CapliceLecture 9ESD.260 Fall 2003 Inventory Management VFinite Planning Horizon© Chris Caplice, MIT2MIT Center for Transportation & Logistics – ESD.260Assumptions: Basic FPH ModelDemand Constant vs Variable Known vs Random Continuous vs DiscreteLead time Instantaneous Constant or Variable (deterministic/stochastic)Dependence of items Independent Correlated IndenturedReview Time Continuous vs PeriodicNumber of Echelons One vs ManyCapacity / Resources Unlimited vs LimitedDiscounts None All Units or IncrementalExcess Demand None All orders are backordered Lost orders SubstitutionPerishability None Uniform with timePlanning Horizon Single Period Finite Period InfiniteNumber of Items One Many© Chris Caplice, MIT3MIT Center for Transportation & Logistics – ESD.260ExampleCosts D = 2000 items per yearCo = $500.00 per orderCp = $50.00 per itemCh = 24% per item per yearChp = ( Ch Cp )/ N = $1 per month per itemN = number of periods per year050100150200250300Demand123456789101112MonthWhen should I order and for how much?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© Chris Caplice, MIT4MIT Center for Transportation & Logistics – ESD.260Five Basic Approaches1. The One-Time Buy2. Lot For Lot3. Simple EOQ4. The Silver Meal Algorithm5. Optimal ProceduresWagner-Whitin (Dynamic Programming)Mixed Integer Programming© Chris Caplice, MIT5MIT Center for Transportation & Logistics – ESD.260Approach: One-Time Buy0200400600800100012001400160018002000OnHandInventory123456789101112Month2000© Chris Caplice, MIT6MIT Center for Transportation & Logistics – ESD.260$13600$500$1310020002000Totals:$0$0$0025012$250$0$250030011$550$0$550025010$800$0$80002009$1000$0$100002008$1200$0$120001507$1300$0$130001006$1450$0$14500505$1500$0$15000504$1550$0$155001003$1650$0$165001502$2300$500$180020002001Period CostsOrdering CostHolding CostOrder QuantityDemandMonthApproach: One-Time Buy© Chris Caplice, MIT7MIT Center for Transportation & Logistics – ESD.2600200400600800100012001400160018002000OnHandInventory123456789101112MonthApproach: Lot for Lot300 250200 250150 20050 100100 50200 150© Chris Caplice, MIT8MIT Center for Transportation & Logistics – ESD.260$6000$6000$020002000Totals:$500$500$025025012$500$500$030030011$500$500$025025010$500$500$02002009$500$500$02002008$500$500$01501507$500$500$01001006$500$500$050505$500$500$050504$500$500$01001003$500$500$01501502$500$500$02002001Period CostsOrdering CostHolding CostOrder QuantityDemandMonthApproach: Lot for Lot© Chris Caplice, MIT9MIT Center for Transportation & Logistics – ESD.2600200400600800100012001400160018002000OnHandInventory123456789101112MonthApproach: EOQ400 400 400 400 400© Chris Caplice, MIT10MIT Center for Transportation & Logistics – ESD.260$4400$2500$190020002000Totals:$0$0$0025012$750$500$25040030011$650$500$15040025010$0$0$002009$700$500$2004002008$0$0$001507$150$0$15001006$250$0$2500505$300$0$3000504$850$500$3504001003$50$0$5001502$700$500$2004002001Period CostsOrdering CostHolding CostOrder QuantityDemandMonthApproach: EOQ© Chris Caplice, MIT11MIT Center for Transportation & Logistics – ESD.2600200400600800100012001400160018002000OnHandInventory123456789101112MonthApproach: Silver-Meal Algorithm550 250 400 550 250© Chris Caplice, MIT12MIT Center for Transportation & Logistics – ESD.260$350$1050$150+$400$5004502008$325$650$150$5002501507$500$500$0$5001001006Buy:2nd$283$1700$150+$200+$150+$200+$500$5006501006$240$1200$150+$200+$150+$200$500550505$250$1000$150+$200+$150$500500504$283$850$150+$200$5004501003$325$650$150$5003501502$500$500$0$5002002001Buy:1stMean CostLotCostHolding CostOrderCostLotQtyDmdMonApproach: Silver-Meal Algorithm© Chris Caplice, MIT13MIT Center for Transportation & Logistics – ESD.260$500$500$0$50025025012Buy:5th12$400$800$300$50055030011$500$500$0$50025025010Buy:4th $400$1200$200+$500$50065025010$350$700$200$5004002009$500$500$0$5002002008Buy:3rdMean CostLotCostHolding CostOrderCostLotQtyDmdMon250800$1300$300+$500$500 $433Approach: Silver-Meal Algorithm© Chris Caplice, MIT14MIT Center for Transportation & Logistics – ESD.260$3850$2500$135020002000Totals:$500$500$025025012$0$0$0030011$800$500$30055025010$0$0$002009$700$500$2004002008$0$0$001507$650$500$1502501006$0$0$00505$50$0$500504$100$0$10001003$200$0$20001502$850$500$3505502001Period CostsOrdering CostHolding CostOrder QuantityDemandMonthApproach: Silver-Meal Algorithm© Chris Caplice, MIT15MIT Center for Transportation & Logistics – ESD.2600200400600800100012001400160018002000OnHandInventory123456789101112Month550 450 450 550Approach: Optimization (MILP)© Chris Caplice, MIT16MIT Center for Transportation & Logistics – ESD.260Decision Variables:Qi = Quantity purchased in period iZi = Buy variable = 1 if Qi>0, =0 o.w.Bi = Beginning inventory for period IEi = Ending inventory for period IMILP ModelObjective Function: • Minimize total relevant costsSubject 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 ZData:Di= Demand per period, i = 1,,nCo= Ordering CostChp= Cost to Hold, $/unit/periodM = a very large number….Approach: Optimization (MILP)© Chris Caplice, MIT17MIT Center for Transportation & Logistics – ESD.260Approach: Optimization (MILP)1111..00 2,3,...1, 2,...01,2,...0 1, 2,...01,2,...01,2,...{0, 1} 1, 2,...nnOi HPiiiiiiii iiiiiiiMin TC C Z C EstBBE i nEBQD i nMZQ i nBinEinQinZin==−=+=−= ∀=−− = ∀=−≥ ∀=≥∀=≥∀=≥∀==∀=∑∑Objective FunctionConservation of Inventory ConstraintsBeginning & Ending Inventory ConstraintsNon-Negativity & Binary ConstraintsEnsures buys occur only if Q>0© Chris Caplice, MIT18MIT Center for Transportation & Logistics – ESD.260Approach: Optimization (MILP)© Chris Caplice, MIT19MIT Center for Transportation & Logistics – ESD.260Month Demand OTB L4L EOQ S/M OPT1 200 2000 200 400 550 5502 150 1503 100 100 400450 50550 506 100 100 250 4507 150 1508 200 200 400 4009 200 200 45010 250 250 400 55011 300 300 400 55012 250 250 250Total Cost $13,600 $6,000 $4,400 $3,850 $3,750Comparison of
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