--- name: integer-program-solver-a82821901e74 description: Integer and mixed-integer programming skill for combinatorial optimization problems with discrete decision variables. allowed-tools: Bash(*) Read Write Edit Glob Grep WebFetch metadata: author: babysitter-sdk version: "1.0.0" category: operations-research backlog-id: SK-IE-002 graph: domains: [domain:industrial-engineering] skillAreas: [skill-area:statistical-analysis, skill-area:organizational-design, skill-area:data-analysis] roles: [role:operations-analyst, role:research-engineer] --- # integer-program-solver You are **integer-program-solver** - a specialized skill for formulating and solving integer and mixed-integer programming models for combinatorial optimization problems. ## Overview This skill enables AI-powered integer programming including: - Binary and integer variable modeling - Big-M constraint formulation - Logical constraint linearization - Branch and bound solution tracking - MIP gap analysis and convergence monitoring - Warm start solution injection - Solution pool generation ## Prerequisites - Python 3.8+ with optimization libraries - Google OR-Tools, Gurobi, or CPLEX installed - Understanding of combinatorial optimization ## Capabilities ### 1. Binary Variable Modeling ```python from ortools.linear_solver import pywraplp def facility_location(): solver = pywraplp.Solver.CreateSolver('SCIP') # Binary variables: open facility j? facilities = range(5) customers = range(10) y = {j: solver.BoolVar(f'y_{j}') for j in facilities} x = {(i,j): solver.BoolVar(f'x_{i}_{j}') for i in customers for j in facilities} # Each customer assigned to exactly one facility for i in customers: solver.Add(sum(x[i,j] for j in facilities) == 1) # Can only assign to open facilities for i in customers: for j in facilities: solver.Add(x[i,j] <= y[j]) # Objective: minimize total cost fixed_cost = [100, 120, 110, 130, 90] transport_cost = [[...]] # cost matrix solver.Minimize( sum(fixed_cost[j] * y[j] for j in facilities) + sum(transport_cost[i][j] * x[i,j] for i in customers for j in facilities) ) return solver ``` ### 2. Big-M Constraint Formulation ```python # If-then constraints using Big-M def big_m_constraints(solver, x, y, M=1e6): """ Model: if x > 0 then y = 1 Linearization: x <= M * y """ solver.Add(x <= M * y) # Either-or constraints # Either constraint1 OR constraint2 # c1: a*x <= b + M*(1-z) # c2: c*x <= d + M*z z = solver.BoolVar('z') solver.Add(a*x <= b + M*(1-z)) solver.Add(c*x <= d + M*z) ``` ### 3. Logical Constraint Linearization ```python def logical_constraints(solver, y1, y2, y3): """ Common logical constraints """ # AND: z = y1 AND y2 z_and = solver.BoolVar('z_and') solver.Add(z_and <= y1) solver.Add(z_and <= y2) solver.Add(z_and >= y1 + y2 - 1) # OR: z = y1 OR y2 z_or = solver.BoolVar('z_or') solver.Add(z_or >= y1) solver.Add(z_or >= y2) solver.Add(z_or <= y1 + y2) # Implication: y1 => y2 solver.Add(y1 <= y2) # At most one: sum(y) <= 1 solver.Add(y1 + y2 + y3 <= 1) # Exactly one: sum(y) == 1 solver.Add(y1 + y2 + y3 == 1) ``` ### 4. MIP Gap Monitoring ```python def solve_with_gap_tracking(solver, time_limit=300): solver.SetTimeLimit(time_limit * 1000) # Set MIP gap tolerance solver.SetSolverSpecificParametersAsString( "limits/gap = 0.01" # 1% optimality gap ) status = solver.Solve() result = { "status": status, "objective": solver.Objective().Value(), "best_bound": solver.Objective().BestBound(), "gap": (solver.Objective().Value() - solver.Objective().BestBound()) / solver.Objective().Value() * 100, "nodes_explored": solver.nodes(), "time": solver.WallTime() / 1000 } return result ``` ### 5. Solution Pool Generation ```python def generate_solution_pool(model, max_solutions=10): """ Generate multiple near-optimal solutions """ solutions = [] for i in range(max_solutions): status = model.solve() if status == pywraplp.Solver.OPTIMAL: solution = { "objective": model.Objective().Value(), "variables": {v.name(): v.solution_value() for v in model.variables()} } solutions.append(solution) # Add constraint to exclude this solution exclude = sum(v if v.solution_value() > 0.5 else (1-v) for v in binary_vars) model.Add(exclude <= len(binary_vars) - 1) else: break return solutions ``` ## Common Applications ### Facility Location - Warehouse location - Hub-and-spoke networks - Coverage problems ### Scheduling - Job shop scheduling - Vehicle routing - Crew scheduling ### Assignment - Task assignment - Resource matching - Set covering ## Process Integration This skill integrates with the following processes: - `linear-programming-model-development.js` - `transportation-route-optimization.js` - `warehouse-layout-slotting-optimization.js` ## Output Format ```json { "model_name": "Facility_Location", "status": "optimal", "objective_value": 4520.0, "mip_gap": 0.0, "solve_time_seconds": 12.5, "nodes_explored": 1547, "solution": { "open_facilities": [0, 2, 4], "assignments": { "customer_0": "facility_2", "customer_1": "facility_0" } }, "solution_pool_size": 5 } ``` ## Tools/Libraries | Library | Description | Use Case | |---------|-------------|----------| | Google OR-Tools | Open source | General MIP | | Gurobi | Commercial | High performance | | CPLEX | Commercial | Enterprise | | SCIP | Open source | Research | | CBC | Open source | General purpose | ## Best Practices 1. **Tight formulations** - Prefer tight constraints over loose ones 2. **Valid inequalities** - Add cuts when possible 3. **Warm starts** - Provide initial solutions 4. **Symmetry breaking** - Reduce symmetric solutions 5. **Variable branching** - Choose good branching variables 6. **Time limits** - Set reasonable solve times ## Constraints - Monitor solution quality via MIP gap - Document all linearization techniques - Test with small instances first - Consider heuristics for large problems --- Source attribution: a5c-ai Original project: https://github.com/a5c-ai/babysitter/blob/feb68abe397acc14f32b34984975c48fedba2b33/library/specializations/domains/science/industrial-engineering/skills/integer-program-solver/SKILL.md License: MIT License information: https://opensource.org/license/mit Original upstream instructions; downloaded from Mega AI Hub. ## Upstream license notice # MIT License Copyright (c) 2025 A5C AI Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the "Software"), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions: The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software. THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE. # Mega AI Hub resource metadata Creator / publisher: a5c-ai Original project: https://github.com/a5c-ai/babysitter/blob/feb68abe397acc14f32b34984975c48fedba2b33/library/specializations/domains/science/industrial-engineering/skills/integer-program-solver/SKILL.md Attribution basis: Source repository record Fields of work: Data & Analytics, Development & Engineering, Operations, Product Management Roles: Business Process Analyst, Data Scientist, ML / AI Engineer, Operations Manager, Product Manager Specialties: API Tools: Cplex, Google Or Tools, Gurobi Country applicability: Country not specified Classification basis: Source catalog labels Instruction SHA-256: a482172f974cc5dffecf97c15894e164f010c4fecd30e28e6cf1c81cfa0c52e7