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A model for spatio-temporal network planning
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Computers & Geosciences 31 (2005) 135–143 www.elsevier.com/locate/cageo
A model for spatio-temporal network planning Ed Nash, Phil James, David Parker School of Civil Engineering and Geosciences, University of Newcastle, Newcastle-upon-Tyne NE1 7RU, UK Received 4 June 2004; received in revised form 10 June 2004; accepted 10 June 2004
Abstract Temporal GIS research has tended to focus on representing a single history through a series of states. For planning future work involving alternative scenarios a branching model of time may be required, however for large systems such models soon become highly complex. In this paper we introduce the temporal topology model which allows sections of work and the spatial, temporal and logical relationships between them to be represented efficiently together with the associated costs. We then discuss how this model could be used for analysis to determine an optimal plan, illustrated with a case study involving cycle network planning, and briefly describe some practical results which have been obtained. r 2004 Elsevier Ltd. All rights reserved. Keywords: Temporal model; Topology; Decision support
1. Introduction GIS is used today for holding a wide range of data relating to spatial networks—from utility and communications infrastructure companies to transport operators the network is recorded and modelled within a GIS. This data is then used for a variety of tasks from basic inventory to operations management, and as a basis for planning the future of the network (Waters, 1999). For many of these tasks a spatio-temporal model may be advantageous, for network planning the temporal aspect is likely to be of even greater importance. However, most temporal GIS research appears focussed on recording historical or transaction-time data, or both in bitemporal systems (Worboys, 1995). Such data, however, has different characteristics to data relating to future plans. Corresponding author.
E-mail addresses: [email protected] (E. Nash), [email protected] (P. James), [email protected] (D. Parker).
In this paper we present a spatio-temporal model for network planning allowing multivariate optimisation incorporating spatial, temporal, financial and other aspects. We show how this model could be used as part of a decision-support process applied to a scenario of planning a network of cycle paths and discuss some possibilities for design optimisation using the model.
2. Temporal models In general, GIS systems do not incorporate a temporal model; they represent one state of the real world, usually the present. Most temporal GIS research (as outlined in, e.g. Peuquet, 2001) attempts to extend the representation to include a series of states, either of the database (a snapshot model) or of individual objects within the database (e.g. the ST-object model (Worboys, 1992)), allowing the changes in the real world to be recorded. This reproduces a linear model of time in which there exists only one possible time-line, with all
0098-3004/$ - see front matter r 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.cageo.2004.06.012
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A
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F time→
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multiple records of the same features and near-identical branches (Fig. 2A). The following section shows the development of the temporal topology model which attempts to overcome such problems, allowing an efficient representation of these situations, together with metadata to enable optimal scenarios to be chosen.
X time→
3. The temporal topology model Fig. 1. Linear (A) and branching (B) models of time. In A only one sequence of events is possible, in B many sequences (worlds) are possible, but only one will occur (the real world).
events having a fixed location upon it (Fig. 1A). Whilst this is generally suitable for recording what has previously happened in the study area, where there is no ambiguity as to what has occurred, it is of limited value in modelling what may happen in the future where one of a number of scenarios may occur. In order to represent this situation of ‘‘alternative futures’’ (Worboys, 1995), some form of branching time model (Fig. 1B) is required. In such a model, there exist many possible time-lines, branching from one or more points in time; usually these are multiple possibilities for the future branching from the present time, although alternative pasts converging are also possible. Only one of these time-lines is generally considered valid, i.e. only one time-line will actually be followed—of the many conceivable worlds, one is the real world (Kripke, 1959). The philosophical and logical basis for such a model has long been established, stretching back to Aristotle (ca. 350 BC), although there remain many views on appropriate semantics (see e.g. Reynolds 2002 or Bennett and Galton, 2004) which, whilst pertinent to how any branching-time model is interpreted, are beyond the scope of this work. Facilities exist within some software allowing evaluation of alternative scenarios, usually through use of database alternatives and without any explicit temporal model being included (e.g. GE Network Solutions’ Design Manager1 is one such GIS-based tool). Whilst a branching model of time can represent immediate alternative scenarios well, it rapidly becomes more complex the greater the time span covered, with many branches forming. Particularly if the scenarios being considered involve what, conceptually, are the same events (e.g. construction of a new bridge) occurring at different times and/or in different orders, a true branching-time representation, such as perhaps could be constructed using standard database versioning techniques, would become somewhat complex with 1 Smallworld Design Manager. 2003. http://www.gepower.com/ prod_serv/products/gis_software/en/downloads/design_manager_ overview.pdf
The ‘‘temporal topology’’ model was developed from extending a simple branching model of time to allow branches to re-join—i.e. considering that given two events, B and C which may occur in either order BC or order CB, the end result is the same. Whilst temporally this is obviously not the case, spatially it may well be— e.g. if events B and C are construction of two different sections of network then the end spatial result is that both are present, regardless of the order in which they were constructed. Additionally, if there is a choice of time-lines followed by a common section, these could also be considered re-joining branches (e.g. D or E then F in Fig. 2B), similar to the concept of ‘‘ultimately converging time’’ discussed in Prior (1967). For instance, if D and E were alternative designs for a section of network, each capable of handling the same capacity and linking the same two points, it could be considered irrelevant which one is chosen to link to section F since the result is a connected network of the required capacity. This leads to the hypothesis that all that is required to model multiple scenarios is the relationships between different sections of those scenarios—the ‘‘temporal topology’’ between events—whether these relationships are temporal or logical. The form of these relationships is discussed further after the components of the temporal topology model are introduced. The underlying theory of the temporal topology model is that planned designs can be separated into self-contained blocks of work (events) which have a fixed spatial location, some associated cost data and an initially undefined temporal location. Linkages between events are defined as relationships which may be temporal (e.g. one event must occur before another), logical (e.g. two events being mutually exclusive), costbased (e.g. if two events both occur, an additional cost is incurred) or a combination of these. Costs may be of any type—financial, temporal or abstract variables such as desirability. Events may or may not be mandatory. Mandatory events could be considered to be those occurring on all possible time-lines (e.g. A, B, C & F in Fig. 2), and non-mandatory events those which only occur on a subset of these (e.g. D and E in Fig. 2). This allows ‘essential’ work to be modelled as mandatory events, and alternative designs for individual sections or ‘extensions’ to the core work to be modelled as nonmandatory events, all of which can be considered
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Fig. 2. True branching time (A) as it could be represented using standard database versioning. Effect of allowing branches to re-join, (B) would be to reduce duplication of representations of events.
