MultiCriteria Decision Analysis (MCDA/MCDM)
MultiCriteria Decision Analysis (MCDA), also known as MultiCriteria DecisionMaking (MCDM), is about making decisions when multiple criteria (or objectives) need to be considered together in order to rank or choose between alternatives.
This is a comprehensive overview of MCDA / MCDM in general. With a practical emphasis as well as solid theoretical underpinnings, this is an essential resource for practitioners and academics alike. Whether you are new to the area or already familiar with it, this article is for you!
A similarly comprehensive article on pairwise comparison and pairwise ranking is also available.
A 3minute introduction to MCDA / MCDM
What is MultiCriteria Decision Analysis used for?
There are many thousands, if not millions, of possible applications for MultiCriteria Decision Analysis (MCDA), also known as Multiple Criteria Decision Making and MultiCriteria DecisionMaking (MCDM). Most decisions made by individuals and groups that involve ranking or choosing between alternatives (including people) are amenable to MCDA / MCDM.
Here are some mainstream examples of applications from the worlds of business, nonprofits, government, health, education and personal decisionmaking:
 Shortlisting job applicants
 Selecting projects or investments for funding
 Picking microfinance or aid programs for support
 Prioritizing local or central government spending
 Prioritizing patients for access to health care (e.g. NZ health system success story)
 Ranking researchers or students for research grants or scholarships
 Choosing a new home, car or smartphone, etc
Common to these examples and all MCDA applications in general is that they involve alternatives (including people) being ranked or chosen based on considering multiple criteria together. Some applications also include the allocation of budgets or other scarce resources across alternatives, with the objective of maximizing ‘value for money’.
Traditional intuitive decisionmaking compared to MCDA
Of course, considering multiple criteria when ranking or choosing between alternatives is a natural approach for making decisions that is as old and fundamental as human history (see famous quotes). However, ‘traditional’ intuitive decisionmaking – how most people make their everyday decisions – typically involves evaluating the criteria and the tradeoffs between them in an intuitive or holistic fashion.
In contrast, MCDA / MCDM, a subdiscipline of operations research with foundations in economics, psychology and mathematics, is concerned with formally structuring and solving decision problems. Most MCDA methods, which are increasingly supported by specialized software (e.g. 1000minds), involve the explicit weighting of criteria and the tradeoffs between them.
Overall, MCDA is intended to reduce biases from decisionmakers relying on their ‘gut feeling’, and also group decisionmaking failures (e.g. ‘groupthink’), that almost inevitably afflict intuitive approaches. By making the weights and associated tradeoffs between the criteria explicit in a structured way, MCDA results in better decisionmaking.
How does MultiCriteria Decision Analysis work?
MCDA / MCDM, in essence, involves these four key components:

Alternatives (or individuals) to be ranked or chosen from
Any number of alternatives may be included in the MCDM – depending on the application, ranging from a minimum of two (otherwise there wouldn’t be a choice to make) up to 10s, 100s, 1000s or even millions of alternatives.

Criteria by which the alternatives are evaluated and compared
For most applications, fewer than a dozen criteria is usually sufficient, with 58 fairly typical, which may be quantitative or qualitative in nature.

Weights representing the relative importance of the criteria
As explained later in the article, there is a variety of methods available for determining the weights on the criteria, representing their relative importance.

Decisionmakers and potentially other stakeholders, whose preferences are to be represented
Again, depending on the application, any number of decisionmakers and potentially other stakeholders may be involved, ranging from just one (e.g. you!) up to many 1000s of people.
MCDA is about getting these four things right! Do so and you’ll be more likely to make the ‘right’ decision (though, of course, there are no guarantees, as things can change or the unexpected happen).
Oneoff versus repeated applications
MCDA / MCDM tools can be used for oneoff (e.g. ranking applicants applying for a job or prioritizing new business projects) or repeated applications (e.g. prioritizing patients as they present for treatment).
Oneoff applications involve ranking particular alternatives (or individuals) that are already known to, or under consideration by, the decisionmaker. In these applications, the number of alternatives is usually in at most the 10s or 100s – e.g. 250 people applying for a job, or 80 business projects to be prioritized.
In contrast, repeated applications involve ranking alternatives in a pool that is continually changing over time, involving potentially many 1000s of alternatives. For example, in health and education applications, new patients or students (‘alternatives’) may need to be prioritized – e.g. for treatment or scholarships – on an ongoing basis (e.g. hourly or daily), including potentially in ‘real time’.
The dynamism of repeated applications means that the MCDA process needs to be capable of including potentially all hypothetically possible alternatives (including people, e.g. patients or students) that might ever occur. Accordingly, MCDA outputs are increasingly incorporated into information systems (e.g. as used by New Zealand’s Ministry of Health).
Overview of the MCDA / MCDM process
Most MCDA applications are based, at least implicitly, on the process represented in Figure 1 below reproduced from
As represented in the diagram, the iterative nature of the process, with multiple possible feedbacks and cycles, serves to emphasize that MCDA is intended to function as a tool to help people, individually or in groups, to reach a decision – i.e. their decision (made by humans), not the tool’s decision.
As well as more transparent and consistent decisionmaking, MCDA can also be used to facilitate the participation of a wide range of stakeholders, systematically taking their preferences into account. MCDA results can also be used to communicate and justify the final decision to stakeholders.
A simple example
As an example, imagine that you are an employer who wants to rank applicants applying for a job vacancy that you have to fill.
In short, there are three fundamental things you need to do to create a MCDA model (a more detailed, sixstep process is explained later below):
 Specify relevant criteria for evaluating and comparing the job applicants
 Determine weights for the criteria, representing their relative importance to you
 Rate the applicants on the criteria, resulting in a ranking
In this simple example, identifying the relevant criteria is probably quite straightforward. Most jobs would include these three criteria (and others too perhaps): qualifications, experience and references.
