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恶化饮食问题

在本部分中，我们将展示如何解决一个称为老虎饮食的经典问题，它以诺贝尔经济学奖获得者 George Stigler 命名，他计算了一种低成本方法来满足给定食物的基本营养需求。尽管计算最佳营养的概念最近流行，但他将其认定为数学练习，而不是饮食建议。

Stigler 饮食要求满足以下最低要求：

营养成分列表

营养	每日建议摄入量
热量	3,000 卡路里
蛋白质	70 克
钙	0.8 克
熨斗	12 毫克
维生素 A	5000 国际元
硫胺素（维生素 B1）	1.8 毫克
核黄素（维生素 B2）	2.7 毫克
烟酸	18 毫克
抗甲酸（维生素 C）	75 毫克

Stigler 评估的这组食物反映了当时的情况（1944 年）。下面的营养数据是按美元计算的，而不是按单位计算的，因此我们的目标是确定要在每种食品上花费多少美元。

商品列表

商品	单位	1939 年价格（美分）	热量（千卡）	蛋白质（克）	钙（克）	铁（毫克）	维生素 A (KIU)	硫胺素 (mg)	核黄素（毫克）	烟酸（毫克）	抗甲酸 (mg)
小麦粉（浓味）	10 磅	36	土耳其里拉	1411	2	365	0	55.4	33.3	441	0
芥末色	1 磅	14.1	11.6	418	0.7	54	0	3.2	1.9	68	0
小麦谷物（浓缩）	28 盎司。	24.2	11.8	377	14.4	175	0	14.4	8.8	114	0
玉米片	8 盎司。	7.1	11.4	252	0.1	56	0	13.5	2.3	68	0
玉米粉	1 磅	4.6	36.0	897	1.7	99	30.9	17.4	7.9	106	0
粗玉米粉	24 盎司。	8.5	28.6	680	0.8	80	0	10.6	1.6	110	0
米饭	1 磅	7.5	21.2	460	0.6	41	0	2	4.8	60	0
燕麦卷	1 磅	7.1	25.3	907	5.1	341	0	37.1	8.9	64	0
白面包（浓缩）	1 磅	7.9	15.0	488	2.5	115	0	13.8	8.5	126	0
全麦面包	1 磅	9.1	12.2	484	2.7	125	0	13.9	6.4	160	0
黑麦面包	1 磅	9.1	12.4	439	1.1	82	0	9.9	3	66	0
磅蛋糕	1 磅	土耳其里拉	8.0	130	0.4	31	土耳其里拉	2.8	3	17	0
苏打饼干	1 磅	15.1	12.5	288	0.5	50	0	0	0	0	0
牛奶	1 夸脱	11	6.1	310	10.5	18	16.8	4	16	7	177
淡奶（罐）	14.5 盎司。	6.7	8.4	422	15.1	9	26	3	23.5	11	60
黄油	1 磅	30.8	10.8	9	0.2	3	44.2	0	0.2	2	0
橄榄油	1 磅	16.1	土耳其里拉	17	0.6	6	55.8	0.2	0	0	0
蛋	1 个低电耗模式。	32.6	2.9	238	1.0	52	18.6	2.8	6.5	1	0
奶酪（切达）	1 磅	24.2	7.4	448	16.4	19	28.1	0.8	10.3	4	0
乳状	1/2 分	14.1	3.5	49	1.7	3	土耳其里拉	0.6	2.5	0	17
花生酱	1 磅	土耳其里拉	15.7	661	1.0	48	0	9.6	8.1	471	0
蛋黄酱	1/2 分	16.7	8.6	18	0.2	8	2.7	0.4	0.5	0	0
克里斯科	1 磅	20.3	20.1	0	0	0	0	0	0	0	0
猪油	1 磅	9.8	41.7	0	0	0	0.2	0	0.5	5	0
西冷牛排	1 磅	39.6	2.9	166	0.1	34	0.2	2.1	2.9	69	0
圆牛排	1 磅	36.4	2.2	214	0.1	32	0.4	2.5	2.4	87	0
烤肋排	1 磅	29.2	3.4	213	0.1	33	0	0	2	0	0
查克烤肉	1 磅	土耳其里拉	3.6	309	0.2	46	0.4	1	4	120	0
盘子	1 磅	14.6	8.5	404	0.2	62	0	0.9	0	0	0
肝（牛肉）	1 磅	26.8	2.2	333	0.2	139	169.2	6.4	50.8	316	525
羊腿腿	1 磅	27.6	3.1	245	0.1	20	0	2.8	3.9	86	0
羊肉排（肋骨）	1 磅	土耳其里拉	3.3	140	0.1	15	0	1.7	2.7	54	0
猪排	1 磅	土耳其里拉	3.5	196	0.2	30	0	17.4	2.7	60	0
烤猪腰肉	1 磅	24.2	4.4	249	0.3	37	0	18.2	3.6	79	0
培根	1 磅	25.6	10.4	152	0.2	23	0	1.8	1.8	71	0
烟熏火腿	1 磅	27.4	6.7	212	0.2	31	0	9.9	3.3	50	0
咸猪肉	1 磅	16	土耳其里拉	164	0.1	26	0	1.4	1.8	0	0
烤鸡	1 磅	30.3	1.8	184	0.1	30	0.1	0.9	1.8	68	46
小牛肉排	1 磅	42.3	1.7	156	0.1	24	0	1.4	2.4	57	0
三文鱼，粉红色（罐）	16 盎司。	13	5.8	705	6.8	45	3.5	1	4.9	209	0
同类	1 磅	4.4	5.8	27	0.5	36	7.3	3.6	2.7	5	544
香蕉	1 磅	6.1	4.9	60	0.4	30	17.4	2.5	3.5	28	498
柠檬	1 个低电耗模式。	26	1.0	21	0.5	14	0	0.5	0	4	952
橙子	1 个低电耗模式。	30.9	2.2	40	1.1	18	11.1	3.6	1.3	10	1998 年
青豆	1 磅	7.1	2.4	138	3.7	80	69	4.3	5.8	37	862
卷心菜	1 磅	3.7	2.6	125	4.0	36	7.2	9	4.5	26	5369
胡萝卜	1 批	4.7	2.7	73	2.8	43	188.5	6.1	4.3	89	608
芹菜	1 根	7.3	0.9	51	3.0	23	0.9	1.4	1.4	9	313
生菜	1 头	8.2	0.4	27	1.1	22	112.4	1.8	3.4	11	449
洋葱	1 磅	3.6	5.8	166	3.8	59	16.6	4.7	5.9	21	1184
土豆	15 磅	34	14.3	336	1.8	118	6.7	29.4	7.1	198	2522
菠菜	1 磅	8.1	1.1	106	0	138	918.4	5.7	13.8	33	2755
红薯	1 磅	5.1	9.6	138	2.7	54	290.7	8.4	5.4	83	1912
桃子（罐）	第 2 期 1/2	16.8	3.7	20	0.4	10	21.5	0.5	1	31	196
梨（罐）	第 2 期 1/2	20.4	3.0	8	0.3	8	0.8	0.8	0.8	5	81
菠萝（罐）	第 2 期 1/2	21.3	2.4	16	0.4	8	2	2.8	0.8	7	399
芦笋（能）	第二	土耳其里拉	0.4	33	0.3	12	16.3	1.4	2.1	17	272
青豆（罐）	第二	10	1.0	54	2	65	53.9	1.6	4.3	32	431
猪肉和豆子（罐）	16 盎司。	7.1	7.5	364	4	134	3.5	8.3	7.7	56	0
玉米（罐）	第二	10.4	5.2	136	0.2	16	12	1.6	2.7	42	218
豌豆（罐）	第二	13.8	2.3	136	0.6	45	34.9	4.9	2.5	37	370
西红柿（罐）	第二	8.6	1.3	63	0.7	38	53.2	3.4	2.5	36	1253
番茄汤（罐）	10 1/2 盎司。	7.6	1.6	71	0.6	43	57.9	3.5	2.4	67	862
桃子（干）	1 磅	15.7	8.5	87	1.7	173	86.8	1.2	4.3	55	57
西梅（干）	1 磅	9	12.8	99	2.5	154	85.7	3.9	4.3	65	257
葡萄干（干燥）	15 盎司。	9.4	13.5	104	2.5	136	4.5	6.3	1.4	24	136
豌豆（干）	1 磅	7.9	20.0	1367	4.2	345	2.9	28.7	18.4	162	0
干青豆	1 磅	8.9	17.4	1055	3.7	459	5.1	26.9	38.2	93	0
海军蓝豆子（干豆）	1 磅	5.9	26.9	1691	11.4	792	0	38.4	土耳其里拉	217	0
咖啡	1 磅	22.4 岁	0	0	0	0	0	4	5.1	50	0
茶	1/4 磅	17.4	0	0	0	0	0	0	2.3	42	0
可可棕	8 盎司。	8.6	8.7	237	3	72	0	2	11.9	40	0
巧克力	8 盎司。	16.2	8.0	77	1.3	39	0	0.9	3.4	14	0
糖	10 磅	51.7	34.9	0	0	0	0	0	0	0	0
玉米糖浆	24 盎司。	13.7	14.7	0	0.5	74	0	0	0	5	0
糖蜜	18 盎司。	13.6	9.0	0	10.3	244	0	1.9	7.5	146	0
草莓果酱	1 磅	20.5	6.4	11	0.4	7	0.2	0.2	0.4	3	0

由于所有营养成分都已按价格标准化，因此我们的目标是最大限度地减少食物总和。

1944 年，Stigler 计算了他所能找到的最佳答案，并提到了以下悲伤表情：

...目前还没有任何直接的方法是在线性条件下找到线性函数的最小值。

他发现了一种每年花费 39.93 美元（1939 美元）的饮食。1947 年，Jack Laderman 使用单工法（后来又发现了一项最近的发明！）确定了最佳解决方案。9 名职员在桌面计算器上花了 120 个工作日才能得出答案

使用线性求解器的解决方案

以下部分介绍一个解决 Stigler 饮食问题的程序。

导入线性求解器封装容器

导入 OR-Tools 线性求解器封装容器，即 [GLOP](/optimization/mip/glop0 线性求解器的接口)，如下所示。

Python

from ortools.linear_solver import pywraplp

C++

#include <array>
#include <memory>
#include <string>
#include <utility>  // std::pair
#include <vector>

#include "absl/flags/flag.h"
#include "absl/log/flags.h"
#include "ortools/base/init_google.h"
#include "ortools/base/logging.h"
#include "ortools/linear_solver/linear_solver.h"

Java

import com.google.ortools.Loader;
import com.google.ortools.linearsolver.MPConstraint;
import com.google.ortools.linearsolver.MPObjective;
import com.google.ortools.linearsolver.MPSolver;
import com.google.ortools.linearsolver.MPVariable;
import java.util.ArrayList;
import java.util.List;

C#

using System;
using System.Collections.Generic;
using Google.OrTools.LinearSolver;

问题数据

以下代码会针对最低营养要求创建一个数组 nutrients，并为任何解决方案中的营养数据表创建一个 data 数组。

Python

# Nutrient minimums.
nutrients = [
    ["Calories (kcal)", 3],
    ["Protein (g)", 70],
    ["Calcium (g)", 0.8],
    ["Iron (mg)", 12],
    ["Vitamin A (KIU)", 5],
    ["Vitamin B1 (mg)", 1.8],
    ["Vitamin B2 (mg)", 2.7],
    ["Niacin (mg)", 18],
    ["Vitamin C (mg)", 75],
]

# Commodity, Unit, 1939 price (cents), Calories (kcal), Protein (g),
# Calcium (g), Iron (mg), Vitamin A (KIU), Vitamin B1 (mg), Vitamin B2 (mg),
# Niacin (mg), Vitamin C (mg)
data = [
    # fmt: off
  ['Wheat Flour (Enriched)', '10 lb.', 36, 44.7, 1411, 2, 365, 0, 55.4, 33.3, 441, 0],
  ['Macaroni', '1 lb.', 14.1, 11.6, 418, 0.7, 54, 0, 3.2, 1.9, 68, 0],
  ['Wheat Cereal (Enriched)', '28 oz.', 24.2, 11.8, 377, 14.4, 175, 0, 14.4, 8.8, 114, 0],
  ['Corn Flakes', '8 oz.', 7.1, 11.4, 252, 0.1, 56, 0, 13.5, 2.3, 68, 0],
  ['Corn Meal', '1 lb.', 4.6, 36.0, 897, 1.7, 99, 30.9, 17.4, 7.9, 106, 0],
  ['Hominy Grits', '24 oz.', 8.5, 28.6, 680, 0.8, 80, 0, 10.6, 1.6, 110, 0],
  ['Rice', '1 lb.', 7.5, 21.2, 460, 0.6, 41, 0, 2, 4.8, 60, 0],
  ['Rolled Oats', '1 lb.', 7.1, 25.3, 907, 5.1, 341, 0, 37.1, 8.9, 64, 0],
  ['White Bread (Enriched)', '1 lb.', 7.9, 15.0, 488, 2.5, 115, 0, 13.8, 8.5, 126, 0],
  ['Whole Wheat Bread', '1 lb.', 9.1, 12.2, 484, 2.7, 125, 0, 13.9, 6.4, 160, 0],
  ['Rye Bread', '1 lb.', 9.1, 12.4, 439, 1.1, 82, 0, 9.9, 3, 66, 0],
  ['Pound Cake', '1 lb.', 24.8, 8.0, 130, 0.4, 31, 18.9, 2.8, 3, 17, 0],
  ['Soda Crackers', '1 lb.', 15.1, 12.5, 288, 0.5, 50, 0, 0, 0, 0, 0],
  ['Milk', '1 qt.', 11, 6.1, 310, 10.5, 18, 16.8, 4, 16, 7, 177],
  ['Evaporated Milk (can)', '14.5 oz.', 6.7, 8.4, 422, 15.1, 9, 26, 3, 23.5, 11, 60],
  ['Butter', '1 lb.', 30.8, 10.8, 9, 0.2, 3, 44.2, 0, 0.2, 2, 0],
  ['Oleomargarine', '1 lb.', 16.1, 20.6, 17, 0.6, 6, 55.8, 0.2, 0, 0, 0],
  ['Eggs', '1 doz.', 32.6, 2.9, 238, 1.0, 52, 18.6, 2.8, 6.5, 1, 0],
  ['Cheese (Cheddar)', '1 lb.', 24.2, 7.4, 448, 16.4, 19, 28.1, 0.8, 10.3, 4, 0],
  ['Cream', '1/2 pt.', 14.1, 3.5, 49, 1.7, 3, 16.9, 0.6, 2.5, 0, 17],
  ['Peanut Butter', '1 lb.', 17.9, 15.7, 661, 1.0, 48, 0, 9.6, 8.1, 471, 0],
  ['Mayonnaise', '1/2 pt.', 16.7, 8.6, 18, 0.2, 8, 2.7, 0.4, 0.5, 0, 0],
  ['Crisco', '1 lb.', 20.3, 20.1, 0, 0, 0, 0, 0, 0, 0, 0],
  ['Lard', '1 lb.', 9.8, 41.7, 0, 0, 0, 0.2, 0, 0.5, 5, 0],
  ['Sirloin Steak', '1 lb.', 39.6, 2.9, 166, 0.1, 34, 0.2, 2.1, 2.9, 69, 0],
  ['Round Steak', '1 lb.', 36.4, 2.2, 214, 0.1, 32, 0.4, 2.5, 2.4, 87, 0],
  ['Rib Roast', '1 lb.', 29.2, 3.4, 213, 0.1, 33, 0, 0, 2, 0, 0],
  ['Chuck Roast', '1 lb.', 22.6, 3.6, 309, 0.2, 46, 0.4, 1, 4, 120, 0],
  ['Plate', '1 lb.', 14.6, 8.5, 404, 0.2, 62, 0, 0.9, 0, 0, 0],
  ['Liver (Beef)', '1 lb.', 26.8, 2.2, 333, 0.2, 139, 169.2, 6.4, 50.8, 316, 525],
  ['Leg of Lamb', '1 lb.', 27.6, 3.1, 245, 0.1, 20, 0, 2.8, 3.9, 86, 0],
  ['Lamb Chops (Rib)', '1 lb.', 36.6, 3.3, 140, 0.1, 15, 0, 1.7, 2.7, 54, 0],
  ['Pork Chops', '1 lb.', 30.7, 3.5, 196, 0.2, 30, 0, 17.4, 2.7, 60, 0],
  ['Pork Loin Roast', '1 lb.', 24.2, 4.4, 249, 0.3, 37, 0, 18.2, 3.6, 79, 0],
  ['Bacon', '1 lb.', 25.6, 10.4, 152, 0.2, 23, 0, 1.8, 1.8, 71, 0],
  ['Ham, smoked', '1 lb.', 27.4, 6.7, 212, 0.2, 31, 0, 9.9, 3.3, 50, 0],
  ['Salt Pork', '1 lb.', 16, 18.8, 164, 0.1, 26, 0, 1.4, 1.8, 0, 0],
  ['Roasting Chicken', '1 lb.', 30.3, 1.8, 184, 0.1, 30, 0.1, 0.9, 1.8, 68, 46],
  ['Veal Cutlets', '1 lb.', 42.3, 1.7, 156, 0.1, 24, 0, 1.4, 2.4, 57, 0],
  ['Salmon, Pink (can)', '16 oz.', 13, 5.8, 705, 6.8, 45, 3.5, 1, 4.9, 209, 0],
  ['Apples', '1 lb.', 4.4, 5.8, 27, 0.5, 36, 7.3, 3.6, 2.7, 5, 544],
  ['Bananas', '1 lb.', 6.1, 4.9, 60, 0.4, 30, 17.4, 2.5, 3.5, 28, 498],
  ['Lemons', '1 doz.', 26, 1.0, 21, 0.5, 14, 0, 0.5, 0, 4, 952],
  ['Oranges', '1 doz.', 30.9, 2.2, 40, 1.1, 18, 11.1, 3.6, 1.3, 10, 1998],
  ['Green Beans', '1 lb.', 7.1, 2.4, 138, 3.7, 80, 69, 4.3, 5.8, 37, 862],
  ['Cabbage', '1 lb.', 3.7, 2.6, 125, 4.0, 36, 7.2, 9, 4.5, 26, 5369],
  ['Carrots', '1 bunch', 4.7, 2.7, 73, 2.8, 43, 188.5, 6.1, 4.3, 89, 608],
  ['Celery', '1 stalk', 7.3, 0.9, 51, 3.0, 23, 0.9, 1.4, 1.4, 9, 313],
  ['Lettuce', '1 head', 8.2, 0.4, 27, 1.1, 22, 112.4, 1.8, 3.4, 11, 449],
  ['Onions', '1 lb.', 3.6, 5.8, 166, 3.8, 59, 16.6, 4.7, 5.9, 21, 1184],
  ['Potatoes', '15 lb.', 34, 14.3, 336, 1.8, 118, 6.7, 29.4, 7.1, 198, 2522],
  ['Spinach', '1 lb.', 8.1, 1.1, 106, 0, 138, 918.4, 5.7, 13.8, 33, 2755],
  ['Sweet Potatoes', '1 lb.', 5.1, 9.6, 138, 2.7, 54, 290.7, 8.4, 5.4, 83, 1912],
  ['Peaches (can)', 'No. 2 1/2', 16.8, 3.7, 20, 0.4, 10, 21.5, 0.5, 1, 31, 196],
  ['Pears (can)', 'No. 2 1/2', 20.4, 3.0, 8, 0.3, 8, 0.8, 0.8, 0.8, 5, 81],
  ['Pineapple (can)', 'No. 2 1/2', 21.3, 2.4, 16, 0.4, 8, 2, 2.8, 0.8, 7, 399],
  ['Asparagus (can)', 'No. 2', 27.7, 0.4, 33, 0.3, 12, 16.3, 1.4, 2.1, 17, 272],
  ['Green Beans (can)', 'No. 2', 10, 1.0, 54, 2, 65, 53.9, 1.6, 4.3, 32, 431],
  ['Pork and Beans (can)', '16 oz.', 7.1, 7.5, 364, 4, 134, 3.5, 8.3, 7.7, 56, 0],
  ['Corn (can)', 'No. 2', 10.4, 5.2, 136, 0.2, 16, 12, 1.6, 2.7, 42, 218],
  ['Peas (can)', 'No. 2', 13.8, 2.3, 136, 0.6, 45, 34.9, 4.9, 2.5, 37, 370],
  ['Tomatoes (can)', 'No. 2', 8.6, 1.3, 63, 0.7, 38, 53.2, 3.4, 2.5, 36, 1253],
  ['Tomato Soup (can)', '10 1/2 oz.', 7.6, 1.6, 71, 0.6, 43, 57.9, 3.5, 2.4, 67, 862],
  ['Peaches, Dried', '1 lb.', 15.7, 8.5, 87, 1.7, 173, 86.8, 1.2, 4.3, 55, 57],
  ['Prunes, Dried', '1 lb.', 9, 12.8, 99, 2.5, 154, 85.7, 3.9, 4.3, 65, 257],
  ['Raisins, Dried', '15 oz.', 9.4, 13.5, 104, 2.5, 136, 4.5, 6.3, 1.4, 24, 136],
  ['Peas, Dried', '1 lb.', 7.9, 20.0, 1367, 4.2, 345, 2.9, 28.7, 18.4, 162, 0],
  ['Lima Beans, Dried', '1 lb.', 8.9, 17.4, 1055, 3.7, 459, 5.1, 26.9, 38.2, 93, 0],
  ['Navy Beans, Dried', '1 lb.', 5.9, 26.9, 1691, 11.4, 792, 0, 38.4, 24.6, 217, 0],
  ['Coffee', '1 lb.', 22.4, 0, 0, 0, 0, 0, 4, 5.1, 50, 0],
  ['Tea', '1/4 lb.', 17.4, 0, 0, 0, 0, 0, 0, 2.3, 42, 0],
  ['Cocoa', '8 oz.', 8.6, 8.7, 237, 3, 72, 0, 2, 11.9, 40, 0],
  ['Chocolate', '8 oz.', 16.2, 8.0, 77, 1.3, 39, 0, 0.9, 3.4, 14, 0],
  ['Sugar', '10 lb.', 51.7, 34.9, 0, 0, 0, 0, 0, 0, 0, 0],
  ['Corn Syrup', '24 oz.', 13.7, 14.7, 0, 0.5, 74, 0, 0, 0, 5, 0],
  ['Molasses', '18 oz.', 13.6, 9.0, 0, 10.3, 244, 0, 1.9, 7.5, 146, 0],
  ['Strawberry Preserves', '1 lb.', 20.5, 6.4, 11, 0.4, 7, 0.2, 0.2, 0.4, 3, 0],
    # fmt: on
]

