Post on 15-Dec-2014
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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!