Location Assignment of Capacitated Services in Smart Cities
Master’s Thesis Econometrics, Operations Research and Actuarial Studies Supervisor: prof. dr. R.H. Teunter
Co-assessor: dr. De Jonge
Master’s Thesis
Econometrics, Operations Research and Actuarial Studies (EORAS)
Location Assignment of Capacitated Services in Smart Cities
G. Hoekstra
s2552108
Abstract
We consider the problem of distributing various services over urban areas. Although the literature on smart cities is evolving, only a few studies have been conducted on the assignment of services to access points. This paper introduces the Multi-Service Capacitated Facility Location Problem (MSCFLP). In the MSCFLP, multiple services need to be distributed over some urban area and cost savings are obtained by installing multiple services on the same location. The aim of the MSCFLP is to find an efficient location assignment of multiple services such that total costs, consisting of fixed opening and fixed service costs, are minimised while satisfying all demand requirements under a capacity constraint. Furthermore, we introduce the Multi-Service Capacitated Facility Location Problem with Partial Covering (MSCFLP-PC), which extends the MSCFLP by allowing for, but penalising partial satisfaction of demand. We propose two heuristic approaches of which one is shown to find good solutions. We present some practical results by solving the problem as an Integer Linear Programming Problem on various real urban areas. Among other things, we show that simultaneous optimisation of the locations assignment of services yields large cost benefits. Experiments indicate that partial covering is an effective means to lower costs. This study is the first to solve the location assignment problem considered and it is the first to formulate and solve the partial covering extension of that problem.
March 30, 2018
Defense date:
April 19, 2018
Supervisors:
prof. dr. R.H. Teunter
dr. De Jonge
Acknowledgement
First of all, I would like to thank my TNO1 supervisor, Frank Phillipson. Thanks for introducing
me to the problem and making me familiar with it. Thanks also for your insightful and detailed feedback that you gave me every time. Next, I would like to thank prof. dr. Ruud Teunter, who has been my supervisor from the University of Groningen. Thanks for your guidance during my project. In particular, thanks for showing me the importance of ‘killing my babies’ at the start of this project and for helping me to actually do it. It definitely improved my work.
Next, I would like to thank my colleagues at TNO for the pleasant working atmosphere and for letting me be part of the team. Thanks to the other interns of the CSR department for having so much fun together. I really enjoyed spending time with you and going on ‘intern uitjes’ every now and then. Special thanks to Irina and Bor for helping me to improve my writing and layout. Next, I would like to thank my family and Jorrit for their support during my stay at TNO and during the rest of my study. Special thanks to my eldest sister Maaike for checking my thesis. Well, Sietske: now it is your turn.
Gerbrich Hoekstra The Hague, March 2018
List of Abbreviations
CFLP . . . Capacitated Facility Location Problem CPLP . . . Capacitated Plant Location Problem FLP . . . Facility Location ProblemMCCFLP . . . Multi-Commodity Facility Location Problem
MSCFLP . . . Multi-Service Capacitated Facility Location Problem
MSCFLP-PC. . . Multi-Service Capacitated Facility Location Problem with Partial Covering MSLSCP . . . Multi-Service Location Set Covering Problem
PCIP . . . Partial Covering 0-1 Integer Program
PCUFLP . . . Partial Coverage Uncapacitated Facility Location Problem PSCP . . . Partial Set Covering Problem
SCP . . . Set Covering Problem
SSCFLP . . . Single Source Capacitated Facility Location Problem UFLP . . . Uncapacitated Facility Location Problem
Contents
1 Introduction 1
2 Literature Review 3
2.1 Capacitated Facility Location Problem . . . 3
2.2 Set Covering Problem and Partial Covering . . . 4
2.3 Smart City Planning . . . 5
3 Problem Formulation 7 3.1 Case Description . . . 7
3.2 Mathematical Formulation . . . 8
3.3 Model Implementation . . . 10
4 Solution Approach 13 4.1 Sequential Solving Heuristic . . . 13
4.2 Ordered Sequential Solving Heuristic with Updating . . . 14
5 Extended Model 15 5.1 Uncapacitated Universal Access Location . . . 15
6 Experimental Design 19 6.1 Software and Hardware . . . 19
6.2 Input Parameters . . . 19 6.2.1 Locations . . . 19 6.2.2 Services . . . 20 6.2.3 Demand Points . . . 21 6.3 Instances . . . 22 6.4 Stopping Criteria . . . 22 7 Computational Results 27 7.1 Order Selection of the Ordered Sequential Solving Heuristic with Updating . . . . 27
7.2 Base Model Results . . . 28
7.2.1 Sequential Solving Heuristic versus Exact Method . . . 28
7.2.2 Performance of the Ordered Sequential Solving Heuristic with Updating . . 31
7.3 Expectations on Extended Model Results . . . 32
7.3.1 Expensive Demand Points - Reduced Sets . . . 32
7.3.2 Base Model Results of the Reduced Demand Point Sets . . . 33
7.4 Extended Model Results . . . 34
7.5.1 Opening Costs of the Locations . . . 37
7.5.2 Penalty Parameters . . . 40
8 Discussion 43 9 Bibliography 45 A Alternative Partial Covering Problem Formulations 49 A.1 Capacitated Universal Access Location . . . 49
A.2 k-out of n . . . 50
A.3 Penalty Constraints . . . 51
A.4 Goal Programming . . . 53
Chapter 1
Introduction
Data management is of growing importance, due to the increasing interest in real-time data. Of particular interest is the data generation and provision in cities. An undesirable example showing the interest in real-time data are the recent terrorist attacks in some Western countries. Whenever such an attack takes place it is of great importance that the situation is controlled by governmental authorities, and that the terrorists are identified and tracked as soon as possible. Data management can also improve the quality of everyday life. Through efficient data collection and provision, different services can be offered to inhabitants of a city, which may ease their lives. As an example, data on the weather conditions can be collected, which could be used to warn people for upcoming bad weather conditions.
In cities, data is gathered regarding various subjects in order to monitor and manage the lives of its inhabitants. Besides the extraordinary pressure on data management in general, cities face an additional challenge in their monitoring process, as an expanding number of people live in cities. Moreover, nowadays the majority of world’s population live in urban areas, and it is expected that urbanisation will continue in the upcoming decades. This evolution shows that efficient networks need to be designed for urban areas to be able to manage all observed data. By successfully implementing applications in urban areas, ‘smart cities’ arise in which data collection and provision can be used to improve the quality of life. The smart city concept is defined by many authors. In general terms, according to Calvillo et al. [9], it is intended to cope with or reduce problems like mobility, and energy supply which arise from urbanisation and population growth. Another common understanding of the various interpretations of the smart city concept is the use of Information and Communications Technologies (ICT).
Data collection and provision with the purpose of improving the quality of life in cities is, however, not new. Especially, data on weather conditions has been collected for a long time. How-ever, nowadays research on smart city technologies goes beyond traditional subjects like weather conditions, including research on smart car parking systems (e.g. Andersen et al. [4]) and electric vehicle charging station placement (e.g. Lam et al. [36]) for example. Another service that could be provided is Smart Vehicle Communication (SVC). SVC provides real-time data to road users to inform them on weather conditions, traffic jams, routing information, and other aspects relevant to traffic. In order to provide such a service, road-side units are necessary. These units on the roadside enable passing vehicles to communicate with each other, in such a way that evolving networks of vehicles arise, commonly referred to as Vehicular Ad Hoc Networks (VANETs).
