1. Inventory Management II Lot Sizing Lecture 5 ESD.260 Fall 2003 Caplice

2. Outline EOQ Model Sensitivity Analysis Discounting All Units Incremental One Time Special MIT Center for Transportation & Logistics - ESD.260 2 © Chris Caplice, MIT

3. Assumptions: Basic EOQ Model Demand Constant vs Variable Known vs Random Continuous vs Discrete Lead time Instantaneous Constant or Variable (deterministic/stochastic) Dependence of items Independent Correlated Indentured Review Time Continuous vs Periodic Number of Echelons One vs Many Capacity / Resources Unlimited vs Limited Discounts None All Units or Incremental Excess Demand None All orders are backordered Lost orders Substitution Perishability None Uniform with time Planning Horizon Single Period Finite Period Infinite Number of Items One Many MIT Center for Transportation & Logistics - ESD.260 3 © Chris Caplice, MIT

4. Notation TC = Total Cost (dollar/time) D = Average Demand (units/time) Co = Ordering Cost (dollar/order) Ch = Holding Cost (dollars/dollars held/time) Cp = Purchase Cost (dollars/unit) Q = Order Quantity (units/order) T = Order Cycle Time (time/order) MIT Center for Transportation & Logistics - ESD.260 4 © Chris Caplice, MIT

5. Lot Sizing: Many Potential Policies MIT Center for Transportation & Logistics - ESD.260 5 © Chris Caplice, MIT Inventory On Hand Time Objective: Pick the policy with the lowest total cost I(t)

6. What is the total cost? TC = Purchase + Order + Holding + Shortage Cost Which costs are relevant? Purchase Costs Ordering Costs Holding Costs Shortage Costs MIT Center for Transportation & Logistics - ESD.260 6 © Chris Caplice, MIT

7. Economic Order Quantity (EOQ) MIT Center for Transportation & Logistics - ESD.260 7 © Chris Caplice, MIT

8. Simple EOQ Example Annual demand of widgets is 2,000. The cost of ordering is $500. Widgets are procured for $50 each and are sold for $95 each. Holding cost is estimated to be 25%. What is the EOQ, Total Cost, and Cycle Time? Q* = 400 units TC* = $5,000 / year T* = 0.20 years or 73 days MIT Center for Transportation & Logistics - ESD.260 8 © Chris Caplice, MIT

9. Total Cost versus Lot (Order) Size MIT Center for Transportation & Logistics - ESD.260 9 © Chris Caplice, MIT Lot Size Annual Cost Annual Cost vs. Order Quantity

10. The Effect of Non-Optimal Q Q DC o /Q C h C p Q/2 TC 2000 $500 $12,500 $13,000 500 $2,000 $3,125 $5,125 400 $2,500 $2,500 $5,000 200 $5,000 $1,250 $6,250 20 $50,000 $125 $50,125 MIT Center for Transportation & Logistics - ESD.260 10 © Chris Caplice, MIT So, how sensitive is TC to Q?

11. Sensitivity of EOQ How does TC change with Q? MIT Center for Transportation & Logistics - ESD.260 11 © Chris Caplice, MIT

12. Sensitivity of EOQ Sensitivity TC wrt Q MIT Center for Transportation & Logistics - ESD.260 12 © Chris Caplice, MIT

13. Sensitivity of EOQ What if our estimate of demand was off? E = Actual Demand / Estimated Demand Q = Estimated EOQ Q* = Actual EOQ, given actual demand that occurred MIT Center for Transportation & Logistics - ESD.260 13 © Chris Caplice, MIT

14. Sensitivity of EOQ to Parameter Error So, for D = 2000 units/year, if the actual demand was … DEQ*/Q'TC'/TC* 2000.100.321.74 1,0000.500.711.06 1,5000.750.871.01 1,8000.900.951.00 2,0001.00 3,0001.501.221.02 4,0002.001.411.06 20,00010.003.161.74 MIT Center for Transportation & Logistics - ESD.260 14 © Chris Caplice, MIT

