A distance function to support optimized selection decisions

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Abstract

Decision-makers often want to see a diverse collection of good solutions in addition to a solution that is in some mathematical sense an optimal solution to a problem. The purpose of the objective function is to quantify the notion “good”, while the purpose of this paper is to exhibit a suitable function for quantifying the notion “diverse”. We focus on the case where important aspects of the solutions are best represented as matrices or sets of vectors, such as when the solution involves selections. We establish a distance function and its connections with related distance functions used in optimization and psychology. A real-world application illustrates its use for decision support.

Introduction

Optimization problem formulations must be an abstraction from the complexities of the real world. Consequently, decision-makers often want to see a diverse collection of good solutions in addition to a solution that is in some mathematical sense an optimal solution. For example, if the decision-maker has been shown a few trial solution vectors (perhaps including an optimal solution) and there are a few more good solution vectors that could be shown, the next solution to be displayed must be selected based on some criteria. The solution with the second best objective function value may well be extremely similar to one that has already been displayed and hence would waste the time of the decision-maker. It would be better to find a good solution that would diversify the set that has been displayed. The purpose of the objective function is to quantify the notion “good”, while the purpose of this paper is to exhibit good dissimilarity functions for quantifying the notion “diverse”.

In some settings, well-known families of norms, such as Lk, can be used or extended to provide the basis for a metric [13]. When the solution vectors, or important portions of them, are binary then Hamming distances or similar metrics are sometimes appropriate. These methods provide metrics or dissimilarity functions in a space of numeric vectors.

In contrast, here, we focus on the case where important aspects of the candidate solutions are best represented as matrices or sets of heterogenous vectors. This is the case, for example, when part or all of the problems is to assign values to binary variables indicating the selection of an object. The objects available for selection have attributes that are important for considering the differences between solutions.

An example of this situation is provided by the effort to develop decision support technology for selection of highway construction projects in Norway. For this problem, there is a set of hundreds of potential highway projects each of which has 50 or 60 attributes that define and describe the project. Many of the attributes are real valued, but some are categorical. We can represent this set of n potential projects each described by p attributes as an n×p matrix. A solution to the problem of selecting the optimal set of projects can be represented by a binary n-vector where each element indicates whether the corresponding project is included in the solution. A more descriptive solution representation is a matrix containing the attributes of the selected projects. In Section 6, we provide details. This practical example is concerned with multiple criteria optimization (MCO) and illustrates important uses of a distance function.

Readers familiar with the MCO literature will notice this paper presents ideas that are potentially complimentary with filtering, which is often discussed in the context of MCO. Among the many works on the topic of MCO, see, e.g., [1], [10], [13]. For continuous optimization problems, the set of undominated solutions can be infinite, so MCO researchers have long suggested that dissimilarity functions or metrics are needed to reduce the number of undominated solutions to be shown. In addition, we note that, in practice, decision-makers can also be interested solutions close to the efficient frontier if they are different enough from those on the frontier. When the problem involves selections, the identification of solutions that are “different enough” in the eyes of the decisions maker can be enhanced by the use of dissimilarity functions that take into account the attributes of the objects selected using functions that we define in this paper.

Our research is quite different from sensitivity analysis. For optimization problems, there is a rich literature concerning the sensitivity of the optimum solution to changes in the problem data even for discrete optimization problems; see, e.g., Ref. [4] or Ref. [12]. Our work augments this by providing means to characterize dissimilarity between solutions. Our work is also different from recent work in the heuristic search literature, which offers some other approaches [2] for generating a varied set of solution vectors.

In order to introduce our notation, we consider the classic selection optimization, namely the knapsack problem:maxcx(K)subject toax≤bx∈{0,1}nwhere c and a are row vectors of length n given as data and x is a column vector of n variables; the budget b is a scalar provided as data. Generally, though, there are additional n vectors of length p each describing the object that will be selected if the corresponding element of x is one. The values of the corresponding elements of c and a may be included in these vectors but there is usually also a lot of other information about the objects being selected. All of this information is not used to find the solution to the optimization problem, but may be critical if one is to make statements about similarities and differences between solution vectors. So for mathematical programming solutions to the problem, we need only the binary vector representation of solutions while to consider solution variety we need to think of these solutions as representing matrices or sets of vectors of the attributes of the objects selected.

Creation of a distance function in the space of solutions using the vector representation is straightforward. For example, Hamming distances can be used. However, such a metric is unsatisfying because it ignores the nature of the projects that differ. A metric based on the attributes of the objects selected is rendered difficult in part by the fact that the dimension of the matrices of attributes (or the cardinality of sets of attribute vectors) can vary from solution to solution.

