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_archived/operations_research/lesson_01.md

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-# Operations Research
-#### Enrico Malucelli
-##### 6 October 2015
-## Introduction
-### What is Operations Research?
-Is a method of taking decisions using math
-
-Typically we solve optimization problems daily withoud being conscious
-
-        Confucius
-
-        I hear and forget,
-        I see and remember,
-        I do and understand.
-####Interesting sites:
-
-
-### What kind of decisions?
-- __Strategic__ Level: design a railroad
-- __Tactical__ level: how many trains to use
-- __Operational__ level: schedule the trains
-
-### How OR approaches the problems
-- __Decisions__
-- __Rules__ Mathematical constraints
-- __Objective__ Maximize the income
-#### Solution to phone building example:
-- plot the constraints in a 2D graph
-- select the region below all the constraints
-- Find the optimal solution analyzing the vertexes of the polygon.
-
-(In this case the vertex were integers but they may be real numbers and the solution is required to be integer.)
-### Challenges
-http://challenge.roadef.org/2016/en/index.php

_archived/operations_research/lesson_02.md

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-# Operations Research
-#### Enrico Malucelli
-##### 6 October 2015
-## Introduction
-### Exercise: How to invest a budget
-
-###Steps
-#### 1. decisions
-#### 2. constraints
-
-####Solution
-#####Variables
-$x_i$= amount of money invested
-(i=1...5), $x_i \ge 0$
-#####constraints
-$\sum x_i \le 137$
-
-$x_1 \le 40$
-
-$x_2 \le 12$
-
-$x_3 \le 130$
-
-$x_4 \le 5$
-
-$x_5 \le 400$
-#####target
-$x_1/40*3.2+x_2/12*1.5+x_3/130*4.2+x_4/5*0.7+x_5/400*17$
-
-####Alternative solution
-$y_1$=share of bond
-$y_1=x_1/availability$
-$0 \le y_1\le1$
-$i=1$
-
-$$40y_1+12y_2+130y_3+5y_4+400y_5\le137$$
-$$max32y_1+15y_2+4.2y_3+0.7y_4+17y_5$$
-
-We can divide each bonus for the total of the earning so we can get an index of the best investment.
-$$8\%y_1+125\%y_2+3.2\%y_3+14\%y_4+4.25\%y_5$$
-
-We notice that the $y_2$ is the best investment, and we should buy as much as we can of that kind.
-
-example:
-$y_4=1$
-$y_2=1$
-$y_1=1$
-$y_5=...$
-####Algorithm generalization
-$$\sum c_i y_i$$
-$$\sum a_i y_i \le b$$
-$$0\le y_i \le 1 i=1 \mu$$
-Sort indicates by non decreasing $C_i/a_i$ cosider the variables in the Order:
-$$y_i=\begin{cases} 1,& if &b \ge a_i\\
-b/a_i,& if &b < a_i
-\end{cases}$$
-$$b-y_ia_i$$
-
-###Logical variables
-Can assume only the values $\{0,1\}$
-####example:
-A burglar sneaks into a jewelry and his bag can hold 8kg, what must he steal?
-
-|| object | 1 | 2 | 3 | 4 | 5 | 6 |
-|-------|--------|---|---|---|---|---|---|
-| value | $v_i$  | 7 | 2 | 4 | 5 | 4 | 1 |
-|weight|$w_i$|5|3|2|3|1|1|
-####solution
-Contrary to the previous problem (in which we had infinite solutions, like any fraction), in this problem we have at most $2^6$ solutions, maybe not all of them are feasible.
-
-x_i = 1 if the burglar has taken, 0 otherwise
-
-$$max \sum v_i x_i$$
-$$\sum w_i x_i \le b(=8)$$
-
-If we apply the same algorithm as before, we can calculate the value_for_weight factor of every object and choose the best ones.
-#####However this is not the best solution
-####Logical variable, __Complement__
-$$x'_i=1-x_i$$
-####Other types of variables
-####Integer Variables
-####Discrete value variables
-In most cases we select the value of a variable within a __finite set__
-e.g.: the memory size of a computer, the capacity of a telecommunication link.
-$$x\in \{ v_1,v_2,...,v_r\}$$
-#####formulas
-$$s_i=\begin{cases}
-1,& x=v_i\\
-0,& \end{cases}$$
-###example: diet
-|food|pasta|rice|steak|carrots|potatoes|pear|
-|----|-----|----|-----|-------|--------|----|
-|calories per hg|300|250|200|70|180|100
-####Minimum required calories: __700__
-We can transform all the inequalities to be less or equal
-
-And later we need to transform our inequalities into equalities

