Let's look at a practical use case where linear programming can be used to solve a real-world problem. Let's assume that we want to maximize the profits of a state-of-the-art factory that manufactures two different types of robots:
- Advanced model (A): This provides full functionality. Manufacturing each unit of the advanced model results in a profit of $4,200.
- Basic model (B): This only provides basic functionality. Manufacturing each unit of the basic model results in a profit of $2,800.
There are three different types of people needed to manufacture a robot. The exact number of days needed to manufacture a robot of each type are as follows:
| Type of Robot | Technician | AI Specialist | Engineer |
| Robot A: advanced model | 3 days | 4 days | 4 days |
| Robot B: basic model | 2 days | 3 days | 3 days |
The factory runs on 30-day cycles. A single AI specialist is available for 30 days in a cycle. Each of the two engineers will take 8 days off in 30 days. So, an engineer is available only for 22 days in a cycle. There is a single technician available for 20 days in a 30-day cycle.
The following table shows the number of people we have in the factory:
| Technician | AI Specialist | Engineer | |
| Number of people | 1 | 1 | 2 |
| Total number of days in a cycle | 1 x 20 = 20 days | 1 x 30 = 30 days | 2 x 22 = 44 days |
This can be modeled as follows:
- Maximum profit = 4200A + 2800B
- This is subject to the following:
- A ≥ 0: The number of advanced robots produced can be 0 or more.
- B ≥ 0: The number of basic robots produced can be 0 or more.
- 3A + 2B ≤ 20: These are the constraints of the technician's availability.
- 4A+3B ≤ 30: These are the constraints of the AI specialist's availability.
- 4A+ 3B ≤ 44: These are the constraints of the engineers' availability.
First, we import the Python package named pulp, which is used to implement ;linear programming:
import pulpThen, we call the LpProblem function in this package to instantiate the problem class. We name the instance Profit maximising problem:
# Instantiate our problem class
model = pulp.LpProblem("Profit maximising problem", pulp.LpMaximize)Then, we define two linear variables, A and B. Variable A represents the number of advanced robots that are produced and variable B represents the number of basic robots that are produced:
A = pulp.LpVariable('A', lowBound=0, cat='Integer')
B = pulp.LpVariable('B', lowBound=0, cat='Integer')We define the objective function and constraints as follows:
# Objective function
model += 5000 * A + 2500 * B, "Profit"
# Constraints
model += 3 * A + 2 * B <= 20
model += 4 * A + 3 * B <= 30
model += 4 * A + 3 * B <= 44We use the solve function to generate a solution:
# Solve our problem
model.solve()
pulp.LpStatus[model.status]Then, we print the values of A and B and the value of the objective function:
And here we come to the end of this chapter! Let's summarize what we have learned.