A manufacturing company plans to begin automating production and reduc

2,3

Each robot will result in annual labor-cost savings of €150,000 beginning with the year of its installation and will have an upfront cost of €400,000.
So, one time cost 400,000 and savings per year 150000. Additional cost of 250000 in the first year of installation

For robot installation in Year 1 of the program, the company will budget €600,000, plus the total amount of labor costs that will be saved by robotic production in that year.
Amount available is 600000+150000x, where x is the number of robots installed that year.

The robot-installation budget for each subsequent year will consist of the total amount of labor costs that will be saved by robotic production in that year plus any money left over from the previous year's robot-installation budget.
Amount available is 150000x+ any money left over, where x is the number of robots installed that year.

First year: Let x be the number of robots installed. Cost incurred = 400000x and money available = 600000+150000x
Thus, \(600000+150000x­\geq 400000x......250000x\leq 600000.....x\leq 2.4\)
So, 2 robots installed and 600000+150000*2-400000*2 or 100000 carried forward to next year

Second year: Let y be the number of robots installed. Cost incurred = 400000y and money available = 100000+150000*(2+y)
Thus, \(100000+150000(2+y)\geq 400000y......250000y\leq 400000.....y\leq 1.6\)
So, 1 robot installed and 100000+150000(2+1)-400000*1 or 150000 carried forward to next year

Third year: Let z be the number of robots installed. Cost incurred = 400000z and money available = 150000+150000*(2+1+z)
Thus, \(150000+150000(3+z)\geq 400000z......250000z\leq 600000.....z\leq 2.4\)
So, 2 robot installed 

First year : 2
Third year : 2+1+2 or 5­

Each time a robot is installed, we do the following:
REDUCE the budget by €400 (since each robot costs €400 to install)
INCREASE the budget by €150 (since each robot saves €150 in labor costs)
Implication:
Net cost for each installation = 400 spent - 150 saved = €250

1st year --> €600
Here, we can install 2 robots (at a net cost of €500), allowing us to apply the remaining €100 to next year's budget.

2nd year --> €100 carried over + €300 in labor savings for the 2 robots already installed = €400
Here, we can install 1 more robot (at a net cost of €250), allowing us to apply the remaining €150 to next year's budget.

3rd year --> €150 carried over + €450 in labor savings for the 3 robots already installed = €600
Here, we can install 2 more robots (at a net cost of €500), bringing the total number of robots to 5.

­

Year 1
Budget + x (150 - 400)
600 - x (250)
Greatest X would be 2
Remaining Budget = 100

Year 2
Budget + saving from existing machine + a (-250)
100 + 300 - a (250)
greatest a would be 1
Remaining Budget = 150
Existing machine 3

Year 3
150 + 3(150) - b (250)
600 - b(250)
Greatest B would be 2

Hence:
x : 2
y : existing machine in Year 3 = 2 + 1 + 2 = 5

­The  trick to solving the question lies in the following sentence:  The robot-installation budget for each subsequent year will consist of the total amount of labor costs that will be saved by robotic production in that year plus any money left over from the previous year's robot-installation budget.
 

The thing to remember here is that as your fleet of robots increases, your saving per year increases. 

Robot Cost = 400k
Saving per robot = 150k
So to buy a robot you need an additional 250k since a new robot will anyway bring its own saving of 150k along. Keep this 250k figure in mind.

First year - Budget is 600k 
You have an additional 600k so you can buy 2 robots because you have an additional 2 * 250k. Then, 100k will be leftover for next year.
No of Robots in first year = 2 Robots

Second year - 100k (leftover from last year) + 300K (savings from 2 existing robots) = 400k 
You can buy only 1 additional robot for 250k with the 400k. Then, 150k will be leftover for next year.
No of Robots in second year = 3 Robots

Third year - 150k (leftover form last year) + 450k (savinsg from 3 existing robots) = 600k
You can buy 2 additional robots for 2*250k.  Then, 100k will be leftover for next year.
No of Robots in third year = 5 Robots

ANSWER - 2, 5
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