More challenging is likely to be determining accurate weights for the criteria, reflecting their relative importance, so that when you assess the job applicants on the criteria, they are accurately ranked from ‘best’ to ‘worst’.
To keep the example as simple as possible, suppose there are just two applicants, Romeo and Juliet. After reading their CVs and interviewing them, suppose you rated them on the three criteria as in Table 1 (where, notice, two ratings are qualitative and one is quantitative).
Criterion  Applicant rating  

Romeo  Juliet  
Qualifications  poor  good 
Experience  4 years  2 years 
References  good  good 
Given that Romeo and Juliet are the same with respect to their references (both ‘good’), whether Romeo is ‘better’ or ‘worse’ than Juliet depends on whether qualifications is more important than experience, or vice versa.
A weightedsum model
An obvious and commonsense approach for aggregating the performance of each alternative (here, job applicant) across multiple criteria is to create a ‘weightedsum model’. Such models are also known as ‘points systems’, and other names are mentioned later below too.
This approach involves weighting the criteria to reflect their relative importance, where the weights sum to 1, and scoring each alternative according to its rating on each criterion, usually in the range 0100. Each alternative’s performance across the criteria is aggregated via a linear (i.e. additive) equation to produce a ‘total score’, also in the range 0100, with the alternatives ranked by their scores.
An example of criterion weights and scores for ranking job applicants appears in Table 2.
Criterion  Criterion weight  Level  Criterion score 

Qualifications  0.50  poor  0 
fair  20  
good  80  
excellent  100  
Experience  0.32  ≤ 2 years  0 
3 years  50  
> 3 years  100  
References  0.18  poor  0 
good  100 
Applying this model to Romeo’s and Juliet’s ratings on the three criteria (Table 1) generates their total scores (out of a theoretical maximum of 100 points):
 Romeo’s total score (for ‘poor’, ‘> 3 years’, ‘good’):
(0.5 x 0) + (0.32 x 100) + (0.18 x 100) = 50
points  Juliet’s total score (‘good’, ‘≤ 2 years’, ‘good’):
(0.5 x 80) + (0.32 x 0) + (0.18 x 100) = 58
points
Based on their total scores, Juliet is ranked ahead of Romeo for the job.
Notice that the hypothetically best candidate, who would have the highest possible rating on each criterion, would get: (0.5 x 100) + (0.32 x 100) + (0.18 x 100) = 100
points.
It should be obvious that even for this simplest of possible examples the ranking of alternatives depends on the criteria and weights used. Different criteria or weights would have resulted in different total scores and possibly the opposite ranking, or an equal ranking. (The weights above were made up for the sake of this simple illustration.)
The successful implementation of MCDA / MCDM – typically for applications involving more criteria or alternatives than this simple example – depends fundamentally on applying valid and reliable criteria and also weights reflecting decisionmakers’ preferences (and potentially other stakeholders’ preferences too, depending on the application). As discussed later below, specialized methods are available for eliciting preferences.
Origins of MCDA
Historically, the first example of a formal approach related to MultiCriteria DecisionMaking was described by Benjamin Franklin in 1772 (
Franklin humbly begins his letter with this sentence (which also serves to reassure us that although as individuals we may have different preferences, we can still use the same decisionmaking methods):
Dear Sir,
In the affair of so much importance to you, wherein you ask my advice, I cannot, for want of sufficient premises, advise you what to determine, but if you please I will tell you how.
Franklin’s method (his “how”) involves first tabulating the ‘pros’ and ‘cons’ of the alternative being evaluated relative to another alternative (e.g. the status quo), and then successively trading them off – in effect, weighting them – in order to identify the better alternative. Though this simple approach is effective for decisions involving choosing between two alternatives, it does not scale up for more alternatives.
More than 200 years later, more technically sophisticated methods for choosing between more than just two alternatives, and potentially involving multiple decisionmakers, were presented by Ralph Keeney and Howard Raiffa in their seminal book, Decisions with Multiple Objectives: Preferences and Value Tradeoffs (
In 1979, Stanley Zionts helped popularize the abbreviation MCDM (for Multiple Criteria DecisionMaking) with his article for managers: “MCDM – If not a Roman numeral, then what?” (
Other significant events from MCDA’s / MCDM’s history are discussed in the book by Köksalan, Wallenius & Zionts (2011).
MCDA software
Nowadays, MultiCriteria Decision Analysis is increasingly supported by specialized software (mostly webbased). Before the advent of the World Wide Web in the 1990s, most MCDA / MCDM software was based on spreadsheets, such as the early example described by
MCDA software frees “the facilitator/analyst and decisionmaker from the technical implementation details, allowing them to focus on the fundamental value judgments” (
MCDA software is especially useful for applications involving many alternatives and criteria, and when the methods for determining the weights on the criteria (and scoring the alternatives on the criteria) are technically sophisticated.
Also, software capable of surveying potentially 1000s of people is useful for applications based on eliciting and analyzing the preferences of members of the general population (e.g. nationwide surveys).
MCDA software, ranging from the most basic to the state of the art, is catalogued in these three overviews:
 Wikipedia’s “comparison of decisionmaking software” – an uptodate list exclusively focused on MCDM software

OR/MS Today’s “Decision Analysis Software Survey” (
Beekman 2020 ) – an uptodate, vendorcompleted survey but with a broader focus on ‘decision analysis’ (DA) software in general (i.e. MCDA / MCDM is a subset of decision analysis) 
Weistroffer & Li’s (2016) survey of “Multiple Criteria Decision Analysis Software” – an exhaustive review of all commercially and academically available MCDM / MCDA software
If you would like to experience a prominent example of MCDA software, 1000minds (implementing the PAPRIKA method, explained later below), you can create yourself a 1000minds account.
