Chapter 1. Introduction. Sales Forecasting of Grameenphone Business Solutions SIM BUS 650 1

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1 Chapter Introduction Sales Forecasting of Grameenphone Business Solutions SIM BUS 65

2 . INTRODUCTION The leading telecom operator of Bangladesh Grameenphone Ltd., from the initiation of its Business Segment, has been catering to Corporate and SME sub-segment through exclusive product offering - Business Solutions Postpaid and Prepaid. To cater these sub segments the product line has been modified and aligned with market offers. Business Solutions Prepaid and Postpaid are premium product and highly distinctive than the regular mass market product to give superior product experience to the valued Corporate and SME client. The main mode of sales channel is direct sales through its highly trained Key Account Manager. Later on Indirect Sales Channel has been introduced to cater to the Small SME/ SOHO sub-segment. The sales of Business Solutions SIM are being made against agreement. This study attempts to find out the sales forecast for Grameenphone Business Solutions SIM for total Business Segment..2 OBJECTIVE AND RATIONALE OF SALES FORECASTING Sales forecasting is a self-assessment tool for a company. Regular sales forecast helps to sustain the pulse of a business. It can make the difference between just surviving and being highly successful in business. It is a vital cornerstone of a company's budget. The future direction of the company may rest on the accuracy of your sales forecasting. Companies that implement accurate sales forecasting processes realize important benefits such as increased revenue, increased customer retention, decreased costs and increased efficiency. A sales forecast is a prediction based on past sales performance and an analysis of expected market conditions. The true value in making a forecast is that it forces to look at the future objectively. The company that takes note Sales Forecasting of Grameenphone Business Solutions SIM BUS 65 2

3 of the past stays aware of the present and precisely analyzes that information to see into the future. Conducting a sales forecast will provide a business with an evaluation of past and current sales levels and annual growth, and allow comparing the company to industry norms. It will also help to establish policies so that one easily can monitor your prices and operating costs to guarantee profits, and make aware of minor problems before they become major problems. Our objective is therefore to conduct sales forecast of Business Solutions SIM of Grameenphone Ltd. applying the most suitable and useful forecasting tools. Grameenphone Ltd. may sell out its inventory due to low sales forecast or may build up excess inventory due to high sales forecast. Stock out or over supply of inventory during period where seasonal variations or trend exists may result in huge loss if forecast is not accurate. Precise and dynamic forecasting of Business Solution SIM will ensure smooth business operation through uninterrupted product supply, thereby hitting the quarterly target and required growth. Customer usage pattern can be determined through forecasting. Customer loyalty and reputation in the market can be sustained through accurate forecast of Business Solution SIM sales. Sales Forecasting of Grameenphone Business Solutions SIM BUS 65 3

4 .3 PROBLEM STATEMENT To precisely predict future sales volume of Grameenphone Business Solutions SIM considering past sales trend applying various forecasting techniques to determine sustainability of growth rate and profit..4 SCOPE AND LIMITATIONS OF THE STUDY The forecast has been calculated based on the last 66 months Sales trend (January 26- June 2) of the Business Solution SIM of Grameenphone Ltd. One of the limitations of the study is only quantitative forecasting techniques have been applied. Also we could not use any future market intelligence information..5 COMPANY OVERVIEW Starting its operations on March 26, 997, the Independence Day of Bangladesh, Grameenphone pioneered the then breakthrough initiative of mobile to mobile telephony and became the first and only operator to cover 98% of the country s people with network Since its inception Grameenphone has built the largest cellular network in the country with over 3, base stations in more than 7 locations. Presently, nearly 98 percent of the country's population is within the coverage area of the Grameenphone network. Grameenphone has always been a pioneer in introducing new products and services in the local market. GRAMEENPHONE LTD. was the first company to introduce GSM technology in Bangladesh when it launched its services in March 997. Sales Forecasting of Grameenphone Business Solutions SIM BUS 65 4

5 Grameenphone was also the first operator to introduce the pre-paid service in September 999. It established the first 24-hour Call Center, introduced value-added services such as VMS, SMS, fax and data transmission services, international roaming service, WAP, SMS-based push-pull services, EDGE, personal ring back tone and many other products and services. The entire Grameenphone network is also EDGE/GPRS enabled, allowing access to high-speed Internet and data services from anywhere within the coverage area. There are currently nearly 2.6 million EDGE/GPRS users in the Grameenphone network. Today, Grameenphone is the leading telecommunications service provider in Bangladesh with more than 33 million subscribers as of May 2. COMPANY HIGHLIGHTS Grameenphone has so far invested more than BDT 6, crore to build the network infrastructure Grameenphone is one of the largest taxpayers in the country, having contributed more than BDT 8,5 crore in direct and indirect taxes to the Government Exchequer over the years. There are now more than 6 GP Service Desks across the country covering nearly all upazilas of all districts and 82 Grameenphone Centers in all the divisional cities Grameenphone has more than 5 full and temporary employees. 3, people are directly dependent on Grameenphone for their livelihood, working for the Grameenphone dealers, retailers, scratch card outlets, suppliers, vendors, contractors and others. Sales Forecasting of Grameenphone Business Solutions SIM BUS 65 5

