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Discrete event system modeling of demand responsive transportation systems operating in real time
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ABSTRACT: Demand responsive transportation is a variable route service of passengers or freight from specific origin(s) to destination(s) in response to the request of users. Operational planning of DRT system encompasses the methods to provide efficient service to the passengers and to the system operators. These methods cover the assignments of vehicles to transportation requests and vehicle routings under various constraints such as environmental conditions, traffic and service limitations. Advances in the information and communication technologies, such as the Internet, mobile communication devices, GIS, GPS, Intelligent Transportation Systems have led to a significantly complex and highly dynamical decision making environment. Recent approaches to DRT operational planning are based on "closed information loop" to achieve a higher level of automation, increased flexibility and efficiency.Intelligent and effective use of the available information in such a complex decision making environment requires the application of formal modeling and control approaches, which are robust, modular and computationally efficient. In this study, DRT systems are modeled as Discrete Event Systems using Finite Automata formalism and DRT real time control is addressed using Supervisory Control Theory. Two application scenarios are considered; the first is based on aircharter service and illustrates uncontrolled system model and operational specification synthesis. The automatic synthesis of centralized and modular supervisors is demonstrated. The second scenario is a mission critical application based on emergency evacuation problem. Decentralized supervisory control architecture suitable for accommodating the realtime contingencies is presented.Conditions for parallel computation of local supervisors are specified and the computational advantages of alternative supervisory control architectures are discussed. Discrete event system modeling and supervisory control theory are well established and powerful mathematical tools. In this dissertation, they are shown to be suitable for expressing the modeling and control requirements of complex and dynamic applications in DRT. The modeling and control approaches described herein, coupled with the mature body of research literature in Discrete Event Systems and Supervisory Control Theory, facilitate logical analysis of these complex systems and provide the necessary framework for development of intelligent decision making tools for real time operational planning and control in a broad range of DRT applications.
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Advisor: Ali Yalcin, Ph.D.
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Intelligent transportation systems
Supervisory control
Air charter service
Aero medical evacuation
Local supervisors
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D iscrete E vent S ystem M odeling O f D emand R esponsive T ransportation S ystems O perating I n R eal T ime By Daniel Y. Yankov A D issertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philo sophy Department of Industrial and Management Systems Engineering College of Engineering University of South Florida Date of Approval: March 1 8 2008 Major Professor: Ali Yalcin, Ph.D. Natasha Jonoska, Ph.D. Alia ks e i Savachkin, Ph.D. Kimon Va lavanis, Ph.D. Susana Lai Yuen, Ph.D. Key words: intelligent transportation systems supervisory control air charter service, aero medical evacuation, local supervisors, concurrent subsystems Copyright 2008 Daniel Yankov
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A cknowledgements I would like to thank my advisor Dr. Ali Yalcin for his support and mentoring throughout the course of this research. I would like to thank my committee members Dr. Natasha Jonoska, Dr. Kimon Valavanis, Dr. Alex Savachkin and Dr. Susana Lai Yuen for their kind reviews and support, and for the wisdom they possess which they allowed me to benefit. Last but not the least, I would like to thank my family and friends, without them the journey would neither start nor complete.
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i T able of C ontents List of Tables v List of Figures vi Abstract x Chapter One Introduction 1 Chapter Two Related Literature Review 6 2.1. DRT r elated OR p roblems 6 2.2. Heuristic a pproaches in DARP 11 2.2.1. Heuristics for SDARP 12 2.2.1.1 Insertion heuristics for SDARP 12 2.2.1.2 Parallel insertion heuristics for SDARP 13 2.2.1.3 Met aheuristic approaches for SDARP 1 3 2.2.1.4 Two or three phase approaches for SDARP 1 5 2.2.2. Heuristics for DDARP 16 2.2.2.1 Heuristics performing global search 17 2.2.2.1.1 Dynamic constructive techniques 1 8 2.2.2.1.2 Dynamic iterative techniques 20 2.2.2.2 Heuristics performing local search 2 1 2.2.2.2. 1 Parallel metaheuristics 2 1 2.2.2.2.2 Clustering and locating 21 2.3. Simulation a pproaches in DRT 23 2.4 Intelligent t ransportation s ystems (ITS) a pproaches in DRT 30
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ii 2.4.1. ITS approaches in centralized DRT systems 31 2.4.2. ITS approaches in decentralized DRT systems 3 5 Chapter Three Research Motivation, Problem Domain, Research Goal and Objectives 38 3.1. Research m otivation 38 3.2 Research p roblem d omai 39 3.3. Research g oal and o bjectives 41 Chapter Four Discrete Event Systems and Supervisory Control 47 4.1. Discrete e vent s ystems 4 8 4.1.1 FA modeling of DES 48 4.1.2 Language and langua ge characteristics 49 4.1.3 Operations on languages 50 4.1.4 Unary operations on automata 50 4.1.5 Composition operations on automata 51 4.1.6 Analysis of DES 52 4.2 Supervisory c ontrol 55 4.2.1 Controlled DES 55 4.2.2 Controllability theorem and realization of supervisors 57 4.2.3 Modular supervisory control 60 4.2.4 Decentralized supervisory control 6 2 4.2.4.1 Conjunctive decentralized architecture 63 4.2.4.2 Disjunctive decentralized architecture 64 4.2.4.3 General decentralized architecture 65 4.2.5 Nonblocking decentralized supervisory control 66 4.2.5.1 Nonblocking conjunctive decentralized supervisor 6 6 4.2.5.2 Nonblocking disjunctive decentralized supervisor 67 4.2.5.3 Nonblocking general decentralized supervisor 68
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iii Chapter Five Taxonomy of DRT Systems, DRT Modeling with FA and Illust rative Example 70 5 .1 Taxonomy of DRT s ystems 70 5.1.1 Origin and d estination c onsiderations 70 5.1.2 Vehicle f leet c haracteristics 7 1 5.1.3 Transportation d emand c haracteristics 7 2 5.2 Modeling of DRT s ystems with FA 7 2 5.3 Illustrative e xample of a s mall a ir charter s ervice o peration 7 7 5.3.1 Problem description of small air 7 8 5.2.2 DES modeling of a small air charter DRT system 7 9 5.3.2.1 Computation of centralized supervisor 80 5.3.2.2 Computation of modular supervisor 8 3 5.2.2.3 Computational complexity of supervisor synthesis 9 2 Chapter Six Decentralized Supervisory Control of Concurrent DES 9 4 6.1 Decentralized control of concurrent DESs 9 4 6.2 Decentralized supervisor of separate groups of vehicles passengers 9 8 6.3 Illustrative Example of ARE Servic e in D DARP MADO Environment 100 6.3.1 Problem description of ARE Service in D DARP MADO environment 10 2 6.3.2 DES modeling of a small emergency DRT system in D DARP MADO environment 10 4 6.3.2.1 C omputation of the supervisor of one vehicle one passenger (LS11) 10 8 6.3.2.2 Computation of the supervisor of one vehicle two passengers (LS 12 ) 11 3 6.3.2.3 Computation of the local supervisor of one vehicle one passenger (LS 32 ) 11 8
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iv 6.3.2.4 Computation of the local supervisor of one vehicle two passengers (LS 113 ) 12 2 6.3.2.5 Computation of the local supervisor of one vehicle two passengers in case of a closed MTF (LS 448 ) 12 8 6.3.2.6 Generating the global SC of the emergency DRT 13 5 6.3.2.7 Computational complexity of decentralized supervisor 13 6 Chapter Seven Contribution of the Study and Future Research 13 7 7.1 Summary of the completed work and contribution of the study 13 7 7 2 Future r esearch 13 9 7. 2 .1 Application of timed DES (TDES) 13 9 7. 2 .2 Application of hybrid DES (HDES) 14 1 References 14 3 Bibliography 14 7 About the Author 14 9
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v List of Tables Table 1 Summary of the discussed heuristics for SDAP 1 6 Table 2 Summary of the discussed heuristics for DDAP 2 3 Tab le 3 The set of all the events of the small air charter 7 9 Table 4 The set of all the events of a small emergency DRT system 10 5 Table 5 Considered requests, assigned vehicles and LS s 13 5
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vi List of Figures Figure 3.1 A map of MTFs and patient pick up locations in Tampa bay area. 40 Figure 3.2 Framework for real time DRT control. 42 Figure 4.1 The feedback loop of supervisory control. 56 Figure 4.2 Modular supervisory control with two supervisors. 60 Figure 4.3 Conjunctive supervisory control with two supervisors. 64 Figure 4.4 Disjunctive supervisory control with two supervisors. 65 Figure 4.5 General decentralized supervisory control with two supervisors 65 Figure 5.1 Simple air taxi DRT system operating at four airports. 7 3 Figure 5.2 Automaton pjet j the possible locations and flights of jet j 7 4 Figure 5.3 Automaton trip j the maximum allowed flight within a trip. 7 5 Figure 5.4 Automaton veh u st j vehicle j in unpredicted stoppages. 7 5 Figure 5.5 Automaton cap j jet j may pickup at most two passengers. 7 6 Figure 5.6 Automaton prior i gives priority of reassigned passengers. 7 6 Figure 5.7 Automaton pjet 1 80 Figure 5.8 Automaton p ass 1 8 1 Figure 5.9 Automaton trip 1 8 1 Figure 5.10 Automaton paspd 1 8 2 Figure 5.11 Superviso r CS 1 8 3
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vii Fig ure 5.12 Automata 8 4 Figure 5.13 Parallel synchronization of automata pjet 1 and pjet 2 8 5 Figure 5.14 Automata 8 5 Figure 5.15 Automata 8 6 Figure 5.16 Automata 8 7 Figure 5.17 Supervisor MS 1 8 8 Figure 5.18 Specm 1 synchronization of jet 1 and passenger 1 8 9 Figure 5.19 Specm 2 synchronization of jet 2 and passenger 2 90 Figure 5.20 Language SP 2 9 1 Figure 6 1 DES split in subsystems of vehicles and passengers. 9 9 Figure 6.2 Structure of the global system and local control. 10 1 Figu re 6.3 Region R with 5 origins and 3 destinations. 10 2 Figure 6.4 Automaton pasn 1 10 8 Figure 6.5 Automaton pveh 1 10 8 Figure 6.6 Automaton fd 1 10 9 Figure 6.7 Automaton trips 1 11 1 Figure 6.8 Automaton vehdil 11 11 1 Figure 6.9 Automaton paspd 11 11 2 Figure 6.10 Automaton of LS 11 11 3 Figure 6.11 Automaton pasn 12 11 4 Figure 6.12 Automaton pasn 2 11 4 Figure 6.13 Automaton pveh 12 11 4
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viii Figure 6.14 Automaton trips 12 11 5 Figure 6.15 Automata 11 6 Figure 6.16 Automata 11 7 Figure 6.17 Automaton pasn 23 11 8 Figure 6.18 Automaton pveh 3 11 9 Figure 6.19 Automaton fd 3 1 20 Figure 6.20 Automaton trips 3 1 20 Figure 6.21 Automaton vehdil 23 12 1 Figure 6.22 Automaton paspd 23 12 1 Figure 6.23 Supervisor LS 32 12 2 Figure 6.24 Automata 12 3 Figure 6.25 Automaton pveh 13 12 3 Figure 6.26 Automaton fd 13 12 4 Figure 6.27 Automaton trips 13 12 4 Figure 6.28 Automata 12 5 Figure 6.29 Automata 12 6 Figure 6.30 Automaton LS 113 12 7 Figure 6.31 Automata 12 8 Figure 6.32 Automaton pveh 4 12 9 Figure 6.33 Automaton fd 4 1 30 Figure 6.34 Automaton fstat 3 1 30
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ix Figure 6.35 Automaton trips 48 13 1 Figure 6.36 Automata 13 2 Figure 6.37 Automata 13 3 Figure 6.38 Automaton fstat 3 13 4 Fig ure 6.3 9 Automaton of LS 448 13 5
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x Discrete Event System Modeling of Demand Responsive Transportation S ystems Operating in Real Time Daniel Y. Yankov ABSTRACT D emand responsive transportation is a variable route service of passengers or freight from specific origin(s) to destination(s) in response to the request of users. Operational planning of DRT system encompasses the methods to provide efficient service to the passengers and to the system operators. These methods cover the assignments of vehicles to transportation requests and vehicle routings under various constraints such as environmental c onditions, traffic and service limitations. Advances in the information and communication technolog ies, such as the Internet, mobile communication devices, GIS, GPS, Intelligent Transportation Systems have led to a significantly complex and highly dynamical decision making environment. nformation Intelligent and effective use of the available information in such a complex decision making environment requires the application of formal modeling and control appro aches which are robust, modular and computationally efficient
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xi In this study DRT systems are modeled as Discrete Event System s using Finite Automata formalism and DRT real time control is addressed using S upervisory C ontrol T heory. Two application scenarios are considered; the first is based on air charter service and illustrates uncontrolled system model and operational specification synthesis. The automatic synthesis of centralized and modular supervisors is demonstr ated. The second scenario is a mission critical application based on emergency evacuation problem. Decentralized supervisory control architecture suitable for accommodating the real time contingencies is pr esented. C onditions for parallel computation of lo cal supervisors are specified and the computational advantages of alternative supervisory control architectures are discussed. Discrete event system modeling and supervisory control theory are well established and powerful mathematical tools. In this disse rtation, they are shown to be suitable for expressing the modeling and control requirements of complex and dynamic applications in DRT. The modeling and control approaches described herein, coupled with the mature body of research literature in D iscrete E v ent S ystems and S upervisory C ontrol T heory, facilitate logical analysis of th ese complex systems and provide the necessary framework for development of intelligent decision making tools for real time operational planning and control in a broad range of DRT applications.
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1 C hapter One Introduction DRT passenger services are public transportation services characterized by flexible routing and scheduling of relatively small capacity vehicles (occupancy of up to 20 persons) to provide shared occupancy and personalized transportation on demand. The role of DRT services has changed dramatically in recent years For example, rural transit which is a wide spread DRT service, was limited t o a type of social service transportation for a specific set of clients who primarily traveled in groups to common meal sites, work centers for the disabled, or clinics in larger communities. Service schedules and passenger assignments were develop ed and a ugmented manually in a preset calendar. Due to the lack of advanced communication and information technologies, the early DRT systems tended to operate as advanced reservation systems with some service providers requiring users to mak e a reservation at least 24 hours in advance of their travel. Since it took hours to build the schedule, any last minute changes could wreak havoc with the operational plan ning of the dispatch office. Nevertheless, given these parameters, a manual schedul ing s ystem worked for small DRT systems Lave at al. (1996) report that the advanced reservation DRT operation has been associated with significantly low system productivity. Despite the problems, such DRT systems allocated capacity easily and are less co mplex to implement than real time
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2 reservation systems. However, for a number of the passenger groups such as job commuters and clinic patients, the 24 hour preplanned schedule is not viable. They need a system that can take their request when they are rea dy. Workers and commuters especially need a system that is reliable and robust. Although DRT service is user friendly because of its door to door capability and semiprivate, comfortable vehicles, its adoption has not been widespread due to the relatively high cost of operation. DRT is a labor intensive mode of transportation with costs comparable to the taxicab, due to inherently low passenger productivity (passengers per vehicle hour) (Lave at al. 1996). As a result, DRT service is most commonly offered by social service agencies to transport their clients, by transit districts experiencing high enough passenger transportation demand, or by counties and cities for persons with special needs or qualifying conditions. T he 1990 Americ ans with Disabilities A ct (ADA) requires every U.S. transit agency operating in fixed route transit to provide complementary DRT for persons with disabilities within their service areas without advanced reservation Thus, the ADA mandate is causing an expansion of the number of DRT services and growth in the size of the existing services. This growth motivate d the search for more cost effective means of operating DRT systems One promising means of improving the cost effective perfo rmance of demand responsive transit is the use of latest developments of information and communication technologies. Contemporary DRT systems accept telephone or internet requests for both immediate and advance reservation service; develop a continually changing set of vehicle
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3 schedules tha t accommodate these trip requests, and route vehicles to the appropriate passenger origin and destination location s in accordance with the schedule. Because both the trip requests and the vehicle scheduling and routing decisions occur in real time, DRT con trol problem bec o me s complex even in systems where small number of vehicles and trip requests a re involved. One of the most critical, complex and dynamic application domain of DRT service is the military aeromedical regulation and evacuation (ARE) of patie nts to medical treatment facilities (MTFs). Doctrinally during both wartime and piece, patients requiring extended treatment must be evacuated by air to a suitable MTF. The process of routing and scheduling the required aeromedical evacuation flights (miss ions) and assigning patients to suitable missions is a critical part of the evacuation planning and execution, Sadeh and Kott (1996). The major ch alleng e in the design of any DRT operation is the choice between the level of efficiency and level of quality of the service. Service quality ranges from the most costly exclusive ride taxi service, in which only one person rides at a time, to trips in which vehicles are shared, and each passeng er may have to ride longer than is needed for his /her trip while the v ehicle drops and picks up other riders. Assigning many passengers to a vehicle results in increased efficiency due to minimizing the total distance traveled by the vehicle and smaller vehicle fleet required. However, high passenger loads lower the quality of the service by i ncrea sing the average ride time and the variability of promised pickup and arrival times. These trade offs are usually determined by specifying
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4 minimum service levels in terms of the longest ride times allowable and the maximum lateness for a promised pickup or arrival. With every service request the system operator obtains the parameters of the desired trip from the pa ssenger pickup point, drop off point, desired pickup or delivery time, number of passengers, and any special requireme nts ( e.g. wheelchair accessibility) and then communicate s to the pa ssenger whether the system is able to accommodate the trip request with these specific parameters and, if so, when a vehicle will arrive. The process of scheduling individual service reque sts while the customer is on the phone or using the Internet is called real time or online scheduling This term refers to a scheduling system in which some means of accepting or denying a trip request is based on available system capacity and, if a request is accepted, an estimated time of arrival of the vehicle is given to the requester, usually within a specified time window. With the online service the request s are accepted during the travel of the vehicles and are to be inserted into their curren t schedules. as well. Therefore with the online communication DRT service experience s real time operational dynamic s that necessitates higher level of automation, flexibility and in tegration of the system development. To achieve such a development, more formal approaches of system design must be applied. We represent DRT systems as discrete event systems (DES s ) where system models capture both the low level dynamics (such as infrast ructure conditions, current status of vehicles) and high level dynamics (such as service demand requests) of system
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5 evolution. Supervisory Control Theory based on Finite Automata formalism is applied to provide real time control of DRT service as superviso ry control of DES The remainder of this dissertation is o rganized as follows: Chapter Two presents a literature review of the operational planning methodologies f o r DRT service Chapter Three introduces the research problem domain motivation, research go a l and objectives. Chapter Four presents an introduction to Discrete Event Systems, Finite Automata formalism and Supervisory Control Theory T he possible architectures of decentralized supervisors an d the conditions for their nonblocking behavior are discussed In Chapter Five first a taxonomy of DRT systems is introduc ed and a framework of DRT operation modeling as DES is presented. A simple air charter system is used to illustrate the system modeling and the synthesis of centralized and modular super visors Chapter Six discusses the decentralized control of concurrent DESs The computation of the local supervisors and the synthesis of the global one are illustrated with the control of a small aeromedical eva cuation system In Chapter S even the complet ed work is summarized and the future research issues are discussed.
