Modelling the Shortest Path for Inner Warehouse Travelling Using the Floyd–Warshall Algorithm

Abstract

Order picking is referred as a critical process of selecting items requested by a customer in a warehouse. Meeting the demand of every customer is the main objective in this area. Large warehouses pose a challenge since the order-picking process is slowed considerably by the lengthy time it takes to transport items across the warehouse. Throughout the study, the system is hoped to develop proper procedures in the order-picking process. In handling this scenario, the decision-makers need to take any possible action to ensure the warehouses can keep operating and meeting the requirements and satisfaction of the customers. Due to this, the study’s main objective is to determine whether the Floyd–Warshall algorithm or the dynamic programming method will give the most accurate shortest path and minimum travel distance for order pickers. Two data sets (nine nodes and nineteen nodes) are used to determine the optimal path and minimum travel distance for the order picker to meet and satisfy customer orders for the warehouse. The two models were modified and applied to address real-world case studies from the automotive manufacturing company in Malaysia. The results show a big difference between the total distance by 113.48% for 19 nodes. Through this finding, the company may choose which method suits their preferences. Concurrently, this study may also contribute to problem-solving issues in any warehouse operation with a similar procedure.

1. Introduction

In any manufacturing organisation, warehouses play an exceptional role in the performance of tasks required by logistics chains. A well-functioning and smooth-running warehouse operation is vital in achieving success or it could also destroy one’s business. However, getting it right is no simple task. Proper function is an essential condition ensuring the smooth flow of material goods in logistics systems, which forces them to apply appropriate technologies and to manage warehouse processes in an effective manner [1]. This is aligned with the objective of a warehouse which is to satisfy customers with effective resource utilisation and delivering the right product, at the right place, and at the right time in good condition [2]. In addition, efficient management is pinnacle in making sure the demand of customers could always be achieved [3].

Coming in the new era, orders made now are not limited to a traditional way of going to the shop but also can be purchased through the internet. Once the order is added to cart and paid, customers expect to have fast return in the delivery of items. However, in order to provide efficient delivery to the customers, organisations need to manage wisely their activities in the warehouse to better match supply with customer demand.

Among the main activities performed in the warehouse are receiving, removing, storing, picking orders, sorting, and shipping [4]. Ref. [5] stated that order-picking activities are one of the most researched areas that relate to the optimisation of a warehouse. Furthermore, this activity is the most labour-intensive process that requires up to 55% of the total costs of an item in the warehouse [6,7]. The labour-intensive and wide change in nature of warehouse operations results in order picking becoming a main activity that is related to the satisfaction of the customer [4]. Late or inaccurate deliveries may have a significant impact on customer satisfaction as this may affect the customer’s emotions and increase the cost [8].

Currently, human pickers are responsible for picking thousands of orders [6]. A main issue in reducing and prolonging order retrieval time is worker’s fatigue [8]. This activity will affect the company since humans have their limitations. Their work will be disrupted, and they need to work overtime once they are sick or on leave. In addition, the management has to pay more as a result, such as more salary, more utilities, and more maintenance on machinery. Thus, this supports that the proper handling of tasks and allocation for each order picker (using human work force) is yet a complex study area. Hence, a better picking system is important and may result in a better service level by providing a shorter picking distance, a minimum labour cost by retaining an optimal number of pickers, or both [4].

Therefore, to address the disparity and to provide the significantly shortest path with a minimum travel distance, this paper presents a Floyd–Warshall algorithm and dynamic programming (DP) method aimed at finding the most accurate shortest path and minimum travel distance for order pickers with no budget restrictions. The main objective is that the models used are able to find the shortest path and able to collect all the items requested with the shortest distance as the top priority. However, the network size is only focused on one zone due to the limited number of pickers and each picker has a limited picking capacity. Hence, this study leads to easier management in the order picker and their respective schedule, reduces the need for overtime hours, reduces the travelling time of the picker in the warehouse, removes the need to hire temporary workers, and increases employers’ productivity through fully utilised resources in such warehouses.

The remainder of this paper is arranged as follows: Introduction, Literature Review, Methodology, and Analysis of the Results. The next section delivers an overview of past studies related to the Floyd–Warshall algorithm and DP model, highlighting the gaps and deepening the understanding of the models, especially in managing warehouse operations. Section 3 discusses the applications of both models and how the calculation is conducted involving nine nodes. Section 4 explains deliberately the procedure of the model applied in the automotive manufacturing company in the real-world case, with 19 nodes, by using automated software. Reviews on the entire procedure and computational results related to the model solution are also shown, and a sensitivity analysis is performed. Section 5 concludes and provides future recommendations from the study.

2. Literature Review

The research interest in this area has significantly grown over time; however, there are still many things can be explored. This can be observed by looking at the evolution of the problem focus and the involved elements, applications, and solution techniques. The evolution and all the related elements are described in this section. Then, all this literature will be treated as groundwork that assists in tackling such problems in the area under study.

2.1. Warehouse Management

The role of warehouse management is normally to enhance picking productivity and efficacy. One big warehouse is divided into several different zones, and these include managing warehouse operations which are order segmentation, order allocation, order-picking tasks, bulk storage for large-scale quantities of goods that do not require frequent access for picking, and delivery and receiving distribution. The delivery areas serve as hubs where items are consolidated and prepared for delivery to various destinations. Receiving areas are sometimes placed in the same area as the delivery area.

Some companies may need bulk storage if they involved with bulky items and their product can be kept for longer. Finally, the order-picking area consists of retrieving individual items from storage on the basis of customer orders. Nowadays, with the advancement of technology, online shopping has also contributed to increase demand and slow-moving consignments preparation from the warehouse. Not only that, the costs involved in managing such a procedure make it a costly operation. In handling this scenario, the decision-makers need to take any possible action to ensure the warehouses can keep operating and meeting the requirements and satisfaction of the customers. This includes providing better service, quality items, maximum profit with minimum service level, and some may downsize their operation in warehouses. Despite of all the activities, complexity may arise in maintaining the process efficiently [9,10]. However, different operations may vary between warehouse operations and a solution that works well for one company may not necessarily be effective for another, even if they are in the same industry. Ref. [11] mentioned that the order-picking process itself accounts for almost 50–75% of the total operating cost in a typical warehouse. Thus, it is essential to close the gap in order to create a solution that may accommodate all types of order picking in warehouse operations.

