Optimisation of Buffer Allocations in Manufacturing Systems: A Study on Intra and Outbound Logistics Systems Using Finite Queueing Networks

Abstract

Optimal buffer allocations can significantly improve system throughput by managing variability and disruptions in manufacturing or service operations. Organisations can minimise waiting times and bottlenecks by strategically placing buffers along the flow path, leading to a smoother and more efficient production or service delivery process. Determining the optimal size of buffers poses a challenging dilemma, as it involves balancing the cost of buffer allocation, system throughput, and waiting times at each service station. This paper presents a framework that utilises finite queueing networks for performance analysis and optimisation of topologies, specifically focusing on buffer allocations. The proposed framework incorporates a finite closed queuing network to model the intra-logistics material transfer process and a finite open queueing network to model the outbound logistics process within a manufacturing setup. The generalised expansion method (GEM) is employed to calculate network performance measures of the system, considering the blocking phenomenon. Discrete event simulation (DES) models are constructed using simulation software, integrating optimisation configurations to determine optimal buffer allocations to maximise system throughput. The findings of this study have significant implications for decision-making processes and offer opportunities to enhance the efficiency of manufacturing systems. By leveraging the proposed framework, organisations can gain valuable insights into supply chain performance, identify potential bottlenecks, and optimise buffer allocations to achieve improved operational efficiency and overall system throughput.

1. Introduction

The evolutionary trajectory of supply chain systems has given rise to the realisation that it is no longer a mere chain but a complex network [1]. Consequently, a novel term has been introduced to refer to conventional supply chains as demand networks [2]. The contemporary supply chain eco-system has undergone significant transformations over time, and supply chain systems are now widely regarded as sophisticated network systems. This evolution can be attributed to several factors, including the growth of trade and the economy, technological disruptions, increased product specialisation, and changing customer landscapes [3]. These changes have led to unprecedented transparency for all stakeholders, promoting excellence in the supply chain domain [4].

The supply chain industry has been transforming significantly due to increasing transparency across supply chains and networks. Despite this progress, organisations still require a more structured and systematic flow of information within and across their supply chains to better understand and have situational awareness of their systems [5]. Supply chain mapping is considered an ideal tool for visualising the network concerning different layers across and through the system [6]. The supply chain mapping is developed for several reasons, including performance measurement [7,8], re-configurations to achieve continuous improvements [9], integration with technological advances [10,11], and the ability to foresee supply chain risks and challenges posed by natural disasters, geopolitical instabilities, and pandemics [12]. According to MacCarthy et al. [6] supply chain mapping can be developed to clarify information at different levels, and Figure 1 illustrates a classification of hierarchies of supply chain mappings according to the relevant domains and focus areas.

Global value chain maps serve as valuable tools for organisations to position themselves within the global structure and gain an understanding of the industries and markets in which they operate [13]. These maps provide a macro-level perspective and offer a high-level understanding of supply chain systems. A supply network map encompasses the supply chains that make up an industry, including the roles of regulatory bodies and other facilitating institutions, to gain an overall understanding of the industry’s structure [14]. Supply chain maps are subsets of supply network maps that provide greater visibility into each element, such as inbound and outbound logistics, making the organisation the focal point of observation [15].

Moreover, supply chain maps are developed to illustrate the flow of products and services from the organisation’s perspective. Value stream maps (VSM) provide information on the material, and information flows throughout the different stages of production, culminating in the final delivery of the product to the customer. VSMs have become a popular tool under lean manufacturing systems to identify and manage non-value-adding activities in the production processes [16,17,18]. Process maps are a conventional mapping tool used to understand the sequence of processes or tasks necessary to produce one unit of product or service [19]. Industry practitioners use time and motion studies and process maps interchangeably.

Value stream mapping is a widely used tool in the manufacturing and service industry to visualise and analyse the flow of materials and information and to comprehend the underlying business processes from an organisation-centric perspective. VSM is particularly effective in promoting cross-functional communication and collaboration within an organisation while providing opportunities for continuous improvement. Additionally, VSM is recognised for facilitating lean manufacturing practices, which help identify and eliminate waste and non-value-adding activities. The resulting efficiency improvements can help organisations track and optimise lead time, ultimately enhancing their overall performance. VSM provides the basis for developing supply chain maps, whereas it gets fed by the process mapping systems.

The interconnection between process topological design and VSMs lies in their ability to provide crucial insights into the structure and flow of activities within a system, particularly within the context of supply chains. Process topological design encompasses the layout, configuration, and connectivity of components or nodes in a system, aiming to attain desired performance objectives. Concurrently, VSMs offer a comprehensive understanding of the material and information flows within a specific process or value stream, aiding in identifying waste, inefficiencies, and improvement opportunities.

Buffer allocation, an integral aspect of topology design, plays a significant role in managing and optimising the flow of materials and information. In manufacturing systems and logistics networks, buffer allocation decisions are often made within the broader context of topology design. This entails determining the appropriate number, size, and placement of buffers at strategic locations throughout the system, aiming to ensure a balanced flow of goods or materials, minimise bottlenecks, and optimise system performance [20,21,22]. Poor buffer allocations in manufacturing and service settings can significantly affect operational performance. These consequences include reduced system throughput, as inadequate buffer allocations lead to congestion, bottlenecks, and delays in material transfers or service delivery. Longer lead times are another result, as insufficient buffer capacity increases waiting times and queuing delays at service stations [23]. Poor buffer allocations can also lead to higher inventory levels, as buffers are not appropriately sized or located, resulting in increased holding costs and potential obsolescence. Moreover, resource utilisation becomes imbalanced, with some areas being utilised and others underutilised, leading to inefficiencies and wasted resources [24]. Inefficient resource allocation is a consequence, with misalignment between resource capacity and demand patterns. Ultimately, better buffer allocations can result in customer satisfaction due to delays, disruptions, and unreliable performance. To mitigate these consequences, careful analysis and optimisation of buffer allocations, considering demand patterns, process variability, and resource capacities, are essential. Implementing a well-designed buffer allocation strategy can help minimise these negative consequences and improve overall operational performance [25].

By integrating VSMs into the process topological design, the current state of the process is visually represented, enabling the identification of critical paths, information flows, material flows, and interdependencies between different stages or workstations. This understanding of the process flow and interdependencies is then utilised to design an optimised process topological configuration. Buffer allocation decisions align with the identified bottlenecks, aiming to strategically allocate buffers or storage capacities to manage and optimise the flow of materials and information throughout the system.

Consequently, process topological design and value stream maps are interconnected, providing essential insights into the structure and flow of activities within a system. Buffer allocation, an integral component of topology design, facilitates efficient material and information flow management by strategically placing buffers at appropriate locations to balance the system, minimise bottlenecks, and optimise overall system performance [26].

This study aims to develop a process mapping approach that utilises finite queueing networks to analyse material logistics systems’ performance within manufacturing facilities. Specifically, the focus is on the intra-logistics process, which involves transporting raw materials to different feeding points based on demand, and the outbound logistics process, which encompasses preparing finished goods for shipment to diverse customers. This research uses finite queueing network models to estimate key performance metrics such as waiting time, resource utilisation, and throughput. Finite queueing networks have proven valuable tools for modelling operations in manufacturing setups and determining appropriate buffer sizes. These networks capture the flow of entities (such as materials or customers) through a series of interconnected queues, representing the various stages or service stations within the manufacturing process. By incorporating factors such as arrival rates, service times, and queue capacities, finite queueing networks enable researchers and practitioners to simulate and analyse real-life scenarios more accurately [27,28].

Compared to infinite systems, which assume an infinite capacity for queues, finite queueing networks provide a more realistic representation of operational constraints and resource limitations. Due to physical and cost constraints, manufacturing setups typically have finite capacities for machines, workstations, and buffers. Researchers can model scenarios closely resembling manufacturing environments by considering these finite capacities [29].

By leveraging finite queueing networks, researchers can simulate different buffer allocation strategies, evaluate their impact on performance metrics such as throughput, waiting times, and resource utilisation, and ultimately determine the optimal buffer sizes for optimal operations. These models allow for experimentation and optimisation, enabling decision-makers to make informed choices regarding buffer allocations that balance operational efficiency, cost-effectiveness, and customer satisfaction [30,31].

This research seeks to make a scholarly contribution by delving into optimising buffer allocations in the material transfer process involving a homogeneous fleet of trucks. The study concentrates on two crucial material handling processes: Inter-facility transfer and outbound logistics. Notably, the significance of this study lies in its pioneering approach to buffer allocation problems within an organisational context, taking into account both ends of the logistics process. This research unfolds in two distinct phases, with the initial phase focusing on inter-facility material transfer and the second phase dedicated to outbound logistics. While prior studies have explored buffer allocation problems, they have predominantly concentrated on a singular logistics process. Consequently, this study takes a critical stride towards an integrated analysis of logistics processes within an organisational setup, specifically concerning buffer allocation problems.

The remaining sections of this study are organised as follows. Section 2 extensively reviews the existing literature about process mapping applications, buffer allocation problems, and the methodologies employed in previous studies. This section provides a comprehensive examination of prior research to establish the foundation for the current study. Section 3 explains the methodology adopted for the present investigation, outlining the key steps and procedures involved. Section 4 offers a meticulous analysis of the numerical experiments conducted using the proposed approach. This section incorporates a case study to demonstrate the method’s effectiveness in practical scenarios. Section 5 concludes the study by summarising the key findings and contributions.

2. Literature Review

This section comprehensively examines the applications of finite queueing network models within process improvement, presenting significant insights gleaned from a sequence of empirical investigations conducted on real-world industrial operations. Moreover, studies related to buffer allocation problems and the employed methodologies, resulting outcomes, and significant ramifications are elucidated in detail. Moreover, particular emphasis is placed on underscoring the scholarly contributions of this study in advancing the optimisation of manufacturing processes, specifically pertaining to material logistics.

2.1. Process Maps and Applications

In a study conducted by Eleftheriadis and Myklebust [32], process maps were used to model the manufacturing process with process owner details to implement a zero-defect manufacturing environment. Similarly, Sharma [33] utilised process maps to analyse the welding process in a manufacturing facility and formulated a multi-criteria optimisation problem to make the process more sustainable. The optimisation problem was solved using meta-heuristics.

Araki et al. [34] and Mutua et al. [35] used process maps integrated with an optimisation problem to determine the fabrication parameters in a laser melting process. Bajaj et al. [36] and Wang et al. [37] utilised process maps in laser melting manufacturing processes to achieve reduced defect rates and identify the root causes of quality issues. Ponticelli et al. [38] presented a fuzzy genetic algorithm to optimise process control functions in a metal foam manufacturing facility. The process map was used to configure the operational parameters to improve the process and reduce defect rates.

He et al. [39] utilised process maps in a polymer dispensing system to determine the process planning parameters. Sabzi et al. [40] used process maps for alloy design and process optimisation. The authors presented a framework to design alloy production and determine the optimal solidification and thermal formation and then used a genetic algorithm to solve the optimisation problem. In a study conducted by Mejri et al. [41], process mapping was used to visualise a detergent manufacturing process. The authors used process re-engineering and innovation to optimise the production process while eliminating uncertainty.

In the additive manufacturing field, process maps are widely used to improve current processes and enhance efficiency, as demonstrated in studies conducted by Gui et al. [42], Akbari et al. [43], and Patel et al. [44]. Kumbhar et al. [19] used process maps integrated with digital twin models to detect bottlenecks in manufacturing systems for the diagnosis and improvement of the system.

2.2. Order Releasing and Scheduling Using Heuristic Based on Drum Buffer (DRB) Rope Technique

Bisogno et al. [45] explored the application of the theory of constraints (TOC), specifically the DBR technique, to improving process performance in healthcare services. Customising DBR for scheduled patient flows and introducing DBR for unscheduled patient flows, the study provided rules to control patient throughput, considering the trade-off between minimising flow times and maximising throughput volumes. Simulation experiments demonstrated the impact of TOC DBR rules on this trade-off.

Thürer and Stevenson [46] examined bottleneck shiftiness and order release methods, revealing that downstream bottleneck shifts had adverse effects on performance while upstream shifts had minimal impact. The distance between the actual and assumed bottlenecks had negligible performance implications, offering valuable insights for DBR and similar release methods. Thürer and Stevenson [47] further explored different sequencing and dispatching rules to enhance the performance of the DBR scheduling mechanism. The researchers found that prioritising the shortest bottleneck processing time during high load periods significantly improved performance.

Researchers are now exploring flow shop configurations, focusing on bottleneck management and connections to the theory of constraints (TOC). The drum buffer rope (DBR) methodology has been developed to address these challenges as a production planning and control tool within this context [48]. Yue et al. [49] presented a heuristic approach based on the DBR method to address the challenges of order releasing and multi-item scheduling in MTO production systems. The proposed approach aims to enhance productivity and efficiency in MTO companies by considering dynamic customer demands and capacity-constrained resources. Through experimentation on different problem types based on due date tightness and product demand, the performance of the proposed heuristic is compared with other well-known heuristic methods from the literature. The results indicate that the proposed DBR-based heuristic outperforms the competitors, particularly when an optimal buffer size is adopted, offering significant improvements in order release and scheduling effectiveness.

2.3. Multi-Level Rolling Horizon Planning and Scheduling Using DBR Approach Considering Material Constraints

The advent of Industry 4.0 concepts has led to a growing trend of digitization in industries. Consequently, traditional planning and scheduling approaches need to be revised to address the challenges posed by this new digital landscape.

Saif et al. [50] proposed a drum buffer rope-based heuristic algorithm (DBR-HA) for efficient planning and scheduling in mixed-model production industries aiming to implement Industry 4.0. The algorithm considered shifting bottleneck resources and maximised the use of capacity constraint resources, providing effective plans and schedules for each planning horizon. Lin et al. [51] integrated TOC and ERP in production planning to improve quality and cost efficiency. Utilising the DBR method, the study synchronised production, identified bottlenecks, allocated resources effectively, and enhanced production performance. A practical case demonstrated the successful integration and implementation of TOC and ERP in production planning.

