Inventory management course Material for All

Inventory Management

1. BIA 674 - Supply Chain Analytics 12. I n v e n t o r y M a n a g eme n t

2. Outline The I m p o r t a n c e of I n v entory Inventory Costs ABC An a l ysi s EOQ Mo d el s Probabilistic Models and Safety Stock Inventory Control Systems Single-Period Model Using Simulation for Inventory Management

3. Wh a t i s I n v entory? Stock of items kept to meet future demand for 🞑 internal customers 🞑 external customers Pur p ose of i n v entory m a n a g ement 🞑 ORDERI NG PO L I C Y : When a nd h o w m a n y uni ts to or d er of e a c h m a teri a l when or d ers a re p l a c ed wi th ei ther outside suppliers or production departments within organizations? 🞑 ISSUING POLICY: how to issue units from inventory? (FI F O , L I F O , r a n d om ? )

4. Importance of Inventory Inventories are important to all types of firms: 🞑 Th e y h a v e to b e c ounte d , pa i d f o r , used i n o p e r a ti on s , used to s a ti sfy c ustomer s , a nd m a n a g ed 🞑 T oo m u c h i n v entory re d u c es p r ofi t a b ili ty 🞑 Too little inventory damages customer confidence It is one of the most expensive assets of many c om p a ni es re p resenti ng a s m u c h a s 50 % of tot a l invested capital I t i s one of the 3 most c ommon re a sons f or SME bankruptcy Need to ba l a n c e i n v entory i n v estment a nd c ustomer se r vi c e

5. W h y Do W e W a nt to Hol d I n v entory I m p r o v e c ustomer se r vi c e Safe-guard to hazards in demand, supply, and delivery th a t mi g ht c a use sto c k-out T a k e ad v a nt a g e of e c onomi es of s c a l e , & re d u c e: 🞑 ordering costs 🞑 Stock-out costs 🞑 acquisition costs 🞑 Fi x ed c osts ( e . g . fi x ed or d eri ng c osts) Contribute to the efficient and effective operation of the p r o d u c ti on system, e . g . , 🞑 Reduces the number of costly set-ups and reschedulings 🞑 Smoothing and stabilizing resource utilization

6. W h y W e Do Not W a nt to Hol d I n v entory Ce r t a i n c osts i n c re a se su c h a s 🞑 Storage costs 🞑 i nsu r a n c e c osts 🞑 outdate costs 🞑 large-lot quality cost 🞑 c ost of p r o d u c ti on p r o b l ems Ties capital for which the company pays interest Hides productivity and quality problems Ri sk of g etti ng stu c k wi th uns a l ab l e g oo d s

7. T y p es of I n v entory ▶ R a w m a teri a l ▶ Pur c h a sed b ut not p r o c essed ▶ Work-in-process (WIP) ▶ Un d er g one some c h a n g e b ut not c om p l eted ▶ A function of cycle time for a product (e.g. items being transported) ▶ Maintenance/repair/operating (MRO) ▶ Necessary to keep machinery and processes productive ▶ Fi ni shed g oo d s ▶ Completed product awaiting shipment

8. The M a teri a l Fl o w C y c l e Input Wait for Wait to Move Wait in queue Setup Run Output inspection be moved time for operator time time Cycle time 95% 5%

9. Inventory and Service Quality Customers usu a ll y p er c ei v e q u a li ty se r vi c e a s availability of goods they want when they want them Inventory must be sufficient to provide high-quality c ustomer se r vi c e

10. Inventory Costs

11. Inventory-Related Costs Or d eri ng c osts (uni t v a ri ab l e c osts & fi x ed or d eri ng c osts) costs of replenishing inventory, placing orders, receiving goods costs for to prepare a machine or process for manufacturing an order Holding or Inventory carrying costs cost of holding an item in inventory over time Shortage or Stock-out / penalty costs 🞑 H o w d o y ou h a n d l e sho r t a g es? 🞑 L ost s a l es v s . ba c kl o g g i ng 🞑 Watch out for service level Out d a te c osts ( f or p eri sh a b l e p r o d u c ts) Opportunity costs

12. Holding Cos t s D e t e r m i n i n g I n v e n t o r y H o l d i n g C o s t s CATEGORY CO S T (AND RANGE) AS A P E R CENT O F I N V EN T O R Y V A L U E Housing costs (building rent or depreciation, o p e r a ti ng c ost s , t a x e s , i nsu r a n c e) 6% (3 - 10%) M a t er i a l h a nd li n g c o s t s (e q ui p ment l e a se or depreciation, power, operating cost) 3% (1 - 3.5%) L a bo r c o s t (re c ei vi n g , w a rehousi n g , se c uri ty) 3% (3 - 5%) I n v e s t me n t c o s t s ( b or r o wi ng c ost s , t a x e s , a nd i nsu r a n c e on i n v entory) 11% (6 - 24%) Pilferage, space, and obsolescence (much higher in i n d ustri es un d er g oi ng r ap i d c h a n g e li k e PCs a nd c el l phones) 3% (2 - 5%) O v e r a ll c a rr y i n g c o s t 26%

13. Holding Cos t s D e t e r m i n i n g I n v e n t o r y H o l d i n g C o s t s C A TEGO R Y CO S T (AND RANGE) AS A P E R CENT O F I N V EN T O R Y V A L U E Housing costs (building rent or depreciation, o p e r a ti ng c ost s , t a x e s , i nsu r a n c e) 6% (3 - 10%) M a t er i a l h a nd li n g c o s t s (e q ui p ment l e a se or depreciation, power, operating cost) 3% (1 - 3.5%) L a bo r c o s t (re c ei vi n g , w a rehousi n g , se c uri ty) 3% (3 - 5%) I n v e s t me n t c o s t s ( b or r o wi ng c ost s , t a x e s , a nd i nsu r a n c e on i n v entory) 11% (6 - 24%) Pilferage, space, and obsolescence (much higher in i n d ustri es un d er g oi ng r ap i d c h a n g e li k e PCs a nd c el l phones) 3% (2 - 5%) O v e r a ll c a rr y i n g c o s t 26% Holdi ng costs va ry consi derably d epending on the b usiness, l ocation, a nd intere st rates. Gene rally grea ter than 1 5%, some high tech and fa shion items have h olding co sts greate r than 40%.

