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# The Base Stock Model

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The Base Stock Model 1 Assumptions ? Demand occurs continuously over time ? Times between consecutive orders are stochastic but independent and identically distributed (i. i. d.

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) ? 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 2 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. 4 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) 5 Inventory Balance Equation ? Inventory position = on-hand inventory + inventory onorder – backorder level 6 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 7 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. 8 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. 11 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) ? 12 The Optimal Base-Stock Level The optimal value of R is the smallest integer that satisfies Y (R ? 1) ? Y ( R) ? 0. 13 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) ? ? ? ? 14 Y ( R ? 1) – Y ( R) ? 0 ? ?b ? (h ? ) 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 15 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 ? If X can be approximated by a normal distribution, then: R * ? E ( D ) L ? z b /( b ? h ) V ar ( X ) Y ( R *) ? ( h ? b ) V ar ( X )? ( z b /( b ? h ) ) ? In the case where X has the Poisson distribution with mean ? L R * ? ? L ? z b /( b ? h ) ? L Y ( R *) ? ( h ? b ) ? L ? ( z b /( b ? h ) ) 17 Example 2 If X has the geometric distribution with parameter ? , 0 ? ? ? 1 P ( X ? x ) ? ? x (1 ? ? ). ? E[X ] ? 1? ? Pr( X ? x ) ? ? x Pr( X ? x ) ? 1 ? ? x ? 1 18 Example 2 (Continued…)

The optimal base-stock level is the smallest integer R* that satisfies Pr( X ? R * ) ? b b? h ln[ b ] b ? h ? 1 ln[ ? ] ? 1? ? R * ? 1 b ? ? R* ? b? h b ? ? ln[ ]? ? * b? h ? R ?? ? ln[ ? ] ? ? ? ? 19 Computing Expected Backorders ? It is sometimes easier to first compute (for a given R), E[I ] ? ? R x? 0 ( R ? x ) Pr( X ? x ) 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)] 20

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