SuperFun Toys
Question Description
Rephrase the final paragraph (bold) using the the SuperFun Toys data below to decide what quantity would be ordered the profit for the 3 sales scenarios; include two references
SuperFun Toys, Inc., is a company that sells a range of new and innovative children's toys. It has been determined that the pre-holiday season appears to be the best time to introduce a new toy in time for the December holiday. To do this manufacturing should start in June or July to reach the store shelves by October.
In this case study, our team is asked to help SuperFun Toys, who plans to introduce a new product called Weather Teddy, which is a talking teddy bear made by a company in Taiwan. In the case study, we will perform statistical analysis and the profit projections which is typically done by the product management group. The team will provide management with an analysis of the stock-out probabilities for various order quantities, an estimate of the profit potential, and help make an order quantity recommendation. The case study will be analyzed by answering the following questions.
Use the sales forecaster's prediction to describe a normal probability distribution that can be used to approximate the demand distribution
Per Black, J. (2017), the normal distribution is described or characterized by two parameters: the mean, μ, and the standard deviation, σ. The values of μ and σ produce a normal ddistribution. The density function of the normal distribution is f(x)=1σ2π√e −1/2[(x−μ)/σ)] 2
Since the expected demand is 20,000 with a 95% probability that demand would be between 10,000 units and 30,000 units, then the demand distribution is assumed to be normally distributed with a mean = 20,000 and standard deviation = (30,000 – 20,000) /2 = 5000.The mean is 20,000 units, and the standard deviation is 5,102 units.
Assumptions:
Let X be the demand for the toy. Then X follows the normal distribution with mean μ = 20000 and standard deviation σ. Then
P(10000 < X < 30000) = 0.95
P((10000-20000)/σ < (X-20000)/σ < (30000-20000)/σ) = 0.95
From tables of areas under the standard normal curve (30000-20000)/σ = 1.96
σ = (30000-20000)/1.96 =10000/1.96 = 5102
Sketch the distribution and show its mean and standard deviation. Hint: To find the standard deviation, think Empirical Rule covered in Week 1
The empirical rule is the statistical rule stating that for a normal distribution almost all data will fall within three standard deviation of the mean. Broken down, the empirical rule shows that 68% will fall within the first standard deviation, 95% within the first two standard deviations, and 99.7% will fall within the first three standard deviations of the distribution's average. In the case of Superfun Toys, a standard deviation of 5099.02 was calculated and was used to show the deviation between order quantities. The sketch below illustrates each percentage breakdown for the order quantities.
Mean | 21,000.00 |
stdev | 5,099.02 |
Compute the probability of a stock-out for the order quantities suggested by members of the management team (i.e. 15,000; 18,000; 24,000; 28,000)
The Members of the management team have suggested some quantities of the Stock-out. The quantities are 15,000, 18,000 24,000 and 28,000. When these quantities of stock-out are computed the probability of 15,000 is 34.13 percent, 18,000 is 15.54 percent, 24,000 is 21.19 percent, and 28,000 is 0.59 percent of occurring. This means out all the four quantities given by the members of management 15,000 of stock-out has the highest probability of occurring. On the other hand, 28,000 of stock-out has the least probability of occurring in this scenario. The probability of stock out with an order of K units is P(X > K) = P(Z > (K-20000)/5102), where Z is distributed as standard normal.
Figure 2.
Order (K) | (K-20000)/5102 | P(X > K) |
15000 | 1 | .3413 |
18000 | .4 | .1554 |
24000 | .8 | .2119 |
28000 | 1.6 | 0.0548 |
Compute the projected profit for the order quantities suggested by the management team using three scenarios: pessimistic in which sales are 10,000 units, most likely case in which sales are 20,000 units, and optimistic in which sales are 30,000 units
The management of SuperFun Toy Company has requested a profit projection for Weather Teddy for three very different scenarios:pessimistic, in which sales would be 10,000 units; most likely, which would be for 20,000 units; and optimistic, which would be for 30,000 units.Even under the best of conditions, making a decision about production and the unknowns that come with that can be a gamble.
“Decision making under uncertainty occurs when it is unknown which states of nature will occur, and the probability of a state of nature occurring is also unknown" (Black, pg. 708, 2017).There are several different decision making approaches, and SuperFun Toy Company will cover three:pessimistic, most likely, and optimistic.
The pessimistic approach is known as maximum criterion.“The assumption is that the worst will happen and attempts must be made to minimize the damage” (Black, pg. 708, 2017).Pessimistic can also be described as worse-case-scenario.
The optimistic approach is known as maximax criterion, “in which the decision maker bases action on a notion that the best things will happen” (Black, pg. 707, 2017).This is known as best-case-scenario.
In the middle of maximum criterion and maximax criterion we will find the Hurwicz Criterion, and can be considered a most-likely scenario, in which the projected sales will more than likely meet the projected profit.The following is a chart with sales projections for each case scenario:
Projected Profit for Order Quantities and Scenarios | |||||
Order | Pessimistic=10000 | Most Likely=20000 | Optimistic=30000 | ||
15000 | 8x10000-11x5000= $25000 | 8x15000= $120000 | 8x150000= $120000 | ||
18000 | 8x10000-11x8000= $-8000 | 8x18000 = $144000 | 8x18000=$144000 | ||
24000 | 8x10000-11x140000=$ -74000 | 8x20000-11x4000= $116000 | 8x24000= $192000 | ||
28000 | 8x10000-11x18000=$ -118000 | 8x20000-11x8000= $72000 | 8x28000= $224000 |
One of SuperFun's managers felt the profit potential was so great the order quantity should have a 70% chance of meeting demand and only a 30% chance of any stock- outs. What quantity would be ordered under this policy, and what is the projected profit under the three sales scenarios?
One of Super Fun’s manager felt that the profit potential was so great that the order quantity should have a 70% chance of meeting demand and only a 30% chance of any stock-outs. What quantity would be order under this policy, and what is the projected profit under the three sales scenarios?
The order quantity to meet 70% demand is found by solving
P (X < K) =0.70
P (Z < (K-20000)/ 5102 ) = 0.70
(K-20000)/5102 = 0.5244
K = 20000 + 5102 * 0.5244 = 20000 + 2675 = 22675
The order quality should have a 70% chance of meeting demand and only a 30%chance ofany stock –outs. Let y denote the order quality corresponding to these restrictions.
Probability that demand is greater that or equal to y= 0.70
P (X ≥ y)=0.70
P (X-μ/σ ≥ y- μ/σ)=0.70
P (Z>y-20000/5102)=0.70
P (X-μ/σ ≥ y-20000/5102)=0.70
1-P (Z > y-20000/5102) =0.70
P(Z≤ y-20000/5102) = 0.30
From normal area tables, we know that P (Z≤−0.5244)=0.30
So we have y-20000/5102= -0.5244
Y=-0.5244×5102+20000
Y=17324
So the quantity to be ordered is 17324 units per approximately
Worst case in which sales equal 10000 units
Expected sales price =10000 x24+240000
Suggested order quantities is 17324
Cost price is 17324 x16=277184
Sale is only 10000
So the stock out is 17324-10000=7324 units
Conclusion
Our team believes that we have presented SuperFun Toys, Inc with enough statistics to make an informed decision on whether to start manufacturing Weather Teddy.
Reference
Black, J. (2017). Business Statistics: For Contemporary Decision Making. John Wiley and Sons, Ince
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