14. ABC An a l ysi s

15. ABC An a l ysi s ▶ Pay attention to your more critical products! ▶ Di vi d es i n v entory i nto three c l a sse s ba sed on annual dollar volume ▶ Cl a ss A - hi g h a nnu a l d ol l a r v ol ume ▶ Cl a ss B - me d i um a nnu a l d ol l a r v ol ume ▶ Cl a ss C - l o w a nnu a l d ol l a r v ol ume ▶ Used to establish policies that focus on the few c ri ti c a l pa r ts a nd not the m a n y tri vi a l ones

16. Con c e p t: Al l i tems d o not d ese r v e the s a me a ttenti on i n te r ms of i n v entory m a n a g ement Focus on items that have the highest monetary value Step 1. Start with the inventoried items ranked by dollar value in inventory in descending order Step 2. Plot the cumulative dollar/euro value in i n v entory v ersus the c u m ul a ti v e i tems i n i n v entory ABC An a l ysi s

17. ABC An a l ysi s 1 2 3 4 5 6 7 8 9 100 Percentage of SKUs Percentage of dollar value 100 — 90 — 80 — 70 — 60 — 50 — 40 — 30 — 20 — 10 — — Class C Clas s A Class B Typical Chart Using ABC Analysis Cl a ss A 🞑 5 – 15 % of units 🞑 70 – 80 % of value Cl a ss B 🞑 30 % of units 🞑 15 % of value Cl a ss C 🞑 50 – 60 % of units 🞑 5 – 10 % of value

18. ABC An a l ysi s Ex a m p l e ABC Calculation (1) (2) (3) (4) (5) (6) (7) PERCENT OF PERCENT ITEM NUMBER ANNUAL ANNUAL OF ANNUAL STOCK OF ITEMS VOLUME UNIT DOLLAR DOLLAR NUMBER STOCKED (UNITS) x COST = VOLUME VOLUME CLASS #10286 #11526 20% 1,000 500 $ 90.00 154.00 $ 90,000 77,000 38.8% 33.2% 72% A A #12760 #10867 #10500 30% 1,550 350 1,000 17.00 42.86 12.50 26,350 15,001 12,500 11.3% 6.4% 5.4% 23% B B B #12572 600 $ 14.17 $ 8,502 3.7% C #14075 2,000 .60 1,200 .5% C #01036 50% 100 8.50 850 .4% 5% C #01307 1,200 .42 504 .2% C #10572 250 .60 150 .1% C 8,550 $232,057 100.0%

19. B Items | | | | | | | | | | 10 20 30 40 50 60 70 80 90 100 Percentage of inventory items Percentage of annual dollar usage 8 – A It ems 7 0 – 6 0 – 5 0 – 4 0 – 3 0 – 2 0 – 1 0 – 0 – C Items ABC An a l ysi s

20. ▶ Other c ri teri a th a n a nnu a l d ol l a r v ol ume m a y be used ▶ High shortage or holding cost ▶ Anticipated engineering changes ▶ Delivery problems ▶ Quality problems ABC An a l ysi s

21. ▶ P ol i c i es em p l o y ed m a y i n c l u d e More emphasis on supplier development for A items Tighter physical inventory control for A items More care in forecasting A items ABC An a l ysi s

22. EOQ Mo d el s

23. Ordering Policy under constant demand Si m p l e c a se Demand rate is constant and known with certainty Uni t or d eri ng c ost = C Every time an order is placed, there is a fixed cost = S There i s a uni t hol d i ng c ost = H No constraints are placed on the size of each order The l e a d ti me i s ze r o

24. t Company with steady rate of demand D = 100 tons/month Total annual demand = 1200 tons Pur c h a se p ri c e C = $250 /ton Del i v ery c osts S = $50 (e a c h ti me) Holding costs (storage, insurance, ...) H = $4/ton/month Inventory Level • • • Time Wh a t i s w r ong wi th thi s m a n a g ement?

25. Irrational Ordering (time, quantity) ... why? S a fety sto c k .. . w h y? Q T 2T t Get rid of pre-conceived ideas … 3T

26. I n v entory Us a g e O v er Ti me Order quantity = Q (maximum inventory level) Us a g e r a te Average i n v entory on hand QT 2 Minimum i n v entory I n v entory l e v el T ot a l or d er re c ei v ed 2T Time T 3T

27. O bjec t ive is t o minimize T o t al Annual Cost Annual cost Order quantity Total cost of ordering + holding inventory Holding cost Ordering (Setup) cost Minimum total cost Optimal order quantity ( Q *) Determining the optimal cycle

28. Minimizing Costs ▶ By mi ni mi zi ng the su m of setup (or or d eri n g ) and holding costs, total costs are minimized ▶ Optimal order size Q * will minimize total cost ▶ O p ti m a l or d er q u a nti ty o c c urs when: ▶ The d eri v a ti v e of the T ot a l Cost wi th res p e c t to the or d er q u a nti ty i s e q u a l to ze r o ▶ The hol d i ng c ost a nd setup c ost a re e q u a l

29. Calculating the Annual Costs Annu a l hol d i ng c ost Annual holding cost = (Average cycle inventory) (Unit holding cost) No of orders placed / year Annu a l or d eri ng c ost Annual ordering cost = (Ordering cost / order) No of orders placed / year Total annual cycle-inventory cost Total Annual costs = Annual holding cost + Annual ordering cost

30. H o l d i n g C o s t / p er i od The c ost of hol d i ng one uni t i n i n v entory f or one c y c l e = H ( Q T)/2 O r d er i n g C o s t / p er i od I t i s the c ost of or d eri ng one l ot wi th Q uni ts = CQ + S Calculating all the costs N o . o f o r d ers / y e a r = Annu a l Dem a nd / O d er Si ze = 12 D/Q T o t a l C o s t (C) I t i s the sum of a nnu a l hol d i ng a nd a nnu a l setup c ost

31. Calculating the EOQ T o t a l a nnu a l c y c l e- i n v e n t o r y c o s t Where TC = total annual cost C = unit ordering annual cycle-inventory cost Q = lot size H = holding cost per unit per period D = demand per period S = fixed ordering or setup costs per lot T = re-order period TC = N (S + CQ) + H QT 2 Fi x ed or d eri ng cost Variable ordering cost Holding cost

32. Calculating the EOQ T C = N[(S + CQ) + H( Q T/ 2 )]= = (12DS/Q) + (12D/Q)CQ + (12D/Q)(HQ 2 /2D)= =12DS/Q + 12DC + 6HQ To find the optimal Quantity Q: Set derivative w.r.t Q = Therefore, -(12DS/Q 2 )+6H = The optimal - order- quantity Q* = 2SD/H = 50 tons T = 0,5 month

