One of the most useful groups of mathematical tools for decision-making is Multiple Criteria Decision Analysis or MCDA. As the name suggests, MCDA is designed to deal with situations where we are choosing among alternatives that can be evaluated on multiple criteria. Many decision problems can be formulated in this way. For example, if we are deciding which phone to buy we need to consider criteria such as price, screen size, storage capacity, or color. Let's explore what the process of solving such problems looks like.

MCDA Decision Process

We can break the process of analyzing a decision using MCDA into several steps. These steps differ slightly depending on which book you read, but the general idea remains the same:

1. Define your problem

2. Define a set of criteria

3. Define a set of alternatives from which to choose

4. Evaluate each alternative against each criterion

5. Apply an MCDA method to solve your problem

Let's take a closer look at each of these five steps:

1. Define your Problem

Before we can start thinking about criteria, possible alternatives, or a specific MCDA method, we must first define the problem we want to solve. We have already seen one such definition: choosing which phone to buy. Other decision problems I have recently encountered include choosing a new programming language to learn, choosing stocks to invest in, or choosing a new project to work on.

Given these examples, one might be tempted to think that the only category of problems you can solve with MCDA is choosing one or more alternatives from a predefined set. Some authors call this a choice problematic. It is only one of many different problematics than can be addressed with MCDA. When we want to sort our alternatives into different categories such as "good", "bad", and "not feasible", we are solving a sorting problematic problem. Sometimes we need not only to find the best alternative but to rank or order all of our options according to how "good" they are. These are ranking problematic problems. Or we may simply want to describe the problem at hand more formally so that we can understand it better or discuss it with others. This is description problematic.

2. Define a Set of Criteria

Criteria represent different characteristics or points of view that we need to consider when making a decision. More often than not we need to take more than one criterion into account. It might be a bit unreasonable to choose a phone to buy based on price alone. We want to know what we are getting for our money and take other characteristics into account. Multiple criteria decision problems are very common. That's why knowing about MCDA methods can be very useful.

Ideally, we want a set of criteria that is both complete and non-redundant. Completeness means that we include all important criteria and that we haven't overlooked any important aspect or viewpoint. Importance is subjective though. When choosing a laptop not care whether a laptop has a SIM card slot or not. However, for a constantly traveling businessperson, this information is very important.

Redundancy describes the opposite problem. Our set of criteria may be complete, but it may now contain multiple criteria that give us the same or very similar information. Suppose you are looking for an apartment to rent. Your criteria may include both a base price and a full price that includes additional services such as heating and water. There is quite a lot of redundancy between these two criteria. It may be safe to drop one of these two criteria and simplify the problem.

There is much more to criteria than just completeness and non-redundancy. We will discuss this topic in depth in a future article.

3. Define a Set of Alternatives

Making a list of all our options is one of the least studied and understood steps in the decision-making process. In the simpler case, we face a closed decision problem. The set of alternatives is finite and does not change over time. Voting in an election is a typical example. There is usually a point in time after which no new candidate can enter the race and our choices become very clear.

But we more frequently encounter open problems. In an open problem, the list of alternatives is not fixed. New choices may be added over time, while others become unavailable. Laptop manufacturers are constantly introducing new models. Apartments get bought and built. To make matters worse, our set of alternatives can be virtually infinite. There are no limits to choosing a new business idea or a blog post topic.

When faced with an open decision problem, it makes sense to spend some time exploring the space of alternatives, for example by browsing the web. But there are diminishing marginal returns to this process. At some point, the cost of additional time spent searching for more data outweighs the benefit of discovering a slightly better alternative. This is a good moment to stop and impose an artificial closure on the set of options. For example, when buying a phone or a laptop, we might limit our search to a single online store or even a small subset of brands.

There are also decision problems where we need to find an optimal mix of different elements. Building a stock portfolio is a good example. Not only do we want to choose stocks to invest in, but we also need to decide on how much to invest in each stock. It is difficult to represent our options as a set of distinct alternatives, so MCDA methods are of limited use. Instead, we can address such problems by methods of mathematical programming (and nowadays also by machine learning and AI) which can be a subject of a separate series of articles.

4. Evaluate each Alternative against each Criterion

Evaluating alternatives can be quite easy when a criterion has a clear numerical value. We can find a phone's price, storage capacity, screen size, or color on the web. It is equally easy to find answers to "yes/no" criteria such as the presence of a MicroSD card slot.

But in many decision problems, we encounter criteria that are much harder to evaluate. In my company, one of the criteria we use to select a project to work on is our personal interest. How can we express this interest in a way that allows us to use MCDA methods? One solution would be to use a numerical scale, say from 1 (not interesting) to 5 (very interesting). We will discuss other approaches in future articles.

Regardless of our choice of representation, our ranking may still be inconsistent, influenced by our current mood, or by the fact that we just read a blog post related to a project we are currently trying to evaluate. No matter how hard we try, it is virtually impossible to eliminate bias and subjectivity from decision-making. MCDA methods can mitigate the problem, but not eliminate it.

It is worth noting that there are also MCDA methods that embrace this kind of uncertainty, instead of seeing it as a problem. Such methods are typically based on probability or fuzzy set theory.

5. Apply an MCDA Method to Solve your Problem

There are many MCDA methods to choose from. Their suitability depends on the size of the problem (number of alternatives and criteria), the types of criteria, or the problematic. Some methods can be easily implemented in Microsoft Excel or a similar editor, while others require specialized software. At the same time, it is not uncommon for multiple methods to be applicable to the same problem and even to produce very similar results.

In future articles, I will introduce some of most common MCDA methods, and discuss their applications, requirements, strengths and weaknesses. We will start our journey with Even Swaps. This method is very easy to use and it doesn't require any complicated mathematical formulas.

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Let me finish by recommending a few books and resources related to the topic of decision-making that I used as sources for this article or enjoyed reading in general:

• Decision Theory: A Brief Introduction by Sven Ove Hansson,

• Multiple Criteria Decision Analysis: State of the Art Surveys edited by José Figueira, Salvatore Greco and Matthias Ehrogott,

• Thinking, Fast and Slow by Daniel Kahneman,

• Misbehaving: The Making of Behavioral Economics by Richard Thaler,

• and finally the Farnam Street blog by Shane Parrish.