Journal of Management Research
Year : 2000, Volume : 1, Issue : 1
First page : ( 38) Last page : ( 46)
Print ISSN : 0972-5814.

Effect of setup cost on inventory and service level

Dr. Gupta H D

Faculty of Management Studies University of Delhi, Delhi.

Abstract

In a batch-manufacturing situation, a reduction in setup cost results in higher marginal reduction in working stock, total variable cost, and safety stock. It also increases the average annual service level. It makes strong economic sense to reduce setup cost from the existing level to the minimum possible level, as each successive reduction brings in higher savings. When setup cost approaches zero, there is a marked reduction in inventory and total variable cost.

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Keywords

Setup Cost, Service Level, Inventory, Safety Stock.

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Section I

1. Introduction

Economic lot size for batch production can help an organisation to cut its inventory to the barest minimum level. The decrease in setup cost reduces inventory and increases service level. Also, irrespective of original setup cost and quantum of its reduction the decrease in setup cost results in marginally increasing returns. When setup cost is reduced to very low levels, i.e. when it approaches zero, reduction ininventory level and total variable cost is very large. Also, slopes of working stock curve, safety stock curve and total cost curve increase dramatically near the zero setup cost. Essential theoretical framework for this is developed in Section II of the paper. Section III demonstrates this with the help of a numerical illustration. Results of this illustration are contained in Tables 1, 2A, 2b and 3.

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Section II

2. Theoretical Framework

Inventory in a batch-manufacturing situation can be divided into two parts. One part depends upon the batch produced and is called working stock. Second part is maintained to take care of fluctuations in demand during the supply lead time and is known as safety stock. The safety stock depends upon the lead time, variability in the lead time and variability of the daily demand during lead time, and desired service level. The lead time in batch manufacturing consists of time spent in initiating the production of the batch, processing time of the batch and time required to make available the output for consumption. If assumption is made that the batch can only be used after it is completely processed (instantaneous replenishment) and that other assumptions of Harris Formula also hold, then average inventory, I, is given by:

2.1 Effect of Reduction in Setup Cost on Batch Size and Variable Cost

Let us define the following:

Then

( 1 )

Reduction in average working stock I_q - I_q' is

( 2 )

By differentiating I_q (equation 1) with respect to K we get the slope of the average working stock curve.

( 3 )

similarly

( 4 )

From equations 3 and 4 it follows that slope of average working stock curve and annual total variable cost curve is proportional to (I/K)^½. This means as K decreases, slope of the average working stock curve increases and as K approaches zero increase in the slope of the curve is rather steep. This shows that the continuous reduction in setup cost results in higher marginal reduction in average working stock. Also, the reduction in average working stock becomes markedly high when setup cost is near zero value. The variable cost component C_q also varies in the similar manner with respect to the changes in setup cost.

2.2 Effect of Reduction in Setup Cost on Safety Stock

The safety stock depends upon the variability of demand during the lead time and service level. Reduction in setup cost reduces the lot size thereby reducing the processing time of the batch. If we assume that the setup cost is proportional to setup time, the reduction in setup cost reduces setup time also. This results in reduction of the variability of demand during the lead time and consequently safety stock requirment for a given service level.

Let us define the following:

Then

( 5 )

( 6 )

As

( 7 )

Also

Where Z is number of standard deviations for a certain service level.

Then reduction in safety stock is I_s - I_s'

( 8 )

(Figure 1 is a typical curve showing variation in safety stock with respect to K based on the data of numerical illustration of Section III.)

Also

( 9 )

Differentiating I_s with respect to K, we get slope of the safety stock curve.

( 10 )

2.3 Effect of reduction in setup cost on total variable cost C

( 11 )

By substituting the values of D, h, σ_d, Z, t_s, R_p, t_i and K (numerical illustration of section III) in equations 3, 10, and 11, we obtain the values of slopes of working stock curve, safety stock curve and total variable cost curve. Table 1 contains the values of slopes of these curves for various values of K between 0.1 to 1000.