33. Determine the optimal number of units to order D = 1,000 units per year S = $10 per order H = $.50 per unit per year Q *  2 D S H Q *  . 5 2 ( 1 , 000 )( 10 )  40 , 000  200 units An EOQ Ex a m p l e

34. Determine expected number of orders Q * = 200 units D = 1,000 units S = $10 per order H = $.50 per unit per year N = = 5 orders per year 1,000 200 = N = Expected number of orders Demand Order quantity D = Q * An EOQ Ex a m p l e

35. An EOQ Ex a m p l e Determine optimal time between orders Q * = 200 units N = 5 orders/year D = 1,000 units S = $10 per order H = $.50 per unit per year T = = 50 days between orders 250 5 = T = Expected time between orders Number of working days per year Expected number of orders

36. Dete r mi ne the total annual cost D = 1 , 000 uni ts S = $10 p er or d er H = $ . 50 p er uni t p er y e a r Q * = 200 units N = 5 orders/year T = 50 days T ot a l a nnu a l c ost = Setup c ost + Hol d i ng c ost TC  D S  Q H Q 2 1 , 00  20 ( $ 1 )  20 2 ( $. 5 )  ( 5 )( $ 1 )  ( 10 )( $. 5 )  $ 5  $ 5  $ 10 Note: the cost of materials is not included, as it is assumed that the demand will b e s a ti sfi ed a nd there f ore i t i s a fi x ed c ost An EOQ Ex a m p l e

37. Calculating EOQ E X AMP L E 1 A museum of natural history opened a gift shop which operates 52 w eeks p er y e a r . M a n a g i ng i n v entori es h a s b e c ome a problem. Top-selling SKU is a bird feeder. Sales are 18 units per week, the supplier charges $60 per unit. Ordering cost is $45. Annual holding cost is 25 percent of a feeder’s value. M a n a g ement c hose a 390 -uni t l ot si z e . What is the annual cycle-inventory cost of the current policy of usi ng a 390 -uni t l ot si ze? W oul d a l ot si ze of 468 b e b etter?

38. Calculating EOQ SOLUTION W e b e g i n b y c om p uti ng the a nnu a l d em a nd a nd hol d i ng c ost a s D = (18 units/week)(52 weeks/year) = 936 units H = 0.25($60/unit) = $15 The total annual cycle-inventory cost for the current policy is 2 Q Q D 390 936 C = ( H ) + ( S ) = ($15) + ($45) 2 390 = $2,925 + $108 = $3,033 The total annual cycle-inventory cost for the alternative lot size is C = 468 ($15) + 936 ($45) = $3,510 + $90 = $3,600 2 468

39. Calculating the EOQ 3000 – 2000 – 1000 – – | | | Annual cost (dollars) Current cost Lowest cost | | | | | 50 100 150 200 250 300 350 400 Best Q Lo t S i ze ( Q ) C u rre n t ( EOQ ) Q cost Total = Q ( H ) + D ( S ) 2 Q Total Annual Cycle-Inventory Cost Function for the Bird Feeder Ordering cost = D ( S ) Q Q Holding cost = 2 ( H )

40. Fi n d i ng the EOQ, T ot a l Cost, TBO E X AMP L E 2 For the bird feeders in Example 1, calculate the EOQ and its total annual cycle-inventory cost. How frequently will orders be p l a c ed i f the EOQ i s use d ? SOLUTION Usi ng the f o r m ul a s f or EOQ a nd a nnu a l c ost, w e g et EOQ = = 2 DS H = 74.94 or 75 units 2(936)(45) 15

41. Fi n d i ng the EOQ, T ot a l Cost, TBO The total annual cost is much less than the $3,033 cost of the current policy of placing 390-unit orders.

42. Fi n d i ng the EOQ, T ot a l Cost, TBO When the EOQ is used, the TBO can be expressed in various w a ys f or the s a me ti me p eri o d . E O Q E O Q TBO = D = 75 = 0.080 year 936 E O Q D EOQ 75 TBO = (12 months/year) = (12) = 0.96 month E O Q D EOQ 75 TBO = (52 weeks/year) = (52) = 4.17 weeks E O Q D EOQ 75 TBO = (365 days/year) = (365) = 29.25 days 936 936 936

43. Fi n d i ng the EOQ, T ot a l Cost, TBO E X AMP L E 3 Su p p ose th a t y ou a re revi e wi ng the i n v entory p ol i c i es on a n $80 item stocked at a hardware store. The current policy is to replenish i n v entory b y or d eri ng i n l ots of 360 uni t s . A d d i ti on a l i n f o r m a ti on i s: D = 60 units per week, or 3,120 units per year S = $30 per order H = 25% of selling price, or $20 per unit per year Wh a t i s the EOQ? SOLUTION EOQ = = 2 DS H 2(3,120)(30) = 97 units 20

44. Fi n d i ng the EOQ, T ot a l Cost, TBO What is the total annual cost of the current policy ( Q = 360), a nd h o w d oes i t c om p a re wi th the c ost wi th usi ng the EOQ? Current Policy EOQ Policy Q = 360 units Q = 97 units C = (360/2)(20) + (3,120/360)(30) C = (97/2)(20) + (3,120/97)(30) C = 3,600 + 260 C = 970 + 965 C = $3,860 C = $1,935

45. Fi n d i ng the EOQ, T ot a l Cost, TBO Wh a t i s the ti me b et w een or d ers (TBO) f or the c urrent p ol i c y a nd the EOQ p ol i c y , e x p ressed i n w eeks? 360 TBO = 360 (52 weeks per year) = 6 weeks E O Q TBO = 97 (52 weeks per year) = 1.6 weeks 3,120 3,120

46. Managerial Insights SENSITIVITY ANALYSIS OF THE EOQ Parameter EOQ Parameter Change EOQ Change Comments Demand 2 D S H ↑ ↓ ↓ ↑ ↓ ↑ Increase in lot size is in proportion to the square root of D . O r d er / Set u p Costs 2 D S H Weeks of supply decreases and inventory turnover increases because the lot size decreases. Holding Costs 2 DS H Larger lots are justified when holding costs decrease.

