Summary
In the paper a design of the probabilistic model of reserves is discussed. The ob-jective was an elaboration of the optimal strategy of materials reserves management in the series manufacturing. The authors discuss the probabilistic model of reserves aiming at the elaboration of the best management of primary material in industrial company. The model has been verified as far as it was feasible using numerical data concerning various wood products including furniture of series production. The pre-sented probabilistic model of reserves allows the formation12 of optimum strategy of the (R, Z) type of primary materials management with due consideration to profits coming from low cost components in working and furniture industrial company. This paper also describes the model of dynamic management system of manufacturing company and presents arguments for changing processes inside a company in the light of change in the stream of incoming orders for finished products. The results of computer simulation of designed model, show some dynamic characteristics across divisions of manufacturing company during production process. In the model of pro-duction facility, different departments were taken into account. The departments are involved in activities related to flow of information, orders, materials and prefabri-cates, production processes as well as finished products storage and shipment Keywords: model of dynamic management system
1. Introduction
Cleverness and effective management of the entire manufacturing company, requires a system capable of performing dynamic analysis of managing operation. This activity is determined by a need to adjust operation of facility to constantly changing demands of market34. The analysis of information flow in manufacturing facility, and subsequent flow of decisions, materials and
1 Jagas, J. (2004): Produktywno pracy. Opole: Uniwersytet Opolski. ISBN 978-83-7641-056-2.
2 Farlow, S.J. - Haggard, G.M. (1987): Applied mathematics for Management, Life Science and Social Sciences. New York: RANDOM HOUSE. ISBN 0-397-35160-6.
3
Jagas, J. (2004): Produktywno pracy. Opole: Uniwersytet Opolski. ISBN 978-83-7641-056-2.
4
products is one of the most important aspects of entrepreneurial activity56. The management method presented in this paper is used to expose weak links in organisation structure that influence efficiency of the entire manufacturing company789. The weak links can be also identified on lower level of organisational structure, for example, in the process of product manufacturing. On such level, inputs may include consumable tooling, machining fluids, scrap and chips headed for recy-cling, etc.101112. The analysis takes into account the interactions between streams of accepted orders, materials, production orders, production processes, finished products, money and personnel within a company. Instrument for simulation of the interactions is model that reflects the specific characteristics of entire system. The goal of this study was to build the dynamic model of manu-facturing company managing operation, and analyze the reasons for changing processes inside a company due to changes in the stream of incoming orders for finished products. The departments taken into account in the model are involved in flow of information, orders, materials and prefabri-cates, production as well as finished products storage and shipment.
2. Description of the model
The analogue symbolic diagram of model (Fig. 1) represents general structure of manufactur-ing company, and reflects principle managerial activities. In the symbolic description of production facility model the following departments were included: orders department (ZF), supply department (ZM), store of materials (MM), production department (P) and stock of fin-ished products (MP). The assumptions of company operations are as follows:
The volume of production should be high enough to maintain stock of finished products at the level of K (i.e. three) times higher than the average of orders (ZS) for products. That demand is achieved by shift from production orders (ZP) to production preparation. The production orders (ZP), as amount, can be represented by the following equation:
ZP = S1+ ZS (1)
S1 – auxiliary variable representing the difference between required and actual levels of
stock of finished products, ZS – average amount of received orders.
5 Hilier, F.S. - Lieberman, G.J. (1998): Introduction to stochastic models in operations research. New York: McGraw-Hill, Inc. ISBN 0-03-894995-7.
6 Wołowiec T.: ZOZ a wybrane aspekty zarzdzania zapasami w warunkach sezonowoci sprzeday usług medycznych. „Antidotum zarzdzanie w opiece zdrowotnej 2002 nr 10. ISSN 1230-0969, s. 13–30.
7 Hilier, F.S. - Lieberman, G.J. (1997): Introduction to operations research. New York: McGraw-Hill, Inc. ISBN 0-07-028908-5.
