Review. ESD.260 Fall 2003

Transcription

1 Review ESD.260 Fall

2 Demand Forecasting 2

3 Accuracy and Bias Measures 1. Forecast Error: e t = D t -F t 2. Mean Deviation: MD = 3. Mean Absolute Deviation 4. Mean Squared Error: 5. Root Mean Squared Error: 6. Mean Percent Error: 7. Mean Absolute Percent Error: n t = 1 n e MSE = t MPE = n t = 1 n n t = 1 e 2 t e t Dt n MAD = RMSE = MAPE = n t = 1 n t = 1 n n t = 1 e n e t e 2 t t Dt n 3 MD cancels out the over and under good measure of bias not accuracy MAD fixes the cancelling out, but statistical properties are not suited to probability based dss MSE fixes cancelling out, equivalent to variance of forecast errors, HEAVILY USED statistically appropriate measure of forecast errors RMSE easier to interpret (proportionate in large data sets to MAD) MAD/RMSE = SQRT(2/pi) for e~n Relative metrics are weighted by the actual demand MPE shows relative bias of forecasts MAPE shows relative accuracy Optimal is when the MSE of forecasts -> Var(e) thus the forecsts explain all but the noise. What is good in practice (hard to say) MAPE 10% to 15% is excellent, MAPE 20%- 30% is average CLASS? 3

4 The Cumulative Mean Generating Process: D t = L + n t where: n t ~ iid (µ = 0, σ 2 = V[n]) Forecasting Model: F t+1 = (D 1 +D 2 +D D t ) / t 4 Stationary model mean does not change pattern is a constant Not used in practice is anything constant? Thought though is to use as large a sample side as possible to 4

5 The Naïve Forecast Generating Process: D t = D t-1 + n t where: n t ~ iid (µ = 0, σ 2 = V[n]) Forecasting Model: F t+1 = D t 5 5

6 The Moving Average Generating Process: D t = L + n t ; t < t s D t = L + S + n t ; t t s where: n t ~ iid (µ = 0, σ 2 = V[n]) Forecasting Model: F t+1 = (D t + D t-1 + +D t-m+1 ) / M where M is a parameter 6 6

7 Exponential Smoothing F t+1 = α D t + (1-α) F t Where: 0 < α < 1 An Equivalent Form: F t+1 = F t + αe t 7 7

8 Holt's Model for Trended Data Forecasting Model: F t+1 = L t+1 + T t+1 Where: L t+1 = αd t + (1-α)(L t + T t ) and: T t+1 = β(l t+1 -L t ) + (1- β)t t 8 8

9 Winter's Model for Trended/Seasonal Data F t+1 = (L t+1 + T t+1 ) S t+1-m L t+1 = α(d t /S t ) + (1- α)(l t + T t ) T t+1 = β(l t+1 -L t ) + (1- β)t t S t+1 = γ(d t+1 /L t+1 ) + (1- γ) S t+1-m 9 9

10 Notes from Homework 1 Problem 1 Did not used the model which yielded the lowest MSE Remove outliers Problem 2 Setting initial values for level (L) and trend (T) The more data you use, the more accurate are these initial values Penalty for waiting too long If initial values are off by a lot, the model will take a longer time to adjust itself Problem 3 Initializing seasonality indexes 10 10

11 Inventory Management 11

12 Bottomline Inventory is not bad. Inventory is good. Inventory is an important tool which, when correctly used, can reduce total cost and improve the level of service performance in a logistics system

13 Fundamental Purpose of Inventory To Reduce Total System Cost To buffer uncertainties in: - supply, - demand, and/or - transportation the firm carries safety stocks. To capture scale economies in: -purchasing, - production, and/or - transportation the firm carries cycle stocks

14 Dimensions of Inventory Modeling Demand Constant vs Variable Known vs Random Continuous vs Discrete Lead time Instantaneous Constant or Variable (deterministic/stochastic) Dependence of items Independent Correlated Indentured Review Time Continuous Periodic Discounts None All Units or Incremental Excess Demand None All orders are backordered Lost orders Substitution Perishability None Uniform with time Planning Horizon Single Period Finite Period Infinite 14 14

