CC BY
8
DOI: 10.29141/2658-5081-2021-22-2-7 JEL classification: D24; G31; M29
Natalya V. Kireeva Socio-Economic Institute (branch of the Academy of Labour and Social
Relations), Chelyabinsk, Russia Evgenia S. Zambrzhitskaya Nosov Magnitogorsk State Technical University, Magnitogorsk, Chelyabinsk oblast, Russia
Elena A. Makarova OOO "Russkiy Khleb Food Group", Magnitogorsk, Chelyabinsk oblast,
Russia
Graph models for evaluating production capacities of enterprises
Abstract. The known methods for evaluating production capacities of industrial enterprises mostly submit approximate estimates and are typical of the planned economy. Currently, their practical application is limited, because the product range of companies is no longer fixed. Models created using the graph theory and matrix calculus are capable of overcoming these limitations and providing relevant information support for management decisions. The paper focuses on developing a method for evaluating the production capacity of an enterprise using graph models. Methodologically, the research relies on the graph theory and industrial engineering; applies methods of analysis and synthesis, matrix modelling. The central idea of the proposed models is that an enterprise is a depersonalized production system made up of certain links forming a production chain. To perform relevant calculations the graph model is aligned with the matrix model, which reckons with the main parameters of the production system: technological relationships, product mix, time and material consumption rates, production capacity of each link. The key difference between the graph model and currently existing approaches lies in abandoning the principle of a bottleneck link and switching to the concept of a limiting link, as well as using conditional units of product range. Testing the proposed models on a case of a bakery enterprise proves the efficiency of the method for assessing the production capacity. The developed graph model allows for sound management of production capacities due to the understanding of the flexibility of a product range and technological relationships.
Keywords: production capacity; production system; product mix; graph theory; matrix modelling; enterprise.
Acknowledgements: The paper is funded by the grant of the President of the Russian Federation no. MK-2003.2020.6 "Financial mechanisms of stimulating municipal development in the context of sustaining national economic growth".
For citation: Kireeva N. V., Zambrzhitskaya E. S., Makarova E. A. (2021). Grafo-matrichnye modeli otsenki proizvodstvennoy moshchnosti predpriyatiya [Graph models for evaluating production capacities of enterprises]. Journal of New Economy, vol. 22, no. 2, pp. 134-154. DOI: 10.29141/2658-5081-2021-22-2-7 Received January 11, 2021.
Н. В. Киреева Уральский социально-экономический институт (филиал) образовательного
учреждения профсоюзов высшего образования «Академия труда и социальных отношений», г. Челябинск, Российская Федерация Е. С. Замбржицкая Магнитогорский государственный технический университет им. Г. И. Носова, г. Магнитогорск, Челябинская область, Российская Федерация Е. А. Макарова Продовольственная группа «Русский хлеб», г. Магнитогорск, Челябинская область, Российская Федерация
Графо-матричные модели оценки производственной мощности предприятия
Аннотация. Известные методы оценки производственных мощностей промышленного предприятия можно охарактеризовать как укрупненные, свойственные плановой экономике. В настоящее время возможности их практического применения существенно ограничены, поскольку ассортиментная структура выпускаемой продукции уже не является фиксированной. Преодолеть эти ограничения и обеспечить информационную поддержку управленческих решений способны модели, созданные на основе теории графов и матричных вычислений. Статья посвящена разработке методики оценки производственных мощностей предприятия с помощью графо-матричных моделей. Методологическую базу исследования составили теория организации производства и теория графов. В работе использовались методы анализа и синтеза, матричное моделирование. Суть предлагаемых моделей заключается в том, что предприятие понимается как некая обезличенная производственная система, состоящая из определенных звеньев, которые складываются в производственную цепочку. Для выполнения соответствующих расчетов графовая модель увязывается с матричной, которая учитывает основные параметры производственной системы: технологические связи; ассортиментную структуру продукции; нормы расхода материала и времени; производственные мощности каждого звена. Принципиальным отличием графо-матричных моделей от существующих в научной практике подходов является отказ от принципа «узкого» звена и переход к понятию «лимитирующего» звена, а также использование условных ассортиментных единиц. Апробация данных моделей на хлебобулочном предприятии показала действенность и эффективность предлагаемых подходов к оценке производственной мощности. Предложенная графо-матричная модель позволяет обеспечить эффективное управление производственными мощностями благодаря учету подвижности ассортиментной структуры продукции и технологических связей.
Ключевые слова: производственная мощность; производственная система; ассортиментная структура; теория графов; матричные модели; предприятие.
Для цитирования: Киреева Н. В., Замбржицкая Е. С., Макарова Е. А. (2021). Графо-матричные модели оценки производственной мощности предприятия // Journal of New Economy. Т. 22, № 2. С. 134-154. DOI: 10.29141/2658-5081-2021-22-2-7 Дата поступления: 11 января 2021 г.
Introduction
The development of information technologies and systems encourages the use of mathematical methods and means to address the issues of managing industrial enterprises. Patently, computing facilities used for economic analysis substantially determine the calculation methods, specialists' mindset, and shape economics in general. Until recently, the practice has mostly relied on very simple mathematical calculations, appropriate to the capabilities of arithmetic.
The majority of modern enterprises are now introducing and operating powerful information systems capable of providing high quality information support for managerial decisions at various levels. These corporate information systems can accumulate a wealth of information, which is processed using different tools, including a spreadsheet package.
The graph theory and matrix calculus are of particular interest for the purposes of assessing the production capacity of an industrial enterprise, though the use of these techniques is only possible when the enterprise maintains a high level of IT development and automation of accounting systems. It is noteworthy that the use of the aforementioned mathematical methods is justified only if there are complex technological relationships.
