Design of Manufacturing Lines Using the Reconfigurability Principle

Abstract

Nowadays, many factories face changes on the global market and manufacturing is unpredictable. This fact creates a demand for developing new concepts of the factory which can represent a solution to these changes. This study presents a way for designing these new factory concepts, particularly a concept of the reconfigurable manufacturing lines. The methodology in this study uses characteristics of reconfigurable manufacturing systems for developing an algorithm for designing the basic factory layout. The methodology also combines classical math operations for designing the production layout with such approaches as simulation, cluster analysis, and LCS algorithm. This combination method with LCS algorithm and an entirely different approach to the design of the manufacturing line, has not yet been used. The accuracy of this methodology is then verified through the results of the complete algorithm containing these features. The main purpose of this study was to find new approaches to designing the reconfigurable factory layout. This article is presenting new ways that differ from the classical design method. The article suggests the new way is possible and the new systems also need new ways for designing and planning.

1. Introduction

The reconfigurability is a new approach enabling a cost-effective production and a fast response to the market changes. According to [1,2,3], the reconfigurable manufacturing system (RMS) is an adaptive system being able to adapt its production capacity due to the demand fluctuation and to adapt its functions for manufacturing new products. The reconfigurable manufacturing system (RMS) is designed for a quick change of the structure of the hardware and software elements in the selected product family. Such manufacturing systems are designed as modular ones and utilise the reconfigurable manufacturing machines and equipment. They frequently work on the basis of the plug and produce approaches that enable fast integration and utilisation of the latest technologies.

The structure of the reconfigurable systems has to consist of the hardware and software modules that can be easily and quickly connected from the initial system design. The modular structure of the system and the ability to respond quickly to the changes require the RMS to fulfil six basic characteristics: customisation, being able to create scales, convertibility, modularity, integrability and diagnosability. These characteristics can be implemented no only to a level of the system but also to the level of individual devices or the company as a whole.

The aspect of the system reconfiguration can be divided on the basis of several approaches, for example, into two types—the hard (physical) and soft (logical) reconfiguration [4]. An example of the hard configuration process can be the adding/removing of a machine (module) or a change of materials handling system. The soft reconfiguration process is achieved through re-programming the equipment, re-planning (or re-designing) the manufacturing process or adding/removing the staff (the employees, the shift). The reconfiguration can be also achieved by selecting a suitable product design that subsequently serves as means for achieving the target [1]. Another approach of classification is dividing RMS on the static and dynamic systems [5,6]. The static systems are based especially on the so called building blocks where another group of the dynamic systems is typical by reconfiguring through advanced robotic systems.

These reconfiguration concepts of the reconfigurable manufacturing systems are only the initial stage of the development of the production systems in the future. The manufacturing system of the future and their organisation will be closer to the emergent activities of the live organisms than the current automated lines. The manufacturing systems as a whole can be understood as multi-factor systems with a dynamic structure. It means that the production effectiveness depends on a large scale of factors changing in time. From this point of view it is thus necessary to view the manufacturing systems of the future as live organisms. Based on these predictions of the system development, it is possible to assume that the change be realised especially in the area of the large-series productions manufacturing complex products. As a matter of fact, the current manufacturing systems will not be able to work in the assigned manufacturing takt because of the customers’ requirements that change quickly [7,8,9].

However, the production of the complex products often requires various operations—press working, welding, painting and testing whose arrangement will probably remain the same. But we can expect a principal change especially in the area of the work organisation. The current lines with a static position will be replaced by a quantity of autonomous workstations and just the conceptions of the static and dynamic reconfigurable lines are the basis of these workplaces. The quantity of these workstations can be called the competency islands. It is possible to assume in a short time that these workplaces will be created by the cooperative robots and there will be mutual cooperation—people —robots. However, from a long-term point of view we assume full automation of the individual competency islands.

The basic assumption of such manufacturing systems will be their design as small, highly flexible manufacturing units with a sufficient demand. Also the activities of the system will be organised chaotically or will lack planning from the point of view of an observer. However, each module of such a system will be controlled through the logic of a higher level management that will enable the module to behave relatively autonomously. In reality it will be an organised chaos. The integrated modules will also require the capability of mutual communication in real time during which they will share all necessary data and information.

The basis of the principle of the competency island’s activity will be especially the alteration of the selected product to an intelligent entity. The product being able to organise its processing will change the overall planning approach in the companies. However, the new organisation of planning and management will request new approaches in the area of the production planning and control [10].

Therefore this study presents the results of the investigation on the possibility of realising a computing model for production planning methodology of the manufacturing line utilising the principles of reconfiguration. The design of these types of the production lines could be the initial phase for creating the competency islands described in the previous part. However, before the next phase and implementing the given planning process it is necessary to verify and validate the designed methodology using the principle of the reconfigurable manufacturing lines. This area of the possibility of realising a computing model can be represented as a scientific hypothesis that investigates the possibilities of creating such a methodology. In our study the scientific problem is thus solved through a mathematical model that is subsequently verified on a particular case and its goal is to verify the validity of our hypothesis. The simulation was realised through the simulation software Tecnomatix Plan Simulation and a parametric simulation model was created for the verification purposes.

2. Materials and Methods

2.1. The Methodology for Designing the Reconfigurable Manufacturing Lines

The proposed methodology for designing the reconfigurable line can be divided into several partial stages. The mutual integration of the stages creates the overall concept of the reconfigurable manufacturing line (RML) requiring a fast adaptation to the frequently changing product portfolio and fluctuating production volume in dependence on the customer requirements. The methodology contains a few basic stages, as shown in Figure 1, that will be described in the next sections.

2.1.1. Input Data Stage

The first part of the designed algorithm is the generating of the input data in the stage Input Data that replace the real data from the customer. The data is saved in the database and is updated after each completion of production of a particular product $kn$ of the product family α, if it was changed. Because of updating the parameters a new product or a change of the customer requirements for the already entered products in the system can be recorded.

2.1.2. Stage of Verifying the Capacity and Operational Availability (Verification)

In the framework of the stage of verifying the capacity and operational availability we gradually carry out a two-stage verification of all products from the quantity P. The first part is the conformity verification between the product operations and operations that can be realised by the manufacturing system equipment as it shown Algorithm 1. The verification requires calculating the total number of the common elements $S{p}_{kn}$. After determining the quantity of the common elements it is necessary to calculate the conformity for a particular product $K_{kn}.$ The conformity result must not be lower than 100% because the manufacturing system can manufacture only such a product for which it is able to ensure processing all its operations. In the case the percentage conformity is lower, an agreement with the customer is necessary. The next step of the verification stage is to determine the critical ratio (CR) for the product kn. The critical ratio (CR) is a rule used in priority sequencing of work waiting for processing at a work centre. If this rule is used for sequencing, the job with the lowest CR is scheduled first. A CR less than 1 indicates that the job will be late, and a CR greater than 1 indicates that the job can be completed ahead of the due date if no unexpected delay occurs. The critical ratio in this case serves as an identifier of the resulting manufacturing time for individual products. Before calculating the critical ratio we have to calculate the values of the manufacturing volume ${\overline{n}}_{kn}$, increased by 0% of the technologically inevitable scrap. The next step is to calculate the value of the real time necessary for manufacturing every product $R{e}_{t_{kn}}$. Through the value of the time proportion defined by the customer and the time that is really necessary for manufacturing the product we can determine the resulting production time $t_{kn}$. If the really necessary manufacturing time is longer than the time defined by the customer for the transfer, we will shorten the resulting time for the product to be produced. If the real time for manufacturing the product is shorter than the time defined by the customer, the resulting time $t_{kn}$ is given a lower value. The last possibility is—if the result of the critical ratio is 1, then the resulting time $t_{kn} \mathrm{is}$ equal with both times [4].

Algorithm 1 Algorithm of input data stages and verifying the capacity and operational availability

01:  stage Input Data()

02:    generate value ($NAM{E}_{kn}$, $ZA$, $POC{P}_{kn}$, $T_{kn}$, $n_{kn}$, $C{r}_{t_{kn}}$, ${\xi }_{kn}$, $P$)

03:    call Verification($NAM{E}_{kn}$, $ZA$, $POC{P}_{kn}$, $T_{kn}$, $n_{kn}$, $C{r}_{t_{kn}}$, ${\xi }_{kn}$, $P$)

04:  stage Verification($NAM{E}_{kn}$, $ZA$, $POC{P}_{kn}$, $T_{kn}$, $n_{kn}$, $C{r}_{t_{kn}}$, ${\xi }_{kn}$, $P$)

05:     $S{p}_{kn}=Consistency Elements of Matrix NAM{E}_{kn} and ZA$

06:    $K_{kn}=\frac{POC{P}_{kn}}{S{p}_{kn}}\ast 100 [\%]$

07:       if $K_{kn}<100$ then

08:       call $\mathrm{Customer}(n_{kn},C{r}_{t_{kn}}$)

09:       else

10:      ${\overline{n}}_{kn}=\frac{n_{kn}}{1-\frac{{\xi }_{kn}}{100}} \left[p.\right]$

11:        $R{e}_{t_{kn}}={\overline{n}}_{kn}\ast {\left({{\displaystyle \sum }}_{m=POC{P}_{kn}}^{n=1}{t}_{ij}\right)}_{kn} \left[min.\right]$

12:        $CR=\frac{C{r}_{t_{kn}}}{R{e}_{t_{kn}}} \left[1\right]$

13:         if $CR<1$ then

14:         $t_{kn}=R{e}_{t_{kn}}$

15:         elseif $CR=1$ then

16:          $t_{kn}=R{e}_{t_{kn}}=C{r}_{t_{kn}}$

17:         else

18:          $t_{kn}=C{r}_{t_{kn}}$

19:         end if

20:        $w_{k{n}_i}=⎡\frac{{\left(t_{ij}\right)}_{kn}\ast {\overline{n}}_{kn} }{ t_{kn}}⎤\left[1\right]$

21:        $q_{kn}={{\displaystyle \sum }}_{i=1}^m{w}_{k{n}_i}\left[p.\right]$

22:         if $q_{kn}<ZQ$ then

23:          call $\mathrm{Customer}(n_{kn},C{r}_{t_{kn}}$)

24:         elseif $q_{kn}=ZQ$ then

25:          call $\mathrm{Customer}(n_{kn},C{r}_{t_{kn}}$)

26:         else

27:          call Verification in Time$(POC{P}_{kn}, T_{kn}, {\overline{n}}_{kn}, ZQ, t_{kn}$)

28:         end if

28:    end if

The last part of the verification stage is to determine the necessary capacities of the equipment for the products $w_{k{n}_i}$. The calculated sum of the theoretical value of the necessary number of devices $q_{kn}$ is subsequently compared with the value of the total number of the system’s devices ZQ. By comparing the values, we will obtain a preliminary overview about the possibilities of manufacturing the products. If the sum of the equipment for each product is not equal with the total defined number of the workplaces of the system, so the system has not a sufficient amount of devices for the time $t_{kn}$. If there is equality $t_{kn}=C{r}_{t_{kn}}$ of the resulting time and the time defined by the customer of transferring the product, it is inevitable to consult the problem with the customer. The stage Customer replaces the decision-making of a real customer in the algorithm. However, if the time $C{r}_{t_{kn}}$ defined by the customer is higher, there is a reserve for manufacturing the product $kn$ and we do not contact the customer. If the value calculated does not exceed the amount of the devices in the system, the calculation continues for the verification stage concerning the product manufacturability in time [11].

2.1.3. Verification of Products’ Manufacturability in Time

After verifying the products in the previous stage, it is necessary to specify the next decision-making criterion—the manufacturability the process is shown in Algorithm 2. This criterion will enable to decide about the manufacturing possibilities of the products and creation of the product family during individual iterations. This criterion is the minimal time of making a product at a maximal utilisation of the available system resources $Mi{n}_{t_{kn}}$. If we are able to define this minimal time, we can guarantee the possibility of manufacturing a particular product in the next iteration. The basic assumption for determining the minimal time is the validity of the algorithm condition which says that the number of the allocated resources $P_{prir}$ has to be lower or equal to the number of all available resources of the system ZQ.

The sequence of determining the minimal time depicted in the algorithm begins with calculating the total number of the machines ${\dot{q}}_{kn}$.

The basis for determining the number of machines is the value x that is increased on the interval $\langle 0,t_{kn}\rangle$ by the lowest possible selected value of the increment $x=1$.

Algorithm 2 Algorithm of product verifiability in time

01:  stage Verification in Time($POC{P}_{kn}, T_{kn}, {\overline{n}}_{kn}, ZQ, t_{kn}, x$)

02:   ${\dot{q}}_{kn}={{\displaystyle \sum }}_{i=1}^m⎡\frac{t_{ij}\ast {\overline{n}}_{kn} }{ t_{kn}-x}⎤\left[p.\right]$

03:  $P_{prir}=call$$\mathrm{Assignment}\mathrm{Submodule}(Q, PO, XA, n$)

04:    if $P_{prir}<{\dot{q}}_{kn}$ then

05:    $x=x-1$

06:     $Mi{n}_{t_{kn}}=t_{kn}-x$$\left[min.\right]$

07:     $Z_{t_{kn}}=t_{kn}-Mi{n}_{t_{kn}}$$\left[min.\right]$

08:     call Priority of Products($NAM{E}_{kn}$, $POC{P}_{kn}$, $t_{kn}$, $max {\Omega }_{SPT}$$,max {\Omega }_{CA}$)

09:   else

10:     $x=x+1$

11:     call Verification in Time($POC{P}_{kn}, T_{kn}, {\overline{n}}_{kn}, ZQ, t_{kn}, x$)

12:   end if

The value of the number of the machines ${\dot{q}}_{kn}$ is subsequently verified in the assignment stage. The essence of this stage consists in assigning the necessary operations to particular existing machines of the system, namely in dependence on the need of the amount of the equipment and operation type. After each assignment of the calculated number of machines the condition introduced in the algorithm is verified. The condition says that in the case if the number of the assigned machines $P_{prir}$ is not lower than the amount of the calculated devices, it will be necessary to increase the increment value x by 1. This process repeats until the condition is fulfilled. After fulfilling the condition it is necessary to calculate the last value x in the case of which all devices could be assigned. We will calculate it by subtracting the value 1 from x. The determined value x subsequently is equal to the minimal time when the available resources of the system $Mi{n}_{t_{kn}}$ are maximally utilised [4,5,6,7,8,9,10,11,12].

2.1.4. Determining the Product Priority

The Algorithm 3. shows stage which is the basic part for creating the product family $\alpha$ because the calculation of the priorities of the products will determine their overall sequence for the assignment Ω. The products will thus be assigned to the product family directly on the basis of their availability therefore we chose two determining criteria for determining the product priority. The first criterion is the assessment of the products based on the calculated times $t_{kn}$, when the products with lower processing time have the priority. This criterion is based on the SPT rule (short processing time). For acquiring the quantity of the SPT we will arrange the values of the times $t_{kn}$ from the lowest up to the highest ones. The second criterion is the product similarity based on the types of the operations that are defined by their technology or assembly. The basis for determining the priority of this criterion is the utilisation of two stages—the cluster analysis (CA) and the algorithm longest common subsequence (LCS). The result of these stages is the sequence quantity CA [13,14].

Algorithm 3 Algorithm for determining the product priority

01:  stage Priority of Products($SUMp, NAM{E}_{kn}$, $POC{P}_{kn}$, $t_{kn}$, $max {\Omega }_{SPT}$$,max {\Omega }_{CA}$)

02:   call $\mathrm{LCS}\mathrm{Submodule}(NAM{E}_{kn,\dots km}, NAM{E}_{\alpha }$)

03:  $orde{r}_{C{A}_{kn}}=$ call $\mathrm{CA}\mathrm{Submodule}(S{p}_{kn,km},POC{P}_{kn}, POC{P}_{km}$)

04:  $\mathbf{sort}the value t_{kn} from low to high$

05:   $\mathbf{determine}the order SPT for orde{r}_{SP{T}_{kn}} for each kn$

06:  $\mathbf{determine}the order CA for orde{r}_{C{A}_{kn}} for each kn$

07:   ${\Omega }_{SPT}=\frac{max {\Omega }_{SPT}}{SUMp} \left[1\right]$

08:   ${\Omega }_{CA}=\frac{max {\Omega }_{CA}}{SUMp} \left[1\right]$

09:   ${\Omega }_{SP{T}_{kn}}=max {\Omega }_{SPT}-\left({\Omega }_{SPT}\ast orde{r}_{SP{T}_{kn}}\right)\left[1\right]$

10:   ${\Omega }_{C{A}_{kn}}=max {\Omega }_{CA}-\left({\Omega }_{CA}\ast orde{r}_{C{A}_{kn}}\right)\left[1\right]$

11:   ${\Omega }_{kn}={\Omega }_{SP{T}_{kn}}+{\Omega }_{C{A}_{kn}}\left[1\right]$

