Question

3-26. The following tableau represents a specific simplex iteration. All variables are nonnegative. The tableau is not optimal for either maximization or minimization. Thus, when a nonbasic variable enters the solution, it can either increase or decrease $z$ or leave it unchanged, depending on the parameters of the entering nonbasic variable. Basic $x_1$ $x_2$ $x_3$ $x_4$ $x_5$ $x_6$ $x_7$ $x_8$ Solution $z$ 0 -5 0 4 -1 -10 0 0 620 $x_8$ 0 3 0 -2 -3 -1 5 1 12 $x_7$ 0 1 1 3 1 0 3 0 6 $x_1$ 1 -1 0 0 6 -4 0 0 0 (a) Categorize the variables as basic and nonbasic, and provide the current values of all the variables. *(b) Assuming that the problem is of the maximization type, identify the nonbasic variables that have the potential to improve the value of $z$. If each such variable enters the basic solution, determine the associated leaving variable, if any, and the associated change in $z$. Do not use the Gauss-Jordan row operations. (c) Repeat part (b) assuming that the problem is of the minimization type. (d) Which nonbasic variable(s) will not cause a change in the value of $z$ when selected to enter the solution?

          3-26. The following tableau represents a specific simplex iteration. All variables are nonnegative.
The tableau is not optimal for either maximization or minimization. Thus, when a nonbasic
variable enters the solution, it can either increase or decrease $z$ or leave it unchanged,
depending on the parameters of the entering nonbasic variable.
Basic
$x_1$
$x_2$
$x_3$
$x_4$
$x_5$
$x_6$
$x_7$
$x_8$
Solution
$z$
0
-5
0
4
-1
-10
0
0
620
$x_8$
0
3
0
-2
-3
-1
5
1
12
$x_7$
0
1
1
3
1
0
3
0
6
$x_1$
1
-1
0
0
6
-4
0
0
0
(a) Categorize the variables as basic and nonbasic, and provide the current values of all
the variables.
*(b) Assuming that the problem is of the maximization type, identify the nonbasic
variables that have the potential to improve the value of $z$. If each such variable
enters the basic solution, determine the associated leaving variable, if any, and the
associated change in $z$. Do not use the Gauss-Jordan row operations.
(c) Repeat part (b) assuming that the problem is of the minimization type.
(d) Which nonbasic variable(s) will not cause a change in the value of $z$ when selected
to enter the solution?

3-26. The following tableau represents a specific simplex iteration. All variables are nonnegative.
The tableau is not optimal for either maximization or minimization. Thus, when a nonbasic
variable enters the solution, it can either increase or decrease z or leave it unchanged,
depending on the parameters of the entering nonbasic variable.
Basic
x1
x2
x3
x4
x5
x6
x7
x8
Solution
z
0
-5
0
4
-1
-10
0
0
620
x8
0
3
0
-2
-3
-1
5
1
12
x7
0
1
1
3
1
0
3
0
6
x1
1
-1
0
0
6
-4
0
0
0
(a) Categorize the variables as basic and nonbasic, and provide the current values of all
the variables.
*(b) Assuming that the problem is of the maximization type, identify the nonbasic
variables that have the potential to improve the value of z. If each such variable
enters the basic solution, determine the associated leaving variable, if any, and the
associated change in z. Do not use the Gauss-Jordan row operations.
(c) Repeat part (b) assuming that the problem is of the minimization type.
(d) Which nonbasic variable(s) will not cause a change in the value of z when selected
to enter the solution?

Added by Amy J.

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