##### Ergonomic Chairs. A large company with offices in four cities

 Mathematics Tutor: None Selected Time limit: 1 Day

Feb 16th, 2015

A) According to the Hamilton's method, we need to follow the following 4 steps for appropriation :

Step 1 :

we need to find the Standard Divisor which will be Equal to [(Total employees in all four cities)/Available chairs]

So standard Divisor = 2664/148 = 18

Step 2 :

we need to calculate the quota for each City

Quota of City A =( No of employees in City A) / (Standard Divisor) = 757/18 = 42.06

Quota of City B =( No of employees in City B) / (Standard Divisor) = 295/18 = 16.39

Quota of City C =( No of employees in City C) / (Standard Divisor) = 636/18 = 35.33

Quota of City D =( No of employees in City D) / (Standard Divisor) = 976/18 = 54.22

Step 3 :

The Quotas of each city should be rounded down to the next lowest Integer

Temporary Seats Quota of City A after rounding down : 42

Temporary Seats Quota of City B after rounding down : 16

Temporary Seats Quota of City C after rounding down : 35

Temporary Seats Quota of City D after rounding down : 54

Summing up these Temporary Seats Quota = (42 + 16 + 35+ 54) = 147

Difference in total number of Seats and the Temporary Seats is 148 - 147 = 1.

Step 4 :

The difference number of seats should be added 1 each to the City based on the decreasing value of the Fractional Values. Here we have only 1 Seat and the Highest fraction value is for City B, So the Final apportion of chairs using Hamilton method is :

Final apportion of chairs for city A : 42

Final apportion of chairs for city B: 16 + 1 = 17

Final apportion of chairs for city C : 35

Final apportion of chairs for city D : 54

B) Alabama Paradox : An increase in the total number of chairs to be apportioned causes a city to lose a chair.

In this case, the Number of chairs have been increased from 148 to 149. So we have to check if any city is loosing the chair after appropriation

a) If any of the cities are loosing the chairs after appropriation, then we can say that Alabama paradox occurred.

b) If no city is loosing the chairs after appropriation, then we can say that Alabama paradox has not occurred

So let us Repeat the Hamilton method with 149 chairs

Step 1 :

we need to find the Standard Divisor which will be Equal to [(Total employees in all four cities)/Available chairs]

So standard Divisor = 2664/149 = 17.88

Step 2 :

we need to calculate the quota for each City

Quota of City A =( No of employees in City A) / (Standard Divisor) = 757/17.88 = 42.34

Quota of City B =( No of employees in City B) / (Standard Divisor) = 295/17.88 = 16.50

Quota of City C =( No of employees in City C) / (Standard Divisor) = 636/17.88 = 35.57

Quota of City D =( No of employees in City D) / (Standard Divisor) = 976/17.88 = 54.59

Step 3 :

The Quotas of each city should be rounded down to the next lowest Integer

Temporary Seats Quota of City A after rounding down : 42

Temporary Seats Quota of City B after rounding down : 16

Temporary Seats Quota of City C after rounding down : 35

Temporary Seats Quota of City D after rounding down : 54

Summing up these Temporary Seats Quota = (42 + 16 + 35+ 54) = 147

Difference in total number of Seats and the Temporary Seats is 148 - 147 = 1.

Step 4 :

The difference number of seats should be added 1 each to the City based on the decreasing value of the Fractional Values. Here we have only 1 Seat and the Highest fraction value is for City B, So the Final apportion of chairs using Hamilton method is :

Final apportion of chairs for city A : 42

Final apportion of chairs for city B: 16 + 1 = 17

Final apportion of chairs for city C : 35

Final apportion of chairs for city D : 54

Appropriation of chairs in all the 4 cities remained same even after increasing the new chairs from 148 to 149

Since None of the cities lost chairs after appropriation even if the number of chairs increase from 148 to 149, Alabama paradox did not occur in this scenario.

Feb 16th, 2015

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Feb 16th, 2015
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Feb 16th, 2015
Dec 7th, 2016
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