1 ANALYSIS OF INVENTORY MODEL Notes 2 of 2 By: Prof. Y.P. Chiu 2011 / 09 / 01

2 § I12 : Inventory model: when demand rate λ when demand rate λ is not constant is not constant Periodic review ~ A general model for Production Planning Terms: Periods: 1,2,3,….N i : demand rate in period i h : holding cost / item / period K : setup cost c : unit cost : cost of producing enough items for period i thru. j at beginning of period i

3 (B) Formula...[Eq.12.1] …...[Eq.12.2] Lowest cost from period i to N that will satisfies demand § I12 : Inventory model: when demand rate λ when demand rate λ is not constant is not constant

4 (C) Demand 1Q 2Q 3Q 4Q P 1 P 2 P 3 P 4 λ 1 λ 2 λ 3 λ 4 X 2 X 3 X 4 [Eg.12.1] ～ When demand rate λ is not constant ～ § I12 : Inventory model: when demand rate λ when demand rate λ is not constant is not constant

5 Use [Eq.12.2] [Eg.12.1] ～ When demand rate λ is not constant ～ § I12 : Inventory model: when demand rate λ when demand rate λ is not constant is not constant

6 [Eg.12.1] ～ When demand rate λ is not constant ～ § I12 : Inventory model: when demand rate λ when demand rate λ is not constant is not constant

7 C 1 =Min

8 [Answer] To produce enough items from 1st period to 2nd period, then To produce enough items from 3rd period to 4th period. In other words, production plan is: “ to produce 5000 items at the beginning of the first period, then to produce 5500 items at the beginning of the 3 rd period ”. [ Eg.12.1 ] When demand rate λis not constant § I12 : Inventory model: when demand rate λis not constant when demand rate λis not constant

K=$20 C=$0.1 h=$0.02 §. I12: Problems & Discussion Preparation Time : 15 ~ 20 minutes Discussion : 10 ~ 20 minutes #C.4 #C.5

10 § I13 : Inventory Model: Resource-Constrained Multiple Product System [Ex.13.1] Item λ j C j $50 $350 $85 K j $100 $150 $50

11 [Eq.13.1] Must run under budget → EOQ 1 = 172, EOQ 2 = 63, EOQ 3 = 61 $50(172) + $350(63) + $85(61) = $35,835 (over-budget) Adjusting Factor …..Eq.13.2] (Budget) § I13 : Resource-Constrained Multiple Product System ……[Eq.13.1]

12 §. I13: Problems & Discussion Preparation Time : 25 ~ 30 minutes Discussion : 15 ~ 25 minutes ( # C.6 ) ( # C.6 ) ( # N4.38(a) )

13 Check : w i / h i w 1 / h 1 = 9 / 12.5 = 0.72 w 2 / h 2 = 12 / 87.5 = 0.14 w 3 / h 3 = 18 / = Diff. Simple solution obtained by a proportional scaling of the EOQ values will not be optimal. Find Lagrange multiple ? § I13 : Inventory Model: Resource-Constrained Multiple Product System [Ex.13.2]

14 …..….[Eq.13.3] …….[Eq.13.4] § I13 : Inventory Model: Resource-Constrained Multiple Product System [Ex.13.2]◇

15 (a) For proportional Not optimal! § I13 : Inventory Model: Resource-Constrained Multiple Product System◇

16 (b) Find Lagrangean Function Find ∴ § I13 : Inventory Model: Resource-Constrained Multiple Product System◇

17 §. I13.1: Problems & Discussion Preparation Time : 15 ~ 20 minutes Discussion : 10 ~ 20 minutes ( # N4.26 ; N4.28 ) ( # N4.26 ; N4.28 )

