A Comparative Study of Simulated Annealing and Self-Organising Map Batching for Solving the Order Batching Problem in Warehouses

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IN

DEGREE PROJECT ELECTRONICS AND COMPUTER ENGINEERING,

FIRST CYCLE, 15 CREDITS STOCKHOLM SWEDEN 2020,

A Comparative Study of Simulated

Annealing and Self-Organising

Map Batching for Solving the

Order Batching Problem in

Warehouses

FRANS TEGELMARK

NILS STREIJFFERT

KTH ROYAL INSTITUTE OF TECHNOLOGY

SCHOOL OF ELECTRICAL ENGINEERING AND COMPUTER SCIENCE

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Simulated Annealing and

Self-Organising Map Batching

for Solving the Order

Batching Problem in

Warehouses

FRANS TEGELMARK & NILS STREIJFFERT

Degree Project in Computer Science, DD142X Date: June 8, 2020

Supervisor: Jörg Conradt Examiner: Pawel Herman

School of Electrical Engineering and Computer Science

Swedish title: En jämförande studie mellan self-organising map batching och simulated annealing för att lösa order batching problemet för lager

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Abstract

Warehouses need efficient picking strategies. Batching orders is one such strategy, but creating efficient batches out of orders is computationally hard. We compare two algorithms to find approximate solutions to the batching problem; the general global optimisation algorithm Simulated Annealing and the batching specific algorithm Self-Organising Map Batching (SOMB).

The results demonstrate that for a small number of orders, Simulated An- nealing performs better than SOMB, but takes longer to run. For a large number of orders, SOMB performs better than Simulated Annealing and generates the result quicker. Which algorithm is the best at batching orders depends on the number of orders that are being batched.

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iv

Sammanfattning

Lager behöver effektiva orderplocks-stragier. Att gruppera ordrar i större set är en strategi, men att skapa effekiva ordergrupper är beräkningsintensivt. I det här arbetet jämför vi två algoritmer som ger approximativa lösningar på problemet; den generella globala optimeringsalgoritmen Simulated Annealing (sv.

simulerad avsvalning) och den ordergrupperings-specifika algoritmen Self- Organising Map Batching (SOMB).

Resultatet visar att för en liten mängd ordrar så genererar Simulated Anne- aling bättre lönsingar än SOMB, men kräver en längre beräkningstid. För sto- ra mängder ordrar så generar SOMB bättre lösningar än Simulated Annealing och har en kortare beräkningstid. Vilken algoritm som är bäst på att gruppera ordrar beror därmed på antalet ordrar som ska grupperas.

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1 Introduction 2

1.1 Purpose . . . 3

1.2 Problem Statement . . . 3

1.3 Delimitations and Scope . . . 3

2 Background 4 2.1 The Travelling Salesperson Problem . . . 4

2.2 The Order Batching Problem . . . 4

2.3 Self-Organising Map . . . 5

2.4 Self-Organising Map Batching . . . 5

2.5 Simulated Annealing . . . 6

2.6 2-Opt . . . 7

2.7 Related Work . . . 7

3 Method 9 3.1 Experiments . . . 9

3.2 Evaluation . . . 10

3.3 Experiment Setup . . . 10

3.3.1 Test Data . . . 10

3.3.2 Warehouses . . . 10

3.3.3 Environment . . . 13

3.4 Self-Organising Map Batching . . . 13

3.5 Simulated Annealing . . . 14

3.5.1 Initial State . . . 14

3.5.2 Generating Neighbour States . . . 14

3.5.3 Calculating the Energy of a State . . . 15

3.5.4 Cooling Schedule . . . 15

4 Results 16

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CONTENTS 1

5 Discussion 29

5.1 The Running Time of the Algorithms . . . 29 5.2 Hyperparameters . . . 30 5.3 Further Research . . . 30

6 Conclusions 32

Bibliography 33

A Warehouses 35

A.1 Warehouse 1 . . . 35 A.2 Warehouse 2 . . . 36 A.3 Warehouse 3 . . . 37

B Simulated Annealing Schedules 38

C Graphs 40

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Introduction

Warehouses have always played a crucial role in the supply chain, yet with the growth of e-commerce, their importance has increased. They are no longer just a place for storage of goods; they are centres for value-creation Today’s warehouses often have assembly and packaging centres to fulfil customers orders directly on the premise [1].

