08/09/2024

# Decision support 101

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## Help with the decision

Any industrial problem consists, under certain conditions, of maximizing profit or minimizing expenditure. In this context, profit and expenditure do not always refer to a monetary variable, it can also be expressed by time, distance or others.

Very often, the problem is stated in a raw way, that is to say by a text or a specification. The manufacturer was not an expert in the field of writing a mathematical problem, the specifications include all kinds of data, useful or not for its modelling.

Even in the context where you yourself are a sponsor, you may not know the extent of your problem, and you discover in the course of the water the various constraints and variables to deal with.

Another problem arises once the mathematical modeling carried out: what computer tool was used to solve this problem? what algorithm Choose ? his simulation? her complexity ? its optimality?

Building and solving an industrial problem therefore requires rigour, flexibility of mind and following a precise modeling approach.

## Optimization and decision support

A problem D is decision if the answer is binary: Yes or No. We note Yes(D) the set of instances to which we answer Yes.

Consider the following decision problem: let a graph G weighted, is there a tree covering? Yes(D) = {non-cyclic connected subgraph of G}. The problem: existence of a tree covering of weight ≤ k is also a decision problem. The optimization problem is to find the value k such that it is minimal.

In order to better understand the two concepts, we will take an example: You want to take a tour of Europe, visiting a certain number of cities in a period of 6 months. In addition, you want to stay a certain amount of time in each place so that you can visit the tourist areas and admire the scenery.

This kind of problem has different ways of being modeled depending on what you want to do: be the fastest, favor densely touristic areas, etc. It is necessary to select a decision among a set of possible decisions so as to optimize the chosen criterion.

Modeling includes a search for minimum or maximum, so it is a optimization. Decision support problems all contain the following three points:

• The type of decision: what we want to do (here we are looking for an optimization)
• Possible decisions: what can be done (the definition field)
• The selection criterion: how one chooses (the modeling of the problem).

The problem studied is placed in a certain context which will be translated into parameters. All the relationships between them are represented in the model. This can either take the form of a mathematical model or a graph.

The modeling is only a schematic representation of the problem, only the elements deemed relevant are retained in the construction of the decision. It proceeds by simplifications and omissions.

The model's environment can also play a role. Whether deterministic or with uncertainty, it is present via probability laws, stochastics, etc. within constraints.

The selection criterion can lead to different solutions depending on the parameter put forward. In some cases, the model has only one selection criterion, this is called operational research.

All models consist of three basic components:

• Result variables are outputs. The reflect the level of efficiency of the system. These are dependent variables.
• Decision variables describe alternative actions.
• Uncontrollable variables are factors that affect the outcome but are not under the control of the decision maker. Either these factors are fixed or they may vary.

Components are linked together by expressions math in a framework of quantitative models. A principle of choice is a criterion that describes the acceptability of an approach solution.

A model can be a normative model or a descriptive model. In the first, the chosen solution is clearly the best of all the possible alternatives. To find it, you have to examine all the alternatives and prove that the one selected is indeed the best (we talk about optimization). Descriptive models study alternative actions under different input and process configurations. Not all alternatives are verified, only a given set are.

## Solution and decision support

Once the model has been created and a solution has been found, it is important to analyze it in order to validate the model. The latter was only a schematic representation of the problem, it may not be suitable for the intended purpose. A solution highlights the validity of the decision choices and model choices. Only the decision-maker/sponsor can validate the approach taken.

The diagram of the decision support process is as follows: