Nearly optimal solutions of linear programming models provide useful information when some of the relevant objectives and constraints are not explicitized in the models. This paper presents a three steps framework to study nearly optimal solutions of linear programming models developed for land use exploration. The first step is to define low dimensional vectors called ‘aspects’ to summarize the solutions. The second step is to generate a group of optimal and of nearly optimal solutions. Three methods are proposed for generating nearly optimal solutions. Method i proceeds by minimization of sums of decision variables that are non-zero in the optimal solution and in previously generated nearly optimal solutions. Method ii proceeds by maximization of sums of randomly selected decision variables. Method iii is targeted at searching nearly optimal solutions with very different values for the aspects. Finally, the third step of the framework is to present graphically the values of the aspects of the generated solutions. The framework is illustrated with a case study in which a linear programming model developed for land use exploration at the European level is presented. First, an optimal solution is calculated with the model by minimizing nitrogen loss with constraints 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. Each solution is summarized by three different aspects that represent the allocations of the agricultural area among two regions, among five types of crop rotation, and among five production orientations respectively. Graphical 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 allocated to the north of the European Community varies from 10.9 to 50.2×10 6 ha among the 52 generated nearly optimal solutions, whereas this area is equal to 26.5×10 6 ha in the optimal solution. The comparison of methods i , ii , and iii shows that the solutions generated with method iii are quite more 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 according to issues that cannot be quantified in a model.
Ecological Modelling – Elsevier
Published: Jun 30, 2000
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