# Quick Start¶

This tutorial show how to create a scikit-criteria `Data`

structure,
and how to feed them inside different multicriteria decisions
algorithms.

## Conceptual Overview¶

The multicriteria data are really complex thing; mostly because you need
at least 2 totally disconected vectors to decribe your problem: A
alternative matrix (`mtx`

) and a vector that indicated the optimal
sense of every criteria (`criteria`

); also maybe you want to add
weights to your criteria

The `skcteria.Data`

object need at least the first two to be created
and also accepts the weights, the names of the criteria and the names of
alternatives as optional parametes.

## Your First Data object¶

First we need to import the `Data`

structure and the `MIN`

, `MAX`

contants from scikit-criteria:

```
In [2]:
```

```
from skcriteria import Data, MIN, MAX
```

Then we need to create the `mtx`

and `criteria`

vectors.

The `mtx`

must be a **2D array-like** where every column is a
criteria, and every row is an alternative

```
In [3]:
```

```
# 2 alternatives by 3 criteria
mtx = [
[1, 2, 3], # alternative 1
[4, 5, 6], # alternative 1
]
mtx
```

```
Out[3]:
```

```
[[1, 2, 3], [4, 5, 6]]
```

The `criteria`

vector must be a **1D array-like** whit the same number
of elements than columns has the alternative mattrix (`mtx`

) where
every component represent the optimal sense of every criteria.

```
In [4]:
```

```
# let's says the first two alternatives are
# for maximization and the last one for minimization
criteria = [MAX, MAX, MIN]
criteria
```

```
Out[4]:
```

```
[1, 1, -1]
```

as you see the `MAX`

and `MIN`

constants are only aliases for the
numbers `-1`

(minimization) and `1`

(maximization). As you can see
the contantes usage makes the code more readable.

Now we can combine this two vectors in our scikit-criteria data.

```
In [5]:
```

```
data = Data(mtx, criteria)
data
```

```
Out[5]:
```

ALT./CRIT. | C0 (max) | C1 (max) | C2 (min) |
---|---|---|---|

A0 | 1 | 2 | 3 |

A1 | 4 | 5 | 6 |

As you can see the otput of the `Data`

structure is much more friendly
as the plain python lists.

To change the generic names of the alternatives (A0 and A1) and the
criteria (C0, C1 and C2); let’s asume that our Data is about cars (*car
0* and *car 1*) and their characteristics of evaluation are *autonomy*
(`MAX`

), *confort* (`MAX`

) and *price* (`MIN`

).

To feed this information to our `Data`

structure we have the params:
`anames`

that accept the names of alternatives (must be the same
number as row the `mtx`

has), and `cnames`

the criteria names (with
the same number of elements as columns has the `mtx`

)

```
In [6]:
```

```
data = Data(mtx, criteria,
anames=["car 0", "car 1"],
cnames=["autonomy", "confort", "price"])
data
```

```
Out[6]:
```

ALT./CRIT. | autonomy (max) | confort (max) | price (min) |
---|---|---|---|

car 0 | 1 | 2 | 3 |

car 1 | 4 | 5 | 6 |

In our final step let’s asume we know in our case, that the importance
of the autonomy is the *50%*, the confort only a *5%* and the price is
*45%*. The param to feed this to the structure is called `weights`

and
must be a vector with the same elemts as criterias has your alternative
matrix (number of columns)

```
In [7]:
```

```
data = Data(mtx, criteria,
weights=[.5, .05, .45],
anames=["car 0", "car 1"],
cnames=["autonomy", "confort", "price"])
data
```

```
Out[7]:
```

ALT./CRIT. | autonomy (max) W.0.5 | confort (max) W.0.05 | price (min) W.0.45 |
---|---|---|---|

car 0 | 1 | 2 | 3 |

car 1 | 4 | 5 | 6 |

## Manipulating the Data¶

The data object are inmutable, if you want to modify it you need create a new one. All the numerical data (mtx, criteria, and weights) are stored as numpy arrays, and the alternative and criteria names as python tuples.

You can acces to the different parts of your data, simply by typing
`data.<your-parameter-name>`

for example:

```
In [8]:
```

```
data.mtx
```

```
Out[8]:
```

```
array([[1, 2, 3],
[4, 5, 6]])
```

```
In [9]:
```

```
data.criteria
```

```
Out[9]:
```

```
array([ 1, 1, -1])
```

```
In [10]:
```

```
data.weights
```

```
Out[10]:
```

```
array([ 0.5 , 0.05, 0.45])
```

```
In [11]:
```

```
data.anames, data.cnames
```

```
Out[11]:
```

```
(('car 0', 'car 1'), ('autonomy', 'confort', 'price'))
```

If you want (for example) change the names of the cars from `car 0`

and `car 1`

; to `VW`

and `Ford`

you must copy from your original
Data

```
In [12]:
```

```
data = Data(data.mtx, data.criteria,
weights=data.weights,
anames=["VW", "Ford"],
cnames=data.cnames)
data
```

```
Out[12]:
```

ALT./CRIT. | autonomy (max) W.0.5 | confort (max) W.0.05 | price (min) W.0.45 |
---|---|---|---|

VW | 1 | 2 | 3 |

Ford | 4 | 5 | 6 |

**Note:**A more flexible data manipulation API will relased in future versions.

## Plotting¶

The Data structure suport some basic rutines for ploting. Actually 5 types of plots are supported:

- Radar
Plot
(
`radar`

). - Histogram
(
`hist`

). - Violin
Plot
(
`violin`

). - Box Plot
(
`box`

). - Scatter
Matrix
(
`scatter`

).

