LINEAR PROGRAMMING UNDER UNCERTAINTY:CONCLUSIONS

CONCLUSIONS

The values used for the parameters of a linear programming model generally are just estimates. Therefore, sensitivity analysis needs to be performed to investigate what happens if these estimates are wrong. The fundamental insight of Sec. 5.3 provides the key to per- forming this investigation efficiently. The general objectives are to identify the sensitive parameters that affect the optimal solution, to try to estimate these sensitive parameters more closely, and then to select a solution that remains good over the range of likely val- ues of the sensitive parameters. Sensitivity analysis also can help guide managerial decisions that affect the values of certain parameters (such as the amounts of the resources to make available for the activities under consideration). These various kinds of sensitivity analysis are an important part of most linear programming studies.

With the help of Solver, spreadsheets also provide some useful methods of performing sensitivity analysis. One method is to repeatedly enter changes in one or more parameters of the model into the spreadsheet and then click on the Solve button to see immediately if the optimal solution changes. A second is to use ASPE in your OR Courseware to systematically check on the effect of making a series of changes in one or two parameters of the model. A third is to use the sensitivity report provided by Solver to identify the allowable range for the coefficients in the objective function, the shadow prices for the functional con- straints, and the allowable range for each right-hand side over which its shadow price re- mains valid. (Other software that applies the simplex method, including various software in your OR Courseware, also provides such a sensitivity report upon request.)

Some other important techniques also are available for dealing with linear programming problems where there is substantial uncertainty about what the true values of the parameters will turn out to be. For problems that have only hard constraints (constraints that must be satisfied), robust optimization will provide a solution that is virtually guar- anteed to be feasible and nearly optimal for all plausible combinations of the actual values for the parameters. When dealing with soft constraints (constraints that actually can be violated a little bit without serious complications), each such constraint can be replaced by a chance constraint that only requires a very high probability that the original constraint will be satisfied. Stochastic programming with recourse is designed for dealing with problems where decisions are made over two (or more) stages, so later decisions can use updated information about the values of some of the parameters.