A modified Lotka-Volterra model for competition forecasting in Taiwan\'s retail industry

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A modiﬁed Lotka–Volterra model for competition forecasting in Taiwan’s

retail industry

Hui-Chih Hung

⇑

, Yun-San Tsai, Muh-Cherng Wu

Department of Industrial Engineering and Management, National Chiao Tung University, 1001 University Rd., Hsinchu 30010, Taiwan, Republic of China

a r t i c l e

i n f o

Article history: Received 10 May 2013

Received in revised form 17 August 2014 Accepted 13 September 2014

Available online 22 September 2014

Keywords: Competition Seasonal patterns

Modiﬁed Lotka–Volterra model Bass model

Mean absolute error

a b s t r a c t

The retail industry is an important component of the supply chain of the goods and services that are consumed daily and competition has been increasing among retailers worldwide. Thus, forecasting the degree of retail competition has become an important issue. However, seasonal patterns and cycles in the level of retail activity dramatically reduce forecasting accuracy. This paper attempts to develop an improved forecasting methodology for retail industry competition subject to seasonal patterns and cycles. Using market share data and the moving average method, a modiﬁed Lotka–Volterra model with an additional constraint on the summation of market share is proposed. Furthermore, the mean absolute error is used to measure the forecasting accuracy of the market share. Real Taiwanese retail data from 1999 is used to validate the forecasting accuracy of our modiﬁed Lotka–Volterra model. Our methodology successfully mitigates errors from seasonal patterns and cycles and outperforms other benchmark models. These benchmarks include the Bass and Lotka–Volterra models for revenue or market share data, with or without using the moving average method. Our methodology assists the retail industry in the development of management strategies and the determination of investment timing. We also demon-strate how the Lotka–Volterra model can be used to forecast the degree of industry competition.

1. Introduction

The retail industry is the ﬁnal element of the supply chain that provides goods and services to consumers. According toDeloitte (2013), aggregate retail sales were $4.27 trillion US dollars in 2011 for the world’s top 250 retailers. Moreover, the 2006–2011 compound annual growth rate of retail sales is 5.4%.

With the new wave of globalization and associated supply chains, the retail industry directly connects manufacturers and consumers, and is thus a source of supply and demand data. In recent years, this key position in the supply chain attracts more retailers to join. With increasing in the number of retailers, the degree of retail competition raises. For example, two Australian retailers were newly included in the top 25 retailers worldwide in 2009 (Deloitte, 2011). As a result, competition forecasts for the retail industry have become an increasingly important issue in supply chain management.

The growth of the retail industry usually reﬂects the develop-ment of a country. A good example is Taiwan. Per capita gross domestic product has grown from $2700 US dollars in 1982 to over

$20,000 US dollars in 2011 (Directorate-General of Budget and Statistics (2013)). The retail industry in Taiwan has experienced many transformations during this period. One major transforma-tion is that the retailers gain the power of changing manufacturer and consumer behaviors in Taiwan. New types of retailing were capitalized and harnessed to take advantage of the demand fore-casting and price control. As a result, a variety of retail types has emerged and these retailers coexist with a high degree of competition in Taiwan. These new retail types are well developed in Taiwan and have appeared in retail industries in China and Philippines (Goldman, 2001).

There are four major retail types in Taiwan: supermarkets (e.g. Wellcome), hypermarkets (e.g. Costco), convenience stores (e.g. 7-Eleven), and traditional stores. Supermarkets are midsize food providers and are generally located within communities. They sell food and other daily necessities within walking distance of cus-tomers. Hypermarkets sell large quantities of goods to customers during one-visit shopping. They are usually located in suburban areas with ample parking. Convenience stores are open for extended hours and are located close to consumers to allow pur-chases of necessities and services at any time. The services include ticket sales, bill payments, deliveries, etc. Traditional stores are one of the oldest retail styles in Taiwan, existing since the 1940s. They are usually run as a family business in old communities and

http://dx.doi.org/10.1016/j.cie.2014.09.010

⇑Corresponding author. Tel.: +886 3 5712121/57305; fax: +886 3 5729101. E-mail addresses: hhc@nctu.edu.tw (H.-C. Hung), yunsan_tsai@yahoo.com

(Y.-S. Tsai),mcwu@mail.nctu.edu.tw(M.-C. Wu).

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provide warm and friendly service. In Taiwan, the same or similar products are sold in all four retail types. Unavoidable competition among these four retail types has attracted considerable attention. There are three focuses of retail competition in Taiwan. First, people enjoy searching for low prices. Because of rapid urbaniza-tion and fast-paced lifestyle, more people are changing their shop-ping habits from cost-oriented to convenience-oriented behavior. Of the four retail types, we classify convenience stores and tradi-tional stores as convenience-oriented submarket. This is because convenience stores and traditional stores in Taiwan have the high-est density (in an area of 35,980 km2_{with a population of 23} mil-lion, there are more than 10,000 convenience stores and traditional stores (Taiwan Institute of Economic Research, 2012)). In contrast, supermarkets and hypermarkets are classiﬁed as cost-oriented submarket. The competition between targeting convenience-oriented and cost-convenience-oriented shopping styles highlights the two opposing forces in Taiwan’s retail industry.

Second, in the convenience-oriented submarket, people used to shop in traditional stores for everyday necessities. However, as the economy grows, more people prefer to shop in convenience stores for everyday necessities. The competition between convenience stores and traditional stores implies the future trend of the new retail types in Taiwan.

Third, in the cost-oriented submarket, people used to shop in their neighborhood supermarkets for family necessities subject to the cost of private transportation in Taiwan. Recently, people have started to shop in hypermarkets following the American style of shopping in which all requirements are purchased in one visit. The competition between supermarkets and hypermarkets are indicative of the trends in family retail shopping in Taiwan.

Competition forecasts for the retail industry show long-term consumer trends. Accurate forecasts can help managers identify the growth and recession potential of different business models and their speed of adopting new strategies. For retail managers, accurate forecasts can help develop strategies to maintain market share in the years ahead. For investors, it can identify future trends and investment targets. Thus, an accurate forecast of retail compe-tition is necessary for retailers and investors.

Traditionally, the Bass and Lotka–Volterra models have been used to forecast innovation diffusion and competition levels. The Bass model was designed to describe the process of market diffu-sion. The Lotka–Volterra model was widely used to investigate the competitive relationships among ﬁrms by a set of differential equations. Some studies have compared the forecasting capability of the Bass and Lotka–Volterra models for new products or tech-nologies. Also, shipment amounts or revenue data are often used in Bass and Lotka–Volterra models in previous studies.

When investigating the retail competition in Taiwan, we input the revenue data to both Bass and Lotka–Volterra models (see Sec-tion4). Unsatisfactory forecasting errors cause us to consider the characteristics of retail industry. That is, the revenue data is usu-ally mixed with confounding factors such as economic growth, inﬂation, and cyclical and seasonal patterns. To mitigate these impacts, we think that some premodiﬁcation on the dataset before selecting a forecasting model is necessary. Furthermore, different datasets may require different models and methods of evaluation. Unfortunately, previous studies have not addressed these issues.

In this paper, we are interested in competition forecasts for the retail industry. Our goal is to develop a methodology for more pre-cise forecasting. For the above issues, we premodify the revenue data into the relative market share before selecting a forecasting model. In addition, we develop a new model based on relative mar-ket share that merges the existing Bass and Lotka–Volterra models. Regarding evaluation of the forecasts of the market share data, we use the mean absolute error (MAE) to measure forecasting accu-racy. Real data from the retail industry from 1999 to 2012 is used

to examine the performance of our methodology. Major improve-ments in forecasting accuracy were obtained to provide a better picture of the competition in Taiwan’s retail industry.

This paper is organized as follows. In Section2, we review the Bass model and the Lotka–Volterra model, and then compare them. In Section3, a modiﬁed Lotka–Volterra model is proposed for more accurate forecasting of retail competition. In Section 4, real data from Taiwan’s retail industry is used to examine retail competition and validate the performance of our methodology. Finally, we summarize our results and discuss several directions for future research in Section5.

2. Literature review

In this section, we review the Bass and Lotka–Volterra models and variations on them.

2.1. Bass model

The Bass model was ﬁrst proposed byBass (1969)and models the diffusion process of a new product among adopters and poten-tial buyers in a market. The diffusion rate of a new product in a market can be described by the following differential equation:

dNðtÞ

dt ¼ ðp þ qNðtÞÞðM NðtÞÞ; ð1Þ

where N(t) is the cumulative number of adopters at time t, and M is the potential market size. The parameter p is the coefﬁcient of innovation, which shows the possibility of new demand by mass media. The parameter q is the coefﬁcient of imitation, which shows the possibility of new demand by oral propagation.

The Bass model has been widely applied in the field of new product/technology development. For example, Sneddon, Soutar, and Mazzarol (2011) investigated the diffusion of wool-testing technologies in Australia using the Bass model. Seol, Park, Lee, and Yoon (2012)adopted the Bass model to forecast the diffusion of new digital broadcasting services in South Korea.Tsai, Li, and Lee (2010)considered the effect of price factors on the coefficient of imitation and modified the Bass model to study the diffusion and evolution of the new liquid crystal display TVs. Heinz, Graeber, and Praktiknjo (2013)studied the diffusion process of fuel cells and hydrogen producers with the Bass model. They verified that the two-sided market effect can accelerate the diffusion of hydrogen economy significantly. Dalla Valle and Furlan (2014) consider the diffusion of nuclear energy in developing countries and adopted the generalized Bass model to estimates the depletion time of uranium.

