Minimizing order picker route length in a warehouse with dynamic item flows: a simulation study

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Minimizing order picker route length in a

warehouse with dynamic item flows:

a simulation study

By: Karim Charif

s2876736

University of Groningen

Faculty of economics and business

In partial fulfilment of the requirements for the degree of M.Sc. Technology and operations management

Word count: 8222

Supervisor: dr. N.D. Van Foreest

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Abstract

Purpose – Order picking costs consists for a large part of order picker travel.

Therefore, travel distances should be minimized by placing items with a high pick frequency in the front of the warehouse and items with a low pick frequency in the back. However, sometimes demand patterns of items change, thus the location should change accordingly. This process is called dynamic slotting. The goal of this research is to investigate how dynamic slotting compares to static slotting, and what effect the number of item rearrangements has on the efficiency of dynamic slotting.

Method – The research method is analytical quantitative research with simulation as

a tool to conduct experiments. Historical pick frequencies of one warehouse are used to investigate the effect of multiple dynamic slotting scenarios on order picker route length under the return routing method and S-shape routing method.

Findings – In a warehouse where demand fluctuates, static slotting is not desirable.

For both the return routing method and S-shape routing method, the dynamic slotting scenarios manage to decrease average route length. However, when increasing the efforts of re-slotting, the benefits do not increase at the same rate.

Limitations – Due to the complex warehouse design of the case company, a simplified

warehouse model is used for the simulation.

Contribution – This research makes a distinction between route length in aisles and

cross aisles. Furthermore, offers first insights in dynamic slotting under the S-shape routing method. Lastly this research shows that re-slotting a smaller number of items is more efficient than re-slotting many items.

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Preface

In front of you lies the research paper “Minimizing order picker route length in a

warehouse with dynamic item flows: a simulation study”. This paper is the result of 5

intensive months of work and is the final step in fulfilling the partial requirements to obtain a master’s degree in Technology and Operations Management. This work was definitely the most challenging part of my journey.

Three years ago I started this journey as I was looking for a new challenge. Not knowing what to expect I left Brabant with three goals in mind: (1) have a great time in Groningen, (2) fulfilling the Pre-MSc program to gain admission to the MSc program Technology and Operations Management, and (3) acquiring my master’s degree. Looking back at the journey, I can proudly say that the first and second goals are definitely fulfilled. I had a great time in Groningen. However, with ups, downs and some setbacks. This resulted in achieving my second goal and third goal with some delays. Being delayed is painful, however, abandoning my goals was never an option. Now I am close to partially fulfilling my final goal. I look forward to the day that my final goal will be fulfilled.

Fulfilling my final goal would never be possible without the help of other people. Therefore, I want to thank my supervisors dr. N.D. Van Foreest and prof. dr. K.J. Roodbergen for their guidance and feedback. Furthermore, my family for supporting me. As a fact I know they are extremely proud of me for starting this journey, regardless of the outcome. I also want to thank my friends in Brabant, for whom I had less time during my study. But the times when we met again where great. Also many thanks to my friends in Groningen for making this an awesome journey.

Karim Charif

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1. Introduction

In operating a warehouse, the order picking process often accounts for the largest part of operating expenses (Frazelle, 2002; Tompkins et al., 2003). To cut down on these expenses, it becomes increasingly important that the order picking process is performed in an efficient way. The efficiency of the process greatly depends on the location of items (Le Duc & de Koster, 2005; Bindi et al., 2009). In an optimal warehouse slotting, fast moving items are slotted in the front and slow moving items in the back. However, due to changes in demand, an optimal warehouse slotting can deteriorate overtime (Kofler, 2014).

When warehouses face stochastic demand, it is important that item locations change dynamically to reflect changing product flows (Gu et al., 2007; Kim and Smith, 2012). This is important to continuously assure an optimal slotting and to keep travel distances minimized. Having optimal travel distances is important as often more than 50% of order picking time is spent on travel (Chan and Chan, 2011).

Determining the optimal slotting is called the storage location assignment problem (Gu et al., 2007; Dijkstra and Roodbergen, 2017). Research concerning the storage location assignment problem is mainly done in environments with static demand patterns (Gu et al., 2007; Kofler, 2014). Nonetheless, the storage location assignment problem in dynamic demand settings is an equally important topic.

The storage location assignment problem under dynamic demand is studied by multiple researchers with the focus on minimizing order picking time or order picker route length. Sadiq et al., (1996) considers fluctuations in demand due to changes in the product mix. Items that are often ordered together are slotted near each other to decrease order picking time. Kim and Smith (2012) use a similar strategy by re-arranging a pick area before each pick wave based on the likelihood that an item appears in the same carton. This results in shorter order picking times.

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Research shows that dynamically rearranging items has positive effects on warehouse performance. However, multiple researchers (Kallina and Lynn, 1976; Gu et al., 2005; Kim, 2009; Kofler, 2014) point out that the efforts of re-arranging items in a warehouse should not outrun the benefits.

Previous research on dynamic slotting mainly focus on total route length. Furthermore no research is done on dynamic slotting when the routing method is S-shaped. Therefore this paper addresses the dynamic storage location assignment problem from the perspective of travel in aisles and cross aisles. Furthermore, next to the return routing method, this paper also investigates dynamic slotting under the S-shape routing method. Lastly, the effect of re-arranging a small number of items versus many items is investigated.

The goal of this research is to investigate how dynamic slotting compares to

static slotting under the return routing method and S-shape routing method, and what effect the number of item rearrangements has on the efficiency of dynamic slotting. To investigate this matter, a simulation study is conducted with data collected

from one company.

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2. Theoretical background

This chapter starts with an overview of relevant slotting policies. Subsequently an explanation of dynamic slotting is given and methods for dynamic slotting are presented. Lastly, an overview of research concerning the storage location assignment problem in dynamic demand environments is given. This research concerns the use of slotting policies in a dynamic manner to decrease order picker travel distance.

2.1 Slotting policies

In warehouse management, the goal of slotting is to determine the best location to store an item (Kofler et al., 2011). Having items slotted at the appropriate location results in an optimized warehouse which leads to an increase in order picking efficiency (Pohl et al., 2011).

Various slotting policies are addressed in literature. The simplest forms of slotting are random slotting, in which an item is randomly slotted and static slotting, in which an item has a dedicated slot. The major benefits of these policies are its ease of implementation (Petersen, 1999; Kofler et al., 2011) and using space in an efficient manner (Francis et al., 1992). However, according to Kofler et al., (2011), random slotting leads to longer order picker travel times.