C++

// Nutrient minimums.
const std::vector<std::pair<std::string, double>> nutrients = {
    {"Calories (kcal)", 3.0}, {"Protein (g)", 70.0},
    {"Calcium (g)", 0.8},     {"Iron (mg)", 12.0},
    {"Vitamin A (kIU)", 5.0}, {"Vitamin B1 (mg)", 1.8},
    {"Vitamin B2 (mg)", 2.7}, {"Niacin (mg)", 18.0},
    {"Vitamin C (mg)", 75.0}};

struct Commodity {
  std::string name;  //!< Commodity name
  std::string unit;  //!< Unit
  double price;      //!< 1939 price per unit (cents)
  //! Calories (kcal),
  //! Protein (g),
  //! Calcium (g),
  //! Iron (mg),
  //! Vitamin A (kIU),
  //! Vitamin B1 (mg),
  //! Vitamin B2 (mg),
  //! Niacin (mg),
  //! Vitamin C (mg)
  std::array<double, 9> nutrients;
};

std::vector<Commodity> data = {
    {"Wheat Flour (Enriched)",
     "10 lb.",
     36,
     {44.7, 1411, 2, 365, 0, 55.4, 33.3, 441, 0}},
    {"Macaroni", "1 lb.", 14.1, {11.6, 418, 0.7, 54, 0, 3.2, 1.9, 68, 0}},
    {"Wheat Cereal (Enriched)",
     "28 oz.",
     24.2,
     {11.8, 377, 14.4, 175, 0, 14.4, 8.8, 114, 0}},
    {"Corn Flakes", "8 oz.", 7.1, {11.4, 252, 0.1, 56, 0, 13.5, 2.3, 68, 0}},
    {"Corn Meal",
     "1 lb.",
     4.6,
     {36.0, 897, 1.7, 99, 30.9, 17.4, 7.9, 106, 0}},
    {"Hominy Grits",
     "24 oz.",
     8.5,
     {28.6, 680, 0.8, 80, 0, 10.6, 1.6, 110, 0}},
    {"Rice", "1 lb.", 7.5, {21.2, 460, 0.6, 41, 0, 2, 4.8, 60, 0}},
    {"Rolled Oats", "1 lb.", 7.1, {25.3, 907, 5.1, 341, 0, 37.1, 8.9, 64, 0}},
    {"White Bread (Enriched)",
     "1 lb.",
     7.9,
     {15.0, 488, 2.5, 115, 0, 13.8, 8.5, 126, 0}},
    {"Whole Wheat Bread",
     "1 lb.",
     9.1,
     {12.2, 484, 2.7, 125, 0, 13.9, 6.4, 160, 0}},
    {"Rye Bread", "1 lb.", 9.1, {12.4, 439, 1.1, 82, 0, 9.9, 3, 66, 0}},
    {"Pound Cake", "1 lb.", 24.8, {8.0, 130, 0.4, 31, 18.9, 2.8, 3, 17, 0}},
    {"Soda Crackers", "1 lb.", 15.1, {12.5, 288, 0.5, 50, 0, 0, 0, 0, 0}},
    {"Milk", "1 qt.", 11, {6.1, 310, 10.5, 18, 16.8, 4, 16, 7, 177}},
    {"Evaporated Milk (can)",
     "14.5 oz.",
     6.7,
     {8.4, 422, 15.1, 9, 26, 3, 23.5, 11, 60}},
    {"Butter", "1 lb.", 30.8, {10.8, 9, 0.2, 3, 44.2, 0, 0.2, 2, 0}},
    {"Oleomargarine", "1 lb.", 16.1, {20.6, 17, 0.6, 6, 55.8, 0.2, 0, 0, 0}},
    {"Eggs", "1 doz.", 32.6, {2.9, 238, 1.0, 52, 18.6, 2.8, 6.5, 1, 0}},
    {"Cheese (Cheddar)",
     "1 lb.",
     24.2,
     {7.4, 448, 16.4, 19, 28.1, 0.8, 10.3, 4, 0}},
    {"Cream", "1/2 pt.", 14.1, {3.5, 49, 1.7, 3, 16.9, 0.6, 2.5, 0, 17}},
    {"Peanut Butter",
     "1 lb.",
     17.9,
     {15.7, 661, 1.0, 48, 0, 9.6, 8.1, 471, 0}},
    {"Mayonnaise", "1/2 pt.", 16.7, {8.6, 18, 0.2, 8, 2.7, 0.4, 0.5, 0, 0}},
    {"Crisco", "1 lb.", 20.3, {20.1, 0, 0, 0, 0, 0, 0, 0, 0}},
    {"Lard", "1 lb.", 9.8, {41.7, 0, 0, 0, 0.2, 0, 0.5, 5, 0}},
    {"Sirloin Steak",
     "1 lb.",
     39.6,
     {2.9, 166, 0.1, 34, 0.2, 2.1, 2.9, 69, 0}},
    {"Round Steak", "1 lb.", 36.4, {2.2, 214, 0.1, 32, 0.4, 2.5, 2.4, 87, 0}},
    {"Rib Roast", "1 lb.", 29.2, {3.4, 213, 0.1, 33, 0, 0, 2, 0, 0}},
    {"Chuck Roast", "1 lb.", 22.6, {3.6, 309, 0.2, 46, 0.4, 1, 4, 120, 0}},
    {"Plate", "1 lb.", 14.6, {8.5, 404, 0.2, 62, 0, 0.9, 0, 0, 0}},
    {"Liver (Beef)",
     "1 lb.",
     26.8,
     {2.2, 333, 0.2, 139, 169.2, 6.4, 50.8, 316, 525}},
    {"Leg of Lamb", "1 lb.", 27.6, {3.1, 245, 0.1, 20, 0, 2.8, 3.9, 86, 0}},
    {"Lamb Chops (Rib)",
     "1 lb.",
     36.6,
     {3.3, 140, 0.1, 15, 0, 1.7, 2.7, 54, 0}},
    {"Pork Chops", "1 lb.", 30.7, {3.5, 196, 0.2, 30, 0, 17.4, 2.7, 60, 0}},
    {"Pork Loin Roast",
     "1 lb.",
     24.2,
     {4.4, 249, 0.3, 37, 0, 18.2, 3.6, 79, 0}},
    {"Bacon", "1 lb.", 25.6, {10.4, 152, 0.2, 23, 0, 1.8, 1.8, 71, 0}},
    {"Ham, smoked", "1 lb.", 27.4, {6.7, 212, 0.2, 31, 0, 9.9, 3.3, 50, 0}},
    {"Salt Pork", "1 lb.", 16, {18.8, 164, 0.1, 26, 0, 1.4, 1.8, 0, 0}},
    {"Roasting Chicken",
     "1 lb.",
     30.3,
     {1.8, 184, 0.1, 30, 0.1, 0.9, 1.8, 68, 46}},
    {"Veal Cutlets", "1 lb.", 42.3, {1.7, 156, 0.1, 24, 0, 1.4, 2.4, 57, 0}},
    {"Salmon, Pink (can)",
     "16 oz.",
     13,
     {5.8, 705, 6.8, 45, 3.5, 1, 4.9, 209, 0}},
    {"Apples", "1 lb.", 4.4, {5.8, 27, 0.5, 36, 7.3, 3.6, 2.7, 5, 544}},
    {"Bananas", "1 lb.", 6.1, {4.9, 60, 0.4, 30, 17.4, 2.5, 3.5, 28, 498}},
    {"Lemons", "1 doz.", 26, {1.0, 21, 0.5, 14, 0, 0.5, 0, 4, 952}},
    {"Oranges", "1 doz.", 30.9, {2.2, 40, 1.1, 18, 11.1, 3.6, 1.3, 10, 1998}},
    {"Green Beans", "1 lb.", 7.1, {2.4, 138, 3.7, 80, 69, 4.3, 5.8, 37, 862}},
    {"Cabbage", "1 lb.", 3.7, {2.6, 125, 4.0, 36, 7.2, 9, 4.5, 26, 5369}},
    {"Carrots", "1 bunch", 4.7, {2.7, 73, 2.8, 43, 188.5, 6.1, 4.3, 89, 608}},
    {"Celery", "1 stalk", 7.3, {0.9, 51, 3.0, 23, 0.9, 1.4, 1.4, 9, 313}},
    {"Lettuce", "1 head", 8.2, {0.4, 27, 1.1, 22, 112.4, 1.8, 3.4, 11, 449}},
    {"Onions", "1 lb.", 3.6, {5.8, 166, 3.8, 59, 16.6, 4.7, 5.9, 21, 1184}},
    {"Potatoes",
     "15 lb.",
     34,
     {14.3, 336, 1.8, 118, 6.7, 29.4, 7.1, 198, 2522}},
    {"Spinach", "1 lb.", 8.1, {1.1, 106, 0, 138, 918.4, 5.7, 13.8, 33, 2755}},
    {"Sweet Potatoes",
     "1 lb.",
     5.1,
     {9.6, 138, 2.7, 54, 290.7, 8.4, 5.4, 83, 1912}},
    {"Peaches (can)",
     "No. 2 1/2",
     16.8,
     {3.7, 20, 0.4, 10, 21.5, 0.5, 1, 31, 196}},
    {"Pears (can)",
     "No. 2 1/2",
     20.4,
     {3.0, 8, 0.3, 8, 0.8, 0.8, 0.8, 5, 81}},
    {"Pineapple (can)",
     "No. 2 1/2",
     21.3,
     {2.4, 16, 0.4, 8, 2, 2.8, 0.8, 7, 399}},
    {"Asparagus (can)",
     "No. 2",
     27.7,
     {0.4, 33, 0.3, 12, 16.3, 1.4, 2.1, 17, 272}},
    {"Green Beans (can)",
     "No. 2",
     10,
     {1.0, 54, 2, 65, 53.9, 1.6, 4.3, 32, 431}},
    {"Pork and Beans (can)",
     "16 oz.",
     7.1,
     {7.5, 364, 4, 134, 3.5, 8.3, 7.7, 56, 0}},
    {"Corn (can)", "No. 2", 10.4, {5.2, 136, 0.2, 16, 12, 1.6, 2.7, 42, 218}},
    {"Peas (can)",
     "No. 2",
     13.8,
     {2.3, 136, 0.6, 45, 34.9, 4.9, 2.5, 37, 370}},
    {"Tomatoes (can)",
     "No. 2",
     8.6,
     {1.3, 63, 0.7, 38, 53.2, 3.4, 2.5, 36, 1253}},
    {"Tomato Soup (can)",
     "10 1/2 oz.",
     7.6,
     {1.6, 71, 0.6, 43, 57.9, 3.5, 2.4, 67, 862}},
    {"Peaches, Dried",
     "1 lb.",
     15.7,
     {8.5, 87, 1.7, 173, 86.8, 1.2, 4.3, 55, 57}},
    {"Prunes, Dried",
     "1 lb.",
     9,
     {12.8, 99, 2.5, 154, 85.7, 3.9, 4.3, 65, 257}},
    {"Raisins, Dried",
     "15 oz.",
     9.4,
     {13.5, 104, 2.5, 136, 4.5, 6.3, 1.4, 24, 136}},
    {"Peas, Dried",
     "1 lb.",
     7.9,
     {20.0, 1367, 4.2, 345, 2.9, 28.7, 18.4, 162, 0}},
    {"Lima Beans, Dried",
     "1 lb.",
     8.9,
     {17.4, 1055, 3.7, 459, 5.1, 26.9, 38.2, 93, 0}},
    {"Navy Beans, Dried",
     "1 lb.",
     5.9,
     {26.9, 1691, 11.4, 792, 0, 38.4, 24.6, 217, 0}},
    {"Coffee", "1 lb.", 22.4, {0, 0, 0, 0, 0, 4, 5.1, 50, 0}},
    {"Tea", "1/4 lb.", 17.4, {0, 0, 0, 0, 0, 0, 2.3, 42, 0}},
    {"Cocoa", "8 oz.", 8.6, {8.7, 237, 3, 72, 0, 2, 11.9, 40, 0}},
    {"Chocolate", "8 oz.", 16.2, {8.0, 77, 1.3, 39, 0, 0.9, 3.4, 14, 0}},
    {"Sugar", "10 lb.", 51.7, {34.9, 0, 0, 0, 0, 0, 0, 0, 0}},
    {"Corn Syrup", "24 oz.", 13.7, {14.7, 0, 0.5, 74, 0, 0, 0, 5, 0}},
    {"Molasses", "18 oz.", 13.6, {9.0, 0, 10.3, 244, 0, 1.9, 7.5, 146, 0}},
    {"Strawberry Preserves",
     "1 lb.",
     20.5,
     {6.4, 11, 0.4, 7, 0.2, 0.2, 0.4, 3, 0}}};

