1. NATCOR Stochastic Modelling Course Inventory Control – Part 2
Adam Letchford
Department of Management Science
Lancaster University Management School

2. Continuous Review System (cont.)
The continuous review or (Q,r) system can be extended to handle some complications that arise in practice.

3. Variations I: minimising total cost
The approach given does not actually minimise the total costs!
To do that, we would need to know or estimate the cost of a stockout.
So suppose that a stockout costs Cs.
If the service level is , the expected number of stockouts per year is D/Q(1- ).
So the expected stockout cost is Cs D/Q(1- ).

4. Variation I: minimise total cost
Then, one must minimise
(CoD/Q) + Ch(0.5Q + b) + CsD/Q(1- ).
This can be solved approximately with an iterative method, in which one alternately fixes Q and finds the best r, and then fixes r and finds the best Q, until some convergence criterion is met.
(Since both b and  depend on r.) 4 4

5. Variation II: limit unmet demand
An alternative measure of the quality of an inventory policy is the expected proportion of demand that is lost due to stockouts.
For a given re-order level r, and a given value of the LTD, the unmet demand is 5 5

6. Variations II: limit unmet demand
Therefore, the expected proportion of lost demand is
where f(x) is the pdf of the LTD.
To determine the smallest value r such that this quantity remains acceptable, one can use standard numerical integration techniques. 6 6

7. Variations III: non-Normal demand
In reality, the demand per unit time period may be far from Normal.
If it is Poisson, then things remain easy (since the LTD will also be Poisson).
In general, however, there may be no easy way to characterise the LTD distribution.
In that case, one can just build a histogram, using historical data. 7 7

8. Variations IV: random lead times
There may be variability in the lead time itself, as well as in the rate of demand.
This makes it very hard to characterise the LTD distribution analytically.
Again, one may have to resort to building a histogram using past data. 8 8

9. Variations V: using forecasts
Instead of being given the demand distribution itself, one may be given only a forecast of the lead time demand.
In that case, instead of using the standard deviation of the LTD, one can use the historical root-mean-square forecast error. 9 9

10. Variations VI: back-orders
Up to now, we assumed that demand is lost during a stock-out.
Sometimes, however, customers are willing to wait until the next delivery arrives, and then collect their order.
This is called “back-ordering”.
In this case, it may pay to have a smaller value for r (possibly even a negative value!). 10 10

11. Periodic Review Systems
Sometimes it is not desirable or practical to monitor the stock level continuously.
In some situations, one can only review stock periodically, e.g., at the start of each week.
This leads to so-called “Periodic Review” policies.

12. The (T, R) Model Suppose that stock is reviewed every T periods.
Each time, we place an order to raise stock up to a fixed level R, called the replenishment or restocking level.
That is, if the level is x, then we order R – x.
Note that, here, the order sizes vary, rather than the intervals between orders. 12

13. Stock variation in (T, R) model

14. Determining R for Fixed T
If T is fixed, what is the right value for R?
In deterministic case, it is D(T+L).
In the stochastic case, we need to add some safety stock, to guard against stochastic demand variations in an interval of T + L time units.
So work out distribution of demand in that period, choose the buffer stock b that gives the desired service level, then set R to D(T+L) + b.

15. Expected Average Stock Level
For optimal R, the expected average stock level is b + DT/2
D(T+L) + b R
DT + b DT b T

16. The (T, Q, r) Model
In some situations, one is given a fixed time interval T and a fixed order quantity Q.
Then, one can use the following policy.
Review every T periods.
If the stock level is below a re-order level r, then place an order of size Q.
Otherwise do not place an order. 16

17. Stock variation in (T, Q, r) model

18. The (T, Q, r) model: value of r
Computing the correct value of r is not easy!
Even in the case of deterministic demand, it is not obvious how to compute the average annual ordering and stock-holding costs, for a given value of r.
With stochastic demand, things are even worse, since the probability that an order takes place is then dependent both on r and on stochastic variation in demand. 18

19. The (T, Q, r) model (cont.)
One way to estimate the optimal r is to assume that orders take place roughly half the time.
Then, in the deterministic case, one can set r to D(L + T/2).
In the stochastic case, one can add buffer stock to guard against stochastic variation in demand over the “average” time period L + T/2.
(Or just use simulation to find a good r!) 19

20. The (T, S, s) Model
An even more complicated periodic review model is the following.
Review every T periods.
If the stock level is below a re-order level s, then place an order to raise stock up to a replenishment level S.
Otherwise do not place an order. 20

21. The (T, S, s) Model That is, if the stock level is x, then order:
S – x if x < s
0 if x  s.
Here, both S and s are parameters to be determined. 21

22. Stock variation in (T, S, s) model

23. The (T, S, s) model (cont.)
There is no analytic formula for the optimal values of S and s, even in the deterministic case.
One way to get approximate values is to assume that the average order size is (S – s) + DT/2…
… and then assume that this equals the EOQ.
This leaves an optimisation problem in just one variable.
(Or just use simulation!) 23