Final answer:
To minimize costs for the Gotham City Police Department, an LP model is formulated with decision variables for workers on consecutive and non-consecutive shifts, constraints for staffing requirements, and an objective function to minimize total hourly costs of $144 and $216 for each type of shift.
Step-by-step explanation:
Linear Programming for Minimizing Costs
To formulate a linear programming (LP) model to minimize the cost of meeting the daily workforce demands at Gotham City Police Department, we must identify our decision variables, constraints, and the objective function. Let's denote:
- x1: number of workers working consecutive shifts (12 A.M. to 12 P.M. or 12 P.M. to 12 A.M.)
- x2: number of workers working non-consecutive shifts
The LP constraints based on the workers needed are:
- 15 workers from 12 A.M. to 6 A.M.
- 5 workers from 6 A.M. to 12 P.M.
- 12 workers from 12 P.M. to 6 P.M.
- 6 workers from 6 P.M. to 12 A.M.
Since consecutive shifts are paid at $12 per hour and non-consecutive shifts are paid at $18 per hour, for two 6-hour shifts, the costs become $144 for consecutive shifts and $216 for non-consecutive shifts. Thus, the objective function for cost minimization is:
Z = 144x1 + 216x2
The constraints in terms of decision variables become:
- x1 + x2 ≥ 15 (for 12 A.M. to 6 A.M.)
- x1 + x2 ≥ 5 (for 6 A.M. to 12 P.M.)
- x1 + x2 ≥ 12 (for 12 P.M. to 6 P.M.)
- x1 + x2 ≥ 6 (for 6 P.M. to 12 A.M.)
- x1, x2 ≥ 0
This LP model minimizes the total cost of staffing while ensuring that the shift requirements are met. It can be solved using standard linear programming software or techniques like the simplex method.