Genetic algorithm optimisation of an agent-based ... - Semantic Scholar

Environment and Planning B: Planning and Design 2007, volume 34, pages 1051 ^ 1070

DOI:10.1068/b32068

Genetic algorithm optimisation of an agent-based model for simulating a retail market Alison J Heppenstall, Andrew J Evans, Mark H Birkin

School of Geography, University of Leeds, Leeds LS2 9JT, England; e-mail: [email protected]; [email protected]; [email protected] Received 1 April 2005; in revised form 9 June 2006; published online 9 March 2007

Abstract. Traditionally, researchers have used elaborate regression models to simulate the retail petrol market. Such models are limited in their ability to model individual behaviour and geographical influences. Heppenstall et al presented a novel agent-based framework for modelling individual petrol stations as agents and integrated important additional system behaviour through the use of established methodologies such as spatial interaction models. The parameters for this model were initially determined by the use of real data analysis and experimentation. This paper explores the parameterisation and verification of the model through data analysis and by use of a genetic algorithm (GA). The results show that a GA can be used to produce not just an optimised match, but results that match those derived by expert analysis through rational exploration. This may suggest that despite the apparent nonlinear and complex nature of the system, there are a limited number of optimal or near optimal behaviours given its constraints, and that both user-driven and GA solutions converge on them.

1 Introduction The retail petrol market comprises a variety of different petrol retailers ranging from large multinationals to small independent outlets. Due to the commercial nature of the petrol market, there is little published information on strategies that retailers may employ. However, a recent study by Ning and Haining (2003) showed that changes in petrol prices at a local level are influenced by the location of competitors and the amount of custom. At a higher (national) level, influences such as taxation and crude oil prices become significant. All these factors interact to produce the price at the pump. Traditionally, researchers have employed empirical regression techniques to model the retail petrol market, focusing on the relationship between the petrol price and one or more variables, normally crude oil price (Galoetti et al, 2003; Reilly and Witt, 1998). However, this approach is incapable of modelling the numerous spatial and temporal influences found in the retail petrol market. An alternative method, a hybrid agentbased model, has been constructed (Heppenstall et al, 2005). This approach uses agents to model the decisions and strategies taken by individual petrol stations whilst a spatial interaction (SI) model is used to represent the effects of consumer behaviour and predict sales and station profit. This framework was successful, with initial results reported in Heppenstall et al (2006) supporting the use of this methodology for a dynamic, locally interacting system such as the retail petrol market. Within this paper, we examine the expert-led method used to select model parameter values based on numerical experiments and real data analysis. A genetic algorithm (GA) is presented as an alternative and objective method for optimising model parameters. The results of using the GA for optimising the model parameters are compared with the results of the first method and conclusions drawn about the validity of each approach.

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A J Heppenstall, A J Evans, M H Birkin

2 Agent-based modelling There is no universal definition of an agent (Franklin and Graesser, 1996), and with an ever-increasing list of agents appearing (Nwana, 1996) the most useful characterisation comes from Wooldridge and Jennings (1997): `Àn agent is an encapsulated computer system that is situated in some environment and that is capable of flexible, autonomous action in that environment in order to meet its design objectives.'' This approach is well suited to geographical applications where there is a discrete set of spatially distributed entities which interact with each other, and their environment. As alluded to in the introduction, the retail petrol price market exists within a complex network of subsystems, subsystem components, interactions, and organisational relationships. In simplistic terms, changes in petrol prices at a local level are influenced by the location of competitors and the amount of custom. At a higher/national level, influences such as taxation and crude oil prices become significant. All these factors interact to affect the final price. Agent-based models are a natural metaphor for complex systems such as the retail petrol market. As petrol stations are discrete entities with their own price setting strategies, it is easy for individual petrol stations to implement different price setting strategies to model, for example, the pricing policies of different multinationals (such as BP and Esso) or supermarket chains. 2.1 Assigning rules

Rules could be assigned to stations on the basis of the company running them, geographically, or simply given identically to all the stations. The rules in this model are based on industry knowledge and implemented after experimentation with differing parameters (parameters are listed in table 1). The interplay of different rule sets allows the stations to compete and implement behaviour with the aim of profit maximisation. Table 1. Name, definition, and value of each parameter used within the model. Parameter

Definition

Òptimal' value

b

Coefficient controlling the impact of distance on consumers' propensity to buy fuel at a petrol station

0.0003

l

Coefficient controlling the impact of price on consumers' propensity to buy fuel at a petrol station

0.7

fixedCosts

Fixed daily costs, eg, salaries

£80

costToProduce

The amount per litre that it costs the station to produce and sell the petrol

66 p

changeInProfit

The level of profit under which the station will not change its strategy

£40

Overprice

The amount a retailer believes they can be more expensive than their neighbours without losing customers (cf, the actions of the SI model which determines whether this is actually true)

5p

Undercut

The amount by which petrol stations can undercut competitors, eg, 1 p, 2 p

1p

Neighbourhood

Radius around each petrol station within which it looks to compete with other stations' prices

5 km

Genetic algorithm optimisation of an agent-based model

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The first set of rules that were derived for the petrol agents were reliant on the assumption that remaining competitively priced would result in profit maximisation. After assessing their own price and their neighbour's price, the following rules were operated: 1. If my profit is increasing then carry on with the current price-change strategy. 2. If my profit is falling, either: (a) increase the price or (b) decrease the price. 3. If my profit between the last two iterations has not changed much, then keep the price constant until the profit falls again. Initial experimentation detailed in Heppenstall et al (2005) showed that using these rules alone was ineffective at recreating the patterns and trends in the real data: the petrol agents were not competitive enough. Even though petrol consumption is relatively insensitive to price changes and consumers tend to buy petrol on the way to complete other tasks, the model still needs consumer choice to accurately reflect real profit-driven trends. 2.2 Spatial interaction models

A solution that would account for both consumer behaviour and a strong profit-based strategy within the agent model was the incorporation of a spatial interaction model and a travel network. Traditionally, SI models 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 this research, the SI model was 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 (see Heppenstall et al, 2005, for further details). The use of an SI model replaces the computationally difficult alternative of modelling millions of individual consumers. 3 Model and data overview Within the model, individual petrol stations are created as agent-objects. The stations are located in geographical space. In this paper all the petrol stations in West Yorkshire were modelled. The price-setting behaviour of each station is governed by a predefined rule set that is directed at maximising profit levels. Customer behaviour is modelled using an SI model. In the SI model, the volume of petrol sold by a particular station to the people in a particular census `ward' (10 000 people) depends exponentially on ÿbd and ÿlp where d is the distance between the station and the ward and p is the price of the petrol at the station. Parameters b and l therefore determine the sensitivity of consumers to distance and price. The parameters that formed the basis of the agent rules were decided upon after analysis of two separate sources of information; the real price data for petrol retailers with a known geography over a period of three months and published information on retailer behaviour (see table 1 for list and definition of parameters). The rules were formed on the basis of this information and experimentation. The real data consisted of daily petrol price readings taken throughout the months of July, August, and September 1999 for petrol stations within West, North, and South Yorkshire (henceforth referred to as the Yorkshire region). Unleaded petrol prices were recorded at a range of stations from large multinationals to independent retailers. Analysis revealed price differences between individual stations and broad geographical areas, as well as the extent of the local competitive environment. Information from published sources

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(The Monopolies and Mergers Commission, 1990; Ning and Haining, 2003) and expert knowledge supplied details on profit margins and amounts or volumes of petrol sold in different locations such as rural, urban, and supermarket-controlled areas. 4 Assigning parameter values 4.1 b and k

In the Yorkshire region, the average distance from a petrol station to its nearest neighbour is 1423 m. This is partly skewed by large rural distances. Within a neighbourhood of this size each petrol station would actually have, on average, four neighbours. For b to exert an influence on the system, there needs to be significant competition between stations. To enable this, 1000 m was added onto the average distance giving a value of 2400 m. This gives an average of seven stations within the neighbourhood. Taking this value, a consumer located 2400 m from the garage would be half as likely to buy petrol at the station as from an equivalently priced station situated adjacent to the consumer. Taking 2400 m as the distance at which sales drop by half produces a b value of 0.0003. Parameter l was calculated by using a combination of the average and maximum price change per day, derived from the real data. The mean absolute price change (calculated over those stations where the price altered) was 0.27 p per day, the maximum price change was 2.6 p. However, only a small percentage of stations changed their prices by any amount greater than 1.0 p. On the basis of this, 1.0 p was chosen. Assuming a 1 p price rise leads to a halving in the sales, this translates to a l value of 0.7. As can be seen, in both the case of b and l, informed assumptions were utilised; however, detailed experimentation (Heppenstall et al, 2006) revealed that the model is relatively insensitive to both the values. 4.2 fixedCosts and costToProduce

Initial experimentation concluded that station profit decreased with increasing fixedCosts and costToProduce as one might expect. However, detailed data pertaining to a retailer's profit are commercially sensitive and therefore not available within the public domain. The Monopolies and Mergers Commission's (1990) report usefully highlighted that the profit margin is very tight due to the constraining factors of fuel tax and duty. From this we can assume that the costToProduce parameter will be a higher proportion of the actual price than the fixedCosts parameter. One detailed breakdown of a station's costs was published by The Monopolies and Mergers Commission (1990) report. Breaking down a price of 74.9 p per litre: 57 p accounts for the duty and tax, 13.9 p covers the cost of the product and the remaining 4 p goes to the retailer. It can be assumed that the greater proportion of the 13.9 p covers the costToProduce (minus tax) and the remainder per litre covers the fixedCosts. This is estimated at 13 p for costToProduce and 0.9 p for fixedCosts. The total costToProduce in this example was 57 p 13 p 70 p (these are the variable costs). Given, on average, 8 000 litres per day is sold from the average station (based on data from Catalist.com accessed 2003), the 0.9 p per litre that goes to fixedCosts gives an estimate for total fixedCosts of »80. However, petrol in the real price data was, on average, 4 p cheaper than that quoted by the Commission's report, therefore the costToProduce is taken as 66 p rather than 70 p. The fixedCosts remain at »80. 4.3 changeInProfit

