Engineering Optimization: Theory and Practice, Fourth Edition

Problems 241

x 1 −x 2 ≤ 11

x 1 ≥ 0 , x 2 unrestricted in sign

(a)Write the dual of this problem.

(b)Find the optimum solution of the dual.

(c)Verify the solution obtained in part (b) by solving the primal problem graphically.

4.8 A water resource system consisting of two reservoirs is shown in Fig. 4.4. The flows and
storages are expressed in a consistent set of units. The following data are available:

Quantity Stream 1 (i=1) Stream 2 (i=2)

Capacity of reservoiri 9 7

Available release from

reservoiri

9 6

Capacity of channel

below reservoiri

4 4

Actual release from

reservoiri

x 1 x 2

The capacity of the main channel below the confluence of the two streams is 5 units.

If the benefit is equivalent to $2× 106 and $3× 106 per unit of water released from

reservoirs 1 and 2, respectively, determine the releasesx 1 andx 2 from the reserovirs to

maximize the benefit. Solve this problem using duality theory.

4.9 Solve the following LP problem by the dual simplex method:

Minimizef= 2 x 1 + 9 x 2 + 24 x 3 + 8 x 4 + 5 x 5

Figure 4.4 Water resource system.