Warehouse layout, how products are stored and grouped, the way workers are organised and move around the facility are all aspects of warehouse operations that can be optimised [2]. Hsieh and Huang [3] found that order picking accounts for over 55 percent of overall operating cost for warehouses. The order picking problem can be broken down into three sub-problems: an item’s storage location, order batching and picker routing problem [4]. This thesis is concerned with optimising order batching, which is a well researched NP- hard problem. That the problem is NP-hard means that the problems cannot be solved in polynomial time (under the assumption that P 6= NP)

Solving the Order Batching Problem means finding the most efficient way of putting orders or parts of orders together forming a so-called batch [4].

One approach to solving the batching problem is using the probabilistic approximating algorithm Simulated Annealing (SA). It has been found to save over 5 000 km in travel distance for order pickers in real-world warehouses over three months in comparison with the previously used routing strategy [5]. Another approach to solving the batching problem is Self-Organising Map Batching (SOMB), it uses an Artificial Neural Network (ANN) to cluster similar orders with the hope that this reduces the total travel distance for order pickers.

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CHAPTER 1. INTRODUCTION 3

1.1 Purpose

Since order picking constitutes a large percentage of the operating cost of a warehouse, it must be done as efficiently as possible. Self-organising-map batching is a relatively new approach to solving the batching problem. We, therefore, want to compare it to Simulated Annealing, which is a more traditional approach to solving the batching problem. The objective of the thesis is, therefore, to determine which algorithm is better at solving the batching problem in warehouses.

1.2 Problem Statement

Which algorithm is best at minimising the total travel distance for the ware- house Order Batching Problem out of Simulated Annealing and Self Organis- ing Map Batching?

1.3 Delimitations and Scope

The objective of this thesis is to compare two specific implementations of Sim- ulated Annealing and Self-Organising Map Batching, not to compare the algorithms in general.

For this thesis, the carrying capacity of an order picker is defined as the batch volume limit. We assume that all warehouses have a rectangle shape and that they are divided into cells. The cells can either be paths, shelves or walls.

All tours in the warehouse start and end at the same cell, henceforth known as the drop off point. All path cells are reachable from the drop off point. The distances in the warehouse are measured in the number of traversed nodes. All orders are known in advance as well as the shortest path between each path cell. The distance of a batch is approximated using the local search algorithm 2-Opt.

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Background

This chapter provides an overview of the Travelling Salesperson Problem, the Order Batching Problem, Self-Organising Maps, Self-Organising Map Batch- ing, Simulated Annealing and 2-Opt. The chapter also includes an overview of related works.

2.1 The Travelling Salesperson Problem

The Travelling Salesperson Problem (TSP) is defined as given a set of cities {c¹, c2, ..., cN} and for each pair {cⁱ, cj} of distinct cities a distance d(cⁱ, cj).

Find an order ⇡ of the cities that minimises the quantity also called tour length given by Equation 2.1 [6]. The Travelling Salesperson Problem is NP-hard [6].

N 1X

i=1

d(c⇡(i), c⇡(i+1))) + d(c⇡(N ), c⇡(1)) (2.1)

2.2 The Order Batching Problem

The purpose of a warehouse is to store items until they are needed. When an item is requested, the retrieval process must be as efficient as possible. A common way of structuring order retrieval in warehouses is that set of orders are split into several subsets, called batches. Batches are then assigned to pickers who each retrieve all the items in a single batch at a time and then drop off all the items at the drop of point where they are later packed and shipped.

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CHAPTER 2. BACKGROUND 5

Since up to 55 percent of the operating cost of a warehouse comes from item retrieval [3] it is crucial to find the most efficient way to group orders.

To avoid having the pickers go back to the drop off point too often, the batches should all be as large as possible within the capacity limits of the pickers. It is this problem that the Order Batching Problem aims to solve, which combination of orders results in the shortest total travel distance for the item picking agents. The batching problem is computationally expensive and is solved using approximation methods.

For every batch, there exists a minimum distance that an agent needs to travel to retrieve it. Finding this distance is an NP-hard problem since it means solving TSP. This step is not necessary for all batching algorithms to generate their solution. However, it is required to evaluate the performance of the output.

