2
Abstract
3
Acknowledgements
Groningen, April 2016 This thesis was submitted as the closing piece to fulfil the requirements set by the University of Groningen for the degree of Master of Science in Technology and Operations
Management.
The period in which I have written this thesis was turbulent. Changing subjects after investing a lot of time and effort, losing a friend and the transfer overseas of Laura. I would like to thank everyone that supported me during this period, but I would like to express my gratitude to a few.
First, I would like to thank PostNL and Jurre Visser for supplying the dataset and giving me the opportunity to use their data for this thesis. Secondly, Rick Beukeboom for sharing his interview he conducted at PostNL.
Then I would like to express my gratitude towards my two supervisors.
Dr. Nicky van Foreest for his critical feedback and additions to my proposed forecasting model and being my second supervisor and Dr. Stefano Fazi, I would also like to express my gratitude to you. Without your feedback and guidance during this period, the thesis would have had a different level of quality.
I also would like to express to my own family and my girlfriend’s family for the many discussions and the unlimited support when I needed it. Also, I would like to thank Johan for all being a discussion partner and getting my thoughts and ideas aligned and of course for your company and meals, when Laura was still here and when she transferred to the Rochester. And finally, of course, I would like to thank Laura for her unlimited support, advice and love.
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List of Abbreviations
ARIMA Auto Regressive Integrated Moving Average
AR Auto Regression
ETA Estimated time of Arrival
HW Holt-Winter
LSP Logistical Service Provider
LT Long Term Forecast
MA Moving Average
MAPE Mean absolute percentage error PDS Parcel Delivery Service
PL-F Planning Load-Factor RC Roller container
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Table of content
Introduction ... 7 1.1 Research Aim ... 8 1.2 Thesis Structure ... 8 Theoretical background ... 10 2.1 Forecasting capacity ... 10 2.2 Holt-Winters method ... 122.2.1 Holt-Winters multiplicative method ... 12
2.3 Autoregressive integrated moving average (ARIMA) model ... 13
2.3.1 ARIMA model for time series data ... 13
2.4 Persistence scaling factor ... 14
2.5 Theoretical and practical relevance ... 14
Case ... 16
Methodology ... 18
4.1 A model for capacity planning ... 18
4.1.1 Holt-Winters multiplicative stochastic mathematical model ... 18
4.1.2 Autoregressive integration moving average ... 19
4.1.3 Persistence scaling factor ... 21
4.1.4 Warm up period ... 21
4.2 Validation ... 21
4.3 Generalizability ... 22
Results ... 23
5.1 Data Description ... 23
5.2 Holt-Winters & ARIMA ... 25
5.3 Persistence scaling factor ... 26
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IntroductionOnline retailing has become more popular than ever before. European citizens have changed their traditional way of shopping in favour of online shopping. This shift is illustrated by the 16% growth in online retailing (e-commerce) in 2014 and is expected to increase further in Europe (Ecommerce Europe, 2015) and especially in the Netherlands (CBS, 2014). As a consequence, the steep growth in e-commerce has caused a significant rise in demand for parcel delivery (Morganti, Seide, Blanquart, Dablanc, & Lenz, 2014). Traditionally, transportation and parcel delivery was performed by Logistical service providers(LSP)(Coyle, Bardi, & Langley, 1988). These LSP’s delivered did not discriminate on shapes and sizes. Resulting in a sub-industry being developed that mainly delivers packages that are small in size, and can be handled by a single courier called a parcel delivery service(PDS)(Morlok, E.K., Nitzberg, B.F., Balasubramaniam, K., Sand, 2000).
The industry is known for the high level of competition. To gain a competitive advantage, PDS’s need to guarantee a high service quality (Cronin & Taylor, 1992) or face a competitive disadvantage (Fabien, 2005). While being pressured to keep costs as low as possible(Accenture, 2015). Service quality is an essential part of obtaining customer satisfaction (Cronin & Taylor, 1992; Parasuraman, Zeithaml, & Berry, 1985) and the perceived reliability of the PDS is derived from that satisfaction(Lien Yee & Daud, 2011). And to ensure high perceived reliability a PDS has to deliver a parcel within the estimated time of arrival (ETA) i.e. on-time delivery. However, these are not the only factors that determine the success of a PDS. Another factor is the rigorous competition on price. Therefore, PDS companies are constantly redesigning their services, to keep costs as low as possible, to cope with market(price) changes and to keep up with the competition and to maintain financial health (Accenture, 2015). Therefore, PDS providers are constantly adapting lean management to reduce or eliminate any waste of time, effort or money.
The scientifically challenging factor, to adapt lean in a PDS, is the demand variability because parcel volume patterns are time and location dependent. To overcome this factor capacity planning can be used, however, classical planning techniques would not be possible in this situation because they assume demand is known upfront. Combining this with forecasting methods would overcome this problem. In capacity planning, inventory management is seen as an important role and has proven to have a significant impact on both customer reliability and total logistical costs (Korpela & Tuominen, 1996). Hence, by applying capacity planning and forecasting the demand for PDS, by taking into account the variability of volume, capacity planning and forecasting can increase its efficiency and result in a cost reduction(Barahona & Bermon, 2005).
In this thesis, the scope of the case is limited to the PDS division of PostNL, known as PostNL Parcels. Moreover, as forecasting of capacity planning can be done at multiple levels, Strategic, Tactical or Operational, in this thesis we focus on the operational level of capacity planning on the number of trucks PostNL Parcels needs to transport all the packages between their hubs located in the Netherlands. The current operational forecasting method they use is a combination of both qualitative and quantitative forecasting. This combination of methods could potentially mean that PostNL Parcels has a suboptimal forecast on the number of trucks they hire. Hence, generating waste of capacity and costs.
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& Bermon, 2005). Moreover, capacity planning is done using demand forecasting methods. Themethods used for forecasting demand can both be qualitative, based on experience, or more quantitative methods such as time series, or a combination of both (J. S. Armstrong, 1983). In the food industry forecasting is used to determine demand for perishable goods (Veiga, Veiga, & Catapan, 2014). Also, in the energy market forecasting is used to cope with variability of demand (Suganthi & Samuel, 2012). As demand can differ due to seasonal influences, it is important to have a model that can cope with this variability. Williams & Hoel discussed the seasonal effect in forecasting in traffic(Williams & Hoel, 2003). Although the potential benefits of forecasting in capacity planning, the possible application of forecasting models on the logistical problem, as such at PostNL Parcels, has not been described so far.
While there is the suspicion that the current method is subject to improvement. Therefore, the current forecast method needs to be accessed by studying its efficacy. In this thesis, a quantitative method is utilized. To test the applicability of inventory forecasting methods on capacity demand forecasting a case study is being used. The subject of the case study is PostNL. Using raw data which was supplied by PostNL Parcels. This data is analysed on patterns and series. And an extension of the Holt-Winters and ARIMA method will be created to suit the PostNL Parcels case. This method only needs to address the forecasting process of capacity demand in the highest transport tier of PostNL Parcels.
1.1 Research Aim
The aim of this thesis is therefore to reduce waste through the reduction of overcapacity by changing the forecasting methods of capacity planning and reaching the goal of reducing the overall total cost of hiring trucks in the highest transport tier of PostNL Parcels.
