The Base Stock Model

1. The Base Stock 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 no fixed cost associated with placing an order • Orders that cannot be fulfilled immediately from on-hand inventory are backordered

3. The Base-Stock Policy • Start with an initial amount of inventory R. Each time a new demand arrives, place a replenishment order with the supplier. • An order placed with the supplier is delivered L units of time after it is placed. • Because demand is stochastic, we can have multiple orders (inventory on-order) that have been placed but not delivered yet.

4. The Base-Stock Policy • The amount of demand that arrives during the replenishment leadtime L is called the leadtime demand. • Under a base-stock policy, leadtime demand and inventory on order are the same. • When leadtime demand (inventory on-order) exceeds R, we have backorders.

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 Balance Equation • Inventory position = on-hand inventory + inventory on-order – backorder level

7. Inventory Balance Equation • Inventory position = on-hand inventory + inventory on-order – backorder level • Under a base-stock policy with base-stock levelR, inventory position is always kept at R (Inventory position = R ) IP = I+X - B = R E[I] + E[X] – E[B] = R

8. Leadtime Demand • Under a base-stock policy, the leadtime demand X is independent of R and depends only on L and D with E[X]= E[D]L (the textbook refers to this quantity asq). • The distribution of X depends on the distribution of D.

9. I = max[0, I – B]= [I – B]+ • B=max[0, B-I] = [ B - I]+ • Since R = I + X – B, we also have • I – B = R – X • I = [R – X]+ • B =[X – R]+

10. E[I] = R – E[X]+ E[B] = R – E[X]+ E[(X – R)+] • E[B] = E[I] + E[X]– R = E[(R – X)+] + E[X]– R • Pr(stocking out) = Pr(X R) • Pr(not stocking out) = Pr(X  R-1) • Fill rate = E(D) Pr(X R-1)/E(D) = Pr(X R-1)

11. Objective Choose a value for R that minimizes the sum of expected inventory holding cost and expected backorder cost, Y(R)= hE[I] + bE[B], where h is the unit holding cost per unit time and b is the backorder cost per unit per unit time.

12. The Cost Function

13. The Optimal Base-Stock Level The optimal value of R is the smallest integer that satisfies

15. Choosing the smallest integer R that satisfies Y(R+1) – Y(R)  0 is equivalent to choosing the smallest integer R that satisfies

16. Example 1 • Demand arrives one unit at a time according to a Poisson process with mean l. If D(t) denotes the amount of demand that arrives in the interval of time of length t, then • Leadtime demand, X, can be shown in this case to also have the Poisson distribution with

17. The Normal Approximation • If X can be approximated by a normal distribution, then: • In the case where X has the Poisson distribution with mean lL

18. Example 2 If X has the geometric distribution with parameter r , 0 r 1

19. Example 2 (Continued…) The optimal base-stock level is the smallest integer R* that satisfies

20. Computing Expected Backorders • It is sometimes easier to first compute (for a given R), and then obtain E[B]=E[I] + E[X] – R. • For the case where leadtime demand has the Poisson distribution (with mean q = E(D)L), the following relationship (for a fixed R) applies E[B]= qPr(X=R)+(q-R)[1-Pr(XR)]