This paper highlights the problems of mathematical modelling for a specific element of the logistic supply chain, that is, the manufacturing system. The complex manufacturing system consisting of a determined number of parallel subsystems is modelled. The fact that the same manufacturing procedure can be carried out in various locations is emphasised. Control algorithms as well as manufacturing strategies are explained. The equations of state are introduced. The twostage criterion lets us use the result data generated by the simulator of the production system as the initial data for further processing; however, the main goal remains to minimise the time of the course of production. The precisely elaborated case study implements initial data obtained by preceding simulation procedures carried out in manufacturing systems consisting of three, four, and five subsystems.
Nowadays, design and optimisation of logistic and production systems is a demanding and important area. Businesses are pushed to improve their performance, decrease production prices, and increase variability of their products or shorten delivery times. If a company fails to adapt to these requirements, it cannot normally survive in the contemporary market environment. To survive in the highly competitive global economy, manufacturing systems must be able to adapt to new circumstances [
Production systems have various structures; however, they are defined by certain common features. One of them is the tool replacement problem. Tools are replaced either individually or within replacement of the socalled tool stand. Usually, there is a tendency to replace only fully worn out tools. A problem which occurs in the discussed system is the need to replace the stand with tools even if the tools in it are not fully worn out. So far manufacturing problems of this type have been solved on condition that either fully worn out tools or the unnecessary ones are subject to replacement [
The main goal of this article is to define and simulate the mathematical model for the evaluation of the distribution of production in several plants within one complex company to achieve maximum production in the shortest possible time and then minimise costs of manufacturing. Individual companies may be geographically located in different countries with different labour costs and a production time unit. Individual companies are considered as autonomous systems with the same production possibilities. The model takes into account three variants consisting of 3, 4, or 5 companies which can make orders according to vacancies. To compare all possible variants, the mathematical methods of permutations without repetition are used. The basic criteria for the comparison of efficiencies of the production system as a whole are the time and cost of manufacturing time unit.
Complexity has always been a part of each manufacturing environment. Therefore, there has always been a need to classify them as complex systems where numerous phenomena take place [
The proposed model represents one of the possible solutions for optimising production in case of dislocated companies which are at disposal. It is significant as it is possible to use it for a particular realistic case described by the defined structure and, of course, it can be treated as one of the scientific bases for solving similar examples (e.g., adaptation of the model for other structures, implementation of other decisionmaking algorithms, and expansion of other evaluation parameters, for example, from the set of key performance indicators (KPI) of the company).
Symbols used for logistic system modelling presented in this section are explained in Symbols shown at the end of the paper. They remain in accordance with the standards of mathematical models used for the description and simulation of logistic, production, trade, and other systems.
It is assumed that manufacturing processes take place in the complex system which consists of
The separate manufacturing plant has a serial form and consists of work stands assembled in sequence. The stands are equipped with tools which are subject to wear throughout the manufacturing process. When a tool in a stand is completely worn out, the manufacturing process is continued with the use of another tool in the same stand as long as either its life or the manufacturing route allows it. It is assumed that tools cannot be replaced separately. When all tools in the given stand are worn out or the manufacturing process cannot be continued through this stand, the discussed stand is subject to replacement. There are buffer stores between the work stands.
The manufacturing plant described above can be multiplied which results in creating exactly the same manufacturing system in various locations. It is assumed that
Let us assume orders to be made are shown in the matrix of orders:
If the
Products are made from various types of charges which leads to introducing the adjustment matrix of charges to products:
The discussed manufacturing system takes the form shown in the matrix of structure:
The route matrix is introduced:
The base life matrix is introduced:
Let us associate the matrix of coefficients with the previous matrix of life:
At the same time
The base coefficient for the calculation purpose
The life matrix for the
At the same time
The structure vector of base capacity of buffer stores is introduced:
It is assumed that each
Let us introduce the matrix of coefficients of the buffer stores:
The state of the buffer store is expressed as follows:
The manufacturing system is always defined in its current state shown in the base matrix of state:
The state of the system is recalculated and shown in the matrix of state:
The base flow capacity of the manufacturing system is defined in the base matrix of flow capacity:
The flow capacity of the system is recalculated and shown in the matrix of flow capacity:
The flow capacity of the
The matrix of manufacturing times takes the following form:
The vector of replacement times takes the following form:
Let us introduce the vector of unit time costs:
The state of the complex manufacturing system changes after either any decision concerning producing any
The state of the
The state of the base
The order matrix
The order matrix is modified after every decision about production:
There is a need to seek satisfactory solutions by making orders in various locations as they generate various manufacturing costs. Comparing costs allows us to determine where the
This algorithm chooses the
This algorithm chooses the
This algorithm chooses the order matrix element characterised by the maximal value of
This algorithm chooses the order matrix element characterised by the minimal value
In order to make orders in the
The immediate strategy: a new order is introduced into the manufacturing system as soon as such a possibility emerges.
The delayed strategy: a new order is introduced into the manufacturing system only then when the preceding one leaves the system.
There are a few criteria to be taken into account; however, the most decisive one in terms of manufacturing under time pressure seems to be the minimal manufacturing time criterion so it is chosen for further consideration. As orders are made in various plants there are corresponding aspects of this problem to be analysed. One of them is a different location of each plant which is connected with different manufacturing costs. It is assumed that each manufacturing process is carried out in accordance with the twostage criteria specified below. First of all, the criterion of minimising the manufacturing time is introduced:
Having generated the best solution from the point of view of minimising time, there is a need to minimise manufacturing costs by means of the criterion of minimising manufacturing costs:
In order to verify correctness of the assumptions presented in the paper hereby it is necessary to carry out the simulation process with the use of sample data. Apart from obtaining results concerning the minimal manufacturing time it is possible to present other associating results for each simulation process, that is, total replacement time of tools in work stands, lost flow capacity of tools due to premature replacement, and remaining capacity of tools in case of completing making the whole order.
The following general assumptions for the case study are taken into account:
All available manufacturing plants are at the same initial state as follows:
There is enough charge material for making the whole order matrix elements.
The simulator used for simulating making order receives data drawn at random from a specified range:
The simulator implemented for processing the data either generates them at random or accepts them from the file or the operator of the system introduces them manually. It is possible to alternate data if necessary. In our case the data are obtained from the ready file which was subject to thorough verification [
Sample initial data introduction for
The data in Figure
Results of simulation experiments.
Number of plants  Criterion  Immediate manufacturing  Delayed manufacturing  

