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3050.0 | Easy | The Zhang family has 6 children, Harry, Hermione, Ron, Fred, George, and Ginny. The cost of taking Harry is $1200, Hermione is $1650, Ron is $750, Fred is $800, George is $800, and Ginny is $1500. Which children should the couple take to minimize the total cost of taking the children?
They can take a maximum of 4 children on the upcoming trip.
Ginny is the youngest, so the Zhang family will definitely take her.
If the couple takes Harry, they will not take Fred because Harry doesn't get along with him.
If the couple takes Harry, they will not take George because Harry doesn't get along with him.
If they take George, they must also take Fred.
If they take George, they must also take Hermione.
Although this will cost them a lot of money, the Zhang family has decided to take at least three children. | IP |
135000.0 | Easy | The Li family plans to invest their retirement fund in commercial real estate. Property 1 has an annual income of $12,500, Property 2 has an annual income of $35,000, Property 3 has an annual income of $23,000, and Property 4 has an annual income of $100,000. The decision to be made is whether to buy or not buy each property, not the quantity, as there is only one property per property. Help them decide which properties to purchase to maximize their annual income.
Property 1 costs $1.5 million, Property 2 costs $2.1 million, Property 3 costs $2.3 million, and Property 4 costs $4.2 million. The Li family's budget is $7 million.
If they purchase Property 4, then they cannot purchase Property 3. | IP |
30400.0 | Easy | A farmer needs to decide how many cows, sheep, and chickens to raise in order to maximize profit. The farmer can sell cows, sheep, and chickens for $500, $200, and $8 respectively. The feed costs for each cow, sheep, and chicken are $100, $80, and $5 respectively. Profit is the difference between the selling price and the feed cost. Cows, sheep, and chickens produce 10, 5, and 3 units of manure per day respectively. Due to limited time for farm employees to clean the farm, they can clean a maximum of 800 units of manure per day. Additionally, due to the limited size of the farm, the farmer can raise a maximum of 50 chickens. Furthermore, the farmer must have at least 10 cows to meet customer demand. The farmer must also have at least 20 sheep. Finally, the total number of animals cannot exceed 100. | MIP |
23000.0 | Easy | A company wants to hire new employees for their team. The salary requirements of candidates A, B, C, D, and E are $8100, $20000, $21000, $3000, and $8000 respectively. They need to decide whether to hire each candidate. The team wants to minimize the total amount paid to the candidates.
They want to hire a maximum of 3 new employees.
The team has a limited budget of $35,000. They need to ensure that the total payment to the selected candidates does not exceed the budget.
The qualifications of the five candidates are as follows:
Candidate A: Bachelor's degree;
Candidate B: Master's degree;
Candidate C: PhD degree;
Candidate D: No degree;
Candidate E: No degree.
They will select at least one candidate with a master's or PhD degree.
The work experience of the five candidates is as follows:
Candidate A: 3 years of work experience;
Candidate B: 10 years of work experience;
Candidate C: 4 years of work experience;
Candidate D: 3 years of work experience;
Candidate E: 7 years of work experience.
They want the total work experience of the selected candidates to be at least 12 years.
Due to the similar professional skills of candidates A and E, the company will choose at most one of them.
They will hire at least 2 new employees. | IP |
180000 | Medium | Tom and Jerry have just bought a farm in Sunshine Valley and are considering using it to grow corn, wheat, soybeans, and sorghum. The profit from planting one acre of corn is $1500, one acre of wheat is $1200, one acre of soybeans is $1800, and one acre of sorghum is $1600. To maximize profit, how many acres of land should Tom and Jerry use to plant each crop?
The total area of Tom and Jerry's farm is 100 acres.
The area of land used for planting corn should be at least twice the area of land used for planting wheat.
The area of land used for planting soybeans should be at least half the area of land used for planting sorghum.
The area of land used for planting wheat must be three times the area of land used for planting sorghum. | LP |
1600.0 | Easy | The Lee family has 5 children, Alice, Bob, Charlie, Diana, and Ella. The cost of taking Alice is $1000, Bob is $900, Charlie is $600, Diana is $500, and Ella is $700. Which children should the couple take to minimize the total cost of taking the children?
They can take a maximum of 3 children together on the upcoming trip.
Bob is the youngest, so the Lee family will definitely take him.
If the couple takes Alice, they will not take Diana because Alice and Diana do not get along.
If the couple takes Bob, they will not take Charlie because Bob and Charlie do not get along.
If they take Charlie, they must also take Diana.
If they take Diana, they must also take Ella.
Although it will cost them a lot of money, the Lee family has decided to take at least two children. | IP |
90000.0 | Easy | The Zhang family has decided to invest in several different restaurants. Restaurant A has an annual income of $15,000, Restaurant B has an annual income of $40,000, Restaurant C has an annual income of $30,000, and Restaurant D has an annual income of $50,000. They need to decide whether to purchase each restaurant, and each restaurant can only be purchased once. Help them decide which restaurants to purchase to maximize their annual income.
The cost of Restaurant A is $1.6 million, the cost of Restaurant B is $2.5 million, the cost of Restaurant C is $1.8 million, and the cost of Restaurant D is $3 million. The Zhang family's investment budget is $6 million.
If they purchase Restaurant D, they cannot purchase Restaurant A. | IP |
600 | Easy | A company plans to transport goods between a city and suburb and needs to choose the most environmentally friendly mode of transportation. The company can choose from the following three options: motorcycles, small trucks, and large trucks. Each motorcycle trip produces 40 units of pollution, each small truck trip produces 70 units of pollution, and each large truck trip produces 100 units of pollution. The company's goal is to minimize total pollution.
The company can only choose two modes of transportation from these three options.
Due to certain road restrictions, the number of motorcycle trips cannot exceed 8.
Each motorcycle trip can transport 10 units of products, each small truck trip can transport 20 units of products, and each large truck trip can transport 50 units of products. The company needs to transport at least 300 units of products.
The total number of trips must be less than or equal to 20. | IP |
9800.0 | Easy | A furniture factory needs to determine how many tables, chairs, and bookshelves to produce in order to maximize profit. The factory can sell tables for $200 each, chairs for $50 each, and bookshelves for $150 each. The manufacturing costs for each table, chair, and bookshelf are $120, $20, and $90, respectively. Profit is the difference between the selling price and the manufacturing cost. Tables, chairs, and bookshelves each occupy 5, 2, and 3 square meters of warehouse space, respectively. Due to limited warehouse space, the total space cannot exceed 500 square meters. Additionally, due to market demand, the factory needs to produce at least 10 tables and 20 bookshelves. Finally, the total number of items produced by the furniture factory cannot exceed 200. | LP |
38000 | Easy | A company needs to decide whether to hire some of the five candidates to join their research and development team. The salary requirements for candidates F, G, H, I, and J are $12,000, $15,000, $18,000, $5,000, and $10,000 respectively. The company wants to minimize the total amount paid to the candidates while staying within the budget.
The company has a budget of $40,000 and wants to hire a maximum of 4 new employees.
The skill levels of the candidates are as follows:
Candidate F: Level 2
Candidate G: Level 3
Candidate H: Level 4
Candidate I: Level 1
Candidate J: Level 2
The company needs to ensure that the total skill level of the hired employees is at least 8.
The project management experience in years for each candidate is as follows:
Candidate F: 1 year
Candidate G: 2 years
Candidate H: 2 years
Candidate I: 5 years
Candidate J: 4 years
They want the total project management experience in the team to be at least 8 years.
Due to the similar technical backgrounds of candidates G and J, the company can only choose one of them at most. | IP |
25000 | Medium | A toy company manufactures three types of tabletop golf toy, each requiring different manufacturing techniques. The advanced type requires 17 hours of processing and assembly labor, 8 hours of inspection, and a profit of 300 yuan per unit. The intermediate type requires 10 hours of labor, 4 hours of inspection, and a profit of 200 yuan. The basic type requires 2 hours of labor, 2 hours of inspection, and a profit of 100 yuan. There are 1000 hours of processing labor and 500 hours of inspection available. Furthermore, market forecasts indicate that the demand for the advanced type does not exceed 50 units, for the intermediate type does not exceed 80 units, and for the basic type does not exceed 150 units. Determine the company's production plan to maximize profit. | IP |
734 | Easy | A factory produces three types of products: A, B, and C. Each unit of product A requires 1 hour of technical preparation, 10 hours of direct labor, and 3 kilograms of material. Each unit of product B requires 2 hours of technical preparation, 4 hours of labor, and 2 kilograms of material. Each unit of product C requires 1 hour of technical preparation, 5 hours of labor, and 1 kilogram of material. The available technical preparation time is 100 hours, labor time is 700 hours, and material is 400 kilograms. The company offers larger discounts for bulk purchases, as shown in Table 1-22. Determine the company's production plan to maximize profit.
Table 1-22
\begin{tabular}{cc|cc|cc}
\hline \multicolumn{2}{c|}{Product A} & \multicolumn{2}{c|}{Product B} & \multicolumn{2}{c}{Product C} \\
\hline Sales volume (units) & Profit (yuan) & Sales volume (units) & Profit (yuan) & Sales volume (units) & Profit (yuan) \\
\hline $0 \sim 40$ & 10 & $0 \sim 50$ & 6 & $0 \sim 100$ & 5 \\
\hline $40 \sim 100$ & 9 & $50 \sim 100$ & 4 & Above 100 & 4 \\
\hline $100 \sim 150$ & 8 & Above 100 & 3 & & \\
\hline Above 150 & 7 & \multicolumn{2}{|c|}{} & & \\
\hline
\end{tabular} | IP |
53 | Hard | A certain 24-hour convenience store requires a certain number of salespersons during each time period as follows: 2:00-6:00 - 10 people, 6:00-10:00 - 15 people, 10:00-14:00 - 25 people, 14:00-18:00 - 20 people, 18:00-22:00 - 18 people, 22:00-2:00 - 12 people. Salespersons work at 2:00, 6:00, 10:00, 14:00, 18:00, and 22:00, and work continuously for 8 hours. Determine the minimum number of salespersons needed to meet the requirements. | IP |
20240 | Hard | A farm has 100 hectares of land and 15,000 yuan available for production development. The farm has a labor force of 3,500 person-days in autumn and winter, and 4,000 person-days in spring and summer. If the labor force is not fully utilized, they can work outside. The wage for hired labor is 2.1 yuan/person-day in spring and summer, and 1.8 yuan/person-day in autumn and winter. The farm grows three crops: soybeans, corn, and wheat, and raises cows and chickens. There is no need for additional investment when planting crops, but for animal farming, it costs 400 yuan per cow and 3 yuan per chicken. When raising cows, 1.5 hectares of land are needed to grow grass for each cow, and it requires 100 person-days of labor in autumn and winter, and 50 person-days in spring and summer. The net income per cow per year is 400 yuan. When raising chickens, no land is needed, but it requires 0.6 person-days of labor in autumn and winter, and 0.3 person-days in spring and summer. The net income per chicken per year is 2 yuan. The farm has a chicken coop that can accommodate up to 3,000 chickens, and a cowshed that can accommodate up to 32 cows. The labor requirements and net income per hectare per year for the three crops are shown in Table 1-9.
Table 1-9
\begin{tabular}{l|c|c|c}
\hline \multicolumn{1}{c}{ Crop } & Soybeans & Corn & Wheat \\
\hline Person-days required in autumn and winter & 20 & 35 & 10 \\
Person-days required in spring and summer & 50 & 75 & 40 \\
Net income per hectare per year (yuan/hm^2) & 175 & 300 & 120 \\
\hline
\end{tabular}
Determine the farm's operating plan to maximize net income per year. | MIP |
-99999 | Hard | A strategic bomber squadron has been ordered to destroy enemy military targets. It is known that there are four key areas, and destroying any one of them will achieve the objective. To complete this mission, the limits are set at $48000 \mathrm{~L}$ of gasoline, 48 heavy bombs, and 32 light bombs. When the aircraft carries heavy bombs, it can fly $2 \mathrm{~km}$ per liter of gasoline, and when it carries light bombs, it can fly $3 \mathrm{~km}$ per liter of gasoline. It is also known that each aircraft can only carry one bomb at a time, and each bombing mission consumes gasoline for the round trip (when empty, it can fly $4 \mathrm{~km}$ per liter of gasoline), as well as $100 \mathrm{~L}$ for takeoff and landing each time. The relevant data is shown in Table 1-17.
Table 1-17
\begin{tabular}{c|c|c|c}
\hline \multirow{2}{*}{ Key Area } & \multirow{2}{*}{ Distance from Base $/ \mathrm{km}$} & \multicolumn{2}{|c}{ Probability of Destruction } \\
\cline { 3 - 4 } & 450 & Per Heavy Bomb & Per Light Bomb \\
\hline 1 & 480 & 0.10 & 0.08 \\
2 & 540 & 0.20 & 0.16 \\
3 & 600 & 0.15 & 0.12 \\
4 & 0.25 & 0.20 \\
\hline
\end{tabular}
To maximize the probability of destroying enemy military targets, how should the bombing plan be determined? Establish a linear programming model for this problem. | IP |
4700 | Medium | A wood storage and transportation company has a large warehouse for storing and selling wood. Due to the fluctuation in wood prices each quarter, the company purchases wood at the beginning of each quarter, sells a portion within the same quarter, and stores the remaining amount for future sales. It is known that the maximum wood storage capacity of the company's warehouse is 200,000 $\mathrm{m}^3$, and the storage cost is $(a+b u)$ yuan per $\mathrm{m}^3$, where $a=70$, $b=100$, and $u$ is the storage time in quarters. The buying and selling prices for each quarter and the projected maximum sales volume are shown in Table 1-18.
Table 1-18
\begin{tabular}{c|c|c|c}
\hline Quarter & Buying Price (10,000 yuan/$10,000 \mathrm{m}^2$) & Selling Price (10,000 yuan/$10,000 \mathrm{m}^2$) & Projected Maximum Sales Volume ($10,000 \mathrm{m}^3$) \\
\hline Winter & 410 & 425 & 100 \\
Spring & 430 & 440 & 140 \\
Summer & 460 & 465 & 200 \\
Autumn & 450 & 455 & 160 \\
\hline
\end{tabular}
Due to the unsuitability of long-term wood storage, all inventory wood should be sold by the end of each autumn. Establish a linear programming model for this problem to maximize the company's annual profit. | LP |
3 | Easy | A convenience store plans to open several chain stores in a new residential area in the northwest suburbs of the city. To facilitate shopping, the distance from any residential area to one of the chain stores should not exceed $800 \mathrm{~m}$. Table 5-1 gives the newly built residential areas and the various areas within a radius of $800 \mathrm{~m}$ of each residential area. How many chain stores should the supermarket build in the above-mentioned areas and in which areas should they be built?
