Solution to Case Problem Specialty Toys

Solution to Case Problem Specialty Toys 10/24/2012
I. Introduction: The Specialty Toys Company faces a challenge of deciding how many units of a new toy should be purchased to meet anticipated sales demand. If too few are purchased, sales will be lost; if too many are purchased, profits will be reduced because of low prices realized in clearance sales. Here, I will help to analyze an appropriate order quantity for the company.
II. Data Analysis: 1. 20,0 00 .025 10,0 00 30,0 00 .025 .95 20,0 00 .025 10,0 00 30,0 00 .025 .95

Since the expected demand is 2000, thus, the mean µ is 2000.
Through Excel, we get the z value given a 95% probability is
1. 96. Thus, we have: z= (x-µ)/ ? =(30000-20000)/ ? =1. 96, so we get the standard deviation ? =(30000-20000)/1. 96=5102. The sketch of distribution is above. 95. 4% of the values of a normal random variable are within plus or minus two standard deviations of its mean.
2. At order quantity of 15,000, z= (15000-20000)/5102=-0. 98, P(stockout) = 0. 3365 + 0. 5 = 0. 8365 At order quantity of 18,000, z= (18000-20000)/5102=-0. 39, P(stockout) = 0. 1517 + 0. 5= 0. 6517 At order quantity of 24,000, z= (24000-20000)/5102=0. 8, P (stockout) = 0. 5 – 0. 2823 = 0. 2177 At order quantity of 28,000, z= (28000-20000)/5102=1. 57, P (stockout) = 0. 5 – 0. 4418 = 0. 0582
3. Order Quantity = 15,000

Unit Sales
 Total Cost
 Sales at $24
 Sales at $5
 Profit

10,000
 240,000
 240,000
 25,000
 25,000

20,000
 240,000
 360,000
 0
 120,000

30,000
 240,000
 360,000
 0
 120,000

Order Quantity = 18,000

 Unit Sales
 Total Cost
 Sales at $24
 Sales at $5
 Profit

 10,000
 288,000
 240,000
 40,000
 -8000

20,000
 288,000
 432,000
 0
 144,000

30,000
 288,000
 432,000
 0
 144,000

Order Quantity = 24,000

 Unit Sales
 Total Cost
 Sales at $24
 Sales at $5
 Profit

 10,000
 384,000
 240,000
 70,000
 -74,000

 20,000
 384,000
 480,000
 20,000
 116,000

 30,000
 384,000
 576,000
 0
 192,000

Order Quantity =28,000

 Unit Sales
 Total Cost
 Sales at $24
 Sales at $5
 Profit

 10,000
 448,000
 240,000
 90,000
 -118,000

 20,000
 448,000
 480,000
 40,000
 72,000

 30,000
 448,000
 672,000
 0
 224,000

4. According to the background information, we get the sketch of distribution above. Since z= (Q-20,000)/5102 =0. 52, so we get Q=20,000+0. 2*5102=22,653. Thus, the quantity would be ordered under this policy is 22,653. The projected profits under the three sales scenarios are below: Order Quantity =22,653

 Unit Sales
 Total Cost
 Sales at $24
 Sales at $5
 Profit

 10,000
 362,488
 240,000
 63,265
 -59,183

 20,000
 362,488
 480,000
 13,265
 130,817

 30,000
 362,488
 543,672
 0
 181,224

5. From the information we get above, I would recommend an order quantity that can maximize the expected profit, and it can be calculated by the formula below: P(Demand<=Q) = C1/(C1+C2).
The probability here suggests the demand that is less than or equal to the recommended order quantity, and C1 is the cost of stock out loss per unit, and C2 is the cost of unsold inventory per unit.
According to the background information, Specialty will sell Weather Teddy for $24 per unit and the cost is $16 per unit. So, we can see that C1= $24 – $16 = $8. If inventory remains after the holiday season, Specialty will sell all surplus inventory for $5 a unit. So, C2 = $16 – $5 = $11.
Therefore, we get P(Demand <=Q) = 8/(8+11) = 0. 4211. And the sketch of distribution is below: Since P= 0. 4211, we know z = (Q-20000)/5102 = -0. 2, then we get Q= 20000-0. 2*5102 =18980. And the projected profits obtained under the 3 scenarios are below: Order Quantity =18,980

 Unit Sales
 Total Cost
 Sales at $24
 Sales at $5
 Profit

 10,000
 303,680
 240,000
 44,900
 -18,780

 20,000
 303,680
 455,520
 0
 151,840

 30,000
 303,680
 455,520
 0
 151,840

So the Order Quantity I recommend is 18,980.