Complexity

Complexity / 2019 / Article

Research Article | Open Access

Volume 2019 |Article ID 7237585 | https://doi.org/10.1155/2019/7237585

Robert Bucki, Petr Suchánek, "Comparative Simulation Analysis of the Performance of the Logistics Manufacturing System at the Operative Level", Complexity, vol. 2019, Article ID 7237585, 36 pages, 2019. https://doi.org/10.1155/2019/7237585

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

Academic Editor: Lucia Valentina Gambuzza
Received14 Jan 2019
Revised27 Mar 2019
Accepted01 Apr 2019
Published07 May 2019

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 [3638], 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.

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 inwhere 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 inwhere 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 inwhere 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 inwhere 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 inwhere 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 inwhere 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 inwhere 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 inwhere 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 inwhere 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