15 Lot sizing 15

16 Cycle Stock & Safety Stock On Hand Cycle Stock Cycle Stock Cycle Stock Safety Stock Time 16 16

17 Lot Sizing: Many Potential Policies Inventory On Hand I(t) Q Objective: Pick the policy with the lowest total cost T 17 Time 17

18 Relevant Costs What makes a cost relevant? Components Purchase Cost Ordering Cost Holding Cost Shortage Cost 18 18

19 Notation TC = Total Cost (dollar/time) D = Average Demand C o = Ordering Cost (dollar/order) C h = Holding Cost (units/time) (dollars/dollars held/time) Cp = Purchase Cost (dollars/unit) Q = Order Quantity (units/order) T = Order Cycle Time (time/order) 19 19

20 Economic Order Quantity (EOQ) D Q TC ( Q ) = C o + C h C p Q 2 TC [ Q ] = C o D C hc + Q 2 p Q Q 2DC = TC * = 2 DC o C h C p CC * o h p 20 From TC [Q] to Q* Take the derivate and set it to 0 20

21 The Effect of Non-Optimal Q Q DC o /Q $2,000 C h C p Q/2 $12,500 $3,125 TC $13,000 $5, $2,500 $2,500 $5, $5,000 $1,250 $6, $50,000 $125 $50,125 So, how sensitive is TC to Q? 21 21

22 Total Cost versus Lot (Order) Size Annual Cost vs. Order Quantity $25,000 $20,000 Annual Cost $15,000 $10,000 $5,000 $ Lot Size Minimum point is relatively flat : there is a range / small changes in parameters may change the optimal Q 22

23 Insights from EOQ There is a direct trade off between lot size and total inventory Total cost is relatively insensitive to changes Very robust with respect to changes in: Q rounding of order quantities D errors in forecasting C h, C o, C p errors in cost parameters Thus, EOQ is widely used despite its highly restrictive assumptions 23 23

24 Introduce Discounts to Lot Sizing Types of discounts All units discount Incremental discount One time only discount How will different discounting strategies impact your lot sizing decision? 24 24

25 All Units Discount Unit Price Price Break Quantity [C pi ] [PBQI] $ $ $

26 All Units Discount Need to introduce purchase cost into TC function TC[Q, C pi ] = DC pi + Co D Q + C h C 2 pi Q 26 26

27 All Units Discount: Method Same Example: D=2000 Units/yr C h =.25 C o = C pi Price Breaks: $50 for 0 to <500 units $45 for 500 to <1000 units $40 for units 1 C pi $40.00 $45.00 $ PBQ EOQ[C pi] Q pi 1000 $80, DC pi $90,000 $100,000 6 CoD/Qpi $1,000 $2,000 $2,500 7 ChCpiQpi/2 $5,000 $2,812 $2,500 8 TC[Qpi] $86,000 $94,812 $105, Method : Start with lowest price ($40) Find EOQ at that price point and price break quantity (EOQ cpi + PBQ) Find Qpi = max [ PBQ, EOQcpi ] Find total cost using new price point ( TCqpi ) Go to next price point If the EOQ was 1,200 the optimal quantity fall between the range, I can t do better. So we can stop the calculations 27

28 Incremental Discount 28 Insight : As oppose to the previous where there is a range The cost I have to incur to be able to get to the next price level is like a fixed cost 28

29 Incremental Discount Index i=3 i=2 i=1 1 C pi $40.00 $45.00 $ PBQ i F i $7500 $ EOQ[C pi ] Q pi C pe $44.19 $ DC pe $88, $100,000 8 C o D/Q pi $ $2,500 9 (C h C pe Q pi )/2 $9, $2, TC[Q pi ] $98, $105, Cpe (eq uivalent price) Quantity Cpe 0 <= Q <= 500 $ <= Q <= 1000 [ $50*(500) + $45*(Q-500) ] / Q 1000 < Q [ $50*500 + $45*(Q-500) +$40*(Q-1000) ] / Q Method Start with i=1 Find fixed cost F1= 0 Fi= Fi-1 + (Cpi-1 Cpi) * PBQi EOQ at Cpi If EOQ cpi is within range, then Qpi Otherwise, stop go to the next I Find Cpe = [ Cpi * Qpi * Fi ] / Qpi Find TC Next I 29