On the whole, the order-picking system may be performed by humans or automated machines. However, the manual order-picking system is totally using human labour without the aid of technologies. In a manual order-picking system, workers need to physically navigate the warehouse aisles to locate, retrieve, and assemble items required for each order. This process is suitable for warehouses with lower order volumes, simpler inventory configurations, and flexible storage arrangements. As mentioned in [12], manual order picking is still considered to be the main practice in a warehouse. Researchers estimate almost 80% of all order-picking processes are still operated manually. Though it may cost less, this process may cause fatigue, boredom, complexity, and missing information that leads to errors.

In contrast, the use of automated machines in order-picking systems is increasing and slowly becoming common in warehouse operations. This technology is able to handle larger orders and makes them easier to handle. In fact, due to increase in demand through e-commerce, there is a need to change the order-picking system. Sometimes, there will be a delay in delivering consignments to customers. Thus, distributors of courier services might work overtime to deliver the consignments, and this will go beyond their own working hours. Through improvements in material handlings in warehouse operations, robotics and cloud technologies, RFID, autonomous guided vehicles (AGVs), cranes, and mobile robots have become vital in warehouses [13]. However, the reason why many companies still rely on manual order picking is due to the motor skills and cognitive abilities of humans that cannot be replicated by the automated machineries [12]. Due to this, the conventional technique of order picking by humans cannot be totally ignored but needs to be improved.

2.2. Floyd–Warshall Algorithm and Dynamic Programming Model

This study is concerned with minimising the travel distance of order picking in a warehouse using the Floyd–Warshall algorithm and DP model. By having the shortest distance to complete the task, the order picker can work faster. Furthermore, the activity of the order picker in the warehouse will be more systematic, reduce fatigue, demotivation, and errors, and lead the company to be more cost-effective and adaptable to changing business needs. This approach also benefits the consumer as they will receive their order in a shorter waiting time. The warehouse under study is a local automotive warehouse. The data that is available for this warehouse is the number of items that need to be picked by the order picker based on a normal 8 h working day, by taking into consideration their resting time, limited number of pickers, and picking capacity.

The adjusted algorithm and model aim to determine the shortest path with limited picking capacity for each picker. The Floyd–Warshall algorithm is the association of mathematical and programming optimisation. It is one of the variants of DP to find shortest path with positive and negative edge weight [14]. This algorithm will generate an interconnected decision, which is a method that performs problems with the solution, and it is possible to obtain more than one solution. Ref. [15] stated that the Floyd–Warshall algorithm is one of the popular algorithms that is used in calculating the shortest path between all pairs of vertices in a directed network. This algorithm is one of the variants of dynamic programming and the correct result can be achieved when there are no negative cycles involved [15].

Ref. [16] stated that the Floyd–Warshall Algorithm was imposed by Robert W. Floyd in 1967. The idea is to find each pair of vertices u and v, and whether u can reach v, since Warshall was interested in the weaker question of reachability. It was also realised that the same technique could be used to compute the shortest paths with only minor variations. The Floyd–Warshall algorithm has a directed and weighing directed graph (V, E), in which V refers to the node or point of where to start and where to end, while for E, this is the adjacent line between the two nodes. Ref. [16] also explained that the number of side weights on a path is the weight of the path and the side on E is allowed to have a negative weight, but it is not allowed for this graph to have a cycle of negative weights. The smallest weights of all points that connect a pair of nodes will be calculated at the same time.

Another option of solving the problem is by using a system. TORA is a software (TORA version 2.00, Feb 2006) that consists of an algorithm that has a set of mathematical instruments or programs [17]. It is an optimisation system in the area which is very easy to use. Furthermore, TORA version 2.00, Feb 2006 is a user-friendly software because it is a menu-based and windows-based system [18]. This software deals with the following algorithms, which are solutions of simultaneous linear equations, linear programming, transportation model, integer programming, network model, and others. In the network model part, it contains the Floyd–Warshall algorithm and Dijsktra’s algorithm.

In a study by [19], they aimed to find a way to determine the minimum travel distance from the KNUST fire station to all areas in the area by using Dsijktra’s algorithm and an extension of Floyd’s algorithm to find the shortest distance from the KNUST fire station to all areas there. Ref. [19] stated that Dijkstra’s algorithm is a single-source shortest algorithm, such that it finds the shortest path from a starting point to other nodes while the Floyd–Warshall algorithm computes the shortest path between nodes. The improved Floyd algorithm can work with any difficulty while calculating the shortest distance.

Last but not least, Refs. [14,20] stated that increasing numbers of vehicle users would need a higher number of garages. This is especially important when cars have problems in places they do not normally visit. Instead of relying on the insurance of the properties, consumers also need to find other solutions to solve the problem. Therefore, information on nearby garages is essential. The Floyd–Warshall algorithm can be used to calculate the shortest distance from an existing area to a nearby garage. This can be implemented to navigate the consumer through the Global Positioning System (GPS) and Location-Based Services (LBS). This algorithm successfully identified the shortest path to the nearest garage through categorisation and radius by the selected user. Hence, in this study, the adjusted mathematical model is solved using a dynamic programming algorithm called the Floyd–Warshall algorithm.

3. Methodology

Managing the inner warehouse transportation processes such as congestion and response time are vital and the results can be used for future research. Response time in this case refers to the moment an order is placed until the order is successfully collected at the packaging point before it is delivered to the customer. This also shows that the response time is influenced by the strategic location and routing of the transportation. According to [21], a proper routing method in order picking efficiently helps to reduce between 17% and 34% of the travelling distance. Order picking in a warehouse system is known to be a crucial part in the supply chain. Any underperformance or unreliable system may lead to dissatisfaction in the customers.

