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70 CHAPTER 2 AN INTRODUCTION TO LINEAR PROGRAMMING

We see that X takes an optimal value of £200 000 and Y of £300 000 with the objective function taking a
value of £54 000. Putting this back into the context of the problem we can inform the investment advisor
that, based on the information given, their client can obtain a maximum annual return of £54 000 by
investing £200 000 in the Internet fund and £300 000 in the Blue Chip fund. We see further from the Excel
output that the first and third constraints are binding but the second constraint is not. The first constraint
related to the maximum available funding of £500 000. All of this is being invested at the optimal solution
(slack is zero). Similarly, for the third constraint. This relates to the maximum risk rating of 240. With the
recommended investment mix, this rating is exactly met – in other words we are at the upper limit of the
client’s risk tolerance with this investment mix. On the other hand, the second constraint is non-binding with
a slack of £150 000. This relates to the advisor’s requirement that no more than £350 000 be invested in the
Internet fund.
For the advisor’s second client we have a similar problem but we now change the maximum risk rating to
320. The Excel solution is shown in Exhibit 2.2.

Target Cell (Max)
Name Original Value Final Value
Values OF 0 55 000

Adjustable Cells
Name Original Value Final Value

Value X 0 350 000
Value Y 0 150 000

Constraints

Name Cell Value Status Slack
LHS 500 000 Binding 0
LHS 350 000 Binding 0
LHS 270 Not Binding 50

We now have an optimal solution of £350 000 invested in the Internet fund with the remainder
of £150 000 in the Blue Chip fund generating an annual return of £55 500. The first constraint is
binding (we are investing all the available amount) as is the second (we are investing the maximum into
the Internet fund). The third constraint is non-binding – we are well below the client’s maximum risk rating
(the slack value is shown as 50 but it is difficult to interpret this given the complexity of the actual
constraint).
On comparison with client 1 we see that client 2’s portfolio has invested more in the higher-return – but also
higher-risk – Internet fund. Client 1 was prevented from investing more than £200 000 in this fund because of
the risk rating constraint which was binding for this client.
We might also want to point out to the investment advisor that the model results are only as good as the
data that we used, particularly the projected annual returns for the two types of fund.

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