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6 Chapter Two Related L iterature R eview The presented literature survey first reviews the fundaments of O perations R esearch p roblems related to DRT operation covers the developed heuristic approaches for solving these problems, and reviews the recent methods in DRT operational planning and real time control We limit our review to deterministic DRT problems and solution approaches and do not cover stochastic methods. The emphasis of the review is on highlighting the advantages of the decentralized methods over the centralized ones in the operational planning of dynamical and complex DRT systems. 2 .1. DRT r elated OR p roblems The OR literature contains numerous studies addressing DRT related problems. In m ost of the works the Vehicle Routing Problem with Pickup and Delivery (VRPPD) represents the mathematical fundaments of DRT and henceforth is of great interest to our study Since the most practical applications of the VRPPD include restrictions on the time at which each location may be visited by a vehicle, it is convenient to pre sent a slightly more general variant of the problem, called the VRPPD with time windows (VRPPDTW). Cordeau at al. (2004) discuss that VRPPDTW is NP hard, because it
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7 generalizes the Traveling Salesman Problem (TSP), which is known to be NP hard. In the pres ence of time windows, even finding a feasible solution to the problem is NP hard since the feasibility problem for the TSP with time windows is itself NP complete. The Dial a Ride Problem (DARP) is a particular case of the VRPPD arising in contexts where p assengers are transported, either in groups or individ ually, between specified origin and destination location s. The most common DARP application arises in door to door transportation services for elderly or handicapped people. In their recent survey Corde au and Laporte (2007) review the developed OR models and algorithms on the DARP. The goal of the DARP solutions is to plan a set of minimum cost vehicle routes capable of accommodating as many service requests as possible, under a set of constraints. The m ain emphasis is on the human satisfaction, and the reduction of passenger inconvenience should be balanced against minimizing the system operating costs. Dial a ride services may operate in static or dynamic mode. In the static case, all transportation re quests are known a priori while in the dynamic case requests are accepted throughout the entire period of service (e.g. a shift) and vehi cle routes are adjusted in real time to meet demand. In practice pure dynamic DARPs rarely exist since a subset of req uests is often known when planning starts Cordeau and Laporte (2007) present two formulations of the DARP a three index integer formulation in case of heterogeneous vehicle fleet, and a two index formulation for the case of homogeneous fleet. The objectives in the static algorithms of multi veh icle DARP vary in minimizing the fleet size, total route duration, total service cost, total distance traveled by vehicles
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8 and by passengers, total service time, time window violations and/or minimizing linear combinations of some of these factors. Cordea u and Laporte (2007) discuss th at the distinction between static and dynamic DARPs is often blurred in practice since the service requests are often cancelled and, as a result, transporters may allow the introduction of new requests in a solution designed for a static problem. The difficulty then is to design seed vehicle routes for these requests with sufficient slack time and capacity to accommodate future dynamic demand. The objectives in the dynamic algorithms of multi vehicle DARP vary in maximizing th e number of served passengers, minimizing the route lengths, ride times and time violations. A special case of the DARP is the Dial a Flight Problem ( DAFP) introduce d by Cordeau at al. (2004 ). seat on Pas sengers select the destinations, time of arrival and the time window for travel. The static DAFP (SDAFP) is concerned with the scheduling of the single passenger requests for air transportation during a given time period (usually a single day ) Each reques t specifies an origin airport, the earliest acceptable departure time, a destination airport and the latest acceptable arrival time at the destination A homogeneous fleet of airplanes operable by a single pilot is available to provide the requested air t ransportation. Each airplane and pilot ha s a home base where they have to return at the end of each planning period In the dynamic DAFP (DDAFP) passengers book seats online as they do with airline service, except there are no fixed schedules. T he set of requests for air transportation arrives during the time of operation and with each request the service provider must immediately decide whether it is feasible to accept the request given the
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9 available resources and the commitments already made. I f it is fe asible to accept the request, the provider will want to decide whether it is desirable to accept i t, i.e. whether it will increase the profit of operation The latter decision is especially complex as it depends on the requests that will arrive in the futu re. A more complex variant of DDAFP a flight schedule and have to be incorporated into the current schedule. Cordeau at al. (2004) present an IP formulation of the SD AFP. It is a time discretized multicommodity network flow model which becomes large quickly and even solving medium size instances ( e.g., involving 15 to 30 airports and 5 to 10 airplanes ) require specialized solution approaches. In DDAFP the operator has to decide in real time given a set of already accepted requests, whether an incoming request can be served or not. Cordeau at al. (2004) suggest that fast heuristics will have to be part of that decision technology. In case the heuristics fail to accept a request quickly, a customer may be given the option of receiving final notification of acceptance or rejection in short time (e.g. 30 minutes ) to allow time for optimization based techniques to try and accommodate the request. Sadeh and Kott (1996) study the application of dynamic transportation planning technologies to the class of complex transportation planning problems, called Dynamic Dial A Ride Problem with Multiple Acceptable Destinations and/or Origins (D DARPMADO). Their work was motivated by the military Aeromedical Regulation and Evacuation (ARE) of patients to Medical Treatment Facilities (MTFs) The problem domain is highly dynamic, complex and critical. There has been very limited experience
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10 with this approach to handling patients other than in peace time. The first Persian Gulf war was the first significant armed conflict in which this concept has been put to a serious test. The results were far from satisfactory about 60% of the patients ended up at the wrong destinations and half in the w rong country Sadeh and Kott (1996) The integrated medical regulation/evacuation problem requires the dynamic identification of appropriate MTFs for new patients and the planning/scheduling of aeromedical evacuation operations to transport these patients from their current locations to the selected MTFs. This is a large scale, highly dynamic planning and scheduling problem that can involve hundreds or even thousands of simultaneous patient movement requests. Despite the similarities with DDAFP, D DARPMADO is more complex and hard to control. Each patient has one or several medical requirements that constrain the type of MTF to which he or she can be evacuated and a ready time prior to which evacuation cannot start. Additional constraints can include a maxi mum altitude above which the evacuation aircraft cannot take the patient, a maximum number of hours that a patient can spend in a flight before requiring an overnight rest, a maximum number of stops the patient can tolerate during evacuation, etc ., (Sadeh and Kott 1996) The most challenging aspect in planning and scheduling medical evacuation operations has to do with the dynamics of a domain in which requirements and constraints continuously change over time. The authors clearly point out that the dynamic transportation problem domain is in many ways more complex than VRP/DARPs traditionally discussed in the literature. The D DARP MADO model expands DARP along two main directions:
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11 T here may be multiple acceptable destination and / or origin locations for a g iven demand; B oth the demands and the resources can change dynamically, while the initial schedule is being executed. Dial (1995) introduced the concept of the Autonomous Dial A Ride Transit (ADART) service based on fully automated command and control, or der entry and routing and scheduling systems implemented on computers on board vehicles. The approach outwits possible large size of DRT system with applying distribut ed communication between the passengers and the vehicles and negligible central managemen t intervention. T he system is fully automated the only human intervention in the process is the customer requestin g service. Furthermore, the routing and scheduling are not done at the central dispatching centre, but are distributed among vehicles through an auction mechanism. In this section the OR problems related to DRT were introduced. The next two sections review the methods for solutions of DARP and DRT service optimization. 2 .2. Heuristic a pproaches in DARP Abundant research work has been done for both static and dynamic modes of DARP. Heuristics is the most widely used approach to provide fast and quality solutions for both subproblems of DARP namely scheduling the passengers and routing of the vehicles. Sc heduling subproblem concerns the assignments of passengers to the vehicles, and routing subproblem consists of search for the shortest sequence of visit s the origin and destination locations of all the passengers scheduled to each vehicle.
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12 2.2.1 Heuristi cs for SDARP Based on the applied techniques the following four types of heuristics approaches for SDARP can be distinguished. For each of them one or two representative works are cited. 2.2.1.1 Insertion heuristics for SDARP One of the first insertion heuristics for the multiple vehicle version of the S DARP i s presented by Jaw et al. (1986). In the problem formulation, customers booking in advance can specify the origin and destination locations and either a desired pick up time or desired delivery time. The actual pick up or delivery time of a customer is allowed to deviate from the desired one, but constraints of a fixed maximum wait time window and a maximum ride time that a passenger may spend in the vehicle are imposed The objective function of the model is the weighted sum of disutility to the customers and to the operator. The heuristic selects users in order of earliest feasible pickup time and gradually inserts them into vehicle routes so as to yield the least possible increase of the objective function. However, Wong and Bell (2006) note that the sequence or order of the requests to be inserted into the schedule s has a significant impact on the performance of insertion heuristics A commonly used technique to reduce the computation time in the insertion heuristics for the S DARP is the cluster ing of the users t o be served by the same vehicle prior to the routing. Such clustering leads to two phase approaches. In the first phase, groups (cluster s) of customers to be served within the same area at approximately the
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13 same time are formed, and the algorithms search for optimal combination of the clusters to form feasible vehicle routes. In the second phase, each vehicle route is reoptimized with a si ngle vehicle algorithm. 2.2.1.2 Parallel insertion heuristics for SDARP Another way to speed up the computation time is through the use of parallel computing. Toth and Vigo (1996) developed a parallel insertion procedure on the assigning of the request s to routes Then the method performs intra route and inter route e xchanges of passengers in search for better solution s Diana and Dessouky (2004) adopted the operating scenario of Jaw et al. (1986) and presented a new parallel insertion heuristic for S DARP with time windows. They developed a route initialization procedure by inserting an initial request to each of the vehicles, taking the spatial and temporal effects into account. A parallel regret insertion heuristic is then used for the rest of the re quests not inserted into the initialization. Instead of ranking the requests by certain criteria (e.g., earliest pick up time or latest delivery time) as in classic insertion heuristics, the regret insertion builds up an incremental cost matrix for each of the unassigned requests when assigned to each of the existing vehicle routes. A regret cost, which is a measure of the potential difficulty if a request is not immediately assigned, is calculated for each request, and the algorithm seeks the request with the largest regret cost, and inserts it into the existing schedules. The regret insertion algorithm requires at each step a feasibility check for the insertion of each un schedul ed request in all the rout es. The whole procedure is repeated until all request s are inserted. The algorithm is successfully implemented for a r eal case of up to 1000 service requests.
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14 2.2.1.3 Met aheuristic approach es for SDARP Because of their ability to find close to optimal solutions, m etaheuristic algorithms have been sought to solve the SDARP. Tabu search stands out as a very powerful tool for the DARP since it is highly flexible and efficient. Flexibility stems from the capacity of handling a large number of variants within the same search framework. Efficiency is associated with solution quality. It is now clear that t abu search is capable of consistently generating high quality solutions on a large variety of routing problems. T he negative side of t abu search algorithms is th at their running time can be rather high. Cordeau and Laporte (2003 ) formulated and solved the static case applying sequential t abu search Instead of measuring disutility by the deviation between the actual pick up/drop off times and the user desired ones, their model allows users to specify a time windo w of a fixed width on their inbound or outbound trips, with an upper limit on the travel time for any user. In general, t he insertion heuristics are computationally fast, but may not p rovide as good solution as met aheu ristics. On the other hand, met aheuri stics may not be computationally feasible when a large number of requests need to be scheduled in a dynamic environment, and they usually require extensive computational tests to set up a number of parameters that are highly case sensitive. Thus, in many o f the approaches both methods are combined the insertion part provides fast and rough solution, whi ch is being improved with a met aheuristic local search. Such a combination leads to two or three phase heuristic approaches.
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15 2.2.1.4 Two or three phase approaches for SDARP Toth and Vigo (1997) are among the first who improve d their solution method obtained after parallel insertion phase through the execution of a local search based on tabu thresholding optimization procedure In their recent study Wong and Bell (2006) modified the parallel insertion heuristic in to a three phase method In the first phase t rip characteristics are calculated and trips are ranked with a particular order for insertion. Next a parallel insertion is performed to iteratively insert the requests into the existing routes. An optional local search procedure based on tabu search is used to further optimize the objective function. Cordeau and Laporte (2007) conclude that excellent heuristics have already been developed for the SDARP, which allow solving instances with several hundreds of users within reasonable times and it should be possible to apply decomposition techniques for larger instances involving, t wo or three thousand users. Therefore, it is expected that more emphasi s be put on the DDARP This involves the construction of an initial solution for a limited set of requests known in advance and the design of features capable of determining whether a new request should be served or not and if so, how existing routes shoul d be modified to accommodate it. In addition it should be possible to update a partially built solution to deal with cancellations and other unforeseen events such as traffic delays and vehicle breakdowns. A brief summary of the reviewed heuristic algorithms for SDARP is presented in Table 1.
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16 Table 1. Summary of the discussed heuristics for SDAP. Reference Objective Time Windows Constraints Algorithm Jaw at al. (1986) Minimize nonlinear combination of to tal disutility function On pick up or on delivery Vehicle capacity; Maximum ride time Insertions Toth and Vigo (1996) Minimize total service cost On pick up and on delivery Vehicle capacity; Maximum ride time Parallel insertion and route exchange Diana and Desouky (2004) Minimize weighted sum of distance, excess ride time, vehicle idle time Lower bound on pick up time, upper bound on delivery time Vehicle capacity; Maximum ride time; Maximum waiting time Parallel regret insertion Cordeau and Laporte (2003) Minimize total route length On pick up or on delivery Vehicle capacity; Maximum route duration; Tabu search Toth and Vigo (1996) Minimize total service cost On pick up and on delivery Vehicle capacity; Maximum ride time Parallel insertion with tabu threshold search Wong and Bell (2006) Minimize weighted sum total operation time, passenger delay, penalty for unsatisfied demand On pick up or on delivery Heterogeneous fleet capacity max wait time; max ride time Three phase: ranking of trips; parallel insertion; local optimization. 2 .2.2 Heuristics for DDARP In the D DARP, operational constraints are the same as in the SDARP and the primary goal is to satisfy as many requests as possible with the available fleet of vehicles. As it was discus sed in S ection I I .1 in some DRT systems if enough time is available, the
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17 operato rs may a pply static approaches in DDARP optimization. Requests are dealt with one at a time in a first come, first served fashion. Whenever a request can be served without vio lating any of the constraints, it is accepted and becomes a part of the problem. A s the planning horizon goes on, the degree of flexibility decreases and the last requests to be releas ed are likely to be rejected. Transportation systems that provide dynami c dial a ride service are more flexible and can react to unpredicted events, but usually have tight real time constraints on the reoptimization algorithm. Moreover they require a monitoring system able to track the position of vehicles, their current load, and the state of the transportation network with a certain frequency. Dynamic dial a ride systems are more competitive than traditional transportation systems, but they need very good scheduling policy and route optimization. Based on the applied search techniques two general types of heuristics approaches for DDARP can be distinguished executing global and local search. For each of them we describe the most common methods and cite one or two representative works. 2 .2.2 .1 Heuristics performing global search In this approach the heuristic algorithms perform search for near optimal scheduling and routing over the whole domain. The two main types covered are constructive and iterative heuristics and dynamic insertion heuristi cs.
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18 2 .2.2 .1.1 Dynamic constructive techniques In dynamic constructive methods the process begins with an incomplete or empty solution and constructs the missing elements of the solution. Typical examples are rebuilding new solutions from scratch, inser tion techniques partial revision the matchup scheduling approach, the conflict propagation approach and truth/reason maintenance approach. Sadeh and Kott (1996) review two general dynamic replanning/rescheduling methods applicable for VRPTW and DARP. The y discuss the possibilities for dynamic re routing and rescheduling using constructive and iterative repair techniques. The authors envisioned two main concerns applying constructive approaches i n large scale domains with highly dynamic demand such as the ARE domain First, the computational requirements of such an approach could be pr ohibitive b y the time a new solution has been constructed, additional contingencies may have occurred, rendering the new solution obsolete. Second, in situations where it is possible to build a brand new solution each time a contingency occurs, this approach may still be undesirable because it introduces too many disruptions. The authors suggest that it is preferable to restrict solution revisions to small parts of the domain because of two reasons to avoid difficulties in communicating new solutions in real time and adapting the system to new solutions Madsen et al. (1995) present a dynamic heuristic s algorithm for passenger DARP with multiple capacities and multiple obj ectives as well a s updating capability The model is based on the procedure introduced by Jaw et al. (1986) R outes are pre planned for the requests known at the beginning of the day, and the new requests can be
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19 dynamically inserted throughout the day. T ra vel time updates and vehicle breakdowns can be considered. T he developed insertion algorithm can be efficient enough to be implemented in a dynamic environment for online scheduling. The model was tested with 300 customers and 24 vehicle instance over a day operation, and the authors report that good quality solutions were generated in short time. One of the challenges when optimizing dynamic transportation is to make good short term decisions without adverse long term effect. Mitrovic Minic et al. (2004) considered the dynamic problem with a double horizon based heuristic considering a short term and long term horizon. The short term goal is to find the shortest route length, similar to the objective function of the static optimization problem. The routing decisions are taken with a constructive heuristic searching for the cheapest insertion procedure. T he long term goal is to minimize the linear combination of routes and travel time so that future requests are easily accommodated. Actually this is a mixed approach, because the solution can be improved through a longer term consideration, performed with a local tabu search heuristic. To obtain a better schedul e the advanced dynamic wa iting strategy is applied. The available waiting time in a route is split into a few large waiting intervals which are arranged along the whole route. The route is partitioned into segments, each containing consecutive locations that are reasonably close t o each other in the plane. The s egments may change dynamically as new locations are inserted in a route or removed from it. The simulated test results with 100 and 500 requests show the superior performance of double horizon heuristic over the classical ro lling horizon heuristics.
