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 !

MGMT E-5050

Characteristics

Streamlined version of the transportation algorithm

A Transportation Algorithm Tableau

Warehouse1

Warehouse 2

Warehouse 3

FactoryA

FactoryB

FactoryC

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 DESTINATION

A Transportation Algorithm Solution

Warehouse1

Warehouse 2

Warehouse 3

FactoryA

FactoryB

FactoryC

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 = $20.00

An Assignment Algorithm Tableau

Warehouse1

Warehouse 2

Warehouse 3

FactoryA

FactoryB

FactoryC

$3

$4

$9

$7

$12 $15

$17

$8

$5

FromTo

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

An Assignment Algorithm Tableau

Worker1

Worker 2

Worker 3

JobA

JobB

JobC

$3

$4

$9

$7

$12 $15

$17

$8

$5

FromTo

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

An Assignment Algorithm Solution

Worker1

Worker 2

Worker 3

JobA

JobB

JobC

$3

$4

$9

$7

$12 $15

$17

$8

$5

FromTo

THE OPTIMAL SOLUTION - TOTAL COSTS ARE 20.00

Characteristics

Guarantees an optimal solution since it is a linear programming model

Characteristics Also known as the Hungarian Method ,

Flood’s Technique , and the Reduced Matrix Method

NAMED AFTER MERRILL MEEKS FLOOD,

FAMED OPERATIONS RESEARCHERINDUSTRIAL ENGINEERPh.D, Princeton , 1935

Characteristics

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

Assignment Examples

COURSES

TERRITORIES

TABLES

CLIENTS

MECHANICS

SALESPERSONS

WAITSTAFF

CONSULTANTS

AUTOMOBILES

INSTRUCTORS

HISTORY

“

Eugene Egervary

Denes KonigFundamentalmathematics

developed at the University of

Budapestin 1932

The Assignment Algorithmis also called the

Hungarian Method in their honor

HISTORY

Developed in its current form by Harold Kuhn, PhD

Princeton, at Bryn Mawr College in 1955

( 1925 - )

Model Assumptions

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

Model Assumptions

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

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

Model Assumptions

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

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

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

Possible Performance Criteria

• Profit maximization

• Cost minimization

• Idle time minimization

• Job completion time minimization

The Assignment Matrix

Worker1

Worker2

Worker3

Worker4

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 StepsSTEP ONE - ROW REDUCTION

SUBTRACT THESMALLEST NUMBER IN EACH ROW FROM

ALL THE OTHERNUMBERS IN

THAT ROW

The Assignment Matrix

Worker1

Worker2

Worker3

Worker4

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

THE SMALLEST NUMBER IN EACH ROW

The Assignment Matrix

Worker1

Worker2

Worker3

Worker4

Job A $0 $5 $2 $8Job B $0 $3 $8 $2Job C $2 $0 $4 $7Job D $2 $0 $1 $1

The Assignment Matrix

Worker1

Worker2

Worker3

Worker4

Job A $0 $5 $2 $8Job B $0 $3 $8 $2Job C $2 $0 $4 $7Job D $2 $0 $1 $1

Assignment Algorithm StepsSTEP TWO - COLUMN REDUCTION

SUBTRACT THESMALLEST NUMBER

IN EACH COLUMNFROM ALL THE

OTHER NUMBERS IN THAT COLUMN

The Assignment Matrix

Worker1

Worker2

Worker3

Worker4

Job A $0 $5 $2 $8Job B $0 $3 $8 $2Job C $2 $0 $4 $7Job D $2 $0 $1 $1

THE SMALLEST NUMBER IN EACH COLUMN

The Assignment Matrix

Worker1

Worker2

Worker3

Worker4

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

THE SMALLEST NUMBER IN EACH COLUMN

The Assignment Matrix

Worker1

Worker2

Worker3

Worker4

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

ROW AND COLUMN REDUCTION PRODUCE THE REDUCED MATRIX

IT IS ALSO CALLED AN OPPORTUNITY COST MATRIX

Assignment Algorithm StepsSTEP THREE - ATTEMPT ALL ASSIGNMENTS

ATTEMPT TO MAKEALL THE REQUIRED

MINIMUM COSTASSIGNMENTS

ONLY THOSECELLS

CONTAINING“ 0 ”

