1. Define the problem; select response; suggest variables.

2. Are the proposed variables fundamental to the problem, and are they variables? Are they measurable/countable? Can one get a complete set of observations at the same time? Ordinary regression analysis does not assume that the independent variables are measured without error. However, they are conditioned on whatever errors happened to be present in the independent data set.

3. Is the problem potentially solvable?

4. Find correlation matrix and first regression runs (for a subset of data).
   Find the basic statistics, correlation matrix.
   How difficult is the problem? Compute the Variance Inflation Factor:
   VIF = 1/(1 -r_ij),     for all i, j.
   For moderate VIF's, say between 2 and 8, you might be able to come-up with a ‘good' model.

   Inspect r_ij's; one or two must be large. If all are small, perhaps the ranges of the X variables are too small.

5. Establish goal; prepare budget and time table.

   a. The final equation should have Adjusted R² = 0.8 (say).
   b. Coefficient of Variation of say; less than 0.10
   c. Number of predictors should not exceed p (say, 3), (for example for p = 3, we need at least 30 points). Even if all the usual assumptions for a regression model are satisfied, over-fitting can ruin a model's usefulness. The widely used approach is the data reduction method to deal with the cases where the number of potential predictors is large in comparison with the number of observations.
   d. All estimated coefficients must be significant at m = 0.05 (say).
   e. No pattern in the residuals

6. Are goals and budget acceptable?

B. Development of the Model:

1. Collect date; check the quality of date; plot; try models; check the regression conditions.

2. Consult experts for criticism.
   Plot new variable and examine same fitted model. Also transformed Predictor Variable may be used.

3. Are goals met?
   Have you found "the best" model?

C. Validation and Maintenance of the Model:

1. Are parameters stable over the sample space?

2. Is there a lack of fit?
   Are the coefficients reasonable? Are any obvious variables missing? Is the equation usable for control or for prediction?

3. Maintenance of the Model.
   Need to have control chart to check the model periodically by statistical techniques.

Regression Analysis Process
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You might like to use Regression Analysis with Diagnostic Tools in performing regression analysis.

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Transfer Functions Methodology

It is possible to extend regression models to represent dynamic relationships between variables via appropriate transfer functions used in the construction of feedforward and feedback control schemes. The Transfer Function Analyzer module in SCA forecasting & modeling package is a frequency spectrum analysis package designed with the engineer in mind. It applies the concept of the Fourier integral transform to an input data set to provide a frequency domain representation of the function approximated by that input data. It also presents the results in conventional engineering terms.

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Testing for and Estimation of Multiple Structural Changes

The tests for structural breaks that I have seen are designed to detect only one break in a time series. This is true whether the break point is known or estimated using iterative methods. For example, for testing any change in level of the dependent series or model specification, one may use an iterative test for detecting points in time by incorporating level shift
(0,0,0,0,...,1,1,1,1,1) variables to account for a change in intercept. Other causes are the change in variance and changes in parameters.

Further Reading:
Bai J., and P. Perron, Testing for and estimation of multiple structural changes, Econometrica, 66, 47-79, 1998.
Clements M., and D. Hendry, Forecasting Non-Stationary Economic Time Series, MIT Press, 1999.
Maddala G., and I-M. Kim, Unit Roots, Cointegration, and Structural Change, Cambridge Univ. Press, 1999. Chapter 13.
Tong H., Non-Linear Time Series: A Dynamical System Approach, Oxford University Press, 1995.

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Box-Jenkins Methodology

Introduction

Forecasting Basics: The basic idea behind self-projecting time series forecasting models is to find a mathematical formula that will approximately generate the historical patterns in a time series.

Time Series: A time series is a set of numbers that measures the status of some activity over time. It is the historical record of some activity, with measurements taken at equally spaced intervals (exception: monthly) with a consistency in the activity and the method of measurement.

Approaches to time Series Forecasting: There are two basic approaches to forecasting time series: the self-projecting time series and the cause-and-effect approach. Cause-and-effect methods attempt to forecast based on underlying series that are believed to cause the behavior of the original series. The self-projecting time series uses only the time series data of the activity to be forecast to generate forecasts. This latter approach is typically less expensive to apply and requires far less data and is useful for short, to medium-term forecasting.

Box-Jenkins Forecasting Method: The univariate version of this methodology is a self- projecting time series forecasting method. The underlying goal is to find an appropriate formula so that the residuals are as small as possible and exhibit no pattern. The model- building process involves a few steps, repeated as necessary, to end up with a specific formula that replicates the patterns in the series as closely as possible and also produces accurate forecasts.

Box-Jenkins Methodology

Box-Jenkins forecasting models are based on statistical concepts and principles and are able to model a wide spectrum of time series behavior. It has a large class of models to choose from and a systematic approach for identifying the correct model form. There are both statistical tests for verifying model validity and statistical measures of forecast uncertainty. In contrast, traditional forecasting models offer a limited number of models relative to the complex behavior of many time series, with little in the way of guidelines and statistical tests for verifying the validity of the selected model.

Data: The misuse, misunderstanding, and inaccuracy of forecasts are often the result of not appreciating the nature of the data in hand. The consistency of the data must be insured, and it must be clear what the data represents and how it was gathered or calculated. As a rule of thumb, Box-Jenkins requires at least 40 or 50 equally-spaced periods of data. The data must also be edited to deal with extreme or missing values or other distortions through the use of functions such as log or inverse to achieve stabilization.

Preliminary Model Identification Procedure: A preliminary Box-Jenkins analysis with a plot of the initial data should be run as the starting point in determining an appropriate model. The input data must be adjusted to form a stationary series, one whose values vary more or less uniformly about a fixed level over time. Apparent trends can be adjusted by having the model apply a technique of "regular differencing," a process of computing the difference between every two successive values, computing a differenced series which has overall trend behavior removed. If a single differencing does not achieve stationarity, it may be repeated, although rarely, if ever, are more than two regular differencing required. Where irregularities in the differenced series continue to be displayed, log or inverse functions can be specified to stabilize the series, such that the remaining residual plot displays values approaching zero and without any pattern. This is the error term, equivalent to pure, white noise.

Pure Random Series: On the other hand, if the initial data series displays neither trend nor seasonality, and the residual plot shows essentially zero values within a 95% confidence level and these residual values display no pattern, then there is no real-world statistical problem to solve and we go on to other things.

Model Identification Background

Basic Model: With a stationary series in place, a basic model can now be identified. Three basic models exist, AR (autoregressive), MA (moving average) and a combined ARMA in addition to the previously specified RD (regular differencing): These comprise the available tools. When regular differencing is applied, together with AR and MA, they are referred to as ARIMA, with the I indicating "integrated" and referencing the differencing procedure.

Seasonality: In addition to trend, which has now been provided for, stationary series quite commonly display seasonal behavior where a certain basic pattern tends to be repeated at regular seasonal intervals. The seasonal pattern may additionally frequently display constant change over time as well. Just as regular differencing was applied to the overall trending series, seasonal differencing (SD) is applied to seasonal non-stationarity as well. And as autoregressive and moving average tools are available with the overall series, so too, are they available for seasonal phenomena using seasonal autoregressive parameters (SAR) and seasonal moving average parameters (SMA).

Establishing Seasonality: The need for seasonal autoregression (SAR) and seasonal moving average (SMA) parameters is established by examining the autocorrelation and partial autocorrelation patterns of a stationary series at lags that are multiples of the number of periods per season. These parameters are required if the values at lags s, 2s, etc. are nonzero and display patterns associated with the theoretical patterns for such models. Seasonal differencing is indicated if the autocorrelations at the seasonal lags do not decrease rapidly.

B-J Modeling Approach to Forecasting
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Referring to the above chart know that, the variance of the errors of the underlying model must be invariant, i.e., constant. This means that the variance for each subgroup of data is the same and does not depend on the level or the point in time. If this is violated then one can remedy this by stabilizing the variance. Make sure that there are no deterministic patterns in the data. Also, one must not have any pulses or one-time unusual values. Additionally, there should be no level or step shifts. Also, no seasonal pulses should be present.

The reason for all of this is that if they do exist, then the sample autocorrelation and partial autocorrelation will seem to imply ARIMA structure. Also, the presence of these kinds of model components can obfuscate or hide structure. For example, a single outlier or pulse can create an effect where the structure is masked by the outlier.

Improved Quantitative Identification Method

Relieved Analysis Requirements: A substantially improved procedure is now available for conducting Box-Jenkins ARIMA analysis which relieves the requirement for a seasoned perspective in evaluating the sometimes ambiguous autocorrelation and partial autocorrelation residual patterns to determine an appropriate Box-Jenkins model for use in developing a forecast model.

ARMA (1, 0): The first model to be tested on the stationary series consists solely of an autoregressive term with lag 1. The autocorrelation and partial autocorrelation patterns are examined for significant autocorrelation often early terms and to see whether the residual coefficients are uncorrelated; that is the value of coefficients are zero within 95% confidence limits and without apparent pattern. When fitted values are as close as possible to the original series values, then the sum of the squared residuals will be minimized, a technique called least squares estimation. The residual mean and the mean percent error should not be significantly nonzero. Alternative models are examined comparing the progress of these factors, favoring models which use as few parameters as possible. Correlation between parameters should not be significantly large and confidence limits should not include zero. When a satisfactory model has been established, a forecast procedure is applied.

ARMA (2, 1): Absent a satisfactory ARMA (1, 0) condition with residual coefficients approximating zero, the improved model identification procedure now proceeds to examine the residual pattern when autoregressive terms with order 1 and 2 are applied together with a moving average term with an order of 1.

Subsequent Procedure: To the extent that the residual conditions described above remain unsatisfied, the Box-Jenkins analysis is continued with ARMA (n, n-1) until a satisfactory model reached. In the course of this iteration, when an autoregressive coefficient (phi) approaches zero, the model is reexamined with parameters ARMA (n-1, n-1). In like manner, whenever a moving average coefficient (theta) approaches zero, the model is similarly reduced to ARMA (n, n-2). At some point, either the autoregressive term or moving average term may fall away completely, and the examination of the stationary series is continued with only the remaining term, until the residual coefficients approach zero within the specified confidence levels.

Model Selection in B-J Approach to Forecasting
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Seasonal Analysis: In parallel with this model development cycle and in an entirely similar manner, seasonal autoregressive and moving average parameters are added or dropped in response to the presence of a seasonal or cyclical pattern in the residual terms or a parameter coefficient approaching zero.

Model Adequacy: In reviewing the Box-Jenkins output, care should be taken to insure that the parameters are uncorrelated and significant, and alternate models should be weighted for these conditions, as well as for overall correlation (R²), standard error, and zero residual.

Forecasting with the Model: The model must be used for short term and intermediate term forecasting. This can be achieved by updating it as new data becomes available in order to minimize the number of periods ahead required of the forecast.

Monitor the Accuracy of the Forecasts in Real Time: As time progresses, the accuracy of the forecasts should be closely monitored for increases in the error terms, standard error and a decrease in correlation. When the series appears to be changing over time, recalculation of the model parameters should be undertaken.

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Autoregressive Models

The autoregressive model is one of a group of linear prediction formulas that attempt to predict an output of a system based on the previous outputs and inputs, such as:

Y(t) = b₁ + b₂Y(t-1) + b₃X(t-1) + e_t,

where X(t-1) and Y(t-1) are the actual value (inputs) and the forecast (outputs), respectively. These types of regressions are often referred to as Distributed Lag Autoregressive Models, Geometric Distributed Lags, and Adaptive Models in Expectation , among others.

A model which depends only on the previous outputs of the system is called an autoregressive model (AR), while a model which depends only on the inputs to the system is called a moving average model (MA), and of course a model based on both inputs and outputs is an autoregressive-moving-average model (ARMA). Note that by definition, the AR model has only poles while the MA model has only zeros. Deriving the autoregressive model (AR) involves estimating the coefficients of the model using the method of least squared error.

Autoregressive processes as their name implies, regress on themselves. If an observation made at time (t), then, p-order, [AR(p)], autoregressive model satisfies the equation:

X(t) = F₀ + F₁X(t-1) + F₂X(t-2) + F₂X(t-3) + . . . . + F_pX(t-p) + e_t,

where e_t is a White-Noise series.

The current value of the series is a linear combination of the p most recent past values of itself plus an error term, which incorporates everything new in the series at time t that is not explained by the past values. This is like a multiple regressions model but is regressed not on independent variables, but on past values; hence the term "Autoregressive" is used.

Autocorrelation: An important guide to the properties of a time series is provided by a series of quantities called sample autocorrelation coefficients or serial correlation coefficient, which measures the correlation between observations at different distances apart. These coefficients often provide insight into the probability model which generated the data. The sample autocorrelation coefficient is similar to the ordinary correlation coefficient between two variables (x) and (y), except that it is applied to a single time series to see if successive observations are correlated.

Given (N) observations on discrete time series we can form (N - 1) pairs of observations. Regarding the first observation in each pair as one variable, and the second observation as a second variable, the correlation coefficient is called autocorrelation coefficient of order one.

Correlogram: A useful aid in interpreting a set of autocorrelation coefficients is a graph called a correlogram, and it is plotted against the lag(k); where is the autocorrelation coefficient at lag(k). A correlogram can be used to get a general understanding on the following aspects of our time series:

1. A random series: if a time series is completely random then for Large (N), will be approximately zero for all non-zero values of (k).

2. Short-term correlation: stationary series often exhibit short-term correlation characterized by a fairly large value of 2 or 3 more correlation coefficients which, while significantly greater than zero, tend to get successively smaller.

3. Non-stationary series: If a time series contains a trend, then the values of will not come to zero except for very large values of the lag.

4. Seasonal fluctuations: Common autoregressive models with seasonal fluctuations, of period s are:
   X(t) = a + b X(t-s) + e_t
   and
   X(t) = a + b X(t-s) + c X(t-2s) +e_t
   where e_t is a White-Noise series.

Partial Autocorrelation: A partial autocorrelation coefficient for order k measures the strength of correlation among pairs of entries in the time series while accounting for (i.e., removing the effects of) all autocorrelations below order k. For example, the partial autocorrelation coefficient for order k=5 is computed in such a manner that the effects of the k=1, 2, 3, and 4 partial autocorrelations have been excluded. The partial autocorrelation coefficient of any particular order is the same as the autoregression coefficient of the same order.

Fitting an Autoregressive Model: If an autoregressive model is thought to be appropriate for modeling a given time series then there are two related questions to be answered: (1) What is the order of the model? and (2) How can we estimate the parameters of the model?

The parameters of an autoregressive model can be estimated by minimizing the sum of squares residual with respect to each parameter, but to determine the order of the autoregressive model is not easy particularly when the system being modeled has a biological interpretation.

One approach is, to fit AR models of progressively higher order, to calculate the residual sum of squares for each value of p; and to plot this against p. It may then be possible to see the value of p where the curve "flattens out" and the addition of extra parameters gives little improvement in fit.

Selection Criteria: Several criteria may be specified for choosing a model format, given the simple and partial autocorrelation correlogram for a series:

1. If none of the simple autocorrelations is significantly different from zero, the series is essentially a random number or white-noise series, which is not amenable to autoregressive modeling.

2. If the simple autocorrelations decrease linearly, passing through zero to become negative, or if the simple autocorrelations exhibit a wave-like cyclical pattern, passing through zero several times, the series is not stationary; it must be differenced one or more times before it may be modeled with an autoregressive process.

3. If the simple autocorrelations exhibit seasonality; i.e., there are autocorrelation peaks every dozen or so (in monthly data) lags, the series is not stationary; it must be differenced with a gap approximately equal to the seasonal interval before further modeling.

4. If the simple autocorrelations decrease exponentially but approach zero gradually, while the partial autocorrelations are significantly non-zero through some small number of lags beyond which they are not significantly different from zero, the series should be modeled with an autoregressive process.

5. If the partial autocorrelations decrease exponentially but approach zero gradually, while the simple autocorrelations are significantly non-zero through some small number of lags beyond which they are not significantly different from zero, the series should be modeled with a moving average process.

6. If the partial and simple autocorrelations both converge upon zero for successively longer lags, but neither actually reaches zero after any particular lag, the series may be modeled by a combination of autoregressive and moving average process.

The following figures illustrate the behavior of the autocorrelations and the partial autocorrelations for AR(1) models, respectively,

AR1 Autocorrelations and Partial Autocorrelations
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Similarly, for AR(2), the behavior of the autocorrelations and the partial autocorrelations are depicted below, respectively:

AR2 Autocorrelations and Partial Autocorrelations
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Adjusting the Slope's Estimate for Length of the Time Series: The regression coefficient is biased estimate and in the case of AR(1), the bias is -(1 + 3 F₁) / n, where n is number of observations used to estimate the parameters. Clearly, for large data sets this bias is negligible.

Stationarity Condition: Note that an autoregressive process will only be stable if the parameters are within a certain range; for example, in AR(1), the slope must be within the open interval (-1, 1). Otherwise, past effects would accumulate and the successive values get ever larger (or smaller); that is, the series would not be stationary. For higher order, similar (general) restrictions on the parameter values can be satisfied.

Inevitability Condition: Without going into too much detail, there is a "duality" between a given time series and the autoregressive model representing it; that is, the equivalent time series can be generated by the model. The AR models are always invertible. However, analogous to the stationarity condition described above, there are certain conditions for the Box-Jenkins MA parameters to be invertible.

Forecasting: The estimates of the parameters are used in Forecasting to calculate new values of the series, beyond those included in the input data set and confidence intervals for those predicted values.

