3.4-15. Oxbridge University maintains a powerful mainframe computer for research

Question

3.4-15. Oxbridge University maintains a powerful mainframe computer for research use by its faculty, Ph.D. students, and research associates. During all working hours, an operator must be available to operate and maintain the computer, as well as to per- form some programming services. Beryl Ingram, the director of the computer facility, oversees the operation. It is now the beginning of the fall semester, and Beryl is con- hours to working fronted with the problem of assigning different her operators. Because all the operators are currently enrolled in the university, they are available to work only a limited number of hours each day, as shown in the following table. Maximum Hours of Avallability Operators Wage Rate | Mon. Tue, Wed- Thurs. Fri. K. C D. H H. B 6 $25/hour 6 $26/hour 0 $24/hour $23/hour $28/hour $30/hour 6 8 6 4 8 6 K. S 2 There are six operators (four undergraduate students and two graduate students). They all have different wage rates because of mputers and in their pro- gramming ability. The above table shows their wage rates, along with the maximum number of hours that each can work each day Each operator is guaranteed a certain minimum number of hours per week that will maintain an adequate knowledge of the operation. This level is set arbitrarily at 8 hours per week for the undergraduate students (K. C., D. H., H. B., and S. C) and 7 hours differences in their experience with co per week for the graduate students (K. S. and N. K.).

Explanation / Answer

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 subtract surplus variable S1 and add artificial variable A1

2. As the constraint 2 is of type '' we should subtract surplus variable S2 and add artificial variable A2

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,surplus,artificial variables

Positive maximum Cj-Zj is 12M-8 and its column index is 3. So, the entering variable is x3.

Minimum ratio is 134 and its row index is 2. So, the leaving basis variable is A2.

The pivot element is 8.

Entering =x3, Departing =A2, Key Element =8

R2(new)=R2(old)÷8

R1(new)=R1(old)-4R2(new)

R3(new)=R3(old)-4R2(new)

R4(new)=R4(old)

R5(new)=R5(old)-4R2(new)

Positive maximum Cj-Zj is 6M-8 and its column index is 1. So, the entering variable is x1.

Minimum ratio is 53 and its row index is 3. So, the leaving basis variable is S1.

The pivot element is 6.

Entering =x1, Departing =S1, Key Element =6

R3(new)=R3(old)÷6

R1(new)=R1(old)-6R3(new)

R2(new)=R2(old)

R4(new)=R4(old)

R5(new)=R5(old)-6R3(new)

Positive maximum Cj-Zj is 13 and its column index is 4. So, the entering variable is x4.

Minimum ratio is 4 and its row index is 3. So, the leaving basis variable is x1.

The pivot element is 512.

Entering =x4, Departing =x1, Key Element =512

R3(new)=R3(old)×125

R1(new)=R1(old)

R2(new)=R2(old)-58R3(new)

R4(new)=R4(old)

R5(new)=R5(old)

Since all Cj-Zj0

Hence, optimal solution is arrived with value of variables as :
x1=0,x2=0,x3=34,x4=4,x5=0,x6=0

Max Z=-38

Min Z=38

But this solution is not feasible
because the final solution violates the 1st constraint   6 x1    + 4 x3 + 5 x4 + 3 x5     25.

and the artificial variable A1 appears in the basis with positive value 2

Min Z = 8 x1 + 8 x2 + 8 x3 + 8 x4 + 7 x5 + 7 x6 subject to 6 x1 + 4 x3 + 5 x4 + 3 x5 25 6 x2 + 8 x3 + 5 x4 26 6 x1 + 4 x3 + 5 x4 + 3 x5 23 6 x2 + 8 x5 + 6 x6 28 6 x1 + 4 x3 + 5 x4 + 2 x6 30 and x1,x2,x3,x4,x5,x60;

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