/* Original Stigler's 1939 diet problem in Picat. 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 ---------------- ----------------- ----------- Wheat Flour 370 lb. $13.33 Evaporated Milk 57 cans 3.84 Cabbage 111 lb. 4.11 Spinach 23 lb. 1.85 Dried Navy Beans 285 lb. 16.80 ---------------------------------------------- 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 . For the original GAMS model stigler1939.gms see [3]. References: 1. George J. Stigler, "The Cost of Subsistence," J. Farm Econ. 27, 1945, pp. 303-14. 2. National Research Council, "Recommended Daily Allowances," Reprint and Circular Series No. 115, January, 1943. 3. Erwin Kalvelagen, "Model building with GAMS," Chapter 2, "Building linear programming models," pp. 128-34. """ Solution according GLPK: """ Commodity Unit Quantity Cost ------------------------- ---------- ---------- ---------- Wheat Flour (Enriched) 10 lb. 29.95 $ 10.78 Liver (Beef) 1 lb. 2.58 $ 0.69 Cabbage 1 lb. 110.70 $ 4.10 Spinach 1 lb. 22.58 $ 1.83 Navy Beans, Dried 1 lb. 377.81 $ 22.29 ----------------- Total: $ 39.69 """ This Picat model was created by Hakan Kjellerstrand, hakank@gmail.com See also my Picat page: http://www.hakank.org/picat/ */ % import util. import mip. main => go. go => stigler, nl. stigler => data(Data), NumCommodities = Data.length, % days in a year Days = 365.25, % nutrients NumNutrients = 9, allowance(Allowance), % decision variables % dollars of food to be purchased daily X = new_list(NumCommodities), X :: 0.0..1.0, XCost = new_list(NumCommodities), XCost :: 0.0..100.0, Quant = new_list(NumCommodities), Quant :: 0.0..1000.0, % total food bill TotalCost #= Days * sum(X), Cost #= sum(X), % cost per day % nutrient balance foreach(N in 3..NumNutrients+2) sum([Data[C,N] * X[C] : C in 1..NumCommodities]) #>= Allowance[N-2] end, foreach(C in 1..NumCommodities) XCost[C] #= Days * X[C], % Quant[C] #= 100.0*Days*X[C] / Data[C,1] % / is integer divistion Quant[C]*Data[C,1] #= 100.0*Days*X[C] end, Vars = X ++ XCost ++ Quant, solve($[min(Cost)], Vars), % println(x=X), println(cost=Cost), % println(xcost=XCost), % println(quant=Quant), println(totalCost=TotalCost), commodities(Commodities), foreach({Comm,Q,C} in zip(Commodities,Quant,XCost), Q > 0) printf("%-35w: %10.2f $ %5.2f\n", Comm,Q,C) end, nl. allowance(Allowance) => Allowance = [3.0, 70.0, 0.8, 12.0, 5.0, 1.8, 2.7, 18.0, 75.0]. commodities(Commodities) => 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."]. % set N := % 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 % ; % price and weight is the two first columns data(Data) => % 1..num_nutrients+2, 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] ].