Integer programming in SciPy - Cheat Sheet & Quick Revision

Recall & Review

beginner

What is integer programming in simple terms?

Integer programming is a way to solve math problems where some or all answers must be whole numbers, like counting apples or people.

beginner

What does the scipy.optimize library help with in integer programming?

It helps find the best solution to problems with limits and whole number rules using tools like milp for mixed-integer linear programming.

beginner

Why do we sometimes need integer solutions instead of decimals?

Because some things can't be split, like people, cars, or machines. You can't have 2.5 cars, so answers must be whole numbers.

intermediate

What is the difference between linear programming and integer programming?

Linear programming allows decimal answers, but integer programming requires some or all answers to be whole numbers.

intermediate

How does the milp function in scipy.optimize work?

It solves problems where you want to maximize or minimize a goal, with rules, and some answers must be integers. You give it the goal, rules, and which answers are integers.

What type of numbers does integer programming require for some variables?

A Whole numbers

B Any real numbers

C Only decimals

D Complex numbers

Integer programming requires some or all variables to be whole numbers (integers).

Which Python library provides tools for integer programming?

A seaborn

B matplotlib

C pandas

D scipy.optimize

scipy.optimize includes functions like milp for integer programming.

What does MILP stand for?

A Matrix Integer Linear Programming

B Multiple Integer Linear Problems

C Mixed-Integer Linear Programming

D Minimal Integer Linear Procedure

MILP means Mixed-Integer Linear Programming, where some variables are integers and others can be decimals.

Why might integer programming be harder than regular linear programming?

A Because it uses less memory

B Because integers limit possible solutions

C Because decimals are easier to count

D Because it ignores constraints

Integer constraints reduce possible answers, making the problem more complex to solve.

In integer programming, what is a common real-life example?

A Scheduling workers in shifts

B Measuring temperature

C Calculating speed

D Predicting weather

Scheduling workers requires whole numbers of people, a classic integer programming problem.

Explain integer programming and why it is useful in real life.

Think about problems where you can't split items or people.

Describe how you would use scipy.optimize's milp function to solve an integer programming problem.

Focus on inputs needed for milp and what it returns.

Practice

1.

What is the main purpose of integer programming in scipy?

easy

A. To perform statistical hypothesis testing

B. To solve differential equations numerically

C. To find the best solution where some variables must be whole numbers

D. To visualize data with plots

Solution

1. Step 1: Understand integer programming concept

   Integer programming is used to find optimal solutions where some or all variables are restricted to integers (whole numbers).

2. Step 2: Match with scipy usage

   In scipy, integer programming helps solve optimization problems with integer constraints, unlike other tasks like plotting or statistics.

3. Final Answer:

   To find the best solution where some variables must be whole numbers -> Option C

4. Quick Check:

   Integer programming = whole number solutions [OK]

Hint: Integer programming means variables are whole numbers [OK]

Common Mistakes:

• Confusing integer programming with plotting or statistics
• Thinking it solves differential equations
• Assuming variables can be fractional

2.

Which of the following is the correct way to specify integer variables in scipy.optimize.linprog?

from scipy.optimize import linprog

result = linprog(c, A_ub=A, b_ub=b, integrality=...)

easy

A. integrality=True # boolean for all integer

B. integrality=[1, 0, 1] # 1 means integer, 0 means continuous

C. integrality='integer' # string to specify all integer

D. integrality=None # default no integer constraints

Solution

1. Step 1: Recall integrality parameter usage

   The integrality argument takes a list or array indicating which variables are integers (1) or continuous (0).

2. Step 2: Check options

   integrality=[1, 0, 1] # 1 means integer, 0 means continuous correctly uses a list with 1s and 0s. Options A, B, and D use incorrect types.

3. Final Answer:

   integrality=[1, 0, 1] # 1 means integer, 0 means continuous -> Option B

4. Quick Check:

   integrality list = integer flags [OK]

Hint: Use list of 1/0 to mark integer variables [OK]

Common Mistakes:

• Passing a string or boolean instead of list
• Leaving integrality as None to expect integers
• Confusing integrality with other parameters

3.