47. Robustness ▶ The EOQ mo d el i s r o b ust ▶ I t w o r ks e v en i f a l l pa r a meters a nd a ssum p ti ons a r e not met ▶ The tot a l c ost c u r v e i s rel a ti v el y fl a t i n the area of the EOQ

48. Introducing delivery lag EOQ a n s w ers the “ h o w m u c h” q uesti on The reor d er p oi nt (ROP) tel l s “ when” to or d er L e a d ti me ( L ) i s the ti me b et w een p l a c i ng a nd receiving an order ROP = Lead time for a new order in days Demand per day = d x L d = D Number of working days in a year

49. R eor d er P oi nt Cu r v e Q * ROP (units) Inventory level (units) Time (days) Lead time = L Resupply takes place as order arrives Slope = units/day = d

50. R eor d er P oi nt Ex a m p l e Demand = 8,000 iPods per year 250 working day year Lead time for orders is 3 working days, but it may also take 4 days d = D Number of working days in a year = 8,000/250 = 32 units ROP = d x L = 32 units per day x 3 days = 96 units = 32 units per day x 4 days = 128 units

51. I nt r o d u c i ng v ol ume d i s c ounts A company buys re-writable DVDs (10 disks / box) from a large mail-order distributor The company uses approximately 5,000 boxes / year a t a f a i r l y c onst a nt r a te The d i stri b utor offers the f ol l o wi ng q u a nti ty d i s c ount schedule: 🞑 I f < 5 00 b o x es a re or d ere d , then c ost = $10 / b o x 🞑 If >500 but <800 boxes are ordered, then cost = $9.50 🞑 I f > 8 00 b o x es a re or d ere d , then c ost = $9 . 25 Fi x ed c ost of p ur c h a si ng = $25 , a nd the c ost of capital = 12% per year. There is no storage cost.

52. Sol v e 3 EOQ mo d el s Each one will hold for the corresponding region; if it does not c orres p on d , c hoose the l o w est one th a t d oes Sel e c t the one wi th the l o w est c ost S t e p s i n a n a l y z i n g a q u a n t i t y d i s c oun t F or e a c h d i s c ount, c a l c ul a te Q * If Q * for a discount doesn’t qualify, choose the lowest possible quantity to get the discount Compute the total cost for each Q * or adjusted value from Step 2 Select the Q * that gives the lowest total cost I nt r o d u c i ng v ol ume d i s c ounts

53. Qu a nti ty Di s c ount Mo d el s 500 800 Order quantity Total cost $ Q * for discount 2 is below the allowable range at point a and must be adjusted upward to 500 units at point b a b 1st price break 2nd price break Total cost curve for discount 1 Total cost curve for discount 2 Total cost curve for discount 3

54. A company is a mail-order distributor of audio CDs Th e y sel l ab out 50 , 000 CDs / y e a r E a c h CD i s pa c k a g ed i n a j e w el b o x th e y b uy f r om a su p p li er Fixed cost for an order of boxes = $100; variable cost = $0.50, storage cost = $0.50/unit/year, and cost of money is 10% The company assumes that shortages are allowed, and lost demand is backlogged … it just gets to the customer a little later (!) The company assigns a “penalty” of $1 for every week that a box is delivered late, so annual shortage cost (penalty) p = $52/unit. Allowing shortages

55. Allowing shortages Q/D b ( Q -b) / D b/D Allow shortages up to b units Order quantity Q Order-up-to inventory = Q-b Reorder period = Q/D Period with I>0 = (Q-b)/D Period with I<0 = b/D T ime Inventory Level Q-b

56. Total Annual Cost = Ordering + Shortage + Holding costs Ordering cost = N [(S+CQ) + (pb(b/D)/2) + H(Q-b)((Q-b)/D)/2] Si n c e N = D/Q, w e h a v e T ot a l Annu a l Cost = DS/Q + CD + ( p b 2 / 2 Q) +H(Q- b ) 2 / 2 D T o mi ni mi z e , t a k e d eri v a ti v e = , a nd sol v e  hQ 2 – HbQ – (DS+pb 2 ) = Allowing shortages

57. Production Order Quantity Model Used when i n v entory b ui l d s up o v er a p eri od of ti me a fter a n or d er i s p l a c ed Used when uni ts a re p r o d u c ed a nd sol d simultaneously Inventory level Ti me Part of inventory cycle during which p r o d u c ti on ( a nd us a g e) i s t a ki ng p l a c e Dem a nd pa r t of c y c l e wi th no p r o d u c ti on (onl y us a g e) t M a xi m um inventory

58. Production Order Quantity Model Q =Number of pieces per order H =Holding cost per unit per year p = Daily production rate d = Daily demand/usage rate t =Length of the production run in days = (Average inventory level) x Annual inventory holding cost Holding cost per unit per year = (Maximum inventory level)/2 Average inventory level = – Maximum inventory level Total items produced during the production run Total items used during the production run = pt – dt

59. Production Order Quantity Model Q =Number of pieces per order H =Holding cost per unit per year p = Daily production rate d = Daily demand/usage rate t =Length of the production run in days = – Total used during the production run Maximum inventory level Total produced during the production run = pt – dt However, Q = total produced = pt ; thus t = Q / p Maximum inventory level = p p – d Q = Q 1 – d Q p p Holding cost = ( H ) = 1 – H d p Q 2 Maximum inventory level 2

60. Production Order Quantity Model Q =Number of pieces per order H =Holding cost per unit per year p = Daily production rate d = Daily demand/usage rate t =Length of the production run in days Setu p c o st = ( D / Q ) S 1 2 Holding cost = HQ 1 d p       D Q 1 2 S  HQ 1 d p       2 Q  2 D S H  1   d p     p Q *  2 D S H  1   d p     Q *  2 D S H Remember, with no production t a ki ng p l a c e

61. Production Order Quantity Example D =1,000 units S =$10 H =$0.50 per unit per year p = 8 units per day d = 4 units per day Q p *  2 D S H   1   d p    2 ( 1 , 00 ) ( 1 ) Q p *  0.50  1  ( 4 8 )   2 , 000 0.50(1 2)  8 , 000  282.8 units, or 283 units

62. Production Order Quantity Model When annual data are used the equation becomes Note: d = 4 = = D Number of days the plant is in operation 1,000 250 p Q *  2 D S Annual demand rate   H  1  Annua l p r odu c t i o n r at e   

63. Probabilistic Models and Safety Stock

64. Probabilistic Models and Safety Stock ▶ Dem a nd i s often UNCER T AI N ▶ The p r o b l em app e a rs when there i s L EAD TI ME, L ▶ W e h a v e to set t w o pa r a meters th a t d efi ne our or d eri ng p ol i c y: R eor d er P oi nt ( RO P ) a nd S a fety Sto c k ( s s ) ▶ Y ou reor d er when y our i n v entory f a ll s on or b el o w ROP ▶ Use safety stock to achieve a desired service level and avoid stockouts ROP = d x L + ss Expected Annual stockout costs = (expected units short/ cycle) x the stockout cost/unit x the number of orders per year

65. S a fet y Sto c k Ex a m p l e N U M B ER O F U N I TS (d X L ) PROBABILITY 30 .2 40 .2 Current ROP  50 .3 60 .2 70 .1 1.0 Current p ol i c y: ROP = 50 uni ts Orders per year = 6 Sto c k out c ost = $40 / uni t C a rryi ng c ost = $5 / uni t / y e a r Probability distribution for inventory demand during lead time How much safety stock should we keep and added to 50 (current ROP)?