6 OWNERSHIP STRUCTURE It is a joint venture enterprise between Telenor (55.8%), the largest telecommunications service provider in Norway with mobile phone operations in 2 other countries, and Grameen Telecom Corporation (34.2% ), a non-profit sister concern of the internationally acclaimed micro-credit pioneer Grameen Bank. The other % shares belong to general retail and institutional investors. Fig : Ownership Structure of GrameenPhone MISSION Leading the industry and exceed customer expectations by providing the best wireless services, making life and business easier. VISION We exist to help our customers get the full benefit of communications services in their daily lives. We want to make it easy for customers to get what they want, when they want it. We're here to help. Sales Forecasting of Grameenphone Business Solutions SIM BUS 65 6

7 VALUES Make It Easy Keep Promises Be Inspiring Be Respectful BRAND PROMISE Stay Close COMPETITION As at 3 March 2, Grameenphone had a market share of 43.8%. In addition to Grameenphone, there are five other mobile operators in Bangladesh. These operators and their market shares as at 3 March 2 are: Banglalink (27.6%), Robi (8.2%), Airtel Bangladesh, previously Warid (6.3%), Citycell (2.4%) and Teletalk (.6%). Competition among operators is intense and tariff levels are among the lowest in the world. Sales Forecasting of Grameenphone Business Solutions SIM BUS 65 7

8 Chapter 2 Methodology Sales Forecasting of Grameenphone Business Solutions SIM BUS 65 8

9 2. METHODOLOGY To forecast the monthly Sales of Grameenphone Business Solution SIM, we have collected actual monthly sales data for the year 26 till Jun DATA SOURCES The data has been collected from Business Sales department of Grameenphone Ltd. They use a tool called Business Solutions Customer Database (BSCDB) to maintain all sales related data and information starting from lead generation to final agreement. We have used BSCDB generated sales data in prior approval from respective concerns of Business Sales Department. 2.3 FORECASTING TECHNIQUES We have applied Time Series Forecasting techniques on our collected data to identify any trend, seasonality, irregular variations and random variations within the data set. Smoothing constant value has been determined through trial and error basis every time.: Trend Analysis Moving Average Single Exponential Smoothing Double Exponential Smoothing Time Series Decomposition We have done sales forecast based on some other appropriate forecasting techniques which were not covered within our course. Sales Forecasting of Grameenphone Business Solutions SIM BUS 65 9

10 Winters Method ARIMA Method To calculate the forecasting accuracy, we have used the following techniques: Mean Absolute Deviation (MAD) Mean Squared Error (MSE) Mean Absolute Percent Error (MAPE) Tracking Signal (TS), and Trigg s Tracking Signal After conducting the forecast applying different techniques, we have found out the forecasting accuracy and the appropriate techniques for forecasting the sales of Business Solutions SIM. 2.4 STATISTICAL TOOL In order to get the forecasting output of Business Solutions SIM, we have used MINITAB Software. Sales Forecasting of Grameenphone Business Solutions SIM BUS 65

11 Chapter 3 Analysis Sales Forecasting of Grameenphone Business Solutions SIM BUS 65

12 Sales Volume 3. ANALYSIS 3.. TIME SERIES PLOT 4 Time Series Sales Plot for Business Solutions SIM Time Fig 5: Time Series Plot Time Series Plot of Business Solutions SIM Sales shows data with both trends and seasonality. An interesting finding is that data values are following common trend of fluctuations since 26 till June 2 incorporating both trend (gradual upward and downward movement) and seasonality (short-term regular variations). The fluctuation is mainly due to quarterly target based sales. The Key Account Managers used to mature majority sales at the last month of each quarter in order to fulfill quarterly target. Moreover, Regular Campaigns and Eid Campaigns, Customer events and development programs have always played an important role in causing sales growth time to time. Declining sales values whereas have been occurred mainly due to Sales Forecasting of Grameenphone Business Solutions SIM BUS 65 2

13 aggressive market competition, emphasize on quality sales to reduce churn, competition with mass market product, less opportunity in segmented approach, difficulty in capturing addressable market in non-metro areas etc. 3.2 DESCRIPTION OF THE FORECASTING METHODS The objective of this paper is to compare the results of several forecasting methods to detect which model appears to be most appropriate for the given time series data. The computation method of different forecasting model is done with the help of Minitab statistical software and Microsoft office. Different types of time series models such as moving average (MA), linear trend, single exponential smoothing model, double exponential model, Time Series Decomposition, Winter s method have been considered. Throughout the analysis one software applications (Minitab) & Ms office were used to facilitate calculations and plotting data. We discussed in brief details below each of the above models along with proper judgment of their use, the applications with result. TREND ANALYSIS Trend analysis represents a picture about the position and pattern of data. It deals with the consistency and ups and downs of the obtained data. It can be parabolic trend, Exponential trend or growth curve.a simple plot of data often can reveal the existence and nature of a trend. A linear trend model is used to predict future values of monthly sales data in equation, Ft = a+bt, () Sales Forecasting of Grameenphone Business Solutions SIM BUS 65 3

14 Where, t = Specified number of time periods from t =, Ft is the forecast for period t, a is the value of Ft at t=, b is the slope of the line. Applying the curve estimation function the time series data is plotted and compared to several models to find an appropriate trend line. A straight line is mapped to the observations which are plotted against a single independent variable, time, on the horizontal axis. We obtain the best fitted linear trend equation. MOVING AVERAGE METHOD Moving averages rank among the most popular techniques for the preprocessing of time series. They are used to filter random "white noise" from the data, to make the time series smoother or even to emphasize certain informational components contained in the time series. Moving average (MA) forecast uses a number of the most recent actual data values in generating a forecast. Under this method effect of extreme data values is neutralized by other observations depending upon the number of periods used. The MA forecast for time period t (Ft) is computed using the following equation: F t MA n n Ai i n Where, Ai is the actual value in period i and n is the number of periods (span or data points) in the MA. Figure shows MA model with a span of 3 and 5 periods along with the original data. Sales Forecasting of Grameenphone Business Solutions SIM BUS 65 4

