Periodic inventory system control decisions under risk.

Pasic, Mugdim ; Bijelonja, Izet ; Kadric, Edin 等

1. INTRODUCTION

Optimal inventory control is one of the crucial business functions, since business low-cost strategy can never be achieved without good inventory management (Ballou, 2004; Wild, 2002). Inventory is one of the most expensive assets of many companies, representing a significant percentage of total invested capital. High level service and smooth operations should be met at the minimum inventory cost. These goals are mutually contradictor, and should be balanced (Heizer & Render, 2006).

The most profitable policy does not consider individual optimization of one of these goals, but they must be jointly considered and optimized in order to achieve the optimal operations results. Those responsible for inventory decision making must take into account all these facts, in order to make decisions based on relevant evaluations of any possible alternative, and of course, consequences of applications of any of it. In general, inventory management functions are contained in the Enterprise Resource Planning (ERP) system, which provides ways to analyze the demand history, make forecasting evaluation, and suggest safety stock levels.

Today ERP systems are reasonable and sophisticated tools to forecast the demand of fast moving items, but most are ill-equipped to deal with the demand of slow moving items such as spare parts (Razi & Tarn, 2003). This is due to the fact that many inventory models available today show a number of difficulties in attempting to apply them to a spare part inventory management. Spare parts inventory management has some specific characteristics: high price, irregular demand hard to forecast, long and stochastic lead times and customers (internal or external) want those parts as soon as possible (Humphrey, 1998; Fortuin, 1999).

The model developed in this paper deals with spare parts inventory control and is generally based on previously developed inventory model (Razi & Tarn, 2003). In this model a specific cost of an item is used to determine the target stock level for that item. Items are grouped based on annual demand and lead time and a common group demand distribution is generated. The authors use a fixed number of review periods, and it was emphasized, that the number of review periods is the decision that must be made by an inventory manager.

In this paper, instead of using a fixed length of review period, developed model simulates a series of length of review periods. The length of review period is automatically chosen and depends on trade off between total inventory cost and the customer serves level. In this case a review period managerial decision making does not depend on manager's personal experience, intuition or approximate estimations. The model and software module are tested and verified on a real life example. Model developed in this paper shows excellent performances.

2. MATHEMATICAL MODEL

Mathematical model is defined by two parameters: number of review periods, T, and maximum inventory level, S. Total cost, TC, is composed of four components: total review cost, total ordering cost, total holding cost, and total penalty cost. Total cost, TC, can be calculated from the following equation:

TC(T, S) =

= [C.sub.r] x T + [C.sub.o] x T + I x c x [S - [bar.x]/2] + [pi] x T x [summation over (x>S)] (x - S) x p(x) * p(x) (1)

where T is number of review periods, [C.sub.r] is cost per review, [C.sub.o] is cost per order, I is annual holding cost expressed as a percent of an item cost, c is an item cost, [pi] is a penalty cost per item, and p(x) is probability that demand is equal to x.

Any unsatisfied demand is assumed as lost sales, which means that backordering is not allowed, because lack of a spare part will surely result in a smaller production, and thus in a smaller revenue.

Number of review periods T represents total number of reviews that will be conducted in specified time period H. Number of review periods T is function of time period H, in which transactions are done, and length of review period r. Number of reviews T, within time period H, can be calculated from the equation:

T = H/r (2)

For example, if time period H is two years, and review period r is one month, then the number of reviews T is 24. Length of review period r is a linear function of the lead time.

Mean demand, [bar.x] , over time period H, represents ratio of sum of all mean demands [[bar.x].sub.i] of all reviews [T.sub.i] and total number of reviews T, and can be calculated using equation:

[bar.x] = 1/T [T.summation over (i=1)] [[bar.x].sub.i] (3)

It is assumed that demand is random, stochastic and that follows Poisson distribution. Poisson distribution is a discrete distribution, determined by one parameter only, its mean. The probability, p(x), that demand is equal to x, if the mean demand over all review periods is equal to [bar.x], can be calculated from equation:

p(x) = [e.sup.-[bar.x]] x [[bar.x].sup.x]/x! (4)

Service level represents probability that quantities on hand, during the lead time, will be sufficient to satisfy expected demands. Service level, SL, is calculated using following equation:

SL = [1 - [summation over (x>S)] p(x)] x 100 (5)

Figure 1 illustrates algorithm for periodic inventory control which is composed of four blocks and one loop.

