Probabilistic Multi Criteria Decision Analysis

A tool to estimate scores of alternatives and their uncertainties based on a Multi Criteria Decision Analysis (MCDA) approach.

Table of Contents

• Contributing
• MCDA quick overview and applications
• Input and output file paths
• Input information needed in the configuration file
• Output
• Requirements
• Running ProMCDA
• Running the tests
• Toy example
• A high level summary
• General information and references

Contributing

We welcome contributions from the community! Before contributing, please read our Contribution Guidelines to learn about our development process, coding standards, and how to submit pull requests.

MCDA quick overview and applications

A MCDA approach is a systematic framework for making decisions in situations where multiple criteria or objectives need to be considered. It can be applied in various domains and contexts. Here are some possible usages of an MCDA approach:

• Environmental impact assessment: assess the environmental consequences of various projects or policies by considering multiple environmental criteria.
• Project selection: choose the most suitable project from a set of alternatives by considering multiple criteria such as cost, risk, and strategic alignment.
• Healthcare decision-making: decide on the allocation of healthcare resources, such as funding for medical treatments or the purchase of medical equipment, while considering criteria like cost-effectiveness, patient outcomes, and ethical considerations. It applies to other decision-making problems too.
• Investment portfolio optimization: construct an investment portfolio that balances criteria like risk, return, and diversification.
• Personnel recruitment: evaluate job candidates based on criteria like qualifications, experience, and cultural fit.
• Location analysis: choose the optimal location for a new facility or business by considering factors like cost, accessibility, and market potential.
• Public policy evaluation: assess the impact of proposed policies on various criteria, such as economic growth, social welfare, and environmental sustainability.
• Product development: prioritize features or attributes of a new product by considering criteria like cost, market demand, and technical feasibility.
• Disaster risk management: identify high-risk areas and prioritize disaster preparedness and mitigation measures based on criteria like vulnerability, exposure, and potential impact.
• Energy planning: make decisions about energy resource allocation and investments in renewable energy sources, taking into account criteria like cost, environmental impact, and reliability.
• Transportation planning: optimize transportation routes, modes, and infrastructure investments while considering criteria like cost, time efficiency, and environmental impact.
• Water resource management: optimize water allocation for various uses, including agriculture, industry, and municipal supply, while considering criteria like sustainability and equity.
• Urban planning: decide on urban development projects and land-use planning based on criteria such as social equity, environmental impact, and economic development.

These are just a few examples of how MCDA can be applied across a wide range of fields to support decision-making processes that involve multiple, often conflicting, criteria. The specific application of MCDA will depend on the context and the goals of the decision-maker.

Before using ProMCDA, we suggest you first to get familiar with the main steps and concepts of an MCDA method: the normalization of the criteria values; their polarities and weights; and the aggregation of information into a composite indicator. In MCDA context an alternative is one possible course of action available; an indicator, or criterion, is a parameter that describes the alternatives. The variability of the MCDA scores are caused by:

• the sensitivity of the algorithm to the different pairs of normalization/aggregation functions (--> sensitivity analysis);
• the randomness that can be associated to the weights (--> robustness analysis);
• the uncertainty associated with the indicators (--> robustness analysis).

Here we define:

• the sensitivity analysis as the one aimed at capturing the output score stability to the different initial modes;
• the robustness analysis as the one aimed at capturing the effect of any changes in the inputs (their uncertainties) on the output scores.

The tool can be also used as a simple (i.e. deterministic) MCDA ranking tool with no robustness/sensitivity analysis (see below for instructions).

Input and output file paths

All input and output files are saved by default in the input_files and output_files folders located in the project root directory. The two variables, which define the default input and output paths, DEFAULT_INPUT_DIRECTORY_PATH and DEFAULT_OUTPUT_DIRECTORY_PATH are defined in utils_for_main.py and utils_for_plotting.py. Users can set custom environmental variables in their environment to override the default directory paths. In the terminal, users can override one or both default variables by typing:

export PROMCDA_INPUT_DIRECTORY_PATH=/path/to/custom/input/directory
export PROMCDA_OUTPUT_DIRECTORY_PATH=/path/to/custom/output/directory

Please ensure that you execute the above command(s) from within the project environment. The variable names PROMCDA_INPUT_DIRECTORY_PATH and PROMCDA_OUTPUT_DIRECTORY_PATH are fixed as defined in utils_for_main.py and utils_for_plotting.py.

This allows the user to customize the directory paths without modifying the source code of the Python package, and using relative paths in the configuration file.

