Time-series methods of forecasting

         Forecasting is a method or a technique for estimating future aspects of a business or the operation. It is a method for translating past data or experience into estimates of the future. It is a tool, which helps management in its attempts to cope with the uncertainty of the future. Forecasts are important for short-term and long-term decisions. Businesses may use forecast in several areas: technological forecast, economic forecast, demand forecast. There two broad categories of forecasting techniques: quantitative methods (objective approach) and qualitative methods (subjective approach). Quantitative forecasting methods are based on analysis of historical data and assume that past patterns in data can be used to forecast future data points. Qualitative forecasting techniques employ the judgment of experts in specified field to generate forecasts. They are based on educated guesses or opinions of experts in that area. There are two types of quantitative methods: Times-series method and explanatory methods.

         Time-series methods make forecasts based solely on historical patterns in the data. Time-series methods use time as independent variable to produce demand. In a time series, measurements are taken at successive points or over successive periods. The measurements may be taken every hour, day, week, month, or year, or at any other regular (or irregular) interval. A first step in using time-series approach is to gather historical data. The historical data is representative of the conditions expected in the future. Time-series models are adequate forecasting tools if demand has shown a consistent pattern in the past that is expected to recur in the future. For example, new homebuilders in US may see variation in sales from month to month. But analysis of past years of data may reveal that sales of new homes are increased gradually over period of time. In this case trend is increase in new home sales. Time series models are characterized of four components: trend component, cyclical component, seasonal component, and irregular component. Trend is important characteristics of time series models. Although times series may display trend, there might be data points lying above or below trend line. Any recurring sequence of points above and below the trend line that last for more than a year is considered to constitute the cyclical component of the time series—that is, these observations in the time series deviate from the trend due to fluctuations. The real Gross Domestics Product (GDP) provides good examples of a time series tat displays cyclical behavior. The component of the time series that captures the variability in the data due to seasonal fluctuations is called the seasonal component. The seasonal component is similar to the cyclical component in that they both refer to some regular fluctuations in a time series. Seasonal components capture the regular pattern of variability in the time series within one-year periods. Seasonal commodities are best examples for seasonal components. Random variations in times series is represented by the irregular component. The irregular component of the time series cannot be predicted in advance. The random variations in the time series are caused by short-term, unanticipated and nonrecurring factors that affect the time series.

            Smoothing methods (stable series) are appropriate when a time series displays no significant effects of trend, cyclical, or seasonal components. In such a case, the goal is to smooth out the irregular component of the time series by using an averaging process. The moving averages method is the most widely used smoothing technique. In this method, the forecast is the average of the last “x” number of observations, where “x” is some suitable number. Suppose a forecaster wants to generate three-period moving averages. In the three-period example, the moving averages method would use the average of the most recent three observations of data in the time series as the forecast for the next period. This forecasted value for the next period, in conjunction with the last two observations of the historical time series, would yield an average that can be used as the forecast for the second period in the future. The calculation of a three-period moving average is illustrated in following table. Based on the three-period moving averages, the forecast may predict that 2.55 million new homes are most likely to be sold in the US in year 2008.

Year	Actual sale(in million)	Forecast(in million)	Calculation
2003	4
2004	3
2005	2
2006	1.5	3	(4+3+2)/3
2007	1	2.67	(3+2+3)/3
2008		2.55	(2+3+2.67)/3

Example: Three-period moving averages

In calculating moving averages to generate forecasts, the forecaster may experiment with different-length moving averages. The forecaster will choose the length that yields the highest accuracy for the forecasts generated. Weighted moving averages method is a variant of moving average approach. In the moving averages method, each observation of data receives the same weight. In the weighted moving averages method, different weights are assigned to the observations on data that are used in calculating the moving averages. Suppose, once again, that a forecaster wants to generate three-period moving averages. Under the weighted moving averages method, the three data points would receive different weights before the average is calculated. Generally, the most recent observation receives the maximum weight, with the weight assigned decreasing for older data values.