15 Note that in a moving average, as each new actual value becomes available, the forecast is updated by adding the newest value and dropping the oldest and then recomposing the average, consequently the forecast moves by reflecting only the most recent values. In computing a moving average, including a moving total column which gives the sum of the n most recent values from which the average will be computed. To update the moving total: subtract the oldest value and add that amount to the moving total for each update. While the approach is quite simple to use and understand, there is also a significant disadvantage. Since the MA forecast uses the averages of previous periods, it will lag behind the actual demand when a trend exists in the data. As the number of periods used in the average calculation increases, the lag in the forecast also increases. In addition, subjectivity is introduced into the approach through the selection of n. It is also critical to understand that the longer the period chosen to calculate the average, the smoother the series since fewer MA values are computed and graphically displayed. SINGLE EXPONENTIAL SMOOTHING Single exponential smoothing smoothes data by computing exponentially weighted averages and provides short-term forecasts. Under this method each new forecast is based on the previous forecast plus a percentage of the difference between that forecast and the actual value of the series at that point. The required model for forecast for period t is: Ft = Ft- + α (At- Ft-) Sales Forecasting of Grameenphone Business Solutions SIM BUS 65 5

16 Where, Ft is the forecast for period t, Ft- is the forecast for the previous period, α is the smoothing constant, and At- is the actual sales of the product at (t-)th month. The smoothing constant α determines the relative weight placed on the current observation while (-α) is the weight placed on past observation. The choice of α is rather subjective ( α ). If the desire is to only smooth a series by eliminating unwanted cyclical and irregular variations, a small value for α should be chosen. On the other hand, if a larger α is chosen, future short-term directions may be more accurately predicted and the forecast is more responsive to changing conditions. We demonstrate these phenomena by choosing two different values for α (.2 and.3). It is however seen that 6 period future forecasts are same but these values can be updated periodically when actual data is available. DOUBLE EXPONENTIAL SMOOTHING Double exponential smoothing provides short-term forecasts for the time series data. It works well when a trend is present. This method utilized two estimates for level and trend components. It is also called trend-adjusted exponential smoothing as it employs a level component and a trend component at each period. Using two smoothing constants, it updates the components at each period. The double exponential smoothing equations are: TAFt+ = St + Tt Where, St = is the smoothed forecasts and, Tt = is the current trend estimate. St = TAFt + α (At - TAFt) The formula for both smoothed forecasts and the current trend estimate is given below: Sales Forecasting of Grameenphone Business Solutions SIM BUS 65 6

17 Tt = Tt- + γ (TAFt - TAFt- Tt-) Where, α & γ are two smoothing constants. If the first observation is numbered one, then level and trend estimates at time zero must be initialized in order to proceed. The initialization method used to determine how the smoothed values are obtained in one of two ways: with optimal weights or with specified weights. Figure shows the forecasts and actual data using double exponential smoothing with ( =.2, =.3) and ( =.3, =.4). TIME SERIES DECOMPOSITION MODEL The simple seasonal method is the most basic method of computing the seasonal factors for a given series of data. A widely used scheme to estimate the initial values of the seasonal factors involves simply dividing the observation in each period by the average for the season (Montgomery et. al., 99). Time series seasonal decomposition model used a centered moving average with a length equal to the length of the seasonal cycle. When the seasonal cycle length is an even number, a two-step moving average is required to synchronize the moving average correctly. It divides the moving average into multiplicative model to obtain what are often referred to as raw seasonal values. For corresponding time periods in the seasonal cycles; this model determines the median of the raw seasonal values. This model uses the seasonal indices to seasonally adjust the data and fits a trend line to the seasonally adjusted data using least squares regression. The data is de-trended by dividing the data with the trend component. Sales Forecasting of Grameenphone Business Solutions SIM BUS 65 7

18 WINTER S METHOD Winters Method smoothes data by Holt-Winters exponential smoothing and provides short to medium-range forecasting. One can use this procedure when both trend and seasonality are present, with these two components being either additive or multiplicative and hence this model may be interpreted as a type of triple exponential smoothing. Winters Method calculates dynamic estimates for three components: level, trend, and seasonal. The Holt-Winters model is multiplicative when the level and seasonal components are multiplied together. Multiplicative model incorporates the magnitude of the seasonal pattern which depends on the magnitude of the data. In other word, the magnitude of the seasonal pattern increases as the data values increase, and decreases as the data values decrease. Winters method employs a level component, a trend component, and a seasonal component at each period. It uses three weights, or smoothing parameters, to update the components at each period. Smoothing parameters, for the level, trend, and seasonal components take values between and. Regardless of the component, large weights result in more rapid changes in that component; small weights result in less rapid changes. The components in turn affect the smoothed values and the predicted values. Initial values for the level and trend components are obtained from a linear regression on time. Initial values for the seasonal component are obtained from a dummy-variable regression using detrended data. The equations involving level, trend seasonal components and forecasts are given below: Sales Forecasting of Grameenphone Business Solutions SIM BUS 65 8