Initial experimentation showed that the system was sensitive to variability in the changeInProfit level. Values over »50 led rapidly to a steady solution, whilst values under »30 resulted in the system staying in flux, with prices at stations continually varying by several pence. The median value, »40, was chosen to ensure that the system


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ran for around thirty days (enough time for the system to operate the rules) before reaching equilibrium (matching the rate of real fluctuations). 4.4 Neighbourhood

The average distance between stations in the Yorkshire region was calculated at 1423 m; this equated to an average of four neighbours. However, to ensure that stations within rural areas have some neighbours to compete with, the Neighbourhood needed to be considerably larger. Esso promise to match any competitors' price within 5 km/3 miles in their Price Watch Policy, and this is thought to be approximately true of all pricewatching stations. A Neighbourhood of 5 km gave an average of eighteen neighbours, and initial experimentation showed this to produce realistic results. 4.5 Undercut

A small value for the Undercut parameter may result in the removal of the variations that the real price data show to occur; too large a value may result in the parameter having no effect as other controls swamp it. To find a suitable value, the difference in the cheapest and second cheapest stations was calculated for each neighbourhood. For a 5 km Neighbourhood, 157 of the neighbourhoods had an Undercut value of less than 1 p. The standard deviation (an indicator of price variation) for the area was also calculated, the range of which was between 0.32 and 0.90. On the basis of these two sets of analyses, 1.0 p was selected as the Undercut value. This value was large enough to allow natural variations to occur, but not too large to be ineffectual. 4.6 Overprice

To find a suitable Overprice value, the maximum price change within the real data was calculated and found to range between 2.0 and 4.6 p. To prevent both unrealistically large price variations and the suppression of naturally occurring variations, Overprice was set to 5 p. 5 Performance of parameters To test the ability of the model to reproduce the patterns and trends within the real data, two series of experiments were run. In the first set, the model was initialised with all the real data prices for the Yorkshire region and run for a period of ten days (this was to allow comparison with real data over a similar period of time). After ten days, the mean price and standard deviation for all the stations were calculated (figure 1). The results in figure 1 show that the model is replicating the trends of the real data accurately. 73.0

Model results Real data

72.5

Mean price (p)

72.0 71.5 71.0 70.5 70.0 69.5 69.0 27/07

29/07

31/07

Date

02/08

04/08

06/08

Figure 1. Comparison of the mean price and standard deviation of the model results and real (1999) data.

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(a) Model data

County boundary Yorkshire region

Price (p) 74.89

Petrol station 67.90

25

12.5

0

25 km

(b) Real data (27 July 1999) Figure 2. Comparison of (a) the model performance on the Yorkshire region with (b) the real data for 27 July 1999. The price surface is interpolated from the point station locations for ease of viewing.


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The model is slightly overpredicting prices compared with the real data, but the standard deviation indicates that the model is mirroring the price variations well. Real prices in rural areas were found to be higher than in urban areas (figure 2). To test the ability of the model to reproduce these spatial differences, a second experiment was run. Each station within the Yorkshire region was given a starting price of 71 p, the parameter values in table 1, and the model run. Assessing the results in figure 2, it can be clearly seen that the model captures the main rural and urban trends within the Yorkshire region. The rural area of North Yorkshire is sustaining prices on average 2 p higher than in the more urban areas of West and South Yorkshire. Within West and South Yorkshire, the intraurban variations have also been recreated: for example, Leeds, Huddersfield, Barnsley, and Sheffield have lower prices than those in their suburban surroundings. This corresponds with the patterns found in the real data. 6 Further validation of parameters A further test of the suitability of these parameter values can be achieved by testing the ability of the model to make long-term predictions using the 1999 data. The SI component of the model calculates the profit made at each station; on the basis of this, strategy is determined. A test of how well the model is predicting these profit levels (and thus if the parameter values selected are giving accurate results) is to assess whether stations with low profit levels were still operating in 2004. It is hypothesised that stations with low or negative profit levels are most likely to have been driven out of business. A survey of stations was taken in June 2004. This provided a list of stations that had closed since the original data were collected in 1999. These stations were overlaid on a profitability map generated by the model initialised with the real data from 27 July 1999 and using the parameter values presented in table 1. Of the forty-three stations surveyed, fourteen have closed since 1999. Figure 3 shows a strong correlation between sites that the model identified as not being profitable and stations that have closed. A Wilcoxon rank-sum test was performed with the null hypothesis ``there is no difference between the mean profits of stations that are open and closed''. The test gave a result of p 0:000812, showing that there is a smaller than 0.1% chance of the null hypothesis being true. The null hypothesis can therefore be rejected. As clearly evident in figure 3, there is a definite difference in the profit between stations that are open and closed. One of the possible factors that catalysed stations going out of business is the influence of supermarkets and their aggressive pricing strategies. As evident from figure 3, a high proportion of low-profit stations located near supermarkets have closed. The correlation between low profitability and out-of-business stations provides an alternative assessment of the performance of the model and the parameters that does not rely directly on the price data for validation. As well as suggesting that the profit-maximisation ethos of the rules is useful for predicting profit levels and prices, the results support the choice of the derived model parameters. However, despite the success of the model, the subjective manner in which parameter values were assigned is open to criticism. Initial experimentations with the variables showed that the retail petrol market is characterised by nonlinear relationships with multiple local optima. This suggests that there are many different potential solutions, and experimenting with one parameter at a time may not result in an optimal set of parameters. There are a range of techniques available that can be used to find the global optima, ranging from conventional search techniques (for example, hill climbing and brute force searches), to artificial intelligence techniques (such as evolutionary