This will be accomplished by I) accessing the current forecasting method, II) by identifying patterns or series in historical data on demand and II) create a novel quantitative forecasting model to minimize the costs by forecasting the future capacity demand and III) optimizing the model to minimize total cost.
The main research question is;
To what extent can a quantitative forecasting model estimate the future demand at PostNL Parcels and can an additional application certain method on this forecast lower the projected total cost per route in comparison to PostNL Parcels current performance?
The main question will be answered through the usage of the following sub-questions. S.Q. 1 How is the forecasting done at this moment at PostNL Parcels
S.Q. 2 What are possible candidates to replace the current model of PostNL Parcels S.Q. 3 What is the current performance of PostNL Parcels forecasting method?
S.Q. 4 How do other quantitative forecasting methods perform compared to PostNL Parcels current forecast?
S.Q. 5 What addition method could lower the total costs (per route)?
1.2 Thesis Structure
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will be discussed in chapter 6. Chapter 7 highlights the conclusions drawn in the discussion and10
Theoretical backgroundIn traditional capacity planning, a wide variation of models is used for capacity planning. The models differ from each other on the basis of assumptions how capacity is determined and if it can be altered over time.
Literature makes distinctions between two types of capacity planning. Capacity that can be altered over time and capacity that cannot be altered after acquiring. The latter type, which is being used in this thesis, describes a capacity level that remains effective until the end of a planning horizon. In other words, a manufacturer or consumer of capacity can only decide whether they want to expand the capacity at a certain point in time, but cannot reduce the capacity within the forecasting horizon. This area of capacity expansion is researched actively. Van Mieghem created an overview of capacity expansion literature (Van Mieghem, 2003). One of those cited papers is from Ahmed et al.. Ahmed et al. look into the acquisition of capacity planning problem, in which they study the acquisition of a single product with multiple resources. They model uncertainty of demand by using a scenario trees (Ahmed, King, & Parija, 2003). In a later study Ahmed et al. apply branch and bound method to two-stage capacity planning problem (Ahmed, Tawarmalani, & Sahinidis, 2004). Barahona et al. apply a tool to study a purchasing problem in the process of semiconductor production (Barahona & Bermon, 2005). In both papers, a two-stage decision process is being used. First, the acquisition tool is chosen, when demand is uncertain and secondly, when demand is realised or known the acquired resources are allocated to manufacture the goods needed. For both models the goal is to reduce demand uncertainty.
2.1 Forecasting capacity
To further decrease the demand uncertainty, in capacity planning, forecasting is used(Cakanyildirim, Roundy, & Wood, 2002; Cakanyildirim & Roundy, 2002). The importance of forecasting has been recognized in many fields of research. In logistic literature, the strategic importance of forecasting has been recognized at all levels of the company (Gattorna, 1992). In strategic capacity planning, inventory management has played a significant role and has proven to have an enormous impact on both customer reliability and total logistical costs (Korpela & Tuominen, 1996). Therefore, forecasting is a crucial part of capacity planning. The aim of the forecast is to estimate the quantity that is needed at a particular point in the future, whether this is a short or long term (Sandelands, 2013).
The leading role of forecasting is further emphasized in the evolutionary logistical stages by McGinnis & Kohn (1990). According to this model, logistics of a company go through three separate stages. In the first phase, company logistics are not seen as key aspects of a company and is only regarded as a necessity such basic logistical activities e.g. outbound transportation. In the second phase, logistics become more complex and are responsible for the timely order procession and internal warehousing and in the third and last phase, the responsibilities of the logistics spread to production planning and demand forecasting (McGinnis & Kohn, 1990). To enable this production planning and coordinate logistics in the future, forecasting is needed. Stock and Lambert defined two justifications why to forecast. The first is to create an effective logistical system and the second is to enable management to make educated projections of the future (Stock & Lambert, 2001).
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1. Qualitative forecasting; these types of forecasting make use of expert opinions such asthe interview Delphi method (Dalkey & Helmer, 1963)
2. Causal methods, these methods are best suited for firms that are regularly characterized by ups and downs due to causal factors or drivers. An example is seasonal businesses like summer tourism (Song, Wong, & Chon, 2003)
3. Time series forecasting is a statistical method that is used to identify systematically or seasonality/cyclical patterns in data and/or trends of growth or decline. These models assume that the future is similar to the past. Forecasting methods for time series are moving averages, exponential smoothing, extended smoothing, and adaptive smoothing. An example is Holt-Winters Multiplicative method (Holt, 2004)
Currently, PostNL Parcels uses a combination qualitative and the two-stage capacity planning methods of Ahmed et al. Adapting a causal method would not be appropriate as they deliver their service all year long, whereas a causal method would be more applicable to seasonal services. Utilizing a time series forecasting model, therefore, would have the preference. In literature, more than 70 different forecast methods of time series or quantitative forecasting exist , which can be further differentiated into linear and nonlinear (Kerkkänen, Korpela, & Huiskonen, 2009).
Although all models following the same basic concept, theoretical frameworks, and/or paradigms are utilized from different areas. The reasoning that there are so many methods is because there is no universal quantitative model for all situations and circumstances (Petropoulos, Makridakis, Assimakopoulos, & Nikolopoulos, 2014). Therefore, any of those methods can potentially be best in a particular situation (Makridakis et al., 1982) and, so far, no has been able to characterise what decisive factors are for choosing a specific method(Petropoulos et al., 2014). However, many methods were tested empirically in recent decades and Armstrong & Morwitz summarized the most commonly used methods (S. Armstrong & Morwitz, 2000). These methods were characterized for which data they are typically suited, their forecasting horizon and the how long the developed model could be utilized (Table 1). Choosing the correct forecasting method is essential, by understanding the data (first analyses) the right technique can be selected.
Model Type Most Suited Data Types Forecast Horizon
Shelf Life of Model
Moving Averages(Slack, Chambers, & Johnston, 2013)
Seasonality Short Short
Exponential Smoothing(De Gooijer & Hyndman, 2006)
No Trend, Varying Levels Short Short
Holt's Method(Holt, 2004) Varying Trends, Varying Levels, No Seasonality
Short Short
Holt‐Winter's Method(Segura & Vercher, 2001)
Varying Trends, Varying Levels, and Seasonality
Short to
medium
Medium ARIMA(Box & Jenkins, 1994) Varying Trends, Varying Levels, and
Seasonality
Short to
medium
Long
Table 1; Time Series forecasting methods
Holt-12
Winters Method as the Box-Jenkins ARIMA method. These will be described in the followingsub-chapters.
2.2 Holt-Winters method
For centuries, people have been using techniques to smooth out data. The most simplistic approach is the unweighted moving average(MA), which uses the mean as a smoothing mechanism and weights every past observation as equal. A further development of this technique led to exponential smoothing. In contrary to MA, exponential smoothing assigns exponentially decreasing weights as the observation gets older. In 1957, Holt further extended the method, the Holt method, which is on one of the most commonly used method (single) exponential smoothing to date. In the 1950’s the Office of Naval Research, Planning and Control of Industrial Operations in the United States requested a method that could cope with the trend or seasonal patterns whereas the exponential smoothing could not do this while being cheap to be implemented (Holt, 2004). Charles Holt proposed a method that combined the exponential smoothing with the method of exponentially weighted moving average. This method could be used, not only to smooth the level of a variable but also to smooth the trend, seasonality and other components of a prediction (Holt, 2004). Moreover, as it was relatively cheap and easy to implement in the current environment, the new method replaced the then most used method fairly quick.