Services differ in their placement requirements, as they do not face similar demand patterns or service locations. For example, WiFi is likely to be requested throughout the whole day at parking slots or houses, while updates on weather conditions are more likely to be requested a number of
times per day from only a selected set of locations in a city. This example demonstrates that it is unsatisfactory to group all services together, as then various services are provided at locations where they are not required. Thus, an efficient location assignment of all different services is needed such that the requirements for all services are met. Fortunately, most urban areas already have an existing network: the lighting system. This network of lamppost is well documented, connected to power sources and available in many cities, which makes it a convenient network choice.
In this thesis the Multi-Service Capacitated Facility Location Problem (MSCFLP) is introduced and the mathematical formulation is provided. In the MSCFLP, an efficient location assignment for several services needs to be established throughout some urban area in order to satisfy the demand of its citizens. The locations of these citizens, which potentially require services, are denoted as demand points. Each service can be installed at different locations in the city, where each candidate access location is in fact a lamppost. Since lampposts need some adaptions before they can be equipped with services, a fixed opening cost is associated with this operation. Another fixed service cost is associated with the supply of services at locations, which does not dependent on the number of demand points that is served from the location. Each service can only serve some maximum number of demand points. The aim of the MSCFLP is to find an efficient location assignment of services for some urban area such that the total costs, consisting of fixed opening and fixed service costs, are minimised without violating capacity restrictions, while satisfying all demand requirements.
The MSCFLP as presented in this thesis arises in the context of service provision, in which no costs can be assigned to individual users. It is challenging to provide all services in an efficient way such that various services are provided from the same location. By doing so, the opening cost of the location can be shared over the different services. Therefore, it is beneficial to optimise the location assignments of services simultaneously such that an efficient distribution network is established.
Besides the base model we will introduce an extension of the MSCFLP. This problem extends the MSCFLP by introducing partial covering to it and it will be referred to as the Multi-Service Ca-pacitated Facility Location Problem with Partial Covering (MSCFLP-PC). Rather than satisfying demand of all points, only a fraction of the demand points needs to be served in the MSCFLP-PC. It yields a more efficient location assignment in which demand points, that are expensive to serve, are excluded from the service set. In general, Facility Location Problems (FLPs) are studied in a setting in which all customer demand need to be satisfied. However, Charikar et al. [12] argued that very distant customers could influence the final solution disproportionately, yielding unsatis-factory solutions. Serving such customers drives up the total costs and lacks to improve the service level to the majority of customers. Another motivation for partial covering concerns the fact that ‘satisfying all requirements’ could imply that the problem becomes unsolvable (see Shi et al. [55]). The contribution of our research is threefold. The first contribution is the formal description and formulation of the MSCFLP. It reduces the current gap in literature on smart city planning. Although smart cities receive more and more attention in scientific literature, literature on smart city planning is scarce. Secondly, a formal description and formulation of the MSCFLP-PC is provided, which can improve location assignments even further. Our third contribution is the design of an efficient heuristic algorithm, which can provide good solutions in a relatively short amount of time.
Chapter 2
Literature Review
In this chapter literature related to the Multi-Service Capacitated Facility Location Problem (MSCFLP) is reviewed. Related research areas are discussed, including some solution methods. First, the Capacitated Facility Location Problem (CFLP) is discussed in Section 2.1. To a lesser extent, the MSCFLP is related to the Set Covering Problem (SCP). A discussion on literature on this problem, and a review on partial covering problems is included in Section 2.2. Lastly, literature on smart cities and smart city planning is evaluated in Section 2.3. To the best of our knowledge, the MSCFLP itself has not been studied in literature before.
2.1
Capacitated Facility Location Problem
The CFLP is an extension of the extensively studied and well-known FLP. In the FLP, (customer) demand has to be fulfilled by some facilities (warehouses or plants). A decision has to be made on the assignment of the (existing) facilities to the available locations such that the customers can be served. In general, opening a facility comes at some fixed cost, and transportation costs are associated with serving a customer from a facility. The aim of the FLP is to find a set of facilities such that customer demand is fulfilled, while the total costs, consisting of fixed opening costs and transportation costs, are minimised. In general, the classical FLP can be categorised into two classes: Simple or Uncapacitated Facility Location Problems (UFLPs), and Capacitated Fa-cility Location Problems (CFLPs), commonly referred to as Capacitated Plant Location Problems (CPLPs) (Hinojosa et al. [30]). When facilities have some upper bound on the amount of demand they can fulfil, the problem is referred to as a CFLP. When no upper bounds on demands are considered, the problem is referred to as the UFLP. Amongst others, Krarup and Pruzan [34] and Cornu´ejols et al. [14] proved that both kinds of problems belong to the set of NP-hard problems, as it is impossible to solve these problems in polynomial time.
The MSCFLP is related to the CFLP, but it is neither a special case nor a generalisation of it. The CFLP considers several locations and decides on which locations facilities should be opened. Similarly, the MSCFLP considers (lamppost) locations, and decides which of these locations should be opened, and with which services it should be equipped. However, the problem differentiates from a CFLP by its cost structure, and it considers multiple services instead of a single service. Moreover, in the MSCFLP, serving a customer from a location does not entail any individual costs. In contrast, a fixed opening cost has to be incurred when at least one customer is served from a location. Nevertheless, the MSCFLP is the most similar to (extensions of) the CFLP compared to all other already existing problems. We therefore discuss key findings on the CFLP.
Numerous exact and heuristic methods for the CFLP have been proposed in literature. An effi-cient branch and bound heuristic was proposed by Akinc and Khumawala [1]. Klose and G¨ortz [33] implemented a branch and price algorithm, and Wentges [64] modified the Benders’ decomposition algorithm. Guignard and Spielberg [29] developed a direct dual method to solve the mixed plant location problem, in which some plants are uncapacitated. The ADD heuristic designed by Kuehn and Hambuger [35], and the DROP heuristic designed by Feldman et al. [18] are generalised by Jacobsen [31] to solve CFLPs. An approximation algorithm is developed by Mahdian et al. [40] for both the UFLP and the CFLP. However, one of the most applied and outstanding solution methods is Lagrangian Relaxation, which is a widely used solution technique for optimisation problems. We refer to Sridharan [56] and Magnanti and Wong [39] for a review of the various solution techniques for the CFLP.
A special case of the CFLP is the Single Source Capacitated Facility Location Problem (SS-CFLP) in which every customer is served from exactly one facility. In general, all decision variables in this problem are integers, which complicates the problem compared to the CFLP in which the supply variables are continuous. Several authors have devoted attention to the problem, including Fisk [21], Barcelo and Casanovas [7], Klincewicz and Luss [32], Pirkul [46], and Guastaroba and Speranza [28]. Lagrangian heuristics are a successful and commonly used approach to generate solutions to the SSCFLP, as stated by R¨onnqvist et al. [50].
Another variant of the CFLP is the Multi-Commodity Facility Location Problem (MCCFLP). The MCCFLP extends the CFLP by including multiple commodities (e.g. services, products). The aim of the MCCFLP is to find for every commodity a set of locations and a set of customers such that total costs, which consist of fixed opening costs and travelling costs, are minimised. There are some important differences between the MCCFLP and the MSCFLP. Contrary to the MSCFLP, a customer (zone) can have demand for various commodities, and a customer can be served from any location in the MCCFLP. Moreover, no travelling costs or other costs per demand point are defined in the MSCFLP.