Input information needed in the configuration file

A configuration file is needed to runProMCDA. The configuration file collects all the input information to run ProMCDA in your specific study case. You find a configuration.json file as an example in this directory: please modify it for your needs. In the following, the entries of the configuration file are described.

Path to the input matrix, a table where rows represent the alternatives and columns represent the indicators. Be sure that the first column of the input matrix contains the names of the alternatives.

Be sure that there are no duplicates among the rows. If the values of one or more indicators are all the same, the indicators are dropped from the input matrix because they contain no information. Examples of input matrix:

• input matrix without uncertainties for the indicators (see an example here: tests/resources/input_matrix_without_uncert.csv);
• input matrix with uncertainties for the indicators (see an example here: tests/resources/input_matrix_with_uncert.csv).

The example input matrix with uncertainties is designed for an example where the PDFs describing the indicators are respectively: uniform; exact; normal (or lognormal); exact; and normal (or lognormal). Please modify it for your specific needs.

If the input matrix without uncertainties has any values of any indicators that are negative, those values are rescaled between [0,1]. This is needed because some normalization functions (as for example target) cannot handle negative values properly.

The input matrix with uncertainties has the following characteristics:

• if an indicator is described by an exact probability density function (PDF), one needs only a column with its values;
• if an indicator is described by a uniform PDF, one needs two columns with the lowest and highest values (in this order);
• if an indicator is described by a normal PDF, one needs two columns with the mean and standard deviation values (in this order);
• if an indicator is described by a lognormal PDF, one needs two columns with the log(mean) and log(standard deviation) values (in this order);
• if an indicator is described by a Poisson PDF, one needs only one colum with the rate.

If any mean value of any indicator is equal or smaller of its standard deviation - in case of a normal or lognormal PDF - ProMCDA breaks to ask you if you need to investigate your data further before applying MCDA. In fact, your data shows a high variability regarding some indicators. If you want to continue anyway, negative sampled data will be rescaled into [0,1], as in the case without uncertainty.

List of polarities for each indicator, "+" (the higher the value of the indicator the better for the evaluation) or "-" (the lower the value of the indicator the better).

The configuration file can trigger a run with or without sensitivity analysis; this is set in the sensitivity_on parameter (yes or no); if no is selected, then the pair normalization/aggregation should be given in normalization and aggregation. If yes is selected, then the normalization/aggregation pair is disregarded.

Similarly, a run with or without uncertainty on the indicators or on the weights (i.e. with robustness analysis) can be triggered by setting the robustness_on parameter to yes or no. If robustness_on is set to yes, then the uncertainties might be on the indicators (on_indicators) or on the weights (on_single_weights or on_all_weights). In the first case (on_single_weights=yes) one weight at time is randomly sampled from a uniform distribution; in the second case (on_all_weights=yes) all weights are simultaneously sampled from a uniform distribution. If there is no uncertainty associated to the weights, then the user should provide a list of weights in the given_weights parameter for the indicators. The sum of the weights should always be equal to 1 or the values will be normalised internally. Depending on the different options, information not needed are disregard. Sensitivity and robustness analysis can be run together. If robustness analysis is selected, it can run either on the weights or on the indicators, but not on both simultaneously.

If robustness analysis is selected, a last block of information is needed: the Number of Monte Carlo runs, "N" (default is 0, then no robustness is considered; N should be a sufficient big number, e.g., larger or equal than 1000). The number of cores used for the parallelization; and a List of marginal distributions for each indicator; the available distributions are:

• exact, "exact",
• uniform distribution, "uniform"
• normal distribution, "normal"
• lognormal distribution, "lnorm"
• Poisson distribution, "poisson"

Output

The user gives the path to output file (e.g. path/output_file.csv). In the output file the scores (normalised or rough) and the ranks relative to the alternatives can be found in the form of CSV tables. If the weights are iteratively sampled, multiple tables are saved in a PICKLE file as an object dictionary. Plots of the scores are saved in PNG images. The configuration.json file is saved in the output directory too; the information stored in the configuration settings are useful in case one runs multiple tests and needs to review them.

All file names of the output objects are followed by a time stamp that group them with the specific test setting.

To retrieve a PICKLE file in Python one can:

import pickle

# Replace 'your_file.pickle' with the path to your PICKLE file
file_path = 'your_file.pickle'

# Load the PICKLE file
with open(file_path, 'rb') as file:
    data = pickle.load(file)

# Now, 'data' contains the object or data stored in the PICKLE file

Note that in case of a robustness analysis, the error bars in the plots are only shown for non-normalized scores. This is because when one calculates the standard deviation after rescaling, the denominator used in the standard deviation formula becomes smaller. This results in a higher relative standard deviation compared to the mean that is solely an artificial effect.