concurrently. Additionally, constraints may be introduced to limit the possible solutions to those where a cost falls below a maximum value, or an event occurs at a specified time. The key component of the temporal topology model is the definition of relationships between events. These relationships could be considered as Boolean expressions, which if they return true when applied to a proposed sequence of events are considered to be met. A sequence of events which meets all relationships (and constraints, which could be similarly specified) is therefore considered to be valid—i.e. a time-line which could occur and would thus appear in a re-constructed branching model of time. Note that, in the development of the model, and the current discussion, it is assumed that events do not occur simultaneously. This models the situation where a single ‘agent’ performs each block of work sequentially—which, although unlikely to be the case in reality, simplifies to some extent the analysis of optimal plans, discussed later. Although, all blocks of work and relationships between them are represented in the model, the assumption is therefore that what may occur, and therefore what a plan for a proposed network build will consist of, is a monotonic sequence of events. Allen (1984) defined 13 temporal relationships, although, of these, six are inverses, and, of the remaining seven (before, equal, meets, overlaps, during, starts and finishes), five represent situations where multiple events occur simultaneously. This leaves two fundamental temporal relationships for temporal topology; before (o) and meets (m). Temporal relationships can therefore be expressed in the form AoB, interpreted that should A and B both occur in the same time-line, it will only be considered valid if A occurs before B. Thus, time-lines such as C-A-D, B-C-D, A-D-B, C-A-B, etc. would be valid due to either only one of events A and B occurring, and/or A occurring before B, whereas time-lines such as B-A, B-D-A, D-C-B-A, etc. would be invalid. Similarly, logical relationships can be expressed using Boolean algebra, where the presence and absence of an event in a given sequence, regardless of ordering, can be interpreted as true (1) or false (0) respectively, and the same definition of validity applied. For example, to ensure
that of two events, A and B, at most one may occur on all valid time-lines, a logical relationship A NAND B (symbolically AmB) could be introduced. Thus, all timelines containing A or B, or neither A nor B would satisfy this relationship, and all time-lines containing both A and B would not. Using conditional operators allows relationships combining logical and temporal aspects (e.g. IF A before B THEN C, symbolically (AoB)-C), providing a compact and powerful way to describe the linkages between events. Cost relationships can also be described through use of conditional operators, e.g. IF A AND B THEN increase financial cost by £x (symbolically (A4B)-+£x). Table 1 outlines basic temporal and logical operators, suggested symbology and semantics within temporal topology relationships. Although temporal topology systems could be represented entirely symbolically, this is hard to interpret manually. A standard branching model of time showing all valid time-lines could be reconstructed from the set of events, relationships and constraints, but, although this may be easily interpreted, it re-introduces the weaknesses of this model as previously discussed. It is therefore suggested that a good graphical representation of a temporal topology system can be obtained through arbitrarily arranging the events as nodes in 2D-space and adding relationships as links connecting these nodes (Fig. 3A). Adding all potential links between events, whether valid or not, and some start (S) and finish (F) ‘pseudo-nodes’ for paths produces a network (Fig. 3B) that could be analysed using methods from graph theory, with a valid path through the network from S to F representing a valid sequence of events, i.e. a valid time-line.
4. Example application of temporal topology The following example shows how the temporal topology model could be applied to a realistic scenario of planning improvements for cyclists along the route of a busy road. For simplicity only a small section is covered, showing a selection of different solutions, and how they could be broken down into events and
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Table 1 Temporal and logical operators for temporal topology relationships Operator
AoB AmB A4B A3B :A AB AmB AkB (A)-B (A)2(B) a
Result for events occurring — A B AB BA A-B
B-A
1 1 0 0 1 0 1 1 1 1
0 0 1 1 0 0 0 0 1 1
1 1 0 1 0 1 1 0 0 0
1 1 0 1 1 1 1 0 1 0
1 1 1 1 0 0 0 0 1 1
0 0 1 1 0 0 0 0 1 1
1 0 1 1 0 0 0 0 1 1
Semantics in relation to temporal topology
before: meets: AND: OR: NOT: XOR: NAND: NOR: IF: IFF:
Should both A and B occur, A must occur before B Should both A and B occur, A must occur immediately before B Both A and B must occur Either A or B or both must occur A must not occura Either A or B but not both must occur Either A or B may occur, both must not occur Neither A nor B nor both may occura If A occurs, B must occur If either A or B occurs, the other must also occur
These relationships are illogical in relation to temporal topology except as part of an if or iff relationship.