19 L T S t t t t Yˆ ( Y / S ( L t t t t t p [ L L t ( Y / L ) ( ) S t T t ) ( )[ L ] ( ) T ) S t p t t p t T t ] Where, L t is the level at time t, α is the weight for the level, T t is the trend at time t, γ is the weight for the trend, S t is the seasonal component at time t, δ is the weight for the seasonal Yˆ component, p is the seasonal period, Y t is the data value at time t, t is the fitted value, or oneperiod-ahead-forecast at time t. AUTO-REGRESSIVE INTEGRATED MOVING AVERAGE (ARIMA) MODEL: ARIMA models are, in theory, the most general class of models for forecasting a time series which can be stationarized by transformations such as differencing and logging. In fact, the easiest way to think of ARIMA models is as fine-tuned versions of random-walk and randomtrend models: the fine-tuning consists of adding lags of the differenced series and/or lags of the forecast errors to the prediction equation, as needed to remove any last traces of autocorrelation from the forecast errors. The acronym ARIMA stands for "Auto-Regressive Integrated Moving Average." Lags of the differenced series appearing in the forecasting equation are called "auto-regressive" terms, lags of the forecast errors are called "moving average" terms, and a time series which needs to be differenced to be made stationary is said to be an "integrated" version of a stationary series. Random-walk and random-trend models, autoregressive models, and exponential smoothing Sales Forecasting of Grameenphone Business Solutions SIM BUS 65 9

20 models (i.e., exponential weighted moving averages) are all special cases of ARIMA models.a nonseasonal ARIMA model is classified as an "ARIMA(p,d,q)" model, where: p is the number of autoregressive terms, d is the number of non-seasonal differences, and q is the number of lagged forecast errors in the prediction equation. AR, MA, AND ARIMA MODELING OF TIME SERIES DATA An Autoregressive (AR) Process Let Yt represent Sales at time t. If we model Yt as where δ is the mean of Y and where ut is an uncorrelated random error term with zero mean annconstant variance σ2 (i.e., it is white noise), then we say that Yt follows a first-order autoregressive, or AR(), stochastic process.. Here the value of Y at time t depends on its value in the previous time period and a random term; the Y values are expressed as deviations from their mean value. In other words, this model says that the forecast value of Y at time t is simply some proportion (= α) of its value at time (t ) plus a random shock or disturbance at time t; again the Y values are expressed around their mean values. But if we consider this model, Sales Forecasting of Grameenphone Business Solutions SIM BUS 65 2

21 then we say that Yt follows a second-order autoregressive, or AR(2), process. That is, the value of Y at time t depends on its value in the previous two time periods, the Y values being expressed around their mean value δ. In general, we can have in which case Yt is a pth-order autoregressive, or AR(p), process. A Moving Average (MA) Process The AR process just discussed is not the only mechanism that may have generated Y. Suppose we model Y as follows: where μ is a constant and u, as before, is the white noise stochastic error term. Here Y at time t is equal to a constant plus a moving average of the current and past error terms. Thus, in the presentcase, we say that Y follows a first-order moving average, or an MA(), process. But if Y follows the expression then it is an MA(2) process. More generally, Sales Forecasting of Grameenphone Business Solutions SIM BUS 65 2

22 is an MA(q) process. In short, a moving average process is simply a linear combination of white noise error terms. An Autoregressive and Moving Average (ARMA) Process Of course, it is quite likely that Y has characteristics of both AR and MA and is therefore ARMA. Thus, Yt follows an ARMA(, ) process if it can be written as because there is one autoregressive and one moving average term. Here θ represents a constant term. In general, in an ARMA( p, q) process, there will be p autoregressive and q moving average terms. An Autoregressive Integrated Moving Average (ARIMA) Process If we have to difference a time series d times to make it stationary and then apply the ARMA(p, q) model to it, we say that the original time series is ARIMA(p, d, q), that is, it is an autoregressive integrated moving average time series, where p denotes the number of autoregressive terms, d the number of times the series has to be differenced before it becomes stationary, and q the number of moving average terms. Thus, an ARIMA(2,, 2) time series has to be differenced once (d = ) before it becomes stationary and the (first-differenced) stationary time series can be modeled as an ARMA(2, 2) process, that is, it has two AR and two MA terms. Of course, if d = (i.e., a series is stationary to begin with), ARIMA(p, d =, q) = ARMA(p, q). An ARIMA(p,, ) process means a purely AR(p) stationary process; an ARIMA(,, q) means Sales Forecasting of Grameenphone Business Solutions SIM BUS 65 22

23 a purely MA(q) stationary process. Given the values of p, d, and q, one can tell what process is being modeled. To identify the appropriate ARIMA model for a time series, we need to begin by identifying the order(s) of differencing needing to stationarize the series and remove the gross features of seasonality, perhaps in conjunction with a variance-stabilizing transformation such as logging or deflating. If we stop at this point and predict that the differenced series is constant, we have merely fitted a random walk or random trend model. (The random walk model predicts the first difference of the series to be constant, the seasonal random walk model predicts the seasonal difference to be constant, and the seasonal random trend model predicts the first difference of the seasonal difference to be constant--usually zero.) However, the best random walk or random trend model may still have autocorrelated errors, suggesting that additional factors of some kind are needed in the prediction equation. THE BOX JENKINS (BJ) METHODOLOGY The million-dollar question obviously is: Looking at a time series, how does one know whether it follows a purely AR process (and if so, what is the value of p) or a purely MA process (and if so, what is the value of q) or an ARMA process (and if so, what are the values of p and q) or an ARIMA process, in which case we must know the values of p, d, and q. The BJ methodology comes in handy in answering the preceding question. To use the Box Jenkins methodology, we must have either a stationary time series or a time series that is stationary after one or more differencings. The method consists of four steps Sales Forecasting of Grameenphone Business Solutions SIM BUS 65 23

