Industrial products inventory management using deterministic and probabilistic models.

Zapciu, Miron ; Zapciu, Georgeta ; Georgescu, Luminita 等

1. INTRODUCTION

Having the stocks in industrial production requires costs with supply or production, costs of specific storage (storage, maintenance, handling, conservation, etc.) and others losses due to depreciation in special cases. To manage stocks are managed stock entries and exits or supplies in stock. Supplies of products could be modelled using three possible situations of exits from stock (Krajewski & Ritzman, 2001): deterministic (calculated as the known functions); random with a known statistical distribution; unknown variable of stock exit.

The main activities involved in inventory management are the following: the size of lots of products, stock level, the volume of replenishment orders, the storage costs, penalty costs, costs of the order etc. For an optimization of inventory levels, the industrial management requires most often, a criterion of minimum total cost of these stocks. Items on which we have intervention are: optimal level of stocks, the optimal volume of orders for replenishment, or the optimal time for replenishment, frequency of replenishment, minimum total cost. For optimum management of stocks are using different mathematical models, shared depending on the nature of demand in deterministic and probabilistic models.

2. DETERMINISTIC MODELS FOR INVENTORY MANAGEMENT

2.1 Storage model of constant demand, constant replenishment period and lack of stock

This model considers the following assumptions: storing a single type of product whose consumption is a linear function during time, demand of the product is N pieces for a period [theta], replenishment of stock is performed instantaneously (in theory) at equal time (T) and equal quantity (q). Unit cost of storage in unit time is [c.sub.s], the cost to launch a replenishment order is the parameter [c.sub.lan] (the total cost of the replenishment operation that does not depend on q) and is not allowed lack of stocks. Graphical representation of this model is made in the Fig. 1.

Under the assumptions of this model, for each time period T are performed following costs (equation 1):

C(q) = [c.sub.lan] + 0.5 * qT [c.sub.s] (1)

[FIGURE 1 OMITTED]

Considering the stock is equal with the average quantity of the products q/2 and the equal periods T is calculating the number of replenishment,y using the equation:

y = N / q = [theta]/T (2)

For this model objective function includes all costs related to inventory management. Given the values of y in the equation (2) is obtaining the following expression for the objective "total cost" function:

C(q) = 1/q N [C.sub.lan] + 1/2 q c, [theta] (3)

The quantity [q.sub.opt] of optimal replenishment, the optimal number of replenishment [y.sub.opt] and the optimal time between two replenishments [T.sub.opt] are calculated using the equation (4). Since C"(q)> 0 extreme value is a minimum (because q has only positive values).

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

2.2 Storage model using different types of products

This type of simple model is very close to the real situations from industrial companies. Practical situations of most enterprises are those in which stocking more products are made thanks to the diversification need, resulting from the necessity to better meet customer requirements (example: in automotive industry). For examples of this model is built the stock model for the case using different products.

Assumptions are: k of Pi products are stored; [P.sub.i] = 1, 2, ..., k whose consumption is given by linear functions of time and the request of [P.sub.i] is [N.sub.i]; for a period of time [theta], replenishment is performed instantaneously at the intervals [T.sub.i], in equal quantity of [q.sub.i]; unit storage costs are [c.sub.si]; the costs of delivery [c.sub.lan,i], with i = 1,2 k.

In this case is not admissible for lack of any stock of the [P.sub.i] products. According to equation (3), the total cost of storage for the product [P.sub.i] is calculated using equation (5):

[C.sub.i]([q.sub.i]) = 1/[q.sub.i][N.sub.i][c.sub.lan,i] + 1/2[q.sub.i][theta][c.sub.si] (5)

[FIGURE 2 OMITTED]

Graphical representation for this type of model, where consumption of products varies over time and the level of replenishment is less than the maximum allowed, is shown in the Figure 2.

It should be noted that with the changing profile of manufacturing in volume production or methods of organization and management of production and supply is very important to reconsider and to study in depth the management of stocks (Zapciu, 2006).

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

The equation (6) allows obtaining the following important parameters: [q.sub.opt]--optimal batch; [y.sub.opt]--optimal frequency of orders, [C.sub.opt]--optimum cost.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

Optimal management is reprezented by the minimum cost. This parameter is determined by the equation (7).

3. PROBABILISTIC MODEL FOR THE COSTS OF THE INVENTORY MANAGEMENT

Using a probabilistic model for the stocks, was analyzed two significant cases: first case is the stock model for a product with stochastic market demand, with neglected cost to depose the product, with supplementary cost when it is no stock available; the second case considers a product with stochastic market demand, with knew deposed cost and penalty cost when the product is not always available.

The second stock cost model studied was more complex and it is to close to the most real industrial cases. When the demand of the product is a continuous function f(x), the total cost of the stocks C(,), is calculated using the relations:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

where:

[c.sub.s] is unitary cost of stock; [c.sub.p]--unitary penalty cost for unavailability of the product; s--instantaneously level of the stock.

C(s) function get the minimum in the point [s.sub.opt] when stock unavailability L(s) = [rho], according to the equations:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

To exemplify the application of this model to stock administration is presented a case study calculus; the model considers a stocked product with stochastically and continuously demand, with storage cost and penalty cost when the product is not available.

Considering the example when the function for the demand of the product is [F.sub.x]) = (-1/2 x +3), considering the realization of 10 products batch size, unitary cost to stock the product is 1.5 [micro]m, unitary penalty cost when the product is not available is equal with 0.5 [micro]m, consequently, optimal value for the stock cost is 162.9 [micro]m.

In the case when the function of the demand of the product has a sinus component, for example g(x) = (-0.5x+3) +sinx, the total stock cost in this case is f(x) = 141.1 [micro]m.

4. CONCLUSION

Inventory management is one of the issues particularly important if the production requires a lot of products to be in stock. New result of this paper is mathematical models presented that can be used in real cases of industrial production. For example, applying the models in the case study from an SME's roumanian company, considering the data entry [c.sub.s] = 1.5 and [c.sub.p] = 0.5, the total stock costs of 20 batch products where: C(s) = 154.4 for the model with random discrete demand and C(s) = 162.9 for the model with random continues demand.

Using a flexible manufacturing system, the product flow could be variable and more adapted to the market product demand. Practically, the stock level is calculated taking into account that the total cost for a product series to be minimum.

Mathematical model presented for the cost of the stock could be used for any other industrial case, but with the remark that must be knew the time variation of the function for the market demand of the product.

Next step will be finding new hybrid stock management models more adapt to the industrial company and helping to become competitive into a difficult market.

5. REFERENCES

Ispas, C.; Zapciu, M. & Mohora C. (2003). Quality ofthe product in the context of flexible manufacturing, Proceedings of the International Symposium on Quality and Standardisation, pp. 236-242, Gheorghe G (Ed.) ISBN 9738177-42-1, Bucharest, October 2003, Bren Edition, Bucharest

Krajewski, L.; Ritzman, L. (2001). Operations Management: Strategy and Analysis, 2nd edition, Addison-Wesley, ISBN 1-890367-01-X, Manchester

Zapciu, G.V. (2006). Contributions about the application of project management concepts in material flow optimisation in industrial companies. PhD Thesis, University POLITEHNICA of Bucharest

Zapciu, G.V. & Zapciu, M. (2005). Taking into account the risk in logistics problems, Proceedings of International Conference Energy-Environment--CIEM, pp. 423-428, ISBN 975-86948-5-X., Bucharest November 2005, Bren Edition, Bucharest

Zapciu, M.; Cote^, E.C. & Stefan, S. (2001). Techniques of management, Bren Edition, ISBN 973-8143-21-7, Bucharest

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