Correct the expression of Problem 14, 17 and 31; Correct the answer of Problem 6 and 36

IndustryOR.json

@@ -1,9 +1,9 @@
 {"en_question": "A factory produces two types of food, I and II, and currently has 50 skilled workers. It is known that one skilled worker can produce $10 \\ \\mathrm{kg} / \\ \\mathrm{h}$ of food I or $6 \\ \\mathrm{kg} / \\ \\mathrm{h}$ of food II. According to contract bookings, the weekly demand for these two foods will rise sharply, as shown in Table 1-11. Therefore, the factory has decided to train 50 new workers by the end of the 8th week. It is known that a worker works $40 \\ \\mathrm{h}$ per week, and a skilled worker can train up to three new workers in two weeks (during the training period, both the skilled worker and the trainees do not participate in production). The weekly wage of a skilled worker is 360 yuan, the weekly wage of a trainee during the training period is 120 yuan, and after training, the wage is 240 yuan per week, with the same production efficiency as skilled workers. During the transition period of training, many skilled workers are willing to work overtime, and the factory has decided to arrange some workers to work $60 \\ \\mathrm{h}$ per week, with a weekly wage of 540 yuan. If the booked food cannot be delivered on time, the compensation fee for each week of delay per $ \\ \\mathrm{kg}$ is 0.5 yuan for food I and 0.6 yuan for food II. Under these conditions, how should the factory make comprehensive arrangements to minimize the total cost?\n\nTable 1-11\n\n| Week | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |\n|------|---|---|---|---|---|---|---|---|\n| I    | 10000 | 10000  | 12000  | 12000  | 16000  | 16000  | 20000  | 20000  |\n| II   | 6000 | 7200 | 8400 | 10800 | 10800 | 12000  | 12000  | 12000  |", "en_answer": "219816.0", "difficulty": "Medium", "id": 1}
 {"en_question": "A fighter jet is a crucial combat tool, but to make a fighter jet effective, enough pilots are required. Therefore, a portion of the produced fighter jets, besides those used directly in combat, must be allocated for training pilots. Given that the annual production of fighter jets is \\(a_j(j=1, 2)\\), with \\(a_1=10\\) and \\(a_2=15\\), furthermore, each fighter jet can train 5 pilots per year. How should the annual production of fighter jets be allocated to maximize their contribution to air defense over \\(n\\) years?", "en_answer": "125", "difficulty": "Medium", "id": 2}
-{"en_question": "For the manufacturing industry, formulating an appropriate product production plan and human resource management scheme is very important for reducing operating costs, inventory costs, stockout costs, and labor costs. Specifically, for products with large demand fluctuations, companies need to accurately predict demand and formulate corresponding production plans. A certain company produces a type of foldable table, the raw material cost for each product is 90 Yuan, and the labor hours are 5 hours. The unit price of the product is 300 Yuan. The company has 1,000 workers at the beginning of January and holds an inventory of 15,000 units. The regular hourly wage for workers is 30 Yuan, with a normal daily working time of 8 hours. Any extra working time is considered overtime, with an overtime hourly wage of 40 Yuan. The number of working days per month is calculated as 20 days. Suppose the company's available production machines are sufficient, and the capacity is not limited by machine hours. Each worker's overtime hours do not exceed 20 hours per month. Holding inventory will generate corresponding inventory costs. If a stockout occurs due to insufficient capacity, there will be certain stockout costs. Additionally, the company can also make up for the stockout part through outsourcing. The outsourcing cost per product, monthly inventory cost, and monthly stockout cost are 200 Yuan, 15 Yuan, and 35 Yuan, respectively. The company's employees are temporary workers, and the company can flexibly decide the number of people to hire and fire each month. The cost of hiring and firing a single employee is 5,000 Yuan and 8,000 Yuan, respectively. Suppose the company has used the existing forecasting model to predict the demand from January to June, as shown in Table 4.1.\n\nTable 4.1: Demand Forecast\n\n| Month | 1   | 2   | 3   | 4   | 5   | 6   |\n|:-----:|:---:|:---:|:---:|:---:|:---:|:---:|\n| Demand Forecast | 20000 | 40000 | 42000 | 35000 | 19000 | 18500 |\n\nPlease formulate a production plan, worker work schedule, and hire/fire plan for the company to maximize the total net profit over 6 months while ensuring that the company holds at least 10,000 units of products by the end of June.", "en_answer": "10349205.0", "difficulty": "Hard", "id": 3}
+{"en_question": "A company specializing in foldable tables needs to create an optimal production and human resources plan for a six-month period (January to June) to maximize its total net profit. The plan must detail monthly in-house production levels, outsourcing quantities, and workforce management (hiring/firing).\n\n**Initial Conditions (at the start of January):**\n- Initial Workforce: 1,000 employees\n- Initial Inventory: 15,000 units\n\n**Revenue and Cost Structure:**\n- **Sales Price:** 300 Yuan per unit sold.\n- **Raw Material Cost:** 90 Yuan per unit, applicable *only* to units produced in-house.\n- **Outsourcing Cost:** 200 Yuan per unit for finished tables acquired from a third-party supplier. This is an all-inclusive cost.\n- **Inventory Holding Cost:** 15 Yuan per unit for any inventory held at the end of a month.\n- **Backorder Cost:** 35 Yuan per unit for any unfulfilled demand (stockout) carried over to the next month.\n\n**Labor and Production Parameters:**\n- **Labor Requirement:** Each in-house unit requires 5 labor hours to produce.\n- **Regular Labor:** Each worker provides 160 regular working hours per month (8 hours/day * 20 days/month). The company pays a regular wage of 30 Yuan/hour for these 160 hours, regardless of full utilization.\n- **Overtime Labor:** Workers can perform overtime. Total overtime hours per month for the entire workforce cannot exceed 20 hours per worker. The overtime wage is 40 Yuan/hour.\n- **Workforce Management:** The company can hire or fire workers each month. The cost to hire a new worker is 5,000 Yuan, and the cost to fire a worker is 8,000 Yuan.\n\n**Demand and Fulfillment Logic:**\n- Unfulfilled demand from one month is back-ordered and must be met in subsequent months.\n- The company fulfills orders (both current demand and backorders) using available inventory from the previous month, current in-house production, and outsourced units.\n\n**Terminal Condition (at the end of June):**\n- The ending inventory must be at least 10,000 units.\n- All backorders must be cleared (i.e., ending backorders must be zero).\n\n**Forecasted Demand:**\n| Month | January | February | March | April | May | June |\n|:---:|:---:|:---:|:---:|:---:|:---:|:---:|\n| Demand Forecast | 20,000 | 40,000 | 42,000 | 35,000 | 19,000 | 18,500 |\n\nBased on this information, formulate the optimal six-month operational plan.", "en_answer": "10349920.0", "difficulty": "Hard", "id": 3}
