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**Chapter 10: Inventory Types of Inventory and Demand Availability**

Cost vs. Service Tradeoff Pull vs. Push Reorder Point System Periodic Review System Joint Ordering Number of Stocking Points Investment Limit Just-In-Time

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**Chapter 10: Inventory Skip the following:**

Single-Order Quantity: pp Lumpy Demand: pp , Box Application: pp , Poisson Distribution: pp )

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**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!

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**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.

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Types of Inventories Regular (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 Obsolete/Shrinkage stock: out-of-date, lost, stolen, etc.

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**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.

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**Performance Measures Turnover ratio: Availability: Annual demand**

Service Level = SL Fill Rate = FR Weighted Average Fill Rate = WAFR Annual demand Turnover ratio = Average inventory

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**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 i SLi = 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 i Annual demand for item i SLi = 1 -

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**Measuring Availability: FR and WAFR**

For 1 order of several items: FRj = Fill Rate for order j FRj = 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 ...

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**WAFR Example Example: 3 items**

I1 (SL=0.98); I2 (SL = 0.90); I3 (SL = 0.95) Order Frequency FR Freq.xFR I I1,I2,I x0.90x0.90= I1,I x0.95= I1,I2,I x0.90x0.95= WAFR =

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**Fundamental Tradeoff $ Level of Service Level of Service vs. Cost**

Revenue

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**Fundamental Tradeoff Level of Service (availability) vs. Cost**

Higher service levels -> More inventory. -> Higher cost. Higher service levels -> Better availability. -> Fewer stockouts. -> Higher revenue.

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**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.

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**Fundamental Cost Tradeoff**

Inventory carrying cost vs. Order & Stockout cost Larger inventory -> Higher carrying costs. Larger inventory -> Fewer larger orders. -> Lower order costs. Larger inventory -> Better availability. -> Few stockouts. -> Lower stockout costs.

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**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 enough True for only 3% of stockouts.

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**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.

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**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.

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**Push Inventory Control**

TRi = Total requirements for warehouse i NRi = Net requirements at i Total excess = Amount available - NR for all warehouses Demand % = (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 i z is from Appendix A

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**Push Inventory Control Example**

Allocate 60,000 cases of product among two warehouses based on the following data. Current Forecast Forecast Warehouse Inventory Demand Error SL , ,000 5, , ,000 3, ,000

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**Push Inventory Control Example**

Current Forecast Forecast Demand Warehouse Inventory Demand Error SL % , , , , , , ,000 TR1 = 20, x 5,000 = 26,400 TR2 = 15, x 3,000 = 21,150 NR1 = 26, ,000 = 16,400 NR2 = 21, ,000 = 16,150 Total Excess = 60, , ,150 = 27,450 Allocation for 1 = 16, ,450 x (0.5714) = 32,086 cases Allocation for 2 = 16, ,450 x (0.4286) = 27,914 cases

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**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:

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**Pull Inventory Control - Repetitive Ordering**

For perpetual (continual) demand. Treat each stocking point independently. Consider 1 product art 1 location. Reorder Periodic Determine: Point System Review System How much to order: Q M-qi When to (re)order: ROP T

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**Reorder Point System Order amount Q when inventory falls to level ROP.**

Constant order amount (Q). Variable order interval.

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**Reorder Point System Each increase in inventory is size Q. LT1 LT2 LT3**

Place 1st order Place 2nd order Place 3rd order Receive 3rd order Receive 1st order Receive 2nd order Each increase in inventory is size Q.

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**Reorder Point System LT1 LT2 LT3 Place 1st order Place 2nd order**

Place 3rd order Receive 3rd order Receive 1st order Receive 2nd order Time between 1st & 2nd order Time between 2nd & 3rd order

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**Periodic Review System**

Order amount M-qi every T time units. Constant order interval (T=20 below). Variable order amount.

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**Periodic Review System - T=20 days**

LT1 LT2 LT3 Place 3rd order Place 2nd order Receive 3rd order Place 1st order Receive 1st order Receive 2nd order Each increase in inventory is size M-amount on hand. (M=90 in this example.)

