I haven't taken a math class in years and am having a great deal of trouble. I got this problem from a professor and need help approaching it (basically, I haven't the slightest clue). Anything you guys are willing to do to advise, I would really appreciate it. Thank you so much in advance.
A squadron of long-range bombers has been tasked with destroying at least one
of two bridges in order to impede the retreat of enemy forces. There are twenty
aircraft available and the fuel consumption of each is 8 km/gallon. Owing to an
acute fuel shortage, only 15,000 gallons of fuel are available for this mission. The
following table describes the other parameters involved.
Bridge #/Distance from base (in km)/Probability of destruction
For their survival, the aircraft also need to maintain a reserve of 50 gallons of
How many aircraft should be assigned to each target (given that each plane attacks
only one target) so as to maximize the probability of success of the mission?
1. Define, in words, variable symbols x and y for the decision variables in the
2. Formulate the two constraints due to fuel limitations and aircraft availability.
There is a third set of constraints – what is it?
3. Find the probability of not destroying at least one bridge. Note that, for
each plane, the probability of not hitting bridge 1, say, is (1-0.12) = 0.88.
Also, if there are many aircraft assigned to bridge 1, we may assume as
a first approximation that the individual probabilities of destruction are
independent, so that the probabilities ________________ (fill in the blank).
Similarly, probabilities of destruction for aircraft going to different bridges
are also independent, so the probabilities of destroying (or not destroying)
the two bridges again ______________. This gives the total probability of not
hitting the bridge as a product P – and this is what is to be minimized. Write
down this product.
4. As will be explained later, minimizing P is equivalent to maximizing –Log P.
We will call this Z and say that we wish to maximize Z = .128x + .174y (this
is the negative of the logarithm of P in part 3. And you can proceed to solve
the problem regardless of whether or not you have done part 3.)
5. Solve the linear programming problem by using a graphical method,
identifying the feasible region and the value of Z at each corner point of this
region. State your solution, rounded off to the nearest integer, in words
the squadron leader can understand.
6. Explain why the optimal point is the corner point you found in 5., by plotting
the lines Z = 1, 2 and 4 on the feasible region and seeing how Z increases
through this region.
7. When you round off your answer, is the pair of optimal values for x and
y still in the feasible region?