Table 5-1
\begin{tabular}{c|c}
\hline Area Code & Areas within $800 \mathrm{~m}$ radius \\
\hline A & A C E G H I \\
B & B H I \\
C & A C G H I \\
D & D J \\
E & A E G \\
F & F J K \\
G & A C E G \\
H & A B C H I \\
I & A B C H I \\
J & D F J K L \\
K & F J K L \\
L & J K L \\
\hline
\end{tabular} | IP |
37000 | Easy | A product can be processed on any one of the four devices A, B, C, or D. The setup completion cost for each device when it is activated, the unit cost of production for the product, and the maximum processing capacity for each device are known as shown in Table 5-7. How can the total cost be minimized to produce 2000 units of the product? Try to establish a mathematical model.
Table 5-7
\begin{tabular}{c|c|c|c}
\hline Device & Setup Completion Cost (in yuan) & Production Cost (in yuan per unit) & Maximum Processing Capacity (units) \\\hline A & 1000 & 20 & 900 \
B & 920 & 24 & 1000 \
C & 800 & 16 & 1200 \
D & 700 & 28 & 1600 \
\hline
\end{tabular} | MIP |
12 | Easy | There are three different products that need to be processed on three machine tools. Each product must be processed first on machine 1, then sequentially on machine 2 and 3. The order of processing the three products should remain the same on each machine. Assume \( t_{ij} \) represents the time it takes to process the \( i \)th product on the \( j \)th machine.\n\n**Given:**\n\n- \( t_{11} = 5 \) minutes, \( t_{12} = 4 \) minutes, \( t_{13} = 3 \) minutes\n- \( t_{21} = 6 \) minutes, \( t_{22} = 5 \) minutes, \( t_{23} = 4 \) minutes\n- \( t_{31} = 7 \) minutes, \( t_{32} = 6 \) minutes, \( t_{33} = 5 \) minutes
| IP |
4 | Hard | A master's student majoring in Operations Research at a certain university is required to take two math courses, two operations research courses, and two computer courses out of a total of seven courses: calculus, operations research, data structures, management statistics, computer simulation, computer programming, and forecasting. Some courses belong to only one category: calculus belongs to the math category, and computer programming belongs to the computer category. However, some courses are cross-categories: operations research belongs to both the operations research and math categories, data structures belong to both the computer and math categories, management statistics belongs to both the math and operations research categories, computer simulation belongs to both the computer and operations research categories, and forecasting belongs to both the operations research and math categories. For courses that belong to two categories, taking the course can be considered as taking one course in each category. In addition, some courses require prerequisite courses: computer simulation or data structures must be taken after taking computer programming, management statistics must be taken after taking calculus, and forecasting must be taken after taking management statistics. The question is: how many and which courses should a master's student take at minimum to meet the above requirements? | IP |
43700 | Hard | Red Star Plastic Factory produces 6 types of plastic containers, with their capacities, demands, and variable costs shown in Table 5-11.
Table 5-11
\begin{tabular}{c|c|c|c|c|c|c}
\hline Container Code & 1 & 2 & 3 & 4 & 5 & 6 \
\hline Capacity $(\mathrm{cm}^3)$ & 1500 & 2500 & 4000 & 6000 & 9000 & 12000 \
Demand/units & 500 & 550 & 700 & 900 & 400 & 300 \
Variable Cost/(¥/unit) & 5 & 8 & 10 & 12 & 16 & 18 \
\hline
\end{tabular}
Each type of container is produced using different dedicated equipment, with a fixed cost of ¥1200. When the quantity of a certain container cannot meet the demand, containers with larger capacities can be used as substitutes. How should production be organized to minimize the total cost while meeting the demand? | MIP |
6800 | Medium | A production base needs to extract raw materials from warehouses A and B every day for production. The required amounts of raw materials are: at least 240 units of material A, at least 80 kg of material B, and at least 120 tons of material C. It is known that each truck from warehouse A can transport 4 units of material A, 2 kg of material B, and 6 tons of material C, with a freight cost of $200 per truck. Each truck from warehouse B can transport 7 units of material A, 2 kg of material B, and 2 tons of material C, with a freight cost of $160 per truck. The question is: to meet the production needs, how many trucks should the production base send to warehouses A and B each day to minimize the total freight cost? | LP |
135.27 | Easy | A factory plans to produce three types of products, I, II, and III, which require processing on devices A, B, and C, respectively. The data is shown in Table 2-3.
Table 2-3
\begin{tabular}{|c|c|c|c|c|}
\hline Device Code & I & II & III & Monthly Device Capacity \\
\hline A & 8 & 2 & 10 & 300 \\
\hline B & 10 & 5 & 8 & 400 \\
\hline C & 2 & 13 & 10 & 420 \\
\hline Unit Product Profit (in thousands) & 3 & 2 & 2.9 & \\
\hline
\end{tabular}
How can the factory fully utilize the capacity of the devices to maximize profit? | LP |
150 | Easy | The number of drivers and crew members required for each time period of a certain day and night service bus route is shown in Table 1-17.
Table 1-17
\begin{tabular}{c|c|c}
\hline Shift & Time & Number Required \\
\hline 1 & $6: 00 \sim 10: 00$ & 60 \\
\hline 2 & $10: 00 \sim 14: 00$ & 70 \\
\hline 3 & $14: 00 \sim 18: 00$ & 60 \\
\hline 4 & $22: 00 \sim 22: 00$ & 50 \\
\hline 5 & $22: 00 \sim 2: 00$ & 20 \\
\hline 6 & $2: 00 \sim 6: 00$ & 30 \\
\hline
\end{tabular}
Assuming the drivers and crew members start working at the beginning of each time period and work continuously for eight hours, how many drivers and crew members should be equipped for this bus route at least? Provide the linear programming model for this problem. | LP |
1030 | Easy | There are two coal mines, A and B, with monthly coal supplies of at least 80 tons and 100 tons, respectively. They are responsible for supplying coal to three residential areas, which require 55 tons, 75 tons, and 50 tons of coal per month, respectively. The distances between mine A and these three residential areas are 10 kilometers, 5 kilometers, and 6 kilometers. The distances between mine B and these three residential areas are 4 kilometers, 8 kilometers, and 15 kilometers. How should these two coal mines distribute the coal to the three residential areas to minimize the ton-kilometers of transportation? | LP |
57 | Easy | There are two products, A and B, which both require two chemical reaction processes: one before and one after. Each unit of product A requires 2 hours for the front process and 3 hours for the back process. Each unit of product B requires 3 hours for the front process and 4 hours for the back process. There are 16 hours available for the front process and 24 hours available for the back process.
For each unit of product B produced, two units of byproduct C are generated, which do not require any additional cost. Up to 5 units of byproduct C can be sold, while the remaining units must be disposed of, with a disposal cost of $2 per unit.
Selling one unit of product A yields a profit of $4, selling one unit of product B yields a profit of $10, and selling one unit of byproduct C yields a profit of $3.
To maximize the total profit obtained, establish the linear programming model for this problem. | LP |
16.0 | Easy | Mary is planning her dinner for tonight. Each 100 grams of okra contains 3.2 grams of fiber, each 100 grams of carrots contains 2.7 grams of fiber, each 100 grams of celery contains 1.6 grams of fiber, and each 100 grams of cabbage contains 2 grams of fiber. How many grams of each food should Mary purchase to maximize her fiber intake?
She is considering choosing either salmon, beef, or pork as her protein source.
She is also considering selecting at least two vegetables from okra, carrots, celery, and cabbage.
Salmon is priced at $4 per 100 grams, beef at $3.6, and pork at $1.8. Okra is priced at $2.6 per 100 grams, carrots at $1.2, celery at $1.6, and cabbage at $2.3. Mary has a budget of $15 for this meal.
The total intake of food is 600 grams. | LP |
16.0 | Easy | A new company is opening in a small town and needs to decide how to deliver its products to customers in surrounding cities. The company can choose from three modes of transportation: cars, trucks, and buses. Each car trip generates 100 units of pollution, each truck trip generates 80 units of pollution, and each bus trip generates 110 units of pollution. The company needs to minimize the total amount of pollution.
The company can only choose a maximum of two out of the three modes of transportation.
Due to restrictions on road transportation, the number of truck trips cannot exceed 10.
Each car trip can transport 25 units of products, each truck trip can transport 30 units of products, and each bus trip can transport 40 units of products. The company needs to transport at least 500 units of products.
The number of bus trips should not exceed one-fifth of the total number of trips.
The total number of trips must be less than or equal to 15. | MIP |
4685100 | Medium | An Italian transportation company needs to transport some empty containers from its 6 warehouses (located in Verona, Perugia, Rome, Pescara, Taranto, and La Spezia) to major national ports (Genoa, Venice, Ancona, Naples, Bari). The inventory of containers in each warehouse is as follows:
| | Empty Containers |
| :---: | :---: |
| Verona | 10 |
| Perugia | 12 |
| Rome | 20 |
| Pescara | 24 |
| Taranto | 18 |
| La Spezia | 40 |
The demand at the ports is as follows:
| | Container Demand |
| :---: | :---: |
| Genoa | 20 |
| Venice | 15 |
| Ancona | 25 |
| Naples | 33 |
| Bari | 21 |
The transportation will be done by a fleet of trucks. The transportation cost for each container is directly proportional to the distance traveled by the truck, at a rate of 30 euros per kilometer. Each truck can transport a maximum of 2 containers. The distances are as follows:
| | Genoa | Venice | Ancona | Naples | Bari |
| :---: | :---: | :---: | :---: | :---: | :---: |
| Verona | $290 \mathrm{~km}$ | $115 \mathrm{~km}$ | $355 \mathrm{~km}$ | $715 \mathrm{~km}$ | $810 \mathrm{~km}$ |
| Perugia | $380 \mathrm{~km}$ | $340 \mathrm{~km}$ | $165 \mathrm{~km}$ | $380 \mathrm{~km}$ | $610 \mathrm{~km}$ |
| Rome | $505 \mathrm{~km}$ | $530 \mathrm{~km}$ | $285 \mathrm{~km}$ | $220 \mathrm{~km}$ | $450 \mathrm{~km}$ |
| Pescara | $655 \mathrm{~km}$ | $450 \mathrm{~km}$ | $155 \mathrm{~km}$ | $240 \mathrm{~km}$ | $315 \mathrm{~km}$ |
| Taranto | $1010 \mathrm{~km}$ | $840 \mathrm{~km}$ | $550 \mathrm{~km}$ | $305 \mathrm{~km}$ | $95 \mathrm{~km}$ |
| La Spezia | $1072 \mathrm{~km}$ | $1097 \mathrm{~km}$ | $747 \mathrm{~km}$ | $372 \mathrm{~km}$ | $333 \mathrm{~km}$ |
Write a mathematical program to find the transportation policy that minimizes cost and solve it using COPTPY. | LP |
5004 | Medium | A project consists of the following 7 activities, with their durations (in days) as follows: $A(4), B(3), C(5), D(2), E(10), F(10), G(1)$. The following priorities are also given: $A \\rightarrow G, D ; E, G \\rightarrow F; D, F \\rightarrow C ; F \\rightarrow B$. The daily cost of work is 1000 euros; in addition, a special machinery must be rented from the start of activity $A$ to the end of activity $B$, with a daily cost of 5000 euros. Formulate this problem as a linear programming problem and solve it using COPTPY. | LP |
42.1 | Hard | On Danzig Street, vehicles can park on both sides of the street. Mr. Edmunds, who lives at number 1, is organizing a party with about 30 attendees who will arrive in 15 cars. The length of the i-th car, denoted as $\lambda_i$, is given in meters as follows:
i 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
$\lambda_i$ 4 4.5 5 4.1 2.4 5.2 3.7 3.5 3.2 4.5 2.3 3.3 3.8 4.6 3
To avoid disturbing the neighbors, Mr. Edmunds wants to arrange the parking on both sides of the street in such a way that the total length of the street occupied by his friends' cars is minimized. Please provide a mathematical programming formulation and solve this problem using AMPL.
If the vehicles on one side of the street cannot occupy more than 15 meters exactly, how would the program change? | IP |
8800 | Easy | A certain store has formulated a purchasing and selling plan for a certain product from July to December. It is known that the store's warehouse capacity cannot exceed 500 units. By the end of June, there are already 200 units in stock. From then on, the store will purchase once at the beginning of each month. Assuming the purchase and selling prices of the product for each month are as shown in Table 1-21, how many units should be purchased and sold each month to maximize total revenue?
Table 1-21
\begin{tabular}{c|cccccc}
\hline Month & 7 & 8 & 9 & 10 & 11 & 12 \\
\hline Purchase & 28 & 24 & 25 & 27 & 23 & 23 \\
\hline Sale & 29 & 24 & 26 & 28 & 22 & 25 \\
\hline Sale & 29 & 24 & 26 & 28 & 22 & 25 \\
\hline
\end{tabular} | LP |
1360 | Medium | A trading company specializes in wholesale business of a certain kind of miscellaneous grain. The company has a warehouse with a capacity of 5000 units. On January 1st, the company has a stock of 1000 units of miscellaneous grain and a capital of 20000 yuan. The estimated prices of the grain for the first quarter are shown in Table 1-8.
Table $1-8$
\begin{tabular}{c|c|c}
\hline Month & Purchase Price/(yuan/unit) & Selling Price $/$ (yuan $/$ unit) \\
\hline 1 & 2.85 & 3.10 \\
2 & 3.05 & 3.25 \\
3 & 2.90 & 2.95 \\
\hline
\end{tabular}
The purchased grain arrives in the same month but can only be sold in the following month, and it is required to be paid upon delivery. The company wants to have a stock of 2000 units at the end of the quarter. What buying and selling strategies should be adopted to maximize the total profit over the three months? | LP |
770.0 | Medium | There are 8 villages in Tuanjie Township, with their respective coordinates and the number of elementary school students shown in Table 5-14.
