Integer programming in SciPy - Interactive Code Practice

Practice - 5 Tasks

1fill in blank

easy

Complete the code to import the integer linear programming solver from scipy.

SciPy

from scipy.optimize import [1]

Drag options to blanks, or click blank then click option'

Alinprog

Bminimize

Croot

Dcurve_fit

Attempts:

3 left

💡 Hint

Common Mistakes

Importing 'minimize' instead of 'linprog'.

Using 'root' which is for root finding, not optimization.

The linprog function from scipy.optimize is used for linear programming, including integer programming when specifying integer constraints.

2fill in blank

medium

Complete the code to define the objective function coefficients for minimization.

SciPy

c = [1]

Drag options to blanks, or click blank then click option'

A1, 2, 3

B{1, 2, 3}

C[1, 2, 3]

D(1, 2, 3)

Attempts:

3 left

💡 Hint

Common Mistakes

Using a tuple or set instead of a list.

Not enclosing the numbers in any brackets.

The objective function coefficients must be a list or array. Here, a list [1, 2, 3] is used.

3fill in blank

hard

Fix the error in the code to specify the integer constraint for the variables.

SciPy

bounds = [(0, None), (0, None), (0, None)]
integrality = [1]

Drag options to blanks, or click blank then click option'

A[0, 0, 0]

B[True, True, True]

C[int, int, int]

D[1, 1, 1]

Attempts:

3 left

💡 Hint

Common Mistakes

Using boolean True instead of 1.

Using data types like int instead of numeric flags.

The integrality parameter expects a list of 1s and 0s indicating which variables are integers (1) or continuous (0). Here, all variables are integers, so [1, 1, 1] is correct.

4fill in blank

hard

Fill both blanks to define inequality constraints matrices and vectors.

SciPy

A_ub = [1]
b_ub = [2]

Drag options to blanks, or click blank then click option'

A[[1, 2, 3], [4, 5, 6]]

B[7, 8]

C[[7, 8]]

D[1, 2, 3]

Attempts:

3 left

💡 Hint

Common Mistakes

Mixing up matrix and vector shapes.

Using a single list for both constraints and bounds.

A_ub is a 2x3 matrix representing coefficients of inequality constraints, and b_ub is a vector with the upper bounds for these constraints.

5fill in blank

hard

Fill all three blanks to call the solver with integer constraints and print the solution.

SciPy

result = linprog(c=[1], A_ub=[2], b_ub=[3], bounds=bounds, integrality=integrality, method='highs')
print(result.x)

Drag options to blanks, or click blank then click option'

Ac

BA_ub

Cb_ub

Dbounds

Attempts:

3 left

💡 Hint

Common Mistakes

Passing bounds instead of b_ub or A_ub.

Mixing up the order of parameters.

The linprog function requires the objective coefficients c, inequality matrix A_ub, and inequality bounds b_ub as arguments. Passing these correctly allows the solver to find the integer solution.

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