GRAPHICAL METHOD
Maximize
Constraints:
- 3
The optimal solution is and
SIMPLEX METHOD
Maximize z = 3x1 + 2x2
subject to
-x1 + 2x2 ≤ 4
3x1 + 2x2 ≤ 14
x1 – x2 ≤ 3
x1, x2 ≥ 0
-x1 + 2x2 + x3 = 4
3x1 + 2x2 + x4 = 14
x1 – x2 + x5 = 3
x1, x2, x3, x4, x5 ≥ 0
Maximize z = 3x1 + 2x2 + 0x3 + 0x4 + 0x5
x1 = 0, x2 = 0, z = 0
x3 = 4, x4 = 14, x5 = 3
a11 = -1, a12 = 2, a13 = 1, a14 = 0, a15 = 0, b1 = 4
a21 = 3, a22 = 2, a23 = 0, a24 = 1, a25 = 0, b2 = 14
a31= 1, a32 = -1, a33 = 0, a34 = 0, a35 = 1, b3 = 3
Calculating values for the index row (zj – cj)
z1 – c1 = (0 X (-1) + 0 X 3 + 0 X 1) – 3 = -3
z2 – c2 = (0 X 2 + 0 X 2 + 0 X (-1)) – 2 = -2
z3 – c3 = (0 X 1 + 0 X 0 + 0 X 0) – 0 = 0
z4 – c4 = (0 X 0 + 0 X 1 + 0 X 0) – 0 = 0
z5 – c5 = (0 X 0 + 0 X 0 + 0 X 1) – 0 = 0
x3 row
a11 = -1 – 1 X ((-1)/1) = 0
a12 = 2 – (-1) X ((-1)/1) = 1
a13 = 1 – 0 X ((-1)/1) = 1
a14 = 0 – 0 X ((-1)/1) = 0
a15 = 0 – 1 X ((-1)/1) = 1
b1 = 4 – 3 X ((-1)/1) = 7
x4 row
a21 = 3 – 1 X (3/1) = 0
a22 = 2 – (-1) X (3/1) = 5
a23 = 0 – 0 X (3/1) = 0
a24 = 1 – 0 X (3/1) = 1
a25 = 0 – 1 X (3/1) = -3
b2 = 14 – 3 X (3/1) = 5
x1 row
a31 = 1/1 = 1
a32 = -1/1 = -1
a33 = 0/1 = 0
a34 = 0/1 = 0
a35 = 1/1 = 1
b3 = 3/1 = 3
Table 2
Calculating values for the index row (zj – cj)
z1 – c1 = (0 X 0 + 0 X 0 + 3 X 1) – 3 = 0
z2 – c2 = (0 X 1 + 0 X 5 + 3 X (-1)) – 2 = -5
z3 – c3 = (0 X 1 + 0 X 0 + 3 X 0) – 0 = 0
z4 – c4 = (0 X 0 + 0 X 1 + 3 X 0) – 0 = 0
z5 – c5 = (0 X 1 + 0 X (-3) + 3 X 1) – 0 = 3
Key column = x2 column
Minimum (7/1, 5/5) = 1
Key row = x4 row
Pivot element = 5
x4 departs and x2 enters.
Calculating values for table 3
x3 row
a11 = 0 – 0 X (1/5) = 0
a12 = 1 – 5 X (1/5) = 0
a13 = 1 – 0 X (1/5) = 1
a14 = 0 – 1 X (1/5) = -1/5
a15 = 1 – (-3) X (1/5) = 8/5
b1 = 7 – 5 X (1/5) = 6
x2 row
a21 = 0/5 = 0
a22 = 5/5 = 1
a23 = 0/5 = 0
a24 = 1/5
a25 = -3/5
b2 = 5/5 = 1
x1 row
a31 = 1 – 0 X (-1/5) = 1
a32 = -1 – 5 X (-1/5) = 0
a33 = 0 – 0 X (-1/5) = 0
a34 = 0 – 1 X (-1/5) = 1/5
a35 = 1 – (-3) X (-1/5) = 2/5
b3 = 3 – 5 X (-1/5) = 4
DUALITY AND SENSITIVITY
TRANSPORTATION MODEL
NWC/R: