OR-Tools Ruby

OR-Tools - operations research tools - for Ruby

Installation

Add this line to your application’s Gemfile:

gem "or-tools"

Installation can take a few minutes as OR-Tools downloads and builds.

Getting Started

Higher Level Interfaces

Scheduling
Seating
Traveling Salesperson Problem (TSP)
Sudoku

MathOpt

Basic example

Linear Optimization

Solving an LP Problem

Integer Optimization

Solving a MIP Problem

Constraint Optimization

CP-SAT Solver
Solving a CP Problem
Cryptarithmetic
The N-queens Problem
Setting Solver Limits

Assignment

Solving an Assignment Problem
Assignment with Teams of Workers
Linear Sum Assignment Solver

Routing

Traveling Salesperson Problem (TSP)
Vehicle Routing Problem (VRP)
Capacity Constraints
Pickups and Deliveries
Time Window Constraints
Resource Constraints
Penalties and Dropping Visits
Routing Options

Bin Packing

The Knapsack Problem
Multiple Knapsacks
Bin Packing Problem

Network Flows

Maximum Flows
Minimum Cost Flows
Assignment as a Min Cost Flow Problem

Scheduling

Employee Scheduling
The Job Shop Problem

Other Examples

Sudoku
Wedding Seating Chart
Set Partitioning

Higher Level Interfaces

Scheduling

Specify people and their availabililty

people = [
  {
    availability: [
      {starts_at: Time.parse("2025-01-01 08:00:00"), ends_at: Time.parse("2025-01-01 16:00:00")},
      {starts_at: Time.parse("2025-01-02 08:00:00"), ends_at: Time.parse("2025-01-02 16:00:00")}
    ],
    max_hours: 40 # optional, applies to entire scheduling period
  },
  {
    availability: [
      {starts_at: Time.parse("2025-01-01 08:00:00"), ends_at: Time.parse("2025-01-01 16:00:00")},
      {starts_at: Time.parse("2025-01-03 08:00:00"), ends_at: Time.parse("2025-01-03 16:00:00")}
    ],
    max_hours: 20
  }
]

Specify shifts

shifts = [
  {starts_at: Time.parse("2025-01-01 08:00:00"), ends_at: Time.parse("2025-01-01 16:00:00")},
  {starts_at: Time.parse("2025-01-02 08:00:00"), ends_at: Time.parse("2025-01-02 16:00:00")},
  {starts_at: Time.parse("2025-01-03 08:00:00"), ends_at: Time.parse("2025-01-03 16:00:00")}
]

Run the scheduler

scheduler = ORTools::BasicScheduler.new(people: people, shifts: shifts)

The scheduler maximizes the number of assigned hours. A person must be available for the entire shift to be considered for it.

Get assignments (returns indexes of people and shifts)

scheduler.assignments
# [
#   {person: 2, shift: 0},
#   {person: 0, shift: 1},
#   {person: 1, shift: 2}
# ]

Get assigned hours and total hours

scheduler.assigned_hours
scheduler.total_hours

Feel free to create an issue if you have a scheduling use case that’s not covered.

Seating

Create a seating chart based on personal connections. Uses this approach.

Specify connections

connections = [
  {people: ["A", "B", "C"], weight: 2},
  {people: ["C", "D", "E", "F"], weight: 1}
]

Use different weights to prioritize seating. For a wedding, it may look like:

connections = [
  {people: knows_partner1, weight: 1},
  {people: knows_partner2, weight: 1},
  {people: relationship1, weight: 100},
  {people: relationship2, weight: 100},
  {people: relationship3, weight: 100},
  {people: friend_group1, weight: 10},
  {people: friend_group2, weight: 10},
  # ...
]

If two people have multiple connections, weights are added.

Specify tables and their capacity

tables = [3, 3]

Assign seats

seating = ORTools::Seating.new(connections: connections, tables: tables)

Each person will have a connection with at least one other person at their table.

Get tables

seating.assigned_tables

Get assignments by person

seating.assignments

Get all connections for a person

seating.connections_for(person)

Get connections for a person at their table

seating.connections_for(person, same_table: true)

Traveling Salesperson Problem (TSP)

Create locations - the first location will be the starting and ending point

locations = [
  {name: "Tokyo", latitude: 35.6762, longitude: 139.6503},
  {name: "Delhi", latitude: 28.7041, longitude: 77.1025},
  {name: "Shanghai", latitude: 31.2304, longitude: 121.4737},
  {name: "São Paulo", latitude: -23.5505, longitude: -46.6333},
  {name: "Mexico City", latitude: 19.4326, longitude: -99.1332},
  {name: "Cairo", latitude: 30.0444, longitude: 31.2357},
  {name: "Mumbai", latitude: 19.0760, longitude: 72.8777},
  {name: "Beijing", latitude: 39.9042, longitude: 116.4074},
  {name: "Dhaka", latitude: 23.8103, longitude: 90.4125},
  {name: "Osaka", latitude: 34.6937, longitude: 135.5023},
  {name: "New York City", latitude: 40.7128, longitude: -74.0060},
  {name: "Karachi", latitude: 24.8607, longitude: 67.0011},
  {name: "Buenos Aires", latitude: -34.6037, longitude: -58.3816}
]

Locations can have any fields - only latitude and longitude are required

Get route

tsp = ORTools::TSP.new(locations)
tsp.route # [{name: "Tokyo", ...}, {name: "Osaka", ...}, ...]

Get distances between locations on route

tsp.distances # [392.441, 1362.926, 1067.31, ...]

Distances are in kilometers - multiply by 0.6214 for miles

Get total distance

tsp.total_distance

Sudoku

Create a puzzle with zeros in empty cells

grid = [
  [0, 6, 0, 0, 5, 0, 0, 2, 0],
  [0, 0, 0, 3, 0, 0, 0, 9, 0],
  [7, 0, 0, 6, 0, 0, 0, 1, 0],
  [0, 0, 6, 0, 3, 0, 4, 0, 0],
  [0, 0, 4, 0, 7, 0, 1, 0, 0],
  [0, 0, 5, 0, 9, 0, 8, 0, 0],
  [0, 4, 0, 0, 0, 1, 0, 0, 6],
  [0, 3, 0, 0, 0, 8, 0, 0, 0],
  [0, 2, 0, 0, 4, 0, 0, 5, 0]
]
sudoku = ORTools::Sudoku.new(grid)
sudoku.solution

It can also solve more advanced puzzles like The Miracle

grid = [
  [0, 0, 0, 0, 0, 0, 0, 0, 0],
  [0, 0, 0, 0, 0, 0, 0, 0, 0],
  [0, 0, 0, 0, 0, 0, 0, 0, 0],
  [0, 0, 0, 0, 0, 0, 0, 0, 0],
  [0, 0, 1, 0, 0, 0, 0, 0, 0],
  [0, 0, 0, 0, 0, 0, 2, 0, 0],
  [0, 0, 0, 0, 0, 0, 0, 0, 0],
  [0, 0, 0, 0, 0, 0, 0, 0, 0],
  [0, 0, 0, 0, 0, 0, 0, 0, 0]
]
sudoku = ORTools::Sudoku.new(grid, anti_knight: true, anti_king: true, non_consecutive: true)
sudoku.solution

And this 4-digit puzzle

grid = [
  [0, 0, 0, 0, 0, 0, 0, 0, 0],
  [0, 0, 0, 0, 0, 0, 0, 0, 0],
  [0, 0, 0, 0, 0, 0, 0, 0, 0],
  [3, 8, 4, 0, 0, 0, 0, 0, 0],
  [0, 0, 0, 0, 0, 0, 0, 0, 0],
  [0, 0, 0, 0, 0, 0, 0, 0, 0],
  [0, 0, 0, 0, 0, 0, 0, 0, 0],
  [0, 0, 0, 0, 0, 0, 0, 0, 0],
  [0, 0, 0, 0, 0, 0, 0, 0, 2]
]
sudoku = ORTools::Sudoku.new(grid, x: true, anti_knight: true, magic_square: true)
sudoku.solution

MathOpt

Basic Example

Guide

# build the model
model = ORTools::MathOpt::Model.new("getting_started_lp")
x = model.add_variable(-1.0, 1.5, "x")
y = model.add_variable(0.0, 1.0, "y")
model.add_linear_constraint(x + y <= 1.5)
model.maximize(x + 2 * y)

# solve
result = model.solve

# inspect the solution
puts "Objective value: #{result.objective_value}"
puts "x: #{result.variable_values[x]}"
puts "y: #{result.variable_values[y]}"

Linear Optimization

Solving an LP Problem

Guide

# declare the solver
solver = ORTools::Solver.new("GLOP")

# create the variables
x = solver.num_var(0, solver.infinity, "x")
y = solver.num_var(0, solver.infinity, "y")
puts "Number of variables = #{solver.num_variables}"

# define the constraints
solver.add(x + 2 * y <= 14)
solver.add(3 * x - y >= 0)
solver.add(x - y <= 2)
puts "Number of constraints = #{solver.num_constraints}"

# define the objective function
solver.maximize(3 * x + 4 * y)

# invoke the solver
status = solver.solve

# display the solution
if status == :optimal
  puts "Solution:"
  puts "Objective value = #{solver.objective.value}"
  puts "x = #{x.solution_value}"
  puts "y = #{y.solution_value}"
else
  puts "The problem does not have an optimal solution."
end

Integer Optimization

Solving a MIP Problem

Guide

# declare the MIP solver
solver = ORTools::Solver.new("CBC")

# define the variables
infinity = solver.infinity
x = solver.int_var(0, infinity, "x")
y = solver.int_var(0, infinity, "y")

puts "Number of variables = #{solver.num_variables}"

# define the constraints
solver.add(x + 7 * y <= 17.5)
solver.add(x <= 3.5)

puts "Number of constraints = #{solver.num_constraints}"

