minimi*es not only parking fees ut alsototal completion time. 6n general, if a schedule frees up space fast, the scheduleminimi*es total completion time. Schedule = minimi*es ma#imum lateness.

n j C C C C C ++++=∑ 321 jC ∑


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Sche!ulin$ O+-ectie"

• As it may e oser!ed from 7#amples ' and 0, differentschedule may e etter "ith respect to different criterion(scheduling o:ecti!e).

• So, it5s !ery important to set up a suitale schedulingo:ecti!e in order to get a suitale schedule.

• There are many scheduling o:ecti!es and differentsituation calls for a different o:ecti!e.

• The ne#t slide pro!ides a rief list of scheduling o:ecti!esdi!ided into four groups.

• See Section %., pp. =03&=0= for a discussion on ho"different situation requires a different schedule.


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Sche!ulin$ O+-ectie"

• Conformance to prescried deadlines 9 +eet customer due dates, minimi*e :o lateness,minimi*e ma#imum lateness, minimi*e numer of tardy :os

• esponse time or lead time 9 +inimi*e mean completion time, minimi*e a!erage timein the system

• 7fficient utili*ation of resources

9 +a#imi*e machine or laor utili*ation, minimi*e idletime, ma#imi*e throughput, minimi*e the length of timethe shop is open, minimi*e utilities and "ages

• Costs

9 +inimi*e "ork&in&process in!entory, minimi*e o!ertime


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?esson '3

eading8

9 Sections %.0&%.3, pp. ='>&=' (=th

7d.), pp. =/>&=/ (th 7d.)

9 Section %., pp. =03&=0= (=th 7d.), pp. ='0&='= (th 7d.)

7#ercises8

9 %.', %.0, %.3 (parts c, d, e) pp. =0=&=0 (=th 7d.), p.='3 (th 7d.)


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LESSON 16: SCHEDULING 7ITH

PRIORIT8 SE9UENCING RULES

Outline

• Sequencing

• Sequencing ules

• Sequencing ule 7#ample

• emarks


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Seuencin$

• As it is discussed in ?esson ', scheduling andsequencing are similar terms. Scheduling pro!ides adetail plan o!er time. Sequencing does not refer to

time at all.• Sometimes, a unique schedule follo"s from a gi!en

sequence. 6n such a case, it5s enough to sol!e thesequencing prolem and not "orry aout the detailscheduling prolem. $or e#ample, in ?esson 0,

7#ample ', Schedule ' follo"s from sequencing -o' efore -o 0 and Schedule 0 follo"s fromsequencing -o 0 efore -o '.


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Seuencin$ Rule"

• The first sequencing rule, that comes naturally toe!eryone5s mind is the first&come first&ser!ed ($C$S)rule. 1e oser!e this rule se!eral times a day "hen

"e !isit a ank for a teller5s ser!ice, "ait in a grocerystore check&out, or cross the Amassador ridge. Therule is simple, ut not al"ays the est.

• 6t5s certainly not desirale that a customer "ho has topay for :ust one ag of read must "ait in a queue

ehind another customer "ith a cart full of groceries.So, the e#press lines are estalished. This is somesort of implementation of the shortest processing time(SPT) rule.


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Seuencin$ Rule"

• ou get the most !alue of money if you pay the ills onthe due dates. The simplest rule that comes to mind inpresence of due dates is the earliest due date (7DD)

rule "hich requires that the :os e done in the order in"hich the :os are due.

• Often in manufacturing, items are put in a stack. Thelast item arri!ing is put on the top and processed first.The last&come first&ser!ed (?C$S) rule is also oser!ed

in ele!ators. The person arri!ing last must step out first.• The critical ratio (C) rule comines SPT and 7DDrule. The C rule is e#plained on the ne#t slide.


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Seuencin$ Rule": Critical Ratio

• C G time remaining @ "ork remaining

• 7ach time a :o is scheduled, C is recalculated fore!ery unscheduled :o.

• C selection criteria

9 -os "ith the smallest C are run first. 9 -os "ith negati!e C are scheduled first.

9 6f there is more than one :o "ith negati!e C,then those :os are sequenced in SPT order.

due date & today5s dateremaining processing time

=


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Seuencin$ Rule E5a('le

A '/

; '/'

C 0

D %'0

7 >%

Processing Due-o Time Date

Suppose that :os "ill eprocessed on a singlemachine. The :os are readyfor processing at time .The other :o characteristicsare as sho"n in the tale onleft.

0=t


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61

• $irst, see ho" many alternati!es are there. 6f there isone :o, there is :ust ' (G'B) alternati!e sequence.

• 6f there are t"o :os ' and 0, there are 0 (G0B)

alternati!e sequences,', 0 and

0, '

• 6f there are three :os ', 0 and 3, there are > (G3B)alternati!e sequences,

',0,3 ',3,0

0,',3 0,3,'

3,',0 3,0,'



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62

• 6n general, if there are n :os, then there are nBsequences.

• So, for e#ample, for :os, there are B G '0/

sequences.• The sequencing rules such as $C$S, SPT, 7DD,

?C$S and C pro!ide a specific sequence. Often,these simple rules pro!ide good and useful results.

• The sequencing rules $C$S, SPT, 7DD, ?C$S andC "ill no" e applied and !arious measures "ill ecomputed.



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#ir"t&Co(e #ir"t&Sere!

