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Chapter 10: Inventory

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. Chapter 10: Inventory. Skip the following:

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Chapter 10: Inventory

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  1. 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

  2. Chapter 10: Inventory • Skip the following: • Single-Order Quantity: pp. 322-323 • Lumpy Demand: pp. 344-345, • Box 10.23 Application: pp. 347-348, • Poisson Distribution: pp. 356-357)

  3. Inventory • Inventory includes: • 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: • 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 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 • Obsolete/Shrinkage stock: out-of-date, lost, stolen, etc.

  6. Types of Demand • 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: • Service Level = SL • Fill Rate = FR • Weighted Average Fill Rate = WAFR Annual demand Turnover ratio= Average inventory

  8. Expected number of units out of stock/year for item i SLi = 1 - Annual demand for item i 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.

  9. 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 ...

  10. WAFR Example • Example: 3 items • I1 (SL=0.98); I2 (SL = 0.90); I3 (SL = 0.95) Order Frequency FR Freq.xFR I1 0.4 0.98 0.392 I1,I2,I2 0.1 0.98x0.90x0.90=0.7938 0.07938 I1,I3 0.2 0.98x0.95=0.931 0.1862 I1,I2,I3 0.3 0.98x0.90x0.95=0.8379 0.25137 WAFR = 0.90895

  11. Revenue $ Cost Level of Service Fundamental Tradeoff • Level of Service vs. Cost

  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: • 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 cost • Larger 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 enough • True for only 3% of stockouts.

  16. Pull vs. Push Systems • Pull: • 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 = Forecast demand at i + Safety stock at i = Forecast demand at i + z x Forecast error at i 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 %) = TRi - Current inventory at i z 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 Forecast Warehouse Inventory Demand Error SL 1 10,000 20,000 5,000 0.90 2 5,000 15,000 3,000 0.98 35,000

  20. Push Inventory Control Example Current Forecast Forecast Demand Warehouse InventoryDemand Error SL% 1 10,000 20,000 5,000 0.90 0.5714 2 5,000 15,000 3,000 0.98 0.4286 35,000 TR1 = 20,000 + 1.28 x 5,000 = 26,400 TR2 = 15,000 + 2.05 x 3,000 = 21,150 NR1 = 26,400 - 10,000 = 16,400 NR2 = 21,150 - 5,000 = 16,150 Total Excess = 60,000 - 16,400 - 16,150 = 27,450 Allocation for 1 = 16,400 + 27,450 x (0.5714) = 32,086 cases Allocation for 2 = 16,150 + 27,450 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 Periodic Determine: Point System Review System How much to order: Q M-qi When 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 Place 1st order LT1 LT2 LT3 Place 2nd order Place 3rd order Receive 3rd order Receive 1st order Receive 2nd order Each increase in inventory is size Q.

  25. Reorder Point System Place 1st order LT1 LT2 LT3 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

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

  28. Periodic Review System - T=20 days Place 1st order LT3 LT1 LT2 Place 3rd order Receive 3rd order Place 2nd order Receive 1st order Receive 2nd order Time between 1st & 2nd order (20 days) Time between 2nd & 3rd order (20 days)

  29. 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.

  30. 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

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

  32. Inventory ROP Time Constant demand and lead time 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

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

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

  35. Q* = IC Q* D S + IC TC = 2 Q* Example D = 10,000/year S = $61.25/order I = 20%/year C = $50/item 2DS 2(10,000)(61.25) = = 350 units/order (0.2)(50) 350 10,000 (61.25) + (0.2)(50) = 2 350 = 1750 + 1750 = $3500/year 10,000 N = = 28.57 orders/year 350 350 = 0.035 years = 1.82 weeks T = 10,000

  36. 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 10,000 D Q = = 384.6 units/order (10% over EOQ) = 26 N 384.6 10,000 Q D (61.25) + (0.2)(50) = S + IC TC = 2 2 Q 384.6 = 1592.56 + 1923.00 = $3515.56/year Q = 384.6 is 9.9% over EOQ, but TC is only 0.4% over optimal cost!!!

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

  38. Total Cost Carrying Cost Order Cost Model is Robust Changing Q by 20% increases cost by a few percent.

  39. 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*.

  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. Slope=7 Inventory Slope=-3 Time Produce Q Do not produce Inventory Level 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.

  42. 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)

  43. Inventory Level Inventory Maximum inventory Time Do not produce Produce Q 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.

  44. Inventory Maximum inventory Time Inventory Level 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

  45. Inventory Maximum inventory Time Maximum Inventory Level 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

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

  47. p 2DS Q = IC p-d Q D S + IC TC = 2 Q Non-instantaneous Resupply Equations N = D/Q p-d p Length of a production run = Q/p Length of time between runs = Q/d

  48. 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 = = 158.11 units 0.2x6000 60-20 Q/p = 158.11/60 = 2.64 days Q/d = 158.11/20 = 7.91 days TC = 63,246 + 63,246 = $126,492/year Every 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 days Then: Q = 160 units Q/p = 2.67 days TC = 62,500 + 64,000 = $126,500/year 8 10.7 0 16 18.7 24 2.7 Production runs Produce other products

  50. 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 + 80,000 = $130,000/year TC is only 2.8% over minimum TC! 10 20 0 Production runs Produce other products

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