Department of Science and Technology Institutionen för teknik och naturvetenskap
Optimization of maintenance
system
Matilda Andersson
Fredrik Wandfelt
2013-12-17
Optimization of maintenance
system
Examensarbete utfört i Logistik
vid Tekniska högskolan vid
Linköpings universitet
Matilda Andersson
Fredrik Wandfelt
Handledare Valentin Polishchuk
Examinator Tobias Andersson Granberg
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Abstract
This report presents an optimization of the allocation of maintenance resources for Air Navigation Service (ANS) equipment of which LFV is responsible for the maintenance. The purpose the authors have worked after is to research ways of minimizing travelling time linked to maintenance visits for ANS equipment, this report includes the suggestions where the maintenance facilities should be placed in order to minimize the total travelling time. The report describes the problem background and presents the customer, LFV. It includes a chapter on some of the theories used for facility location and routing, and also presents methods for reducing the total travelling time used for maintenance visits annually.
The authors have worked with a given set of airports in Sweden. Information about the general work with maintenance as well as the annual demand of maintenance, including the frequency of visits, for each airport included in this project was received by Pär Oberger, the task expert and contact at LFV for this report.
A model for facility location based on the p-median model have been created and used when solving the problem, it was written in AMPL and solved with the CPLEX solver. The model was modified with two additional constraints regulating the minimum annual working time and the maximum distance for one-way travelling. The model, and the modified versions of it, is a sufficient way of minimizing travelling time. The results are presented in maps and in tables, some of these tables are in the report and the others are included as appendices.
The solutions from running the model with the constraints regulating one-way driving to a maximum of four hours and a minimum of 1100 hour annual working time gives an optimal solution when placing eight facilities. The authors propose a solution with fewer facilities instead with regards to the cost of establishing facilities although this cost is not known. The authors deems that a solution with five facilities is better since the benefit of three additional facilities, in term of lower total distance, do not compensate for the assumed cost of establishing them. These five facilities are located at ESGG, ESNO, ESOK, ESPA and ESSP. The authors also deem that a model with both routing and facility location combined would have been able to present a solution better suited for this reports presented problem.
Table of contents
1 Introduction ... 1
1.1 Purpose ... 1
1.2 Delimitations ... 1
1.3 Materials and methods ... 2
1.4 Analysis of sources ... 2
1.5 Structure of Report ... 3
2 Problem Background ... 4
3 Literature Study ... 5
3.1 Facility Location ... 5
3.2 The Structure of a Facility Location Model ... 5
3.2.1 Facilities ... 5
3.2.2 Locations ... 5
3.2.3 Customers ... 6
3.2.4 Distances ... 6
3.3 Facility Location Models ... 6
3.3.1 The p-Median Facility Location Model ... 6
3.3.2 The p-Center Facility Location Model... 7
3.3.3 Fixed Charged Location Problem ... 8
3.3.4 Overview of Facility Location Models ... 8
3.4 Routing ... 9
3.4.1 Travelling Salesman Problem ... 9
3.4.2 Vehicle Routing Problem ... 9
4 Models for Facility Location and Routing ... 11
4.1 Facility Location Model ... 11
4.2 Adding Constraints to Facility Location Model ... 12
4.2.1 Requiring Annual Working Time ...12
4.2.2 Restricting Maximum Driving Time ...13
4.3 Routing Found by Heuristic ... 13
5 Results From Facility Location Model ... 15
5.1 Facility Location Model ... 16
5.2 Facility Location Model Requiring Annual Working Time ... 19
5.3 Facility Location Model Including two Constraints ... 21
6 Results from Routing ... 24
7 Comparision and Discussion ... 26
8 Conclusions ... 29 Appendix 1. Location Indicator Declaration ... Appendix 2. Results From Basic FAcility Location Model ... Appendix 3. Visualization of Basic Facility Location Model Results ... Appendix 4. Results for Requiring Annual Working Time ... Appendix 5. Results for Final Version of Model DrivningTime ≤4... Appendix 6. Results for Final Version of Model DrivningTime ≤5... Appendix 7. Results for Final Version of Model DrivningTime ≤6... Appendix 8. Results for Final Version of Model DrivningTime ≤7... Appendix 9. Results for Final Version of Model DrivningTime ≤8... Appendix 10. Distance Matrix ... Appendix 11. Ampl Code of Final Model ...
List of Figures
Figure 1. Visualization of allocation for p=6 from basic facility location model ... 17
Figure 2. Visualization of allocation for p=6 from facility location model with working time constraint ... 20
Figure 3. Visualization of result from one run of final model ... 22
Figure 4. Visualization of routes for facility ESSP ... 24
Figure 5. Bar chart comparison between all models and scenarios tested ... 27
List of Tables Table 1. Comparison between facility location models ... 8
Table 2. Results from basic facility location model ... 16
Table 3. Results from facility location model with working time constraint ... 19
Table 4. Travelling time results from facility location model including two constraints ... 21
Table 5. Results over routing for facility ESSP ... 25
Glossary
AIS Air Information Service ANS Air Navigation Service
ANSP Air Navigation Service Provider ATCC Air Traffic Control Center ATS Air Traffic Service
CNS Communication, Navigation, Surveillance Service MET Meteorological Service for Air Navigation
TSP Travelling Salesman Problem
1 INTRODUCTION
This chapter describes the problem, the background and the purpose of the project. It presents the methods that the authors have worked by as well as an analysis for the sources used. Initially a brief presentation of Air Navigation Services (ANSs) is made, and the importance of functional equipment within ANS.
Within Air Navigation Service several sub services functions (MET, AIS, CNS and ATS), all with the purpose to serve air traffic. Air traffic worldwide uses the services provided by ANS when pilots file their flight plan, when they take part of meteorological information, or when navigating between navigational aids or fixes, among other things. To be able to provide these services, and keep them reliable with updated information, ANS uses different sorts of equipment and systems.
Every individual equipment station, or system, within ANS has its own maintenance scheme, which is based on a periodic plan for preventive maintenance; that is to reduce the risk that the equipment or system malfunctions. In the maintenance scheme, a time buffer is also included to allow for unplanned repairs.
In Sweden there is a difference between the user of equipment, the one responsible for maintenance of equipment and the one executing the maintenance. The responsibility of the three mentioned areas may be divided between different companies; the assignments to the maintenance responsibility are depending on the winner of contract for maintenance at each location and for each equipment unit. LFV is one of the companies in Sweden that can be responsible for maintenance of equipment at different locations in Sweden.
1.1 Purpose
The purpose of the project is to research ways to reduce travelling time connected to maintenance visits for ANS equipment. The aim of the project is to suggest an allocation of maintenance resources for a specified set of ANS equipment in Sweden and to create a model that can propose facility location, based on minimizing the travelling time linked to maintenance missions over a year. The aim is also to create a model that may propose routes via which the maintenance should be executed.