A '/

; '/ 'C 0

D % '0

7 > %

A!erage

Start Processing Completion DueSequenceTime Time Time Date Tardiness


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64

Shorte"t Proce""in$ Ti(e

C 0

A '/7 > %

D % '0

; '/ '

A!erage



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Earlie"t Due Date

C 0

7 > % A '/

D % '0

; '/ '

A!erage



66/252

La"t&Co(e #ir"t&Sere!

7 / > > % /D > % '= '0 0

C '= 0 '> ''

; '> '/ 0> ' ''

A 0> 3' '/ 0'

A!erage '%.>/

Start Processing Completion DueSequence Time Time Time Date Tardiness

?C$S rule8 The :os are arranged in last&come first&ser!ed

order.


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67

• Enlike $C$S, SPT, 7DD and ?C$S, the C sequenceis otained y using an iterati!e procedure. Then,!arious measures are computed using the C

sequence.• The C rule is applied in t"o phases

9 Phase 68

• $ind the C sequence using an iterati!e

procedure. 6n each iteration, one :o is assignedto a position. $irst a :o is assigned to the firstposition, then to the second position, and so on.

Critical Ratio


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68

• 6nitially, the current time is set to *ero. 6n each iterationthe current time is augmented y the processing timeof the :o assigned in the pre!ious iteration.

• Then, C is computed for e!ery unassigned :o. SeeSlide for the C formula.

• The C rule is applied to select the :o that "ill eassigned. See Slide the C selection criteria.

9 Phase 668• Iarious performance measures are computed using

the C sequence otained in Phase 6.

Critical Ratio

Phase I


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69

Critical Ratio

6teration '

Current

ime

Job !rocessin"

ime

#ue

#ate

Critica$

%atio

&ssi"n'

& 5 10

( 10 15

C 2 5

# 8 12

E 6 8

Phase I


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70

Critical Ratio

6teration 0

Current

ime

Job !rocessin"

ime

#ue

#ate

Critica$

%atio

&ssi"n'

& 5 10

( 10 15

C 2 5

# 8 12

E 6 8

Phase I


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71

Critical Ratio

6teration 3

Current

ime

Job !rocessin"

ime

#ue

#ate

Critica$

%atio

&ssi"n'

& 5 10

( 10 15

C 2 5

# 8 12

E 6 8

Phase I


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72

Critical Ratio

6teration =

Current

ime

Job !rocessin"

ime

#ue

#ate

Critica$

%atio

&ssi"n'

& 5 10

( 10 15

C 2 5

# 8 12

E 6 8

Phase II


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Critical Ratio

7 > %

C 0

A '/

D % '0; '/ '

A!erage


C rule8 The :os are arranged in order otained y the

iterati!e procedure sho"n on Slides 0F&30.


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Su((ary

$C$S '%.>/ .> 3J 03

SPT 16;%./ 3J '>J

7DD '.// .>J 3J 1>=

?C$S '%.>/ ./ = 0'

C '.%/ >.0 = '>J

J ;est !alues


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Re(ar"

• Oser!e that the makespan is the same for e!eryschedule. This is e#pected for a single machineprolem if ready times are all *ero (static scheduling).

$or a multi&machine prolem, makespan may edifferent from one schedule to another.

• Total completion time and mean completion time areequi!alent o:ecti!es. Since mean completion time isotained from the total completion time y di!iding

the total completion time y the numer of :os, 9 if a schedule minimi*es total completion time, it

also minimi*es mean completion time.


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Re(ar"

• Similarly, total flo" time and mean flo" time areequi!alent o:ecti!es. 6f a schedule minimi*es totalflo" time, it also minimi*es mean flo" time.

• 6n case of static scheduling, completion time of a :o isthe same as its flo" time. So, the follo"ing o:ecti!esare equi!alent (if a schedule minimi*es one, it alsominimi*es all other)

9 Total completion time

9 +ean completion time

9 Total flo" time

9 +ean flo" time


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Re(ar"

• The o:ecti!e of minimi*ing in!entory carrying costssuch as parking fees in ?esson 0, 7#ample 0, isequi!alent to minimi*ing total completion time. At this

point, re!isit this e#ample on slides &'/ of ?esson 0.Check that the total completion time is 'K=G forSchedule 3 and 3K=GF for Schedule =. Thus,Schedule 3 minimi*es not only parking fees, ut alsototal completion time. This holds in general.+inimi*ing in!entory carrying costs is equi!alent tominimi*ing total completion time, mean completiontime, total flo" time and mean flo" time.


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Re(ar"

• ?ateness and tardiness are closely related terms. 6f aschedule minimi*es ma#imum lateness, the schedulealso minimi*es ma#imum tardiness. Ho"e!er, the

con!erse is not true. 6f a schedule minimi*es ma#imumtardiness, the schedule does not necessarily minimi*ema#imum lateness. Thus, ma#imum lateness andma#imum tardiness are not equi!alent. Plus, it5s moreinteresting to minimi*e ma#imum lateness ecause if

ma#imum lateness is minimi*ed, ma#imum tardiness isalso minimi*ed.

• The 7arliest Due Date (7DD) rule minimi*es ma#imumlateness and ma#imum tardiness (for the single machinestatic scheduling prolem).


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Re(ar"

• Total tardiness and mean tardiness are equi!alent o:ecti!es.

• Total lateness and mean lateness are equi!alent o:ecti!es.

