Lot sizing of spare parts

M. Bošnjakovića,*, M. Cobovićb

a University of Applied Sciences in Slavonski Brod, Dr. Mile Budaka 1, HR-35000 Slavonski Brod, Croatiab University of Applied Sciences in Slavonski Brod, Dr. Mile Budaka 1, HR-35000 Slavonski Brod, Croatia

*Corresponding author. E-mail address: mladen.bosnjakovic@vusb.hr

Abstract

Spare part demand could significantly vary over a time. Even though there are periods without demand. Commonly used lot sizing policies like Economic-Order-Quantity, Lot-For-Lot and Period Order Quantity do not take these effects into account. This research compares these policies with dynamic models, within which lot sizes are based on minimizing total inventory cost. Appropriate example is used to compare results within static and dynamic inventory models applied to spare parts. Results show that the dynamic inventory models give the lower total inventory cost.

Keywords: lot sizing, spare parts, dynamic models, static models

1. Introduction

Modern industry applications require the availability and reliability of machines, which ensures, among other things, the availability of spare parts and components at the time of their needs. As the intensity of wear of individual parts of the machine is very different and often unpredictable, it is necessary to stock a certain amount of spare parts. However, ordering1 and inventory holding2 costs are affecting performance. It is therefore necessary to find the optimal order size that will minimize total costs, while at the same time ensure availability of spare parts at the time of their needs.

To find the optimal ordering plan, there are different mathematical models, but the question is which of them give the best result in the issue of procurement of spare parts (HM. Wagner, 2004., R. Kleber, K. Inderfurth, 2009.).

In general, for solving this problem we can use static and dynamic programming inventory models.

Figure 1. Inventory models

2. Static Lot-Sizing Models

2.1 Economic Order Quantity (EOQ)

The best known and the simplest model is the EOQ model, which was developed in 1915 by FW Harris. EOQ is based on the following assumptions3:

Known and constant demand in time Known and constant lead time4 over time Instantaneous receipt of spares No quantity discounts

1 This is the sum of the fixed costs that are incurred each time a number of spare parts is ordered. These costs are not associated with the quantity ordered but primarily with the physical activities required to process the order. These activities are: specifying the order, selecting a supplier, issuing the order to the supplier, receiving the ordered goods, handling, checking, storing and payment. It is also called setup cost.2 Holding costs express the costs (direct or indirect) to keeping parts on stock in a warehouse (warehouse space, refrigeration, insurance, etc. usually not related to the unit cost).3 These assumptions do not hold all in the case of spare parts. 4 The lead time is the time needed to get the spare part as indicated by the supplier. It starts from the moment the supplier is informed until he delivers the part on site.

Inventory modelsStatic modelsEconomic Order

Quantity

Period Order Quantity

Lot for Lot

Dynamic modelsWagner-Within

Least Period Cost

Least Unit Cost

Part-Period Balancing

Constant ordering and holding costs over time No stock-outs are allowed.It is necessary to know the following values for the

optimization:D - Annual demand in units of the spare partCn - Fixed cost per order h - Holding cost per unit per year

Optimal lot size is determined by the equation:

Q∗¿√ 2 D Cn

h (1)

2.2 Period Order Quantity (POQ)

The procedure of POQ model is following: Calculate EOQ using average demand Calculate time supply and round it to the nearest

integer In each replenishment, order to cover that many

periods’ demandOrder interval is constant, but ordered quantities

could be different.

2.3 Lot-For-Lot Model (LFL)

Spare parts are ordered precisely when needed. Each period is ordering a lot to satisfy only that period’s demand. Lot-for-Lot is among the most popular with practitioners since it is simple and produces the least remnant work-in-process inventory. However, setup costs can be excessive if too many small lot sizes result.

3. Dynamic Lot-Sizing Models

Dynamic lot-sizing models are used within the demand which vary during a period of time. Furthermore, all of the models described in this chapter take assumptions:

Demand during period t is Pt and can be anticipated.