Table 5-14
\begin{tabular}{c|c|c|c}
\hline Village Code & \multicolumn{2}{|c|}{Coordinate Position} & \multirow{2}{*}{Number of Elementary School Students} \
\cline {2-4} 1 & $x$ & $y$ & 60 \
2 & 10 & 0 & 80 \
3 & 12 & 3 & 100 \
4 & 14 & 15 & 120 \
5 & 16 & 13 & 80 \
6 & 18 & 9 & 60 \
7 & 8 & 6 & 40 \
8 & 6 & 12 & 80 \
\hline
\end{tabular}
Considering the economies of scale for the schools, it is planned to build one elementary school in each of the two villages. Where should the two schools be built to minimize the walking distance for the elementary school students? (The walking distance for the students is calculated based on the Euclidean distance between the two villages.) | NLP |
14 | Medium | Now we need to determine 4 out of 5 workers to each complete one of the four tasks. Since each worker has different skill sets, the amount of time required for each worker to complete each task is also different. The time required for each worker to complete each task is shown in Table 5-2.
Table 5-2
\begin{tabular}{|c|c|c|c|c|}
\hline Task Time Required & $A$ & $B$ & $C$ & $D$ \\
\hline Worker & & & & \\
\hline I & 9 & 4 & 3 & 7 \\
\hline II & 4 & 6 & 5 & 6 \\
\hline III & 5 & 4 & 7 & 5 \\
\hline IV & 7 & 5 & 2 & 3 \\
\hline V & 10 & 6 & 7 & 4 \\
\hline
\end{tabular}
Try to find a work assignment plan that minimizes the total working hours. | IP |
246 | Medium | A company produces two products, microwaves and water heaters, which are manufactured in two workshops, Workshop A and Workshop B. It is known that, excluding purchased components, producing one microwave requires 2 hours of processing in Workshop A and 1 hour of assembly in Workshop B. Producing one water heater requires 1 hour of processing in Workshop A and 3 hours of assembly in Workshop B. After being produced, both products need to go through inspection and sales processes. It is known that the inspection and sales cost for each microwave is $30, and for each water heater is $50. Workshop A has 120 hours of available production time per month, with a cost of $80 per hour. Workshop B has 150 hours of available production time per month, with a cost of $20 per hour. It is estimated that an average of 80 microwaves and 50 water heaters can be sold per month in the next year. Based on these facts, the company has set the following monthly planning objectives:
First priority: Inspection and sales costs should not exceed $4500 per month.
Second priority: Sell no less than 80 microwaves per month.
Third priority: Ensure full utilization of production hours in Workshop A and Workshop B (weighting factor determined by the hourly cost ratio of each workshop).
Fourth priority: Workshop A overtime should not exceed 20 hours per month.
Fifth priority: Sell no less than 50 water heaters per month.
Please determine the monthly production plan that the company should establish in order to achieve the above objectives. | LP |
165 | Medium | A company has three factories that produce the same product, and now they need to transport the products from the three factories to four customers. The supply capacity of the factories, the demand of the customers, and the unit transportation cost from the three factories to the four customers are shown in Table 3.4 (the number in the top right corner of the table is the unit transportation cost).
Table 3.4
| Factory | | | | | Customer | | | | Supply |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| | 1 | | 2 | | 3 | | 4 | | |
| 1 | | 5 | | 2 | | 6 | | 7 | 30 |
| 2 | | 3 | | 5 | | 4 | | 10 | 20 |
| 3 | | 4 | | 3 | | 2 | | 13 | $\Delta 0$ |
| Demand | 20 | | 10 | | 45 | | 25 | | 100 |
Now, a transportation plan needs to be made to meet the following requirements:
$p_{1}$: The order quantity of Customer 4 must be fully satisfied first.
$p_{2}$: The degree of satisfaction of the order quantity for the remaining customers should not be less than 80%.
$p_{3}$: The quantity of products transported from Factory 3 to Customer 1 should be at least 15 units.
$p_{4}$: Due to route restrictions, Factory 2 should preferably not be allocated to Customer 4.
$p_{5}$: The degree of satisfaction of the demands of Customer 1 and Customer 3 should be balanced as much as possible.
$p_{6}$: Minimize the total transportation cost.
Try to establish a goal programming model for the above problem. | LP |
16 | Hard | The planning of the operation and use of arrival and departure lines at large passenger railway stations in our country is mainly carried out by dispatchers, and the adjustment of the operation plan is not flexible. In the case of a fixed scale of arrival and departure lines at the station, optimizing and improving the technical operations of trains can help improve the throughput capacity of railway stations. In order to adapt to the characteristics of railway operation and improve the efficiency of station technical operations, the feasibility and real-time performance of adjusting and optimizing station operation plans need to be continuously improved. The allocation and use of approach tracks and arrival and departure lines in the throat area of the station are key links in the preparation and adjustment of operation plans, which need to be continuously studied to meet the needs of on-site operations. The arrival and departure line of a railway refers to the route where trains stop at the station, pick up and drop off passengers, and perform dispatch operations. The arrival and departure line is usually a line or multiple lines parallel or connected to the main line, used for train entry and exit, as well as train stopping and dispatching. The allocation and preparation of arrival and departure lines for railway trains refers to the reasonable arrangement of the operation plan and stopping tracks of railway trains on different arrival and departure lines based on the needs of train operation and the conditions of arrival and departure lines.
### Question Description
There are multiple optimization objectives for the allocation of railway train arrival and departure lines, such as:
- Minimizing the dwell time of trains, reducing the waiting time for trains to park, and improving the efficiency of train operation;
- Maximizing the speed of trains, reducing the travel time of trains, and improving the efficiency of train operation;
- Maximizing the throughput capacity of the station, allowing the station to accommodate and dispatch more trains at the same time, and meet the travel needs of more passengers;
- Minimizing the delay rate of trains, reducing train delays, and improving the punctuality and reliability of trains;
- Maximizing passenger satisfaction and providing a convenient travel experience for passengers;
- Other optimization objectives.
Please model the medium-term arrangement of approach tracks in the train station based on the job shop model. Use the COPT (Cardinal Optimizer) solver to solve this problem. Output the train timetable for each train, as well as the arrival and departure timetable for each station.
### Data Introduction
We assume that a total of 16 trains are operated, and model the approach tracks and track selection in Station A. A track refers to a lane in the station where trains can stop. Each station has a certain number of tracks depending on its size, used to park different trains. The station boundary refers to the boundary of the station area, which is the dividing point between entering and leaving the station. An approach track refers to the path a train takes when entering the station and reaching a track. Generally, there are multiple approach tracks connecting different tracks in the station. Trains choose different approach tracks before entering the station, and then stop on the corresponding tracks.
### Train Data
We consider 16 trains on the line, each with different speeds and stopping information. Each column in the table represents: train number, train speed, train stop status. "1" indicates that the train stops at the station, "0" indicates that the train does not stop at the station.
| trainNO | speed | A | B | C | D | E | F | G |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| G1 | 350 | 1 | 0 | 0 | 0 | 0 | 1 | 1 |
| G3 | 350 | 1 | 1 | 1 | 0 | 1 | 0 | 1 |
| G5 | 350 | 1 | 1 | 0 | 1 | 0 | 1 | 1 |
| G7 | 350 | 1 | 1 | 0 | 0 | 1 | 0 | 1 |
| G9 | 350 | 1 | 1 | 0 | 0 | 0 | 1 | 1 |
| G11 | 350 | 1 | 1 | 0 | 0 | 1 | 0 | 1 |
| G13 | 300 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| G15 | 300 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| G17 | 300 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| G19 | 300 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| G21 | 300 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| G23 | 300 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| G25 | 300 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| G27 | 300 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| G29 | 300 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| G31 | 300 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
### Station Data
There are 5 tracks in Station A, among which tracks I, II, and III take 4 minutes to reach the station boundary, and tracks IV and V take 5 minutes to reach the station boundary. The time to the station boundary refers to the time it takes for a train to start from a track and pass through the station boundary, including the entire process.
|track|time to boundary (min)|
|:---:|:---:|
|I|4|
|II|4|
|III|4|
|IV|5|
|V|5|
The optional approach tracks and conflict relationships within Station A are as follows. Approach track conflict refers to the occurrence of collisions or operational impacts if two trains choose the same or intersecting approach tracks. To avoid conflicts, at any time, each approach track or any pair of intersecting approach tracks should only allow one train to use them.
|path|from|conflict|
|:---:|:---:|:---|
|1|I|2, 3, 4, 5, 6, 7|
|2|I|1, 3, 4, 5, 6, 7|
|3|II|1, 2, 4, 5, 6, 7|
|4|III|1, 2, 3, 5, 6, 7|
|5|IV|1, 2, 3, 4, 6, 7|
|6|IV|1, 2, 3, 4, 5, 7|
|7|V|1, 2, 3, 4, 5, 6|
Here we assume that the approach tracks conflict with each other pairwise, and at a more detailed level of approach track arrangement, approach tracks to the same boundary may not conflict.
### Other Data
- Since Station A is the starting station, the minimum dwell time is 20 minutes.
- The safe interval between any two trains is 3 minutes. | MIP |
146.0 | Hard | Mary is planning tonight's dinner and wants to choose either chicken, salmon, or tofu as a source of protein in addition to the vegetables she had previously considered. Chicken contains 23 grams of protein per 100 grams, salmon contains 20 grams of protein per 100 grams, and tofu contains 8 grams of protein per 100 grams. Chicken is priced at $3 per 100 grams, salmon is priced at $5 per 100 grams, and tofu is priced at $1.5 per 100 grams. Considering Mary's budget of $20, how should she choose the ingredients to maximize her protein intake? Additionally, the total weight of the food should not exceed 800 grams, and she needs to choose at least three types of vegetables and one source of protein. | IP |
1000.0 | Easy | A manufacturing company needs to transport 1800 units of products from the warehouse to three different sales points. The company has four transportation options to choose from: trucks, vans, motorcycles, and electric vehicles. Due to the high energy consumption of vans and electric vehicles, the company wants to choose only one of these two transportation options. Trucks generate 100 units of pollution per trip, vans generate 50 units of pollution, motorcycles generate 10 units of pollution, and electric vehicles generate 0 units of pollution. The total pollution generated by all trips must not exceed 2000 units. Trucks must be used at least 10 times. Trucks, vans, motorcycles, and electric vehicles can carry 100 units, 80 units, 40 units, and 60 units of products per trip, respectively. The company needs to ensure that the total quantity of products transported is at least 1800 units. | MIP |
1581550 | Medium | A company plans to produce three products, $A_{1}, A_{2}, A_{3}$, within a four-month timeframe (January to April). The demand for the products is shown in the table below:
| Demand | January | February | March | April |
| :---: | :---: | :---: | :---: | :---: |
| $A_{1}$ | 5300 | 1200 | 7400 | 5300 |
| $A_{2}$ | 4500 | 5400 | 6500 | 7200 |
| $A_{3}$ | 4400 | 6700 | 12500 | 13200 |
The prices, production costs, production quotas, activation costs, and minimum batch sizes (refer to the definitions in Exercise 4.3) are as follows:
| Product | $A_{1}$ | $A_{2}$ | $A_{3}$ |
| :---: | :---: | :---: | :---: |
| Unit Price | $\\$ 124$ | $\\$ 109$ | $\\$ 115$ |
| Activation Cost | $\\$ 150000$ | $\\$ 150000$ | $\\$ 100000$ |
| Production Cost | $\\$ 73.30$ | $\\$ 52.90$ | $\\$ 65.40$ |
| Production Quota | 500 | 450 | 550 |
| Minimum Batch Size | 20 | 20 | 16 |
January has 23 production days, February has 20, March has 23, and April has 22. The activation status of the production lines can be changed each month. The minimum batch size is calculated on a monthly basis.
In addition, storage space can be rented monthly at a cost of $\\$ 3.50$ for $A_{1}$, $\\$ 4.00$ for $A_{2}$, and $\\$ 3.00$ for $A_{3}$. Each product takes up the same amount of storage space. The total available capacity is 800 units.
Write a mathematical program to maximize revenue. | MIP |
2.78195 | Hard | There are 10 tasks that must be run on 3 CPUs, with frequencies of 1.33, 2, and $2.66 \mathrm{GHz}$ respectively (each processor can only run one task at a time). The basic instructions (in billions of instructions, BI) for each task are as follows:
| Process | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| BI | 1.1 | 2.1 | 3 | 1 | 0.7 | 5 | 3 |
Arrange the tasks onto the processors to minimize the completion time of the last task. Use COPTPY to solve the problem. | IP |
10000 | Hard | A company that produces only one product has 40 workers. Each worker produces 20 units per month. The demand for the semester changes according to the following table:
| Month | 1 | 2 | 3 | 4 | 5 | 6 |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| Demand (units) | 700 | 600 | 500 | 800 | 900 | 800 |
To adjust production based on demand, the company can offer some (paid) additional working hours (each worker can produce a maximum of 6 extra units per month, with a unit cost of 5 euros), use warehouse space (10 euros per unit per month), and hire or lay off employees (the number of employees can vary by a maximum of $\\pm 5$ per month, with a hiring cost of 500 euros per person and a layoff cost of 700 euros per person).
Initially, the warehouse space is empty, and we require it to be empty at the end of the semester as well. Develop a mathematical program to maximize revenue and solve it using COPTPY. How does the objective function change when all variables are relaxed to continuous? | MIP |
153 | Hard | A traveling salesman must visit 7 customers at 7 different locations, and the (symmetric) distance matrix is as follows:
| | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| 1 | - | 86 | 49 | 57 | 31 | 69 | 50 |
| 2 | | - | 68 | 79 | 93 | 24 | 5 |
| 3 | | | - | 16 | 7 | 72 | 67 |
| 4 | | | | - | 90 | 69 | 1 |
| 5 | | | | | - | 86 | 59 |
| 6 | | | | | | - | 81 |
Develop a mathematical program to determine the visiting order starting from location 1 and ending at location 1, in order to minimize the traveling distance, and solve it using COPTPY. | IP |
103801 | Hard | The mass of a rocket is $m$, and it must reach a height of $H$ within time $T$ when launched at sea level. Let $y(t)$ be the height of the rocket at time $t$, and let $u(t)$ be the force acting on the rocket in the vertical direction at time $t$. Assuming $u(t)$ cannot exceed a given value $b$, the mass of the rocket $m$ remains constant throughout the process, and the gravitational acceleration $g$ remains constant in the interval $[0, H]$. Discretize the time $t \in [0, T]$ into $n$ intervals and propose a linear program to determine the force $u(t_k)$ at each moment $k \leq n$ to minimize the total energy consumption. Solve this problem using the following data: $m=2140~\mathrm{kg}$, $H=23~\mathrm{km}$, $T=1~\mathrm{min}$, $b=10000~\mathrm{N}$, $n=20$. | LP |
8505 | Medium | A company has three types of products, I, II, and III. The contract orders for each quarter of the next year are shown in Table 1-23. At the beginning of the first quarter, there is no inventory, and it is required to have 150 units of each product in inventory at the end of the fourth quarter. It is known that the company has 15,000 hours of production time per quarter, and it takes 2 hours, 4 hours, and 3 hours to produce one unit of products I, II, and III, respectively. Due to the replacement of production equipment, product I cannot be produced in the second quarter. It is specified that if the products cannot be delivered on time, a compensation of $20 per unit per quarter is required for products I and II, and $10 for product III. If the produced products are not delivered in the same quarter, a storage fee of $5 per unit per quarter is incurred. How should the company arrange production to minimize the total compensation and storage fee?