Prasetyaningsih et al. [52] addressed an imbalance problem in a shoe company’s production lines by applying the TOC approach, utilising linear programming for production planning optimisation, and DBR for flow control. By considering the output gap between production lines, buffers were determined, and three solution alternatives were proposed, resulting in a substantial reduction of the imbalance issue at the shoe production lines. Telles et al. [53] examined the effects of implementing DBR on the efficiency of three engineering-to-order (ETO) production lines in an aerospace manufacturing context. The analysis, conducted longitudinally through a case study, utilised data envelopment analysis (DEA), the Wilcoxon test, and analysis of variance (ANOVA) to assess the impact of DBR. The findings revealed a significant efficiency increase of up to 19% following the implementation of DBR. Liu et al. [54] incorporated the theory of constraints and leveraged the DBR mechanism to develop multi-level planning and scheduling strategies in the context of mixed-model production. Additionally, the researchers propose a multi-level planning heuristic (MLPH) that employs DBR and priority rules to achieve efficient planning and scheduling. The approach considers material requirement planning and optimally utilises capacity-constrained resources (CCR) to the fullest extent.

2.4. Buffer Allocation Problems

Cruz et al. [55] presented an original methodology to address the buffer allocation and throughput trade-off problem in finite queueing networks. A specialised multi-objective genetic algorithm approximates the Pareto optimal set of solutions. Computational results demonstrate the efficiency and efficacy of the proposed methodology, highlighting the impact of the coefficient of service time variation on buffer allocation decisions and the dependence of buffer allocation on target throughput. Smith et al. [56] addressed optimising topological network design for multi-server queueing networks. Series, merge, and split topologies are examined using approximation and iterative search methods to estimate performance and determine optimal buffer allocation. The impact of the coefficient of variation on buffer allocation is highlighted, and computational results illustrate emerging buffer patterns within different topologies.

G. Wang et al. [57] compared three layout structures in automotive engine shops to maximise profit based on throughput and buffer investment costs. The analysis utilises queueing network models with finite buffers and unreliable machines, considering heterogeneous service times and exponentially distributed failure and repair times. The study employed approximation methods and a gradient search approach to determine the optimal buffer allocation. Zhang et al. [58] modelled mould manufacturing using OQNs with finite buffers. Based on queuing theory approximations, the proposed methods effectively evaluate system performance and demonstrate feasibility for large-scale practical problems. The findings provide valuable insights for system design, resource planning, buffer allocation, and capacity configuration. MacGregor Smith [59] presented an approach to determine the buffer allocation of a finite system using a queueing network decomposition methodology combined with a nonlinear sequential quadratic programming algorithm. The joint optimisation problem maximises throughput by treating the population as a constraint. Li et al. [60] developed analytical formulas to estimate the throughput of a reliable production line with exponential service times and finite buffers. The formulas apply to approximately balanced lines with similar processing times and provide upper bounds for throughput.

Cruz et al. [61] presented a study of optimising performance in finite single-server acyclic queueing networks using a multi-objective methodology to minimise buffer areas and service rates while maximising throughput. The proposed approach utilises a simulated annealing algorithm to redistribute buffer spaces and improve throughput without compromising overall capacity. Pedrielli et al. [62] introduced the discrete event optimisation (DEO) methodology for the simultaneous simulation and optimisation of manufacturing systems for buffer allocation in a production line. DEO utilises mathematical programming to approximate mixed integer linear programming models, providing a formal approach for simulation optimisation in queueing systems.

Yu et al. [63] addressed the optimisation of buffer allocation in AMHS in intelligent manufacturing workshops, highlighting the need for more integration between production and MHS in existing research. The study proposes a flow shop model with capacitated batch transports, utilising an OQN with blocking. An approximation method is presented to compute performance measures, and an iterative optimisation algorithm is developed for determining optimal buffer allocation. Hu et al. [64] presented a study identifying the optimal task assignment policy for maximising long-term average throughput. It provides insights into the preferred server coordination policies based on buffer allocation and numerical comparisons and generalisations for longer queueing lines.

A co-occurrence network (Figure 2) in the relevant literature provides insights into their interconnectedness and identifies important research trends and relationships. For example, the co-occurrence of keywords such as manufacturing, material handling, and machine shop practice suggests a strong association between these concepts in buffer allocation and queueing networks. Performance evaluation and throughput are often explored together, indicating the significance of evaluating system performance in terms of throughput. The co-occurrence of keywords such as system analysis, topology, and blocking implies the importance of studying the system’s structure and characteristics, including network topology and the impact of blocking on performance. Moreover, keywords such as integer programming and genetic algorithms suggest utilising optimisation techniques for solving buffer allocation problems in queueing networks. These methods are commonly employed to find optimal solutions by balancing conflicting objectives.

Uniform and Non-Uniform Buffers

Uniform or homogeneous buffers possess equivalent capacities or sizes throughout the queueing network. Conversely, non-uniform buffers, also called heterogeneous buffers, exhibit varied capacities or sizes across different buffers in the queueing network, enabling each buffer to accommodate different entities. From an optimisation standpoint, the suitability of uniform buffers in a queueing network relies on the specific goals and requirements of the analysed system [65]. While uniform buffers can simplify the analysis, they may only sometimes represent the optimal choice regarding performance or resource utilisation. Uniform buffers foster balanced resource utilisation across the network, which can be advantageous in scenarios prioritising uniformity to ensure fairness and prevent bottlenecks. However, in certain situations, non-uniform buffers may offer superior resource utilisation. For instance, when demand or arrival rates significantly differ across different network parts, allocating larger buffers to high-demand areas can avert congestion and optimise resource allocation. Uniform buffers may not be optimal for minimising queueing delays. In the presence of heterogeneous arrival rates or service times, non-uniform buffers facilitate the allocation of buffer capacity based on demand at each network stage. By allowing larger buffers at bottleneck points or areas with high variability in arrival rates, non-uniform buffers can diminish queueing delays and enhance the overall system’s performance [66]. The cost associated with buffer size plays a vital role in optimisation. Uniform buffers simplify resource allocation and buffer management but can result in over-provisioning in specific network segments.

Conversely, appropriately sizing non-uniform buffers based on demand patterns can optimise resource utilisation and potentially reduce costs by eliminating unnecessary buffer capacity. Non-uniform buffers offer greater flexibility and adaptability to dynamic changes within the system. In situations where the demand pattern or service times vary, non-uniform buffers enable better adjustment to changing requirements. This flexibility leads to improved performance and responsiveness compared to uniform buffers, which may need to be more adaptable to varying conditions [67].

2.5. Solving Finite Queueing Networks

Solving finite queueing networks presents more significant challenges than solving infinite networks due to several key factors. Firstly, finite networks have a complex and expansive state space that grows exponentially with the number of nodes, buffers, and entities. This complexity hinders direct analysis and solutions using traditional methods [68]. Secondly, interactions and dependencies among nodes in finite networks introduce intricate dynamics, making it more challenging to understand entity flow, bottlenecks, and the influence of routing strategies compared to the simplified assumptions in infinite networks. Additionally, evaluating performance measures such as queue lengths, waiting times, and resource utilisation becomes more complex due to interdependencies among nodes and the variability of system parameters [69]. Non-exponential service time distributions in practical scenarios further complicate the analysis, as traditional methods assume exponential distributions [70]. Adapted analytical techniques or alternative methodologies are required to address this challenge. Finally, as the size and complexity of finite queueing networks increase, analytical solutions may become intractable, necessitating numerical methods [71], approximations [72,73,74,75], or simulations [20,76] that demand significant computational resources and time. In light of these complexities, researchers employ a range of methodologies and techniques, such as approximation methods, decomposition techniques, numerical methods, and simulations, to overcome the challenges and gain meaningful insights from finite queueing networks.

