examples/notebook/contrib/stigler_contrib.ipynb
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First, you must install ortools package in this colab.
%pip install ortools
Original Stigler's 1939 diet problem Google or-tools.
From GLPK:s example stigler.mod ''' STIGLER, original Stigler's 1939 diet problem
The Stigler Diet is an optimization problem named for George Stigler, a 1982 Nobel Laureate in economics, who posed the following problem: For a moderately active man weighing 154 pounds, how much of each of 77 foods should be eaten on a daily basis so that the man's intake of nine nutrients will be at least equal to the recommended dietary allowances (RDSs) suggested by the National Research Council in 1943, with the cost of the diet being minimal?
The nutrient RDAs required to be met in Stigler's experiment were calories, protein, calcium, iron, vitamin A, thiamine, riboflavin, niacin, and ascorbic acid. The result was an annual budget allocated to foods such as evaporated milk, cabbage, dried navy beans, and beef liver at a cost of approximately $0.11 a day in 1939 U.S. dollars.
While the name 'Stigler Diet' was applied after the experiment by outsiders, according to Stigler, 'No one recommends these diets for anyone, let alone everyone.' The Stigler diet has been much ridiculed for its lack of variety and palatability, however his methodology has received praise and is considered to be some of the earliest work in linear programming.
The Stigler diet question is a linear programming problem. Lacking any sophisticated method of solving such a problem, Stigler was forced to utilize heuristic methods in order to find a solution. The diet question originally asked in which quantities a 154 pound male would have to consume 77 different foods in order to fulfill the recommended intake of 9 different nutrients while keeping expense at a minimum. Through 'trial and error, mathematical insight and agility,' Stigler was able to eliminate 62 of the foods from the original 77 (these foods were removed based because they lacked nutrients in comparison to the remaining 15). From the reduced list, Stigler calculated the required amounts of each of the remaining 15 foods to arrive at a cost-minimizing solution to his question. According to Stigler's calculations, the annual cost of his solution was $39.93 in 1939 dollars. When corrected for inflation using the consumer price index, the cost of the diet in 2005 dollars is $561.43. The specific combination of foods and quantities is as follows:
Stigler's 1939 Diet
Food Annual Quantities Annual Cost
Total Annual Cost $39.93
The 9 nutrients that Stigler's diet took into consideration and their respective recommended daily amounts were:
Table of nutrients considered in Stigler's diet
Nutrient Daily Recommended Intake
Calories 3,000 Calories Protein 70 grams Calcium .8 grams Iron 12 milligrams Vitamin A 5,000 IU Thiamine (Vitamin B1) 1.8 milligrams Riboflavin (Vitamin B2) 2.7 milligrams Niacin 18 milligrams Ascorbic Acid (Vitamin C) 75 milligrams
Seven years after Stigler made his initial estimates, the development of George Dantzig's Simplex algorithm made it possible to solve the problem without relying on heuristic methods. The exact value was determined to be $39.69 (using the original 1939 data). Dantzig's algorithm describes a method of traversing the vertices of a polytope of N+1 dimensions in order to find the optimal solution to a specific situation.
(From Wikipedia, the free encyclopedia.)
Translated from GAMS by Andrew Makhorin [email protected].
For the original GAMS model stigler1939.gms see [3].
References:
George J. Stigler, 'The Cost of Subsistence,' J. Farm Econ. 27, 1945, pp. 303-14.
National Research Council, 'Recommended Daily Allowances,' Reprint and Circular Series No. 115, January, 1943.
Erwin Kalvelagen, 'Model building with GAMS,' Chapter 2, 'Building linear programming models,' pp. 128-34. '''
This model was created by Hakan Kjellerstrand ([email protected]) Also see my other Google CP Solver models: http://www.hakank.org/google_or_tools/
import sys
from ortools.linear_solver import pywraplp
def main(sol="CBC"):
