Uncertainty in the Drug Production Problem

Return to the drug production problem described here.

Uncertainty model

We now assume a very small variation in some data in the problem. Specifically, we assume that the content of the active agent in the raw materials are subject to variation, with a margin of relative error of 0.5% (raw material I) and 2 % (raw material II).

The possible values of the coefficients are shown as intervals, in the following table:

Contents of raw materials:

mbox{ begin{tabular}{||c||c||} hlinehline Raw material& Content of agent A, & g per kg hlinehline RawI &[0.00995, 0.01005] hline RawII & [0.0196, 0.0204] hlinehline end{tabular} }

Impact on solution

Recall the solution to the nominal problem (when uncertainty is ignored):

$p^ast = -8819.658, ;; x_{rm Raw I} = 0, ;; x_{rm Raw II} = 438.789, ;;x_{rm Drug I} = 17.552, ;;x_{rm Drug II} = 0.$

The uncertainty affects the constraint on the balance of the active agent. In the nominal problem, this constraint was

$0.01 cdot x_{rm Raw I} + 0.02 cdot x_{rm Raw II} - 0.05 cdot x_{rm Drug I} - 0.600 cdot x_{rm Drug II} ge 0 .$

At optimum, this constraint is active. Therefore, even with a tiny error in the first and second coefficient, the constraint becomes invalid.

An adjustment policy

To remedy the problem, there is a simple solution: adjust the levels of production of the drugs, so as to satisfy the balance constraint. Let us adjust the production of Drug I, since that of Drug II is zero according to the original plan.

Clearly, if the actual content of active ingredient increases, the balance constraint will remain valid. In such a case, there is nothing to adjust, and the original production plan is still valid, and optimal. The balance constraint does become invalid only if “nature is against us”, that is when the level of active agent is less than originally thought.

Since the original optimal production plan recommends to purchase only the raw material II, a change in the corresponding coefficient (nominally set at 0.02 ) to the lesser value 0.0196 results, if we are to adopt the above simple “adjustment policy”, in a variation in the amount of production of Drug I from 17552 packs (the nominal value) to the ( less) value of 17201 packs. Accordingly, the cost function will decrease from the nominal value of 8,820 to the 21% (!) less value 6,929 .

This shows that for this problem, even a tiny variation in a single coefficient can result in a substantial decrease in the profit predicted by the model.

If we are to believe that the uncertain coefficients are actually random, and take their extreme values with 1/2 probability each, then the expected value of the cost (still with the above adjustment policy) will be also random, with expected value (8,820+6,929)/2 = 7,8745 . Thus, the expected loss due to random uncertainty is still high: 11% .