Results 1 to 8 of 8

Math Help - business math linear programming

  1. #1
    Newbie
    Joined
    Nov 2006
    Posts
    2

    business math linear programming

    1.) George Johnson recently inherited a large sum of money; he wants to use a portion of this money to set up a trust fund for his two children. The trust fund has two investment options: (1) a bond fund and (2) a stock fund. The projected returns over the life of the investments are 6% for teh bond fund and 10% for the stock fund. Whatever portion of the inheritance he finally decides to commit to the trust fund, he wants to invest at lesat 30% of that amount in the bond fund. In addition, he wants to select a mix that will enable him to obtain a total return of at least 7.5%.
    a. Formulate a linear programming model that can be used to determine the percentage that should be allocated to each of hte possible investment alternatives.
    b. Solve the problem using the graphical soluiton procedure.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Grand Panjandrum
    Joined
    Nov 2005
    From
    someplace
    Posts
    14,972
    Thanks
    4
    Quote Originally Posted by pauline112 View Post
    1.) George Johnson recently inherited a large sum of money; he wants to use a portion of this money to set up a trust fund for his two children. The trust fund has two investment options: (1) a bond fund and (2) a stock fund. The projected returns over the life of the investments are 6% for teh bond fund and 10% for the stock fund. Whatever portion of the inheritance he finally decides to commit to the trust fund, he wants to invest at lesat 30% of that amount in the bond fund. In addition, he wants to select a mix that will enable him to obtain a total return of at least 7.5%.
    a. Formulate a linear programming model that can be used to determine the percentage that should be allocated to each of hte possible investment alternatives.
    b. Solve the problem using the graphical soluiton procedure.
    Let b be the percentage invested in the bond func, and s be the percentage
    invested in the stock fund.

    Then: b+s<=100

    (the total investment cannot be more than 100% of the available funds
    <= here indicates that there is no requirement that he invest all of the
    inheritance)

    Also: b>=0.30(s+b)

    (b must be at least 30% of the invested funds).

    Finally he wants the return to be >=7.5%, so:

    0.06b + 0.10s>=7.5.

    These three inequalities together with the requirments that s>=0, b>=0
    represent the problem statement, that is:

    b+s<=100
    b>=0.30(s+b)
    0.06b + 0.10s>=7.5
    s>=0
    b>=0

    Which may be rearranged into:

    -b-s>=-100
    0.7b-0.3s>=0
    0.06b + 0.10s>=7.5
    s>=0
    b>=0

    But as these stand they do not constitute a linear program as there is
    no objective function specified that is to be maximised (or minimised).
    Though a solution to the problem as posed will be the entire feasible region
    (if there is one).

    (We can probably assume that we are to maximise the return 0.06b + 0.10s
    but we have not been asked to).

    The attached plot shows the feasible region (shaded) and which vertex of
    the region maximises 0.06b + 0.10s.

    RonL
    Attached Thumbnails Attached Thumbnails business math linear programming-gash.jpg  
    Last edited by CaptainBlack; November 6th 2006 at 07:09 AM.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Super Member

    Joined
    May 2006
    From
    Lexington, MA (USA)
    Posts
    11,683
    Thanks
    615
    Hello, Pauline!

    This linear programming problem has a feature
    . . that makes it quite unusual . . .


    George Johnson recently inherited a large sum of money.
    He wants to use a portion of this money to set up a trust fund for his two children.
    The trust fund has two investment options: (1) a bond fund and (2) a stock fund.
    The projected returns over the life of the investments are 6% for the bond fund
    and 10% for the stock fund.

    Whatever portion of the inheritance he finally decides to commit to the trust fund,
    he wants to invest at lesat 30% of that amount in the bond fund.
    In addition, he wants a mix with a total return of at least 7.5%.

    a. Formulate a linear programming model that can be used to determine the percentage
    that should be allocated to the two investment alternatives.

    b. Solve the problem using the graphical soluiton procedure.

    Let B = percentage invested in the bond fund. . B \geq 0
    Let S = percentage invested in the stock fund. . S \geq 0

    So we have: .  B + S \:=\:100 [1]

    We are told that: B \geq 30 [2]


    The return on the bond fund is: 6B dollars.
    The return on the stock fund is; 10S dollars.
    . . The total return is: 6B + 10S dollars.

    The total return is to be at least 7.5% of the total investment: 7.5(B + S)

    Hence, we have: . 6B + 10S \:\geq \:7.5(B + S)\quad\Rightarrow\quad S \:\geq\:\frac{3}{5}B [3]

    Graph these statements on a B\text{-}S coordinate system.
    We are already limited to the first quadrant.

    Inequaltiy [2]: B \geq 30
    This is the vertical line B = 30 and the region to the right.