Java

// Nutrient minimums.
List<Object[]> nutrients = new ArrayList<>();
nutrients.add(new Object[] {"Calories (kcal)", 3.0});
nutrients.add(new Object[] {"Protein (g)", 70.0});
nutrients.add(new Object[] {"Calcium (g)", 0.8});
nutrients.add(new Object[] {"Iron (mg)", 12.0});
nutrients.add(new Object[] {"Vitamin A (kIU)", 5.0});
nutrients.add(new Object[] {"Vitamin B1 (mg)", 1.8});
nutrients.add(new Object[] {"Vitamin B2 (mg)", 2.7});
nutrients.add(new Object[] {"Niacin (mg)", 18.0});
nutrients.add(new Object[] {"Vitamin C (mg)", 75.0});

List<Object[]> data = new ArrayList<>();
data.add(new Object[] {"Wheat Flour (Enriched)", "10 lb.", 36,
    new double[] {44.7, 1411, 2, 365, 0, 55.4, 33.3, 441, 0}});
data.add(new Object[] {
    "Macaroni", "1 lb.", 14.1, new double[] {11.6, 418, 0.7, 54, 0, 3.2, 1.9, 68, 0}});
data.add(new Object[] {"Wheat Cereal (Enriched)", "28 oz.", 24.2,
    new double[] {11.8, 377, 14.4, 175, 0, 14.4, 8.8, 114, 0}});
data.add(new Object[] {
    "Corn Flakes", "8 oz.", 7.1, new double[] {11.4, 252, 0.1, 56, 0, 13.5, 2.3, 68, 0}});
data.add(new Object[] {
    "Corn Meal", "1 lb.", 4.6, new double[] {36.0, 897, 1.7, 99, 30.9, 17.4, 7.9, 106, 0}});
data.add(new Object[] {
    "Hominy Grits", "24 oz.", 8.5, new double[] {28.6, 680, 0.8, 80, 0, 10.6, 1.6, 110, 0}});
data.add(
    new Object[] {"Rice", "1 lb.", 7.5, new double[] {21.2, 460, 0.6, 41, 0, 2, 4.8, 60, 0}});
data.add(new Object[] {
    "Rolled Oats", "1 lb.", 7.1, new double[] {25.3, 907, 5.1, 341, 0, 37.1, 8.9, 64, 0}});
data.add(new Object[] {"White Bread (Enriched)", "1 lb.", 7.9,
    new double[] {15.0, 488, 2.5, 115, 0, 13.8, 8.5, 126, 0}});
data.add(new Object[] {"Whole Wheat Bread", "1 lb.", 9.1,
    new double[] {12.2, 484, 2.7, 125, 0, 13.9, 6.4, 160, 0}});
data.add(new Object[] {
    "Rye Bread", "1 lb.", 9.1, new double[] {12.4, 439, 1.1, 82, 0, 9.9, 3, 66, 0}});
data.add(new Object[] {
    "Pound Cake", "1 lb.", 24.8, new double[] {8.0, 130, 0.4, 31, 18.9, 2.8, 3, 17, 0}});
data.add(new Object[] {
    "Soda Crackers", "1 lb.", 15.1, new double[] {12.5, 288, 0.5, 50, 0, 0, 0, 0, 0}});
data.add(
    new Object[] {"Milk", "1 qt.", 11, new double[] {6.1, 310, 10.5, 18, 16.8, 4, 16, 7, 177}});
data.add(new Object[] {"Evaporated Milk (can)", "14.5 oz.", 6.7,
    new double[] {8.4, 422, 15.1, 9, 26, 3, 23.5, 11, 60}});
data.add(
    new Object[] {"Butter", "1 lb.", 30.8, new double[] {10.8, 9, 0.2, 3, 44.2, 0, 0.2, 2, 0}});
data.add(new Object[] {
    "Oleomargarine", "1 lb.", 16.1, new double[] {20.6, 17, 0.6, 6, 55.8, 0.2, 0, 0, 0}});
data.add(new Object[] {
    "Eggs", "1 doz.", 32.6, new double[] {2.9, 238, 1.0, 52, 18.6, 2.8, 6.5, 1, 0}});
data.add(new Object[] {"Cheese (Cheddar)", "1 lb.", 24.2,
    new double[] {7.4, 448, 16.4, 19, 28.1, 0.8, 10.3, 4, 0}});
data.add(new Object[] {
    "Cream", "1/2 pt.", 14.1, new double[] {3.5, 49, 1.7, 3, 16.9, 0.6, 2.5, 0, 17}});
data.add(new Object[] {
    "Peanut Butter", "1 lb.", 17.9, new double[] {15.7, 661, 1.0, 48, 0, 9.6, 8.1, 471, 0}});
data.add(new Object[] {
    "Mayonnaise", "1/2 pt.", 16.7, new double[] {8.6, 18, 0.2, 8, 2.7, 0.4, 0.5, 0, 0}});
data.add(new Object[] {"Crisco", "1 lb.", 20.3, new double[] {20.1, 0, 0, 0, 0, 0, 0, 0, 0}});
data.add(new Object[] {"Lard", "1 lb.", 9.8, new double[] {41.7, 0, 0, 0, 0.2, 0, 0.5, 5, 0}});
data.add(new Object[] {
    "Sirloin Steak", "1 lb.", 39.6, new double[] {2.9, 166, 0.1, 34, 0.2, 2.1, 2.9, 69, 0}});
data.add(new Object[] {
    "Round Steak", "1 lb.", 36.4, new double[] {2.2, 214, 0.1, 32, 0.4, 2.5, 2.4, 87, 0}});
data.add(
    new Object[] {"Rib Roast", "1 lb.", 29.2, new double[] {3.4, 213, 0.1, 33, 0, 0, 2, 0, 0}});
data.add(new Object[] {
    "Chuck Roast", "1 lb.", 22.6, new double[] {3.6, 309, 0.2, 46, 0.4, 1, 4, 120, 0}});
data.add(
    new Object[] {"Plate", "1 lb.", 14.6, new double[] {8.5, 404, 0.2, 62, 0, 0.9, 0, 0, 0}});
data.add(new Object[] {"Liver (Beef)", "1 lb.", 26.8,
    new double[] {2.2, 333, 0.2, 139, 169.2, 6.4, 50.8, 316, 525}});
data.add(new Object[] {
    "Leg of Lamb", "1 lb.", 27.6, new double[] {3.1, 245, 0.1, 20, 0, 2.8, 3.9, 86, 0}});
data.add(new Object[] {
    "Lamb Chops (Rib)", "1 lb.", 36.6, new double[] {3.3, 140, 0.1, 15, 0, 1.7, 2.7, 54, 0}});
data.add(new Object[] {
    "Pork Chops", "1 lb.", 30.7, new double[] {3.5, 196, 0.2, 30, 0, 17.4, 2.7, 60, 0}});
data.add(new Object[] {
    "Pork Loin Roast", "1 lb.", 24.2, new double[] {4.4, 249, 0.3, 37, 0, 18.2, 3.6, 79, 0}});
data.add(new Object[] {
    "Bacon", "1 lb.", 25.6, new double[] {10.4, 152, 0.2, 23, 0, 1.8, 1.8, 71, 0}});
data.add(new Object[] {
    "Ham, smoked", "1 lb.", 27.4, new double[] {6.7, 212, 0.2, 31, 0, 9.9, 3.3, 50, 0}});
data.add(new Object[] {
    "Salt Pork", "1 lb.", 16, new double[] {18.8, 164, 0.1, 26, 0, 1.4, 1.8, 0, 0}});
data.add(new Object[] {"Roasting Chicken", "1 lb.", 30.3,
    new double[] {1.8, 184, 0.1, 30, 0.1, 0.9, 1.8, 68, 46}});
data.add(new Object[] {
    "Veal Cutlets", "1 lb.", 42.3, new double[] {1.7, 156, 0.1, 24, 0, 1.4, 2.4, 57, 0}});
data.add(new Object[] {
    "Salmon, Pink (can)", "16 oz.", 13, new double[] {5.8, 705, 6.8, 45, 3.5, 1, 4.9, 209, 0}});
data.add(new Object[] {
    "Apples", "1 lb.", 4.4, new double[] {5.8, 27, 0.5, 36, 7.3, 3.6, 2.7, 5, 544}});
data.add(new Object[] {
    "Bananas", "1 lb.", 6.1, new double[] {4.9, 60, 0.4, 30, 17.4, 2.5, 3.5, 28, 498}});
data.add(
    new Object[] {"Lemons", "1 doz.", 26, new double[] {1.0, 21, 0.5, 14, 0, 0.5, 0, 4, 952}});
data.add(new Object[] {
    "Oranges", "1 doz.", 30.9, new double[] {2.2, 40, 1.1, 18, 11.1, 3.6, 1.3, 10, 1998}});
data.add(new Object[] {
    "Green Beans", "1 lb.", 7.1, new double[] {2.4, 138, 3.7, 80, 69, 4.3, 5.8, 37, 862}});
data.add(new Object[] {
    "Cabbage", "1 lb.", 3.7, new double[] {2.6, 125, 4.0, 36, 7.2, 9, 4.5, 26, 5369}});
data.add(new Object[] {
    "Carrots", "1 bunch", 4.7, new double[] {2.7, 73, 2.8, 43, 188.5, 6.1, 4.3, 89, 608}});
data.add(new Object[] {
    "Celery", "1 stalk", 7.3, new double[] {0.9, 51, 3.0, 23, 0.9, 1.4, 1.4, 9, 313}});
data.add(new Object[] {
    "Lettuce", "1 head", 8.2, new double[] {0.4, 27, 1.1, 22, 112.4, 1.8, 3.4, 11, 449}});
data.add(new Object[] {
    "Onions", "1 lb.", 3.6, new double[] {5.8, 166, 3.8, 59, 16.6, 4.7, 5.9, 21, 1184}});
data.add(new Object[] {
    "Potatoes", "15 lb.", 34, new double[] {14.3, 336, 1.8, 118, 6.7, 29.4, 7.1, 198, 2522}});
data.add(new Object[] {
    "Spinach", "1 lb.", 8.1, new double[] {1.1, 106, 0, 138, 918.4, 5.7, 13.8, 33, 2755}});
data.add(new Object[] {"Sweet Potatoes", "1 lb.", 5.1,
    new double[] {9.6, 138, 2.7, 54, 290.7, 8.4, 5.4, 83, 1912}});
data.add(new Object[] {"Peaches (can)", "No. 2 1/2", 16.8,
    new double[] {3.7, 20, 0.4, 10, 21.5, 0.5, 1, 31, 196}});
data.add(new Object[] {
    "Pears (can)", "No. 2 1/2", 20.4, new double[] {3.0, 8, 0.3, 8, 0.8, 0.8, 0.8, 5, 81}});
data.add(new Object[] {
    "Pineapple (can)", "No. 2 1/2", 21.3, new double[] {2.4, 16, 0.4, 8, 2, 2.8, 0.8, 7, 399}});
data.add(new Object[] {"Asparagus (can)", "No. 2", 27.7,
    new double[] {0.4, 33, 0.3, 12, 16.3, 1.4, 2.1, 17, 272}});
data.add(new Object[] {
    "Green Beans (can)", "No. 2", 10, new double[] {1.0, 54, 2, 65, 53.9, 1.6, 4.3, 32, 431}});
data.add(new Object[] {"Pork and Beans (can)", "16 oz.", 7.1,
    new double[] {7.5, 364, 4, 134, 3.5, 8.3, 7.7, 56, 0}});
data.add(new Object[] {
    "Corn (can)", "No. 2", 10.4, new double[] {5.2, 136, 0.2, 16, 12, 1.6, 2.7, 42, 218}});
data.add(new Object[] {
    "Peas (can)", "No. 2", 13.8, new double[] {2.3, 136, 0.6, 45, 34.9, 4.9, 2.5, 37, 370}});
data.add(new Object[] {
    "Tomatoes (can)", "No. 2", 8.6, new double[] {1.3, 63, 0.7, 38, 53.2, 3.4, 2.5, 36, 1253}});
data.add(new Object[] {"Tomato Soup (can)", "10 1/2 oz.", 7.6,
    new double[] {1.6, 71, 0.6, 43, 57.9, 3.5, 2.4, 67, 862}});
data.add(new Object[] {
    "Peaches, Dried", "1 lb.", 15.7, new double[] {8.5, 87, 1.7, 173, 86.8, 1.2, 4.3, 55, 57}});
data.add(new Object[] {
    "Prunes, Dried", "1 lb.", 9, new double[] {12.8, 99, 2.5, 154, 85.7, 3.9, 4.3, 65, 257}});
data.add(new Object[] {"Raisins, Dried", "15 oz.", 9.4,
    new double[] {13.5, 104, 2.5, 136, 4.5, 6.3, 1.4, 24, 136}});
data.add(new Object[] {
    "Peas, Dried", "1 lb.", 7.9, new double[] {20.0, 1367, 4.2, 345, 2.9, 28.7, 18.4, 162, 0}});
data.add(new Object[] {"Lima Beans, Dried", "1 lb.", 8.9,
    new double[] {17.4, 1055, 3.7, 459, 5.1, 26.9, 38.2, 93, 0}});
data.add(new Object[] {"Navy Beans, Dried", "1 lb.", 5.9,
    new double[] {26.9, 1691, 11.4, 792, 0, 38.4, 24.6, 217, 0}});
data.add(new Object[] {"Coffee", "1 lb.", 22.4, new double[] {0, 0, 0, 0, 0, 4, 5.1, 50, 0}});
data.add(new Object[] {"Tea", "1/4 lb.", 17.4, new double[] {0, 0, 0, 0, 0, 0, 2.3, 42, 0}});
data.add(
    new Object[] {"Cocoa", "8 oz.", 8.6, new double[] {8.7, 237, 3, 72, 0, 2, 11.9, 40, 0}});
data.add(new Object[] {
    "Chocolate", "8 oz.", 16.2, new double[] {8.0, 77, 1.3, 39, 0, 0.9, 3.4, 14, 0}});
data.add(new Object[] {"Sugar", "10 lb.", 51.7, new double[] {34.9, 0, 0, 0, 0, 0, 0, 0, 0}});
data.add(new Object[] {
    "Corn Syrup", "24 oz.", 13.7, new double[] {14.7, 0, 0.5, 74, 0, 0, 0, 5, 0}});
data.add(new Object[] {
    "Molasses", "18 oz.", 13.6, new double[] {9.0, 0, 10.3, 244, 0, 1.9, 7.5, 146, 0}});
data.add(new Object[] {"Strawberry Preserves", "1 lb.", 20.5,
    new double[] {6.4, 11, 0.4, 7, 0.2, 0.2, 0.4, 3, 0}});

C#

// Nutrient minimums.
(String Name, double Value)[] nutrients =
    new[] { ("Calories (kcal)", 3.0), ("Protein (g)", 70.0),    ("Calcium (g)", 0.8),
            ("Iron (mg)", 12.0),      ("Vitamin A (kIU)", 5.0), ("Vitamin B1 (mg)", 1.8),
            ("Vitamin B2 (mg)", 2.7), ("Niacin (mg)", 18.0),    ("Vitamin C (mg)", 75.0) };