 {"en_question": "A farmer needs to decide how many cows, sheep, and chickens to raise in order to achieve maximum profit. The farmer can sell cows, sheep, and chickens for $500, $200, and $8 each, respectively. The feed costs for each cow, sheep, and chicken are $100, $80, and $5, respectively. The profit is the difference between the selling price and the feed cost. Each cow, sheep, and chicken produces 10, 5, and 3 units of manure per day, respectively. Due to the limited time the farm staff has for cleaning the farm each day, they can handle up to 800 units of manure. Additionally, because of the limited farm size, the farmer can raise at most 50 chickens. Furthermore, the farmer must have at least 10 cows to meet customer demand. The farmer must also raise at least 20 sheep. Finally, the total number of animals cannot exceed 100.", "en_answer": "30400.0", "difficulty": "Easy", "id": 4}
 {"en_question": "Mary is planning her dinner tonight. Every 100 grams of okra contains 3.2 grams of fiber, every 100 grams of carrots contains 2.7 grams of fiber, every 100 grams of celery contains 1.6 grams of fiber, and every 100 grams of cabbage contains 2 grams of fiber. How many grams of each type of food should Mary buy to maximize her fiber intake?\n\nShe is considering choosing one among salmon, beef, and pork as a protein source.\n\nShe also considers choosing at least two kinds of vegetables among okra, carrots, celery, and cabbage.\n\nThe price of salmon is $4 per 100 grams, beef is $3.6 per 100 grams, pork is $1.8 per 100 grams. The price of okra is $2.6 per 100 grams, carrots are $1.2 per 100 grams, celery is $1.6 per 100 grams, and cabbage is $2.3 per 100 grams. Mary has a budget of $15 for this meal.\n\nThe total food intake should be 600 grams.", "en_answer": "1600.0", "difficulty": "Easy", "id": 5}
-{"en_question": "The contract reservations for the next year for products I, II, and III of a certain factory in each quarter are shown in Table 1-10.\n\nTable 1-10\n| Product | 1    | 2    | 3    | 4    |\n|---------|------|------|------|------|\n| I       | 1500 | 1000 | 2000 | 1200 |\n| II      | 1500 | 1500 | 1200 | 1500 |\n| III     | 1000 | 2000 | 1500 | 2500 |\n\nAt the beginning of the first quarter, there is no inventory for these three products, and it is required to have 150 units in stock for each product by the end of the fourth quarter. It is known that the factory has 15,000 production hours per quarter, and each unit of products I, II, and III requires 2, 4, and 3 hours respectively. Due to a change in equipment, product I cannot be produced in the second quarter. It is stipulated that if the products cannot be delivered on time, a compensation of 20 yuan per unit per quarter delay is required for products I and II, while for product III, the compensation is 10 yuan. Additionally, for products produced but not delivered in the current quarter, the inventory cost is 5 yuan per unit per quarter. How should the factory schedule production to minimize the total cost of compensation and inventory?", "en_answer": "24250.0", "difficulty": "Hard", "id": 6}
+{"en_question": "The contract reservations for the next year for products I, II, and III of a certain factory in each quarter are shown in Table 1-10.\n\nTable 1-10\n| Product | 1    | 2    | 3    | 4    |\n|---------|------|------|------|------|\n| I       | 1500 | 1000 | 2000 | 1200 |\n| II      | 1500 | 1500 | 1200 | 1500 |\n| III     | 1000 | 2000 | 1500 | 2500 |\n\nAt the beginning of the first quarter, there is no inventory for these three products, and it is required to have 150 units in stock for each product by the end of the fourth quarter. It is known that the factory has 15,000 production hours per quarter, and each unit of products I, II, and III requires 2, 4, and 3 hours respectively. Due to a change in equipment, product I cannot be produced in the second quarter. It is stipulated that if the products cannot be delivered on time, a compensation of 20 yuan per unit per quarter delay is required for products I and II, while for product III, the compensation is 10 yuan. Additionally, for products produced but not delivered in the current quarter, the inventory cost is 5 yuan per unit per quarter. How should the factory schedule production to minimize the total cost of compensation and inventory?", "en_answer": "10755.00", "difficulty": "Hard", "id": 6}
 {"en_question": "An Italian transportation company needs to move some empty containers from its 6 warehouses (located in Verona, Perugia, Rome, Pescara, Taranto, and Lamezia) to major national ports (Genoa, Venice, Ancona, Naples, Bari). The container inventory at the warehouses is as follows:\n\n|  | Empty Containers |\n|:---:|:---:|\n| Verona | 10 |\n| Perugia | 12 |\n| Rome | 20 |\n| Pescara | 24 |\n| Taranto | 18 |\n| Lamezia | 40 |\n\nThe demand at the ports is as follows:\n\n|  | Container Demand |\n|:---:|:---:|\n| Genoa | 20 |\n| Venice | 15 |\n| Ancona | 25 |\n| Naples | 33 |\n| Bari | 21 |\n\nThe transport is carried out by a fleet of trucks. The cost to transport each container is proportional to the distance traveled by the trucks, with a rate of 30 euros per kilometer. Each truck can carry up to 2 containers. The distances are as follows:\n\n|  | Genoa | Venice | Ancona | Naples | Bari |\n|:---:|:---:|:---:|:---:|:---:|:---:|\n| Verona | $290 \\mathrm{~km}$ | $115 \\mathrm{~km}$ | $355 \\mathrm{~km}$ | $715 \\mathrm{~km}$ | $810 \\mathrm{~km}$ |\n| Perugia | $380 \\mathrm{~km}$ | $340 \\mathrm{~km}$ | $165 \\mathrm{~km}$ | $380 \\mathrm{~km}$ | $610 \\mathrm{~km}$ |\n| Rome | $505 \\mathrm{~km}$ | $530 \\mathrm{~km}$ | $285 \\mathrm{~km}$ | $220 \\mathrm{~km}$ | $450 \\mathrm{~km}$ |\n| Pescara | $655 \\mathrm{~km}$ | $450 \\mathrm{~km}$ | $155 \\mathrm{~km}$ | $240 \\mathrm{~km}$ | $315 \\mathrm{~km}$ |\n| Taranto | $1010 \\mathrm{~km}$ | $840 \\mathrm{~km}$ | $550 \\mathrm{~km}$ | $305 \\mathrm{~km}$ | $95 \\mathrm{~km}$ |\n| Lamezia | $1072 \\mathrm{~km}$ | $1097 \\mathrm{~km}$ | $747 \\mathrm{~km}$ | $372 \\mathrm{~km}$ | $333 \\mathrm{~km}$ |\n\nWrite a mathematical program to find the minimum cost transportation policy and solve it using COPTPY.", "en_answer": "904590.0", "difficulty": "Medium", "id": 7}
 {"en_question": "Now, we need to determine 4 out of 5 workers to complete one of the four tasks respectively. Due to each worker's different technical specialties, the time required for them to complete each task varies. The hours required by each worker to complete each task are shown in Table 5-2.\n\nTable 5-2\n| Worker | $A$ | $B$ | $C$ | $D$ |\n|--------|-----|-----|-----|-----|\n| I      | 9   | 4   | 3   | 7   |\n| II     | 4   | 6   | 5   | 6   |\n| III    | 5   | 4   | 7   | 5   |\n| IV     | 7   | 5   | 2   | 3   |\n| V      | 10  | 6   | 7   | 4   |\n\nTry to find a job assignment plan that minimizes the total working hours.", "en_answer": "14", "difficulty": "Medium", "id": 8}
 {"en_question": "Haus Toys can manufacture and sell toy trucks, toy airplanes, toy boats, and toy trains. The profit for each truck sold is $5, each airplane $10, each boat $8, and each train $7. How many types of toys should Haus Toys manufacture to maximize profits?\n\nThere are 890 units of wood available. Each truck requires 12 units, each airplane 20 units, each boat 15 units, and each train 10 units.\n\nThere are 500 units of steel available. Each airplane requires 3 units, each boat 5 units, each train 4 units, and each truck 6 units.\n\nIf Haus Toys manufactures trucks, they will not manufacture trains.\n\nHowever, if they manufacture boats, they will also manufacture airplanes.\n\nThe number of toy boats manufactured cannot exceed the number of toy trains manufactured.", "en_answer": "623.0", "difficulty": "Hard", "id": 9}
