Overview of Quantitative Forecasting Methods on Sales of Naphthenic oils

Transcription

1 Overview of Quantitative Forecasting Methods on Sales of Naphthenic oils ALI HADIZADEH Production Economics Master s thesis Department of Management and Engineering LIU-IEI-TEK-A-11/1237 SE

2 2 P a g e Overview of Quantitative Forecasting Methods on Sales of Naphthenic oils Ali Hadizadeh [email protected] Master Thesis Subject category: Technology Linkoping University, Institute of Technology, Department of management and engineering SE Linkoping Examiner: Mathias Henningsson [email protected] Supervisor: Martin Kylinger [email protected]

3 3 P a g e Abstract An overview of mathematical forecasting methods has been presented in this research. First, the data of revenue and sales volume of 15 products is collected, cleaned, and prepared for further studies by applying data mining steps. Second, ABC analysis is applied to narrow down the domain of research to only critical products. A literature review of the most common and applicable quantitative methods of forecasting are addressed in the next phase and finally the implementation and numerical results is presented. Having considered the most well known quantitative forecasting methods, exponential smoothing and ARIMA give the best result based on MSE and ARIMA gives the best result based on MAPE, while multiple regression model with ARIMA error gives the best perspective to forecasters on finding the most effective factors in sales and revenue. Having calculated the mean of MAPE for all forecasting methods, we were interested to see if the forecasting method is a significant factor and if the difference between the average values obtained by ARIMA is statically different from other methods. To do so, we run the randomized block design method and by drawing the main effect plot we have come to this conclusion that forecasting method can be considered as a significant factor and by running Two Sample T Test in MINITAB and presenting 9% confidence interval, the ARIMA method outperforms other methods significantly. Keywords: quantitative forecasting methods, Data analysis, ABC analysis, Mean Square Error (MSE), Mean Absolute Percentage Error (MAPE)

4 4 P a g e Acknowledgement I would like to thank first and foremost my supervising tutors, Dr. Martin Kylinger and Dr. Mathias Henningson for their abundant help and productive advices. I am also grateful to my parents and friends especially Faezeh Malekian who consistently supported me during this period.

5 5 P a g e Table of Contents Chapter 1: Introduction and Motivation Introduction Aim and Scope Structure of Thesis Nynas history Nynas supply chain Nynas products Limitations Thesis outline Chapter 2: Data Cleaning and Classification Data analysis ABC analysis Chapter 3: Basics of Quantitative Forecasting Quantitative methods Accuracy measures Non-parametric trend test of product demand data Chapter 4: Research Methodology Chapter 5: Implementation and Numerical Results ABC Implementation Non-parametric trend analysis Visual analysis Output of decomposition Output of moving average Output of exponential smoothing Output of ARIMA method Output of multiple regression method Output of multiple regression method with ARIMA error Chapter 6: Comparison of Numeric Results Chapter 7: Conclusion References... 89

6 6 P a g e Table of Figures and Tables FIGURE 1.1 SUPPLY CHAIN NETWORK AT NYNAS FIGURE 2.1 THE TRANSFORMATION OF RAW DATA TO KNOWLEDGE, SOURCE: KRZYSZTOF J ET AL (27) TABLE 2.1 POLICIES ASSIGNED TO EACH CATEGORY OF PRODUCTS... 2 FIGURE 3.1 DYNAMIC DATA SERIES FIGURE 3.2 STATIC TIME SERIES FIGURE 3.3 CONSTANT SEASONAL VARIANCE FIGURE 3.4 INCREMENTAL SEASONAL VARIANCE TABLE 3.1 DIFFERENT TYPES OF EXPONENTIAL SMOOTHING FIGURE 3.5 TIME SERIES PLOT OF STATIONARY DATA TABLE 3.2 PATTERN OF ACF AND PACF FOR BASIC AR AND MA MODELS FIGURE3.8 RESIDUAL PLOT TABLE 5.1 LIST OF REVENUES IN DECREASING ORDER TABLE 5.2 CUMULATIVE DISTRIBUTION OF REVENUE FOR ALL PRODUCTS TABLE 5.3 CLASSIFICATION OF PRODUCTS BASED ON REVENUE TABLE 5.4 VOLUME AND REVENUE PROPORTION OF PRODUCTS IN DIFFERENT CLASSES FIGURE % OF THE TOTAL REVENUE IS MADE BY ONLY 17% OF THE TOTAL PRODUCTS (X AXIS INDICATES PRODUCTS) TABLE 5.5 UNIVARIATE TEST FOR THE 23 TOP RANKED PRODUCTS TABLE 5.6 UNIVARIATE TEST OF THE 5 SELECTED PRODUCTS FIGURE 5.2 TIME SERIES PLOT FOR P FIGURE 5.3 SEASONAL PLOT OF P1 FROM 24 TO FIGURE 5.4 SHIFT OF P1 SALES FROM ONE YEAR TO ANOTHER FIGURE 5.5 TIME SERIES PLOT FOR P1 IN MINITAB FIGURE 5.6 COMPONENT ANALYSIS OF DECOMPOSITION METHOD FOR P FIGURE 5.7 TIME SERIES DECOMPOSITION PLOT FOR P FIGURE 5.8 RESIDUAL PLOT... 6 TABLE 5.7 GENERATED FORECAST... 6 TABLE 5.8 MEAN SQUARE DEVIATION FOR THE 5 PRODUCTS FIGURE 5.9 FITTING PLOT OF MOVING AVERAGE WITH LENGTH TABLE 5.9 ACCURACY MEASURES BASED ON DIFFERENT LENGTH FIGURE 5.1 LINEARITY TESTS OF RESIDUALS FIGURE 5.11 RESIDUAL PLOTS OF P TABLE 5.1 FORECAST IN ADDITION TO UPPER AND LOWER BOUNDS FIGURE 5.12 FORECAST LINE WITH INTERVALS TABLE 5.11 FORECAST WITH INTERVALS TABLE 5.12 MEAN SQUARE DEVIATION FOR OTHER CRITICAL PRODUCTS TABLE 5.13 COMPARISON OF SES, HOLT, AND HOLT-WINTERS BASED ON MSD AND MAPE FIGURE 5.13 SES PLOT IN ADDITION TO THE FORECAST LINES FIGURE 5.14 RESIDUAL PLOTS FIGURE 5.15 RESIDUAL TREND PLOT TABLE 5.14 GENERATED FORECAST WITH BOUNDARIES AND MEAN SQUARE DEVIATION FIGURE 5.16 SALES PLOT OF P FIGURE 5.17 SALES PLOT FOR THE DIFFERENCED DATA SERIES TABLE 5.15 MSE COMPARISON IN DIFFERENT ARIMA METHODS TABLE 5.16 LEVEL OF SIGNIFICANCE FOR ARIMA PARAMETERS TABLE 5.17 BOX-PIERCE ANALYSIS FIGURE 5.19 RESIDUAL ANALYSIS FIGURE 5.2 RESIDUAL TESTS FIGURE 5.21 PLOT OF RESIDUALS TABLE 5.18 FORECASTS OF P1... 7

7 7 P a g e TABLE 5.19 GENERATED FORECAST FOR PRODUCT TABLE 5.2 GENERATED FORECAST FOR PRODUCT TABLE 5.21 GENERATED FORECAST FOR PRODUCT TABLE 5.22 GENERATED FORECAST FOR PRODUCT TABLE 5.23 CORRELATION AMONG SELECTED POTENTIAL VARIABLES FIGURE 5.23 SALES PLOT OF NYTRO TABLE 5.24 CORRELATION BETWEEN THE RESPONSE VARIABLE AND SEASON INDICES FIGURE 5.24 FITTED LINEAR TEST FIGURE 5.25 FITTED QUADRATIC TEST FIGURE 5.26 FITTED CUBIC TEST TABLE 5.25 REGRESSION ANALYSIS (P1 VS T & T2) TABLE 5.27 ANALYSIS OF VARIANCE AND THE REGRESSION EQUATION OF PRODUCT 1 (NYTRO) TABLE 5.28 MSE AND CRITICAL FACTORS IN REGRESSION MODEL FOR DIFFERENT PRODUCTS TABLE 5.29 MSE OF 5 CRITICAL PRODUCTS IN MULTIPLE REGRESSION WITH ARIMA MODEL FIGURE 6.1 TIME SERIES PLOT OF P1 AND P TABLE 6.1 COMPARISON OF DIFFERENT FORECASTING METHODS BASED ON MSE... 8 TABLE 6.2 COMPARISON OF DIFFERENT FORECASTING METHODS BASED ON MAPE... 8 TABLE 6.3 AVERAGE MSE FOR 5 CRITICAL PRODUCTS TABLE 6.4 AVERAGE MAPE FOR 5 CRITICAL PRODUCTS TABLE 6.5, THE RESULT OF MAPE FOR DIFFERENT PRODUCTS WITH SUM AND AVERAGE TABLE 6.6 ANALYSIS OF VARIANCE FOR FORECASTING METHOD AS THE MAIN FACTOR AND PRODUCTS AS BLOCK83 FIGURE 6.2 MAIN EFFECT PLOT FOR DIFFERENT METHODS AGAINST AVERAGE OF MAPE TABLE 6.7 TWO-SAMPLE T FOR ARIMA VS DECOMPOSITION TABLE 6.8 TWO-SAMPLE T FOR ARIMA VS MOVING AVERAGE TABLE 6.9 TWO-SAMPLE T FOR ARIMA VS EXPONENTIAL SMOOTHING... 85

8 8 P a g e Chapter 1: Introduction and Motivation

9 9 P a g e In today s business, forecasting is one of the most important tools used for operation strategies especially make to stock environments in which the main goal of forecasting is to ensure that the level of materials needed for production satisfies customer s demands without resulting any overcapacity situation and extra inventory. On the other hand, forecast should not create any shortage for the manufacturer whose main role is to fulfill customer s orders Introduction The term of forecasting is used when it is aimed to estimate the value of variables in future. It is a tool for mangers to make better plans and decisions. Forecasting can be applied for different situations as follows: 1- Inventory and production plan. [Sales, product demand, production schedule ] 2- Investment and financial information. [Interest rate, share price] 3- Economic. [Economy growth, inflation] Many case studies have revealed that inaccurate forecasts can make a catastrophic cost for companies. It causes big problems form late customer deliveries to increased inventories and higher costs. According to the research which has been made by D. R Rice Company the companies that can overcome the forecast hurdle have seen numbers such as: 1. 3% increases in on-time deliveries 2. 5% reduction in inventories 3. A staggering 8% in customer lead times 4. 2% reduction in total business costs Thus, applying the right techniques of forecasting is a big concern for supply chain managers. Generally, forecasting can be developed for different levels of product from finished goods and raw materials to components and service parts. Forecasting should not be static, rather must be reviewed by forecasters on a regular base. In this way, future information on trends, seasonality, and other external and internal elements can be extracted and employed to give a better result. There are different ways to classify forecasting problems. One method is to consider the time scale on which forecasting is applied. In other words, it depends on how far we go toward future and predict. Based on this classification, three main categories are created namely, short-term, medium-term, and long-term, each of which might have different meaning according to the situations in which forecasting is applied. For instance in energy industry, 5 to 1 years is considered short-term while in forecasting consumer demand 2 years would be a long-term forecast. Typically, there are three levels of decisions in the supply chain namely, operational, tactical, and strategic. Operational decisions encompass short-term decisions from 3-6 months such as inventory control, production planning and distribution. Tactical decisions which are appropriate for the interval between 6 months to 2 years such as decisions concern staff and facility changes. Lastly, strategic decisions covering decisions lasting for more than 2 years such

