Inventory Management II

1. Chapter 7 Inventory Management II Stochastic Models

2. Supplier Customer Supply-Demand Management Relationship Relationship "Make, Move, Store" Management Management Plant "Buy" "Sell" Plant Warehouse Customers Suppliers Plant Managing the store side of a business

3. Key questions • What is the cost trade-off captured by the single period model (SPM)? • In what settings is SPM analysis appropriate, & how can an order quantity be determined? • What is the definition of base stock level, & how does a base stock policy answer the questions of when & how much to order? • In what settings may a base stock policy be reasonable, & how can base stock be determined? • How do (Q, R) & (I, S) policies answer the questions of when & how much to order, when is each policy appropriate, & how can the values of policy parameters be determined ? • What is the difference between fill rate & service level, and how do these performance measures depend on policy parameters? • How can the return on investments to reduce uncertainty be estimated? • What are the managerial insights from the chapter?

4. Road map • Fundamental Problem Type #2 • Single Period Model • Policies for When & How Much • Summary

5. Trade-off Fundamental problem #1 From Chapter 6 • Need to pick a number • Economic incentives for picking a large number • E.g., transaction costs • Economic disincentives if number is too large • E.g., inventory carrying costs

6. Trade-off Fundamental problem #2 • Need to pick a number • There are costs if the number is too high • Excess costs • There are costs if the number is too low • Shortage costs

7. Trade-off A common trade-off • How many newspapers to order at a corner newsstand when demand is uncertain? • Cost of unsold newspapers versus cost of lost profit if not enough • How much capacity to reserve for a new line of summer wear when market potential is uncertain? • Cost of reserving unneeded capacity versus premium cost of secondary supplier if there is not enough capacity • How many extra seats to sell (overbook) when the # of no shows is uncertain? • Cost of compensating passengers to give up ticket versus lost opportunity to make a profit on an empty seat

8. Road map • Fundamental Problem Type #2 • Single Period Model • Policies for When & How Much • Summary

9. Single Period Model How many newspapers to order? • You are a newsvendor with some information • D = demand on Monday ~ Uniform[50,100] • Selling price = p = $1.00/paper • Purchase cost = c = $0.80/paper • Salvage value/paper (what you can get for a leftover paper) = s • What would you order if s = $0.80, s = $0.60, s = $0.60+?

10. Single Period Model Concept of service level • For each order quantity Q, there is an associated service level (SL) • Relationship: P[D Q] = SL • Service level is the probability that we will have enough papers to satisfy all demand • E.g., what is SL if Q = 100, Q = 75 in the example? • What is the “best” long run service level if the excess cost per unit matches the shortage cost per unit?

11. Single Period Model Balance point Optimal service level = service level with least expected cost Suppose ce = excess cost rate = 1, cs = shortage cost rate = 4: balance point at 4/5 = 80%  SL* = cs/(cs+ce) = optimal probability that will satisfy all demand = optimal service level = 4/5 = 0.8 1 4 0 .8 1

12. Single Period Model Finding Q, given a service level target Want Q consistent with P[DQ] = SL 1. Continuous distributions for demand 1a. Uniform distribution, D ~ U[a,b] Area = SL Q = a + SL(b-a) a Q b

13. Single Period Model Finding Q, given a service level target Want Q consistent with P[DQ] = SL 1. Continuous distributions for demand 1b. Normal distribution, D ~ N(D, D2) Area = SL Q = D + zSLD

14. Single Period Model Finding Q, given a service level target 2. Discrete distributions for demand 2a. General discrete distribution xP[D=x] P[Dx] 10 10% 10% 20 30% 40% 30 30% 70% 40 30% 100% If SL* = 70%, then Q = ? If SL* = 75%, then Q = ? Round up: Q = 40 gives lower expected cost than Q = 30

15. Single Period Model What have we just seen anyway? Analysis using the single period model Characteristics • Order or production quantity decision for a single period • Demand is unpredictable • There is cost for guessing high & for guessing low, & cost is proportional to deviation A fundamental & widespread problem type • Traced back to economist Edgeworth (1888) who applied to bank cash-flow planning • Spotting arbitrage opportunities in the stock market, dynamic product pricing, yield management, civil engineering,...

16. Single Period Model Example – Sport Obermeyer Fisher, Hammond, Obermeyer, & Raman, Harvard Business Review cs = shortage cost/unit (e.g., gross margin) = $14.50 ce = excess cost/unit (e.g., cost less mark down) =$5.00 D ~ N(1200, 1302) SL* = cs/(cs+ce) = 14.5/(14.5 + 5)  0.743 From probability table, z0.74215 = 0.65 Q* = D + zSL*D = 1200 + 0.65(130) = 1,285  produce 1,285 parkas for upcoming season

17. Road map • Fundamental Problem Type #2 • Single Period Model • Policies for When & How Much • Base stock • (Q, R) • (I, S) • Summary

18. When & How Much - Base Stock Base stock policy A simple policy for answering when & how much • Whenever demand occurs, place an order to replenish the demand quantity • Is this a push or pull approach? • Base stock level = on hand + on order • If the policy defined above is strictly followed, then the base stock level is the same at any time • The answer to when? • Whenever demand occurs • The answer to how much? • Whatever demand was

19. When & How Much - Base Stock Example – cash management • Background from Chapter 6: transfer fee = $25, high & low yield rates = 8% & 1%, D = $900,000/year • What concession did we get from the bank to make this policy a viable option? • D = $4,000/day (225 working days), L = lead time = 3 days • Suppose there is no variability in demand • Suggestions for the base stock level? • A picture on the board...