• Ho"e!er, total lateness and total tardiness are different.

• Total lateness is total completion time minus the sum of thedue dates. Since sum of the due dates is a constant (samefor all schedules), minimi*ing total lateness is equi!alent tominimi*ing total completion time.

• So, the equi!alence list gro"s8


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Re(ar"

• The follo"ing o:ecti!es are equi!alent8

9 Total completion time

9 +ean completion time

9 Total flo" time (if ready times are all *ero) 9 +ean flo" time (if ready times are all *ero)

9 Total lateness

9 +ean lateness

9 6n!entory carrying costs• Shortest processing time (SPT) rule minimi*es all of the

ao!e o:ecti!es (for the single machine static schedulingprolem).


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Re(ar"

• Priority rules are not a!ailale for minimi*ing numerof tardy :os. The ne#t lesson "ill discuss aprocedure for minimi*ing numer of tardy :os.

• +inimi*ing total tardiness is difficult and not co!ered.


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82


?esson '=

eading8

9 Section %.=, pp. ='&=03 (=th

7d.), pp. =/&='0 (th

7d.)

7#ercises8

9 %.=, %., pp. =0=&=0 (=th 7d.), pp. ='3&='= (th 7d.)


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83

LESSON 1?: SINGLE 0ACHINE

SCHEDULING

Outline

• Total Completion time

• +a#imum ?ateness

• 4umer of Tardy -os

• $or"ard and ;ack"ard Scheduling

• Precedence Constraints


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85

Sin$le 0achine Sche!ulin$

• +inimi*ing makespan is not included in this lesson.;ecause in case of our single machine prolem, themakespan is constant o!er all sequences. Ho"e!er,

minimi*ing makespan is important and "ill ediscussed for t"o or more machines. ;ecause,!arious costs are directly proportional to makespan.This includes "orker5s "ages, utilities, o!erheads etc.

• The last topic of the lesson is a procedure "hich isapplied "hen there are some precedenceconstraints.


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86

Total Co('letion Ti(e

• Different schedule pro!ides different !alues of totalcompletion time. +inimi*ing total completion timemeans finding a schedule that pro!ides minimum

total completion time.• 1hy is it important to minimi*e total completion timeL

9 See ?esson 3, Slide =/ for some equi!alences

9 ?ess total completion time means a :o stays in

the system ("aiting time K processing time) for ashorter duration.


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87


9 So, manufacturing lead time (the time et"eenorder placement and order deli!ery) is less.

9 6f the system minimi*es total flo" time (a related

o:ecti!e), the indi!idual customers are likely toe#perience a faster ser!ice ("aiting time K ser!icetime).

9 Since the :os stay in the system for a shorter

duration, the in!entory carrying costs are less. Tosee an e#ample, re!isit 7#ample 0, ?esson 0,Slides &'/.


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88


• Sequence the :os in the Shortest Processing Time(SPT) order if there is a single machine and theo:ecti!e is to minimi*e

9 Total (or, mean) completion time (or, flo" time) 9 Total (or, mean) "aiting time

9 Total (or, mean) lateness

9 6n!entory carrying costs

• e!isit 7#ample 0, ?esson 0, Slides &'/. There aret"o cars. Car ' has a processing time of ' hour andCar 0 has a processing time of 3 hours.


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89


• Since Car ' has a smaller processing time than Car0, the SPT rule requires that Car ' e processedefore Car 0. This is done y Schedule 3.

• So, Schedule 3 is a SPT schedule and it has a totalcompletion time G 'K= G hours.

• Schedule = is a non&SPT schedule and it has a totalcompletion time G 3K= G F.

• Thus, the SPT schedule, Schedule 3 minimi*es totalcompletion time.


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90

0a5i(u( Latene""

• Different schedule pro!ides different !alues ofma#imum lateness. +inimi*ing ma#imum latenessmeans finding a schedule that pro!ides minimumma#imum lateness.

• 1hy is it important to minimi*e ma#imum latenessL

9 6t5s a alanced approach. The productiondepartment likes to minimi*e costs "hich arerelated to minimi*ing makespan and total

completion time. Ho"e!er, the marketingdepartment likes to pro!ide faster ser!ice and setearlier due dates. +inimi*ing ma#imum latenesspro!ides a alance.


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91

0a5i(u( Latene""

9 ;y minimi*ing ma#imum lateness one gets someinsight if the due dates are realistic (if the duedates can e met using the gi!en resources).

• 6f the minimum ma#imum lateness is positi!e,

the due dates are not realistic. So, themarketing department should promise longerlead times and the production departmentshould e allocated more resources.

• 6f the minimum ma#imum lateness is negati!e,the marketing department can promise shorterlead times and carry out some promotionalacti!ities to generate more demand.


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92

0a5i(u( Latene""

• Sequence the :os in the 7arliest Due Date (7DD)order if there is a single machine and the o:ecti!e isto minimi*e

9 ma#imum lateness 9 ma#imum tardiness

• e!isit 7#ample ', ?esson 0, Slides &%. There aret"o :os. -o ' is a due after = hours and -o 0 is

due after 0 hours.


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93

0a5i(u( Latene""

• Since -o 0 is due efore -o ', the 7DD rulerequires that -o 0 is done efore -o '. This is doney Schedule 0.

• So, Schedule 0 is an 7DD schedule and it has ama#imum lateness of *ero (oth :o is completedright "hen it5s due).

