1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
|
/* 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 2.0 365
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;
|
