Consider the following problem: Maximize Z= 2x1 + 4x2 + x3 - x4 subject to x1 +
ID: 3184731 • Letter: C
Question
Consider the following problem:
Maximize Z= 2x1 + 4x2 + x3 - x4
subject to x1 + 3x2 + x4 = 4
2x1 + x2 + x3 x4 3
3x1 + 3x3 + 2x4 5
x1 0, x2 0, x3 0 x4 unrestricted
A.) It is given that x1 = 7/3 and y2 = 1/3. Using complementary slackness, find the optimal solution x = (x1, x2, x3, x4, x5, x6, x7) and y = (y1, y2, y3, y4, y5, y6, y7) without resolving the problem again. Clearly show your work.
B.) If your are offered an opportunity of purchasing an additional unit of resource 1 (currently b1 = 4 for 1.25, should you buy it? Why or why not?
C.) If your are offered an opportunity of purchasing an additional unit of resource 3 (currently b3 = 5 for 0.35, should you buy it? Why or why not?
Explanation / Answer
Maximize Z= 2x1 + 4x2 + x3 - x4
subject to x1 + 3x2 + x4 = 4
2x1 + x2 + x3 x4 3
3x1 + 3x3 + 2x4 5
x1 0, x2 0, x3 0 x4 unrestricted
Let slack variables t1 t2 t3 t4 t5 t6 t7 t8,
Then initial table:
Tableau #1
x1 x2 x3 x4 t1 t2 t3 t4 t5 t6 t7 t8 z
1 3 0 1 1 0 0 0 0 0 0 0 0 4
1 2 1 -1 0 1 0 0 0 0 0 0 0 3
3 3 0 2 0 0 -1 0 0 0 0 0 0 10
1 0 0 0 0 0 0 -1 0 0 0 0 0 0
0 1 0 0 0 0 0 0 -1 0 0 0 0 0
0 0 1 0 0 0 0 0 0 -1 0 0 0 0
0 0 0 1 0 0 0 0 0 0 -1 0 0 0
1 3 0 1 0 0 0 0 0 0 0 -1 0 4
-2 -4 -1 1 0 0 0 0 0 0 0 0 1 0
Tableau #2
x1 x2 x3 x4 t1 t2 t3 t4 t5 t6 t7 t8 z
0 3 0 1 1 0 0 1 0 0 0 0 0 4
0 2 1 -1 0 1 0 1 0 0 0 0 0 3
0 3 0 2 0 0 -1 3 0 0 0 0 0 10
1 0 0 0 0 0 0 -1 0 0 0 0 0 0
0 1 0 0 0 0 0 0 -1 0 0 0 0 0
0 0 1 0 0 0 0 0 0 -1 0 0 0 0
0 0 0 1 0 0 0 0 0 0 -1 0 0 0
0 3 0 1 0 0 0 1 0 0 0 -1 0 4
0 -4 -1 1 0 0 0 -2 0 0 0 0 1 0
Tableau #3
x1 x2 x3 x4 t1 t2 t3 t4 t5 t6 t7 t8 z
0 0 0 1 1 0 0 1 3 0 0 0 0 4
0 0 1 -1 0 1 0 1 2 0 0 0 0 3
0 0 0 2 0 0 -1 3 3 0 0 0 0 10
1 0 0 0 0 0 0 -1 0 0 0 0 0 0
0 1 0 0 0 0 0 0 -1 0 0 0 0 0
0 0 1 0 0 0 0 0 0 -1 0 0 0 0
0 0 0 1 0 0 0 0 0 0 -1 0 0 0
0 0 0 1 0 0 0 1 3 0 0 -1 0 4
0 0 -1 1 0 0 0 -2 -4 0 0 0 1 0
Tableau #4
x1 x2 x3 x4 t1 t2 t3 t4 t5 t6 t7 t8 z
0 0 -1 2 1 -1 0 0 1 0 0 0 0 1
0 0 1 -1 0 1 0 1 2 0 0 0 0 3
0 0 -3 5 0 -3 -1 0 -3 0 0 0 0 1
1 0 1 -1 0 1 0 0 2 0 0 0 0 3
0 1 0 0 0 0 0 0 -1 0 0 0 0 0
0 0 1 0 0 0 0 0 0 -1 0 0 0 0
0 0 0 1 0 0 0 0 0 0 -1 0 0 0
0 0 -1 2 0 -1 0 0 1 0 0 -1 0 1
0 0 1 -1 0 2 0 0 0 0 0 0 1 6
Tableau #5
x1 x2 x3 x4 t1 t2 t3 t4 t5 t6 t7 t8 z
0 0 -1 0 1 -1 0 0 1 0 2 0 0 1
0 0 1 0 0 1 0 1 2 0 -1 0 0 3
0 0 -3 0 0 -3 -1 0 -3 0 5 0 0 1
1 0 1 0 0 1 0 0 2 0 -1 0 0 3
0 1 0 0 0 0 0 0 -1 0 0 0 0 0
0 0 1 0 0 0 0 0 0 -1 0 0 0 0
