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Operations ManagementUncertain DemandThe Magnitude of Shortages (Out of Stock)What are the Reasons?Consumer ReactionWhat can be done to minimize shortages?Slide 7Optimal Service Level: The Newsvendor ProblemNews Vendor ModelSlide 10Slide 11Slide 12Compute the Average DemandSlide 14Slide 15Cumulative ProbabilitiesNumber of Units Sold, SalvagesTotal Revenue for Different Ordering PoliciesExample 2: Denim WholesalerSlide 20Example 2: Marginal AnalysisSlide 22Slide 23Slide 24Slide 25Slide 26Slide 27Slide 28Slide 29Analytical Solution for the Optimal Service LevelSlide 31Marginal Value: The General FormulaSlide 33Marginal Value: Uniform distributionSlide 35Type-1 Service LevelMarginal Value: Normal DistributionSlide 38Slide 39Slide 40Slide 41Slide 42Slide 43Additional ExampleAdditional Example - SolutionSlide 46Slide 47Slide 481 Managing Flow Variability: Safety InventoryOperations ManagementSession 23: Newsvendor Model2 Managing Flow Variability: Safety InventoryUncertain DemandUncertain Demand: What are the relevant trade-offs?–Overstock•Demand is lower than the available inventory•Inventory holding cost–Understock•Shortage- Demand is higher than the available inventory–Why do we have shortages?–What is the effect of shortages? Session 23Operations Management23 Managing Flow Variability: Safety InventoryThe Magnitude of Shortages (Out of Stock) Session 23Operations Management34 Managing Flow Variability: Safety InventoryWhat are the Reasons? Session 23Operations Management45 Managing Flow Variability: Safety InventoryConsumer Reaction Session 23Operations Management56 Managing Flow Variability: Safety InventoryWhat can be done to minimize shortages?Better forecastProduce to order and not to stock–Is it always feasible?Have large inventory levelsOrder the right quantity–What do we mean by the right quantity?Session 23Operations Management67 Managing Flow Variability: Safety InventoryUncertain DemandWhat is the objective?–Minimize the expected cost (Maximize the expected profits).What are the decision variables?–The optimal purchasing quantity, or the optimal inventory level. Session 23Operations Management78 Managing Flow Variability: Safety InventoryOptimal Service Level: The Newsvendor ProblemCost of Holding Extra InventoryImproved ServiceOptimal Service Level under uncertaintyThe Newsvendor ProblemThe decision maker balances the expected costs of ordering too much with the expected costs of ordering too little to determine the optimal order quantity.How do we choose what level of service a firm should offer?Session 23Operations Management89 Managing Flow Variability: Safety InventoryNews Vendor ModelAssumptions–Demand is random–Distribution of demand is known–No initial inventory–Set-up cost is equal to zero–Single period–Zero lead time–Linear costs:•Purchasing (production)•Salvage value•Revenue•GoodwillSession 23Operations Management910 Managing Flow Variability: Safety InventoryOptimal Service Level: The Newsvendor ProblemCost =1800, Sales Price = 2500, Salvage Price = 1700Underage Cost = 2500-1800 = 700, Overage Cost = 1800-1700 = 100What is probability of demand to be equal to 130?What is probability of demand to be less than or equal to 140?What is probability of demand to be greater than 140?What is probability of demand to be equal to 133?Session 23Operations Management1011 Managing Flow Variability: Safety InventoryOptimal Service Level: The Newsvendor ProblemWhat is probability of demand to be equal to 116?What is probability of demand to be less than or equal to 116?What is probability of demand to be greater than 116?What is probability of demand to be equal to 113.3?Session 23Operations Management1112 Managing Flow Variability: Safety InventoryOptimal Service Level: The Newsvendor ProblemWhat is probability of demand to be equal to 130?What is probability of demand to be less than or equal to 145?What is probability of demand to be greater than 145?Session 23Operations Management1213 Managing Flow Variability: Safety InventoryCompute the Average DemandN1i)( Demand AverageiiXxPXAverage Demand = +100×0.02 +110×0.05+120×0.08 +130×0.09+140×0.11 +150×0.16+160×0.20 +170×0.15 +180×0.08 +190×0.05+200×0.01Average Demand = 151.6 How many units should I have to sell 151.6 units (on average)? How many units do I sell (on average) if I have 100 units?Session 23Operations Management1314 Managing Flow Variability: Safety InventorySuppose I have ordered 140 Unities.On average, how many of them are sold? In other words, what is the expected value of the number of sold units? When I can sell all 140 units? I can sell all 140 units if x ≥ 140Prob(x ≥ 140) = 0.76The expected number of units sold –for this part- is(0.76)(140) = 106.4Also, there is 0.02 probability that I sell 100 units 2 unitsAlso, there is 0.05 probability that I sell 110 units5.5Also, there is 0.08 probability that I sell 120 units 9.6Also, there is 0.09 probability that I sell 130 units 11.7106.4 + 2 + 5.5 + 9.6 + 11.7 = 135.2Session 23Operations Management1415 Managing Flow Variability: Safety InventorySuppose I have ordered 140 Unities.On average, how many of them are salvaged? In other words, what is the expected value of the number of salvaged units? 0.02 probability that I sell 100 units. In that case 40 units are salvaged 0.02(40) = .80.05 probability to sell 110 30 salvaged 0.05(30)= 1.5 0.08 probability to sell 120 20 salvaged 0.08(20) = 1.60.09 probability to sell 130 10 salvaged 0.09(10) =0.9 0.8 + 1.5 + 1.6 + 0.9 = 4.8Total number Sold 135.2 @ 700 = 94640Total number Salvaged 4.8 @ -100 = -480Expected Profit = 94640 – 480 = 94,160Session 23Operations Management1516 Managing Flow Variability: Safety InventoryCumulative ProbabilitiesSession 23Operations Management1617 Managing Flow Variability: Safety InventoryNumber of Units Sold, SalvagesSession 23Operations [email protected] [email protected] Managing Flow Variability: Safety InventoryTotal Revenue for Different Ordering PoliciesSession 23Operations Management1819 Managing Flow Variability: Safety InventoryExample 2: Denim WholesalerThe demand for denim is:–1000 with probability 0.1–2000 with probability 0.15–3000 with probability 0.15–4000 with probability 0.2–5000 with probability 0.15–6000 with probability 0.15–7000 with probability 0.1Unit Revenue (r

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