Need help: stats/probability problem

[danaf]

Hi all,

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.

-Danaf

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
1/1020/0.12
2/4000/0.16

For their survival, the aircraft also need to maintain a reserve of 50 gallons of
fuel each.
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
problem

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?

 

[JeffM]

Hi all,

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.

-Danaf

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
1/1020/0.12
2/4000/0.16

For their survival, the aircraft also need to maintain a reserve of 50 gallons of
fuel each.
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
problem

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?

Hi Dana

I am not sure that I shall have time to help you through this entire problem, but I am sure that others will jump in if I am not available.

First, the strategy for solving a big problem is to break it into smaller problems. Don't let the apparent complexity spook you.

Second, it is almost always a good idea to start by identifying the relevant variables, write down a brief description of what each variable is, and assign a unique letter to stand for the value of each variable. You write all this down so that you do not need to clutter your mind and memory with unnecessary details.

In the case of THIS Problem, you are specifically asked to identify the "decision variables," meaning the ultimate answers that are required to solve the problem. This is how I approach every problem: what question or questions am I ultimately trying to answer. So, without worrying yet about how to find the ultimate answer or answers, what do you think the question is asking you to find? That is, please tell me briefly what you must find and assign letters to stand for each.

 

[danaf]

Jeff,

Thank you for the head start. I think I'm starting to make a little headway. Here is what I have so far:

1) x = # of aircraft deployed to bridge 1; y = # of aircraft deployed to bridge 2
2) x + y <_ (less than or equal to) 20; 305x + 1050y <_ 15000; x>_0; y>_0
3) P-.12^x(.16^y)

Does this look sound so far. My next move is to set the equations of question 2 equal to each other...

-Danaf

 

[JeffM]

Jeff,

Thank you for the head start. I think I'm starting to make a little headway. Here is what I have so far:

1) x = # of aircraft deployed to bridge 1; y = # of aircraft deployed to bridge 2 Fine

Now you proceed to create inequalities for your constraints in terms of your decision variables. Quite right.

2) x + y <_ (less than or equal to) 20; Right on the button. You cannot send more than 20 planes.

305x + 1050y <_ 15000; Again, you are doing fine. Fuel required to get to bridge #1 is 1020 / 8 = 127.5. Fuel to return = 127.5. Reserve required = 50. So 2 * 127.5 + 50 = 255 + 50 = 305. Fuel required to get to bridge 2 is 2 * 4000 / 8 + 50 = 1000 + 50 = 1050. Thus total fuel required is 305x + 1050y, and that cannot exceed 15,000.

x>_0; y>_0 Good, you did not forget the non-negativity constraints.

Finally, you SHOULD label your objective variable just as you did your decision variables. You did not do this. The objective variable is what you want to maximize. Identifying and labeling variables is a basic step in preventing confusion (not just in linear programming.) The way the problem has been set up, identifying an objective variable takes some thought. You COULD decide to use as your objective variable the probability of success.

So, S = probability of success = probability of destroying at least one bridge. In which case, you would try to maximize S. However, S turns out to be quite cumbersome to work with, and your teacher has done some work for you to make the math easier. First he defines
P = the probability of failure = the probability of not destroying at least one bridge = the probability of destroying no bridge.

Your teacher may have inadvertently confused things by writing P followed by a dash, which looks like P followed by a minus.

3) P-.12^x(.16^y) This is wrong. 0.12 is the probability of one plane destroying bridge 1 and 0.16 is the probability of one plane destroying bridge 2. What you need to do is to calculate P = what? Remember P is the probability of no plane hitting a bridge.

Does this look sound so far. My next move is to set the equations of question 2 equal to each other...

-Danaf

Dana, this problem demands understanding linear programming, logs, and probabilities. You have not told us anything about your math training so it is very hard to give sensible advice. Do you know enough probability theory to write an equation for P in terms of x and y.

Hint: the probability of one plane sent to bridge 1 not hitting the bridge is 1 - 0.12 = 0.88. Do you know why?