prob_id stringlengths 8 18 ⌀ | description stringlengths 443 3.51k ⌀ | problem_type stringlengths 2 61 ⌀ | problem_size stringclasses 3
values | reference_model stringlengths 855 2.34k ⌀ | produced_model stringlengths 587 13.8k ⌀ | converged? stringclasses 2
values | notes stringclasses 3
values | generated_opt_code stringlengths 1.59k 2.86k ⌀ | gen_groud_truth stringlengths 2 13 ⌀ | groud_truth stringlengths 5 13 ⌀ | score_var float64 4 10 ⌀ | score_obj float64 2 10 ⌀ | score_con float64 3 10 ⌀ | score_total stringlengths 4 15 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
prob_023 | HAPPY pharmaceutical produces the drug Panacea from four chemicals. Today they must produce 1,000 lb of the drug. The three active ingredients in Panacea are A, B, and C. By weight, at least 8% of Panacea must consist of A, at least 4% of B, and at least 2% of C. The cost per pound of each chemical and the amount of ea... | LP | toy | Set:
1. Chemical
The set of chemicals, \( C = \{1, 2, 3, 4\} \)
2. Ingredient
The set of ingredients, \( I = \{1, 2, 3\} \)
Parameter:
1. ChemicalID
['1', '2', '3', '4']
2. IngredientID
['A', 'B', 'C']
3. ChemicalCost
[8, 10, 11, 14]
4. IngredientContent
# units/lb
[
[0.03, 0.02, 0.01],
[0.06, 0.04, 0.01],
[0.10, 0.03... | │ FINAL ANSWER │
│ I'll now create a complete mathematical model for this optimization problem using standard mathematical notation. ... | no | null | """
Chemical blending cost-minimization LP, implemented with Gurobi (gurobipy).
Minimize Z = 8x1 + 10x2 + 11x3 + 14x4
Subject to:
x1 + x2 + x3 + x4 = 1000
0.03x1 + 0.06x2 + 0.10x3 + 0.12x4 >= 80
0.02x1 + 0.04x2 + 0.03x3 + 0.09x4 >= 40
0.01x1 + 0.01x2 + 0.04x3 + 0.04x4 >= 20
x2 >= 100
x1, x2, x3... | 10637.5 | 10,637.50 | 10 | 10 | 10 | 10.00 |
prob_032 | A manufacturing company operates three production facilities (1, 2, and 3) that produce two types of components (X and Y). Each facility generates waste during production, which can be treated to reduce environmental impact. Treating the waste incurs a cost and reduces the amount of pollutants emitted. It costs $12 to ... | LP | toy | Set:
1. facility
The set of production facilities, \( F = \{1, 2, 3\} \)
2. pollutant
The set of pollutants, \( P = \{1, 2\} \) # 1: pollutant X, 2: pollutant Y
Parameter:
1. TreatmentCost
[12, 8, 18]
2. PollutantReduction
[
[0.15, 0.35], # facility 1
[0.25, 0.20], # facility 2
[0.30, 0.40] # facility 3
]
3. Pollution... | │
│ **Decision Variables:** │
│ Let $ t_1 $, $ t_2 $, and $ t_3 $ represent the tons of waste treated from facilities 1, 2, and 3 respectively. ... | no | null | """
Waste-treatment cost-minimization LP, implemented with Gurobi (gurobipy).