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20 2 .2.2 .1.2 Dynamic iterative techniques Dynamic iterative repair techniques traverse in the domain of complete, possible infeasible solutions, eliminate constraint violations and try to improve the quality of the solutions. S adeh and Kott (1996) review two main iterative approaches interchange approaches and constraint directed repair An interchange procedure iteratively considers possible interchanges in the neighborhood of the current solution. If a given interchange impr oves the quality of the solution, it is performed and a new solution is obtained. The procedure can be applied until a solution is found that can no longer be improved. In their simplest form, interchange procedures are only allowed to move from one feasib le solution to another. By allowing the procedure to wander into infeasible regions of the search space, it is possible to eventually reach better solutions. If applied in their simplest form, interchange procedures usually get stuck in local optima. A number of techniques have been developed to allow the procedure to transition to neighboring solutions that are not as good as the current one in the hope of eventually reaching better solutions. Examples of such techniques include genetic algorithm proce dures, simulated annealing or constraint directed repair procedures reviewed by Sadeh and Kott (1996) Iterative improvement methods that exclude infeasible solutions can still be used to reoptimize solutions when favorable contingencies occur that make th e problem easier and offer opportunities for improving the quality of the existing solution (e.g. c ancellation of a request, addition of a new vehicle, duration of a trip is shorter than expected, etc.) In the face of contingencies that invalidate an exis ting solution (e.g. a transportation asset becoming unavailable for some period of time), iterative techniques
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21 require heuristics t o decide which part of the solution to restore, similar to constructive techniques. Thus, in large scale system s with highly dynamic demand, both constructive and iterative techniques result in low efficiency if they search the entire domain for better solution. 2 .2.2.2 Heuristics performing local search In addition to the NP hardness of the problem, the solution of a dynamic dial a ride system is time critical, because it must be performed in real time and repeated every time when significant variations of data occur. Therefore, some researchers seek for approximation, not for optimization. Two representative examples of appro aches based on local search are reviewed in this section parallel metaheuristics and clustering and locating 2.2.2.2.1 Parallel metaheuristics To improv e the computation efficiency of metaheuristics, Attanasio et al. (2004) implemented a family of parallel tabu search heuristic s. Their work is an extension of the method by Cordeau and Laporte (2003 ) to the dynamic case First a static solution is constructed on the basis of the requests known at the beginning of the planning horizon. When a new request arrives, the algorithm performs a feasibility check for solution that can include the new service request. If the new request can be accepted, the algorithm performs a post optimization i .e., it tries to improve the current solution. The c omputational experiments indicate that parallel computing can be beneficial in solving real time vehicle routing problems. Moreover, the penalty mechanism of the objective
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22 function turns out to provide th e best results while the choice of the initial static solution seems to be irrelevant. 2.2.2.2.2 Clustering and locating Colorni and Righini (2001) develop a two phase model, based on clustering and local search rather than a constructive mechanism. The a lgorithm computes the ordered sequence of pick up and destination points, and leaves the drivers to follow their own routes through the area. Local search algorithm is performed to find a better sequence of points in its neighborhood. The neighborhood of a solution is the set of all solutions that can be obtained from the current one by removing a customer, which is scheduled but not The authors do not provide results from the simulation experiments, i nstead discuss that the level of service of the system is dependent on the following parameters: number of overlapping time windows of the requests, tightness of time windows, computational time, planning horizon, and number of vehicles with their capaciti es. The quality of solutions produced by modern heuristics is strongly related to running time. Thus, i f sufficient time is given, the algorithms attain near optimal or even optimal solutions, as borne out by empirical studies Diana and Dessouky (2004) However, the time availabl e for decision making in a real time service in highly dynamic environment is often short and a different approach is needed in such contexts. A brief summary of the reviewed heuristic algorithms for DDARP is presented in Table 2.
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23 Table 2. Summary of the discussed heuristics for DDAP Reference Objective Time Windows Constraints Algorithm Madsen at al. (1995) Multi criteria On pick up or on delivery Vehicle capacity; Maximum route duration; Maximum deviation of ride time Insertion heuristic performing global search Mitrovic Minic at al. (2004) Minimize total route length Time window from start to end service of request All request to be served; pairing and preceding constraints Double horizon insertion Attanasio et al. (2004) Minimize time windows constraints, route duration and riding times On pick up and on delivery Upper bound of the ride times Three phase insertion with tabu search for optimality Colorni and Righini (2001) Maximize number of served customers; Minimize total traveled distance Time window from start to end service of request Vehicle capacity; preceding constraints Iterative clustering algorithm based on local search After the OR transportation problems and the heuristic approaches of DARP were introduced, in the next two sections some of the applied approaches in DRT service are presented. 2 .3. S imulation a pproaches in DRT In this section we briefly review some practical applications of the heuristic methods discussed in the previous sections into DRT real time operations. As it was discussed in C hapter O ne i n DRT operation passeng ers and service provide r usually have oppo site interests passengers need quick and reliable service,
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24 while the provider would like to have more passengers served by the same vehicle driving in the shortest possible route between the pickup and drop off locations. To cope with these conflicting requirements in real time some researchers developed dynamic multi objective heuristic methods. Dessouky and Adam (1996) propose a real time scheduling algorithm for DRT service that considers vehicle location, vehicle capac ity and passenger demand. The algorithm tries to optimize three conflicting objectives minimum total travel distance of vehicles, minimum total travel time of passengers and minimum total lateness of passenger pickup or drop off. The limiting assumptions are that the number of vehicles is given in any shift and the vehicles operate under a fixed schedule. At first step the algorithm determines the schedule based on the calculated total cost of service and at second step the solution is improved either wit hin the schedule of the same vehicle, or with reassigning the passengers to different vehicles. The performance of the heuristic is simulated with data generated from real para transit service. A service request is considered for scheduling 10 min before t he desired pick up time, and a i s considered to be on time if it arrived no later than 15 minutes of the schedule for the advance reservation requests and 1 hour for the immediate requests. The authors conclude that when the DRT system's workload is low, i t will operate similar ly to a taxi service (depending on the selection of the penalties in the objectives ). As soon as the workload increases over a given limit, ridesharing is the preferred alternative of the heuristic. Horn (2002) introduc es a software s cheduling and dispatching system called L2sched for passenger DRT. Demand is realized as a stream of service requests, which
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25 are scheduled as they arrive. Each service request applies to a group of one or more passengers and includes the locations and time windows for pick up and drop offs. Travel requirements are temporally elaborated to allow a long sighted view of fleet management and exploit system optimization. Scheduling objectives are designed to obtain efficient fleet utilization while satisfying th e service requirements of each request. Thus, the software applies the centralized approach in routing and scheduling. Each vehicle provides real time information about arrivals, departures, trip cancellations and breakdowns. The software provides dynamic scheduling and routing as an extension of the current system plan. Typically the difference between the current and the next plan is induced with a small change in scheduling and/or routing, e.g. assignment of additional request and inclusion a new trip. T hus, the optimal system plan does not change radically, but evolves over time. This evolution is implemented in a three tier optimization strategy: least cost insertions of new requests ; search for local improvements in the neighborhood of the passenger; p eriodic reoptimization of the planned routes. A so also proposed for governing the relocation of idle vehicles. A set of locations, known as cab ranks are specified in advance and the heuristic chooses the cab rank wher e the idle vehicle should be dispatched. To m ake a decision, the heuristic exploits information about future patterns of demand at each cab rank. The performance of the software is tested in simulated environments. Two major conditions with two levels are considered single and shared riding; immediate service or reservations in advance. Initial experiments show that in single ride mode the system accommodates approximately 95% of the demand with an upper limit of 15 min on waiting time. In a
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26 case of share d riding and advanced reservations the number of possible implementations is significantly greater. The CPU execution time varies from 2:12 to 6:06 min in single hiring and immediate service, 2:31 26:01 min in single riding and advanced reservation, 2:24 6:18 in shared riding and immediate service, and 2:31 46:40 min in shared riding and advanced reservation. The test results show that the proposed software produces fast and quality solutions in both single riding cases, but in shared riding and in ca se of high rate of contingencies, the centralized optimization does not perform well. To reduce the limitations of the centralized approach, Uchimura, Takahashi and Saitoh ( 2002 ) introduce a hierarchical model of three level transit operation system, called local initiative for neighborhood circulation (LINC) The first two levels provide regular transportation between the cities in the metropolitan area and between the communities within the cities respectively. The third level provides a dial a ride service on passengers in a given area within the communities and the neighborhoods using small vans. Thus, the third level is a feeder service to both Level 1 and 2. To achieve better reliability d rivers are given freedom to follow any route between the s tations in Level 2 The system has the following operational characteristics: 10 15 min reservation; coverage area 1.5 2 sq mi with approximately 10,000 people; unlimited origins and destinations within the area; ADA accessible vehicles with maximum capa city of 20 passengers. To meet these service characteristics, the LINC system should select in real time the routes with the shortest overall trip time and minimum on board time for most of the passengers. To track the origins and destinations of the reque sts in real time and to inform the passengers about the time of pickups, GIS with GPS will be used. Since the
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27 combinatorial optimization would determine the economical route and the optimum scheduling in very long time, the authors have developed a heurist ic based on genetic algorithms (GA) to obtain near optimal solution in real time. The heuristic follows a search procedure based on Dijkstra algorithm to d etermin e the minimum cost of vehicle route s. The heuristics is tested with simulated instances of 10 passengers, which are solved in short processing time (approximately 40 s). However, the model does not incorporate any constraints such as traffic congestions, unmet service demand and multiple vehicle service. It was observed that i n many DRT system s in order to circumvent the undesirable feature of taxicab systems and to avoid traffic congestions driver s are allowed to deviate from their direct route s between the destination points. This strategy increases the average riding times but also increas es the flexibility to serve other passengers increases the average occupancy and productivity of the vehicles, and hence decreases average waiting times. Since DRT is a service operation, it is expected that the main stress is on re, a reasonable objective can be of maximizing the sum of passenger and operator surplus. Such an objective function recognizes the separate roles of customers and provid ers and the trade off of increasing operational costs and increasing service quality. Gillen and Raffaillac (2002) present an algorithm to measure the contribution of automatic vehicle location (AVL) to both passenger satisfaction and system efficiency. The model accurately predicts the average waiting and total time in the system and the average total distance traveled. A similar problem is faced by the recently food delivery service which takes orders for groceries over the
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28 internet and commits to delivery to the order's address within a given time frame and telemarket logistics which is discussed with the next study Both systems are of single origin with multiple destina tions. Sheu (2006) presents a dynamic customer group based resource allocation methodology for the use in demand responsive city logistics distribution operations. The motivating example comes from the resource allocation problem resulting from tele shopp ing service to manage the corresponding inventories and to provide quick responsive door to door logistics services to the corresponding end customers. Thus dynamic allocation of logistics resources defines the feasibility of an efficient demand responsi ve city logistics distribution system by enhancing the resource utility as well as by shortening the pre route work process time in quick response to changes in cu s tomer demands. In his review Sheu (2006) notes that some multi resource allocation problems are formulated with globally optimized procedures under strong assumptions in the problem definition, demand and/ or supply side, and thus lead to too simplified models In addition, global optimization programming approaches may have difficulties in searching optimal solutions in large scale distribution networks and high customer demand. Furthermore, these globally optimized models may not have the capabiliti es of updating and grouping customer orders dynamically in quick response of customer orders. For all these reasons the author formulates the dynamic logistics resource allocation model with sequential mechanism The proposed methodology is composed of five sequential operational phases: order processing, customer grouping, customer group ranking, container assignment, and vehicle assignment. The whole procedure is executed each
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29 time when the database of customer entries is input to trigger a new logistics distribution mission The methodology is tested in a simulated environment of 136 orders serv ed in one day by 14 vehicles with different capacities Two generalizations can be made from the obtained results: first, the algorithm assigns the large sized and medium sized vehicles to grouped customer orders and small sized vehicles for short distance and miscellaneous goods delivery. Second, different customer groups can be consolidated, and then served by the same vehicle avoiding extra loading and dispatching. Sheu (2006) discusses that appropriate customer order grouping and resource assignment pri or to vehicle dispatching do improve the performance of city logistics systems in reducing the operational costs and average lead time. The implementation of a novel route guidance technology with the proposed dynamic resource allocation method reduce s the expected delivery time associated with each customer group, which is critical in stimulating the customer satisfaction with the improved average lead time. There is still a great potential for integrating more elaborate vehicle routing algorithms for quic k responsive logistics distribution operations. Such an integrated customer group based logistics distribution operation appears even more important to provide efficient goods delivery service in a large scale logistics network under time varying traffic n etwork conditions. In the last two sections some of the simulation approaches in DRT operations were introduced. All of them adopted centralized approaches, where the control and decision making i s done through the objective(s) that maximize the global utility of the whole system (i.e. benefit for the service provide r and convenience for the clients). These approaches are usually implemented as heuristic procedures that extend basic graph
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30 search algorithms, acting over large data coll ections that describe the entities of the domain problem (service requests, vehicles and schedules). A key as pect when applying these approaches is the identification of in order to allow the generation o point of view. However, this is not always feasible because not all the clients share the same desires, nor appreciate them with the same importance. From the review of heuristic and simulation approaches the followi ng general deficiencies of centralized DRT planning method s are observe d: Computational complexity, i.e. the models suffer to adjust the schedules and routes in real time; Difficulties in planning of large scale and highly dynamic problems; Inability to re spond in case of missing information about a service request or current status of a vehicle; Low utilization of the vehicle fleet due to special requirements such as handicapped people transportation; Possible high cost of operation, in some instants close to taxi service. To address some of these deficiencie s, some researchers perform met aheuristic local search instead of global one Cordeau and Laporte (2007), or search for approximation rather than optimization of the solutions Attanasio et al. (2004). 2 .4 Intelligent Transportation Systems ( ITS ) a pproaches in DRT The advance in the information and communication technologies, such as Internet, Geographic Information Systems (GIS), Global Positioning Systems (GPS),
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31 Artificial Intelligence (AI) and the availability of low cost mobile communication devices have led to a significant changes in DRT operation al planning The real time reservations become easier to manage and simultaneously the systems can operate in more complex and highly dynamic decision making environment. Th e increase in automation has caused the shift to online reservation system, hence, requiring service providers to have real time scheduling and dynamic dispatching capabilities. In a dynamic dispatching mode, the schedules and routes of vehicles are modified in real time to account for any trip cancellations or any new orders. To be effective, real time scheduling and dynamic dispatching systems require immediate information and data on the location and status of each vehicle. By taking into account real time information conce rn ing passenger demand, vehicle location, and road conditions, real time scheduling can give the best assignment of vehicles to riders and route selection. Hence, real time scheduling and routing ha ve the potential to improve service efficiency, to reduce the cost of transit providers and to improve customer satisfaction ITS offer a number of newly developed approaches for DRT operational planning and control To increase the service thro ugh increased system efficiency, two types of advanced technological responses have been implemented: AVL and dynamic scheduling Kihl at al. ( 1996 ). AVL can track and report in real time the location of all vehicles in the fleet as frequently as every other second. With the aid of a real time display map generated by an AVL system, trips can be inserted by the dispatcher and directly posted to the closest vehicle. The most utilized method of AVL is GPS. The main disadvantage of AVL is the high cost. Dy namic s cheduling is time specific, rather than
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32 location specific like AVL. Unlike AVL, it does not report the actual location of the vehicle, but rather it a between points. Based on the deci sion making process concerning service requests the ITS approaches applied in DRT operation can be split in two main groups centralized and decentralized, which are reviewed in the next two sections. 2 .4.1 ITS approaches in c entralized DRT systems In DRT systems that adopt centralized approaches, the control and decision making i s done through the objective(s) that maximize the global utility of the whole system (i.e. benefit for the operator and convenience for the clients). T o adapt DRT o perations i n advance or to meet the current demand in real time Finn and Breen (1996) introduce the telematics approach Telematics can be broadly defined as the integration of telecommunications and informatics systems. It consist of a communication platform (eithe r by wire or by air) and ITS Telematics DRT systems are based on the integration of information and telecommunication ( ITC ) technologies vehicle location systems, dispatch centers, communications, booking, and reservation systems. In addition, optimization systems are included to determine the routing, vehicle size, assigned passenger based on cost, passenger requirements, a nd fleet ability. The most utilized telematics technologies include the following components: Communications between the vehicle s and dispatch cent er s (or depot s) across the area of coverage.
