MULTI-PERIOD MODELS: PERFORMANCE MEASURES. LM6001 Inventory management

Transcription

1 MULTI-PERIOD MODELS: PERFORMANCE MEASURES LM6001 Inventory management 1

2 TODAY S AGENDA Given the system parameters Continuous review (s,q): Lot size, reorder point (ROP) Periodic review (R,S): Order-up-to level (OUTL) Determine key performance measures, e.g., Average inventory Average backorder Expected service level Fill rate In-stock probability Annual average cost 2

3 Continuous (s,q) Periodic (R,S) Effective demand, D e D L D L+R (μ e, σ e ) (μ 1 L, σ 1 L ) ( μ 1 (L + R), σ 1 (L + R) ) Performance measure z = (s μ e )/σ e z = (S μ e )/σ e (Expected) order size Q തQ = μ 1 R = E D R Cycle stock SS = E[IL at end of period (just before replenishment arrives)] (Order size)/2 E s D e = s μ e E S D e = S μ e E[Backorder at end of period], തB E D e s + = σ e L(z) E D e S + = σ e L(z) E[On-hand inv at end of period] E s D e + = SS + തB SS E S D e + = SS + തB SS E[On-hand avg inv] = (Beginning+Ending)/2 (Cycle stock) + E[On-hand inv at end of period] Cycle SL, in-stock probability P(D e s) P(D e S) Fill rate, 1 E[Backorder in one cycle] E[Demand during one cycle] 1 ത B Q B 1 ത തQ = 1 B ത μ 1 R (Expected) order frequency E D 1 /Q 1 P(D R 0) R Decision rule s = μ e + k σ e S = μ e + k σ e Cycle SL, a Fill rate k = L 1 k = Φ 1 (α) 1 β Q σ e k = L 1 1 β (μ 1 R) σ e 3

4 E[on-hand at end of cycle] A cycle is defined as the duration which two successive replenishment orders are received. Effective demand is the demand during lead time (DDLT) 4

5 SAFETY STOCK Recall Inventory on-order, IO: # of units that we ordered in previous periods that we have not yet received. IP = IL + IO Inventory on-hand I = IL + = max IL, 0 Backorder: Total amount of demand that has occurred but has not been satisfied B = IL = max( IL, 0) Safety stock is defined as the average inventory level at the end of cycle IL = IL + - IL (Avg IL at the end) = (Avg On-hand at the end) (Avg Backorder at end) SS = (Avg on-hand) (Avg backorder) Thus, Avg on-hand = SS + (Avg Back order) 5

6 Recall Inventory on-hand I = IL + = max IL, 0 Backorder: Total amount of demand that has occurred but has not been satisfied B = IL = max( IL, 0) Safety stock is defined as the average inventory level at the end of cycle IL = IL + - IL (Avg IL at the end) = (Avg On-hand at the end) (Avg Backorder at end) SS = (Avg on-hand) (Avg backorder) Thus, Avg on-hand = SS + (Avg Back order) Cycle Avg IL at end of cycle on-hand backorder

7 AVERAGE OF A FUNCTION AVERAGE ON-HAND INV 7

8 Cycle service level, or in-stock probability, is the probability that we are in stock in each cycle. P(D L ROP) Fill rate is the expected fraction of demand served immediately from stock. 1 തB E demand in one cyle = 1 ത B Q Over 10 cycles: 2 out of 10 result in stockout; so, the in-stock probability is 8/10=80% Total demand is 907, and the backorder is 27, so the demand served from stock is (907-27), and the fill rate is = = 97.03% 8

9 ROP 9

10 Cycle stock Safety stock Average backorder at end of cycle Average on-hand inventory over time Cycle service level (in-stock probability) Fill rate Order frequency Average backorder cost Average holding cost Average setup cost ROP Lot size (Q) 10

11 Originally stock recording systems used to include stock control levels as minimum and maximum stock levels. The use of a minimum stock level for order control is not sensible, since the minimum occurs immediately before delivery. Items have to be ordered will in advance of this, so control is through a reorder point, not a minimum stock. However, a minimum stock level is vital to ensure that there is warning of low stocks. In a stock control system, there is a need for both reorder point and minimum stock level, i.e., safety stock. 11