# define the objective
solver.maximize(x + 10 * y)

# call the solver
status = solver.solve

# display the solution
if status == :optimal
  puts "Solution:"
  puts "Objective value = #{solver.objective.value}"
  puts "x = #{x.solution_value}"
  puts "y = #{y.solution_value}"
else
  puts "The problem does not have an optimal solution."
end

Constraint Optimization

CP-SAT Solver

Guide

# declare the model
model = ORTools::CpModel.new

# create the variables
num_vals = 3
x = model.new_int_var(0, num_vals - 1, "x")
y = model.new_int_var(0, num_vals - 1, "y")
z = model.new_int_var(0, num_vals - 1, "z")

# create the constraint
model.add(x != y)

# call the solver
solver = ORTools::CpSolver.new
status = solver.solve(model)

# display the first solution
if status == :optimal || status == :feasible
  puts "x = #{solver.value(x)}"
  puts "y = #{solver.value(y)}"
  puts "z = #{solver.value(z)}"
else
  puts "No solution found."
end

Solving a CP Problem

Guide

# declare the model
model = ORTools::CpModel.new

# create the variables
var_upper_bound = [50, 45, 37].max
x = model.new_int_var(0, var_upper_bound, "x")
y = model.new_int_var(0, var_upper_bound, "y")
z = model.new_int_var(0, var_upper_bound, "z")

# define the constraints
model.add(x*2 + y*7 + z*3 <= 50)
model.add(x*3 - y*5 + z*7 <= 45)
model.add(x*5 + y*2 - z*6 <= 37)

# define the objective function
model.maximize(x*2 + y*2 + z*3)

# call the solver
solver = ORTools::CpSolver.new
status = solver.solve(model)

# display the solution
if status == :optimal || status == :feasible
  puts "Maximum of objective function: #{solver.objective_value}"
  puts "x = #{solver.value(x)}"
  puts "y = #{solver.value(y)}"
  puts "z = #{solver.value(z)}"
else
  puts "No solution found."
end

Cryptarithmetic

Guide

# define the variables
model = ORTools::CpModel.new

base = 10

c = model.new_int_var(1, base - 1, "C")
p = model.new_int_var(0, base - 1, "P")
i = model.new_int_var(1, base - 1, "I")
s = model.new_int_var(0, base - 1, "S")
f = model.new_int_var(1, base - 1, "F")
u = model.new_int_var(0, base - 1, "U")
n = model.new_int_var(0, base - 1, "N")
t = model.new_int_var(1, base - 1, "T")
r = model.new_int_var(0, base - 1, "R")
e = model.new_int_var(0, base - 1, "E")

letters = [c, p, i, s, f, u, n, t, r, e]

# define the constraints
model.add_all_different(letters)

model.add(c * base + p + i * base + s + f * base * base + u * base +
  n == t * base * base * base + r * base * base + u * base + e)

# define the solution printer
class VarArraySolutionPrinter < ORTools::CpSolverSolutionCallback
  attr_reader :solution_count

  def initialize(variables)
    super()
    @variables = variables
    @solution_count = 0
  end

  def on_solution_callback
    @solution_count += 1
    @variables.each do |v|
      print "%s=%i " % [v.name, value(v)]
    end
    puts
  end
end

# invoke the solver
solver = ORTools::CpSolver.new
solution_printer = VarArraySolutionPrinter.new(letters)
status = solver.search_for_all_solutions(model, solution_printer)

puts
puts "Statistics"
puts "  - status          : %s" % status
puts "  - conflicts       : %i" % solver.num_conflicts
puts "  - branches        : %i" % solver.num_branches
puts "  - wall time       : %f s" % solver.wall_time
puts "  - solutions found : %i" % solution_printer.solution_count

The N-queens Problem

Guide

# declare the model
board_size = 8
model = ORTools::CpModel.new

# create the variables
queens = board_size.times.map { |i| model.new_int_var(0, board_size - 1, "x%i" % i) }

# create the constraints
board_size.times do |i|
  diag1 = []
  diag2 = []
  board_size.times do |j|
    q1 = model.new_int_var(0, 2 * board_size, "diag1_%i" % i)
    diag1 << q1
    model.add(q1 == queens[j] + j)
    q2 = model.new_int_var(-board_size, board_size, "diag2_%i" % i)
    diag2 << q2
    model.add(q2 == queens[j] - j)
  end
  model.add_all_different(diag1)
  model.add_all_different(diag2)
end

# create a solution printer
class SolutionPrinter < ORTools::CpSolverSolutionCallback
  attr_reader :solution_count

  def initialize(variables)
    super()
    @variables = variables
    @solution_count = 0
  end

  def on_solution_callback
    @solution_count += 1
    @variables.each do |v|
      print "%s = %i " % [v.name, value(v)]
    end
    puts
  end
end

# call the solver and display the results
solver = ORTools::CpSolver.new
solution_printer = SolutionPrinter.new(queens)
status = solver.search_for_all_solutions(model, solution_printer)
puts
puts "Solutions found : %i" % solution_printer.solution_count

Setting Solver Limits

Guide

# create the model
model = ORTools::CpModel.new

# create the variables
num_vals = 3
x = model.new_int_var(0, num_vals - 1, "x")
y = model.new_int_var(0, num_vals - 1, "y")
z = model.new_int_var(0, num_vals - 1, "z")

# add an all-different constraint
model.add(x != y)

# create the solver
solver = ORTools::CpSolver.new

# set a time limit of 10 seconds.
solver.parameters.max_time_in_seconds = 10

# solve the model
status = solver.solve(model)

# display the first solution
if status == :optimal
  puts "x = #{solver.value(x)}"
  puts "y = #{solver.value(y)}"
  puts "z = #{solver.value(z)}"
end

Assignment

Solving an Assignment Problem

Guide

# create the data
costs = [
  [90, 80, 75, 70],
  [35, 85, 55, 65],
  [125, 95, 90, 95],
  [45, 110, 95, 115],
  [50, 100, 90, 100]
]
num_workers = costs.length
num_tasks = costs[0].length

# create the solver
solver = ORTools::Solver.new("CBC")

# create the variables
x = {}
num_workers.times do |i|
  num_tasks.times do |j|
    x[[i, j]] = solver.int_var(0, 1, "")
  end
end

# create the constraints
# each worker is assigned to at most 1 task
num_workers.times do |i|
  solver.add(num_tasks.times.sum { |j| x[[i, j]] } <= 1)
end

# each task is assigned to exactly one worker
num_tasks.times do |j|
  solver.add(num_workers.times.sum { |i| x[[i, j]] } == 1)
end

# create the objective function
objective_terms = []
num_workers.times do |i|
  num_tasks.times do |j|
    objective_terms << (costs[i][j] * x[[i, j]])
  end
end
solver.minimize(objective_terms.sum)

# invoke the solver
status = solver.solve

# print the solution
if status == :optimal || status == :feasible
  puts "Total cost = #{solver.objective.value}"
  num_workers.times do |i|
    num_tasks.times do |j|
      # test if x[i,j] is 1 (with tolerance for floating point arithmetic)
      if x[[i, j]].solution_value > 0.5
        puts "Worker #{i} assigned to task #{j}. Cost = #{costs[i][j]}"
      end
    end
  end
else
  puts "No solution found."
end

Assignment with Teams of Workers

Guide

# create the data
costs = [
  [90, 76, 75, 70],
  [35, 85, 55, 65],
  [125, 95, 90, 105],
  [45, 110, 95, 115],
  [60, 105, 80, 75],
  [45, 65, 110, 95]
]
num_workers = costs.length
num_tasks = costs[1].length

team1 = [0, 2, 4]
team2 = [1, 3, 5]
team_max = 2

# create the solver
solver = ORTools::Solver.new("CBC")

# create the variables
x = {}
num_workers.times do |i|
  num_tasks.times do |j|
    x[[i, j]] = solver.bool_var("x[#{i},#{j}]")
  end
end

# add the constraints
# each worker is assigned at most 1 task
num_workers.times do |i|
  solver.add(num_tasks.times.sum { |j| x[[i, j]] } <= 1)
end

# each task is assigned to exactly one worker
num_tasks.times do |j|
  solver.add(num_workers.times.sum { |i| x[[i, j]] } == 1)
end

# each team takes at most two tasks
solver.add(team1.flat_map { |i| num_tasks.times.map { |j| x[[i, j]] } }.sum <= team_max)
solver.add(team2.flat_map { |i| num_tasks.times.map { |j| x[[i, j]] } }.sum <= team_max)

# create the objective
solver.minimize(
  num_workers.times.flat_map { |i| num_tasks.times.map { |j| x[[i, j]] * costs[i][j] } }.sum
)

# invoke the solver
status = solver.solve

# display the results
if status == :optimal || status == :feasible
  puts "Total cost = #{solver.objective.value}"
  num_workers.times do |worker|
    num_tasks.times do |task|
      if x[[worker, task]].solution_value > 0.5
        puts "Worker #{worker} assigned to task #{task}. Cost = #{costs[worker][task]}"
      end
    end
  end
else
  puts "No solution found."
end

Linear Sum Assignment Solver

Guide

# create the data
costs = [
  [90, 76, 75, 70],
  [35, 85, 55, 65],
  [125, 95, 90, 105],
  [45, 110, 95, 115],
]
num_workers = costs.length
num_tasks = costs[0].length

# create the solver
assignment = ORTools::LinearSumAssignment.new

# add the constraints
num_workers.times do |worker|
  num_tasks.times do |task|
    if costs[worker][task]
      assignment.add_arc_with_cost(worker, task, costs[worker][task])
    end
  end
end

# invoke the solver
status = assignment.solve

# display the results
case status
when :optimal
  puts "Total cost = #{assignment.optimal_cost}"
  assignment.num_nodes.times do |i|
    puts "Worker #{i} assigned to task #{assignment.right_mate(i)}. Cost = #{assignment.assignment_cost(i)}"
  end
when :infeasible
  puts "No assignment is possible."
when :possible_overflow
  puts "Some input costs are too large and may cause an integer overflow."
end

