The president of a consulting firm wants to minimize the total number of hours it will take to complete four projects for a new client. Accordingly, she has estimated the time it should take for each of her top consultants-Charlie, Gerald, Johnny, and Rick-to complete any of the four projects, as follows:
Project Hours
$17171222\begin{array} { | l | l | l | l | l | } \hline \text { Consultant } & \text { A } & \text { B } & \text { C } & \text { D } \\\hline \text { Charlie } & 13 & 16 & 11 & 18 \\\hline \text { Gerald } & 13 & 15 & 10 & 12 \\\hline \text { Johnny } & 15 & 11 & 20 & 15 \\\hline \text { Rick } & 17 & 17 & 12 & 22 \\\hline\end{array}$ a. In how many different ways can she assign these consultants to these projects?
b. What is the total number of hours required by the following arbitrary assignment?
Charlie-B; Gerald-A; Johnny-D; Rick-C
c. What is the optimal assignment of consultants to projects? (Use the assignment method; SHOW YOUR WORK!)
d. For the optimal schedule, what is the total number of hours it will take these consultants to complete these projects?
e. What is the significance, if any, of the fact that Gerald is the best performer at all four projects?

Question

The president of a consulting firm wants to minimize the total number of hours it will take to complete four projects for a new client. Accordingly, she has estimated the time it should take for each of her top consultants-Charlie, Gerald, Johnny, and Rick-to complete any of the four projects, as follows:
Project Hours

17171222\begin{array} { | l | l | l | l | l | } \hline \text { Consultant } & \text { A } & \text { B } & \text { C } & \text { D } \\\hline \text { Charlie } & 13 & 16 & 11 & 18 \\\hline \text { Gerald } & 13 & 15 & 10 & 12 \\\hline \text { Johnny } & 15 & 11 & 20 & 15 \\\hline \text { Rick } & 17 & 17 & 12 & 22 \\\hline\end{array}

a. In how many different ways can she assign these consultants to these projects?
b. What is the total number of hours required by the following arbitrary assignment?
Charlie-B; Gerald-A; Johnny-D; Rick-C
c. What is the optimal assignment of consultants to projects? (Use the assignment method; SHOW YOUR WORK!)
d. For the optimal schedule, what is the total number of hours it will take these consultants to complete these projects?
e. What is the significance, if any, of the fact that Gerald is the best performer at all four projects?

Lizzie Rojas · Accepted Answer

Final Answer :

(a) 4!, or 24 ways. (b) 16 + 13 + 15 + 12 = 56 hours. (c) Table appears below. (d) 48 hours. (e) Each person must get exactly one assignment. Note that Gerald does not get the assignment at which he is absolutely most productive (Project C). The optimum assignment is about opportunity cost, not absolute high or low values.

0010\begin{array} { | l | c | c | c | c | } \hline & \text { Project A } & \text { Project B } & \text { Project C } & \text { Project D } \\\hline \text { Charlie } & 1 & 0 & 0 & 0 \\\hline \text { Gerald } & 0 & 0 & 0 & 1 \\\hline \text { Johnny } & 0 & 1 & 0 & 0 \\\hline \text { Rick } & 0 & 0 & 1 & 0\\\hline\end{array}

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