12 S R R R = Review period R S = Order-up-to level (OUTL) Effective demand is the demand during lead time plus review 12

13 Continuous review Cycle service level, or in-stock probability, is the probability that we are in stock in each cycle. P(D L ROP) Fill rate is the expected fraction of demand served immediately from stock. 1 തB E demand in one cyle = 1 ത B Q Periodic review In-stock probability is the probability that we are in stock in each cycle. P D L+R OUTL Fill rate is the expected fraction of demand served immediately from stock. 1 തB E demand during review period = 1 ത B μ 1 R 13

14 Weekly demand is normally distributed Mean of weekly demand μ 1 = 85 Standard deviation of weekly demand σ 1 = 17.5 Lead time L=3 weeks 1. Find ROP for 90% CSL 2. Find ROP for 96% FR. Suppose lot size Q=200 Effective demand Mean, AVGL, μ L = μ 1 L = 85 3 = 255 Standard deviation, STDL, σ L = σ 1 L = = Given 90% CSL, α = 0.90 Safety factor k=φ 1 (0.90)=1.28 SS = k σ L =1.28(30.31)=38.84 ROP = μ L + SS= = Given 96% FR, β = Safety factor k=l 1 1 β Q = L σ L SS = k σ L =0.31(30.31)=9.40 ROP = μ L + SS= = = L 1 (0.2639)=

15 Weekly demand is normally distributed Mean of weekly demand μ 1 = 85 Standard deviation of weekly demand σ 1 = 17.5 Lead time L=3 weeks. (1 year = 48 weeks) Suppose Q=200, s=rop=278. Calculate Effective demand Mean, AVGL, μ L = μ 1 L = 85 3 = 255 Standard deviation, STDL, σ L = σ 1 L = = ROP, s =278. Then, z = s μ L σ L = Avg on-hand inventory, in-transit inventory 2. Avg backorder 3. Order frequency 4. Service level: CSL, FR = L(z) = L(0.76)= Lot size, Q = 200. Then Cycle stock (avg inventory) = Q/2 = 200/2 = 100. SS = s μ L = = 23 E[backorder at end of cycle] തB = σ L L z = (30.31)( ) = 3.92 units E[on-hand inv] = (cycle stock) + SS + തB = = units E[in-transit inv] = ((85)(52))(3/52) = μ L = 255 units (Recall Little s Law L = λw where L is avg # of customers in system, λ avg no of arrivals entering the system, W avg time a customer spends in system) Order freq = μ 1 /Q=85/200= times per week (or 48*0.4250=20.4 times/yr) CSL = Φ z = Φ(0.76)=0.776=77.6% FR = 1 തB /Q = /200= = 98.01% 15

16 Assume that 1 year = 12 months = 48 weeks Monthly demand is normally distributed with mean 85 units and standard deviation 26 units. The replenishment lead time is 6 weeks. Ordering cost (THB/order) Holding cost factor (/Year) 27% Unit cost (THB/unit) Unit Holding cost (THB/unit/Yr) Given 90% CSL, find safety stock and ROP. 2. Suppose that the order quantity the company is using for this SKU is 500 units. The company wants to maintain 90% CSL. a) What is the average inventory level? b) What is the average holding cost per year? Also average setup cost per year 3. Suppose that the lead time is reduced to 3 weeks. Assume order quantity of 50 and CSL of 90%. Repeat problem 1 and The order quantity the company is using for this SKU is 50 units Suppose lead time is 6 weeks. You want 99% SL. What is the average inventory level? 16

