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

Highlights

• We study a generalization of the Skill VRP, by proposing a MILP model.

• We address time windows, precedence and synchronization constraints.

• We manage a special device to perform special operations with a qualified technician.

• We present lower bounding (LB) techniques and quite effective valid inequalities.

• We report computational tests showing that LB techniques may produce good lower bounds.

Abstract

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.

Introduction

Resource constrained routing and scheduling problems are very relevant in theory and practice, and thus they have attracted considerable attention by the scientific literature. In these problems, customers have specific requests that can be met by specialized resources, like technicians, vehicles and devices. In addition, the travel and the delivery of products or services to the customer locations have to be designed. The resources are usually classified as renewable (e.g. vehicles), non-renewable (e.g. spare parts), or doubly constrained (e.g. electric energy).

According to a recent review paper (Paraskevopoulos et al., 2017), the most popular problems in this field are the Skill VRP and the Technician Routing and Scheduling Problem. The Skill VRP has been introduced in Cappanera et al. (2011) and further investigated in Cappanera et al. (2013). There, technicians with given skills must perform routes to serve customers, each requiring a set of skills to be served. The aim is to minimize the total routing costs while satisfying the constraints related to the required levels of skill. An extension to the Skill VRP has been studied in Schwarze et al. (2015). Another generalization has been proposed in Paraskevopoulos et al. (2015), where the service of each customer requires one or more resources (e.g. technicians and equipment). There, the resources of each type are limited, and each of them must be assigned to a single route.

The Technician Routing and Scheduling Problem, introduced in Pillac et al. (2013), can be seen as another generalization of the Skill VRP dealing with multiple resources. Specifically, each technician has a set of skills and may in addition carry a set of tools and spare parts, while each customer requires a subset of them. Tools are renewable resources, whereas spare parts are non-renewable. The goal is to design minimum duration routes for the technicians so that each customer request is served by one technician with the required skills, tools and spare parts. Also in this case the resources of each type are limited, and each of them is carried out by a single technician. Variants to the Technician Routing and Scheduling Problem will be discussed in the section devoted to the literature overview. Interestingly, one of these problems has been the object of a recent challenge facilitated by VeRoLog, the EURO Working Group on Vehicle Routing and Logistics Optimization (Dullaert et al., 2018). Specifically, special tools for measuring milk quality have to be delivered to customers, at their request, over a multiple day time horizon. After the measurement, the tools have to be picked up again. The scheduling of these deliveries along the time horizon, and the routing for the planned deliveries and pickups, are the key decisions to address. The studied problem is indeed a simplification of a richer version faced by the service provider. In fact, as outlined in Kheiri et al. (2019), the real problem includes also the scheduling and the routing of some inspectors, who should visit the customers while the measuring tools are present. Such inspectors are different than the technicians in charge of transporting the measuring tools, since they must possess specific qualifications. The need of technicians with different levels of skill, and the simultaneous presence of qualified technicians and devices, with the treatment of the underlying synchronization issues, would make the problem an even greater challenge to address.

The aim of this paper is to investigate in this direction of research, by addressing the management of technicians with different levels of qualification, and the simultaneous presence of a special device and of a properly qualified technician at some customer locations. Such aspects make the problem particularly significant in application contexts such as Home Care and Field Service. The relevance of considering special devices in Home Care is well described in Aimonino Ricauda et al. (2008), where the authors present a case study to evaluate hospital readmission rates and mortality in elderly patients with chronic obstructive pulmonary disease. The treatment provided at home to such patients requires in fact to consider two types of materials: consumable materials, i.e. non-renewable (e.g. oxygen replenishment), and specialized equipment, i.e. renewable (e.g. Doppler ultrasonograph and electrocardiograph). See also Regis-Hernández et al. (2019) for the description of a dimensioning problem in Home Care where the importance of managing devices is well outlined.

More in detail, with respect to the kernel Skill VRP problem presented in Cappanera et al. (2011) and in Cappanera et al. (2013), here multiple visits per customer are allowed, i.e. more than one technician may visit a customer in order to operate his required services, although each operation must be performed by a single technician. Furthermore, the generalization considers precedence constraints among the operations, time windows at the customers, and maximum workday duration constraints, which are not addressed in Cappanera et al. (2011) and in Cappanera et al. (2013). Finally, the handling of a special device, which is required to perform some special operations, and which must be present at the customer location together with a technician qualified to use it, to the best of our knowledge constitutes an original contribution to the literature on Resource constrained routing and scheduling problems. The special device is moved by the technicians during their own tours, independently of their qualifications, and therefore non standard synchronization between the movement of the special device and the tours of the technicians must be imposed, giving rise to a peculiar tour for the special device, defined as a composition of some technician tour fragments. The resulting Skill VRP generalization appears thus to be relevant from an operational perspective, and it may constitute a basic block for interesting multiple day planning extensions.

The plan of the paper is the following. Section 2 proposes an overview of papers dealing with some of the characteristics of the generalized Skill VRP. Section 3 describes the studied problem, while Section 4 presents the Mixed Integer Linear Programming (MILP) model proposed for its mathematical formulation. Some simple, but computationally quite effective, valid inequalities to enhance the Linear Programming (LP) relaxation of the MILP model are presented in Section 5. Finally, Section 6 presents some lower bounding techniques, based on the proposed MILP formulation, and the results of computational experiments, showing that some lower bounding techniques, when enhanced via the proposed valid inequalities, may rapidly produce good lower bounds. The returned percentage optimality gaps, estimated also thanks to simple matheuristics, 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 for the generalized Skill VRP, and also as valuable tools to evaluate the efficacy of more sophisticated heuristic approaches to the problem. Section 7 concludes the paper.