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33 Vehicle location systems for effective system management and pa ssenger information systems. The most practical form is GPS Network M a nagement and Control Systems dispatch cent er s which have substantial data collection and processing capabilities, combined with the decision and communication mechanisms to implement needed interventions. Booking and rese rv ation systems b y combining integrated databases of services with real time knowledge of network state, it is possible to operate a more dynamic booking service, and to use the network control communication system to advise the vehicle driver of seat availability. Ticket and fare collection systems can be linked to the booking and reservation systems to automatically generate travel documents. Currently, t he greatest potential for the fare collection is smart cards. Passenger information services allow potential users to determine the available service offer. All data is normally held i n a centralized database with links to the systems of the individual operators. The construction of the database is to be designed to allow rapid retrieval of information. The presented trial DRT system by Wipke (1996) utilizes most of the above discussed components GPS to locate the vehicles, two way communications between the vehicles and a central computer server, and advanced d ispatching and routing software to control the movement of vehicles within the fleet. To provide passenger information service, the developed advanced web site allows visitors to see all the updates of vehicle position on a map every 20 seconds. The projec t demonstrates how a
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34 fixed route, fixed schedule shuttle service can be converted to be demand responsive with increased efficiency. The proposed concept is based on three essential telematic elements: Precise location of t he vehicle s through GPS and two w ay electronic communicator ; Advanced mapping software to take current vehicle locations and directions o f travel, and the incoming passenger requests for rides ; O ptimization routines in real time to determine which vehicle should make the pickup and the o ptimal route to take. Thus, DRT service overcomes many of the disadvantage s of public transport by using state of the art ITC technologies, GPS and system optimization to arrange pick ups and drop offs from the desired locations. Casey at al. (2000) report on an Advanced Public Transportation System (APTS) project. The purpose of the project is to apply ITS technolog ies that will improve the intermodal transportation services in a rural area with seasonal variability of demand While the p aratr ansit/dial a ride system serves residents only, because of the summer tourist pattern o f the area the fixed route services experience significant seasonal changes in demand. The system utilizes GPS to provide real time information on vehicle locations and /or expected arrival times available to customers in the three ways by phone calls, via the internet and at video monitors position ed at transit or public centers. Mobile d ata t erminals are used to send messages between dispatchers and drivers, and to st ore data collected on board the vehicle s A GIS based decision support system
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35 integrated with an Internet based travel planner performs the scheduling of the passenger s. This tool assist s the client agencies and individual customers in planning their trips by displaying vehicle routes and schedules that can serve a desired trip origin/destination and time. In addition to making real time information available, the APTS is able to increase the number of handled customer calls (including information requests) as a result of reducing the time required for other tasks. Without APTS callers sometimes give up service because of the long waiting time to communicate to the system dispatcher. 2 .4.2 ITS approaches in d ecentralized DRT systems In DRT systems applying decentralized decision making approach vehicle fleet is represented as a communit y of agents that perform low level planning, scheduling, execution, and control tasks As opposite to centralized evaluations, optimization can be don e with less information and, as consequence, the planning solution s could be far fr om the optimal for the whole system. This might be the main reason why very few researchers apply decentralized approach in their studies of DRT operations. Cubillos at al. (2004) present a mixed multi agent system (MAS) approach to perform distributed operational planning of DRT service. The method combines the best features of both centralized and decentralized decision making approaches. The model is structured as a two l ayer architecture: the Internet layer which provides the interface with the vehicles, clients and other systems, and the Planning layer which encapsulates the assignment and s cheduling services. The model involv es a negotiation process to
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36 solve the trade offs between the passenger s and the service provide r, incorporating the client only in the final decision making. The underlying MAS framework allows the implementation of different scheduling policies, and evaluates the insertion s of the trip s The adopte d policy finds all the feasible ways in which a new customer can be inserted utility according to an objective function. The advantage of this approach is in avoiding t he estimation of the utility function, because the client is involved only in the final decision process. This is the most utilized approach in the online search engines of transportation service. In his decentralized ADART technology, Dial (1995) introduces a fully automated dispatching (FAD) system, which can field a customer requests, schedule and optimally route a vehicle without human intervention. Every vehicle is autonomous and when board computer receives a customer request, it inserts this request into and plans the optimal route to accomplish the schedule. Furthermore, the computer may pass the request off to another vehicle computer collectively assign s ing to the responsible vehicle, thus leaving each computer to solve only a small optimization problem. A ll routing and scheduling problems in parallel. Th u s the huge system problem is decomposed into several e asier small problems and all of them are solved simultaneously This enables an ADART operation to keep up with even largest demand surges. In addition, each vehicle computer can operate in virtual
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37 ignorance of the states of the other vehicles, while at t he same time cooperating with the other computers towards minimizing the total cost of service. After reviewing the simulation and ITS approaches in real time DRT control we can note the following disadvantages of the centralized systems: The developed h euristics are not invariant to the sequences of the service requests to be inserted into the vehicle schedules; With approaching the end of the planning horizon, the degree of freedom of flexibility of inserting the last requests decreases; In a highly dynamic environment by the time a new solution is constructed, additional contingencies occur, causing too frequent disruptions of the determined assignments and schedules; In large scale systems with highly dynamic demand the developed heuristics work wit h low efficiency if search over the entire domain for better solution; The proposed simulation products do not produce quality real time solutions in case of high rate of contingencies and multi shared vehicles. To reduce the low efficiency of the central ized systems in areas with heavy traffic contingenc ies some of the DRT operators give their drivers freedom to select the actual routes between the pickup and drop off locations, Colorni and Righini (2001), or between the stationery bus stops Uchimura, Takahashi and Saitoh ( 2002 ). Thus, the actual routings of the vehicles are determined individually, not by a central processor. This partial decentralization of the routings saves computational time and reduces the information exchange between the vehicles and the operating center.
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38 Chapter Three Research Motivation P roblem Domain Research Goal and Objectives 3 .1 Research Motivation DRT operation al planning where transport request s are accepted and scheduled for service, and vehicles are routed/rerouted in real time has changed significantly due to the recent advances in Intelligent Transportation Systems. However, the high level of dynamics associated with real time communication between the system operator and passengers, and system operator and vehicles require fast processing of a nu mber of parameters. Some of these parameters consider the passenger requests; others characterize the vehicle routing s and the environmental conditions. Some of these groups of data might be unrelated to each other. In addition, some system related information may or may not be available continuously based on the reliability of the technological infrastructure. Thus, the intelligent and effective processing of the available information in such a comple x decision making environment requires the use of formal modeling, analysis and control approaches which are robust, modular, and/or decentralized. Robustness will provide that the system behaves in the desired manner in the unpredictable and quickly chang ing environment. Modularity will provide independent modeling of the assignments to the vehicles vehicle routings and re routings and environmental conditions I n case of a conflict or other unpredicted
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39 situ ation, only the modules that co ver the particular request will be affected. The decentralization will r educe the computational efforts, improve the tractability of the solution and allow parallel computations 3 .2 Research P roblem Domain In this research we aim to provide real time control of DRT operations in a complex transportation problem referred to as Dynamic Dial A Ride Problem with Multiple Acceptable Destinations and Origins (D DARP MADO). A highly dynamical and critical application domain of D DARP MADO is the military Aer omedical Regulation and Evacuation (ARE) of patients to Medical Treatment Facilities (MTFs). In this problem the origin of the service requests can be any location within the affected region and the destination of the demand can be assumed to be one or more locations known a pr iori (such as MTFs) Routing and scheduling operations in such a domain require the dynamic coordination and (re)allocation of a large number of resources subject to a wide variety of constraints. Key assets/resources and associated constraints include vehicle s (airplanes or helicopters ) and their characteristics (e.g. capacity, length of travel, fueling requirements, etc.), pilot and medical crews and restrictions on the nu mber of hours they can work in any given day, airports and the ir different characteristics (e. g. capacity, types of aircraft they can accommodate, etc.), number of hospital beds at MTFs and the types of patients each MTF can accommodate, etc. For example, in case of a natural or man made disaster in Tampa bay area the community authorities may appoint consider four) hospitals to serve as
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40 temporary MTFs Tampa General Hospital (TGH), St Joseph Hospital (SJH), Town & Country Hospital (TCH), and University Community Hospital (UCH) Fig3.1 Helicopters, light jets or heavy duty land transporters can be used to transport patients (passengers) to the MTFs, which provide shelter and first aid. Fig. 3.1 A map of MTFs and patient pick up locations in Tampa bay area If patients can be accommodated at more than one possible MTF, the problem is with multiple acceptable destinations. In case t he patients can get to different designated areas to be pic ked (the dark spots on Fig. 3.1), we talk about multiple acceptable origins. A special case of D DARP MADO is when patients can be picked from any possible location.
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41 T he most challenging aspect in planning and scheduling of medical evacuation operations is the high dynamics of the domain in which requirements and constraints continuously change over time. As it was discussed in S ection I I .1 ARE imposes two general extensions in DRT operations: M ultiple acceptable destination and / or origin locations for a given demand; the solution to this problem must include assignment s of each demand to a destination and / or origin location s ; B oth the demands and the r esources can change dynamically while the initial route and schedule are being executed. T he proposed solution method must be capable of rea l time revision of the assignments of patient s to resources and routes and schedules of vehicl es 3 3 Research Goal and Objectives In this study we propose the representation of DRT systems as a Discrete Event System s (DES s ) where the model captures both the low level dynamics (such as infrastructure conditions, current status of vehicles and limitations) and high level dynamics (such as service demand requests) of system evolution in a modular manner. The mathematical foundation of DES theory facilitates logi cal analysis of these complex systems and provides the necessary framework for the de velopment of real time scheduling and intelligent decision making tools
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42 The real time control of DRT is developed as SC of DES, which synthesizes the supervisor(s) i.e the acceptable behaviors of all the elements of the system. Fig. 3. 2 outlines the framework of the online DRT control structure. Fig. 3. 2 Framework for real time DRT control In Fig.3.2, DRT control system takes input data from the passenger request interface and the physical environment (vehicle fleet and service area with its conditions) When a new se rvice request is received the assignment controller checks if it is feasible to accept this passenger. If the request cannot be accept ed because of operational limits, the system sends a signal of rejected request. If the request is feasible the routing PASSENGER REQUEST INTERFACE SC Assignments and Routings PHYSICAL ENVIRONMENT Request A ssignment & Routing Traffic Surveillance & Vehicle Availability Vehicle Fleet Interface DRT CONTROL SYSTEM Data information (in both levels) Low er task level signal information (issue 1) Process level: logical event feedback and control information
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43 supervisor generates the possible routings of the vehicles to serve the request. In case of more than one possible assignments and /or routings, the system may use an optimizer or rule based logic (e.g. Route P lanner and Task assignment) to select the prefe rred vehicle. In any case, the information to the selected vehicle is sent through the Vehicle Fleet Interface, and the passenger is informed for the service. During operation the system ( e.g. Traffic Survei l lance and Vehicle Availability) receives feedbac k information for the current conditions of the physical environment (vehicles breakdowns, traffic congestions, etc.). To the best of our knowledge, Seow Pasquier and Hong (1999) are the first researchers who proposed the appl ication of Supervisory Contr ol Theory ( SCT ) to the modeling and real time operational control of the class of land DRT systems. The main advantages of SCT for online service control of DRT system s over the heuristic and simulation methods for operational planning are: Possibilities t o consider service of a new request without affecting the already scheduled requests; Possible modularity and decentralization of the supervised control, which allows autonomous service operational control of the vehicles and parallel computation of the ir supervisors ; Dealing with unobserved events th at may occur in complex systems.
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44 In this research we provide several supervisory contro l l er synthesis method ologies applicable in real time control of large scale DRT systems operating in ARE environment Within this goal are the following objectives : Model the uncontrolled system behavior and specifications of ARE problem using Finite Automata (FA) ; Synthesize centralized supervisory control ler to d e monstrate the decision making of accepting or rejecting service requests ; Synthesize general supervisor from the independent modular supervisors of the different specifications ; Apply decentralized supervisory control to compute in parallel the local supervisors of concurrent groups of vehicles and passengers; synthesize the global supervisor of the entire system To accomplish th ese objectives we apply and extend the DES modeling framework in the study of Seow and Pasquier (2004) of DRT supervisory control of the land transportation model in the following four main direction s: Extend m odular SC with additional specifications which are characteristics of ARE problem domain : maximum length of the routes ( e.g. flights) ; finite set of origin destina tions of the requests. In modular SC the action of the centra l supervisor S is represented as a combination of the control actions of two or more supervisors. The advantage of this method is in the simplified procedure to check the feasibility of any
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45 service request. If a given request cannot be accepted by one of t he supervisor s there is no need to check for the rest of the supervisors. D evelop d ecentralized SC: A decentralized that jointly control a distributed system, Cassandras and Lafortune (1999). In a decentralized DRT syste m each vehicle and its assigned passengers form a subsystem Thus, v ehicles of each subsystem do not interfere with the routings and assignments from any other subsystem Hence, local supervisors of each subsystem may not observ e and do not control the behavior of the rest of the subsystems Since the formed subsystems operate simultaneously they form concurrent DESs, which are independent to each other. Thus, all the local supervisors can be synthesized in parallel. In the cen tralized planning approach (see S ection II.4), the scheduling and routing of the entire system is updated with any new request or change in the domain. Despite efficient heuristics and communication technologies, the permanent update of all the passenger a ssignments and vehicle routings take computational time, which cannot be neglected in real time planning of a complex problem like the emergency aeromedical evacuation. In addition, the heuristics need all the relevant information of the requests to comput e the passenger assignments and calculate the vehicle routings.
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46 The results of this research are expected to overcome the disadvantages centralized control and achieve a methodology for synthesis of robust, modular and decentralized real time control of c oncurrent systems.
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47 Chapter Four Discrete Event Systems and Supervisory Control In this chapter we introduce the basic concepts of DES, supervisory control theory (SCT) and their representation with finite automata (FA). 4 .1. Discrete E vent S ystems DES s are dynamic systems driven by event occurrence s usually at irregular intervals These events take the systems from one state to another. Such systems arise in a variety of context s such as information and communication networks complex and multimode production processes and robotics logistics and vehicular traffic These applications require control and coordination to ensure the orderly flow of events. As controlled (or controllable) dynamic systems, DES s qualify for a proper subject for control theory (CT) CT for DES considered in this study is based on FA concepts. The essential concepts and modeling of DES can be found in Cassandras and Lafortune (1999) and the fundamentals of the FA theory and supervisory control theory ( SC T ) in Wonham (2006). In the following review of DES modeling and SCT background till S ection 4 .2.3 we adopt the formalism of Cassandras and Lafortune (1999) and Sections 4.2.4 and 4.2.5 are based on the study of Yoo and Lafortune (2002)
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48 4 .1.1 FA model ing of DES An automaton is a device that is capable of representing a sequence of events according to well defined rules. Automata are used as a modeling formalism since they are easy to use, intuitive, amenable to all the unary and composition operations, and easy to analyze. A DES can be modeled as a five tuple automaton A i.e. where Q is a set of states, is a non empty set of events (alphabet), is a transition function, is the initial state and is the set of marke d states ( i.e. states indicating the completion of the tasks or sequences of tasks from a control perspective) A transition in the automaton A is any element of and may be denoted simply by the triple where If the transition function is partial, only a proper subset of can occur, and a more flexible and economical representation of DES is provided by a generator G i.e. If is defined, then we say that is eligible at q in G and denote it as The set of all feasible events that can be executed at state q is denoted by i.e. Finite state automata are graphic ally described by directed transition graphs. In order to represent an automaton, a state is identified by a node (represented by a circle ) of the graph whose edges are labeled by
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49 transition labels (represented by an arrow, e.g. ) The initial state is labeled with an entering arrow while a marked state is labeled with an emit ting arrow When is also a marked state, it is labeled with a double arrow 4 .1.2 Language and l anguage characteristics A language L defined over an event set is a set of finite length strings formed from events in The set contains all possible finite sequences, or strings, over plus the null string The definition of can be extended to as follows: behavior may then be described by two languages: L(A) the prefix closed language generated by automaton A and L m (A) the language marked by automaton A Formally and The language generated by automaton A can be interpreted as the set of all the sequences of events that take the system from its initial state to some reachable state in A The language marked by A can be interpreted as the set of all the strings that take the system from its initial state to some marked state i.e. final state or a state of satisfactory completion. By definition, is the subset of strings in L(A) which end s in any of the final states Q m Thus, if an automaton A represents a DES, then Q m represents
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50 completed tasks executed with the physical process of the DES. If automaton A models a behavioral specification K then is the behavior of interest. 4 .1.3 Operations on languages The following three operations on languages are essential in language composition : Concatenation : If then a string s is in if it can be written as the concatenation of a string in L a with a string in L b is : Prefix closure : The prefix closure of L is the language denoted by consisting of all the prefixes of all the strings in L If then L is said to be prefix closed if any prefix of any string in L is also an element of L i.e. Language closure : a language is said to be closed if 4 .1.4 Unary operations on automata The following three operations on automata are essential in FA theory: Accessible states : The set of all the states that can be reached from the initial state is called the accessible states subset. Let Q a denotes the accessible states subset, and is described as:
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51 Co accessible states : The set of all the states q from which some marked state can be reached is called the co accessible states s ubset. The co accessible states subset denoted by Q ca Trim automaton : an automaton that is both accessible and co accessible is said to be trimmed. 4 .1.5 Composition operations on automata The following two composition operations on automata are of great importance in S CT: Product of two automata A 1 and A 2 is the accessible automaton A , where In the product, the transitions of the two automata are synchronized on a common event, i.e. It is verified that and Parallel composition of two automata A 1 and A 2 is the automaton A , where
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52 In the parallel composition a common event can only be executed if both automata execute it simultaneously. The rest of the events can be executed whenever possible. If then there are no synchronized transitions and is the concurrent behavior of the two automata. This is also called the shuffle of A 1 and A 2 4 .1.6 Analysis of DES One of the key reasons for applying finite state automata (FSA) to model DES is their flexibility and amenability to analysis for answering various questions about the behavior of the system. The computational complexity of navigating the state transition diagram of a deterministic automaton if there is no need of iterat ions is linear of the state space, i.e. where n is the state space, If iterations are necessary, the complexity typically is In the next subsections the most often encountered analysis problems for DES are reviewed. Safety properties are concerned with the reachability of certain undesired states, i.e. the presence of certain undesirable strings or substrings in the language ge nerated by the automaton. A DES model of a system is usually built in two steps: first automaton models of the components of the system are defined; next the complete system model is obtained by either product and/ or parallel composition of the constituent automata. The safety questions are
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53 posed on this complete automaton. The algorithms that answer all these safety questions are quite straightforward and described in Cassandras and Lafortune (1999) : To determine if a given state q 2 is reachable from anot her state q 1 one has to check if q 2 is accessible from q 1 being initial state. To determine if a given substring s 1 is possible in the automaton, one has to try to execute s 1 from all the accessible states. To test the inclusion i s equivalent to testing The intersection is implemented by taking the product of A and B Blocking properties are concerned with the coaccessibility of states to the set of marked states. An automaton A is said to be blocking if and nonblocking if This implies that for every string there is at least one string such that In o ther words, an automaton is non blocking if every string starting from the initial state can be completed to some string that leads to a marked state. To determine if a given accessible automaton A is blocking, one has to check if all the states of A are coaccessible. If there are states that are not coacce s sible, A is b locking, otherwise it is nonblocking. If A can reach a state q where then q is said to be a deadlock state. Deadlock states can be found by examining the active event sets of the states. A can also reach an unmarked state p which is strongly connected to a set of unmarked states P i.e these states are reachable from one another but there is no
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54 transition going out of P In such a case there is always at least one transition that can be executed but A can never reach any of the ma rked states. This situation is called a livelock Unobservable events are events that occur in the system but are not seen or observed by an outside observer of the system behavior. For example, fault events that do not cause any immediate change in the sensor readings are unobservable events. If the transitions caused by all the unobservable events are labeled by then a nondeterministic automaton model of the system will be obtained. In order to keep the determinism, the event set is partitioned into two disjoint sets: o the set of observable events, and uo the set of unobservable events. Recall from section IV.1.1 a n automaton with a partial transi tion function is called a generator ( G ) With the structure ( G, o ) the natural projection is define d as follows : In other words P erases only the unobservable events. If G obs denotes the minimum deterministic automaton equivalent to the generator of interest G we have that:
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55 The state of G obs reached after string will contain all the states of G that can be reached after any of the strings in In words, the state of G obs is the union of all the states of G consistent with the observable events occurred so far (i.e. string s ). 4 .2 Supervisory control In supervisory co ntrol of a given DES the behavior of the system must be modified by feedback control to achieve a given set of specifications If a generator G models a DES, then it is said that G represents the uncontrolled behavior of the system. The premise is that this behavior is not satisfactory and must be modified by control; modifying the behavior is restricting to a subset of To alter the behavior of G we need a supervisor S S observes some (possibly all) of the events that G generates and tells G which of the defined events are allowed. Thus, the two key considerations are that S is limited in terms of observing the events executed by G and S is also limited in disabling feasible events of G Therefo re, we consider the observable events in those that S can ob serve and controllable events in those that S can disable. 4 .2.1 Controlled DES Let a DES be modeled by a pair of languages L and L m where L is the set of all strings that can be generated by the system and is the set of marked strings that represent the completion of some tasks by the DES. Assume that both L and L m a re the languages generated by
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56 T he event set is partitioned in two disjoint subsets: where c is the set of controllable events that can be prevented from occurring by a supervisor S and uc is the set of uncontrollable events that cannot be prevented from happening. The adjoined supervisor S interacts with generator G in a feedback manner, as depicted in F ig. 4. 1. Fig. 4. 1 The feedback loop of supervisory control Let all the events in be obser ved by S Thus, in F ig. 4. 1 s represents all the strings of the events executed by G so far and observed by S The control pattern means that the transition function can be controlled by S in the sense that c can be dynamically enabled or disabled so that the modeled system exhibits a desired language Formally, S is any map Thus, for each generated by G the set of enabled events that G can execute at its current state i f S is said to be admissible if for all , i.e. S is not allowed to disable a feasible uncontrollable event. Given G and an admissible S the resulting closed loop system is denoted by S / G (i.e. S controlling G ). The controlled system S / G is a DES, characterized with its generated and marked languages The generated language is defined recursively as follows:
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57 Since always is nonempty and closed. The marked language is defined as follows : The DES S / G is said to be blocking if and nonblocking when Since always holds, the nonblocking condition is also equivalent to 4 .2.2 Controllability theorem and realization of supervisors The key existence result for supervisors in the presence of uncontrolled events is specified by the C ontrollability T heorem (CT h ) : Let a DES is modeled by the generator where is the set of uncontrolled events, and There exist a supervisor S such that if and only if This condition is called the controllability condition The proof of the theorem is presented in Cassandras and Lafortune (1999) CT h is utilized to define when a language is controllable with respect to another given language. Thus, if K and are languages over event set and K is said to be controllable with respect to M and uc if Since
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58 controllability i s a property of prefix closure, K is controllable if and only if is controllable. Suppose a language is controllable with respect to G and From the proof of CT it follows that the supervisor S of the controlled system S / G is defined by for and results in Cassandras and Lafortune (1999) To build an automaton realization of S it suffices to build an automaton that marks Let R be such an automaton, i.e. where R is trim, and R can be connected to G by product operation and the result is the desired behavior of the system S / G ; Similarly, Note that R is defined over the same event set thus Hence the control action S (s) is encoded into the transition structure of R i.e. In the latter, and denote the active event set and transition function of respectively
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59 The interpretation with the control paradigm is as follows: Let G is in state q and R is in state p following the execution of a string and G generates an event that is enabled. The same event is also present in the active event s et of R at p Thus, R also executes If and are the new states of G and R after execution of the set of enabled events of G after string is given by the active event set of R at With this procedure R is called the standard realization of S Consider the reverse question if there is a given automaton C and we form the product can that be interpreted as controlling G by C ? The supervisor S for G induced by C can be defined as; Therefore, if and only if is controllable with respect to and uc i.e. The resulting closed loop behavior is defined with the languages: If a given language L sublanguage of L that is controllable, denoted by Cass andras and Lafortune (1999) present two effective algorithms to calculate in prefix close case and in general case.
