correct problem 36

IndustryOR.json

@@ -33,7 +33,7 @@
 {"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}
 {"en_question": "A fabric dyeing plant has 3 dyeing vats. Each batch of fabric must be dyed in sequence in each vat: first, the second, and third vats. The plant must color five batches of fabric of different sizes. The time required in hours to dye batch $i$ in vat $j$ is given in the following matrix:\n\n$$\n\\left(\\begin{array}{ccc}\n3 & 1 & 1 \\\\\n2 & 1.5 & 1 \\\\\n3 & 1.2 & 1.3 \\\\\n2 & 2 & 2 \\\\\n2.1 & 2 & 3\n\\end{array}\\right)\n$$\n\nSchedule the dyeing operations in the vats to minimize the completion time of the last batch.", "en_answer": "14.1", "difficulty": "Medium", "id": 35}
-{"en_question": "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: In a certain area, there are a certain number of customers and a distribution center or warehouse. Customers are generally distributed in different locations, and each customer has a certain amount of delivery needs. The distribution center or warehouse needs to dispatch a fleet of vehicles and design an appropriate delivery plan to meet the delivery needs of all customers. The goal of VRP is to optimize efficiency while meeting all customer needs. The efficiency is usually measured in the form of an objective function, which varies with the enterprise's requirements. Common objective functions include minimizing the total distance traveled by vehicles, minimizing the total delivery time, and minimizing the number of vehicles used. In addition to meeting customers' delivery needs, VRP usually needs to consider various constraints, giving rise to various variants. For example, if the vehicle's load must not exceed its maximum capacity, the problem becomes the Capacitated Vehicle Routing Problem (CVRP); if each customer's delivery needs must be met within a specific time window, the problem changes to the Vehicle Routing Problem with Time Windows (VRPTW).\n\nThe Vehicle Routing Problem with Time Windows (VRPTW) is a classic problem developed from VRP. In reality, there are many application scenarios for VRPTW. Each customer point generally has a service time window. For example, some logistics centers need to store packages during less busy times, some large supermarkets need to replenish stocks during non-business hours, and real-time delivery methods such as food delivery require delivery within specific time windows. Time windows are divided into soft time windows and hard time windows. Hard Time Windows (HTW) mean that vehicles must arrive at or before the delivery point within the time window and cannot arrive late. If they arrive early, they must wait until the time window starts to perform the service, such as in supermarket restocking and logistics center warehousing. Soft Time Windows (STW) mean that vehicles do not have to arrive within the original hard time window but should try to arrive within it. If vehicles arrive early, there is a cost for early arrival; if they arrive late, there is a penalty, as seen in scenarios such as food delivery, school bus pick-up and drop-off, and industrial delivery.\n\nThe 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 an area. Vehicles need to depart from and return to the distribution center, and routes should be continuous, with each customer being served by only one vehicle. Vehicles also have a capacity constraint. Each customer has a specific service time window and will only accept service within this window. Vehicles can arrive at customer points early and wait for the time window to open, or they can arrive within the time window to provide service. Vehicles can only start providing service within the time window, and the service time is known. The distribution center needs to arrange a reasonable delivery plan to complete the delivery tasks while minimizing travel costs. Since VRPHTW does not allow delays, it primarily emphasizes travel costs on the routes, similar to VRP. Please now proceed with operations research modeling for VRPHTW.", "en_answer": "1644.63", "difficulty": "Hard", "id": 36}
+{"en_question": "The Vehicle Routing Problem (VRP) was first proposed by Dantzig and Ramser in 1959. It is a classic combinatorial optimization problem. The basic VRP can be described as follows: in a certain area, there is a number of customers and a distribution center or depot. Customers are generally located at different positions, and each has a specific demand for goods. The distribution center needs to dispatch a fleet of vehicles and design appropriate delivery routes to fulfill the demands of all customers. The objective of VRP is to optimize a certain benefit metric while satisfying all customer demands. The benefit metric is usually presented as an objective function, which varies according to the company's requirements. Common objective functions include minimizing the total distance traveled by vehicles, minimizing the total delivery time, or minimizing the number of vehicles used. In addition to satisfying customer demands, VRP often needs to consider various other constraints, leading to several variants. For example, if the vehicle's load cannot exceed its maximum capacity, the problem becomes the Capacitated Vehicle Routing Problem (CVRP). If each customer's delivery must be made within a specific time frame, the problem becomes the Vehicle Routing Problem with Time Windows (VRPTW).\n\nThe Vehicle Routing Problem with Time Windows (VRPTW) is a classic variant of the VRP. There are many real-world applications of VRPTW, as customer locations often have service time windows. For instance, some logistics centers need to stock parcels during off-peak hours, and large supermarkets need to replenish goods outside of business hours. Real-time delivery services like food delivery also require strict delivery time windows. Time windows can be categorized as hard or soft. A Hard Time Window (HTW) means that a vehicle must arrive at the delivery point within or before the time window; late arrivals are not permitted. If a vehicle arrives early, it must wait until the time window opens to begin service. This is common in scenarios like supermarket restocking and logistics center inbound operations. A Soft Time Window (STW) means that a vehicle is not strictly required to arrive within the time window, but it is encouraged to do so. A penalty is incurred for early or late arrivals. This is applicable in scenarios such as meal delivery, school bus services, and industrial deliveries.