Case Study:

Sigma Pie Company

Additional Restrictions

MATP4700/ DSES4770 Math Models of Operations Research

Fall 2008

Students may work in groups of up to three people. You may consult only your textbooks, your notes, and me.

Due: Friday, December 5. (20 points)
SigmaPi now want to make the model a better representation of their operations. In particular, they want to make two modifications:

• Fixed charge for orders of apples:
  The linear charge-per-carton purchase cost described in the first part of the project is a simplification of the true cost, which has a fixed charge for any positive order, together with a linear term in the number of cartons of apples purchased. These costs are as follows:

  	Fixed charge	Charge per carton
  Washington	$40000	$300
  Michigan	$10000	$400
  New Zealand	$40000	$100

  Thus, if 250 cartons of apples are ordered from New Zealand in a particular month then the cost is $40000 + 250 $\times$ $100 = $65000.

• Philadelphia over Birmingham:
  SigmaPi has a special agreement with its distributor in Philadelphia regarding generic pies. In particular, in any month, SigmaPi will sell no pies in Birmingham unless it can meet demand in Philadelphia.

Add these restrictions to the model and resolve it. Hand in your new model file, highlighting the changes you made. What is the new optimal value? How many cartons and bushels are bought? What is the unmet demand?

AMPL hints:

• A variable can be declared binary. For example:

  var make {SET} binary;
  

• A variable can be declared integer as well as nonnegative and bounded. For example:

  var make {SET} >= 0, <= 100, integer;
  

• You will probably need to set up a constraint that refers explicitly to the number of generic pies sold in Birmingham in month $t$. This variable can be written in the model file as

  sell['generic','Birmingham',t]
  

  where $t$ runs over the set of months.
• cplex is available for Windows or Mac OSX: see

  http://www.rpi.edu/~mitchj/ILOG11.htm

  It is also available on RCS. Instructions for using it on RCS are available from

  http://www.rpi.edu/~mitchj/cplex11.html

  From a unix machine, use ssh rcs-sun.rpi.edu or ssh rcs-ibm.rpi.edu to access RCS. From a Windows machine, you can use SecureCRT; you need to connect to rcs-sun or rcs-ibm to ensure that cplex and ampl are available.

  Alternatively, you can submit your model to the NEOS server at

  http://www-neos.mcs.anl.gov/neos/

  I would suggest using either the solver cbc or the solver MINTO. Note that you will need to use a commands file if you want to get output that is nicely formatted and easy to interpret.

  A simple commands file is the following:

  solve;
  display buy;
  display sell;