1.2 Delimitations
The authors will only let airports housing ANS equipment from the given specified set to be potential bases. This allows for the locations to be airports, instead of working with coordinates and the shortest route there between which could result in placing a base in an inhabitable area.
In reality every technician needs to be specially trained for each individual equipment system and station that he/she will handle. The authors will assume that every technician is omniscient, and therefore not take into consideration the competence needed in each region and for each route.
The travelling time is considered to be symmetric and constant. This means that the time from location
i-j is the same as the time from location j-i and that no consideration is taken to current traffic situation. Nor yet will a buffer be added for traffic situations that can occur which might exceed the travelling time set in the input. Travelling time shall also only be by car and not by airplane or ferry. In the case a technician needs to repair something unexpected on an airport, it often means that the airport is closed and not accessible by airplane. The authors will also therefore assume that one technician is already located at the airport ESSV (Visby), which is the case in reality as well, due to the inconvenience of accessibility. ESSV is therefore excluded from the database.
ESSA (Stockholm-Arlanda airport) and ESMS (Malmö-Sturup airport) houses both an air traffic control tower as an ATCC (Air Traffic Control Center). This, among other things, creates a large need for maintenance since so many systems and equipment stations are located at the same place. It is
maintenance need for these airports, although they will be available to act as potential locations for facilities for the other airports.
1.3 Materials and methods
The activities in the project have been to find information on how maintenance work is performed, investigate the current situation on how the resources are being used, build a database with distances and maintenance demand for each location, create a model that optimizes the use of resources and write a report on the findings.
The authors met with an expert at LFV to get a better understanding of the current situation, to get access to data concerning the locations of equipment subject for maintenance, as well as to get data about the frequency and duration of maintenance in order to create the database to use as basis for the model.
The distances, expressed in travelling time, between the locations was extracted from Google maps. The locations was plotted and saved on one and the same map, the authors then used Google maps route planning service between every possible set of locations. The time, expressed in hours, was then written in Excel as a matrix. The travelling time chosen in the search between each location was the first suggested by Google Maps, this was done to value the travelling times in the same way. The authors have gathered information for theories and methods concerning facility location and network optimization from literature on network location and several articles published in journals and science compendiums. The theories and methods have been assessed between each other and
compared to the problem specific for this report in order to find a method suitable to solve it.
The authors chose to use AMPL for programming the model, due to the knowledge already possessed in the programming language from a previous course at Linköpings University. After a method was decided, its mathematical function was used as a basis when formulating the problem in AMPL. The files needed for the model were programed with information from the earlier mentioned database and knowledge from the literature research. Once the files were created the program was run in AMPL with the CPLEX solver to troubleshoot errors to the code. To ensure that the model was calculating correct during the whole process, a scaled down program was run at first, and once it worked more information was added both to model and database. When a functioning program was available the model was run with different constraints to get results for analysis. Tables with results from AMPL with the CPLEX solver were written in Excel to structure and allow further calculations, such as travelling time between single airports and its facility, using data from the database and the output from the facility location model.
The results from the models were presented in tables, figures, and in text depending on its relevance. A visual presentation of the result was made as maps using Google Maps Engine, a service for adding symbols on maps.
1.4 Analysis of sources
The travelling times used as data in the model between the airports are taken from Google Maps via their route planning service. Google Maps was chosen due to its easy access. Google is a large worldwide company giving a serious impression. Google Maps have up-to-date maps over the world, also at detailed level, the data retracted from Google Maps is considered as trustworthy and correct. The origins of the sources to the research are mostly articles taken from educational webpages and published literature and are mainly retracted from ScienceDirect.com. The webpage articles have been printed in renowned journals and papers within the mathematical and technical field and the literatures editors are professors and renown within their field of work, therefore the authors value them as trustworthy and correct sources.
1.5 Structure of Report
Firstly a description of LFV and a brief overview of how they work with maintenance today is presented as well as the problem this thesis is based on. Chapter 3 presents a literature study of the subject facility location and routing as well as theories within these subjects chosen to work with. Thereafter in chapter 4 the model for facility location is presented including the two constraints the facility location model is developed with. The structure of working with routing is also presented in chapter 4. In chapter 5 follow the results, and analysis of these, from all the versions of the facility location model. The result and analysis of routing are presented in chapter 6. In chapter 7 a discussion is held and a comparison between all versions of the facility location model is made, and finally in chapter 8 conclusions are drawn by the authors, based on the results of this project.
Throughout the report the ICAO location indicators will be used for the airports subject for research in this report. A declaration connecting the code to airports official name is found in appendix 1.
2 PROBLEM BACKGROUND
Sweden was one of the world’s first countries to liquidate the monopoly for the ANS (Air Navigation Service). When the market was free for others than the state to provide ANS, it opened up for
negotiation for e.g. who would provide the ATS (Air Traffic Service). Although who will supply the equipment to provide ATS, and who will be responsible for managing this equipment, have faced competition for a longer period (Oberger, 2013).
LFV is a Swedish public enterprise with approximately 1 300 employees, and one of Sweden’s ANSP (Air Navigation Service Provider). Their main task is to provide ANS for both civil and military operators, primarily in Sweden. LFV is as well an overall supplier of aerodrome specific services, they provide, among other things, to manage and be responsible for the technical equipment used for ANS, e.g. meteorological and navigational equipment. This is run by the department System- och Utveckling, who also manages the communication, navigation and surveillance services used within ANS. Today LFV is responsible for the management of maintenance of approx. 1 800 individual equipment units in Sweden that is used for ANS. These units are scattered between approx. 150 locations in Sweden. An example of a location can be an airport in a big city or a navigational aid in the middle of a far away forest. LFV calculated that in 2012 more than 60 000 hours were used for the maintenance of this equipment (Oberger, 2013).
Only looking at airports and premises, LFV is today responsible for maintenance at 42 airports (40 towers and 2 ATCCs), as well as the air traffic management school Entry Point North and LFV’s headquarter. At these locations, both the amount of equipment units that LFV is responsible for, and the amount of maintenance that each unit require, differs. It can be an inequality of only one unit at a location needing maintenance once a year, to the majority of e.g. an air traffic control tower’s equipment needing supervision daily. There are two kinds of maintenance, planned and variable. The planned is done on a scheduled basis within certain time limits for each unit, while the variable is executed when needed, that is when there is a problem with any of the equipment units (Oberger, 2013).