A few years later, one of the students of Holt empirically tested the method on other time series and found out it was surprisingly accurate (Holt, 2004). Winters published these results in the 1960s, and the newly developed method was the called Holt-Winters method (Hyndman, Koehler, Ord, & Snyder, 2008). The method quickly found its way into commercial software systems for forecasting and 50 years on, it is still being used in many types of research (De Gooijer & Hyndman, 2006).
The Holt-Winters method extension of the exponential smoothing, also known as the method of Winters or seasonal exponential smoothing, considers α, β and γ as the three smoothing parameters per seasonal cycle of any given length (Hyndman et al., 2008). There are two different methods of Holt-Winters, differing by how the seasonality is modelled and the amplitude and variations of the seasonality, classified as additive and multiplicative (Figure 1).
Figure 1; Holt-winters Methods
2.2.1 Holt-Winters multiplicative method
As it is probable that PostNL does have strong varying seasonality the Holt-Winters multiplicative method (HWMM) is used. The HWMM can be modelled by using the following equations (Hyndman et al., 2008):
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ŷ / ∶ ŷ /Equation 4
The Holt-Winters method consists of forecast equation (Equation 4) and three smoothing equations; Level (Equation 1), Trend (Equation 2), and the Seasonal component (Equation 3), with smoothing parameters α, β, and γ which are restricted between {0-1} (non-integer). m is the expressed period length.
2.3 Autoregressive integrated moving average (ARIMA) model
Next to the Holt-Winters model, the ARIMA method will be used in this thesis. ARIMA is a method that is highly comparable to a lot of exponential smoothing methods, except for the multiplicative form of Holt-Winters (Holt, 2004) and therefore also interesting competitive model. The ARIMA model is the non-stationary version of the ARMA model. ARMA stands for Auto Regressive Moving Average. ARMA is a method made popular by Box and Jenkins in the late 70’s by applying the method of time series analyses and forecasting (Box & Jenkins, 1994). It soon gained popularity, although the internals was sophisticated, it used a similar approach to other methods, making it easy to understand (Bovet, 1991). As with other methods, the forecast involved two steps; 1) the analysis of the time series and 2) selecting the best model(regression) that fits the data set (Box & Jenkins, 1994; Gattorna, 1992).
2.3.1 ARIMA model for time series data
Some steps had to be taken to create an ARIMA model. This section will be a walkthrough of the ARIMA model. The regression part of the ARIMA model is quite basic.
It utilizes an which is the predictive variable or outcome variable, an, . . that represents the linear regression coefficients and an independent variable . . that influences the regression coefficient and represents the error (Equation 5).
The regression model takes the form:
0 1 1 1 1 ⋯ 1 1
Equation 5
If, however, these independent variables are defined as , … . . Then Equation 5 becomes regressive variant Equation 6.
0 1 1 2 ⋯
Equation 6
Compared to Equation 5 the independent variables became regressive and now predictor variable is called autoregressive (AR). In the same manner it is also possible to integrate error (Equation 7) transforming the AR into a moving average model (ARMA).
In this situation AR is connected to the MA model to form a time series models called autoregressive moving average models (ARMA)
0 1 1 2 ⋯
Equation 7
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number of times that the series needs to be differentiated to produce a stationary series. Afterthe differentiation, to establish the degree of seasonality the non-seasonal component (i) or disruption needs to be extracted. After which the stationary series can be described as an ARMA model and subsequently determine possible specifications of p and q (Box & Jenkins, 1994).
2.4 Persistence scaling factor
In every forecasted method, as mentioned before, the forecast is not hundred percent correct, and there will always be a discrepancy between the forecast and the actual realisation. This discrepancy is apparent in many markets. In the market for renewable energy, such wind, and solar/PV energy, futures of generated energy are being traded (Bathurst, Weatherill, & Strbac, 2002), however with the nature of this product, there are many variabilities. This variability may cause a lower or higher amount of generated energy at the delivery date. This discrepancy leads to extra costs, so-called imbalance costs (Bathurst et al., 2002).
These imbalance costs, for example, a shortage of capacity at the point of delivery, result in a penalty. This shortage penalty cost is known as Top-up costs, but the opposite is also possible. This is called a spillover; this overcapacity that will have a lower market value than the acquisition costs and can be considered waste. Because of these facts, companies aim to minimize these imbalance costs. The most used policy in energy production is Persistence Scaling Factor(PSF) (Bathurst et al., 2002). Applying a PSF policy, in the case of wind energy they have the tendency to over forecast, a modification is made to the forecast by adapting a scaling factor to adjust the forecast. Theoretically, a PSF can have any value above 0, but normally the factor is in the vicinity of 1.
In the renewable energy generation a 0,9 PSF is applied, meaning that the forecasted generation is lowered by 10%. A PSF policy has some commonalities with the planning load factor(PL-F policy of PostNL Parcels. In the PL-F policy, a safety margin is deducted from the capacity of the server, in this case the TTC, and does not influence the forecast, whereas a PSF policy does not influence the capacity of the server as it modifies the forecast. The main benefit of applying a PSF policy instead of PL-F policy is the ability to change the capacity of a server (changing truck type) without changing the impact.
2.5 Theoretical and practical relevance
This thesis will adopts a two-stage decision model similar to the one used by Ahmed et al. and Barahona et al.. However, in contrary to their models in this study the acquisition method is different. In our model, the (transportation) capacity needed is not owned by PostNL Parcels but is acquired by external parties. The external parties give PostNL Parcel the right to use their capacity for a given period of time and a predetermined price. The contracts PostNL Parcels has with these parties are short term. Because of short-term contracts, they can reserve different amounts of capacity for various periods(Yazlali & Erhun, 2007). These periods can have any length shorter than one year. However, the supplier of capacity can offer contracts with greater or smaller planning horizons with a better price to tempt PostNL Parcels to commit earlier. Combining these theories with the research aim, to reduce waste through the reduction of overcapacity by changing the forecasting methods of demand and reaching the goal of reducing the overall total cost of hiring trucks in the highest transport tier of PostNL Parcels, a theoretical and practical relevance is constructed.
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problem. This relevance will be accomplished by developing a forecasting model withcorrection for seasonal and trend variation, which has been widely discussed in inventory forecasting literature. However, the potential application in logistical capacity forecasting has not been reviewed in the logistics literature. Moreover, the models are designed for continues data, whereas the data will show data gaps and this will be overcome in an extension of the models. Also, the potential impact of the cross-industry application of Persistence Scaling Factor has not been recognized. These additions to literature will add rigor and represent the theoretical relevance (Hevner, March, Park, & Ram, 2004).
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CasePostNL is the provider of postal and parcel services in the Netherlands and operates the largest mail and parcel distribution network in the Benelux. Internationally PostNL operates under name TNT Post, which it also used in the Netherlands until TNT NV was demerged into TNT Post and TNT Express. Regarding volume, PostNL delivers over 1.1 million items to 200 countries on a daily basis (PostNL, 2016). One of the unique selling points of PostNL is its delivery reliability, and this has earned them a ‘Certificate of Excellence’. Other key performance differentiators include exceptional delivery quality, customer-specific solutions with integrated tracking, and an unwavering focus on customer service.