Planning orders is done for a specific timetable (planning horizons): t=1, 2... T

No shortage is allowed. No limitations in warehouse nor in ordered

quantity. The time necessary for the order realization is

ignored (equals zero) or it is constant Warehouse expenses depend upon the level of

supplies at the beginning of a period. The cost of ordering Cn, and holding costs ht,

Model objective is to determine the quantity of ordering xt that minimize the inventory cost during T periods.

In addition, it is supposed that the following data is known:

Pt - Demand by periodsCn(t) - The ordering cost (usually Cn(t)=const.=Cn)ht - Inventory holding cost per unit (for unit that

remain at the end of a period t)T - Analyzed number of periods (usually it is 12

months T=12, i.e. one year)

Mathematical definition of the problem:TCt

*- Cost of an optimal ordering plan for the first t periods

Zm,t - The cost of satisfying demands in periods m to t by ordering in period m for the periods up to t.Ym,t  - The cost of satisfying demands for periods 1 to t:

• By having in mind the optimal ordering plan in periods 1 to m-1

• Ordering in period m (m≤t ) for periods m to t

Y m, t=TCm−1¿ +Zm, t (2)

TC t¿=min(Y m ,t ) (1≤m≤ t )

(3)

Boundary conditions: Ordering is performed only when the inventory

level is zero, Ordered quantity exactly corresponds to the

demands in observed time periods, State of supplies x ordered quantity = zero

The following means that is never optimal to order if there are any quantity on stock,

If it is optimal to order in the period m to satisfy the demand for periods m to t, it is also optimal to order in the period m for the periods (m, m+1, …., t).

Horizon theorem:If it is in solving t periods optimal to order in the

period m to meet the demand in the period t, then in resolving w periods (w>t) it is optimal to deliver order in the period m or later:

If zt*=1 for the t period than zt

*=1 for w periods (w>t) and the ordering plan for the period t remains unchanged (frozen)

If zt*=0 for t periods then zt

*=0 or 1 for w periods (w>t)

where zt* is a binary variable (= 1 if the order is issued

in period t, otherwise = 0)

3.1 Wagner-Within Model (W-W)

The goal of this model is to determine the replenishment plan so that the ordering and holding cost for certain period is minimal. Thus, the Wagner-Whitin model for Zm,t and Ym,t takes the total inventory cost.

The optimal ordering plan procedure is as follows:a) Try to set inventory status demand to zero at the

beginning and end of the period T , i.e. I1=0 and IT+1=0

b) Start with the first period i.e. t=1. All demand must be satisfied z*= (1,-,-,…,-). Calculate TC1

*=Y1,1=Z1,1=Cn

c) Setup t=t+1. If t >T End of procedure.

d) Calculate Y m, t=TCm−1¿ +Zm, t

for all m which

correspond to unfrozen zm

e) Calculate TC t¿=min(Y m ,t ), m≤t ,

and try to

determine z*=(z1*,…,zt

*)f) If zt

*=1, frozen z* for the period (z1*,…,zt

*)g) Return to the item c)

Efficient computer implementation of the algorithm was presented in 1985 by James R. Evans.

3.2 Least Period Cost Model (LPC)

Whenever the demand is positive model find the order size that will cover the next "n" periods, where "n" is set to minimize the average cost per unit time. (E. Silver, H. Meal, 1973.)

The optimal ordering plan procedure is as follows:a) Let the current period be t=1. For t=1, 2,…, T

calculate average ordering and holding cost, if all items are ordered in the periodt :

ACt=1t [Cn+∑

τ=2

t

Pτ (∑u=2

τ

h)] (4)

where ACt is the average setup and holding cost per time unit (monthly) and Pτ is the demand in period τ.

b) Select the period t in which t is ACt < ACt+1. That period should be noted as the period t*.

c) Order in period 1 for the period t*.d) Subtract t* from the T and repeat the process

from the beginning

3.3 Least Unit Cost Model (LUC)

Whenever the demand is positive model find the order size that will cover the next "n" periods, where "n" is set to minimize the average ordering and holding cost per unit. The procedure for finding the optimal ordering plan in the period t=1, 2,…, T is as follows:

a) Let the current period be t=1. For t=1,2,…,T calculate the average ordering and holding cost per quantity unit, if all items are ordered in the period t:

UCt=1

∑τ

t

Pτ

[Cn+∑τ=2

t

P τ(∑u=2

τ

h)] (5)

Where:UCt - Average ordering and holding cost of

inventory per quantity unit. Pτ - The demand in period τ

b) Select the period in which t is UCt<UCt+1.

