Optimizing the logistics capacity planning of PostNL parcels using the newsvendor model and option contracting

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Optimizing the logistics capacity planning of PostNL

parcels using the newsvendor model and option

contracting

Jochem Pfeiffer 26-07-2016 Student number: 2203235

MSc thesis Technology and Operations Management

University of Groningen Faculity of Economics and Business

Department of Operations

Supervisor: dr. ir. S. Fazi

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Abstract

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1. Introduction

In the last decade European B2C e-commerce turnover has been growing rapidly. This trend continues in the upcoming years with an expected annual European growth rate of around 15% (Ecommerce 2015). Due to the explosion of e-commerce in the past ten years, the demand for parcel delivery services has increased significantly (Ducret 2014; McWilliams 2010). To satisfy this demand the parcel delivery service industry arose (Ducret 2014). Companies which op-erate in this industry are, for example, United Parcel Service (UPS), Federal Express (FedEx), Deutsche Post (DHL) and PostNL. In general parcel deliv-ery service companies (henceforth PDS) are capable of delivering small, light parcels, quickly and accurately all over the world (Ducret 2014; McWilliams 2010). Today most PDSs have integrated the parcel supply chain from outbound to inbound by relying on sophisticated networks. For the most part, these are hub-and-spoke networks combined with depot networks (Ducret 2014). The parcel delivery market is a global high competitive market (Lin and Lee 2009). To achieve a competitive advantage PDSs should guarantee a high service level to customers. According to Lin and Lee (2009), the two most important criteria in order to satisfy a high service level and which customers have when selecting a PDS are: fast on-time delivery and fair competitive freight rates. Optimizing the logistics capacity planning can be used to guarantee a high service level while costs are minimized. High utilization of planned capacity reduces costs while the right amount of planned capacity guarantees a high service level. In this way the optimization of the logistics capacity planning results in a compet-itive advantage.

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expensive the order (capacity) will be. However, demand information may be to uncertain if one orders too soon. When there are multiple opportunities to order one should make a trade-off between improved demand information and increasing costs (Wang et al. 2012).

This thesis focuses on optimizing the logistics capacity planning of a PDS in order to gain such a competitive advantage. In this thesis PostNL is used as a case study to illustrate the logistics capacity planning process that PDSs are facing. PostNL distributes parcels between their sorting and distribution centers by means of trailer truck combinations (henceforth TTC). The PostNL parcel devision is outsourcing this transportation process from a separate division of PostNL, this means that they need to ‘order’ TTCs for each route for each night to fulfill transportation demand. There are 3 opportunities for PostNL to order the TTCs. The order price of a TTC at each order opportunity increases as the realization gets closer. The first two order opportunities work with a forecast of the expected demand, the first one usually less accurate than the second one. PostNL needs to decide how much TTCs to order at each order opportunity to satisfy demand at minimum costs. Currently PostNL uses a rule of thumb based general planning load factor policy. This might work in the overall situation, however the result might be that when making an order the individual route characteristics are not taken into account. Which could lead to an increase in costs. A route specific planning load factor policy could possibly reduce costs while maintaining high service levels. A more detailed description of the case is given in chapter 2.

In literature there are several papers which discuss the trade-off between im-proved demand information and increased ordering cost. The (extended) newsven-dor model is generally used as a tool to solve these kind of problems. They all take optimal inventory control as their focus. There are several differences made which include multiple ordering opportunities (Wang et al. 2012), the evolution of forecasting (Zheng et al. 2014; Wang et al. 2012; Gurnani and Tang 1999; Sethi et al. 2001), flexibility in changing orders (Huang et al. 2005), the use of different suppliers in the same model (Yan et al. 2009) and orders with capac-ity constraints (Zheng et al. 2016; Miltenburg and Pong 2007). The thesis of Beukeboom (2015) researched the same problem and case as in this thesis using stochastic mathematical programming. The similarities in terms of problems justify the use of the newsvendor model in the PostNL case.

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done by improving their current policy by introducing route specific policies. Secondly, option contracting is introduced to investigate if such alternative con-tracting could minimize costs even further. Option concon-tracting is the type of contracting in which a buyer is able to purchase options in addition to the initial order, in order to obtain the flexibility of adjusting its order in a later stage (Nosoohi and Nookabadi 2016). The scope is limited to the individual transportation routes between the parcel distribution centers of PostNL. In the research a hybrid model is used to determine the optimal capacity planning policy. A hybrid model is a mathematical model which combines simulation and analytic models (Shanthikumar and Sargent 1983). The purpose of using a hybrid model in problem solving would usually be to reduce model developing cost in case of complex problems and/or to reduce the amount of computa-tional effort required to obtain acceptable answers (Shanthikumar and Sargent 1983). The newsvendor problem is mostly used for decision making in perish-able goods inventory or production capacity planning (Chen et al. 2016). The TTCs of PostNL may be considered as perishable goods, because they can be used only once at a given moment in time. The application to minimize costs in logistics capacity planning by means of a newsvendor approach has not been described so far. Besides comparison to the current policy of PostNL, the results are also compared with the results of Beukeboom (2015). Beukeboom (2015) researched the same problem and case as this thesis, which will be discussed in more detail in chapter 3.1. The hybrid models used in this thesis are ca-pable of handling one route at a time and measure 4 KPI’s: average cost per day, average service level, average TTC utilization and average RC service level.

The main research question and sub questions to accomplish this is are:

Can a newsvendor model be adopted to optimize PostNL logistics capacity planning policy?

• What is the optimum policy according to the hybrid model and what are the differences compared with the current policy when minimizing costs? • How does the hybrid model optimum policy differ from the policy derived

by the stochastic mathematical model from Beukeboom (2015).

• What is the effect of guaranteeing a higher service level on average mini-mum cost and how does the earlier derived optimini-mum policy change? • To what extend can option contracting optimize the logistics capacity

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1.1 Structure

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2. Case: PostNL

This chapter discusses the general characteristics of the decision process of PostNL in further depth. Also the current policy applied by PostNL in the decision process is discussed. The case and its characteristics are derived from Beukeboom (2015).

According to the PostNL annual report 2015 (PostNL 2015), PostNL has over 49.000 employees which are mostly stationed in the Benelux. The company exists out of three business segments, parcels is one of them. PostNL operates the biggest parcel network in the Benelux with 18 parcel sorting and distribution centers in the Netherlands. In terms of volume, PostNL delivered 156 million parcels in 2015 with an expected growth of around 10% in 2016. PostNL is a customer focused company.

2.1 Decision process characteristics

Each night TTCs transport parcels between distribution centers of PostNL. The transport between two particular distribution centers are called routes. Parcels are put into roller containers (henceforth RC) which are put into a TTC before transportation. Each TTC has a maximum capacity of 56 RCs. So the amount TCCs needed per transportation night per route is the rounded up to an integer of the demand of RCs to be transported divided by the maximum TTC capacity of 56, which is showed in 2.1.

Required amount of T T Cs = dAmount of RCs

56 e (2.1)

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stage costs e230,- and one TCC in the third stage costs e260,-. Due to time constraints the availability of TCCs in the third stage cannot be guaranteed and therefore it is assumed a maximum of three TTCs can be ordered in stage three. Once ordered, TTCs are not refundable and prior decisions cannot be altered. This means when it turns out that overcapacity is booked this is a pure waste of resources.

2.2 Current policy

To take uncertainty of the forecast into account a fixed planning load factor (PLF) is used to decide in each stage on how much TTCs PostNL has to order. In practice this means that the RC capacity of a TTC is virtually increased or decreased by the PLF. A safety margin is built in if a load factor < 100% is used. For example, if a PLF of 0.9 is used the RC capacity of a TTC becomes 50.4 instead of 56. The PLF is applied on the forecasts of the stages. In this way unexpected higher realization demand can be taken into account. However, if demand stays the same 10% of the available capacity is wasted. This could mean that one or more TCCs will not be used. The consequences are even higher if it turns out the realization demand is lower than the prior forecasts. A similar method is used in the hotel and aviation industry. There a load factor of < 100% is used to take into account no shows.