OPPORTUNITYCOSTS ARE

CANDIDATESFOR MINIMUM

COSTASSIGNMENTS

The Assignment Matrix

Worker1

Worker2

Worker3

Worker4

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

The Assignment Matrix

Worker1

Worker2

Worker3

Worker4

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

ATTEMPT TO MAKE FOUR MINIMUM COST ASSIGNMENTS

NON-PERMITTED ASSIGNMENT - X

X

X X

The Assignment Matrix

Worker1

Worker2

Worker3

Worker4

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

JOB “ B “ WAS NOT ABLE TO BE ASSIGNED

NON-PERMITTED ASSIGNMENT - X

X

X X

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

IF ALL REQUIREDASSIGNMENTSCANNOT BE

MADE, USE THE “H” - FACTORTECHNIQUE

IT CREATESMORE “ 0 “

CELLS, WHICHIN TURN,

INCREASES THECHANCES OFMAKING ALL

THE REQUIREDASSIGNMENTS

The Assignment Matrix

Worker1

Worker2

Worker3

Worker4

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

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 “

The Assignment Matrix

Worker1

Worker2

Worker3

Worker4

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

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

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

The Assignment Matrix

Worker1

Worker2

Worker3

Worker4

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

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

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

The Assignment Matrix

Worker1

Worker2

Worker3

Worker4

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

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

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

The Assignment Matrix

Worker1

Worker2

Worker3

Worker4

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

THE “ H “ FACTOR IS THE LOWEST UNCOVERED NUMBER

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

The Assignment Matrix

Worker1

Worker2

Worker3

Worker4

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

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

The Assignment Matrix

Worker1

Worker2

Worker3

Worker4

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

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

The Assignment Matrix

Worker1

Worker2

Worker3

Worker4

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

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

The Assignment Matrix

Worker1

Worker2

Worker3

Worker4

Job A 0 4 0 6Job B 0 2 6 0Job C 3 0 3 6Job D 3 0 0 0

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

Assignment Algorithm StepsSTEP FIVE - RE-ATTEMPT ALL REQUIRED ASSIGNMENTS

RE-ATTEMPT ALLREQUIRED

ASSIGNMENTSAFTER USING

THE “ H “ - FACTORTECHNIQUE

SOMETIMESTHE “ H “FACTOR

TECHNIQUEMUST BE

EMPLOYEDMORE THAN

ONCE, INORDER TO

CREATEENOUGH“ ZERO “CELLSTO DOTHIS

The Assignment Matrix

Worker1

Worker2

Worker3

Worker4

Job A 0 4 0 6Job B 0 2 6 0Job C 3 0 3 6Job D 3 0 0 0

THE 1st OPTIMAL SOLUTION

NON - PERMISSABLE ASSIGNMENT : X

XX

X X

The Assignment Matrix

Worker1

Worker2

Worker3

Worker4

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

THE 1st OPTIMAL SOLUTION

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

The Assignment Matrix

Worker1

Worker2

Worker3

Worker4

Job A 0 4 0 6Job B 0 2 6 0Job C 3 0 3 6Job D 3 0 0 0

THE 2nd OPTIMAL SOLUTION

NON - PERMISSABLE ASSIGNMENT : X

XX

X X

The Assignment Matrix

Worker1

Worker2

Worker3

Worker4

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

THE 2nd OPTIMAL SOLUTION

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

Alternate Optimal SolutionsWHY BOTHER ?

The “Alternate Solution” 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 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 withQM for Windows

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 Algorithm

Templateand

Sample Data

The Assignment Algorithm