Inventory Management with Demand Uncertainty

Transcription

1 Annie Chen Questions? Inventory Management with Demand Uncertainty Intro to OM, Recitation 4 Annie Chen June 12, 2014

2 Announcements Process Improvement Analysis (individual) Due Sat, June 14, 11pm (PDF on Stellar). Guidelines in syllabus! Sport Obermeyer Case (team) Due Fri, June 20, beginning of class Hard copy & PDF on Stellar only; NO Excel sheets. Q&A: submit questions by TONIGHT (June 12)! Readings for class on June Questions? See syllabus and Study.Net. 2

3 Recitation Outline Single order opportunity Newsvendor model Example: FIFA t-shirts Example: Commencement programs Multiple order opportunities Service level Continuous review: (R,Q) policy Example: Cambridge Chowda Co. Periodic review: (T,S) policy Example: Bottled Water Vendor Risk pooling Example: Bottled Water Vendor Take Two Summary & comparison 3

4 Newsvendor Given: Probabilistic forecast: Cumulative distribution F(x) = Prob(demand x) Overage cost c o ($ per unit) Underage cost c u ($ per unit) Decision: What is the optimal order quantity Q* such that the expected total cost (overage + underage) is minimized? Solution: Prob(demand Q*) critical ratio 4

5 5 FIFA T-shirts The FIFA store is making a one-time order for the limited edition T-shirt on sale only the opening game. The Marketing Dept came up with the following forecast. If 21 thousand T-shirts were ordered, what is the probability that the store will run short of T-shirts? (a) 15% (b) 20% (c) 56% (d) 29% (e) Other

6 6 FIFA T-shirts The FIFA store is making a one-time order for the limited edition T-shirt on sale only the opening game. Ordering each T-shirt costs $6. Each T-shirt is sold for $30 at the opening game. In order to keep this a limited edition available only to those who attended the opening game, all unsold T-shirts will be destroyed with zero salvage value after the game. What are the overage and underage costs? c o =$6, c u =$24 What is the optimal order quantity Q*? Solution 1: Compute expected cost for all possible Q Solution 2:

7 Commencement Programs MIT is deciding how many copies of the program to print for the commencement ceremony. There are 2,000 students "walking". Each of them received 4 tickets to bring guests. From historical data, 20% of the students use 4 tickets, 40% of the students use 3 tickets, 30% of the students use 2 tickets, 5% of the students use 1 ticket, 5% don't use any tickets (i.e. comes by him/herself). Let X be the total number of students and guests attending the commencement. What is the mean and standard deviation of X? Normal distribution! X = Normal(7300, 45.3) 7

8 8 Commencement Programs MIT is deciding how many copies of the program to print for the commencement ceremony. Printing each copy in advance costs 5 cents. If MIT runs out of pre-printed copies, new copies can be made quickly at 20 cents each. Leftover copies will be sold to the recycling company at 1 cent each after the ceremony. What are the overage and underage costs? c o = 4, c u = 20 What is the optimal pre-printing quantity?

9 Recitation Outline Single order opportunity Newsvendor model Example: FIFA t-shirts Example: Commencement programs Multiple order opportunities Service level Continuous review: (R,Q) policy Example: Cambridge Chowda Co. Periodic review: (T,S) policy Example: Bottled Water Vendor Risk pooling Example: Bottled Water Vendor Take Two Summary & comparison Questions? 9

10 Service level Service level α%: probability of not having a stock-out. If we carry R units of inventory, the probability of not stocking out is α%. Prob(x) α% AVG Safety factor z: number of standard deviations required as safety stock. R = AVG + zσ zσ R x Safety stock 10

11 11 Inventory-service tradeoff Highly nonlinear relationship when the service level is high (i.e., need a lot of safety stock!). Operation point shows how a company is doing. Improve service level Reduce safety stock

