1 / 14

Previously

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. inventory. reorder times.

altessa
Download Presentation

Previously

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Previously • Optimization • Probability Review • pdf, cdf, E, Var • Poisson, Geometric, Normal, Binomial, … • Inventory Models • Newsvendor Problem • Base Stock Model

  2. Agenda • Projects • Order Quantity Model • aka Economic Order Quantity (EOQ) • Markov Decision Processes

  3. inventory reorder times time Order Quantity Model • Continuous review (instead of periodic) • Ordering costs vs. • Inventory costs Q: When to reorder?

  4. Order Quantity Model (12.7) • Constant demand rate A/year • Inventory • No backlogging • Replenishment lead time L years(Time between ordering more and delivery) • Order placement cost $K(Independent of order size) • Holding cost H/unit/year • Q: Reorder point r? Order quantity q?

  5. order quantity q slope = -A … inventory time reorder point r lead time L Order Quantity Model

  6. order quantity q slope = -A reorder point r inventory … time lead time L Economic Order Quantity (EOQ) • r=AL • time between orders =q/A • orders per year = A/q • ordering cost per year = KA/q • holding cost per year: H(q/2)

  7. q -A inventory r … time L Economic Order Quantity (EOQ) • ordering cost per year = KA/q • holding cost per year: H(q/2) • total cost C(q) = KA/q + Hq/2 max C(q) s.t. q≥0 • C’(q) = H/2 - KA/q2, C’(q*)=0 • q* = (2AK/H)1/2 (cycle stock)

  8. 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

  9. 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 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)

  10. 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 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)

  11. 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

  12. 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

  13. MDP f(i) = largest expected current + future profitif currently in state i f(i,k) = largest expected current+future profitif currently in state i, will do action k f(i) = maxk f(i,k) 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)]

  14. 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

More Related