Learn Transportation Problems and Analytics Solutions | Integer Programming & Discrete Decision Models

Transportation problems are a classic analytics challenge where you must decide how to distribute goods from multiple sources (such as warehouses) to multiple destinations (such as stores) while minimizing total shipping costs. In these problems, each source has a fixed supply of goods available, and each destination has a specific demand that must be met. The cost to ship a unit of goods from each source to each destination is known and may vary depending on the route or distance. The objective is to determine the quantity of goods shipped along each route so that all supply and demand constraints are satisfied, and the overall cost is minimized. This structure is common in logistics, inventory management, and retail analytics, where efficient distribution directly impacts profitability and service levels.


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import numpy as np
from scipy.optimize import linprog

# Supply at each warehouse
supply = [20, 30, 25]
# Demand at each store
demand = [10, 25, 20, 20]

# Cost matrix (warehouses x stores)
costs = np.array([
    [8, 6, 10, 9],
    [9, 12, 13, 7],
    [14, 9, 16, 5]
])

num_warehouses = len(supply)
num_stores = len(demand)

# Decision variables: x[i][j] is units shipped from warehouse i to store j
# Flattened into a single vector for linprog
c = costs.flatten()

# Build constraints
A_eq = []
b_eq = []

# Supply constraints: sum of shipments from each warehouse <= supply
for i in range(num_warehouses):
    row = [0] * (num_warehouses * num_stores)
    for j in range(num_stores):
        row[i * num_stores + j] = 1
    A_eq.append(row)
    b_eq.append(supply[i])

# Demand constraints: sum of shipments to each store >= demand
for j in range(num_stores):
    row = [0] * (num_warehouses * num_stores)
    for i in range(num_warehouses):
        row[i * num_stores + j] = 1
    A_eq.append(row)
    b_eq.append(demand[j])

# Bounds: variables must be integers >= 0
bounds = [(0, None) for _ in range(num_warehouses * num_stores)]

# Solve as a linear program (relaxing integer constraints for demonstration)
result = linprog(
    c,
    A_eq=A_eq,
    b_eq=b_eq,
    bounds=bounds,
    method='highs'
)

# Reshape solution to warehouse x store grid
shipment_plan = np.round(result.x).reshape((num_warehouses, num_stores))
print("Optimal shipment plan (units from each warehouse to each store):")
print(shipment_plan)
print("Total minimum shipping cost:", result.fun)

The solution to this transportation model provides a shipment plan that specifies exactly how many units to send from each warehouse to each store. By satisfying both the supply constraints (not shipping more than each warehouse can provide) and the demand constraints (ensuring each store receives what it needs), the model guarantees that logistics requirements are met. The cost minimization objective ensures that the chosen shipment plan is the most economical possible, directly supporting analytics goals such as reducing operational expenses and improving supply chain efficiency. This approach allows you to make data-driven decisions that align with business objectives, ensuring resources are allocated optimally across your network.

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Section 2. Chapter 3

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              123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960
            
import numpy as np
from scipy.optimize import linprog

# Supply at each warehouse
supply = [20, 30, 25]
# Demand at each store
demand = [10, 25, 20, 20]

# Cost matrix (warehouses x stores)
costs = np.array([
    [8, 6, 10, 9],
    [9, 12, 13, 7],
    [14, 9, 16, 5]
])

num_warehouses = len(supply)
num_stores = len(demand)

# Decision variables: x[i][j] is units shipped from warehouse i to store j
# Flattened into a single vector for linprog
c = costs.flatten()

# Build constraints
A_eq = []
b_eq = []

# Supply constraints: sum of shipments from each warehouse <= supply
for i in range(num_warehouses):
    row = [0] * (num_warehouses * num_stores)
    for j in range(num_stores):
        row[i * num_stores + j] = 1
    A_eq.append(row)
    b_eq.append(supply[i])

# Demand constraints: sum of shipments to each store >= demand
for j in range(num_stores):
    row = [0] * (num_warehouses * num_stores)
    for i in range(num_warehouses):
        row[i * num_stores + j] = 1
    A_eq.append(row)
    b_eq.append(demand[j])

# Bounds: variables must be integers >= 0
bounds = [(0, None) for _ in range(num_warehouses * num_stores)]

# Solve as a linear program (relaxing integer constraints for demonstration)
result = linprog(
    c,
    A_eq=A_eq,
    b_eq=b_eq,
    bounds=bounds,
    method='highs'
)

# Reshape solution to warehouse x store grid
shipment_plan = np.round(result.x).reshape((num_warehouses, num_stores))
print("Optimal shipment plan (units from each warehouse to each store):")
print(shipment_plan)
print("Total minimum shipping cost:", result.fun)

Everything was clear?

Thanks for your feedback!

Section 2. Chapter 3