Routing

Traveling Salesperson Problem (TSP)

Guide

# create the data
data = {}
data[:distance_matrix] = [
  [0, 2451, 713, 1018, 1631, 1374, 2408, 213, 2571, 875, 1420, 2145, 1972],
  [2451, 0, 1745, 1524, 831, 1240, 959, 2596, 403, 1589, 1374, 357, 579],
  [713, 1745, 0, 355, 920, 803, 1737, 851, 1858, 262, 940, 1453, 1260],
  [1018, 1524, 355, 0, 700, 862, 1395, 1123, 1584, 466, 1056, 1280, 987],
  [1631, 831, 920, 700, 0, 663, 1021, 1769, 949, 796, 879, 586, 371],
  [1374, 1240, 803, 862, 663, 0, 1681, 1551, 1765, 547, 225, 887, 999],
  [2408, 959, 1737, 1395, 1021, 1681, 0, 2493, 678, 1724, 1891, 1114, 701],
  [213, 2596, 851, 1123, 1769, 1551, 2493, 0, 2699, 1038, 1605, 2300, 2099],
  [2571, 403, 1858, 1584, 949, 1765, 678, 2699, 0, 1744, 1645, 653, 600],
  [875, 1589, 262, 466, 796, 547, 1724, 1038, 1744, 0, 679, 1272, 1162],
  [1420, 1374, 940, 1056, 879, 225, 1891, 1605, 1645, 679, 0, 1017, 1200],
  [2145, 357, 1453, 1280, 586, 887, 1114, 2300, 653, 1272, 1017, 0, 504],
  [1972, 579, 1260, 987, 371, 999, 701, 2099, 600, 1162, 1200, 504, 0]
]
data[:num_vehicles] = 1
data[:depot] = 0

# create the distance callback
manager = ORTools::RoutingIndexManager.new(data[:distance_matrix].length, data[:num_vehicles], data[:depot])
routing = ORTools::RoutingModel.new(manager)

distance_callback = lambda do |from_index, to_index|
  from_node = manager.index_to_node(from_index)
  to_node = manager.index_to_node(to_index)
  data[:distance_matrix][from_node][to_node]
end

transit_callback_index = routing.register_transit_callback(distance_callback)
routing.set_arc_cost_evaluator_of_all_vehicles(transit_callback_index)

# run the solver
assignment = routing.solve(first_solution_strategy: :path_cheapest_arc)

# print the solution
puts "Objective: #{assignment.objective_value} miles"
index = routing.start(0)
plan_output = String.new("Route for vehicle 0:\n")
route_distance = 0
while !routing.end?(index)
  plan_output += " #{manager.index_to_node(index)} ->"
  previous_index = index
  index = assignment.value(routing.next_var(index))
  route_distance += routing.arc_cost_for_vehicle(previous_index, index, 0)
end
plan_output += " #{manager.index_to_node(index)}\n"
puts plan_output

Vehicle Routing Problem (VRP)

Guide

# create the data
data = {}
data[:distance_matrix] = [
  [0, 548, 776, 696, 582, 274, 502, 194, 308, 194, 536, 502, 388, 354, 468, 776, 662],
  [548, 0, 684, 308, 194, 502, 730, 354, 696, 742, 1084, 594, 480, 674, 1016, 868, 1210],
  [776, 684, 0, 992, 878, 502, 274, 810, 468, 742, 400, 1278, 1164, 1130, 788, 1552, 754],
  [696, 308, 992, 0, 114, 650, 878, 502, 844, 890, 1232, 514, 628, 822, 1164, 560, 1358],
  [582, 194, 878, 114, 0, 536, 764, 388, 730, 776, 1118, 400, 514, 708, 1050, 674, 1244],
  [274, 502, 502, 650, 536, 0, 228, 308, 194, 240, 582, 776, 662, 628, 514, 1050, 708],
  [502, 730, 274, 878, 764, 228, 0, 536, 194, 468, 354, 1004, 890, 856, 514, 1278, 480],
  [194, 354, 810, 502, 388, 308, 536, 0, 342, 388, 730, 468, 354, 320, 662, 742, 856],
  [308, 696, 468, 844, 730, 194, 194, 342, 0, 274, 388, 810, 696, 662, 320, 1084, 514],
  [194, 742, 742, 890, 776, 240, 468, 388, 274, 0, 342, 536, 422, 388, 274, 810, 468],
  [536, 1084, 400, 1232, 1118, 582, 354, 730, 388, 342, 0, 878, 764, 730, 388, 1152, 354],
  [502, 594, 1278, 514, 400, 776, 1004, 468, 810, 536, 878, 0, 114, 308, 650, 274, 844],
  [388, 480, 1164, 628, 514, 662, 890, 354, 696, 422, 764, 114, 0, 194, 536, 388, 730],
  [354, 674, 1130, 822, 708, 628, 856, 320, 662, 388, 730, 308, 194, 0, 342, 422, 536],
  [468, 1016, 788, 1164, 1050, 514, 514, 662, 320, 274, 388, 650, 536, 342, 0, 764, 194],
  [776, 868, 1552, 560, 674, 1050, 1278, 742, 1084, 810, 1152, 274, 388, 422, 764, 0, 798],
  [662, 1210, 754, 1358, 1244, 708, 480, 856, 514, 468, 354, 844, 730, 536, 194, 798, 0]
]
data[:num_vehicles] = 4
data[:depot] = 0

# define the distance callback
manager = ORTools::RoutingIndexManager.new(data[:distance_matrix].length, data[:num_vehicles], data[:depot])
routing = ORTools::RoutingModel.new(manager)

distance_callback = lambda do |from_index, to_index|
  from_node = manager.index_to_node(from_index)
  to_node = manager.index_to_node(to_index)
  data[:distance_matrix][from_node][to_node]
end

transit_callback_index = routing.register_transit_callback(distance_callback)
routing.set_arc_cost_evaluator_of_all_vehicles(transit_callback_index)

# add a distance dimension
dimension_name = "Distance"
routing.add_dimension(transit_callback_index, 0, 3000, true, dimension_name)
distance_dimension = routing.mutable_dimension(dimension_name)
distance_dimension.global_span_cost_coefficient = 100

# run the solver
solution = routing.solve(first_solution_strategy: :path_cheapest_arc)

# print the solution
max_route_distance = 0
data[:num_vehicles].times do |vehicle_id|
  index = routing.start(vehicle_id)
  plan_output = String.new("Route for vehicle #{vehicle_id}:\n")
  route_distance = 0
  while !routing.end?(index)
    plan_output += " #{manager.index_to_node(index)} -> "
    previous_index = index
    index = solution.value(routing.next_var(index))
    route_distance += routing.arc_cost_for_vehicle(previous_index, index, vehicle_id)
  end
  plan_output += "#{manager.index_to_node(index)}\n"
  plan_output += "Distance of the route: #{route_distance}m\n\n"
  puts plan_output
  max_route_distance = [route_distance, max_route_distance].max
end
puts "Maximum of the route distances: #{max_route_distance}m"

Capacity Constraints

Guide

data = {}
data[:distance_matrix] = [
  [0, 548, 776, 696, 582, 274, 502, 194, 308, 194, 536, 502, 388, 354, 468, 776, 662],
  [548, 0, 684, 308, 194, 502, 730, 354, 696, 742, 1084, 594, 480, 674, 1016, 868, 1210],
  [776, 684, 0, 992, 878, 502, 274, 810, 468, 742, 400, 1278, 1164, 1130, 788, 1552, 754],
  [696, 308, 992, 0, 114, 650, 878, 502, 844, 890, 1232, 514, 628, 822, 1164, 560, 1358],
  [582, 194, 878, 114, 0, 536, 764, 388, 730, 776, 1118, 400, 514, 708, 1050, 674, 1244],
  [274, 502, 502, 650, 536, 0, 228, 308, 194, 240, 582, 776, 662, 628, 514, 1050, 708],
  [502, 730, 274, 878, 764, 228, 0, 536, 194, 468, 354, 1004, 890, 856, 514, 1278, 480],
  [194, 354, 810, 502, 388, 308, 536, 0, 342, 388, 730, 468, 354, 320, 662, 742, 856],
  [308, 696, 468, 844, 730, 194, 194, 342, 0, 274, 388, 810, 696, 662, 320, 1084, 514],
  [194, 742, 742, 890, 776, 240, 468, 388, 274, 0, 342, 536, 422, 388, 274, 810, 468],
  [536, 1084, 400, 1232, 1118, 582, 354, 730, 388, 342, 0, 878, 764, 730, 388, 1152, 354],
  [502, 594, 1278, 514, 400, 776, 1004, 468, 810, 536, 878, 0, 114, 308, 650, 274, 844],
  [388, 480, 1164, 628, 514, 662, 890, 354, 696, 422, 764, 114, 0, 194, 536, 388, 730],
  [354, 674, 1130, 822, 708, 628, 856, 320, 662, 388, 730, 308, 194, 0, 342, 422, 536],
  [468, 1016, 788, 1164, 1050, 514, 514, 662, 320, 274, 388, 650, 536, 342, 0, 764, 194],
  [776, 868, 1552, 560, 674, 1050, 1278, 742, 1084, 810, 1152, 274, 388, 422, 764, 0, 798],
  [662, 1210, 754, 1358, 1244, 708, 480, 856, 514, 468, 354, 844, 730, 536, 194, 798, 0]
]
data[:demands] = [0, 1, 1, 2, 4, 2, 4, 8, 8, 1, 2, 1, 2, 4, 4, 8, 8]
data[:vehicle_capacities] = [15, 15, 15, 15]
data[:num_vehicles] = 4
data[:depot] = 0

manager = ORTools::RoutingIndexManager.new(data[:distance_matrix].size, data[:num_vehicles], data[:depot])
routing = ORTools::RoutingModel.new(manager)

distance_callback = lambda do |from_index, to_index|
  from_node = manager.index_to_node(from_index)
  to_node = manager.index_to_node(to_index)
  data[:distance_matrix][from_node][to_node]
end

