# Formulation average ordering headway qd average

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Formulation Average ordering headway = Q/D’ Average inventory Expected backorder per cycle Expected cost per cycle= c f + c v Q + c h I Q / D’ + c b B ( R ) Cost per unit time F.O.C. w.r.t. R and Q Solution approach: start with some initial value, solve R and Q iteratively to convergence. Puzzle: what does the policy look like if the fixed ordering cost diminishes? ( ) ( ) ( ) R B R x R f x dx = f ( , ) '/ ' ( ' / 2) ' ( ) / v h b g Q R c D Q c D c R D Q c D B R Q τ = + + + + 2 2 f ( , ) '/ 2 ' ( ) / 0 h b g Q R Q c D Q c c D B R Q = − + + = '/ ( ) 0 h b R g R c c D Q f x dx = = 1/2 f 1 [2 '( ( )) / ] [1 ( ')] b h h b Q D c c B R c R F Qc c D = + = Q R τ Re-order point ' 2 I R D Q τ = + Safety stock 10 LSA Classical Inventory Models - Overview Stochastic, periodical review demand random and iid; rate D (item/period) in period 1, 2, …, m ; fixed lead time, τ >0 holding cost c h , fixed and variable ordering costs c f and c v (\$/item), backorder penalty c b (\$/item) Inventory at the end of horizon may be sold at price c s The inventory position (in-stock + in transition) is reviewed in each period, and within each period the following events happen sequentially: (review inventory – place order – order arrival – demand observed – demand filled/backordered – holding/backorder costs charged) Without proof, we give out the following results (Karlin & Scarf, 1961): (s, S) policy: if we try to minimize the total expected costs for all i=1, 2, …, m periods, then there exist scalar values s i and S i (s i ≤ S i ) such that the optimal ordering policy in period i has the following simple structure: (1) if inventory position at the beginning of period I i-1 > s i , do not order; (2) Otherwise, order quantity S i -I i-1 . Base-stock (order-up-to) policy: if c f =0, then s i = S i , such that the optimal ordering policy in period i has the following simple structure: (1) if inventory position I i-1 > S i , do not order; (2) Otherwise, order quantity S i -I i-1 .

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6 11 (R, Q) policy R Q (s, S) policy
LSA: Chapter 3 Lot size problem with variable demand Allow demand rate D’ to change deterministically with time • D(t): cumulative number of items demanded by time t • D’(t): variable demand rate at time t D’ Lecture Outline • Dynamic inventory (lot size) problem • Solution methods • Dynamic programming (two formulation methods) • Shooting method • Continuous approximation Special Case: rent costs dominate Multi-period, deterministic inventory model with time-varying demand ( ) max t D 0 t 1 t 2 t 3 t 4 t max t time Cum’l items ( ) t D ( ) t R LSA Lot size problem with variable demand – Discrete Model Deterministic, time-varying demand D t (in period t = 1, 2, …, T=t max ) no lead time, initial inventory = 0 cost include Ordering/transportation cost c f + c v v t (\$) inventory holding cost c h (\$/item-time) sequence of events within each period place order - receive replenishment - fulfill demand - pay holding cost Formulation (Wagner-Whiten, 1958) Min t [c f (v t ) + c h I t + c v v t ] s.t. I t = I t-1 + v t – D t , for all t I 0 =0 I t , v t 0, for all t Observations: (1) in an optimal solution, v t I t-1 =0, for all t the arrival curve always

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