10 1 P a g e as research and development and product design change. This classification is imperative since different forecasting methods should be used in each class. For example, in the case of sales forecasting, we may be interested to forecast sales for the next 6 months or 5 years while each of them requires a suitable forecasting method. Forecasting can also be classified into other categories based on the type and genre of forecasting techniques. These classes are depicted as follows. 1- Qualitative forecast with no mathematical method and merely based on the experts experience and skills. 2- Regression models in which forecast (response) variable is linearly related to a number of other independent variables. 3- Multiple equation also known as econometrics which are a variety of dependent variables interacting with each other in number of equations. 4- Time series methods that is a single variable whose future value is related to its past and it changes over time. In this master thesis, we are motivated to introduce and analyze the abovementioned forecasting methods and apply them for the NYNAS supply chain and finally come up with the best models for this case study Aim and Scope The current thesis aims to have an overview on important quantitative forecasting methods available in literature for the products in a continuous flow production in accordance with the NYNAS case study. In this research, the general research question is which quantitative forecasting method(s) should be applied for this project and why? Nowadays, having a precise forecast plays a significant role on companies financial success. For businesses with a high level of turbulence in costs, competitors, and other dominant parameters, to have an appropriate model for forecasting is a leading character. Through the reviewed literature, we come up with appropriate methods of sales forecasting that could be practically used in the petroleum industry. In each section of this study, specific methods and techniques required for forecasting will be reviewed and some aspects of this problem are taken into consideration. This study provides us with a framework for data treatment that can be applied prior to any further analysis on the result of any research. Data analysis as one of the major phases before any process and implementation should be applied. In this paper, it is tried to describe important steps of data cleaning and especially ABC analysis. The scope of this research is defined as follows. 1- This research only considers naphthenic oils in NYNAS supply chain. 2- Among naphthenic oils, only those selected as financially critical products are studied.

11 11 P a g e 3- Among all critical products, all with full availability of data have been processed. 4- Among different types of forecasting methods, only quantitative techniques are applied Structure of Thesis The data for 15 products is extracted from NYNAS. Data is cleaned, integrated, and formatted. Most of the missing values are recovered and derived attributes are produced in access file. Pareto (ABC) analysis is made based on the revenue that each product makes and 23 critical products are selected. Data accuracy, correctness, completeness, and relevance are checked. A sample of 5 products is selected from the 25 critical products based on the data completeness. Decomposition, moving average, exponential smoothing, multiple regression, ARIMA, and regression with ARIMA error are the methods applied on the data of selected 5 products The results are recorded and the best methods based on the accuracy measures for this case study are selected. Analysis of Variance and T statistic test are run to statistically approve the significance of forecasting method factor and the difference of the mean of the best forecasting method with the mean of other methods Nynas history NYNAS business started in 192s as the first oil refinery company in nynäshmn Sweden. This company played a magnificent role during the World War II in supplies of oil substitute within the country s borders. After the world war by increasing the national demand for expanding road networks and the request for bitumen and other oil derivatives, it had a huge physical and financial growth. NYNAS policies in 196s and 197s have encountered major ups and downs. During 196 s they broadened the range of products from bitumen to fuels, lubricants, solvents, diesel etc, but in 197s when the oil crisis arrived they narrowed down their production by focusing on some special products. This decision soon urged NYNAS to be an international player in oil industry. This happened with investment in hydrogenation technology to renovate and boost the production of naphthenic. Since 199, NYNAS started to develop and enlarge the company in its specialty areas. That s how they increased their new sales companies in many other countries and simultaneously turned to one of the biggest bitumen companies in Britain. This specialization process has led NYNAS to be a world leader in the market of naphthenic specialty oils and one of the great producers of bitumen in the Europe.

12 12 P a g e 1.5. Nynas supply chain The company s supply chain is constructed on 4 layers including refineries, hubs, depots and customer areas. The crude oil is transported to the refineries by suppliers and will be processed and converted to specific raw products in refineries. These components could be shipped directly to the customers as raw materials or be blended in hubs and transported to different depots and different customers s as new raw products. The supply chain network is depicted in Fig 1.1 Mainly, the finished products are in the category of bitumen or naphthenic. In this study, the whole analysis is made on naphthenic products. Figure 1.1 Supply chain network at NYNAS 1.6. Nynas products There are two main categories of products available in Nynas namely, bitumen and naphthenic oils. Bitumen and its derivatives are much older and their history of production goes back to the time of Nynas foundation. However Naphthenic oils are much newer products with higher profit margin. Due to the higher profitability and based on the company s policy, naphthenic oils have a worldwide supply chain whereas bitumen merely meets local demands within Scandinavian, Nordic and some other European countries.

13 13 P a g e 1.7. Limitations The data and information obtained for this study is a combination of quantitative and qualitative records. An interview was arranged with a sales analyst in Nynas and the raw data has been collected over 4 months. The data had many missing values and ambiguities such as large outliers. This certainly jeopardizes the validity of this research. More importantly, due to the lack of enough interaction with the company, we could not lighten all imprecision of this research. Although, some of the missing values have been recovered by interpolation technique, we were forced to eliminate some of the products because of the huge number of lost information in the raw data. Also, the data available for this study only covers 6 years which is not enough for running a strong validity test. The analysis is only made for Naphthenic oils and the research does not consider another important product family, bitumen. Considering all these limitations, the major quantitative forecasting methods will be implemented in this research Thesis outline To do this research, all monthly data about sales volume, customer segments, sale channels and other potentially useful information at NYNAS, from 24 to 29 have been collected, cleaned and got ready to be processed. ABC analysis as a major tool for selecting, sorting and prioritizing products is accomplished. Having selected the most critical products, quantitative forecasting methods are reviewed to find a best model that answers the main question of this research which is to present the best quantitative forecasting model for the sales of naphthenic oils. Considering many criteria, the best model is suggested in the conclusion part of this study. The rest of this study is as follows. In Chapter 2, the methods of data cleaning and analysis will be explained. In Chapter 3, a theoretical review of quantitative forecasting methods is provided. Research methodology is presented in Chapter 4. In Chapter 5, the reviewed methods are applied on the real data collected from NYNAS and finally the best method based on the accuracy measures has been selected in chapter 6. Conclusion is also presented in Chapter 7

14 14 P a g e Chapter 2: Data Cleaning and Classification

15 15 P a g e Before analyzing the raw data, they should be checked and cleaned. In this study some steps of Data Mining (DM) are applied for the preparation of data. At first, a more comprehensive concept called the knowledge discovery process in which data mining is considered as the major core, will be reviewed. Having cleaned and prepared the data, ABC analysis will be implemented to narrow down the scope of this research only to those products which are financially critical for the company. At the end of this section, theoretical overview of most important quantitative forecasting method will be presented Data analysis Knowledge discovery process is defined as the nontrivial process of identifying valid, novel, potentially useful, and ultimately understandable patterns in data. Knowledge discovery concerns the entire knowledge extraction process, including how data are stored and accessed, how to use efficient and scalable algorithms to analyze massive datasets, how to interpret and visualize the results, and how to model and support the interaction between human and machine. Figure 2.1 The transformation of raw data to knowledge, Source: Krzysztof J et al (27) Typical inputs include data in various formats, such as numerical and nominal data stored in databases or flat files. The output is the generated new knowledge usually described in terms of rules, patterns, classification models, associations, trends, statistical analysis, etc. Krzysztof J et al (27) During this process, data turns into information, knowledge, and finally wisdom. In the following, we ll see how this transformation works but first, data, information and knowledge should be explained Data are any facts, numbers, or text that can be processed by a computer. The patterns, associations, or relationships among data can provide information. For example, analysis of sale transaction data at NYNAS can create information on which products to be sold and when.

16 16 P a g e Information can be turned to knowledge based on historical patterns and future trends. For example, in this study, list of information on products sales can be analyzed for fostering efforts to provide knowledge of purchasing behavior. Thus, a manufacturer could determine which items are most liable for promotional efforts. Data Mining is to make sense of large amount of mostly unsupervised data in some domain Krzysztof J et al (27). In Data Mining, data is analyzed from different views and then summarized into useful information in which the relationships are completely identified. This information could be used to increase profits and cut costs significantly. Technically it is defined as the process of discovering patterns and correlations among many areas in data base. Data mining steps are introduced in many literatures. Here The CRISP-DM methodology has been used. It was first established in the late 199s through the cooperation of four companies: Integral Solutions Ltd. (a provider of commercial data mining solutions), NCR (a database provider), DaimlerChrysler (an automobile manufacturer), and OHRA (an insurance company). The CRISP-DM consortium (2) introduces 6 steps of data mining as: 1. Business understanding 2. Data understanding 3. Data preparation 4. Modeling 5. Evaluation 6. Deployment The First four steps which have been applied for this research are described as follows: Business understanding The main part of business understanding is the determination of business objective. It could be defined as what the customers really want to be accomplished. In this step, all information about business situation should be recorded. The success criteria for the business should also be determined. For NYNAS, the objective could be defined as reaching to a best quantitative method to forecast the sales of critical products. Our success criterion is to give a useful insight into the level of demand sensitivity for A-category products to some potential factors such as prices, firm s policy, etc. For the forecasting project, we first have to find exactly where the forecaster is situated in the supply chain and how she is exploiting the flow of information from market to factory in the

17 17 P a g e forecasting process. Forecasters should determine how many products are needed from depots or hubs to customers. The important issue is the role of sellers in forecasting and how they predict customers demand. Sellers may forecast a specific number of products for a specific number of customers, but the company might produce less or more than the amount of request. They experience stock out many times and we may need to find out that this stock out or extra inventory is because of the poor forecast or is due to the impact of other potential factors. We also need to find out how is the forecast getting updated. How the feedbacks are influencing the forecasts of the next months. In discussion with the supply chain manager, it is understood that yearly budgeted volumes and consideration of Bitumen production are also taken into account in forecasting. The question is how this phenomenon impacts demand. Data understanding Data collection and verification of data quality are the major parts in this step. The quality of data plays an important role in future calculations. The high quality data is a data that users have coverage for all needed materials. High quality data is a data that has accuracy, correctness, completeness and relevance. Accuracy is the degree of closeness of measurements of a quantity to its actual value. Correctness is declared when we have a correct output for each input Data completeness is an indication showing if all the data necessary to meet the current and future information demand are available in the resource Data relevance shows how much something is connected and applicable to a specific issue After collecting data from NYNAS, all of the above cited four steps should be checked before and during data analysis. One method is to use several methods to collect and process data in order to investigate the quality of the data. Data Preparation In order to prepare the final data set and make them ready to be fed into the next phases, they should be cleaned, integrated, and formatted. Attributes (columns) and records (rows) should be selected in this step too. This is what we have done on our data. All of the missing values are recovered, and derived attributes are produced in the access file. Information is combined from multiple tables to create new values. ABC analysis as a major tool for selecting, sorting and prioritizing products is also accomplished in this phase. This analysis will be explained thoroughly in this study.

