Page 101 - Microsoft Word - 00 Prelims.docx

Planning with limiting factors

Example 2, continued

We do not know the maximum value of the objective function; however, we
can draw an iso-contribution (or 'profit') line that shows all the combinations of
x and y that provide the same total value for the objective function.

If, for example, we need to maximise contribution $4x + $8y, we can draw a
line on a graph that shows combination of values for x and y that give the
same total contribution, when x has a contribution of $4 and y has a
contribution of $8.

Any total contribution figure can be picked, but a multiple of $4 and $8 is
easiest. .Optimum corner is Corner C, the intersection of: 8x + 10y = 11,000
and y = 600.

At this corner, x = 625 and y = 600. The optimum production plan is to
produce 625 units of Product X and 600 units of Product Y; The contribution at
this point is maximised.

C = (625 × $4) + (600 × $8) = $7,300.

For example, assume 4x + 8y = 4,000. This contribution line could be found by
joining the points on the graph x = 0, y = 500 and x =1,000 and y = 0.

Instead, we might select a total contribution value of 4x + 8y = $8,000. This
contribution line could be found by joining the points on the graph x = 0, y =
1,000 and x = 2,000 and y = 0.

When drawing both of these contribution lines on a graph, we find that the two
lines are parallel and the line with the higher total contribution value for values
x and y ($8,000) is further away from the origin of the graph (point 0).

This can be used to identify the solution to a linear programming problem.
Draw the iso-contribution line showing combinations of values for x and y that
give the same total value for the objective function.

95