_archived/soft_computing/lesson_01.md

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-# Soft Computing
-#### Andrea Bonarini
-##### 5 October 2015
-
-#### What is Soft Computing?
-
-The term was introduced by Lotfi Zadeh, the father of fuzzy sets,
-to denote programming techniques not related to *traditional* programming languages:
-- Fuzzy systems
-- Neural networks
-- Stochastic systems
-
-##### definition
-A set of techniques to model systems (input-output mapping) by approximating them.
-
-##### Modelling technique: generalization
-Considers a small sample of the entity to make an approximate model,
-
-
-#### Model
-
-- Is a *representation* of some entity, defined for a *specific purpose*.
-
-- Is limited to aspects of the entity represented which are *relevant for the purpose*
-
-- A model is different to the represented entity ex: map vs land
-
-- Models lead to *approximation, uncertainty, imprecision*.
-
-##### Approximation
-The model features are similar to the real ones but not the same.
-##### Uncertainty
-We are not sure that the features of the model are the same of the entity.
-##### Imprecision
-The model features values are not precise.
-
-#### Techniques
-
-- ##### Fuzzy sets:
-    Correct model in a finite number of points, smooth transistion (approximation) among them.
-
-    ex: in a thermal control system we fix the normal working point and the critical one, and the system approximates the behaviour between.
-
-- ##### Neural networks
-    input-output samples, learning algorithms to define output values for unknown values.
-
-- ##### Genetic algorithms
-    Optimal solution, obtained by evaluating populations of tentative solutions and combining their parts (sort of copying from nature).
-
-#### Applications:
-
-From washing machines to helicopters, to rice cookers
-
-### Fuzzy Sets
-
-Were created in 1965 by Lotfi Zadeh
-
-The principle is to make computation with words.
-
-#### What is a fuzzy set?
-Is a set whose membership function can range on the interval [0,1].
-
-On contrary to __crisp sets__ that admit only {0,1}
-
-###### A membership function defines a set
-###### Defines the degree of membership of an element to the set
-μ: U -> [0,1]
-
-#### How to define membership functions
-- select a variable
-    - distance
-- define a range of the variable
-    - [0..10]
-- identify labels
-    - close, medium, far
-- for each label identify the charachteristic points
-    - 0, max, middle values, where MF=1, ...
-- identify function shapes
-    - linear
-- check
-
-#####Membership functions can also overlap,
-and this quality is useful for example in classification of a noisy input,
-in fact with overlapping MF we have a *smooth transition* from a label to another
-
-#### MF and concepts
-
-__MFs__ define __fuzzy sets__
-
-__Labels__ denote __fuzzy sets__
-
-__Fuzzy sets__ can be considered as conceptual representations
-
-#### Symbol grounding
-Reason in terms of concepts and grounds them to reality.

_archived/soft_computing/lesson_02.md

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-# Soft Computing
-#### Andrea Bonarini
-##### 8 October 2015
-
-#### Membership Function and concepts
-
-#### Propertise of a frame of cognition
-each fuzzy set is called a __granule__
-- __Coverage__:each value belong at least to a value
-
-### Robustness
-
-### $\alpha$-cuts
-
-### Support of a fuzzy set
-
-### Height of a fuzzy set
-
-### Strange MF
-
-####Singleton
-A fuzzy set with one member
-
-### Standard operators of fuzzy sets
-
-__Complement__
-
-$\mu_f(x)=1-\mu_f(x)$
-
-__Union__
-
-The maximum of two fuzzy sets
-
-__Intersection__