Statistical Methods and Applications

PAOLO DELL'OLMO

General
Managers worldwide, beyond their personal experience, rely more and more on the use of
quantitative decision models which allow to take advantage of today’s data availability. Morover,
new computational tools, including algorithms, cloud computing and distributed processing, make
it possible to both develop and compute analytical models in a very short time, meeting the
requirement of practical applications and often using real time data. Data Driven Decision Making
is the new paradigm for managers to make better, evidence based, more rational, transparent and
reliable decisions.
In this context, the primary educational objective of the course is students' learning of the main
decision problems that arise in real world and the quantitative methods to model them and to
feed them with adequate data. Students must also be able to correctly use, for decision-making
and management purposes, computer tools to analyze data generated by real problems in
different contexts (e.g. service management, marketing, transportation, operations management
and production, and finance) through the analysis of several case studies.

Specific objectives

a) Knowledge and ability to understand
After attending the course the students know and classify the main decision problems arising in
real world organization and the main analytical methods (decision and optimization models and
algorithms) to be used to support a Manager during his/her decision process.

b) Ability to apply knowledge and understanding
At the end of the course the students are able to formalize real problems in terms of decision
problems and to apply the specific methods taught in the course to solve them. They are also able
to classify the type of problem to it the most appropriate quantitative method, experimenting the
effectiveness for decisional purposes also on real problems.

c) Autonomy of judgment
Students develop critical skills through the application of modeling, decision analysis and multi
objective optimization methodologies to a broad set of practical problems. They also develop the
critical sense through the comparison between alternative solutions to the same problem
obtained using methods of analysis and realistic scenarios different from each other. They learn to
critically interpret the results obtained by applying the procedures to real data sets.

d) Communication skills
Students, through the study and the carrying out of practical exercises, acquire the technical-
scientific language of the course, which must be properly used both in the intermediate and final
written tests and in the oral tests. Communication skills are also developed through group
activities.

e) Learning ability
Students who pass the exam have learned methods of decision analysis and multiobjective
optimization that allow them to face, decision-making problems and optimization on complex
organizations.

Ability to classify decision problems and identify the adequate models and tools to face them.
Ability to investigate data sets to understand their quality with respect to the decision making problem
Ability to Model Decision Maker Preferences
Knowledge of Multi attribute decision making methods
Capability of mathematical programming with multiple criteria
Knowledge of applications like Recommendation System, GoogleAds, Kidney allocations and others

Knowledge of the main elements of analysis, geometry, design and analysis of algorithms

Programma
Part I Introduction
Role of Data and Models in Decision Making Process
Platform and Network architectures for DDDM
Analytics Tools for DM
Choice and Optimization Models
Part II Decision Models
Combinatorial Representation of Preferences
Ordinal Value Functions
Multi-Attribute Value Theory and Machine Learning Models
Multicriteria Decision Making Methods
Part III Multi-Objective Optimization Models
Linear Programming with Multiple Criteria
Goal Programming
Multi-Objective Combinatorial Optimization
Data Sensitivity Analysis in the Objective Space
Part IV Multiple Decisor Makers and Agent Based Decision Models
Combinatorial Models for Collective Choice
Aggregation of Preferences
Metric Approach to Collective Choice
Game Theory Models

Part V Case Studies
Finance, Sports and Games, Transportation, Healthcare, Management Operations, Crime, Internet, Netflix, ecc.

1. D. Bertsimas, and R. Freund. Data, Models, and Decisions: The Fundamentals of Management Science. Dynamic Ideas, Wiley, 2004. ISBN: 9780975914601.
2. M. Ehrgott, Multicriteria Optimization, Springer, 2005.
3. A. Ishizaka, P. Nemery, Multi-criteria Decision Analysis: Methods and Software, ISBN: 978-1-119-97407-9, WIley, 2013.
4. Software manuals available on line and on the e-learning platform

Lesson with exercises

Attendance to the course is strongly recommended

The exam consists of a written test with some exercises inspired by those carried out during the course and some open-ended questions on the topics covered in class.

Exercise 1 (points 6/30)
Given the following bi-objective linear programming problem:
Max (-x1+5/3x2, -x2)
2x1 - x2 ≤ 6
2x1 ≤ 6
2x2 ≤ 6

x1, x2 ≥ 0
x1, x2 ∈ R
Questions
1. Represent the feasible region X of the problem in the decision space.
2. Determine the inverse transformation φ-1.
3. Represent the feasible region Y of the problem in the objective space.
4. Determine the optimal solution of the objectives taken individually and calculate the ideal point.
5. Check if the ideal point belongs to Y
6. Identify the Pareto Frontier.

Exercise 2 (points 6/30)
A company is planning the manufacturing of three products A, B, C, with respect to three criteria (Profit, Employment, Capital). Profit should be greater or equal to 125, Employment should be equal to 40, Capital should be less or equal to 40. Deviations from these targets are penalized according to the following weights:
Criteria Weight
Profit 5
Employment 2(+), 4 (-)
Capital 3

Formulate this problem as a Goal Programming model.

Exercise 3 (points 6/30)
Write and comment the mathematical formulation of the Median Order Problem related to the aggregation of a system of preferences represented by a set of partial orders.
Exercise 4 (points 6/30)
The Program manager of a multinational must provide a ranking of a set of projects (A, B, C, D, E, F, G) for which preferences have been acquired based on technical aspects. You are called by the company's marketing director as expert analysts to produce a support tool to build the rankings.
You are given the set of preferences and indifferences on (A, B, C, D, E, F, G) reported below:
• B is preferred to A, D and C, but is almost equivalent to E, F, G
• G is preferred to F, E, D, C, A
• F is preferred to E, D, C, A
• E is preferred to D and C, but is almost equivalent to A and B
• D is preferred to C, and both are almost equivalent to A
Questions:
1. What type of relationship models the locations considered (pairwise) equivalent?
2. What type of relationship describes preferences between projects?
3. Can you find an ordinal value function to construct the ranking of alternatives? What function(s) are needed to analytically model the preference relationship? Write the function(s)
Exercise 5 (points 6/30)

Four judges (1, 2, 3, 4) assign individually a total order to four alternatives (A, B, C, D). Find the order of the Group applying the Condorcet method, and the Borda count. Then, formulate and comment the problem of finding a total order at minimum total geometric distance from total orders of the three judges.
Alternative Judge 1 Judge 2 Judge 3 Judge 4
A A D B A
B B B A C
C C C C D
D D A D B


QUESTIONS

Question 1 (up to points 15/30)
Describe the basic steps of the Promethee method
Question 2 (up to points 15/30)
Describe the main aspects of the UTA+ method. Write the LP formulation adopted by UTA+?
What is the meaning of the variables?
Way it is used the Kendall coefficient? Explain.