# Matematica | OPERATIONS RESEARCH

## Matematica OPERATIONS RESEARCH

 0512300034 DIPARTIMENTO DI MATEMATICA EQF6 MATHEMATICS 2021/2022

 OBBLIGATORIO YEAR OF COURSE 2 YEAR OF DIDACTIC SYSTEM 2018 AUTUMN SEMESTER
SSD CFU HOURS ACTIVITY TYPE OF ACTIVITY MAT/09 7 56 LESSONS SUPPLEMENTARY COMPULSORY SUBJECTS
 FRANCESCO CARRABS T RAFFAELE CERULLI
Objectives
KNOWLEDGE AND UNDERSTANDING
KNOWLEDGE ABOUT THE MAIN CONCEPT OF THE MATHEMATICAL FORMULATION OF AN OPTIMIZATION PROBLEM. FORMULATION OF THE OPTIMIZATION PROBLEMS BY USING THE LINEAR PROGRAMMING MODELS (WHEN POSSIBLE). KNOWLEDGE OF THE MAIN ALGORITHMS TO SOLVE CONTINUOUS LINEAR PROGRAMMING PROBLEMS. KNOWLEDGE OF THE BASIC CONCEPTS CONCERNING GRAPH THEORY. KNOWLEDGE OF THE CLASSICAL NETWORK OPTIMIZATION PROBLEMS AND THE ALGORITHMS USED TO SOLVE THEM.

APPLYING KNOWLEDGE AND UNDERSTANDING
ABILITY TO FORMULATE AN OPTIMIZATION PROBLEM WITH A LINEAR PROGRAMMING MODEL.
ABILITY TO SOLVE LINEAR PROGRAMMING PROBLEMS.
ABILITY TO USE EXCEL SOFTWARE TO SOLVE LINEAR PROGRAMMING PROBLEMS.
ABILITY TO FORMULATE NETWORK OPTIMIZATION PROBLEMS AND TO IDENTIFY THE MORE APPROPRIATE ALGORITHM TO SOLVE THEM.
Prerequisites
THE COURSE REQUIRES A KNOWLEDGE OF THE BASIC NOTIONS OF LINEAR ALGEBRA AND ANALYTICAL GEOMETRY AND THE CAPACITY OF SOLVING SYSTEMS OF LINEAR EQUATIONS AND PERFORMING OPERATIONS ON VECTORS AND MATRICES.
Contents
1. LINEAR PROGRAMMING (PL) (LECTURES: 12H; EXERCISE 4H)
- BASIC OPERATIONS ON MATRICES AND VECTORS, SYSTEM OF LINEAR EQUATIONS;
- FORMULATION OF REAL PROBLEMS;
- HYPERPLANES, HALF-SPACES, POLYHEDRA, EXTREME DIRECTIONS OF A POLYHEDRON, REPRESENTATION THEOREM.
- USE OF GEOGEBRA SOFTWARE TO DRAW THE FEASIBLE REGION AND SOLVE THE LINEAR PROGRAMMING PROBLEM WITH TWO VARIABLES.
2. THE SIMPLEX METHOD (LECTURES: 8H; EXERCISE 4H)
- EXTREME POINTS OF A POLYHEDRON AND BASIC FEASIBLE SOLUTIONS, OPTIMALITY AND UNBOUNDEDNESS CONDITIONS, SIMPLEX METHOD ALGEBRA, DEGENERATE BASIC FEASIBLE SOLUTION AND CYCLING PHENOMENON, SIMPLEX CONVERGENCE;
- STARTING BASIC FEASIBLE SOLUTION: TWO-PHASES METHOD AND BIG-M METHOD.
3. THE DUALITY THEORY (LECTURES: 8H; EXERCISE 4H)
- DUAL PROBLEM FORMULATION, THEOREM OF WEAK DUALITY, THEOREM OF STRONG DUALITY, COMPLEMENTARY SLACKNESS THEOREM, COMPUTATION OF THE DUAL OPTIMAL SOLUTION BY USING THE COMPLEMENTARY SLACKNESS CONDITIONS, PROPERTIES OBTAINED BY THE ORTHOGONALITY CONDITIONS OF THE COMPLEMENTARY SLACKNESS THEOREM, PRIMAL-DUAL RELATION;
- ECONOMIC INTERPRETATION OF DUAL VARIABLES;
- SENSITIVITY ANALYSIS: POST-OPTIMALITY ANALYSIS, OPTIMAL POINT VARIATION, OPTIMAL SOLUTION VALUE VARIATION BY CHANGING DATA;
- USE OF THE EXCEL SOFTWARE TO SOLVE LINEAR PROGRAMMING PROBLEMS.
4. NETWORK OPTIMIZATION PROBLEMS (LECTURES: 12H; EXERCISE 4H)
FORMULATIONS AND ALGORITHMS FOR THE FOLLOWING NETWORK OPTIMIZATION PROBLEMS:
- MINIMUM COST FLOW;
- TRANSPORTATION;
- MAX FLOW;
- SHORTEST PATH;
- MINIMUM SPANNING TREE.
TOTALLY UNIMODULAR MATRICES AND THEIR IMPACT ON THE COMPLEXITY OF SOME OPTIMIZATION PROBLEMS ON GRAPHS.
Teaching Methods
THE COURSE IS ORGANIZED IN 56 HOURS OF FRONTAL LESSONS (7 CFU), USING PROJECTED SLIDES. AT THE END OF EACH TOPIC, SOME EXAMPLES AND CLASSROOM EXERCISES ARE PRESENTED. DURING THE CLASSROOM EXERCISES, THE STUDENTS FACE SOME EXERCISES TO SOLVE BY USING THE TECHNIQUES PRESENTED IN THE THEORETICAL LECTURES. THE RESOLUTION OF THE EXERCISES, WHICH IS CARRIED OUT UNDER THE SUPERVISION OF THE TEACHER, SEEKS TO DEVELOP AND STRENGTHEN THE STUDENT’S CAPACITY OF IDENTIFYING THE MOST APPROPRIATE TECHNIQUES TO SOLVE THEM. METHODS TO PRODUCE A CLEAR AND ACCURATE PRESENTATION OF THE ACHIEVED RESULTS ARE ALSO PROPOSED.
Verification of learning
THERE ARE NOT MIDTERM EXAMINATIONS FOR THIS COURSE.
THE FINAL EXAM IS DESIGNED TO EVALUATE AS A WHOLE: THE KNOWLEDGE AND UNDERSTANDING OF THE CONCEPTS PRESENTED IN THE COURSE, AS WELL AS THE ABILITY TO APPLY SUCH KNOWLEDGE FOR THE RESOLUTION OF LINEAR PROGRAMMING PROBLEMS.
THE EXAM CONSISTS OF A WRITTEN TEST AND AN ORAL INTERVIEW.
- THE WRITTEN TEST IS DESIGNED TO ASSESS THE ABILITY TO SOLVE OPTIMIZATION PROBLEMS AND NORMALLY LASTS 120 MINUTES. IT CONSISTS OF 4 OR 5 EXERCISES, AND POSSIBLY OPEN-ENDED QUESTIONS, WHICH ARE GIVEN A SCORE. THE SUM OF THESE SCORES IS 30. TYPICAL TOPICS OF THE EXERCISES CONCERN: THE GRAPHICAL RESOLUTION OF PL PROBLEMS AND THE COMPUTATION OF THE EXTREME DIRECTIONS OF THE POLYHEDRON, THE FORMULATION OF OPTIMIZATION PROBLEMS, THE RESOLUTION OF A PL PROBLEM BY USING THE SIMPLEX METHOD, THE CONSTRUCTION OF THE DUAL PROBLEM, THE SENSITIVITY ANALYSIS AND THE RESOLUTION OF GRAPH PROBLEMS PRESENTED IN THE COURSE. THE SCORE OF THE WRITTEN EXAM IS EQUAL TO THE SUM OF THE SCORES ASSIGNED BY THE TEACHER TO THE EXERCISES SOLVED BY THE STUDENT. IS ADMITTED TO THE ORAL EXAMINATION THE STUDENT THAT GAINS A SCORE OF AT LEAST 18/30.
- WITH THE ORAL INTERVIEW, ARE EVALUATED THE KNOWLEDGE ABOUT THE MODELLING AND SOLVING OF LINEAR PROGRAMMING PROBLEMS. THE INTERVIEW INCLUDES THE PRELIMINARY DISCUSSION OF THE WRITTEN TEST AND VARIOUS QUESTIONS REGARDING THE TOPICS OF THE COURSE PROGRAM. THE MINIMUM ASSESSMENT LEVEL (18) IS AWARDED WHEN THE STUDENT SHOWS A FRAGMENTARY KNOWLEDGE OF THEORETICAL CONTENTS AND A LIMITED ABILITY TO FORMULATE OPTIMIZATION PROBLEMS AND TO APPLY ALGORITHMS TO SOLVE THEM. THE MAXIMUM ASSESSMENT LEVEL (30) IS ATTRIBUTED WHEN THE STUDENT SHOWS A COMPLETE AND IN-DEPTH KNOWLEDGE OF THE COURSE TOPICS AND A REMARKABLE ABILITY TO IDENTIFY THE MOST APPROPRIATE METHODS TO SOLVE THE OPTIMIZATION PROBLEMS FACED.
THE FINAL GRADE IS DEFINED BY THE TEACHER ACCORDING TO THE RESULTS OF THE TWO TESTS. IN ANY CASE, THE FINAL GRADE CANNOT EXCEED THE WRITTEN TEST GRADE BY MORE THAN 6 POINTS.

