Assignment Problems

Hazırlayanlar: Ali Evren Erdin

Arzu Çalık

Hilal Demirhan

INDEX

IntroductionDescription Of The Assignment

ProblemsUses of The Assignment ProblemsSimple ExamplesThe ArticleExplanation of the ArticleThe Solution of the Problem in Lingo

Description of the Assignment Problems

• The problems that their goal is to find an optimal assignment of agents to tasks without assigning an agent more than once and ensuring that all tasks are completed

What can be the objectives?

Minimize the total time to complete set of tasks

Maximize skill ratings

Minimize the cost of the assignments

Or Etc.

What are the Applications of

Assignment Problems?

Assigning employees to tasks

Assigning machines to production jobs

Assign fleets of aircrafts to particular trips

Assigning school buses to routes

Networking computers

A Simple Example...

• An assignment problem seeks to minimize the total cost assignment of m workers to m jobs, given that the cost of worker i performing job j is cij.

• It assumes all workers are assigned and each job is performed.

The network Representation of Example (continued...)

2222

3333

1111

2222

3333

1111cc1111

cc1212

cc1313

cc2121 cc2222

cc2323

cc3131 cc3232

cc3333

AgentsAgents TasksTasks

Mathemetical Explanation

• LP Formulation

Min ∑∑cijxij i j

s.t. ∑ xij = 1 for each agent i

j

∑xij = 1 for each task j

i

xij = 0 or 1 for all i and j

“An Application of Genetic Algorithm Methods for Teacher Assignment

Problems”

The ARTICLE

What is the Problem??

“ What are the most suitable teacher and course assignments?”

Which teacher? Which Course?

What is Genetic Algorithm?

• The Genetic Algorithm is optimization procedure based on the natural law of evolution!

• The Key Idea of Genetic Algorithm is Survival of the Fittest!

• It is an Heuristic Approach based on Darwin’s Theory of Evolution

• Teacher Assignment Problem include multiple constraints

Teachers willingness need to be considered,

There should be a fair distribution of over time

• Teacher satisfaction has to be maximized

• One course should not be appointed to different teachers.

• There are 20 teachers.• There are 45 courses. Each course has two

classes: A and B.• Each teacher have an upper and minimum

workhour limits• Each Teacher rank the courses that they want

to teach

The Datas for the Problem

The Questionnarie

20 points

19 points

minlimit upperlimit

The objection function for the problem will be:

• Upper And Lower Limits for teacher work Hours

The Lingo Formulation

SETSSETS::

teachers / A B C D E F G H I J K L M N O P Q R S T /: upperlimit, minlimit;

courses / C1A C2A ....................C45A

C1B C2B ....................C45B /: hours;

chromosomes ( teachers, courses ) : willingness, match;

ENDSETSENDSETS

SETSSETS::

teachers / A B C D E F G H I J K L M N O P Q R S T /: upperlimit, minlimit;

courses / C1A C2A ....................C45A

C1B C2B ....................C45B /: hours;

chromosomes ( teachers, courses ) : willingness, match;

ENDSETSENDSETS

DATA:DATA:

willingness = (The matrix taken from willingness = (The matrix taken from thethe given table B1 )given table B1 )

hours = 4 4 5 3 3 3 3 3 3 4 4 2 3 3 3 2 4 3 3 3 3 3 3 hours = 4 4 5 3 3 3 3 3 3 4 4 2 3 3 3 2 4 3 3 3 3 3 3 3 3 3 3 3 2 3 3 3 2 2 3 3 3 3 3 3 3 2 3 3 3 4 3 3 3 3 3 2 3 3 3 2 2 3 3 3 3 3 3 3 2 3 3 3 4 4 5 4 5 3 3 3 3 3 3 4 4 2 3 3 3 2 4 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 2 3 3 3 2 4 3 3 3 3 3 3 3 3 3 3 3 2 3 3 3 3 2 3 3 3 2 2 3 3 3 3 3 3 3 2 3 3 3;3 3 3 2 2 3 3 3 3 3 3 3 2 3 3 3;

minlimit = 12 12 11 12 14 12 14 12 12 12 14 12 12 minlimit = 12 12 11 12 14 12 14 12 12 12 14 12 12 12 12 9 12 12 4 12;12 12 9 12 12 4 12;

upperlimit = 13 13 12 18 15 18 15 18 18 18 15 18 upperlimit = 13 13 12 18 15 18 15 18 18 18 15 18 18 18 18 15 13 13 11 13;18 18 18 15 13 13 11 13;

ENDDATAENDDATA

Matrix of Willingness

Courses Teachers

C1A C2A C3A C4A C5A

A 0 0 0 0 0

B 0 0 0 0 0

C 16 15 0 0 0

D 12 19 20 11 0

E 16 15 14 0 0

J=1

OBJECTIVE FUNCTION

MAX= @SUM(chromozomes(i,j):

willingness(i,j)*match(i,j));

CONSTRAINTS

@FOR(chromozomes(i,j): @BIN(match(i,j)));

@FOR(courses(j):@SUM(chromozomes(i,j):

match(i,j))=1);

@FOR(teachers(i):@SUM(courses(j):match(i,j)*

hours(j))<=upperlimit(i));

@FOR(teachers(i):@SUM(courses(j):match(i,j)*

hours(j))>=minlimit(i));

CONSTRAINTS

Objective value

REPORT

Variable ValueReduced

Cost

MATCH( A, C26A) 1 -19

MATCH( A, C27A) 1 -18

MATCH( A, C26B) 1 -19

MATCH( A, C27B) 1 -18

The teacher Ais going to teach :

• C26A, B• C27A, B

courses.

REDUCED COSTS

• Negative reduced cost value(-19) means;

The objective value will increase 19

units.

REPORT

Variable Value Reduced Cost

MATCH( T, C7A) 1 -20

MATCH( T, C34A) 1 -16

MATCH( T, C7B) 1 -20

MATCH( T, C34B) 1 -16

MATCH( T, C38B) 1 -17

THANKS!