Among the first papers that consider multiple commodities in the context of location problems are Warszawski [62] and Warszawski and Peer [63]. Geoffrion and Graves [26] extend this research by including capacity limitations for both plants and distribution centers. Following this work, several studies have been conducted on variations and generalisations of the problem, including the work by Pirkul and Jayaraman [47], Hinojosa et al. [30], Canel et al. [10], and Melo et al. [41]. Furthermore, Li et al. [37] combined the CFLP with a multi-commodity Min-Cost Flow Problem, and more recently, Li et al. [38] considered the Multi-Product FLP in a two-stage supply chain setting.
2.2
Set Covering Problem and Partial Covering
An extension of the SCP is the Partial Set Covering Problem (PSCP), where Daskin and Owen [15] were among the first ones to define the problem. The PSCP aims at determining a minimum set of supply locations to cover a prespecified fraction of the population. In their modelling procedure a so-called service set is introduced, that corresponds to the set of demand locations for which demand will be covered. A different modelling approach is introduced by Gandhi et al. [24] for their k-partial SCP. Similar to the PSCP, the k-partial SCP aims at determining a minimum number of sets such that at least k elements (e.g. demand locations) are covered. Partial covering problems are extensively studied, because they can be applied to various real life applications as argued by Amini et al. [3]. Recently, Takazawa et al. [57] introduced the Partial Covering 0-1 Integer Program (PCIP), which is a generalisation of the PSCP. Contrary to the PSCP, demand and supply parameters are not restricted to be binary in the PCIP.
Partial covering has been of interest in other problems as well, including FLPs, maximal covering models, and Partial Vertex Cover Problems. Vasko et al. [59] introduce the Partial Coverage Uncapacitated Facility Location Problem (PCUFLP). The aim of the PCUFLP is to determine which warehouses should be opened in order to satisfy the demand of some of the locations, such that the total costs, representing the summation of fixed opening costs and variable costs, are minimised. In line with this research, Monabbati [42] developed the Uncapacitated Facility Location Problem with self-serving demands (UFLP-SS). It generalises the UFLP by introducing demand-side servers. Such a server is located at one of the demand points, and can exclusively serve the corresponding demand point. A demand point that can be served from both a demand-side server, and the facilities, is called a self-serving demand point. To the best of our knowledge, partial covering has not been previously researched in the context of CFLPs.
Maximal covering models are another stream in literature in which the concept of partial cov-ering has been investigated. Church and ReVelle [13] formulated the Maximal Covcov-ering (Location) Problem, which aims at maximising demand coverage within the specified service distances by using a fixed number of facilities. In this context, partial covering is not considered a relaxation of some demand covering constraint as in other partial covering problems, but rather the main criteria to be maximised. In a similar fashion, partial covering problems can be considered as optimisation versions of the Dominating Set Problem, and the Vertex Cover Problem as argued by Fomin et al. [22]. Given the relevance of partial covering in related problems, it is believed that it is worthwhile to investigate the added value of partial covering in our problem.
2.3
Smart City Planning
The concept of smart city planning is found in literature concerning the planning of access networks. In general, access networks connect users and their supplier by means of cables, wires, and other technological equipment. Such networks are generally studied in the context of telecommunication networks.
The planning of access networks is studied by several authors, including Sarkar and Mukherjee [52]. In this work, the Hybrid Wireless-Optical Broadband Access Network (WOBAN) is con-sidered of which the planning and setup are studied. A WOBAN consists of both wireless and wired connections. The design of the topology of such a network is considered by Filippini and Cesana [19]. However, we only consider the planning of wireless services and therefore literature on WOBANs is not of great value for our research. Similarly, concepts from the field of urban planning cannot be applied to smart city planning, as they consider more elemental aspects of cities such as protection of the environment and infrastructure.
Unfortunately, literature on smart city planning is relatively scarce, which is surprising, as the availability of services and the corresponding service network are crucial for smart cities. Most
recent studies on smart cities consider the general concept, the different definitions of this concept (e.g. Albino et al. [2]), the combination of the Internet of Things (e.g. Georgakopoulos and Jayaraman [27]), or the big data challenges in smart cities (e.g. Rathore et al. [49]). Smart city planning is studied by Yamamura et al. [65], however, their work focuses on the planning of the energy system of cities. A complete system including multiple types of service deployments is proposed in Rathore et al. [48]. They developed a system, which makes use of big data for urban planning and smart city evolution. However, their developed four-tier architecture does not show how the various services should be distributed. A planning model is proposed in Peralta et al. [45]. It takes into account scalability and uncertainty at different time stages. However, it does not consider the joint deployment of various services.
Chapter 3
Problem Formulation
In this chapter the problem formulation of the MSCFLP is provided. First, the problem setting is described in Section 3.1. Thereafter, in Section 3.2 the Integer Linear Program is presented. The problem formulation as it has been implemented in the software program is provided in Section 3.3. The problem formulation of the MSCFLP-PC is provided in Chapter 5.
3.1
Case Description
This thesis considers an urban area in which data has to be collected and shared at the same point in time. In order to collect data or share data, a location assignment of the various service boxes needs to be established. A service box can provide service for only a single service. When a location is equipped with some service, it is said to be a ‘service access point’ of the service. Depending on the context in which it will be used, the term ‘service’ either refers to the service itself or to one of the service access points (i.e., one of the service boxes). Various services such as a WiFi service and an Alarm service are considered. For every service, it has to be decided which locations should become service access points such that the service requirements are met. A location that is a service access point for at least one service, is referred to as an ‘access location’. Such a location is said to be opened and will be equipped with one or multiple services. As mentioned in Chapter 1, locations of lampposts are considered as candidate access locations.
Every service has its own (unique) set of points, which potentially have demand for the service. The geographical location of a potential user for some service is referred to as a ‘demand point’ of the particular service. A demand point does not need to have demand at all times. Instead, it will require service only at certain moments in time. The set of demand points is unique in the sense that it can vary across the various service types. As an example, WiFi might be requested more often and on different locations than a weather sensor. It is likely that WiFi is requested on housing addresses throughout the entire day, while statistics on the weather are preferred to be conducted only at fixed moments in time. As a result, each demand point requires service for either one of the services and the sets of demand points are disjoint.
When a demand point is served from a specific service access point at some access location, it is said that a ‘connection’ is made between the access location and the demand point. A connection can be made between the access location and the demand point when the service is present at the access location, and the demand point is located in the neighbourhood of the access location. The reason is that services do not have an unlimited reach, in fact each service only has some limited range in which it can serve demand points.
Next to the limited range, a service access point can only serve a maximum number of demand points. Thus, services are restricted in both range and in the number of demand points they can serve. This problem characteristic will most likely imply that in a dense network it is no longer optimal that demand points can be served by only one access location. In particular, multiple neighbouring locations will be opened, and the demand point will be served by either one of the access locations, whichever has capacity left. In this case, the ranges of different service access points overlap in such a way that the service requirements are met. Opening a location comes at some positive fixed cost, and equipping it with a service comes at some positive fixed cost as well. No costs are associated with connections. The sum of both the fixed opening costs of the locations and the service costs is used as the optimality criterion.