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**Periodic Review System - T=20 days**

LT1 LT2 LT3 Place 3rd order Place 2nd order Receive 3rd order Place 1st order Receive 1st order Receive 2nd order Time between 1st & 2nd order (20 days) Time between 2nd & 3rd order (20 days)

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**Optimal Inventory Control**

For perpetual (continual) demand. Treat each stocking point independently. Consider 1 product art 1 location. Reorder Periodic Determine: Point System Review System How much to order: Q M-qi When to (re)order: ROP T Find optimal values for: Q & ROP or for M & T.

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**Inventory Variables D = demand (usually annual) d = demand rate**

S = order cost ($/order) LT = (average) lead time I = carrying cost k = stockout cost (% of value/unit time) P = probability of being in C = item value ($/item) stock during lead time sd = std. deviation of demand sLT = std. deviation of lead time s’d = std. deviation of demand during lead time Q = order quantity N = number of orders/year TC = total cost (usually annual) ROP = reorder point T = time between orders

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**Simplest Case - Constant demand and lead time**

No variability in demand and lead time (sd = 0, sLT = 0). Will never have a stock out. Inventory Time ROP Q Suppose: d = 4/day and LT = 3 days Then ROP = 12 (ROP = d x LT)

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**Constant demand and lead time**

Inventory Time ROP Q TC = Order cost + Inventory carrying cost Order cost = N x S = (D/Q) x S Carrying cost = Average inventory level x C x I = (Q/2) x C x I

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**Economic Order Quantity (EOQ)**

Inventory Time ROP Q TC = Q D S + IC 2 Select Q to minimize total cost. Set derivative of TC with respect to Q equal to zero. 0 = - Q2 D S + 2 IC 2DS Q = IC

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**Optimal Ordering 2DS Economic order quantity: Q* = IC**

Inventory Time ROP Q Q* = IC 2DS Economic order quantity: Optimal number of orders/year: Optimal time between orders: Optimal cost: D Q* Q* D TC = Q* D S + IC 2

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**Example D = 10,000/year S = $61.25/order I = 20%/year C = $50/item 2DS**

Q* = IC 2DS 2(10,000)(61.25) = = 350 units/order (0.2)(50) TC = Q* D S + IC 2 10,000 350 = (61.25) + (0.2)(50) 350 2 = = $3500/year 10,000 N = = orders/year 350 350 T = = years = 1.82 weeks 10,000

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**Example - continued Q* = 350 units/order N = 28.57 orders/year**

T = 1.82 weeks This is not a very convenient schedule for ordering! Suppose you order every 2 weeks: T = 2 weeks, so N = 26 orders/year D 10,000 Q = = = units/order (10% over EOQ) N 26 D Q 10,000 384.6 S + IC = (61.25) + (0.2)(50) TC = Q 2 384.6 2 = = $ /year Q = is 9.9% over EOQ, but TC is only 0.4% over optimal cost!!!

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**Model is Robust Q* = 350 TC = $3500/year Total Cost Carrying Cost**

Order Cost

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**Model is Robust Changing Q by 20% increases cost by a few percent.**

Carrying Cost Order Cost Total Cost

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Model is Robust A 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*.

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**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”.

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**Inventory Level Suppose: p = 10/day (while producing this product).**

Time Produce Q Do not produce Slope=7 Slope=-3 Suppose: 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.

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**Variables D = demand (usually annual) d = demand rate**

S = setup cost ($/setup) p = production rate I = carrying cost (% of value/unit time) C = item value ($/item) Assume d and p are constant (no variability). Q = production quantity (in each production run) N = number of production runs (setups)/year TC = total cost (usually annual) Also want: Length of a production run (for example, in days) Length of time between runs (cycle time)

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**Inventory Level Inventory pattern repeats:**

Maximum inventory Time Produce Q Do not produce Inventory 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.

