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Single Period Inventory Models Yossi Sheffi Mass Inst of Tech Cambridge, MA

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Outline Single period inventory decisions Calculating the optimal order size Numerically Using spreadsheet Using simulation Analytically The profit function For specific distribution Level of Service Extensions: Fixed costs Risks Initial inventory Elastic demand

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Single Period Ordering Seasonal items Perishable goods News print Fashion items Some high tech products Risky investments

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Selling Magazines Weekly demand: Total: 4023 magazines Average: 77.4 Mag/week Min: 51; max: 113 Mag/week

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Detailed Histogram Average=77.4 Mag/wk Frequency (Wks/Yr) Demand (Mag/Wk) Cumm Freq. (Wks/Yr)

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Histogram Demand (Mag/Wk) Frequency (Wks/Yr) Cumm Freq. (Wks/Yr) Cummulative Frequency More

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The Ordering Decision (Spreadsheet) Assume: each magazine sells for: $15 Cost of each magazine: $8 Order Exp.Profit: d/wk Prob Newsboy Framework – Panel 1: basic Scenario

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Expected Profits Profit Order Size

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Optimal Order (Analytical) The optimal order is Q* At Q* the probability of selling one more magazine is the probability that demand is greater than Q* The expected profit from ordering the (Q*+1)st magazine 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 profit from ordering one more magazine is zero: (REV-COST) x Pr( Demand > Q*) – COST x Pr( Demand Q*) = 0

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Optimal Order The critical ratio: Frequency (Wks/Yr Cummulative Frequency Demand (Mag/week) More

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Salvage Value Salvage value = $4/Mag. Demand (Mag/week) Order Size Cummulative Frequency Frequency (Wks/Yr) Profit

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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 meeting customers demand. The lost sales cost can include lost of good will and actual penalties for low service. The cost of buying the merchandise in the first place.

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The Profit Function

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The Profit Function – Simple Case Optimal Order: and:

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Level of Service Cycle Service – The probability that there will be a stock-out during a cycle Cycle Service = F(Q) Fill Rate - The probability that a specific customer will encounter a stock-out

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Level of Service Service Level Order Cycle Service Fill Rate REV=$15 COST=$7

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Normal Distribution of Demand Order Size Expected Profit

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Incorporating Fixed Costs With fixed costs of $300/order: REV=$15 COST=$7 Expected Profi Order Size

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Risk of Loss 300/(15-7)=37.5. Below that there is no possibility to make money even if we sell ALL the order. REV=$15 COST=$7 Order Quantity Probability of Loss

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Ordering with Initial Inventory Note: the cost of Q0 is irrelevant Given initial Inventory: Q 0, how to order? 1. Calculate Q* as before 2. If Q 0 < Q*, order (Q*< Q 0 ) 3. If Q 0 Q*, order 0 With fixed costs, order only if the expected profits from ordering are more than the ordering costs 1. Set Q cr as the smallest Q such that E[Profits with Q cr ]>E[Profits with Q*]-F 2. If Q 0 < Q cr, order (Q*< Q 0 ) 3. If Q 0 Q cr, order 0

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Ordering with Fixed Costs and Initial Inventory If initial inventory is LE 46, order up to 80 If initial inventory is GE 47, order nothing REV=$15 COST=$7 Example: F = $150 Expected Profit Initial Inventory

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μ = D ( P ); σ = f (μ) Procedure: 1. Set P 2. Calculate μ 3. Calculate σ Calculate optimal expected profits as a function of P. Elastic Demand Expected Profit Function Profit Price Rev = $15 Cost= $8 μ(p)=165-5*p σ = μ/2 P* = $22 Q*= 65 Mag μ(p)=56 Mag σ = 28 Exp. Profit=$543

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Any Questions? Yossi Sheffi

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