17 Assume that 1 year = 12 months = 48 weeks Monthly demand is normally distributed with mean 85 units and standard deviation 26 units. The replenishment lead time is 6 weeks. Ordering cost (THB/order) Holding cost factor (/Year) 27% Unit cost (THB/unit) Unit Holding cost (THB/unit/Yr) Given 90% CSL, find safety stock and ROP. 2. Suppose that the order quantity the company is using for this SKU is 10,000 units. The company wants to maintain 90% CSL. [Ans: ROP=169] a) What is the average inventory level? [Ans: /2 = ] b) What is the average holding cost per year? Average setup cost per year = 500*((12*85)/500)=500*(2.04)=1020 THB/yr 3. Suppose that the lead time is reduced to 3 weeks. Assume order quantity of 50 and CSL of 90%. Repeat problem 1 and The order quantity the company is using for this SKU is 50 units Suppose lead time is 6 weeks. You want 99% SL. What is the average inventory level? 17

18 If you wish to divide demand distribution from a long period length (e.g., a month) into n short periods (e.g., a week), then E[demand in the short period] = E[demand in the long period]/n Stdev of demand in short period = (Stdev of demand in long period)/ n If you wish to combine a demand distribution from n short period lengths (e.g., a week ) into one long period (e.g., a three-week period), then E[demand in the long period] = E[demand in the long period]*n Stdev of demand in long period = (Stdev of demand in long period)* n These assume demands in each period are independent and identically distributed. Monthly demand is normally distributed with mean 85 units and standard deviation 26 units. The replenishment lead time is 6 weeks. Find mean and standard deviation of demand during lead time. (1 month = 4 weeks) Sol 1 LT = 6 wks = 6/4 = 1.5 month μ L = = units σ L = = units Sol 2 Demand in one week has mean 85/4=21.25 units/week, standard deviation 26/ 4=13 units/week. μ L = 21.25(6) = units σ L = 13 6 = units 18

19 Weekly demand is normally distributed Mean of weekly demand μ 1 = 85 Standard deviation of weekly demand σ 1 = 17.5 Effective demand Mean, AVGLR, μ L+R = μ 1 L + R = 85 5 =425 Standard deviation, STDLR, σ L+R = σ 1 (L + R) = = Expected order size = μ 1 R=(85)(2)=170 Desired CSL α = 0.90 Safety factor k=φ 1 (0.90)=1.28 SS = k σ L+R =1.28(39.13)=50.15 OUTL = μ L+R + SS= = Periodic review with lead time L=3 weeks and review period R=2 weeks 1. Given 90% CSL, find OUTL 2. Given 96% FR, find OUTL Desired FR β = Safety factor k=l 1 1 β E[ order size ] σ L+R = L 1 ( )=0.58 SS = k σ L+R =0.58(39.13)=22.70 OUTL = μ L+R + SS= =

20 Weekly demand is normally distributed Mean of weekly demand μ 1 = 85 Standard deviation of weekly demand σ 1 = 17.5 Periodic review with lead time L=3 weeks and review period R=2 weeks Given OUTL, S=450, find performance measures Effective demand Mean, AVGLR, μ L+R = μ 1 L + R = 85 5 =425 Standard deviation, STDLR, σ L+R = σ 1 (L + R) = = Expected order size = μ 1 R=(85)(2)=170 Expected order size = μ 1 R=(85)(2)=170 Then, z = S μ L+R σ L+R =( )/39.13=0.64 L(z) = L(0.64)= Cycle stock = E(order size)/2 = 170/2 = 85. SS = S μ L+R = =25 E[backorder at end of cycle] തB = σ L+R L z =(39.13)( )= E[on-hand avg inv] = (cycle stock) + SS + തB = = = CSL = Φ z = Φ(0.64)=0.7385=73.85% FR = 1- തB /E(order size) = /170=0.9630=96.30% Order frequency =(1/R)P(D R > 0) = (1/(2/48))*[1 Φ( 170/(17.5 2))] =24*[1-Φ( 6.869)] = 24 times/year 20

21 Assume that 1 year = 12 months = 48 weeks Monthly demand is normally distributed with mean 85 units and standard deviation 26 units. The replenishment lead is 6 weeks. Length of review period is 1 week (say every Saturday) Ordering cost (THB/order) Holding cost factor (/Year) 27% Unit cost (THB/unit) Unit Holding cost (THB/unit/Yr) Given 90% CSL, find safety stock and OUTL. 2. The company wants to maintain 90% CSL. a) What is the average inventory level? b) What is the observed fill rate? c) What is the average annual cost? 3. Suppose that the length of the review period time increases to 2 weeks. Assume CSL of 90%. Repeat problem 2. 21

