Exercise 3 (MIP heuristics)

1) Fractional Diving (slide 20)
Consider the Capacitated Lot Sizing Problem with Setup Costs in cls.lp. 
Do fractional diving by hand using cplex.

cplex> read cls.lp    // reads problem
cplex> opt            // solves lp relaxation
cplex> dis sol var -  // showes non-zero variable values
cplex> ch bo z4 u 0   // changes the upper bound of varaible z4 to 0, i.e., fixes z4 to 0
                      // use l 1 to fix lower bound to 1, i.e, fix variable to 1
cplex> opt            // resolves lp
cplex> dis sol var -
cplex> ch bo ...

For each step report the fractional varaibles, the variable to fix and the fixed value.
Are any of the variable types better to fix than others? y better than z?
Report the final solution and objective value.


2) Feasibility Pump (slide 26-27)
Run algorithm on slide 26.

Run the cls.lp to get an initial lp feasible point. 
Use the lp solution to construct a rounded integer feasible solution.
Construct lp as described on slide 27.
Note, that there are only binary (y,z) and continuoes (x) variables - no integer.

In each step report the rounded solution and the feasibility pump lp solution.
Report the final solution and the objective value.


3) Local Branching (slide 42-44)
Use the feasibility pump solution to construct the local branching sub-MIP for k=2.

Report the local branch cut, the objective cutoff cut, and the final solution and objective value.


4) RINS (slide 45-46)
Use the feasibility pump solution to construct the RINS sub-MIP.

Report the fixed variables, and the final solution.