In the current policy each stage has its own PLF. Stage 1 uses a PLF of 100%, stage 2 uses 95.5% and stage 3 uses 100%. So in stage 2 the capacity of TCCs is virtually decreased as a safety margin for unexpected higher demand realization. This also means that PostNL orders all the TCCs needed to fulfill the first forecast, while this forecast is usually the most uncertain which make it possible that non-refundable overcapacity is ordered. Below two examples are given which illustrate the decision process. Equations 2.2, 2.3 and 2.4 illustrate a situation when current policy of PostNL turns out to be good. Equations 2.5, 2.6 and 2.7 illustrate a situation when the current policy of PostNL turns out to be bad. In the examples Q1/Q2/Q3stands for the amount of TTCs ordered

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Good example Q1= d RC F orecast1 T T C RC capacity · P LF1 e (2.2) = d 600 56 · 1.00e = 11 Q2= max d RC F orecast2 T T C RC capacity · P LF2 e − Q1, 0 (2.3) = maxd 630 56 · 0.955e − 11, 0 = 1 Q3= max d RC demand T T C RC capacitye − Q2− Q1, 0 (2.4) = maxd670 56 e − 1 − 11, 0 = 0 Bad example Q1= d RC F orecast1 T T C RC capacity · P LF1 e (2.5) = d 600 56 · 1.00e = 11 Q2= max d RC F orecast2 T T C RC capacity · P LF2 e − Q1, 0 (2.6) = maxd 630 56 · 0.955e − 11, 0 = 1 Q3= max d RC demand T T C RC capacitye − Q2− Q1, 0 (2.7) = maxd500 56 e − 1 − 11, 0 = 0 (−3) The examples show that when the forecast in stage 2 is inaccurate and realiza-tion is actually lower, this could lead to wasted resources (3 TTCs in this case). The same could happen the other way around. If the actual demand is higher then the earlier demand forecast then more expensive stage 3 TTCs must be ordered, in addition there is a risk of not having enough stage 3 TTCs available to satisfy demand. This could lead to lower service levels.

PostNL uses the current PLFs for all their routes. However this policy could be too general. Routes may have specific characteristics in the amount of demand and forecasting accuracy. This could mean that route specific PLF policies re-duce cost while the same service level is guaranteed.

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Stage 1 3 weeks before realization

Demand forecast 1 €170,- per TCC

PLF of 100%

Stage 2

16 hours before realization Demand forecast 2 €220,- per TCC PLF of 95.5% Stage 3 Realization Demand known €260,- per TCC (≤3) PLF of 100%

Decision 1 Decision 2 Decision 3

Decisions

Stages

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3. Theoretical background

This chapter gives an overview of the literature which is relevant to the PostNL logistics capacity planning problem.

3.1 Thesis of Beukeboom (2015)

Beukeboom (2015) researched the same problem and case as this thesis. A stochastic multistage model is developed to determine optimal PLFs for the PostNL case. To take into account the uncertainty in demand scenarios are used as input for the model. Historical data from PostNL is used to create the scenarios. First a histogram is made from the historical forecast deviations (forecast errors) between stage 1 and 2 and between stage 2 and 3. These histograms are divided in 3 equal parts of which the average forecast deviation is determined, these are the scenarios. The probability is determined by the number of historical measurements for that scenario divided by the total number of historical measurements. In total 9 scenarios are created for each route, which are put together in a scenario tree. The model developed is capable of determining the optimal PLFs for particular day given a particular forecast using the scenario tree of a particular route. This requires the model to be run every day to determine the PLFs for that particular day. When the stage 3 capacity constraint is removed the PLFs are not dependent on the forecasted amount anymore and general PLFs can be determined for each route. In the research it is found that route specific PLFs are able to decrease costs significantly and that a more wait-and-see policy is required to minimize costs (less stage 1 and 2 TTC orders and more stage 3 TTC orders). Suggested in further research is to apply a different approach to the PostNL case, namely the newsvendor model.

3.2 Newsvendor model

3.2.1 Classic newsvendor model

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a lot of extensions have been developed on the original model, which are called extended newsvendor models. The most interesting for the logistic capacity de-cision problem will be discussed further on.

The newsvendor model is one of the most popular analytical models in decision science and operations management (Chen et al. 2016). The standard newsven-dor problem concerns a single period inventory management decision problem in which the decision maker has to decide on the optimal stocking quantity for a single product to be ordered (Chen et al. 2016). For the optimal quantity in order to minimize the expected total costs a trade-off is made between the expected costs of having to much in stock (overage cost) and the expected costs of having not enough in stock (underage cost). The single period newsvendor model alone is not applicable in the postNL case due to the fact that there are more stages.

3.2.2 Relevant extensions

The PostNL decision process can be described as an extended newsvendor prob-lem with the following characteristics: multiple ordering points with increased costs, non-refundable orders, forecast (demand) updates and guaranteeing a high overall service level. There are several relevant extensions of the newsven-dor model in literature which take into account the above characteristics.

Zheng et al. (2014) extend the newsvendor model by introducing an emergency order besides the original order. The model is solved by means of dynamic pro-gramming. The emergency order has a capacity constraint. Which is also the case in the last stage of the PostNL decision process. The idea is that the emer-gency order can be used to place an order based on a more accurate demand forecast. Two types of the Martingale model of forecast evolution (MMFE) are used to assume the differences in successive demand forecasts. Assumed is that the differences are either normally distributed or log normally distributed. A special case is analyzed assuming the emergency order is done using perfect information (e.g. actual demand). This is also the case in the PostNL decision process. However PostNL uses 3 stages in their decision process which makes the model of Zheng et al. (2014) inapplicable in this case.

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in-creasing ordering costs. The model is solved by means of dynamic programming. The market demand is a random variable and is realized after all the ordering periods. In all the ordering periods a forecast update is given and an order is placed. According to the analysis dynamic ordering based on forecast evolu-tion is significantly more valuable in boosting profit when the opevolu-tion of having multiple ordering options is available. They find that multiple orders are most valuable when the market is highly uncertain. This is explained by the fact that the multi ordering strategy provides extra flexibility for the newsvendor to respond to unexpected market conditions. Also when the ordering options are widely spread on the time line multiple orders offer more value due to the advantage of the low-cost of early orders and the low-uncertainty of late orders. Multiple orders are also valuable when no option clearly dominates others if they are considered individually. They let uncertainty diminish linearly over time and show that a state dependent base stock level provides the optimal ordering (up to) policy. There is no capacity constraint modeled (in the last stage), at PostNL this is the case which make the model of Wang et al. (2012) inapplicable.

Gurnani and Tang (1999) extend the newsvendor model by incorporating two order opportunities of which the second one has an uncertain cost by means of a nested newsvendor model. Forecast updates are used in the ordering op-portunities. Interesting is the use of a bivariate normal distribution to link the forecast to the actual demand. This way of forecasting differs from the other papers which use MMFE to model forecasting evolution. Assuming that in the PostNL decision process higher forecasts have a higher probability to realize higher demand, it could be possible to make use of a bivariate normal distri-bution to model the PostNL decision process. They also state that when there is a cost associated with obtaining market information, the decision maker has to trade-off the cost versus the benefit of gathering market information. The model of Gurnani and Tang (1999) uses two stages which make it inapplicable with the PostNL case which uses three stages, also the uncertain cost in the second stage are known in the PostNL case.

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does not have these characteristics which makes the model inapplicable.