In view of the all above, the research aims to develop a new method for evaluating production capacity using graph models. In order to achieve the purpose, the following objectives are set: 1) to review existing methods and reveal their drawbacks under current conditions; 2) suggest new methodological approaches to calculation of production capacity based on the graph models; 3) test these approaches using the case of a real production enterprise and assess their analytical possibilities in terms of benefits for management.
Overview of the methods for calculating production capacity and their insufficiency under current conditions
Until recently, to calculate the production capacity the bottleneck method has been applied the most often. According to this method, the production capacity is estimated based on the capacity of the key machinery, processing lines, etc. [Aleksandrova, Garshina, 2011, p. 99; Voronina, 2004, p. 79; Yakobson, Zambrzhitskaya, 2017, p. 115; Semushkina, Shishkarev, 2017a, p. 33]. Normally, this parameter of the fixed assets is specified in the relevant technical documentation. The production capacity of a production site is determined by the smallest capacity among all the equipment installed at the site. In the same way, the shop floor capacity corresponds to the minimum capacity among all production sites comprising it. The capacity of the whole enterprise is calculated based on the capacity of the shop floor with the minimum output [Zaitsev, 2006, p. 7; Mets et al., 1986, p. 383; Kislitsyna, Sherman, Yamoleev, 2017, p. 1723; Pilyugina, Mishchenko, 2017, p. 110; Semushkina, Shishkarev, 2017b, p. 33].
Economists actively debate the ways to promote the efficiency of the production system [Morrison, 1986, p. 51; Ceryan, Koren, 2009, p. 404; Fraser, 2014, p. 18]. Some of the scholars focus on improving the performance of the bottleneck segment [Rose et al., 1995, p. 56; Goldratt, 2009, p. 333; Haan, Naubs, Overboom, 2012, p. 157; Georgiadis, Politou, 2013, p. 689; Teerasopon-pong, Sopadang, 2021] and do not pay due attention to its measurement, considering this issue obvious and well-researched both in theory and in practice. In particular, Pacheco, Pergher, Jung, Scwenberg ten Caten [2014, p. 353] have elaborated 28 practice-oriented strategies for debottle-necking. According to the authors, one of the most promising strategy is the production capacity lease, which aims to split the bottleneck operation into smaller sub-operations with subsequent redistribution. Particularly noteworthy are the studies on the Japanese production management practices of the top corporations [Shingo, 1996; Jayaram, Das, Nicolae, 2010, p. 280; Schonberger, 2007, p. 403; Lee et al., 2010, p. 3747; Zhao, Song, Li, 2018; Yang, Fukuyama, Song, 2019, p. 94]. These scholars scrutinise the practices of the top corporations on the matter under consideration.
The aforementioned approach to the assessment and management of production capacity is quite reasonable and straightforward in terms of its practical implementation. However, its efficiency raises doubts, since it is oriented to a fixed (planned, pre-defined) product mix, which is becoming less and less common in the real-world production conditions. A fixed range of products is typical of the planned economy (for instance, of the USSR) or large enterprises operating under long-term contracts in a stable economic environment [Revutskiy, 2005, p. 9]. The majority of present-day industrial enterprises manufacture multiple products and have a flexible product mix. The product mix flexibility is a factor that necessitates a re-examination of the traditional methods used to calculate production capacity [Gorelik, 2007, p. 111; Danilov, Voynova, Ryzhova, 2012, p. 42; Danilov, Zambrzhitskaya, Ryzhova, 2012, p. 19; Danilov, Ryzhova, Voynova, 2012, p. 80].
The diagnosed problem can be approached using the graph theory and matrix calculus. The details of the developed approach are presented below.
Graph models as a basis for developing new methods for calculating production capacity
As has been already noted above, modern information technologies and systems allow the adoption of complex models to provide information support for managerial decision-making [Pesch et al., 1999, p. 265; Bello, 2014, p. 16; Provatar, Pichugin, 2014, p. 55; Meporiya, 2015, p. 54; Revutskiy, 2002].
The matter of using mathematical models to manage the production capacities of industrial enterprises is being actively discussed in the scholarly literature [Al-Hakim, 1991, p. 1701]. The proposed mathematical models differ principally depending on a type of manufacturing processes: machine tools; batch apparatus; machines with the capacity determined by the size of the occupied production site, etc. Within the scope of this research, we focus on enterprises running continuous stage-to-stage production that have several processing departments. Such type of enterprises is exemplified by manufacturers of food, textile, iron and steel, wood processing industries, mechanical engineering.
Continuous stage-to-stage production implies that the production process has several processing stages consecutively passed through by raw materials. In terms of managerial accounting and analysis, continuous stage-to-stage production matches continuous operation costing and accounting, which is implemented in two forms:
• with semi-finished products (semi-finished products of own production are taken into account on a separate account 21);
• without semi-finished products (semi-finished goods are not accounted for separately; they are accounted for as a work in progress on the account 20).
To calculate the production capacity of the continuous stage-to-stage production systems we suggest using the graph models. An attempt to represent a production system in the form of a network graph was made by Fayngold and Kuznetsov [2002, p. 9]. In their work Principles for calculating production capacity and equipment load, the researchers distinguish between simple and complex production processes. They propose representing the simple process by a linear structure, and the complex one by a network structure.
The central idea of the proposed graph models for calculating production capacity is that a manufacturing enterprise is depersonalized and perceived as a production system made up of production links forming a production chain and creating a production network [Danilov, Ryzhova, Voynova, 2012, p. 80]. All of the above is implemented by means of a graph model of the production system, which is complemented by an appropriate matrix for the convenience of calculations and possibility of accounting for counter material flows. The matrix reckons with the following parameters of the production system:
• technological relationships and possibilities to change them;
• product mix;
• direct consumption rates of products per product;
• production capacity of the links by product.
The algorithm for calculating the production capacity of a manufacturing enterprise with a continuous stage-to-stage production process based on the proposed graph model of the production system may include the following basic steps.