Minimize Z = 12t1 + 8t2 + 18t3
Subject to:
0.15t1 + 0.25t2 + 0.30t3 >= 25 (Pollutant X reduction)
0.35t1 + 0.20t2 + 0.40t3 >= 35 (Pollutant Y reduction)
t1, t2, t3 >= 0
"""
import gurobipy as gp
from gurobipy import GRB
# -... | 1269.57 | 1,269.57 | 10 | 10 | 10 | 10.00 |
prob_033 | A fruit processing company, NatureBounty, specializes in producing two product lines: packaged fresh apples and bottled apple juice. NatureBounty grades apples on a scale of 1 (poor) to 10 (excellent). The company currently has 90,000 lbs of grade 7 apples and 110,000 lbs of grade 4 apples in inventory. To maintain pro... | LP | toy | Set:
1. Product
The set of products, \( P = \{1, 2\} \) # 1: packaged fresh apples, 2: bottled apple juice
2. Grade
The set of apple grades the company has, \( G = \{7, 4\} \)
Parameter:
1. Inventory
[90000, 110000]
2. Revenue
[0.35, 1.10]
3. VariableCost
[0.12, 0.80]
4. AverageGrade
[5, 6]
5. MaxProcessingCapacity
15... | │ Let x be the pounds of grade 7 apples used for juice production, and y be the pounds of grade 4 apples used for juice production. Let a be the pounds of grade 7 apples │
│ used for packaging, and b be the pounds of grade 4 apples used for packaging. ... | no | null | import gurobipy as gp
from gurobipy import GRB
# Create a new model
model = gp.Model("Apple_Production_Optimization")
# Define Variables
# By default, gurobipy variables have a lower bound (lb) of 0.0,
# which automatically satisfies the x, y, a, b >= 0 constraints.
x = model.addVar(vtype=GRB.CONTINUOUS, name="x") #... | 42,900.00 | 42,900.00 | 10 | 10 | 10 | 10.00 |
prob_030 | LogiTrans operates two distribution centers to manage the shipment of three types of goods to various regional hubs. Each distribution center can be staffed by up to seven logistics coordinators at a time. Coordinators are paid $500 per week at distribution center 1 and $900 per week at distribution center 2. Activatin... | ILP | toy | Set:
1. Center
The set of distribution centers, \( C = \{1, 2\} \)
2. GoodsType
The set of goods types, \( G = \{1, 2, 3\} \)
Parameter:
1. FixedCost
[1000, 2000]
2. StaffCost
[500, 900]
3. MaxStaffNum
[7, 7]
4. ProcessRate
[
[20, 25],
[18, 22],
[15, 20]
]
5. Demand
[120, 150, 200]
Decision vari... | │
│ # Let me define the math_model variable first and then continue building it │
│ ... | no | null | import gurobipy as gp
from gurobipy import GRB
def solve_logitrans_model():
# Create a new Gurobi model
model = gp.Model("LogiTrans")
# 1. Define Decision Variables
# x1, x2: Integer variables with a lower bound of 0 and an upper bound of 7
x1 = model.addVar(vtype=GRB.INTEGER, lb=0, ub=7, name="x1... | 11,000.00 | 11,000.00 | 10 | 10 | 10 | 10.00 |
prob_031 | Global Freight Solutions (GFS), a logistics company, needs to transport goods across three regions using services from three shipping providers. Each provider offers different cost structures and service mixes for large, medium, and small shipments, as detailed in Table as follows:
| Supplier | Cost Per Shipment ($) |... | ILP | toy | Set:
1. Supplier
The set of suppliers, \( S = \{1, 2, 3\} \)
2. ShipmentType
The set of shipment types, \( T = \{1, 2, 3\} \) # 1: large, 2: medium, 3: small
Parameter:
1. CostPerShipment
[5.2, 4.7, 3.5]
2. Percent
[[45, 35, 20], [30, 45, 25], [15, 20, 65]]
3. Demand
[500, 300, 300]
4. MaxShipment
[700, 700, 700]
Dec... |
╭─ ★ Final Answer ──────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────╮
│ ... | yes! | null | import gurobipy as gp
from gurobipy import GRB
# Initialize the model
m = gp.Model("GFS_Freight_Optimization")
