ILOG CPLEX 11.0 User's Manual

The model is now restored to feasibility, and the new optimum has an overall cost of 310, atangible improvement of 25 over the previous optimum, using employees {2,3,5,6,8,10};employee 7 has been removed in favor of employee 8. Is that enough monetary benefit tooffset whatever reasons there were for separating employees 4 and 8? That is not a decisionthat can be made here; but at least this model provides some quantitative basis towardmaking that decision. Additionally, a check of the service variable shows that its solutionvalue is back up to 3.2, a further benefit from relaxing constraint c9. Perhaps this decisionshould have been made sooner, the first time constraint c9 appeared in a conflict.The solution of 310 could be investigated further by changing the upper bound of cost to be305, for example. The conflict resulting from this change consists of the skills constraintplus the constraint requiring at least one manager on duty. At this point, the analysis hasreached a conclusion, unless management or the model formulator wishes to challenge thepolicy.More about the Conflict RefinerPresolve proved the infeasibility of that simplified example in A Model for the ConflictRefiner on page 394. However, a minimal conflict can be refined from an infeasible modelregardless of how the infeasibility was found. The infeasibility may have been proven bypresolve, by the continuous optimizers, or by the mixed integer optimizer.A minimal conflict on a nontrivial model can take longer to refine than the associatedoptimization algorithm would have taken either to prove the infeasibility or to solve a similarmodel instance that was feasible. One reason that refining a minimal conflict may takelonger is that multiple passes (that is, iterations) are performed; each iteration solves asubmodel to decide its feasibility status. Another reason is that even if the full model isquickly detected to be infeasible, the infeasibility of the submodels may be less blatant andthus require more time to analyze. It can happen that a number of refinement iterationsproceed quickly, and then suddenly no further progress is seen for quite a long time. Thisdeceptive appearance of lack of progress may be especially noticeable in the case of mixedinteger models, where a proof of infeasibility may become a quite difficult mathematicalproblem.If the user sets a resource limit, such as a time limit, an iteration limit, or node limit, forexample, or if a user interrupts the process interactively, the conflict that is available at thattermination will be the best (that is, the most refined) that was achievable at that point. Evena nonminimal conflict may be more useful than the full model for discovering the cause ofinfeasibility. The status of a bound or constraint in such a nonminimal conflict may beproved, meaning that the conflict refiner had sufficient resources to prove participation ofbound or constraint in the conflict, or the status may be possible, meaning that the conflictrefiner has not yet proven whether the bound or constraint is necessarily part of a minimalconflict.ILOG CPLEX 11.0 — USER’ S MANUAL 401