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60 4 .2.3 Modular supervisory control In modular control, the control action of a supervisor S is given by combination of the control action of two or more supervisors. Consider the case of two supervisors S 1 and S 2 each defined for G the modular supervisor is determined as Thus, an event is enabled if and only if it is en abled by both S 1 and S 2 Fig. 4. 2 depicts the architecture of a modular supervisory control with two supervisors. Fig. 4. 2 Modular supervisory control with two supervisors The closed loop behavior under modular control is formalized with the following languages: Modular supervisory control is introduced as a solution to the problem of state space increase faced by the centralized supervisory control. The idea is in presenting as the intersection of the active event sets of R 1 and R 2 i.e.
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61 Then, if the standard realizations R 1 and R 2 of S 1 and S 2 have n 1 and n 2 states respectively, the model needs to store a total of states instead of The modular supervisory control problem (MSCP) with a g iven a DES G with event set uncontrollable event set and admissible language w here is to find a modular supervisor S mod (according to the architecture in Figure 4. 2 ) such that To solve MSCP, first we b uild the standard realizations R i of S i such that Next, take S mod to be the modular supervisor such that With this choice of modular supervisor S mod the desired solution is W on h am and Ramadge (1988) defined two languages to be nonconflicting if If are the individual nonblocking supervisors for G then S mod1n is nonblocking if and only if every is a nonconflicting language, i.e. This statement is proved in section IV.2.5.
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62 4 .2.4 D ecentralized supervisory control Decentralized control represents the situation where there are several local supervisors that are jointly controlling a given system that is inherently distributed Such decentralized control architectures arise in a variety of network systems such as mobile communications, automated vehicular systems, and integrated sensor networks. There are two main advantages of the decentralization improved computational trac tability of the control and possibility of partial observation of the event set. Consider a DES controlled by n local supervisors and the i th having states, A global supervisor with the same control action wil l require states. Let the complexity of designing a supervisor with m states is Then the complexity of designing a global supervisor is while the complexity of designing n local supervisors is Lin and Wonham (1988) report that in a typical case and the ratio explodes to Let the memory requirement for implementation of a supervisor is Then the memory required to implement a centralized supervisor is while the memory required to implement n local supervisors is Typically is a linear function of m i.e. Lin and Wonham also report that with the same values of n and m i the ratio Another distinguishing feature of the decentralized from the modular control architecture is the possibility that the individual supervisors can be partial observation supervisors and moreover their respective sets of observable and controllable events need
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63 not be the same. To formulate the decentralized supervisory control problem consider a set of n partial observation supervisors, each associated with a different projection P i jointly control ling the given DES G with event set Four set s of events are associated with G : c uc o and u o With each supervisor S i we have: the set of controllable events where the set of observable events where and the natural projection corresponding to The domain of partial observation supervisor can be extended from to and Here we briefly review the three architectures of decentralized supervision: conjunctive disjunctive and general described by Yoo and Lafortune (2002). 4 .2.4.1 Conjunctive decentralized architecture Similarly to modular control, the net control action of conjunctive architecture is the intersection of the sets of the events enabled by each supervisor, i.e. For the conjunctive architecture, a local decision rule of S i enables by default the set Fig. 4. 3 depicts the architecture of conjunctive decentralized supervisory control with two supervisors. The prefix closed language generated by the conjunctive supervisor is expressed as follows: ;
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64 The marked langu age is defined as: Fig. 4. 3 Conjunctive decentralized supervisory control with two supervisors 4 .2.4.2 Disjunctive decentralized architecture For the disjunctive architecture, a local decision rule of S i disables by default the set which is controllable by the other supervisors. The disjunctive supervisor S disj is defined as follows: Fig. 4. 4 depic ts the architecture of disjunctive decentralized supervisory control with two supervisors. The prefix closed language generated by the disjunctive supervisor is expressed as follows: ; Analogously, the marked language of the disjunctive supervisor is
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65 Fig. 4. 4 Disjunctive decentralized supervisory control with two supervisors 4 .2.4.3 General decentralized architecture In the general architecture the set of controllable events c is partitioned into two subsets c,e and c,d : Here c,e is the set of controllable events for which the default setting is enablement, while c,d is the set of controllable events for which the default setting is disablement. Fig. 4. 5 General decentralized supervisory control with two supervisors Fig. 4. 5 depicts the architecture of disjunctive decentralized supervisory control w ith two supervisors. The generalized decentralized supervisor S gen d is defined as
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66 follows: where P ce and P cd are the following projection mappings: and The prefix closed language generated in the general architecture is : ; The marked language is Yoo and Lafortune (2002). It is importan t to note that when the sets c,i are mutually disjoint, the three architectures (general, disjunctive and conjunctive) are the same. The reason is that each controllable event is controlled by only one supervisor i.e. the event is enabled if and only if the corresponding supervisor enables it. 4 .2. 5 Nonblocking decentralized supervisory control In this section we present the conditions under which the conjunctive and disjunctive decentralized supervisors are nonblocking. Recall from Section IV.1.6 that a language generated by G is nonblocking if and from Section IV.2.1 Thus, is said to be nonblocking supervisor if i.e. S is nonblocking for G if every state trajectory of the closed loop process can be extended to reach the set of marked states of G
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67 4 2.5.1 Nonblocking conjunctive decentralized supervisor W on h am and Ramadge (1988) prove that the conjunctive supervisor is nonblocking if and only if the marked languages and are nonconflicting ( Proposition 4.2) Here we re state the proof from Wonham and Ramadge : is nonblocking With the extension of the nonblocking property for finite number of languages, the above proof is valid for finite set of languages. Hence, the conjunction decentralized supervisor is nonblocking with respect to G if all individual supervisors are nonconflicting. 4 .2.5.2 Nonblocking disjunctive decentralized supervisor Theorem 3.4.1 by Wonham (2006) states that there exist a nonblocking supervisory controller for G if and only if is controllable with respect to G and is L m ( G ) closed. Thus, to prove that is nonblocking we have to
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68 show that (i) S disj is controllable with respect to G and (ii) S disj is L m ( G ) closed. Let then Proof : (i) need to show that if and are controllable, then is also controllable. (ii) need to show that if and are L m ( G ) closed, then is also L m ( G ) closed. With the extension of the nonblocking property for finite number of languages, the above proof is valid for finite set of languages. Hence, the disjunction decentralized supervisor is nonblocking with respect to G if all individual supervisors are controllable and L m ( G ) closed. 4 .2.5.3 Nonblocking general decentralized supervisor Based on the above two proofs, in case of a general decentralized supervisor S gend i.e. we may say that S gend is nonblocking
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69 with respect to G if all individual conjunctive supervisors are nonconflicting and all individual disjunctive supervisors are controllable and L m ( G ) closed.
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70 Chapter Five Taxonomy of DRT Systems, DRT Modeling with FA and I llustrative E xample In this chapter we present an approach of modeling DRT system s as DES s and their real time control with centralized and modular supervisor s To facilitate the formalism of modeling and analysis of the systems we first present taxonomy of the DRT systems according to their characteristics relevant to DES representation 5 .1 Taxonomy of DRT s ystems Every DRT system is determined with the following three component characteristics: origin/destination characteristics, vehicle fleet characteristics, and transportation demand characteristics. Based on these component s, DRT systems can be classified in the manner described below. 5.1 1 Origin and d estination c onsiderations Many to one these systems transport passengers or freight from many origin locations to one destination location. Typical examples are systems with single commodity PDP, e.g. an armor ed vehicle that transports money from local branches to the head office of a bank; on demand air charter (taxi) service
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71 utilizi ng Dial a Flight Problem (DAFP) pick ing passengers from small airports and transferring them to a larger airport ( hub ) Many to few these systems serve more than one, but a fixed number of origin or destination locations; Example include n commodity PDP, where n types of goods are considered and each commodity requires single pickup and delivery node, military and ARE service, th e emergency services like police patrols, ambulance fleet management. Many to many these systems serve large and usually random number of origin and destination locations. Typical examples are based on Urban Courier Service Problem (UCSP), taxi cab service. 5.1 2 Vehicle f leet c haracteristics Systems with fleet of vehicles where no capacity constraints are considered like postal and courier service, emergency fire fighting. Systems with a homogeneous fleet with the same load capacity and speed capabilities like taxi cabs, shuttle vans. Systems with a heterogeneous fleet with the different capacities and/or speed capabilities like air charters operating with different size airplanes. Systems with constraints on length or duration of vehicle route s e.g. range of an aircraft, pilot shift restrictions in air taxi systems. Systems where the fleet is located at one central or multiple depots. After the end of service all the vehicles must return back to the depot(s) like in taxi operators.
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72 Systems where vehicles are subject to unpredicted stoppages or re routings like caught in a traffic jam, detours, or breakdown. Examples include all the land transportation systems operating in urban areas. 5.1.3 Transportation d emand c haracteristics Systems wi th a priori known static demand that accept service reservations made in advance. Classical example s are the school bus se rvice and fixed route dial a ride systems working with advance reservations Systems with dynamic service demand where every customer request is eligible for immediate consideration and requires real time adjustments of the already established routes and schedules. Typical example is a courier service system. Systems where some groups of passengers are given priority over the rest or h ave special service requirements. Examples include service of people with disabilities, air charter transportation of special cargos. 5 2 Modeling of DRT s ystems with FA T he application of SCT in DRT control where it is required to provide autom ated system update in real time is based on the following three groups of model s, Seow and Pasquier ( 2005 ): Plant models of the with FA ; Specifications models of control objectives (behaviors) to be specified with FA ; A supervisory controller to be synthesized.
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73 The taxonomy presented in Section V.1 is used here as a guideline to present the plant and specifications automata modeling various feathers of a DRT system operation To model a system with origins an d destination locations from a fixed finite set, the origins/destinations can be presented as states, and the travels between every two destinations as events. For example a small air taxi system covering the demand over four airports A, B, C, and D is pre sented in Fig. 5.1 Any airport is reachable from the other airports by the used jets. Fig. 5.1 Simple air taxi DRT system operating at four airports All possible flights of jet j are depicted in automaton pjet j of Fig. 5.2 T he set of states of pjet j represent s all the possible locations of the jet i.e. the four airports D, A, B, and C respectively and the set of transitions i.e. the events represent the flight s of jet j between the corresponding airports (e.g. jDA means that j is in flight from the depot D to airport A)
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74 Fig. 5.2 Automaton pjet j the possible locations and flights of jet j In modeling systems with depot(s), where fleet starts and ends its operation, specific automata must assure that all the allowed sequences of transitions of the fleet (events) start and end at the state representing the depot(s). To model a system with co nstraints on the length of vehicle routes, automata of vehicle behavior should limit the number of the possible consecutive transitions. For example in the air taxi system described in Fig. 5.1 depot and each jet is allowed max imum three flights per trip, the automaton trip j in Fig. 5.3 guarantees that all the flights start and end at D and jet j performs a t most three flights per trip. State 0 represents j being located at depot D, state 1 all the possible locations of j aft er the first flight from D, and state 2 all the possible locations of j after one more flight. Since ea ch sequence of transitions ends at state 0 the max allowed flights are three.
