Linear programming (linprog) in SciPy - Mini Project: Build & Apply

Linear Programming with scipy.linprog

📖 Scenario: You are managing a small factory that produces two products: chairs and tables. You want to maximize your profit while considering the limits on materials and labor hours.

🎯 Goal: Build a linear programming model using scipy.optimize.linprog to find the best number of chairs and tables to produce to maximize profit.

📋 What You'll Learn

Create variables for the number of chairs and tables

Set up the profit coefficients for each product

Define the constraints for materials and labor

Use scipy.optimize.linprog to solve the problem

Print the optimal number of chairs and tables to produce

💡 Why This Matters

🌍 Real World

Factories and businesses use linear programming to maximize profits or minimize costs while respecting resource limits.

💼 Career

Understanding linear programming is useful for roles in operations research, supply chain management, and data science.

1

Set up profit coefficients and variables

Create a list called profit_coeffs with values -20 and -30 representing the negative profit per chair and table respectively (negative because linprog minimizes).

SciPy

# Create the list profit_coeffs with values -20 and -30
# Your code here

Hint

Remember, linprog minimizes, so use negative profits to maximize.

2

Define constraints for materials and labor

Create a list of lists called constraints_matrix with these rows: [1, 2] for wood usage and [3, 2] for labor hours. Also create a list called constraints_limits with values 100 and 90 representing the maximum wood and labor available.

SciPy

profit_coeffs = [-20, -30]
# Create constraints_matrix and constraints_limits as described
# Your code here

Hint

Each row in constraints_matrix corresponds to a resource limit.

3

Use linprog to solve the optimization problem

Import linprog from scipy.optimize. Then create a variable called result by calling linprog with profit_coeffs as the objective, constraints_matrix as A_ub, and constraints_limits as b_ub. Use default bounds.

SciPy

profit_coeffs = [-20, -30]
constraints_matrix = [[1, 2], [3, 2]]
constraints_limits = [100, 90]
# Import linprog and solve the problem, store in result
# Your code here

Hint

Use c= for objective coefficients, A_ub= and b_ub= for inequality constraints.

4

Print the optimal production quantities

Print the string Optimal chairs: followed by the first value in result.x and the string Optimal tables: followed by the second value in result.x. Use two separate print statements.

SciPy

from scipy.optimize import linprog

profit_coeffs = [-20, -30]
constraints_matrix = [[1, 2], [3, 2]]
constraints_limits = [100, 90]

result = linprog(c=profit_coeffs, A_ub=constraints_matrix, b_ub=constraints_limits)
# Print optimal chairs and tables
# Your code here

Hint

Access the solution with result.x and print the values.

Practice

1. What is the main purpose of the linprog function in scipy.optimize?

easy

A. To find the best solution for a problem with linear constraints and objective

B. To perform nonlinear regression analysis

C. To generate random linear equations

D. To plot linear graphs

Solution

1. Step 1: Understand the purpose of linear programming

   Linear programming is used to find the best (optimal) solution under given linear constraints and objectives.

2. Step 2: Identify what linprog does

   The linprog function in scipy.optimize solves linear programming problems by minimizing a linear objective function subject to linear constraints.

3. Final Answer:

   To find the best solution for a problem with linear constraints and objective -> Option A

4. Quick Check:

   Purpose of linprog = find best solution [OK]

Hint: Remember: linprog solves linear optimization problems [OK]

Common Mistakes:

• Confusing linprog with plotting functions
• Thinking linprog handles nonlinear problems
• Assuming linprog generates random data

2. Which of the following is the correct way to import the linprog function from scipy.optimize?

easy

A. import scipy.optimize.linprog

B. import linprog from scipy.optimize

C. from scipy import linprog.optimize

D. from scipy.optimize import linprog

Solution

1. Step 1: Recall Python import syntax

   To import a specific function from a module, use from module import function.

2. Step 2: Apply to linprog in scipy.optimize

   The correct syntax is from scipy.optimize import linprog.

3. Final Answer:

   from scipy.optimize import linprog -> Option D

4. Quick Check:

   Correct import syntax = from scipy.optimize import linprog [OK]

Hint: Use 'from module import function' to import specific functions [OK]

Common Mistakes:

• Using 'import linprog from ...' which is invalid syntax
• Trying to import submodules as functions
• Using dot notation incorrectly in import statements

3. What will be the output of the following code snippet?

from scipy.optimize import linprog
c = [-1, -2]
A = [[2, 1], [1, 1]]
b = [20, 16]
res = linprog(c, A_ub=A, b_ub=b)
print(res.x.round(2))

medium

A. [0. 0.]

B. [10. 0.]

C. [8. 8.]

D. [0. 16.]

Solution

1. Step 1: Understand the problem setup

   The objective is to minimize -1*x1 - 2*x2, which is equivalent to maximizing x1 + 2*x2, with constraints 2*x1 + x2 <= 20 and x1 + x2 <= 16.

