1) How much of each type of resource is left over? MAX Z = 9000X 1 + 12000X 2 X1

Question

1) How much of each type of resource is left over?

MAX Z = 9000X1 + 12000X2

X1         X2                       RHS       Dual

Maximize           9000      12000                               

Solution->          4            6    108000

Variable              Value                  Reduced Cost    Original Val        Lower Bound Upper Bound

X1                        4                           0                           9000                    6000                    12000

X2                        6                           0                           12000                  9000                    18000

Constraint         Dual Value         Slack/Surplus    Original Val        Lower Bound Upper Bound

Constraint 1       750                      0                           64                        57.1429              68

Constraint 2       1200                    0                           50                        45                        52.7273

Constraint 3       0                           12                        120                      108                      Infinity

Constraint 4       0                           3                           7                           4                          Infinity

Constraint 5       0                           1                           7                           6                          Infinity

Variable              Status                 Value

X1                        Basic                    4

X2                        Basic                    6

slack 1                 NONBasic           0

slack 2                 NONBasic           0

slack 3                 Basic                    12

slack 4                 Basic                    3

slack 5                 Basic                    1

Optimal Value (Z)                          108000

Explanation / Answer

Solution:
Problem is

The problem is converted to canonical form by adding slack, surplus and artificial variables as appropiate

1. As the constraint 1 is of type '?' we should add slack variable S1

2. As the constraint 2 is of type '?' we should add slack variable S2

3. As the constraint 3 is of type '?' we should add slack variable S3

4. As the constraint 4 is of type '?' we should add slack variable S4

5. As the constraint 5 is of type '?' we should add slack variable S5

After introducing slack variables

Positive maximum Cj-Zj is 12000 and its column index is 2. So, the entering variable is x2.

Minimum ratio is 7 and its row index is 5. So, the leaving basis variable is S5.

? The pivot element is 1.

Entering =x2, Departing =S5, Key Element =1

R5(new)=R5(old)

R1(new)=R1(old)-8R5(new)

R2(new)=R2(old)-5R5(new)

R3(new)=R3(old)-8R5(new)

R4(new)=R4(old)

Positive maximum Cj-Zj is 9000 and its column index is 1. So, the entering variable is x1.

Minimum ratio is 2 and its row index is 1. So, the leaving basis variable is S1.

? The pivot element is 4.

Entering =x1, Departing =S1, Key Element =4

R1(new)=R1(old)÷4

R2(new)=R2(old)-5R1(new)

R3(new)=R3(old)-15R1(new)

R4(new)=R4(old)-R1(new)

R5(new)=R5(old)

Positive maximum Cj-Zj is 6000 and its column index is 7. So, the entering variable is S5.

Minimum ratio is 1 and its row index is 2. So, the leaving basis variable is S2.

? The pivot element is 5.

Entering =S5, Departing =S2, Key Element =5

R2(new)=R2(old)÷5

R1(new)=R1(old)+2R2(new)

R3(new)=R3(old)-22R2(new)

R4(new)=R4(old)-2R2(new)

R5(new)=R5(old)-R2(new)

Since all Cj-Zj?0

Hence, optimal solution is arrived with value of variables as :
x1=4,x2=6

Max Z=108000

Max Z = 9000 x1 + 12000 x2 subject to 4 x1 + 8 x2 ? 64 5 x1 + 5 x2 ? 50 15 x1 + 8 x2 ? 120 x1 ? 7 x2 ? 7 and x1,x2?0;

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