2.2. Lotka–Volterra model

In the field of ecology,Lotka (1925)investigated the competi-tion and mutualism of two species and was the first to model predator–prey interactions using a set of logistic equations. Volterra (1926)then adopted Lotka’s model with real data to study fish catches in the Adriatic Sea. In their models, the following two differential equations are used to describe how two species’ population growth rates interact over time:

dxðtÞ

dt ¼ ða1þ b1xðtÞ þ c1yðtÞÞxðtÞ; ð2Þ dyðtÞ

dt ¼ ða2þ b2yðtÞ þ c2xðtÞÞyðtÞ; ð3Þ

where x(t) and y(t) represent the populations of two competing species at time t. The two terms, dx(t)/dt and dy(t)/dt , represent the growth of the two populations over time. Moreover, x2and y2 represent internal self-interaction in the same species and xy

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represents the mutual inﬂuences of the two species. The model includes three basic parameters that affect the growth rates of both species. Parameter aiis the logistic parameter of growth for species i. Parameter biis the limitation parameter of growth for species i. Parameter ciis the interaction parameter with the other species for species i.

The Lotka–Volterra model has long been applied in the ﬁeld of ecology. For example,Geijzendorffer, Van der Werf, Bianchi, and Schulte (2011)used the Lotka–Volterra model to predict long-term coexistence patterns of grassland species. In the ﬁeld of biodiver-sity, Roques and Chekroun (2011) explored the competition of multiple species using the Lotka–Volterra model and examined the degree of chaos and the risk of extinction.

Beyond the ﬁeld of ecology, the Lotka–Volterra model has also been adopted widely for the analysis of competitive behaviors in a market. For example,Lee, Lee, and Oh (2005)investigated the trade values of two Korean stock exchanges, Korean Stock Exchange and Korean Securities Dealers Automated Quotation. They used Lotka–Volterra models to study competitive behaviors and forecast the future trends for the two stock exchanges.Kim, Lee, and Ahn (2006)estimated the dynamic competition of mobile phone subscription in Korea with the Lotka–Volterra model and showed the commensalism relationship.Kreng and Wang (2011) used the Lotka–Volterra model to examine the competitive rela-tionships between liquid–crystal display and plasma display panel televisions in Taiwan and showed equilibrium.Lin (2013)adopted the Grey system theory to predict the diffusion of mobile cellular broadband and ﬁxed broadband in Taiwan. The author then used the Lotka–Volterra model to analyze the competitive relationship and show the commensalism.Duan, Zhu, and Fan (2013)revised the Lotka–Volterra model to study the evolution of wind and photovoltaic solar technologies worldwide. With estimation and simulation, mutualism relationship was found in most of countries and the possible reasons were analyzed.

2.3. From the Bass model to the Lotka–Volterra model

In recent decades, many researchers have constructed models of new-product development based on the Bass model. However, the Bass model does not consider the interaction between the new product and other competing products. In contrast, the Lotka–Volterra model incorporates interactions between competi-tors and has been used to examine models of competition for products, technologies, and industries.

Some researchers have focused on competitive behaviors and compared the Bass and Lotka–Volterra models in terms of their forecasting capabilities with respect to new-product/technology diffusion. For example,Chiang (2012)explored the predator–prey

relationship between 200 mm and 300 mm silicon wafer technolo-gies in Taiwan.Chiang and Wong (2011)examined the shipment data of notebook and desktop computers and investigated the competitive diffusion relationship.Tsai and Li (2009)split Taiwan’s integrated circuit (IC) industry into IC design, manufacturing, and packaging/testing industries, and studied the interindustrial com-petition and cooperative effects on clustering formation.Chang, Li, and Kim (2014)investigated the saturated mobile phone market with churn effects in Korea and studied the performance of differ-ent diffusion models.

The above studies adopted both the Bass and Lotka–Volterra models to examine competition and diffusion and compared both models with data on shipment quantities or foreign direct invest-ment. All of these authors thought that the Lotka–Volterra model embedded the competitive relationships of different products/ technologies and mainly observed smaller forecast errors. As a result, they suggested that the Lotka–Volterra model might be more suitable for forecasting than the Bass model.

3. Methodology

We now propose a methodology to forecast competition among the different retail types. Our methodology involves four parts: (1) data selection, (2) data processing, (3) model selection, and (4) forecasting evaluation.

3.1. Data selection and processing

These previous studies usually used raw revenue or shipment data. However, these data may be inﬂuenced by economic growth, inﬂation, and industry long-term trends that have no relationship with competition forecasting. For illustration,Fig. 1shows monthly revenues of convenience stores and traditional stores in Taiwan from 1999 to 2012 (Department of Statistics, 2013). We observe the upward trends that may create major problems for traditional forecasting models.

To eliminate these confounding factors and describe the compe-tition between these two retail types, we ﬁrst transform the raw monthly revenue data into relative market shares. Take the conve-nience-oriented submarket in April 2012 as an example. There are two retail types in the convenience-oriented submarket, conve-nience stores and traditional stores, with monthly revenue of NT$21.25 billion and NT$13.19 billion, respectively. The total monthly revenue of the convenience-oriented submarket is NT$34.44 billion. In this submarket, the convenience stores take the relative market share of 61.7% and the traditional stores take the relative market share of 38.3%. The market shares of both retail types are summed up to be 100%.Fig. 2shows the monthly market

NT$ (Billion)

Convenience stores Traditional stores

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shares of convenience stores and traditional stores in the convenience-oriented submarket of Taiwan.

InFig. 2, some of the trends that may create problems for fore-casting models have been eliminated. However,Fig. 2still shows ﬂuctuations and volatility in the market share data. This is because the retail industry is easily affected by business cycle, seasonal, and weather factors such as festival celebrations, seasonal promotions, and typhoons, respectively. These factors induce ﬂuctuations in the data and lead to inaccurate forecasting. For better data processing, the moving average method is recommended. Fig. 3presents a 12-month moving average of market share for convenience stores and traditional stores in Taiwan.

3.2. Model selection

When using the market share data, one notable constraint is that the total market share is ﬁxed at 100%. For the Bass model based on market share data, deﬁne M = 1 as the total market share and yðtÞ as the cumulative market share of a retail type at time t. The Bass model given by Eq.(1)can be rewritten as follows:

dyðtÞ

dt ¼ ðp þ qyðtÞÞð1 yðtÞÞ ¼ C1þ B1y þ A1y

2_; _ð4Þ

where C1= p , B1= q p and A1= q.

For the Lotka–Volterra model based on market share data, we focus on a pair of retail types as the submarket and let xðtÞ and

yðtÞ be their market shares at time t. As a result, the total market share for the pair of retail types is 100%. We can rewrite Eqs.(2) and (3)with the constraint xðtÞ þ yðtÞ ¼ 1 s follows:

dxðtÞ dt ¼ ða1þ b1xðtÞ þ c1yðtÞÞxðtÞ ¼ ðða1þ c1Þ þ ðb1 c1ÞxÞx ¼ B2x þ A2x2; ð5Þ where B2= a1+ c1and A2= b1 c1.

From the derivation of the above formulas, we find that with the constraint of 100% total market share, the Bass and Lotka– Volterra models degenerate to the same model. This new model is named the ‘modified Lotka–Volterra model’. The modified Lotka–Volterra model is formulated as follows:

dxðtÞ dt ¼ BxðtÞ þ AðxðtÞÞ 2_; yðtÞ ¼ 1 xðtÞ: 8 > < > : ð6Þ

Note that the modiﬁed Lotka–Volterra model can only be used to study a pair of participants with the total market shares of 1. With-out the total market share constraint, applying the Lotka–Volterra model directly on market share data may result in impractical fore-casting results.

3.3. Forecasting evaluation

To assess forecasting ability, the mean absolute percentage error (MAPE) has been widely used to estimate forecast errors. The formula for the MAPE is:

MAPE ¼1 n Xn t¼1 yðtÞ ^yðtÞ yðtÞ ; ð7Þ

where yðtÞ is the actual value, ^yðtÞ is the forecast value, and n is the number of forecast periods. The MAPE is used to measure the errors between the actual and forecast values and is expressed in percent-age terms. Unfortunately, small y(t) values may result in mislead-ingly large MAPE values and actual values close to zero can generate inﬁnitely large MAPEs.

To solve this problem, we propose the MAE for evaluation of forecasting models using market share data. The formula for the MAE is:

Fig. 2. Monthly market share of convenience stores vs. traditional stores in the convenience-oriented submarket of Taiwan.

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MAE ¼1 n

Xn t¼1

jðyðtÞ ^yðtÞÞj ð8Þ

where yðtÞ is the actual value, ^yðtÞ is the forecast value, and n is the number of forecast periods.

The MAE estimates the absolute difference between the actual and forecast values. Furthermore, it can handle small actual values. When applied to market share data, the MAE generates the average absolute error in percentage terms. Thus, to ensure the compara-bility of our forecasting evaluations, we compare the MAEs of our market share forecasts with the MAPEs of our revenue forecasts.

4. Numerical study

As mentioned in Section 1, we are interested in the three focuses of retail competition in Taiwan, the competition between convenience stores and traditional stores (in the convenience-oriented submarket), the competition between supermarkets and hypermarkets (in the cost-oriented submarket), and the competi-tion between convenience-oriented and cost-oriented submarkets. In this section, we collect retail industry data in Taiwan to imple-ment our methodology and other existing forecasting models, which include the Bass and Lotka–Volterra models for revenue or market share data, with or without using the moving average method. Our goals are to examine the three focuses of retail competition in Taiwan, to illustrate our methodology, and to com-pare our methodology with benchmark models.

4.1. Methodology implementation

We ﬁrst consider the competition between convenience stores and traditional stores in the convenience-oriented submarket and implement our methodology for purposes of illustration. The monthly revenue data for convenience stores and traditional stores are collected from the Department of Statistics, Ministry of Economic Affairs in Taiwan. From January 1999 to April 2012, we

obtained 160 monthly observations (Fig. 1). Then, the market share data was generated from the 160 monthly observations individually.

Also, we adopted a 12-month moving average method. This is because the retail industry is mainly affected by weather, seasonal patterns, and festival celebrations. These factors are usually cycli-cal within a yearly period. After the 12-month moving average was calculated, 149 monthly observations remained.

To verify the forecasting capabilities of our model over various time periods, we considered six scenarios of different forecast for-ward periods. For each scenario, data in the forecast forfor-ward period were used as testing data to evaluate forecasting accuracy. The remaining data were used as estimation data for the model parameters. For example, in Scenario 1, the last 10 months of data were used as testing data to verify the forecasting accuracy. The remaining 150 months of data were used to estimate the model parameters. After calculation of the 12-month moving average, 149 monthly observations remained. For the 12-month moving average data, the last 10 months of data were used to verify the forecasting accuracy. The remaining 139 months of data were used to estimate the model parameters. The details of the six scenarios are listed inTable 1.

The modiﬁed Lotka–Volterra model was adopted in implement-ing our methodology. The market share data of convenience stores and traditional stores were applied to estimate the parameters and the forecast errors were calculated using the MAE.

For comparison, the traditional Bass model was implemented. The Bass model estimates the diffusion of a single product, so we examined the diffusion of convenience stores and traditional stores using two separate Bass models. The parameters of the two Bass models were ﬁtted separately and the forecast errors evaluated individually. For the revenue data, the errors were calculated as MAPE. For the market share data, the errors were calculated as MAE.

The Lotka–Volterra model was implemented for comparison. The revenue data of convenience stores and traditional stores were Table 1

The six scenarios of different forward forecast periods.

Scenario Without 12-month moving average (160 months) With 12-month moving average (149 months)

Number of testing periods Number of estimation periods Number of testing periods Number of estimation periods

1 10 months 150 months 10 months 139 months

Table 2

Forecast errors of convenience stores vs. traditional stores without the 12-month moving average.

Scenario Number of testing periods Number of estimation periods Retail types Forecast errors (without 12-month moving average) Revenue (MAPE) Market share (MAE)

Bass model LV model Bass model Modiﬁed LV model

1 10 150 Convenience 5.66% 4.03% 0.67% 0.64% Traditional 9.33% 5.32% 0.67% 0.64% 2 20 140 Convenience 9.96% 9.32% 1.79% 1.79% Traditional 12.15% 8.75% 1.79% 1.79% 3 40 120 Convenience 13.39% 15.25% 2.58% 2.82% Traditional 11.88% 10.23% 2.58% 2.82% 4 60 100 Convenience 13.71% 13.95% 2.99% 2.57% Traditional 12.25% 9.61% 2.99% 2.57% 5 80 80 Convenience 18.58% 19.49% 3.86% 3.80% Traditional 12.09% 9.68% 3.78% 3.80% 6 100 60 Convenience 25.03% 25.44% 5.60% 4.50% Traditional 13.07% 14.54% 4.68% 4.50%

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used to ﬁt the model parameters and the errors were calculated as MAPE.

We adopt the nonlinear regression method with ordinary least square approach to ﬁt these models. The software used is Statisti-cal Analysis System (SAS, version 9.1) and the hardware is a personal computer with AMD Athlon II X2 250 CPU at 3 GHz

frequency and 4 GB memory. For each scenario, it takes less than one second to ﬁnish the ﬁtting and forecasting.

To validate the contribution of the moving average method, all our estimations were conducted without and with the 12-month moving average. The results without and with the 12-month mov-ing average are reported inTables 2 and 3, respectively.

Table 3

Forecast errors of convenience stores vs. traditional stores with the 12-month moving average.

Scenario Number of testing periods Number of estimation periods Retail types Forecast errors (with 12-month moving average) Revenue (MAPE) Market share (MAE)

1 10 139 Convenience 2.53% 2.40% 0.50% 0.37% Traditional 0.27% 0.51% 1.30% 0.37% 2 20 129 Convenience 4.75% 4.29% 0.54% 0.31% Traditional 1.43% 1.85% 1.38% 0.31% 3 40 109 Convenience 4.39% 3.67% 1.03% 0.72% Traditional 5.82% 3.89% 1.40% 0.72% 4 60 89 Convenience 7.29% 7.48% 0.33% 1.52% Traditional 4.94% 1.74% 1.39% 1.52% 5 80 69 Convenience 3.92% 2.66% 2.48% 0.56% Traditional 4.32% 5.08% 2.98% 0.56% 6 100 49 Convenience 6.10% 4.42% 3.53% 2.56% Traditional 9.87% 11.81% 2.50% 2.56% Table 4

Forecast errors of supermarkets vs. hypermarkets without the 12-month moving average.

