Simple Applications in Linear Problems

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Linear Programming (LP) is a mathematical technique used to optimize (maximize or minimize) a linear objective function subject to linear constraints, aiming to find the best outcome under given conditions. Key components include the Objective Function (the goal to optimize), Constraints (limitations or restrictions), and Decision Variables (the choices to adjust). LP problems generally involve minimizing or maximizing \(Z = c_1 \times 1 + ... + c_n \times n\) subject to linear inequalities and non-negativity constraints.

Applications include Production Scheduling, where the objective is to optimize the production mix to maximize profit or minimize cost based on resource constraints, and Transportation Problems, which aim to minimize total shipping cost from sources to destinations subject to supply and demand constraints.

Solving methods range from the graphical method for problems with 2 variables to the simplex algorithm, special transportation methods (like Northwest Corner, Least Cost, MODI), and the use of optimization software or solvers.

Graphical Method (2 variables)

A small workshop makes two types of furniture: chairs and tables. Each chair requires 2 hours of carpentry and 1 hour of painting. Each table requires 1 hour of carpentry and 1 hour of painting. The workshop has 6 hours of carpentry time and 4 hours of painting time available each day. Each chair gives a profit of $30, and each table gives a profit of $20.

Task:

• Formulate the problem as a linear program.
• Plot the feasible region and determine the optimal number of chairs and tables to maximize profit using a graphical method.

Answer

Formulation of the Problem

Let:

• \(x\) = number of chairs produced
• \(y\) = number of tables produced

Objective: Maximize profit

\(\text{Maximize } Z = 30x + 20y\)

Subject to constraints:

• Carpentry time:
  \(2x + y \leq 6\)

• Painting time:
  \(x + y \leq 4\)

• Non-negativity:
  \(x \geq 0, \quad y \geq 0\)

Plot the Feasible Region

We plot the constraints: - \(2x + y = 6\) => (x-intercept at 3, y-intercept at 6) - \(x + y = 4\) => (x-intercept at 4, y-intercept at 4)

import numpy as np
import matplotlib.pyplot as plt

# Define the constraints
x = np.linspace(0, 4, 100)

# Constraint lines
y1 = (6 - 2*x)  # from 2x + y <= 6
y2 = (4 - x)    # from x + y <= 4

# Plotting
plt.figure(figsize=(8,6))
plt.plot(x, y1, label=r'$2x + y \leq 6$')
plt.plot(x, y2, label=r'$x + y \leq 4$')

# Feasible region
plt.fill_between(
    x, 
    0, 
    np.minimum(y1, y2), 
    where=(np.minimum(y1, y2)>=0), 
    color='lightblue', 
    alpha=0.5)

# Label axes
plt.xlim(0, 4)
plt.ylim(0, 4)
plt.xlabel('Chairs (x)')
plt.ylabel('Tables (y)')
plt.title('Feasible Region for Furniture Workshop')
plt.axhline(0, color='black')
plt.axvline(0, color='black')
plt.legend()

# Define corner points manually
points = np.array([
    [0, 0],
    [3, 0],    # from 2x + y = 6
    [2, 2],    # intersection point
    [0, 4],    # from x + y = 4
])

# Objective function: 30x + 20y
profits = [30*x + 20*y for x, y in points]

# Find max profit
max_profit_index = np.argmax(profits)
optimal_point = points[max_profit_index]
optimal_profit = profits[max_profit_index]

# Plot points
for i, (pt, profit) in enumerate(zip(points, profits)):
    plt.plot(pt[0], pt[1], 'ro')
    plt.text(
        pt[0]+0.1, 
        pt[1]+0.1, 
        f'P{i}\n({pt[0]}, {pt[1]})\nProfit=${profit}')

# Highlight optimal point
plt.plot(
    optimal_point[0], 
    optimal_point[1], 'go', 
    markersize=10, 
    label='Optimal Solution')
plt.legend()

plt.grid(True)
plt.show()

# Print solution
print(f"Optimal production plan: {int(optimal_point[0])} chairs and {int(optimal_point[1])} tables")
print(f"Maximum profit: ${optimal_profit}")

Optimal production plan: 2 chairs and 2 tables
Maximum profit: $100

Simplex Algorithm (via scipy.optimize.linprog)

A factory produces 3 products: A, B, and C. Each requires machine hours on 2 machines: M1 and M2.

Product	Profit	M1 Hours	M2 Hours
A	$40	2	1
B	$30	1	2
C	$20	1	1

• M1 is available for 100 hours/week.
• M2 is available for 80 hours/week.

Task:

• Formulate and solve using the Simplex algorithm via scipy.optimize.linprog.
• Determine how many units of A, B, and C to produce to maximize profit.

