Optimizing Witness Cost?

Question

The Mumbai Police Department wants to allocate its personnel for surveillance in a high-security area. There are two types of witnesses: those who can identify a suspect with high accuracy and those who can do so with moderate accuracy. A high-accuracy witness costs ₹30,000 per day, while a moderate-accuracy witness costs ₹15,000 per day. For a specific event, the Police Department has a budget of ₹1,50,000. If the probability of the suspect attending the event is 0.7, and the probability of a high-accuracy witness correctly identifying the suspect is 0.85, while that of a moderate-accuracy witness is 0.6, how many high-accuracy and moderate-accuracy witnesses should be hired to achieve the highest possible probability of identifying the suspect?

Accepted Answer

📌 CONCEPT: The problem can be modeled as a linear programming problem, where the goal is to maximize the probability of identifying the suspect by allocating the budget to hire high-accuracy and moderate-accuracy witnesses.

📐 RULE / FORMULA: The objective function is to maximize the probability of identifying the suspect, P = 0.7 × 0.85 × (number of high-accuracy witnesses) + 0.7 × 0.6 × (number of moderate-accuracy witnesses), subject to the budget constraint, 30000 × (number of high-accuracy witnesses) + 15000 × (number of moderate-accuracy witnesses) ≤ 150000.

💡 WORKED EXAMPLE: Suppose we want to hire x high-accuracy witnesses and y moderate-accuracy witnesses. We can set up the following linear programming problem: Maximize P = 0.7 × 0.85 × x + 0.7 × 0.6 × y, subject to 30000x + 15000y ≤ 150000. The optimal solution can be found by graphing the feasible region and finding the point that maximizes P.

⚠️ COMMON MISTAKE: Students often assume that the objective function is the only constraint, ignoring the budget constraint and the fact that the number of witnesses must be a non-negative integer.

Optimizing Witness Cost?

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