| 1 | /* STIGLER, original Stigler's 1939 diet problem */ |
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| 2 | |
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| 3 | /* The Stigler Diet is an optimization problem named for George Stigler, |
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| 4 | a 1982 Nobel Laureate in economics, who posed the following problem: |
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| 5 | For a moderately active man weighing 154 pounds, how much of each of |
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| 6 | 77 foods should be eaten on a daily basis so that the man's intake of |
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| 7 | nine nutrients will be at least equal to the recommended dietary |
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| 8 | allowances (RDSs) suggested by the National Research Council in 1943, |
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| 9 | with the cost of the diet being minimal? |
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| 10 | |
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| 11 | The nutrient RDAs required to be met in Stigler's experiment were |
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| 12 | calories, protein, calcium, iron, vitamin A, thiamine, riboflavin, |
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| 13 | niacin, and ascorbic acid. The result was an annual budget allocated |
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| 14 | to foods such as evaporated milk, cabbage, dried navy beans, and beef |
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| 15 | liver at a cost of approximately $0.11 a day in 1939 U.S. dollars. |
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| 16 | |
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| 17 | While the name "Stigler Diet" was applied after the experiment by |
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| 18 | outsiders, according to Stigler, "No one recommends these diets for |
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| 19 | anyone, let alone everyone." The Stigler diet has been much ridiculed |
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| 20 | for its lack of variety and palatability, however his methodology has |
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| 21 | received praise and is considered to be some of the earliest work in |
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| 22 | linear programming. |
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| 23 | |
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| 24 | The Stigler diet question is a linear programming problem. Lacking |
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| 25 | any sophisticated method of solving such a problem, Stigler was |
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| 26 | forced to utilize heuristic methods in order to find a solution. The |
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| 27 | diet question originally asked in which quantities a 154 pound male |
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| 28 | would have to consume 77 different foods in order to fulfill the |
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| 29 | recommended intake of 9 different nutrients while keeping expense at |
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| 30 | a minimum. Through "trial and error, mathematical insight and |
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| 31 | agility," Stigler was able to eliminate 62 of the foods from the |
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| 32 | original 77 (these foods were removed based because they lacked |
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| 33 | nutrients in comparison to the remaining 15). From the reduced list, |
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| 34 | Stigler calculated the required amounts of each of the remaining 15 |
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| 35 | foods to arrive at a cost-minimizing solution to his question. |
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| 36 | According to Stigler's calculations, the annual cost of his solution |
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| 37 | was $39.93 in 1939 dollars. When corrected for inflation using the |
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| 38 | consumer price index, the cost of the diet in 2005 dollars is |
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| 39 | $561.43. The specific combination of foods and quantities is as |
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| 40 | follows: |
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| 41 | |
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| 42 | Stigler's 1939 Diet |
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| 43 | |
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| 44 | Food Annual Quantities Annual Cost |
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| 45 | ---------------- ----------------- ----------- |
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| 46 | Wheat Flour 370 lb. $13.33 |
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| 47 | Evaporated Milk 57 cans 3.84 |
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| 48 | Cabbage 111 lb. 4.11 |
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| 49 | Spinach 23 lb. 1.85 |
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| 50 | Dried Navy Beans 285 lb. 16.80 |
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| 51 | ---------------------------------------------- |
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| 52 | Total Annual Cost $39.93 |
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| 53 | |
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| 54 | The 9 nutrients that Stigler's diet took into consideration and their |
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| 55 | respective recommended daily amounts were: |
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| 56 | |
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| 57 | Table of nutrients considered in Stigler's diet |