22 Assume that 1 year = 12 months = 48 weeks Monthly demand is normally distributed with mean 85 units and standard deviation 26 units. The replenishment lead is 6 weeks. Length of review period is 1 week (say every Saturday) Ordering cost (THB/order) Holding cost factor (/Year) 27% Unit cost (THB/unit) Unit Holding cost (THB/unit/Yr) Given 90% CSL, find safety stock and OUTL. [Ans OUTL=193] 2. The company wants to maintain 90% CSL. a) What is the average inventory level? [B + SS + cycle stk = /2= ] b) What is the observed fill rate? [Ans: /21.25 = 92.42%] c) What is the average annual cost? OF = 48*P(D R > 0) = 48*(1-P(D R 0)) = 48*[1-Φ( )] =48*( ) = Avg setup cost per yr = 500*OF = 500*(45.55). Avg holding cost per yr = 87.48*(avg inv) = 87.48*

23 Lead Time, L D L Pipeline inv Review Period, R D R E[order size] L+R D L+R Fill rate Demand in one period, D 1 Order-up-to level (OUTL), S E[backorder] Safety stock E[Inv] Loss Fn CDF In-stock probability 23

24 The holding cost (a.k.a. carrying cost or inventory cost) is the sum of all costs that are proportional to the amount of inventory physically on hand at any point in time. Four main components are 1. Capital cost or opportunity cost (return that company could make on money tied up in inventory) 2. Inventory service cost (e.g., insurances and taxes) 3. Storage cost (warehousing space-related costs that change with level of inventory) 4. Inventory risk or shrinkage cost (e.g., obsolescence, damage) Suppose that Cost of capital 28% Cost of storage 6% Taxes & insurance 2% Breakage & spoilage 1% Total percentage 37% An item valued at 180 THB would have an annual holding cost of h = c r = (180)(.37) = 66.6 THB/unit/year Suppose that we pay for the items upfront and take ownership of them as soon as the order is placed, we should also consider the cost of the 'pipeline inventory.' Holding pipeline inventory is a little cheaper than holding inventory on hand. Cost of carrying inventory on hand includes both the cost of money (which is 28% in the above example) and the cost of storing and managing the inventory in the DC (which accounts for the rest). 24

25 What is the appropriate level of service? May be determined by the downstream customer Retailer may require the supplier, to maintain a specific service level Supplier will use that target to manage its own inventory Facility may have flexibility to choose appropriate level of service Trade-offs Everything else being equal: The higher the service level, the higher the inventory level. For the same inventory level, the longer the lead time to the facility, the lower the level of service provided by the facility. The lower the inventory level, the higher the impact of a unit of inventory on service level and hence on expected profit 25

26 More inventory is needed as demand uncertainty increases for any fixed fill rate. The required inventory is more sensitive to the fill rate level as demand uncertainty increases The tradeoff between inventory and fill rate with Normally distributed demand and a mean of 100. The curves differ in the standard deviation of demand: 60, 50, 40, 30, 20, 10 from top to bottom. 26

27 Expected inventory Reducing the lead time reduces expected inventory, especially as the target fill rate increases Lead time The impact of lead time on expected inventory for four fill rate targets, 99.9%, 99.5%, 99.0% and 98%, top curve to bottom curve respectively. Demand in one period is Normally distributed with mean 100 and standard deviation

28 Inventory Lead time Expected inventory (diamonds) and total inventory (squares), which is expected inventory plus pipeline inventory, with a 99.9% fill rate requirement and demand in one period is Normally distributed with mean 100 and standard deviation 60 Reducing the lead time reduces expected inventory and pipeline inventory The impact on pipeline inventory can be even more dramatic that the impact on expected inventory 28

