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老虎飲食問題

這個頁面中的內容
使用線性解題工具的解決方案

在本節中，我們會介紹如何解決名為 Stigler diet 的經典問題，這個問題以經濟實惠的 George Stigler 名叫 George Stigler 認為，這種問題是以經濟實惠的方式計算到一組食物，以滿足基本營養需求。儘管運算最佳營養的概念近來已成為職業，但他在數學方面的表現卻是數學習題，而不是飲食建議。

Stigler 的飲食要求如下：

營養素清單

營養素	每日建議攝取量
卡路里	3,000 卡
蛋白質	70 公克
鈣	0.8 公克
鐵	12 毫克
維生素 A	5,000 IU
硫胺素 (維他命 B1)	1.8 毫克
核黃素 (維他命 B2)	2.7 毫克
菸鹼酸	18 毫克
Ascorbic 酸 (Vitamin C)	75 毫克

Stigler 評估的食物組合是時間的反射 (1944 年)。下列營養資料是以美元 (而非單位為單位) 計算，因此目標為決定每項食品要支出多少美元。

商品清單

商品	單位	1939 價格 (美分)	卡路里 (大卡)	蛋白質 (g)	鈣 (g)	鐵 (毫克)	維生素 A (KIU)	硫胺素 (毫克)	核黃素 (毫克)	菸鹼 (毫克)	Ascorbic 酸 (mg)
麵粉 (凸版)	68 公斤	36	44.7 號	1411	2	365	0	55.4	33.3	441	0
通心麵	1 磅	14.1	11.6	418	0.7	54	0	3.2	1.9	68	0
麥片麥片 (充實)	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	0.2	56	0	13.5	2.3	68	0
玉米大餐	1 磅	4.6	36.0	897	1.7	99	30.9	17.4	7.9	106	0
哈米尼格里斯	24 盎司。	8.5	28.6	680	0.8	80	0	10.6	1.6	110	0
飯	1 磅	7.5	21.2	460	0.6	41	0	2	4.8	60	0
捲烤燕麥	1 磅	7.1	25.3	907	5.1	341	0	37.1	8.9	64	0
白色麵包 (充實)	1 磅	7.9	15.0	488	2.5	115	0	13.8	8.5	126	0
全麥麵包	1 磅	9.1	12.2	484	2.7	125	0	13.9	6.4	160	0
裸麥麵包	1 磅	9.1	12.4 號	439	1.1	82	0	9.9	3	66	0
英鎊蛋糕	1 磅	24.8 號	8.0	130	0.4	31	18.9 號	2.8	3	17	0
蘇打餅乾	1 磅	15.1	12.5	288	0.5	50	0	0	0	0	0
牛奶	1 季。	11	6.1	310	10.5	18	16.8	4	16	7	177
蒸發牛奶 (可以)	14.5 盎司。	6.7	8.4	422	15.1	9	26	3	23.5	11	60
奶油	1 磅	30.8	10.8	9	0.2	3	44.2	0	0.2	2	0
奧利奧馬加林	1 磅	16.1	20.6 號	17	0.6	6	55.8	0.2	0	0	0
蛋	11:00。	32.6	2.9 次	238	1.0	52	18.6	2.8	6.5	1	0
起司 (Cheddar)	1 磅	24.2	7.4	448	16.4	19	28.1	0.8	10.3	4	0
奶油	1/2 點	14.1	3.5	49	1.7	3	16.9 號	0.6	2.5	0	17
花生醬	1 磅	17.9 號	15.7	661	1.0	48	0	9.6	8.1	471	0
美乃滋	1/2 點	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 號	0	0	0	0	0	0	0	0
拉德	1 磅	9.8	41.7	0	0	0	0.2	0	0.5	5	0
希隆牛排	1 磅	39.6	2.9 次	166	0.2	34	0.2	2.1	2.9 次	69	0
圓形牛排	1 磅	36.4	2.2	214	0.2	32	0.4	2.5	2.4	87	0
拉斯特	1 磅	29.2	3.4	213	0.2	33	0	0	2	0	0
嘟嘟車	1 磅	22.6 號	3.6	309	0.2	46	0.4	1	4	120	0
平板型	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
羊角	1 磅	27.6	3.1	245	0.2	20	0	2.8	3.9	86	0
羊角 (Rib)	1 磅	36.6 號	3.3	140	0.2	15	0	1.7	2.7	54	0
