Linear Optimization(借助ortools)

目录

• 优化问题的要素
• 解决问题步骤
• The Stigler diet

优化问题的要素

• objective，欲优化的量。比如某函数的最大值或最小值。
• constraints, 约束变量。基于问题的特定需求，对可行解进行约束。

线性优化用于计算一组线性关系建模问题的最优解。

谷歌提供的开源库Glop可用来求解该问题。

解决问题步骤

• 声明求解器
• 创建变量
• 定义约束
• 定义目标函数
• 调用求解器
• 展示结果

eg1:

• objective: maximize(x+y)
• constraints:
  • 0 ≤ x ≤ 1
  • 0 ≤ y ≤ 2

from ortools.linear_solver import pywraplp

def main():
  solver = pywraplp.Solver('SolveSimpleSystem',
                           pywraplp.Solver.GLOP_LINEAR_PROGRAMMING)
  # Create the variables x and y.
  x = solver.NumVar(0, 1, 'x')
  y = solver.NumVar(0, 2, 'y')
  # Create the objective function, x + y.
  objective = solver.Objective()
  objective.SetCoefficient(x, 1)
  objective.SetCoefficient(y, 1)
  objective.SetMaximization()
  # Call the solver and display the results.
  solver.Solve()
  print('Solution:')
  print('x = ', x.solution_value())
  print('y = ', y.solution_value())

if __name__ == '__main__':
  main()

运行得

Solution:
x =  1.0
y =  2.0

eg2:

• objective: maximize(3x + 4y)
• constraints:
  • x + 2y ≤ 14
  • 3x – y ≥ 0
  • x – y ≤ 2

"""Linear optimization example"""
from ortools.linear_solver import pywraplp

def main():
  # Instantiate a Glop solver, naming it LinearExample.
  solver = pywraplp.Solver('LinearExample',
                           pywraplp.Solver.GLOP_LINEAR_PROGRAMMING)

# Create the two variables and let them take on any value.
  x = solver.NumVar(-solver.infinity(), solver.infinity(), 'x')
  y = solver.NumVar(-solver.infinity(), solver.infinity(), 'y')

  # Constraint 1: x + 2y <= 14.
  constraint1 = solver.Constraint(-solver.infinity(), 14)
  constraint1.SetCoefficient(x, 1)
  constraint1.SetCoefficient(y, 2)

  # Constraint 2: 3x - y >= 0.
  constraint2 = solver.Constraint(0, solver.infinity())
  constraint2.SetCoefficient(x, 3)
  constraint2.SetCoefficient(y, -1)

  # Constraint 3: x - y <= 2.
  constraint3 = solver.Constraint(-solver.infinity(), 2)
  constraint3.SetCoefficient(x, 1)
  constraint3.SetCoefficient(y, -1)

  # Objective function: 3x + 4y.
  objective = solver.Objective()
  objective.SetCoefficient(x, 3)
  objective.SetCoefficient(y, 4)
  objective.SetMaximization()

  # Solve the system.
  solver.Solve()
  opt_solution = 3 * x.solution_value() + 4 * y.solution_value()
  print('Number of variables =', solver.NumVariables())
  print('Number of constraints =', solver.NumConstraints())
  # The value of each variable in the solution.
  print('Solution:')
  print('x = ', x.solution_value())
  print('y = ', y.solution_value())
  # The objective value of the solution.
  print('Optimal objective value =', opt_solution)
if __name__ == '__main__':
  main()

运行得

Number of variables = 2
Number of constraints = 3
Solution:
x =  5.999999999999998
y =  3.9999999999999996
Optimal objective value = 33.99999999999999