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75 Fig. 5.3 Automaton trip j the maximum allowed flight within a trip. If vehicles are subject to unpredicted stoppages like in traffic jam s or breakdown s, the events that lead to these states are to be introduced in the plant model For example in modeling a land DRT system, if both traffic jams and breakdowns are considered, the automaton veh u st j in Fig. 5.4 describes such a behavior of vehicle j When j is in service (state 1) it may get in a jam (state 3) and after the jam is eli minated it is back in service; if it breakdowns (state 2), after repair it is in initial standby state (state 0). Fig. 5.4 Automaton veh u st j vehicle j in unpredicted stoppages In modeling systems where the vehicles have capacity constraints, the number of passengers on board or loaded cargo units represent different states of the vehicle and the picking up or dropping of a passenger or delivery of a cargo events. In
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76 the air tax i system examp le, if the seating capacity of jet j is two passengers, the automaton cap j in Fig. 5.5 limits the possible pickups and drops off. State 0 represents the jet without passengers on board states 1 and 2 represent the jet with 1 and 2 passengers on board, respectively. Fig. 5.5 Automaton cap j jet j may pickup a t most two passengers If a priorit ization in the service of a group of passengers is needed a group of automata should impose that the service of the rest of the passengers starts after all the passengers with priority have been served. For example the automaton prior i in Fig. 5.6 assures that all the reassigned passengers (event ras ij ) are helped before the remaining passengers that have to be assigned (event ac ij ) for first time Fig. 5.6 Automaton prior i gives priority of reassigned passengers
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77 5 3 Illustrative e xample of a s mall a ir charter s ervice o peration In this s ection we dev e lop a DES model that provides nonblocking behavior of a DRT system, capable of making real tim e decision s regard ing the acceptance of passenger request s The model can also cover the case of service with minimum possible fleet size, i.e. a new vehicle is being acti vated only if none of the currently active vehicles can meet a particular service request In addition, there is a constraint on the length of vehicle service operation during a working shift. An example of a destination specific DRT system is used. It is based on DDAFP defined by Cordeau at al. ( 2004) The system is a small air taxi operator providing on demand air charter service. Such a business encounters an increased interest because of its ability to quick ly respond to the e service. The modeling of this type of service is close to D DARPMADO operations studied by Sadeh and Kott ( 1996 ), and to a large group of emergency and rescue air logistics problems, Shen, Dessouky and Ordonez (2005) The service of an air taxi operator is similar to the emergency ARE problem in the following characteristics: high dynamics of operations that requires immediate decision about the feasibility of a request and real time update of jets routings and schedules ; limited jet capacities with small number of seats or beds; limited length of flights; possible closures of some airports causing unpredicted changes of the flights. The dissimilarities are that the origins of the requests belong to a set of airports in the air taxi service and c ould be anywhere in the covered region in ARE environment, and the available jets are not subject to change or breakdown during service
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78 An air taxi service operates over a given set of airports, which implies that flight and schedule optimizers can be su ccessfully applied. seat on except there are no fixed schedules. 5 3 .1 Problem d escription of a small air Consider an air charter DRT system which covers the demand over a fixed set of four airports ( P = 4) by means of a homogeneous fleet of jets (Fig. 5.1 ). A jet may fly from any to any other airport. One of the airports (D) serves as a depot, where all the jets are kept and after the end of their services must return. The fleet consists of very light jets (VLJ) with seating capacity of two passengers. The system receives randomly initiated passenger r equests ( N is the current number of passengers to serve) with origin and destination locations, and provides real time answers i.e. the dispatcher must decide in real time whether the system can serve a particular request assign t he passenger to a jet, and route or reroute that jet. To formalize the length of service of a jet, we define a flight of a jet within the system to be the route from one airport to another; a trip of a jet to be a sequence of flights which starts and ends at D. To incorporate the limits of pilot duty, VLJ flight range, etc. the following constraint s are included : At most two intermediate stops are allowed during a trip; A jet may complete up to one trip through a working shift. Hence a working shift (i.e. a trip) may include up to three flights.
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79 Let at the beginning of a shift the system receives a request from passenger 1 who wants to fly from airport A to airport C The control procedure needs to compute the possible behavior of jet 1 so that passenger 1 will be picked from its location and transported to the desired destination. 5 3 .2 DES modeling of a small air charter DRT system The set of all the events of the considered system is summarized in T able 3 The pickup and drop off events have two indexes representing the number of the passenger and the number of the jet serving that passenger. The first event is controllable (can be controlled by the operator), while the second one is uncontrollable (whethe r a passenger will reach the final destination depends on airport condition, flight condition, etc. all uncontrolled). Each flight is labeled as a combination of a digit followed by two letters. The digit represents the number of the jet and the letters the origin and the destination correspondingly. All flights are considered as controllable events. Table 3 The set of all the events of the small air charter Process Events c: controllable; u: uncontrollable demand service pick ij Passenger i picked with jet j c drop ij Passenger i transported with jet j u Flights jDA Jet j flies from D to A c jDB Jet j flies from D to B c jDC Jet j flies from D to C c
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80 Table 3 (Continued) jAB Jet j flies from A to B c jAC Jet j flies from A to C c jAD Jet j flies from A to D c jBA Jet j flies from B to A c jBC Jet j flies from B to C c jBD Jet j flies from B to D c jCA Jet j flies from C to A c jCB Jet j flies from C to B c jCD Jet j flies from C to D c 5 3 .2.1 Computation of centralized supervisor We apply the procedure of S ection V.2 and develop the following three model s: P lant model : the plant consists of two automata pjet 1 (Fig. 5.7 ) model s the possible behavior of jet 1 and p ass 1 (Fig. 5.8) describes the behavior of passenger 1 Fig. 5.7 Automaton pjet 1
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81 The set of states Q and the set of events of pjet 1 are the same Q and f or automaton pjet j of Fig.5.2. Fig. 5.8 Automaton p ass 1 In automaton pass 1 p assenger 1 release s a service request at the initial state 0 next it is picked by jet 1 (state 1) and jet 1 drops off passenger 1 (state 2) In this case the plant is obtained by parallel composition of the two automata i.e. S pecification model s: two specifications are considered automaton trip 1 (Fig. 5.9 ) ensures that jet 1 will make up to three flights, and automaton paspd 1 (Fig. 5.10 ) specifies after which flights passenger 1 can be picked and dropped off. Fig. 5.9 Automaton trip 1 The states and events of trip 1 are analogous to the state and event sets of a utomaton trip j of Fig.5.3 In the trip 1 automaton the selfloops (not shown) are adjoined to each state and account for the events that are irrelevant to the specification, but may be
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82 executed in the model. In the graphs of the automata of this section denotes the set of all flights of jet 1 i.e. Fig. 5.10 Automaton paspd 1 In state 0 of paspd 1 jet 1 can fly from the depot D to any location and passenger 1 is at airport A. Only the flights that end at airport A allow the jet to get to the passenger (state 1) and after picking them up (state 2) jet 1 can fly to any location. The flights that end at airport C take the system to state 3, and after dropping off passenger 1 at its destination, the system reache s in the marked state 4. The event sets of paspd 1 1 and 2 denote the flight sets and respectively. The automaton represents the synchronization of both specification automata. It has 12 states and 33 transitions. Synthesis of the centralized supervisor : the intersection of languages marked by Plant 1 and Spec 1 automata provides the centralized supervisor ( CS 1 ), i.e. The described three steps of the procedures are performed with XPTCT software developed by Systems Control Group in the Dept. of Electrical & Computer Engineering at University of Toronto, ( Design Software: XPTCT) The computed CS 1 is controllable with 5 states and 5 tr ansitions: i.e. starting from the initial
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83 position at depot D (state 0) jet 1 must fly to airport A (state 1) pick passenger 1 (state 2) flies to airport C (state 3) drop s off the passenger (state 4 ) and flies back to depot D Fig. 5.11 Fig. 5.11 Supervisor CS 1 5 3 .2.2 Computation of modular supervisor Let at the current time instant jet 1 is at airport A pick ing passenger 1 and a new request is received: passenger 2 wants to fly from airport B to airport D The problem consists of making an immediate decision if it is fe asible to accept the new request given the available resources ( active jet 1 ) and the existing schedule. In the case of the small air charter system, t he control procedure would check if the ac tive jet will be enough to meet the demand and if not the scheduling and routing for two jets should be developed Thus, a new jet is introduced in the system jet 2 To illustrate the modularity of the supervisory synthesis, the three steps of the procedure will be developed in such a way, that two supervisors will be synthesized one controlling system operation with jet 1 only, and one controlling service with both jet 1 and jet 2
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84 Plant model is the synchronization of the following two pairs of automata : (Fig. 5.12 a and 5.12 b ) model s the possible flights of jet 1 and jet 2 respectively Automaton pjet 1 is updated with the current location of jet 1 airport A. Thus, the initial state of pjet 1 at this step is 1, not 0, i.e. a) Automaton pjet 1 b ) Automaton pjet 2 Fig. 5.12 Automata Fig. 5.13 depicts the parallel synchronization of automata pjet 1 and pjet 2 The flights of jet 1 are in continuous line and the flights of jet 2 in dash. To avoid obscurity only the flights in the first row and column are label ed.
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85 Fig. 5.13 Parallel s ynchronization of automata pjet 1 and pjet 2 (Fig. 5.1 4 a and 5.1 4 b), where pass 1 encounters that passenger 1 is to be picked by jet 1 and passenger 2 can be picked by any jet. a) Automaton pass 1 b ) Automaton pass 2 Fig. 5.1 4 Automata
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86 After synchronization, the plan automaton of this case is obtained: has 1 44 states and 1152 transitions. The s pecifications of this case are modeled with the following two pairs of automata: (Fig. 5.15 a and 5.15 b), where trip 1 is updated with the current position of jet 1 and encounters that jet 1 has two flights left i.e. at state 0 jet 1 is at airport A, at state 1 it is either at B or C and at state 2 it is back at depot D Automaton trip 2 is analogous to trip j from Fig. 5.9. a) Automaton trip 1 Here b ) Automaton trip 2 Fig. 5.15 Automata
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87 paspd i ( i = 1, 2) (Fig. 5.16 a and 5.16 b), paspd 1 is updated with the current position of jet 1 and paspd 2 covers the possibilities that passenger 2 can be picked by either jet 1 or jet 2 a ) Automaton paspd 1 Recall that and In addition, , b ) Automaton paspd 2 Fig. 5.16 Automata paspd i ( i = 1, 2) The state space and transitions of the upper rung of paspd 2 cover the case when passenger 2 is picked and dropped off by jet 1 and the bottom rung consider the possibility
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88 that passenger 2 is transported by jet 2 Each of these run gs has three intermediate states (1 3 5 or 2 4 6, respectively) and 53 transitions. Synthesis of the modular supervisor : First, the planning procedure may check if jet 1 can transport both passengers. The required specification for that case Spec 2 is computed with the product of automata trip 1 paspd 1 and paspd 2 i.e. The supervisor MS 1 is obtained as the intersection of Plant 1 and Spec 2 i.e. (fig. 5.17 ). It has 19 states and 18 transitions. Fig. 5.17 Supervisor MS 1 However, t he re is no marked state on the graph. When the system gets to state 12 jet 1 arrive s at depot D and has drop ped off only passenger 2 and when at state 15 jet 1 arrive s at D and has drop ped off only passenger 1 Thus, jet 1 can not transport both
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89 passengers in one shift T o meet the demand the service provider has to use one more jet jet 2 To compute the specification when jet 2 is introduced, the control procedure may use that passenger 1 which is already picked by jet 1 is to be transported by the same jet hence, passenger 2 should be picked by jet 2 Thus, one module of specifications is for passenger 1 jet 1 coordination depicted in Fig. 5.18 and another specification for passenger 2 jet 2 Fig. 5.19 Fig. 5.18 Specm 1 synchronization of jet 1 and passenger 1
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90 Fig. 5.19 Specm 2 synchronization of jet 2 and passenger 2 The modular supervisor s for transportation of passenger 1 S P 1 and for transportation of passenger 2 SP 2 are computed with the intersection s of Plant 2 with Specm 1 (Fig. 5.18) and Specm 2 (Fig. 5.20), respectively Specm 1 provides the only possible way to complete service of passenger 1 and its supervisor is the same as the specification, i.e .
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91 Fig. 5.2 0 Language SP 2 Since both supervisors S P 1 and S P 2 have no common transitions, the modular supervisor MS 2 is their union, It has 23 states and 22 transitions. As a comparison, the centralized supervisor for transportation of two passengers by two j ets calculated with XPTCT software has 55 states and 114 transitions.
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92 5 .2.2. 3 Computational complexity of supervisor synthesis L et the plant generator G be modeled with r states and two supervisors S 1 and S 2 with p 1 and p 2 states respectively, jointly control the system. Cassandras and Lafortune (1999) discuss the significant computational and memory savings of modular control. T he supervision of G can be interpreted as the product If the centralized supervisor is built, we need to store totally states and in modular control only states. In the worst case the computational complexity for centralized supervisor synthesis is and for the modular supervisor. Consider the general case when at a given state of the DRT system the available m number of jets are supposed to transport n passengers. Let the seating capacity of the jets is two passengers and T here will be m 0 jets that have not assigned passengers, m 1 jets that have one passenger assigned, and m 2 jets that have two passengers assigned, i.e. P assengers can be split into two groups: n 0 that are not assigned yet and n 1 that are already assigned to any jet, i.e. Obviously, Thus, for the number of the unassigned passengers n 0 we have :
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93 For all n 0 requests the control procedure is to check for feasibility of service first with the m 1 jets. As we saw in V. 3 .2. 2, this is done as a product of automata trip j and paspd i However, the numbers of states and transitions of automata trip j and paspd i depend on the current location and the number of remaining flights of jet j If jet j is at the depot D and can make a 3 flight trip (e.g. trip 2 of Fig. 5.15 b ), automaton trip j will have four states and 18 transitions plus a selfloop of transitions at each state If jet j is at an airport and has two flights remaining in its trip (e.g. trip 1 of Fig. 5.15a) then trip j will have 3 states and 5 transitions plus a selfloop of transitions at each state. For each unassigned passenger i (from 1 to n 0 ) the corresponding automaton paspd i will have rungs of three states and each of them will generate additional transition s and a selfloop of transitions for each state. In addition, automata cap j of Fig 5.5 with 3 states and transitions should be used to secure that up to two passengers will be assigned to each jet In any real case combination of n 0 n 1 m 0 and m 1 the product of these automata will be large enough to cause computational complexity in the synthesis of the modular supervisor. Thus, we need a procedure that will limit the check for a feasible jet for every new service request. In the next chapter we present such a method, based on decentralized supervisory control.