Step 1. A process flow diagram of the production system is constructed. It necessarily details and specifies all technological relationships, consumption rates and production capacities of the machinery, equipment, processing lines, etc. at an enterprise.
Step 2. A graph model of the production system is developed and built. The production structure of the enterprise is analysed to identify the production links, which are interlinked to form a processing chain operating within a processing network.
Step 3. A matrix is formed. For the graph model constructed in step 2, the main input parameters (parameters of the production system) are set in a matrix form for the purposes of further calculation of the throughput capacity of production links, namely:
• direct consumption rates of products per product;
• production capacity of each production link in terms of manufactured product units and product range of the specific link. If the link is multi-product, i.e. it can process several types of products, then the production capacity of the link is calculated for each type of product individually;
• other parameters of the system if needed and appropriate based on specific managerial decisions regarding the capacity of the production system under consideration.
Step 4. The final product mix of the production system is set (the most probable or planned).
Step 5. The throughput capacities of all production links (Q) are calculated in the units of final products of the production system according to the formula:
where r = (rj)n1 is the vector of product range proportions of the gross (finished) product; b = (bjj)mn is the matrix of the direct consumption rates of products per product; q = (qk,j)in is the matrix of the production capacity of the links by products; E is the identity matrix of the corresponding dimension.
Step 6. Bottleneck links are identified and measures on debottlenecking are designed. This step of the proposed algorithm is taken only when there is a managerial objective to modernise and revamp the production facilities of a particular manufacturing enterprise, otherwise this step is omitted.
Step 7. A limiting link is identified, the throughput capacity of which dictates the production capacity of the whole system. Importantly, in this case we can reasonably dismiss the traditional term "bottleneck", since we adopt a fundamentally different methodological approach. To be more specific, in the traditional methodology, pinpointing the bottleneck of the entire production system involves a successive calculation of the production capacity from the lowest level (level of machinery, equipment, processing lines, etc.) up to the level of an enterprise, thereby the production link with the minimum throughput capacity is determined as the bottleneck. According to the proposed approach, the throughput capacity of each production link is identified and a limiting link for a given product range is pinpointed. Obviously, for a different product range the limiting production link may change:
Q =
l
l-xtf-b^xr'
(1)
Qo = min{Q}. (2)
It is noteworthy that the proposed graph model is more flexible and functional in terms of the real-life manufacturing conditions. In addition, this approach allows using a simulation feature (modelling) in the capacity management, and this can be vitally important at the stage of designing manufacturing systems and enterprises.
Practical evaluation of the proposed method for calculating production capacity using the case of a bakery enterprise
Consider the case of using of the graph models of production systems to calculate the production capacity of a bakery enterprise. Modern bakery industry is multi-product and includes manufacture of bread, biscuits, waffles, and many other products. Moreover, within each group of products there are several dozens of subgroups with separate items within the product range. The bakery industry is also characterized by rather a complicated process flow. For this reason, we will look only at one production shop floor, which produces biscuits.
Figure 1 presents a process flow diagram of the biscuit production at the bakery enterprise under analysis (Step 1).
Fig. 1. Process flow diagram for biscuits Рис. 1. Процесс производства печенья
The production of biscuits at the bakery enterprise consists of the following stages.
Stage 1. Mixing of ingredients and dough preparation in a dough mixing machine.
Stage 2. Forming and depositing. Biscuits are shaped with the use of a dough forming machine and an extruder (depositing machine).
Stage 3. Baking. The biscuits are transported via the conveyor (1) to the oven for the heat treatment of the finished product.
Stage 4. Cooling and glazing. The biscuits are cooled down while travelling on the conveyor (2) and fed into a glazing machine, which coats the biscuits with chocolate glaze and/or forms chocolate glaze patterns depending on the type of biscuit. It is important to note that not all biscuits are glazed, for instance, in our case, the butter biscuits, oat biscuits, and home-style biscuits are not glazed. These biscuits are immediately delivered for packaging.
Stage 5. Packaging. In view of a wide variety of packs both by weight and by technology, this process is handled by several packaging machines. In our example, three types of packaging are used: film packaging, flow pack, and shrink wrapping.
Then a graph model and an appropriate matrix of the production system should be developed (Step 2).
Figure 2 shows the network of technological relationships of the analysed production system, which includes 9 production links and produces 28 types of products. However, only 8 items (no. 21, 22, 23, 24, 25, 26, 27 and 28) are the marketed goods (sold to the end consumers), while the rest are semi-finished products of own production.
1
ql,l = 8 rl = 0
ql,2 = 6 r2 = 0
3
q2,3 = 4.00
bl,3 = 0.75
r3 = 0.00
4
q2,4 = 4.00
bl,4 = 0.75
r4 = 0.00
5
q2,5 = 3.60
bl,5 = 0.75
r5 = 0.00
6
q2,6 = 3.20
bl,6 = 0.75
r6 = 0.00
q3,7 = 3.80 b2,7 = 0.88 r7 = 0.00
8
q3,8 = 4.48 b2,8 = 0.86 r8 - 0.00
q3,9 = 5.62 \ b2,9 = 0.82 r9= 0.00
16
10
q4,10 = b3,10 = 6.00 1.05
rlO = 0.00
11
q4,ll = b4,ll = 6.00 1.04
rll = 0.00
12
q4,12 = b5,12 = 6.51 1.05
rl2 = 0.00
13
q4,13 = b6,13 = 5.28 1.06
rl3 = 0.00
14
q4,14 = b7,14 = 4.75 1.09
rl4 = 0.00
15
q4,15 = b8,15 = 5.25 1.09
rl5 = 0.00
q4,16 = 6.01 b9,16= 1.08 rl6 = 0.00
17
q5,17 = 4.75 bll,17= 0.90 rl7 = 0.00
18
q5,17 = 4.78 bl2,18 = 0.90 rl8 = 0.1
19
q5,17 = 5.60
1 bl3,19 = 0.90
\ rl9 = 0.00
20
q5,17 = 4.98 b 13,20 = 0.75 r20 = 0.00
21
q6,21 = 2.21
b 10,21 = 5.00
r21 = 0.05
+ 22
q6,22 = 2.21
b 17,22 = 5.00
r22 = 0.15
В
23
* q7,23 = 1.15
b 14,23 = 2.00
\ r23 = 0.10
24
q7,24 = 1.15
b 18,24 = 2.00
r24 = 0.03
25
" q7,25 = 1.15
b 15,25 = 2.00
r25 = 0.25
□
* 27
q9,27 = 1.89
b 19,27 = 1.50
r27 = 0.12
+ 28
q9,28 = 1.89 b20,28 = 1.50 r28 = 0.18
26
q8,26 = 0.89 b 16,26 = 1.00 r26 = 0.12
Fig. 2. Structure of the biscuit production system Рис. 2. Структура производственной системы по изготовлению печенья
Figure 2 uses the same symbols as formula 1.