What will be the output of this code snippet?

from scipy.optimize import linprog

c = [-1, -2]
A = [[1, 1]]
b = [3]
integrality = [1, 1]

result = linprog(c, A_ub=A, b_ub=b, integrality=integrality, method='highs')
print(result.x.round())

medium

A. [1. 1.]

B. [1. 2.]

C. [3. 0.]

D. [0. 3.]

Solution

1. Step 1: Understand the problem setup

   The objective is to minimize -x - 2y, which is equivalent to maximizing x + 2y, with constraint x + y ≤ 3 and both x,y integers.

2. Step 2: Find integer values maximizing x + 2y under constraint

   Feasible integer points include (0,3): x+2y=6, (1,2):5, (2,1):4, (3,0):3. Maximum at (0,3), so result.x.round() prints [0. 3.].

3. Final Answer:

   [0. 3.] -> Option D

4. Quick Check:

   Max x+2y with x+y≤3 integer = [0,3] [OK]

Hint: Maximize by checking integer combos under constraints [OK]

Common Mistakes:

• Picking suboptimal integer point like [1,2]
• Misunderstanding objective sign for maximization
• Ignoring non-negativity bounds

4.

Identify the error in this integer programming code using scipy.optimize.linprog:

from scipy.optimize import linprog

c = [1, 1]
A = [[-1, 2]]
b = [4]
integrality = [1, 1]

result = linprog(c, A_ub=A, b_ub=b, integrality=integrality)
print(result.x)

medium

A. No error; code runs correctly

B. Missing method='highs' argument causes solver failure

C. Constraint matrix A has wrong sign for inequality

D. integrality must be a boolean, not a list

Solution

1. Step 1: Check linprog default solver compatibility

   In recent SciPy, the default method is 'highs', which supports integrality for integer programming.

2. Step 2: Identify if any error exists

   integrality=[1,1] is correct format. Parameters c, A_ub, b_ub are valid. No syntax or runtime errors; code runs.

3. Final Answer:

   No error; code runs correctly -> Option A

4. Quick Check:

   Default method='highs' supports integrality [OK]

Hint: Default method='highs' supports integer constraints [OK]

Common Mistakes:

• Assuming default solver lacks integer support
• Passing integrality as boolean instead of list
• Misinterpreting constraint matrix

5.

You want to solve an integer programming problem to maximize profit with variables x and y, where x + 2y ≤ 8, x ≥ 0, y ≥ 0, and both x and y must be integers. Which scipy.optimize.linprog call correctly models this problem?

hard

A. c = [-1, -2]
A = [[1, 2]]
b = [8]
integrality = [1, 1]
linprog(c, A_ub=A, b_ub=b, bounds=[(0, None), (0, None)], integrality=integrality, method='highs')

B. c = [1, 2]
A = [[1, 2]]
b = [8]
integrality = [1, 1]
linprog(c, A_ub=A, b_ub=b, bounds=[(0, None), (0, None)], integrality=integrality, method='highs')

C. c = [-1, -2]
A = [[-1, -2]]
b = [-8]
integrality = [1, 1]
linprog(c, A_ub=A, b_ub=b, bounds=[(0, None), (0, None)], integrality=integrality, method='highs')

D. c = [-1, -2]
A = [[1, 2]]
b = [8]
integrality = [0, 0]
linprog(c, A_ub=A, b_ub=b, bounds=[(0, None), (0, None)], method='highs')

Solution

1. Step 1: Translate maximization to minimization

   Maximize profit = x + 2y is same as minimize -x - 2y, so c = [-1, -2].

2. Step 2: Set constraints and integrality

   Constraint x + 2y ≤ 8 is A = [[1, 2]], b = [8]. Variables are non-negative with bounds (0, None). Both x and y are integers, so integrality = [1, 1].

3. Step 3: Confirm method and parameters

   Use method='highs' to support integer programming.

4. Final Answer:

   The code with c = [-1, -2], A = [[1, 2]], integrality = [1, 1], method='highs' -> Option A

5. Quick Check:

   Maximize -> minimize negative, integrality=1 for integers [OK]

Hint: Maximize by minimizing negative objective with integer flags [OK]

Common Mistakes:

• Using positive c vector for maximization
• Incorrect sign or values in constraints
• Not setting integrality for integer variables
• Omitting method='highs' for integer programming