The default scikit criteria uses the Radar Plot to visualize all the data. Take in account that the radar plot by default convert all the minimization criteria to maximization and push all the values to be greater than 1 (obviously all this options can be overided).

```
In [13]:
```

```
data.plot();
```

You can accesing the different plot by passing as first parameter the name of the plot

```
In [14]:
```

```
data.plot("box");
```

or by using the name as method call inside the `plot`

attribute

```
In [15]:
```

```
data.plot.violin();
```

Every plot has their own set of parameters, but at last every one can receive:

`ax`

: The plot axis.`cmap`

: The color map (More info).`mnorm`

: The normalization method for the alternative matrix as string (Default:`"none"`

).`wnorm`

: The normalization method for the criteria array as string (Default:`"none"`

).`weighted`

: If you want to weight the criteria (Default:`True`

).`show_criteria`

: Show or not the criteria in the plot (Default:`True`

in all except radar).`min2max`

: Convert the minimization criteria into maximization one (Default:`False`

in all except radar).`push_negatives`

: If a criteria has values lesser than 0, add the minimun value to all the criteria (Default:`False`

in all except radar).`addepsto0`

: If a criteria has values equal to 0, add an \(\epsilon\) value to all the criteria (Default:`False`

in all except radar).

Let’s change the colors of the radar plot and show their criteria optimization sense:

```
In [16]:
```

```
data.plot.radar(cmap="inferno", show_criteria=True);
```

## Using this data to feed some MCDA methods¶

Let’s rank our toy data by Weighted Sum Model, Weighted Product Model and TOPSIS

```
In [17]:
```

```
from skcriteria.madm import closeness, simple
```

First you need to create the decission maker.

Most of methods accepts as hyper parameters (parameters of the to
configure the method), the method of normalization of the alternative
matrix (divided by the `sum`

in Weighted Sum and Weighted Product, and
the `vector`

normalization on `Topsis`

) and the method to normalize
the weight array (normally `sum`

); But complex methods has more.

### Weighted Sum Model:¶

```
In [18]:
```

```
# first create the decision maker
# (with the default hiper parameters)
dm = simple.WeightedSum()
dm
```

```
Out[18]:
```

```
<WeightedSum (mnorm=sum, wnorm=sum)>
```

```
In [19]:
```

```
# Now lets decide the ranking
dec = dm.decide(data)
dec
```

```
Out[19]:
```

**WeightedSum (mnorm=sum, wnorm=sum) - Solution:**

ALT./CRIT. | autonomy (max) W.0.5 | confort (max) W.0.05 | price (min) W.0.45 | Rank |
---|---|---|---|---|

VW | 1 | 2 | 3 | 1 |

Ford | 4 | 5 | 6 | 2 |

The result says that the **VW** is better than the **FORD**, lets make
the maths:

**Note:**The last criteria is for minimization and because the WeightedSumModel only accepts maximization criteria by default, scikit-criteria invert all the values to convert the criteria to maximization

```
In [20]:
```

```
print("VW:", 0.5 * 1/5. + 0.05 * 2/7. + 0.45 * 1 / (3/9.))
print("FORD:", 0.5 * 4/5. + 0.05 * 5/7. + 0.45 * 1 / (6/9.))
```

```
VW: 1.46428571429
FORD: 1.11071428571
```

If you want to acces this points, the `Decision`

object stores all the
particular information of every method in a attribute called `e_`

```
In [21]:
```

```
print(dec.e_)
dec.e_.points
```

```
Extra(points)
```

```
Out[21]:
```

```
array([ 1.46428571, 1.11071429])
```

Also you can acces the type of the solution

```
In [22]:
```

```
print("Generate a ranking of alternatives?", dec.alpha_solution_)
print("Generate a kernel of best alternatives?", dec.beta_solution_)
print("Choose the best alternative?", dec.gamma_solution_)
```

```
Generate a ranking of alternatives? True
Generate a kernel of best alternatives? False
Choose the best alternative? True
```

The rank as numpy array (if this decision is a \(\alpha\)-solution)

```
In [23]:
```

```
dec.rank_
```

```
Out[23]:
```

```
array([1, 2])
```

