Presentation on theme: "Chapter 10: Inventory Types of Inventory and Demand Availability"— Presentation transcript:
1 Chapter 10: Inventory Types of Inventory and Demand Availability Cost vs. Service TradeoffPull vs. PushReorder Point SystemPeriodic Review SystemJoint OrderingNumber of Stocking PointsInvestment LimitJust-In-Time
3 Inventory Inventory includes: Located in: Raw materials, Supplies, Components, Work-in-progress, Finished goods.Located in:Warehouses, Production facility, Vehicles, Store shelves.Cost is usually 20-40% of the item value per year!
4 Why Keep Inventories? Positive effects: Negative Effects: Economies of scale in production & transportation.Coordinate supply and demand.Customer service.Part of production.Negative Effects:Money tied up could be better spent elsewhere.Inventories often hide quality problems.Encourages local, not system-wide view.
5 Types of InventoriesRegular (cycle) stock: to meet expected demand between orders.Safety stock: to protect against unexpected demand.Due to larger than expected demand or longer than expected lead time.Lead time=time between placing and receiving order.Pipeline inventory: inventory in transit.Speculation inventory: precious metals, oil, etc.Obsolete/Shrinkage stock: out-of-date, lost, stolen, etc.
6 Types of Demand Terminating: Derived (dependent): Perpetual (continual):Mean and standard deviation (or variance) of demand are known (or can be calculated).Use repetitive ordering.Seasonal or Spike:Order once (or a few time) per season.Lumpy: hard to predict.Often standard deviation > mean.Terminating:Demand will end at known time.Derived (dependent):Depends on demand for another item.
7 Performance Measures Turnover ratio: Availability: Annual demand Service Level = SLFill Rate = FRWeighted Average Fill Rate = WAFRAnnual demandTurnover ratio =Average inventory
8 Measuring Availability: SL Want product available in the right amount, in the right place, at the right time.For 1 item: SLi = Service Level for item iSLi = Probability that item i is in stock.= 1 - Probability that item i is out-of-stock.Expected number of units out of stock/year for item iAnnual demand for item iSLi = 1 -
9 Measuring Availability: FR and WAFR For 1 order of several items: FRj = Fill Rate for order jFRj = Product of service levels for items ordered.For all orders: WAFR (Weighted Average Fill Rate)Sum over all orders of (FRj) x (frequency of order j).FRj = SL1 x SL2 x SL3 x ...
11 Fundamental Tradeoff $ Level of Service Level of Service vs. Cost Revenue
12 Fundamental Tradeoff Level of Service (availability) vs. Cost Higher service levels -> More inventory.-> Higher cost.Higher service levels -> Better availability.-> Fewer stockouts.-> Higher revenue.
13 Inventory Costs Procurement (order) cost: Carrying or Holding cost: To prepare, process, transmit, handle order.Carrying or Holding cost:Proportional to amount (average value) of inventory.Capital costs - for $ tied up (80%).Space costs - for space used.Service and risk costs - insurance, taxes, theft, spoilage, obsolecence, etc.Out-of-stock costs (if order can not be filled from stock).Lost sales cost - current and future orders.Backorder cost - for extra processing, handling, transportation, etc.
14 Fundamental Cost Tradeoff Inventory carrying cost vs. Order & Stockout costLarger inventory -> Higher carrying costs.Larger inventory -> Fewer larger orders.-> Lower order costs.Larger inventory -> Better availability.-> Few stockouts.-> Lower stockout costs.
15 Retail Stockouts On average 8-12% of items are not available! Causes: Inadequate store orders.Not knowing store is out-of-stock.Poor promotion forecasting.Not enough shelf space.Backroom inventory not restocked.Replenishment warehouse did not have enoughTrue for only 3% of stockouts.
16 Pull vs. Push Systems Pull: Push: Treat each stocking point independent of others.Each orders independently and “pulls” items in.Common in retail.Push:Set inventory levels collectively.Allows purchasing, production and transportation economies of scale.May be required if large amounts are acquired at one time.
