Three products can be produced at two machining centers. The products
may be produced in fractional amounts. The linear relationships
describing this situation are listed below. The variables are:
A, B and C are the amounts of the three products in units.
R_{1} and R_{2} are the amounts of raw materials
used in kilograms.
T_{1} and T_{2} are the times used in the
two machining centers.
Linear expressions describing important quantities associated
with the products have been derived and are shown below.
Profit:

P = 2A + 3B + 2.55C  0.6R1  0.8R2

Time required on machine 1:

T_{1 }= 0.5A + 0.8B + 1C (hours)

Time required on machine 2:

T_{2 }= 0.8A + 0.6B + 0.2C (hours)

Raw material 1 used:

R_{1} = 0.1A + 0.2B + 0.075C (kilograms)

Raw material 2 used:

R_{2} = 0.05A + 0.1B + 0.05C (kilograms)

Market Limits:

A _ 100, B _ 200, C _ 100.

The linear programming model when objective is to maximize
profit is:
Max Z = 2A + 3B + 2.5C  0.6R1  0.8R2
subject to:
Machine 1:

0.5A + 0.8B + 1C _ 100 (hours)

Machine 2:

0.8A + 0.6B + 0.2C _ 100 (hours)

RM 1

0.1A + 0.2B + 0.075C  R_{1} = 0

RM 2:

0.05A + 0.1B + 0.05C  R_{2} = 0

Market:

A _ 100, B _ 200, C _ 100.

Nonnegativity: A _ 0, B _ 0, C_ 0, R_{1} _ 0, R_{2}
_ 0.
The following paragraphs describe modifications of the situation.
The modifications are not cumulative. Show the changes in the
model necessary to describe the new situation. Some changes
require the introduction of integer variables while others require
the incorporation of nonlinear functions. In either case add
as few variables as possible.
When the model uses integer variables it must have the linear
form.
When the model is a nonlinear program, indicate whether the
Excel solver would find a global optimum. Justify your conclusion.
a. The revenue for each product is reduced by an advertising
cost as illustrated in the figure for product A. In order to
sell any A, $40 must be expended. If no A is sold, the cost
is not expended. Product B and C have similar advertising costs,
$5 for B and $60 for C.
b. The raw material costs are nonlinear in the following fashion.
Raw Material R1: The unit cost for the first 50 kilograms is
$0.60 per kg.. The unit cost for amounts greater than 50 but
less than 100 is $0.70 per kg. The unit cost for purchases above
100 kg. is $0.75 per kg..
Raw Material R2: The unit cost for the first 50 kilograms is
$0.80 per kg.. The unit cost for amounts greater than 50 but
less than 100 is $0.75 per kg. The unit cost for purchases above
100 kg. is $0.70 per kg..
c. Add the logical restriction that if you produce more than
20 of product A, then you must produce at least 5 of product
B.
d. We specify that you can produce at most two of the three
products.
e. The revenue is a nonlinear function of A, B and C.
The revised objective function expressing the profit is given
below.
Z = 0.01A^{2 }+ 2A  0.02B^{2 }+ 3B
 0.03C^{2} + 2.5C + .01AB + .03BC  0.6R1  0.8R2
f. The time required on machine 1 is a nonlinear function
of the amounts produced. The time used must still be less
than 100 hours
g. The costs of the raw materials are nonlinear functions
of the amounts purchased.
Cost of Raw Material 1: 0.06R_{1} + 0.001R_{1}^{2}.
Cost of Raw Material 2: 0.08R_{2}  0.002R_{2}^{2}