The rectangles with grey borders represent the production links; the rectangles with blue borders correspond to the types of products manufactured. The arrows indicate the technological relationships between the production links of the system. The product range manufactured by the production system under consideration is presented in Table 1.
Table 1. The manufactured product range
Таблица 1. Ассортимент выпускаемой продукции
Item produced (sold to the end consumer) Product number in the graph model of the production system Share in the total output
Sweet biscuits 21 0.05
Sweet biscuits 'Poteshki' 22 0.15
Soft sweet biscuits 23 0.10
Filled sweet biscuits (packaging format no. 1) 24 0.03
Filled sweet biscuits (packaging format no. 2) 25 0.25
Butter biscuits 26 0.12
Oat biscuits 27 0.18
Home-style biscuits 28 0.12
Total - 1.00
Another essential aspect of the proposed methodology is the understanding of the units of measurement for the calculated value of the production capacity of the production system.
Since the enterprise produces multiple products, let us introduce a dummy link, which manufactures one conditional type of product, consumed in the production of all types of finished products of the system in proportions equal to the specified product range proportions.
Figure 3 shows the introduction of the dummy link for the purposes of switching to the equivalent units of the product range as the basic ones for the calculation of the production capacity of the investigated production system. The dummy link (no. 10), the corresponding dummy technological relationships and dummy product (no. 29) are indicated by the dotted lines in Figure 3.
Clearly, in this situation we have replaced the specified product range proportions of the finished products of the biscuit production system by technological relationships between the real production links and the dummy link.
The next step of the proposed method is to form a matrix consistent with the graph model of the production system. A matrix form is quite convenient for handling the tasks of the production capacity management under a flexible product mix, because it allows for prompt recalculation and situation simulation when various parameters of the system change. For instance, changes can occur in such parameters as technological relationships, time and material consumption rates, production capacities of the links by products, and the structure of the final product range.
The matrix of the production system is generated in Microsoft Excel spreadsheet package. At this, we shall accept that if a product is not directly processed by this particular link, then the production capacity of the link for this product is considered to be infinitely large (for example, in Microsoft Excel we will fill such matrix elements with a conditional number of 100,000).
□
q2,3 = 4.00 bl,3 = 0.75 r3 - 0.00
1
ql,l = 8
rl = 0
□
q2,4 = 4.00 bl,4 = 0.75 r4 = 0.00
q2,5 = 3.60 bl,5 = 0.75 i5 = 0.00
q2,6 = 3.20 bl,6 = 0.75 r6 = 0.00
В
q3,7 = 3.80 b2,7 - 0.88 r7 = 0.00
T
q3,8 = 4.48 b2,8 = 0.86 r8 = 0.00
q3,9 = 5.62 b2,9 = 0.82 r9 = 0.00
_10_
q4,10 = 6.00 b3,10= 1.05 rlO - 0.00
11
12
13
14
15
16
q4,ll = 6.00 b4,ll = 1.04 rll = 0.00
q4,12 = 6.51 b5,12 = 1.05 rl2 = 0.00
q4,13 - 5.28 b6,13 - 1.06 rl3 = 0.00
В
В
q4,14 = 4.75 b7,14= 1.09 rl4 = 0.00
q4,15 = 5.25 b8,15 = 1.09 rl5 = 0.00
q4,16 = 6.01 b9,16= 1.08 rl6 = 0.00
17
q5,17 = 4.75 bll,17= 0.90 rl7 = 0.00
18
q5,17 = 4.78 bl2,18 = 0.90 rl8 = 0.t
q5,17 = 5.60 bl3,19 = 0.90 rl9 = 0.00
20
q5,17 = 4.98 b 13,20 = 0.75 r20 - 0.00
/
* 21
q6,21 = 2.21 \
b 10,21 = 5.00
r21 = 0.05
► 22 ,
q6,22 = 2.21 \
b 17,22 = 5.00 i
r22 = 0.15
В
23 i
* q7,23 = 1.15 \
b 14,23 = 2.00
r23 = 0.10
24 ..
q7,24 = 1.15
b 18,24 = 2.00
r24 = 0.03
25
" q7,25 = 1.15
b 15,25 = 2.00
r25 = 0.25
в /
► 27 ;
q9,27 = 1.89
b 19,27 = 1.50
r27 = 0.12 /
► 28 !
q9,28 = 1.89
b20,28 = 1.50
r28 = 0.18
в i
f 26
q8,26 = 0.89
b 16,26 = 1.00
r26 = 0.12
Fig. 3. Structure of the biscuit production system with the added dummy link Рис. 3. Структура производственной системы по изготовлению печенья с фиктивным звеном
The input data used for the matrix representation of the graph built for the biscuit production shop floor are shown in Figures 4 and 5 (Step 3).
Vector of the established product range proportions as an input parameter for further calculations was presented earlier in Table 1 (Step 4).