@@ -11,10 +11,10 @@
 {"en_question": "A company produces two types of small motorcycles, where type A is entirely manufactured by the company, and type B is assembled from imported parts. The production, assembly, and inspection time required for each unit of these two products are shown in Table 3.2.\n\nTable 3.2\n\n| Type | Process | | | Selling Price <br> (Yuan/unit) |\n| :---: | :---: | :---: | :---: | :---: |\n| | Manufacturing | Assembly | Inspection | |\n| Type A (hours/unit) | 20 | 5 | 3 | 650 |\n| Type B (hours/unit) | 0 | 7 | 6 | 725 |\n| Max production capacity per week (hours) | 120 | 80 | 40 | |\n| Production cost per hour (Yuan) | 12 | 8 | 10 | |\n\nIf the company's operational goals and targets are as follows:\n\n$p_{1}$ : The total profit per week should be at least 3000 yuan;\n\n$p_{2}$ : At least 5 units of type A motorcycles should be produced per week;\n\n$p_{3}$ : Minimize the idle time of each process as much as possible. The weight coefficients of the three processes are proportional to their hourly costs, and overtime is not allowed.\n\nTry to establish a model for this problem.", "en_answer": "4136.0", "difficulty": "Medium", "id": 11}
 {"en_question": "Red Star Plastics Factory produces six distinct types of plastic containers. Each container type is characterized by a specific volume, market demand, and unit variable production cost, as detailed in Table 5-11.\n\n**Table 5-11: Container Data**\n| Container Type (Code)             | 1    | 2    | 3    | 4    | 5    | 6     |\n| :------------------------------ | :--- | :--- | :--- | :--- | :--- | :---- |\n| Volume ($\\text{cm}^3$)             | 1500 | 2500 | 4000 | 6000 | 9000 | 12000 |\n| Market Demand (units)           | 500  | 550  | 700  | 900  | 400  | 300   |\n| Unit Variable Production Cost (Yuan/unit) | 5    | 8    | 10   | 12   | 16   | 18    |\n\nThe production of any container type necessitates the use of its dedicated specialized equipment. If the decision is made to **activate** the production equipment for a particular container type (i.e., if the production quantity of that type is greater than zero), a fixed setup cost of 1200 Yuan is incurred for that specific equipment.\n\nShould the production quantity of a certain container type be insufficient to meet its direct demand, the factory has the option to utilize other container types with **larger or equal volume** as substitutes to fulfill this unmet demand. For instance, type 2 containers (volume 2500 $\\text{cm}^3$) can be used to satisfy the demand for type 1 containers (requiring a volume of 1500 $\\text{cm}^3$), but type 1 containers cannot be used for type 2 demand. In this problem, the container type codes are pre-sorted in ascending order of their volumes.\n\n**Question:**\nHow should the factory organize its production? The objective is to develop a production plan that minimizes the total cost—comprising the sum of variable production costs for all containers produced and the fixed costs for all activated equipment—while ensuring that the demand for all container types is fully met.", "en_answer": "43200.00","difficulty": "Hard","id": 12}
 {"en_question": "Tom and Jerry just bought a farm in Sunshine Valley, and they are considering using it to plant corn, wheat, soybeans, and sorghum. The profit per acre for planting corn is $1500, the profit per acre for planting wheat is $1200, the profit per acre for planting soybeans is $1800, and the profit per acre for planting sorghum is $1600. To maximize their profit, how many acres of land should they allocate to each crop? Tom and Jerry’s farm has a total area of 100 acres.\n\nThe land area used for planting corn must be at least twice the land area used for planting wheat.\n\nThe land area used for planting soybeans must be at least half the land area used for planting sorghum.\n\nThe land area used for planting wheat must be three times the land area used for planting sorghum.", "en_answer": "180000.0", "difficulty": "Medium", "id": 13}
-{"en_question": "Mary is planning tonight's dinner. In addition to the vegetables she previously considered, she also wants to choose chicken, salmon, or tofu as a source of protein. Each 100 grams of chicken contains 23 grams of protein, each 100 grams of salmon contains 20 grams of protein, and each 100 grams of tofu contains 8 grams of protein. The price of chicken is $3 per 100 grams, salmon is $5 per 100 grams, and tofu is $1.5 per 100 grams. Considering Mary's budget is $20, how should she choose the ingredients to maximize protein intake? Additionally, the total weight of the food should not exceed 800 grams, and she needs to select at least three vegetables and one source of protein.", "en_answer": "1.5333333333333334", "difficulty": "Hard", "id": 14}
+{"en_question": "Mary is planning tonight's dinner. She wants to choose a combination of protein and vegetables to maximize her protein intake for the meal. Her protein options are chicken, salmon, and tofu, which can be bought in any quantity.\n\n- Chicken: 23g protein, $3.00 cost, per 100g.\n- Salmon: 20g protein, $5.00 cost, per 100g.\n- Tofu: 8g protein, $1.50 cost, per 100g.\n\nShe also wants to choose from a list of five vegetables, sold in 100g packs. She must select at least three different types of vegetables.\n\n- Broccoli (100g pack): 2.8g protein, $1.20 cost.\n- Carrots (100g pack): 0.9g protein, $0.80 cost.\n- Spinach (100g pack): 2.9g protein, $1.50 cost.\n- Bell Pepper (100g pack): 1.0g protein, $1.00 cost.\n- Mushrooms (100g pack): 3.1g protein, $2.00 cost.\n\nMary has two main constraints:\n1. Her total budget is $20.\n2. The total weight of all food must not exceed 800 grams.\n\nHow should Mary choose her ingredients to get the maximum possible amount of protein?", "en_answer": "123.80", "difficulty": "Hard", "id": 14}
 {"en_question": "A certain factory needs to use a special tool over $n$ planning stages. At stage $j$, $r_j$ specialized tools are needed. At the end of this stage, all tools used within this stage must be sent for repair before they can be reused. There are two repair methods: one is slow repair, which is cheaper (costs $b$ per tool) but takes longer ($p$ stages to return); the other is fast repair, which costs $c$ per tool $(c > b)$ and is faster, requiring only $q$ stages to return $(q < p)$. If the repaired tools cannot meet the needs, new ones must be purchased, with a cost of $a$ per new tool $(a > c)$. This special tool will no longer be used after $n$ stages. Determine an optimal plan for purchasing and repairing the tools to minimize the cost spent on tools during the planning period.\\n\\nn = 10  # number of stages\\nr = [3, 5, 2, 4, 6, 5, 4, 3, 2, 1]  # tool requirements per stage, indexing starts at 1\\na = 10  # cost of buying a new tool\\nb = 1   # cost of slow repair\\nc = 3   # cost of fast repair\\np = 3   # slow repair duration\\nq = 1   # fast repair duration", "en_answer": "36", "difficulty": "Medium", "id": 15}