30 One Time Discount 30 Similar to a price increase where we order more right before the price increase 30

31 Let, One Time Discount C pg = One time deal purchase price ($/unit) Q g = One time special order quantity (units) TC sp =TC over time covered by special purchase ($) Then, TC [ / D] = + C Q Q + 2 D g g sp Q g C pg Q g h C pg C o 31 31

32 One Time Discount TCsp[ Q g /D] = C pg Q g + C hc pg Q 2 g Q g D +C o TC nsp [ Q /D] = C g pg Q +C p( Q - Q )+ C h C pg w g w Qw Q w + C h C 2 D p Q ( Q - Q ) Q w g w g +Co 2 D Q w 32 32

33 One Time Discount Q * g ( C p - C = C pg C pg h )D C p Q + C pg w SAVINGS[ Q * g C o C ] = C p pg Q Q * g w

34 Notes from Homework 2 Problem 1 Explore impact of reducing the ordering cost on the total system operating costs. Problem 2 Explored mechanics of prices discounts on lot sizing Critical Cpi how low the price need to be Critical PBQi how low quantity need to be Problem 3 All units discount and added a minimum dollar value 34 34

35 Safety Stock 35

36 Assumptions: Basic EOQ Model Demand Constant vs Variable Known vs Random Continuous vs Discrete Lead time Instantaneous Constant or Variable (deterministic/stochast i c) Dependence of items Independent Correlated Indentured Review Time Continuous vs Periodic Number of Echelons One vs Many Capacity / Resources Unlimited vs Limited Discounts None All Units or Incremental Excess Demand None All orders are backordered All orders are lost Substitution Perishability None Uniform with time Planning Horizon Single Period Finite Period Infinite Number of Items One Many 36 36

37 Fundamental Purpose of Inventory Firm carries safety stock to buffer uncertainties in: - supply, - demand, and/or - transportation 37 37

38 Cycle Stock and Safety Stock On Hand Cycle Stock Cycle Stock Cycle Stock Safety Stock Time What should my inventory policy be? (how much to order when) What should my safety stock be? What are my relevant costs? 38 38

39 Preview: Safety Stock Logic C b P[SO] SL K N[K] E[US] TC[R] R FR 39 39

40 Determining the Reorder Point R = d' + kσ Reorder Point Estimated demand over the lead time (F k = t ) ( R d ') σ Safety Stock k = SS factor σ = RMSE Note We usually pick k for desired stock out probability Safety Stock = R d 40 40

41 Define Some Terms Safety Stock Factor (k) Amount of inventory required in terms of standard deviations worth of forecast error Stockout Probability = P[d > R] The probability of running out of inventory during lead time Service Level = P[d R] = 1- P[SO] The probability of NOT running out of inventory during lead time 41 41

42 Service Level and Stockout Probability Prob (x ) Normal Distribution Where d ~ iid N(d =10, σ=25) 1.8% 1.6% SS = kσ 1.4% Probability of 1.2% Stockout 1.0% 0.8% 0.6% 0.4% 0.2% 0.0% Demand Reorder Point Service Level Forecasted Demand (d ) 42 42

43 Cumulative Normal Distribution Cumulative Normal Distribution Where d ~ iid N(d =10,σ=25) 120.0% 100.0% Probability of Stockout = 1-SL Prob (x) 80.0% 60.0% 40.0% 20.0% 0.0% Service Level Demand 43 Reorder Point 43

44 Finding SL from a Given K If I select a K = 0.42 Using a Table of Cumulative Normal Probabilities... K then my Service Level is this value (kσ+d, d, σ, TRUE) Or, in Excel, use the function: SL= NORMDIST 44 44

45 Safety Stock and Service Level Example: if d ~ iid Normal (d =100,σ=10) What should my SS & R be? Safety Reorder P[SO] SL k Stock Point

46 So, how do I find Item Fill Rate? Fill Rate Fraction of demand met with on-hand inventory Based on each replenishment cycle OrderQuantity E [ UnitsShort ] FillRate = OrderQuantity But, how do I find Expected Units Short? More difficult Need to calculate a partial expectation: 46 46

