Integrated multi-criteria optimization of production, transport, and storage assets in machine-building systems

1 Introduction

The production processes at machine-building enterprises are structurally complex and involve various classes of tasks (Zhetessova et al., 2025; Lyubomudrov et al., 2019). A primary task within technological production planning is the optimization of load parameters for production facilities, including manufacturing equipment, transportation systems, and storage units (Savelyeva et al., 2024; Khrustaleva et al., 2021).

Existing research on optimizing production capacity utilization parameters directly impacts a company’s competitiveness, profitability, and sustainability in a dynamic market. The existing research and its limitations are presented in Table 1.

TABLE 1. Existing studies and their limitations.

The practical value of these studies lies in increasing production flexibility and reducing costs.

Existing studies typically focus on optimizing the loading parameters of only one type of production asset (e.g., only production equipment or only vehicle routes), ignoring interdependencies. However, these parameters are not considered for complex asset loading.

This article addresses a key research gap by developing a unified, scalable mathematical architecture for end-to-end multi-criteria optimization, which enables simultaneous and dynamic balancing of heterogeneous equipment loading parameters, vehicle routes, and warehouse space allocation under the uncertainty of real-world mechanical engineering production.

The goal of the study is to improve the efficiency of planning the loading parameters of production assets (process equipment, transportation, and storage facilities) based on a multi-criteria analysis of possible system states.

2 Research methodology

2.1 Structural analysis of the process of planning the parameters of loading of production means

The process of planning the capacity utilization of production assets, as a subject of study, is multi-component. Its multi-component nature manifests itself in structural heterogeneity, characterized by the presence of various classes of structural elements:

• - Time management system elements: calendar planning periods (calendar year, quarter, calendar month, workday, work shift);
• - Resource elements: means of production (process equipment, storage systems, freight transportation equipment, etc.);
• - Process elements: sets of actions of various types aimed at implementing corresponding classes of tasks within the production process.

Optimization of equipment loading parameters, as well as transportation and storage facilities, is carried out within the framework of five management levels corresponding to the following calendar planning periods (Figure 1):

• - First level of management: calendar year - P C P p 1;
• - Second level of management: quarter - P C P p n 2;
• - Third level of management: calendar month - P C P p t 3;
• - Fourth level of management: working day - P C P p m 4;
• - Fifth level of management: work shift - P C P p s 5.

Within the first management level, one structural element of the temporary management system is defined, corresponding to the calendar year P C P 1, where P C P 1 is the parametric model of the first-level temporary management system element.

At the first control level, the target values ​​of the failure probabilities or parameters Q i k M a c h P C P 1 and Q i k V e h P C P 1 were not calculated in the equations. This limitation was introduced due to the lack of sufficient reliable statistical data on failures and malfunctions of specific units of process equipment and vehicles at the enterprise. To avoid introducing unreliable reliability parameters into the optimization equations, which could distort the Pareto optimization results, these components were temporarily excluded from calculations at the global annual level.

Within the second management level, four structural elements of the temporary management system are defined, corresponding to quarters. Based on these structural elements, a second-level structural model is formed, which has the following form (Equation 1):where M P C P 2 – a second-level structural model for managing production capital loading parameters;

M P C P 2 = P C P n 2 | n ∈ 0 ; 4 , n ∈ N (1)

P C P n 2 - a parametric model of the nth element of the second-level time management system for scheduling.

Within the third-level of management, twelve structural elements of the time management system are defined, corresponding to a calendar month. Based on these structural elements, a third-level structural model of management has been developed, which has the following form (Equation 2):where M P C P 3 – a third-level structural model of production scheduling parameters;

M P C P 3 = { P C P 1 3 , P C P 2 3 , P C P 3 3 , P C P 4 3 , P C P 5 3 , P C P 6 3 , P C P 7 3 , P C P 8 3 , P C P 9 3 , P C P 10 3 , P C P 11 3 , P C P 12 3 } (2)

P C P t 3 - a parametric model of the t-th element of the third-level time management system for scheduling.

Structurally, the set M P C P 3 has four subsets characterizing time intervals corresponding to quarters.

Within the fourth level of management N C D structural elements of the time management system corresponding to working days are defined. Based on these structural data, a structural model of the fourth level of management is formed, which has the following form (Equation 3):where M P C P 4 – the structural model of the fourth-level scheduling of production capacity loading parameters;

M C P C 4 = P C P 1 4 … P C P m 4 , … , P C P p 4 … P C P N C D 4 (3)

P C P m 4 - the parametric model of the m-th element of the fourth-level time management system.

Structurally, the set M C P C 4 has twelve subsets characterizing time intervals corresponding to calendar months.

Within the fifth-level management, N P S h structural elements of the time management system corresponding to a work shift are defined. Based on these structural data, a structural model of the fifth-level management is formed, which has the following form (Equation 4):where M P C P 5 – structural model of the fifth-level scheduling of production capacity parameters;

M C P C 5 = P C P 1 5 … P C P S 5 , … , P C P p 5 … P C P N P S h 5 (4)

P C P s 5 - a parametric model of the sth element of the fifth-level time management system of scheduling.

Structurally, the set M P C P 5 has N C D subsets characterizing time intervals corresponding to working days.

Thus, the set of elements of the time management system has the following structure (Equation 5):where M T M S P R P S - the set of time management elements of the production asset load planning system.

M T M S P R P S = P C P 1 , M P C P 2 , M P C P 2 , M P C P 3 , M P C P 4 , M P C P 5 (5)

Resource elements of the production capacity planning process include objects used to implement the corresponding process types: where M R e s E l P R P S - а set of resource elements of the production resource planning system.

• Process equipment used in the process of implementing the mechanical processing of the established product range (Equation 6):where M P M a c h - a set of units of process equipment used within a mechanical engineering enterprise; M a c h i k – parametric model of the kth unit of process equipment of the ith type.
• Storage systems used to implement the process of storing production batches and process equipment (Equation 7):where M S t S y s - a set of storage system units used within a mechanical engineering enterprise; S t S y s i k – parametric model of the kth storage system unit of the ith type.
• Vehicles used to transport production batches and process equipment (Equation 8):where M V e h - a set of transport units used within a mechanical engineering enterprise; V e h i k – parametric model of the kth unit of transport of the ith type.
• Loading and unloading equipment used in the process of implementing loading and unloading operations (Equation 9):where M L E q - а set of units of loading and unloading equipment used within a mechanical engineering enterprise; L E q i k – a parametric model of the kth unit of loading and unloading equipment of the ith type. The parametric model of the kth unit of loading and unloading equipment of the ith type has the following structure:
• Installation and clamping devices used in the process of implementing mechanical processing of the established range of products (Equation 10):where M I C D e v - a set of units of installation and clamping device used within the framework of a mechanical engineering enterprise; I C D e v i k – parametric model of the kth unit of technological equipment of the ith type.
• Cutting tool used in the process of implementing mechanical processing of the established range of products (Equation 11):where M C T l - a set of cutting tool units used within a mechanical engineering enterprise; C T l i k – parametric model of the kth unit of the ith type cutting tool.
• Auxiliary equipment used in the manufacturing process of the established range of products (Equation 12):where M A u x E q - a set of units of auxiliary equipment used within a mechanical engineering enterprise; A u x E q i k – parametric model of the kth unit of auxiliary equipment of the ith type. Thus, the set of resource elements has the following structure (Equation 13):

M R e s E l P R P S = M P M a c h , M S t S y s , M V e h , M L E q , M I C D e v , M C T l , M A u x E q (13)

• Within the class of process elements, three subclasses are defined, organized by process type:

The process of mechanical processing of a product:

• 1.1. Sets of basic actions aimed at directly changing the properties of a workpiece (Equation 14):where M B A c M e c h P r – a set of sets of basic actions implemented in the manufacturing process of a given range of products; S e t j B A c - a parametric model of the jth set of basic actions implemented within the framework of the technological operation of mechanical processing of the zth product.
• 1.2. Sets of auxiliary actions aimed at directly changing the properties of the workpiece (Equation 15):where M A u x A c M e c h P r – a set of auxiliary action sets implemented in the manufacturing process of a given range of products; S e t j B A c - a parametric model of the jth set of primary actions implemented within the framework of the technological operation of machining the zth product.
• 1.3. Preparatory and final action sets. Preparatory action sets are aimed at setting up process equipment for processing a batch of products. Final action sets are aimed at completing the batch of products and preparing the process equipment for the next process cycle. These action sets form a set of preparatory and final actions (Equation 16):where M P F A c M e c h P r – a set of preparatory and final action sets implemented during the manufacturing process of a given range of products;

M P r e p A c M e c h ⁡ Pr - a subset of preparatory action sets implemented during the manufacturing process of a given range of products;

M F i n A c M e c h P r - a subset of final action sets implemented during the manufacturing process of a given range of products;

S e t j P r e p A c – a parametric model of the jth preparatory action set implemented during the machining of a batch of product;

S e t j F i n A c – a parametric model of the jth final action set implemented during the machining of a batch of products.

• 2. Cargo transportation process
• 2.1 Basic unit transport operation. In this case, a basic unit transport operation is the movement of a single item of cargo within a single transport segment. These elements form a set of basic unit transport operations implemented within the production process (Equation 17):where M S T r O p T r – a set of basic unit transport operations implemented during the manufacturing process of a defined range of products;

• 1.4. Loading and unloading operations. These elements form a set of loading and unloading operations implemented within the production process (Equation 18):where M L U L O p T r – a set of loading and unloading operations implemented within the production process; M L O p T r – a subset of basic loading operations implemented in the manufacturing process of a given range of products; M U L O p T r – a subset of basic unloading operations implemented in the manufacturing process of a given range of products; L O p s – a parametric model of the sth loading operation; U L O p q - a parametric model of the qth unloading operation.

• 3. Cargo storage process. The basic element of this subclass is the storage operation. Storage operations form a corresponding set of operations implemented within the production process (Equation 19):where M S t O p – a set of storage operations implemented in the manufacturing process of a given range of products;

S t O p t – a parametric model of the t-th storage operation.

Thus, the set of process elements has the following structure (Equation 20):where M P r o c E l P R P S - a set of process elements of the production asset loading planning system;

M P r o c E l P R P S = M M e c h ⁡ Pr , M T r , M S t O p { M B A c M e c h ⁡ Pr , M A u x A c M e c h ⁡ Pr , M P r e p A c M e c h ⁡ Pr , M F i n A c M e c h ⁡ Pr , M S T r O p T r , M L O p T r , M U L O p T r , M S t O p } (20)

M M e c h ⁡ Pr – a subset of mechanical processing operations implemented in the manufacturing process of a given range of products;

M T r – a subset of mechanical processing operations implemented in the manufacturing process of a given range of products.

2.2 Description of the hierarchical model of complex optimization of production means loading parameters

Based on the structural elements (Equations 1–20), a hierarchical model of complex optimization of the loading parameters of production means was developed (Figure 2).

The first management level is tasked with optimizing the utilization parameters of individual units of production assets over a calendar year. This first management level has a structural hierarchy and contains three sublevels, which perform the following:

• Management Sublevel 1.1: Optimizing the utilization parameters of the process equipment fleet over a calendar year by assigning process operations to the corresponding units of process equipment. The system’s state over a calendar year, in terms of utilization parameters of the process equipment fleet, is characterized by a set (Equation 21):where S M a c h C P L 1.1 – a parametric model characterizing the state of the system within the calendar in terms of the loading parameters of the technological equipment fleet; M i k C P L 1.1 / T e c h O p o p t – an optimized subset of technological operations implemented within a calendar year using the kth unit of technological equipment of the ith type.
• Management sublevel 1.2: Optimization of storage system load parameters within a calendar year by assigning storage operations to the corresponding storage system units. The system’s state within a calendar year in terms of storage system load parameters is characterized by a set (Equation 22):where S S t S y s C P L 1.2 – a parametric model characterizing the state of the system within a calendar year in terms of the parameters of the storage system load; M i k C P L 1.2 / S t O p o p t – an optimized subset of storage operations implemented using the kth storage unit of the ith type.
• Management sublevel 1.3: optimization of fleet loading parameters within a calendar year by assigning individual transport operations to the corresponding vehicles. The system’s state within a calendar year in terms of fleet loading parameters is characterized by a set (Equation 23):where S V a h C P L 1.3 – a parametric model characterizing the state of the system within a calendar year in terms of the parameters of the vehicle fleet loading; M i k C P L 1.2 / S T r O p o p t – an optimized subset of unit transport operations implemented using the kth unit of the ith type of vehicle. For the first level of management, the following set of basic structural elements has been defined, characterizing the state of the system within a calendar year (Equation 24):where M P l P r S S C P L 1- a set of parametric models characterizing the system’s state over a calendar year in terms of production asset loading parameters. The established hierarchy within the first control level is determined by the following conditions:
• The process of generating material flows is based on optimized manufacturing routes. Thus, the first stage of optimization is the determination of the process equipment that will implement the corresponding manufacturing operations. The result of this control sublevel is the matrix W i k z p T e c h O p C P L 1.1 / o p t , having the following structure and parameters of structural elements (Equation 25):where W i k z p T e c h O p C P L 1.1 / o p t – an optimized matrix of parameters mapping manufacturing operations to the corresponding units of manufacturing equipment;

w i k z p T e c h O p C P L 1.1 / o p t – the optimized compatibility parameter for the pth manufacturing operation performed during the production of the zth product by the kth unit of manufacturing equipment of type i.

The matrix W i k z p T e c h O p C P L 1.1 / o p t is one of the input data elements used to determine options for the inter-operation storage of product batches, since specific inter-operation storage areas are assigned to the corresponding piece of process equipment. The variability of inter-operation storage is determined based on an analysis of the following parameters: the location of the processing equipment units used to perform the corresponding processing operations; production batch parameters (weight, volume); and storage durations. The result of implementing this sub-level of control is an optimized matrix W i k z t S t O p C P L 1.2 / o p t, which has the following structure and parameters of its structural elements (Equation 26):where W i k z t S t O p C P L 1.2 / o p t – an optimized matrix of parameters mapping storage operations to the corresponding storage system units;

W i k z t S t O p C P L 1.2 / o p t = W i k z t S t O p L P C 1.2 / o p t , W i k z t S t O p C P L 1.2 o p t ∈ 0 ; 1 , W i k z t S t O p C P L 1.2 / o p t ∈ Z (26)

w i k z t S t O p L P C 1.2 / o p t – the optimized parameter for the t-th storage operation performed during the manufacturing of the zth product by the kth unit of process equipment of type i.

Thus, the stage of optimizing storage system loading parameters is the second stage in the first-level control hierarchy.

The matrix W i k z t S t O p C P L 1.2 / o p t is one of the input data elements used to determine the variability in the use of vehicles and loading/unloading equipment for carrying out the corresponding transport operations. This relationship manifests itself in determining the type of transport operation (inter-shop/intra-shop) and the availability of the corresponding types of loading and unloading equipment within the relevant inter-operational warehouses. The type of transport operation directly influences the choice of transport vehicle, as it characterizes the labor intensity of the operation in terms of route duration. The result of implementing this sub-level of management is an optimized matrix W i k r S T r O p C P L 1.3 / o p t. Thus, this stage of optimizing vehicle loading parameters is the third in the hierarchical structure of the first level of management.

The hierarchical relationship between the first and second levels of control is established through control sub-level 1.1, since the second level of control optimizes only the utilization parameters of process equipment. This relationship is established using the matrix W i k z p T e c h O p C P L 1.1 / o p t.

Within the second level of control, technological operations are distributed among the corresponding periods of the second level of calendar planning - P C P τ 2. The system state within the second level of control, in terms of the utilization parameters of the process equipment fleet, is characterized by set (Equation 27):where M P l P r S S L P C 2 - a set of parametric models that characterize the state of the system within the framework of the second level of scheduling with respect to production resource utilization parameters;

M P l P r S S C P L 2 = S M a c h C P L 2 . n n ∈ 1 , 2 , 3 , 4 (27)

S M a c h C P L C 2 . n - a parametric model that characterizes the state of the system during the nth period of the second level of calendar planning in terms of the utilization parameters of the process equipment fleet;

n – a parameter indicating the number of quarters in a calendar year, units.

The result of implementing the second level of control is the matrix W i k С L P 2 / T e c h O p z p j n o p t, W i k С L P 2 / T e c h O p z p j n o p t – an optimized matrix of parameters describing the alignment of technological operations performed using the kth unit of technological equipment of the ith type with the corresponding calendar planning periods. The matrix W i k С L P 2 / T e c h O p z p j n o p t has the following structure and structural element parameters (Equation 28):where w i k С L P 2 / T e c h O p z p j n o p t – the optimized compliance parameter for the pth manufacturing operation performed during the production of the jth production batch of the zth product in the nth calendar planning period (quarter).

W i k С L P 2 / T e c h O p z p j n o p t = W i k С L P 2 / T e c h O p z p j n o p t , W i k С L P 2 / T e c h O p z p j n o p t ∈ 0 ; 1 , W i k С L P 2 / T e c h O p z p j n o p t ∈ N (28)

The relationship between the second and third levels of management is established through the matrix W i k С L P 2 / T e c h O p z p j n o p t. At the third level of management, technological operations are allocated among the corresponding calendar months.

The result of implementing the third level of control is the matrices W i k C P L 3 / T e c h O p z p j t o p t, W i k C P L 3 / T e c h O p z p j t o p t – n optimized matrix of parameters describing the alignment of technological operations performed using the kth unit of technological equipment of the ith type with the corresponding periods of the third level of calendar planning.

The matrix W i k C P L 3 / T e c h O p z p j t o p t has the following structure and structural element parameters (Equation 29):where w i k C P L 3 / T e c h O p z p j t o p t – the optimized alignment parameter for the pth manufacturing operation performed during the production of the jth production batch of the zth product, corresponding to the t-th period of the third level of calendar planning.

