Question

kindly do it in written form on word

2 Optimal Restructuring of Team Size (40 marks)

There are currently three teams (labeled as X, Y, and Z) currently operating in a department. After a successful recruitment cycle, the manager of the department decides to do a re-organisation of the team structure with the aim to maximise overall productivity. Let x, y, and z denote the change of size of teams X, Y, and Z, respectively.2 Let the overall change of productivity of the department be given by

  −2x2 + 4x − y 2 − 2y − 0.5z2 + z

1. Suppose the manager cannot change the size of team Z (i.e., z = 0). Present a graph that visualises how the overall change in productivity of the department changes with the change of size in teams X and Y. Comment on the relationship. [10 marks]

2. Use Solver to find the optimal change to the size of teams X, Y, and Z, if the department had hired exactly

(a) one new staff member

(b) fifteen new staff members

Based on your results comment on whether the manager would prefer to hire one or fifteen new members. Provide some intuition for your answer using the properties of the objective function. [30 marks]

Answer 1

Answer:

Part A)

given

change in Productivity =-2x^2+4x-y^2-2y-0.5z^2+z

Given z=0 so new change in productivity=-2x^2+4x-y^2-2y

From above graph it is clear that as long as there is positive change in X and Y there will be negative change in overall productivity.

part 2)

a) one new staff member

if x=1 then y=0 and z=0 so change in overall productivity =-2*1^2+4*1=2 eq 1

if y=1 then x=0 and z=0 so change in overall productivity =-1^2-2*1=-3 eq 2

if z=1 then z=0 and y=0 so change in overall productivity =-0.5*1^2+1=0.5 eq 3

from equation 1 , 2 and 3 it is clear that change in one unit in X will give maximum change in overall productivity.

b) 15 new staff member

we need to divide these 15 new staff members into x,y and z.

x+y+z=15

and we also need to maximize the change in overall productivity= -2x^2+4x-y^2-2y-0.5z^2+z

So using solver we get

x=3

y=2 and z=10

and maximum change in overall productivity=-54

Since change in Productivity of hiring 15 staff is much lesser than hiring 1 staff. So we will hire only 1 staff.

Change in overall productivity will be negative as there is any new addition of staff member to each of department.