The remainder of the article is devoted to explaining the steps for creating and applying weightedsum models, beginning with a more detailed discussion of weightedsum models (as introduced earlier). Some of this material is adapted from parts of
Weightedsum models
As mentioned earlier, most MCDA / MCDM applications are based on ‘weightedsum models’. Such models are also known in the academic literature as ‘additive multiattribute value models’.
As explained earlier (and see Table 2 again), weightedsum models involve decisionmakers explicitly weighting the criteria for the decision problem being addressed and rating the alternatives on the criteria. Each alternative’s overall performance on the criteria is aggregated via a linear (i.e. additive) equation to produce the alternative’s ‘total score’, with the alternatives ranked by their scores.
Points systems
A common and equivalent representation of the linear equations at the heart of weightedsum models is a schedule of ‘point values’ (sometimes referred to as ‘preference values’) for each criterion. These schedules are also known as ‘points systems’, and also as ‘additive’, ‘linear’, ‘scoring’, ‘pointcount’ and ‘points’ models or systems.
The pointssystem equivalent of the weightedsum model in Table 2 appears in Table 3. In the table, the point value for each level on a criterion represents the combined effect of the criterion’s relative importance (weight) and its degree of achievement as reflected by the level.
In other words, consistent with the linear equations described above, a level’s point value can be obtained by multiplying the criterion weight by the criterion score from the equation. This relationship can be easily confirmed by multiplying the weights by the scores in Table 2, as reported in the fourth column below (for illustrative purposes). Likewise, a point value can be ‘decomposed’ – in the opposite direction – into the criterion’s weight and score.
A pointssystem representation is sometimes preferred because, arguably, a simple schedule of point values is easier to implement than an equation (a set of weights and scores that need to be multiplied together). Also, in tabular form, a points system takes up less space (e.g. compare Table 2 and Table 3).
Criterion  Level  Points  Illustrative calculation 

Qualifications  poor  0  0.5 x 0 
fair  10  0.5 x 20  
good  40  0.5 x 80  
excellent  50  0.5 x 100  
Experience  ≤ 2 years  0  0.32 x 0 
3 years  16  0.32 x 50  
> 3 years  32  0.32 x 100  
References  poor  0  0.18 x 0 
good  18  0.18 x 100 
Applying the points system in Table 3 to Romeo’s and Juliet’s ratings on the three criteria (Table 1) generates their total scores:
 Romeo’s total score (for ‘poor’, ‘> 3 years’, ‘good’):
0 + 32 + 18 = 50
points  Juliet’s total score (‘good’, ‘≤ 2 years’, ‘good’):
40 + 0 + 18 = 58
points
Notice that these total scores are the same as before, thereby confirming the equivalence of Tables 2 and 3. Again, based on their total scores, Juliet is ranked ahead of Romeo for the job.
Notice, again, that the hypothetically best candidate, with the highest possible rating on each criterion, would get: 50 + 32 + 18 = 100
points.
Simplicity and accuracy
In practical terms, a major attraction of weightedsum models – in either their linear equation (Table 2) or pointssystem (Table 3) forms – is the simplicity by which each alternative’s performance on each criterion is aggregated to produce the alternative’s total score. (Nonlinear functions for aggregating the criteria, including multiplicative functions, are also possible but very rare.)
More importantly, such simple models (sometimes referred to as ‘algorithms’ or ‘formulas’) have been found nearly universally in very many studies to be more accurate than the intuitive or holistic judgments of decisionmakers (
According to
surprisingly successful in many applications. We say ‘surprisingly’, because many judges claim that their mental processes are much more complex than the linear summary equations would suggest – although empirically, the equation does a remarkably good job of ‘capturing’ their judgment habits.
Hastie and Dawes (p. 52) also explained:
The mind is in many essential respects a linear weighting and adding device. In fact, much of what we know about the neural networks in the physical brain suggests that a natural computation for such a ‘machine’ is weighting and adding, exactly the fundamental processes that are well described by linear equations.
Also, according to
[Weightedsum models / points systems] are, as a rule, more accurate than human predictors. This is not surprising, as it has been known for decades that human cognitive performance is limited with respect to complex multivariable judgment and prediction tasks (
Meehl 1954 ).
Other approaches to MultiCriteria Decision Analysis
For completeness, it’s worthwhile mentioning that other MCDAbased approaches not based on aggregative functions for combining alternatives’ performance on criteria (i.e. via weights) are also potentially available. These other approaches are available as alternatives to weightedsum models, albeit they are unlikely to be more useful.
Performance matrices
The most basic alternative approach is a simple table, sometimes referred to as a ‘performance matrix’, for reporting the alternatives’ performance on the criteria. In the table/matrix, each alternative is a row and each criterion is a column.
If one alternative dominates the others on all criteria, or where the tradeoffs involved in selecting an alternative are clear and uncontroversial, then decisionmakers can use such a table to reach their decision quickly.
However, most decision problems are more complicated than this!
Most MCDA / MCDM applications involve confronting nontrivial tradeoffs between criteria. Therefore, merely tabulating alternatives’ performance on the criteria is insufficient for most applications. Nonetheless, such a table may be useful at step 3 in the MCDA process outlined in the next section.
Outranking methods
A MCDA / MCDM approach capable of evaluating tradeoffs between criteria, but that is not based on aggregative functions for combining alternatives’ performance on criteria (via weights), is the group of ‘outranking’ methods.
