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Items with irregular and sporadic demand profiles are frequently tackled by companies, given the necessity of proposing wider and wider mix, along with characteristics of specific market fields (i.e., when spare parts are manufactured and sold). Furthermore, a new company entering into the market is featured by irregular customers' orders. Hence, consistent efforts are spent with the aim of correctly forecasting and managing irregular and sporadic products demand. In this paper, the problem of correctly forecasting customers' orders is analyzed by empirically comparing existing forecasting techniques. The case of items with irregular demand profiles, coupled with seasonality and trend components, is investigated. Specifically, forecasting methods (i.e., Holt-Winters approach and (S)ARIMA) available for items with seasonality and trend components are empirically analyzed and tested in the case of data coming from the industrial field and characterized by intermittence. Hence, in the conclusions section, well-performing approaches are addressed.

In the recent competitive environment, where manufacturing and service companies operate in unstable sectors, managing irregular and sporadic demand patterns represents an increasingly frequent and complex issue. Startup productions, multiechelon supply chains or spare parts production, and selling are some examples of market fields characterized by intermittent demand profiles.

The complexity of dealing with these kinds of demand patterns lies in finding the best tradeoff between negative effects related with high storage levels, such as high amount of space and resources for keeping large warehouse areas, high holding costs, as well as high risks and cost due to items obsolescence, and negative effects related with low storage levels, such as lost demand and customers.

Therefore, when treating irregular and sporadic demand patterns, two relevant issues are discussed:

demand forecasting in the future periods,

utilization of demand forecasting obtained for managing stocks. Hence issues related with when and how much it costs to create stocks for satisfying the forecasted customers’ orders are faced.

The focus of this paper is on the first issue, which represents an unforgettable prerequisite for the second one and could become a needful competitive leverage for companies.

Croston [

Successively, modifications of Croston’s approach are proposed. Johnston and Boylan [

Recently, original contributions are published. Willemain et al. [

Since Syntetos et al. [

Forecasting methods for demand patterns with seasonality and trend components are proposed by several authors. The focus in the following brief overview is on two techniques: the Holt-Winters (HW) approach [

HW is an extrapolative technique that isolates level, trend, and seasonal components of a time series regardless of the nature of the time series data being collected. It presents both a multiplicative and an additive version. ARIMA model is an integrated technique of auto-regressive (AR) models and moving average models, capable of finding a fitting function in an iterative way through the Box-Jenkins procedure. In the following, the acronym (S)ARIMA is used in place of ARIMA to specify the possibility that seasonality is present in analysed time series. For a more detailed discussion on the application of (S)ARIMA models see the studies by Jarrett in [

Several authors investigate HW and (S)ARIMA performances in a wide variety of operating conditions [

Hence, the purpose of the paper is to present results obtained by comparing HW and (S)ARIMA forecasting performances when applied to a set of real-life sporadic and irregular time series with seasonality and trend components.

The paper is organized as described in the sequel. A synthesis of the methodology implemented in the experimental analysis and then the first step of the project, concerning the collection and preliminary analysis of data, are presented, respectively, in Sections

As aforementioned, the aim of the paper is the comparison between the Holt-Winters exponential smoothing with (S)ARIMA in cases of erratic and sporadic demands with seasonal and trend components.

Holt-Winters method manages three components of demand per period: a level component, a trend component, and a seasonal component. Each of them is estimated by exponential smoothing and successively opportunely weighted and combined in order to predict demand. In particular, two versions of HW components compositions are available: additive and multiplicative, but the presence of time periods with null demand does not allow the multiplicative version to be applied in this paper [

While HW is simply applied by commercial softwares (in the sequel EViews 5 is adopted), which allow the solution to be achieved without any intervention of the user, (S)ARIMA models require the optimal definition of a set of parameters in accordance with results obtained in fitting tests. In Figure

The Box-Jenkins procedure.

The flow diagram depicted in Figure

Graphics, statistical indexes, and correlograms support this phase. In particular, the more useful indicators are the distribution of the global autocorrelation coefficients (

In order to chose the best (S)ARIMA model avoiding overfitting occurrence (the necessity of testing too many parameters), many techniques and methods have been suggested to add mathematical rigor to the search process, including Akaike’s criterion [

After the identification of the (S)ARIMA model, a diagnostic check must be conducted (see Figure

Several accuracy measures are presented in literature for comparing the performances of forecasting methods. For a more detailed discussion about them, see the study by Makridakis in [

Define

Accuracy measures adopted in this paper are described in (

It represents the Mean Absolute Deviation (MAD) divided by the average demand size. This index, by describing the incidence of the mean absolute forecasting error on the mean existing demand, allows the evaluation of forecasting approaches performance on time series with very different mean values, as introduced by Regattieri et al. [