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94 Chapter Six Decentralized Supervisor y Control of Concurren t DES In S ection 4 .2.4 we introduced the decentralized supervisory control (D SC ) with three modeling architectures and i n S ection 4 .2.5 the nonblocking conditions of the decentralized control architectures were presented. In this chapter we consider the decentralized control of DES with specialization to local supervisory control ( SC ) and concurrent systems. The advantages of the D SC are illustrated in a n example of emergency ARE DRT service. 6 1 Decentralized control of concurrent DES s Concurrent DESs are defined as collection s of components (subsystems) that perform simultaneously and may interact with each other. Consider a DES G composed of n concurrent subsystems G i with event sets Suppose that for each subsystem G i there is a local supervisor S i that observes and controls only the events of The global controlled DES can be obtained as the concurrent operation of the locally controlled subsystems Th us, th e problem of decentralized control of concurrent DESs is to find the conditions under which local synthesis and control for any
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95 specifications of G i do not result in loss of optimality compared to the global supervisor control and control of one subsystem G i never incurs blocking in the other subsystem G j Recall from the Controllability theorem introduced in S ection 4 .2.2 that controllability of the language of the desired behavior is a necessary and sufficient condition for the existence of a supervisor that achieves this behavior for a given DES under the complete observation of the events. In the case of decentralized control when there are n local supervisors observing and contro lling their corresponding set s of events Cieslak at al. (1988) introduce d the condition of co observability if the controlled behavior is given as a prefix closed language. Lafortune at al. (2001) relax ed the conditions of co observab ility for the existence of local supervisor s in the conjunctive, disjunctive and general decentralized architectures. Willner and Heymann (1991) introduce the notion of separability of a language L is said to be separable with respect to (w.r.t.) if there exists a set of languages called a generating set of L such that For a finite set of languages the parallel composition of denoted is defined as Recall, is the natural projection Consider a set of concurrent DES s with event partitions such that ( eq. 6.1)
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96 This assumption means that there is no synchronization between the uncontrolled events of the systems. Willner and Heymann (1991) prove that separability under assumption ( eq. 6.1) is a necessary and sufficient condition that guaranties that the decentralized control can achieve the optimal behavior of the centralized supervisor. Since their work is close ly related to our method, we briefly review it in the remaining part of this section. The model of the global system G is defined by where (with ) and is given by ; We denote for the entire (global) system. Thus, in G an event that belongs to exactly one subsystem can occur asynchronously and independently. If an event belongs to several subsystems, it must occur simultaneously in all of them, in order to occur in the composit e system. I t follows that i f then In particular, if are all disjoint, then G is the shuffle product of G i (see section IV.2.4). Recall from section 4 .2.2 that a global supervisor S achieves optimal ( i.e. less restrictive) behavior of G under the controlled specification C by synthesizing the language In the case of concurrent systems, where each subsystem G i is controlled by its local supervisor S i the concurrent operation
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97 of all controlled subsystems generates a new global system G gl namely The following theorem (Theorem 4.1 in [27]) gives the conditions under which the concurrent control scheme achieves the optimal global behavior. Theorem 1 : Let a global DES where There exist local supervisors S i which observe and control only the events of of each G i such that if and only if K is separable w.r.t. The proof of Theorem 1 is given in Willner and Heymann (1991) The authors introduce an algorithm of polynomial complexity for checking the separability of a language K when the subsets are pairwise disjoint. Since our method is similar to this algorithm, here we introduce it in brief. Let be a deterministic automaton with m states that accepts a language K and are pairwise disjoint sub set of event s et Algorithm 1 (Algorithm 4.1 in [ 36 ]) : (1) For each construct the automaton as defined in step 2. (2) For each pair where and define and define
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98 (2a) Construct the product automaton Define to be the set of all states such that is an accessible state in i.e. (2b) If there exists such that then stop. (3) If all the pairs were checked, then stop. Else repeat step (2) for another pair The concept of Algorithm 1 is as follows. For each pair is the set of all strings such that and and is the set of all such that By constructing the algorithm identifies the set which is the set of all states such that there exists which satisfies and and If then there exists such that and which contradicts separability. The complexity of Algorithm 1 is 6 2 Decentralized supervisor of separate groups of vehicles passengers To avoid the discussed increases of the state space and number of transitions in section 5 .2.2. 3 a decentralized approach of supervisory control can be applied for a D RT system split in separate subsystems ( groups ) of vehicles and passengers. Let al l the n passengers and m vehicle s are split in disjoint groups, such that each group of passengers is to be served only by the ir designated group of vehicles Fig.6.1 depicts the case when groups of two passengers are to be served by two vehicles
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99 Fig. 6 1 DES split in subsystem s of vehicles and passengers In this case the local supervisors of each subsystem are easily computed similarly to computation o f MS 2 in section 5 .2.2.2. Since the event sets of the groups are disjoint, the d ecentralized supervisor of the global DES will be the union of all the local supervisors of each group: The re are two main advantage s of such a decentralization: very limited state space and number of transitions for each local supervisor, and all the local supervisors can be computed in parallel. However, with the decentralized architecture of separate groups if the vehicles are designated only to one group of passengers, some of them will not be utilized with full capacity, e.g. can have assigned one passenger (or generally less than their seating capacity) and thus, the global supervisor is not optima l. As Leduc at al. (2005) discusses, this is the price we have to pay for the advantages that the approach offers. To avoid the possibility of underutilization of the vehicles and thus using the smallest possible fleet, we develop a D SC of dynamic subsystem s of vehicles and passengers. Every vehicle with its assigned passenger(s) is a subsystem of the global DES
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100 and is controlled with its local supervisor. With every new request, all the local supervisors check if their vehicles can serve the new pa ssenger. A new vehicle is to be involved only if n one of the active vehicles can serve received request. In the next section we demonstrate the method with an example for control of DRT system for emergency evacuation. 6 3 Illustrative Example of ARE Ser vice in D DARP MADO Environment In the present section we develop a D SC model capable for real time nonblocking control of a DRT system offer ing emergency service in ARE environment. The system operates under D DARPMADO conditions over a region of natural or man made disaster providing emergency evacuation of passengers from their origins to specific destinations (MTFs) as defined by Sadeh and Kott ( 1996 ). The DRT system is modeled as a global DES system consisting of a set of conc urrent subsystems. Each subsystem is a DES modeling the behavior of a vehicle ( e.g. a helicopter or a VLJ) and its assigned passenger(s) The local supervisor s (LSs) of the particular subsystem s are capable in real time decision making of accepting passeng er requests and routing or rerouting the vehicle s If there is no interaction between the vehicles and passengers, the event sets of all the subsystems are disjoint and the global supervisor (GS) of the entire system is constructed as a conjunction of all i.e. where M is the number of the vehicles Fig. 6. 2
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101 Fig.6. 2 Structure of the global system and local control Similar to the model of S ection 5 .3.2 this model controls the service with minimum possible fleet size, and satisfie s the same constraint on the length of vehi cle service during a working shift. The emergency evacuation DRT service in ARE environment differs from the air charter service in the following characteristics: Vehicles do not have specific depots to be kept, i.e. they may stand by at any MTF and do no t have to conclude their service at a given depot ; Some of the MTFs can be closed during service and the vehicles with passengers whose destinations are closed should be redirected to other ones; Some of the passengers may have more than one possible desti nation, i.e. they may be transported to e ither one of two different MTFs. The common features of both problems are high dynamics of operations that requires immediate decision about the feasibility of a request and real time update of jets rou t ings and schedules; limited car riage capacities with s mall number of seats or beds; li mited length of flights the available vehicle s are not subject to change or breakdown during service LS 1 LS 2 LS M M GLOBAL DRT SYSTEM
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102 The most challenging question of D DARPMADO problem is the set of the possible origins of the service requests they may belong to a large but finite set of location s or could be any point in the covere d region. In the next sections we solve the problem with a finite set of origins. In Chapter Seven we discuss th e challenges and possible way s to solve the problem with infinite many origins of requests. 6 3 .1 Problem description of ARE Service in D DARP MADO e nvironment Consider a DRT system which covers the demand for emergency evacuation of people over a reg ion R with a fleet of four jets (Fig. 6 3 ). T here are five origin locations in R where passengers can release service requests and can be picked up and three MTF destinations Fig. 6. 3 Region R with 5 origins and 3 destinations
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103 Let t he fleet consists of vehicles (VLJ s and/or helicopters) with limited seating capacities vehicles1 2 and 4 can accommodate two passengers and vehicle 3 three passengers The system receives randomly initiated passenger requests for transportation from one of the origins to some of the destinations. Because of the limits of the software used for verification of the model (XPTCT software) we will review the modeling of only the first nine requests i.e. In this model we keep t he same definitions of a flight of a jet and a trip of a jet as in the model in section 5 .3.2 In addition, the same constraints of at most two intermediate stops during a trip and up to one trip through a working shift per vehicle are to be satisfi ed Thus, a working shift (i.e. a trip) include s up to three flights. However, i f the destination facility of some of the passengers on board a given vehicle is clo sed, then the vehicle is assumed to have a traveling resource to make an emergency flight to another MTF i.e. destination. The main difference of the two models is in the way the ir SC s are implemented. In the model of section 5 3.2 the centralized and mod ular SC were computed. With this model we demonstrate the synthesis of distributed SC of concurrent systems. Each of the vehicles and its assigned passengers form a subsystem which performs concurrently with the other subsystems of the rest of the fleet an d their passengers. Since there is no interaction between the vehicles and the passengers assigned to different vehicles, the DSC of the sub system s is conjunctive. To utilize a minimum number of vehicles, with every new released request the procedure checks if any of the activated vehicles can be assigned to th at passenger If
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104 adding the origin and destination locations in the route of a vehicle with enough seating capacity do es not violate the constraints, the LS of that subsystem provides t he updated language of desired behavior of these vehicle and passenger. If the control procedure finds the updated language of the LS j of the subsystem of some vehicle j to be feasible, i.e. that vehicle can accommodate the request a nd the passenger is assigned to vehicle j There is no need for the procedure to check for the rest of LS j Let at the beginning of the working shift vehicle 1 and vehicle 2 are positioned at F 1 vehicle 3 is at F 2 and vehicle 4 is at F 3 Let the DRT system follows some simple rules for the initial activating of the vehicles: if a request is released from either O 1 or O 2 and th ere is no active vehicle, vehicle 1 is activated; if the request comes from either O 3 or O 4 vehicle 3 is activated, and if the request is released from O 5 vehicle 4 is activated A t the beginning of the working shift the system receives a request from passenger 1 who need s to be transported from O 1 to F 1 Since there are no active vehicles, vehicle 1 is activated. The control procedure needs to compute the possible behavior of vehicle 1 so that passenger 1 will be picked from its location ( O 1 ) and transported to the desired destination ( F 1 ) 6 3 .2 DES modeling of a small emergency DRT system in D DARP MADO enviro nment The set of all the events of the considered system is summarized in Table 4 Any vehicle can be in active state or waiting at a MTF. The assignment, pickup drop off and emergency drop off events have two indexes i represents the number of the
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105 passenger and j the number of the vehicle serving that passenger. Facilities can be open or closed Each flight is labeled as a combination of a digit followed by two letters. The digit represents the number of the vehicle and the letters the origin a nd the destination correspondingly. There are two un controllable events when a vehicle lands at a facility and ( atf j ) and when a facility is closed ( closed k ) Table 4 The set of all the events of a small emergency DRT system Process Events c: controllable; u: uncontrollable Vehicle status act j Vehicle j is in service c atf j Vehicle j is landed at a MTF u demand service pasgn ij Passenger i assigned to vehicle j c pick ij Passenger i picked by vehicle j c drop ij Passenger i dropped by vehicle j c edrop ij Passenger i dropped in emergency by vehicle j c Facility status open k MTF k is open for passenger acceptance c closed k MTF k is closed for passengers u Flights j F 1 O 1 Vehicle j flies from F 1 to O 1 c j F 1 O 2 // from F 1 to O 2 c j F 2 O 3 // from F 2 to O 3 c j F 2 O 4 // from F 2 to O 4 c j F 3 O 5 // from F 3 to O 5 c
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106 Table 4 (Continued) j O 1 O 2 // from O 1 to O 2 c j O 1 O 3 // from O 1 to O 3 c j O 1 O 4 // from O 1 to O 4 c j O 1 O 5 // from O 1 to O 5 c j O 1 F 1 // from O 1 to F 1 c j O 1 F 2 // from O 1 to F 2 c j O 1 F 3 // from O 1 to F 3 c j O 2 O 1 // from O 2 to O 1 c j O 2 O 3 // from O 2 to O 3 c j O 2 O 4 // from O 2 to O 4 c j O 2 O 5 // from O 2 to O 5 c j O 2 F 1 // from O 2 to F 1 c j O 2 F 2 // from O 2 to F 2 c j O 2 F 3 // from O 2 to F 3 c j O 3 O 1 // from O 3 to O 1 c j O 3 O 2 // from O 3 to O 2 c j O 3 O 4 // from O 3 to O 4 c j O 3 O 5 // from O 3 to O 5 c j O 3 F 1 // from O 3 to F 1 c j O 3 F 2 // from O 3 to F 2 c j O 3 F 3 // from O 3 to F 3 c
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107 Table 4 (Continued) j O 4 O 1 // from O 4 to O 1 c j O 4 O 2 // from O 4 to O 2 c j O 4 O 3 // from O 4 to O 3 c j O 4 O 5 // from O 4 to O 5 c j O 4 F 1 // from O 4 to F 1 c j O 4 F 2 // from O 4 to F 2 c j O 4 F 3 // from O 4 to F 3 c j O 5 O 1 // from O 5 to O 1 c j O 5 O 2 // from O 5 to O 2 c j O 5 O 3 // from O 5 to O 3 c j O 5 O 4 // from O 5 to O 4 c j O 5 F 1 // from O 5 to F 1 c j O 5 F 2 // from O 5 to F 2 c j O 5 F 3 // from O 5 to F 3 c Emergency flights j eF 1 F 2 Emergency flight of vehicle j from F 1 to F 2 c j eF 1 F 3 Emergency flight of vehicle j from F 1 to F 3 c j eF 2 F 1 Emergency flight of vehicle j from F 2 to F 1 c j eF 2 F 3 Emergency flight of vehicle j from F 2 to F 3 c j eF 3 F 1 Emergency flight of vehicle j from F 3 to F 1 c j eF 3 F 2 Emergency flight of vehicle j from F 3 to F 2 c
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108 6 3 .2.1 Computation of the supervisor of one vehicle one passenger ( LS 11 ) Formalization of the plant model : having only one passenger at O 1 we use vehicle 1 by default. T he plant consists of the following three automata: pasn 1 (Fig. 6. 4 ) represents the possible behavior of passenger 1 it releases its service request (state 0) passenger 1 is assigned to vehicle 1 (state 1), passenger 1 is picked by vehicle 1 (state 2) and dropped off (state 3). F ig.6. 4 Automaton pasn 1 pveh 1 (Fig. 6. 5 ) presents the possible behavior of vehicle 1 Fig. 6. 5 Automaton pveh 1 In this chapter, denotes all the flight events of vehicle 1 i.e. and denotes all the flights of vehicle 1 ending at any
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109 MTF, i.e. After vehicle 1 is activated from its stand by position at F 1 (state 1), it can fly either to O 1 or O 2 (states 2 and 3, respectively). Any new flight except those ending at the MTFs keeps the vehicle in these states. When a flight from is executed, the system i s in state 4, and if vehicle 1 l a nds at a MTF, the system is in state 5. f d 1 (Fig. 6. 6 ) coordinat es the flight s wit h which vehicle 1 ends its trips. For example, the vehicle may lend at any MTF through O 1 e.g. if it has visited O 1 with the previous flight e.g. one of these flights has been executed: Fig.6. 6 Automaton f d 1
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110 From the initial state (0), with every possible flight of vehicle 1 that goes to O 1 the system gets to state 1, and every flight of vehicle 1 that goes to O 2 takes the system to state 2. Similarly, every flight between all the five origin location s takes the system to the corresponding state e.g. state 5 represents that vehicle 1 is in O 5 From any state 1 to 5 vehicle 1 can fly to any MTF, thus bringing the system to the marked state 6. Automaton f d 1 becomes necessary in modeling the emergency DRT system because in automaton pveh 1 which describes the possible behavior of vehicle 1 all the flights among the origin locations ar e modeled with two states 2 and 3 and all the flights to the MTFs take the system to one state 4. This simplicity in representation of the possible flights does not consider where exactly the vehicle is, like in the small air charter model (section V.2 .2) and reduces the state space. However, the price for it is the necessary additional automaton to secure that after all the flights the vehicle gets to the final MTF with the correct sequence of flights. Thus, the plant automaton of the model is the pa rallel composition of three automata i.e. It is comprised of 56 states and 338 transitions. Formalization of the specifications: the following three automata specify the desired behavior of vehicle 1 and passenger 1 : S imilarly to the small air charter model, we use an automaton trips 1 (Fig. 6. 7 ) to ensure that vehicle 1 makes up to three flights per trip.
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111 Fig. 6. 7 Automaton trips 1 vehdil 1 1 (Fig. 6. 8 ) en sures the diligent service of vehicle 1 i.e. passenger 1 can be assigned to vehicle 1 if the vehicle is activated or not ( state 0 ), and if it is not, it must be activated, (state 2 ), next passenger 1 must be picked up (state 3 ), after vehicle 1 gets at the facility (state 4 ), passenger 1 can be dropped off (state 5 ). Fig. 6. 8 Automaton vehdil 1 1 paspd 1 (Fig. 6. 9 ) specifies that vehicle 1 can pick up passenger 1 right after a flight to O 1 (state 1), and passenger 1 can be dropped off when the vehicle gets to F 1
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112 Fig. 6. 9 Automaton paspd 1 1 With we denote all the flights of vehicle 1 which go to O 1 i.e. and denotes all the flights of vehicle 1 which end up in F 1 i.e. The cross product of trips 1 vehdil 1 and paspd 1 generates the specification automaton, i.e. which consists of 24 states and 132 transitions. Synthesis of the supervisor of vehicle 1 passenger 1 ( LS 11 ): the intersection of the languages marked by and automata produces LS 11 (Fig. 6. 10 ) i.e. Again, we use the XPTCT software to compute all the languages of the automata in the three steps.
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113 Fig. 6. 10 Automaton of LS 11 The synthesized LS 11 is controllable with 15 states and 19 transitions Starting from the initial state (0), passenger 1 has to be assigned to vehicle 1 (state 1), after vehicle 1 is activated (state 2), there are two possible routes through O 1 (states 3 5 7, 8, 9, 10 12) and through O 2 (states 4 6 11 12). In either way, pass enger 1 is picked up when the vehicle is at O 1 (states 3 or 11) and when vehicle 1 gets to F 1 (state 12) it is at facility (state 13). At the facility passenger 1 can be dropped off (state 14), marked state. 6 3 .2. 2 Computation of the supervisor of one vehicle two passengers ( LS 12 ) Let at a given time instant passenger 1 is assigned to vehicle 1 vehicle 1 is activated and is tak ing off from F 1 when another service request arrives passenger 2 has to be transferred from O 3 to F 2 The operational planning procedure is to check if the active
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114 vehicle 1 is capable to meet the second request. If it is, then passenger 2 has to be assigned to the same vehicle, and if not, a new vehicle is to be activated. Formalization of the plant mode l : with two passengers the plant consists of the following three automata: pasn 12 (Fig.6. 1 1 ) describes the updated behavior of passenger 1 being already assigned it has to be picked and dropped off; and pasn 2 (Fig.6. 1 2 ) describes the possible behavior of passenger 2 Fig.6. 1 1 Automaton pasn 12 Fig.6. 1 2 Automaton pasn 2 pveh 12 (Fig.6. 1 3 ) describes the updated possible behavior of vehicle 1 the new initial state is at F 1 Fig. 6. 1 3 Automaton pveh 12
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115 there is no change in automaton f d 1 (Fig. 6. 6 ) that coordinates the flights w it h which vehicle 1 ends its trips. Thus, the plant is computed with the parallel composition of the four automata, i.e. Formalization of the specifications: the following five automata specify the desired behavior of vehicle 1 and both passengers : trips 12 (Fig. 6.1 4 ) is the updated automaton of trips 1 and ensures that vehicle 1 has no more than two flights remaining. Fig. 6.1 4 A utomaton trips 12 two automata (Fig. 6.1 5 ) secure diligent service for both passengers by vehicle 1 vehdil 1 1 (Fig. 6.1 5 a ) is the updated automaton of vehdil 1 and encounters the remaining events that must be executed for service of passenger 1 and vehdil 21 (Fig. 6.1 5 b ) is the corresponding automaton to ensure diligent service for passenger 2 by the same vehicle.
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116 a) A utomaton vehdil 1 1 b) A utomaton vehdil 21 Fig. 6.1 5 Automata two automata (Fig. 6.1 6 ) are needed to specify when each passenger can be picked and dropped off by vehicle 1 : paspd 11 (Fig. 6.1 6 a) covers the picking and dropping of passenger 1 since it is assigned but not picked yet and vehicle 1 is activated the change compared with paspd 1 from (Fig. 6. 9 ) is the elimination of events act 1 and pasgn 21 and a selfloop that consider s the necessary events for the service of the other passenger ; paspd 21 (Fig. 6.1 6 b ) is the corresponding automaton for passenger 2
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117 a ) Automaton paspd 1 1 b) A utomaton paspd 21 Fig. 6.1 6 Automata With we denote all the flights of vehicle 1 that end at O 3 i.e. and denotes all the flights of vehicle 1 that end at F 2 i.e. Thus, the specification of the service of the two passengers with one vehicle is computed with the cross product of trips 1 2 vehdil 1 1 vehdil 21 paspd 1 1 and paspd 21 i.e.