3.3 Fleet sizing

The PostNL case has characteristics of a vehicle fleet sizing problem. One should determine the optimal fleet size to be able to meet demand while minimizing costs. List et al. (2003) create a formulation and solution procedure for fleet sizing under uncertainty. Future demands and operating conditions contain this uncertainty. In their solution procedure multiple routes are included (network) in order to optimize them simultaneously. The focus is on robust optimization by using a risk measure. The scope of this thesis are the individual routes of PostNL, which makes List et al. (2003) inapplicable. The same applies to Desrochers and Verhoog (1991), in which a heuristic is developed with the ob-jective to minimize the routing costs by determining the optimal fleet size.

3.4 Option contracting

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Nosoohi and Nookabadi (2016) develop a two-stage mathematical model which determines how much to order and how much additional options must be ordered in order to maximize profits when demand is uncertain. All types of options are used. The two-stage mathematical model of Wang and Tsao (2006) only uses bidirectional options to determine an optimal policy. Liu et al. (2013) develop a two-stage mathematical model to determine the optimal option contracting policy for shipping containers. The newsvendor critical quantile is also used in the model. All types of options are used and results are compared. All the models mentioned are only applicable in two-stage settings, which make them inapplicable in the PostNL case.

3.5 Forecasting

As described in chapter 2, PostNL creates forecasts and uses these for their logistics capacity planning. Forecasting is used in a lot of other industries to predict future behavior. According to Chen and Ou (2009) when done well, forecasting can increase the operational performance, profits and competitive position of a business. However, when forecasting is done bad the opposite is often true. According to Wang et al. (2012) initial demand forecasts that are available for planning are highly inaccurate due to the unpredictable nature of customer preferences and constantly changing market trends. Over time, new information about demand gradually becomes available and the uncertainty resolves as the selling season approaches. Which result in a more accurate forecast. In general, PostNL faces the same phenomenon which means the overall first forecast is usually more inaccurate then the overall second forecast. An example is visualized in figure 3.1 where the overall forecast accuracy of the second forecast is considerably higher (blue points closer to the orange line) than the overall forecast accuracy of the first forecast.

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400 500 600 700 800 900 1000 1100 400 500 600 700 800 900 1000 1100 Fo re cas t Actual demand (a) Forecast 1 400 500 600 700 800 900 1000 1100 400 500 600 700 800 900 1000 1100 Fo recas t Actual demand (b) Forecast 2

Figure 3.1: Historical forecast accuracy PostNL route HT-ELT

3.5.1 Forecast accuracy measures

The absolute forecast error is the difference between the actual value and the forecast value for the corresponding period and showed in equation 3.1. Where Fi is the forecast value and Ai is the actual value.

Ei= Ai− Fi (3.1)

According to de De Gooijer and Hyndman (2006) there are a lot of different mea-sures to determine overall forecast accuracy. The most commonly known and used measures are the MSE (mean squared error) which is showed in equation 3.2 and MAPE (mean absolute percentage error) which is showed in equation 3.3. M SE = 1 n n X i=1 (Fi− Ai)2 (3.2) M AP E = 1 n n X i=1 Ai− Fi Ai (3.3)

Because the MSE measure is absolute, it cannot be used to make comparisons of series on different scales. The MAPE measure is relative, so it does not have this disadvantage. However, one must be keep in mind that the MAPE measure is more sensitive when quantities are low. Which could make it unsuitable for low quantity forecasting. Cachon and Terwiesch (2009) propose another forecast accuracy measure. They call it the A/F ratio which gives a relative forecast error and is showed in equation 3.4.

A/F ratio =Ai Fi

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4. Methodology

In this chapter the methodology is discussed. First, to be able to perform the numerical analysis mathematical models are derived from the case description. Which are described in more detail below. The models make it possible to an-alyze the TTC capacity planning process of PostNL. The models consists out of 3 stages, in which each a decision must be made on how much TTCs should be ordered. Option contracting models allow to use options next to the normal ordering process. In appendix A all the models are written in full mathematical notation. The models are programmed in R using R studio in order to be able to run the simulation. Second, the application of forecast error in the newsven-dor model which is also used in the hybrid models is explained in more depth. Third, the historical data analysis is performed which will be used as input for the mathematical models.

4.1 Mathematical models

This section explains the mathematical models in depth. These models are pro-grammed using the R programming language in the R Studio software package. R studio is able to run the simulation parts in the hybrid models. It also takes care of the analytical newsvendor part by programming the newsvendor critical quantile equations. Random seed 756373 is used to generate all random num-bers. Besides the development of the mathematical models below the model from Beukeboom (2015) is implemented in AIMMS in order to compare PLFs. The stage 3 capacity constraint is removed due to the fact that this constraint will cause a different P LF1 and P LF2 for each specific day (dependent on the

amount of forecasted RCs in stage 1 and 2). All the mathematical models in-cluding its parameters are fully written in mathematical notation in appendix A.

In total there are 4 KPIs which the models are able to measure. Besides the average cost per day there are service level, RC service level and TTC utilization. They are defined as follows:

• Service level is defined as the percentage of times no more than 3 extra TTCs need to be ordered in stage 3.

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• TTC utilization is defined as the average utilized percentage of all ordered TTCs. The utilization of a TTC is 0% if the TTC is not used at all and the utilization of a TTC is 100% if the TTC transports at least one RC.

4.1.1 Model assumptions

In order to fit the mathematical models to the PostNL case and to be able to handle its complexity the following assumptions are made and used in developing the mathematical models.

• Future forecast errors are identical to the historical forecast errors • Forecast errors are normally distributed and their parameters are known • Demand (forecast 1) is normally distributed and its parameters are known • The forecast error distributions are independent (no correlation)

• In each stage only its forecast, historical distributions (forecast errors and demand (forecast 1)) and decisions from previous stages are known • Each TTC is identical and has a capacity of 56 RCs

• There is a maximum of 3 TTCs available in stage 3, using more will decrease the service level

• Once a TTC is ordered there is no refund possible (except for the hybrid put and bidirectional option models)

• The cost to order a trailer per stage is known and is the same for each route

• The cost of the different types of options is known

4.1.2 Basic hybrid model

This model is used to derive P LF1 and P LF2 in order to minimize the average

cost per day per route. In the hybrid approach two A/F ratio distributions are used. The first one is actually derived from the historical F orecast2

F orecast1 ratio, so the forecast accuracy between stage 1 and 2 (equation 4.6). The second one is derived from the historical Actual demand_{F orecast}

2 ratio, so the forecast accuracy be-tween stage 2 and 3 (equation 4.7). From the historical A/F ratios the mean and standard deviation are determined in order to use them as the forecast error distributions. µ12 and σ12 are the mean and standard deviation of the

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and standard deviation of the forecast error distribution between stage 2 and 3. Besides the forecast error distributions the F orecast1distribution is determined

by historical data to be used in simulating the amount of RCs to be transported per route (equation 4.5). The way the forecast errors are used and evolve from stage to stage is the same as in Beukeboom (2015). Beukeboom (2015) also uses the forecast error between stage 1 and 2 and the forecast error between stage 2 and 3 to create scenarios, which are used in the stochastic model. This makes comparison of the results possible.

To determine P LF2, the newsvendor approach in section 4.2 is used. First the

critical quantile z is determined. In stage 2 the underage cost is the additional cost when it turns out that an extra TTC is needed in stage 3, which is the TTC ordering cost in stage 3 subtracted by the TTC ordering cost in stage 2 (260 − 220 = 40). In stage 2 the overage cost is the cost when it turns out that a TTC ordered in stage 2 is not used in stage 3 (which is a pure waste of resources). This is the full stage 2 TTC ordering price (220). Equation 4.2 shows the calculation of the stage 2 critical quantile. The optimal adjustment factor can be determined using the historical forecast error distribution combined with the stage 2 critical quantile (equation 4.3). Last, the optimal adjustment factor is converted to P LF2using equation 4.4.

z = Φ−1

_{U nderage cost}

U nderage cost + Overage cost (4.1) = Φ−1 40 40 + 220 (4.2) A∗23= µ23+ Φ−1 ₄₀ 40 + 220 σ23 (4.3) P LF2= 1 A∗₂₃ (4.4)

The simulation is used to determine P LF1, in which P LF1 is a variable. The

variable is set as a sequence of numbers with a small interval between them. P LF1 is optimal when the mean average cost per day is minimum. A

descrip-tion of how the simuladescrip-tion works will be given below. The simuladescrip-tion works the same for all the other models.

