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

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Agenda Projects Order Quantity Model –aka Economic Order Quantity (EOQ) Markov Decision Processes

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

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Order Quantity Model (12.7) Constant demand rateA/year Inventory –No backlogging –Replenishment lead timeL years (Time between ordering more and delivery) –Order placement cost$K (Independent of order size) –Holding costH/unit/year Q: Reorder point r? Order quantity q?

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

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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) slope = -A time inventory … order quantity q reorder point r lead time L

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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/q 2, C’(q*)=0 q* = (2AK/H) 1/2 (cycle stock) -A time inventory … q r L

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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

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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 i0123456+ p(i)0.050.400.200.100.030.010.00 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)

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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 i0123456+ p(i)0.050.400.200.100.030.010.00 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)

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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) ≥ 0for all i

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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

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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) = max k f(i,k) f(i,k) = R(i,k) + ∑ j P(j|i,k) f(j) f(i) = max k [R(i,k) + ∑ j P(j|i,k) f(j)]

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MDP as LP f(i) = max k [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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