short-term scheduling planning, scheduling, and controlling intermittent- flow operations in the...

SHORT-TERM SCHEDULINGSHORT-TERM SCHEDULING

PLANNING, SCHEDULING, AND CONTROLLINGINTERMITTENT- FLOW OPERATIONS IN THE MANUFACTURING AND SERVICE SECTORS

Applied Management Science for Decision Making, 1e Applied Management Science for Decision Making, 1e © 2012 Pearson Prentice-Hall, Inc. Philip A. Vaccaro , PhD© 2012 Pearson Prentice-Hall, Inc. Philip A. Vaccaro , PhD

Each job or job batch travels through a series of work centers, each of which performs a particular task.

Intermittent-Flow OperationsIntermittent-Flow Operations

SALIENT FEATURESSALIENT FEATURES

Work centers are grouped by majorprocessing function.



Each job or job batch has its own unique route through the system.





Job processing times at each workcenter are estimated based on similar past jobs and worker

experience


Work centers are grouped by function.

Each job or job batch has its own unique route

through the system.

Job processing times at each work center are

estimated based on similar past jobs and worker

experience.

Workers are highly skilled and flexible.


An An Intermittent-Flow Intermittent-Flow OperationOperation

Work Center

A

WorkCenter

B

WorkCenter

E

WorkCenter

C

WorkCenter

D

Work Center

F

cutting drilling inspecting

sanding painting packaging

JOBENTERS

HEREAS

RAWWOODAND

HARDWARE

FINISHEDJOB

LEAVESHERE

CARPENTRY SHOP EXAMPLECARPENTRY SHOP EXAMPLE

Evaluating all incoming jobs to see which workcenters they must pass through in order to

be completed.

Short-Term SchedulingShort-Term SchedulingTHE BIG PICTURETHE BIG PICTURE

Arranging all jobs scheduled for each work centerin a specific processing order chosen to meet

the shop’s performance goals.



Evaluating all incoming jobs to see which work centers they must pass through in order to be completed.

Arranging all jobs scheduled for each work center in a specific processing order chosen to meet shop performance goals.

Developing detailed start / finish times for each job at each work center.

Short-Term Scheduling Short-Term Scheduling HistoryHistory

Developed by Henry Gantt, a school teacher by training and later an engineer.

Refined and expanded on the existing body of production

and cost control techniques.

Developed the Gantt Chart in 1914 for scheduling and con-

trolling production and major projects.

Henry Laurence GanttHenry Laurence Gantt1861 - 19191861 - 1919

Gantt Chart Gantt Chart AccomplishmentsAccomplishments

WORLD WAR I NAVAL SHIPS ( 1917 ) HOOVER DAM ( 1931 ) MANHATTAN PROJECT ( 1942 ) INTERSTATE HIGHWAY SYSTEM ( 1956 )

Dr. Robert Oppenheimer and General Kenneth Nichols

THE GANTT CHART WAS THE PRECURSORTHE GANTT CHART WAS THE PRECURSOROF PERT/CPM: TODAY’S POPULAR TECHNIQUEOF PERT/CPM: TODAY’S POPULAR TECHNIQUE

FOR MANAGING MAJOR PROJECTS INFOR MANAGING MAJOR PROJECTS INGOVERNMENT AND INDUSTRYGOVERNMENT AND INDUSTRY

Short-Term Scheduling Short-Term Scheduling StepsSteps

I. Aggregate Planning

II. Loading

III. Priority Sequencing

IV. Detailed Scheduling

V. Dispatching

Aggregate PlanningAggregate Planning

Identify a quasi-unit that best reflects the

firm’s overall output of goods and services

Multiply the quasi-unit forecast by the quasi-

unit resource requirements.

BRIEF REVIEWBRIEF REVIEW

THESE STEPS WILL DETERMINE THE NUMBER AND TYPE OF WORK CENTERS, EQUIPMENT, PERSONNEL, PARTS,

AND SUPPLIES THAT THE FIRM MUST INVEST IN

Number of Vehicles to be Repaired( in quasi-units )

Average Consumption of Resources,Human and Non-Human

( per quasi-unit )

Total Resource RequirementsOver Life of the Aggregate Plan

Number of Mechanics Number of Vehicle Lifts

Size of the Parts Department Number and Types of Equipment

AUTOMOBILEAUTOMOBILEDEALERSHIPDEALERSHIP

REPAIR FACILITYREPAIR FACILITYEXAMPLEEXAMPLE



II. Loading



V. Dispatching

LOADINGLOADING

ALSO KNOWN AS SHOPLOADING or MACHINE LOADING

The assignment and commitmentof arriving jobs to one or more

work centers for the day, week, ormonth, based on job needs.

