1

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.

3

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]+

9

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 R1 ( x R 1) Pr( X x ) h R E[ D]L (h b) x R ( x R) Pr( X x) h (h b) x R1 ( x R 1) ( x R) Pr( X x) h (h b) x R1 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) bh

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) bh

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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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