8
Wołowiec T. Suseł A.: Conception of the optimal and safe strategy of material reserves management on the basis of the probabilistic model. /in/ JASEK R. (red.) Internet, Competititiveness and Organizational Security. lin: Tomas Bata University in Zlin 2010. ISBN: 978-83-61645-16-0. s. 467–472.
9 ukowski, P. (2009): Managing Model of the Storing Material in an Economic Organization, Foundry Journal - Science and Practice, vol. 59, No. 7-8, p. 414–419.
10
Wołowiec T. Suseł A.: Conception of the optimal and safe strategy of material reserves management on the basis of the probabilistic model. /in/ JASEK R. (red.) Internet, Competititiveness and Organizational Security. lin: Tomas Bata University in Zlin 2010. ISBN: 978-83-61645-16-0. s. 467–472.
11 Wołowiec T. Suseł A. Uwarunkowania płodnoci imigrantek oraz nie-imigrantek w USA, Acta Universitatis Lodziensis, „Folia Oeconomica”, 2009, nr 231. ISSN 0208-6018. s. 447–467.
12 ukowski, P. (2009): Koncepcja optymalnej strategii zarzdzania zapasami materiałów w przedsibiorstwie przemysło-wym, (in) Uwarunkowania rozwoju systemów zarzdzania, (ed.). H. Hawanie, W. Iwaszkiewicz. Bielsko-Biała: Akademia Humanistyczno-Ekonomiczna. ISBN 978-83-9332397-0-3.
The needed level of product is the average volume of orders (ZS) and multiple by the amplification coefficient K. The coefficient K is a number of weeks during which the re-quired level of stock of finished products would be sufficient for shipment of products with speed equal to the average speed of received orders. Auxiliary variable S1 is defined by the
following formula:
S1 = K
⋅
ZS – MP (2)K – amplification coefficient, ZS – average amount of received orders, MP – actual amount (level) of stock of finished products.
1) In case when the level of store of materials (MM) falling below the level defined (S2) by
amount of received orders (ZF), incoming orders should be temporarily stopped (ZZF). If the level of store of materials (MM) is below (S3) than defined minimum value (Min),
transfer of production orders (ZR) to production department (P) lines should be stopped as well.
S3 = MM – Min (3)
2) The stream of amount of orders for purchase of materials (ZZM) shifted to supply department equals the stream of amount of orders for finished products (R) in manufacturing company. Information processing ability of office are also taken into account.
This ability is defined as the average times of transition for: averaging of orders – TZS, supply – TZM, production – TP, distribution – TZF. Then, the condition 1 – TMP is met.
Figure 1. Analogue-symbolic model of manufacturing company Source: Own work.
P(x) ZZF S2 1 R 2 2 11 12 S1 S3 4 4 7 6 6 P(x) ZR x ZP ZF 3 K ZMP 8 WP TP TZF TZS Z R ZP R ZZF ZZF = ZR Min MZM ZZM TMP Z 1(T) Deliver y orders materi als and prefabricated elements ZM 13 Supply department MM 14 Materi als store Creat e of pr oduction orders Orders portfolio Market of manuf aktures pr oducts Archi ve of executive an orders ZS 5 Aver agi ng of orders P 9 Production department MP 10 Finished products stock orders information materi als T Z M
3. Building of the management model
Two streams of information come from market through sales department (Fig. 1). The first information carries orders about the specific types of products, whereas the second one about the volume of these orders.
In the first decision making point, R, amount of stream of incoming orders for finished prod-ucts is described. The orders are coming in batches. This amount is assumed to be represented by the step function and shown in the equation:
R = R0 + Z · 1 (T) (4)
The second decision making point can be represented by the function P*:
< ≥ = ZF MM if ZF MM if P , 0 , 1 * (5) hence: ZZF = P* · R (6)
The third decision making point, ZF, presents amount of received orders for products. The amount of orders portfolio ZF can be described as:
ZF = ZF0+ DT (ZZF – ZMP) (7)
The fourth decision making point, ZMP, regulates:
- amount of products taken from production taken from stock of finished products (MP) - amount of completed orders transmitted from orders portfolio to archive.