Gantt Chart for LoadingGantt Chart for Loading

WEEKLY SCHEDULE – DEPARTMENT 3985: MODEL SHOP SCHEDULE 3/16 - 22WEEKLY SCHEDULE – DEPARTMENT 3985: MODEL SHOP SCHEDULE 3/16 - 22

WORKWORK

CENTERCENTERMONMON TUESTUES WEDWED THURTHUR FRIFRI SATSAT

Machining

Fabrication

Assembly

Testing

Packaging

D

D

D

D

A

B

B

B

C

C

C

E

E

E

F

F

The The Gantt Chart for LoadingGantt Chart for Loading

Shows the specific jobs assigned to each work center for the day, week, or month at a glance.

The color bars show the estimated processing times for each job at each work center.

The The Gantt Chart for LoadingGantt Chart for Loading

Does not show the exact start and finish times for each job at each center.

Does not show the exact order in which each job will be processed at each center.



II. Loading



V. Dispatching

PRIORITY SEQUENCINGPRIORITY SEQUENCING

THE ORDER IN WHICH JOBSWAITING AT EACH WORK

CENTER WILL BEPROCESSED

THE ORDER OF PROCESSING SELECTED WILL BEST MEETMANAGEMENT’S SHOP GOALS

Priority Sequence RulesPriority Sequence Rules

• Average job completion time

• Number of setups• Setup costs• Work-in-process inventory

levels• Utilization of equipment• Idle time• Idle time costs• Shop productivity• Customer delivery time

WILL HAVE AN IMPACT ON THE FOLLOWING ( AND MORE )WILL HAVE AN IMPACT ON THE FOLLOWING ( AND MORE )

A Few A Few Priority Sequence Priority Sequence RulesRulesOVER 36 TO CHOOSE FROMOVER 36 TO CHOOSE FROM

• SPTSPT shortest processing time

• FIFOFIFO first in - first out

• LIFOLIFO last in - first out

• SSSS static slack

• CRCR critical ratio

• FISFSFISFS first in system - first served

( ALSO KNOWN AS DD , DUE DATE )

Priority Sequence RulesPriority Sequence RulesNEW PERFORMANCE CRITERIANEW PERFORMANCE CRITERIA

1. Average job completion time2. Labor or machine utilization3. Average number of jobs in the system4. Average number of late days per job

We evaluate priority sequencerules using one or more of the

following criteria:

Possible Job Shop GoalsPossible Job Shop Goals

Internal Shop Efficiency

Customer Service

Mix of Both

Internal Shop EfficiencyInternal Shop Efficiency

To promote this goal, the firm should evaluate priority sequence rules that: - maximize utilization of labor and equipment

- minimize the average number of jobs in the system, that is, the work-in-process inventory

Customer ServiceCustomer Service

To promote this goal, the firm

should evaluate priority

sequence rules that:

- minimize average job lateness ( tardiness )

Efficiency & Customer ServiceEfficiency & Customer Service

To promote both of these goals,

the firm should evaluate

priority sequence

rules that:

- minimize average job completion time

There are currently about a dozen priority sequence rules that support the goal of internal shop efficiency.

We evaluate those dozen rules using two particular performance criteria only. 1. “MAXIMIZE UTILIZATION of workers and equipment” 2. “MINIMIZE WORK-in-PROCESS INVENTORY” We select the rule that best satisfies those two criteria.

TheTheconnectionconnection

betweenbetweenshop goalsshop goals

andandprioritypriority

sequencesequencerulesrules

There are currently about a dozen priority sequence rules that support the goal of customer service.

We evaluate those dozen rules using one particular performance criterion only: “MINIMIZE JOB LATENESS”

We select the rule that best satisfies that criterion.





There are currently about a dozen priority sequence rules that support the goal of efficiency and service.

We evaluate those dozen rules using one particular performance criterion only. “ MINIMIZE AVERAGE JOB COMPLETION TIME”

We select the rule that best satisfies that criterion.





Priority Sequence Rule EvaluationPriority Sequence Rule EvaluationTEXT EXAMPLETEXT EXAMPLE

Assume that this job shop has only one work center.Assume that this job shop has only one work center.


