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Minimize the cost of conducting investigations of FP and FN

The over-arching objective of this assignment is to minimize the cost of conducting investigations of False-Positives (FP) and False-Negatives (FN) in the context of credit card fraud detection. An FP occurs when the front-end analytics identifies a suspicious set of data that has attributes of potential fraud, but is not. A FN occurs when the front-end analytics fails to identify data that is a real fraud event.

I have set this problem up to be solved using linear programming. The Objective Function (OF) is to minimize the current cost of investigating FP and FN that sum to the thresholds, e.g., the threshold is front-end analytics that identifies the number of FP and FN’s to investigate. The higher the threshold the fewer the number of investigations and vice versa. The threshold has been set arbitrarily (i.e.,there are no front-end analytics).

So, the OF becomes: FN + FP = Threshold

My issue and problem is to identify a set of constraints that minimizes the cost of these investigations. I am not sure how to do this. One issue that is giving me trouble is that the constraints I have used thus far are not allowing me to obtain convergence in the LP model. I believe this is because my constraints are not independent of the OF. This, I believe, results in circular reasoning.

I am therefore seeking assistance and a proposal to assist me with:

1. How to set up the constraints to solve for this OF.

2. To be able to relate the Thresholds to the objective function

3.. How to solve the problem when my initial values change

4.. I need to know how to solve this in either Excel Solver or in SAS

I need to be able to write up (in English) how a solution to this problem can be achieved by Thursday of this week with implementation to proceed afterwords.

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