The fourth decision making point can be represented by the equation:
ZMP = ZF/TZF (8)
The fifth decision making point, ZS, describes the average value of incoming orders stream, R, (with influence of the second decision station, ZZF). The fifth decision making point is repre-sented by the equation:
ZS = ZS0 + DT (ZZF – ZS0) (9)
The sixth decision making point, ZP, determines (S1):
- amount of orders transmitted to production, - amount of materials transmitted to production.
The amount of production orders ZP is described by the equation:
ZP = ZS + S1/TP (10)
The process of transmission of orders to production is stopped when the level of stock of ma-terials is minimal. This decision is taken at the seventh decision making point, ZP. It makes certain of fulfiling the assumption 2. This point can be presented by the function F*:
< ≥ = 0 , 0 0 , 1 3 3 * S if S if F (11) and
ZR = F* · ZP (12) The eighth decision making point, WP, defines time for production. The time of transmission, P, is considered as permanent and accumulating element of manufacturing company abilities which seems to be equaled amount of production orders. The influence of the eighth decision making point can be presented by the equation:
WP = P/TP (13)
The ninth decision making point defines current amount of production orders portfolio and is defined as:
P = P0 + DT (ZR – WP) (14)
The tenth decision making point, MP, defines the level of finished products. Its input is the stream of production WP, and output is the stream of finished products delivering a market. The difference between these two streams remains as stock of finished products. The influence of the tenth decision making point:
MP = MP0 + DT (WP – ZMP) (15)
The eleventh decision making point, ZZM, defines the value of stream of orders for purchase of materials through supply department. In accordance with the condition 3, it equals the stream of orders for products. It is expressed by the equation:
ZZM = R (16)
The twelfth decision making point, MZM, presents the stream of materials transfering to ma-terials store in a supply department. The amount of this stream is regulated based on the information about orders number accumulated in the supply department. This point can be pre-sented as the equation:
MZM = ZM/TZM (17)
The thirteenth decision making point, ZM, accumulates the difference between incoming stream of orders for purchase of materials, ZZM, and the stream of completed orders. This is equal to the stream of purchased materials, MZM, transfering to store of materials through the supply department.
ZM = ZM0+ DT (ZZM – MZM) (18)
The fourteenth decision making point, MM, gathers the difference between the stream of pur-chased materials, MZM, and the stream of their transmission to production ZP. The influence of this decision point presents the equation:
MM = MM0 + DT (MZM – ZP) (19)
where:
R – stream of orders for products, R0 – initial amount of orders, Z – amount of incoming
or-ders, ZF – new amount of received orders portfolio, ZF0 – former amount of incoming orders, ZS
– average amount of incoming orders, ZS0 – former amount of orders, P – amount of production
orders, P0 – former value of production orders portfolio, MM – level of store of materials, MM0 –
former amount (level) of store of materials, ZM – amount of orders of the supply department, ZM0
– former amount of orders of the supply department, MP – finished products stock, MP0 – former
amount (level) of stock of finished products, ZZF – amount of incoming orders, ZMP – amount of completed orders, ZR – transmission of orders to production, WP – transmission of orders from production equal to production value – amount of production stream, ZZM – stream of orders for
purchase of materials, MZM – stream of materials, DT – time period, TZF – orders realisation lead time (assumed to be a constant), TP – time of transmission of company production abilities, TZM – time of materials purchase.
4. Results of simulation
The simulation results of the above described two cases are illustrated in Fig. 2 and Fig. 3. The figures show that with the increase of product orders to the level of 20 units/week, disruptions in the company operation appear. That results in the fluctuations in production level and as well as in the stream of materials required for production. These fluctuations cease at about 16th weeks after the increase of orders for products.
Figure 2. Simulation results for level of materials in store while increase of orders 40 units/week and initial level of materials 270 units
Source: Own studies.
When the increase of product orders to the level of 40 units/week was simulated, disruptions in the company operation can be observed even in 25th week, but the amplitude of fluctuations is within 10% of average value as well.
This graph shows also the reason of disruptions in the company operation at very long period of time. The intersection of lines MM and ZF (in accordance with the assumption 2) is the graph-ical presentation of periodic suspending of acceptance of orders. This effect is proportional to market demand fluctuations and causes additional disruptions in the company operation1314.