The index of the row of the best alternative (if this decision is a \(\gamma\)-solution)

```
In [24]:
```

```
dec.best_alternative_, data.anames[dec.best_alternative_]
```

```
Out[24]:
```

```
(0, 'VW')
```

And the kernel of the non supered alternatives (if this decision is a \(\beta\)-solution)

```
In [25]:
```

```
# this return None because this
# decision is not a beta-solution
print(dec.kernel_)
```

```
None
```

### Weighted Product Model¶

```
In [26]:
```

```
dm = simple.WeightedProduct()
dm
```

```
Out[26]:
```

```
<WeightedProduct (mnorm=sum, wnorm=sum)>
```

```
In [27]:
```

```
dec = dm.decide(data)
dec
```

```
Out[27]:
```

**WeightedProduct (mnorm=sum, wnorm=sum) - Solution:**

ALT./CRIT. | autonomy (max) W.0.5 | confort (max) W.0.05 | price (min) W.0.45 | Rank |
---|---|---|---|---|

VW | 1 | 2 | 3 | 2 |

Ford | 4 | 5 | 6 | 1 |

As before let’s do the math (remember the weights are now exponets)

```
In [28]:
```

```
print("VW:", ((1/5.) ** 0.5) * ((2/7.) ** 0.05) + ((1 / (3/9.)) ** 0.45))
print("FORD:", ((4/5.) ** 0.5) * ((5/7.) ** 0.05) + ((1 / (6/9.)) ** 0.45))
```

```
VW: 2.05953437557
FORD: 2.0796708665
```

As wee expected the **Ford** are little better than the **VW**. Now lets
theck the `e_`

object

```
In [29]:
```

```
print(dec.e_)
dec.e_.points
```

```
Extra(points)
```

```
Out[29]:
```

```
array([-0.16198384, 0.02347966])
```

As you note the points are differents, this is because internally to avoid undeflows Scikit-Criteria uses a sums of logarithms instead products. So let’s check

```
In [30]:
```

```
import numpy as np
print("VW:", 0.5 * np.log10(1/5.) + 0.05 * np.log10(2/7.) + 0.45 * np.log10(1 / (3/9.)))
print("FORD:", 0.5 * np.log10(4/5.) + 0.05 * np.log10(5/7.) + 0.45 * np.log10(1 / (6/9.)))
```

```
VW: -0.161983839762
FORD: 0.0234796582871
```

### TOPSIS¶

```
In [31]:
```

```
dm = closeness.TOPSIS()
dm
```

```
Out[31]:
```

```
<TOPSIS (mnorm=vector, wnorm=sum)>
```

```
In [32]:
```

```
dec = dm.decide(data)
dec
```

```
Out[32]:
```

**TOPSIS (mnorm=vector, wnorm=sum) - Solution:**

ALT./CRIT. | autonomy (max) W.0.5 | confort (max) W.0.05 | price (min) W.0.45 | Rank |
---|---|---|---|---|

VW | 1 | 2 | 3 | 2 |

Ford | 4 | 5 | 6 | 1 |

The TOPSIS add more information into the decision object.

```
In [33]:
```

```
print(dec.e_)
print("Ideal:", dec.e_.ideal)
print("Anti-Ideal:", dec.e_.anti_ideal)
print("Closeness:", dec.e_.closeness)
```

```
Extra(anti_ideal, ideal, closeness)
Ideal: [ 0.48507125 0.04642383 0.20124612]
Anti-Ideal: [ 0.12126781 0.01856953 0.40249224]
Closeness: [ 0.35548671 0.64451329]
```

Where the `ideal`

and `anti_ideal`

are the normalizated sintetic
better and worst altenatives created by TOPSIS, and the `closeness`

is
how far from the *anti-ideal* and how closer to the *ideal* are the real
alternatives

Finally we can change the normalization criteria of the alternative
matric to `sum`

(divide every value by the sum opf their criteria) and
check the result:

```
In [34]:
```

```
dm = closeness.TOPSIS(mnorm="sum")
dm
```

```
Out[34]:
```

```
<TOPSIS (mnorm=sum, wnorm=sum)>
```

```
In [35]:
```

```
dm.decide(data)
```

```
Out[35]:
```

**TOPSIS (mnorm=sum, wnorm=sum) - Solution:**

ALT./CRIT. | autonomy (max) W.0.5 | confort (max) W.0.05 | price (min) W.0.45 | Rank |
---|---|---|---|---|

VW | 1 | 2 | 3 | 2 |

Ford | 4 | 5 | 6 | 1 |

The rankin has changed so, we can compare the two normalization by plotting

```
In [36]:
```

```
import matplotlib.pyplot as plt
f, (ax1, ax2) = plt.subplots(1, 2, sharey=True)
ax1.set_title("Sum Norm")
data.plot.violin(mnorm="sum", ax=ax1);
ax2.set_title("Vector Norm")
data.plot.violin(mnorm="vector", ax=ax2);
f.set_figwidth(15)
```