\n\nThe Vehicle Routing Problem with Hard Time Windows (VRPHTW) can be described as follows: within a region, there is a set of customer locations and a central depot. Vehicles must start from the depot and return to the depot, following continuous paths. Each customer must be served by exactly one vehicle, and vehicles have a limited capacity. Each customer has a specific service time window, and service is only accepted within this window. A vehicle can arrive at a customer location early and wait for the time window to open, or it can arrive within the time window to provide service. Service can only begin within the time window, and the service duration is known. The distribution center must arrange an optimal delivery plan to both complete the delivery tasks and minimize travel costs. Because VRPHTW does not allow for delays, it, like the VRP, primarily emphasizes the minimization of travel costs along the routes.\n\n Now Considering a city logistics company named 'Speedy Logistics' has a central depot and 15 customer locations in a city. The company needs to plan the daily delivery routes for its 4 trucks, each with a maximum capacity of 80 units.\n\n**Specific Requirements:**\n\n1.  **Delivery Task**: The demands of all 15 customers must be met, and each customer must be served by exactly one vehicle.\n2.  **Vehicle Constraints**: The company can dispatch a maximum of 4 trucks. All trucks must start from the central depot and return to it after completing their delivery tasks.\n3.  **Capacity Constraint**: The total demand of all customers on a single route must not exceed the truck's maximum capacity (80 units).\n4.  **Time Window Constraint**: Each customer has a strict 'hard time window.' Service must begin within this specified time window. Early arrivals must wait, and late arrivals are not permitted.\n5.  **Service Time**: A fixed service time of 10 minutes is required at each customer location for unloading.\n6.  **Optimization Objective**: The company's primary cost is fuel and vehicle wear, so the main objective is to **minimize the total distance traveled by all vehicles**.\n\n**Data Details:**\n\n* **Central Depot (Depot 0)**:\n    * Coordinates: (50, 50)\n    * Operating Time Window: [0, 1200] (minutes)\n* **Customer Locations (Customers 1-15)**: The coordinates, demand, service time window, and service duration for each customer are shown in the table below.\n\n| Customer ID | Coordinates (x, y) | Demand (units) | Time Window (minutes) | Service Duration (minutes) |\n| :--- | :--- | :--- |:--- | :--- |\n| 1 | (87, 81) | 19 | [211, 271] | 10 |\n| 2 | (12, 15) | 25 | [120, 180] | 10 |\n| 3 | (92, 3) | 15 | [205, 265] | 10 |\n| 4 | (25, 75) | 28 | [98, 158] | 10 |\n| 5 | (70, 20) | 12 | [101, 161] | 10 |\n| 6 | (5, 95) | 22 | [155, 215] | 10 |\n| 7 | (65, 80) | 18 | [105, 165] | 10 |\n| 8 | (30, 5) | 26 | [115, 175] | 10 |\n| 9 | (80, 60) | 11 | [95, 155] | 10 |\n| 10 | (40, 90) | 17 | [122, 182] | 10 |\n| 11 | (10, 40) | 29 | [85, 145] | 10 |\n| 12 | (95, 45) | 14 | [135, 195] | 10 |\n| 13 | (20, 30) | 16 | [91, 151] | 10 |\n| 14 | (75, 5) | 23 | [108, 168] | 10 |\n| 15 | (55, 25) | 10 | [88, 148] | 10 |\n\nNow, please provide an operations research model for this VRPHTW.","en_answer": "680.78","difficulty": "Hard","id": 36}
 {"en_question": "A factory produces two types of microcomputers, A and B. Each type of microcomputer requires the same two production processes. The processing time, profit from sales, and the maximum weekly processing capacity for each type are shown in Table 3.1.\n\nTable 3.1\n\n| Process | Model |  | Maximum Weekly Processing Capacity |\n| :---: | :---: | :---: | :---: |\n|  | $\\\\mathrm{A}$ | $\\\\mathrm{B}$ |  |\n| I (hours / unit) | 4 | 6 | 150 |\n| II (hours / unit) | 3 | 2 | 70 |\n| Profit ($ per unit) | 300 | 450 |  |\n\nThe expected values for the factory's operational goals are as follows:\n\n$p_{1}$: The total weekly profit must not be less than $10,000.\n\n$p_{2}$: Due to contractual requirements, at least 10 units of Model A and at least 15 units of Model B must be produced per week.\n\n$p_{3}$: The weekly production time for Process I should be exactly 150 hours, and the production time for Process II should be fully utilized, with potential overtime if necessary.\n\nTry to establish the mathematical programming model for this problem.", "en_answer": "11250.0", "difficulty": "Medium", "id": 37}
 {"en_question": "There are three different products to be processed on three machine tools. Each product must first be processed on machine 1, then sequentially on machines 2 and 3. The order of processing the three products on each machine should remain the same. Assuming $t_{ij}$ represents the time to process the $i$-th product on the $j$-th machine, how should the schedule be arranged to minimize the total processing cycle for the three products? The timetable is as follows:\n| Product | Machine 1 | Machine 2 | Machine 3 |\n|---------|-----------|-----------|-----------|\n| Product 1 | 2           | 3           | 1           |\n| Product 2 | 4           | 2           | 3           |\n| Product 3 | 3           | 5           | 2           |", "en_answer": "14", "difficulty": "Easy", "id": 38}
 {"en_question": "A company plans to transport goods between the city and the suburb and needs to choose the most environmentally friendly transportation method. The company can choose from the following three methods: motorcycle, small truck, and large truck. 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.\n\nThe company can only choose two out of these three transportation methods.\n\nDue to certain road restrictions, the number of motorcycle trips cannot exceed 8.\n\nEach 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.\n\nThe total number of trips must be less than or equal to 20.", "en_answer": "600.0", "difficulty": "Easy", "id": 39}