Project describes the use of Analytic hierarchy process (AHP) by taking bollywood songs of different era and finding the best song out of the listed options based on different parameters.
Predicting Salary Using Data Science: A Comprehensive Analysis.pdf
Analytic hierarchy process (AHP)
1. ADVANCED MANAGEMENT
SCIENCE
PROJECT ON ANALYTICAL
HEIRARCHY PROCESS AND
FABRICS AND FALL FASHION
Submitted to:
DR. G.N. Patel
Submitted by:
UDIT JAIN
13DM206
2. INTRODUCTION
Analytical heirarchy process is a structured technique for organizing and analysing complex
decisions, based on mathematics and psychology. It is designed for situations in which ideas,
feelings, and emotions affecting the decision process are quantified to provide a numeric
scale for prioritizing the alternatives. Decision Making involves setting priorities and the
AHP is the methodology for doing that.
Rather than prescribing a "correct" decision, the AHP helps decision makers find one that
best suits their goal and their understanding of the problem. It provides a comprehensive and
rational framework for structuring a decision problem, for representing and quantifying its
elements, for relating those elements to overall goals, and for evaluating alternative solutions.
Users of the AHP first decompose their decision problem into a hierarchy of more easily
comprehended sub-problems, each of which can be analyzed independently. The elements of
the hierarchy can relate to any aspect of the decision problem—tangible or intangible,
carefully measured or roughly estimated, well or poorly understood—anything at all that
applies to the decision at hand.
Once the hierarchy is built, the decision makers systematically evaluate its various elements
by comparing them to one another two at a time, with respect to their impact on an element
above them in the hierarchy. In making the comparisons, the decision makers can use
concrete data about the elements, but they typically use their judgments about the elements'
relative meaning and importance. It is the essence of the AHP that human judgments, and not
just the underlying information, can be used in performing the evaluations.
3. The AHP converts these evaluations to numerical values that can be processed and compared
over the entire range of the problem. A numerical weight or priority is derived for each
element of the hierarchy, allowing diverse and often incommensurable elements to be
compared to one another in a rational and consistent way. This capability distinguishes the
AHP from other decision making techniques.
In the final step of the process, numerical priorities are calculated for each of the decision
alternatives. These numbers represent the alternatives' relative ability to achieve the decision
goal, so they allow a straightforward consideration of the various courses of action.
USES AND APPLICATION
The Analytic Hierarchy Process (AHP) is most useful where teams of people are working on
complex problems, especially those with high stakes, involving human perceptions and
judgments, whose resolutions have long-term repercussions.[3] It has unique advantages
when important elements of the decision are difficult to quantify or compare, or where
communication among team members is impeded by their different specializations,
terminologies, or perspectives.
Decision situations to which the AHP can be applied include:
Choice – The selection of one alternative from a given set of alternatives, usually
where there are multiple decision criteria involved.
Ranking – Putting a set of alternatives in order from most to least desirable
Prioritization – Determining the relative merit of members of a set of alternatives, as
opposed to selecting a single one or merely ranking them
Resource allocation – Apportioning resources among a set of alternatives
Benchmarking – Comparing the processes in one's own organization with those of
other best-of-breed organizations
Quality management – Dealing with the multidimensional aspects of quality and
quality improvement
Conflict resolution – Settling disputes between parties with apparently incompatible
goals or positions
4. RANKING OF SONGS
In our case we have to rank the selected songs according to our preferences.
