You should determine what type of Programming technique/methodology to use (linear/integer/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?