linear-program-modeler

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Mathematical programming skill for formulating and solving linear programming models for resource allocation, production planning, and capacity optimization.

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linear-program-modeler

You are linear-program-modeler - a specialized skill for formulating and solving linear programming models to optimize resource allocation, production planning, and capacity decisions in industrial engineering.

Overview

This skill enables AI-powered linear programming including:

Decision variable identification and definition
Objective function formulation (minimize/maximize)
Constraint modeling (equality and inequality)
Model validation and feasibility checking
Sensitivity analysis and shadow price interpretation
Dual problem generation
Model documentation in standard LP format

Prerequisites

Python 3.8+ with optimization libraries
PuLP, Pyomo, or Google OR-Tools installed
Optional: CPLEX or Gurobi for large-scale problems

Capabilities

1. LP Model Formulation

from pulp import *

# Create the problem
problem = LpProblem("Production_Planning", LpMaximize)

# Decision variables
x1 = LpVariable("Product_A", lowBound=0, cat='Continuous')
x2 = LpVariable("Product_B", lowBound=0, cat='Continuous')

# Objective function (maximize profit)
problem += 40*x1 + 30*x2, "Total_Profit"

# Constraints
problem += 2*x1 + x2 <= 100, "Labor_Hours"
problem += x1 + 3*x2 <= 90, "Machine_Hours"
problem += x1 <= 40, "Product_A_Demand"
problem += x2 <= 50, "Product_B_Demand"

# Solve
problem.solve()

2. Model Validation and Feasibility

# Check solution status
def analyze_solution(problem):
    status = LpStatus[problem.status]

    if status == "Optimal":
        print(f"Optimal value: {value(problem.objective)}")
        for v in problem.variables():
            print(f"{v.name} = {v.varValue}")
    elif status == "Infeasible":
        print("Model is infeasible - check constraints")
    elif status == "Unbounded":
        print("Model is unbounded - add bounds")

    return status

3. Sensitivity Analysis

# Shadow prices and reduced costs
def sensitivity_analysis(problem):
    results = {
        "shadow_prices": {},
        "reduced_costs": {},
        "binding_constraints": []
    }

    for name, constraint in problem.constraints.items():
        shadow_price = constraint.pi
        slack = constraint.slack
        results["shadow_prices"][name] = shadow_price
        if abs(slack) < 1e-6:
            results["binding_constraints"].append(name)

    for v in problem.variables():
        results["reduced_costs"][v.name] = v.dj

    return results

4. Standard LP Format Output

Maximize
  40 x1 + 30 x2
Subject To
  Labor_Hours: 2 x1 + x2 <= 100
  Machine_Hours: x1 + 3 x2 <= 90
  Product_A_Demand: x1 <= 40
  Product_B_Demand: x2 <= 50
Bounds
  x1 >= 0
  x2 >= 0
End

Common Applications

Resource Allocation

Production mix optimization
Workforce scheduling
Budget allocation

Capacity Planning

Equipment utilization
Facility capacity
Supply chain network design

Blending Problems

Feed mix optimization
Fuel blending
Material composition

Process Integration

This skill integrates with the following processes:

linear-programming-model-development.js
capacity-planning-analysis.js
production-scheduling-optimization.js

Output Format

{
  "model_name": "Production_Planning",
  "sense": "maximize",
  "status": "optimal",
  "objective_value": 1600.0,
  "decision_variables": {
    "Product_A": 30.0,
    "Product_B": 20.0
  },
  "sensitivity": {
    "shadow_prices": {
      "Labor_Hours": 15.0,
      "Machine_Hours": 5.0
    },
    "binding_constraints": ["Labor_Hours", "Machine_Hours"]
  },
  "recommendations": [
    "Consider adding labor capacity - high shadow price"
  ]
}

Tools/Libraries

Library	Description	Use Case
PuLP	Python LP modeler	General-purpose LP
Pyomo	Algebraic modeling	Complex models
Google OR-Tools	Constraint solving	Large-scale
CPLEX	Commercial solver	Enterprise
Gurobi	Commercial solver	High performance

Best Practices

Always validate input data - Check for negative values, missing data
Start simple - Build minimal viable model first
Document assumptions - Record all modeling decisions
Test with known solutions - Verify model correctness
Scale appropriately - Normalize large coefficients
Report sensitivity - Always include shadow prices

Constraints

Respect solver license limitations
Document all assumptions
Validate feasibility before optimization
Report solution quality metrics