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and C are located further away, in which the latter category is located furthest away. Essentially it is a hybrid form of random slotting and turnover based slotting as items are assigned to zones, however within the zones items are slotted randomly. It results in getting the benefits of both random slotting and turnover based slotting (Francis et al., 1992; Mantel et al., 2007). Furthermore, class based slotting results in lower order picking and handling costs compared to dedicated slotting (Muppani and Adil, 2008).

Sometimes it can be beneficial to not only look at individual items but consider items that are frequently ordered together and slot these near each other. These are affinity based slotting policies, also known as correlated slotting. The rationale behind this policy is that it minimizes total travel time. However, as mentioned by Kofler et al., (2011), this is not always the case. It depends amongst others on the warehouse layout and the order picker routing method. Correlated slotting is first seen in the work of Frazelle and Sharp (1989) and subsequently similar methods are proposed by other researchers, such as Mantel et al., (2007) who propose order oriented slotting and Kofler et al., (2011) who introduces the pick-frequency/part-affinity (PF/PA) score which combines slotting by affinity (correlations) and slotting by pick frequency.

2.2 Dynamic slotting

Previous paragraph gave an overview of slotting policies. These policies are not necessary dynamic as items do not necessary change slots over time. However, it can occur in warehouses that flows of incoming and outgoing items are not static (Kim, 2009). This means that patterns of item flows change periodically due to factors such as lifecycle, seasonality, turnover rate, pricing/marketing plans and promotions (Gu et al., 2007; WERC, 2007; Kim and Smith, 2012). When facing this dynamic demand state, it becomes necessary to regularly re-slot items in a warehouse. This is done to ensure that the quality of slotting does not decline over time, as demand volatility greatly affects slotting (Kim, 2009; Kofler et al., 2015).

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dynamic partial slotting, the main decisions to be made are determining the items to be re-slotted, their new slot locations and when to schedule re-slotting (Kim, 2009). In dynamic whole slotting, the pick area is emptied entirely and replenished with selected items to the optimal determined slots. This also occurs in-between pick waves. Similar strategies are mentioned by Kofler et al., (2011). These strategies are healing and re-warehousing. The healing strategy aims to gradually improve the slotting by rearranging a few items daily or slot incoming goods dynamically. Re-warehousing is a more rigorous method where a warehouse is completely emptied and slotted from scratch (Kofler et al., 2011). The main difference with dynamic whole slotting is that re-warehousing only occurs a few times a year due to the impact of re-re-warehousing on a warehouse (Kofler, 2014). The assumption here is that dynamic whole slotting only occurs at a smaller forward pick area, and re-warehousing a much larger area.

Re-slotting, may it be partial or full, re-warehousing or healing, comes with its costs. Especially if re-slotting occurs dynamically, it becomes important that the costs and efforts of re-slotting do not outrun the benefits for order picking (Kallina and Lynn, 1976; Gu et al., 2005; Kim, 2009; Kofler, 2014). As mentioned before, the decisions concerning the selection of items to be re-slotted, the new locations and the scheduling of re-slotting should be made thoughtfully (Gu et al., 2005: Kim, 2009). According to Kallina and Lynn (1976), experimentation and simulation are suitable tools to evaluate these re-slotting decisions.

2.3 Storage location assignment problem

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part of research is done concerning automated storage and retrieval systems, such as publications from Christofides and Colloff (1973), Jaikumar and Sokomon (1990), Muralidharan et al., (1995) and Carlo and Giraldo (2010). These publications are excluded as they mainly focusses on increasing throughput or decreasing crane waiting time. The focus of this research is reducing order picker travel distances in a manual order picking warehouse.

Minimizing order picking time in dynamic demand environments is first seen in the work of Sadiq et al., (1996). The research focusses on a manual forward picking area where item demand patterns change due to evolving lifecycles and changes in the product mix. Sadiq et al., (1996) also takes into account that items throughout the product mix are often ordered together. A heuristic is proposed that considers these changes in product mix, life cycle and correlated demand to make re-slotting decisions. The heuristic is shown outperforming the cube per order index rule in minimizing order picking time (Sadiq et al., 1996).

Pierre et a al., (2004) propose a dynamic variant of class based slotting to decrease order picking time. In this dynamic variant, the ABC classification is re-evaluated frequently to cope with changes in demand. Overtime this results in misclassified items that must be re-slotted to the appropriate class. When an item is selected for a class change, it is not re-slotted directly. Pierre et al., (2004) propose a re-slotting scheme that has three criteria. The priority of re-slotting an item to a new slot increases when: (1) the item has been in the wrong class for multiple periods, (2) the pick frequency increases or decreases significantly, or (3) the demand variation is low. By means of a simulation study Pierre et al., (2004) show that the proposed dynamic ABC policy outperforms both static slotting and random slotting in average order picking time.

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Kofler et al., (2011) aim to decrease travel by re-slotting items based on pick frequency and part affinity. Results show that compared to random slotting, slotting by pick frequency decreased order picker travel by 61%. Subsequently, also taking part affinity into consideration reduced order picker travel by another 3 to 4%. Furthermore Kofler et al., (2011) shows that iteratively re-slotting a small number of items (healing) based on the proposed PF/PA heuristic can rival re-warehousing over a longer period. Emptying and re-slotting a warehouse (re-warehousing) outperforms healing activities when the re-warehousing occurred recently. However, gradually the optimal slotting deteriorates again due to fluctuations in demand. This is because re-warehousing activities only take place a few times a year (Kofler et al., 2011).

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3. Methodology

This chapter embodies the methodology of this study. This chapterexplains the research method and underpins why simulation is chosen as an appropriate tool to conduct experiments. Furthermore, information is given on the case company and some remarks on the data acquired from the company.

3.1 Research method

To investigate the effects of dynamic partial slotting and dynamic whole slotting on order picker travel distance under the return routing method and S-shape routing method, a simulation model is coded in Python. As this study is quantitatively oriented, the methodology for this study is analytical quantitative research with the use of simulation as a tool to conduct experiments.

