Linear Programming Solver

Overview

The Linear Programming Solver skill provides comprehensive capabilities for formulating and solving linear optimization problems. It supports resource allocation, production planning, scheduling, and other business optimization challenges through efficient solver integration and solution analysis.

Capabilities

• LP model formulation assistance
• Solver integration (GLPK, CBC, CPLEX, Gurobi)
• Sensitivity analysis (shadow prices, reduced costs)
• Infeasibility diagnosis
• Unboundedness detection
• Integer programming support
• Multi-objective LP (goal programming)
• Solution interpretation

Used By Processes

• Prescriptive Analytics and Optimization
• Resource Allocation
• Supply Chain Optimization

Usage

Problem Formulation

# Define LP problem
lp_problem = {
    "name": "Production Planning",
    "sense": "maximize",  # or "minimize"
    "decision_variables": {
        "product_A": {"type": "continuous", "lower_bound": 0, "upper_bound": 1000},
        "product_B": {"type": "continuous", "lower_bound": 0, "upper_bound": 800},
        "product_C": {"type": "integer", "lower_bound": 0}  # integer variable
    },
    "objective": {
        "expression": "50*product_A + 40*product_B + 60*product_C",
        "description": "Maximize total profit"
    },
    "constraints": [
        {
            "name": "labor_hours",
            "expression": "2*product_A + 3*product_B + 4*product_C <= 2400",
            "description": "Total labor hours available"
        },
        {
            "name": "machine_time",
            "expression": "3*product_A + 2*product_B + 3*product_C <= 2000",
            "description": "Machine time capacity"
        },
        {
            "name": "raw_material",
            "expression": "product_A + product_B + product_C <= 1200",
            "description": "Raw material availability"
        },
        {
            "name": "demand_A",
            "expression": "product_A >= 100",
            "description": "Minimum demand for product A"
        }
    ]
}

Solver Configuration

# Solver settings
solver_config = {
    "solver": "CBC",  # or "GLPK", "CPLEX", "GUROBI"
    "time_limit": 300,  # seconds
    "mip_gap": 0.01,  # 1% optimality gap for MIP
    "threads": 4,
    "presolve": True,
    "cuts": "automatic"
}

Sensitivity Analysis

# Request sensitivity information
sensitivity_config = {
    "shadow_prices": True,
    "reduced_costs": True,
    "allowable_ranges": True,
    "what_if": [
        {"constraint": "labor_hours", "change": 100},
        {"objective_coeff": "product_A", "change": 5}
    ]
}

Common LP Problem Types

Problem Type	Objective	Key Constraints
Production Planning	Maximize profit	Capacity, demand
Transportation	Minimize cost	Supply, demand
Assignment	Minimize cost/time	One-to-one matching
Blending	Minimize cost	Quality specs, availability
Network Flow	Min cost/max flow	Flow balance, capacity
Portfolio	Maximize return	Risk, budget, diversification

Input Schema

{
  "problem_definition": {
    "name": "string",
    "sense": "maximize|minimize",
    "decision_variables": "object",
    "objective": {
      "expression": "string",
      "description": "string"
    },
    "constraints": ["object"]
  },
  "solver_config": {
    "solver": "string",
    "time_limit": "number",
    "mip_gap": "number"
  },
  "analysis_options": {
    "sensitivity": "boolean",
    "what_if": ["object"],
    "report_format": "string"
  }
}

Output Schema

{
  "status": "Optimal|Infeasible|Unbounded|TimeLimit",
  "objective_value": "number",
  "solution": {
    "variable_name": "number"
  },
  "sensitivity": {
    "shadow_prices": {
      "constraint_name": {
        "value": "number",
        "allowable_increase": "number",
        "allowable_decrease": "number"
      }
    },
    "reduced_costs": {
      "variable_name": {
        "value": "number",
        "allowable_increase": "number",
        "allowable_decrease": "number"
      }
    }
  },
  "infeasibility_analysis": {
    "conflicting_constraints": ["string"],
    "suggested_relaxations": ["object"]
  },
  "what_if_results": ["object"],
  "solve_time": "number"
}

Sensitivity Interpretation

Metric	Meaning	Use
Shadow Price	Value of relaxing constraint by 1 unit	Prioritize constraint relief
Reduced Cost	Cost of forcing non-basic variable into solution	Evaluate non-optimal alternatives
Allowable Range	Range where basis stays optimal	Assess stability of solution

Best Practices

1. Verify model formulation with simple test cases
2. Check units consistency in coefficients
3. Analyze infeasibility before debugging manually
4. Use shadow prices to guide resource acquisition
5. Consider integer programming only when necessary (harder to solve)
6. Validate solution against business constraints
7. Document model assumptions clearly

Infeasibility Handling

When model is infeasible:

1. Identify Irreducible Infeasible Subset (IIS)
2. Relax constraints using elastic variables
3. Prioritize constraint satisfaction
4. Use goal programming for conflicting objectives

Integration Points

• Feeds into Optimization Specialist agent
• Connects with Sensitivity Analyzer for robustness
• Supports Constraint Satisfaction Solver for hybrid problems
• Integrates with Decision Visualization for solution display