18 18 P a g e Modeling This section is the key segment of our research and the conclusion which is made at the end of the study is based on the results obtained in this phase. Different methods and techniques are chosen and applied for the same problem. Each modeling technique might have specific assumptions, (i.e. no missing value checked in the previous section could be one of these assumptions). Having considered these assumptions, we need to check the quality and validity of the selected techniques. This is called generating the test design. Error rate is one of the criteria for checking the quality of the methods. One or more years of data could also be used for testing the validity of the methods. In the upcoming segments, testing and developing of these models, will be explained but first ABC analysis is explicated ABC analysis The ABC analysis or 8/2 rule is a tool stating that a 2% of a given population stands for 8% of a particular characteristic. As mentioned earlier in this research, it is one of the steps of data preparation. ABC analysis is usually accompanied with this renowned speech: the ABC tool is used to identify the vital few from the trivial many. based on the definition above the ABC analysis classifies products or whatever under study into 3 or more levels, A, B, C. Generally, the A segment stands for almost 8% of the total spend in a group, the B segment stands for the following 15% of the total spend and the C segment represents the remaining (final 5% of total spend) EIPM (24) Before we go through the whole mechanism we have to note that ABC analysis does not lead to a straight solution. It only explores the important areas for future opportunities. In inventory management, Different types of criteria might be used for creating ABC categories. the most common criterion is the amount of produced /spent dollar for each item, it helps managers to find the few A sections that require the most special attention and this is all because of the high amount of money spent in this category. That s why small difficulties in managing this segment might lead to a large cost for the company. Categorizing products can be accomplished for many reasons. In many cases, managers need to know what products are the most critical ones to invest. The aim of ABC analysis in this study is to find the demand of the most critical products and extract their pattern and behavior over time. But the question is how do we define the concept of criticality? Today, ABC analysis is a much vaster concept; managers know that focusing merely on monetary issues like amount of dollar- usage or costs cannot present the criticality and importance of monitoring of different products. Also in some cases, products with low investment bring about the high level of risk for the company. These risks should first be determined. They can be understood differently in various areas. For example, for a production company, lead time, safety stock, and consumer sensitivity could be considered as risk factors. Having defined the risk elements, we have to identify which items (especially in C segments) are risky for the business. In this research, due to the lack of enough information about the

19 19 P a g e inventory management and production systems, these critical factors are not studied and the estimation is merely based on the amount of revenue for each product. However, a useful method with the assumption of having all available information will be suggested later in this section. Considering this method, we can have a more precise ABC analysis for the demand forecasting. The following 6-step procedure is carried out to run the ABC analysis. 1 Identify the objective and the analysis criterion 2 Collect data about the analyzed population 3 Sort out the list by decreasing impact 4 Calculate the accumulated impact and the percentage 5 Identify the classes 6 Analyze the classes and take appropriate decisions Source: EIPM (24) The above-cited steps will be thoroughly explained in the methodology section Criteria selection What we ve analyzed so far was about considering a single criterion which is the amount of dollar obtained in the sale process. However, Special attention must be paid to other factors influencing the product demand directly or indirectly. One of the first applications of ABC analysis was in inventory management methods applied in General Electric by Dickie, H.F (1951). They used a simple concept of concentration on significant few and spending less on the trivial many. Categorizing items into classes of A, B and C is generally accomplished by considering some criteria in inventory control. This criterion as already mentioned, is often the amount of dollar spent for the items. However, there are some other criteria that affect other aspects of inventory management and impress the company s financial success. For inventory managers, these criteria could be the impact of shortages, the confidence of supply, the number of obsolescent items, and so forth. Some of these criteria may even have a bigger influence than the amount of spent dollars has. Many researchers have considered these factors in their research. For example, Villarreal et al (198) use item costs, costs of subassemblies for each item, and the lead time in management of capital goods inventory as their criteria. Flores et al (1986) analyzed different non-financial criteria such as lead time, availability, sustainability, and criticality. They have found that the factor criticality is covering almost the most facets of maintenance inventory items. They define criticality as the seriousness of stock-out, the speed of purchasing process, availability of substitutes, and so on. Their main effort during their study was to see how they

20 2 P a g e can identify the degrees of criticality in a practical way. Flores, et.al (1987) explain the necessity of using multiple criteria in categorizing each product in a year later. The theoretical way of using different criteria is to have ABC categories for both the amount of spent dollars and criticality independently. However, this will make the large number of combinations for which different management policies are needed. Thus, to shorten this process, non-critical costly items and critical low cost items are located in the same category. In the first phase, managers rank the items based on the amount of spent money and assign them to A, B and C classes. In the second phase, they were asked to consider all factors concerning the criticality such as impact of an outage, ease of replacement, lead time, availability, etc and rank items accordingly. In the third phase, they are asked to see if they can combine criticality with the spent money and assign them to new categories. They gathered data from the maintenance inventory records. Critical items are assigned to category I, non-critical items are assigned to category III and those in between are categorized in class II. Many possible combinations could be made and for each combination different policies are required. However, based on what Flores et.al (1986) suggest these combinations are reduced to a manageable number. They use a simple mechanical procedure to combine different criteria and finally provide three initial categories of items: AA, BB and CC. This process is done by merging AI, AII and BI to AA, AIII, CI and BII to BB and BIII, CII and CIII to CC. A specific policy will be determined for each category. These policies are to cover four important areas: inventory record accuracy, order quantity, safety stock and the classification of the item itself. After assigning items to each class, the managers will be asked if they are eager to accept these policies shown in table 2.5 or not, for example if item #3 is assigned to class BB, the managers will be asked to see if the policies in four areas are appropriate for this item. If not, they will re-categorize items and only use the policies as a guideline. The interesting point of reclassification is that we might see a new category of items named DD which stands for don t stock items. Having reviewed the items, the managers may find some obsolete items or some items less critical even than C items, so stocking them is not a smart option. The policy of the company is to scrap or sell out all DD items in the first year; this will help to empty the warehouse spaces which lead to less money consumption and more capital generation in the future. Decision/category AA BB CC Counting frequency Monthly Every 6 month yearly Order quantity Small for costly items Medium EOQ based Large quantities Safety stock Medium for critical items Large for critical items Low or none Reclassify review Every 6 month Every 6 month yearly Source: Flores et al. (1986) Table 2.1 Policies assigned to each category of products

21 21 P a g e The order quantities are based on EOQ (Economic Order Quantity) and safety stock is derived from the criticality of the item. Due to the lack of enough information about the criticality of products this method is beyond the scope of this research. Thus, categorization of products is done only by considering the factor of income for each product in this study.

22 22 P a g e Chapter 3: Basics of Quantitative Forecasting

23 23 P a g e This study aims to present an overview of different quantitative forecasting methods for forecasting the sales of some of the critical petroleum products at NYNAS. In forecasting, there are many methods some of which are reviewed in this section and finally the best model based on the study objective is suggested. Before spelling out the different techniques of forecasting, the role and position of forecasting in the business and the factors affecting this phenomenon should be articulated. Spyros Makridakis et al (1998) describes the distinction between uncontrollable external events which originates from national economy, governments, customers and competitors and controllable internal events such as marketing or manufacturing decisions in firms. He mentions that the success of a company depends on controlling both factors. While forecasting is applied directly for the former, decision making is accomplished for the latter and planning is applied for both. Sales projection is a significant part of forecasting in each business. As if error in sales projection happens it can prompt a series of reactions on budget determination, operating costs, cash flows, inventory levels, pricing and so on. That makes forecasters try their best to select the best possible method that predicts demand precisely. Quantitative methods are accomplished when we have enough quantitative information. It is classified into two major types, time series and explanatory forecasting. Time series forecasting is based on the persistence of historical patterns such as growth in sale, while explanatory forecasting is based on understanding how different variables such as prices, firm s policy etc, are influencing sales. Similarly, Aunupindi et al (26) have classified forecasting methods into two major categories as subjective and objective methods. Subjective methods apply forecasts based on experience and judgment while the objective forecasts do so based on data analysis. The objective methods are classified into smaller groups per se. Casual models (such as explanatory forecasts) and time series analysis are considered as two basic objective methods. Casual models assume that other factors such as price, personal income, etc are effective in addition to the behavior of data, while time series analysis relies merely on past data. As Spyros Makridakis et al (1998) explain quantitative forecasting can be applied when three conditions exist: 1) Past information should be accessible 2) Past information can be quantified as numerical data 3) some aspects of past pattern keep on in future (assumption of continuity) The main question of this study is to find out which quantitative method(s) should be applied for this project. Before answering this question, a deeper understanding about these two models and their pros and cons should be obtained.

24 24 P a g e Explanatory vs. time series Explanatory forecasting demonstrates explicative relations with some independent variables. The purpose of this model is to find the form of this relationship and apply them to forecast future. Of course, these relationships are not precise, and variables in the model cannot account for all the changes in the dependant variable. So an error term should be considered to represent the randomness and unexplained behavior beyond the effective variables of the model. In contrast to explanatory forecasting, Time series forecasting does not look for potential effective factors rather, predicting future is merely based on the values of variables or/and errors in the past with the aim of finding pattern in the historical data and infer future. Makridakis et al (1998) point out 2 reasons why forecasters choose time series over other methods. First the system is not understood or even if it is understood it might be extremely hard to measure. Second, forecasters might only care about what will happen and not to know why it happens. Thus, the advantage of time series is based on the ease of use, while explanatory variables are used when policy and decision making is needed. In this project, several methods and techniques of quantitative forecasting including both explanatory and time series have been studied. Time series is a series of observation over time. In forecasting, we are eager to see how this series will continue in future. There is a variety of techniques in time series. The most important thing that helps us to find the right method is the type of data patterns and what is visualized when the data is plotted. At one glance, the data pattern can be dynamic or static. In static pattern, past data continues its behavior in future. While in dynamic pattern we cannot follow a stable behavior in past and future. In the following dynamic and static pattern are shown. 2 Time Series Plot of y 15 y Index Figure 3.1 Dynamic Data Series

25 25 P a g e 8 Time Series Plot of y y Index Figure 3.2 Static Time Series In static pattern, we might have some specific behavior as follows: 1) stationary 2) seasonal 3) cyclical 4) and trend Figure 3.2 is a static pattern that has trend and seasonality. Many data series have the combination of these behaviors at the same time and due to the large number of different patterns in a time series, the solving procedure can be very challenging Quantitative methods In this study, very popular methods of forecasting have been reviewed. Then, the candidate approaches for this case based on their applicability in forecasting will be selected. 6 different methods are used to forecast the demand of naphthenic products at NYNAS, namely, Decomposition, Moving average, Exponential smoothing, Autoregressive moving average, Multiple regression model, and Multiple regression with ARIMA error. Having analyzed the performance of these methods through different accuracy measures, one can decide which approach can be used for forecasting of a specific product. Among the aforementioned methods, the first four models are considered in the time series category, the fifth method is explanatory and the last one is the combination of explanatory and time series forecasting.

