Analysis of the spare parts stocking model used in preventive maintenance.

Popa, Ionut ; Lupescu, Octavian ; Scurtu, Ramona Popa 等

1. INTRODUCTION

Because the corrective maintenance activities of technological equipments also imply the replacement of the used pieces with new spare parts, the stocking theory contributes, corroborated with the "history" of the failures appearance (Kelly, 2006), to the creation of a preventive maintenance system that will attenuate, from economically point of view, the company budget necessary for equipments maintenance activities.

2. APPLICATION AREA

A stocking theory problem exists when the maintenance activities are solved using spare parts and their necessary minimum quantity must be controlled. (Lupescu et.al., 2008).

In the most cases, the stocking construction can generate some direct and indirect expenses (Popa et.al., 2008), fact that justify the necessity of an optimized maintenance program in the company, that can ensure a minimum level of the expenses, necessary for acquisition, transport, storage of the spare parts.

3. METHOD USED

The utilization of a stocking spare parts determinist model with gradually provision, when no absence in stock may be accepted, suppose the following work hypothesis:

--the provision rate is bigger than the request rate;

--the absence in stock may not be accepted. (15 [not equal to] 0);

--considering [T.sub.3] = 0, results also that s = 0 and [T.sub.4] = 0 (Rusu, 2001); to be possible the cost function determination, the overage expenses can be evaluated on time units. So, a graphical representation of the model is reproduced in figure 1.

[FIGURE 1 OMITTED]

The demands number to be launched can be determinate through [theta]/T rapport, either through N/Q (Lupescu et.al., 2008). From the presented graphic results the cost function f(T), depending by a single variable [T.sub.2] that allow a minimum and can be determined with the following relation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

in which: k--provision rate, that is the number that will be aquisitioned on a time unit; [c.sub.s]--stocking cost of a product, on a time unit; [c.sub.1]--launching cost of a provision order, that does not depend by the products number that will be manufactured.

Annulling the f derivate reported to [T.sub.2] will result:

[T.sup.*.sub.2] = [square root of 2[c.sub.l](k - r)/k x r x [c.sub.s] (2)

Because on [0,[T.sub.1] interval one produce spare parts with a (k) production rate and then, one use pieces for the preventive maintenance activities with a (r) request rate, on arrive finally to a stock accumulation gived by relation (3):

S = (k - r)[T.sub.1] (3)

On the [[T.sub.1], [T.sub.1] + [T.sub.2]] interval the fabrication is stopped, and the stock. S begins to be completely used with a (r) request rate, according to relation (4):

S = r x [T.sub.2] (4)

From (k - r)[T.sub.2] = r x [T.sub.2] result that the production time is:

[T.sup.*.sub.1] = r/k-r] x [T.sup.*.sub.2] [right arrow] [T.sup.*.sub.l] = [square root of 2[c.sub.l] x r/k(k - r) x [c.sub.l]] (5)

From here, we obtain the maximum S stock size:

[S.sup.*] = r x [T.sup.*.sub.2] (6)

The [T.sup.*] period can be than determined with relation (7) as:

[T.sup.*] [square root of 2[F.sub.l]k/[C.sub.s](k-r) (7)

The economical quantity of the spare parts lot, that has to be provided in the [0, T] time period, becomes:

[Q.sup.*] = r x [T.sup.*] [right arrow] [Q.sup.*] = [square root of [c.sub.l] x r/[c.sub.s](1 - r/k) (8)

We can also establish the minimum expenses, as:

f([T.sup.*.sub.2]) = [f.sub.min] = [square root of 2[c.sub.l] x [c.sub.s](1 - r/k) x r (9)

Considering a particularly case in witch, the total request for a year is known, but one work with a certain number of L < 365 days on year, we observe that only the period [T.sup.*] is changing and then, the relation (9) becomes:

[T.sup.*] = [[square root of 2[c.sub.l]/[c.sub.o] x N(1 - r/k).sup.xL] (days) (10)

The series number provided will be N/[Q.sup.*].