LFV is the service provider responsible for maintenance of the ANS equipment. LFV has contracted the company Eltel Networks to perform the maintenance. LFV specifies to Eltel Networks what maintenance that is to be done when and where, based on this Eltel Networks decides where their technician bases are located and create the routing and scheduling for maintenance. Eltel Networks also plan the need for technicians for the different units; each equipment unit needs a technician specifically trained and authorized for its specific system. LFV also specify a maximum time limit, expressed in hours, within which the maintenance should commence. Technicians from Eltel Networks also work with other missions than those ordered from LFV (Oberger, 2013).
The geographical outspread of units in need of maintenance, the need to match technician competence to each system, as well as the planned hours alone for maintenance are all contributing factors to why it can be a challenge to find and develop an efficient plan for maintenance. Interesting topics to investigate, in terms of reducing the resources used for maintenance, is where to place the technicians, which units should be included in their remit and how should their routes be planned. From LFV’s point of view it is interesting to research facility location and routing in order to gain better understanding of the complexity of the problem, as well as it gives a chance to see if their specifications need adjustments in order to maintain a good relation to the technician company (Oberger, 2013).
3 LITERATURE STUDY
To be able to find a solution to the given problem a literature study within the subject of facility location and routing has been conducted. This chapter will present general theories and methods used when trying to solve a facility location problem and routing problems, as well as the general structure of a facility location model. The chapter also presents three chosen, by the authors, facility location models, firstly optimization in general is described shortly below.
Optimization can be seen as to make something as perfect, as efficient or most useful as possible based on a set of input variables, or a given domain. Within mathematics optimization is often used for determining the maximum (or minimum) values of a specified function, counting with the constraints that may be set to it (Dictionary, 2014).
3.1 Facility Location
The concept of facility location is to find where to place resources with regards to determined factors. The first location model was published in 1909 by Alfred Weber, but the subject became more popular as a research topic in the 1960s (Scaparra & Scutellà). A location model can consist of many things, but some elements are common for almost every model. There is often a set of customers spread over a geographical area, with a demand for a resource. This demand should be met by the facilities serving the customers. In order to serve the customers, a location needs to be chosen for the facility as well as placing the facility there.
Facility location is an important part of a company’s strategic planning. A facility location model will provide the decision maker with valuable information on where to place a facility (Owen et al. 1998).
3.2 The Structure of a Facility Location Model
Scaparra & Scutellà (2001) describes three basic parts of a model; facilities, locations and customers. Revelle (2005) also describes the above mentioned but add distances between facilities and customers as a fourth component that characterize a location problem. In addition to these theories there are many different parameters defining each location problem. A problem can be weighted or unweighted, a term describing how the distances are calculated. A demand-weighted distance will weight the distance to a node with high demand higher than to a node, with the same distance, but with lower demand. The problem can also be characterized as a capacitated or an uncapacitated problem. This definition determines if there should be any restrictions on the capacity at a facility. If there is a restriction to a facility’s capacity the problem is capacitated, an uncapacitated problem assumes that a facility can deliver all the demand that is requested from it. The objective function of a model is used to measure the benefit of a solution, and can either minimize or maximize a value. Whether the objective function minimizes or maximizes depends on the solution wanted. Costs are generally minimized but distance can be both maximized or minimized depending on the nature of the problem.
3.2.1 Facilities
The term facility is used for the resource that needs to be placed in order to serve the customers. A facility is many times a plant, a factory, a station or a store but can also be something less apparent like a component on a circuit board. When placing a facility many things must be taken into
consideration; the number to be placed, the available capacity each facility and the cost of establishing a facility at a certain location etc. The number of facilities to be located depends on the problem. There are both single-facility problems and multi-facility problems, where to latter is more complex
(Scaparra & Scutellà, 2001).
3.2.2 Locations
When forming a location model the possible locations where a facility can be placed at must be decided. Where the facilities can be placed depends on whether the problem is continuous, discrete or in a network (Scaparra & Scutellà, 2001).
A continuous location problem differs from the others by allowing location of a facility at any point in the plane. As a result of this, the model requires that all distances can be calculated from coordinates that represent a customer. These coordinates can later be used to measure the distance between all customers and the proposed locations of facilities in order to minimize the total distance (Klose & Drexl, 2005).
A discrete location problem is similar of a continuous problem in the aspect that it allows location to take place anywhere in the space. The difference is that a discrete problem can only place facilities on allowed areas, therefore a discrete problem demands candidate positions for location to be presented before solving the problem (ReVelle & Eislet, 2005). An example of a discrete location problem can be siting of train stations that only can take place on places where enough passengers are available. In a network location problem there are nodes and arcs building a network. Demand is regularly connected to nodes, but demand can also occur on arcs (Arabani & Farahani, 2012). The arc distances are used to calculate the shortest path in the network. The facilities can only be located in the network and at least one optimal solution is to place them in the nodes (ReVelle & Eiselt, 2005).
3.2.3 Customers
In the other end of the problem are the customers. A description of a customer can be e.g. a person available to buy a product from a facility or a warehouse getting their deliveries from a facility. It is important to know the customers’ distribution, demand and behaviour when forming a location model. Distribution is describing where the customers are located, for example at a specific point or spread evenly over a specified area. Demand is a number of how much service are needed from the facilities. The behaviour will describe how the customers interact with the facilities available. There can be cases where the customer does not fill their demand from the closest facility, e.g. grocery shopping, and there can be cases when the customer is forced to choose the closest facility, e.g. a hospital (Scaparra & Scutellà, 2001).
3.2.4 Distances
A facility location model needs a way to measure the distances between customers and facilities in order to measure the benefit of a solution. The distances can be in either length or time. For a continuous problem, coordinates is often used to calculate distances instead of using point-to-point distances, distances calculated from coordinates requires less data storage. In a network based problem all distances must be calculated with consideration of the possible routes that the network allows, for example a network of roads or train tracks (Revelle & Eiselt, 2005).
3.3 Facility Location Models
There are many different location models used for facility location. Focus in this chapter lays on facility location on networks by describing the p-median and p-center models, two commonly used methods for suggesting placing of facilities in a network. The fixed charge location theory is also presented, and a comparison between the three location theories is made. Characteristic for the p-median and p-center models are the determination of p: the number of facilities to be placed in the network.