18 § I14: The Newsboy Model [Eg. 14.1] On consecutive Sundays, Mac, the owner of a local newsstand, purchases a number of copies of The Computer Journal, a popular weekly magazine. He pays 25 cents for each copy and sells each for 75 cents. Copies he has not sold during the week can be returned to his supplier for 10 cents each. The supplier is able to salvage the paper for printing future issues. Mac has kept careful records of the demand each week for the Journal. (This includes the number of copies actually sold plus the number of customer requests that could not be satisfied.) The observed demands during each of the last 52 weeks were

19 § I14: The Newsboy Model There is no discernible pattern to these data, so it is difficult to predict the demand for the Journal in any given week. However, we can represent the demand experience of this item as a frequency histogram, which gives the number of times each weekly demand occurrence was observed during the year. The histogram for this demand pattern appears in the following Figure. [Eg. 14.1] Consider the example of Mac’s newsstand. From past experience, we saw that the weekly demand for the Journal is approximately normally distributed with mean μ=11.73 and standard deviation σ= 4.47.

20 § I14: The Newsboy Model C o : Overage Cost D : Demand C u : Underage Cost.….[Eq.14.1].…..…………..[Eq.14.1.a] …...[Eq.14.1.b]

21 …...[Eq.14.1.b] …...[Eq.14.1.c] …( Critical Ratio).[Eq.14.2] § I14: The Newsboy Model

Q→ Expected Cost Function for Newsboy Model Fig.14.1 § I14: The Newsboy Model

23 [Eg. 14.1] Purchase cost $ 0.25 Sell for $ 0.75 Salvage $ 0.10 Demand has mean = (μ) Standard deviation = 4.74 (σ) Use [Eq.14.2] C u : Underage Cost = = 0.5 C o : Overage Cost = = 0.15 § I14: The Newsboy Model

24 f(x) Q* X → 76.9% 76.9% → Z = 0.74 ∴ Buy 15 copies every week. § I14: The Newsboy Model

25 §. I14: Problems & Discussion ( # N5.8 ; N 5.9 ) ( # N5.8 ; N 5.9 ) # N 5.2T # N 5.2T # N 5.11 # N 5.11 Preparation Time : 15 ~ 20 minutes Discussion : 10 ~ 15 minutes

26 § I15 : ( R,Q ) model Q (A)Safety stock (B)Demand in lead time =λ ． τ (C)Reorder point = s+ λ ． τ = R◇

27 § I16 : ( s, S ) model s S t μ : starting inventory in any period s : reorder point ◆ If μ < s, order S – μ If μ ≧ s, do not order. μ1μ1 μ2μ2 μ3μ3 μ4μ4 μ5μ5

28 § I17: Stochastic Inventory Model Inventory Model (1) : SINGLE PERIOD MODEL WITH No setup Cost c = unit cost p = sell price or shortage cost for unsatisfied demand per unit where p > c h = holding cost or cost of excess supply per unit Q = quantities ordered D= Demand ( A random variable) d = demand (actual)◇

29 (A) RISK OF BEING “ SHORT “ (  Shortage cost incurred ) (B) RISK OF HAVING AN “ EXCESS “ ( Wasted unit cost & holding cost )  Recall Q = quantities ordered. D = actual demand, then amount sold:  Expected Cost : Discrete R.V § I17: (1) : Single Period Model with No setup Cost No setup Cost

30  Continuous R.V Let L(Q) = Expected [shortage+holding costs] & Minimum can be obtained.   § I17 (1) : Single Period Model with No setup Cost No setup Cost where C u = p – c ; C o = c + h 57 ◆

31 § I18 : Another way to look at SINGLE PERIOD INVENTORY SINGLE PERIOD INVENTORY  Let Q* be the smallest Q for which The Optimal quantity to order Q*, is the smallest integer such that the above function being satisfied. ◆ 2 投影片 60 投影片 60

32 Suppose Demand of a certain single period product is a random variable and which follows Probability density function: Cumulative probability function: [Eg. 18.1] ◇

33 (A) let c = $100 p = $200 h = $ -25 (salvage value) 代入 [Eg. 18.1]◇ ~ about COSTS