W i k C P L 3 / T e c h O p z p j t o p t = W i k C P L 3 / T e c h O p z p j t o p t , W i k C P L 3 / T e c h O p z p j t o p t ∈ 0 ; 1 , W i k C P L 3 / T e c h O p z p j t o p t ∈ N (29)

The system’s state within the third level of management, in terms of the utilization parameters of the process equipment fleet, is characterized by set (Equation 30):where M P l P r S S C P L 3 - a set of parametric models that characterize the state of the system within the third level of scheduling with respect to production resource utilization parameters;

M P l P r S S C P L 3 = S M a c h C P L 3 . t t ∈ 1 ; 12 , t ∈ N = = { S M a c h C P L 3.1 , S M a c h C P L 3.2 , S M a c h C P L 3.3 , S M a c h C P L 3.4 , S M a c h C P L 3.5 , S M a c h C P L 3.6 , S M a c h C P L 3.7 , S M a c h C P L 3.8 , S M a c h C P L 3.9 , S M a c h C P L 3.10 , S M a c h C P L 3.11 , S M a c h C P L 3.12 } (30)

S M a c h C P L 3 . t – a parametric model that characterizes the state of the system within the t-th period of the third level of calendar planning in terms of the utilization parameters of the process equipment fleet;

t - a parameter indicating the number of calendar planning periods within the third level of management, units.

The fourth and fifth management levels have a hierarchical structure and each contain two sub-levels. Management sub-level 4.1 has a hierarchical relationship with the third management level, and management level 4.2 has a hierarchical relationship with management level 1.3.

Within control level 4.1, the utilization of process equipment units is optimized within the respective periods of the fourth level of calendar planning. The relationship between the third control level and control level 4.1 is established via the matrix W i k C P L 3 / T e c h O p z p j t o p t.

Within management level 4.2, the utilization of transportation units is optimized within the corresponding periods of the fourth level of calendar planning. The relationship between management levels 1.3 and 4.2 is established using the matrix W i k r S T r O p C P L 1.3 / o p t.

The state of the system at the fourth control level, in terms of the utilization parameters of the technological equipment and vehicle fleets, is characterized by set (Equation 31):where M P l P r S S C P L 4 - a set of parametric models that characterize the state of the system within the fourth level of scheduling with respect to production resource utilization parameters;

M P l P r S S C P L 4 = { S M a c h C P L 4 . m m ∈ 1 ; N C D , N C D ∈ N , S V e h C P L 4 . m m ∈ 1 ; N C D , N C D ∈ N (31)

S M a c h C P L 4 . m – a parametric model that characterizes the state of the system within the mth period of the fourth level of calendar planning in terms of the utilization parameters of the process equipment fleet;

S V e h C P L 4 . m – a parametric model that characterizes the state of the system within the mth period of the fourth level of calendar planning in terms of the vehicle fleet’s workload parameters;

m - a parameter indicating the number of calendar planning periods within the fourth level of management, units.

The result of implementing the fourth level of control is the matrices W i k C P L 4.1 / T y p e A c z p j n m o p t, V i k C P L 4.2 / S T r O p t m o p t , which have the following structure and structural element parameters (Equations 32 and 33): where W i k C P L 4.1 / T y p e A c z p j n m o p t – a matrix of parameters characterizing the number of actions of a specific type performed for batches of products within the corresponding period of the fourth level of calendar planning; V i k C P L 4.2 / S T r O p t m o p t – a matrix of parameters mapping individual transport operations to the corresponding periods of the fourth level of calendar planning; w i k C P L 4.1 / T y p e A c z p j n m o p t – parameters characterizing the number of actions of the nth type performed for the jth production lot of the zth product during the pth process operation within the mth period of the fourth level of calendar planning, pcs.; v i k C P L 4.2 / S T r O p t m o p t – a compliance parameter for the t-th unit transport operation, performed using the kth vehicle of type i, in the m-th period of the fourth level of calendar planning.

W i k C P L 4.1 / T y p e A c z p j n m o p t = W i k C P L 4.1 / T y p e A c z p j n m o p t , W i k C P L 4.1 / T y p e A c z p j n m o p t ∈ 0 ; N z j p r B , W i k C P L 4.1 / T y p e A c z p j n m o p t ∈ Z (32)

V i k C P L 4.2 / S T r O P t m o p t = V i k C P L 4.2 / S T r O P t m o p t , V i k C P L 4.2 / S T r O P t m o p t ∈ 0 ; 1 , V i k C P L 4.2 / S T r O P t m o p t ∈ N (33)

Management sub-level 5.1 has a hierarchical relationship with management level 4.1, and management level 5.2 has a hierarchical relationship with management level 4.2.

Within management sub-level 5.1, the utilization of process equipment units is optimized within the corresponding periods of the fifth level of calendar planning. The relationship between control levels 4.1 and 5.1 is established using the matrix W i k C P L 4.1 / T y p e A c z p j n m o p t . At control level 5.2, the utilization of transportation vehicles is optimized within the corresponding periods of the fifth level of calendar planning. The relationship between control levels 4.2 and 5.2 is established via the matrix V i k C P L 4.2 / S T r O p t m o p t.

The system’s state within the fifth level of management, in terms of the utilization rates of process equipment and transportation fleets, is characterized by set (Equation 34):where M P l P r S S C P L 5 - a set of parametric models characterizing the state of the system within the fifth level of calendar planning in terms of production equipment utilization parameters; S M a c h C P L 5 . s – a parametric model characterizing the state of the system within the t-th period of the fifth level of calendar planning in terms of process equipment fleet utilization parameters; S V e h C P L 5 . s – a parametric model characterizing the state of the system within the t-th period of the fifth level of scheduling with respect to the utilization parameters of the process equipment fleet; s - a parameter characterizing the number of scheduling periods within the fifth level of control, units.

M P l P r S S C P L 5 = { S M a c h C P L 5 . s s ∈ 1 ; N W S h , N W S h ∈ N , S V e h C P L 5 . s s ∈ 1 ; N W S h , N W S h ∈ N (34)

The result of implementing the fifth level of control is the matrices W i k C P L 5.1 / T y p e A c z p j n s o p t, V i k C P L 5.2 / S T r O p t s o p t , which have the following structure and structural element parameters (Equations 35 and 36): where W i k C P L 5.1 / T y p e A c z p j n s o p t – a matrix of parameters characterizing the number of actions of a specific type performed for batches of products within the corresponding period of the fifth level of calendar planning; V i k C P L 5.2 / S T r O p t s o p t – a matrix of parameters mapping individual transport operations to the corresponding periods of the fifth level of calendar planning; w i k C P L 5.1 / T y p e A c z p j n s o p t – parameters characterizing the number of operations of the nth type performed for the jth production batch of the zth product during the pth process operation within the mth period of the fifth level of calendar planning, units.; v i k C P L 5.2 / S T r O p t s o p t – a parameter indicating the compliance of the t-th unit transport operation, performed using the kth unit of a vehicle of the ith type, with the m-th period of the fifth level of calendar planning.

W i k C P L 5.1 / T y p e A c z p j n s o p t = W i k C P L 5.1 / T y p e A c z p j n s o p t , W i k C P L 5.1 / T y p e A c z p j n s o p t ∈ 0 ; N j P r B , W i k C P L 5.1 / T y p e A c z p j n s o p t ∈ Z (35)

V i k C P L 5.2 / S T r O p t s o p t = V i k C P L 5.2 / S T r O p t s o p t , V i k C P L 5.2 / S T r O p t s o p t ∈ 0 ; 1 , V i k C P L 5.2 / S T r O p t s o p t ∈ N (36)

2.3 Formulating the problem of optimizing the utilization of production resources

Optimization of the utilization parameters for the fleet of process equipment and vehicles, as well as storage systems, within the first level of control is performed based on the vector optimization criteria presented in Supplementary Appendix 1. The following optimization criteria have been defined for the corresponding sub-levels of control (Equations 37–39):

Sub − level of control 1.1 : K e f M e c c h C P L 1 → ⁡ max ⁡ , P 1 … n G r M a c h C P L 1 → ⁡ min ⁡ , Q i f M e c c h C P L 1 → ⁡ min (37)

Sub − level of control 1.2 : K e f S t S y s C P L 1 → ⁡ max ⁡ , P 1 … k S t S y s C P L 1 → ⁡ min (38)

Sub − level of control 1.3 : R a v V e h C P L 1 → ⁡ min ⁡ , L o s g e n V a h a v C P L 1 → ⁡ min ⁡ , R a v V e h C P L 1 → ⁡ max ⁡ , L o s g e n V e h a v C P L 1 → ⁡ min (39)

Control parameter sets have been defined for sublevels 1.1, 1.2, and 1.3, which have the following structures (Equations 40–42):

U P l P r C P L 1.1 = T e c h O p z p | T e c h O p z p ∈ M i k P C L 1.1 / T e c h O p a c p (40)

U P l P r C P L 1.2 = S t O p s | S t O p s ∈ M i k P C L 1.2 / S t O p a c p (41)

U P l P r C P L 1.3 = S T r O p r | S T r O p r ∈ M i k P C L 1.3 / S T r O p a c p (42)

The following restrictions have been defined for control level 1.1 (Equations 43–47):

K e f M a c h C P L 1 ≥ K e f M a c h min C P L 1 , K e f M a c h C P L 1 ∈ 0 ; 1 , K e f M a c h min C P L 1 ∈ 0 ; 1 (43)

P n G r M a c h C P L 1 ≤ P n G r M a c h max C P L 1 , P n G r M a c h C P L 1 ∈ 0 ; 1 , P n G r M a c h max C P L 1 ∈ 0 ; 1 (44)

K i k M a c h ≥ K i k M a c h min , K i k M a c h ∈ 0 ; 1 , K i k M a c h min ∈ 0 ; 1 (45)

Q i k M a c h C P L p 1 ≤ Q max M e c h C P L 1 (46)

∑ z = 1 N i k Pr ∑ j = 1 N z P r B ∑ p = 1 N z i k T e c h O p T z j p T e c h O p ≤ T e f M e c h i k C P L 1 , N z i k T e c h O p ∈ N , N z P r B ∈ N , N i k Pr ∈ N (47)

A description of the variables is provided in Supplementary Supplementary Appendix 2.