3.3.1 The p-Median Facility Location Model
The p-median problem has its origin in Hakimis papers published in 1964 and 1965. Hakimis papers investigated the p-median problem, the minimum total distance location of p facilities on a network of
n demand nodes. The optimization problem Hakimi was trying to solve was to minimize the total length of cables by locating switching centres in a communication network. At this time Hakimi did not found a method for solving the problem although he proved that at least one optimal solution was to locate all p facilities at the network’s nodes, this solution is often referred to as the Hakimi
Theorem. The term p-median was created by Hakimi but the formulation as a zero-one problem was made by Revelle and Swain (1970). The formulation of a general p-median problem is
��� � = � × � × � � = , � = , . . , � � � , � = , . . , � � = , . . , � � = � � ∈ { , }, � = , … , � � = , . . , � � ∈ { , }, � = , . . , �
where xj are one if a facility is located at node j and zero otherwise. If location i is assigned to be served by facility j then the allocation variable yij are one, and otherwise zero. The objective function (1) is based on the demand, v, of each node and the cost, c, to travel between node i and node j. Constraint (2) declares that a connection between a location node i and a facility j must be made, as well that a location i can only be allocated to one facility. To ensure that a node i is allocated only to a designated facility, constraint (3) is declared which only allows a connection between node i and node
j to be made (yij=1) if j is set as a facility (xj=1). Constraint (4) declares that the summed amount of facilities, location variables xj, must be the same as the set value for p, so that the model locates as many facilities as the user require. (5) and (6) are the zero-one constraints for variables xj and yij. The p-median problem is a location problem on a network. To solve the problem a set number of facilities must be decided, this will determine how facilities should be placed in the network to minimize the total travelling cost. The travel cost can be expressed in e.g. distance or travelling times to and from nodes in the network. When the model has placed the facilities it will calculate the travel cost to all nodes that need service from each facility in order to measure the benefit of the solution (ReVelle & Eiselt, 2005). The p-median model will not consider the capacity of a facility or any cost related to the location of the facilities (Current et.al., 2004).
3.3.2 The p-Center Facility Location Model
The p-center model is similar to the p-median model. The difference between the two is the relation between the customers and the facilities. The p-center model’s objective function (1) will minimize the maximum distance D between a facility and its assigned node. D will also be a constraint to the maximum length between a customer and a facility (7), defining the maximum allowed distance between a facility and a customer served by the facility in question. This ensures that no connection between any node and facility can be used if it is longer than D. Constraints (3), (4), (5) and (6) can also be found in the p-median model, which are described in chapter 3.3.1. The p-center model can be divided into two sub-problems, vertex p-center and absolute p-center problems. A vertex p-center problem can only place facilities on the nodes on the network, the absolute p-center problem allows the facilities to be placed anywhere on the network (Scaparra & Scutellà 2001). Below you find a general formulation of the model.
��� � � = , � = , . . , � � � , � = , . . , � � = , . . , � � = � � ∈ { , }, � = , … , � � = , . . , � � ∈ { , }, � = , . . , � � � � , � = , . . , �
A practicable use for a p-center model can be to locate emergency resources that must be able to reach all citizens within a maximum waiting time. The p-center model, in similarity to the p-median model, does not consider the cost related to location of a facility and capacity restrictions for facilities.
3.3.3 Fixed Charged Location Problem
A problem that considers cost of establishing facilities and capacity restrictions is the fixed charge location problem. It will solve the location problem by minimizing the total costs of facilities and travel, and as a result of this it will also present the optimal number of facilities to be placed in the solution. An effect of the capacity restrictions is that a customer does not necessarily get its demand fulfilled by the closest facility. If the closest facility to a customer already delivers all its capacity to another location this situation appears (Current et al. 2004).
The fixed charge location problem can be suitable to a company that wants to establish a distribution network at the lowest possible cost.
3.3.4 Overview of Facility Location Models
Table 1 below shows a simple comparison between the location models presented in the previous chapters. The comparison is made with regards to inputs, available constraints and the output.
Table 1. Comparison between facility location models
Type of model
Inputs Available Constraints What is minimized?
p-median Number of facilities to be used,
distances between nodes, demand at nodes.
Only one facility serving a node, all nodes must be assigned to a facility
Total weighted distance of travel
p-center Number of facilities to be used,
distances between nodes, demand at nodes.
Same as p-median + Facility must be placed at node (optional)
The maximum weighted distance of travel between a node and its assigned facility Fixed
Charge Location
Cost for locating facility at each site, capacity of a possible facility at each location, cost for travel between nodes.
Demand at node can not exceed supply at facility
Total cost for travel and facilities
As seen in Table 1 the inputs for the p-median and the p-center problem are related to distances and demand. The fixed charge location problem has inputs related to costs and demand. It is possible to add more constraints to all problems described in the table above but the mentioned constraints are in the original formulation for each problem. The objective functions for each of the three problems are all minimizing functions. The median problem will minimize the total distance travelled, the p-center will minimize the maximum distance of the single connections between nodes and facilities in the network and the fixed charge location problem will minimize the total costs for establishing facilities and travel.
3.4 Routing
If a set of facilities, or cities, is known and it is left to find the best route between these, this can be formulated as finding the optimized circuits within a network. Problems with optimized circuits have been relevant since the 1800th century when preachers set out on month long journeys between cities to preach their missions. These journeys were tweaked, and in a sense optimized, changing the order in which the cities where visited, to reduce the length of travel summed over the whole journey. During the years many professions have had the need of finding an optimal route, or circuit, to minimize their travelling time e.g. circuit riders in different jurisdictions (judges and lawyers in the 1500th century), messengers, airline operators or tourist wanting to fit in all the grand attractions during their visit. In the 1930’s the name ‘Travelling Salesman Problem’ started to merge and be used to the problem concerning optimized circuits. No one knows the true origin of the name for the
concept, but it is believed to have originated from Princeton University, even though the problem itself has existed in centuries (Applegate et.al., 2007).
3.4.1 Travelling Salesman Problem
The TSP (Travelling Salesman Problem) is to find the cheapest way to travel between all locations in a given set, and returning back to the starting point, given the cost that exists travelling between each pair of locations. It can be thought of a salesman wanting to visit each city in his region, and wanting to do this finding the shortest possible route, or circuit. According to Applegate et.al. (2007) the TSP is “one of the most intensely investigated problems in computational mathematics”. The idea and formulation of the problem is easily understandable, but when handling with problems with more than 50 locations it becomes impracticable to solve since there is no simple and efficient algorithm for the problem. The potential solutions for a given set of points, n, to visit is equal to (n-1)!. The running time proportional to finding this amount of solutions makes the problem impracticable to solve to optimization, and to check all potential tours when dealing with problems for n≥50 is out of the question, even if mustering the world’s computer resources according to Applegate et.al. (2007). Studies and experiments on TSP solving’s have though been made within psychology between groups of people. Applegate et.al. (2007) describes an experiment made by Vickers, where one group of people was given the task to find the shortest tour between a given set of cities, at the same time another group of people was given the task to find the most graphically pleasing tour between an identical set of cities. The results were that the tours found in each group where equal in quality and optimality.