26 26 P a g e Decomposition Decomposition method is one of the oldest and at the same time most reliable techniques of time series. As mentioned above, the pattern in each data series in many instances can be decomposed into some sub-patterns so that each time series component can be specified separately. This will definitely increase the level of comprehension of the series behavior and also the forecasting accuracy. As has been said, among all time series methods, decomposition is one of the oldest methods that usually classifies pattern into two major components for describing the economic series. These two components are the trend-cycle factor and the seasonal factor. Data = pattern + error = f (trend-cycle, seasonality, error) One important assumption about decomposition method is if the parameters describing the time series are almost constant over time (static pattern), the decomposition approach will be an appropriate method to characterize the behavior of the data and forecast. In view of the fact that what type of variation pattern a time series might have, two major types of decomposition may be used. In case of increasing seasonal variation, (i.e. the variation of data series increases when the mean increases) the multiplicative decomposition method and in case of constant variation, the additive decomposition approach will be used. In the following figures, both states are depicted. Time Series Plot of y y Index Figure 3.3 Constant Seasonal Variance

27 27 P a g e 12 Time Series Plot of y y Index Figure 3.4 Incremental Seasonal Variance Multiplicative decomposition and additive decomposition methods are stated in the following formula: Multiplicative decomposition model Additive decomposition model [Makridakis et al (1998)] Where stands for the value of time series at time t depicts the trend factor, is the seasonal factor, would be the cyclical factor and demonstrates the irregular factor all at time t One important fact about decomposition method is that this approach is useful only when its forecasting components (trend, seasonality, etc) keep their pattern constant over time and so is not recommended for data series with dynamic pattern. That s why if the pattern changes, the classical decomposition method does not function very well. Furthermore, estimating the trend cycle is the most difficult process since simple functions for describing the trends such as trend line or other parametric techniques do not visualize them adequately. In this case Makridakis et al (1982) suggest that we use the Holt s method to estimate the trend of seasonally adjusted data obtained from decomposition and then add the seasonal component for the final forecast.

28 28 P a g e Moving average Moving average is a procedure of making the series of averages of different subsets of the data set so that when each new observation becomes available, the oldest observation is removed and the new average can be calculated by adding the latest observation to the equation. The result can be used as the forecast of the next period. For each averaging process, the number of data points should be fixed over time. This is shown with the value of k in the formula. Selecting a larger number of periods will produce a smoother forecasting. Moving average of order k [Makridakis et al (1998)] Although this method is widely used in industry, there are some disadvantages in applying this method. It is not an appropriate method for forecasting when the assumption of an underlying constant process is not met. In other words, when there is trend or seasonality pattern in the data series, this method is not recommended. That s how this technique is not used often as a forecasting procedure under such situations. Also, exponential smoothing generally gives more superior results than the moving average. Exponential smoothing Moving average is a method of averaging when all observations are equally weighted while exponential smoothing devotes unequal set of weights to all past observations and as these weights are exponentially decaying from the latest to the oldest observations, this method is called exponential smoothing. The procedure is that we take a weighted average of past data by using weights that decay in an exponential manner. There are different methods of exponential smoothing used in industries and especially in inventory management. In the following some of these approaches are explained:

29 29 P a g e Single exponential smoothing ) If this equation is expanded by replacing with its components, the following output will be obtained [Makridakis et al (1998)] And this proves the fact that weights are decreasing as we go toward the oldest observations. Thus, this method follows the pattern much better than the moving average approach. As indicated in the first equation, the new forecast is the sum of the last forecast and an adjustment of the forecasting error in the last prediction. This adjustment is shown by α in the formula above. When describing moving average, we mentioned that selecting larger number of periods makes a smoother forecasting. The same will happen when using a small adjustment coefficient (close to ) The value of α has a significant impact on the forecasting accuracy and we ll have different results and especially different forecast errors by using dissimilar α. obviously, by choosing a small value of (α), the primary forecasts are more dominant than when a larger (α) is selected and this could be simply understood by another indication form of single exponential smoothing as follows: In other words, the larger (α) gives a slight smoothing in forecast, whereas, a small value of (α) gives a considerable smoothing. It is better to find the optimum value for α so that the MSE or MAPE or other accuracy measures which will be described later in this chapter, get minimized. This could be done by try and error or some more sophisticated algorithms. Single exponential smoothing is useful for the pattern that has no trend, seasonality or cyclical behavior. If there is a trend in the pattern, forecast will be lagged behind the trend and also this lag will be larger for smaller α. One major advantage of this method against the moving average methods is that this technique requires a small amount of data storage and also computations. Thus, it is faster and more attractive when the number of items is large.

30 3 P a g e Single exponential smoothing with adjusted α It has the same procedure as single exponential smoothing. However, instead of α, there is α. It means this approach allows that the value of α be modified when the pattern changes and this makes the aforementioned method more attractive comparing to the single exponential smoothing. The whole formula is as follows: Where α α [Bowerman et al. (25)] stands for a smoothed approximation of the forecasting error. Similar to exponential smoothing, is calculated as a weighted average of and the last forecasting. is similar except, it is the estimate of the absolute forecast error. Even if we gain a worse forecasting result from this method than the single exponential smoothing, this approach is still more attractive since it minimizes the risk of severe errors and makes minimum administrative concerns. Especially, when there are a large number of items and there is no seasonality and trend pattern in the data series. Linear exponential smoothing (Holt s method) If there is a trend pattern in the data, the suggested method is Holt s method. This time, there are two smoothing constants as follows [Bowerman et al. (25)]

31 31 P a g e When and respectively denote the estimate of the level and slope of the series at time t. handles the forecast for the coming observations. The weights α and β can be estimated by minimizing MSE similar to what it is done in single exponential smoothing. Holt-Winters method So far, it is declared that that moving average and single exponential smoothing are suitable methods for forecasting when there are no trends or seasonality and Holt s method is also an appropriate approach when there is an increasing or decreasing trend in the pattern. However, the recommended method of exponential smoothing when there are both trend and seasonality pattern in the data series is Holt-Winters method which has been extended by Winters in 196. Despite other two methods, this method has three smoothing constants for the level, trend and seasonality. Based on the type of seasonality whether it is additive or multiplicative, there are two sorts of Holts-Winters models which are described below Holt-Winters for multiplicative seasonality [Bowerman et al. (25)] As can be seen is added to the equations and it stands for the seasonal component. Holt-Winters for additive seasonality

32 32 P a g e [Bowerman et al. (25)] This concept can be used for generalizing the application of exponential smoothing methods in different environments. In this way we can consider different combinations of trend and seasonality. It is explained more in the next step. Exponential smoothing categorization The general formula of the exponential smoothing is written as: [Makridakis et al (1998)],, are the variables and their form will change according to the pattern of data. Pegels (1969) classified smoothing exponential as shown in table 3.1 and he shows that for each cell, specific parameters should be selected. For example, if there is an additive trend with additive seasonal component (B-2 cell) then:,, Trend No seasonality additive multiplicative None A-1 A-2 A-3 Additive B-1 B-2 B-3 multiplicative C-1 C-2 C-3 Source: Pegels 1969 Table 3.1 Different types of exponential smoothing The major advantage of exponential smoothing compared to more advanced forecasting methods such as ARIMA or the appealing decomposition methods is its simplicity and low cost. By using ARIMA or even decomposition we might be able to get more accurate results, (note that many exponential smoothing methods are the special cases of ARIMA models) but due to the low storage requirement of exponential smoothing and also its simplicity, this method is a common smoothing approach among forecasters in forecasting product demands.

33 33 P a g e Autoregressive integrated moving average (ARIMA) Before going through the concept of ARIMA models, it is important to define stationary pattern as the perquisite for applying ARMA methods, Sample Auto Correlation (SAC), and Sample Partial Autocorrelation (SPAC) as the tools to recognize if the data is stationary. Stationary pattern is a pattern in which the mean and variance is constant over time. One stationary plot is shown in figure 3.5. Time Series Plot of stationary data 2 1 stationary data Index Figure 3.5 Time series plot of stationary data When the mean is not constant, for example, when there is an upward or downward trend or there is a combination of them, differencing is an appropriate method to make the data pattern constant. While, when the variance is not constant, logarithmic transformation, square root or other types of exponential transformations could be employed for stabilizing the variance. We have to bear in mind that this approach only works when the variance is increasing or decreasing by increasing or decreasing of the level (mean) of the pattern. Otherwise, creating a stationary environment would be much more difficult. SAC and SPAC are appropriate tools to see if there is any pattern left in the errors after applying a forecasting model and a good way to improve the forecasting methods. The definitions of both are thoroughly explained in section 3.2, and their application is explained further in this section. ARIMA method is formed from two main parts namely, Auto-regression and moving average:

34 34 P a g e Auto-regression is a regression equation whose explanatory variables are previous values of the forecast variable as indicated in the following equation in which are the time lagged values of the forecast variable. [Makridakis et al (1998)] The second part of ARMA model is actually regressing against the past value of errors and is indicated as follows This term is called moving average but is totally different from the regular moving average that has been reviewed in the previous sections. It is indeed the moving average of the error series. The coefficients and are unknown parameters. The similar equations can be added for describing the seasonality of pattern, however, in this case, the time lagged values are based on S order (the length of seasonality). In other words, will change to and will turn to These two models separately or together can be used to form a general class of time series which is called ARMA model but only for stationary environments. In other words, data should be constant in mean and variance and has independent error term. To overcome this dilemma and to generalize this method for non-stationary pattern, we can add the degree of first differencing to this formula and make data stationary. That s how the model will change from ARMA to ARIMA. Autoregressive and moving average both can be in different orders and it is shown by ARIMA (p,d,q).(p,d,q) s in which p stands for the order of non-seasonal/seasonal auto regressive part, d exhibits the degree of non-seasonal/seasonal differencing and q is the order of the nonseasonal/seasonal moving average. A very useful technique to recognize SAC and SPAC and finally appropriate ARIMA model is to verify the behavior of Auto Correlation Function (ACF) and Partial Auto Correlation Function (PACF) after differencing (or any other transferring methods). Bowerman et al (25) consider three major types of dying down for SAC and SPAC as 1) Damped exponential fashion 2) Damped sine-wave fashion 3) Combination of 1 and 2 4) In the following, these forms plus the cut off shape which are the most common states of SAC and SPAC are indicated.

35 35 P a g e Damped sine-wave dying down correlation 1.5 Damped exponential dying down correlation 1-1 Damped exponential dying down with ascillaton correlation Cuts off after lag correlation Figure 3.6 Typical SAC and SPAC pattern [Bowerman et al (25)] Based on the types of SAC and SPAC pattern explained above, we can assign a specific ARIMA method with the following instructions. process ACF PACF AR(1) Exponential decaying Positive if and alternating on negative side when Spike at lag 1 and then cuts off after lag 1: positive spike if and negative spike if AR(p) MA(1) MA(q) Exponential decaying or sinusoidal damp based on the size and sign of Spike at lag 1 and cuts off after lag 1 Positive spike when and negative if Spikes at lag 1 to q and then cuts off after lag q Source: Makridakis 1998 Table 3.2 Pattern of ACF and PACF for basic AR and MA models Spikes at lags 1 to p and cuts off after lag p Exponential decaying Positive if and alternating on negative side when Exponential decaying or sinusoidal damp based on the size and sign of

36 36 P a g e Note that the seasonal part of AR and MA is recognized in the seasonal lags of ACF and PACF. For example, if we consider a ARIMA(1,,) (1,,) 4 then we have to expect a seasonal spike (spike at lag 4) in the PACF plot and no other significant spikes in other lags based on the Autoregression behavior explained in the table above. So the potential SPAC pattern would be as shown in figure 3.7. SPAC plot SAC plot correlation corelation -1-1 Figure 3.7 SAC and SPAC with Seasonality features As we can see, on the SPAC plot there is a spike at lag 1 and lag 4 and then cuts off after these two lags and this proves this model should be ARIMA(1,,)(1,,) 4 The main advantage of ARIMA models is that it gives an optimal solution under some circumstances and provides us with a comprehensive family of models. Although there are more disadvantages than advantages but the advantages overweigh disadvantages. In the following, some of the disadvantages are mentioned. 1. ARIMA classification is hard and time consuming and many models have no structural interpretation 2. It is hard to be explained to others 3. Outliers may badly distort the estimation 4. 2 models may have a very similar historical data but very different forecasts