Assume that this work center can only process one Assume that this work center can only process one job at a time.job at a time.


Assume that processing time can be labor Assume that processing time can be labor or machine time.or machine time.


Assume it is the 1Assume it is the 1stst day of the month. day of the month.


Assume five ( 5 ) jobs are waiting to be done.Assume five ( 5 ) jobs are waiting to be done.

Priority Sequence Rule ExamplePriority Sequence Rule ExampleTHE FIVE JOBS THE FIVE JOBS

JOBSJOB

PROCESS TIMES

JOB

DEADLINES( tentative )

A 5 Days 10th Day

B 10 Days 15th Day

C 2 Days 5th Day

D 8 Days 12th Day

E 6 Days 8th Day

Shortest Processing Time Shortest Processing Time (SPT)(SPT)THE JOB PROCESSING ORDERTHE JOB PROCESSING ORDER

JOB PROCESSJOB PROCESS

ORDERORDER


TIMETIME

COMPLETIONCOMPLETION

TIME TIME

JOB DEADLINEJOB DEADLINE( tentative )( tentative )

JOBJOB

LATENESSLATENESS

C 2 Days 2nd Day 5th Day 0 Days

A 5 Days 7th Day 10th Day 0 Days

E 6 Days 13th Day 8th Day 5 Days

D 8 Days 21st Day 12th Day 9 Days

B 10 Days 31st Day 15th Day 16 Days

5 Jobs 31 Days 74 Days - 30 Days

11

Completion TimeCompletion Time, or , or Flow TimeFlow Time = Job Waiting Time + Job Processing Time = Job Waiting Time + Job Processing Time

First-In , First Out First-In , First Out ( FIFO )( FIFO )THE JOB PROCESSING ORDERTHE JOB PROCESSING ORDER


ORDERORDER


TIMETIME


TIME TIME


JOBJOB

LATENESSLATENESS


B 10 Days 15th Day 15th Day 0 Days

C 2 Days 17th Day 5th Day 12 Days

D 8 Days 25th Day 12th Day 13 Days

E 6 Days 31st Day 8th Day 23 Days

11



First-in-System , First-Served First-in-System , First-Served (FSFS)(FSFS)JOBS ARE LINED UP BY JOBS ARE LINED UP BY DUE DATEDUE DATE


ORDERORDER


TIMETIME


TIME TIME


JOBJOB

LATENESSLATENESS

C 2 Days 2nd Day 5th Day 0 Days



D 8 Days 21st Day 12th Day 9 Days



11


Static Slack Static Slack ComputationsComputations

JOB DEADLINE DATE – CURRENT DATE – JOB PROCESS TIMEJOB DEADLINE DATE – CURRENT DATE – JOB PROCESS TIME

JOB A : 10th day – 1st day – 5 days = 4 days

JOB B : 15th day – 1st day – 10 days = 4 days

JOB C : 5th day – 1st day – 2 days = 2 days

JOB D : 12th day – 1st day – 8 days = 3 days

JOB E : 8th day – 1st day – 6 days = 1 day

THE JOB WITH THE SMALLEST

STATIC SLACK IS DONEFIRST


ORDERORDER


TIMETIME


TIME TIME


JOBJOB

LATENESSLATENESS


C 2 Days 8th Day 5th Day 3 Days


A 5 Days 21st Day 10th Day 11 Days



11

Static Slack - Static Slack - (( SS )SS )


Critical Ratio Critical Ratio ComputationsComputations

DEADLINE DATE – CURRENT DATEDEADLINE DATE – CURRENT DATE

REMAINING PROCESSING TIMEREMAINING PROCESSING TIME

JOB A : ( 10th – 1st ) / 5 days = 1.80

JOB B : ( 15th – 1st ) / 10 days = 1.40

JOB C : ( 5th – 1st ) / 2 days = 2.00

JOB D : ( 12th – 1st ) / 8 days = 1.37

JOB E : ( 8th – 1st ) / 6 days = 1.16

AS THE CRITICAL RATIOGETS SMALLER,THE JOB GETS

A HIGHERPRIORITY

JOB PROCESS

ORDER

JOB PROCESS

TIME

COMPLETION

TIME

JOB DEADLINE( tentative )

JOB

LATENESS



B 10 Days 24th Day 15th Day 9 Days


C 2 Days 31st Day 5th Day 26 Days


11

Critical Ratio - Critical Ratio - CRCR


Summary TabulationsSummary Tabulations

Priority

Sequence

Rule

Total Job

Processing Time

( in days )