Blocking Phenomenon

According to Smith [77], three types of blocking can occur in a two-stage queueing network: blocking after service (BAS), blocking before service (BBS), and repetitive service blocking (RSB). In the case of BAS, a customer or entity remains on the server until they are serviced, and they can only proceed to the downstream block if there is an empty space (buffer) available. BAS ensures that customers are not prematurely removed from the system, allowing them to complete their service before moving forward in the network. In the case of BBS, customers or entities are prevented from entering the server or queue before receiving service due to factors such as limited buffer space or predetermined constraints. This blocking occurs at the server’s entrance, resulting in a loss of service opportunity. On the other hand, RSB refers to a situation where a customer repeatedly receives service from a server, preventing other waiting customers from being served. RSB typically arises when a customer requires multiple service instances consecutively, effectively monopolising the server’s attention and delaying service for other waiting entities.

2.6. Scholarly Contributions and Future Extensions of the Study

This study aims to make a scholarly contribution by focusing on optimising buffer allocations in the material transfer process involving a homogeneous fleet of trucks. Specifically, the study addresses the optimal buffer allocation in two material handling processes: The inter-facility transfer and outbound logistics processes. These material handling processes were modelled using finite queueing networks. The networks are designed to accommodate constrained and unconstrained populations with generalised service times. By utilising the GEM approximation approach, the performance measures of the processes are evaluated, including cycle time, resource utilisation rates, and throughput levels. Given that the optimal buffer allocation problem is NP-hard, this study adopts a simulation optimisation approach to solve it. This approach considers the complex interactions and dependencies between various factors involved in the buffer allocation problem. In future extensions of this research, a comprehensive analysis of all logistics processes will be conducted to gain a holistic understanding of the supply chain across all stages. This integrated approach will provide valuable insights for optimising the overall performance and efficiency of the supply chain.

3. Methodology

This section elucidates the methodology implemented in the present study, focusing on modelling inter-facility and outbound logistics processes as finite queueing networks. The GEM approximation technique is elaborated upon, enabling the estimation of network performance measures. Additionally, a mathematical model is formulated to optimise buffer allocation within the network, followed by an explanation of the solution approaches employed to address the optimisation problem.

3.1. Modelling the Material Transfer Processes Using a Finite Queueing Network

The study adopts a rigorous technical approach to modelling material transfer operations involving a homogeneous fleet of trucks. The trucks carrying raw materials from storage to feeding points are represented as network jobs. Each sub-process within the material transfer operations is depicted as a node in the network. The inter-facility material transfer process encompasses a restricted number of assigned trucks, while the outbound logistics process allows for an unrestricted number of customer trucks. Therefore, the study uses finite CQN and finite OQN for the abovementioned cases. Different raw materials or product types are represented through multi-class jobs.

The stochastic nature of the service time at each node captured within the queues is modelled as M/G/1 queues, assuming a generally distributed service time. The queue discipline follows a first-come, first-serve policy. All stations within the network consist of a single-server configuration, and the movement between stations is modelled with an infinite number of servers.

The presence of limited buffer capacity at each node gives rise to the potential occurrence of blocking. In this study, the blocking after service (BAS) strategy is employed, wherein a job remains in the current node even after completion of service if it cannot be transferred to the succeeding node due to insufficient available space. Consequently, exponential service time distributions are unsuitable for this analysis, as they assume memoryless service durations following a constant exponential distribution. Within an exponential setting, jobs serviced at a given node promptly depart to the next node, assuming adequate space is available.

3.2. Generalized Expansion Method (GEM)

The GEM has emerged as a powerful approach for solving finite queueing networks and has demonstrated several advantages compared to other methodologies. GEM offers accurate performance analysis by considering finite capacities, blocking phenomena, and service time distributions [78]. It provides a robust framework for analysing complex systems with multiple interconnected queues, enabling researchers to derive analytical expressions for performance measures. Regarding implementation cost, GEM offers an advantageous position as it relies on existing mathematical techniques and equations without requiring extensive computational resources or specialised software [79]. Additionally, GEM is relatively easy to use, providing researchers with clear analytical insights into system behaviour and facilitating sensitivity analysis. While other methodologies, such as simulation-based approaches, may offer more flexibility in capturing system complexities and stochastic behaviour, they often require significant computational resources and expertise in simulation software [80]. Overall, GEM is a valuable methodology for accurate performance analysis of finite queueing networks, offering a cost-effective and user-friendly alternative to other methodologies.

The GEM, as proposed by Kerbache and Smith [73], comprises three fundamental steps: network reconfiguration, parameter estimation, and feedback elimination (Figure 3). This study employs GEM as an approximation technique to analyse the finite queueing network and the blocking phenomenon.

The first step, network reconfiguration, involves restructuring the original queueing network by introducing auxiliary nodes (holding nodes) and links to account for the blocking effects. This reconfiguration enables the representation of blocking occurrences within the network model. The second step, parameter estimation, aims to estimate the unknown parameters of the modified queueing network. This estimation step involves determining the routing probabilities, blocking probabilities, squared coefficient values, service times, and other relevant parameters crucial for accurately modelling the system behaviour.

The final step, feedback elimination, focuses on eliminating the feedback loops that may arise due to the reconfiguration. Feedback loops can lead to complexity in the analysis and may hinder the accurate estimation of performance measures. Removing these feedback loops can simplify and analyse the modified queueing network more efficiently. The following mathematical notations were employed to derive equations for the performance measures of the finite queueing network using GEM.

Notations and Definitions

$M$     Number of nodes

${\mathsf{\Lambda }}_j$     Input rate node i (i = 1,2,… M)

$T$     Number of product classes t (t = 1,2,… T)

${\mathrm{B}}_i$     Buffer capacity at node i excluding those in service

${\mathrm{K}}_i$     Buffer capacity at node i including those in service

${\alpha }_j$     The probability of jobs leaving the system from node j (j = 1,2,…, M)

$C_j$     Cost of a buffer space in node j

$L_{sj}$     Mean number of jobs (queue length) at queue s in node j

${\lambda }_h$     Arrival rate to holding node h

${\mu }_h$     Mean service rate at holding node h

$P_{Kj}$    Blocking probability that node j is at capacity $K_i$

${P^{\prime }}_{Kj}$    Feedback blocking probability

$\Theta$      Mean throughput of the system

${\Theta }_j$      Output rate at node j

$U_j$     Utilisation rate at node j

$S^2$     Squared Coefficient of variation of the service time

$\rho =\frac{\lambda }{\mu }$     Server busy time as a proportion of arrival and service rate

${\rho }_h=\frac{{\lambda }_h}{{\mu }_h}$     Server busy time as a proportion of arrival and service rate at node h

$\beta$      Maximum budget allocated for buffer space allocation

$D_s$     Mean waiting time at queue s

$N$      Network population (for CQN)

An M/G/1 queue is a queueing system where customer arrivals follow a Poisson process (M), service times have a general distribution (G), and there is a single server (1) serving customers one at a time.