24 . Identification of the model 2. Parameter estimation of the chosen model YES (Go to step 4) 4. Forecasting 3. Diagnostic checking ( Are the estimated rasiduals white noise?) NO (Return to step ) Figure: The Box-Jenkins Methodology Step. Identification. That is, we have to find out the appropriate values of p, d, and q. Sales Forecasting of Grameenphone Business Solutions SIM BUS 65 24

25 Step 2. Estimation. Having identified the appropriate p and q values, the next stage is to estimate the parameters of the autoregressive and moving average terms included in the model. Sometimes this calculation can be done by simple least squares but sometimes we will have to resort to nonlinear (in parameter) estimation methods..step 3. Diagnostic checking. Having chosen a particular ARIMA model, and having estimated its parameters, we next see whether the chosen model fits the data reasonably well, for it is possible that another ARIMA model might do the job as well. This is why Box Jenkins ARIMA modeling is more an art than a science; considerable skill is required to choose the right ARIMA model. One simple test of the chosen model is to see if the residualsestimated from this model are white noise; if they are, we can accept the particular fit; if not, we must start over. Thus, the BJ methodology is an iterative process. Step 4. Forecasting. One of the reasons for the popularity of the ARIMA modeling is its success in forecasting. In many cases, the forecasts obtained by this method are more reliable than those obtained from the traditional econometric modeling, particularly for short-term forecasts. Sales Forecasting of Grameenphone Business Solutions SIM BUS 65 25

26 Sales Volume 3.3 ANALYSIS OF FORECASTING TECHNIQUES TREND ANALYSIS Linear Trend Analysis Linear Trend Model Yt = *t Variable Actual Fits Forecasts Accuracy Measures MAPE 66 MAD 652 MSD Time Fig 6: Linear Trend Analysis Trend Analysis - Linear: The plot of Linear Trend Analysis shows an increasing trend. However, the demand for the Business Solutions SIM will not be increasing that much for the following year; as a result, it is going to largely affect future sales figures Sales Forecasting of Grameenphone Business Solutions SIM BUS 65 26

27 Sales Volume Quadratic Trend Analysis Quadratic Trend Model Yt = *t - 3.*t** Variable Actual Fits Forecasts Accuracy Measures MAPE 58 MAD 597 MSD Time Fig 7: Quadratic Trend Analysis Trend Analysis-Quadratic: The fitted trend equation, Yt = *t 3.*t2 shows a decreasing trend of data. Hence, the demand of Business Solutions SIM will gradually decrease throughout the following year. However, if we observed MAD, MSD and MAPE values, we could say that Quadratic Trend Analysis is giving more accurate forecast than Linear Trend Analysis. Sales Forecasting of Grameenphone Business Solutions SIM BUS 65 27

28 Sales Volume MOVING AVERAGE 3 Period Moving Average 4 3 Variable Actual Fits Forecasts 95.% PI Moving Average Length 3 2 Accuracy Measures MAPE 49 MAD 5627 MSD Time Fig 8: Three period moving average As we know, Moving Average forecast tends to smooth and lag changes in the data, both the 3- periods and 5-periods Moving Average of Business Solutions SIM Sales data smoothing the time series data to reduce the effects of random variation and reveal any underlying trend or seasonality. From the above Moving Average Trend it is clearly visible that short-term regular variation (seasonality) is present along with long-term movement in data (gradual upward and downward trend). The forecast is however quite stable during the next 6 months period. Sales Forecasting of Grameenphone Business Solutions SIM BUS 65 28

29 Sales Volume Considering comparatively lower value of MAPE, MAD and MSE we have found that 5-periods Moving Average is giving reliable forecast. This is also true in comparison with both Linear and Quadratic Trend Analysis. SINGLE EXPONENTIAL METHOD 4 3 Single Exponential Smoothing Single Exponential Method Variable Actual Fits Forecasts 95.% PI 2 Smoothing Constant Alpha.2 Accuracy Measures MAPE 57 MAD 63 MSD Time Fig 9: Single Exponential Smoothing (α=.2) Sales Forecasting of Grameenphone Business Solutions SIM BUS 65 29

30 Sales Volume 4 3 Single Exponential Smoothing Single Exponential Method Variable Actual Fits Forecasts 95.% PI 2 Smoothing Constant Alpha.3 Accuracy Measures MAPE 59 MAD 659 MSD Time Fig : Single Exponential Smoothing (α=.3) The above Single Exponential Smoothing Forecasting is smoothing the time series data of Business Solution SIM sales where the mean is either stationary or changes only slowly with time. The weight declines as the lag between the current time increases. As this is smoothing method which relies on previous values, the smoothed value lags the current value. Due to comparatively small smoothing value of α =.2, fluctuations are heavily damped and the smoothed value tends towards the mean. On the hand, for comparatively large smoothing constant of α =.3, fluctuations are significant and the smoothed value tends towards the current value. Moreover, the stable forecasted data is better in case of Alpha value (α=.2) because of Sales Forecasting of Grameenphone Business Solutions SIM BUS 65 3

