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Inventory Management and Risk Pooling

Inventory Management and Risk Pooling. Tokyo University of Marine Science and Technology Mikio Kubo. Why the inventory management is so important?. GM reduced their inventory and transportation costs by 26% using a decision support tool that optimizes their fright shipment schedule.

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Inventory Management and Risk Pooling

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  1. Inventory Managementand Risk Pooling Tokyo University of Marine Science and Technology Mikio Kubo

  2. Why the inventory management is so important? • GM reduced their inventory and transportation costs by 26% using a decision support tool that optimizes their fright shipment schedule. • In 1994, IBM struggled with shortages in the ThinkPad line due to ineffective inventory management • The average inventory levels of Japanese super markets are 2 weeks in food, 4 weeks in non-food products.

  3. Two basic laws in inventory management • The first lawThe demand forecast is always wrong. • The second lawThe aggregation of inventories reduces the total amount of inventories • Other reasons for holding inventories are: • Economy of scale in production and/or transportation (lot-sizing inventories) • Uncertainty of the lead time • To catch up with the seasonal demand (seasonal inventories)

  4. Economic ordering quantity modelInventory of Beers Demand ratio per day =10 cans. Inventory holding cost is 10 yen per day per can. Ordering cost is 300 yen. Stock out is prohibited. What is the best ordering policy of beers?

  5. Economic order quantity (EOQ) model Demand Ordering Quantity Inventory Cycle time Time

  6. EOQ formula • Fixed ordering cost ＝３００ yen • Demand＝１０ cans/day • Inventory＝１０ yen/can・day

  7. Economic Ordering Quantity (EOQ) model • d (units/day): Demand per day. • Q (units): Ordering quantity （variable） • K ($): Fixed ordering cost • h ( $/(day・unit) ）: Inventory holding cost Objective：Find the minimum cost ordering policy • Constraints： • Backorder is not allowed． • The lead time, the time that elapses between the placement of an order and its receipt, is zero. • Initial inventory is zero. • The planning horizon is long (infinite).

  8. Inventory level d：demand speed Q h×Area Cycle time (T days) = [ ] Time Total cost over T days = Ordering cost +Inventory Cost = f(Q)= Cost per day =

  9. EOQ formula • minimize f(Q) • ∂f(Q)/∂Q = • ∂2f(Q)/∂Q2 = • f(Q) is [ ] function. • Q* = • f(Q* )=

  10. Swimsuit Production using Excel =125*MIN($D$1,$B2) =D2+E2-F2-G2 =20*MAX($D$1-B2,0) • Fixed production cost 100000 $ • Variable production cost 80$/unit • Selling price 125 $/unit • Salvage value 20$/unit When the company produced 9000 units, the expected profit is 293450$.

  11. Swimsuit Production (Continued)

  12. Effect of initial inventory • If initial inventory is 5000 units. • Do not produce:225000+5000×80(pink line) • Produce up to 12000 units: 370700+5000×80(blue line)

  13. Truncated Normal distribution with mean100 and standard deviation 100 Probability density function Demand

  14. Service Level and Critical Ratio • Service Level：The probability with which stock-out does not happen.Optimal service level=Critical ratio

  15. Service level and density function Critical ratio=0.99 The area (probability) that rhe demand is below 333 is Set to 0.99. ３３３

  16. Inverse of cumulative distribution function Excel NORMINV(0.99,100,100) ０．９９ ３３３

  17. Service level and safety stock ratio Safety stock ratio NORMINV(service level,0,1) Service level

  18. Base stock level • Base stock level：target level of inventory position 333=100×1+2.33×100×SQRT(1)

  19. TV set example • Lead time =2 weeks • Service level =97%-> safety stock ratio =[ ] • Average demand in a week (note that 1 month =4.4 week) ＝[ ] • Standard deviation in a week ＝[ ] • Base stock level＝[ ]； Week of Supply？ =[ ] =STDEV(B2:M2) =AVERAGE(B2:M2)

  20. (s,S) Policy • Fixed cost of an order （K）-> determine the ordering quantity Q using EOQ model • (s,S) policy：If the inventory position is below a re-ordering point s, order the amount so that it becomes an order-up-to level S • サプライ・チェインの設計と管理 p.58 事例　秋葉原無線

  21. TV set example (Continued) • Fixed cost of ordering （K）=4500$ • Price =250$，interest rate = 18 % /year （1 year = 52 weeks）->Inventory holding cost/week =[ ] • Q is determined by EQO formulaQ= [ formula ] = [ ] • inventory position （S）=[ ]

  22. When the lead time L is a random variable • Lead time L: Normal distribution with mean （AVGL） and standard deviation （STDL） • Remark that the assumption that L follows a normal distribution is not realistic.

