### A Generalized Network Model

The manager notes that there are shipping losses. As a first assumption
assume that every shipping link loses 5% of the amount shipped. |

We introduce an additional parameter called the arc gain to handle losses
or gains that occur along an arc. The arc gain is an arc parameter that
multiplies the flow at the beginning of the arc to obtain the flow at the
end of the arc. Figure 5 illustrates the effect of the gain on flow. We
model the 5% loss from Phoenix to Chicago with a gain of 0.95 on the arc.
With the gain factor, although 200 leave Phoenix, only 190 arrive at Chicago.

Figure 5. Modeling losses with a gain factor

The solution with loses in Fig. 6 is significantly different than those
previously presented. The flows are no longer integer, the extra demand
at Chicago is no longer satisfied, Austin does not produce to its full
capacity, more arcs are used to provide materials lost during shipping
and the profit is considerably reduced.

Figure 6. The solution with shipping losses represented
by gain factors. z = -494.3.

Gains are very for useful modeling. When all arc gains are 1, the model
is a pure network flow model. When some gains are other than 1, the model
is a generalized network flow model. Integer solutions cannot be guaranteed
for the generalized model.