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Background: Logistics operations are integral to manufacturing systems, particularly in the transportation processes that occur not only between facilities and stakeholders but also between warehouses and workstations within a facility. The design of functional areas and allocating goods to appropriate zones within the warehouse management system (WMS) are critical activities that substantially influence the efficiency of manufacturing logistics operations. Methods: This study develops a mixed-integer programming (MIP) model to optimize material flow and product routing in manufacturing. The model identifies efficient pathways, assigns products to routes, and determines the required material-handling equipment. It is implemented in Python (3.11.5) using the Pyomo (6.7.3) package and the CBC solver (2.10.11), with sensitivity analysis performed on constraints and decision variables to evaluate robustness. Results: The findings indicate that Material Flow 3 and Material-Handling Equipment 1 represent the optimal configurations for managing the majority of goods within the manufacturing system. Conclusions: The proposed mathematical model supports the decision-making process by enabling adjustments to the proportions of functional areas within the manufacturing logistics system, ensuring operational efficiency and flexibility in response to changing demands. Furthermore, the study offers managerial insights and suggests directions for future research.

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1. Introduction

The manufacturing sector has consistently played a pivotal role in the global economy, contributing 16% to the global gross domestic product (GDP) in 2018 [1]. Within this sector, the domain of manufacturing logistics is integral to ensuring the efficiency, cost-effectiveness, and responsiveness of production processes to market demands. Manufacturing logistics traditionally encompasses the planning, scheduling, and control functions that are critical to the operation of a manufacturing system, as described by Wu et al. (1999) [2]. These functions facilitate the seamless execution of key activities such as the acquisition of raw materials, their transformation into finished goods, the movement of products through various stages of production, and the management of inventory storage. The sequence of manufacturing activities is meticulously detailed in a production plan, which is designed to optimize the efficiency and output of the manufacturing process (Kiran, 2019) [3]. This plan is comprehensive, encompassing capacity planning, aggregate planning, and the intricate details of routing, scheduling, loading, dispatching, expediting, and the progress reporting of work-in-progress (WIP) inventory. The complexity of these processes is further magnified by the existence of multiple functional areas within a manufacturing system, each staffed by teams possessing specialized skills in areas such as planning, production, finance, sales, marketing, and research and development.

The scope of manufacturing logistics extends well beyond the confines of the factory floor, beginning with the determination of customer demands for end products and culminating in the fulfilment of these demands. This expansive scope introduces significant challenges, particularly within the realm of inbound logistics. Inbound logistics encompasses the processes involved in acquiring and transporting raw materials and components from suppliers to the manufacturing facility. This segment of the supply chain is critical, as it directly influences material availability, the efficiency of production schedules, and ultimately, the capability to meet customer demands in a timely manner. Optimization of the inbound logistics supply chain within the manufacturing sector thus emerges as a crucial area of focus. Effective inbound logistics ensures the timely arrival of the correct materials at the correct location, in the appropriate quantities. Achieving this precision necessitates the resolution of several challenges, including variability in supplier lead times, fluctuations in demand, transportation delays, and the coordination complexities associated with managing multiple suppliers. Addressing these challenges is essential not only for enhancing the overall efficiency of the manufacturing process but also for reducing costs and maintaining a competitive advantage in the marketplace.

In this context, the optimization of inbound logistics in the manufacturing supply chain transcends mere operational efficiency and becomes a strategic imperative. This optimization requires a nuanced understanding of the interdependencies among various logistical components, the application of advanced planning and forecasting techniques, and the deployment of innovative technologies that provide enhanced visibility and control over the supply chain. By concentrating on these aspects, manufacturers can significantly improve production processes, reduce lead times, lower inventory costs, and more effectively respond to the dynamic demands of the global market. This study addresses material flow management and product route allocation within a manufacturing environment through the application of a mixed-integer programming (MIP) model. The model incorporates key internal logistics metrics, including transportation costs, material-handling costs, capacity constraints, and order quantities. A sensitivity analysis is conducted on both constraints and decision variables to evaluate their impact. Furthermore, the study explores managerial implications and proposes directions for future research.

2. Literature Review

In the context of production environments, the management of inbound logistics operations is paramount for ensuring the efficiency and sustainability of production systems. As a result, numerous studies have been conducted and published on this topic. In the literature review section, these studies are comprehensively analyzed through a full-text screening approach.

Martel (2005) provided a critical review of mathematical programming models within the production logistics literature, proposing a mixed-integer programming method [4]. Berman and Wang (2006) introduced a mixed-integer nonlinear model for addressing inbound logistics strategy selection, utilizing a greedy heuristic approach combined with a Lagrangian relaxation-based branch-and-bound algorithm to manage the problem’s complexity [5]. Cochran and Ramanujam (2006) took a different approach, developing an integer programming model and heuristic algorithm to optimize carrier-mode logistics within inbound logistics systems [6]. Chan and Chan (2007) contributed the first simulation-based study in this domain, presenting a case study focused on managing inbound logistics operations in a small- and medium-sized enterprise (SME) [7]. Bozer and Carlo (2008) applied the Simulated Annealing (SA) method to optimize door assignments in inbound and outbound logistics, with the aim of minimizing material-handling workload [8]. Expanding on the scope of objective functions, Chan and Tang (2007) explored inbound logistics operations from the perspective of minimizing stocking time [9].