豬排	1 磅	30.7 號	3.5	196	0.2	30	0	17.4	2.7	60	0
豬肉烤餅	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
煙燻燻爐	1 磅	27.4	6.7	212	0.2	31	0	9.9	3.3	50	0
鹽豬	1 磅	16	18.8 號	164	0.2	26	0	1.4	1.8	0	0
烘烤雞	1 磅	30.3	1.8	184	0.2	30	0.2	0.9	1.8	68	46
小牛肉排	1 磅	42.3	1.7	156	0.2	24	0	1.4	2.4	57	0
鮭魚，粉紅色 (可以)	16 盎司。	13	5.8	705	6.8	45	3.5	1	4.9	209	0
對等的	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
檸檬	11:00。	26	1.0	21	0.5	14	0	0.5	0	4	952
柳橙	11:00。	30.9	2.2	40	1.1	18	11.1	3.6	1.3	10	1998
四季豆	1 磅	7.1	2.4	138	3.7	80	69	4.3	5.8	37	862
高麗菜	1 磅	3.7	2.6	125	4.0	36	7.2	9	4.5	26	5369
紅蘿蔔	1 組	4.7	2.7	73	2.8	43	188.5	6.1	4.3	89	608
名人	1 個跟蹤	7.3	0.9	51	3.0	23	0.9	1.4	1.4	9	313
生菜	1 頭	8.2	0.4	27	1.1	22	112.4	1.8	3.4	11	449
洋蔥	1 磅	3.6	5.8	166	3.8	59	16.6	4.7	5.9	21	1184
馬鈴薯	68 公斤	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	1108.4	5.7	13.8	33	2755
番薯	1 磅	5.1	9.6	138	2.7	54	290.7	8.4	5.4	83	1912
桃子 (可以)	2021 年 1 月 2 日	16.8	3.7	20	0.4	10	21.5	0.5	1	31	196
梨子 (可以)	2021 年 1 月 2 日	20.4	3.0	8	0.3	8	0.8	0.8	0.8	5	81
鳳梨 (可以)	2021 年 1 月 2 日	21.3	2.4	16	0.4	8	2	2.8	0.8	7	399
蘆筍 (可以)	第 2 號	27.7 號	0.4	33	0.3	12	16.3	1.4	2.1	17	272
四季豆 (可以)	第 2 號	10	1.0	54	2	65	53.9	1.6	4.3	32	431
豬肉和豆類 (可以)	16 盎司。	7.1	7.5	364	4	134	3.5	8.3	7.7	56	0
玉米 (可以)	第 2 號	10.4	5.2	136	0.2	16	12	1.6	2.7	42	218
豌豆 (可以)	第 2 號	13.8	2.3	136	0.6	45	34.9	4.9	2.5	37	370
番茄 (坎)	第 2 號	8.6	1.3	63	0.7	38	53.2	3.4	2.5	36	1253
番茄湯 (嘴)	10 1/2 盎司。	7.6	1.6	71	0.6	43	57.9	3.5	2.4	67	862
桃子 (乾燥)	1 磅	15.7	8.5	87	1.7	173	86.8	1.2	4.3	55	57
淡紫色、乾燥	1 磅	9	12.8	99	2.5	154	85.7	3.9	4.3	65	257
雨林乾燥	15 盎司。	9.4	13.5	104	2.5	136	4.5	6.3	1.4	24	136
豌豆乾燥	1 磅	7.9	20.0	1367	4.2	345	2.9 次	28.7	18.4	162	0
利馬豆，乾燥	1 磅	8.9	17.4	1055	3.7	459	5.1	26.9	38.2	93	0
海軍藍豆類，乾燥	1 磅	5.9	26.9	1691	11.4	792	0	38.4	24.6 號	217	0
咖啡	1 磅	22.4 號	0	0	0	0	0	4	5.1	50	0
茶	6.8 公斤	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	68 公斤	51.7 號	34.9	0	0	0	0	0	0	0	0
玉米糖漿	24 盎司。	13.7	14.7	0	0.5	74	0	0	0	5	0
莫拉西斯	18 盎司。	13.6	9.0	0	10.3	244	0	1.9	7.5	146	0
草莓保護區	1 磅	20.5	6.4	11	0.4	7	0.2	0.2	0.4	3	0

