The Use of Hybrid Agent Based Systems to Model ...

The Use of Hybrid Agent Based Systems to Model Petrol Markets Alison J. Heppenstall1*, Andrew J. Evans1*, Mark Birkin1, James MacGill2, and David O’Sullivan2 1

School of Geography, University of Leeds, Leeds, West Yorkshire, UK GeoVista Center, Pennsylvania State University, Pennsylvania, USA * Corresponding authors: http://www.geog.leeds.ac.uk/

2

1. Introduction Petrol is one of the most valuable oil derived commodities, valued by retailers and customers alike. Despite pressures on natural resources there is a rising demand for petrol associated with an ever increasing individual mobility. At the end of 2002, there were 11707 sites retailing over 36 billion litres of motor fuel per annum in the UK. This equates to an average of approximately 1,350 litres of fuel consumed by each vehicle per year. Consumers are becoming ever more aware of petrol prices; internet sites in the UK, such as the Automobile Association site, enable consumers to have an almost perfect knowledge of prices within their area. Consumer sensitivity to petrol prices was clearly demonstrated in the UK during August and September 2000 when there was a “Petrol Crisis”, consisting of consumer blockages of refineries and protests in reaction to soaring fuel taxes. This sensitivity and knowledge has created both a highly competitive and rapidly responding market, with organisations employing various strategies to maximise profits. Many processes combine at different temporal and spatial scales to affect each petrol station’s prices. For example: internal costs (cost of production; fixed costs like staff pay), external influences (price of crude oil and levels of taxation) and effects of locality (rural versus urban areas). More importantly, in the UK, competition within the local neighbourhood exerts a great influence on the setting of prices. For example, ESSO have explicitly stated they use a price-watching scheme whereby individual petrol stations attempt to undercut local competitors. The literature concerned with the examination of petrol prices and its relationship with other variables is vast. These studies generally differ in one or more of the following aspects: the variables considered; the country under scrutiny; the time frequency and period of data used; the focus on wholesale or retail prices and, finally, the model employed (Galoetti et al. 2003). Examples include Bacon (1991), Manning (1991), Shin (1994), Reilly & Witt (1998) and Mitchell et al. (2000). High frequency price cycles have been thoroughly investigated by Castanias & Johnson (1993), Noel (2002) and Eckert (2002; 2003). Typically, the models developed by researchers to represent the relationship between petrol and a variable are empirical and mathematical. They suffer from a number of problems, chiefly: the parameters are all on the same scale (behaviours executed at the ‘micro’ level are not tied to ‘global’ level variables like oil prices); the parameters are often difficult to estimate and lack realism; very little, if any, account of any geographical effects is taken, and, finally, mathematical models by their nature only consider quantitative parameters and therefore miss out on qualitative, behavioural information. Within this paper, we present a series of three multi-agent and hybrid models that seek to rectify some of these problems. The models are behavioural and work at the scale of individual customers and petrol stations, but take into account quantitative and global parameters.

2. Petrol Stations as Agents The three systems were developed in Java. Each has a similar model for the petrol stations. Individual petrol stations are created as agent-objects and supplied with knowledge of their initial starting price, production costs, and the prices of those stations within their neighbourhood. The prices are either set with real data or idealised data depending on whether the systems are being used to examine real or abstract dynamics. Each agent views the prices of neighbouring stations and applies a series of rules to adjust its own prices. The geographical distribution of the stations can, again, be either real or idealised. In this paper we present the results of real data experiments modelling the West Yorkshire region of the UK (Figure 1).

Key West Yorkshire SCOTLA AN ND D

Country Outline

¯ 150 75

0

E N G L A N D WALES

Key

150

Kilometers

County Boundary

A-Road

Urban Area

Petrol Station

¯ 10

5

0

10

Motorway Kilometers

(a)

(b)

Figure 1: (a) Location of West Yorkshire within the UK. (b) Showing the distribution of petrol stations and extent of urban areas within West Yorkshire.

The rules can be assigned to stations on the basis of the company running them, geographically, or simply given identically to all the stations. They are based on industry knowledge and implemented after experimentation with differing parameters. The interplay of different rule sets allows the stations to compete and implement behaviour with the aim of profit maximisation. The stations update their prices synchronously once per day. The parameters that form the basis of the rules are: • • • • •

The allowable minimum and maximum price. The amount each station might try to undercut its competitors by. The amount each station can be priced above its competitors before it takes action. The neighbourhood size in which stations are regarded as competitors. The station profitability compared with the last time iteration.