Fig. 3. Schematic representation of a temporal topology system showing either relationships (A) or a full network with all links (B).
relationships. Additionally, only the centrelines of the proposed routes are shown; a full model would include the associated crossings, signs, road markings, etc. which would be required, and the installation of which would be modelled as temporal topology events. This scenario (Fig. 4) is based around a busy main road (hatched), which is located in a walled cutting, and a station. The intent is to have a continuous cycle route running along or parallel to the main road, with an additional requirement to provide direct access to the station from both east and west. Some possible routes near the station are shown, consisting of a variety of dedicated cycle paths, converting footpaths to dual-use, signing routes along quiet roads and marking out cycle lanes along existing roads. These different options have differing implementation costs, in terms of financial, temporal and ease of installation. Additionally, some solutions may be more desirable than others (e.g. dedicated cycle routes may be preferable to on-road lanes in terms of environment and safety for the cyclist), and the route should be overall as short and as straight as possible. The options could be considered as being split in to three distinct routes—a predominantly quiet road route to the south of the busy road, a mixture of route types to
the north of the busy road and with-traffic cycle lanes on the busy road itself. There is an additional option to the northeast of the study area of a traffic-free route heading north then east. Two bridges to the east of the station allow the busy road to be crossed safely, and also allow crossover between the route options, e.g. the westerly section north of the road could be matched with the easterly section south of the road to produce an optimal route. However, only the construction of one complete east–west route will be considered possible, i.e. building more than one parallel east–west section is not allowed. To the northeast of the station there exist two local alternatives, of converting the pavement to a shared foot/cycle-way or using the quiet road. It is thought that to save costs initially, the quiet road could be used for this section, with the pavement being converted later if necessary—although doing one then the other would be more expensive overall. It would not however be allowable to first change the pavement to dual use, and later remove this and use the quiet road. Fig. 5 shows this scenario broken down into the structure of temporal topology events. The only constraint is that the station must be connected to both the east and west (therefore either A, P or K are required to link to the west, and F, I, O or M to the east). The relationships between events are shown in Fig. 6, based on only one complete route being constructed. An additional relationship, illustrating a purely temporal relationship, is that since construction of the ramps (event N) would entail works at the edge of the busy road, the cycle lanes along the busy road (events O and P) may only be built after the construction of the ramps.
5. Analysis of temporal topology systems The aim of temporal topology analysis is to determine an optimal set and order of events based on the specified
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Fig. 4. Example scenario of possible cycle routes.
Fig. 5. Scenario from Fig. 4 broken down into temporal topology events each representing an independent section of cycle network.
costs and satisfying all the given relationships and constraints. This gives the events a relative temporal location and discards all events deemed not to be in the optimal set. From the duration of the events in the sequence, and a given start or end time, an absolute temporal location for each can then be calculated if required. Each individual solution could therefore be considered as a single time-line, with the whole set of valid solutions constituting a branching time-line. Currently it is assumed that there is no overlap between
events, i.e. they occur sequentially, although the model could be extended to allow simultaneous events. The other assumption that may be made in many cases is that having spatially close or spatial-topologically connected events occurring temporally close or adjacent is desirable—i.e. that building continuous sections of networks is advantageous. How optimality is defined will affect the result of the analysis. The least-cost solution for one individual variable is likely to give unacceptable results for other
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Fig. 6. Definitions of events, their possible costs and relationships within example scenario from Figs. 4 and 5.
factors under consideration, as illustrated in a GIS context by Carver (1991). Aggregating all weighted costs to give a single variable is also likely to be unsatisfactory due to the artificial nature of the combination of different classes of variable, e.g. calculating weightings between financial and abstract variables (Horn, 1997). The most satisfactory option is therefore to perform a true multivariate optimisation producing a solution, or set of solutions, to which the contribution of each individual variable can be seen, allowing the decisionmaker to make a more informed decision (Cohon, 1978). This however is likely to be a more complex, and therefore time-consuming process, and so single-variable methods are considered here first. 5.1. Single-variable optimisation The intent of the network representation of a temporal topology system, as shown in Fig. 3B is that it is possible to analyse it using established graph theory methods such as shortest-path analysis. Thus, a variation on an algorithm such as Dijkstra’s (1959) could be used to perform the optimisation with the list of nodes crossed on a ‘shortest path’ between S and F pseudoevent nodes representing the optimal order of events. Some modifications to the standard algorithm are necessary to take account of constraints and relationships in the temporal topology model, and costs must be expressed in a suitable form for the algorithm (e.g. as positive weightings on the links for Dijkstra’s algorithm). However, analysis of the required modifications (which falls beyond the scope of this paper) shows that they make the original efficient algorithm inefficient, with the problem appearing to be NP-complete, similar to the ‘shortest-path with constraints’ problem which is
shown NP-complete (e.g. in Garey and Johnson, 1979). Indeed, the problem could be considered as a ‘travellingsalesman’ type problem where mandatory events constitute nodes that must be visited during a circuit from/ to a pseudo-event node. Efficient exact methods are therefore unlikely, although a shortest-path approach has been adopted for practical tests to investigate whether on actual temporal topology systems the performance is acceptable. 5.2. Multivariate optimisation The optimisation theory used here for multivariate optimisation is that of Pareto optimisation (from Pareto, 1906) wherein a solution is considered to be optimal if there exists no other feasible solution that has equal or better costs across all variables considered. This usually results in a set of good solutions being produced (Foncesa and Fleming, 1995), from which the decisionmaker can choose. In order to search the solution-space for Pareto-optimal orderings of events, and assuming from the analysis of single-variable methods that no efficient exact approach is possible, two approaches have been taken. For small systems it may be feasible to perform an exhaustive search of all possible solutions, thus finding the true Pareto-optimal set. However, for large systems this would become impractical due to the large number of solutions to be generated and tested. Therefore, a genetic algorithm (after Holland, 1975) is being used as a suitable approximation method which has been used elsewhere for multivariate optimisation problems with considerable success (e.g. Foncesca and Fleming, 1995; Van Veldhuizen and Lamont, 2000). This is an algorithm that mimics the principles of natural evolution and is suited to problems where there
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is a large and poorly understood solution space and no easy analytical or linear approach (Mitchell, 1996). The fundamental idea is that combining sections of good solutions (in this case short series of events forming part of a possible solution) with sections of other good solutions will produce further good, or better, solutions. Thus, given sufficient iterations where only the best solutions survive to the next generation, a set of good approximations to the best solution will be produced.
