pulp.tar.gz

pulp

We make optimization decisions every day. To improve the performance of your applications, computers need to do the same. This course will help you optimize computing functions through the open-source Python library PuLP.

In this course, you’ll learn about linear programming, an optimization approach that maximizes or minimizes functions through defined algebra, and the PuLP library along with its various optimizers available via APIs. You’ll identify the variables and constraints of computing functions and define the optimization goals. You’ll then learn and practice common linear programming algorithms. Finally, you’ll be introduced to potential applications in the real world.

By the end of the course, you will be able to optimize computing functions using Python and PuLP.

Master Linear Programming in Python with PuLP

# Overview of the problem

Balancing a chemical equation essentially means respecting the conservation of mass and ensuring that the same number of atoms of an element are present on the left-hand side (reactants) and the right-hand side (products). Simply put, if you have X grams of an element as a reactant, the product will also have X grams of that element in the products (ideal conditions).


In the figure above, we can notice the following:

- In the unbalanced equations, there are 2 atoms of H and Cl in the products. However, there is only a single atom of H and Cl in the reactants. This does not respect the law of conservation of mass.

- In the balanced equation, we are using 2 atoms of H and Cl in the reactants. The number of atoms in the products remains unchanged. As a result, the equation is balanced because it no longer violates the law of conservation of mass.

The equation above was easy to solve and can be done manually. However, things get complicated as the number of elements increases and the number of reactants and products increases. For example, the equation below can be solved manually but will require a fair amount of trial and error before that can happen.

$$ Al_{2}(CO_{3})_{3} + H_{3}PO_{4} \rightarrow AlPO_{4} + CO_{2} + H_{2}O $$


# Applying constraint optimization for balancing equations

So, how can we use constraint optimization to balance our equations?

Let’s consider the first equation:
$$Zn + HCl \rightarrow ZnCl_{2} + H_{2}$$

The coefficient of each reactant/product can be thought of as a variable. 

Let’s consider the first equation again:

$$ x_{1}Zn + x_{2}HCl \rightarrow x_{3}ZnCl_{2} + x_{4}H_{2}$$


An `x``1`, `x``2`, `x``3`, and `x``4` are the variable coefficients. Each variable must be an integer greater than or equal to 1.

Each element adds a constraint. 
- For zinc, this constraint is added:
$$ x_{1}*1 + x_{2}*0 = x_{3}*1 + x_{4}*0$$

$$ \rightarrow x_{1} = x_{3}$$
$$\rightarrow x_{1} - x_{3} = 0$$


- For chlorine, this constraint is added:

$$ x_{1}*0 + x_{2}*1 = x_{3}*2 + x_{4}*0$$

$$ \rightarrow x_{2} = 2x_{3}$$
$$\rightarrow x_{2} - 2x_{3} = 0$$


- For hydrogen, this constraint is added:

$$ x_{1}*0 + x_{2}*1 = x_{3}*0 + x_{4}*1$$

$$ \rightarrow x_{2} = x_{4}$$
$$\rightarrow x_{2} - x_{4} = 0$$

Learn how to balance chemical equations using linear programming.  

Problem: Balancing Chemical Equations, Part I

Learn how to balance chemical equations using linear programming.