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Assingment Problem3

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  • 1. Assignment Problems Hazırlayanlar: Ali Evren Erdin Arzu Çalık Hilal Demirhan
  • 2. INDEX
    • Introduction
    • Description Of The Assignment Problems
    • Uses of The Assignment Problems
    • Simple Examples
    • The Article
    • Explanation of the Article
    • The Solution of the Problem in Lingo
  • 3. 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
  • 4. What can be the objectives?
    • M inimize the total time to complet e set of tasks
    • M aximize skill ratings
    • M inimize the cost of the assignments
    • Or Etc.
  • 5. What are the Applications of Assignment Problems?
    • A ssigning e mployees to tasks
    • Assigning machines to production jobs
    • A ssign fleets of aircrafts to particular t rips
    • A ssigning school buses to routes
    • N etworking computers
  • 6. 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 c ij .
    • It assumes all workers are assigned and each job is performed.
  • 7. The network Representation of Example (continued...) 2 3 1 2 3 1 c 11 c 12 c 13 c 21 c 22 c 23 c 31 c 32 c 33 Agents Tasks
  • 8. Mathemetical Explanation
    • LP Formulation
    • Min ∑∑ c ij x ij
    • i j
    • s.t. ∑ x ij = 1 for each agent i
    • j
    • ∑ x ij = 1 for each task j
    • i
    • x ij = 0 or 1 for all i and j
  • 9. “ An Application of Genetic Algorithm Methods for Teacher Assignment Problems” The ARTICLE
  • 10. What is the Problem??
    • “ What are the most suitable teacher and course assignments ?”
    • Which teacher?
    • Which Course?
  • 11. 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
  • 12.
    • 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
  • 13.
    • 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
  • 14. The Questionnarie
  • 15. 20 points 19 points minlimit upperlimit
  • 16.  
  • 17.  
  • 18. The objection function for the problem will be :
  • 19.
    • Upper And Lower Limits for teacher work Hours
  • 20. The Lingo Formulation
  • 21. SETS : teachers / A B C D E F G H I J K L M N O P Q R S T /: upperlimit, minlimit; c ourses / C1A C2A .................... C45A C1B C2B .................... C45B /: hours; chromosomes ( teachers, courses ) : willingness, match; ENDSETS
  • 22. DATA: willingness = (The matrix taken from the 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 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 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; minlimit = 12 12 11 12 14 12 14 12 12 12 14 12 12 12 12 9 12 12 4 12; upperlimit = 13 13 12 18 15 18 15 18 18 18 15 18 18 18 18 15 13 13 11 13; ENDDATA
  • 23. Matrix of Willingness J=1 0 0 14 15 16 E 0 11 20 19 12 D 0 0 0 15 16 C 0 0 0 0 0 B 0 0 0 0 0 A C5A C4A C3A C2A C1A Courses Teachers
  • 24. OBJECTIVE FUNCTION
    • MAX= @SUM(chromozomes(i,j):
    • w illingness(i,j) *m atch(i,j));
  • 25. CONSTRAINTS
    • @FOR(chromozomes(i,j):
    • @BIN(match(i,j)));
    • @FOR(courses(j):
    • @SUM(chromozomes(i,j): match(i,j))=1);
  • 26.
    • @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
  • 27. Objective value
  • 28. REPORT -18 1 MATCH( A, C27B) -19 1 MATCH( A, C26B) -18 1 MATCH( A, C27A) -19 1 MATCH( A, C26A) Reduced Cost Value Variable
  • 29. The teacher A is going to teach :
    • C 26 A , B
    • C 2 7 A, B
    • courses.
  • 30. REDUCED COSTS
    • Negative reduced cost value
    • (-19) means;
    • T he objective value will increase 19 unit s .
  • 31. REPORT -17 1 MATCH( T, C38B) -16 1 MATCH( T, C34B) -20 1 MATCH( T, C7B) -16 1 MATCH( T, C34A) -20 1 MATCH( T, C7A) Reduced Cost Value Variable
  • 32. THANKS!