    Inequality [3]: S \geq \frac{3}{5}B
    This is the slanted line S = \frac{3}{5}B and the region above it.

    So far, we have:
    Code:
            S
            |       :::::::::::::::::
            |       ::::::::::::::::
            |       :::::::::::::::
            |       ::::::::::::::
            |       ::::::::::::*
            |       ::::::::*
            |       ::::*
            |       *
            |   *   :
          --*-------+-------------- B
            |      30

    But statement [1] is an equation: B + S \:=\:100

    There is no "region" . . . the points are on the slanted line.
    Code:
            |
        100 *       :
            | *     :
            |   *   :
            |     * :
            |       oA          *
            |       : *     *
            |       :   oB
            |       *     *
            |   *   :       *
          --*-------+---------*----
            |      30        100

    So the "region" is the line segment from A(30,70) to (62.5,37.5)

    We find that the maximum return is at point A(30,70)

    Therefore, George should invest 30% in bonds and 70% in stocks.


    Follow Math Help Forum on Facebook and Google+

  4. #4
    Grand Panjandrum
    Joined
    Nov 2005
    From
    someplace
    Posts
    14,972
    Thanks
    4
    Quote Originally Posted by Soroban View Post
    Hello, Pauline!

    This linear programming problem has a feature
    . . that makes it quite unusual . . .


    Let B = percentage invested in the bond fund. . B \geq 0
    Let S = percentage invested in the stock fund. . S \geq 0

    So we have: .  B + S \:=\:100 [1]

    We are told that: B \geq 30 [2]
    This last inequality is not what the condition specifies, it actually says:

    "Whatever portion of the inheritance he finally decides to commit to the trust fund, he wants to invest at lesat 30% of that amount in the bond fund"

    which is that 0.3*(s+b)<=b

    RonL
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Super Member

    Joined
    May 2006
    From
    Lexington, MA (USA)
    Posts
    11,683
    Thanks
    615
    Hello, Captain!


    I believe we're both saying the same thing.

    You said: . B \:\geq \:0.3(B + S)

    Since B + S \:=\:100 (they're percentages): . B \,\geq\,0.3(100) \,=\,30

    Follow Math Help Forum on Facebook and Google+

  6. #6
    Grand Panjandrum
    Joined
    Nov 2005
    From
    someplace
    Posts
    14,972
    Thanks
    4
    Quote Originally Posted by Soroban View Post
    Hello, Captain!


    I believe we're both saying the same thing.

    You said: . B \:\geq \:0.3(B + S)

    Since B + S \:=\:100 (they're percentages): . B \,\geq\,0.3(100) \,=\,30
    But we (however bizarre it may seem) not told that b+s=100
    We just have b+s<=100.

    (with one interpretation of what we are to maximise it is true that
    at the optimum b+s=100, so with that interpretation the optimum
    will be the same, but they are then different problems which happen
    to have the same solution

    Or have I slipped into pedantic mode without noticing?

    RonL
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Newbie
    Joined
    Sep 2007
    Posts
    2
    so which one is right here?
    Follow Math Help Forum on Facebook and Google+

  8. #8
    Grand Panjandrum
    Joined
    Nov 2005
    From
    someplace
    Posts
    14,972
    Thanks
    4
    Quote Originally Posted by livingtwo View Post
    so which one is right here?
    Probably neither as the problem is ambiguous, it is not clear if the 7.5%
    return is on the investment or the inheritance.

    There I believe the main difference between the ywo methods is that
    CaptainBlack is calculating for a return of 7.5% on the inheritance while
    Soroban is calculating for a return of 7.5% on the investment (as far as
    I can tell at this distance in time)

    You will note that Soroban says B+S=100, that is he is either assuming all
    of the inheritance is invested or that B and S represents the split of what
    is invested. CaptainBlack says b+s<=100, that is they are the percentages
    of the inheritance invested in each option.

    Of course if you had been asked to maximise the monetary return all of the
    inheritance would have been invested as the diagram attached to CB's post
    shows that the optimum mix would be 70-30 which is the same as Soroban's
    optimum.


    RonL
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 2
    Last Post: February 15th 2010, 03:09 PM
  2. Replies: 1
    Last Post: November 17th 2008, 03:18 AM
  3. Linear programming (quantitative methods for business)
    Posted in the Business Math Forum
    Replies: 0
    Last Post: October 25th 2008, 03:14 PM
  4. Business Math
    Posted in the Business Math Forum
    Replies: 0
    Last Post: April 18th 2008, 05:07 PM
  5. linear programming, quantitive math
    Posted in the Pre-Calculus Forum
    Replies: 1
    Last Post: February 24th 2008, 07:50 PM

Search Tags


/mathhelpforum @mathhelpforum