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Math Help - Operations analysis word problem

  1. #1
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    Operations analysis word problem

    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?
    Last edited by danaf; November 15th 2012 at 12:13 PM. Reason: spelling error
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  2. #2
    MHF Contributor
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    Re: Operations analysis word problem

    Hey danaf.

    To get you started, you should consider that a lot of mathematics is about translating the problem from words into constraints.

    So you have aircraft and fuel as your things that can vary (i.e. things that can change). You have also been given the constraints in word form.

    Constraints are either written as inequalities or equalities and they can be as complex as you can represent them algebraically.

    These ones a very simple and some involve inequalities (i.e. >, <, >=, <=) while some involve equalities.

    I've hinted towards what the variables are, now consider how you can take these variables (involving x and y) and get constraints (involving algebra and equalities/inequalities) that tie them together.

    If you will be doing a fair amount of mathematics and analysis then this is something that you will have do without thinking.

    It certainly is a pathetic example as well.
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