13 ukowski, P. (2009): Managing Model of the Storing Material in an Economic Organization, Foundry Journal - Science and Practice, vol. 59, No. 7-8, p. 414-419.
14 [9] ukowski, P. (2009): Koncepcja optymalnej strategii zarzdzania zapasami materiałów w przedsibiorstwie przemy-słowym, (in) Uwarunkowania rozwoju systemów zarzdzania, (ed.). H. Hawanie, W. Iwaszkiewicz. Bielsko-Biała: Akademia Humanistyczno-Ekonomiczna. ISBN 978-83-9332397-0-3.
ϯϱϬ ϮϬϬ ϭϱϬ ϭϬϬ ϱϬ ϭϬϬ ϭϱϬ ϮϬϬ ϯϬϬ ϮϱϬ ϱ ϭϬ ϭϱ ϮϬ Ϯϱ ϯϬ ϯϱ ϰϬ ϰϱ dŝŵĞǁĞĞ ŬƐ W W & DD DDϬ &Ϭ5. Conclusions
The investigation of the dynamic model of manufacturing company management was con-ducted using data from the study of manufacturing company. The result of simulation show that, with the increase of product orders, disruptions in a company appear. The disruptions result in the fluctuations in the level of production and stream of materials and prefabricates required for production. These fluctuations cease after various periods of time. The length of periods of time depends on the increase of orders and operating settings that are internal to the operation of manu-facturing facility.
6. Case model – introduction
In order to ensure the production continuity, the reserves of different kinds of based material are generated and maintained. In the furniture industry, wide range and community of decisions, as well as, economic influence of wrong decisions concerning material reserves are of such im-portance, that they prove a need of overworking an optimal strategy of based material reserves management on the basis of the probabilistic model with an application of computer techniques15.
Figure 3. Change in the level of material reserves in a store: Z – value of order of special kind of material, R – value of safety reserve, L – period of delivery
Source: Base on [1].
The level of wood material reserves in a store of furniture industry's differs during time, de-pending on input and output of these materials. Input of these materials depends on the amount of supplied material and time between deliveries, whereas output depends on the level of production consumption (Fig. 1). A period of time between subsequent material deliveries (delivery cycle) fluctuates, as well as production consumption of different kinds of materials. The problem of based material management could be a source of opposite tendencies in a factory. There are groups of different factors (technical, organizational, economical, financial etc.) which influence high or low levels of material reserves. High levels of material reserves are concerned with the great costs
15 Jagas, J. (2004): Produktywno pracy. Opole: Uniwersytet Opolski. ISBN 978-83-7641-056-2.
L L
Cycle 2 Time
Z
Z Z
of storage, but low levels influence costs as well, because of the lack of production consumption ensuing (the continuity not ensured). The level of based material reserves is maintained on certain rigidly determined levels. Keeping of high or low levels of materials in a store is a source of additional costs1617. The main objective of reserves management optimization should focus on analysis of costs of these materials, while criterion of optimization should be the minimization of value of expected medium costs of buying, storing and lack of production consumption ensuring of based materials. As a result of main (criterion) function optimization there is to ensure the levels of based materials needed for the regular production program realization, with the lowest level of costs181920.
7. Constructing of the probabilistic model
In order to construct the probabilistic model of reserves management such symbols are intro-duced:
D – mean production consumption for special kind of material in certain time (e.g. medium years consumption),
Z – value of order of special kind of material,
D/Z – mean number of orders of special kind of material in certain time (e.g. during one year), R – value of safety reserve (for special kind of material),
K – constant costs of orders, h – single cost of storing,
p – single cost in case of lack of needed level of reserve for special kind of material in a store, L – period (time) of delivery (time for order realisation),
v – production consumption for special kind of material in period of delivery,
E(v) – wanted value of production consumption for special kind of material in period of deliv-ery,
g(v) – probability distribution of production consumption for special kind of material in period of delivery (function, of random variable density probability),
b – mean level of lacking stores in period of delivery,
B – mean level of lacking stores for special material in certain investigation time, E(B) – expected value of mean level of lacking stores for special material in certain time, A – uniform distribution – upper limit of function,
E – operator of expected value,
F(Z,R)– main function of model with decisive variables Z and R.
