APPLICATIONS OF VECTOR OPTIMIZATION PROBLEMS BY

APPLICATIONS OF VECTOR OPTIMIZATION

PROBLEMS

BY

VANDANA BAGLA

UNDER THE SUPERVISION OF

DR. ANJANA GUPTA

Delhi Technological University (D.T.U.), DELHI

Submitted

in fulfillment of the requirement of Research Summary

to the

TEERTHANKER MAHAVEER UNIVERSITY

MORADABAD

SUMMARY

OF

RESEARCH WORK

The research work titled “Applications of Vector Optimization Problems”

focuses on various perspectives of formation and solutions of Multi Criteria

Decision Making (MCDM) problems. Over the last few decades, various new

approaches have been developed and the methodologies of decision making

processes have been improving gradually. Decision-making approaches are

broadly employed for the selection of the best compromise solution, as any

alternative cannot usually excel in all the criteria under consideration simul-

taneously. Besides, the real criteria values by which a decision is made, the

selection of the best solution also depends on the decision maker’s individual

preferences. Traditional decision making processes and models are falling be-

hind in the fast pace of expedition. Although mathematical advances have

been made in this area, these research areas are still contemporary and inno-

vative. In this study, integrated solution methodologies for general MCDM

problems are developed based on various MCDM approaches. Model-based

MCDM is investigated, with the focus being on improving and applying ap-

proaches based on interactive multi objective optimization methods. In ad-

dition to the methodological aspects, there has been a focus on applying the

interactive multi-objective optimization to the application areas that contain

complex processes and conflicting targets, which have gathered increasing in-

terest of modeling and optimization during recent years. The contribution of

this thesis lies in a state of decision making from the applications in various

scenarios taken up during the research. In this work, a new approach to de-

cision making has been introduced that facilitates fast and reliable decisions.

It enables pair-wise comparisons with necessary sagacity to obtain a clear

and unambiguous conclusion.

3

Organization Of The Thesis

The thesis consists of eight chapters followed by

References and Bibliography

4

Contents

1 Introduction 8

1.1 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.2 Introduction to MCDM Methods . . . . . . . . . . . . . . . . . . . 9

1.2.1 Analytical Hierarchy Process (AHP) . . . . . . . . . . . . . 10

1.2.2 Preference Ranking Organization Method for Enrichment Eval-

uation (PROMETHEE) . . . . . . . . . . . . . . . . . . . . 10

1.2.3 Technique for Order Preference by Similarity to Ideal Solution

(TOPSIS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.2.4 Rank Order Centroids (ROC) . . . . . . . . . . . . . . . . . 12

1.2.5 Ratio Method . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.2.6 Lexicographic Approach . . . . . . . . . . . . . . . . . . . . 13

1.2.7 Weighted Penalty Method . . . . . . . . . . . . . . . . . . . 13

1.3 Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2 AHP Based Assignment Model For Allotting Parking Slots To Dif-

ferent Localities 15

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2 Problem Description and Mathematical formulation . . . . . . . . . 16

2.3 Solution Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3 Site Selection Using Optimization Techniques 18

5

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.2 Problem Description and Mathematical Formulation . . . . . . . . . 19


4 Leader Culling Using AHP - PROMETHEE Methodology 21

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4.2 Problem Description . . . . . . . . . . . . . . . . . . . . . . . . . . 22


5 Agronomic Business Promotion 23

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

5.2 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . 23


6 A Qualitative Assessment of Educational Software 25

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

6.2 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . 26


7 Improving Consistency of Comparison Matrices in Analytical Hi-

erarchy Process 27

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

7.2 Research Methodology . . . . . . . . . . . . . . . . . . . . . . . . . 28

8 Conclusion of the Thesis 29

8.1 Summary of the Content and Conclusions . . . . . . . . . . . . . . 29

8.2 Summarizing the Findings and Limitations . . . . . . . . . . . . . . 32

8.3 Suggested Implications (Research Contribution) . . . . . . . . . . . 33

8.4 Recommendations for Further Work (Research) . . . . . . . . . . . 34

6

References 35

Bibliography 43

Applications of Vector Optimization Problems

7

Chapter 1

Introduction

This chapter is introductory and covers review of literature since 20th century till

date. A brief preamble of Multi Criteria Decision Making (MCDM) methodologies

is also included.

1.1 Literature Review

The study of decision problems has a long history and in the last few decades, it

has been one of the major research field in optimization discipline. A preeminent

work has been carried out in 20th century in field of optimization theory. The

mathematical modeling of these decision making problems gained momentum with

monumental contributions of various eminent economists and mathematicians.

In context of MCDM problems, Major advances have been recorded since 1980s.

Saaty(1980) introduced the Analytic Hierarchy Process (AHP), which brought evo-

lutionary advances in MCDM approaches. Saaty being placed in Fortune magazine,

is visibly one of the most successful people in MCDM. AHP, developed by Saaty

is one of the most popular techniques used by the researchers and practitioners.

Another useful technique, Technique for Order Preference by Similarity to Ideal So-

lution (TOPSIS) was developed by Hwang and Yoon(1981) and it is a popular ap-

8

proach to MCDM problems. In early 1980s, Brans(1982) laid fundamentals of Pref-

erence Ranking Organization Method for Enrichment Evaluation (PROMETHEE),

which is a part of outranking methods used as decision-aid in various MCDM prob-

lems. Considerable work was carried out for more than a decade to bring out various

versions of PROMETHEE by Brans et al.(1982; 1984; 1988; 1992; 1994; 1996).

Saaty & Takizawa(1986) revised AHP in their study so that it could handle the

non-linear hierarchies. Edwards & Barron(1994) first propounded Multi Attribute

Global Inference of Quality(MAGIQ) which was based on rank order centroids to

assign attributes’ weights. Carlsson & Fuller(1995) in mid 1990s introduced inter-

dependency concept into MCDM which later on was improved by Ostermark(1997).

Saaty(1996; 1999) developed the Analytical Network Process (ANP) by the end of

20th century, which is a more general form of AHP. Numerous other credible ap-

proaches have been cultivated and fostered till date, by eminent researchers in the

specified field.

1.2 Introduction to MCDM Methods

MCDM models are divided into two groups of Multiple Attribute Decision Making

(MADM) and Multiple Objective Decision Making (MODM). With regard to the

subject under consideration, the problems of decision making in this research cover

both MADM and MODM models. By considering the various techniques of MCDM

Models, some of the MCDM methods used in the present work are: Analytical Hier-

archy Process (AHP), Technique for Order Preference by Similarity to Ideal Solution

(TOPSIS), Preference Ranking Organization Method for Enrichment Evaluation

(PROMETHEE), Multi Attribute Global Inference of Quality (MAGIQ), Simple

Additive Weighting (SAW), Ratio Method, etc.

9

1.2.1 Analytical Hierarchy Process (AHP)

The formulation of AHP given by Saaty(1980) emerged as a paradigm for decision

making scenarios categorized under MADM. It is based on the well-defined mathe-

matical structure of consistent matrices and their associated eigenvector’s ability to

generate true or approximate weights. AHP can be considered to be both a descrip-

tive and prescriptive model of decision making and it is perhaps, the most widely

used decision making approach in the world today. Its validity is based on the

many thousands of actual applications in which the AHP results are accepted and

used by the aspirants. It is an approach to decision making that involves structur-

ing multiple judgment criteria into a hierarchy, assessing the relative importance of

these criteria, comparing alternatives for each criterion, and determining an overall

ranking of the alternatives. It provides a comprehensive and rational framework for

structuring a decision problem for representing and quantifying its elements. The

outcome of AHP is a prioritized weighting of each decision alternative. 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.

1.2.2 Preference Ranking Organization Method for Enrich-

ment Evaluation (PROMETHEE)

PROMETHEE, an outranking decision making technique, was introduced by Pro-

fessor Jean-Pierre Brans(1982) in 1982. At that time he proposed only the ba-

sic PROMETHEE I (partial ranking) and PROMETHEE II (complete ranking)

versions. Soon thereafter he started working with Bertrand Mareschal(1984) and

they proposed Graphical Analysis for Interactive Assistance (GAIA) method as

10

a descriptive extension of PROMETHEE in 1988. The visual interactive module

GAIA provides a marvelous graphical representation supporting the PROMETHEE

methodology and it is one of a very few effective descriptive MADM methods till

date. PROMCALC (later PROMCALC-GAIA) was one of the first truly interac-

tive software with a strong emphasis on user interface, graphical representations

and sensitivity analysis. The PROMETHEE method is based on mutual compar-

ison of each alternative pair with respect to each of the selected criteria. In order

to perform alternative ranking by the PROMETHEE method, it is necessary to

define preference function P (a, b) for alternatives a and b after defining the crite-

ria. Alternatives a and b are evaluated according to the criteria functions. It is

considered that alternative a is better than alternative b according to criterion f , if

f(a) > f(b). The decision maker has the authority to assign the preference to one

of the alternatives on the basis of such comparison. The preference can take values

on the scale from 0 to 1.

1.2.3 Technique for Order Preference by Similarity to Ideal

Solution (TOPSIS)

TOPSIS is a MADM method to identify solutions from a finite set of alternatives.

As suggested by Hwang and Yoon(1981), a MADM problem may be viewed as a

geometric system in which the m alternatives that are evaluated by n attributes

are similar to m points in a n dimensional space. Therefore, the most preferable

alternative should be the point in that space that is nearest to the ideal solution

and farthest from the worst solution.

11

1.2.4 Rank Order Centroids (ROC)

This method is a simple way of giving weights to a number of items ranked according

to their importance. The decision-makers usually can rank items much more easily

than giving weights to them. This method takes those ranks as inputs and converts

them to weights for each of the items. The term ‘Rank Order Centroid’ was coined

by Barron and Barrett(1996) who also argued for its use in MADM problems. The

idea is to convert ranks into values that are normalized on a 0.0 to 1.0 interval scale.

As proposed by James D. McCaffrey(2005), ROC is a MADM technique based on a

hierarchical decomposition of comparison attributes. It has features similar to AHP

and is used to assign a single, overall measure of quality to each member of a system

having arbitrary number of comparison attributes. This technique uses rank order

centroids to assign normalized numeric weights to the comparison attributes as an

overall measure of quality with reference to a set of evaluation criteria. MAGIQ

was originally developed to validate the results of AHP by adopting a comparatively

simplified course of action and the technique has produced highly useful results in

practice.

1.2.5 Ratio Method

The Ratio Method is another simple way of calculating weights for a number of

critical factors. A decision-maker should first rank all the items according to their

importance in the preferred domain. The next step is giving weight to each item

based on its rank in the interval [10, 90]. Here lowest ranked item will be given

a weight of 10 and rests of the items are rated in multiples of 10 based on the

preferences given by decision maker. For example if item II is five times more

important to item I (the lowest ranked item), then item II is provided with a

rating 50. The last step is normalizing these raw weights as proposed by Weber &

Borcherding(1993).

12

1.2.6 Lexicographic Approach

Multi-objective optimization consists of optimizing a number of objectives that

are usually conflicting. One way to tackle MODM problems is the lexicographic

method. Here each objective is optimized one at a time subject to a pre-defined

ordering established by the decision makers. It is important that the decision-maker

must express preferences in order to establish the ordering and the performance of

the method is vulnerable to the priorities given to various objectives. The single ob-

jective problem taking into account the most prioritized objective is solved, subject

to given constraints by usual methods ignoring rest of the objectives. The problem

is reconstituted considering the next prioritized objective with the previous solution

as an added constraint. The process is repeated until all the objectives are dealt one

by one adding the previous solutions in the constraint inequations. The approach

is useful while dealing with few objectives (two or three) but the procedure may be

lengthy when number of objectives exceed.

1.2.7 Weighted Penalty Method

While solving the MODM problems for a variety of parameters, scalarization is

the most practical and feasible approach. Several possible solutions are generated

depending on the priorities provided to various objectives. In the last decades,

the main focus was on finding one optimal solution to such problems by interactive

methods(1999; 1995) but now due to availability of much advanced application soft-

ware, it is possible to represent the whole efficient set without much manual efforts.

The decision maker gets a better insight to the problem structure by visualizing

the whole solution set at a stretch. A wide variety of scalarization methods exist

based on which one can assemble a MODM problem into prioritized single objec-

tive problem. In this work, a Weighted Penalty Cost Approach to put restrictions

13

on assignments, apart from achieving a prioritized efficient solution by exercising

priorities to preemptive factors, has been employed.

1.3 Publications

The subject matter of the thesis is published / under publication in the form of

following research papers worked out by the author.

I. Vandana Bagla & Anjana Gupta (2011). Analytical Hierarchy Process Based

Assignment Model for Allotting Parking Slots to Different Localities, Journal of

Multi-Criteria Decision Analysis, Wiley Online Library, 18, 173 - 185.

II. Vandana Bagla, Anjana Gupta & Deepika Kukreja (2011). A Qualitative As-

sessment of Educational Software, International Journal of Computer Applications,

36(11), 1 - 7.

III. Vandana Bagla, Anjana Gupta & Bhavna Sharma (2012), Leader Culling

Using AHP-PROMETHEE Methodology, presented in Third International Confer-

ence on Computational Intelligence Applications (ICCIA -2012) held on Feb 11 -

12, 2012 in Sandip Institute of Technology & Research Centre, Nashik-4222013 and

proceedings published in International Journal of Computer Application (IJCA).

IV. Vandana Bagla & Anjana Gupta (2012), Agronomic Business Promotion

published in the book titled “Business Intelligence and Data Warehousing”, Narosa

publications, 132 - 143, ISBN 978-81-8487-212-5.

V. Vandana Bagla , Anjana Gupta & Aparna Mehra (2013), Improving Consis-

tency of Comparison Matrices in Analytical Hierarchy Process, accepted for publi-

cation in International Journal of Current Engineering and Technology.

VI. Vandana Bagla & Anjana Gupta (2013), Site Selection Using Optimiza-

tion Techniques, accepted for publication in International Journal of Scientific &

Engineering Research.

14

Chapter 2

AHP Based Assignment Model

For Allotting Parking Slots To

Different Localities

2.1 Introduction

This chapter propounds a multi-criteria decision-making assignment model to allo-

cate appropriate parking slots to various localities. An algorithm is developed that

allows the consumers to choose the most suitable car parking according to their

requirements. Due to growing population and a consequent increase in number of

vehicles, the problem of parking personal vehicles is becoming explosive. Parking

has been widely recognized to be an important transportation policy issue in the

broad context of the transportation-land use relationship. All the aspects of sharing

of parking slots are visualized subject to land availability. An effective resolution

is suggested by framing a bi-objective problem to minimize the weighted cost and

maximum distances traveled by the residents. MADM technique, Analytic hier-

archy process (AHP) has been used to rank various relating attributes. MODM

15

model ‘Weighted Penalty Method’ is used for scalarization purpose.

2.2 Problem Description and Mathematical for-

mulation

Suppose there are m localities and n available parking slots. Each of the locality

is to be allocated with a parking slot subject to the conditions that development

cost of the prescribed parking slot should be minimized, traveling distance should

be minimum and the prescribed parking slot should justify the requirements of the

locality. For m > n , available parking slots can be shared by aspiring localities

and for m ≤ n , no sharing is required .

Let the localities li (i = 1, . . . ,m) are to be allocated with pj (j = 1, . . . , n)

parking slots. Suppose ai be the requirement of a particular locality li (i = 1, . . . ,m)

and bj be the capacity of the parking slot pj. The set up cost and weight of the

parking slot pj (j = 1, . . . , n) be denoted by cj and wj respectively.

Let ρj =cj∑j cj

(1 − wj) denote the weighted cost of parking slot pj . Here we

take the weights (1 − wj) as our purpose is to minimize the weighted set up cost.

The distance between the locality li and parking slot pj is denoted by dij.

Then above described model is formulated as the following two objective non

linear programming problem.

min (C(x), D(x))

C(x) =m∑

i=1

n∑j=1

ρjxij

D(x) = max{dij : xij = 1 (i = 1, . . . ,m ; j = 1, . . . , n)}

(P ) subject to the constraints

16

n∑j=1

xij = 1 (i = 1, . . . ,m) (2.1)

m∑i=1

xij ≤ β 1 ≤ β ≤ m; β ∈ N ; (j = 1, . . . , n) (2.2)

m∑i=1

aixij ≤ bj ( j = 1, . . . , n) (2.3)

xij = 0 or 1 (i = 1, . . . ,m ; j = 1, . . . , n) (2.4)

It is to be noted that β = 1 if m ≤ n whereas 1 < β ≤ m for m > n.

2.3 Solution Procedure

The above formulated problem is a bi-objective integer programming problem. The

prioritized two-objective problem is reduced to an equivalent single-objective inte-

ger programming problem by using scalarization techniques. Surveys are conducted

and corresponding positive reciprocal matrices are constructed at each level of hier-

archy. MADM technique, AHP is used to calculate the weights for different criteria

associated with a parking slot. The quandary is scalarized using MODM model

‘Weighted Penalty Method’ and subsequently solved using Hungarian Method to

assignment problem and 0− 1 integer programming.

17

Chapter 3

Site Selection Using Optimization

Techniques

3.1 Introduction

This chapter extends the study conducted in chapter 2 by analyzing a three objec-

tive problem in relevance to site selection model. Almost all investment decisions

involve multiple, diverse and complex set of social and financial factors which are

quite hard to be overcome by mere intuition. A number of allocation problems

have been solved in the recent past; see Ignizio(1982), Ignizio and Cavalier(1994).

In Multi objective Programming Problems, there may not be a single solution that

simultaneously optimizes each objective to its fullest. In each case an objective

must have reached a point such that, while attempting to optimize it further, other

objectives suffer as a result. Finding such a solution and quantifying how much

better this solution is compared to many other such solutions, is the goal when

setting up and solving a Multi-objective Optimization Problem; see Azarm(1994),

Prakash and Gupta(2006), Serafini(1985) and Steuer(1986). Here we envision a site

selection model for commercial activities which efficiently explores a multi-criteria

18

decision-making model involving three objectives. Maximization of profit and mini-

mization of set-up cost are the major objectives . Third objective is to rank various

attributes such as capacity, neighborhood, connectivity, transport availability and

proximity which are considerably important factors for a flourishing business.

3.2 Problem Description and Mathematical For-

mulation

Suppose a corporate body/M.N.C. deals in m different business/outlets and there

are n available sites (m ≤ n). Problem is to allocate a suitable site out of the

available sites to each business/outlet. While doing this, the two major objectives

of the company are to maximize the overall profit and to minimize the overall cost.

At the same time, the company wants to prioritize the sites carrying more weights

for flourishing business.

Let pij (i = 1, . . . ,m; j = 1, . . . , n) denote the expected profit , when ith business

is set up at jth site. Also let cj be the overall cost and wj be the weight of jth site

(j = 1, . . . , n) calculated using AHP. Let ρj denote the normalized cost of the jth

site. Then (1− ρj) = rj denotes the reversed cost of the jth site. It is to be noted

that we have reversed the cost as our objective is to minimize the cost and on the

contrary, we are dealing with a maximization problem. Then the above described

model is formulated as the following three objective problem.

(A) Maximize Z(X) = (P (x), R(x), W (X))

P (x) =m∑

i=1

n∑j=1

pijxij , R(x) =m∑

i=1

n∑j=1

rjxij , W (x) =m∑

i=1

n∑j=1

wjxij

19

subject to the constraints:

n∑j=1

xij = 1 (i = 1, . . . ,m) (3.1)

m∑i=1

xij ≤ 1 (j = 1, . . . , n) (3.2)

xij = 0 or 1 (i = 1, . . . ,m ; j = 1, . . . , n) (3.3)


The solution procedure consists of two phases. In the first phase, weights are allo-

cated to various available sites based on various important aspects such as neigh-

boring locality, broad or narrow connecting roads, area/capacity of the available

sites, proximity factors such as competitive business rivals in the nearby areas,

transport availability such as metro or other public transports. To accomplish this,

an approach of Analytic Hierarchy Process (AHP) given by Saaty (1980) is used.

In the second phase, the three objective problem seeking to maximize the profit,

minimize the set up cost born by the investors and to maximize qualitative stan-

dards is untangled using Lexicographic approach and thereafter solved using integer

programming. A comparative solution is projected using scalarization method.

20

Chapter 4

Leader Culling Using AHP -

PROMETHEE Methodology

4.1 Introduction

This chapter proposes the use of improved PROMETHEE methodology for candi-

date selection eligibility and has the potential to re-shape and influence the way

the general public perceive social justice for a rational decision. PROMETHEE

is a MCDM method well adapted to problems where a finite number of alterna-

tives are to be ranked considering several, sometimes conflicting, criteria. It has

successively been applied in many fields especially in the investment analysis and

performance evaluation. Albadvi(2007), Babic and Plazibat(1998), Bouri(2000),

Mareschal(1988), Mareschal and Brans(1988; 1991) and Vranegl(1996) , all applied

PROMETHEE as a decision making tool to solve the different problems in the field

of finance.

The developed methodology facilitates the selection procedure in a rational and

transparent way that can be examined and understood by the concerned voters.

The objective quality of this method keeps inconsistency of decision makers’ op-

21

posing views within reasonable limits. The study is substantiated using Simple

Additive Weighting (SAW) technique. The factors considered here explore personal

as well as professional aspects of the aspiring candidates. The employed methodol-

ogy utilizes systematic approach to screen and ultimately select the most suitable

candidate.

4.2 Problem Description

A group of voters is asked to select a candidate among a set of contesting candidates.

Each voter has a personal ranking of the candidates according to his/her preferences.

The development of a ranking procedure requires a consensus on a set of criteria

for evaluation, upon which the selection of candidates will be based. Potentially

large number criteria are to be judged on a valid scale that is acceptable to all the

participants. The most critical phase in designing the selection process model is

to structure the decision problem. The main goal is to select the best candidate,

according to a set of criteria for evaluation. As conflicting views may arise among

different communities in determining the most important criteria of evaluation in a

given decision making setting, a general survey was conducted to develop the main

criteria, sub criteria and categories for each sub criteria for achieving the goal.


Various surveys were conducted to rate each attribute to others in a series of pair

wise comparisons. An approach of AHP is applied to find out the weights of each

criterion at different levels of hierarchy. The evaluation process finally generates

the global weights for each requisite criterion of interest. After having weight allo-

cations to the set of perceived criteria via AHP, the problem of selection of suitable

candidate is submitted to PROMETHEE II methodology for a pertinent ranking.

22

Chapter 5

Agronomic Business Promotion

5.1 Introduction

Present chapter critically analyzes the agronomic market opportunities for the in-

vestors so as to formulate better merchandise programs. Rural markets in India

constitute a wide and untapped market for many products and services which are

being marketed for the urban masses. The urban market reaching almost to a sat-

uration level, demand from rural India has been on a high trajectory. It provides

a promising business as many new products have already made their entry into the

rural consumer basket. Thus, Indian rural markets have caught the attention of

many companies, advertisers and multinational companies. Contemporary desider-

atum is what kind of investments is required to unlock long-term value from rural

markets.