The following constraints have been defined for control level 1.2 (Equations 48–54):

K e f S t S y s C P L p 1 ≥ K e f S t S y s min C P L 1 , K e f S t S y s C P L 1 ∈ 0 ; 1 , K e f S t S y s min C P L 1 ∈ 0 ; 1 (48)

P n G r S t S y s C P L 1 ≤ P n G r S t S y s max C P L 1 , P n G r S t S y s C P L 1 ∈ 0 ; 1 , P n G r S t S y s max C P L 1 ∈ 0 ; 1 (49)

K i k S t S y s ≥ K i k S t S y s min , K i k S t S y s ∈ 0 ; 1 , K i k S t S y s min ∈ 0 ; 1 (50)

∑ z = 1 N Pr ∑ j = 1 N Z P r B S z j P r B · t a v S t i 24 ≤ S i k S t S y s / max · N W D C P S p 1 , S z j P r B > 0 , S z j Pr ⁡ B ∈ Q , N z P r B ∈ N , N Pr ∈ N , t a v S t i > 0 , t a v S t i ∈ Q , S i k S t S y s / max > 0 , S i k S t S y s / max ∈ Q , N W D C P S p 1 ∈ N (51)

∑ z = 1 N Pr ∑ j = 1 N Z P r B m z j P r B · t a v S t i 24 ≤ M i k S t S y s · N W D C P S p 1 , m z j P r B > 0 , m z j P r B ∈ Q , M i k S t S y s > 0 , M i k S t S y s ∈ Q (52)

m z j P r B ≤ M i k S t S y s (53)

S z j P r B ≤ S i k S t S y s / ⁡ max (54)

A description of the variables is provided in Supplementary Appendix 2.

The following constraints have been defined for control level 1.3 (Equations 55–62):

R a v V e h C P L 1 ≤ R max V e h C P L 1 , R a v V e h C P L 1 ∈ 0 ; 1 , R max V e h C P L 1 ∈ 0 ; 1 (55)

L o s g e n V e h a v C P L 1 ≤ L o s g e n V e h max C P L 1 , L o s g e n V e h a v C P L 1 ∈ 0 ; 1 , L o s g e n V e h max C P L 1 ∈ 0 ; 1 (56)

K G V e h C P L 1 ≥ K G V e h min , K G V e h C P L 1 ∈ 0 ; 1 , K G V e h min ∈ 0 ; 1 (57)

Q i k V e h C P L 1 ≤ Q max V e h C P L 1 (58)

Κ i k V e h ≥ K i k V e h min , Κ i k V e h ∈ 0 ; 1 , K i k V e h min ∈ 0 ; 1 (59)

∑ z = 1 N i k Pr ∑ j = 1 N z P r B ∑ f = 1 N z j i k T r O p T z j f T r O p ≤ T e f V e h i k C P S P 1 , T z j f T r O p ≥ 0 , T z j f T r O p ∈ Q , T e f V e h i k C P L 1 ≥ 0 , T e f V e h i k C P L 1 ∈ Q , N z j i k T r O p ∈ N , N z P r B ∈ N , N i k Pr ∈ N (60)

m z j P r B ≤ G i k V e h , G i k V e h ≥ 0 , G i k V e h ∈ Q (61)

S z j P r B ≤ S i k V e h / ⁡ max (62)

A description of the variables is provided in Supplementary Appendix 2.

The optimization of utilization parameters for process equipment and vehicles across 2–4 management levels, as well as the fifth management level under a three-shift work schedule, is performed based on the vector optimization criterion (Equation 63):where F P l P r C P L _ n – vector optimization criterion for the nth control level; U P l P r C P L _ n - the set of load control parameters for the corresponding types of production resources within the nth level of calendar planning; R a v M P r / C P L _ n – a parameter characterizing the range of average values of the utilization coefficients for the corresponding types of production assets within the nth calendar planning period; I P / a v M P r / C P L _ n – a parameter characterizing the average level of unevenness in the distribution of the utilization of the corresponding types of production resources across the calendar planning periods of the nth management level; I L o s M P r / C P L _ n - a parameter characterizing the average level of unevenness in the distribution of losses due to underutilization of the corresponding types of production assets across the calendar planning periods of the nth management level; n, where n ∈ {2, 3, 4} – a parameter characterizing the number of the management level.

F P l P r C P L _ n = R a v M P r / C P L _ n U P l P r C P L _ n , I P / a v M P r / C P L _ n U P l P r C P L _ n , I L o s M P r / C P L _ n U P l P r C P L _ n (63)

For the relevant levels of control, the optimization criteria are (Equation 64):

R a v M P r / C P L n → ⁡ min ⁡ , I P / a v M P r / C P L n → ⁡ min ⁡ , I LOS M P r / C P L n → ⁡ min (64)

A description of the parameter sets for the 2–5 levels of calendar planning is provided in Supplementary Appendix 3.

The target values R a v M P r / C P L _ n , I P / a v M P r / C P L _ n , I L o s M P r / C P L _ n are calculated using Equations 65, 66 through (Equation 67):

R a v M P r / C P L _ n = K max L d / M e c h − K min L d / M e c h (65)

I P / a v M P r / C P L _ n = ∑ n = 1 N C P L _ n K m M e c h / C P L _ n − K M e c h / a v 2 N C P L _ n N M P r (66)

I L o s M P r / C P L _ n = ∑ n = 1 N C P L _ n l o s m M e c h / C P L _ n − l o s m M e c h / a v 2 N C P L _ n (67)

A description of the variables is provided in Supplementary Appendix 2.

The following constraints have been defined for optimizing the utilization parameters of the process equipment fleet within management levels 2, 3, 4.1, and 5.1 (Equations 68–72):

R a v M e c h / C P L n ≤ R max M e c h / C P L n , R a v M e c h / C P L n ∈ 0 ; 1 , R max M e c h / C P L n ∈ 0 ; 1 (68)

I P / a v M e c h / C P L n ≤ I P / a v M e c h / C P L n max , I P / a v M e c h / C P L n ∈ 0 ; 1 , I P / a v M e c h / C P L n max ∈ 0 ; 1 (69)

I L O S M e c h / C P L n ≤ I L O S M e c h C P L n max , I L O S M e c h / C P L n ∈ 0 ; 1 , I L O S M e c h C P L n max ∈ 0 ; 1 (70)

K m M e c h / C P L n ≥ K i k M e c h min , K m M e c h / C P L n ∈ 0 ; 1 , K i k M e c h / min ∈ 0 ; 1 (71)

∑ z = 1 N i k Pr ∑ j = 1 N z P r B ∑ p = 1 N z i k T r O p T z j p T e c h O p ≤ T e f M e c h i k C P L n , T e f M e c h i k C P L n ≥ 0 , T e f M e c h i k C P L n ∈ Q (72)

A description of the variables is provided in Supplementary Appendix 2.

The following constraints have been defined for optimizing the vehicle fleet loading parameters within management levels 4.2, 5.2 (Equations 73–79):

R a v V e h C P L n ≤ R max V e h / C P L n , R a v V e h / C P L n ∈ 0 ; 1 , R max V e h / C P L n ∈ 0 ; 1 (73)

I P / a v V e h / C P L n ≤ I P / a v V e h / C P L n max , I P / a v V e h / C P L n ∈ 0 ; 1 , I P / a v V e h / C P L n max ∈ 0 ; 1 (74)

I L O S V e h / C P L n ≤ I L O S V e h / C P L n max , I L O S V e h / C P L n ∈ 0 ; 1 , I L O S V e h / C P L n max ∈ 0 ; 1 (75)

K m V e h / C P L n ≥ K i k V e h / min , K n V e h / C P L n ∈ 0 ; 1 , K i k V e h / min ∈ 0 ; 1 (76)

∑ z = 1 N i k Pr ∑ j = 1 N z P r B ∑ p = 1 N z i k T r O p T z j f T r O p ≤ T e f V e h i k C P S p n , T e f V e h i k C P S p n ≥ 0 , T e f V e h i k C P S p n ∈ Q (77)

∑ z = 1 N i k Pr ∑ j = 1 N z i k P r B m z j P r B N n T r O p ≤ G i k V e h , G i k V e h ≥ 0 , G i k V e h ∈ Q , (78)

∑ z = 1 N i k Pr ∑ j = 1 N z i k P r B S z j P r B N n T r O p ≤ S i k V e h / ⁡ max (79)

A description of the variables is provided in Supplementary Appendix 2.

