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Inventory Management with Demand Uncertainty 15.734 Intro to OM, Recitation 4 Annie Chen June 12, 2014 Annie Chen Questions?
Inventory Management with Demand Uncertainty
15.734 Intro to OM, Recitation 4
Annie Chen
June 12, 2014
Annie Chen
Questions?
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)! http://goo.gl/As7VEY
• Readings for class on June 20-21 – See syllabus and Study.Net.
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Questions?
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
Newsvendor Multi-order: Service Level (R,Q) (T,S) Risk Pooling Summary
Newsvendor
• Given: – Probabilistic forecast: Cumulative distribution F(x) = Prob(demand ≤ x) – Overage cost co ($ per unit) – Underage cost cu ($ per unit)
• Decision: What is the optimal order quantity Q* such that the expected total cost (overage + underage) is minimized?
• Solution:
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critical ratio Prob(demand ≤ Q*)
Newsvendor Multi-order: Service Level (R,Q) (T,S) Risk Pooling Summary
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
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Newsvendor Multi-order: Service Level (R,Q) (T,S) Risk Pooling Summary
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? co=$6, cu=$24
• What is the optimal order quantity Q*?
Solution 1: Compute expected cost for all possible Q
Solution 2:
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Newsvendor Multi-order: Service Level (R,Q) (T,S) Risk Pooling Summary
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? X = Normal(7300, 45.3) 7
Normal distribution!
Newsvendor Multi-order: Service Level (R,Q) (T,S) Risk Pooling Summary
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? • What is the optimal pre-printing quantity?
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Newsvendor Multi-order: Service Level (R,Q) (T,S) Risk Pooling Summary
co = 4¢, cu = 20¢
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 9
Newsvendor Multi-order: Service Level (R,Q) (T,S) Risk Pooling Summary
Questions?
α%
AVG x
Prob(x)
R
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%.
• Safety factor z: number of standard deviations required as safety stock. R = AVG + zσ
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zσ
Newsvendor Multi-order: Service Level (R,Q) (T,S) Risk Pooling Summary
Safety stock
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.
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Newsvendor Multi-order: Service Level (R,Q) (T,S) Risk Pooling Summary
Improve service
level
Reduce safety stock
(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(DLT ,σLT)
• Requirement: service level safety factor z • Decisions:
– Reorder point R: inventory needed to maintain service level during lead time
R = DLT + zσLT
– Order quantity Q: use EOQ formula
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Time
Inventory
Q
L
R
Newsvendor Multi-order: Service Level (R,Q) (T,S) Risk Pooling Summary
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: Demand during lead time: DLT = 2*1250 = 2,500 units, σLT = √2*176.7 Safety stock: zσLT =1.96*√2*176.7 = 490 units R = DLT + zσLT = 2,990 units.
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R = DLT + zσLT
Newsvendor Multi-order: Service Level (R,Q) (T,S) Risk Pooling Summary
(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< = Normal(DT< ,σT<) • Requirement: service level safety factor z • Decision:
– Order-up-to level S: inventory needed to maintain service level during lead time and review period
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S = DT< + z σT<
Newsvendor Multi-order: Service Level (R,Q) (T,S) Risk Pooling Summary
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? T + LT = 7+2 = 9
DT< = 9*100 = 900, σT< = √9*25 = 75
S = DT< + zσT< = 900 + 174.75 = 1,075 units.
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S = DT< + z σT<
(Lecture 6, p.15)
Newsvendor Multi-order: Service Level (R,Q) (T,S) Risk Pooling Summary
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 16
Questions?
Newsvendor Multi-order: Service Level (R,Q) (T,S) Risk Pooling Summary
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
• (2) the inventory were held together at a central location? safety stock = z*√9*√2*25 = 247
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Individual σT<
Total σT<
Newsvendor Multi-order: Service Level (R,Q) (T,S) Risk Pooling Summary
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: D1&2 = Normal(2μ, √2σ) Safety stock = √2zσ = 1.41zσ
• Similarly, when serving n locations: – Served separately: safety stock = nzσ – Served together: safety stock = √nzσ
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Sum variances, not std:
Newsvendor Multi-order: Service Level (R,Q) (T,S) Risk Pooling Summary
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 19
Questions?
Newsvendor Multi-order: Service Level (R,Q) (T,S) Risk Pooling Summary
Inventory Management under demand uncertainty
Newsvendor (R,Q) policy (T,S) policy
Single order Multiple order opportunies
Minimize expected cost (overage + underage)
Meet service level by holding safety stock
Continuous review (T varies)
Periodic review (T is given & fixed)
R = DLT + zσLT
S = DT< + zσT<
Risk pooling: reduces demand variability reduces safety stock!
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Newsvendor Multi-order: Service Level (R,Q) (T,S) Risk Pooling Summary
Questions?
