The document discusses assignment problems and their applications. It provides examples of assignment problems involving assigning workers to jobs or tasks. The goal is typically to minimize costs or maximize satisfaction. The document then summarizes a research article that uses a genetic algorithm to solve the specific problem of optimally assigning teachers to courses while considering various constraints.

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

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!