A squadron of 16 bombers needs to penetrate air defenses to reach its target. They can
either fly low and expose themselves to the air defense guns, or fly high and expose themselves to
surface-to-air missiles. In either case, the air defense firing sequence proceeds in three stages. First,
they must detect the target, then they must acquire the target (lock on target), and finally they
must hit the target. Each of these stages may or may not succeed. The probabilities are as follows:
The gums can fire 20 shells per minute, and the missile installation can fire three per minute. The
AD Type Pdetect Pacquire Phit
Low 0.90 0.80 0.05
High 0.75 0.95 0.70
proposed flight path will expose the planes for one minute if they fly low, and five minutes if they
fly high.
(a) Determine the optimal flight path (low or high). The objective is to maximize the number of
bombers that survive to strike the target.
(b) Each individual bomber has a 70% chance to destroy the target. Use the results of part (a)
to determine the chances of success (target destroyed) for this mission.
(c) Determine the minimum number of bombers necessary to guarantee a 95% chance of mission
success.
(d) Perform a sensitivity analysis with respect to the probability p = 0.7 that an individual
bomber can destroy the target. Consider the number of bombers that must be sent to
guarantee a 95% chance of mission success.
(e) Bad weather reduces both Pdetect and p, the probability that a bomber can destroy the target.
If all of these probabilities are reduced in the same proportion, which side gains an advantage
in bad weather?
To determine the optimal flight path, we need to compare the survival rates of the bombers flying low and flying high. We can calculate the expected number of surviving bombers for each flight path option and select the one with the higher survival rate.
Let's calculate the expected number of surviving bombers for each flight path:
For the low flight path:
The probability of detection is Pdetect = 0.90.
The probability of acquiring the target is Pacquire = 0.80.
The probability of hitting the target is Phit = 0.05.
The gums can fire 20 shells per minute.
Therefore, the survival rate for each bomber flying low is:
Survival rate low = Pdetect * Pacquire * Phit^20
For the high flight path:
The probability of detection is Pdetect = 0.75.
The probability of acquiring the target is Pacquire = 0.95.
The probability of hitting the target is Phit = 0.70.
The missile installation can fire 3 missiles per minute.
Therefore, the survival rate for each bomber flying high is:
Survival rate high = Pdetect * Pacquire * Phit^3
Now, we can calculate the expected number of surviving bombers for each flight path. Since the firing sequence proceeds in three stages, we raise the survival rate to the power of three:
Expected number of surviving bombers for low flight = (Survival rate low) ^ 3
Expected number of surviving bombers for high flight = (Survival rate high) ^ 3
Compare these values and select the flight path with the higher expected number of surviving bombers.
Answer (a): Determine the optimal flight path:
Compare the expected number of surviving bombers for low flight and high flight and select the flight path with the higher value.
Moving on to answer (b): Each individual bomber has a 70% chance to destroy the target.
Since each bomber has a 70% chance of destroying the target, we can multiply the expected number of surviving bombers (from the chosen flight path) by 0.70 to determine the chances of success (target destroyed) for this mission.
Answer (b): To determine the chances of success (target destroyed) for this mission, multiply the expected number of surviving bombers from the chosen flight path by 0.70.
Now let's move on to answer (c): Determine the minimum number of bombers necessary to guarantee a 95% chance of mission success.
To guarantee a 95% chance of mission success, we need to find the minimum number of bombers that ensures the chances of success (target destroyed) are at least 95%.
For this calculation, multiply the expected number of surviving bombers (from the chosen flight path in part (a)) by 0.70 until the result is greater than or equal to 0.95.
Answer (c): To guarantee a 95% chance of mission success, multiply the expected number of surviving bombers from the chosen flight path by 0.70 until the result is greater than or equal to 0.95.
Moving on to answer (d): Perform a sensitivity analysis with respect to the probability p = 0.7 that an individual bomber can destroy the target. Consider the number of bombers that must be sent to guarantee a 95% chance of mission success.
In this part, we will explore how changing the probability p (the chance of an individual bomber destroying the target) affects the number of bombers required to guarantee a 95% chance of mission success.
By varying the value of p and calculating the minimum number of bombers required for a 95% chance of success, we can perform this sensitivity analysis.