3.2
Mathematical Formulation
The problem described in Section 3.1 is related to the problems studied by Vos [61] and Verhoek [60]. Thereafter, the mathematical model is partly deducted from both works. Recall that Vos [61] introduced the MSLSCP, without taking into account capacity constraints. Therefore, a capacity constraint is added compared to his mathematical formulation. Verhoek [60] does take capacity limitations into account. In her work, two constraint sets are introduced to model the capacity of a service. First, services are modelled to serve only a maximum number of demand points at the same time. Second, the total available bandwidth of a service is modelled to be bounded from above. Thus, services are capacitated in two dimensions in this approach. However, when the demand distribution is known, the bandwidth constraint can be reformulated into a constraint on the (average) number of connections and thus the constraints boil down to a similar restriction. In turn, we chose to use a different formulation of the problem compared to Verhoek [60]. We incorporate capacity restrictions by only one set of constraints restricting the total number of offered connections by a location to be no larger than some prespecified maximum number.
In Table 3.1 an overview of the notation is presented. The problem consists of a set of demand points, locations, and services. Each demand point i ∈ Gu has some potential demand dui for service u ∈ F . A demand point i ∈ Gu is characterised by its location and its demand for service
u ∈ F . Similarly, a location j ∈ L is characterised by its location, and its connected services. A service u ∈ F is characterised by its range and its capacity ηu
j, which is defined as the maximum
Table 3.1: Parameters and decision variables for the MSCFLP. General Notation
L Set of all locations F Set of all services
Gu Set of all demand points for service u ∈ F
|X | Cardinality of the set X Indices
i Demand point, i ∈ Gu for service u ∈ F j Location, j ∈ L u Service, u ∈ F Parameters au ij = (
1, if demand point i ∈ Gucan be served from location j ∈ L for service u ∈ F
0, otherwise cu
j Cost of equipping location j ∈ L with service u ∈ F , cuj > 0
fj Cost of opening location j ∈ L, fj> 0
dui Demand of demand point i ∈ Gu for service u ∈ F , dui ∈ N
ηju Maximum number of connections access location j ∈ L can release for service u ∈ F , ηuj ∈ N+
M A large number, M > 0 Decision variables
su
ij The number of connections made between access location j ∈ L and demand
point i ∈ Gu, su ij ∈ N
xu j =
(
1, if access location j ∈ L is a service access point for service u ∈ F 0, otherwise
yj =
(
1, if location j ∈ L is an access location 0, otherwise
The Integer Linear Program (ILP) for the MSCFLP is formulated as follows:
minX j∈L X u∈F cujxuj + X j∈L fjyj, (3.2.1) subject to xuj ≤ yj ∀j ∈ L, ∀u ∈ F , (3.2.2) X i∈Gu suij ≤ ηu jx u j ∀j ∈ L, ∀u ∈ F , (3.2.3) X j∈L suij ≥ du i ∀i ∈ G u, ∀u ∈ F , (3.2.4) suij ≤ au ijM ∀i ∈ G u, ∀j ∈ L, ∀u ∈ F , (3.2.5) suij ∈ N ∀i ∈ Gu, ∀j ∈ L, ∀u ∈ F , (3.2.6) xuj ∈ {0, 1} ∀j ∈ L, ∀u ∈ F , (3.2.7) yj∈ {0, 1} ∀j ∈ L. (3.2.8)
Objective (3.2.1) minimises the total costs, which is defined as the sum of the opening costs of the services and the opening costs of the access locations. Constraint (3.2.2) ensures that only access locations can be equipped with services. Capacity restrictions are taken into account by including Constraint (3.2.3). It limits the number of connections that an access location can release for a specific service. When an access location is not equipped with some service, the capacity of this service is set equal to zero, which ensures that for the service at this location no connections can be made. Constraint (3.2.4) ensures that the demand of every demand point is satisfied, and Constraint (3.2.5) implies that a connection can only be established between demand point i ∈ Gu and access location j ∈ L for service u ∈ F , when the demand point is located in the range of the service (auij = 1). Lastly, constraints (3.2.6)-(3.2.8) specify the solution space. The problem formulation consists of
2|L||F | + |G|(|L| + 1) constraints and (3.2.9)
|G||L| + |L|(|F | + 1) variables. (3.2.10)
3.3
Model Implementation
In the software program, the model is implemented in a slightly different formulation than the one presented in equations (3.2.1)-(3.2.8). The difference lies in the modulation of the connection variables su
ij. In the given formulation, the variable suij is defined for all combinations of i ∈ Gu
and j ∈ L subject to Constraint (3.2.5). When a demand point is not within range of a service for some location, this constraint implies that the solution space of the corresponding variable su
ij
consists of the single element zero. Thus, it is reasonable to define su
ij only for those combinations
of demand points and locations for which the demand point lies within the range of the location implying that the solution space of the corresponding connection variable consists of both elements zero and one.
In mathematical terms this implies the following. For some demand point i ∈ Gu having
demand for service u ∈ F let the set of locations j ∈ L, for which i is within the range, be denoted by Lu
i. That is, let the set Lui be defined as,
Lu i = {j|a
u
ij = 1}, (3.3.1)
for some demand point i ∈ Gu having demand for service u ∈ F . Then for some demand point i ∈ Gu define su
ij only for locations j ∈ L such that j ∈ Lui. By doing so, Constraint (3.2.5)
becomes redundant and can be removed from the problem formulation. Therefore, the equivalent problem formulation as implemented in the software program can be stated as given in equations (3.3.2)-(3.3.8). The definitions of the decision variables and parameters are the same as those provided in Table 3.1.
xuj ≤ yj ∀j ∈ L, ∀u ∈ F , (3.3.3) X i∈Gu suij≤ ηjux u j ∀j ∈ L, ∀u ∈ F , (3.3.4) X j∈Lu i suij≥ dui ∀i ∈ G u, ∀u ∈ F , (3.3.5) suij∈ N ∀i ∈ Gu, ∀j ∈ L, ∀u ∈ F , (3.3.6) xuj ∈ {0, 1} ∀j ∈ L, ∀u ∈ F , (3.3.7) yj∈ {0, 1} ∀j ∈ L. (3.3.8)
This problem formulation has
2|L||F | + |G| constraints and (3.3.9) X u∈F X i∈Gu Lu i + |L|(|F | + 1) variables. (3.3.10)
Compared to the linear program given by equations (3.2.1)-(3.2.8) this problem formulation reduces the number of variables by |G||L| − P
u∈F
P
i∈Gu
Lu
i variables, and the number of constraints by |L||G|
constraints.
Chapter 4
Solution Approach
The MSCFLP considers the location assignments of multiple services. The problem can be solved ‘as a whole’ by an exact method, but large running times are expected, and thus two alternative solution methods for the MSCFLP are suggested. Their performance will be evaluated relative to an exact method. Both heuristic approaches make use of the CPLEX solver to generate solutions. The first heuristic optimises the location assignments of the services for each service individually. The solutions of these single service optimisations are combined to yield a solution to the MSCFLP. The second heuristic extends this approach by updating some of the cost parameters. This heuristic uses a specific optimisation order to generate a solution to the MSCFLP.
