MCDM: Multiple Criteria Decision Making - A Boring Introduction
Decision Making, in general, is a vast area of study for a multitude of subject fields. It can be simply interpreted as the process of the selection of a sensible choice from a set of possible options. When it comes to humans, the decision-making process or the reasoning is complex and is affected by many factors which can be both internal and external. Human reasoning is directly affected by human psychology. However, this behavior is extremely difficult to be correctly mimicked by a computer. A computer’s decision-making process is solely based on logic, unlike humans.
Multiple Criteria Decision Making
When taking a decision, there might not always be a finite number of choices or there might be many alternatives to the original decision. There is also some possibility for not having a suitable choice for the criterion. Multiple Criteria Decision Making (MCDM) is an approach designed for the evaluation of problems with a finite or an infinite number of choices.
The basic process of decision-making can be broken into several stages as shown in Figure 01 below.
These steps can be briefly described as follows.
- Identifying the objective/goal of the decision-making process
This step is straightforward and involves the correct identification of the goal or the final output of the decision making process. - Selection of criteria
The criteria should be consistent with the decision and also should be independent of each criterion. They should also be represented on the same and a measurable scale and should be inter-related with the alternatives. - Selection of alternatives
When selecting alternatives, attributes such as availability and comparability must be taken into consideration. The alternatives also should be realistic and practical. - Selection of the Weighing Methods
The weighting of criteria specifies the importance of them. This weight can be determined using the two methods (Compensatory and Outranking) methods discussed before in this section. - Aggregation
This step will separate the best alternative from the available options. This could be a mathematical function or an average.
MCDM problems can further be divided into two major categories namely Compensatory Decision Making and Outranking Decision Making.
Compensatory Decision Making
This method is based on a rational model which evaluates the choices on different criteria. In this method, important attributes of an alternative choice outweigh the less important attributes. One of the very good real-world examples of this particular technique is the selection of a car by a consumer. There can be a car which has a low price, higher mileage and low acceleration. In this case, the price and the mileage can be identified as positive attributes and the acceleration can be identified as a negative attribute. The consumer might buy this car considering these positive attributes while neglecting the negative attribute. Examples of this method are Analytic Hierarchy Process (AHP), Fuzzy Multi-Criteria Decision Making Process (FDM).
Outranking Decision Making
This method which has been originated in France is based on the concept of outranking. Usually, this method is used to discard some alternatives to the problem. It involves two steps. The first step is the construction of outranking rules and the second step is an exploitation procedure which embellishes on the recommendations gathered in the first place. Some of the examples are Elimination and Choice Expressing Reality (ELECTRE), Preference Ranking Organization Method for Enrichment of Evaluations (PROMETHUS).
The following are a few of commonly used MCDM methods which belong in both the categories of Compensatory and Outranking MCDM.
Analytic Hierarchy Process (AHP) Method
The AHP is a widely used MCDM method which is utilized in many practical scenarios to derive solutions. This approach decomposes a problem into distinct hierarchies and evaluates each hierarchy by comparing them in a pairwise manner. The hierarchy in this method is a linear one with a finite number of levels. The goals or the potential solutions of the problem are in the top level of the hierarchy, and the criteria along with sub-criteria are in the intermediate level. The alternatives of the criteria are in the bottom level. The criteria and alternatives are compared in a pairwise manner with a scale factor of one to nine, and the weights of the criteria are determined to calculate the global weights for the alternatives. Figure 02 is a very good example of the AHP technique which involves choosing a suitable person out of a group.
One of the advantages of the AHP method is that its support provided for Qualitative Evaluation as well as Quantitative Evaluation. Being applicable to group decision-making environments is another advantage of this method. However, the AHP method has been criticized for its imprecision in the comparison process as well as the inability to handle the inherent uncertainty. Also, the difficulty to add or take out an alternative or a criterion is a major drawback of using this method.
Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) Method
The TOPSIS approach is an effective way of determining the best alternative by calculating the relative closeness of re-sampled and weighted criteria to an ideal alternative. This involves a simple concept of minimizing the distance to the negative-ideal alternative and maximizing the distance to the positive-ideal alternative. The positive-ideal alternative represents the best possible solution to the problem. These proximity values are calculated using the square root of squared distances in the attribute space.
The TOPSIS model is considered to be an appealing and an easily understandable method which utilises a unique way of approaching MCDM problems. Another major advantage of using this method is that it does not limit the number of criteria identified in the decision-making process. However, the TOPSIS method fails to address the rank-reversal problem which is the alteration in the ranking of alternatives when a non-optimal alternative is introduced.
Elimination and Choice Expressing Reality (ELECTRE) Method
The ELECTRE method is a widely studied MCDM technique and it currently has evolved into many versions namely ELECTRE I, II, III, IV, and TRI. These distinct versions are similar in the context but differ based on the type of the problem being solved, and ELECTRE III is considered to be the most suitable method for outranking-based problems.
Similar to the AHP method, this technique uses Concordance and Discordance Indexes to perform a pairwise comparison among alternatives. The Concordance Index reflects that one alternative is better than another alternative in relation to the sum of weights. The Discordance Index reflects the difference of a pair of alternatives divided by the maximum difference over all pairs.
The above figure shows the basic process of ELECTRE method, and the alternatives are outranked by defining a threshold for the indexes. One of the disadvantages of this method is that it consumes more time to complete the entire process.
Preference Ranking Organization Method for Enrichment Evaluations (PROMETHEE) Method
One of the outranking methods used in MCDM is the PROMETHEE method. This approach utilises a preferential function to derive the preference difference between alternative pairs on each criterion. The preference function is designed in a manner which reflects the difference of the preferences of the decision maker’s perspective. The value of this function varies from zero to one. The difference is nil when the value is zero, and an alternative strictly outranks another when the value is one. PROMETHEE has two different versions namely PROMETHEE I and PROMETHEE II. The traditional PROMETHEE I uses the partial ranking method which involves the exclusion of some alternatives which cannot be compared. Conversely, PROMETHEE II uses a complete ranking method to compare all the alternatives.
One of the advantages of using PROMETHEE is the support it provides for group-level decision-making. Both PROMETHEE I and PROMETHEE II can handle both quantitative and qualitative criteria. The ability to handle fuzzy and uncertain information is another advantage of using this method. Despite the usefulness of this method, it is one of the few MCDM methods which are affected by the rank reversal issue which transpires when a fresh alternative is introduced. PROMETHEE does not provide any guideline for the weighting of the criteria which leads the decision-maker to decide suitable weights.
The Rank-reversal Problem in MCDM Methods
One of the common issues which could occur in an MCDM technique is the rank reversal issue. This phenomenon can occur when the decision maker tries to introduce a new alternative in the process of selecting the best alternative. Many research and studies have provided alternative approaches to overcome this issue. However, this issue only occurs when a new alternative is introduced in the alternative selection phase and could be overcome when there is a static set of alternatives in an MCDM problem.
Fuzzy Multi Criteria Decision Making
Fuzzy Logic is a multi-valued logical system which is used in a multitude of applications such as household machines, medical instruments, decision-making systems and industrial process control. Sometimes a decision might need to be more significant than being precise, and the distinction between precision and significance is reflected in figure below.
The traditional MCDM methods such as AHP can be problematic in some situations since they use exact values to express the opinion of decision makers through the comparison of alternatives. To overcome such problems, Fuzzy Multi Criteria Decision Making was introduced. The basic idea is to define a range for a judgement rather than defining a fixed value for each judgement since the decision makers find it more easy and accurate to define interval judgments. There is plenty of articles and research papers related to fuzzy MCDM on the internet.
This article provides a concise overview of a vast domain which has been discussed in a multitude of research publications so it might seem difficult to grasp everything explained at once. However, I hope it might at least provide the basic idea of MCDM and its importance.
(This article was originally published on www.dilee.me)
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