由於營養素均已根據價格正規化，我們的目標是盡可能減少食物總和。

1944 年，Sigler 計算出的最佳答案，並註明悲傷：

...似乎不是直接尋找受線性函式最小計算的直接方法。

他發現一件飲食計畫，每年花費 $39.93 美元，價值 1939 美元。1947 年，Jack Laderman 使用 Simplex 方法 (也就是近期的發明！) 來判斷最佳解決方案。總共花了 120 天，每天要靠書桌計算機上的 9 位工作人員才得出答案。

使用線性解題工具的解決方案

以下各節將說明如何解決 Stigler 飲食問題。

匯入線性解題工具包裝函式

匯入 OR-Tools 線性求解器包裝函式，這個介面用於 [GLOP](/optimization/mip/glop0 線性解析器，如下所示)。

PythonC++JavaC#

from ortools.linear_solver import pywraplp

#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"

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;

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

問題資料

下列程式碼會針對最低營養素需求建立陣列 nutrients，並為任何解決方案中的營養資料表建立陣列 data。

PythonC++JavaC#

# 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
]

// 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}}};

// 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}});

// 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 })
};

宣告到達網頁解題工具

以下程式碼會例項化 MPsolver 包裝函式。

PythonC++JavaC#

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

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

// 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;
}

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

建立變數

下列程式碼會建立問題變數。

PythonC++JavaC#

# 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())

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();

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());

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()}");

方法 MakeNumVar 會為資料表的每一列建立一個變數 food[i]。如前所述，營養資料為每美元，因此 food[i] 是可用於在 i 商品上花費的金額。

定義限制條件

Stigler 節食的限制要求所有食物提供的營養素總量必須至少達到每個營養素的最低要求。接下來，我們會將這些限制撰寫為與陣列 data 和 nutrients 以及變數 food[i] 相關的不等式。

首先，食物 j 提供的營養 i 量為 data[j][i+3] (由於營養資料從 data 的第四欄開始，因此我們會在欄索引中加入 3)。由於食物 j 的花費金額是 food[j]，因此食品 j 提供的營養 i 量為 $data[j][i+3] \cdot food[j]$ 。最後，由於營養 i 的最低要求為 nutrients[i][1]，因此我們可以編寫限制 i，如下所示：

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

以下程式碼定義這些限制。

PythonC++JavaC#

# 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())

// 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();

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());

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 (對應 C++ 方法 MakeRowConstraint) 會建立問題的限制。針對每個 i、constraint(nutrients[i][1], solver.infinity)

這會建立限制，其中 food[j] (定義下一個) 變數的線性組合大於或等於 nutrients[i][1]。線性運算式的係數是由方法 SetCoefficient 定義，如下所示：SetCoefficient(food[j], data[j][i+3]

這會將 food[j] 的係數設為 data[j][i+3]。

總而言之，程式碼定義了上述 (1) 所示的限制。

建立目標

下列程式碼定義了問題的目標函式。

PythonC++JavaC#

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

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

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

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

目標函式是食物總費用，也就是變數 food[i] 的總和。

方法 SetCoefficient 會設定目標函式的係數，在本例中為 1。最後，SetMinimization 宣告這是最小化問題。

叫用求解工具

下列程式碼會叫用解題工具。

PythonC++JavaC#

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

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

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

Solver.ResultStatus resultStatus = solver.Solve();

Glop 能在 300 毫秒以內，以一般電腦解決問題：

顯示解決方案

下列程式碼顯示解決方案。

PythonC++JavaC#

# 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])
    )

// 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) << ")";
}

// 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] + ")");
}

// 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})");
}

以下是程式的輸出內容。

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

完整程式程式碼

以下顯示 Stigler 飲食計畫的完整程式碼。

PythonC++JavaC#

"""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()

// 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;
}

// 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() {}
}

// 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");
    }
}