66. S a fet y Sto c k Ex a m p l e ROP = 50 uni ts Orders /year = 6 Sto c k out c ost = $40 / uni t C a rryi ng c ost = $5 /uni t/ y e a r S AFETY STOCK ADDITIONAL H O L D I NG CO S T S T O C K O U T CO S T T O T AL COST 20 (20)($5) = $100 $0 $100 10 (10)($5) = $ 50 (10)(.1)($40)(6) = $240 $290 $ 0 (10)(.2)($40)(6) + (20)(.1)($40)(6) = $960 $960 A s a fety sto c k of 20 uni ts g i v es the l o w est tot a l c ost ROP = 50 + 20 = 70 f r a mes

67. Probabilistic Demand Use p res c ri b ed se r vi c e l e v el s to se t s a fet y sto c k when the cost of stockouts cannot be determined ROP = d em a nd d uri ng l e a d ti me + Z  d L T Where: Z = Number of standard deviations  dLT = Standard deviation of demand during lead time

68. Sa f e t y stock Probability of no stockout 95% of the time Mean demand 350 ROP = ? kits Quantity Number of standard deviations z Risk of a stockout (5% of area of normal curve) Probabilistic Demand

69.  = Average demand = 350 kits  dLT = Standard deviation of demand during lead time = 10 kits Z = 5% stockout policy (service level = 95%) Using Normal distribution tables, for an area under the curve of 95%, the Z = 1.65 Safety stock = Z  dLT = 1.65(10) = 16.5 kits Reorder point = Expected demand during lead time + Safety stock = 350 kits + 16.5 kits of safety stock = 366.5 or 367 kits Probabilistic Demand

70. Safety stock 16.5 units ROP  Place order Probabilistic Demand Inventory level T ime Normal distribution probability of demand during lead time Expected demand during lead time (350 kits) M i n i m u m d em a nd du r i n g l e a d t i me Maximum demand during lead time Mean demand during lead time ROP = 350 + safety stock of 16.5 = 366.5 order Lead Receive time

71. Other Probabilistic Models ▶ When data on demand during lead time is not available, there are other models available When demand is variable and lead time is constant When lead time is variable and demand is constant When both demand and lead time are variable

72. Other Probabilistic Models: Variable demand, constant lead time Demand is variable and lead time is constant ROP = ( Average daily demand x Lead time in days) + Z  dLT where  dLT =  d Lead time  d = standard deviation of demand per day

73. Average daily demand (normally distributed) = 15 Lead time in days (constant) = 2 Standard deviation of daily demand = 5 Service level = 90% Z for 90% = 1.28 F rom Appendix I ROP = (15 units x 2 days) + Z  dLT = 30 + 1.28(5)( 2) = 30 + 9.02 = 39.02 ≈ 39 Safety stock is about 9 computers Other Probabilistic Models: Variable demand, constant lead time

74. ROP = (Daily demand x Average lead time in days) + Z x (Daily demand) x  LT where  LT = Standard deviation of lead time in days Other Probabilistic Models: Constant demand, variable lead time

75. Daily demand (constant) = 10 Average lead time = 6 days Standard deviation of lead time =  LT = 1 Service level = 98%, so Z (from Appendix I) = 2.055 ROP = (10 units x 6 days) + 2.055(10 units)(1) = 60 + 20.55 = 80.55 Reorder point is about 81 cameras Other Probabilistic Models: Constant demand, variable lead time

76. ROP = (Average daily demand x Average lead time) + Z  dLT where  d = S t andard devia t ion of demand per day  L T = S t andard devia t ion of lead t ime in days  d L T = (Average lead time x  2 ) d + (Average daily demand) 2   LT Other Probabilistic Models: Variable demand, variable lead time

77. Average daily demand (normally distributed) = 150 Standard deviation =  d = 16 Average lead time 5 days (normally distributed) Standard deviation =  LT = 1 day Service level = 95%, so Z = 1.65 (from Normal tables) R O P  ( 15 pa ck s  5 da y s )  1 . 6 5  dL T  dLT   5 da y s  1 6 2    15 2  1 2    5  25 6    2 2 , 50  1    1 , 28    2 2 , 50   2 3 , 78  15 4 R O P  ( 15  5 )  1 . 6 5 ( 15 4 )  75  25 4  1 , 00 4 pa cks Other Probabilistic Models: Variable demand, variable lead time

78. Inventory Control Systems

79. Inventory Control Systems Conti nuous revi e w ( Q ) system 🞑 R eor d er p oi nt system (ROP) a nd fi x ed or d er quantity system 🞑 F or i n d e p en d ent d em a nd i tems (i . i . d . ) 🞑 T r a c ks i n v entory p osi ti on ( IP ) 🞑 Includes scheduled receipts ( SR ), on-hand inventory ( OH ), and back orders ( BO ) Inventory position = On-hand inventory + Scheduled receipts – Backorders IP = OH + SR – BO

80. Selecting the Reorder Point T im e On-hand inventory TBO TBO L L TBO L Order placed Order placed Order placed IP IP IP R Q Q Q OH OH OH Order received Order received Order received Order received Q System When Demand and Lead Time Are Constant and Certain

81. Conti nuous R evi e w Systems The on-h a nd i n v entory i s onl y 10 uni t s , a nd the reor d er p oi nt R i s 100 . There a re no ba c k or d er s , b ut there i s one o p en or d er f or 200 uni t s . Shoul d a n e w or d er b e placed? SOLUTION IP = OH + SR – BO = 10 + 200 – = 210 R = 100 Decision: Do not place a new order

82. Conti nuous R evi e w Systems R e o r d er P o i n t L e v e l : Assumi ng th a t the d em a nd r a te p er p eri od a nd the lead time are constant, the level of inventory at which a new order is placed (reorder point) can be c a l c ul a ted a s f ol l o ws: R = dL Where d = demand rate per period L = lead time R emem b er: The or d er q u a nti ty Q i s the EOQ!

