Mathematics and Computers in Simulation 27 (1985) 95-105 North-Holland
A MICRO-SIMULATION P.G. GIPPS CSIRO
MODEL FOR PEDESTRIAN
and B. MARKSJO
Division of Building Research, Highett,
Victoria 3190, Australia
The ability to predict how changes in the walking environment will affect the pedestrian flow is important to the designers of buildings and other constructed facilities. These changes can act on an individual pedestrian directly by diverting him from his preferred route. and indirectly through their effect on the other pedestrians. If the behaviour of individuals can be adequately modelled, and the appropriate distribution of pedestrian types is employed, their corporate behaviour be realistic. This paper presents a model for the interactions between pedestrians which is intended for use in a graphical computer simulation. The program runs on a microcomputer and uses interactive colour graphics to display the operation of the model and assist in the validation and verification of the model. 1
The need control uedestrian movements and around buildings an important of design. many buildings as and hospitals, pedestrian offices, traffic constrained to corridors, and have or no about the they take a particular origin In malls or on the hand, objects as benches, Eountains kiosks or stands frequently prevent following lines between origins and While objects provide of interest which people likely to talking watching the traffic, they involve pedestrians a choice route. From viewpoint of operat.ors of facilities these fulfil a role in the speed pedestrians and dispersing them, pedestrians that walk fast *are to be attracted window displays. if too impediments to movement the mall be unable handle the at times peak usage. the ability predict the of a in a or an area to behaviour of neighbours is important estimating the of changes the walking environment. the ability predict flows and around facilities is important 0378-4754/85/$3.30
0 1985, IMACS/Elsevier
models of flow are the most limited to quasi-steady flow in corridors Cll, Predtechenskii Milinskii C21 Fruin C31). many buildings pedestrian flows are transient vary over short time-intervals. Such in flows arise from such as a lift disgorging its passengers, or a set of traffic signals outside the building allowing pedestrians to cross the road and enter the building. Consequently, it is desirable to be able to model the behaviour of pedestrians in more detail than is provided by macroscopic models. One of the simplest ways to handle the stochastic nature of such situations is to perform a micro-simulation of the pedestrian movements. This paper presents a simulation model for pedestrian flows and interactions designed to possess the following properties: The model should mimic real pedestrians. The pedestrians should be of a number of different types, and it should be possible to change their characteristics and numbers to suit the situation being investigated. The parameters in the model should correspond to obvious characteristics of pedestrians whenever possible. Parameters that do not conform to this requirement should be set within the program and be transparent to the user.
Science Publishers B.V. (North-Holland)
P.G. Gipps, B. Murksjii / Micro-simulation
The model should operate satisfactorily in a discrete formulation. Since decisions and movements in reality are being made in parallel in a continuous space-time framework in a fashion that is not possible in a simulation. the errors generated by resorting to sequential decisions in a discrete or partially discrete framework should not be too gross. The model should be easy to upgrade to more detailed descriptions of behaviour if necessary. Approximations to real behaviour which are satisfactory in one context are not necessarily acceptable in another. However, making the model more complex than necessary for general use is likely to incur time penalties while processing the simulation. Consequently the basic model should be simple. but nevertheless relatively easy to modify or refine. The operation of the simulation should be suitable for real time graphical monitoring. Many potential users are more likely to be interested in ‘seeing’ what conditions certain layouts produce rather than poring through tables of figures describing them. The simulation is tackled at the level of the individual pedestrian under the hypothesis that if the behnviour of individuals is modelled adequately, and the appropriate distribution of pedestrian types employed, the corporate behaviour of the simulated pedestrians will be realistic. Further, by working at the level of the individual it is possible to collect data on individual travel times and diversions, and subsequently to analyse the variability between different types of pedestrian. However in order to simulate pedestrian flows at the level of the individual, it is necessary to be able to model the way in which pedestrians select their routes and move along them. The present model separates these two aspects of pedestrian behaviour into independent sub-models which can be treated sequentially. That is,the pedestrian selects a route or part of a route , and t.hen endeavours to follow it as consistently as possible. This separation of pedestrian behaviour into these two components permits the development of efficient mathematical criteria at later stages.
While the subject of the present paper is the interactions between pedestrians, it is necessary to discuss the general principles of route selection so that the relationship between route selection and pedestrian Unless the interactions can be appreciated. relationship is understood the criteria and behaviour associated with pedestrians while following their route may seem too limlted.
