Linear Programming Investment Problem


Now we’re going to work on an investment
problem you have twelve thousand dollars to invest and three different funds from
which to choose the municipal bond fund has a 7% return the local bank CD’s have
an 8% return and the high risk account has an expected I hope for return of 12%
to minimize risks we decide not to invest any more than two thousand
dollars in the high-risk account for tax reasons you need to invest at least
three times as much in the municipal bonds as in the bank CD’s assuming the
year-end yields are as expected what is the optimal investment amounts so again
the first thing we’re going to do is identify the variables we’re going to
say B is the amount we invest in the bonds C is the amount we invest in the
CDs and H is the amount invested in the high-risk account our objective function
what we want to do is we want to maximize our revenue and letter to
maximize our revenue we’re going to take seven percent times the amount invested
in bonds eight percent times the amount invested in the CDs and twelve percent
times the amount invested in the high-risk subject to the total available
is the bond the amount invested in bonds CDs and the high risks has to be less
than or equal to twelve thousand dollars it says that we have to invest less than
we want to invest you wouldn’t decide to invest less than twenty two thousand
dollars in high risk the tax requirements to invest at least three
times as much your municipals as CD so bonds has to be greater than or equal to
three times the amount invested in CDs since we want our variables all on the
left-hand side of the inequality we subtract receipt from both sides so B
minus three C has to be greater than or equal to zero again all the variables
the B the C and the H have to be positive numbers so we make them greater
than or equal to zero setting this up we initialize the values of zeros for all
of our variables the amount invested in the bonds the CDs
and the high-risk has to be no less than zero we put in the 7% the 8% in the 12%
which are the returns for these investment we put the amount for the
bonds the CDs the high-risk has to be great less than or equal to twelve
thousand dollars high risk that’s one has to be less than or equal to two
thousand the mountain bonds that’s one minus three times c0 for the high risk
has to be greater than or equal to zero bonds has to be greater than or equal to
zero C DS is greater than or equal to zero and high risks has to be greater
than or equal to zero now we’re going to set up our equations in order to
evaluate the algebraic expressions so again we have that the 7% times the
amount invested in bonds and we want that to be a fixed sell so we can copy
it to the constraints plus the the return on the CDs which is the 8% times
the amount invested in CDs and again we go to make that a big sell and lastly
the 12% times the amount invested in the C in the high risk again which is zero
and again we want that to be a fixed sell we notice that that is zero and
then we’re going to copy this down and we want to make sure that it’s correct
so we click on the cell and we notice we have one times zero one times zero one
times zero which is what we want we’ll try one more and we have one times zero
negative 3 times zero and zero times zero so it looks like it’s correct now
we’re going to go to data and we’re going to open up solver in order to be
able to determine how we’re going to solve this particular system of
inequality equations where it says set objective we click on cell K 13 14
because that is where we evaluated the objective function it goes automatically
to maximize and we want to maximize the
values associated with the amount we put in the bonds the CDs and the high-risk
funds and make sure you’re in the correct window so let’s do that again
the objective function is in cell K 14 we’re maximizing go to the window for
the B C and H that’s the zeros and we have subject to our constraints now
looking at the constraints we have the information over here that we are going
to add the constraints so we’re going to add the constraint for the total amount
invested where we have 0 has to be less than or equal to the $12,000 we add the
high risk constraint which says 0 has to be less than or equal to the $2,000 we
add the tax requirement constraint which says 0 has to be greater than or equal
to the 0 that we had there we add the constraints for the the trivial
constraints so we have that the B has to be greater than or equal to 0 we add the
C again has to be greater than or equal to 0 let’s put the box over there and
lastly we want to make sure that we have the H all right and I want to make sure
that it’s in the right place so I have the 20 and this should also be 20 and
lastly we want the H to be greater than or equal to let’s get that correct
greater than or equal to again 0 and we finish we press ok now I want to watch
to make sure that I did everything correctly so I have the K on the left
hand side of the inequality I have the J on the right hand side and we have row
16 and 17 or less than or equal to everything else should be greater than
or equal to I noticed that 20 I have the wrong side
so I click on here I change it and I’m just going to change the sign to greater
than or equal to I press ok now all the constraints look correct
I go to simplex I ask it to solve the equation and we are noticing that we
should invest $7,500 in municipal bonds $2,500 in CDs and $2,000 in the
high-risk account for the maximum return or Rev or profit of nine hundred and
sixty-five dollars

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