Download presentation

Presentation is loading. Please wait.

Published byMalakai Walston Modified over 2 years ago

1
Stochastic Models Inventory Control

2
Inventory Forms u Form u Raw Materials u Work-in-Process u Finished Goods

3
Inventory Function u Safety Stock u inventory held to offset the risk of unplanned demand or production stoppages u Decoupling Inventory u buffer inventory required between adjacent processes with differing production rates u Synchronized Production u In-transit (pipeline) u materials moving forward through the value chain u order but not yet produced/received

4
Inventory Function u Cycle Inventory u orders in lot size not equal to demand requirements to lower per unit purchase costs u Decoupling Inventory u buffer inventory required between adjacent processes with differing production rates u Synchronized Production u In-transit (pipeline) u materials moving forward through the value chain u order but not yet produced/received

5
Inventory Function u Seasonal Inventory u produce in low demand periods to meet the needs in high demand periods u anticipatory - produce ahead of planned downtime u In-transit (pipeline) u materials moving forward through the value chain u order but not yet produced/received u Cycle Inventory u orders in lot size not equal to demand requirements to lower per unit purchase costs

6
Inventory Costs u Item Cost (C) u Order Cost (S) u Process Setup Costs u Holding Costs (H) u Function of time in inventory, average inventory level, material handling, utilities, overhead,... u Often calculated as a % rate of inventory cost (iC) u Stockout or Shortage Costs (s) u reflects costs associated with lost opportunity

7
Economic Order Quantity Q r time L T 1 D order arrives Q = reorder quantity r = reorder point D = demand rate L = leadtime T = inventory cycle

8
Economic Order Quantity Q = reorder quantity r = reorder point D = demand rate L = leadtime T = inventory cycle Q r time L T Avg. Inventory

9
2 Inventory Cost TRC= Total Relevant Cost = Order Cost + Holding Costs = cost per cycle TCU = Total Relevant Cost per Unit Time = TRC/T SHQT 1 2 S T HQ 1

10
Cost per unit Time But, T= length of a cycle Q D TCU SD Q HQ 1 2

11
Cost per Unit Time

12
Note: that minimal per unit cost occurs when holding cost = order cost (per unit time)

13
Economic Order Quantity Find min TCU TCU QQ SD Q HQ 0 1 2 () -SDQ0 2 1 2 H Q*Q* SD iC SD H 22

14
Example The monthly demand for a product is 50 units. The cost of each unit is $500 and the holding cost per month is estimated at 10% of cost. It costs $50 for each order made. Compute the EOQ. Q*Q* *50*50.1*500 2 = 10 Sol:

15
Optimal Inventory Cost Recall TCU SD Q HQ 1 2 TCU * SD Q*Q* HQ * 1 2 TCU * SD H 2 H 1 2 H 2

16
Optimal Inventory Cost TCUHSDiCSD * 22Example: 2*.1*500*50*50TCU * = $500 per month

17
Orders per year HD N D Q S * 2 N = number of orders per year Example: Example: D = 50 / month, Q * = 10 D = 50 x 12 = 600 / yr. H =.1x12x500 = $600 / unit-yr N 600*600 2*50 = 60

18
Cycle Time T Q D S HD * 2 T = cycle time Example: Example: D = 50 units/month, Q * = 10 T 2*50 50*50 10 50 =.2 months = 6 days

19
Reorder Point L= lead time r = reorder point = inventory depleted in time L = L*D Example: Example: Lead time for company is 2 days. Demand is 50 units per month or 1.67 units/day. r= 2*1.67 = 3.33 Reorder at 4 units

20
Lead Time Example 2: Example 2: Suppose our lead time is closer to 8 days. r = 8*1.67 = 13.33 but, recall we only order 10 units at a time r = 13.33 - 10 = 3.33

21
Example (cont.) Reorder at 4 units 1 cycle ahead. 10 4 time L T order arrives reorder

22
Sensitivity Q*Q* SD H 2 Recall that Now suppose we deviate by p amount so that Q = Q * (1+p). What affect does this have on total cost? Let PCP = Percentage Cost Penalty

