Nearly Optimal Linear Programming Solutions: Some Conceptual Issues and a Farm Management Application

Abstract

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...

A framework to study nearly optimal solutions of linear programming models developed for agricultural land use exploration

“…Methods have been developed to enumerate all the extreme points of a polytope (Mattheiss and Rubin, 1980) but, when the number of decision variables is too high, the calculations are intractable. As a rule of thumb, Burton et al (1987) fixed the upper limit on model size for a complete enumeration of the extreme points at 50 decision variables and 50 constraints. Thus, in practice, one can generate only a part of the possible extreme nearly optimal solutions.…”

Nearly optimal linear programming as a guide to agricultural planning

“…The MGA approach has been used by civil engineers in water resource planning problems (Chang, Brill and Hopkins, 1982;Harrington and Gidley, 1985). More recently, MGA has been used in a farm management application to select a marketing strategy for calf producers (Burton et al, 1987).…”

“…A number of enumeration algorithms have been developed to identify the extreme points on a convex polytope (Dyer and Proll, 1977;Dyer, 1983). Burton et al (1987) also provide a method of computing the number of possible extreme points for a given problem. An enormous number of alternative solutions can be generated for most empirical problems using these types of procedures.…”

The North American Hard Wheat Milling Industry Under Free Trade: Spatial Equilibrium and Nearly Optimal Solutions

“…MGA has been used by civil engineers for water resource planning problems iChang, Hopkins 1982: Harrington andGidley 1985) and more recently in a farm management application to select a marketing strategy for calf producers (Burton et al 1987). The MGA approach is based on the premise that modeling should be a tool of the decision maker rather than a replacement for the decision maker (Gidley and Bari 1986).…”