transit_callback_index = routing.register_transit_callback(distance_callback)

routing.set_arc_cost_evaluator_of_all_vehicles(transit_callback_index)

demand_callback = lambda do |from_index|
  from_node = manager.index_to_node(from_index)
  data[:demands][from_node]
end

demand_callback_index = routing.register_unary_transit_callback(demand_callback)
routing.add_dimension_with_vehicle_capacity(
  demand_callback_index,
  0,  # null capacity slack
  data[:vehicle_capacities],  # vehicle maximum capacities
  true,  # start cumul to zero
  "Capacity"
)

solution = routing.solve(first_solution_strategy: :path_cheapest_arc)

total_distance = 0
total_load = 0
data[:num_vehicles].times do |vehicle_id|
  index = routing.start(vehicle_id)
  plan_output = String.new("Route for vehicle #{vehicle_id}:\n")
  route_distance = 0
  route_load = 0
  while !routing.end?(index)
    node_index = manager.index_to_node(index)
    route_load += data[:demands][node_index]
    plan_output += " #{node_index} Load(#{route_load}) -> "
    previous_index = index
    index = solution.value(routing.next_var(index))
    route_distance += routing.arc_cost_for_vehicle(previous_index, index, vehicle_id)
  end
  plan_output += " #{manager.index_to_node(index)} Load(#{route_load})\n"
  plan_output += "Distance of the route: #{route_distance}m\n"
  plan_output += "Load of the route: #{route_load}\n\n"
  puts plan_output
  total_distance += route_distance
  total_load += route_load
end
puts "Total distance of all routes: #{total_distance}m"
puts "Total load of all routes: #{total_load}"

Pickups and Deliveries

Guide

data = {}
data[:distance_matrix] = [
  [0, 548, 776, 696, 582, 274, 502, 194, 308, 194, 536, 502, 388, 354, 468, 776, 662],
  [548, 0, 684, 308, 194, 502, 730, 354, 696, 742, 1084, 594, 480, 674, 1016, 868, 1210],
  [776, 684, 0, 992, 878, 502, 274, 810, 468, 742, 400, 1278, 1164, 1130, 788, 1552, 754],
  [696, 308, 992, 0, 114, 650, 878, 502, 844, 890, 1232, 514, 628, 822, 1164, 560, 1358],
  [582, 194, 878, 114, 0, 536, 764, 388, 730, 776, 1118, 400, 514, 708, 1050, 674, 1244],
  [274, 502, 502, 650, 536, 0, 228, 308, 194, 240, 582, 776, 662, 628, 514, 1050, 708],
  [502, 730, 274, 878, 764, 228, 0, 536, 194, 468, 354, 1004, 890, 856, 514, 1278, 480],
  [194, 354, 810, 502, 388, 308, 536, 0, 342, 388, 730, 468, 354, 320, 662, 742, 856],
  [308, 696, 468, 844, 730, 194, 194, 342, 0, 274, 388, 810, 696, 662, 320, 1084, 514],
  [194, 742, 742, 890, 776, 240, 468, 388, 274, 0, 342, 536, 422, 388, 274, 810, 468],
  [536, 1084, 400, 1232, 1118, 582, 354, 730, 388, 342, 0, 878, 764, 730, 388, 1152, 354],
  [502, 594, 1278, 514, 400, 776, 1004, 468, 810, 536, 878, 0, 114, 308, 650, 274, 844],
  [388, 480, 1164, 628, 514, 662, 890, 354, 696, 422, 764, 114, 0, 194, 536, 388, 730],
  [354, 674, 1130, 822, 708, 628, 856, 320, 662, 388, 730, 308, 194, 0, 342, 422, 536],
  [468, 1016, 788, 1164, 1050, 514, 514, 662, 320, 274, 388, 650, 536, 342, 0, 764, 194],
  [776, 868, 1552, 560, 674, 1050, 1278, 742, 1084, 810, 1152, 274, 388, 422, 764, 0, 798],
  [662, 1210, 754, 1358, 1244, 708, 480, 856, 514, 468, 354, 844, 730, 536, 194, 798, 0]
]
data[:pickups_deliveries] = [
  [1, 6],
  [2, 10],
  [4, 3],
  [5, 9],
  [7, 8],
  [15, 11],
  [13, 12],
  [16, 14],
]
data[:num_vehicles] = 4
data[:depot] = 0

manager = ORTools::RoutingIndexManager.new(data[:distance_matrix].size, data[:num_vehicles], data[:depot])
routing = ORTools::RoutingModel.new(manager)

distance_callback = lambda do |from_index, to_index|
  from_node = manager.index_to_node(from_index)
  to_node = manager.index_to_node(to_index)
  data[:distance_matrix][from_node][to_node]
end

transit_callback_index = routing.register_transit_callback(distance_callback)
routing.set_arc_cost_evaluator_of_all_vehicles(transit_callback_index)

dimension_name = "Distance"
routing.add_dimension(
  transit_callback_index,
  0,  # no slack
  3000,  # vehicle maximum travel distance
  true,  # start cumul to zero
  dimension_name
)
distance_dimension = routing.mutable_dimension(dimension_name)
distance_dimension.global_span_cost_coefficient = 100

data[:pickups_deliveries].each do |request|
  pickup_index = manager.node_to_index(request[0])
  delivery_index = manager.node_to_index(request[1])
  routing.add_pickup_and_delivery(pickup_index, delivery_index)
  routing.solver.add(routing.vehicle_var(pickup_index) == routing.vehicle_var(delivery_index))
  routing.solver.add(distance_dimension.cumul_var(pickup_index) <= distance_dimension.cumul_var(delivery_index))
end

solution = routing.solve(first_solution_strategy: :parallel_cheapest_insertion)

total_distance = 0
data[:num_vehicles].times do |vehicle_id|
  index = routing.start(vehicle_id)
  plan_output = String.new("Route for vehicle #{vehicle_id}:\n")
  route_distance = 0
  while !routing.end?(index)
    plan_output += " #{manager.index_to_node(index)} -> "
    previous_index = index
    index = solution.value(routing.next_var(index))
    route_distance += routing.arc_cost_for_vehicle(previous_index, index, vehicle_id)
  end
  plan_output += "#{manager.index_to_node(index)}\n"
  plan_output += "Distance of the route: #{route_distance}m\n\n"
  puts plan_output
  total_distance += route_distance
end
puts "Total Distance of all routes: #{total_distance}m"

Time Window Constraints

Guide

data = {}
data[:time_matrix] = [
  [0, 6, 9, 8, 7, 3, 6, 2, 3, 2, 6, 6, 4, 4, 5, 9, 7],
  [6, 0, 8, 3, 2, 6, 8, 4, 8, 8, 13, 7, 5, 8, 12, 10, 14],
  [9, 8, 0, 11, 10, 6, 3, 9, 5, 8, 4, 15, 14, 13, 9, 18, 9],
  [8, 3, 11, 0, 1, 7, 10, 6, 10, 10, 14, 6, 7, 9, 14, 6, 16],
  [7, 2, 10, 1, 0, 6, 9, 4, 8, 9, 13, 4, 6, 8, 12, 8, 14],
  [3, 6, 6, 7, 6, 0, 2, 3, 2, 2, 7, 9, 7, 7, 6, 12, 8],
  [6, 8, 3, 10, 9, 2, 0, 6, 2, 5, 4, 12, 10, 10, 6, 15, 5],
  [2, 4, 9, 6, 4, 3, 6, 0, 4, 4, 8, 5, 4, 3, 7, 8, 10],
  [3, 8, 5, 10, 8, 2, 2, 4, 0, 3, 4, 9, 8, 7, 3, 13, 6],
  [2, 8, 8, 10, 9, 2, 5, 4, 3, 0, 4, 6, 5, 4, 3, 9, 5],
  [6, 13, 4, 14, 13, 7, 4, 8, 4, 4, 0, 10, 9, 8, 4, 13, 4],
  [6, 7, 15, 6, 4, 9, 12, 5, 9, 6, 10, 0, 1, 3, 7, 3, 10],
  [4, 5, 14, 7, 6, 7, 10, 4, 8, 5, 9, 1, 0, 2, 6, 4, 8],
  [4, 8, 13, 9, 8, 7, 10, 3, 7, 4, 8, 3, 2, 0, 4, 5, 6],
  [5, 12, 9, 14, 12, 6, 6, 7, 3, 3, 4, 7, 6, 4, 0, 9, 2],
  [9, 10, 18, 6, 8, 12, 15, 8, 13, 9, 13, 3, 4, 5, 9, 0, 9],
  [7, 14, 9, 16, 14, 8, 5, 10, 6, 5, 4, 10, 8, 6, 2, 9, 0],
]
data[:time_windows] = [
  [0, 5],  # depot
  [7, 12],  # 1
  [10, 15],  # 2
  [16, 18],  # 3
  [10, 13],  # 4
  [0, 5],  # 5
  [5, 10],  # 6
  [0, 4],  # 7
  [5, 10],  # 8
  [0, 3],  # 9
  [10, 16],  # 10
  [10, 15],  # 11
  [0, 5],  # 12
  [5, 10],  # 13
  [7, 8],  # 14
  [10, 15],  # 15
  [11, 15],  # 16
]
data[:num_vehicles] = 4
data[:depot] = 0

manager = ORTools::RoutingIndexManager.new(data[:time_matrix].size, data[:num_vehicles], data[:depot])
routing = ORTools::RoutingModel.new(manager)

time_callback = lambda do |from_index, to_index|
  from_node = manager.index_to_node(from_index)
  to_node = manager.index_to_node(to_index)
  data[:time_matrix][from_node][to_node]
end

transit_callback_index = routing.register_transit_callback(time_callback)
routing.set_arc_cost_evaluator_of_all_vehicles(transit_callback_index)
time = "Time"
routing.add_dimension(
  transit_callback_index,
  30,  # allow waiting time
  30,  # maximum time per vehicle
  false,  # don't force start cumul to zero
  time
)
time_dimension = routing.mutable_dimension(time)

