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: 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.

Expert avatar
Timmothy
4.8
99 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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