Methods in this tradition include:
 PROMETHEE (
Vincke & Brans 1985 ) – an acronym for “Preference Ranking Organization METHod for Enrichment of Evaluations”  GAIA (
Brans & Mareschal 1994 ) – for “Geometrical Analysis for Interactive Aid”  ELECTRE family of methods (
Roy 1991 ) – for “ELimination Et Choix Traduisant la REalité” (which translates as “elimination and choice expressing reality”)
In essence, outranking methods involve decisionmakers pairwise ranking all the alternatives relative to each other on each criterion in turn and then combining these pairwiseranking results but not via weights. The objective is to obtain measures of support for judging each alternative to be the topranked alternative overall.
Relative to weightedsum models, outranking methods are rarely used. Their unpopularity is probably mostly due to their complexities and the nonintuitive nature of their inputs and algorithms relative to weightedsum models.
The steps for creating and applying weightedsum models, including a survey of methods for determining weights, is explained next.
Steps for performing MultiCriteria Decision Analysis
The steps for creating and applying weightedsum models (or points systems), likely to be useful for most applications, are summarized in Table 4 and are discussed in turn next. These steps are consistent with the overall MCDA / MCDM process represented in the diagram near the start of the article.
Given the wide range of possible MCDA applications (as mentioned earlier), the steps, as presented here, are necessarily generic. Equivalent processes for specific applications are available in the literature, such as prioritizing patients for access to health care (
Although the steps are presented in sequence below, they do not necessarily need to be performed in that order.
Also, earlier steps, such as step 1 (structuring the decision problem) and step 2 (specifying criteria), can be revisited during the process as new insights into the particular application emerge and revisions and refinements become desirable.
Though, in principle, the steps can be performed ‘by hand’ (e.g. supported by spreadsheets), many of them, and often entire processes, are supported by MCDM software (as discussed above). Such software is especially useful for applications involving many alternatives and criteria, and when the scoring and weighting (step 4) methods are technically sophisticated.
Step  Brief description 

1. Structuring the decision problem 
Identify objectives, alternatives, decisionmakers, any other stakeholders, and the output required. 
2. Specifying criteria 
Specify criteria for the decision that are relevant to decisionmakers (and, potentially, other stakeholders) and preferentially independent. 
3. Measuring alternatives’ performance 
Gather information about the performance of the alternatives on the criteria. 
4. Scoring alternatives and weighting the criteria 
Convert the alternatives’ performances on the criteria into scores on the criteria, and determine weights for the criteria. 
5. Applying scores and weights to rank alternatives 
Multiply alternatives’ scores on the criteria by the weights and sum to get total scores, by which the alternatives are ranked. 
6. Supporting decisionmaking 
Use the MCDA outputs to support decisionmaking – i.e. ranking or selecting alternatives (depending on the application). 
1. Structuring the decision problem
The first step of the MCDA / MCDM process involves structuring and framing the decision problem being addressed (and see the first stage in the earlier diagram again).
In particular, it’s important to clarify the overarching objectives of the decisionmaking exercise, and to think about the overall outcomes that are desired.
Related issues include:
 Identifying the alternatives under consideration
 Whether the decision is a oneoff or repeated application (as discussed earlier)
 The type of output required from the MCDM, e.g. a ranking or a selection
These elements should be validated with stakeholders.
2. Specifying criteria
The criteria specified for the decision should be valid, reliable and relevant to decisionmakers and other stakeholders. They should also be preferentially independent, as explained below.
The criteria should be structured to avoid doublecounting and minimize overlap between them. For example, having criteria for attractiveness and also beauty when choosing a car would invalidate the weighting methods discussed below and their results.
Depending on the application, the criteria can be identified from: similar decisions, reviews of the literature, facilitated discussions, expert opinions, focus groups and surveys. As in the previous step, stakeholders should be involved in identifying and validating the criteria.
For example, 1000minds offers ‘noise audits’ whereby participants use their intuition to rank real or imaginary alternatives to inspire a discussion of the criteria (as well as to demonstrate how inconsistent decisionmakers are and hence the need for a better approach than relying on intuition).
Preferential independence
Part of the requirement that the criteria are valid and reliable includes that they are preferentially independent. This property is most easily explained using an example: Imagine that you are looking for a car to buy, and your criteria include ‘power source’ (i.e. electric versus fossil fuels) and ‘type of car’ (modern, classic or vintage).
If your preference for a car to be powered by electricity rather than fossil fuels does not depend on whether the car is modern, classic or vintage, then these two criteria for choosing a car – ‘power source’ and ‘type of car’ – are preferentially independent.
If they are not (i.e. if your criteria are preferentially dependent), then you need to respecify them in such a way that they are (or don’t use these criteria).
Criteria can be preferentially independent despite being highly correlated: e.g. vintage cars almost always run on petrol instead of electricity, and this correlation between the two criteria, ‘power source’ and ‘type of car’, is fine when it comes to specifying them for choosing between cars.
Similarly, in another example, if you were choosing a phone based on the criteria of weight and screen size, given current technology, very light phones have to have small screens. This correlation between weight and screen size is fine as far as their inclusion as criteria is concerned: independence of criteria in relation to the decisionmaker’s preferences is what matters, not independence of criteria in relation to real alternatives – we’re allowed to dream of the perfect phone (or car, etc)!
3. Measuring alternatives’ performance
Data about the alternatives’ performance in terms of each of the criteria can be gathered in a variety of ways, ranging from expert opinions to rapid literature reviews, to full systematic reviews and modeling exercises. These data can be presented in a ‘performance matrix’ or table (as discussed earlier), if useful.