Table 1-23
\begin{tabular}{c|c|c|c|c}
\hline \multirow{2}{*}{Product} & \multicolumn{4}{|c}{Contract Orders for Each Quarter} \\
\cline { 2 - 5 } & 1 & 2 & 3 & 4 \\
\hline I & 1500 & 1000 & 2000 & 1200 \\
II & 1500 & 1500 & 1200 & 1500 \\
III & 1000 & 2000 & 1500 & 2500 \\
\hline
\end{tabular} | LP |
5069500 | Medium | The market demand for products I and II is as follows: Product I requires 10,000 units per month from January to April, 30,000 units per month from May to September, and 100,000 units per month from October to December. Product II requires 15,000 units per month from March to September and 50,000 units per month for the other months. The production cost for these two products is as follows: Product I costs $5 per unit to produce from January to May and $4.50 per unit to produce from June to December. Product II costs $8 per unit to produce from January to May and $7 per unit to produce from June to December. The total production capacity for both products should not exceed 120,000 units per month. Product I occupies 0.2 cubic meters per unit, while Product II occupies 0.4 cubic meters per unit. The warehouse capacity of the factory is 15,000 cubic meters. If the warehouse is not sufficient, the factory can rent additional space from an external facility. If occupying 1 cubic meter of the factory's storage space incurs a cost of $1 per month, and renting storage space from an external facility incurs an additional cost of $1.5 per month, how should the factory arrange its production to minimize the total production cost including storage fees while meeting market demand? | MIP |
105.52 | Medium | A factory produces two types of food, I and II. There are currently 50 skilled workers. It is known that one skilled worker can produce food I at a rate of 10 kg/h or food II at a rate of 6 kg/h. According to the contract, the demand for these two types of food will sharply increase each week, as shown in Table 1-11. To meet this demand, the factory plans to train 50 new workers by the end of the 8th week, with production in two shifts. It is known that one worker works 40 hours per week, and a skilled worker can train no more than three new workers in two weeks (during the training period, skilled workers and trainees do not participate in production). Skilled workers earn a weekly salary of 360 yuan, trainees earn a weekly salary of 120 yuan during the training period, and a weekly salary of 240 yuan after training is completed. Their production efficiency is the same as that of skilled workers. During the training transition period, many skilled workers are willing to work overtime. The factory decides to arrange for some workers to work 60 hours per week at a salary of 540 yuan per week. If the ordered food cannot be delivered on time, there is a compensation fee of 0.50 yuan per kg for food I and 0.60 yuan per kg for food II. Under the above conditions, how should the factory make comprehensive arrangements to minimize the total cost?
Table 1-11
\begin{tabular}{|c|c|c|c|c|c|c|c|c|}
\hline Food Week & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\
\hline I & 10 & 10 & 12 & 12 & 16 & 16 & 20 & 20 \\
\hline II & 6 & 7. 2 & 8.4 & 10.8 & 10.8 & 12 & 12 & 12 \\
\hline
\end{tabular} | MIP |
-99999 | Medium | A large paper manufacturing company has 10 paper mills and supplies 1000 customers. These paper mills use three interchangeable machines and four different raw materials to produce five types of paper. The company wants to develop a plan to determine the quantity of each type of paper produced on each machine in each mill, and to determine which type of paper is supplied to which customers and in what quantity, in order to minimize total transportation costs. The following information is known:
$D_{j k}$ - Customer j requires k units of paper each month;
$r_{k l m}$ - The amount of raw material m required to produce one unit of paper of type k on machine l;
$R_{i m}$ - The amount of raw material m available each month at mill i;
$c_{k l}$ - The machine time required to produce one unit of paper of type k on machine l;
$c_{i l}$ - The machine time available each month at mill i for machine type l;
$P_{i k l}$ - The cost of producing one unit of paper of type k on machine type l at mill i;
$T_{i j k}$ - The cost of transporting one unit of paper of type k from mill i to customer j.
Develop a linear programming model for this problem. | LP |
76 | Medium | A certain factory needs a special tool in $n$ planning stages. In the $j$-th stage, $r_j$ specialized tools are required. At the end of each stage, the tools used in that stage must be sent for repair before they can be used again. There are two types of repairs: slow repair, which is cheaper (costs $b$ dollars per repair) but takes longer (requires $p$ stages to retrieve the tool), and fast repair, which is more expensive (costs $c$ dollars per repair, where $c>b$) but faster (requires $q$ stages to retrieve the tool, where $q<p$). If the repaired tools cannot meet the requirements, new tools need to be purchased at a cost of $a$ dollars per tool ($a>c$). Furthermore, these specialized tools will not be used after $n$ stages. Determine an optimal plan for purchasing and repairing tools to minimize the cost spent on tools during the planning period.
n = 10 # number of stages
r = [0] + [3, 5, 2, 4, 6, 5, 4, 3, 2, 1] # tool requirements per stage, indexing starts at 1
a = 10 # cost of buying a new tool
b = 1 # cost of slow repair
c = 3 # cost of fast repair
p = 3 # slow repair duration
q = 1 # fast repair duration | MIP |
44480.0 | Medium | A company has three tasks that require the recruitment of skilled workers and laborers. The first task can be completed by a skilled worker alone or by a team consisting of one skilled worker and two laborers. The second task can be completed by either a skilled worker or a laborer alone. The third task can be completed by a team of five laborers or by a skilled worker leading three laborers. It is known that the weekly wages for skilled workers and laborers are 100 yuan and 80 yuan, respectively. They work 48 hours per week, but their actual effective working time is 42 hours and 36 hours, respectively. To complete these three tasks, the company needs a total effective working time of 10,000 hours for the first task, 20,000 hours for the second task, and 30,000 hours for the third task. The number of workers that can be recruited is limited to a maximum of 400 skilled workers and 800 laborers. Establish a mathematical model to determine the number of skilled workers and laborers to recruit in order to minimize the total wage expenditure. | MIP |
13400 | Medium | Xuri Company has signed delivery contracts for 5 products $(i=1, \cdots, 5)$ for the next year from January to June. It is known that the order quantity (in units), unit selling price (in yuan), unit cost (in yuan), and production hours required for each product are $D_i, S_i, C_i, a_i$, respectively. The normal production hours and maximum allowable overtime hours for each month from January to June are shown in Table 1-19.
Table 1-19
\begin{tabular}{c|r|r|r|r|r|r}
\hline Month & \multicolumn{1}{|c|}{1} & \multicolumn{1}{|c|}{2} & \multicolumn{1}{|c}{3} & \multicolumn{1}{|c}{4} & \multicolumn{1}{c}{5} & \multicolumn{1}{c}{6} \\
\hline Normal Production Hours $/ \mathrm{h}$ & 12000 & 11000 & 13000 & 13500 & 13500 & 14000 \\
\hline Maximum Allowable Overtime Hours $/ \mathrm{h}$ & 3000 & 2500 & 3300 & 3500 & 3500 & 3800 \\
\hline
\end{tabular}
However, the cost of each product produced during overtime hours increases by $C_i^{\prime}$ yuan. Due to production preparation and delivery requirements, product 1 is scheduled to start production in March, product 3 needs to be delivered by the end of April, and product 4 can be produced starting from February and must be fully delivered by the end of May. If products 3 and 4 are delayed in delivery, a penalty of $p_3$ yuan and $p_4$ yuan per month of delay will be imposed, respectively. All products must be delivered by the end of June. Please design a production plan for the company that guarantees completion of the contracts and maximizes profit, and establish a mathematical model. | MIP |
527.9999999999999 | Medium | Hongsheng Factory produces three types of products, I, II, and III, which all undergo processing in two steps, A and B. Step A has two machines, A1 and A2, while step B has three machines, B1, B2, and B3. It is known that product I can be processed on either type of A machine, product II can be processed on any type of A machine but can only be processed on machine B2 in step B, and product III can only be processed on machine B2 in step B. Products II and III can only be processed on machines A2 and B2 in step A and B, respectively. The processing time per unit product and other relevant data are shown in Table 1-20. How should the production plan be arranged to maximize profit for the factory?
Table 1-20
\begin{tabular}{|c|c|c|c|c|c|}
\hline \multirow{2}{*}{ Machine } & \multicolumn{3}{|c|}{ Products } & \multirow{2}{*}{ Effective Machine Hours } & \multirow{2}{*}{\begin{tabular}{c}
Machine Processing Fee \\
/(元/h)
\end{tabular}} \\
\hline & I & II & III & & \\
\hline A1 & 5 & 10 & $2+4$ & 6000 & 0.05 \\
\hline A2 & 7 & 9 & 12 & 10000 & 0.03 \\
\hline B1 & 6 & 8 & & 4000 & 0.06 \\
\hline B2 & 4 & & 11 & 7000 & 0.11 \\
\hline B3 & 7 & & & 4000 & 0.05 \\
\hline Raw Material Cost/(元/件) & 0.25 & 0.35 & 0.50 & & \\
\hline Selling Price/(元/件) & 1.25 & 2.00 & 2.80 & & \\
\hline
\end{tabular} | MIP |
-99999 | Medium | Jiali Company needs to recruit personnel from three different majors to work in the branch offices located in Donghai City and Nanjiang City. The demand for personnel from different majors in these two branch offices is shown in Table 4-3. After statistical analysis of the applicants, the company divides them into 6 categories, and Table 4-4 lists the majors that each category of personnel is competent in, the majors they prefer to prioritize, and the cities they prefer to work in. The company considers personnel arrangements based on the following three priorities:
$p_1$: All three types of required personnel are satisfied;
$p_2$: Among the recruited personnel, 8000 people satisfy their preferred majors;
$p_3$: Among the recruited personnel, 8000 people satisfy their preferred cities.
Based on this, establish a mathematical model for goal programming.
Table 4-3
\begin{tabular}{c|c|c}
\hline Branch Location & Major & Demand \\
\hline \multirow{3}{*}{Donghai City} & 1 & 1000 \\
& 2 & 2000 \\
\hline \multirow{4}{*}{Nanjiang City} & 3 & 1500 \\
& 1 & 2000 \\
& 2 & 1000 \\
& 3 & 1000 \\
\hline
\end{tabular}
Table 4-4
\begin{tabular}{c|c|c|c|c}
\hline Category & Personnel & Competent Majors & Preferred Majors & Preferred Cities \\
\hline 1 & 1500 & 1,2 & 1 & Donghai \\
2 & 1500 & 2,3 & 2 & Donghai \\
3 & 1500 & 1,3 & 1 & Nanjiang \\
4 & 1500 & 1,3 & 3 & Nanjiang \\
5 & 1500 & 2,3 & 3 & Donghai \\
6 & 1500 & 3 & 3 & Nanjiang \\
\hline
\end{tabular} | Others |
21 | Medium | There are $m$ production points for a certain commodity, where the output of the $i$th point $(i=1, \cdots, m)$ is $a_i$. The commodity is sold to $n$ demand points, where the required quantity at the $j$th demand point is $b_j$ $(j=1, \cdots, n)$. It is known that $\sum_i a_i \geqslant \sum_j b_j$. It is also known that when transporting from each production point to a demand point, it must go through one of the $p$ intermediate transfer stations, and if the $k$th intermediate transfer station is used, regardless of the amount of transfer, a fixed cost of $f_k$ is incurred, and the maximum capacity limit for the $k$th intermediate transfer station is $q_k$ $(k=1, \cdots, p)$. Let $c_{i k}$ and $c_{k j}$ represent the transportation costs per unit of commodity from $i$ to $k$ and from $k$ to $j$, respectively. Determine a transportation plan for the commodity that minimizes the total cost. | MIP |
1000.0 | Easy | There are 10 different parts that can be processed on equipment A, equipment B, or equipment C, with the processing costs per unit shown in Table 5-6. Furthermore, as long as parts are processed on the mentioned equipment, regardless of processing one or multiple parts, the one-time setup costs are $d_A$, $d_B$, and $d_C$ respectively. If the following requirements are to be met:
(1) Each of the 10 parts should be processed once.
(2) If the first part is processed on equipment A, then the second part should be processed on equipment B or C. Conversely, if the first part is processed on equipment B or C, then the second part should be processed on equipment A.
(3) Parts 3, 4, and 5 must be processed on equipment A, B, and C respectively.
(4) The number of parts processed on equipment C should not exceed 3.
Please establish an integer programming mathematical model for this problem, with the objective of minimizing the total cost.
Table 5-6
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|}
\hline Equipment $\quad$ Part & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\
\hline A & $a_1$ & $a_2$ & $a_3$ & $a_4$ & $a_5$ & $a_6$ & $a_7$ & $a_8$ & $a_9$ & $a_{10}$ \\
\hline B & $b_1$ & $b_2$ & $b_3$ & $b_4$ & $b_5$ & $b_6$ & $b_7$ & $b_8$ & $b_9$ & $b_{10}$ \\
\hline C & $c_1$ & $c_2$ & $c_3$ & $c_4$ & $c_5$ & $c_6$ & $c_7$ & $c_8$ & $c_9$ & $c_{10}$ \\
\hline
\end{tabular} | IP |
770.0 | Easy | Changjiang Comprehensive Shopping Mall has an area of $5000 \mathrm{~m}^2$ for rent and plans to attract tenants from the following 5 types of stores. It is known the area occupied by each store, the minimum and maximum number of stores to be opened in the mall, and the annual expected profit (in ten thousand yuan) for each store at different numbers of stores opened, as shown in Table 5-12. Each store is required to pay the mall 20% of their annual profit as rent. What is the optimal number of each type of store the mall should rent in order to maximize the total rental income?
