FUZZY DEMAND FORECASTING. THE pDF MODEL

Nguyen Nhu Phong

Department of Industrial Systems Engineering, HCMUT

ABSTRACT

Demand Forecasting is an important function of production systems. Traditional demand forecasting models are quantitative-based models, the outputs of these models must be adjusted by experts and the adjustment is based on qualitative input factors. The PDF model is a forecasting model which has both quantitative and qualitative inputs; this model is more effective than traditional models in terms of model development and performance.

Keywords: Demand Forecast, Quantitative-Based Models, Qualitative Forecasting Models, Fuzzification, Inference Engine, De-fuzzification, Expert Rule base.

1.                   Introduction

Forecasting is the prediction or estimation of the occurrences of uncertain future events of levels of activity. Planners of any production systems are interested in timing, magnitude, and effects of future evens that influence their operations. Forecasting is the window into the future.

Forecasting can be used to predict revenues, costs, profits, prices, technological changes, …In organizational environment, forecasting most often pertains to estimate future demand. This paper is focused on demand forecasting.

Traditional demand forecasting models are quantitative models based on statistical techniques and available present data. All statistical forecasting techniques assume to some extent that forces that existed in the past will persist in the future.

Forecasts are never perfect, absolute accuracy in predicting events and activity levels is unachievable. Management should look at all available information then make decision they feel best. Any forecast should have a subjective review before they are used.

Many environmental factors influence the demand for an organization‘s products. Some of them are general business conditions, state of the economy, competitor actions and reactions, governmental legislative actions, technology innovations, marketplace trends including product life cycles, style and faction, changing consumer demand.

To improve the performance of the forecasting model, the output of the quantitative models must be adjusted by experts based on qualitative factors. This paper constructs a model that integrates the qualitative input factors into the forecasting model.

The proposed model is particularly applicable to changing business environment that is full of qualitative variables. Fuzzy logic principles are judged to be suitable to incorporate all of these uncertain and vague variables.

2.                   The PDF model

The PDF model is illustrated as in the following figure. Where:

                W: the output of the model, that is forecasting variable.

X: the quantitative input, that is the data in the past.

                Y: the qualitative inputs that is all the factors that influence the demand.

                Z: the output of the quantitative model, that is the forecast of traditional models.

                K: the coefficient of modification.

Fig 1: The PDF model.

The historical data is used in the traditional quantitative model to forecast the value of the forecasting variable, but the output is not the final result. Qualitative models will take all the qualitative factors that affect the forecasting variable, infers the value of the coefficient of modification with the support of opinion, judgment, experience, and expertise of experts. The final value of the forecasting variable is defined as follow:

W=KZ

2.1.              The quantitative forecasting model

It is important to have a good backcast in order to get a good forecast. The quantitative model is a statistical model that forecasts the future demand by a set of data from the past.

Statistical forecasting is a model-based information system, and there are many models from which to choose. These models can be Time Series Analysis Models like Last Period Demand, Arithmetic Average, and Moving Average, Exponentially Weighting Moving Average, without or with trend or seasonal corrections Regression Analysis. The choice of a statistical model is often constrained by available data. If the demand pattern shows several components like trend, seasonal, cyclic, more advanced techniques are needed.

Planners can use Box-Jenkins Model, a systematic and complicated time series analysis based model. Planners can also use models based on modern approaches like neuron network, genetic algorithm.

The performance of the using model can be accessed by forecast errors, like Mean Absolute Deviation, Mean Squared Error, Standard Deviation Of Regression, Mean Absolute Percent Error, Mean Error, Mean Percent Error.

2.2.              The qualitative forecasting model

Qualitative models will take all the qualitative factors that affect the forecasting variable, infers the value of the coefficient of modification with the support of opinion, judgment, experience, and expertise of experts. The model is shown in the following figure. Where:

                X: qualitative input factors

                V: fuzzified input, that is fuzzy sets

                W: Implied fuzzy sets

                K: the coefficient of modification.