An Illustrative Numerical Example: The analyst at Aron Company has a time series of readings for the monthly sales to be forecasted. The data are shown in the following table:

Aron Company Monthly Sales ($1000) t X(t) t X(t) t X(t) t X(t) t X(t) 1 50.8 6 48.1 11 50.8 16 53.1 21 49.7 2 50.3 7 50.1 12 52.8 17 51.6 22 50.3 3 50.2 8 48.7 13 53.0 18 50.8 23 49.9 4 48.7 9 49.2 14 51.8 19 50.6 24 51.8 5 48.5 10 51.1 15 53.6 20 49.7 25 51.0

By constructing and studying the plot of the data one notices that the series drifts above and below the mean of about 50.6. By using the Time Series Identification Process JavaScript, a glance of the autocorrelation and the partial autocorrelation confirm that the series is indeed stationary, and a first-order (p=1) autoregressive model is a good candidate.

X(t) = F₀ + F₁X(t-1) + e_t,

where e_t is a White-Noise series.

Stationary Condition: The AR(1) is stable if the slope is within the open interval (-1, 1), that is:

| F₁| < 1

is expressed as a null hypothesis H₀ that must be tested before forecasting stage. To test this hypothesis, we must replace the t-test used in the regression analysis for testing the slope with the t-test introduced by the two economists, Dickey and Fuller. This test is coded in the Autoregressive Time Series Modeling JavaScript.

The estimated AR(1) model is:

X(t) = 14.44 + 0.715 X(t-1)

The 3-step ahead forecasts are:

X(26) = 14.44 + 0.715 X(25) = 14.44 + 0.715 (51.0) = 50.91
X(27) = 14.44 + 0.715 X(26) = 14.44 + 0.715 (50.91) = 50.84
X(28) = 14.44 + 0.715 X(27) = 14.44 + 0.715 (50.84) = 50.79

Notice: As always, it is necessary to construct the graph and compute statistics and check for stationary both in mean and variance, as well as the seasonality test. For many time-series, one must perform, differencing, data transformation, and/or deasonalitization prior to using this JavaScript.

Further Reading:
Ashenfelter , et al., Statistics and Econometrics : Methods and Applications , Wiley, 2002.

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Introduction

The five major economic sectors, as defined by economists, are agriculture, construction, mining, manufacturing and services. The first four identified sectors concern goods, which production dominated the world's economic activities. However, the fastest growing aspect of the world's advanced economies includes wholesale, retail, business, professional, education, government, health care, finance, insurance, real estate, transportation, telecommunications, etc. comprise the majority of their gross national product and employ the majority of their workers. In contrast to the production of goods, services are co-produced with the customers. Additionally, services should be developed and delivered to achieve maximum customer satisfaction at minimum cost. Indeed, services provide an ideal setting for the appropriate application of systems theory, which, as an interdisciplinary approach, can provide an integrating framework for designing, refining and operating services, as well as significantly improving their productivity.

We are attempting to 'model' what the reality is so that we can predict it. Statistical Modeling, in addition to being of central importance in statistical decision making, is critical in any endeavor, since essentially everything is a model of reality. As such, modeling has applications in such disparate fields as marketing, finance, and organizational behavior. Particularly compelling is econometric modeling, since, unlike most disciplines (such as Normative Economics), econometrics deals only with provable facts, not with beliefs and opinions.

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Modeling Financial Time Series

Time series analysis is an integral part of financial analysis. The topic is interesting and useful, with applications to the prediction of interest rates, foreign currency risk, stock market volatility, and the like. There are many varieties of econometric and multi-variate techniques. Specific examples are regression and multi-variate regression; vector auto-regressions; and co- integration regarding tests of present value models. The next section presents the underlying theory on which statistical models are predicated.

Financial Modeling: Econometric modeling is vital in finance and in financial time series analysis. Modeling is, simply put, the creation of representations of reality. It is important to be mindful that, despite the importance of the model, it is in fact only a representation of reality and not the reality itself. Accordingly, the model must adapt to reality; it is futile to attempt to adapt reality to the model. As representations, models cannot be exact. Models imply that action is taken only after careful thought and reflection. This can have major consequences in the financial realm. A key element of financial planning and financial forecasting is the ability to construct models showing the interrelatedness of financial data. Models showing correlation or causation between variables can be used to improve financial decision-making. For example, one would be more concerned about the consequences on the domestic stock market of a downturn in another economy, if it can be shown that there is a mathematically provable causative impact of that nation's economy and the domestic stock market. However, modeling is fraught with dangers. A model which heretofore was valid may lose validity due to changing conditions, thus becoming an inaccurate representation of reality and adversely affecting the ability of the decision-maker to make good decisions.

The examples of univariate and multivariate regression, vector autoregression, and present value co-integration illustrate the application of modeling, a vital dimension in managerial decision making, to econometrics, and specifically the study of financial time series. The provable nature of econometric models is impressive; rather than proffering solutions to financial problems based on intuition or convention, one can mathematically demonstrate that a model is or is not valid, or requires modification. It can also be seen that modeling is an iterative process, as the models must change continuously to reflect changing realities. The ability to do so has striking ramifications in the financial realm, where the ability of models to accurately predict financial time series is directly related to the ability of the individual or firm to profit from changes in financial scenarios.

Univariate and Multivariate Models: The use of regression analysis is widespread in examining financial time series. Some examples are the use of foreign exchange rates as optimal predictors of future spot rates; conditional variance and the risk premium in foreign exchange markets; and stock returns and volatility. A model that has been useful for this type of application is called the GARCH-M model, which incorporates computation of the mean into the GARCH (generalized autoregressive conditional heteroskedastic) model. This sounds complex and esoteric, but it only means that the serially correlated errors and the conditional variance enter the mean computation, and that the conditional variance itself depends on a vector of explanatory variables. The GARCH-M model has been further modified, a testament of finance practitioners to the necessity of adapting the model to a changing reality. For example, this model can now accommodate exponential (non-linear) functions, and it is no longer constrained by non-negativity parameters.

One application of this model is the analysis of stock returns and volatility. Traditionally, the belief has been that the variance of portfolio returns is the primary risk measure for investors. However, using extensive time series data, it has been proven that the relationship between mean returns and return variance or standard deviation are weak; hence the traditional two-parameter asset pricing models appear to be inappropriate, and mathematical proof replaces convention. Since decisions premised on the original models are necessarily sub-optimal because the original premise is flawed, it is advantageous for the finance practitioner to abandon the model in favor of one with a more accurate representation of reality.

Correct specification of a model is of paramount importance, and a battery of mis-specification testing criteria has been established. These include tests of normality, linearity, and homoskedasticity, and these can be applied to a variety of models. A simple example, which yields surprising results in the Capital Asset Pricing Model (CAPM), one of the cornerstones of elementary economics is the application of the testing criteria to data concerning companies' risk premium shows significant evidence of non-linearity, non-normality and parameter non-constancy. The CAPM was found to be applicable for only three of seventeen companies that were analyzed. This does not mean, however, that the CAPM should be summarily rejected; it still has value as a pedagogic tool, and can be used as a theoretical framework. For the econometrician or financial professional, for whom the misspecification of the model can translate into sub-optimal financial decisions, the CAPM should be supplanted by a better model, specifically one that reflects the time-varying nature of betas. The GARCH-M framework is one such model.

Multivariate linear regression models apply the same theoretical framework. The principal difference is the replacement of the dependent variable by a vector. The estimation theory is essentially a multivariate extension of that developed for the univariate, and as such can be used to test models such as the stock and volatility model and the CAPM. In the case of the CAPM, the vector introduced is excess asset returns at a designated time. One application is the computation of the CAPM with time-varying covariances. Although, in this example the null hypothesis that all intercepts are zero cannot be rejected, the misspecification problems of the univariate model still remain. Slope and intercept estimates also remain the same, since the same regression appears in each equation.

Vector Autoregression: General regression models assume that the dependent variable is a function of past values of itself and past and present values of the independent variable. The independent variable, then, is said to be weakly exogenous, since its stochastic structure contains no relevant information for estimating the parameters of interest. While the weak exogenicity of the independent variable allows efficient estimation of the parameters of interest without any reference to its own stochastic structure, problems in predicting the dependent variable may arise if "feedback" from the dependent to the independent variable develops over time. (When no such feedback exists, it is said that the dependent variable does not Granger-cause the independent variable.) Weak exogenicity coupled with Granger non-causality yields strong exogenicity which, unlike weak exogenicity, is directly testable. To perform the tests requires utilization of the Dynamic Structural Equation Model (DSEM) and the Vector Autoregressive Process (VAR). The multivariate regression model is thus extended in two directions, by allowing simultaneity between the endogenous variables in the dependent variable, and explicitly considering the process generating the exogenous variables in the dependent variable, and explicitly considering the process generating the exogenous independent variables.

Results of this testing are useful in determination of whether an independent variable is strictly exogenous or is predetermined. Strict exogenicity can be tested in DSEMs by expressing each endogenous variable as an infinite distributed lag of the exogenous variables. If the independent variable is strictly exogenous, attention can be limited to distributions conditional on the independent variable without loss of information, resulting in simplification of statistical inference. If the independent variable is strictly exogenous, it is also predetermined, meaning that all of its past and current values are independent of the current error term. While strict exogenicity is closely related to the concept of Granger non-causality, the two concepts are not equivalent and are not interchangeable.

It can be seen that this type of analysis is helpful in verifying the appropriateness of a model as well as proving that, in some cases, the process of statistical inference can be simplified without losing accuracy, thereby both strengthening the credibility of the model and increasing the efficiency of the modeling process. Vector autoregressions can be used to calculate other variations on causality, including instantaneous causality, linear dependence, and measures of feedback from the dependent to he independent and from the independent to the dependent variables. It is possible to proceed further with developing causality tests, but simulation studies which have been performed reach a consensus that the greatest combination of reliability and ease can be obtained by applying the procedures described.

Co-Integration and Present Value Modeling: Present value models are used extensively in finance to formulate models of efficient markets. In general terms, a present value model for two variables y1 and x1, states that y1 is a linear function of the present discounted value of the expected future values of x1, where the constant term, the constant discount factor, and the coefficient of proportionality are parameters that are either known or need to be estimated. Not all financial time series are non-integrated; the presence of integrated variables affects standard regression results and procedures of inference. Variables may also be co-integrated, requiring the superimposition of co-integrating vectors on the model, and resulting in circumstances under which the concept of equilibrium loses all practical implications, and spurious regressions may occur. In present value analysis, cointegration can be used to define the "theoretical spread" and to identify co-movements of variables. This is useful in constructing volatility-based tests.

One such test is stock market volatility. Assuming co-integration, second-order vector autoregressions are constructed, which suggest that dividend changes are not only highly predictable but are Granger-caused by the spread. When the assumed value of the discount rate is increased, certain restrictions can be rejected at low significance levels. This yields results showing an even more pronounced "excess volatility" than that anticipated by the present value model. It also illustrates that the model is more appropriate in situations where the discount rate is higher. The implications of applying a co-integration approach to stock market volatility testing for financial managers are significant. Of related significance is the ability to test the expectation hypotheses of interest rate term structure.

Mean absolute error is a robust measure of error. However, one may also use the sum of errors to compare the success of each forecasting model relative to a baseline, such as a random walk model, which is usually used in financial time series modeling.

Further Reading:
Franses Ph., and D. Van Dijk, Nonlinear Time Series Models in Empirical Finance, Cambridge University Press, 2000.
Taylor S., Modelling Financial Time Series, Wiley, 1986.
Tsay R., Analysis of Financial Time Series, Wiley, 2001.

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Econometrics and Time Series Models

Econometrics models are sets of simultaneous regressions models with applications to areas such as Industrial Economics, Agricultural Economics, and Corporate Strategy and Regulation. Time Series Models require a large number of observations (say over 50). Both models are used successfully for business applications ranging from micro to macro studies, including finance and endogenous growth. Other modeling approaches include structural and classical modeling such as Box-Jenkins approaches, co-integration analysis and general micro econometrics in probabilistic models; e.g., Logit, and Probit, panel data and cross sections. Econometrics is mostly studying the issue of causality; i.e. the issue of identifying a causal relation between an outcome and a set of factors that may have determined this outcome. In particular, it makes this concept operational in time series, and exogenetic modeling.

Further Readings:
Ericsson N., and J. Irons, Testing Exogeneity, Oxford University Press, 1994.
Granger C., and P. Newbold, Forecasting in Business and Economics, Academic Press, 1989.
Hamouda O., and J. Rowley, (Eds.), Time Series Models, Causality and Exogeneity, Edward Elgar Pub., 1999.

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Simultaneous Equations

The typical empirical specification in economics or finance is a single equation model that assumes that the explanatory variables are non-stochastic and independent of the error term.

This allows the model to be estimated by Least Squares Regression (LSR) analysis, such an empirical model leaves no doubt as to the assumed direction of causation; it runs directly from the explanatory variables to the dependent variable in the equation.

The LSR analysis is confined to the fitting of a single regression equation. In practice, most economic relationships interact with others in a system of simultaneous equations, and when this is the case, the application of LSR to a single relationship in isolation yields biased estimates. Therefore, it is important to show how it is possible to use LSR to obtain consistent estimates of the coefficients of a relationship.

The general structure of a simultaneous equation model consists of a series of interdependent equations with endogenous and exogenous variables. Endogenous variables are determined within the system of equations. Exogenous variables or more generally, predetermined variables, help describe the movement of endogenous variables within the system or are determined outside the model.

Applications: Simultaneous equation systems constitute a class of models where some of the economic variables are jointly determined. The typical example offered in econometrics textbooks is the supply and demand model of a good or service. The interaction of supply and demand forces jointly determine the equilibrium price and quantity of the product in the market. Due to the potential correlation of the right-hand side variables with the error term in the equations, it no longer makes sense to talk about dependent and independent variables. Instead we distinguish between endogenous variables and exogenous variables.

Simultaneous equation estimation is not limited to models of supply and demand. Numerous other applications exist such as the model of personal consumption expenditures, the impact of protectionist pressures on trade and short-term interest rate model. Simultaneous equation models have natural applications in the banking literature Due to the joint determination of risk and return and the transformation relationship between bank deposits and bank assets.

Structural and Reduced-Form Equations: Consider the following Keynesian model for the determination of aggregate income based on a consumption function and an income identity:

C = b₁ + b₂Y + e
Y = C + I,

Where:

C is aggregate consumption expenditure in time period t,
I is aggregate investment in period t,
Y is aggregate income in period t, and
e is a disturbance (error) term with mean zero and constant variance.

These equations are called Structural Equations that provide a structure for how the economy functions. The first equation is the consumption equation that relates consumption spending to income. The coefficient b₂ is the marginal propensity to consume which is useful if we can estimate it. For this model, the variables C and Y are the endogenous variables. The other variables are called the exogenous variables, such as investment I. Note that there must be as many equations as endogenous variables.

Reduced-Form Equations: On the condition that I is exogenous, derive the reduced-form equations for C and Y.

Substituting for Y in the first equation,

C = b₁ + b₂ (C + I) + e.

Hence

C = b₁ / (1 - b₂) + b₂ I / (1 - b₂) + e / (1 - b₂),

and

Y = b₁ / (1 - b₂) + I / (1 - b₂) + e / (1 - b₂),

Now we are able to utilize the LSR analysis in estimating this equation. This is permissible because investment and the error term are uncorrelated by the fact that the investment is exogenous. However, using the first equation one obtains an estimate slope b₂ /(1 - b₂), while the second equation provides another estimate of 1 /(1 - b₂). Therefore taking the ration of these reduced-form slopes will provide an estimate for b.

An Application: The following table provides consumption capital and domestic product income in US Dollars for 33 countries in 1999.

Country C I Y Country C I Y Australia 15024 4749 19461 South Korea 4596 1448 6829 Austria 19813 6787 26104 Luxembourg 26400 9767 42650 Belgium 18367 5174 24522 Malaysia 1683 873 3268 Canada 15786 4017 20085 Mexico 3359 1056 4328 China-PR 446 293 768 Netherlands 17558 4865 24086 China-HK 17067 7262 24452 New Zealand 11236 2658 13992 Denmark 25199 6947 32769 Norway 23415 9221 32933 Finland 17991 4741 24952 Pakistan 389 79 463 France 19178 4622 24587 Philippines 760 176 868 Germany 20058 5716 26219 Portugal 8579 2644 9976 Greece 9991 2460 11551 Spain 11255 3415 14052 Iceland 25294 6706 30622 Sweden 20687 4487 26866 India 291 84 385 Switzerland 27648 7815 36864 Indonesia 351 216 613 Thailand 1226 479 1997 Ireland 13045 4791 20132 UK 19743 4316 23844 Italy 16134 4075 20580 USA 26387 6540 32377 Japan 21478 7923 30124

By implementing the Regression Analysis JavaScript, two times, once for (C and I), and then for (Y and I), the estimated coefficient b₂, the marginal propensity to consume, is 0.724.

Further Readings:
Dominick, et al, Schaum's Outline of Statistics and Econometrics, McGraw-Hill, 2001.
Fair R., 1984, Specification, Estimation, and Analysis of Macroeconometric Models , Harvard University Press), 1984.

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Prediction Interval for a Random Variable

In many applied business statistics, such as forecasting, we are interested in construction of statistical interval for random variable rather than a parameter of a population distribution. For example, let X be a random variable distributed normally with estimated mean and standard deviation S, then a prediction interval for the sample mean with 100(1- a)% confidence level is:

- t . S (1 + 1/n)    ,     + t . S (1 + 1/n)

This is the range of a random variable with 100(1- a)% confidence, using t-table. Relaxing the normality condition for sample mean prediction interval requires a large sample size, say n over 30.

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Census II Method of Seasonal Analysis

Census-II is a variant of X-11. The X11 procedure provides seasonal adjustment of time series using the Census X-11 or X-11 ARIMA method. The X11 procedure is based on the US Bureau of the Census X-11 seasonal adjustment program, and it also supports the X-11 ARIMA method developed by Statistics Canada.

Further Readings:
Ladiray D., and B. Quenneville, Seasonal Adjustment with the X-11 Method, Springer-Verlag, 2001.

