Case Study:

Sigma Pie Company Sensitivity Analysis

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: Tuesday, November 4. (40 points)
SigmaPi's model included various restrictions on its supply chain and estimated values for several parameters. It would like to know how the schedule found in Part 1 depends on the precise values of these restrictions and parameter choices. Use sensitivity analysis to answer the following questions. Answer them without re-solving the linear program. Include printouts of relevant parts of your AMPL output. Assume that each part is independent of the others. Procedures for obtaining shadow prices and other relevant information are given below. If you prefer to use them, my model and data file are available on the course website.

Note: You may want to solve a modified problem to check your answer, or to try to determine where to look for sensitivity information. That is OK, but I don't want to see those results! You must give me the information using only sensitivity information.

1. The optimal schedule sells generic apple pies in only six of the seven distribution centers in Week 5. What is the reduced cost for generic pies in this week in the seventh distribution center? What does this imply?
2. If the required safety stock for apples in Sacramento was reduced to 18 cartons, how would the optimal value be changed? (You may assume that the set of basic variables in the optimal solution does not change.)
3. SigmaPi is considering reducing the price of name-brand apple pies in Birmingham in week 4 as a promotion. How far can they reduce the price without the set of basic variables in the optimal solution changing?
4. What would be the benefit to SigmaPi of being able to purchase an extra 10 cartons of apples each week from their supplier in Washington, for the same unit purchase and shipping costs? (You may assume that the set of basic variables in the optimal solution does not change.)
5. Now consider increasing the supply of apples in Washington to 500 cartons per week. Would the increase in profit be 25 times as large as that in Question 4? If not, do you think the improvement would be more or less than 25 times as large? Justify your answer.
6. SigmaPi is considering hiring some temporary workers in order to expand the capacity of the factory in Buffalo. It can hire a crew of workers for a total of $90000 for the six weeks, and this crew would increase capacity by 50 cartons per week. Is it a good idea for SigmaPi to expand capacity in this way?
7. Now consider making the change described in Question 6 at the factory in Sacramento instead. Is this worthwhile? (You may need to resolve a slightly modified problem for this question. If so, justify why.)
8. A customer wishes to place an order for an additional 100 boxes of name-brand apple pies in Week 3 in Fresno. These pies would be in addition to the usual demand for 2000 pies. SigmaPi would like to reduce the price slightly for special orders of this type. What is the minimum amount that SigmaPi should be willing to accept? Explain why this is so. (Note: On this part, it is advisable to check your answer by modifying the data file.)

AMPL hints:

• To find the value of a dual variable corresponding to a constraint called fabcap, type

  display fabcap;

  To find the slack in the constraint, type

  display fabcap.slack;

• To find the reduced cost of a variable called SELL in the optimal tableau, type

  display SELL.rc;

• If you want to read in a new model file chips.mod and data file chips.dat, you can use the reset command:

  ampl: reset;
  ampl: model chips.mod;
  ampl: data chips.dat;

• If you want to reset the whole data file and read in a new data file chips.dat, but you want to keep the model file as before, you can type

  ampl: reset data;
  ampl: data chips.dat;

• If you want to change one parameter, you can use the let command as follows:

  $(OS)~$ampl
  ampl: model sample/steel.mod;
  ampl: data sample/steel.dat;
  ampl: solve;
  MINOS 5.4: optimal solution found.
  2 iterations, objective 192000
  ampl: let rate["bands"]:=250;
  ampl: solve;
  MINOS 5.4: optimal solution found.
  1 iterations, objective 217200

  This changes the rate for bands and then resolves the problem.
• One way to find bounds on parameters in AMPL is to use the cplex option of sensitivity. This will let you find ranges in which parameters can be changed without the set of optimal basic variables changing. First make sure CPLEX is the solver:

  ampl: option solver cplex;

  Once you've established cplex as the solver, enter the following command:

  ampl: option cplex_options 'sensitivity';

  Solve the problem again, and then you can then use the suffices up and down on variable names, as described for reduced costs and slacks.

• All the variables and their reduced costs can be displayed simultaneously by using the command

   
  display {j in 1.._nvars} (_varname[j],_var[j],_var[j].rc);
  

  (See page 249 of the AMPL text.)

  All the constraints and their shadow prices can be displayed simultaneously by using the command

  display {j in 1.._ncons} (_conname[j],_con[j]);
  

  (Again, see page 249 of the AMPL text.)

  These commands can be used with the other suffices available when using the cplex sensitivity option.

• You can use sum and other AMPL commands after solving the problem to find values of various combinations of the primal and dual variables and the parameters.