12:   $\Omega =set of values {\Omega }_{kn} sort from high to low$

13:   call Family Time$(NAM{E}_{kn, \dots , km}$, $t_{kn\dots , km},T_{kn\dots , km}$, ${\overline{n}}_{kn, \dots , km}$$,NAM{E}_{\alpha }={({\alpha }_{ij})}_{m=Q}^{n=1}$)

Based on the defined quantities SPT and CA, it is then possible to determine the sequence of their values $orde{r}_{SP{T}_{kn}}$ and $orde{r}_{C{A}_{kn}}$ based on the following logic:

$$\begin{array}{l}ifsetSPT=\left\{k1,k2,\dots \right\}=>orde{r}_{SP{T}_{k1}};orde{r}_{SP{T}_{k2}}=0;1;\dots \\ ifsetCA=\left\{k2,k1,\dots \right\}=>orde{r}_{SP{T}_{k2}};orde{r}_{SP{T}_{k1}}=0;1;\dots \end{array}$$

(1)

The input parameters from previous stages enter again the stage of determining the product priority, however, the values of the defined weights $max {\Omega }_{SPT},\max {\Omega }_{CA}$ are important. They define which of the criteria will be more important for defining the overall sequence Ω. The determination of the previous values of individual maximal weights is dependent, in this case, on the following equation:

$$1=max{\Omega }_{SPT}+max{\Omega }_{CA}$$

(2)

Through defining the real sequence from the point of view of the time criteria SPT and similarity CA we can unite the values of their weights and the result will be the overall product sequence Ω. However, first of all it is necessary to calculate the unit index of the weight ${\Omega }_{SPT}$ for both criteria. The final phase of determining the sequence Ω for creating the product family is the calculation of the values of the weights ${\Omega }_{kn}$ for both criteria and also for the individual products kn. These values of both criteria are subsequently added up for each product independently and the resulting values are ordered from the highest to the lowest one. By ordering the results we achieve finally the total sequence Ω.

2.1.5. Creating the Product Family and the Stage of the Resource Availability

The main part of the designed methodology is the stage of creating the product family and the stage of the resource availability. As mentioned in the previous chapter, these stages are created by several partial steps. These steps can be united into three main parts that will be described in the next sections. The first part is the creation of a cluster of products α and defining the production volume for the defined operations of the manufacturing configuration. The second part is the calculation of the necessary quantity of the machines $q_{\alpha }$ for the designed configuration and its subsequent verification through the simulation software. The third part specifies the necessary backup of the equipment for the verified configuration containing two or more products. The products are assigned gradually in dependence on the designed total sequence of assignment Ω and the decision about the amount of the non-assigned equipment of the system. The last step is the verification of the configuration in the assignment stage that will determine the volume of the assigned machines for the products of the family and will send the values $P_{prir}$ to the decision-making stage.

Creating Product Family—Part 1

The clustering of the products into a product family is the basic part of the designed methodology assuming their mutual similarity from the point of view of the operation types (production technology) and this process show Algorithm 4. The creation of the family requires not only finding the same operations for manufacturing the products, it is also necessary to determine the longest mutual sequence of the same operations. This operation sequence is defined by the stage LCS [15]. The algorithm of the LCS stage will find and determine this operation sequence of a couple of products, in the case of an already created cluster of two or more products it creates a sequence between the couple product—the created cluster α. This mutual sequence of the same operations can be written as the matrix $LC{S}_{\alpha }$ that is the main entrance parameter for the next parts of the algorithm.

Algorithm 4 Creating the Product Family and the Resource Availability—Part 1

01:  stage Family Time($NAM{E}_{kn, \dots , km}$, $t_{kn\dots , km},T_{kn\dots , km}$, ${\overline{n}}_{kn, \dots , km}, NAM{E}_{\alpha }={({\alpha }_{ij})}_{m=Q}^{n=1}$)

02:   $(LC{S}_{\alpha }={(lc{s}_{do})}_{m=S{p}_{\alpha }}^{n=1}, S{p}_{\alpha }$$)=\mathbf{call}\mathrm{LCS}\mathrm{Submodule}(NAM{E}_{kn,\dots km}, NAM{E}_{\alpha }$)

03:   rename elements of matrix $NAM{E}_{kn, \dots , km}$$\to OPA={(op{a}_{io})}_{m=POC{P}_{kn}}^{n=1}, OPB={(op{b}_{jo})}_{m=Q}^{n=1}$

04:  $\mathbf{rename}\mathrm{elements}\mathrm{of}\mathrm{matrix}NAM{E}_{\alpha }$$\to OPB={(op{b}_{jo})}_{m=Q}^{n=1}$

05:   $\mathbf{rename}\mathrm{elements}\mathrm{of}\mathrm{matrix}{T}_{kn\dots , km}$$\to TA={(t{a}_{io})}_{m=POC{P}_{kn}}^{n=1}, TB={(t{b}_{jo})}_{m=Q}^{n=1}$

06:   $\mathbf{rename}\mathrm{elements}\mathrm{of}\mathrm{matrix}{N}_{\alpha }$$\to TB={(t{b}_{jo})}_{m=Q}^{n=1}$

07:    $\mathbf{for}k=1$ to $POC{P}_{kn, \dots , km}$

08:     $\mathbf{if}POC{P}_{kn, \dots , km}=k$ then

09:       $T\alpha =Minimal Value from \{t_{kn}, \dots , t_{km}\} \left[min.\right]$

10:       $x_{kn}=\frac{T\alpha \ast {\overline{n}}_{kn}}{t_{kn}} \left[p.\right]$

11:       $n{\alpha }_i={{\displaystyle \sum }}^{ }\left(x_{kn},\dots ,x_{km}\right) for operation i \left[p.\right]$

12:       $L{c}_i=count of kn for operation i \left[p.\right]$

13:       $n_z={{\displaystyle \sum }}_{i=1}^{Pk{n}_{\alpha }}{x}_{kn}\left[p.\right]$

14:       call Family Sources()

15:     else

16:       $\mathbf{if}op{a}_{io}=lc{s}_{do}$ then

17:        $\mathbf{if}op{b}_{jo}=lc{s}_{do}$ then

18:         $i=i+1$

19:         $j=j+1$

20:         $d=d+1$

21:         $lc{s}_{do}={\alpha }_{kj} \left[1\right]$

22:         $t{\alpha }_{kj}=t{a}_{io}+t{b}_{jo} \left[min.\right]$

23:       else

24:         $j=j+1$

25:         $op{b}_{jo}={\alpha }_{kj} \left[1\right]$

26:         $t{\alpha }_{kj}=t{b}_{jo} \left[min.\right]$

27:       end if

28:      else

29:       $\mathbf{if}k=S{p}_{\alpha }$ then

30:         $j=j+1$

31:         $op{b}_{jo}={\alpha }_{kj} \left[1\right]$

32:         $t{\alpha }_{kj}=t{b}_{jo}\left[min.\right]$

33:       else

34:         $i=i+1$

35:         $op{a}_{io}={\alpha }_{kj} \left[1\right]$

36:         $t{\alpha }_{kj}=t{a}_{io} \left[min.\right]$

37:       end if

38:      end if

39:     end if

40:    next k

Several products or an already existing cluster enter the connection process during creating a cluster. Their operations are generally defined by the matrices $NAM{E}_{kn, \dots , km}$ for the products and the matrix $NAM{E}_{\alpha }$ for the cluster. However, a general designation is to be created for the connection process, i.e., for the product entering the cluster and a product or a set of products fulfilling the conditions of assigning the resources. The same process of renaming is necessary for the matrix T_kn expressing the operation times for the product and also for the matrix of operation times $N_{\alpha }$ of the created cluster α. Owing to the renamed matrix elements it is possible to clarify the principle of the clustering algorithm function in a simpler way. The first step defines the condition comparing the operation of the entering product $op{a}_{io}$ with the matrix element of the common elements $lc{s}_{do}$. If the operations are the same, another condition continues and compares again the matrix element $op{a}_{io}$ with the element of the matrix of common elements $lc{s}_{do}$. If all three elements are identical the operation is assigned to the new final matrix $NAM{E}_{\alpha }$ and it is necessary to add up the operation time $t{a}_{io}$ and $t{b}_{jo}$ of these identical operations.

However, if the element $op{a}_{io}$ is not identical with the element $lc{s}_{do}$, it is necessary to verify the variable k with the number of elements of the matrix designated as $S{p}_{\alpha }.$ If there are no common elements any more, only the operations from the second matrix $OPB$ are assigned, if not all common elements are assigned, the elements from $OPA$ are assigned to the matrix $NAM{E}_{\alpha }$. The whole assignment cycle takes place until the variable equals the value of the maximal number of the $POC{P}_{kn}$ operations from the products selected.