Answer (d): Vary the value of p and calculate the minimum number of bombers required for a 95% chance of mission success. Observe how changing the probability p affects the required number of bombers.
Lastly, for answer (e): Bad weather reduces both Pdetect and p, the probability that a bomber can destroy the target. If all of these probabilities are reduced in the same proportion, which side gains an advantage in bad weather?
If all probabilities, including Pdetect and p, are reduced in the same proportion due to bad weather, the side that gains an advantage will depend on the relative reduction in the survival rates for low and high flight paths.
To determine which side gains an advantage, calculate the expected number of surviving bombers for low and high flight paths using the adjusted reduced probabilities. Compare the new values and determine which flight path has the higher expected number of surviving bombers.
Answer (e): Calculate the expected number of surviving bombers for low and high flight paths using the adjusted reduced probabilities due to bad weather. Compare the new values and determine which flight path has the higher expected number of surviving bombers.
(a) To determine the optimal flight path (low or high), we need to compare the expected number of surviving bombers for each option.
If the squadron flies low, the expected number of surviving bombers can be calculated as follows:
Number of surviving bombers if flying low = Number of bombers * P(low)
P(low) = Pdetect(low) * Pacquire(low) * Phit(low)
Pdetect(low) = 0.90 (probability of detecting the target when flying low)
Pacquire(low) = 0.80 (probability of acquiring the target when flying low)
Phit(low) = 0.05 (probability of hitting the target when flying low)
Therefore, P(low) = 0.90 * 0.80 * 0.05
If the squadron flies high, the expected number of surviving bombers can be calculated as follows:
Number of surviving bombers if flying high = Number of bombers * P(high)
P(high) = Pdetect(high) * Pacquire(high) * Phit(high)
Pdetect(high) = 0.75 (probability of detecting the target when flying high)
Pacquire(high) = 0.95 (probability of acquiring the target when flying high)
Phit(high) = 0.70 (probability of hitting the target when flying high)
Therefore, P(high) = 0.75 * 0.95 * 0.70
Since the exposure time is different for each flight path, we need to adjust the number of bombers accordingly. If flying low, the squadron is exposed for 1 minute, and if flying high, the squadron is exposed for 5 minutes. Therefore, we multiply the number of bombers for the low flight path by 5.
Now, we can compare the expected number of surviving bombers for each flight path.
Expected number of surviving bombers if flying low = 16 * P(low)
Expected number of surviving bombers if flying high = 5 * 16 * P(high)
The optimal flight path is the one that yields a higher expected number of surviving bombers.
(b) To determine the chances of success (target destroyed) for this mission, we need to consider the probability of a bomber surviving and successfully hitting the target.
Chances of success = P(Surviving and Hitting Target)
Chances of success = (Number of surviving bombers) * Probability of hitting the target
If we use the optimal flight path determined in part (a), we can calculate the chances of success as follows:
Chances of success = (Expected number of surviving bombers) * 0.70
(c) To determine the minimum number of bombers necessary to guarantee a 95% chance of mission success, we need to find the number of bombers that result in a chances of success equal to or higher than 95%.
Let's assume the number of bombers as N.
Chances of success = N * 0.70 >= 0.95
Solving for N:
N >= 0.95 / 0.70
Therefore, the minimum number of bombers necessary to guarantee a 95% chance of mission success is the smallest integer greater than or equal to 1.357, which is 2.
(d) To perform a sensitivity analysis with respect to the probability p = 0.7, we need to determine the number of bombers that must be sent to guarantee a 95% chance of mission success for different values of p.
Let's assume p as the probability that an individual bomber can destroy the target.
Using the formula from part (c):
Number of bombers >= 0.95 / (P(Surviving and Hitting Target))
Number of bombers >= 0.95 / (Number of surviving bombers * p)
For p = 0.7, the minimum number of bombers necessary to guarantee a 95% chance of mission success is 2, as calculated in part (c).
(e) In bad weather, both Pdetect and p, the probability that a bomber can destroy the target, are reduced in the same proportion.
Since both flight paths are affected in the same way, neither side gains an advantage in bad weather. The advantage remains the same irrespective of the flight path in bad weather conditions.