According to Ciolek C41 the route selected should possess as many as, of the following properties as possible, (a) (b) (c) (d) (e!
The route should be the shortest path connecting the origin and destination. The route should not lead to collisions with fixed obstacles. The route should minimize the number of sharp and rapid changes in direction. The route should offer pedestrians some stimulation: people tend to walk at the side The route should not be located too close to a wall
These ideals are not mutually consistent and their relative importance is likely to vary between different classes of pedestrians. Two extreme examples are: lunrh-time shoppers who are likely to place little weight on the shortest route with minimum sharp turns but a lot on stimulation, and peak-hour commuters who are intent on getting to their office or catching their train and have no time to waste on shop windows. Consequently, any model for route choice must take account of the differing behaviour of various classes of pedestrians.. Because a pedestrian cannot necessarily see his final destination from his startins and may in any case choose to deviate point, from a direct path, route selection is based around the concept of intermediate destinations (or nodes) generated by the objects in the open area. The model uses the physical layout to generate a number of A pedestrian walking between his nodes. origin and destination moves from one node When he is within a short to another. distance of the node to which he is walking, he has to make a decision about the The choice is limited by following node.
P. G. Gipps, B. [email protected]
the requirement that the next node must not be hidden from his present position by a That is, a straight line fixed obstacle. between the present node and the next does not intersect any obstacle.
model for pedestrian
In order to ensure that all pedestrians walking between a particular origin destination pair do not take the same route, the selection of successive nodes in a pedestrian’s route is stochastic. The probability of selecting a particular node to succeed the current one depends on the relative attractions of the candidates.
A number of general propositions can be advanced concerning the location of nodes. The function of nodes is to serve as points where pedestrians can change their general direction and thus negotiate obstacles. In consequence, nodes are associated with places where pedestrians may need to change Once within a circular zone direction. surrounding a node, a pedestrian is free to select the next node on his path. For instance, when a pedestrian walking from A to B in Fig. 1 reaches the circular zone around B. he has to choose between C and D for his next node. For simplicity, the radius of this circular zone is refered to as the release distance. Nodes are associated with origins and and with obstacles. All destinations, obstacles are represented as a series of straight line segments and a node generated near each of the resulting corners. (Even a circle can be represented by an n-sided polygon, where n may be made as large as necessary to provide the desired level of approximation.) The model locates the node associated with a corner on the bisector of the the re-entrant angle formed by the two edges, some B metres from the corner (Fig.1). The release distance is taken as r metres. This approach is simpler than that adopted by Lozano-Perez and Wesley CSI and Jarvis C61 for finding minimal length collision-free paths by ‘growing’ polyhedra obstacles to account for the size and shape of vehicles, and achieves this simplicity by relying on the capacity of the pedestrians that are eventually simulated to make the minor course corrections necessary to avoid obstacles.
When a pedestrian walking from A to B reaches the circular zone around B, he has to choose betwee> C and D for his next node. The radius of the zone (release distance) is Y and B is located some 6 metres along the bisector of the re-entrant angle at the corner of an obstacle. *
An approach using a function of path length to calculate the probability of selecting a particular node looks promising. The use of distance in the selection procedure means that the relative likelihood of the shortest path being selected can be controlled more directly, while the triangle inequality tends to eliminate unnecessary sharp changes in direction: ideals !a) and cc) respectively. In its simplest form, the method employs the minimum path length to the pedestrian’s final destination as the criterion. The distances between pairs of mutually visible nodes are used to obtain a set of minimum distances from each node to the destination. The shortest distances from the current node to his final destination via each of the alternatives can then be derived.
P.G. Gipps, 8. [email protected]
The attraction of this method is that perceived distance can be used rather than For instance, shoppers or actual distance. sightseers can be made to perceive the distance of a link beside a wall as being shorter than it actually is so that they are encouraged to select paths beside walls (or shop windows). Further, if the phrase ‘the shortest path’ in ideal (a) is replaced by ‘the shortest perceived path’, where the perceived distance is a function of the type of pedestrian and the sources of stimulation, it is possible to make the desirable route properties mutually consistent.
