The Assignment Algorithm A loading technique for committing two or more jobs to two or more workers or machines in a single work center. With one job assigned to each processor only ! MGMT E-5050

Characteristics Streamlined version of  Streamlined version of the transportation algorithm the transportation algorithm

A Transportation Algorithm Tableau Warehouse 1 Warehouse 2 Warehouse 3 Factory A Factory B Factory C 3 3 $3 $4 $9 $7 $12$15 $17 $8 $5 From To Demand Availability ONE UNIT SHIPPED FROM EACH SOURCE - ONE UNIT RECEIVED AT EACH DESTINATION

A Transportation Algorithm Solution Warehouse 1 Warehouse 2 Warehouse 3 Factory A Factory B Factory C 3 3 $3 $4 $9 $7 $12$15 $17 $8 $5 From To Demand Availability THE OPTIMAL SOLUTION - TOTAL COST = $20.00

An Assignment Algorithm Tableau Warehouse 1 Warehouse 2 Warehouse 3 Factory A Factory B Factory C $3 $4 $9 $7 $12$15 $17 $8 $5 From To THE “DEMAND “ ROW & “AVAILABILITY ” COLUMN ARE ELIMINATED

An Assignment Algorithm Tableau Worker 1 Worker 2 Worker 3 Job A Job B Job C $3 $4 $9 $7 $12$15 $17 $8 $5 From To  SHOWS ONLY THE COSTS OF PERFORMING EACH JOB UNDER EACH WORKER  ASSIGNABLE JOBS AND WORKERS CAN REPLACE FACTORIES AND WAREHOUSES

An Assignment Algorithm Solution Worker 1 Worker 2 Worker 3 Job A Job B Job C $3 $4 $9 $7 $12$15 $17 $8 $5 From To THE OPTIMAL SOLUTION - TOTAL COSTS ARE 20.00

Characteristics  Guarantees an optimal solution since it is a solution since it is a linear programming linear programming model model

Characteristics  Also known as the Hungarian Method, Flood’s Technique, and the Reduced Flood’s Technique, and the Reduced Matrix Method Matrix Method NAMED AFTER MERRILL MEEKS FLOOD, FAMED OPERATIONS RESEARCHER INDUSTRIAL ENGINEER Ph.D, Princeton, 1935

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

Assignment Examples COURSES TERRITORIES TABLES CLIENTS MECHANICS SALESPERSONS WAITSTAFF CONSULTANTS AUTOMOBILES INSTRUCTORS

HISTORY “ Eugene Egervary Denes Konig Fundamental mathematics developed at the University of Budapest in 1932 The Assignment Algorithm is also called the Hungarian Method in their honor

HISTORY 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 1955 College in 1955 ( )

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

Model Assumptions Employed only when all workers or machines  Employed only when all workers or machines are capable of processing all arriving jobs are capable of processing all arriving jobs  Dictates that only 1 job be assigned to each worker / machine, and vice-versa worker / machine, and vice-versa

Model Assumptions Employed only when all workers or machines  Employed only when all workers or machines are capable of processing all arriving jobs are capable of processing all arriving jobs  Dictates that only 1 job be assigned to each worker / machine, and vice-versa worker / machine, and vice-versa  Total number of arriving jobs must equal the total number of available workers / machines total number of available workers / machines

Possible Performance Criteria Profit maximization Cost minimization Idle time minimization Job completion time minimization

The Assignment Matrix Worker 1 Worker 2 Worker 3 Worker 4 Job A $20$25$22$28 Job B $15$18$23$17 Job C $19$17$21$24 Job D $25$23$24 These cells contain the labor costs of a particular worker performing a particular job

Assignment Algorithm Steps STEP ONE - ROW REDUCTION SUBTRACT THE SMALLEST NUMBER IN EACH ROW FROM ALL THE OTHER NUMBERS IN THAT ROW

The Assignment Matrix Worker 1 Worker 2 Worker 3 Worker 4 Job A $20$25$22$28 Job B $15$18$23$17 Job C $19$17$21$24 Job D $25$23$24 THE SMALLEST NUMBER IN EACH ROW