TSP is probably one of the most studied problems within combinatorial optimization, but yet to date no answer has been settled to the theoretical question whether or not there is a good algorithm for it. It has, thanks to the many studies about it, functioned as an engine of discovery of other general-purpose techniques in applied mathematics (Applegate et.al., 2007). Some examples of areas that has its origins in TSP is Mixed-Integer programming, the Branch-and-Bound method, Heuristic-search algorithm, the Vehicle Routing Problem as well as Adelman’s DNA computing. Below a method originating from the TSP is presented, that the authors have chosen to look closer on.
3.4.2 Vehicle Routing Problem
TSP that it allows more than one vehicle (“salesman”) in the network as well as each customer (location) to be visited more than once in the route. Each customer can only belong to one route, and the routes need to start and end at the same location (Gu et.al., 2006; Cordeau, Gendreau & Laporte, 1997). The VRP can e.g. be found in grocery delivery.
4 MODELS FOR FACILITY LOCATION AND ROUTING
This chapter will present the models developed by the authors. The first two subchapters describe the facility location model including the modifications made to it from its original form as a p-median model. The third, and last, subchapter presents the working way for developing a method to find routes that can minimize travelling times further.The problem is based on 34 airports with various needs for maintenance. Each airport needs a certain number of visits for maintenance per year and each visit takes from approximately 1 to 4 hours. One solution is to have maintenance resources at every airport to fulfil the need for these visits, but that is also an expensive solution in terms of cost. Therefore a solution with facilities (“technicians base”) that allows technicians to travel to airports, by car, is preferable.
Since the travelling times between some airports exceed the hours of a normal workday, it is necessary to divide the airports into smaller groups. The groups will be decided based on the distance between the airports. In each group the airports can then be connected in a network of their own, to be able to find the best routing for maintenance visits. The model will focus on solving the facility location problem, but will get constraints added to allow for better conditions for when implementing routes. The basic version of the model will allow for groups of airports to be found, as well as presenting the one airport in each group that will serve as facility (technician base).
The basic version of the facility location model will be formulated as a p-median problem. The number of facilities, p, is selected by the user and thus set in the model as a static number. The p-median model focuses on minimizing the travel times, which is very similar to the main purpose of this project. P-median is also a suitable formulation for the model since it does not take into
consideration a cost for establishing a facility, a cost the authors do not have access to in this specific scenario.
4.1 Facility Location Model
The facility location problem was formulated in AMPL and is described below, found in parenthesis is the connection to the mathematical formulation described last in this chapter. Following inputs, outputs, constraints and objective function are declared:
Inputs
Airports housing equipment where LFV is responsible for the maintenance (Airports and possible facilities)
The travel time by car between said airports (Distance)
Sum of maintenance visits over one year for each location (Visits) Number of facilities to be located (technician bases, “p”)
Outputs
Location of facilities
Airports allocated to be served by each individual facility Constraints
An airport must, by one arc, connect to only one facility (2) All airports must be connected to a facility (2 & 3)
Sum of located facilities must be same as number of facilities sought, p (4) Objective function
The objective is to find locations for facilities with suitable groups of airports whose demand of visits can be satisfied by their facility and travel the shortest way.
The mathematical formulation of the model is
��� � = ������ × �������� × � � = , � ∈ �������� � � , � ∈ �������� � ∈ �������� ���������� � = � � ∈ { , }, � ∈ �������� � ∈ �������� ���������� � ∈ { , }, � ∈ �������� ����������
where xj and yij are the location respectively allocation variable. xj are one if a facility is located at node j and zero otherwise. If location i is assigned to be served by facility j then the allocation variable
yij are one, and otherwise zero. Constraints (5) and (6) are these variables zero-one constraints. The objective function will calculate the distance travelled from facilities to airports annually. The number of facilities will be decided by setting p, which is equal to the number of facilities to place. The function will calculate all possible solutions in order to minimize the total travelling time. Each airport must be assigned to a facility; the distance between facility and airport will be multiplied by the number of visits needed for each airport. This will generate a total travelling time for each airport and its assigned facility. If a facility is assigned to several airports the total travelling time is summed up for the facility in question. The sum of all travelling time from all facilities to their assigned airports will be the output of the objective function. The assignment between facilities and airports will also be an output.
One constraint will force all airports to be assigned to a facility while another constraint restricts each airport so it can be connected to one facility only. One constraint will also force all facilities to actually be placed in the network since possible facilities are only allowed to be chosen from the databank that consists of the airports subject for maintenance, including ESSA and ESMS.
The basic version of the model will calculate the travelling time one-way multiplied with the number of visits for each airport. Calculating with travelling time both ways will give no change in the result for location of facilities and its assigned airports, although it will affect the total travelling time.
4.2 Adding Constraints to Facility Location Model
To find a solution allowing a more practical use, two constraints were added to the basic model. Firstly a requirement of a minimum annual working time for each facility was formulated and secondly a restriction to the maximum travel time between a facility and each of its assigned nodes was formulated. To do this two constraints were implemented; “SuffiForEmploy” and “DrivingTime”, including another post in the input; annual maintenance need for each individual airport.
4.2.1 Requiring Annual Working Time
To ensure that a facility can provide enough annual working time for a technician in each group of airports the constraint SuffiForEmploy regulating this was added. The constraint will require the sum of driving and performing maintenance to be at least the value of S hours every year for each group of
airports. No upper limit for working time is set since more than one technician can work at each facility.
To be able to run this constraint with the model the annual maintenance need for each airport was added as a new input. The formulation for the constraint SuffiForEmploy is:
����������� + ������ × �������� × × � � × � , � ∈ �������� ����������
The new constraint will result in a change of the original model’s objective function. Instead of calculating the distance one-way it needs to be multiplied by a factor of two. This gives a solution representing actual travelling time from airport to facility and back and present a maximum result, rather than presenting a travelling time that are unrealistically low. The new objective function is formulated:
min � = ������ × �������� × × �
4.2.2 Restricting Maximum Driving Time
With the constraint restricting annual working time active, the model will assign airports further away to reach the minimum set for the constraint, when p is set to higher numbers. Because the maintenance hours are constant and the driving time is variable, the model will increase travelling time to find more working hours to meet the SuffiForEmploy constraint. To avoid that the model does this the constraint DrivingTime was formulated to regulate the driving time between a facility and each of its assigned airports. The formulation for the constraint is:
� × �������� �, � ∈ ��������
� ∈ �������� ����������
where t is a set variable representing the maximum driving time in hours to be set by the user.
4.3 Routing Found by Heuristic
To ensure a feasible solution to the routing for the facilities, a heuristic approach was used. The authors created a table including all allowed, according to the constraints described below, routes for a given facility and its assigned airports to serve. To find a feasible set of routes to use, among the ones found, a trial-and-error approach was adapted. As a part of the heuristics working method the authors used their intuitive judgment when deciding the final set of routes to present, no improvements to the first found feasible set of routes where attempted to be made.