max_order max_flow  max_order min_flow  min_order max_flow  min_order min_flow  1000 sim  max_order max_flow  max_order min_flow  min_order max_flow  min_order min_flow  1000 sim  
3  Total manufacturing time  2045  2045  2140  2140 

2196  2196  2223  2223  2124 
Total replacement time  1487  1487  1576  1576  1395  1392  1392  1503  1503 


Lost flow capacity  7,00  7,00  4,50  4,50  9,83  7,17  7,17  4,50  4,50 


Remaining capacity  185,33  185,33 


133,83  189,17  189,17 


130,50  


4  Total manufacturing time  1553  1553  1613  1613 

1648  1648  1704  1704  1609 
Total replacement time  1286  1286  1337  1337 

1236  1236  1338  1338  1119  
Lost flow capacity  6,17  6,17  5,67  5,67 

8,17  8,17  5,67  5,67  1,17  
Remaining capacity  258,17  258,17  251,67  251,67  192,00 


251,67  251,67  193,83  


5  Total manufacturing time  1320  1317  1326  1358 

1381  1373  1390  1416  1308 
Total replacement time  1129  1151  1137  1159 

1163  1159  1158  1172  1003  
Lost flow capacity  7,50  8,00  5,83  7,00 

7,17  6,33  6,83  6,83  1,50  
Remaining capacity  303,83 

278,50  293,33  239,83  318,17  332  296,5  308,50  240,00 
In the case of the threeplant system the total minimal manufacturing time (1997 time units) is obtained by means of carrying out 1000 simulations with the use of the socalled immediate strategy; however, it does not guarantee achieving the lowest value of the total replacement time as well as the lowest value of the lost flow capacity and the highest value of the remaining capacity.
In the case of the fourplant system the total minimal manufacturing time (1509 time units) is also obtained by means of carrying out 1000 simulations with the use of the socalled immediate strategy. This time, it achieves the lowest value of the total replacement time as well as the lowest value of the lost flow capacity. Unfortunately, it delivers the worst value of the remaining capacity.
In the case of the fiveplant system the total minimal manufacturing time (1242 time units) is again obtained by means of carrying out 1000 simulations with the use of the immediate strategy. It also reaches the lowest value of the total replacement time as well as the lowest value of the lost flow capacity. Again, it delivers the worst value of the remaining capacity. In conclusion, all best results are obtained by means of the immediate manufacturing strategy.
Let us assume that the results shown in Table
Manufacturing times in autonomous systems.

