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Lecture 6 Newsvendor examples and (Q,R) systemsl

Lecture 6 Newsvendor examples and (Q,R) systemsl -...

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Newsvendor Model Let’s Practice 1 s o s c c c Q x Q F + = = *) Pr( ) (
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A retail outlet sells a seasonal product for $10 per unit. The cost of the product is $8 per unit. All units not sold during the regular season are sold for half the retail price in an end-of-season clearance sale. Assume that the demand for the product is normally distributed with μ = 500 and σ = 100. a. What is the recommended order quantity? b. What is the probability of a stockout? Example 1: c. To keep customers happy and returning to the store later, the owner feels that stockouts should be avoided if at all possible. What is your recommended quantity if the owner is willing to tolerate a 0.15 probability of stockout? d. Using your answer to part (c), what is the goodwill cost you are assigning to a stockout?
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Solution to Example 4: a . Selling price=$10, Purchase price=$8 Salvage value=10/2=$5 c s =10 - 8 = $2, c o = 8-10/2 = $3 Single-Period Models (Continuous Demand) G(Q*)= = = 0.4 Now, find the Q so that G(Q*) = 0.4 or, area (1) = 0.4 3 2 2 + s o s c c c + Probability Demand μ =500 Area=0.40 z = 0.255 σ = 100 (1)
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So, Q * = μ + z σ =500+(-0.255)(100)= 474.5 units. b. P(stockout)=P(demand Q )=1-P(demand< Q )=1- p =1-0.4=0.6 Single-Period Models (Continuous Demand) Probability Demand μ =500 Area=0.60 z = 0.255 σ = 100
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c . P(stockout)=Area(3)=0.15 Single-Period Models (Continuous Demand) Probability σ = 100 Area=0.15 Q * = μ + z σ =500+(1.035)(100)=603.5 units. Demand μ =500 z = 1.035
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d. p =P(demand< Q )=1-P(demand Q ) =1-P(stockout)=1-0.15=0.85 For a goodwill cost of g c u =10 - 8+ g = 2+ g , c o = 8-10/2 = $3 Now, solve g in p = = =0.85 u c 2 + g Single-Period Models (Continuous Demand) Hence, g =$15. u o c c + 3 2 + + ) ( g
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Example 2: The J&B Book Shop orders a book monthly . Unsold books are kept on the shelves for future sales. Assuming that customers who request the book when they are out of stock will wait until the following month. The J&B Book Shop buys the books for $1.15 and sells it for $2.75. They estimate a loss-of-goodwill cost of $50 cents each time a book is backordered. Demand follows a normal distribution with μ=18 and σ =6. The J&B Book Shop uses a 20 percent annual interest rate to determine its holding cost, 1. How many books should they purchase at the beginning of each month? 2. Solve the problem assuming sales are lost Backorder case: c o = Holding cost Lost sales case: c s =Loss-of-goodwill cost + lost profit
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Example 2 solution a. Backorder case: Selling price=$2.75, Purchase price=$1.15 loss-of-goodwill cost = $.50 D ----- N(18,16 2 ) c s = loss-of-goodwill cost = $.50 c o = ( i% * c /12)=(0.2*1.150/12=0.0192 8 9630 . 0 0192 . 0 5 . 0 5 . 0 *) ( = + = + = s o s c c c Q G Q * = μ + z σ =18+(1.79)(6)=29 units From tables
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Example 2 solution a. Lost sales case: Selling price=$2.75, Purchase price=$1.15 loss-of-goodwill cost = $.50 D ----- N(18,16 2 ) c s = loss-of-goodwill cost + lost profit= .50 + 1.6=2.1 c o = ( i% * c /12)=(0.2*1.150/12=0.0192 9 9909 . 0 0192 . 0 1 . 2 1 . 2 *) ( = + = + = s o s c c c Q G Q * = μ + z σ =18+(2.36)(6)=32 units From tables
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Newsvendor Takeaways Inventory is a hedge against demand uncertainty.
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