The sophistication and intensity of the datagathering activity depend on the availability of relevant evidence, the decision problem, and also other practical factors such as the resources available for the job.
MCDA / MCDM is capable of including both quantitative and qualitative data. It is also capable of combining subjective judgments (in the absence of ‘harder’ data) with more traditional scientific evidence in the same application. For example,1000minds has surveying tools for collecting decisionmakers’ ratings of the alternatives on the criteria, and for aggregating results.
4. Scoring alternatives and weighting the criteria
Scoring alternatives and weighting criteria are sometimes considered separate steps. They are treated together here because they are intrinsically linked. They can be performed sequentially, simultaneously or iteratively, depending on the application and methods (or software) used.
Scoring alternatives on the criteria
Scoring the alternatives on the criteria involves converting each alternative’s performance on each criterion into a numerical score. The scores are usually normalized so the worst performance on the criterion gets a score of zero and the highest performance gets 100.
Scores on the criteria can be implemented using a continuous scale (0100). Or, alternatively, as in the example at the beginning of the article, two or more mutuallyexclusive and exhaustive levels (e.g. low, medium, high) can be employed, each with a point value (e.g. medium = 60).
Weighting the criteria
Weighting the criteria involves determining their weights, representing their relative importance to decisionmakers. The weights are usually normalized to sum across the criteria to one (100%).
Overall, it’s very important that the alternative scores and criterion weights are as valid and reliable as possible.
Otherwise, if the scores or weights are ‘wrong’ (inaccurate), then the ‘wrong’ decision, as determined by the ranking of the alternatives’ total scores, is almost certain – even if valid and reliable criteria have been specified (step 2) and the alternatives’ performances have been accurately measured (step 3).
A variety of methods for scoring alternatives on the criteria and weighting the criteria are available, as discussed in the next section.
5. Applying scores and weights to rank alternatives
Having scored the alternatives on the criteria and weighted the criteria, it’s easy to calculate their ‘total scores’ (usually performed automatically by MCDA software).
With reference to the scoring and weighting methods explained below:
 For the direct methods and, in effect, the PAPRIKA (choicebased) method, each alternative’s scores on the criteria are multiplied by the weights, and then the weighted scores are summed across the criteria to get each alternative’s total score (e.g. see Table 2 and Table 3 above).
 For the conjoint analysis (or DCE) method, the regression technique estimates each alternative’s value (or ‘utility’) or its probability of being preferred by decisionmakers.
6. Supporting decisionmaking
The final step of the MCDA / MCDM process is to use the results, usually presented in tables like the one above, or charts, to support your decisionmaking – i.e. depending on your application, for ranking or selecting alternatives.
Where appropriate, the MCDM inputs, process and results can be used to communicate and explain the final decision to stakeholders.
As for any analysis based on uncertain inputs, it’s important to perform sensitivity analysis to check the robustness of your results to plausible variations, e.g. in the alternatives’ performance (step 3) and their scores, or on the criterion levels, or in the criterion weights (step 4).
Overall, decisionmakers need to understand the MCDM results, including any significant limitations of the analysis. These results can be used to support decisionmakers in reaching and, if need be, defending their decision.
It is important to remember, as emphasized at the beginning of the article, that MCDM is intended as a tool to help people, individually or in groups, to reach a decision – their decision (the humans’ decision), not the tool’s decision.
Also, in some applications, other considerations in addition to the MCDM results may also be relevant for the decision being made.
For example, decisions with budgetary implications, such as investment or project decisionmaking (e.g. CAPEX or OPEX), involve comparing the benefits of each alternative, as represented by their MCDM total score, relative to their costs. Other factors, such as strategic or legal factors, might also be worthwhile considering.
Valueformoney assessments
The MCDA / MCDM process explained above, resulting in a decisionmaking tool – a weightedsum model, or points system, comprising criteria and weights – can be complemented by a framework for making resourceallocation decisions based explicitly on “value for money” and efficiency, as summarized here.
For the simple example of project decisionmaking subject to a budget, consider the “valueformoney” (VfM) chart in Figure 2.
The vertical axis of the VfM chart represents each project’s benefit, expressed in terms of a total score, i.e. based on the criteria and weights derived earlier. The horizontal axis displays each project’s estimated cost.
As an illustration, the chart can be used to think about the following questions.
Suppose you had a budget of $8 million.
Which of the 15 projects in the chart – costing more than $16 million, in total – represent high quality spending, e.g. with high benefitcost ratios?
Which projects would you reject because they represent low quality spending, e.g. that are “dominated” by the “efficient” projects?
Thinking about the first question should draw your focus of attention to the chart’s topleft quadrant, i.e. projects with relatively high Benefit (total score) and low Cost.
Possible tradeoffs between Cost and Benefit are along the gold line, known as the “efficiency” or “Pareto” frontier: higher Cost can be compensated for by higher Benefit.
The size and color of the bubbles in the chart can also be used to represent other potentially important considerations, such as confidence in cost estimates (e.g. high, medium, low), synergies, political support, etc.
In general, applying this “value for money” framework involves decisionmakers considering alternative affordable combinations to arrive at their “optimal portfolio” of projects.
In operations research, this combinatorial optimization problem is an example of the famous “knapsack problem” (Dantzig & Mazur 2007).
Scoring and weighting methods
As noted at step 4 above, a variety of methods for scoring alternatives on the criteria and weighting the criteria are available, often supported by MCDA software (as discussed earlier). For the reasons explained below, these methods can be broadly classified as direct and choicebased methods respectively.
Also, many of the methods explained below involve pairwise comparisons of alternatives or criteria, including pairwise ranking. [A comprehensive overview of pairwise comparisons and pairwise rankings, which, like the present article, is intended as an essential resource for practitioners and academics (including students), is available here.]
