Temporal constraints and device management for the Skill VRP: Mathematical model and lower bounding techniques

We study a generalization of the Skill VRP that incorporates time windows aspects, precedence and synchronization constraints. Specifically, we are given a logistic network where nodes correspond to customers, and where each customer requires a set of (partially ordered) operations. A set of technicians is available to perform such operations, and each technician is qualified to execute only a subset of them, depending on his skill. By referring to a specific context such as Health Care, customers are patients while technicians are caregivers. In a Field Service context, instead, customers are usually referred to as clients while technicians as field technicians. The innovative aspect is that some operations may require a special device, which must be transported at the customer site and must be present at the customer location together with a technician qualified to use it. Given technician dependent traveling costs, we address the problem of defining the tours for the technicians and for the special device, while respecting the skill compatibility between customers and technicians, and the time windows, precedence and synchronization constraints. We propose a Mixed Integer Linear Programming (MILP) model for the generalized Skill VRP, and present some lower bounding techniques based on the proposed formulation. Preliminary computational experiments show that some lower bounding techniques may rapidly produce good lower bounds, thanks to quite effective valid inequalities. The returned percentage optimality gaps, estimated also thanks to a simple matheuristic, are in fact quite small for several scenarios of medium to large size, by encouraging the use of the proposed lower bounding techniques both as building blocks for designing exact approaches, and also as valuable tools to evaluate the efficacy of more sophisticated heuristic approaches to the problem.