Chapter 9. NONLINEARITIES AND APPROXIMATION. MAD Models. Minimize the sum of absolute deviations The deviation is defined as ei = Yi - ∑X ji B j Similar to regression, except we minimize sum of the absolute values of the error term and not the sum of the squares of the error terms. .
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Predict the price based on quantity of oranges
and quantity of juice sold.
Here, I've let the Bj be unrestricted in sign,
because Solver will let me do that. I could
add more columns and define negative Bj.
McCarl and Spreen Section 9.1.3
Co + CjXj
do + djXj
S.T. aijXj le bi
Xj ge 0 and the denominator strictly positive
Yo = [do +djXj]-1
Yj = YoXj
(Xj = Yj/Yo)
Max CoYo + CjYj
s.t. -biYo + aijYj le 0
doYo + djYj = 1
Yo, Yj GE 0
Solve this and do the reverse
transformation to get the X values
1.8 X1 + 1.7 X2
10 + 4X1 + 4.1 X2
s.t. 1.5X1 + X2 le 6
3.0X1 + 4X2 le 20
X1, X2 ge 0
Max 1.8 Y1 + 1.7Y2
s.t -6Yo + 1.5Y1 + Y2 le 0
-20Yo + 3Y1 + 4Y2 le 0
10Yo + 4Y1 + 4.1Y2 = 1
Yo, Y1, Y2 ge 0
X1 = Y1/Yo = 1 1/3
X2 = Y2/Yo = 4
In some cases, the level of return to an
activity is not constant as the activity level
increases. Because of the law of diminishing
returns, returns to additional units of an
enterprise often decrease as more units
level of activity j
level of activity j
Max Profit = R1X1 + R2X2 + R3X3
X1 LE A
X2 LE (B-A)
And X2 and X3 use the same (or higher) levels
of all resources. Because R1 > R2 > R3, X1
will enter solution to its full level A before X2 is
selected and so on.
Totals: A-635; B-100, C-50
This method of solving separable programming
is sometimes called the "delta method."
An alternative, presented in McCarl and
Spreen chapter 9, is sometimes called the
"lambda method" or "grid point approximation."
McCarl and Spreen present a different
application of the gridpoint method in
chapter 9, section 2.1. Also, they
show you how to extend this method
to functions of more than one variable.
The more complicated material won't
appear on the final but you may want
to read it on your own.
McCarl and Spreen, 15.1.6
We will come back to this
material. If you have already
seen integer programming, you
may want to look at it now.
Some functions cannot be approximated
using the delta or lambda methods
because the more "attractive" section
of the function takes place at larger
numbers. If, for example, unit
profits increase as output increases or
costs (in a cost min) decrease as a variable
gets larger, those techniques don't work.
for a max, P0<P1<P1<P3
The Zi are 0-1 integers
Modification of our original 3 product
situation so that the profit from A
increases as output goes up.
We are skipping the intermediate steps.
Profit from 635 units of A is $6200, not
$6254.7 as it would appear from this answer.
Profit from the first activity is now calculated