# Create the solver.
print("Solver: ", sol)
solver = pywraplp.Solver.CreateSolver(sol)
if not solver:
return
#
# data
#
# commodities
num_commodities = 77
C = list(range(num_commodities))
# days in a year
days = 365.25
# nutrients
num_nutrients = 9
N = list(range(num_nutrients))
nutrients = [
"calories", # Calories, unit = 1000
"protein", # Protein, unit = grams
"calcium", # Calcium, unit = grams
"iron", # Iron, unit = milligrams
"vitaminA", # Vitamin A, unit = 1000 International Units
"thiamine", # Thiamine, Vit. B1, unit = milligrams
"riboflavin", # Riboflavin, Vit. B2, unit = milligrams
"niacin", # Niacin (Nicotinic Acid), unit = milligrams
"ascorbicAcid" # Ascorbic Acid, Vit. C, unit = milligrams
]
commodities = [["Wheat Flour (Enriched)", "10 lb."], ["Macaroni", "1 lb."],
["Wheat Cereal (Enriched)",
"28 oz."], ["Corn Flakes", "8 oz."], ["Corn Meal", "1 lb."],
["Hominy Grits", "24 oz."], ["Rice", "1 lb."],
["Rolled Oats", "1 lb."], ["White Bread (Enriched)", "1 lb."],
["Whole Wheat Bread", "1 lb."], ["Rye Bread", "1 lb."],
["Pound Cake", "1 lb."], ["Soda Crackers", "1 lb."],
["Milk", "1 qt."], ["Evaporated Milk (can)", "14.5 oz."],
["Butter", "1 lb."], ["Oleomargarine", "1 lb."],
["Eggs", "1 doz."], ["Cheese (Cheddar)", "1 lb."],
["Cream", "1/2 pt."], ["Peanut Butter", "1 lb."],
["Mayonnaise", "1/2 pt."], ["Crisco", "1 lb."],
["Lard", "1 lb."], ["Sirloin Steak", "1 lb."],
["Round Steak", "1 lb."], ["Rib Roast", "1 lb."],
["Chuck Roast", "1 lb."], ["Plate", "1 lb."],
["Liver (Beef)", "1 lb."], ["Leg of Lamb", "1 lb."],
["Lamb Chops (Rib)", "1 lb."], ["Pork Chops", "1 lb."],
["Pork Loin Roast", "1 lb."], ["Bacon", "1 lb."],
["Ham - smoked", "1 lb."], ["Salt Pork", "1 lb."],
["Roasting Chicken", "1 lb."], ["Veal Cutlets", "1 lb."],
["Salmon, Pink (can)", "16 oz."], ["Apples", "1 lb."],
["Bananas", "1 lb."], ["Lemons", "1 doz."],
["Oranges", "1 doz."], ["Green Beans", "1 lb."],
["Cabbage", "1 lb."], ["Carrots", "1 bunch"],
["Celery", "1 stalk"], ["Lettuce", "1 head"],
["Onions", "1 lb."], ["Potatoes", "15 lb."],
["Spinach", "1 lb."], ["Sweet Potatoes", "1 lb."],
["Peaches (can)", "No. 2 1/2"], ["Pears (can)", "No. 2 1/2,"],
["Pineapple (can)", "No. 2 1/2"], ["Asparagus (can)", "No. 2"],
["Grean Beans (can)", "No. 2"],
["Pork and Beans (can)", "16 oz."], ["Corn (can)", "No. 2"],
["Peas (can)", "No. 2"], ["Tomatoes (can)", "No. 2"],
["Tomato Soup (can)", "10 1/2 oz."],
["Peaches, Dried", "1 lb."], ["Prunes, Dried", "1 lb."],
["Raisins, Dried", "15 oz."], ["Peas, Dried", "1 lb."],
["Lima Beans, Dried", "1 lb."], ["Navy Beans, Dried", "1 lb."],
["Coffee", "1 lb."], ["Tea", "1/4 lb."], ["Cocoa", "8 oz."],
["Chocolate", "8 oz."], ["Sugar", "10 lb."],
["Corn Sirup", "24 oz."], ["Molasses", "18 oz."],
["Strawberry Preserve", "1 lb."]]