20. When & How Much - Base Stock Optimal base stock level S* Back to fundamental problem #2 • With uncertainty • LD = demand during lead-time (uncertain) • Mechanics are same as single period model • S* satisfies P[LD S*] = SL* where SL* = cs/(cs + ce) • E.g., LD ~ N(12000, 10002), ce = 0.07, cs = 0.08 • SL* = 0.08/(0.08+0.07) = 0.5333 & z0.53586= 0.09 • S* = LD + zSL*(LD) = 12000 + 0.09(1000) = 12,090

21. When & How Much Transition to alternative policies Two common variants of a pure base stock policy • Appropriate when there are benefits from fewer transactions, e.g., as a bar owner, it’s not especially practical to place an order every time someone buys a beer 1.Continually track sales - inventory reaches 10 cases, place an order for 15 cases • A (Q, R) policy, e.g., Q = 15 cases, R = 10 cases 2. Every Friday, we place an order for what was sold during the week (to bring on-hand + on-order up to 25 cases) • An (I, S) policy, e.g., I = 7 days, S = 25 Continuous review policy Periodic review policy

22. Road map • Fundamental Problem Type #2 • Single Period Model • Policies for When & How Much • Base stock • (Q, R) • (I, S) • Summary

23. When & How Much – (Q,R) (Q, R) policy • Continually track sales • When inventory reaches 10 cases, we place an order for 15 cases—is an example of a (Q,R) policy • Q = 15 = order quantity - reflects a balance of scale economies with associated costs • Fundamental problem #1 • R = 10 = reorder point - raises the number guessing issue • Fundamental problem #2

24. When & How Much – (Q,R) Questions • Suppose demand is constant at $4,000/day in cash management problem (recall L = 3 days) • Suggestions for R assuming we aren’t worrying about the possibility of a negative balance? • What if demand is uncertain?

25. When & How Much – (Q,R) Concept of fill rate • Reorder point could based on • service level target, or • a fill rate target • Fill rate = proportion of demand shipped on time (e.g., shipped from stock) • A common performance measure in industry • Lands’ End: 90% (ship within 24 hours) • Frito Lay: 99.5% (scanner data  forecasts R’s)

26. When & How Much – (Q,R) Cash management example • Q = $25,000, D = $4,000/day, target FR = 98% • This means an average of (0.02)(4000) = $80 written out of negative balance per day • Q = $25,000 = average demand between receipt of funds • For 98% fill rate target, this means*… • Q(1-FR) = 25,000(0.02) = $500 = average negative balance when cash arrives *As long as there is no chance that demand during the leadtime will be greater than R+Q

27. When & How Much – (Q,R) Example continued Suppose the estimated distribution of demand during lead-time is: xP[LD=x] What is the fill rate if R = $12,000? $10,000 10% $11,000 15% Average negative balance = 10%(0) + 15%(0)+ $12,000 50% 50%(0) + 15%(1000) + 10%(2000) = $350 $13,000 15% $14,000 10% 350/25000 = 1.4% FR = 98.6% Another view: Q = $25,000  36 transactions/year (36 = 900K/25K)  (36)($350) = $12,600 = average amount/year written out of negative balance  $12,600/$900,000 = 1.4% FR = 98.6%

28. When & How Much – (Q,R) Example continued Suppose LD ~ N(LD,LD2), LD=12,000, LD2=4,0002 Reorder point can be expressed as R = LD + buffer stock1 = LD + zLD What value of z is consistent with FR = 98%? 1buffer stock & safety stock mean the same thing

29. When & How Much – (Q,R) Example continued We can use the unit normal loss table (G(z) in Appendix 4) • From normal, R = LD + zLD & LD = 12000LD = 4000 • Just need the value of z • Can use the approximation formula: Q(1-FR)/LDG(z) = standardized average number of units short upon receipt of order • Know Q = 25000, FR = 0.98, & LD = 4000 • So Q(1-FR)/LD = $25000(1-0.98)/4000 = 500/4000 = 0.125 = G(z) • From table, G(0.78) = 0.1245, G(0.77) = 0.12669, so z = 0.78 or 0.77 • Plugging it in: R = LD + zLD = 12000 + (0.78)4000 = $15,120