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Measuring for Accuracy

The most straightforward way of evaluating the accuracy of forecasts is to plot the observed values and the one-step-ahead forecasts in identifying the residual behavior over time.

The widely used statistical measures of error that can help you to identify a method or the optimum value of the parameter within a method are:

Mean absolute error: The mean absolute error (MAE) value is the average absolute error value. Closer this value is to zero the better the forecast is.

Mean squared error (MSE): Mean squared error is computed as the sum (or average) of the squared error values. This is the most commonly used lack-of-fit indicator in statistical fitting procedures. As compared to the mean absolute error value, this measure is very sensitive to any outlier; that is, unique or rare large error values will impact greatly MSE value.

Mean Relative Percentage Error (MRPE): The above measures rely on the error value without considering the magnitude of the observed values. The MRPE is computed as the average of the APE values:

Relative Absolute Percentage Error_t = 100|(X_t - F_t )/X_t|%

Durbin-Watson statistic quantifies the serial correlation of serial correlation of the errors in time series analysis and forecasting. D-W statistic is defined by:

D-W statistic = S₂ⁿ (e_j - e_j-1)² / S₁ⁿ e_j²,

where e_j is the j^th error. D-W takes values within [0, 4]. For no serial correlation, a value close to 2 is expected. With positive serial correlation, adjacent deviates tend to have the same sign; therefore D-W becomes less than 2; whereas with negative serial correlation, alternating signs of error, D-W takes values larger than 2. For a forecasting where the value of D-W is significantly different from 2, the estimates of the variances and covariances of the model's parameters can be in error, being either too large or too small.

In measuring the forecast accuracy one should first determine a loss function and hence a suitable measure of accuracy. For example, quadratic loss function implies the use of MSE. Often we have a few models to compare and we try to pick the "best". Therefore one must be careful to standardize the data and the results so that one model with large variance does not 'swamp' the other model.

An Application: The following is a set of data with some of the accuracy measures:

Periods	Observations	Predictions
1	567	597
2	620	630
3	700	700
4	720	715
5	735	725
6	819	820
7	819	820
8	830	831
9	840	840
10	999	850
Some Widely Used Accuracy Measures
Mean Absolute Errors		20.7
Mean Relative Errors (%)		2.02
Standard Deviation of Errors		50.91278
Theils Statatistic		0.80
Mc Laughlins Statatistic		320.14
Durbin-Watson Statatistic		0.9976

You may like checking your computations using Measuring for Accuracy JavaScript, and then performing some numerical experimentation for a deeper understanding of these concepts.

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The Best Age to Replace Equipment

The performance of almost everything declines with age such as machines. Although routine maintenance can keep equipment working efficiently, there comes a point when the repairs are too expensive, and it is less expensive to buy a replacement.

Numerical Example: The following table shows the cost of replacing a ($100000) machine, and the expected resale value, together with the running cost (in $1000) for each year.

Data for Decision on the Age of Replacing Equipment Age of machine 0 1 2 3 4 5 Resale value 100 50 30 15 10 5 Running cost 0 5 9 15 41 60

 

The manager must decide on the best age to replace the machine. Computational aspects are arranged in the following table:

Computational and Analysis Aspects Age of machine 1 2 3 4 5 Cumulative running cost 5 14 29 70 130 Capital cost (100-resale cost) 50 70 85 90 95 Total cost over the age 55 84 114 160 225 Average cost over the age 55 42 38 40 45

The analysis of the average cost over the age plot indicates that it follows parabola shape as expected with the least cost of $38000 annually. This corresponds to the decision of replacing the machine at the end of the third year.

Mathematical Representation: We can construct a mathematical model for the average cost as a function of its age.

We know that we want a quadratic function that best fits; we might use Quadratic Regression JavaScript to estimate its coefficients. The result is:

Average cost over the age = 3000(Age)² -20200(Age) + 71600,       for 1 £ Age£ 5.

Its derivative is: 6000(Age) - 20200 which, vanishes at Age = 101/30. That is, the best time for replacement is at the end of 3 years and 4.4 months.

You may like to use Optimal Age for Equipment Replacement JavaScript for checking your computation and perform some experiments for a deeper understanding.

Further Reading
Waters D., A Practical Introduction to Management Science, Addison-Wesley, 1998.

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Pareto Analysis: The ABC Inventory Classification

Vilfredo Pareto was an Italian economist who noted that approximately 80% of wealth was owned by only 20% of the population. Pareto analysis helps you to choose the most effective changes to make. It uses the Pareto principle that, e.g., by doing 20% of work you can generate 80% of the advantage of doing the entire job. Pareto analysis is a formal technique for finding the changes that will give the biggest benefits. It is useful where many possible courses of action are competing for your attention. To start the analysis, write out a list of the changes you could make. If you have a long list, group it into related changes. Then score the items or groups. The first change to tackle is the one that has the highest score. This one will give you the biggest benefit if you solve it. The options with the lowest scores will probably not even be worth bothering with because solving these problems may cost you more than the solutions are worth.

Application to the ABC Inventory Classification: The aim is in classifying inventory according to some measure of importance and allocating control efforts accordingly. An important aspect of this inventory control system is the degree of monitoring necessary. Demand volume and the value of items vary; therefore, inventory can be classified according to its value to determine how much control is applied.

The process of classification is as follow: Determine annual dollar usage for each item; that is, the annual demand times the cost for each item, and orders them in decreasing order. Compute the total dollar usage. Compute % dollar usage for each item. Rank the items according to their dollar % usage in three classes:
A = very important B = moderately important and C = least important.

Numerical Example: Consider a small store having nine types of products with the following cost and annual demands:

Product name P1 P2 P3 P4 P5 P6 P7 P8 P9 Cost ($100) 24 25 30 4 6 10 15 20 22 Annual demand 3 2 2 8 7 30 20 6 4

Compute the annual use of each product in terms of dollar value, and then sort the numerical results into decreasing order, as is shown in the following table. The total annual use by value is 1064.

Annual use by value 300 300 120 88 72 60 50 42 32 Product name P6 P7 P8 P9 P1 P3 P2 P5 P4 % Annual use 28 28 11 8 7 6 5 4 3 Category A B C

Working down the list in the table, determine the dollar % usage for each item. This row exhibits the behavior of the cumulative distribution function, where the change from one category to the next is determined. Rank the items according to their dollar % usage in three classes: A = very important, B = moderately important, and C = least important.

The ABC Inventory Classification JavaScript constructs an empirical cumulative distribution function (ECDF) as a measuring tool and decision procedure for the ABC inventory classification. The following figure depicts the classification based upon the ECDF of the numerical example:

ABC inventory classification
Click on the image to enlarge it and THEN print it

Even with this information, determination of the boundary between categories of items is often subjective. For our numerical example, Class A-items require very tight inventory control, which is more accurate forecasting, better record-keeping, lower inventory levels; whereas Class C-items tend to have less control.

The Impacts of the ABC Classification on Managerial Policies and Decisions

Policies and decisions that might be based on ABC classification include the following :

• Purchasing resources expended should be much higher for A-items than for C-items.
• Physical inventory control should be tighter for A-items; perhaps they belong in more secure area, with the accuracy of their records being verified more frequently.
• Forecasting A-items may warrant more care than forecasting other items.

  Better forecasting, physical control, supplier reliability, and an ultimate reduction in safety stock and inventory investment can all result from ABC analysis.

• Inventory systems require accurate records. Without them, managers cannot make precise decisions about ordering, scheduling and shipping.
• To ensure accuracy, incoming and outgoing record keeping must be good, as must be stockroom security.

  Well-organized inventory storage will have limited access, good housekeeping, and storage areas that hold fixed amounts of inventory. Bins, shelf space, and parts will be labeled accurately.

• Cycle counting: Even though an organization may have gone to substantial efforts to maintain accurate inventory records, these records must be verified through continuing audits - are known as cycle counting. Most cycle counting procedures are established so that some of each classification is counted each day. A-items will be counted frequently, perhaps once a month. B-items will be counted less frequently, perhaps once a quarter. C- items will be counted even less frequently, perhaps once every 6 months. Cycle counting also has the following advantages: Eliminating the shutdown and interruption of production necessary of annual physical inventories. Eliminating annual inventory adjustments. Providing professional personnel to audit the accuracy of inventory. Allowing the cause of the errors to be identified and remedial action to be taken. Maintaining accurate inventory records.

You might like to use the ABC Inventory Classification JavaScript also for checking your hand computation.

Further Reading
Koch R., The 80/20 Principle: The Secret to Success by Achieving More with Less, Doubleday, 1999.

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Cost/Benefit Analysis: Economic Quantity

Cost-benefit analysis is a process of finding a good answer to the following question: Given the decision-maker's assessment of costs and benefits, which choice should be recommended? The cost-benefit analysis involves the following general steps: Specify a list of all possible courses of actions. Assign a value (positive or negative) to the outcome for each action, and determine the probability of each outcome. Compute the expected outcome for each action. Take the action that has the best-expected outcome.

Economic Quantity Determination Application: The cost-benefit analysis is often used in economics for the optimal production strategy. Costs are the main concern, since every additional unit adds to total costs. After start-up production cost, the marginal cost of producing another unit is usually constant or rising as the total number of unit increases. Each additional product tends to cost as much or more than the last one. At some point, the additional costs of an extra product will outweigh the additional benefits.

The best strategy can be found diagrammatically by plotting both benefits and costs on the horizontal axis and choosing the point where the distance between them is the greatest, as shown in the following figure:

Alternatively, one may plot net profit and find the optimal quantity where it is at its maximum as depicted by the following figure:

Finally, one may plot the marginal benefit and marginal cost curves and choosing the point where they cross, as shown in the following figure:

Notice that, since "net gains" are defined as (benefits-costs), at the maximum point its derivative vanishes; therefore, the slope of the net gains function is zero at the optimal quantity. Therefore, to determine the maximum distance between two curves, the focus is on the incremental or marginal change of one curve relative to another. If the marginal benefit from producing one more unit is larger than the marginal cost, producing more is a good strategy. If the marginal benefit from producing one more product is smaller than the additional cost, producing more is a bad strategy. At the optimum point, the additional benefit will just offset the marginal cost; therefore, there is no change in net gains; i.e., the optimal quantity is where its

Marginal benefit = Marginal cost

You might like to use Quadratic Regression JavaScript to estimate the cost and the benefit functions based on a given data set.

Further Reading:
Brealey R., and S. Myers, Principles of Corporate Finance, McGraw, 2002.

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Introduction

Inventory control is concerned with minimizing the total cost of inventory. In the U.K. the term often used is stock control. The three main factors in inventory control decision-making process are:

• The cost of holding the stock; e.g., based on the interest rate.
• The cost of placing an order; e.g., for raw material stocks, or the set-up cost of production.
• The cost of shortage; i.e., what is lost if the stock is insufficient to meet all demand.

The third element is the most difficult to measure and is often handled by establishing a "service level" policy; e. g, a certain percentage of demand will be met from stock without delay.

Production systems and Inventory Control: In a production process, it is expected to obtain the minimum levels of work-in-process (WIP), possible, satisfying its demands and due dates. Working under these conditions, lead times, inventory levels and processing costs can be reduced. However, the stochastic nature of production, i.e. the arrival of demands and the uncertainty of a machine failure produce inevitable increases of WIP levels. For these and other reasons, many new heuristic production control policies have been developed, introduced and applied in order to control production in existing plants.

Production control systems are commonly divided into push and pull systems. In push systems, raw materials are introduced in the line and are pushed from the first to the last work station. Production is determined by forecasts in a production-planning center. One of the best-known push systems is material requirement planning (MRP) and manufacturing resources planning (MRPII), both developed in western countries. Meanwhile, in pull systems production is generated by actual demands. Demands work as a signal, which authorizes a station to produce. The most well-known pull systems are Just in time (JIT) and Kanban developed in Japan.

Comparing what both systems accomplish, push systems are inherently due-date driven and control release rate, observing WIP levels. Meanwhile, pull systems are inherently rate driven and control WIP level, observing throughput.

Both push and pull systems offer different advantages. Therefore, new systems have been introduced that adopts advantages of each, as a result obtaining hybrid (push-pull) control policies. The constant work in process and the two-boundary control are the best know hybrid systems with a push-pull interface. In both systems, the last station provides an authorization signal to the first one in order to start production, and internally production in pushed from one station to another until the end of the line as finished good inventory.

Simulation models are tools developed to observe systems behavior. They are used to examine different scenarios allowing evaluating the performance measure for deciding on the best policy.

Supply Chain Networks and Inventory Control: A supply chain is a network of facilities that procure raw materials, transform them into intermediate goods and then final products, and deliver the products to customers through a distribution system. To achieve an integrated supply chain management, one must have a standard description of management processes, a framework of relationships among the standard processes, standard metrics to measure process performance, management practices that produce best-in-class performance, and a standard alignment to software features and functionality, together with a users’ friendly computer-assisted tools.

Highly effective coordination, dynamic collaborative and strategic alliance relationships, and efficient supply chain networks are the key factors by which corporations survive and succeed in today's competitive marketplace. Thus all existing supply chain management models rightly focus on inventory control policies and their coordinating with delivery scheduling decisions.

Inventory control decisions are both problem and opportunity for at least three parties Production, Marking, and Accounting departments. Inventory control decision-making has an enormous impact on the productivity and performance of many organizations, because it handles the total flow of materials. Proper inventory control can minimize stock out, thereby reducing capital of an organization. It also enables an organization to purchase or produce a product in economic quantity, thus minimizing the overall cost of the product.

Further Readings:
Hopp W., and M. Spearman, Factory Physics Examines operating policies and strategic objectives within a factory.
Louis R., Integrating Kanban With Mrpii: Automating a Pull System for Enhanced Jit Inventory Management, Productivity Press Inc, 1997. It describes an automated kanban principle that integrates MRP into a powerful lean manufacturing system that substantially lowers inventory levels and significantly eliminates non-value-adding actions.

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Economic Order Quantity (EOR) and Economic Production Quantity (EPQ)

Inventories are, e.g., idle goods in storage, raw materials waiting to be used, in-process materials, finished goods, individuals. A good inventory model allows us to:

• smooth out time gap between supply and demand; e.g., supply of corn.
  
• contribute to lower production costs; e.g., produce in bulk
  
• provide a way of "storing" labor; e.g., make more now, free up labor later
  
• provide quick customer service; e.g., convenience.

For every type of inventory models, the decision maker is concerned with the main question: When should a replenishment order be placed? One may review stock levels at a fixed interval or re-order when the stock falls to a predetermined level; e. g., a fixed safety stock level.

 

Keywords, Notations Often Used for the Modeling and Analysis Tools for Inventory Control
Demand rate: x	A constant rate at which the product is withdrawn from inventory
Ordering cost: C1	It is a fixed cost of placing an order independent of the amount ordered. Set-up cost
Holding cost: C2	This cost usually includes the lost investment income caused by having the asset tied up in inventory. This is not a real cash flow, but it is an important component of the cost of inventory. If P is the unit price of the product, this component of the cost is often computed by iP, where i a percentage that includes opportunity cost, allocation cost, insurance, etc. It is a discount rate or interest rate used to compute the inventory holding cost.
Shortage cost: C3	There might be an expense for which a shortage occurs.
Backorder cost: C4	This cost includes the expense for each backordered item. It might be also an expense for each item proportional to the time the customer must wait.
Lead time: L	It is the time interval between when an order is placed and when the inventory is replenished.

The widely used deterministic and probabilistic models are presented in the following sections.

The Classical EOQ Model: This is the simplest model constructed based on the conditions that goods arrive the same day they are ordered and no shortages allowed. Clearly, one must reorder when inventory reaches 0, or considering lead time L

The following figure shows the change of the inventory level with time:

The figure shows time on the horizontal axis and inventory level on the vertical axis. We begin at time 0 with an order arriving. The amount of the order is the lot size, Q. The lot is delivered all at one time causing the inventory to shoot from 0 to Q instantaneously. Material is withdrawn from inventory at a constant demand rate, x, measured in units per time. After the inventory is depleted, the time for another order of size Q arrives, and the cycle repeats.

The inventory pattern shown in the figure is obviously an abstraction of reality in that we expect no real system to operate exactly as shown. The abstraction does provide an estimate of the optimum lot size, called the economic order quantity (EOQ), and related quantities. We consider alternatives to those assumptions later on these pages.

	Ordering		Holding
Total Cost =	C1x/Q	+	C2/(2Q)

The Optimal Ordering Quantity = Q* = (2xC₁/C₂) ^1/2, therefore,
The Optimal Reordering Cycle = T* = [2C₁/(xC₂)]^1/2

Numerical Example 1: Suppose your office uses 1200 boxes of typing paper each year. You are to determine the quantity to be ordered, and how often to order it. The data to consider are the demand rate x = 1200 boxes per year; the ordering cost C₁ = $5 per order; holding cost C₂ = $1.20 per box, per year.

The optimal ordering quantity is Q* = 100 boxes, this gives number of orders = 1200/100 = 12, i.e., 12 orders per year, or once a month.

Notice that one may incorporate the Lead Time (L), that is the time interval between when an order is placed and when the inventory is replenished.

Models with Shortages: When a customer seeks the product and finds the inventory empty, the demand can be satisfied later when the product becomes available. Often the customer receives some discount which is included in the backorder cost.

A model with backorders is illustrated in the following figure:

In this model, shortages are allowed some time before replenishment. Regarding the response of a customer to the unavailable item, the customer will accept later delivery which is called a backorder. There are two additional costs in this model; namely, the shortage cost (C₃), and the backorder cost (C₄).

Since replenishments are instantaneous, backordered items are delivered at the time of replenishment and these items do not remain in inventory. Backorders are as a negative inventory; so the minimum inventory is a negative number; therefore the difference between the minimum and maximum inventory is the lot size.

	Ordering		Holding		Shortage + Backorder
Total Cost =	xC1/Q	+	(Q-S)2C2/(2Q)	+	xSC3/Q + S2C4/(2Q)

If xC₃ ² < 2C₁C₂, then

Q* = M/(C₂C₄), and S* = M/(C₂C₄ +C₄²) - (xC₃)/(C₂ + C₄), where,
M = {xC₂C₄[2C₁(C₂ + C₄) - C₃²]}^1/2

Otherwise,

Q* = (2xC₁/C₂)^1/2, with S* = 0.

However, if shortage cost C₃ = 0, the above optimal decision values will reduce to:

Q* = [2xC₁(C₂ + C₄)/(C₂C₄)]^1/2, and , S* = [2xC₁C₂/(C₂C₄ + C₄²)]^1/2

Numerical Example 2: Given C₃ = 0, and C₄ = 2 C₂, would you choose this model? Since S* = Q*/3 under this condition, the answer is, a surprising "Yes". One third of orders must be back-ordered.

Numerical Example 3: Consider the numerical example no. 1 with shortage cost of C₄ = $2.40 per unit per year.

The optimal decision is to order Q* = 122 units, allowing shortage of level S = 81.5 units.

Production and Consumption Model: The model with finite replenishments is illustrated in the following figure:

Rather than the lot arrives instantaneously, the lot is assumed to arrive continuously at a production rate K. This situation arises when a production process feeds the inventory and the process operates at the rate K greater than the demand rate x.

The maximum inventory level never reaches Q because material is withdrawn at the same time it is being produced. Production takes place at the beginning of the cycle. At the end of production period, the inventory is drawn down at the demand rate x until it reaches 0 at the end of the cycle.

	Ordering		Holding
Total Cost =	xC1/Q	+	(K-x)QC2/(2K)

Optimal Run Size Q* = {(2C₁xK)/[C₂(K - x)] }^1/2
Run Length = Q*/K
Depletion Length = Q*(K-x)/(xK)
Optimal Cycle T* = {(2C₁)/[C₂x(1 - x/K)] }^1/2

Numerical Example 3: Suppose the demand for a certain energy saving device is x = 1800 units per year (or 6 units each day, assuming 300 working days in a year). The company can produce at an annual rate of K = 7200 units (or 24 per day). Set up cost C₁ = $300. There is an inventory holding cost C₂ = $36 per unit, per year. The problem is to find the optimal run size, Q.

Q* = 200 units per production run. The optimal production cycle is 200/7200 = 0.0278 years, that is 8 and 1/3 of a day. Number of cycle per year is 1800/200 = 9 cycles.

Production and Consumption with Shortages: Suppose shortages are permitted at a backorder cost C4 per unit, per time period. A cycle will now look like the following figure:

If we permit shortages, the peak shortage occurs when production commences at the beginning of a cycle. It can be shown that:

Optimal Production = q* = {[(2C₁x)/C₂][K/(K- x)][(C₂+C₄)/C₄]}^1/2

Period per Cycle Is: T = q/x

Optimal Inventory Is: Q* = t₂(K-x)

Optimal Shortage Is: P* = t₁(K-x);

Total Cost Is: TC = {[(C₂t₂² + C₄t₁²)(K-x)] + [(2C₁x)/K] }/ {2(t₁+t₂)},

where,

t₁ = {[2xC₁C₂]/[C₄K(K-x)(C₂+C₄)]}^1/2,

and

t₂ = {[2xC₁C₄]/[C₂K(K-x)(C₂+C₄)]}^1/2

You may like using Inventory Control Models JavaScript for checking your computation. You may also perform sensitivity analysis by means of some numerical experimentation for a deeper understanding of the managerial implications in dealing with uncertainties of the parameters of each model

Further Reading:
Zipkin P., Foundations of Inventory Management, McGraw-Hill, 2000.

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Optimal Order Quantity Discounts

The solution procedure for determination of the optimal order quantity under discounts is as follows:

• Step 1: Compute Q for the unit cost associated with each discount category.

• Step 2: For those Q that are too small to receive the discount price, adjust the order quantity upward to the nearest quantity that will receive the discount price.

• Step 3: For each order quantity determined from Steps 1 and 2, compute the total annual inventory price using the unit price associated with that quantity. The quantity that yields the lowest total annual inventory cost is the optimal order quantity.

Quantity Discount Application: Suppose the total demand for an expensive electronic machine is 200 units, with ordering cost of $2500, and holding cost of $190, together with the following discount price offering:

Order Size	Price
1-49	$1400
50-89	$1100
90+	$900

The Optimal Ordering Quantity:

Q* = (2xC1/C2) ^1/2 = [ 2(2500)(200)/190] ^1/2 = 72.5 units.

The total cost is = [(2500)(200)/72.5] + [(190)(72.5)/2] + [(1100)(200)] = $233784

The total cost for ordering quantity Q = 90 units is:

TC(90) = [(2500)(200)/90] + [(190)(90)/2] + [(900)(200)] = $233784,

this is the lowest total cost order quantity.

Therefore, should order Q = 90 units

You may like using Inventory Control Models JavaScript for checking your computation. You may also perform sensitivity analysis by means of some numerical experimentation for a deeper understanding of the managerial implications in dealing with uncertainties of the parameters in the model

Further Reading:
Zipkin P., Foundations of Inventory Management, McGraw-Hill, 2000.

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Finite Planning Horizon Inventory

We already know from our analysis of the "Simple EOQ" approach that any fixed lot size will create "leftovers" which increase total cost unnecessarily. A better approach is to order "whole periods worth" of stock. But the question is should you order one (period worth), or two, or more? As usual, it depends. At first, increasing the buy quantity saves money because order costs are reduced since fewer buys are made. Eventually, though, large order quantities will begin to increase total costs as holding costs rise.

The Silver-Meal Method: The Silver-Meal Algorithm trades-off ordering and holding costs by analyzing the problem "one buy at a time". The idea is should the first buy cover period 1, periods 1 and 2, periods 1, 2, and 3, and so forth. To answer this question, the procedure considers each potential buy quantity sequentially and calculates the "average cost per period covered" as the sum of the ordering and holding costs implied by the potential buy divided by the number of periods which would be covered by such an order. Simply put, the decision rule is: "Add the next period's demand to the current order quantity unless the average cost per period covered would not be reduced, that is, as long as the average cost per period covered by the order would be reduced by adding an additional period worth to the order, we will do so. If adding an additional period worth to the order would not reduce the average cost per period covered, then we will consider that the order size is determined, and we will begin to calculate the next order using the same procedure. We will continue one order at a time until every period has been covered with an order.

Silver-Meal Logic: Increase T, the number of periods covered by next replenishment order, until the total relevant costs per period (over the periods covered by the order) start to decrease. The Silver-Meal method is a "near optimal" heuristic which builds order quantities by taking a marginal analysis approach. Start with the first period in which an order is required. Calculate the average per-period cost of ordering for the next t periods: AC_i, i = 1, 2, ... Select the smallest i* that satisfies AC_i* < AC_i*+1

Notice that this method assumes that AC_i/i initially decreases then increases, and never decreases again as t increases, but this is not always true. Moreover, solution is myopic so it may leave only one, two, or a few periods for the final batch, even if the setup cost is high. However, Extensive numerical studies show that the results are usually within 1 or 2 percent of optimal (using mixed-integer linear programming) if horizon is not extremely short.

Finite Planning Horizon Inventory Application: Suppose the forecasted demand for a row material in a manufacturing process for the beginning of the next twelve periods is:

Period	1	2	3	4	5	6	6	8	9	10	11	12
Demand	200	150	100	50	50	100	150	200	200	250	300	250

The ordering cost is $500, the unit price is $50 and the holding cost is $1 per unit per period. The main questions are the usual questions in general inventory management, namely: What should be the order quantity? and When should the orders placed? The following table provides the detailed computations of the Silver-Meal approach with the resulting near optimal ordering strategy:

Period	Demand	Lot QTY	Holding Cost	Lot Cost	Mean Cost
First Buy
1	200	200	0	500	500
2	150	350	150	650	325
3	100	450	150+2(100)=350	850	283
4	50	500	150+200+3(50)=500	1000	250
5	50	550	150+200+150+4(50)=700	1200	240
6	100	650	15+200+150+200+5(100)=1200	1700	283
Second Buy
6	100	100	0	500	500
7	150	250	150	650	325
8	200	450	150+2(200)=550	1050	350
Third Buy
8	200	200	0	500	500
9	200	400	200	700	350
10	250	650	200 + 2(250)=700	1200	400
Fourth Buy
10	250	250	0	500	500
11	300	550	300	800	400
12	250	800	300+2(250)=800	1300	433
Fifth Buy
12	250	250	0	500	500
Solution Summary
Period	Demand	Order QTY	Holding $	ordering $	Period Cost
1	200	550	350	500	850
2	150	0	200	0	200
3	100	0	100	0	100
4	50	0	50	0	50
5	50	0	0	0	0
6	100	250	150	500	650
7	150	0	0	0	0
8	200	400	200	500	700
9	200	0	0	0	0
10	250	550	300	500	800
11	300	0	0	0	0
12	250	250	0	500	500
Total	2000	2000	1350	2500	3850

Wagner and Whitin Approach: It is a considerably more laborious procedure than Silver-Meal which is based on the principles of dynamic programming.

Mixed Integer Linear Programming: The Finite Planning Horizon Inventory decision can be formulated and solved exactly as an integer program. Zero-one integer variables are introduced to accommodate the ordering costs. Decision Variables are: quantity purchased in period i , buy variable = 1 if Qi is positive, = 0 o.w., Beginning inventory for period i, Ending inventory for period i. The objective is to minimize the total overall costs, subject to mixed-integer linear constraints.

Formulating the above application as an mixed-integer linear program, the optimal solution is:

Order 550 at the beginning of period 1
Order 450 at the beginning of period 6
Order 450 at the beginning of period 9
Order 550 at the beginning of period 11
The optimal total cost is $3750.

Conclusions: Optimal solutions trade-off ordering and holding costs across time periods based on the certainty of the demand schedule. In practice, the procedure would be re-run each month, with a new month added on the end, and the old month eliminated. Only the most immediate orders would be placed; the later orders would be held. In this sort of "rolling horizon" application, short-term look-ahead procedures like Silver-Meal typically can out-perform the "optimal" approaches, particularly if updates are made to demand forecasts within the planning horizon.

Further Reading:
Zipkin P., Foundations of Inventory Management, McGraw-Hill, 2000.

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Inventory Control with Uncertain Demand

Suppose you are selling a perishable item (e.g., flower bunches in a florist shop) having random demands X. Your decision under uncertainty is mainly the following question: How many should I order to maximize my profit?

Your profit is:

• P ´ X - (D-X) ´ L,    for any X less than D, and
  
• P ´ D,                   for any X at least equal to D

where D is the daily order, P is your unit profit, and L is the loss for any left over item.

It can be shown that the optimal ordering quantity D* with the largest expected daily profit is a function of the Empirical Cumulative Distribution Function (ECDF) = F(x). More specifically, the optimal quantity is X* where F(x) either equals or exceeds the ratio P/(P + L) for the first time.

The following numerical example illustrates the process. Given P = $20, L = $10, suppose you have taken records of the past frequency of the demand D over a period of time. Based on this information one can construct the following table.

Optimal Ordering Quantity

Demand Daily (x)	Relative Frequency	ECDF	Expected Profit
0	0.0067	0.0067	0.000
1	0.0337	0.0404	19.800
2	0.0842	0.1246	38.59
3	0.1404	0.2650	54.85
4	0.1755	0.4405	66.90
5	0.1755	0.6160	73.37
6	0.1462	0.7622	75.20
7	0.1044	0.8666	72.34
8	0.0653	0.9319	66.34
9	0.0363	0.9682	58.38
10	0.0181	0.9863	49.34
11	0.0082	0.9945	39.75
12	0.0034	0.9979	19.91
13	0.0013	0.9992	19.98
14	0.0005	0.9997	10.00
15	0.0002	1	-29.99

The critical ratio P/(P + L) = 20/30 = 0.6667, indicating D* = X* = 6 units.

To verify this decision, one may use the following recursive formula in computing:

Expected profit [D+1] = Expected profit [D] - (P + L)F(x) + P

The daily expected profit using this formula computed and recorded in the last column of the above table with the optimal daily profit is $75.20.

You may like using Single-period Inventory Analysis JavaScript to check your computation.

Further Reading:
Silver E., D. Pyke, and R. Peterson, Inventory Management and Production Planning and Scheduling, Wiley, 1998.

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Managing and Controlling Inventory

Inventory models give answers to two questions. When should an order be placed or a new lot be manufactured? And how much should be ordered or purchased?

Inventories are held for the following reasons:

• To meet anticipated customer demand with large fluctuations.
• To protect against shortages.
• To take advantage quantity discounts.
• To maintain independence of operations.
• To smooth production requirements.
• To guard against price increases.
• To take advantage of order cycles.
• To overcome the variations in delivery times.
• To guard against uncertain production schedules.
• To count for the possibility of large number of defects.
• To guard against poor forecasts of customer demand.

A Factors-Guideline for Developing a "good" Inventory System

• A system to keep track of inventory by reviewing continuously or periodically.
• A reliable forecast of demand.
• Reasonable estimates of:
  • Holding costs
  • Ordering costs
  • Shortage costs
  • Lead Time
• Interest on loans to purchase inventory or opportunity costs because of funds tied up in inventory.
• Taxes, and insurance costs.
• Ordering and setup costs.
• Costs of holding an item in inventory.
• Cost of funds tied up in inventory.
• Transportation & shipping cost.
• Receiving and inspection costs.
• Handling & storage cost.
• Accounting and auditing cost.
• Storage costs such as rent, heating, lighting, and security.
• Depreciation cost.
• Obsolescence cost.

How to Reduce the Inventory Costs?

• Cycle inventory.
• Streamline ordering/production process.
• Increase repeatability.
• Safety Stock inventory.
• Better timing of orders.
• Improve forecasts.
• Reduce supply uncertainties.
• Use capacity cushions instead.
• Anticipation inventory.
• Match production rate with demand rate.
• Use complementary products.
• Off-season promotions.
• Creative pricing.
• Pipeline inventory.
• Reduce lead time.
• More responsive suppliers.
• Decrease lot size when it affects lead times.

The ABC Classification The ABC classification system is to grouping items according to annual sales volume, in an attempt to identify the small number of items that will account for most of the sales volume and that are the most important ones to control for effective inventory management.

Types of Inventory Control Reviews: The inventory level for different products can be monitored either continuously or on a periodic basis.

• Continuous review systems: Each time a withdrawal is made from inventory, the remaining quantity of the item is reviewed to determine whether an order should be placed

• Periodic review systems: The inventory of an item is reviewed at fixed time intervals, and an order Is placed for the appropriate amount

Advantage and Disadvantage of Fixed-Period Model:

• Do not have to continuously monitor inventory levels.
• Does not require computerized inventory system.
• Low cost of maintenance.
• Higher inventory carrying cost.
• Orders placed at fixed intervals.
• Inventory brought up to target amount.
• Amounts ordered may vary.
• No continuous inventory count is needed; however there is a possibility of being out of stock between intervals.
• Useful when lead time is very short.

Cash Flow and Forecasting: Balance sheets and profit and loss statements indicate the health of your business at the end of the financial year. What they fail to show you is the timing of payments and receipts and the importance of cash flow.

Your business can survive without cash for a short while but it will need to be "liquid" to pay the bills as and when they arrive. A cash flow statement, usually constructed over the course of a year, compares your cash position at the end of the year to the position at the start, and the constant flow of money into and out of the business over the course of that year.

The amount your business owes and is owed is covered in the profit and loss statement; a cash flow statement deals only with the money circulating in the business. It is a useful tool in establishing whether your business is eating up the cash or generating the cash.

Working Capital Cycle: Cash flows in a cycle into, around and out of a business. It is the business's life blood and every manager's primary task is to help keep it flowing and to use the cash flow to generate profits. If a business is operating profitably, then it should, in theory, generate cash surpluses. If it doesn't generate surpluses, the business will eventually run out of cash and expire.

Each component of working capital, namely inventory, receivable and payable has two dimensions, time, and money. When it comes to managing working capital -- Time is money. If you can get money to move faster around the cycle, e.g. collect moneys due from debtors more quickly or reduce the amount of money tied up, e.g. reduce inventory levels relative to sales, the business will generate more cash or it will need to borrow less money to fund working capital. As a consequence, you could reduce the cost of interest or you will have additional money available to support additional sales growth. Similarly, if you can negotiate improved terms with suppliers e.g. get longer credit or an increased credit limit, you effectively create free finance to help fund future sales.

The following are some of the main factors in managing a “good” cash flow system:

• If you collect receivable (debtors) faster then you release cash from the cycle
• If you collect receivable slower, then your receivable soak up cash
• If you get better credit, in terms of duration or amount from suppliers then you increase your cash resources.
• If you sift inventory faster then you free up cash. If you move inventory slower then you consume more cash.

Further Reading:
Schaeffer H., Essentials of Cash Flow, Wiley, 2002.
Silver E., D. Pyke, and R. Peterson, Inventory Management and Production Planning and Scheduling, Wiley, 1998.
Simini J., Cash Flow Basics for Nonfinancial Managers, Wiley, 1990.

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Modeling Advertising Campaign

Introduction: A broad classification of mathematical advertising models results in models based on concept of selling with some assumed advertising/sales response functions and those based on marketing using the theory of consumer buying behavior.

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Selling Models

Selling focuses on the needs of seller. Selling models are concerned with the sellers need to convert the product into cash. One of the most well known selling models is the advertising/sales response model (ASR) that assumes the shape of the relationship between sales and advertising is known.

The Vidale and Wolfe Model: Vidale and Wolfe developed a single-equation model of sales response to advertising based on experimental studies of advertising effectiveness. This sales behavior through time relative to different levels of advertising expenditure for a firm, consistent with their empirical observation, has been developed.