• Schedule ' is a non&7DD schedule and it has ama#imum lateness of 0 hours (-o ' has a latenessof 0 hours and -o ' has a lateness of 9' hour 6.e.,-o ' is early y ' hour).

• Thus, the 7DD schedule, Schedule 0 minimi*esma#imum lateness.


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94

Nu(+er o Tar!y *o+"

• Different schedule pro!ides different numer of tardy :os (:os "ith a positi!e lateness). +inimi*ingnumer of tardy :os means finding a schedule that

meets as many due dates as possile.• 1hy is it important to minimi*e numer of tardy :osL

9 Sometimes, a product is useless if it5s completedafter its due date. $or e#ample, con!ocation

go"ns, "edding dresses and irthday cakes muste deli!ered efore their due dates. Spaceshuttles must lea!e on&time, else the mission "illnot e successful.


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95

Nu(+er o Tar!y *o+"

Steps

'. Arrange the :os in the 7arliest Due Date (7DD)order.

0. epeat the follo"ing as long as there is any tardy :o8

6f the :o is the first tardy :o, consider the first :os and remo!e the one "ith the largest processingtime.

3. Append all the tardy :os, if any, in the end in anyorder.

th−i i


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96

A F ; % 'F

C = '% D > ' 7 > 0'

Nu(+er o Tar!y *o+": E5a('le

Processing Completion Due

-o Time Time Date


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97


Ste' 1


-o Time Time Date

Arrange the :os in the 7DD order and find if any is tardy

A F ; % 'F

C = '% D > ' 7 > 0'


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98



-o Time Time Date

Ste' 2&1

7liminate the longest :o efore the first one tardy andarrange the others in the 7DD order.

A F ; % 'F

C = '% D > ' 7 > 0'


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99



-o Time Time Date

Ste' 2&2


A F ; % 'F

C = '% D > ' 7 > 0'


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100



-o Time Time Date

Ste' 2&4


A F ; % 'F

C = '% D > ' 7 > 0'


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101



-o Time Time Date

Ste' 2&6


A F ; % 'F

C = '% D > ' 7 > 0'


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102



-o Time Time Date

Append the pre!iously eliminated A, ; in the end and stop.

Ste' 4


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103

• 4ote8 The optimal schedule may change if all theprocessing times increase y the same amount.

• Preprocessing8 if e!ery :o requires some

setup@deli!ery time, add the set setup@deli!ery time toall :os efore applying the algorithm.

• Such preprocessing is not necessary for scheduling :os if the o:ecti!e is to minimi*ing makespan, mean

flo" time, ma#imum lateness, etc.

Nu(+er o Tar!y *o+": A Note


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104

• The procedure discussed for minimi*ing the numerof tardy :os assigns :os to position ' first, then toposition 0, and so on. 6n a for"ard scheduling

procedure the assignment starts from position ' andcontinues in the for"ard direction

• The procedure "hich "ill e discussed ne#t,schedules in the opposite direction starting from the

last position. 6n a ack"ard scheduling procedure theassignment starts from the last position andcontinues in the ack"ard direction.

#or)ar! an! ,ac)ar! Sche!ulin$


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105

La)ler@" Al$orith( or Prece!ence Con"traint"

• The algorithm is applicale for single machineprolems "ith the o:ecti!e of minimi*ing

9 +akespan

9 +a#imum lateness 9 +a#imum tardiness


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106

• Consider a single machine prolem "ith precedenceconstraints and minimi*ing ma#imum latene"" o:ecti!e (other o:ecti!es pre!iously stated may e

minimi*ed similarly)



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107

• General "che(e: The algorithm first assigns a :o tothe last position, then a :o to the position ne#t tolast, and so on.

• Can!i!ate -o+ or a 'o"ition: Due to precedenceconstraints, not all the :os are candidates for aposition. $or e#ample, if a :o has a successor, the

:o cannot e assigned to the last position. Hence,candidates for the last position are the ones "ithoutany successor.



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108

• 7hich -o+ to a""i$n

'. 7liminate all the :os "hich are pre!iously assigned

(to later positions)0. 6dentify the candidates & :os that ha!e no successor

or ha!e successors all pre!iously assigned (to laterpositions)

3. Among all the candidates, schedule the one "ith theminimum latene"".


C


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109

A F ; % 'F C = '%

D > ' 7 > 0'

Processing Due ?ateness

-o Time Date CandidateL 6f scheduled

A ; D

7

Completion time if scheduledG FK%K=K>K>G3'

C


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110

A ; D

A F ; % 'F C = '%

D > '



Completion time if scheduledG FK%K=K>G0 E


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111

A F ; % 'F

D > '



Completion time if scheduledG FK%K>G0' EC

A ; D


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112

A F ; % 'F



Completion time if scheduledG FK%G' ECD

A ;


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113

A F es F&G&0



Completion time if scheduledG F ECD

A

,


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114


• The algorithm is descried in the conte#t ofminimi*ing ma#imum lateness. To get the algorithmfor minimi*ing

9 ma#imum tardiness, replace MlatenessN yMtardinessN


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115


?esson '

eading8

9 Section %.>, pp. =0&=3' (=th 7d.), pp. ='=&=' (th 7d.)

7#ercises8

9 %.>, %.F, p. =3' (=th

7d.), pp. ='&=0/ (th

7d.)