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**Inventory Level TC = Setup cost + Inventory carrying cost**

Time Maximum inventory TC = Setup cost + Inventory carrying cost Setup cost = N x S = (D/Q) x S Carrying cost = Average inventory level x C x I = (Max. inventory/2) x C x I

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**Maximum Inventory Level**

Time Maximum inventory Length of a production run = Q/p (days) Max. inventory = (p-d) x Q/p = Q Carrying cost = IC p-d p Q p-d 2 p

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**Optimum Production Run Size: Q**

Time Maximum inventory Inventory TC = Q D S + IC 2 p-d p Select Q to minimize total cost. Set derivative of TC with respect to Q equal to zero. 2DS p Q = IC p-d

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**Non-instantaneous Resupply Equations**

IC 2DS p-d p N = D/Q TC = Q D S + IC 2 p-d p Length of a production run = Q/p Length of time between runs = Q/d

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**Non-instantaneous Resupply Example**

D=5000/year assume 250 days/year I = 20%/year S = $2000/setup C = $6000/unit p=60/day First, calculate d=5000/250 = 20/day 2x5000x2000 60 Q = = units 0.2x6000 60-20 Q/p = /60 = 2.64 days Q/d = /20 = 7.91 days TC = 63, ,246 = $126,492/year Every 7.91 days begin a 2.64 day production run.

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**Adjust Values to Fit Business Cycles**

Change cycle length to 8 days -> Q/d = 8 days Then: Q = 160 units Q/p = 2.67 days TC = 62, ,000 = $126,500/year 2.7 8 10.7 16 18.7 24 Production runs Produce other products

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**Cost is Insensitive to Small Changes**

Change cycle length to 10 days=2 weeks (+26%) Then: Q/d = 10 days Q = 200 units Q/p = 3.33 days TC = 50, ,000 = $130,000/year TC is only 2.8% over minimum TC! 10 20 Production runs Produce other products

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**Scheduling Multiple Products**

Suppose 3 products are produced on the same equipment. Optimal values are: P1: Q/d = Q/p = 2.64 P2: Q/d = 13.4 Q/p = 4.8 P2: Q/d = 25.8 Q/p = 5.9 Adjust cycle lengths to a common value or multiple. For example 8 days P1: Q/d = 8 -> Q/p = 2.7 P2: Q/d = 12 -> Q/p = 4.3 P2: Q/d = 24 -> Q/p = 5.5 Now schedule 3 runs of P1, 2 runs of P2 and 1 run of P3 every 24 days.

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**Scheduling Multiple Products - continued**

P1: Q/d = 8 -> Q/p = 2.7 P2: Q/d = 12 -> Q/p = 4.3 P2: Q/d = 24 -> Q/p = 5.5 Now schedule 3 runs of P1, 2 runs of P2 and 1 run of P3 every 24 days. 2.7 7 9.7 15.2 17.9 22.2 24 P1 P2 P3 P1 P2 P3 Idle

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**Reorder Point System - Variability**

Order amount Q when inventory falls to level ROP. If demand or lead time are larger than expected -> stockout

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**Variability Variability in demand and lead time may cause stockouts.**

d = mean demand sd = std. deviation of demand LT = mean lead time sLT = std. deviation of lead time s’d = std. deviation of demand during lead time s’d = LT x sd2 + d2 x sLT2

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Safety Stock Use safety stock to protect against stockouts when demand or lead time is not constant. Safety stock = z x s’d z 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’d Average Inventory Level (AIL) = regular stock + safety stock Q AIL = + z x s’d 2

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**Special Cases 1. Constant lead time, variable demand: sLT = 0**

2. Constant demand, variable lead time: sd = 0 3. Constant demand, constant lead time: sd = 0, sLT = 0 s’d = LT x sd2 = sd LT s’d = d2 x sLT2 = dsLT s’d = 0

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**Total Cost TC = Order cost + Regular stock carrying cost**

+ Safety stock carrying cost + Stockout cost TC = Q D S + IC 2 + ICz s’d + k s’d E(z) k = out-of-stock cost per unit short s’d E(z) = expected number of units out-of-stock in one order cycle E(z) = unit Normal loss integral P -> z (from Appendix A) -> E(z) (from Appendix B)

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**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. Dk QIC P = 1 - (1) 2D[s + ks’dE(z) Q = (2) IC

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**Reorder Point Example D = 5000 units/year d = 96.15 units/week**

S = $10/order sd = 10 units/week C = $5/unit I = 20% per year LT = 2 weeks (constant) sLT = 0

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**Reorder Point Example - Case 1**

D = 5000 units/year d = units/week S = $10/order sd = 10 units/week C = $5/unit I = 20% per year LT = 2 weeks (constant) sLT = 0 k = $2/unit; P is not given Iterate to find optimal P. 2x5000x10 Q = = units 0.2x5 s’d = sd = 14.14 LT = 10 2