Table 5-12:
\begin{tabular}{c|c|c|c|c|c|c|c}
\hline
\multirow{2}{*}{Code} & \multirow{2}{*}{Store Type} & Store Area $/ \mathrm{m}^2$ & \multicolumn{2}{|c|}{Number of Stores} & \multicolumn{3}{|c}{Profit per Store at Different Numbers} \\
\cline{3-8}
& & & Minimum & Maximum & 1 & 2 & 3 \\
\hline
1 & Jewelry & 250 & 1 & 3 & 9 & 8 & 7 \\
2 & Shoes and Hats & 350 & 1 & 2 & 10 & 9 & - \\
3 & Department Store & 800 & 1 & 3 & 27 & 21 & 20 \\
4 & Bookstore & 400 & 0 & -2 & 16 & 10 & - \\
5 & Restaurant & 500 & 1 & 3 & 17 & 15 & 12 \\
\hline
\end{tabular} | IP |
32.436 | Easy | Suppose an animal needs at least $700 \mathrm{~g}$ of protein, $30 \mathrm{~g}$ of minerals, and $100 \mathrm{mg}$ of vitamins per day. There are 5 types of feed to choose from, and the contents of each nutrient per gram and the unit price of each feed are shown in Table 1-5. Try to establish a linear programming model that not only meets the animal's growth needs but also minimizes the cost of selecting feed.
Table 1-6
\begin{tabular}{|c|c|c|c|c||c|c|c|c|c|}
\hline Feed & \begin{tabular}{c}
Protein \\
$/ \mathrm{g}$
\end{tabular} & \begin{tabular}{c}
Minerals \\
$/ \mathrm{g}$
\end{tabular} & \begin{tabular}{c}
Vitamins \\
$/ \mathrm{mg}$
\end{tabular} & \begin{tabular}{c}
Price \\
$/$ (yuan $/ \mathrm{kg})$
\end{tabular} & Feed & \begin{tabular}{c}
Protein \\
$/ \mathrm{g}$
\end{tabular} & \begin{tabular}{c}
Minerals \\
$/ \mathrm{g}$
\end{tabular} & \begin{tabular}{c}
Vitamins \\
$/ \mathrm{mg}$
\end{tabular} & \begin{tabular}{c}
Price \\
$/($ yuan $/ \mathrm{kg})$
\end{tabular} \\
\hline 1 & 3 & 1 & 0.5 & 0.2 & 4 & 6 & 2 & 2 & 0.3 \\
\hline 2 & 2 & 0.5 & 1 & 0.7 & 5 & 18 & 0.5 & 0.8 & 0.8 \\
\hline 3 & 1 & 0.2 & 0.2 & 0.4 & & & & & \\
\hline
\end{tabular} | LP |
1146.6 | Hard | A factory produces three types of products: I, II, and III. Each product requires two processing steps, A and B. The factory has two types of equipment for step A, represented by A1 and A2, and three types of equipment for step B, represented by B1, B2, and B3. Product I can be processed on any type of A equipment and any type of B equipment. Product II can be processed on any type of A equipment, but for step B, it can only be processed on B1 equipment. Product III can only be processed on A2 and B2 equipment. The unit processing time, raw material cost, selling price, equipment available time, and equipment cost at full load for each type of equipment are shown in Table 1-18. Determine the optimal production plan to maximize the profit for the factory.
Table 1-18
\begin{tabular}{c|c|c|c|c|c}
\hline \multirow{2}{*}{ Equipment } & \multicolumn{3}{|c|}{ Product } & \multirow{2}{*}{ Equipment Available Time } & \multirow{2}{*}{\begin{tabular}{c}
Equipment Cost at \\
Full Load (CNY)
\end{tabular}} \\
\cline { 2 - 4 } & I & II & III & 6000 & 300 \\
$A_1$ & 5 & 10 & & 10000 & 321 \\
$A_2$ & 7 & 9 & 12 & 4000 & 250 \\
$B_1$ & 6 & 8 & & 7000 & 783 \\
$B_2$ & 4 & & 11 & 4000 & 200 \\
$B_3$ & 7 & & & & \\
\hline Raw Material Cost (CNY / unit) & 0.25 & 0.35 & 0.50 & & \\
Unit Price (CNY / unit) & 1.25 & 2.00 & 2.80 & & \\
\hline
\end{tabular} | MIP |
4500 | Medium | A certain brand of wine is made by blending three grades of wine. The daily supply and unit cost of these three grades of wine are shown in Table 4-8.
Table 4-8
\begin{tabular}{c|c|c}
\hline Grade & Daily Supply (kg) & Cost (元/kg) \\
\hline I & 1500 & 6 \\
I & 2000 & 4.5 \\
II & 1000 & 3 \\
\hline
\end{tabular}
Assume that there are three brands (red, yellow, blue) for this type of wine, and the blending ratios and selling prices for each brand are shown in Table 4-9. The decision maker specifies that the wines for each brand must be blended strictly according to the specified ratios, followed by maximizing profit, and ensuring that the red brand produces at least 2000 kg per day. Formulate the mathematical model.
Table 4-9
\begin{tabular}{c|c|c}
\hline Brand & Blending Requirements & Selling Price (元/kg) \\
\hline Red & \begin{tabular}{l}
I less than 10\% \\
I more than 50\%
\end{tabular} & 5.5 \\
\hline Yellow & \begin{tabular}{l}
II less than 70\% \\
I more than 20\%
\end{tabular} & 5.0 \\
\hline Blue & \begin{tabular}{l}
I less than 50\% \\
I more than 10\%
\end{tabular} & 4.8 \\
\hline
\end{tabular} | LP |
2924 | Easy | A product consists of three components, which are produced by four workshops. The total number of production hours available for each workshop is limited. Table 1.4 shows the productivity of the three components. The objective is to determine how many hours should be allocated to each component in each workshop in order to maximize the number of completed products. Represent this problem as a linear programming problem.
Table 1.4
| Workshop | Production Capacity <br> (hours) | Productivity (units/hour) | | |
| :---: | :---: | :---: | :---: | :---: |
| | | Component 1 | Component 2 | Component 3 |
| A | 100 | 10 | 15 | 5 |
| B | 150 | 15 | 10 | 5 |
| C | 80 | 20 | 5 | 10 |
| D | 200 | 10 | 15 | 20 | | LP |
11250 | Medium | A factory produces two models of microcomputers, A and B. Each model requires two identical processes. The processing time, sales profit, and maximum processing capacity of the factory per week for each microcomputer model are shown in Table 3.1.
Table 3.1
| Process | Model | Maximum Processing Capacity |
| :---: | :---: | :---: |
| | A | B | |
| Process I (hours per unit) | 4 | 6 | 150 |
| Process II (hours per unit) | 3 | 2 | 70 |
| Profit (RMB per unit) | 300 | 450 | |
If the factory's desired objectives and priority levels are as follows:
$p_{1}$: Total profit per week should not be less than 10,000 RMB;
$p_{2}$: Due to contractual requirements, at least 10 units of model A and 15 units of model B must be produced per week;
$p_{3}$: The weekly production time for Process I should be exactly 150 hours, and the production time for Process II should be maximized, even if overtime is necessary.
Establish the goal programming model for this problem. | LP |
1250 | Medium | A factory produces two models of microcomputers, A and B. Each model requires two identical processes. The processing time, sales profit, and maximum processing capacity per week for each microcomputer are shown in Table 3.1.
Table 3.1
| Process | Model A | Model B | Maximum Processing Capacity |
| :---: | :---: | :---: | :---: |
| I (hours per unit) | 4 | 6 | 150 |
| II (hours per unit) | 3 | 2 | 70 |
| Profit (RMB per unit) | 300 | 450 | |
If the factory's desired objectives and priority levels are as follows:
$p_{1}$: The total weekly profit should not be less than 10,000 RMB.
$p_{2}$: Due to contractual requirements, at least 10 units of model A and 15 units of model B should be produced per week.
$p_{3}$: It is desired that the weekly production time for process I is exactly 150 hours, and the production time for process II is maximized, even with overtime if necessary.
If the products produced during overtime in process II result in a decrease of 20 RMB in profit per unit for model A and 25 RMB for model B, and the maximum overtime for process II is 30 hours per week, with this objective as level $p_{4}$, establish an objective programming model for this problem. | MIP |
0.0 | Hard | The current problem the company is facing is how to use the minimum number of containers to transport the current goods, while considering the weight, volume, specific packaging requirements, and inventory limitations of the goods. Professional modeling and analysis are needed to provide a transportation strategy for a batch of goods to ensure the maximum utilization of limited container space.
Currently, the company has a batch of goods that need to be transported. Each container can hold a maximum of 60 tons of goods, and each container used must contain at least 18 tons of goods. The dimensions of the container are 15.5 meters in length, 2.8 meters in width, and 2.8 meters in height. The goods that need to be loaded include five types: A, B, C, D, and E, with 120, 90, 300, 90, and 120 units respectively. The weight of type A is 0.5 tons, type B is 1 ton, type C is 0.4 tons, type D is 0.6 tons, and type E is 0.65 tons. The dimensions of goods A, B, C, D, and E are 0.65*1*1, 0.2*2*2, 0.35*2*1, 0.2*3*1, and 0.15*5*1 meters respectively. Additionally, to meet specific usage requirements, whenever loading goods A, at least 1 unit of goods C must be loaded at the same time, while loading goods C alone does not require loading goods A at the same time. Furthermore, considering the demand restrictions of goods D, we need to load at least 12 units of goods D in each container.
Create an operations research model so that the company can use the minimum number of containers to transport this batch of goods. | IP |
6105 | Medium | A company produces two types of small motorcycles, with Type A being completely manufactured by the company and Type B being assembled from imported parts. The manufacturing, assembly, and inspection times required for these two products are shown in Table 3.2.
Table 3.2
| Type | Manufacturing | Assembly | Inspection | Selling Price <br> (yuan per unit) |
| :---: | :---: | :---: | :---: | :---: |
| | | | | |
| Type A (hours per unit) | 20 | 5 | 3 | 650 |
| Type B (hours per unit) | 0 | 7 | 6 | 725 |
| Maximum weekly production capacity (hours) | 120 | 80 | 40 | |
| Hourly production cost (yuan) | 12 | 8 | 10 | |
If the company's expected goals and priority levels are as follows:
$p_{1}$: Total weekly profit is at least 3000 yuan;
$p_{2}$: At least 5 units of Type A motorcycles are produced weekly;
$p_{3}$: Minimize the idle time of each process, with the coefficients of the three processes being proportional to their hourly costs, and no overtime is allowed.
Formulate an objective programming model for this problem. | LP |
58 | Medium | Consider the task of delivering goods from a central warehouse to $m$ different sales locations. In each delivery, each sales location receives its order. Feasible routes have been specified for each delivery person, and each delivery person can transport up to $r$ types of orders. Assume there are $n$ feasible routes, and each route specifies the sales locations to deliver goods to. Let $c_{j}$ be the cost of the $j$th route. It is possible to have duplicates, so that more than one delivery person can arrive at the same sales location. Represent this problem as an integer model. | IP |
770.0 | Medium | Consider the allocation of $n$ factories to $n$ locations, where the transportation quantity between factory $i$ and factory $j$ is $d_{ij}$, and the unit transportation cost from location $p$ to location $q$ is $c_{pq}$. To minimize the total transportation cost, represent this problem as an integer model. | IP |
9337440 | Hard | For manufacturing companies, it is important to develop appropriate production plans and human resource management strategies to reduce operating costs, inventory costs, stockout costs, and labor costs. In particular, for products with high demand fluctuations, companies need to accurately forecast demand and develop corresponding production plans. A company produces a folding table, with a raw material cost of 90 yuan per unit and 5 labor hours. The unit price of the product is 300 yuan. At the beginning of January, the company has 1000 workers and holds 15,000 units of inventory. The normal wage for workers is 30 yuan per hour, and the normal working hours per worker per day is 8 hours. Overtime hours are paid at an hourly rate of 40 yuan. The number of working days per month is calculated as 20 days. Assuming the company has sufficient available production machines and its production capacity is not limited by machine hours. The overtime hours per worker per month do not exceed 20 hours. Holding inventory incurs corresponding inventory costs. If there is a stockout due to insufficient capacity, there will be certain stockout costs. In addition, the company can make up for the shortage through outsourcing. The outsourcing cost per unit, monthly inventory cost, and monthly stockout cost are 200 yuan, 15 yuan, and 35 yuan, respectively. The company's employees are all temporary workers, and the company can flexibly decide the number of employees to hire and dismiss each month. The cost of hiring and dismissing a single employee is 5000 yuan and 8000 yuan, respectively. Assuming the company has used existing forecasting models to predict the demand from month 1 to month 6, as shown in Table 4.1.
Table 4.1: Demand Forecast
| Month | 1 | 2 | 3 | 4 | 5 | 6 |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| Demand Forecast | 20000 | 40000 | 42000 | 35000 | 19000 | 18500 |
Please develop a production plan, worker schedule, and plan for hiring and dismissing employees for the company to maximize its total net profit over 6 months, while ensuring that the company holds at least 10,000 units of product by the end of June. | MIP |
1644.63 | Hard | The Vehicle Routing Problem (VRP) was first proposed by Dantzig in 1959 (Dantzig and Ramser 1959). It is a very classic combinatorial optimization problem. The basic VRP can usually be described as follows: in a certain area, there are a certain number of customers and a distribution center or warehouse. The customers are generally distributed in different locations, and each customer has a certain amount of goods delivery demand. The distribution center or warehouse needs to dispatch a fleet of vehicles and design a suitable delivery plan to fulfill all customer's goods delivery demands. The objective of VRP is to maximize efficiency while meeting all customer demands. The measurement of efficiency is usually presented in the form of an objective function. The objective function varies with the company's requirements, and common objective functions include minimizing the total distance traveled by vehicles, minimizing the total delivery time, and minimizing the number of vehicles used, etc. In addition to meeting customer delivery demands, VRP generally needs to consider various other constraints, resulting in various variants. For example, if the vehicle's load cannot exceed its maximum capacity (capacity constraint), the problem becomes the Capacitated Vehicle Routing Problem (CVRP). If the delivery demand of each customer must be delivered within a specific time window, the problem becomes the Vehicle Routing Problem with Time Windows (VRPTW).
