Linear Programming Exercise

An aluminum supplier has been asked to examine the economics of reprocessing recovered automobile aluminum to product aluminum for the next generation of automobiles - in effect, to examine the economics of "closed loop recycling."

The materials available for producing the aluminum required for a new automobile are as follows (Note that Cast 2 and Wrought 1 are recovered from an old automobile):

Raw Materials
Availability
Scrap
Cost
Si Content
Fe Content
Cu Content
Mg Content
Cast 2 120
Yes
$0.90 17.600% 1.475% 5.750% 0.670%
Wrought 1 80
Yes
$1.00 0.900% 0.740% 0.880% 0.800%
Primary 1 1000
No
$1.10 0.500% 0.500% 0.500% 0.500%
Primary 2 1000
No
$1.50 0.100% 0.100% 0.100% 0.100%
Mg 1000
No
$3.00 1.000% 1.000% 1.000% 97.000%
Cu 1000
No
$2.00 1.000% 1.000% 97.000% 1.000%
Si 1000
No
$2.00 97.000% 1.000% 1.000% 1.000%
Fe 1000
No
$0.50 1.000% 97.000% 1.000% 1.000%

The materials to be produced are as follows (note that the minimum production requirements constitute the materials absolutely required for the production of the next generation automobile. Production beyond that amount is limited by the size of other markets for the materials):

Alloy  
Production Requirements
Sales Price
Product Composition Specifications
Si
Fe
Cu
Mg
Al-380* Min 0
$1.17
7.50% 0.00% 3.00% 0.25%
Al-380* Max 200 9.50% 2.00% 4.00% 0.50%
Al - 390 Min 120
$1.10
16.00% 0.00% 4.00% 0.45%
Al - 390 Max 200 18.00% 1.30% 5.00% 0.65%
Al - 6111 Min 100
$1.25
0.60% 0.00% 0.50% 0.50%
Al - 6111 Max 200 1.10% 0.40% 0.90% 1.00%

  1. Assuming that all of the scrap materials must be consumed, what is the cost minimizing production plan? (i.e., how much of each resource is required, how much of each product is produced, and how much does it cost to make the output)
  2. Suppose there were no constraint on scrap consumption. How would your production plan change?
  3. By how much does the production cost change compared to part "a" if you must produce one unit of alloy 380*?
  4. By how much will your costs change if the minimum production of Al-6111 is increased by 1 unit? By how much will your costs change if the minimum production of Al-390 is increased by 1 unit?
  5. By how much will your costs change if you are only obliged to consume 95% of the total mass of scrap material available?
  6. Suppose that you decide to solve the problem as a profit maximizer rather than a cost minimizer. What is the profit maximizing production plan? Is your result different than in part a)? Why or why not?
  7. Given the profit maximizing production plan, examine the shadow prices on the composition constraints for the output alloys. What do their differences imply about the development strategies for new aluminum alloys? Which solution would you use to guide future alloy development?
Note: Please review the following sites for tools that can be used to solve this problem. For Excel users, the demonstration version of What's Best is probably the best solution. MIT users may also wish to check out resources available on Athena. Finally, if you have an old enough version of Excel or Lotus 123, you may have a "solver" already.

http://www-fp.mcs.anl.gov/otc/Guide/SoftwareGuide/Categories/linearprog.html

http://web.mit.edu/afs/athena.mit.edu/org/c/cwis/computing.html