The three parameters in this model are:

• The sales decay constant (l): the sales decay constant is defined as the rate at which sales of the product decrease in the absence of advertising.
• The saturation level (m): the saturation level of a product is defined as the practical limit of sales that can be captured by the product.
• The sales response constant (r): the sales response constant is defined as the addition to sales per round of advertising when sales are zero.

The fundamental assumptions in this model are as follows:

1. Sales would decrease in the absence of advertising;
2. The existence of a saturation level beyond which sales would not increase;
3. The rate of response to advertising is constant per dollar spent;
4. The advertising is assumed to be operative only on those potential customers who are not customers at the present time; i.e., advertising merely obtains new customers and does not make existing customers increase their volume of purchase;
5. The effectiveness of different media is negligible over one another;
6. On the basis of empirical evidence, it is assumed that the sales phenomenon can be represented mathematically by the relation:
   dS/dt = r A(t)[(m-S)/m - l S(t)

   where dS/dt is the instantaneous change in the rate of sales at time t, S is the rate of sales at time t, A(t) is the rate of advertising at time t, r is the sales response constant, l is the sales decay constant and m is the saturation level of sales.

This equation suggests that the change or increase in the rate of sales will be greater the higher the sales response constant; the lower the sales decay constant l, the higher the saturation level, and the higher the advertising expenditure.

The three parameters r, l, and m are constant for a given product and campaign.

The sales response, r, is assessed by measuring the increase in the rate of sales resulting from a given amount of advertising in a test area with controlled conditions. The sales decay constant l, is assessed by measuring the decline in sales in a test area when advertising is reduced to zero. The saturation level of sales, m, is assessed from market research information on the size of the total market. Note that the term (m-S)/S is the sales potential remaining in the market which can be captured by advertising campaigns.

The model can be rearranged and written as:

dS/dt + [r A(t)/m + l)] S(t) = r A(t)

The following depict a typical sales response to an advertising campaign.

A Typical Sales Response
Click on the image to enlarge it and THEN print it

A Typical Advertising Campaign
Click on the image to enlarge it and THEN print it

The advertising campaign has a constant rate A(t) =A of advertising expenditure maintained for duration T, after which A is almost zero:

             æ A          for 0 £ t £ T,
A(t) =   ç
            è 0          for t >T

While many marketing researchers have aligned the ASR approach as an established school in advertising modeling, nevertheless they readily admit the most aggravating problem is the assumption on the shape of the ASR function. Moreover, ASR models do not consider the need and motives leading to consumer behavior. It is well established that marketing managers are concerned about delivering product benefit, changing brand attitudes, and influencing consumer perceptions. Marketing management realizes that advertising plans must be based on the psychological and social forces that condition consumer behavior; that is, what goes on inside the consumer's head.

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Buying Models

Modern business firms have oriented their advertising campaigns into a fully consumer buying behavior approach rather than selling. This approach is based on the marketing wisdom: in order to sell something the marketer must know what the potential buyer wants and/or wants to hear. There has been considerable discussion in marketing literature about "consumer behavior". This discussion centers around the need for marketing to be consumer-oriented, to be concerned with the idea of satisfying the needs of the consumer by means of the product and the whole cluster of factors associated with creating, delivering, and finally consuming it. This is now possible by considering the needed technological advances such as "brain-storming".

Modeling Consumer Choice: When the modular and the decision maker come up with a good model of customer choice among discrete options, they often implement their model of customer choice. However, one might take the advantage of using multi-method object -oriented software (e.g., AnyLogic ) that the practical problem can be modeled at multiple levels of aggregation, where, e.g., the multi-nominal logit of discrete choice methods are represented by object state-chart transitions (e.g. from "aware" state to "buy" state) -- the transition is the custom probability function estimated by the discrete choice method. Multi-level objects representing subgroups easily represent nesting. Moreover, each object can have multiple state-charts.

The consumer buying behavior approach to advertising modeling presumes that advertising influences sales by altering the taste, preference and attitude of the consumer, and the firm's effort in communication that results in a purchase.

Since there are a multitude of social-psychological factors affecting buying behavior, some of them complex and unknown to the advertiser, it is preferable to consider the probabilistic version of consumer buying behavior model. Because of the diminishing effect of advertising, often an advertising pulsing policy as opposed to the constant policy may increase the effectiveness of advertising, especially on the impact of repetition in advertising.

The operational model with additional characteristics is often derived by optimal advertising strategy over a finite advertising duration campaign. The prescribed strategies are the maximizer of a discounted profit function which includes the firm's attitude toward uncertainty in sales. The needed operational issues, such as estimation of parameters and self-validating, are also recommended.

The marketing literature provides strong evidence that consumers do substitute rules of their own for information about product quality, perceived value, and price. The lower search costs associated with the rules, for example, may more than offset the monetary or quality losses. Generally, consumers tend to perceive heavily advertised brands to be of higher quality. The psychological studies have discovered that human-being is an "attitudinal being" and evaluates just about everything they come into contact with through "revision of all values". Consider the question "How do you feel abut this particular brand?" Surely, the answer depends on the degree to which you like or dislike, value or disvalue, the brand. The crux of the consumer behavior model then is that the marketer attempt to recognize consumer as an attitudinal being who constantly revises all values, even within a given segment. Once the goal-directed behavior is manifested, the consumer experiences the consequences of his or her behavior. He or she uses this experience as a source of learning in which he or she revises his or her total attitude toward the product or service. Several researchers have expressed the fact that attitude alone determines subsequent behavior.

A summary flow chart of a simple model is shown in the following figure:

Consumer Behavior
Click on the image to enlarge it and THEN print it

The structure of the decision process of a typical consumer concerning a specific brand X, contains three functional values namely attitude A(t), level of buying B(t) and communication C(t).

Nicosia's Model: The Nicosia model's dynamic state equations are described by the following two linear algebraic/differential equations:

B¢(t) = dB(t)/dt = b[A(t) - bB(t)]

A¢(t) = dA(t)/dt = a[B(t) - aA(t)] + C(t)

Where:

B(t) = the Buying behavior; i.e., purchase rate at time t.

A(t) = The consumers' Attitude toward the brand which results from some variety of complex interactions of various factors, some of which are indicated in the above Figure.

C(t) = The impact of communication (advertising campaign) made by the business firm. This may be any stimuli, a new package design or in general an advertisement of a particular brand. A successful marketing strategy is to develop product and promotional stimuli that consumers will perceive as relevant to their needs. Consumer needs are also influenced by factors such as consumer past experience and social influences. Because of the diminishing effect of advertising, we may consider C(t) to be a pulse function, as opposed to the constant advertising policy.

a, b, a, and b are the 'personality' parameters of the equations of the model. These parameters are assumed to be constant with respect to time.

The main major drawbacks of the above descriptive models are:

1) That the advertising rate is constant over time. It is clear that the return on constant advertising is diminishing with time and hence it is not related to the volume of sales; therefore further expenditures on advertising will not bring abut any substantial increase in the sales revenues. The term "advertising modeling" has been used to describe the decision process of improving sales of a product or a service. A substantial expense in marketing is advertising expenses. The effect of repetitions of a stimulus on the consumer's ability to recall the message is a major issue in learning theory. It is well established that advertising must be continuous to stop it being forgotten.

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The Advertising Pulsing Policy

The advertising pulsing policy (APP) is a policy with a high constant advertising intensity, alternating with periods with no advertising, as shown in the following figure:

Advertising Pulsing Policy
Click on the image to enlarge it and THEN print it

APP may be preferable to one of constant advertising over the campaign duration.

2) That the advertising horizon is an infinite time. This infinite horizon decreases the models' use since budget planning for advertising expenditures seldom has an infinite horizon. However, in the Nicosia's model it is not clear how to generate the sales response function when advertising is discontinued.

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Internet Advertising

It is a fact of business that in order to make money, you have to spend it first. For most business it is the spending on advertising. And for the online business, there is no shortage of options to choose from.

Most websites offer some kind of graphic or text advertising, and there are a bewildering variety of mailing lists, newsletters, and regular mailings. However, before deciding where to advertise, one must think of why advertising?

For many companies the aim of an advert is to increase sales to make more money. For others, it might be increase in profile, increasing brand awareness, and testing new pricing strategies or new markets. Therefore, it is necessary to know exactly what it is to be achieved. This determines where to advertise.

When selecting a site to advertise, the main factor is to ask how large the targeted audience is and the price to pay for. The price could a flat fee, a cost-per-click, pay per exposure, or some other arrangement including the cost of a professional designer to create and maintain the ad, and the duration of campaign.

Banner Advertising: If you have spent any time surfing the Internet, you have seen more than your fair share of banner ads. These small rectangular advertisements appear on all sorts of Web pages and vary considerably in appearance and subject matter, but they all share a basic function: if you click on them, your Internet browser will take you to the advertiser's Web site.

Over the past few years, most of us have heard about all the money being made on the Internet. This new medium of education and entertainment has revolutionized the economy and brought many people and many companies a great deal of success. But where is all this money coming from? There are a lot of ways Web sites make money, but one of the main sources of revenue is advertising. And one of the most popular forms of Internet advertising is the banner ad.

Because of its graphic element, a banner ad is somewhat similar to a traditional ad you would see in a printed publication such as a newspaper or magazine, but it has the added ability to bring a potential customer directly to the advertiser's Web site. This is something like touching a printed ad and being immediately contacted the advertiser's store! A banner ad also differs from a print ad in its dynamic capability. It stays in one place on a page, like a magazine ad, but it can present multiple images, include animation and change appearance in a number of other ways. Different measures are more important to different advertisers, but most advertisers consider all of these elements when judging the effectiveness of a banner ad. Cost per sale is the measure of how much advertising money is spent on making one sale. Advertisers use different means to calculate this, depending on the ad and the product or service. Many advertisers keep track of visitor activity using Internet cookies . This technology allows the site to combine shopping history with information about how the visitor originally came to the site.

Total Cost:
Cost per Thousand Impressions:
Number of Exposures:

Entering numerical values for any two input cells then click on
Calculate to get the numerical value for the other one.

Like print ads, banner ads come in a variety of shapes and sizes with different cost and the effectiveness. The main factors are the total cost, the cost per thousand impressions (CPM), and number of ads shown, i.e., the exposures. By entering two of these factors, the above JavaScript calculates the numerical value of the other one.

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Predicting Online Purchasing Behavior

Suppose that a consumer has decided to shop around several retail stores in an attempt to find a desired product or service. From his or her past shopping experience, the shopper may know:

• the assortment size of each store,
• the search cost per visit, and
• the price variation among the stores.

Therefore it is necessary to analyze the effects of the assortment size, the search cost, and the price variation on the market shares of existing retail stores. An element of this analysis is to consider the optimal sequence of stores and the optimal search strategy from the shopper's search in order to estimate the market share of each store in the market area. The analysis might explain:

• why shoppers visit bigger stores first,
• why they visit fewer stores if the search cost is relatively higher than the product price, and
• why they shop around more stores if the price variation among the stores is large.

There are different types of predictors to the purchasing behavior at an online store too. Often the Logit Modeling is used to predict whether or not a purchase is made during the next visit to the web site to find the best subset of predictors. The main four different categories in predicting online purchasing behavior include:

• general clickstream behavior at the level of the visit,
• more detailed clickstream information,
• customer demographics, and
• historical purchase behavior.

Although the model might includes predictors from all four categories indicating that clickstream behavior is important when determining the tendency to buy, however one must determine the contribution in predictive power of variables that were never used before in online purchasing studies. Detailed clickstream variables are the most important ones in classifying customers according to their online purchase behavior. Therefore, a good model enables e-commerce retailers to capture an elaborate list of customer information.

Link Exchanging: The problem with exchanging links is two-fold. The first, and more important one, is the fact that link exchanging does not have as strong an effect as it once had. The second problem with exchanging is the cosmetic effect it has on your website. Visitors that come to your website do not want to see a loosely collected arrangement of links to sites that may or may not be similar to your topic. They came to your website to see what you have to offer.

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Concluding Remarks

More realistic models must consider the problem of designing an optimal advertising (say, pulsing policy) for a finite advertising campaign duration. Since there are multitudes of social-psychological factors affecting purchase, some of them complex and unknown to the advertiser, the model must be constructed in a probabilistic environment. Statistical procedure for the estimation of the market parameters must be also applied. The prescribed strategy could be the maximizer of a discounted profit function. Recognizing that the marketing managers are concerned with economic and risk implications of their decision alternative, the profit function should include the decision maker's attitude toward perceived risk.

Web Advertising: Investors constantly preach the benefit of diversifying a portfolio to reduce the risk of investment fluctuations. The same strategy needs to be taken with developing your website's marketing strategy. Diversify the sources of your traffic. Becoming over-reliant on any single type of traffic sets your website up for failure if that type of traffic happens to fail for some reason.

Further Readings:
Arsham H., A Markovian model of consumer buying behavior and optimal advertising pulsing policy, Computers and Operations Research, 20(1), 35-48, 1993.
Arsham H., Consumer buying behavior and optimal advertising strategy: The quadratic profit function case, Computers and Operations Research, 15(2), 299-310, 1988.
Arsham H., A stochastic model of optimal advertising pulsing policy, Computers and Operations Research, 14(3), 231-239, 1987.
Gao Y., (Ed.), Web Systems Design and Online Consumer Behavior, Idea Group Pub., Hershey PA, 2005.
Wang Q., and Z. Wu, A duopolistic model of dynamic competitive advertising, European Journal of Operational Research, 128(1), 213-226, 2001

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Markov Chains

Several of the most powerful analytic techniques with business applications are based on the theory of Markov chains. A Markov chain is a special case of a Markov process, which itself is a special case of a random or stochastic process.

In the most general terms, a random process is a family, or ordered set of related random variables X(t) where t is an indexing parameter; usually time when we are talking about performance evaluation.

There are many kinds of random processes. Two of the most important distinguishing characteristics of a random process are: (1) its state space, or the set of values that the random variables of the process can have, and (2) the nature of the indexing parameter. We can classify random processes along each of these dimensions.

1. State Space:
   • continuous-state: X(t) can take on any value over a continuous interval or set of such intervals
   • discrete-state: X(t) has only a finite or countable number of possible values {x₀, x₁ … ,x_i,..}
     A discrete-state random process is also often called a chain.

2. Index Parameter (often it is time t):
   • discrete-time: permitted times at which changes in value may occur are finite or countable X(t) may be represented as a set {X _i}
   • continuous-state: changes may occur anywhere within a finite or infinite interval or set of such intervals

		Change in the States of the System
		Continuous	Discrete
Time	Continuous	Level of water behind a dam	Number of customers in a bank
	Discrete	Weekdays' range of temperature	Sales at the end of the day
A Classification of Stochastic Processes

The state of a continuous-time random process at a time t is the value of X(t); the state of a discrete-time process at time n is the value of X _p. A Markov chain is a discrete-state random process in which the evolution of the state of the process beginning at a time t (continuous-time chain) or n (discrete-time chain) depends only on the current state X(t) or X_p, and not how the chain reached its current state or how long it has been in that state. We consider a discrete time finite-state Markov chain {X_t, t= 0, 1, 2, } with stationary (conditional) transition probabilities:

P [X_t+1 = j | X_t = i ]

where i, and j belong to the set S.

Let P = p_ij denote the matrix of transition probabilities. The transition probabilities between t and t + 1 are noted by pⁿ _ij and the transition matrix P_n = Pⁿ

A Typical Markov Chain with Three States and
Their Estimated Transitional Probabilities

Elements of a Markov Chain: A Markov chain consists of

• A finite number of states
• A recurrent state to which the chain returns with probability
• A state which is not recurrent called a transient state.
• A possible set of closed and absorbed states
• A probabilistic transition function from state to state
• The initial state S₀ with probability distribution P₀.

Diagrammatic Representation of Transient, Closed and Absorbed States

In the above Figure, state A is an absorbing state. Once the process enters this state, it does not leave it. Similarly, the states Dl, D2, and D3 represent a closed set. Having entered Dl, the process can move to D2 or D3 but cannot make a transition to any other state. In contrast, the states Bl, B2 and B3 represent a transient set, linking the absorbing state A to the closed set D.

Two Special Markov Chains:

• The Gambler's Ruin Chain: This chain is a simple random walk on S with absorbing barriers.
• The Random Walk Chain: This chain is a simple random walk on S with reflecting barriers.

The Main Result: If limit of pⁿ _ij = p_j exists as n approaches, then the limiting or stationary distribution of the chain P = {p_j can be found by solving the following linear system of equation: P P = P

Numerical Example: The following represents a four-state Markov chain with the transition probability matrix:

P=	|.25	.20	.25	.30|
	|.20	.30	.25	.30|
	|.25	.20	.40	.10|
	|.30	.30	.10	.30|

Note that the sum of each column in this matrix is one. Any matrix with this property is called a matrix probability or a Markov matrix. We are interested in the following question:

What is the probability that the system is in the i^th state, at the n^th transitional period?

To answer this question, we first define the state vector. For a Markov chain, which has k states, the state vector for an observation period n, is a column vector defined by

x(n) =	x1
	x2
	.
	.
	xk

where x_i = probability that the system is in the i^th state at the time of observation. Note that the sum of the entries of the state vector has to be one. Any column vector x,

x =	x1
	x2
	.
	.
	xk

where x₁+ x₂+ …. +x_k = 1

[x₁,x₂,…. ,x_k] is called a probability vector.

