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OPERATIONS RESEARCH

Francesco CARRABS OPERATIONS RESEARCH

0512300034
DEPARTMENT OF MATHEMATICS
EQF6
MATHEMATICS
2024/2025

OBBLIGATORIO
YEAR OF COURSE 2
YEAR OF DIDACTIC SYSTEM 2018
AUTUMN SEMESTER
CFUHOURSACTIVITY
756LESSONS
FRANCESCO CARRABS T Curriculum
RAFFAELE CERULLI Curriculum
ExamDate
RICERCA OPERATIVA14/01/2025 - 09:00
RICERCA OPERATIVA14/01/2025 - 09:00
RICERCA OPERATIVA05/02/2025 - 09:00
RICERCA OPERATIVA05/02/2025 - 09:00
RICERCA OPERATIVA20/02/2025 - 09:00
RICERCA OPERATIVA20/02/2025 - 09:00
Objectives
OPERATIONS RESEARCH IS A SECTOR OF MATHEMATICS THAT DEALS WITH QUANTITATIVELY MODELING AND WITH THE RESOLUTION OF DECISION-MAKING PROBLEMS THAT ARISE IN VARIOUS AREAS OF THE REAL WORLD SUCH AS: ECONOMICS, FINANCE, PRODUCTION PLANS, TRANSPORT LOGISTICS AND HEALTHCARE. THE OPERATIONS RESEARCH COURSE AIMS TO PROVIDE THE KNOWLEDGE NECESSARY FOR THE RESOLUTION OF DECISION-MAKING PROBLEMS THROUGH THE FORMULATION OF CONTINUOUS LINEAR PROGRAMMING MODELS (PL) AND THEIR RELATIVE RESOLUTION ALGORITHMS. FURTHER ALGORITHMS ARE PRESENTED FOR GRAPH OPTIMIZATION PROBLEMS.

KNOWLEDGE AND UNDERSTANDING
AT THE END OF THE COURSE, THE STUDENT WILL KNOW:
- THE TOOLS FOR THE FORMULATION OF REAL PROBLEMS THROUGH THE USE OF LP MATHEMATICAL MODELS;
- THE GRAPHICAL METHOD FOR SOLVING TWO-VARIABLE LP PROBLEMS;
- THE THEORETICAL ASPECTS BEHIND THE FUNCTIONING AND THE CORRECTNESS OF THE SIMPLEX METHOD;
- HOW THE SIMPLEX METHOD WORKS TO SOLVE LP PROBLEMS;
- THE DUALITY THEORY OF THE LP PROBLEMS;
- THE SENSITIVITY ANALYSIS OF THE OPTIMAL SOLUTIONS;
- THE MATHEMATICAL MODELS AND THE SOLVING ALGORITHMS FOR SOME CLASSICAL LP PROBLEMS DEFINED ON GRAPHS.

APPLYING KNOWLEDGE AND UNDERSTANDING
AT THE END OF THE COURSE, THE STUDENT WILL BE ABLE TO:
- FORMULATE (WHEN POSSIBLE) OPTIMIZATION PROBLEMS THROUGH LP MATHEMATICAL MODELS;
- GRAPHICALLY SOLVE LP PROBLEMS IN TWO VARIABLES;
- APPLY THE SIMPLEX METHOD TO SOLVE LP PROBLEMS;
- BUILD THE DUAL OF AN LP PROBLEM AND, THROUGH ITS OPTIMAL SOLUTION, COMPUTE THE SHADOW PRICES ASSOCIATED WITH THE PRIMAL CONSTRAINTS;
- CARRY OUT THE SENSITIVITY ANALYSIS ON THE OPTIMAL SOLUTIONS;
- USE THE EXCEL SOFTWARE TO COMPUTE THE OPTIMAL SOLUTION OF THE LP PROBLEMS AND TO CARRY OUT THE SENSITIVITY ANALYSIS ON THIS SOLUTION;
- MODEL LP PROBLEMS ON GRAPHS AND SOLVE THEM USING THE ALGORITHMS PRESENTED IN THE COURSE.

