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Calculadora del método simplex
Resolver problemas de optimización usando el método simplex
La calculadora resolverá el problema de optimización dado usando el algoritmo simplex. Agregará variables de holgura, excedentes y artificiales, si es necesario. En el caso de variables artificiales, se utiliza el método de la Gran M o el método de dos fases para determinar la solución de partida. Los pasos están disponibles.
Tu aportación
Maximiza $$$Z = 3 x_{1} + 4 x_{2}$$$, sujeto a $$$\begin{cases} x_{1} + 2 x_{2} \leq 8 \\ x_{1} + x_{2} \leq 6 \\ x_{1} \geq 0 \\ x_{2} \geq 0 \end{cases}$$$.
Solución
El problema en forma canónica se puede escribir de la siguiente manera:
$$Z = 3 x_{1} + 4 x_{2} \to max$$$$\begin{cases} x_{1} + 2 x_{2} \leq 8 \\ x_{1} + x_{2} \leq 6 \\ x_{1}, x_{2} \geq 0 \end{cases}$$Agregue variables (holgura o excedente) para convertir 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 el cuadro 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 más negativo $$$-4$$$ 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 proporción más pequeña.
Divide la fila $$$1$$$ entre $$$2$$$: $$$R_{1} = \frac{R_{1}}{2}$$$.
| Basic | $$$x_{1}$$$ | $$$x_{2}$$$ | $$$S_{1}$$$ | $$$S_{2}$$$ | Solución |
| $$$Z$$$ | $$$-3$$$ | $$$-4$$$ | $$$0$$$ | $$$0$$$ | $$$0$$$ |
| $$$x_{2}$$$ | $$$\frac{1}{2}$$$ | $$$1$$$ | $$$\frac{1}{2}$$$ | $$$0$$$ | $$$4$$$ |
| $$$S_{2}$$$ | $$$1$$$ | $$$1$$$ | $$$0$$$ | $$$1$$$ | $$$6$$$ |
Agregue 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 |
| $$$Z$$$ | $$$-1$$$ | $$$0$$$ | $$$2$$$ | $$$0$$$ | $$$16$$$ |
| $$$x_{2}$$$ | $$$\frac{1}{2}$$$ | $$$1$$$ | $$$\frac{1}{2}$$$ | $$$0$$$ | $$$4$$$ |
| $$$S_{2}$$$ | $$$1$$$ | $$$1$$$ | $$$0$$$ | $$$1$$$ | $$$6$$$ |
Reste 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 más negativo $$$-1$$$ 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 proporción más pequeña.
Multiplique la fila $$$2$$$ por $$$2$$$: $$$R_{2} = 2 R_{2}$$$.
| Basic | $$$x_{1}$$$ | $$$x_{2}$$$ | $$$S_{1}$$$ | $$$S_{2}$$$ | Solución |
| $$$Z$$$ | $$$-1$$$ | $$$0$$$ | $$$2$$$ | $$$0$$$ | $$$16$$$ |
| $$$x_{2}$$$ | $$$\frac{1}{2}$$$ | $$$1$$$ | $$$\frac{1}{2}$$$ | $$$0$$$ | $$$4$$$ |
| $$$x_{1}$$$ | $$$1$$$ | $$$0$$$ | $$$-1$$$ | $$$2$$$ | $$$4$$$ |
Agregue la fila $$$3$$$ a la fila $$$1$$$: $$$R_{1} = R_{1} + R_{3}$$$.
| Basic | $$$x_{1}$$$ | $$$x_{2}$$$ | $$$S_{1}$$$ | $$$S_{2}$$$ | Solución |
| $$$Z$$$ | $$$0$$$ | $$$0$$$ | $$$1$$$ | $$$2$$$ | $$$20$$$ |
| $$$x_{2}$$$ | $$$\frac{1}{2}$$$ | $$$1$$$ | $$$\frac{1}{2}$$$ | $$$0$$$ | $$$4$$$ |
| $$$x_{1}$$$ | $$$1$$$ | $$$0$$$ | $$$-1$$$ | $$$2$$$ | $$$4$$$ |
Reste la fila $$$3$$$ multiplicada por $$$\frac{1}{2}$$$ de la fila $$$2$$$: $$$R_{2} = R_{2} - \frac{R_{3}}{2}$$$.
| 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 alcanza 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 logra en $$$\left(x_{1}, x_{2}\right) = \left(4, 2\right)$$$A.