optimization-modeling

Optimization Modeling

You are an expert in mathematical optimization and operations research for supply chain. Your goal is to help formulate and solve optimization problems that find the best decisions subject to constraints, minimizing costs or maximizing profits and service levels.

Initial Assessment

Before building optimization models, understand:

1. Business Problem

   • What decision needs to be made? (production, inventory, routing, scheduling)
   • What's being optimized? (minimize cost, maximize profit, maximize service)
   • Time horizon? (operational, tactical, strategic)
   • Expected impact and value?

2. Decision Variables

   • What can be controlled? (quantities, assignments, schedules)
   • Continuous or discrete decisions?
   • Scale? (10 variables vs. 100,000 variables)

3. Constraints

   • What limits exist? (capacity, budget, time, demand requirements)
   • Hard constraints (must satisfy) vs. soft constraints (preferences)?
   • How many constraints? (dozens vs. millions)

4. Data Availability

   • All parameters known with certainty?
   • Uncertainty or variability?
   • Data quality and completeness?

5. Technical Environment

   • Optimization expertise in team?
   • Solver access? (commercial vs. open-source)
   • Computational resources?
   • Integration requirements?

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Optimization Problem Types

Linear Programming (LP)

Characteristics:

• Continuous decision variables
• Linear objective function
• Linear constraints
• Fast to solve (polynomial time)

Supply Chain Applications:

• Production planning
• Transportation optimization
• Blending problems
• Network flow optimization

Example: Production Planning

from pulp import *
import pandas as pd

def production_planning_lp(products, resources, demand, capacity, costs):
    """
    Determine optimal production quantities

    Decision: How much to produce of each product?
    Objective: Minimize total production cost
    Constraints: Resource capacity, meet demand

    products: list of product names
    resources: list of resource names
    demand: dict {product: demand_quantity}
    capacity: dict {resource: available_capacity}
    costs: dict {(product, resource): cost_per_unit}
    """

    # Create problem
    prob = LpProblem("Production_Planning", LpMinimize)

    # Decision variables: production quantity for each product
    production = LpVariable.dicts("Produce",
                                  products,
                                  lowBound=0,
                                  cat='Continuous')

    # Objective: Minimize total cost
    prob += lpSum([costs[product] * production[product]
                   for product in products]), "Total_Cost"

    # Constraints
    # 1. Meet demand
    for product in products:
        prob += production[product] >= demand[product], \
                f"Meet_Demand_{product}"

    # 2. Resource capacity
    # Assume resource_usage[product][resource] is known
    for resource in resources:
        prob += lpSum([resource_usage[product][resource] * production[product]
                      for product in products]) <= capacity[resource], \
                f"Capacity_{resource}"

    # Solve
    prob.solve(PULP_CBC_CMD(msg=1))

    # Extract results
    results = {
        'status': LpStatus[prob.status],
        'total_cost': value(prob.objective),
        'production_plan': {
            product: production[product].varValue
            for product in products
        }
    }

    return results

# Example data
products = ['Product_A', 'Product_B', 'Product_C']
resources = ['Labor', 'Machine', 'Material']

demand = {
    'Product_A': 100,
    'Product_B': 150,
    'Product_C': 80
}

capacity = {
    'Labor': 1000,  # hours
    'Machine': 800,  # hours
    'Material': 5000  # units
}

costs = {
    'Product_A': 50,
    'Product_B': 75,
    'Product_C': 60
}

resource_usage = {
    'Product_A': {'Labor': 2, 'Machine': 3, 'Material': 10},
    'Product_B': {'Labor': 3, 'Machine': 2, 'Material': 15},
    'Product_C': {'Labor': 1.5, 'Machine': 4, 'Material': 12}
}

result = production_planning_lp(products, resources, demand, capacity, costs)
print(f"Optimal Cost: ${result['total_cost']:,.2f}")
print("Production Plan:")
for product, qty in result['production_plan'].items():
    print(f"  {product}: {qty:.2f} units")