16 Farlow, S.J. - Haggard, G.M. (1987): Applied mathematics for Management, Life Science and Social Sciences. New York: Random House. ISBN 0-397-35160-6.
17
Hilier, F.S. - Lieberman, G.J. (1998): Introduction to stochastic models in operations research. New York: McGraw-Hill, Inc. ISBN 0-03-894995-7.
18
Wołowiec T.: ZOZ a wybrane aspekty zarzdzania zapasami w warunkach sezonowoci sprzeday usług medycznych. „Antidotum zarzdzanie w opiece zdrowotnej 2002 nr 10. ISSN 1230-0969, s. 13–30.
19
Wołowiec T. Suseł A.: Conception of the optimal and safe strategy of material reserves management on the basis of the probabilistic model. /in/ Jasek R. (red.) Internet, Competititiveness and Organizational Security. lin: Tomas Bata University in Zlin 2010. ISBN: 978-83-61645-16-0. s. 467–472.
20
Wołowiec T. Suseł A. Uwarunkowania płodnoci imigrantek oraz nie-imigrantek w USA, Acta Universitatis Lodziensis, „Folia Oeconomica”, 2009, nr 231. ISSN 0208-6018. s. 447–467.
It should be noted, that by the end of delivery cycle expected level of reserve of certain based material in a store is R–E(v) but when a particular order is completed (on the beginning of cycle) such level is Z+R–E(v). Expected medium level of reserves for certain kind material in the cycle (if v R) is equal to:
( ) ( ) ( ( )) ( ) − + = − + − + v E R Z v E R v E R Z 2 2 (1)
If, v > R , then medium level of lacking reserve of certain based material (b) in a store is calculated: (v R) ( )g v dv b R
∞ − = (2)Expected in a certain time (e.g. during one year) medium level of lacking stocks special based material (B) in a store is obtained by the equation:
( ) Z D b B E = ⋅ (3)
Having calculated the certain levels of special material reserves, we can calculate appropri-ate costs, multiplicating appropriappropri-ate single costs h and p by the expressions (1) and (3).
Thus, the main function of optimisation in the probabilistic model of material reserve man-agement at a certain time can be expressed by the formula:
( ) [ ] ( ) min 2 , = + + − + b → Z D p v E R Z h Z D K R Z F E (4)
The first component of sum represents mean cost of constants, the second one is mean cost of storing reserves, whereas the third component means cost in case of lack kind of certain reserve. Changes in the level of quantities of basic values (Z and R ) in the main function will be minimal when Z and R are optimal2122. Then, the dependencies should determined from which optimal values Zo and Ro could be calculated. It is necessary, the first partial derivatives ofthe function (1) in a comparison with Z and R values will be zero.
21
Hilier, F.S. - Lieberman, G.J. (1998): Introduction to stochastic models in operations research. New York: McGraw-Hill, Inc. ISBN 0-03-894995-7.
22
Hilier, F.S. - Lieberman, G.J. (1997): Introduction to operations research. New York: McGraw-Hill, Inc. ISBN 0-07-028908-5.
Figure 4. Scheme of iteration researches Z0 and R0 Source: Own studies.
Optimal values of order and store level R expressions could be found:
( ) h pb K D Z 0 = 2 + (5) ( ) ∞ = Ro pD hZ dv v g 0 (6) Should be underlined that the expressions (5) and (6) can not be used for direct calculating of optimal values (Zo, Ro). (Fig. 2.). Thus, an iterative method (process) of seeking Zo and Ro values in the finite number of steps was elaborated2324. The conditions of required convergence of the iterative method (existence of the solution of problem) satisfies the inequality as follows:
( )
[
]
h v pE K D h pD + > 2 (7) i. e. m w Z Z > (Fig. 4). 23Farlow, S.J. - Haggard, G.M. (1987): Applied mathematics for Management, Life Science and Social Sciences. New York: Random House. ISBN 0-397-35160-6.