# ---------------------------------------------------------
# 1. Decision Variables
# ---------------------------------------------------------
# Variables are continuous (real numbers).
# lb=0 handles the ... | 6,551.11 | 6,553.70 | 10 | 10 | 10 | 10.00 |
prob_042 | LogiFlow Solutions manages the distribution of two types of packages: Standard and Express. Each package must be transported entirely by air freight or entirely by ground shipping. A total of 150 cubic meters of air freight capacity and 210 cubic meters of ground shipping capacity are available. A Standard package requ... | ILP | toy | Set:
1. Package
The set of packages, \( P = \{1, 2\} \) # 1: Standard, 2: Express
2. ShippingMethod
The set of shipping methods, \( S = \{1, 2\} \) # 1: Air freight, 2: Ground shipping
Parameter:
1. AirFreightCapacity
150 # in cubic meters
2. GroundShippingCapacity
210 # in cubic meters
3. AirFreightRequire... | # Decision variables: │
│ # Let S_a = number of Standard packages transported by air freight ... | no | null | import gurobipy as gp
from gurobipy import GRB
# Initialize the model
m = gp.Model("Package_Routing_Optimization")
# ---------------------------------------------------------
# 1. Decision Variables (Integer & Non-negative)
# ---------------------------------------------------------
S_a = m.addVar(vtype=GRB.... | 730 | 730.00 | 10 | 10 | 10 | 10.00 |
prob_055 | A logistics company needs to optimize the loading of 5 different types of cargo items (A, B, C, D, E) onto a single truck with a maximum weight capacity of 2000 kg. Each item has a base profit value and weight, but certain item combinations generate additional synergy profits when transported together (e.g., perishable... | NLP | toy | Set:
1. Items
The set of cargo items, \( I = \{A, B, C, D, E\} \)
Parameter:
1. Weight
The weight of each item \( w_i \) where \( i \in I \):
\( w = \{A:400, B:600, C:300, D:500, E:700\} \)
2. BaseProfit
The base profit of each item \( c_i \) where \( i \in I \):
\( c = \{A:120, B:180, C:90, D:150, E:210\} \... | │
│ **Mathematical Model:** │
│ │
│ Let's define the decision variables: ... | no | At first treated/classified the problem as MILP and then fixed itself to produce an NLP model | import gurobipy as gp
from gurobipy import GRB
# Initialize the model
m = gp.Model("Cargo_Loading_Synergy")
# ---------------------------------------------------------
# 1. Data Definitions
# ---------------------------------------------------------
items = ['A', 'B', 'C', 'D', 'E']
weights = {
'A': ... | 665 | 665.00 | 10 | 10 | 10 | 10.00 |
prob_081 | A large manufacturing enterprise needs to ship a consignment of 1,000 tons of goods from its factory in City F to a distribution center in City D. There are three potential transportation routes available. Each route is composed of segments using different modes of transport (road or rail). The transportation cost for ... | NLP | toy | Set:
1. Route
The set of available transportation routes, \( R = \{1, 2, 3\} \)
Parameter:
1. TotalTonnage
# The total amount of goods to be shipped
100
2. LinearCostPerTon
# The base transportation cost per ton for each route, excluding congestion fees. This is calculated by summing the costs of all segments for a gi... | │
│ complete_formulation = """ │
│ Let me carefully formulate the complete mathematical model for this optimization problem. │
│ ... | no | null | import gurobipy as gp
from gurobipy import GRB
def solve_transportation_problem():
# Create a new model
model = gp.Model("Transportation_Optimization")
# Add Decision Variables
# The non-negativity (lb=0.0) is the default in Gurobi.
# The capacity constraints (x <= 1000) are set using the... | 670,001.80 | 670,003.80 | 10 | 5 | 5.5 | 6.83 |
prob_090 | A shared mobility company plans to deploy a fleet of 1,000 electric scooters across five key hotspot zones in a city. To ensure service stability and a positive user experience, the company aims to achieve a balanced deployment that aligns with the demand profile of each zone.
The company's goal is to minimize the net... | NLP | toy | Set:
1. Zones
The set of hotspot zones, \( I = \{\text{Commercial Center, University City, Tech Park, Tourist Area, Transit Hub}\} \).
Parameters:
1. TotalScooters
The total number of scooters to be deployed.