With the advent of Industry 4.0 in 2011, researchers have increasingly focused on integrating this perspective into the management of manufacturing logistics operations. Liu and Sun (2011) reviewed the management of information flow in Internet of Things (IoT)-based Vendor-Managed Inventory (VMI) systems, particularly in automotive manufacturing applications [10]. Kilic and Durmusoglu (2013) developed a mixed-integer linear programming (MILP) model and heuristic algorithm for managing milk-run operations in inbound logistics [11]. Similarly, Aljuneidi and Bulgak (2016) proposed a MILP model for the design of sustainable manufacturing systems [12]. Knoll et al. (2016) provided a review of inbound logistics processes, forecasting future developments using machine learning methods [13]. Rocco and Morabito (2016) offered a production and inbound logistics planning model specific to the tomato industry in Brazil, while Zhong et al. (2016) addressed an RFID-based shopfloor logistics problem in automotive manufacturing through IoT applications [14,15].

The literature has also seen a growing emphasis on green and sustainable solution approaches. Chan et al. (2017) reviewed mathematical models for managing sustainable manufacturing systems [16]. Maas et al. (2017) proposed a linear programming model aimed at minimizing both total cost and CO₂ emissions in the inbound logistics activities of an automotive manufacturer [17]. Mei et al. (2017) developed a Clark–Wright (C–W) algorithm-based MIP model for optimizing route management in milk-run systems within an automotive manufacturing context [18]. Falsafi et al. (2018) introduced a decision-support system for managing and monitoring inbound logistics activities in the automotive sector [19]. Hakim et al. (2018) developed a mixed-integer nonlinear programming (MINLP) model to minimize the costs associated with inbound logistics in automotive manufacturing [20]. Vieira Takita and Cabral Leite (2017) applied the Value Flow Mapping (VFM) method to inbound logistics activities in an automotive manufacturing environment in Brazil [21]. Wu et al. (2018) proposed a MIP model combined with a double-population adaptive genetic algorithm (DPAGA) to optimize workstation-driven inbound logistics activities in the automotive sector [22]. Fink and Benz (2019) focused on flexibility planning within inbound logistics operations, particularly in automotive manufacturing [23]. Pitakaso and Sethanan (2019) published a case study on the inbound logistics management of sugarcane machinery, where they developed a MIP model and an Adaptive Large Neighborhood Search (ALNS)-based metaheuristic algorithm [24].

The years 2020 and 2021 marked a notable increase in publications concerning the application of machine learning and heuristic algorithms in inbound logistics management. Albadrani et al. (2020) reviewed various machine learning algorithms—such as K-nearest neighbors (KNN), decision trees, Support Vector Machine (SVM), and Artificial Neural Network (ANN)—and their impact on inbound logistics management [25]. Calabrò et al. (2020) introduced a novel Ant Colony Optimization (ACO) algorithm for route optimization in inbound logistics within a freight transport and logistics company [26]. Chen et al. (2021) published a model based on intelligent scheduling methods for managing integrated inbound logistics (IIL) operations [27]. Marques et al. (2020) developed a matheuristic algorithm to determine the optimal routing plan for inbound logistics in mill manufacturing [28]. Ranjbaran et al. (2020) presented a MILP model and heuristic algorithms for planning inbound logistics in automotive manufacturing [29]. Zhou and Chen (2020) addressed inbound logistics optimization in the automotive sector through a MIP model [30]. More recent studies have explored innovative approaches to inbound logistics management. Sarabi and Darestani (2021) applied fuzzy MULTIMOORA and BWM methods for selecting logistics service providers in the mining equipment manufacturing sector [31]. Mourato et al. (2021) proposed a kanban-card-based lean logistics framework to enhance internal material flows in a bus manufacturing company [32]. Raja Santhi and Muthuswamy (2022) provided a comprehensive review of blockchain technology’s role in manufacturing supply chain management [33]. Daroń (2022) introduced a simulation-based methodology for evaluating order-picking systems (OPS) in manufacturing environments [34]. Valtonen and Lehtonen (2023) proposed another simulation-based methodology focused on the resilience of military logistics through additive manufacturing [35]. Zhang et al. (2023) provided an overview of combinatorial optimization problems in manufacturing environments, emphasizing the critical role of Deep Reinforcement Learning (DRL) in the digitalization of manufacturing systems [36].

Recent studies have highlighted significant advancements in optimizing logistics and manufacturing operations through the integration of mathematical modeling and metaheuristic approaches. Famularo et al. (2024) proposed intelligent multi-layer control systems for the logistics operations of autonomous vehicles in manufacturing systems [37]. Guo et al. (2024) introduced a data-driven optimization framework for the manufacturing and service operations of a home appliance manufacturer in China [38]. Zhao et al. (2024) introduced a bi-objective mixed-integer programming (MIP) model aimed at optimizing inbound logistics operations within a steel manufacturing environment. Their study notably considers the number of rolling units and energy consumption as critical factors within the objective functions, reflecting a comprehensive approach to efficiency and sustainability [39]. Similarly, Liu et al. (2024) developed a multi-objective model for dual-service integrated scheduling of manufacturing and logistics (DISML). To solve this complex problem, they employed an improved Non-Dominated Sorting Genetic Algorithm-II (INSGA-II), which was implemented and validated within a cloud manufacturing environment, demonstrating its effectiveness in managing integrated manufacturing and logistics services [40]. In the context of in-plant logistics, Facchini et al. (2024) proposed an analytical model for IoT-based milk-run routing operations in the automotive manufacturing industry. Their approach specifically addresses the Vehicle Routing Problem with Time Windows (VRPTW) and incorporates capacity constraints, offering a dynamic and practical solution validated in a real-world setting [41].

Based on the literature reviewed, it is evident that most studies have concentrated on developed economies and heavy industries, particularly the automotive sector. However, there is a noticeable gap in research addressing the specific challenges and opportunities present in emerging markets and developing economies. Additionally, the existing studies predominantly focus on standardized logistics models, which may not adequately address the needs of industries increasingly moving towards mass customization. The rise of mass customization highlights a significant need for research into how inbound logistics can be tailored to support the specific and varying requirements of particular manufacturing environments. In response to these gaps, this study introduces a mixed-integer programming (MIP) model aimed at optimizing inbound logistics specifically for the inner door shelves of refrigerator door manufacturing. The model is designed with consideration of the unique necessities and constraints of the manufacturing environment and processes involved, thereby providing a more customized approach to inbound logistics that aligns with the evolving demands of the manufacturing industry.

