bushindo

Winoc sells four types of products.  The resources needed to produce one unit of each and the sales prices is listed below:

 

Resource                       Product 1                         Product 2                       Product 3                         Product 4

Raw Material                         2                                       3                                    4                                      7

Hours of labor                       3                                       4                                    5                                      6

Sales Price ($)                      4                                       6                                    7                                      8

 

Currently, 4,600 units of raw material and 5,000 labor hours are available.To meet customer demands, exactly 950 total units must be produced.  Customers also demand that at least 400 units of product 4 be produced.    What is the solution that maximizes Winoc's sales revenue?

Here's my try

 

Let a, b, c, and d stand for the fraction of a project that A, B, C, and D can do within each day, respectively.

Spoiler for

It is straightforward to construct a linear equation
 
Bx = y
 
where B is a 4x4 matrix given from the OP, and y = (1,1,1,1)'. Solving for x, we get the following values for A, B, C, and D, respectively.
 
[1,] 0.02614608
[2,] 0.03372183
[3,] 0.04013209
[4,] 0.01705517
 
Units are project/day. From here, it is simple to compute how much each individual contributes to a project.

 

I like your answer.  It is interesting how in America this problem is solved differently from my home country.

 

I'm intrigued. How do people from your country solve this problem?

 

 

Spoiler for

 

 

 

Anna (A), BIll (B), Cindy ©, and Dante (D) work on a project.
Together, A, B, and C can complete it in 10 days.
Together, B, C, and D can complete it in 11 days.
Together, C, D, and A can complete it in 12 days.
Together, D, A, and B can complete it in 13 days.
 
Who is the best performer?  Prove your answer.

 

Spoiler for

In each scenario, 1 worker is excluded and everyone else contributes.
 
Therefore the best worker is the one without whom the work takes the longest. That would mean C is the best worker since without him it takes 13 days to complete the task.
 

 

How many days does each person contribute to the total project days?
 

 
Individuals don't contribute to the project's time.
A can do a project, alone, in a days, but a does not increase the time of a project.
Rather, 1/a contributes to the reciprocal of the project's time.
 
1/a +1/b+1/c = 1/10, etc.
 

Allow me to rephrase cause I disagree: Including someone on a project and excluding someone else directly affects the amount of days a project takes. So in terms of making a three man team we can determine the day load attributed to each person (as well as rank them by productivity). In terms of proportional pay for effectiveness this is very important as we in manufacturing projects pay individuals incentives based on their individual contribution to group projects.
 

So, by your later question, are you asking for how long each individual would take to complete the project working on his/her own? [edit from here] This would then, of course, make that proportional payment easier, using [actual time]/[individual time] to determine the proportion of total payment each individual should receive.
 

essentially, i am asking how much of the duration is caused by each individual on the three man team if we assume that a person is just as productive no matter who they work with
 

Anna (A), BIll (B), Cindy ©, and Dante (D) work on a project.

Together, A, B, and C can complete it in 10 days.

Together, B, C, and D can complete it in 11 days.

Together, C, D, and A can complete it in 12 days.

Together, D, A, and B can complete it in 13 days.

 

Who is the best performer?  Prove your answer.

 

Spoiler for

:

32piz.JPG

Make the star cut: 3 x 5 around plus the middle=16 ,

two points added cut though 5 chunks with red

cut through 7 chunks with green and cut through

4 chunks with blue.

          

 

Spoiler for So you are saying that

... it takes 7 points to get 32 slices.

 

Correct. The star awaits the formula.

 

Recursive formula