15. Insights from EOQ There is a direct trade off between lot size and total inventory Total cost is relatively insensitive to changes Very robust with respect to changes in: Q –rounding of order quantities D –errors in forecasting C h, C o, C p –errors in cost parameters Thus, EOQ is widely used despite its highly restrictive assumptions MIT Center for Transportation & Logistics - ESD.260 15 © Chris Caplice, MIT

16. Introduce Discounts to Lot Sizing Types Types of discounts All units discount Incremental discount One time only discount How will different discounting strategies impact your lot sizing decision? MIT Center for Transportation & Logistics - ESD.260 16 © Chris Caplice, MIT

17. All Units Discount Unit Price [Cpi ] Price Break Quantity [PBQ I ] $50.000 $45.00500 $40.001000 MIT Center for Transportation & Logistics - ESD.260 17 © Chris Caplice, MIT

18. All Units Discount Need to introduce purchase cost into TC function MIT Center for Transportation & Logistics - ESD.260 18 © Chris Caplice, MIT

19. All Units Discount Annual Costs MIT Center for Transportation & Logistics - ESD.260 19 © Chris Caplice, MIT Dollars Lot Size

20. All Units Discount Annual Costs MIT Center for Transportation & Logistics - ESD.260 20 © Chris Caplice, MIT Dollars Lot Size

21. All Units Discount: Method Same Example: D=2000 Units/yr C h =.25 C o =$500 C pi Price Breaks: $50 for 0 to <500 units $45 for 500 to <1000 units $40 for 1000+ units MIT Center for Transportation & Logistics - ESD.260 21 © Chris Caplice, MIT 1C pi $40.00$45.00$50.00 2PBQ10005000 3EOQ[C pi ]447421400 4Q pi 1000500400 5DC pi $80,000$90,000$100,000 6C 0 D/Q pi $1,000 $2,500 7C h C pi Q pi /2$5,000 $2,500 8TC[Q pi ]$86,000$94,812$105,000

22. Incremental Discount Treatment of the Incremental Discount Cost of Each Unit in the Lot Cost of the Unit Lot Unit MIT Center for Transportation & Logistics - ESD.260 22 © Chris Caplice, MIT

23. Incremental Discount MIT Center for Transportation & Logistics - ESD.260 23 © Chris Caplice, MIT Annual Cost Lot Size

24. Incremental Discount MIT Center for Transportation & Logistics - ESD.260 24 © Chris Caplice, MIT Indexi=3i=2i=1 1C pi $40.00$45.00$50.00 2PBQ i 10005000 3FiFi $7500$2500$0 4EOQ[C pi ]17891033400 5Q pi 1789400 6C pe $44.19$50.00 7DC pe $88,384.57$100,000 8C o D/Q pi $558.97$2,500 9(C h C pe Q pi )/2$9,882.50$2,500 10TC[Q pi ]$98,826.04$105,000

25. One Time Discount MIT Center for Transportation & Logistics - ESD.260 25 © Chris Caplice, MIT

26. One Time Discount MIT Center for Transportation & Logistics - ESD.260 26 © Chris Caplice, MIT Let, C pg = One time deal purchase price ($/unit) Q g = One time special order quantity (units) TC sp =TC over time covered by special purchase ($) Then,

27. One Time Discount MIT Center for Transportation & Logistics - ESD.260 27 © Chris Caplice, MIT

28. One Time Discount MIT Center for Transportation & Logistics - ESD.260 28 © Chris Caplice, MIT

29. One Time Discount MIT Center for Transportation & Logistics - ESD.260 29 © Chris Caplice, MIT A quick numerical example... D = 2000 units / year C o = $500.00 / order C h = 25% C p = $50.00 / uni Q* = 400 units TC* = $5000.00 / year C pg = $45.00 / unit Q* g = 1333 Savings* = $2448.25