Consider our example -- suppose the initial state vector x⁰ is:

x(0) =		1
		0
		0
		0

In the next observation period, say end of the first week, the state vector will be

x(1)=	Px(0) =		.25
			.20
			.25
			.30

At the end of 2^nd week the state vector is Px¹

x(2) =	Px(1) =	|.25	.20	.25	.30|	|.25|	=	|.2550 |
		|.20	.30	.25	.30|	|.20|	=	|.2625 |
		|.25	.20	.40	.10|	|.25|	=	|.2325 |
		|.30	.30	.10	.30|	|.30|	=	|.2500 |

Note that we can compute x² directly using x⁰ as

x⁽²⁾ = Px⁽¹⁾ = P(Px⁽⁰⁾) = P² x⁽⁰⁾

Similarly, we can find the state vector for 5^th, 10^th, 20^th, 30^th, and 50^th observation periods.

x(5)=	P5x(0) =		.2495
			.2634
			.2339
			.2532

x(10)=	P10x(0) =		.2495
			.2634
			.2339
			.2532

x(20)=	P20x(0) =		.2495
			.2634
			.2339
			.2532

x(30) =		.2495
		.2634
		.2339
		.2532

x(50) =		.2495
		.2634
		.2339
		.2532

The same limiting results can be obtained by solving the linear system of equations P P = P using this JavaScript. It suggests that the state vector approached some fixed vector, as the number of observation periods increase. This is not the case for every Markov Chain. For example, if

P =	0	1
	1	0

, and

x(0) =	1
	0

We can compute the state vectors for different observation periods:

x(1) =	|0|	, x(2) =	|1|	, x(3) =	|0|	, x(4) =	|1|	,......., x(2n) =	|1|	, and x(2n+1) =	|0|
	|1|		|0|		|1|		|0|		|0|		|1|

These computations indicate that this system oscillates and does not approach any fixed vector.

You may like using the Matrix Multiplications and Markov Chains Calculator-I JavaScript to check your computations and to perform some numerical experiment for a deeper understanding of these concepts.

Further Reading:
Taylor H., and S. Karlin, An Introduction to Stochastic Modeling, Academic Press, 1994.

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Leontief's Input-Output Model

The Leontief Input-Output Model: This model considers an economy with a number of industries. Each of these industries uses input from itself and other industries to produce a product.

In the Leontief input-output model, the economic system is assumed to have n industries with two types of demands on each industry: external demand (from outside the system) and internal demand (demand placed on one industry by another in the same system). We assume that there is no over-production, so that the sum of the internal demands plus the external demand equals the total demand for each industry. Let xi denote the i'th industry's production, ei the external demand on the ith industry, and aij the internal demand placed on the i^th industry by the j^th industry. This means that the entry aij in the technology matrix A = [aij] is the number of units of the output of industry i required to produce 1 unit of industry j's output. The total amount industry j needs from industry i is aijxj. Under the condition that the total demand is equal to the output of each industry, we will have a linear system equation to solve.

Numerical Example: An economic system is composed of three industries A, B, and C. They are related as follows:

Industry A requires the following to produce $1 of its product:
$0.10 of its own product;
$0.15 of industry B's product; and
$0.23 of industry C's product.

Industry B requires the following to produce $1 of its product:
$0.43 of industry A's product and
$0.03 of industry C's product.

Industry C requires the following to produce $1 of its product:
$0.02 of its own product.
$0.37 of industry B's product and

Sales to non-producing groups (external demands) are:
$20 000 for industry A, $30 000 for industry B, $25 000 for industry C

What production levels for the three industries balance the economy?

Solution: Write the equations that show the balancing of the production and consumption industry by industry X = DX + E:

Production	Consumption
by			by A		by B		by C		external
Industry A:	x1	=	.10x1	+	.43x2			+	20 000
Industry B:	x2	=	.15x1			+	.37x3	+	30 000
Industry C:	x3	=	.23x1	+	.03x2	+	.02x3	+	25 000

Now solve this resulting system of equations for the output productions Xi, i = 1, 2, 3.

You may like using the Solving System of Equations Applied to Matrix Inversion JavaScript to check your computations and performing some numerical experiment for a deeper understanding of these concepts.

Further Reading:
Dietzenbacher E., and M. Lahr, (Eds.), Wassily Leontief and Input-Output Economics, Cambridge University, 2003.

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Risk as a Measuring Tool and Decision Criterion

One of the fundamental aspects of economic activity is a trade in which one party provides another party something, in return for which the second party provides the first something else, i.e., the Barter Economics.

The usage of money greatly simplifies barter system of trading, thus lowering transactions costs. If a society produces 100 different goods, there are [100(99)]/2 = 4,950 different possible, "good-for-good" trades. With money, only 100 prices are needed to establish all possible trading ratios.

Many decisions involve trading money now for money in the future. Such trades fall in the domain of financial economics. In many such cases, the amount of money to be transferred in the future is uncertain. Financial economists thus deal with both risk (i.e., uncertainty) and time, which are discussed in the following two applications, respectively.

Consider two investment alternatives, Investment I and Investment II with the characteristics outlined in the following table:

- Two Investments -
Investment I			Investment II
Payoff %	Prob.		Payoff %	Prob.
1	0.25		3	0.33
7	0.50		5	0.33
12	0.25		8	0.34

Here we have to two multinomial probability functions. A multinomial is an extended binomial. However, the difference is that in a multinomial case, there are more than two possible outcomes. There are a fixed number of independent outcomes, with a given probability for each outcome.

The Expected Value (i.e., averages):

Expected Value = m = S (X_i ´ P_i),     the sum is over all i's.

Expected value is another name for the mean and (arithmetic) average.

It is an important statistic, because, your customers want to know what to “expect”, from your product/service OR as a purchaser of “raw material” for your product/service you need to know what you are buying, in other word what you expect to get:

The Variance is:

Variance = s² = S [X_i² ´ P_i] - m²,     the sum is over all i's.

The variance is not expressed in the same units as the expected value. So, the variance is hard to understand and to explain as a result of the squared term in its computation. This can be alleviated by working with the square root of the variance, which is called the Standard (i.e., having the same unit as the data have) Deviation:

Standard Deviation = s = (Variance) ^½

Both variance and standard deviation provide the same information and, therefore, one can always be obtained from the other. In other words, the process of computing standard deviation always involves computing the variance. Since standard deviation is the square root of the variance, it is always expressed in the same units as the expected value.

For the dynamic process, the Volatility as a measure for risk includes the time period over which the standard deviation is computed. The Volatility measure is defined as standard deviation divided by the square root of the time duration.

Coefficient of Variation: Coefficient of Variation (CV) is the absolute relative deviation with respect to size provided is not zero, expressed in percentage:

CV =100 |s/| %

Notice that the CV is independent from the expected value measurement. The coefficient of variation demonstrates the relationship between standard deviation and expected value, by expressing the risk as a percentage of the expected value.

You might like to use Multinomial for checking your computation and performing computer-assisted experimentation.

Performance of the Above Two Investments: To rank these two investments under the Standard Dominance Approach in Finance, first we must compute the mean and standard deviation and then analyze the results. Using the Multinomial for calculation, we notice that the Investment I has mean = 6.75% and standard deviation = 3.9%, while the second investment has mean = 5.36% and standard deviation = 2.06%. First observe that under the usual mean-variance analysis, these two investments cannot be ranked. This is because the first investment has the greater mean; it also has the greater standard deviation; therefore, the Standard Dominance Approach is not a useful tool here. We have to resort to the coefficient of variation (C.V.) as a systematic basis of comparison. The C.V. for Investment I is 57.74% and for Investment II is 38.43%. Therefore, Investment II has preference over the Investment I. Clearly, this approach can be used to rank any number of alternative investments. Notice that less variation in return on investment implies less risk.

You might like to use Performance Measures for Portfolios in check your computations, and performing some numerical experimentation.

As Another Application, consider an investment of $10000 over a 4-year period that returns T(t) an the end of year t, with R(t) being statistically independent as follow:

R(t)	Probability
$2000	0.1
$3000	0.2
$4000	0.3
$5000	0.4

Is it an attractive investment given the minimum attractive rate of return (MARR) is I =20%?

One may compute the expected return: E[R(t)] = 2000(0.1) +….= $4000

However the present worth, using the discount factor [(1+I)ⁿ -1]/[I(1+I)ⁿ] = 2.5887, n=4, for the investment is:
4000(2.5887) - 10000 = $354.80.

Not bad. However, one needs to know its associated risk. The variance of R(t) is:
Var[R(t)] = E[R(t)²] - {E[R(t)]}² = $²10⁶.
Therefore, its standard deviation is $1000.

A more appropriate measure is the variance of the present value is:
Var(PW) = S Var[R(t)]. (1+I)^-2t = 106 [0.6944+. . .. + 0.2326] = 1.7442(10⁶),
therefore, its standard deviation is $1320.68.

Are you willing to invest?

Diversification may reduce your risk: Diversifying your decision may reduce the risk without reducing the benefits you gain from the activities. For example, you may choose to buy a variety of stocks rather than just one by using the coefficient of variation ranking.

The Efficient Frontier Approach: The efficient frontier is based on of the mean-variance portfolio selection problem. The behavior of efficient frontier and it difficulty depends on correlated risk assets. It is not an easy task to extend the efficient frontier analysis to treat the continuous-time portfolio problem in particular under transaction costs for a finite planning horizon. For more information visit Optimal Business Decisions.

For some other financial economics topics visit Maths of Money: Compound Interest Analysis.

Further Reading:
Elton E., Gruber, M., Brown S., and W. Goetzman, Modern Portfolio Theory and Investment Analysis, John Wiley and Sons, Inc., New York, 2003.

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Break-even and Cost Analyses for
Planning and Control of the Business Process

A Short Summary: A firm's break-even point occurs when at a point where total revenue equals total costs.

Break-even analysis depends on the following variables:

1. Selling Price per Unit: The amount of money charged to the customer for each unit of a product or service.
2. Total Fixed Costs: The sum of all costs required to produce the first unit of a product. This amount does not vary as production increases or decreases, until new capital expenditures are needed.
3. Variable Unit Cost: Costs that vary directly with the production of one additional unit.
4. Total Variable Cost The product of expected unit sales and variable unit cost, i.e., expected unit sales times the variable unit cost.
5. Forecasted Net Profit: Total revenue minus total cost. Enter Zero (0) if you wish to find out the number of units that must be sold in order to produce a profit of zero (but will recover all associated costs)

Clearly, each time you change a parameter in Break-Even Analysis, the break-even volume changes, and so do your loss/profit profile.

Total Cost: The sum of the fixed cost and total variable cost for any given level of production, i.e., fixed cost plus total variable cost.

Total Revenue: The product of forecasted unit sales and unit price, i.e., forecasted unit sales times the unit price.

Break-Even Point: Number of units that must be sold in order to produce a profit of zero (but will recover all associated costs). In other words, the break-even point is the point at which your product stops costing you money to produce and sell, and starts to generate a profit for your company.

One may use break-even analysis to solve some other associated managerial decision problems, such as:

• setting price level and its sensitivity
• targeting the "best" values for the variable and fixed cost combinations
• determining the financial attractiveness of different strategic options for your company

The graphic method of analysis helps you in understanding the concept of the break-even point. However, the break-even point is found faster and more accurately with the following formula:

B_E = F_C / (U_P - V_C)

where:

B_E = Break-even Point, i.e., Units of production at B_E point,
F_C = Fixed Costs,
V_C = Variable Costs per Unit
U_P = Unit Price

Therefore,

Break-Even Point = Fixed Cost / (Unit Price - Variable Unit Cost)

Introduction: Break-even analyses are an important technique for the planning, management and control of business processes. In planning they facilitate an overview of the individual effects of alternative courses of action on a firm’s goals. In particular they provide a means of judging and comparing alternatives by reference to satisfying goals or critical goal optimal. Break-even analyses also furnish decision criteria in that they indicate the minimum output volumes below which satisfying levels cannot be attained.

The addition of a time-dimension to break-even analyses is also useful in some cases from the standpoint of managerial intervention. Milestones can then be set as a basis for measuring the profitability of previous activities. When separate break-even analyses are undertaken for each product or product group, weaknesses, and therefore the points at which managerial intervention should begin, become evident.

In the control of the business process the importance of break-even analysis lies in the fact that it uncovers the strengths and weaknesses of products, product groups or procedures, or of measures in general. Achieved profit can then be judged by reference to the extent to which actual output deviates from the projected break-even point. The consequential analyses of such a deviation provide information for planning.

Break-even points are the managerial points of the profitability evaluation of managerial action. The planning, management and control of output levels and sales volumes, and of the costs and contribution margins of output levels, constitute the best-known applications. The importance of preparation in break-even analyses is ultimately reinforced by the fact that the same data can be used for other planning, management and control purposes, for example, budgeting.

The applicability of the results of break-even analysis depends to a large extent upon the reliability and completeness of the input information. If the results of break-even analyses are to be adequately interpreted and used, the following matters in particular must be clearly understood: the implicitly assumed structure of the goods flow; the nature and features of the goals that are to be pursued; the structure of cost, outlay and sales revenue functions.

Costing and break-even analysis: Break-even analysis is decision-making tool. It helps managers to estimate the costs, revenues and profits associated with any level of sales. This can help to decide whether or not to go ahead with a project. Below the break-even level of output a loss will be made; above this level a profit will be made. Break-even analysis also enables managers to see the impact of changes in price, in variable and fixed costs and in sales levels on the firm’s profits.

To ascertain the level of sales required in order to break even, we need to look at how costs and revenues vary with changes in output.

Rachel Hackwood operates as a sole trader. She sells sandwiches from a small shop in the center of a busy town. The fixed costs per month, including rent of the premises and advertising total $600. The average variable cost of producing a sandwich is 50 cents and the average selling price of one sandwich is $1.50. The relationship between costs and revenues is as follows:

MONTHLY OUTPUT (SANDWICHES)	FIXED COSTS ($)	VARIABLE COSTS ($)	TOTAL COSTS ($ FC+VC)	TOTAL REVENUE ($)	PROFIT/ LOSS ($)
0	600	0	600	0	(600)
200	600	100	700	300	(400)
400	600	200	800	600	(200)
600	600	300	900	900	0
800	600	400	1,000	1,200	200
1,000	600	500	1,100	1,500	400

The loss is reduced as output rises and she breaks even at 600 sandwiches per month. Any output higher than this will generate a profit for Rachel.

To show this in a graph, plot the total costs and total revenue. It is also normal to show the fixed cost. The horizontal axis measures the level of output. At a certain level of output, the total cost and total revenue curves will intersect. This highlights the break-even level of output.

The level of break even will depend on the fixed costs, the variable cost per unit and the selling price. The higher the fixed costs, the more the units will have to be sold to break even. The higher the selling price, the fewer units need to be sold.

For some industries, such as the pharmaceutical industry, break even may be at quite high levels of output. Once the new drug has been developed the actual production costs will be low, however, high volumes are needed to cover high initial research and development costs. This is one reason why patents are needed in this industry. The airline and telecommunications industries also have high fixed costs and need high volumes of customers to begin to make profits. In industries where the fixed costs are relatively small and the contribution on each unit is quite high, break-even output will be much lower.

Uses and limitations of break-even for decision making: The simple break-even model helps managers analyze the effects of changes in different variables. A manager can easily identify the impact on the break even level of output and the change in profit or loss at the existing output.

However, simple break-even analysis also makes simplifying assumptions; for example, it assumes that the variable cost per unit is constant. In reality this is likely to change with changes in output. As a firm expands, for example, it may be able to buy materials in bulk and benefit from purchasing economies of scale. Conversely, as output rises a firm may have to pay higher overtime wages to persuade workers to work longer hours. In either case, the variable costs per unit are unlikely to stay constant.

Another simplifying assumption of the model is that fixed costs are assumed to remain fixed at all levels of output. In fact, once a certain level of output is reached a firm will have to spend more money on expansion. More machinery will have to be purchased and larger premises may be required, this means that the fixed costs are likely to stepped-function.

To be effective, break eve charts must e combined with the manager’s own judgment. Will a particular output really be sold at this price? How will competitors react to change in price or output levels? What is likely to happen to costs in the future? The right decision can only be made if the underlying assumptions of the model are relevant and the manager balances the numerical findings with his or her own experience.

Can a firm reduce its break-even output? Not surprisingly, firms will be eager to reduce their break even level of output, as this means they have to sell less to become profitable. To reduce the break even level of output a firm must do one or more of the following:

• Increase the selling price
• Reduce the level of fixed costs
• Reduce the variable unit cost

Should a firm accept an order at below cost price? Once a firm is producing output higher than the break even level then the firm will make a profit for that time period. Provided the output is sold at the standard selling price, and then any extra units sold will add to this profit. Each additional unit sold will increase profit by an amount equal to the contribution per unit. The firm, providing it has necessary capacity and working capital, as those factors will increase profit, might welcome any extra orders in this situation.

Consider if a customer asks to buy additional units but is only willing to pay a price below the unit cost. Intuitively we would probably reject this order on the grounds that selling output at below cost price will reduce the firm’s total profits. In fact, rejecting this deal as loss making might be a mistake, depending on the level of sales.

	($)	($)
Sales Revenue (200 x $150) Materials Labor Other direct costs Indirect overheads Profit	80,000 80,000 20,000 70,000	300,000       250,000 50,000

An order is received from a new customer who wants 300 units but would only be willing to pay $100 for each unit. From the costing data in the table above, we can calculate the average cost of each unit to be $250,000/2,000 units = $125. Therefore, it would appear that accepting the order would mean selling the firm would lose $25 on each unit sold.

The order would, however, in fact add to the firm’s profits. The reason for this is that the indirect costs are fixed over the range of output 0-2500 units. The only costs that would increase would be the direct cost of production, i.e. labor, materials and other direct costs. The direct cost of each unit can be found by dividing the total for direct costs by the level of output. For example, the material cost for 2,000 units is $80,000. This means that the material cost for each unit would be $80,000/2,000 = $40. If we repeat this for labor and other direct costs then the cost of production an extra unit would be as follows:

	DIRECT COST PER UNIT ($)
Materials	40
Labour	40
Other direct costs	10
Marginal Costs	90

Each extra unit sold would, therefore, generate an extra $10 contribution (selling price – direct costs). Hence, accepting the order would actually add to the overall profits for the firm by $3,000*(300*$10 contribution). Providing the selling price exceeds the additional cost of making the product, and then this contribution on each unit will add to profits.