Robinson (2004) mentions that operations systems are prone to complexity and variability. This is also seen in warehouses where many processes are interconnected. This makes operating a warehouse complex. Concerning the order picking process, multiple sources of variability can occur, such as variability in the items that need to be picked or variability in the quantity that need to be picked. Predicting the performance of a system that is prone to complexity or variability is therefore difficult (Robinson, 2014). However, a simulation model can offer the opportunity to predict the performance of order picking when a warehouse is prone to demand variability. Furthermore, as mentioned by Robinson (2014), simulation is suitable for comparing system alternatives. Concerning this research such system alternatives are static slotting, dynamic partial slotting and dynamic whole slotting.

Robinson (2004) also mentions multiple advantages of a simulation study compared to doing experiments in a real system. The main advantage is that simulation is a cost-effective and time efficient tool. Doing experiments for dynamic slotting in a real warehouse setting would be a time consuming and expensive process. Furthermore, when using a simulation model, the experimental factors can be easily controlled and changed. This offers the opportunity to repeat the simulation for a multitude of scenarios (Robinson, 2004).

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3.2 Case company

Historical order pick frequencies are collected from company X. Company X is a manufacturer of scales and analytical instruments. These are mainly laboratory scales, scales for food retail and industrial scales. The distribution is carried out in the Netherlands by a third party. The warehouse carries around 20.000 items, which are analytical instruments, finished scales, semi-finished scales, spare parts and a large array of necessities such as ink and printing paper for certain types of scales. Not all stock keeping units are active. Manufacturing is carried out outside the Netherlands. However, some value-added services are performed in the distribution centre. In this case, the customer order decoupling point is shifted closer to the customer.

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4. The model

This chapter presents the simulation model design and validation. Furthermore, key performance indicators (KPIs) are presented. These are the numerical output of the model in which performance of order picking is measured.

4.1 Model design

The simulation model uses monthly historical pick frequencies of 720 SKUs to calculate the probability that an item occurs on a picklist. Based on these probabilities, pick lists are generated, each consisting of 20 items. Each item on the list has an location assigned to it. The location consist of an aisle parameter and a slot parameter. The model calculates the route length for each pick list, which consists of distance in aisles and distance in cross aisles. The exact manner of calculation differs per routing method and will be explained in the KPI section.

The simulation model is based on a warehouse layout (Fig. 1) which is often studied in literature (Dijkstra and Roodbergen, 2017). Due to the complexity of the warehouse that the case company operates (multiple blocks and multiple layers), the choice is made to use this simplified warehouse layout. Furthermore, complexity concerning the calculation of route length is avoided by using this model. Nonetheless, the warehouse layout is still suitable to investigate dynamic slotting. The warehouse layout and parameters are adapted from the publication of Dijkstra and Roodbergen (2017). The parameters can be found in table 1 on the next page.

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Table 1. Model parameters

Parameter Description

m Number of aisles

n Number of storage locations per aisle

Wa Width of an aisle

Wc Distance from middle of a cross aisle to the head of an aisle

f Distance between two adjacent storage locations

i Index of aisles

j Index for location in an aisle

m 15 n 24 Wa 2 Wc 0,5 f 1 i   j      

4.2 Key performance indicators

Model output will be measured by multiple KPIs. These KPIs are travel distance in aisles, travel distance in cross aisles and pick frequencies. All KPIs will be calculated under the return routing method and S-shape routing method (Fig. 2). As mentioned by Kofler (2014), the routing method also impacts travel distances. Therefore, a second routing method is included. The equations to calculate average route length are also adapted from the publication of Dijkstra and Roodbergen (2017) and are used as follows:

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In accordance with the return routing policy (Fig. 2), the average route length is calculated by equation 1. From the middle of the cross aisle till the head of the aisle an order picker must travel 𝑤𝑐. The order picker subsequently travels to slot 𝑗. Regardless of the number of picks in an aisle, the travel distance is determined by the pick at the slot which is furthest away. The distance between each adjacent slot is given by 𝑓. This means that the distance travelled from the head of the aisle until the furthest slot that needs to be visited is 𝑗𝑓. However, when the final slot is reached, the order picker stands in front of the middle of the slot. Therefore, 1₂ 𝑓 needs to be subtracted from the travelled distance. The entire distance is multiplied by two as the order picker can only exit the aisle from the front. This calculation should be done for each aisle 𝑖 that is visited and summed up (Dijkstra and Roodbergen, 2017).

𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑡𝑟𝑎𝑣𝑒𝑙𝑙𝑒𝑑 𝑖𝑛 𝑎𝑖𝑠𝑙𝑒𝑠 = ∑ 2(𝑤_𝑐 + 𝑗𝑓 − 𝑛

𝑗=1

1

2𝑓) (1)

Concerning the S-shape policy (Fig. 2), one extra parameter is defined. This is parameter 𝑎 which is the number of visited aisles. If the number of aisles that need to be visited is even, equation 2 is used. The calculation is simply the number of visited aisles multiplied by the aisle length. If the number of visited aisles is uneven, equation 3 is used. In this situation, routing in the final aisle will be performed in the same manner as in the return routing method. Thus, from the number of aisles visited one is subtracted. Subsequently, multiplied by aisle length and the return routing calculation for the last aisle is added up. Again, as can be seen the model uses the equations by Dijkstra and Roodbergen (2017), however slightly adapted to be easily implemented in the simulation model.

𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑡𝑟𝑎𝑣𝑒𝑙𝑙𝑒𝑑 𝑖𝑛 𝑎𝑖𝑠𝑙𝑒𝑠 𝑖𝑓 𝑒𝑣𝑒𝑛 = 𝑎(2𝑤_𝑐 + 𝑛𝑓) (2)

𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑡𝑟𝑎𝑣𝑒𝑙𝑙𝑒𝑑 𝑖𝑛 𝑎𝑖𝑠𝑙𝑒𝑠 𝑖𝑓 𝑢𝑛𝑒𝑣𝑒𝑛 = (𝑎 − 1)(2𝑤_𝑐+ 𝑛𝑓) + 2(𝑤_𝑐+ 𝑗𝑓 −1

2𝑓) (3)

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explained as follows. The furthest aisle that needs to be visited is subtracted by one, as the starting point of an order picker is in front of aisle one. That means that no distance in a cross aisle is made when aisle one is the furthest. If aisle two is the furthest, the order picker travels only the distance of one width of an aisle 𝑊𝑎, and so on. 𝑊_𝑎 is multiplied by two because in every scenario the width of an aisle must be travelled twice. With the return routing method in the same cross aisle, with the S-shape routing method in the parallel cross aisle.

𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑡𝑟𝑎𝑣𝑒𝑙𝑙𝑒𝑑 𝑖𝑛 𝑐𝑟𝑜𝑠𝑠 𝑎𝑖𝑠𝑙𝑒 = 2𝑊_𝑎(𝑖 − 1) (4)

These are calculations that the model makes for one picklist, depending on the routing method used in the simulation. Running the model, the Python code adds up the total route length for the simulation run, based on the number of picklists simulated. Next to route length, the model also records pick frequencies. These consist of pick frequencies in aisles and pick frequencies in slots. For this, no equation is needed, the simulation model records for every pick lists which aisles and slots are visited.

Lastly, some model assumptions need to be made for the model: (1) no congestion, (2) slots are always full, the simulation model does not take replenishments into account, and (3) the slot on the left side of an order picker is the same distance as a slot on the right side of an order picker.

4.3 Model validation

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5. Results

In this chapter, first an overview of the experimental design is given and subsequently the results of the experiments are presented. This consists of; average route length for both routing methods, SKUs shifts in warehouse locations and lastly the efficiency of re-slotting.

5.1 Experimental design

As mentioned by Kim (2009) the main decisions to be made for dynamic slotting are the set of items to be relocated, their new locations and when to schedule relocations. These decisions are important in the experimental design. Six experiments are conducted (table 2), each for the duration of 12 months with monthly changing pick frequencies. Appendix B gives an in depth explanation of the experimental design.

Table 2. Experiments

Return routing method S-shape routing method

Static slotting (turnover based) Static slotting (turnover based)

Dynamic partial slotting (class based) Dynamic partial slotting (class based) Dynamic whole slotting (turnover based) Dynamic whole slotting (turnover based)

Due to the use of the return routing method and the S-shape routing method, it is important to note that the assignment of slots depends on the routing method. Popular items in a warehouse under the return routing method have different locations than with the S-shape routing method. See Fig. 3 for optimal storage experiments conducted by Dijkstra and Roodbergen (2017).

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The static slotting scenario is the base case in the experimental design. For the first month, items are slotted in a turnover based manner. Thus, the item with the highest pick frequency is located in aisle 1, slot 1. The item with the second highest pick frequency also corresponds to parameter aisle 1 and slot 1, as there is a slot on the left side and one on the right side of an aisle. Subsequently, the item with the third highest pick frequency is located in aisle 1, slot 2, and so on. The assigned parameters do not change during the entire 12 months of simulation. The probabilities of an item occurring on a picklist do change monthly. E.g. an item with a low pick frequency corresponds aisle and slot parameters that represent a slot far in the back of the warehouse. In the second month, the probability of that item occurring on a picklist increases. However, the aisle and slot parameter stay the same. In this manner a scenario is created where demand changes but items are slotted in a static manner.

In the dynamic partial slotting scenario, for the first month, items are slotted again on popularity in classes as seen in Fig. 3, A=20, B=30 and C=50. In the subsequent months, the probability of a certain item occurring on a picklist changes. Based on these changes, the warehouse is shuffled monthly. A location change only occurs if the class of an item changes. Thus, if demand changes within the boundaries of the classes, items will not be re-slotted. If a class change occurs, the item will be located randomly in its new class, which is in accordance with class based strategies as explained in the literature section.

For the dynamic whole slotting scenario, the pick area is emptied monthly and items are subsequently rearranged and re-slotted based on their turnover.

Due to the nature of the data (monthly pick frequencies), it is only possible to rearrange the warehouse monthly. The three scenarios explained above will be simulated with the return routing method and the S-shape routing method. Appendix B also gives an explanation on how the coded simulation model uses the data from the experimental design.

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5.2 Average route length for return routing method

Table 3 shows the average route length per picklist for the return routing method. Dynamic partial slotting was able to reduce the average route length per pick list by 8.46 meter. This is an overall decrease of 3.60%. The reduction is mainly seen in the travel within aisles, which decreases by 5.31%. Travel within the cross aisle increased by 2.03%. Dynamic whole slotting performs the best. Compared to static slotting, on average the route length is reduced by 18.54 meter for each picklist. This is an overall a reduction of 7.89% in travel. The reduction is mainly observed in travel within aisles, which is reduced by 10.54%. Within the cross aisle, an increase in travel of 0.50% is observed. Total travel distances for the return routing method can be found in appendix C.

Table 3. Average route length per picklist for return routing

method (20 items). Month Static slotting Partial re-slotting Whole re-slotting Jan 232.74 227.18 218.15 Feb 233.34 223.86 212.24 Mar 231.80 226.37 216.41 Apr 231.89 218.99 210.90 May 229.81 223.04 213.63 June 226.59 220.70 212.92 July 234.67 221.20 210.68 Aug 230.29 219.48 209.14 Sept 232.48 222.82 211.51 Oct 232.20 225.93 214.46 Nov 232.50 226.48 214.82 Dec 226.52 217.24 207.45

5.3 Shift in warehouse locations return routing method

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and decrease of 0.71% at slot 19. This show that over 12 months, dynamically re-arranging items results in more items being picked in the front of the warehouse. Monthly pick frequencies for static slotting can be found in appendix D and for dynamic partial slotting in appendix E.

Fig. 4. Decrease in picks at furthest slots and increase at front slots.

The same is observed for dynamic whole slotting as can be seen in Fig. 5. However, now the shift is even larger. Compared to static slotting, an increase of 2.48% is seen at slot 1 and a decrease of 0.37% at slot 24. This means that order pickers travel less to slots in the back. Monthly pick frequencies for static slotting can be found in appendix D and for dynamic whole slotting in appendix F.

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5.4 Average route length for S-shape routing method

Table 4 shows the average route length per picklist for the S-shape routing method. Dynamic partial slotting reduces the average route length per pick list by 7.97 meter. This is an overall decrease of 3.18%. The reduction is seen in the travel within aisles which decreases by 3.36%. Travel within cross aisles decreased by 2.40%. Again, dynamic whole slotting performs the best. Compared to static slotting, on average, the route length is reduced by 15.43 meter for each picklist. This is an overall a reduction of 6.19% in travel. In this scenario, the reduction in travel is mainly observed in the cross aisles with a reduction in travel of 7.35%. Travel in aisles is reduced by 5.93%. Total travel distances for the S-shape routing method can be found in appendix G.