Objective : Choose the best song among the selected five
Criteria:
Alternatives:
Since five criteria (N) have been chosen for selecting the best song hence there will be six
(N+1) pairwise matrix comparisons. The following are the six matrices:
Pairwise comparison of criteria (Cr)
Pairwise comparison of songs for Lyrics (Lr)
Pairwise comparison of songs for Music (M)
Pairwise comparison of songs for Singer (S)
Pairwise comparison of songs for Location (L)
Pairwise comparison of songs for Actor (A)
In order to find out the best song analytical hierarchy process has been used. The three main
methods used for decision making i.e finding out the weights are:
Eigen value method
Iteration method
Lyrics
Music
Singer
Location
Actor
O hasina zulfo wali ohz
Chup gaye sare nazare cgs
Malish tel malish mtm
ude jab jab zulfe ujj
Aaj mausamb bada amb
5. Linear programming method
The scale used for pairwise comparison is a nine point scale the detail of which is as follows:
1 Equally preferred
2 Equally to Moderately preferred
3 Moderately preferred
4 Moderately to Strongly preferred
5 Strongly preferred
6 Strongly to Very Strongly preferred
7 Very Strongly preferred
8 Very strongly to Extremely preferred
9 Extremely preferred
METHODOLOGY
1. HEIRARCHICAL DECOMPOSITION
Best Song
Lyrics
Ohz hasina
julfo
Akhiyo Key
jharokhe
sar jab jab
Ude jab jab
Music
Ohz hasina
julfho
Akhiyo key
jharokhe
Sar jab jab
Ude jab jab
Singer
Ohz hasina
julfho
Akhiyo key
jharokhe
Sar jab jab
Ude jab
jab
Location
Ohz hasina
julfho
Akhiyo key
jharokhe
Sar jab jab
Ude jab jab
Actor
Ohz hasina
julfho
Akhiyo key
jharokhe
Sar jab jab
Ude jab jab
6. 2. PAIRWISE MATRIX EVALUATION
Since there are six matrices for which weightage is to be found out hence two matrices have
been solved by using each method. The different methods used for different matrices are –
1. Eigen Value Method
a. Pairwise comparison of criteria (Cr)
b. Pairwise comparison of songs for Location (D)
2. Iteration Method
a. Pairwise comparison of songs for lyrics (L)
b. Pairwise comparison of songs for music (M)
3. Linear Programming Method
a. Pairwise comparison of songs for singer (S)
b. Pairwise comparison of songs for costume (C)
Finally after obtaining the weightages of the criteria matrix and the alternatives for each
criterion, the final rankings of the songs can be found out as follows -
Criteria Weightages Rankings
Songs [ ] × [ ] = [ ]
Let us analyse the six above mentioned matrixes with the help of three methods.
A. Eigen Value Method
Let A = [aij] be the pairwise comparison matrix and w be the Eigen vector i.e. the weights for
each criteria or song. Mostly the consistency of A does not hold for the pairwise comparison
matrices entered by us and hence we find out λmax (slightly greater than n) and find out the
weightages by dividing Axw by λmax.
8. Normalized
0.486486 0.318584 0.580645 0.439024 0.217391
0.162162 0.106195 0.048387 0.195122 0.26087
0.243243 0.530973 0.290323 0.292683 0.391304
0.054054 0.026549 0.048387 0.04878 0.086957
0.054054 0.017699 0.032258 0.02439 0.043478
Ranking
2.219866 0.412965
0.767011 0.142688
1.95371 0.363452
0.263999 0.049112
0.170843 0.031782
Nmax 5.375429
B. Iteration Method
Let A = [aij] be the pairwise comparison matrix. Calculate A2 and find the row sum. The row
sums are then normalised which forms the weights of the criteria or alternatives. We further
square the obtained pairwise matrix, find the row sum and normalise them to obtain the
weights. We will continue to do this until the weights obtained in two subsequent steps are
similar up to 4 digits (or more if more accuracy is desired). This normalised matrix then
corresponds to the weights of the criteria or the alternatives.
3. Pairwise comparisonofsongs for Music (M) by iteration method
music
Ohz cgs mtm ujj amb
Ohz 1 3 5 0.142857143 9
Cgs 0.333333333 1 0.125 7 3
Mtm 0.2 8 1 5 6
Ujj 7 0.142857143 0.2 1 4
Amb 0.111111111 0.333333333 0.166666667 0.25 1
12. A˄32
Rowsum
2.61738E+12 1.01094E+13 6.04071E+12 1.3543E+12 1.56769E+13 3.57987E+13
5.81986E+11 2.24798E+12 1.34323E+12 3.01139E+11 3.48597E+12 7.96031E+12
1.13809E+12 4.39583E+12 2.62668E+12 5.88869E+11 6.81666E+12 1.55661E+13
5.60297E+12 2.16415E+13 1.29313E+13 2.89922E+12 3.35603E+13 7.66353E+13
3.96977E+11 1.53338E+12 9.16222E+11 2.05412E+11 2.37785E+12 5.42984E+12
1.4139E+14
Normalized
0.2532
0.0563
0.1101
0.5420
0.0384
C. Linear Programming Method
Let A = [aij] be the pairwise comparison matrix. We can obtain the weights of the criteria or
alternatives by converting the comparison matrix into an Linear Programming problem. The
objective function of the LP problem then becomes –
Objective Function: ∑ ∑ 𝑍ij𝑛
𝑗=𝑖+1
𝑛−1
𝑖=1
Constraints:
Xi – Xj – Yij = ln aij (i)
Zij – Yij >= 0 (ii)
Zij – Yji >= 0 for i ≠ j (iii)
Xi – Xj >= 0 i,j = 1,2,3… n ∀ aij > 1 (iv)
Xi – Xj >= 0 i,j = 1,2,3… . n for all k ∀ aik ≥ ajk (v)
X1 = 0 (vi)
After obtaining the values of the decision variables the weights of the criteria or alternatives can be
found out. The weights are the antilog of Xi.