6. Implementation and testing of a temporal topology application A decision-support application has been implemented based on the temporal topology theories outlined above. This implementation is built in the GE Smallworld GIS framework, using a metadata database to record the assignment of real world objects to temporal topology events and the relationships between these events. From this database, schematics of the temporal topology network can be produced which can then be analysed for single-variable optimisation using customised versions of the standard Smallworld network follower tools. An analysis engine using the Pareto optimal principle with a genetic algorithm or an exhaustive search has been written in the Smallworld Magik object-oriented language. Both these tools can read the data from both the metadata and GIS databases to perform the analysis. Once temporal topology analysis is finished, the sequence of events comprising each optimal solution can be viewed using the standard version-management tools in the GIS so that the planner can make a final decision.
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The scenario described in Section 4 has been used for some preliminary testing of this application, and of the ideas presented in this paper. This has shown that, whilst the temporal topology model as presented in Section 3 is usable and capable of representing a reasonably complex scenario in a fairly compact and straightforward manner, the analysis techniques outlined in Section 5 are of limited use in production of optimised solutions. For instance, although running a shortest-path trace to minimise spatial distance (i.e. attempting to produce a continuous network) took only 10 s, it resulted in a solution of C-B-D-E-R-Q-N-P-O-J-H-G, which as can be seen in Fig. 7 is a continuous network but in practical terms would probably not be considered an optimal solution due to the high level of redundancy. Using the shortest-path trace with other cost classes, however, failed as the physical memory limit, used for storing the paths found thus far, was reached before a path to the end was found. Even for this relatively small system of 18 events, an exhaustive search was found to be somewhat impractical, requiring approximately 1016 potential solutions to be generated and tested which with the current implementation and extrapolation of trial testing would take around 107 years. Processing using a genetic algorithm produced a set of solutions in a more reasonable timescale (around 5 h), but produced a set containing a large number of valid non-dominated solutions (over 100 in this case), a common problem with multiobjective techniques (Graves et al., 1992). Without the use of clustering techniques, such as those used by Zittler and Thiele (1999), to reduce the size of the Pareto-optimal set to a reasonable level, this form of analysis is therefore also of limited practical value. However, of these 100 solutions about six could be considered good by a more subjective measure of being
Fig. 7. Proposed cycle network produced from a network trace to minimise spatial distance, i.e. produce a continuous network.
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Table 2 Sample results and costs resulting from optimisation by network trace and genetic algorithm of system described in Fig. 6 Method/Result ID Trace Genetic algorithm
SD GA2 GA23 GA27 GA65 GA73 GA79
Events
Sp. dist. (km)
Cost (£000)
Dur. (h)
Des. (Sd)
Exp. (Se)
C-B-D-E-R-Q-N-P-O-J-H-G R-E-N-P-L-M J-M-N-D-C-P-E-R H-L-J-R-G-F-N-E-P-Q E-R-N-D-B-H-P-I-G J-H-M-N-D-C-P-E-R E-N-L-R-Q-P-M
0.000 0.013 0.103 0.251 0.143 0.146 0.062
305 119 120 119 127 120 119
30.46 13.16 13.86 13.74 16.35 13.97 13.22
36 19 25 33 32 28 22
30 16 18 20 24 19 17
Sp. dist.: Spatial distance (km, kilometres), Cost: Financial cost (£000, thousand Pound Stirling), Dur.: Duration (h, hours), Des.: Desirability (d, 5-+5), Exp.: Expediency (e, 5-+5).
continuous (i.e. no gaps in the network) and without redundancy (i.e. no extraneous sections), suggesting that this method is worth pursuing as a means of network design optimisation. These six solutions and their costs, together with the spatial distance trace result discussed previously are summarised in Table 2.
7. Conclusions This paper has presented a model, temporal topology, which aims to provide an efficient and useful method to represent and analyse alternative scenarios for network planning. The need for such a model was outlined, noting that most TGIS research has considered recording historical changes and has therefore concentrated upon linear models of time which cannot sensibly be used for modelling multiple scenarios. The perceived problems with standard branching models of time were highlighted and it was suggested that these could be overcome by recording, instead of all branches of time, just the relationships between events using a combination of temporal and Boolean logics. Through a small example, it was demonstrated that, using this technique, a number of alternative plans for a network could sensibly be managed. Some possibilities for optimisation of designs through analyses using this model were then introduced. Following this, a brief description of an implementation of an application based on the temporal topology model was given and some results obtained using this implementation with the example previously given were shown. Although, these optimisation results were perhaps disappointing, they do not detract from the validity of the model—indicating rather that further work is required on perfecting suitable optimisation methods before the model becomes truly useful for practical purposes.
References Allen, J.F., 1984. Towards a general theory of action and time. Artificial Intelligence 23 (2), 123–154. Aristotle, ca. 350 BC. De Interpretatione (On Interpretation). Bennett, B., Galton, A.P., 2004. A unifying semantics for time and events. Artificial Intelligence 153 (1–2), 13–58. Carver, S.J., 1991. Integrating multi-criteria evaluation with geographical information systems. International Journal of Geographical Information Systems 5 (3), 321–339. Cohon, J.L., 1978. Multiobjective Programming and Planning. Academic Press, New York, NY 333pp. Dijkstra, E., 1959. A note on two problems in connexion with graphs. Numerische Mathematik 1, 269–271. Foncesa, C.M., Fleming, P.J., 1995. An overview of evolutionary algorithms in multiobjective optimization. Evolutionary Computation 3 (1), 1–16. Graves, S.B., Ringuest, J.L., Bard, J.F., 1992. Recent developments in screening methods for nondominated solutions in multiobjective optimization. Computers & Operations Research 19 (7), 683–694. Holland, J.H., 1975. Adaptation in Natural and Artificial Systems, second ed. University of Michigan Press, 1992; MIT Press, Cambridge, MA, 211pp. Horn, J., 1997. Multicriteria decision making and evolutionary computation. In: Ba¨ck, T., Fogel, D.B., Michalewicz, Z. (Eds.), The Handbook of Evolutionary Computation. IOP Publishing Ltd. & Oxford University Press, pp. F1.9:1–F1.9:15. Kripke, S.A., 1959. A completeness theorem in modal logic. The Journal of Symbolic Logic 24 (1), 1–14. Mitchell, M., 1996. An Introduction to Genetic Algorithms. MIT Press, Cambridge, MA 205pp. Pareto, V., 1906. Manual of Political Economy. Macmillan Press Ltd., London 504pp (1971 Edition translated from French Edition of 1927). Peuquet, D.J., 2001. Making space for time: issues in space– time data representation. GeoInformatica 5 (1), 11–32. Prior, A., 1967. Past, Present and Future. Oxford University Press, London 217pp. Reynolds, M., 2002. Axioms for branching time. Journal of Logic and Computation 12 (4), 679–697.
