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

Due: Friday, November 12. (40 points)
DDD’s model included data for responses to 40 compounds, and values for two parameters C and ϵ. It would like to know how the solution found in Part 1 depends on the precise values of the proportions and responses for the compounds, and on the 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 run 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 analysis.

1. Let C = 0.1, ϵ = 0.05, and use all 40 samples to find w and b. What do you estimate the optimal value to be if y₁₄ = -11.509? What if y₁₄ = -11.5? Now set y₁₄ = -11.5 and solve the problem; you should get an answer different from your estimate — how do you explain this?
2. Let C = 0.1, ϵ = 0.05, and use all 40 samples to find w and b. What do you estimate the optimal value to be if x_1,1 = 5.351?
3. Let C = 0.1, ϵ = 0.05, and use all 40 samples to find w and b. Assume C is increased to 0.101. What is your estimate of the change in the optimal value?
4. Let C = 0.1, ϵ = 0.05, and use all 40 samples to find w and b. Assume ϵ is increased to 0.051. What is your estimate of the change in the optimal value?
5. Let C = 0.1, ϵ = 0.05, and use all 40 samples to find w and b. DDD have tested a new compound containing each factor with a proportion value x_ij equal to 5.5.
   1. The first time DDD tested the compound, the response was y = -10.6. Use sensitivity analysis to show that this extra result does not change the optimal solution.
   2. DDD measured the response again and got a value of y = -10.7. Use sensitivity analysis to show that this may result in a change to the optimal solution. If the optimal value changes, will it increase or decrease?
6. Let C = 0 and ϵ = 0.3. Let the training data be samples 1,…, 20, and so the training problem then has optimal value 0. Can you use sensitivity analysis to conclude that there are alternative optimal solutions to the problem?

   Can you construct an argument based on degrees of freedom? (To keep the optimal value equal to zero with a small change Δw, Δb, what must the changes in the errors ν_i satisfy?)

7. Let C = 0 and ϵ = 0.3, and use all 40 samples to find w and b. The problem then has optimal value 0. It may be possible to decrease ϵ and still obtain an optimal value of 0. Can you use sensitivity analysis to get a lower bound on how far ϵ could be decreased while still getting an optimal value of 0?

   How does your answer change if you use an interior point method to solve the linear progam, which gives a solution in the center of the face of optimal solutions? To use the interior point solver in cplex, enter the commands:

   ampl: option cplex_options ’baropt’;
   ampl: option cplex_options ’crossover = 0’;

   and then solve the problem.

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.