Question
Suppose a cleaning products distribution company has three factories (F1, F2, F3) and needs to supply these products to four stores (T1, T2, T3, T4). Each factory has a certain production capacity and each store has a specific demand for products. The objective is to allocate products from factories to stores while minimizing transportation costs. Data: Production capacity of each factory: F1: 100 units F2: 150 units F3: 200 units Demand of each store: T1: 80 units T2: 120 units T3: 150 units T4: 100 units Transportation costs (in $ per unit) between factories and stores: T1 T2 T3 T4 F1 4 6 8 10 F2 5 3 7 11 F3 9 4 5 2 Aim: Minimize total transportation costs by meeting factory production capacity and store demand.
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Answer to a math question Suppose a cleaning products distribution company has three factories (F1, F2, F3) and needs to supply these products to four stores (T1, T2, T3, T4). Each factory has a certain production capacity and each store has a specific demand for products. The objective is to allocate products from factories to stores while minimizing transportation costs. Data: Production capacity of each factory: F1: 100 units F2: 150 units F3: 200 units Demand of each store: T1: 80 units T2: 120 units T3: 150 units T4: 100 units Transportation costs (in $ per unit) between factories and stores: T1 T2 T3 T4 F1 4 6 8 10 F2 5 3 7 11 F3 9 4 5 2 Aim: Minimize total transportation costs by meeting factory production capacity and store demand.
Sigrid
4.5111 Answers
1. Organizar los datos en una tabla.
\begin{array}{cccc} & T1 & T2 & T3 & T4 \\F1 & 4 & 6 & 8 & 10 \\F2 & 5 & 3 & 7 & 11 \\F3 & 9 & 4 & 5 & 2 \\\end{array}
2. Configurar las restricciones de capacidad y demanda:
x_{11} + x_{12} + x_{13} + x_{14} \leq 100 \quad (F1: 100 \text{ unidades})
x_{21} + x_{22} + x_{23} + x_{24} \leq 150 \quad (F2: 150 \text{ unidades})
x_{31} + x_{32} + x_{33} + x_{34} \leq 200 \quad (F3: 200 \text{ unidades})
x_{11} + x_{21} + x_{31} = 80 \quad (T1: 80 \text{ unidades})
x_{12} + x_{22} + x_{32} = 120 \quad (T2: 120 \text{ unidades})
x_{13} + x_{23} + x_{33} = 150 \quad (T3: 150 \text{ unidades})
x_{14} + x_{24} + x_{34} = 100 \quad (T4: 100 \text{ unidades})
3. Resolver el problema con el método de transporte:
- Asignamos unidades primero minimizando los costos más bajos y revisamos las restricciones
\begin{array}{cccc|c} & \text{T1} & \text{T2} & \text{T3} & \text{T4} & \text{Suma} \\\hline\text{F1} & 0 \ (4) & 0 \ (6) & 100 \ (8) & 0 \ (10) & 100 \\\text{F2} & 80 \ (5) & 40 \ (3) & 30 \ (7) & 0 \ (11) & 150 \\\text{F3} & 0 \ (9) & 80 \ (4) & 20 \ (5) & 100 \ (2) & 200 \\\hline\text{Suma} & 80 & 120 & 150 & 100 & \\\end{array}
4. Cálculo del costo total:
Costo = 80 \times 5 + 40 \times 3 + 30 \times 7 + 100 \times 8 + 80 \times 4 + 20 \times 5 + 100 \times 2
Costo = 400 + 120 + 210 + 800 + 320 + 100 + 200
Costo = 1900
El costo total mínimo es \$1900.
2. Configurar las restricciones de capacidad y demanda:
3. Resolver el problema con el método de transporte:
- Asignamos unidades primero minimizando los costos más bajos y revisamos las restricciones
4. Cálculo del costo total:
El costo total mínimo es \$1900.
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