The simulation uses the following equations. The number of iterations used are the number of days simulated. For each day three decisions must be made: the amount of TTC’s to order in stage 1 (Q1), stage 2 (Q2) and stage 3 (Q3). The

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for example if the simulation simulates 1000000 days, a vector is created with 1000000 F1’s. The same applies for 12, 23, F2 and D.

F1∼ N (µD, σD) (4.5) 12∼ N (µ12, σ12) (4.6) 23∼ N (µ23, σ23) (4.7) F2= F112 (4.8) D = F223 (4.9) Decision stage 1 Q1= d F1 T · P LF1 e (4.10)

Q1is the first decision made by the model, which is the amount of TTCs ordered

in the first stage. The input is a random number F1 to simulate the amount

forecast of RCs in stage 1 (equation 4.5). T = 56, the RC capacity of each TTC. And P LF1 is the planning load factor of stage 1, which is a variable in

the simulation. For every F1a variable sequence of P LF1 is applied and Q1 is

determined. For every simulated day this decision is made the amount of times the length of the variable sequence of P LF1.

Decision stage 2 Q2= max d F2 T · P LF2 e − Q1, 0 (4.11)

Q2 is the second decision made by the model, which is the amount of TTCs

ordered in the second stage. The inputs are the generated random number F2

(equation 4.8) and the decision of stage 1 (Q1). P LF2is the planning load factor

of stage 2, which is already determined optimal by the newsvendor model (equa-tion 4.4). This means that there is not a sequence of variables used here. The optimal amount of TTCs is reduced by the earlier ordered TTCs in stage 1(Q1).

There will be no TTCs ordered in stage 2 (Q2 = 0), if the ordered amount in

stage 1 (Q1) is larger than the optimal amount in stage 2 (d_{T ·P LF}F2 ₂e). For

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Q3 is the third decision made by the model, which is the amount of TTCs

or-dered in the third stage. This is however not really a decision since the demand (D) must be met. The inputs are the generated random number D (equation 4.9) and the decisions of stage 1 (Q1) and 2 (Q2). The amount of TTCs needed

to satisfy demand (dD_Te) is reduced by the earlier ordered TTCs in stage 1 and 2. There will be no TTCs ordered in stage 3 (Q3 = 0), if the sum of ordered

amount in stage 1 and 2 is larger than the amount needed to satisfy the demand. For every simulated day this decision is made the amount of times the length of the variable sequence of P LF1.

In this basic hybrid model there is no hard capacity/service level constraint, so from the simulation the P LF1 that is optimal is the one which minimizes the

average cost/day.

T otal cost simulated day = Q1C1+ Q2C1+ Q3C3 (4.13)

The equation above determines the total cost of a simulated day. For each value in the sequence of variable P LF1 this is done each day. Afterwards the mean

cost per day per value of P LF1 can be determined in order to determine the

optimal P LF1 that will minimize the average cost/day.

Set (P LF1) contains the average cost per day per element of P LF1 (n =

number of simulated days):

Average cost/day = Pn

i=1Q1C1+ Q2C1+ Q3C3

n (4.14)

The objective is to minimize the average cost/day. By selecting the element of set P LF1 with the lowest average cost/day.

M in z = Average cost/day ∀ P LF1 (4.15)

A schematic overview of the hybrid approach is given in figure C.1. The hori-zontal arrows between the stages show how the forecasts evolve during the sim-ulation. The bended arrows show where the determined optimal P LF per stage is based on. Because of the dependence between the stages, P LF1 is based on a

combination of the F orecast2

F orecast1 simulation decision and the

Actual

F orecast2 newsvendor

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4.1.3 Hybrid service level model

The basis of this model is the same as the one in the previous section (4.1.2). Stage 1 stays the same, but an extension is made in the second stage in order to be able to guarantee a certain service level α. Next to the newsvendor optimal adjustment factor and the P LF2from the previous model a second adjustment

factor is determined which is required to guarantee a certain service level α. Equation 4.19 shows the PLF in stage 2 which can guarantee service level α.

α = service level (4.16) zα= Φ−1(α) (4.17) Aα23= µ23+ zασ23 (4.18) P LF₂α= 1 Aα 23 (4.19)

In stage 2 a decision must be made on whether to choose for the newsvendor order (like in the basic model to minimize the expected cost) or the service level order using the PLF derived in equation 4.19 to meet a required service level α. Equation 4.20 shows this decision. Notice the −M3, which represents

the maximum extra TTC capacity available to order in stage 3. This capacity is subtracted from the service level order because the service level α could be met if it turns out that maximum 3 additional TTCs need to be ordered in the third stage. The newsvendor order will be chosen if the service level order subtracted by the stage 3 capacity is smaller than the newsvendor order, which will minimize the expected cost. On low volume routes the newsvendor order will be chosen more frequently than the service level order because the 3 additional trailers available in stage 3 is relatively much. Of course this also depends on the required service level α.

Q2= max maxd F2 T · P LFα 2 e − M3, d F2 T · P LF2 e− Q1, 0 (4.20)

The objective of this model is to minimize the average cost/day (just as the basic model in section 4.1.2) combined with a minimum required overall service level. In this model not only an optimal P LF1 needs to be determined but also

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4.1.4 Hybrid option contracting model

The basis of this model is the same as the one in section (4.1.2). An extension is made in the first and second stage in order to be able to buy options in the first stage and to utilize them in the second stage. An option load factor (OLF) is introduced which determines the amount of options to be bought/utilized in the first/second stage. The price of the options in stage 1 is set in advance. And in this thesis the utilization cost/refund of an option in stage 2 is the same as the stage 1 TTC order cost. OLF1 is determined by simulation and

OLF2 can be determined by the newsvendor model (just like P LF2). There

are 3 different sub models which each represent there own type of option: call, put and bidirectional. For each sub model OLF2 is the same because the cost

structures do not differ. C_uo= C3− OC2,

underage cost of using an option in stage 2 C_oo= OC2,

overage cost of using an option in stage 2 z₂₃o = Φ−1 _Co u Co u+ Coo

(newsvendor critical quantile) (4.21)

Ao₂₃= µ23+ z23o σ23 (4.22) OLF2= 1 Ao 23 (4.23) Decision stage 1

For every sub model the buying of options in stage 1 is the same. OLF1should

be lower than P LF1in order to be able to buy options in the model. O1 is the

amount of options bought in the first stage. The ratio between OLF1and P LF1

can be used to determine the percentage of the forecast which will be bought as options. Q1= d F1 T · P LF1 e (4.24) O1= max d F1 T · OLF1 e − Q1, 0 (4.25) Decision stage 2

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for the price of stage 1. In the bidirectional model both ways are possible. O2

is the amount of the earlier bought options that is being utilized in stage 2. In the bidirectional model options become call if d F2

T ·OLF2e − Q1> Q1− d F2 T ·OLF2e, and put if otherwise.

Call options O2= min maxd F2 T · OLF2 e − Q1, 0 , O1 (4.26) Q2= max d F2 T · P LF2 e − Q1− O2, 0 (4.27) (4.28) Put options O2= min maxQ1− d F2 T · OLF2 e, 0, O1 (4.29) Q2= max d F2 T · P LF2 e − (Q1− O2), 0 (4.30) (4.31) Bidirectional options O2= min maxd F2 T · OLF2 e − Q1, Q1− d F2 T · OLF2 e, O1 (4.32) Q2=    maxd F2 T ·P LF2e − Q1− O2, 0 , if call maxd F2 T ·P LF2e − (Q1− O2), 0 , if put (4.33) (4.34) Decision stage 3

The sub models are also different in the third stage and in the way how the costs of a particular day are calculated.