1 10 139 Supermarket 12.68% 8.85% 2.72% 2.61% Hypermarket 14.07% 8.65% 2.57% 2.61% 2 20 129 Supermarket 15.72% 13.93% 3.30% 3.29% Hypermarket 14.28% 11.09% 3.24% 3.29% 3 40 109 Supermarket 20.09% 22.00% 3.97% 3.85% Hypermarket 14.50% 15.38% 3.69% 3.85% 4 60 89 Supermarket 23.24% 25.44% 4.23% 3.65% Hypermarket 13.96% 15.00% 3.94% 3.65% 5 80 69 Supermarket 21.69% 22.56% 3.37% 3.53% Hypermarket 12.59% 13.21% 3.37% 3.53% 6 100 49 Supermarket 22.34% 22.75% 3.58% 3.72% Hypermarket 11.92% 12.34% 4.01% 3.72% Table 5

Forecast errors of supermarkets vs. hypermarkets with the 12-month moving average.

1 10 139 Supermarket 0.15% 0.92% 0.08% 0.08% Hypermarket 1.46% 1.41% 0.08% 0.08% 2 20 129 Supermarket 2.95% 0.56% 0.20% 0.21% Hypermarket 6.08% 4.62% 0.21% 0.21% 3 40 109 Supermarket 13.64% 14.08% 1.35% 1.37% Hypermarket 6.44% 1.81% 1.52% 1.37% 4 60 89 Supermarket 15.14% 8.87% 3.68% 2.43% Hypermarket 11.63% 12.64% 3.09% 2.43% 5 80 69 Supermarket 5.14% 19.52% 2.60% 2.60% Hypermarket 8.89% 7.89% 3.21% 2.60% 6 100 49 Supermarket 5.70% 10.53% 4.14% 4.05% Hypermarket 6.92% 6.04% 4.06% 4.05%

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4.2. Additional implementations

We now consider the competition between supermarkets and hypermarkets in the cost-oriented submarket. Similar to

Section4.1, the results without and with the 12-month moving average are reported inTables 4 and 5, respectively.

We then consider the competition between convenience-oriented and cost-convenience-oriented submarkets. Similar to Section4.1, the Table 6

Forecast errors of convenience-oriented vs. cost-oriented submarkets without 12-month moving average.

1 10 150 Convenience-oriented 6.22% 4.69% 2.14% 1.94% Cost-oriented 14.75% 7.42% 1.94% 1.94% 2 20 140 Convenience-oriented 10.30% 9.80% 2.07% 1.80% Cost-oriented 16.20% 12.03% 1.81% 1.80% 3 40 120 Convenience-oriented 12.03% 15.22% 2.26% 2.18% Cost-oriented 17.60% 17.01% 2.26% 2.18% 4 60 100 Convenience-oriented 13.12% 13.71% 2.28% 1.85% Cost-oriented 18.42% 16.35% 2.08% 1.85% 5 80 80 Convenience-oriented 16.43% 19.66% 2.95% 2.72% Cost-oriented 16.23% 16.26% 2.95% 2.72% 6 100 60 Convenience-oriented 22.76% 22.98% 3.40% 3.16% Cost-oriented 15.77% 15.71% 3.40% 3.16% Table 7

Forecast errors of convenience-oriented vs. cost-oriented submarkets with 12-month moving average.

1 10 150 Convenience-oriented 1.64% 1.64% 0.20% 0.20% Cost-oriented 0.16% 0.34% 0.15% 0.20% 2 20 140 Convenience-oriented 3.69% 3.38% 0.30% 0.30% Cost-oriented 3.37% 3.82% 0.43% 0.30% 3 40 120 Convenience-oriented 4.59% 8.60% 0.54% 0.21% Cost-oriented 2.59% 4.43% 0.54% 0.21% 4 60 100 Convenience-oriented 4.38% 7.71% 2.38% 2.38% Cost-oriented 13.87% 9.82% 2.38% 2.38% 5 80 80 Convenience-oriented 2.10% 6.39% 1.10% 1.10% Cost-oriented 10.46% 6.21% 1.10% 1.10% 6 100 60 Convenience-oriented 5.87% 18.58% 3.33% 1.10% Cost-oriented 11.41% 9.22% 6.90% 1.10% Table 8

Average forecast errors.

Competition Retail types With 12-month moving average Average forecast errors

Revenue (MAPE) Market share (MAE)

Bass model LV model Bass model Modiﬁed LV model Convenience stores vs. traditional stores Convenience stores No 14.39% 14.58% 2.92% 2.69%

Yes 4.83% 4.15% 1.40% 1.01%

Traditional stores No 11.80% 9.69% 2.75% 2.69%

Yes 4.44% 4.15% 1.83% 1.01%

Supermarket vs. hypermarket Supermarket No 19.29% 19.26% 3.53% 3.44%

Yes 7.12% 9.08% 2.01% 1.79%

Hypermarket No 13.55% 12.61% 3.47% 3.44%

Yes 6.90% 5.74% 2.03% 1.79%

Convenience-oriented vs. cost-oriented Convenience-oriented No 13.48% 14.34% 2.52% 2.28%

Yes 3.71% 7.72% 1.31% 0.88%

Cost-oriented No 16.50% 14.13% 2.41% 2.28%

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results without and with the 12-month moving average are reported inTables 6 and 7, respectively.