Therefore, the Methodology section focuses on identifying and accomplishing a particular task based on the specified models. Previously, Dijkstra’s algorithm and the Travelling Salesman Problem (TSP) were being tested and compared with the DP method, in which the DP method was found to have more advantages than the other methods despite the distance calculated by the DP method being sometimes farther. More often, the DP method was able to avoid redundant calculations, which results in significant performance improvements [3]. Furthermore, the DP method ensures that the optimal solution is found by considering all possible combinations. This means that no nodes are left out. This has been verified in [22], where the Dijkstra’s algorithm despite producing shorter travelling distance, sometimes omitted some nodes.

In this study, the Floyd–Warshall algorithm and DP model are tested for their performance. The primary purpose of the Floyd–Warshall algorithm is to compute the shortest paths between all pairs of nodes in a graph. It may give the shortest distance between locations; however, there is a need for an analysis of whether it directly handles order assignment based on the order picker’s picking capacities. Meanwhile, the DP model is known to be able to solve optimisation problems where decisions need to be made consecutively. This model may be incorporated with other algorithms, such as the TSP and Knapsack algorithm in finding the optimal solution.

The flow of order picking in the warehouse is best described in Figure 1. The flow starts with pickers receiving the order notes from customers and reading them. At the same time, the picker will count the number of items and mark the items needed. Next, the order picker starts to search for the desired items from the picking location. After the items are located, the identified items are picked from the storage rack. In this identifying process, the pickers also need to count the number of items and tally them with the order lists. The picking flow and movement along the aisles continue until all the items are fully collected. Similarly, in Figure 1, the additional part is the system also calculates the total travel time or total distance for each picker starting from entering the warehouse until the picker completes the picking process and exits the warehouse. This is to ensure the whole process in completing a day’s order is within the normal working hours.

The process starts with the order notes, and the order picker will start picking accordingly. The items needed are referred to as nodes and the nearest node to the picker is denoted as the starting point. Then, from the starting point, the picker will identify the next location near to the current station and the process continues until all the items are collected. The shortest distance and path are provided for the pickers to help them minimise the waiting time of the customers. The procedure stops when the picker leaves the warehouse.

3.1. Study Settings

3.1.1. Demand and Order Patterns

This study used data collection from a warehouse of an automotive manufacturing company situated in Serendah, Selangor. This company manufactures parts including body parts, suspension, engine parts, modular assemblies, engineering plastic parts, and lamp assemblies. In this study, secondary data will be used to test for the suitability of the model improved later on. The data include the complete layout of the warehouse, number of machines or transport used to transport the materials, number of exits and entrances of the warehouse for each transport to collect supplies, number of shelves, and the station for each stop will be identified. The warehouse operates with two shifts; however, for this study, only the day shift will be considered. The floor plan for the study area is as in [23] which reflects the current practice of the company. Upon solving the problem, a few assumptions are made based on the study area:

• All the order pickers are well experienced in the routes and the picking area;
• All the order pickers need to collect all the items in their picking list;
• Data collected only involve a normal working shift and normal working hours;
• Data collected do not involve holidays, special seasons, sales, or after 5 p.m.;
• The aisles in the zone are equal in length;
• The length of the aisle is measured as 1 cm = 10 m;
• All the items needed are available on the rack;
• Once the items reach 25, pickers need to place them in the packaging station, and continue to pick the remaining items again.

3.1.2. Current Practice in the Automotive Manufacturing Company

In this automotive company, there are mainly two floors in the warehouse. On the ground floor is where all the big parts are placed on shelves. Should any order be made by a customer, an order picker will use a forklift to deliver the big parts onto a pallet. Next, this pallet will be transferred to the delivery truck. There are 10 forklifts available in the automotive company. This number of forklifts is hoped to reduce the waiting time to retrieve the order.

On the other hand, Zone 1, located on the first floor, is where all the small parts are placed. Here, the order retrieved will be picked manually by the order pickers. There are seven order pickers all together. The procedure starts with collecting order forms from customers. Next, the order picker will print the order form and collect the needed item accordingly. The order picker will standby at their particular shelf and supposedly they are already familiar with the area. This helps to save time when picking the order and thus, the response time can be reduced. Once the order is completed, the items are placed in the ‘Standby for Delivery’ area to be finalised, wrapped, and tagged before they are handed to the delivering units. The floor plan as described in [24] is the current layout of the warehouse.

This study focused on finding the shortest route for the order picker to pick an order in Zone 1. Zone 1 is considered a critical area since the number of pickers involved is only seven pickers and they have to collect numerous items and orders within a limited time. In addition, the pickers need to pick all the items manually and complete the order within normal working hours. The layout plan for Zone 1 is shown in Figure 2. There are four shelves with four front subaisles (A, B, C, D) and four end subaisles (E, F, G, H). Each aisle is an open-ended route. Each shelf is loaded with items ready to be picked. In this case, an order picker is free to go to any side of the shelves and may return to the same point. However, the objective is to find the shortest route to help minimise the order-picking distance for the order picker.

Basically, the order picker starts their working operation with collecting information on the number of orders from the customer. They will gather at the main platform in each zone to receive a delivery order (DO). A DO form is based on the order made online by the customer. Once the form is received, the parts are identified. Next, the order picker starts from the platform to their respected item area.

In order to apply the algorithm, a random order is picked. Under the purpose of this study, nodes are defined as an item placed in the warehouse, while an edge is an aisle of an item stored in a location that serves as the directional link between them. Essentially, there are four aisles and the numbers 1, 2, …, 19 are the items that need to be picked. These pickers need to visit all these places and can only stop once all the items are collected. In addition, pickers are limited to only picking 25 items for one trip. For every trip, when the items collected equal to 25, they have to go back to the packaging station, leave the items, and start to collect the next desired item. The procedure will continue until all the items are collected. Thus, based on this structure, a network layout is developed for the purpose of identifying the shortest path. This is to make it easier to calculate the shortest path for the pickers.