Table 1 presents a detailed summary of manufacturing logistics studies, emphasizing a wide range of research objectives such as cost minimization, system resilience, and flexibility assessment, with a predominant focus on the automotive sector. Deterministic methodologies, including MIP and LP, are the most employed, providing exact solutions for well-defined problems. Nevertheless, heuristic and metaheuristic approaches are gaining traction in addressing complex, nonlinear challenges. Recent studies highlight the integration of emerging technologies such as IoT, blockchain, and machine learning, signaling a shift toward innovative, adaptive, and real-time logistics solutions. Future research could explore hybrid methodologies, stochastic modeling, and underrepresented sectors, such as healthcare and renewable energy logistics while incorporating sustainability objectives like CO₂ reduction and energy efficiency. Notably, the first row in Table 1—This study—focuses on minimizing transportation and material-handling costs within the household appliances sector using an MIP model. Its deterministic and exact methodology provides a robust framework for resolving logistics inefficiencies in this industry. By prioritizing cost optimization, a critical component of supply chain management, the study enhances operational efficiency and reduces logistics expenditures. Furthermore, its focus on the household appliances sector, which is traditionally underexplored in the literature, underscores the potential for industry-specific logistics optimization and lays the groundwork for future research in diverse manufacturing contexts.

3. The Experimental Investigation

This study examines the inbound logistics operations of a multinational household appliance manufacturer based in Turkey. The company specializes in the production of household appliances, including refrigerators, washing machines, dishwashers, and ovens, catering to both domestic markets and international export destinations. Specifically, the research focuses on a case study conducted within the company’s refrigerator manufacturing facility. However, the proposed model is adaptable and can be applied to other facilities, considering the contextual conditions appropriately.

The primary inbound logistics operation analyzed in this study pertains to the transfer of inner door shelves for refrigerators, as illustrated in Figure 1. Within this context, four potential material routes (Material Routes 1, 2, 3, and 4) have been identified by the decision-makers as feasible options for the manufacturing process. Each route originates at Manufacturing Module 1 and terminates at Manufacturing Module 2, reflecting the operational preferences and strategic considerations of the decision-makers for the logistics system. These routes represent the only viable alternatives under the given constraints and objectives of the manufacturing environment.

The material routes depicted in Figure 2 are as explained as follows:

Material Route 1 represents a variation of the cross-docking operation, where the process begins at Manufacturing Module 1, and the plates are directly transferred to Manufacturing Module 2 for further processing without intermediate storage.

Material Route 2 also originates at Manufacturing Module 1 but introduces an intermediate storage stage at the Facility Storage Unit. The work-in-progress (WIP) inventory is temporarily stored here before being transported to Manufacturing Module 2 for further processing. This approach provides additional flexibility in managing production schedules and inventory levels.

Material Route 3 employs a more complex logistics process, where the WIP inventory is first transferred to the Facility Storage Unit. From there, smaller batches are sequentially moved to the Storage Unit of Manufacturing Module 2. After this intermediate storage step, the materials are finally transferred to Manufacturing Module 2 for processing. This route is particularly suitable for managing high-volume operations with batch-based workflows and limited storage capacity at the destination.

Material Route 4 involves a direct transfer from Manufacturing Module 1 to the Storage Unit of Manufacturing Module 2 in smaller, pre-defined batches. These batches are subsequently processed in Manufacturing Module 2. This route balances direct transfers with batch management, making it suitable for operations requiring smaller batch sizes.

In this study, the research objectives are as follows:

Identifying the most suitable material route for each work-in-progress (WIP) item.
Determining the most appropriate material-handling equipment for each WIP item.
Evaluating which WIP items should be directly transferred from Manufacturing Module 1 to Manufacturing Module 2.
Assessing which WIP items should be transferred via the Facility Storage Unit and the Storage Unit of Manufacturing Module 2.
Determining the quantity and types of material-handling equipment required for each product.
Calculating the overall material-handling and transportation costs associated with the operations.
Determining the proportion of available space allocated to functional areas.

3.1. Mathematical Modeling

Heragu et al. (2005) proposed a model for warehouse design and product allocation; however, this model was originally tailored for the supply chain context between a warehouse and its stakeholders [42]. In this study, the model has been adapted for a manufacturing environment, with revisions made to the objective function and decision variables to suit the new context. The sets and parameters of the revised mathematical model are defined as follows:

$i$	Denotes the type of product, where i = 1, 2, …, I.
$j$	Represents the type of material route, with j = 1, 2, …, J.
$k$	Indicates the type of material-handling equipment, where k = 1, 2, …, K.
${A D}_{i}$	The annual demand for product i, in units.
$A_{i}$	The ordering cost associated with product i, expressed in Turkish Lira (₺).
$P_{i}$	The unit production cost of product i, also expressed in Turkish Lira (₺).
$p_{i}$	The average percentage of waiting time incurred in the factory’s storage unit if product i is assigned to material flow j = 3.
$q_{i j} = 1$	Equals 1 when product i is assigned to routes j = 1, 2, or 4; and equals $⌈r_{i}⌉ + 1$ when product i is assigned to route j = 3, where $r_{i}$ is the ratio of the size of the unit load in the storing unit of the factory to that in the storing unit of the Manufacturing Module 2, $⌈r_{I}⌉$ denotes the smallest integer greater than or equal to $r_{i} .$
a, b, c	Represents the levels of vertical space availability in each storage area, where a is the Buffer Zone of Manufacturing Module 1, b is the Factory Storage Unit, and c is the Storage Unit of Manufacturing Module 2.
r	The carrying cost per inventory unit, in Turkish Lira (₺).
$H_{i j k}$	The handling cost for a unit load of product i in material flow j using material-handling equipment k, in Turkish Lira (₺).
${C S}_{i j}$	The annual storage cost for a unit load of product i in material route j, in Turkish Lira (₺).
$S_{i}$	The space required to store a unit load of product i, measured in square meters (m²).
$T S$	The total available storage space, in square meters (m²).
${O Q}_{i}$	The order quantity for product i, in unit loads.
$T_{i}$	The dwell time per unit load of product i, in years.
${L L}_{b f}$ , ${U L}_{b f}$	The lower and upper storage space limits for the Buffer Zone of the Manufacturing Module 1, in square meters (m²).
${L L}_{f r}$ , ${U L}_{f r}$	The lower and upper storage space limits for the Factory Storage Unit, in square meters (m²).
${L L}_{w s}$ , ${U L}_{w s}$	The lower and upper storage space limits for the Storage Unit of Manufacturing Module 2, measured in square meters (m²).
${N D}_{k}$	The number of material-handling equipment units required for equipment type k.
${I C}_{k}$	The investment cost for material-handling equipment k, expressed in Turkish Lira (₺).
$N_{i j k}$	The average number of unit loads of product i handled in material route j using material-handling equipment k.
${L U L}_{i j k}$	The average time spent on loading and unloading product i using material-handling equipment k in material route j, in minutes.
$d_{i j k}$	The average distance covered while loading and unloading product i using material-handling equipment k in material route j, in meters.
$L_{k j}$	The average percentage of time that material-handling equipment k travels loaded in material route j.
${S P}_{k j}$	The average speed of material-handling equipment k in material route j, in meters per minute (m/min).
${O P}_{k j}$	The operating cost of material-handling equipment k in material flow j, in Turkish Lira per minute (₺/min).
${c a p}_{k}$	The capacity of material-handling equipment k, in units per year.

The decision variables of the proposed model are defined as follows:

$X_{i j k}$	1 if product i is assigned to route j using material-handling equipment k; otherwise, 0.
$θ, β, γ$	The proportion of available space allocated to each functional area, including the Buffer Zone, Storage Unit of the Factory, and Storage Unit of Manufacturing Module 2.

(1) $\min 2 \sum_{i = 1}^{I} \sum_{j = 1}^{J} \sum_{k = 1}^{K} q_{i j} H_{i j k} {A D}_{i} X_{i j k} + \sum_{i = 1}^{I} \sum_{j = 1}^{J} \sum_{k = 1}^{K} (q_{i j} {C S}_{i j} {O Q}_{i} X_{i j k} / 2)$

Equation (1) represents the objective function that aims to minimize the annual handling cost associated with the average quantity of products assigned to a specific material flow, as well as the annual storage cost.

(2) $q_{i j} = 1 i f j = 1, 2, 4, \forall i$

(3) $q_{i j} = ⌈\frac{{P L}_{i t}}{{P L}_{i t + 1}}⌉ + 1 i f j = 3, \forall i$

Equations (2) and (3) ensure that when product i is assigned to material routes j = 1, 2, or 4, it is handled accordingly, and when assigned to route j = 3, it is handled $⌈r_{i}⌉ + 1$ times. Here, $r_{i}$ represents the ratio of the unit load size in the Facility Storage Unit to that in the Storage Unit of Manufacturing Module 2, and $⌈r_{i}⌉$ denotes the smallest integer greater than or equal to $r_{i}$ .

(4) $\sum_{j = 1}^{J} \sum_{k = 1}^{K} X_{i j k} = 1, \forall i$

Equation (4) ensures that each product i is assigned to exactly one type of route j and material-handling equipment k.

(5) $\sum_{i = 1}^{I} \sum_{k = 1}^{K} ({O Q}_{i} S_{i} X_{i 1 k} / 2) - a θ T S \leq 0$

(6) $\sum_{i = 1}^{I} \sum_{k = 1}^{K} ({O Q}_{i} S_{i} X_{i 2 k} + p_{i} {O Q}_{i} S_{i} X_{i 3 k}) / 2 - b β T S \leq 0$

(7) $\sum_{i = 1}^{I} \sum_{k = 1}^{K} ((1 - p_{i}) {O Q}_{i} S_{i} X_{i 3 k} + {O Q}_{i} S_{i} X_{i 4 k}) / 2 - c γ T S \leq 0$

Equations (5)–(7) guarantee that the space capacities of the functional areas are not exceeded.

(8) $H_{i j k} = {T C}_{i j k} / N_{i j k}, \forall i, j, k$

(9) ${T C}_{i j k} = {N D}_{k} {I C}_{k} N_{i j k} (\frac{{L U L}_{i j k} + (d_{i j k} / L_{k j} {S P}_{k j})}{\sum_{i = 1}^{I} N_{i j k} ({L U L}_{i j k} + (d_{i j k} / L_{k j} {S P}_{k j}))}) + N_{i j k} {O P}_{k j} ({L U L}_{i j k} + d_{i j k} / L_{k j} {S P}_{k j}), \forall i, j, k$

Equations (8) and (9) calculate the labor and non-labor-based handling cost for product i in material flow j using material-handling equipment k.

(10) ${N D}_{k} = \sum_{i = 1}^{I} \sum_{j = 1}^{J} \frac{X_{i j k} N_{i j k}}{{c a p}_{k}}, \forall k$

Equation (10) determines the required quantity of material-handling equipment k.

