mathematical optimisation Operations Research (OR) is a mathematical discipline that consists of the use of mathematical modelling and optimisation techniques to solve complex problems, maximising or minimising one or more objectives, taking into account the constraints and limitations of the problem.

The purpose is to be able to describe a process to be studied so that we can understand the impact of different decisions. In this way, the results of each decision can be quantified and those that meet the objectives can be selected.

When solving an optimisation problem, the first step to take is mathematical modelling, which takes into account all the possible cases we face. We have decision variables, their behaviour and a measurable objective that depends on the variables. The next step, the optimisation of the model, allows us to obtain the value of the decision variables that maximise or minimise what we are studying: hedging strategies, inventory optimisation, selection of the best transport route, product optionality, mix and optimisation of portfolios, etc. The objective of maximising or minimising a function is to find the value of its variables such that no value of the function for the rest of the variables is greater (maximise) or smaller (minimise).

There are an infinite number of mathematical modelling techniques and optimisation algorithms that can be implemented to maximise or minimise functions. One type of technique or another will be used depending on the nature and complexity of the problem to be solved.

Some of the most commonly used mathematical models are integer linear programming, mixed integer programming, constraint programming, heuristic and metaheuristic techniques, and local search or stochastic programming. If we talk about optimisation algorithms, the most commonly used are the SIMPLEX method, the branching and pruning algorithm, heuristic and metaheuristic algorithms, and decomposition methods, among many others.

## Impact of mathematical optimisation

After numerous success stories over the years, in this section, we briefly break down the main benefits and impacts that some of our clients have had after implementing optimisation systems. Some of these benefits are related to the decisions taken and others to the decision-making process itself.

## Increased operational efficiency

Some business decisions involve such a volume of variables and possibilities that it is impossible for a human to detect all possible solutions and make the best decision. For example, in the commuter problem, the complexity increases exponentially as the size of the problem increases, and there can be millions of solutions.

Optimisation models and algorithms are able to find the best solution among all the possible solutions, taking into account all the variables involved in the problem, in a reasonable time. That is why they provide great value to companies. Finding the fastest route, for example, will reduce mileage, fuel use and CO2 emissions, which directly contributes to cost savings and a reduced carbon footprint.

### Decision support and improved responsive

Such techniques help decision-makers to rely on real data when making a decision. In this way, they can take the model’s recommendation into account and respond more quickly to the decision. This translates into greater time efficiency, allowing decision-makers to concentrate on analysing results.

Saving time is extremely important in today’s ever-changing environment, where decisions must be made quickly to adapt to new paradigms.

### Optimisation in dimensioning and resource utilisation

By using mathematical models and algorithms, companies can maximise the use of their resources, e.g. by optimally planning the personnel at the point of sale. In this way they can have the right number of employees at any given time, neither more (overstaffing) nor less (understaffing). Reducing costs and reducing problems derived from the incorrect management of human resources.

Each of the above benefits brings great value and contributes directly to the success of an organisation. Operations research allows better decisions to be made and business processes to be improved in any area and in any business sector. It is undoubtedly a tool that can make a difference with the competition in such a saturated and competitive market.

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