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 losses 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
The figure below shows the generalized model solved
in Excel.
