You should determine what type of Programming
technique/methodology to use: linear programming, integer
programing, or non-linear programming. After formulation, you
should solve to obtain the optimal solution. You should then
interpret and make any relevant recommendations. All the
instructions are included in the problem itself. One important
hint: you will need to use normal distribution formulations. If you
use Excel, the built-in formulas are very relevant.

The Problem Information:

An ED is trying to budget for their weekly supply and staffing
needs. They are basing their needs on the weekly number of expected
patients, which is normally distributed. Part of their mission is
to determine how many patients they should be able to handle in a
week, so they can set their mean and variance for patient traffic
for each week. The minimum number of patients seen in any week is
three standard deviations from the mean; as well, the maximum
number of patients seen in a week is 3 standard deviations above
the mean. They know from the past that when they experience fewer
than 25% of their expected patient load that they incur a shortage
cost. If they treat the upper 10% of their determined distribution,
they incur an overage cost. Total shortage costs are calculated by
multiplying the per-unit shortage cost by the average number of
patients who show up when short. Likewise, the total overage costs
are calculated by multiplying the per-unit overage cost by the
number of patients who show up when over expectations. Finally,
multiplying the number showing up between short and over limits by
the normal cost per patient will allow for the determination of
total normal costs.

The average short is calculated as the arithmetic mean between
the lowest number of patients seen in a week and the 25th
percentile. This average is then considered the number short and
multiplied by the per-unit short cost to determine the total short
cost. Alternatively, the average over is calculated as the
arithmetic mean between the highest number of patients seen in a
week and the 90th percentile; costs are calculated similarly to the
short case. The average number of normal patients is the arithmetic
mean between the 25th and 90th percentiles and is then multiplied
by the per-unit normal cost. The total budget cost for the ED is
the proportional sum of all the costs.

Management has told the ED that for planning purposes, 70% of
the time the department cannot see any more than 140 patients.
Management has allowed the ED staff to determine how much to invest
in supply and labor that will impact the under and over costs. The
per-unit underage cost can be set between $150 and $300 while the
per-unit overage cost is allowed to be set between $255 and $400.
The normal cost is $120 times the ratio of per-unit over costs to
per-unit under costs. Additionally, this normal cost ratio requires
that the per-unit over cost cannot be great than 1 and one-half
times the per-unit under cost.

Finally, the mean number of weekly patients must be at least 9.2
times greater than the standard deviation while the actual variance
of the patient distribution can be no larger than 150.

REQUIREMENTS

Formulate this scenario so that the ED can complete their weekly
budget. Identify the 1) decision variables, 2) the objective
function (maximization, minimization, or other of what desired
outcome, and 3) the relevant constraints.

Interpret the results and provide recommendations. How much is
the optimal solution going to cost the ED in terms of:

1) the per unit and total under cost, and how many patients
reside within the under amount

2) the per unit and total over cost, and how many patients
reside within the over amount

3) the per unit an total normal cost, and how many patients
reside within the normal range

Identify the binding constraints and interpret any shadow
prices. If you are unable to do so, explain why.

Revaluate the situation for management so that all budget values
are stated in optimal whole numbers. How does this solution differ
from the originally stated problem?