The last part of the described algorithm is the calculation of the production volume for the individual production levels $n{\alpha }_i$ that will produce certain types of operations given by the matrix $NAM{E}_{\alpha }$. Every manufacturing level is thus determined for an independent type of operation and a various quantity of the products will be processed in its framework. Therefore from the capacity point of view a particular manufacturing volume has to be calculated for each level. Also the durations $t_{kn}$ differ from product to product therefore it is necessary to determine the common time $T_{\alpha }$ during which the ratio of the product quantity of the product family is produced. The common time of the product family is actually a minimal time from the quantity of all manufacturing times of the products that are assigned to the product family. After the time $T_{\alpha }$ is over, the production of the product belonging to these values will be finished and the system configuration can be re-configurated again for another product family [16].

The calculation of the manufacturing volume for individual operational levels will be preceded by a calculation of the manufacturing volume for the product per determined time period $T_{\alpha }$ regarding to the original time $t_{kn}.$ However, these manufacturing volumes are determined for the product, therefore it is necessary to calculate the manufacturing volume for each operational level independently depending on the types of the products and operations that are assigned to the manufacturing levels.

In the next part of the algorithm we will design the manufacturing configuration compared with the simulation model and the criterion of this model will require fulfilling the conditions for processing the given manufacturing volume. Because of this condition it is necessary to calculate the whole assigned manufacturing volume by the sum of all re-calculated manufacturing volumes for the products. This volume $n_z$ will be a comparison criterion for fulfilling the conditions of the configuration being tested.

Creating the Product Family—Part 2

The basis for the production layout design is to determine the necessary amount of equipment for a particular operation of the given manufacturing/operational level and Algorithm 5. show this process. The number of devices of the manufacturing level is dependent on the calculated takt and rhythm of the whole line. The takt and rhythm of production are the values giving the average time interval between the transfer of two parts that follow each other. However, the rhythm of line operation takes into account the production time reduced by the total time of the losses due to the technical or organisational causes. For the given calculation we will not assume the time of losses due to the organisation breaks and therefore it is not included in the algorithm calculation. The failure of the equipment due to technical causes is not the subject of this calculation but it is taken into consideration in the modules of the line backups. The last factor that can be defined as a loss of time is the time of the line reconfiguration. However, this time does not affect the calculation of the capacities and there we take it into account even for determining the total iteration time.

The calculated number of devices need not be able to process the necessary products per the time defined and therefore the designed configuration is to be verified. For the verification purposes we created a correction coefficient $K_p$ that reduces the value of the line rhythm and the number of devices is growing in this way. The rhythm of the line operation is calculated on the basis of an adapted formula of the line rhythm $r_{{\alpha }_{i_p}}$.

Through the determined rhythm of the line operation it is possible to calculate the synchronisation coefficient ${\vartheta }_{\alpha i_p}$ for every operation. But the synchronisation coefficient formula is to be adapted. As a matter of fact, several products of the product family pass through the manufacturing level. The total time calculated for the product family makes is thus to be divided by the number of these products passing through the given operation $L{c}_i$.

By rounding up the coefficient value we will obtain an integral number defining the total amount of the necessary workplaces for the given operation $w_{\alpha i_p}$.

The next part of calculating the algorithm is the calculation of the sum of the total number of the line workplaces $q_{{\alpha }_{i_p}}$. This value will be an assessing criterion of feasibility for the configuration of the product family cluster α in the next module [17,18].

Algorithm 5 Creating the Product Family and the Resource Availability—Part 2

01:   stage Family Sources($T\alpha$$,T_{kn\dots , km}$$,L{c}_i$, $n{\alpha }_i$$,n_z$)

02:      do while $n_z>n{s}_p$

03:      $Kp=1+p \left[1\right]$

04:      $r_{{{\alpha }_i}_p}=\frac{T\alpha }{n{\alpha }_i\ast K_p } \left[min/p.\right]$

05:      ${\vartheta }_{\alpha i_p}=\frac{t{\alpha }_{ij}}{L{c}_i\ast r_{{{\alpha }_i}_p}}, j=1 \left[1\right]$

06:      $w_{\alpha i_p}=⎡{\vartheta }_{\alpha i_p}⎤ \left[1\right]$

07:      $q_{{\alpha }_p}={{\displaystyle \sum }}_{i=1}^m{w}_{{\alpha }_{i_p}}\left[p.\right]$

08:      ${\tau }_{\alpha i_p}=\frac{t{\alpha }_{ij}}{w_{{\alpha }_i}\ast r_{{\alpha }_i}\ast L{c}_i}$ $\left[1\right]$

09:      $\left(T_{sim},n{s}_p={{\displaystyle \sum }}_{i=1}^{Pk{n}_{\alpha }}n{s}_{k{n}_p}\right)=$

10:      call Simulation Submodule($POC{P}_{kn}, T_{kn}, {\overline{n}}_{kn}, ZQ, t_{kn}$)

11:      $p=p+0,02 \left[1\right]$

12:     loop

13:    call Family Backup($T\alpha$$,{\lambda }_n$, ${\mu }_n, {\Phi }_n,$ $T_n$, $P_{p_n}$, ${{\displaystyle \sum }}^{ }{\left(D_{f-g}\right)}_n, {{\displaystyle \sum }}^{ }{\left(U_{f-g}\right)}_n$, ${\Delta }_{t{z}_i}$)

Besides the calculation of the total number of the workplaces we can also calculate the time utilisation of the operational level ${\tau }_{\alpha i_p}$ and assess the utilisation of the workplace in the future. The results of the utilisation can be also compared with the results of the sub-model simulation stage.

As already mentioned, the designed configuration is to be verified in the next step, namely in the simulation stage. The simulation was realised through the simulation software Tecnomatix Plan Simulation and a parametric simulation model was created for the verification purposes. However, especially the results concerning the quantity of the processed products $n{s}_p \mathrm{are}$ an important aspect of the realised simulation.

Based on the precious part we determined the total sum of all assigned manufacturing volumes $n_z$ for the individual products $kn$; this value will be compared with the value of all processed products $kn$ achieved by simulation. If this condition is valid, the manufacturing configuration with the calculated amount of the devices $q_{{\alpha }_p}$ is suitable and enables to manufacture the given production volume.

If this condition is not valid, it is necessary to utilise the described correction coefficient that will be cumulatively increased always when the condition about the increment value is not fulfilled. The increment value was stated at the value of $p=0.02$ based on several experimental calculations. Owing to simulation we calculated and verified the configuration up to defining the configuration that is capable to process the given production volume during the defined time $T\alpha$.

Creating the Product Family—Part 3

The calculation of the backup for the machines is the last stage of the algorithm of creating the product family shown in Algorithm 6. The calculation of the equipment backups is based on the data about the failure rate occurring during the production of the previous product family. If a failure developed in the set of equipment in the previous iteration there is a higher probability that this failure will repeat and therefore it is necessary to create reserves in the form of a backup machine. In general we can differentiate two types of the machine backup—the cold and warm backup. In the case of the cold backup the equipment is inactive when no failure occurs in the line, however, when a failure develops that device is put into operation. The equipment cold backup needs a certain run-up time designated as ${\Delta }_{t{z}_i}$. Compared with cold backup the warm backup in an uninterrupted operation and no run-up time is necessary, however, the possible failures of the warm back-up can be a problem.