LESSON 1>:


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0ULTIPLE 0ACHINE SCHEDULING

Outline

• Planning and +onitoring "ith


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117

• Planning and +onitoring "ith


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119

T)o 0achine #lo) Sho'

• $lo" shop is introduced in ?esson ', in Slides '%&'.• ecall that a flo" shop is suitale for a make&to&stock

or assemle&to&stock production system "herestandard products are produced in high !olume.

• Here, "e discuss the special cases of t"o&and three&machine systems.

+ ' + 07 n t e r 7 # i t

A C o n c e p t u a l I i e " o f A T " o & + a c h i n e $ l o " S h o p


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120


• The main characteristic of a t"o&machine flo" shopsystem is that e!ery :o first !isits +achine ' andthen +achine 0.

• 7#amples8

9 Customi*ing and painting 9 +achining and polishing

9 +oulding and aking

9 epair and testing

9 Typing and proofing (of chapters of a ook)

9 e!ie" and data entry (of claims)

9 Checkups y a nurse and a doctor (of patients)


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121


• 1e continue to assume that 9 e!ery machine can process one :o at a time.

9 e!ery :o can e processed y one machine at atime and

• So, in terms of a


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123


< r i n d i n g m a c h i n e

0 = > % ' / ' 0

D a y s

' =

? a t h e m a c h i n e - o A ' 0

- o A ' 0

- o ; 0 3

- o ; 0 3

' >


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124


• $or most o:ecti!es listed in ?esson 0, the sameorder on oth machines is optimal. So, if +achine 'processes -o ' efore -o 0, then +achine 0 "illalso process -o ' efore -o 0. 1hen the :o&order

is the same on all machines, the schedule is called apermutation schedule

• See the on


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125


• 6t5s etter to keep track of starting, processing andfinishing times of e!ery :o on e!ery machine.Consider an e#ample.

• E5a('le 18 7ach of the t"o :os A and ; must e

processed on +achine ' efore +achine 0. Theprocessing times are sho"n elo"8

A '> ; 'F

+achine +achine-o Center ' Center 0

S


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126

• Sche!ule1: Suppose that each machine processes-o A efore -o ;. Then, the starting, processing,and finishing times are as follo"s8

• Processing starts at time / (-o A on +achine ').$inishing time G Starting time K Processing time. So,-o A finishes at time /K'> G '> on +achine '.

• -o A can start on +achine 0 only after time '>, "henit5s completed on +achine '.


-o Start Process $inish Start Process $inish A;

+achine ' +achine 0

16

9

5

17

T 0 hi #l Sh


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127

• -o ; can start on +achine ' only after time '>,"hen +achine ' ecomes idle.

• The starting time of -o ; on +achine 0 is thema#imum of finishing times of -o A on +achine 0

(+achine 0 ecomes idle) and -o ; on +achine '(-o ecomes free).

• So, Starting time of -o ; on +achine 0 G ma#(finishing time of -o A on +achine 0, finishing time

of -o ; on +achine ') G ma# (0', 0) G 0.• Starting time of every job, except the first one, on

Machine 2 is a maximum of two numbers.


T 0 hi #l Sh


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128

• The finishing time of the last :o (-o ;) on the lastmachine (+achine 0) is the makespan. So, heremakespan G =0.

• 1e shall discuss -ohnson5s rule, "hich can minimi*e

makespan (repetiti!e application of the rule yields aschedule "ith least makespan)

• $irst, oser!e that a different sequence pro!ides adifferent makespan.


T 0 hi #l Sh


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129

• Sche!ule 2: Suppose that each machine processes-o ; efore -o A.

• The process starts at time / (-o ; on +achine ').

•Starting time of -o A on +achine 0 G ma# (0>, 0) G0>.

• +akespan G $inishing time of -o A (the last :o) on+achine 0 (the last machine) G 3' (impro!edBBB)


-o Start Process $inish Start Process $inish

; 'F A '>

+achine ' +achine 0


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T)o&0achine #lo) Sho'

*ohn"on@" Rule


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131

'. ?ist time required to process each :o at each machine. Setup a one&dimensional matri# to represent desired sequence"ith of slots equal to of :os.

0. Select smallest processing time at either machine. 9 6f that time is on machine ', put the :o as near to

eginning of sequence as possile.

9 6f smallest time occurs on machine 0, put the :o as nearto the end of the sequence as possile.

3. emo!e the :o from the list.

=. epeat steps 0&3 until all slots in matri# are filled Q all :osare sequenced.

*ohn"on " Rule


*ohn"on@" Rule E5a('le


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*ohn"on " Rule E5a('le

A >; '>

C % 0D 'F7 = >


The minimum processing time, 0, is gi!en y -o C on+achine 0. So, Schedule -o C in the end.


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134

A >; '>

D 'F



C7

-o 7 is remo!ed. 4o", there is a tie. +inimum processingtime is gi!en y -os A and ;. ;reak ties aritrarily.Schedule one of -o A or -o ;.




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135

; '>

D 'F



C A7

-o A is chosen aritrarily. -o A is scheduled in theeginning, ecause its minimum time is gi!en on +achine '.;eginning means position 0 ecause position ' is taken.




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136

D 'F



C; A7

4e#t, -o ; is scheduled. 6t5s scheduled in the end, ecauseits minimum processing time is gi!en on +achine 0.




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137



C;D A7

The sequencing is complete after assigning the remaining-o D to the remaining position. 4e#t, the makespan iscomputed.