 {"en_question": "A store plans to formulate the purchasing and sales plan for a certain product for the first quarter of next year. It is known that the warehouse capacity of the store can store up to 500 units of the product, and there are 200 units in stock at the end of this year. The store purchases goods once at the beginning of each month. The purchasing and selling prices of the product in each month are shown in Table 1.3.\n\nTable 1.3\n\n| Month | 1 | 2 | 3 |\n| :---: | :---: | :---: | :---: |\n| Purchasing Price (Yuan) | 8 | 6 | 9 |\n| Selling Price (Yuan) | 9 | 8 | 10 |\n\nNow, determine how many units should be purchased and sold each month to maximize the total profit, and express this problem as a linear programming model.", "en_answer": "4100.0", "difficulty": "Easy", "id": 16}
-{"en_question": "Certain strategic bomber groups are tasked with destroying enemy military targets. It is known that the target has four key parts, and destroying any one of them will suffice. The fuel consumption limit for completing this task is 48,000 liters, with a maximum of 48 heavy bombs and 32 light bombs. When carrying heavy bombs, each liter of fuel allows a distance of 2 km, whereas with light bombs, each liter allows 3 km. Additionally, each aircraft can only carry one bomb per trip, and each bombing run requires fuel not only for the round trip (each liter of fuel allows 4 km when the aircraft is empty) but also 100 liters for both takeoff and landing per trip. Relevant data is shown in Table 1-17.\n\nTable 1-17\n| Key Part | Distance from Airport (km) | Probability of Destruction per Heavy Bomb | Probability of Destruction per Light Bomb |\n|----------|----------------------------|-----------------------------------------|------------------------------------------|\n|          |                            |                                         |                                          |\n| 1        | 450                        | 0.10                                    | 0.08                                     |\n| 2        | 480                        | 0.20                                    | 0.16                                     |\n| 3        | 540                        | 0.15                                    | 0.12                                     |\n| 4        | 600                        | 0.25                                    | 0.20                                     |\n\nTo maximize the probability of destroying the enemy military target, how should the bombing plan be determined? Establish a linear programming model for this problem.", "en_answer": "1.1911788832813004e-09", "difficulty": "Hard", "id": 17}
+{"en_question": " Certain strategic bomber groups are tasked with destroying enemy military targets. It is known that the target has four key parts, and destroying at least two of them will suffice. \n\nResources and constraints:\n\nBomb stockpile: A maximum of 28 heavy bombs and 12 light bombs can be used.\nFuel limit: Total fuel consumption must not exceed 10,000 liters.\nFuel consumption rules: Same as before (depend on bomb type, flight distance, and takeoff/landing).\n\nTable 1-17\n| Key Part | Distance from Airport (km) | Probability of Destruction per Heavy Bomb | Probability of Destruction per Light Bomb |\n|----------|----------------------------|-----------------------------------------|------------------------------------------|\n|          |                            |                                         |                                          |\n| 1        | 450                        | 0.03                                    | 0.08                                     |\n| 2        | 480                        | 0.10                                    | 0.11                                     |\n| 3        | 540                        | 0.05                                    | 0.12                                     |\n| 4        | 600                        | 0.05                                    | 0.09                                     |\n\nHow should the bombing plan be determined to maximize the probability of success? What is the maximum probability of success? ", "en_answer": "0.5765", "difficulty": "Hard", "id": 17}
 {"en_question": "A textile factory produces two types of fabrics: one for clothing and the other for curtains. The factory operates two shifts, with a weekly production time set at 110 hours. Both types of fabrics are produced at a rate of 1000 meters per hour. Assuming that up to 70,000 meters of curtain fabric can be sold per week, with a profit of 2.5 yuan per meter, and up to 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 in formulating its production plan:\n\n$p_{1}$ : The weekly production time must fully utilize 110 hours;\n\n$p_{2}$ : Overtime should not exceed 10 hours per week;\n\n$p_{3}$ : At least 70,000 meters of curtain fabric and 45,000 meters of clothing fabric must be sold per week;\n\n$p_{4}$ : Minimize overtime as much as possible.\n\nFormulate a model for this problem.", "en_answer": "227500.0", "difficulty": "Medium", "id": 18}
 {"en_question": "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 store needs to minimize the total cost of the order.\n\nAdditionally, each order from manufacturer A will include 15 chairs, while each order from manufacturers B and C will include 10 chairs. The number of orders must be an integer. The store needs to order at least 100 chairs.\n\nEach order from manufacturer A will include 15 chairs, while each order from manufacturers B and C will include 10 chairs. The store needs to order at most 500 chairs.\n\nIf the store decides to order chairs from manufacturer A, it must also order at least 10 chairs from manufacturer B.\n\nFurthermore, if the store decides to order chairs from manufacturer B, it must also order chairs from manufacturer C.", "en_answer": "4000.0", "difficulty": "Easy", "id": 19}
 {"en_question": "Bright Future Toys wants to build and sell robots, model cars, building blocks, and dolls. The profit for each robot sold is $15, for each model car sold is $8, for each set of building blocks sold is $12, and for each doll sold is $5. How many types of toys should Bright Future Toys manufacture to maximize profit?\nThere are 1200 units of plastic available. Each robot requires 30 units of plastic, each model car requires 10 units of plastic, each set of building blocks requires 20 units of plastic, and each doll requires 15 units of plastic.\n\nThere are 800 units of electronic components available. Each robot requires 8 units of electronic components, each model car requires 5 units of electronic components, each set of building blocks requires 3 units of electronic components, and each doll requires 2 units of electronic components.\n\nIf Bright Future Toys manufactures robots, they will not manufacture dolls.\n\nHowever, if they manufacture model cars, they will also manufacture building blocks.\n\nThe number of dolls manufactured cannot exceed the number of model cars manufactured.", "en_answer": "956.0", "difficulty": "Easy", "id": 20}
@@ -28,7 +28,7 @@
