An-Najah N. University Faculty of Engineering and Information Technology Department of Management Information systems Operations Research and Applications MIS: Prepared by Mr.Maher Abubaker Fall 2015/2016 Resources Daniel J. Epstein Department of Industrial and Systems Engineering University of Southern California Andrew and Erna Viterbi School of Engineering Operations Research: An Introduction, 9/EHamdy A. Taha, University of ArkansasISBN-10: X ISBN-13: ©2011 Prentice Hall Cloth, 832 ppPublished 08/29/ Introduction/ page#downlaoddiv ™ INFORMS – ™ ORMS - ™ Science of Better -

The Assignment Problem A special case of the transportation problem which is a special case of a linear program

The Problem There are n workers and n tasks to be performed. The time it takes worker i to perform task j is c ij. Which task should be assigned to which workers? I want the easy task.

The Model This is just the transportation problem with the right hand side values equal to one! animated

Some Applications workers to tasks jobs to machines facilities to locations Truck drivers to customer pick-up points Umpire crews to baseball games Judges to court dockets State inspectors to construction sites These are terrific applications.

A combinatorial problem If there are n workers and n tasks there are n! (factorial) possible assignments. Example: Workers are Al, Art, Alice, and Ann. There are four tasks: 1,2,3, & =4! AlArtAliceAnn If n = 10 then 10! = 3,628,800

An Example - assign a construction project (building) to a contractor Contractor Building Bids (in $10,000)

The Algorithm (Flood’s or the Hungarian Method) 1. Subtract the smallest cost element in each row from every element in that row. 2. Subtract the smallest cost element in each column from every element in that column. 3. Test for optimality by drawing the minimum number of lines that will cover every zero cell (no diagonal lines). If the minimum = n, a feasible assignment involving only zero cells is possible. Go to step Select the smallest element not having a line through it. Subtract this amount from all elements not covered by a line;and add this amount to all elements at the intersection of lines.Go to step Solution is optimum; make assignments using zero cells so that all constraints are satisfied.

I bet it works by magic. Subtract a constant ‘a’ from row k: since Some Magic animated

Let’s solve the problem-1!

Let’s solve the problem-2! Subtract 44 from row 1

Let’s solve the problem-3! Subtract 56 from row 2

Let’s solve the problem-4! Subtract 85 from row 3 and 42 from row 4

Let’s solve the problem-5! Subtract 2 from column 2 and 4 from column 3 Is there a feasible solution using the zero cells? 3 = 4 animated

Let’s solve the problem-6! Minimum uncovered element animated

Let’s solve the problem-7! animated

Let’s solve the problem-8! Optimal assignment is now possible using the zero cells! animated

Let’s solve the problem-9! X X X X animated

Another Example The Match Maker, a computerized dating service that attempts to bring two compatible people together, has to match the following individuals: Mandy, Mollie, and Martha with Bill, Bob, Bruno, and Bruce. The ladies have ranked each man on a scale of 1 to 10 with the higher number being the more preferred.

The Rankings-1

The Rankings-2

Convert to a minimization problem

Test for optimality-1 animated

Test for optimality-2 Minimum uncovered cell animated

Test for optimality-3 animated

Test for optimality-4 Minimum uncovered cell animated

Test for optimality-5 animated

Test for optimality-6 animated

Test for optimality-7 X X X X animated

Scheduling Umpires Night game to a day game Need a travel day off No crew assigned to same team more than 2 series Flight scheduling problems 7! = 5040 animated

The Assignment Problem This has been another delightful OR experience that both invigorates and excites the mind. Don’t you agree? return