Presentation on theme: "Single Period Inventory Models"— Presentation transcript:
1 Single Period Inventory Models Yossi SheffiMass Inst of TechCambridge, MA
2 Outline Single period inventory decisions Calculating the optimal order size� NumericallyUsing spreadsheetUsing simulation� AnalyticallyThe profit function� For specific distributionLevel of ServiceExtensions:� Fixed costs� Risks� Initial inventory� Elastic demand
3 Single Period Ordering Seasonal itemsPerishable goodsNews printFashion itemsSome high tech productsRisky investments
9 Optimal Order (Analytical) The optimal order is Q*At Q* the probability of selling one more magazineis the probability that demand is greater than Q*The expected profit from ordering the (Q*+1)stmagazine is: If demand is high and we sell it: (REV-COST) x Pr( Demand is higher than Q*) If demand is low and we are stuck: (-COST) x Pr( Demand is lower or equal to Q*)The optimum is where the total expected profitfrom ordering one more magazine is zero:� (REV-COST) x Pr( Demand > Q*) – COST x Pr( Demand≤ Q*) = 0
10 Optimal Order The “critical ratio”: Cummulative Frequency Frequency (Wks/YrMoreDemand (Mag/week)
11 Salvage Value Salvage value = $4/Mag. Cummulative Frequency ProfitCummulative FrequencyFrequency (Wks/Yr)Order SizeDemand (Mag/week)
12 The Profit Function Revenue from sold items Revenue or costs associated with unsold items. These may include revenue from salvage or cost associated with disposal.Costs associated with not meetingcustomers’ demand. The lost sales cost can include lost of good will and actualpenalties for low service.The cost of buying the merchandise in the first place.
19 Risk of Loss REV=$15 COST=$7 Probability of Loss Order Quantity 300/(15-7)=37.5. Below that there is no possibility to make money even if we sell ALL the order.
20 Ordering with Initial Inventory Given initial Inventory: Q0, how to order?1. Calculate Q* as before2. If Q0 < Q*, order (Q*< Q0 )3. If Q0 ≥ Q*, order 0With fixed costs, order only if the expectedprofits from ordering are more than theordering costs1. Set Qcr as the smallest Q such thatE[Profits with Qcr]>E[Profits with Q*]-F2. If Q0 < Qcr, order (Q*< Q0 )3. If Q0 ≥ Qcr, order 0Note: the cost of Q0 is irrelevant
21 Ordering with Fixed Costs and Initial Inventory REV=$15COST=$7Ordering with Fixed Costs and Initial InventoryExample: F = $150Expected ProfitInitial Inventory•If initial inventory is LE 46, order up to 80•If initial inventory is GE 47, order nothing
22 Elastic Demand μ =D(P); σ = f(μ) Procedure: 1. Set P 2. Calculate μ Expected Profit FunctionElastic Demandμ =D(P); σ = f(μ)Procedure:1. Set P2. Calculate μ3. Calculate σ4.5. Calculate optimal expected profits as afunction of P.ProfitPriceP* = $22Q*= 65 Magμ(p)=56 Magσ= 28Exp. Profit=$543Rev = $15Cost= $8μ(p)=165-5*pσ= μ/2