Question 2 (2+2+3+3+3=13 points) – Newsvendor model with Normal demand Consider

Question

Question 2 (2+2+3+3+3=13 points) – Newsvendor model with Normal demand Consider a product satisfying assumptions of the newsvendor model. Demand ~ Normal[=1000, =200]. Purchase cost: $100/unit. For every unit sold, there is a profit of $30 per unit. (Revenue = cost + profit = 100 + 30 = $130). Unsold units are disposed off at a loss of value: $20 per Unit (therefore V=$80).
a.) What is the mean demand?
b.) What is the newsvendor critical ratio?
c.) What is the optimal order quantity Q*? (Use Standard Normal Distribution Function Table Round up to the nearest integer if you get a non-integer.)
d.) If we order Q* as obtained in (c), what is the expected lost sales? (Use Standard Normal Loss Function Table)
e.) If we order Q* as obtained in (c), what is the expected profit?

Explanation / Answer

Mean demand = 1000 units

Critical ratio = Profit per unit / ( Profit / unit + Loss/unit) = 30/ ( 30 + 20) = 30/50 = 0.6

Thus probability of stockout < = Critical ratio ( 0.6)

Corresponding Z value = NORMSINV ( 0.6) in excel = 0.253

Therefore , Optimal order quantity

= Mean demand + Z value x standard deviation of demand

= 1000 + 0.253 x 200

= 1000 + 50.6

= 1050.6

Expected Lost sales = Loss function ( Z value) x Standard deviation of demand

                                     = Loss ( 0.253) x 200

                                      = 0.2863 x 200

                                       = 57.26

Expected lost sales = 57.26 units

Expected profit

= Profit/ unit x Expected sales – Loss through salvage/ unit x Expected Leftover inventory

Now,

Expected sales = Average demand – Expected lost sales = 1000 – 57.26 = 942.74

Therefore ,

Expected Leftover inventory

= Optimal order quantity – Expected sales

= 1050.60 – 942.74

= 107.86

Hence ,

Expected profit

= Profit / unit x Expected sales – Loss through salvage/unit x Expected leftover inventory

= $30 x 942.74 – $ 20 x 107.86

= $28282.2 - $2157.2

= $26125

Therefore , Expected profit = $26125

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