The Assignment Matrix Worker 1 Worker 2 Worker 3 Worker 4 Job A $0$5$2$8 Job B $0$3$8$2 Job C $2$0$4$7 Job D $2$0$1

The Assignment Matrix Worker 1 Worker 2 Worker 3 Worker 4 Job A $0$5$2$8 Job B $0$3$8$2 Job C $2$0$4$7 Job D $2$0$1

Assignment Algorithm Steps STEP TWO - COLUMN REDUCTION SUBTRACT THE SMALLEST NUMBER IN EACH COLUMN FROM ALL THE OTHER NUMBERS IN THAT COLUMN

The Assignment Matrix Worker 1 Worker 2 Worker 3 Worker 4 Job A $0$5$2$8 Job B $0$3$8$2 Job C $2$0$4$7 Job D $2$0$1 THE SMALLEST NUMBER IN EACH COLUMN

The Assignment Matrix Worker 1 Worker 2 Worker 3 Worker 4 Job A $0$5$1$7 Job B $0$3$7$1 Job C $2$0$3$6 Job D $2$0 THE SMALLEST NUMBER IN EACH COLUMN

The Assignment Matrix Worker 1 Worker 2 Worker 3 Worker 4 Job A $0$5$1$7 Job B $0$3$7$1 Job C $2$0$3$6 Job D $2$0 ROW AND COLUMN REDUCTION PRODUCE THE REDUCED MATRIX IT IS ALSO CALLED AN OPPORTUNITY COST MATRIX

Assignment Algorithm Steps STEP THREE - ATTEMPT ALL ASSIGNMENTS ATTEMPT TO MAKE ALL THE REQUIRED MINIMUM COST ASSIGNMENTS ONLY THOSE CELLS CONTAINING “ 0 ” OPPORTUNITY COSTS ARE CANDIDATES FOR MINIMUM COST ASSIGNMENTS

The Assignment Matrix Worker 1 Worker 2 Worker 3 Worker 4 Job A 0517 Job B 0371 Job C 2036 Job D 2000 THE OPPORTUNITY COST MATRIX WE CAN NOW DROP THE DOLLAR SIGNS

The Assignment Matrix Worker 1 Worker 2 Worker 3 Worker 4 Job A 0517 Job B 0371 Job C 2036 Job D 2000 ATTEMPT TO MAKE FOUR MINIMUM COST ASSIGNMENTS NON-PERMITTED ASSIGNMENT - X X XX

The Assignment Matrix Worker 1 Worker 2 Worker 3 Worker 4 Job A 0517 Job B 0371 Job C 2036 Job D 2000 JOB “ B “ WAS NOT ABLE TO BE ASSIGNED NON-PERMITTED ASSIGNMENT - X X XX

Assignment Algorithm Steps STEP FOUR - EMPLOY THE “H”-FACTOR TECHNIQUE IF ALL REQUIRED ASSIGNMENTS CANNOT BE MADE, USE THE “H” - FACTOR TECHNIQUE IT CREATES MORE “ 0 “ CELLS, WHICH IN TURN, INCREASES THE CHANCES OF MAKING ALL THE REQUIRED ASSIGNMENTS

The Assignment Matrix Worker 1 Worker 2 Worker 3 Worker 4 Job A 0517 Job B 0371 Job C 2036 Job D 2000 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 Worker 1 Worker 2 Worker 3 Worker 4 Job A 0517 Job B 0371 Job C 2036 Job D 2000 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 Worker 1 Worker 2 Worker 3 Worker 4 Job A 0517 Job B 0371 Job C 2036 Job D 2000 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 Worker 1 Worker 2 Worker 3 Worker 4 Job A 0517 Job B 0371 Job C 2036 Job D 2000 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 Worker 1 Worker 2 Worker 3 Worker 4 Job A 0517 Job B 0371 Job C 2036 Job D 2000 THE “ H “ FACTOR IS THE LOWEST UNCOVERED NUMBER THE “ H “ FACTOR EQUALS “ 1 “ IN THIS PARTICULAR PROBLEM