data[:time_windows].each_with_index do |time_window, location_idx|
  next if location_idx == 0
  index = manager.node_to_index(location_idx)
  time_dimension.cumul_var(index).set_range(time_window[0], time_window[1])
end

data[:num_vehicles].times do |vehicle_id|
  index = routing.start(vehicle_id)
  time_dimension.cumul_var(index).set_range(data[:time_windows][0][0], data[:time_windows][0][1])
end

data[:num_vehicles].times do |i|
  routing.add_variable_minimized_by_finalizer(time_dimension.cumul_var(routing.start(i)))
  routing.add_variable_minimized_by_finalizer(time_dimension.cumul_var(routing.end(i)))
end

solution = routing.solve(first_solution_strategy: :path_cheapest_arc)

time_dimension = routing.mutable_dimension("Time")
total_time = 0
data[:num_vehicles].times do |vehicle_id|
  index = routing.start(vehicle_id)
  plan_output = String.new("Route for vehicle #{vehicle_id}:\n")
  while !routing.end?(index)
    time_var = time_dimension.cumul_var(index)
    plan_output += "#{manager.index_to_node(index)} Time(#{solution.min(time_var)},#{solution.max(time_var)}) -> "
    index = solution.value(routing.next_var(index))
  end
  time_var = time_dimension.cumul_var(index)
  plan_output += "#{manager.index_to_node(index)} Time(#{solution.min(time_var)},#{solution.max(time_var)})\n"
  plan_output += "Time of the route: #{solution.min(time_var)}min\n\n"
  puts plan_output
  total_time += solution.min(time_var)
end
puts "Total time of all routes: #{total_time}min"

Resource Constraints

Guide

data = {}
data[:time_matrix] = [
  [0, 6, 9, 8, 7, 3, 6, 2, 3, 2, 6, 6, 4, 4, 5, 9, 7],
  [6, 0, 8, 3, 2, 6, 8, 4, 8, 8, 13, 7, 5, 8, 12, 10, 14],
  [9, 8, 0, 11, 10, 6, 3, 9, 5, 8, 4, 15, 14, 13, 9, 18, 9],
  [8, 3, 11, 0, 1, 7, 10, 6, 10, 10, 14, 6, 7, 9, 14, 6, 16],
  [7, 2, 10, 1, 0, 6, 9, 4, 8, 9, 13, 4, 6, 8, 12, 8, 14],
  [3, 6, 6, 7, 6, 0, 2, 3, 2, 2, 7, 9, 7, 7, 6, 12, 8],
  [6, 8, 3, 10, 9, 2, 0, 6, 2, 5, 4, 12, 10, 10, 6, 15, 5],
  [2, 4, 9, 6, 4, 3, 6, 0, 4, 4, 8, 5, 4, 3, 7, 8, 10],
  [3, 8, 5, 10, 8, 2, 2, 4, 0, 3, 4, 9, 8, 7, 3, 13, 6],
  [2, 8, 8, 10, 9, 2, 5, 4, 3, 0, 4, 6, 5, 4, 3, 9, 5],
  [6, 13, 4, 14, 13, 7, 4, 8, 4, 4, 0, 10, 9, 8, 4, 13, 4],
  [6, 7, 15, 6, 4, 9, 12, 5, 9, 6, 10, 0, 1, 3, 7, 3, 10],
  [4, 5, 14, 7, 6, 7, 10, 4, 8, 5, 9, 1, 0, 2, 6, 4, 8],
  [4, 8, 13, 9, 8, 7, 10, 3, 7, 4, 8, 3, 2, 0, 4, 5, 6],
  [5, 12, 9, 14, 12, 6, 6, 7, 3, 3, 4, 7, 6, 4, 0, 9, 2],
  [9, 10, 18, 6, 8, 12, 15, 8, 13, 9, 13, 3, 4, 5, 9, 0, 9],
  [7, 14, 9, 16, 14, 8, 5, 10, 6, 5, 4, 10, 8, 6, 2, 9, 0]
]
data[:time_windows] = [
  [0, 5],  # depot
  [7, 12],  # 1
  [10, 15],  # 2
  [5, 14],  # 3
  [5, 13],  # 4
  [0, 5],  # 5
  [5, 10],  # 6
  [0, 10],  # 7
  [5, 10],  # 8
  [0, 5],  # 9
  [10, 16],  # 10
  [10, 15],  # 11
  [0, 5],  # 12
  [5, 10],  # 13
  [7, 12],  # 14
  [10, 15],  # 15
  [5, 15],  # 16
]
data[:num_vehicles] = 4
data[:vehicle_load_time] = 5
data[:vehicle_unload_time] = 5
data[:depot_capacity] = 2
data[:depot] = 0

manager = ORTools::RoutingIndexManager.new(data[:time_matrix].size, data[:num_vehicles], data[:depot])
routing = ORTools::RoutingModel.new(manager)

time_callback = lambda do |from_index, to_index|
  from_node = manager.index_to_node(from_index)
  to_node = manager.index_to_node(to_index)
  data[:time_matrix][from_node][to_node]
end

transit_callback_index = routing.register_transit_callback(time_callback)

routing.set_arc_cost_evaluator_of_all_vehicles(transit_callback_index)

time = "Time"
routing.add_dimension(
  transit_callback_index,
  60,  # allow waiting time
  60,  # maximum time per vehicle
  false,  # don't force start cumul to zero
  time
)
time_dimension = routing.mutable_dimension(time)
data[:time_windows].each_with_index do |time_window, location_idx|
  next if location_idx == 0
  index = manager.node_to_index(location_idx)
  time_dimension.cumul_var(index).set_range(time_window[0], time_window[1])
end

data[:num_vehicles].times do |vehicle_id|
  index = routing.start(vehicle_id)
  time_dimension.cumul_var(index).set_range(data[:time_windows][0][0], data[:time_windows][0][1])
end

solver = routing.solver
intervals = []
data[:num_vehicles].times do |i|
  intervals << solver.fixed_duration_interval_var(
    time_dimension.cumul_var(routing.start(i)),
    data[:vehicle_load_time],
    "depot_interval"
  )
  intervals << solver.fixed_duration_interval_var(
    time_dimension.cumul_var(routing.end(i)),
    data[:vehicle_unload_time],
    "depot_interval"
  )
end

depot_usage = [1] * intervals.size
solver.add(solver.cumulative(intervals, depot_usage, data[:depot_capacity], "depot"))

data[:num_vehicles].times do |i|
  routing.add_variable_minimized_by_finalizer(time_dimension.cumul_var(routing.start(i)))
  routing.add_variable_minimized_by_finalizer(time_dimension.cumul_var(routing.end(i)))
end

solution = routing.solve(first_solution_strategy: :path_cheapest_arc)

time_dimension = routing.mutable_dimension("Time")
total_time = 0
data[:num_vehicles].times do |vehicle_id|
  index = routing.start(vehicle_id)
  plan_output = String.new("Route for vehicle #{vehicle_id}:\n")
  while !routing.end?(index)
    time_var = time_dimension.cumul_var(index)
    plan_output += "#{manager.index_to_node(index)} Time(#{solution.min(time_var)},#{solution.max(time_var)}) -> "
    index = solution.value(routing.next_var(index))
  end
  time_var = time_dimension.cumul_var(index)
  plan_output += "#{manager.index_to_node(index)} Time(#{solution.min(time_var)},#{solution.max(time_var)})\n"
  plan_output += "Time of the route: #{solution.min(time_var)}min\n\n"
  puts plan_output
  total_time += solution.min(time_var)
end
puts "Total time of all routes: #{total_time}min"

Penalties and Dropping Visits

Guide

data = {}
data[:distance_matrix] = [
  [0, 548, 776, 696, 582, 274, 502, 194, 308, 194, 536, 502, 388, 354, 468, 776, 662],
  [548, 0, 684, 308, 194, 502, 730, 354, 696, 742, 1084, 594, 480, 674, 1016, 868, 1210],
  [776, 684, 0, 992, 878, 502, 274, 810, 468, 742, 400, 1278, 1164, 1130, 788, 1552, 754],
  [696, 308, 992, 0, 114, 650, 878, 502, 844, 890, 1232, 514, 628, 822, 1164, 560, 1358],
  [582, 194, 878, 114, 0, 536, 764, 388, 730, 776, 1118, 400, 514, 708, 1050, 674, 1244],
  [274, 502, 502, 650, 536, 0, 228, 308, 194, 240, 582, 776, 662, 628, 514, 1050, 708],
  [502, 730, 274, 878, 764, 228, 0, 536, 194, 468, 354, 1004, 890, 856, 514, 1278, 480],
  [194, 354, 810, 502, 388, 308, 536, 0, 342, 388, 730, 468, 354, 320, 662, 742, 856],
  [308, 696, 468, 844, 730, 194, 194, 342, 0, 274, 388, 810, 696, 662, 320, 1084, 514],
  [194, 742, 742, 890, 776, 240, 468, 388, 274, 0, 342, 536, 422, 388, 274, 810, 468],
  [536, 1084, 400, 1232, 1118, 582, 354, 730, 388, 342, 0, 878, 764, 730, 388, 1152, 354],
  [502, 594, 1278, 514, 400, 776, 1004, 468, 810, 536, 878, 0, 114, 308, 650, 274, 844],
  [388, 480, 1164, 628, 514, 662, 890, 354, 696, 422, 764, 114, 0, 194, 536, 388, 730],
  [354, 674, 1130, 822, 708, 628, 856, 320, 662, 388, 730, 308, 194, 0, 342, 422, 536],
  [468, 1016, 788, 1164, 1050, 514, 514, 662, 320, 274, 388, 650, 536, 342, 0, 764, 194],
  [776, 868, 1552, 560, 674, 1050, 1278, 742, 1084, 810, 1152, 274, 388, 422, 764, 0, 798],
  [662, 1210, 754, 1358, 1244, 708, 480, 856, 514, 468, 354, 844, 730, 536, 194, 798, 0]
]
data[:demands] = [0, 1, 1, 3, 6, 3, 6, 8, 8, 1, 2, 1, 2, 6, 6, 8, 8]
data[:vehicle_capacities] = [15, 15, 15, 15]
data[:num_vehicles] = 4
data[:depot] = 0