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Put options Q3= max dD Te − (Q1− O2) − Q2, 0 (4.37) T C = Q1C1+ Q2C1+ Q3C3+ O1OC1− O2OC2 (4.38) Bidirectional options Q3=    maxdD Te − Q1− O2− Q2, 0 , if call maxdD Te − (Q1− O2) − Q2, 0 , if put (4.39) T C =    Q1C1+ Q2C1+ Q3C3+ O1OC1+ O2OC2, if call Q1C1+ Q2C1+ Q3C3+ O1OC1− O2OC2, if put (4.40)

The objective of this model is to minimize the average cost/day (just as the basic model in section 4.1.2). In this model not only an optimal P LF1needs to

be determined but also OLF1 in order to determine the amount of options to

buy. A combined model of the hybrid service level model and the hybrid option contracting models is also developed. These models are showed in appendix A.2.6, A.2.8 and A.2.4.

4.2 Forecast error and the newsvendor model

As mentioned in the previous section the newsvendor model is used in the second stage to decide on the amount of TTCs to order. In this section the underlying theory is discussed in more depth. Given that the historical A/F ratios reflect the historical forecast accuracy, it is possible that the current forecast accuracy is comparable Cachon and Terwiesch (2009). This means that one could as-sume the future forecast accuracy is equal to the historical forecast accuracy if the forecasting method stays the same and there are not any other significant changes which could influence the forecasting process.

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cumulative empirical distribution function is plotted. Figure 4.2b shows that the empirical distribution has a good fit with the orange line, which is a normal distribution with µ = 0.96 and σ = 0.09.

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 0,7 0,8 0,9 1 1,1 1,2 P ro ba bi lt y A/F ratio (a) Empirical CDF 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 0,7 0,8 0,9 1 1,1 1,2 P roba bi lt y A/F ratio (b) Fitted normal CDF

Figure 4.1: CDF of historical A/F ratio of forecast 2, PostNL route HT-ELT

As with the empirical cumulative distribution function the same can be done with the empirical probability density function (histogram). Figure 4.2a shows the histogram derived from the historical A/F ratios and figure 4.2b shows the fitted normal distribution as in figure 4.1b.

A/F ratio Density 0.7 0.8 0.9 1.0 1.1 1.2 0 1 2 3 4 5

(a) Empirical PDF (histogram)

A/F ratio Density 0.7 0.8 0.9 1.0 1.1 1.2 0 1 2 3 4 5 (b) Fitted normal PDF

Figure 4.2: PDF of historical A/F ratio forecast 2, PostNL route HT-ELT

According to the fitted distribution in figures 4.1 and 4.2, the expected actual realization is 0.96 · F2 with an uncertainty of 0.09 · F2. Assumed is that the

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equation 4.41.

z = Φ−1

_{U nderage cost}

U nderage cost + Overage cost

(4.41)

Overage cost is defined as the cost of buying one unit more than demand during the selling season. Generally this is the unit cost subtracted by the salvage revenue. Underage cost is defined as the cost of buying one unit short than demand during the selling season. Generally this is the shortage or stock out cost (Hill 2011). When U nderage cost > Overage cost, the critical value will be positive. And when U nderage cost < Overage cost, the critical value will be negative. So when the cost of having one product to much is higher (lower) than having one product to little the critical value is negative (positive). The optimal order quantity according to the classical newsvendor model is given in equation 4.42, where µ is the mean demand and σ the standard deviation of the demand. Here z is also known as the safety factor, zσ as the safety stock and Q∗ as the optimal base-stock level.

Q∗= µ + zσ (4.42)

The optimal order quantity will be lower than µ when the cost of having to much is higher than the cost of having to little due to a negative safety stock. Also the safety stock will get bigger if uncertainty (σ) rises. In order to apply the newsvendor model in a forecasting context an adjustment needs to be made (Cachon and Terwiesch 2009). This adjustment is showed in equation 4.43. In which µ is the fitted mean historical A/F ratio, σ is the fitted standard deviation of the historical A/F ratio and F is the forecast.

Q∗= µF + zσF (4.43)

= F (µ + zσ) (4.44)

A more general way to determine the optimal base-stock quantity is to first use equation 4.45 to determine an optimal forecast adjustment factor. The product of the adjustment factor and the forecast (equation 4.46) will give the same optimal base-stock quantity as given in equation 4.43/4.44.

A∗= µ + zσ (4.45)

Q∗= F A∗ (4.46)

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input for the hybrid model later on.

4.3 Analysis of historical data

The historical data is analyzed to be able to use it as input for the model. Per route there are 3 types of historical data which must be derived and analyzed, two forecast error distributions and one demand (forecast 1) distribution. As-sumed is that all distributions are normally distributed. This assumption must be checked.

4.3.1 Data set description

The data set provided by PostNL contains the historical data of 17 routes. The timespan of the data is from 16-03-2015 till 21-02-2016 and contains working days only (Monday to Friday). Each route/day combination contains a long-term forecast, short-long-term forecast and realization of RC demand. There are 2 routes (WVN-BD and WVN-WIL) that are not operational anymore due to the fact that these routes do not have data available till the end of the timespan. These routes will not be covered by the analysis in this thesis. All routes have one week gaps in the available data with a total of 7 weeks of data per route. Of the remaining 15 routes there are 2 routes (AMF-WIL and HT-WIL) which have an additional large gap (9 and 11 weeks) in the available data. These routes will also not be covered by the analysis of this thesis due to the large data gaps and because these are relatively small routes. The historical data of the 13 remaining routes is further analyzed.

4.3.2 Removing outliers and fitting distributions

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5. Numerical results

In this chapter the numerical results will be discussed. First the basic hybrid model results are compared with the current policy and with the results deter-mined by the stochastic programming model from Beukeboom (2015). Second, to take the stage 3 TTC capacity into account the routes which performed be-low a service level of 99.5 are further analyzed to determine the policy needed to guarantee a higher service level. Third, the use of option contracting in the PostNL case is analyzed.

5.1 Basic model PLFs

5.1.1 Derived results and determined PLFs

In table 5.1 the results from the simulation using the current policy are showed. Table 5.2 shows the results from the hybrid model. And table 5.3 shows the simulation results when using the PLFs derived by the stochastic model from Beukeboom (2015). The stage 3 capacity constraint in the model from Beuke-boom (2015) is removed in order to make comparison possible.

Route P LF1 P LF2 Cost/day TTC utilization Service level RC service level

AMF-HGL 1 0.955 2079.59 93.51 100.00 100.00 AMF-LW 1 0.955 1488.72 92.63 100.00 100.00 AMF-ZL 1 0.955 2596.36 93.84 100.00 100.00 HT-BD 1 0.955 1495.50 88.63 100.00 100.00 HT-ELT 1 0.955 2467.98 87.55 100.00 100.00 HT-SON 1 0.955 2212.52 86.48 100.00 100.00 WVN-GS 1 0.955 1761.10 95.28 100.00 100.00 WVN-HBD 1 0.955 2247.43 92.40 100.00 100.00 WVN-HW 1 0.955 1859.72 85.72 99.94 100.00 WVN-OPM 1 0.955 2135.59 93.15 99.99 100.00

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Route P LF1 P LF2 Cost/day TTC utilization Service level RC service level AMF-HGL 1.0409 1.0763 2002.38 97.80 99.94 100.00 AMF-LW 1.0055 1.1144 1426.68 96.80 99.91 100.00 AMF-ZL 0.964 1.0901 2468.32 97.73 99.19 99.98 HT-BD 1.0016 1.2001 1370.41 96.17 99.65 99.99 HT-ELT 1.195 1.1265 2293.93 97.12 98.79 99.97 HT-SON 1.2307 1.1143 2030.39 97.39 99.66 99.99 WVN-GS 1.0057 1.0502 1717.17 97.89 100.00 100.00 WVN-HBD 1.0926 1.0764 2166.36 97.60 99.74 99.99 WVN-HW 1.2468 1.1645 1740.89 95.56 97.26 99.88 WVN-OPM 1.0061 1.0934 2060.00 96.82 99.07 99.97