(11) $θ + β + γ = 1$

Equation (11) ensures that the total proportion of available space allocated to each functional area does not exceed 1.

(12) ${L L}_{b f} \leq a θ T S \leq {U L}_{b f}$

(13) ${L L}_{f r} \leq b β T S \leq {U L}_{f r}$

(14) ${L L}_{w s} \leq c γ T S \leq {U L}_{w s}$

Equations (12)–(14) are the balance constraints for the available spaces within the functional areas.

(15) $θ, β, γ \geq 0$

(16) $X_{i j k} ϵ \{0, 1\}, \forall i, j, k$

Equation (15) stipulates the non-negativity of the proportion values for each functional area, and Equation (16) defines the binary variables.

The data utilized in this case study have been sourced from Karagoz (2021), and the following assumptions have been established [43]:
The total available storage space is predetermined.
The average dwell time of a product on the shelf is known.
The annual demand for each product (AD_i) is specified.
Decision-makers have previously identified the types of available material-handling equipment, considering their capacity and fixed costs, to assess their impact on handling costs.
The investment costs of material-handling equipment, as well as labor costs, are known.
The unit load sizes of the products stored on pallets are specified.
The available vertical storage levels within the storage units are known.

Dwell time refers to the average duration for which a product is stored on a shelf. It is assumed that a basic Economic Order Quantity (EOQ) model is employed to determine the optimal order quantity, ${O Q}_{i}$ . The time between two replenishments is given by $\frac{{O Q}_{i}}{{A D}_{i}}$ , and the average dwell time per unit load of product i can be calculated as $T_{i} = \frac{{O Q}_{i}}{2 {A D}_{i}}$ , which also implies that $\frac{{O Q}_{i}}{2} = {A D}_{i} T_{i}$ .

Additionally, it is assumed that the unit load size of the product i transferred to a functional area is equal to the size received from the supplier, except in the case of Material Route 3. In this route, the unit load size of product i changes when it is transferred to the storage unit of the Storage Unit of Manufacturing Module 2. A pallet load in the Facility Storage Unit is broken down into smaller batches according to the production plan and then transferred to the Storage Unit of Manufacturing Module 2.

3.2. Results and Sensitivity Analysis

The mathematical model was solved to optimality using Python (version 3.11.5), utilizing the Pyomo package (version 6.7.3) and the CBC solver (version 2.10.11) within the JupyterLab integrated development environment (version 3.6.3). Pyomo provides a flexible and robust framework for modeling mathematical optimization problems, while CBC serves as an efficient, open-source solver specifically designed for medium-sized MIP problems. The compatibility between Pyomo and CBC facilitates seamless implementation and execution of the optimization models. While CBC may not achieve the computational efficiency of commercial solvers like Gurobi or CPLEX for handling large-scale or highly complex problems, its open-source nature and cost-effectiveness make it a suitable and practical option for the research objectives of this study.

Table 2 presents the optimal values of the decision variables obtained from the mathematical model. The analysis indicates that the annual cost associated with the material transfer operations for the refrigerator inner door shelves amounts to ₺12,437,475,447.17. This outcome directly addresses the research objectives of “Identifying the most suitable material route for each work-in-progress (WIP) item” and “Calculating the overall material-handling and transportation costs associated with the operations”. The model evaluates 19 distinct types of inner door shelves, each assigned to a specific material route with the corresponding material-handling equipment. The allocation of products across the four material routes is as follows: six products are assigned to Material Route 1, four to Material Route 2, seven to Material Route 3, and two to Material Route 4. These allocations reveal that Material Route 3 is the most frequently utilized, indicating it may be the most optimal choice for the majority of products based on the model’s criteria. This finding aligns with the research objectives of “Evaluating which WIP items should be directly transferred from Manufacturing Module 1 to Manufacturing Module 2” and “Assessing which WIP items should be transferred via the Facility Storage Unit and the Storage Unit of Manufacturing Module 2”.

The optimal solution of the mathematical model indicates that one unit of each type of material-handling equipment is required. The results reveal that Material-Handling Equipment 1 was utilized 8 times, while Equipment 2 and Equipment 3 were used 5 and 6 times, respectively. These findings align with the research objectives of “Determining the most appropriate material-handling equipment for each WIP item” and “Determining the quantity and types of material-handling equipment required for each product”. Furthermore, the results demonstrate a specific allocation of available storage space: 91.41% of the total storage capacity is allocated to the Buffer Zone (θ), while 4.00% and 4.59% are allocated to the Storage Unit of the Factory (β) and the Storage Unit of Manufacturing Module 2 (γ), respectively. This distribution reflects the spatial requirements and utilization strategies for the various functional areas within the manufacturing environment. It ensures an optimal balance between storage capacity and operational efficiency, directly addressing the research objective of “Determining the proportion of available space allocated to functional areas”.

Table 2 presents the optimal results for the decision variables, including $X_{i j k}$ , ${N D}_{k}$ , Functional Area Rates, and the Objective Function. Regarding $X_{i j k}$ , the first row indicates that Product 1 (i = 1) should be assigned to Material Flow 2 (j = 2) using Material-Handling Equipment 1 (k = 1). This allocation is repeated for all 19 products (I = 19), determining the optimal material flow and equipment assignments. The variable ${N D}_{k}$ specifies the quantity of material-handling equipment required, with the results indicating that one unit is necessary for each type of equipment. The Functional Area Rates reveal that over 91% of the total storage area should be allocated to the Buffer Zone (θ), emphasizing its critical role in ensuring efficient material flow within the system.