System  

3 plants  4 plants  5 plants  
1  1968  1466 

2 

1489 

3  1957  1498  1196 
4  × 

1235 
5  ×  ×  1168 


Total manufacturing time in the system 



The times shown in Table
Time scheduling in case of implementing the time minimisation criterion.
Order number 





Plant number  1  5  3  4  4 
Start  119  442  0  813  0 
End  353  640  363  1235  161 


Order number 





Plant number  2  5  2  1  4 
Start  900  0  624  540  462 
End  1242  134  917  722  698 


Order number 





Plant number  3  5  1  3  1 
Start  733  901  708  344  336 
End  996  1168  970  503  554 


Order number 





Plant number  3  5  2  1  4 
Start  979  622  0  0  675 
End  1196  922  334  134  828 


Order number 





Plant number  3  2  4  1  5 
Start  488  319  144  909  120 
End  751  650  479  1242  469 
The sequences for minimising the total manufacturing time are given in Table
Sequences leading to minimisation the total manufacturing time.
Number of plant  Sequence in the 3plant system  Production time 

1  
1968 
2  
1997 
3  
1957 


Number of plant  Sequence in the 4plant system  Production time 


1 

1466 
2 

1489 
3 

1498 
4 

1509 


Number of plant  Sequence in the 5plant system  Production time 


1 

1242 
2 

1242 
3 

1196 
4 

1235 
5 

1168 
The basis for improving the sample time scheduling approach is the minimal total manufacturing time obtained by means of implementing the immediate manufacturing strategy: 1997 for the 3plant system, 1509 for the 4plant system, and 1242 for the 5plant system (see Table
Results of the simulation process for
The simulation results are associated with system plants; however, it is possible to move the manufacturing process to any other plant in the system as they perform the same production operations. Following this kind of reasoning it was assumed that it is possible to transfer the sequence of orders adjusted to a certain plant on the basis of the simulation results to any other plant in the discussed system. To carry out this process it is necessary to implement permutation without repetitions in order to find the solution which can minimise manufacturing costs to meet the criterion of minimising costs. It became possible only by comparing the results obtained by the method using the permutations without repetitions. In case of 3 plants there are
Manufacturing costs in the threeplant system (all data).
Plant  Unit cost  Combination number of production sequences  

1  2  3  4  5  6  
1  5 

9840 

9840 

9985 

9985 

9785 

9785 
2  6 

11982 

11742 

11808 

11742 

11808 

11982 
3  4 

7828 

7988 

7828 

7872 

7988 

7872 


Total costs  29650 

29621  29599  29581  29639 
Manufacturing costs in the fourplant system (all data).
Plant  Unit cost  Combination number of production sequences  

1  2  3  4  5  6  
1  5 

7330 

7330 

7330 

7330 

7330 

7330 
2  6 

8934 

8934 

8988 

8988 

9054 

9054 
3  4 

5992 

6036 

6036 

5956 

5956 

5992 
4  7 

10563 

10486 

10423 

10563 

10486 

10423 


Total costs  32819  32786  32777  32837  32826  32799  


Plant  Unit cost  Combination number of production sequences  
7  8  9  10  11  12  


1  5 

7445 

7445 

7445 

7445 

7445 

7445 
2  6 

8988 

8988 

9054 

9054 

8796 

8796 
3  4 

6036 

5864 

5864 

5992 

5992 

6036 
4  7 

10262 

10563 

10486 

10262 

10563 

10486 


Total costs  32731  32860  32849  32753  32796  32763  


Plant  Unit cost  Combination number of production sequences  
13  14  15  16  17  18  


1  5 

7490 

7490 

7490 

7490 

7490 

7490 
2  6 

9054 

9054 

8796 

8796 

8934 

8934 
3  4 

5864 

5956 

5956 

6036 

6036 

5864 
4  7 

10423 

10262 

10563 

10423 

10262 

10563 


Total costs  32831  32762  32805  32745 

32851  


Plant  Unit cost  Combination number of production sequences  
19  20  21  22  23  24  


1  5 

7545 

7545 

7545 

7545 

7545 

7545 
2  6 

8796 

8796 

8934 

8934 

8988 

8988 
3  4 

5956 

5992 

5992 

5864 

5864 

5956 
4  7 

10486 

10423 

10262 

10486 

10423 

10262 


Total costs  32783  32756  32733  32829  32820  32751 
Manufacturing costs in the fiveplant system (chosen data).
Plant  Unit cost  Combination number of production sequences  