23
Sensitivity Q*Q* SD H 2 Recall that Now suppose we deviate by p amount so that Q = Q * (1+p). What affect does this have on total cost? Let PCP = Percentage Cost Penalty PCP TCUQ Q Q x ()() () * * 100

24
Senstivity (cont.) TCUQ SD Qp HQp() () () * * 1 1 2 1TCU SD Q HQ 1 2 Recall Miracle 1 Occurs 2TCUQSDHp p ()() 1 2 1 1 1

25
Sensitivity (cont.) Recall TCUHSD * 2 2SDHp p () 1 2 1 1 1 PCP = HSD2 2 x 100 = 50 p p () 1 1 1 100

26
Sensitivity (cont.) PCP = 50 p p () 1 1 1 100 Miracle 2 Occurs PCP p p 50 1 2

27
Example Recall that Q * = 10. Suppose now that a minimum order of 15 is introduced. Compute the percentage cost penalty (increase). p 1510 5. PCP 505 15 83 2 (.).. Total relevant costs increase 8.3%

28
Example 2 Suppose demand forecast increases by 25% so that D = 50(1.25) = 62.5. Then TCU * *.**2 62550 559 or TCU * increases by 11.8%

29
Shortages ImIm r time L T T1T1 T2T2 Q Q-R T 1 = time inventory carriedH = holding cost T 2 = time of stockoutS = order cost I m = max inventory level p = cost per unit short per unit time

30
Inventory Costs TCR = order + holding + shortages SHI m TpQImIm T 1 2 1 2 12 () Miracle 3 Occurs HDS Q H p p * 2 H R H p p * 2 +

31
Example Suppose we allow backorders for our previous example. We estimate that the cost of a backorder is $1 per unit per day ($30 / month). Then *50*50 Q 50 * 30 2 = 16.3 = 17 units

32
Production Model (ELS) ImIm time T1T1 1 P-D T2T2 T Q = batch size order quantity D= demand rate P = production rate P-D = replenish rate during T 1 S = setup costs H = holding cost /unit-time I m = max inventory level

33
Production Model (ELS) ImIm time T1T1 1 P-D T2T2 T T= cycle length = T 1 + T 2 = Q/D T 1 = length of production run = Q/P T 2 = depletion time = I m /D I m = max inventory level = (P-D)T 1 = (P-D)Q/P

34
Costs TC = total costs per cycle = order + holding S HPDQT P 1 2 () TCU= Cost per unit time TCT / S T HPDQ P ()1 2 SD Q HPDQ P () 2

35
Optimal Q* (ELS) TCU Q SD Q HPD P 0 2 2 () Solving for Q, Q SDP HPD EOQ D P * () 2 1

36
Max Inventory I m = max inventory = (P-D)T 1 = (P-D)Q*/P IQ D P ELS D P m * 11

37
Summary ImIm time T1T1 1 P-D T2T2 T Q SDP HPD EOQ * () 2 D P 1 IQ D P m * 1 D P 1

38
Probabilistic Models ImIm time L R=B+LD B Q* D= demand rate B= buffer stock R= reorder point = B+LD D L = actual demand from time of order to time of arrival

39
Probabilistic Model Let = max risk level for out of stock condition Idea: we want to set a buffer level B so that the probability of running out of stock is < . > P{out of stock} = P{demand in D L > R} = P{D L > B+LD}

40
Example Prob: Suppose S=$100, H=$.02/day, L=2 days. D = daily demand N(100, 10). From EOQ model, Q* = 1,000 units Find: Buffer level, B, such that probability of out of stock <.05.

41
Solution D L = demand for 2 days = D 1 + D 2 Question: D 1 & D 2 are identically independently distributed normal variates with mean and standard deviation =10. What can we say about the distribution of D L ?