Based on the two-moment approximation technique, a closed-form approach can be used to calculate the blocking probability using the following equation [73,81].

$$P_{Kj}=\frac{{\rho }^{\frac{(2+\sqrt{\rho s^2}-\sqrt{\rho }+2(K_j-1)}{(2+\sqrt{\rho S^2}-\sqrt{\rho })}}(\rho -1)}{{\rho }^{\frac{(2+\sqrt{\rho s^2}-\sqrt{\rho }+2(K_j-1)}{(2+\sqrt{\rho S^2}-\sqrt{\rho })}}-1}$$

(1)

According to Zhang et al. [82], the output rate of node j can be calculated using the following for an OQN system.

$${\Theta }_j={\Lambda }_j\left(1-P_{Kj}\right)+{\lambda }_h{(1-{P^{\prime }}_{Kj})}^{{\rho }_h}·{(1-P_{Kj})}^{{\rho }_j}$$

(2)

According to Kerbache and Smith [26], the total throughput of node j in a multi-class finite CQN with total population of N.

$${\Theta }_j=\frac{{\sum }_{i=1}^M{\sum }_{t=1}^T({\lambda }_{ij}·\left(1-P_{Kj}\right))}{N}$$

(3)

According to Kerbache and Smith and Kerbachea and Smith [72,73] the utilisation rate at node j in OQN and CQN can be calculated using the following equations, respectively.

$$U_j=\frac{{\lambda }_j}{({\Theta }_j·K_j)}$$

(4)

$$U_j=\frac{{\Theta }_j}{{\mu }_j}$$

(5)

3.3. Formulation of Optimisation Problem

Buffer allocation in queueing networks offers the potential to optimise multiple objectives by considering various performance measures such as throughput, waiting time, resource utilisation, and system stability. However, achieving a harmonious equilibrium among these objectives poses challenges due to their inherent trade-offs.

The complexity of the buffer allocation problem arises from the need to determine the optimal allocation of finite buffer resources across different nodes within the network. The task involves striking a delicate balance in buffer sizes to maximise overall system performance while simultaneously addressing multiple objectives. For instance, increasing buffer sizes can alleviate congestion and enhance throughput, but it may also lead to prolonged waiting times. Conversely, reducing buffer sizes may enhance responsiveness but elevate blocking probabilities and diminish overall throughput.

In order to address the buffer allocation problem in a finite queueing network, an optimisation framework is formulated with a carefully designed objective function and constraints [83].

$$Maximise\Theta (K,\mu )$$

(6)

$$Maximise\Theta (N,K)$$

(7)

Constraints

$$\sum _{i=1}^M{K}_i·C_i\le \beta$$

(8)

$$L_i\le K_i\forall i=\mathrm{1,2},..M$$

(9)

$${1\le K}_i\in \mathbb{Z}+\forall i=\mathrm{1,2},..M$$

(10)

Equation (6) presents the system throughput as a function of arrival rates and buffer sizes in the OQN context. Similarly, Equation (7) demonstrates the throughput as a function of network population and buffer sizes in CQN. In this study, we will maintain constant arrival rates for OQN and network population for CQN to determine the optimal buffer allocation for each node, aiming to maximise throughput. However, we will conduct various scenarios in both cases, considering different arrival rates, network populations, and compositions to assess their impacts on the system’s overall throughput. It is important to note that the objective function in both cases is subject to constraints. For instance, Equation (8) establishes an upper bound on the total number of buffers allocated due to budget limitations. Additionally, in a steady state, the average queue length should not exceed the buffer sizes of a given node (Equation (9)), and the number of buffers allocated must always be a positive integer with a minimum value of one (Equation (10)).

The optimisation problem of buffer allocation, known for its NP-Hard nature, poses a significant challenge in finding an optimal solution in a reasonable computation time [26,84]. Over the years, various solution approaches have been developed to tackle this intricate problem. Traditional methods, such as mathematical modelling and optimisation algorithms, have been widely employed to derive optimal buffer allocation strategies. However, due to the complexity and combinatorial nature of the problem, these approaches often require more computation and may only sometimes yield optimal solutions. As a result, simulation optimisation approaches have gained prominence in recent years. By combining the power of simulation and optimisation techniques, these approaches provide robust solutions with high accuracy, making them increasingly popular for addressing the buffer allocation optimisation problem.

The buffer allocation problem in this study was optimised using the built-in optimisation engine of AnyLogic software. The authors utilised a personal computer with 8GB of RAM and a core i3 7th Gen processor to execute the simulation experiments. The maximum iteration count was set to 5000 runs to ensure convergence towards optimal results. Figure 4 comprehensively illustrates the sequential steps in this study’s AnyLogic optimisation solution approach.

The OptQuest optimisation engine in AnyLogic simulation software solves larger combinatorial problems by efficiently navigating search trees and identifying fruitful directions towards optimal solutions. It utilises advanced techniques such as genetic algorithms, simulated annealing, and tabu search to explore the solution space and prune unproductive paths [85].

4. Case Study—Numerical Experiments

To demonstrate buffer allocation problems, this section presents a case study that utilizes numerical experiments to investigate and evaluate the proposed methodology in inter-facility material transfer operations and outbound logistics processes. These experiments aim to validate the effectiveness and efficiency of the methodology in optimising the buffer allocation problem within the intra-organisational material flows and the corresponding outbound logistics activities. We conducted the case study using a manufacturing facility with multiple workstations as the testbed. Applying the analytical method based on GEM, we formulated a finite queueing network model that considered various parameters, such as arrival rates, service times, and buffer capacities. By solving the GEM equations, we obtained analytical expressions for performance measures. Simultaneously, we developed a simulation model to replicate the dynamics of the facility and conducted multiple numerical experiments to collect data on system performance metrics. The results from the analytical method and simulation were compared to assess the effectiveness of the proposed methodology in optimising buffer allocation. This empirical illustration demonstrates the practical application of the proposed methodology. It showcases its efficacy in addressing the buffer allocation problem in the context of intra-organisational material flows and outbound logistics processes.

4.1. Inter-Facility Material Transfer Process

SM is a specialised steel manufacturing firm that produces steel rebars for the domestic market. To manufacture steel billets, which serve as the primary inputs for producing steel rebars, SM employs an electric arc furnace (EAF) system. The billet production process involves utilising various materials, including scrap materials, hot briquette iron (HBI), and specific additives such as carbon, chromium, and deoxidisers such as aluminium or silicon. Daily, SM acquires scrap materials from the domestic market, which are stored in different forms and purities in the SM storage yard. These forms include heavy metal scrap and shredded scrap. Additionally, SM incorporates HBI and direct reduced iron (DRI) as raw materials in the billet production process. The selection of billet grades is determined by evaluating the quality and percentage of the scrap materials. To simplify our discussions in subsequent sections, we will utilise the notations A, B, and C to represent heavy metal scrap, shredded scrap, and HBI/DRI materials.

In the context of SM, the production of steel billets takes place on a daily basis, using materials A, B, and C. These materials are transported each day by a fleet of homogeneous trucks to the facility situated within the billet plant. Figure 5 illustrates the inter-facility material transfer operations undertaken to meet the demand requirements of the billet plant. Figure 6 presents the layout of the storage area and the billet plant, visually representing their spatial arrangement.

As depicted in Figure 6, except for the loading and unloading service stations, all other stations are shared among the three types of material trucks. Truck servicing follows a first-come, first-served basis. The node details and average service time for each service station are presented in

Table 1. Node details and average service times.