 {"en_question": "Someone has a fund of 300,000 yuan and has the following investment projects in the next three years:\n(1) Investment can be made at the beginning of each year within three years, with an annual profit of 20% of the investment amount, and the principal and interest can be used for investment in the following year;\n(2) Investment is only allowed at the beginning of the first year, and it can be recovered at the end of the second year, with the total principal and interest amounting to 150% of the investment amount, but the investment limit is no more than 150,000 yuan;\n(3) Investment is allowed at the beginning of the second year within three years, and it can be recovered at the end of the third year, with the total principal and interest amounting to 160% of the investment amount, and the investment limit is 200,000 yuan;\n(4) Investment is allowed at the beginning of the third year within three years, and it can be recovered in one year with a profit of 40%, and the investment limit is 100,000 yuan.\nChapter One: Linear Programming and Simplex Method\nTry to determine an investment plan for this person that maximizes the principal and interest at the end of the third year.", "en_answer": "580000", "difficulty": "Medium", "id": 28}
 {"en_question": "Jieli Company needs to recruit three types of professionals to work in the two regional branches located in Donghai City and Nanjiang City. The demand for different professionals in these regional branches is shown in Table 4-3. After assessing the situation of the applicants, the company has categorized them into 6 types. Table 4-4 lists the specialties each type of person can handle, the specialty they prefer, and the city they prefer to work in. The company's personnel arrangement considers the following three priorities:\n$p_1$: All three types of professionals needed are fully met;\n$p_2$: 8000 recruited personnel meet their preferred specialty;\n$p_3$: 8000 recruited personnel meet their preferred city.\nTry to establish a mathematical model for goal planning accordingly.\n\nTable 4-3\n| Branch Location | Specialty | Demand |\n|-----------------|-----------|--------|\n| Donghai City    | 1         | 1000   |\n| Donghai City    | 2         | 2000   |\n| Nanjiang City   | 3         | 1500   |\n| Nanjiang City   | 1         | 2000   |\n| Nanjiang City   | 2         | 1000   |\n| Nanjiang City   | 3         | 1000   |\n\nTable 4-4\n\n| Type | Number of People | Suitable Specialty | Preferred Specialty | Preferred City |\n|------|------------------|--------------------|---------------------|----------------|\n| 1    | 1500             | 1,2                | 1                   | Donghai        |\n| 2    | 1500             | 2,3                | 2                   | Donghai        |\n| 3    | 1500             | 1,3                | 1                   | Nanjiang       |\n| 4    | 1500             | 1,3                | 3                   | Nanjiang       |\n| 5    | 1500             | 2,3                | 3                   | Donghai        |\n| 6    | 1500             | 3                  | 3                   | Nanjiang       |", "en_answer": "11500.0", "difficulty": "Medium", "id": 29}
 {"en_question": "Suppose a certain animal needs at least $700 \\mathrm{~g}$ of protein, $30 \\mathrm{~g}$ of minerals, and $100 \\mathrm{mg}$ of vitamins daily. There are 5 types of feed available, and the nutritional content and price per gram of each type of feed are shown in Table 1-5:\nTry to formulate a linear programming model that meets the animal's growth needs while minimizing the cost of selecting the feed.\nTable 1-6\n| Feed | Protein (g) | Minerals (g) | Vitamins (mg) | Price (¥/kg) | Feed | Protein (g) | Minerals (g) | Vitamins (mg) | Price (¥/kg) |\n|------|-------------|--------------|---------------|--------------|------|-------------|--------------|---------------|--------------|\n| 1    | 3           | 1            | 0.5           | 0.2          | 4    | 6           | 2            | 2             | 0.3          |\n| 2    | 2           | 0.5          | 1             | 0.7          | 5    | 18          | 0.5          | 0.8           | 0.8          |\n| 3    | 1           | 0.2          | 0.2           | 0.4          |      |             |              |               |              |", "en_answer": "32.43589743589744", "difficulty": "Easy", "id": 30}
-{"en_question": "A factory produces three types of products, I, II, and III. Each product requires two processing stages, A and B. It is assumed that the factory has two types of equipment to complete the A processing stage, denoted as A1 and A2; and two types of equipment to complete the B processing stage, denoted as B1 and B2. Product I can be processed on any type of A equipment, but for the B processing stage, it can only be processed on B1 equipment; Product III can only be processed on A2 and B2 equipment. The processing time per piece, raw material cost, sales price of products, effective machine hours for various equipment, and costs of equipment at full capacity are given in Table 1-18. The requirement is to arrange the optimal production plan so that the factory's profit is maximized.\n\nTable 1-18\n| Equipment   | Product I | Product II | Product III | Effective Machine Hours | Cost of Equipment at Full Capacity (Yuan) |\n|-------------|-----------|------------|-------------|------------------------|-------------------------------------------|\n| A1          | 5         | 10         |             | 10000                 | 321                                       |\n| A2          | 7         | 9          | 12          | 4000                  | 250                                       |\n| B1          | 6         | 8          |             | 7000                  | 783                                       |\n| B2          | 4         |            | 11          | 4000                  | 200                                       |\n| B3          | 7         |            |             |                       |                                           |\n| Raw Material Cost (Yuan/Piece) | 0.25      | 0.35       | 0.50       |                        |                                           |\n| Unit Price (Yuan/Piece)        | 1.25      | 2.00       | 2.80       |                        |                                           |", "en_answer": "1217.3999999999999", "difficulty": "Hard", "id": 31}
+{"en_question": "A factory produces three types of products: I, II, and III. Each product must undergo two processing stages, A and B. The factory has two types of equipment to complete stage A (A1, A2) and three types of equipment to complete stage B (B1, B2, B3).\n\nThe production rules are as follows:\n- Product I can be processed on any type of A equipment (A1 or A2) and any type of B equipment (B1, B2, or B3).\n- Product II can be processed on any type of A equipment (A1 or A2), but for stage B, it can only be processed on B1 equipment.\n- Product III can only be processed on A2 equipment for stage A and B2 equipment for stage B.\n\nThe detailed data for processing time per piece, costs, sales price, and machine availability is provided in the table below. The objective is to determine the optimal production plan to maximize the factory's total profit.\n\nData Table\n| Equipment | Product I | Product II | Product III | Effective Machine Hours | Full - load Equipment Cost (Yuan) | Processing Cost per Machine Hour (Yuan/hour) |\n| :--- | :--- | :--- | :--- | :--- | :--- | :--- |\n| A1 | 5 | 10 | - | 6000 | 300 | 0.05 |\n| A2 | 7 | 9 | 12 | 10000 | 321 | 0.03 |\n| B1 | 6 | 8 | - | 4000 | 250 | 0.06 |\n| B2 | 4 | - | 11 | 7000 | 783 | 0.11 |\n| B3 | 7 | - | - | 4000 | 200 | 0.05 |\n| Raw Material Cost (Yuan/piece) | 0.25 | 0.35 | 0.5 | - | - | - |\n| Unit Price (Yuan/piece) | 1.25 | 2 | 2.8 | - | - | - |", "en_answer": "1146.39", "difficulty": "Hard", "id": 31}