(String Name, String Unit, double Price, double[] Nutrients)[] data = new[] {
    ("Wheat Flour (Enriched)", "10 lb.", 36, new double[] { 44.7, 1411, 2, 365, 0, 55.4, 33.3, 441, 0 }),
    ("Macaroni", "1 lb.", 14.1, new double[] { 11.6, 418, 0.7, 54, 0, 3.2, 1.9, 68, 0 }),
    ("Wheat Cereal (Enriched)", "28 oz.", 24.2, new double[] { 11.8, 377, 14.4, 175, 0, 14.4, 8.8, 114, 0 }),
    ("Corn Flakes", "8 oz.", 7.1, new double[] { 11.4, 252, 0.1, 56, 0, 13.5, 2.3, 68, 0 }),
    ("Corn Meal", "1 lb.", 4.6, new double[] { 36.0, 897, 1.7, 99, 30.9, 17.4, 7.9, 106, 0 }),
    ("Hominy Grits", "24 oz.", 8.5, new double[] { 28.6, 680, 0.8, 80, 0, 10.6, 1.6, 110, 0 }),
    ("Rice", "1 lb.", 7.5, new double[] { 21.2, 460, 0.6, 41, 0, 2, 4.8, 60, 0 }),
    ("Rolled Oats", "1 lb.", 7.1, new double[] { 25.3, 907, 5.1, 341, 0, 37.1, 8.9, 64, 0 }),
    ("White Bread (Enriched)", "1 lb.", 7.9, new double[] { 15.0, 488, 2.5, 115, 0, 13.8, 8.5, 126, 0 }),
    ("Whole Wheat Bread", "1 lb.", 9.1, new double[] { 12.2, 484, 2.7, 125, 0, 13.9, 6.4, 160, 0 }),
    ("Rye Bread", "1 lb.", 9.1, new double[] { 12.4, 439, 1.1, 82, 0, 9.9, 3, 66, 0 }),
    ("Pound Cake", "1 lb.", 24.8, new double[] { 8.0, 130, 0.4, 31, 18.9, 2.8, 3, 17, 0 }),
    ("Soda Crackers", "1 lb.", 15.1, new double[] { 12.5, 288, 0.5, 50, 0, 0, 0, 0, 0 }),
    ("Milk", "1 qt.", 11, new double[] { 6.1, 310, 10.5, 18, 16.8, 4, 16, 7, 177 }),
    ("Evaporated Milk (can)", "14.5 oz.", 6.7, new double[] { 8.4, 422, 15.1, 9, 26, 3, 23.5, 11, 60 }),
    ("Butter", "1 lb.", 30.8, new double[] { 10.8, 9, 0.2, 3, 44.2, 0, 0.2, 2, 0 }),
    ("Oleomargarine", "1 lb.", 16.1, new double[] { 20.6, 17, 0.6, 6, 55.8, 0.2, 0, 0, 0 }),
    ("Eggs", "1 doz.", 32.6, new double[] { 2.9, 238, 1.0, 52, 18.6, 2.8, 6.5, 1, 0 }),
    ("Cheese (Cheddar)", "1 lb.", 24.2, new double[] { 7.4, 448, 16.4, 19, 28.1, 0.8, 10.3, 4, 0 }),
    ("Cream", "1/2 pt.", 14.1, new double[] { 3.5, 49, 1.7, 3, 16.9, 0.6, 2.5, 0, 17 }),
    ("Peanut Butter", "1 lb.", 17.9, new double[] { 15.7, 661, 1.0, 48, 0, 9.6, 8.1, 471, 0 }),
    ("Mayonnaise", "1/2 pt.", 16.7, new double[] { 8.6, 18, 0.2, 8, 2.7, 0.4, 0.5, 0, 0 }),
    ("Crisco", "1 lb.", 20.3, new double[] { 20.1, 0, 0, 0, 0, 0, 0, 0, 0 }),
    ("Lard", "1 lb.", 9.8, new double[] { 41.7, 0, 0, 0, 0.2, 0, 0.5, 5, 0 }),
    ("Sirloin Steak", "1 lb.", 39.6, new double[] { 2.9, 166, 0.1, 34, 0.2, 2.1, 2.9, 69, 0 }),
    ("Round Steak", "1 lb.", 36.4, new double[] { 2.2, 214, 0.1, 32, 0.4, 2.5, 2.4, 87, 0 }),
    ("Rib Roast", "1 lb.", 29.2, new double[] { 3.4, 213, 0.1, 33, 0, 0, 2, 0, 0 }),
    ("Chuck Roast", "1 lb.", 22.6, new double[] { 3.6, 309, 0.2, 46, 0.4, 1, 4, 120, 0 }),
    ("Plate", "1 lb.", 14.6, new double[] { 8.5, 404, 0.2, 62, 0, 0.9, 0, 0, 0 }),
    ("Liver (Beef)", "1 lb.", 26.8, new double[] { 2.2, 333, 0.2, 139, 169.2, 6.4, 50.8, 316, 525 }),
    ("Leg of Lamb", "1 lb.", 27.6, new double[] { 3.1, 245, 0.1, 20, 0, 2.8, 3.9, 86, 0 }),
    ("Lamb Chops (Rib)", "1 lb.", 36.6, new double[] { 3.3, 140, 0.1, 15, 0, 1.7, 2.7, 54, 0 }),
    ("Pork Chops", "1 lb.", 30.7, new double[] { 3.5, 196, 0.2, 30, 0, 17.4, 2.7, 60, 0 }),
    ("Pork Loin Roast", "1 lb.", 24.2, new double[] { 4.4, 249, 0.3, 37, 0, 18.2, 3.6, 79, 0 }),
    ("Bacon", "1 lb.", 25.6, new double[] { 10.4, 152, 0.2, 23, 0, 1.8, 1.8, 71, 0 }),
    ("Ham, smoked", "1 lb.", 27.4, new double[] { 6.7, 212, 0.2, 31, 0, 9.9, 3.3, 50, 0 }),
    ("Salt Pork", "1 lb.", 16, new double[] { 18.8, 164, 0.1, 26, 0, 1.4, 1.8, 0, 0 }),
    ("Roasting Chicken", "1 lb.", 30.3, new double[] { 1.8, 184, 0.1, 30, 0.1, 0.9, 1.8, 68, 46 }),
    ("Veal Cutlets", "1 lb.", 42.3, new double[] { 1.7, 156, 0.1, 24, 0, 1.4, 2.4, 57, 0 }),
    ("Salmon, Pink (can)", "16 oz.", 13, new double[] { 5.8, 705, 6.8, 45, 3.5, 1, 4.9, 209, 0 }),
    ("Apples", "1 lb.", 4.4, new double[] { 5.8, 27, 0.5, 36, 7.3, 3.6, 2.7, 5, 544 }),
    ("Bananas", "1 lb.", 6.1, new double[] { 4.9, 60, 0.4, 30, 17.4, 2.5, 3.5, 28, 498 }),
    ("Lemons", "1 doz.", 26, new double[] { 1.0, 21, 0.5, 14, 0, 0.5, 0, 4, 952 }),
    ("Oranges", "1 doz.", 30.9, new double[] { 2.2, 40, 1.1, 18, 11.1, 3.6, 1.3, 10, 1998 }),
    ("Green Beans", "1 lb.", 7.1, new double[] { 2.4, 138, 3.7, 80, 69, 4.3, 5.8, 37, 862 }),
    ("Cabbage", "1 lb.", 3.7, new double[] { 2.6, 125, 4.0, 36, 7.2, 9, 4.5, 26, 5369 }),
    ("Carrots", "1 bunch", 4.7, new double[] { 2.7, 73, 2.8, 43, 188.5, 6.1, 4.3, 89, 608 }),
    ("Celery", "1 stalk", 7.3, new double[] { 0.9, 51, 3.0, 23, 0.9, 1.4, 1.4, 9, 313 }),
    ("Lettuce", "1 head", 8.2, new double[] { 0.4, 27, 1.1, 22, 112.4, 1.8, 3.4, 11, 449 }),
    ("Onions", "1 lb.", 3.6, new double[] { 5.8, 166, 3.8, 59, 16.6, 4.7, 5.9, 21, 1184 }),
    ("Potatoes", "15 lb.", 34, new double[] { 14.3, 336, 1.8, 118, 6.7, 29.4, 7.1, 198, 2522 }),
    ("Spinach", "1 lb.", 8.1, new double[] { 1.1, 106, 0, 138, 918.4, 5.7, 13.8, 33, 2755 }),
    ("Sweet Potatoes", "1 lb.", 5.1, new double[] { 9.6, 138, 2.7, 54, 290.7, 8.4, 5.4, 83, 1912 }),
    ("Peaches (can)", "No. 2 1/2", 16.8, new double[] { 3.7, 20, 0.4, 10, 21.5, 0.5, 1, 31, 196 }),
    ("Pears (can)", "No. 2 1/2", 20.4, new double[] { 3.0, 8, 0.3, 8, 0.8, 0.8, 0.8, 5, 81 }),
    ("Pineapple (can)", "No. 2 1/2", 21.3, new double[] { 2.4, 16, 0.4, 8, 2, 2.8, 0.8, 7, 399 }),
    ("Asparagus (can)", "No. 2", 27.7, new double[] { 0.4, 33, 0.3, 12, 16.3, 1.4, 2.1, 17, 272 }),
    ("Green Beans (can)", "No. 2", 10, new double[] { 1.0, 54, 2, 65, 53.9, 1.6, 4.3, 32, 431 }),
    ("Pork and Beans (can)", "16 oz.", 7.1, new double[] { 7.5, 364, 4, 134, 3.5, 8.3, 7.7, 56, 0 }),
    ("Corn (can)", "No. 2", 10.4, new double[] { 5.2, 136, 0.2, 16, 12, 1.6, 2.7, 42, 218 }),
    ("Peas (can)", "No. 2", 13.8, new double[] { 2.3, 136, 0.6, 45, 34.9, 4.9, 2.5, 37, 370 }),
    ("Tomatoes (can)", "No. 2", 8.6, new double[] { 1.3, 63, 0.7, 38, 53.2, 3.4, 2.5, 36, 1253 }),
    ("Tomato Soup (can)", "10 1/2 oz.", 7.6, new double[] { 1.6, 71, 0.6, 43, 57.9, 3.5, 2.4, 67, 862 }),
    ("Peaches, Dried", "1 lb.", 15.7, new double[] { 8.5, 87, 1.7, 173, 86.8, 1.2, 4.3, 55, 57 }),
    ("Prunes, Dried", "1 lb.", 9, new double[] { 12.8, 99, 2.5, 154, 85.7, 3.9, 4.3, 65, 257 }),
    ("Raisins, Dried", "15 oz.", 9.4, new double[] { 13.5, 104, 2.5, 136, 4.5, 6.3, 1.4, 24, 136 }),
    ("Peas, Dried", "1 lb.", 7.9, new double[] { 20.0, 1367, 4.2, 345, 2.9, 28.7, 18.4, 162, 0 }),
    ("Lima Beans, Dried", "1 lb.", 8.9, new double[] { 17.4, 1055, 3.7, 459, 5.1, 26.9, 38.2, 93, 0 }),
    ("Navy Beans, Dried", "1 lb.", 5.9, new double[] { 26.9, 1691, 11.4, 792, 0, 38.4, 24.6, 217, 0 }),
    ("Coffee", "1 lb.", 22.4, new double[] { 0, 0, 0, 0, 0, 4, 5.1, 50, 0 }),
    ("Tea", "1/4 lb.", 17.4, new double[] { 0, 0, 0, 0, 0, 0, 2.3, 42, 0 }),
    ("Cocoa", "8 oz.", 8.6, new double[] { 8.7, 237, 3, 72, 0, 2, 11.9, 40, 0 }),
    ("Chocolate", "8 oz.", 16.2, new double[] { 8.0, 77, 1.3, 39, 0, 0.9, 3.4, 14, 0 }),
    ("Sugar", "10 lb.", 51.7, new double[] { 34.9, 0, 0, 0, 0, 0, 0, 0, 0 }),
    ("Corn Syrup", "24 oz.", 13.7, new double[] { 14.7, 0, 0.5, 74, 0, 0, 0, 5, 0 }),
    ("Molasses", "18 oz.", 13.6, new double[] { 9.0, 0, 10.3, 244, 0, 1.9, 7.5, 146, 0 }),
    ("Strawberry Preserves", "1 lb.", 20.5, new double[] { 6.4, 11, 0.4, 7, 0.2, 0.2, 0.4, 3, 0 })
};

声明 LP 求解器

以下代码会实例化 MPsolver 封装容器。

Python

# Instantiate a Glop solver and naming it.
solver = pywraplp.Solver.CreateSolver("GLOP")
if not solver:
    return

C++

// Create the linear solver with the GLOP backend.
std::unique_ptr<MPSolver> solver(MPSolver::CreateSolver("GLOP"));

Java

// Create the linear solver with the GLOP backend.
MPSolver solver = MPSolver.createSolver("GLOP");
if (solver == null) {
  System.out.println("Could not create solver GLOP");
  return;
}

C#

// Create the linear solver with the GLOP backend.
Solver solver = Solver.CreateSolver("GLOP");
if (solver is null)
{
    return;
}

创建变量

以下代码会创建问题的变量。

Python

# Declare an array to hold our variables.
foods = [solver.NumVar(0.0, solver.infinity(), item[0]) for item in data]

print("Number of variables =", solver.NumVariables())

C++

std::vector<MPVariable*> foods;
const double infinity = solver->infinity();
for (const Commodity& commodity : data) {
  foods.push_back(solver->MakeNumVar(0.0, infinity, commodity.name));
}
LOG(INFO) << "Number of variables = " << solver->NumVariables();

Java

double infinity = java.lang.Double.POSITIVE_INFINITY;
List<MPVariable> foods = new ArrayList<>();
for (int i = 0; i < data.size(); ++i) {
  foods.add(solver.makeNumVar(0.0, infinity, (String) data.get(i)[0]));
}
System.out.println("Number of variables = " + solver.numVariables());

C#

List<Variable> foods = new List<Variable>();
for (int i = 0; i < data.Length; ++i)
{
    foods.Add(solver.MakeNumVar(0.0, double.PositiveInfinity, data[i].Name));
}
Console.WriteLine($"Number of variables = {solver.NumVariables()}");

MakeNumVar 方法会为表的每一行创建一个变量 food[i]。如前所述，营养数据是按美元计算的，因此 food[i] 是为商品支出的金额 i。

定义限制条件

Stigler 饮食的限制要求所有食物提供的营养素总量至少达到每种营养素的最低要求。接下来，我们将这些约束条件编写为涉及数组 data 和 nutrients 以及变量 food[i] 的不等式。

首先，食物 j 每一美元提供的营养量 i 为 data[j][i+3]（由于营养数据从 data 的第四列开始，因此我们在列索引中添加了 3）。由于食物 j 的消费金额为 food[j]，因此食物 j 提供的营养量 i 为\(data[j][i+3] \cdot food[j]\)。最后，由于营养成分 i 的最低要求为 nutrients[i][1]，因此我们可以按以下方式编写限制条件 i：

\( \sum_{j} data[j][i+3] \cdot food[j] \geq nutrients[i][1] \;\;\;\;\; (1) \)

以下代码定义了这些约束条件。

Python

# Create the constraints, one per nutrient.
constraints = []
for i, nutrient in enumerate(nutrients):
    constraints.append(solver.Constraint(nutrient[1], solver.infinity()))
    for j, item in enumerate(data):
        constraints[i].SetCoefficient(foods[j], item[i + 3])

print("Number of constraints =", solver.NumConstraints())

C++

// Create the constraints, one per nutrient.
std::vector<MPConstraint*> constraints;
for (std::size_t i = 0; i < nutrients.size(); ++i) {
  constraints.push_back(
      solver->MakeRowConstraint(nutrients[i].second, infinity));
  for (std::size_t j = 0; j < data.size(); ++j) {
    constraints.back()->SetCoefficient(foods[j], data[j].nutrients[i]);
  }
}
LOG(INFO) << "Number of constraints = " << solver->NumConstraints();

Java

MPConstraint[] constraints = new MPConstraint[nutrients.size()];
for (int i = 0; i < nutrients.size(); ++i) {
  constraints[i] = solver.makeConstraint(
      (double) nutrients.get(i)[1], infinity, (String) nutrients.get(i)[0]);
  for (int j = 0; j < data.size(); ++j) {
    constraints[i].setCoefficient(foods.get(j), ((double[]) data.get(j)[3])[i]);
  }
  // constraints.add(constraint);
}
System.out.println("Number of constraints = " + solver.numConstraints());

C#

List<Constraint> constraints = new List<Constraint>();
for (int i = 0; i < nutrients.Length; ++i)
{
    Constraint constraint =
        solver.MakeConstraint(nutrients[i].Value, double.PositiveInfinity, nutrients[i].Name);
    for (int j = 0; j < data.Length; ++j)
    {
        constraint.SetCoefficient(foods[j], data[j].Nutrients[i]);
    }
    constraints.Add(constraint);
}
Console.WriteLine($"Number of constraints = {solver.NumConstraints()}");

Python 方法 Constraint（对应于 C++ 方法 MakeRowConstraint）会为问题创建约束条件。对于每个 i，constraint(nutrients[i][1], solver.infinity)

这样就会创建一个约束条件，其中变量 food[j]（接下来定义）的线性组合大于或等于 nutrients[i][1]。线性表达式的系数由 SetCoefficient 方法定义，如下所示：SetCoefficient(food[j], data[j][i+3]

这会将 food[j] 的系数设置为 data[j][i+3]。

综上所述，代码定义了上面 (1) 中表示的约束条件。

创建目标

以下代码定义了问题的目标函数。

Python

# Objective function: Minimize the sum of (price-normalized) foods.
objective = solver.Objective()
for food in foods:
    objective.SetCoefficient(food, 1)
objective.SetMinimization()

C++

MPObjective* const objective = solver->MutableObjective();
for (size_t i = 0; i < data.size(); ++i) {
  objective->SetCoefficient(foods[i], 1);
}
objective->SetMinimization();

Java

MPObjective objective = solver.objective();
for (int i = 0; i < data.size(); ++i) {
  objective.setCoefficient(foods.get(i), 1);
}
objective.setMinimization();

C#

Objective objective = solver.Objective();
for (int i = 0; i < data.Length; ++i)
{
    objective.SetCoefficient(foods[i], 1);
}
objective.SetMinimization();

目标函数是食物的总费用，即变量 food[i] 的总和。

SetCoefficient 方法可设置目标函数的系数，在本例中为 1。最后，SetMinimization 声明这属于最小化问题。

调用求解器

以下代码会调用求解器。

Python

print(f"Solving with {solver.SolverVersion()}")
status = solver.Solve()

C++

const MPSolver::ResultStatus result_status = solver->Solve();

Java

final MPSolver.ResultStatus resultStatus = solver.solve();

C#

Solver.ResultStatus resultStatus = solver.Solve();

Glop 可在普通计算机上以不到 300 毫秒的速度解决问题：

显示解决方案

以下代码将显示解决方案。

Python

# Check that the problem has an optimal solution.
if status != solver.OPTIMAL:
    print("The problem does not have an optimal solution!")
    if status == solver.FEASIBLE:
        print("A potentially suboptimal solution was found.")
    else:
        print("The solver could not solve the problem.")
        exit(1)