Table 4. Average route length per picklist for S-shape routing

method (20 items). Month Static slotting Partial re-slotting Whole re-slotting Jan 251.23 244.33 238.19 Feb 250.11 240.79 233.07 Mar 248.31 242.75 236.24 Apr 246.70 237.95 231.93 May 248.13 240.88 234.81 June 245.65 240.42 233.74 July 251.57 240.19 231.02 Aug 248.51 239.51 231.19 Sept 249.87 241.53 232.21 Oct 251.52 243.43 236.08 Nov 251.49 243.94 235.60 Dec 244.85 236.62 228.76

5.5 Shift in warehouse positions S-shape routing

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example, an increase in pick activity of 1.64% at aisle 3 and a decrease of 0.90% at aisle 12 (Fig. 6). Pick frequencies for static slotting can be found in appendix H and for dynamic partial slotting in appendix I.

Fig. 6. Decrease in picks at furthest aisles and increase at nearest aisles.

For dynamic whole slotting (Fig. 7), the shift is larger. More items are picked in the first few aisles. For example compared to static slotting, the pick frequency at aisle 1 increased by 2.91% and decreased by 0.55% at aisle 15. Thus order picking increases at the nearest aisles and decrease at the furthest aisles. Pick frequencies for static slotting can be found appendix H and for dynamic full slotting in appendix J.

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5.6 Efficiency of dynamic slotting

As seen in the simulation results, dynamic whole slotting outperforms dynamic partial slotting in reducing average route length. However, in the dynamic whole slotting scenario, a lot more items are arranged. Details on the number of arrangements can be found in appendix K. Fig. 8 shows the monthly efficiency of slotting. The efficiency can be explained as follows. When offsetting the number of re-arrangements (effort) against the reduction in travel distance (benefits), the reduction in route length for 1 rearrangement is calculated. Comparing this between dynamic partial slotting and dynamic full slotting, it becomes evident that for one relocation dynamic partial slotting performs better. Thus, is more efficient.

Fig. 8. Efficiency ratio of dynamic slotting

To give a better idea of the impact of re-slotting many items or a small amount of items, table 5 shows this difference in efficiency.

Table 5. Efficiency of re-slotting

Slotting strategy and routing method Reduction # of rearrangements Partial re-slotting for return routing 1 meter 21

Full re-slotting for return routing 1 meter 39 Partial re-slotting for S-shape routing 1 meter 22 Full re-slotting for S-shape routing 1 meter 47 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 Tr av el re du ct io n 1 r ea rr an ge m en t

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6. Discussion

When a warehouse faces fluctuating demands for multiple items, for a longer time period, both dynamic partial slotting and dynamic whole slotting can reduce average route length for an order picker. As observed in this research, travel distances are not optimized when item locations do not change although demand changes. Thus, indeed as mentioned by Kim and Smith (2012) and WERC (2007), items locations in a warehouse should reflect changes in demands.

The overall results concerning the return routing method are in line with the results of researchers who also investigate dynamic slotting strategies (Sadiq et al., 1996; Kofler, 2011; Kim and Smith, 2012). Although their slotting strategy is different (affinity based), their research shows that dynamically relocating items does decrease order picker travel. However, it seems that relocating items dynamically in combination with clustering them based on correlations decreases travel distances even more. Especially, the PF/PA score by Kofler (2011) results in high savings. This is because the research of Kofler (2011) takes into account changes in pick frequency, and part affinity. This research only considers pick frequency. Furthermore, one should keep in mind that most researchers compare dynamic slotting to random slotting. Comparing these two extremes result in large percentages of decrease in route length. In this research, the dynamic slotting comparison is made with static slotting where the first month is slotted in a turnover based manner. Thus the base case scenario is already a pretty good slotting. This is especially the case when subsequent months have variations in demand that are not extreme, which is the case with the data in this paper. Therefore it can give a distorted view when comparing the savings in travel with other researchers.

Pierre et al., (2004), observe a decrease in travel when using their proposed dynamic ABC strategy. This corresponds to the results of dynamic partial slotting in this research. For both instances a small number of items is dynamically re-slotted.

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frequencies that shift towards the front of the aisles. This means that order picking occurs less in the back of the warehouse and more in the front. When looking at the return routing strategy and the optimal locations determined by Dijkstra and Roodbergen (2017), one can observe that popular items span the first slots in all 15 aisles. Thus, an order picker must shift a lot between aisles and travel more in the cross aisle. Especially when over a longer timeframe popular items are often slotted in the front.

To the knowledge of the author of this paper, there is no research concerning dynamic SLAP that also considers the S-shape routing method. Research is done concerning the return routing method (Kofler et al., 2011; Kofler, 2014; Kofler et al., 2015), the nearest neighbour algorithm (Pierre et al., 2004), sequencing zone visitation (Kim and smith, 2012) and some methods special for automated warehouses.

The results for the S-shape routing method show a similar trend as the results for the return routing method. Compared to static slotting, dynamic partial slotting manages to decrease the average route length by a bit and dynamic whole slotting manages to reduce it even more. However, in terms of percentage, the overall decrease in travel for the S-shape routing method is not as big as the decrease in travel for the return routing method. Even though the used data and pick frequencies are the same and the only difference are the locations in the warehouse, no good explanation can be found on why dynamic slotting seems less effective when the routing method is S-shape. Perhaps, because in general the S-shape routing method has higher average route lengths compared to return routing method and when slotting is optimized, sometimes an order picker still has to travel an entire aisle to go to the next aisle. Which could be an explanation why dynamic slotting seems less effective when the routing method is S-shape. However, the should be researched more in depth.

An interesting but logical finding for the S-shape routing method is that most of the travel is decreased in the cross aisles. This is due to the nature of the routing strategy and the optimal location of items in the warehouse. Less is walked in the cross aisles as popular items are dynamically shifted to the first few aisles.

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7. Conclusion

This final chapter gives the conclusion of this research. Furthermore, the limitations are presented and directions for further research are given. Lastly, the theoretical and managerial implications will be discussed.

7.1 Main conclusion

The research goal was to investigate how dynamic slotting compares to

static slotting under the return routing method and S-shape routing method, and what effect the number of item rearrangements has on the efficiency of dynamic slotting. The conducted experiments show that when item demand fluctuates in a

warehouse, dynamic slotting can be used to keep route length optimized. This is true for the return routing method and S-shape routing method. Dependant on the routing method, route length is mainly decreased in the aisles or in the cross aisles.