Node #	Process	Aver. Service Time (min)	Node #	Process	Aver. Service Time (min)
1	Gate operation	1.5	7	Weigh bridge 2	1.5
2	Weigh bridge1	1.5	8	Unloading—A	2.5
3	Loading—A	6	9	Unloading—B	2
4	Loading—B	4	10	Unloading—C	3.5
5	Loading—C	7	11	Return trip	6
6	Quality check	2

, denoted in minutes. Additionally,

Table 2. Full truckload weights.

Raw Material (Product Class)	Full Truck Load (Tons)
A	7
B	3
C	9

provides the weight of a full truckload of each material.

Each service station within the system is managed by a single server, implying that each station has the capacity to serve only one truck at any given time.

In order to evaluate and optimise inter-facility material transfer operations, this study conducted performance measurements and buffer allocation optimisation exercises. The experiments were designed based on scenarios identified in

Table 3. Scenarios used for optimisation and performance analysis experiments.

Scenario #	Number of Trucks			Total Number of Trucks
	A	B	C
Scenario 1	3	3	3	9
Scenario 2	4	4	4	12
Scenario 3	6	6	6	18
Scenario 4	4	5	6	15
Scenario 5	6	4	5	15

. Additionally, various parameters were considered, including the cost of buffer space (C_i) and the maximum allotted budget for buffer space allocation (β), with values of EUR 200 and EUR 5000, respectively. All experiments were conducted to simulate a 24 h time-period operation.

4.1.1. Development of the DES Model for Inter-Facility Material Transfer Operations

The DES model for the inter-facility material transfer operations was developed based on the abovementioned details, as illustrated in Figure 7. Anylogic University edition (version 8.8.1) simulation software was employed. The model represents all service stations (nodes) as finite queue capacity service blocks. The decision variables, namely the buffer sizes for each node, were introduced as parameters in the optimisation configuration. The objective function was formulated to calculate the total throughput of the system. The trucks carrying three different types of materials were incorporated into the system through three sources. Each service block is coupled with a single resource pool to ensure the single-server phenomenon. Furthermore, the model accounted for the blocking phenomenon known as blocking after service (BAS), which occurs when a truck will not leave the station even after servicing if there is no space in the next node.

4.1.2. Optimal Buffer Allocation

Table 4. Optimal buffer allocations in each scenario.

Scenario #	Optimal Buffer Allocation	Throughput (Tons of Materials Transferred)	Total # of Buffers Used
Scenario 1	4,1,4,3,1,3,2,2,1,1	3101	22
Scenario 2	1,1,3,2,2,4,3,2,2,1	3656	21
Scenario 3	1,3,2,4,1,1,2,7,2,1	3677	24
Scenario 4	1,1,1,1,1,10,1,1,1,1	3628	18
Scenario 5	2,1,6,2,1,5,1,1,3,3	3794	25

presents the optimal buffer allocations obtained for each scenario investigated in the study. The allocations differ across scenarios, indicating the significance of tailoring buffer allocation strategies to specific operational conditions.

A uniform truck allocation was employed for all material types in the initial three scenarios. Furthermore, an analysis of the results reveals that an increase in the number of trucks allocated to each material type and the total number of trucks corresponded to an increase in throughput, quantified as the total tons of materials transferred. However, it is worth noting that this increase in throughput was accompanied by a potential decrease in the average throughput per truck. In scenarios 4 and 5, different truck allocations were implemented for each material type while maintaining a constant total of 15 trucks. Consequently, distinct optimal buffer allocations were obtained for these scenarios, resulting in disparate throughput values. These findings underscore the influence of the number and allocation of trucks on buffer optimisation and subsequent material transfer efficiency in inter-facility operations.

Furthermore, in scenario 4, we observed a near-uniform allocation of buffers across the service stations, except for a single station. This finding underscores the significance of conducting a comprehensive analysis to determine each case’s optimal buffer allocation strategy.

Additionally, noteworthy observations can be drawn from scenarios 1 and 2. Despite an increase in the total number of trucks employed (from 9 to 12), the total number of buffers utilised decreased from 22 to 21. This suggests that an increase in truck quantity does not necessarily correlate with a proportional increase in buffer requirements. Similarly, in scenario 4, where the total number of buffers used was 18, an equivalent number of trucks in scenario 5 necessitated 25 buffers to achieve the optimal throughput. This discrepancy highlights the varying impact of truck allocations and compositions on buffer requirements, considering that certain service stations are exclusively utilised by trucks carrying a single product type. In contrast, others are shared among multiple product types.

The decision to set the maximum iterations to 5000 was carefully considered to strike a balance between ensuring convergence towards optimal results and computational efficiency. The figures presented in Appendix A (Figure A1, Figure A2, Figure A3 and Figure A4) demonstrate that convergence typically occurs well before reaching the maximum of 5000 iterations, indicating the efficiency of the proposed method in achieving optimal solutions in most experimental examples. However, to account for potential problem size and complexity variations, we adopted a conservative approach by setting a maximum of 5000 iterations. For larger-scale problems, the convergence might occur more slowly than in the experimented examples, making the maximum iteration limit crucial in capturing longer convergence trends. This choice ensures the robustness of our approach, enabling effective handling of a broader range of problem sizes and complexities.

These findings accentuate the importance of considering the interplay between truck allocations, the composition of product types, and their corresponding buffer allocations when optimising material transfer operations. A holistic approach is essential to comprehensively assess and determine each unique scenario’s most effective and efficient buffer allocation strategy.

4.1.3. Utilisation Rates Comparison

Efficient utilisation of servers is paramount to optimising system performance and ensuring smooth operations in closed queueing networks. The utilisation rate, representing the ratio of time a server is busy to the total time, is a crucial performance indicator for evaluating server allocation strategies.

Table 5. Server utilisation rates in identified scenarios using analytical and simulation method.

Scenario #	Scenario 1			Scenario 5
Node #	GEM Method	Simulation Model	Diff.	GEM Method	Simulation Model	Diff.
Node1	63.30%	66%	−2.70%	75.9%	79%	−3.1%
Node2	63.30%	66%	−2.70%	77.8%	78%	−0.2%
Node3	68.90%	66%	2.90%	72.9%	78%	−5.1%
Node4	69.80%	73%	−3.20%	92.4%	99%	−6.6%
Node5	54.10%	55%	−0.90%	54.1%	58%	−3.9%
Node6	85.80%	82%	3.80%	87.5%	98%	−10.5%
Node7	87.80%	84%	3.80%	87.1%	99%	−11.9%
Node8	28.70%	32%	−3.30%	37.5%	44%	−6.5%
Node9	35.80%	30%	5.80%	25.0%	30%	−5.0%
Node10	38.10%	43%	−4.90%	50.0%	51%	−1.0%

presents the comparison of server utilisation rates obtained through the GEM-based analytical method and simulation model. The results demonstrate the accuracy and reliability of the GEM-based analytical method in estimating server utilisation rates, with average differences between the analytical method and simulation models ranging from 1% to 6% in scenario one and from 1% to 7% in scenario 5, except for nodes 6 and 7, where the utilisation rates are close to 1. However, it should be noted that when utilisation rates approach 1, the GEM-based method may exhibit slightly higher deviations from the simulation results. These findings highlight the effectiveness of the GEM-based method in calculating server utilisation rates in finite closed queueing networks while acknowledging the need for caution when utilisation rates are close to 1.

$$Difference=(Analyticalmeth.-Simulationmeth.)$$

(11)

Analysing the scenarios provides valuable insights into the system dynamics and potential bottlenecks. Scenario 1 demonstrates a relatively stable system, exhibiting lower utilisation rates than scenario 5. In scenario 5, several nodes approach or reach utilisation rates near 1.0, indicative of bottleneck situations where system performance stagnates. Notably, scenario 5 (18 trucks) exhibits different truck quantities and compositions compared to scenario 1 (9 trucks), which has the fewest number of trucks.