2.3 Self-Organising Map

Self-Organising Map (SOM) is a technique that uses an artificial neural network (ANN) that is trained using unsupervised learning to reduce the dimen- sion of a data set. The technique was developed by the Finish professor Teuvo Kohonen in 1982 [7].

The idea behind the technique is to simulate how the brain is structured, where cells with similar functionality group together, for example, cells related to taste and sight. When the brain receives different stimuli, different areas of the cortex reacts, and a topological mapping is formed through the neural system learning process. This training, when completed, makes the neural system capable of correctly reflecting the received stimulation [8]. It is the formation of this mapping through the neural system’s learning process that enables SOM to reduce the dimensionality of the input data.

2.4 Self-Organising Map Batching

The Self-Organising Map Batching (SOMB) algorithms groups similar orders together into clusters using a SOM [8]. The SOM is trained on orders from the same distribution as those that will later be batched. The batches are created one by one by taking the biggest yet unbatched order and then placing other orders that are similar to it on the SOM in the same batch until the volume limit of pickers is reached. At that point, the next batch is started, and the process continues until all orders are in a batch. With a fixed trained SOM

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this algorithm is deterministic.

This process is greedy and only guided by the SOM as heuristic. This makes it fast and it can proceed without the complication or cost of evaluating the quality of the batches. These upsides are also the downsides. The greedy approach means it doesn’t reconsider alternative solutions and without any way to evaluate batches it isn’t really possible to modify it to do so. Similarity between orders is loosely the amount of items that are close to some items in the other order. Items that are close to paths that will be travelled between items that are in a batch are thus not considered similar. These paths are computed after the SOMB algorithm completes which makes them unavailable for the algorithm.

2.5 Simulated Annealing

Simulated Annealing (SA) is a probabilistic algorithm for finding the optimal global solution for a function. The algorithm derives its name the process of slowly cooling a physical system in order to obtain states with globally minimum energy [9]. Simulated Annealing is a heuristic which means that it is not guaranteed to find the optimal global solution.

The algorithm starts with an initial temperature and a random state. Af- ter the initial setup, the algorithm starts its main loop where it first chooses a neighbour state of the current state and calculates the energy of the neighbour. Then the algorithm decides if it should swap the current state with the neighbour state. The decision is based on the difference in energy between the two states and the current temperature. If the energy of the neighbour state is lower then the current state, the neighbour state is set as the current state. If the energy of the neighbour state is higher than the current state, the decision to keep the neighbour state is decided by a temperature-dependent probability function. The higher the current temperature is, the higher the chance that a state with a higher temperature is kept. After the decision has been made, the system temperature is decreased, and the loop restarts. The loop repeats until a predetermined final temperature has been reached.

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CHAPTER 2. BACKGROUND 7

2.6 2-Opt

The 2-opt algorithm is a local search algorithm for TSP, that was originally proposed in 1958 by Croes [10]. However, the primary move of the algorithm had already been suggested two years earlier in 1956 by Flood [11]. The algorithm aims to remove any instances of the tour crossing over itself. The algorithm works by deleting two edges in the tour and reconnecting the nodes.

A visualisation of the 2-Opt algorithm being applied to a TSP tour can be found in Figure 2.1.

Figure 2.1: A 2-Opt move: original tour on the left and resulting tour on the right [6].

2.7 Related Work

In an article by Matusiak et al. [5] from 2013, the authors proposed a joint order batching and picker routing method to solve the combined routing and order- batching problem. They used the A*-algorithm to solve the routing problem and Simulated Annealing to solve the batching problem. The approach was tested on sets of three orders and was found to be only 1.2 percent worse than the optimal solution.

In their thesis from 2019 Ardö and Lindholm [12] compared two meta- heuristics Simulated Annealing and a Genetic algorithm for solving the order- batching problem. They found that Simulated Annealing performed better or equally as good as the Genetic algorithm at minimising the total travel distance.

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They also found that Simulated Annealing finds the shortest solution faster than the Genetic algorithm.

Theys et al. [13] compared the Lin–Kernighan–Helsgaun (LKH) TSP heuristic with existing dedicated order picker heuristics and found an average saving in route distance of up to 47 percent when using LKH.

De Koster and Van Der Poort [14] compared optimal and heuristic solutions for routing pickers in warehouses. They found that the optimal solution reduced the travel time per route between 7 and 34 percent compared with the heuristic. However, they also found that the reduction in time strongly depends on the layout and operation of the warehouse.