Figure 3 is the network layout representing Zone 1. The representation of routes within the warehouse network plays a vital role in achieving this goal. In this study, the warehouse network contains nodes and edges. Nodes denote junctions or intersections within the warehouse, typically corresponding to the locations of shelves where orders are stored. An edge represents the aisle connecting two adjacent nodes, facilitating the movement of order pickers between pick locations. Each edge is assigned with weights, where the weights must be nonnegative since the weight is in distance or time. The length is measured in centimetres (cm). In this study, the length of the aisle is 1310 cm. The distance between two neighbouring subaisles is 360 cm. There are 17 units for each aisle. Every unit should be filled with items. Each unit is denoted by x, and hence this can be expressed as 17x = 1310 cm. Thus, the distance for each unit is given by $\frac{1310}{17}=77.06$ cm. The black circles indicate there is no item, but pickers still need to walk through them to go to the next aisle.

Under this study, a sample order is selected based on the automotive manufacturing warehouse. As in

Table 1. Sample order made by customers.

Node	Items	Order Number
		Order 1	Order 2	Order 3	Order 4	Order 5	Order 6	Order 7	Order 8	Order 9
1	A								150
2	B									90
3	C		90
4	D		60				60
5	E		60				60
6	F				60
7	G			60				60
8	H				30		30		30
9	I		30				30		30
10	J	40				40		40		40
11	K			40				40	40
12	L		250
13	M				250
14	N				30				30
15	O		30				30
16	P		90	120
17	Q		60	120
18	R		60		60				60
19	S	60				60		60
	Total Daily Order	100	730	340	430	100	210	200	340	130

, nine orders were made by customers and nineteen items were involved. Every customer made specific orders with different types of items needed. For instance, ORDER 1 only requires ITEM J (40 items), ITEM S (60 items), and ITEM T (70 items). The process begins by giving the picking list to the order pickers to pick. The limitation is that each picker is only able to pick 25 items per trip. Say the picker needs to pick ITEM J, the picker needs to go twice in order to complete the desired item. In the first round, the picker picks 25 items and goes back to the packaging station. In the second round, the picker goes back to ITEM J and picks the remaining items (15 items) and he can choose to continue to the next nodes to add another 10 items or go back to the packaging station. Item A to S is then denoted as node 1 to 19 in Table 1 and Figure 3.

Table 1 indicates a sample order of 2580 items which must be gathered by 7 pickers from 19 nodes. Nodes are defined by n and for n = 19, each picker must pick close to 369 items. Due to the limited picking capacity for each picker (25 items per picking), each picker must make a maximum of 15 delivery trips, until all of the orders have been entirely processed. All these items need to be completed by the end of normal working hours. Later, figure of Shortest path for n = 9. will be a basis in achieving the shortest distance for order picking by using the Floyd–Warshall algorithm and DP method [3]. In the purpose of understanding the algorithm, the calculation of Floyd–Warshall is firstly completed manually on a small node, n = 9. Basically, for n = 9, the total number of items to be picked is 930 items. Each picker is expected to pick at most 133 items. The procedure of solving the minimum travel time is explained thoroughly in the next section.

3.2. Phase I: Process of the Floyd–Warshall Algorithm

There are five steps in the Floyd–Warshall algorithm. The shortest path can be defined as the undirected graph, where the edge (E) is the vertices that connect the line between the two vertices $\left|V\right| = n ,$ and at least one cycle exists on the path. The algorithm consists of two square matrices known as matrix $D_j$ and matrix $S_j$ for $j = 0 , 1 , \dots , n$ which hold the shortest path between two arbitrary nodes i and j. The Floyd–Warshall algorithm requires the $D_j$ and $S_j$ matrices to be calculated up till $n + 1$ starting by $D_0$ and $S_0$.

3.2.1. Step 1

All loop paths within the node need to be deleted and every path that starts and ends in the same node is not required in the selection. This is because it does not connect more than one node. The value for it is infinity (∞) which means it cannot be passed through.

3.2.2. Step 2

One or more parallel lines between the two nodes must be removed to leave one path of the least weight. Route selection coordination is possible when parallel paths are diverted while making the selected route as a single path.

3.2.3. Step 3

Define the initial distance matrix $D^{\left(0\right)} = {d_{ij}}^{\left(0\right)}$ as in Equation (1), sequence number matrix $S^{\left(0\right)} = {s_{ij}}^{\left(0\right)}$ in Equation (2), and $w_{i j }$ is the vertices’ adjacent line between two nodes, where:

Let the matrices be $n \times n$ matrices where j is the stage number and n are the total number of nodes.

$${d_{ij}}^{\left(0\right)}= \{\begin{array}{c}{w}_{ij}, v_i and v_j are adjacent \\ \infty , others\end{array}\left(i, j = 1, 2, 3, \dots , n\right)$$

(1)

$${s_{ij}}^{\left(0\right)}= \{\begin{array}{c}{w}_{ij}, v_i and v_j are adjacent \\ -, others\end{array}\left(i, j = 1, 2, 3, \dots , n\right)$$

(2)

$${d_{ij}}^{\left(k\right)}= \min \left({d_{ij}}^{\left(k-1\right)}, {d_{ik}}^{\left(k-1\right)} + {d_{kj}}^{\left(k-1\right)}\right) \mathrm{for} k \ge 1$$

(3)

Equation (3) represents the determination of the shortest distance matrix using the Floyd–Warshall algorithm. From the formula, $w_{i j}$ refers to the distance while $v_i$ and $v_j$ represent the two locations or nodes.

3.2.4. Step 4

Construct iterative matrix $D^{\left(k\right)} = \left({d_{i j}}^{\left(k\right)}\right)$ and $S^{\left(k\right)} =\left({s_{i j}}^{\left(k\right)}\right)$. Figure of Shortest path for n = 19 illustrates the flow of the algorithm. The minimum distance between $w\left(i, j\right)$ and $w\left(i, k\right) + w\left(k, j\right)$ is chosen as the shortest distance.

