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/* 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 <mao@gnu.org>.
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. */
set C;/* commodities */
check card(C) = 77;/* there must be 77 commodities */
set N;/* nutrients */
param data{c in C, {"price", "weight"} union N};/* nutritive values per dollar of expenditure */
param allowance{n in N};/* recommended daily allowance for a moderately active man */
var x{c in C}, >= 0;/* dollars of food to be purchased daily */
s.t. nb{n in N}: sum{c in C} data[c,n] * x[c] >= allowance[n];/* nutrient balance */
minimize cost: sum{c in C} x[c];/* total food bill */
solve;
param days := 365.25;/* days in a year */
param commodity{c in C}, symbolic;
param unit{c in C}, symbolic;
printf "\n";printf "MINIMUM COST ANNUAL DIET\n";printf "\n";printf " Commodity Unit Quantity Cost \n";printf "------------------------- ---------- ---------- ----------\n";printf{c in C: x[c] != 0} "%-25s %10s %10.2f $%7.2f\n", commodity[c], unit[c], 100 * days * x[c] / data[c,"price"], days * x[c];printf " -----------------\n";printf " Total: $%7.2f\n", days * sum{c in C} x[c];printf "\n";
data;
param : C : commodity unit :=flour "Wheat Flour (Enriched)" "10 lb."macaroni "Macaroni" "1 lb."cereal "Wheat Cereal (Enriched)" "28 oz."cornflakes "Corn Flakes" "8 oz."cornmeal "Corn Meal" "1 lb."grits "Hominy Grits" "24 oz."rice "Rice" "1 lb."oats "Rolled Oats" "1 lb."whitebread "White Bread (Enriched)" "1 lb."wheatbread "Whole Wheat Bread" "1 lb."ryebread "Rye Bread" "1 lb."poundcake "Pound Cake" "1 lb."crackers "Soda Crackers" "1 lb."milk "Milk" "1 qt."evapmild "Evaporated Milk (can)" "14.5 oz."butter "Butter" "1 lb."margarine "Oleomargarine" "1 lb."eggs "Eggs" "1 doz."cheese "Cheese (Cheddar)" "1 lb."cream "Cream" "1/2 pt."peanutbutter "Peanut Butter" "1 lb."mayonnaise "Mayonnaise" "1/2 pt."crisco "Crisco" "1 lb."lard "Lard" "1 lb."sirloinsteak "Sirloin Steak" "1 lb."roundsteak "Round Steak" "1 lb."ribroast "Rib Roast" "1 lb."chuckroast "Chuck Roast" "1 lb."plate "Plate" "1 lb."liver "Liver (Beef)" "1 lb."lambleg "Leg of Lamb" "1 lb."lambchops "Lamb Chops (Rib)" "1 lb."porkchops "Pork Chops" "1 lb."porkroast "Pork Loin Roast" "1 lb."bacon "Bacon" "1 lb."ham "Ham - smoked" "1 lb."saltpork "Salt Pork" "1 lb."chicken "Roasting Chicken" "1 lb."veal "Veal Cutlets" "1 lb."salmon "Salmon, Pink (can)" "16 oz."apples "Apples" "1 lb."bananas "Bananas" "1 lb."lemons "Lemons" "1 doz."oranges "Oranges" "1 doz."greenbeans "Green Beans" "1 lb."cabbage "Cabbage" "1 lb."carrots "Carrots" "1 bunch"celery "Celery" "1 stalk"lettuce "Lettuce" "1 head"onions "Onions" "1 lb."potatoes "Potatoes" "15 lb."spinach "Spinach" "1 lb."sweetpotato "Sweet Potatoes" "1 