Algorithm 6 Creating the Product Family and the Resource Availability—Part 3

01:   stage Family Backup($T\alpha$$,{\lambda }_n$, ${\mu }_n,{\Phi }_n,$ $T_n$, $P_{p_n}$, ${{\displaystyle \sum }}^{ }{\left(D_{f-g}\right)}_n,{{\displaystyle \sum }}^{ }{\left(U_{f-g}\right)}_n$, ${\Delta }_{t{z}_i}$)

02:    $T_{\beta }={{\displaystyle \sum }}^{ }{\left(U_{f-g}\right)}_n+{{\displaystyle \sum }}^{ }{\left(D_{f-g}\right)}_n\left[min.\right]$

03:    ${\lambda }_i=\frac{{{\displaystyle \sum }}_{n=1}^m{\lambda }_n}{m}\left[min{.}^{-1}\right]$

04:    ${\mu }_i=\frac{{{\displaystyle \sum }}_{n=1}^m{\mu }_n}{m}\left[min{.}^{-1}\right]$

05:   ${\Phi }_i=\frac{{{\displaystyle \sum }}_{n=1}^m{\Phi }_n}{m}\left[min.\right]$

06:    $T_i=\frac{{{\displaystyle \sum }}_{n=1}^m{T}_n}{m}\left[min.\right]$

07:    $P_{p_i}=⎡\frac{{{\displaystyle \sum }}_{n=1}^m{P}_{p_n}}{m}⎤\left[1\right]$

08:    ${{\displaystyle \sum }}^{ }{\left(U_{f-g}\right)}_i=\frac{{{\displaystyle \sum }}_{n=1}^m{{\displaystyle \sum }}^{ }{\left(U_{f-g}\right)}_n}{m}\left[min.\right]$

09:    ${{\displaystyle \sum }}^{ }{\left(D_{f-g}\right)}_i=\frac{{{\displaystyle \sum }}_{n=1}^m{{\displaystyle \sum }}^{ }{\left(D_{f-g}\right)}_n}{m}\left[min.\right]$

10:    $K_{vi}=\frac{T_i}{T_i+{\Phi }_i}\left[1\right]$

11:    choose Device with $Min\left(K_{vi}\right)$

12:    $\overline{T_i^s}=\frac{2{\lambda }_i+{\mu }_i}{{\lambda }_i^2}\left[min.\right]$

13:    $\overline{{\Phi }_i^s}=\frac{2{\lambda }_i+{\mu }_i}{2{\mu }_i\left({\mu }_i+{\lambda }_i\right)}\left[min.\right]$

14:    $K_{vi}^s=\frac{\overline{T_i^s}}{\overline{T_i^s}+\overline{{\Phi }_i^s}}\left[1\right]$

15:    $\overline{T_i^t}=\frac{3{\lambda }_i+{\mu }_i}{2{\lambda }_i^2}\left[min.\right]$

16:    $\overline{{\Phi }_i^t}=\frac{3{\lambda }_i+{\mu }_i}{2{\mu }_i\left({\mu }_i+2{\lambda }_i\right)}\left[min.\right]$

17:    $K_{vi}^t=\frac{\overline{T_i^t}}{\overline{T_i^t}+\overline{{\Phi }_i^t}}\left[1\right]$

18:    ${{\displaystyle \sum }}^{ }{\left(U_{f-g}\right)}_z=T_{\alpha }-{{\displaystyle \sum }}^{ }{\left(U_{f-g}\right)}_i-P_{p_i}\ast {\Delta }_{t{z}_i}\left[min.\right]$

19:    ${{\displaystyle \sum }}^{ }{\left(D_{f-g}\right)}_z={{\displaystyle \sum }}^{ }{\left(U_{f-g}\right)}_i+P_{p_i}\ast {\Delta }_{t{z}_i}\left[min.\right]$

20:    $K_{v{z}_{i,{\Delta }_t}}^s=\frac{{{\displaystyle \sum }}^{}{\left(U_{f-g}\right)}_z}{{{\displaystyle \sum }}^{}{\left(U_{f-g}\right)}_z+{{\displaystyle \sum }}^{}{\left(D_{f-g}\right)}_z}\left[1\right]$

21:    $K_{v{i}_{{\Delta }_t}}^s=K_{v{z}_{i,{\Delta }_t}}^s+K_{vi}+\left(1-K_{vi}^s\right)\left[1\right]$

22:     if $K_{v{i}_{{\Delta }_t}}^s>K_{vi}^t$ then

23:      determine $Warmstandby\left(q_{\alpha }=q_{\alpha }+1\right)$

24:     else

25:      determine $Coldstandby\left(q_{\alpha }=q_{\alpha }+1\right)$

26:     end if

27:    $P_{prir}$ = call $\mathrm{Assignment}\mathrm{Submodule}(Q,PO,XA,n$)

28:    call $\mathrm{Family}\mathrm{Configuration}(T_{sim}$, $P_{prir}$, $q_{\alpha },ZQ$, $t_{kn}$, Ω)

The backup equipment should possess substitutability especially for the operation level. The highest failure rate occurred in the previous iteration. The most suitable variant is a device providing a high variability level when a high failure rate exists at several workplaces. In the next parts we will aim at determining the backup of the equipment for a particular operational level regardless of the variability rate of the backup device.

The calculation of the equipment backup is defined by several input parameters, they are mutually derivable and therefore they need not be unconditionally given. The basic input data is in this case the mean time between the failures $T_n$ and the mean time between repairs ${\Phi }_n$ for a particular machine that were determined from the previous iteration.

As already mentioned in the previous section, the data is determined independently because of the reconfiguration after each iteration for every equipment. In the case of a newly formed product family the equipment can belong to a different operational level and in this way a change of the total value of the levels’ failure rate for a new line configuration develops. That is why it is important to calculate an average value of the given data of particular devices for the given operational level i. The average value of each parameter is calculated as a share of the total number of the equipment at the given level that was assigned by the stage of assignment sub-model.

Subsequently we can define the utilisation coefficient $K_{vi}$ out of these average values for each operational level i. As it is depicted in the previous algorithm, the next step is to define the minimal value of the utilisation coefficient from all calculated coefficients. The lowest value of the utilisation coefficient determines the devices and operational level with the highest failure rate during creating the previous iterations. The machines realising just this operation will be backed up by the same backup device [19].

The final configuration design also requires determining the type of the backup and therefore in the next step we will calculate the coefficient of the warm and cold backup. The calculated cold backup coefficient does not include the necessary runup time and therefore it is necessary to re-calculate the coefficient with the stated value of ${\Delta }_{t{z}_i}$.

The majority of the devices needs a short runup time defined by the value ${\Delta }_{t{z}_i}$ that is not part of the calculation of the utilisation coefficient for the cold backup. If we assume that ${\Delta }_{t{z}_i}$ is part of every runup when a failure occurs it is possible to say that it is a coefficient of the equipment failures and the defined runup time. This product can be deducted from the total operational time of the equipment operation and the total time $T_{\alpha }$—the result will be the value of the operation of the backup equipment. Based on the assumed operation times and the inactivity we can calculate the assumed coefficient of the backup utilisation with the defined runup time.

If we calculate the utilisation coefficient of the backup device $K_{v{z}_{i,{\Delta }_t}}^s$ with the defined run-up time and this value will be added to the original utilisation coefficient $K_{vi}$ of the operation level i we will obtain a value that is distorted by the missing failure rate of the backup equipment. In the case of utilising the cold backup $K_{vi}^s$ we can assume that this value contains the failure rate of the operational level devices as well as the failure rate of the backup equipment. However, the value calculated does not involve the value of the given delay; we can say that the rest of the utilisation coefficient value up to 100% is the value of the missing failure rate. This rest is to be deducted from the sum of the coefficients of the backup device and the coefficient of the operational level. In this way we will obtain the real value of the utilisation coefficient of the cold backup with the given run-up time for the defined operational level. This stage is depicted in Figure 2.

The last step is to assess the condition that compares the calculated cold and warm backup coefficients. In both cases it is necessary to reserve the devices for the system, however, the difference lies in the method of utilising the backup device during manufacturing the product family. In case the cold backup coefficient is higher, the equipment backed up by the cold backup principle will be utilised. In an opposite case the principle of the backup device through the warm backup is chosen [20].

2.1.6. Final Configuration of the Product Family

The product family created by the previous stages can contain one or more products depending on fulfilling the criteria included in the next stage of the final configuration the process show Algorithm 7. This stage fulfils several functions. The check of the conditions verifying the product family manufacturability and the assessment of the times for transferring products for the next iteration belong to the most important ones. The stage algorithm can be divided into two basic parts—in the first one we carry out the verification of the designed product family.

Several values of the parameters enabling to assess the possibility of manufacturing the family products were acquired from the previous stages. The first parameter is the number of the necessary devices $q_{\alpha }$ for the designed product family involving one or several products. The number of the assigned machines $P_{prir}$ out of the total capacity of the necessary devices of the configurations is the next value. The value of the total number of the manufacturing system machines ZQ is also necessary for the evaluation. The first step of the verification is the assessment of the condition evaluating the possibility of assigning the available devices to all required machines.

$$q_{\alpha }?P_{prir}$$

(3)

where symbol ? represents the way of concrete output of decision block in the algorithm. This representation was used because many decision blocks provide two and more possible results. Not only two as for “<” or “>”.

In the case of this condition only two situations can arise. The first situation will develop if the number of the assigned equipment of the system equals with all devices. The other situation develops if it is not possible to assign all machines that are required, and the designed product family is unrealisable for the given iteration. In the case the product family contains two or more products, it is necessary for the last assigned product given by the sequence $\Omega$ to be replaced by another one in a defined sequence. This can be realised if the product given by the sequence $\Omega$ that has not been assigned to a product family in the current iteration still exists.

