The Base Stock Model

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The Base Stock Model

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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

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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.

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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.

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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)

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Inventory Balance Equation
 Inventory position = on-hand inventory + inventory onorder – backorder level

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Inventory Balance Equation
 Inventory position = on-hand inventory + inventory onorder – 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

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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.

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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]+

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 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) 10

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.

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The Cost Function

Y (R)  hE[ I ]  bE[ B]  h( R  E[ X ]  E[B])  bE[ B]  h( R  E[ X ])  (h  b) E[ B]  h( R  E[ D]L)  (h  b)E ([ X  R] )  h( R  E[ D]L)  (h  b) x  R ( x  R) Pr( X  x) 

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The Optimal Base-Stock Level

The optimal value of R is the smallest integer that satisfies Y (R  1)  Y ( R)  0.

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Y ( R  1) - Y ( R)  h  R  1  E[ D]L   (h  b) x R1 ( x  R  1) Pr( X  x ) h  R  E[ D]L   (h  b) x  R ( x  R) Pr( X  x)  h  (h  b) x R1  ( x  R  1)  ( x  R)  Pr( X  x)  h  (h  b) x  R1 Pr( X  x)  h  (h  b) Pr( X  R  1)  h  (h  b) 1  Pr( X  R)   b  (h  b) Pr( X  R)   



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Y ( R  1) - Y ( R)  0  b  (h  b) Pr( X  R)  0 b  Pr( X  R)  bh

Choosing the smallest integer R that satisfies Y(R+1) – Y(R)  0 is equivalent to choosing the smallest integer R that satisfies b Pr( X  R)  bh
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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 ( t) x e t P r( D ( t )  x )  , x  0. x!

 Leadtime demand, X, can be shown in this case to also have the Poisson distribution with ( L ) x e  L P r( X  x )  , E [ X ]   L , and V ar ( X )   L . x! 16

The Normal Approximation...
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