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 level R, 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 as  ).  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

14

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. 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 L

18 Example 2 If X has the geometric distribution with parameter , 0    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  = E( D ) L ), the following relationship (for a fixed R ) applies E [ B ]=  Pr( X = R )+(  - R )[1-Pr( X  R )]