3.2.5. Step 5

Stop the calculation of $D_i$ and $S_j$ if $D^{\left(k+1\right)} = D^{\left(k\right)}$.

The calculation of $D_j$ and $S_j$ for remaining $j = 1, 2, \dots , n$ must be continued. Iteration of $D_j$ and $S_j$ are based on previous matrices, $D_{j-1}$ and $S_{j-1}$, respectively until $D_n$ and $S_n$ are yielded. The iteration ends if and only if $D^{\left(k+1\right)} = D^{\left(k\right)}$, otherwise return to Step 2. The procedure of the algorithm is also described in Figure 4.

The calculation of the shortest path for order pickers involved all aisles that contain items that have been ordered by the customers. Based on the items according to Figure 3, the distance and sequence matrix are built. As mentioned earlier, for the purpose of understanding the Floyd–Warshall algorithm, only a small sample of nine nodes are selected. These nodes are calculated manually in order to gain a deeper knowledge of the algorithm. The process of obtaining the shortest path and distance matrix is the same throughout the whole data of 19 nodes (n = 19).

Firstly, the nine nodes are selected. Based on Figure 3, node 1 until node 9 are placed in the third aisle (C) and fourth aisle (D).

Table 2. Initial distance matrix for sample data under study.

Node	1	2	3	4	5	6	7	8	9
1		385.3	INF	INF	INF	INF	INF	INF	360
2	385.3		231.18	INF	INF	INF	INF	INF	INF
3	INF	231.18		231.18	INF	INF	INF	INF	INF
4	INF	INF	231.18		462.3	INF	INF	INF	INF
5	INF	INF	INF	462.36		745.3	INF	INF	INF
6	INF	INF	INF	INF	745.3		539.42	INF	INF
7	INF	INF	INF	INF	INF	539.42		231.18	INF
8	INF	INF	INF	INF	INF	INF	231.18		154.12
9	360	INF	INF	INF	INF	INF	INF	154.12

shows the initial distance matrix involving the nine nodes. Based on the table, the number of iterations, k, is the process to find the value from node i to node j where k = 0, 1, …, n. For k = 0 and for every node i = j, then the space in the matrix is left unvalued since it is invalid to be passed. On the other hand, values filled with numbers in the matrix are the distances between two consecutive nodes. Whereas values with ‘INF’ refer to the nodes that cannot be passed directly by the order picker.

The purpose of this matrix in Table 2 is to show the updated value that is a minimum from one node to another which will be replaced by using the Floyd–Warshall algorithm. The distance matrix calculation is conducted once the matrix has been constructed and no INF is left. The calculation starts with the first iteration, where k = 1. The iteration stops until all the nodes have been processed; when n = 9.

The value of Row 1 and Column 1 will remain constant. The calculation starts with Row 2 and Column 1 until k = 9. An example of the calculation is shown in the next procedure. All these are repeated steps to find distance, D, by finding the minimum distance from node i to node j. For instance, to find the distance between node 2 and node 3, the minimum value between the direct distance between node 2 and node 3 is then compared with an alternative. As seen in the procedure, the distance between the two nodes may also be calculated by finding the sum of the minimum distance from node 2 to node 1 and node 1 and node 3. However, since node 1 cannot reach node 3 directly, the iteration is denoted as infinity (∞). For any value added with infinity, no solution can be found. Thus, the minimum distance from node 2 to node 3 is just 132.94 cm.

$$\begin{array}{ll}{D_{2, 3}}^{\left(1\right)}& =\min \left\{d\left(2,3\right), d\left(2,1\right)+d\left(1,3\right)\right\}\hfill \\ & =\min \left\{231.18, 385.3+\infty \right\}\hfill \\ & =231.18\hfill \end{array}$$

$$\begin{array}{ll}{D_{2, 4}}^{\left(1\right)}& =\min \left\{d\left(2,4\right), d\left(2,1\right)+d\left(1,4\right)\right\}\hfill \\ & =\min \left\{\infty , 385.3+\infty \right\}\hfill \\ & =\infty \hfill \end{array}$$

$$\begin{array}{ll}{D_{2, 5}}^{\left(1\right)}& =\min \left\{d\left(2,5\right), d\left(2,1\right)+d\left(1,5\right)\right\}\hfill \\ & =\min \left\{\infty , 385.3+\infty \right\}\hfill \\ & =\infty \hfill \end{array}$$

$$\begin{array}{ll}{D_{2, 6}}^{\left(1\right)}& =\min \left\{d\left(2,6\right), d\left(2,1\right)+d\left(1,6\right)\right\}\hfill \\ & =\min \left\{\infty , 385.3+\infty \right\}\hfill \\ & =\infty \hfill \end{array}$$

$$\begin{array}{ll}{D_{2, 7}}^{\left(1\right)}& =\min \left\{d\left(2,7\right), d\left(2,1\right)+d\left(1,7\right)\right\}\hfill \\ & =\min \left\{\infty , 385.3+\infty \right\}\hfill \\ & =\infty \hfill \end{array}$$

$$\begin{array}{ll}{D_{2, 8}}^{\left(1\right)}& =\min \left\{d\left(2,8\right), d\left(2,1\right)+d\left(1,8\right)\right\}\hfill \\ & =\min \left\{\infty , 385.3+\infty \right\}\hfill \\ & =\infty \hfill \end{array}$$

$$\begin{array}{ll}{D_{2, 9}}^{\left(1\right)}& =\min \left\{d\left(2,9\right), d\left(2,1\right)+d\left(1,9\right)\right\}\hfill \\ & =\min \left\{\infty , 385.3+360\right\}\hfill \\ & =745.3\hfill \end{array}$$

Once the values for row 2 and column 1 are successfully calculated, then the distance matrix will be updated based on the calculation obtained.