0 0 0 1 0 0 0 0 0 0 -1 0 0 0
0 0 -1 0 0 -1 0 0 1 0 2 -1 0 1
0 0 1 0 0 2 0 0 0 0 -1 0 1 6
Tableau #6
x1 x2 x3 x4 t1 t2 t3 t4 t5 t6 t7 t8 z
0 0 0.2 0 1 0.2 0.4 0 2.2 0 0 0 0 0.6
0 0 0.4 0 0 0.4 -0.2 1 1.4 0 0 0 0 3.2
0 0 -0.6 0 0 -0.6 -0.2 0 -0.6 0 1 0 0 0.2
1 0 0.4 0 0 0.4 -0.2 0 1.4 0 0 0 0 3.2
0 1 0 0 0 0 0 0 -1 0 0 0 0 0
0 0 1 0 0 0 0 0 0 -1 0 0 0 0
0 0 -0.6 1 0 -0.6 -0.2 0 -0.6 0 0 0 0 0.2
0 0 0.2 0 0 0.2 0.4 0 2.2 0 0 -1 0 0.6
0 0 0.4 0 0 1.4 -0.2 0 -0.6 0 0 0 1 6.2
Tableau #7
x1 x2 x3 x4 t1 t2 t3 t4 t5 t6 t7 t8 z
0 0 0.2 0 1 0.2 0.4 0 2.2 0 0 0 0 0.6
0 0 0.4 0 0 0.4 -0.2 1 1.4 0 0 0 0 3.2
0 0 -0.6 0 0 -0.6 -0.2 0 -0.6 0 1 0 0 0.2
1 0 0.4 0 0 0.4 -0.2 0 1.4 0 0 0 0 3.2
0 1 0 0 0 0 0 0 -1 0 0 0 0 0
0 0 -1 0 0 0 0 0 0 1 0 0 0 0
0 0 -0.6 1 0 -0.6 -0.2 0 -0.6 0 0 0 0 0.2
0 0 0.2 0 0 0.2 0.4 0 2.2 0 0 -1 0 0.6
0 0 0.4 0 0 1.4 -0.2 0 -0.6 0 0 0 1 6.2
Tableau #81
x1 x2 x3 x4 t1 t2 t3 t4 t5 t6 t7 t8 z
0 0 0 0 1 0 0 0 0 0 0 1 0 0
0 0 0.272727 0 0 0.272727 -0.454545 1 0 0 0 0.636364 0 2.81818
0 0 -0.545455 0 0 -0.545455 -0.0909091 0 0 0 1 -0.272727 0 0.363636
1 0 0.272727 0 0 0.272727 -0.454545 0 0 0 0 0.636364 0 2.81818
0 1 0.0909091 0 0 0.0909091 0.181818 0 0 0 0 -0.454545 0 0.272727
0 0 -1 0 0 0 0 0 0 1 0 0 0 0
0 0 -0.545455 1 0 -0.545455 -0.0909091 0 0 0 0 -0.272727 0 0.363636
0 0 0.0909091 0 0 0.0909091 0.181818 0 1 0 0 -0.454545 0 0.272727
0 0 0.454545 0 0 1.45455 -0.0909091 0 0 0 0 -0.272727 1 6.36364
Tableau #9
x1 x2 x3 x4 t1 t2 t3 t4 t5 t6 t7 t8 z
0 0 0 0 1 0 0 0 0 0 0 1 0 0
0 0 0.272727 0 -0.636364 0.272727 -0.454545 1 0 0 0 0 0 2.81818
0 0 -0.545455 0 0.272727 -0.545455 -0.0909091 0 0 0 1 0 0 0.363636
1 0 0.272727 0 -0.636364 0.272727 -0.454545 0 0 0 0 0 0 2.81818
0 1 0.0909091 0 0.454545 0.0909091 0.181818 0 0 0 0 0 0 0.272727
0 0 -1 0 0 0 0 0 0 1 0 0 0 0
0 0 -0.545455 1 0.272727 -0.545455 -0.0909091 0 0 0 0 0 0 0.363636
0 0 0.0909091 0 0.454545 0.0909091 0.181818 0 1 0 0 0 0 0.272727
0 0 0.454545 0 0.272727 1.45455 -0.0909091 0 0 0 0 0 1 6.36364
Tableau #10
x1 x2 x3 x4 t1 t2 t3 t4 t5 t6 t7 t8 z
0 0 0 0 1 0 0 0 0 0 0 1 0 0
0 0 0.5 0 0.5 0.5 0 1 2.5 0 0 0 0 3.5
0 0 -0.5 0 0.5 -0.5 0 0 0.5 0 1 0 0 0.5
1 0 0.5 0 0.5 0.5 0 0 2.5 0 0 0 0 3.5
0 1 0 0 0 0 0 0 -1 0 0 0 0 0
0 0 -1 0 0 0 0 0 0 1 0 0 0 0
0 0 -0.5 1 0.5 -0.5 0 0 0.5 0 0 0 0 0.5
0 0 0.5 0 2.5 0.5 1 0 5.5 0 0 0 0 1.5
0 0 0.5 0 0.5 1.5 0 0 0.5 0 0 0 1 6.5
So ,
Optimal Solution: z = 6.5; x1 = 3.5, x2 = 0, x3 = 0, x4 = 0.5
For second case use b1=4, for 1.25(z) Not.
For another case use b3= 5, for 0.35(z) not