Correct problem 17

IndustryOR.json

@@ -14,7 +14,7 @@
 {"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 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": " 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: 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. \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 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 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": "1146.56", "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}