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138

7 A

D;C

C;D A7

7ach triplet ao!e sho"s the starting, processing, and finishingtimes of an operation. -ohnson5s rule guarantees that the ao!eschedule gi!es the est !alue (==) of makespan.






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139


+0

> '0 '% 0= 3>

Time

=0 =%

+'

3/

The


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E5ten"ion o *ohn"on@" Rule


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141

• The e#tension of -ohnson5s rule does not guarantee anoptimal makespan for all three&machine flo" shop cases.Ho"e!er, the e#tension guarantees an optimal makespan

• if the largest processing time on the second machine isnot larger than the smallest processing times on

'. +achine ' or

0. +achine 3 or

3. ;oth• 6n Case ' +achine ' dominates +achine 0,

• 6n Case 0 +achine 3 dominates +achine 0 and

• 6n Case 3 oth +achines ' and 3 dominate +achine 0

To A Three 0achine #lo) Sho'


T A Th 0 hi #l Sh


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142

• Some e#amples "hen the rule applies are gi!en in thene#t slide8

• 7#ample (a)8 The largest processing time on +achine 0

G ma# (, 3, =, 0, 3) G ≤ G min (>, , , %, F) Gsmallest processing time on +achine '. So, +achine 'dominates +achine 0 and the e#tension of -ohnson5srule applies.

•7#ample ()8 The largest processing time on +achine 0G ma# (>, 3, 0, =, ) G > ≤ > G min (F, %, >, , %) Gsmallest processing time on +achine 3. So, +achine 3dominates +achine 0 and the e#tension of -ohnson5srule applies.



T A Th 0 hi #l Sh


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143-ohnson5s rule applies in each of the ao!e cases

-o ' 0 3 -o ' 0 3 -o ' 0 3

' > F ' > > F ' >

0 3 3 0 3 % 0 3 3 = % 3 = 0 > 3 F = %

= % 0 = = = = > 0 F

F 3 3 % F 3

+achine

(c)

+achine

(a)

+achine

()



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144

• 7#ample (c)8 The e#tension of -ohnson5s rule appliesecause oth +achine ' and 3 dominate +achine 0(check).

• 1hene!er the rule applies, the follo"ing is done8• Step '8 Create a fictitious t"o&machine flo" shop prolem

"ith t"o fictitious machines +'5 and +05. 7!ery :o isassigned processing times on these t"o fictitious

machines in the follo"ing "ay8• +'58 Processing time of a :o on fictitious machine +'5is the sum of the processing times of that :o on theoriginal machines +' and +0.



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145

• +058 Processing time of a :o on fictitious machine +05is the sum of the processing times of that :o on theoriginal machines +0 and +3.

• Step 08 Apply -ohnson5s rule on the fictitious t"o&machineprolem "ith machines +'5 and +05.

• Step 38 Sequence the :os on the original three machinesusing the optimal sequence otained from Step 0.



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146

An optimal (not unique) sequence is4e#t, compute makespan using the original 3 machines.

-o ' 0 3

' > . F

0 3 3

3 . = %

= % 0 =

. F 3 .

Three&+achine Prolem

+achine

-o ' 0

'

0

3

=

+achine

$ictitious T"o&+achine Prolem



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147

7ach triplet ao!e sho"s the starting, processing, and finishingtimes of an operation. -ohnson5s rule guarantees that the ao!eschedule gi!es the est !alue (=') of makespan.

+achine +achine +achine-o Center ' Center 0 Center 3



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148

• +ore on the pre!ious slide8• Starting time of every job, except the first one, on

Machine 2 is a maximum of two numbers.

• 7#ample8 Starting time of -o ' on +achine 0 G ma#

(finishing time of -o 3 on +achine 0, finishing time of-o ' on +achine ') G ma# (, '') G ''.

• Starting time of every job, except the first one, onMachine is a maximum of two numbers.

• 7#ample8 Starting time of -o ' on +achine 3 G ma#(finishing time of -o 3 on +achine 3, finishing time of-o ' on +achine 0) G ma# ('F, '>) G 'F.


Gantt Chart


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149

• Esing the starting and finishing times of each operation,"e can dra" a


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150

The '0 '% 0= 3>Time

=0

+0

3/

+'

Gantt Chart


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151

• 4ote that since a :o cannot e processed on more thanone machine at the same time, a !ertical line (indicating atime) must not cut through more than one o#corresponding to the same :o. $or e#ample, there are

three o#es corresponding to -o 3 one for +', one for+0 and the other for +3. Ho"e!er, these 3 o#esrepresent 3 operations "hich are processed in 3 distincttime periods. As a result, the o#es do not share any!ertical line

• Similarly, since a machine cannot process more than on :o at the same time, the o#es do not o!erlap.


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LESSON 1B: T7O&*O,

*O, SHOP AND #LO7 SHOP SCHEDULING


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153

*O, SHOP AND #LO7 SHOP SCHEDULING

Outline

• T"o&-o -o Shop and $lo" Shop Prolem

• Steps of the Solution Procedure• 6nterpretation

•


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154

T)o *o+ *o+ Sho' an! #lo) "ho' Pro+le(

• So far, "e ha!e discussed scheduling prolems "itha limited numer of machines. 6n this lesson, "ediscuss a solution procedure for a schedulingprolem "ith an unlimited numer of machines.