Table 5.2: Hybrid basic model results

Route P LF1 P LF2 Cost/day TTC utilization Service level RC service level

AMF-HGL 1.0518 1.0705 2002.74 97.99 99.95 100.00 AMF-LW 1.0410 1.0912 1428.74 97.27 99.95 100.00 AMF-ZL 0.9889 1.0698 2471.91 98.09 99.47 99.99 HT-BD 1.0481 1.1775 1372.49 96.83 99.76 99.99 HT-ELT 1.2438 1.1333 2297.59 97.95 98.51 99.96 HT-SON 1.2686 1.1107 2032.57 97.96 99.68 99.99 WVN-GS 1.0114 1.0477 1717.28 97.99 100.00 100.00 WVN-HBD 1.0957 1.0600 2167.10 97.53 99.84 100.00 WVN-HW 1.2770 1.1879 1741.96 96.16 96.45 99.83 WVN-OPM 1.0146 1.0733 2060.95 96.82 99.40 99.98

Table 5.3: Simulated results using PLFs from stochastic model (Beukeboom 2015)

Table 5.2 and 5.3 show that each route has their specific P LF1 and P LF2.

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5.1.2 Comparison of results

In table 5.4 and 5.5 the results from the previous section are compared.

PLFs Cost/day TTC utilization Service level RC service level Route ∆P LF1 ∆P LF2 ∆ ∆(%) ∆ ∆ ∆ AMF-HGL -0.041 -0.081 -77.21 -3.71 4.29 -0.06 0.00 AMF-LW -0.006 -0.119 -62.04 -4.17 4.17 -0.09 0.00 AMF-ZL 0.036 -0.095 -128.04 -4.93 3.89 -0.81 -0.02 HT-BD -0.002 -0.205 -125.09 -8.36 7.53 -0.35 -0.01 HT-ELT -0.195 -0.132 -174.05 -7.05 9.56 -1.21 -0.03 HT-SON -0.231 -0.119 -182.14 -8.23 10.91 -0.34 -0.01 WVN-GS -0.006 -0.055 -43.93 -2.49 2.61 0.00 0.00 WVN-HBD -0.093 -0.081 -81.07 -3.61 5.20 -0.26 -0.01 WVN-HW -0.247 -0.170 -118.83 -6.39 9.84 -2.68 -0.12 WVN-OPM -0.006 -0.098 -75.59 -3.54 3.67 -0.91 -0.03

Table 5.4: Hybrid results compared with current policy

PLFs Cost/day TTC utilization Service level RC service level Route ∆P LF1 ∆P LF2 ∆ ∆(%) ∆ ∆ ∆ AMF-HGL 0.011 -0.006 -0.37 -0.02 -0.19 -0.01 0.00 AMF-LW 0.035 -0.023 -2.06 -0.14 -0.47 -0.04 0.00 AMF-ZL 0.025 -0.020 -3.59 -0.15 -0.36 -0.29 -0.01 HT-BD 0.047 -0.023 -2.08 -0.15 -0.66 -0.10 0.00 HT-ELT 0.049 0.007 -3.66 -0.16 -0.83 0.28 0.01 HT-SON 0.038 -0.004 -2.18 -0.11 -0.58 -0.02 0.00 WVN-GS 0.006 -0.003 -0.12 -0.01 -0.11 0.00 0.00 WVN-HBD 0.003 -0.016 -0.74 -0.03 0.07 -0.10 0.00 WVN-HW 0.030 0.023 -1.08 -0.06 -0.60 0.81 0.04 WVN-OPM 0.009 -0.020 -0.95 -0.05 0.00 -0.32 -0.01

Table 5.5: Hybrid results compared with Beukeboom (2015)

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which is caused by ordering less stage 1 and stage 2 TTCs. Only one route does not have a decrease in service level and in some routes the service level decreases rather much (HT-ELT and WVN-HW). In general the RC service level is only decreased minimally.

Table 5.5 shows the differences between the stochastic model without a stage 3 capacity constraint from Beukeboom (2015) and the basic hybrid model. The differences are rather small. This makes sense because the same data is used to derive forecast error distributions and the scenarios. The differences could possibly be smaller if more scenarios were derived than the original 9 from Beukeboom (2015) because possibly it would better fit with the forecast error distributions.

5.2 Service level model

In this section the hybrid service level model is used in combination with the routes which have a service level below 99.5% using the PLFs determined by the basic hybrid model. These routes are the most interesting to be further analyzed by the hybrid service level model in order to see what the additional costs will be if one would want to guarantee a certain service level. The following routes are included: HT-ELT, AMF-ZL, WVN-OPM and WVN-HW.

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Service level 98.79 99 99.25 99.5 99.75 99.95 Cost/day 2293.93 2294.21 2295.11 2297.37 2303.33 2330.24 α 0 92.3 95.3 97.3 98.7 99.7 P LF1 1.195 1.193 1.193 1.193 1.193 1.182 Table 5.6: HT-ELT

For the HT-ELT route the average cost per/day without guaranteeing a certain service level is 2293.93 with an overall service level of 98.79% (table 5.6). Using the current policy a service level of 100% is guaranteed, but on average 2467.98 is paid each day. If assumed that a service level of 99.75% (once every 400 days more than 3 stage 3 TTCs are needed) is sufficient, the average cost per day is 2303.33. P LF1is slightly lower, which means that in the first stage more TTCs

are ordered. The parameter α in the model needs to be set to 98.7 in order to get this result. Table 5.6 also shows the average cost per day for other service levels, the required P LF1 and parameter α.

0,946 0,949 0,952 0,955 0,958 0,961 0,964 2460 2470 2480 2490 2500 2510 2520 99,2 99,3 99,4 99,5 99,6 99,7 99,8 99,9 100 P LF 1 Av er ag e co st /da y Service level Average cost/day PLF1 Figure 5.2: AMF-ZL Service level 99.19 99.25 99.5 99.75 99.95 Cost/day 2468.32 2468.33 2469.30 2473.08 2493.69 α 0 92.1 96.5 98.5 99.7 P LF1 0.964 0.963 0.963 0.961 0.954 Table 5.7: AMF-ZL

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service level is 2468.32 with an overall service level of 99.19% (table 5.7). Using the current policy a service level of 100% is guaranteed, but on average 2596.36 is paid each day. If assumed that a service level of 99.75% (once every 400 days more than 3 stage 3 TTCs are needed) is sufficient, the average cost per day is 2473.08. P LF1is slightly lower, which means that in the first stage more TTCs

are ordered. The parameter α in the model needs to be set to 98.5 in order to get this result. Table 5.7 also shows the average cost per day for other service levels, the required P LF1 and parameter α.