Answer

Formulate the Linear Program

Let: - \(x_1\) = number of units of product A - \(x_2\) = number of units of product B - \(x_3\) = number of units of product C

Objective: Maximize profit

\(\text{Maximize } Z = 40x_1 + 30x_2 + 20x_3\)

But scipy.optimize.linprog minimizes by default, so we minimize (-Z):

\(\text{Minimize } Z = -40x_1 -30x_2 -20x_3\)

Subject to constraints:

• Machine 1 hours:
  \(2x_1 + 1x_2 + 1x_3 \leq 100\)

• Machine 2 hours:
  \(1x_1 + 2x_2 + 1x_3 \leq 80\)

• Non-negativity:
  \(x_1, x_2, x_3 \geq 0\)

from scipy.optimize import linprog

# Coefficients for the objective function (negative because linprog minimizes)
c = [-40, -30, -20]

# Coefficients for the inequalities
A = [
    [2, 1, 1],  # M1 constraint
    [1, 2, 1],  # M2 constraint
]

# Right-hand side values
b = [100, 80]

# Bounds for variables (all must be >= 0)
x_bounds = (0, None)
bounds = [x_bounds, x_bounds, x_bounds]

# Solve the linear program
result = linprog(c, A_ub=A, b_ub=b, bounds=bounds, method='highs')

# Print results
if result.success:
    x1, x2, x3 = result.x
    max_profit = -result.fun
    print(f"Optimal production plan:")
    print(f"Product A: {x1:.2f} units")
    print(f"Product B: {x2:.2f} units")
    print(f"Product C: {x3:.2f} units")
    print(f"Maximum profit: ${max_profit:.2f}")
else:
    print("No solution found.")

Optimal production plan:
Product A: 40.00 units
Product B: 20.00 units
Product C: 0.00 units
Maximum profit: $2200.00

Transportation Method - Northwest Corner Rule (manual or pandas/numpy)

A company has 3 factories (S1, S2, S3) and 4 distribution centers (D1, D2, D3, D4).

Supplies:

• S1: 30 units
• S2: 40 units
• S3: 20 units

Demands:

• D1: 20 units
• D2: 30 units
• D3: 25 units
• D4: 15 units

Cost Matrix:

	D1	D2	D3	D4
S1	8	6	10	9
S2	9	7	4	2
S3	3	4	2	5

Task: - Use the Northwest Corner Method to construct an initial feasible solution manually or via a custom function in Python. - Display the allocation matrix and compute the total transportation cost.

Answer

Apply Northwest Corner Method (Manual)

We start from the top-left corner and allocate as much as possible.

• Start at (S1, D1):
  • Demand D1 = 20
  • Supply S1 = 30
  • Allocate 20 units (fulfills D1).
  • Remaining supply S1 = 10, D1 fulfilled.
• Move to (S1, D2):
  • Demand D2 = 30
  • Supply S1 = 10
  • Allocate 10 units (uses all of S1).
  • Remaining demand D2 = 20, S1 exhausted.
• Move to (S2, D2):
  • Demand D2 = 20
  • Supply S2 = 40
  • Allocate 20 units.
  • Remaining supplyS2 = 20, D2 fulfilled.
• Move to (S2, D3):
  • Demand D3 = 25
  • Supply S2 = 20
  • Allocate 20 units (uses all of S2).
  • Remaining demand D3 = 5, S2 exhausted.
• Move to (S3, D3):
  • Demand D3 = 5
  • Supply S3 = 20
  • Allocate 5 units.
  • Remaining supply S3 = 15, D3 fulfilled.
• Move to (S3, D4):
  • Demand D4 = 15
  • Supply S3 = 15
  • Allocate 15 units (both S3 and D4 are fulfilled).

Allocation Matrix

	D1	D2	D3	D4
S1	20	10	0	0
S2	0	20	20	0
S3	0	0	5	15

Compute Total Transportation Cost

\((20 \times 8) + (10 \times 6) + (20 \times 7) + (20 \times 4) + (5 \times 2) + (15 \times 5) = 525\)

Python

import numpy as np

# Cost matrix
cost = np.array([
    [8, 6, 10, 9],
    [9, 7, 4, 2],
    [3, 4, 2, 5]
])

supply = [30, 40, 20]
demand = [20, 30, 25, 15]

# Initialize allocation matrix
alloc = np.zeros((3,4), dtype=int)

i, j = 0, 0

while i < 3 and j < 4:
    alloc_qty = min(supply[i], demand[j])
    alloc[i, j] = alloc_qty
    supply[i] -= alloc_qty
    demand[j] -= alloc_qty
    
    if supply[i] == 0:
        i += 1
    else:
        j += 1

# Calculate total cost
total_cost = np.sum(alloc * cost)

print("Allocation Matrix:")
print(alloc)
print(f"Total Transportation Cost: ${total_cost}")

Allocation Matrix:
[[20 10  0  0]
 [ 0 20 20  0]
 [ 0  0  5 15]]
Total Transportation Cost: $525

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