The creation of routes was made with following inputs, output and constraints: Inputs
Group of airports with their allocated facility Travelling time by car between airports in the group
Sum of maintenance visits over one year for each airport, à 2,5 hours each per visit Output
The sum of travelling time over a day is maximum 8 hours
The sum of working hours (travelling time and maintenance time) for each route cannot exceed 12 hours
A route must start and end at the facility Objective function
cost = sum of travelling time for each route
The objective is to find feasible routes between each facility and its allocated airports, which fulfil the visit demand for each airport and have a reduced total travelling time.
It is an assumption that the length of all individual maintenance visits are the same; 2,5 hours. This number is based on a calculation of the average maintenance durance, and is a value used to represent a maintenance visit so that a route will not consist of only driving between airports and back to the facility.
5 RESULTS FROM FACILITY LOCATION MODEL
This chapter presents the results after running the facility location model with the constraints as described in chapter 4, results from each version of the model are presented in its own subchapter. The results presented in this chapter have been derived from running the model in AMPL with the CPLEX solver. Working time presented in each result is calculated as the sum of travelling time and
maintenance time. In addition to p, the number of facilities to place, one should keep in mind that it is assumed that ESMS, ESSA and ESSV already functions as facilities to meet their own demand respectively. This means the total of facilities is an additional 3 to the number set on p.
For the facility location model described in chapter 5.1 a p-median formulation has been used, whereas the versions of the facility location model described in chapter 5.2 and chapter 5.3 are based on the first p-median formulation but have both added constraints and configured objective functions.
5.1 Facility Location Model
The basic facility location model was run with values 1-10 set for p. Table 2 summarize the outputs for p=1,2,..,10, not including the airports allocated to each facility. In Table 2 the objective function, z, is multiplied with two to allow for a comparison with the outputs from the other versions of the model. Added in the table is also the travelling time, both ways, and the total working time for each group of airports calculated afterwards in a summarizing excel-file (see appendix 2).
Table 2. Results from basic facility location model
p Facilities Driving h z×2 Working h p Facilities Driving h z×2 Working h
1 ESCM 24603,38 24603,38 39089,38 8 ESGG 1082,64 3733,18 4254,64 2 ESGJ 7528,28 12662,96 16456,28 ESMQ 90,04 1002,04 ESNU 5134,68 10692,68 ESNN 667,44 2595,44 3 ESGJ 3504,76 10388,6 9420,76 ESNQ 0 552 ESNU 5125,68 10681,68 ESNU 1289,5 4367,5 ESOW 1758,16 4772,16 ESOK 0 1298 4 ESGJ 3504,76 8475,32 9420,76 ESOW 593,5 2431,5 ESNN 1389,36 4341,36 ESSP 10,06 10,06 ESNS 1828,38 4434,38 9 ESGG 1082,64 2960,96 4254,64 ESOW 1752,82 4764,82 ESMQ 90,04 1002,04 5 ESGG 1300,2 6869,32 4616,2 ESNN 667,44 2595,44 ESNN 1389,36 4341,36 ESNO 344,54 2204,54 ESNS 1828,38 4434,38 ESNQ 0 552 ESOK 744,28 2688,28 ESNS 172,74 1390,74 ESSP 1607,1 5275,1 ESOK 0 1298 6 ESGG 1300,2 5608,52 4616,2 ESOW 593,5 2301,5 ESNN 667,44 2595,44 ESSP 10,06 1848,06 ESNQ 0 552 10 ESGG 1082,64 2297,28 4254,64 ESNU 1289,5 4367,5 ESMQ 90,04 1002,04 ESOK 744,28 2688,28 ESNN 3,76 1301,76 ESSP 1607,1 5275,1 ESNO 344,54 2204,54 7 ESGG 1300,2 4658,7 4616,2 ESNQ 0 552 ESNN 667,44 2595,44 ESNS 172,74 1390,74 ESNQ 0 552 ESNZ 0 630 ESNU 1289,5 4367,5 ESOK 0 1298 ESOK 0 1298 ESOW 593,5 2301,5 ESOW 593,5 2301,5 ESSP 10,06 1848,06 ESSP 808,06 3414,06
In Figure 1 the output for p=6 is presented, for visualizations for all the solutions (except p=1) see appendix 3. The locations designated as facility for a group of airports are visualized as a star, as seen in Figure 1, the allocated airports to each facility are circles with the same colour as their facility’s icon. ESSA (Stockholm-Arlanda), ESMS (Malmö-Sturup) and ESSV (Visby) airport are included on the map as facilities with only themselves to serve.
Figure 1. Visualization of allocation for p=6 from basic facility location model
A p-median formulation for the model was chosen since it matches this project’s problem considering what it have as available inputs, and what the objective is. Advantages is that it is a model that minimizes the travelling time and the cost between nodes can be expressed in different ways
depending on what suits the user best. Also there is no need for an establishing cost since it is all about the travelling time and distance. To use the p-median model as a solution the user needs to keep in mind that the travelling time is minimized at the cost of establishing remote facilities that only is responsible for themselves. The travelling time will decrease until all nodes are a facility with themselves to serve.
When running the projects problem as a p-median problem the results show a great decrease in
travelling time when increasing the number of facilities from placing one to five facilities, see Table 2. When placing six to ten facilities the decrease continues with lower pace. Therefore the benefit of adding one more facility is greater when the existing number of facilities is quite low. Since the cost for adding facilities is unknown but assumed to be of some significance, the number of facilities placed should be reduced with this in consideration. If this were not considered a solution with the same number of facilities as locations to serve would be optimal, as it would give the lowest total travelling time.
For the solutions p=2,3,4,5 the model places the facilities at locations with a high visit frequency. A recurring event for all runs when the value for p increases from 2 to 5 is that the model swaps one facility for two other facilities in the same area. The two new facilities divides the responsibility for maintenance for the airports previously served by the first facility.
When p=6, ESNQ becomes a facility with no travel (see Figure 1), and for p=7 ESOK becomes a facility with no travel. These allocations reoccurs for p=8,9,10. For p=6,7,8 the solutions are all similar, the model does not rearrange much but simply adds a new location as facility in every
scenario to an existing solution. When p=9 a change in the allocations occurs when ESNU no longer is assigned as facility and in its place ESNO and ESNS are assigned, taking over and dividing between them the allocated airports ESNU used to be responsible for. This indicates that allocating 6 or more facilities results in the model finding airports to isolate from groups to minimize the travelling time, to isolate an airport results in it serving itself as facility. The facilities from p=6 is also facilities for all solutions until p=9, which shows that these locations are good choices for facilities.