                Z: Decision rules

Fig 2. The qualitative forecasting model

a.                   Fuzzification

The fuzzification block fuzzify all of the input and output variables of the system. The out put variable is the coefficient of modification K, the input variable are all the environmental factors influence the demand for an organization‘s products. They should be identified by experts and may be general business conditions, state of the economy, competitor actions and reactions, governmental legislative actions, technology innovations, marketplace trends including product life cycles, style and faction, changing consumer demand.

To fuzzify a variable x, we must first define the domain of value of the variable, [x_min, x_max], then on the domain we define the linguistic states of the variable. The domain of value of the variable is often identified by experts.

The PDF model defines 5 linguistic states of any variable, that is very low – VL, low – L, medium – M, high – H, very high – VH. Each state of the variable is modeled by a triangular fuzzy number as shown in the following figure.

Fig 3: Triangular fuzzy numbers

b.                   Expert rule base

The rule-base contains expert’s rules that govern the translation of input variables into output variable. If the system has n input variables and each variable has 5 linguistic states, then there are N=5ⁿ rules with the format as follow

If (x₁ = X₁ⁱ and … and x_n = X_nⁱ) then K = Kⁱ

Where:   X₁ⁱ is a state of x₁

                X_nⁱ is a state of x_n

Kⁱ is a state of output variable K

The rule base has set up a fuzzy relation between input and output variables:

R_i: (X₁ⁱ Ç… ÇX_nⁱ ) Þ Kⁱ

R= È_i=1-NR_i

c.                    Inference engine

The inference engine infers the state of the output variable at specific states of the input variables:

K₀=X₀°R

Where:   K₀ - the state of the output variable K, that is a fuzzy set

X₀ – the specific states of the input variables

° - the composition operator.

R – fuzzy relation defined by the rules identified in 2b.

There are 2 type of composition, that is max-min and max-prod. The PDF model uses the max-min composition to infer the state of output variable from the states of input variables. The PDF model also uses the min operator for logic and in the integration of input variables.

d.                   Defuzzification

After inference, the state of the output variable K₀ is a fuzzy set. This fuzzy set is defuzzied to get a crisp value, using to modify the forecast. There are many types of defuzzification, for the sake of simplification in computation, the PDF model uses the

Max-membership function defuzzification.

3.                   The PDF model development

Steps in the PDF model development:

1. Construct the quantitative forecasting model, best suitable with the available backcast data.
2. Identify all input factors that influence the demand.
3. Define the domains of value and fuzzy states of all input and output variables.
4. Construct the fuzzy relation between input and output variables by defining all rules.
5. Construct the inference engine by defining the composition method, that is max-min composition.
6. Define the method of defuzzification, that is Max-membership function defuzzification.

4.                   The PDF model implication

After construction, the PDF model is used to forecasting by the following steps:

1. Use the backcast data and the quantitative model to forecast the output variable.
2. Access the current states of all input variables, that are all fuzzy sets.
3. Infer the state of coefficient of modification, that is a fuzzy set.
4. Defuzzify the output fuzzy set to get the crisp value of the coefficient of modification.
5. Modify the forecast of the output variable by the coefficient of modification.

Future conditions of the environment are always changing, the PDF model keeps tracking with the changing conditions by making on-line adjustment as in the following steps

1. Forecasting the demand in the next period
2. Receiving the actual demand of the period
3. Calculating the forecasting error for the period
4. Making decisions on the forecasting error
   1. If the error is acceptable, keep the model for forecasting the next period
   2. If the error is not acceptable, accessing the conditions then make adjustment on the parameters of the model.

5.                   Conclusions

The paper uses fuzzy logic to construct a forecasting model which has both quantitative and qualitative inputs. With the on-line adjustment during the implication phase, the proposed model is an adaptive model which is particularly applicable to changing business environment that are full of qualitative variables.

The model has been used in some master thesises and showing the effectiveness in terms of model development and performance.

REFERENCES

1. Celia Frank, Amar Raheja, Les Sztandera, Ashish Garg “ A Fuzzy Forecasting Model for Apparel Sales”
2. Ruey-Chyn Tsaur “Hybrid Forecasting Model for Product Life Cycle”
3. Arnold F. Shapiro “Fuzzy Regression Models”
4. S. Aly, I. Vrana “Fuzzy Expert Marketing-Mix Model”