17 Push Inventory Control Acquire a large amount.Allocate amount among stocking points (warehouses) based on:Forecasted demand and standard deviation.Current stock on hand.Service levels.Locations with larger demand or higher service levels are allocated more.Locations with more inventory on hand are allocated less.
18 Push Inventory Control TRi = Total requirements for warehouse iNRi = Net requirements at iTotal excess = Amount available - NR for all warehousesDemand % = (Forecast demand at i)/(Total forecast demand)Allocation for i = NRi + (Total excess) x (Demand %)= Forecast demand at i + Safety stock at i= Forecast demand at i + z x Forecast error at i= TRi - Current inventory at iz is from Appendix A
19 Push Inventory Control Example Allocate 60,000 cases of product among two warehouses based on the following data.Current Forecast ForecastWarehouse Inventory Demand Error SL, ,000 5,, ,000 3, ,000
20 Push Inventory Control Example Current Forecast Forecast DemandWarehouse Inventory Demand Error SL %, , ,, , , ,000TR1 = 20, x 5,000 = 26,400TR2 = 15, x 3,000 = 21,150NR1 = 26, ,000 = 16,400NR2 = 21, ,000 = 16,150Total Excess = 60, , ,150 = 27,450Allocation for 1 = 16, ,450 x (0.5714) = 32,086 casesAllocation for 2 = 16, ,450 x (0.4286) = 27,914 cases
21 Pull Inventory Control - Repetitive Ordering For perpetual (continual) demand.Treat each stocking point independently.Consider 1 product at 1 location.Determine:How much to order:When to (re)order:
22 Pull Inventory Control - Repetitive Ordering For perpetual (continual) demand.Treat each stocking point independently.Consider 1 product art 1 location.Reorder PeriodicDetermine: Point System Review SystemHow much to order: Q M-qiWhen to (re)order: ROP T
23 Reorder Point System Order amount Q when inventory falls to level ROP. Constant order amount (Q).Variable order interval.
24 Reorder Point System Each increase in inventory is size Q. LT1 LT2 LT3 Place 1storderPlace 2ndorderPlace 3rdorderReceive3rd orderReceive1st orderReceive2nd orderEach increase in inventory is size Q.
25 Reorder Point System LT1 LT2 LT3 Place 1st order Place 2nd order Place 3rdorderReceive3rd orderReceive1st orderReceive2nd orderTime between1st & 2nd orderTime between2nd & 3rd order
26 Periodic Review System Order amount M-qi every T time units.Constant order interval (T=20 below).Variable order amount.
27 Periodic Review System - T=20 days LT1LT2LT3Place 3rdorderPlace 2ndorderReceive3rd orderPlace 1storderReceive1st orderReceive2nd orderEach increase in inventory is size M-amount on hand.(M=90 in this example.)
29 Optimal Inventory Control For perpetual (continual) demand.Treat each stocking point independently.Consider 1 product art 1 location.Reorder PeriodicDetermine: Point System Review SystemHow much to order: Q M-qiWhen to (re)order: ROP TFind optimal values for: Q & ROP or for M & T.