The Vehicle Routing Problem with Time Windows (VRPTW) is a classic problem developed from VRP. There are many applications of VRPTW in reality, and each customer point generally has a service time window. For example, some logistics centers need to deliver packages during low-traffic hours, some large supermarkets need to restock during non-business hours, and real-time delivery methods such as food delivery require specific delivery time windows. Time windows can be classified as hard time windows and soft time windows. Hard Time Windows (HTW) means that vehicles must arrive at or before the time window, and cannot delay their arrival. If they arrive early, they must wait until the time window starts to provide service, such as restocking supermarkets and delivering goods from logistics centers, etc. Soft Time Windows (STW) means that vehicles are not required to arrive within the original hard time window, but they should try to arrive within the hard time window. If the vehicle arrives early, a penalty value will be deducted, and if it arrives late, a certain penalty value will be imposed, such as food delivery, school bus pick-up and drop-off, industrial distribution, etc.
The Vehicle Routing Problem with Hard Time Windows (VRPHTW) can be described as follows: there are a certain number of customer points and a distribution center in the region. Vehicles need to depart from the distribution center and return to the distribution center. The route should be continuous, and each customer should be served by one and only one vehicle. The vehicles also have a capacity. Each customer has a specific service time window and can only be served within the time window. Vehicles can arrive at the customer point ahead of time and wait for the time window to open, then provide service to the customer. They can also arrive at the customer point within the time window to provide service. Vehicles can only start serving customers within the time window, and the service time is known. The distribution center needs to arrange a reasonable delivery plan to both complete the delivery tasks and minimize travel costs. VRPHTW, like VRP, mainly emphasizes the travel costs on the routes because delays are not allowed. Now, please model the VRPHTW in operations research.
### Example
r101.txt = r\"""
0 35 35 0 0 230 0
1 41 49 10 161 171 10
2 35 17 7 50 60 10
3 55 45 13 116 126 10
4 55 20 19 149 159 10
5 15 30 26 34 44 10
6 25 30 3 99 109 10
7 20 50 5 81 91 10
8 10 43 9 95 105 10
9 55 60 16 97 107 10
10 30 60 16 124 134 10
11 20 65 12 67 77 10
12 50 35 19 63 73 10
13 30 25 23 159 169 10
14 15 10 20 32 42 10
15 30 5 8 61 71 10
16 10 20 19 75 85 10
17 5 30 2 157 167 10
18 20 40 12 87 97 10
19 15 60 17 76 86 10
20 45 65 9 126 136 10
21 45 20 11 62 72 10
22 45 10 18 97 107 10
23 55 5 29 68 78 10
24 65 35 3 153 163 10
25 65 20 6 172 182 10
26 45 30 17 132 142 10
27 35 40 16 37 47 10
28 41 37 16 39 49 10
29 64 42 9 63 73 10
30 40 60 21 71 81 10
31 31 52 27 50 60 10
32 35 69 23 141 151 10
33 53 52 11 37 47 10
34 65 55 14 117 127 10
35 63 65 8 143 153 10
36 2 60 5 41 51 10
37 20 20 8 134 144 10
38 5 5 16 83 93 10
39 60 12 31 44 54 10
40 40 25 9 85 95 10
41 42 7 5 97 107 10
42 24 12 5 31 41 10
43 23 3 7 132 142 10
44 11 14 18 69 79 10
45 6 38 16 32 42 10
46 2 48 1 117 127 10
47 8 56 27 51 61 10
48 13 52 36 165 175 10
49 6 68 30 108 118 10
50 47 47 13 124 134 10
51 49 58 10 88 98 10
52 27 43 9 52 62 10
53 37 31 14 95 105 10
54 57 29 18 140 150 10
55 63 23 2 136 146 10
56 53 12 6 130 140 10
57 32 12 7 101 111 10
58 36 26 18 200 210 10
59 21 24 28 18 28 10
60 17 34 3 162 172 10
61 12 24 13 76 86 10
62 24 58 19 58 68 10
63 27 69 10 34 44 10
64 15 77 9 73 83 10
65 62 77 20 51 61 10
66 49 73 25 127 137 10
67 67 5 25 83 93 10
68 56 39 36 142 152 10
69 37 47 6 50 60 10
70 37 56 5 182 192 10
71 57 68 15 77 87 10
72 47 16 25 35 45 10
73 44 17 9 78 88 10
74 46 13 8 149 159 10
75 49 11 18 69 79 10
76 49 42 13 73 83 10
77 53 43 14 179 189 10
78 61 52 3 96 106 10
79 57 48 23 92 102 10
80 56 37 6 182 192 10
81 55 54 26 94 104 10
82 15 47 16 55 65 10
83 14 37 11 44 54 10
84 11 31 7 101 111 10
85 16 22 41 91 101 10
86 4 18 35 94 104 10
87 28 18 26 93 103 10
88 26 52 9 74 84 10
89 26 35 15 176 186 10
90 31 67 3 95 105 10
91 15 19 1 160 170 10
92 22 22 2 18 28 10
93 18 24 22 188 198 10
94 26 27 27 100 110 10
95 25 24 20 39 49 10
96 22 27 11 135 145 10
97 25 21 12 133 143 10
98 19 21 10 58 68 10
99 20 26 9 83 93 10
100 18 18 17 185 195 10
\""" | MIP |
40 | Hard | As one of the main arteries of urban transportation, the city subway carries a huge passenger flow and its operation and management are very complex. Among them, crew scheduling is one of the key links in subway operation management and also the starting point and difficulty of crew management. The subway transportation volume is large, with complex and changing routes, and frequent operation intervals. Crew drivers face huge, continuous, and high-intensity transportation tasks. It is important to arrange the driver's shift and duty mode reasonably and achieve the optimal allocation of personnel to ensure the efficient and stable operation of the subway. Due to the increasing complexity of the subway operation network, traditional manual scheduling methods can no longer meet the high standard operation requirements. Automated and intelligent crew scheduling has become an inevitable trend.
The general process of crew scheduling for subway operating companies is as follows: export train operation section and time information tables based on the current operation chart, manually divide the crew duty roster based on these information tables, compile the crew scheduling master table based on the duty roster, and finally allocate the crew groups based on the scheduling master table to form the final scheduling table. The entire process relies on manual compilation and mainly has the following three major problems:
1. Low scheduling efficiency: Crew scheduling business rules are complex, and the workload is large. The efficiency of manual compilation is very low, requiring at least one week. The manpower and time costs are high.
2. Difficulty in adjustment: When encountering unexpected situations such as train failures and temporary leave of crew members, it is difficult to quickly adjust the scheduling plan based on manual methods, affecting the efficiency of operation management.
3. Uneven task assignment: Manual scheduling has subjective limitations and strong dependence on human experience. If the scheduling is unreasonable, it may lead to uneven task assignment, unfair scheduling, and low crew satisfaction.
This subway line includes 40 platforms, among which A, M, and Z platforms are available for driver transfers and handovers. Please consider 200 driving tasks and their attributes such as time, location, and mileage, and consider classic constraints in crew scheduling scenarios, such as maximum continuous driving time constraint, mileage limit constraint, attendance location constraint, etc. Compare different mixed integer modeling methods to minimize the number of drivers as the goal and output the optimal task chain combination (i.e. the allocation and connection relationship of all tasks) for all tasks.
### Example
Mileage Information Table.csv = \"""
Departure Station,Arrival Station,Mileage
A,M,17.1
Z,M,26.4
\"""
Task Information Table.csv = \"""
Train Number,Train ID,Pickup Time,Pickup Location,Drop-off Time,Drop-off Location
331,M16,10:03:11,M,10:57:43,Z
1,L06,10:03:43,Z,10:58:32,M
163,N23,10:04:10,A,10:39:11,M
332,N27,10:04:32,M,10:40:10,A
333,N20,10:09:11,M,11:03:43,Z
2,L07,10:09:43,Z,11:04:32,M
164,L11,10:10:10,A,10:45:11,M
334,L01,10:10:32,M,10:46:10,A
335,M17,10:15:11,M,11:09:43,Z
3,M13,10:15:43,Z,11:10:32,M
165,N24,10:16:10,A,10:51:11,M
336,N28,10:16:32,M,10:52:10,A
337,N21,10:21:11,M,11:15:43,Z
4,L08,10:21:43,Z,11:16:32,M
166,N25,10:22:10,A,10:57:11,M
338,L02,10:22:32,M,10:58:10,A
339,M18,10:27:11,M,11:21:43,Z
5,M14,10:27:43,Z,11:22:32,M
167,N26,10:28:10,A,11:03:11,M
340,L03,10:28:32,M,11:04:10,A
341,N22,10:33:11,M,11:27:43,Z
6,L09,10:33:43,Z,11:28:32,M
168,L12,10:34:10,A,11:09:11,M
342,N29,10:34:32,M,11:10:10,A
343,N23,10:39:11,M,11:33:43,Z
7,M15,10:39:43,Z,11:34:32,M
169,N27,10:40:10,A,11:15:11,M
344,L04,10:40:32,M,11:16:10,A
345,L11,10:45:11,M,11:39:43,Z
8,N19,10:45:43,Z,11:40:32,M
170,L01,10:46:10,A,11:21:11,M
346,L05,10:46:32,M,11:22:10,A
347,N24,10:51:11,M,11:45:43,Z
9,L10,10:51:43,Z,11:46:32,M
171,N28,10:52:10,A,11:27:11,M
348,N30,10:52:32,M,11:28:10,A
349,N25,10:57:11,M,11:51:43,Z
10,M16,10:57:43,Z,11:52:32,M
172,L02,10:58:10,A,11:33:11,M
350,L06,10:58:32,M,11:34:10,A
351,N26,11:03:11,M,11:57:43,Z
11,N20,11:03:43,Z,11:58:32,M
173,L03,11:04:10,A,11:39:11,M
352,L07,11:04:32,M,11:40:10,A
353,L12,11:09:11,M,12:03:43,Z
12,M17,11:09:43,Z,12:04:32,M
174,N29,11:10:10,A,11:45:11,M
354,M13,11:10:32,M,11:46:10,A
355,N27,11:15:11,M,12:09:43,Z
13,N21,11:15:43,Z,12:10:32,M
175,L04,11:16:10,A,11:51:11,M
356,L08,11:16:32,M,11:52:10,A
357,L01,11:21:11,M,12:15:43,Z
14,M18,11:21:43,Z,12:16:32,M
176,L05,11:22:10,A,11:57:11,M
358,M14,11:22:32,M,11:58:10,A
359,N28,11:27:11,M,12:21:43,Z
15,N22,11:27:43,Z,12:22:32,M
177,N30,11:28:10,A,12:03:11,M
360,L09,11:28:32,M,12:04:10,A
361,L02,11:33:11,M,12:27:43,Z
16,N23,11:33:43,Z,12:28:32,M
178,L06,11:34:10,A,12:09:11,M
362,M15,11:34:32,M,12:10:10,A
363,L03,11:39:11,M,12:33:43,Z
17,L11,11:39:43,Z,12:34:32,M
179,L07,11:40:10,A,12:15:11,M
364,N19,11:40:32,M,12:16:10,A
365,N29,11:45:11,M,12:39:43,Z
18,N24,11:45:43,Z,12:40:32,M
180,M13,11:46:10,A,12:21:11,M
366,L10,11:46:32,M,12:22:10,A
367,L04,11:51:11,M,12:45:43,Z
19,N25,11:51:43,Z,12:46:32,M
181,L08,11:52:10,A,12:27:11,M
368,M16,11:52:32,M,12:28:10,A
369,L05,11:57:11,M,12:51:43,Z
20,N26,11:57:43,Z,12:52:32,M
182,M14,11:58:10,A,12:33:11,M
370,N20,11:58:32,M,12:34:10,A
371,N30,12:03:11,M,12:57:43,Z
21,L12,12:03:43,Z,12:58:32,M
183,L09,12:04:10,A,12:39:11,M
372,M17,12:04:32,M,12:40:10,A
373,L06,12:09:11,M,13:03:43,Z
22,N27,12:09:43,Z,13:04:32,M
184,M15,12:10:10,A,12:45:11,M
374,N21,12:10:32,M,12:46:10,A
375,L07,12:15:11,M,13:09:43,Z
23,L01,12:15:43,Z,13:10:32,M
185,N19,12:16:10,A,12:51:11,M
376,M18,12:16:32,M,12:52:10,A
377,M13,12:21:11,M,13:15:43,Z
24,N28,12:21:43,Z,13:16:32,M
186,L10,12:22:10,A,12:57:11,M
378,N22,12:22:32,M,12:58:10,A
379,L08,12:27:11,M,13:21:43,Z
25,L02,12:27:43,Z,13:22:32,M
187,M16,12:28:10,A,13:03:11,M
380,N23,12:28:32,M,13:04:10,A
381,M14,12:33:11,M,13:27:43,Z
26,L03,12:33:43,Z,13:28:32,M
188,N20,12:34:10,A,13:09:11,M
382,L11,12:34:32,M,13:10:10,A
383,L09,12:39:11,M,13:33:43,Z
27,N29,12:39:43,Z,13:34:32,M
189,M17,12:40:10,A,13:15:11,M
384,N24,12:40:32,M,13:16:10,A
385,M15,12:45:11,M,13:39:43,Z
28,L04,12:45:43,Z,13:40:32,M
190,N21,12:46:10,A,13:21:11,M
386,N25,12:46:32,M,13:22:10,A
387,N19,12:51:11,M,13:45:43,Z
29,L05,12:51:43,Z,13:46:32,M
191,M18,12:52:10,A,13:27:11,M
388,N26,12:52:32,M,13:28:10,A
389,L10,12:57:11,M,13:51:43,Z
30,N30,12:57:43,Z,13:52:32,M
192,N22,12:58:10,A,13:33:11,M
390,L12,12:58:32,M,13:34:10,A
391,M16,13:03:11,M,13:57:43,Z
31,L06,13:03:43,Z,13:58:32,M
193,N23,13:04:10,A,13:39:11,M
392,N27,13:04:32,M,13:40:10,A
393,N20,13:09:11,M,14:03:43,Z
32,L07,13:09:43,Z,14:04:32,M
194,L11,13:10:10,A,13:45:11,M
394,L01,13:10:32,M,13:46:10,A
395,M17,13:15:11,M,14:09:43,Z
33,M13,13:15:43,Z,14:10:32,M
195,N24,13:16:10,A,13:51:11,M
396,N28,13:16:32,M,13:52:10,A
397,N21,13:21:11,M,14:15:43,Z
34,L08,13:21:43,Z,14:16:32,M
196,N25,13:22:10,A,13:57:11,M
398,L02,13:22:32,M,13:58:10,A
399,M18,13:27:11,M,14:21:43,Z
35,M14,13:27:43,Z,14:22:32,M
197,N26,13:28:10,A,14:03:11,M
400,L03,13:28:32,M,14:04:10,A
401,N22,13:33:11,M,14:27:43,Z
36,L09,13:33:43,Z,14:28:32,M
198,L12,13:34:10,A,14:09:11,M
402,N29,13:34:32,M,14:10:10,A
403,N23,13:39:11,M,14:33:43,Z
37,M15,13:39:43,Z,14:34:32,M
199,N27,13:40:10,A,14:15:11,M
404,L04,13:40:32,M,14:16:10,A
405,L11,13:45:11,M,14:39:43,Z
38,N19,13:45:43,Z,14:40:32,M
200,L01,13:46:10,A,14:21:11,M
406,L05,13:46:32,M,14:22:10,A
407,N24,13:51:11,M,14:45:43,Z
39,L10,13:51:43,Z,14:46:32,M
201,N28,13:52:10,A,14:27:11,M
408,N30,13:52:32,M,14:28:10,A
409,N25,13:57:11,M,14:51:43,Z
40,M16,13:57:43,Z,14:52:32,M
202,L02,13:58:10,A,14:33:11,M
410,L06,13:58:32,M,14:34:10,A
411,N26,14:03:11,M,14:57:43,Z
41,N20,14:03:43,Z,14:58:32,M
203,L03,14:04:10,A,14:39:11,M
412,L07,14:04:32,M,14:40:10,A
413,L12,14:09:11,M,15:03:43,Z
42,M17,14:09:43,Z,15:04:32,M
204,N29,14:10:10,A,14:45:11,M
414,M13,14:10:32,M,14:46:10,A
415,N27,14:15:11,M,15:09:43,Z
43,N21,14:15:43,Z,15:10:32,M
205,L04,14:16:10,A,14:51:11,M
416,L08,14:16:32,M,14:52:10,A
417,L01,14:21:11,M,15:15:43,Z
44,M18,14:21:43,Z,15:16:32,M
206,L05,14:22:10,A,14:57:11,M
418,M14,14:22:32,M,14:58:10,A
419,N28,14:27:11,M,15:21:43,Z
45,N22,14:27:43,Z,15:22:32,M
207,N30,14:28:10,A,15:03:11,M
420,L09,14:28:32,M,15:04:10,A
421,L02,14:33:11,M,15:27:43,Z
46,N23,14:33:43,Z,15:28:32,M
208,L06,14:34:10,A,15:09:11,M
422,M15,14:34:32,M,15:10:10,A
423,L03,14:39:11,M,15:33:43,Z
47,L11,14:39:43,Z,15:34:32,M
209,L07,14:40:10,A,15:15:11,M
424,N19,14:40:32,M,15:16:10,A
425,N29,14:45:11,M,15:39:43,Z
48,N24,14:45:43,Z,15:40:32,M
210,M13,14:46:10,A,15:21:11,M
426,L10,14:46:32,M,15:22:10,A
427,L04,14:51:11,M,15:45:43,Z
49,N25,14:51:43,Z,15:46:32,M
211,L08,14:52:10,A,15:27:11,M
428,M16,14:52:32,M,15:28:10,A
429,L05,14:57:11,M,15:51:43,Z
50,N26,14:57:43,Z,15:52:32,M
212,M14,14:58:10,A,15:33:11,M
430,N20,14:58:32,M,15:34:10,A
\""" | MIP |
623.0 | Hard | Haus Toys can manufacture and sell toy trucks, toy planes, toy boats, and toy trains. The profit from selling one truck is $5, from one plane is $10, from one boat is $8, and from one train is $7. How many types of toys should Haus Toys manufacture to maximize profit?