4.1
Sequential Solving Heuristic
In the MSCFLP the location assignments of various services are simultaneously optimised. A simple heuristic approach is to optimise the location assignments of the various services independently and combine the solutions of these single service optimisations to yield a solution to the MSCFLP. That is, during each step of the heuristic only one of the services is considered and the location assignment of this service is optimised regardless of any other location assignment. Since the various steps are independent, the order in which the steps are performed does not affect the solution (quality).
When the MSCFLP is solved for only one single service the problem is similar, but not equiva-lent to a CFLP. Recall that contrary to the CFLP, no individual costs such as transportation costs are defined for demand points in the MSCFLP. Instead, a fixed opening cost is associated with equipping a location with a service, which is independent of the number of demand points that will be served from this location. In a MSCFLP with a single service, an access location will always be a service access point of the considered service in an efficient solution. Stated differently, the decision variables yj and xuj are equal in an efficient solution and thereby the MSCFLP could be
modelled by excluding either one of these variables. The cost of equipping a location j with some service u is then equal to fj+ cuj.
This heuristic approach is expected to result in short computation times. However, the method is likely to yield non-optimal solutions, since the location assignments of the various services are determined one at a time. The general structure of the Sequential Solving Heuristic (SSH) is provided in Algorithm 1. The solutions of the various steps are combined in order to obtain the overall multi-service location assignment and to obtain the set of access locations (line 5 of Algorithm 1). The overall objective of this combined solution is not simply equal to the sum of the intermediate objective values. This is due to the fact that a location can be equipped with
Algorithm 1 General structure of the Sequential Solving Heuristic (SSH) 1: for u ∈ F do
2: Optimise the location assignment of service u ∈ F 3: end for
4: Compose the MSCFLP solution 5: Set yj= min 0, P u∈F xu j
for every location j ∈ F 6: Objective value of the MSCFLP solution is P
j∈L P u∈F cu jxuj + P j∈L fjyj
services during multiple steps, but in reality it only needs to be opened when it is equipped with the first service.
4.2
Ordered Sequential Solving Heuristic with Updating
The Ordered Sequential Solving Heuristic with Updating (OSSHU) extends the Sequential Solving Heuristic (SSH) by updating some of the cost parameters in between the various optimisations. Contrary to the SSH, the intermediate steps (i.e., optimisations) of the OSSHU are not indepen-dent, since the current step is based on the solution of the previous step(s). During the first step of the heuristic the location assignment of the first service is optimised. At this stage, the cost of equipping some location j with service u is equal to fj+ cuj. When this first step is completed, a
set of access locations (and thereby a set of service access points for the first service) is obtained. This information is used in the next step in which the second service is considered.
For the second step, the opening cost of the current set of access locations is set to zero (i.e., fj = 0 for access location j). Thus, equipping some location with the second service, that is already
a service access point for the first service, comes at a cost of only cuj. Contrary to these access locations, equipping one of the other locations with a service comes at the original cost of fj+ cuj.
Similar cost updates are performed before the last step is executed.
Because of the intermediate cost updates, the steps are no longer independent, which implies that the order in which the various services are considered affects the overall solution (quality). Furthermore, due to the intermediate cost updates the steps point to an overall solution which prefers locations that are a service access point for multiple services. Thereby, the OSSHU is likely to result in better solutions than the SSH. The general structure of the OSSHU is provided in Algorithm 2.
Algorithm 2 General structure of the Ordered Sequential Solving Heuristic with Updating (OS-SHU)
1: Select some order O of services u ∈ F 2: for the kth element of O do
3: Optimise the location assignment of service k ∈ F 4: Set fj= 0 for every location j ∈ F such that xkj = 1
5: end for
6: Compose the MSCFLP solution 7: Set yj= min 0, P u∈F xu j
for every location j ∈ F 8: Objective value of the MSCFLP solution is P
Chapter 5
Extended Model
In this chapter we introduce a generalisation of the MSCFLP as formulated in Section 3.2. It is called the Multi-Service Capacitated Facility Location Problem with Partial Covering, abbreviated as MSCFLP-PC. The goal of this problem is to ‘cover’ the demand of only a fractional part of the demand points, rather than covering all demand points as in the base problem while minimising total costs. This generalisation is similar to the PSCP, which is a generalisation of the SCP.
There have been various studies on the combination of partial covering and SCPs (e.g. Gandhi et al. [24]), and partial covering in the context of FLPs (e.g. Monabbati [42]). As a result, various model approaches are suggested in literature. The interested reader is referred to Chapter A of the Appendix for a discussion on some alternative problem formulations. We will refer to the selected problem formulation as the ‘Uncapacitated Universal Access Location’ and we will elaborate on it in the next section. It is an intuitive approach which penalises uncovered demand points in the solution. This penalty cost can be interpreted as a lost sales cost. Since the universal access location is uncapacitated and can serve any demand point, a feasible solution always exists.
5.1
Uncapacitated Universal Access Location
The Uncapacitated Universal Access Location modelling approach is inspired by the work of Mon-abbati [42] and Vasko et al. [59], who deal with related partial covering problems. Both studies put an upper bound on the number of unserved demand points by introducing an additional service location or service possibility. As stated in Chapter 2, Monabbati [42] introduces demand-side servers as a second service option. When a server is present at such a demand-side, the demand point can either be served by the server, or by one of the facilities as in general FLPs. Vasko et al. [59] use a different formulation of the problem to incorporate partial covering. However, similar to Monabbati [42] an additional service possibility is introduced. In their work, a ‘universal warehouse’ is introduced, which is a ‘dummy’ warehouse that allows a feasible solution to not cover all demand points. When a demand point is ‘served’ by this additional warehouse, it implies that the demand point is not served by another (real) warehouse, and thus it is left unserved in the solution. This warehouse can serve all demand points, and its fixed opening cost is equal zero. The variable cost of serving a demand point by the dummy warehouse is similar to the lost sales cost of the demand point. In line with the optimality criterion of the MSCFLP, the optimality criterion of the MSCFLP-PC is the total costs of the solution, which is the sum of both the total opening costs of the locations and service, and the total penalty costs.
The model proposed by Vasko et al. [59] served as an inspiration for two suitable problem formulations of our problem. First, a universal access location can be introduced, which has
infinite capacity and range, and a variable cost equal to the lost sales cost of a demand point. Due to its infinite capacity and range, the universal access location can serve all demand points. This implies that for relatively low lost sales costs, the majority of the demand points are served by the universal access location, which might be considered as unsatisfactory in some situations. Setting the variable cost of the universal access location relatively high, optimal solutions of the extended problem are equivalent to optimal solutions of the base problem. The second problem formulation based on Vasko et al. [59] can be found in Section A.1 of the Appendix.
The mathematical formulation is as follows. First of all, let the universal access location be denoted by j0, and replace the demand covering constraint by the system of constraints given in Constraint set (5.1.1), where suij0 is a binary variable that is equal to one when demand point i
is served by the universal access location. That is, when a demand point i ∈ Gu having demand
for service u ∈ F is served by the universal access location (i.e., su
ij0 = 1), the demand point is
unserved in the solution. In turn, the objective function should be rewritten to Equation (5.1.2), which includes the lost sales cost pu
i for all demand points i ∈ Guand for all services u ∈ F . As the
fixed opening cost of both the location and the services is zero for the universal access location, the universal access location is only part of the objective function through the penalty cost summation.