Su [15] designed an automated control mechanism for the planning of order picking by combining a neural network with Simulated Annealing. Al- though the author of the paper did not fully test the performance of the system, the limited test that where performed showed promising results.

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Chapter 3 Method

This chapter describes the experiments performed to answer the research question. The chapter also contains information about the test setup, how the test data was generated and implementation details for both Simulated Annealing and Self-Organising Map Batching.

3.1 Experiments

Twelve experiments were designed to help answer the research question, see Table 3.1 for an overview of the experiments. Each experiment was run 15 times per warehouse for Simulated Annealing and once for SOMB. The reason for running Simulated Annealing multiple times for each experiment and SOMB once is that Simulated Annealing is non-deterministic while SOMB is deterministic.

Experiment Number of orders Max items per order Item volume limit Batch volume limit

1 10 5 20 132

2 10 10 20 241

3 10 15 20 350

4 10 20 20 458

5 100 5 20 132

6 100 10 20 241

7 100 15 20 350

8 100 20 20 458

9 500 5 20 132

10 500 10 20 241

11 500 15 20 350

12 500 20 20 458

Table 3.1: An overview of the experiments.

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The batch volume limit depends on the maximum number of items allowed in an order and is four times the average order volume. The reason for picking this volume limit is that we want to test the capability of the algorithm to batch orders. If the volume limit is too small, the algorithms would not be able to find any orders to combine into batches. If the order limit is too big, then the algorithms could combine all the orders into one or two batches.

3.2 Evaluation

The resulting batches from each algorithm are compared by calculating the total tour distance for each set of batches. Since calculating the tour distance for a batch means solving TSP, we can not calculate the exact distance for each batch. We instead approximate the tour distance using the local search algorithm 2-Opt. The algorithm is run for 30 seconds for each set of batches to produces a comparable result.

3.3 Experiment Setup

This section provides an overview of the warehouses and the test environment.

3.3.1 Test Data

The warehouse shelves are filled with items, and then orders are generated with a weighted distribution where shelves that are closer to the drop off point will be accessed more often. We chose to make a distribution where 80 percent of the item types that are ordered are selected from the 20 percent items on the closest shelves. The reason for choosing this distribution is to simulate a warehouse sorted with high throughput items closer to the drop off point. Since we are using a SOM, which organises structure in data, order data generated with a uniform distribution and no structure is unlikely to produce good results.

3.3.2 Warehouses

A warehouse is a rectangle, divided into cells that can be either paths, shelves or walls. Each warehouse has one drop off point where the agent that picks the items in the orders begin and end their trip. The warehouses are stored in text files, where the first two lines are integers representing the warehouse width and height. The following lines are the warehouse layout placed inside a square

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CHAPTER 3. METHOD 11

of number signs (#). Paths are encoded as spaces; shelves can be encoded with four different characters (see table 3.2) depending on from which side the shelf is accessible. Walls can be any character not used to encode paths or shelves or X, which is used to mark the drop-off point in the warehouse. The drop-off point is also a path that the agent can traverse.

Character Access direction Access coordinates ] From the cell to the left (x 1, y) [ From the cell to the right (x + 1, y) v From the cell above (x, y + 1)

^ From the cell below (x, y 1)

Table 3.2: Characters used to encode shelves in the text representation of the warehouse.

The warehouses are converted into graphs using the NetworkX Python library [16]. The graph representations of the warehouses allow us to calculate the shortest distance between all the items using NetworkX’s implementation of the A* algorithm. The calculated distances are stored in a distance matrix.

This allows us to have an instant lookup time for the shortest path between items in the warehouse. We also calculate the shortest distance between all the items and the drop off point.

Three warehouses were designed for this thesis. Warehouse one and three has a traditional warehouse layout with long rows and few crossing aisles, and their layout is based on the Traditional Layout 2 from the article An Overview of Warehouse Optimization [17]. Warehouse number two is based on the Flying-V Layout from the same article. Graph representations of the ware- houses can be seen in Figures 3.1-3.3. The text file representation of the warehouses can be found in appendix A. For more information about the warehouse layouts, see Table 3.3.

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Figure 3.1: Warehouse 1. The blue circles are path nodes, the black squares are shelves and the red node is the drop off point. An agent can move between two path nodes if there is an edge between them.