Next is the calculation for k = 1, row 3:

$$\begin{array}{ll}{D_{3, 2}}^{\left(1\right)}& =\min \left\{d\left(3,2\right), d\left(3,1\right)+d\left(1,2\right)\right\}\hfill \\ & =\min \left\{231.18, \infty +385.3\right\}\hfill \\ & =231.18\hfill \end{array}$$

$$\begin{array}{ll}{D_{3, 4}}^{\left(1\right)}& =\min \left\{d\left(3,4\right), d\left(3,1\right)+d\left(1,4\right)\right\}\hfill \\ & =\min \left\{231.18, \infty +\infty \right\}\hfill \\ & =231.18\hfill \end{array}$$

$$\begin{array}{ll}{D_{3, 5}}^{\left(1\right)}& =\min \left\{d\left(3,5\right), d\left(3,1\right)+d\left(1,5\right)\right\}\hfill \\ & =\min \left\{\infty , \infty +\infty \right\}\hfill \\ & =\infty \hfill \end{array}$$

$$\begin{array}{ll}{D_{3, 6}}^{\left(1\right)}& =\min \left\{d\left(3,6\right), d\left(3,1\right)+d\left(1,6\right)\right\}\hfill \\ & =\min \left\{\infty , \infty +\infty \right\}\hfill \\ & =\infty \hfill \end{array}$$

$$\begin{array}{ll}{D_{3, 7}}^{\left(1\right)}& =\min \left\{d\left(3,7\right), d\left(3,1\right)+d\left(1,7\right)\right\}\hfill \\ & =\min \left\{\infty , \infty +\infty \right\}\hfill \\ & =\infty \hfill \end{array}$$

$$\begin{array}{ll}{D_{3, 8}}^{\left(1\right)}& =\min \left\{d\left(3,8\right), d\left(3,1\right)+d\left(1,8\right)\right\}\hfill \\ & =\min \left\{\infty , \infty +\infty \right\}\hfill \\ & =\infty \hfill \end{array}$$

$$\begin{array}{ll}{D_{3, 9}}^{\left(1\right)}& =\min \left\{d\left(3,9\right), d\left(3,1\right)+d\left(1,9\right)\right\}\hfill \\ & =\min \left\{\infty , \infty +360\right\}\hfill \\ & =\infty \hfill \end{array}$$

If the value in the matrix does not change, the calculation continues by considering k = 2, 3, …, n until the matrix obtains the shortest value or no changes between the previous calculation. Therefore, the matrix that has no changes with its preceding matrix is the shortest distance between nodes i and j. The procedure continues until all nodes have been processed.

Table 3. Full distance matrix for sample data, n = 9.

Node	1	2	3	4	5	6	7	8	9
1	0	385.3	616.48	847.66	1310	1284.75	745.3	514.12	360
2	385.3	0	231.18	462.36	924.72	1670.02	1130.6	899.42	745.3
3	616.48	231.18	0	231.18	693.54	1438.84	1361.78	1130.6	976.48
4	847.66	462.36	231.18	0	462.3	1207.66	1592.96	1361.78	1207.66
5	1310	924.72	693.54	462.36	0	745.3	1284.72	1515.9	1670
6	1284.75	1670.02	1438.84	1207.66	745.3	0	539.42	770.6	924.72
7	745.3	1130.6	1361.78	1592.96	1284.72	539.42	0	231.18	385.3
8	514.12	899.42	1130.6	1361.78	1515.9	770.6	231.18	0	154.12
9	360	745.5	976.48	1207.66	1670	924.72	385.3	154.12	0

shows the complete calculation from k = 1, 2, …, 9. Hence, the matrix obtained in the end is the shortest distance between i and j.

3.3. Phase II: Process of DP Method

Following the same data collection as mentioned in the Current Scenario section, the DP method now is used to calculate the shortest distance for the order pickers. The shortest path is obtained using Excel Solver in Microsoft Excel (ME) 2013. The procedure for the DP method can be referred in [24]. Since the calculation is a bit tedious to do manually, the help of this Excel Solver will save time and give accurate results.

4. Results and Discussion

The results are presented in two phases as described in the methodology. The first phase is on the detailed application of the Floyd–Warshall algorithm in constructing the distance matrix of the shortest path. Next, phase II describes the results obtained using the DP method.

4.1. Phase I: Floyd–Warshall Algorithm

The calculation of the shortest path for order pickers involved all aisles that contained items that have been ordered by customers. The network consists of nodes and edges, which are the directional links between pairs of nodes. Hence, the distance and sequence matrices are built referred from the network layout of the warehouse. The algorithm was tested on two sets of data, small and large data sets. The size of the data is based on the number of nodes for stored data, n. The first set is n = 9 and the second set is n = 19.

4.1.1. Result Using Floyd–Warshall for n = 9

Based on the manual calculation using the Floyd–Warshall algorithm, the shortest path for the pickers is obtained. Assuming every picker needs to collect the same amount of items, the shortest path is achieved via nodes: 1→9→8→7→6→5→4→3→2 with the total distance of 29.54 km. The suggested path is as shown in Figure 5.

4.1.2. Result Using Floyd–Warshall for n = 19

The same procedure is applied as in n = 9 in the TORA software. The TORA software can be used for many solutions and algorithms. The software works by the user keying in the number of nodes involved in the study. After entering all the values referred from the distance matrix in the system, click on the ‘Solve Problem’ button, and hence, the software will generate the result. Say, let the starting node (i = 19) be equal to the ending node (j = 19). Therefore, the shortest path obtained is 132.94 m and the shortest path is achieved via path 19→18→19.

There are many possible shortest paths given by the software for the final solution of n = 19. Firstly, the order picker needs to identify the first node to be visited. Then, the picker will continue picking until all the items have been collected. The items collected will be based on the suggested routes from the TORA software and by considering the limited items to be carried out by each picker. Next, Figure 6 shows the result for the large data set, n = 19, which is 1-2-3-4-5-6-7-8-9-10-11-12-13-14-15-16-17-18-19 with 39.3821 kilometres (km).