Additionally, consistent observations reveal that the last three nodes consistently exhibit low utilisation rates. This suggests the need for process redesign to balance the workload across both streams rather than exert excessive pressure on certain segments. Potential solutions include:

• Subdividing specific processes into two sub-processes,
• merging idle stations to consolidate workloads, or
• employing additional servers to alleviate the burden on busy servers.

These interventions can contribute to better resource utilisation, reduced congestion, and improved overall system efficiency. Further analysis and targeted optimisation strategies should be considered to address these observed bottlenecks and enhance the performance of the queueing network.

4.1.4. Sensitivity Analysis

Sensitivity analysis is a valuable tool used in decision-making processes to evaluate the influence of various factors on the outcomes of a system or model. Sensitivity analysis provides insights into the robustness and stability of decision-making frameworks by systematically varying these factors and observing the resulting changes in the outputs. In this section, we employ sensitivity analysis techniques to investigate the impact of critical factors on our decision-making model. Through this analysis, we aim to identify the most influential factors, understand their effects on decision outcomes, and enhance the reliability and effectiveness of our decision-making processes.

In this sensitivity analysis section, we examine the variations in utilisation rates, throughput, and buffer allocations resulting from introducing an additional server to node 6 in scenarios 1 and 3. We aim to assess the impact on the system’s performance by introducing this additional server. The results of this analysis are presented in

Table 6. Impact of adding an extra server in node 6.

Scenario #	Node Utilisation Rate		Throughput		Total # of Buffers
	1 Server	2 Servers	1 Server	2 Servers	1 Server	2 Servers
Scenario 1	84%	42%	3101	3126	22	10
Scenario 3	99%	67%	3677	4199	24	18

, which provides insights into the changes observed in utilisation rates, throughput, and buffer allocations under these specific conditions. Through this examination, we gain a deeper understanding of how the system responds to introducing an extra server and its implications for overall system performance in scenarios 1 and 3.

Table 6 unveils intriguing insights regarding the factors influenced by the introduction of an additional server in node 6. In scenario 1, the results demonstrate a slight increase in throughput, reaching 3126 tons, representing a modest improvement of approximately 0.8% compared to the original case. Conversely, scenario 3 exhibits a more substantial boost in throughput, surging to 4199 tons with the inclusion of a dual server in node 6, signifying a significant increase of approximately 14% from the baseline. Furthermore, both scenarios showcase a reduction in the total number of servers utilised and a decrease in the utilisation rate of node 6. These findings emphasise the interconnected nature of design factors within the system, whereby modifications in certain factors influence other performance metrics to varied extents. Consequently, this analysis provides valuable insights for informed decision making when designing system topologies, enabling practitioners to consider the interplay of factors and their implications on system performance.

4.1.5. Comparison of Analytical and Simulation Methodologies

Analytical and simulation methodologies offer unique advantages when comparing methodologies for studying buffer allocations and system performance. Analytical methodologies provide fast and cost-effective analysis, with mathematical equations and formulas enabling quick calculations of system performance measures. They offer closed-form solutions and accessible interpretation, making them suitable for straightforward systems and providing valuable insights. In contrast, simulation methodologies excel at modelling complex and dynamic systems, capturing real-world complexities and uncertainties more accurately. Although more time consuming and resource intensive, simulation models allow for detailed process flows, interactions, and random variations, providing flexibility and realism. The choice between methodologies depends on research goals, system complexity, available resources, and the trade-off between accuracy, time, and cost. Researchers often employ both methods to gain comprehensive insights into buffer allocations and system performance. This integrated approach ensures a balance between efficiency, accuracy, and the ability to capture the intricacies of the studied systems.

4.2. Out-Bound Logistics Process

SM organisation serves two customer types: Standard customers and spot customers. Standard customers, often large manufacturing companies, have long-term contracts, such as contracts of affreightment with SM, ensuring a stable flow of predictable orders. SM can efficiently plan and allocate resources for these known orders, benefiting from established trust and coordination. In contrast, spot customers have short-term or one-time orders, often with short notice and varying requirements. SM must promptly respond to these demands, adapting quickly to their unpredictability. Despite their shorter engagement, spot customers contribute to SM’s growth and resource optimisation.

The outbound logistics process for customer trucks carrying steel rebars within the manufacturing company’s premises begins with the gate entry process. This involves recording the entry of trucks and initiating necessary security and safety checks. For spot order trucks, an additional process of creditworthiness evaluation is conducted between the gate entry and the order processing process. This evaluation verifies the customer’s ability to pay back by assessing their financial stability and credit history. Figure 8 shows the flowchart of the whole outbound logistics process for both standard and spot customers.

After the gate entry and creditworthiness evaluation, the subsequent processes are initiated. Process 2, order processing, involves receiving and verifying the customer’s order for steel rebars, confirming pricing and payment details, and generating an order confirmation. The order processing process for standard and spot customers differs in terms of time requirements. The order processing process takes an average amount of time for standard customers whose orders are previously known to the company. Since the company has established relationships with standard customers and has their order details on record, the process can be streamlined and expedited. However, when it comes to spot customers who place orders on short notice or for one-time purchases, the order processing process tends to take longer. The company must allocate additional time for spot customers, as their orders require thorough verification and may require additional documentation and checks. The extended processing time for spot customers is necessary to ensure these special orders’ accuracy, compliance, and proper handling.

In process 3, steel rebar preparation, the requested steel rebars are retrieved from the inventory or production area within the manufacturing company and checked for compliance with the required specifications. This process includes packaging and labelling and focuses on securely packaging the steel rebars for transport. This may involve bundling the rebars, placing them on pallets, and labelling the packages with relevant information. In process 4, documentation and compliance, the necessary shipping documents, invoices, and paperwork are prepared, ensuring compliance with internal procedures and legal requirements. Process 5, loading and staging, entails loading the packaged steel rebars onto the customer trucks within the manufacturing company’s premises. This process includes utilising appropriate equipment, such as forklifts or cranes, to ensure the safe and efficient loading of the rebars onto the trucks. Process 6 involves inspecting the loaded trucks to verify that the steel rebars are adequately secured and that the trucks are in suitable transportation conditions. Any identified concerns or issues are addressed during this truck inspection process.

The truck dispatch process is the final step in outbound logistics, closely coupled with the delivery confirmation. The truck dispatch process comes into play once the trucks carrying steel rebars are loaded and ready to depart from the manufacturing company’s premises.

Table 7. Average service time for each outbound logistics processes.

Process	Aver. Service Time (min)	Process	Aver. Service Time (min)
Gate operation	2	Documentation and compliance	2
Creditworthiness evaluation	3	Loading and staging	12
Order processing—standard	7	Truck inspection	2
Order processing—spot	10	Truck dispatch and delivery confirmation	2

shows the average service time of sub-processes of outbound logistics operations.

4.2.1. DES Model for SM’s Outbound Logistics Operation

A DES model, as illustrated in Figure 9, has been constructed to emulate the outbound logistics process of SM. This simulation model incorporates a finite OQN framework within the Anylogic software platform. Subsequently, optimisation configurations are inputted into the model to ascertain the optimal buffer sizes for each service station.