If the values of the number of the assigned and required devices equal the availability of the system equipment will be verified. The second condition compares the amount of the equipment required by the designed product family with the total number of the devices that are available in the manufacturing system. In this condition only two possible situation develop again due to the fact that a case in the framework of which the number if the required machines is higher than the total number of the devices of the system logically cannot develop (regarding to the previous condition).

However, in the case when the number of the equipment assigned to the family $q_{\alpha }$ is lower than the total amount of the devices in the system the product family can be assigned another product given by the sequence $\Omega$. The next step is then repeated testing the condition of the product family with the assigned product. The assignment of a new product is possible only in the case if there is still a product that was not assigned to a product family during the current iteration.

$$q_{\alpha }?ZQ$$

(4)

During a gradual verification of all assigned products several variants of the product families are continuously created. Out of these variants of the product families just the variant fulfilling both conditions is chosen. In this way are verified all products and their selection is determined by the condition of the operation similarity and the total processing time.

Another part of the algorithm is the assessment of the production feasibility that is not assigned to the current iteration. However, it is necessary to define the total processing time of the family $T_I$ before this verification and it is also necessary to define the input parameters that are defined in the next sections.

The designed product family is always created as a temporary configuration for a group of products whose duration is defined by the minimal processing time of the product of the family $T_{\alpha }$. During this time period only the product with a minimal selected time $t_{kn}$ will be completed—the time $T_{\alpha }$. The other products of the family will be manufactured only in this time period (duration). The completion of the unfinished products of the family and other non-assigned products will continue in the next iteration and thus in the reconfiguration of the system. The reconfiguration of the manufacturing line system is not part of the times of processing the products therefore it is necessary to take it into account by the iteration time. The time that will be necessary for the process of changing the configuration system $T_r$ is added to the total production time. The next time affecting the total time of processing a product family is the simulation time $T_{sim}$ during which the production of all assigned $n{s}_p$ in the simulation is realised. The simulation time need not equal the given time $T_{\alpha }$, therefore a simple condition for determining the total production time of the product family $T_I$ taking the reconfiguration time into account can be defined.

$$T_r+T_{sim}?T_r+T_{\alpha }$$

(5)

The assessment of the product not assigned to a product family is the last part of the algorithm. Two main conditions—the comparison of the total production time of the family products $T_I$ with the final time necessary for transferring the product $C{r}_{t_{kn}}$ and the residual time $Z_{t_{kn}}$ by which the manufacturing process can be reduced.

Algorithm 7 Creating the final configuration of the product family

01:   stage $\mathrm{Family}\mathrm{Configuration}(T_{sim}$, $P_{prir}$, $q_{\alpha }, ZQ$, $t_{kn}$, Ω)

03:    if $q_{\alpha }>P_{prir}$ then

04:     if $0<count of not verified kn$ then

04:      replace $last kn next kn from \Omega$

04:      call $\mathrm{Family}\mathrm{Configuration}(T_{sim}$, $P_{prir}$, $q_{\alpha }, ZQ$, $t_{kn}$, Ω)

04:    else

01:      call $\mathrm{Remaining}\mathrm{products}(T_r$, $T_{\alpha }$,$C{r}_{t_{kn}}$, $Z_{t_{kn}}$)

05:    end if

04:   else

04:    if $q_{\alpha }<ZQ$ then

04:     if $0<count of not verified kn$ then

04:       assign $last kn next kn from \Omega$

04:       call $\mathrm{Family}\mathrm{Configuration}(T_{sim}$, $P_{prir}$, $q_{\alpha }, ZQ$, $t_{kn}$, Ω)

04:      else

01:       call $\mathrm{Remaining}\mathrm{products}(T_r$, $T_{\alpha }$,$C{r}_{t_{kn}}$, $Z_{t_{kn}}$)

05:      end if

04:     else

01:      call $\mathrm{Remaining}\mathrm{products}(T_r$, $T_{\alpha }$,$C{r}_{t_{kn}}$, $Z_{t_{kn}}$)

05:     end if

05:    end if

01:   stage $\mathrm{Remaining}\mathrm{products}(T_r$, $T_{\alpha }$,$C{r}_{t_{kn}}$, $Z_{t_{kn}}$)

03:    if $T_r+T_{sim}<T_r+T_{\alpha }$ then

03:     $T_I=T_{sim}+T_r$

04:    else

03:     $T_I=T_{\alpha }+T_r$

05:    end if

05:    if $0<count of not assigned kn$ then

05:     if $T_I>C{r}_{t_{kn}}$then

05:      call Customer()

05:     else

05:      if $T_I>Z_{t_{kn}}$ then

05:       call $\mathrm{Customer}\mathrm{Decision}(C{r}_{t_{kn}}$)

05:      else

05:       load $kn$ for next iteration

05:      end if

05:     end if

05:    else

05:      call $\mathrm{Manufacture}\mathrm{Line}(T_{\alpha },T_I,T_r,q_{\alpha },C{r}_{t_{kn}}$, $Z_{t_{kn}}, ZQ$,$P_{prir}$)

05:    end if

01:   stage $\mathrm{Customer}\mathrm{Decision}(C{r}_{t_{kn}}$)

05:    if $Is a possible a change of parameters for kn?$ then

05:     $t_{kn}=t_{kn}+T_I$

05:     load $kn$ for next iteration

05:      call $\mathrm{Remaining}\mathrm{products}(T_r$, $T_{\alpha }$,$C{r}_{t_{kn}}$, $Z_{t_{kn}}$)

05:    else

05:      remove $kn$

05:      call $\mathrm{Remaining}\mathrm{products}(T_r$, $T_{\alpha }$,$C{r}_{t_{kn}}$, $Z_{t_{kn}}$)

05:    end if

In the case that all products were assigned to the product family, it is possible to continue the further stage and the next criteria will not be verified. However, the residual products are verified on the basis of the first condition that says that if the time given by the customer $C{r}_{t_{kn}}$ is shorter than the time of the iteration $T_I$ an agreement with the customer is necessary. The products that possess the time $C{r}_{t_{kn}}$ that is higher than the product family pass to the next condition that compares the iteration time with the residual production time $Z_{t_{kn}}$. The residual time is a time by which it is possible to shorten the manufacturing process of the product for the next iteration. The verified product after shortening by this value will still have enough resources and can be produced during the original time $t_{kn}$. After exceeding this boundary a time shorter or the same as the minimal production time $Mi{n}_{t_{kn}}$ with a maximal utilisation of the resources remains for manufacturing. The product thus cannot be manufactured because unfinished products that will be processed in the next iteration remain in the product family. Similarly as it was in the previous case, an agreement with the customer is necessary. Only in the case of an inequality of $C{r}_{t_{kn}}>t_{kn}$ the customer need not be contacted. If there is no reserve production time, the customer will decide about the prolongation of the time necessary for manufacturing a product or rejecting the production. In the case of prolongation the processing time will be increased by the value of the time iteration and the original time necessary for processing the product.

If the customer rejects the change of conditions, the product cannot be manufactured and it has to be removed from the assigned data. The assessment of the customer’s requirement is, owing to the method described, a rapid process and in this way the customer receives the feedback immediately after entering the product order. It is also possible to test the development of the iterations for the actually assigned products and the customers are informed in advance about the possibility of manufacturing their products with the actual time of transfer and requirement adaptations and to predict the various states of the future iterations.

2.1.7. Manufacturing Line

After verifying all criteria of the previous stage the created product family continues with the stage manufacturing line Algorithm 8. The main part of this stage should be a complex simulation that will take into consideration the equipment failure rate, possible fluctuations during iterations and their influence on manufacturing the products. The possibility of determining a suitable sequence of processing the products of the product family that could significantly contribute to further optimisation of the designed line should be part of the stage. The simulation could be also utilised for predicting the future states and on the basis of the historical data about the manufactured products also the iteration for the potential products the customers can order could be simulated. The selection of the optimal solution suitable for the current conditions would be realised on the basis of criteria defined in advance. However, the stage system optimisation is not thoroughly described because it part of the research in the future.

The next stage involved in the algorithm is the creation of the dynamic layout and already the name shows the overall concept of the methodology designed for the reconfigurable manufacturing and assembly lines. This methodology of designing the lines creates a configuration limited on a certain time interval—after the time interval is over the layout is modified. The problem of the line adaptability can be solved by two concepts. The dynamically reconfigurable system is the first concept. This system is able to arrange particular parts of the line according to the required needs; however, these parts have to of a modular character. The second concept is a statically reconfigurable system reaching the change through re-planning the material flow.

