# Mathematics

## Logistics and Management Mathematics Group (LMMG)

### Research Focus

#### Distance Metric Optimisation

LMMG members are interested in the development of the range of techniques that rely on the minimisation of one or more distance metric. This includes compromise programming, goal programming, and reference point method formulations. Linkages between these techniques have been explored and the issue of variant and choice of metric is being invested both by conventional and evolutionary means. The production of appropriate small sets of sufficiently diverse solutions for decision maker consideration is another topic of interest.

#### Goal Programming

LMMG members are actively involved in the development of theory of goal programming. Methods that allow for more intelligent modelling and solution of goal programmes are of particular interest. This has resulted in contributions to Pareto efficiency detection and restoration, interactive methods, preference modelling techniques, fuzzy goal programming formulations, and linkages with other decision analysis techniques such as the analytical hierarchy process (AHP). Many of the resulting algorithms have been brought together to form the GPSYS intelligent goal programming package.

LMMG members are interested in high quality applications of goal programming to real-life situations. The LMMG has been or is currently involved in applying goal programming to a range of fields including educational planning, finance, fisheries economics, healthcare, diet planning, and analysis of socio-economic behaviour.

#### Modern heuristic methods: heuristics, meta-heuristics, hyper-heuristics

Recently, there have been tremendous efforts on using modern heuristic approaches to solve many NP-Hard combinatorial optimisation and scheduling problems. Many real problems are too complicated with too large search spaces to be solved exactly within realistic timescales, lacking the structures required to allow mathematical programming methods to be used effectively. Modern heuristic methods solve really difficult problems reasonably well in a reasonable computation time.

#### Multiple Objective Evolutionary Algorithms (MOEA’s)

The MMG is particularly interested in MOEA’s that can be applied to multi-objective problems arising in the field of Operational Research. That is, highly constrained models normally with between three and seven objectives. Adaptations to commonly used MOEA’s for this purpose have been investigated. The use of genetic algorithms to solve models arising in the field of distance metric optimisation is also of interest.

#### Reverse Logistics

Reverse logistics is primarily concerned with the care for products and packaging material after they have been used. Due to recent changes in environmental laws and economic incentives, new but still underdeveloped solutions like material recycling and component and product reuse have emerged. Indeed, there is growing evidence that environmental considerations strongly and increasingly favour these above the use of extractive resources. It implies that many products need to flow at their end-of-life from the users back to producers or recovery companies and ultimately, a large fraction of it should be reused in an economic application.

Take-back, reuse, and recycling structures will need to be put forward that will not jeopardise economic development and social welfare; structures that will function in a competitive economic environment; structures that support the principle of sustainable development. Unsolved questions, both fundamental and practical, arise and need to be addressed.

#### Transport and vehicle routing

The group is especially interested in: the mathematical development of decision support systems to aid company and personal (dynamic) vehicle routing; performance evaluation and improvement of meta-heuristics, including Genetic Algorithms, Guided Local Search; developing hybrid approaches from the fields of Operational Research and Artificial Intelligence; and travel speed prediction models.