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SYS 6050
2012 Fall Semester
Homework 5
4 November 2012
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Budget for Tamiflu
The Al Jaber Base clinic is considering a change in budget for
flu drugs for the upcoming 4 months. While
increasing the budget should reduce the number of sick, we
assume that at some point the number of
sick will continue to rise as the budget rises. We developed a
cost function from the CDC flu model'sbase scenario results
(roughly parabolic). This function approximates the number of
sickdays versus the
effects of a variable budget.
Changing the budget (negative means reduce it, positive means
increase) could have an effect on the
number of sickdays during this time, and can be calculated using
the following equation:
Where x is the change in budget, f(x) is the total sickdays and
is the uncertainty. We will minimize
objective function to get x_star, and minimize the sensitivity
function to get x_hat.
We nominally choose
to be 3. This corresponds to approximately 291 sickdays for the
status quo
budget.
We take the partials of each at zero to find their individual
maxima:
x_star =10 2/3 (business as usual)
x_hat =-3/2 (conservative)
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We now use the -constraint optimization method:
Min[ Subject to:
And >=0
The Lagrangian:
Using
and the non-negativity constraint for :
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Table 1: Non-inferior Solutions
Table 1 show non-inferior solutions and tradeoff values ()
between -1
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Analysis
We will now return to x_star and x_hat, the two functions are
associated with business-as-usual and
conservative approaches. The business as usual approach will
minimize the number of sick, while the
conservative approach will minimize the sensitivity to cost
changes. The x_hat and x_star functions arederived with respect to
and graphed below.
Figure 2: Changes in cost for business-as-usual and conservative
cases versus alpha
Adjusting the sensitivity To study the sensitivity of changes to
, we evaluate f1(*) at x_star and x_hat with our nominal value
for
and a 50% change in , or (-.5).
( )
( ) | | Which is approximately a 47% change
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0
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f1(x_star, alpha)
f1(x_hat, alpha)
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( )
( ) || Which is about a .006% change. This is a very stable
solution. This is an interesting finding in that x_hat
only cut the budget by $1.5K. However, it is stable in a region
that we wish to avoid (i.e. adding more
sickdays).
Figure 3: Sensitivity Graph: changes in cost with change in
alpha
Recommendations
We can see that following a conservative (x_hat) policy will
result in a very stable solution (.006%
change versus 47%), but the number of sickdays is significantly
higher. Since goal of the policy needs to
be to reduce the number of workdays lost to illness, we
recommend that any budget changes be
between no change and an additional $10.66K. However, it is
important for the decisionmaker to
understand that the "business as usual" path, although it
decreases the number of lost sickdays, is
susceptible to higher sensitivity to uncertainty.