model for pedestriun
(a 1 S m
P (b) S m
The generation of paths from a multiplicity of nodes allows the simulated pedestrians to depart from the physically shortest route at points of interest, The movement of pedestrians between nodes can now be treated as the simple task of getting to the next node as efficiently as possible, subject to a number of constraints. The possibility of deliberate diversion does not arise until the pedestrian gets within the release distance of his immediate destination. 3
MOVEMENT ALONGA LINK
Macroscopic studies of pedestrian flows in corridors have produced the predictable conclusion that the denser the packing of pedestrians, the slower is the mean speed. The main differences between the studies have been the nature of the relationship and the values of the parameters. In the context of micro-simulation this result can be interpereted as: The presence of other nearby pedestrians will cause a particular (For clarity in pedestrian to slow down. the discussion this particular pedestrian will be referred to as the subject, to distinguish him/her from the rest of the pedestrians.) Consider the situation illustrated in Fig. 2(a), in which another pedestrian, P, is located close to the subject, S, and between the subject and his intended destination, 0. If the subject were to take a normal step along his intended path he would collide with the other pedestrian (Fig. 2(b)). Consequently he is forced to step short tFig.2!c)). However, if the subject deviates from the straight line to his destination (Fig 2(d)), he may still be
Illustration of another pedestrian, P, being located close to the subiect, S, and in the path of the subject’s destination, DI (a). If the subject were to take a normal step along his intended path he would collide with the The other pedestrian (b). alternatives are for the subject to step short (c) or deviate from the straight line cd).
P. G. Gipps, B. [email protected]
forced to step short but nonetheless be able to move closer to his destination than if he had adhered to a straight line. This suggests that the model of pedestrian behaviour on a link should attempt to maximize the speed of the subject towards his destination, and that any deviations from a straight line should arise solely from attempts to achieve this goal. The model hypothesizes the existence of repulsive forces between pedestrians so that as the subject approaches another pedestrian the ‘potential energy’ of his position rises and the ‘kinetic energy’ of his speed drops. The repulsive forces also deflect him from a straight line. The situation is loosely analogous to that of a body moving through gravitational fields generated by a number of other objects, except that the forces are repulsive rather than attractive, and not necessarily symmetric about the bodies concerned. In theory the subject should use all the available information, which includes his own position, s, the location of his destination, d, and the location, p,, and velocity, vi, of other pedestrians when selecting his own speed and direction of movement. However, if the location of the subject is allowed to vary continuously in the x-y domain and through time, the solution to this problem will be too time consuming to solve on a regular basis in a simulation. In practice, this continuous control mechanism has to operate on a discrete basis with the subject being moved by some increment each time. A further complicating factor arises from the number of pedestrians present in the simulation. Even though the influence of other pedestrians is likely to decline rapidly with distance, it is still necessary to consider the separation between the subject and each of the other pedestrians to determine whether that pedestrian needs to be included in the formulation. Consequently the time taken to carry out the simulation is likely to vary as the square of the number of pedestrians. The solution adopted was to tackle the simulation on a ‘particle in a cell’ basis. The study region is divided into squares 0.5 metres on each side by a rectangular grid.
model for pedestrian
Each cell can be occupied by at most one pedestrian, and a score is assiuned to each cell on the basis of its proximity to pedestrians. This score represents the repulsive effects of nearby pedestrians, and has to be balanced auainst the gains made by the subiect in moving towards his destination. Where the fields of two uedestrians overlap, the score in each cell is the sum of the scores generated by the pedestrians individually. In a simple model, in which the subject does not take account of the directions in which other pedestrians are travelling, the scores would be symmetric around a pedestrian. For example, the cell occupied by a pedestrian might be given a score of 1000, the cells with a side in common with this a score of 40, the cells with a corner in common a score of 13> and more distant cells still loiier scores. The very high repulsive score in the cell occupied by a pedestrian serves as a absolute barrier to another pedestrian The scores in the joining him in the cell. surrounding cells are approximately inversely