29 Wal-Mart has consistently improved its annual inventory turns (approximately) the last two decades. While a number of a factors could explain this dramatic improvement, reductions in its lead time is surely a significant factor. These reductions were achieved through numerous initiatives. To improve the lead time Wal-Mart receives from its suppliers to its DCs, Wal-Mart build electronic linkages with its suppliers. These linkages ensure that no time is wasted in order transmission and order processing. Furthermore, they allow Wal-Mart to share demand data with suppliers so that suppliers can ensure they have enough capacity to meet Wal-Mart s needs on a timely basis. (Lead times can be quite long if a supplier runs out of critical components or if the supplier runs out of capacity.) Next, Wal-Mart designed its DCs and logistics so that inventory spends very little time in the DCs. For example, a popular product such as Crest toothpaste generally spends less than eight hours in a Wal-Mart distribution center Through a process called cross-docking, inventory is moved from in-bound trucks directly to out-bound trucks, that is, it is never actually put on a shelf in the warehouse (Nelson 1999). Finally, via computerized replenishment and control of its own delivery fleet of vehicles, Wal- Mart s lead time from its DCs to its stores is as fast as it can be. As a result of the combined impact of these initiatives, Wal-Mart is able to sell much of its inventory even before it must pay for that inventory, a rather enviable situation for any retailer. 29

30 1. Annual average setup cost (order freq)*(setup cost) = (OF)*(K) OF units/year = (annual demand rate)/e[order size] = d/e[order size] Setup cost = K THB/order Fixed ordering cost + (TL transportation cost) 2. Annual average holding cost (unit holding cost)*(avg on-hand inv) Unit holding cost, h THB/unit/year h = (unit cost)*(interest rate) = c*r unit cost = (variable ordering cost) + (LTL transportation cost) Avg on-hand inv = (Cycle stock) + (SS + E[backorder]) Avg in-transit (pipeline) inventory 3. Annual average backorder cost 4. Annual acquisition cost No discount: (unit cost)*(annual demand rate) = c*d All-unit discount; unit cost depends on order quantity 30

31 b 1 cost per stockout occasion (THB/occasion) Annual avg backorder cost = b 1 (order freq)*(stockout prob) THB b 1 occasion * OF cycle year * stockout prob occasion cycle Recall E[Backorder at end of cycle] = σ e L(z) := തB b 2 cost per unit short (THB/unit) Annual avg backorder cost = b 2 (order freq)* തB b 2 THB unit * OF cycle year * തB unit cycle b 3 cost per unit short per yr (THB/unit/yr) 31

32 Average yearly demand is d = 200 units/year Unit holding cost = (24%)(2) = 0.48 THB/unit/year Cost per stockout occasion b 1 = 300 THB Demand during lead time μ L = 50 units, σ L = 21 units The order quantity is given as Q = 129 units ROP = s units b1 = cost per stockout occasion THB/occasion c = unit cost 2.00 THB/unit h = unit holding cost 0.48 THB/unit/year d = annual demand rate unit/year Q = order quantity unit OF = d/q 1.55 times/year mul units sigmal units stockout prob annual avg backorder cost 3.67 THB/year annual avg holding cost THB/year annual cost THB/year Yearly average cost Yearly average setup cost = K*OF = K*(d/Q) Yearly backorder cost = b 1 *OF*(stockout prob) = b 1 d/q P(D L > s) Yearly holding cost = h*(q/2 + SS) = h*(q/2 + (s-μ L )) We want to choose ROP (s) to minimize yearly average cost Using Excel Solver (GRGNonlinear), we find that the optimal ROP is s =

33 Average yearly demand is d = 200 units/year Unit holding cost = (24%)(2) = 0.48 THB/unit/year Cost per stockout occasion b 1 = 300 THB Demand during lead time μ L = 50 units, σ L = 21 units The order quantity is given as Q = 129 units Yearly average cost Yearly average setup cost = K*OF = K*(d/Q) Yearly backorder cost = b 1 *OF*(stockout prob) = b 1 d/q P(D L > s) Yearly holding cost = h*(q/2 + SS) = h*(q/2 + (s-μ L )) ROP = s units b1 = cost per stockout occasion THB/occasion c = unit cost 2.00 THB/unit h = unit holding cost 0.48 THB/unit/year d = annual demand rate unit/year Q = order quantity unit OF = d/q 1.55 times/year mul units sigmal units stockout prob annual avg backorder cost 3.67 THB/year annual avg holding cost THB/year annual cost THB/year We want to choose ROP (s) to minimize yearly average cost Using Excel Solver (GRGNonlinear), we find that the optimal ROP is s = Given that the setup cost is K = 120 THB/order and that the required CSL = 99.9%. Determine the optimal pair of (s,q). 33

