Question

A supermarket is equipped with the following four fire safety systems: (1) a smoke detection system, (2) an automatic sprinkler system, (3) a smoke exhaust system, and (4) a fire alarm system.

Tasks: Conduct reliability analysis by answering the questions below. Please provide enough significant digits especially when the answer is very close to zero or one.

7. Calculate the system probability that at least three of the four fire systems operate. Do NOT draw a block diagram.

Assumed reliabilities

Smoke detection system 95%

Automatic sprinkler system 95%

Smoke exhaust system 85%

Fire alarm system 98%

Answer 1

7.

The four fire systems are,

(1) smoke detection system

(2) an automatic sprinkler system

(3) a smoke exhaust system

(4) a fire alarm system.

The following operational combinations are possible for system success:

1. All 4 fire systems operate.
2. (1) fails while (2),(3),(4) operate.
3. (2) fails while (1),(3),(4) operate.
4. (3) fails while (1),(2),(4) operate.
5. (4) fails while (1),(2),(3) operate.

The reliability of the system is given by:

R = R1*R2*R3*R4 + (1-R1)*R2*R3*R4 + R1*(1-R2)*R3*R4 + R1*R2*(1-R3)*R4 + R1*R2*R3*(1-R4)

Given,

R1 = 0.95

R2 = 0.95

R3 = 0.85

R4 = 0.98

On calculation, R = 0.9789

So the system probability that at least three of the four fire systems operate is  97.89%.