Total Job

Completion Time

( in days )

Total Job

Lateness

( in days )

SPT 31 74 30

FIFO 31 93 48

FSFS 31 75 28

SS 31 82 34

CR 31 104 56

Average Job Completion TimeAverage Job Completion Time

Total Flow Time / Number of JobsTotal Flow Time / Number of Jobs

SPT Rule 74 days / 5 jobs = 14.8 days

FIFO Rule 93 days / 5 jobs = 18.6 days

FSFS (DD) 75 days / 5 jobs = 15.0 days

STATIC SLACK 82 days / 5 Jobs = 16.4 days

CRITICAL RATIO 104 days / 5 Jobs = 20.8 days

Labor / Machine UtilizationLabor / Machine Utilization

Total Processing Time / Total Flow TimeTotal Processing Time / Total Flow Time

SPT Rule 31 days / 74 days = 42.00%

FIFO Rule 31 days / 93 days = 33.3%

FSFS (DD) 31 days / 75 days = 41.33%

STATIC SLACK 31 days / 82 days = 37.8%

CRITICAL RATIO 31 days / 104 days = 29.81%

Average Number of Jobs in the SystemAverage Number of Jobs in the System

Total Flow Time / Total Processing TimeTotal Flow Time / Total Processing Time

SPT Rule 74 days / 31 days = 2.39 jobs

FIFO Rule 93 days / 31 days = 3.0 jobs

FSFS (DD) 75 days / 31 days = 2.42 jobs

STATIC SLACK 82 days / 31 days = 2.65 jobs

CRITICAL RATIO 104 days / 31 days = 3.35 jobs

Average Number of Late Days per JobAverage Number of Late Days per Job

SPT Rule 30 days / 5 jobs = 6.0 days

FIFO Rule 48 days / 5 jobs = 9.6 days

FSFS ( DD ) 28 days / 5 jobs = 5.6 days

STATIC SLACK 34 days / 5 jobs = 6.8 days

CRITICAL RATIO 56 days / 5 jobs = 11.2 days

Total Late Days / Number of JobsTotal Late Days / Number of Jobs

Performance SummaryPerformance Summary

RULERULE

AVERAGEAVERAGE

COMPLETECOMPLETE

TIMETIMEUTILIZATIONUTILIZATION

AVER. NO.AVER. NO.

JOBS INJOBS IN

SYSTEMSYSTEM

AVERAGEAVERAGE

JOB JOB

LATENESSLATENESS

NUMBERNUMBER

OF JOBSOF JOBS

LATELATE

MAXIMUMMAXIMUM

JOBJOB

LATENESSLATENESS

SPTSPT 14.8 Days

42.00% 2.39 Jobs

6.0 Days 3 16 Days

FIFOFIFO 18.6 Days

33.33% 3.00 Jobs

9.6 Days 3 23 Days

SSSS 16.4 Days

37.80% 2.65 Jobs

6.8 Days 4 16 Days

CRCR 20.8 Days

29.81% 3.35 Jobs

11.2 Days

4 26 Days

DDATEDDATE 15.0 Days

41.33% 2.42 Jobs

5.6 Days 3 16 Days

PRIORITY SEQUENCE RULE EVALUATIONPRIORITY SEQUENCE RULE EVALUATION

Postscript

The SPT rule always minimizes average job completion time ( flow time ) and the average number of jobs in the system..

Postscript

The DDATE rule always minimizes average job lateness and total job lateness.

Postscript

FIFO ( FCFS ) does not score well on most criteria but neither does it score poorly. It does however appear fair to customers which is important in service sector systems.

Job Shop w / QMJob Shop w / QM forfor WindowsWindows

We Scroll To

JOB SHOP SCHEDULING

There Are Five ( 5 ) JobsTo Be Processed

There Is Only One ( 1 )Machine In Each

Work Center

Jobs Labeled A,B,C, etc.

The Data Input TableAppears

We Schedule Under TheSPT

Priority Sequence Rule.

It is the 1st Day of the Month.