The graph represents the 19 nodes connected by edges, illustrating the shortest path between nodes. The nodes are labelled from 1 to 19. The image highlights the shortest path, considering the distances between nodes. Arrows and lines indicate the direction of the shortest path in the graph.

Table 4. Shortest path for an order picker when n = 19.

Route	Distance (cm)	Number of Order Pickers
1-2-3-4-5	797.64	1
5-6-7-8	1091.17	1
8-9-10-11-12	1024.70	1
12-13-14-15-16-17-18	891.76	3
Total Distance	4036.45	7

shows the suggested path the pickers need to follow to pick the items with the lowest travel distance. The route highlighted in yellow shows the routes that need 3 order pickers since they involve 1190 items. Other than that, each route is sufficient with only one order picker.

4.2. Phase II: DP Method

4.2.1. Result Using DP Method for n = 9

The process starts with identifying the node that is assumed to be the block farthest from the depot (packaging point), that is $b_{i1},$ thus we start with two partial routes ${{\displaystyle L}}_1^a$ and ${{\displaystyle L}}_1^b$. ${{\displaystyle L}}_1^a$ starts at node $b_{i1}$ and ends at node $a_{i1}$. This process involves transition $t_1$ with a total distance of 1310 cm. This can also be written as $c\left(t_1\right) = 1310$. On the other hand, ${{\displaystyle L}}_1^b$ means we start and end at the same node $b_{i1}$ and consists of transition $t_3$. In this case, the total distance or $c\left(t_3\right) = 1695.32$.

Next, there are two possibilities for creating ${{\displaystyle L}}_2^a$, namely ${{\displaystyle L}}_1^a + t_a + t_4$ and ${{\displaystyle L}}_1^b + t_b + t_1$. The calculation for ${{\displaystyle L}}_1^a + t_a + t_4$ is $1310 + 360 +$$\left[154.12+231.18+462.36+231.18\right] \times 2$ and ${{\displaystyle L}}_1^b + t_b + t_1 = 1695.32 + 360 + 1310$ which results in 3827.68 cm and 3365.32 cm, respectively. The shortest value between the two will be chosen as the next possible shortest path. In this case, the current $c\left(t_3\right) = 3365.32$. The same procedure needs to be repeated to find ${{\displaystyle L}}_2^b.$ The possible path for ${{\displaystyle L}}_2^b$ can be achieved via ${{\displaystyle L}}_2^b = {{\displaystyle L}}_1^a + t_a + t_1$ and ${{\displaystyle L}}_2^b = {{\displaystyle L}}_1^b + t_b + t_3$. The calculation can be seen clearer using the diagram below:

Thus, from the findings, $c\left(t_1\right) = 2980$ is chosen as the shortest distance from node 1 to node 9. The same process is repeated to find ${{\displaystyle L}}_3^a$ and ${{\displaystyle L}}_3^b$. For later cases where n = 19, the same process is repeated until all nodes have been processed. The detail for the calculation of n = 19 is explained in the next paragraph. Thus, for n = 9, the shortest path is obtained via 1→2→3→4→5→6→7→8→9 with the total distance of 29.8 km. The suggested path is shown in Figure 7.

The graph represents the nine nodes connected by edges, illustrating the shortest path between nodes. The nodes are labelled from 1 to 9. The image highlights the shortest path, considering the distances between nodes. Arrows and lines indicate the direction of the shortest path in the graph.

4.2.2. Result Using DP Method for n = 19

In the DP method, the calculation follows the same procedure as in n = 9. However, the calculation for the last subaisle, r, is different. Here, determine ${{\displaystyle L}}_r^b$ by comparing the previous shortest value of ${{\displaystyle L}}_3^a$ and ${{\displaystyle L}}_3^b$. From the findings, ${{\displaystyle L}}_3^b = 4650$ and ${{\displaystyle L}}_3^b = 5651.76$. In this step, there is no need to find ${{\displaystyle L}}_r^a$ since all the items have been picked in a block and we just need to continue to end the picking point $b_{i3}$. The value of $r = 4$ represents the right-most aisle in the layout. Thus, ${{\displaystyle L}}_4^b$ can be calculated as follows:

Based on the value obtained, it is clearly shown that ${{\displaystyle L}}_4^b$ can be achieved via ${{\displaystyle L}}_3^a + t_b + t_3$, with the total distance of 7167.68 cm. However, this distance does not consider the amount of picking items the picker needs to pick. The calculation is extended by using the DP method in MS Excel Solver. Excel Solver is more advanced than MS Excel because it considers a big data set (maximum nodes = 100). It also provides the optimal shortest paths for pickers (start at node 1→2→3→4→5→6→7→8→9…→19 to the end). Thus, for n = 19, the result is represented in Figure 8.

In the graph representation, the 19 nodes are connected by edges, illustrating the shortest path between nodes. The nodes are labelled from 1 to 19. The image highlights the shortest path, taking into account the distances between nodes. Arrows and lines indicate the direction of the shortest path in the graph.

Based on Figure 8, this is the current shortest path for an unlimited number of pickings. The DP method produces results that are the shortest due to its flexibility in choosing paths. As part of the DP concept, the picker will follow the path that has already been organised in the warehouse. In this case, all impossible solutions can be eliminated. Furthermore, the problem is solved stage by stage. Figure 9 shows the suggested shortest path with a limited picking capacity of seven order pickers.

4.3. Comparison Analysis between Floyd–Warshall and DP

In phase I, the Floyd–Warshall algorithm is run by using TORA while the DP model in phase II is solved by using MS Excel Solver. The total distance took a longer time by using the DP method; however, in phase I, the number of items to be picked by each picker was not considered in the algorithm. In Floyd–Warshall, the algorithm is only able to find the shortest distance. On the other hand, some pickers might have to pick more items than others. This may lead to dissatisfaction and may also demotivate the workers. For the DP method, an equal number of items are divided among pickers. On the other hand, the distance travelled for each picker is longer and they need to maintain their fitness to go through all the items.