In addition to Table 2, Figure 3 illustrates the optimal material flow within the manufacturing environment, including the allocation of products and material-handling equipment. The figure highlights that Material-Handling Equipment 1 (k = 1) is the most frequently utilized among the available equipment types. Furthermore, the majority of WIP products are allocated to Material Flow 3 (j = 3), indicating that storing partial pallets in the Facility Storage Unit and the Storage Unit of Manufacturing Module 2 is the most cost-effective strategy for the majority of products. Additionally, Material Route 1 (j = 1) is identified as a feasible option for a significant number of products. This route operates as a form of cross-docking, directly transferring materials between modules without intermediate storage, further demonstrating its utility for specific product allocations. These results provide valuable insights into optimizing material flow and equipment utilization to achieve operational efficiency.

The sensitivity analysis of the objective function, with θ = 0.91 held constant, reveals that the optimal balance of the parameters β and γ significantly affects the objective value (Figure 4). The results indicate that the objective function, which represents a combined cost metric, initially decreases as β decreases from 0.09 to 0.04 and γ increases from 0 to 0.05. The minimum objective value is achieved at β = 0.04 and γ = 0.05 (₺ 12,439,475,298.44). Beyond this point, further adjustments in γ lead to a slight increase in the objective value, suggesting diminishing returns. This nonlinear interaction highlights the importance of carefully tuning β and γ to achieve cost minimization in the model.

The sensitivity analysis with θ = 0 explores the impact of varying the parameters β and γ on the objective function value (Figure 5). The analysis reveals that the objective function, representing a cost metric, remains relatively stable at approximately (₺ 12,482,271,657.76) for most combinations of β and γ. However, a significant increase is observed when β = 1 and γ = 0, as well as when β = 0 and γ = 1, indicating potential nonlinear interactions between these parameters. The minimum objective function value is identified at β = 0.6 and γ = 0.4, suggesting that a balanced adjustment between the Storage Unit of the Factory, and Storage Unit of Manufacturing Module 2 yields the most optimal result in minimizing costs under these conditions.

The sensitivity analysis explores the effect of varying parameters θ, β, and γ on the objective function value (Figure 6). The results indicate that the objective function remains stable at approximately (₺ 12,437,475,468.56) for most combinations of these parameters. However, a slight increase is observed as θ approaches 0, especially when β = 0.45 and γ = 0.45. The minimum objective function value is achieved at θ = 0.8, β = 0.1, and γ = 0.1, suggesting that higher values of θ with lower values of β and γ yield a more optimal result for minimizing costs. This analysis highlights the importance of fine-tuning these parameters to achieve the best cost-effective outcomes in the model.

4. Conclusions and Managerial Implications

This case study focused on optimizing the allocation of refrigerator inner door shelves to various material routes and material-handling devices ( $X_{i j k}$ ), as well as the distribution of available space among three key functional areas within a refrigerator manufacturing facility: the Buffer Zone (θ), the Storage Unit of the Factory (β), and the Storage Unit of Manufacturing Module 2 (γ). Sensitivity analysis was employed to assess the impact of different space allocation proportions on the objective function, which represents the total cost of material flow and storage. The analysis revealed that specific combinations of space allocations (θ, β, and γ) can lead to cost minimization. In particular, scenarios where θ is relatively high (e.g., 0.8) and β and γ are lower (e.g., 0.1) resulted in the lowest costs, with the objective function reaching its minimum value of approximately ₺12,437,475,447.17 when θ = 0.91, β = 0.04, and γ = 0.05. This suggests that prioritizing a greater proportion of space for the Buffer Zone, while balancing the space allocations between the two storage units, is crucial for achieving cost efficiency. The results also underscore the complex interaction between θ, β, and γ. Although certain parameter ranges may not individually affect costs significantly, their combined effects can be substantial. Therefore, precise calibration of these proportions is essential to manage the complexities of inter-unit operations and achieve optimal performance.

The optimal results suggest that all material routes and material-handling equipment should be utilized to minimize costs, highlighting that different routes and equipment types are more suitable for certain inner door shelf plates depending on factors such as order quantity and frequency. Future models could incorporate these factors from expert perspectives, such as Multi-Criteria Decision-Making (MCDM) scores, to enhance the decision-making process.

The insights from this study are valuable for managers aiming to optimize material flow and storage within manufacturing facilities. It is essential to avoid extreme allocations—over-allocating space to one area at the expense of others—, which could lead to higher handling costs, delays, and inefficiencies in material flow. Maintaining an optimal balance in space allocation is vital for ensuring operational efficiency and cost savings. Given the complex relationships identified, managers should adopt a flexible approach to space allocation. Regular sensitivity analyses should be conducted to reassess optimal space allocations in response to fluctuations in production demand, product mix, and market conditions. This adaptive strategy will enable more responsive and efficient management of inter-unit operations.

Future research could expand this analysis by applying multi-objective optimization methods that consider not only cost minimization but also other factors such as lead time reduction, labor productivity, and energy efficiency. Such an approach would provide a more comprehensive framework for optimizing inter-unit operations in manufacturing settings. Additionally, further studies could explore the impact of modifying the physical layout of the facility or reconfiguring storage units on the efficiency of material flow. This might include assessing the potential benefits of advanced automation technologies, such as automated guided vehicles (AGVs) and robotics, for optimizing inter-unit transportation. Future research could also expand the optimization model to incorporate additional material routes and functional areas or address more complex constraints, such as safety regulations, labor availability, and environmental considerations. These extensions would allow for more robust and practical applications of the model in real-world manufacturing environments.

The adoption of the MIP approach in this study was guided by the specific characteristics of the allocation problem. With a primary focus on cost minimization within a well-defined problem space, MIP served as an optimal framework for decision-making. The problem’s moderate complexity and relatively small dataset size further validated this choice, rendering the use of heuristic methods unnecessary. Moreover, the research objective centered on identifying the optimal scenario, as opposed to analyzing system behavior through simulations, reinforcing the suitability of MIP for this case study. For larger datasets with increased complexity, researchers may consider employing more comprehensive approaches such as heuristics, meta-heuristics, or math-heuristics to effectively address the problem. Additionally, simulation-based methodologies, such as system dynamics, could be applied to analyze system behavior under stochastic conditions, providing valuable insights into dynamic and uncertain environments.