Three stages forecasting
As in many business practises ordering in advance is cheaper, then at the last moment. PostNL Parcels tries to order their TTC as upfront as possible using three ordering stages. The downside of ordering upfront is the possibility of over ordering and subsequently have overcapacity on the day of transport. This negative aspect is due to the non-refundable nature of the hiring process, and this overcapacity is a pure waste regarding lean management.
The time between the different stages is set. The first forecast of the amount of RC’s is done three weeks in advance of the transportation night. Based on this forecast the initial amount of TTC needed is calculated and ordered. Currently, this forecast 1 TTC costs approximately €170,-. The second forecast is done 16 hours before the transportation and should give a better forecast than the first as more accurate information is available. These second stage TTC cost €220,-, an increase of almost 30%. At the last phase, the actual amount of RC’s is known. The planner checks whether the amount of TTC is sufficient, if not he or she can still order extra trucks at a cost of €260,-. As these trucks are ordered at such a late stage of the delivery process the availability cannot be guaranteed and, therefore, a maximum of four third-stage TTC can be ordered per route.
Current situation
Currently, PostNL Parcels bases its amount of trailers it needs per night on probability forecasting model. This model is a static model that enables PostNL Parcel to estimate their needed vehicle fleet size several in three stages upfront. The number of truck-trailer combinations (TTC) is determined by the number of roller containers (RCs) that are transported on a predetermined route (e.g. ‘Amersfoort-Hengelo). The maximum RC capacity of one TTC is 56 per cycle and thus making the calculations for the amount of needed TTC the integer amount of RC’s divided by 56. The forecasting of RC’s is subjected to variance. The exact amount of RCs that need to be transported is only known at the last minute. PostNL Parcels is therefore continuously balancing between constraining costs and upholding a high service quality and consumer satisfaction.
Current approach by PNP
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This method is also used in other industry such as aviation and hospitality. In these sectors, itis common to use a load factor over 100%, due to the common no show of passengers or guests. The usage of a load factor creates the risk of needing to sell no if more than maximum capacity do show up and negatively influence the service quality and reliability.
The current forecasting horizon and capacity planning of PostNL Parcels are illustrated in Figure 2. This flowchart gives an overview of the forecasting period; PostNL Parcels applies to the three stages of the forecasting process method.
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MethodologyThe following four steps have been performed in this research.
1. Data gathering. Both primary and secondary data collection techniques are used in the thesis to get quantitative and qualitative information for analysis. An interview with the company and management has been done to gain the primary data. This interview data contains the relevant information, such as standard working procedure. As described in the Case description. PostNL Parcels shared the raw data, containing data expressed in the roller container data and trucks predictions of fourteen routes, has been analysed and converted to the desirable form and unit to formulate the mathematical models. This data will be used as input and comparative data for step 4.
2. The methods will be turned into models, which are usable in the case context. As the models cannot be used as-is. Adaptations will be made, and the operation of the model will be verified.
3. In the test and validation process, the models have been tested in the actual environment. Subsequently, the result of the model has been compared to the actual outcome to eliminate all flaws. Consequently, the validity of the model is assured that the models represent the real processes.
These steps will be further discussed in the following paragraphs in more detail.
4.1 A model for capacity planning
The goal is to find an optimal, or most efficient, way of using limited resources to achieve the objective of the situation. This goal may be maximizing the profit or minimizing costs. For the given problem, we formulate a mathematical description called a mathematical model to represent the situation. The model consists of following components:
• Decision variables or parameters: These variables represent unknown quantities • Objective function: The goal of the problem is expressed as a mathematical
expression in decision variables.
• Constraints: The limitations or requirements of the problem are expressed as inequalities or equations in decision variables.
In the following two paragraphs the HW and ARIMA model will be constructed and in 4.1.3, the PSF addition is described that is applied on top of the described models.
4.1.1 Holt-Winters multiplicative stochastic mathematical model
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Parameters: ∶ 1 Equation 8 ∶ ∗ 1 Equation 9 ∶ / 1 Equation 10 ŷ ∶ ŷ Equation 11 Objective function: ∑ ∗ ∀Objective function constraints:
= Forecasted number of RC
/56 = initial forecast divided by TTC capacity
/56 = Total TTC need minus forecasted number of TTC 4 = Maximum extra trucks at Stage 3 is 4
forecasting period; 15 3 , 0 170 = Cost of stage 1 TTC 260 = Cost of stage 3 TTC , , 0, 1 , 0 , 0, ∈ IN
4.1.2 Autoregressive integration moving average
In a similar way as the HW model, the ARIMA will be explained. As mentioned in the theoretical background ARIMA is using the parameters of p,d,q.
p is the number of autoregressive terms,
d is the number of non-seasonal differences needed for stationarity, and q is the number of lagged forecast errors in the prediction equation.
The problem with the standard ARIMA model is that it assumes linearity. Meaning that the method takes on that the trend will have a static coefficient that is negative or positive over time. However, the initial analysis of the raw data suggests an exponential distribution of the AR function. This problem is on of the reasons that theoretical model cannot directly be implemented and, therefore, the model will need adjustment to suit the purpose of this research. The adjustment to cope with the linearity assumption, whereas the data suggests an exponential spread during the week., is the following. The linear parameter was replaced by a
2 function. Secondly, the trend function was decoupled from the function of time t=1, 2,
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reasoning for this adjustment is the repetitive cycle of the exponential function between theweeks. Therefore the trend parameter will be expressed as 2 (Equation 13). Sets: Rt Route AMF HGL, … … WVN RD St Stages 1,3 1 3 Variables: 1 , 2 , 3 , 4 , 5 ; 15 3 ∗ ∗ 6 , … . Parameters ; ∶ŷ ∗ Equation 12 ; 2 Equation 13 ; ∑ Equation 14 & ; / Equation 15 ; 2 Equation 16 ; 5 Equation 17 Objective function: ∑ ∗ ∀
Objective function constraints:
ŷ ∗ = Forecasted number of RC = Realisation of RC
/56 = initial forecast divided by TTC capacity
/56 = Total TTC need minus forecasted number of TTC 4 = Maximum extra trucks at Stage 3 is 4
, 0
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260 = Cost of stage 3 TTC, 0,
4.1.3 Persistence scaling factor
The following paragraph shows the application of the PSF on both models. The outcome is an approximate adjustment factor because it has been calculated as the average of the minimum expected imbalance cost policy.
To find the optimal persistence scaling factors. Scenarios have to be written and need to be calculated. In the scenarios the forecasting protocol of Holt-Winters and ARIMA are the same. The PSF is applied after the initial forecast of HW and ARIMA. The individual scenarios will be generated through a scripted scenario generation. The tool used for the scenario generation is the Solver of Microsoft Excel.
The objective function minimizes the value of z because PSF does not influence the TR nor the C function the object function of both models is unaltered;
∑ ∗ ∀
The forecasting function, however, do need to be altered. The original forecasting function needs to be multiplied by the PSF
Holt-Winters; ∗ ; ŷ ∗ ∗
ARIMA; ∗ ;: ŷ ∗ ∗ ∗
Next to the object function, the objective function constraints mentioned in for 4.1.1 & 4.1.2 are the same, with exception that that the following two are added to the constraint list:
0 1,5
4.1.4 Warm up period
In the two models, of HW and ARIMA, a warm up period or ramp up period is used. The warm up time is the time a data model or simulation needs to have run before results can be utilized that are generated from it. In this thesis, warm up period is used to forecast all the sequent data. In the two models, a warm up period of six weeks is chosen. In the HW model, in the first three weeks the seasonal effect is calculated and in the next three weeks, the trend and forecast normalized. The following results are generated by these figures.
