1. The (Q, r) Model

2. Assumptions Demand occurs continuously over time
Times between consecutive orders are stochastic but independent and identically distributed (i.i.d.)
Inventory is reviewed continuously
Supply leadtime is a fixed constant L
There is a fixed cost associated with placing an order
Orders that cannot be fulfilled immediately from on-hand inventory are backordered

3. The (Q, r) Policy
Start with an initial amount of inventory R. When inventory level reaches level r, place an order in the amount Q = R-r to bring inventory position back up to level R. Thereafter whenever inventory position drops to r, place an order of size Q.
The base-stock policy is the special case of the (Q, r) policy where Q = 1.

4. Inventory versus Time Q
Inventory
Time l r

5. Notation I: inventory level, a random variable
B: number of backorders, a random variable
X: Leadtime demand (inventory on-order), a random variable
IP: inventory position
E[I]: Expected inventory level
E[B]: Expected backorder level
E[X]: Expected leadtime demand
E[D]: average demand per unit time (demand rate)

6. Inventory Position
Inventory position = on-hand inventory + inventory on-order – backorder level
Under the (Q, r) policy, inventory position, IP, takes on values takes on values r+1, r+2, ..., r+Q
The time IP remains at any value is the time between consecutive demand arrivals. Since the times between consecutive arrivals are independent and identically distributed, the long run fraction of time IP remains at any value is the same for all values

7. Inventory Position (Continued…)
IP is equally likely to take on values r+1, r+2, ..., r+Q, or equivalently, IP is uniformly distributed with
Pr(IP = i)=1/Q
where i = r+1, ..., r+Q.

8. Net Inventory
Net inventory IN=I–B increases when a delivery is made and decreases when demand occurs.
Let D(t, t+L] refer to the amount of demand that takes place in the interval (t, t+L].
All the inventory that was on order at t will be delivered in the interval (t, t+L]

9. Net Inventory (Continued…)
IN(t+L)=IN(t) + IO(t) – D(t, t+L]
IN(t+L)=IP(t) – D(t, t+L]
IN=IP – X

10. Inventory and Backorders
I=IN+=(IP -X)+
B=IN-=(IP-X)- =(X - IP)+
E[B]=E[(X - IP)+]
E[I]=E[(IP - X)+]

11. Probability of Stocking Out
Since IP and X are independent, we can condition on IP to obtain

12. Probability of Not Stocking Out
out for the base-stock policy
with base-stock x

13. It can be shown that
Expected backorder level under a base-stock policy with base-stock level r+1
Expected backorder level under a base-stock policy with base-stock level r+Q+1

14. Service Level Approximations
Type I Service:
Type II Service:

15. Expected Backorders and Inventory
To emphasize the dependency of inventory and backorder levels on the choice of Q and R, we let I(Q, r) and B(Q, r) denote respectively inventory level I and backorder level B when we use a (Q, r) policy.

16. Expected Backorders and Inventory (Continued…)
Conditioning on the value IP also leads to

17. E[IN] = E[IP] – E[X] = r +(Q+1)/2 –E[D]L
Since IN = IP – X, we have
E[IN] = E[IP] – E[X] = r +(Q+1)/2 –E[D]L
E[I(Q,r)] = r +(Q+1)/2 –E[D]L + E[B(Q,r)]

18. The Expected Total Cost
h: inventory holding cost per unit per unit time
b: backorder cost per unit per unit time.
A: ordering cost per order

19. The Expected Total Cost (Continued…)

20. Objective
We want to choose r and Q so that expected total cost (the sum of expected ordering cost, inventory holding cost and backorder cost per unit time) is minimized.

21. Solution Approach
Y(Q, r) is jointly convex Q and in r. Therefore, an efficient computational search can be implemented to solve for Q* and r*, the optimal values of Q and r, respectively.

22. An Approximate Solution Approach
1. Approximate E[B(Q,r)] by E[B(r)]
2. Assume demand is continuous
3. Treat Q and r as continuous variables

23. An Approximate Solution Approach
If the distribution of leadtime demand is approximated by a normal distribution, then the optimal reorder point can be approximated by

24. Model with Backorder Costs per Occurrence

25. Model with Backorder Costs per Occurrence
Y(Q, r) = AE[D]/Q + hE[I(Q,r)]+ kE[D](1 - S(Q,r))

26. Model with Backorder Costs per Occurrence
Using the approximation
leads to
Y(Q,r)  AE[D]/Q+h(r+(Q + 1)/2 – E[D]L+ E[B(r)])+
kE[D]E[B(r)]/Q

27. Model with Backorder Costs per Occurrence
Using the approximation
leads to
Y(Q,r)  AE[D]/Q + h(r + (Q + 1)/2 – E[D]L+ E[B(r)])+
kE[D]E[B(r)]/Q
An approximate solution is then given by

28. Insights from the (Q, r) Model
Main Insights:
Increasing the reorder point r increases safety stock but provides a greater buffer against stockouts
Increasing the order quantity increases cycle stock but reduces ordering (or setup) costs
Increasing the order quantity leads to a decrease in the reorder point

29. Insights from the (Q, r) Model
Main Insights:
Increasing the reorder point r increases safety stock but provides a greater buffer against stockouts
Increasing the order quantity increases cycle stock but reduces ordering (or setup) costs
Increasing the order quantity leads to a decrease in the reorder point
Other Insights:
Increasing L tends to increase the optimal reorder point
Increasing the variability of the demand process tends to increase the optimal reorder point
Increasing the holding cost tends to decrease the optimal order quantity and reorder point

30. The (R, r) Model This is usually called the (S, s) model
Each demand order can be for multiple units
Demand orders are stochastic
A replenishment order is placed to bring inventory position back to R
Decision variables are R (instead of Q) and r

31. Dealing with Lead Time Variability
L: replenishment lead time (a random variable)
E(L): expected replenishment leadtime
Var(L): variance of lead time
D: demand per period
E(D) = expected demand per period
Var(D): variance demand
E(X)=E(L)E(D)
Var(X)=E(L)Var(D) + E(D)2Var(L)

32. Dealing with Lead Time Variability (Continued…)
Under the Normal approximation, the optimal reorder can be obtained as

33. Expected Leadtime Demand

34. Variance of Leadtime Demand

35. Variance of Leadtime Demand

36. Variance of Leadtime Demand (Continued..)