37 37 P a g e Explanatory forecast In this type of forecast, the response variable or variables are articulated as the function of one or several factors that affect its outcomes. These factors known as explanatory variables can be both independent and dependant on time. One of the recognized methods in this sort of forecasting is named Multiple Regression Model in which there is one forecast variable and multiple explanatory variables and the goal is to estimate a function that relates these explanatory variables to the forecast variable. Another well-known method in this genre of forecasting is Econometrics model and is an appropriate method when there is more than one independent variable or one foreseeable explanatory variable to forecast. So, it calls for having more than one equation to handle the regression model. In the following we describe these two approaches in more detail. Multiple Regression Model This is a method for explaining the relationship between multiple independent predictor variables and a dependant response variable so that the response variable is a function of several predictor variables accompanying with regression coefficients along with the constant term. The general form of the Multiple Regression Model is shown in the following [Freedman (25)] (i = 1,2, k) are regression coefficients that are fixed but unknown and reflect the amount of change in the response variable with one unit change in the predictor variable. Usually, least square procedure is applied to find the unknown variables (, ) Assumptions Like many other forecasting methods, this model must be used under specific assumptions. We have to note that if these assumptions are valid, then the multiple regression model is potentially good and gives exact forecast. While, if any particular pattern is seen in the error data series, it means these assumptions are not valid and regression model does not cover all information in the data series. Main assumptions are listed as follows: 1) For any combination of the error term (ε) has zero mean

38 38 P a g e 2) For any combination of the error term (ε) has a constant variance and different populations of error term for different combination of has an equal variance: 3) The population of the error terms for different combination of has a normal distribution 4) Different values of the error terms are statistically independent from each other. In other words, pattern should be random visually [Freedman (25)] Time related explanatory variables Time series regression is a type of regression whose explanatory variables are time related. Trend, seasonality or trading variation could be expressed as examples for explanatory variables in this type of forecasting. Here some popular time-related explanatory variables are mentioned 1) Time (T) which depicts the linear trend in the regression model 2) Seasonal dummy variables, which is a zero-one integer variable with the coefficients that stand for the effect of that season and is the mean difference between the effect of the i th month and the base month. 3) Holliday effect, such as the effect of Christmas and the Easter holiday. (the effect of Christmas is shown by the coefficient of December since it always happens in December) 4) Interventions as exogenous influences such as a specific policy, advertising expenditures, etc that effects just in a particular time and is modeled by an integer dummy variable so that the is for when that specific policy is not introduced yet and 1 stands for starting time of the implementation of that policy. Note that the high correlation among different explanatory variables (multi-collinearity) makes the estimated interval for the coefficients wider and this leads to have an unstable coefficients (high standard error of the coefficients) and consequently not very good regression model. Thus, the correlation among the explanatory variables should also be verified very well. Another important effect of multi-collinearity is that it makes the coefficient estimates unreliable so that different packages might give different solution. However, multi-collinearity doesn t affect the model ability to forecast.

39 39 P a g e Selecting variables One of the most important phases in the regression analysis is to find appropriate explanatory factors affecting the response variable. This process starts with providing a long list of potential variables that have impacts on the response variable and finally shortening this list to a smaller one by using specific tools and some skills of experts. Finally, the regression model will be formed by regressing the response varibale to the selected explanatory variables from this short list. Three important criteria for making the long list is experts knowledge, data availability and the trade-off between time and cost. Econometrics As described earlier, multiple regression is a subset of a special case named Econometric models. While Multiple Regression consists of one equation, Econometric methods include several concurrent regression equations. Thus, it is a set of linear equations with a number of independent variables. Sometimes, this term also covers the simple and multiple regression models too, but here when we discuss econometrics the target is the one with more than one equation. Econometric methods forecast future economic developments by applying mathematical and statistical models, analyzing past economic trends and forecasting the impacts of economic changes on the pattern of data. It is based on the models that highlight the association among different business variables. This procedure is very much dependant on the forecaster s judgment and the available data to be used. Studies reveal that the econometric forecasts that employ forecaster s input demonstrates real trends in a better way than those merely rely on economic models. Difficulties in use of Econometric models It should be noted that all explanatory variables in regression methods are exogenous to the system while in the real world cases the explanatory variables can be the function of the other explanatory variables or even the response variable. So the basic concept in the econometric model is everything should be a function of everything else and the main question in solving econometric models is where to stop these interdependencies since the accuracy in forecasting will not be necessarily increase by including additional variables and equations Although econometric models have many advantages such as a great potential in dealing with interdependencies, understanding of economic systems and evaluating different policies, there

40 4 P a g e are also many obstacles in the application of econometric models that prevent us to use it as one of the forecasting methods for this case. Some of these disadvantages are articulated as: 1) More complicated econometric models do not necessarily provide a better forecast than simpler time series methods. 2) Arranging the different equations and estimating their parameters are more difficult than other methods. 3) The required volume of data plus the cost of computation and human resources is significantly larger. 4) The nature of econometric models is very problem-oriented and this induces the request for the involvement of more experienced and knowledgeable econometricians, which consequently may lead to the more forecasting costs. Regression with ARIMA error In this approach the Multiple Regression model is integrated with the ARIMA method and the result is a regression model with ARIMA errors that covers all advantages of both methods. Previously, we described the multiple regression model with the following formula. Considering the main assumption of this method which is the error term (c) should be an uncorrelated white noise, we present a model with correlated errors in this section. Actually, the ARIMA method will take care of the auto-correlation of the errors and regression method is describing the explanatory relationship. Our final model would be in the following form [Makridakis et al (1998)] In which is modeled as an ARIMA process and we have to note that it is not the white noise anymore. To explain more, suppose the error term is auto correlated and can be described by an ARMA (1,1). Thus, the model can be described as follows. [Pankratz (1991)] In this case, is assumed to be a white noise. In this new series, due to the specific forms of error, parameters will be obtained by SAS-ARIMA procedure with maximum likelihood instead of using the classical least squares methods. The likelihood method is called Marquardt s method

41 41 P a g e via nonlinear least squares estimation which is shown in SAS Online Doc TM Version 7-1 (1999) Pankratz (1983) describes 5 steps in the modeling process of regression with ARIMA method. In this procedure, an estimated ARIMA method might be selected first as a proxy model and afterwards the errors will be examined. These steps are described as follows: 1) Use proxy AR(1) or AR(2) for errors and for fit the regression model with appropriate regressors 2) See the errors; if they re not stationary and differencing makes them stationary, then difference the response and explanatory variables and fit the model with differenced variables by keeping the same proxy model for errors 3) If you are dealing with the stationary errors, select a suitable ARMA model for the error series 4) Refit the entire model by using the new ARMA model for errors 5) Do the final check that residuals are similar to white noise ARIMA vs. Regression with ARIMA model Comparing the results obtained by these two methods, we have come to this conclusion that the forecast is almost the same. However, the prediction intervals are narrower in the ARIMA regression models with linear trend rather than the ARIMA models obtained by the first differencing. On the other hand, one important assumption in applying the Regression with ARIMA error is to consider that the trend will continue over time, but in many data sets we see a dynamic pattern so the trend is not expected to continue. That s how forecasters will prefer ARIMA over the Regression models with ARIMA errors 3.2. Accuracy measures Here we ll examine how to assess the suitability of forecasting methods. In many forecasting models, forecasting accuracy is defined as the goodness of fit and how to reproduce the data which are known earlier in future. Makridakis et al. (1998). In this section several accuracy measures are defined and later will be used in implementation of the aforementioned forecasting methods. If Y t is defined as the observation at time t and F t is defined as forecast at this time, then the error at time t would be defined as:

42 42 P a g e Now, by considering observations and forecasting for n periods we have n error terms which are used in the following equations to reflect the accuracy of forecasts: [Makridakis et al (1998)] ME is usually small since the positive and negative errors counteract with each other. So it is a measure to see if there is a systematic forecast bias rather to see the error size. This dilemma has been solved by taking the absolute or square value of the errors in MAE and MSE measures. In these two methods, the errors turn to the positive values. MAE is more easily interpretable even for non-statisticians but MSE is better handled mathematically, especially in statistical optimization. Makridakis et al. (1998). However, if we decide to compare different forecasting methods in different scales, for example the monthly data with the yearly data, then we cannot compare their MSE or MAE since their performance depends on their scales. In this case it is recommended to use mean percentage error or mean absolute percentage error which are described as follows: [Makridakis et al (1998)] MPE could be very small for the same reason as ME is small. So MAPE is more recommended. Also the output of this measure is a percentage value and it is more understandable than MSE. More importantly, in contrast to MSE, it does not depend on the unit of measurement. However, it should be kept in mind that MAPE can be properly used when we have a meaningful scale and no zero in the time series. Considering all the above mentioned matters, we use MAPE as our main accuracy measures in this research. Forecasting methods are also compared based on MSE.

43 43 P a g e Sample Auto Correlation Function (SAC) SAC is an appropriate tool to see if there is any pattern left in the errors after applying a forecasting model and a good way to improve the forecasting methods. If the autocorrelation among data are significantly large then it means there is a pattern in errors. Usually, it is considered to check the autocorrelation with the critical value of. If the autocorrelation is larger than this value or smaller than its symmetry, it means there is a correlation in errors and patterns are not well extracted from the data series. The only way that shows the forecast method has captured all information perfectly is that the errors get close to the white noise situation. This should be standardized as follows: is the standard error of and is a t statistic. So, if there is a correlation [Bowerman et al (25)] Sample Partial Autocorrelation Function (SPAC) This quantity is considered as the sample autocorrelation of time series observations separated by a lag of k units by considering the elimination of the effect of intervening observations.

44 44 P a g e Residual Plots The last and in the mean time the simplest way to measure the forecasting model is to plot the time series of residuals in exactly the same way as we perform on the time series plot of original data series. Many unusual observations, patterns and information could be extracted only by observing the behavior of time series visually. As previously mentioned the error terms should be a white noise. [i.e. they must be independent and normally distributed with zero mean and finite variance]. In residual analysis, the aim is to see whether or not the error terms have the above cited characteristics. As indicated in figure 3.8, residual plots represent 4 plots, namely, normal probability of the residuals, the plot of residuals versus the fitted values, the plot of histogram of the residuals, and the plot of residuals versus the order of the data. Residual Plots for C1 Normal Probability Plot of the Residuals Residuals Versus the Fitted Values 99 1 Percent Residual Residual Fitted Value Histogram of the Residuals 1 Residuals Versus the Order of the Data Frequency Residual Residual Observation Order 35 4 Figure3.8 Residual Plot In the following each of these plots are defined and explained what behavior is needed for a white noise residual. Normal probability plot of the residuals This plot is a graphical method for assessing whether or not the data set is roughly normally distributed. The data is plotted against the theoretical normal distribution and the points must show an almost straight line and any deviation from this line shows the deviation from normality. Residual plots versus the fitted value This plot indicates the amount of errors made for each fitted value. Obviously, it should make a distribution of points dotted randomly around zero and apart from of the size of the fitted value.