Figure 3.2: Warehouse 2: the Flying-V warehouse layout. The blue circles are path nodes, the black squares are shelves and the red node is the drop off point.

An agent can move between two path nodes if there is an edge between them.

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Figure 3.3: Warehouse 3. The blue circles are path nodes, the black squares are shelves and the red node is the drop off point. An agent can move between two path nodes if there is an edge between them.

Warehouse Dimensions Types of items

1 13⇥ 9 56

2 85⇥ 30 1512

3 31⇥ 30 530

Table 3.3: Overview of warehouse layout.

3.3.3 Environment

All test were performed on an Amazon Web Service (AWS) Amazon Elas- tic Compute Cloud (EC2) instance of type c5.4xlarge [18]. The instance is equipped with an Intel Xeon Platinum 8124M CPU (3.00GHz), 16 vCPU (16 threads) and 32GB of RAM. The experiments were run in parallel, one experiment per thread.

3.4 Self-Organising Map Batching

Our implementation of SOMB for this thesis is written in Python. The SOM’s neural network is trained using a matrix where the rows correspond to order

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IDs and the columns represents if an item ID exits in that order. If a cell is zero, it means that the item is not part of that order, if it is one the item is part of the order.

Items that are on shelves that are close to items that are part of orders that have been batched do not require us to extend our picking tour if they are included in a batch. The order picker would have to pass by these shelves to complete orders that are currently in the batch.

Items that are nearby are given a score between zero and one in the training matrix. The score is calculated by the formula 1 M/D where M is the maximum distance that is considered "close" and D is the distance we have to walk to reach this item ID from the closest position we will have to go to pick an item that is in the order. Thus items that are close in warehouse space will be considered similar items when batching the orders.

3.5 Simulated Annealing

The Simulated Annealing algorithm was implemented using the Python li- brary simanneal (version 0.5.0) written by Matthew Perry [19]. The library exposes two functions move which creates neighbour states and energy, which calculates the energy of a state.

3.5.1 Initial State

The initial state is generated by placing each order into an empty batch. The reason for placing one order per batch is to make it as easy as possible to generate neighbour states.

3.5.2 Generating Neighbour States

The generate neighbour function contains two functions that are called with a certain probability. The first function picks a random order and tries to put it into a random batch or a place it into an empty batch. The chance that an order is placed in an empty batch is one divided by the number of batches. The order is also placed into an empty batch if it does not fit inside any of the current batches. The second function tries to switch two orders from different batches.

The second function is called 90 percent of the time. The ratio between the functions was set after running several tests.

The reason that the second function is called more often than the first one is that after a while, it becomes harder to find a batch that has space to fit a

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random order and placing single order in a batch is always less efficient.

3.5.3 Calculating the Energy of a State

The energy of a state is defined as the total distance a picking agent would have to travel to complete the batches in the state. Since finding the shortest travel distance for each batch means solving TSP this is an NP-hard problem.

We, therefore, approximate the travel distance for each batch using the local search algorithm 2-Opt. The algorithm is run once for each combination of nodes in the tour. This approximation will not find the shortest path for a batch;

however, it is sufficient to approximate the energy of a state.

3.5.4 Cooling Schedule

The cooling schedules used in the experiments were found by first by using the simanneal library’s built-in schedule finder and then manually tuning the parameters with a few tests. The goal of the tuning was to find a schedule for each experiment that produced good results and took on average 30 seconds to run. A complete list of cooling schedules can be found in Appendix B.

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Results

Table 4.1 shows the results for the 12 experiments for each warehouse. Simu- lated Annealing performs better than SOMB for smaller order sizes, but SOMB performs better than SA for larger order sizes.

Figures 4.1-4.9 compares the average distance produced by Simulated An- nealing over 15 test runs with SOMB, for the three different warehouses.

Tables 4.2 - 4.4 shows the variance in the distances produced by Simulated Annealing over 15 test runs. The distance for the batches produced by Simu- lated Annealing has a small spread, and for some of the smaller order sizes, all runs converged to the same final distance. The derivation of the results generally increased with the order size.

Table 4.5 shows the time it took for SOMB to generate a batch for each experiment. The results show that the duration increases with the number of orders that are being batched.

Figures 4.10-4.18 compare the best tour distances found over time for the average Simulated Annealing run, the best Simulated Annealing run and SOMB for a selected number of experiments. See the complete set of graphs in Ap- pendix C.