34 Example CSCMP: A regional retailer, Value Dime and Five (VDF) has one DC that serves 500 stores. It only sells one SKU of toilet paper. It replenishes stores in case pack quantities, and each case contains 80 rolls. VDF only buys it by the truckload, which holds Q = 560 cases and pays 40 USD/case. The transportation cost per truckload is 400 USD. All other costs of ordering associated with purchasing, accounts payable, receiving about 50 USD/order. Fixed order cost K = = 450 USD/order Demand during LT~normal mean of μ L =80 cases and a standard deviation of σ L =30 cases. Lead time is 1 day. Lead time 1 day Inventory carrying cost rate 25% /yr. In-transit mul inventory carrying cost rate 23% /yr sigmal Unit carrying cost = (0.25)(40) = 10 USD/case/yr Unit in-transit carrying cost = (0.23)(40) = 9.2 USD/case/yr The back-order cost is about b 2 =5 USD/case. VDF open every day of the yr; i.e., # of days in 1 yr=365. Average daily demand for DC is 80 cases/day. Average yearly demand d=(80)(365)=29200 cases/yr 80 units 30 units fixed ordering cost 50 USD/order c = unit cost 40 THB/case holding cost factor for inv 0.25 per year holding cost factor for in-transit 0.23 per year backorder cost per unit short 5 USD/unit TL cost 400 USD/truckload 1 TL = 560 case Q ROP 560 case 100 case 34

35 Input parameters Cost given Q = 560, ROP = 100 Lead time 1 day mul 80 units sigmal 30 units fixed ordering cost 50 USD/order c = unit cost 40 THB/case holding cost factor for inv 0.25 per year holding cost factor for in-transit 0.23 per year backorder cost per unit short 5 USD/unit TL cost 400 USD/truckload 1 TL = 560 case Q ROP 560 case 100 case What-If using 1, 2, 3, 4 truckloads? # of truckload order quantity annual avg cost 31, , , , , d = annual demand rate 29, case/year OF = d/q setup cost = TL cost + fixed ordering cost THB/time annual ordering and transportation cost 23, THB/year cycle stock z = (ROP-muL)/sigmaL 0.67 L(z) 0.15 E[backorder per cycle] = sigmal*l(z) := B 4.53 unit annual avg backorder cost = b2*of*b 1, THB/year SS = ROP-muL on-hand inventory annual avg holding cost for on-hand in-transit stock = mul annual avg holding cost for in-transit annual avg cost 20 units units 3000 THB/year 80 units 736 THB/year 28, THB/year 35

36 36 PERIODIC REVIEW: CONTROLLING ORDERING COSTS Input parameters Wkly demand AVG 100 fixed ordering cost 275 STD 75 annual holding cost 12.5 Lead time,l 8 # of weeks in one yr 52 Review period Target in-stock prob OUTL cycle stock safety stock Avg backorder Avg ending inv Avg on-hand inv order freq (times/yr) annual fixedordering c 3,561 6,938 3,561 1,787 inventory cost 10,408 8,470 10,408 14,134 total cost 13,969 15,408 13,969 15,922 Suppose that the review period is four weeks, R=4 weeks. Effective demand, demand during L+R =8+4 =12 weeks, has mean μ L+R = μ 1 L + R =100(12)=1200 and standard deviation σ L+R = σ 1 12= The order frequency 1 P(D R 0) 1 Φ(( )/(75 4) = = /week or OF = 52(0.2490) = /year. R 4 The annual fixed ordering cost K*(OF) = 275(12.95)=3561 THB/yr. The expected order size is തQ = E D R = μ 1 R=100(4)=400, and the cycle stock is തQ/2=400/2 = 200. The desired in-stock probability is 99.25%. Safety factor z=φ 1 (0.9925)=2.43. SS =2.43σ L+R =2.43(259.81) = 632 OULT =μ L+R +SS = =1832. E[backorder] തB = σ L+R L(z)= L(2.43) = ( )= E[ending inv] ҧ I=SS+ തB= = E[On-hand inv] = (cycle stock) + E[ending inv] = = The annual holding cost h*e[on-hand inv] = 12.5( )= The expected total cost = K*(OF) + h*e[on-hand inv] = = 13,969.37