Typical curves representing the slopes of safety stock curve, working stock curve and total variable stock curve are shown in Figure 1. From these curves it is clear that when setup cost decreases slope of these curves go on increasing. Also there is a large increase in the slope when K approaches zero.

2.4 Effect of Reduction in Setup Cost on Service Level

Reduction in setup cost reduces lot size and therefore, for a given annual demand D, increases the number of lots produced in a year. As number of lots increases, safety stock is required as many more number of times in a year. This aparently would seem to reduce average service level over the year. However in practice, reduction in K and increase in number of lots achieves normally higher service level.

Let us define the following:

Then

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Section III

Numerical Illustrations

Let

Then

Lead time L = ts + tp + ti = 10 + 10 + 0.1 = 20.1 days Safety Stock Is = = = 17,933 units Average Inventory I = 50,000 + 17,933 = 67,933 units Total Variable Cost C = = 20,000 + 0.2x17,933 = $23,587

Average Service Level

Let setup cost be decreased to $ 900 from $ 1,000

K' = $900 Q' = N' = number of lots per year when K = $ 900 = Iq' = 47,434 units Decrease in working stock Iq- Iq' is = 50,000 - 47,434 = 2,566 units tp' = 94,868/10,000 = 9.4846 days ts' = K'/K = = 9 days L' = 9 + 9.4868 + 0.1 = 18.5868 days Is' = = = 17,245 units Decrease in safety stock is Is- Is' = 17,933 - 17,245 = 688 units safety stock C = = 18973.7 + 3449 = $ 22,423 Reduction in total variable cost is C-C' = 23,587 - 22,423 = $1164 Average service level when K is $ 900 = = 99.997649%

In this manner incremental reduction in working stock, safety stock and total variable cost as well as change in average service level can be calculated by varying values of K. By keeping increments in K same, incremental variations in I_q, I_s and C can be analysed. Tables 2A & 2B contain values of I_q, I_s, C and incremental variation in I_q, I_s and C for different values of K. Figure 2 graphically shows variation in I_q, I_s and C with respect to K. Table 3 contains average service level and probability of shortage for different values of K.

4.1 Limitations

In the present work it has been assumed that setup cost and setup time are directly related. No research studies were found to substantiate or contradict this assumption. As setup cost mainly depends upon the setup time of the equipment and related manpower, this assumption is logical and is therefore made.

4.2 Conclusion

Reduction in setup cost is essential in reducing inventories, and total variable cost associated with inventories including that of the safety stocks.

Each successive reduction in setup cost provides higher and higher returns. In the vicinity of zero setup cost, reduction in inventory and total variable costs are very large.

In the typical case of Section III, when setup cost decreases from $ 1,000 to $ 10, average service level goes on increasing from 99.997588% to 99.998560% (Table 3). Further reduction in K decreases service level from 99.998560% to 99.997588% (at K = $ 0.1). However, when t_i is zero, then S_LA is 99.997600% when K is $ 1000, and as K decreases from $ 1000 to $ 0.1, S_LA goes on increasing. When K is $ 0.1 average service level is 99.998788%. Also probability of shortage becomes almost half from 0.0024% to 0.0012%. From this, conclusion can be drawn that reduction in setup cost has favourable effect on average service level. However when t_i is 0.1 day, average service level starts decreasing after reaching a maximum value of 99.998560% at K = $ 10. This is because when K becomes very small, value of t_i becomes a substantial component of lead time. At this stage in order to improve average service level reduction in setup time and batch processing time is not enough. The time required to make available the batch for consumption should also be reduced.

4.3 End Notes

In the paper standard expressions for economic lot size, working stock, average inventory, lead time, safety stock and service level are used and these are available in standard texts.

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Figures

Figure 1: TopBack

Figure 2: TopBack

Tables