Complexity
Volume 2019 (2019), Article ID 7237585, 36 pages
https://doi.org/10.1155/2019/7237585

Research Article

Comparative Simulation Analysis of the Performance of the Logistics Manufacturing System at the Operative Level

Robert Bucki and Petr Suchánek

Department of Informatics and Mathematics, Silesian University in Opava, School of Business Administration in Karvina, Karviná 73340, Czech Republic

Correspondence should be addressed to Petr Suchánek; suchanek@opf.slu.cz

Received 14 January 2019; Revised 27 March 2019; Accepted 1 April 2019; Published 7 May 2019

Academic Editor: Lucia Valentina Gambuzza

Copyright © 2019 Robert Bucki and Petr Suchánek. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The article focuses on one of the current problems of manufacturing systems which consist of individual machines equipped with dedicated tools that are replaced when they are worn out. It is assumed that the machines are located within the reach of the robotic arm which carries out transport operations of semifinished products to designated production machines and storage containers in accordance with the production time period. The aim is to find such an arrangement of production activities, respectively, production paths for a given set of orders that will be effective from the time and cost point of view. Moreover, the whole issue is solved with regard to possible failures of individual stands, overfilling of some tanks, etc. The theory and practice of creating and using simulators as tools for the definition and verification of production plans are used to solve this issue. The starting point is the creation of a mathematical simulation model with the necessary but acceptable degree of simplification. The mathematical simulation model is tested on sample data in a feasibility study to perform a detailed usability analysis of the model. The output of the article is a simulation model for which, based on the analysis of simulation results, patterns of possible use in specific types of enterprises are given.

1. Introduction

A key trend in the area of business activities is the effort to optimise and thus increase the efficiency of individual processes. The reason for this is a long-established relationship between the efficiency of internal and external processes, their management and the competitiveness of the business [1]. The current rapid development of the business environment requires that process efficiency measurements and evaluations are carried out at increasingly shorter intervals and that the result of their optimisation meets the principles of a lean company concept [2] to comply with standard business process management approaches [3].

Analyses are performed using data and information that are the outputs of enterprise information systems. Not all of these systems are capable of providing the necessary information in sufficient quality and within a short time, and, in particular, knowledge of interrelationships and relationships about the current status of each business process, the enterprise as a whole, and enterprise behavior in the corporate environment. Companies are aware of this fact and therefore are willing to invest considerable funds in modern information systems and their modules [4]. As it can be seen from a wide range of sources and from real practice an adequate information system allowing a sufficient number of modern management support functionalities is one of the key conditions for promoting competitiveness [5] and after its introduction or implementation, its positive benefits can be observed in the first weeks after its introduction [6]. One of the conditions of a sufficiently strong competitiveness is meeting the needs of all business activities of the company in accordance with the principles of the so-called lean company concept [7] and constantly monitoring compliance with all its principles [8].

To sum up, we can say that choosing an appropriate information system has long been an important factor for businesses of all types. Due to the usually high investment in acquisition or expansion of the information system, it is necessary to pay close attention to the input analysis so that the resulting functionality meets all expectations [9]. Completely new requirements are emerging with the development of Industry 4.0. Automation, robotics, implementation of cobots, etc. require completely new approaches in the field of digitisation and management of enterprise informatics [10]. Also, business information systems must be tailored to this [11] as they are expected to have the highest possible degree of automation which is dependent on fast data processing [12]. This technological development necessitates a focus on human resources which must be sufficiently competent to be able to integrate into the company’s comprehensive information system and respond in the right way at the right time [13]. The synergy of people and technology is the key prerequisite for the smooth running of the business. Therefore, an important aspect is the training of human resources, so that workers at all levels of the company (from employees to top management) are able to control, manage, and use new technologies [14]. A corporate information system as a comprehensive integrated business system can be described as optimal when the management system outputs show a high degree of efficiency, for example, through the appropriate sorting and ordering of orders, continuous production lines, correct production rerouting in the event of a machine failure, etc. This happens either automatically or depends on a human decision. Increasing decision-making efficiency can be achieved by implementing software modules to simulate system behavior after regulatory intervention [15]. Simulators, as we call these software modules, are a very powerful tool to reduce the failure of managerial decisions and thus reduce the costs that make bad decisions generally. Ideally, simulators are separate modules integrated into information systems so that they can easily define mathematical simulation models using current databases. Of course, it is also possible to implement the simulation approach in control systems with artificial intelligence support and fully automatic control systems [16] with which we will meet more and more often within Industry 4.0.

Based on the above, one of the key areas of informatics is the problem of creation of simulators. The main idea and goal is to create simulators that can be used to create plans and to validate these proposed plans before starting their subactivities. The basic starting point for creating a simulator is a mathematical model that represents, with a corresponding degree of simplification, the structural, functional and data architecture of the system [17]. In particular, in a manufacturing company, a simulation model can represent the flow of resources and products during the manufacturing process. It is advisable to implement the creation of a mathematical model of complex manufacturing system within a project whose starting point is a detailed feasibility study containing precise assumptions and specifications. The examples that can be found in professional and scientific sources include development of the mathematical model for the scheduling of production process and the allocation of the automatic guided vehicle in the flexible manufacturing system [18] or creating a mathematical programming model for a Kanban-controlled cellular manufacturing system with configuration capability [19]. The next step while creating a simulator is its programming followed by testing. Testing and functional verification of the simulator are important at every stage of the simulator’s development. The above-mentioned feedback activities require adequate data, the source of which is the information systems of particular companies. The specific simulator procedure used in this article is presented in Section 3. A variety of methods, methodologies, and approaches can be used to create mathematical simulation models. The use of the process-oriented approach is presented by, e.g., [20, 21], whereas examples of value-oriented approach specifically for the supply chain modelling area are presented by, e.g., [22, 23]. Furthermore, there is a growing interest in the use of multiagent systems. The background to this topic is, inter alia, described in [24], the example in the form of negotiation model and tactics of manufacturing enterprise supply chain based on multiagent is presented in [25], and the example presenting agent-based modelling and simulation of decision-making behavior of e-retailers is shown in [26]. Another useful theme for modelling is using discrete or continuous Petri nets, for example, to model and simulate a logistics production system [27], discrete manufacturing systems [28], and flexible manufacturing systems [29]. Modelling and simulation are implemented in many cases to optimise the logistics system. A fuzzy approach can be used for the purposes of optimisation in relation to the problem of simulation presented, e.g., in [30], for a production system, e.g., in [31]. The heuristic approach appears to be effective in case of the so-called hard systems and can be applied, for example, in conjunction with time series characterising selected systems [32], for e-commerce processes [33], for resource planning and management needs in flexible manufacturing systems [34], or, for example, to optimise manufacturing and distributed systems [35]. Hybrid approaches for modelling and optimising manufacturing and other systems can be found, e.g., in [36–38], the use of neural networks, e.g., in [39]. It is possible to find the whole series of specific examples of theoretical background and examples of the so-called good practice. The essence is always to create corresponding models with simulation outputs usable for decision-making processes, preferably with the widest possible elements of intelligence [40].

The optimisation of production logistics as a complex of all direct and indirect activities of the company plays an increasingly important role in relation to the production of goods, production costs, used technologies, production organisation, etc. [41]. The diversity and complexity of production processes can be better modelled by providing an integrated view of different modelling approaches [42]. Logistics systems in manufacturing plants are often very extensive and their optimisation is conditioned by the evaluation of a number of parameters and their context. Specifically, when we focus on production logistics, which is the main subject of this article, we conclude that logistics financial costs are a budget cost item for all types of businesses. Therefore, the goal of every business is to minimise these costs. The aim of the article is to present a computer science approach to solving the problem of modelling a model company that works with random logistic costs, which alternate according to the current set of requirements flowing from individual orders. A heuristic approach is used for the solution of this model example which can be easily implemented to optimise the production logistics system. It is based on the experience of the authors who present examples of theoretical and practical bases, for example, in [43], [44], [45], or [46] who provides satisfactory decision support.

Logistics systems may be similar in nature; however, even a small difference in assumptions can lead to the need for preparing a new information solution to the problem of meeting customers’ needs. Moreover, logistics systems are characterised by various limitations and their criteria are not often the same. These factors require autonomous solutions leading to forming new specification assumptions and a project resulting in offering the adequate simulator for solving specific logistics tasks.

Due to the fact that there are a number of different influencing factors in production and logistics systems the heuristic approach allows a number of experiments to be carried out on a continuous basis in order to achieve a satisfactory solution at the operation level which will ultimately result in a reduction in financial costs. The study presented in the article concerns the problem of manufacturing logistics which is important from the point of view of minimising manufacturing costs. This kind of approach is omnipresent in the contemporary world which is becoming increasingly globalised. Companies which either wish to continue to exist or become competitive need to implement more and more sophisticated solutions.

The main goal of the article is to introduce the production model of a production company usable for production strategy simulations. Heuristic algorithms that always determine a given production strategy are defined as part of the model. The goals of the paper also include preparing the specification assumptions and the model for creating a simulator of the complex manufacturing system. Subsequently, this model is to be implemented to carry out a series of simulation experiments which are then subject to the evaluation process in the case study section.

The article is divided into the main sections, i.e., Introduction, Methodology, Analysis, Results and Discussion, and Conclusion. The Introduction section presents the state-of-the-art modelling and simulation of logistics systems. The Methodology section is devoted to presenting the way in which the paper was prepared in order to meet the research objectives. The Analysis section shows the mathematical model step by step as well as the control of the system. Finally, Results and Discussion section includes the study case where the way of simulating manufacturing processes is shown in detail which is followed by adequate discussion. The Conclusion section explains the research novelty, research limitations, contributions of the study for academics and practices, and recommendation for future studies.

2. Methodology

The article focuses on modelling the manufacturing system. The input of the model is the so-called order matrix in which orders from individual customers are stored. Each element of this matrix represents the customer and their specific requirement. Each order has predetermined resources. Resources are passed by a specified route in a production system assembled from a set of workstations performing specific operations on semifinished products and the final product is progressively made by individual machines wherein the manufacturing route is determined by a heuristic algorithm. There are storage capacities and storage facilities for intermediates, called buffers, along the production line. Buffer capacity is predefined and there is a possibility of temporary limitation of the site’s operation in case of insufficient storage space. Although we consider series production, specifications, project, and subsequent software assume the possibility of extending the system to parallel-arranged production lines. To satisfy the lean approach requirements, heuristic algorithms select a product to meet that criterion. The principle of operation forms the basis for creating a simulator of a modelled manufacturing system. However, there are manufacturing strategies that determine the starting point of creating an order matrix. In addition, costs are taken into account for demonstration needs. The costs are divided into those that can be strictly determined before the start of the production process and those that can be at least predicted before the manufacturing process.

To analyse the problem of minimising logistics costs it is necessary to follow the iterative model for creating the simulation tool. First of all, analysis of the real environment is the basis for the specification assumptions which lets the project of the simulator be prepared. Testing and subsequent validation of the simulator precede its implementation into the case study procedures. Figure 1 illustrates the research process of the study in detail.

Figure 1: The iterative model for creating the simulator of the logistics system.

The simulator of the logistics manufacturing system was designed in the form of three integrated modules: the data entry module, the simulation module, and the results module. In addition, each of the modules includes different working modes. The data entry module modes differ with the source of data; they can be input with a keyboard, generated randomly with or without the seed value or loaded directly from a file. The simulation module allows the operator of the system to perform a single simulation and experiments at random or with a set of heuristics as well as step-by-step simulations. The results module allows the operator to analyse and compare results depending on the number of performed simulations as well as their type. The simulator was written in the C# programming language, .NET Framework 4 and Microsoft Visual C# 2012 Express programming environment. A DataGridFunctions library was created for the needs of the system. The library consists of two classes including the set of static methods working on arrays and DataGridView objects. It implements exception handling allowing an easier detection of incorrect input data. During the system testing the administrator and user manuals were created, including instructions for the installation process and the typical work with the system, showing its protection mechanisms against incorrect input data. Results of practical examples were analysed thoroughly. Moreover, they were confronted with the simulation results which allowed verifying the correctness of the system calculations. Testing the simulator required creating the set of initial data from the range of reasonable values based on thorough observations carried out in a few manufacturing plants. However, for the purpose of the simulation process there was a need for the initial data simplification. The verification result was positive so all the values returned by the system may now be considered to be correct. The data for the subsequent simulation process were generated within the assumed ranges. The data generation process can be replicated without any limitations. Once the data is generated it can be optionally modified in order to test alternative possibilities of making order matrix elements.

3. Analysis

3.1. Mathematical Model

Symbols used for logistic system modelling presented in this section are explained in detail in the symbol list at the end of this paper. They remain in accordance with the standards of mathematical models used for the description and simulation of logistics, production, trade and other systems.

It is assumed that the manufacturing system consists of the defined number of logistics manufacturing subsystems which are placed at various locations. Each -th subsystem, is equipped with the same number of machines which use the same type of tools. Each -th subsystem is also arranged in an identical way consisting of the charge zone, manufacturing hall, storing place for tools, storing place for semiproducts, and dispatching zone from which products are passed to the order matrix. It is also assumed that any element of the order matrix can be made on certain conditions in any manufacturing -th subsystem if it is available at a required moment.

It is necessary to assume that customers set orders which are gathered together mathematically in the following matrix of orders (1):

where is the state of the -th order set by the -th customer at the -th stage whereas is the total number of system decisions necessary to produce all order matrix elements approved for manufacturing at the stage .

However, in a real manufacturing subsystem machines are placed at different locations within the range of the robotic arm in accordance with the predefined location arrangement. Each machine performs only one type of operation with the use of a predefined tool.

Charges of a certain type which are necessary for the manufacturing process are supplied on the demand basis with no delay. Each -th product type, , is associated with a certain type of charge.

It is further assumed that there is the dedicated robotic arm in each π-th subsystem, which is responsible for the following operations:

(i)taking the charge material from the entrance gate of the logistics subsystem to the determined available machine;(ii)transporting semiproducts from the preceding machine to the determined subsequent one;(iii)transporting ready products to the dispatching gate respectively to the store for ready products;(iv)transporting semiproducts to their dedicated storing zone and back to the manufacturing zone;(v)manipulating with semiproducts in their storing zone;(vi)replacing a used tool with a new one.

The information point of view makes us present the sample manufacturing system in its simplified form in which each existing machine is to be associated with a certain defined point. This leads to creating the general matrix of structure of the manufacturing system which takes the following form:

where is a machine located in the -th row of the -th column in the -th subsystem dedicated to performing a manufacturing operation on the -th order for the -th customer with the use of a certain type of tool.

At the same time, if the -th order for the -th customer can be made in the machine placed in the -th row of the -th column in the -th subsystem; otherwise.

Let us present the matrix of adjustment of tools to operations in the following form:

where represents the -th tool able to perform the -th operation.

At the same time if the -th tool can perform the -th operation; otherwise.

It is assumed that a robotic arm is equipped with the required function enabling it to carry out the adequate operation. When a tool needs to be replaced, the machine is excluded from the manufacturing process for the time of replacement.

It is assumed that a certain machine is equipped with a dedicated tool only which is shown by means of the adjustment matrix (4):

where is the adjustment of the -th tool to the machine placed in the -th row of the -th column.

At the same time, if the adjustment is possible; otherwise.

Let us introduce the vector of buffer zones in the -th manufacturing subsystem (5):

where is the buffer zone behind the machine placed in the -th column in the -th manufacturing subsystem.

At the same time, if the buffer zone behind the machine in the -th row of the -th column which exists in the -th manufacturing subsystem exists; otherwise. Moreover, for calculation reasons it is assumed that the flow capacity of each buffer store is sufficient to ensure uninterrupted progress of the manufacturing process.

The matrix of production times in the -th manufacturing system is introduced in

where is the production time of the -th type of product for the -th customer in the machine placed in the -th row of the -th column in the -th subsystem.

The matrix of assembling times of tools in the -th subsystem is introduced in

where is the assembling time of the tool in the machine placed in the -th row of the -th column in the -th subsystem.

The matrix of disassembling times of tools in the -th subsystem is introduced in

where is the disassembling time of the tool in the machine placed in the -th row of the -th column in the -th subsystem.

The matrix of transportation times for semiproducts from machines to the key manipulation point in the -th subsystem is introduced in

where is the transportation time of products from the machine placed in the -th row of the -th column to the key point in the -th manufacturing system.

At the same time, if there is a transport operation in the -th manufacturing system; otherwise.

The matrix of transportation times for semiproducts from the key point to machines in the -th subsystem is introduced in

where is the transportation time of products from the key point to the machine placed in to the machine placed in the row of the column in the -th manufacturing system, . Moreover, in case of the product leaving column it is assumed it is directed to the key transport point before sending it to the store for ready products .

The matrix of transportation times for semiproducts from machines to their buffer stores in the π-th subsystem is introduced in

where is the transportation time of a product from the machine placed in the -th row of the -th column to the -th buffer store in the -th manufacturing system.

If there is no transport operation carried out to the -th buffer store due to its nonexistence then .

The matrix of transportation times for semiproducts from buffer stores to machines in the -th subsystem is introduced in

where is the transportation time of a product from the -th buffer store to machine placed in the -th row of the column in the -th manufacturing system.

If there is no transport operation carried out from the -th buffer store due to its nonexistence then .

The matrix of unit transportation times for charge material from the store of charges to machines in the -th subsystem is introduced in

where is the unit transportation time of the charge material to the machine placed in the -th row of the -th column in case of making the -th order for the -th customer in the -th manufacturing system.

If the charge material is not transported to the machine placed in the -th row of the -th column in case of making the -th order for the -th customer in the -th manufacturing system then .

The matrix of unit transportation times for ready products to the product store in the π-th subsystem is introduced in

where is the unit transportation time of the ready product from the machine placed in the -th row of the -th column to the product store in case of making the -th order for the -th customer in the -th manufacturing system.

If the ready product is not transported to the product store from the machine placed in the -th row of the -th column in case of making the -th order for the -th customer in the -th subsystem then .

The matrix of times of storing charge materials in the -th subsystem takes the following form:

where is the time of storing the -th charge material for making the -th product for the -th customer in the charge store for the -th plant.

The matrix of times of storing semiproducts in buffer stores in the -th subsystem takes the following form:

where is the average storing time of the -th semiproduct made for the -th customer in the available buffer store in the -th manufacturing subsystem.

The matrix of times of storing ready products in the π-th subsystem takes the following form:

where is the storing time of the -th ready product made for the -th customer in the ready product store of the -th manufacturing system.

The matrix of parallel manufacturing times is introduced in

where

  is the parallel manufacturing time of making the -th order for the -th customer and the -th order for the -th customer in the -th subsystem with the use of the -th criterion;  is the moment of beginning making the -th order for the -th customer and the -th order for the -th customer in the -th subsystem with the use of the -th criterion;  is the moment of finishing making the -th order for the -th customer and the -th order for the -th customer in the -th subsystem with the use of the -th criterion.

At the same time if then and if then . The matrix of times of replacement operations is introduced in

where is the replacement operation time of the -th tool in the machine placed in the -th row of the -th column in the -th subsystem.

At the same time, if the replacement operation of the -th tool in the machine placed in the -th row of the -th column in the -th subsystem takes place; otherwise.

Moreover, there is a need to introduce the matrix of transportation times of new tools which are stored in the storing place for tools in the -th subsystem (20):

where is the transportation time of the new -th type tool from the storing place to the machine placed in the -th row of the -th column in the -th subsystem. At the same time, if the transportation to the discussed machine takes place; otherwise.

Additionally, there is a need to introduce the matrix of transportation times for worn out tools to the storing place for used tools in the -th subsystem (21):

where is the transportation time of the worn out -th type tool from the machine placed in the -th row of the -th column to the storing place for worn out tools in the -th subsystem. At the same time, if the transportation from the discussed machine to the storing place for worn out tools takes place; otherwise.

The base life matrix of tools used in the manufacturing system of a new brand set of tools used to manufacture elements of the order matrix is given in

where is the base number of units of the -th product which can be manufactured in the machine in the -th row of the -th column before the -th dedicated tool is completely worn out and requires an immediate replacement. It is assumed that the life of a new tool is the same in each -th subsystem.

Let us introduce the matrix of state of the -th manufacturing subsystem (23):

where is the number of units of the -th product already made in the machine in the -th row of the -th column with the use of the -th tool in the -th subsystem at the -th state; if the tool is regarded as brand new.

Let us introduce the capacity matrix of the π-th manufacturing subsystem (24):

where is the number of the -th product units which still can be made in the machine in the -th row of the -th column with the use of the -th tool in the -th subsystem at the -th state. If the flow capacity of the machine does not allow to make the whole -th unit, then . If there is any remaining flow capacity in the machine in the -th row of the -th column but the subsequent unit of the -th product cannot be made fully in this machine in the -th subsystem, then the replacement process is carried out automatically.

On the basis of the above assumptions we can determine the flow capacity of the machine in the -th row of the -th column in the -th subsystem for the -th element of the order matrix (25):

Let us introduce the matrix of routes (for the case when the order of manufacturing operations is obligatory) (26):

where is the number of the -th row through which the -th semiproduct for the -th customer is passed throughout the manufacturing process.

Throughout the course of manufacturing process the state of the system changes. Therefore, there is a need to analyse the state of the order matrix, tools in machines, charge materials, storing zones, etc. Equations of state illustrate changes in the manufacturing system after each decision made by the operator of the system.

The state of the order matrix changes after each production decision (27):

The order matrix is modified at each k-th stage as follows:

The state of the -th manufacturing system is subject to change if a production decision is made (29):

The state of the -th tool in the -th machine in the -th column changes in case of manufacturing the -th product in the -th subsystem:

The state of the machine in the -th row in the -th column changes in case of replacement as follows:

Let us introduce the matrix of unit manufacturing costs in the π-th manufacturing subsystem (32):

where is the unit cost of manufacturing the -th order for the -th customer in the -th machine in the -th row in the -th manufacturing subsystem.

The cost of manufacturing the order in the -th manufacturing subsystem is calculated as follows:

Let us introduce the matrix of probability coefficients of failure of tools in the -th manufacturing subsystem (34):

where is the probability coefficient of failure of tools in case of making the -th order for the -th customer in the -th manufacturing subsystem.

Let us introduce the matrix of possible maintenance costs due to failure of the -th manufacturing subsystem (35):

where is the unit cost of possible maintenance procedures due to failure of a tool while making the -th order for the -th customer in the -th manufacturing subsystem.

The costs of possible failure maintenance in case of making the order in the -th manufacturing subsystem are calculated as follows (36):

The matrix of unit assembling costs of tools in the π-th manufacturing subsystem is introduced in

where is the unit assembling cost of the tool in the machine placed in the -th row of the -th column in the -th subsystem.

The assembling costs of tools in case of making the order in the -th manufacturing subsystem are calculated as follows:

The matrix of unit disassembling costs of tools in the -th manufacturing subsystem is introduced (39):

where is the unit disassembling cost of the tool in the machine placed in the -th row of the -th column in the -th subsystem.

The disassembling costs of tools in case of making the order in the -th manufacturing subsystem are calculated as follows:

The matrix of unit transportation costs for semiproducts from machines to the key point in the -th manufacturing subsystem is introduced in

where is the unit transportation time of products from the machine placed in the -th row of the -th column to the key point in the -th subsystem.

The transportation costs for semiproducts from machines to the key point in case of making the order in the -th manufacturing subsystem are calculated as follows:

The matrix of unit transportation costs for semiproducts from the key point in the -th manufacturing subsystem is introduced in

where is the unit transportation cost of products from the key point to the machine placed in placed in the row i of the column in the -th subsystem, .

The transportation costs for semiproducts from the key point to machines in case of making the order in the -th manufacturing subsystem are calculated as follows:

The matrix of unit transportation costs for semiproducts transported from machines to buffer stores in the -th manufacturing subsystem is introduced in

where is the unit transportation cost of a product transported from the machine placed in the -th row of the -th column to the -th buffer store in the -th subsystem.

The transportation costs for semiproducts transported to buffer stores from machines in case of making the order in the -th manufacturing subsystem are calculated as follows:

The matrix of unit costs of storing charge materials in the -th manufacturing subsystem takes the following form:

where is the storing cost of the -th charge material for making the -th product for the -th customer in the machine placed in the -th manufacturing subsystem.

The matrix of unit costs of storing semiproducts in the buffer stores in the -th manufacturing subsystem takes the following form:

where is the storing cost of the -th semiproduct made for the -th customer in the available buffer stores in the -th manufacturing subsystem.

Costs of storing semiproducts in the buffer stores in the -th manufacturing subsystem can be presented as follows:

The matrix of unit costs of storing ready products in the -th manufacturing subsystem takes the following form:

where is the unit storing cost of the -th ready product made for the -th customer in the ready product store in the -th manufacturing subsystem.

The total storing costs in the ready product store in the -th manufacturing subsystem are presented as follows:

The matrix of maintenance costs in the -th manufacturing subsystem takes the following form:

where is the maintenance cost of the machine placed in the -th row of the -th column in the -th manufacturing subsystem (per one unit of the order made in this machine).

The matrix of standstill costs in the -th manufacturing subsystem takes the following form:

where is the unit standstill costs in case of making the -th product for the m-th customer in the machine placed in the -th row of the -th column in the -th manufacturing subsystem (per one unit of the order matrix).

The matrix of costs of implementing heuristic algorithms in the π-th manufacturing subsystem takes the following form:

where is the cost of implementing heuristic algorithms in case of making the -th product made for the -th customer in the machine placed in the -th row of the -th column in the -th manufacturing subsystem (per one unit of the order matrix).

The matrix of hidden operating costs in the -th manufacturing subsystem takes the following form:

where is the hidden unit cost in case of making the -th product for the -th customer in the machine placed in the -th row of the -th column of the -th manufacturing subsystem (per one unit of the order matrix).

The matrix of unit transportation costs for semiproducts transported from buffer stores to machines in the -th manufacturing subsystem is introduced in

where is the transportation cost of a product from the -th buffer store to machine placed in the -th row of the column in the -th subsystem.

The unit transportation costs for semiproducts transported from buffer stores to machines in case of making the order in the -th manufacturing subsystem are calculated as follows:

The matrix of unit transportation costs for the charge material from the store of charges to machines in the -th manufacturing subsystem is introduced in

where is the unit transportation cost of the charge material to the machine placed in the -th row of the -th column in case of making the -th order for the -th customer in the -th subsystem.

The total transportation costs of charge material transported from the charge store to machines in case of making the order in the -th manufacturing subsystem are calculated as follows:

The matrix of unit transportation costs of ready products to the product store in the -th manufacturing subsystem is introduced in

where is the unit transportation cost of the ready product transported the product store in case of making the -th order for the -th customer in the -th subsystem.

The transportation cost of a ready product transported to the product store in case of making the order in the -th manufacturing subsystem are calculated as follows:

The matrix of parallel manufacturing costs in the -th manufacturing subsystem is introduced in

where is the manufacturing cost of making the -th order for the -th customer and the -th order for the -th customer in a parallel way in the -th subsystem with the use of the -th criterion.

At the same time if then and if then .

The matrix of costs of replacement operations in the -th manufacturing subsystem is introduced in

where is the replacement operation cost of the -th tool in the machine placed in the -th row of the j-th column in the -th subsystem.

At the same time, if the replacement operation of the -th tool in the machine placed in the -th row of the -th column in the -th subsystem takes place; otherwise.

The matrix of transportation costs for new tools which are stored in the storing place for tools in the -th manufacturing subsystem takes the following form:

where is the transportation cost of the new -th type tool from the storing place to the machine placed in the -th row of the -th column in the -th subsystem.

At the same time, if the transportation to the discussed machine takes place; otherwise.

The matrix of transportation costs for worn out tools to the storing place for used tools in the -th manufacturing subsystem takes the following form:

where is the transportation cost of the worn out -th type tool from the machine placed in the -th row of the -th column to the storing place for worn out tools in the -th subsystem. At the same time, if the transportation from the discussed machine to the storing place for worn out tools takes place; otherwise.

Let us introduce the cost of the replacement process in the -th manufacturing subsystem (66):

where

  is the unit cost of the replacement process in the -th manufacturing subsystem,  is the total replacement time in the -th manufacturing subsystem.

Let us introduce the cost of the lost flow capacity in the π-th manufacturing subsystem (67):

where

  is the unit cost of the lost flow capacity after completing the manufacturing process in the -th manufacturing subsystem,  is the number of lost flow capacity units after completing the manufacturing process in the -th manufacturing subsystem.

Let us introduce the fixed manufacturing cost in the π-th manufacturing subsystem (68):

where is the unit cost of making the -th order for the -th customer in the -th manufacturing subsystem; is the coefficient of stable costs in the -th manufacturing subsystem.

For the purpose of the paper costs are divided into two sample groups:

(a)direct costs consisting of costs which can be calculated straight away as they depend on the amount of the order, i.e., costs of manufacturing, charge storing, charge transportation, ready product transportation, ready product storing, and parallel manufacturing,(b)indirect costs consisting including costs of failures, semiproduct storing in buffer stores, assembling of tools, replacement operations, semiproduct transportation into and out of buffer stores, replacement operations, transportation of a new tool in and out, maintenance, standstill, heuristic implementing, and hidden operations. The approach cost of the -th manufacturing subsystem is calculated as follows:

The company cost of the π-th manufacturing subsystem is calculated as follows:

The logistics costs of the -th manufacturing subsystem are calculated as follows:

It can also be assumed that the longer the manufacturing process lasts the higher the costs can be generated increasing the base manufacturing costs for the given -th approach. These costs are the sum of manufacturing, parallel manufacturing, replacement and lost flow capacity costs for each -th approach. The minimal manufacturing costs are sought for in accordance with the given criterion from the set of criteria also including approaches involving the use of one heuristic only. However, another manufacturing approach can generate lower costs but at the same time it does not meet other requirements such as the minimal manufacturing time, minimal replacement time and minimal lost flow capacity.

It is also assumed that prolonged manufacturing generates costs which can be presented by means of the matrix of prolonged manufacturing in the -th subsystem (72):

where

  is the unit cost of prolonged manufacturing in the -th subsystem.  is the time of prolonged manufacturing according for the -th approach in the -th subsystem.

At the same time,

where

  is the time of completing making the order by the manufacturing approach minimising the manufacturing time in the -th subsystem;  is the time of completing making the order by the -th approach in the -th subsystem.

Each -th manufacturing subsystem can be evaluated on the basis of many factors. For the purpose of the article flexibility of the -th manufacturing subsystem is subject to analysis (74):

where

  is flexibility of the -th manufacturing subsystem,  is state of the -th manufacturing subsystem,  is usage of the -th manufacturing subsystem.

3.2. Control

To control the simulation process there is a need to take the following elements into account and combine them subsequently:

(1)heuristic algorithms;(2)manufacturing strategies;(3)control approaches.

3.2.1. Heuristic Algorithms

In order to choose an element of the matrix and subsequently direct it to the manufacturing zone there is a need to choose one from a wide range of available heuristic algorithms. Sample heuristic algorithms are considered for demonstration purposes; however, their number may be much bigger.

Let be the element to be made in the manufacturing zone at the -th stage, where , , . On this basis the following heuristic algorithms are put forward:

3.2.2. Manufacturing Strategies

It is assumed that there are the following manufacturing strategies:

(i)The manufacturing process continues according to the chosen heuristic algorithm until each element chosen for production is made and only then the next determined element is subject to manufacturing; i.e., it is allowed to enter the manufacturing zone.(ii)The continuity of the manufacturing process is guaranteed by using every possible machine so a decision about manufacturing is made immediately. The priority is to put charge material into the machine first. The robotic arm can make all other transportation tasks after delivering the charge material to all determined machines. Only then can semiproducts be transported to the subsequent determined machines.

Let us introduce the matrix of order priority (76):

where

  is the priority of the manufacturing task for the -th customer who ordered the -th product the stage .

The priority element takes the following values: for the high priority and otherwise.

3.2.3. Control Approaches

It is assumed that there are the following control approaches to be discussed:

(i)The first method assumes that the sequence of manufactured orders remains unchanged throughout the whole manufacturing process, i.e., the manufacturing sequence is determined. An element of the order matrix to be made is chosen with the use of heuristic algorithms. In this case the system is treated as the serial manufacturing system and a need to implement the matrix of routes arises.(ii)The second method is based on the principle that any element of the order matrix can be made without the need to follow any sequence restriction of manufacturing operations (the sequence is optional), i.e., a machine is chosen according to a specified heuristic algorithm in order to meet defined criteria.(iii)The third method mixes two previous methods as some order matrix elements require a defined manufacturing sequence whereas others do not.

In case of the first approach the sequence of operations on the n-th product is predefined and the order of making products cannot be alternated. Searching for solutions minimising the total order making time consists in minimising the standstill time of the manufacturing system as a whole or its individual elements (machines). Moreover, the manufacturing time cannot be increased so there is a need to search for other solutions such as, e.g., minimising the total manufacturing time, minimising the replacement time of tools, minimising lost flow capacity of tools. It is assumed that exactly at the moment when a certain n-th order frees the manufacturing space another order element enters the manufacturing zone.

The modelled system requires introducing manufacturing criteria according to which the manufacturing procedures are controlled as follows:

(1)In order to minimise the total manufacturing time the time criterion of the production process is formulated (77):

(2)In order to minimise the total replacement time the criterion of the replacement process is formulated (78):

(3)In order to minimise the total replacement time the criterion of lost flow capacity of tools is formulated (79):

 The losses during the manufacturing process influence the costs which increase then.

4. Results and Discussion

To validate the above iteration approach to creating the dedicated software and the subsequent simulation process certain sample data and assumption are taken into account for the purpose of the calculation procedure. The main goals of the case study include

(i)comparing the results of simulations for strategies and available heuristics;(ii)analysing timescales of making order for chosen strategies: Q_pr_ii_at random, Q_repl_ii_at random, Q_lfc_ii_at random, Q_pr_ii_heuristic_2, Q_pr_i_heuristic_1;(iii)analysing the problem of the system elasticity depending on the system state and the system usage;(iv)analysing results of the manufacturing process simulation for 5000 simulations;(v)presenting and analysing the histogram of the multisimulation process for 5000 repetitions for the initial order for the strategy generating the minimal manufacturing time;(vi)analysing the parallel and subsequent simulation results;(vii)searching for the best solution of manufacturing and prolonged manufacturing costs;(viii)the comparative cost analysis of the five discussed approaches. The case study analysis requires assuming the following data:

The sample data necessary to carry out the simulation of the manufacturing are given as follows:

(i)manufacturing routes (Table 1);(ii)manufacturing times (Table 2);(iii)unit manufacturing costs (Table 3);(iv)order matrix; order priority; coefficient of failures of tools; storing times and unit storing costs (Table 4);(v)initial state of tools and life of tools (Table 5);(vi)times and costs of assembling and disassembling of tools, transport of semiproducts to the key point and buffer zones, transport of charge, and transport of ready products (Table 6);(vii)times and unit costs of replacement operations, transport of tools to and from machines for replacement; costs of maintenance, standstill, implementing heuristics, and predicted hidden costs (Table 7).

Table 1: The flow of products, manufacturing routes.

1,n

2,n

3,n

4,n

5,n

1,1
1
3
2
1
4
2,1
1
2
2
2
4
3,1
3
3
4
4
1
4,1
3
3
4
3
1
5,1
4
1
2
1
1

1,2
3
1
1
1
3
2,2
1
4
2
1
4
3,2
3
4
4
4
4
4,2
3
1
2
2
3
5,2
3
4
1
3
4

1,3
3
1
4
1
2
2,3
2
1
4
1
1
3,3
2
4
3
1
2
4,3
2
1
2
2
2
5,3
4
4
4
4
4

1,4
1
2
1
3
1
2,4
2
4
3
4
2
3,4
1
2
4
1
4
4,4
2
3
4
2
3
5,4
1
4
1
4
3

1,5
4
1
4
1
3
2,5
4
2
2
2
1
3,5
1
3
4
2
1
4,5
2
1
3
4
2
5,5
4
3
1
3
2

Table 2: Manufacturing times.

1,n

2,n

3,n

4,n

5,n

1,1
9
6
9
8
9
2,1
8
7
8
9
7
3,1
6
6
6
9
9
4,1
8
6
9
7
6
5,1
7
8
9
8
8

1,2
7
7
6
7
9
2,2
7
9
8
8
7
3,2
6
7
8
8
9
4,2
7
9
9
8
7
5,2
9
7
6
8
7

1,3
9
9
9
7
9
2,3
8
8
9
6
7
3,3
8
8
9
9
7
4,3
6
6
8
7
6
5,3
8
6
7
8
8

1,4
8
9
6
6
6
2,4
8
7
8
8
7
3,4
8
8
9
8
9
4,4
8
9
9
6
7
5,4
9
8
9
8
9

1,5
9
8
6
9
6
2,5
6
8
8
9
7
3,5
8
6
8
9
8
4,5
8
9
8
7
8
5,5
9
6
7
9
8

Table 3: Unit manufacturing costs.

1,n

2,n

3,n

4,n

5,n

1,1
1
2
1
3
2
2,1
1
2
1
3
2
3,1
1
2
1
3
2
4,1
1
2
1
3
2
5,1
1
2
1
3
2

1,2
2
3
1
2
3
2,2
2
3
1
2
3
3,2
2
3
1
2
3
4,2
2
3
1
2
3
5,2
2
3
1
2
3

1,3
2
1
3
1
2
2,3
2
1
3
1
2
3,3
2
1
3
1
2
4,3
2
1
3
1
2
5,3
2
1
3
1
2

1,4
3
2
2
1
3
2,4
3
2
2
1
3
3,4
3
2
2
1
3
4,4
3
2
2
1
3
5,4
3
2
2
1
3

1,5
2
1
2
2
3
2,5
2
1
2
2
3
3,5
2
1
2
2
3
4,5
2
1
2
2
3
5,5
2
1
2
2
3

Table 4: Order matrix; order priority; coefficient of failures of tools; storing times; and unit storing costs.

(a)

29
24
26
27
22

9
25
10
7
10

23
24
28
19
27

7
16
6
9
14

31
12
14
12
16

(b)

0
0
0
0
0

0
0
0
0
0

0
0
0
0
0

0
0
0
0
0

0
0
0
0
0

(c)

5
6
3
9
9

6
5
7
5
4

5
5
7
1
1

5
6
7
7
4

6
8
3
5
4

(d)

7
8
8
8
8

8
7
8
7
8

8
7
8
7
8

7
8
7
8
7

8
7
8
8
8

(e)

13
12
19
15
23

25
17
14
23
31

24
9
18
17
21

19
29
31
7
14

15
8
17
19
30

(f)

4
5
5
5
4

5
4
4
5
5

5
5
5
4
4

4
5
4
5
5

5
4
5
5
4

(g)

5
5
5
5
5

5
5
5
5
5

5
5
5
5
5

5
5
5
5
5

5
5
5
5
5

(h)

1
2
1
1
1

1
2
1
2
2

2
1
2
2
1

1
2
1
2
2

1
4
2
5
1

(i)

25
24
28
12
11

14
17
13
31
33

25
19
23
26
21

31
18
14
7
14

17
19
30
19
30

(j)

6
7
7
6
7

7
6
6
6
5

6
5
6
7
7

7
6
6
5
6

5
6
7
7
7

Table 5: Initial state of tools and life of tools.

(a)

5
6
1
3
6

2
6
3
0
2

4
3
0
2
6

2
2
2
1
1

(b)

6
8
7
8
9

6
7
6
8
6

6
6
7
9
9

7
6
7
8
9

Table 6: Times and costs of assembling and disassembling of tools, transport of semiproducts to the key point and buffer zones, transport of charge, and transport of ready products.

6
9
7
8
7
5
4
3
5
3

6
8
9
9
7
3
4
4
5
3

7
7
8
7
6
3
4
3
4
4

7
8
6
6
6
4
3
3
5
4

6
7
9
7
6
4
3
3
4
4

8
7
6
8
7
4
3
3
5
4

8
7
7
6
9
3
4
4
3
3

7
7
6
6
9
4
3
4
5
4

9
6
6
6
7
2
3
2
3
2

7
7
9
7
7
2
2
3
2
1

7
8
9
8
9
2
1
2
3
2

6
6
8
8
6
2
2
1
2
1

8
6
8
9
8
1
2
2
2
2

7
7
8
8
8
1
2
2
3
2

9
6
6
8
8
2
2
1
2
1

9
6
9
6
9
1
2
3
1
2

6
9
6
0
0
1
2
1
0
0

9
8
9
0
0
2
1
2
0
0

8
8
9
0
0
1
2
1
0
0

8
9
8
0
0
2
1
2
0
0

6
6
7
0
0
2
2
1
0
0

8
8
6
0
0
2
1
2
0
0

8
8
8
0
0
1
2
1
0
0

7
7
7
0
0
2
1
2
0
0

9
0
0
0
0
2
0
0
0
0

6
0
0
0
0
2
0
0
0
0

9
0
0
0
0
2
0
0
0
0

6
0
0
0
0
2
0
0
0
0

0
0
0
0
6
0
0
0
0
2

0
0
0
0
7
0
0
0
0
2

0
0
0
0
7
0
0
0
0
2

0
0
0
0
8
0
0
0
0
2

Table 7: Times and unit costs of replacement operations, transport of tools to and from machines for replacement; costs of maintenance, standstill, implementing heuristics, and predicted hidden costs.

5
6
8
7
6
5
6
8
7
6

7
4
6
7
6
7
4
6
7
6

4
7
6
5
7
4
7
6
5
7

6
5
6
6
7
6
5
6
6
7

6
7
7
9
4
4
5
9
10
3

4
5
7
6
7
8
3
4
8
2

5
8
7
6
8
5
6
8
6
6

7
6
7
7
6
5
4
7
7
3

5
6
5
8
5
3
2
4
5
3

5
6
11
4
7
5
2
5
3
5

6
7
6
5
9
4
3
5
2
3

6
7
8
4
3
4
5
7
6
4

9
11
13
9
10
3
4
4
4
3

12
14
9
15
14
4
4
3
4
4

13
18
9
12
13
3
3
4
4
4

11
19
12
20
9
4
4
3
3
3

3
2
3
3
3
45
39
34
48
64

2
3
2
2
3
34
29
42
46
37

3
2
3
3
2
30
37
34
37
36

2
3
3
3
3
46
35
47
40
20

Sample data for the simulation process are not drawn from any external sources which would require special permission from any provider. They are defined by the authors with the sole purpose of demonstrating the functionality and usability of the mathematical model as well as the simulator created especially for the purpose of this paper. However, the sample data used by the authors in the paper partly result from observations of production processes taking place in the automotive manufacturing industry especially while making complex orders for identical products set by various customers.

There are two approaches to the problem of manufacturing. The first one requires waiting for the preceding order to leave the system, strategy i. Another one enables the operator of the system to start manufacturing immediately when such an opportunity emerges, strategy ii. First of all, the simulation process is carried out twice with the use of the implemented data for strategies i. and ii. The compared results of the simulation process are shown in Table 8.

Table 8: Comparison of results for strategies and available heuristics.