34 (B) To verify minimum G(Q*) = = $22,143 FIND c = $100 p = $200 h = $ -25   [Eg. 18.1]◇

35 Further discussion Further discussion (a) [ $200*9.245=$1849 ] [ $-25*16.245=-$406 ] [Eg 18.1] (b) (c)

36 §. I18: Problems & Discussion Preparation Time : 25 ~ 30 minutes Discussion : 15 ~ 25 minutes ( # C.7 ) ( # C.8 ) ( # C.7 ) ( # C.8 ) ( # C.9 ) ( # C.9 ) ( # C.9.1.c ) Advance Topics Follow...

37 § I19 : Single Period with § I19 : Single Period with “ Initial stock x ” “ Initial stock x ” Conclusion: Conclusion: then order up to Q* (i.e. order Q*-x) Do not order◇ [Eg 19.1] Let us suppose that in Example 14.1, Mac has received 6 copies of the Journal at the beginning of the week from another supplier. The optimal police still calls for having 15 copies on hand after ordering, so now he would order the difference 15-6 = 9 copies. ( Set Q*=15 and u=6 to get the order quantity of Q*-u=9.)

38 § I20 : Single Period with § I20 : Single Period with Ordering ( set-up ) cost “K” Ordering ( set-up ) cost “K” (s,S) policy : if on-hand Inv. x < s, order up to S. if on-hand Inv. x ≧ s, don’t order. ◆ 1 g-t-60 * *

39 c=20 p=45 h=-9 k=800 When Demand Dist.~Exponential Suppose ordering cost for the single period product described in Eg. #C.7 is $800 [Eg 20.1] ◆ 1-g-s-61 ◆2◆2 ◆ 1-g-s-63

40 §. I20: Problems & Discussion Preparation Time : 25 ~ 30 minutes Discussion : 15 ~ 25 minutes ( # C.10 )

41 § I21: ∞ periods with starting § I21: ∞ periods with starting inventory x units. inventory x units. Di = demand for period i 2 nd period purchases what’s being used in the previous period Assumptions : 1. Backorders; and assumes (a) that each unit left over at the end of the final period can be salvaged with a return of the initial purchase cost c. (b) if there is a shortage at the end of the final period, this shortage is met by an emergency shipment with the same unit purchase cost c. 2. Demand Distribution & Costs are the same in all periods. 3. α= discount rate =

42 § I 21 : ∞periods § I 21 : ∞periods (continued) ◆

43 …...[ Eq.21.1 ] g-s-30 ◆ g-b-48 g-b-48 § I 21 : ∞periods § I 21 : ∞periods (continued)

44 And if p (shortage cost) = ＄ 15 [ backorder case, p may be less than c, cost just for handling backordering ] c = 35, h = 1 and discount rate α=0.995 Suppose Demand of a certain multiple period product is a random variable and it follows Therefore, [Eg 21.1]

45 §. I21: Problems & Discussion Preparation Time : 25 ~ 30 minutes Discussion : 15 ~ 25 minutes ( # C.11 ) ( # C.11 ) ( # C.12 ) ( # C.12 )

46 §I 22 : Interpretation of Co, Cu §I 22 : Interpretation of Co, Cu Define …...[ Eq.22.1 ] ◆ g-s-66 g-s-66 ◆ ◆ b-t-67 b-t-67

47 §.I 23 : ∞ periods with backordered backordered◇ ◆ g-s-68 g-s-68

48◇ §.I 23 : ∞ periods with backordered backordered …... [ Eq.23.1 ] [Eg. 23.1] Let us return to Mac’s newsstand, described previously. Suppose that Mac is considering how to replenish the inventory of a very popular paperback thesaurus that is ordered monthly. Copies of the thesaurus unsold at the end of a month are still kept on the shelves for future sales. ◆ b-t-51 b-t-51 ◆ g-s-57 g-s-57 ◆ g-s-43 g-s-43