# Display the amounts (in dollars) to purchase of each food.
nutrients_result = [0] * len(nutrients)
print("\nAnnual Foods:")
for i, food in enumerate(foods):
    if food.solution_value() > 0.0:
        print("{}: ${}".format(data[i][0], 365.0 * food.solution_value()))
        for j, _ in enumerate(nutrients):
            nutrients_result[j] += data[i][j + 3] * food.solution_value()
print("\nOptimal annual price: ${:.4f}".format(365.0 * objective.Value()))

print("\nNutrients per day:")
for i, nutrient in enumerate(nutrients):
    print(
        "{}: {:.2f} (min {})".format(nutrient[0], nutrients_result[i], nutrient[1])
    )

C++

// Check that the problem has an optimal solution.
if (result_status != MPSolver::OPTIMAL) {
  LOG(INFO) << "The problem does not have an optimal solution!";
  if (result_status == MPSolver::FEASIBLE) {
    LOG(INFO) << "A potentially suboptimal solution was found";
  } else {
    LOG(INFO) << "The solver could not solve the problem.";
    return;
  }
}

std::vector<double> nutrients_result(nutrients.size());
LOG(INFO) << "";
LOG(INFO) << "Annual Foods:";
for (std::size_t i = 0; i < data.size(); ++i) {
  if (foods[i]->solution_value() > 0.0) {
    LOG(INFO) << data[i].name << ": $"
              << std::to_string(365. * foods[i]->solution_value());
    for (std::size_t j = 0; j < nutrients.size(); ++j) {
      nutrients_result[j] +=
          data[i].nutrients[j] * foods[i]->solution_value();
    }
  }
}
LOG(INFO) << "";
LOG(INFO) << "Optimal annual price: $"
          << std::to_string(365. * objective->Value());
LOG(INFO) << "";
LOG(INFO) << "Nutrients per day:";
for (std::size_t i = 0; i < nutrients.size(); ++i) {
  LOG(INFO) << nutrients[i].first << ": "
            << std::to_string(nutrients_result[i]) << " (min "
            << std::to_string(nutrients[i].second) << ")";
}

Java

// Check that the problem has an optimal solution.
if (resultStatus != MPSolver.ResultStatus.OPTIMAL) {
  System.err.println("The problem does not have an optimal solution!");
  if (resultStatus == MPSolver.ResultStatus.FEASIBLE) {
    System.err.println("A potentially suboptimal solution was found.");
  } else {
    System.err.println("The solver could not solve the problem.");
    return;
  }
}

// Display the amounts (in dollars) to purchase of each food.
double[] nutrientsResult = new double[nutrients.size()];
System.out.println("\nAnnual Foods:");
for (int i = 0; i < foods.size(); ++i) {
  if (foods.get(i).solutionValue() > 0.0) {
    System.out.println((String) data.get(i)[0] + ": $" + 365 * foods.get(i).solutionValue());
    for (int j = 0; j < nutrients.size(); ++j) {
      nutrientsResult[j] += ((double[]) data.get(i)[3])[j] * foods.get(i).solutionValue();
    }
  }
}
System.out.println("\nOptimal annual price: $" + 365 * objective.value());

System.out.println("\nNutrients per day:");
for (int i = 0; i < nutrients.size(); ++i) {
  System.out.println(
      nutrients.get(i)[0] + ": " + nutrientsResult[i] + " (min " + nutrients.get(i)[1] + ")");
}

C#

// Check that the problem has an optimal solution.
if (resultStatus != Solver.ResultStatus.OPTIMAL)
{
    Console.WriteLine("The problem does not have an optimal solution!");
    if (resultStatus == Solver.ResultStatus.FEASIBLE)
    {
        Console.WriteLine("A potentially suboptimal solution was found.");
    }
    else
    {
        Console.WriteLine("The solver could not solve the problem.");
        return;
    }
}

// Display the amounts (in dollars) to purchase of each food.
double[] nutrientsResult = new double[nutrients.Length];
Console.WriteLine("\nAnnual Foods:");
for (int i = 0; i < foods.Count; ++i)
{
    if (foods[i].SolutionValue() > 0.0)
    {
        Console.WriteLine($"{data[i].Name}: ${365 * foods[i].SolutionValue():N2}");
        for (int j = 0; j < nutrients.Length; ++j)
        {
            nutrientsResult[j] += data[i].Nutrients[j] * foods[i].SolutionValue();
        }
    }
}
Console.WriteLine($"\nOptimal annual price: ${365 * objective.Value():N2}");

Console.WriteLine("\nNutrients per day:");
for (int i = 0; i < nutrients.Length; ++i)
{
    Console.WriteLine($"{nutrients[i].Name}: {nutrientsResult[i]:N2} (min {nutrients[i].Value})");
}

以下是程序的输出。

make rpy_stigler_diet
"/usr/bin/python3.11" ortools/linear_solver/samples/stigler_diet.py
Number of variables = 77
Number of constraints = 9

Annual Foods:
Wheat Flour (Enriched): $10.774457511918223
Liver (Beef): $0.6907834111074193
Cabbage: $4.093268864842877
Spinach: $1.8277960703546996
Navy Beans, Dried: $22.275425687243036

Optimal annual price: $39.6617

Nutrients per day:
Calories (kcal): 3.00 (min 3)
Protein (g): 147.41 (min 70)
Calcium (g): 0.80 (min 0.8)
Iron (mg): 60.47 (min 12)
Vitamin A (KIU): 5.00 (min 5)
Vitamin B1 (mg): 4.12 (min 1.8)
Vitamin B2 (mg): 2.70 (min 2.7)
Niacin (mg): 27.32 (min 18)
Vitamin C (mg): 75.00 (min 75)

Advanced usage:
Problem solved in  1  milliseconds
Problem solved in  14  iterations

程序的完整代码

Stigler 饮食计划的完整代码如下所示。

Python

"""The Stigler diet problem.

A description of the problem can be found here:
https://en.wikipedia.org/wiki/Stigler_diet.
"""
from ortools.linear_solver import pywraplp


def main():
    """Entry point of the program."""
    # Instantiate the data problem.
    # Nutrient minimums.
    nutrients = [
        ["Calories (kcal)", 3],
        ["Protein (g)", 70],
        ["Calcium (g)", 0.8],
        ["Iron (mg)", 12],
        ["Vitamin A (KIU)", 5],
        ["Vitamin B1 (mg)", 1.8],
        ["Vitamin B2 (mg)", 2.7],
        ["Niacin (mg)", 18],
        ["Vitamin C (mg)", 75],
    ]

    # Commodity, Unit, 1939 price (cents), Calories (kcal), Protein (g),
    # Calcium (g), Iron (mg), Vitamin A (KIU), Vitamin B1 (mg), Vitamin B2 (mg),
    # Niacin (mg), Vitamin C (mg)
    data = [
        # fmt: off
      ['Wheat Flour (Enriched)', '10 lb.', 36, 44.7, 1411, 2, 365, 0, 55.4, 33.3, 441, 0],
      ['Macaroni', '1 lb.', 14.1, 11.6, 418, 0.7, 54, 0, 3.2, 1.9, 68, 0],
      ['Wheat Cereal (Enriched)', '28 oz.', 24.2, 11.8, 377, 14.4, 175, 0, 14.4, 8.8, 114, 0],
      ['Corn Flakes', '8 oz.', 7.1, 11.4, 252, 0.1, 56, 0, 13.5, 2.3, 68, 0],
      ['Corn Meal', '1 lb.', 4.6, 36.0, 897, 1.7, 99, 30.9, 17.4, 7.9, 106, 0],
      ['Hominy Grits', '24 oz.', 8.5, 28.6, 680, 0.8, 80, 0, 10.6, 1.6, 110, 0],
      ['Rice', '1 lb.', 7.5, 21.2, 460, 0.6, 41, 0, 2, 4.8, 60, 0],
      ['Rolled Oats', '1 lb.', 7.1, 25.3, 907, 5.1, 341, 0, 37.1, 8.9, 64, 0],
      ['White Bread (Enriched)', '1 lb.', 7.9, 15.0, 488, 2.5, 115, 0, 13.8, 8.5, 126, 0],
      ['Whole Wheat Bread', '1 lb.', 9.1, 12.2, 484, 2.7, 125, 0, 13.9, 6.4, 160, 0],
      ['Rye Bread', '1 lb.', 9.1, 12.4, 439, 1.1, 82, 0, 9.9, 3, 66, 0],
      ['Pound Cake', '1 lb.', 24.8, 8.0, 130, 0.4, 31, 18.9, 2.8, 3, 17, 0],
      ['Soda Crackers', '1 lb.', 15.1, 12.5, 288, 0.5, 50, 0, 0, 0, 0, 0],
      ['Milk', '1 qt.', 11, 6.1, 310, 10.5, 18, 16.8, 4, 16, 7, 177],
      ['Evaporated Milk (can)', '14.5 oz.', 6.7, 8.4, 422, 15.1, 9, 26, 3, 23.5, 11, 60],
      ['Butter', '1 lb.', 30.8, 10.8, 9, 0.2, 3, 44.2, 0, 0.2, 2, 0],
      ['Oleomargarine', '1 lb.', 16.1, 20.6, 17, 0.6, 6, 55.8, 0.2, 0, 0, 0],
      ['Eggs', '1 doz.', 32.6, 2.9, 238, 1.0, 52, 18.6, 2.8, 6.5, 1, 0],
      ['Cheese (Cheddar)', '1 lb.', 24.2, 7.4, 448, 16.4, 19, 28.1, 0.8, 10.3, 4, 0],
      ['Cream', '1/2 pt.', 14.1, 3.5, 49, 1.7, 3, 16.9, 0.6, 2.5, 0, 17],
      ['Peanut Butter', '1 lb.', 17.9, 15.7, 661, 1.0, 48, 0, 9.6, 8.1, 471, 0],
      ['Mayonnaise', '1/2 pt.', 16.7, 8.6, 18, 0.2, 8, 2.7, 0.4, 0.5, 0, 0],
      ['Crisco', '1 lb.', 20.3, 20.1, 0, 0, 0, 0, 0, 0, 0, 0],
      ['Lard', '1 lb.', 9.8, 41.7, 0, 0, 0, 0.2, 0, 0.5, 5, 0],
      ['Sirloin Steak', '1 lb.', 39.6, 2.9, 166, 0.1, 34, 0.2, 2.1, 2.9, 69, 0],
      ['Round Steak', '1 lb.', 36.4, 2.2, 214, 0.1, 32, 0.4, 2.5, 2.4, 87, 0],
      ['Rib Roast', '1 lb.', 29.2, 3.4, 213, 0.1, 33, 0, 0, 2, 0, 0],
      ['Chuck Roast', '1 lb.', 22.6, 3.6, 309, 0.2, 46, 0.4, 1, 4, 120, 0],
      ['Plate', '1 lb.', 14.6, 8.5, 404, 0.2, 62, 0, 0.9, 0, 0, 0],
      ['Liver (Beef)', '1 lb.', 26.8, 2.2, 333, 0.2, 139, 169.2, 6.4, 50.8, 316, 525],
      ['Leg of Lamb', '1 lb.', 27.6, 3.1, 245, 0.1, 20, 0, 2.8, 3.9, 86, 0],
      ['Lamb Chops (Rib)', '1 lb.', 36.6, 3.3, 140, 0.1, 15, 0, 1.7, 2.7, 54, 0],
      ['Pork Chops', '1 lb.', 30.7, 3.5, 196, 0.2, 30, 0, 17.4, 2.7, 60, 0],
      ['Pork Loin Roast', '1 lb.', 24.2, 4.4, 249, 0.3, 37, 0, 18.2, 3.6, 79, 0],
      ['Bacon', '1 lb.', 25.6, 10.4, 152, 0.2, 23, 0, 1.8, 1.8, 71, 0],
      ['Ham, smoked', '1 lb.', 27.4, 6.7, 212, 0.2, 31, 0, 9.9, 3.3, 50, 0],
      ['Salt Pork', '1 lb.', 16, 18.8, 164, 0.1, 26, 0, 1.4, 1.8, 0, 0],
      ['Roasting Chicken', '1 lb.', 30.3, 1.8, 184, 0.1, 30, 0.1, 0.9, 1.8, 68, 46],
      ['Veal Cutlets', '1 lb.', 42.3, 1.7, 156, 0.1, 24, 0, 1.4, 2.4, 57, 0],
      ['Salmon, Pink (can)', '16 oz.', 13, 5.8, 705, 6.8, 45, 3.5, 1, 4.9, 209, 0],
      ['Apples', '1 lb.', 4.4, 5.8, 27, 0.5, 36, 7.3, 3.6, 2.7, 5, 544],
      ['Bananas', '1 lb.', 6.1, 4.9, 60, 0.4, 30, 17.4, 2.5, 3.5, 28, 498],
      ['Lemons', '1 doz.', 26, 1.0, 21, 0.5, 14, 0, 0.5, 0, 4, 952],
      ['Oranges', '1 doz.', 30.9, 2.2, 40, 1.1, 18, 11.1, 3.6, 1.3, 10, 1998],
      ['Green Beans', '1 lb.', 7.1, 2.4, 138, 3.7, 80, 69, 4.3, 5.8, 37, 862],
      ['Cabbage', '1 lb.', 3.7, 2.6, 125, 4.0, 36, 7.2, 9, 4.5, 26, 5369],
      ['Carrots', '1 bunch', 4.7, 2.7, 73, 2.8, 43, 188.5, 6.1, 4.3, 89, 608],
      ['Celery', '1 stalk', 7.3, 0.9, 51, 3.0, 23, 0.9, 1.4, 1.4, 9, 313],
      ['Lettuce', '1 head', 8.2, 0.4, 27, 1.1, 22, 112.4, 1.8, 3.4, 11, 449],
      ['Onions', '1 lb.', 3.6, 5.8, 166, 3.8, 59, 16.6, 4.7, 5.9, 21, 1184],
      ['Potatoes', '15 lb.', 34, 14.3, 336, 1.8, 118, 6.7, 29.4, 7.1, 198, 2522],
      ['Spinach', '1 lb.', 8.1, 1.1, 106, 0, 138, 918.4, 5.7, 13.8, 33, 2755],
      ['Sweet Potatoes', '1 lb.', 5.1, 9.6, 138, 2.7, 54, 290.7, 8.4, 5.4, 83, 1912],
      ['Peaches (can)', 'No. 2 1/2', 16.8, 3.7, 20, 0.4, 10, 21.5, 0.5, 1, 31, 196],
      ['Pears (can)', 'No. 2 1/2', 20.4, 3.0, 8, 0.3, 8, 0.8, 0.8, 0.8, 5, 81],
      ['Pineapple (can)', 'No. 2 1/2', 21.3, 2.4, 16, 0.4, 8, 2, 2.8, 0.8, 7, 399],
      ['Asparagus (can)', 'No. 2', 27.7, 0.4, 33, 0.3, 12, 16.3, 1.4, 2.1, 17, 272],
      ['Green Beans (can)', 'No. 2', 10, 1.0, 54, 2, 65, 53.9, 1.6, 4.3, 32, 431],
      ['Pork and Beans (can)', '16 oz.', 7.1, 7.5, 364, 4, 134, 3.5, 8.3, 7.7, 56, 0],
      ['Corn (can)', 'No. 2', 10.4, 5.2, 136, 0.2, 16, 12, 1.6, 2.7, 42, 218],
      ['Peas (can)', 'No. 2', 13.8, 2.3, 136, 0.6, 45, 34.9, 4.9, 2.5, 37, 370],
      ['Tomatoes (can)', 'No. 2', 8.6, 1.3, 63, 0.7, 38, 53.2, 3.4, 2.5, 36, 1253],
      ['Tomato Soup (can)', '10 1/2 oz.', 7.6, 1.6, 71, 0.6, 43, 57.9, 3.5, 2.4, 67, 862],
      ['Peaches, Dried', '1 lb.', 15.7, 8.5, 87, 1.7, 173, 86.8, 1.2, 4.3, 55, 57],
      ['Prunes, Dried', '1 lb.', 9, 12.8, 99, 2.5, 154, 85.7, 3.9, 4.3, 65, 257],
      ['Raisins, Dried', '15 oz.', 9.4, 13.5, 104, 2.5, 136, 4.5, 6.3, 1.4, 24, 136],
      ['Peas, Dried', '1 lb.', 7.9, 20.0, 1367, 4.2, 345, 2.9, 28.7, 18.4, 162, 0],
      ['Lima Beans, Dried', '1 lb.', 8.9, 17.4, 1055, 3.7, 459, 5.1, 26.9, 38.2, 93, 0],
      ['Navy Beans, Dried', '1 lb.', 5.9, 26.9, 1691, 11.4, 792, 0, 38.4, 24.6, 217, 0],
      ['Coffee', '1 lb.', 22.4, 0, 0, 0, 0, 0, 4, 5.1, 50, 0],
      ['Tea', '1/4 lb.', 17.4, 0, 0, 0, 0, 0, 0, 2.3, 42, 0],
      ['Cocoa', '8 oz.', 8.6, 8.7, 237, 3, 72, 0, 2, 11.9, 40, 0],
      ['Chocolate', '8 oz.', 16.2, 8.0, 77, 1.3, 39, 0, 0.9, 3.4, 14, 0],
      ['Sugar', '10 lb.', 51.7, 34.9, 0, 0, 0, 0, 0, 0, 0, 0],
      ['Corn Syrup', '24 oz.', 13.7, 14.7, 0, 0.5, 74, 0, 0, 0, 5, 0],
      ['Molasses', '18 oz.', 13.6, 9.0, 0, 10.3, 244, 0, 1.9, 7.5, 146, 0],
      ['Strawberry Preserves', '1 lb.', 20.5, 6.4, 11, 0.4, 7, 0.2, 0.2, 0.4, 3, 0],
        # fmt: on
    ]

