Deploying Neural-Network-Based Models for Dynamic Pricing in ...

Deploying Neural-Network-Based Models for Dynamic Pricing in Supply Chain Management Yevgeniya Kovalchuk Department of Computing and Electronic Systems, University of Essex [email protected]

Maria Fasli Department of Computing and Electronic Systems, University of Essex [email protected]

Abstract

which includes a number of interrelated activities such as: negotiating with suppliers for raw materials, competing for customer orders, managing inventory, scheduling production, and delivering goods to customers. When dealing with customers, the problem is to find optimal selling prices, which should be high enough to allow for profit and at the same time low enough to be attractive to customers. The problem can be seen as defined in the first price sealed bid reverse auctions: find the largest price a seller could have offered to a customer and still won. In our work we assume that customers place their orders with the manufacturer who proposes the lowest price. We then solve the task of predicting the lowest prices proposed by all market participants based on time-series of these prices. As opposed to traditional statistical and technical analysis methods, learning approaches can react to market irregularities more successfully. Among others, a Neural Networks learning technique was chosen to develop a number of predictive models which differ in the number of observables they consider and the methods of data transformation and normalisation applied over inputs. These models are compared in terms of their accuracy and predictive abilities. To evaluate the performance of the models, we tested them in the Trading Agent Competition (TAC) SCM game, which offers a realistic simulated environment for studying SCM strategies [6]. The rest of this paper is organized as follows. An overview of related work is provided first. The proposed models for predicting winning bidding prices are described next followed by the experimental results. The paper closes with conclusions and a discussion of future work.

With the advent of e-Commerce, enterprises can no longer rely on static business strategies. They have to be able to cope in dynamic and uncertain electronic environments, especially when developing their pricing strategies. In such environments, prices are determined dynamically through a competitive bidding process depending on the market situation, competitor strategies and/or customer preferences. The ability to predict winning bidding prices is crucial. This paper introduces a number of neuralnetwork-based models for performing time-series forecasts of customer offer prices which could result in winning orders in the context of supply chain management (SCM). Different data transformation and normalisation methods are explored in the models, as well as the impact of the number of historical data points included in time-series on the accuracy of the prediction. Experiments in the Trading Agent Competition SCM game show the potential of the proposed algorithms for predicting prices in competitive and dynamic environments.

1. Introduction The forecasting of financial time-series is a challenging task which has been attracting a lot of attention. Many different techniques have been proposed to solve the task for various financial instruments. However, not many works have been dedicated to exploring the problem in relation to electronic environments, where the process of forming prices occurs dynamically. Examples of such environments include online auctions, airline industry, and supply chain management among others. This paper is devoted to the problem of predicting customer offer prices which could result in securing orders in the context of supply chain management (SCM). The SCM is a complex process

2. Related work The problem of dynamic pricing has become very popular in recent years. A detailed survey on

different models for dynamic pricing in electronic business is presented in [17]. Application of the Estimation of Distribution Algorithms, Genetic Algorithms and Simulated Annealing Algorithms to this task can be found in [21]. A number of works has been dedicated to solving the task in the context of SCM. In particular, the TAC SCM game participants offer different approaches to finding optimal customer bidding prices. An overview of the strategies applied to the problem of predicting winning offer prices up to 2004 is provided in [20]. The paper also presents a comparison of different learning algorithms for accomplishing the task in the context of the TAC SCM. M5 trees and BoosTexter has been shown to give the minimum root mean squared error. Below we present strategies implemented by the TAC agents in their up-to-date versions. SouthamptonSCM [9] applies a fuzzy reasoning inference mechanism to determine offer prices according to the agent’s inventory level, the market demand and the time in the game. TacTex uses additive regression with decision stumps [18]. In an earlier version of the agent, the developers used linear regression on six data points to generate a linear function which is modified then by the day factor [19]. The day factor measures the effect of the due date on offer acceptance. A similar approach is implemented in Botticelli [3] and CMieux [2]. The latter computes a linear least squares fit for the selling prices of each product over the past several game days. Additionally, the agent enforces lower and upper bounds on the predictions to ensure that the prediction remains relatively conservative. The agent maintains the probability distribution for each PC type mapping bidding prices to the likelihood of winning orders with these prices. The distributions are learned off-line using data from previously played games to build a regression tree. Under certain assumptions they have shown that this pricing problem can be reduced to the continuous knapsack problem [1]. Mertacor [15] has selected the M5 data mining algorithm applied to historical data from past games in order to choose which attributes influence offer prices. It also uses two on-line modelling mechanisms in order to handle unexpected circumstances that may arise with regard to selling prices. The agent applies the k-Nearest Neighbours algorithm then to find the probability of offer acceptance for each bid placed. The probability of winning customer offers is also used in the bidding strategies implemented by MinneTAC [13] and DeepMaize [14]. RedAgent [12] uses an internal marketplace structure with competing bidders to set offer prices. The agent computes offer prices as a