  23. Non-stationary demand case Derive a formula for determining the safety stock level When the demand is NOT stationary. For each period t=1,2…, Customer Retailer Ordering quantity q[t] Inventory I[t] Demand D[t]

  24. Discrete time model(Periodic ordering system) Lead time L Items ordered at the end of period t will arrive at the beginning of period t+L+1. 2) Demand D[t] occurs t t+1 t+2 t+3 t+4 １） Arrive the items ordered in period t-L-1 3) Forecast demand F[t+1]4) Order q[t] Arrive the itemsin period t+L+1 （L=3)

  25. Demand process • Mean d • A parameter that represents the un-stability of demand process a (0<a<1) • Forecast error e[t], t=1,2,… D[1]= d+e[1] D[t]= D[t-1] -(1-a) e[t-1] +e[t], t=2,3,…

  26. Exceld=100,a=0.9,e[t]=[-10,10] (uniform r.v.) A B C １ ２ ３ ４ ５ ６ ７ =100+B2 =RAND()*(-20)+10 =C3 =A2-（１－C2）*B2+B3

  27. Demand processa=0.9

  28. Demand process a=0.5

  29. Demand processa=0.1

  30. Ordering quantity q[t] • Forecast future demands (exponential smoothing method）F[1]=dF[t]=a D[t-1] + (1-a) F[t-1], t=2,3,… • Ordering quantity: At the end of period t, order the amount q[t]=D[t]+(L+1) (F[t+1]-F[t]) ,t=1,2,… where q[t]=d, t<=0.

  31. Forecast and ordering amounta=0.5

  32. Inventory I[t] • Inventory flow conservation equation:Final inventory (period t)=Final inventory (period t-1)-Demand＋Arrival VolumeI[0]=A Safety Stock LevelI[t] =I[t-1] –D[t] +q[t-L-1],t=1,2,…

  33. Example using Excel A B C D E F G 1 2 3 4 5 6 7 8 =C7*A6+(1-C7)*D6 =A6+(E6+1)*(D7-D6) =G5-A6+F2

  34. Inventory process: a=0.5

  35. Relationship between demand and forecast

  36. Expansion of demand and forecast

  37. Expansion of inventory

  38. Derived formula • e[t]： mean =０，S.D. =σ, normal distribution • Expected value of inventory • Standard deviation

  39. Safety stock • z：Safety stock ratio • When a=0 （stationary）: • When a=1（random walk）:

  40. Echelon Inventory Relailer Echelon lead time（２ weeks） Echelon inventoryof warehouse Warehouse Supplier Echelon inventory position os warehouse

  41. Multi echelon model For each period t=1,2… Lead time L2 Lead time L1 Customer Warehouse (or Supplier) Retailer Order q2[t] Inventory I2[t] Inventory I1[t] Demand D1[t] Demand in the second level D2[t] ＝ordering quantity of the retailer q1[t] = Demand+Lead time ×（Forecast Error） = D1[t]+(L1+1) (F1[t+1]-F1[t])

  42. Expansion of 2nd level demand (1) D2[t]=D1[t]+(L1+1) (F1[t+1]-F1[t])

  43. Expansion of 2nd level demand (2) Same as the first level demand！

  44. Inventory in the 2nd level

  45. When the inventory is controlled by the warehouse (supplier) • Warehouse (or supplier) controls the echelon inventory are controlled EI[t] • Echelon lead time Ｌ１＋Ｌ２(=EL) Customer Warehouse (or Supplier) Retailer Echelon lead time L1+L2

  46. When the inventories are controlled by the retailer and the warehouse separately Retailer Warehouse (or Supplier)

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