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Published byChristiana Wiggins Modified over 8 years ago
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1 The Base Stock Model
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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
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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.
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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.
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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)
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6 Inventory Balance Equation Inventory position = on-hand inventory + inventory on- order – backorder level
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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
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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.
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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 ] +
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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)
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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.
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12 The Cost Function
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13 The Optimal Base-Stock Level The optimal value of R is the smallest integer that satisfies
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15 Choosing the smallest integer R that satisfies Y ( R +1) – Y ( R ) 0 is equivalent to choosing the smallest integer R that satisfies
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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
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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
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18 Example 2 If X has the geometric distribution with parameter , 0 1
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19 Example 2 (Continued…) The optimal base-stock level is the smallest integer R * that satisfies
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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 )]
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