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| 58 | |
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| 59 | Nutrient Daily Recommended Intake |
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| 60 | ------------------------- ------------------------ |
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| 61 | Calories 3,000 Calories |
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| 62 | Protein 70 grams |
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| 63 | Calcium .8 grams |
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| 64 | Iron 12 milligrams |
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| 65 | Vitamin A 5,000 IU |
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| 66 | Thiamine (Vitamin B1) 1.8 milligrams |
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| 67 | Riboflavin (Vitamin B2) 2.7 milligrams |
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| 68 | Niacin 18 milligrams |
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| 69 | Ascorbic Acid (Vitamin C) 75 milligrams |
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| 70 | |
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| 71 | Seven years after Stigler made his initial estimates, the development |
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| 72 | of George Dantzig's Simplex algorithm made it possible to solve the |
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| 73 | problem without relying on heuristic methods. The exact value was |
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| 74 | determined to be $39.69 (using the original 1939 data). Dantzig's |
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| 75 | algorithm describes a method of traversing the vertices of a polytope |
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| 76 | of N+1 dimensions in order to find the optimal solution to a specific |
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| 77 | situation. |
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| 78 | |
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| 79 | (From Wikipedia, the free encyclopedia.) */ |
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| 80 | |
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| 81 | /* Translated from GAMS by Andrew Makhorin <mao@gnu.org>. |
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| 82 | |
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| 83 | For the original GAMS model stigler1939.gms see [3]. |
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| 84 | |
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| 85 | References: |
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| 86 | |
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| 87 | 1. George J. Stigler, "The Cost of Subsistence," J. Farm Econ. 27, |
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| 88 | 1945, pp. 303-14. |
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| 89 | |
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| 90 | 2. National Research Council, "Recommended Daily Allowances," Reprint |
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| 91 | and Circular Series No. 115, January, 1943. |
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| 92 | |
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| 93 | 3. Erwin Kalvelagen, "Model building with GAMS," Chapter 2, "Building |
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| 94 | linear programming models," pp. 128-34. */ |
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| 95 | |
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| 96 | set C; |
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| 97 | /* commodities */ |
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| 98 | |
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| 99 | check card(C) = 77; |
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| 100 | /* there must be 77 commodities */ |
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| 101 | |
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| 102 | set N; |
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| 103 | /* nutrients */ |
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| 104 | |
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| 105 | param data{c in C, {"price", "weight"} union N}; |
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| 106 | /* nutritive values per dollar of expenditure */ |
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| 107 | |
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| 108 | param allowance{n in N}; |
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| 109 | /* recommended daily allowance for a moderately active man */ |
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| 110 | |
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| 111 | var x{c in C}, >= 0; |
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| 112 | /* dollars of food to be purchased daily */ |
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| 113 | |
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| 114 | s.t. nb{n in N}: sum{c in C} data[c,n] * x[c] >= allowance[n]; |
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| 115 | /* nutrient balance */ |
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| 116 | |
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| 117 | minimize cost: sum{c in C} x[c]; |
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| 118 | /* total food bill */ |
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| 119 | |
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| 120 | solve; |
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| 121 | |
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| 122 | param days := 365.25; |
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| 123 | /* days in a year */ |
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| 124 | |
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| 125 | param commodity{c in C}, symbolic; |
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| 126 | |
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| 127 | param unit{c in C}, symbolic; |
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| 128 | |