In the ARIMA, the equivalent warm up time is used to establish a stable seasonal effect ( and to determine the exponential function of the trend within the week. After this warm up time the, forecasts are generated based on the warm-up period. The data fits of the ARIMA model can be found in Appendix
4.2 Validation
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values can arise on days with no transport (holidays). This problem can be avoided by excludingvalues zero’s from the analysis. However, this artificial solution limits the application of the method in various situations (J. S. Armstrong, 1983), because of the heavy penalty on the positive errors compared to negative errors (De Gooijer & Hyndman, 2006).Nevertheless, these authors also show that more scientific papers and discussions are needed about the other symmetric measures proposed so far. In practice, a MAPE value lower than 10% may suggest a forecast potentially very good, lower than 20%, potentially good and above 30%, potentially inaccurate (Thomopoulos, 2015). The MAPE can be expressed as following:
∑ 100 Equation 18 Where: 4.3 Generalizability
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Results5.1 Data Description
PostNL Parcels provided the raw database file for this thesis. The database contains historical data of fourteen routes within the Netherlands. These are not all the routes, on this level of transport, but are the carrying the most volume. All the routes between distribution hubs are abbreviated in two to four letters e.g. Amersfoort-Hengelo is AMF-HGL. These abbreviations are used in the file to indicate the specific route.
The data of the current forecast is expressed on three different sheets. The first sheet includes the first stage forecasts, long term, on all fourteen routes. All the forecasts are data stamped. The included data stretches from 23-3-2015 (wk13) until 25-9-2015(wk39) at the most. Week 17, 18, 19, 27, 28 and 30 do not contain forecasts. These missing values reduce the weeks containing data to 21. The forecasts are expressed in non-integer amount of trucks needed. The second tab includes the second stage forecasts, day prognoses, on all fourteen routes. And uses the same date and route setup. The included data stretches from 16-3-2015 (wk12) until 15-9-2015(wk38) at the most. The only missing values in this forecast represent holidays and other no transport dates. The second stage forecasts are expressed in non-integer amount of RC’s.
The third tab includes the third stage, actual demand, on all fourteen routes. And uses the same date and route setup. The included data stretches from 16-3-2015 (wk12) until 11-9-2015(wk37) at the most. The only missing values in this forecast represent holidays and no transportation dates. The actual values are expressed in integer amount of RC’s. In Table 2 is an overview of the databases raw data description.
Long term Day Prognoses Realisation
Start week 13 12 12
End week 39 38 37
Unit Trucks RC RC
Integer no no yes
Table 2; Data Description
Other input data
Next to the raw data, also, an overview is given of what the hiring costs of a TTC are at any of the three stages. The costs are based on the yearly agreed upon tariff card and are a tariff that includes a truck, trailer, and driver.
The agreed upon tariffs are First-stage TTC €170,- Second-stage TTC €220,- Third-stage TTC €260,-
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are point forecast and have the tendency to deviate from the actual realisation. The modelledthe variables as normally distributed around the mean ( ) over time ( 1,2,3 … . ) variables with an expected standard deviation ( ).
The ladder of powers is applied to test the assumption of normality (Tukey, 1977). This tool checks the data normality or skewness of the realised RC’s and what is needed to transform the data into normally distributed data. With the exception of the HT-BD route, which was skewed to the right, all routes were normally distributed and did not need any transformation in Table 8 a graphical visualisation of the distribution is given with the corresponding significance level. After the assumption of normality was proven. The accuracy of the current forecasts of PostNL Parcels was tested. As mentioned in the in 3 the case description PostNL Parcels currently is using two forecasting periods, a long-term forecast (LT) and a sixteen hours forecast (Day), and provides the data of the actual RC’s at (Real).
A univariate summary was generated of each route to give a detailed insight of the data. This provides an overview of the distribution of RC’s per route and per day within a route.
The amount of RC’s distributed per differs a lot per route. The lowest amount of RC’s is transported on the WVN-BD route with ~150 RC’s per day, whereas five times the amount is transported on the AMF-ZL route (~700 RC’s). The median amount of RC’s transported on a non-specific day is approximately 500.
Within the week, there is also a clear pattern of distribution of the amount of RC’s that is transported. On all routes about a quarter of all transport that will be done during the week is done on Monday. On Tuesday, this is about a fifth of the total RC’s, and this steadily decreased to a sixth that is transported on Friday.
In Table 3 is an overview of the average transport percentages during the week on all routes. In Appendix B, a (visual) overview is given of the distribution within the week per route.
Weekday Percentage Monday 24% Tuesday 21% Wednesday 19% Thursday 18% Friday 17%
Table 3; Week distribution (in % of total)
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5.2 Holt-Winters & ARIMAAfter analysing the data offered by PostNL Parcels. The standard forecasting models were coded into Excel and adjustments were made, as described in the methodology section, to work with the data. One of these modifications was needed to cope with the sudden drops of capacity demand caused by holidays. These holidays are represented as zero transport days in the models. In the Holt-Winter model, this impacts both the seasonal correction as well as the trend. Causing an unwanted over- or under- estimation around the holidays. The disruptive effect is clearly shown when the holiday is the case on a Monday. In the event of a holiday on a Monday in the HW model, the undermining effect is causing an underestimate on that Tuesday leading to extra Stage 3 trucks and therefore costs. The same principle accounts for the ARIMA method when it is the day after a holiday. If the holiday is on another day then Monday the effect is less disruptive.
A graphical overview of the RC’s to transport shows this phenomenon. Figure 3 is the graphical representation of the AMF-HGL route with the realised transport (blue line), the Holt-winters forecast (orange) and the ARIMA (grey). In this graph, the steep drops represent the holidays. And after every drop, both the HW and ARIMA forecast are on a lower level then the realisation. This situation is representative for all other routes, and they are also present the same kind of underestimation.
Figure 3; AMF-HGL total RC's
To further analyse the over and under estimation of the models and the current, the errors were visualized. In the visualization of the AMF-HGL errors seen in Figure 4 & Figure 5.
Comparing these two figures would suggest that the current forecasting is more stable and would have a lower average error relative to those of the HW and ARIMA model.
When comparing the numbers the average error on the current forecast is around 1 TTC (45RC’s). The HW model also keeps the error below 1 TTC (53RC). However, the ARIMA model has an error of 88RC’s equivalent to 1,6 TTC.
0 200 400 600 800 1000
Total RC to transport
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Figure 4; AMF-HGL LT/Day Error
Figure 5; AMF-HGL HW/ARIMA Error
On all routes, the same analyse was performed. The outcomes of these analyses are summarized in Appendix E. The expressed error values are the deviation from the actual demand for RC’s transporting capacity. A positive value represents an overestimate, whereas a negative value represents an underestimation. On average, the error is less than the capacity then one full TTC. What is interesting is that the model of PostNL Parcels overestimated on almost every route on average.
