Calculadora del método símplex

La calculadora resolverá el problema de optimización dado utilizando el algoritmo símplex. Añadirá variables de holgura, de exceso y artificiales, si es necesario. En caso de variables artificiales, se utiliza el método de la Gran M o el método de dos fases para determinar la solución inicial. Los pasos están disponibles.

Función objetivo:

¿Maximizar?

Restricciones:

Separados por comas.

¿Todas las variables son no negativas?

Método de solución inicial artificial:

Si la calculadora no pudo calcular algo, ha identificado un error o tiene una sugerencia o comentario, por favor contáctenos.

Tu entrada

Maximizar $$$Z = 3 x_{1} + 4 x_{2}$$$, sujeto a $$$\begin{cases} x_{1} + 2 x_{2} \leq 8 \\ x_{1} + x_{2} \leq 6 \\ x_{2} \geq 0 \\ x_{1} \geq 0 \end{cases}$$$.

Solución

El problema en forma canónica puede escribirse como sigue:

$$Z = 3 x_{1} + 4 x_{2} \to max$$$$\begin{cases} x_{1} + 2 x_{2} \leq 8 \\ x_{1} + x_{2} \leq 6 \\ x_{1}, x_{2} \geq 0 \end{cases}$$

Agregue variables (de holgura o de exceso) para transformar todas las desigualdades en igualdades:

$$Z = 3 x_{1} + 4 x_{2} \to max$$$$\begin{cases} x_{1} + 2 x_{2} + S_{1} = 8 \\ x_{1} + x_{2} + S_{2} = 6 \\ x_{1}, x_{2}, S_{1}, S_{2} \geq 0 \end{cases}$$

Escriba la tabla del símplex:

Basic	$$$x_{1}$$$	$$$x_{2}$$$	$$$S_{1}$$$	$$$S_{2}$$$	Solución
$$$Z$$$	$$$-3$$$	$$$-4$$$	$$$0$$$	$$$0$$$	$$$0$$$
$$$S_{1}$$$	$$$1$$$	$$$2$$$	$$$1$$$	$$$0$$$	$$$8$$$
$$$S_{2}$$$	$$$1$$$	$$$1$$$	$$$0$$$	$$$1$$$	$$$6$$$

La variable entrante es $$$x_{2}$$$, porque tiene el coeficiente $$$-4$$$ más negativo en la fila Z.

Basic	$$$x_{1}$$$	$$$x_{2}$$$	$$$S_{1}$$$	$$$S_{2}$$$	Solución	Ratio
$$$Z$$$	$$$-3$$$	$$$-4$$$	$$$0$$$	$$$0$$$	$$$0$$$
$$$S_{1}$$$	$$$1$$$	$$$2$$$	$$$1$$$	$$$0$$$	$$$8$$$	$$$\frac{8}{2} = 4$$$
$$$S_{2}$$$	$$$1$$$	$$$1$$$	$$$0$$$	$$$1$$$	$$$6$$$	$$$\frac{6}{1} = 6$$$

La variable saliente es $$$S_{1}$$$, porque tiene la razón mínima.

Divide la fila $$$1$$$ por $$$2$$$: $$$R_{1} = \frac{R_{1}}{2}$$$.

Basic	$$$x_{1}$$$	$$$x_{2}$$$	$$$S_{1}$$$	$$$S_{2}$$$	Solución
$$$Z$$$	$$$-3$$$	$$$-4$$$	$$$0$$$	$$$0$$$	$$$0$$$
$$$x_{2}$$$	$$$\frac{1}{2}$$$	$$$1$$$	$$$\frac{1}{2}$$$	$$$0$$$	$$$4$$$
$$$S_{2}$$$	$$$1$$$	$$$1$$$	$$$0$$$	$$$1$$$	$$$6$$$

Suma la fila $$$2$$$ multiplicada por $$$4$$$ a la fila $$$1$$$: $$$R_{1} = R_{1} + 4 R_{2}$$$.

Basic	$$$x_{1}$$$	$$$x_{2}$$$	$$$S_{1}$$$	$$$S_{2}$$$	Solución
$$$Z$$$	$$$-1$$$	$$$0$$$	$$$2$$$	$$$0$$$	$$$16$$$
$$$x_{2}$$$	$$$\frac{1}{2}$$$	$$$1$$$	$$$\frac{1}{2}$$$	$$$0$$$	$$$4$$$
$$$S_{2}$$$	$$$1$$$	$$$1$$$	$$$0$$$	$$$1$$$	$$$6$$$

Resta la fila $$$2$$$ de la fila $$$3$$$: $$$R_{3} = R_{3} - R_{2}$$$.