The managerial implications of this study extend beyond warehouse design and material flow allocation to encompass the integration of Industry 4.0 principles, such as automated guided vehicles (AGVs), robotics, and real-time data analytics, within manufacturing environments. These advanced technologies enhance flexibility, precision, and responsiveness, enabling managers to dynamically optimize space allocations and material routes based on real-time production data and demand variations. The deployment of smart sensors and IoT-enabled devices in storage units and transportation systems further supports predictive maintenance and efficient resource utilization. These applications not only enhance operational efficiency but also minimize downtime and waste, contributing to sustainable manufacturing practices. By leveraging Industry 4.0 technologies, manufacturing facilities can modernize traditional material-handling and storage processes, transitioning to more agile, data-driven, and cost-effective systems.

The proposed MIP model focuses on a limited number of products and aims to minimize material-handling, transportation, storage, and fixed costs. However, in more complex manufacturing environments, additional factors could significantly influence the optimization process. These factors include the presence of multiple manufacturing units, additional storage areas, demand fluctuations, and potential breakdowns. Moreover, critical aspects such as time efficiency and resource utilization, which are essential for operational effectiveness in manufacturing systems, could be incorporated into future MIP models for a more comprehensive analysis. In addition, sustainability considerations should also be integrated into future models to address the growing importance of environmentally conscious manufacturing practices. This could include optimizing energy consumption, reducing CO₂ emissions from transportation and material-handling operations, and incorporating waste management strategies. Sustainable resource utilization, such as minimizing material wastage and adopting circular economy principles, could further enhance the model’s applicability in meeting modern industrial and environmental standards. Due to constraints in data availability from the company, these additional factors were not included in the current mathematical model. Nevertheless, they present valuable opportunities for further research, enabling more robust, realistic, and sustainable modeling of manufacturing logistics systems.

Author Contributions

Conceptualization, S.K. and Y.K.; methodology, S.K.; software, S.K.; validation, S.K.; formal analysis, S.K.; investigation, S.K.; resources, S.K.; data curation, S.K.; writing—original draft preparation, S.K. and Y.K.; writing—review and editing, S.K. and Y.K.; visualization, S.K.; supervision, S.K.; project administration, S.K. and Y.K. All authors have read and agreed to the published version of the manuscript.

Data Availability Statement

The data presented in this study are available upon request from the corresponding author due to company restrictions.

Conflicts of Interest

The authors declare no conflicts of interest.

Footnotes

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

Figures and Tables

Figure 1. The inner door shelf plates of a refrigerator arranged on a pallet.

Figure 2. Framework for material flow within the manufacturing environment.

Figure 3. Framework for optimal material flow within the manufacturing environment.

Figure 4. Sensitivity analysis of objective function with respect to β and γ (θ = 0.91).

Figure 5. Sensitivity analysis of objective function with respect to β and γ (θ = 0.00).

Figure 6. Sensitivity analysis of objective function with respect to β, γ, and various θ values.

Table 1

Comprehensive summary of manufacturing logistics literature.