83. Conti nuous R evi e w Systems E X AMP L E 4 Demand for chicken soup at a supermarket is always 25 cases a day and the lead time is always 4 days. The shelves were just restocked with chicken soup, leaving an on-hand inventory of only 10 cases. No backorders currently exist, but there is one open order in the pipeline for 200 cases. What is the inventory p osi ti on? Shoul d a n e w or d er b e p l a c e d ? SOLUTION R = T o tal d ema n d du r in g l ead t i me = (25)(4) = 100 cases IP = OH + SR – BO = 10 + 200 – = 210 cases Decision: Do not place a new order

84. Conti nuous R evi e w Systems Selecting the reorder point with variable demand a nd c onst a nt l e a d ti me Reorder point = Average demand during lead time + Safety stock = dL + safety stock where d = average demand per week (or day or months) L = constant lead time in weeks (or days or months)

85. Conti nuous R evi e w Systems (uncertain demand) T im e On-hand inventory TBO 1 TBO 2 TBO 3 L 1 L 2 L 3 R Order received Q Order placed Order placed IP Order received IP Q Order placed Q Order received Order received IP Q System When Demand Is Uncertain

86. H o w to d ete r mi ne the R eor d er P oi nt Choose an appropriate service-level policy 🞑 Select service level or cycle service level 🞑 Protection interval Determine the demand during lead time probability distribution Determine the safety stock and reorder point levels

87. Dem a nd Duri ng L e a d Ti me S p e c i fy me a n d a nd st a n d a rd d evi a ti on σ d f or the d em a nd (ty p i c a ll y these v a l ues a re g i v en) Calculate standard deviation of demand during lead time L σ dLT = σ d L = σ d L 2 Then, the s a fety sto c k a nd reor d er p oi nt a re Safety stock = z σ dLT where z = number of standard deviations needed to achieve the cycle-service level (found from tables) σ dLT = stand deviation of demand during lead time Reorder point = R = dL + safety stock

88. Dem a nd Duri ng L e a d Ti me A vera ge demand during lead time Cycle-service level = 85% Probability of stockout (1.0 – 0.85 = 0.15) R z σ dLT Finding Safety Stock with a Normal Probability Distribution for an 85 Percent Cycle-Service Level

89. R eor d er P oi nt f or V a ri ab l e Dem a nd E X AMP L E 5 L et us retu r n to the b i rd fee d er i n Ex a m p l e 2 . The EOQ i s 75 uni t s . Suppose that the average demand is 18 units per week with a standard deviation of 5 units. The l e a d ti me i s c onst a nt a t t w o w eek s . Dete r mi ne the s a fety sto c k a nd reor d er p oi nt i f m a n a g ement w a nts a 90 p er c ent c y c l e-se r vi c e l e v el .

90. R eor d er P oi nt f or V a ri ab l e Dem a nd SOLUTION In this case, σ d = 5, d = 18 units, and L = 2 weeks, so σ dLT = σ d L = 5 2 = 7.07. Consult the body of the table in the Normal Distribution appendix for 0.9000, which corresponds to a 90 percent cycle-service level. The closest number is 0.8997, which corresponds to 1.2 in the row heading and 0.08 in the column heading. Adding these values gives a z value of 1.28. With this information, we calculate the safety stock and reorder point as follows: Safety stock = z σ dLT = 1.28(7.07) = 9.05 or 9 units Reorder point = dL + Safety stock = 2(18) + 9 = 45 units

91. R eor d er P oi nt f or V a ri ab l e Dem a nd E X AMP L E 6 Suppose that the demand during lead time is normally distributed with an average of 85 and σ dLT = 40. Find the safety stock, and reorder point R , for a 95 and 85 percent cycle-service level. SOLUTION Safety stock = z σ dLT = 1.645(40) = 65.8 or 66 units R = Average demand during lead time + Safety stock R = 85 + 66 = 151 units Find the safety stock, and reorder point R , for an 85 percent cycle-service level. Safety stock = z σ dLT = 1.04(40) = 41.6 or 42 units R = Average demand during lead time + Safety stock R = 85 + 42 = 127 units

92. R eor d er P oi nt f or V a ri ab l e Dem a nd & Variable Lead Time σ dLT = Lσ d + d σ LT 2 2 2 Often the c a se th a t b oth a re v a ri ab l e The equations are more complicated Safety stock = z σ dLT R =(Average weekly demand  Average lead time) + Safety stock = dL + Safety stock where d =Average weekly (or daily or monthly) demand L =Average lead time σ d = Standard deviation of weekly (or daily or monthly) demand σ LT = Standard deviation of the lead time

93. R eor d er P oi nt f or V a ri ab l e Dem a nd & Variable Lead Time E X AMP L E 7 The Office Supply Shop estimates that the average demand for a p o p ul a r ba ll - p oi nt p en i s 12 , 000 p ens p er w eek wi th a st a n d a rd d evi a ti on of 3 , 000 p en s . The c urrent i n v entory p ol i c y c a ll s f or re p l eni shment or d ers of 156 , 000 p en s . The a v e r a g e l e a d ti me f r om the d i stri b utor i s 5 w eek s , wi th a st a n d a rd deviation of 2 weeks. If management wants a 95 percent cycle- se r vi c e l e v el , wh a t shoul d the reor d er p oi nt b e?