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Motorway A road

Profit (») 105 981

Petrol station

Closed Unsurveyed

New supermarket

53

2.5

1.25

0

2.5

km

Figure 3. Map showing the profitability of stations in Leeds using the models with data from 1999. Also shown for comparison are the status (open or closed) of the stations in 2004.

algorithms and tabu searches). Conventional techniques are often incapable of optimising nonlinear, multimodal functions (see Pham and Karaboga, 2000, for a detailed discussion). In contrast, evolutionary algorithms can find the globally optimal solution in complex multidimensional space. 7 Genetic algorithms GAs are a branch of evolutionary algorithms that are particularly useful for multidimensional optimisation problems where there are several variables to be optimised (Reeves and Rowe, 2003). Modelled on natural evolution, GAs use genetic operators (selection, recombination, and mutation) to manipulate individuals in a population over several generations to improve the model. In a GA, the properties of each individual are represented in an encoded form known as a chromosome (or genome), in our case, rule parameters are encoded as a series of numbers. Chromosomes are combined or mutated to breed new individuals, and the `fittest' of these (that is, those creating the best model) pass to the next generation. Recombination (or crossover) of two chromosomes models the sexual reproduction occurring in nature. Here, an offspring's chromosome is created by joining segments chosen alternately from each of two parents' chromosomes which


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are of fixed length. Selection is the process of choosing the chromosomes to be recombined. Mutation is the alteration of one or more parts of the chromosome with a random probability. There are several advantages to using GAs over traditional techniques for optimisation (Pham and Karaboga, 2000). One of the most significant is parallelism. GAs are capable of considering many points at once during the search process. This reduces the chance of converging to local optima in the solution space, that is, suboptimal solutions overall. During the search process, GAs use probabilistic rather than deterministic rules. This allows them to outperform conventional optimisation techniques on difficult, discontinuous, and multimodal functions (Reeves and Rowe, 2003). Additionally, GAs do not require derivative information or other auxiliary knowledge which may not be easily available; only the definition of `fitness' and corresponding levels influence the directions of search. One of the strengths of using GAs is the ability to hybridise with other techniques. Examples can be found within the literature of GAs hybridised with neural networks (Harp and Samad, 1991; Whitley, 1988), fuzzy logic (See, 1999), and multiagent systems (Choi et al, 2004; Dosi et al, 1999; Wu et al, 2003). 7.1 Operation of the GA

The GA (and indeed the model in general), was built using the Java programming language. A flowchart illustrating the operation of the GA is given in figure 4. Using this method, each of the genes (parameters) presented in table 2 will be optimised within their ranges. During the operation of the GA, values are randomly selected from these ranges and assigned to each gene. Fitness was judged by calculation of the standardised root mean square error (SRMSE) between the real and model data (following recommendations by Knudsen and Fotheringham, 1986). Table 2. Range of allowed values assigned to each gene. Gene (parameter)

Range of values

b l costToProduce fixedCosts changeInProfit Undercut Overprice Neighbourhood

0.000003 ± 0.003 0.001 ± 1.5 60.0 p ± 70.0 p 100 p ± 10000.0 p 2000 p ± 5000 p 0.1 p ± 5.0 p 0.1 p ± 5.0 p 1000 m ± 10 000 m

One of the key factors in the success of a GA is finding the correct balance between the amount of exploration and exploitation required (Flake, 2001). Factors include the initial size of population and the methods used for selection, recombination, and mutation. A poor selection of control parameters can result in a corresponding poor performance in the GA. Several studies have concentrated on developing optimal parameter settings for GAs (see De Jong, 1975; Greffenstette, 1986; Schaffer et al, 1989). Each study presents a different set of optimal parameters; for example, suggested values for population size range between 50 ^ 100 (De Jong, 1975), 20 ^ 30 (Schaffer et al, 1989), and 30 (Greffenstette, 1986). These values are obviously dependent on the problem under investigation. On the basis of recommendations within the literature, selection by ranking, intermediate recombination, and a mutation rate of 1=n (n being the number of chromosomes) were chosen.

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In addition, a convergence factor (size of mutation at each generation) of 0.99 and a selection pressure (related to the probability of the fittest chromosome being selected) of two were chosen. This was explained in section 6.2.

3

Start

3

Create initial random chromosomes, calculate their fitness, and rank them.

3

3

3

Write out the best solution so far.

3

Start the next generation. Copy numKeep best chromosomes.

3

Select pairs of parents. Breed new chromosomes.

3

For each new chromosome, mutate numMutate variants. Calculate fitness and keep the best one.

3

Rank the new generation according to its fitness.

3

NO

3

Number of generations completed?

YES

Write out the best solution. END.

Figure 4. Flowchart summarising the operation of the GA. The number of chromosomes kept at the end of each generation is denoted numKeep, and numMutate is the number of chromosomes mutated.