13. 5. Pairwise comparisonofsongs for Location (L) by using Linear
programming
Location
ohz cgs Mtm ujj
Ohz 1.00 0.20 4.00 0.50
Cgs 5.00 1.00 7.00 5.00
Mtm 0.25 0.14 1.00 0.33
Ujj 2.00 0.20 3.00 1.00
Sum = 8.25 1.54 15.00 6.83
Normalised
Singer Ohz Cgs mtm ujj
Average
(w)
ohz 0.12 0.13 0.27 0.07 0.15
cgs 0.61 0.65 0.47 0.73 0.61
mtm 0.03 0.09 0.07 0.05 0.06
ujj 0.24 0.13 0.20 0.15 0.18
0.598441 Ci = 0.095105
2.666589 Ri = 0.99
S x w
=
0.243962
0.776325 Ci/Ri= 0.096066
Nmax
=
4.285316
14.
15. 6. Pairwise comparisonofsongs for Actor (A) by using linear
programming method
actor
ohz cgs mtm ujj
ohz 1.00 3.00 0.25 0.20
cgs 0.33 1.00 0.14 0.17
mtm 4.00 7.00 1.00 2.00
ujj 5.00 6.00 0.50 1.00
Sum = 10.33 17.00 1.89 3.37
Singer Ohz cgs mtm ujj
Average
(w)
Ohz 0.10 0.18 0.13 0.06 0.12
cgs 0.03 0.06 0.08 0.05 0.05
mtm 0.39 0.41 0.53 0.59 0.48
Ujj 0.48 0.35 0.26 0.30 0.35
0.468201 Ci = 0.068195
0.219607 Ri = 0.99
2.02213
1.494646 Ci/Ri= 0.068884
4.204584
16.
17. RESULTS
Given below are the results for the criteria and the songs based on each criterion –
1. Weights for criteria
2 Weights for Lyrics
3 Weights for Music
4 Weights for Singer
Weight
0.405625
0.146843
0.360729
0.058322
0.028482
Weight
0.412965
0.142688
0.363452
0.049112
0.031782
Weight
0.2590
0.1904
0.3094
0.2194
0.0219
Weight
0.2532
0.0563
0.1101
0.5420
0.0384
18. 5 Weights for Location
6 Weights for Actor
Final Result (Ranking)
lyrics music singer Location actor
O hsina julfho 0.427524 0.2590 0.2532 0.14 0.08
Akhiyo ke jharokhe 0.135754 0.1904 0.0563 0.69 0.08
sar jo tera 0.354691 0.3094 0.1101 0.03 0.42
ude jab jab 0.052575 0.2194 0.5420 0.14 0.42
Criteria Ranking
0.427524 0.317446 1
0.135754 0.142563 4
0.354691 0.24677 3
0.052575 0.264031 2
0.029456
Weight
0.14
0.69
0.03
0.14
Weight
0.08
0.08
0.42
0.42
19. FABRICS AND FALL FASHION
PART - 1
Our main objective function is to maximize profits. For this we have to find the optimal
number of units which we can sell for each product.
Given the price of Raw material and Labour and Machinery cost we can find the profit per
unit sold for each item as selling price of items are given. This calculation is done above.
In the Given question we have three type of constraints, one is upper limit of material to be
used, second is the maximum production according to the demand and third is minimum no.
of production.
Hence keeping in mind we have formed our Objective Function and constraints below.