manager = ORTools::RoutingIndexManager.new(data[:distance_matrix].size, data[:num_vehicles], data[:depot])
routing = ORTools::RoutingModel.new(manager)

distance_callback = lambda do |from_index, to_index|
  from_node = manager.index_to_node(from_index)
  to_node = manager.index_to_node(to_index)
  data[:distance_matrix][from_node][to_node]
end

transit_callback_index = routing.register_transit_callback(distance_callback)

routing.set_arc_cost_evaluator_of_all_vehicles(transit_callback_index)

demand_callback = lambda do |from_index|
  from_node = manager.index_to_node(from_index)
  data[:demands][from_node]
end

demand_callback_index = routing.register_unary_transit_callback(demand_callback)
routing.add_dimension_with_vehicle_capacity(
  demand_callback_index,
  0,  # null capacity slack
  data[:vehicle_capacities],  # vehicle maximum capacities
  true,  # start cumul to zero
  "Capacity"
)

penalty = 1000
1.upto(data[:distance_matrix].size - 1) do |node|
  routing.add_disjunction([manager.node_to_index(node)], penalty)
end

assignment = routing.solve(first_solution_strategy: :path_cheapest_arc)

dropped_nodes = String.new("Dropped nodes:")
routing.size.times do |node|
  next if routing.start?(node) || routing.end?(node)

  if assignment.value(routing.next_var(node)) == node
    dropped_nodes += " #{manager.index_to_node(node)}"
  end
end
puts dropped_nodes

total_distance = 0
total_load = 0
data[:num_vehicles].times do |vehicle_id|
  index = routing.start(vehicle_id)
  plan_output = "Route for vehicle #{vehicle_id}:\n"
  route_distance = 0
  route_load = 0
  while !routing.end?(index)
    node_index = manager.index_to_node(index)
    route_load += data[:demands][node_index]
    plan_output += " #{node_index} Load(#{route_load}) -> "
    previous_index = index
    index = assignment.value(routing.next_var(index))
    route_distance += routing.arc_cost_for_vehicle(previous_index, index, vehicle_id)
  end
  plan_output += " #{manager.index_to_node(index)} Load(#{route_load})\n"
  plan_output += "Distance of the route: #{route_distance}m\n"
  plan_output += "Load of the route: #{route_load}\n\n"
  puts plan_output
  total_distance += route_distance
  total_load += route_load
end
puts "Total Distance of all routes: #{total_distance}m"
puts "Total Load of all routes: #{total_load}"

Routing Options

Guide

routing.solve(
  solution_limit: 10,
  time_limit: 10, # seconds,
  lns_time_limit: 10, # seconds
  first_solution_strategy: :path_cheapest_arc,
  local_search_metaheuristic: :guided_local_search,
  log_search: true
)

Bin Packing

The Knapsack Problem

Guide

# create the data
values = [
  360, 83, 59, 130, 431, 67, 230, 52, 93, 125, 670, 892, 600, 38, 48, 147,
  78, 256, 63, 17, 120, 164, 432, 35, 92, 110, 22, 42, 50, 323, 514, 28,
  87, 73, 78, 15, 26, 78, 210, 36, 85, 189, 274, 43, 33, 10, 19, 389, 276,
  312
]
weights = [[
  7, 0, 30, 22, 80, 94, 11, 81, 70, 64, 59, 18, 0, 36, 3, 8, 15, 42, 9, 0,
  42, 47, 52, 32, 26, 48, 55, 6, 29, 84, 2, 4, 18, 56, 7, 29, 93, 44, 71,
  3, 86, 66, 31, 65, 0, 79, 20, 65, 52, 13
]]
capacities = [850]

# declare the solver
solver = ORTools::KnapsackSolver.new(:branch_and_bound, "KnapsackExample")

# call the solver
solver.init(values, weights, capacities)
computed_value = solver.solve

packed_items = []
packed_weights = []
total_weight = 0
puts "Total value = #{computed_value}"
values.length.times do |i|
  if solver.best_solution_contains?(i)
    packed_items << i
    packed_weights << weights[0][i]
    total_weight += weights[0][i]
  end
end
puts "Total weight: #{total_weight}"
puts "Packed items: #{packed_items}"
puts "Packed weights: #{packed_weights}"

Multiple Knapsacks

Guide

# create the data
data = {}
data[:weights] = [48, 30, 42, 36, 36, 48, 42, 42, 36, 24, 30, 30, 42, 36, 36]
data[:values] = [10, 30, 25, 50, 35, 30, 15, 40, 30, 35, 45, 10, 20, 30, 25]
data[:num_items] = data[:weights].length
data[:all_items] = data[:num_items].times.to_a
data[:bin_capacities] = [100, 100, 100, 100, 100]
data[:num_bins] = data[:bin_capacities].length
data[:all_bins] = data[:num_bins].times.to_a

# declare the solver
solver = ORTools::Solver.new("CBC")

# create the variables
x = {}
data[:all_items].each do |i|
  data[:all_bins].each do |b|
    x[[i, b]] = solver.bool_var("x_#{i}_#{b}")
  end
end

# each item is assigned to at most one bin
data[:all_items].each do |i|
  solver.add(data[:all_bins].sum { |b| x[[i, b]] } <= 1)
end

# the amount packed in each bin cannot exceed its capacity
data[:all_bins].each do |b|
  solver.add(data[:all_items].sum { |i| x[[i, b]] * data[:weights][i] } <= data[:bin_capacities][b])
end

# maximize total value of packed items
objective = solver.objective
data[:all_items].each do |i|
  data[:all_bins].each do |b|
    objective.set_coefficient(x[[i, b]], data[:values][i])
  end
end
objective.set_maximization

# call the solver and print the solution
status = solver.solve

if status == :optimal
  puts "Total packed value: #{objective.value}"
  total_weight = 0
  data[:all_bins].each do |b|
    bin_weight = 0
    bin_value = 0
    puts "Bin #{b}\n\n"
    data[:all_items].each do |i|
      if x[[i, b]].solution_value > 0
        puts "Item #{i} - weight: #{data[:weights][i]}  value: #{data[:values][i]}"
        bin_weight += data[:weights][i]
        bin_value += data[:values][i]
      end
    end
    puts "Packed bin weight: #{bin_weight}"
    puts "Packed bin value: #{bin_value}"
    puts
    total_weight += bin_weight
  end
  puts "Total packed weight: #{total_weight}"
else
  puts "The problem does not have an optimal solution."
end

Bin Packing Problem

Guide

# create the data
data = {}
weights = [48, 30, 19, 36, 36, 27, 42, 42, 36, 24, 30]
data[:weights] = weights
data[:items] = (0...weights.length).to_a
data[:bins] = data[:items]
data[:bin_capacity] = 100

# create the mip solver with the CBC backend
solver = ORTools::Solver.new("CBC")

# variables
# x[i, j] = 1 if item i is packed in bin j
x = {}
data[:items].each do |i|
  data[:bins].each do |j|
    x[[i, j]] = solver.int_var(0, 1, "x_%i_%i" % [i, j])
  end
end

# y[j] = 1 if bin j is used
y = {}
data[:bins].each do |j|
  y[j] = solver.int_var(0, 1, "y[%i]" % j)
end

# constraints
# each item must be in exactly one bin
data[:items].each do |i|
  solver.add(data[:bins].sum { |j| x[[i, j]] } == 1)
end

# the amount packed in each bin cannot exceed its capacity
data[:bins].each do |j|
  sum = data[:items].sum { |i| x[[i, j]] * data[:weights][i] }
  solver.add(sum <= y[j] * data[:bin_capacity])
end

# objective: minimize the number of bins used
solver.minimize(data[:bins].sum { |j| y[j] })

# call the solver and print the solution
status = solver.solve

if status == :optimal
  num_bins = 0
  data[:bins].each do |j|
    if y[j].solution_value == 1
      bin_items = []
      bin_weight = 0
      data[:items].each do |i|
        if x[[i, j]].solution_value > 0
          bin_items << i
          bin_weight += data[:weights][i]
        end
      end
      if bin_weight > 0
        num_bins += 1
        puts "Bin number #{j}"
        puts "  Items packed: #{bin_items}"
        puts "  Total weight: #{bin_weight}"
        puts
      end
    end
  end
  puts
  puts "Number of bins used: #{num_bins}"
  puts "Time = #{solver.wall_time} milliseconds"
else
  puts "The problem does not have an optimal solution."
end

Network Flows

Maximum Flows

Guide

# define the data
start_nodes = [0, 0, 0, 1, 1, 2, 2, 3, 3]
end_nodes = [1, 2, 3, 2, 4, 3, 4, 2, 4]
capacities = [20, 30, 10, 40, 30, 10, 20, 5, 20]

# declare the solver and add the arcs
max_flow = ORTools::SimpleMaxFlow.new

start_nodes.length.times do |i|
  max_flow.add_arc_with_capacity(start_nodes[i], end_nodes[i], capacities[i])
end

# invoke the solver and display the results
if max_flow.solve(0, 4) == :optimal
  puts "Max flow: #{max_flow.optimal_flow}"
  puts
  puts "  Arc    Flow / Capacity"
  max_flow.num_arcs.times do |i|
    puts "%1s -> %1s   %3s  / %3s" % [
      max_flow.tail(i),
      max_flow.head(i),
      max_flow.flow(i),
      max_flow.capacity(i)
    ]
  end
  puts "Source side min-cut: #{max_flow.source_side_min_cut}"
  puts "Sink side min-cut: #{max_flow.sink_side_min_cut}"
else
  puts "There was an issue with the max flow input."
end