42
Prob. Review Suppose we have a random variable X L given by X = Y 1 + Y 2 Then E[X] = E[Y 1 ] + E[Y 2 ] If Y 1 & Y 2 are independent, then xx 2 1 2 2 2 12 2 cov(,)yy xx 2 1 2 2 2

43
200 Solution (cont.) Recall D L = demand for 2 days = D 1 + D 2 ~ N( L, L ) Then L = E[D L ] = E[D 1 ] + E[D 2 ] = 200 LDD 22222 12 10 200 L 14.

44
Solution (cont.) D L ~ N(200, 14.14) B + LD} = P{D L > B + L } P D B ll LL PZ B L

45
Solution (cont.) Recall for our problem that =.05 and L =14.14. Then,.. 05 14 PZ B 0 Z =1.645 =.05 B 14 1645.. B = 23.3

46
Summary For D ~ N(100,10), L = 2 days, Q* = 1,000 units, and = risk level =.05 D L ~ N( L =200, L =14.14) B = Z L = 23.3 R = B + D L = B + L = 23.3 + 200 = 224

47
Optional Replacement In the continuous review model, an order of Q* is made whenever inventory level reaches the reorder point R. We can also utilize periodic review systems with variable order quantities. The two most common are Optional Replacement (s,S) P system ImIm time L R=B+LD B

48
Optional Replacement At t=1, inventory level is above minimum stock level s, no order is made. At t =2, inventory level is below s, order up to S s = R = B+DL S = Q* S time s 12345

49
P System Order up to Target level T at each review interval P. Let D P+L = demand in review period + lead time P+L = standard deviation of demand in period P+L = level of risk associated with a stockout T = D P+L + Z P+L T time 12345

50
Newsboy Problem Often inventory for a single product is met only once; e.g. News Stand (can’t sell day old papers) Pet Rocks Christmas Trees If Q > D, incur costs for Q but revenue only for D If Q < D, incur opportunity costs in form of lost sales

51
Newsboy (cont.) Objective: Determine best order quantity which maximizes expected profit Payoff Matrix: R ij = payoff for order quantity Q i and demand level D j P = profit per unit sold L = loss per unit not sold R PQifQD PDLQDifQD ij iij iijij ()

52
Newsboy (cont.) Expected Payoff: EPQPR iD j m ij j () 1 where EP(Qi) = expected payoff for order quantity Qi P = probability of demand level j Rij = payoff for order quantity Qi and demand level Dj D j

53
Example; Newsboy Boy Scout troop 53 plans to sell Christmas trees to earn money. Each tree costs the troop $10 and can be sold for $25. They place no value on lost sales due to lack of trees, L=0. Demand schedule is shown below. DemandP{demand} 1000.10 1200.15 1400.25 1600.25 1800.15 2000.10

54
Example (cont.) Payoff Matrix P = Profit = $25 - $10 per tree sold L = Loss = $-10 per tree not sold Demand Level Order Q100120140160180200 1001,500 1201,3001,800 1401,1001,6002,100 1609001,4001,9002,400 1807001,2001,7002,2002,700 2005001,0001,5002,0002,5003,000

55
Example (cont.) Expected Payoff: Demand Level 0.10.150.25 0.150.1Expected Order Q100120140160180200Payoff 100150225375 225150 1,500 120130270450 270180 1,750 140110240525 315210 1,925 16090210475600360240 1,975 18070180425550405270 1,900 20050150375500375300 1,750 Order quantity has largest expected payoff of $1,975 order 160 trees

Similar presentations

OK

1 OM3 Chapter 12 Managing Inventories © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible.

1 OM3 Chapter 12 Managing Inventories © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible.

© 2018 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on social media on business Ppt on world environment day logo Carta post ppt online Ppt on bank lending process Ppt on metro train in delhi Ppt on shell scripting basics Ppt on biodegradable and non biodegradable meaning Ppt on fire fighting training Ppt on recycling waste materials Cathode ray tube display ppt online