# price and weight are the two first columns
data = [
[36.0, 12600.0, 44.7, 1411.0, 2.0, 365.0, 0.0, 55.4, 33.3, 441.0, 0.0],
[14.1, 3217.0, 11.6, 418.0, 0.7, 54.0, 0.0, 3.2, 1.9, 68.0, 0.0],
[24.2, 3280.0, 11.8, 377.0, 14.4, 175.0, 0.0, 14.4, 8.8, 114.0, 0.0],
[7.1, 3194.0, 11.4, 252.0, 0.1, 56.0, 0.0, 13.5, 2.3, 68.0, 0.0],
[4.6, 9861.0, 36.0, 897.0, 1.7, 99.0, 30.9, 17.4, 7.9, 106.0, 0.0],
[8.5, 8005.0, 28.6, 680.0, 0.8, 80.0, 0.0, 10.6, 1.6, 110.0, 0.0],
[7.5, 6048.0, 21.2, 460.0, 0.6, 41.0, 0.0, 2.0, 4.8, 60.0, 0.0],
[7.1, 6389.0, 25.3, 907.0, 5.1, 341.0, 0.0, 37.1, 8.9, 64.0, 0.0],
[7.9, 5742.0, 15.6, 488.0, 2.5, 115.0, 0.0, 13.8, 8.5, 126.0, 0.0],
[9.1, 4985.0, 12.2, 484.0, 2.7, 125.0, 0.0, 13.9, 6.4, 160.0, 0.0],
[9.2, 4930.0, 12.4, 439.0, 1.1, 82.0, 0.0, 9.9, 3.0, 66.0, 0.0],
[24.8, 1829.0, 8.0, 130.0, 0.4, 31.0, 18.9, 2.8, 3.0, 17.0, 0.0],
[15.1, 3004.0, 12.5, 288.0, 0.5, 50.0, 0.0, 0.0, 0.0, 0.0, 0.0],
[11.0, 8867.0, 6.1, 310.0, 10.5, 18.0, 16.8, 4.0, 16.0, 7.0, 177.0],
[6.7, 6035.0, 8.4, 422.0, 15.1, 9.0, 26.0, 3.0, 23.5, 11.0, 60.0],
[20.8, 1473.0, 10.8, 9.0, 0.2, 3.0, 44.2, 0.0, 0.2, 2.0, 0.0],
[16.1, 2817.0, 20.6, 17.0, 0.6, 6.0, 55.8, 0.2, 0.0, 0.0, 0.0],
[32.6, 1857.0, 2.9, 238.0, 1.0, 52.0, 18.6, 2.8, 6.5, 1.0, 0.0],
[24.2, 1874.0, 7.4, 448.0, 16.4, 19.0, 28.1, 0.8, 10.3, 4.0, 0.0],
[14.1, 1689.0, 3.5, 49.0, 1.7, 3.0, 16.9, 0.6, 2.5, 0.0, 17.0],
[17.9, 2534.0, 15.7, 661.0, 1.0, 48.0, 0.0, 9.6, 8.1, 471.0, 0.0],
[16.7, 1198.0, 8.6, 18.0, 0.2, 8.0, 2.7, 0.4, 0.5, 0.0, 0.0],
[20.3, 2234.0, 20.1, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0],
[9.8, 4628.0, 41.7, 0.0, 0.0, 0.0, 0.2, 0.0, 0.5, 5.0, 0.0],
[39.6, 1145.0, 2.9, 166.0, 0.1, 34.0, 0.2, 2.1, 2.9, 69.0, 0.0],
[36.4, 1246.0, 2.2, 214.0, 0.1, 32.0, 0.4, 2.5, 2.4, 87.0, 0.0],
[29.2, 1553.0, 3.4, 213.0, 0.1, 33.0, 0.0, 0.0, 2.0, 0.0, 0.0],
[22.6, 2007.0, 3.6, 309.0, 0.2, 46.0, 0.4, 1.0, 4.0, 120.0, 0.0],
[14.6, 3107.0, 8.5, 404.0, 0.2, 62.0, 0.0, 0.9, 0.0, 0.0, 0.0],
[26.8, 1692.0, 2.2, 333.0, 0.2, 139.0, 169.2, 6.4, 50.8, 316.0, 525.0],
[27.6, 1643.0, 3.1, 245.0, 0.1, 20.0, 0.0, 2.8, 3.0, 86.0, 0.0],
[36.6, 1239.0, 3.3, 140.0, 0.1, 15.0, 0.0, 1.7, 2.7, 54.0, 0.0],