Direct methods include:
 Direct rating and points allocation (
Bottomley, Doyle & Green 2000 )  SMART (
Edwards 1977 ) – an acronym for “Simple MultiAttribute Rating Technique”  SMARTER (
Edwards & Barron 1994 ) – for “Simple MultiAttribute Rating Technique Exploiting Ranks”  Analytic Hierarchy Process (AHP) (
Saaty 1977 )  Swing weighting
 ‘Bisection’ and ‘difference’ methods (
Winterfeldt & Edwards 1986 )
Choicebased methods include:
 PAPRIKA (
Hansen & Ombler 2008 ) – an acronym for “Potentially All Pairwise RanKings of all possible Alternatives”  Conjoint analysis (or discrete choice experiments, DCE) (
McFadden 1974 ) (Green, Krieger & Wind 2001 )
These eight direct and two choicebased methods are explained in detail in turn below. First, though, the main difference between the two groups of methods – direct and choicebased – are discussed.
Direct versus choicebased methods
Direct methods for scoring alternatives on the criteria involve decisionmakers directly expressing how they feel about the relative performance of each alternative on a criterion relative to the other alternatives.
These expressions of a decisionmaker’s preferences are usually measured using either an interval scale (e.g. alternatives are rated on a 0100 scale on a criterion) or a ratio scale (e.g. “alternative A is three times as important as alternative B on a criterion”).
Similarly, direct methods for weighting the criteria involve decisionmakers (directly) expressing how they feel about the relative importance of the criteria, again usually represented in terms of either an interval scale (e.g. criteria are rated on a 0100 scale) or a ratio scale measurement (e.g. “criterion A is three times as important as criterion B”).
Direct methods have the superficial appeal of directly asking the decisionmaker about how they feel about the relative importance of the criteria or alternatives, from which scores for the alternatives on each criterion or criterion weights are derived (i.e. directly).
Also, when you read the direct methods’ explanations below, hopefully, you’ll to be struck by their algorithmic and computational simplicity, such that they can be easily implemented ‘by hand’ (e.g. supported by spreadsheets), whereas the choicebased methods require specialized software (which come with other useful features, of course).
Unfortunately, however, these direct methods have the major disadvantage of being based on preferenceelicitation tasks that are cognitively difficult for most people, and so the validity and reliability of their results is questionable. For example, expressing how strongly you feel about one criterion (or alternative) relative to another one on a scale ranging from “equally preferred” (ratio = 1) to “extreme importance” (ratio = 9) is not a natural human mental activity that many of us have much experience of.
As an example, suppose you’re an employer looking to hire someone for a job. How easy would you find thinking about a question like this:
On a ninepoint scale ranging from “equally preferred” (ratio = 1) to “extreme importance” (ratio = 9), how much more important to you is a person’s qualifications than their experience?
How confident would you be in the accuracy (validity and reliability) of your answer?
In contrast, the choicebased methods explained below involve decisionmakers expressing their preferences usually by answering relatively simple questions based on choosing between two or more alternatives defined on some or all of the criteria – an example of such a question appears in Figure 3 below. From these rankings or choices, scores and weights are derived (indirectly) using regressionbased techniques or other quantitative methods.
The advantage of these choicebased methods is that choosing between things is a natural type of decision activity experienced by all of us in our daily lives. We make such judgments dozens, if not hundreds, of times every day (e.g. “would you like scrambled or poached eggs for breakfast?”). Therefore, choicebased methods are likely to be more valid and reliable than direct methods, as well as being easier for decisionmakers.
In the words of
The advantage of choicebased methods is that choosing, unlike scaling, is a natural human task at which we all have considerable experience, and furthermore it is observable and verifiable.
The eight direct and two choicebased methods are briefly explained in turn below (and more detailed information is available from their references above). Given the advantages of choicebased methods discussed above, you might like to skip to that subsection.
Direct methods
Each of the following eight methods is explained in terms of scoring alternatives on the criteria, and weighting the criteria.
Direct rating
Scoring alternatives on the criteria
For each criterion, each alternative is rated on a point scale ranging from 0 to 100 in proportion to its performance on the criterion relative to the other alternatives being considered. This rating exercise is typically performed using a visual analogue scale (VAS) or equivalent. (A VAS looks like a household thermometer, either standing up or on its side.)
The alternatives’ scores on the criterion are usually normalized so that the alternative with the lowest (or ‘worst’) score gets 0 and the one with the highest (‘best’) score gets 100.
Weighting the criteria
Each criterion is rated on a point scale ranging from 0 to 100 – e.g. using a VAS (as above) – in proportion to its importance relative to the other criteria.
Weights for the criteria are calculated from the ratios of the point ratings, which are normalized to sum to one across the criteria.
Points allocation
Scoring alternatives on the criteria
For each criterion, a total of 100 points is allocated across all the alternatives being considered in proportion to their relative performances on the criterion.
The alternatives’ scores on the criterion are usually normalized so that the alternative with the lowest (or ‘worst’) score gets 0 and the alternative with the highest (‘best’) score gets 100.
Weighting the criteria
A total of 100 points is allocated across the criteria in proportion to their relative importance.
Weights for the criteria are calculated from the ratios of the point ratings, which are normalized to sum to one across the criteria.
SMART
SMART is an acronym for “Simple MultiAttribute Rating Technique”.
Scoring alternatives on the criteria
For each criterion, the lowestranked (or ‘worst’) alternative is identified and given a value of 10 points. The other alternatives being considered are rated relative to this alternative by also assigning points to them in proportion to their relative performances on the criterion.
