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Assingment Problem3
 

Assingment Problem3

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    Assingment Problem3 Assingment Problem3 Presentation Transcript

    • Assignment Problems Hazırlayanlar: Ali Evren Erdin Arzu Çalık Hilal Demirhan
    • 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
    • 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?
      • M inimize the total time to complet e set of tasks
      • M aximize skill ratings
      • M inimize the cost of the assignments
      • Or Etc.
    • 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
    • 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.
    • 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
    • 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
    • “ 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
    • 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
    • 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
    • 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
    • OBJECTIVE FUNCTION
      • MAX= @SUM(chromozomes(i,j):
      • w illingness(i,j) *m atch(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 -18 1 MATCH( A, C27B) -19 1 MATCH( A, C26B) -18 1 MATCH( A, C27A) -19 1 MATCH( A, C26A) Reduced Cost Value Variable
    • The teacher A is going to teach :
      • C 26 A , B
      • C 2 7 A, B
      • courses.
    • REDUCED COSTS
      • Negative reduced cost value
      • (-19) means;
      • T he objective value will increase 19 unit s .
    • 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
    • THANKS!