Basic	$$$x_{1}$$$	$$$x_{2}$$$	$$$S_{1}$$$	$$$S_{2}$$$	Solución
$$$Z$$$	$$$-1$$$	$$$0$$$	$$$2$$$	$$$0$$$	$$$16$$$
$$$x_{2}$$$	$$$\frac{1}{2}$$$	$$$1$$$	$$$\frac{1}{2}$$$	$$$0$$$	$$$4$$$
$$$S_{2}$$$	$$$\frac{1}{2}$$$	$$$0$$$	$$$- \frac{1}{2}$$$	$$$1$$$	$$$2$$$

La variable entrante es $$$x_{1}$$$, porque tiene el coeficiente $$$-1$$$ más negativo en la fila Z.

Basic	$$$x_{1}$$$	$$$x_{2}$$$	$$$S_{1}$$$	$$$S_{2}$$$	Solución	Ratio
$$$Z$$$	$$$-1$$$	$$$0$$$	$$$2$$$	$$$0$$$	$$$16$$$
$$$x_{2}$$$	$$$\frac{1}{2}$$$	$$$1$$$	$$$\frac{1}{2}$$$	$$$0$$$	$$$4$$$	$$$\frac{4}{\frac{1}{2}} = 8$$$
$$$S_{2}$$$	$$$\frac{1}{2}$$$	$$$0$$$	$$$- \frac{1}{2}$$$	$$$1$$$	$$$2$$$	$$$\frac{2}{\frac{1}{2}} = 4$$$

La variable saliente es $$$S_{2}$$$, porque tiene la razón mínima.

Multiplica la fila $$$2$$$ por $$$2$$$: $$$R_{2} = 2 R_{2}$$$.

Basic	$$$x_{1}$$$	$$$x_{2}$$$	$$$S_{1}$$$	$$$S_{2}$$$	Solución
$$$Z$$$	$$$-1$$$	$$$0$$$	$$$2$$$	$$$0$$$	$$$16$$$
$$$x_{2}$$$	$$$\frac{1}{2}$$$	$$$1$$$	$$$\frac{1}{2}$$$	$$$0$$$	$$$4$$$
$$$x_{1}$$$	$$$1$$$	$$$0$$$	$$$-1$$$	$$$2$$$	$$$4$$$

Suma la fila $$$3$$$ a la fila $$$1$$$: $$$R_{1} = R_{1} + R_{3}$$$.

Basic	$$$x_{1}$$$	$$$x_{2}$$$	$$$S_{1}$$$	$$$S_{2}$$$	Solución
$$$Z$$$	$$$0$$$	$$$0$$$	$$$1$$$	$$$2$$$	$$$20$$$
$$$x_{2}$$$	$$$\frac{1}{2}$$$	$$$1$$$	$$$\frac{1}{2}$$$	$$$0$$$	$$$4$$$
$$$x_{1}$$$	$$$1$$$	$$$0$$$	$$$-1$$$	$$$2$$$	$$$4$$$

Resta a la fila $$$2$$$ la fila $$$3$$$ multiplicada por $$$\frac{1}{2}$$$: $$$R_{2} = R_{2} - \frac{R_{3}}{2}$$$.

Basic	$$$x_{1}$$$	$$$x_{2}$$$	$$$S_{1}$$$	$$$S_{2}$$$	Solución
$$$Z$$$	$$$0$$$	$$$0$$$	$$$1$$$	$$$2$$$	$$$20$$$
$$$x_{2}$$$	$$$0$$$	$$$1$$$	$$$1$$$	$$$-1$$$	$$$2$$$
$$$x_{1}$$$	$$$1$$$	$$$0$$$	$$$-1$$$	$$$2$$$	$$$4$$$

Ninguno de los coeficientes de la fila Z es negativo.

Se ha alcanzado el óptimo.

Se obtiene la siguiente solución: $$$\left(x_{1}, x_{2}, S_{1}, S_{2}\right) = \left(4, 2, 0, 0\right)$$$.

Respuesta

$$$Z = 20$$$A se alcanza en $$$\left(x_{1}, x_{2}\right) = \left(4, 2\right)$$$A.

Calculadora del método símplex

Resuelve problemas de optimización utilizando el método símplex

Tu entrada

Solución

Respuesta