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| 129 | printf "\n"; |
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| 130 | printf "MINIMUM COST ANNUAL DIET\n"; |
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| 131 | printf "\n"; |
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| 132 | printf " Commodity Unit Quantity Cost \n"; |
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| 133 | printf "------------------------- ---------- ---------- ----------\n"; |
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| 134 | printf{c in C: x[c] != 0} "%-25s %10s %10.2f $%7.2f\n", commodity[c], |
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| 135 | unit[c], 100 * days * x[c] / data[c,"price"], days * x[c]; |
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| 136 | printf " -----------------\n"; |
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| 137 | printf " Total: $%7.2f\n", |
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| 138 | days * sum{c in C} x[c]; |
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| 139 | printf "\n"; |
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| 140 | |
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| 141 | data; |
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| 142 | |
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| 143 | param : C : commodity unit := |
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| 144 | flour "Wheat Flour (Enriched)" "10 lb." |
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| 145 | macaroni "Macaroni" "1 lb." |
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| 146 | cereal "Wheat Cereal (Enriched)" "28 oz." |
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| 147 | cornflakes "Corn Flakes" "8 oz." |
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| 148 | cornmeal "Corn Meal" "1 lb." |
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| 149 | grits "Hominy Grits" "24 oz." |
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| 150 | rice "Rice" "1 lb." |
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| 151 | oats "Rolled Oats" "1 lb." |
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| 152 | whitebread "White Bread (Enriched)" "1 lb." |
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| 153 | wheatbread "Whole Wheat Bread" "1 lb." |
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| 154 | ryebread "Rye Bread" "1 lb." |
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| 155 | poundcake "Pound Cake" "1 lb." |
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| 156 | crackers "Soda Crackers" "1 lb." |
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| 157 | milk "Milk" "1 qt." |
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| 158 | evapmild "Evaporated Milk (can)" "14.5 oz." |
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| 159 | butter "Butter" "1 lb." |
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| 160 | margarine "Oleomargarine" "1 lb." |
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| 161 | eggs "Eggs" "1 doz." |
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| 162 | cheese "Cheese (Cheddar)" "1 lb." |
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| 163 | cream "Cream" "1/2 pt." |
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| 164 | peanutbutter "Peanut Butter" "1 lb." |
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| 165 | mayonnaise "Mayonnaise" "1/2 pt." |
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| 166 | crisco "Crisco" "1 lb." |
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| 167 | lard "Lard" "1 lb." |
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| 168 | sirloinsteak "Sirloin Steak" "1 lb." |
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| 169 | roundsteak "Round Steak" "1 lb." |
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| 170 | ribroast "Rib Roast" "1 lb." |
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| 171 | chuckroast "Chuck Roast" "1 lb." |
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| 172 | plate "Plate" "1 lb." |
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| 173 | liver "Liver (Beef)" "1 lb." |
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| 174 | lambleg "Leg of Lamb" "1 lb." |
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| 175 | lambchops "Lamb Chops (Rib)" "1 lb." |
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| 176 | porkchops "Pork Chops" "1 lb." |
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| 177 | porkroast "Pork Loin Roast" "1 lb." |
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| 178 | bacon "Bacon" "1 lb." |
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| 179 | ham "Ham - smoked" "1 lb." |
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| 180 | saltpork "Salt Pork" "1 lb." |
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| 181 | chicken "Roasting Chicken" "1 lb." |
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| 182 | veal "Veal Cutlets" "1 lb." |
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| 183 | salmon "Salmon, Pink (can)" "16 oz." |
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| 184 | apples "Apples" "1 lb." |
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| 185 | bananas "Bananas" "1 lb." |
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| 186 | lemons "Lemons" "1 doz." |
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| 187 | oranges "Oranges" "1 doz." |
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| 188 | greenbeans "Green Beans" "1 lb." |
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| 189 | cabbage "Cabbage" "1 lb." |
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| 190 | carrots "Carrots" "1 bunch" |
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| 191 | celery "Celery" "1 stalk" |
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| 192 | lettuce "Lettuce" "1 head" |
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| 193 | onions "Onions" "1 lb." |
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| 194 | potatoes "Potatoes" "15 lb." |
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| 195 | spinach "Spinach" "1 lb." |