MAKING JUDGMENTS
AT THE END OF THE COURSE, THE STUDENT WILL BE ABLE TO:
- DEVELOP AND ADAPT THE MODELS, PRESENTED DURING THE COURSE, TO SPECIFIC PROBLEMS;
- ANALYSE AND CORRECTLY UNDERSTAND THE MEANING OF THE RESULTS OBTAINED FROM SOLVING A PROBLEM.

COMMUNICATION SKILLS
AT THE END OF THE COURSE, THE STUDENT WILL BE ABLE TO:
-DESCRIBE, CLEARLY AND CONCISELY, THE MATHEMATICAL MODELS THAT HE DEFINED TO FORMULATE REAL PROBLEMS;
- EXPLAIN HOW THE ALGORITHMS USED TO SOLVE THE OPTIMIZATION PROBLEM WORK;
- DESCRIBE THE RESULTS OBTAINED FROM SOLVING THE PROBLEM;
- DISCUSS ISSUES RELATING TO THE RESOLUTION OF OPTIMIZATION PROBLEMS WITH OTHER INTERLOCUTORS.

LEARNING SKILLS
AT THE END OF THE COURSE, THE STUDENT WILL BE ABLE TO:
- APPLY THE KNOWLEDGE ACQUIRED TO CONTEXTS DIFFERENT FROM THOSE PRESENTED DURING THE COURSE;
- DELVE DEEPER INTO THE TOPICS COVERED USING TEACHING MATERIAL DIFFERENT FROM THOSE USED DURING THE COURSE;
- LEARN, EVEN INDEPENDENTLY, FURTHER KNOWLEDGE OF APPLIED MATHEMATICS PROBLEMS.
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. PL PROBLEMS IN STANDARD AND CANONICAL FORM. TRANSFORMATION RULES BETWEEN STANDARD AND CANONICAL FORM.
- GRAPHICAL RESOLUTION OF PL PROBLEMS. DEFINITION OF HYPERPLANES, HALF-SPACES, POLYHEDRAL, CONVEX FUNCTION AND CONVEX SET. CORRESPONDENCE BETWEEN THE LOCAL AND GLOBAL OPTIMUM FOR THE PL PROBLEMS (THEOREM AND PROOF).
- FINDING THE EXTREME DIRECTIONS OF A POLYHEDRON, REPRESENTATION THEOREM. RESOLUTION OF PL PROBLEMS BY REPRESENTATION THEOREM.

2. THE SIMPLEX METHOD (LECTURES: 8H; EXERCISE 4H)
- EXTREME POINTS OF A POLYHEDRON AND BASIC FEASIBLE SOLUTIONS. CORRESPONDENCE BETWEEN BASIC FEASIBLE SOLUTIONS AND EXTREME POINTS (THEOREM AND PROOF). 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.

THE ORAL INTERVIEW IS USUALLY SCHEDULED WITHIN ONE WEEK OF THE WRITTEN EXAM AND IT IS COMMUNICATED TOGETHER WITH THE PUBLICATION OF THE RESULTS OF THE WRITTEN TEST ON THE TEACHER'S WEBSITE.
TO PROVIDE MORE TIME FOR THE PREPARATION OF THE ORAL INTERVIEW, IT IS ALLOWED FOR THE STUDENT TO TAKE THE ORAL INTERVIEW IN ANY CALL OF THE SAME EXAM SESSION. THE STUDENT WHO WANTS TO TAKE ADVANTAGE OF THIS POSSIBILITY MUST NOTIFY THE TEACHER BY EMAIL IMMEDIATELY AFTER THE PUBLICATION OF THE RESULTS OF 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
- EMAIL: FCARRABS@UNISA.IT, RAFFAELE@UNISA.IT
- 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/001227/RISORSE
- OFFICE HOURS FOR STUDENTS ARE AVAILABLE ON THE WEBPAGE: HTTPS://DOCENTI.UNISA.IT/001227/HOME
Lessons Timetable

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