PREV

NEXT

# In aHead Research, we believe mathematical optimization is necessary for AI-based solutions.

### How do we use mathematical optimization?

In aHead Research, we provide our experience and skills to the business world, facing optimization issues with mathematical solutions. Our technical and scientific knowledge of the several optimization methods allows us to offer adaptive solutions to the complexity of the various systems and contexts.

### Combinatorial optimization

An optimization issue with discrete variables – for example, when the variables describe a choice among a finite, yet enormous number of options, is known as a problem of combinatorial optimization. The areas of application are several, from production plant to combinatorial online auctions. Our solutions to combinatorial optimization issues support managers in complex decision-making processes regarding costs, profit, or overall revenues. Among the specific cases of aHead we have the optimal assortment of a store, primary and secondary distribution of wares in the supply chain, the planning of production in plants and scheduling of activities.

### Combinatorial optimization

An optimization issue with discrete variables – for example, when the variables describe a choice among a finite, yet enormous number of options, is known as a problem of combinatorial optimization. The areas of application are several, from production plant to combinatorial online auctions. Our solutions to combinatorial optimization issues support managers in complex decision-making processes regarding costs, profit, or overall revenues. Among the specific cases of aHead we have the optimal assortment of a store, primary and secondary distribution of wares in the supply chain, the planning of production in plants and scheduling of activities.

### Continuous and mixed / integer optimization

Optimization problems with continuous and integer variables, for instance, when the variables represent measures that could have continuous or only integer values, are known respectively as continuous and mixed / integer optimization problems. Such problems are found in most business areas -revenue management, online trading, balancing offer/demand of electricity. A special mention here goes to the training of supervised machine learning models (Deep Neural Network, Support Vector Machines) and unsupervised (clustering). In fact, in order to perform the training of the parameters and to set up any machine-learning model, optimization – typically continuous, mixed / integer and stochastic – algorithms are needed.

### Continuous and mixed / integer optimization

Optimization problems with continuous and integer variables, for instance, when the variables represent measures that could have continuous or only integer values, are known respectively as continuous and mixed / integer optimization problems. Such problems are found in most business areas -revenue management, online trading, balancing offer/demand of electricity. A special mention here goes to the training of supervised machine learning models (Deep Neural Network, Support Vector Machines) and unsupervised (clustering). In fact, in order to perform the training of the parameters and to set up any machine-learning model, optimization – typically continuous, mixed / integer and stochastic – algorithms are needed.

### Stochastic optimization

The methods of stochastic optimization are based on algorithms that incorporate probability into the problem’s data, in the objective function, in the limitations or in the algorithm itself. In such algorithms, variability is always considered, and the goal is to identify and manage the risk to which such variables are exposed. For instance, in the case of the optimization of a shop’s assortment of products, the intrinsically variability of demand cannot be ignored, as it relies on several factors: temperatures and weather, seasonal products and sales, consumer behavior, etc. Stochastic optimization finds great application in the supply chain as it offers and intelligent method for dealing with unforeseen situations and uncertainty regarding the several steps – reordering products for the point of sale, reducing stock-outs and overstocks, optimizing space in the warehouse and moving goods. Moreover, the Operation Research Scientists in aHead Research are experienced in modeling and solving stochastic optimization problems for revenue management, trading, risk management and balancing demand and offer of electricity.

### Stochastic optimization

The methods of stochastic optimization are based on algorithms that incorporate probability into the problem’s data, in the objective function, in the limitations or in the algorithm itself. In such algorithms, variability is always considered, and the goal is to identify and manage the risk to which such variables are exposed. For instance, in the case of the optimization of a shop’s assortment of products, the intrinsically variability of demand cannot be ignored, as it relies on several factors: temperatures and weather, seasonal products and sales, consumer behavior, etc. Stochastic optimization finds great application in the supply chain as it offers and intelligent method for dealing with unforeseen situations and uncertainty regarding the several steps – reordering products for the point of sale, reducing stock-outs and overstocks, optimizing space in the warehouse and moving goods. Moreover, the Operation Research Scientists in aHead Research are experienced in modeling and solving stochastic optimization problems for revenue management, trading, risk management and balancing demand and offer of electricity.

### Heuristics and meta-heuristics

There are problems characterized by hundreds of thousands, if not millions, of variables and limitations. In such cases, traditional method may be too slow give the computational complexity or may take longer times to find an exact solutions. For such kind of problems, in aHead Research we resort to heuristics and meta-heuristics, algorithms that can find an approximate, yet acceptable solution to the problem. At a later stage, such solution is fine-tuned to the problem through the application of specific algorithms, able to transform it via operators such as mutation and crossover. Such kind of heuristics allows for the obtainment of a solution that balances the optimization goals with completeness, accuracy, and rapid execution.

An example of a real application is the composition of outbound pallets in a distribution centrer. Every day, hundreds or thousands of pallets leave distribution centers and deciding their composition is a difficult process, in that is needs to account for factors such as weight and volume of each item, optimizing each pallet’s saturation and minimizing movement in the warehouse with the aim of reducing costs.

### Heuristics and meta-heuristics

There are problems characterized by hundreds of thousands, if not millions, of variables and limitations. In such cases, traditional method may be too slow give the computational complexity or may take longer times to find an exact solutions. For such kind of problems, in aHead Research we resort to heuristics and meta-heuristics, algorithms that can find an approximate, yet acceptable solution to the problem. At a later stage, such solution is fine-tuned to the problem through the application of specific algorithms, able to transform it via operators such as mutation and crossover. Such kind of heuristics allows for the obtainment of a solution that balances the optimization goals with completeness, accuracy, and rapid execution.

An example of a real application is the composition of outbound pallets in a distribution centrer. Every day, hundreds or thousands of pallets leave distribution centers and deciding their composition is a difficult process, in that is needs to account for factors such as weight and volume of each item, optimizing each pallet’s saturation and minimizing movement in the warehouse with the aim of reducing costs.

### Multi-goal optimization

For case where the optimization needs to account for more goals that sometimes are in contrast with each other, we aim at identifying the minimal set of solutions, from which to pick the best one based on a specific, examined need. All such solutions are equally optimal and their multeplicity is due to the best solution of a specific goal being in contrast with at least another one. An example of such process is the optimization of goods distribution in the supply chains. For such kind of process, it is usually ideal to optimize the saturation of the trucks and minimize their movement and their stoppage times, while respecting all time limits of the node of the chain, means capacity and products compatibility. The solution is transformed according to the priority attributed to each individual goal, to fit the client’s specific needs.

### Multi-goal optimization

For case where the optimization needs to account for more goals that sometimes are in contrast with each other, we aim at identifying the minimal set of solutions, from which to pick the best one based on a specific, examined need. All such solutions are equally optimal and their multeplicity is due to the best solution of a specific goal being in contrast with at least another one. An example of such process is the optimization of goods distribution in the supply chains. For such kind of process, it is usually ideal to optimize the saturation of the trucks and minimize their movement and their stoppage times, while respecting all time limits of the node of the chain, means capacity and products compatibility. The solution is transformed according to the priority attributed to each individual goal, to fit the client’s specific needs.