 {"en_question": "A product consists of three components produced by four workshops, each with a limited number of production hours. Table 1.4 below provides the production rates of the three components. The objective is to determine the number of hours each workshop should allocate to each component to maximize the number of completed products. Formulate this problem as a linear programming problem.\n\nTable 1.4\n\n| Workshop | Production Capacity (hours) | Production Rate (units/hour) |   |   |\n| :------: | :-------------------------: | :--------------------------: | - | - |\n|          |                             | Component 1 | Component 2  | Component 3 |\n|    A     |           100               |      10      |      15     |      5      |\n|    B     |           150               |      15      |      10     |      5      |\n|    C     |           80                |      20      |      5      |      10     |\n|    D     |           200               |      10      |      15     |      20     |", "en_answer": "2924.0", "difficulty": "Easy", "id": 32}
 {"en_question": "A wealthy noble passed away, leaving the following inheritance:\n\n- A painting by Caillebotte: $25000\n- A bust of Diocletian: $5000\n- A Yuan dynasty Chinese vase: $20000\n- A 911 Porsche: $40000\n- Three diamonds: each $12000\n- A Louis XV sofa: $3000\n- Two very precious Jack Russell racing dogs: each $3000 (will stipulates they must not be separated)\n- A sculpture from 200 AD: $10000\n- A sailing boat: $15000\n- A Harley Davidson motorcycle: $10000\n- A piece of furniture once belonging to Cavour: $13000,\n\nwhich must be shared between two sons. How to formulate a mathematical program and solve it using COPTPY to minimize the difference in value between the two parts?", "en_answer": "1000.0", "difficulty": "Medium", "id": 33}
 {"en_question": "The current problem faced by the company is how to use the fewest number of containers to pack the currently needed goods for transportation, while considering the weight of the goods, specific packaging requirements, and inventory limitations. Professional modeling and analysis are needed for a batch of goods’ transportation strategy to ensure maximum utilization of the limited container space.\n\nThe company currently has a batch to be transported, with each container able to hold a maximum of 60 tons of goods and each container used must load at least 18 tons of goods. The goods to be loaded include five types: A, B, C, D, and E, with quantities of 120, 90, 300, 90, and 120 respectively. The weights are 0.5 tons for A, 1 ton for B, 0.4 tons for C, 0.6 tons for D, and 0.65 tons for E. Additionally, to meet specific usage requirements, every time A goods are loaded, at least 1 unit of C must also be loaded, but loading C alone does not require simultaneously loading A; and considering the demand limitation for D goods, each container must load at least 12 units of D.\n\nEstablish an operations research model so that the company can use the fewest number of containers to pack this batch of goods.", "en_answer": "7.0", "difficulty": "Hard", "id": 34}
@@ -44,7 +44,7 @@
 {"en_question": "Given that there are $m=2$ production points for a certain type of material, where the output at the $i$-th point $(i=1,2)$ is $a_i$, $a_1 = 100$, and $a_2 = 150$. This material is to be shipped to $n=2$ demand points, where the demand at the $j$-th point $(j=1, 2)$ is $b_j$, $b_1 = 80$, and $b_2 = 120$. It is known that $\\sum_i a_i \\geqslant \\sum_j b_j$. It is also known that when shipping from production points to demand points, it must pass through one of the $p=2$ intermediate marshaling stations. If the $k$-th $(k=1, 2)$ intermediate marshaling station is used, a fixed cost $f_k$ is incurred regardless of the transshipment volume, where $f_1 = 10$ and $f_2 = 15$. The $k$-th intermediate marshaling station has a maximum transshipment capacity limitation $q_k$, where $q_1 = 100$ and $q_2 = 100$. Let $c_{i k}$ and $c'_{k j}$ denote the unit transportation cost from $i$ to $k$ and from $k$ to $j$, respectively, where $c_{11}=2$, $c_{12}=3$, $c_{21}=4$, $c_{22}=1$, $c'_{11}=3$, $c'_{12}=2$, $c'_{21}=1$, and $c'_{22}=4$. Try to determine a transportation plan for this material that minimizes the total cost.", "en_answer": "685", "difficulty": "Medium", "id": 44}
 {"en_question": "A factory produces three types of products, A, B, and C. Each unit of product A requires 1 hour for technical preparation, 10 hours of direct labor, and 3 kg of materials. Each unit of product B requires 2 hours for technical preparation, 4 hours of labor, and 2 kg of materials. Each unit of product C requires 1 hour for technical preparation, 5 hours of labor, and 1 kg of materials. The available technical preparation time is 100 hours, labor time is 700 hours, and materials are 400 kg. The company offers larger discounts for bulk purchases, as detailed in Table 1-22. Determine the company's production plan to maximize profit.\nTable 1-22\n| Product A       |           | Product B       |           | Product C       |           |\n|:---------------|:---------:|:---------------|:---------:|:---------------|:---------:|\n| Sales Volume (pieces) | Profit (yuan) | Sales Volume (pieces) | Profit (yuan) | Sales Volume (pieces) | Profit (yuan) |\n| 0 ~ 40         | 10        | 0 ~ 50         | 6         | 0 ~ 100        | 5         |\n| 40 ~ 100       | 9         | 50 ~ 100       | 4         | Above 100      | 4         |\n| 100 ~ 150      | 8         | Above 100      | 3         |                |           |\n| Above 150      | 7         |                |           |                |           |", "en_answer": "734", "difficulty": "Easy", "id": 45}
 {"en_question": "A university computer lab hires 4 undergraduates (designated 1, 2, 3, and 4) and 2 graduate students (designated 5 and 6) for duty answering questions. The maximum duty hours from Monday to Friday and the hourly wage for each person are shown in Table 5-9.\n\nTable 5-9\n| Student ID | Wage (CNY/h) | Monday | Tuesday | Wednesday | Thursday | Friday |\n|------------|--------------|--------|---------|-----------|----------|--------|\n| 1          | 10.0         | 6      | 0       | 6         | 0        | 7      |\n| 2          | 10.0         | 0      | 6       | 0         | 6        | 0      |\n| 3          | 9.9          | 4      | 8       | 3         | 0        | 5      |\n| 4          | 9.8          | 5      | 5       | 6         | 0        | 4      |\n| 5          | 10.8         | 3      | 0       | 4         | 8        | 0      |\n| 6          | 11.3         | 0      | 6       | 0         | 6        | 3      |\n\nThe lab operates from 8:00 AM to 10:00 PM, and there must be one and only one student on duty during open hours. It is also required that each undergraduate must work at least 8 hours per week, and each graduate student must work at least 7 hours per week. Additionally, supplement the following requirements: each student can work no more than 2 shifts per week, and no more than 3 students can be scheduled for duty each day. Based on these conditions, establish a new mathematical model.", "en_answer": "472.3", "difficulty": "Medium", "id": 46}