• 1e assume that t"o :os requires processing ysome m machines. $or each :o, the sequence ofmachines is kno"n. 6t5s not assumed that each :omust !isit the machines in the same order. So, the

procedure is applicale to :o shop. Ho"e!er, theprocedure is same for the flo" shop "hen each :omust !isit the machines in the same order.

T)o&*o+ *o+ Sho' an! #lo) "ho' Pro+le(


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155

T)o *o+ *o+ Sho' an! #lo) "ho' Pro+le(

• The procedure is est descried y an 7#ample8

• 7#ample '8 A engineering faculty requires arefundale deposit from each student. To get the

deposit ack at the end of the term, e!ery studentmust otain a clearance from each la used. Peterand Patricia needs clearance from +achine la (A)and Computer la (;). Peter "ants to !isit +achinela first and Patricia Computer la. After getting

certificates, each "ill !isit an administrati!e assistant(C) "ho "ill issue a Mno claimN certificate. Then, each"ill !isit the cashier (D) "ho "ill return the deposit.



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156

' '

• Peter and Patricia estimate the follo"ing processingtimes

Peter Patricia Acti!ity Time Acti!ity Time

A '/ ;

; A

C 0 C 0/

D '/ D


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158

' '

• Step 8 Esing only three special types of linesegments, and not crossing the rectangles, identify apath from (/,/) to the upper right corner (a, b), "herea G total time on x&a#is and b G total time on y&a#is.

• Step >8 Compute clock times along the path. Thelength of the path is the clock time at the upper rightcorner (a, b).

• Step F8 epeat Steps = and for other paths

• Step %8 $ind out the shortest path, interpret theshortest path and list acti!ities o!er time.

• Step 8 Dra" a


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159

'/

0/

3/

'/ 0/ 3/ =/ /

Carte"ian Coor!inate Sy"te(

Ste' 2: #in! Peter@" an! Patricia@"

Start an! En! Ti(e"


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160

Start an! En! Ti(e"

• The start and end times are computed elo" separately forPeter and Patricia. 7ach starts at time *ero. 4otice that thetimes are not clock times, ut Peter5s and Patricia5scumulati!e times. The clock times "ill e computed later.

Peter5s Time Patricia5s Time

Acti!ity Start Process 7nd Acti!ity Start Process 7nd

A ;

; A C C

D D

Note a = 50, b = 35 according to the notations on slide 6

Ste' 4: I!entiy Coor!inate" o Corner" o

Rectan$le" Re're"entin$ Actiitie"


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161


• $or each acti!ity, a rectangle "ill e dra"n on the graphsho"n in Step '. $or each rectangle, the lo"er, left cornerrepresents start of the acti!ity and the upper right cornerend of the acti!ity. So, one corner is otained from Peterand Patricia5s start time of that acti!ity and another corner isotained from Peter and Patricia5s end times of that acti!ity.

• $or e#ample, lo"er left corner of rectangle C is (','/)ecause Peter starts acti!ity C at time ' and Patricia starts

acti!ity C at time '/. Epper left corner of rectangle C is(=/,3/) ecause Peter ends C at time =/ and Patricia endsC at time 3/.

Ste' 4: I!entiy Coor!inate" o Corner" o



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162


• Esing the same reasoning, "e find the follo"ing coordinatesof the corners of A, ;, C and D.

ectangle ?o"er ?eft Corner Epper ight Corner A

;

C

D

Ste' 6: Dra) Rectan$le"


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163

'/

0/

3/

'/ 0/ 3/ =/ /

Peters Time

P a t r i c i a O s

T i m e

' $

Ste' ?: U"in$ only Three S'ecial Ty'e" o

Line Se$(ent" an! Not Cro""in$ the


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164

$ $

Rectan$le" #in! A Path #ro(


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165



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166



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167



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168

Cloc Ti(e" Alon$ the Path

• The length of the path is the clock time at (a,b)G(/,3).

• The clock time is set to / at (/,/).

• Clock times are computed at other reak points as follo"s8

• $or a =&degree line, ad!ance clock time y the line5spro:ection on the x&a#is (or, equi!alently, the pro:ection onthe y&a#is).

• $or e#ample, there is a =&degree line et"een (/,/) and

(,). 6ts pro:ection on the hori*ontal a#is is units and thesame is its pro:ection on the !ertical a#is.

Ste' >: Co('ute



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169


• The line represents that oth Peter and Patricia are usyfor units of time. So, it mo!es units hori*ontally and units !ertically. So, clock time ad!ances y units.

• Thus, the clock time at (,)G clock time at (/,/)K

G / K

G

Ste' >: Co('ute



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170


• $or each hori*ontal or !ertical line, ad!ance clock timey the length of the line segment. $or e#ample, thereis a hori*ontal line et"een (,) and ('/,). Thelength of the line is . So, the clock time at ('/,)

G clock time at (,)K

G K

G '/

• The ne#t fe" slides sho"s clock times at all reakpoints till (a,b) G (/, 3). The length of the path is theclock time at (a,b).

Ste' >: Co('ute



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171


?ength of the !ertical line G clock time G >/ K G >

'/

0/

3/

'/ 0/ 3/ =/ /

Peters Time

P a t r i c i a O s

T i m e

A

;

C

D(a,b)

(/,/)

Ste' B: Another Path


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172

'/

0/

3/

'/ 0/ 3/ =/ /

Peters Time

P a t r i c i a O s

T i m e

A

;

C

D(a,b)

(/,/) '/

'

3

=/

F/

Check the clock times and length of the path

Ste' %: Inter'retation


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173

• The shortest path is the one sho"n on Slide 0 as itslength is >, less than F/ "hich is the length of the otherpath sho"n on Slide 0>.