Other issues concerned with accepting the order: It will also help the firm to utilize any spare capacity that is currently lying idle. For example, if a firm is renting a factory, then this will represent an indirect cost for the firm. It does not matter how much of the factory is used, the rent will remain the same.

By accepting this order the firm may also generate sales with new customers or, via word-of-mouth, with other customers. The firm will have to decide whether the attractions of extra orders and higher sales outweigh the fact that these sales are at a lower selling price than normal. It will want to avoid having too many of its sales at this discounted price, as this lower price may start to be seen as normal. Customers already paying the higher price may be unhappy and demand to be allowed to buy at this lower price.

Although the lower price is above the marginal cost of production, it may be that the firm does not cover its indirect and direct costs if too many are sold at the low price. Tough the contribution sold on these discounted units is positive; sales still have to be high enough to allow for enough unit contributions to cover the indirect costs.

Orders at Below Cost Price
Click on the image to enlarge it and THEN print it

Buying in products: Increasing profit can be achieved either by increasing the selling price, which depends on the impact on sales, or reducing costs can increase profits. One possible way to reduce costs for a firm that uses manufactured goods would be if an alternative supplier could be found who can manufacture and sell products (or part of the products, such as components) for a lower price than the present costs of the firm producing these for it self. If this is the case then the firm will have a choice of whether to continue making the products or to buy them in from a supplier.

Considerations: When making this decision a firm would probably consider the possible impact on its workforce. If production is being reduced there is likely to be a reduction in the size of the workforce needed. Unless the firm can retrain the workers for other functions within the firm, such as sales, redundancies are likely to occur. This could lead to industrial action or reduction in productivity as seeing co-workers their jobs may demotivate employees.

The firm will also have to ensure that the supplier of the product is reliable. If they are located some distance away then the lead-time for delivery will become an important factor. Problems with delivery could lead to production bottlenecks, whereby overall production is halted or orders cannot be met due to unreliable suppliers. This is a particular problem if the firm is adopting just-in-time (JIT) production techniques.

The quality of the products will also have to be monitored closely. Depending on the size of the order, the firm may be able to demand their own specifications for the order. On the other hand, if the firm is only a small customer of the supplier, it may have to accept the supplier’s own specifications.

If the firm does decide to buy in components or products from another supplier, it may close down all or part of the production facilities, unless alternative uses can be found, such as producing goods for other firms. If closures do take place this will save the firm fixed costs in the long-term, although the firm may be committed to paying some of these for the next few months. For example, rent or insurance may be payable annually without rebate if the service is no longer required.

Contribution and full costing: When costing, a firm can use either contribution (marginal) costing, whereby the fixed costs are kept separate, or it can apportion overheads and use full costing. If the firm uses full costing then it has to decide how the overheads are to be apportioned or allocated to the different cost centers.

Methods of allocating indirect costs: One of the easiest ways to allocate indirect costs is to split the overheads equally between the different cost centers. However, although easier to decide, splitting the indirect cost equally may not be as fair as it initially appears.

Methods of allocating indirect costs: Chase Ltd. produces office furniture. It has decided to classify its different products as profit centers. The direct costs incurred in the production of each product are as follows:

	COMPUTER WORKSTATION	SWIVEL CHAIR	STANDARD DESK
Material costs	$20	$15	$10
Labor Costs	$25	$8	$12
Packaging and finishing	$5	$7	$3
TOTAL DIRECT COSTS	$50	$30	$25

Along with the direct costs of production there are also indirect costs that are not specifically related to the production procedure. These total $90,000. Further data relating to Chase Ltd. is as follows:

	COMPUTER WORKSTATION	SWIVEL CHAIR	STANDARD DESK
Annual Output	5,000	3,000	4,000
Selling price	$75	$45	$35

We can produce a costing statement that highlights the costs and revenues that arise out of each profit center:

	COMPUTER WORKSTATION ($)	SWIVEL CHAIR ($)	STANDARD DESK ($)
Sales Revenue Materials Labor Packaging ang finishing Total direct costs Contribution	375,000 100,000 125,000 25,000 250,000 125,000	135,000 45,000 24,000 21,000 90,000 45,000	140,000 40,000 48,000 12,000 100,000 40,000

If a firm wishes to work out the profit made by each profit center then the overheads will have to be allocated to each one. In the example below, overheads are allocated equally:

	COMPUTER WORKSTATION ($)	SWIVEL CHAIR ($)	STANDARD DESK ($)
Sales Revenue Materials Labor Packaging and finishing Indirect costs Total costs Profit	375,000 100,000 125,000 25,000 30,000 280,000 95,000	135,000 45,000 24,000 21,000 30,000 120,000 15,000	140,000 40,000 48,000 12,000 30,000 130,000 10,000

It is worth noting that the firm’s overall profit should not be any different whether it uses contribution of full costing. All that changes is how it deals with the costs-either apportioning them out to the cost or profit centers for full costing or deducting them in total from the total contribution of the centers for contribution costing. If the indirect costs are allocated, the decision about how to allocate them will affect the profit or loss of each profit center, but it will not affect the overall profit of the firm.

Allocation rules: Allocating overheads equally is the simplest and quicker means of apportioning indirect costs, but many managers do use other allocation rules. In some cases they also use different allocation rules for different types of indirect costs-this is known as absorption costing. Although these do not attempt to allocate the indirect costs accurately (in the sense that indirect costs cannot clearly be allocated to different cost centers), they attempt to take account of relevant factors that might affect the extent to which different cost centers incur the indirect costs. For example, overall heating costs might be allocated according to the floor space of different departments.

Typical Allocation Rules include:

• Typical indirect costs are connected with the staff of the firm, and then allocating overheads on the basis of labor costs may be suitable. Example of staff costs would include canteen expenses or the costs associated with running the human resources department.
• For manufacturing firms, the basis of allocating indirect costs may be related to the materials costs incurred by each cost center. This will depend on the costs centers within the organization
• If a firm is operating in an industrial sector using expensive equipment, then the overheads may be allocated on the basis of the value of machinery in each cost center. This is because maintenance, training and insurance costs may be related to the value of machinery in a loose way.

In some ways these rules are no more or less accurate than dividing their indirect costs equally although they may appear to be intuitively appealing and in some sense feel fairer. Consequences of unfair overhead allocation: We can rationalize over the reason chosen for the basis of overhead allocation; however, we must realize that no method is perfect. Costs being apportioned require a method to be chosen independently, precisely because there is no direct link between the cost and the cost center. The method chosen can have unfortunate effects on the organization as a whole. If the firm uses departments as cost centers then it is possible that using absorption costing could lead to resentment by staff. This can be illustrated through the following example.

Hopkinson Ltd. has decided to allocate fixed overheads using labor costs as the basis of allocation. Fixed overheads for the organization total $360,000 and will be allocated on the basis of labor costs (i.e. in the ratio 2:3:4) between the three branches.

	A ($)	B ($)	C ($)
Sales Revenue Labor Costs Materials Costs Other Direct Costs	165,000 40,000 20,000 10,000	240,000 60,000 30,000 10,000	300,000 80,000 40,000 10,000
Fixed overheads Profit/loss	80,000 15,000	120,000 20,000	160,000 10,000

Allocating overheads in this way gives the result that branch B generates the highest profit and branch C is the least profitable. The staff at branch C may be labeled as poor performers. This could lead to demotivation, rivalry between branches and lower productivity. Staff at branch C may also be worried that promotions or bonuses may not be available to them due to rating lowest out of three branches. However, this result is arrived at only because the high fixed overheads were allocated in these ways. If we ignored the fixed costs and considered contribution only, the following results occur:

	A ($)	B ($)	C ($)
Sales Revenue Labor Costs Materials Costs	165,000 40,000 20,000	240,000 60,000 30,000	300,000 80,000 40,000
Other direct costs Contribution	10,000 95,000	10,000 140,000	10,000 170,000

Based on contribution costing, branch C provides the biggest input into earning money for the firm.

The problems that can occur when allocating overheads can lead arguments between managers over how they should be divided up. To boost their particular division’s performance, managers will eager to change a method that shifts some of their indirect costs onto another division.

In some ways, however, it does not matter what rules are used to allocate indirect costs. Whichever rule is used is inaccurate (by definition indirect costs cannot be clearly be associated with a particular cost center) but the actual process of allocating overheads makes everyone aware of their importance and of the need to monitor and control them. Furthermore, provided the rules are not changed over time, managers will be able to analyze the trend profit figures for different departments, products or regions. A significant increase in indirect costs will decrease the profits of all business units to some degree, regardless of how these costs are allocated. If the indirect costs continue to rise, all the managers will be able to notice this trend in their accounts.

If we use the full costing method of allocating indirect overheads then we can illustrate how this information may be used to make a strategic decision in terms of closing down an unprofitable business.

In the following question, we will look at the costing data for Beynon’s Ltd., as small family chain of bakeries. The chain is owned and managed as a family concern, with the father, James Beynon, has been convinced of the merits of segmental reporting. He is worried because his youngest son, who he considers to be inexperienced in retail management, runs the branch. Consider the following breakdown of costs:

	HIGHFIELDS ($)	BRWONDALE ($)	NORTON ($)
Sales Revenue Staffing costs Supplies Branch running Marketing Central admin.	22,000 7,000 5,000 1,000 2,000 4,000	17,000 8,000 4,000 1,000 2,000 4,000	26,000 9,000 6,000 1,000 2,000 4,000
Other direct costs Contribution	19,000 3,000	19,000 (2,000)	22,000 4,000

The marketing and central administration costs incorporate many of the overall costs associated with running the bakery chain. They are indirect and not related to any one branch in particular. These have been allocated equally across all three branches, as it seemed to be the fairest method of cost allocation.

The data in the above appears to confirm the father’s belief that in the long-term interest of the firm, he may have to close down the Browndale branch and concentrate his efforts on the other two branches. If we use contribution costing, however, we see a different picture:

	HIGHFIELDS ($)	BRWONDALE ($)	NORTON ($)
Sales Revenue Staffing costs Supplies Branch running	22,000 7,000 5,000 1,000	17,000 8,000 4,000 1,000	26,000 9,000 6,000 1,000
Total costs Profit (loss)	19,000 3,000	19,000 (2,000)	22,000 4,000

As we can see, all three branches make a positive contribution to the overall profits. The reason why the father wished to close down the branch was that it appeared to be making a loss. However, it is quite the reverse; if the branch was closed then, the positive contribution from the branch would be lost and overall profits would fall. This is because the indirect costs of production do not vary with output and, therefore, closure of a section of the firm would not lead to immediate savings. This may mean that closing the branch would be a mistake on financial grounds.

This mistake is made due to a misunderstanding of nature of cost behavior. If the branch is closed then the only costs that would be saved are the costs directly related to the running of the branch: the staffing costs, the supplies and the branch running costs. The costs are indirect in nature, in this example the marketing and central administration costs, would still have to be paid as they are unaffected by output. For this decision to be made, we should use contribution as a guide for deciding whether or not to close a branch.

The Beynon’s Ltd. example highlighted that contribution is a guide to keeping a branch open that, if we used full costing, could make a loss. This can also be applied to the production of certain product lines, or the cost effectiveness of departments. On financial grounds, contribution is therefore, a better guide in making decisions.

	Total ($)
Overall Contribution Indirect Costs Profit	23,000 18,000 5,000

Continuing production even if the contribution is negative: It is possible that a section of a firm, be it a product line or branch, is kept open even though on financial grounds that particular section is making a negative contribution to the overall profit levels of organization. The reason for this is that closing down a section of a business is likely to lead a firm shedding labor that becomes surplus. The workers employed in that section may no longer be required.

If alternative employment cannot be found within the firm then these workers may be redundant. This could impose redundancy costs upon the firm. It is likely that trade unions will be involved that may oppose any redundancies. This could lead to industrial action by workers in other sections of the firm. It may also lead to bad publicity in the media, which may affect the level of sales and profits. In this situation, a business may let natural wastage occur in staff involved, rather than make job cuts, or it may simply decide to keep the section going. Even if there is industrial unrest, the effect of closure on overall morale within the firm could be very important. It is likely that the remaining employees will be demotivated on seeing c0-workers being made redundant. This could lead to unrest, and declining productivity.

In the case of a loss-making product, a firm may decide to keep this in production if it has been recently launched. In the early years of product life cycle, sales are likely to be lower than they are expected to be in later years and, as a result, the contribution may be negative. Sales will hopefully eventually rise and the revenues arising from sales will eventually outweigh the costs of running this new product.

Complementary products: A loss-making product may also be kept in production because the firm produces complementary products. In this situation a firm may be willing to incur negative contribution in order to maintain or even boost the sales of its other products. Examples of complementary products include:

• Pottery firms – dinner plates, saucers and cups
• Textile firms – bed sheets, pillowcases and duvet covers

An Example A firm is producing garden furniture, selling parasols, tables and chairs. These form the basis of different cost centers for the firm as they are produced in different sections. The firm has produced the following contribution costing statement:

	PARASOLS	TABLES	CHAIRS
Sales Revenue Labor Costs Material Costs Other direct costs Contribution	7,000 2,000 2,000 1,000 5,000 2,000	5,000 1,000 500 2,000 3,500 1,500	4,000 1,000 2,000 1,500 4,500 (500)

As you can see from the data in table 5.13, the chairs are making a negative contribution and would appear to be lowering the overall profits for the firm. Closing down production of the chairs would appear to lead to higher profits. The profits may be boosted further if the production of the chair producing facility saved some of the indirect costs.

It is important to consider the impact on the sales of other products. With a firm selling garden equipment is likely that the three separate products will be purchased together as they form part of a matching set. If the production of one of these complementary products is halted, then it is likely to adversely affect the sales of the other products. This could mean that discontinuing the production of a product with a negative contribution leads to lower overall profits.

You may like using the Break-Even Analysis and Costing Analysis JavaScript for performing some sensitivity analysis on the parameters for investigation of their impacts on your decision making.

Further Reading:
Schweitzer M., E. Trossmann, and G. Lawson, Break-Even Analyses: Basic Model, Variants, Extensions, Wiley, 1991.

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Modeling the Bidding Process in Competitive Markets

Due to deregulation in most market such as the electrical power markets, the cost minimization utilities used by electric utilities are being replaced by bidding algorithms. Every firm is trying to maximize their profit subject to the price determined by suppliers, consumers and other participants. Finding an optimized bidding policy in a competitive electricity market has become one of the main issues in electricity deregulation. There are many factors that can affect the behavior of market participants, such as the size of players, market prices, technical constraints, inter-temporal linkages, etc. Several of these factors are purely technical and the others are strictly economical. Thus there is a need to develop a methodology combining both issues in a structured way. Daily electricity markets can be classified according to the market power that one or more players can exercise: monopolistic, oligopolistic, or perfectly competitive.

The most competitive oligopolistic models can be categorized as follows: Nash-Cournot models, Bertrand models, Supply function equilibrium models, Quantity leadership models, and Price leadership models. Each one of these models uses different strategic variables, such as price and quantity, producing results that are sometimes close to a monopoly and other times close to perfect competition.

Nash-Cournot models have been widely studied to model competitive markets. However, these models are based on certain assumptions, such as fixing the quantity offered by the competitors finding the equilibrium if all players hold this assumption.

Further Reading:
Varian H.R., Microeconomics Analysis, Norton, New York, 1992.

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Product’s Life Cycle Analysis and Forecasting

The stage in a product's life cycle conventionally, divided into four stages as depicted in the following figure:

Product’s Life Cycle
Click on the image to enlarge it and THEN print it

Design and Introduction: This stage mainly concerns the development of a new product, from the time is was initially conceptualized to the point it is introduced on the market. The enterprise having first an innovative idea will often have a period of monopoly until competitors start to copy and/or improve the product (unless a patent is involved).

      Characteristics:

• cost high, very expensive
• no sales profit, all losses
• sales volume low

      Type of Decisions:

• amount of development effort
• product design
• business strategies

• optimal facility size
• marketing strategy including distribution and pricing

      Related Forecasting Techniques:

• Delphi method
• historical analysis of comparable products
• input-output analysis
• panel consensus

• consumer survey
• market tests

Growth and Competitive Turbulence: If the new product is successful (many are not), sales will start to grow and new competitors will enter the market, slowly eroding the market share of the innovative firm.

      Characteristics:

• costs reduced due to economies of scale
• sales volume increases significantly
• profitability

      Type of Decisions:

• facilities expansion
• marketing strategies
• production planning

      Related Forecasting Techniques:

• statistical techniques for identifying turning points
• market surveys
• intention-to-buy survey

Maturity: At this stage, the product has been standardized, is widely accepted on the market and its distribution is well established.

      Characteristics:

• costs are very low
• sales volume peaks
• prices tend to drop due to the proliferation of competing products
• very profitable

      Type of Decisions:

• promotions, special pricing
• production planning
• inventory control and analysis

      Related Forecasting Techniques:

• time series analysis
• causal and econometric methods
• market survey
• life cycle analysis

Decline or Extinction: As the product is becoming obsolete, eventually, the product will be retired, event that marks the end of its life cycle.

      Characteristics:

• sales decline
• prices drop
• profits decline

Forecasting the Turning Points: To be able to forecast a major change in growth that is about to occur allows managers to develop plans without the pressure of having to immediately react to unforeseen changes. For example, the turning point is when growth will go from positive to negative. It is these turning points that help managers develop plans early.

Consider the Mexican economy, since it is directly related to US economy, a dramatic changes in US economic climate can lead to a major turning point in Mexican economy, with some lagged-time (i.e., delay).