24
ukowski, P. (2009): Koncepcja optymalnej strategii zarzdzania zapasami materiałów w przedsibiorstwie przemysło-wym, (in) Uwarunkowania rozwoju systemów zarzdzania, (ed.). H. Hawanie, W. Iwaszkiewicz. Bielsko-Biała: Akademia Humanistyczno-Ekonomiczna. ISBN 978-83-9332397-0-3.
Z w Z m Z2 Z3 Z o Z 1 = Z r R 1 R R 2 R 3 R o o E q u a t i o n 5 E q u a t i o n 6
8. Algorithm of setting the optimal values
Beginning the iterative process with the first probable meaning of Z value equals h
DK /
2 , with the increase of iteration numbers the value of Zi increases, when Rivalue
de-creases. Hence, the iterative process is quickly convergent.
It is recommended to use computer techniques to calculate the R0 and Z0
(
0lim
i)
i
R
R
∞ →=
. For this purpose the computer software of operative scheme of R0 and Z0 was written. This
program does calculations until the difference Ri+1 – Ri values is adequately low (e.g. 0,00001). It
means, that two calculated values are similar. For the optimal value R0 we use then Ri+1 value,
because R0≅ Ri+1. The optimal value of Z0 were estimated on the basis of R0(Ri+1) (Fig. 3, Fig. 4).
It should be noted, that in case when there is even distribution of probability of produc-tion consumpproduc-tion of bases materials, the expressions (5) and (6) should be solved directly, i. e. optimal values of R0 and Z0 could be presented as follows:
R A p K h D A h 0 1 1 2 = − − − (8) Z D K h D A h 0 2 = − ( ) (9)
The formulas (8), (9) are based on case of even probability of production consumption of mate-rials: ∈ ∉ = [ , 0 ] , 1 [ , 0 ] , 0 ) ( A v if A A v if v g (10)
The integral in the expression (6) can be presented by using elementary functions. In gen-eral case, the presented simplification is not possible and iterative process of estimation R0 and Z0
should be then employed.
9. Empirical verification of the model (on real data)
The constructed probabilistic model is under empirical verification. The practical verification of the constructed model was performed for every main kind of based material used during serial production of furniture (kitchen sets, combined set, chairs, armchairs). In the calculations numeri-cal data was used from the 6-year periods of production activities of furniture factories in Poland. The verification showed, that worked out the iterative method of solving the model of reserves leads to estimation of optimal values of reserves R0and order Z0 with minimal costs, and in this
way leads to determination of optimal (R, Z) type strategy of based material reserves management in a factory25.
25
Hilier, F.S. – Lieberman, G.J. (1997): Introduction to operations research. New York: McGraw-Hill, Inc. ISBN 0-07-028908-5.
No solution Start Data input D, K, h, p, ε Α h pD Zw = h A p K D Zm ) 2 ( 2 + = m w Z Z < 0 : 1 = R 1 : = i h DK Zi 2 = − = + pD hZ A R i i 1 1 2 2 1 2 1 R A A R b i i i = − + + + h pb K D Z i i ) ( 2 1 + = + ε < − + i i R R 1 1 1, + + i i Z R Printing: Stop 1 := i+ i No solution
Determination of the values of the matrix elements
R0(I + 1) Z0(I + 1)
R0(I + 1) - R0(I) < ε
Printing of the matrix elements values R0(I +1) Z0(I + 1) Stop m w Z Z < R(1) = 0 I = 1 Values determination Z0(1) Start Data input D, K, h, p, ε Α Values determination Zw; Zm I = I + 1
Figure 6. R0 and Z0 optimal values searching block diagram with uniform distributio Source: Own studies.
10. Optimal strategy formation
The main rule of optimal strategy of reserves management is: when the level of reserves of special based material in a store reaches value R0, the order should be equal to the value of Z0, so
that mean costs concerned with reserves with time under consideration will be minimal.
