Linear Programming : Simplex Method

1. Consider the simplex tableau

x y u v w M
[ 1 0 3 0 0 0 | 10]
[ 0 0 1 0 1 0 | 0]
[ 0 1 -6 0 0 0 | 3]
[ 0 0 8 1 0 0 | 7]
[ 0 0 5 0 0 1 4]

The tableau above is the final one in a problem to minimize -x + 2y. The minimum value of -x + 2y is:

a. -10
b. -4
c. 0
d. 4
e. none of the above

2. The inequality 2x - y + 5z (greater than or equal to symbol) -6 is equivalent to the inequality

a. 2x - y + 5z (less than or equal to symbol) -6
b. 2x - y + 5z (greater than or equal to symbol) 6
c. -2x + y - 5z (less than or equal to symbol) -6
d. -2x + y -5z (greater than or equal to symbol) 6
e. none of the above

3. Consider the following linear programming problem: A workshop of Peter's Potters makes vases and pitchers. Profit on a vase is $3; profit on a pitcher is $4. Each vase requires ½ hour of labor, each pitcher requires 1 hour of labor. Each item requires 1 unit of time in the kiln. Labor is limited to 4 hours per day and kiln time is limited to 6 units per day. Initial and final tableaux are shown in finding the production plan which will maximize profits: (x = number of vases and y = number of pitchers made per day).

x y u v M
[ -1 1 1 0 0 | 4 ]
[ 2 | ]
[ 1 1 0 1 0 | 6 ]
[-3 -4 0 0 1 | 0 ]
(initial)

x y u v M
[ 0 1 2 -1 0 | 2 ]
[ 1 0 -2 2 0 | 4 ]
[0 0 2 2 1 | 20 ]
(final)

If kiln time were decreased by one unit per day, determine the optimal performance schedule.

a. x = 3, y = 4
b. x = 4, y = 2
c. x = 6, y = 1
d. x = 2, y = 3
e. none of the above

4. In a linear programming problem in a standard from, the initial and final tableaux are given as below:

x y u v M
[ 1 3 1 0 0 | 50 ]
[ 1 5 0 1 0 | 70 ]
[-6 -24 0 0 1 | 0 ]
(initial)

x y u v M
[ 0 1 2 -5 -3 | 20 ]
2 2
-1 1
[ 1 0 2 2 0 | 10 ]
[0 0 3 3 1 | 360]
(final)

Given that x (greater than or equal to sign) 0 and y (greater than or equal to sign) 0, if h units were added to the first resource, the maximum value of the objective function is

a. 360
b. 360 + 3h
c. 360 + h
d. 360 + h(to the 5th power)/2
e. none of the above

5. To solve the linear programming problem

Minimize 50x + 70y subject to:

x + y (greater than or equal to symbol) 6
3x + 5y (greater than or equal to symbol) 24 we can use the simplex method on its dual.
x (greater than or equal to symbol) 0 , y (greater than or equal to symbol) 0
The objective function of the dual is

a. u + 3v
b. u + 5v
c. 6u + 24v
d. 70u + 50v
e. none of the above