sum of 3 terms: a base price of the PC, an estimated discounted profit for the product (the difference between base price and order price, discounted according to the number of days left until the order expires), and a discounted penalty. PackaTAC [7] sets prices according to the market state taking into consideration the lowest and highest previous day prices and the current demand level. Algorithms for customer price prediction that were implemented in the context of the TAC Classic game are reviewed in [23]. The techniques discussed there are divided into three categories: historical averaging, machine learning, and competitive equilibrium analysis. The Neural Networks (NN) machine learning technique has been successfully used for solving forecasting tasks in the domains of finance and business, often outperforming other learning techniques. In [4], the authors discuss application of classical regression models, NNs, fuzzy logic, and fractal theory for forecasting time-series of dollar/peso exchange rate, U.S./Mexico exchange rates and prices of onions and tomatoes in the U.S. market. The researchers conclude that the regression models show the poorest performance, and also that NNs outperform fuzzy logic when forecasting in the short-term, while fuzzy logic outperforms NNs when forecasting in the long term. The authors of [24] report empirical evidence of applicability of NNs to the prediction of foreign exchange rates and discuss issues on the frequency of sampling, choice of network architecture and forecasting periods, and measures for evaluating the predictive power of a model. In [10], the researchers propose several methods for predicting online auction prices using regression, decision trees (C5.0), and NNs. Their binary classifier based on NNs demonstrated the highest prediction accuracy (96%). An overview of successful NN-based models applied to various business domains such as marketing, retail, banking and finance, insurance, telecommunication, and operations management is provided in [22]. The results reported in [8] suggest that the window size and network architecture do have an important effect on the quality of NN-based time-series forecasts. The paper offers a heuristics for estimating the correct window size using false nearest neighbor method and the singular-value analysis. In our paper we further explore the issue for the case of predicting customer order prices. To the best of the authors' knowledge, no previous work has been dedicated to developing and comparing NN-based models for dynamic price prediction in the domain of SCM.

Table 1. Summary of the NN predictive models Predictive Model

Actual Price Five (APF) Actual Price Ten (APT) Differential Transformation Five (DTF) Differential Transformation Ten (DTT) Rational Transformation Five (RTF) Rational Transformation Five Fixed (RTFF) Rational Transformation Ten (RTT) Statistical Actual Price Ten (SAPT)

Input Number

Hidden Number

Lag window

Transformation Method

Normalisation Method

5 10 5 10 5 5 10 10

3 5 3 5 3 3 5 5

5 10 6 11 6 6 11 10

None None Differential Differential Rational Rational Rational Statistical

Linear Varied Linear Varied Linear Varied Linear Varied Linear Varied Linear Fixed Linear Varied Non-linear

Table 2. Average relative error (std. dev. in parentheses) and HIT percentage of models’ prediction