The operation of dynamic slotting can be an intensive process. Therefore the efforts of dynamic slotting should not outrun the benefits. This research shows that an increased effort of re-slotting, does not mean that the decrease in route length will grow with the same rate. Re-slotting a smaller number of items is found to be more efficient than re-slotting many items.

7.2 Limitations and further research

Due to the nature of the data, which are monthly pick frequencies the warehouse slotting could only be re arranged monthly. The dynamic partial slotting and dynamic whole slotting strategies often occur in-between pick waves. Data on the order picking process from pick wave to pick wave was not available. Nonetheless, re-arranging a pick area monthly still could be a valid strategy.

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Furthermore, using the simplified warehouse layout, means that the simulation model is also validated with the used warehouse model from literature. Therefore the model validation lacks the white-box and black-box validation as no comparison with a real world problem is made. In this case, the simulation model represents the warehouse model used from literature, and not a real world model.

Concerning the missing link with the real world, the model does not use certain distributions, for example incoming orders or order picker process time. This data was not available. Therefore, this study simulates pick lists based on assigned probabilities and calculates the travel distances. A time factor is not included. Therefore, the simulation runs for a certain number of pick lists and not for a certain time period. Therefore it could be argued that this study leans more towards a quantitative analysis than a simulation.

Two interesting directions for further research can be given. When optimizing a warehouse by dynamically re-locating items, the results show that there is a shift in pick frequency from locations in the back in the warehouse to the front. This is of course the goal of dynamically re-arranging items. However, studies do not consider the increase in congestion that could occur in the front of the warehouse. An interesting direction is to investigate the effect of dynamic slotting on congestion.

Further research can be done by not only looking at the relative distance from an entrance/exit point when re-arranging items. But also look at the distance from a replenishment point. If an item is shifted closer an entrance/exit point, the distance from a replenishment point changes and maybe increases. This could be inefficient, especially if replenishments occur often.

7.3 Theoretical implications

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7.4 Managerial implications

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Appendix A: Model validation

First, a small number of picklists are generated. The Python code is slightly changed so that it prints the pick list (items with assigned aisle and slot) and the calculated average route length. Subsequently the formulas are used to calculate the average route length by hand. This is done for the return routing method and S-shape routing method. The output for both situations where correct. This means that all the equations to calculate travel distance are modelled correctly.

Further validation is done by designing an experiment in which the behaviour of the route length under the return routing method and S-shape routing method is predictable. In the experiment, equal probabilities are divided in such a way that picks only occur in the front (1), the middle (2) and the back (3) of the warehouse (Fig. 9). Each picklist has 20 items and every scenario is calculated for 8000 picklists.

Fig. 9. Experimental design for model validation.

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Fig. 10. Validation results return routing method.

in accordance with the s-shape routing method, the outcome of the experiment is also predictable. Regardless off scenario 1, 2 or 3. The only difference in travel is at aisle 15. In each simulated scenario all aisles are travelled with equal probability which means that in accordance with the S-shape routing method the first 14 aisles are travelled entirely and the last aisle concerning the return routing method. Thus, as expected, the difference in average route length for each scenario is small as the only travel difference occurs in aisle 15. Travel in cross aisles is roughly the same, only decimal differences are observed. As the difference is small and visually difficult to see, lines are added (Fig. 11).

Fig. 11. Validation results S-shape routing method.

For the return routing method and S-shape routing method, the results show that the model is coded accurately and represent the warehouse model from literature

0 100 200 300 400 500 600 700 1 2 3 Av er ag e ro ut e le ng th

Aisle Cross aisle

0 50 100 150 200 250 300 350 400 1 2 3 Av er ag e ro ut e le ng th

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Appendix B: Experiments and simulation model background

This appendix provides a background of the experimental design and the simulation model. First, the raw data is presented and the implementation of aisle (𝑖) and slot (𝑗) parameters is explained. Subsequently, a step by step explanation is given on how the static slotting, dynamic partial slotting and dynamic whole slotting scenarios are designed. Lastly, the simulation model is explained step by step and pseudocode is presented.

1. Raw data

Input data is an important part of this research. Microsoft excel is used to prepare the data and define the scenarios. The preparation of data mainly consists of the decisions that need to be made for dynamic slotting. Namely determining the items that need to be relocated, their new locations and the timing of relocations (Kim, 2009).

Table 6 provides a small selection of the data acquired from company X. The entire dataset consists of monthly pick frequencies for 720 items. Each column represents pick frequencies for one month. In column C, the data is filtered descending as a starting point. The data from 01_2017 and up is used to design the scenarios. Next section will explain how in optimal conditions the items that are picked the most are at the top of the dataset. Therefore, the situation which is created now is that 12_2016 is the optimal slotting and in subsequent months the slotting deteriorates and need to be optimized.

Table 6. Raw data

A B C D E F G H Material Description 12_2016 01_2017 02_2017 03_2017 04_2017 05_2017 1 Ink ribbon 6766 2500 2200 2800 2800 2600 2 Paperroll-Set LCP 6200 6900 4150 2300 2650 3900 3 Folded box 340x205x115 4650 500 2000 3000 2000 1501 4 Versandschachtel 34x28x22.5 4600 3251 3244 3786 2705 3795 5 Sachets buffer pH 4.01 4589 2400 2150 1850 1750 2300

6 Softening Point Cups Disposable (Set of 3000 4715 4372 6074 5152 4404

7 Sachets buffer pH 7.00 2907 3100 2300 2400 3200 3200

8 Versandschachtel 420x280x350 2367 2500 2400 3000 2800 3400

9 Sachets buffer pH 9.21 2300 1800 2405 3001 2102 2902

10 Dropping Point Cups Disposable (Set of 5 2200 365 1481 2917 1442 1926

11 Versandschachtel 40x29x56 2086 405 650 1727 1475 1736

12 Tips LTS 250 µL 960/10 GPS-L250 1950 1959 2258 2260 1958 1962

13 TEST REPORT FORM 1847 1683 2145 2009 1519 1941

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2. Implementing aisle and slot parameters

Calculating route length is based on parameters that are explained in chapter 4. Within these parameters the aisle (𝑖) and slot (𝑗) parameters are an index of numbers of which each number results in a different route length. This section shows the implementation of aisle and slot parameters in the dataset. For this, the return routing method is used as an example. Figure 12 shows the first 4 aisles of the optimal slotting determined by Dijkstra and Roodbergen (2017). These locations are translated to the dataset in table 7.