Which we denote as period t*.c) Order the required quantity for the period 1 up

to t*.d) Repeat the procedure for the period t=t*+1,

t*+2, t*+3, …,T

3.4 Part-Period Balancing Model (PPB)

The basic idea of this model is to equalize the holding cost in the period 1 to t with the cost of ordering during the period 1 (U. Wemmerlov, 1983.).

The optimal ordering plan procedure is as follows:a) Let the current period be t=1. Then calculate

holding cost for t=1, 2,…, T if ordering for

periods 1 to t is done in period t:

PPCt=∑τ=2

t

Pτ (∑u=2

τ

h)(6)

b) Select a value for t that is PPCt closest to the

value of the setup cost Cn. Denote this period t*.c) Order the required amount for the period 1 to t*.d) Repeat the procedure for the period t=t*+1,

t*+2, t*+3, …,T

4. Ordering plan calculation

4.1 The input data

Spare parts demand often tends to be "lumpy," that is, discontinuous and no uniform, with periods of zero demand. According this assumption appropriate test data are used in evaluation of certain inventory models.

Table 1. The spare part demand Period 1 2 3 4 5 6 7 8 9 10 11 12 Total

Demand 22 62 0 35 124 68 25 0 120 70 44 30 600

In this test ordering (setup) cost per order is 30,00 € and holding cost per unit and period is 0,2 €.

4.2 The test results

The figures 2. to 9. show the calculation results of the ordering plan for particular model. Calculation is done according to given procedures.

All values in the figures are given in Euros (€).

Figure 2. Lot-for-lot lot sizes

Figure 3. Economic Order Quantity lot sizes

Figure 4. Period order quantity lot sizes

Figure 5. Least unit cost lot sizes

Figure 6. Part-period balancing lot sizes

Figure 7. Least period cost lot sizes

Figure 8. Wagner-Within lot sizes

Figure 9. Comparison of the total cost

5. Conclusion

Spare parts demand tends to be "lumpy," that is, discontinuous and no uniform, with periods of zero demand.

In general, dynamic models give better result than static models for approximately 20%. The results of dynamic methods depend on the value and mutual respect of input data, and especially about the relationship between the ordering and holding cost. However, as it is evidently from the example and additional analysis, the best result in determining the optimal lot size of spare parts gives Wagner-Whitin method.

References

[1] HM. Wagner, Comments on “Dynamic version of the economic lot-size model”. Management Science, Vol. 50, No 12, December 2004, pp. 1775-1777

[2] S. Chand, “A note on dynamic lot-sizing in a rolling horizon environment”. Decision Sciences, Vol. 13, 1982, pp 113-119

[3] J.D. Blackburn, R. A. Millen, Heuristic lot-sizing performance in a rolling schedule environment. Decision Sciences, Vol.11, 1980, pp 691-701

[4] R. Kleber, K. Inderfurth, A Heuristic Approach for Integrating Product Recovery into Post PLC Spare Parts Procurement. Springer Berlin Heidelberg, 2009., ISBN 978-3-642-00141-3, pp. 209-214

[5] E. Silver, H. Meal, A heuristic for selecting lot size requirements for the case of a deterministic time varying demand rate and discrete opportunities for replenishment. Production and Inventory Management Journal, Vol. 14, No 2 1973., pp. 64–74

[6] U. Wemmerlov, The part-period balancing algorithm and its look ahead-look back feature: A theoretical and experimental analysis of a single stage lot-sizing procedure. Journal of Operations Management, Vol. 4, No 1, 1983., pp. 23–39

[7] James R. Evans, An Efficient Implementation of the Wagner-Whitin Algorithm for Dynamic Lot-Sizing. Journal of Operations Management, Vol. 5, No. 2, , 1985., pp. 229-235