ARE HIT

Actual difference

APF

APT

DTF

DTT

RTF

RTFF

RTT

SAPT

0.0249 (0.034) -

0.0064 (0.006) 0.882

0.0128 (0.011) 0.882

0.0151 (0.012) 0.847

0.0059 (0.005) 0.898

0.0062 (0.006) 0.890

0.0060 (0.005) 0.901

0.0061 (0.006) 0.895

0.0062 (0.006) 0.884

3. Proposed Methods In the TAC SCM game there are six agents who act as product manufacturers competing for supplier components and customers orders for finished personal computers (PCs). Customers send requests for quotes (RFQs) to all agents for the 16 types of PCs that can be manufactured on a daily basis. Agents make offers and according to the game rules, customers accept the lowest offers proposed among all agents. Information on competitors’ offer prices is not available to TAC agents. However, the lowest and the highest order prices for each PC type from the previous day are reported daily. Analysis of the games played against highly competitive agents revealed that the difference between the lowest and the highest order prices tends to be very low. Thus, knowledge of these prices for the next day can assist an agent in bidding more successfully. Accordingly, we based our bidding strategy on predicting the lowest and highest order prices one day in advance. We build 8 different predictors for each PC type to perform the task based on the time-series of the prices. All the models use neural networks (NN) with one hidden layer, Backpropagation training algorithm and the sigmoid activation function [16]. The models differ in the number of historical data included in the series (window size) and also in their transformation and normalization methods applied over inputs. Table 1 summarises these differences. Three different data transformation methods were tested in the models. These methods were introduced in

order to render the time-series approximately stationary. Stationarised series are relatively easy to predict, as their statistical properties (mean, variance, autocorrelation, etc.) remain the same over time. We consider the following data transformations: Differential Transformation x d = xt − xt −1 (1) Rational Transformation

x r = ln(

xt ) xt −1

(2)

Statistical Transformation

xs = x=

1 N

N

∑x , t

xt − x

σ

σ2 =

t =1

,

1 N (xt − x )2 ∑ N − 1 t =1

(3)

where xt and xt-1 are consecutive data values in a series,

x is the mean of the series values and σ 2 is their

variance. After performing the transformation, we normalize data to be in the interval (0; 1) that is the range of the sigmoid function, which we use as an activation function. We applied three different formulas for performing data normalisation: Linear Varied

xilv = (

xi − x min ) x max − x min

(4)

Linear Fixed

xilf = (

xi − x min ) ⋅ 0 .8 + 0 .1 x max − x min

(5)

Non-linear

xinl =

1 1 + e − xi

(6)

where xmin and xmax are the minimum and maximum values for the corresponding type of data transformation, which were set according to the ones observed in the games. We experimented with the number of hidden units for each model and found that it does not substantially influence the prediction performance. Thus, we set this number similarly for all the models. In contrast, we set the learning rate and number of iterations individually for each model and PC type according to the dynamics of prediction error during the training process. To evaluate the performance of the proposed algorithms, we played 3 sets of games in the TAC SCM simulated environment. The first set of 15 games we run to collect data for initial training of the models. In the next 15 games we used the predictors and collected data for further training of the models. The last set of 15 games was played to evaluate the models’ performance. We chose the following highly competitive agents to be our competitors: TacTex 2007 [19], PhantAgent 2006, Maxon 2006, SouthamptonSCM 2006 [9], and CrocodileAgent 2005. The agents’ binary code is publicly available (http://www.sics.se/tac/). In the games played, we set our offer prices to the average predicted by all the models.

4. Experimental results The proposed predictive models are compared based on the average relative error (ARE) and Hit Percentage (HIT) [11] calculated on average for all PC types across all the games played. The ARE was calculated as follows:

ARE = xi’

1 N

N

∑ abs( x i =1

1 N

i

(7)

N

∑x

− xi' )

i

i =1

are the actual and predicted values where xi and observed in a case; N is the number of cases observed in all games. HIT demonstrates whether a model follows the trend of the observed time-series: the higher the value, the more accurately the model follows the trend. It is defined as:

⎧1 if Δx i > 0 ∧ Δxi' > 0 ∨ ⎪ Δx i < 0 ∧ Δxi' < 0 1 N ⎪ HIT = (8) ∑⎨ N − 1 i =2 ⎪0 if Δxi > 0 ∧ Δxi' < 0 ∨ ⎪ Δx i < 0 ∧ Δxi' > 0 ⎩ where N is the number of data points observed in all games and Δxi = xi − xi −1 is the difference of two