ARTICLE IN PRESS E. Nash et al. / Computers & Geosciences 31 (2005) 135–143 Van Veldhuizen, D.A., Lamont, G.B., 2000. Multiobjective evolutionary algorithms: analyzing the state-of-the-art. Evolutionary Computation 8 (2), 125–147. Waters, N., 1999. Transportation GIS: GIS-T. In: Longley, P.A., Goodchild, M.F., Maguire, D.J., Rhind, D.W. (Eds.), Geographical Information Systems, second ed. Wiley, New York, pp. 824–827. Worboys, M.F., 1992. A unified model of spatial and temporal information. In: Proceedings of the Sixth
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A model for spatio-temporal network planning Ed Nash, Phil James, David Parker School of Civil Engineering and Geosciences, University of Newcastle, Newcastle-upon-Tyne NE1 7RU, UK Received 4 June 2004; received in revised form 10 June 2004; accepted 10 June 2004
Abstract Temporal GIS research has tended to focus on representing a single history through a series of states. For planning future work involving alternative scenarios a branching model of time may be required, however for large systems such models soon become highly complex. In this paper we introduce the temporal topology model which allows sections of work and the spatial, temporal and logical relationships between them to be represented efficiently together with the associated costs. We then discuss how this model could be used for analysis to determine an optimal plan, illustrated with a case study involving cycle network planning, and briefly describe some practical results which have been obtained. r 2004 Elsevier Ltd. All rights reserved. Keywords: Temporal model; Topology; Decision support
1. Introduction GIS is used today for holding a wide range of data relating to spatial networks—from utility and communications infrastructure companies to transport operators the network is recorded and modelled within a GIS. This data is then used for a variety of tasks from basic inventory to operations management, and as a basis for planning the future of the network (Waters, 1999). For many of these tasks a spatio-temporal model may be advantageous, for network planning the temporal aspect is likely to be of even greater importance. However, most temporal GIS research appears focussed on recording historical or transaction-time data, or both in bitemporal systems (Worboys, 1995). Such data, however, has different characteristics to data relating to future plans. Corresponding author.
E-mail addresses: [email protected] (E. Nash), [email protected] (P. James), [email protected] (D. Parker).
In this paper we present a spatio-temporal model for network planning allowing multivariate optimisation incorporating spatial, temporal, financial and other aspects. We show how this model could be used as part of a decision-support process applied to a scenario of planning a network of cycle paths and discuss some possibilities for design optimisation using the model.
2. Temporal models In general, GIS systems do not incorporate a temporal model; they represent one state of the real world, usually the present. Most temporal GIS research (as outlined in, e.g. Peuquet, 2001) attempts to extend the representation to include a series of states, either of the database (a snapshot model) or of individual objects within the database (e.g. the ST-object model (Worboys, 1992)), allowing the changes in the real world to be recorded. This reproduces a linear model of time in which there exists only one possible time-line, with all
0098-3004/$ - see front matter r 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.cageo.2004.06.012
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A
B
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F time→
Y W U (B)
V
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multiple records of the same features and near-identical branches (Fig. 2A). The following section shows the development of the temporal topology model which attempts to overcome such problems, allowing an efficient representation of these situations, together with metadata to enable optimal scenarios to be chosen.
X time→
3. The temporal topology model Fig. 1. Linear (A) and branching (B) models of time. In A only one sequence of events is possible, in B many sequences (worlds) are possible, but only one will occur (the real world).
