Question

For the last 50 weeks, the demand for a product was observed to be as given in the table below. For example, 250 units were demanded for 10 weeks in the span of 50 weeks (not necessarily in one single stretch!). The unit price of the product $500 and normally sells for $750, If the product is not sold during that week, it can be sold at a reduced price of $300 per unit. If it is out of stock, the lost goodwill amounts $150/unit.  

Demand	Frequency
240	5
250	10
260	20
270	10
280	5

1. Calculate the probabilities for the various demands and find also the cumulative probabilities.
2. What is the appropriate value of ordering quantity? Please explain the steps.
3. If the unit selling price was increased to $900, how would your answer in part ‘b’ change?

Answer 1

a.

The probability distribution of the demand is calculated as follows:

Demand	Frequency	Probability	Cumulative Probability (less than demand)
240	5	5/50 = 0.1	0.1
250	10	10/50 = 0.2	0.1+0.2 = 0.3
260	20	20/50 = 0.4	0.3+0.4 = 0.7
270	10	10/50 = 0.2	0.7+0.2 = 0.9
280	5	5/50 = 0.1	0.9+0.1 = 1
Total	50	1

b.

Given problem is solved by applying Newsvendor problem method as follows:

Sales price = p = $750/unit

Purchase Cost = c = $500/unit

Salvage cost of unsold unit = s = $300/Unit

Stock-out cost = $150/unit

Cu = under-stocking cost = cost of shortage (underestimate demand) = Sales price/unit – Cost/unit + stock-out cost

Cu = $750 – $500 + 150 = $400 per unit

Co = Over-stocking cost = Cost of overage (overestimate demand) = Cost/unit – Salvage value/unit

Co = $500 – $300 = $200 per unit

The service level or probability of not stocking out, is set at,

Service Level = critical ratio = Cu/( Cu + Co) = 400/(400 + 200)

Service Level = critical ratio = 0.67

Select the order quantity whose Cumulative probability is immediate greater than Critical ratio = 0.67 to maximize expected profit.

The cumulative prob. of order quantity less than 260 units is 0.7, immediate greater probability than Critical ratio.

Thus, to maximize the profit, the Optimal Order Quantity = Qo = 260 units

Part c.

If the price is increase to $900/unit,

Cu = under-stocking cost = cost of shortage (underestimate demand) = Sales price/unit – Cost/unit + stock-out cost

Cu = $900 – $500 + 150 = $550 per unit

Co = Over-stocking cost = Cost of overage (overestimate demand) = Cost/unit – Salvage value/unit

Co = $500 – $300 = $200 per unit

The service level or probability of not stocking out, is set at,

Service Level = critical ratio = Cu/( Cu + Co) = 550/(550 + 200)

Service Level = critical ratio = 0.73

Select the order quantity whose Cumulative probability is immediate greater than Critical ratio = 0.73 to maximize expected profit.

The cumulative prob. of order quantity less than 270 units is 0.73, immediate greater probability than Critical ratio.

Thus, to maximize the profit, the Optimal Order Quantity = Qo = 270 units

If the price is increased t $900 from $750, the order quantity also changes from 260 units to 270 units.