Manual without using the computer Question (B7) The Kleen City Police Department

Question

Manual without using the computer

Question (B7) The Kleen City Police Departments has the following minimal daily requirements for policemen: Time of day (24-hour clock) 2-6 6-10 10-14 14-18 18-22 22-2 Minimal Number of policemen Required During Period 20 50 80 100 40 30 Period 2 4 Note, you are to consider Period 1 as following immediately after Period 6. Each policeman works eight consecutive hours. Let xi, denote the number of men starting work in Period t every day. The Police Department seeks, a daily manpower schedule that employs the least number of policemen provided that each of the above requirements are met. Formulate a linear programming model to find an optimal schedule.

Explanation / Answer

Let x1, x2, ... , x6 be the policement starting the work in period 1, 2, ..., 6 respectively.

As period 1 follows after period 6, and each policeman works for 8 hours i.e. 2 periods, policemen available in period 1 will be x6 (continuing from period 6) + x1 (starting in period 1) and that has to be at least 20.

Similarly, policemen available in period 2 will be x1 (continuing from period 1) + x2 (starting in period 2) and that has to be at least 50.

Similar constraints will be there for the period 3 to 6.

As the objective is to employ least number of policemen, linear programming model to find an optimal schedule can be formulated as follows:

Decision Variables: x1, x2, x3, x4, x5, x6

Objective Function: Minimize x1 + x2 + x3 + x4 + x5 + x6

Constraints: A total of 6 constraints (1 for each period) as follows:

x6 + x1 >= 20

x1 + x2 >= 50

x2 + x3 >= 80

x3 + x4 >= 100

x4 + x5 >= 40

x5 + x6 >= 30

Related Questions

Navigate

• Browse (All)
• Browse M
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.