/* 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. */ 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 2.0 365 macaroni 14.1 3217 11.6 418 .7 54 cereal 24.2 3280 11.8 377 14.4 175 cornflakes 7.1 3194 11.4 252 .1 56 cornmeal 4.6 9861 36.0 897 1.7 99 grits 8.5 8005 28.6 680 .8 80 rice 7.5 6048 21.2 460 .6 41 oats 7.1 6389 25.3 907 5.1 341 whitebread 7.9 5742 15.6 488 2.5 115 wheatbread 9.1 4985 12.2 484 2.7 125 ryebread 9.2 4930 12.4 439 1.1 82 poundcake 24.8 1829 8.0 130 .4 31 crackers 15.1 3004 12.5 288 .5 50 milk 11.0 8867 6.1 310 10.5 18 evapmild 6.7 6035 8.4 422 15.1 9 butter 20.8 1473 10.8 9 .2 3 margarine 16.1 2817 20.6 17 .6 6 eggs 32.6 1857 2.9 238 1.0 52 cheese 24.2 1874 7.4 448 16.4 19 cream 14.1 1689 3.5 49 1.7 3 peanutbutter 17.9 2534 15.7 661 1.0 48 mayonnaise 16.7 1198 8.6 18 .2 8 crisco 20.3 2234 20.1 0 .0 0 lard 9.8 4628 41.7 0 .0 0 sirloinsteak 39.6 1145 2.9 166 .1 34 roundsteak 36.4 1246 2.2 214 .1 32 ribroast 29.2 1553 3.4 213 .1 33 chuckroast 22.6 2007 3.6 309 .2 46 plate 14.6 3107 8.5 404 .2 62 liver 26.8 1692 2.2 333 .2 139 lambleg 27.6 1643 3.1 245 .1 20 lambchops 36.6 1239 3.3 140 .1 15 porkchops 30.7 1477 3.5 196 .2 80 porkroast 24.2 1874 4.4 249 .3 37 bacon 25.6 1772 10.4 152 .2 23 ham 27.4 1655 6.7 212 .2 31 saltpork 16.0 2835 18.8 164 .1 26 chicken 30.3 1497 1.8 184 .1 30 veal 42.3 1072 1.7 156 .1 24 salmon 13.0 3489 5.8 705 6.8 45 apples 4.4 9072 5.8 27 .5 36 bananas 6.1 4982 4.9 60 .4 30 lemons 26.0 2380 1.0 21 .5 14 oranges 30.9 4439 2.2 40 1.1 18 greenbeans 7.1 5750 2.4 138 3.7 80 cabbage 3.7 8949 2.6 125 4.0 36 carrots 4.7 6080 2.7 73 2.8 43 celery 7.3 3915 .9 51 3.0 23 lettuce 8.2 2247 .4 27 1.1 22 onions 3.6 11844 5.8 166 3.8 59 potatoes 34.0 16810 14.3 336 1.8 118 spinach 8.1 4592 1.1 106 .0 138 sweetpotato 5.1 7649 9.6 138 2.7 54 peaches 16.8 4894 3.7 20 .4 10 pears 20.4 4030 3.0 8 .3 8 pineapple 21.3 3993 2.4 16 .4 8 asparagus 27.7 1945 .4 33 .3 12 cannedgrbn 10.0 5386 1.0 54 2.0 65 porkbeans 7.1 6389 7.5 364 4.0 134 corn 10.4 5452 5.2 136 .2 16 peas 13.8 4109 2.3 136 .6 45 tomatoes 8.6 6263 1.3 63 .7 38 tomatosoup 7.6 3917 1.6 71 .6 43 driedpeach 15.7 2889 8.5 87 1.7 173 prunes 9.0 4284 12.8 99 2.5 154 raisins 9.4 4524 13.5 104 2.5 136 driedpeas 7.9 5742 20.0 1367 4.2 345 limabeans 8.9 5097 17.4 1055 3.7 459 navybeans 5.9 7688 26.9 1691 11.4 792 coffee 22.4 2025 .0 0 .0 0 tea 17.4 652 .0 0 .0 0 cocoa 8.6 2637 8.7 237 3.0 72 chocolate 16.2 1400 8.0 77 1.3 39 sugar 51.7 8773 34.9 0 .0 0 cornsirup 13.7 4996 14.7 0 .5 74 molasses 13.6 3752 9.0 0 10.3 244 strawberry 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 0 macaroni .0 3.2 1.9 68 0 cereal .0 14.4 8.8 114 0 cornflakes .0 13.5 2.3 68 0 cornmeal 30.9 17.4 7.9 106 0 grits .0 10.6 1.6 110 0 rice .0 2.0 4.8 60 0 oats .0 37.1 8.9 64 0 whitebread .0 13.8 8.5 126 0 wheatbread .0 13.9 6.4 160 0 ryebread .0 9.9 3.0 66 0 poundcake 18.9 2.8 3.0 17 0 crackers .0 .0 .0 0 0 milk 16.8 4.0 16.0 7 177 evapmild 26.0 3.0 23.5 11 60 butter 44.2 .0 .2 2 0 margarine 55.8 .2 .0 0 0 eggs 18.6 2.8 6.5 1 0 cheese 28.1 .8 10.3 4 0 cream 16.9 .6 2.5 0 17 peanutbutter .0 9.6 8.1 471 0 mayonnaise 2.7 .4 .5 0 0 crisco .0 .0 .0 0 0 lard .2 .0 .5 5 0 sirloinsteak .2 2.1 2.9 69 0 roundsteak .4 2.5 2.4 87 0 ribroast .0 .0 2.0 0 0 chuckroast .4 1.0 4.0 120 0 plate .0 .9 .0 0 0 liver 169.2 6.4 50.8 316 525 lambleg .0 2.8 3.0 86 0 lambchops .0 1.7 2.7 54 0 porkchops .0 17.4 2.7 60 0 porkroast .0 18.2 3.6 79 0 bacon .0 1.8 1.8 71 0 ham .0 9.9 3.3 50 0 saltpork .0 1.4 1.8 0 0 chicken .1 .9 1.8 68 46 veal .0 1.4 2.4 57 0 salmon 3.5 1.0 4.9 209 0 apples 7.3 3.6 2.7 5 544 bananas 17.4 2.5 3.5 28 498 lemons .0 .5 .0 4 952 oranges 11.1 3.6 1.3 10 1993 greenbeans 69.0 4.3 5.8 37 862 cabbage 7.2 9.0 4.5 26 5369 carrots 188.5 6.1 4.3 89 608 celery .9 1.4 1.4 9 313 lettuce 112.4 1.8 3.4 11 449 onions 16.6 4.7 5.9 21 1184 potatoes 6.7 29.4 7.1 198 2522 spinach 918.4 5.7 13.8 33 2755 sweetpotato 290.7 8.4 5.4 83 1912 peaches 21.5 .5 1.0 31 196 pears .8 .8 .8 5 81 pineapple 2.0 2.8 .8 7 399 asparagus 16.3 1.4 2.1 17 272 cannedgrbn 53.9 1.6 4.3 32 431 porkbeans 3.5 8.3 7.7 56 0 corn 12.0 1.6 2.7 42 218 peas 34.9 4.9 2.5 37 370 tomatoes 53.2 3.4 2.5 36 1253 tomatosoup 57.9 3.5 2.4 67 862 driedpeach 86.8 1.2 4.3 55 57 prunes 85.7 3.9 4.3 65 257 raisins 4.5 6.3 1.4 24 136 driedpeas 2.9 28.7 18.4 162 0 limabeans 5.1 26.9 38.2 93 0 navybeans .0 38.4 24.6 217 0 coffee .0 4.0 5.1 50 0 tea .0 .0 2.3 42 0 cocoa .0 2.0 11.9 40 0 chocolate .0 .9 3.4 14 0 sugar .0 .0 .0 0 0 cornsirup .0 .0 .0 5 0 molasses .0 1.9 7.5 146 0 strawberry .2 .2 .4 3 0 ; param allowance := calories 3 protein 70 calcium .8 iron 12 vitaminA 5 thiamine 1.8 riboflavin 2.7 niacin 18 ascorbicAcid 75 ; end;