It represents the arithmetic Mean of the Sum of the Squares of the forecasting Errors (MSE) divided by the average demand size. Low values of MSE/

It represents the Mean Error (ME) divided by the average demand size. This index permits to define the estimation behavior of forecasting methods and specifically to understand whether an overestimation or an underestimation of the prediction data occurs:

Specifically, in this proposed paper, the goodness of forecasting is evaluated by computing MAD/

Twelve data series describing demand of twelve spare parts have been collected from real industrial applications, each of them composed by 36 time periods. In detail, the data are related to several high-value minuteria products, like precision screws and small spare parts for transmission and hydraulic units. They are all characterized by erratic patterns because of their variability in demand sizes while some data series are sporadic too due to the presence of time periods in which demand does not occur. Therefore, two coefficients are computed (CV and ADI) in accordance with definitions reported by Willemain et al. In [

In the following sections forecast will concern five and twelve periods ahead; thus CV and ADI are calculated both for 31 time periods, from period 1 to period 31 (

The analysis based on the Box-Jenkins procedure (see Figure

Table

Data collection and preliminary analysis.

Series | Group | ddp | Seasonality | Trend | ||||
---|---|---|---|---|---|---|---|---|

s1 | 1 | 1.80 | 1.35 | 1.79 | 1.26 | reject | x | |

s2 | 1.54 | 1.11 | 1.45 | 1.09 | Geometric ( | x | ||

s3 | 1.30 | 1.19 | 1.35 | 1.09 | Neg. Binomial (1, | x | ||

s4 | 1.09 | 1.48 | 1.01 | 1.33 | Neg. Binomial (3, | x | ||

s5 | 1.22 | 1.19 | 1.21 | 1.14 | Neg. Binomial (2, | x | ||

s6 | 1.25 | 1 | 1.30 | 1 | reject | x | ||

s7 | 2.40 | 1.29 | 2.36 | 1.33 | reject | x | ||

s8 | 2 | 1.11 | 1.41 | 1.33 | 1.50 | reject | x | x |

s9 | 2.38 | 1.35 | 1.77 | 1.14 | Neg. Binomial (2, | x | x | |

s10 | 1.63 | 1.41 | 1.69 | 1.41 | Neg. Binomial (1, | x | x | |

s11 | 1.28 | 1 | 1.42 | 1 | Neg. Binomial (1, | x | x | |

s12 | 1.30 | 1 | 1.34 | 1 | Neg. Binomial (2, | x | x |

The software AutoFit has been used. It evaluates all the best fitting distribution functions in descending order of ranking. Sometimes it does not find any fitting function. Such cases are traced in column ddp of Table

Time series are grouped into two sets: Group 1 and Group 2. The former includes series from s1 to s7, mainly characterized by seasonal component, while the latter includes series from s8 to s12, with both seasonal and consistent trend components.

The implementation of the Holt-Winters method does not require any discretional intervention of the user because the commercial software adopted in this paper finds the best smoothing parameters in an iterative way. For this reason, the main portion of this section is focused on (S)ARIMA models identification.

In order to reduce the number of tested (S)ARIMA models, the differentiation orders (

Since a (S)ARIMA model is uniquely defined by seven parameters

Note that each (S)ARIMA model could generate negative forecasted values, which are practically inconsistent. Thus, a null demand is imposed every time a negative value is forecasted.

In order to compare the different (S)ARIMA models, their forecasting performance is evaluated in terms of MAD/

Selected (S)ARIMA models for 5 and 12 time periods ahead.

Series | Group | 5 time periods ahead | 12 time periods ahead |
---|---|---|---|

s1 | 1 | (3,1,1) _{4} | (2,1,3) _{4} |

s2 | (2,1,2) _{4} | (1,1,3) _{4} | |

s3 | (3,1,1) _{4} | (3,1,2) _{4} | |

s4 | (1,1,2) _{4} | (1,1,3) _{4} | |

s5 | (2,1,2) _{4} | (3,1,2) _{4} | |

s6 | (1,1,2) _{4} | (3,1,2) _{4} | |

s7 | (2,1,2) _{4} | (1,1,1) _{4} | |

s8 | 2 | (3,1,2) _{4} | (1,1,1) _{4} |

s9 | (2,1,2) _{4} | (2,1,2) _{4} | |

s10 | (1,1,2) _{4} | (2,1,3) _{4} | |

s11 | (2,1,2) _{4} | (1,1,1) _{4} | |

s12 | (2,1,2) _{4} | (3,1,2) _{4} |

Subsequently, selected (S)ARIMA models are also compared with those of HW in terms of MSE/

The results obtained by HW method are directly comparable with those achieved through selected (S)ARIMA model.

