Multiple Optimal Solutions in Linear Programming Models: Reply

McCarl and Nelson (MCN) give me the opportunity to reaffirm the importance of dealing properly with multiple optimal solutions in linear programming (LP) and to clarify several misconceptions that still linger on with regard to this topic. McCarl, who is the author of the only empirical work to be found in the literature on multiple optimal solutions, takes issue with my statement that "all empirical studies using LP have neglected the consequences of this aspect of mathematical programming." Because I cited this rare article in the original note, my statement is not contradictory. Although MC discusses multiple optimal solutions, his procedure is intended to circumvent what he perceives as a difficulty, rather than the opportunity to improve the realism, of the LP solution. In the context I proposed, a researcher should not avoid multiple optimal solutions (as, by means of perturbation procedures, McCarl suggested in his note), but rather seek them out as a measure of the informational content of his empirical problem. While I can easily emend my statement to reaccount for McCarl's work, I find it more difficult and indeed impossible to accept MCN's proposition that, "Actually, there have been many papers which deal with this situation." One of their quotes is Schurle and Erven, who investigate the sensitivity of risk frontiers by recomputing the solution after excluding an enterprise activity. This peculiar procedure may seem of interest to MCN as a way to gain some insight of the empirical problem. For one good reason, however, I would not classify this paper under the heading of multiple optimal solutions in LP. In fact, by eliminating one activity at a time, the structure of the various problems solved by Schurle and Erven is not the same. Indeed, these authors exhibit an attitude toward multiple optimal solutions which is similar to that of MCN. They write: "The usefulness and uniqueness of frontiers as a decision aid would be reduced substantially if these near optimal solutions include substantially different enterprise combinations" (p. 506). I can easily agree that in the presence of alternative quasi-optimal solutions, the uniqueness of frontiers may be destroyed. But there is no reason to think that the usefulness of those frontiers as decision aids will be diminished when alternative solutions include different enterprise combinations. Personally, I take the opposite view: alternative optimal solutions with substantially different enterprise combinations convey a great deal of information about the problem's structure. One only needs to adapt the procedure properly for decision making to account for this richness of information. Surprisingly, MCN suggest that multiple objective and goal programming are methods to study alternative optimal solutions in LP. I must disagree since multiple objective programming is a rather different animal from LP, and goal programming (when linear) may or may not exhibit multiple optimal solutions. Furthermore, MCN have cited LP bibliographies but, except for the McCarl's note I originally quoted, they have failed to point out another paper dealing empirically with multiple optimal solutions in LP.