These parameters can be built into rules or strategies such as: “Am I more expensive by X amount than the competition in my neighbourhood (Ykm)? If yes, drop my price by Z”. Such rules are utilized under the following conditions: 1. If the profit is increasing then agents carry on with the current price change strategy. 2. If the profit is falling, agents pick a strategy to either: a. Increase the price.

b. Decrease the price. 3. If the profit between the last two iterations hasn't changed much, then keep the price constant until the profit falls again. The petrol stations are designed to be only interested in their own profit. Secondary aims, such as attaining market share or behaviours such as inter-station collusion are currently ignored. Future research will investigate these areas.

3. Customer models The models vary chiefly in how profitability is assessed. The three systems have an increasing complexity of customer model. Plainly, an individual-level agent based model of each potential customer would be a vast computational and behavioural modelling effort, so the three systems utilise simplified models of customers. The systems are: 1) A pure Agent based model in which the profitability is decided by simple price comparison. Stations attempt to undercut competitors while maximizing price. 2) A hybrid Spatial Interaction Model (SIM) - Agent system in which customers travel to petrol stations depending on the distance to them and their prices. Stations aim to maximize their profit based on customer numbers, price, and a fixed volume of fuel that is sold to each customer. Where real data is used, the customers’ initial locations are based on population and car ownership data from the UK census. 3) A hybrid SIM-Network-Agent system. This is the same as the hybrid SIM-Agent model, except that the customers are initially distributed on the basis of journey to work data along potential routes prior to the SIM running. This introduces an “intervening opportunities” style component.

4. Pure Agent model In the experiments conducted with the basic Agent model, all the petrol stations within West Yorkshire (Figure 1b) were initialised with the real data for one day and the same ruleset. The simulations were run to equilibrium (defined as the stage when all the prices within the area remained static for 3 days). The predicted days’ data was then compared with the real data for those days. Figure 2 shows the mean per-station price difference (in UK pence) between the real and agent data over the course of the data set.

Figure 2. Comparison of the mean price difference (UK pence) within West Yorkshire between the Agent results and the real data. The bars are standard deviations of the differences for each petrol station. Note that days 4 to 8 are missing from the dataset.

The model under-predicts the prices, a difference that increases over time. However, the model only took two days (iterations) to reach equilibrium. This is clearly not enough time for the model to operate the rule set effectively. Any variations in the difference between the real and model data after equilibrium were entirely due to changes within the real data. To examine whether the model could generate observed geographical variations, for example different price levels between rural and urban areas, the model was run with all the stations’ prices initially set to the average level from day one of the real dataset. As can be seen by comparing Figure 3a and Figure 3b, the model does not capture the system dynamics adequately enough to replicate these variations. Key

Key

Price Value (p)

County Boundary Motorway

71.00

A-Road Petrol Station

71.00

Price Value (p) 73.89

67.90

(b) Real Data

(a) Agent Model

Key

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Price Value (p) 67.59

67.19

62.99

63.31

¯ 10

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10

Kilometers

(c) Hybrid Model

(d) Hybrid Network Model

Figure 3: Price distributions for the various models ten days after runs started with all stations initialized at 71 pence: (a) Agent model, (c) hybrid SIM-Agent model and (d) hybrid SIM-NetworkAgent model. Data from day ten of the real dataset (b) is included for comparison. Price surfaces are interpolated from point station prices for ease of visualization.

5. Hybrid SIM-Agent model Spatial Interaction Models essentially calculate the amount of people travelling to multiple locations, based on levels of attractiveness and distance-costs. Traditionally, they are used for “what-if?” scenarios (Birkin et al., 2002), for example, what will happen to the sales of supermarket X if supermarket Y locates nearby? Within the petrol models discussed here, they are used to calculate customer numbers and, hence, volume of sales information by assuming a constant, average, customer fuel requirement. This information is then used to calculate the amount of profit that is being made. Each petrol station implements profit maximising strategies based on this information. The SIM model is in two parts: Equation 1 calculates the relative amounts of fuel sold by each station to the population of each Ward (UK Census area of ~400 to 35,000 people). The amount sold decays exponentially with the distance between the Ward centroid and the station, and also with price. The exponential fall off is traditional in SIM models and well matched by many retailing examples (Birkin, et al., 1996). Equation (2) ensures that the total volume of each fuel sold to each Ward equals the demand H i F m (the amount of fuel type m required by car owners in ward i per day). This gives the actual amount sold in each ward. λ and β are two coefficients which are usually determined by calibrating the model to real data.