The simulation-optimisation experiments conducted in this study involve using various customer arrival rates to represent distinct scenarios. The identified customer arrival rates and the ratio between standard and spot customers are presented in

Table 8. Identified scenarios for optimisation experiments.

Scenario #	Arrival Rate (per Hour)	Stand.: Spot
Scenario 1	4	80 to 20
Scenario 2	4	90 to 10
Scenario 3	5	90 to 10
Scenario 4	8	90 to 10

. The objective of the optimisation study is to maximise the throughput, which corresponds to the number of successfully fulfilled customer orders. A constraint is imposed with the budget limit for buffer spaces, with each space costing EUR 100. The total budget allocated for buffer spaces is EUR 8000. The optimisation experiment is executed over 12 h, with a maximum of 1000 iteration runs. These experimental settings align with the methodology employed in the previous section.

4.2.2. Optimal Buffer Allocation for Outbound Logistics Process

Table 9. Optimal buffer allocations for identified scenarios with order processing efficiency.

Scenario #	Optimal Buffer Allocation	Throughput (# of Orders Processed)	Aver. Order Processing Time (min)	Total Number of Buffer Spaces Used
Scenario 1	1,1,5,1,7,1,4,3	49	14.69	23
Scenario 2	2,2,2,3,3,1,1,2	49	14.69	16
Scenario 3	1,1,1,1,1,1,1,1	51	14.12	8
Scenario 4	5,12,3,3,20,7,2,16	57	12.63	68

presents the optimal buffer allocations obtained across the previously identified scenarios. Several noteworthy observations can be drawn from the results presented below. In scenarios 1 and 2, where the arrival rate of trucks remains the same but the customer composition differs, different buffer allocations are observed while maintaining the same throughput. On the other hand, scenario 3 exhibits an optimal uniform buffer allocation. In the final scenario, a higher throughput and lower order processing time are achieved; however, this is accompanied by significantly greater utilisation of buffer spaces. As a result, it is crucial for logistics practitioners in the SM domain to consider various factors such as arrival rate, customer order composition, order processing time, and buffer costs in order to manage their operations effectively.

4.2.3. Cycle Time Comparison with Analytical and Simulation Method

Table 10. Comparison of cycle times using analytical and simulation methods.

Scenario #	Cycle Time (min)
	GEM Method	Simulation Model	Diff.
Scenario 1	13.471	14.69	−8.30%
Scenario 2	13.851	14.69	−5.71%
Scenario 3	12.981	14.12	−8.07%
Scenario 4	12.135	12.63	−3.92%

presents the estimated cycle time (sojourn time) for various scenarios using the analytical method and simulation model. The difference between the results is calculated using Equation (12), ranging between approximately ±4% and ±8%. The table demonstrates that the analytical method, employing the GEM approximation, offers reliable and robust solutions for analysing finite OQNs. The accuracy and effectiveness of the analytical method make it a valuable tool for studying and evaluating system performance in real-world applications.

$$Differnce=\frac{(Analyticalmeth.-Simulationmeth.)}{Analyticalmeth.}$$

(12)

4.3. Managerial Insights

The analysis of buffer allocation in a steel manufacturing company’s inter-facility material transfer operation and outbound logistics process reveals several key managerial insights. Firstly, the number and allocation of trucks significantly impact buffer optimisation and material transfer efficiency. By increasing the number of trucks allocated to each material type and the total number of trucks, the overall throughput in total tons of materials transferred can be increased. However, a potential decrease in the average throughput per truck may accompany this increase. Therefore, managers must carefully balance the number of trucks allocated to different material types to achieve optimal performance.

Different optimal buffer allocations were obtained when different truck allocations were implemented for each material type while maintaining a constant total number of trucks, leading to varying throughput values. This finding highlights the importance of considering the quantity and composition of trucks when determining buffer requirements. Furthermore, the analysis reveals that an increase in the total number of trucks does not necessarily result in a proportional increase in buffer requirements. Different scenarios with equivalent numbers of trucks may require significantly different buffer allocations, depending on the truck composition and the specific service stations involved. Thus, managers should consider the specific characteristics of each scenario and carefully analyse the interplay between truck allocations, customer order compositions, and buffer requirements to optimise material transfer operations.

The analysis also emphasises the significance of resource utilisation rates in identifying potential bottlenecks and improving system performance. By evaluating the utilisation rates of servers at various nodes or service stations, managers can pinpoint areas with high levels of occupancy that may hinder system efficiency. In particular, the analysis identifies nodes where utilisation rates approach or reach 1.0, indicating potential bottlenecks. These bottlenecks signal the need for reallocation or additional resources to ensure optimal system performance. Strategies such as process subdivision, workload consolidation, or adding servers can help balance the workload and enhance resource utilisation, reducing congestion and improving overall system efficiency.

Sensitivity analysis further enhances decision making by assessing the impact of critical factors on system outcomes. The analysis demonstrates the influence on throughput, buffer allocations, and utilisation rates by introducing an additional server in specific scenarios. The results indicate that including an extra server can improve throughput and decrease server utilisation, thereby enhancing overall system performance. These findings highlight the system’s interconnected nature of design factors and provide valuable insights for informed decision making when designing system topologies.

Overall, the managerial insights derived from the analysis underscore the importance of considering factors such as truck allocations, customer order compositions, buffer requirements, and resource utilisation rates in optimising inter-facility material transfer operations and outbound logistics processes. By carefully evaluating and balancing these factors, managers can effectively manage their operations, improve system performance, and enhance overall efficiency in the steel manufacturing company’s supply chain.

5. Conclusions

This paper has presented a framework for optimising buffer allocation in inter-facility material transfer and outbound logistics processes. The study has demonstrated the significance of strategic buffer placement and sizing in improving supply chain efficiency and performance. By utilising finite queueing networks and the generalised expansion method (GEM), the framework allows for the modelling, analysis, and optimisation of buffer allocations in manufacturing systems.

Resource utilisation rates are crucial in identifying bottlenecks and improving system efficiency. Evaluating server utilisation at different nodes allows managers to allocate resources effectively by employing process subdivision or workload consolidation strategies.

This study’s significance lies in its pioneering approach to buffer allocation problems, considering both ends of logistics processes. While previous studies focused on single logistics processes, this research integrates inter-facility transfer and outbound logistics, paving the way for a holistic examination of buffer allocation in complex setups. Future research can explore integrated analyses encompassing inbound, intra, and outbound logistics to understand buffer allocation across the entire logistics network comprehensively.

The study has specific limitations that require acknowledgement. Firstly, it concentrates on a single-server environment, overlooking the intricacies of multi-server setups. Secondly, assuming homogeneous jobs or customers may only partially represent real-world situations where variations exist. Thirdly, the study adopts a first-come, first-serve queue discipline, neglecting potential priority-based service considerations. Lastly, the assumption of uniform buffer costs at all stations disregards the possibility of varying costs at different locations. These limitations indicate potential areas for future research to enhance the applicability of buffer allocation models.

In conclusion, this study provides valuable insights into the buffer allocation process for manufacturing companies to optimise inter-facility material transfer and outbound logistics. The proposed framework offers opportunities for improving operational efficiency and overall performance in manufacturing supply chains.