IN THE EVENT THAT THE EXAM SHOULD TAKE PLACE REMOTELY DUE TO THE COVID-19 EMERGENCY, THE TEACHER CAN DECIDE (BY PROMPTLY INFORMING THE STUDENTS), TO ABOLISH THE WRITTEN TEST. IN THIS CASE, DURING THE ORAL EXAM STUDENTS WILL ALSO BE ASKED TO SOLVE ONE OR MORE EXERCISES OF THE TYPE PROPOSED IN THE WRITTEN TEST.
Texts
- M.S. BAZARAA, J.J. JARVIS & H.D. SHERALI, LINEAR PROGRAMMING AND NETWORK FLOWS, FOURTH EDITION, JOHN WILEY, 2010.
- SLIDES AVAILABLE HERE: HTTPS://DOCENTI.UNISA.IT/020511/RISORSE
OTHER RESOURCES:
HILLIER FREDERICK S., RICERCA OPERATIVA, MCGRAW-HILL EDUCATION, 2010.
More Information
- THE COURSE LANGUAGE IS ITALIAN.
- THE ATTENDANCE OF THE LESSONS IS HIGHLY RECOMMENDED.
- OTHER SUPPORTING MATERIAL IS AVAILABLE ON THE WEBPAGE: HTTPS://DOCENTI.UNISA.IT/020511/RISORSE
- OFFICE HOURS FOR STUDENTS ARE AVAILABLE ON THE WEBPAGE: HTTPS://DOCENTI.UNISA.IT/020511/HOME
- EMAIL PROF. CERULLI: raffaele@unisa.it
- EMAIL PROF. CARRABS: fcarrabs@unisa.it
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