    # Instantiate a Glop solver and naming it.
    solver = pywraplp.Solver.CreateSolver("GLOP")
    if not solver:
        return

    # Declare an array to hold our variables.
    foods = [solver.NumVar(0.0, solver.infinity(), item[0]) for item in data]

    print("Number of variables =", solver.NumVariables())

    # Create the constraints, one per nutrient.
    constraints = []
    for i, nutrient in enumerate(nutrients):
        constraints.append(solver.Constraint(nutrient[1], solver.infinity()))
        for j, item in enumerate(data):
            constraints[i].SetCoefficient(foods[j], item[i + 3])

    print("Number of constraints =", solver.NumConstraints())

    # Objective function: Minimize the sum of (price-normalized) foods.
    objective = solver.Objective()
    for food in foods:
        objective.SetCoefficient(food, 1)
    objective.SetMinimization()

    print(f"Solving with {solver.SolverVersion()}")
    status = solver.Solve()

    # Check that the problem has an optimal solution.
    if status != solver.OPTIMAL:
        print("The problem does not have an optimal solution!")
        if status == solver.FEASIBLE:
            print("A potentially suboptimal solution was found.")
        else:
            print("The solver could not solve the problem.")
            exit(1)

    # Display the amounts (in dollars) to purchase of each food.
    nutrients_result = [0] * len(nutrients)
    print("\nAnnual Foods:")
    for i, food in enumerate(foods):
        if food.solution_value() > 0.0:
            print("{}: ${}".format(data[i][0], 365.0 * food.solution_value()))
            for j, _ in enumerate(nutrients):
                nutrients_result[j] += data[i][j + 3] * food.solution_value()
    print("\nOptimal annual price: ${:.4f}".format(365.0 * objective.Value()))

    print("\nNutrients per day:")
    for i, nutrient in enumerate(nutrients):
        print(
            "{}: {:.2f} (min {})".format(nutrient[0], nutrients_result[i], nutrient[1])
        )

    print("\nAdvanced usage:")
    print(f"Problem solved in {solver.wall_time():d} milliseconds")
    print(f"Problem solved in {solver.iterations():d} iterations")


if __name__ == "__main__":
    main()

C++

// The Stigler diet problem.
#include <array>
#include <memory>
#include <string>
#include <utility>  // std::pair
#include <vector>

#include "absl/flags/flag.h"
#include "absl/log/flags.h"
#include "ortools/base/init_google.h"
#include "ortools/base/logging.h"
#include "ortools/linear_solver/linear_solver.h"

namespace operations_research {
void StiglerDiet() {
  // Nutrient minimums.
  const std::vector<std::pair<std::string, double>> nutrients = {
      {"Calories (kcal)", 3.0}, {"Protein (g)", 70.0},
      {"Calcium (g)", 0.8},     {"Iron (mg)", 12.0},
      {"Vitamin A (kIU)", 5.0}, {"Vitamin B1 (mg)", 1.8},
      {"Vitamin B2 (mg)", 2.7}, {"Niacin (mg)", 18.0},
      {"Vitamin C (mg)", 75.0}};

  struct Commodity {
    std::string name;  //!< Commodity name
    std::string unit;  //!< Unit
    double price;      //!< 1939 price per unit (cents)
    //! Calories (kcal),
    //! Protein (g),
    //! Calcium (g),
    //! Iron (mg),
    //! Vitamin A (kIU),
    //! Vitamin B1 (mg),
    //! Vitamin B2 (mg),
    //! Niacin (mg),
    //! Vitamin C (mg)
    std::array<double, 9> nutrients;
  };

  std::vector<Commodity> data = {
      {"Wheat Flour (Enriched)",
       "10 lb.",
       36,
       {44.7, 1411, 2, 365, 0, 55.4, 33.3, 441, 0}},
      {"Macaroni", "1 lb.", 14.1, {11.6, 418, 0.7, 54, 0, 3.2, 1.9, 68, 0}},
      {"Wheat Cereal (Enriched)",
       "28 oz.",
       24.2,
       {11.8, 377, 14.4, 175, 0, 14.4, 8.8, 114, 0}},
      {"Corn Flakes", "8 oz.", 7.1, {11.4, 252, 0.1, 56, 0, 13.5, 2.3, 68, 0}},
      {"Corn Meal",
       "1 lb.",
       4.6,
       {36.0, 897, 1.7, 99, 30.9, 17.4, 7.9, 106, 0}},
      {"Hominy Grits",
       "24 oz.",
       8.5,
       {28.6, 680, 0.8, 80, 0, 10.6, 1.6, 110, 0}},
      {"Rice", "1 lb.", 7.5, {21.2, 460, 0.6, 41, 0, 2, 4.8, 60, 0}},
      {"Rolled Oats", "1 lb.", 7.1, {25.3, 907, 5.1, 341, 0, 37.1, 8.9, 64, 0}},
      {"White Bread (Enriched)",
       "1 lb.",
       7.9,
       {15.0, 488, 2.5, 115, 0, 13.8, 8.5, 126, 0}},
      {"Whole Wheat Bread",
       "1 lb.",
       9.1,
       {12.2, 484, 2.7, 125, 0, 13.9, 6.4, 160, 0}},
      {"Rye Bread", "1 lb.", 9.1, {12.4, 439, 1.1, 82, 0, 9.9, 3, 66, 0}},
      {"Pound Cake", "1 lb.", 24.8, {8.0, 130, 0.4, 31, 18.9, 2.8, 3, 17, 0}},
      {"Soda Crackers", "1 lb.", 15.1, {12.5, 288, 0.5, 50, 0, 0, 0, 0, 0}},
      {"Milk", "1 qt.", 11, {6.1, 310, 10.5, 18, 16.8, 4, 16, 7, 177}},
      {"Evaporated Milk (can)",
       "14.5 oz.",
       6.7,
       {8.4, 422, 15.1, 9, 26, 3, 23.5, 11, 60}},
      {"Butter", "1 lb.", 30.8, {10.8, 9, 0.2, 3, 44.2, 0, 0.2, 2, 0}},
      {"Oleomargarine", "1 lb.", 16.1, {20.6, 17, 0.6, 6, 55.8, 0.2, 0, 0, 0}},
      {"Eggs", "1 doz.", 32.6, {2.9, 238, 1.0, 52, 18.6, 2.8, 6.5, 1, 0}},
      {"Cheese (Cheddar)",
       "1 lb.",
       24.2,
       {7.4, 448, 16.4, 19, 28.1, 0.8, 10.3, 4, 0}},
      {"Cream", "1/2 pt.", 14.1, {3.5, 49, 1.7, 3, 16.9, 0.6, 2.5, 0, 17}},
      {"Peanut Butter",
       "1 lb.",
       17.9,
       {15.7, 661, 1.0, 48, 0, 9.6, 8.1, 471, 0}},
      {"Mayonnaise", "1/2 pt.", 16.7, {8.6, 18, 0.2, 8, 2.7, 0.4, 0.5, 0, 0}},
      {"Crisco", "1 lb.", 20.3, {20.1, 0, 0, 0, 0, 0, 0, 0, 0}},
      {"Lard", "1 lb.", 9.8, {41.7, 0, 0, 0, 0.2, 0, 0.5, 5, 0}},
      {"Sirloin Steak",
       "1 lb.",
       39.6,
       {2.9, 166, 0.1, 34, 0.2, 2.1, 2.9, 69, 0}},
      {"Round Steak", "1 lb.", 36.4, {2.2, 214, 0.1, 32, 0.4, 2.5, 2.4, 87, 0}},
      {"Rib Roast", "1 lb.", 29.2, {3.4, 213, 0.1, 33, 0, 0, 2, 0, 0}},
      {"Chuck Roast", "1 lb.", 22.6, {3.6, 309, 0.2, 46, 0.4, 1, 4, 120, 0}},
      {"Plate", "1 lb.", 14.6, {8.5, 404, 0.2, 62, 0, 0.9, 0, 0, 0}},
      {"Liver (Beef)",
       "1 lb.",
       26.8,
       {2.2, 333, 0.2, 139, 169.2, 6.4, 50.8, 316, 525}},
      {"Leg of Lamb", "1 lb.", 27.6, {3.1, 245, 0.1, 20, 0, 2.8, 3.9, 86, 0}},
      {"Lamb Chops (Rib)",
       "1 lb.",
       36.6,
       {3.3, 140, 0.1, 15, 0, 1.7, 2.7, 54, 0}},
      {"Pork Chops", "1 lb.", 30.7, {3.5, 196, 0.2, 30, 0, 17.4, 2.7, 60, 0}},
      {"Pork Loin Roast",
       "1 lb.",
       24.2,
       {4.4, 249, 0.3, 37, 0, 18.2, 3.6, 79, 0}},
      {"Bacon", "1 lb.", 25.6, {10.4, 152, 0.2, 23, 0, 1.8, 1.8, 71, 0}},
      {"Ham, smoked", "1 lb.", 27.4, {6.7, 212, 0.2, 31, 0, 9.9, 3.3, 50, 0}},
      {"Salt Pork", "1 lb.", 16, {18.8, 164, 0.1, 26, 0, 1.4, 1.8, 0, 0}},
      {"Roasting Chicken",
       "1 lb.",
       30.3,
       {1.8, 184, 0.1, 30, 0.1, 0.9, 1.8, 68, 46}},
      {"Veal Cutlets", "1 lb.", 42.3, {1.7, 156, 0.1, 24, 0, 1.4, 2.4, 57, 0}},
      {"Salmon, Pink (can)",
       "16 oz.",
       13,
       {5.8, 705, 6.8, 45, 3.5, 1, 4.9, 209, 0}},
      {"Apples", "1 lb.", 4.4, {5.8, 27, 0.5, 36, 7.3, 3.6, 2.7, 5, 544}},
      {"Bananas", "1 lb.", 6.1, {4.9, 60, 0.4, 30, 17.4, 2.5, 3.5, 28, 498}},
      {"Lemons", "1 doz.", 26, {1.0, 21, 0.5, 14, 0, 0.5, 0, 4, 952}},
      {"Oranges", "1 doz.", 30.9, {2.2, 40, 1.1, 18, 11.1, 3.6, 1.3, 10, 1998}},
      {"Green Beans", "1 lb.", 7.1, {2.4, 138, 3.7, 80, 69, 4.3, 5.8, 37, 862}},
      {"Cabbage", "1 lb.", 3.7, {2.6, 125, 4.0, 36, 7.2, 9, 4.5, 26, 5369}},
      {"Carrots", "1 bunch", 4.7, {2.7, 73, 2.8, 43, 188.5, 6.1, 4.3, 89, 608}},
      {"Celery", "1 stalk", 7.3, {0.9, 51, 3.0, 23, 0.9, 1.4, 1.4, 9, 313}},
      {"Lettuce", "1 head", 8.2, {0.4, 27, 1.1, 22, 112.4, 1.8, 3.4, 11, 449}},
      {"Onions", "1 lb.", 3.6, {5.8, 166, 3.8, 59, 16.6, 4.7, 5.9, 21, 1184}},
      {"Potatoes",
       "15 lb.",
       34,
       {14.3, 336, 1.8, 118, 6.7, 29.4, 7.1, 198, 2522}},
      {"Spinach", "1 lb.", 8.1, {1.1, 106, 0, 138, 918.4, 5.7, 13.8, 33, 2755}},
      {"Sweet Potatoes",
       "1 lb.",
       5.1,
       {9.6, 138, 2.7, 54, 290.7, 8.4, 5.4, 83, 1912}},
      {"Peaches (can)",
       "No. 2 1/2",
       16.8,
       {3.7, 20, 0.4, 10, 21.5, 0.5, 1, 31, 196}},
      {"Pears (can)",
       "No. 2 1/2",
       20.4,
       {3.0, 8, 0.3, 8, 0.8, 0.8, 0.8, 5, 81}},
      {"Pineapple (can)",
       "No. 2 1/2",
       21.3,
       {2.4, 16, 0.4, 8, 2, 2.8, 0.8, 7, 399}},
      {"Asparagus (can)",
       "No. 2",
       27.7,
       {0.4, 33, 0.3, 12, 16.3, 1.4, 2.1, 17, 272}},
      {"Green Beans (can)",
       "No. 2",
       10,
       {1.0, 54, 2, 65, 53.9, 1.6, 4.3, 32, 431}},
      {"Pork and Beans (can)",
       "16 oz.",
       7.1,
       {7.5, 364, 4, 134, 3.5, 8.3, 7.7, 56, 0}},
      {"Corn (can)", "No. 2", 10.4, {5.2, 136, 0.2, 16, 12, 1.6, 2.7, 42, 218}},
      {"Peas (can)",
       "No. 2",
       13.8,
       {2.3, 136, 0.6, 45, 34.9, 4.9, 2.5, 37, 370}},
      {"Tomatoes (can)",
       "No. 2",
       8.6,
       {1.3, 63, 0.7, 38, 53.2, 3.4, 2.5, 36, 1253}},
      {"Tomato Soup (can)",
       "10 1/2 oz.",
       7.6,
       {1.6, 71, 0.6, 43, 57.9, 3.5, 2.4, 67, 862}},
      {"Peaches, Dried",
       "1 lb.",
       15.7,
       {8.5, 87, 1.7, 173, 86.8, 1.2, 4.3, 55, 57}},
      {"Prunes, Dried",
       "1 lb.",
       9,
       {12.8, 99, 2.5, 154, 85.7, 3.9, 4.3, 65, 257}},
      {"Raisins, Dried",
       "15 oz.",
       9.4,
       {13.5, 104, 2.5, 136, 4.5, 6.3, 1.4, 24, 136}},
      {"Peas, Dried",
       "1 lb.",
       7.9,
       {20.0, 1367, 4.2, 345, 2.9, 28.7, 18.4, 162, 0}},
      {"Lima Beans, Dried",
       "1 lb.",
       8.9,
       {17.4, 1055, 3.7, 459, 5.1, 26.9, 38.2, 93, 0}},
      {"Navy Beans, Dried",
       "1 lb.",
       5.9,
       {26.9, 1691, 11.4, 792, 0, 38.4, 24.6, 217, 0}},
      {"Coffee", "1 lb.", 22.4, {0, 0, 0, 0, 0, 4, 5.1, 50, 0}},
      {"Tea", "1/4 lb.", 17.4, {0, 0, 0, 0, 0, 0, 2.3, 42, 0}},
      {"Cocoa", "8 oz.", 8.6, {8.7, 237, 3, 72, 0, 2, 11.9, 40, 0}},
      {"Chocolate", "8 oz.", 16.2, {8.0, 77, 1.3, 39, 0, 0.9, 3.4, 14, 0}},
      {"Sugar", "10 lb.", 51.7, {34.9, 0, 0, 0, 0, 0, 0, 0, 0}},
      {"Corn Syrup", "24 oz.", 13.7, {14.7, 0, 0.5, 74, 0, 0, 0, 5, 0}},
      {"Molasses", "18 oz.", 13.6, {9.0, 0, 10.3, 244, 0, 1.9, 7.5, 146, 0}},
      {"Strawberry Preserves",
       "1 lb.",
       20.5,
       {6.4, 11, 0.4, 7, 0.2, 0.2, 0.4, 3, 0}}};

  // Create the linear solver with the GLOP backend.
  std::unique_ptr<MPSolver> solver(MPSolver::CreateSolver("GLOP"));

  std::vector<MPVariable*> foods;
  const double infinity = solver->infinity();
  for (const Commodity& commodity : data) {
    foods.push_back(solver->MakeNumVar(0.0, infinity, commodity.name));
  }
  LOG(INFO) << "Number of variables = " << solver->NumVariables();

  // Create the constraints, one per nutrient.
  std::vector<MPConstraint*> constraints;
  for (std::size_t i = 0; i < nutrients.size(); ++i) {
    constraints.push_back(
        solver->MakeRowConstraint(nutrients[i].second, infinity));
    for (std::size_t j = 0; j < data.size(); ++j) {
      constraints.back()->SetCoefficient(foods[j], data[j].nutrients[i]);
    }
  }
  LOG(INFO) << "Number of constraints = " << solver->NumConstraints();

  MPObjective* const objective = solver->MutableObjective();
  for (size_t i = 0; i < data.size(); ++i) {
    objective->SetCoefficient(foods[i], 1);
  }
  objective->SetMinimization();

  const MPSolver::ResultStatus result_status = solver->Solve();

  // Check that the problem has an optimal solution.
  if (result_status != MPSolver::OPTIMAL) {
    LOG(INFO) << "The problem does not have an optimal solution!";
    if (result_status == MPSolver::FEASIBLE) {
      LOG(INFO) << "A potentially suboptimal solution was found";
    } else {
      LOG(INFO) << "The solver could not solve the problem.";
      return;
    }
  }

  std::vector<double> nutrients_result(nutrients.size());
  LOG(INFO) << "";
  LOG(INFO) << "Annual Foods:";
  for (std::size_t i = 0; i < data.size(); ++i) {
    if (foods[i]->solution_value() > 0.0) {
      LOG(INFO) << data[i].name << ": $"
                << std::to_string(365. * foods[i]->solution_value());
      for (std::size_t j = 0; j < nutrients.size(); ++j) {
        nutrients_result[j] +=
            data[i].nutrients[j] * foods[i]->solution_value();
      }
    }
  }
  LOG(INFO) << "";
  LOG(INFO) << "Optimal annual price: $"
            << std::to_string(365. * objective->Value());
  LOG(INFO) << "";
  LOG(INFO) << "Nutrients per day:";
  for (std::size_t i = 0; i < nutrients.size(); ++i) {
    LOG(INFO) << nutrients[i].first << ": "
              << std::to_string(nutrients_result[i]) << " (min "
              << std::to_string(nutrients[i].second) << ")";
  }