49 Assume that customers who request copies of the thesaurus when they are out of stock will wait until the following month. (Back-ordered allowed) Mac buys the thesaurus for $1.15 and sells it for $2.75. Mac estimates a loss-of-goodwill cost of 50 cents each time a demand for a thesaurus must be back-ordered. Monthly demand for the book is fairly closely approximated by a Normal distribution with mean 18 and standard deviation 6. Mac uses a 20 percent annual interest rate to determine his holding cost. How many copies of the thesaurus should he purchase at the beginning of each month? [Eg 23.1] Solution: using [Eq.23.1] The overage cost in this case is just the cost of holding, which is (1.15)(0.20) / 12 = The underage cost is just the loss-of-goodwill cost, which is assumed to be 50 cents. Hence, the critical ratio is 0.5/( )= From the Table of Normal Dist., corresponds to a z value of The optimal value of the order-up-to point Q*=σZ+u=(6)(1.79)+18 = =29.

50 §. I23: Problems & Discussion Preparation Time : 25 ~ 30 minutes Discussion : 15 ~ 25 minutes ( # C.13 )

51 § I 24: ∞ periods with Lost Sales § I 24: ∞ periods with Lost Sales ◆ b-t-69 b-t-69 ◆ b-t-69b-t-69 ◆ g-s-68 g-s-68

52 …... [ Eq.24.1 ] § I 24: ∞ periods with Lost Sales § I 24: ∞ periods with Lost Sales [Eg. 24.1] Assume that a local bookstore also stocks the thesaurus and that customers will purchase the thesaurus there, if Mac is out of stock. In this case excess demands are lost rather than back-ordered. ( Lost Sales case ) Determining the order-up-to points (which will be different from that obtained assuming full back-ordering of demand.) ◆ g-t-48 g-t-48

53 In the lost sales case the underage cost should be interpreted as the loss-of- goodwill cost (i.e. $0.50) plus the lost profit (ie. $2.75-$1.15=$1.60). Therefore, the underage cost is $0.50+$1.60=$2.10. The critical ratio is 2.1/( )=0.9909, giving a Z value of 2.36, the optimal value of Q in the lost sales case is Q*=σZ+u = (6)(2.36)+18 = = 32. Versus backordering allowed Q*= 29 Solution: using [Eq.24.1] [Eg. 24.1]

54 §. I24: Problems & Discussion Preparation Time : 25 ~ 30 minutes Discussion : 15 ~ 25 minutes ( # C.14 ) ( # C.14 )

55 § I25: Markov Model in Stochastic Inventory Stochastic Inventory Management Management A camera store stocks a particular model camera that can be ordered weekly. Let D 1, D 2, …, represent the Demands for this camera during the first week, second week, …, respectively. It is assumed that the D t are independent and identically distributed random variables having a known probability distribution. Let X 0 represents the number of cameras on hand at the outset, X 1 the number of cameras on hand at the end of week one, X 2 the number of cameras on hand at the end of week two, and so on. Assume that X 0 = 3. On Saturday night the store places an order that is delivered in time for the opening of the store on Monday.◇ [Eg 25.1]

56 The store uses (s,S) ordering policy, where (s,S) = (1,3). It is assumed that sales are lost when demand exceeds the inventory on hand. Demand ( i.e. D t ) has a Poisson distribution with λ= 1. If the ordering cost K=$10, each camera costs the store $25 to own it and the holding is $0.8 per item per week, while unsatisfied demand is estimated to be $50 per item short per week. Find the long-run expected total inventory costs per week? Demonstration follows C.15 Please see C.15. § I25: Markov Model (cont’d) [Eg 25.1]

57 §. I25: Problems & Discussion Preparation Time : 25 ~ 30 minutes Discussion : 15 ~ 25 minutes ( # C.15 ) ( # C.15 )

58 The End of Lecture Notes 2 of 2