The solution with p=5 is considered the best for the basic facility location model since it gives every facility a sufficient amount of total working hours for all facilities when calculating this in excel. Also each allocated facility is assigned to serve more airports than just itself. When p is 6 or greater at least one facility drops below recommended values in working hours, which conflicts with the request of enough working time for an employment at a facility. If this request is ignored a solution with p=6 or
p=7 can also be considered as good. For p=8,9,10 it is likely that the cost for extra facilities exceeds the cost savings for shorter travelling times.
5.2 Facility Location Model Requiring Annual Working Time
The result after running the model with the constraint SuffiForEmploy activated, regulating the annual amount of working hours (see chapter 4.2.1), is presented in Table 3. The minimum working hours per year, S, allowed for each group of airport was set to be ≥ 1100 hours for all scenarios run. The value 1100 hours represents 25 hours a week and was selected after a recommendation from the expert at LFV. This since the technicians also have other missions to see to except maintenance done for LFV, which allows the sum of hours to be less than an 100% employment. Table 3 summarize the outputs for p=1,2,..,10, not including the airports allocated to each facility. Added is the travelling time both ways and working time for each group of airports calculated afterwards in a summarizing excel-file (see appendix 4).
Table 3. Results from facility location model with working time constraint
p Facilities Driving h z Working h p Facilities Driving h z Working h
1 ESCM 24603,38 24603,38 31846,38 8 ESGG 484,58 4671,72 1504,58 2 ESGJ 7528,28 12662,96 11992,28 ESGJ 784 1689 ESNU 5134,68 7913,68 ESNN 667,44 1631,44 3 ESGJ 3504,76 10388,6 6462,76 ESNO 344,54 1274,54 ESNU 5125,68 7903,68 ESNS 1306,74 2191,74 ESOW 1758,16 3265,16 ESOK 366,56 1100,56 4 ESGJ 3504,76 8475,32 6462,76 ESOW 579,3 1421,3 ESNN 1389,36 2865,36 ESSP 138,56 1101,56 ESNS 1828,38 3131,38 9 ESGG 457,94 4653,42 1439,94 ESOW 1752,82 3258,82 ESGJ 786,66 1692,66 5 ESGG 1300,2 6869,32 2958,2 ESNN 663,68 1626,68 ESNN 1389,36 2865,36 ESNO 344,54 1274,54 ESNS 1828,38 3131,38 ESNS 470,62 1103,62 ESOK 744,28 1716,28 ESOK 380,9 1126,9 ESSP 1607,1 3441,1 ESOW 581,3 1424,3 6 ESGG 1264,14 5960,1 2912,14 ESPE 829,04 1107,04 ESNN 667,44 1631,44 ESSP 138,74 1100,74 ESNQ 694,48 1100,48 10 ESGG 295,26 5038,94 1101,26 ESNU 992 2413 ESGJ 517 1334 ESOK 744,28 1716,28 ESMK 714,18 1100,18 ESSP 1597,76 3429,76 ESNN 663,68 1626,68 7 ESGG 1082,64 5149,36 2668,64 ESNO 344,54 1274,54 ESNN 663,68 1626,68 ESNQ 693,54 1100,54 ESNO 344,54 1274,54 ESNU 610,54 1102,54 ESNS 1306,74 2191,74 ESOK 499,2 1426,2 ESOK 366,74 1100,74 ESSB 549,9 1113,9 ESOW 581,3 1424,3 ESSP 151,1 1102,1 ESSP 803,72 2105,72
Figure 2 visualizes the output for p=6 as an example for comparison with the basic facility location solution. The allocated facilities are shown as stars on the map, while the airports they are assigned to serve are showed as circles in an identical colour as their facility.
Figure 2. Visualization of allocation for p=6 from facility location model with working time constraint
The SuffiForEmploy constraint was added as a reversed capacity constraint with inspiration from the fixed charge location model, this is the only part of the fixed charge location model being used. Consideration of cost for establishing facilities is not made since this is not the projects purpose. The model's objective function is changed as described in chapter 4.2.1 but still a version of the basic facility location model.
With setting only a minimum limit to working hours a situation where there is enough work for one technician but not enough for two can appear. There can be a grey zone between the maximum working hours for one employment and until the working hours increase enough for one full-time employment and one that just reach the minimum limit. This has not been considered by the authors when recommending locations, but is left for the final user to consider.
When running the model with the new constraint SuffiForEmploy the solutions does not differ from the basic facility location model solutions until p=5. When the model no longer allows an annual working time of less than 1100 hours, solutions with facilities only serving themselves is no longer possible for p=6,7,8,9,10 since this does not give enough hours.
The SuffiForEmploy model works satisfactory until p has increased to p=6. This is also when the basic facility location model starts locating ESNQ as an isolated facility (see Table 3). When p=6 the
SuffiForEmploy model starts adding long routes to facilities with little hours. This results in the facility at ESNQ serving airports in southern Sweden to gain total working hours.
The phenomena to allocate airports further away from the facility to gain working time reoccurs for
p=7,8,9 and 10, although the solutions are less drastic as for p=6 (see appendix 4). ESNQ is not allocated as facility again until p=10. In the model’s solutions for p=7,8,9 ESOK is allocated as facility with the responsibility over airports that geographically are very close to other assigned facilities such as the ones in the area near Gothenburg or airports in the south of Sweden. Although the driving time from ESOK to the airports are not unreasonable, it is an impractical solution since both ESSP and ESGG are facilities located closer to the airports in question.
For p=10 the model allocate some facilities that does not get assigned with responsibility for themselves. When the model is assigned to find ten facilities ESNU is allocated as facility but maintenance responsibility is assigned to the ESNO facility.
5.3 Facility Location Model Including two Constraints
The model was run with the two constraints SuffiForEmploy and DrivingTime activated. Constraint SuffiForEmploy was set to be ≥1100 in all scenarios, the value for DrivingTime time was set to be ≤ 3,4,5,6,7 and 8. The model was run for each of these values with the different values on p=1,2,..,10. The objective function for each run, including which value set for parameters p and DrivingTime, is presented in Table 4. The value written in bold in Table 4 is the minimum value found for z for each DrivingTime set. For more details see appendix 5 to 9.
Table 4. Travelling time results from facility location model including two constraints
Driving Time s p t 3 4 5 6 7 8 1 X X X X X X 2 X X X 16258.08 12662.96 12662.96 3 X 11114.64 11114.64 10569.52 10388.6 10388.6 4 X 8942.34 8942.34 8475.32 8475.32 8475.32 5 X 7338.36 7338.36 6869.32 6869.32 6869.32 6 X 6157.54 6157.54 5970.3 5970.3 5970.3 7 X 5487.22 5487.22 5149.36 5149.36 5149.36 8 X 5023.42 5018.62 4671.88 4671.72 4671.88 9 X 5107.56 4978.5 4814.04 4814.04 4814.04 10 X 5302.16 5173.1 5154.72 5153.3 5153.3
Figure 3 visualizes the output for p=6 and DrivingTime ≤4 as an example for comparison with the other maps presented. The allocated facilities are shown as stars on the map, while the airports they are assigned to serve are showed as circles in an identical colour as their facility.