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| 196 | sweetpotato "Sweet Potatoes" "1 lb." |
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| 197 | peaches "Peaches (can)" "No. 2 1/2" |
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| 198 | pears "Pears (can)" "No. 2 1/2" |
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| 199 | pineapple "Pineapple (can)" "No. 2 1/2" |
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| 200 | asparagus "Asparagus (can)" "No. 2" |
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| 201 | cannedgrbn "Grean Beans (can)" "No. 2" |
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| 202 | porkbeans "Pork and Beans (can)" "16 oz." |
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| 203 | corn "Corn (can)" "No. 2" |
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| 204 | peas "Peas (can)" "No. 2" |
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| 205 | tomatoes "Tomatoes (can)" "No. 2" |
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| 206 | tomatosoup "Tomato Soup (can)" "10 1/2 oz." |
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| 207 | driedpeach "Peaches, Dried" "1 lb." |
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| 208 | prunes "Prunes, Dried" "1 lb." |
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| 209 | raisins "Raisins, Dried" "15 oz." |
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| 210 | driedpeas "Peas, Dried" "1 lb." |
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| 211 | limabeans "Lima Beans, Dried" "1 lb." |
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| 212 | navybeans "Navy Beans, Dried" "1 lb." |
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| 213 | coffee "Coffee" "1 lb." |
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| 214 | tea "Tea" "1/4 lb." |
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| 215 | cocoa "Cocoa" "8 oz." |
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| 216 | chocolate "Chocolate" "8 oz." |
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| 217 | sugar "Sugar" "10 lb." |
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| 218 | cornsirup "Corn Sirup" "24 oz." |
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| 219 | molasses "Molasses" "18 oz." |
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| 220 | stawberry "Strawberry Preserve" "1 lb." |
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| 221 | ; |
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| 222 | |
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| 223 | set N := |
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| 224 | calories /* Calories, unit = 1000 */ |
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| 225 | protein /* Protein, unit = grams */ |
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| 226 | calcium /* Calcium, unit = grams */ |
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| 227 | iron /* Iron, unit = milligrams */ |
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| 228 | vitaminA /* Vitamin A, unit = 1000 International Units */ |
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| 229 | thiamine /* Thiamine, Vit. B1, unit = milligrams */ |
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| 230 | riboflavin /* Riboflavin, Vit. B2, unit = milligrams */ |
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| 231 | niacin /* Niacin (Nicotinic Acid), unit = milligrams */ |
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| 232 | ascorbicAcid /* Ascorbic Acid, Vit. C, unit = milligrams */ |
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| 233 | ; |
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| 234 | |
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| 235 | param data |
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| 236 | : price weight calories protein calcium iron := |
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| 237 | # aug. 15 edible |
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| 238 | # 1939 per $1 |
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| 239 | # (cents) (grams) (1000) (grams) (grams) (mg.) |
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| 240 | flour 36.0 12600 44.7 1411 2.0 365 |
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| 241 | macaroni 14.1 3217 11.6 418 .7 54 |
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| 242 | cereal 24.2 3280 11.8 377 14.4 175 |
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| 243 | cornflakes 7.1 3194 11.4 252 .1 56 |
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| 244 | cornmeal 4.6 9861 36.0 897 1.7 99 |
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| 245 | grits 8.5 8005 28.6 680 .8 80 |
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| 246 | rice 7.5 6048 21.2 460 .6 41 |
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| 247 | oats 7.1 6389 25.3 907 5.1 341 |
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| 248 | whitebread 7.9 5742 15.6 488 2.5 115 |
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| 249 | wheatbread 9.1 4985 12.2 484 2.7 125 |
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| 250 | ryebread 9.2 4930 12.4 439 1.1 82 |
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| 251 | poundcake 24.8 1829 8.0 130 .4 31 |
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| 252 | crackers 15.1 3004 12.5 288 .5 50 |
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| 253 | milk 11.0 8867 6.1 310 10.5 18 |
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| 254 | evapmild 6.7 6035 8.4 422 15.1 9 |
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| 255 | butter 20.8 1473 10.8 9 .2 3 |
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| 256 | margarine 16.1 2817 20.6 17 .6 6 |
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| 257 | eggs 32.6 1857 2.9 238 1.0 52 |
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| 258 | cheese 24.2 1874 7.4 448 16.4 19 |
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| 259 | cream 14.1 1689 3.5 49 1.7 3 |
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| 260 | peanutbutter 17.9 2534 15.7 661 1.0 48 |