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CHAPTER 4. RESULTS 17

Experiment Algorithm Warehouse 1 Warehouse 2 Warehouse 3

1 SA 138 782 282

SOMB 172 962 316

2 SA 148 1 048 429

SOMB 198 1 298 552

3 SA 116 1 002 638

SOMB 156 1 166 688

4 SA 126 1 588 732

SOMB 224 1 906 894

5 SA 782 5 743 2 943

SOMB 1 102 7 366 3 854

6 SA 1 081 8 966 4 527

SOMB 1 450 9 862 5 380

7 SA 1 067 11 344 6 322

SOMB 1 370 11 174 6 066

8 SA 1 118 14 813 8 166

SOMB 1 214 13 242 6 890

9 SA 7 931 42 060 27 325

SOMB 5 816 34 246 20 282

10 SA 12 588 72 396 43 532

SOMB 6 224 46 844 26 352

11 SA 16 016 93 615 55 948

SOMB 6 628 53 978 29 634

12 SA 19 629 111 237 67 550

SOMB 7 018 61 534 32 304

Table 4.1: Comparison of the average distance produced by Simulated An- nealing over 15 runs, with the result from SOMB.

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Experiment Median 1st Quartile 2nd Quartile Min Max

1 138 138 138 138 138

2 148 148 148 148 148

3 116 116 116 116 116

4 126 126 126 126 126

5 782 776 788 762 802

6 1 080 1 059 1 097 1 052 1 122

7 1 054 1 047 1 093 1 022 1 122

8 1 098 1 063 1 170 966 1 266

9 7 974 7 869 8 009 7 526 8 152

10 12 478 12 363 12 728 12 132 13 344

11 16 018 15 898 16 257 15 550 16 298

12 19 678 19 367 19 813 19 112 20 220

Table 4.2: Variance in Simulated Annealing result for warehouse 1 over 15 runs.

Experiment Median 1st Quartile 2nd Quratile Min Max

1 782 782 782 782 782

2 1 048 1 480 1 480 1 480 1 480

3 1 002 1 002 1 002 1 002 1 002

4 1 586 1 586 1 586 1 586 1 614

5 5 758 5 729 5 763 5 678 5 824

6 8 968 8 871 9 024 8 680 9 402

7 11 138 11 190 11 452 1 0804 12 238

8 14 952 14 678 15 087 1 3684 15 592

9 42 282 41 353 42 515 40 624 43 340

10 72 400 72 042 72 928 70 808 73 762

11 93 826 93 429 94 114 90 358 94 716

12 111 237 110 682 111 823 109 654 112 492

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Experiment Median 1st Quartile 2nd Quratile Min Max

1 282 282 282 282 282

2 428 428 428 428 428

3 638 638 638 638 638

4 732 732 732 732 732

5 2 926 2 850 3 012 2 810 3 118

6 4 548 4 436 4 597 4 244 4 819

7 6 320 6 215 6 450 5 902 6 720

8 8 126 8 012 8 290 7 726 8 904

9 27 214 26 943 27 579 26 664 28 726

10 43 716 43 109 43 949 42 292 44 442

11 55 962 55 606 56 246 55 146 57 144

12 67 624 67 246 67 925 66 480 68 530

Experiment Warehouse 1 Warehouse 2 Warehouse 3

1 2.4 3.9 2.5

2 2.6 5.3 3.2

3 1.9 3.6 3.3

4 2.9 6.2 3.3

5 34.6 50.0 41.6

6 37.6 57.1 48.5

7 35.6 57.7 51.1

8 38.2 63.9 54.7

9 483.0 621.5 551.0

10 559.4 729.8 652.2

11 611.1 762.8 726.6

12 656.6 829.8 701.5

Table 4.5: Run time for SOMB to find a batching solution for each warehouse.

The durations are given in milliseconds (ms).

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Figure 4.1: Average distance produced by Simulated Annealing compared with SOMB for experiment 1-4 for warehouse 1.

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Figure 4.10: Comparison of the best tour distance found over time for the average Simulated Annealing run, the best Simulated Annealing run and SOMB.

Config: Warehouse 1, number of orders: 10, max items per order: 15.

