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 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 we consider a major enterprise logistics provider, 'Global Logistics', is responsible for providing precise material delivery services for multiple high-end office buildings and shops in a city's central business district (CBD). Due to traffic control in the CBD and the specific receiving requirements of the customers, the delivery task is highly challenging.\n\n**Specific Requirements:**\n\n1.  **Delivery Task**: There are 20 customers requiring delivery service on the day, and the demands of all customers must be met.\n2.  **Vehicle Constraints**: The company can use at most 5 trucks, and the capacity of each truck is 200 units.\n3.  **Capacity Constraint**: The total demand of all customers on a single route must not exceed the truck's maximum capacity (200 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**: Due to the complex handover procedures at customer sites, a fixed service time of 90 minutes is required for unloading, handover, and paperwork at each customer location.\n6.  **Optimization Objective**: While satisfying all constraints, the company's objective is to **minimize the total distance traveled by all vehicles** to reduce operational costs.\n\n**Data Details:**\n\n* **Central Depot (Depot 0)**:\n    * Coordinates: (40, 50)\n    * Operating Time Window: [0, 1236] (minutes)\n* **Customer Locations (Customers 1-20)**: 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 | (45, 68) | 10 | [912, 967] | 90 |\n| 2 | (45, 70) | 30 | [825, 870] | 90 |\n| 3 | (42, 66) | 10 | [65, 146] | 90 |\n| 4 | (42, 68) | 10 | [727, 782] | 90 |\n| 5 | (42, 65) | 10 | [15, 67] | 90 |\n| 6 | (40, 69) | 20 | [621, 702] | 90 |\n| 7 | (40, 66) | 20 | [170, 225] | 90 |\n| 8 | (38, 68) | 20 | [255, 324] | 90 |\n| 9 | (38, 70) | 10 | [534, 605] | 90 |\n| 10 | (35, 66) | 10 | [357, 410] | 90 |\n| 11 | (35, 69) | 10 | [448, 505] | 90 |\n| 12 | (25, 85) | 20 | [652, 721] | 90 |\n| 13 | (22, 75) | 30 | [30, 92] | 90 |\n| 14 | (22, 85) | 10 | [567, 620] | 90 |\n| 15 | (20, 80) | 40 | [384, 429] | 90 |\n| 16 | (20, 85) | 40 | [475, 528] | 90 |\n| 17 | (18, 75) | 20 | [99, 148] | 90 |\n| 18 | (15, 75) | 20 | [179, 254] | 90 |\n| 19 | (15, 80) | 10 | [278, 345] | 90 |\n| 20 | (30, 50) | 10 | [10, 73] | 90 |\n\nNow, please provide an operations research model for this VRPHTW.","en_answer": "160.8159","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 we consider a major enterprise logistics provider, 'Global Logistics', is responsible for providing precise material delivery services for multiple high-end office buildings and shops in a city's central business district (CBD). Due to traffic control in the CBD and the specific receiving requirements of the customers, the delivery task is highly challenging.\n\n**Specific Requirements:**\n\n1.  **Delivery Task**: There are 20 customers requiring delivery service on the day, and the demands of all customers must be met.\n2.  **Vehicle Constraints**: The company can use at most 5 trucks, and the capacity of each truck is 200 units.\n3.  **Capacity Constraint**: The total demand of all customers on a single route must not exceed the truck's maximum capacity (200 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**: Due to the complex handover procedures at customer sites, a fixed service time of 90 minutes is required for unloading, handover, and paperwork at each customer location.\n6.  **Optimization Objective**: While satisfying all constraints, the company's objective is to **minimize the total distance traveled by all vehicles** to reduce operational costs.\n\n**Data Details:**\n\n* **Central Depot (Depot 0)**:\n    * Coordinates: (40, 50)\n    * Operating Time Window: [0, 1236] (minutes)\n* **Customer Locations (Customers 1-20)**: 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 | (45, 68) | 10 | [912, 967] | 90 |\n| 2 | (45, 70) | 30 | [825, 870] | 90 |\n| 3 | (42, 66) | 10 | [65, 146] | 90 |\n| 4 | (42, 68) | 10 | [727, 782] | 90 |\n| 5 | (42, 65) | 10 | [15, 67] | 90 |\n| 6 | (40, 69) | 20 | [621, 702] | 90 |\n| 7 | (40, 66) | 20 | [170, 225] | 90 |\n| 8 | (38, 68) | 20 | [255, 324] | 90 |\n| 9 | (38, 70) | 10 | [534, 605] | 90 |\n| 10 | (35, 66) | 10 | [357, 410] | 90 |\n| 11 | (35, 69) | 10 | [448, 505] | 90 |\n| 12 | (25, 85) | 20 | [652, 721] | 90 |\n| 13 | (22, 75) | 30 | [30, 92] | 90 |\n| 14 | (22, 85) | 10 | [567, 620] | 90 |\n| 15 | (20, 80) | 40 | [384, 429] | 90 |\n| 16 | (20, 85) | 40 | [475, 528] | 90 |\n| 17 | (18, 75) | 20 | [99, 148] | 90 |\n| 18 | (15, 75) | 20 | [179, 254] | 90 |\n| 19 | (15, 80) | 10 | [278, 345] | 90 |\n| 20 | (30, 50) | 10 | [10, 73] | 90 |\n\nNow, please provide an operations research model for this VRPHTW.","en_answer": "175.37","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}