Figure 3. Visualization of result from one run of final model
The DrivingTime constraint was added with inspiration from the p-center model. When the p-center model finds the minimum maximum distance in the network the model forces the objective function not to be greater than this distance. The authors modified this and forced the maximum distance to not be greater than a decided number instead. The only way to prevent the model locating facilities with airports far away is to restrict how far away it can search for solutions. There are different ways for restricting distance between facility and location, but to base it on travelling time matches this purpose since it also allows for a stand-by time in the case of unplanned maintenance visits.
The final model with two constraints was run with values for maximum driving time set to be <= 3,4,5,6,7 and 8 (see Table 4). There is no solution when p=1 in any of the scenarios, since the driving time must be greater to reach all locations from one facility, the DrivingTime constraint must be set to 10 hours to give a p=1 solution. The model can not find an allowed solution for when the driving time needs to be 3 hours or less, which was expected since few airports can provide enough working hours that the constraint requires for an employment while at the same time being located within 3 hours driving time from a facility. The constraint DrivingTime is restricting the travelling time one-way. To run the model and allow maximum driving time to be up to 9 hours will reduce the motive to why the maximum driving time constraint was added since a maximum 8 hours allows for a reasonable working day for a technician. Even if one can assume that there exists a possibility to spend the night, it is preferred to find solutions where the technicians can drive to an airport and come back on the same day.
The solutions for DrivingTime set to max 4 hours and 5 hours are from p=3 to p=7 identical (see appendix 5 and 6), p=1 and p=2 does not have a solution since it is not possible to reach all locations within the allowed driving time. For p=8 the objective function for DrivingTime ≤4 is 5023,42 hours and for DrivingTime ≤5 is 5018,62 hours. The facilities are the same in both solutions but the
the DrivingTime constraint. This allows the model to find new combinations of groups, resulting in a slight improvement of less than 5 hours of the objective function. The solution for p=8 is the optimal found for DrivingTime ≤4, while p=9 is the optimal found for DrivingTime ≤5.
When DrivingTime ≤4 the objective function increases for p=9 and p=10. When DrivingTime ≤5 it increases for p=10. With DrivingTime ≤5 the model does not allocate facilities with assigned responsibility for themselves anymore for p=9 and p=10, this to gain working time via increasing unnecessary travelling time. This reoccurs for the other DrivingTime values set, with the exception for DrivingTime ≤4.
The solution for DrivingTime set to max 6 hours differs from solutions with other DrivingTimes set for p=2 and p=3. Unlike when DrivingTime is set to ≤4 and ≤5 it gives an allowed solution for p=2 and when p=3 it gives a solution with total travelling time decreased with approx. 500 hours. When DrivingTime is set to ≤7 and ≤8, the model finds cheaper solutions for p=2 and p=3 compared with when the driving time is maximum 6 hours. For p=4,5,6,7 and DrivingTime ≤6, the solutions are identical to the ones when DrivingTime is set to ≤7 and to ≤8.
The optimal solution for DrivingTime set to ≤6,7,8 is found when p=8, although there is a difference between the three. The solutions for p=8 when the constraint DrivingTime is set to maximum 6 and 8 hours are exactly the same, 4671,88 hours with identical assignment of airports to facilities. When DrivingTime ≤7 the result is slightly lower, 4671,72 hours and a difference in the assignment occurs. Both solutions are allowed for all three DrivingTimes, to why the model does not choose the cheaper solution earlier the authors have not found an explanation for. It can be speculated that the difference in assignment originates from the branch-and-bound algorithm in AMPL with the CPLEX solver, the difference in travelling time is though considered negligible.
When p=9 or greater, and DrivingTime set to ≤6,7,8, the travelling time starts to increase. This can be explained by that the SuffiForEmploy constraint increases the travelling time to fill the 1100 hours by creating impracticable assignments. For p=10 only the objective function was controlled so that it was not a better solution than the others already given, because of this no further analysis on the
6 RESULTS FROM ROUTING
This chapter presents the results of routing by heuristics and after the premises described in chapter 4.3. Figure 4 shows a visualization of the routing for one group of airports and their facility, derived from the results when p=6 and the constraints SuffiForEmploy and DrivingTime added. DrivingTime was set to ≤ 4 and SuffiForEmploy to ≥1100 in this scenario. The facility chosen is ESSP, visualized as a star in Figure 4, with its seven assigned airports to serve visualized as circles in an identical colour.
Figure 4. Visualization of routes for facility ESSP
Details for the set routes are presented in Table 5. The travelling time when following this plan for routes sums up to a total of 1324,08 hours for one year. The sum of travelling time when driving back and forth for all visits from facility ESSP is 1607,1 hours (see appendix 5).
Table 5. Results over routing for facility ESSP Route Colour in Figure 4 Driving time for route Planned frequency of route Annual driving time ESSP-ESOW-ESSB-ESSP Green 4,96 96 476,16 ESSP-ESSL-ESIA-ESSP Red 4,62 1 4,62 ESSP-ESSL-ESKN-ESSP Blue 2,51 1 2,51 ESSP-ESMQ-ESSP Black 5,7 140 798 ESSP-ESOW-ESSP Purple 4,24 8 33,92 ESSP-ESCM-ESSP Yellow 5 1 5 ESSP-ESKN-ESSB-ESSP Pink 3,87 1 3,87 ESSP - 0 115 0
Sum total driving
time: 1324,08
A feasible solution to the routing of one facility might have looked different if it would have been done by programming instead of by reasoning and choosing the best routes within limits for given constraints. The working method used by the authors allows for a weighing between two solutions based on possible maintenance duration. If the airport has a high maintenance visit demand it can possibly more often need longer time for a single maintenance than the rated average of 2,5 hours, in which case a route with a bigger buffer for maintenance is preferred. With this in mind a solution can be chosen that might only be near the most optimal when seen in the numbers, but it might be more practicable since it can allow for a buffer if it is suspected that the maintenance visits will sometimes take longer than the average.
The routes found for the ESSP facility shows that the travelling time can be reduced (see Table 5). The model, except for the basic facility location version, calculates that each maintenance visit will be a back-and-forth trip. Routing can allow the technician to travel in a circuit, which is cheaper than going back-and-forth for every maintenance visit. The travelling times for the ESSP facility group shows that the routings annual travelling time is 82% of the model’s.