Similarly, your time series might be compared to key national economic data to identify leading indicators that can give you advance warning -- before changes occur in consumer-buying behavior. Currently, the U.S. government publishes data for over ten leading indicators that change direction before general changes in the economy.

They do not want to be taken by surprise and ruined. They are anxious to learn in time when the turning points will come because they plan to arrange their business activities early enough so as not to be hurt by, or even to profit from.

Further Readings:
Ross Sh., An Elementary Introduction to Mathematical Finance: Options and other Topics, Cambridge University Press, 2002. It presents the Black-Scholes theory of options as well as introducing such topics in finance as the time value of money, mean variance analysis, optimal portfolio selection, and the capital assets pricing model.
Ulrich K., and S. Eppinger, Product Design and Development, McGraw-Hill, 2003.
Urban G., and J. Hauser, Design and Marketing Of New Products, Prentice Hall, 1993.
Zellner A., Statistics, Econometrics and Forecasting, Cambridge University Press, 2004.

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Learning and The Learning Curve

Introduction: The concept of the learning curve was introduced to the aircraft industry in 1936 when T. P. Wright published an article in the February 1936 Journal of the Aeronautical Science. Wright described a basic theory for obtaining cost estimates based on repetitive production of airplane assemblies. Since then, learning curves (also known as progress functions) have been applied to all types of work from simple tasks to complex jobs like manufacturing.

The theory of learning recognizes that repetition of the same operation results in less time or effort expended on that operation. Its underlying concept is that, for example the direct labor man-hours necessary to complete a unit of production will decrease by a constant percentage each time the production quantity is doubled. If the rate of improvement is 20% between doubled quantities, then the learning percent would be 80% (100-20=80). While the learning curve emphasizes time, it can be easily extended to cost as well.

Psychology of Learning: Based on the theory of learning it is easier to learn things that are related to what you already know. The likelihood that new information will be retained is related to how much previous learning there is that provides "hooks" on which to hang the new information. In other words, to provide new connectivity in the learner's neural mental network. For example, it is a component of my teaching style to provide a preview of the course contents and review of necessary topics form prerequisites courses (if any) during the first couple of class meeting, before teaching them to course topics in detail. Clearly, change in one's mental model happens more readily when you have a mental model similar to the one you're trying to learn; and that it will be easier to change a mental model after you become more consciously aware.

A steep learning curve is often referred to indicate that something is difficult to learn. In practice, a curve of the amount learned against the number of trials (in experiments) or over time (in reality) is just the opposite: if something is difficult, the line rises slowly or shallowly. So the steep curve refers to the demands of the task rather than a description of the process.

The following figure is of a fairly typical of a learning curve. It depicts the fact that the learning curve does not proceed smoothly: the plateaus and troughs are normal features of the process.

A Typical Learning Curve
Click on the image to enlarge it and THEN print it

The goal is to make the "valley of despair" as Shallow and as Narrow as possible.

To make it narrow, you must give plenty of training, and follow it up with continuing floor support, help desk support, and other forms of just-in-time support so that people can quickly get back to the point of competence. If they stay in the valley of despair for too long, they will lose hope and hate the new software and the people who made them switch.

Valley of Despair Characteristics:

• Who's dumb idea was this?
• I hate this
• I could do better the old way
• I cannot get my work done

Success Characteristic:

• How did I get along without this?

To make it as shallow as possible, minimize the number of things you try to teach people at once. Build gradually, and only add more to learn once people have developed a level of competence with the basic things.

In the acquisition of skills, a major issue is the reliability of the performance. Any novice can get it right occasionally, but it is consistency which counts, and the progress of learning is often assessed on this basis.

Need to train workers in new method based on the facts that the longer a person performs a task, the quicker it takes him/her:

1. Learn-on-the-job approach:
   • learn wrong method
   • bother other operators, lower production
   • anxiety
2. Simple written instructions: only good for very simple jobs
3. Pictorial instructions: "good pictures worth 1000 words"
4. Videotapes: dynamic rather than static
5. Physical training:
   • real equipment or simulators, valid
   • does not interrupt production
   • monitor performance
   • simulate emergencies

Factors that affect human learning:

1. Job complexity - long cycle length, more training, amount of uncertainty in movements, more C-type motions, simultaneous motions
2. Individual capabilities- age, rate of learning declines in older age, amount of prior training, physical capabilities, active, good circulation of oxygen to brain

Because of the differences between individuals, their innate ability, their age, or their previous useful experience then each turner will have his/her own distinctive learning curve. Some possible, contrasting, curves are shown in the following figure:

An Individuals Differences Classification
Click on the image to enlarge it and THEN print it

Individual C is a very slow learner but he improves little by little. Individual B is a quick learner and reaches his full capacity earlier than individuals A or C. But, although A is a slow learner, he eventually becomes more skilled than B.

Measuring and Explaining Learning Effects of Modeling: It is already accepted that modeling triggers learning, this is to say the modeler's mental model changes as effect of the activity "modeling". In "systems thinking" it also includes the way people approach decision situations by studying attitude changes model building.

Modeling the Learning Curve: Learning curves are all about ongoing improvement. Managers and researchers noticed, in field after field, from aerospace to mining to manufacturing to writing, that stable processes improve year after year rather than remain the same. Learning curves describe these patterns of long-term improvement. Learning curves help answer the following questions.

• How fast can you improve to a specific productivity level?
• What are the limitations to improvement?
• Are aggressive goals achievable?

The learning curve was adapted from the historical observation that individuals who perform repetitive tasks exhibit an improvement in performance as the task is repeated a number of times.

With proper instruction and repetition, workers learn to perform their jobs more efficiently and effectively and consequently, e.g., the direct labor hours per unit of a product are reduced. This learning effect could have resulted from better work methods, tools, product design, or supervision, as well as from an individual’s learning the task.

A Family of Learning Curves Funtions: Of the dozens of mathematic concepts of learning curves, the four most important equations are:

• Log-Linear: y(t) = k t^b
• Stanford-B: y(t) = k (t + c)^b
• DeJong: y(t) = a + k t^b
• S-Curve: y(t) = a + k (t + c)^b

The Log-Linear equation is the simplest and most common equation and it applies to a wide variety of processes. The Stanford-B equation is used to model processes where experience carries over from one production run to another, so workers start out more productively than the asymtote predicts. The Stanford-B equation has been used to model airframe production and mining. The DeJong equation is used to model processes where a portion of the process cannot improve. The DeJong equation is often used in factories where the assembly line ultimately limits improvement. The S-Curve equation combines the Stanford-B and DeJong equations to model processes where both experience carries over from one production run to the next and a portion of the process cannot improve.

An Application: Because of the learning effect, the time required to perform a task is reduced when the task is repeated. Applying this principle, the time required to perform a task will decrease at a declining rate as cumulative number of repetitions increase. This reduction in time follows the function: y(t) = k t ^b, where b = log(r)/log (2), i.e., 2^b = r, and r is the learning rate, a lower rate implies faster learning, a positive number less than 1, and k is a constant.

For example, industrial engineers have observed that the learning rate ranges from 70% to 95% in the manufacturing industry. An r = 80% learning curve denotes a 20% reduction in the time with each doubling of repetitions. An r = 100% curve would imply no improvement at all. For an r = 80% learning curve, b = log(0.8)/log(2) = -0.3219.

Numerical Example: Consider the first (number if cycles) and the third (their cycle times) columns for the following data set:

# Cycles	Log # Cycles	Cycle Time	Log Cycle Time
1	0	12.00	1.08
2	0.301	9.60	0.982
4	0.602	7.68	0.885
8	0.903	6.14	0.788
16	1.204	4.92	0.692
32	1.505	3.93	0.594

To estimate y = k t^b one must use linear regression on the logarithmic scales, i.e., log y = log(k) + b log(t) using a data set, and then computing r = 2^b. Using the Regression Analysis JavaScript, for the above data, we obtain:

b = Slope = -0.32, y-Intercept = log(k) = 1.08

log y = log(k) + b log(t)

b = -0.32

k = 10^1.08 = 12

y(t) = 12 t ^-0.32

r = 2^b = 2^-0.32 = 80%

Conclusions: As expected while number of cycles doubles, cycle time decreases by a constant %, that is, the result is a 20% decrease or 80% learning ratio or 80% learning curve with a mathematical model y(t) = 12 t ^-0.32

Further Readings:
Dilworth J., Production and Operations Management: Manufacturing and Non-manufacturing, Random House Business Division, 2003.
Krajewski L., and L. Ritzman, Operations Management: Strategy and Analysis, Addison-Wesley Publishing Company, 2004.

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Economics and Financial Ratios and Price Indices

Economics and finance use and analysis ratios for comparison and as a measuring tool and decision process for the purpose of evaluating certain aspects of company's operations. The following are among the widely used ratios:

Liquidity Ratios: Liquidity ratios measure a firm's ability to meet its current obligations, for example:

• Acid Test or Quick Ratio = (Cash + Marketable Securities + Accounts Receivable) / Current Liabilities

• Cash Ratio = (Cash Equivalents + Marketable Securities) / Current Liabilities

Profitability Ratios: Profitability ratios profitability ratios measure management's ability to control expenses and to earn a return on the resources committed to the business, for example:

• Operating Income Margin = Operating Income / Net Sales

• Gross Profit Margin = Gross Profit / Net Sales

Leverage Ratios: Leverage ratios measure the degree of protection of suppliers of long-term funds and can also aid in judging a firm's ability to raise additional debt and its capacity to pay its liabilities on time, for example:

• Total Debts to Assets = Total Liabilities / Total Assets

• Capitalization Ratio= Long-Term Debt /(Long-Term Debt + Owners' Equity)

Efficiency: Efficiency activity or turnover ratios provide information about management's ability to control expenses and to earn a return on the resources committed to the business, for example:

• Cash Turnover = Net Sales / Cash

• Inventory Turnover = Cost of Goods Sold / Average Inventory

Price Indices

Index numbers are used when one is trying to compare series of numbers of vastly different size. It is a way to standardize the measurement of numbers so that they are directly comparable.

The simplest and widely used measure of inflation is the Consumer Price Index (CPI). To compute the price index, the cost of the market basket in any period is divided by the cost of the market basket in the base period, and the result is multiplied by 100.

If you want to forecast the economic future, you can do so without knowing anything about how the economy works. Further, your forecasts may turn out to be as good as those of professional economists. The key to your success will be the Leading Indicators, an index of items that generally swing up or down before the economy as a whole does.

	Period 1		Period 2
Items	q1 = Quantity	p1 = Price	q1 = Quantity	p1 = Price
Apples	10	$.20	8	$.25
Oranges	9	$.25	11	$.21

we found that using period 1 quantity, the price index in period 2 is

($4.39/$4.25) x 100 = 103.29

Using period 2 quantities, the price index in period 2 is

($4.31/$4.35) x 100 = 99.08

A better price index could be found by taking the geometric mean of the two. To find the geometric mean, multiply the two together and then take the square root. The result is called a Fisher Index.

In USA, since January 1999, the geometric mean formula has been used to calculate most basic indexes within the Comsumer Price Indeces (CPI); in other words, the prices within most item categories (e.g., apples) are averaged using a geometric mean formula. This improvement moves the CPI somewhat closer to a cost-of-living measure, as the geometric mean formula allows for a modest amount of consumer substitution as relative prices within item categories change.

Notice that, since the geometric mean formula is used only to average prices within item categories, it does not account for consumer substitution taking place between item categories. For example, if the price of pork increases compared to those of other meats, shoppers might shift their purchases away from pork to beef, poultry, or fish. The CPI formula does not reflect this type of consumer response to changing relative prices.

The following are some of useful and widely used price indices:

Geometric Mean Index:

G_j = [Õ (p_i/p₁) ] ^{(V1 / SV_i}),     i = 1, 2,..., j,

where p_i is the price per unit in period i and q_i is the quantity produced in period n, and V _i = p_i q_i the value of the i units, and subscripts 1 indicate the reference period of n periods.

pi
qi
The Geometric Mean Index =

Replace the numerical example values with your own pairs
of data, and then click on the Calculate button.

Harmonic Mean Index:

H_j = (SV_i) / [(SV_i . p_j )/ p_i ],     i = 1, 2,..., j,

where p_i is the price per unit in period i, q_n is the quantity produced in period i, and V _i = p_i q_i the value of the i units, and subscripts 1 indicate the reference period of n periods.

Laspeyres' Index:

L_j = S (p_iq₁) S (p₁q₁),      the first sum is over     i = 1, 2,..., j while the second one is over all     i = 1, 2,..., n,

where p_i is the price per unit in period i and q_i is the quantity produced in period I, and subscripts 1 indicate the reference period of n periods.

Paasche's Index:

P_j = S (p_iq_i) S (p₁q_i),     the first sum is over     i = 1, 2,..., j while the second one is over all     i = 1, 2,...., n,

where p_i is the price per unit in period i and q_i is the quantity produced in period I, and subscripts 1 indicate the reference period of n periods.

Fisher Index:

F_j = [Laspeyres' index_j . Paasche's index_j]^1/2

For more economics and financial ratios and indices, visit the Index Numbers and Ratios with Applications site

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JavaScript E-labs Learning Objects

This section is a part of the JavaScript E-labs learning technologies for decision making.

Each JavaScript in this collection is deigned to assisting you in performing numerical experimentation, for at least a couple of hours as students do in, e.g. Physics labs. These leaning objects are your statistics e-labs. These serve as learning tools for a deeper understanding of the fundamental statistical concepts and techniques, by asking "what-if" questions.

Decision Making in Economics and Finance:

• ABC Inventory Classification -- an analysis of a range of items, such as finished products or customers into three "importance" categories: A, B, and C as a basis for a control scheme. This pageconstructs an empirical cumulative distribution function (ECDF) as a measuring tool and decision procedure for the ABC inventory classification.
• Inventory Control Models -- Given the costs of holding stock, placing an order, and running short of stock, this page optimizes decision parameters (order point, order quantity, etc.) using four models: Classical, Shortages Permitted , Production & Consumption, Production & Consumption with Shortages.
• Optimal Age for Replacement -- Given yearly figures for resale value and running costs, this page calculates the replacement optimal age and average cost.
• Single-period Inventory Analysis -- computes the optimal inventory level over a single cycle, from up-to-28 pairs of (number of possible item to sell, and their associated non-zero probabilities), together with the "not sold unit batch cost", and the "net profit of a batch sold".

Probabilistic Modeling:

• Bayes' Revised Probability -- computes the posterior probabilities to "sharpen" your uncertainties by incorporating an expert judgement's reliability matrix with your prior probability vector. Can accommodate up to nine states of nature.
• Decision Making Under Uncertainty -- Enter up-to-6x6 payoff matrix of decision alternatives (choices) by states of nature, along with a coefficient of optimism; the page will calculate Action & Payoff for Pessimism, Optimism, Middle-of-the-Road, Minimize Regret, and Insufficient Reason.
• Determination of Utility Function -- Takes two monetary values and their known utility, and calculates the utility of another amount, under two different strategies: certain & uncertain.
• Making Risky Decisions -- Enter up-to-6x6 payoff matrix of decision alternatives (choices) by states of nature, along with subjective estimates of occurrence probability for each states of nature; the page will calculate action & payoff (expected, and for most likely event), min expected regret , return of perfect information, value of perfect information, and efficiency.
• Multinomial Distributions -- for up to 36 probabilities and associated outcomes, calculates expected value, variance, SD, and CV.
• Revising the Mean and the Variance -- to combine subjectivity and evidence-based estimates. Takes up to 14 pairs of means and variances; calculates combined estimates of mean, variance, and CV.
• Subjective Assessment of Estimates -- (relative precision as a measuring tool for inaccuracy assessment among estimates), tests the claim that at least one estimate is away from the parameter by more than r times (i.e., a relative precision), where r is a subjective positive number less than one. Takes up-to-10 sample estimates, and a subjective relative precision (r<1); the page indicates whether at least one measurement is unacceptable.
• Subjectivity in Hypothesis Testing -- Takes the profit/loss measure of various correct or incorrect conclusions regarding the hypothesis, along with probabilities of Type I and II errors (alpha & beta), total sampling cost, and subjective estimate of probability that null hypothesis is true; returns the expected net profit.

Time Series Analysis and Forecasting

• Autoregressive Time Series -- tools for the identification, estimation, and forecasting based on autoregressive order obtained from a time series.
• Detecting Trend & Autocrrelation in Time Series -- Given a set of numbers, this page tests for trend by Sign Test, and for autocorrelation by Durbin-Watson test.
• Plot of a Time Series -- generates a graph of a time series with up to 144 points.
• Seasonal Index -- Calculates a set of seasonal index values from a set of values forming a time series. A related page performs a Test for Seasonality on the index values.
• Forecasting by Smoothing -- Given a set of numbers forming a time series, this page estimates the next number, using Moving Avg & Exponential Smoothing, Weighted Moving Avg, and Double & Triple Exponential Smoothing, &and Holt's method
• Runs Test for Random Fluctuations -- in a time series.
• Test for Stationary Time Series -- Given a set of numbers forming a time series, this page calculates the mean & variance of the first & second half, and calculates one-lag-apart & two-lag-apart autocorrelations. A related page: Time Series' Statistics calculates these statistics, and also the overall mean & variance, and the first & second partial autocorrelations.

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The Copyright Statement: The fair use, according to the 1996 Fair Use Guidelines for Educational Multimedia, of materials presented on this Web site is permitted for non-commercial and classroom purposes only.
This site may be mirrored intact (including these notices), on any server with public access. All files are available at http://home.ubalt.edu/ntsbarsh/Business-stat for mirroring.

Kindly e-mail me your comments, suggestions, and concerns. Thank you.

Professor Hossein Arsham   

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This site was launched on 2/18/1994, and its intellectual materials have been thoroughly revised on a yearly basis. The current version is the 9^th Edition. All external links are checked once a month.

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EOF: Ó 1994-2015.

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