: In a TV filming studio there is Planning the programs to record next month. They commonly record artistic and commercial programs. In On average, an artistic space requires 8 hours of recording, It costs $18,000 and produces an income of $38,000. While a commercial requires 1.6 hours of recording, It costs $8,000 and produces an income of $14,000. The studio management works 10 hours a day and for the Next month you have 26 working days. There is a commitment to record at least 10 spaces artistic. It is also established that the total time spent to commercial does not exceed 40%. Determine: to. Pose the mathematical model. Analyze the reason for these restrictions or limitations. b. Solve the problem using the Simplex method and interpret it.

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Answer to a math question : In a TV filming studio there is Planning the programs to record next month. They commonly record artistic and commercial programs. In On average, an artistic space requires 8 hours of recording, It costs $18,000 and produces an income of $38,000. While a commercial requires 1.6 hours of recording, It costs $8,000 and produces an income of $14,000. The studio management works 10 hours a day and for the Next month you have 26 working days. There is a commitment to record at least 10 spaces artistic. It is also established that the total time spent to commercial does not exceed 40%. Determine: to. Pose the mathematical model. Analyze the reason for these restrictions or limitations. b. Solve the problem using the Simplex method and interpret it.

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Timmothy

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88 Answers

a. Planteamiento del modelo matemático:

Definamos las siguientes variables:
- x: número de espacios artísticos a grabar
- y: número de comerciales a grabar

El objetivo es maximizar el ingreso total, que está compuesto por los ingresos de los espacios artísticos y los comerciales. Por lo tanto, la función objetivo será:

Maximizar: 38000x + 14000y

Sin embargo, tenemos las siguientes restricciones:
1. Se deben grabar al menos 10 espacios artísticos:
x ≥ 10
2. El tiempo total dedicado a comerciales no puede exceder el 40%:
1.6y ≤ 0.4(10)(8) = 32

Además, debemos tomar en cuenta las restricciones de tiempo disponibles:
- Cada espacio artístico requiere 8 horas de grabación, por lo que el tiempo total utilizado para grabar los espacios artísticos será de 8x horas.
- Cada comercial requiere 1.6 horas de grabación, por lo que el tiempo total utilizado para grabar los comerciales será de 1.6y horas.

La gerencia trabaja 10 horas diarias y hay 26 días laborables, por lo que el tiempo máximo disponible para grabar es de 260 horas (10 horas/día * 26 días).

Por lo tanto, también tenemos las siguientes restricciones de tiempo:
- Tiempo total utilizado para grabar los espacios artísticos: 8x ≤ 260
- Tiempo total utilizado para grabar los comerciales: 1.6y ≤ 260

b. Resolución del problema usando el método Simplex.

Para resolver este problema utilizando el método Simplex, necesitamos convertir las restricciones en igualdades. Agregamos variables de holgura y de exceso para convertir las desigualdades en igualdades.

Las restricciones convertidas son:

x - s1 = 10 (Restricción 1)
1.6y ≤ 32 (Restricción 2)
8x + s2 = 260 (Restricción 3)
1.6y + s3 = 260 (Restricción 4)

La tabla Simplex correspondiente sería:

| | x | y | s1 | s2 | s3 | RHS |
|-----|----|----|----|-----|-----|-------|
| s1 | 1 | 0 | 1 | 0 | 0 | 10 |
| s2 | 0 | 1 | 0 | 8 | 0 | 260 |
| s3 | 0 | 1.6| 0 | 0 | 1 | 260 |
| Z | -38| -14| 0 | 0 | 0 | 0 |

Aplicando el método Simplex, obtenemos que la solución óptima es:
x = 10
y = 20
Ingreso total = 38000(10) + 14000(20) = $780,000

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