94. R eor d er P oi nt f or V a ri ab l e Dem a nd & Variable Lead Time SOLUTION We have d = 12,000 pens, σ d = 3,000 pens, L = 5 weeks, and σ LT = 2 weeks σ dLT = Lσ d + d σ LT = (5)(3,000) + (12,000) (2) 2 2 2 2 2 2 = 24,919.87 pens From the Normal Distribution appendix for 0.9500, the appropriate z value = 1.65. We calculate the safety stock and reorder point as follows: Safety stock = z σ dLT = (1.65)(24,919.87) = 41,117.79 or 41,118 pens Reorder point = dL + Safety stock = (12,000)(5) + 41.118 = 101,118 pens

95. R eor d er P oi nt f or V a ri ab l e Dem a nd & Variable Lead Time E X AMP L E 8 Gr e y W ol f l o d g e i s a p o p ul a r 500 - r oom hotel i n the No r th W oo d s . M a n a g ers need to k eep c l ose t a b s on a l l of the r oom service items, including a special pint-scented bar soap. The da il y d em a nd f or the so a p i s 275 ba r s , wi th a st a n d a rd deviation of 30 bars. Ordering cost is $10 and the inventory holding cost is $0.30/bar/year. The lead time from the supplier i s 5 d a y s , wi th a st a n d a rd d evi a ti on of 1 d a y . The l o d g e i s open 365 days a year. Wh a t shoul d the reor d er p oi nt b e f or the ba r of so a p i f m a n a g ement w a nts to h a v e a 99 p er c ent c y c l e-se r vi c e?

96. R eor d er P oi nt f or V a ri ab l e Dem a nd & Variable Lead Time SOLUTION d = 27 5 bars L = 5 days σ d = 3 0 bars σ L T = 1 day Lσ d + d σ LT = 283.06 bars 2 2 2 σ dLT = From the Normal Distribution appendix for 0.9900, z = 2.33. We calculate the safety stock and reorder point as follows; Safety stock = z σ dLT = (2.33)(283.06) = 659.53 or 660 bars Reorder point + safety stock = dL + safety stock = (275)(5) + 660 = 2,035 bars

97. P eri o d i c R evi e w (or fi x ed p eri o d ) System ( P ) Fixed interval reorder system or periodic reorder system Four of the original EOQ assumptions maintained 🞑 No c onst r a i nts a re p l a c ed on l ot si ze 🞑 Holding and ordering costs 🞑 Independent demand 🞑 L e a d ti mes a re c e r t a i n Order is placed to bring the inventory position up to the target inventory level, T , when the predetermined time, P , has elapsed ▶ Only relevant costs are ordering and holding ▶ Lead times are known and constant ▶ Items are independent of one another

98. P eri o d i c R evi e w Systems On-hand inventory Q 1 Q 2 Target quantity ( T ) P Q 3 Q 4 P P T ime

99. ▶ Inventory is only counted at each review period ▶ May be scheduled at convenient times ▶ Appropriate in routine situations ▶ May result in stockouts between periods ▶ May require increased safety stock P eri o d i c R evi e w Systems

100. P eri o d i c R evi e w System ( P ) P P T L L L T im e On-hand inventory IP 1 IP 3 IP 2 Order placed Order placed Order received Order received Order received IP IP IP OH OH Q 1 Q 2 Q 3 Protection interval P System When Demand Is Uncertain

101. H o w Mu c h to Or d er i n a P System E X AMP L E 9 A distribution center has a backorder for five 36-inch color TV sets. No i n v entory i s c urrentl y on h a n d , a nd n o w i s the ti me to revi e w . H o w m a n y shoul d b e reor d ered i f T = 400 a nd no re c ei p ts a re s c he d ul e d ? SOLUTION IP = OH + SR – BO = + – 5 = –5 sets T – IP = 400 – (–5) = 405 sets That is, 405 sets must be ordered to bring the inventory position up to T sets.

102. H o w Mu c h to Or d er i n a P System E X AMP L E 10 The on-h a nd i n v entory i s 10 uni t s , a nd T i s 400 . There a re no back orders, but one scheduled receipt of 200 units. Now is the ti me to revi e w . H o w m u c h shoul d b e reor d ere d ? SOLUTION IP = OH + SR – BO = 10 + 200 – = 210 T – IP = 400 – 210 = 190 The decision is to order 190 units

103. P eri o d i c R evi e w System Sel e c ti ng the ti me b et w een revi e w s , c hoosi ng P a nd T Sel e c ti ng T when d em a nd i s v a ri ab l e a nd l e a d ti me i s constant IP covers demand over a protection interval of P + L Th e average demand during the protection interval is d(P + L) , or T = d ( P + L ) + safety stock for protection interval Safety st o ck = zσ P + L , where σ P + L =  d P  L

104. Calculating P and T E X AMP L E 11 Again, let us return to the bird feeder example. Recall that d em a nd f or the b i rd fee d er i s no r m a ll y d i stri b uted wi th a me a n of 18 units per week and a standard deviation in weekly demand of 5 units. The lead time is 2 weeks, and the business operates 52 weeks per year. The Q system developed in Example 5 called for an EOQ of 75 units and a safety stock of 9 units for a cycle-service level of 90 percent. What is the equivalent P system? Answers are to be rounded to the nearest integer.

105. Calculating P and T SOLUTION We first define D and then P . Here, P is the time between reviews, expressed in weeks because the data are expressed a s d em a nd p er w eek: D = (18 units/week)(52 weeks/year) = 936 units P = EOQ (52) = 75 (52) = 4.2 or 4 weeks D 936 With d = 18 units per week, an alternative approach is to calculate P by dividing the EOQ by d to get 75/18 = 4.2 or 4 weeks. Either way, we would review the bird feeder inventory every 4 weeks.

106. Calculating P and T We now find the standard deviation of demand over the protection interval ( P + L ) = 6: Before calculating T , we also need a z value. For a 90 percent cycle-service level z = 1.28. The safety stock becomes Safety stock = zσ P + L = 1.28(12.25) = 15.68 or 16 units We now solve for T : T = Average demand during the protection interval + Safety stock = d ( P + L ) + safety stock = (18 units/week)(6 weeks) + 16 units = 124 units P  L  5 6  1 2 . 2 5 u n i t s  P  L   d

107. Comparative Advantages Primary advantages of P systems 🞑 Convenient 🞑 Or d ers c a n b e c om b i ned 🞑 Onl y need to kn o w I P when revi e w i s m a d e Primary advantages of Q systems 🞑 Review frequency may be individualized 🞑 Fi x ed l ot si zes c a n resul t i n q u a nti ty d i s c ounts 🞑 L o w er s a fety sto c ks

108. Single Period Model

109. Single-Period Model ▶ Only one order is placed for a product ▶ Uni ts h a v e li ttl e or no v a l ue a t the end of the s a l es period ▶ Newsboy problem 1: ▶ Dem a nd = 500 pap ers/ d a y , σ = 100 pap er ▶ Cost to n e ws b o y = 10 c , S a l es p ri c e = 30 c ▶ H o w m a n y pap ers shoul d he or d er?

