## 2 Background  
In planning large infrastructure projects, engineers are often faced with discrete, multi-criteria decision or choice problems. These arise when there are n feasible options (a\({}_{1}\), a\({}_{2}\),..., a\({}_{n}\)) each of which must be assessed on m criteria (c\({}_{1}\), c\({}_{2}\),..., c\({}_{m}\)), typically related to cost, performance, sustainability, environmental impact, reliability, aesthetic value, expandability, risk, health, safety, etc. A two dimensional (n by m) array of assessments can be generated, in which each element, s\({}_{ij}\), contains the assessment of option a\({}_{i}\) for criterion c\({}_{j}\). Sometimes, all s\({}_{ij}\) elements of the array are numbers but this is not always the case and some criteria are more naturally assessed qualitatively, for instance aesthetic value. Many methods have been developed to support decision making based on the information in this array of assessments. One way of classifying these methods is:  
\"Compensatory analysis methods\" [1] that produce a single score for each decision or design option so they can be ranked, or at least the best or a set of better options can be identified. The principal differences between the methods in this category are mainly on how the assessment information is used to generate the single score for each option and in what additional information is required for this, e.g., importance weights and how such information is obtained. Such methods reduce all aspects of the problem to a single one-dimensional index or score, usually financial, i.e., related to costs, or to the concept of utility [2] or some other index, that, for each option, that combines its performance under all the criteria and allows options to be directly compared. In these methods, the performance of a particular design option or choice evaluated for each of the criteria are effectively traded-off amongst each other, i.e., a bad performance in one criterion can be offset by good performances in other criteria. The type of trade-off is common for all the compensatory methods and these include cost-benefit analysis (CBA) [3], simple additive weighting (SAW) [4], multi-attribute utility theory (MAUT) [5], the Analytic Hierarchy Proces (AHP) [6] and the Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) [7]. In contrast, \"non-compensatory methods\" [8], sometimes called \"outranking methods\" focus on the options and do a series of individual comparisons of pairs of options to determine, from the assessment array, whether one of them is better, worse, or equivalent to the other. This is done for all possible pairs of options, producing a matrix of outranking relationships which can be used to select options. The approach maintains the multi-dimensionality of the criteria and allows them to influence the final choice or ranking of option(s) without an explicit trade-off of one criterion against another. Concordance-Discordance based methods such as the various forms of ELECTRE [9; 10] and PROMETHEE [11] are examples of the second group, and there are a number of other methods following a similar approach [12]. The important difference between this and the first group is there is no explicit trading-off or ranking of the importance of the individual criteria. The third category is \"other\" methods that do not fit neatly into either of the first two categories, such as decision-rule-based techniques [13]. These tend to be less widely used than the first two groups so will not be discussed further here. The technical details of the individual methods are not described here, and, for these, the reader is referred to the key citations listed above and in Table 1. Instead, their use in practice is inferred from a search of published literature, described below. Note that fuzzy number versions of many of these methods have been developed. It is interesting to consider what each approach assumes about the preferences and internal decision processes of the decision maker(s) and which represents these best.