Previously • Optimization • Probability Review – pdf, cdf, E, Var – Poisson, Geometric, Normal, Binomial, … • Inventory Models – Newsvendor Problem – Base Stock Model
Agenda • Projects • Order Quantity Model – aka Economic Order Quantity (EOQ) • Markov Decision Processes
Order Quantity Model • Continuous review (instead of periodic) • Ordering costs vs. • Inventory costs inventory Q: When to reorder? reorder times time
Order Quantity Model (12. 7) • Constant demand rate • Inventory A/year – No backlogging – Replenishment lead time L years – Order placement cost $K – Holding cost H/unit/year (Time between ordering more and delivery) (Independent of order size) • Q: Reorder point r? Order quantity q?
Order Quantity Model slope = -A inventory order quantity q reorder point r … lead time L time
Economic Order Quantity (EOQ) • r=AL reorder point r • • slope = -A inventory order quantity q time between orders =q/A orders per year = A/q ordering cost per year = KA/q holding cost per year: H(q/2) … lead time L time
Economic Order Quantity (EOQ) max C(q) s. t. q≥ 0 • C’(q) = H/2 - KA/q 2, C’(q*)=0 • q* = (2 AK/H)1/2 (cycle stock) inventory • ordering cost per year = KA/q • holding cost per year: H(q/2) • total cost C(q) = KA/q + Hq/2 q -A … r L time
Summary of Inventory Models • Newsvendor model • Base stock model – safety stock • Order quantity model – cycle stock • Growth with square-root of demand • 12. 8 covers order quantity + uncertain demand
Markov Decision Processes (9. 10 -9. 12) Junk Mail example (9. 12) • $1. 80 to print and mail a catalog • $25 profit if you buy something • 5% probability of buying if new customer expected profit = -$1. 80 + 5%*$25 = -$0. 55 • but you might be a profitable repeat customer i 0 1 2 3 4 5 6+ p(i) 0. 05 0. 40 0. 20 0. 10 0. 03 0. 01 0. 00 p(i) probability of an order if received i catalogs since last order (i=1 means ordered from last catalog sent, i=0 means new customer)
Junk Mail Example • Give up on customers with 6 catalogs and no orders • 7 states i=0, …, 6+ • f(i) = largest expected current+future profit from a customer in state i i 0 1 2 3 4 5 6+ p(i) 0. 05 0. 40 0. 20 0. 10 0. 03 0. 01 0. 00 p(i) probability of an order if received i catalogs since last order (i=1 means ordered from last catalog sent, i=0 means new customer)
LP Form Idea: f(i) decision variables piecewise linear function min f(0)+…+f(6) s. t. f(i) ≥ -1. 80+p(i)[25+f(1)]+[1 -p(i)]f(i+1) for i=1. . 5 f(0) ≥ -1. 80+p(i)[25+f(1)] f(i) ≥ 0 for all i
Markov Decision Processes (MDP) • • States i=1, …, n Possible actions in each state Reward R(i, k) of doing action k in state i Law of motion: P(j | i, k) probability of moving i j after doing action k
MDP f(i) = largest expected current + future profit if currently in state i f(i, k) = largest expected current+future profit if currently in state i, will do action k f(i) = maxk f(i, k) = R(i, k) + ∑j P(j|i, k) f(j) f(i) = maxk [R(i, k) + ∑j P(j|i, k) f(j)]
![MDP as LP f(i) = maxk [R(i, k) + ∑j P(j|i, k) f(j)] Idea:](https://present5.com/presentation/0e4ce2d36ce0120961b99d5667cf5f70/image-14.jpg "MDP as LP f(i) = maxk [R(i, k) + ∑j P(j|i, k) f(j)] Idea:")
MDP as LP f(i) = maxk [R(i, k) + ∑j P(j|i, k) f(j)] Idea: f(i) decision variables piecewise linear function min ∑j f(i) s. t. f(i) ≥ R(i, k) + ∑j P(j|i, k) f(j) for all i, k
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