As stated before, the production capacity of a bakery enterprise regarding the biscuit production will be calculated in Microsoft Excel, which requires the involvement of the functions that
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28
1 0 0 0.75 0.75 0.75 0.75 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
2 0 0 0 0 0 0 0.88 0.86 0.82 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
3 0 0 0 0 0 0 0 0 0 1.05 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
4 0 0 0 0 0 0 0 0 0 0 1.04 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
5 0 0 0 0 0 0 0 0 0 0 0 1.05 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
6 0 0 0 0 0 0 0 0 0 0 0 0 1.06 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
7 0 0 0 0 0 0 0 0 0 0 0 0 0 1.09 0 0 0 0 0 0 0 0 0 0 0 0 0 0
8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1.09 0 0 0 0 0 0 0 0 0 0 0 0 0
9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1.08 0 0 0 0 0 0 0 0 0 0 0 0
10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5.00 0 0 0 0 0 0 0
11 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.90 0 0 0 0 0 0 0 0 0 0 0
12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.90 0 0 0 0 0 0 0 0 0 0
13 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.90 0.75 0 0 0 0 0 0 0 0
14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2.00 0 0 0 0 0
15 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2.00 0 0 0
16 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1.00 0 0
17 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5.00 0 0 0 0 0
18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2.00 0 0 0 0
19 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1.50 0
20 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1.50
21 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
22 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
23 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
24 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
25 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
26 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
27 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
28 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Fig. 4. Matrix of direct consumption rates of products per product (b) Рис. 4. Матрица прямых норм расхода продуктов на продукт (Ь)
1 8 6 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000
2 1000 1000 4 4 3.6 3.2 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000
3 1000 1000 1000 1000 1000 1000 3.8 4.48 5.62 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000
4 1000 1000 1000 1000 1000 1000 1000 1000 1000 6 6 6.51 5.28 4.75 5.25 6.01 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000
5 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 4.75 4.78 5.6 4.98 1000 1000 1000 1000 1000 1000 1000 1000
6 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 2.21 2.21 1000 1000 1000 1000 1000 1000
7 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1.15 1.15 1.15 1000 1000 1000
8 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 0.89 1000 1000
9 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1.89 1.89
Fig. 5. Matrix of the production capacity of the links by products (q) Рис. 5. Матрица производственных мощностей звеньев по продуктам (q)
allow for matrix calculus. Importantly, the use of these functions is not the only tool for the given task. It can be also performed by writing macros in the Visual Basic for Applications (VBA). The second option is more efficient in the real-life manufacturing environment, because in this case:
• calculation is more formalised and requires fewer skills from the end user (an economist) who performs the calculation of the production capacity in a matrix form;
• calculation requires less labour input from the specialists involved (is less labour-intensive), because most of the operations are done automatically, for instance, dimensioning of the matrices and generating the relevant identity matrices needed for further calculations, etc.;
• the number of human factor mistakes is reduced to a minimum.
Generally, we can conclude that using the functions of a spreadsheet package is justified at the stage of scientific experiments aimed at developing new methodological approaches. In real production conditions, the approach that relies on the procedures written using Excel VBA is more efficient from the practical point of view.
The calculation of production capacity based on a graph model of the production system (Step 5) is performed using MS Excel functions in the following sequence.
1. Calculation of the pass-through consumption rates of products per product (ft'):
h = (E - b)-1 = MINVERSE (E - b). (3)
The result of the calculation of the pass-through consumption rates of products per product in the biscuit production system is shown in Figure 6.
2. Calculation of the pass-through consumption rates of products per conditional units of the product range (h):
h = (E - b)-1 x r = MMULT (MINVERSE (E - b)); r. (4)
The result of the calculation of the pass-through consumption rates of products per conditional unit of the product range is shown in Figure 7.
It is noteworthy that the calculation of the consumption rates of products per conditional unit of the product range results in a column vector, which has been transposed and presented as a row vector for ease of perception.
3. Calculation of the direct consumption rates of the links' time per unit of each product type
(t):
t = Rev(q)=±-. (5)
The result of the calculation is shown in Figure 8.
4. Calculation of the direct consumption rates of the links' time per conditional unit of the product range (t"):
t" = t x h = MMULT(t; h). (6)
The result of the calculation is shown in Figure 9.
5. Evaluation of the throughput capacities of the links of the production system (Q):
c-f (7)
The result of the calculation is shown in Figure 10.
Finally, the production capacity of the system (Q0) is calculated (Step 7):
Qo = min{Q}. (8)
The result of the calculation is shown in Figure 11.