-{"en_question": "A certain farm has 100 hectares of land and 15,000 yuan in funds for production development. The labor force situation on the farm is 3,500 person-days in autumn and winter, and 4,000 person-days in spring and summer. If the labor force itself is not fully utilized, they can work externally, earning 2.1 yuan/person-day in spring and summer and 1.8 yuan/person-day in autumn and winter.\n\nThe farm cultivates three types of crops: soybeans, corn, and wheat, and also raises dairy cows and chickens. Crop cultivation requires no specialized investment, but raising animals involves an investment of 400 yuan per dairy cow and 3 yuan per chicken. Raising dairy cows requires allocating 1.5 hectares of land per cow to grow feed, and involves 100 person-days in autumn and winter, and 50 person-days in spring and summer per cow. The annual net income is 400 yuan per dairy cow. Raising chickens does not use land, requires 0.6 person-days in autumn and winter, and 0.3 person-days in spring and summer per chicken. Annual net income is 2 yuan per chicken. The current chicken coop can accommodate up to 3,000 chickens, and the cow barn can accommodate up to 32 dairy cows. The labor and income requirements for the three types of crops per year are shown in Table 1-9.\n\nTable 1-9\n| Item           | Soybean | Corn | Wheat |\n|----------------|---------|------|-------|\n| Person-days (Autumn/Winter) | 20      | 35   | 10    |\n| Person-days (Spring/Summer) | 50      | 75   | 40    |\n| Annual Net Income (Yuan/hectare) | 175     | 300   | 120   |\n\nDetermine the farm's operating plan to maximize annual net income.", "en_answer": "20241.615384615387", "difficulty": "Hard", "id": 47}
+{"en_question": "A certain farm has 100 hectares of land and 15,000 yuan in funds for production development. The labor force situation on the farm is 3,500 person-days in autumn and winter, and 4,000 person-days in spring and summer. If the labor force itself is not fully utilized, they can work externally, earning 2.1 yuan/person-day in spring and summer and 1.8 yuan/person-day in autumn and winter.\n\nThe farm cultivates three types of crops: soybeans, corn, and wheat, and also raises dairy cows and chickens. Crop cultivation requires no specialized investment, but raising animals involves an investment of 400 yuan per dairy cow and 3 yuan per chicken. Raising dairy cows requires allocating 1.5 hectares of land per cow to grow feed, and involves 100 person-days in autumn and winter, and 50 person-days in spring and summer per cow. The annual net income is 400 yuan per dairy cow. Raising chickens does not use land, requires 0.6 person-days in autumn and winter, and 0.3 person-days in spring and summer per chicken. Annual net income is 2 yuan per chicken. The current chicken coop can accommodate up to 3,000 chickens, and the cow barn can accommodate up to 32 dairy cows. The labor and income requirements for the three types of crops per year are shown in Table 1-9.\n\nTable 1-9\n| Item           | Soybean | Corn | Wheat |\n|----------------|---------|------|-------|\n| Person-days (Autumn/Winter) | 20      | 35   | 10    |\n| Person-days (Spring/Summer) | 50      | 75   | 40    |\n| Annual Net Income (Yuan/hectare) | 175     | 300   | 120   |\n\nDetermine the farm's operating plan to maximize annual net income. Please note that Labor days are calculated in whole days, fractions are not allowed.", "en_answer": "20241.615384615387", "difficulty": "Hard", "id": 47}
 {"en_question": "A factory produces two models of microcomputers, A and B. Each model requires the same two processes. The processing time, sales profit, and the factory’s maximum weekly processing capacity for each model are shown in Table 3.1.\n\nTable 3.1\n\n| Process | Model | | Maximum Weekly Processing Capacity |\n| :---: | :---: | :---: | :---: |\n| | $A$ | $B$ | |\n| I (hours/unit) | 4 | 6 | 150 |\n| II (hours/unit) | 3 | 2 | 70 |\n| Profit (yuan/unit) | 300 | 450 | |\n\nGiven the factory's business goals:\n\n$p_{1}$: The total weekly profit should not be less than 10,000 yuan;\n\n$p_{2}$: Due to contract requirements, at least 10 units of model A and at least 15 units of model B must be produced each week;\n\n$p_{3}$: The processing time for Process I should be exactly 150 hours per week, and the processing time for Process II should ideally be fully utilized, with potential for appropriate overtime;\n\n$p_{4}$: If products are produced during overtime in Process II, the profit per unit is reduced by 20 yuan for model A and 25 yuan for model B, and the maximum overtime for Process II is 30 hours per week. Formulate the mathematical model for this problem.", "en_answer": "11250.0", "difficulty": "Medium", "id": 48}
 {"en_question": "A factory needs to rent a warehouse to store materials in the next 4 months. The required warehouse area for each month is listed in Table 1-14.\n\nTable 1-14\n| Month | 1    | 2    | 3    | 4    |\n|-------|------|------|------|------|\n| Required warehouse area (㎡) | 1500 | 1000 | 2000 | 1200 |\n\nThe longer the rental contract period, the greater the discount on warehouse rental fees. The specific data is listed in Table 1-15.\n\nTable 1-15\n| Contract rental period (months) | 1  | 2  | 3  | 4  |\n|---------------------------------|----|----|----|----|\n| Rental fees for warehouse area during the contract period (yuan)", "en_answer": "56000.0", "difficulty": "Easy", "id": 49}
 {"en_question": "A store has formulated a purchase and sales plan for a certain product from July to December. It is known that the warehouse capacity must not exceed 500 units, with 200 units in stock at the end of June. Thereafter, purchases are made at the beginning of each month. Assume the purchase and selling prices of this product for each month are shown in Table 1-21. How much should be purchased and sold each month to maximize the total revenue?\n\nTable 1-21\n| Month | 7  | 8  | 9  | 10 | 11 | 12 |\n|-------|----|----|----|----|----|----|\n| Buy   | 28 | 24 | 25 | 27 | 23 | 23 |\n| Sell  | 29 | 24 | 26 | 28 | 22 | 25 |", "en_answer": "9100", "difficulty": "Easy", "id": 50}
@@ -82,7 +82,7 @@
 {"en_question": "Changjiang Comprehensive Shopping Mall has 5000 m² of space for lease and plans to attract the following 5 types of stores as tenants. The table below shows the area occupied by each type of store for one shop, the minimum and maximum number of shops for each type within the mall, and the expected annual profit (in ten thousand yuan) per store for different numbers of stores. Each store pays 20% of its annual profit as rent to the mall. Question: How many of each type of store should the mall lease to maximize total rental income?\n\nTable 5-12\n\n| Code | Store Type | Area per Shop / m² | Min | Max | 1 Store | 2 Stores | 3 Stores |\n|------|------------|--------------------|-----|-----|---------|----------|----------|\n| 1    | Jewelry    | 250                | 1   | 3   | 9       | 8        | 7        |\n| 2    | Shoes & Hats | 350              | 1   | 2   | 10      | 9        | -        |\n| 3    | General Merchandise | 800      | 1   | 3   | 27      | 21       | 20       |\n| 4    | Bookstore  | 400                | 0   | 2   | 16      | 10       | -        |\n| 5    | Catering   | 500                | 1   | 3   | 17      | 15       | 12       |", "en_answer": "28", "difficulty": "Easy", "id": 82}