Minimum Cost Flows

Guide

# define the data
start_nodes = [ 0, 0,  1, 1,  1,  2, 2,  3, 4]
end_nodes   = [ 1, 2,  2, 3,  4,  3, 4,  4, 2]
capacities  = [15, 8, 20, 4, 10, 15, 4, 20, 5]
unit_costs  = [ 4, 4,  2, 2,  6,  1, 3,  2, 3]
supplies = [20, 0, 0, -5, -15]

# declare the solver and add the arcs
min_cost_flow = ORTools::SimpleMinCostFlow.new

start_nodes.length.times do |i|
  min_cost_flow.add_arc_with_capacity_and_unit_cost(
    start_nodes[i], end_nodes[i], capacities[i], unit_costs[i]
  )
end

supplies.length.times do |i|
  min_cost_flow.set_node_supply(i, supplies[i])
end

# invoke the solver and display the results
if min_cost_flow.solve == :optimal
  puts "Minimum cost #{min_cost_flow.optimal_cost}"
  puts
  puts "  Arc    Flow / Capacity  Cost"
  min_cost_flow.num_arcs.times do |i|
    cost = min_cost_flow.flow(i) * min_cost_flow.unit_cost(i)
    puts "%1s -> %1s   %3s  / %3s       %3s" % [
      min_cost_flow.tail(i),
      min_cost_flow.head(i),
      min_cost_flow.flow(i),
      min_cost_flow.capacity(i),
      cost
    ]
  end
else
  puts "There was an issue with the min cost flow input."
end

Assignment as a Min Cost Flow Problem

Guide

# create the solver
min_cost_flow = ORTools::SimpleMinCostFlow.new

# create the data
start_nodes = [0, 0, 0, 0] + [1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4] + [5, 6, 7, 8]
end_nodes = [1, 2, 3, 4] + [5, 6, 7, 8, 5, 6, 7, 8, 5, 6, 7, 8, 5, 6, 7, 8] + [9, 9, 9, 9]
capacities = [1, 1, 1, 1] + [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1] + [1, 1, 1, 1]
costs = [0, 0, 0, 0] + [90, 76, 75, 70, 35, 85, 55, 65, 125, 95, 90, 105, 45, 110, 95, 115] + [0, 0, 0, 0]
supplies = [4, 0, 0, 0, 0, 0, 0, 0, 0, -4]
source = 0
sink = 9
tasks = 4

# create the graph and constraints
start_nodes.length.times do |i|
  min_cost_flow.add_arc_with_capacity_and_unit_cost(
    start_nodes[i], end_nodes[i], capacities[i], costs[i]
  )
end

supplies.length.times do |i|
  min_cost_flow.set_node_supply(i, supplies[i])
end

# invoke the solver
if min_cost_flow.solve == :optimal
  puts "Total cost = #{min_cost_flow.optimal_cost}"
  puts
  min_cost_flow.num_arcs.times do |arc|
    if min_cost_flow.tail(arc) != source && min_cost_flow.head(arc) != sink
      if min_cost_flow.flow(arc) > 0
        puts "Worker %d assigned to task %d.  Cost = %d" % [
          min_cost_flow.tail(arc),
          min_cost_flow.head(arc),
          min_cost_flow.unit_cost(arc)
        ]
      end
    end
  end
else
  puts "There was an issue with the min cost flow input."
end

Scheduling

Employee Scheduling

Guide

# define the data
num_nurses = 4
num_shifts = 3
num_days = 3
all_nurses = num_nurses.times.to_a
all_shifts = num_shifts.times.to_a
all_days = num_days.times.to_a

# create the variables
model = ORTools::CpModel.new

shifts = {}
all_nurses.each do |n|
  all_days.each do |d|
    all_shifts.each do |s|
      shifts[[n, d, s]] = model.new_bool_var("shift_n%id%is%i" % [n, d, s])
    end
  end
end

# assign nurses to shifts
all_days.each do |d|
  all_shifts.each do |s|
    model.add(model.sum(all_nurses.map { |n| shifts[[n, d, s]] }) == 1)
  end
end

all_nurses.each do |n|
  all_days.each do |d|
    model.add(model.sum(all_shifts.map { |s| shifts[[n, d, s]] }) <= 1)
  end
end

# assign shifts evenly
min_shifts_per_nurse = (num_shifts * num_days) / num_nurses
max_shifts_per_nurse = min_shifts_per_nurse + 1
all_nurses.each do |n|
  num_shifts_worked = model.sum(all_days.flat_map { |d| all_shifts.map { |s| shifts[[n, d, s]] } })
  model.add(num_shifts_worked >= min_shifts_per_nurse)
  model.add(num_shifts_worked <= max_shifts_per_nurse)
end

# create a printer
class NursesPartialSolutionPrinter < ORTools::CpSolverSolutionCallback
  attr_reader :solution_count

  def initialize(shifts, num_nurses, num_days, num_shifts, sols)
    super()
    @shifts = shifts
    @num_nurses = num_nurses
    @num_days = num_days
    @num_shifts = num_shifts
    @solutions = sols
    @solution_count = 0
  end

  def on_solution_callback
    if @solutions.include?(@solution_count)
      puts "Solution #{@solution_count}"
      @num_days.times do |d|
        puts "Day #{d}"
        @num_nurses.times do |n|
          working = false
          @num_shifts.times do |s|
            if value(@shifts[[n, d, s]])
              working = true
              puts "  Nurse %i works shift %i" % [n, s]
            end
          end
          unless working
            puts "  Nurse #{n} does not work"
          end
        end
      end
      puts
    end
    @solution_count += 1
  end
end

# call the solver and display the results
solver = ORTools::CpSolver.new
a_few_solutions = 5.times.to_a
solution_printer = NursesPartialSolutionPrinter.new(
  shifts, num_nurses, num_days, num_shifts, a_few_solutions
)
solver.search_for_all_solutions(model, solution_printer)

puts
puts "Statistics"
puts "  - conflicts       : %i" % solver.num_conflicts
puts "  - branches        : %i" % solver.num_branches
puts "  - wall time       : %f s" % solver.wall_time
puts "  - solutions found : %i" % solution_printer.solution_count

The Job Shop Problem

Guide

# create the model
model = ORTools::CpModel.new

jobs_data = [
  [[0, 3], [1, 2], [2, 2]],
  [[0, 2], [2, 1], [1, 4]],
  [[1, 4], [2, 3]]
]

machines_count = 1 + jobs_data.flat_map { |job| job.map { |task| task[0] }  }.max
all_machines = machines_count.times.to_a

# computes horizon dynamically as the sum of all durations
horizon = jobs_data.flat_map { |job| job.map { |task| task[1] }  }.sum

# creates job intervals and add to the corresponding machine lists
all_tasks = {}
machine_to_intervals = Hash.new { |hash, key| hash[key] = [] }

jobs_data.each_with_index do |job, job_id|
  job.each_with_index do |task, task_id|
    machine = task[0]
    duration = task[1]
    suffix = "_%i_%i" % [job_id, task_id]
    start_var = model.new_int_var(0, horizon, "start" + suffix)
    duration_var = model.new_int_var(duration, duration, "duration" + suffix)
    end_var = model.new_int_var(0, horizon, "end" + suffix)
    interval_var = model.new_interval_var(start_var, duration_var, end_var, "interval" + suffix)
    all_tasks[[job_id, task_id]] = {start: start_var, end: end_var, interval: interval_var}
    machine_to_intervals[machine] << interval_var
  end
end

# create and add disjunctive constraints
all_machines.each do |machine|
  model.add_no_overlap(machine_to_intervals[machine])
end

# precedences inside a job
jobs_data.each_with_index do |job, job_id|
  (job.size - 1).times do |task_id|
    model.add(all_tasks[[job_id, task_id + 1]][:start] >= all_tasks[[job_id, task_id]][:end])
  end
end

# makespan objective
obj_var = model.new_int_var(0, horizon, "makespan")
model.add_max_equality(obj_var, jobs_data.map.with_index { |job, job_id| all_tasks[[job_id, job.size - 1]][:end] })
model.minimize(obj_var)

# solve model
solver = ORTools::CpSolver.new
status = solver.solve(model)

# create one list of assigned tasks per machine
assigned_jobs = Hash.new { |hash, key| hash[key] = [] }
jobs_data.each_with_index do |job, job_id|
  job.each_with_index do |task, task_id|
    machine = task[0]
    assigned_jobs[machine] << {
      start: solver.value(all_tasks[[job_id, task_id]][:start]),
      job: job_id,
      index: task_id,
      duration: task[1]
    }
  end
end

# create per machine output lines
output = String.new("")
all_machines.each do |machine|
  # sort by starting time
  assigned_jobs[machine].sort_by! { |v| v[:start] }
  sol_line_tasks = "Machine #{machine}: "
  sol_line = "           "

  assigned_jobs[machine].each do |assigned_task|
    name = "job_%i_%i" % [assigned_task[:job], assigned_task[:index]]
    # add spaces to output to align columns
    sol_line_tasks += "%-10s" % name
    start = assigned_task[:start]
    duration = assigned_task[:duration]
    sol_tmp = "[%i,%i]" % [start, start + duration]
    # add spaces to output to align columns
    sol_line += "%-10s" % sol_tmp
  end

  sol_line += "\n"
  sol_line_tasks += "\n"
  output += sol_line_tasks
  output += sol_line
end