11  12  13  14  15  16  
1  5 

6210 

6210 

6210 

6210 

6210 

6210 
2  6 

7176 

7176 

7410 

7410 

7410 

7410 
3  4 

4968 

4968 

4672 

4672 

4968 

4968 
4  7 

8645 

8176 

8694 

8372 

8372 

8176 
5  8 

9344 

9880 

9568 

9936 

9344 

9568 


Total costs  36343  36410  36554  36600 

36332  


Plant  Unit cost  Combination number of production sequences  
17  18  19  20  21  33  


1  5 

6210 

6210 

6210 

6210 

6210 

6210 
2  6 

7410 

7410 

7008 

7008 

7008 

7410 
3  4 

4784 

4784 

4968 

4968 

4784 

4968 
4  7 

8176 

8694 

8372 

8645 

8645 

8372 
5  8 

9936 

9344 

9880 

9568 

9936 

9344 


Total costs  36516  36442  36438  36399  36583 

In the 3plant system total manufacturing costs are minimised in the case of the combination of production sequence number 2 by 80 monetary units comparing them with the best result for 1000 simulations (see Table
In the 4plant system total manufacturing costs are minimised in the case of the combination of production sequence number 17 by 97 monetary units comparing them with the best result for 1000 simulations (see Table
On the basis of comparison results shown in Table
Manufacturing costs in case of the immediate strategy in the 5plant system.
Plant number  Production sequence  End time  Unit cost  Total costs 

Best results for 1000 simulations  
1 

1242  5  6210 
2 

1242  6  7452 
3 

1196  4  4784 
4 

1235  7  8645 
5 

1168  8  9344 


Total manufacturing costs 




Permutations without repetitions, combination 15  
1 

1242  5  6210 
2 

1235  6  7410 
3 

1242  4  4968 
4 

1196  7  8372 
5 

1168  8  9344 


Total manufacturing costs 




Permutations without repetitions, combination 33  
1 

1242  5  6210 
2 

1235  6  7410 
3 

1242  4  4968 
4 

1196  7  8372 
5 

1168  8  9344 


Total manufacturing costs 

The results obtained on the basis of the data implemented for the purpose of the case study let us come to a conclusion that under the very theoretical conditions specified in the article it can be expected that the greater number of manufacturing plants in the system leads to bigger financial savings.
It can be expected that carrying out more simulation experiments for a certain number of manufacturing plants in the system may deliver even a shorter total manufacturing time which is the time in the system characterised by the longest manufacturing time in its subsystem. This result forms the basis for improving the total manufacturing costs by means of seeking for such a combination of adjusting the sequence of production to plants which minimise the total manufacturing costs. It also seems reasonable to implement other criteria, that is, either the criterion of minimising lost capacity or the criterion of maximising the remaining capacity or the criterion of minimising the replacement time. Nevertheless, the assumptions for the mentioned criteria differ from the ones presented in the case study, emphasising both the criterion of minimising the manufacturing time and the criterion of minimising manufacturing costs. Moreover, it is necessary to analyse the behaviour of the system in case of various initial states obtained at random from data characterised by the same range for drawing.
The number of the subsystem,
The state of the
The base capacity of the buffer store placed behind the
The cost of one unit time of operating the
The vector of unit time costs
The structure vector of base capacity of buffer stores
The state of the buffer store
The number of the
The route matrix
The
The matrix of structure
The number of base elements of a product which can be made by the
The base life matrix
The stage number resulting from the preceding decision in the system,
The number of the customer,
The type of order,
The base capacity of the
The base matrix of flow capacity
Matrix of flow capacity
The criterion of minimising manufacturing costs
The criterion of minimising the manufacturing time
The algorithm of the maximal flow capacity of the
The algorithm of the minimal flow capacity of the
The algorithm of the maximal order
The algorithm of the minimal order
The base state of the
The state of the
The
The
The number of units of the
The amount of the
The
The matrix of orders in the
The manufacturing time of one unit of the
The replacement time of the
The matrix of manufacturing times
The vector of replacement times
The coefficient of flow capacity of the production plant
The matrix of coefficients of the buffer stores
The coefficient of the
The coefficient determining how many units of the
The matrix of coefficients
The adjustment of the
The adjustment matrix of charges to products.
The authors declare that they have no conflicts of interest.
This paper was supported by Project SGS/19/2016 (Silesian University in Opava, School of Business Administration in Karvina): Advanced Mining Methods and Simulation Techniques in the Business Process Domain.