A heuristic approach to the overnight security service problem

doi:10.1016/S0305-0548(02)00070-9

Computers & Operations Research

Volume 30, Issue 9, August 2003, Pages 1269-1287

https://doi.org/10.1016/S0305-0548(02)00070-9 Get rights and content

Abstract

This paper introduces the overnight security service problem. The model obtained is a single-objective mixed integer programming problem. It is $NP$ -hard in the strong sense, and exact approaches are not practicable when solving real-life instances. Thus, the model is solved heuristically, through a decomposition in two subproblem. The former is a capacitated clustering problem, the latter is a multiple travelling salesman problem with time windows. Both have a radius constraint which is unusual. The computational results prove the robustness of the approach used. Moreover, a detailed discussion of the results shows that the management objectives are satisfied, providing lower costs, a strong guarantee on the level of service and several different solutions.

Scope and purpose

This paper addresses the management of an overnight security service. Its model includes a number of informal objectives and operational constraints due to practical management requests. Once formalized, it combines a clustering problem and a routing problem, both challenging to solve exactly. Therefore, this paper proposes a heuristic approach to generate good solutions in a reasonable amount of time. The approach achieves a strong improvement both in the quality of service and in the cost. As well, it generates a wide variety of solutions, as required for security reasons. The results prove fairly robust with respect to modelling approximations and to unexpected events.

Introduction

Private security patrols have become a common feature of modern night life. As the police cannot provide a continuous surveillance everywhere, companies and citizens have got used to hiring private guards in order to keep under control their firms, factories, houses, and so on. Much has changed from the classical figure of the night-guard, riding on his bicycle along the streets: nowadays, security is a business, employing higher and higher technology and management skills. The security companies offer a wide variety of services to meet the needs of their customers, such as the escort and keeping of valuables, the daytime watch of banks and commercial centers, the opening and closure of offices.

This paper deals with a real-world case, provided by a company operating in Milan, Italy. It focuses on the overnight patrolling of streets and the inspection of buildings and yards. The guards who perform these routine inspections are also charged with the response to alert signals, which strongly influence the organization of their duty. The present structure of the service is rather rigid: each guard is assigned to a distinct, well-defined zone. A higher degree of flexibility is, therefore, surely desirable. On the other hand, each guard is independent from the other ones and freely decides his own route and schedule, in accordance to a few general principles. The management would prefer to have a tighter control on the behavior of the guards in order to reduce the costs and to guarantee a better service to the customers.

This work describes the practical objectives and constraints characterizing the service and their reduction to a mathematical model. A heuristic approach solves the model, by decomposing it into a capacitated clustering problem and a multiple travelling salesman problem with time windows (m-TSPTW). The results show that it is possible to strongly decrease the costs and, in the meantime, to improve the quality standards and the level of security. A special stress is laid on the practical requirements of the management. For instance, obtaining a great number of good solutions is more important than determining a single optimal one. Solutions should be robust with respect to modelling approximations and unexpected events. In particular, they must be flexible enough to sustain the alarm signals, whenever and wherever they arise.

To our knowledge, this problem is proposed here for the first time, and no literature on the subject is available. Nevertheless, related problems are known: the stochastic vehicle routing problem deals with unpredictable requests (see [1] in [2] and [3], [4]), whereas in moving facility or generalized routing problems (such as the covering tour problem [5], [6]) requests can be satisfied by keeping close to given locations, instead of visiting them.

Section 2 describes the tasks of the guards and the present structure of the service, pointing out its lack of flexibility. Section 3 discusses the features which need the stronger improvements according to the management. In Section 4, the model is described, showing how to turn most of the quality requirements and management goals into a single-objective mathematical formulation. In Section 5, three lower bounds on the number of guards required are presented. Section 6 proposes a heuristic decomposition of the problem and describes the algorithms used to solve the subproblems. Section 7 deals with the data set. Section 8 presents the experimental campaign and discusses how further aspects of the problem, neglected in the formulation, have been addressed. A thorough discussion of the results and conclusions close the paper.

