Linear Programming Solver Skill

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

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

Infeasibility Handling

When model is infeasible:

Identify Irreducible Infeasible Subset (IIS)
Relax constraints using elastic variables
Prioritize constraint satisfaction
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

Agent Skills: Linear Programming Solver

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