6. Consider the newsvendor problem seen in class. A product costs $20 and sells

Question

6.    Consider the newsvendor problem seen in class. A product costs $20 and sells for $50. It has a salvage value of 4$ if it does not sell. (8 points)

a)    What is the stock-in and stock-out probability?

b)    Assume the demand is uniformly distributed from 0 to 1,000, as in the exercise seen in class. What is the optimal number of products to buy?

c)     Given that you bought a number of products following b), what are the net profits if you observe a demand of 500?

d)    Given that you bought a number of products following b), what are the net profits if you observe a demand of 1,000? What is the stock-out cost?

Explanation / Answer

a)    What is the stock-in and stock-out probability?

Cost -20

Price -50

Salvage price =4

Overage cost is the per-unit cost of over ordering. Underage cost is the per-unit opportunity cost of ordering below demand.

Cost of Overage (Co) = cost - salvage = $`20 - $4 = $ 16

Cost of Underage (Cu) = Price - cost = $ 50- $ 20 = $ 30

In -stock probability is the probability that demand is less than inventory quantity i.e. we can safely meet the demand. F(Q) is the probability the demand is Q or lower.

In-stock probability/critical ratio =P(demand<quantity)= F(Q )=Cu/(Cu + Co) =30/(30+16) = 0.6522

(F(Q) also gives us the target service level )

Stockout probability = 1 - F(Q) = 1-0.6522=0.3478

b)    Assume the demand is uniformly distributed from 0 to 1,000, as in the exercise seen in class. What is the optimal number of products to buy?

STEPS

Uniform distribution form 0(A) to 1000(B)

Mean , u=              (A + B)/2==(0+1000)/2 =500

Standard Deviation, s =    sqrt[(BA)2/12] = sqrt( ( 1000-0)2 /12) =288.66

c)     Given that you bought a number of products following b), what are the net profits if you observe a demand of 500?

Expected profit =[( Price - Cost) X Expected sales)- [(Cost - Salvage value) X Expected leftover inventory]

We order Q= 615.46 units

Expected demand,(mean)µ= 500(given)

STEP 1

[Standard Deviation, s =    sqrt[(BA)2/12] = sqrt( ( 1000-0)2 /12) =288.66 ]

STEP 2

STEP 3

Expected Leftovers = Q – Expected Sale = 615.46- 433.49 =181.97 units

STEP 4

Expected profit =[ ( 50 - 20) * 433.49 ]- [(20 - 4) * 181.97] = 2911.52 = $10093.18

d)    Given that you bought a number of products following b), what are the net profits if you observe a demand of 1,000? What is the stock-out cost?

Expected profit =[( Price - Cost) X (Expected sales)]- [(Cost - Salvage value) X Expected leftover inventory]

We order Q= 615.46 units

Expected demand,(mean)µ= 1000(given)

STEP 1

[Standard Deviation, s =    sqrt[(BA)2/12] = sqrt( ( 1000-0)2 /12) =288.66 ]

STEP 2

STEP 3

Expected Leftovers = Q – Expected Sale = 615.46- 603.76 =11.7 units

STEP 4

Expected profit =[( 50 - 20) X (603.76)] - [(20 - 4) * 11.7] = $17925.6

Stockout cost= expected lost sales* cost of underage

= 396.24 *30 = $11887.2

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