P j∈{L,j0} su ij≥ dui, ∀i ∈ Gu, ∀u ∈ F su ij0 ∈ {0, 1}, ∀i ∈ Gu, ∀u ∈ F (5.1.1)
In turn, the ILP for the MSCFLP-PC is formulated as follows:
minX j∈L X u∈F cujxuj + X j∈L fjyj+ X u∈F X i∈Gu puisuij0, (5.1.2) subject to xuj ≤ yj ∀j ∈ {L, j0}, ∀u ∈ F , (5.1.3) X i∈Gu suij ≤ ηu jx u j ∀j ∈ L, ∀u ∈ F , (5.1.4) X j∈{L,j0} suij ≥ dui ∀i ∈ G u, ∀u ∈ F , (5.1.5) suij ≤ a u ijM ∀i ∈ G u , ∀j ∈ L, ∀u ∈ F , (5.1.6) suij ∈ N ∀i ∈ Gu, ∀j ∈ {L, j0}, ∀u ∈ F , (5.1.7) xuj ∈ {0, 1} ∀j ∈ {L, j0}, ∀u ∈ F , (5.1.8) yj ∈ {0, 1} ∀j ∈ {L, j0}. (5.1.9)
The universal access location is uncapacitated, and thus no capacity constraint needs to be taken into account for it. Similarly, due to its infinite range, suij0 is not bounded from above by auij0
for any i ∈ Gu, and thus Constraint (5.1.6) does not need to be defined for location j0. Stated differently, auij0 is equal to one for all i ∈ Gu. The problem formulation consists of
2|L||F | + |G|(|L| + 1) constraints and (5.1.10)
|G||L| + |L|(|F | + 1) variables. (5.1.11)
has infinite capacity and range, a solution in which all demand points are served by the universal access location is always a feasible solution. Furthermore, the approach including its penalty parameters is intuitive and can easily be implemented in the model. This implementation requires the addition of |G| + |F | + 1 new variables and |F | + 1 constraints. In the current formulation in which the universal access location is uncapacitated, there is no lower bound on the service levels. A constraint could be implemented such that a minimum service level is imposed. However, as this would imply that a feasible solution is no longer guaranteed, it is decided to not include such a constraint. Similar to the MSCFLP, the model is slightly differently implemented than the formulation above in order to lower the number of connection variables.
Chapter 6
Experimental Design
In this chapter we discuss the experimental design of the various experiments which have been conducted. In Section 6.1 we describe the software and hardware that are used to implement and solve the problems. An overview of the various input parameters is given in Section 6.2. We present the set of access locations, the set of studied services, and the demand point selection. In Section 6.3 the various instances are discussed. In the last section we elaborate on the stopping criteria.
6.1
Software and Hardware
The implementation of the model and the numerical experiments are carried out in MATLAB version R2016b. This is a programming language published by MathWorks, which allows for a wide range of computations, and other data processing. The problems are solved by use of the external solver CPLEX. IBM ILOG CPLEX Optimisation Studio (COS) is a solver developed by IBM. It is an optimisation software package for solving linear programs, mixed integer programs, and quadratic programs. The free student 12.7.1 version of the package has been used to generate the results. The experiments are performed on a DELL E7240 laptop with an Intel(R) Core(TM) i5-4310U CPU 2.00 GHz 2.60 GHz processor. The laptop is operational on a 64-bit operating system.
6.2
Input Parameters
In this section all parameter values and the various characteristics of the instances will be discussed. We describe the parameter settings for the locations, services, and the demand points.
6.2.1
Locations
As described in Chapter 1, the lighting system is used as a network in which different access locations can be opened. Every lamppost is a candidate access location that can be equipped with services. The set of all access locations is denoted by L. Data on the locations of lampposts is publicly available for many cities of the Netherlands. It is accessible via Dataplatform1, which is an
initiative of Civity. The instances describe various subareas of the city of Amsterdam. Instances are specified by the region enclosed by a certain boundary. The locations of lampposts of the instances
1https://www.dataplatform.nl/
are the union of two datasets of Dataplatform, as the locations of lampposts of the Dutch main roads are included in a separate documentation. The fixed opening cost of a location is taken to be equal to fj = 5, 000 for every location j ∈ L. A mapping of all locations of some subarea of the
city of Amsterdam can be found in Figure 6.1.
Figure 6.1: Mapping of all locations in some subarea of the city of Amsterdam. The locations are given by the lampposts represented in blue. The black contour marks the boundary of the region.
6.2.2
Services
Three services will be considered for the instances. The various services and their parameter values are based on the works of Vos [61] and Verhoek [60]. As previously stated, a service is characterised by its range and its capacity. The capacity is defined as the maximum number of connections which can be established at the same time. It is assumed that every service has a circular coverage area. This area is dependent on the range of the service. However, in reality there is no specific range such that the demand point could not be served by the access location anymore, when it is located just further away from the location than this range. In turn, the range of a service is defined such that the offered service is of sufficient quality throughout the coverage area. Given the range, the coverage matrix with elements au
ij can be filled.
A service is denoted by u ∈ F , where F is the set of all services. For every service, information is provided in Table 6.1 on the range, capacity, and opening cost. Recall that this is the cost of opening a service box of the considered service on an access location. The first service is a WiFi service. It has a range of 100 meters, can serve up to a maximum of 30 demand points, and its opening cost is equal to 300. The second service is a Smart Vehicle Communication (SVC) technique, which aims at providing data to drivers. It has a range of 200 meters, a capacity of 15 connections, and an opening cost of 300. The last service is an Alarm service, which has an unlimited capacity. This service has a range of 300 meters, and opening cost equal to 150. The Alarm service aims at providing a loud signal to warn humans about dangers. As the service provision is independent on the number of humans within the range of the access location, the capacity of the service is unlimited.
places, as a missed notification means that someone is not informed of some dangerous situation. Thus, a relatively high value is chosen for the penalty parameter of the Alarm service.
Table 6.1: Overview of parameter values of the services. Range Capacity Open. cost Penalty cost
Service ηu j cuj pui u = 1: WiFi 100 30 ∀j ∈ L 300 ∀j ∈ L 500 ∀i ∈ G1 u = 2: SVC 200 15 ∀j ∈ L 350 ∀j ∈ L 250 ∀i ∈ G2 u = 3: Alarm 300 ∞ ∀j ∈ L 150 ∀j ∈ L 1,500 ∀i ∈ G3
6.2.3
Demand Points
Up to now, only the locations and the various services are discussed. Besides these two characteris-tics, instances have different demand points. Recall that by construction (copying original demand points that require multiple services), every demand point requires only one service, and in turn, every service has its own disjoint set of demand points. Although sets of demand points differ across the various instances, every set is generated by the same procedure. The demand points are generated within the boundary that specifies the test area. An overview of all demand points classified per service in some subarea of the city of Amsterdam is given in Figure 6.2.