Finally, we summarizeTables 2–7. For each retail type inTables 2–7, we calculate the average forecast errors among the six scenar-ios. The results are listed inTable 8and visualized inFigs. 4–9.

4.3. Analysis and comparison of proposed methodology

4.3.1. Analysis of data selection

The results reported in Sections4.1 and 4.2indicate that the forecasts using the market share data were signiﬁcantly better than the forecasts using the revenue data for both the Bass and Lot-ka–Volterra models. Consider the competition between conve-nience stores and traditional stores as an example (seeTable 2). When using the revenue data in the Bass model, the forecast errors ranged from 5.66% to 25.03% for the six scenarios. When using the market share data in the Bass model, the forecast errors were much smaller: between 0.67% and 5.6% for the six scenarios. Similarly, for the Lotka–Volterra model, using revenue data generates forecast errors of between 4.03% and 25.44%, whereas using market share

data (the modiﬁed Lotka–Volterra model) gave forecast errors of between 0.64% and 4.5% for the six scenarios. The results using the 12-month moving average were also similar (see Table 3). When forecasting the competition between supermarkets and hypermarkets and between convenience-oriented and cost-ori-ented submarkets, the forecasts of the market share data outper-formed the forecasts of the revenue data for both the Bass and Lotka–Volterra models (seeTables 4–7). This result can also be ver-iﬁed for the average forecast errors (seeTable 8andFigs. 4–9).

Inflation and economic growth may raise the prices of necessi-ties and goods, which increases the revenue of retail industry. As a result, the revenue data may not accurately measure the actual growth of each specific retail type. When market share data are used, the market shares of different retail types must sum to 100%. This removes the market fluctuations and helps us to observe the relationships between different retail types clearly. Thus, market share data are more suitable than revenue data for forecasting retail competition.

4.3.2. Analysis of data processing

From the results reported in Sections4.1 and 4.2, we ﬁnd that the moving average method improves forecast accuracy. Consider 0% 5% 10% 15% Bass model LV model Bass model Modified LV model Revenue (MAPE) Market share

(MAE) Average forecasting errors

Convenience stores

Without 12-month moving average With 12-month moving average

Fig. 4. Average forecasting errors for convenience stores.

0% 5% 10% 15% Bass model LV model Bass model Modified LV model Revenue (MAPE) Market share

Traditional stores

Fig. 5. Average forecasting errors for traditional stores.

0% 5% 10% 15% 20% Bass model LV model Bass model Modified LV model Revenue (MAPE) Market share

Supermarkets

Fig. 6. Average forecasting errors for supermarkets.

0% 5% 10% 15% Bass model LV model Bass model Modified LV model Revenue (MAPE) Market share

Hypermarkets

Fig. 7. Average forecasting errors for hypermarkets.

Convenience-oriented submarket Without 12-month moving average With 12-month moving average

Fig. 8. Average forecasting errors for convenience-oriented submarket.

Cost-oriented submarket

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the competition between convenience stores and traditional stores as an example. When market share data in the Bass model is used, the forecast errors ranged from 0.67% to 5.6% for the six scenarios (seeTable 2). When the moving average method is used, the fore-cast errors fell to between 0.33% and 3.53% for the six scenarios (seeTable 3). Similarly, when the market share data in the modi-ﬁed Lotka–Volterra model is used, the forecast errors ranged from 0.64% to 4.5% for the six scenarios (seeTable 2). The forecast errors using the moving average method decreased to between 0.31% and 2.56% for the six scenarios (seeTable 3). The value of the moving average method can also be found in the analysis of the competi-tion between supermarkets and hypermarkets and between conve-nience-oriented and cost-oriented submarkets (see Tables 4–7). This result was also observed in the average forecast errors (see Table 8andFigs. 4–9).

Over the course of a year, the consumption behavior of custom-ers changes with the seasons and traditional festivals. In addition, promotions and sales are used to increase short-term proﬁts. The use of the moving average method removes short-term ﬂuctuation in retail sales associated with the seasons, festivals, promotions, and other factors. As these factors occur over an annual cycle for the retail industry, the 12-month moving average is recommended to mitigate the impact of those events to increase forecast accuracy.

4.3.3. Model selection

From the results reported in Sections4.1 and 4.2, we confirm that our modified Lotka–Volterra model outperforms other bench-mark models for both the revenue and bench-market share data, even with or without the moving average method, including the Bass and standard Lotka–Volterra models. Take the competition between convenience stores and traditional stores as an example (see Table 3). With the 12-month moving average, the forecast errors of the modified Lotka–Volterra model range from 0.31% to 2.56% for the six scenarios, which are smaller than those of the Bass model with revenue data (errors ranged between 0.27% and 9.87%) and with market share data (errors ranged between 0.33% and 3.53%). Moreover, the forecast errors are smaller than in the Lot-ka–Volterra model with revenue data (errors ranged between 0.51% and 11.81%). Similar results were found without the 12-month moving average (Table 2). In examining the competition between supermarkets and hypermarkets and between conve-nience-oriented and cost-oriented submarkets (see Tables 4–7), the advantages of the modified Lotka–Volterra model were also confirmed. This result was also observed for the average forecast errors (Table 8andFigs. 4–9).