Particulars
Price
Per Yard
($)
Tailored
Wool
Slacks
(x1)
Cashmer
e
Sweater
(x2)
Silk
Blouse
(x3)
Silk
Camisol
e
(x4)
Tailored
Skirt
(x5)
Wool
Blazer
(x6)
Velvet
Pants
(x7)
Cotton
Sweater
(x8)
Cotton
Miniskir
t
(x9)
Velvet
Shirt
(x10)
Button-
Down
Blouse
(x11)
Wool (in yards) 9.00 3.00 2.50
Acetate (in yards) 1.50 2.00 1.50 1.50 2.00
Cashmere (in yards) 60.00 1.50
Silk (in yards) 13.00 1.50 0.50
Rayon (in yards) 2.25 2.00 1.50
Velvet (in yards) 12.00 3.00 1.50
Cotton (in yards) 2.50 1.50 0.50
30.00 90.00 19.50 6.50 6.75 24.75 39.00 3.75 1.25 18.00 3.38
160.00 150.00 100.00 60.00 120.00 140.00 175.00 60.00 40.00 160.00 90.00
300.00 450.00 180.00 120.00 270.00 320.00 350.00 130.00 75.00 200.00 120.00
110.00 210.00 60.50 53.50 143.25 155.25 136.00 66.25 33.75 22.00 26.63
Material Cost (C1)
Labor andMachine Cost (C2)
Price (P)
Profit Per Unit Sold(Profit = P - C1 - C2)($)
22. PART - 2
Now if the Constraint is given that the Velvet material could not be returned back then whole of velvet
material available should be used.
We can see in the above solution that the Velvet shirts are produced to the maximum limit but velvet
shirts are not being produced in order to maximize the profit.
Hence in the above constraints we shall change only one constraint: Velvet Material Constraints:
3x7+1.5x10=20000
In above constraint we have removed the less than (<) condition as we now have to use the whole
material.
23. Solving the Equations and Finding the Maximum Profit by Using
EXCEL SOLVER.
As by fully utilizing the Velvet She is gaining lesser profit hence she should not change her change her
decision and stick to the first plan of production. The balance material she can keep with herself.
Tailored
Wool
Slacks
(x1)
Cashmer
e
Sweater
(x2)
Silk
Blouse
(x3)
Silk
Camisole
(x4)
Tailored
Skirt
(x5)
Wool
Blazer
(x6)
Velvet
Pants
(x7)
Cotton
Sweater
(x8)
Cotton
Miniskirt
(x9)
Velvet
Shirt
(x10)
Button-
Down
Blouse
(x11)
4200 4000 7000 15000 3177.78 5000 3666.67 0 60000 6000 15763
110.00 210.00 60.50 53.50 143.25 155.25 136.00 66.25 33.75 22.00 26.63
Constraints LHS Sign RHS
3.00 2.50 0 <= 45000
2.00 1.50 1.50 2.00 0 <= 28000
1.50 0 <= 9000
1.50 0.50 0 <= 18000
2.00 1.50 0 <= 30000
3.00 1.50 0 = 20000
1.50 0.50 0 <= 30000
1.00 0 <= 7000
1.00 0 <= 4000
1.00 0 <= 12000
1.00 0 <= 15000
1.00 0 <= 5500
1.00 0 <= 5000
1.00 0 <= 6000
1.00 -1.00 0 <= 0
1.00 -1.00 0 <= 0
1.00 0 >= 4200
1.00 0 >= 3000
1.00 0 >= 2800
1.00 0 >= 0
1.00 0 >= 0
1.00 0 >= 0
1.00 0 >= 0
1.00 0 >= 0
1.00 0 >= 0
1.00 0 >= 0
1.00 0 >= 0
1.00 0 >= 0
1.00 0 >= 0
1.00 0 >= 0
Non Negativity :x7
Non Negativity :x8
Non Negativity :x9
Non Negativity :x10
Non Negativity :x11
Non Negativity :x1
Non Negativity :x2
Non Negativity :x3
Non Negativity :x4
Non Negativity :x5
Non Negativity :x6
Max. Production - Velvet Shirt
Min. Production - Silk Camisole
Min. Production - Cotton
Min. Production - Tailored Wool
Min. Production - Wool Blazer
Min. Production - Tailored Skirt
Max. Production - Tailored Wool
Max. Production - Cashmere
Max. Production - Silk Blouse
Max. Production - Silk Camisole
Max. Production - Velvet Pants
Max. Production - Wool Blazer
Acetate - Material Conatraint
Cashmere - Material Conatraint
Silk - Material Conatraint
Rayon - Material Conatraint
Velvet - Material Conatraint
Cotton - Material Conatraint
Particulars Max Profit ($)
No.ofUnits to be Produced
6834822.22
Objective Funtion : Maximize Profit
Wool - Material Conatraint