events having a fixed location upon it (Fig. 1A). Whilst this is generally suitable for recording what has previously happened in the study area, where there is no ambiguity as to what has occurred, it is of limited value in modelling what may happen in the future where one of a number of scenarios may occur. In order to represent this situation of ‘‘alternative futures’’ (Worboys, 1995), some form of branching time model (Fig. 1B) is required. In such a model, there exist many possible time-lines, branching from one or more points in time; usually these are multiple possibilities for the future branching from the present time, although alternative pasts converging are also possible. Only one of these time-lines is generally considered valid, i.e. only one time-line will actually be followed—of the many conceivable worlds, one is the real world (Kripke, 1959). The philosophical and logical basis for such a model has long been established, stretching back to Aristotle (ca. 350 BC), although there remain many views on appropriate semantics (see e.g. Reynolds 2002 or Bennett and Galton, 2004) which, whilst pertinent to how any branching-time model is interpreted, are beyond the scope of this work. Facilities exist within some software allowing evaluation of alternative scenarios, usually through use of database alternatives and without any explicit temporal model being included (e.g. GE Network Solutions’ Design Manager1 is one such GIS-based tool). Whilst a branching model of time can represent immediate alternative scenarios well, it rapidly becomes more complex the greater the time span covered, with many branches forming. Particularly if the scenarios being considered involve what, conceptually, are the same events (e.g. construction of a new bridge) occurring at different times and/or in different orders, a true branching-time representation, such as perhaps could be constructed using standard database versioning techniques, would become somewhat complex with 1 Smallworld Design Manager. 2003. http://www.gepower.com/ prod_serv/products/gis_software/en/downloads/design_manager_ overview.pdf
The ‘‘temporal topology’’ model was developed from extending a simple branching model of time to allow branches to re-join—i.e. considering that given two events, B and C which may occur in either order BC or order CB, the end result is the same. Whilst temporally this is obviously not the case, spatially it may well be— e.g. if events B and C are construction of two different sections of network then the end spatial result is that both are present, regardless of the order in which they were constructed. Additionally, if there is a choice of time-lines followed by a common section, these could also be considered re-joining branches (e.g. D or E then F in Fig. 2B), similar to the concept of ‘‘ultimately converging time’’ discussed in Prior (1967). For instance, if D and E were alternative designs for a section of network, each capable of handling the same capacity and linking the same two points, it could be considered irrelevant which one is chosen to link to section F since the result is a connected network of the required capacity. This leads to the hypothesis that all that is required to model multiple scenarios is the relationships between different sections of those scenarios—the ‘‘temporal topology’’ between events—whether these relationships are temporal or logical. The form of these relationships is discussed further after the components of the temporal topology model are introduced. The underlying theory of the temporal topology model is that planned designs can be separated into self-contained blocks of work (events) which have a fixed spatial location, some associated cost data and an initially undefined temporal location. Linkages between events are defined as relationships which may be temporal (e.g. one event must occur before another), logical (e.g. two events being mutually exclusive), costbased (e.g. if two events both occur, an additional cost is incurred) or a combination of these. Costs may be of any type—financial, temporal or abstract variables such as desirability. Events may or may not be mandatory. Mandatory events could be considered to be those occurring on all possible time-lines (e.g. A, B, C & F in Fig. 2), and non-mandatory events those which only occur on a subset of these (e.g. D and E in Fig. 2). This allows ‘essential’ work to be modelled as mandatory events, and alternative designs for individual sections or ‘extensions’ to the core work to be modelled as nonmandatory events, all of which can be considered
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B
D
F
E
F
C
A D
F
B
C
D
C
B
E
F time→
E (A)
B A
C
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F time→
(B)
Fig. 2. True branching time (A) as it could be represented using standard database versioning. Effect of allowing branches to re-join, (B) would be to reduce duplication of representations of events.
concurrently. Additionally, constraints may be introduced to limit the possible solutions to those where a cost falls below a maximum value, or an event occurs at a specified time. The key component of the temporal topology model is the definition of relationships between events. These relationships could be considered as Boolean expressions, which if they return true when applied to a proposed sequence of events are considered to be met. A sequence of events which meets all relationships (and constraints, which could be similarly specified) is therefore considered to be valid—i.e. a time-line which could occur and would thus appear in a re-constructed branching model of time. Note that, in the development of the model, and the current discussion, it is assumed that events do not occur simultaneously. This models the situation where a single ‘agent’ performs each block of work sequentially—which, although unlikely to be the case in reality, simplifies to some extent the analysis of optimal plans, discussed later. Although, all blocks of work and relationships between them are represented in the model, the assumption is therefore that what may occur, and therefore what a plan for a proposed network build will consist of, is a monotonic sequence of events. Allen (1984) defined 13 temporal relationships, although, of these, six are inverses, and, of the remaining seven (before, equal, meets, overlaps, during, starts and finishes), five represent situations where multiple events occur simultaneously. This leaves two fundamental temporal relationships for temporal topology; before (o) and meets (m). Temporal relationships can therefore be expressed in the form AoB, interpreted that should A and B both occur in the same time-line, it will only be considered valid if A occurs before B. Thus, time-lines such as C-A-D, B-C-D, A-D-B, C-A-B, etc. would be valid due to either only one of events A and B occurring, and/or A occurring before B, whereas time-lines such as B-A, B-D-A, D-C-B-A, etc. would be invalid. Similarly, logical relationships can be expressed using Boolean algebra, where the presence and absence of an event in a given sequence, regardless of ordering, can be interpreted as true (1) or false (0) respectively, and the same definition of validity applied. For example, to ensure
that of two events, A and B, at most one may occur on all valid time-lines, a logical relationship A NAND B (symbolically AmB) could be introduced. Thus, all timelines containing A or B, or neither A nor B would satisfy this relationship, and all time-lines containing both A and B would not. Using conditional operators allows relationships combining logical and temporal aspects (e.g. IF A before B THEN C, symbolically (AoB)-C), providing a compact and powerful way to describe the linkages between events. Cost relationships can also be described through use of conditional operators, e.g. IF A AND B THEN increase financial cost by £x (symbolically (A4B)-+£x). Table 1 outlines basic temporal and logical operators, suggested symbology and semantics within temporal topology relationships. Although temporal topology systems could be represented entirely symbolically, this is hard to interpret manually. A standard branching model of time showing all valid time-lines could be reconstructed from the set of events, relationships and constraints, but, although this may be easily interpreted, it re-introduces the weaknesses of this model as previously discussed. It is therefore suggested that a good graphical representation of a temporal topology system can be obtained through arbitrarily arranging the events as nodes in 2D-space and adding relationships as links connecting these nodes (Fig. 3A). Adding all potential links between events, whether valid or not, and some start (S) and finish (F) ‘pseudo-nodes’ for paths produces a network (Fig. 3B) that could be analysed using methods from graph theory, with a valid path through the network from S to F representing a valid sequence of events, i.e. a valid time-line.