30. What is impact if able to cut uncertainty in half while FR is unchanged? FR = 0.98 = 1 - 0.02  1 – LDG(z)/Q LD = 4000 to LD = 2000  G(z) = 0.125 to G(z) = 0.25 (for constant FR) G(z) = .25  z  0.32 safety stock = zLD= (0.32)(2000) = 640 From 3120 to 640  80% reduction in safety stock What is impact if able to cut uncertainty in half while SS is unchanged? safety stock = zLD= (0.78)(4000) = 3120 LD = 4000 to LD = 2000  z = .78 to z = 1.56 (for constant SS) z = 1.56  G(z) = G(1.56) = 0.02552 From G(.78) = 0.1245 to G(1.56) = 0.02552  79.5% reduction in “standardized” backorder rate FR 1 – LDG(z)/Q = 1 – (2000)(0.02552)/25000 From FR = 98% to FR = 99.8%, or 90% reduction in backorder rate When & How Much – (Q,R) Example 2 – estimating ROI • In previous example, • safety stock = R - LD = zLD= 15,120 – 12,000 = 3,120 • LD = 4000, z = 0.78, Q = 25,000, & fill rate = FR = 98% Remaining step is compare cost with savings and/or profit increase

31. When & How Much – (Q,R) Fill rate - buffer stock Q = 750, LD = 1,000,& buffer stock = zLD = z(1000) Managerial Insight:Each percentage increase in fill rate requires an increasing incremental investment in inventory

32. Road map • Fundamental Problem Type #2 • Single Period Model • Policies for When & How Much • Base stock • (Q, R) • (I, S) • Summary

33. (I, S) policy When & How Much – (I,S) • Periodic review policy • Every Friday, we place an order to bring on-hand + on-order up to 25 cases—is an example of a (I,S) policy • I = 7 days = order interval - reflects a balance of scale economies with associated costs • Fundamental problem #1 • S = 25 cases = order-up-to quantity - raises the number guessing issue • Fundamental problem #2

34. Cash management example When & How Much – (I,S) Transfer is automatically made every 6.25 working days • I = 6.25 days, D = $4,000/day, target FR = 98% • Average transfer amount = 6.25 days  $4000/day = $25,000 • FR = 98% means an average of (0.02)(4000) = $80 written out of negative balance per day • ID = 6.25(4000) = $25,000 = average demand between receipt of funds • For 98% fill rate target, this means*… • ID(1-FR) = 25,000(.02) = $500 = average negative balance when cash arrives

35. Example continued When & How Much – (I,S) I = 6.25 days, L = leadtime 3 days, I + L = 9.25 days ILD = random demand during the order interval plus leadtime (9.25 days) Suppose ILD ~ N(ILD,ILD2), ILD= 49,000, ILD2 = 8,1002 Order-up-to quantity can be expressed as S = ILD + buffer stock1 = ILD + zILD What value of z is consistent with FR = 98%? 1buffer stock & safety stock mean the same thing

36. Example continued When & How Much – (I,S) We can use the unit normal loss table (G(z) in Appendix 4) • From normal, S = ILD + zILD & ILD = 49000ILD = 8100 • Just need the value of z • Can use the approximation formula: (ID)(1-FR)/ILDG(z) = standardized average number of units short upon receipt of order • Know ID = 25000, FR = 0.98, & ILD = 8100 • SoID(1-FR)/ILD = $25000(1-0.98)/8100 = 500/8100 = 0.0617 = G(z) • From table, G(1.15) = 0.0621, G(1.16) = 0.06086, so z = 1.15 or 1.16 • Plugging it in: S = ILD + zILD = 49000 + (1.15)8100 = $58,315

37. (Q, R) Q = 25,000, e.g., balances $25/transaction with 7% loss of interest in low yield account R = 15,120, e.g., to hit target of 98% fill rate Safety stock = R - LD = 15120 – 12000 = 3,120 (I, S) I = 6.25 days, e.g., balances $25/transaction with 7% loss of interest in low yield account S = 58,315, e.g., to hit target of 98% fill rate Safety stock = S - ILD = 58315 – 49000 = 9,315 Example continued When & How Much – (I,S) (Q, R) versus (I, S) • (I, S) higher safety stock (& more expensive) for same FR • Illustrates the cost of not continuously monitoring, so why use? • One reason - combine orders to gain delivery efficiencies, e.g., all products ordered every Friday & delivered together

38. Road map • Fundamental Problem Type #2 • Single Period Model • Policies for When & How Much • Summary

39. SL* = cs/(cs+ce) Fundamental E.g., shows up in setting order quantities setting base stock levels setting reorder points setting overbooking levels Number guessing – burned if guess too high or too low Reducing demand uncertainty leads to reduced inventory investment Buffer stock = zLD Each percentage increase in fill rate requires an increasing incremental investment in inventory Ouch!! Summary Managerial insights

40. Back to key questions • What is the cost trade-off captured by the single period model (SPM)? • In what settings is SPM analysis appropriate, & how can an order quantity be determined? • What is the definition of base stock level, & how does a base stock policy answer the questions of when & how much to order? • In what settings may a base stock policy be reasonable, & how can base stock be determined? • How do (Q, R) & (I, S) policies answer the questions of when & how much to order, when is each policy appropriate, & how can the values of policy parameters be determined ? • What is the difference between fill rate & service level, and how do these performance measures depend policy parameters? • How can the return on investments to reduce uncertainty be estimated? • What are the managerial insights from the chapter?