The first aisle starts with the most popular items which are the first 5 slots on the left and 5 slots on the right. This translates to item 1 till 10 on the dataset. Subsequently, the next popular items are in aisle 2, again 5 slots on the left and 5 slots on the right. This translates to item 11 till 20 in the dataset. In the dataset, 720 slots are defined in this manner, from optimal slot locations descending to less optimal locations.

Table 7. Aisles and slots in dataset

A B C D

Material Description Aisle i Slot j

1 _{Ink ribbon} ₁ ₁

2 _{Paperroll-Set LCP} ₁ ₁

3 _{Folded box 340x205x115} ₁ ₂

4 _{Versandschachtel 34x28x22.5} ₁ ₂

5 _{Sachets buffer pH 4.01} ₁ ₃

6 _{Softening Point Cups Disposable (Set of} ₁ ₃

7 _{Sachets buffer pH 7.00} ₁ ₄

8 _{Versandschachtel 420x280x350} ₁ ₄

9 _{Sachets buffer pH 9.21} ₁ ₅

10 _{Dropping Point Cups Disposable (Set of 5} ₁ ₅

11 _{Versandschachtel 40x29x56} ₂ ₁

12 _{Tips LTS 250 µL 960/10 GPS-L250} ₂ ₁

13 _{TEST REPORT FORM} ₂ ₂

14 _{Cond standard 84 µs/cm, 250 mL} ₂ ₂ 24 Ais le 1 24 24 Ais le 2 24 24 Ais le 3 24 24 Ais le 4 23 23 23 23 23 23 23 22 22 22 22 22 22 22 21 21 21 21 21 21 21 20 20 20 20 20 20 20 19 19 19 19 19 19 19 18 18 18 18 18 18 18 17 17 17 17 17 17 17 16 16 16 16 16 16 16 15 15 15 15 15 15 15 14 14 14 14 14 14 14 13 13 13 13 13 13 13 12 12 12 12 12 12 12 11 11 11 11 11 11 11 10 10 10 10 10 10 10 9 9 9 9 9 9 9 8 8 8 8 8 8 8 7 7 7 7 7 7 7 6 6 6 6 6 6 6 5 5 5 5 5 5 5 4 4 4 4 4 4 4 3 3 3 3 3 3 3 2 2 2 2 2 2 2 1 1 1 1 1 1 1

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3. Static slotting scenario

Slots closer to the entrance/exit point of the warehouse should have an higher probability to be visited. This can be interpreted as items with high pick frequencies should be slotted closer to the entrance/exit point.

First, the base case scenario (static slotting) is defined. Column E and F show the probability than an aisle and slot will be visited. For this example, the probabilities are multiplied by 100 to make the differences more visible. As shown in the raw data section, the column with the pick frequencies of 12_2016 was sorted in a descending manner. Thus, the probabilities of an aisle and slot being visited were optimal. In the subsequent months, due to changes in demand, the probabilities are not optimal as the slotting deteriorates (Table 8). E.g. slot 4 in aisle 2 has a higher probability to be visited than slot 2 in aisle 1. This is the case throughout the entire dataset. Every month is simulated in this un-optimized manner.

Table 8. Static slotting dataset

A _B _C _D _E _F

Material Description Aisle Slot 01_2017 02_2017

1 _{Ink ribbon} ₁ ₁ _{1.54854183 1.35158381}

2 _{Paperroll-Set LCP} ₁ ₁ _{4.27397545 2.54957855}

3 _{Folded box 340x205x115} ₁ ₂ _{0.30970837 1.22871255}

4 _{Versandschachtel 34x28x22.5} ₁ ₂ _{2.0137238 1.99297176}

5 _{Sachets buffer pH 4.01} ₁ ₃ _1.48660016 _1.320866

6 _{Softening Point Cups Disposable (Set of} ₁ ₃ _{2.92054989 2.68596565}

7 _{Sachets buffer pH 7.00} ₁ ₄ _{1.92019187 1.41301944}

8 _{Versandschachtel 420x280x350} ₁ ₄ _{1.54854183 1.47445507}

9 _{Sachets buffer pH 9.21} ₁ ₅ _{1.11495012 1.47752685}

10 Dropping Point Cups Disposable (Set of 5 1 5 0.22608711 0.90986165

11 Versandschachtel 40x29x56 2 1 0.25086378 0.39933158

12 Tips LTS 250 µL 960/10 GPS-L250 2 1 1.21343738 1.38721647

13 TEST REPORT FORM 2 2 1.04247836 1.31779422

14 Cond standard 84 µs/cm, 250 mL 2 2 1.07035211 1.31042194

15 Tips LTS 20 µL 960/10 GPS-L10 2 3 0.73772533 1.15744723

16 GENDER CHANGER 9D-SUB M/M 2 3 0.61260315 0.58179539

17 Faltschachtel 385x360x125 2 4 0.6664924 0.80112059

…… …… …… …… …… ……

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4. Dynamic partial slotting scenario

The static slotting scenario that was addressed in previous section is not optimized. Throughout the dataset, many aisle and slot parameters that result in a higher route length have a high probability to be visited and on the other hand, many aisle and slot parameters that result in a lower route length have a low probability to be visited. The aim of the dynamic partial slotting experiment is to improve this by shuffling the probabilities. This can be interpreted as shuffling items in a warehouse by shifting items with high pick frequencies towards the front of a warehouse and low pick frequencies towards the back.

To determine which items change location in the dynamic partial slotting scenario, a few steps are taken, which are conducted in the dataset.

➢ First, column F is defined in the static slotting dataset (Table 9).

Table 9. Category column

A B C D E F

Material Description Aisle Slot 01_2017 Category

1 Ink ribbon 1 1 1.54854183

2 Paperroll-Set LCP 1 1 4.27397545

3 Folded box 340x205x115 1 2 0.30970837

4 Versandschachtel 34x28x22.5 1 2 2.0137238

5 Sachets buffer pH 4.01 1 3 1.48660016

6 Softening Point Cups Disposable (Set of 1 3 2.92054989

8 Versandschachtel 420x280x350 1 4 1.54854183

10 Dropping Point Cups Disposable (Set of 5 1 5 0.22608711

11 Versandschachtel 40x29x56 2 1 0.25086378

……. ………. ………. ………. ………

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➢ The second step is to sort the probabilities in a descending manner with the entire dataset range selected. Subsequently, A, B and C categories are pasted in column F. The dataset has 720 locations of which 144 are A, 216 B and 360 C. Thus the distribution is A=20, B=30 and C=50. This would be the ideal slotting, as probabilities are optimally assigned to the slots and in the appropriate categories.

