Engineering Optimization: Theory and Practice, Fourth Edition

Problems 169

3.44 Reduce the system of equations

2 x 1 + 3 x 2 − 2 x 3 − 7 x 4 = 2

x 1 +x 2 −x 3 + 3 x 4 = 12

x 1 −x 2 +x 3 + 5 x 4 = 8

into a canonical system withx 1 ,x 2 , andx 3 as basic variables. From this derive all other

canonical forms.

3.45 Maximizef=^240 x^1 +^104 x^2 +^60 x^3 +^19 x^4

subject to

20 x 1 + 9 x 2 + 6 x 3 +x 4 ≤ 20

10 x 1 + 4 x 2 + 2 x 3 +x 4 ≤ 10

xi≥ 0 , i=1 to 4

Find all the basic feasible solutions of the problem and identify the optimal solution.

3.46 A progressive university has decided to keep its library open round the clock and gathered
that the following number of attendants are required to reshelve the books:

Time of day Minimum number of

(hours) attendants required

0–4 4

4–8 7

8–12 8

12–16 9

16–20 14

20–24 3

If each attendant works eight consecutive hours per day, formulate the problem of finding

the minimum number of attendants necessary to satisfy the requirements above as a LP

problem.

3.47 A paper mill received an order for the supply of paper rolls of widths and lengths as
indicated below:

Number of rolls Width of roll Length

ordered (m) (m)

1 6 100

1 8 300

1 9 200

The mill produces rolls only in two standard widths, 10 and 20 m. The mill cuts the

standard rolls to size to meet the specifications of the orders. Assuming that there is no