0,993 0,996 0,999 1,002 1,005 1,008 2055 2065 2075 2085 2095 2105 99,1 99,2 99,3 99,4 99,5 99,6 99,7 99,8 99,9 100 P LF1 Av er ag e co st /da y Service level Average cost/day PLF1 Figure 5.3: WVN-OPM Service level 99.07 99.25 99.5 99.75 99.95 Cost/day 2060.00 2060.25 2061.46 2065.50 2084.76 α 0 92.9 96.4 98.5 99.9 P LF1 1.0061 1.006 1.006 1.004 0.998 Table 5.8: WVN-OPM

For the WVN-OPM route the average cost per day without guaranteeing a certain service level is 2060.00 with an overall service level of 99.19% (table 5.8). Using the current policy a service level of 99.99% is guaranteed, but on average 2135.59 is paid each day. If assumed that a service level of 99.75% (once every 400 days more than 3 stage 3 TTCs are needed) is sufficient, the average cost per day is 2065.50. P LF1 is slightly lower, which means that in the first

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1,19 1,2 1,21 1,22 1,23 1,24 1,25 1730 1750 1770 1790 1810 1830 1850 97,4 97,92 98,44 98,96 99,48 100 P LF 1 Av er ag e co st /da y Service level Average cost/day PLF1 Figure 5.4: WVN-HW Service level 97.26 98 98.5 99 99.25 99.5 99.75 99.95 Cost/day 1740.89 1742.02 1743.88 1747.94 1751.37 1757.60 1767.41 1817.59 α 0 89.3 92.9 95.7 96.7 98.1 99.1 99.9 P LF1 1.2468 1.242 1.242 1.242 1.242 1.237 1.232 1.209 Table 5.9: WVN-HW

For the WVN-HW route the average cost per day without guaranteeing a cer-tain service level is 1740.89 with an overall service level of 97.26% (table 5.9). Using the current policy a service level of 99.94% is guaranteed, but on average 1859.72 is paid each day. If assumed that a service level of 99.75% (once every 400 days more than 3 stage 3 TTCs are needed) is sufficient, the average cost per day is 1757.60. P LF1 is lower, which means that in the first stage more

TTCs are ordered. The parameter α in the model needs to be set to 98.5 in order to get this result. Table 5.9 also shows the average cost per day for other service levels, the required P LF1and parameter α.

Figures 5.1, 5.2, 5.3 and 5.4 show a plot of the all the results from the hy-brid service level model. The average cost per day increases exponentially for all routes. P LF1 seems to do the same, only negatively. Which makes sense,

because more TTCs are needed to guarantee a higher service level. And it is cheaper to buy these TTCs in the first stage. The plot of P LF1is not smooth,

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The numerical results show that a higher service level can be guaranteed with a relatively low increase in cost per day when using the hybrid service level model compared to the derived PLFs from the basic hybrid model. And a relatively high decrease in cost per day is possible if a service level lower than 100% is sufficient when using the hybrid service level model compared to the current policy. Only at service level 99.95 of route WVN-HW the cost increase is higher than the cost decrease. This is summarized in table 5.10. Where the current PLFs and the ones derived in section 5.1, table 5.2 are compared with the results from the hybrid service level model.

Route Service level

99 99.25 99.5 99.75 99.95 HT-ELT ∆ current -173.77 (-7.04%) -172.87 (-7.00%) -170.61 (-6.91%) -164.65 (-6.67%) -137.74 (-5.58%) ∆ hybrid 0.28 (0.01%) 1.18 (0.05%) 3.44 (0.15%) 9.4 (0.41%) 36.31 (1.58%)

AMF-ZL ∆ current n/a -128.03

(-4.93%) -127.06 (-4.89%) -123.28 (4.75%) -102.67 (-3.95%) ∆ hybrid n/a 0.01 (0.00%) 0.98 (0.04%) 4.76 (0.19%) 25.37 (1.03%)

WVN-OPM ∆ current n/a -75.34

(-3.53%) -74.13 (-3.47%) -70.09 (-3.28%) -50.83 (-2.28%) ∆ hybrid n/a 0.25 (0.01%) 1.46 (0.07%) 5.5 (0.27%) 24.76 (1.20%) WVN-HW ∆ current -111.78 (-6.01%) -108.35 (-5.83%) -102.12 (-5.49%) -92.31 (-4.96%) -42.13 (-2.27%) ∆ hybrid 7.05 (0.40%) 10.48 (0.60%) 16.71 (0.96%) 26.52 (1.52%) 76.7 (4.41%) Table 5.10: Difference average cost/day guaranteeing a certain service level

5.3 Option contracting

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are bought in the first stage and how many TTCs are bought in the first stage. Second the cost is fixed on the initial minimal average cost per day (option cost of 50), because it is arguable if the TTC subcontractor agrees to decreasing option prices if he makes less revenue. The cost/refund of utilizing an option in stage 2 is the same as the cost of an TTC in stage 1, which is 170.

0 0.5 1 1.5 2 2.5 3 3.5 4 5 10 15 20 25 30 35 40 45 50 % co st decre as e Option cost Call Put Bi (a) WVN-HW 0 0.5 1 1.5 2 2.5 3 3.5 4 5 10 15 20 25 30 35 40 45 50 % co st decre as e Option cost Call Put Bi (b) HT-ELT

Figure 5.5: Percentage cost decrease for a given option cost

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 5 10 15 20 25 30 35 40 45 50 % ser vice leve l in cre as e Option cost Call Put Bi (a) WVN-HW 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 5 10 15 20 25 30 35 40 45 50 % ser vice leve l in cre as e Option cost Call Put Bi (b) HT-ELT

Figure 5.6: Percentage service level increase for a given option cost

0 5 10 15 20 25 30 35 40 45 50 5 10 15 20 25 30 35 40 45 50 % f o recas t 1 in op tio ns Option cost Call Put Bi (a) WVN-HW 0 5 10 15 20 25 30 35 40 45 50 5 10 15 20 25 30 35 40 45 50 % f o recas t 1 in op tio ns Option cost Call Put Bi (b) HT-ELT

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60 65 70 75 80 85 90 95 100 105 110 5 10 15 20 25 30 35 40 45 50 % f o recas t 1 in T TC s Option cost Call Put Bi (a) WVN-HW 60 65 70 75 80 85 90 95 100 105 110 5 10 15 20 25 30 35 40 45 50 % f o recas t 1 in T TC s Option cost Call Put Bi (b) HT-ELT

Figure 5.8: Percentage of TTCs ordered in stage 1 for a given option cost

Figures 5.5, 5.6, 5.7 and 5.8 show the numerical results of the introduced option contracting. For the WVN-HW route options are bought if the cost of a call option is less than 35, if the cost of a put option is less then 25 and if the cost of a bidirectional option is less then 50. For the HT-ELT route options are bought if the cost of a call option is less then 30, if the cost of a put option is less then 25 and if the cost of a bidirectional option is less then 45. Figure 5.5 shows that the cost decrease faster for the WVN-HW route, which indicate that more uncertain routes could benefit more from option contracting. The figure also shows that bidirectional options decrease cost further than call and put options, because bidirectional options are twice as flexible. Call options decrease cost just minimally more than put options, this difference can be explained by the fact that P LF1 and P LF2 are >1 which means that less than the forecasted

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for the put options. 0 0.5 1 1.5 2 2.5 5 10 15 20 25 30 35 40 45 50 % ser vice leve l in cre as e Option cost Call Put Bi (a) WVN-HW 0 0.5 1 1.5 2 2.5 5 10 15 20 25 30 35 40 45 50 % ser vic e leve l in creas e Option cost Call Put Bi (b) HT-ELT

Figure 5.9: Percentage service level increase for a given option cost when average cost/day is fixed 0 20 40 60 80 100 120 140 160 180 5 10 15 20 25 30 35 40 45 50 % f o re ca st 1 in o pt io ns Option cost Call Put Bi (a) WVN-HW 0 20 40 60 80 100 120 140 160 180 5 10 15 20 25 30 35 40 45 50 % f o recas t 1 in o pt io ns Option cost Call Put Bi (b) HT-ELT

Figure 5.10: Percentage of options bought in stage 1 for a given option cost when average cost/day is fixed

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6. Discussion

In this chapter the research insights, limitations and suggestions for further re-search are discussed.