Contributing Author(s)	Research Objectives	Target Industry	Research Methodology	Parameter Classification	Solution Approach
This study	Minimization of Transportation and Material-Handling Costs	Household Appliances	MIP	Deterministic	Exact
Martel (2005) [4]	Maximization of Operating Income	Demonstrative	MINLP	Deterministic	Exact
Berman and Wang (2006) [5]	Minimization of Transportation and Inventory Costs	Automotive	MINLP, GH	Deterministic	Heuristic
Cochran and Ramanujam (2006) [6]	Minimization of Origin, Freight, and Destination Costs	Electronics	IP, B-D HEU	Deterministic	Heuristic
Chan and Chan (2007) [7]	Management of the Cellular Manufacturing Process	Small- and Medium-Sized Enterprises (SMEs)	SIM	Deterministic	Exact
Bozer and Carlo (2008) [8]	Minimization of Material-Handling Workload	Freight Transportation	MIP, SA	Deterministic	Metaheuristic
Chan and Tang (2009) [9]	Minimization of Stock Level	Demonstrative	SIM	Deterministic	Exact
Liu and Sun (2011) [10]	Analysis of VMI and IoT Applications in Inbound Logistics	Automotive	IoT	Deterministic	Exact
Kilic and Durmusoglu (2013) [11]	Minimization of Inventory and Transportation Costs	Demonstrative	MILP, HEU	Deterministic	Heuristic
Aljuneidi and Bulgak (2016) [12]	Minimization of of Maintenance, Machine Installation/Removal, Production, Inventory, Setup, and Operational Costs	Demonstrative	MILP	Deterministic	Exact
Knoll et al., (2016) [13]	Future inbound logistics process prediction	Demonstrative	ML	Deterministic	Exact
Rocco and Morabito (2016) [14]	Minimization of Transportation, Inventory, and Procurement Costs	Tomato Processing	LP	Deterministic	Exact
Zhong et al., (2016) [15]	Visualization of the Cloud Logistics Shop Floor	Automotive	IoT	Deterministic	Exact
Chan et al., (2017) [16]	Review	Review	MM	-	-
Maas et al., (2017) [17]	Minimization of Fixed, Inventory, Transportation Costs, and CO₂ Emissions	Automotive	LP	Deterministic	Exact
Mei et al., (2017) [18]	Minimization of Total Distance Traveled	Automotive	MIP	Deterministic	Exact
Vieira Takita and Cabral Leite (2017) [21]	Minimization of Total Logistics Costs	Automotive	VFM	Deterministic	Exact
Falsafi et al., (2018) [19]	Management of Disruptive Orders	Automotive	LP, SIM	Deterministic	Exact
Hakim et al., (2018) [20]	Minimization of Material, Transportation, and Administrative Costs	Automotive	MINLP	Deterministic	Exact
Wu et al., (2018) [22]	Minimization of Overall Logistics Costs	Automotive	MIP, DPAGA	Deterministic	Heuristic
Fink and Benz (2019) [23]	Assessment of Flexibility	Automotive	MM	Deterministic	Exact
Pitakaso and Sethanan (2019) [24]	Maximization of Total Serviced Area	Sugarcane	MIP, ALNS	Deterministic	Metaheuristic
Albadrani et al., (2020) [25]	A Review of ML Applications in Inbound Logistics	Review	ML	-	-
Calabrò et al., (2020) [26]	Minimization of Total Distance Traveled	Freight Transportation	SIM, ACO	Deterministic	Heuristic
Marques et al., (2020) [28]	Minimization of Logistics Costs and Revenue Maximization	Forest Industry	MIP, MHEU	Deterministic	Matheuristic
Ranjbaran et al., (2020) [29]	Minimization of Transportation Costs	Automotive	MILP, GES	Deterministic	Heuristic
Zhou and Chen (2020) [30]	Minimization of Pallet Truck Utilization	Automotive	MIP	Deterministic	Exact
Chen et al., (2021) [27]	Minimization of Total Vehicle Mileage	Automotive	MIP, GA	Deterministic	Heuristic
Sarabi and Darestani (2021) [31]	Optimization of Logistics Service Provider Selection	Mining Equipment	MULTIMOORA, BWM	Fuzzy	Approximate
Mourato et al., (2021) [32]	Enhancement of Material Reception and Positioning in the Warehouse	Automotive	AR	Deterministic	Exact
Raja Santhi and Muthuswamy (2022) [33]	A Review of Blockchain Applications in Manufacturing Logistics	Automotive	Review	-	-
Daroń (2022) [34]	Minimization of Logistics Operating Costs	Demonstrative	SIM	Deterministic	Exact
Valtonen and Lehtonen (2023) [35]	Design of Resilient Systems in Military Maintenance Logistics	Military	SIM	Deterministic	Exact
Zhang et al., (2023) [36]	A Review of DRL in Manufacturing Systems	Manufacturing	Review, DRL	-	-
Famularo et al., (2024) [37]	Minimization of Logistics Task Completion Time	Automotive	TCPN, DRL, MPC	Hybrid	Approximate
Guo et al., (2024) [38]	A Review of Data in Household Appliances Manufacturing and Service	Household Appliances	Review	-	-
Zhao et al., (2024) [39]	Minimization of Rolling Units and Energy Consumption	Stainless Steel Manufacturing	MIP, SA	Deterministic	Metaheuristic
Liu et al., (2024) [40]	Minimization of Maximum Delivery Time, Manufacturing Costs, and Logistics Costs	Cloud Manufacturing	MINLP, INSGA-II	Deterministic	Heuristic
Facchini et al., (2024) [41]	Minimization of Total Route Distance	Automotive	MIP	NP	Exact

ACO: Ant Colony Optimization, ALNS: Adaptive Large Neighborhood Search, AR: Action Research, B-D HEU: Bischoff–Dowsland (B–D) Heuristic, BWM: Best–Worst Method, DRL: Deep Reinforcement Learning, DPAGA: Double-Population Adaptive Genetic Algorithm, GES: Grouping Evolution Strategy, GH: Greedy Heuristic, INSGA-II: Non-Dominated Sorting Genetic Algorithm-II, IoT: Internet of Things, IP: Integer Programming, LP: Linear Programming, ML: Machine Learning, MIP: Mixed-Integer Programming, MILP: Mixed-Integer Linear Programming, MM: Mathematical Modeling, MINLP: Mixed-Integer Nonlinear Programming, MULTIMOORA: Multiple Objective Optimizations on the basis of Ratio Analysis plus full Multiplicative Form, MPC: Model Predictive Control, NP: Non-Deterministic Polynomial, SA: Simulated Annealing, SIM: Simulation, TCPN: Timed Colored Petri Nets, VFM: Value Flow Mapping, VMI: Vendor-Managed Inventory.

Table 2

The optimal values of the decision variables.

i	j	k	i	j	k	i	j	k	i	j	k	k		Functional Area Rates		Objective Function (₺)
1	2	1	6	4	3	11	2	2	16	2	3	1	1	θ	0.9141	12,437,475,447.17
2	3	2	7	3	3	12	1	1	17	3	1	2	1	β	0.0400
3	3	2	8	1	3	13	3	3	18	3	1	3	1	γ	0.0459
4	1	2	9	2	1	14	1	3	19	3	2
5	1	1	10	4	1	15	1	1

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Optimization of Material Flow and Product Allocation in Inter-Unit Operations: A Case Study of a Refrigerator Manufacturing Facility

Abstract

Full text

1. Introduction

2. Literature Review

3. The Experimental Investigation

3.1. Mathematical Modeling

3.2. Results and Sensitivity Analysis

4. Conclusions and Managerial Implications

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Optimization of Material Flow and Product Allocation in Inter-Unit Operations: A Case Study of a Refrigerator Manufacturing Facility

Content area

Abstract

Full text

1. Introduction

2. Literature Review

3. The Experimental Investigation

3.1. Mathematical Modeling

3.2. Results and Sensitivity Analysis

4. Conclusions and Managerial Implications

Looking for business discipline content? Try using ProQuest One Business...