45 45 P a g e It must be checked if this residuals do not make any funnel or shifting shape as the fitted value increases. Histogram of residuals It is an informal way to test normality by comparing the histogram of the sample residual data with a normal probability curve. Clearly, this histogram should be bell shaped and similar to the normal distribution. The plot of residuals versus the order of the data Another way to recognize the non-independence form of the error term is residual versus order plot. It helps to understand if there is any correlation between error terms that are near each other in sequence. Normally, residuals demonstrating normal random noise around the residual = line proves that there is no ongoing correlation Non-parametric trend test of product demand data Here, the non-parametric trend test for the monotonic trends in uni-variate time series will be discussed. For obtaining this goal, the Mann-Kendall test will be used in this study which is broadly used to distinguish increasing or decreasing trends in time series data. To run this test, the Multi-test program has been used which facilitates the trend analysis in multiple time series. This program can implement the trend test in both individual and group of data series. The benefit of this method is clearly understood when there is a large number of data points in the series, in this case the trend detection and also the level shifts could not be visually identified. For example, this approach can be used to analyze the trend behavior of the daily inventory data with thousands of data points in several years. Siegel et al. (1988) address the advantages of nonparametric test against many other types of trend tests as follows. 1- If the sample size is small, there may be no option for using non-parametric tests unless the population distribution is already identified. 2- The number of assumptions about data in non-parametric tests is fewer and it is more relevant to particular situations. 3- Non-parametric test usually treats data that are measured in a nominal scale i.e., are classificatory or categorical. 4- Non-parametric tests can treat samples made up of observations from a number of different populations while parametric tests are not able to do so. 5- Non-parametric tests are easier to learn and apply with more direct interpretation rather than parametric tests.

46 46 P a g e Theoretical Concept If we consider a set of all observations as in the time series and make all possible pairs from these observations as and then calculate the number of harmonious pars which is defined as and non-harmonious pairs as the difference between these two sets of pairs is defined as an indicator of the increasing or decreasing trend in the time series. So if we define the number of pairs when as P and the number of pairs when as M then the sampling distribution of is normalized when n > 1 as follows (T is roughly normal when n > 1): [K. Hamed (28)] And the equals to which enables the computation of p values. When the estimated Z is larger than then, we can conclude there is a trend in the series. For investigating of overall upward or downward trend we can use the following T statistic as the sum of all t statistics of different individual time series:

47 47 P a g e Chapter 4: Research Methodology

48 48 P a g e In order to implement forecasting tools in NYNAS, all raw data for sales volume, sales channels, and customer segments from 24 to 29 are collected from the company s data base and stored in MS Access file. As already mentioned, the raw data is not appropriate for running analysis and it should be cleaned beforehand. As explained earlier, there are more than 2 naphthenic oils products at NYNAS. As a result, reviewing 6 forecasting methods for all these products is am exhausting work and is beyond the scope of this study. Therefore, ABC analysis as a major tool for selecting, sorting, and prioritizing products is run to narrow down the research scope and limit it only to those products that are critically important for the company and make the most income over the last years. Based on this approach, 23 products have been selected. It means that 23 products in the company make the 8% of the firm s income. This method has 6 steps introduced in the following. ABC Analysis EIPM (24) toolbox introduces the 6 steps of ABC analysis as follows. 1- Identify the objective and the analysis criterion 2- Collect data about the analyzed population 3- Sort out the list by decreasing impact 4- Calculate the accumulated impact and the percentage 5- Identify the classes 6- Analyze the classes and take appropriate decisions However, due to the lack of interaction with the company, many ambiguities and missing values were not explained well and consequently recovering them was not possible. Accordingly, in this research, it is mostly tried to select the products that have no or at least very few outliers and missing values. To obtain this goal, 5 products among these 23 critical products have been selected for the final analysis. A few number of missing values for the data of the 5 selected products has been recovered by using interpolation technique. In order to specify any increasing or decreasing trend in data series of products, non-parametric trend test is run for the selected 23 naphthenic goods. However, quantitative forecasting and further analysis are implemented merely on the 5 selected products. Time series plot and seasonal plot are used as a measure to understand the trend and seasonal behavior of sales over the last years. 6 major quantitative forecasting methods, namely, Decomposition, Moving average, Exponential smoothing, ARIMA method, Multiple regression, and Multiple regression with ARIMA errors are used and run in MINITAB 14 and SAS 9.3 and results are obtained. As already mentioned, each of the cited methods has their own pros and cons. For instance, in decomposition, parameters describing the time series should be constant over time. In other words, in order for the decomposition to characterize the behavior of the data and forecast, a static pattern should be viewed and not a dynamic pattern. Moving average is the second

49 49 P a g e quantitative method that is applied on the data series. This method has been widely used in industry. However, it is mostly used to study the behavior of the data series rather to forecast. It is not recommended for data patterns with trend and seasonality and due to the large data storage, usually, the exponential smoothing is selected over this method. Since it requires a small amount of data storage and computation, it is faster and more appealing when the size of data is large. [Makridakis et al (1998)], [Bowerman et al. (25)] ARIMA method is the most general class of models in time series forecasting. The term generality is due the fact that many other quantitative methods of forecasting such as exponential smoothing, autoregressive models and random walk models are all special cases of ARIMA method. However, this model is mostly appropriate for the large data series. Multiple regression model is the next method used in this study. Similar to other quantitative methods, it is established based on some assumptions whose fulfillment is necessary for the applicability of the method. As mentioned earlier, the error terms should be independently and normally distributed with zero mean and constant variance. Otherwise, a pattern might be seen in the error data series and this reveals that the regression model does not cover all information in the data series and the validity of the model will be jeopardized. The last method applied in this study is the multiple regression with ARIMA error. Obviously, it has a benefit over a multiple regression model. In multiple regression model, it is assumed that the errors are uncorrelated. However, in many time series cases, this assumption cannot be correct. In this case, ARIMA model will be used to model the correlation of errors. This combination of regression with ARIMA can help to exploit both methods advantages in forecasting. For this study, in order to collect the required data, a combination of qualitative and quantitative techniques has been applied. An interview was arranged with a sales analyst and the raw data and other required information have been obtained within 4 months. As already declared, this data contained a large amount of outliers, missing values, and other types of ambiguity. Also, due to the lack of enough interaction with the company, many of these ambiguities are still available and this can jeopardize the validity of this research. To prevent this problem, it is tried to cope with the data of products that have the least above-mentioned ambiguities. This, in addition to the goal of narrowing down the research scope, has made us to select only 5 products among the 23 critical products for forecasting and analysis. Furthermore, the data under study only covers 6 years which is not enough for some of the applied quantitative methods. Having considered all the aforementioned limitations, the major quantitative forecasting methods will be implemented in the next chapter.

50 5 P a g e Chapter 5: Implementation and Numerical Results

51 51 P a g e In the following, 6 steps of ABC analysis are thoroughly explained and executed for the NYNAS case study ABC Implementation Identify the objective and the analysis criterion For this research, we are interested to find the future demand of each product. Since there are a large number of products manufactured in the company, we would like to categorize them in different classes and do the demand analysis for the most important products. The main question is how these products will be classified. As discussed in the previous section, the products are categorized based on the amount of money they make. We choose to undertake an ABC analysis, classified by the annual sale value. Collect data about the analyzed population Now, the required data are to be collected. It includes the list of different items and annual sale revenue. Sort out the list by decreasing impact The list of products is sorted out in the decreasing order. Annual sales value is the criterion and it is demonstrated in table 5.1 Products P1 P2 P3 P4 P5 P6 P7 P22 P23 P24 P25 P26 P54 P55 Annual Revenue

52 52 P a g e P56 P57 P148 P149 P15 Table 5.1 List of revenues in decreasing order Calculate the accumulated impact and the percentage In this step, as indicated in table 5.2, the summation of annual sales value and the percentage of the accumulated spending compared to the total are calculated Products Annual Revenue Ratio Sum of Ratio P P P P P P P P P P P P P P P P P E P E P E-7 Table 5.2 Cumulative distribution of revenue for all products 1

53 53 P a g e Identify the classes We make three different categories. Note that our aim is not to get the exact 2% and 8%, but to understand which segments make the most profit and consequently need the most attention. This is shown in Table 5.3. Products Annual Revenue Ratio Sum of Ratio Category P A P A P A P A P A P A P A. P A P A P B P B P B P B P B P C P C P E C P E C P E-7 1 C Table 5.3 Classification of products based on revenue Table 5.4 shows the volume and revenue proportion of the products in class A, B, and C. The result of this analysis shows that only 17% of total products make 77% of the revenue in the company, this result is plotted also in the figure 5.1.

54 54 P a g e Segment Annual value (29) Accumulated volume Accumulated % value A % 79% B % 16.5% C % 4.5% Table 5.4 Volume and Revenue proportion of products in different classes Figure % of the total revenue is made by only 17% of the total products (X axis indicates products) Analyze the classes and take appropriate decisions In the final step, the final analysis will be accomplished. The decisions could be which products or product families are the most critical goods or we can investigate how many suppliers are involved in the production of (A) category products and so on.

55 55 P a g e 5.2. Non-parametric trend analysis This analysis has been done for the first ranked 23 products based on the ABC analysis and the result is shown in table 5.5. Variable Product MK statistic p-value (twosided) Significance code Slope (change/index) Median Demand Product Demand Product Demand Product Demand Product Demand Product Demand Product Demand Product Demand Product Demand Product Demand Product Demand Product Demand Product Demand Product Demand Product Demand Product Demand Product Demand Product Demand Product Demand Product Demand Product Demand Product Demand Product Demand Product Table 5.5 Univariate test for the 23 top ranked products As can be seen in table 5.5, for products 7, 12 and 16 we have an increasing trend with a large slope and for products 5 and 22 we have a large decreasing trend. Product 11 has a slight positive trend and products 18 and 21 have a slight negative trend while other products do not show any trend. As mentioned already in the methodology section, since the forecasting is implemented on the selected 5 products among the 23 products above, the Mann-Kendall test is run for them and the result is indicated in table 5.6.

56 56 P a g e Variable Product MK statistic p-value (twosided) Significance code Slope (change/index) Median Demand P Demand P Demand P Demand P Demand P No Trend Table 5.6 Univariate test of the 5 selected products As demonstrated above, no trend has been detected for the above products and this can help in finding a best forecasting method for the sales volume of these products Visual analysis The time series plot is an appropriate tool to observe the behavior of time series. It gives a lot of information about the nature of the data series under study. For instance, in the case of production, management might find it necessary to take options such as overtime or extra shift works into consideration with the aim to handle peak periods or reload stocks during slack periods. The time series plot of the one sample of critical products is depicted in the following figure: 25 Sales plot of P Figure 5.2 Time series plot for P1 As indicated in figure 5.2, the data series has a constant variance and an increasing trend up to the 57 th observation and there is a constant mean afterwards. In this plot, it is clear that this increasing behavior in sales stops at the end of 28 and then we have a big shift in the mean value. This could be explained by the introduction of financial crisis in 29. Appropriate method of forecasting in many instances has been selected by considering the behavior of the data pattern. In the following we do the empirical study and implementation of

57 57 P a g e the above mentioned forecasting methods, and finally, based on the appropriate accuracy measures, the best models will be selected. Seasonal plot One of the important factors in demand planning is to observe seasonal variations. One way to see if seasonal patterns exist is to use seasonal plots (Figure 5.3). For monthly data, for example, the response variable which is the sales volume of the mentioned product could be plotted over month and grouped by year. As a result, each year s time series plot overlaps over different years and the change of seasonal indices in different periods can be easily recognized. Figure 5.3 Seasonal plot of P1 from 24 to 29 Apart from the sale reduction in 29 which is explainable due to the financial crisis, we see that the level of sales is almost increasing over the years. This is clearly shown in the figure below (Figure 5.4) with highlighting the level shifts from one year to another (slight trend). This figure also allows the easy assessment of consecutive seasonal cycles.