31 Sales Volume relatively lower MAPE, MAD and MSD value than Alpha value (α=.3).as per the previous observation, we should not take this model for further analysis of residual analysis and accuracy check as the Moving Average demonstrates rather good fit. DOUBLE EXPONENTIAL SMOOTHING 4 3 Double Exponential Smoothing Double Exponential Method Variable Actual Fits Forecasts 95.% PI 2 Smoothing Constants Alpha (level).2 Gamma (trend).3 Accuracy Measures MAPE 57 MAD 6682 MSD Time Fig : Double Exponential Smoothing (α=.2, γ=.3) Sales Forecasting of Grameenphone Business Solutions SIM BUS 65 3

32 Sales Volume Double Exponential Smoothing Double Exponential Method Variable Actual Fits Forecasts 95.% PI Smoothing Constants Alpha (level).3 Gamma (trend).2 Accuracy Measures MAPE 6 MAD 6868 MSD Time Fig 2: Double Exponential Smoothing (α=.3, γ=.2) As we know the double exponential smoothing forecast is based on the assumption of a model consisting of a constant plus a linear trend, the Business Solutions SIM Sales trend however is comprising of both short-term regular variations and long-term movement (gradual upward and downward trend) in sales volume that means data with both seasonally and trend We have tried with two values of Alpha (level).2 and Gamma (trend).3 and again Alpha (level).3 and Gamma (trend).2 Where alpha is low which is.2, forecast does not adjust (response) rapidly, but where alpha is higher like.3, forecast follows the actual more closely. In both the cases the forecasted values are showing increasing trend. However, Double Exponential Smoothing Sales Forecasting of Grameenphone Business Solutions SIM BUS 65 32

33 forecast with Alpha (level).2 and Gamma (trend).3 has been able to provide better sales forecast due to comparatively lower MAPE, MAD and MSD value. Forecast accuracy however is very poor compared to Moving Average forecast. MERITS & DEMERITS OF SINGLE EXPONENTIAL SMOOTHING & DOUBLE EXPONENTIAL SMOOTHING Single exponential smoothing works well for time series without an overall trend. Single Exponential smoothing is suitable for forecasting mean values that remain fairly stable. However, in the presence of an overall trend, the smoothed values tend to lag behind the raw data. This is where double exponential smoothing comes in. The problem with single and double exponential smoothing is that either one makes a totally different assumption about the underlying model: while single exponential smoothing assumes that the system will stay steady at the last (smoothed) value, double exponential smoothing assumes that the system will continue to grow linearly at the most recent (smoothed) rate. However, both will miss drastic departures from the previous behavior. Sales Forecasting of Grameenphone Business Solutions SIM BUS 65 33

34 Sales Volume TIME SERIES DECOMPOSITION MODEL: Seasonal Decomposition Additive Model Variable Actual Fits Trend Forecasts Accuracy Measures MAPE 44 MAD 4977 MSD Time Fig 3: Seasonal Decomposition: Additive Model Sales Forecasting of Grameenphone Business Solutions SIM BUS 65 34

35 Sales Volume Seasonal Decomposition Multiplicative Model Variable Actual Fits Trend Forecasts Accuracy Measures MAPE 44 MAD 496 MSD Time Fig 4: Seasonal Decomposition: Multiplicative Model The above output of Time Series Decomposition shows how Additive and Multiplicative models are decomposing time series data into trend, cycle, seasonal, and irregular components, and returning seasonally adjusted data if desired. In the multiplicative model, the components (Trend Component and Seasonality, Randomness) have been multiplied together to estimate demand whereas in The Additive Model, those components have been added together. The effects of such addition and multiplication of components are quite visible in the above output. The additive model is useful when the seasonal variation is relatively constant over time. The multiplicative model is useful when the seasonal variation increases over time. The plot of Sales Forecasting of Grameenphone Business Solutions SIM BUS 65 35

36 decomposition shows a consistent increasing trend of sales volume. The forecasted values have also clearly adopted the seasonality component. It means that there will be opportunity for higher Business Solutions SIM sales for those similar periods with seasonal effects in the following year. From this analysis and along with forecast accuracy measurement (MAPE, MAD and MSD value), we can say that the performance of this method may be considered as the satisfied one and also reliable. Multiplicative model in this regard is providing better accuracy measures considering lower MAD and MSD value though the MAPE is same for both cases. MERITS & DEMERITS OF TIME SERIES DECOMPOSITION TECHNIQUES: This is useful for data sets with seasonal patterns, but uses complicated procedures to depersonalized data sets. Decomposition can be risky because errors in the components multiply when the forecasts are recombined. Sales Forecasting of Grameenphone Business Solutions SIM BUS 65 36

37 Sales Volume WINTERS METHOD Winters' Method Additive Method Variable Actual Fits Forecasts 95.% PI Smoothing Constants Alpha (level).2 Gamma (trend).3 Delta (seasonal).4 Accuracy Measures MAPE 36 MAD 428 MSD Time Fig 5: Winters Method: Additive Method Sales Forecasting of Grameenphone Business Solutions SIM BUS 65 37

38 Sales Volume Winters' Method Multiplicative Method Variable Actual Fits Forecasts 95.% PI Smoothing Constants Alpha (level).2 Gamma (trend).3 Delta (seasonal).4 Accuracy Measures MAPE 33 MAD 473 MSD Time Fig 6: Winters Method: Multiplicative Method The time series plot of Winters method (Triple Exponential Smoothing) have been used as Business Solutions SIM sales data contain a trend and a seasonal pattern. Slowdowns or speedups in demand, consumer behaviour can all be accommodated. The above output shows an gradual upward and downward trend respectively. The fitted values whereas adopted the similar seasonal variations. Accuracy measures are quite good. The forecasted values are also reflecting seasonality and better value compared to recent sales volume trend. Accuracy measurement is better in the model with Alpha =.2, Gamma =.3 and Delta=.4.The Additive Winters Model is providing better accuracy considering MAD and MSD, whereas MAPE value is better for Sales Forecasting of Grameenphone Business Solutions SIM BUS 65 38