Also, the total projected costs were compared. Among the fourteen, the HW model performed better on 5 and then ARIMA model on 4 routes. Comparing the HW with the ARIMA model, the HW model has the lowest total project costs on almost every route. The differences between the projected total costs are minimal.
5.3 Persistence scaling factor
In the third stage of the methodology. A series of simulations was created to see if is possible to minimize imbalance costs by applying a persistence scaling factor(Bathurst et al., 2002). In this policy, a modification is made to the forecast by adapting a scaling factor to adjust the forecast. After the introduction of the PSF excel generated scenarios according to the coded script and calculated the total costs of these scenarios. The goal of the script was to minimize total projected costs.
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In the case of the AMF-HGL route, the forecast had to be underestimated by 6% to come to thelowest projected total costs which were around 240.000 over the modelled period. This equalled to a lower projected total costs (1-4%) compared to the original model without PSF.
All other routes also applied an underestimation to come to the lowest projected total cost. The highest category of underestimation included 6 routes. On these 6 routes, the forecast was decreased by 12-18% percent. On 5 routes the best performing scenario had underestimated the forecast by 7-10% and the last 2 routes by only 2-3%. An overview of the optimal PSF’s is given in Table 4
AMF-HGL AMF-KHM AMF-LW
94% 85% 82% AMF-ZL HT-BD HT-BORN 87% 88% 98% HT-ELT HT-SON WVN-BD 91% 97% 90% WVN-GS WVN-HBD WVN-HW 93% 92% 86% WVN-OPM WVN-RD 84% 91% Table 4; PSF overview
After the application and calculation of the PSF factors, the difference in total projected costs was compared. In all cases, the total projected costs were reduced. The ARIMA model benefitted more from the under-estimating PSF factor than the HW model (Table 5).
HW TC HW Gain ARIMA TC ARIMA Gain AMF‐HGL 3020 1% 8820 4% AMF‐KHM 3560 2% 6440 3% AMF‐LW 2960 2% 4320 2% AMF‐ZL 3800 1% 2420 1% HT‐BD 2960 2% 3400 2% HT‐BORN 460 0% 0 0% HT‐ELT 1840 1% 5100 2% HT‐SON 1840 1% 1440 1% WVN‐BD 660 1% 2320 3% WVN‐GS 1800 1% 2880 1% WVN‐HBD 2100 1% 2320 1% WVN‐HW 2180 1% 5420 3% WVN‐OPM 10580 4% 7220 3% WVN‐RD 1080 0% 3840 2%
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In comparing, the altered models with the current model of PostNL Parcels the differencedecreased, but the in most cases the current model kept the lowest projected costs on average. In the case of the errors, the adaptation of a PFC policy led to a higher average error. An updated version of Appendix E is presented in Appendix F.
5.4 MAPE
To validate the results that were generated in the previous paragraphs. The MAPE scores of all routes were calculated In practice, a MAPE value lower than 10% may suggest a forecast potentially very good, lower than 20%, potentially good and above 30%, potentially inaccurate (Thomopoulos, 2015). According to the tool, the HW and ARIMA in all cases performed better than the current model.
Both the LT and Day on average generated a potentially good forecast, whereas the alternative HW and ARIMA and the version with PSF applied scored better. The average scores are presented in Table 6. A more detailed overview of outcomes per route can be found in Appendix I. Method MAPE LT 16% Day 19% HW 10% Hwalt 9% ARIMA 9% ARIMA ALT 9%
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DiscussionThis research was conducted to answer the following research question;
To what extent can a quantitative forecasting model estimate the future demand at PostNL Parcels and can an additional application certain method on this forecast lower the projected total cost per route in comparison to PostNL Parcels current performance?
To answer this main question. Five sub-questions were formulated to guide the thesis toward this answer. These five subs questions will be dealt with in this chapter.
S.Q. 1 How is the forecasting done at this moment at PostNL Parcels
S.Q. 2 What are possible candidates to replace the current model of PostNL Parcels S.Q. 3 What is the current performance of PostNL Parcels forecasting method?
S.Q. 4 How do other quantitative forecasting methods perform compared to PostNL Parcels current forecast?
S.Q. 5 What addition method could lower the total costs (per route)?
How is the forecasting done at this moment at PostNL Parcels?
PostNL currently uses a probabilistic forecasting (Wachs, 2007) this approach combines a quantitative base with a qualitative “educated” guess what will happen. The goal of PostNL Parcels is aimed at; to order their TTC as upfront as possible. The TTC orders that are made are non-revocable, and these contracts cannot be voided, which leads to pure waste regarding lean management. The upside of ordering, however, means that the TTC’s are cheaper in total costs and that the TTC’s are available. PostNL Parcels promises a next day delivery. To upkeep a high level of service reliability, PostNL Parcels further applies a load factor to the used trucks. This load factor reduces the actual loading capacity of the TTC’s and creates a safety margin on every single truck and thus route. The safe margin will be used when the forecast is underestimated compared to the actual demand and thus reduces the probability of under capacity.
The identified problem is the probability of over ordering and having over capacity on the day of transport. Therefore, there is the suspicion that the current method is subject to improvement.
What are possible candidates to replace the current model of PostNL Parcels?
In literature, the types of forecasting methods were looked up. The models that are described in literature come in all shapes and complexities, but all have the same goal. To fulfil the need to forecast scenario in the future. There are typically three types of forecasting methods (Sandelands, 2013);
1. Qualitative forecasting; (Dalkey & Helmer, 1963) 2. Causal methods; (Song et al., 2003)
3. Time series forecasting, (Holt, 2004)
A possible replacement was sought in the Time series forecasting. The argument is that quantitative data was available to create a quantitative model instead of a partial quantitative forecast. Also, the process of PostNL Parcels is not periodical operations as is the case with causal methods, but continues.
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We have utilized four different approaches to model and forecast the number of roller containersper day per route. These are; the exponential smoothing methods of Holt and Winters with a seasonality of 14 days with and without PSF applied so a model that considers the day of the week, the month and smoothens the holiday effect and ARIMA models with and without PSF which combines autoregressive and moving average models on possibly integrated data. The choice of these models is based on literature and data analysis applied to the number of Roller containers The exponential smoothing methods, and the ARIMA models are based on correlations between successive days, the difference between the number of Roller containers on a day and a seasonality factor.
Both of models, however, could not be implemented directly on the data. This problem was due to the limitations of methods. To overcome these challenges an extension had to be created. The limitation that had to be overcome in the Holt-Winters method was the missing values(Chatfield & Yar, 1988; Cipra, 1989, 2006; Hanzák, 2012). The models assume a continuum of data. Therefore to overcome this challenge, the modelled HW model makes use of a smoothed trend and seasonality in decrease the influence of time-outs. These missing values have a disruptive effect on the seasonality of the HW method and subsequently the forecast. This problem was countered by interpolating the previous seasonal smoothing and trend correction when confronted with missing values. Other approaches to overcome this issue is to “clean” the data (Gelper, Fried, & Croux, 2009). However, this would reduce the data set in this study if a bigger dataset would be available this would be an option.
The ARIMA model, as introduced by the Box-Jenkins(Box & Jenkins, 1994), uses stationary data and assumes linearity over time(Hyndman et al., 2008). After the initial analyses, the data showed a repetitive exponential function. The original ARMA and ARIMA would not be suited for this type of data(Marriot & Newbold, 1998). Therefore, the applied ARIMA model was adapted to suit the data. This adaptation was made by detaching the trend line from the continues timeline and attaching it to the weekday variable. Moreover, whereas the standard model generated a data fitted linear regression line based on the warm-up period. An exponential regression was generated through a data fit to replace the linear regression line.