58 58 P a g e Figure 5.4 Shift of P1 sales from one year to another 5.4. Output of decomposition Additive decomposition model with the seasonal length equal to 12 is chosen for running the decomposition method. 25 Time Series Plot of P P Index Figure 5.5 Time series plot for P1 in MINITAB

59 59 P a g e In figure 5.6 component analysis (seasonally adjusted and de-trended data plot) has been shown. Original Data Component Analysis for P1 Additive Model Detrended Data 2 1 Data 1 Detr. Data Index Index 56 7 Seas. Adj. Data Seasonally Adjusted Data Index 7 Seas. Adj. and Detr. Data Seasonally Adj. and Detrended Data Index Figure 5.6 Component Analysis of Decomposition method for P1 It can be seen in the figure above that the method does not recognize any specific seasonality and trend. This confirms our first comment regarding the applicability and appropriateness of decomposition method for static pattern. Since the data series does not show any stable behavior in past and future, this method is not recommended for the forecasting of product 1. The same happens for the rest of the products from P2 to P5. By plotting their data series, it is indicated that they have dynamic pattern as well. In Figure 5.7, the lack of usefulness of decomposition method for fitting seasonality of product 1 in a dynamic pattern is quite clear. Time Series Decomposition Plot for P1 Additive Model Variable Actual Fits Trend Accuracy Measures MAPE 214 MAD 435 MSD P Index Figure 5.7 Time Series Decomposition Plot for P1

60 6 P a g e To check the validity of this method, the residual plots of product 1 is shown in Figure 5.8. They almost demonstrate normal behavior and the time series and the scatter plot of residuals are almost indicating random behavior. Residual Plots for P1 Normal Probability Plot of the Residuals Residuals Versus the Fitted Values Percent Residual Residual Fitted Value 14 Frequency Histogram of the Residuals Residual Residuals Versus the Order of the Data Residual Observation Order Figure 5.8 Residual Plot The demand forecast in addition to the mean square deviation as an accuracy measure for the forecasts of 5 critical products have been shown in table 5.7 and 5.8 Date P1 P2 P3 P4 P5 1/ / / / / / / / / / / / Table 5.7 Generated forecast

61 61 P a g e Product P1 P2 P3 P4 P5 MSD Table 5.8 Mean square deviation for the 5 products 5.5. Output of moving average The data series doesn t show any outstanding seasonality. As already mentioned, this method is more useful when there is no trend or seasonality in the data sereis. For the non-seasonal data series it is common to set the moving average length short. It also depends on the level of the noises of the series. The larger the length, the smoother but less sinsitive outpust. In the following, the fiiting plot of moving average method with actual data has been depicted Moving Average Plot for P1 Variable Actual Fits Moving Average Length 5 P Accuracy Measures MAPE 27 MAD 44 MSD Index Figure 5.9 Fitting Plot of Moving Average with length 5 Different lengths are compared in table 5.9. Obviously, the moving average with the length of 5 gets the best forecasting result. Length MAPE MSD Table 5.9 Accuracy measures based on different length

62 62 P a g e Normality Test Figure 5.1 and 5.11 almost indicate normal behavior and stationary pattern for the error data series. 1 5 Axis Title Axis Title RESI1 Figure 5.1 Linearity tests of residuals Residual Plots for P1 Normal Probability Plot of the Residuals Residuals Versus the Fitted Values Percent Residual Residual Fitted Value 2 12 Histogram of the Residuals 1 Residuals Versus the Order of the Data Frequency Residual Residual Observation Order Figure 5.11 Residual plots of P1 In table 5.1, the sales forecast with length 5 plus the upper and lower bounds with 95% prediction intervals are demonstrated.

63 63 P a g e Date [Month/Year] Forecast Upper band lower band 1/21 to 12/ Table 5.1 Forecast in addition to upper and lower bounds Also, forecast line with intervals is indicated in figure Moving Average Plot for P1 Variable Actual Fits Forecasts 95.% PI P Moving Average Length 5 Accuracy Measures MAPE 27 MAD 44 MSD Index Figure 5.12 Forecast line with intervals The results for other critical products are as follows. product Forecast Upper limit Lower limit Table 5.11 Forecast with intervals The MSD results of applied moving average methods are shown in table Product MSD Length Table 5.12 Mean square deviation for other critical products

64 64 P a g e 5.6. Output of exponential smoothing As explained earlier in this study, there are different approaches within the category of exponential smoothing. Three basic methods in exponential smoothing are single exponential smoothing (SES), Holt s method and Holt-Winters method which are appropriate respectively for the patterns with constant mean and variance, fixed increasing or decreasing trends and finally with trend and seasonality. For the first critical product, we can hardly see any significant trend, though there is a slight one until the 56 th observation and a big shift afterwards. Also there is not any significant seasonality. So SES method performs better than others. In table 5.13 results are compared: Methods SES (optimum α =.28) Holt (optimum α =.63 and β =.3) Holt-Winters (α, β and γ =.2) MAPE MSD Table 5.13 Comparison of SES, Holt, and Holt-Winters based on MSD and MAPE As we were expecting SES outperforms other methods Single Exponential Smoothing Plot for P1 Variable Actual Fits Forecasts 95.% PI P Smoothing Constant Alpha Accuracy Measures MAPE 219 MAD 437 MSD Index Figure 5.13 SES plot in addition to the forecast lines The residuals show more randomness and normality attributes compared to the moving average approach. (Figure 5.14 and 5.15)

65 65 P a g e Residual Plots for P Normal Probability Plot of the Residuals 99 1 Residuals Versus the Fitted Values Percent Residual Residual Fitted Value Histogram of the Residuals 1 Residuals Versus the Order of the Data Frequency Residual Residual Observation Order Figure 5.14 Residual Plots Axis Title Axis Title Residuals Figure 5.15 Residual Trend plot Generated forecast for 12 months for the 5 critical products is depicted in table Date product Forecast Lower bound Upper bound Method MSD 1/21 to 12/21 P SES /21 to 12/21 P SES /21 to 12/21 P SES /21 to 12/21 P SES /21 to 12/21 P SES Table 5.14 Generated forecast with boundaries and mean square deviation

66 66 P a g e 5.7. Output of ARIMA method Earlier it was emphasized that when dealing with the ARIMA model, the important perquisite is to make the series stationary. Looking at the time series plot of the first critical product, the one thing that comes to mind is if there is any possibility to make this series more stationary. 25 Sales plot of P Figure 5.16 Sales plot of P1 Getting the first difference of the series, the data pattern will change. This plot is demonstrated as follows: sales plot for the Differenced data series for P1 First Differenced Figure 5.17 Sales plot for the Differenced data series Now, we have to fit the best model of ARIMA based on the data pattern and the plot of SAC and SPAC for the transformed data. It is shown in figure 5.18

67 67 P a g e Autocorrelation Function for first (with 5% significance limits for the autocorrelations) Partial Autocorrelation Function for first (with 5% significance limits for the partial autocorrelations) Autocorrelation Partial Autocorrelation Lag Lag Figure 5.18 SAC and SPAC plot of the transformed data Based on the SAC and SPAC of the transformed series the best ARIMA method that would be suggested is ARIMA (1,1,). (No seasonality seen in the time series plot) Table 5.15 proves the validity of our claim; Method ARIMA(1,1,) ARIMA(,1,1) ARIMA(1,1,1) MSE Table 5.15 MSE comparison in different ARIMA methods As can be seen from the P value of the parameters, the constant term is not significant, so the final estimation would be as follows: Type Coef SE Coef T P MA Constant Differencing: 1 regular difference Number of observations: Original series 7, after differencing 69 Residuals: SS = (backforecasts excluded) MS = 2934 DF = 67 Table 5.16 Level of significance for ARIMA parameters Also, the Box-Pierce chi-square statistics result shows the residuals are stationary since the p values of different lags are insignificant

68 68 P a g e Modified Box-Pierce (Ljung-Box) Chi-Square statistic Lag Chi-Sq DF P-Value Table 5.17 Box-Pierce Analysis The residuals almost indicate the stationary behavior. The scatter plot of errors vs. fitted value and also the time series plot of residuals show an approximate randomness. Considering residual plots in figure 5.19, 5.2, and 5.21, we can come to this conclusion that the errors are nearly normally distributed, and SAC and SPAC plot of residuals show no correlation among residuals Residual 1 5 Fitted values Fitted value Figure 5.19 Residual analysis

69 69 P a g e Residual Plots for P Normal Probability Plot of the Residuals 99 1 Residuals Versus the Fitted Values Percent Residual Residual Fitted Value 2 16 Histogram of the Residuals 1 Residuals Versus the Order of the Data Frequency Residual Residual Observation Order Figure 5.2 Residual Tests Plot of Residuals RESI1 Figure 5.21 Plot of residuals ACF and PACF of residuals show that there is no correlation among errors. (Figure 5.22) PACF of Residuals for NYTRO 11GX (with 5% significance limits for the partial autocorrelations) ACF of Residuals for NYTRO 11GX (with 5% significance limits for the autocorrelations) Partial Autocorrelation Autocorrelation Lag Lag Figure 5.22 SAC and SPAC plot of residuals

70 7 P a g e Estimated forecast for 12 month in 21 (product 1) and other four critical products is shown in tables 5.18 to Date (month/year) Forecast Lower bound Upper bound 1/ / / / / / / / / / / / Table 5.18 Forecasts of P1 Date (month/year) Forecast Lower bound Upper bound ARIMA Type 1/ / / / / / / / / / / / Table 5.19 Generated forecast for product 2 ARIMA(,1,1) Product Product 2 MSE

71 71 P a g e Date (month/year) Forecast Lower bound Upper bound ARIMA Type 1/ / / / / / / / / / / / Table 5.2 Generated forecast for product 3 ARIMA(1,1,1) Product Product 3 MSE Date (month/year) Forecast Lower bound Upper bound ARIMA Type 1/ / / / / / / / / / / / Table 5.21 Generated forecast for product 4 ARIMA(1,1,) Product Product 4 MSE 15966

72 72 P a g e Date (month/year) Forecast Lower bound Upper bound 1/ / / / / / / / / / / / Table 5.22 Generated forecast for product 5 ARIMA Type ARIMA(,1,1) Product Product 5 MSE Output of multiple regression method In the following, the best possible variables for the regression model will be selected and the linear regression model will be presented. Variable selection As discussed earlier, the first part in applying this method is to select the potential variables that might be influencing the response variable. Apart from on what company and industry we are focusing, there are some general factors that can influence the demand of specific products including price of the good, price of the complementary goods, price of substitutes, consumer s income, consumer s taste for the good, consumers expectation about the future prices, advertising, and taxation. For this case, the selected factors are as follows: Time (linear, quadratic, cubic), Seasonality, Average oil price, Average oil price with lag 1 and 2, Differenced price of oil and also Price of the product under study. The value of elasticity of the product could also be considered as another potential variable that may have impact on the volume of sale. In table 5.22, correlations among the selected variables are depicted. Due to the high correlation among the predictors, namely, average oil price, difference of oil price and lag of oil price, one of them should be selected as an explanatory variable. With this regard, the average of oil price

73 73 P a g e will be used because of the largest correlations with the response variable (volume of sale). On the other hand, the product price demonstrates its effect with one lag. This is reasonable since buyers will not react in the same period in which the prices are announced. In this case, lag 1 (one period later) makes the biggest correlation and will be selected as one of the explanatory variables. Correlations: P1, AvgOfClose o, diff of oil, lag of oil p,... Product sale AvgOfClose o.567. average oil p diff oil p lag of oil p diff oil p lag of oil p product price lag 1 price of p diff of product p product price lag 1 price of p lag 1 nytro.562. diff of product p Cell Contents: Pearson correlation P-Value Table 5.23 Correlation among selected potential variables As indicated in the sales plot of the product in figure 5.23, no significant seasonality could be recognized. This issue will be proved by doing correlation test shown in table 5.24 and checking the p value of the regression for different seasons (dummy variables).