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| 261 | mayonnaise 16.7 1198 8.6 18 .2 8 |
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| 262 | crisco 20.3 2234 20.1 0 .0 0 |
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| 263 | lard 9.8 4628 41.7 0 .0 0 |
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| 264 | sirloinsteak 39.6 1145 2.9 166 .1 34 |
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| 265 | roundsteak 36.4 1246 2.2 214 .1 32 |
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| 266 | ribroast 29.2 1553 3.4 213 .1 33 |
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| 267 | chuckroast 22.6 2007 3.6 309 .2 46 |
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| 268 | plate 14.6 3107 8.5 404 .2 62 |
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| 269 | liver 26.8 1692 2.2 333 .2 139 |
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| 270 | lambleg 27.6 1643 3.1 245 .1 20 |
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| 271 | lambchops 36.6 1239 3.3 140 .1 15 |
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| 272 | porkchops 30.7 1477 3.5 196 .2 80 |
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| 273 | porkroast 24.2 1874 4.4 249 .3 37 |
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| 274 | bacon 25.6 1772 10.4 152 .2 23 |
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| 275 | ham 27.4 1655 6.7 212 .2 31 |
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| 276 | saltpork 16.0 2835 18.8 164 .1 26 |
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| 277 | chicken 30.3 1497 1.8 184 .1 30 |
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| 278 | veal 42.3 1072 1.7 156 .1 24 |
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| 279 | salmon 13.0 3489 5.8 705 6.8 45 |
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| 280 | apples 4.4 9072 5.8 27 .5 36 |
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| 281 | bananas 6.1 4982 4.9 60 .4 30 |
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| 282 | lemons 26.0 2380 1.0 21 .5 14 |
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| 283 | oranges 30.9 4439 2.2 40 1.1 18 |
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| 284 | greenbeans 7.1 5750 2.4 138 3.7 80 |
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| 285 | cabbage 3.7 8949 2.6 125 4.0 36 |
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| 286 | carrots 4.7 6080 2.7 73 2.8 43 |
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| 287 | celery 7.3 3915 .9 51 3.0 23 |
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| 288 | lettuce 8.2 2247 .4 27 1.1 22 |
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| 289 | onions 3.6 11844 5.8 166 3.8 59 |
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| 290 | potatoes 34.0 16810 14.3 336 1.8 118 |
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| 291 | spinach 8.1 4592 1.1 106 .0 138 |
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| 292 | sweetpotato 5.1 7649 9.6 138 2.7 54 |
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| 293 | peaches 16.8 4894 3.7 20 .4 10 |
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| 294 | pears 20.4 4030 3.0 8 .3 8 |
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| 295 | pineapple 21.3 3993 2.4 16 .4 8 |
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| 296 | asparagus 27.7 1945 .4 33 .3 12 |
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| 297 | cannedgrbn 10.0 5386 1.0 54 2.0 65 |
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| 298 | porkbeans 7.1 6389 7.5 364 4.0 134 |
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| 299 | corn 10.4 5452 5.2 136 .2 16 |
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| 300 | peas 13.8 4109 2.3 136 .6 45 |
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| 301 | tomatoes 8.6 6263 1.3 63 .7 38 |
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| 302 | tomatosoup 7.6 3917 1.6 71 .6 43 |
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| 303 | driedpeach 15.7 2889 8.5 87 1.7 173 |
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| 304 | prunes 9.0 4284 12.8 99 2.5 154 |
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| 305 | raisins 9.4 4524 13.5 104 2.5 136 |
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| 306 | driedpeas 7.9 5742 20.0 1367 4.2 345 |
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| 307 | limabeans 8.9 5097 17.4 1055 3.7 459 |
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| 308 | navybeans 5.9 7688 26.9 1691 11.4 792 |
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| 309 | coffee 22.4 2025 .0 0 .0 0 |
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| 310 | tea 17.4 652 .0 0 .0 0 |
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| 311 | cocoa 8.6 2637 8.7 237 3.0 72 |
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| 312 | chocolate 16.2 1400 8.0 77 1.3 39 |
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| 313 | sugar 51.7 8773 34.9 0 .0 0 |
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| 314 | cornsirup 13.7 4996 14.7 0 .5 74 |
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| 315 | molasses 13.6 3752 9.0 0 10.3 244 |
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| 316 | stawberry 20.5 2213 6.4 11 .4 7 |
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| 317 | |
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| 318 | : vitaminA thiamine riboflavin niacin ascorbicAcid := |
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| 319 | # (1000 IU) (mg.) (mg.) (mg.) (mg.) |
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| 320 | flour .0 55.4 33.3 441 0 |
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| 321 | macaroni .0 3.2 1.9 68 0 |
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| 322 | cereal .0 14.4 8.8 114 0 |
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| 323 | cornflakes .0 13.5 2.3 68 0 |
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| 324 | cornmeal 30.9 17.4 7.9 106 0 |
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| 325 | grits .0 10.6 1.6 110 0 |
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| 326 | rice .0 2.0 4.8 60 0 |
