Multi-criteria decision making methodologies are particularly appropriated to address supplier evaluation and selection problems since, in addition to involving several dimensions (here translated into criteria) on which to evaluate the suppliers, some of these dimensions are quantitative in nature, while others are qualitative and thus subjective. More often than not, these criteria are conflicting and hence their simul- taneous optimization is not possible. Furthermore, the number of possible suppliers, here termed alternatives, is small and each has its own known characteristics. MCDM is known for being able to incorporate Decision Makers (DMs) subjectivity through their involvement in the analysis and evaluation processes [2]. Thus, assisting them in making their own choice by providing a ranking of the alternative solutions using the DMs preferences and evaluations. Therefore, supplier selection belongs to the class of multi-criteria decision making problems in which the company needs to identify the top priorities to select the “best” supplier, based on its needs and suppliers infor- mation [1]. Thus, it is better addressed by a two stage procedure, whereby a set of potential interesting solutions is first identified, followed by an analysis along several dimensions of each of the alternative solutions. The main goal of this analysis is to assist the decision maker in the process of identifying the most preferred solution(s) from a set of possible ones (explicitly or implicitly defined). The alternative solutions differ in the extent in which they achieve the objectives, since none of them has the best performance for all objectives; otherwise the choice would be trivial.
The use of MCDM in real-world problems is a non-linear recursive process involv- ing several steps, which vary in number with the specific approach. Nevertheless, it is possible to outline the following critical steps (common to most approaches and studies, although not always corresponding to the same number of steps):
1. Structuring – begins with a contextualization of the problem and then moves onto the definition of the alternatives and criteria.
2. Evaluation – includes the evaluation of each alternative on each of the criterion defined, as well as the relative importance of the criteria.
3. Preparation of recommendations – this step involves obtaining a global score for each alternative making use of the individual scores of each one of them on each criterion and of the criteria weights (both obtained in the previous step). The set of alternatives is then ranked based on the overall scores. The process may also involve a sensitivity analysis of the results to changes in scores or criteria, in order to infer on the robustness of the outcome of the MCDM.
It is important to note that the results yielded by a MCDM process are not prone to generalisations, in the sense that they only apply to the set of alternatives that were evaluated [6].
It follows a brief description of the Analytic Hierarchy Process (AHP) as this is the approach used in this study, although others exist.
9.2.1
AHP – Analytic Hierarchy Process
The origin of the AHP goes back to the 70s and was first proposed by Thomas L. Saaty [16]. This methodology is based on mathematics and psychology and pro- vides a framework for structuring decision problems by allowing for the represen- tation and quantification of all problem elements. The AHP considers a discrete set of possible solutions, which are then analysed and ranked according to their value regarding a set of relevant characteristics, i.e., criteria, previously identified.
The main reasons behind the wide applicability and acceptability of AHP are: (i) easy to understand – its simple and intuitive nature allows for the decision-makers to understand how the recommendations are obtained; (ii) active participation - the decision makers are involved in every step of the decision analysis; (iii) hierarchy – helps in identifying the importance of the factors involved in the problem; (iv) sub- jectiveness - its ability to deal with subjective assessments (feelings and judgments), which are then converted into numerical values that reflect the decision makers opin- ion and preferences; (v) pairwise comparison – it is easier to perform and it helps to articulate the relative importance of criteria and to quantify the relative contribu- tions of the alternatives on the criteria; (vi) inconsistency measure – helps to avoid inconsistent judgments, since by identifying them the decision makers can repeatedly work through their inconsistent judgments until they obtain acceptable results. Nev- ertheless, some drawbacks have also been pointed out [13,18]: (i) the compensatory effect of trade-offs, and (ii) the time involved in the process of gathering the DM’s evaluations. The first is a consequence of the additive aggregation and allows for the degradation of performance on a criterion to be compensated by good performance in other(s); while the second is due to the substantial number of pairwise compar- isons when dealing with a large number of criteria (over ten) and/or alternatives. For
Table 9.1 Scale of relative scores [16,17], even values have intermediate meaning
Score value (ai j) Interpretation
1 i and j are equally important
3 i is slightly more important than j
5 i is more important than j
7 i is strongly more important than j
9 i is absolutely more important than j
Table 9.2 Average random consistency [16]
Size of the matrix (n) 3 4 5 6 7 8
Random consistency (RI) 0.58 0.90 1.12 1.24 1.32 1.41
these reasons, the AHP is sometimes complemented with another MCDM method, see e.g., [14].
AHP basic idea is converting subjective assessments of relative importance into a set of overall scores and weights. The assessments are relative and subjective, since they are provided and reflect the opinion of a specific decision maker and are based on pairwise comparisons.
The pairwise comparisons are performed by asking the decision maker to compare pairs of criteria and to indicate, for each pairwise comparison, their relative contri- bution to the overall goal using the nine-point intensity scale proposed by [16,17], see Table9.1. The comparisons are then represented through a squared matrix n× n, where n is the number of criteria. Each matrix element ai jrepresents the importance of criterion i relative to criterion j . If ai j > 1 then criterion i is more important than criterion j ; otherwise criterion j is more important. Moreover, ai j × aj i = 1.
However, care must be taken when using the pairwise comparisons as they may lead to inconsistencies in the judgments. Therefore, a consistency check needs to be performed by computing the Consistency Ratio (CR) as [16,17]:
C R=C I
R I, with C I =
λmax− n
n− 1 , (9.1)
whereλmaxdenotes the maximum eigenvalue of the pairwise comparison matrix, n represents the matrix size (in this case, the number of criteria), and the acronyms C R,
C I and R I stand for Consistency Ratio, Consistency Index, and Random Consistency
Index, respectively. RI is obtained from a randomly generated pairwise comparison matrix of size n (see Table9.2). If C R> 0.10 the decision maker’s judgments reveal an unacceptable degree of inconsistency and, thus, the entries of the pairwise com- parison matrix need to be amended before proceeding to the next step of the AHP.
The output of AHP consists of a set of priorities, or weights, one for each criteria and a set of relative performance scores, for each alternative in each criterion. The
former, i.e., the vector of priorities or criteria weights, is obtained by finding the first eigenvector of the pairwise comparisons matrix; while the latter can be obtained by separate pairwise comparisons on the set of alternatives in each criterion or by simply rating each alternative for each criterion, by identifying the grade that best describes them. Afterwards, a global score is obtained for each alternative by computing for each the weighted sum of the performance in each criterion.