Strategy
Heuristic
Total production time
Effective production time
Robotic arm work
Total replacement time
Lost flow capacity
Final flow capacity
Total transportation time
System state indicator
System wear indicator
System flexibility indicator

i.
1
5422
4208
2117
1214
10,37
64,47
903
0,48
0,74
0,60

2
5458
4227
2140
1231
7,78
70,06
909
0,43
0,73
0,56

3
5422
4208
2117
1214
10,37
64,47
903
0,48
0,74
0,60

4
5422
4208
2117
1214
10,37
64,47
903
0,48
0,74
0,60

ii.
1
4478
2828
2396
1650
10,37
64,47
746
0,49
0,91
0,67

2
4466
2844
2364
1620
7,88
77,95
744
0,43
0,92
0,63

3
4478
2828
2396
1650
10,37
64,47
746
0,49
0,91
0,67

4
4478
2828
2396
1650
10,37
64,47
746
0,49
0,91
0,67

It is visible that the winning strategy in terms of the total production time and effective production time is strategy ii. However, in terms of the robotic arm work as well as the total replacement time strategy i. is much more effective. The lost flow capacity is on par for both strategies nevertheless, it is worth noticing that heuristic 2 is the winning one in both approaches and the best of all in strategy i. The total transportation time is shorter in case of strategy i. where heuristic 2 is the best again. The final flow capacity is identical for all implemented heuristics other than for heuristic 2 which is better for strategy i. The system flexibility indicator is the best again for heuristic 2; however, it is the best for strategy ii. using heuristic 2.

Time scaling for Q_pr_ii_heuristic_2 is shown in Figure 2 and for Q_pr_i_heuristic_1 in Figure 3 and Q_pr_ii_at random in Figure 4.

Figure 2: The timescale of making order Z⁰ for Q_pr_ii_heuristic_2.

Figure 3: The timescale of making order Z⁰ for Q_pr_i_heuristic_1.

Figure 4: Time scaling of parallel manufacturing of the order Z⁰ for Q_pr_ii_at random.

Figure 2 presents making order Z⁰ for Q_pr_ii_heuristic_2 which enables parallel manufacturing. This approach eliminates standstill of machines and however, as seen in Table 14, generates extra costs due to parallel manufacturing tasks. The time scale shows that in the early manufacturing stages orders do not cumulate and only two of them are made at the same time. With the pass of manufacturing orders accumulate in the production system which leads to intensifying manufacturing of even four elements of the order matrix at the same time.

Figure 3 presents making order Z⁰ for Q_pr_i_heuristic_1 which excludes parallel manufacturing. The obvious thing emerging from this approach is the fact that the manufacturing process lasts much longer than in case of Q_pr_ii_heuristic_2; however, it does not generate excessive parallel manufacturing costs.

Figure 4 presents making order Z⁰ for Q_pr_ii_at random which involves parallel manufacturing. The time scale results from carrying out 5000 simulation. The time scale characterised by the shortest order making time is presented in accordance with Table 14. Making orders is rather distributed evenly which results in making more than often at the same time in logistics system. Four times three orders are made at the same period of time.

Each simulation is also characterised by its elasticity which is the function of the system state and its usage. For the purpose of demonstration Figure 5 presents system elasticity for Q_pr_i_heuristic_1 whereas Figure 6 shows system elasticity for Q_pr_ii_heuristic_2.

Figure 5: System elasticity for Q_pr_i_heuristic_1.

Figure 6: System elasticity for Q_pr_ii_heuristic_2.

As system elasticity is the function of system state and its usage it is visible how these two factors influence it. The system elasticity is high when its usage is high and the system imitates the lasting manufacturing process. At the same time the lasting manufacturing process leads to increasing its state values which as a result leads to the necessity of replacement worn out tool. As a consequence each replacement process decreases the state of the system enabling resuming the manufacturing procedures. After carrying out 5000 simulations of the manufacturing process for Q_pr_ii_at random the results are arranged presented from the point of view of the criterion in Table 9.

Table 9: Results of the manufacturing process simulation for 5000 simulations.

Simulation no.
Production time
Replacement time of tools
Lost flow capacity
Final flow capacity

2737
4356
1573
7,89
63,95

1523
4393
1384
6,68
72,15

1547
4432
1470
10,00
64,83

4428
4445
1623
7,98
106,85

As seen in Table 9 the best total manufacturing time (expressed in time units) is obtained in the simulation process no. 2737. However, this simulation procedure does not guarantee the minimal replacement time of tools which is much worse than the minimal replacement time of tools obtained in the simulation process no. 1523. This difference which equals 189 time units is quite immense and may lead to financial losses; therefore, it does make sense to consider another kind of approach presented in the paper hereby. Moreover, the simulation process no. 1523 offers the minimal value of the lost flow capacity. Table 10 presents more detailed results from the point of view of . The results are segregated increasing according to the stated criterion where the minimal manufacturing time is searched for.

Table 10: Results of the manufacturing process simulation for .

Simulation
Total production time
Tools replacement time
Lost flow capacity
Final flow capacity

2737
4356
1573
7,89
63,95

527
4367
1536
7,07
68,77

2082
4373
1542
6,71
78,13

4623
4377
1502
8,23
74,61

4854
4384
1632
7,01
80,83

239
4384
1586
4,57
91,27

1182
4386
1537
6,65
88,18

4387
4386
1535
5,85
66,98

2643
4386
1514
8,12
82,72

1912
4387
1554
8,16
74,68

2205
4387
1558
8,97
72,86

4395
4388
1497
7,92
51,92

As seen in Table 11 the most minimal manufacturing time does not guarantee either the minimal tools replacement time or the lost flow capacity or the final flow capacity. Simulation no. 2737 out of the set of 5000 simulations delivered the satisfactory solution satisfying . We can assume that the probability of finding a better result in terms of is extremely low; however, it is not excluded.

Table 11: Results of the combined manufacturing simulation process for , , and in case of 5000 simulation procedures.

Simulation no.
2737
1523
2271

Total manufacturing time
4356
4393
4504

Total replacement time
1573
1384
1579

Lost flow capacity
7,89
6,68
2,39

Total transportation time
726
732
750

System state indicator
0,44
0,50
0,47

System wear indicator
0,93
0,92
0,91

System flexibility indicator
0,64
0,68
0,65

Final flow capacity
64
72,2
85,5

Figure 7 presents results of 5000 multisimulation procedures in which 232 different results were obtained. The shortest manufacturing time equals 4356 time units whereas the longest manufacturing time equals 4672 time units. The spread equals 316 time units. The most repeated result equals 4464 time units and was obtained 64 times which is 1,3% of all simulation results.

Figure 7: The histogram of the multisimulation process for 5000 repetitions for order for strategy ii. with choosing heuristics at random.

Table 12 presents assumed time unit cost depending on the parallel manufacturing process in case of strategy ii. For example, if manufacturing of the order was ongoing and at one moment of this process manufacturing of the order began in a parallel way it is assumed that . Nevertheless, as the company should be treated as one complex logistics system it was decided to calculate its uniform parallel manufacturing cost which in fact is its average parallel manufacturing cost, i.e., = 3,52

Table 12: Time unit costs depending on parallel manufacturing.

m,n
1,1
1,2
1,3
1,4
1,5
2,1
2,2
2,3
2,4
2,5
3,1
3,2
3,3
3,4
3,5
4,1
4,2
4,3
4,4
4,5
5,1
5,2
5,3
5,4
5,5
Σ

1,1
x
3
3
4
3
4
2
3
2
4
3
2
3
4
4
3
4
4
5
2
3
4
3
4
5
3,38

1,2
3
x
3
2
4
3
2
3
4
4
3
4
4
5
2
3
4
3
4
5
4
3
2
3
4
3,38

1,3
3
3
x
3
4
4
3
4
4
5
2
3
4
3
3
4
3
4
5
4
3
2
3
4
2
3,42

1,4
4
2
3
x
4
5
2
3
4
3
3
4
3
4
5
4
5
2
3
4
3
3
4
3
4
3,50

1,5
3
4
4
4
x
4
3
3
4
3
4
5
4
5
2
3
4
3
4
5
4
3
3
4
3
3,67

2,1
4
3
4
5
4
x
4
3
4
5
4
5
2
3
4
3
3
2
4
3
2
3
4
5
2
3,54

2,2
2
2
3
2
3
4
x
4
3
3
4
3
4
5
4
5
2
3
4
4
3
4
5
4
5
3,54

2,3
3
3
4
3
3
3
4
x
3
2
4
3
2
3
4
4
3
4
4
3
3
4
3
4
5
3,38

2,4
2
4
4
4
4
4
3
3
x
4
4
3
4
4
5
2
3
4
3
3
4
3
2
3
2
3,38

2,5
4
4
5
3
3
5
3
2
4
x
3
4
5
4
5
2
3
4
3
4
5
4
3
5
3
3,75

3,1
3
3
2
3
4
4
4
4
4
3
x
3
4
5
4
5
2
3
4
3
4
5
4
3
2
3,54

3,2
2
4
3
4
5
5
3
3
3
4
3
x
4
3
4
4
5
2
3
4
3
3
2
3
4
3,46

3,3
3
4
4
3
4
2
4
2
4
5
4
4
x
2
3
4
4
3
4
4
5
2
3
4
5
3,58

3,4
4
5
3
4
5
3
5
3
4
4
5
3
2
x
3
4
5
4
5
2
3
4
3
4
3
3,75

3,5
4
2
3
5
2
4
4
4
5
5
4
4
3
3
x
5
4
5
2
3
4
3
4
2
3
3,63

4,1
3
3
4
4
3
3
5
4
2
2
5
4
4
4
5
x
3
3
4
3
4
5
4
2
4
3,63

4,2
4
4
3
5
4
3
2
3
3
3
2
5
4
5
4
3
x
3
4
4
5
2
3
4
3
3,54

4,3
4
3
4
2
3
2
3
4
4
4
3
2
3
4
5
3
3
x
3
2
3
4
4
3
4
3,29

4,4
5
4
5
3
4
4
4
4
3
3
4
3
4
5
2
4
4
3
x
3
4
3
3
4
3
3,67

4,5
2
5
4
4
5
3
4
3
3
4
3
4
4
2
3
3
4
2
3
x
2
3
4
4
3
3,38

5,1
3
4
3
3
4
2
3
3
4
5
4
3
5
3
4
4
5
3
4
2
x
4
3
3
4
3,54

5,2
4
3
2
3
3
3
4
4
3
4
5
3
2
4
3
5
2
4
3
3
4
x
4
5
4
3,5

5,3
3
2
3
4
3
4
5
3
2
3
4
2
3
3
4
4
3
4
3
4
3
4
x
3
4
3,33

5,4
4
3
4
3
4
5
4
4
3
5
3
3
4
4
2
2
4
3
4
4
3
5
3
x
5
3,67

5,5
5
4
2
4
3
2
5
5
2
3
2
4
5
3
3
4
3
4
3
3
4
4
4
5
x
3,58

Σ
3,38
3,38
3,42
3,50
3,67
3,54
3,54
3,38
3,38
3,75
3,54
3,46
3,58
3,75
3,63
3,63
3,54
3,29
3,67
3,38
3,54
3,50
3,33
3,67
3,58
3,52

Table 13 presents simulation results of parallel manufacturing for each order matrix element for Q_pr_ii_at random, Q_repl_ii_at random, and Q_lfc_ii_at random, where the best results satisfying the stated criteria are chosen, as well as for Q_pr_ii_heuristic_2 in terms of parallel manufacturing and simulation results of the manufacturing process for Q_pr_i_heuristic_1. The beginning and finishing moments of the manufacturing process are given.

