The Base Stock Model

The Base Stock Model 1 Assumptions ? Demand occurs perpetually over time ? Times between consecutive orders are stochastic but independent and identically distributed (i. i. d. ) ? Inventory is reviewed continuously ? Supply leadtime is a resolved constant L ? There is no fixed cost associated with placing an order ? Orders that cannot be fulfilled immediately from on-hand stock certificate are backordered 2 The Base-Stock Policy ? stand out with an initial amount of fund R. Each time a new quest arrives, place a replenishment order with the provider. 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 NotationI inventory direct, a random variable B number of backorders, a random variable X Leadtime demand (inventory on-order), a random variable IP inventory position EI Expected inventory level EB Expected backorder level EX Expected leadtime demand ED average demand per unit time (demand roam) 5 Inventory brace 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 EI + EX EB = 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 EX= EDL (the textbook refers to this quantity as ? ). ? The distribution of X depends on the distribution of D. 8 I = max0, I B= I B+ B=max0, B-I = B I+ Since R = I + X B, we also have IB=RX I = R X+ B =X R+ 9 ? EI = R EX + EB = R EX + E(X R)+ ?EB = EI + EX R = E(R X)+ + EX 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 attribute cost and expected backorder cost, Y(R)= hEI + bEB, 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 ? EB) ? bE B ? h( R ? E X ) ? (h ? b) E B ? h( R ? E DL) ? (h ? b)E ( X ? R? ? h( R ? E DL) ? (h ? b)? x ? R ( x ? R) Pr( X ? x) ? 12 The Optimal Base-Stock Level The optimal value of R is the smallest whole number that satisfies Y (R ? 1) ? Y ( R) ? 0. 13 Y ( R ? 1) Y ( R) ? h ? R ? 1 ? E DL ? ? (h ? b)? x? R? 1 ( x ? R ? 1) Pr( X ? x ) ? h ? R ? E DL ? ? (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 whole number 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 fit in to a Poisson process with compressed ?. 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 causa 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 distrib ution 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 ? ? ). ? EX ? 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 work up (for a given R), EI ? ? R x? 0 ( R ? x ) Pr( X ? x ) and then obtain EB=EI + EX R. ? For the case where leadtime demand has the Poisson distribution (with mean ? = E(D)L), the following relationship (for a fixed R) applies EB= ? Pr(X=R)+(? -R)1-Pr(X? R) 20