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Inventory Management: Safety Inventory ( I ) 【本著作除另有註明外，採取創用 CC 「姓名標示 －非商業性－相同方式分享」台灣 3.0 版授權釋出】創用 CC 「姓名標示 －非商業性－相同方式分享」台灣 3.0 版 第六單元： Inventory Management: Safety Inventory ( I ) 郭瑞祥教授 1

Safety Inventory – Demand uncertainty – Supply uncertainty ►S►Safety Inventory is inventory carried for the purpose of satisfying demand that exceeds the amount forecasted for a given period. ►P►Purposes of holding safety inventory Average Inventory Time Safety Inventory Cycle Inventory 2

Planning Safety Inventory ► Appropriate level of safety inventory is determined by ctions to improve product availability while reducing safety inventory 》 Uncertainty of both demand and supply – Uncertainty increases, then safety inventory increases. 》 Desired level of product availability Desired level of product availability – increases, then safety inventory increases. 3

Measuring Demand Uncertainty  k i=1 DiDi P= CV=  P=KD D k  > Coefficient of variation – The total demand during k period is normally distributed with a mean of P and a standard deviation of  : –If demand in each period is independent and normally distributed with a mean of D and a standard deviation of  D, then   i 2 +2  Cov(i,j) i=1 i>j k   i 2 + 2  i  j i=1 i>j k ► Uncertainty within lead time – Assume that demand for each period i, i=1,….,k is normally distributed with a mean D i and standard deviation  i.  i 2 i=1 k + 2  i  j i>j D k  4

Measuring Product Availability ►Order fill rate ►Product fill rate ( f r ) ►Cycle service level (CSL) –T–The fraction of replenishment cycles that end with all the customer demand being met –T–The CSL is equal to the probability of not having a stockout in a replenishment cycle –A–A CSL of 60 percent will typically result in a fill rate higher than 60% – The fraction of product demand that is satisfied from product in inventory – It is equivalent to the probability that product demand is supplied from available inventory –T–The fraction of orders that are filled from available inventory –O–Order fill rates tend to be lower than product fill rates because all products must be in stock for an order to be filled CoolCLIPS 網站 5

►P►Product fill rate ( f r ) ►O►Order fill rate ►C►Cycle service level (CSL) Measuring Product Availability -- Page 5 On-hand inventory Order received Unfilled demand Filled demand 0 –Don't run out of inventory in 6 out of 10 replenishment cycles –An order for a total of 100 palms and has 90 in inventory –Customer may order a palm along with a calculator. The order is filled only if both products are available. → CSL = 60% → fill rate > 60% → fill rate of 90% –In the 40% of the cycles where a stockout does occur, most of the customer demand is satisfied from inventory Cycle Microsoft 。 6

►A replenishment policy consists of decisions regarding –When to reorder –How much to reorder. ►Continuous review –Inventory is continuously tracked and an order for a lot size Q is placed when the inventory declines to the reorder point (ROP). Replenishment Policies ►Periodic review –Inventory status is checked at regular periodic intervals and an order is placed to raise the inventory level to a specified threshold, i.e. order up to level (OUL). QP 7

►A replenishment policy consists of decisions regarding –When to reorder –How much to reorder. ►Continuous review –Inventory is continuously tracked and an order for a lot size Q is placed when the inventory declines to the reorder point (ROP). Replenishment Policies ►Periodic review –Inventory status is checked at regular periodic intervals and an order is placed to raise the inventory level to a specified threshold. QP 8

Continuous Review System ► Other names are: Reorder point system, fixed order quantity system ► Decision rule ► The remaining quantity of an item is reviewed each time a withdrawal is ► made from inventory, to determine whether it is time to reorder. ►Inventory position 》 IP = inventory position 》 OH = on-hand inventory 》 SR = scheduled receipts (open orders) 》 BO = units backordered or allocated IP = OH+SR-BO – Whenever a withdrawal brings IP down to the reorder point (ROP), place an order for Q (fixed) units. 9

Time On-hand inventory Order received ROP OH IP TBO 2 TBO 3 L2L2 L3L3 Order received OH Q IP Order placed ROP = average demand during lead time + safety stock Continuous Review System ROP Order placed L1L1 TBO 1 10

Time On-hand inventory TBO 2 TBO 3 L2L2 L3L3 Order received OH Q IP Order placed ROP = average demand during lead time + safety stock Continuous Review System Order placed L1L1 TBO 1 FIX Order received ROP OH IP 11

Time On-hand inventory TBO 2 TBO 3 L2L2 L3L3 ROP = average demand during lead time + safety stock Continuous Review System L1L1 TBO 1 Order received OH Q IP Order placed Order placed Order received ROP IP OH 12

Example Given the following data  Average demand per week, D = 2,500  Standard deviation of weekly demand, s D =500  Average lead time for replacement, L = 2 weeks  Reorder point, ROP = 6,000  Average lot size, Q = 10,000 =ROP-DL=6,000-5,000=1,000►Safety inventory,ss ►Cycle inventory ►Average inventory ►Average flow time =Q/2=10,000/2=5,000 =5,000+1,000=6,000 = Average inventory / Throughput=6,000/2,500 =2.4weeks 13

