Abstract

Abstract

Abstract

Abstract

Abstract
When planning a selling season, a car rental company must decide on the number and type of vehicles in the fleet to meet demand. The demand for the rental products is uncertain and highly price-sensitive, and thus capacity and pricing decisions are interconnected. Moreover, since the products are rentals, capacity "returns". This creates a link between capacity with fleet deployment and other tools that allow the company to meet demand, such as upgrades, transferring vehicles between locations or temporarily leasing additional vehicles. We propose a methodology that aims to support decision-makers with different risk profiles plan a season, providing good solutions and outlining their ability to deal with uncertainty when little information about it is available. This matheuristic is based on a co-evolutionary genetic algorithm, where parallel populations of solutions and scenarios co-evolve. The fitness of a solution depends on the risk profile of the decision-maker and its performance against the scenarios, which is obtained by solving a mathematical programming model. The fitness of a scenario is based on its contribution in making the scenario population representative and diverse. This is measured by the impact the scenarios have on the solutions. Computational experiments show the potential of this methodology regarding the quality of the solutions obtained and the diversity and representativeness of the set of scenarios generated. Its main advantages are that no information regarding probability distributions is required, it supports different decision-making risk profiles, and it provides a set of good solutions for an innovative complex application.

Abstract
In this paper, we explore the use of reference values (predictors) for the optimal objective function value of hard combinatorial optimization problems, instead of bounds, obtained by data mining techniques, and that may be used to assess the quality of heuristic solutions for the problem. With this purpose, we resort to the rectangular two-dimensional strip-packing problem (2D-SPP), which can be found in many industrial contexts. Mostly this problem is solved by heuristic methods, which provide good solutions. However, heuristic approaches do not guarantee optimality, and lower bounds are generally used to give information on the solution quality, in particular, the area lower bound. But this bound has a severe accuracy problem. Therefore, we propose a data mining-based framework capable of assessing the quality of heuristic solutions for the 2D-SPP. A regression model was fitted by comparing the strip height solutions obtained with the bottom-left-fill heuristic and 19 predictors provided by problem characteristics. Random forest was selected as the data mining technique with the best level of generalisation for the problem, and 30,000 problem instances were generated to represent different 2D-SPP variations found in real-world applications. Height predictions for new problem instances can be found in the regression model fitted. In the computational experimentation, we demonstrate that the data mining-based framework proposed is consistent, opening the doors for its application to finding predictions for other combinatorial optimisation problems, in particular, other cutting and packing problems. However, how to use a reference value instead of a bound, has still a large room for discussion and innovative ideas. Some directions for the use of reference values as a stopping criterion in search algorithms are also provided.