consecutive points. The experimental results are provided in Table 2 and Figure 1 illustrates an example from one game of the lowest order prices, actual and predicted by all the models for PC of type 2 (PC2). Analysis of individual cases showed that the performance of the models varies across different PC types and games. This can be explained by that the models correspond differently to different market settings in the games and different conditions over the trading period [5]. This also means that the time-series of customer order prices really reflect the current situation in the SCM environment and the competitors’ reaction to it. As can be seen from Table 2 and Figure 1, all the implemented predictors follow the trend. The Differential Transformation Ten (DTT) model demonstrates the highest accuracy of prediction (ARE=0.0059, HIT=0.898), followed by the Rational Transformation Five Fixed (RTFF) model (ARE=0.0060, HIT=0.901) and the Rational Transformation Ten (RTT) model (ARE=0.0061, HIT=0.895). Surprisingly, the Differential Transformation Five (DTF) model appears to be the worst among all the proposed models. This indicates that the differential transformation method is sensitive to the lag window (the number of data points included in the time-series): the more observables considered, the more accurate the prediction produced. In contrast, all three models which are based on the rational transformation method showed relatively similar results, making this transformation method robust to the lag window. Another observation can be made from Figure 2, where actual and average predicted by all models prices are compared1: if the curves of the predicted prices are shifted one day back, then the actual and predicted lines match each other almost perfectly. This leads to the conclusion that the predicted prices are closer to the previous day’s reported prices rather than the ones observed on the current day. One could suggest that in the games with highly competitive bidding prices (i.e., when the difference between them is very low), the strategy of just following the previous reported prices could be sufficient for making 1 Average predicted values are used to make the figure clearer; one can get similar pictures by replacing average predicted prices by prices predicted by any of the proposed models.

assumptions on the current day’s winning order prices. However, our results do not prove this hypothesis. In particular, we calculated the differences between the two following data points of actual prices across all games (second column in Table 2). It appears that its rate is much higher than the rate of relative errors demonstrated by our predictors. This provides evidence for the effectiveness of the proposed methods. 2000

1800

1600

1400

1200

1000

800

1

11 APF

21 31 41 APT

51 61 71

DTF

DTT

81 91 101 111 121 131 141 151 161 171 181 191 201 RTF

RTFF

RTT

SAPT

AP

Actual

Fig. 1. Lowest order prices for PC2: predicted by all models, average predicted and actual (axis X – day in the game; axis Y – order price) 2000 1800 1600 1400 1200 1000 800 1

11 21 31 41 51 61 71 81 91 101 111 121 131 141 151 161 171 181 191 201 Average predicted

Actual

Fig. 2. Lowest order prices for PC2: average, predicted by all models, and actual (axis X – day in the game; axis Y – order price)

5. Conclusions and future work Configuring prices dynamically is one of the most challenging and interesting research problems. With the development of e-Commerce, it became also crucial. This paper investigates the problem of predicting winning bid prices in the first price sealed bid reverse auctions. In particular, 8 different neuralnetwork based models are proposed to predict customer order prices based on the time-series of these prices in the context of Supply Chain Management (SCM), which is a very complex, stochastic and everchanging environment. The experiments in the TAC SCM online game show that the models are capable to

respond to dynamic changes in real-time markets: all of them follow the trend (high HIT rate) and provide low prediction error. In particular, the experimental results prove that more accurate solutions are achieved when following predicted prices rather than relying on the prices known from the previous day. Different data transformation and normalisation methods for preparing neural networks inputs are investigated in the predictive models. The best performance is achieved by applying the differential and rational transformation methods. Among all the models, we distinguish the Differential Transformation Ten (DTT) and the Rational Transformation Ten (RTT) models, which consider prices for eleven previous days, and also the Rational Transformation Five Fixed (RTFF) model, which takes only six previous data points. These models have the lowest average relative error and the highest HIT rate. It appears that the differential transformation method is sensitive to the number of historical data points included in the time-series: in contrast to the DTT model, the Differential Transformation Five (DTF) model, which considers only 6 previous data points, showed the lowest accuracy of prediction. The models which do not apply any transformation methods (APF and APT) also showed relatively poor performance. The prediction accuracy of the proposed models varies for different PC types and across the games played. This difference in the models’ performance leads to the conclusion that different models might work better in different market conditions, which in turn depends on the strategies applied by competitors. As part of future work we aim to develop a metamodel, which can consolidate the results obtained from individual models and find the best solution for the current market situation. Another task for future work is to test other neural networks architectures and learning algorithms. Levenberg-Marquardt learning algorithm is one of them. We are also going to apply other learning techniques to perform forecasts of customer order prices and compare the results obtained with the results reported in this paper.

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