Evaluating Cycle Service Level and Safety Inventory ► CSL=Function ( ROP,D L,  L ) CSL= Prob (Demand during lead time of L weeks  ROP) z=F s -1 (CSL) ss=z L  D Demand during lead time is normally distributed with a mean of D L and a standard deviation of  L ROP=DL+Z L  D CSL 14

Finding Safety Stock with a Normal Probability Distribution for an 85 Percent CSL Safety stock = z  L Average demand during lead time Probability of stockout (1.0 - 0.85= 0.15) ROP CSL = 85% ? z Lz L 1 2 3 4:->ROP 15

Evaluating Cycle Service Level and Safety Inventory  CSL=Function ( ROP,D L,  L ) CSL= Prob (Demand during lead time of L weeks  ROP) z=F s -1 (CSL) ss=z L  D Demand during lead time is normally distributed with a mean of D L and a standard deviation of  L ROP=DL+Z L  D 16

Example Given the following data  Q = 10,000  ROP = 6,000  L = 2 weeks  D=2,500/week, σ D =500 2x2,500=5,000►D L =DL= =F(ROP, DL,  L )=F(6000,5000,707) =NORMDIST(6000,5000,707,1)=0.92 = 2 x500=707 ►CSL=Proability of not stocking out in a cycle ►  L = L  D 17

Normal Distribution in Excel Commands (Page 12) ► ► ► ► 18

Normal Distribution in Excel (Demo) 臺灣大學 郭瑞祥老師 19

Example Given the following data  Q = 10,000  ROP = 6,000  L = 2 weeks  D=2,500/week,  D=500  CSL=0.9 2x2,500=5,000►D L =DL= =F(ROP, DL, sL )=F(6000,5000,707) =NORMDIST(6000,5000,707,1)=0.92 = 2 x500=707 ►ss=F s -1 (CSL ) ►  L = L  D 20

Example Given the following data  D=2,500/week    D =500  L = 2 weeks  Q = 10,000,  CSL=0.9 2x2,500=5,000►D L =DL= =1.282x707=906 = 2 x500=707 ►ss=F s -1 (CSL)x  L =NORMDIST(CSL)x  L ►  L = L  D ►ROP= 2x2,500+906=5,906 21

Example Given the following data 5D=2,500/week    D =500 5L = 2 weeks 5Q = 10,000, 5CSL=0.9 2x2,500=5,000>D L =DL= =1.282x707=906 = 2 x500=707  ss=F s -1 (CSL)x  L =NORMDIST(CSL)x  L   L = L  D >ROP= 2x2,500+906=5,906 臺灣大學 郭瑞祥老師 22

Periodic Review System ►O►Other names are: fixed interval reorder system or periodic reorder system. ►D►Decision Rule Review the item’s inventory position IP every T time periods. Place an order equal to (OUL-IP) where OUL is the target inventory, that is, the desired IP just after placing a new order. ►T►The periodic review system has two parameters: T and OUL. ►H►Here Q varies, and time between orders (TBO) is fixed. 23

On-hand inventory Periodic Review System OUL Time Order placed IP L T L L Order received OH Q2Q2 IP Order placed Q1Q1 Q3Q3 Order placed T Protection interval OH IP 1 IP 3 IP 2 OUL 24

►The new order must be large enough to make the inventory position, IP, last not only beyond the next review, which is T periods from now, but also for one lead time (L) after the next review. IP must be enough to cover demand over a protection interval of T + L. ►OUL = Finding OUL + Safety stock for protection interval Average demand during protection interval 25

►Administratively convenient (such as each Friday) Selecting the Reorder Interval (T ) ►E►Example: Suppose D = 1200 /year and EOQ = 100 ►A►Approximation of EOQ 26

Example Given the following data  D=2,500/week   D =500  L = 2 weeks  T= 4weeks  CSL=0.9 (4+2)x2,500=15,000►D T+L =(T+L)D= =1,570 ►ss=F s -1 (CSL)x  T+L =F s -1 (0.9)x  T+L ►OUL= D T+L +ss = 1,5000+1,570=16,570 ►D T+L = T+L  D= (4+2) x500=1,225 27

Periodic System versus Continuous System 28

Evaluating Fill Rate Given a Replenishment Policy f (x) is density function of demand distribution during the lead time f r =1 - = ► In the case of normal distribution, we have ESC X=ROP    (X-ROP) f(x)dx ► For a continuous review policy Expected shortage per replenishment cycle (ESC) is the average units of demand that are not satisfied from inventory in stock per cycle Q Q-ESC Q ESC 29

Evaluating Fill Rate Given a Replenishment Policy f (x) is the density function of demand distribution during the lead time f r =1 - = ► In the case of normal distribution, we have ESC X=ROP    (X-ROP) f(x)dx ► For a continuous review policy Expected shortage per replenishment cycle (ESC) is the average units of demand that are not satisfied from inventory in stock per cycle ESC Q Q-ESC Q 30