39 Multiplicative Winters model. By considering the overall observation, we can say that this model performance in predicting or forecasting the actual demand of the SIM is quite good. MERITS & DEMERITS OF WINTERS METHOD The method is popular because it is simple, has low data-storage requirements, and is easily automated. It also has the advantage of being able to adapt to changes in trends and seasonal patterns in sales when they occur. The motivation behind using the adaptive technique, as opposed to the non-adaptive technique is that, the time series may change is behavior and the model parameters should adapt to this change in behavior. The value of Look back size also play an important role in the performance of the adaptive model, hence these parameters were also varied for forecasting the different time series. It produces prediction intervals, which affect safety-stock calculations, among other things. AN AUTOREGRESSIVE INTEGRATED MOVING AVERAGE (ARIMA) PROCESS The Box-Jenkins Methodology We will look at these four steps of The Box-Jenkins Methodology in some details. Throughput we will use Sales data to illustrate the various points. IDENTIFICATION The chief tools in identification are the autocorrelation function (ACF), the partial autocorrelation function (PACF), and the resulting correlograms, which are simply the plots of ACFs and PACFs against the lag length. Sales Forecasting of Grameenphone Business Solutions SIM BUS 65 39

40 Partial Autocorrelation Autocorrelation In Figure7 and Figure8 we show the correlogram and partial correlogram of the Sales series Figure 7: Autocorrelation Function for Sales (with 5% significance limits for the autocorrelations) To model the data using the ARIMA method, a stationary mean is necessary by definition. The auto-correlation function plot (ACF) shows large positive auto correlations that dominate the plot. This suggests a non-stationary mean Lag Here the PACF coefficients values are not fluctuating around a constant mean then it is reasonable to believe that the time series is non-stationary Figure 8: Partial Autocorrelation Function for sales (with 5% significance limits for the partial autocorrelations) Lag Since the Business Solution Sales data series is not stationary, we have to make it stationary before we can apply the Box Jenkins methodology. In Figure 9 and 2 we plotted the second differences of sales.unlike Figure 7 and 8 we do not observe any trend in this series, perhaps suggesting that the second-differenced Sales data series is stationary. We can also see this visually from the estimated ACF and PACF correlograms given in Figure and Now we have a much different pattern of ACF and PACF. The ACF coefficients dampen out very Sales Forecasting of Grameenphone Business Solutions SIM BUS 65 4

41 Autocorrelation Partial Autocorrelation slowly, where third-order ACF coefficient is quite large; near to.the PACF coefficients are significant up to lag 3 and dies out oscillating. Figure 9: ACF for second differenced series of Sales (with 5% significance limits for the autocorrelations) Figure 2: PACF for second differenced series of Sales (with 5% significance limits for the partial autocorrelations) Lag Lag Now how do the correlograms given in above figures enable us to find the ARMA pattern of the Sales time series? (We will now consider only the second-differenced sales series because it is stationary.) One way of accomplishing this is to consider the ACF and PACF and the associated correlograms of a selected number of ARMA processes, such as AR(), AR(2), MA(), MA(2), ARMA(, ), ARIMA(2, 2), and so on. Since each of these stochastic processes exhibits typical patterns of ACF and PACF, if the time series under study fits one of these patterns we can identify the time series with that process. Of course, we will have to apply diagnostic tests to find out if the chosen ARMA model is reasonably accurate. To study the properties of the various standard ARIMA processes would consume a lot of space. What we plan to do is to give general guidelines (see Table 3.); the references can give the details of the various stochastic processes. Sales Forecasting of Grameenphone Business Solutions SIM BUS 65 4

42 Table 3.: Patterns ACF and PACF- General rules ACF PACF Nonstationary series Very slow to decay still not fully decayed after a large number of periods Often Spikes at lags to p; nothing thereafter AR Exponential decay; either positive or alternating Spike (either +ve or ve) at lag ; nothing thereafter AR(p) Exponential decay or damped sine wave Spikes at lags to p; nothing thereafter MA Spike (either +ve or ve: opposite Exponential decay; neg if is to ) at lag ; nothing thereafter +ve; alternating (starting +ve) if ve MA(q) Spikes at lags to q; nothing thereafter Exponential decay or damped sine wave Notice that the ACFs and PACFs of AR(p) and MA(q) processes have opposite patterns; in the AR(p) case the ACF declines geometrically or exponentially but the PACF cuts off after a Sales Forecasting of Grameenphone Business Solutions SIM BUS 65 42

43 certain number of lags, whereas the opposite happens to an MA(q) process. Geometrically, these patterns are shown in Figure 2 Figure: 2 Since in practice we do not observe the theoretical ACFs and PACFs and rely on their sample counterparts, the estimated ACFs and PACFs will not match exactly their theoretical counterparts. What we are looking for is the resemblance between theoretical and sample ACFs and PACFs so that they can point us in the right direction in constructing ARIMA models. And that is why ARIMA modeling requires a great deal of skill, which of course comes from practice. Sales Forecasting of Grameenphone Business Solutions SIM BUS 65 43