37 37 PERIODIC REVIEW: CONTROLLING ORDERING COSTS Review period Target in-stock prob OUTL cycle stock safety stock Avg backorder Avg ending inv Avg on-hand inv order freq (times/yr) annual fixedordering c 12,996 6,938 4,717 3,561 2,856 2,382 2,042 1,787 inventory cost 7,482 8,470 9,458 10,408 11,358 12,296 13,222 14,134 total cost 20,478 15,408 14,175 13,969 14,214 14,678 15,264 15,922 Our best option is to set the period length to four weeks. A shorter period length results in too many orders so the extra ordering costs dominate the reduced holding costs. A longer period suffers from too much inventory.

38 PERIODIC REVIEW: CONTROLLING ORDERING COSTS 38 Wkly demand Input parameters AVG 100 fixed ordering cost 275 STD 75 annual holding cost 12.5 Lead time,l 8 # of weeks in one yr 52 Although this analysis has been done in the context of the OUT model, it may very well remind you of another model, the EOQ model. Recall that in the EOQ model, there is a fixed cost per order/batch K = 275 THB, a holding cost per unit per unit of time h = 12.5 THB/unit/yr, and demand occurs at a constant flow rate d. In this case, yearly demand d = (52)(100)=5200 unit/yr. 2Kd = 2(275)(5200) h Review period Target in-stock prob OUTL cycle stock safety stock Avg backorder Avg ending inv Avg on-hand inv order freq (times/yr) annual fixedordering c 12,996 6,938 4,717 3,561 2,856 2,382 2,042 1,787 inventory cost 7,482 8,470 9,458 10,408 11,358 12,296 13,222 14,134 total cost 20,478 15,408 14,175 13,969 14,214 14,678 15,264 15,922 The EOQ is =478. This implies a cycle time of EOQ/d = 478/ yr = (478/5200)*52 = 4.78 weeks: An order should be submitted every 4.78 weeks. The key difference between our model and the EOQ model is that here we have random demand whereas the EOQ model assumes demand occurs at a constant rate. Even though the OUT model and the EOQ models are different, the EOQ model gives a very good recommendation for the period length.

39 39 PERIODIC REVIEW & JOINT REPLENISHMENT EOQ formula gives us an easy way to check if our period length is reasonable. One advantage of this approach is that we submit orders on a regular schedule. This is a useful feature if we need to coordinate the orders across multiple items. For example, since we incur a fixed cost per truck shipment, we generally deliver many different products on each truck, because no single product s demand is large enough to fill a truck. In that situation, it is quite useful to order items at the same time so that the truck can be loaded quickly and we can ensure a reasonably full shipment (given that there is a fixed cost per shipment, it makes sense to utilize the cargo capacity as much as possible). Therefore, we need only ensure that the order times of different products align.

40 40 CONTROLLING ORDERING COSTS Instead of using fixed order intervals, as in the OUT model, we could control ordering costs by imposing a minimum order quantity. For example, we could wait for Q units of demand to occur and then order exactly Q units. With such a policy, we would order on average every Q/d units of time, but due to randomness in demand, the time between orders would vary. Not surprisingly, the EOQ quantity provides an excellent recommendation for that minimum order quantity. Important insight is that it is possible to control ordering costs by 1) restricting to a periodic schedule of order, or 2) restricting to a fixed order quantity. With the first option, there is little variability in the timing of orders, which facilitates the coordination of orders across multiple items, but the order quantities are variable (which may increase handling costs). With the second option, the order quantities are not variable (we always order Q), but the timing of those orders varies.