[

Sˆijm = δ jm exp − βd ij − λp mj

S

m ij

=

]

Sˆijm H i Fm Sˆ m

∑

j

(1)

(2)

ij

where: −

S ijm is the amount of fuel m sold by garage j to ward i.

−

δ jm is 1 where garage j sells fuel m and 0 otherwise.

− −

dij is the distance between ward i and garage j. p mj is the price of fuel m at garage j.

− −

Hi is the number of households within the ward i. Fm is the amount of fuel of type m required per car per day.

This model assumes that the monetary costs of collecting the fuel are of no significance to the customer – this is reasonable as people are rarely driving just to get petrol. Interactions between the Agent model and Spatial Interaction Model can be summarised as: 1. 2. 3. 4.

The agent model and population are initialised with the real data. The fuel price and station location data is passed to the SIM. The SIM calculates fuel sales for each station and passes the information to them. The stations use this information to choose the rules to implement based on their profitability. 5. Each agent calculates its new price and the simulation returns to step 3 until equilibrium or a set time limit is reached.

The results (Figure 4) from the model would appear to suggest the Hybrid SIM-Agent model performs slightly better than naïve Agent model after the latter has reached equilibrium.

Figure 4. Comparison of the mean price difference (UK pence) within West Yorkshire between the Agent and Hybrid SIM-Agent results and the real data. The bars are standard deviations of the differences for each petrol station.

Given a constant price surface, this model can generate the known geographical variations and patterns within the real data better than the naïve Agent model (Figure 3c). Additionally, this model has a sounder theoretical basis than the agent model with each station reacting competitively to consumer actions.

6. The SIM-Network-Agent model The SIM model clearly models the dynamics of the system well, but suffers from having an over-simplified notion of when people buy fuel. Most people tend to buy fuel on the way to somewhere (Plummer et al., 1998) and, in most cases, this will be on the way to work. The naïve SIM-Agent model, however, represents people who travel directly from home to the nearest petrol station where fuel is cheap. To rectify this disparity, an additional model was added, which reallocates people to Wards on the basis of travel to work data. The model takes in the population leaving each Ward and going to each of the other Wards from census data tables. It then uses Dijkstra’s Algorithm to calculate the shortest distance through the intervening network of Wards for each set of Ward-to-Ward movements. Car users are then redistributed to the home, destination and intervening Wards evenly. This new population is fed into the SIM. Future work will use petrol journey data to weight the home populations of the simple SIM-Agent model against these new travelling populations appropriately. For now we present the initial results of a purely travelling population. The averaged results using the hybrid SIM-Network-Agent model (Figure 5) are almost identical to that of the hybrid SIM-Agent model. However, the hybrid SIM-Network-Agent model is much better than either of the other models at generating the geographical variation in the data, convincingly generating the observed rural-urban differences in price (Figure 3d).

Figure 5. Comparison of the mean price difference (UK pence) within West Yorkshire between the Agent and Hybrid SIM-Agent and SIM-Network-Agent results and the real data. The bars are standard deviations of the differences for each petrol station.

7. Conclusions In this paper we have presented two hybrid agent based models of a petrol market. These models were found to outperform a naïve agent model that did not account for important customer movement behaviour. Despite the good performance of the hybrid models, further experimentation with the assignment of different rules to individual brands is required to reproduce the interactions that occur within real markets. Multi-agent systems provide an ideal framework in which a retail market can be accurately described. Individual agents (petrol stations) can be successfully supplied with detailed knowledge of the market by the attachment of more specialised models, such as Spatial Interaction Models. This type of model can be extremely valuable when modelling trends at a regional level. The outcome of local interactions can be easily seen at a global level and behavioural and quantitative data can be easily combined. This flexibility, coupled with the upsurge in readily available computing power makes agent based modelling a valuable tool for studying market forces and dynamics.

Acknowledgements The work reported within this paper is funded by the Economic and Social Research Council and GMap. All digitised boundary data are UK Crown and ED-Line Copyright. The Network Model components were developed during an exchange visit to The GeoVista Center funded by the Worldwide Universities Network.

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