In Table

Comparison between (S)ARIMA and HW based on MAD/

Series | Group |
MAD/ |
MAD/ | ||

(S)ARIMA | HW | (S)ARIMA | HW | ||

s1 | 1 | 9.3% | |||

s2 | 14.2% | ||||

s3 | 13.2% | 33.5% | |||

s4 | 7.3% | 21.1% | |||

s5 | 4.8% | 7.6% | |||

s6 | 1.8% | 4.1% | |||

s7 | 1.4% | 5.5% | |||

s8 | 2 | 21.6% | 40.9% | ||

s9 | 66.7% | 92.9% | |||

s10 | 26.7% | 32.5% | |||

s11 | 11.1% | 24.6% | |||

s12 | 10.7% | 21.7% |

In Figures

MAD/

MAD/

MSE/

ME/

In cases of forecasts on 5 time periods ahead and for time series belonging to Group 1 (from s1 to s7), the Holt-Winters method gives comparable results in respect of the best (S)ARIMA model found. In series s2, s5, s6, (S)ARIMA outperforms; in series s3, s4, s7, HW outperforms; in series s1, the same value of MAD/

Increasing the number of the forecasted time periods from 5 to 12, the same guidelines can be traced. For time series belonging to Group 1, (S)ARIMA outperforms in s4, s5, s7, and HW outperforms in s1, s3, s6. The same MAD/

Tables

Comparison between (S)ARIMA and HW based on MSE/

Series | Group |
MSE/ |
MSE/ | ||

(S)ARIMA | HW | (S)ARIMA | HW | ||

s1 | 1 | 27.0% | |||

s2 | 79.6% | 74.2% | |||

s3 | 41.8% | 469.9% | |||

s4 | 35.1% | 48.9% | |||

s5 | 20.2% | 34.5% | |||

s6 | 5.1% | 22.0% | |||

s7 | 434.4% | 10675.5% | |||

s8 | 2 | 424.1% | 953.2% | ||

s9 | 666.7% | 678.6% | |||

s10 | 435.9% | 502.4% | |||

s11 | 463.0% | 1424.8% | |||

s12 | 529.8% | 1369.8% |

Comparison between (S)ARIMA and HW based on ME/

Series | Group |
ME/ |
ME/ | ||

(S)ARIMA | HW | (S)ARIMA | HW | ||

s1 | 1 | 2.9% | |||

s2 | |||||

s3 | 18.7% | ||||

s4 | 0.8% | 9.5% | |||

s5 | |||||

s6 | |||||

s7 | 5.2% | ||||

s8 | 2 | 1.0% | |||

s9 | 33.3% | 64.3% | |||

s10 | |||||

s11 | 24.6% | ||||

s12 | 21.7% |

MSE/

The results obtained corroborate the experimental analysis carried out by Bianchi et al. [

The issue dealt with in the present paper is the comparison between the Holt-Winters method and the (S)ARIMA model for forecasting real life time series. In particular, the analyzed series present a high level of variability in terms of demand size and several null-demand time periods. Moreover, all of the time series reveal a clear seasonality while only several of these present a consistent trend component. On one hand, sporadic and irregular time series are extensively treated in literature, while on the other hand several authors compared the two methods above for seasonal and trendy time series. However, sporadic and irregular time series that present both trend and seasonal components are still neglected. Hence, evaluating the applicability of the (S)ARIMA and the Holt-Winters methods in forecasting sporadic demand time series with seasonality and trend components is the aim of the present paper in order to establish some useful guidelines for practitioners. The methodology applied consists of testing several (S)ARIMA models and then choosing the best model only in terms of forecasting performances, which is subsequently compared with the Holt-Winters method. In fact, statistical software programs also let the user test a robust and complex method like the (S)ARIMA very quickly and therefore the results from the comparison between the two methods can give a guidance for their applicability.

In particular, in the case of seasonality without a consistent trend component, the best (S)ARIMA model found and the Holt-Winters exponential smoothing model give similar results in terms of MAD/

This observation represents a useful decision-making guideline in plant management. In fact, several real contexts present these characteristics, such as startup productions, multi-echelon supply chains or spare parts production, and selling, where demand forecasting constitutes an unforgettable prerequisite for an efficient production or selling management and could become a needful competitive leverage for companies.

As underlined in the introduction, when treating sporadic and irregular time series, two relevant issues refer to forecasting and inventory management. Further researches are addressed in the field of order and inventory management when sporadic demand data series with seasonality and consistent trend components are present. Furthermore, a comparative analysis on reachable performances when previously cited methodologies for sporadic demand forecasting with specifical hypothesis (i.e., demand distribution, time periods with zero demand distribution,

The authors wish to thank Eng. L. Biolchini for supporting this research by means of fruitful discussion and constructive criticism.