12 (R,Q) Policy Continuous review: when the inventory level is R, order Q. Data: Probabilistic forecast AVG and STD Leadtime L demand during lead time = Normal(D LT,σ LT ) Requirement: service level safety factor z Decisions: Reorder point R: inventory needed to maintain service level during lead time R = D LT + zσ LT Order quantity Q: use EOQ formula Inventory Q R L Time 12

13 Cambridge Chowda Co. Demand: D = Normal (AVG=60,000, STD=1,224) (cases per year) Fixed ordering cost: K = 200 ($ per order) Variable ordering cost: c = 4 ($ per case Holding cost: h = 0.96 ($ per case per year) EOQ formula: The company operates for 48 weeks a year. It takes L = 2 weeks for an order to be delivered. The desired service level is 97.5% (z=1.96). What is the reorder point R? Convert to weekly demand: R = D LT + zσ LT Demand during lead time: D LT = 2*1250 = 2,500 units, σ LT = 2*176.7 Safety stock: zσ LT =1.96* 2*176.7 = 490 units R = D LT + zσ LT = 2,990 units. 13

14 (T,S) Policy Periodic review: at each period T, order up to S. Data: Probabilistic forecast AVG and STD Leadtime L Review period T demand during T&LT = Normal(D T&LT,σ T&LT ) Requirement: service level safety factor z Decision: Order-up-to level S: inventory needed to maintain service level during lead time and review period S = D T&LT + z σ T&LT 14

15 Bottled Water Vendor A vendor sells bottled water 7 days a week. On average, 100 bottles is sold per day, with a standard deviation of 25. He wishes to maintain a 99% service level (z=2.33). He reviews inventory and places an order on Friday before closing. New bottles arrive on Monday before opening. What should be his order-up-to level S? S = D T&LT + z σ T&LT T + LT = 7+2 = 9 (Lecture 6, p.15) D T&LT = 9*100 = 900, σ T&LT = 9*25 = 75 S = D T&LT + zσ T&LT = = 1,075 units. 15

16 Recitation Outline Single order opportunity Newsvendor model Example: FIFA t-shirts Example: Commencement programs Multiple order opportunities Service level Continuous review: (R,Q) policy Example: Cambridge Chowda Co. Periodic review: (T,S) policy Example: Bottled Water Vendor Risk pooling Example: Bottled Water Vendor Take Two Summary & comparison Questions? 16

17 Bottled Water Vendor Take Two A vendor sells bottled water at two locations. Daily demand is identical: AVG = 100, STD = 25. He wishes to maintain a 99% service level (z=2.33). Leadtime L = 2 days. Review period T = 7 days. What is the total safety stock if (1) the inventory were held separately at the two locations? safety stock = 2*z* 9*25 = 350 Individual σ T&LT (2) the inventory were held together at a central location? safety stock = z* 9* 2*25 = 247 Total σ T&LT 17

18 Risk Pooling Pool together demand from multiple different locations (i.e., served with the same pool of inventory) Variability in demand is mitigated Less safety stock is needed! When serving 2 locations, each with demand D=Normal(μ,σ) Served separately: Safety stock = zσ + zσ = 2zσ Served together: D 1&2 = Normal(2μ, 2σ) Safety stock = 2zσ = 1.41zσ Similarly, when serving n locations: Served separately: safety stock = nzσ Served together: safety stock = nzσ Sum variances, not std: 18

19 Recitation Outline Single order opportunity Newsvendor model Example: FIFA t-shirts Example: Commencement programs Multiple order opportunities Service level Continuous review: (R,Q) policy Example: Cambridge Chowda Co. Periodic review: (T,S) policy Example: Bottled Water Vendor Risk pooling Example: Bottled Water Vendor Take Two Summary & comparison Questions? 19

20 Inventory Management under demand uncertainty Newsvendor (R,Q) policy (T,S) policy Single order Minimize expected cost (overage + underage) Multiple order opportunies Meet service level by holding safety stock Continuous review (T varies) Periodic review (T is given & fixed) R = D LT + zσ LT S = D T&LT + zσ T&LT Questions? Risk pooling: reduces demand variability reduces safety stock! 20