For the WiFi service, the home addresses located inside the boundary are taken as demand points. All houses are assigned a demand of one. As the second service is a SVC technique, which aims at providing data to drivers, the demand points are generated on the roads inside the boundary. Data on Dutch roads is taken from the OpenStreetMap (OSM) project. In the data set, roads are represented by sequences of points, which simulate the curvature of a road. In turn, a curved road is represented by more points per distance unit than a straight road, so part of these road points are deleted. Road points are removed in such a way that the distance between a road point and its closest neighbour does not exceed a maximum limit of 100 meters. Similarly, road points are deleted when their closest neighbouring point is less than 10 meters away. This implies that the distance between a road point and its closest neighbour will in general be in between 10 and 100 meters. These road points are taken as demand points for the SVC technique. Contrary to the WiFi service, not every road point has a demand of one. In fact, a road point is assigned a demand one, two, or three, depending on its characteristics. Demand points referring to so called ‘A-roads’ are assigned a demand of three, simulating the fact that these important highways are in general congested. These roads are labelled as motorways, and freeways in the original documentation. Less important roads are national and regional roads. These roads are labelled primary and secondary roads, and a demand of two is assigned to demand points on such roads. All other roads are of least importance, and in turn are assigned a demand equal to one.
The last service is the Alarm service. As it has an infinite capacity, serving its demand points can be approached as a covering problem instead of some capacitated supply problem. In line with this approach, the demand points of the alarm are intersections of a grid. It is indicated in Murray and Church [44] that this approach works best with regard to computational efficiency. For more information on the generation of the grid, we refer to Section 6.3 of Vos [61]. Similar to the WiFi service, a demand of one is assigned to every demand point. However, the optimal solution is the same for other demand values, as the Alarm service has an unlimited capacity. Thus, for efficiency reasons, a demand of one is assigned.
One final remark: some demand points are not located within the range of at least one location, which implies that these demand points cannot be served and thus Constraint (3.2.4) would be
violated. Therefore, we exclude such demand points from the set of demand points, as otherwise no feasible solution exists for the MSCFLP. Since Constraint (3.2.4) is relaxed in the extended model, inclusion of these demand points will not lead to infeasibility of the MSCFLP-PC. Moreover, it is optimal in the MSCFLP-PC to not serve these demand points for any value of the penalty parameter, so the optimal solution is not affected by these demand points. This implies that they can be excluded from the set of demand points without any loss of generality.
Figure 6.2: Overview of the demand points per service of some small area in Amsterdam.
6.3
Instances
The MSCFLP and the MSCFLP-PC will be solved for a number of instances. These instances are subareas of the city of Amsterdam. By taking subareas with a varying number of locations and demand points, we tried to test the models on different kinds of urban areas, ranging from quiet suburbs to more crowded neighborhoods. As an example, some of the instances include motorways and freeways, while others only contain less important roads. In total, nine instances are considered of which instances 1-7 are the small instances. Instance 6, which is the largest instance of the first seven instances in terms of the total number of locations, and in terms of surface, contains only 0.6% of the locations of Amsterdam, and spans only 0.8% of the surface of the city. Similarly, instance 7, which contains the most home addresses of the first instances, only contains 0.5% of all home addresses located in Amsterdam. Instances 8 and 9 are relatively big instances compared to the rest. They contain significantly more locations and demand points, and span a larger surface. However, they are still relatively small compared to the size of the entire city. Instance 9, for example, spans only 3.8% of the surface of the whole city, and contains only 7.2% of all addresses. Mappings of the various instances can be found in figures B.1-B.9 in Chapter B of the Appendix. In Table 6.2 some statistics are provided for all instances. The table contains information on the number of locations and the number of demand points per service.
6.4
Stopping Criteria
Table 6.2: Overview of the instances. The second column denotes the number of locations, which is denoted by |L|. Columns 3-5 display the total number of demand points for the WiFi, SVC, and Alarm service, respectively. Column 6 provides the total number of demand points.
No. Locations No. Demand points Tot. Inst. |L| |G1| |G2| |G3| |G| 1 33 47 3 9 59 2 77 73 8 15 96 3 99 260 9 13 282 4 102 462 8 15 485 5 400 111 20 25 156 61 782 21 93 104 218 7 516 1,241 42 46 1,329 8 6,079 8,106 397 326 8,829 91 6,981 10,122 528 431 11,081
1 For these two instances some WiFi or SVC demand points are excluded from the initial set of demand points,
as these demand points could not be served from any location. Per service at most five demand points are excluded.
addition of the connection variables su
ij. This conclusion was obtained by solving two similar
‘uncapacitated’ problems. As a first problem (P1), the MSCFLP is considered without the capacity constraints provided in Equation (3.3.4). Omitting this constraint and rewriting the demand coverage constraints of Equation (3.3.5) results in superfluous connection variables. This implies that the total number of variables of the model reduces tremendously. As a second problem (P2), the MSCFLP is considered with non-binding capacity parameters. Selecting the capacity parameters to be greater than the total number of demand points implies that the problem is similar to the uncapacitated version of the MSCFLP. The results of both problems are provided in Table 6.3 and show an increasing difference in the running times between the two equivalent problems. Thus, even without limiting capacity bounds, the running time of the CPLEX solver increases due to the inclusion of the connection variables.
To gain more insights in the running times of the solver the MSCFLP has been solved on instance 3 for various values of the WiFi capacity parameter η1j. The capacity of the second service is set to infinity as well in this set of experiments. The set of considered capacity parameters of the WiFi service is ηj1∈ [3, ..., 260] for all j ∈ L. That is, the maximum value considered is equal to the total number of demand points for the WiFi service, which is denoted by |G1|. Any larger
capacity value yields the same results. For η1
j < 3, the problem is infeasible for instance 3. In
Figure 6.3 the running time of the solver has been plotted against the capacity parameter of the WiFi service. A running time limit of 60 seconds was incurred. Next to the running time, the ‘detection time’ has been plotted. For some value of the WiFi capacity, this line shows the point in time at which the final solution has been found. When the running time is shorter than 60 seconds, the difference between the detection time and the running time of the solver is the amount of time that the solver needs to prove optimality. For capacity values less than 115 the solver did find the optimal solution or could prove optimality of the best found solution for only some capacity values. Thus, in order to determine whether the best solution found is also optimal for the other capacity values, a second experiment was conducted. The capacity parameter was taken equal to 25, and the problem was solved again, but this time with a time limit of 28,800 seconds (i.e., 8 hours). The solver terminated after 8 hours, but the same solution was obtained. From this observation, it can be concluded that for η1
j = 25, the solver can find the optimal solution relatively fast (i.e.,
0.72 seconds), but it requires a large amount of time to prove optimality. It is believed that this
Table 6.3: Objective values, gaps, and corresponding running times for both the MSCFLP with-out connection variables (denoted by P1) and for the MSCFLP with non-binding capacity limits (denoted by P2).
P1 P2
Inst. Obj. Gap (%) Time (s) Obj. Gap (%) Time (s)
1 5,800 0.00 0.00 5,800 0.00 0.03 2 16,400 0.00 0.02 16,400 0.00 2.41 3 21,700 0.00 0.02 21,700 0.00 14.36 4 27,350 0.00 0.03 27,350 0.00 31.70 5 27,850 0.00 0.08 27,850 0.00 40.38 6 99,850 0.00 0.13 99,850 0.00 416.44 7 92,800 0.00 1.24 103,9001 14.74 43,200.00 8 678,7001 3.72 43,200.00 914,5501 37.80 43,200.00 9 853,2501 4.27 43,200.00 1,125,5501 40.47 43,200.00
1 The solver is terminated after 43,200 seconds (i.e., 12 hours).
holds true for other capacity parameters as well.
