Capital rationingThere is no mathematical reason why the optimal solution should not completely
exclude one of the projects. Note that the above formulation assumes that any unin-
vested funds cannot be carried forward from one year to be invested in the next, but
it does assume that inflows from existing projects can be reinvested. It need not make
these assumptions, however.
The optimal solution can be derived manually or by computer through the LP tech-
nique, finding the maximum value of the objective function subject to the various con-
straints. In addition to providing the proportion of each project that should be
undertaken to achieve the maximum NPV, the LP output will also show, directly or
indirectly, the following:l the value of the maximum NPV;
l how much it is worth paying for additional investment funds in order to undertake
more investment and so increase the NPV; and
l how much capital would be needed in each year before that year’s shortage of
investment funds ceased to be a constraint.LP will give results that assume that the projects can be partially undertaken.
As with the profitability index approach to single-period constraints, this will not
be practical to apply in many real-life situations because many business investments
cannot be made in part. Real-life instances of businesses going into partnership on
large projects can be found. Some of these are clearly the pooling of expertise, but
some provide examples of businesses undertaking only part of a project, because
capital is rationed.
The oil business British Petroleum plc, the food and retailing business Associated
British Foods plcand the US chemicals giant DuPonthave joined forces to build a bio-
fuel plant near Hull, in the north of England. The plant, which will cost £200 million,
will be in action by the end of 2009 when it will turn wheat grown locally into a sub-
stitute for petrol to be used to power motor vehicles. This is probably an example both
of pooling of expertise and of reluctance of the three businesses concerned to commit
all of the finance necessary to fund the plant.
Most texts on operations research and quantitative methods in business explain
how to arrive at a solution to Example 5.7 through linear programming.Integer programming
As we have just seen, LP will often provide a solution that assumes that projects can
be undertaken partially. This may not be practical in many cases; a business may have
to undertake a particular project in full or not at all. In these circumstances an altern-
ative technique, integer programming(IP), can be applied. IP derives the optimal
combination of complete projects. This combination may well not be able to use all of
the scarce finance.‘