Write an AMPL model file and a data file for the LP formulation of BT-ILO Proble

Question

Write an AMPL model file and a data file for the LP formulation of BT-ILO Problem 1.16 (BT-ILO pages 37–38, discussed in Homework 2). Solve the LP in AMPL, and record the optimal solution and the optimal objective function value. Your submission must include the model and data files, as well as the output from the AMPL run.

A manager of an oil refinery has 8 million barrels of crude oil A and 5 million barrels of crude oil B allocated for production during the coming month. These resources can be used to make either gasoline, which sells for $38 per barrel, or home heating oil, which sells for $99 per barrel. There are three production processes with the characteristics in table 1. All quantities in the table are in barrels. For example, with the first process, 3 barrels of crude A and 5 barrels of crude B are used to produce 4 barrels of gasoline and 3 barrels of heating oil. The costs in this table refer to variable and allocated overhead costs, and there are no separate cost items for the cost of the crudes. Formulate (and solve with AMPL) a linear programming problem that would help the manager maximize net revenue over the next month.

Explanation / Answer

We introduce variables x1, x2 and x3 for the three production processes.
Since the amount of crude A is limited by 8 million barrels, we get the constraint
   3x1 + x2 +5x3 8.
Similarly, we get the constraint
   5x1 + x2 +3x3 5
for crude B.
The net revenue (in million $) is given as
   38 ·(4x1 + x2 +3x3)+99 ·(3x1 + x2 +4x3)51x1 11x2 40x3.
This can be simplified to
398x1 +126x2 +470x3

Thus the linear program is given as:
max ,Z=398x1 +126x2 +470x3

subject to:

3x1 + x2 +5x3 8
5x1 + x2 +3x3 5
x1,x2,x3 0

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