•The interpretation of a line segment is done as follo"s8• Hori*ontal line segment8 Only Peter is usy and

Patricia idle. Peter5s acti!ity is the one on the ao!e orelo" the line segment.

• Iertical line segment8 Only Patricia is usy and Peteridle. Patricia5s acti!ity is the one on the left or right ofthe line segment.



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174

• =&degree line segment8 ;oth are usy.

• Peter5s acti!ity is one on the ao!e or elo" theline segment.

• Patricia5s acti!ity is the one on the left or right ofthe line segment.

• Esing the ao!e guidelines, the shortest path sho"non Slide 0 is interpreted on the ne#t slide. Oser!e

that e!ery line segment is interpreted separately.



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175

Start 7nd ?ine Segment Peter5s Patricia5s

Clock Time Clock Time Type Acti!ity Acti!ity

/

'/'/ '

' =/

=/ /

/ >/

>/ >


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177

?esson 'F

eading8

9 Section %.F, pp. =3F&==/ (=th 7d.), pp. =0&=0F (th

7d.)

7#ercise8

9 %.', %.'>, pp. =='&==0 (=th 7d.), pp. =0%&=0 (th 7d.)

LESSON 1%: STOCHASTIC SCHEDULING


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Outline

• A e!ie" on Proaility Distriutions

• Stochastic Scheduling• Static Analysis8 Single +achine

• Static Analysis8 Parallel +achines

• Static Analysis8 T"o&+achine $lo" Shop

• Dynamic Analysis8 Selection of Disciplines• Dynamic Analysis8 The c µ rule

A Reie) on Pro+a+ility Di"tri+ution"


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• $irst, "e shall re!ie" "hat is meant y a proailitydistriution. 6t5s easier to understand proaility distriutions"hen there is a finite numer of e!ents. $or e#ample, "hen acoin is tossed, there are only t"o possile e!ents a head or atail. 1hen a coin is tossed t"ice, there can e follo"ing

e!ents 9 oth heads (0H)

9 one head and one tail ('H'T)

9 the first one head and the second one tail or

9 the first one tail and the second one head

9 t"o tails (0T)



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• A discrete proaility distriution is a tale, a formula, or agraph that lists all possile e!ents and proailities adiscrete random !ariale can assume.

• 6f a coin is tossed t"ice, the e!ents and correspondingproailities are as follo"s8

7!ent Proaility

T"o heads '@=

$irst one head, second one tail '@=

$irst one tail, second one head '@=T"o tails '@=



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• So, "e get the follo"ing proaility !alues87!ent Proaility

0H '@= G /.0

'H'T '@=K'@=G/./

0T '@= G /.0• The ao!e proaility !alues are sho"n on a graph in the

ne#t slide. This graph is a discrete proaility distriution.



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Pro+a+ility Di"tri+ution

/

/.0

/.

/.F

0H 'H'T 0T

Eent

P r o + a + i l i t y



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• The proaility distriution in case of a coin tossinge#ample is discrete. The numer of e!ents is finite. Somerandom numers such as the ones for processing times,arri!al times, etc. assume an infinite numer of !alues.

The proaility distriutions of such random numers arecontinuous.

• A continuous proaility distriution is similar to a discreteproaility distriution. 6n the follo"ing, some differences

are discussed. Coin tossing "ill e used as an e#ample ofa discrete distriution and processing time of acontinuous distriution.



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• 1hile the numer of heads in the coin tossing e#amplemay assume a finite numer of !alues, a processing timemay assume an infinite numer of !alues. So, it5s notpossile to list all the !alues that the processing time may

assume.• 1hile the numer of heads in the coin tossing e#ample

may assume a particular !alue, the proaility that aprocessing time "ill assume a particular !alue is *ero. $or

e#ample, there is almost *ero proaility that a :o "ill edone in e#actly 3 minutes, not a fraction of a second moreor less than 3 minutes. Ho"e!er, proaility that aprocessing time "ill lie in a gi!en range may e non&*ero.



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• A graphical description of a proaility distriution ofprocessing time gi!es the proaility !alues in terms ofthe area under the cur!e.

Area G proaility

• $or e#ample, The proaility that the processing time "illlie et"een 0 and % minutes is the area under the cur!e

from processing time G 0 to processing time G % (see thene#t slide).

• The total area under the cur!e G './/



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186

! (

t )

P r o c e s s i n g T i m e , t

0/ = > % ' /

P r o a i l i t y ( 0 ≤ t ≤ % )

G S h a d e d a r e a



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• The processing times, arri!al times, etc. are oftenassumed to follo" either a normal distriution or ane#ponential distriution.

• Nor(al Di"tri+ution: Consider the time required y

Har!ey5s to ser!e a customer. A customer !isits Har!ey5sduring lunch hours. On each !isit, he measures the lengthof the time et"een order placement and deli!ery. Heoser!es that the time is more or less 3 minutes.

Sometimes, it5s more than 3 minutes. Almost the samenumer of times it5s less than 3 minutes. Sometimes, it5

chapter 08 lecture 12 to 19 w08 431 scheduling

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