Evaluating Fill Rate Given a Replenishment Policy f (x) is density function of demand distribution during the lead time f r =1 - = ► In the case of normal distribution, we have ESC X=ROP    (X-ROP) f(x)dx ► For a continuous review policy Expected shortage per replenishment cycle (ESC) is the average units of demand that are not satisfied from inventory in stock per cycle ESC Q Q-ESC Q 31

Evaluating Fill Rate Given a Replenishment Policy f (x) is density function of demand distribution during the lead time f r =1 - = ► In the case of normal distribution, we have ESC X=ROP    (X-ROP) f(x)dx ► For a continuous review policy Expected shortage per replenishment cycle (ESC) is the average units of demand that are not satisfied from inventory in stock per cycle ESC Q Q-ESC Q 32

Proof WIKIPEDIA 33

 L dz Proof Substituting Z=(X-D L )/  L and dx=  L dz, we have 34

Proof Substituting Z=(X-D L )/  L and dx=  L dz, we have 35

Proof 36

Proof 37

Proof dw=2zdz/2 dw=zdz 38

Proof ESC derivation 0 39

Proof 40

Evaluating Fill Rate Given a Replenishment Policy f (x) is density function of demand distribution during the lead time f r =1 - = ► In the case of normal distribution, we have ESC X=ROP    (X-ROP) f(x)dx ► For a continuous review policy Expected shortage per replenishment cycle (ESC) is the average units of demand that are not satisfied from inventory in stock per cycle ESC Q Q-ESC Q 41

Example For a continuous review system with the following data  Lot size,Q=10,000   D L =5,000    L = 707 ►ss=ROP-DL=6,000-5,000=1,000 ►ESC= -1,000[1-NORMDIST(1000/707,0,1,1)] f r = =0.9975 10,000 10,000-25 +707xNORMDIST(1000/707,0,1,1) =25 42

Excel-Demo For a continuous review system with the following data 5Lot size,Q=10,000   D L =5,000    L = 707 >ss=ROP-DL=6,000-5,000=1,000 >ESC= -1,000[1-NORMDIST(1000/707,0,1,1)] f r = =0.9975 10,000 10,000-25 +707xNORMDIST(1000/707,0,1,1) =25 臺灣大學 郭瑞祥老師 43

Factors Affecting Fill Rate ►Safety inventory Fill rate increases if safety inventory is increased. This also increases the cycle service level. ►Lot size Fill rate increases with the increase of the lot size even though cycle service level does not change. 44

Factors Affecting Fill Rate -- Page 42 f r = 1- ESC/Q CSL = F(ROP, D L, s L ) is independent of Q ►Safety inventory Fill rate increases if safety inventory is increased. This also increases the cycle service level. ► Lot size Fill rate increases on increasing the lot size even though cycle service level does not change. 45

Evaluating Safety Inventory Given Desired Fill Rate 5If desired fill rate is fr = 0.975, how much safety inventory should be held? ESC = (1 - fr)Q = 250 Solve 46

Excel-Demo 臺灣大學 郭瑞祥老師 47

Evaluating Safety Inventory Given Desired Fill Rate 5If desired fill rate is fr = 0.975, how much safety inventory should be held? ESC = (1 - fr)Q = 250 Solve 48

Evaluating Safety Inventory Given Fill Rate Fill RateSafety Inventory 97.5%67 98.0%183 98.5%321 99.0%499 99.5%767 The required safety inventory grows rapidly with an increase in the desired product availability (fill rate). 49

Two Managerial Levers to Reduce Safety Inventory Safety inventory increases with an increase in the lead time and the standard deviation of periodic demand. ►Reduce the underlying uncertainty of demand (  D ) ►Reduce the supplier lead time (L) –If lead time decreases by a factor of k, safety inventory in the retailer decreases by a factor of. –If  D is reduced by a factor of k, safety inventory decreases by a factor of k. –The reduction in  D can be achieved by reducing forecast uncertainty, such as by sharing demand information through the supply chain. –It is important for the retailer to share some of the resulting benefits to the supplier. 50

Impact of Supply (Lead time) Uncertainty on Safety Inventory ►Assume demand per period and replenishment lead time are normally distributed D:Average demand per period  D :Standard deviation of demand per period (demand uncertainty) L: Average lead time for replenishment S L :Standard deviation of lead time (supply uncertainty) ►Consider continuous review policy, we have: Demand during the lead time is N(D L,  L 2 ) 51

Example SLSL σLσL ss(units)ss(days) 615,0587.72 512,5706.44 410,0875.17 37,6163.90 25,1722.65 12,8281.45 01,3230.68 ► Suppose we have Required safety inventory, ► A reduction in lead time uncertainty can help reduce safety inventory 19,298 16,109 12,927 9,760 6,628 3,625 1,695 52

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