A company sells spare parts for cars and uses a t,S-replenishment policy for a c

Question

A company sells spare parts for cars and uses a t,S-replenishment policy for a certain product. Replenishment orders are issued every 4 weeks, delivery lead time is 2 weeks. Average weekly demand is normally distributed with mean 5 units and standard deviation 2 units. At the start of a period the physical stock amounts to 12 units. There are not backorders and no outstanding deliveries. The manager issues an order of 23 units.

a) How large is the order-up-to level in this situation?

b) Which fill-rate can be achieved?

c) If the manager wanted to achieve a fill-rate of 98.3%, how many units must he order?

d) Explain the relationship between inventories, service levels and order-up-to level (S)

Explanation / Answer

Ordering/ Review period = T = 4 weeks
Lead time = L = 2 weeks
Mean demand = = 5
Std. dev. of demand = = 2

Order-up-to level (S) = x (T+L) + Z x x (T+L) -----------------------(Expression 1)

Given S = 23+12 = 35 ------(a)

So, 35 = 5*(4+2) + Z*2*sqrt(4+2)
or, 35 = 30 + Z*4.9
or, Z = 1.021

At, this Z, the correspeonding loss function, L(Z) = 0.08

Fill Rate = 1 - (L(Z) x x (T+L) / x (T+L)) ----------------- (Expression 2)

or, Fill Rate = 1 - (0.08*2*sqrt(6)/(5*6)) = 98.69% --------(b)

For a fill rate of 98.3%

0.983 = 1 - 1 - (L(Z)*2*sqrt(6)/(5*6))
or, L(Z) = 0.104

Corresponding Z will be 0.88

Order-up-to level (S) = x (T+L) + Z x x (T+L) = 5*6 + 0.88*2*sqrt(6) = 34
Order size = 34 - 12 = 22 units ------(c)

The rlationship is simple. if order up-to level is more, we will have higher stock position resulting in higher inventory. As a result, service level will improve. Loss function will reduce and hence the fill rate will also improve. These conclusions can be made easily for the expressions (1) and (2).

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