There are 890 units of wood available. Manufacturing one truck requires 12 units of wood, one plane requires 20 units of wood, one boat requires 15 units of wood, and one train requires 10 units of wood.
There are 500 units of steel available. Manufacturing one plane requires 3 units of steel, one boat requires 5 units of steel, one train requires 4 units of steel, and one truck requires 6 units of steel.
If Haus Toys manufactures trucks, then they will not manufacture trains.
However, if they manufacture boats, they will also manufacture planes.
The number of toy boats manufactured cannot exceed the number of toy trains manufactured. | IP |
240000.0 | Easy | Vicky and David just bought a farm in the Yarra Valley and are considering using it to grow apples, pears, oranges, and lemons. The profit from growing one acre of apples is $2000, of pears is $1800, of oranges is $2200, and of lemons is $3000. In order to maximize profit, how many acres of land should they use to grow each fruit?
Vicky and David just bought a farm in the Yarra Valley and are considering using it to grow apples, pears, oranges, and lemons. The total land area is 120 acres.
The land area for growing apples should be at least twice the land area for growing pears.
The land area for growing apples should be at least three times the land area for growing lemons.
The land area for growing oranges must be twice the land area for growing lemons.
Vicky and David are not willing to grow more than two types of fruit. | IP |
21.0 | Easy | A farmer needs to transport 1000 units of fresh agricultural products from the farm to a nearby market. The farmer has three transportation options: horses, bicycles, and hand carts. Due to the physical exertion required for bicycles and hand carts, the farmer only wants to choose one of these two transportation methods. Horses produce 80 units of pollution per trip, bicycles produce 0 units of pollution, and hand carts produce 0 units of pollution. The total pollution generated by all trips must not exceed 1000 units. The farmer needs to use horses for at least 8 trips. Horses, bicycles, and hand carts can each carry 55 units, 30 units, and 40 units of products per trip, respectively. The farmer needs to ensure that the total quantity of transported products is at least 1000 units. | MIP |
365.0 | Easy | A furniture store can choose to order chairs from three different manufacturers, A, B, and C. The cost of ordering each chair from manufacturer A is $50, from manufacturer B is $45, and from manufacturer C is $40. The furniture store needs to minimize the total cost of the orders.
Additionally, each order from manufacturer A will include 15 chairs, while orders from B and C will include 10 chairs each. The order quantities can only be integers. The furniture store needs to order at least 100 chairs.
Each order from manufacturer A will include 15 chairs, while orders from manufacturers B and C will include 10 chairs each. The furniture store needs to order a maximum of 500 chairs.
If the furniture store decides to order chairs from manufacturer A, it must also order at least 10 chairs from manufacturer B.
Additionally, if the furniture store decides to order chairs from manufacturer B, it must also order chairs from manufacturer C. | IP |
960.0 | Easy | Bright Future Toys wants to build and sell robots, model cars, building blocks, and dolls. The profit from selling one robot is $15, from one model car is $8, from one set of building blocks is $12, and from one doll is $5. How many types of toys should Bright Future Toys manufacture to maximize profit?
There are 1200 units of plastic available. Manufacturing one robot requires 30 units of plastic, one model car requires 10 units of plastic, one set of building blocks requires 20 units of plastic, and one doll requires 15 units of plastic.
There are 800 units of electronic components available. Manufacturing one robot requires 8 units of electronic components, one model car requires 5 units of electronic components, one set of building blocks requires 3 units of electronic components, and one doll requires 2 units of electronic components.
If Bright Future Toys manufactures robots, then they will not manufacture dolls.
However, if they manufacture model cars, they will also manufacture building blocks.
The number of dolls manufactured cannot exceed the number of model cars manufactured. | MIP |
15000 | Easy | A restaurant needs to order tables from three different suppliers, A, B, and C. The cost of ordering each table from supplier A is $120, from supplier B is $110, and from supplier C is $100. The restaurant wants to minimize the total cost of the orders.
In addition, each order from supplier A will include 20 tables, while orders from suppliers B and C will include 15 tables each. The quantity of orders can only be whole numbers. The restaurant needs to order at least 150 tables.
Each order from supplier A will include 20 tables, while orders from suppliers B and C will include 15 tables each. The restaurant needs to order a maximum of 600 tables.
If the restaurant decides to order tables from supplier A, it must also order at least 30 tables from supplier B.
Furthermore, if the restaurant decides to order tables from supplier B, it must also order tables from supplier C. | MIP |
368999.99999999994 | Medium | A company plans to produce three products $A_{1}, A_{2}, A_{3}$. It can produce for 22 days in a month. The table below gives the maximum demand (in units of 100 kg), prices ($\$/100 \mathrm{~Kg}$), production costs (per 100 kg of product), and production quotas (the maximum number of units of 100 kg of product that can be produced in a day if all production lines are dedicated to that product).
| Product | $A_{1}$ | $A_{2}$ | $A_{3}$ |
| :---: | :---: | :---: | :---: |
| Maximum Demand | 5300 | 4500 | 5400 |
| Selling Price | $\$ 124$ | $\$ 109$ | $\$ 115$ |
| Production Cost | $\$ 73.30$ | $\$ 52.90$ | $\$ 65.40$ |
| Production Quota | 500 | 450 | 550 |
The fixed activation cost for each production line is as follows:
| Product | $A_{1}$ | $A_{2}$ | $A_{3}$ |
| :---: | :---: | :---: | :---: |
| Activation Cost | $\$ 170000$ | $\$ 150000$ | $\$ 100000$ |
The minimum production batch sizes are as follows:
$$
\begin{array}{c|ccc}
\text { Product } & A_{1} & A_{2} & A_{3} \\
\hline \text { Minimum Batch Size } & 20 & 20 & 16
\end{array}
$$
Please formulate an operations research model to determine the production plan that maximizes total revenue while accommodating the fixed activation costs and minimum production batch size restrictions. | MIP |
0.0 | Medium | A wealthy nobleman has passed away, leaving behind the following inheritance:
- A painting by Caillebotte: $25,000
- A bust of Diocletian: $5,000
- A Yuan Dynasty Chinese vase: $20,000
- A 911 Porsche: $40,000
- Three diamonds: $12,000 each
- A Louis XV sofa: $3,000
- Two very valuable Jack Russell racing dogs: $3,000 each (the will states they cannot be separated)
- A sculpture from 200 AD: $10,000
- A sailboat: $15,000
- A Harley Davidson motorcycle: $10,000
- A piece of furniture that once belonged to Cavour: $13,000
It must be divided between two sons. How can a mathematical program be formulated and solved using COPTPY to minimize the difference in value between the two parts? | IP |
435431000 | Medium | The independent country of Carelland primarily exports four commodities: steel, engines, electronic components, and plastics. The Finance Minister of Carelland (i.e., the Minister of Economy) wants to maximize exports and minimize imports. The unit prices of steel, engines, electronics, and plastics on the world market, expressed in the local currency (Klunz), are: 500, 1500, 300, and 1200, respectively. Producing 1 unit of steel requires 0.02 units of engines, 0.01 units of plastics, 250 Klunz of other imported goods, and 6 person-months of work. Producing 1 unit of engines requires 0.8 units of steel, 0.15 units of electronic components, 0.11 units of plastics, 300 Klunz of imported goods, and 1 person-year. Producing 1 unit of electronics requires 0.01 units of steel, 0.01 units of engines, 0.05 units of plastics, 50 Klunz of imported goods, and 6 person-months. Producing 1 unit of plastics requires 0.03 units of engines, 0.2 units of steel, 0.05 units of electronic components, 300 Klunz of imported goods, and 2 person-years. The production limit for engines is 650,000 units, and the production limit for plastics is 60,000 units. The total available labor force per year is 830,000. Write a mathematical program to maximize the gross domestic product and solve the problem using AMPL. | LP |
7.1 | Medium | A textile dyeing factory has 3 dyeing vats. The fabric must be dyed in each vat in order: vat 1, vat 2, vat 3. The factory needs to dye five batches of fabric of different sizes. The time required to dye batch i in vat j, denoted as $s_{ij}$ in hours, is given in the matrix below:
$$
\left(\begin{array}{ccc}
3 & 1 & 1 \\
2 & 1.5 & 1 \\
3 & 1.2 & 1.3 \\
2 & 2 & 2 \\
2.1 & 2 & 3
\end{array}\right)
$$
Arrange the dyeing operations in the vats to minimize the end time of the last batch. | IP |
4316659.199999999 | Medium | A person has a capital of 500,000 yuan and has the following investment projects in the next three years:
(1) It is possible to invest at the beginning of each year for three years, with a profit of 20% of the investment amount each year.
(2) Only allowed to invest at the beginning of the first year, and can be recovered at the end of the second year, with a total interest of 150% of the investment amount. However, the investment limit for this type of investment is not more than 120,000 yuan.
(3) Allowed to invest at the beginning of the second year, and can be recovered at the end of the second year, with a total interest of 160% of the investment amount. The investment limit for this type of investment is 150,000 yuan.
(4) Allowed to invest at the beginning of the third year, and can be recovered within one year with a profit of 40%. The investment limit for this type of investment is 100,000 yuan.
Try to determine an investment plan that maximizes the total interest at the end of the third year for this person. | LP |
530 | Medium | A hospital requires the following number of nurses during different time periods within a 24-hour day: 2:00-6:00 - 10 nurses, 6:00-10:00 - 15 nurses, 10:00-14:00 - 25 nurses, 14:00-18:00 - 20 nurses, 18:00-22:00 - 18 nurses, 22:00-2:00 - 12 nurses. Nurses work in 6 batches at 2:00, 6:00, 10:00, 14:00, 18:00, and 22:00, and work continuously for 8 hours. Determine if the hospital should hire contract nurses who work the same hours as regular nurses. If the salary for regular nurses is $10/hour and for contract nurses is $15/hour, how many contract nurses should the hospital hire? | MIP |
978400 | Easy | A person has a capital of 300,000 yuan and has the following investment projects for the next three years:
(1) Each year at the beginning of the year, the investment can be made, and the profit each year is 20% of the investment amount, and the principal and interest can be used together for the next year's investment;
(2) Only allowed to invest at the beginning of the first year, can be recovered at the end of the second year, and the total principal and interest is 150% of the investment amount, but the investment limit for this type of investment does not exceed 150,000 yuan;
(3) Allowed to invest at the beginning of the second year within three years, can be recovered at the end of the third year, and the total principal and interest is 160% of the investment amount, with an investment limit of 200,000 yuan;
(4) Allowed to invest at the beginning of the third year within three years, recoverable within one year, with a profit of 40% and an investment limit of 100,000 yuan.
Chapter 1 Linear Programming and Simplex Method
Try to determine an investment plan that maximizes the total principal and interest at the end of the third year for this person. | LP |
4848 | Medium | A candy factory processes raw materials A, B, and C into three different types of candies: A, B, and C. The content of A, B, and C in each type of candy, the cost of raw materials, the monthly limit of each raw material, and the processing fee and selling price of the three types of candies are shown in Table 1-7.