Table 13: Parallel and subsequent simulation results - the beginning and finishing moments of the manufacturing process.

Start
End
Start
End
Start
End
Start
End
Start
End

1,1
29
82
2595
3028
972
1362
3706
4305
268
618
4760
5090

1,2
24
82
0
243
0
243
2042
2446
1815
2117
2624
2867

1,3
26
79
1701
2044
168
486
1720
2120
1318
1650
3496
3788

1,4
27
78
1460
1789
1254
1533
1480
1860
782
1061
3788
4067

1,5
22
74
869
1209
766
1078
3182
3594
2412
2855
2136
2370

2,1
9
71
426
586
3257
3574
1962
2221
3821
4182
257
384

2,2
25
86
3202
3576
3642
4199
2218
2661
1567
1879
3204
3496

2,3
10
71
4104
4356
678
874
2825
3129
3640
3971
503
633

2,4
7
83
4026
4262
3073
3365
254
404
3986
4335
73
159

2,5
10
63
168
340
1647
1807
3601
4001
3742
4077
633
773

3,1
23
69
2183
2487
2918
3303
3432
3905
2250
2651
2370
2624

3,2
24
84
1074
1489
3419
3958
0
340
1994
2422
2867
3204

3,3
28
74
2353
2744
2421
2840
970
1387
550
892
4453
4760

3,4
19
93
1973
2271
2015
2384
318
648
2618
3079
1861
2136

3,5
27
80
597
993
1744
2140
528
990
1025
1409
4067
4453

4,1
7
62
3571
3770
2217
2443
3988
4363
4051
4403
159
257

4,2
16
87
2922
3220
3894
4343
2576
2946
2827
3220
1462
1663

4,3
6
61
2878
3078
4045
4393
3388
3674
4115
4466
0
73

4,4
9
87
517
665
3339
3663
2742
3031
3906
4270
384
503

4,5
14
79
3438
3711
2281
2564
820
1123
3130
3514
1096
1286

5,1
31
72
3636
4080
402
788
1212
1693
0
343
5090
5422

5,2
12
82
1337
1605
2659
2958
3066
3390
3388
3742
773
932

5,3
14
67
3067
3356
3137
3486
4052
4504
3254
3635
1286
1462

5,4
12
96
3899
4198
2790
3075
2456
2806
3519
3885
932
1096

5,5
16
80
261
493
1494
1699
2914
3286
2976
3386
1663
1861

Earliest start:
168

168

254

268

0

Latest start:
4104

4045

4052

4115

5090

Table 14: Costs of parallel manufacturing.

1,1
29
150
528,00
108
380,16
317
1115,84
68
239,36

1,2
24
75
264,00
75
264,00
228
802,56
123
432,96

1,3
26
71
249,92
84
295,68
158
556,16
83
292,16

1,4
27
88
309,76
39
137,28
140
492,80
36
126,72

1,5
22
135
475,20
106
373,12
206
725,12
237
834,24

2,1
9
69
242,88
235
827,20
179
630,08
276
971,52

2,2
25
138
485,76
305
1073,60
205
721,60
64
225,28

2,3
10
0
0,00
108
380,16
215
756,80
229
806,08

2,4
7
158
556,16
228
802,56
86
302,72
284
999,68

2,5
10
79
278,08
63
221,76
295
1038,40
256
901,12

3,1
23
134
471,68
230
809,60
304
1070,08
239
841,28

3,2
24
152
535,04
312
1098,24
86
302,72
172
605,44

3,3
28
149
524,48
181
637,12
175
616,00
110
387,20

3,4
19
88
309,76
167
587,84
120
422,40
252
887,04

3,5
27
124
436,48
125
440,00
170
598,40
91
320,32

4,1
7
134
471,68
162
570,24
311
1094,72
288
1013,76

4,2
16
153
538,56
298
1048,96
204
718,08
244
858,88

4,3
6
156
549,12
0
0,00
242
851,84
0
0,00

4,4
9
68
239,36
244
858,88
206
725,12
284
999,68

4,5
14
140
492,80
143
503,36
153
538,56
260
915,20

5,1
31
181
637,12
110
387,20
213
749,76
75
264,00

5,2
12
145
510,40
168
591,36
208
732,16
223
784,96

5,3
14
154
542,08
229
806,08
0
0,00
247
869,44

5,4
12
172
605,44
157
552,64
230
809,60
245
862,40

5,5
16
67
235,84
52
183,04
220
774,40
256
901,12

Sum:
2980
10489,60
3929
13830,08
4871
17145,92
4642
16339,84

Average:
119,2
419,58
157,16
553,2032
194,84
685,8368
185,68
653,5936

Table 14 presents the simulation results of parallel manufacturing for Q_pr_ii_at random, Q_repl_ii_at random, and Q_lfc_ii_at random as well as for Q_pr_ii_heuristic_2. The first three results were obtained by means of carrying out 5000 simulations for every approach and choosing the best result satisfying each criterion. Both the parallel manufacturing time for each order and the cost of parallel manufacturing for each order are given. The costs are summed for every approach and their average value is calculated.

As seen in Table 14 the lowest parallel manufacturing cost is obtained by implementing Q_pr_ii_at random whereas Q_lfc_ii_at random proves to be the most expensive when implemented.

Table 15 presents costs in case of prolonged manufacturing for sample data. The regular intervals of 10 costs units are taken into account for illustration reasons. The simulations were carried out for:

(1)Q_pr_ii_at random is the best result of 5000 simulations for the minimal manufacturing time criterion with the use of strategy ii. where heuristics are drawn at random.(2)Q_repl_ii_at random is the best result of 5000 simulations for the minimal replacement time criterion with the use of strategy ii. where heuristics are drawn at random.(3)Q_lfc_ii_at random is the best result of 5000 simulations for the minimal lost flow capacity criterion with the use of the strategy ii. where heuristics are drawn at random.(4)Q_pr_ii_heuristic_2: strategy ii. is implemented for heuristic 2.(5)Q_pr_i_heuristic_1: strategy i. is implemented for heuristic 1.

Table 15: Prolonged manufacturing costs for sample data.

Q_pr_ii_at random
Q_repl_ii_at random
Q_lfc_ii_at random
Q_pr_ii_heuristic_2
Q_pr_i_heuristic_1

10,00
90989,09
96400,77
110836,30
108291,36
79712,00

15,00
90989,09
96585,77
111576,30
108841,36
85042,00

20,00
90989,09
96770,77
112316,30
109391,36
90372,00

25,00
90989,09
96955,77
113056,30
109941,36
95702,00

30,00
90989,09
97140,77
113796,30
110491,36
101032,00

35,00
90989,09
97325,77
114536,30
111041,36
106362,00

40,00
90989,09
97510,77
115276,30
111591,36
111692,00

45,00
90989,09
97695,77
116016,30
112141,36
117022,00

50,00
90989,09
97880,77
116756,30
112691,36
122352,00

55,00
90989,09
98065,77
117496,30
113241,36
127682,00

60,00
90989,09
98250,77
118236,30
113791,36
133012,00

As seen in Table 15 and subsequently in Figure 8 the minimal prolonged manufacturing costs are generated by Q_pr_i_heuristic_1 which does not satisfy the minimal manufacturing time criterion. These costs equal the costs generated in case of implementing Q_pr_ii_at random when = 90989,09307692 (see Table 16) and then start growing when the unit manufacturing cost increases making the manufacturing process unprofitable if this strategy is unchanged.

Table 16: Prolonged manufacturing costs, intersection points.

Q_pr_ii_ at random
Q_repl_ii_ at random
Q_lfc_ii_ at random
Q_pr_ii_ heuristic_2
Q_pr_i_ heuristic_1

20,57888656
90989,09307692
96792,18495670
112401,97290374
109455,03444509
90989,09307692

26,21843164
90989,09307692
97000,84812439
113236,62557449
110075,38440308
97000,84812439

39,89472481
90989,09307692
97506,87097200
115260,71696492
111579,77665272
111579,77665272

43,90446372
90989,09307692
97655,23131138
115854,15832244
112020,84793196
115854,15832244

Figure 8: Prolonged manufacturing costs.

Figure 8 is the graphic representation of prolonged manufacturing. Each of the five approaches is described by its function showing the dependence of the base manufacturing costs on the unit cost.

Manufacturing and prolonged manufacturing costs are included in Figure 8. The vertical axis presents the values which are the sum of the base manufacturing costs and prolonged manufacturing costs. The horizontal axis presents the values adequate to the unit cost of the prolonged manufacturing. The obvious thing that emerges is the fact that Q_pr_i_heuristic_1 is profitable till the moment when the unit cost of the prolonged manufacturing reaches the value of 20,57888656 and the cost is on par with the costs of the manufacturing approach Q_pr_ii_at random. The sum of the real manufacturing time and prolonged manufacturing time reach the value of 90989,09307692. Table 16 analyses prolonged manufacturing costs and shows intersection points of functions representing implemented manufacturing approaches included in Figure 8.

Table 16 enables us to notice that Q_pr_i_heuristic_1 matches

Q_pr_ii_at random for = 20,57888656

Q_repl_ii_at random for = 26,21843164

Q_lfc_ii_at random for = 43,90446372

Q_pr_ii_heuristic_2 for = 39,89472481

Finally, the comparative cost analysis of the five discussed approaches is carried out. Initial data for this analysis result from calculations taking into account data included in Tables 1–7. The input data for the analysis include 25 components shown in Tables 17 and 18.

Table 17: Intermediate results of the simulation process, costs for (part 1).