1 1.00 0 0.75 0.75 0.75 0.75 0 0 0 0.79 0.78 0.79 0.80 0 0 0 0.70 0.71 0.72 0.60 3.94 3.51 0 1.42 0 0 1.07 0.89
2 0 1.00 0 0 0 0 0.88 0.86 0.82 0 0 0 0 0.96 0.94 0.88 0 0 0 0 0 0 1.92 0 1.87 0.89 0 0
3 0 0 1.00 0 0 0 0 0 0 1.05 0 0 0 0 0 0 0 0 0 0 5.25 0 0 0 0 0 0 0
4 0 0 0 1.00 0 0 0 0 0 0 1.04 0 0 0 0 0 0.94 0 0 0 0 4.68 0 0 0 0 0 0
5 0 0 0 0 1.00 0 0 0 0 0 0 1.05 0 0 0 0 0 0.95 0 0 0 0 0 1.89 0 0 0 0
6 0 0 0 0 0 1.00 0 0 0 0 0 0 1.06 0 0 0 0 0 0.95 0.80 0 0 0 0 0 0 1.43 1.19
7 0 0 0 0 0 0 1.00 0 0 0 0 0 0 1.09 0 0 0 0 0 0 0 0 2.18 0 0 0 0 0
8 0 0 0 0 0 0 0 1.00 0 0 0 0 0 0 1.09 0 0 0 0 0 0 0 0 0 2.18 0 0 0
9 0 0 0 0 0 0 0 0 1.00 0 0 0 0 0 0 1.08 0 0 0 0 0 0 0 0 0 1.08 0 0
10 0 0 0 0 0 0 0 0 0 1.00 0 0 0 0 0 0 0 0 0 0 5.00 0 0 0 0 0 0 0
11 0 0 0 0 0 0 0 0 0 0 1.00 0 0 0 0 0 0.90 0 0 0 0 4.50 0 0 0 0 0 0
12 0 0 0 0 0 0 0 0 0 0 0 1.00 0 0 0 0 0 0.90 0 0 0 0 0 1.80 0 0 0 0
13 0 0 0 0 0 0 0 0 0 0 0 0 1.00 0 0 0 0 0 0.90 0.75 0 0 0 0 0 0 1.35 1.13
14 0 0 0 0 0 0 0 0 0 0 0 0 0 1.00 0 0 0 0 0 0 0 0 2.00 0 0 0 0 0
15 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1.00 0 0 0 0 0 0 0 0 0 2.00 0 0 0
16 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1.00 0 0 0 0 0 0 0 0 0 1.00 0 0
17 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1.00 0 0 0 0 5.00 0 0 0 0 0 0
18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1.00 0 0 0 0 0 2.00 0 0 0 0
19 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1.00 0 0 0 0 0 0 0 1.50 0
20 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1.00 0 0 0 0 0 0 0 1.50
21 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1.00 0 0 0 0 0 0 0
22 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1.00 0 0 0 0 0 0
23 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1.00 0 0 0 0 0
24 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1.00 0 0 0 0
25 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1.00 0 0 0
26 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1.00 0 0
27 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1.00 0
28 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1.00
Fig. 6. Matrix of the pass-through consumption rates of products per product (h') Рис. б. Матрица сквозных норм расхода продуктов на продукты (h')
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 1.067 0.767 0.263 0.702 0.057 0.401 0.218 0.545 0.130 0.250 0.675 0.054 0.378 0.200 0.500 0.120 0.750 0.060 0.270 0.180 0.050 0.150 0.100 0.030 0.250 0.120 0.180 0.120
Fig. 7. Vector of the pass-through consumption rates of products per conditional unit of the product range (h) Рис. 7. Вектор сквозных норм расхода продукции на условную ассортиментную единицу продукции (h)
1
2
3
4
5
6
7
8 9
0.13 0.17 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
0.00 0.00 0.25 0.25 0.28 0.31 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
0.00 0.00 0.00 0.00 0.00 0.00 0.26 0.22 0.18 0.00 0.00 0.00 0.00 0.00
0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.17 0.17 0.15 0.19 0.21
0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
0.00 0.00 0.00 0.00 0.00 0.17 0.00 0.21
0.00 0.00 0.00 0.00
0.19
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00 0.00 0.00
0.00 0.00 0.00 0.00 0.21 0.00 0.00 0.00 0.00
0.00 0.00 0.00 0.00 0.18 0.00 0.00 0.00 0.00
0.00 0.00 0.00 0.00 0.20 0.00 0.00 0.00 0.00
0.00 0.00 0.00 0.00 0.00 0.45 0.00 0.00 0.00
0.00 0.00 0.00 0.00 0.00 0.45 0.00 0.00 0.00
0.00 0.00 0.00 0.00 0.00 0.00 0.87 0.00 0.00
0.00 0.00 0.00 0.00 0.00
0.00 0.00 0.00 0.00 0.00
0.00 0.00 0.00 0.00 0.00
0.00 0.00 0.00 0.00 0.00
0.00 0.00 0.00 0.00 0.00
0.00 0.00 0.00 0.00 0.00
0.87 0.87 0.00 0.00 0.00
0.00 0.00 1.12 0.00 0.00
0.00 0.00 0.00 0.53 0.53
Fig. 8. Matrix of the direct consumption rates of the links' time per unit of each product type (t) Рис. 8. Матрица прямых норм расхода времени звеньев на единицу каждого вида продукции (f)
1 0.26 1 3.83 1 3.83
2 0.38 2 2.62 2 2.62
3 0.20 3 4.95 3 4.95
4 0.39 4 2,55 4 2.55
5 0.25 5 3.92 5 3.92
6 0.09 6 11.04 6 11.04
7 0.33 7 3.03 7 3.03
8 0.13 8 7.41 8 7.41
9 0.16 9 6.30 9 6.30
Fig. 9. Vector of the direct consumption rates of the links' time per conditional unit of the product range (t") Рис. 9. Вектор прямых норм расхода времени звеньев на условную ассортиментную единицу продукции (Г)
Fig. 10. Vector of the throughput capacities of the production system's links (Q) Рис. 10. Вектор пропускных способностей звеньев производственной системы (Q)
Fig. 11. Identification of the limiting link
of the biscuit production system Рис. 11. Определение лимитирующего звена производственной системы по изготовлению печенья
Accordingly, the production capacity of the system under investigation amounts to 2.55 conditional units of the product range. For the purposes of the managerial decision-making this result should be decomposed in terms of the given product range (Table 2).
Table 2. Production capacity of the biscuit enterprise specified by product
Таблица 2. Производственная мощность предприятия по изготовлению печенья в разрезе выпускаемой номенклатуры
Item produced Quantity, tonnes
Sweet biscuits 0.13
Sweet biscuits 'Poteshki' 0.38
Soft sweet biscuits 0.26
Filled sweet biscuits (packaging format no.1) 0.08
Filled sweet biscuits (packaging format no.2) 0.64
Butter biscuits 0.31
Oat biscuits 0.46
Home-style biscuits 0.31
Total 2.55
The considered case of forming a graph model for a shop floor of a bakery enterprise that makes a range of biscuits allows understanding the possibilities provided by the proposed model for managerial decision-making at various levels.