# finally print the solution found
puts "Optimal Schedule Length: %i" % solver.objective_value
puts output

Other Examples

Sudoku

Example

# create the model
model = ORTools::CpModel.new

cell_size = 3
line_size = cell_size**2
line = (0...line_size).to_a
cell = (0...cell_size).to_a

initial_grid = [
  [0, 6, 0, 0, 5, 0, 0, 2, 0],
  [0, 0, 0, 3, 0, 0, 0, 9, 0],
  [7, 0, 0, 6, 0, 0, 0, 1, 0],
  [0, 0, 6, 0, 3, 0, 4, 0, 0],
  [0, 0, 4, 0, 7, 0, 1, 0, 0],
  [0, 0, 5, 0, 9, 0, 8, 0, 0],
  [0, 4, 0, 0, 0, 1, 0, 0, 6],
  [0, 3, 0, 0, 0, 8, 0, 0, 0],
  [0, 2, 0, 0, 4, 0, 0, 5, 0]
]

grid = {}
line.each do |i|
  line.each do |j|
    grid[[i, j]] = model.new_int_var(1, line_size, "grid %i %i" % [i, j])
  end
end

# all different on rows
line.each do |i|
  model.add_all_different(line.map { |j| grid[[i, j]] })
end

# all different on columns
line.each do |j|
  model.add_all_different(line.map { |i| grid[[i, j]] })
end

# all different on cells
cell.each do |i|
  cell.each do |j|
    one_cell = []
    cell.each do |di|
      cell.each do |dj|
        one_cell << grid[[i * cell_size + di, j * cell_size + dj]]
      end
    end
    model.add_all_different(one_cell)
  end
end

# initial values
line.each do |i|
  line.each do |j|
    if initial_grid[i][j] != 0
      model.add(grid[[i, j]] == initial_grid[i][j])
    end
  end
end

# solve and print solution
solver = ORTools::CpSolver.new
status = solver.solve(model)
if status == :optimal
  line.each do |i|
    p line.map { |j| solver.value(grid[[i, j]]) }
  end
end

Wedding Seating Chart

Example

# From
# Meghan L. Bellows and J. D. Luc Peterson
# "Finding an optimal seating chart for a wedding"
# https://www.improbable.com/news/2012/Optimal-seating-chart.pdf
# https://www.improbable.com/2012/02/12/finding-an-optimal-seating-chart-for-a-wedding
#
# Every year, millions of brides (not to mention their mothers, future
# mothers-in-law, and occasionally grooms) struggle with one of the
# most daunting tasks during the wedding-planning process: the
# seating chart. The guest responses are in, banquet hall is booked,
# menu choices have been made. You think the hard parts are over,
# but you have yet to embark upon the biggest headache of them all.
# In order to make this process easier, we present a mathematical
# formulation that models the seating chart problem. This model can
# be solved to find the optimal arrangement of guests at tables.
# At the very least, it can provide a starting point and hopefully
# minimize stress and arguments.
#
# Adapted from
# https://github.com/google/or-tools/blob/stable/examples/python/wedding_optimal_chart_sat.py

# Easy problem (from the paper)
# num_tables = 2
# table_capacity = 10
# min_known_neighbors = 1

# Slightly harder problem (also from the paper)
num_tables = 5
table_capacity = 4
min_known_neighbors = 1

# Connection matrix: who knows who, and how strong
# is the relation
c = [
  [1, 50, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0],
  [50, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0],
  [1, 1, 1, 50, 1, 1, 1, 1, 10, 0, 0, 0, 0, 0, 0, 0, 0],
  [1, 1, 50, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0],
  [1, 1, 1, 1, 1, 50, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0],
  [1, 1, 1, 1, 50, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0],
  [1, 1, 1, 1, 1, 1, 1, 50, 1, 0, 0, 0, 0, 0, 0, 0, 0],
  [1, 1, 1, 1, 1, 1, 50, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0],
  [1, 1, 10, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0],
  [0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 50, 1, 1, 1, 1, 1, 1],
  [0, 0, 0, 0, 0, 0, 0, 0, 0, 50, 1, 1, 1, 1, 1, 1, 1],
  [0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1],
  [0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1],
  [0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1],
  [0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1],
  [0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1],
  [0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1]
]

# Names of the guests. B: Bride side, G: Groom side
names = [
  "Deb (B)", "John (B)", "Martha (B)", "Travis (B)", "Allan (B)",
  "Lois (B)", "Jayne (B)", "Brad (B)", "Abby (B)", "Mary Helen (G)",
  "Lee (G)", "Annika (G)", "Carl (G)", "Colin (G)", "Shirley (G)",
  "DeAnn (G)", "Lori (G)"
]

num_guests = c.size

all_tables = num_tables.times.to_a
all_guests = num_guests.times.to_a

# create the cp model
model = ORTools::CpModel.new

# decision variables
seats = {}
all_tables.each do |t|
  all_guests.each do |g|
    seats[[t, g]] = model.new_bool_var("guest %i seats on table %i" % [g, t])
  end
end

pairs = all_guests.combination(2)

colocated = {}
pairs.each do |g1, g2|
  colocated[[g1, g2]] = model.new_bool_var("guest %i seats with guest %i" % [g1, g2])
end

same_table = {}
pairs.each do |g1, g2|
  all_tables.each do |t|
    same_table[[g1, g2, t]] = model.new_bool_var("guest %i seats with guest %i on table %i" % [g1, g2, t])
  end
end

# Objective
model.maximize(model.sum((num_guests - 1).times.flat_map { |g1| (g1 + 1).upto(num_guests - 1).select { |g2| c[g1][g2] > 0 }.map { |g2| colocated[[g1, g2]] * c[g1][g2] } }))

#
# Constraints
#

# Everybody seats at one table.
all_guests.each do |g|
  model.add(model.sum(all_tables.map { |t| seats[[t, g]] }) == 1)
end

# Tables have a max capacity.
all_tables.each do |t|
  model.add(model.sum(all_guests.map { |g| seats[[t, g]] }) <= table_capacity)
end

# Link colocated with seats
pairs.each do |g1, g2|
  all_tables.each do |t|
    # Link same_table and seats.
    model.add_bool_or([seats[[t, g1]].not, seats[[t, g2]].not, same_table[[g1, g2, t]]])
    model.add_implication(same_table[[g1, g2, t]], seats[[t, g1]])
    model.add_implication(same_table[[g1, g2, t]], seats[[t, g2]])
  end

  # Link colocated and same_table.
  model.add(model.sum(all_tables.map { |t| same_table[[g1, g2, t]] }) == colocated[[g1, g2]])
end

# Min known neighbors rule.
all_guests.each do |g|
  model.add(
    model.sum(
      (g + 1).upto(num_guests - 1).
      select { |g2| c[g][g2] > 0 }.
      product(all_tables).
      map { |g2, t| same_table[[g, g2, t]] }
    ) +
    model.sum(
      g.times.
      select { |g1| c[g1][g] > 0 }.
      product(all_tables).
      map { |g1, t| same_table[[g1, g, t]] }
    ) >= min_known_neighbors
  )
end

# Symmetry breaking. First guest seats on the first table.
model.add(seats[[0, 0]] == 1)

# Solve model
solver = ORTools::CpSolver.new
solution_printer = WeddingChartPrinter.new(seats, names, num_tables, num_guests)
solver.solve(model, solution_printer)

puts "Statistics"
puts "  - conflicts    : %i" % solver.num_conflicts
puts "  - branches     : %i" % solver.num_branches
puts "  - wall time    : %f s" % solver.wall_time
puts "  - num solutions: %i" % solution_printer.num_solutions

Set Partitioning

Example

# A set partitioning model of a wedding seating problem
# Authors: Stuart Mitchell 2009

max_tables = 5
max_table_size = 4
guests = %w(A B C D E F G I J K L M N O P Q R)

# Find the happiness of the table
# by calculating the maximum distance between the letters
def happiness(table)
  (table[0].ord - table[-1].ord).abs
end

# create list of all possible tables
possible_tables = []
(1..max_table_size).each do |i|
  possible_tables += guests.combination(i).to_a
end

solver = ORTools::Solver.new("CBC")

# create a binary variable to state that a table setting is used
x = {}
possible_tables.each do |table|
  x[table] = solver.int_var(0, 1, "table #{table.join(", ")}")
end

solver.minimize(possible_tables.sum { |table| x[table] * happiness(table) })

# specify the maximum number of tables
solver.add(x.values.sum <= max_tables)

# a guest must seated at one and only one table
guests.each do |guest|
  tables_with_guest = possible_tables.select { |table| table.include?(guest) }
  solver.add(tables_with_guest.sum { |table| x[table] } == 1)
end

status = solver.solve

puts "The chosen tables are out of a total of %s:" % possible_tables.size
possible_tables.each do |table|
  if x[table].solution_value == 1
    p table
  end
end

History

View the changelog

Contributing

Everyone is encouraged to help improve this project. Here are a few ways you can help:

Report bugs
Fix bugs and submit pull requests
Write, clarify, or fix documentation
Suggest or add new features

To get started with development:

git clone https://github.com/ankane/or-tools-ruby.git
cd or-tools-ruby
bundle install
bundle exec rake compile
bundle exec rake test

Resources

OR-Tools Reference

or-tools

Runtime

OR-Tools Ruby

Installation

Getting Started

Higher Level Interfaces

Scheduling

Seating

Traveling Salesperson Problem (TSP)

Sudoku

MathOpt

Basic Example

Linear Optimization

Solving an LP Problem

Integer Optimization

Solving a MIP Problem

Constraint Optimization

CP-SAT Solver

Solving a CP Problem

Cryptarithmetic

The N-queens Problem

Setting Solver Limits

Assignment

Solving an Assignment Problem

Assignment with Teams of Workers

Linear Sum Assignment Solver

Routing

Traveling Salesperson Problem (TSP)

Vehicle Routing Problem (VRP)

Capacity Constraints

Pickups and Deliveries

Time Window Constraints

Resource Constraints

Penalties and Dropping Visits

Routing Options

Bin Packing

The Knapsack Problem

Multiple Knapsacks

Bin Packing Problem

Network Flows

Maximum Flows

Minimum Cost Flows

Assignment as a Min Cost Flow Problem

Scheduling

Employee Scheduling

The Job Shop Problem

Other Examples

Sudoku

Wedding Seating Chart

Set Partitioning

History

Contributing