The numerical results show that route specific PLFs could reduce costs sig-nificantly, because routes have each there individual characteristics which re-quires an individual policy in order to minimize costs. This also shows that the newsvendor model can be adopted as part of a hybrid model in the PostNL case, however the newsvendor model alone does not make a hard stage 3 TTC capacity constraint possible. Small differences are observed between the hybrid model an the stochastic model of Beukeboom (2015), which could be explained by the limited number of scenarios used in the stochastic model resulting in a bad fit with historical data. The results show that when historically a route forecasts systematically to high (low) the PLF should be adjusted to be able to order systematically lower (higher) amounts of TTCs. If taking the hybrid results and one would want to guarantee a certain service level, this could be done at a relatively low increase in price (compared to the current policy). The results also show that guaranteeing a higher service level causes P LF1 to drop,

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move the risk/uncertainty to the subcontractor. If he agrees on partly taking over the risk but does not want the average cost per day to decrease, more resources are available to buy additional options. Which could increase the ser-vice level by up to 2%. However the amount of options that can be bought with the additional resources increase exponentially, which at a certain option price could be unfeasible to the subcontractor.

A limitation of this research are that the PLFs minimize overall costs, individual days are not considered. This is required when the stage 3 capacity constraint is used, which make the PLFs dependent on the forecasted amounts. Another limitation is not taking into account the possible correlation between forecast errors, which could make the results less validate in the real-life situation. Pos-sible trends in forecast errors over time are also not taken into account, which could also make the results less validate in the real-life situation. Also routes are considered as individual, with each the same cost structure and stage 3 TTC constraint. In real-life routes could differ in cost structure and in stage 3 TTC constraint depending for example on the location of a specific distribution center.

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7. Conclusion

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Bibliography

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A. Mathematical models

A.1 Parameters

A.1.1 Basic parameters

C1= 170, T T C order cost stage 1

OC1, cost to buy option in stage 1which can be used in stage 2

OC2, cost/ref und to use option in stage2

Q1, amount of T T Cs ordered stage 1

T = 56, T T C RC capacity M3= 3, available T T C in stage 3

P LF1, planning load f actor stage 1 (simulation)

P LF2, planning load f actor stage 2 (newsvendor)

F1, RC f orecast stage 1 F2, RC f orecast stage 2 D, RC realization stage 3 12= F2 F1

, relative f orecast error between stage 1 and 2 23=

D F2

, relative f orecast error between stage 2 and 3

µD= 1 N N X i=1

Di, mean of historical RC demand f orecast stage 1

σD= v u u t 1 N N X i=1 (Di− µD)2,

standard deviation of historical RC demand f orecast 1 µ12= 1 N N X i=1 12,

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σ12= v u u t 1 N N X i=1 (12− µ12)2,

std dev of historical relative f orecast error between stage 1 and 2 µ23= 1 N N X i=1 23,

mean of historical relative f orecast error between stage 2 and 3

σ23= v u u t 1 N N X i=1 (12− µ23)2,

std dev of historical relative f orecast error between stage 2 and 3

A.1.2 Simulation parameters

F1∼ N (µD, σD) (A.1)

12∼ N (µ12, σ12) (A.2)

23∼ N (µ23, σ23) (A.3)

F2= F112 (A.4)

D = F223 (A.5)

A.1.3 Service level parameters

α = service level (A.6)

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A.1.4 Option parameters

OLF1, option load f actor stage 1 (simulation) (A.10)

C_uo= C3− OC2,

underage cost of using an option between stage 2 and 3 Coo= OC2,

overage cost of using an option between stage 2 and 3 z₂₃o = Φ−1 _Co u Co u+ Coo

(newsvendor critical quantile) (A.11)

Ao₂₃= µ23+ z23o σ23 (A.12) OLF2= 1 Ao 23 (A.13)

A.2 Models

A.2.1 Basic hybrid model

Q1= d F1 T · P LF1 e (A.14) Q2= max d F2 T · P LF2 e − Q1, 0 (A.15) Q3= max dD Te − Q2− Q1, 0 (A.16) T C = Q1C1+ Q2C1+ Q3C3 (A.17)

A.2.2 Hybrid service level model

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A.2.3 Basic call hybrid model

Q1= d F1 T · P LF1 e (A.22) O1= max d F1 T · OLF1 e − Q1, 0 (A.23) O2= min maxd F2 T · OLF2 e − Q1, 0 , O1 (A.24) Q2= max d F2 T · P LF2 e − Q1− O2, 0 (A.25) Q3= max dD Te − Q1− Q2− O2, 0 (A.26) T C = Q1C1+ Q2C1+ Q3C3+ O1OC1+ O2OC2 (A.27)

A.2.4 Hybrid call service level model

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A.2.5 Basic put hybrid model

Q1= d F1 T · P LF1 e (A.34) O1= max d F1 T · OLF1 e − Q1, 0 (A.35) O2= min maxQ1− d F2 T · OLF2 e, 0, O1 (A.36) Q2= max d F2 T · P LF2 e − (Q1− O2), 0 (A.37) Q3= max dD Te − (Q1− O2) − Q2, 0 (A.38) T C = Q1C1+ Q2C1+ Q3C3+ O1OC1− O2OC2 (A.39)

A.2.6 Hybrid put service level model

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A.2.7 Basic bidirectional hybrid model

Q1= d F1 T · P LF1 e (A.46) O1= max d F1 T · OLF1 e − Q1, 0 (A.47) O2= min maxd F2 T · OLF2 e − Q1, Q1− d F2 T · OLF2 e, O1 (A.48) Q2=    maxd F2 T ·P LF2e − Q1− O2, 0 , if call maxd F2 T ·P LF2e − (Q1− O2), 0 , if put (A.49) Q3=    maxdD Te − Q1− O2− Q2, 0 , if call maxdD Te − (Q1− O2) − Q2, 0 , if put (A.50) T C =    Q1C1+ Q2C1+ Q3C3+ O1OC1+ O2OC2, if call Q1C1+ Q2C1+ Q3C3+ O1OC1− O2OC2, if put (A.51)

A.2.8 Hybrid bidirectional service level model

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B. Historical data analysis

Forecast error 12 Forecast error 23

Route Data points Outliers Remaining Data points Outliers Remaining

AMF-HGL 189 6 183 210 6 204 AMF-KHM 189 1 188 210 4 206 AMF-LW 189 3 186 210 6 204 AMF-ZL 189 2 187 210 2 208 HT-BD 189 1 188 210 12 198 HT-BORN 189 0 189 210 2 208 HT-ELT 189 6 183 210 9 201 HT-SON 189 3 186 210 7 203 WVN-GS 189 6 183 210 4 206 WVN-HBD 189 7 182 210 8 202 WVN-HW 189 8 181 210 6 204 WVN-OPM 189 5 184 210 8 202 WVN-RD 189 0 189 210 10 200 Table B.1: Outliers

Forecast 1 Forecast error 12 Forecast error 23

Route Mean SD Mean SD Mean SD

AMF-HGL 580.99 103.48 1.0326 0.0976 0.9938 0.0634 AMF-KHM 567.14 96.32 0.9551 0.1241 1.0169 0.0981 AMF-LW 368.16 65.51 1.1129 0.1595 0.9910 0.0918 AMF-ZL 670.48 96.85 1.1147 0.0970 0.9906 0.0718 HT-BD 337.94 80.87 1.2116 0.2111 0.9355 0.1002 HT-BORN 355.38 160.13 1.4599 0.7490 0.9423 0.1517 HT-ELT 738.03 115.58 0.9460 0.1183 0.9676 0.0783 HT-SON 677.34 128.39 0.9091 0.1051 0.9715 0.0726 WVN-GS 476.10 74.45 1.0509 0.1026 1.0147 0.0612 WVN-HBD 652.60 110.26 0.9871 0.1034 0.9993 0.0690 WVN-HW 521.41 92.30 0.9509 0.1890 0.9818 0.1207 WVN-OPM 542.74 96.25 1.1024 0.1595 1.0014 0.0851 WVN-RD 619.31 115.68 0.8806 0.1575 1.0296 0.0859

Optimizing the logistics capacity planning of PostNL parcels using the newsvendor model and option contracting