These stages and the solution of the RMS system is not part of the mathematical solution and their description serves only for specifying the given stage. The purpose of these stages is to define the state of the real or simulated manufacturing system after completing a particular product family during the time $T_{\alpha }$ through the output variables. The values of these variables subsequently define if new products were ordered or there is a change of the given orders of the products. The conditions of the algorithm based on the output variables will assess whether the stage will be completed or we will return to the initial stages of the algorithm and the whole process will repeat with new data. The stage is completed only in the case if there are no products that are to be processed any more.

Algorithm 8 Manufacture Line Reconfiguration

01:   stage $\mathrm{Manufacture}\mathrm{Line}(T_{\alpha },T_I,T_r,q_{\alpha },C{r}_{t_{kn}}$, $Z_{t_{kn}}, ZQ$,$P_{prir}$)

02:    call$\mathrm{System}\mathrm{Optimization}(T_{\alpha },T_I,T_r,q_{\alpha },C{r}_{t_{kn}}$, $Z_{t_{kn}}, ZQ$,$P_{prir}$)

03:    call$\mathrm{Generating}\mathrm{Dynamic}\mathrm{Layout}(T_{\alpha },T_I,T_r,q_{\alpha },C{r}_{t_{kn}}$, $Z_{t_{kn}}, ZQ$,$P_{prir}$)

04:     if $Was all product kn in databse manufacture?$ then

05:     else

06:      if $Was changed product kn in database?$ then

07:       call Verification($NAM{E}_{kn}$, $ZA$, $POC{P}_{kn}$, $T_{kn}$, $n_{kn}$, $C{r}_{t_{kn}}$, ${\xi }_{kn}$, $P$)

08:     else

09:       call Verification in Time($POC{P}_{kn}, T_{kn}, {\overline{n}}_{kn}, ZQ, t_{kn}, x$)

10:      end if

11:     end if

3. Results and Discussion

From the point of view of this solution design it is necessary to define its capability through the data limiting the real conditions. When we speak about the input data it is necessary to realise that the described model is designed for the RMS systems. During the verification process of the designed solution, 10 production orders were defined. Their manufacturing parameters belong to the systems of series and small-series production. A certain variability of the manufactured product portfolio is also assumed and therefore some assigned products have a very small rate of similarity of the technological operations.

We determined also the basic data for processing for all depicted semi-products. The time defined by the customer for processing the product and the required manufacturing volume belong to this data. The overall amount of the operations of a particular product is an additional value resulting from the matrices $NAM{E}_{kn}$ or $T_{kn}$. The following table shows these values. The specified values are listed in the following

Table 1. The input values for verification.

Products (P)			Time Necessary for Product Transfer Defined by Customer (min.)						Produced Product Volume Defined by Customer (pcs)						Total Number of Product Operations
K1			$C{r}_{t_{k1}}$				1200		$n_{k1}$				450		$POC{P}_{k1}$		7
K2			$C{r}_{t_{k2}}$				800		$n_{k2}$				500		$POC{P}_{k2}$		8
K3			$C{r}_{t_{k3}}$				3000		$n_{k3}$				2500		$POC{P}_{k3}$		9
K4			$C{r}_{t_{k4}}$				900		$n_{k4}$				800		$POC{P}_{k4}$		7
K5			$C{r}_{t_{k5}}$				2000		$n_{k5}$				15		$POC{P}_{k5}$		7
K6			$C{r}_{t_{k6}}$				5000		$n_{k6}$				2000		$POC{P}_{k6}$		8
K7			$C{r}_{t_{k7}}$				2500		$n_{k7}$				480		$POC{P}_{k7}$		6
K8			$C{r}_{t_{k8}}$				1000		$n_{k8}$				500		$POC{P}_{k8}$		8
K9			$C{r}_{t_{k9}}$				5000		$n_{k9}$				1200		$POC{P}_{k9}$		6
K10			$C{r}_{t_{k10}}$				1500		$n_{k10}$				500		$POC{P}_{k10}$		6
Products (P)
	K1		K2		K3		K4		K5		K6		K7		K8		K9		K10
i	$T_{k1} \left(\min .\right)$	$NAM{E}_{k1}$	$T_{k2}\left(\min .\right)$	$NAM{E}_{k2}$	$T_{k3} \left(\min .\right)$	$NAM{E}_{k3}$	$T_{k4} \left(\min .\right)$	$NAM{E}_{k4}$	$T_{k5} \left(\min .\right)$	$NAM{E}_{k5}$	$T_{k6} \left(\min .\right)$	$NAM{E}_{k6}$	$T_{k7} \left(\min .\right)$	$NAM{E}_{k7}$	$T_{k8} \left(\min .\right)$	$NAM{E}_{k8}$	$T_{k9} \left(\min .\right)$	$NAM{E}_{k9}$	$T_{k10} \left(\min .\right)$	$NAM{E}_{k10}$
1	2	A	6	A	6	N	6	A	6	F	2	K	7	C	3	A	6	D	5	C
2	4	B	7	C	10	J	4	B	4	G	5	G	6	D	1	B	4	C	2	D
3	5	C	5	D	8	G	2	C	2	D	3	J	9	E	2	C	8	B	4	G
4	6	D	5	E	7	H	7	D	3	B	4	F	10	F	2	D	4	F	6	R
5	4	E	5	F	5	T	5	F	5	C	5	T	2	G	3	E	5	G	2	F
6	2	F	4	G	4	J	4	G	1	D	4	L	5	H	4	F	4	A	5	C
7	3	G	3	H	2	H	3	R	2	E	3	N			7	G
8			2	J	5	I					4	E			8	H
9					4	K

.

The further input values are the types of individual operations $NAM{E}_{kn}$ and the time necessary for their performance for a particular product $T_{kn}$. The values of the operation types and the processing times are the values defined on the basis of the production technology of the assigned product. The table depicts all current values of the parameters of the product quantity P.

In the case of calculating a configuration for a non-existing manufacturing system the requirements on the system outfit are defined after determining the number of the necessary devices for the designed configuration. However, the goal of the reconfigurable line design is not to create a firm quantity of equipment for particular products for a longer time period. The reconfigurable system should fulfil the function of changing its capacity in time and its value should be based on its continuous maximal utilisation. The most suitable solution for determining this value is to define an average value of the future values of the system capacities in the framework of which the system will be maximally utilised. The estimation of the initial capacity of the future system configuration is possible through the analyses of the market development and the future demand for the manufactured products. The influence of the financial, capacity, or dimensional limitations of the existing or future product can be included to the estimation of the initial capacities of the designed reconfigurable line.

The independent parts of the system (manufacturing equipment, conveyors, etc.,) are another aspect of the reconfigurable manufacturing system design. This system should be created by independent devices possessing a certain rate of variability that would enable their convertibility and modularity. These values of the initial equipment capacity as well as the extent of their variability were to be defined for verifying the designed solution.

Table 2. The extent of the operational variability and the number of devices for the entire system.

All Realisable Operations of the Manufacturing System	$ZA$$={(o_{ij})}_m^{n=1}$	$\left[ABCDEFGHIJKLMNOPQRSTUVWXYZ\right]$
Total Initial Number of System Equipment	$ZQ$ (pcs)	50

shows the extent of the operational variability and the number of devices for the whole system.

3.1. Calculating the Final Configuration of the Product Family

Based on the results of the individual algorithms described in the part Material and Methods we calculate the configuration of the final sequence. However, the calculation of this sequence was preceded by the calculation of the unsuitable sequences. The definition of the suitable sequence is realised in the stage of the product family final configuration. It means the individual sequences of the united products enter the stage of the product family final configuration. The designed cluster then enters the first condition that compares the number of the necessary devices $q_{\alpha }$ with the total number of assigned machines $P_{prir}$. In the case that the amount of the necessary devices is higher than the number of the assigned ones, the last assigned product is replaced by the next one in the sequence. If these values are equal, we verify the condition that will compare the number of the necessary devices $q_{\alpha }$ with the total amount of the system ZQ devices. For this condition the number of the assigned devices is smaller or equal to the number of the system devices. If the amount of the necessary equipment is smaller than the number of the equipment in the system, we will assign a next product and the newly designed configuration is verified. The situation when we transfer or replace the products from the given sequence $\Omega$ is called sequence. During the 9 sequences the products in the product family were assigned and replaced, subsequently each sequence was assessed and we chose the sequence in the framework of which the configuration fulfils all defined conditions. The following

Table 3. The final appreciation of each sequence.