47 Expected Units Short Consider both continuous and discrete cases Looking for expected units short per replenishment cycle. E [ US ] = ( x R ) p [ x ] = ( ) x= R E [ US ] x R f ( x ) dx R o x o o P[x] 1/ What is E[US] if R=5? x 47 For normal distribution we have a nice result: E[US] = σn[k] Where N[k] = Normal Unit Loss Function Found in tables or formula 47

48 The N[k] Table A Table of Unit Normal Loss Integrals K

49 Item Fill Rate OrderQuantity E [ UnitsShort ] FillRate = = FR OrderQuantity EUS [ ] σ N [ k] FR = 1 = 1 Q Q ( 1 ) FR Q Nk [ ] = So, now we can look for σ the k that achieves our desired fill rate

50 Finite Horizon Planning 50

51 Approach: One-Time Buy On 1200 Hand 1000 Inventory Month

52 Approach: One-Time Buy Month Demand Order Quantity Holding Cost Ordering Cost Period Costs $1800 $ $1650 $ $1550 $ $1500 $ $1450 $ $1300 $ $1200 $ $1000 $ $800 $ $550 $ $250 $ Totals: $13100 $

53 Approach: Lot for Lot On 1200 Hand 1000 Inventory Month

54 Approach: Lot for Lot Month Demand Order Quantity Holding Cost Ordering Cost Period Costs Totals: $6000 $

55 Approach: EOQ On Hand Inventory Month

56 Approach: EOQ Month Totals: Demand Order Quantity Holding Cost $200 $50 $350 $300 $250 $150 $200 $150 $250 $1900 Ordering Cost $2500 Period Costs $700 $50 $850 $300 $250 $150 $700 $650 $750 $

57 Approach: Silver-Meal Algorithm On Hand Inventory Month

58 Approach: Silver-Meal Algorithm Mon Dmd Lot Qty Order Cost Holding Cost Lot Cost Mean Cost 1st nd Buy: Buy: $150 $150+$200 $150+$200+$150 $150+$200+$150+$200 $150+$200+$150+$200+ $150 $150+$400 $650 $850 $1000 $1200 $1700 $650 $1050 $325 $283 $250 $240 $283 $325 $

59 Approach: Silver-Meal Algorithm Mon Dmd Lot Order Holding Cost Lot Mean Qty Cost Cost Cost 3rd Buy: $200 $700 $ $200+ $1200 $400 4th Buy: $300 $800 $ $300+ $1300 $433 5th Buy:

60 Approach: Silver-Meal Algorithm Month Totals: Demand Order Quantity Holding Cost $350 $200 $100 $50 $150 $200 $300 $1350 Ordering Cost $2500 Period Costs $850 $200 $100 $50 $650 $700 $800 $

61 Approach: Optimization (MILP) On Hand Inventory Month

62 Approach: Optimization (MILP) Decision Variables: Qi = Quanti ty purchased in period i Zi = Buy variab le = 1 if Q i >0, =0 o.w. Bi = Beginning inventory for period I Ei = Endi ng inventory for period I Data: D i = Demand per period, i = 1,,n C o = Ordering Cost C hp = Cost to Hold, $/ unit/period M = a very large number. MILP Model Objective Function: Minimize total relevant costs Subject To: Beginning inventory for period 1 = 0 Beginning and ending inventories must match Conservation of inventory within each period Nonnegativity for Q, B, E Binary for Z 62 62

63 Approach: Optimization (MILP) Min TC = C Z + st.. B = 0 1 i 1 n O i HP i i = 1 i = 1 B E = 0 i = 2, 3,... n i i i i i MZ Q 0 i = 1, 2,... n i i B 0 i = 1, 2,... n i E 0 i = 1, 2,... n i Q 0 i = 1, 2,... n i i n C E E B Q = D i = 1, 2,... n Z = {0,1} i = 1, 2,... n Objective Function Beginning & Ending Inventory Constraints Conservation of Inventory Constraints Ensures buys occur only if Q>0 Non-Negativity & Binary Constraints 63 63

64 Comparison of Approaches Month Demand OTB L4L EOQ S/M OPT Total Cost $13,600 $6,000 $4,400 $3,850 $3,

65 Notes from Homework 3 Problem 1 Critique an item being ordered Did not know the backorder cost (5 or 10) Problem 2 Split between back order and lost sales Problem 3 Silver-Meal vs. MILP Problem 4 MRP Problem 5 Padded lead time 65 65