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Determining the optimal level of seryice (product availability)

The levei of product availability is measured using CSL (cycle Service Level) or the fillrate. A

company can use a high level of product availability to improve its responsiveness and thus

attracting the customers, which in turn, increases the revenue. However, a high level of product

availability requires large inventories and hence increased costs. Therefore the company must

strike a balance between the level of availability and the cost of inventory.

In this note, we focus on the products that are ordered repeatedly and safety stock (SS) is used

to increase the level of availability and decrease the probability of stocking out between

successive del iveries.

Most safety stock decisions are based on a more or less arbitrary level of protection (95% CSL

in the textbook examples is a case in point). How do we set this level? If the inventory control

manager increases the level of safety inventory, more orders are satisfied from the stock,

resuiting in lower backlogs (or unmet demand). This decreases the backlogging cost (or lost

sales cost). However, the cost of holding inventory increases. The manager must pick a level of

safely inventory that minimizes the backlogging (or lost sales) cost and the holding cost.

The fbllowing analysis (approximation) shows how to minimize cost by considering the cost

and saving of a one-unit change in the safety stock level.

Suppose that we considered reducing safety stock (SS) by one unit. Wat will it save? If SS is

lowered the entire inventory curve moves down, indicating a lower average inventory. The

saving is therefore Cs (holding cost per unit per unit). Wat will it cost? The answer depends on

the expected number of times per year we are out of stock. Each time we were out of stock last

year, if SS had been lower we would have had greater shorlages. Thus, lowering SS by one unit

wtll cost us one more unit of shortage for each out-of-stock condition; hence the average cost

pcr year is

8/13/2019 CSL Based on Shortage Cost

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(1 sholtagc) * (Expected number of stockout occasions per year) x(cost per urut shortage, C6)

Il'rlr, .',,-r ir lt.s. thrrr .rrirrs. rve should reduce SS. Ilcosr exceeds the savinus. SS should be htgher. Therefore, at

the optimal SS, the cost of marginal change should equal saving, i.e.,

(1 shortaue) * (F:xpected number of stockout occasions per year) *(cost per urut shortage, Cd = Cu

'I'l'rus, thc safe fi, stock ierrel (55; should be set in such a way that:

li.r1r.'1'1.. .1 rrLrrrrb.-r , rf srockout occrstons per yexr - C'f C,

-> (l :xpcctccl nlrnrber of stockout occasions per cycie)+(No. of cycles per year) =C'

f C,

=> (Ir.xpectccl number of stockout occasions per cycle)* E(D)/Q -C'

f C,

= ) Irxpccted nr-rmber of stockout occasions per cycle = * - +  ) E(D) Cs

Sir-rcc stocl<orit cither arises or doesn't arise in any cyc1e, therefore,

lixpcctcc'l number r-rf stockout occasions per cycle

= 1 { P(stockiti-rt,.rLises in a cvcle) + 0 * P(stockout doesn't arise in a cycle)

= I),str,t'ii,ut rris.'s in I cicJe) = #-#

T'hcrctirre, (-SI- = P(stockoutdoesn'tariseinacycle) = 1*P(stockoutarisesinacycle) =, ,b-#

Note: 'l'his is arr approxirnation since we said lowering SS by 1 urut rvill cost us 1 more urut of shortage for each

stocliot-lt occasiorl. IIere, we chd not account for the extra stockout occasions that wrl,1 arise on lowering SS by 1

urut. Thesc extra stockout occasions rvI-1 happen when the inventory position just before repierushment was exactly

0.