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Single Period Inventory Models Yossi Sheffi Mass Inst of Tech Cambridge, MA Outline Single period inventory decisions Calculating the optimal order size Numerically Using spreadsheet Using simulation Analytically The profit function For specific distributions Level of Service Extensions: Fixed costs Risks Initial inventory Elastic demand Single Period Ordering Selling Magazines Weekly demand: 90 57 99 79 48 87 86 95 73 92 79 105 78 67 66 76 58 89 67 73 71 102 87 70 113 52 89 87 64 78 50 107 Total: 4023 magazines Average: 77.4 Mag/week Min: 51; max: 113 Mag/week 66 84 70 80 79 62 54 78 97 91 67 51 75 71 88 79 89 66 62 80 Detailed Histogram 100% 4 90% Frequency (Wks/Yr) 70% 60% 50% 2 40% 30% 1 20% 10% 0 0% 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 Demand (Mag/Wk) Average=77.4 Mag/wk Cumm Freq. (Wks/Yr) 80% 3 16 100% 14 90% 80% 70% 10 60% 8 50% 6 40% 30% 4 20% 90 10 0 11 0 12 0 13 0 M or e 80 0% 70 0 60 10% 50 2 40 Frequency (Wks/Yr) 12 Demand (Mag/week) Cummulative Frequency Histogram The Ordering Decision (Spreadsheet) Assume: each magazine sells for: $15 Cost of each magazine: $8 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 d/wk Prob. 40 0.00 $140 50 0.04 $140 60 0.10 $140 70 0.21 $140 80 0.29 $140 90 0.19 $140 100 0.10 $140 110 0.06 $140 120 0.02 $140 130 0.00 $140 Exp. Profit: $140 $210 $210 $210 $210 $210 $210 $210 $210 $210 $210 $210 $280 $280 $280 $280 $280 $280 $280 $280 $280 $280 $280 $200 $350 $350 $350 $350 $350 $350 $350 $350 $350 $350 $120 $270 $420 $420 $420 $420 $420 $420 $420 $420 $414 $40 $190 $340 $490 $490 $490 $490 $490 $490 $490 $464 -$40 $110 $260 $410 $560 $560 $560 $560 $560 $560 $482 -$120 $30 $180 $330 $480 $630 $630 $630 $630 $630 $457 -$200 -$50 $100 $250 $400 $550 $700 $700 $700 $700 $403 -$280 -$130 $20 $170 $320 $470 $620 $770 $770 $770 $334 -$360 -$210 -$60 $90 $240 $390 $540 $690 $840 $840 $257 -$440 -$290 -$140 $10 $160 $310 $460 $610 $760 $910 $177 -$520 -$370 -$220 -$70 $80 $230 $380 $530 $680 $830 $97 -$600 -$450 -$300 -$150 $0 $150 $300 $450 $600 $750 $17 -$680 -$530 -$380 -$230 -$80 $70 $220 $370 $520 $670 -$63 Order: Expected Profits $600 $500 Profit $400 $300 $200 $100 $0 -$100 20 40 60 80 100 120 Order Size 140 160 Optimal Order (Analytical) The optimal order is Q* At Q* - what is the probability of selling one more magazine ? 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 Pr( Demand ≤ Q*) = REV-COST REV Optimal Order REV-COST 15 − 8 = = 0.47 REV 15 100% 14 90% 80% 12 70% 10 60% 8 50% 6 40% 30% 4 20% 90 10 0 11 0 12 0 13 0 M or e 80 0% 70 0 60 10% 50 2 Demand (Mag/week) Cummulative Frequency Pr( Demand ≤ Q*) = 16 40 Frequency (Wks/Yr The “critical ratio”: Salvage Value Salvage value = $4/Mag. Critcal Ratio = REV − COST 15 − 8 = = 0.64 REV − SLV 15 − 4 $600 $500 40% 30% 4 20% Demand (Mag/week) 13 0 M or e 90 10 0 11 0 12 0 80 0% 70 0 60 10% 50 2 