A biased random key genetic algorithm for the field technician scheduling problem

Highlights

• A problem that service companies often face is tackled: the scheduling of technicians.

• The sum of priority values associated with the tasks performed each day is maximized.

• A BRKGA combined with elaborated decoders and with an unusual elite set is proposed.

• Numerical experiments show that the proposed methods provide high quality solutions.

Abstract

This paper addresses a problem that service companies often face: the field technician scheduling problem. The problem considers the assignment of a set of jobs or service tasks to a group of technicians. The tasks are in different locations within a city, with different time windows, priorities, and processing times. Technicians have different skills and working hours. The main objective is to maximize the sum of priority values associated with the tasks performed each day. Due to the complexity of this problem, constructive heuristics that explore specific characteristics of the problem are developed. A customized Biased Random Key Genetic Algorithm (BRKGA) is also proposed. Computational tests with 1040 instances are presented. The constructive heuristics outperformed a heuristic of the literature in 90% of the instances. In a comparative study with optimal solutions obtained for small-sized problems, the BRKGA reached 99% of the optimal values; for medium- and large-sized problems, the BRKGA provided solutions that are on average 3.6% below the upper bounds.

Introduction

This paper analyses the field technician scheduling problem (FTSP), which service companies often face [1], [2], [3], especially in telecommunications [4], [5], [6], [7]. These services are generally maintenance or installation services that need to take into account several parameters: technicians with different skills and working hours, time windows of tasks, cost of displacement and travel time, priority or urgency of service, due dates of the orders, etc. There are several possible objectives for the FTSP, such as maximizing the number of performed tasks, minimizing the completion time of all tasks, minimizing costs or total displacement, minimizing the number of technicians, etc. In this paper we will primarily address the maximization of the total priority of the performed tasks.

According to Kovacs et al. [4] and Pillac et al. [8], the FTSP is an extension of the vehicle routing problem with time windows (VRPTW), which is NP-hard. As far as we know, the research on FTSP is not extensive and most papers focus on heuristic methods. Tsang and Voudouris [7] and Xu and Chiu [9] were among the first authors to study the problem. After 2005, the problem received increased attention in the literature, particularly in 2007, when the French Operational Research Society proposed the technician and task scheduling problem as the subject of the 2007 challenge in collaboration with France Telecom. This company needed to deal with a large increase in the number of services while having a limited number of technicians. Services were divided into three groups of priorities and the objective function minimized a weighted sum of the completion time of the last task of each group. Technicians had different levels of skills and teams could be organized to perform the services. Furthermore, some tasks could be outsourced and some tasks had precedence constraints; on the other hand, the time windows to perform the tasks were not imposed. Hashimoto et al. [5] applied the Greedy Randomized Adaptive Search Procedure (GRASP) metaheuristic and Cordeau et al. [6] developed constructive heuristics and customized the Adaptive Large Neighborhood Search (ALNS) heuristic to solve this problem. Kovacs et al. [4] studied a similar problem considering tasks with the same priority but with different time windows. The objective function intended to minimize the costs of displacement and outsourcing. The authors also applied the ALNS heuristic.

Other research studies also considered the formation of teams of technicians. In Dohn et al. [10] the number of assigned tasks in a day is maximized subject to restrictions related to the teams and tasks time windows and with a limited number of teams. The authors introduced a branch-and-price approach to address the problem and found 11 optimal solutions of the 12 realistic instances. Li et al. [3] intended to minimize the total number of workers and total displacement in the port of Singapore by allocating different types of workers in teams. In this problem each job must be carried out in a preestablished time window. The authors developed two constructive heuristics associated with simulated annealing to solve the problem. In Overholts II et al. [11] the daily maintenance of military equipment in USA is scheduled to maximize the weighted sum of the maintenance tasks performed. A mixed integer linear programming model was developed to solve the problem. The authors presented a detailed sensitivity analysis on security criteria and quality of daily maintenance services.