  LOG(INFO) << "";
  LOG(INFO) << "Advanced usage:";
  LOG(INFO) << "Problem solved in " << solver->wall_time() << " milliseconds";
  LOG(INFO) << "Problem solved in " << solver->iterations() << " iterations";
}
}  // namespace operations_research

int main(int argc, char** argv) {
  InitGoogle(argv[0], &argc, &argv, true);
  absl::SetFlag(&FLAGS_stderrthreshold, 0);
  operations_research::StiglerDiet();
  return EXIT_SUCCESS;
}

Java

// The Stigler diet problem.
package com.google.ortools.linearsolver.samples;
import com.google.ortools.Loader;
import com.google.ortools.linearsolver.MPConstraint;
import com.google.ortools.linearsolver.MPObjective;
import com.google.ortools.linearsolver.MPSolver;
import com.google.ortools.linearsolver.MPVariable;
import java.util.ArrayList;
import java.util.List;

/** Stigler diet example. */
public final class StiglerDiet {
  public static void main(String[] args) {
    Loader.loadNativeLibraries();
    // Nutrient minimums.
    List<Object[]> nutrients = new ArrayList<>();
    nutrients.add(new Object[] {"Calories (kcal)", 3.0});
    nutrients.add(new Object[] {"Protein (g)", 70.0});
    nutrients.add(new Object[] {"Calcium (g)", 0.8});
    nutrients.add(new Object[] {"Iron (mg)", 12.0});
    nutrients.add(new Object[] {"Vitamin A (kIU)", 5.0});
    nutrients.add(new Object[] {"Vitamin B1 (mg)", 1.8});
    nutrients.add(new Object[] {"Vitamin B2 (mg)", 2.7});
    nutrients.add(new Object[] {"Niacin (mg)", 18.0});
    nutrients.add(new Object[] {"Vitamin C (mg)", 75.0});

    List<Object[]> data = new ArrayList<>();
    data.add(new Object[] {"Wheat Flour (Enriched)", "10 lb.", 36,
        new double[] {44.7, 1411, 2, 365, 0, 55.4, 33.3, 441, 0}});
    data.add(new Object[] {
        "Macaroni", "1 lb.", 14.1, new double[] {11.6, 418, 0.7, 54, 0, 3.2, 1.9, 68, 0}});
    data.add(new Object[] {"Wheat Cereal (Enriched)", "28 oz.", 24.2,
        new double[] {11.8, 377, 14.4, 175, 0, 14.4, 8.8, 114, 0}});
    data.add(new Object[] {
        "Corn Flakes", "8 oz.", 7.1, new double[] {11.4, 252, 0.1, 56, 0, 13.5, 2.3, 68, 0}});
    data.add(new Object[] {
        "Corn Meal", "1 lb.", 4.6, new double[] {36.0, 897, 1.7, 99, 30.9, 17.4, 7.9, 106, 0}});
    data.add(new Object[] {
        "Hominy Grits", "24 oz.", 8.5, new double[] {28.6, 680, 0.8, 80, 0, 10.6, 1.6, 110, 0}});
    data.add(
        new Object[] {"Rice", "1 lb.", 7.5, new double[] {21.2, 460, 0.6, 41, 0, 2, 4.8, 60, 0}});
    data.add(new Object[] {
        "Rolled Oats", "1 lb.", 7.1, new double[] {25.3, 907, 5.1, 341, 0, 37.1, 8.9, 64, 0}});
    data.add(new Object[] {"White Bread (Enriched)", "1 lb.", 7.9,
        new double[] {15.0, 488, 2.5, 115, 0, 13.8, 8.5, 126, 0}});
    data.add(new Object[] {"Whole Wheat Bread", "1 lb.", 9.1,
        new double[] {12.2, 484, 2.7, 125, 0, 13.9, 6.4, 160, 0}});
    data.add(new Object[] {
        "Rye Bread", "1 lb.", 9.1, new double[] {12.4, 439, 1.1, 82, 0, 9.9, 3, 66, 0}});
    data.add(new Object[] {
        "Pound Cake", "1 lb.", 24.8, new double[] {8.0, 130, 0.4, 31, 18.9, 2.8, 3, 17, 0}});
    data.add(new Object[] {
        "Soda Crackers", "1 lb.", 15.1, new double[] {12.5, 288, 0.5, 50, 0, 0, 0, 0, 0}});
    data.add(
        new Object[] {"Milk", "1 qt.", 11, new double[] {6.1, 310, 10.5, 18, 16.8, 4, 16, 7, 177}});
    data.add(new Object[] {"Evaporated Milk (can)", "14.5 oz.", 6.7,
        new double[] {8.4, 422, 15.1, 9, 26, 3, 23.5, 11, 60}});
    data.add(
        new Object[] {"Butter", "1 lb.", 30.8, new double[] {10.8, 9, 0.2, 3, 44.2, 0, 0.2, 2, 0}});
    data.add(new Object[] {
        "Oleomargarine", "1 lb.", 16.1, new double[] {20.6, 17, 0.6, 6, 55.8, 0.2, 0, 0, 0}});
    data.add(new Object[] {
        "Eggs", "1 doz.", 32.6, new double[] {2.9, 238, 1.0, 52, 18.6, 2.8, 6.5, 1, 0}});
    data.add(new Object[] {"Cheese (Cheddar)", "1 lb.", 24.2,
        new double[] {7.4, 448, 16.4, 19, 28.1, 0.8, 10.3, 4, 0}});
    data.add(new Object[] {
        "Cream", "1/2 pt.", 14.1, new double[] {3.5, 49, 1.7, 3, 16.9, 0.6, 2.5, 0, 17}});
    data.add(new Object[] {
        "Peanut Butter", "1 lb.", 17.9, new double[] {15.7, 661, 1.0, 48, 0, 9.6, 8.1, 471, 0}});
    data.add(new Object[] {
        "Mayonnaise", "1/2 pt.", 16.7, new double[] {8.6, 18, 0.2, 8, 2.7, 0.4, 0.5, 0, 0}});
    data.add(new Object[] {"Crisco", "1 lb.", 20.3, new double[] {20.1, 0, 0, 0, 0, 0, 0, 0, 0}});
    data.add(new Object[] {"Lard", "1 lb.", 9.8, new double[] {41.7, 0, 0, 0, 0.2, 0, 0.5, 5, 0}});
    data.add(new Object[] {
        "Sirloin Steak", "1 lb.", 39.6, new double[] {2.9, 166, 0.1, 34, 0.2, 2.1, 2.9, 69, 0}});
    data.add(new Object[] {
        "Round Steak", "1 lb.", 36.4, new double[] {2.2, 214, 0.1, 32, 0.4, 2.5, 2.4, 87, 0}});
    data.add(
        new Object[] {"Rib Roast", "1 lb.", 29.2, new double[] {3.4, 213, 0.1, 33, 0, 0, 2, 0, 0}});
    data.add(new Object[] {
        "Chuck Roast", "1 lb.", 22.6, new double[] {3.6, 309, 0.2, 46, 0.4, 1, 4, 120, 0}});
    data.add(
        new Object[] {"Plate", "1 lb.", 14.6, new double[] {8.5, 404, 0.2, 62, 0, 0.9, 0, 0, 0}});
    data.add(new Object[] {"Liver (Beef)", "1 lb.", 26.8,
        new double[] {2.2, 333, 0.2, 139, 169.2, 6.4, 50.8, 316, 525}});
    data.add(new Object[] {
        "Leg of Lamb", "1 lb.", 27.6, new double[] {3.1, 245, 0.1, 20, 0, 2.8, 3.9, 86, 0}});
    data.add(new Object[] {
        "Lamb Chops (Rib)", "1 lb.", 36.6, new double[] {3.3, 140, 0.1, 15, 0, 1.7, 2.7, 54, 0}});
    data.add(new Object[] {
        "Pork Chops", "1 lb.", 30.7, new double[] {3.5, 196, 0.2, 30, 0, 17.4, 2.7, 60, 0}});
    data.add(new Object[] {
        "Pork Loin Roast", "1 lb.", 24.2, new double[] {4.4, 249, 0.3, 37, 0, 18.2, 3.6, 79, 0}});
    data.add(new Object[] {
        "Bacon", "1 lb.", 25.6, new double[] {10.4, 152, 0.2, 23, 0, 1.8, 1.8, 71, 0}});
    data.add(new Object[] {
        "Ham, smoked", "1 lb.", 27.4, new double[] {6.7, 212, 0.2, 31, 0, 9.9, 3.3, 50, 0}});
    data.add(new Object[] {
        "Salt Pork", "1 lb.", 16, new double[] {18.8, 164, 0.1, 26, 0, 1.4, 1.8, 0, 0}});
    data.add(new Object[] {"Roasting Chicken", "1 lb.", 30.3,
        new double[] {1.8, 184, 0.1, 30, 0.1, 0.9, 1.8, 68, 46}});
    data.add(new Object[] {
        "Veal Cutlets", "1 lb.", 42.3, new double[] {1.7, 156, 0.1, 24, 0, 1.4, 2.4, 57, 0}});
    data.add(new Object[] {
        "Salmon, Pink (can)", "16 oz.", 13, new double[] {5.8, 705, 6.8, 45, 3.5, 1, 4.9, 209, 0}});
    data.add(new Object[] {
        "Apples", "1 lb.", 4.4, new double[] {5.8, 27, 0.5, 36, 7.3, 3.6, 2.7, 5, 544}});
    data.add(new Object[] {
        "Bananas", "1 lb.", 6.1, new double[] {4.9, 60, 0.4, 30, 17.4, 2.5, 3.5, 28, 498}});
    data.add(
        new Object[] {"Lemons", "1 doz.", 26, new double[] {1.0, 21, 0.5, 14, 0, 0.5, 0, 4, 952}});
    data.add(new Object[] {
        "Oranges", "1 doz.", 30.9, new double[] {2.2, 40, 1.1, 18, 11.1, 3.6, 1.3, 10, 1998}});
    data.add(new Object[] {
        "Green Beans", "1 lb.", 7.1, new double[] {2.4, 138, 3.7, 80, 69, 4.3, 5.8, 37, 862}});
    data.add(new Object[] {
        "Cabbage", "1 lb.", 3.7, new double[] {2.6, 125, 4.0, 36, 7.2, 9, 4.5, 26, 5369}});
    data.add(new Object[] {
        "Carrots", "1 bunch", 4.7, new double[] {2.7, 73, 2.8, 43, 188.5, 6.1, 4.3, 89, 608}});
    data.add(new Object[] {
        "Celery", "1 stalk", 7.3, new double[] {0.9, 51, 3.0, 23, 0.9, 1.4, 1.4, 9, 313}});
    data.add(new Object[] {
        "Lettuce", "1 head", 8.2, new double[] {0.4, 27, 1.1, 22, 112.4, 1.8, 3.4, 11, 449}});
    data.add(new Object[] {
        "Onions", "1 lb.", 3.6, new double[] {5.8, 166, 3.8, 59, 16.6, 4.7, 5.9, 21, 1184}});
    data.add(new Object[] {
        "Potatoes", "15 lb.", 34, new double[] {14.3, 336, 1.8, 118, 6.7, 29.4, 7.1, 198, 2522}});
    data.add(new Object[] {
        "Spinach", "1 lb.", 8.1, new double[] {1.1, 106, 0, 138, 918.4, 5.7, 13.8, 33, 2755}});
    data.add(new Object[] {"Sweet Potatoes", "1 lb.", 5.1,
        new double[] {9.6, 138, 2.7, 54, 290.7, 8.4, 5.4, 83, 1912}});
    data.add(new Object[] {"Peaches (can)", "No. 2 1/2", 16.8,
        new double[] {3.7, 20, 0.4, 10, 21.5, 0.5, 1, 31, 196}});
    data.add(new Object[] {
        "Pears (can)", "No. 2 1/2", 20.4, new double[] {3.0, 8, 0.3, 8, 0.8, 0.8, 0.8, 5, 81}});
    data.add(new Object[] {
        "Pineapple (can)", "No. 2 1/2", 21.3, new double[] {2.4, 16, 0.4, 8, 2, 2.8, 0.8, 7, 399}});
    data.add(new Object[] {"Asparagus (can)", "No. 2", 27.7,
        new double[] {0.4, 33, 0.3, 12, 16.3, 1.4, 2.1, 17, 272}});
    data.add(new Object[] {
        "Green Beans (can)", "No. 2", 10, new double[] {1.0, 54, 2, 65, 53.9, 1.6, 4.3, 32, 431}});
    data.add(new Object[] {"Pork and Beans (can)", "16 oz.", 7.1,
        new double[] {7.5, 364, 4, 134, 3.5, 8.3, 7.7, 56, 0}});
    data.add(new Object[] {
        "Corn (can)", "No. 2", 10.4, new double[] {5.2, 136, 0.2, 16, 12, 1.6, 2.7, 42, 218}});
    data.add(new Object[] {
        "Peas (can)", "No. 2", 13.8, new double[] {2.3, 136, 0.6, 45, 34.9, 4.9, 2.5, 37, 370}});
    data.add(new Object[] {
        "Tomatoes (can)", "No. 2", 8.6, new double[] {1.3, 63, 0.7, 38, 53.2, 3.4, 2.5, 36, 1253}});
    data.add(new Object[] {"Tomato Soup (can)", "10 1/2 oz.", 7.6,
        new double[] {1.6, 71, 0.6, 43, 57.9, 3.5, 2.4, 67, 862}});
    data.add(new Object[] {
        "Peaches, Dried", "1 lb.", 15.7, new double[] {8.5, 87, 1.7, 173, 86.8, 1.2, 4.3, 55, 57}});
    data.add(new Object[] {
        "Prunes, Dried", "1 lb.", 9, new double[] {12.8, 99, 2.5, 154, 85.7, 3.9, 4.3, 65, 257}});
    data.add(new Object[] {"Raisins, Dried", "15 oz.", 9.4,
        new double[] {13.5, 104, 2.5, 136, 4.5, 6.3, 1.4, 24, 136}});
    data.add(new Object[] {
        "Peas, Dried", "1 lb.", 7.9, new double[] {20.0, 1367, 4.2, 345, 2.9, 28.7, 18.4, 162, 0}});
    data.add(new Object[] {"Lima Beans, Dried", "1 lb.", 8.9,
        new double[] {17.4, 1055, 3.7, 459, 5.1, 26.9, 38.2, 93, 0}});
    data.add(new Object[] {"Navy Beans, Dried", "1 lb.", 5.9,
        new double[] {26.9, 1691, 11.4, 792, 0, 38.4, 24.6, 217, 0}});
    data.add(new Object[] {"Coffee", "1 lb.", 22.4, new double[] {0, 0, 0, 0, 0, 4, 5.1, 50, 0}});
    data.add(new Object[] {"Tea", "1/4 lb.", 17.4, new double[] {0, 0, 0, 0, 0, 0, 2.3, 42, 0}});
    data.add(
        new Object[] {"Cocoa", "8 oz.", 8.6, new double[] {8.7, 237, 3, 72, 0, 2, 11.9, 40, 0}});
    data.add(new Object[] {
        "Chocolate", "8 oz.", 16.2, new double[] {8.0, 77, 1.3, 39, 0, 0.9, 3.4, 14, 0}});
    data.add(new Object[] {"Sugar", "10 lb.", 51.7, new double[] {34.9, 0, 0, 0, 0, 0, 0, 0, 0}});
    data.add(new Object[] {
        "Corn Syrup", "24 oz.", 13.7, new double[] {14.7, 0, 0.5, 74, 0, 0, 0, 5, 0}});
    data.add(new Object[] {
        "Molasses", "18 oz.", 13.6, new double[] {9.0, 0, 10.3, 244, 0, 1.9, 7.5, 146, 0}});
    data.add(new Object[] {"Strawberry Preserves", "1 lb.", 20.5,
        new double[] {6.4, 11, 0.4, 7, 0.2, 0.2, 0.4, 3, 0}});


    // Create the linear solver with the GLOP backend.
    MPSolver solver = MPSolver.createSolver("GLOP");
    if (solver == null) {
      System.out.println("Could not create solver GLOP");
      return;
    }

    double infinity = java.lang.Double.POSITIVE_INFINITY;
    List<MPVariable> foods = new ArrayList<>();
    for (int i = 0; i < data.size(); ++i) {
      foods.add(solver.makeNumVar(0.0, infinity, (String) data.get(i)[0]));
    }
    System.out.println("Number of variables = " + solver.numVariables());

    MPConstraint[] constraints = new MPConstraint[nutrients.size()];
    for (int i = 0; i < nutrients.size(); ++i) {
      constraints[i] = solver.makeConstraint(
          (double) nutrients.get(i)[1], infinity, (String) nutrients.get(i)[0]);
      for (int j = 0; j < data.size(); ++j) {
        constraints[i].setCoefficient(foods.get(j), ((double[]) data.get(j)[3])[i]);
      }
      // constraints.add(constraint);
    }
    System.out.println("Number of constraints = " + solver.numConstraints());

    MPObjective objective = solver.objective();
    for (int i = 0; i < data.size(); ++i) {
      objective.setCoefficient(foods.get(i), 1);
    }
    objective.setMinimization();

    final MPSolver.ResultStatus resultStatus = solver.solve();