1000
2. Attractiveness
A dictionary mapping each zone \(i \in I\) to its attractiveness value.
\(\{\text{Comm... | │ # Decision variables │
│ # x_CC, x_UC, x_TP, x_TA, x_TH ≥ 0 and integers (since we can't deploy fractio... | no | null | null | null | null | 10 | 10 | 5 | 8.33 |
prob_039 | HealthSupply Co. manages a medical supply chain that must fulfill demand for two types of products, Vaccine X and Vaccine Y, over the next two quarters, with demands of 100 doses of Vaccine X and 200 doses of Vaccine Y in Quarter 1, and 300 doses of Vaccine X and 100 doses of Vaccine Y in Quarter 2. The company can pro... | MILP | small | Set:
1. Product
The set of products, \( P = \{1, 2\} \) # 1: Vaccine X, 2: Vaccine Y
2. Quarter
The set of quarters, \( Q = \{1, 2\} \) # 1: Quarter 1, 2: Quarter 2
Parameter:
1. Demand
[[100, 200], [300, 100]] # Demand for each product in each quarter
2. MaxProductionPerQuarter
[1500, 1500]
3. RawMaterialCost
[300, 4... | │
│ Output: │
│ You responded with: ... | no | null | null | null | null | 10 | 10 | 5.9 | 8.63 |
prob_073 | A regional distribution center needs to optimize its daily delivery routes for 5 stores located at different distances. The center has 3 available delivery trucks with capacities of 8, 10, and 12 tons respectively. The daily demands of the stores are: Store1 (3 tons), Store2 (4 tons), Store3 (5 tons), Store4 (2 tons), ... | MILP | small | Set:
1. Trucks
The set of available trucks, \( T = \{1, 2, 3\} \)
2. Stores
The set of stores to be served, \( S = \{1, 2, 3, 4, 5\} \)
Parameter:
1. TruckCapacities
[8, 10, 12] # Capacity of each truck in tons
2. StoreDemands
[3, 4, 5, 2, 6] # Demand of each store in tons
3. Distances
[15, 20, 25, 30, 35] # Distance ... | │
│ Mathematical Model: │
│ ... | no | null | null | null | null | 4 | 3 | 3 | 3.33 |
prob_024 | A commodities trader oversees a secure vault facility with a maximum capacity of 500 kilograms, dedicated to storing refined platinum. At the beginning of Quarter 1, the trader holds 150 kilograms of platinum. Each quarter, the trader assesses the current inventory, sells any amount of platinum up to the available stoc... | LP | small | Set:
1. Quarter
The set of quarters, \( Q = \{1, 2, ..., QuarterNum\} \)
Parameter:
1. QuarterNum
6
2. VaultCapacity
500
3. InitialInventory
150
4. SellingPrice
[30000, 35000, 40000, 36000, 38000, 42000]
5. PurchasePrice
[32000, 36000, 38000, 33000, 39000, 40000]
Decision variable:
1. SellingAmount
... | │
│ # Decision Variables: │
│ # Let s_t ∈ ℝ_+ be the amount of platinum sold in quarter t (kg), for t = 1,... | no | null | import gurobipy as gp
from gurobipy import GRB
# --- Parameters ---
num_quarters = 6
initial_inventory = 150 # kg
vault_capacity = 500 # kg
# Selling prices ($/kg) for Q1 to Q6
p_s = [30000, 35000, 40000, 36000, 38000, 42000]
# Purchase prices ($/kg) for Q1 to Q6
p_b = [32000, 36000, 38000, 33000, 3... | 13,300,000.00 | 13,300,000.00 | 8.6 | 10 | 8 | 8.87 |
prob_026 | GreenWeave specializes in transforming raw materials such as organic cotton, jute, and linen into eco-friendly textiles: organic cotton fabric, jute fabric, and linen fabric. Organic cotton fabric can be manufactured by processing either organic cotton or jute. Linen fabric can be produced from any of the raw materials... | LP | small | Set:
1. RawMaterials
The set of raw materials, \( R = \{"cotton", "jute", "linen"\} \)
2. Products
The set of products, \( P = \{"cotton_fabric", "jute_fabric", "linen_fabric"\} \)
3. ProducedFrom
The set of products and their corresponding raw materials, \( PF = \{"cotton_fabric": ("cotton", "jute"), "linen_fabr... | │
│ Classification: Linear Programming (LP) │
│ ... | no | null | import gurobipy as gp
from gurobipy import GRB
# Initialize the model
m = gp.Model("Fabric_Production")
# --- Decision Variables ---
# x1: tons of organic cotton used for organic cotton fabric
x1 = m.addVar(vtype=GRB.CONTINUOUS, name="x1")
# x2: tons of jute used for organic cotton fabric
x2 = m.addVar(vtyp... | 545,357.14 | 545,357.14 | 10 | 10 | 10 | 10.00 |
prob_076 | A mining company operates three ore processing plants (P1, P2, P3) with monthly processing capacities of 8000, 10000, and 12000 tons respectively. The company sources ore from four different mines (M1, M2, M3, M4) with available quantities of 6000, 9000, 7000, and 5000 tons per month.