The Bass model was originally designed for the diffusion of a single product. Although it can mimic the diffusion of a single industry, it may not be suitable for describing the competition between different retail types. However, the Lotka–Volterra model describes the interactions among the populations of different spe-cies. When applying it to the retail industry, the revenue data are generally treated as the populations of the different retail types. However, the revenue data may be affected by market fluctuations, thus reducing forecast accuracy. In our modified Lotka–Volterra model, we adopted the market share data instead of revenue data, which led to significant improvements in the accuracy of forecasting.

In addition, our modiﬁed Lotka–Volterra model produces fore-cast errors that are symmetric and increase as the estimation per-iod decreases. The forecast errors are symmetric because the forecast values for the competing retail types sum to 100%. As a result, the forecast values must have the same MAEs. On the con-trary, the Bass and Lotka–Volterra models generate unstable results. Consequently, our modiﬁed Lotka–Volterra model is rec-ommended for competition forecasting in Taiwan’s retail industry.

4.4. Synergy of the three enhancements

In Section4.3, three extensions were implemented to improve the accuracy of forecasting for the retail industry. First, market share data are more suitable than revenue data when analyzing the competition between retail types. Second, using a 12-month moving average helps to eliminate the short-term noise over the course of a year. Third, the modiﬁed Lotka–Volterra model can not only mimic the interaction between different retail types but also eliminate the effects of general market ﬂuctuations.

These three extensions work together to create synergy and dramatically improve the accuracy of short- and long-term fore-casting. That is, the use of any of these two extensions provides better results than using the two methods independently. Take the example of the competition between convenience stores and traditional stores (seeTables 2 and 3).The use of both the market share data and the moving average method provides better results than using each independently. When using only market share data, the forecast errors of the Bass model ranged between 0.67% and 5.6%, and when using only the moving average method, the forecast errors of the Bass model ranged between 0.27% and 9.87% for the six scenarios. When using the market share data and the moving average together, the forecast errors of the Bass model ranged between 0.33% and 3.53% for the six scenarios.

Finally, the use of all three extensions provides even better results than using any two. Consider the following (seeTables 2 and 3). When using the market share data and the moving average in the modiﬁed Lotka–Volterra model, the forecast errors ranged between 0.31% and 2.56% for the six scenarios. Similarly, the syn-ergy of using all three extensions can be observed clearly in the competition between supermarkets and hypermarkets and between convenience-oriented and cost-oriented submarkets (seeTables 4–7). This result is also observed in the average forecast errors (seeTable 8andFigs. 4–9).

5. Conclusions and future research

This research focused on the competition between the different retail types and a proposed methodology to improve forecasting of the level of competition, which has four aspects. First, regarding data selection, market share data are recommended for competi-tion forecasting in retail industry. As the characteristics of retail industry, revenue data is usually mixed with confounding factors such as inﬂation and economic growth. Market share data, which can mitigate some volatility associated with inﬂation and eco-nomic growth, is able to perform better in forecasting.

Second, regarding data transformation, the moving average method can effectively lower forecast errors. This is because the moving average method can smooth seasonal patterns and cycles. Third, regarding model selection, our modiﬁed Lotka–Volterra model, which contains the total market share constraint, is able to attain the best forecasting results. Fourth, regarding forecasting evaluation, MAE is recommended to evaluate our modiﬁed Lotka– Volterra model with market share data. It is because market share data might contain actual values very close to zero. Some minor forecast errors on these small actual values can result in mislead-ingly large MAPE values. MAE not only can handle small actual val-ues but also generates the average absolute error in percentage terms.

Revenue data from Taiwan’s retail industry for a 160-month period are used to verify the performance and accuracy of our pro-posed methodology. Our numerical results indicate signiﬁcant improvement in forecasting capabilities.

For future research, we are interested in the following two directions. First, given rising fuel prices and ongoing economic

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growth, the role of online retailing becomes more important for the modern retail industry. As a result, the competition between online and brick-and-mortar retailing has attracted more attention. We would like to apply our methodology to study the impact of this new retail type.

Second, rather than simply removing seasonal patterns and cycles, we are interested to capturing these features of the data in our model. Some methods, such as the X11 and X12-ARIMA pro-grams, use seasonal adjustments and iterative approaches to esti-mate seasonality and trends in the data. Other methods, such as the Hodrick–Prescott ﬁlter, can separate trends and cycles by add-ing smoothadd-ing parameters. We are interested in these possible extensions to the model and would like to develop a better proce-dure to capture seasonal patterns and cycles.

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