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118 Synthesis of the supervisor of vehicle 1 passenger 1 and passenger 2 ( LS 12 ): the intersection of the languages marked by and automata produces LS 12 i.e. However, LS 12 is empty (i.e. it has zero states and zero events) because it is infeasible for a vehicle to visit two different origin locations ( O 1 and O 3 ) and two different destinations ( F 1 and F 2 ) in one trip. Therefore, we need another vehicle to be involved by default it will be vehicle 3 currently located at MTF 2 6 3 .2. 3 Computation of the local supervisor of one vehicle one passenger ( LS 32 ) As vehicle 3 is getting involved, a need arises for another supervisor LS 32 Since at this moment passenger 2 will be the only passengers of vehicle 3 LS 32 will be analogous to LS 11 one vehicle one passenger. Here w e briefly de scribe the synthesis of LS 2 3 to demonstrate the difference in the notations and indexes The s ynthesis of the plant is the parallel composition of the following three automata: p asn 23 (Fig. 6.1 7 ) represents the possible behavior of passenger 2 Fig. 6.1 7 Automaton pasn 23 pveh 3 (Fig. 6.1 8 ) describes the possible behavior of vehicle 3
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119 Fig. 6.1 8 Automaton pveh 3 Similarly to LS 11 here denotes all the flight events of vehicle 3 i.e. and denotes all the flights of vehicle 3 ending at any MTF, i.e. fd 3 (Fig. 6.1 9 ) coordinates the flights with which vehicle 3 has to end its trips ;
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120 Fig. 6.1 9 Automaton fd 3 Thus, The s ynthesis of the specifications is the cross product of the following three automata: t rips 3 (Fig. 6. 20 ) limits the number of the flights in the trip of vehicle 3 Fig. 6. 20 Automaton trips 3
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121 vehdil 2 3 (Fig.6. 2 1 ) ensures diligent service for passenger 2 by vehicle 3 Fig. 6. 2 1 Automaton vehdil 2 3 paspd 23 (Fig.6. 2 2 ) specifies after which flights passenger 2 can be picked and dropped off by vehicle 3 Fig. 6. 2 2 Automaton paspd 23 Here denotes all the flight events of vehicle 3 going to O 3 i.e. and denotes all the flight events of vehicle 3 going to F 2 i.e. Hence, the automaton of specifications of the service of passenger 2 by vehicle 3 is computed: The local supervisor LS 3 2 (Fig. 6. 2 3 ) of vehicle 3 serving passenger 2 can be synthesized:
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122 Fig. 6. 2 3 Supervisor LS 3 2 6 3 .2. 4 Computation of the local supervisor of one vehicle two passengers ( LS 113 ) Let at the current time instant vehicle 1 has picked passenger 1 passenger 2 is assigned to vehicle 3 which is activated, took off from its initial location F 2 and another service request is received: passenger 3 has to be transferred from O 2 to F 1 or F 2 Now the control procedure checks from all the active vehicles if any of them is capable to meet this request. In the remaining part of this section we will demonstrate that vehicle 1 can transfer passenger 3 without violation of its current routing and sched uling. Considering the current state of vehicle 1 and both passengers, the plant of the subsystem, Plant 113 is composed with the following four automata: a pair of automata (Fig. 6.2 4 ) model the updated behavior of both passengers pasn 1 (Fig. 6.2 4 a) is updated, generating the only remaining event of
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123 service of passenger 1 and pasn 3 (Fig. 6.2 4 b) models the necessary behavior of passenger 3 a) Automaton pasn 1 b ) Automaton pasn 3 Fig. 6.2 4 Automata pveh 1 3 (Fig. 6.2 5 ) describes the updated possible behavior of vehicle 1 Fig. 6.2 5 Automaton pveh 1 3 fd 1 3 (Fig. 6. 2 6 ) synchronizes the all flight events with the necessary end flights to the possible destinations
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124 Fig. 6.2 6 Automaton fd 1 3 Hence, It has 56 states and 438 transitions. The specification automaton of the subsystem vehicle 1 and passenger 1 and passenger 3 Spec 113 is computed with the product of the following five automata: trips 13 (Fig. 6.2 7 ) specifies that up to two flights remain in the trip of vehicle 1 Fig. 6.2 7 Automaton trips 13
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125 two automata (Fig. 6.2 8 ) ensure diligent service of the vehicle for both passengers vehdil 11 (Fig.6.2 8 a ) is the updated automaton vehdil 1 that covers service for passenger 1 and vehdil 1 3 (Fig.6.2 8 b) is the corresponding automaton for passenger 3 a) Automaton vehdil 11 b) Automaton vehdil 1 3 Fig.6.2 8 Automata two analogous automata (Fig. 6.2 9 ) specify when both passenger s can be picked and dropped off since passenger 1 is already picked, paspd 11 (Fig. 6. 2 9 a) covers only its dropping off while paspd 1 3 (Fig. 6. 2 9 b) ensures both picking and dropping of passenger 3
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126 a) Automaton paspd 11 b) Automaton paspd 1 3 Fig. 6.2 9 Automata Thus, It contains 20 states and 123 transitions. The local supervisor of the subsystem, LS 113 (Fig. 6. 30 ) is then computed, i.e.
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127 Fig. 6. 30 Automaton LS 113 One can note that LS 113 is a part of LS 11 (Fig.6. 10 ) with added new states and events for picking and drop off passenger 3 S tarting from state 5 of LS 11 which corresponds to state 0 in Fig. 6. 30 LS 113 assigns passenger 3 to vehicle 1 ( state 1 of Fig. 6. 30 ) then travels to O 2 (states 7 and 2 of LS 11 and LS 113 correspondingly ), picks passenger 3 ( state 3 of LS 113 ), travels to F 1 (states 11 and 4 of LS 11 and LS 113 ), arrives at a facility (states 13 and 5 of LS 11 and LS 113 ), where the passengers are dropped off (states14 of LS 11 and 6 8 of LS 113 ). With the so far developed cases of distributed SC of operation of emergency DRT system in sections 6 3 .2.1 through 6 3 .2.4 we modeled subsystems and obtained the local supervisors of one vehicle one passenger ( LS 11 and LS 32 ) one vehicle two passengers with infeasible operation ( LS 1 2 ) and one vehicle two passengers with feasible service ( LS 11 3 ) In these four cases all the resources (i.e. vehicles and MTFs) were available during the entire operation. Howev er, as it was discussed in S ections 3 .1 and 6 .1, one of the most critical features of the emergency DRT service in ARE environment is that some of the resources may become suddenly unavailable during service. In the next section we
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128 demonstrate the system control in case when one of the MTFs is closed and cannot accept any vehicles to l a nd. 6 3 .2. 5 Computation of the local supervisor of one vehicle two passengers in case of a closed MTF ( LS 448 ) Consider the following possible development of our system passenger 4 and passenger 8 have released service requests for transportation from O 5 to F 1 or F 3 and from O 1 to F 3 respectively. Both have been assigned to vehicle 4 which have been routed from its initial location F 3 to visit O 5 picked passenger 4 from O 5 traveled to O 1 and right after picking up passenger 8 the system receives a signal that the desired destination of vehicle 4 F 3 is closed. In modeling such a scenario we utilize the emergency flight events, which have not been used so far. The plant of the subsystem, Plant 448 is composed as the parallel composition of the following five automata: A pair of automata (Fig. 6.3 1 ) model the possible behaviors of both passengers pasn 4 (Fig. 6.3 1 a) describes the behavior of passenger 4 and pasn 8 (Fig. 6.3 1 b) of passenger 8 respectively a) Automaton pasn 4 b ) Automaton pasn 8 Fig. 6.3 1 Automata
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129 pveh 4 (Fig. 6. 3 2 ) describes the uncontrolled behavior of vehicle 4 Fig. 6. 3 2 Automaton pveh 4 Similarly to the notations in automata pveh 1 and pveh 3 denotes all the flight events of vehicle 4 i.e. In addition, denotes all the flights of vehicle 4 that end at F 1 without the emergency flights, i.e. denotes all the flights of vehicle 4 that end at F 2 without the emergency flights, i.e. and denotes all the flights of vehicle 4 that end at F 3 without the emergency flights, i.e. fd 4 (Fig.6. 3 3 ) ensures that all flight events of vehicle 4 are bound with the necessary terminal flights to the three possible destinations. The emergency flight events keep the system at the marked state 5.
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130 Fig.6. 3 3 Automaton fd 4 fstat 3 (Fig. 6.3 4 ) specifies that F 3 is closed Fig.6.3 4 Automaton fstat 3 Therefore, It has 64 states and 544 transitions. The specification automaton Spec 448 is synthesized with the cross product of the following six automata:
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131 trips 48 (Fig.6.3 5 ) limits the number of the allowed flights in the trip being at O 1 (state 0) vehicle 4 has one more flight to end the trip if it is from the sets and the system gets to state 1, where it is considered that the vehicle is at facility (state 3), if the flight is from set the system gets to state 2, where some emergency flight to F 1 or F 2 must be executed. Fig. 6.3 5 Automaton trips 48 two analogous automata (Fig. 6.3 6 ) ensure the diligent service of vehicle 4 for both passengers. Automaton vehdil 44 (Fig. 6.3 6 a) models the service for passenger 4 and vehdil 4 8 (Fig. 6.3 6 b) for and passenger 8 respectively being already picked (state 0), the vehicle has to get to a MTF (state 1) in order to do drop off or emergency drop off the passengers (state 2). a ) Automaton vehdil 44
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132 b) A utomaton vehdil 48 Fig. 6.3 6 Automata a pair of automata (Fig. 6.3 7 ) specif ies when the passenger s can be dropped off. Automaton paspd 44 (Fig. 6.3 7 a) covers the dropping of passenger 4 and paspd 4 8 (Fig. 6.3 7 b) of passenger 8 respectively Both passengers can be dropped off either at their regular or emergency destination s a) Automaton paspd 44
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133 b) Automaton paspd 4 8 Fig. 6.3 7 Automata fstat 3 (Fig. 6.3 8 ) specifies when vehicle 4 gets at a n open MTF providing F 3 is closed at the initial state 0 the system receives a signal (event closed 3 ) and gets in state 1, next only the flight events from sets and and the emergency flights 4eF 3 F 1 and 4eF 3 F 2 can take the system to state 2, where the vehicle is considered at a MTF and gets to the marked state 3. Fig. 6.3 8 Automaton fstat 3
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134 Therefore, It includes 16 states and 43 transitions. The local supervisor of the subsystem vehicle 4 passenger 4 and passenger 8 LS 448 (Fig. 6.3 9 ) is computed: At the initial state 0 vehicle 4 is at O 1 and is routed to F 3 The two branches going out of state 0 are determined from the exact receiving of the event closed 3 i f it comes before the take off, the system gets to state 1, and if vehicle 4 takes off first, the system gets to state 2. Then the signal for closed F 3 comes during the flight to F 3 (state 5). Fig. 6.3 9 Automaton of LS 448 In the first branch the vehicle may fly to all MTFs, but if it goes to F 1 or F 2 (states 3 and 4, respectively) it is considered at a n open MTF, and can drop both passengers ( passenger 8 is always dropped off in emergency). If vehicle 4 flies to F 3 (state 5), it joins the second branch and has to make one more emergency flight to F 1 or F 2 (state 8) before it gets to a MTF and consecutively drops off the pas sengers.
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135 6 3 .2. 6 Generating the global SC of the emergency DRT W ith the models for LSs of the received passenger requests considered in sections 6 .3.2.1 through 6 .3.2.5 we covered the basic cases of and vehicle routings of the emergency DRT problem described in 6 3 1. Because of the limitations of the applied XPTCT software in terms of the number of states and events we were able to verify the modeling of 9 passeng er requests served with 5 vehicles. Table 5 shows all the requests, their assignments to the vehicles and the controlling LS s. Table 5 Considered requests, assigned vehicles and LS s Request Passenger# Origin destination Facility Assigned vehicle LS 1 passenger 1 O 1 F 1 vehicle 1 LS 113 2 passenger 2 O 3 F 2 v ehicle 3 LS 3259 3 passenger 3 O 2 F 1 or F 2 vehicle 1 LS 113 4 passenger 4 O 5 F 1 or F 3 v ehicle 4 LS 448 5 passenger 5 O 4 F 2 v ehicle 3 LS 3259 6 passenger 6 O 5 F 1 or F 2 v ehicle 2 LS 267 7 passenger 7 O 2 F 2 v ehicle 2 LS 267 8 passenger 8 O 1 F 3 v ehicle 4 LS 448 9 passenger 9 O 4 F 2 or F 3 v ehicle 3 LS 3259 Although dynamically formed, every LS controls a group of a vehicle with its assigned passengers, which does not interact with the other groups. Thus, there are no
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136 shared events (i.e. transitions) between the groups except of a closing or opening a MTF (e.g. open k closed k ) if F k is a common destinati on. Since there are no limits in the number of vehicles to l a nd at any F k open k and closed k do not cause any interaction or dependency between the corresponding LS s. Therefore, the general SC of the global DRT system, SC gen can be computed as the union of all the LS s, i.e. 6 3 .2. 7 C omputational complexity of decentralized supervisor Recall from Section 5 .2.2. 3 that in the worst case the computational complexity of modular supervisory control is In decentralized synthesis if the service of a given passenger with the vehicle is developed as an independent module, the computational complexity of the corresponding LS has the same upper limit as in the modular control. The main advantage of decentra lization is that if the subsystems are disjunctive, all the LS s can be computed in parallel and the nonblocking property of the SC gen is still guaranteed
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137 Chapter Seven Co ntribution of the Study and Future Research 7 1 Summary of the c ompleted w ork and contribution of the study In this study DRT systems are modeled as DES using Finite Automate formalism, and DRT operational planning and real time control are addressed using discrete event supervisory control theory. DES modeling and supervisory c ontrol theory are well established and powerful mathematical tools. In this dissertation, they are shown to be suitable for expressing the modeling and control requirements associated with the complex and dynamic applications in DRT. The modeling and contr ol approaches described herein, coupled with the mature body of research literature in discrete event systems and supervisory control theory, facilitates logical analysis of these complex systems and provides the necessary framework for the development of real time scheduling and intelligent decision making tools for operational planning in a broad range of DRT applications. To this extent, this work includes several significant contributions to the field of DRT systems modeling and operational control. To establish a systematic approach to the study of DRT systems, a taxonomy of the identifying features of DRT application domains is presented. This taxonomy is based on origin/destination characteristics, fleet characteristics, and demand characteristics.
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138 Within this taxonomy, several characteristics associated with DRT systems such as capacity constraints, route lengths etc. are modeled using Finite Automata. The representation of systems specifications and characteristics associated with DRT are straight forward to express in spoken languages, however correct mathematical representation of these features is not without challenge. Two application scenarios are considered; the first is based on air taxi service operation and illustrates uncontrolled system m odel and operational specification synthesis. Based on the uncontrolled system model and the specifications models, the automatic synthesis of centralized and modular supervisors are demonstrated. The second scenario is a mission critical application based on the emergency aero medical evacuation problem. In this scenario, decentralized supervisory control architecture suitable for accommodating the real time contingencies associated with this application is presented. The conditions for parallel computatio n of local supervisors are specified and the computational advantages of alternative supervisory control architectures are discussed. The alternative control architectures utilized in this work exhibit varying degrees of suitability to different applicatio n domains within DRT systems. Centralized control schemes suffer from exponentially increasing computational complexity and are only suitable for sm all sized static systems ( as illustrated with the air taxi service application). Decentralized control schem es provide a robust control solution to highly dynamic applications, such as the emergency evacuation. Furthermore, it is shown that, following the appropriate design procedures, the decentralized architectures still manage to maintain desirable supervisor y control characteristics such as nonblocking and are computationally tractable for a subset of the DRT application domains.
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139 7 2 Future R esearch The research should continue with modeling and control of many to many type of DRT system wh ere the origin and destination locations of the service requests can be anywhere over the covered region The main challenge is to control the allowed length of travel of the vehicles. There are two possible approach es to cope with this problem : 7 2 .1 A pplication of timed DES (TDES) In this approach the length of travel will be controlled with the limits of time it can take. In TDES both logical behavior and timing information are considered in In modeling TDES first a FA called activity transition graph denoted with G atg is introduced to describe the untimed behavior of the system. w here A is the finite set of activities, is the finite set of events, is the parti al activity transition function, a 0 is the initial activity and is the set of marked activities. Timing information is introduced into with the following way: each event is given a lower time bound and upper time bound such that and N denote the nonnegative integers. T he set of events is decomposed into two subsets : and where is the set of prospective and is the set of remote events. The lower time
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140 bound typically represents a delay in control, while the upper time bound is a hard deadline. For each the time interval is defined as follows: TDES is defined as a FA where the state set Q is defined as A state is of the form where activity and timer Timer encounters the passage of global time for each The set is given as by a subset of The event set is defined as where event tick represents the passage of one time unit. The state transition function is defined as follows for each and is defined i.e. if and only if one of the following condition s hold: and and and and and In DRT system, every flight and travel of a vehicle will have its lower and upper bounds, i.e. the limits of beginning and end of service. However, t he main disadvantage of TDES approach is the very large state space, caused by tracking all the states at any tick of time. To improve the efficiency of the model, Saadatpoor and Wonham (2007) propose instead of language control, state based predicates in compressed form represented with binary decision diagra ms (BDDs) In this approach, the structure in the
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141 states in the form of event timers of the modeled TDES can help reduce the size of BDDs. 7 2 .2 Application of hybrid DES (HDES) In HDES modeling some of the state variables are discrete and some are continuous. The dynamic behavior of discrete state systems is usually simpler to represent, but the mathematical tools to formally express and solve the state equations may be more complex. In contrast continuous state models ultimately reduce to the analysis of differential equations, for which many mathematical techniques are available. The type of supervisory control problems that is of interest in HDES arises whenever a continuous system is to be controlled by a discrete process such as a digital computer program. The continuous process to be controlled, together with any continuous controllers, is identified as the Plant and is typically described by differential/difference equations. The Controller includes a discrete decision process that is typically a represented by FA The Interface makes it possible for these different processes to communicate with each other. This control framework is quite flexible and can describe modern engineerin g systems where a computer process is used to control and coordinate several physical processes over a computer network. It can also describe a switching control system where a continuous plant is controlled by different continuous controllers over a numbe r of operating regions. The discrete event controllers for hybrid systems are based on discrete abstractions of the continuous dynamics. Applications have been primarily in the c ontinuous process industry and transportation service The a dvantage of this a pproach is that it generalizes well known concepts from digital control design. One
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142 of the main characteristics of the SC approach has been the emphasis and explicit identification of the interface issues between the continuous and discrete dynamics. These interface issues are the cornerstone of any HDES study. Koutsoukos at al. (2000) present a detailed framework for hybrid systems modeling and synthesis of SC for continuous Plant and discrete Controller (supervisor) The developed Interface consists of a generator and an actuator. The generator converts the continuous time output (state s ) of the Plant to an asynchronous, symbolic input for the supervisor The actuator sends the appropriate control signal into the Plant In HDES modeling of DRT service different continuous controllers can provide controllers can complexity of the interface to coordinate the behav iors of all the system elements.
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149 About the Author Technical University, Sofia, Bulgaria in 1993 and a M.S. in Industrial Engineering from Rochester Institute of Technology, Rochester, NY in 2004. He has wide engineering experience in project management, tool design and quality control. Daniel entered the Ph.D. program at the University of South Florida in 2004 While in the Ph.D. program at the University of South Florida, Mr. Yankov was actively involved in a project for simulatio n and optimization of security check points at major commercial airports H e also made a paper presentation at A nnual meeting in Seattle, WA