lb."peaches "Peaches (can)" "No. 2 1/2"pears "Pears (can)" "No. 2 1/2"pineapple "Pineapple (can)" "No. 2 1/2"asparagus "Asparagus (can)" "No. 2"cannedgrbn "Grean Beans (can)" "No. 2"porkbeans "Pork and Beans (can)" "16 oz."corn "Corn (can)" "No. 2"peas "Peas (can)" "No. 2"tomatoes "Tomatoes (can)" "No. 2"tomatosoup "Tomato Soup (can)" "10 1/2 oz."driedpeach "Peaches, Dried" "1 lb."prunes "Prunes, Dried" "1 lb."raisins "Raisins, Dried" "15 oz."driedpeas "Peas, Dried" "1 lb."limabeans "Lima Beans, Dried" "1 lb."navybeans "Navy Beans, Dried" "1 lb."coffee "Coffee" "1 lb."tea "Tea" "1/4 lb."cocoa "Cocoa" "8 oz."chocolate "Chocolate" "8 oz."sugar "Sugar" "10 lb."cornsirup "Corn Sirup" "24 oz."molasses "Molasses" "18 oz."strawberry "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 */;
param data: price weight calories protein calcium iron :=# aug. 15 edible# 1939 per $1# (cents) (grams) (1000) (grams) (grams) (mg.)flour 36.0 12600 44.7 1411 2.0 365macaroni 14.1 3217 11.6 418 .7 54cereal 24.2 3280 11.8 377 14.4 175cornflakes 7.1 3194 11.4 252 .1 56cornmeal 4.6 9861 36.0 897 1.7 99grits 8.5 8005 28.6 680 .8 80rice 7.5 6048 21.2 460 .6 41oats 7.1 6389 25.3 907 5.1 341whitebread 7.9 5742 15.6 488 2.5 115wheatbread 9.1 4985 12.2 484 2.7 125ryebread 9.2 4930 12.4 439 1.1 82poundcake 24.8 1829 8.0 130 .4 31crackers 15.1 3004 12.5 288 .5 50milk 11.0 8867 6.1 310 10.5 18evapmild 6.7 6035 8.4 422 15.1 9butter 20.8 1473 10.8 9 .2 3margarine 16.1 2817 20.6 17 .6 6eggs 32.6 1857 2.9 238 1.0 52cheese 24.2 1874 7.4 448 16.4 19cream 14.1 1689 3.5 49 1.7 3peanutbutter 17.9 2534 15.7 661 1.0 48mayonnaise 16.7 1198 8.6 18 .2 8crisco 20.3 2234 20.1 0 .0 0lard 9.8 4628 41.7 0 .0 0sirloinsteak 39.6 1145 2.9 166 .1 34roundsteak 36.4 1246 2.2 214 .1 32ribroast 29.2 1553 3.4 213 .1 33chuckroast 22.6 2007 3.6 309 .2 46plate 14.6 3107 8.5 404 .2 62liver 26.8 1692 2.2 333 .2 139lambleg 27.6 1643 3.1 245 .1 20lambchops 36.6 1239 3.3 140 .1 15porkchops 30.7 1477 3.5 196 .2 80porkroast 24.2 1874 4.4 249 .3 37bacon 25.6 1772 10.4 152 .2 23ham 27.4 1655 6.7 212 .2 31saltpork 16.0 2835 18.8 164 .1 26chicken 30.3 1497 1.8 184 .1 30veal 42.3 1072 1.7 156 .1 24salmon 13.0 3489 5.8 705 6.8 45apples 4.4 9072 5.8 27 .5 36bananas 6.1 4982 4.9 60 .4 30lemons 26.0 2380 1.0 21 .5 14oranges 30.9 4439 2.2 40 1.1 18greenbeans 7.1 5750 2.4 138 3.7 80cabbage 3.7 8949 2.6 125 4.0 36carrots 4.7 6080 2.7 73 2.8 43celery 7.3 3915 .9 51 3.0 23lettuce 8.2 2247 .4 27 1.1 22onions 3.6 11844 5.8 166 3.8 59potatoes 34.0 16810 14.3 336 1.8 118spinach 8.1 4592 1.1 106 .0 138sweetpotato 5.1 7649 9.6 138 2.7 54peaches 16.8 4894 3.7 20 .4 10pears 20.4 4030 3.0 8 .3 8pineapple 21.3 3993 2.4 16 .4 8asparagus 27.7 1945 .4 33 .3 12cannedgrbn 10.0 5386 1.0 54 2.0 65porkbeans 7.1 