proportional to the square of the separation of pedestrians in the two cells. The function used is 1/C(A-0.4)2+0.0151 where A metres is the distance separating the centres of the two cells, 0.4 metres is slightly less than the diameter of a pedestrian, and 0.015 is an arbitary constant to moderate fluctuations in scores In a more , close to the pedestrian. elaborate model, the ‘potential’ shadow cast by a pedestrian need not be symmetric, but can make use of size, asymmetry and intensitv to convey direction and speed information. The cell to which the subject moves is determined by calculating a net benefit for the move to each of the accessible cells, and selecting that cell with the maximum The benefit is obtained by benefit. subtracting the cost of moving closer to other pedestrians (as measured by the score in the cell) from the gain the subject obtains by moving closer to his destination. The need to mimic the behaviour of real pedestrians imposes limits on the length of single moves by the simulated pedestrians, If the feasible moves are too large (one metre or more), there exists a possibility that the subject may leapfrog a pedestrian
P.G. Gipps, B. Murksjij
This is not immediately adjacent to him. conventional behaviour, and to avoid its occurrence the model constrains the subject to remain stationary or move to one of the eight cells surrounding his current Thus the subject has to select position. the cell with the maximum net benefit from the set of nine surrounding and including the subject’s present position. If the measure of the gain obtained by moving to a particular cell is a function of the change in separation of the subject from his destination, the longer step-length assocated with diagonal moves will cause the subject to move diagonally in the absence of other pedestrians unless the destination is in the same row or column as the subiect’s current ccl 1. This preference for diagonal moves will persist even when other pedestrians are present and will distort the subject’s path, Pedestrians tend to follow a diagonal until they reached the row or column containing their destination and then proceed parallel to an axis (Fig. 3(a)). The tendency toward diagonal moves can be overcome by making the gain from the move dependent on the angle of deviation from the desired path rather than the distance of the target cell from the destination. This produces a sequence of moves similar to that in Fig. 3(b). Despite the numerous corners in this trajectory it is effectively straight since it can be viewed as a discrete representation of a perfectly straight line within the limits of accuracy imposed by the cellular structure. The obvious functions to assess the gain of a particular move are trigonometric since they can be obtained directly from the coordinates of the subject, his destination and target cell. The function has to be positive when the deviation is less than n/2 and negative when the deviation is greater than n/2. The selected function. P(a). is defined by P(a,)
= K cos cr, lcos
is a constant of proportionality to enable the gain of moving in straight line to be balanced against the costs of approaching other pedestrians too closely, and
model for pedestriun flows
D+es+t'+"a: i o+n
0: i,ti,t Figure
When the gain obtained by moving to a particular cell is a function of the distance moved towards the subject’s destination. pedestrians will tend to follow a diagonal until they reached the row or column containing their destination and then proceed parallel to an axis (a). This tendency toward diagonal moves can be overcome by making the gain dependent on the angle of deviation from the desired path Cb).
u, is the angle by which the pedestrian deviates from a straight line to his immediate destination when
From elementary geometry, angle between two vectors
the cosine of the is the inner
P. G. Gipps, B. Marksjti / Micro-simulation product product
of the two vectors divided of their module. Thus.
(xi-g).(d-g) cos u, = ----------IX,-_sl
model for pedestrian flows
where, is the location X, is the location and is the location D.
Xl -s d
of the target
of the subject,
of the destination,
i = 5, 6, 7,
i = 5, 6, 7,
and in consequence P(u,)
= - P(u,-,)
The calculations involved in selecting the next target cell can be performed relatively rapidly and efficiently by some minor rearrangements and constraints. Consider the nine target cells shown in Fig.4. If the numbering of the target cells remains constant with respect to the location of the subject and the direction of the axes, xl-~ will be constant for given i. Consequently x,-s can be stored rather than recalculated each time. Further, from Fig.4 it can be seen that -
Designated target cells with a 0.5 metre grid. The subject occupies cell 0. i=O
While a positive multiple of cos u has the same general attributes. P(a) is computationally more attractive as it can be calculated without taking a square root. Consequently, in the absence of any theoretical reasons for preferring the cosine, P(a) was adopted as the objective function. For completeness P(a) is defined to be zero when the subject remains stationary.