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7.2 Selection

Holland (1975) commented that ``the very essence of good GA design is retention of diversity''. Selection by ranking was chosen primarily because this technique gives any of the population the chance of becoming parents. The fitter the chromosomes, the more likely they are to be parents. The probability P of a chromosome being selected is linearly related to its ranking, k, (which is determined by its fitness), that is 1 2S ÿ 1k ÿ 1 2ÿS , (1) Pk N N ÿ 1 where N is the number of chromosomes, k is the position in the ranking of the chromosome (k 1 is the least fit, k N is the most fit), and S is the selection pressure. The selection pressure is the probability of the best chromosome being selected compared with the average probability of a chromosome being selected. For the linear ranking used here S must be in the range [1, 2]. A value of 2 ensures the maximum chance of the fittest chromosomes being selected. 7.3 Intermediate recombination

In intermediate recombination, the variable values of the offspring are randomly chosen from between the values of the parents. This has the advantage of preventing `super genes' from dominating the search space and thus producing a suboptimal solution. Therefore, in producing any generation, the best solutions from the previous generations are copied across and the rest of the population is generated from random recombinations of the previous generations. For each gene X the value of the offspring, Xoff is Xoff aX1 1 ÿ aX2 ,

(2)

where X1 and X2 are the gene (parameter) values of the parents and a is uniformly distributed in the range [ÿd : 1 d ]. This means the offspring value for each gene is a linear combination of the two parent genes. A value of d 0 would ensure that the offspring value was somewhere between the two parent values. However, over time this results in a shrinkage of the range of values covered by the various chromosomes. A value of d 0:25 ensures (statistically) that the variable area of the offspring is the same as the variable area spanned by the variables of the parents. 7.4 Mutation

Adding mutations increases the gene's search space. This potentially allows the fitness of the gene to be improved. Initially, the range of mutation values for each gene was specified, and the actual mutation was chosen uniformly from within this range. After each generation, the size of the range was decreased by a fixed factor to aid convergence on an optimal solution. At each generation, several different mutated versions of each chromosome were created and their fitness calculated (see below for details). Only the best mutation from each chromosome was kept. This prevents too rapid a loss of diversity within the population resulting from several similar genes becoming dominant. 8 Model parameter variation A set of exploratory runs was undertaken to assess the impact of varying different GA-related parameters on the performance of the GA and to generate a benchmark set. Table 3 presents the values chosen on the basis of these. The GA was then run to derive the model parameters. To ensure that the GArelated parameters did not skew the model parameters chosen, a series of GA runs were undertaken to generate the model parameters, each varying a single GA-related parameter (table 4). Each experiment was repeated at least five times to give an idea of the variability in the solution.

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Table 3. Values of control parameter selected after benchmark runs. Control parameter

Value

Number Number Number Number

100 100 1 3

of of to of

generations (runs) chromosomes (rule sets) keep between generations mutations to generate

Table 4. Summary of the various experiments conducted with the genetic algorithm (GA). In each case, the default benchmark GA-related parameters (see table 3) are used except for the variation shown. Experiment

Variation

1 2 3 4 5 6

default (see values in table 3) 200 chromosomes 50 generations 500 generations 0 mutations 3 mutations

Figure 5 shows each model parameter value as a function of the SRMSE. For each parameter there is a range of possible solutions that correspond to different local optima. In addition, each parameter value has a cluster of results situated around the lowest value of the SRMSE. This clustering around one particular value for each parameter suggests that there is indeed a global optimum for this problem. The number of points in the clusters also suggests that, provided an experiment with a given set of GA-related parameters is repeated 5 ^ 10 times, there is a high probability that a good approximation to this optimum will be found. In general, these clusters are not located near the edges of the allowed parameter range, suggesting that the specified ranges are wide enough so as not to interfere with the search. 9 Comparison of parameter values Table 5 details the optimal parameters derived by modeller-driven experimentation and the optimal parameters suggested by the GA. The GA was run five times to give an indication of the spread of values produced; figure 6 shows the spread of values for each parameter as a function of the SRMSE. Table 5. Comparison of parameter values derived using modeller-driven experimentation and the genetic algorithm (GA). Parameter

Modeller-derived value

GA value

Standard deviation (5 GA runs)

b l fixedCosts (£) costToProduce (p) changeInProfit (p) Overprice (p) Undercut (p) Neighbourhood (km)

0.0003 0.7 80.00 66.00 40.00 5 1 5.0

0.0002 0.4 12.73 65.75 1.0 4.4 0.8 3.6

2.04 6 10 ÿ4 0.176 15.9 2.27 22.8 0.73 1.12 1.21


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The GA values for b and l were both reasonably close to the parameters derived by modeller-driven experimentation. As evident in figures 6(a) and (b), neither of these values produced definite clusters. This means that reasonable solutions can be produced from a wide range of values and do not greatly effect other variables. 0.8 0.6

6 l

b (610ÿ4 )

8

4

0.2

2

(a)

0

(b)

100

Cost to produce (p)

Fixed cost (£)

80 60 40 20 0

(d)

40

(e)

0

Maximum undercut (p)

50

No change profit (£)