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Chapter 5 Discussion

Simulated Annealing is allowed to peek at the measure which we are optimising for and can potentially continue improving its output until we are satisfied (or it gets stuck in a local minimum). SOMB uses a heuristic and greedily groups orders but still manages to make decent batches in a short amount of time. It seems reasonable that Simulated Annealing would have the potential to outperform SOMB, but for a large number of orders, the time complexity of the algorithms for this task favours the batching specialised heuristically driven SOMB.

5.1 The Running Time of the Algorithms

While SOMB was much faster than Simulated Annealing at batching, to get the final result, we used the approximate TSP solver on each output to be able to compare them. The time spent on this part is not taken into account when comparing Simulated Annealing and SOMB since it is not part of the batching problem that we are investigating. In practice, this step is mandatory, though, since the pickers need to know in which order to collect the items. Depending on how much time is spent on this part, saving a few seconds on the batching process might not make much sense when the full run time of the program takes minutes. The amount of time we spend on the TSP solver was probably not necessary for minimising route distances though. Our priority was to let the randomness of the TSP solver to stabilise so that the comparison of the batching results was not due to the TSP solver not running long enough.

For an actual warehouse, the running time of the TSP solver must be taken into account. If the solver is allowed to take minutes, the batching process would reasonably also not need to take less than a second and Simulated An-

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nealing could be used to improve the results.

5.2 Hyperparameters

The parameters for both SOMB and Simulated Annealing could be changed for better comparisons. With low limit numbers (size or volume) there is no optimisation space since in the limit each order forms a batch and becomes the only order in that batch. With no limitation on how much can be put in each batch, both algorithms can just put all orders in one single optimal batch (since merging batches always produces better a shorter roundtrip if the TSP problem of all those locations is solved), and then we get the same output from both algorithms and cannot make any interesting comparisons. The volume limits were chosen to be something we thought would strike the right balance between the number of orders per batch and the number of batches.

Other parameters and implementation details for each algorithm could also have been chosen differently to produce other results. The values chosen seemed to produce good enough results. Other values may have affected the comparison a bit, but it is unlikely that it would change the results in any sig- nificant way. Any change either way would most likely only shift the cut-off point where Simulated Annealing would fail to catch up to SOMB for larger numbers of orders.

5.3 Further Research

Since SOMB can produce excellent results quickly, it would be an interesting idea to combine SOMB and Simulated Annealing. The output from SOMB could be used as the starting state for Simulated Annealing.

The probabilities for the two functions in the Simulated Annealing’s neighbour function are fixed. It would be interesting to see if the Simulated An- nealing algorithm produced better results if the probabilities for the neighbour functions depended on the current temperature. The idea is that in the begin- ning, we want the algorithm to combine batches since fewer batches always produce a better result. Then in the later stage of the algorithm’s run when it is hard to find a new batch for an order, we want to swap orders between batches.

Two interesting parameters that would make this research more applicable to real-world warehouse optimisation would be to introduce the notion of order priority and deadlines. Sometimes we perhaps need to accept more extended

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CHAPTER 5. DISCUSSION 31

tours for the order pickers to guarantee that orders are completed in time.

Another interesting area of research is to use more sophisticated heuristics for calculating the energy of a state than 2-Opt. More exact measurements of the energy of a state would make it possible for Simulated Annealing to make a more informed decision about if it should accept a state. The usefulness of a more exact energy measurement must be balanced with how fast the approximation is because the Simulated Annealing algorithm must get to evaluate as many neighbour states as possible.

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Conclusions

If Simulated Annealing is limited in run-time to the time, it takes SOMB to complete its batching SOMB will always perform better than Simulated An- nealing. The output of Simulated Annealing is then still at a stage that is close to the random initialisation. However, result quality per time spent batching is not necessarily the most important metric for a warehouse in practice. The great strength of SOMB is its short run-time; it is also a part of its weakness.

Unlike Simulated Annealing, we cannot choose to run it for longer to improve the quality of the output even if we have the time to do so.

The research question this thesis set out to answer is: Which algorithm is best at minimising the total travel distance for the warehouse Order Batching Problem out of Simulated Annealing and Self Organising Map Batching? The answer to this questions depends on how quickly one needs the results, how valuable improvement in output quality is compared to computation time and how many orders the warehouse needs to complete.