The Stigler diet

from __future__ import print_function
from ortools.linear_solver import pywraplp

def main():
  # Commodity, Unit, 1939 price (cents), Calories, Protein (g), Calcium (g), Iron (mg),
  # Vitamin A (IU), Thiamine (mg), Riboflavin (mg), Niacin (mg), Ascorbic Acid (mg)
  data = [
    ['Wheat Flour (Enriched)', '10 lb.', 36, 44.7, 1411, 2, 365, 0, 55.4, 33.3, 441, 0],
    ['Macaroni', '1 lb.', 14.1, 11.6, 418, 0.7, 54, 0, 3.2, 1.9, 68, 0],
    ['Wheat Cereal (Enriched)', '28 oz.', 24.2, 11.8, 377, 14.4, 175, 0, 14.4, 8.8, 114, 0],
    ['Corn Flakes', '8 oz.', 7.1, 11.4, 252, 0.1, 56, 0, 13.5, 2.3, 68, 0],
    ['Corn Meal', '1 lb.', 4.6, 36.0, 897, 1.7, 99, 30.9, 17.4, 7.9, 106, 0],
    ['Hominy Grits', '24 oz.', 8.5, 28.6, 680, 0.8, 80, 0, 10.6, 1.6, 110, 0],
    ['Rice', '1 lb.', 7.5, 21.2, 460, 0.6, 41, 0, 2, 4.8, 60, 0],
    ['Rolled Oats', '1 lb.', 7.1, 25.3, 907, 5.1, 341, 0, 37.1, 8.9, 64, 0],
    ['White Bread (Enriched)', '1 lb.', 7.9, 15.0, 488, 2.5, 115, 0, 13.8, 8.5, 126, 0],
    ['Whole Wheat Bread', '1 lb.', 9.1, 12.2, 484, 2.7, 125, 0, 13.9, 6.4, 160, 0],
    ['Rye Bread', '1 lb.', 9.1, 12.4, 439, 1.1, 82, 0, 9.9, 3, 66, 0],
    ['Pound Cake', '1 lb.', 24.8, 8.0, 130, 0.4, 31, 18.9, 2.8, 3, 17, 0],
    ['Soda Crackers', '1 lb.', 15.1, 12.5, 288, 0.5, 50, 0, 0, 0, 0, 0],
    ['Milk', '1 qt.', 11, 6.1, 310, 10.5, 18, 16.8, 4, 16, 7, 177],
    ['Evaporated Milk (can)', '14.5 oz.', 6.7, 8.4, 422, 15.1, 9, 26, 3, 23.5, 11, 60],
    ['Butter', '1 lb.', 30.8, 10.8, 9, 0.2, 3, 44.2, 0, 0.2, 2, 0],
    ['Oleomargarine', '1 lb.', 16.1, 20.6, 17, 0.6, 6, 55.8, 0.2, 0, 0, 0],
    ['Eggs', '1 doz.', 32.6, 2.9, 238, 1.0, 52, 18.6, 2.8, 6.5, 1, 0],
    ['Cheese (Cheddar)', '1 lb.', 24.2, 7.4, 448, 16.4, 19, 28.1, 0.8, 10.3, 4, 0],
    ['Cream', '1/2 pt.', 14.1, 3.5, 49, 1.7, 3, 16.9, 0.6, 2.5, 0, 17],
    ['Peanut Butter', '1 lb.', 17.9, 15.7, 661, 1.0, 48, 0, 9.6, 8.1, 471, 0],
    ['Mayonnaise', '1/2 pt.', 16.7, 8.6, 18, 0.2, 8, 2.7, 0.4, 0.5, 0, 0],
    ['Crisco', '1 lb.', 20.3, 20.1, 0, 0, 0, 0, 0, 0, 0, 0],
    ['Lard', '1 lb.', 9.8, 41.7, 0, 0, 0, 0.2, 0, 0.5, 5, 0],
    ['Sirloin Steak', '1 lb.', 39.6, 2.9, 166, 0.1, 34, 0.2, 2.1, 2.9, 69, 0],
    ['Round Steak', '1 lb.', 36.4, 2.2, 214, 0.1, 32, 0.4, 2.5, 2.4, 87, 0],
    ['Rib Roast', '1 lb.', 29.2, 3.4, 213, 0.1, 33, 0, 0, 2, 0, 0],
    ['Chuck Roast', '1 lb.', 22.6, 3.6, 309, 0.2, 46, 0.4, 1, 4, 120, 0],
    ['Plate', '1 lb.', 14.6, 8.5, 404, 0.2, 62, 0, 0.9, 0, 0, 0],
    ['Liver (Beef)', '1 lb.', 26.8, 2.2, 333, 0.2, 139, 169.2, 6.4, 50.8, 316, 525],
    ['Leg of Lamb', '1 lb.', 27.6, 3.1, 245, 0.1, 20, 0, 2.8, 3.9, 86, 0],
    ['Lamb Chops (Rib)', '1 lb.', 36.6, 3.3, 140, 0.1, 15, 0, 1.7, 2.7, 54, 0],
    ['Pork Chops', '1 lb.', 30.7, 3.5, 196, 0.2, 30, 0, 17.4, 2.7, 60, 0],
    ['Pork Loin Roast', '1 lb.', 24.2, 4.4, 249, 0.3, 37, 0, 18.2, 3.6, 79, 0],
    ['Bacon', '1 lb.', 25.6, 10.4, 152, 0.2, 23, 0, 1.8, 1.8, 71, 0],
    ['Ham, smoked', '1 lb.', 27.4, 6.7, 212, 0.2, 31, 0, 9.9, 3.3, 50, 0],
    ['Salt Pork', '1 lb.', 16, 18.8, 164, 0.1, 26, 0, 1.4, 1.8, 0, 0],