(c)

0.4

30 20 10

(f)

0.0 70 68 66 64 62 60 4 3 2 1 0

(g)

Distance scale (61000 m)

Maximum overprice (p)

5 4 3 2 1 0.008

0.010 0.012 SRMSE

Experiment 1

Experiment 2

0.014

(h)

Experiment 3

9 7 5 3 1 0.008 Experiment 4

0.010 0.012 SRMSE Experiment 5

0.014

Experiment 6

Figure 5. Variation in model parameters plotted against standard root mean square error (SRMSE) for simulations with different genetic algorithm control parameters: (a) b, (b) l, (c) fixedCosts, (d) CostToProduce, (e) changeInProfit, (f ) Undercut, (g) Overprice, (h) Neighbourhood. The various experiments are explained in table 4.

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The standard deviation (SD) values in table 5 also support this conclusion. The SD for b is the same size as the optimal value. For l the SD is smaller, but still 40% of the optimal value. The optimal values for b and l were converted into distance and price scales. These give scales of 3.5 km and 1.7 p over which the sales predicted by the SI model be reduced by a half. 10

0.8

Default

6

0.2

(b) Cost to produce (p)

Default

30 20 10 0

(d)

50 40

(e)

0

Maximum undercut (p)

Fixed cost (£)

0 40

No change profit (£)

(c)

30 20 10 Default

(f)

4 3 2 1 0 0.008

0.0 70

Default

68 66 64 62 Default

3

2

1

0 10

Default 0.01 0.012 SRMSE

0.014

Distance scale (61000 m)

Maximum overprice (p)

5

(g)

0.4

4 2

(a)

Default

0.6 l

b (610ÿ4 )

8

(h)

Default

8 6 4 2 0 0.008

0.01 0.012 SRMSE

0.014

Figure 6. Variation in model parameters plotted against root mean square error (SRMSE) for simulations with different genetic algorithm control parameterss: (a) b, (b) l, (c) fixedCosts, (d) CostToProduce, (e) changeInProfit, (f ) Undercut, (g) Overprice, (h) Neighbourhood.


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The GA gave the Neighbourhood parameter a slightly smaller value than the modeller-derived values. The values are also slightly more clustered than those for b and l, with an SD of 1.2 km and an optimal value of 3.6 km [figure 6(h)]. The Overprice parameter was well clustered and in good agreement with the modellerderived value [figure 6(g)]. The clustering for the Undercut value [figure 6(h)] was also quite tightly clustered. The optimal value of 0.8 p is still close to the value of 1 p used in the values derived by the modeller. The optimal Undercut parameter is much smaller than the Overprice parameter. There is a large difference between the GA-predicted value (»1) and the modellerderived value (»40) for the changeInProfit [figure 6(e)]. In experimentation with the modeller-derived parameters, the simulations were run for thirty days to achieve equilibrium. However, the GA only runs the model for ten days and therefore the long-term equilibrium is not important in determining the optimal changeInProfit value. For short-term model predictions, a smaller changeInProfit value provides a greater scope for stations to change their price in the model. There is also a very large SD (»22.80) for the parameter, indicating a large degree of uncertainty in the optimal value. The system modelled here is highly complex with many nonlinear processes contributing to price setting at each station. One of the interesting finds here was the control over the stability of the system that this parameter appeared to exert. This is an area for further research. The costToProduce and fixedCosts parameters are closely linked (profit will decrease as a result of an increase in either or both of these parameters) and they will therefore be assessed together. The overall cost is a combination of these two values. The GA-optimal value for costToProduce almost matched the modellerderived value. However, the GA values were not tightly clustered [figure 6(d)] with an SD of 2.27 p, again suggesting that a reasonable solution can be obtained from a range of values. The costToProduce value is strongly related to the fixedCosts value. The GA-predicted value (»12.73) is considerably smaller than the value derived by the modeller (»80). This was one of the parameters that little information was known about. However, in figure 6(c), the best solutions for the fixedCosts are not tightly clustered ö they span a »20 range. This suggests that this parameter does not have as great an effect on the overall solution. The fixedCosts parameter will affect the profit level the same way at all stations and so it will have less effect on the changes in profit (which control the pricing strategy). In summary, the values derived on the basis of real data analysis and modeller experimentation provided a good set of parameter values for the model. The parameters produced by the GA are not constrained to agree with the modeller-driven parameters, they are merely chosen to minimise the SRMSE of the model. It is reassuring that the parameters obtained by optimising the model using a GA are in close agreement with those obtained independently on the basis of features of the real petrol market. This suggests that the model is behaving in a realistic manner and that the assumptions made in deriving the parameter values using real data analysis and modeller experimentation were reasonable. 10 Comparison of performances Identical simulations were run with the model using the modeller-driven and GA-optimised parameters. The results were examined using a combination of statistical and visual comparisons. Statistically, the difference between the real and model price at each petrol station was calculated over a period of ten days (27 July ^ 6 August). Assessment of the mean price over time [figure 7(a)] shows that the modeller-driven parameters, on average, perform better than the GA parameters.