Consider a brutally simple example where the problem is to select one or two automobiles from the list given in Table 1. Maybe the knapsack formulation (K) is used to select potential solution vectors or perhaps a more complicated generalization with a non-linear objective function and logical constraints. The optimization problem is not of interest here. Our interest is in characterizing the similarity between pairs of selections. Suppose the autos have been numbered in the order shown. To remove optimization considerations, suppose further that the following three vectors indicating selection of two autos each have the same, or about the same, objective function value:x(1)=(0,1,0,0,1)x(2)=(1,0,1,0,0)x(3)=(0,1,0,1,0)

We will provide details in Section 4, but it is easy to see that Hamming distances and Lk metrics will indicate that vector two, x(2), is as far from vector one, x(1), as it is from vector three, x(3). However, this does not make intuitive sense given that the attributes of the cars indicated by the first two vectors are very similar, while the third is quite a bit different from the second. The first selects the four-door civic and the two-door corolla, while the second selects the two-door civic and the four door corolla. In all other ways, the selections are very similar. Solution vector three is the only one that includes a cressida, which is older and has a higher initial price in addition to being a different model. This small example will be the subject of sample calculations for our method in Section 5.

In Section 2, we discuss these issues in detail by reviewing some related similarity and distance functions described in the psychology literature. These methods are extended to measure the distance between sets of vectors in Section 3. We briefly contrast our proposal with metrics often proposed for filtration in Section 4. The description in Section 6 of a major real-world implementation affords the opportunity to illustrate application of our distance function for an important problem. The paper closes with some concluding remarks.

Section snippets

Categorical and set features

The problems of computing similarity and distance are complimentary and the literature is intertwined. We will use the terms similarity and distance function in their broad, intuitive sense and reserve the word metric for those functions f(·) that obey four properties for all vector, set or matrix triplets x, y and z:

  • 1.

    f(x, x)=0 (or at least f(x, x)=f(y, y)),

  • 2.

    f(x, y)>0 if xy,

  • 3.

    f(x, y)=f(y, x) and

  • 4.

    f(x, y)+f(y, z)≥f(x, z).

We will strive to identify distance functions with the first three properties,

A general method for sets of vectors

Our interest is in finding the distance between sets of vectors, which we also refer to as portfolios. In the Norwegian highway example, each vector gives the attributes of a particular project and a portfolio of projects is a collection of these vectors. We will use p to denote the number of attributes that each project has. To compare two portfolios, we will need to specify a function for computation of element-wise comparisons. We use this function to build up a generalization of Tversky's

Contrast with traditional metrics

A simple way to compute the difference between two categorical or discrete vectors is to count mismatches, which is the so-called Hamming distance. This distance function can easily be projected onto the zero–one interval by dividing the distances by the vector length. When the categorical variables are binary and indicate selection, it is sometimes more reasonable to scale by the number of vector elements that are not zero in both vectors. This idea is provided by Jacard's similarity

Example calculations

In order to illustrate the computations and some of the concepts, consider the very small, hypothetical example where there are five candidate projects as given in Table 1 in Section 1. If a subset of these projects (or “objects”) is to be selected using the knapsack formulation (K), then some of these attributes might be part of the data for the optimization problem. For example, the column labeled “cost” might be some or all of the data used for the cost vector, a. However, our interest is

Major example

This example is a recurring problem for NPRA, the Norwegian Public Roads Administration, the governmental body in charge of the main roads in Norway. Every 4 years, they have to select which road development projects to pursue, from a much larger set of worthy projects (details are in Ref. [5]). The potential road development projects can be evaluated on several objectives. For most projects, the potential socioeconomic value can be evaluated, i.e. the overall gain for the society. For many

Conclusions

Many users of decision support systems want to see a diverse set of good solutions in addition to an optimal solution. While research in the area of optimization algorithms has produced an impressive array of methods, there is a need for new and better distance functions to support creation of decision support tools. This paper provides a distance function for selection problems that takes into account the nature of the objects selected. Traditional metrics such as the Hamming distances and

Acknowledgements

This work was partially funded by the Norwegian Ministry of Transport and Communications under the POT program.

Arne Løkketangen is a professor in informatics/optimization at Molde University College in Molde, Norway. He received his Ph.D. from the University of Bergen in 1995. Løkketangen has published in a wide range of journals, and is on the editorial board of several international journals.

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Arne Løkketangen is a professor in informatics/optimization at Molde University College in Molde, Norway. He received his Ph.D. from the University of Bergen in 1995. Løkketangen has published in a wide range of journals, and is on the editorial board of several international journals.

David L. Woodruff is Professor of Management in the Graduate School of Management at the University of California, Davis. He received a Ph.D. from Industrial Engineering Department at Northwestern University. He has served in various society and editorial posts, including Chair of the INFORMS Computing Society.

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