1. Inventory Management V Finite Planning Horizon Lecture 9 ESD.260 Fall 2003 Caplice

2. Assumptions: Basic FPH Model MIT Center for Transportation & Logistics - ESD.260 2 © Chris Caplice, MIT 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

3. Example Demand Month When should I order and for how much? Costs D = 2000 items per year Co = $500.00 per order Cp = $50.00 per item Ch = 24% per item per year Chp = ( Ch Cp )/ N = $1 per month per item N = number of periods per year More Assumptions Demand is required and consumed on first day of the period Holding costs are not charged on items used in that period Holding costs are charged for inventory ordered in advance of need MIT Center for Transportation & Logistics – ESD.260 3 © Chris Caplice, MIT

4. Five Basic Approaches 1. The One-Time Buy 2. Lot For Lot 3. Simple EOQ 4. The Silver Meal Algorithm 5. Optimal Procedures Wagner-Whitin (Dynamic Programming) Mixed Integer Programming MIT Center for Transportation & Logistics - ESD.260 4 © Chris Caplice, MIT

5. Approach: One-Time Buy On Hand Inventory Month MIT Center for Transportation & Logistics - ESD.260 5 © Chris Caplice, MIT 2000

6. Approach: One-Time Buy MIT Center for Transportation & Logistics - ESD.260 6 © Chris Caplice, MIT MonthsDemand Order Quantity Holding Cost Ordering Cost Period Costs 12002000$1800$5002300 21500$1650$01650 31000$1550$01550 4500$1500$0$1500 5500$1450$0$1450 61000$1300$0$1300 71500$1200$0$1200 82000$1000$0$1000 92000$800$0$800 102500$550$0$550 113000$250$0$250 122500$0 Totals:2000 $13100$500$13600

7. Approach: Lot for Lot On Hand Inventory MIT Center for Transportation & Logistics - ESD.260 7 © Chris Caplice, MIT 20015010050 100150200 250300250 Month

8. Approach: Lot for Lot MIT Center for Transportation & Logistics - ESD.260 8 © Chris Caplice, MIT MonthDemand Ordering Quantity Holding Cost Ordering Cost Period Costs 1200 $0$500 2150 $0$500 3100 $0$500 450 $0$500 550 $0$500 6100 $0$500 7250150$0$500 8200 $0$500 9200 $0$500 10250 $0$500 11300 $0$500 12250 $0$500 Yotals:2000 $0$6000

9. Approach: EOQ On Hand Inventory Month 400 MIT Center for Transportation & Logistics - ESD.260 9 © Chris Caplice, MIT

10. Approach: EOQ MIT Center for Transportation & Logistics - ESD.260 10 © Chris Caplice, MIT Month Demand Order Quantity Holding Cost Ordering Cost Period Costs 1 200 400$200$500$700 2 150 0$50$0$50 3 100 400$350$500$850 4 50 0$300$0$300 5 50 0$250$0$250 6 100 0$150$0$150 7 150 0$0 8 200 400$200$500$700 9 200 0$0 10 250 400$150$500$650 11 300 400$250$500$750 12 250 0$0 Totals: 2000 $1900$2500$4400

11. Approach: Silver-Meal Algorithm On Hand Inventory MIT Center for Transportation & Logistics - ESD.260 11 © Chris Caplice, MIT 550400250550250 Month

12. Approach: Silver-Meal Algorithm MIT Center for Transportation & Logistics - ESD.260 12 © Chris Caplice, MIT MonDmdLot Qty Order Cost Holding CostLot Cost Mean Cost 1 st Buy: 1200 $500$0$500 2150350$500$150$650$325 3100450$500 $150+$200 $850$283 450500$500 $150+$200+$150 $1000$250 550550$500 $150+$200+$150+$200 $1200$240 6100650$500 $150+$200+$150+$200+ $500 $1700$283 2 nd Buy: 6100 $5000 7150250$500150$650$325 8200450$500 $150+$400 $1050$350

13. Approach: Silver-Meal Algorithm MonDmd Lot Qty Order Cost Holding CostLot Cost Mean Cost 3 rd Buy: 8200 $500$0$500 9200400$500$200$700$350 10250650$500 $200+$500 $1200$400 4 th Buy: 10250 $500$0$500 11300550$500$300$800$400 12250800$500$300+$500$1300$433 5 th Buy: 12250 $500$0$500 MIT Center for Transportation & Logistics - ESD.260 13 © Chris Caplice, MIT

14. Approach: Silver-Meal Algorithm MonthsDemand Order Quantity Holding Cost Ordering Cost Period Costs 1200 550$350$500$850 2150 0$200$0$200 3100 0$100$0$100 450 0$50$0$50 550 0$0 6100 250$150$500$650 7150 0$0 8200 400$200$500$700 9200 0$0 10250 550$300$500$800 11300 0$0 12250 $0$500 Totals:2000 $1350$2500$3850 MIT Center for Transportation & Logistics - ESD.260 14 © Chris Caplice, MIT

15. Approach: Optimization (MILP) MIT Center for Transportation & Logistics - ESD.260 15 © Chris Caplice, MIT On Hand Inventory 550 450

16. Approach: Optimization (MILP) MIT Center for Transportation & Logistics - ESD.260 16 © Chris Caplice, MIT Decision Variables: Qi = Quantity purchased in period i Zi = Buy variable = 1 if Qi>0, =0 o.w. Bi = Beginning inventory for period I Ei = Ending inventory for period Data: Di = Demand per period, i = 1,,n Co = Ordering Cost Chp = Cost to Hold, $/unit/period M = a very large number…. MILP Model Objective Function: Minimize total relevant costs Subject To: Beginning inventory for period 1 = 0 Beginning and ending inventories must match Conservation of inventory within each period Nonnegativity for Q, B, E Binary for Z

17. Objective Function Beginning & Ending Inventory Constraints Conservation of Inventory Constraints Ensures buys occur only if Q>0 Non-Negativity & Binary Constraints MIT Center for Transportation & Logistics - ESD.260 17 © Chris Caplice, MIT

18. Approach: Optimization (MILP) MIT Center for Transportation & Logistics - ESD.260 18 © Chris Caplice, MIT

19. Comparison of Approaches MIT Center for Transportation & Logistics - ESD.260 19 © Chris Caplice, MIT Month DemandOTBL4LEOQS/MOPT 1 2002000200400550 2 150 3 100 400 4 50 5 6 100 250450 7 150 8 200 400 9 200 450 10 250 400550 11 300 400550 12 250 Totals Cost $13,600$6,000$4,400$3,850$3,750