For a large number of orders, SOMB produces a better result and is faster than Simulated Annealing. Because of the search space size, the more general optimisation algorithm Simulated Annealing is not able to come close to SOMB in a reasonable time frame. Also, if results are needed immediately or need to be updated continuously at interactive timescales, then SOMB is superior. When the search space is small Simulated Annealing outperforms SOMB. Simulated Annealing can produce better results but has a run-time that is roughly 100 times longer than SOMB. Nevertheless, with the TSP solver already taking a similar amount of time, the relative time gained from using SOMB is diminished. With a physical warehouse where kilometres saved means minutes or hours of less order picking time, a few seconds or minutes of extra run-time will be won back and pay for itself.

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Appendix A

Warehouses

A.1 Warehouse 1

warehouse_1.txt

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#X #

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A.2 Warehouse 2

warehouse_2.txt

85 30

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# #

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# ][ ][ ][ ][ ][ ][ ][ ][ ][ ][ ][ ][ ][ ][ ][ ][ ][ ][ ][ ][ ][ ][ #

# ][ ][ ][ ][ ][ ][ ][ ][ ][ ][ ][ ][ ][ ][ ][ ][ ][ ][ ][ ][ ][ ][ ][ ][ #

# ][ ][ ][ ][ ][ ][ ][ ][ ][ ][ ][ ][ ][ ][ ][ ][ ][ ][ ][ ][ ][ ][ ][ ][ ][ ][ #

# ][ ][ ][ ][ ][ ][ ][ ][ ][ ][ ][ ][ ][ ][ ][ ][ ][ ][ ][ ][ ][ ][ ][ ][ ][ ][ ][ ][ #

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APPENDIX A. WAREHOUSES 37

A.3 Warehouse 3

warehouse_3.txt

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Simulated Annealing Schedules

Experiment Warehouse tmax tmin steps

1 1 270 0.68 1.0⇥ 10⁵

3 1 167 4.1 9.0⇥ 10⁵

7 600 0.55 7.4⇥ 10⁵

2 1 330 0.39 4.7⇥ 10⁴

3 810 1.9 1.6⇥ 10⁴

7 940 0.92 1.4⇥ 10⁴

3 1 297 0.30 1.7⇥ 10⁴

3 1 267 3.3 1.0⇥ 10⁴

7 770 1.2 8 300

4 1 327 0.56 2.5⇥ 10⁴

3 1 267 2.6 4 300

7 730 0.76 4 600

5 1 170 0.14 1.3⇥ 10⁴

3 1 100 0.31 8 000

7 520 0.23 5 700

6 1 227 0.23 5 100

3 937 0.16 2 000

7 1 000 0.35 2 500

7 1 223 0.12 3 500

3 943 0.20 1 320

7 1 200 0.11 1 500

8 1 213 0.20 2 800

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APPENDIX B. SIMULATED ANNEALING SCHEDULES 39

Experiment Warehouse tmax tmin steps

3 953 4.4⇥ 10 ² 860

7 1 220 7.3 ⇥ 10 ² 1 000

9 1 313 1.0⇥ 10 ⁶ 4 000

3 1 333 3.4 ⇥ 10 ⁷ 3 000

7 650 3.7⇥ 10 ⁶ 3 000

10 1 223 6.6⇥ 10 ⁴ 1 200

3 2 800 2.2 ⇥ 10 ⁶ 1 200

7 1 333 3.0 ⇥ 10 ⁶ 1 300

11 1 230 1.3⇥ 10 ³ 1 300

3 2 800 2.2 ⇥ 10 ⁶ 550

7 1 333 6.7 ⇥ 10 ⁵ 700

12 1 800 2.2⇥ 10 ⁴ 800

3 1 037 5.6 ⇥ 10 ⁶ 300

7 500 1.5⇥ 10 ⁴ 330

Table B.1: Simulated Annealing schedules used for the experiments.

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Graphs

Figure C.1: Comparison of the best tour distance found over time for the average Simulated Annealing run, the best Simulated Annealing run and SOMB.

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APPENDIX C. GRAPHS 41

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Figure C.10: Comparison of the best tour distance found over time for the average Simulated Annealing run, the best Simulated Annealing run and SOMB.

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Figure C.25: Comparison of the best tour distance found over time for the average v run, the best Simulated Annealing run and SOMB. Config: Warehouse 3, number of orders: 500, max items per order: 5.

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