The optimization of the utilization rates of process equipment and vehicle fleets within the fifth level of management, under a two-shift work schedule, is performed based on the vector optimization criterion (Equation 80):where F P l P r C P L 5 . n – vector optimization criterion for the fifth control level under a two-shift work schedule; U P l P r C P L 5 . n - the set of load control parameters for the corresponding types of production facilities within the fifth control level under a two-shift work schedule; R a v M P r / C P L 5 . n s , s + 1 – a parameter characterizing the range of average values of the utilization coefficients for the corresponding types of production assets for the calendar planning periods of the fifth level of control; P L d / a v M P r / C P L 5 . n s , s + 1 – a parameter characterizing the average value of the load unevenness parameters for the technological equipment fleet within the corresponding calendar planning periods of the fifth management level; δ L o s / a v M P r / C P L 5 . n s , s + 1 - a parameter characterizing the magnitude of the deviation of the proportion of costs arising from the downtime of production assets from the average value for the corresponding calendar planning periods of the fifth management level.

F P l P r C P L . n = ( R a v M P r / C P L 5 . n s , s + 1 U P l P r C P L 5 . n , P L d / a v M P r / C P L . n s , s + 1 U P l P r C P L 5 . n , δ L o s / a v M P r / L C P 5 . n s , s + 1 U P l P r C P L 5 . n ) (80)

2.4 Limitations of the study

The theoretical part of the model embodies a system of interdependent (cross-functional) constraints designed to eliminate the isolation of individual services. Here’s the logic behind their construction. The model is built on a rigid sequence, where the results of the upper sublevel act as constraints for the lower one. At sublevel 1.1, an optimized matrix for assigning technological operations to machines is formed W i k z p T e c h O p C P L 1.1 / o p t. It is then passed to sublevel 1.2 as an input constraint for selecting intershop storage locations, since warehouses are strictly tied to machine locations. The warehouse results W i k z t S t O p C P L 1.2 / o p t in turn impose constraints on transport sublevel 1.3, determining the availability of loading equipment and route durations. Mathematical constraints in the equations are designed to link disparate assets: the processing time of a part on a machine directly dictates the delivery schedule of a transport vehicle, and the physical capacity of the warehouse (volume/weight constraints on batches) limits the production rhythm of the machines. The traditional approach optimizes resources sequentially (first production, then logistics), which leads to local optima at the expense of overall efficiency and creates information gaps. The introduction of end-to-end constraints into a unified system of vector criteria allows equations to find balanced solutions, minimizing downtime at the interfaces between processing, transportation, and storage.

3 Results

As part of this study, the problem of optimizing the utilization rates of the process equipment fleet and the vehicle fleet used in the manufacturing process for a specified product range was solved. The specified product range is manufactured within a single production unit - the production shop. The following types of manufacturing equipment and vehicles are used to carry out the production process within the production unit:

Manufacturing equipment:

2.4 a group of CNC lathes: GrMech 1 = Mach 1 1 , Mach 1 2 , Mach 1 3;

2.5 a group of CNC turning and milling machines: GrMech 2 = Mach 2 1 , Mach 2 2 , Mach 2 3;

1.3 a group of CNC milling machines: GrMech 3 = { Mach 3 1 , Mach 3 2 , Mach 3 3 , Mach 3 4 , Mach 3 5 ;

1.4 Boring machine: Mach 4 1

1.5 Group of cylindrical grinding machines: GrMech 5 = { Mach 5 1 , Mach 5 2 ;

1.6 Group of surface grinders: GrMech 6 = { Mach 6 1 , Mach 6 2 , Mach 6 3 ;

1.7 vehicles: platform electric carts GrVah 1 = { Vah 1 1 , Vah 1 2 , Vah 1 3 .

The production unit operates on a two-shift schedule.

Over the course of a calendar year, the production unit manufactures 83 product lines. For each product line, two to seven production batches are formed, ranging in size from 15 to 55 units. The total number of manufacturing operations carried out by the production unit under analysis during a calendar year is 450.

The results presented in this paper reflect the state of the production system in the 2026 calendar year. Optimization was performed based on a baseline schedule for the utilization of manufacturing equipment and vehicles, developed by specialists from the company’s production planning department.

At the first level of management, the utilization parameters for the fleet of production equipment and vehicles were optimized. During the production planning process, a partial redistribution of production equipment utilization was carried out, caused by:

• The introduction of five new product lines, which are processed on the milling group’s equipment;
• The decommissioning of one milling machine that was used to process seven product lines.

For the seven product lines previously processed on the decommissioned equipment, the structures of the processing routes were optimized. Consequently, there was also a change in the utilization parameters of the surface grinding equipment group.

Within the production unit under analysis, each piece of process equipment is linked to only one inter-operation warehouse, which is dictated by logistical processes. Additionally, within these warehouses, goods are stored on the floor without the use of specialized storage systems. Consequently, there is no flexibility in the location of storage operations. The lack of flexibility in the execution of storage operations is a constraint on the division, resulting from its organizational structure and management system. Given these constraints, optimization at the first management level was performed only for process equipment and vehicles; accordingly, only management sub-levels 1.1 and 1.3 were implemented as part of the study.

As part of management sub-level 1.1, the utilization parameters for the process equipment fleet were optimized within a calendar year. Figure 3 shows a graph of Pareto-optimal solutions for management level 1.1. The graph displays three sets of points characterizing the values of the target indicators for eleven non-dominant variants (Var_1- Var_11) of the distribution of manufacturing operations among the corresponding units of manufacturing equipment. Since the production process under analysis within the division in question underwent partial reengineering due to the decommissioning of one milling machine and the introduction of an additional product range, the change in utilization occurred only for two groups of equipment (milling and surface grinding). Accordingly, the graph (Figure 2) shows curves characterizing the uniformity of the production equipment’s load only for these two groups: P 2 G r M a c h C P L 1 – a target indicator characterizing the degree of load unevenness for the milling equipment group within a calendar year; P 5 G r M a c h C P L 1 - a target indicator characterizing the degree of load unevenness for the surface grinding equipment group within a calendar year.

The status of the production system in terms of the utilization rate of the process equipment fleet within a calendar year prior to optimization is characterized by the target values (Equation 81):

K e f M a c h C L P 1 = 0.642 , P 2 G r M a c h C P L 1 = 0.059 , P 5 G r M a c h C P L 1 = 0.081 (81)

Within Management Level 1.1, an optimization strategy has been defined (Equation 82):

K e f M a c h C P L 1 → ⁡ max (82)

Taking condition (Equation 82) into account, the optimal option for utilizing the production equipment fleet within a calendar year is the first option, Var_1, since it has the maximum value of the target indicator K e f M a c h C P L 1. Thus, the state of the production system in terms of the utilization of the production equipment fleet within a calendar year after optimization is characterized by the following values of the objective functions and utilization coefficients for the corresponding units of production equipment (Equations 83 and 84):

K e f M a c h C P L 1 = 0.739 , P 2 G r M a c h C P L 1 = 0.055 , P 2 G r M a c h C P L 1 = 0.074 (83)

Κ Mach 1 1 ld = 0.542 ; Κ Mach 1 2 ld = 0.603 ; Κ Mach 1 3 ld = 0.658 ; Κ Mach 2 1 ld = 0.842 ; Κ Mach 2 2 ld = 0.833 ; Κ Mach 2 3 ld = 0.852 ; Κ Mach 3 1 ld = 0.769 ; Κ Mach 3 2 ld = 0.797 ; Κ Mach 3 3 ld = 0.835 ; Κ Mach 3 4 ld = 0.83 ; Κ Mach 3 5 ld = 0.795 ; Κ Mach 4 1 ld = 0.409 ; Κ Mach 5 1 ld = 0.389 ; Κ Mach 5 2 ld = 0.338 ; Κ Mach 6 1 ld = 0.524 ; Κ Mach 6 2 ld = 0.511 ; Κ Mach 6 3 ld = 0.428 (84)

As part of management sub-level 1.3, the vehicle fleet utilization parameters were optimized for the calendar year. The state of the production system with regard to vehicle fleet utilization prior to optimization is characterized by the following target values (Equation 85):

R a v V e h C P L 1 = 0.392 , L o s g e n V e h a v C P L 1 = 0.849 , K G V e h C P L 1 = 0.372 (85)

Within management level 1.3, an optimization strategy has been defined (Equation 86):

K G V e h C P L 1 → ⁡ max (86)

Figure 4 shows a graph of Pareto-optimal solutions for control level 1.3. The graph depicts three sets of points representing the values of the objective functions for 23 non-dominant variant (Var_1- Var_23) of the allocation of individual transport operations among the corresponding vehicle units.