What is the current performance of PostNL Parcels forecasting method?
To quantitatively express the potentially suboptimal forecast on the number of trucks they hire, hence, generating waste of capacity and costs, the current situation was analysed.
Initially, the forecasts periods were compared with the realisation. The analyses of the data, confirming if the forecasts are significantly the same as the realisation, in all cases returned false. This is in line with literature that forecasts are inaccurate(J. S. Armstrong, 1983; Makridakis et al., 1982).
Secondly, the errors were examined. Errors can be seen as an indicator of the accuracy of the forecast(J. S. Armstrong & Collopy, 1992). The smallest average error is ten times smaller than the largest error (21vs211). The relative error as a percentage of the total RC’s forecasted or transported, except the HT-BORN route, is comparable. The relative errors fluctuated between the 10% and 20% of the actual demand. Translating these errors to currency, reduction of the error can lead to waste reduction ranging from €7.500,- on the smallest route to €56.000,- on the largest route. This confirms the potential sub performance.
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How do other quantitative forecasting methods perform compared to PostNL Parcels current forecast?
After coding the models without PSF, the parameters were optimized to generate a forecast. The populations of forecasts were tested using an unpaired T-test, as both the forecasts generated by the HW and ARIMA were normally distributed as were the Real as mentioned before. The p-value’s in all cases were similar to the LT or Day forecast. However, the peaks of underestimation resulted in a higher projected total costs on more than half of the routes when HW and ARIMA are applied and do not outperform the current model PostNL Parcels applies. However, the relative differences are small. The “worst” performance of HW was on ZL with -8% compared to the current model and on the ARIMA, it was -8% on AMF-HGL. Both were large volume routes. The best performance for HW and ARIMA was incurred at the HT-BORN route, an average sized route, with +4%. On average the HW model performed 3% less than the current model and the ARIMA model 4%.
Both the HW and the ARIMA model performed better according to the MAPE tool. The forecasts did not have a MAPE score above the 20%. The average error was slightly higher but in most cases this error stayed within the capacity of the TTC not resulting in a different number of TTC’s, on average, needed. However, the inability of both models to cope with outliers leads to peaks in costs. These costs are caused by having to hire stage 3 TTC because the forecast was underestimated on particular days, especially days following a holiday or other timeouts. In the paper Cho in which he compares probabilistic forecasting with time-series methods in the tourism industry(Cho, 2003). Cho also saw that on average the MAPE scores of probabilistic forecasting is comparable as time-series methods. Comparing the time series models, Veiga et al. had similar outcomes comparing time-series models in the market of perishable goods(Veiga et al., 2014).
The model that was created in this thesis follows the conclusions that Holt-Winters and ARIMA show good performance in many cases and different industries (Makridakis et al., 1982; Petropoulos et al., 2014).
What addition method could lower the total costs (per route)?
In the last step, a PSF was applied on the forecast to underestimate the forecasts. The idea of this PSF comes from the market for renewable energy. In the renewables market, the normally use a PSF factor of 90% as projected generated energy is overestimated (Bathurst et al., 2002). Although the optimal PSF factor per route differed, on average the optimal PSF factor is 90%. Thus generalizing that a PSF of 90% will be a good weighting factor to apply.
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The performance of the suggested models and the current performance are almost equal,however the PSF has some flexibility advantages compared to the planning load factor(PLF) as applied by PostNL Parcels. The difference lies in the variable that is reduced. In the PSF the forecast is reduced, whereas PLF reduces the server capacity or the TTC capacity. This limits the potential of using TTC of different capacities, because if using a uniform PLF the capacity safety margin is relative to the size of the TTC capacity. In contrary the PSF does not influence the TTC capacity, thus creating more flexibility in the choice of different capacity TTC’s.
Conclusion
Combining the acquired knowledge throughout the thesis and the answers to the sub-questions. The main question “To what extent can a quantitative forecasting model estimate the future demand at PostNL Parcels and can an additional application certain method on this forecast lower the projected total cost per route in comparison to PostNL Parcels current performance?” can be answered.
Comparing the adapted Holt-Winters and ARIMA forecasting model to the current forecasting method, both forecasting methods generated a lower error than the original model. However, the decrease of the average error is not translated into a true gain regarding less TTC or less overall total projected costs on all routes. The differences between the suggested models and the current model are marginal (-8% - +4%). The further addition of a persistence scaling factor on the generated forecast showed an even further decrease in the average error, leading to a more precise forecast and a lower projected total cost compared to the current model. So the answer to the main question is that the suggested models can produce a very exact forecast, and the addition of PSF can lower the projected costs, but the performance overall is comparable to that of the current model because the differences are marginal. Also, the application of PSF instead of PLF increases the flexibility of types of TTC’s.
7.1 Theoretical implications
The goal of a master thesis was to fill a literature gap of the applicability of inventory management models for capacity planning in a logistical setting. The Holt-Winters model was known not to be able to cope with data gaps. This thesis created an extension on the HW model(Chatfield & Yar, 1988). This was accomplished by implementing a Moving Average on seasonality and the trend parameter. The extension reduced the impact of data gaps on seasonality and trend parameter. Confirming the applicability of the HW model in a logistical setting
A further extension to the Holt-Winters model was made by including a persistence scaling factor. The incorporation of a PSF factor in the HW model increased the overall performance of the HW without a PSF included.
The ARIMA model, as described by Box-Jenkins(Box & Jenkins, 1994), is not usable in the current form in logistical a logistical form. The extension that was created redesigned the trend parameter to cope with exponential time series within a predefined period. This extension to the ARIMA model created new approach with the ARIMA model created theoretical relevance. Also, the application of PSF in the ARIMA model has proven to have a positive effect on the projected total costs.
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in logistical capacity planning. Moreover, the additions to the HW and ARIMA model reducedthe impact of data gaps. These additions to logistical literature are reflected in the rigor cycle of the Information Systems Research (Hevner, March, Park, & Ram, 2004).
7.2 Practical implication
The practical relevance of this research can be seen in the output of this research. The comparative study of the ARIMA and Holt-Winters model aimed at creating a more precise forecast of capacity demand at PostNL and, therefore, support the forecasting decisions that are made on a daily basis while coping with uncertainties.
Comparing the current model with the proposed models, the proposed model reduced the overall projected costs for PostNL Parcels. This reduction results in a better financial position. Replacing the PLF with a PSF would also increase the capacity flexibility for PostNL Parcels. As the current impact of the PLF is relative to the TTC size, whereas the PSF is not influencing the impact. So if PostNL Parcels in the future chooses to use a variation of TTC capacities, they need to increase the amount of PLF’s increasing complexity whereas the PSF needs no adaptation and can be applied uniformly.
7.3 Further research
During the analyses of the data and the limited time available for this research some potential further research surfaced.
The current model HW with PFS could be further enhanced towards a multistage model. The development of such a model may potentially reduce the imbalance cost even further.