Each ore type has different chara... | LP | small | Set:
1. Mines
The set of mines, \( M = \{M1, M2, M3, M4\} \)
2. Plants
The set of processing plants, \( P = \{P1, P2, P3\} \)
Parameter:
1. MineSupply
\( s_m \): Available ore at mine m (tons)
[M1:6000, M2:9000, M3:7000, M4:5000]
2. PlantCapacity
\( c_p \): Processing capacity of plant p (tons)
[P1:8000, P... | # Decision Variables: │
│ # Let x_ij = tons of ore allocated from mine i to plant j ... | no | null | null | null | 22,933.33 | 8 | 9 | 3 | 6.67 |
prob_002 | An airport has two available gates (G1, G2) and needs to assign three flights (F1: 10:00-11:00, F2: 10:30-11:30, F3: 11:15-12:30, F4: 10:45-11:45) to these gates. Flights assigned to the same gate must have non-overlapping time intervals, and the goal is to minimize flight delays, where each 1-minute delay beyond the s... | ILP | small | Set:
1. Gates
The set of gates available at the airport, \( G = \{1, 2\} \)
2. Flights
The set of flights to be assigned to the gates, \( F = \{1, 2, 3, 4\} \)
Parameter:
1. ArrangeStartTime
[600, 630, 645, 675]
2. Duration
[60, 60, 75, 60]
3. CostPerDelyTime
1
4. M: A large positive number
1500
Decision variable:
1.... |
│ **Problem Requirements:** │
│ 1. 2 gates (G1, G2) ... | no | null | null | null | null | 7.5 | 10 | 6.7 | 8.06 |
prob_053 | A retail chain wants to open warehouses in 3 potential locations to serve 5 customer zones. The distances (in km) between each potential warehouse location and customer zone are given in the table below. The goal is to select exactly 2 warehouse locations that minimize the total distance between each customer zone and ... | ILP | small | Set:
1. CustomerZones
The set of customer zones, \( C = \{C1, C2, C3, C4, C5\} \)
2. WarehouseLocations
The set of potential warehouse locations, \( W = \{W1, W2, W3\} \)
Parameter:
1. DistanceMatrix
The distance between warehouse location \( w \in W \) and customer zone \( c \in C \), denoted as \( d_{wc} \)
... | │ Mathematical Model: │
│ │
│ Decision Variables: ... | no | null | import gurobipy as gp
from gurobipy import GRB
# --- Data Definition ---
# Customer zones (i = 1 to 5) and Warehouses (j = 1 to 3)
customers = range(1, 6)
warehouses = range(1, 4)
# Distance matrix extracted from the objective function coefficients
# distances[i, j] represents the cost/distance of assigning ... | 48 | 48.00 | 10 | 10 | 10 | 10.00 |
prob_075 | A distribution center needs to assign 4 delivery trucks to 4 customer locations for same-day shipments. Each truck has different transportation costs to each location due to varying distances and fuel consumption rates. The cost matrix (in USD) for assigning trucks to locations is as follows:
| Truck\Location | Loc1 |... | ILP | small | Set:
1. Trucks
The set of delivery trucks, \( T = \{1, 2, 3, 4\} \)
2. Locations
The set of customer locations, \( L = \{1, 2, 3, 4\} \)
Parameter:
1. CostMatrix
The transportation cost (USD) when truck \( t \) is assigned to location \( l \), represented as \( c_{tl} \):
- \( c_{11} = 120 \), \( c_{12} = 150... | │ This is a Linear Programming problem, specifically a Assignment Problem which is a special case of the Transportation Problem and can be modeled as a Balanced Assignment Problem. │
│ ... | yes! | incorrectly classified the problem at first then corrected itself at the end | import gurobipy as gp
from gurobipy import GRB
# --- Data Definition ---
# Cost matrix c_ij representing transportation costs from Truck i to Location j
costs = [
[120, 150, 180, 90],
[110, 140, 170, 80],
[130, 160, 190, 100],
[140, 170, 200, 110]
]
num_trucks = len(costs)
num_locations... | 560 | 560.00 | 10 | 10 | 10 | 10.00 |
prob_011 | A city plans to establish a low-altitude drone delivery network covering a 100x100 area (with both horizontal and vertical coordinates ranging from [0, 100]). There are 8 demand points within this area, with coordinates at (10, 20), (25, 60), (40, 80), (65, 50), (80, 20), (15, 45), (70, 75), and (90, 90). The goal is t... | NLP | small | Set:
1. DemandNode
The set of demand nodes, \( D = \{1, 2, \dots, DemandNodeNum\} \)
2. Hub
The set of drone logistics hubs, \( H = \{1, 2, \dots, HubNum\} \)
Parameter:
1. DemandNodeNum
8
2. DemandNodeCoordinates
[[10, 20], [25, 60], [40, 80], [65, 50], [80, 20], [15, 45], [70, 75], [90, 90]]
3. HubNum
2
4. HubCoordi... |
│ # Decision variables for hub locations │
│ x1, y1 = symbols('x1 y1') # Coordinates of first hub │
│ x2, y2 = symbols('x2 y2') # Coordinates of second hu... | no | null | null | null | null | 9 | 2 | 9 | 6.67 |
prob_059 | A logistics company needs to pack 6 heavy cylindrical containers into a specialized truck bed. The truck bed has a fixed width of 2.5 meters, and the company wants to find the minimum possible length required to fit all containers. The diameters of the 6 containers (in meters) are: [1.2, 1.1, 1.0, 0.9, 0.8, 0.7].
The ... | NLP | small | # Set
# C: The set of containers to be packed.
Set:
1. Containers
The set of containers, \( C = \{1, 2, 3, 4, 5, 6\} \)