= -------------------_---___-__ 1x4 -s12 ld-si2 --
Thus P(a,) need only be calculated for four of the taruet cells. and the net benefit. B,, the subject obtains by moving to target cell i is given by
2, 3, 4,
6, 7, 8
is the score in target cell i after the repulsive contribution of the subject has been subtracted
Now, the term, Id-al, in equation (3). is common to all the target cells and only needs to be calculated once, while IX,-~I~ is equal to l/2 for diagonal moves, and l/4 for moves along a row or column. Thus if a new benefit function, Bi,, is defined which is related to the old function by the factor, I&-al’, an equivalent for (6) is obtained which does not require any divisions. Further. by replacing distancebased coordinates by cell subscripts and requiring K and the cell scores to be integers, it is possible to reduce all calculations to integer additions, subtractions, and multiplications. 4
OPERATIONOF THE SIMULATION
The simulation operates by considering each pedestrian in turn as the subject and moving him/her to the cell with the maximum net benefit, then proceeding to the next pedestrian. This sequential operation prevents conflicts from arising when two or
more pedestriansseek to enter the one cell at the same time. However, not all pedestrians desired speed, and differences
have the same in speed are
P. G. Gipps, B. [email protected]
obtained by varying the frequency with which particular pedestrians are moved. The model allots pedestrians to five different lists depending on their desired speed. During the simulation all pedestrians on a particular list are processed before the program moves to the next list. The lists are processed in the order - 5, 4, 3, 5, 2, 4, 5, 3, 4, 5, 1, 4, 5, 3, 2 - with the complete cycle taking one second of Thus, on the basis of a simulated time. notional 0.5 metres per move the model can cope with speeds ranging from 0.5 to 2.5 metres/second in steps of 0.5 metres/second. The treatment of the repulsive effects of pedestrians is readily extended to walls and other obstacles. Thus the route selection algorithm can be used to promote routes beside walls and the ‘potential’ scores of cells crossed by the wall (or close to it) can be used to prevent pedestrians from bumping into the wall or getting to close to it. From the computational view, it is only the score in a cell that matters: it is immaterial whether the score was generated by a wall or a pedestrian. This formulation of the model for movement along links produces two classes of The first class consists of parameter. those parameters which are set by the user to characterize the situation being simulated and take values which are directly observable such as mean speed and flows of the various classes of pedestrian. The second class of parameters are transparent to the user and consists of the pattern of scores which compose the shadows of the pedestrian, and the balance between the value of maintaining a straight line and the cost of approaching another pedestrian too closely. 5
One problem associated with the development of any computer program is that of ensuring that it actually functions as it was This is not a major difficulty if intended, the purpose of the program is to perform a number of routine calculations according to An assortment of test established formulae. calculations is usually sufficient to determine whether the program is functioning correctly, and appropriate amendments can be
model for pedestrian flows
made if necessary. is attempting to system for which are tentative or established, the can present major
However, if the program model the operation of some the descriptive formulae experimental rather than task of detecting faults difficulties.
There are three classes of fault which are likely to be encountered when developing a program as part of a research project. The first class of fault consists of errors in translating formulae into programming code. Faults of the second class occur when the hypothesized formulae do not produce the intended result, and as a consequence the computer is unable to reproduce specific behaviour patterns observable in the original. The third class of fault is represented by aberrant behaviour: modes of behaviour which the researcher does not intentionally include in his model, but which are not explicitly excluded because he does not envisage the possibility of the model misbehaving in such a Eashion. Trying to detect an unknown number of unspecified errors in an experimental program by perusing the final output or wading through tables of numbers produced at intermediate points in the program is a difficult and time consuming task which has no guarantee of success. Therefore it is well worth considering other faster and more certain means of detecting faults. For simulation programs, graphical techniques can provide a powerful nonspecific tool for detecting faults in coding and logic. The graphical techniques employed with this model took advantage of the interactive and colour capabilities available on most microcomputers. The graphics were designed to provide a plan view of the area being studied, and to move the pedestrians over the area in accordance with the simulation (Fig. 5). Obstacles, entrances, and exits are drawn as they appear, and by employing a range of colours it was possible to identify individual pedestrians or pedestrian types easily (Fig.6). An example of the third class of fault which was encountered during the development of the model was an instance of one pedestrian leapfrogging another. The cause was readily
P. G. Gipps, B. [email protected]
/ Micro-simulation model for pedestrian flows
recognized as the length of a single move being too long. With hindsight, this is an obvious problem that needs to be dealt with during the formulation of the problem, but it is not one that was anticipated. Nor would it have been easy to detect without the use of graphical techniques. Once detected, however, it was easily remedied and the solution incorporated in the model as part of the original formulation.