Appendices
9.1 Appendix A
Route Weekday Long
term Lt error Day Day error Realisation Real
error p‐value LT Real p‐value Day Real
WVN‐GS Monday 578,31 57,4 624,1 48,44 610,22 37,29 0,3864 0,4432 Tuesday 481,63 55,07 515 81,36 517,4 89,87 0,369 0,4557 Wednesday 462,94 51,1 528,18 153,57 482,38 47,55 0,5098 0,4232 Thursday 432,94 63,16 485,45 136,84 460,65 42,11 0,385 0,5253 Friday 432,31 57,96 430,19 55,04 434,57 47,31 0,4951 0,4241 477,626 56,938 516,584 95,05 501,044 52,826 0,42906 0,4543 WVN‐HBD Monday 720,31 113,82 665,3 64,85 656,11 72,69 0,4457 0,4544 Tuesday 693,06 111,05 646,1 117,37 639,5 107,97 0,4497 0,4557 Wednesday 653,94 105,04 632,41 182,19 584,05 74,48 0,4514 0,3454 Thursday 615,88 108,17 586,7 144,26 559,15 62,38 0,4497 0,4568 Friday 596,31 118,03 534,62 67,69 518,71 69,39 0,4027 0,3327 655,9 111,22 613,026 115,272 591,504 77,382 0,43984 0,409 WVN‐HW Monday 592,13 100,94 546,05 53,4 489,67 56,25 0,4078 0,4544 Tuesday 576 89,76 469,9 105,55 459,3 76,08 0,4497 0,4557 Wednesday 526,88 81,29 455,86 106,86 428,43 62,69 0,7117 0,4579 Thursday 503,75 86,28 411,25 80,59 411,3 43,11 0,4497 0,4099 Friday 483,69 92,68 381,71 54,53 373,19 54,59 0,4497 0,5517 536,49 90,19 452,954 80,186 432,378 58,544 0,49372 0,46592 WVN‐OPM Monday 637,13 98,68 693,65 107,33 684,61 88,74 0,4457 0,3867 Tuesday 564,75 75,41 629,45 127,52 620,15 135,04 0,4497 0,4557 Wednesday 524,81 66,46 616,23 240,35 560,71 94,06 0,4027 0,4579 Thursday 498,06 68,97 547,25 231,89 532,65 76,89 0,4497 0,3775 Friday 489,63 76,69 498,57 113,05 508,14 100,66 0,4497 0,4568 542,876 77,242 597,03 164,028 581,252 99,078 0,4395 0,42692 WVN‐RD Monday 784,06 97,53 658 56,45 627,89 80,96 0,4457 0,4544 Tuesday 652,13 89,54 515,2 108,73 515,25 104,88 0,3958 0,4557 Wednesday 606,38 79,68 498,86 137,52 480,14 85,86 0,4558 0,4579 Thursday 576,44 80,9 453,25 108,5 452,05 71,22 0,4497 0,4568 Friday 561,5 90,56 406,05 51,75 414 63,84 0,6868 0,406 636,102 87,642 506,272 92,59 497,866 81,352 0,48676 0,44616
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9.2 Appendix B AMF‐HGL Normal distributed .000 AMF‐KHM Normal distributed 0.018 AMF‐LW Normal distributed 0.018 AMF‐ZL distributed 0.064 HT‐BD Normal distributed 0.132 HT‐BORN Normal distributed 0.004 HT‐ELT Normal distributed 0.041 HT‐SON Normal distributed 0.05 WVN‐BD Normal distributed 0.000 WVN‐GS Normal distributed 0.009 WVN‐HBD Normal distributed 0.0000 WVN‐HW cubic 0.005 square .00 WVN‐OPM Normal distributed 0.038 WVN‐RD Normal distributed 0.027Table 8; Distributions of real. data
9.3 Appendix C
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AMF‐ZL HT‐BD HT‐BORN
HT‐ELT HT‐SON WVN‐BD
WVN‐GS WVN‐HBD WVN‐HW
WVN‐OPM WVN‐RD
9.5 Appendix E
Current Method PostNL Holt‐Winters Forecast ARIMA Forecast
Current Costs LT Error Day Error HW TC HW Error Diff Cur‐HW ARIMA TC ARIMA Error Diff Cur‐ ARIMA AMF‐HGL 229600 45 33 243760 ‐23 ‐14160 248640 ‐8 ‐19040 AMF‐ KHM 211380 44 14 223180 ‐33 ‐11800 227020 ‐2 ‐15640 AMF‐LW 164000 ‐12 13 172980 ‐39 ‐8980 175700 0 ‐11700 AMF‐ZL 276440 8 21 298700 ‐51 ‐22260 298220 ‐17 ‐21780 HT‐BD 180100 3 34 188220 ‐16 ‐8120 190540 ‐5 ‐10440 HT‐BORN 195130 ‐38 38 187800 ‐16 7330 188300 ‐3 6830 HT‐ELT 267790 117 39 271180 ‐27 ‐3390 272340 ‐22 ‐4550 HT‐SON 240610 128 41 235320 ‐36 5290 237360 20 3250 WVN‐BD 75270 28 22 74340 ‐18 930 75660 ‐2 ‐390 WVN‐GS 202980 13 22 215100 ‐32 ‐12120 215460 ‐11 ‐12480 WVN‐ HBD 243730 80 26 254460 ‐45 ‐10730 255380 3 ‐11650 WVN‐HW 191730 99 18 187400 ‐53 4330 190480 ‐11 1250 WVN‐ OPM 236440 0 22 256480 0 ‐20040 254480 ‐6 ‐18040 WVN‐RD 225100 131 11 216760 ‐38 8340 221260 ‐13 3840
Table 10; Route Data Summary
AMF‐KHM 211380 44 14 219620 ‐55 ‐8240 220580 ‐72 ‐9200 AMF‐LW 164000 ‐12 13 170020 ‐55 ‐6020 171380 ‐63 ‐7380 AMF‐ZL 276440 8 21 294900 ‐78 ‐18460 295800 ‐97 ‐19360 HT‐BD 180100 3 34 185260 ‐40 ‐5160 187140 ‐50 ‐7040 HT‐BORN 195130 ‐38 38 187340 ‐22 7790 188300 ‐11 6830 HT‐ELT 267790 117 39 269340 ‐59 ‐1550 267240 ‐70 550 HT‐SON 240610 128 41 233480 ‐71 7130 235920 ‐35 4690 WVN‐BD 75270 28 22 73680 ‐21 1590 73340 ‐16 1930 WVN‐GS 202980 13 22 213300 ‐50 ‐10320 212580 ‐43 ‐9600 WVN‐HBD 243730 80 26 252360 ‐53 ‐8630 253060 ‐38 ‐9330 WVN‐HW 191730 99 18 185220 ‐73 6510 185060 ‐64 6670 WVN‐OPM 236440 0 22 245900 ‐48 ‐9460 247260 ‐80 ‐10820 WVN‐RD 225100 131 11 215680 ‐69 9420 217420 0 7680
Table 11; Adjusted with PSF HW and ARIMA
9.7 Appendix G
Route Weekday HW HW
Error ARIMA
Arima
Error Realisation Real error p‐value HW
9.8 Appendix H
Route Weekday HWALT HW
ALTError
ARIMA ALT
Arima
ALT Error Realisation Real error p‐value HW ALT
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9.9 Appendix IRoute Forecast
Method P‐value MAPE Route
Forecast
Method P‐value MAPE