Table 1-7
\begin{tabular}{c|ccc|c|c}
\hline Item & A & B & C & Raw Material Cost (元/kg) & Monthly Limit (kg) \\
\hline A & $\geqslant 60 \%$ & $\geqslant 15 \%$ & & 2.00 & 2000 \\
B & & & & 1.50 & 2500 \\
C & $\leqslant 20 \%$ & $\leqslant 60 \%$ & $\leqslant 50 \%$ & 1.00 & 1200 \\
\hline Processing Fee (元/kg) & 0.50 & 0.40 & 0.30 & & \\
Selling Price (元/kg) & 3.40 & 2.85 & 2.25 & & \\
\hline
\end{tabular}
What is the monthly production of each type of candy in kg that maximizes the profit of the factory? | LP |
10755 | Hard | The contract booking numbers for three products, I, II, and III, for each quarter of the next year are shown in Table 1-10.
Table 1-10
\begin{tabular}{c|c|c|c|c}
\hline \multirow{2}{*}{ Product } & \multicolumn{4}{|c}{ Quarter } \\
\cline { 2 - 5 } & 1 & 2 & 3 & 4 \\
\hline I & 1500 & 1000 & 2000 & 1200 \\
II & 1500 & 1500 & 1200 & 1500 \\
III & 1000 & 2000 & 1500 & 2500 \\
\hline
\end{tabular}
At the beginning of the first quarter, there is no inventory for any of the three products. It is required to have 150 units of inventory for each product at the end of the fourth quarter. It is known that the factory has 15,000 hours of production time per quarter, and it takes 2, 4, and 3 hours respectively to produce one unit of products I, II, and III. Due to equipment replacement, product I cannot be produced in the second quarter. It is stipulated that if a product cannot be delivered on time, a compensation of $20 per unit per quarter will be paid for products I and II, and $10 for product III. Additionally, for products produced but not delivered in the same quarter, there is a storage cost of $5 per unit per quarter. How should the factory arrange its production to minimize the total compensation and storage costs? | MIP |
118400 | Easy | A factory needs to rent a warehouse to store materials for the next 4 months. The required warehouse area for each month is listed in Table 1-14.
Table 1-14
\begin{tabular}{c|c|c|c|c}
\hline Month & 1 & 2 & 3 & 4 \\
\hline Required Warehouse Area $/ \mathrm{m}^2$ & 1500 & 1000 & 2000 & 1200 \\
\hline
\end{tabular}
The longer the rental contract period, the greater the discount on warehouse rental fees. The specific data is listed in Table 1-15.
Table 1-15
\begin{tabular}{c|c|c|c|c}
\hline Contract Rental Period $/$ months & 1 & 2 & 3 & 4 \\
\hline \begin{tabular}{c}
Rental Fee for Warehouse \\
Area within the Contract Period $/ \mathrm{m}^2$
\end{tabular} & 28 & 45 & 60 & 73 \\
\hline
\end{tabular}
The warehouse rental contract can be processed at the beginning of each month, and each contract specifies the rental area and period. Therefore, the factory can rent a contract on any month, and each time, they can sign one contract or multiple contracts with different rental areas and rental periods. The overall goal is to minimize the rental fees paid. Try to establish a linear programming mathematical model based on the above requirements. | MIP |
426.0 | Medium | Fighter jets are important combat tools, but in order for them to be effective, there must be enough pilots. Therefore, in addition to a portion of the produced fighter jets being used directly for combat, another portion needs to be allocated for pilot training. It is known that the number of fighter jets produced each year is $a_j(j=1, \cdots, n)$, and each fighter jet can train $k$ pilots per year. How should the production of fighter jets be allocated each year to maximize their contribution to national defense over a period of $n$ years? | LP |
85 | Easy | Traveling Salesman Problem (TSP) with Specific Values\n\nThe famous traveling salesman problem in operations research can be described as follows: a traveling salesman starts from a certain city and goes to visit other \( n \) cities to sell goods. It is required to visit each city exactly once and then return to the original starting city. The distance between city \( i \) and city \( j \) is given as \( d_{ij} \). What kind of route sequence should the salesman choose to minimize the total travel distance?\n\nWe will establish an integer programming model for this problem using specific values for the number of cities and distances between them. Let's assume there are 5 cities (including the starting city) and the distance matrix \( D \) is given as follows:\n\n\[\nD = \begin{bmatrix}\n0 & 10 & 15 & 20 & 25 \\\n10 & 0 & 35 & 25 & 30 \\\n15 & 35 & 0 & 30 & 20 \\\n20 & 25 & 30 & 0 & 15 \\\n25 & 30 & 20 & 15 & 0\n\end{bmatrix}\n\ | IP |
16 | Medium | A certain university computer laboratory hires 4 undergraduate students (code names 1, 2, 3, 4) and 2 graduate students (code names 5, 6) to provide on-duty assistance. It is known that the maximum number of duty hours that each person can be assigned from Monday to Friday, as well as the hourly duty remuneration for each person, are shown in Table 5-9.
Table 5-9:
\begin{tabular}{c|c|c|c|c|c|c}
\hline \multirow{2}{*}{ Student Code } & \multirow{2}{*}{ Remuneration/(¥/h) } & \multicolumn{5}{|c}{ Maximum Duty Time per Day (h) } \
\cline { 3 - 7 } & & Monday & Tuesday & Wednesday & Thursday & Friday \
\hline 1 & 10.0 & 6 & 0 & 6 & 0 & 7 \
\hline 2 & 10.0 & 0 & 6 & 0 & 6 & 0 \
\hline 3 & 9.9 & 4 & 8 & 3 & 0 & 5 \
\hline 4 & 9.8 & 5 & 5 & 6 & 0 & 4 \
\hline 5 & 10.8 & 3 & 0 & 4 & 8 & 0 \
\hline 6 & 11.3 & 0 & 6 & 0 & 6 & 3 \
\hline
\end{tabular}
The laboratory is open from 8:00 AM to 10:00 PM, and there must be one and only one student on duty during the opening hours. It is also stipulated that each undergraduate student must have a minimum of 8 hours of duty per week, and each graduate student must have a minimum of 7 hours of duty per week. Based on the above, the following requirements are added: firstly, each student should not have more than 2 duty shifts per week, and secondly, the number of students assigned to duty each day should not exceed 3. Based on this, establish a new mathematical model. | MIP |
-1900.0 | Easy | Red Bean Clothing Factory uses three specialized machines to produce shirts, short-sleeved shirts, and casual wear. The labor, material, selling price, and variable cost per unit for the three products are given in Table 5-10.
Table 5-10
\begin{tabular}{c|c|c|c|c}
\hline Product Name & Labor per unit & Material per unit & Selling Price & Variable Cost \
\hline Shirt & 3 & 4 & 120 & 60 \
\hline Short-sleeved shirt & 2 & 3 & 80 & 40 \
\hline Casual wear & 6 & 6 & 180 & 80 \
\hline
\end{tabular}
It is known that the factory has a weekly labor capacity of 150 units and a material capacity of 160 units. The fixed costs per week for producing shirts, short-sleeved shirts, and casual wear with the specialized machines are 2000, 1500, and 1000, respectively. Design a weekly production plan for the factory to maximize profit. | LP |
150 | Easy | For a certain daytime and nighttime service bus route, the number of drivers and crew members required during each time period is given in Table 1-2:
Table 1-2
\begin{tabular}{|c|c|c||c|c|c|}
\hline Shift & Time & Number Required & Shift & Time & Number Required \\
\hline 1 & $6: 00 \sim 10: 00$ & 60 & 4 & $18: 00 \sim 22: 00$ & 50 \\
\hline 2 & $10: 00 \sim 14: 00$ & 70 & 5 & $22: 00 \sim 2: 00$ & 20 \\
\hline 3 & $14: 00 \sim 18: 00$ & 60 & 6 & $2: 00 \sim 6: 00$ & 30 \\
\hline
\end{tabular}
Assuming that the drivers and crew members start working at the beginning of each time period and work continuously for 8 hours, how many drivers and crew members should be assigned to this bus route at least? Write down the linear programming model for this problem. | LP |
1146.57 | Medium | A factory produces three products, I, II, and III. Each product goes through two processing procedures, A and B. The factory has two types of equipment, A1 and A2, to complete procedure A, and three types of equipment, B1, B2, and B3, to complete procedure B. Product I can be processed on either type of A equipment or any type of B equipment. Product II can be processed on any type of A equipment, but when completing procedure B, it can only be processed on B1 equipment. Product III can only be processed on A2 and B2 equipment. Given the processing time, raw material cost, product selling price, available equipment operating time, and equipment cost at full load for each type of equipment, as shown in Table 1-4, determine the optimal production plan to maximize profit.
Table 1-4
\begin{tabular}{|c|c|c|c|c|c|}
\hline \multirow{2}{*}{ Equipment } & \multicolumn{3}{|c|}{ Product } & \multirow{2}{*}{ Available Equipment Operating Time } & \multirow{2}{*}{ Equipment Cost at Full Load (in yuan) } \\
\cline { 2 - 4 } & I & II & III & & \\
\hline$A_1$ & 5 & 10 & & 6000 & 300 \\
\hline$A_2$ & 7 & 9 & 12 & 10000 & 321 \\
\hline$B_1$ & 6 & 8 & & 4000 & 250 \\
\hline$B_2$ & 4 & & 11 & 7000 & 783 \\
\hline$B_3$ & 7 & & & 4000 & 200 \\
\hline Raw Material Cost (in yuan per unit) & 0.25 & 0.35 & 0.50 & & \\
\hline Unit Price (in yuan per unit) & 1.25 & 2.00 & 2.80 & & \\
\hline
\end{tabular} | MIP |
20 | Easy | In a rebar workshop, a batch of rebars (with the same diameter) is being produced. There are 90 rebars with a length of 3 meters and 60 rebars with a length of 4 meters. It is known that each rebar used for cutting is 10 meters long. What is the most efficient way to cut the rebars? Establish a linear programming model for this problem. | LP |
5000.0 | Easy | A steel mill has two steelmaking furnaces, each using a different method. The first method requires $a$ hours per furnace and costs $m$ dollars in fuel. The second method requires $b$ hours per furnace and costs $n$ dollars in fuel. Assuming each furnace produces $k$ tons of steel, and now they want to produce at least $d$ tons of steel within $c$ hours, how should they allocate these two methods to minimize fuel costs? Express this problem as a linear programming model. | LP |
22.0 | Hard | A certain restaurant operates 24 hours a day and requires a minimum number of waitstaff as shown in Table 1.1.
Table 1.1
| Time | Minimum Number of Waitstaff |
| :---: | :---: |
| 2am-6am | 4 |
| 6am-10am | 8 |
| 10am-2pm | 10 |
| 2pm-6pm | 4 |
| 6pm-10pm | 8 |
| 10pm-2am | 4 |
Each waitstaff works continuously for 8 hours a day. The goal is to find the minimum number of waitstaff that satisfies the above conditions and represent this problem as a linear programming model. | IP |
770.0 | Easy | Assume a paper mill receives three orders for rolls of paper with specified widths and lengths as shown in Table 1.2.
Table 1.2
| Order Number | Width (m) | Length (m) |
| :---: | :---: | :---: |
| 1 | 0.5 | 1000 |
| 2 | 0.7 | 3000 |
| 3 | 0.9 | 2000 |
The mill produces rolls of paper with two standard widths: 1 meter and 2 meters. Assuming the length of the rolls is unlimited, meaning they can be continuously connected to achieve the required length, how should the rolls be cut to minimize the area lost during cutting? | LP |
9500.0 | Easy | A certain store wants to develop a procurement and sales plan for a certain product for the first quarter of next year. It is known that the store's warehouse can store a maximum of 500 units of this product, and there are 200 units of inventory at the end of this year. The store makes a purchase at the beginning of each month. The unit prices for purchasing and selling this product for each month are shown in Table 1.3.
Table 1.3
| Month | 1 | 2 | 3 |
| :---: | :---: | :---: | :---: |
| Purchase Price (in yuan) | 8 | 6 | 9 |
| Selling Price (in yuan) | 9 | 8 | 10 |
Now, to determine how many units should be purchased and sold each month in order to maximize total profit, express this problem as a linear programming model. | LP |
1360000.0 | Easy | An investor plans to invest his 100,000 yuan, and there are two investment options to choose from. The first investment guarantees a profit of 0.7 yuan for every 1 yuan invested after one year. The second investment guarantees a profit of 2 yuan for every 1 yuan invested after two years, but the investment time must be a multiple of two years. In order to maximize the money earned by the end of the third year, how should the investor invest? Represent this problem as a linear programming problem. | LP |
25 | Medium | A textile factory produces two types of fabrics, one for making clothing and the other for making curtains. The factory operates two shifts, with a weekly production time set at 80 hours. Both types of fabrics are produced at a rate of 1000 meters per hour. Assuming that 70,000 meters of curtain fabric can be sold per week with a profit of 2.5 yuan per meter, and 45,000 meters of clothing fabric can be sold per week with a profit of 1.5 yuan per meter. The factory has the following objectives when formulating its production plan:
$p_{1}$: Use the full 80 hours of production time per week.
$p_{2}$: Overtime hours per week should not exceed 10 hours.
$p_{3}$: Sell no less than 70,000 meters of curtain fabric and 45,000 meters of clothing fabric per week, with profit as the weighting factor.
$p_{4}$: Minimize overtime hours.
Build an objective programming model for this problem. | LP |
5500 | Medium | A certain shoe store employs 5 full-time salespeople and 4 part-time salespeople. Their working hours and salary information are shown in Table 3.3.
Table 3.3
| | Monthly Working Hours | Sales Volume (pairs/hour) | Salary (CNY/hour) | Overtime Pay (CNY/hour) |
| :---: | :---: | :---: | :---: | :---: |
| Full-time | 160 | 5 | 1 | 1.5 |
| Part-time | 80 | 2 | 0.6 | 0.7 |
A profit of 0.5 CNY is made for each pair of shoes sold. The store has the following objectives:
$p_{1}$: Monthly sales volume reaches 5500 pairs;
$p_{2}$: Full-time salespeople do not exceed 100 hours of overtime;
$p_{3}$: All salespeople are fully employed, but full-time employees should be given double consideration;
$p_{4}$: Minimize overtime hours as much as possible.
Try to establish an objective programming model for this problem. | MIP |