Purpose of block 1 is to enable item selection, which will be analyzed, and to collect item data, and after that to define length of time period H, which serves as the basis for estimation of number of review periods T and mean demand [bar.x].

In the preprocessing phase (block 2), periods are created, and then records about demand, in particular periods [T.sub.i], are collected. For given number of review periods T and known demands [[bar.x].sub.i], mean demand [bar.x] is estimated. When mean demand [bar.x] is known, it is possible to create Poisson distribution with mean [bar.x], and to estimate particular probabilities of random demand x.

As a result of execution of the processing phase (block 3), values of maximum inventory level S, total inventory cost TC and service level SL are estimated, for different lengths of review periods r, i.e. number of review periods T, within time period H.

Using loop, blocks 2 and 3 are executed for different lengths of review period r, enabling us to create a set of optimal solutions of total inventory cost TC, for different number of review periods T and maximum inventory levels S.

The main role of the postprocessing phase (block 4) is to show all results of analysis for different lengths of review periods r, i.e. number of periods T, and to select optimal solution for inventory control of selected spare part. Optimal solution represents optimal value of length of review period r, and maximum inventory level S, for which the relationship between total inventory cost, TC, and service level, SL, is acceptable.

3. RESULTS AND INTERPRETATIONS

Testing and verification of mathematical model and software module for periodic inventory control is done using demand records for a critical spare part of Sarajevo Public Transportation Company. This spare part is critical because if it is not available when it is needed, company suffers from unavailability of transportation vehicle. Demand records for this item were available for recent 50 weeks. Total demand for this item is 8 units, item price is 1.968,00 [euro], lead time is 5 days, annual holding cost is 30% of item value, ordering cost is 10,00 [euro] and review cost is 1,00 [euro]. Graphical representation of results is shown on Figure 2. It can be seen from Figure 2. that minimum inventory cost TC of 3.207,15 [euro] and service level SL of 99,94 % are achieved for length of review period r of 15 days and maximum inventory level S of 5 item units.

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

4. CONCLUSION

Theoretical mathematical model and software module are tested and verified using real practical example. Results of tests show that theoretical model and software module are capable of ensuring efficient periodical control of spare parts inventory.

Length of review period, i.e. number of review periods, and maximum inventory level, determined for the case of optimal relationship between total inventory costs and service level, represent significant benefit to optimization of balance between three main aims of inventory control. Testing of the developed model showed excellent results.

5. REFERENCES

Ballou, R. H. (2004). Business logistics/ Supply chain management and logware compact disc package, Pearson Higher Education, ISBN-10: 0131492861, New Jersey

Fortuin, L. & Martin, H. (1999). Control of service parts, International Journal of Operations & Production Management, Vol.19 No.9, pp. 950-971, ISSN 0144-3577

Heizer, J. & Render, B., (2006). Operations Management, Pearson Education, Inc. 0-13-185755-X, New Jersey

Humphrey, A.S.; Taylor, G.D. & Landers, T.L. (1998). Stock level determination and sensitivity analysis in repair/rework operations, International Journal of Operations & Production Management, Vol. 18 No.6, 1998, pp.612-630, DOI: 10.1108/01443579810209566

Razi, M.A. & Tarn, J.M. (2003). An applied model for improving inventory management in ERP systems, Logistics Information Management, Vol. 16, No. 2, pp. 114-124, ISSN 0957-60-53

Wild, R. (2002). Operations management, Continuum International Publishing Group, ISBN-10: 0826449271, London