Order Size 16 0 6 $0 14 0 50% 12 0 8 $100 10 0 60% 80 10 40 Frequency (Wks/Yr 70% $200 60 80% 12 $300 40 14 90% 20 100% Cummulative Frequency 16 Profit $400 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. The Profit Function ∞ E [Sales ] = Qi ∫ Q f ( x )dx + X =Q ∫ x if ( x )dx x =0 Q E [Unsold ] = ∫ (Q − x )if ( x )dx = Q − E [Sales] x =0 ∞ E [Lost Sales ] = ∫ ( x − Q )if ( x )dx = μ − E [Sales ] X =Q E [Profit ] = R iE [Sales ] + S iE [Unsold ] − L iE [Lost Sales ] − C iQ The Profit Function – Simple Case E [Profit ] = R iE [Sales ] − C iQ Optimal Order: d E [Profit ] = (1 − F (Q ))iR − C = 0 dQ d E [Sales ] = 1 − F (Q ) dQ F [Q *] = R −C R and: ⎡R − C ⎤ Q * = F −1 ⎢ ⎥ ⎣ R ⎦ Level of Service Cycle Service – The probability that there will be a stock-out during a cycle Fill Rate - The probability that a specific customer will encounter a stock-out REV=$15 COST=$8 Level of Service 100% 90% 80% Service Level 70% 60% 50% 40% 30% Cycle Service 20% Fill Rate 10% 0% 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 Order Normal Distribution of Demand X ~ N ( μ ,σ ) E [sales ] = Q − σ i( ziΦ( z ) + φ ( z )) z= Q−μ σ E[Profit ] = ( R − C ) • Q − R • σ ⋅ [ z • Φ ( z ) + φ ( z )] $500 ⎛ R−C ⎞ Q* = NORMINV ⎜ ⎟= ⎝ R ⎠ $300 $200 ⎛ 15 − 8 ⎞ = NORMINV ⎜ ⎟ = 76 Mags ⎝ 15 ⎠ $100 Order Size -$200 16 0 14 0 12 0 10 0 80 60 -$100 40 $0 20 Expected Profit $400 REV=$15 COST=$8 Incorporating Fixed Costs With fixed costs of $300/order: $600 $500 $400 $200 $100 -$300 -$400 -$500 160 150 140 130 120 Order Size 110 90 80 70 60 50 100 -$200 40 -$100 30 $0 20 Expected Profi $300 REV=$15 COST=$8 Risk of Loss Probability of Loss 1.00 0.80 0.60 F=$300 0.40 F=0 0.20 0.00 20 40 60 80 100 Order Quantity 120 140 160 Ordering with Initial Inventory Given initial Inventory: Q0, how to order? 1. Calculate Q* as before 2. If Q0 < Q*, order (Q*< Q0 ) 3. If Q0 ≥ Q*, order 0 Cost of initial inventory With fixed costs, order only if the expected profits from ordering are more than the ordering costs 1. Set Qcr as the smallest Q such that with Qcr]>E[Profits with Q*]-F 2. If Q0 < Qcr, order (Q*< Q0 ) 3. If Q0 ≥ Qcr, order 0 E[Profits Ordering with Fixed Costs and Initial Inventory REV=$15 COST=$8 Example: F = $150 $600 Expected Profit $500 $400 $300 $200 $100 $0 Initial Inventory 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 -$100 •If initial inventory is LE 46, order up to 80 •If initial inventory is GE 47, order nothing Elastic Demand ⎝ P $600 $500 $400 Profit µ =D(P); σ = f(µ) Procedure: 1. Set P 2. Calculate µ 3. Calculate σ P −C ⎞ 4. Q * = F −1 ⎛⎜ ⎟ Expected Profit Function $300 $200 $100 $0 10 15 20 25 Price ⎠ 5. Calculate optimal expected profits as a function of P. P* = $22 Rev = $15 Cost= $8 µ(p)=165-5*p σ= µ/2 30 Q*= 65 Mag µ(p)=56 Mag σ= 28 Exp. Profit=$543 Elastic Demand: Numerical Optimization Screenshots removed due to copyright restrictions. Any Questions? ? ? ?? ? ? Yossi Sheffi