One of the basic decisions regarding the capacity management of an enterprise with a continuous stage-to-stage production is the development of a plan for modernising and revamping existing production facilities [Revutsky, 2006, p. 126]. Specifically, the limiting link for the given product range is the oven responsible for baking biscuits, which is one of the key production processes. The debottlenecking of this limiting link requires thorough development of the marketing strategy of the enterprise. Besides, it is obvious that the next limiting link is the dough forming machine, the throughput of which approximates the production capacity of the oven, which can be explained by the simple fact that this equipment is the oldest in the enterprise. All other equipment (dough mixing machine, extruder, and packaging machines) was installed substantially later during a scheduled modernisation driven by the business owners' intention to increase the market share by producing butter biscuits that are especially popular with the end consumers. In other words, the replacement or modernisation/renovation of the oven will not allow the bakery enterprise to increase its output much within the given product range and technological network. For this specific case, a complex solution is needed involving a change in the throughput capacity of several links of the production system. In addition, it is evident that changing the product range structure through specialisation or other similar processes may affect the calculation of the throughput capacities of the production system's links and, consequently, the focus of the programme for the replacement or modernisation/renovation of the equipment may shift. The modelling of production system parameters and fixing them at the defined values of other parameters will enable the company management to take efficient decisions at different levels and of various coverage down to suspending operations.
From this perspective, the task of managing the production capacity of a modern industrial enterprise is complex and involves aligning it with other basic performance indicators, including such key ones as indicators of strategic marketing, production cost, breakeven point, and financial safety margin.
Importantly, in the case considered, the proposed graph model of the production system was applied in the conditions of an operating enterprise. However, this model can be used efficiently at the stage of designing production systems, where the coherence of the above mentioned basic
indicators related to the functioning of industrial enterprises is analysed even more thoroughly, reckons with the market situation and its forecast for the long term.
It is fundamentally important to understand that the use of such mathematical techniques is only justified for complex technological processes.
Conclusion
Within the research, we have developed a method for evaluating production capacity using graph models of production systems. The central idea of the proposed model is that a manufacturing enterprise is depersonalized and perceived as a production system made up of production links, which form a production chain and create a production network. All of this is implemented in the graph model of the production system, which is complemented by a matrix model for ease of calculations and possibility to account for counter material flows. The matrix model reckons with the following parameters of the production system under investigation: technological relationships and possibilities to change them, product range, direct consumption rates of products per products, production capacities of the links by product. Therefore, the graph model directly correlates with the matrix model and is indeed its representation in a matrix form for the convenience of calculations.
The crucial aspect for the practical use of the proposed model is related to the units of measurement of the final result. Here, conditional units of product range were utilized. For the purposes of switching to the conditional units of the product range, a dummy link manufacturing a conditional type of product was introduced into the graph model of the production system. As a result, the specified final product range proportions of the production system were replaced by technological relationships between the real links and the dummy link.
Another essential characteristic of the proposed approach is abandoning the traditional method for calculating the bottleneck link, which involves the successive calculation of production capacity from the lowest level (the level of machines, equipment, processing lines, etc.) up to the level of the enterprise as a whole. This method sequentially determines the link with the minimum throughput capacity calculated in the units of products of the particular equipment / production site /shop floor /enterprise. Such link is identified as the bottleneck for the whole production system. In contrast with this method, the proposed approach based on the use of the graph model of the production system suggest that first, the throughput capacities of each link should be determined and then under a predefined product range the limiting link should be identified, i.e. the link with the minimum throughput capacity measured in conditional units of products.
The efficiency of the proposed approach to capacity management based on the use of the graph models of production systems, and its essence were exemplified and tested at the case of a bakery enterprise with regard to the shop floor producing biscuits.
Production capacity of the biscuits-producing shop floor amounted to 2.55 conditional units of the product range. This result was decomposed in terms of the given product range for the purposes of the managerial decision-making. The basic decision that could be taken regarding the production capacity management based on this information was determined to be the decision concerning modernisation of existing production facilities.
The findings of the analysis of the throughput capacities of the production system's links indicate that under the predefined product range the limiting link of the biscuit production is the oven responsible for one of the main processes, namely, baking. The debottlenecking of this limiting link is pointless without increasing the capacity of the dough forming machine, because its throughput capacity approximates the capacity of the oven, which is due to the fact that they were aligned with each other during the shop floor design. All other equipment, namely dough
mixing machine, extruder, and packaging machines have large reserves of production capacities and may not be taken into consideration by the programme for modernisation and renovation of existing production facilities of the biscuit-producing shop floor. Hence, modelling of the parameters of the production system and fixing them at the specified values of the other parameters will enable management of modern industrial enterprises to take efficient decisions concerning the production down to suspending operations.
Bringing the paper to a close, we shall stress that the proposed graph model for calculating the production capacity of a manufacturing enterprise enables the prompt real-time recalculation of the production capacity based on the existing product range structure as defined by the portfolio of orders. Furthermore, the graph model of the production system allows evaluating the production capacity of an enterprise and analysing its structure, as well as formulating the directions for modernisation of the fixed assets and assessing the company position on target markets.
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Pacheco D. A. de J., Pergher I., Jung C. F, Scwenberg ten Caten C. (2014). Strategies for increasing productivity in production systems. Independent Journal of Management & Production, vol. 5, pp. 344-359. DOI: 10.14807/ijmp.v5i2.134.
Pesch E., Glover F., Bartsch T., Salewski F., Osman I. (1999). Efficient facility layout planning in a maximally planar graph model. International Journal of Production Research, vol. 37, issue 2, pp. 263-283. DOI: 10.1080/002075499191760.
