The LP Procedure |
This example continues to examine the effects of a change in
the cost of crude and the selling price of jet fuel.
Suppose that you know the cost of ARABIAN_LIGHT crude is
likely to increase 30 units, with the effects on oil and
fuel prices as described in Example 3.4.
The analysis in the last example only
accounted for an increase of a little over 4 units (because
the minimum was -4.15891).
Because an increase in the cost of ARABIAN_LIGHT beyond 4.15891
units requires a change in the optimal basis, it may require
a change in the optimal production strategy as well.
This type of analysis, where you want to find how
the solution changes with changes in the objective function
coefficients or right-hand-side vector,
is called parametric programming.
You can answer this question by using the PRICEPHI= option in the PROC LP statement. The following program instructs PROC LP to continually increase the cost of the crudes and the return from jet fuel using the ratios given previously, until the cost of ARABIAN_LIGHT increases at least 30 units.
proc lp sparsedata primalin=solution pricephi=-30; run;
The PRICEPHI= option in the PROC LP statement tells
PROC LP to perform parametric programming on any price change
vectors specified in the problem data set.
The value of the PRICEPHI= option tells PROC LP how
far to change the value of and in what direction.
A specification of PRICEPHI=-30 tells PROC LP to continue pivoting until the
problem has objective function equal to (original
objective function value) - 30
(change vector).
Output 3.5.1 shows the result of this analysis.
The first page is the Price Sensitivity Analysis Summary,
as discussed in Example 3.4.
The next page is an accounting for
the change in basis as a result of decreasing beyond -4.1589.
It shows that BREGA left the basis at an upper bound
and entered the basis at a lower bound.
The interpretation of
these basis changes can be difficult (Hadley 1962; Dantzig 1963).
The last page of output shows the optimal solution at
the displayed value of , namely -30.6878.
At an increase of 30.6878 units in the cost of ARABIAN_LIGHT and the
related changes to the other crudes and the jet fuel, it
is optimal to modify the production of jet fuel as shown
in the activity column. Although this plan is optimal, it
results in a profit of 0.
This may suggest that the ratio of a unit increase in the
price of jet fuel per unit increase in the cost of ARABIAN_LIGHT
is lower than desirable.
The LP Procedure Price Sensitivity Analysis Summary Sensitivity Vector change Minimum Phi -30.68783069 Entering Variable arabian_light Optimal Objective 0 ----Minimum Phi---- Reduced Col Variable Name Status Activity Price Cost 1 arabian_heavy 0 -201.8254 -43.59127 2 arabian_light ALTER 110 -205.6878 0 3 brega 0 -251.0317 -21.36905 4 heating_oil BASIC 42.9 0 0 5 jet_1 BASIC 33.33 330.68783 0 6 jet_2 BASIC 35.09 330.68783 0 7 naphtha_inter BASIC 11 0 0 8 naphtha_light BASIC 3.85 0 0 |
What is the optimal return if is exactly -30?
Because the change in the objective is linear as a function
of
, you can calculate the objective for any value of
between those given by linear interpolation.
For example, for any
between
-4.1589 and -30.6878, the optimal
objective value is
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