Question

2. A computer company looking for a new location for a plant has determined three criteria to use to rate cities. Pair-wise comparisons are given below.

	Recreation Opportunities	Proximity to University	Cost of Living
Recreation Opportunities	1	1/3	1/5
Proximity to University	3	1	1/4
Cost of living	5	4	1

(b) Calculate the consistency ratio.

Answer 1

Ans b) consistency ratio = 0.071571576

We know consistency ratio = Consistency index / random index

Let us calculate the consistency index

Consistency index = (max - n) / (n-1)

n = number of criteria = 3

let us workout the Pair- wise comparison matrix

	Recreational Opportunity	Proximity to university	Cost of living	3rd Root of the product	Sum of all the 3rd root of the product	Priority vecor
Recreational Opportunity	1	0.33	0.2	0.404124002	4.027101915	0.100351074
Proximity to university	3	1	0.25	0.908560296	4.027101915	0.225611449
Cost of living	5	4	1	2.714417617	4.027101915	0.674037478
Sum	9	5.33	1.45
Sum*PV	0.903159666	1.202509021	0.977354342
Lambda Max	3.083023029

The initial matrix is highlighted in Green

Steps to get lambda max

1) Find the nth root of the product - here n = 3 (3 criteria) & the total sum of all the 3rd root of the product (Rows)

we need to find the 3rd root of the product

Recreational Opportunity = (1* 0.33 * 0.2) ^ (1 /3) = 0.404124002

Proximity to university = ( 3*1*0.25)^(1/3) = 0.908560296

Cost of living = (5*4*1)^(1/3) = 2.714417617

The total sum of all the 3rd root of the product = 0.404124002 + 0.908560296 + 2.714417617 = 4.027101915

2) Calculate priority vector (Rows)

PV = 3rd root of the product / Sum of all the 3rd root of the product

Recreational Opportunity = =0.404124002 / 4.027101915 = 0.100351074

Similarly for

Proximity to university = 0.225611449

Cost of living = (5*4*1)^(1/3) = 0.674037478

3) Calculate sum (Columns)

Recreational Opportunity = 1 + 3 + 5 = 9

Proximity to university = 0.33 +1 +4 = 5.33

Cost of living = 0.2 +0.25 + 1 = 1.45

4) Calculate Sum* PV ( Column * Rows)

Recreational Opportunity = 9 * 0.100351074 = 0.903159666

Similarly for

Proximity to university = 1.202509021

Cost of living = 0.977354342

5) Calculate lambda max

max = Sum of ( Sum *PV)

max = 1.202509021 + 0.977354342 + 0.903159666

max = 3.083023029

Consistency index = (max - n) / (n-1)

Consistency index = 3.083023029 - 3 / (3-1)

Consistency index = 0.083023029 / 2

Consistency index = 0.041511514

For n= 3

From the table of random indices

The random index = 0.58

consistency ratio = Consistency index / random index

consistency ratio = 0.041511514/ 0.58

consistency ratio = 0.071571576

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