2. 1 Resource-Constrained Multiple Product System & Stochastic Inventory Model Prof. Yuan-Shyi Peter Chiu Feb. 2012 Material Management Class Note #4

3. 2 § I13 : Inventory Model: Resource-Constrained Multiple Product System [Ex.13.1] Item 1 2 3 λ j 1850 1150 800 C j $50 $350 $85 K j $100 $150 $50

4. 3 [Eq.13.1] Must run under budget → EOQ 1 = 172, EOQ 2 = 63, EOQ 3 = 61 $50(172) + $350(63) + $85(61) = $35,835 (over-budget) Adjusting Factor …..Eq.13.2] (Budget) § I13 : Resource-Constrained Multiple Product System ……[Eq.13.1]

5. 4 §. I13: Problems & Discussion Preparation Time : 25 ~ 30 minutes Discussion : 15 ~ 25 minutes ( # C.6 ) ( # C.6 ) ( # N4.38(a) )

6. 5 Check : w i / h i w 1 / h 1 = 9 / 12.5 = 0.72 w 2 / h 2 = 12 / 87.5 = 0.14 w 3 / h 3 = 18 / 21.25 = 0.847 Diff. Simple solution obtained by a proportional scaling of the EOQ values will not be optimal. Find Lagrange multiple ? § I13 : Inventory Model: Resource-Constrained Multiple Product System [Ex.13.2]

7. 6 …..….[Eq.13.3] …….[Eq.13.4] § I13 : Inventory Model: Resource-Constrained Multiple Product System [Ex.13.2]◇

8. 7 (a) For proportional Not optimal! § I13 : Inventory Model: Resource-Constrained Multiple Product System◇

9. 8 (b) Find Lagrangean Function Find ∴ § I13 : Inventory Model: Resource-Constrained Multiple Product System◇

10. 9 §. I13.1: Problems & Discussion Preparation Time : 15 ~ 20 minutes Discussion : 10 ~ 20 minutes ( # N4.26 ; N4.28 ) ( # N4.26 ; N4.28 )

11. 10 § I25: Markov Model in Stochastic Inventory Stochastic Inventory Management Management A camera store stocks a particular model camera that can be ordered weekly. Let D 1, D 2, …, represent the Demands for this camera during the first week, second week, …, respectively. It is assumed that the D t are independent and identically distributed random variables having a known probability distribution. Let X 0 represents the number of cameras on hand at the outset, X 1 the number of cameras on hand at the end of week one, X 2 the number of cameras on hand at the end of week two, and so on. Assume that X 0 = 3. On Saturday night the store places an order that is delivered in time for the opening of the store on Monday.◇ [Eg 25.1]

12. 11 The store uses (s,S) ordering policy, where (s,S) = (1,3). It is assumed that sales are lost when demand exceeds the inventory on hand. Demand ( i.e. D t ) has a Poisson distribution with λ= 1. If the ordering cost K=$10, each camera costs the store $25 to own it and the holding is $0.8 per item per week, while unsatisfied demand is estimated to be $50 per item short per week. Find the long-run expected total inventory costs per week? C.15 Please see C.15. § I25: Markov Model (cont’d) [Eg 25.1]

13. 12 §. I13.1: Problems & Discussion Preparation Time : 15 ~ 20 minutes Discussion : 10 ~ 20 minutes ~ The End ~ ( # C.16 ) ( # C.16 )