Order (m,n)
Unit
Production
Failures
Assembling of tools
Replacement operations
Disassembling of tools
Transportation of a new tool_in
Transportation of a new tool_out
Semi-product transport →
Semi-product transport ←
Semi-product transport to buffer zone →
Storing charge
Semi-product t storing in buffers
Storing ready products

1,1
82
2378
1015
158
208
134
208
143
77
72
40
52
5
150

1,2
82
1968
1728
142
214
127
261
123
80
72
32
60
10
168

1,3
79
2054
624
136
173
125
213
167
65
91
42
95
5
196

1,4
78
2106
504
132
166
114
150
72
82
70
20
75
5
72

1,5
74
1628
1944
146
206
128
257
159
74
74
50
92
5
77

2,1
71
639
432
167
175
139
133
106
79
80
32
125
5
98

2,2
86
2150
700
154
184
127
184
157
81
72
33
68
10
102

2,3
71
710
560
133
206
129
218
148
72
80
52
56
5
78

2,4
83
581
245
117
182
139
175
149
67
47
36
115
10
186

2,5
63
630
320
162
173
131
138
118
81
79
42
155
10
165

3,1
69
1587
920
118
173
130
183
140
60
79
40
120
10
150

3,2
84
2016
168
93
162
135
165
151
56
81
33
45
5
95

3,3
74
2072
1568
106
195
137
216
165
55
59
50
90
10
138

3,4
93
1767
133
120
175
133
196
135
64
85
30
68
10
182

3,5
80
2160
216
121
195
140
181
119
62
87
38
84
5
147

4,1
62
434
245
95
162
118
170
126
68
89
40
76
5
217

4,2
87
1392
1024
162
186
130
159
130
91
78
44
145
10
108

4,3
61
366
294
156
206
139
157
139
80
75
54
124
5
186

4,4
87
783
504
133
232
151
225
141
62
78
50
35
10
35

4,5
79
1106
392
129
193
139
186
126
73
47
45
70
10
84

5,1
72
2232
1488
161
193
119
200
146
89
71
52
75
5
85

5,2
82
984
672
118
179
126
166
101
68
80
23
32
20
114

5,3
67
938
336
124
182
139
175
151
54
72
41
85
10
210

5,4
96
1152
480
129
199
129
208
121
76
50
21
95
25
133

5,5
80
1280
512
126
210
129
196
110
63
69
38
120
5
210

Σ

35113
17024
3338
4729
3287
4720
3343
1779
1837
978
2157
215
3386

Table 18: Intermediate results of the simulation process, costs for (part 2).

Order (m,n)
Semi-product transport from buffers ←
Maintenance
Standstill
Heuristic implementing
Hidden
Charge transport
Ready product transport
Parallel production _criterion_ manufacturing time
Parallel production _criterion _replacement time
Parallel production criterion_lost capacity
Parallel production _strategy ii_heuristic 2

1,1
40
54
16
13
192
522
464
528,00
380,16
1115,84
239,36

1,2
27
59
19
13
187
432
336
264,00
264,00
802,56
432,96

1,3
34
59
18
14
201
468
364
249,92
295,68
556,16
292,16

1,4
27
58
18
15
209
486
324
309,76
137,28
492,80
126,72

1,5
40
56
19
12
216
264
308
475,20
373,12
725,12
834,24

2,1
32
56
17
13
182
162
144
242,88
827,20
630,08
971,52

2,2
31
55
17
14
190
450
400
485,76
1073,60
721,60
225,28

2,3
42
54
18
13
232
120
120
0,00
380,16
756,80
806,08

2,4
31
74
19
14
180
84
98
556,16
802,56
302,72
999,68

2,5
34
59
18
12
227
120
120
278,08
221,76
1038,40
901,12

3,1
38
73
15
14
218
368
276
471,68
809,60
1070,08
841,28

3,2
29
73
16
15
172
432
384
535,04
1098,24
302,72
605,44

3,3
31
63
20
14
188
336
392
524,48
637,12
616,00
387,20

3,4
34
53
17
15
189
342
304
309,76
587,84
422,40
887,04

3,5
42
64
16
13
239
486
324
436,48
440,00
598,40
320,32

4,1
38
65
16
14
215
126
84
471,68
570,24
1094,72
1013,76

4,2
32
61
18
11
193
288
224
538,56
1048,96
718,08
858,88

4,3
40
61
19
11
198
72
84
549,12
0,00
851,84
0,00

4,4
46
70
18
11
200
108
126
239,36
858,88
725,12
999,68

4,5
36
66
19
13
184
168
196
492,80
503,36
538,56
915,20

5,1
38
50
18
12
239
372
372
637,12
387,20
749,76
264,00

5,2
22
66
18
15
156
216
192
510,40
591,36
732,16
784,96

5,3
35
71
17
14
188
168
224
542,08
806,08
0,00
869,44

5,4
26
74
18
14
190
216
168
605,44
552,64
809,60
862,40

5,5
37
68
19
13
191
192
224
235,84
183,04
774,40
901,12

Σ
862
1562
443
332
4976
6998
6252
10489,60
13830,08
17145,92
16339,84

Finally, the results sought for are presented in Table 19 and Figures 9–13 and include the results of the comparative cost simulation for five approaches:

Table 19: Comparative cost simulation for the analysed approaches.

Direct cost
Indirect cost
Summary logistics. cost
Direct production cost
Parallel production
Replacement cost
Lost flow capacity cost
Base cost

Q_pr_ii_at random
144895,09
48563
193458,09
44217,923
10489,60
36179
102,57
90989,093

Q_repl_ii_at random
146596,29
48563
195159,29
50281,846
13830,08
31832
86,84
96030,766

Q_lfc_ii_at random
156605,98
48563
205168,98
55862,308
17145,92
36317
31,07
109356,298

Q_pr_ii_heuristic_2
155247,12
48563
203810,12
54570,077
16339,84
36179
102,44
107191,357

Q_pr_i_heuristic_1
133582,41
48563
182145,41
32873,000
0,00
36179
134,81
69186,810

Figure 9: Combined logistics costs.

Figure 10: Direct production costs.

Figure 11: Parallel production costs.

Figure 12: Replacement costs.

Figure 13: Lost flow capacity costs.

(1) Q_pr_ii_at random;

(2) Q_repl_ii_at_random;

(3) Q_lfc_ii_at_random;

(4) Q_pr_ii_heuristic_2;

(5) Q_pr_i_heuristic_1.

As seen in Table 19 and Figures 9–13 in terms of combined logistics costs and direct production costs the best is the manufacturing approach Q_pr_i_heuristic_1 whereas in terms of parallel production costs the approach Q_pr_ii_at random is the best as these costs are not generated by Q_pr_i_heuristic_1. On the other hand, when replacement costs are taken into account the approach Q_repl_ii_at random wins; however, the lowest lost flow capacity costs are generated in case of Q_lfc_ii_at random.

5. Conclusions

The article emphasises the problem of modelling the complex manufacturing system. The goal is to find such an arrangement of production activities, respectively, production paths for a given set of orders that is effective from the time and cost point of view. First of all, the mathematical model of the system consisting of necessary assumptions creates the basis for building the simulation tool which is implemented to carry out simulations of the manufacturing process. The proposed system with certain data can be evaluated from the point of view of the implemented strategy, heuristic, total production time, effective production time, robotic arm work, total replacement time, lost flow capacity, final flow capacity, total transportation time, system state indicator, system wear indicator, and system flexibility indicator.

The simulation approach may lead to finding the satisfactory solution from the point of view of either the minimal total production time criterion or the minimal replacement time criterion or the minimal lost flow capacity criterion. However, the best achieved result satisfying one analysed criterion does not usually satisfy the others. This leads to the concept of building the multicriteria models which should be devoted to finding the satisfactory solution. Searching for satisfactory solutions is limited by the number of simulations carried out. The bigger the number of simulations, the greater the probability of finding a better solution. The only limitation is the time and resources of the software and hardware.

Another important issue is to detect when a manufacturing approach stops being profitable and requires alternating into the one which does not generate excessive costs. This case is illustrated by the analysis of prolonged manufacturing costs. The intersection points inform the system operator that it is the right moment to change the manufacturing approach.

Minimising the order making time remains the main criterion to be discussed; however, verification of order making is to be carried out from the point of view of minimising costs which should result in creating a two-criterion model. On the basis of practical observations of manufacturing processes times of operations must be modified in order to make calculations more precise. Creating a simulation tool on the basis of the above assumptions will enable the authors to carry out simulation experiments analysis of which should result in general conclusions.

Another important factor which needs to be considered is how to control the production process in order to minimise the lost flow capacity of tools; i.e., the best option is to replace fully worn out tools only.

In case of considering prolonged manufacturing up till a certain moment the so far manufacturing approach becomes unprofitable. Therefore, the so-called cost intersection point shows the maximal unit cost of prolonged manufacturing.

Moreover, it is important to bear in mind that the model presented in the paper is only one way of showing how complex it is to prepare the information support for logistics business systems. First of all, it is unavoidable to carry out a thorough analysis of the real existing system which results in specification assumptions as well as the project of the system. These steps are the basis for building the proper simulation tool thoroughly tested and verified before being implemented into the real company. This paper focuses mainly on connecting mathematical modelling of the sample hypothetical system which are more than often found in the automotive industry supplying companies as manufacturing tasks are subject to frequent changes; however, costs are to be calculated immediately to enable both the customer and the manufacturer to carry out financial negotiations. There is always a gap to modify the real costs nevertheless; the starting point for subsequent negotiations should be marked.

The research objectives are fully met in the form of the mathematical model of the complex manufacturing system which is the basis for the dedicated simulator. The created simulator is the information tool which enables the simulation of the sample manufacturing process. The results of the simulation process are evaluated in detail in the case study section. Simulation experiments let us draw conclusions concerning functioning of the specific system supplied with sample introductory data. It is necessary to mention that each manufacturing system is autonomous and more than often requires equipping it with the adjusted information tool as the available platform tools lack unavoidable functionalities that need to be implemented. The research in our case focuses on laying foundations for the proper analysis of the modelled problem with the aim of preparing an error-free specification for the dedicated simulator of the manufacturing system. The simulator required thorough testing and validation before it could be employed to solve tasks stated in the paper. However, there are also specification limitations as the modelled system required adequate simplification in order to be able to let us build the simulation tool in its present form. There are numerous possibilities of adding extra functionalities to the specification with the aim of equipping the simulator with them in case of project needs. The approach to solving the problem of modelling a complex manufacturing system with the subsequent simulation experiments seems to be a useful contribution for academics who lecture on modelling and simulation matters. From the practical point of view the article presents a useful method of modelling logistics systems which leads to building a dedicated simulator of the manufacturing system which is an unavoidable information tool for training operators of logistics systems.

Further studies should focus on analysing various logistics systems characterised by their own interdependencies. It is obvious that each of them requires their own specification and project. Nevertheless, it is advisable to focus on the approach which could help develop a simulation tool that is able to carry out simulation experiments in similar types of manufacturing logistics systems.