    ['Roasting Chicken', '1 lb.', 30.3, 1.8, 184, 0.1, 30, 0.1, 0.9, 1.8, 68, 46],
    ['Veal Cutlets', '1 lb.', 42.3, 1.7, 156, 0.1, 24, 0, 1.4, 2.4, 57, 0],
    ['Salmon, Pink (can)', '16 oz.', 13, 5.8, 705, 6.8, 45, 3.5, 1, 4.9, 209, 0],
    ['Apples', '1 lb.', 4.4, 5.8, 27, 0.5, 36, 7.3, 3.6, 2.7, 5, 544],
    ['Bananas', '1 lb.', 6.1, 4.9, 60, 0.4, 30, 17.4, 2.5, 3.5, 28, 498],
    ['Lemons', '1 doz.', 26, 1.0, 21, 0.5, 14, 0, 0.5, 0, 4, 952],
    ['Oranges', '1 doz.', 30.9, 2.2, 40, 1.1, 18, 11.1, 3.6, 1.3, 10, 1998],
    ['Green Beans', '1 lb.', 7.1, 2.4, 138, 3.7, 80, 69, 4.3, 5.8, 37, 862],
    ['Cabbage', '1 lb.', 3.7, 2.6, 125, 4.0, 36, 7.2, 9, 4.5, 26, 5369],
    ['Carrots', '1 bunch', 4.7, 2.7, 73, 2.8, 43, 188.5, 6.1, 4.3, 89, 608],
    ['Celery', '1 stalk', 7.3, 0.9, 51, 3.0, 23, 0.9, 1.4, 1.4, 9, 313],
    ['Lettuce', '1 head', 8.2, 0.4, 27, 1.1, 22, 112.4, 1.8, 3.4, 11, 449],
    ['Onions', '1 lb.', 3.6, 5.8, 166, 3.8, 59, 16.6, 4.7, 5.9, 21, 1184],
    ['Potatoes', '15 lb.', 34, 14.3, 336, 1.8, 118, 6.7, 29.4, 7.1, 198, 2522],
    ['Spinach', '1 lb.', 8.1, 1.1, 106, 0, 138, 918.4, 5.7, 13.8, 33, 2755],
    ['Sweet Potatoes', '1 lb.', 5.1, 9.6, 138, 2.7, 54, 290.7, 8.4, 5.4, 83, 1912],
    ['Peaches (can)', 'No. 2 1/2', 16.8, 3.7, 20, 0.4, 10, 21.5, 0.5, 1, 31, 196],
    ['Pears (can)', 'No. 2 1/2', 20.4, 3.0, 8, 0.3, 8, 0.8, 0.8, 0.8, 5, 81],
    ['Pineapple (can)', 'No. 2 1/2', 21.3, 2.4, 16, 0.4, 8, 2, 2.8, 0.8, 7, 399],
    ['Asparagus (can)', 'No. 2', 27.7, 0.4, 33, 0.3, 12, 16.3, 1.4, 2.1, 17, 272],
    ['Green Beans (can)', 'No. 2', 10, 1.0, 54, 2, 65, 53.9, 1.6, 4.3, 32, 431],
    ['Pork and Beans (can)', '16 oz.', 7.1, 7.5, 364, 4, 134, 3.5, 8.3, 7.7, 56, 0],
    ['Corn (can)', 'No. 2', 10.4, 5.2, 136, 0.2, 16, 12, 1.6, 2.7, 42, 218],
    ['Peas (can)', 'No. 2', 13.8, 2.3, 136, 0.6, 45, 34.9, 4.9, 2.5, 37, 370],
    ['Tomatoes (can)', 'No. 2', 8.6, 1.3, 63, 0.7, 38, 53.2, 3.4, 2.5, 36, 1253],
    ['Tomato Soup (can)', '10 1/2 oz.', 7.6, 1.6, 71, 0.6, 43, 57.9, 3.5, 2.4, 67, 862],
    ['Peaches, Dried', '1 lb.', 15.7, 8.5, 87, 1.7, 173, 86.8, 1.2, 4.3, 55, 57],
    ['Prunes, Dried', '1 lb.', 9, 12.8, 99, 2.5, 154, 85.7, 3.9, 4.3, 65, 257],
    ['Raisins, Dried', '15 oz.', 9.4, 13.5, 104, 2.5, 136, 4.5, 6.3, 1.4, 24, 136],
    ['Peas, Dried', '1 lb.', 7.9, 20.0, 1367, 4.2, 345, 2.9, 28.7, 18.4, 162, 0],
    ['Lima Beans, Dried', '1 lb.', 8.9, 17.4, 1055, 3.7, 459, 5.1, 26.9, 38.2, 93, 0],
    ['Navy Beans, Dried', '1 lb.', 5.9, 26.9, 1691, 11.4, 792, 0, 38.4, 24.6, 217, 0],
    ['Coffee', '1 lb.', 22.4, 0, 0, 0, 0, 0, 4, 5.1, 50, 0],
    ['Tea', '1/4 lb.', 17.4, 0, 0, 0, 0, 0, 0, 2.3, 42, 0],
    ['Cocoa', '8 oz.', 8.6, 8.7, 237, 3, 72, 0, 2, 11.9, 40, 0],
    ['Chocolate', '8 oz.', 16.2, 8.0, 77, 1.3, 39, 0, 0.9, 3.4, 14, 0],
    ['Sugar', '10 lb.', 51.7, 34.9, 0, 0, 0, 0, 0, 0, 0, 0],
    ['Corn Syrup', '24 oz.', 13.7, 14.7, 0, 0.5, 74, 0, 0, 0, 5, 0],
    ['Molasses', '18 oz.', 13.6, 9.0, 0, 10.3, 244, 0, 1.9, 7.5, 146, 0],
    ['Strawberry Preserves', '1 lb.', 20.5, 6.4, 11, 0.4, 7, 0.2, 0.2, 0.4, 3, 0]];