The alternatives’ scores on the criterion are usually normalized so that the alternative with the lowest (or ‘worst’) score gets 0 and the one with the highest (‘best’) score gets 100.
Weighting the criteria
The leastimportant criterion is identified and given a value of 10 points. The other criteria are rated relative to this criterion by also assigning points (of higher value) to them in proportion to their relative performances on the criterion.
Weights for the criteria are calculated from the ratios of the point ratings, which are normalized to sum to one across the criteria.
SMARTER
SMARTER is an acronym for “Simple MultiAttribute Rating Technique Exploiting Ranks”.
Scoring alternatives on the criteria
SMARTER is not usually used for scoring alternatives on criteria (some other method is used instead).
Weighting the criteria
The K criteria are ranked in order of their importance. The mostimportant criterion gets a value of 1, the secondmost important criterion gets a value of 2, and so on down to a value of K for the leastimportant criterion.
Weights for the criteria are calculated using this formula:
w_{k} = (1/K)∑(1/i)
summed across
i = k to K
, where w_{k} is the weight of the k^{th}ranked criterion, k = 1 to K
.
For example, with four criteria, applying the formula above, the weights are: 0.52, 0.27, 0.15, 0.06.
Other, similar methods based on rank orders also exist (
Analytic Hierarchy Process (AHP)
Scoring alternatives on the criteria
For each criterion, the alternatives are scored relative to each other by pairwise comparing them in terms of their “intensity of importance” by asking decisionmakers questions like this: “With respect to criterion 1, on a ninepoint scale ranging from “equally preferred” (ratio = 1) to “extreme importance” (ratio = 9), how much more important to you is alternative A than alternative B?”
Scores for the alternatives are calculated from the ratios using eigenvalue analysis.
Weighting the criteria
Each level in the hierarchy of criteria and subcriteria (and subsubcriteria, etc.), as represented in a ‘value tree’, can be analyzed as a separate decision problem (and then combined multiplicatively).
For each level of the tree, the criteria are pairwise compared and their “intensity of importance” relative to each other is expressed on the abovementioned 19 ratio scale.
Weights for the criteria are calculated from the ratios using eigenvalue analysis, which are normalized to sum to one across the criteria.
Swing weighting
Scoring alternatives on the criteria
Swing weighting is not used for scoring alternatives on criteria (some other method is used instead).
Weighting the criteria
For each criterion, the effects of a ‘swing’ in performance from the worst to the best possible performance is evaluated. The criterion that is judged to be the most important in terms of the swing gets 100 points.
The secondmost important criterion is identified and assigned points relative to the 100 points for the most important criterion. The exercise is repeated for the remaining criteria.
Weights for the criteria are calculated from the ratios of the point ratings, which are normalized to sum to one across the criteria.
Bisection method
Scoring alternatives on the criteria
For each criterion, the lowestranked (or ‘worst’) and the highestranked (or ‘best’) alternatives are identified and rated 0 and 100 respectively. The performance on the criterion that is halfway between these two extremes is therefore worth 50 (midway between 0 and 100) and is defined as such.
The next two midpoints on the criterion relative to the performances worth 0 and 50 and 50 and 100 respectively are then defined. These two endpoints and three midpoints are usually sufficient to trace out the approximate shape of the criterion’s ‘value function’. The remaining alternatives are mapped to the shape of the function.
Weighting the criteria
The bisection method is not used for weighting criteria (some other method is used instead).
Difference method
Scoring alternatives on the criteria
For each criterion, the range of possible values for the performance on the criterion is divided into, say, four or five equal intervals (this method is for criteria that are quantitatively measurable and monotonic).
The intervals are ranked in order of importance. This ranking indicates the shape of the ‘value function’ (e.g. concave or convex) so that it can be traced out approximately. The alternatives are mapped to the shape of the function.
Weighting the criteria
The difference method is not used for weighting criteria (some other method is used instead).
Choicebased methods
In contrast to the direct methods above, the following choicebased methods are based on scoring alternatives on the criteria and weighting the criteria together in the same process.
These methods are algorithmically more sophisticated than the direct methods above, and so they cannot be implemented ‘by hand’ (like most direct methods), and the explanations below are relatively ‘broad brush’.
PAPRIKA
Implemented by 1000minds software, PAPRIKA is an acronym for “Potentially All Pairwise RanKings of all possible Alternatives”.
The PAPRIKA method involves the decisionmaker answering a series of simple pairwise ranking questions based on choosing between two hypothetical alternatives, initially defined on just two criteria at a time. The levels on the criteria are specified so that there is a tradeoff between the criteria.
Figure 3 is an example of a pairwise ranking tradeoff question – involving choosing between ‘projects’ for a business, nonprofit or government organization.
Thus, PAPRIKA’s pairwise ranking questions are based on ‘partialprofiles’ – just two criteria at a time – in contrast to ‘fullprofile’ methods for conducting conjoint analysis (discussed in the next subsection) which involve all criteria together at a time (e.g. six or more).
The obvious advantage of such simple questions is that they are the easiest possible question to think about, and so decisionmakers can answer them with more confidence (and faster) than more complicated questions.
It’s possible for the decisionmaker to proceed to the ‘next level’ of pairwise ranking questions, involving three criteria at a time, and then four, five, etc (up to the number of criteria included). But for most practical purposes answering questions with just two criteria at a time is usually sufficient with respect to the accuracy of the results achieved.
Each time the decisionmaker ranks a pair of hypothetical alternatives (defined on the criteria), all other pairs that can be pairwise ranked via the logical property of ‘transitivity’ are identified and eliminated, thereby minimizing the number of pairwise rankings explicitly performed. For example, if the decisionmaker were to rank alternative A over B and then B over C, then, logically – by transitivity – A must be ranked over C, and so a question about this third pair would not be asked.
