Optimization models do not perfectly represent the real world. Therefore the mathematically optimal solution is not necessarily the "best" decision to implement. Water resources planning models tend to have many solutions that are nearly optimal in terms of objective function value, but which are significantly different decisions. Methods for generating nearly optimal solutions to linear programming models have been devised. The use of such methods is illustrated with an example problem involving the sizing of reservoirs and hydroelectric power plants and demands for municipal water supply and irrigation. The results show that the interactions among the decision variables and the economics of the system determine the range of choice among nearly optimal decisions. The examination of these interactions yields insight into the basic structure of the decision problem. models generally yields more insight into the solution method than into the decision problem itself. This view appears to be fairly widely held [Rogers et al., 1976]. It follows that a more complex model may in fact be less relevant to the real world than a simpler model. This is true not only because the model may not reflect reality, but also because it is too complicated for the decision maker to understand. Haith [1982] observes , *The percentage away from the optimal objective function value is shown m the parentheses under the column headings.
Nearly optimal solutions in linear programming provide useful information to decision makers. Modeling to generate alternatives may be used to generate a set of nearly optimal solutions from which a decision maker may select the desired solution by considering criteria not quantified in the model. The mathematical problem is to find vertices of a convex polytope. A pivoting method of vertex enumeration is used to generate all extreme-point nearly optimal solutions of an example problem involving selection of a marketing strategy for beef calves. Compared to the optimal solution, nearly optimal solutions have more diversity or use less cash or hired labor.Key words: farm management, linear programming, modeling to generate alternatives, nearly optimal solutions. During the last three decades, linear programming (LP) has been used extensively in agricultural economics research and extension. But extensive use does not alter the fact that the LP model is a simplification of reality. The well-known assumptions of additivity, linearity, divisibility, finiteness, and single-value expectations (Heady and Candler) are used to reduce complex real-world situations to mathematical formulations which can be optimized using the simplex method. Typically, LP solvers report only one optimal solution, and the number of decision variables included in a solution cannot exceed the number of mathematical constraints.Because of these limitations, considerable interest has been expressed in the informational value of multiple optimal or nearly optimal LP solutions (NOS). Appreciation is expressed to Larry Padgett. Muhammad Bari. and Jong-i Perng for computer assistance: to Margaret Burton for computer and editorial assistance: and to Anwarul Hoque. Robert Jack. Dennis Smith. Bryan Schurle. Cleve Willis. and an anonymous Journal reviewer for helpful comments on earlier manuscripts. ematics of finding extreme point NOS of an LP model. The usefulness of information provided by NOS is illustrated using a model designed to evaluate management strategies for small, family farms. Single Objectives, Multiple Objectives, and Modeling to Generate AlternativesThe rationale for generating NOS is seen by focusing on the objective function in modeling decision problems. The objective function formalizes the criterion for ranking alternative decisions. For multiple criteria, one can formulate multiple objectives and perform multiobjective programming using methods such as goal programming and generating techniques (see Willis and Perlack 198ra, b).In some situations, the relevant criteria are difficult to quantify. While multiobjective programming may seem appropriate, the practical difficulties of quantifying criteria such as risk aversion, environmental impacts, and the like may limit the usefulness of model results. To cope with such problems, researchers in water resources planning have developed a set of techniques known as "modeling to generate alternatives" (MGA) (e.g., Brill; Chang, Brill, and Hopkins; Hopkins, Brill, and Wong). Other resear...
is a professor of mechanical engineering at Geneva College and has sponsored student teams in competitions since 1994. He has also enjoyed teaching the freshman-level introduction to engineering course, helping students get hands-on experience with a variety of projects over the past 12 years. He loves the outdoors, including hiking, biking, canoeing, and camping, and is blessed to have friends and family that share that love.
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