The Assignment Matrix Worker 1 Worker 2 Worker 3 Worker 4 Job A 0517 Job B 0371 Job C 2036 Job D 2000 ADD THE “ H “ FACTOR TO THE CRISS-CROSSED NUMBERS

The Assignment Matrix Worker 1 Worker 2 Worker 3 Worker 4 Job A 0517 Job B 0371 Job C 3036 Job D 3000 ADD THE “ H “ FACTOR TO THE CRISS-CROSSED NUMBERS

The Assignment Matrix Worker 1 Worker 2 Worker 3 Worker 4 Job A 0517 Job B 0371 Job C 3036 Job D 3000 SUBTRACT THE “ H “ FACTOR FROM ITSELF AND THE UNCOVERED NUMBERS

The Assignment Matrix Worker 1 Worker 2 Worker 3 Worker 4 Job A 0406 Job B 0260 Job C 3036 Job D 3000 SUBTRACT THE “ H “ FACTOR FROM ITSELF AND THE UNCOVERED NUMBERS

Assignment Algorithm Steps STEP FIVE - RE-ATTEMPT ALL REQUIRED ASSIGNMENTS RE-ATTEMPT ALL REQUIRED ASSIGNMENTS AFTER USING THE “ H “ - FACTOR TECHNIQUE SOMETIMES THE “ H “ FACTOR TECHNIQUE MUST BE EMPLOYED MORE THAN ONCE, IN ORDER TO CREATE ENOUGH “ ZERO “ CELLS TO DO THIS

The Assignment Matrix Worker 1 Worker 2 Worker 3 Worker 4 Job A 0406 Job B 0260 Job C 3036 Job D 3000 THE 1 st OPTIMAL SOLUTION NON - PERMISSABLE ASSIGNMENT : X X X XX

The Assignment Matrix Worker 1 Worker 2 Worker 3 Worker 4 Job A $20$25$22$28 Job B $15$18$23$17 Job C $19$17$21$24 Job D $25$23$24 THE 1 st OPTIMAL SOLUTION TOTAL COST = ( $20. + $17. + $17. + $24 ) = $78.00

The Assignment Matrix Worker 1 Worker 2 Worker 3 Worker 4 Job A 0406 Job B 0260 Job C 3036 Job D 3000 THE 2 nd OPTIMAL SOLUTION NON - PERMISSABLE ASSIGNMENT : X X X XX

The Assignment Matrix Worker 1 Worker 2 Worker 3 Worker 4 Job A $20$25$22$28 Job B $15$18$23$17 Job C $19$17$21$24 Job D $25$23$24 THE 2 nd OPTIMAL SOLUTION TOTAL COST = ( $22. + $15. + $17. + $24 ) = $78.00

Alternate Optimal Solutions WHY 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 workers the jobs that they really want to do

The Alternate Solution Case When employed in a shipping environment, alternate routes provide flexibility in the event of bridge, rail, road closures, accidents, and other unforeseen events.

Assignment Algorithm with QM for Windows

We Scroll To The “ ASSIGNMENT “ Module

We Want To Solve A New Problem

The Dialog Box Appears

There Are Four ( 4 ) Jobs To Be Assigned There Are Four ( 4 ) Workers or Machines That Are Available The Objective Function Is To Minimize Total Time or Cost The Jobs Are Labeled A, B, C, etc.

The Workers Are Numbered As 1, 2, 3, 4

THE DATA INPUT TABLE

THE COMPLETED DATA INPUT TABLE INCLUDES THE COST OF PROCESSING EACH JOB BY EACH WORKER

THE OPTIMAL SOLUTION Assign Worker 1 to Job A Assign Worker 2 to Job C Assign Worker 3 to Job D Assign Worker 4 to Job B Total Minimum Cost = $78.00

THE “ TILE “ OPTION all solution windows can be displayed simultaneously and removed one by one after discussion

THE “CASCADE” OPTION All window solutions can be discussed and removed one by one afterwards

The Assignment Algorithm Assignment Algorithm

Template and Sample Data

The Assignment Algorithm