One explanation for this result is the fact that the locations in the test solutions are very similar to each other in the sense that they can serve a similar, or even the same set of demand points, which makes them identical to the solver. In turn, there exist many different solutions, which yield the same objective value, and thus there is no clear direction of steepest ascent. For this reason, two stopping criteria have been implemented.
The stopping criteria are based on a number of experiments. First, the MSCFLP has been solved on instance 3 with a maximum running time of one hour. This experiment showed that after 1.6 seconds neither the current best lower bound nor the current best solution could be improved by the solver. Thus, after 1.6 seconds no improvement in the current gap of 3.7% has been made. However, by inspection of the solution it could be concluded that the current best solution is in fact an optimal solution. This conclusion is based on the following reasoning. Instance 3 contains 260 WiFi demand points, which implies that at least 9 WiFi service access points are needed to generate a feasible solution, since the capacity of the WiFi service is 30. Similarly, at least one SVC service access point, one Alarm access point, and at least nine access locations are necessary. In the current best solution nine WiFi service access points, one SVC service access point, one Alarm access point, and nine locations are opened, thus the current best solution is optimal. Summarising, the current lower bound is not tight, which implies that a maximum gap tolerance of 5% would be appropriate. A gap of 1% would lead to unnecessary high running times, without making any improvements in the current best solution.
Second, the MSCFLP has been solved on a larger instance, namely instance 7 with a maximum running time of one hour. This experiment showed that terminating the solver when the gap of the current best solution is lower than 5% yields unsatisfactory results, since the solution could be significantly improved by slightly expanding the running time of the solver. Thus, next to the maximum gap tolerance limit, it is chosen to implement an additional limit on the number of iterations without improvement. This additional criterion implies that solutions can be improved without increasing the running time unnecessarily.
Concluding, the solver is terminated when the gap between the objective value of the best solution found and the best lower bound is less than or equal to 5%, and for maxI iterations
no improvement of the current solution has been made. The formulation of maxI is provided in
Figure 6.3: Time plot of a set of experiments in which ηj2= η3j = ∞ for all locations j ∈ L. The MSCFLP is solved for η1
j ∈ [3, ..., 260] on instance 3. The solver is terminated when the running
time exceeds the maximum running time of 60 seconds. For η1
j ≥ 162 the detection time is equal
to the running time.
0 50 100 150 200 250 WiFi capacity ( j1) 0 10 20 30 40 50 60 Time (s) Running time Detection time Max. running time
43,200 seconds (i.e., 12 hours) has been implemented.
maxI = 50, 000 +
No. of Constraints + No. of Variables 10, 000 · 500 (6.4.1) = 50, 000 + |G| + |L|(3|F | + 1) + P u∈F P i∈Gu Lu i 10, 000 · 500 (6.4.2)
Chapter 7
Computational Results
We start this chapter with a discussion on the order selection of the Ordered Sequential Solving Heuristic with Updating (OSSHU). Thereafter, the computational results of the base model are presented. Section 7.2 discusses the added value of simultaneous optimisation of the location assignments and the results of the OSSHU. Some expectations on the results of the extended model are drawn in Section 7.3. The results of the extended model are provided in Section 7.4. Finally, a sensitivity analysis is included in Section 7.5.
7.1
Order Selection of the Ordered Sequential Solving
Heuris-tic with Updating
In Section 4.2 we already stated that the order in which the various services are considered is important, as it affects the overall solution (quality). For this reason, a small analysis is conducted on the six permutations of the set of services to determine the selection order. The OSSHU is applied to all instances for every order with a maximum running time of 1,800 seconds (i.e., 0.5 hours) per step. In Table 7.1 information is provided on the objective value of the best solution of all permutations. Furthermore, for every order and every instance, information is provided on the relative cost differences compared to the best solution: the higher the relative cost differences, the worse the solution generated by the order. Just as before, the WiFi service is denoted by 1, the SVC service by 2, and the Alarm service by 3.
Table 7.1 shows that orders 1-4 yield unsatisfactory solutions especially for instance 1. This is due to the fact that instance 1 is small in size, which implies that only one service access point for the SVC and Alarm service is sufficient to serve all corresponding demand points. If during the first step of the heuristic approach either the SVC or Alarm service access point is selected inefficiently, an additional location needs to be opened in one of the subsequent steps, which yields a relatively large cost difference with respect to the best solution.
In instances 2 and 3 it also suffices to have only one service point for the SVC and Alarm services to serve all demand points. However, for these two instances the first four orders do yield good solutions. This result is explained by the fact that these instances span a slightly larger surface, which implies that both the SVC and the Alarm service access point need to be opened at one of the central locations. Since some of the WiFi demand points are also located in the centre of the area, the Alarm and SVC service access points are easily combined with a WiFi service access point. Observing all instances, we can conclude that mainly the first element of the order determines whether the order yields bad solutions, and if so, how bad the solutions will be. As an
Table 7.1: Results generated by using the OSSHU with a maximum running time of 1,800 seconds per step. Column 2 shows the objective value of the best solution and columns 3-8 show the cost differences with respect to this minimum.
Cost differences (%) c.t. the best solution Inst. Best obj. 3-2-1 3-1-2 2-3-1 2-1-3 1-3-2 1-2-3
1 11,100 45.0 45.0 45.0 45.0 0.0 0.0 2 21,700 0.0 1.6 0.0 0.0 1.6 1.6 3 48,200 0.0 0.0 0.0 0.0 0.0 0.0 4 85,650 0.0 0.0 0.2 0.0 0.0 0.0 5 43,150 23.21 23.21 11.61 11.61 0.0 0.0 6 109,700 13.9 18.5 0.0 0.0 14.2 0.6 7 230,600 0.0 0.2 0.1 0.0 0.2 0.2 8 1,882,550 1.41 0.61 3.21 2.51 0.31 0.01 9 2,351,150 3.41 1.91 0.31 0.71 0.21 0.01
1 One or multiple steps of the heuristic are terminated, since the maximum running time was reached.
example, the two orders that consider the Alarm service first (i.e., orders 1 and 2) yield the worst solutions for instance 5 compared to the other orders. Similarly, the two orders that consider the WiFi service first yield the best solution for instance 5. These two orders give good solutions not only for instance 5. In general, the last order results in the best solution. Summarising, based on this small analysis, the last order 1-2-3 (i.e., WiFi-SVC-Alarm) is selected for the OSSHU.
Besides this data driven approach to determine the order, we could also look at the various service specifications. The WiFi service has the smallest range and needs to serve the most demand points (relative to its capacity) in general. Contrary to the WiFi service, the Alarm service has a large range, unlimited capacity, and only needs to service a relatively small number of demand points. Thus, the various specifications of the WiFi service will most likely have a larger effect on the location assignment than the specifications of the Alarm service. Therefore, the order WiFi-SVC-Alarm seems to be appropriate.
7.2
Base Model Results
In this section we will present the results of the base model. The performance of the two heuristic approaches of Chapter 4 are evaluated together with the performance of an exact approach. For the exact approach, CPLEX is used to optimise the location assignments of the various services simultaneously. The results of both heuristics and the exact approach are presented in Table 7.2. The table provides information on the objective value, the running times, and the cost differences of the various approaches. Firstly, we assess the performance of the SSH relative to the exact approach in Section 7.2.1. Secondly, we discuss the performance of the OSSHU relative to the SSH and the exact approach in Section 7.2.2.