Rose E., Odom R., Dunbar R., Hinchman J. (1995). How TOC & TPM work together to build the quality toolbox of SDWTs. Proc. of 17th Int. Electronics Manufacturing Technology Symposium "Manufacturing Technologies-Present and Future", pp. 56-59.
Schonberger R. J. (2007). Japanese production management: An evolution - With mixed results. Journal of Operations Management, vol. 25, issue 2, pp. 403-419. https://doi.org/10.1016/j.jom.2006.04.003.
Shingo S. (1996). The Toyota Production System: From the standpoint ofproduction engineering. Porto Alegre: Bookman.
Teerasoponpong S., Sopadang A. (2021). A simulation-optimization approach for adaptive manufacturing capacity planning in small and medium-sized enterprises. Expert Systems with Applications, vol. 168, art. no. 114451. https://doi.org/10.1016/j.eswa.2020.114451.
Yang G.-L., Fukuyama H., Song Y.-Y. (2019). Estimating capacity utilization of Chinese manufacturing industries. Socio-Economic Planning Sciences, vol. 67, pp. 94-110. DOI: 10.1016/j.seps.2018.10.004.
Zhao T., Song Z.-J., Li T.-J. (2018). Effect of innovation capacity, production capacity and vertical specialization on innovation performance in China's electronic manufacturing: Analysis from the supply and demand sides. PLoS ONE, vol. 13, issue 7, e0200642. https://doi.org/10.1371/journal.pone.0200642.
Information about the authors
Natalya V. Kireeva, Dr. Sc. (Econ.), Associate Prof., Head of Economics Dept., Ural Socio-Economic Institute (branch of the Academy of Labour and Social Relations), 155/1 Svobody St., Chelyabinsk, 454091, Russia
Phone: +7 (3512) 260-07-11, e-mail: veo.chel@gmail.com
Evgenia S. Zambrzhitskaya, Cand. Sc. (Econ.), Associate Prof., Associate Prof. of Economics Dept., Nosov Magnitogorsk State Technical University, 38 Lenina Ave., Magnitogorsk, Chelyabinsk oblast, 455000, Russia
Phone: +7 (912) 805-77-26, e-mail: jenia-v@yandex.ru
Elena A. Makarova, Head of Production Scheduling and Financial Dept., OOO "Russkiy Khleb Food Group", 26 Verkhneuralskoe Road, Magnitogorsk, Chelyabinsk oblast, 455000, Russia Phone: +7 (3519) 25-15-40, e-mail: rushleb-econ@mail.ru
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Morrison C. J. (1986). Productivity measurement with non-static expectations and varying capacity utilization: An integrated approach. Journal of Econometrics, vol. 33, issue 1-2, pp. 51-74. https://doi. org/10.1016/0304-4076(86)90027-8.
Pacheco D. A. de J., Pergher I., Jung C. F, Scwenberg ten Caten C. (2014). Strategies for increasing productivity in production systems. Independent Journal of Management & Production, vol. 5, pp. 344-359. DOI: 10.14807/ijmp.v5i2.134.
Pesch E., Glover F., Bartsch T., Salewski F., Osman I. (1999). Efficient facility layout planning in a maximally planar graph model. International Journal of Production Research, vol. 37, issue 2, pp. 263-283. DOI: 10.1080/002075499191760.
Rose E., Odom R., Dunbar R., Hinchman J. (1995). How TOC & TPM work together to build the quality toolbox of SDWTs. Proc. of 17th Int. Electronics Manufacturing Technology Symposium "Manufacturing Technologies-Present and Future", pp. 56-59.
Schonberger R. J. (2007). Japanese production management: An evolution - With mixed results. Journal of Operations Management, vol. 25, issue 2, pp. 403-419. https://doi.org/10.1016/jjjom.2006.04.003.
Shingo S. (1996). The Toyota Production System: From the standpoint ofproduction engineering. Porto Alegre: Bookman.
Teerasoponpong S., Sopadang A. (2021). A simulation-optimization approach for adaptive manufacturing capacity planning in small and medium-sized enterprises. Expert Systems with Applications, vol. 168, art. no. 114451. https://doi.org/10.1016/j.eswa.2020.114451.
Yang G.-L., Fukuyama H., Song Y.-Y. (2019). Estimating capacity utilization of Chinese manufacturing industries. Socio-Economic Planning Sciences, vol. 67, pp. 94-110. DOI: 10.1016/j.seps.2018.10.004.
Zhao T., Song Z.-J., Li T.-J. (2018). Effect of innovation capacity, production capacity and vertical specialization on innovation performance in China's electronic manufacturing: Analysis from the supply and demand sides. PLoS ONE, vol. 13, issue 7, e0200642. https://doi.org/10.1371/journal. pone.0200642.
Информация об авторах Киреева Наталья Владимировна, доктор экономических наук, доцент, заведующий кафедрой экономики Уральского социально-экономического института (филиала) образовательного учреждения профсоюзов высшего образования «Академия труда и социальных отношений», 454091, РФ, г. Челябинск, ул. Свободы, 155/1
Контактный телефон: +7 (3512) 260-07-11, e-mail: veo.chel@gmail.com
Замбржицкая Евгения Сергеевна, кандидат экономических наук, доцент, доцент кафедры экономики Магнитогорского государственного технического университета им. Г. И. Носова, 455000, РФ, Челябинская область, г. Магнитогорск, пр-т Ленина, 38 Контактный телефон: +7 (912) 805-77-26, e-mail: jenia-v@yandex.ru
Макарова Елена Анатольевна, начальник планово-экономического отдела ООО «Продовольственная группа «Русский хлеб», 455000, РФ, Челябинская область, г. Магнитогорск, Верхнеуральское шоссе, 26
Контактный телефон: +7 (3519) 25-15-40, e-mail: rushleb-econ@mail.ru
© Киреева Н. В., Замбржицкая Е. С., Макарова Е. А., 2021