6389 7.5 364 4.0 134corn 10.4 5452 5.2 136 .2 16peas 13.8 4109 2.3 136 .6 45tomatoes 8.6 6263 1.3 63 .7 38tomatosoup 7.6 3917 1.6 71 .6 43driedpeach 15.7 2889 8.5 87 1.7 173prunes 9.0 4284 12.8 99 2.5 154raisins 9.4 4524 13.5 104 2.5 136driedpeas 7.9 5742 20.0 1367 4.2 345limabeans 8.9 5097 17.4 1055 3.7 459navybeans 5.9 7688 26.9 1691 11.4 792coffee 22.4 2025 .0 0 .0 0tea 17.4 652 .0 0 .0 0cocoa 8.6 2637 8.7 237 3.0 72chocolate 16.2 1400 8.0 77 1.3 39sugar 51.7 8773 34.9 0 .0 0cornsirup 13.7 4996 14.7 0 .5 74molasses 13.6 3752 9.0 0 10.3 244strawberry 20.5 2213 6.4 11 .4 7
: vitaminA thiamine riboflavin niacin ascorbicAcid :=# (1000 IU) (mg.) (mg.) (mg.) (mg.)flour .0 55.4 33.3 441 0macaroni .0 3.2 1.9 68 0cereal .0 14.4 8.8 114 0cornflakes .0 13.5 2.3 68 0cornmeal 30.9 17.4 7.9 106 0grits .0 10.6 1.6 110 0rice .0 2.0 4.8 60 0oats .0 37.1 8.9 64 0whitebread .0 13.8 8.5 126 0wheatbread .0 13.9 6.4 160 0ryebread .0 9.9 3.0 66 0poundcake 18.9 2.8 3.0 17 0crackers .0 .0 .0 0 0milk 16.8 4.0 16.0 7 177evapmild 26.0 3.0 23.5 11 60butter 44.2 .0 .2 2 0margarine 55.8 .2 .0 0 0eggs 18.6 2.8 6.5 1 0cheese 28.1 .8 10.3 4 0cream 16.9 .6 2.5 0 17peanutbutter .0 9.6 8.1 471 0mayonnaise 2.7 .4 .5 0 0crisco .0 .0 .0 0 0lard .2 .0 .5 5 0sirloinsteak .2 2.1 2.9 69 0roundsteak .4 2.5 2.4 87 0ribroast .0 .0 2.0 0 0chuckroast .4 1.0 4.0 120 0plate .0 .9 .0 0 0liver 169.2 6.4 50.8 316 525lambleg .0 2.8 3.0 86 0lambchops .0 1.7 2.7 54 0porkchops .0 17.4 2.7 60 0porkroast .0 18.2 3.6 79 0bacon .0 1.8 1.8 71 0ham .0 9.9 3.3 50 0saltpork .0 1.4 1.8 0 0chicken .1 .9 1.8 68 46veal .0 1.4 2.4 57 0salmon 3.5 1.0 4.9 209 0apples 7.3 3.6 2.7 5 544bananas 17.4 2.5 3.5 28 498lemons .0 .5 .0 4 952oranges 11.1 3.6 1.3 10 1993greenbeans 69.0 4.3 5.8 37 862cabbage 7.2 9.0 4.5 26 5369carrots 188.5 6.1 4.3 89 608celery .9 1.4 1.4 9 313lettuce 112.4 1.8 3.4 11 449onions 16.6 4.7 5.9 21 1184potatoes 6.7 29.4 7.1 198 2522spinach 918.4 5.7 13.8 33 2755sweetpotato 290.7 8.4 5.4 83 1912peaches 21.5 .5 1.0 31 196pears .8 .8 .8 5 81pineapple 2.0 2.8 .8 7 399asparagus 16.3 1.4 2.1 17 272cannedgrbn 53.9 1.6 4.3 32 431porkbeans 3.5 8.3 7.7 56 0corn 12.0 1.6 2.7 42 218peas 34.9 4.9 2.5 37 370tomatoes 53.2 3.4 2.5 36 1253tomatosoup 57.9 3.5 2.4 67 862driedpeach 86.8 1.2 4.3 55 57prunes 85.7 3.9 4.3 65 257raisins 4.5 6.3 1.4 24 136driedpeas 2.9 28.7 18.4 162 0limabeans 5.1 26.9 38.2 93 0navybeans .0 38.4 24.6 217 0coffee .0 4.0 5.1 50 0tea .0 .0 2.3 42 0cocoa .0 2.0 11.9 40 0chocolate .0 .9 3.4 14 0sugar .0 .0 .0 0 0cornsirup .0 .0 .0 5 0molasses .0 1.9 7.5 146 0strawberry .2 .2 .4 3 0;
param allowance :=calories 3protein 70calcium .8iron 12vitaminA 5thiamine 1.8riboflavin 2.7niacin 18ascorbicAcid 75;
end;
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