The constructed probabilistic model of wood reserves management has methodological value because it shows the method of working out an optimal strategy of basal reserves of based material management in the conditions of serial and polyassortment production of furniture and other based products in special conditions of economical practice in the furniture industry.
11. Conclusions
The model has been verified at an attainable scale using the numerical data concerning differ-ent assortmdiffer-ent of materials in the serial production of furniture. The proposed probabilistic model makes it possible to elaborate the optimal strategy of type R, Z in the management of reserves of based materials of the manufacturing company with large-lot production. The strategy is based on minimal expenses connected with a supply of materials. Computer modeling provide sets of information on the optimal strategy of material reserves as well as cut the costs of production.
%LEOLRJUDSK\
[1] Jagas, J. (2004): ProduktywnoĞü pracy. Opole: Uniwersytet Opolski. ISBN 978-83-7641-056-2. [2] Farlow, S.J.- Haggard, G.M. (1987): Applied mathematics for Management, Life Science and Social Sciences. New York: Random House. ISBN 0-397-35160-6.
[3] Hilier, F.S. - Lieberman, G.J. (1998): Introduction to stochastic models in operations research. New York: McGraw-Hill, Inc. ISBN 0-03-894995-7.
[4] Hilier, F.S. - Lieberman, G.J. (1997): Introduction to operations research. New York: McGraw-Hill, Inc. ISBN 0-07-028908-5.
[5] Wołowiec T.: ZOZ a wybrane aspekty zarzdzania zapasami w warunkach sezonowoci sprzeday usług medycznych. „Antidotum zarządzanie w opiece zdrowotnej 2002 nr 10. ISSN 1230–0969, s. 13–30.
[6] Wołowiec T. Suseł A.: Conception of the optimal and safe strategy of material reserves man-agement on the basis of the probabilistic model. /in/ JASEK R. (red.) Internet, Competititiveness and Organizational Security. ĩlin: Tomas Bata University in Zlin 2010. ISBN: 978-83-61645-16-0. s. 467–472.
[7] Wołowiec T. Suseł A. Uwarunkowania płodnoci imigrantek oraz nie-imigrantek w USA, Acta Universitatis Lodziensis, „Folia Oeconomica”, 2009, nr 231. ISSN 0208-6018. s. 447–467. [8] ĩukowski, P. (2009): Managing Model of the Storing Material in an Economic Organization,
Foundry Journal – Science and Practice, vol. 59, No. 7–8, p. 414–419.
[9] ĩukowski, P. (2009): Koncepcja optymalnej strategii zarządzania zapasami materiałów w przedsiĊbiorstwie przemysłowym, (in) Uwarunkowania rozwoju systemów zarządzania, (ed.). H. Hawanie, W. Iwaszkiewicz. Bielsko-Biała: Akademia Humanistyczno-Ekonomiczna. ISBN 978-83-9332397-0-3.
MODEL DYNAMICZNEGO SYSTEMU ZARZĄDZANIA W FIRMIE PRODUKCYJNEJ W WYKORZYSTANIEM MODELI PROPABLIUSTYCZNYCH
Streszczenie
W artykule zaprezentowano propablistyczny model rezerw. Konstrukcja modelu podporzdkowana była budowie optymalnej strategii rezerw materiałowych w pro-dukcji seryjnej. Model zweryfikowano przy uyciu danych liczbowych dotyczcych rónych produktów z drewna, w tym meble z produkcji seryjnej. W artykule przed-stawiono równie model dynamicznego systemu zarzdzania firm produkcyjn. Słowa kluczowe: zarządzanie zapasami, modele propablistyczne
Tomasz Wołowiec
Wyzsza Szkola Biznesu — National-Louis University (WSB-NLU) ul. Zielona 27, 33-300 Nowy Sacz, Poland
e-mail: wolowiec@wsb-nlu.edu.pl Janusz SoboĔ
WyĪsza Szkola Biznesu — National-Louis University (WSB-NLU) ul. Zielona 27, 33-300 Nowy Sącz, Poland
PaĔstwowa WyĪsza Szkoła Zawodowa w Gorzowie Wlkp. ul. Teatralna 25, 66-400 Gorzów Wlkp.
