Download - Assingment Problem3
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Assignment Problems
Hazırlayanlar: Ali Evren Erdin
Arzu Çalık
Hilal Demirhan
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INDEX
IntroductionDescription Of The Assignment
ProblemsUses of The Assignment ProblemsSimple ExamplesThe ArticleExplanation of the ArticleThe Solution of the Problem in Lingo
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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
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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.
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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
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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.
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The network Representation of Example (continued...)
2222
3333
1111
2222
3333
1111cc1111
cc1212
cc1313
cc2121 cc2222
cc2323
cc3131 cc3232
cc3333
AgentsAgents TasksTasks
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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
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“An Application of Genetic Algorithm Methods for Teacher Assignment
Problems”
The ARTICLE
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What is the Problem??
“ What are the most suitable teacher and course assignments?”
Which teacher? Which Course?
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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
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• 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
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• 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
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The Questionnarie
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20 points
19 points
minlimit upperlimit
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The objection function for the problem will be:
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• Upper And Lower Limits for teacher work Hours
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The Lingo Formulation
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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
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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
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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
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OBJECTIVE FUNCTION
MAX= @SUM(chromozomes(i,j):
willingness(i,j)*match(i,j));
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CONSTRAINTS
@FOR(chromozomes(i,j): @BIN(match(i,j)));
@FOR(courses(j):@SUM(chromozomes(i,j):
match(i,j))=1);
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@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
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Objective value
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REPORT
Variable ValueReduced
Cost
MATCH( A, C26A) 1 -19
MATCH( A, C27A) 1 -18
MATCH( A, C26B) 1 -19
MATCH( A, C27B) 1 -18
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The teacher Ais going to teach :
• C26A, B• C27A, B
courses.
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REDUCED COSTS
• Negative reduced cost value(-19) means;
The objective value will increase 19
units.
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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
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THANKS!