Consider the following linear program:

Minimize z = 3x12 + 2x13 + 5x14 + 2x41 + x23 + 2x24 + 6x42 + 4x34 + 4x43,

subject to:

x12 + x13 + x14 x41 8,

x12 x23 x24 + x42 4,

x34 x13 x23 x43 4,

x14 + x34 + x24 x42 x42 x43 5,

all xi j 0.

a) Show that this is a network problem, stating it in general minimum-cost flow form. Draw the associated network and give an interpretation to the flow in this network.

b) Find an initial feasible solution. (Hint. Exploit the triangular property of the basis.)

c) Show that your initial solution is a spanning tree.

d) Solve completely.