Year	Actual sale(in million)	Forecast(in million)	Calculation
2005	2 (.2)
2006	1.5 (.3)
2007	1 (.4)
2008		.42	(2*.2+1.5*.3+1*.4)/3

Example: Weighted three-period moving averages method

A more complex form of weighted moving average is exponential smoothing. I this method the weight fall off exponentially as the data ages. Exponential smoothing takes the previous period’s forecast and adjusts it by a predetermined smoothing constant, ά (called alpha; the value for alpha is less than one) multiplied by the difference in the previous forecast and the demand that actually occurred during the previously forecasted period (called forecast error). Exponential smoothing is mathematically represented as follows: New forecast = previous forecast + alpha (actual demand − previous forecast) Or can be formulated as  F = F + ά(D − F)

         Other time-series forecasting methods are, forecasting using trend projection, forecasting using trend and seasonal components and causal method of forecasting. Trend projection method used the underlying long-term trend of time series of data to forecast its future values. Trend and seasonal components method uses seasonal component of a time series in addition to the trend component. Causal methods use the cause-and-effect relationship between the variable whose future values are being forecasted and other related variables or factors. The widely known causal method is called regression analysis, a statistical technique used to develop a mathematical model showing how a set of variables is related. This mathematical relationship can be used to generate forecasts. There are more complex time-series techniques as well, such as ARIMA and Box-Jenkins models. These are heavier duty statistical routines that can cope with data with trends and the seasonality in them.

         Time series models are used in Finance to forecast stock’s performance or interest rate forecast, used in forecasting weather. Time-series methods are probably the simplest methods to deploy and can be quite accurate, particularly over the short term. Various computer software programs are available to find solution using time-series methods.  

Waiting lines and Queuing system

             Waiting in lines is a part of our everyday life. Waiting in lines may be due to overcrowded, overfilling or due to congestion. Any time there is more customer demand for a service than can be provided, a waiting line forms. We wait in lines at the movie theater, at the bank for a teller, at a grocery store. Wait time is depends on the number of people waiting before you, the number of servers serving line, and the amount of service time for each individual customer. Customers can be either humans or an object such as customer orders to be process, a machine waiting for repair. Mathematical analytical method of analyzing the relationship between congestion and delay caused by it can be modeled using Queuing analysis. Queuing theory provides tools needed for analysis of systems of congestion. Mathematically, systems of congestion appear in many diverse and complicated ways and can vary in extent and complexity.

              A waiting line system or queuing system is defined by two important elements: the population source of its customers and the process or service system. The customer population can be considered as finite or infinite. The customer population is finite when the number of customers affects potential new customers for the service system already in the system. When the number of customers waiting in line does not significantly affect the rate at which the population generates new customers, the customer population is considered infinite. Customer behavior can change and depends on waiting line characteristics. In addition to waiting, a customer can choose other alternative. When customer enters the waiting line but leaves before being serviced, process is called Reneging. When customer changes one line to another to reduce wait time, process is called Jockeying. Balking occurs when customer do not enter waiting line but decides to come back latter.   Another element of queuing system is service system. The number of waiting lines, the number of servers, the arrangements of the servers, the arrival and service patterns, and the service priority rules characterize the service system. Queue system can have channels or multiple waiting lines. Examples of single waiting line are bank counter, airline counters, restaurants, amusement parks. In these examples multiple servers might serve customers. In the single line multiple servers has better performance in terms of waiting times and eliminates jockeying behavior than the system with a single line for each server. System serving capacity is a function of the number of service facilities and server proficiency. In queuing system, the terms server and channel are used interchangeably. Queuing systems are either single server or multiple servers. Single server examples include gas station food mart with single checkout counter, a theater with a single person selling tickets and controlling admission into the show. Multiple server examples include gas stations with multiple gas pumps, grocery stores with multiple cashiers, multiple tellers in a bank. Services require a single activity or services of activities called phases.  In a single-phase system, the service is completed all at once, such as a bank transaction or grocery store checkout counter. In a multiphase system, the service is completed in a series of phases, such as at fast-food restaurant with ordering, pay, and pick-up windows. Queuing system is characterized by rate at which customers arrive and served by service system. Arrival rate specifies the average number of customers per time period. The service rate specifies the average number customers that can be serviced during a time period. The service rate governs capacity of the service system. It is the fluctuation in arrival and service patterns that causes wait in queuing system. Waiting line models that assume that customers arrive according to a Poisson probability distribution, and service times are described by an exponential distribution. The Poisson distribution specifies the probability that a certain number of customers will arrive in a given time period. The exponential distribution describes the service times as the probability that a particular service time will be less than or equal to a given amount of time. A waiting line priority rule determines which customer is served next. A frequently used priority rule is first-come, first-served. Other rules include best customers first, high-test profit customer first, emergencies first, and so on. Although each priority rule has merit, it is important to use the priority rule that best supports the overall organization strategy. The priority rule used affects the performance of the waiting line system.