Requirements

On Windows:

conda create --name <choose-a-name-like-Promcda> python=3.9
activate.bat <choose-a-name-like-Promcda>
pip install -r requirements.txt

On Mac and Linux:

conda create --name <choose-a-name-like-Promcda> python=3.9
source activate <choose-a-name-like-Promcda>
pip install -r requirements.txt

The Python version should de 3.9 or higher.

Running ProMCDA

From the root dir, via command line

• on Windows:

activate.bat <your-env>
python3 -m mcda.mcda_run -c configuration.json

• on Mac and Linux:

source activate <your-env>
python3 -m mcda.mcda_run -c configuration.json

where an example of configuration file can be found in ./configuration.json.

We tested ProMCDA on Windows, Linux and Mac. We identified a possible issue with some Windows machines caused by the library kaleido (used to generate static images) and reported here. If you are experiencing any unexpected behavior of the package, we provide to you also the possibility to run it via Docker.

First, build the docker image:

docker build -t <your-docker-image> .

Then, run the Docker container:

docker run -p 4000:80 <your-docker-image>

Change the configuration file for your needs.

Running the tests

python3 -m pytest -s tests/unit_tests -vv

Toy example

ProMCDA contains a toy example, a simple case to test run the package. The toy example helps you identify the best car models (i.e., the alternatives) you can buy based on a few indicators (e.g., Model, Fuel Efficiency, Safety Rating, Price, Cargo Space, Acceleration, Warranty). In the output_files/toy_example/ directory you can find a few different tests that have been run already, with their relative configuration and output files. The input matrix for the toy example is in input_files/toy_example/car_data.csv. If you want to test the package yourself, you have to modify your configuration file according to the desired experiment. The directory toy_example_utilities contains also a Jupyter notebook to allow you to modify the input matrix easily. The chosen example is very simple and not suitable for a robustness analysis test. Running the robustness analysis requires an input matrix containing information on the uncertainties of the criteria as described above, and it is not under the scope of the toy example.

A high level summary

If no robustness analysis is selected, then:

• the indicator values are normalized by mean of all the possible normalization methods (or by the selected one);
• the normalized indicators are aggregated by mean of all the possible aggregation methods (or by the selected one), by considering their assigned weights (in the given_weights parameter);
• the resulting scores of all the combinations normalization/aggregation (or the selected ones only) are provided in form of a csv table and plots in png format in the output directory.

If the weights are randomly sampled (robustness analysis of the weights), then:

• all weights or one weight at time are randomly sampled from a uniform distribution [0,1];
• the weights are normalized so that their sum is always equal to 1;
• if all weights are sampled together, MCDA calculations receive N-inputs (N being the number of monte_carlo_runs; if the weights are sampled one at time, MCDA will receive (n-inputs x num_weights) inputs;
• iterations 1,2,3 of the first condition follow;
• the results of all the combinations normalization/aggregation (or the one selected) are provided in the form of mean and standard deviation over all the runs (if the weights are iteratively sampled, this applies for num_indicators-times).

If the robustness analysis regards the indicators, then:

• for each indicator, the parameters (e.g., mean and standard deviation) describing the marginal distribution under interest are extracted from the input matrix;
• for each N, and for each indicator, a value is sampled from the relative assigned marginal distribution: therefore, one of N input matrix is created;
• normalizations and aggregations are performed as in points 1,2 of the first case: a list of all the results is created in the output directory;
• mean and standard deviation of all the results are estimated across (monte_carlo_runs x pairs of combinations);
• in this case, no randomness on the weights is allowed, therefore, they need to be assigned to all indicators in the given_weights parameter.

General information and references

The aggregation functions are implemented by following Langhans et al., 2014

The normalization functions minmax, target and standardized can produce negative or zero values, therefore a shift to positive values is implemented so that they can be used also together with the aggregation functions geometric and harmonic (which require positive values).

The code implements 4 normalization and 4 aggregation functions. However, not all combinations are meaningful or mathematically acceptable. For more details refer to Table 6 in Gasser et al., 2020

The standard deviation of rescaled scores with any form of randomness are not saved nor plotted because they cannot bring a statistically meaningful information. In fact, when one calculates the standard deviation after rescaling between (0,1), the denominator used in the standard deviation formula becomes smaller. This results in a higher relative standard deviation compared to the mean. However, the higher relative standard deviation is not indicating a greater spread in the data but rather a consequence of the rescaling operation and the chosen denominator in the standard deviation calculation.