 {"en_question": "A certain restaurant operates around the clock, and the number of waiters needed in 24 hours is shown in Table 1.1.\n\nTable 1.1\n\n| Time        | Minimum Number of Waiters Needed | Time        | Minimum Number of Waiters Needed |\n|:-----------:|:-------------------------------:|:-----------:|:-------------------------------:|\n| $2 \\sim 6$  | 4                                | $14 \\sim 18$| 7                                |\n| $6 \\sim 10$ | 8                                | $18 \\sim 22$| 12                               |\n| $10 \\sim 14$| 10                               | $22 \\sim 2$ | 4                                |\n\nEach waiter works continuously for 8 hours a day. The goal is to find the minimum number of waiters that meet the above conditions and represent this problem as a linear programming model.", "en_answer": "26.0", "difficulty": "Hard", "id": 83}
 {"en_question": "A company hopes to recruit new employees for its team. The salary requirements for 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.\n\nThey hope to hire a maximum of 3 new employees.\n\nThe 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.\n\nThe qualifications of the five candidates are as follows:\nCandidate A: Bachelor's degree;\nCandidate B: Master's degree;\nCandidate C: Doctoral degree;\nCandidate D: No degree;\nCandidate E: No degree.\nThey will select at least one candidate with a Master's or Doctoral degree.\n\nThe work experience of the five candidates is as follows:\nCandidate A: 3 years of work experience;\nCandidate B: 10 years of work experience;\nCandidate C: 4 years of work experience;\nCandidate D: 3 years of work experience;\nCandidate E: 7 years of work experience.\nThey hope the total work experience of the selected candidates is no less than 12 years.\n\nDue to the equivalent professional skills of candidates A and E, the company will choose at most one from the two.\n\nThey will hire at least 2 new employees.", "en_answer": "23000.0", "difficulty": "Easy", "id": 84}
-{"en_question": "A company is producing two products (X and Y). The resources required for the production of X and Y are divided into two parts: machine time for automated processing and craftsman time for manual finishing. The table below shows the number of minutes required for each product:\n\n| Item | Machine Time (minutes) | Craftsman Time (minutes) |\n| :---: | :---: | :---: |\n| X | 13 | 20 |\n| Y | 19 | 29 |\n\nThe company has 40 hours of machine time available in the next working week, but only 35 hours of craftsman time. The cost of machine time is £10 per hour, and the cost of craftsman time is £2 per hour. Idle time for machines and craftsmen incurs no cost. For each product produced (all products produced will be sold), the revenue for product X is £20, and the revenue for product Y is £30. The company has a specific contract that requires 10 units of product X to be produced for a customer each week. Formulate a model for this problem.", "en_answer": "1866.5", "difficulty": "Hard", "id": 85}
+{"en_question": "A company is producing two products (X and Y). The resources required for the production of X and Y are divided into two parts: machine time for automated processing and craftsman time for manual finishing. The table below shows the number of minutes required for each product:\n\n| Item | Machine Time (minutes) | Craftsman Time (minutes) |\n| :---: | :---: | :---: |\n| X | 13 | 20 |\n| Y | 19 | 29 |\n\nThe company has 40 hours of machine time available in the next working week, but only 35 hours of craftsman time. The cost of machine time is £10 per hour, and the cost of craftsman time is £2 per hour. Idle time for machines and craftsmen incurs no cost. For each product produced (all products produced will be sold), the revenue for product X is £20, and the revenue for product Y is £30. The company has a specific contract that requires 10 units of product X to be produced for a customer each week. Formulate a model for this problem.", "en_answer": "1866.37", "difficulty": "Hard", "id": 85}
 {"en_question": "Healthy Pet Foods Company produces two types of dog food: Meaties and Yummies. Each pack of Meaties contains 2 pounds of grains and 3 pounds of meat; each pack of Yummies contains 3 pounds of grains and 1.5 pounds of meat. The company believes it can sell any quantity of dog food that it can produce. Meaties sell for $2.80 per pack, and Yummies sell for $2.00 per pack. The company's production is subject to several constraints. First, a maximum of 400,000 pounds of grains can be purchased each month at a price of $0.20 per pound of grains. A maximum of 300,000 pounds of meat can be purchased each month at a price of $0.50 per pound of meat. Additionally, a special machine is required to produce Meaties, with a monthly capacity of 90,000 packs. The variable costs for mixing and packaging dog food are $0.25 per pack (Meaties) and $0.20 per pack (Yummies). Detailed information is provided in Table B-1.\n\n**Table B-1 Healthy Pet Foods Data**\n\n|                    | Meaties      | Yummies    |\n|--------------------|--------------|------------|\n| Price per pack     | $2.80        | $2.00      |\n| Raw materials      |              |            |\n| - Grains           | 2.0 lbs      | 3.0 lbs    |\n| - Meat             | 3.0 lbs      | 1.5 lbs    |\n| Variable cost      | $0.25/pack   | $0.20/pack |\n| Resources          |              |            |\n| Meaties capacity   | 90,000 packs/month |       |\n| Monthly available grains | 400,000 lbs |      |\n| Monthly available meat | 300,000 lbs |        |\n\nAssume you are the manager of the dog food department at Healthy Pet Foods Company. Your salary is based on the department's profit, so you will try to maximize profit. How should you operate the department to maximize both the profit and your salary?", "en_answer": "77500", "difficulty": "Hard", "id": 86}
 {"en_question": "A transportation company has two types of trucks, Type A and Type B. Type A trucks have 20 cubic meters of refrigerated capacity and 40 cubic meters of non-refrigerated capacity. In contrast, Type B trucks have the same total capacity, but the capacities for refrigerated and non-refrigerated cargo are equal. A grocer needs to rent trucks to transport 3000 cubic meters of refrigerated cargo and 4000 cubic meters of non-refrigerated cargo. The rental cost per kilometer for Type A trucks is £30, while the rental cost per kilometer for Type B trucks is £40. How many of each type of truck should the grocer rent to minimize the total cost?\n\nTry to formulate a model for this problem.", "en_answer": "4170", "difficulty": "Medium", "id": 87}
 {"en_question": "A company uses two machines (Machine 1 and Machine 2) to produce two types of products (liquid fertilizer and solid fertilizer). To produce one unit of liquid fertilizer, it takes 50 minutes on Machine 1 and 30 minutes on Machine 2. To produce one unit of solid fertilizer, it takes 24 minutes on Machine 1 and 33 minutes on Machine 2. At the beginning of the week, there are 30 units of liquid fertilizer and 90 units of solid fertilizer in inventory. The available processing time for Machine 1 this week is expected to be 40 hours, and for Machine 2 it is expected to be 35 hours. The demand for liquid fertilizer this week is estimated at 75 units, and for solid fertilizer at 95 units. The company's policy is to maximize the total number of units of liquid fertilizer and solid fertilizer in inventory at the end of the week.\n\nFormulate a model for this problem.", "en_answer": "1.25", "difficulty": "Medium", "id": 88}