30 Inventory Variables D = demand (usually annual) d = demand rate S = order cost ($/order) LT = (average) lead timeI = carrying cost k = stockout cost(% of value/unit time) P = probability of being inC = item value ($/item) stock during lead timesd = std. deviation of demandsLT = std. deviation of lead times’d = std. deviation of demand during lead timeQ = order quantityN = number of orders/yearTC = total cost (usually annual)ROP = reorder pointT = time between orders
31 Simplest Case - Constant demand and lead time No variability in demand and lead time (sd = 0, sLT = 0).Will never have a stock out.InventoryTimeROPQSuppose: d = 4/day and LT = 3 daysThen ROP = 12 (ROP = d x LT)
32 Constant demand and lead time InventoryTimeROPQTC = Order cost + Inventory carrying costOrder cost = N x S = (D/Q) x SCarrying cost = Average inventory level x C x I= (Q/2) x C x I
33 Economic Order Quantity (EOQ) InventoryTimeROPQTC =QDS + IC2Select Q to minimize total cost.Set derivative of TC with respect to Q equal to zero.0 = -Q2DS +2IC2DSQ =IC
34 Optimal Ordering 2DS Economic order quantity: Q* = IC InventoryTimeROPQQ* =IC2DSEconomic order quantity:Optimal number of orders/year:Optimal time between orders:Optimal cost:DQ*Q*DTC =Q*DS + IC2
35 Example D = 10,000/year S = $61.25/order I = 20%/year C = $50/item 2DS Q* =IC2DS2(10,000)(61.25)== 350 units/order(0.2)(50)TC =Q*DS + IC210,000350=(61.25) + (0.2)(50)3502= = $3500/year10,000N == orders/year350350T == years = 1.82 weeks10,000
36 Example - continued Q* = 350 units/order N = 28.57 orders/year T = 1.82 weeksThis is not a very convenient schedule for ordering!Suppose you order every 2 weeks:T = 2 weeks, so N = 26 orders/yearD10,000Q === units/order (10% over EOQ)N26DQ10,000384.6S + IC=(61.25) + (0.2)(50)TC =Q2384.62= = $ /yearQ = is 9.9% over EOQ, but TC is only 0.4% over optimal cost!!!
37 Model is Robust Q* = 350 TC = $3500/year Total Cost Carrying Cost Order Cost
38 Model is Robust Changing Q by 20% increases cost by a few percent. Carrying CostOrder CostTotal Cost
39 Model is RobustA small change in Q (or N or T) causes very little increase in the total cost.Changing Q by 10% increases cost < 1%.Changing Q changes N=D/Q, T=Q/D and TC.Changing N or T changes Q!A near optimal order plan, will have a very near optimal cost.You can adjust values to fit business operations.Order every other week vs. every 1.82 weeks.Order in multiples of 100 if required rather than Q*.
40 Non-instantaneous Resupply Produce several products on same equipment.Consider one product.p = production rate (for example, units/day)d = demand rate (for example, units/day)Inventory increases slowly while it is produced.Inventory decreases once production stops.Stop producing this product when inventory is “large enough”.
41 Inventory Level Suppose: p = 10/day (while producing this product). TimeProduce QDo not produceSlope=7Slope=-3Suppose: p = 10/day (while producing this product).d = 3/day (for this product).Put p-d = 7 in inventory every day while producing.Remove d = 3 from inventory every day while not producing this product.
42 Variables D = demand (usually annual) d = demand rate S = setup cost ($/setup) p = production rateI = carrying cost(% of value/unit time)C = item value ($/item)Assume d and p are constant (no variability).Q = production quantity (in each production run)N = number of production runs (setups)/yearTC = total cost (usually annual)Also want:Length of a production run (for example, in days)Length of time between runs (cycle time)
43 Inventory Level Inventory pattern repeats: MaximuminventoryTimeProduce QDo not produceInventory pattern repeats:Produce Q units of product of interest.Then produce other products.Every production run of Q units requires 1 setup.Find Q to minimize total cost.
44 Inventory Level TC = Setup cost + Inventory carrying cost TimeMaximuminventoryTC = Setup cost + Inventory carrying costSetup cost = N x S = (D/Q) x SCarrying cost = Average inventory level x C x I= (Max. inventory/2) x C x I
45 Maximum Inventory Level TimeMaximuminventoryLength of a production run = Q/p (days)Max. inventory = (p-d) x Q/p = QCarrying cost = ICp-dpQp-d2p
46 Optimum Production Run Size: Q TimeMaximuminventoryInventoryTC =QDS + IC2p-dpSelect Q to minimize total cost.Set derivative of TC with respect to Q equal to zero.2DSpQ =ICp-d
47 Non-instantaneous Resupply Equations IC2DSp-dpN = D/QTC =QDS + IC2p-dpLength of a production run = Q/pLength of time between runs = Q/d
48 Non-instantaneous Resupply Example D=5000/year assume 250 days/yearI = 20%/yearS = $2000/setupC = $6000/unitp=60/dayFirst, calculate d=5000/250 = 20/day2x5000x200060Q == units0.2x600060-20Q/p = /60 = 2.64 daysQ/d = /20 = 7.91 daysTC = 63, ,246 = $126,492/yearEvery 7.91 days begin a 2.64 day production run.