4. Example application of temporal topology The following example shows how the temporal topology model could be applied to a realistic scenario of planning improvements for cyclists along the route of a busy road. For simplicity only a small section is covered, showing a selection of different solutions, and how they could be broken down into events and
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Table 1 Temporal and logical operators for temporal topology relationships Operator
AoB AmB A4B A3B :A AB AmB AkB (A)-B (A)2(B) a
Result for events occurring — A B AB BA A-B
B-A
1 1 0 0 1 0 1 1 1 1
0 0 1 1 0 0 0 0 1 1
1 1 0 1 0 1 1 0 0 0
1 1 0 1 1 1 1 0 1 0
1 1 1 1 0 0 0 0 1 1
0 0 1 1 0 0 0 0 1 1
1 0 1 1 0 0 0 0 1 1
Semantics in relation to temporal topology
before: meets: AND: OR: NOT: XOR: NAND: NOR: IF: IFF:
Should both A and B occur, A must occur before B Should both A and B occur, A must occur immediately before B Both A and B must occur Either A or B or both must occur A must not occura Either A or B but not both must occur Either A or B may occur, both must not occur Neither A nor B nor both may occura If A occurs, B must occur If either A or B occurs, the other must also occur
These relationships are illogical in relation to temporal topology except as part of an if or iff relationship.
Fig. 3. Schematic representation of a temporal topology system showing either relationships (A) or a full network with all links (B).
relationships. Additionally, only the centrelines of the proposed routes are shown; a full model would include the associated crossings, signs, road markings, etc. which would be required, and the installation of which would be modelled as temporal topology events. This scenario (Fig. 4) is based around a busy main road (hatched), which is located in a walled cutting, and a station. The intent is to have a continuous cycle route running along or parallel to the main road, with an additional requirement to provide direct access to the station from both east and west. Some possible routes near the station are shown, consisting of a variety of dedicated cycle paths, converting footpaths to dual-use, signing routes along quiet roads and marking out cycle lanes along existing roads. These different options have differing implementation costs, in terms of financial, temporal and ease of installation. Additionally, some solutions may be more desirable than others (e.g. dedicated cycle routes may be preferable to on-road lanes in terms of environment and safety for the cyclist), and the route should be overall as short and as straight as possible. The options could be considered as being split in to three distinct routes—a predominantly quiet road route to the south of the busy road, a mixture of route types to
the north of the busy road and with-traffic cycle lanes on the busy road itself. There is an additional option to the northeast of the study area of a traffic-free route heading north then east. Two bridges to the east of the station allow the busy road to be crossed safely, and also allow crossover between the route options, e.g. the westerly section north of the road could be matched with the easterly section south of the road to produce an optimal route. However, only the construction of one complete east–west route will be considered possible, i.e. building more than one parallel east–west section is not allowed. To the northeast of the station there exist two local alternatives, of converting the pavement to a shared foot/cycle-way or using the quiet road. It is thought that to save costs initially, the quiet road could be used for this section, with the pavement being converted later if necessary—although doing one then the other would be more expensive overall. It would not however be allowable to first change the pavement to dual use, and later remove this and use the quiet road. Fig. 5 shows this scenario broken down into the structure of temporal topology events. The only constraint is that the station must be connected to both the east and west (therefore either A, P or K are required to link to the west, and F, I, O or M to the east). The relationships between events are shown in Fig. 6, based on only one complete route being constructed. An additional relationship, illustrating a purely temporal relationship, is that since construction of the ramps (event N) would entail works at the edge of the busy road, the cycle lanes along the busy road (events O and P) may only be built after the construction of the ramps.
5. Analysis of temporal topology systems The aim of temporal topology analysis is to determine an optimal set and order of events based on the specified
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Fig. 4. Example scenario of possible cycle routes.
Fig. 5. Scenario from Fig. 4 broken down into temporal topology events each representing an independent section of cycle network.
costs and satisfying all the given relationships and constraints. This gives the events a relative temporal location and discards all events deemed not to be in the optimal set. From the duration of the events in the sequence, and a given start or end time, an absolute temporal location for each can then be calculated if required. Each individual solution could therefore be considered as a single time-line, with the whole set of valid solutions constituting a branching time-line. Currently it is assumed that there is no overlap between
events, i.e. they occur sequentially, although the model could be extended to allow simultaneous events. The other assumption that may be made in many cases is that having spatially close or spatial-topologically connected events occurring temporally close or adjacent is desirable—i.e. that building continuous sections of networks is advantageous. How optimality is defined will affect the result of the analysis. The least-cost solution for one individual variable is likely to give unacceptable results for other
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Fig. 6. Definitions of events, their possible costs and relationships within example scenario from Figs. 4 and 5.
factors under consideration, as illustrated in a GIS context by Carver (1991). Aggregating all weighted costs to give a single variable is also likely to be unsatisfactory due to the artificial nature of the combination of different classes of variable, e.g. calculating weightings between financial and abstract variables (Horn, 1997). The most satisfactory option is therefore to perform a true multivariate optimisation producing a solution, or set of solutions, to which the contribution of each individual variable can be seen, allowing the decisionmaker to make a more informed decision (Cohon, 1978). This however is likely to be a more complex, and therefore time-consuming process, and so single-variable methods are considered here first. 5.1. Single-variable optimisation The intent of the network representation of a temporal topology system, as shown in Fig. 3B is that it is possible to analyse it using established graph theory methods such as shortest-path analysis. Thus, a variation on an algorithm such as Dijkstra’s (1959) could be used to perform the optimisation with the list of nodes crossed on a ‘shortest path’ between S and F pseudoevent nodes representing the optimal order of events. Some modifications to the standard algorithm are necessary to take account of constraints and relationships in the temporal topology model, and costs must be expressed in a suitable form for the algorithm (e.g. as positive weightings on the links for Dijkstra’s algorithm). However, analysis of the required modifications (which falls beyond the scope of this paper) shows that they make the original efficient algorithm inefficient, with the problem appearing to be NP-complete, similar to the ‘shortest-path with constraints’ problem which is
shown NP-complete (e.g. in Garey and Johnson, 1979). Indeed, the problem could be considered as a ‘travellingsalesman’ type problem where mandatory events constitute nodes that must be visited during a circuit from/ to a pseudo-event node. Efficient exact methods are therefore unlikely, although a shortest-path approach has been adopted for practical tests to investigate whether on actual temporal topology systems the performance is acceptable. 5.2. Multivariate optimisation The optimisation theory used here for multivariate optimisation is that of Pareto optimisation (from Pareto, 1906) wherein a solution is considered to be optimal if there exists no other feasible solution that has equal or better costs across all variables considered. This usually results in a set of good solutions being produced (Foncesa and Fleming, 1995), from which the decisionmaker can choose. In order to search the solution-space for Pareto-optimal orderings of events, and assuming from the analysis of single-variable methods that no efficient exact approach is possible, two approaches have been taken. For small systems it may be feasible to perform an exhaustive search of all possible solutions, thus finding the true Pareto-optimal set. However, for large systems this would become impractical due to the large number of solutions to be generated and tested. Therefore, a genetic algorithm (after Holland, 1975) is being used as a suitable approximation method which has been used elsewhere for multivariate optimisation problems with considerable success (e.g. Foncesca and Fleming, 1995; Van Veldhuizen and Lamont, 2000). This is an algorithm that mimics the principles of natural evolution and is suited to problems where there
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is a large and poorly understood solution space and no easy analytical or linear approach (Mitchell, 1996). The fundamental idea is that combining sections of good solutions (in this case short series of events forming part of a possible solution) with sections of other good solutions will produce further good, or better, solutions. Thus, given sufficient iterations where only the best solutions survive to the next generation, a set of good approximations to the best solution will be produced.