    // Check that the problem has an optimal solution.
    if (resultStatus != MPSolver.ResultStatus.OPTIMAL) {
      System.err.println("The problem does not have an optimal solution!");
      if (resultStatus == MPSolver.ResultStatus.FEASIBLE) {
        System.err.println("A potentially suboptimal solution was found.");
      } else {
        System.err.println("The solver could not solve the problem.");
        return;
      }
    }

    // Display the amounts (in dollars) to purchase of each food.
    double[] nutrientsResult = new double[nutrients.size()];
    System.out.println("\nAnnual Foods:");
    for (int i = 0; i < foods.size(); ++i) {
      if (foods.get(i).solutionValue() > 0.0) {
        System.out.println((String) data.get(i)[0] + ": $" + 365 * foods.get(i).solutionValue());
        for (int j = 0; j < nutrients.size(); ++j) {
          nutrientsResult[j] += ((double[]) data.get(i)[3])[j] * foods.get(i).solutionValue();
        }
      }
    }
    System.out.println("\nOptimal annual price: $" + 365 * objective.value());

    System.out.println("\nNutrients per day:");
    for (int i = 0; i < nutrients.size(); ++i) {
      System.out.println(
          nutrients.get(i)[0] + ": " + nutrientsResult[i] + " (min " + nutrients.get(i)[1] + ")");
    }

    System.out.println("\nAdvanced usage:");
    System.out.println("Problem solved in " + solver.wallTime() + " milliseconds");
    System.out.println("Problem solved in " + solver.iterations() + " iterations");
  }

  private StiglerDiet() {}
}

C#

// The Stigler diet problem.
using System;
using System.Collections.Generic;
using Google.OrTools.LinearSolver;

public class StiglerDiet
{
    static void Main()
    {
        // Nutrient minimums.
        (String Name, double Value)[] nutrients =
            new[] { ("Calories (kcal)", 3.0), ("Protein (g)", 70.0),    ("Calcium (g)", 0.8),
                    ("Iron (mg)", 12.0),      ("Vitamin A (kIU)", 5.0), ("Vitamin B1 (mg)", 1.8),
                    ("Vitamin B2 (mg)", 2.7), ("Niacin (mg)", 18.0),    ("Vitamin C (mg)", 75.0) };

        (String Name, String Unit, double Price, double[] Nutrients)[] data = new[] {
            ("Wheat Flour (Enriched)", "10 lb.", 36, new double[] { 44.7, 1411, 2, 365, 0, 55.4, 33.3, 441, 0 }),
            ("Macaroni", "1 lb.", 14.1, new double[] { 11.6, 418, 0.7, 54, 0, 3.2, 1.9, 68, 0 }),
            ("Wheat Cereal (Enriched)", "28 oz.", 24.2, new double[] { 11.8, 377, 14.4, 175, 0, 14.4, 8.8, 114, 0 }),
            ("Corn Flakes", "8 oz.", 7.1, new double[] { 11.4, 252, 0.1, 56, 0, 13.5, 2.3, 68, 0 }),
            ("Corn Meal", "1 lb.", 4.6, new double[] { 36.0, 897, 1.7, 99, 30.9, 17.4, 7.9, 106, 0 }),
            ("Hominy Grits", "24 oz.", 8.5, new double[] { 28.6, 680, 0.8, 80, 0, 10.6, 1.6, 110, 0 }),
            ("Rice", "1 lb.", 7.5, new double[] { 21.2, 460, 0.6, 41, 0, 2, 4.8, 60, 0 }),
            ("Rolled Oats", "1 lb.", 7.1, new double[] { 25.3, 907, 5.1, 341, 0, 37.1, 8.9, 64, 0 }),
            ("White Bread (Enriched)", "1 lb.", 7.9, new double[] { 15.0, 488, 2.5, 115, 0, 13.8, 8.5, 126, 0 }),
            ("Whole Wheat Bread", "1 lb.", 9.1, new double[] { 12.2, 484, 2.7, 125, 0, 13.9, 6.4, 160, 0 }),
            ("Rye Bread", "1 lb.", 9.1, new double[] { 12.4, 439, 1.1, 82, 0, 9.9, 3, 66, 0 }),
            ("Pound Cake", "1 lb.", 24.8, new double[] { 8.0, 130, 0.4, 31, 18.9, 2.8, 3, 17, 0 }),
            ("Soda Crackers", "1 lb.", 15.1, new double[] { 12.5, 288, 0.5, 50, 0, 0, 0, 0, 0 }),
            ("Milk", "1 qt.", 11, new double[] { 6.1, 310, 10.5, 18, 16.8, 4, 16, 7, 177 }),
            ("Evaporated Milk (can)", "14.5 oz.", 6.7, new double[] { 8.4, 422, 15.1, 9, 26, 3, 23.5, 11, 60 }),
            ("Butter", "1 lb.", 30.8, new double[] { 10.8, 9, 0.2, 3, 44.2, 0, 0.2, 2, 0 }),
            ("Oleomargarine", "1 lb.", 16.1, new double[] { 20.6, 17, 0.6, 6, 55.8, 0.2, 0, 0, 0 }),
            ("Eggs", "1 doz.", 32.6, new double[] { 2.9, 238, 1.0, 52, 18.6, 2.8, 6.5, 1, 0 }),
            ("Cheese (Cheddar)", "1 lb.", 24.2, new double[] { 7.4, 448, 16.4, 19, 28.1, 0.8, 10.3, 4, 0 }),
            ("Cream", "1/2 pt.", 14.1, new double[] { 3.5, 49, 1.7, 3, 16.9, 0.6, 2.5, 0, 17 }),
            ("Peanut Butter", "1 lb.", 17.9, new double[] { 15.7, 661, 1.0, 48, 0, 9.6, 8.1, 471, 0 }),
            ("Mayonnaise", "1/2 pt.", 16.7, new double[] { 8.6, 18, 0.2, 8, 2.7, 0.4, 0.5, 0, 0 }),
            ("Crisco", "1 lb.", 20.3, new double[] { 20.1, 0, 0, 0, 0, 0, 0, 0, 0 }),
            ("Lard", "1 lb.", 9.8, new double[] { 41.7, 0, 0, 0, 0.2, 0, 0.5, 5, 0 }),
            ("Sirloin Steak", "1 lb.", 39.6, new double[] { 2.9, 166, 0.1, 34, 0.2, 2.1, 2.9, 69, 0 }),
            ("Round Steak", "1 lb.", 36.4, new double[] { 2.2, 214, 0.1, 32, 0.4, 2.5, 2.4, 87, 0 }),
            ("Rib Roast", "1 lb.", 29.2, new double[] { 3.4, 213, 0.1, 33, 0, 0, 2, 0, 0 }),
            ("Chuck Roast", "1 lb.", 22.6, new double[] { 3.6, 309, 0.2, 46, 0.4, 1, 4, 120, 0 }),
            ("Plate", "1 lb.", 14.6, new double[] { 8.5, 404, 0.2, 62, 0, 0.9, 0, 0, 0 }),
            ("Liver (Beef)", "1 lb.", 26.8, new double[] { 2.2, 333, 0.2, 139, 169.2, 6.4, 50.8, 316, 525 }),
            ("Leg of Lamb", "1 lb.", 27.6, new double[] { 3.1, 245, 0.1, 20, 0, 2.8, 3.9, 86, 0 }),
            ("Lamb Chops (Rib)", "1 lb.", 36.6, new double[] { 3.3, 140, 0.1, 15, 0, 1.7, 2.7, 54, 0 }),
            ("Pork Chops", "1 lb.", 30.7, new double[] { 3.5, 196, 0.2, 30, 0, 17.4, 2.7, 60, 0 }),
            ("Pork Loin Roast", "1 lb.", 24.2, new double[] { 4.4, 249, 0.3, 37, 0, 18.2, 3.6, 79, 0 }),
            ("Bacon", "1 lb.", 25.6, new double[] { 10.4, 152, 0.2, 23, 0, 1.8, 1.8, 71, 0 }),
            ("Ham, smoked", "1 lb.", 27.4, new double[] { 6.7, 212, 0.2, 31, 0, 9.9, 3.3, 50, 0 }),
            ("Salt Pork", "1 lb.", 16, new double[] { 18.8, 164, 0.1, 26, 0, 1.4, 1.8, 0, 0 }),
            ("Roasting Chicken", "1 lb.", 30.3, new double[] { 1.8, 184, 0.1, 30, 0.1, 0.9, 1.8, 68, 46 }),
            ("Veal Cutlets", "1 lb.", 42.3, new double[] { 1.7, 156, 0.1, 24, 0, 1.4, 2.4, 57, 0 }),
            ("Salmon, Pink (can)", "16 oz.", 13, new double[] { 5.8, 705, 6.8, 45, 3.5, 1, 4.9, 209, 0 }),
            ("Apples", "1 lb.", 4.4, new double[] { 5.8, 27, 0.5, 36, 7.3, 3.6, 2.7, 5, 544 }),
            ("Bananas", "1 lb.", 6.1, new double[] { 4.9, 60, 0.4, 30, 17.4, 2.5, 3.5, 28, 498 }),
            ("Lemons", "1 doz.", 26, new double[] { 1.0, 21, 0.5, 14, 0, 0.5, 0, 4, 952 }),
            ("Oranges", "1 doz.", 30.9, new double[] { 2.2, 40, 1.1, 18, 11.1, 3.6, 1.3, 10, 1998 }),
            ("Green Beans", "1 lb.", 7.1, new double[] { 2.4, 138, 3.7, 80, 69, 4.3, 5.8, 37, 862 }),
            ("Cabbage", "1 lb.", 3.7, new double[] { 2.6, 125, 4.0, 36, 7.2, 9, 4.5, 26, 5369 }),
            ("Carrots", "1 bunch", 4.7, new double[] { 2.7, 73, 2.8, 43, 188.5, 6.1, 4.3, 89, 608 }),
            ("Celery", "1 stalk", 7.3, new double[] { 0.9, 51, 3.0, 23, 0.9, 1.4, 1.4, 9, 313 }),
            ("Lettuce", "1 head", 8.2, new double[] { 0.4, 27, 1.1, 22, 112.4, 1.8, 3.4, 11, 449 }),
            ("Onions", "1 lb.", 3.6, new double[] { 5.8, 166, 3.8, 59, 16.6, 4.7, 5.9, 21, 1184 }),
            ("Potatoes", "15 lb.", 34, new double[] { 14.3, 336, 1.8, 118, 6.7, 29.4, 7.1, 198, 2522 }),
            ("Spinach", "1 lb.", 8.1, new double[] { 1.1, 106, 0, 138, 918.4, 5.7, 13.8, 33, 2755 }),
            ("Sweet Potatoes", "1 lb.", 5.1, new double[] { 9.6, 138, 2.7, 54, 290.7, 8.4, 5.4, 83, 1912 }),
            ("Peaches (can)", "No. 2 1/2", 16.8, new double[] { 3.7, 20, 0.4, 10, 21.5, 0.5, 1, 31, 196 }),
            ("Pears (can)", "No. 2 1/2", 20.4, new double[] { 3.0, 8, 0.3, 8, 0.8, 0.8, 0.8, 5, 81 }),
            ("Pineapple (can)", "No. 2 1/2", 21.3, new double[] { 2.4, 16, 0.4, 8, 2, 2.8, 0.8, 7, 399 }),
            ("Asparagus (can)", "No. 2", 27.7, new double[] { 0.4, 33, 0.3, 12, 16.3, 1.4, 2.1, 17, 272 }),
            ("Green Beans (can)", "No. 2", 10, new double[] { 1.0, 54, 2, 65, 53.9, 1.6, 4.3, 32, 431 }),
            ("Pork and Beans (can)", "16 oz.", 7.1, new double[] { 7.5, 364, 4, 134, 3.5, 8.3, 7.7, 56, 0 }),
            ("Corn (can)", "No. 2", 10.4, new double[] { 5.2, 136, 0.2, 16, 12, 1.6, 2.7, 42, 218 }),
            ("Peas (can)", "No. 2", 13.8, new double[] { 2.3, 136, 0.6, 45, 34.9, 4.9, 2.5, 37, 370 }),
            ("Tomatoes (can)", "No. 2", 8.6, new double[] { 1.3, 63, 0.7, 38, 53.2, 3.4, 2.5, 36, 1253 }),
            ("Tomato Soup (can)", "10 1/2 oz.", 7.6, new double[] { 1.6, 71, 0.6, 43, 57.9, 3.5, 2.4, 67, 862 }),
            ("Peaches, Dried", "1 lb.", 15.7, new double[] { 8.5, 87, 1.7, 173, 86.8, 1.2, 4.3, 55, 57 }),
            ("Prunes, Dried", "1 lb.", 9, new double[] { 12.8, 99, 2.5, 154, 85.7, 3.9, 4.3, 65, 257 }),
            ("Raisins, Dried", "15 oz.", 9.4, new double[] { 13.5, 104, 2.5, 136, 4.5, 6.3, 1.4, 24, 136 }),
            ("Peas, Dried", "1 lb.", 7.9, new double[] { 20.0, 1367, 4.2, 345, 2.9, 28.7, 18.4, 162, 0 }),
            ("Lima Beans, Dried", "1 lb.", 8.9, new double[] { 17.4, 1055, 3.7, 459, 5.1, 26.9, 38.2, 93, 0 }),
            ("Navy Beans, Dried", "1 lb.", 5.9, new double[] { 26.9, 1691, 11.4, 792, 0, 38.4, 24.6, 217, 0 }),
            ("Coffee", "1 lb.", 22.4, new double[] { 0, 0, 0, 0, 0, 4, 5.1, 50, 0 }),
            ("Tea", "1/4 lb.", 17.4, new double[] { 0, 0, 0, 0, 0, 0, 2.3, 42, 0 }),
            ("Cocoa", "8 oz.", 8.6, new double[] { 8.7, 237, 3, 72, 0, 2, 11.9, 40, 0 }),
            ("Chocolate", "8 oz.", 16.2, new double[] { 8.0, 77, 1.3, 39, 0, 0.9, 3.4, 14, 0 }),
            ("Sugar", "10 lb.", 51.7, new double[] { 34.9, 0, 0, 0, 0, 0, 0, 0, 0 }),
            ("Corn Syrup", "24 oz.", 13.7, new double[] { 14.7, 0, 0.5, 74, 0, 0, 0, 5, 0 }),
            ("Molasses", "18 oz.", 13.6, new double[] { 9.0, 0, 10.3, 244, 0, 1.9, 7.5, 146, 0 }),
            ("Strawberry Preserves", "1 lb.", 20.5, new double[] { 6.4, 11, 0.4, 7, 0.2, 0.2, 0.4, 3, 0 })
        };

        // Create the linear solver with the GLOP backend.
        Solver solver = Solver.CreateSolver("GLOP");
        if (solver is null)
        {
            return;
        }

        List<Variable> foods = new List<Variable>();
        for (int i = 0; i < data.Length; ++i)
        {
            foods.Add(solver.MakeNumVar(0.0, double.PositiveInfinity, data[i].Name));
        }
        Console.WriteLine($"Number of variables = {solver.NumVariables()}");

        List<Constraint> constraints = new List<Constraint>();
        for (int i = 0; i < nutrients.Length; ++i)
        {
            Constraint constraint =
                solver.MakeConstraint(nutrients[i].Value, double.PositiveInfinity, nutrients[i].Name);
            for (int j = 0; j < data.Length; ++j)
            {
                constraint.SetCoefficient(foods[j], data[j].Nutrients[i]);
            }
            constraints.Add(constraint);
        }
        Console.WriteLine($"Number of constraints = {solver.NumConstraints()}");

        Objective objective = solver.Objective();
        for (int i = 0; i < data.Length; ++i)
        {
            objective.SetCoefficient(foods[i], 1);
        }
        objective.SetMinimization();

        Solver.ResultStatus resultStatus = solver.Solve();

        // Check that the problem has an optimal solution.
        if (resultStatus != Solver.ResultStatus.OPTIMAL)
        {
            Console.WriteLine("The problem does not have an optimal solution!");
            if (resultStatus == Solver.ResultStatus.FEASIBLE)
            {
                Console.WriteLine("A potentially suboptimal solution was found.");
            }
            else
            {
                Console.WriteLine("The solver could not solve the problem.");
                return;
            }
        }

        // Display the amounts (in dollars) to purchase of each food.
        double[] nutrientsResult = new double[nutrients.Length];
        Console.WriteLine("\nAnnual Foods:");
        for (int i = 0; i < foods.Count; ++i)
        {
            if (foods[i].SolutionValue() > 0.0)
            {
                Console.WriteLine($"{data[i].Name}: ${365 * foods[i].SolutionValue():N2}");
                for (int j = 0; j < nutrients.Length; ++j)
                {
                    nutrientsResult[j] += data[i].Nutrients[j] * foods[i].SolutionValue();
                }
            }
        }
        Console.WriteLine($"\nOptimal annual price: ${365 * objective.Value():N2}");

        Console.WriteLine("\nNutrients per day:");
        for (int i = 0; i < nutrients.Length; ++i)
        {
            Console.WriteLine($"{nutrients[i].Name}: {nutrientsResult[i]:N2} (min {nutrients[i].Value})");
        }

        Console.WriteLine("\nAdvanced usage:");
        Console.WriteLine($"Problem solved in {solver.WallTime()} milliseconds");
        Console.WriteLine($"Problem solved in {solver.Iterations()} iterations");
    }
}