[30.7, 1477.0, 3.5, 196.0, 0.2, 80.0, 0.0, 17.4, 2.7, 60.0, 0.0],
[24.2, 1874.0, 4.4, 249.0, 0.3, 37.0, 0.0, 18.2, 3.6, 79.0, 0.0],
[25.6, 1772.0, 10.4, 152.0, 0.2, 23.0, 0.0, 1.8, 1.8, 71.0, 0.0],
[27.4, 1655.0, 6.7, 212.0, 0.2, 31.0, 0.0, 9.9, 3.3, 50.0, 0.0],
[16.0, 2835.0, 18.8, 164.0, 0.1, 26.0, 0.0, 1.4, 1.8, 0.0, 0.0],
[30.3, 1497.0, 1.8, 184.0, 0.1, 30.0, 0.1, 0.9, 1.8, 68.0, 46.0],
[42.3, 1072.0, 1.7, 156.0, 0.1, 24.0, 0.0, 1.4, 2.4, 57.0, 0.0],
[13.0, 3489.0, 5.8, 705.0, 6.8, 45.0, 3.5, 1.0, 4.9, 209.0, 0.0],
[4.4, 9072.0, 5.8, 27.0, 0.5, 36.0, 7.3, 3.6, 2.7, 5.0, 544.0],
[6.1, 4982.0, 4.9, 60.0, 0.4, 30.0, 17.4, 2.5, 3.5, 28.0, 498.0],
[26.0, 2380.0, 1.0, 21.0, 0.5, 14.0, 0.0, 0.5, 0.0, 4.0, 952.0],
[30.9, 4439.0, 2.2, 40.0, 1.1, 18.0, 11.1, 3.6, 1.3, 10.0, 1993.0],
[7.1, 5750.0, 2.4, 138.0, 3.7, 80.0, 69.0, 4.3, 5.8, 37.0, 862.0],
[3.7, 8949.0, 2.6, 125.0, 4.0, 36.0, 7.2, 9.0, 4.5, 26.0, 5369.0],
[4.7, 6080.0, 2.7, 73.0, 2.8, 43.0, 188.5, 6.1, 4.3, 89.0, 608.0],
[7.3, 3915.0, 0.9, 51.0, 3.0, 23.0, 0.9, 1.4, 1.4, 9.0, 313.0],
[8.2, 2247.0, 0.4, 27.0, 1.1, 22.0, 112.4, 1.8, 3.4, 11.0, 449.0],
[3.6, 11844.0, 5.8, 166.0, 3.8, 59.0, 16.6, 4.7, 5.9, 21.0, 1184.0],
[34.0, 16810.0, 14.3, 336.0, 1.8, 118.0, 6.7, 29.4, 7.1, 198.0, 2522.0],
[8.1, 4592.0, 1.1, 106.0, 0.0, 138.0, 918.4, 5.7, 13.8, 33.0, 2755.0],
[5.1, 7649.0, 9.6, 138.0, 2.7, 54.0, 290.7, 8.4, 5.4, 83.0, 1912.0],
[16.8, 4894.0, 3.7, 20.0, 0.4, 10.0, 21.5, 0.5, 1.0, 31.0, 196.0],
[20.4, 4030.0, 3.0, 8.0, 0.3, 8.0, 0.8, 0.8, 0.8, 5.0, 81.0],
[21.3, 3993.0, 2.4, 16.0, 0.4, 8.0, 2.0, 2.8, 0.8, 7.0, 399.0],
[27.7, 1945.0, 0.4, 33.0, 0.3, 12.0, 16.3, 1.4, 2.1, 17.0, 272.0],
[10.0, 5386.0, 1.0, 54.0, 2.0, 65.0, 53.9, 1.6, 4.3, 32.0, 431.0],
[7.1, 6389.0, 7.5, 364.0, 4.0, 134.0, 3.5, 8.3, 7.7, 56.0, 0.0],
[10.4, 5452.0, 5.2, 136.0, 0.2, 16.0, 12.0, 1.6, 2.7, 42.0, 218.0],
[13.8, 4109.0, 2.3, 136.0, 0.6, 45.0, 34.9, 4.9, 2.5, 37.0, 370.0],
[8.6, 6263.0, 1.3, 63.0, 0.7, 38.0, 53.2, 3.4, 2.5, 36.0, 1253.0],
[7.6, 3917.0, 1.6, 71.0, 0.6, 43.0, 57.9, 3.5, 2.4, 67.0, 862.0],
[15.7, 2889.0, 8.5, 87.0, 1.7, 173.0, 86.8, 1.2, 4.3, 55.0, 57.0],
[9.0, 4284.0, 12.8, 99.0, 2.5, 154.0, 85.7, 3.9, 4.3, 65.0, 257.0],