6. Implementation and testing of a temporal topology application A decision-support application has been implemented based on the temporal topology theories outlined above. This implementation is built in the GE Smallworld GIS framework, using a metadata database to record the assignment of real world objects to temporal topology events and the relationships between these events. From this database, schematics of the temporal topology network can be produced which can then be analysed for single-variable optimisation using customised versions of the standard Smallworld network follower tools. An analysis engine using the Pareto optimal principle with a genetic algorithm or an exhaustive search has been written in the Smallworld Magik object-oriented language. Both these tools can read the data from both the metadata and GIS databases to perform the analysis. Once temporal topology analysis is finished, the sequence of events comprising each optimal solution can be viewed using the standard version-management tools in the GIS so that the planner can make a final decision.
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The scenario described in Section 4 has been used for some preliminary testing of this application, and of the ideas presented in this paper. This has shown that, whilst the temporal topology model as presented in Section 3 is usable and capable of representing a reasonably complex scenario in a fairly compact and straightforward manner, the analysis techniques outlined in Section 5 are of limited use in production of optimised solutions. For instance, although running a shortest-path trace to minimise spatial distance (i.e. attempting to produce a continuous network) took only 10 s, it resulted in a solution of C-B-D-E-R-Q-N-P-O-J-H-G, which as can be seen in Fig. 7 is a continuous network but in practical terms would probably not be considered an optimal solution due to the high level of redundancy. Using the shortest-path trace with other cost classes, however, failed as the physical memory limit, used for storing the paths found thus far, was reached before a path to the end was found. Even for this relatively small system of 18 events, an exhaustive search was found to be somewhat impractical, requiring approximately 1016 potential solutions to be generated and tested which with the current implementation and extrapolation of trial testing would take around 107 years. Processing using a genetic algorithm produced a set of solutions in a more reasonable timescale (around 5 h), but produced a set containing a large number of valid non-dominated solutions (over 100 in this case), a common problem with multiobjective techniques (Graves et al., 1992). Without the use of clustering techniques, such as those used by Zittler and Thiele (1999), to reduce the size of the Pareto-optimal set to a reasonable level, this form of analysis is therefore also of limited practical value. However, of these 100 solutions about six could be considered good by a more subjective measure of being
Fig. 7. Proposed cycle network produced from a network trace to minimise spatial distance, i.e. produce a continuous network.
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Table 2 Sample results and costs resulting from optimisation by network trace and genetic algorithm of system described in Fig. 6 Method/Result ID Trace Genetic algorithm
SD GA2 GA23 GA27 GA65 GA73 GA79
Events
Sp. dist. (km)
Cost (£000)
Dur. (h)
Des. (Sd)
Exp. (Se)
C-B-D-E-R-Q-N-P-O-J-H-G R-E-N-P-L-M J-M-N-D-C-P-E-R H-L-J-R-G-F-N-E-P-Q E-R-N-D-B-H-P-I-G J-H-M-N-D-C-P-E-R E-N-L-R-Q-P-M
0.000 0.013 0.103 0.251 0.143 0.146 0.062
305 119 120 119 127 120 119
30.46 13.16 13.86 13.74 16.35 13.97 13.22
36 19 25 33 32 28 22
30 16 18 20 24 19 17
Sp. dist.: Spatial distance (km, kilometres), Cost: Financial cost (£000, thousand Pound Stirling), Dur.: Duration (h, hours), Des.: Desirability (d, 5-+5), Exp.: Expediency (e, 5-+5).
continuous (i.e. no gaps in the network) and without redundancy (i.e. no extraneous sections), suggesting that this method is worth pursuing as a means of network design optimisation. These six solutions and their costs, together with the spatial distance trace result discussed previously are summarised in Table 2.
7. Conclusions This paper has presented a model, temporal topology, which aims to provide an efficient and useful method to represent and analyse alternative scenarios for network planning. The need for such a model was outlined, noting that most TGIS research has considered recording historical changes and has therefore concentrated upon linear models of time which cannot sensibly be used for modelling multiple scenarios. The perceived problems with standard branching models of time were highlighted and it was suggested that these could be overcome by recording, instead of all branches of time, just the relationships between events using a combination of temporal and Boolean logics. Through a small example, it was demonstrated that, using this technique, a number of alternative plans for a network could sensibly be managed. Some possibilities for optimisation of designs through analyses using this model were then introduced. Following this, a brief description of an implementation of an application based on the temporal topology model was given and some results obtained using this implementation with the example previously given were shown. Although, these optimisation results were perhaps disappointing, they do not detract from the validity of the model—indicating rather that further work is required on perfecting suitable optimisation methods before the model becomes truly useful for practical purposes.
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