Taking condition (Equation 86) into account, the optimal vehicle fleet loading scenario for the calendar year is the first option, Var_1, since it yields the maximum value of the objective function K G V e h C P L 1.

Thus, the state of the production system in terms of the utilization of the vehicle fleet after optimization is characterized by the values of the target indicators (Equation 46) and the utilization coefficients of the corresponding vehicle units (Equations 87 and 88):

R a v V e h C P L 1 = 0.316 , L o s g e n V e h a v C P L 1 = 0.604 , K G V e h C P L 1 = 0.59 (87)

Κ Veh 1 1 ld = 0.49 ; Κ Veh 1 2 ld = 0.432 ; Κ Veh 1 3 ld = 0.658 (88)

At the first level of management, the target values Q i k M a c h C L P 1 and Q i k V e h C P L 1 were not calculated due to the lack of sufficient reliable data on failures of the relevant units of process equipment and vehicles.

At the second level of management, the utilization parameters of the process equipment fleet were optimized for the relevant quarters. The state of the production system in terms of equipment fleet utilization prior to optimization is characterized by the following target indicator values (Equation 89):

R a v M e c h / C P L 2 = 0.199 , I P / a v M e c h / C P L 2 = 0.102 , I L O S M e c h / C P L 2 = 0.059 (89)

An optimization strategy has been defined for this control level (Equation 90):

I P / a v M e c h / C P L 2 → ⁡ min (90)

Figure 5 shows a graph of Pareto-optimal solutions for the second level of control. The graph depicts three sets of points representing the values of the objective functions for eight non-dominant alternatives (Var_1- Var_8) for the allocation of technological operations among the corresponding calendar planning periods (quarters).

Taking condition (Equation 90) into account, the optimal option for loading the production equipment fleet within the relevant quarters is the third option Var_3, since it has the minimum value of the objective function I P / a v M e c h / C P L 2. Thus, the state of the production system in terms of the utilization of the process equipment fleet after optimization is characterized by the values of the target indicators (Equation 91):

R a v M e c h / C P L 2 = 0.13 , I p a v M e c h / C P L 2 = 0.068 , I P / a v M e c h / C P L 2 = 0.027 (91)

As part of the third level of management, the utilization parameters of the production equipment fleet were optimized for each calendar month. This paper presents the results of optimizing the utilization parameters of the production equipment fleet for the three calendar months comprising the first quarter. The state of the production system in terms of equipment fleet utilization prior to optimization is characterized by the following target values (Equation 92):

R a v M e c h / C P L 2 = 0.158 , I p a v M e c h / C P L 2 = 0.079 , I LOS M e c h / C P L 3 = 0.026 (92)

An optimization strategy (Equation 93) has been defined for this control level: (Equation 93):

I P / a v M e c h / C P L 3 → ⁡ min (93)

Figure 6 shows a graph of the Pareto-optimal options for allocating the workload of the production equipment fleet over the 3 months of the first quarter. The graph shows three sets of points representing the values of the objective functions for seven non-dominant alternatives (Var_1- Var_7) for the allocation of manufacturing operations among the corresponding calendar planning periods (calendar months).

Taking condition (Equation 93) into account, the optimal loading option for the process equipment fleet within the relevant calendar months of the first quarter is the sixth option, Var_6, since it has the minimum value of the target indicator I P / a v M e c h / C P L 3. Thus, the state of the production system in terms of the utilization of the process equipment fleet for the relevant calendar planning periods after optimization is characterized by the values of the target indicators (Equation 94):

R a v M e c h / C P L 3 = 0.102 , I P / a v M e c h / C P L 3 = 0.05 , I LOS M e c h / C P L 3 = 0.018 (94)

As part of management level 4.1, the utilization parameters of the process equipment fleet were optimized for the relevant workdays. This paper presents the results of optimizing the utilization parameters of the process equipment fleet for the workdays in the first calendar month. The state of the production system in terms of process equipment fleet utilization prior to optimization is characterized by the following target values (Equation 95):

R a v M e c h / C P L 4 = 0.138 ; I P / a v M e c h / C P L 4 = 0.08 ; I LOS M e c h / C P L 4 = 0.0118 (95)

An optimization strategy (Equation 96) has been defined for this control level:

I P / a v M e c h / C P L 4 → ⁡ min (96)

Figure 7 shows a graph of the Pareto-optimal loadings for the fleet of process equipment for the working days in the first calendar month. The graph shows three sets of points representing the values of the objective functions for eight non-domination alternat (Var_1- Var_8) for the distribution of operation sequences and the number of workpieces processed during the corresponding calendar planning periods (working days).

Taking condition (Equation 96) into account, the optimal loading option for the production equipment fleet is the seventh option Var_7, since it has the minimum value of the objective function I P / a v M e c h / C P L 4. Thus, after optimization, the state of the production system in terms of the utilization of the process equipment fleet for the corresponding calendar planning periods is characterized by the following target indicator values (Equation 97):

R a v M e c h / C P L 4 = 0.1 ; I P / a v M e c h / C P L 4 = 0.0705 ; I LOS M e c h / C P L 4 = 0.0112 (97)

As part of management level 4.2, the utilization parameters for the vehicle fleet were optimized for the relevant workdays. This paper presents the results of optimizing the vehicle fleet utilization parameters for the workdays in the first calendar month. The state of the production system in terms of vehicle fleet utilization for the corresponding calendar planning period prior to optimization is characterized by the following target values (Equation 98):

R a v v e h / C P L 4 = 0.282 , I P / a v v e h / C P L 4 = 0.082 , I LOS v e h / C P L 4 = 0.027 (98)

An optimization strategy (Equation 99) has been defined for this control level:

I P / a v V e h / C P L 4 → ⁡ min (99)

Figure 8 shows a graph of the Pareto-optimal load distribution options for the vehicle fleet on the working days of the first calendar month. The graph shows three sets of points representing the values of the objective functions for six non-dominant alternatives (Var_1- Var_6) for the allocation of individual transport operations among the corresponding periods of the calendar schedule (working days).

Taking condition (Equation 99) into account, the optimal vehicle fleet utilization plan for the relevant working days in the first calendar month is the first option, Var_1, since it has the minimum value of the objective function Var_1, Thus, the state of the production system in terms of the vehicle fleet utilization for the corresponding calendar planning period after optimization is characterized by the values of the objective functions (Equation 100):

R a v v e h / C P L 4 = 0.158 , I P / a v v e h / C P L 4 = 0.073 , I LOS v e h / C P L 4 = 0.022 (100)

As part of management level 5.1, the utilization parameters for the production equipment were optimized for the relevant work shifts. This paper presents the optimization results for the work shifts corresponding to the eleventh workday. The state of the production system in terms of the utilization of the process equipment fleet for the relevant calendar planning period prior to optimization is characterized by the following target values (Equation 101):

R a v M e c h / C P L 5 s , s + 1 = 0.047 , P L d / a v M e c h / C P L 5 s , s + 1 = 0.041 , δ L O S / a v M e c h / C P L 5 s , s + 1 = 0.042 (101)

An optimization strategy (Equation 102) has been defined for this control level:

P L d / a v M e c h / C P L 5 s , s + 1 → ⁡ min (102)

Figure 9 shows a graph of the Pareto-optimal options for allocating the workload of the process equipment fleet across work shifts for the eleventh workday. The graph shows three sets of points representing the values of the objective functions for five non-dominant options (Var_1- Var_5) for the distribution of sets of operations and the number of workpieces processed among the corresponding calendar planning periods (work shifts).

Taking into account condition (Equation 102), the optimal loading option for the fleet of process equipment within the relevant work shifts for the eleventh workday is the first option, Var_1, since it has the minimum value of the objective function P L d / a v M e c h / C P L 5 s , s + 1. Thus, the state of the production system in terms of the utilization of the production equipment fleet for the corresponding calendar planning period after optimization is characterized by the values of the objective functions (Equation 103):

R a v M e c h / C P L 5 s , s + 1 = 0.041 , P L d / a v M e c h / C P L 5 s , s + 1 = 0.027 , P L d / a v M e c h / C P L 5 s , s + 1 = 0.039 (103)

As part of management level 5.2, the vehicle fleet utilization parameters for the relevant work shifts were optimized. This paper presents the optimization results for the work shifts corresponding to the eleventh workday. The state of the production system with regard to the utilization of the vehicle fleet for the relevant calendar planning periods prior to optimization is characterized by the following target values (Equation 104):

R a v v e c h / C P L 5 s , s + 1 = 0.031 , P L d / a v V e c h / C P L 5 s , s + 1 = 0.028 , P L d / a v V e c h / C P L . 1 s , s + 1 = 0.019 (104)

An optimization strategy (Equation 105) has been defined for this control level:

P L d / a v V e h / C P L 5 s , s + 1 → ⁡ min (105)

Figure 10 shows a graph of the Pareto-optimal options for allocating the vehicle fleet’s workload across work shifts for the eleventh workday. The graph shows three sets of points representing the values of the objective functions for four non-dominant alternatives (Var_1- Var_4) for the allocation of individual transport operations among the corresponding calendar planning periods (work shifts).