44 ARIMA Identification for Business Solution SIM Sales Now we have to Examine the Autocorrelation Function (ACF) and Partial Autocorrelation Function (PACF) of stationary series to get some ideas about what ARIMA(p,d,q) models to estimate. The d in ARIMA stands for the number of times the data have been differenced to render to stationary. This was already determined in the previous section. The p in ARIMA (p,d,q) measures the order of the autoregressive component. To get an idea of what orders to consider, we have to examine the partial autocorrelation function. If the timeseries has an autoregressive order of, called AR(), then we should see only the first partial autocorrelation coefficient as significant. If it has an AR(2), then we should see only the first and second partial autocorrelation coefficients as significant. (Note that they could be positive and/or negative; what matters is the statistical significance.) Generally, the partial autocorrelation function PACF will have significant correlations up to lag p, and will quickly drop to near zero values after lag p. The q measures the order of the moving average component. To get an idea of what orders to consider, we examine the autocorrelation function. If the time-series is a moving average of order, called a MA(), we should see only one significant autocorrelation coefficient at lag. This is because a MA() process has a memory of only one period. If the time-series is a MA(2), we should see only two significant autocorrelation coefficients, at lag and 2, because a MA(2) process has a memory of only two periods. Generally, for a time-series that is a MA(q), the autocorrelation function will have significant correlations up to lag q, and will quickly drop to near zero values after lag q. Sales Forecasting of Grameenphone Business Solutions SIM BUS 65 44

45 Returning to the correlogram and partial correlogram of the stationary Sales series for period 26 January - 2 June what do we see? Remembering that the ACF and PACF shown there are sample quantities, we do not have a nice pattern as suggested in Table 3.. Looking at PACF function we can see that the first three partial autocorrelation coefficients are statistically significant. Therefore, AR (2) or AR (3) might be considered. Additionally, the pattern of Autocorrelation function reveals that the ACF coefficients up to lag 3 are statistically significantly different from zero, for they all are outside the 95% confidence bounds. Although, ACFs at lag 6, and 9 are individually statistically significant we do not have to include all the AR terms up to 9; if the autocorrelation coefficient were significant up to lag 3, we could have identified this as MA (3) or MA(2) model. ESTIMATION AND DIAGNOSTIC CHECKING OF THE ARIMA MODEL How do we know that this model is a reasonable fit to the data? There are several ways to diagnose, but we have applied only two methods. i. Diagnostics: One simple diagnostic is to obtain residuals from stationary data and obtain the ACF and PACF of these residuals, say, up to lag 25. The residuals should be stationary white noise. ii. Portmanteau test: This can be done by using the Q-statistics developed by Ljung and Box: Sales Forecasting of Grameenphone Business Solutions SIM BUS 65 45

46 Q( m) T( T 2) m l ˆ 2 l T l If Q is significant, then the series is not a white noise (this occurs whenthe p-value is less than.5). Now, we will estimate four models: ARIMA(,2,), ARIMA(,22), ARIMA(2,2,2), ARIMA(2,2,3),will do diagnostic check and will choose the best one. ARIMA (, 2, ) Model : y t = β y t- + ε t + λ ε t- The parameter estimates of the MA() term λ and AR() term β are statistically significant. However, the autocorrelation check of the residuals tells us that the residuals from this ARIMA(,2,) are not white-noise, with a p-value of.. So, we need a better model. Final Estimates of Parameters Type Coef SE Coef T P AR MA Modified Box-Pierce (Ljung-Box) Chi- Square statistic Lag Chi-Square DF P-Value.... ARIMA (,2,2) Model : y t = β y t- + ε t + λ ε t- + λ 2 ε t-2 Final Estimates of Parameters Type Coef SE Coef T P AR MA MA Modified Box-Pierce (Ljung-Box) Chi- Square statistic Lag Chi-Square DF P- Value.... Here one of the parameter estimates is not statistically significant; the residuals up to lag 2 reject the null hypothesis of white noise, casting some doubt on this model. Sales Forecasting of Grameenphone Business Solutions SIM BUS 65 46

47 ARIMA (2, 2, 2) Model: y t = β y t- + β 2 y t-2 + ε t + λ ε t- + λ 2 ε t-2 This model has statistically significant coefficient estimates.the Chi-Square test statistics for the residuals series indicate whether the residuals are uncorrelated (white noise) or contain additional information that might be utilized by a more complex model. In this case, the test statistics fails to reject the noautocorrelation hypothesis at a high level of significance. (p=.9 for the first 2 lags.) This means that the residuals are white noise, and so this model is adequate for this series. Final Estimates of Parameters Type Coef SE Coef T P AR AR MA MA Modified Box-Pierce (Ljung-Box) Chi- Square statistic Lag Chi-Square DF P-Value ARIMA (2, 2, 3) Model: y t = β y t- + β 2 y t-2 + ε t + λ ε t- + λ 2 ε t-2 + λ 3 ε t-3 Final Estimates of Parameters Type Coef SE Coef T P AR AR MA MA MA Modified Box-Pierce (Ljung-Box) Chi-Square statistic Lag Chi-Square DF P-Value In this model two coefficients are statistically insignificant. Also, the autocorrelation check of the residuals tells us that the residuals from this ARIMA(2,2,3) are not white-noise, since the Chi-Square statistics up to a lag of 2 have p-values less than 5%, meaning we fail to accept the null hypothesis that the autocorrelations up to lag 2 are jointly zero (p-value =.5). Sales Forecasting of Grameenphone Business Solutions SIM BUS 65 47