2. Step 2: Solve constraints to find feasible maximum

   The feasible region vertices include (10,0), which maximizes the objective (x1 + 2*x2 = 10) and satisfies both constraints (2*10 + 0 = 20 <= 20, 10 + 0 = 10 <= 16). Thus res.x.round(2) prints [10. 0.].

3. Final Answer:

   [10. 0.] -> Option B

4. Quick Check:

   Optimal solution = [10, 0] [OK]

Hint: Remember: linprog minimizes; negate objective to maximize [OK]

Common Mistakes:

• Forgetting linprog minimizes, not maximizes
• Mixing up constraint inequalities
• Ignoring variable bounds defaulting to non-negative

4. Identify the error in this code snippet that uses linprog:

from scipy.optimize import linprog
c = [1, 2]
A = [[-1, 1], [3, 4]]
b = [1, 12]
res = linprog(c, A_eq=A, b_eq=b)
print(res.success)

medium

A. Objective coefficients should be negative to minimize

B. Missing variable bounds argument

C. Using A_eq with inequality constraints instead of A_ub

D. Incorrect import statement

Solution

1. Step 1: Check constraint type usage

   The code uses A_eq and b_eq, which define equality constraints, but the constraints given are inequalities (e.g., -1*x1 + x2 <= 1).

2. Step 2: Correct constraint parameter

   For inequality constraints, A_ub and b_ub should be used instead of A_eq and b_eq.

3. Final Answer:

   Using A_eq with inequality constraints instead of A_ub -> Option C

4. Quick Check:

   Use A_ub for inequalities, A_eq for equalities [OK]

Hint: Use A_ub for inequalities, A_eq for equalities [OK]

Common Mistakes:

• Confusing equality and inequality constraint parameters
• Assuming linprog automatically detects constraint types
• Ignoring error messages about constraint shapes

5. You want to minimize the cost function 3x + 4y subject to constraints:
- x + 2y ≥ 8
- 3x + y ≤ 15
- x, y ≥ 0
Which is the correct way to set up the linprog call in Python?

hard

A. c = [3, 4]; A_ub = [[-1, -2], [-3, -1]]; b_ub = [-8, -15]; res = linprog(c, A_ub=A_ub, b_ub=b_ub, bounds=[(0, None), (0, None)])

B. c = [3, 4]; A_ub = [[1, 2], [3, 1]]; b_ub = [8, 15]; res = linprog(c, A_ub=A_ub, b_ub=b_ub, bounds=(0, None))

C. c = [3, 4]; A_ub = [[-1, -2], [3, 1]]; b_ub = [-8, 15]; res = linprog(c, A_ub=A_ub, b_ub=b_ub, bounds=(0, None))

D. c = [3, 4]; A_ub = [[1, 2], [-3, -1]]; b_ub = [8, -15]; res = linprog(c, A_ub=A_ub, b_ub=b_ub, bounds=[(0, None), (0, None)])

Solution

1. Step 1: Convert constraints to ≤ form for linprog

   linprog requires constraints as A_ub * x ≤ b_ub. The first constraint x + 2y ≥ 8 can be rewritten as -x - 2y ≤ -8. The second constraint 3x + y ≤ 15 stays as is.

2. Step 2: Set up matrices and bounds correctly

   So A_ub = [[-1, -2], [-3, -1]], b_ub = [-8, -15]. Bounds for x and y are (0, None) each, so use bounds=[(0, None), (0, None)].

3. Step 3: Match options to correct setup

   c = [3, 4]; A_ub = [[-1, -2], [-3, -1]]; b_ub = [-8, -15]; res = linprog(c, A_ub=A_ub, b_ub=b_ub, bounds=[(0, None), (0, None)]) matches this setup exactly.

4. Final Answer:

   c = [3, 4]; A_ub = [[-1, -2], [-3, -1]]; b_ub = [-8, -15]; res = linprog(c, A_ub=A_ub, b_ub=b_ub, bounds=[(0, None), (0, None)]) -> Option A

5. Quick Check:

   Rewrite ≥ as negative ≤ and set bounds as list of tuples [OK]

Hint: Rewrite ≥ constraints as negative ≤ for linprog [OK]

Common Mistakes:

• Not converting ≥ constraints to ≤ form
• Using single tuple for bounds instead of list of tuples
• Mixing signs in constraint matrices