74 74 P a g e 25 Sales plot of P Figure 5.23 Sales plot of Nytro Correlations: P1, T, jan, feb, mar, apr, may, jun,... P1 T jan feb mar T jan feb mar apr may jun jul aug sep oct

75 75 P a g e nov apr may jun jul aug may jun jul aug sep oct nov sep oct oct nov Cell Contents: Pearson correlation P-Value Table 5.24 Correlation between the response variable and season indices Among all factors above, only T (time) could be considered as one of the effective factors. So far, T, average oil price, and the price of the product with lag 1 have shown some effects on the response variable which is the quantity of sale. Among these variables, T has shown a smallest contribution for making the future value of sales. However, we have to keep in mind that this only considers a linear correlation, while two variables might have zero linear but significant non-linear correlation. That s why the fitted line test for quadratic and cubic states is also examined. The scatter plots with the fitted lines are shown in the following.

76 76 P a g e Figure 5.24 Fitted Linear Test Figure 5.25 Fitted Quadratic Test Figure 5.26 Fitted Cubic Test

77 77 P a g e The regression results are indicated in table It is clear that the following regression model gives the best result: Polynomial Regression Analysis: P1 versus T The regression equation is P1 = T T**2 S = R-Sq = 16.2% R-Sq(adj) = 13.8% Analysis of Variance Source DF SS MS F P Regression Error Total Sequential Analysis of Variance Source DF SS F P Linear Quadratic Table 5.25 Regression analysis (P1 VS T & T2) So far, our multiple regression model is: In which T is time, AO t is the monthly average oil price at time t, Pr t-1 is the price of product p at time t-1 and ε is the error term. The analysis of variance plus the regression equation of Product 1 is shown in table P1 quantity = AvgOfClose price of P1 with 1 lag -.22 T** T 71 cases used, 1 cases contain missing values

78 78 P a g e Predictor Coef SE Coef T P Constant AvgOfClose price of P1 with 1 lag T** T S = R-Sq = 39.1% R-Sq(adj) = 35.4% Table 5.27 Analysis of variance and the regression equation of product 1 (Nytro) The final equation would be: The MSE obtained from the regression model of other 4 products is stored in table Product name P1 P2 P3 P4 P5 MSE The most significant factor (based on P value) Oil price + product price with lag 1 Product price with lag 1 Product price with lag 1 product price with lag 1+ oil price Table 5.28 MSE and critical factors in regression model for different products No significant factor has been found 5.9. Output of multiple regression method with ARIMA error The MSE of the obtained results has been indicated in table 5.29 Product name P1 P2 P3 P4 P5 MSE Table 5.29 MSE of 5 critical products in multiple regression with ARIMA model

79 79 P a g e Chapter 6: Comparison of Numeric Results

80 8 P a g e Clearly, due to the different trend or seasonal behavior of products and since each of them may follow the own pattern, there is no specific method that can be selected for all products. Rather, the best one should be selected based on the accuracy measures for each product. Figure 6.1 clearly indicates the different behavior of the first and second product over time. This emphasizes the nature of problem orientation of forecasting for different products. Time Series Plot of P1 Time Series Plot of P P1 P Index Index Figure 6.1 Time series plot of P1 and P2 The amount of [MSD/MSE] and [MAPE] of different methods are represented in the table 6.1 and 6.2 respectively. Product name P1 P2 P3 P4 P5 Decomposition Moving Average Exponential Smoothing ARIMA Multiple Regression Regression with ARMA error Table 6.1 Comparison of different forecasting methods based on MSE Product name p1 p2 p3 p4 p5 decpmposition moving average exponetial smoothing ARIMA Table 6.2 Comparison of different forecasting methods based on MAPE Table 6.1 demonstrates the amount of MSE/MSD for the 5 selected products and table 6.2 does the similar comparison for the first four methods. This is practically impossible to see the pattern of each product and come up with the best forecasting method for each. Then, it will be more appropriate to get the average of MSD/MSE

81 81 P a g e and also MAPE of each method for 5 products in order to find the best rough method that can be used for the general forecasting. The averages of errors (both MSE and MAPE) for all 5 critical products are expressed in the following table for each method. Product name Average of MSE for 5 products Decomposition Moving Average Exponential Smoothing ARIMA Multiple Regression Regression with ARMA error Product name Table 6.3 Average MSE for 5 critical products Average of MAPE for 5 products Decomposition 57.2 Moving Average 62.6 Exponential Smoothing 64 ARIMA Table 6.4 Average MAPE for 5 critical products As can be seen in table 6.3 and 6.4, exponential smoothing and ARIMA method give the least MSE and ARIMA model gives the least MAPE. However, it can be understood that each method depending on the undergone situation, has its own function and usefulness. For example, if the purpose is to discover the potential factors affecting the sales volume, regression or regression with ARIMA error could be more effective. While, if the objective is to merely determine the output and what will happen or when the system is too complicated to find the potential influencing factors, ARIMA would be a better option of forecasting. However, in real-world problems, the accuracy and performance alone are not the imperative factors. For instance, when dealing with large data volume of sales or demand, the method that can be run in the shorter processing time such as exponential smoothing is more advantageous. But now, one may be interested to see if the the mean of our populations (forecasting methods) are really different or other unexplained factors have influenced this difference. In other words, we might be interested to see whether or not choosing different quantitative forecasting methods has brought about different mean values or this difference is appeared accidently. To obtain this goal, the Completely Randomized Block Design is suggested. This test is done on the 4 methods of decomposition, moving average, Exponential smoothing, and ARIMA. In fact, we are interested to apply the following null hypothesis;

82 82 P a g e To do this, Means square factor and mean square error should be calculated by using the following formula; (D. Montgomery et al 27) Product name P1 P2 P3 P4 P5 SUM AVE Decomposition Moving Average Exponential Smoothing ARIMA SUM AVERAGE Table 6.5, The result of MAPE for different products with SUM and Average For comparing means, parametric statistics always assume that the data we would like to test is normally distributed and has equal variances. Anderson-Darling normality test in MINITAB shows that all data are normally distributed. Also by running the F test, variances are not statistically different. In order to see the effect of forecasting methods, and the blocks (products in our case) in the value of MAPE, we have run the Completely Randomized Block Design test. The result of analysis of variance for MAPE has shown in the table below;

83 83 P a g e Source DF SS MS F P Product Forecasting method ,7 Error Total S = R-Sq = 8% R-Sq (adj) = 68% Table 6.6 Analysis of variance for forecasting method as the main factor and products as block It is indicated that with 9% confidence intervals, our forecasting method is a significant factor and changing to ARIMA model will enhance the performance to a reasonable extent. This is clearly shown in the main effect plot for data means in figure 7 Main Effects Plot (data means) for Methods 6 Mean of Methods ARIMA Decomposition Exponential Smoothing Forecasting methods Moving Average Figure 6.2 Main Effect Plot for different methods against average of MAPE

84 84 P a g e As can be seen from figure 6.2 ARIMA method gives the least average of error. Now, one might be interested to see if this difference with the mean of other populations (forecasting methods) is significantly different or not. To do this we run the null hypothesis for the difference of means with unknown variances and sample size smaller than 3. This is a t statistics test illustrated as follows; The degree of freedom for the t statistics above is: The null hypothesis would be as follows: [D. Montgomery et al (27)] Considering this, we run the Two Sample T test in MINITAB and the result would be as follows: Two-sample T for ARIMA vs Decomposition N Mean StDev SE Mean ARIMA decomposition Difference = mu (ARIMA) - mu (decomposition) Estimate for difference: % CI for difference: ( , ) T-Test of difference = (vs not =): T-Value = P-Value =.125 DF = 4 Table 6.7 Two-sample T for ARIMA vs Decomposition With 9% confidence interval, it is shown that zero is not included in this interval. So, the ARIMA method outperforms significantly. The same analysis has been done for the difference between ARIMA and moving average, and ARIMA and exponential smoothing; the similar results are obtained (Table 6.8 and 6.9);

85 85 P a g e Two-sample T for ARIMA vs Moving average N Mean StDev SE Mean ARIMA moving average Difference = mu (ARIMA) - mu (moving average) Estimate for difference: % CI for difference: ( , ) T-Test of difference = (vs not =): T-Value = P-Value =.174 DF = 4 Table 6.8 Two-sample T for ARIMA vs Moving average Two-sample T for ARIMA vs Exponential Smoothing N Mean StDev SE Mean ARIMA exponential smoo Difference = mu (ARIMA) - mu (exponential smoothing) Estimate for difference: % CI for difference: ( , ) T-Test of difference = (vs not =): T-Value = P-Value =.189 DF = 4 Table 6.9 Two-sample T for ARIMA vs Exponential Smoothing

86 86 P a g e Chapter 7: Conclusion

87 87 P a g e This research has its own limitations. One of these limitations that might affect the result is the amount of data available for the forecasting. For some specific methods such as ARIMA and decomposition, the more data we collect, the more valid results we may obtain. This will certainly influence the conclusion of this study that gives the best score to ARIMA and exponential smoothing. Furthermore, over and under fitting as two important expressions in the forecasting literature are beyond the scope of this research while, at the same time may have a major influence on the validity of the forecasting methods. Having studied and reviewed related literature in the area of quantitative forecasting, we ve come to this conclusion that the econometric method is not a right option for this project mainly due to the limitations that have been mentioned in previous chapters. In Time Series, selecting a particular method for the forecast of all products cannot be a right decision. It is based on the behavior of the data on which the analysis should be made. On the other hand, time series methods are appropriate when the system is not well understood and so complicated or we only care about what will happen rather why it will happen. If our purpose is not only to forecast the demand of potentially critical products but to give special tools to managers to take right decisions in many areas including sales, revenue management, pricing and control of influencing factors, we will need to employ a method that not only eases the forecasting procedures and gets the precise projection of demand volume but also demonstrates effective variables on demand and suggests specific decision for this behavior. To obtain this goal, it is recommended to have a combination of explanatory and time series methods. That s why a more advanced method of forecasting such as regression model with ARIMA errors has been suggested. This method integrates two methods of forecasting namely, Multiple Regression and ARIMA time series and computes the new coefficients with maximum likelihood algorithm. However, regression with ARIMA error did not result in a much better output compared to ARIMA and Exponential smoothing. The reason is that finding appropriate factors to predict the response variable calls for the high level of sophistication in product demand analysis. Unfortunately, due to the lack of adequate interaction with NYNAS, only a short number of factors have been detected. Thus, the R-square value of regression is below 4% for all products. Generally exponential smoothing and ARIMA method might produce more precise forecast. However, by using regression with ARIMA error, we have the ability to interpret the interfering factors and the reason for observing different behaviors. Having calculated the mean of MAPE for all forecasting methods, we were interested to see if the forecasting method is a significant factor and if the difference between the average values obtained by ARIMA is statically different from other methods. To do so, we run the randomized block design method and by drawing the main effect plot we have come to this conclusion that forecasting method can be considered as a significant factor and by running Two Sample T Test

88 88 P a g e in MINITAB and by considering 9% confidence interval, the ARIMA method outperforms other methods significantly.

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