Estimated Processing TimesAre Entered Under“ Production Time”

&Job Deadlines Are

Entered Under“Due Date”

Under the SPT priority sequence rule, the order of processing is:1st Job C , 2nd Job A , 3rd Job E , 4th Job D , 5th Job B

ThePerformance

Criteria

SUMMARY STATISTICSfor

Seven ( 7 ) Priority Sequence Rules

Performance SummaryPerformance Summary

RULERULE

AVERAGEAVERAGE

COMPLETECOMPLETE

TIMETIMEUTILIZATIONUTILIZATION

AVER. NO.AVER. NO.

JOBS INJOBS IN

SYSTEMSYSTEM

AVERAGEAVERAGE

JOB JOB

LATENESSLATENESS

NUMBERNUMBER

OF JOBSOF JOBS

LATELATE

MAXIMUMMAXIMUM

JOBJOB

LATENESSLATENESS

SPTSPT 14.8 Days

42.00% 2.39 Jobs

6.0 Days 3 16 Days

FIFOFIFO 18.6 Days

33.33% 3.00 Jobs

9.6 Days 3 23 Days

SSSS 16.4 Days

37.80% 2.65 Jobs

6.8 Days 4 16 Days

CRCR 20.8 Days

29.81% 3.35 Jobs

11.2 Days

4 26 Days

DDATEDDATE 15.0 Days

41.33% 2.42 Jobs

5.6 Days 3 16 Days

PRIORITY SEQUENCE RULE EVALUATIONPRIORITY SEQUENCE RULE EVALUATION

The Gantt Loading Chart for

One ( 1 ) Work Center

First Come, First ServePriority Sequence Rule

Analysis

Earliest Due DatePriority Sequence Rule

Analysis

Static Slack Priority Sequence Rule

Analysis

Critical RatioPriority Sequence Rule

Analysis



II. Loading



V. Dispatching

Gantt Chart for Detailed SchedulingGantt Chart for Detailed Scheduling

• Conveys all loading gantt chart data

• Indicates the exact start and finish times for each job at each work center

• Allows real-time tracking of all jobs at each work center

• Provides the basis for customer delivery dates

• Indicates scheduled downtime for repair and maintenance

• Indicates time blocks reserved for important

emergency jobs

Gantt Chart for Detailed SchedulingGantt Chart for Detailed Scheduling

WEEKLY SCHEDULE – DEPARTMENT 3985: MODEL SHOP SCHEDULE 3/16 - 22

WORK WORK

CENTERCENTER

MONMON TUESTUES WEDWED THURTHUR FRIFRI SATSAT

MachiningMachining

FabricationFabrication

AssemblyAssembly

TestingTesting

PackagingPackaging

D

D

D

D

A

B

B

B

C

C

C

E

E

E

F

F

- SCHEDULED DOWN TIME FOR MAINTENANCE AND SPECIAL JOBSSCHEDULED DOWN TIME FOR MAINTENANCE AND SPECIAL JOBS

Job Status At-A-Glance

WWorkorkCenterCenter MondayMonday TuesdayTuesday WednesdayWednesday

MACHININGMACHINING

FABRICATIONFABRICATION

ASSEMBLYASSEMBLY

SCHEDULED TIME - JOB “B”SCHEDULED TIME - JOB “B”

SCHEDULED TIME - JOB “M”SCHEDULED TIME - JOB “M”

SCHEDULED TIME - JOB “R”SCHEDULED TIME - JOB “R”

Job Status Job Status At-A-GlanceAt-A-Glance


MACHININGMACHINING


ASSEMBLYASSEMBLY

CURSOR DENOTES REAL TIMECURSOR DENOTES REAL TIME

ACTUAL PROGRESS

ACTUAL PROGRESS

ACTUAL PROGRESS

JOB B

JOB M

JOB R

SCHEDULED TIME

SCHEDULED TIME

SCHEDULED TIME

JOB B IS BEHIND SCHEDULE AS OF LATE MORNING TUESDAY

Job Status At-A-Glance


MACHININGMACHINING


ASSEMBLYASSEMBLY


ACTUAL PROGRESS

ACTUAL PROGRESS

ACTUAL PROGRESS

JOB B

JOB M

JOB R

SCHEDULED TIME

SCHEDULED TIME

SCHEDULED TIME

JOB M IS AHEAD OF SCHEDULE AS OF LATE MORNING TUESDAY

Job Status Job Status At-A-GlanceAt-A-Glance


MACHININGMACHINING


ASSEMBLYASSEMBLY


ACTUAL PROGRESS

ACTUAL PROGRESS

ACTUAL PROGRESS

JOB B

JOB M

JOB R

SCHEDULED TIME

SCHEDULED TIME

SCHEDULED TIME

JOB R IS EXACTLY ON SCHEDULE AS OF LATE MORNING TUESDAY



II. Loading



V. Dispatching

DispatchingDispatchingThe formal release of the completed job to the customerThe formal release of the completed job to the customer

by the last work center that processed it. by the last work center that processed it.