Taking condition (Equation 105) into account, the optimal loading option for the fleet of production equipment within the relevant work shifts for the eleventh workday is the third option, Var_3, since it has the minimum value of the objective function P L d / a v V e h / C P L 5 s , s + 1. Thus, the state of the production system in terms of the utilization of the vehicle fleet for the corresponding calendar planning periods after optimization is characterized by the values of the objective functions (Equation 106):

R a v V e c h / C P L 5 s , s + 1 = 0.026 , P L d / a v V e c h / C P L 5 s , s + 1 = 0.024 , P L d / a v V e c h / C P L . 1 s , s + 1 = 0.015 (106)

4 Discussion

As part of this study, the problem of comprehensively optimizing the utilization parameters of production resources (the fleet of process equipment and the fleet of vehicles) was solved. The application of the developed model made it possible to conduct a comprehensive analysis of the efficiency of production resource utilization within the relevant calendar planning periods. The developed model served as the basis for the creation of a software module used to optimize the utilization parameters of production resources for the production unit under consideration. This software module was developed to automate the calculation of optimal equipment utilization, minimize downtime, and generate work schedules based on input data regarding orders, production capacities, and delivery deadlines for finished products.

As a result of the optimization carried out using the software module, the following results were obtained:

• By redistributing manufacturing operations among the relevant pieces of equipment, the overall efficiency of equipment utilization over the calendar year (K e f M a c h C L P 1) increased by 15.11%, and load imbalance was reduced within the milling (P 2 G r M a c h C P L 1) и surface grinding (P 5 G r M a c h C P L 1) equipment groups by 6.78% and 8.65%, respectively;
• Due to the redistribution of individual transport operations among vehicles, there was a reduction in load unevenness (R a v V e h C P L 1) and in the total losses resulting from underutilization of vehicles over the calendar year (L o s g e n V e h a v C P L 1), by 19.39%, and 28.86% respectively, as well as an increase in the efficiency of vehicle utilization relative to their load capacity (K G V e h C P L 1) by 37.06%;
• Due to the redistribution of process operations among quarters, there was a reduction in the load unevenness of the entire fleet of process equipment (R a v M e c h / C P L 2), as well as in the average load unevenness of individual units of process equipment by 34.68%. and 33,34% respectively (I P / a v M e c h / C P L 2). There was also a 54.24% reduction in the unevenness of the distribution of losses caused by process equipment downtime across quarters (I L o s M e c h / C P L 2) by 54.24%.

For management levels 3, 4, and 5, the optimization results are presented using the examples of the first quarter (third management level), the first calendar month (fourth management level), and the eleventh working day (fifth management level). Based on the optimization results within the respective management levels for the analyzed calendar planning periods, the following outcomes were obtained:

• By redistributing manufacturing operations among the calendar months of the first quarter, the load unevenness of the entire fleet of manufacturing equipment (R a v M e c h / C P L 3) and the load unevenness of individual units of manufacturing equipment (I P / a v M e c h / C P L 3) by 35.45%. and 36,71% respectively. There was also a 30.77% reduction in the unevenness of loss distribution caused by process equipment downtime across the calendar months of the first quarter (I L o s M e c h / C P L 3) by 30.77%;
• By redistributing the sets of operations and the number of parts processed during the corresponding workdays of the first calendar month, the unevenness in the utilization of the entire fleet of manufacturing equipment (R a v M e c h / C P L 4) and the unevenness in the utilization of individual units of manufacturing equipment (I P / a v M e c h / C P L 4) by 27.54%. and 11,88% respectively. There was also a 5.09% reduction in the unevenness of loss distribution caused by equipment downtime (I L o s M e c h / C P L 4).
• By redistributing individual transport operations across the working days of the first calendar month, the load unevenness of the entire vehicle fleet (R a v V e h / C P L 4) and the load unevenness of individual units of process equipment (I P / a v V e h / C P L 4) by 43.58%. and 10.98% respect There was also an 18.52% decrease in the variation in the distribution of losses caused by vehicle downtime between working days in the first calendar month (I L o s V e h / C P L 4).
• By redistributing the sets of operations and the number of parts among the work shifts on the eleventh workday, the load unevenness of the entire fleet of process equipment (R a v M e c h / C P L 5 s , s + 1) and the load unevenness of individual pieces of process equipment (P L d / a v M e c h / C P L 5 s , s + 1) by 12.77% and 36.59%, respectively. There was also a 7.15% reduction in the unevenness of the distribution of losses caused by process equipment downtime (δ L o s / a v M e c h / L C P 5 s , s + 1).
• By redistributing individual transport operations among the work shifts on the eleventh workday, the load unevenness of the entire vehicle fleet (R a v V e h / C P L 5 s , s + 1) and the load unevenness of individual vehicles (P L d / a v V e h / C P L 5 s , s + 1) by 16.13% and 14.28%, respectively. There was also a 21.05% reduction in the unevenness of the distribution of losses caused by vehicle downtime between work shifts on the eleventh workday (P L d / a v V e h / C P L . 1 s , s + 1).

The proposed hierarchical model for managing the process of planning the parameters of production asset loading not only expands the toolkit of scheduling but also overcomes a number of fundamental limitations typical of classical and modern approaches:

• Integration of heterogeneous resource elements: overcoming the narrow specialization of models. Traditional loading models, developed within the framework of scheduling theory and production planning, have historically focused on optimizing a single type of resource: machine tools (Mandal et al., 2010; Tiwari et al., 2010; Ozpeynirci and Azizoglu, 2010; Swarnkar and Tiwari, 2011; Sharma and Singh, 2012; Li et al., 2014; Kumar et al., 2016; Zhang et al., 2020; Motlagha et al., 2013), vehicles (Lian et al., 2015; Zhang et al., 2019; Wang et al., 2020; Zhang et al., 2020; Xin et al., 2023; Przybył et al., 2021), and storage systems (Ruiz et al., 2010; Geng et al., 2020; Istokovic et al., 2020; Li et al., 2020; Centobelli et al., 2016; Ziaee et al., 2016). This approach ensures high accuracy in a narrow subject area, but leads to fragmentation of the production system and the loss of synergy between heterogeneous resources.

In studies on integrated planning, attempts have been made to jointly consider several types of resources; however, these have typically been reduced to sequential or loosely coupled optimization (i.e., first machines, then vehicles) (Herlina et al., 2022; Hrabec et al., 2022; Low et al., 2012; Trstenjak et al., 2024; Dan et al., 2024; Nikolopoulu et al., 2012; Issaa et al., 2017; Lu, Singh et al., 2025; He et al., 2022; Dan et al., 2025; Hanbali et al., 2016; Mota et al., 2025; Sun et al., 2024; Przybył et al., 2021). The proposed model is unique in that it is designed from the outset as a single system, where optimization of the loading parameters for three types of production assets occurs within a coordinated hierarchical structure, allowing for cross-functional constraints and interdependencies to be taken into account in real time. This level of integration aligns with modern trends in integrated supply chain and production planning, but is implemented at a more detailed level-at the level of scheduling the loading of specific types of production assets.

• 2. A clear hierarchy of management sublevels: a transition from time intervals to functional-resource decomposition. Classic scheduling systems (APS - Advanced Planning Systems) and MRP/ERP approaches traditionally structure management levels by time horizons: strategic, tactical, and operational. In this case, the functional differences between resource types are often leveled out, and management is reduced to the allocation of time slots. As a result, the specific operational characteristics of different types of production assets are lost. Research in the field of hierarchical planning (Alford et al., 2009; Baker et al., 2012; Baker rt al., 2018; Bercher et al., 2019; Shivashankar et al., 2012; Smith et al., 2008; Wang et al., 2020) suggests decomposition by levels of detail, but does not always link it to resource types. The proposed model develops this approach by establishing a clear hierarchy of sublevels, where each level corresponds to a specific type of production asset. This allows for a rational optimization sequence that takes into account the technological and operational characteristics of each resource type and maintains end-to-end interconnections between levels. This approach is closer to the principles of multi-level planning in modern production systems, but is adapted to the task of scheduling, taking into account the specific loading characteristics of different types of equipment.
• 3. Modularity and scalability: structural flexibility as the basis for adaptation. Many existing planning models have a rigid architecture, which complicates their adaptation to changing production conditions or an expanded resource range. Studies on flexible manufacturing systems (FMS) and reconfigurable manufacturing systems (Azab et al., 2012; Benyoucef et al., 2016; Bortolini et al., 2013; Koren et al., 2011; Liu et al., 2018; Naderi et al., 2014; Tiwari et al., 2015; Wang et al., 2017; Youssef et al., 2010) emphasize the importance of adaptability, but the emphasis is on physical reconfiguration rather than the structural flexibility of planning models.

