Theory and MethodologyGeneration and presentation of nearly optimal solutions for mixed-integer linear programming, applied to a case in farming system design
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 asin 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 Sα of an LP model includes the vectors x that verify (2) andwhere α is a small and positive scalar corresponding to a tolerable deviation from z*. Usually the decision variables are bounded and, in such case, Sα 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 Sα. 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 Sα. 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.
Section snippets
Nearly optimal solutions for mixed-integer programming
An MILP model can be defined as follows:in which x is the n-dimensional vector of the continuous decision variables, ω is the q-dimensional vector of the integer decision variables, scalar z is the objective function, c1 and c2 are respectively the n-dimensional and the q-dimensional vectors that define the coefficients of the objective function, A1 and A2 are respectively a p×n and a p×q matrices of constraint coefficients, and b is a p-dimensional vector of
Generation of optimal and nearly optimal solutions with branch-and-bound methods
Branch-and-bound methods (Mitten, 1970) are commonly used to generate optimal solutions of MILP models. In this section, we show how to use branch-and-bound methods to generate both the optimal and nearly optimal solutions of MILP models. Two approaches are considered: a two-stage approach and a one-stage approach. In the two-stage approach the optimal solution and the nearly optimal solutions are generated successively whereas, in the one-stage approach, both types of solutions are generated
Presentation of optimal and nearly optimal solutions in the space of the continuous decision variables
The algorithms presented in the previous section can be used to generate an optimal solution and a group of nearly optimal solutions (x1,ω1),…,(xi,ωi),…,(xt,ωt). In this section, we describe two approaches to present the generated solutions in the space of the continuous decision variables.
The model
An MILP model, called MGOPT-CROP Schans, 1996, Rossing et al., 1997, is used to apply the methodology outlined earlier. This model calculates crop rotations by optimizing economic or environmental objectives. The results of the model can be used as a basis to design a theoretical prototype of a farming system. The prototype can then be tested experimentally, and, eventually, disseminated.
Before running MGOPT-CROP, the user defines a farm by its soil type, the crops that can be cultivated, and
Conclusions
A methodology has been described to generate, summarise, and present nearly optimal solutions of MILP models. It has been shown that nearly optimal solutions can be generated with branch-and-bound algorithms. Methods have been developed to summarise groups of nearly optimal solutions and to give information on the variability of the continuous decision variables within the set of nearly optimal solutions. All proposed methods can easily be applied with standard software.
The methodology has been
Acknowledgements
This paper contributes to the Design of Farming Systems sub-programme of the DLO Programme Biological Agriculture. The support of P.A.C.M. van de Sanden of Plant Research International (formerly AB-DLO), Wageningen, is gratefully acknowledged.
References (18)
- et al.
Developing alternative farm plans for cropping system decision making
Agricultural Systems
(1998) - et al.
Application of interactive multiple goal programming techniques for analysis and planning of regional agricultural development
Agricultural Systems
(1988) - et al.
Nearly optimal linear programming as a guide to agricultural planning
Agricultural Economics
(1992) - et al.
A mathematical programming model of a crop-livestock farm system
Agricultural Systems
(1986) - et al.
Multiple criteria and nearly optimal solutions in greenhouse management
Agricultural Systems
(1993) Planning of irrigation distribution and applications systems by mixed-integer linear programming
Agricultural Water Management
(1985)The use of optimization models in public-sector planning
Management Science
(1979)- et al.
Modeling to generate alternatives: The HSJ approach and an illustration using a problem in land use planning
Management Science
(1982) - et al.
Nearly optimal linear programming solutions: Some conceptual issues and a farm management application
American Journal of Agricultural Economics
(1987)
Cited by (8)
Assessing farm innovations and responses to policies: A review of bio-economic farm models
2007, Agricultural SystemsDiet models with linear goal programming: Impact of achievement functions
2015, European Journal of Clinical NutritionGlobal agronomy, a new field of research. A review
2014, Agronomy for Sustainable DevelopmentSustainable agricultural development: Knowledge-based decision support
2009, Technological and Economic Development of EconomyOptimal management of agricultural systems
2008, Studies in Computational Intelligence