Assingment Problem3

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Assingment Problem3

  1. 1. Assignment Problems Hazırlayanlar: Ali Evren Erdin Arzu Çalık Hilal Demirhan
  2. 2. INDEX <ul><li>Introduction </li></ul><ul><li>Description Of The Assignment Problems </li></ul><ul><li>Uses of The Assignment Problems </li></ul><ul><li>Simple Examples </li></ul><ul><li>The Article </li></ul><ul><li>Explanation of the Article </li></ul><ul><li>The Solution of the Problem in Lingo </li></ul>
  3. 3. Description of the Assignment Problems <ul><li>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 </li></ul>
  4. 4. What can be the objectives? <ul><li>M inimize the total time to complet e set of tasks </li></ul><ul><li>M aximize skill ratings </li></ul><ul><li>M inimize the cost of the assignments </li></ul><ul><li>Or Etc. </li></ul>
  5. 5. What are the Applications of Assignment Problems? <ul><li>A ssigning e mployees to tasks </li></ul><ul><li>Assigning machines to production jobs </li></ul><ul><li>A ssign fleets of aircrafts to particular t rips </li></ul><ul><li>A ssigning school buses to routes </li></ul><ul><li>N etworking computers </li></ul>
  6. 6. A Simple Example... <ul><li>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 . </li></ul><ul><li>It assumes all workers are assigned and each job is performed. </li></ul>
  7. 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. 8. Mathemetical Explanation <ul><li>LP Formulation </li></ul><ul><li> </li></ul><ul><li> Min ∑∑ c ij x ij </li></ul><ul><li>i j </li></ul><ul><li>s.t. ∑ x ij = 1 for each agent i </li></ul><ul><li>j </li></ul><ul><li>∑ x ij = 1 for each task j </li></ul><ul><li>i </li></ul><ul><li> x ij = 0 or 1 for all i and j </li></ul>
  9. 9. “ An Application of Genetic Algorithm Methods for Teacher Assignment Problems” The ARTICLE
  10. 10. What is the Problem?? <ul><li>“ What are the most suitable teacher and course assignments ?” </li></ul><ul><li>Which teacher? </li></ul><ul><li>Which Course? </li></ul>
  11. 11. What is Genetic Algorithm? <ul><li>The Genetic Algorithm is optimization procedure based on the natural law of evolution! </li></ul><ul><li>The Key Idea of Genetic Algorithm is Survival of the Fittest! </li></ul><ul><li>It is an Heuristic Approach based on Darwin’s Theory of Evolution </li></ul>
  12. 12. <ul><li>Teacher Assignment Problem include multiple constraints </li></ul><ul><li>Teachers willingness need to be considered, </li></ul><ul><li>There should be a fair distribution of over time </li></ul><ul><li>Teacher satisfaction has to be maximized </li></ul>
  13. 13. <ul><li>One course should not be appointed to different teachers. </li></ul><ul><li>There are 20 teachers. </li></ul><ul><li>There are 45 courses. Each course has two classes: A and B. </li></ul><ul><li>Each teacher have an upper and minimum workhour limits </li></ul><ul><li>Each Teacher rank the courses that they want to teach </li></ul>The Datas for the Problem
  14. 14. The Questionnarie
  15. 15. 20 points 19 points minlimit upperlimit
  16. 18. The objection function for the problem will be :
  17. 19. <ul><li>Upper And Lower Limits for teacher work Hours </li></ul>
  18. 20. The Lingo Formulation
  19. 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
  20. 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
  21. 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
  22. 24. OBJECTIVE FUNCTION <ul><li>MAX= @SUM(chromozomes(i,j): </li></ul><ul><li> w illingness(i,j) *m atch(i,j)); </li></ul><ul><ul><li> </li></ul></ul>
  23. 25. CONSTRAINTS <ul><li>@FOR(chromozomes(i,j): </li></ul><ul><li> @BIN(match(i,j))); </li></ul><ul><li>@FOR(courses(j): </li></ul><ul><li>@SUM(chromozomes(i,j): match(i,j))=1); </li></ul>
  24. 26. <ul><li>@FOR(teachers(i): </li></ul><ul><li>@SUM(courses(j):match(i,j)* </li></ul><ul><li>hours(j))<=upperlimit(i)); </li></ul><ul><li>@FOR(teachers(i): </li></ul><ul><li>@SUM(courses(j):match(i,j)* </li></ul><ul><li>hours(j))>=minlimit(i)); </li></ul>CONSTRAINTS
  25. 27. Objective value
  26. 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
  27. 29. The teacher A is going to teach : <ul><li>C 26 A , B </li></ul><ul><li>C 2 7 A, B </li></ul><ul><li> courses. </li></ul>
  28. 30. REDUCED COSTS <ul><li>Negative reduced cost value </li></ul><ul><li>(-19) means; </li></ul><ul><li>T he objective value will increase 19 unit s . </li></ul>
  29. 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
  30. 32. THANKS!

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