  # Nutrient minimums.
  nutrients = [
      ['Calories (1000s)', 3],
      ['Protein (grams)', 70],
      ['Calcium (grams)', 0.8],
      ['Iron (mg)', 12],
      ['Vitamin A (1000 IU)', 5],
      ['Vitamin B1 (mg)', 1.8],
      ['Vitamin B2 (mg)', 2.7],
      ['Niacin (mg)', 18],
      ['Vitamin C (mg)', 75]]
  # Instantiate a Glop solver, naming it SolveStigler.
  solver = pywraplp.Solver('SolveStigler',
                           pywraplp.Solver.GLOP_LINEAR_PROGRAMMING)
  # Declare an array to hold our nutritional data.
  food = [[]] * len(data)

  # Objective: minimize the sum of (price-normalized) foods.
  objective = solver.Objective()
  for i in range(0, len(data)):
    food[i] = solver.NumVar(0.0, solver.infinity(), data[i][0])
    objective.SetCoefficient(food[i], 1)
  objective.SetMinimization()
  # Create the constraints, one per nutrient.
  constraints = [0] * len(nutrients)
  for i in range(0, len(nutrients)):
    constraints[i] = solver.Constraint(nutrients[i][1], solver.infinity())
    for j in range(0, len(data)):
      constraints[i].SetCoefficient(food[j], data[j][i+3])
  # Solve!
  status = solver.Solve()

  if status == solver.OPTIMAL:
    # Display the amounts (in dollars) to purchase of each food.
    price = 0
    num_nutrients = len(data[i]) - 3
    nutrients = [0] * (len(data[i]) - 3)
    for i in range(0, len(data)):
      price += food[i].solution_value()

      for nutrient in range(0, num_nutrients):
        nutrients[nutrient] += data[i][nutrient+3] * food[i].solution_value()

      if food[i].solution_value() > 0:
        print('%s = %f' % (data[i][0], food[i].solution_value()))

    print('Optimal annual price: $%.2f' % (365 * price))
  else:  # No optimal solution was found.
    if status == solver.FEASIBLE:
      print('A potentially suboptimal solution was found.')
    else:
      print('The solver could not solve the problem.')

if __name__ == '__main__':
  main()