73

0.020

72

0.015

SRMSE in price

Mean price (p)

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71 70 69 27/07

29/07 31/07 02/08 Date (1999)

(a) Mean price

Modeller parameters GA parameters Real data

04/08

06/0

0.010 0.005 0 27/07

(b) SRMSE

29/07

31/07 02/08 Date (1999)

04/08

06/08

Modeller parameters GA parameters Constant price

Figure 7. Comparison of (a) the mean difference and (b) the standard root mean square error (SRMSE) over time between the real data, the model results using the modeller-driven parameters, and the model results using the genetic algorithm (GA) parameters. In each case the model was initialised with the real data from 27 July. Standard deviations in the mean price are represented by vertical bars in (a).

However, both solutions overpredict the real data. On day 9 (5 August), the GA overpredicts by approximately 0.4 p (ie, the model is 0.4 p higher than the real data), whilst the modeller-derived parameters are only 0.1 p overpriced. However, on day 10 (6 August) the GA overprediction price has fallen to 0.05 p. This is due to a marked increase in price between 5 August and 6 August which results in the GA becoming closer to the real data at the end of the period than the modeller-driven parameters. Figure 7(b) shows the SRMSE of the different parameter values over time. The overall model performance for the GA and modeller-driven parameters are very similar over the first part of the ten-day period and also in close agreement with results assuming the prices remain constant at their values on 27 July (denoted by `constant price' on the graph). However, after 29 July the SRMSE of the GA solution decreases and remains lower than the nonoptimised parameters and the constant price data for the rest of the period. This indicates that the GA parameters are better at representing the real data than either of the other two sets of results. Due to its ability to search the parameter space effectively, the GA has evolved an optimal set of parameters. However, a better test is to see whether the spatial variations within the real data are reproduced by the GA or the modeller-driven parameters. Figure 8 presents a comparison of the prices produced at the end of the experimental period with the modeller-driven and GA parameterised model both initialised with all the stations at 71 p. Figure 8(c) shows that the GA parameter values are recreating the rural ^ urban divide seen in the real data [figure 8(a)]. For example, the petrol stations within the predominantly rural area of North Yorkshire are sustaining higher prices than West or South Yorkshire. The main difference lies in the magnitude of the prices. On average, the prices produced by the GA are approximately 1 ^ 2 p lower than the real data. This probably reflects a mild disequilibrium in the real prices; for example, as the result of crude oil price changes.


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(a)

(b)

(c)

County boundary Yorkshire region

Price (p) 74.89 40

Petrol station 67.90

20

0

40

km

Figure 8. (a) Real data for 27 July 1999. (b) and (c) Comparison of the model results for Yorkshire using (b) the modeller-derived parameters and (c) the optimal parameters from the genetic algorithm. In each run the stations were initialised with a constant price of 71 p.

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The solution produced by the GA parameter values has improved on the model by reproducing the lower prices surrounding Scarborough and Harrogate [figures 8(b) and (c)]. It has also successfully identified the lower priced areas (cities) within West and South Yorkshire and the area of higher prices between Wakefield and Barnsley. There is evidence of variation within urban areas, although this is not as prominent as within the real data. Based on assessment of figure 7, the parameters optimised by the GA have improved the spatial distribution of prices compared with the modeller-led values. The ability of the model (using the GA parameter values) to predict profit in the long term was tested by repeating the experiment outlined in section 4. Figure 9 shows similar results to those reported in figure 3. Areas of low station profitability correspond to locations where stations are now closed. This relationship was again formalised by a Wilcoxon rank sum test with a result of p 0:000947. This means that there is a smaller than 0.1% chance of the null hypothesis ``there is no difference between the mean profits of stations that are open and closed'' being true.

Motorway A road

Profit (») 105 981

Closed Unsurveyed

53

New supermarket Petrol station

2.5

1.25

0

2.5

km

Figure 9. Map showing the profitability of stations in Leeds using data from 1999 and the genetic algorithm parameter values. Also shown for comparison are the status (open or closed) of the stations in 2004.


