The figure below describes a production process that makes
three products, A, B and C. The numbers at the top of the figure
provides the maximum sales and unit revenues for the three products.
The numbers at the bottom indicate the raw materials used and
the unit costs of the raw materials.
The network structure shows the processing requirements for
the products. The nodes represent operations the products must
pass through and the arcs represent the inputs to the operations.
Product A requires one unit from operation 1. Product B requires
one unit from operation 2. Product C requires one unit from
operation 3.
The product at operation 1 is made from one unit passing out
of operation 4. The product at operation 2 is made from one
unit each of the products passing out of operations 4
and 5. The product at operation 3 is made from one unit passing
out of operation 5.
A unit at operation 4 is made from one unit each from operations
6 and 7. A unit at operation 5 is made from one unit each from
operations 7 and 8.
Finally, operation 6 requires one unit of raw material D. Operation
7 requires one unit of raw material E. Operation 8 requires
one unit of raw material F.
The numbers adjacent to the nodes are the operation times
in minutes. The operations use time on Machine 1, Machine 2
and Machine 3. Each machine has a weekly capacity of 2400 minutes.
The products share the capacities of the machines. For example,
if 100 units of each product were produced, 1200 minutes would
be used on machine 1. Note that product B uses 10 minutes of
machine 2 because both operations 4 and 5 are needed for one
unit of B.
a. Using a linear programming model to find the production
quantities that maximize profit. Production is limited by the
time available on the three machines.
b. Using the dual variables (shadow prices), determine what
aspects of this situation are limiting the company's profit?
Explain your answer.
c. A new machine of type 2 with 2400 minutes of capacity
can be purchased. If the machine costs $1600 per week will it
be justified? Use the sensitivity results to explain your answer.
d. An additional market of 50 units of B is available if
the revenue of B is reduced to $19 for this market. The original
market of 100 units still yields a unit revenue of $21. Change
the model to incorporate the new information and solve.
e. The plant may remain open for additional time beyond
the 2400 minutes in the week. A premium of $0.25 per minute
is charged for any machine that operates beyond the 2400 minutes.
A premium of $0.30 per minute is charged to keep the plant open
beyond 2400 minutes. The plant must be kept open if one or machines
are in operation.
f. The company adopts two goals. The first is that each
machine should have no more than 200 minutes of idle time. The
second priority goal is to maximize profit. Describe a model
that will give this the answer with one solution by a linear
programming algorithm.
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