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| 327 | oats .0 37.1 8.9 64 0 |
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| 328 | whitebread .0 13.8 8.5 126 0 |
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| 329 | wheatbread .0 13.9 6.4 160 0 |
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| 330 | ryebread .0 9.9 3.0 66 0 |
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| 331 | poundcake 18.9 2.8 3.0 17 0 |
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| 332 | crackers .0 .0 .0 0 0 |
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| 333 | milk 16.8 4.0 16.0 7 177 |
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| 334 | evapmild 26.0 3.0 23.5 11 60 |
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| 335 | butter 44.2 .0 .2 2 0 |
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| 336 | margarine 55.8 .2 .0 0 0 |
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| 337 | eggs 18.6 2.8 6.5 1 0 |
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| 338 | cheese 28.1 .8 10.3 4 0 |
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| 339 | cream 16.9 .6 2.5 0 17 |
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| 340 | peanutbutter .0 9.6 8.1 471 0 |
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| 341 | mayonnaise 2.7 .4 .5 0 0 |
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| 342 | crisco .0 .0 .0 0 0 |
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| 343 | lard .2 .0 .5 5 0 |
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| 344 | sirloinsteak .2 2.1 2.9 69 0 |
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| 345 | roundsteak .4 2.5 2.4 87 0 |
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| 346 | ribroast .0 .0 2.0 0 0 |
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| 347 | chuckroast .4 1.0 4.0 120 0 |
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| 348 | plate .0 .9 .0 0 0 |
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| 349 | liver 169.2 6.4 50.8 316 525 |
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| 350 | lambleg .0 2.8 3.0 86 0 |
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| 351 | lambchops .0 1.7 2.7 54 0 |
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| 352 | porkchops .0 17.4 2.7 60 0 |
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| 353 | porkroast .0 18.2 3.6 79 0 |
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| 354 | bacon .0 1.8 1.8 71 0 |
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| 355 | ham .0 9.9 3.3 50 0 |
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| 356 | saltpork .0 1.4 1.8 0 0 |
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| 357 | chicken .1 .9 1.8 68 46 |
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| 358 | veal .0 1.4 2.4 57 0 |
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| 359 | salmon 3.5 1.0 4.9 209 0 |
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| 360 | apples 7.3 3.6 2.7 5 544 |
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| 361 | bananas 17.4 2.5 3.5 28 498 |
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| 362 | lemons .0 .5 .0 4 952 |
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| 363 | oranges 11.1 3.6 1.3 10 1993 |
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| 364 | greenbeans 69.0 4.3 5.8 37 862 |
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| 365 | cabbage 7.2 9.0 4.5 26 5369 |
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| 366 | carrots 188.5 6.1 4.3 89 608 |
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| 367 | celery .9 1.4 1.4 9 313 |
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| 368 | lettuce 112.4 1.8 3.4 11 449 |
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| 369 | onions 16.6 4.7 5.9 21 1184 |
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| 370 | potatoes 6.7 29.4 7.1 198 2522 |
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| 371 | spinach 918.4 5.7 13.8 33 2755 |
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| 372 | sweetpotato 290.7 8.4 5.4 83 1912 |
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| 373 | peaches 21.5 .5 1.0 31 196 |
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| 374 | pears .8 .8 .8 5 81 |
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| 375 | pineapple 2.0 2.8 .8 7 399 |
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| 376 | asparagus 16.3 1.4 2.1 17 272 |
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| 377 | cannedgrbn 53.9 1.6 4.3 32 431 |
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| 378 | porkbeans 3.5 8.3 7.7 56 0 |
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| 379 | corn 12.0 1.6 2.7 42 218 |
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| 380 | peas 34.9 4.9 2.5 37 370 |
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| 381 | tomatoes 53.2 3.4 2.5 36 1253 |
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| 382 | tomatosoup 57.9 3.5 2.4 67 862 |
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| 383 | driedpeach 86.8 1.2 4.3 55 57 |
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| 384 | prunes 85.7 3.9 4.3 65 257 |
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| 385 | raisins 4.5 6.3 1.4 24 136 |
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| 386 | driedpeas 2.9 28.7 18.4 162 0 |
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| 387 | limabeans 5.1 26.9 38.2 93 0 |
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| 388 | navybeans .0 38.4 24.6 217 0 |
|---|
| 389 | coffee .0 4.0 5.1 50 0 |
|---|
| 390 | tea .0 .0 2.3 42 0 |
|---|
| 391 | cocoa .0 2.0 11.9 40 0 |
|---|
| 392 | chocolate .0 .9 3.4 14 0 |
|---|
| 393 | sugar .0 .0 .0 0 0 |
|---|
| 394 | cornsirup .0 .0 .0 5 0 |
|---|
| 395 | molasses .0 1.9 7.5 146 0 |
|---|
| 396 | stawberry .2 .2 .4 3 0 |
|---|
| 397 | ; |
|---|
| 398 | |
|---|
| 399 | param allowance := |
|---|
| 400 | calories 3 |
|---|
| 401 | protein 70 |
|---|
| 402 | calcium .8 |
|---|
| 403 | iron 12 |
|---|
| 404 | vitaminA 5 |
|---|
| 405 | thiamine 1.8 |
|---|
| 406 | riboflavin 2.7 |
|---|
| 407 | niacin 18 |
|---|
| 408 | ascorbicAcid 75 |
|---|
| 409 | ; |
|---|
| 410 | |
|---|
| 411 | end; |
|---|