             Basic single server model assumes customers are arriving at Poisson arrival rate with exponential service times, and first come, first serviced queue discipline, and infinite queue length, and infinite calling population. By adding additional resources to single server system either service rate can be increased or arrival rate at each server can be decreased with additional cost overhead.  In Single server single-phase system, customer is served once completed. Common examples of single server single-phase are a teller counter in bank, a cashier counter in super market, automated ticketing machine at train station. In single server queuing system wait time or performance of system depends on efficiency of serving person or service machine. Single server single-phase queuing system is most commonly automated system found in our regular life. For example many superstores have replaced manual billing counters with automated machines. Single server multiple-phase system incorporates division of work into phases to keep waiting line moving as completion of whole complete operation might increase wait in a line. Common examples of these systems are automatic or manual car wash, drive through restaurants.

             Waiting line models are important to a business because they directly affect customer service perception and the costs of providing service. If system average utilization is low, that suggests the waiting line design is inefficient. Poor system design can result in over staffing. Long waits suggest a lack of concern by the organization or can be view as a perception of poor service quality. Queuing analysis has changed the way businesses use to run and has increased efficiency and profitability of businesses.

Management science modeling techniques

             Management science is the science for managing and involves decision making. It utilizes what is controllable, and tries to predict what is uncontrollable in order to archive a specific objective. Science is a continuous search; it is a continuing generation of theories, models, concepts, and categories. Management science uses analytical methods to solve problems in areas such as production and operations, inventory management, and scheduling. Typical management science approach is to build a model for the problem being studied, such a model is often a mathematical model. Practical problems are often unstructured and lack clarification in definition of problem which makes mathematical modeling a challenge. Therefore modeling of a problem is important phase in problem solving technique. Once model is built, algorithms are used to solve problem. Various techniques are devised to model problem and solve it for possible solutions.

              Linear programming is one of the widely used modeling techniques. Linear programming problems consist of an objective function (also know as cost function) which has to be minimized or maximized subject to a certain number of constraints. The objective function consists of a certain number of variables. The constraints are linear inequalities of the variables used in the objective function. This technique is closely related to linear algebra and uses inequalities in the problem statement rather than equalities. A linear programming problem can fall in three categories: infeasible, unbounded and an optimal solution. In an infeasible problem values of decision variables do not satisfy constraint condition. A problem is unbounded if the constraints do not sufficiently restrain the objective function so that for any given feasible solution, another feasible solution can be found that makes further improvement to the objective function. In an optimal solution, the objective function has a unique maximum or minimum value. Linear programming problems can be solved using graphical analysis method. Sensitive analysis is extension to solution found in linear programming to find out effect of parameter changes on the optimal solution. The parameters are called coefficients and can be quantity or value used in objective function. In linear programming results are rounded to get reasonable output, however rounded solution might not be feasible and many not give an optimal solution. Therefore, Integer programming model is used with fractional values. Linear programming is also use to solve transportation, transshipment, and assigning problems. Linear programming is widely used in production planning and scheduling. It is very well used in airline industry for aircraft and crew scheduling.

             Probabilistic techniques are another class of modeling approach for problem solving. It is based on application of statistics for probability of uncontrollable events as well as risk assessment of decision. In this technique risk means uncertainty for which the probability of distribution is know. Therefore risk assessment involves study of the outcomes of decisions along with their probabilities. Probability assessment tries to fill gap between what is know and what need to be know for an optimal solution. Therefore, probabilistic models are used to prevent events happening due to adverse uncertainty. Decision analysis and queuing systems are example of probabilistic techniques. The modeling technique use to solve physical problems such as transportation or flow of commodities is Network modeling. Network problems are an abstract representation of processes and activities for a give problem and illustrated by using network branches and nodes. This technique uses most cost effective way to transport the goods, to determine maximum/minimum possible flow from source to destination and to find shortest critical path in large projects. 

            The management science modeling process helps businesses to improve their operations through the use of scientific methods and the development of specialized techniques. It is the process of re searching for an optimal solution to the existing problem. Management science modeling process provides systematic, analytical and general approaches to the problem solving for decision-making, regardless of the nature of the system, product, or service. Management science modeling process is the application of scientific methods to complex organizational problems. Models are aimed at assisting the decision-maker in decision-making process. Management science modeling process is one of the innovative decision making tool of the twentieth century.