# Parameter
# TruckWidth: The fixed width of the truck bed in meters.
# Diameter_c: The diameter of container c.
# Radius_c: The radius of container c, calculated as Diameter_c / 2.
Paramet... | │ **Decision Variables:** │
│ Let $ c_i = (x_i, y_i) $ represent the center coordinates of the $ i $-th container, where $ i = 1, 2, ..., 6 $. ... | no | correctly classified the problem | null | null | null | 9.2 | 10 | 8.7 | 9.30 |
prob_088 | An urban shared e-bike company, "E-Donkey Express," needs to create a nightly battery-swapping plan for its fleet, which is distributed across three zones using its two mobile maintenance vans. The company's objective is to minimize the total nightly cost, which consists of a fixed dispatch fee for each van used and a ... | NLP | small | Set
1. Station
The set of maintenance stations, \( I = \{1, 2\} \) # 1: Van 1, 2: Van 2
2. Zone
The set of zones, \( J = \{1, 2, 3\} \) # 1: Zone A, 2: Zone B, 3: Zone C
Parameter
1. DispatchCost
[300, 320] # Fixed cost for using van 1 and 2
2. VanCapacity
[150, 180] # Max battery capacity for van 1 and 2
3... | │ I'll formulate a complete mathematical model for this optimization problem based on the extracted data. │
│ ... | no | correctly classified the problem | null | null | null | 10 | 10 | 4.4 | 8.13 |
⬆️ Generic ⬆️ | null | null | null | null | null | null | null | null | null | null | null | null | null | Accuracy 87.4% |
⬇️ By Dr Haider ⬇️ | null | null | null | null | null | null | null | null | null | null | null | null | null | Accuracy 91% |
problem--1 | A logistics company operates 5 candidate plants (P1-P5) that can supply 4 warehouses (W1-W4). Each plant has a maximum production capacity, and each warehouse has a fixed demand that must be fully met. Keeping a plant open incurs a fixed annual cost, and shipping goods from a plant to a warehouse incurs a per-unit tran... | classic_fidility_location | null | null | │
│ Decision Variables: │
│ x_p ∈ {0, 1} for p = 1, 2, 3, 4, 5 (1 if plant p is open, 0 otherwise) ... | no | null | null | null | null | 10 | 10 | 10 | 10.00 |
problem--2 | A company produces two products, Pencils and Pens, at two production cities: Detroit and Denver. These products must be shipped to warehouses in three cities: Boston, New York, and Seattle, to meet each city's demand for each product.
Each production city has a limited supply of each product available for shipment, an... | multi_commodity_network_flow | null | null | │ # Decision variables: │
│ # x_ijp = units of product p shipped from production city i to warehouse city j │
│ # where: │
│ ... | no | null | import gurobipy as gp
from gurobipy import GRB
# Initialize the model
m = gp.Model("MultiProductTransportation")
# --- Indices ---
# i: 1 (Detroit), 2 (Denver)
# j: 1 (Boston), 2 (New York), 3 (Seattle)
# p: 1 (Pencils), 2 (Pens)
I = [1, 2]
J = [1, 2, 3]
P = [1, 2]
# --- Data ---
# Cost mapping: (i, j, p) -> cost
co... | 5,500.00 | 5,500.00 | 10 | 10 | 7.69 | 9.23 |
problem--3 | This problem models an AC power flow network with 4 buses connected by transmission lines, arranged in a ring: Bus 1 - Bus 2, Bus 2 - Bus 3, Bus 3 - Bus 4, and Bus 4 - Bus 1. Each bus has a specific role in the network:
Bus 1 is the swing (slack) bus. Its voltage magnitude and voltage angle are fixed, while its real a... | optimal_power_flow | null | null | │ # Now let's define the mathematical model │
│ ... | no | null | null | null | null | 8.2 | 10 | 4.4 | 7.53 |
new_prob1 | A power system operator must economically dispatch three thermal generators to meet a demand of 150 MW during one hour. Transmission losses and ramping limits are ignored.
Generator data:
| Generator | Minimum Output (MW) | Maximum Output (MW) | Cost Function ($/h) |
|---|---:|---:|---|
| G1 | 20 | 100 | 100 + 20 P1 +... | NLP
Economic dispatch | Set:
1. Generators
The set of generators, \( G = \{1,2,3\} \)
Parameter:
1. Demand
\( D = 150 \) MW
2. Minimum output
\( P^{min} = [20,10,0] \) MW
3. Maximum output
\( P^{max} = [100,80,60] \) MW
4. Fixed cost coefficients
\( a = [100,120,150] \)
5. Linear cost coefficients
\( b = [20,25,30] \)
6. Quadratic cost coeff... | │ │
│ Output: ... | no | null | null | null | null | 10 | 10 | 7.3 | 9.10 | |
new_prob1_var1 | A grid coordinator must dispatch three fossil-fuel power plants to meet a demand of 220 MW during one hour. Transmission losses and ramping limits are ignored.