Short-Term SchedulingShort-Term SchedulingTacticsTactics


The Assignment Algorithm

A loading technique for committing two or morejobs to two or more workers or machines

in a single work center.With one job

assigned to eachprocessor only !


CharacteristicsCharacteristics

Streamlined version ofStreamlined version of the the transportation algorithmtransportation algorithm

A Transportation Algorithm Tableau A Transportation Algorithm Tableau

Warehouse

1

Warehouse 2

Warehouse 3

Factory

A

Factory

B

Factory

C

33

$3

$4

$9

$7

$12 $15

$17

$8

$5

FromTo

1

1

1

1 1 1Demand

Availability

ONE UNIT SHIPPED FROM EACH SOURCE - ONE UNIT RECEIVED AT EACH DESTINATIONONE UNIT SHIPPED FROM EACH SOURCE - ONE UNIT RECEIVED AT EACH DESTINATION

A Transportation Algorithm Solution A Transportation Algorithm Solution

Warehouse

1

Warehouse 2

Warehouse 3

Factory

A

Factory

B

Factory

C

33

$3

$4

$9

$7

$12 $15

$17

$8

$5

FromTo

1

1

1

1 1 1Demand

Availability

1

1

1

THE OPTIMAL SOLUTION - TOTAL COST = $20.00THE OPTIMAL SOLUTION - TOTAL COST = $20.00

An Assignment Algorithm TableauAn Assignment Algorithm Tableau

Warehouse

1

Warehouse 2

Warehouse 3

Factory

A

Factory

B

Factory

C

$3

$4

$9

$7

$12 $15

$17

$8

$5

FromTo

THE “THE “DEMANDDEMAND “ ROW & “ “ ROW & “AVAILABILITY AVAILABILITY ” COLUMN ARE ELIMINATED” COLUMN ARE ELIMINATED

An Assignment Algorithm TableauAn Assignment Algorithm Tableau

Worker

1

Worker

2

Worker

3

Job

A

Job

B

Job

C

$3

$4

$9

$7

$12 $15

$17

$8

$5

FromTo

SHOWS ONLY THE COSTS OF PERFORMING EACH JOB UNDER EACH WORKERSHOWS ONLY THE COSTS OF PERFORMING EACH JOB UNDER EACH WORKER ASSIGNABLE JOBS AND WORKERS CAN REPLACE FACTORIES AND WAREHOUSES ASSIGNABLE JOBS AND WORKERS CAN REPLACE FACTORIES AND WAREHOUSES

An Assignment Algorithm SolutionAn Assignment Algorithm Solution

Worker

1

Worker

2

Worker

3

Job

A

Job

B

Job

C

$3

$4

$9

$7

$12 $15

$17

$8

$5

FromTo

THE OPTIMAL SOLUTION - TOTAL COSTS ARE 20.00 THE OPTIMAL SOLUTION - TOTAL COSTS ARE 20.00


Guarantees an optimal Guarantees an optimal solution since it is a solution since it is a linear programminglinear programming modelmodel

CharacteristicsCharacteristics Also known as the Also known as the Hungarian Method ,Hungarian Method ,

Flood’s Technique , Flood’s Technique , and the and the Reduced Reduced Matrix MethodMatrix Method

NAMED AFTER MERRILL MEEKS FLOOD,

FAMED OPERATIONS RESEARCHERINDUSTRIAL ENGINEERPh.D, Princeton , 1935


Determines the most efficientDetermines the most efficient assignment of jobs to workersassignment of jobs to workers and machines or vice-versaand machines or vice-versa

Assignment ExamplesAssignment Examples

COURSESCOURSES

TERRITORIESTERRITORIES

TABLESTABLES

CLIENTSCLIENTS

MECHANICSMECHANICS

SALESPERSONS

WAITSTAFF

CONSULTANTS

AUTOMOBILES

INSTRUCTORS

HISTORYHISTORY

“

Eugene EgervaryEugene Egervary

Denes KonigDenes Konig“

Fundamentalmathematics

developed at the University of

Budapestin 1932

The Assignment Algorithmis also called the

Hungarian Method in their honor

HISTORYHISTORY

Developed in its current form Developed in its current form by Harold Kuhn, PhD by Harold Kuhn, PhD