If a delivery time [tau] exists, time in which one prepare the production starting up, the stock level will be:

[q.sup.*] = [tau] x N/L (11)

4. RESULTS

Following the maintenance activities of a company that produce technological equipments, we applied a determinist model with provision, when the absence in stock may not be accepted, After monitoring the maintenance activities we observed that, the spare parts necessary for an year arrived to 30.000 bench marks, spare parts being necessary to resolve adequately the maintenance activities caused by the apparition of different

"failures" in the equipments working processes. The fabrication of the spare parts capacity is about 150.000 units on year, and the starting cost of a production lot is 1.350 m.u., while the maintening stock cost is about 13,5 m.u. for a piece.

If the absence in stock may not be accepted and the fabrication starts instantaneously, one can resolve: which is the maximum stock level (S*), what interval exist between two consecutive starts of the production (T*), the establishment spare parts quantity that has to be produced for each time (Q*), how many times the production starts on a year (N/[Q.sup.*]).

A particulary case can be considered, the one in which one works only 260 days per year. The question is what stock level will be, if the production starting subsist two days and one works only 260 days per year? The needed time to produce the spare parts lot can be calculated with the following relation:

[T.sup.*.sub.1] [square root of 2[c.sub.l] x r/k(k - r) x [c.sub.l] = 0.0182years = 6,64 days

The necessary time to consume the stock quantity is:

[T.sup.*.sub.2] = [square root of 2[c.sub.l](k - r)/k x r x [c.sub.s] = 0,073 years = 26,65 days

The maximum stock level S can be established with:

[S.sup.*] = r x [T.sup.*.sub.2] = 30.0000,073 = 2190units

The period [T.sup.*] will be determined using relation (7) as:

[T.sup.*] [square root of 2[c.sub.l] x k/[c.sub.s] x r(k - r) = 0,0912years = 33,28days

The economical spare parts lot quantity needed to be provided can be established with relation (8):

[Q.sup.*] = r x [T.sup.*] = 30.000?0,0912 = 2.736units

The minimum expenses can be calculated with relation (9):

[f.sub.min] = [square root of 2[c.sub.l] x [c.sub.s](1 -r/k) x r = 29.577,02 m.u.on year

If one work L = 260 days per year, from relation (10):

[T.sup.*] = [square root of 2[c.sub.l]/[c.sub.o] x N(1 - r/k) x L = 23,71 days

The series number is gived by the relation N/[Q.sup.*]:

30.000/2.736 = 10,96 series on year

The stock level determined with the relation (11) becomes:

[q.sup.*] = [tau] x N/L = 2 x 30.000/260 = 230,76 spare parts

5. CONCLUSIONS AND FUTURE RESEARCHES

The results of the done researches leads for the future to the possibility to establish the stocking expenses and also, the launching expenses of the spare parts, for a certain time interval.

Also, the done researches will be directed in the future to establish the optimum spare parts stock, necessary to the maintenance activities in the industrial companies, using determinist methods and also, the failure "history' of each existing equipment.

6. REFERENCES

Kelly, A. (2006). Plant Maintenance Management Set--Maintenance Systems and Documentation, Elsevier, ISBN--10: 0750 6699 50; ISBN--13: 9780 7506 6995 5, London

Lupescu, O.; Gramescu, Tr.; Popa, I.; Popa, R. (2008),Using stocking strategy in technological equipment life cycle management,Academic Journa' of Manufacturing EngineeringVoX. 6, pp.87-92, ISSN: 1583-7904, Timisoara

Lupescu, Ov.; Leonte, P.; Popa, I.; Murarasu, M. (2008), Researches regarding the analisys methodology of spare parts stocks necessary in maintenance activities, Bulletin of the Polytechnic Institute of Iasi, Tom LIV(LVIII), pp.301-308, ISSN: 1011-2855, Iasi

Popa, I.; Pruteanu, O.; Lupescu, O.; Popa, R. (2008), Equipment maintenance through spare parts stocks when absence in stock may be accepted, Machine--Building and Technosphere of XXI Century/Tom 4, pp.203-206, ISBN: 966-7907-23-6, Donetsk

Rusu, E., (2001), Decizii optime in management, prin metode ale cercetarii operationale, Optimal decisions in management through operational research methods, Publisher Economica, ISBN: 973-590-513-2, Bucuresti