Generation and presentation of nearly optimal solutions for mixed-integer linear programming, applied to a case in farming system design

Abstract

In complex decision problems, some objectives are not well quantified or are not introduced explicitly in optimization models. In view of this inherent limitation of models, solutions that are nearly optimal, i.e. deviating less than a predefined percentage from the optimal value of the quantified objective functions, can constitute interesting alternatives for stakeholders. The aim of this paper is to extend the analysis of nearly optimal solutions to mixed-integer linear programming (MILP) models. Two branch-and-bound algorithms are described to generate nearly optimal solutions, and methods are discussed to summarise and to present nearly optimal solutions. A case study is introduced with an MILP model developed to design farming systems in the Netherlands. This case study shows that the presentation of nearly optimal solutions provides relevant information to resolve complex decision problems.

Introduction

Linear programming models may be used to resolve decision problems that involve one or several objective functions. When the decision problem involves only one objective function, an optimal solution can be calculated with a simplex or a branch-and-bound algorithm. When the problem involves several objectives and when all the objectives are quantified, an optimal solution can be calculated by using multiple criteria optimization methods like goal programming, compromise programming, or the Interactive Multiple Criteria Decision Making approach Romero and Rehman, 1989, De Wit et al., 1988. A problem is more difficult to solve if some of the relevant objectives are not explicit or cannot be quantified because of lack of knowledge. In such cases, it is still possible to derive an optimal solution by optimizing the quantified objective functions (for instance with goal programming). However, the calculated optimal solution does not represent necessarily the best solution for the stakeholder. The true optimal solution is likely to lie in the inferior region with respect to the quantified objectives Brill, 1979, Jeffrey et al., 1992.

Brill (1979) suggested to use optimisation models to generate alternative solutions that are nearly optimal with respect to the objectives included in the model and different with respect to the decision variables. Presentation of nearly optimal solutions provides stakeholders with more choice than the mere calculated optimal solution and allows stakeholders to choose a solution according to their own preferences.

Several authors have described the generation of nearly optimal solutions of linear programming (LP) models (Brill et al., 1982, Burton et al., 1987, Jeffrey et al., 1992, Willis and Willis, 1993, Abdulkadri and Ajibefun, 1998, Makowski et al., 2000). An LP model can be defined as

$$\text{max}\text{{z=c}{}^{\mathrm{\prime }}\text{x}}$$

$$\text{Ax⩽b,}$$

in which x is the n-dimensional vector of the continuous decision variables, scalar z is the objective function, c is the n-dimensional vector of the coefficients that define the objective function, A is an p×n matrix of constraint coefficients, and b is a p-dimensional vector of right-hand side coefficients. The optimal solution, x^*, and the optimal value of the objective function, z^*, are obtained by solving this model. The set of nearly optimal solutions S_α of an LP model includes the vectors x that verify (2) and

$$\text{z⩾(1−α)z}{}^{\mathrm{\ast }}\text{,}$$

where α is a small and positive scalar corresponding to a tolerable deviation from z^*. Usually the decision variables are bounded and, in such case, S_α defines a polytope. Vertex enumeration methods (Matheiss and Rubin, 1980) can be used to generate the nearly optimal solutions corresponding to the extreme points of S_α. However, when the decision variables and constraints are numerous, a vertex enumeration is intractable in practice (Burton et al., 1987). A preferred approach for large LP models is to generate a small group of extreme points of S_α. Several methods for doing this have been developed Brill et al., 1982, Harrington and Gidley, 1985, Makowski et al., 2000.

After having generated nearly optimal solutions, the choice of how to present them to the stakeholders is an important problem. Stakeholders need an overall view of the generated nearly optimal solutions to make their choice. When the decision variables are numerous, the generated nearly optimal solutions cannot be presented directly because each solution is equivalent to a high dimensional vector. Makowski et al. (2000) proposed to summarize nearly optimal solutions of LP models by projection into low dimensional spaces. The authors suggested to define these spaces by using the indices of the decision variables. The different values of a certain index are used in this approach to define a space with a relatively low dimension into which the generated solutions can be projected. The result of a projection of a solution is a low dimensional vector which represents a particular “aspect” of the solution and which can be conveniently presented in a graph or in a short table.

The aim of this paper is to extend the analysis of nearly optimal solutions to mixed-integer linear programming (MILP). MILP models are often used for decision support, notably in problems related to agriculture Yoo, 1985, Morrison et al., 1986, Schans, 1996, Rossing et al., 1997. We first introduce new notations for convenience in Section 2. A methodology to generate, summarise and present nearly optimal solutions for MILP is given in 3 Generation of optimal and nearly optimal solutions with branch-and-bound methods, 4 Presentation of optimal and nearly optimal solutions in the space of the continuous decision variables. Finally, the methodology is applied in Section 5 to an MILP model developed to design farming systems in the Netherlands.