Table 10. Sorting and pasting categories

A B C D E F

2 Paperroll-Set LCP 1 1 4.27397545 a

6 Softening Point Cups Disposable (Set of 1 3 2.92054989 a

4 Versandschachtel 34x28x22.5 1 2 2.0137238 a 7 Sachets buffer pH 7.00 1 4 1.92019187 a 1 Ink ribbon 1 1 1.54854183 a 8 Versandschachtel 420x280x350 1 4 1.54854183 a 5 Sachets buffer pH 4.01 1 3 1.48660016 a 12 Tips LTS 250 µL 960/10 GPS-L250 2 1 1.21343738 a 9 Sachets buffer pH 9.21 1 5 1.11495012 a 14 Cond standard 84 µs/cm, 250 mL 2 2 1.07035211 a

23 Self-adhesive paper roll 58mm diam. 50m 3 2 1.05300845 a

……. ………. ………. ………. ……… ………

➢ The next step is to sort column A ascending. What happens is that the entire dataset is sorted back to the static slotting position, however showing items that are in a wrong category. Table 11 shows a small section of the A category range (1 – 144). In this small section there are a few items that should be in another category. See item 38 which should be in category c, as it has a much lower probability than others.

Table 11. Items in wrong category

A B C D E F

……. ……. ……. ……. ……. …….

37 O2 Membrane Kit T-96 3.1 4 4 0.55623623 a

38 Techn. Buffer pH 4.01 sachets 30x20mL 4 4 0.04955334 c

39 Tips LTS 300 µL 768/8 GPS-L300 4 5 0.12636101 b

40 Platen 3 Inch 4 5 0.51721297 a

41 Crucible 40 uL, Al, 100 pcs 5 1 0.61136431 a

42 O-Ring 20.29x2.62 Si USP VI N5 5 1 0.23661719 a

43 Tips LTS 1000 µL 768/8 GPS-L1000 5 2 0.26758803 a

44 InPro3253i/SG/120 5 2 0.36607529 a

45 Tips LTS 200UL Fltr 960/10 TR-L200F 5 3 0.37784421 a

46 Acc. Plastic sample bottle 50ml 5 3 0.45836838 a

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48 Technical buffer pH 9.21, 6 x 250mL 5 4 0.44969655 a 49 Zapfen 5 5 0.55004206 a 50 AK9/10m 5 5 0.47323438 a 51 PWR,ACCES,AC-PRONG,EU 6 1 0.26572978 a 52 Tips LTS 20 µL Filter 960/5 SR-L10F 6 1 0.19883277 a 53 Battery AM3 1.5V 6 2 0.36855296 a

54 PDX LOWER RECEIVER MOUNTING PIN 6 2 0.4552713 a

55 Beaker PP (100mL) 6 3 0.19078035 a

56 405-DPAS-SC-K8S/120 6 3 0.48190622 a

57 pH electrode InLab Routine Pro 6 4 0.17467552 a

58 Tips 1000 µL 768/8 GPS-1000 6 4 0.24962494 a 59 AK9/5m 6 5 0.23723661 a 60 Tips LTS 10 mL 200/Pkg RC-L10ML 6 5 0.62003615 a 61 pH guidebook E 7 1 0.74330008 a 62 Suction Tube 7 1 0.41253154 a 63 O-ring 10.77x2.62 Si FDA S70R4 7 2 0.44845771 a 64 405-60-SC-P-PA-K19/120/3m 7 2 0.26325211 a

65 Propeller stirrer for Compact stirrer 7 3 0.29732003 a

66 Dispensing Tube 7 3 0.48066738 a 67 HA405-DPA-SC-S8/120 7 4 0.13874935 b 68 Tips LTS 10 µL 960/5 SS-L10 7 4 0.23413952 a 69 Tips LTS 300 µL Filter 768/8 RT-L300F 7 5 0.04273975 c 70 Technical buffer pH 7.00, 250mL 7 5 0.27935695 a 71 Flat seal 8 1 0.06256109 c 72 Tips LTS 1 mL Filter 768/8 RT-L1000F 8 1 0.2849317 a …… ……. ……. ……. ……. …….

➢ Within the dataset, each category has a certain range. A from 1 till 144, B from 145 till 360 and C from 361 till 720. Selecting column F with the corresponding ranges and the COUNTIF function in Excel shows the category switches. See table 12.

Table 12. Category changes

A->B 22 =AANTAL.ALS(F1:F144,"b") A->C 7 =AANTAL.ALS(F1:F144,"c") B->A 27 =AANTAL.ALS(F145:F360,"a") B->C 60 =AANTAL.ALS(F145:F360,"c") C->A 2 =AANTAL.ALS(F361:F720,"a") C ->B 65 =AANTAL.ALS(F361:F720,"b")

This means that exactly:

- 29 items leave a and 29 items enter a - 87 items leave b and 87 items enter b - 67 items leave c and 67 items enter c

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➢ The next step is swapping the probabilities between categories. Three new columns are defined. The IF function in excel is used to fill the columns with the probabilities for each corresponding category (Table 13).

Table 13. Swapping probabilities

=ALS(F1="a",E1,"") =ALS(F1="b",E1,"") =ALS(F1="c",E1,"")

G H I J

Category A Category B Category C

….. ……. ……. …... 37 0.55623623 38 0.04955334 39 0.12636101 40 0.51721297 41 0.61136431 42 0.23661719 43 0.26758803 44 0.36607529 45 0.37784421 46 0.45836838 47 0.38775487 48 0.44969655 49 0.55004206 50 0.47323438 51 0.26572978 52 0.19883277 53 0.36855296 54 0.4552713 55 0.19078035 56 0.48190622 57 0.17467552 58 0.24962494 59 0.23723661 60 0.62003615 61 0.74330008 62 0.41253154 63 0.44845771 64 0.26325211 65 0.29732003 66 0.48066738 67 0.13874935 68 0.23413952 69 0.04273975 70 0.27935695 71 0.06256109 72 0.2849317 ….. …… ….. ……

Minimizing order picker route length in a warehouse with dynamic item flows: a simulation study