SEQUENCE	Products	${\mathit{q}}_{\mathit{\alpha }} \left(\mathbf{pcs}\right)$	${\mathit{P}}_{\mathit{p}\mathit{r}\mathit{i}\mathit{r}} \left(\mathbf{pcs}\right)$	${\mathit{T}}_{\mathit{\alpha }}$$\left(\mathit{m}\mathit{i}\mathit{n}.\right)$	${\mathit{T}}_{\mathit{s}\mathit{i}\mathit{m}}$$\left(\mathit{m}\mathit{i}\mathit{n}.\right)$	STATE	$\mathit{Z}\mathit{Q} \left(\mathbf{Pcs}\right)$	${\mathit{T}}_{\mathit{r}}$$\left(\mathit{m}\mathit{i}\mathit{n}.\right)$	${\mathit{T}}_{\mathit{\alpha }}+{\mathit{T}}_{\mathit{s}\mathit{i}\mathit{m}}$$\left(\mathit{m}\mathit{i}\mathit{n}.\right)$
Sequence 1	K1-K8	32	32	1000	985	add	50	20	1005
Sequence 2	K1-K8-K2	61	50	800	784	replace			804
Sequence 3	K1-K8-K7	45	45	1000	957	add			977
Sequence 8	K1-K8-K7-K3	130	50	1000	986	replace			1006
Sequence 4	K1-K8-K7-K4	97	48	900	890	replace			910
Sequence 6	K1-K8-K7-K5	64	48	350	339	replace			359
Sequence 9	K1-K8-K7-K6	66	50	1000	990	replace			1010
Sequence 7	K1-K8-K7-K9	66	48	1000	850	replace			870
Sequence 5	K1-K8-K7-K10	54	48	1000	999	replace			1019

shows the final assessment of each sequence

The previous table depicts an accurate stage of determining the final product family where the introduced final sequence was the third one; however, it was necessary to verify all assigned products. After verifying the results of all sequences we choose a sequence with the highest number of the assigned machines that is the only one to fulfil the aforementioned conditions. The verification of the final conditions can be depicted by the following Figure 3.

The next step after determining the final configuration for the designed reconfigurable line is the design of the iteration time. The iteration time is determined by the condition that compares the sum of the simulation time $T_{sim}$ with the assumed manufacturing time of the product family$T_{\alpha }$. The simulation time and the time of the product family are the final values of the selected sequence; the reconfiguration time is the assigned assumed value. This value can be accurately specified through the multi-agent simulation that could be part of the application for the complex simulation stage system optimisation. However, no accurate value is necessary for the method verification and therefore it was only estimated. The previous Table 3 contains the values of the necessary parameters for the final sequence. Based on the described condition it is possible to determine that the final iteration time will be equal to the sum of the simulation time $T_{sim}$ and the reconfiguration time $T_r$.

The results show that the product family contains three particular products and the remaining products will be produced in the next iterations. The possibility of manufacturing each product in the next iteration is to be assessed through the last part of the algorithm. Each product has to be verified through a condition that compares the time defined by the customer $C{r}_{t_{kn}}$ with the calculated iteration time $T_I$. If the product has to be handed over to the customer in an earlier deadline than the iteration time ends, it is necessary to agree to new transfer conditions with the customer. Another necessary parameter of the residual time $Z_{t_{kn}}$ is calculated in the framework of the stage Verifiability of Products in Time. If the value $Z_{t_{kn}}$ is smaller than the determined value of the time iteration, it is necessary to change the transfer time again. If the customer agrees with changing the transfer time in both cases the time of the product is adapted and it will be made in the next iterations.

The Figure 4 shows that during the first iteration the final processing of the K8 whose processing time was determined as the minimal time of the product family $T_{\alpha }$ will be realised. The next pieces of the product family $K1$ and $K7$ will hand over such an amount of the manufacturing volume as it was stated during the theoretical time $T_{\alpha }$ but their production will be really completed according to the iteration time and in this way a time reserve for their complete processing develops here.

3.2. Experimental Verification of Methodology for Further Iterations

Further iterations were realised with the same input parameters as the first iteration besides the products whose manufacturing was completed. The quotient of the criteria weight $max {\Omega }_{SPT}=0.1$, $\max {\Omega }_{CA}=0.9$ and also the parameter ${\Delta }_{t{z}_i}=2 min$., belongs to the unchangeable input parameters.

During the first iteration there was the final processing of the product $K8$ that was replaced by a new entering product—its input values are in

Table 4. The input values for the product K8.

Product	K8
$t_{kn} \left(min.\right)$	1300
$C{r}_{t_{kn}} \left(min.\right)$	10,000
$n_{k1} \left(pcs\right)$	100
$T_{k1} \left(\min .\right)$	2	3	4	3	1
$NAM{E}_{k1}$	V	F	G	N	J

.

Based on this data, we repeatedly carried out the whole process of creating the product family including the initial verification stages. Based on the acquired data, as in Figure 5, of the next iteration a new product family was created.

During the previous iteration a quantity of the products $\left\{K2,K3,K4,K5\right\}$ whose manufacturing was moved to the next iteration was developed. The manufacturing time of these products as well as the production of the product $K8$ begins from T0 (point zero of the next iteration). The other products have enough time for manufacturing and therefore they start from the original time. Based on our calculation another product family was created of the product quantity $\left\{K1,K7,K10,K5,K8\right\}$ and the product $K1$ is the finally processed one. The new time $T_{\alpha }$ will be the remaining time for producing the product $K1.$ Based on determining the new value $T_I$ it is possible again to assess the manufacturability of the remaining products. The only product that does not fulfil the manufacturability condition is the product $K3.$ However, we assume a transfer (approved by the customer) of this product to the next iteration. After completing the second iteration and processing the product $K1$ it is necessary to define a new entering product again. A product designated as $K1$ whose input characteristics are also depicted in

Table 5. The input values for the product K1.

Product	K1
$t_{kn} \left(min.\right)$	1500
$C{r}_{t_{kn}} \left(min.\right)$	1500
$n_{k1} \left(pcs\right)$	1000
$T_{k1} \left(\min .\right)$	2	2	2	2	2
$NAM{E}_{k1}$	Z	Q	X	Y	Z

enters in the next iteration.

The product family created in the third iteration contains a quantity of three products $\left\{K2,K8,K1\right\}$ and the product $K2$ is the finally processed one. However, a problem arises with the products $K10,K3$, and $K4$, that is of not having sufficient time $C{r}_{t_{kn}}$ and that there has already been a postponement of the deadline for processing $K3$ and $K4$. If it was not possible to move the transfer deadline of these products, the algorithm enables achieving better conditions for the given products through changing the input parameters. All the previous iteration can be repeated and the individual line configurations can be planned according to the specific requirements of the market environment. The chart of the third iteration for the product family is shown in Figure 6.

The last part that is necessary to assess while creating the iteration is the usability of the manufacturing system equipment. The algorithm conditions are defined for the maximal possible usability of the equipment during creating the configuration and this statement can be verified by Figure 7.

The final diagrams show that the minimal usability of the devices during iteration was 90% when the overall available number of the machines was available. The remaining equipment that was not utilised for setting the final line configuration can be used as the backup devices. These backup devices can be defined according to the next lowest value of the usability coefficient and subsequently they can be added to the necessary operational level as the warm or cold backup. Another possibility of utilising the non-used devices is their outsourcing to the individual manufacturing lines (islands) as a possibility of the overall optimisation of the resources’ usability for the manufacturing enterprise.

4. Conclusions

The main objective of this research was to design a methodology for designing the manufacturing lines using the reconfiguration principles. Through a methodology based on a mathematical model algorithm we created a particular application in the MS Excel and Tecnomatix Plant Simulation environments. The designed application enables to generate relevant results serving for the verification and validation of the methodology. The achieved results provide an opportunity for better understanding the area investigated and also knowledge for creating a superstructure of this system.

The application of this methodology creates a configuration that takes into account the basic reconfiguration principles of its design. Through the partial stages of the algorithm, the adaptability of the manufacturing system at the level of a particular line is also taken into consideration. Another important part of the methodology is the direct verification of each designed configuration by the computer simulation that is part of the whole application concept. Based on the achieved results we can say that the utilised stages and mathematical model enable designing a manufacturing line and utilise the defined principles of reconfigurability. The introduced hypothesis of the possibility of creating a methodology for designing a manufacturing line using the configuration principles is real and the methodology can be created.

The next part of the research for verifying the given solution would be the implementation of the real company data and a subsequent investigation of the effectiveness compared with stage being used now. However, before this verification phase it is necessary to put the designed mathematical model into an algorithm form and to connect it with the input data database of a particular company. But for a particular company it is necessary to define various limitations resulting from the process or technology of manufacturing particular products. This study deals with a general solution and its verification therefore the introduced calculations did not use real data of a particular enterprise.