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11 Conclusion An agent-based system to model retail petrol prices (Heppenstall et al, 2005) was run on the basis of rule sets constructed from simple parameters. The parameters and their values were decided upon on the basis of real data analysis and numerical experimentation. This paper explored two contrasting methodologies for model calibration. The model was tested with these parameter values and the results demonstrated that the main trends and patterns within the real data were being replicated, with the model only slightly overpredicting the prices within the real data. The ability of the model to predict long-term market trends was also tested by making predictions on the profitability of stations in 1999 and examining whether these same stations were trading in 2004. A strong correlation was found between stations predicted as having a low profit and those closed in 2004. This work provided further validation of the parameter values used independent of the price data and time scale used to calibrate and validate the model. The variation within the parameter results (figure 6) shows that there are potentially several sets of optimal calibration values that would produce good results within the model. The values chosen here were selected simply because they produced the best results statistically. The subjective manner in which the model parameters were obtained was discussed and a GA presented as an alternative and objective method for optimising model parameters. A comparison of the GA and modeller-driven parameters were found to be in close agreement with the GA making a minor improvement in the final price prediction though a day-to-day comparison shows little difference. This suggests that the arguments used to derive suitable parameter values were justified. The closeness of the results may suggest that, despite the apparent nonlinear and complex nature of the system, there are a limited number of optimal or near optimal parameter values given the model framework and the constraint to match real price data, and that both the modeller-driven and GA solutions converge on them. This may, in part, be reflected in the general stability of the real retail system, which, in the absence of external forcing, is not chaotic to the point of breaking down. In terms of reproducing spatial trends in pricing, the GA parameters outperformed the modeller-driven parameters. In addition, its success at predicting the long-term profitability of stations showed the parameters from the GA to be dependable. Deriving parameters on the basis of numerical experimentation and real data analysis has been shown to produce reasonable values and good agreement between the model and real data. However, the choice of these values is somewhat subjective. Use of a GA allows parameters in this nonlinear system to be obtained objectively. The success of this method suggests that it may be useful for other similar problems. Acknowledgements. The work reported within this paper was funded by the Economic and Social Research Council and GMap. All digitised boundary data are UK Crown and ED-Line Copyright. References Birkin M, Clarke G, Clarke M P, 2002 Retail Geography and Intelligent Network Planning (John Wiley, Chichester, Sussex) Choi H R, Kim H S, Park B J, 2004, ``Multi-agent based integration scheduling system under supply chain management environment'' Lecture Notes in Computer Science number 3029 249 ^ 263 De Jong K A, 1975 An Analysis of the Behaviour of a Class of Genetic Adaptive Systems PhD thesis, University of Michigan, Ann Arbor, MI Dosi G, Marengo L, Bassanini A, Valente M, 1999, ``Norms as emergent properties of adaptive learning'' Journal of Evolutionary Economics 9(1) 5 ^ 26 Flake G W, 2001 The Computational Beauty of Nature: Computer Explorations of Fractals, Chaos, Complex Systems and Adaptation (MIT Press, Cambridge, MA)

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Franklin S, Graesser A (Eds), 1996, `Ìs it an agent, or just a program?'', in Proceedings of the Third International Workshop on Agent Theories, Architectures and Languages Budapest, Hungary, http://www.cs.memphis.edu/franklin/AgentProg.html Galoetti M, Lanza A, Manera M, 2003, ``Rockets and feathers revisited: an international comparison on European gasoline markets'' Energy Economics 25 175 ^ 190 Grefenstette J J, 1986, `Òptimisation of control parameters for genetic algorithms'' IEEE Transactions on Systems Man and Cybernetics 16 122 ^ 128 Harp S A, Samad T, 1991, ``Genetic synthesis of neural network architecture'', in Handbook of Genetic Algorithms Ed. L Davis (Van Nostrand Reinhold, New York) pp 203 ^ 221 Heppenstall A J, Evans A J, Birkin M H, 2005, `À hybrid multi-agent/spatial interaction model system for petrol price setting'' Transactions in GIS 9(1) 35 ^ 51 Heppenstall A J, Evans A J, Birkin M H, 2006, `Àpplication of multi-agent systems to modelling a dynamic, locally interacting retail market'' Journal of Artificial Societies and Social Simulation 9(3), http://jasss.soc.surrey.ac.uk/9/3/2.html Holland J (Ed.), 1975 Adaptation in Natural and Artificial Systems (MIT Press, Cambridge, MA) Knudsen D C, Fotheringham A S, 1986, ``Matrix comparison, goodness-of-fit and spatial interaction modelling'' International Regional Science Review 10 127 ^ 147 Ning X, Haining R, 2003, ``Spatial pricing in interdependent markets: a case study of petrol retailing in Sheffield'' Environment and Planning A 35 2131 ^ 2159 Nwana H S, 1996, ``Software agents: an overview'' Knowledge Engineering Review 11 205 ^ 244 Pham D T, Karaboga D, 2000 Intelligent Optimisation Techniques (Springer, London) Reeves C R, Rowe J E, 2003 Genetic Algorithms: Principles and Perspectives (Kluwer Academic, New York) Reilly B, Witt R, 1998, ``Petrol price asymmetries revisited'' Energy Economics 20 297 ^ 308 Schaffer J D, Caruana R A, Eshelman L J, Das R, 1989, ``Combinatorial optimisation by genetic algorithms'', in Proceedings of the 3rd International Conference on Genetic Algorithms Ed. J D Schaffer (Morgan Kaufmann, San Mateo, CA) pages 51 ^ 60 See L M, 1999 Geographical Applications of Fuzzy Logic and Fuzzy Hybrid Techniques PhD thesis, School of Geography, University of Leeds The Monopolies and Mergers Commission, 1990 The Supply of Petrol: A Report on the Supply in the United Kingdom of Petrol by Wholesale chapter 4, http://www.competition-commission.org.uk/ rep pub/reports/1990/fulltext/265c4.pdf Whitley D, Kauth 1988, ``GENITOR: a different genetic algorithm'', Department of Computer Science, Colorado State University, Fort Collins, CO Wooldridge M J, Jennings N R (Eds), 1997 Intelligent Agents: Lecture Notes in Artificial Intelligence (Cambridge University Press, Cambridge) Wu C G, Liang Y C, Lee H P, 2003, ``Study on optimisation of agent initial positions in land combat simulation'' Progress in Natural Science 14 257 ^ 261

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