The proposed modular structure allows for the expansion of the system’s functionality by introducing additional control sublevels, each responsible for optimizing the loading parameters of a new type of production equipment. This is consistent with the principles of modular design in information systems and production planning (Li et al., 2012, Liu et al., 2022; Mejía et al., 2018; Wang et al., 2015; Zhang et al., 2020; Van de Ven et al., 2023), but applied to the problem of hierarchical scheduling. This scalability is particularly relevant in the context of digitalization and the implementation of flexible production lines, where the composition of equipment can change dynamically.

• 4. A multi-level system for comprehensive performance assessment: from unified indicators to vector criteria. Traditional approaches to assessing the performance of manufacturing systems often rely on a single set of key performance indicators (KPIs), such as equipment utilization, cycle time, or defect rate (Bang et al., 2019; Dominguez et al., 2019; Kontar et al., 2016; Slack et al., 2019; Van Looy and Shafagatova, 2016). While this simplifies monitoring, it fails to account for the structural heterogeneity of the manufacturing system and can lead to local optima, to the detriment of overall performance.

Research on multi-objective optimization (MCO) and the balanced scorecard (BSC) recognizes the need to consider multiple objectives; however, their application in scheduling remains limited. The proposed multi-level performance assessment system generates unique vector optimization criteria for each management sublevel, allowing for the specifics of a particular type of production asset to be taken into account and ensuring.

Thus, the novelty of the developed hierarchical model for the comprehensive optimization of production asset utilization parameters lies in the following:

• A unified integrated structure for optimizing heterogeneous production resources. Unlike traditional approaches with sequential optimization of individual resource types (machine tools, transport, storage), the model simultaneously accounts for cross-functional constraints and interrelations of three types of production assets within a coordinated hierarchical structure, enabling synergies and avoiding fragmentation of the production system.
• A temporal-resource decomposition of management levels instead of a time-based one. The novelty lies in the transition from classical temporal structuring (strategic, tactical, operational levels) to a hierarchy of sublevels, each of which is strictly associated with a specific type of production asset, preserving and accounting for the technological and operational characteristics of each resource.
• A rational optimization sequence determined by the specificity of the resources. The model formalizes the optimization order at different hierarchical levels, taking into account the process dependencies and operational limitations of specific equipment types, ensuring a logically sound and practically feasible decision-making sequence.
• Modular architecture focused on dynamic resource expansion. Unlike the rigid architectures of existing planning systems, the proposed model has a modular structure, where each new type of production asset is added as a separate management sublevel, ensuring structural flexibility and adaptation to changing production conditions.
• Scalability at the scheduling level. The novelty lies in the application of modular design principles not to strategic or tactical levels, but directly to operational scheduling, allowing the system to scale as the equipment mix changes without redesigning the entire model.
• Vector performance evaluation criteria specific to each resource type. Instead of a single set of KPIs for the entire system, the model generates unique vector optimization criteria for each management sublevel, allowing for the heterogeneity of the production system to be taken into account and local optima to be avoided at the expense of overall efficiency.
• Balanced decisions at all hierarchical levels through a multi-level evaluation system. The proposed criteria system ensures alignment of individual sublevel goals with the overall goals of the production system, generating balanced decisions that consider both local and global performance indicators.
• Comprehensive synthesis of key production planning areas within a single methodology. The model’s novelty lies in its unification of elements of scheduling theory, integrated and hierarchical planning, modular design, and multi-criteria optimization into a holistic scheduling methodology that overcomes the limitations of highly specialized approaches and ensures more balanced management of production asset utilization.

Implementation of the developed optimization model will lead to the following practical consequences:

• Reduction in production cycle time by eliminating interfunctional gaps. Integrated accounting of machine tools, transport, and storage systems within a single structure makes it possible to minimize downtime between operations caused by schedule inconsistencies between individual services. In practice, this translates into reduced holding time for semi-finished products and accelerated order flow through the production chain;
• Increased overall production capacity utilization while reducing peak overloads. The functional-resource hierarchy allows for the redistribution of load from bottlenecks to reserve capacity of related resource types, while vector criteria help balance the load without creating new localized overloads. As a result, equipment utilization increases without increasing the number of emergencies;
• Reduction in the volume of work-in-progress (WIP) and the need for warehouse space. Coordinated planning of transportation and warehouse operations with the production schedule enables the adoption of just-in-time principles at the scheduling level. This directly reduces the volume of work-in-process inventory and frees up warehouse space for other tasks.
• Optimization of transportation routes and reduction of idle runs. Since transportation is considered an equal resource in the overall model, its routes and schedules are generated synchronously with machine operation. In practice, this leads to a reduction in idle runs, lower fuel consumption, and reduced equipment wear.
• Accelerated adaptation of the production system to changes in orders and product range. The modular architecture of the model allows for the prompt addition of new equipment types or the elimination of decommissioned units without restructuring the entire planning system. This significantly reduces response times to changes in the order portfolio or the launch of new products.
• Reduction in the number of missed order fulfillment deadlines. A comprehensive performance assessment and balanced decisions at all levels of the hierarchy enable the creation of realistic schedules that take into account all key constraints. As a result, delivery delays due to unforeseen overloads or shortages of transport vehicles are reduced;
• Simplifying the process of coordinating plans between various departments of the enterprise. A unified methodology and common evaluation criteria create a common language for production, logistics, and warehouse personnel. This reduces the number of iterations in plan coordination, reduces administrative costs, and increases the transparency of decision-making;
• Improving the validity of investment decisions regarding the purchase of new equipment. A multi-level performance evaluation system allows for a quantitative assessment of the contribution of each resource type to the overall system performance. In practice, this helps to more accurately identify bottlenecks and make informed decisions about which equipment types (machines, transport, or storage systems) should be invested in first;
• Reducing the labor intensity of operational management and the number of plan adjustments. Thanks to more accurate and balanced initial planning, the need for manual adjustments during plan execution is reduced. Dispatchers spend less time “putting out fires,” freeing up resources for strategic tasks and reducing the risk of human error;
• Improved transparency and manageability of production processes for management. Vector criteria and a multi-level assessment system provide detailed analytics for each resource type while maintaining the ability to aggregate overall performance. This allows management to make management decisions based on more comprehensive and structured information, as well as more effectively monitor the achievement of production goals.

5 Conclusion

The study, the results of which are presented in this paper, focused on developing a methodological framework for improving the efficiency of the process of planning the loading parameters of production assets (process equipment, transportation and storage facilities). The results of the study are as follows:

• A comprehensive structural and parametric analysis of the process of planning the loading parameters of production assets (process equipment, transportation and storage facilities) was conducted.
• Sets of structural elements of the process of planning the loading parameters of production assets were formed, characterizing its structural and hierarchical complexity in terms of the typical diversity of elements.
• A hierarchical model for managing the process of planning the loading parameters of production assets was developed, within which clear relationships between management levels and sublevels were established. The established hierarchy of levels and sublevels was justified.
• Vector optimization criteria were developed for each management level, which are the basic elements of a multi-level system for assessing the efficiency of the loading parameters of production assets.

Thus, the study allowed for a comprehensive analysis of the process of planning the loading parameters of production assets. The identified structural and hierarchical complexity of the process made it possible to clearly identify its key elements, systematize their diversity, and establish clear relationships between them. The developed hierarchical control model defines a strict sequence for optimizing system elements and creates the basis for a multi-level assessment of its effectiveness. The obtained results form a methodological basis for improving the efficiency of planning production capacity utilization parameters in practice.

Implementation of the developed optimization model will:

• Reduce downtime and improve the efficiency of production capacity utilization;
• Clear distribution of tasks between management levels and sublevels will reduce the time it takes to make decisions on capacity adjustments;
• Implementation of a multi-level performance assessment system will enable the prompt identification of bottlenecks in the production process.
• Improved capacity planning accuracy will reduce costs for unscheduled repairs and maintenance due to equipment overload;
• Automation of calculations based on the developed model will reduce the impact of the human factor and the number of errors in the planning process;
• Improving coordination between departments through a unified production capacity management model will improve overall enterprise productivity;
• Implementation of the proposed model will create the foundation for the gradual digitalization of planning processes and integration with ERP and MES systems, which will ultimately enable the transition to predictive analytics and the automatic generation of load schedules.

The developed hierarchical model for managing the process of planning production asset load parameters differs from previous approaches in several key aspects that enhance its efficiency and adaptability.

Future research will focus on testing the model within structurally diverse multi-industry supply chains, which will help generalize the mathematical dependencies between disparate assets, confirming its viability beyond standard engineering configurations.

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Supplementary material

The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/fmech.2026.1885264/full#supplementary-material

Supplementary material Supplementary material at the source

Footnotes

Edited by: Viet Q. Vu, Thai Nguyen University of Technology, Vietnam

Reviewed by: Lina Gozali, Tarumanagara University, Indonesia

Ahmed Raecky Baihaqy, Sepuluh Nopember Institute of Technology, Indonesia