Symbols

: | The matrix of unit assembling costs of tools in the -th manufacturing subsystem
: | The unit assembling cost of the tool in the machine placed in the -th row of the -th column in the -th subsystem
: | The assembling costs of tools in case of making the order in the -th manufacturing subsystem
: | The matrix of unit disassembling costs of tools in the -th manufacturing subsystem
: | The unit disassembling cost of the tool in the machine placed in the -th row of the -th column in the -th subsystem
: | The disassembling costs of tools in case of making the order in the -th manufacturing subsystem
: | The matrix of unit manufacturing costs in the -th manufacturing subsystem
: | The unit cost of manufacturing the -th order for the -th customer in the -th machine in the -th row in the -th manufacturing subsystem
: | The cost of manufacturing the order in the -th manufacturing subsystem
: | The matrix of possible maintenance costs due to failure of the -th manufacturing subsystem
: | The unit cost of possible maintenance procedures due to failure of tools while making the -th order for the -th customer in the -th manufacturing subsystem
: | The costs of possible failure maintenance in case of making the order in the -th manufacturing subsystem
: | The matrix of unit transportation costs for semiproducts from machines to the key point in the -th manufacturing subsystem
: | The unit transportation time of products from the machine placed in the -th row of the -th column to the key point in the -th subsystem
: | The transportation costs for semiproducts from machines to the key point in case of making the order in the -th manufacturing subsystem
: | The matrix of unit transportation costs for semiproducts from the key point in the -th manufacturing subsystem
: | The unit transportation cost of products from the key point to the machine placed in placed in the row of the column in the -th subsystem
: | The transportation costs for semiproducts from the key point to machines in case of making the order in the -th manufacturing subsystem
: | The matrix of unit transportation costs for semiproducts transported from machines to buffer stores in the -th manufacturing subsystem
: | The unit transportation cost of a product transported from the machine placed in the -th row of the -th column to the -th buffer store in the -th subsystem
: | The transportation costs for semiproducts transported to buffer stores from machines in case of making the order in the -th manufacturing subsystem
: | The matrix of unit costs of storing charge materials in the -th manufacturing subsystem
: | The storing cost of the -th charge material for making the -th product for the -th customer in the machine placed in the -th plant
: | The matrix of unit costs of storing semiproducts in the buffer stores in the -th manufacturing subsystem
: | The storing cost of the -th semiproduct made for the -th customer in the available buffer stores in the -th manufacturing subsystem
: | Costs of storing semiproducts in the buffer stores in the -th manufacturing subsystem
: | The matrix of unit costs of storing ready products in the -th manufacturing subsystem
: | The unit storing cost of the -th ready product made for the -th customer in the ready product store in the -th manufacturing subsystem
: | The total storing costs in the ready product store in the -th manufacturing subsystem
: | The matrix of maintenance costs in the -th manufacturing subsystem
: | The maintenance cost of the machine placed in the -th row of the -th column in the -th manufacturing plant
: | The matrix of standstill costs in the -th manufacturing subsystem
: | The unit standstill costs in case of making the -th product for the -th customer in the machine placed in the -th row of the -th column in the -th manufacturing subsystem
: | The matrix of costs of implementing heuristic algorithms in the -th manufacturing subsystem
: | The cost of implementing heuristic algorithms in case of making the -th product made for the -th customer in the machine placed in the -th row of the -th column in the -th manufacturing subsystem
: | The matrix of hidden operating costs in the -th manufacturing subsystem
: | The hidden unit cost in case of making the -th product for the -th customer in the machine placed in the -th row of the -th column of the -th manufacturing subsystem
: | The base manufacturing costs for the given -th approach
: | The matrix of unit transportation costs for semiproducts transported from buffer stores to machines in the -th manufacturing subsystem
: | The transportation cost of a product from the -th buffer store to machine placed in the -th row of the column in the -th subsystem
: | The unit transportation costs for semiproducts transported from buffer stores to machines in case of making the order in the -th manufacturing subsystem
: | The matrix of unit transportation costs for the charge material from the store of charges to machines in the -th manufacturing subsystem
: | The unit transportation cost of the charge material to the machine placed in the -th row of the -th column in case of making the -th order for the -th customer in the -th subsystem
: | The total transportation costs of charge material transported from the charge store to machines in case of making the order in the -th manufacturing subsystem
: | The matrix of unit transportation costs of ready products to the product store in the -th manufacturing subsystem
: | The unit transportation cost of the ready product transported the product store in case of making the -th order for the -th customer in the -th subsystem
: | The transportation cost of a ready product transported to the product store in case of making the order in the -th manufacturing subsystem
: | The matrix of parallel manufacturing costs in the -th manufacturing subsystem
: | The manufacturing cost of making the -th order for the -th customer and the -th order for the -th customer in a parallel way in the -th subsystem with the use of the -th criterion
: | The matrix of costs of replacement operations in the -th manufacturing subsystem
: | The replacement operation cost of the -th tool in the machine placed in the -th row of the -th column in the -th subsystem
: | The matrix of transportation costs for new tools which are stored in the storing place for tools in the -th manufacturing subsystem
: | The transportation cost of the new -th type tool from the storing place to the machine placed in the -th row of the -th column in the -th subsystem
: | The matrix of transportation costs for worn out tools to the storing place for used tools in the -th manufacturing subsystem
: | The transportation cost of the worn out -th type tool from the machine placed in the -th row of the -th column to the storing place for worn out tools in the -th subsystem
: | The costs of prolonged manufacturing in the -th subsystem
: | The unit cost of prolonged manufacturing in the -th subsystem
: | The cost of the replacement process in the -th manufacturing subsystem
: | The unit cost of the replacement process in the -th manufacturing subsystem
: | The cost of the lost flow capacity in the -th manufacturing subsystem
: | The unit cost of the lost flow capacity after completing the manufacturing process in the -th manufacturing subsystem
: | The company cost of the -th manufacturing subsystem
: | The fixed manufacturing cost of the -th manufacturing subsystem
: | The logistics costs of the -th manufacturing subsystem
: | The approach cost of the -th manufacturing subsystem
: | The matrix of routes
: | The number of the -th row through which the -th semiproduct for the -th customer is passed throughout the manufacturing process
: | The general matrix of structure of the manufacturing system
: | A machine located in the -th row of the -th column in the -th subsystem dedicated to performing a manufacturing operation on the -th order for the -th customer with the use of a certain type of tool
: | The base life matrix of tools used in the manufacturing system of a new brand set of tools used to manufacture elements of the order matrix
: | The base number of units of the -th product which can be manufactured in the machine in the -th row of the -th column before the -th dedicated tool is completely worn out and requires an immediate replacement
: | The adjustment matrix of tools to machines
: | The adjustment of the -th tool to the machine placed in the -th row of the -th column
: | The number of the row in the system,
: | The number of the column in the system,
: | The stage number,
: | The total number of system decisions necessary to produce all order matrix elements approved for manufacturing at the stage
: | The number of the customer,
: | The type of order,
: | The time of completing making the order by the manufacturing approach minimising the manufacturing time in the -th subsystem
: | The capacity matrix of the -th manufacturing subsystem
: | The number of the -th product units which still can be made in the machine in the -th row of the -th column with the use of the -th tool in the -th subsystem at the -th state
: | The criterion of lost flow capacity of tools
: | The time criterion of the production process
: | The criterion of the replacement process
: | The matrix of state of the the -th manufacturing subsystem
: | The number of units of the -th product already made in the machine in the -th row of the -th column with the use of the -th tool in the -th subsystem at the -th state
: | The matrix of assembling times of tools in the -th subsystem
: | The assembling time of the tool in the machine placed in the -th row of the -th column in the -th subsystem
: | The matrix of disassembling times of tools in the -th subsystem
: | The disassembling time of the tool in the machine placed in the -th row of the -th column in the -th subsystem
: | The matrix of production times in the -th manufacturing system
: | The production time of the -th type of product for the -th customer in the machine placed in the -th row of the -th column in the -th subsystem
: | The time of completing making the order by the -th approach in the -th subsystem
: | The total replacement time in the -th manufacturing subsystem
: | The matrix of transportation times for semiproducts from machines to the key manipulation point in the -th subsystem
: | The transportation time of products from the machine placed in the -th row of the -th column to the key point in the -th manufacturing system
: | The matrix of transportation times for semiproducts from the key point to machines in the -th subsystem
: | The transportation time of products from the key point to the machine placed in to the machine placed in the row of the column in the -th manufacturing system
: | The matrix of transportation times for semiproducts from machines to their buffer stores in the -th subsystem
: | The transportation time of a product from the machine placed in the -th row of the -th column to the -th buffer store in the -th manufacturing system
: | The matrix of transportation times for semiproducts from buffer stores to machines in the -th subsystem
: | The transportation time of a product from the -th buffer store to machine placed in the -th row of the column in the -th manufacturing system
: | The matrix of unit transportation times for charge material from the store of charges to machines in the -th subsystem
: | The unit transportation time of the charge material to the machine placed in the -th row of the -th column in case of making the -th order for the -th customer in the -th manufacturing system
: | The matrix of unit transportation times for ready products to the product store in the -th subsystem
: | The unit transportation time of the ready product from the machine placed in the -th row of the -th column to the product store in case of making the -th order for the -th customer in the -th manufacturing system
: | The matrix of times of storing charge materials in the -th subsystem
: | The time of storing the -th charge material for making the -th product for the -th customer in the charge store for the -th plant
: | The matrix of times of storing semiproducts in buffer stores in the -th subsystem
: | The average storing time of the -th semiproduct made for the -th customer in the available buffer store in the -th plant
: | The matrix of times of storing ready products in the -th subsystem
: | The storing time of the -th ready product made for the -th customer in the ready product store of the -th manufacturing system
: | Matrix of parallel manufacturing times in the -th subsystem with the use of the -th criterion
: | The parallel manufacturing time of making the -th order for the -th customer and the -th order for the -th customer in the -th subsystem with the use of the -th criterion
: | The moment of beginning making the -th order for the -th customer and the -th order for the -th customer in the -th subsystem with the use of the -th criterion
: | The moment of finishing making the -th order for the -th customer and the -th order for the -th customer in the -th subsystem with the use of the -th criterion
: | The matrix of times of replacement operations
: | The replacement operation time of the -th tool in the machine placed in the -th row of the -th column in the -th subsystem
: | The matrix of transportation times of new tools which are stored in the storing place for tools in the -th subsystem
: | The transportation time of the new -th type tool from the storing place to the machine placed in the -th row of the -th column in the -th subsystem
: | The matrix of transportation times for worn out tools to the storing place for used tools in the -th subsystem
: | The transportation time of the worn out -th type tool from the machine placed in the -th row of the -th column to the storing place for worn out tools in the -th subsystem
: | The number of units of the -th order for the -th customer is made at the -th stage
: | The matrix of orders at the -th stage
: | The state of the -th order set by the -th customer at the -th stage
: | The element to be made in the manufacturing zone at the -th stage
: | The -th tool,
: | The -th operation,
: | The matrix of probability coefficients of failure of tools in the -th manufacturing subsystem
: | The probability coefficient of failure of tools in case of making the -th order for the -th customer in the -th manufacturing subsystem
: | The time of prolonged manufacturing according for the -th approach in the -th subsystem
: | The matrix of adjustment of tools to operations
: | Adjustment of the -th tool to the -th operation
: | The matrix of order priority
: | The priority of the manufacturing task for the -th customer who ordered the -th product at the -th stage
: | The vector of buffer zones in the -th manufacturing subsystem
: | The buffer zone behind the machine placed in the -th column in the -th manufacturing plant
: | The number of lost flow capacity units after completing the manufacturing process in the -th manufacturing subsystem
: | The coefficient of stable costs in the -th manufacturing subsystem
: | The total number of logistics manufacturing centres
: | The -th logistics manufacturing centre.

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This paper was supported by the Project SGS/8/2018, “Advanced Methods and Procedures of Business Processes Improvement” at the Silesian University in Opava, School of Business Administration in Karvina.

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