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**Case 1 (continued) - Find best P**

316.23(0.2)5 P = 1 - = 5000(2) z = 1.86 E(z) = 2(5000)[10 + 2(14.14)0.0123 Q = = 0.2(5) 321.68(0.2)5 P = 1 - = 5000(2) z = 1.85 E(z) = 2(5000)[10 + 2(14.14)0.0126 Q = = 0.2(5)

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**Case 1 (continued) 321.81(0.2)5 P = 1 - = 0.9678 5000(2)**

z = 1.85 E(z) = z does not change, so STOP Solution: Q = 322 z = E(z) = ROP = d x LT + z x s’d = 96.15(2) (14.14) = TC = = $347.97/year

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**Reorder Point Example - Case 2**

D = 5000 units/year d = units/week S = $10/order sd = 10 units/week C = $5/unit I = 20% per year LT = 2 weeks (constant) sLT = 0 k is not known; P =90% Solution: z = 1.28 2x5000x10 Q = = units s’d = (as in Case 1) 0.2x5 ROP = d x LT + z x s’d = 96.15(2) (14.14) = TC = = $334.33/year

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**Reorder Point Example - Case 3**

D = 5000 units/year d = units/week S = $10/order sd = 10 units/week C = $5/unit I = 20% per year LT = 2 weeks (constant) sLT = 0 k =$2/unit; P =90% Solution: z = 1.28 2x5000x10 Q = = units s’d = (as in Case 2) 0.2x5 ROP = d x LT + z x s’d = 96.15(2) (14.14) = TC = = $355.58/year

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**Reorder Point Example - Case 3**

k =$2/unit; P =90% Solution: Q = ROP = TC = $355.58/year Could use k=$2/unit to find optimal P It would be P = 96.78% as in Case 1! Order size would be slightly larger (322 vs. 316). Cost would be slightly less ($ vs. $355.58).

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**Reorder Point Example - Case 4**

Suppose we keep no safety stock Solution: 2x5000x10 Q = = units 0.2x5 ROP = d x LT = 96.15(2) = TC = = $494.73/year With no safety stock there is a stockout whenever demand during lead time exceeds expected amount (dxLT). Therefore: P = 0.5

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**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

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**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 = 0.90 SL = Service level = % of items in-stock = 1 - % of items out-of-stock = 1 - 2/1000 = 0.998

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**Service Level - Reorder Point**

SL = 1 - % of items out-of-stock Expected number of units out-of-stock/year = 1 - Annual demand (D/Q) x s’d x E(z) = 1 - D s’d E(z) = 1 - Q

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**Service Levels for Cases 1-4**

14.14(.0126) Case 1: SL = 1 - Case 2: SL = 1 - Case 3: SL = 1 - Case 4: SL = 1 - = 322 14.14(.0475) = 316 14.14(.0475) = 316 14.14(.3989) = 316

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**Reorder Point Example - Summary**

Case k P Q ROP TC($/yr) SL Note difference between P and SL!

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**Out-of-Stock for Cases 1-4**

Out-of-stock: 3 items per year and 0.5 orders/year SL = > ( )x5000 = 3 items/year P = > ( )x5000/322 = 0.5 orders/year Case 2 & 3: Out-of-stock: items per year and 1.58 orders/year SL = > ( )x5000 = 10.5 items/year P = > (1-.90)x5000/316 = 1.58 orders/year Case 4: Out-of-stock: 89 items per year and 7.9 orders/year SL = > ( )x5000 = 89 items/year P = > (1-.50)x5000/316 = 7.9 orders/year

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**Lead Time Variability in Example**

D = 5000 units/year d = units/week S = $10/order sd = 10 units/week C = $5/unit I = 20% per year LT = 2 weeks (constant) Suppose sLT = 1.2 (not 0 as before) Now: For constant lead time (sLT = 0) s’d =14.14 Additional safety stock due to lead time variability = z( ) s’d = LT x sd2 + d2 x sLT2 =

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**Optimal Inventory Control**

For perpetual (continual) demand. Treat each stocking point independently. Consider 1 product art 1 location. Reorder Determine: Point System How much to order: Q When to (re)order: ROP

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