The concept of graphical checking is similar to the visual simulation models of Withers and Hurrion C71 in which the user was presented with a dynamic display representing the model. Withers and Hurrion were more concerned with being able to alter the model’s parameters while it was running than with changing the structure of the mode 1. However, the technique can be used also to check the structure of the model. and make the tasks of validation and calibration simpler and more thorough. The development of the graphics was influenced by the desirability of being able to change the scale of the display to suit the aspect of the model being investigated. Consequent on this was the need to be able to control the centre of view. As a result, the graphics display incorporated the interactive capability to pan in or out from the study area as well as move north, south, east, or west. 6
CONCLIJS I ON5
The cellular approach to pedestrian simulation has proved to have a number of attractive properties. The simulation operates by considering each pedestrian in turn and moving him to the cell with the maximum net gain. The cell to which the subject moves is determined by simple calculations, and the only a limited number of cells around the subject need to be Figure
Succession of moves as two pedestrians approach each other. First, the repulsive forces cause the pedestrians to deviate from their paths. Then, when they have passed, the desire to reach their destination restores them to a path close to the original.
P.G. Gipps, B. Marksjij / Micro-simulation
model for pedestrian flows
TIME Yin. 7
l4n R16flT Pln LfFT
U II c
Pm UP Pm DONI CYlRAcr rlrr EXPAND rirr
f I s
2 Notre grid
Example of graphical
tried. Consequently, the time required for the simulation rises in proportion to the number of pedestrians rather than the square of the number of pedestrians. The very high repulsive score in the cell occupied by a pedestrian serves as a barrier to another pedestrian joining him in the cell. The scores in the surrounding cells can be adjusted easily while the program is running and do not require methods of solution that are particular to the function that generated them, and the size, asymmetry and intensity of the ‘shadow’ can be used to convey direction and speed information. Because the operation of the simulation is carried out in integer arithmetic and does not distinguish between the effects of pedestrians and obstacles, computation is rapid.
from the simulation,
The formulation of the model produces two classes of parameter of which only the first (and simpler) class is set by the user. This consists of those parameters which characterize the situation being simulated and take values which are directly observable such as speed and flows. The graphical techniques take advantage of the interactive and colour capabilities available on most micro-computers, to provide a plan view of the area being studied, and to move the pedestrians over the area in accordance with the simulation. Obstacles, entrances and exits are drawn as they appear, and by employing a range of colours it is easy to identify individual pedestrians or pedestrian types. The model is still in an early stage of development, and work planned for the future
P.G. Gipps, B. Marksjii / Micro-simulation
10 t t
t t t 9 8 7 t ttttt-
Designated target 0.25 metre q-id. occupies cell 0.
cells to those shown in Fig. 7, it will be possible to produce smoother trajectories, while increasing the calculation involved j.rt selecting the next rell by less than 50 per cent.
1 2 3 12
model for pedestrian flows
cells with a The subject
involves testing the model against macroscopic observations of flow characteristics in corridors. Such observations already exist in the literature !Toqawa (1955). Predtechenskii and Milinskii (1969) and Fruin (1972)). The next step will be to compare the model with data collected on a microscopic basis. It is also intended to reduce the cell sizes to 0.25 metres per side and allow pedestrians to overlap the sides of the cell thev nominally occupy. This is expected to facilitate the treatment of pedestrians of different physical diameters. Groups of pedestrians walkina together can then be represented as a single super pedestrian. Further. by retaining the 0.5 metre step length but limiting the possible target
:ll Togawa. K. Study on fire escapes on observation of multitude currl?nts, (Japanese Building Research Institute No. 14. Tokyo (1955;.,
C2! Pedtechenskli. V.M. and Milinskii. A.I. Planning for foot traffic flow in buildings. fAmerind Publishina Co. Pvt. Ltd., New (1978?. 1 (‘translated from Russian.) Delhi, C?! Fruit-~, J.J. Designing for pedestrians: a ievel of service concept. [Highway Research Record 355 (1971)). C41 Ciolek. M.T. oedestr 1an areas.
Spatial behaviour in Ekistics 268 (1978’),
L53Lozano-Perez. L. and Wesley. M.A. An nlqorit hm for planning co11 ision-free -paths amono polyhedral obstacles. Communications or the ACM, 22 (101 (19791, 560-570. f6l J,arvis. R.A. Growing polvhedral obstacles for tjlanninq collision-free paths. The Austral ian Computer Journal. 15 ( 1983). 10:-111. 171 Withers, S.J. and Hurrion. R.D. The interactive development of visual simulation model s. Journal of Operational Research. 33 (19821, 973-975.