Section snippets

The problem

The company offers a wide variety of different services, that can be easily reduced to three basic classes: tickets, watches and alarms. They are required both in the night and in the daytime, when the normal activities suspend (during the holidays and the week ends). This paper deals with the night service, which is unrelated to the daytime service and poses the more challenging problems of management.

Ticket service: This is the simplest and cheapest service, as well as the most required,

The management objectives

As in many practical cases, improving the organization of such a service does not involve a single, well-definite, objective, but a set of conflicting and sometimes vague goals. Interviews with the management have revealed a strong interest for the following aspects.

Cost reduction: This mainly concerns the number of the guards, whose wage and equipment constitute the major component of the service cost. Other aspects, such as fuel consumption, are much less important, since higher outlays on a

The model

The problem concerns a number of guards (to be minimized), who move on an urban street network; an operating station, from which the guards leave and to which they return at the end of their duty; a set of locations on which routine inspections (ticket and watch services) must be performed; a set of locations protected by an alarm system. This situation can be modelled by a directed graph G(N,A), whose set of nodes N={0}∪N_tw∪N_a consists of the operating station $(node 0)$ , a subset of “routine

Lower bounds

In order to evaluate the quality of heuristic results, it is proper to compare them to a lower bound on the optimum. Moreover, the algorithm starts with a number of guards equal to the best of the three following bounds.

Capacity bound: Since at most C alarms can be assigned to each guard, these must be at least as many as $LB_{1} = |N_{a} | C .$

Duty time bound: The second lower bound considers the duty time T=l₀−e₀. Request i takes up an amount of time $t ̃_{i}$ given by the service time s_i, the travel time t_ij

The algorithm

The structure of the problem, which combines a capacitated clustering problem and a m-TSPTW, suggests a natural heuristic decomposition: first the alarm requests are partitioned into a number of clusters given by the best lower bound described above and this number is increased until a feasible solution is found. Then, routes based on the operating station are built to visit the routine requests. Each route keeps below a required distance from the corresponding cluster of alarms. If necessary,

The data

The data set refers to the urban area of Milan. A graph, representing the street network of Milan in convenient detail, has been built with ArcView, a well-known geographical information system (GIS): the result has up to 2000 nodes and 5000 arcs (see Fig. 2). The travel times have been estimated by evaluating the shortest path between each pair of nodes, assuming a constant and uniform speed of $36 km/h$ (that is $10 m/s$ ). This is justified, since during the night most of the streets are rather

Computational results

The purpose of the computational results is to give satisfactory qualitative answers to the management from all the points of view considered. The whole campaign is organized into a logical sequence of experiments. First of all, the purpose is to study the combinatorial structure of the problem and the dependence of the number of guards on the most relevant constraints, through the evaluation of lower bounds. Then, the experiments determine the results corresponding to a set of realistic values

Conclusions

This paper introduces the overnight security services as a field of study for operations research. It gives a model, namely a single-objective mixed integer programming formulation, discussing its limitations and proposing some heuristic ways to overcome them. The formulation is in itself an interesting problem, which combines scheduling and clustering constraints in a way never proposed before, as far as we know. The problem is strongly $NP$ -hard and exact approaches are not practicable on

Acknowledgements

The authors would like to thank A. Monti and the company IVRI S.p.A for kindly providing us the data set and the possibility to deal with this problem. Thanks also to G. Rossi for his precious help in the computational experiments.

Last, but not least, we acknowledge the work of the anonymous referees for the precious contribution to the improvement of the quality of the paper.

Roberto Wolfler Calvo received the DrEng degree in Electronical Engineering, and the PhD degree in Information and Automation Engineering (Computer Engineering) from the Politecnico di Milano, Italy. He is currently Associate professor at the Laboratoire d'optimisation des Systemes Industrieles, Universite de technologie de Troyes, France. His main research interests concern vehicle routing, network design, multi-criteria analysis.

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Roberto Cordone received the DrEng degree in Electronical Engineering in 1996 and in 2000 the PhD degree in Information and Automation Engineering from the Politecnico di Milano, Italy. Presently, he is a research assistant at the Politecnico di Milano. His research interests include vehicle routing, network design, computational complexity.

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