The Happy Toy Company’s R&D department is always looking for great ideas for new toys. On average, the R&D department generates about 10 new toy ideas a week. To go from idea to approved product, the idea must go through 5 stages: https://d2vlcm61l7u1fs.cloudfront.net/media%2F9cb%2F9cbcf687-036b-4fb6-b54e-a05363a1ddb4%2FphpLrJfO9.png In initial screening a staff member spends 2 hours considering the idea. In mock-up, the idea is sent to one of many suppliers to produce a physical mock-up of the toy. There is essentially unlimited capacity at this stage, but it takes 4 days to get the mockup built even though multiple mockups can be built simultaneously. In safety testing each idea requires 2 days from a staff member to have the idea checked for safety. In focus groups each idea requires 3 hours from a staff member to have the idea test marketed. Finally, the management team meets for 4 hours per idea to decide if the toy should be added to its line of products sold to customers. At the end of each stage, successful ideas enter the next stage. All other ideas are dropped. The following table summarizes the process: https://d2vlcm61l7u1fs.cloudfront.net/media%2F54e%2F54e0586d-79fb-4cbe-9e67-6cd428988122%2FphpQMorPW.png For example, there are 7 staff members that work in the safety stage. In each stage each idea is processed by only one staff member: for example, once one of the 7 staff members starts working on an idea in the safety stage, the idea completes that stage in 2 days (and either exits or moves to the focus group stage). Everyone works 8 hours per day, 5 days a week. a)On average, how many ideas are in the mock-up stage? b)Which stage has the highest implied utilization (as a %)?

Let us begin from the first stage to answer to the first part of the question i.e.
a) Average number of ideas in mock up stage:
Given:
10 new toy ideas are generated per week
P(success) screening: 60%
Staff at screening stage: 2 & time per idea at screening: 2 hours
Working hour per staff: 8 hours a day & 5 days a week so total 8*5=40 hours a week
Solution:
10 ideas are generated on average out of which 60% pass through the screening stage and reach mock up stage
Thus, in total 60% of 10 ideas i.e. 0.6*10 = 6 ideas pass the screening stage
Capacity of the screening stage to pass ideas= frac{No. of per staff hours in a week*no. of staffs in stage}{time taken by each idea} =frac{40*2}{2}=frac{80}{2}= 40 ideas
As, capacity is enough to pass 6 ideas on an average thus 6 ideas are in mock up stage at an average.
b) Highest implied utilization:
We need to first calculate capacity of each stages with the same formula used in subpart a
1. Screening stage
Capacity=frac{80}{2}= 40 ideas
2. Mock up stage
Capacity= unlimited, given in the question
3. Safety
Capacity=frac{7*40}{2*24}=frac{280}{48}= 5.833 ideas
4.Focus group
Capacity=frac{40}{3}= 13.33 ideas
5. Final decision:
Capacity=frac{40}{4}= 10 ideas
Next we will calculate number of ideas that each stage.
Given Probabilities:
P(success) screening: 60%
P(success) safety: 50%
P(success) focus group:35%
P(success) final decision: 20%
Thus out of 10 ideas generated,
0.6*10= 6 ideas reach mockup stage
Then out of 6, all ideas reach safety i.e. 6
Then 50% of ideas at safety reach focus group i.e. 0.5*6=3 ideas
Then 35% of focus group reach final stage i.e. 0.35*3= 1 idea
Thus number of ideas at each stage:
1. Screening: 10
2. Mock up: 6
3. Safety: 6
4. Focus: 3
5. Final Decision: 1
Thus Implied uitilization of the stage can be calculated by frac{No. of ideas}{Capacity of the stage}frac{No. of ideas}{Capacity of the stage}
So, Implied utilization of
.1. Screening stage = frac{10}{40}= 0.25 = 25%
2. Mock up stage = frac{6}{infinity} =0%
3. Safety stage = frac{6}{5.833}=1.02=102%
4. Focus Group = frac{3}{13.33}=0.225=22.5%
5. Final Decision = frac{1}{10}=0.1=10%
Thus, Safety stage has the highest implied utilization of 102%
 
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