Princeton, at Bryn Mawr Princeton, at Bryn Mawr College in 1955College in 1955

( 1925 - )( 1925 - )

Model AssumptionsModel Assumptions

Employed only when all workers or machinesEmployed only when all workers or machines are capable of processing all arriving jobsare capable of processing all arriving jobs



Dictates that only 1 job be assigned to eachDictates that only 1 job be assigned to each worker / machine , and vice-versaworker / machine , and vice-versa



Dictates that only 1 job be assigned to eachDictates that only 1 job be assigned to each worker / machine , and vice-versaworker / machine , and vice-versa

Total number of arriving jobs must equal theTotal number of arriving jobs must equal the total number of available workers / machinestotal number of available workers / machines

Possible Performance CriteriaPossible Performance Criteria

• Profit maximization

• Cost minimization

• Idle time minimization

• Job completion time minimization

The Assignment MatrixThe Assignment Matrix

Worker

1

Worker

2

Worker

3

Worker

4

Job A $20 $25 $22 $28Job B $15 $18 $23 $17Job C $19 $17 $21 $24Job D $25 $23 $24 $24

These cells contain the labor costs of a particular worker performing a particular job

Assignment Algorithm StepsAssignment Algorithm StepsSTEP ONE - ROWSTEP ONE - ROW REDUCTIONREDUCTION

SUBTRACT THESMALLEST NUMBER IN EACH ROW FROM

ALL THE OTHERNUMBERS IN

THAT ROW


Worker

1

Worker

2

Worker

3

Worker

4


THE SMALLEST NUMBER IN EACH ROW


Worker

1

Worker

2

Worker

3

Worker

4


Assignment Algorithm StepsAssignment Algorithm StepsSTEP TWO - COLUMSTEP TWO - COLUMNN REDUCTIONREDUCTION

SUBTRACT THESMALLEST NUMBER

IN EACH COLUMNFROM ALL THE

OTHER NUMBERS IN THAT COLUMN


Worker

1

Worker

2

Worker

3

Worker

4


THE SMALLEST NUMBER IN EACH COLUMN


Worker

1

Worker

2

Worker

3

Worker

4


ROW AND COLUMN REDUCTION PRODUCE THE REDUCED MATRIX

IT IS ALSO CALLED AN OPPORTUNITY COST MATRIX

Assignment Algorithm StepsAssignment Algorithm StepsSTEP THREE - ATTEMPT STEP THREE - ATTEMPT ALL ASSIGNMENTSALL ASSIGNMENTS

ATTEMPT TO MAKEALL THE REQUIRED

MINIMUM COSTASSIGNMENTS

ONLY THOSECELLS

CONTAINING“ 0 ”

OPPORTUNITYCOSTS ARE

CANDIDATESFOR MINIMUM

COSTASSIGNMENTS


Worker

1

Worker

2

Worker

3

Worker

4

Job A 0 5 1 7Job B 0 3 7 1Job C 2 0 3 6Job D 2 0 0 0

THE OPPORTUNITY COST MATRIX

WE CAN NOW DROP THE DOLLAR SIGNS


Worker

1

Worker

2

Worker

3

Worker

4


ATTEMPT TO MAKE FOUR MINIMUM COST ASSIGNMENTS

NON-PERMITTED ASSIGNMENT - XX

XX

XX XX


Worker

1

Worker

2

Worker

3

Worker

4


JOB “ B “ WAS NOT ABLE TO BE ASSIGNED

NON-PERMITTED ASSIGNMENT - XX

XX

XX XX

Assignment Algorithm StepsAssignment Algorithm StepsSTEP FOUR - EMPLOY THE “STEP FOUR - EMPLOY THE “H”-FACTOR TECHNIQUEH”-FACTOR TECHNIQUE