Based on the shortest path assignments, the total number of travelling distances with respect to order pickers are obtained. The analysis between n = 9 and n = 19 is carried out to find out which method provides significant results. The percentage difference is calculated between the two methods and discussed. For smaller sized nodes, the percentage difference is larger with multiple pickers, indicating that Floyd–Warshall is more advantageous in these scenarios. The differences are minimal when there are fewer pickers, suggesting a similar performance for certain configurations. For instance, when n = 9 and five pickers are involved, the DP method performs slightly better than the DP. Nevertheless, for seven pickers, the difference is still minimal with 3.67%, but the Floyd–Warshall algorithm performs slightly better. Despite that, the results are varied when n = 19. Floyd–Warshall often shows a significant advantage, but in scenarios with four to six pickers, the differences are minimal, indicating that the performance gap narrows as the problem size increases. In most cases, when the number of pickers is small (from one to three pickers), the Floyd–Warshall algorithm shows a notable advantage over the DP method.

As in the DP method, the results obtained when the nodes are small are significant between the number of pickers and the travelling distance. The higher the number of pickers, the shorter the travel distance and path for the pickers. The optimum number of pickers for n = 9 is six with a distance travelled of only 8942.32 m. However, when n = 19, in the DP method, the results are quite the opposite. It can be concluded that the more pickers there are, the more likely it is to have an increase in the distance travelled. This is explained by the route taken by each picker during the collection when equal items are assigned to each picker. Since the number of items to be picked are equal, some pickers need to travel far from one node to the other. Whereas, when there is only one picker, the picker can move from one node to the other consecutively, making it a more comfortable and smoother journey for the picker.

Table 5. Sensitivity analysis on number of pickers for comparison between the Floyd–Warshall algorithm and DP method.

Nodes	No. of Pickers	Total Distance (m)		Percentage (%) Different
		Floyd–Warshall	DP Method
n = 9	1	15,030.18	14,027.88	6.90
	2	19,440	14,073.53	32.03
	3	10,096.02	17,491.68	53.62
	4	7964.8	16,426.92	69.39
	5	12,099.48	11,711.08	3.26
	6	15,026.6	8942.32	50.77
	7	11,690.14	12,126.96	3.67
n = 19	1	34,579.28	42,526.9	20.61
	2	34,503.38	45,171.08	26.78
	3	47,203.42	71,403.4	40.81
	4	61,644.62	62,127.74	0.78
	5	56,124.72	60,026.46	6.72
	6	62,018.34	60,848.8	1.92
	7	77,225.94	103,615	29.18

shows the percentage difference obtained.

On the other hand, the results obtained from the Floyd–Warshall algorithm look more compromising. This may be because the algorithm focuses solely on determining the shortest path between every pair of nodes in a graph. This involves systematically evaluating paths between all pairs of nodes and updating the path as it discovers shorter routes. However, the algorithm does not consider additional constraints or specific problem requirements, such as the number of items to be picked. Its main objective is to provide the shortest distances between nodes, without considering constraints that may affect the route selection. The manufacturing company may choose to adapt any of the methods accordingly to suit their needs and to match the scenario of the warehouse.

In summary, the Floyd–Warshall algorithm generally outperforms the DP method in terms of total distance, with a greater advantage in smaller node sets and complexity in order-picking arrangements. However, as the problem size increases, the difference in performance between the two methods reduces, with the DP method and Floyd–Warshall showing more comparable results in certain scenarios.

5. Conclusions and Recommendation

In this research, the Floyd–Warshall algorithm was compared to the DP method. The Floyd–Warshall algorithm uses TORA to find the shortest distance for order pickers while DP uses MS Excel Solver. This is important to let the future researcher know the preferable method that can be used for the shortest path problem. This study focused on the travel distance of the order pickers in completing a task because without a specific path, the total distance will become greater. The data used in this study were collected from one of the automotive manufacturing companies that consists of three zones. In this study, only Zone 1 was selected. For the first objective, the shortest distance was satisfied by using the Floyd–Warshall algorithm. For the result, the algorithm may be modified to suit the needs of the pickers as well. On the other side, the DP method considered the number of items to be picked by each picker within the normal working hours. This is important in handling peoples’ feelings and satisfaction. For future research, the two methods may be considered in combination.

Determining the shortest path is an important thing for order picking, as it can reduce and save time to satisfy the demands made by customers. This can benefit both parties, including the customers and management of the manufacturing company. Customer satisfaction increases when the waiting time is reduced. If the customers are satisfied with the quality of service, the credibility of the manufacturing company will increase too. The company may choose which scenario may give them the best outcome that boosts their profitability. Additionally, the connection between each area, which can be accessed directly from another, can be determined. Using the TORA software, an optimal solution for the shortest path was determined. Secondly, this software requires the actual distance between nodes to operate, which can be computed using information from the manufacturing company. The TORA software was used to code the program for iteration and optimisation. The calculations were made much faster and contained fewer errors using the TORA software.

The results suggested that the algorithm can arrange a sequence for the delivery locations with the shortest path. In this study, the Floyd–Warshall algorithm proved to be reliable as this algorithm takes into account all possible paths. Even though there are differences in the theoretical estimation of the travel time using the algorithm as compared to real-life travel times, this study and another similar one in the past should help courier services in improving the quality of their service.

The recommendation for this study and for future research is to consider all zones in the warehouse so that more consistent results can be obtained. The application of other methods such as genetic algorithm, ant colony algorithm, and other meta heuristics is also recommended in the shortest path problem. Furthermore, seasonal factors such as an unexpected increase in the number of orders during holidays or any festival season can be considered in future research. In addition, the effect of working hours can also be explored on top of the numbers of order pickers to employ which can be of interest. Moreover, it is also recommended to consider warehouse automation as well as the use of vehicles in the warehouse instead of humans.