Naval Operations Analysis, Additional Problems for Chapter 3

1. A squadron of bombers is attempting to destroy an ammunition depot that has a rectangular shape and is 1250 meters by 950 meters.  The bombing run will proceed down the long center axis of the depot.  If a bomb lands anywhere on the depot, a hit is scored.  Otherwise, the bomb is a miss.  There are 5 bombers in each squadron, each carry 1 bomb.  The aim point is the center of the depot.  The point of impact is assumed to be independent and normally distributed around the aim point with a standard deviation of 600 meters in the down range direction and 300 meters in the cross range direction.
(a) Find the expected number of hits in one run by a squadron.
(b) Simulate the operation and obtain an estimate of the expected number of hits in one run by a squadron.  Use the following random number streams.
 

stream 1	0.089	0.821	0.609	0.540	0.558	0.733	0.788	0.671	0.588
stream 2	0.629	0.823	0.862	0.631	0.019	0.774	0.890	0.885	0.647

2. Suppose a simulation was performed 20 times to obtain an estimate of the parameter X.  The average and sample standard deviation of the observed values was calculated to be 1.414 and 0.011, respectively.  Estimate the number of runs that must be done in order that the standard error does not exceed 0.001.

 

3.  For an upcoming mission, a wave of 5 aircraft is to be launched from an aircraft carrier.  The aircraft will be launched using two available catapults.  The time to launch an aircraft is normally distributed with a mean of 7 minutes and a standard deviation of 2 minutes.  Approximately 20% of the time a mechanical failure is found on an aircraft before it is launched.  For this reason, 2 aircraft are on standby, and the time to launch one of these aircraft is normally distributed with a mean of 10 minutes and a standard deviation of 3 minutes.  The time to discover that an aircraft has a mechanical failure and cannot be launched is exponentially distributed with mean of 4 minutes.  Should it happen that it is impossible to launch 5 of the 7 aircraft, the mission will be canceled and will be considered as a failure.  Simulate the launch of one wave of 5 aircraft and obtain an estimate of the time required to launch the wave.

ANSWERS

1. (a) .61, (b) Either 5 or 6 hits, depending on how stream and direction were paired.
2. 121.
3. First, simulate aircraft failures.  Failure occurs if the random number is less than .20.

   aircraft	1	2	3	4	5	6	7
   random number	.83	.18	.24	.49	.46	.22	.61
   failure?	no	yes	no	no	no	no	no