[9.4, 4524.0, 13.5, 104.0, 2.5, 136.0, 4.5, 6.3, 1.4, 24.0, 136.0],
[7.9, 5742.0, 20.0, 1367.0, 4.2, 345.0, 2.9, 28.7, 18.4, 162.0, 0.0],
[8.9, 5097.0, 17.4, 1055.0, 3.7, 459.0, 5.1, 26.9, 38.2, 93.0, 0.0],
[5.9, 7688.0, 26.9, 1691.0, 11.4, 792.0, 0.0, 38.4, 24.6, 217.0, 0.0],
[22.4, 2025.0, 0.0, 0.0, 0.0, 0.0, 0.0, 4.0, 5.1, 50.0, 0.0],
[17.4, 652.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 2.3, 42.0, 0.0],
[8.6, 2637.0, 8.7, 237.0, 3.0, 72.0, 0.0, 2.0, 11.9, 40.0, 0.0],
[16.2, 1400.0, 8.0, 77.0, 1.3, 39.0, 0.0, 0.9, 3.4, 14.0, 0.0],
[51.7, 8773.0, 34.9, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0],
[13.7, 4996.0, 14.7, 0.0, 0.5, 74.0, 0.0, 0.0, 0.0, 5.0, 0.0],
[13.6, 3752.0, 9.0, 0.0, 10.3, 244.0, 0.0, 1.9, 7.5, 146.0, 0.0],
[20.5, 2213.0, 6.4, 11.0, 0.4, 7.0, 0.2, 0.2, 0.4, 3.0, 0.0]
]
# recommended daily allowance for a moderately active man
allowance = [3.0, 70.0, 0.8, 12.0, 5.0, 1.8, 2.7, 18.0, 75.0]
#
# variables
#
x = [solver.NumVar(0, 1000, "x[%i]" % i) for i in C]
x_cost = [solver.NumVar(0, 1000, "x_cost[%i]" % i) for i in C]
quant = [solver.NumVar(0, 1000, "quant[%i]" % i) for i in C]
# total food bill
total_cost = solver.NumVar(0, 1000, "total_cost")
# cost per day, to minimize
cost = solver.Sum(x)
#
# constraints
#
solver.Add(total_cost == days * cost) # cost per year
for c in C:
solver.Add(x_cost[c] == days * x[c])
solver.Add(quant[c] == 100.0 * days * x[c] / data[c][0])
# nutrient balance
for n in range(2, num_nutrients + 2):
solver.Add(solver.Sum([data[c][n] * x[c] for c in C]) >= allowance[n - 2])
objective = solver.Minimize(cost)
#
# solution and search
#
solver.Solve()
print()
print("Cost = %0.2f" % solver.Objective().Value())
# print 'Cost:', cost.SolutionValue()
print("Total cost: %0.2f" % total_cost.SolutionValue())
print()
for i in C:
if x[i].SolutionValue() > 0:
print("%-21s %-11s %0.2f %0.2f" %
(commodities[i][0], commodities[i][1], x_cost[i].SolutionValue(),
quant[i].SolutionValue()))
print()
print("walltime :", solver.WallTime(), "ms")
if sol == "CBC":
print("iterations:", solver.Iterations())
sol = "CBC"
if len(sys.argv) > 1:
sol = sys.argv[1]
if sol != "GLPK" and sol != "CBC":
print("Solver must be either GLPK or CBC")
sys.exit(1)
main(sol)