Why Linear programming (linprog) in SciPy? - Purpose & Use Cases

The Big Idea

What if a simple tool could instantly find the best way to use your resources without guesswork?

The Scenario

Imagine you run a small bakery and want to decide how many cakes and cookies to bake each day to maximize your profit. You try to calculate this by hand, considering your limited ingredients and time.

The Problem

Doing this manually means juggling many numbers and constraints, which is slow and easy to mess up. If you change one ingredient amount, you must redo all calculations, risking mistakes and wasted resources.

The Solution

Linear programming with linprog lets you describe your problem with simple rules and automatically finds the best solution quickly and accurately, saving time and avoiding errors.

Before vs After

✗ Before

profit = 5*cakes + 3*cookies
if flour >= 2*cakes + 1*cookies and eggs >= cakes + 2*cookies:
    # try different numbers manually

✓ After

from scipy.optimize import linprog
result = linprog(c=[-5, -3], A_ub=[[2,1],[1,2]], b_ub=[flour, eggs], bounds=[(0, None), (0, None)])

What It Enables

It opens the door to solving complex resource allocation problems easily, helping you make the best decisions fast.

Real Life Example

Companies use linear programming to plan production schedules, minimize costs, or optimize delivery routes, all automatically and efficiently.

Key Takeaways

Manual calculations for optimization are slow and error-prone.

linprog automates finding the best solution under constraints.

This method helps make smart decisions in business and beyond.

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