49 Adjust Values to Fit Business Cycles Change cycle length to 8 days -> Q/d = 8 daysThen: Q = 160 unitsQ/p = 2.67 daysTC = 62, ,000 = $126,500/year2.7810.71618.724Production runsProduce other products
50 Cost is Insensitive to Small Changes Change cycle length to 10 days=2 weeks (+26%)Then: Q/d = 10 daysQ = 200 unitsQ/p = 3.33 daysTC = 50, ,000 = $130,000/yearTC is only 2.8% over minimum TC!1020Production runsProduce other products
51 Scheduling Multiple Products Suppose 3 products are produced on the same equipment. Optimal values are:P1: Q/d = Q/p = 2.64P2: Q/d = 13.4 Q/p = 4.8P2: Q/d = 25.8 Q/p = 5.9Adjust cycle lengths to a common value or multiple.For example 8 daysP1: Q/d = 8 -> Q/p = 2.7P2: Q/d = 12 -> Q/p = 4.3P2: Q/d = 24 -> Q/p = 5.5Now schedule 3 runs of P1, 2 runs of P2 and 1 run of P3 every 24 days.
52 Scheduling Multiple Products - continued P1: Q/d = 8 -> Q/p = 2.7P2: Q/d = 12 -> Q/p = 4.3P2: Q/d = 24 -> Q/p = 5.5Now schedule 3 runs of P1, 2 runs of P2 and 1 run of P3 every 24 days.2.779.715.217.922.224P1P2P3P1P2P3Idle
53 Reorder Point System - Variability Order amount Q when inventory falls to level ROP.If demand or lead time are larger than expected -> stockout
54 Variability Variability in demand and lead time may cause stockouts. d = mean demandsd = std. deviation of demandLT = mean lead timesLT = std. deviation of lead times’d = std. deviation of demand during lead times’d =LT x sd2 + d2 x sLT2
55 Safety StockUse safety stock to protect against stockouts when demand or lead time is not constant.Safety stock = z x s’dz is from Standard Normal Distribution Table and is based on P = Probability of being in-stock during lead time.ROP = expected demand during lead time + safety stock= d x LT + z x s’dAverage Inventory Level (AIL) = regular stock + safety stockQAIL =+ z x s’d2
56 Special Cases 1. Constant lead time, variable demand: sLT = 0 2. Constant demand, variable lead time: sd = 03. Constant demand, constant lead time: sd = 0, sLT = 0s’d =LT x sd2= sdLTs’d =d2 x sLT2= dsLTs’d = 0
57 Total Cost TC = Order cost + Regular stock carrying cost + Safety stock carrying cost + Stockout costTC =QDS + IC2+ ICz s’d +k s’d E(z)k = out-of-stock cost per unit shorts’d E(z) = expected number of units out-of-stock in one order cycleE(z) = unit Normal loss integralP -> z (from Appendix A) -> E(z) (from Appendix B)
58 3 Cases 1. Stockout cost k is known; P is not known. -> Calculate optimal P by repeating (1) and (2) until z does not change.2. Stock cost k is not known; P is known.-> Can not use last term in TC.3. Stockout cost k is known; P is known.-> Could use k to calculate optimal P.DkQICP = 1 -(1)2D[s + ks’dE(z)Q =(2)IC
59 Reorder Point Example D = 5000 units/year d = 96.15 units/week S = $10/order sd = 10 units/weekC = $5/unitI = 20% per yearLT = 2 weeks (constant) sLT = 0
60 Reorder Point Example - Case 1 D = 5000 units/year d = units/weekS = $10/order sd = 10 units/weekC = $5/unitI = 20% per yearLT = 2 weeks (constant) sLT = 0k = $2/unit; P is not givenIterate to find optimal P.2x5000x10Q == units0.2x5s’d =sd= 14.14LT= 102
