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

Nearly optimal solutions of linear programming models provide useful inform ation when some of the relevant objectives and constraints are not explicit ized in the models. This paper presents a three steps framework to study ne arly optimal solutions of linear programming models developed for land use exploration. The first step is to define low dimensional vectors called 'as pects' to summarize the solutions. The second step is to generate a group o f optimal and of nearly optimal solutions. Three methods are proposed for g enerating nearly optimal solutions. Method i proceeds by minimization of su ms of decision variables that are non-zero in the optimal solution and in p reviously generated nearly optimal solutions. Method ii proceeds by maximiz ation of sums of randomly selected decision variables. Method iii is target ed at searching nearly optimal solutions with very different values for the aspects. Finally, the third step of the framework is to present graphicall y the values of the aspects of the generated solutions. The framework is il lustrated with a case study in which a linear programming model developed f or land use exploration at the European level is presented. First, an optim al solution is calculated with the model by minimizing nitrogen loss with c onstraints on area, water use, product balances, and manure balances. Then, 52 nearly optimal solutions are generated by using methods i, ii, and iii with a deviation tolerance of 5% from the optimal Value of nitrogen loss. E ach solution is summarized by three different aspects that represent the al locations of the agricultural area among two regions, among five types of c rop rotation, and among five production orientations respectively. Graphica l presentation of these aspects and principal component analysis show that nearly optimal solutions can be very different from the optimal solution in terms of land use allocations. For example, the agricultural area allocate d to the north of the European Community varies from 10.9 to 50.2 x 10(6) h a among the 52 generated nearly optimal solutions, whereas this area is equ al to 26.5 x 10(6) ha in the optimal solution. The comparison of methods i, ii, and iii shows that the solutions generated with method iii are quite m ore contrasted than the solutions generated with methods i and ii. The case study presented in this paper illustrates how our methodological framework can be used to allow a stakeholder to select a satisfactory solution accor ding to issues that cannot be quantified in a model. (C) 2000 Elsevier Scie nce B.V. All rights reserved.