The FTSP was also addressed considering real problems with the assumption that each task can be performed by a single technician. In Tsang and Voudouris [7], the objective was the minimization of the total cost: the cost of displacement of engineers, the cost of overtime, and the cost (or penalty) of tasks not executed. This problem was a real problem faced by British Telecom. Instead of specifying time windows in hours, three types of time windows were adopted for tasks: morning, afternoon, and indifferent. The authors developed two heuristics for the problem: Fast Local Search and Guided Local Search. Tang et al. [2] developed a tabu search metaheuristic for a real problem of United Technologies Corporation. It was a problem of periodic maintenance of pieces of equipment located in buildings that were geographically dispersed. The daily work of each technician has to be scheduled for a period of one or two weeks, aiming to maximize the total reward received from servicing the selected tasks over the scheduling period (in this context, rewards are a value assigned to each task to represent its urgency and not the monetary profit). Pillac et al. [8] analyzed the similarity between the technician routing and scheduling problem and the vehicle routing problem with time windows. The objective of both problems was to minimize the total travel time. The authors developed an ALNS parallel algorithm and validated the algorithm with instances of Solomon [32] for the VRPTW. Recently, Cortés et al. [1] presented a real problem of maintenance of printing machines and digital copiers of a large company in Santiago, Chile. Twenty technicians usually visit seventy customers every day, and according to customer relevance, the company establishes a soft time window for each service, so that the most important customers are served first. The objective function considers the number of times that the time windows are violated, the number of customers served, and the total travel time. A branch-and-price approach associated with constraint programming was developed.

Observe that while Tang et al. [2], Overholts II et al. [11], Cordeau et al. [6], and Hashimoto et al. [5] directly consider the priorities of the tasks, Tsang and Voudouris [7], Li et al. [3], Dohn et al. [10], Kovacs et al. [4], Pillac et al. [8], and Cortés et al. [1] tackle the allowed time windows to perform the tasks. As far as we know, the only research that simultaneously approaches these relevant characteristics of the maintenance services, as well as working hours of technicians, is the one presented by Xu and Chiu [9]. These authors addressed the FTSP aiming to assign a set of jobs, at different locations with time windows, to a group of field technicians with different job skills. The objective was to maximize the sum of priority values associated with the tasks performed in a day and, secondly, the idle time of the employees after returning to the base. A constructive heuristic, a local search algorithm, and a GRASP metaheuristic were proposed to solve this problem.

Motivated by the practical relevance of the FTSP, e.g. [1], [2], [3], [5], [6], [7], [11], and the reduced number of studies that consider time windows and priorities of the tasks concurrently, in the present paper we consider the same problem addressed by Xu and Chiu [9] through the development of methods that explore these specific characteristics. Note that in the problem addressed here, differently from the problem addressed in Cortés et al. [1], these tasks parameters are independent (in fact we consider that the time window of the task is defined by the client) and both are considered in our resolution methods.

Initially, constructive heuristics are proposed; the small computational effort of such strategy is one of the reasons that motivated this study. Then we present a customized Biased Random Key Genetic Algorithm (BRKGA) metaheuristic, a Genetic Algorithm (GA) that uses random keys and does not generate unfeasible solutions. The choice of BRKGA is based on its success in several combinatorial optimization problems: single machine [12], covering problem [13], divisible load scheduling [14], lot sizing [15], telecommunications [16], winner determination in auctions [17], bin packing problems [18], berth allocation problem [19], layout [20], and transportation planning [21]. In special, BRKGA was successfully applied to related routing problems such as the problem of routing and wavelength assignment in optical networks [22], the family travelling salesman problem [23], and a problem of collection of blood samples at clinical laboratories [24], among others. On the other hand, as far as we know, the application of a population-based metaheuristic such as BRKGA to the field technician scheduling problem (FTSP) has never been reported in the literature. This scenario motivates the application of BRKGA to the FTSP. A computational study with 1040 instances was carried out to analyze the performance of the suggested algorithms in comparison with methods of the literature and proposed upper bounds; a comparison with optimal solutions values was also conducted for small problems.

This paper is organized as follows. Section 2 presents the mathematical model of the problem. Section 3 describes the proposed constructive heuristics, while Section 4 explains how the Biased Random Key Genetic Algorithm was customized for the focused problem. Section 5 presents two different schemes for the generation of upper bounds for the FSTP. Section 6 describes the computational experiments and the last section summarizes the main results.