61 Case 1 (continued) - Find best P 316.23(0.2)5P = 1 -=5000(2)z = 1.86 E(z) =2(5000)[10 + 2(14.14)0.0123Q ==0.2(5)321.68(0.2)5P = 1 -=5000(2)z = 1.85 E(z) =2(5000)[10 + 2(14.14)0.0126Q ==0.2(5)
62 Case 1 (continued) 321.81(0.2)5 P = 1 - = 0.9678 5000(2) z = 1.85 E(z) =z does not change, so STOPSolution: Q = 322 z = E(z) =ROP = d x LT + z x s’d = 96.15(2) (14.14) =TC = = $347.97/year
63 Reorder Point Example - Case 2 D = 5000 units/year d = units/weekS = $10/order sd = 10 units/weekC = $5/unitI = 20% per yearLT = 2 weeks (constant) sLT = 0k is not known; P =90%Solution: z = 1.282x5000x10Q == unitss’d = (as in Case 1)0.2x5ROP = d x LT + z x s’d = 96.15(2) (14.14) =TC = = $334.33/year
64 Reorder Point Example - Case 3 D = 5000 units/year d = units/weekS = $10/order sd = 10 units/weekC = $5/unitI = 20% per yearLT = 2 weeks (constant) sLT = 0k =$2/unit; P =90%Solution: z = 1.282x5000x10Q == unitss’d = (as in Case 2)0.2x5ROP = d x LT + z x s’d = 96.15(2) (14.14) =TC = = $355.58/year
65 Reorder Point Example - Case 3 k =$2/unit; P =90%Solution:Q =ROP =TC = $355.58/yearCould use k=$2/unit to find optimal PIt would be P = 96.78% as in Case 1!Order size would be slightly larger (322 vs. 316).Cost would be slightly less ($ vs. $355.58).
66 Reorder Point Example - Case 4 Suppose we keep no safety stockSolution:2x5000x10Q == units0.2x5ROP = d x LT = 96.15(2) =TC = = $494.73/yearWith no safety stock there is a stockout whenever demand during lead time exceeds expected amount (dxLT).Therefore: P = 0.5
67 Reorder Point Example - Summary Case k P Q ROP TC($/year)A small amount of safety stock can save a large amount!Case 4 vs Case 3
68 P and SL Suppose that on average: There are 10 orders/year.Each order is for 100 items (Q=100).We are out-of-stock 2 items per year on one order.P = probability of being in stock during lead time.= 1 - probability of being our-of-stock during lead time.= 1 - 1/10 = 0.90SL = Service level = % of items in-stock= 1 - % of items out-of-stock = 1 - 2/1000 = 0.998
69 Service Level - Reorder Point SL = 1 - % of items out-of-stockExpected number of units out-of-stock/year= 1 -Annual demand(D/Q) x s’d x E(z)= 1 -Ds’d E(z)= 1 -Q
71 Reorder Point Example - Summary Case k P Q ROP TC($/yr) SLNote difference between P and SL!
72 Out-of-Stock for Cases 1-4 Out-of-stock: 3 items per year and 0.5 orders/yearSL = > ( )x5000 = 3 items/yearP = > ( )x5000/322 = 0.5 orders/yearCase 2 & 3:Out-of-stock: items per year and 1.58 orders/yearSL = > ( )x5000 = 10.5 items/yearP = > (1-.90)x5000/316 = 1.58 orders/yearCase 4:Out-of-stock: 89 items per year and 7.9 orders/yearSL = > ( )x5000 = 89 items/yearP = > (1-.50)x5000/316 = 7.9 orders/year
73 Lead Time Variability in Example D = 5000 units/year d = units/weekS = $10/order sd = 10 units/weekC = $5/unitI = 20% per yearLT = 2 weeks (constant)Suppose sLT = 1.2 (not 0 as before)Now:For constant lead time (sLT = 0) s’d =14.14Additional safety stock due to lead time variability= z( )s’d =LT x sd2 + d2 x sLT2=
74 Optimal Inventory Control For perpetual (continual) demand.Treat each stocking point independently.Consider 1 product art 1 location.ReorderDetermine: Point SystemHow much to order: QWhen to (re)order: ROP