# Exercise 1: Simple Exercise
```{admonition} MUDE Exam Information
:class: tip, dropdown
For the exam, you are expected to have a clear understanding behind the Simplex method and how to do it. You will not need to do the calculations by hand in the exam.
```
Consider the following problem:
$$\max \left( Z \right) = 2{x_1} + 2{x_2}$$
such that
$$\begin{cases}2{x_1} + {x_2} \le 4 \\{x_1} + 2{x_2} \le 4 \\{x_1} + {x_2} \le 5 \\{x_1},{x_2} \ge 0 \end{cases}$$
## Task 1
Transform the problem into the augmented form
```{admonition} Solution
:class: tip, dropdown
$$Z - 2{x_1} - 2{x_2} - 0{s_1} - 0{s_2} - 0{s_3} = 0$$
such that
$$\begin{cases}2{x_1} + {x_2} + s_1 = 4\\{x_1} + 2{x_2} + s_2 = 4 \\{x_1} + {x_2} + s_3 = 5 \\{x_1},{x_2},s_1,s_2,s_3 \ge 0 \end{cases}$$
```
## Task 2
```{div} full-width
```
## Task 3
Why is your final table the optimal solution?
```{admonition} Solution
:class: tip, dropdown
This is the optimal solution because all the coefficients are positive in the row of the objective function and this is a maximization problem.
```
## Task 4
Find the graphical solution of this problem. Does it give the same solution as using the Simplex method?
````{admonition} Solution
:class: tip, dropdown
```{figure} https://files.mude.citg.tudelft.nl/Graphical_solution.png
:height: 300px
```
````
**Note:** in this problem and solution the intersection between the gradient and the constraints is not important (see figure), but rather the _direction_ that is important! It is a coincidence that the intersection is the optimal solution.
% START-CREDIT
% source: optimization
```{attributiongrey} Attribution
:class: attribution
This chapter is written by Gonçalo Homem de Almeida Correia, Maria Nogal Macho, Jie Gao and Bahman Ahmadi. {ref}`Find out more here `.
```
% END-CREDIT