IF ALL REQUIREDASSIGNMENTSCANNOT BE

MADE, USE THE “H” - FACTORTECHNIQUE

IT CREATESMORE “ 0 “

CELLS, WHICHIN TURN,

INCREASES THECHANCES OFMAKING ALL

THE REQUIREDASSIGNMENTS


Worker

1

Worker

2

Worker

3

Worker

4


COVER ALL ZEROS WITH THE MINIMUM NUMBER OF LINES - VERTICAL and / or HORIZONTAL

WE CAN COVER THREE ( 3 ) ZEROS WITH A LINE ACROSS ROW “ D “


Worker

1

Worker

2

Worker

3

Worker

4



WE CAN COVER TWO MORE ZEROS WITH A LINE DOWN COLUMN “ 1 “


Worker

1

Worker

2

Worker

3

Worker

4



WE CAN COVER THE REMAINING ZERO WITH A LINE DOWN COLUMN “ 2 “


Worker

1

Worker

2

Worker

3

Worker

4



WE CAN ALTERNATELY COVER THE LAST ZERO WITH A LINE ACROSS ROW “ C “


Worker

1

Worker

2

Worker

3

Worker

4


THE “ H “ FACTOR IS THE LOWEST UNCOVERED NUMBER

THE “ H “ FACTOR EQUALS “ 1 “ IN THIS PARTICULAR PROBLEM


Worker

1

Worker

2

Worker

3

Worker

4


ADD THE “ H “ FACTOR TO THE CRISS-CROSSED NUMBERS


Worker

1

Worker

2

Worker

3

Worker

4


SUBTRACT THE “ H “ FACTOR FROM ITSELF AND THE UNCOVERED NUMBERS

Assignment Algorithm StepsAssignment Algorithm StepsSTEP FIVE - RE-ATTEMPT ALL STEP FIVE - RE-ATTEMPT ALL REQUIRED ASSIGNMENTSREQUIRED ASSIGNMENTS

RE-ATTEMPT ALLREQUIRED

ASSIGNMENTSAFTER USING

THE “ H “ - FACTORTECHNIQUE

SOMETIMESTHE “ H “FACTOR

TECHNIQUEMUST BE

EMPLOYEDMORE THAN

ONCE, INORDER TO

CREATEENOUGH“ ZERO “CELLSTO DOTHIS


Worker

1

Worker

2

Worker

3

Worker

4


THE 1st OPTIMAL SOLUTION

NON - PERMISSABLE ASSIGNMENT : XX

XXXX

XX XX


Worker

1

Worker

2

Worker

3

Worker

4


THE 1st OPTIMAL SOLUTION

TOTAL COST = ( $20. + $17. + $17. + $24 ) = $78.00


Worker

1

Worker

2

Worker

3

Worker

4


THE 2nd OPTIMAL SOLUTION

NON - PERMISSABLE ASSIGNMENT : XX

XXXX

XX XX


Worker

1

Worker

2

Worker

3

Worker

4


THE 2nd OPTIMAL SOLUTION

TOTAL COST = ( $22. + $15. + $17. + $24 ) = $78.00

Alternate Optimal SolutionsAlternate Optimal SolutionsWHY BOTHER ?WHY BOTHER ?

The “The “Alternate Solution”Alternate Solution” Case Case

As a supervisor, you can only recommend a subordinate for a pay raise or promotion.

However, you can give your best workersthe jobs that they really want to do

The Alternate Solution CaseThe Alternate Solution Case

When employed in a shipping environment, alternate routes provide flexibility in the eventof bridge, rail, road closures, accidents, and

other unforeseen events.

Assignment Algorithm withAssignment Algorithm withQMQM forfor WindowsWindows

We Scroll To The

“ ASSIGNMENT “

Module

We Want To Solve ANew Problem

The Dialog Box Appears

There Are Four ( 4 ) JobsTo Be Assigned

There Are Four ( 4 ) Workers or Machines

That Are Available

The Objective FunctionIs To Minimize Total

Time or Cost

The Jobs Are LabeledA, B, C, etc.

The Workers

AreNumbered

As1, 2, 3, 4

THE DATA INPUT TABLE

THE COMPLETEDDATA INPUT TABLE

INCLUDES THE COST OFPROCESSING EACH JOB

BY EACH WORKER

THE OPTIMAL SOLUTION

Assign Worker 1 to Job AAssign Worker 2 to Job CAssign Worker 3 to Job DAssign Worker 4 to Job B

Total Minimum Cost = $78.00

THE “ TILE “ OPTION

all solution windowscan be displayedsimultaneouslyand removedone by one

after discussion

THE “CASCADE” OPTION

All window solutionscan be discussed

and removedone by oneafterwards

TheAssignmentAlgorithm

Assignment AlgorithmAssignment Algorithm

Templateand

Sample Data

The Assignment Algorithm


short-term scheduling planning, scheduling, and controlling intermittent- flow operations in the...

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