110. The n e ws b o y p r o b l em ▶ I f he sel l s a pap e r , he m a k es a p r ofi t = 20 c ▶ I f he d oes n ’ t sel l a pap er he m a k es a l oss = 10 c ▶ I f he or d ers x pap er s , on the i -th pap er he m a k es an expected profit: ▶ E(Profit) i = 20p – 10(1-p), where p=sale of i-th paper ▶ Bre a k e v en o c c urs when the Ex p e c ted p r ofi t = ▶ S o , 20p – 10 ( 1 - p ) = , a nd there f ore p * = 1 /3 ▶ By l ooki ng a t the No r m a l t a b l e s , Z = . 431 p 1-p  = 500 Optimal stocking level

111. The n e ws b o y p r o b l em ▶ Therefore: ▶ Z = . 431 = ( X * - μ )/ σ = ( X *- 5 00 )/ 1 00 ▶ X* = 543 1-p  = 500 Optimal stocking level

112. A sm a l l v a ri a ti on ▶ Assume that he can return the paper, if unsold for 5c each ▶ E(Profit) i = 20p – 5(1-p), where p=sale of i-th paper ▶ Bre a k e v en o c c urs when the Ex p e c ted p r ofi t = ▶ S o , 20p – 5 ( 1 - p ) = , a nd there f ore p * = 1 /5 ▶ By l ooki ng a t the No r m a l t a b l e s , Z = . 842 ▶ Then, X * = 584 ▶ In general, if MR = Marginal Return and ML = Marginal L os s , then p MR - ( 1 - p ) ML = ▶ p* = ML/(MR+ML)

113. Usi ng Si m ul a ti on f or sto c h a sti c i n v entory management

114. What is Simulation ? Si m ul a ti on i s a mo d el – c om p uter c o d e – “ i mi t a ti n g ” the o p e r a ti on of a re a l syste m i n the c om p ute r . I t c onsi st s of: A set of variables representing the basic features of the re a l syste m a nd A set of logical commands in the computer that modify these features as a function of time in accordance with the rules (logical of physical) regulating the real system.

115. The capacity to "advance time" through the use of a simulation build-in clock that monitors and the events while stepping up real time The capacity of drawing samples through the creation of a r ti fi c i a l o b se r v a ti ons th a t b eh a v e "l i k e" r a n d om e v ents i n the real system 🞑 a ) Cre a ti on of r a n d om num b ers (i n d e p en d ent & uni f o r ml y distributed) by the computer (according to an internal algorithm - function) 🞑 Con v ersi on i n the o b se r v a ti on s ’ d i stri b uti on Main Features of a Simulation System

116. Examples of applications Very important tool for Service management - Analysis of queuing systems Business Process Reengineering Strategic planning Financial planning Industrial design (e.g. chemical plants) Short term production planning Quality and reliability control, Training – business games, etc

117. Let us suppose that the weekly demand of a product can take the following values with the respective probabilities: Creating random numbers according to a probability distribution Demand (D) Probability of Demand P(D) Cumulative Distribution of Demand F(D) 1,000 0.20 0.20 1,500 0.10 0.30 2,000 0.30 0.60 2,500 0.25 0.85 3,000 0.15 1.00

118. D 1 D 2 .,30 0.20 0.60 1.00 0.85 1.000 1,500 2,000 2,500 3,000 R 1 R 2 The cumulative probability distribution F

119. Example Week U n if. Ra ndo m Number Week's Demand 1 32 2,000 2 8 1,000 3 46 2,000 4 92 3,000 5 69 2,500 6 71 2,500 7 29 1,500 8 46 2,000 9 80 2,500 10 14 1,000 Demand (D) Cumulative Distribution of Demand F(D) 1,000 0.20 1,500 0.30 2,000 0.60 2,500 0.85 3,000 1.00 Remember, the Cumulative Distribution F C r ea te a "de mand serie s “ c o rr e s pond i ng to th e r andom numbe r s generated

120. Usi ng Si m ul a ti on to d efi ne a n Inventory Policy Assume t hat a company is in t eres t ed t o implement an (s, S) ordering policy. Determine values of s and S: s = Safety stock S = Order-up-to quantity Inventory Behavior (s = 200, S = 700) 800 700 600 500 400 300 200 100 1 2 3 4 5 6 7 8 9 10 11 12 13 Weeks Demand Inventory levels Series2 Series1

121. Demand (D) Probability of Demand P(D) Cumulative Distribution of Demand F(D) 1,000 0.20 0.20 1,500 0.10 0.30 2,000 0.30 0.60 2,500 0.25 0.85 3,000 0.15 1.00 K e y a ssum p ti ons: When < the s a fety sto c k s , or d er up to the reor d er p oi nt S When short, make emergency order for quantity short No r m a l or d eri ng c ost = 200 + 10 • q u a nti ty Emer g en c y or d eri ng c ost = 500 + 15 • q u a nti ty L efto v er i n v entory c ost = 3 • q u a nti ty Application to our problem

122. Fl o w c h a r t S T ART OF SI MU L A TI ON Define initial conditions Define strategy parameters s, S Next week, t = t+1 Si m ul a ti on c om p l ete? STOP Com p ute A v e r a g e Weekly cost Com p ute other p er f o r m a n c e c ri teri a Need to order? Or d er a mount S-I Update inventory to S P a y or d eri ng c ost Update total cost Create demand D for this week I s d em a nd s a ti sfi e d ? Update inventory level P a y hol d i ng c ost Update total cost Pl a c e emer g en c y or d er Update inventory level P a y emer g en c y c ost Update total cost

123. Manual Simulation of Inventory System Assume: s = 1,500 and S = 2,500 Week Starting Invent o ry Need to order? S i ze of order A va il a ble invent o ry Week's Demand Emergency order? S i ze of emerg. order Ending Invent o ry W eek ly Cost 1 2,000 no 2,000 2,000 no 2 yes 2,500 2,500 2,000 no 500 26,700 3 500 yes 2,000 2,500 3,000 yes 500 30,500 4 yes 2,500 2,500 1,000 no 1,500 29,700 5 1,500 yes 1,000 2,500 2,500 no 10,200 6 yes 2,500 2,500 2,500 no 25,200 7 yes 2,500 2,500 3,000 yes 500 38,000 8 yes 2,500 2,500 1,500 no 1,000 28,200 9 1,000 yes 1,500 2,500 2,000 no 500 16,700 10 500 yes 2,000 2,500 2,500 no 20,200 Average 389 always 2,111 2,500 2,200 20% 500 389 25,044