Each time the decisionmaker answers a question, the method adapts with respect to the next question asked (one whose answer is not implied by earlier answers) by taking all preceding answers into account. Thus, PAPRIKA is also a type of adaptive conjoint analysis or discrete choice experiment (DCE) (discussed below).
In the process of answering a relatively small number of questions, the decisionmaker ends up having pairwise ranked all hypothetical alternatives differentiated on two criteria at a time, either explicitly or implicitly (by transitivity). Because the pairwise rankings are consistent, a complete overall ranking of alternatives is defined, based on the decisionmaker’s preferences.
Thus, the PAPRIKA method is capable of identifying potentially all possible pairwise rankings of all possible alternatives representable by the criteria – hence the method’s name: Potentially All Pairwise RanKings of all possible Alternatives (PAPRIKA).
Finally, criterion weights and scores for the criteria’s levels are determined from the explicitly ranked pairs using mathematical methods based on linear programming.
You can easily experience PAPRIKA for yourself by creating a 1000minds software account. If you are interested in learning more about PAPRIKA, more information is available.
Conjoint analysis (or discrete choice experiments, DCE)
As explained above, the PAPRIKA method is a type of adaptive conjoint analysis or discrete choice experiment (DCE). Other methods for performing conjoint analysis (or DCE) are also available (
All conjointanalysis methods involve decisionmakers expressing their preferences by repeatedly choosing from ‘choice sets’ comprising two or more (e.g. up to six) alternatives at a time, and typically based on pairwise ranking alternatives.
Also, depending on the type of conjoint analysis performed, the hypothetical alternatives included in each choice set are defined on some or all of the criteria (or ‘attributes’) involved. As explained in the previous PAPRIKA subsection above, alternatives defined on a subset of criteria, typically two criteria, are known as ‘partial profiles’, whereas alternatives defined on all attributes (e.g. six or more) are ‘full profiles’.
Conjoint analysis is usually implemented in surveys whereby participants are presented with different choice sets to evaluate – e.g. a dozen different choice sets each. After people have ranked their individual choice sets, they are all combined in a single dataset.
Weights for the criteria are calculated from the aggregated rankings across all participants using regressionbased techniques, such as multinomial logit analysis and hierarchical Bayes estimation.
Which method is best?
The eight direct and two direct methods surveyed above all have strengths and weaknesses – such that choosing the ‘best’ method is itself a multicriteria decision problem!
As discussed earlier, the choicebased methods – the PAPRIKA method and conjoint analysis (or discrete choice experiments, DCE) – are likely to be more valid and reliable than the direct methods, as well as being easier for decisionmakers.
Because they tend to be algorithmically more sophisticated than direct methods, choicebased methods need to be implemented using specialized software instead of ‘by hand’ (as is possible for most direct methods). This is not necessarily a disadvantage of course.
In brief,
 How well do the methods elicit tradeoffs between the criteria?
 How much time and other resources are required to implement the methods?
 How high is the cognitive burden imposed on participants?
 Is a skilled facilitator required to implement the methods?
 Is additional data processing and statistical analysis required?
 Are the underlying assumptions valid with respect to decisionmakers’ preferences?
 Will the outputs produced satisfy decisionmakers’ objectives?
Conclusions
As discussed in this article, MCDA (MultiCriteria Decision Analysis), also known as MCDM (MultiCriteria DecisionMaking), is about methods and software for making decisions when multiple criteria (or objectives) need to be considered together in order to rank or choose between alternatives.
MCDA – often supported by specialized MCDA software – is well suited to supporting decisionmaking by individuals, groups, businesses, nonprofits and government organizations in literally many thousands, if not millions, of possible applications.
Most MCDA applications are based on creating and applying weightedsum models (also known as points systems, etc), which involves the explicit weighting of criteria and scoring of alternatives.
In general terms, ‘good practice’ when implementing weightedsum models (points systems) includes:
 Carefully structuring the decision problem being addressed
 Ensuring that appropriate criteria are specified
 Measuring alternatives’ performance accurately
 Using valid and reliable methods for scoring alternatives on the criteria and weighting the criteria,
 Presenting the MCDA results, including sensitivity analysis, to decisionmakers to support their decisionmaking
Other MCDA / MCDM resources
You might find some of these resources useful.
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Groups and websites
Wikipedia articles
Books
E Ballestero & C Romero (2013), Multiple Criteria Decision Making and its Applications to Economic Problems, Springer
V Belton & T Stewart (2002), Multiple Criteria Decision Analysis: An Integrated Approach, Kluwer
J Dodgson, M Spackman, A Pearman & L Phillips (2009), MultiCriteria Analysis: A Manual, Department for Communities and Local Government
J Figueira, S Greco & M Ehrgott, editors (2016), Multiple Criteria Decision Analysis: State of the Art Surveys, 2nd edition, Springer
B Hobbs & P Meier (2000), Energy Decisions and the Environment: A Guide to the Use of Multicriteria Methods, Springer
S Huber, M Geiger & A de Almeida (2019), Multiple Criteria Decision Making and Aiding, Springer
A Ishizaka & P Nemery (2013), MultiCriteria Decision Analysis: Methods and Software, Wiley
M Köksalan, J Wallenius & S Zionts (2011), Multiple Criteria Decision Making: From Early History to the 21st Century, World Scientific
C Romero & T Rehman (2003), Multiple Criteria Analysis for Agricultural Decisions, Elsevier
PL Yu (2013), MultipleCriteria Decision Making: Concepts, Techniques, and Extensions, Springer
Journals
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