Plant data:
| Plant | Minimum Output (MW) | Maximum Output (MW) | Emission Function (kg/h) |
|---|---:|---:|---|
| P1 | 40 | 150 | 300 + 45 P1 + 0.12 P1^2 |
| ... | NLP
Emission Dispatch. | null | **Decision Variables:** │
│ Let P1, P2, and P3 represent the output levels (in MW) of the three fossil-fuel power plants during the hour, where: │
│ - 40 ≤ P1 ≤ 150 ... | no | null | null | null | null | 10 | 10 | 10 | 10.00 | |
new_prob1_var2 | A district heating manager must schedule three thermal boiler units to meet a demand of 110 MW during one hour. Distribution losses and ramping limits are ignored.
Boiler data:
| Boiler | Minimum Output (MW) | Maximum Output (MW) | Cost Function ($/h) |
|---|---:|---:|---|
| B1 | 15 | 65 | 80 + 12 P1 + 0.08 P1^2 |
| B... | NLP
Thermal Load Dispatch | null | Decision Variables: │
│ Let P1, P2, P3 be the output levels of boilers B1, B2, B3 in MW, respectively. │
│ ... | no | null | null | null | null | 10 | 10 | 7.8 | 9.27 | |
new_prob2 | A charging station schedules three electric vehicles over four one-hour time slots. Charging efficiency is assumed to be 100%, and charging power may be fractional.
Electricity prices:
| Time Slot | 1 | 2 | 3 | 4 |
|---|---:|---:|---:|---:|
| Price ($/kWh) | 0.10 | 0.25 | 0.15 | 0.30 |
Vehicle requirements:
| EV | Av... | LP
Smart Grid / EV Smart Charging Optimization | Set:
1. Electric vehicles
\( E = \{EV1,EV2,EV3\} \)
2. Time slots
\( T = \{1,2,3,4\} \)
3. Available vehicle-time pairs
\( A = \{(EV1,1),(EV1,2),(EV1,3),(EV2,2),(EV2,3),(EV2,4),(EV3,1),(EV3,2),(EV3,3),(EV3,4)\} \)
Parameter:
1. Electricity price
\( price = [0.10,0.25,0.15,0.30] \) $/kWh
2. Required energy
... | │ **1. Decision Variables Definition:** │
│ ... | no | null | null | null | null | 9.1 | 10 | 6 | 8.37 | |
new_prob2_var1 | A water facility manager must dispatch three pump stations to meet a water flow demand of 160 ML/day during one hour. Pipe friction losses and ramp rates are ignored.
Pump data:
| Pump | Minimum Output (ML/day) | Maximum Output (ML/day) | Operational Cost Function ($/h) | Activation Cost ($) |
|---|---:|---:|---|---:|... | LP
Pump Scheduling / Unit Commitment | null | │ math_model = """ │
│ Decision Variables: │
│ x1, x2, x3 ∈ {0, 1} (binary variables indicating pump activation) │
│ f1, f2, f3 ≥ 0 (flow rates in ML/day) │
│ │
│ Objective Function: │
│ Minimize Total Cost = 500x3 + (100 + 20f1 + 0.05f1²)x1 + (120 + 25f2 + 0.04f2²)x2 + (80 + 29f3 + 0.03f3²)x3 │
│
... | yes! | null | null | null | null | 7.5 | 10 | 10 | 9.17 | |
new_prob2_var2 | A cloud infrastructure operator must allocate computational workloads across three server clusters to meet a processing demand of 160 THz during one hour. Network latency and bandwidth limits are ignored.
Cluster data:
| Cluster | Minimum Output (THz) | Maximum Output (THz) | Power Function (kW) | Boot Overhead (kWh) ... | LP
Computational Resource Allocation / Data Center Dispatch. | null | │ **Decision Variables:** │
│ - L1, L2, L3: Actual workload allocated to each cluster (THz) ... | no | null | null | null | null | 10 | 10 | 7.7 | 9.23 | |
null | null | null | null | null | null | null | null | null | null | null | null | null | null | Average = 88.5% |
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