5.2 Problem Statement

A marketer trying to market his product or service in the rural areas is faced by

many challenges; the first is posed by the geographic aspects like location, profit cost

revenue, population density and climatic conditions. The second challenge is from

23

the demographic factors like purchasing power, literacy level and denominations in

the region. Third is psychographic segmentation like activities, interests and ethics

of the residents. And last but not the least are behavioristic influences namely usage

rates, brand loyalty and festivity. One of the key issues, which require research, is

the methodology by which we can design a cost effective, efficient, environment

friendly model which is befitting on the above set of criteria.


To evaluate the hierarchy, decision makers are asked to allot rankings to the leveled

criteria according to their requisite priorities. Numeric weights are provided to

all the criteria using MAGIQ technique. TOPSIS has been used to provide final

rankings to available agronomic locations. Various surveys were conducted to rate

each attribute to others in a series of pair wise comparisons. An approach of AHP is

applied to find out the weights of each criterion at different levels of hierarchy. The

evaluation process finally generates the global weights for each requisite criterion of

interest. After having weight allocations to the set of perceived criteria via AHP,

the problem of selection of suitable candidate is submitted to PROMETHEE II

methodology for a pertinent ranking.

24

Chapter 6

A Qualitative Assessment of

Educational Software

6.1 Introduction

This chapter inspects the prerequisite of children’s software evaluation in the light

of dynamic nature of edutainment perspective. The work provides comparative

ranking of educational software for scholastic programs scouting both technical and

non technical aspects. Work was conducted to help educational institutions gain a

deeper understanding of professional decisions they face and reduce their initial state

of uncertainty about the best course of action. The Internet-related technologies

have helped to lay the blueprint for the futuristic software evaluation information in

the 21st century. More studies need to be accomplished in this field, as that done by

Escobedo and Evans(1997), where the ratings assigned by the published software

methods are compared with actual child selection or Tammen and Brock(1997),

where middle school students are asked to identify issues they feel are important

for the evaluation of software programs.

25

6.2 Problem Statement

The problem of evaluation of alternatives submitted to a multi-criteria decision

analysis is a contentious task. Usually there is no optimal solution as no alternative

is the best one on each criterion. Problem is to evaluate the software according

to their credibility on the set of weighted judgment criteria. The most demanding

phase in designing the evaluation model is to structure the decision problem. The

goal is to reckon the software according to a set of criteria for evaluation. The con-

ventional approach to predictive evaluation is to use a checklist exploring technical

and non-technical features of the requisite software.


Rank Order Centroid (ROC) methodology and Ratio Method are used to accomplish

the result. The key advantage of ROC Methodology is its simplicity in surveying

whereas Ratio method requires quantified rating of prioritized alternatives. Ratio

Method has its own advantages when the decision makers influence their prioritized

specifications regarding significances.

26

Chapter 7

Improving Consistency of

Comparison Matrices in

Analytical Hierarchy Process

7.1 Introduction

This chapter designs a contemporary MADM model to assist the decision making

scenarios and is specifically applicable to multi-criteria decision problems associated

with large number of attributes. Antecedent of inconsistencies are analyzed and a

corrective application based approach is introduced to construct nearly consistent

matrices. The analysis extends the previous researches in many ways. Firstly, it

analyzes the root causes of inconsistencies in pairwise comparison matrices and

suggests the constructive approaches to overcome the discrepancies. Secondly, the

model enables us to visualize the impact of various criteria on the final ranking

and determine the level of importance of each criterion based on survey analysis.

Thirdly, implementing above methodology can serve as a paradigm for MCDM

problems for which similar models can be developed, modified and improved. A

27

further advantage concerns this model’s inherent ability to procure instant reporting

of analysis results, where numerous subtle factors are involved.

7.2 Research Methodology

Decision Makers are more likely to be cardinally inconsistent because they cannot

estimate precise measurement values even from a known scale and worse when

they deal with intangibles (a is preferred to b twice and b to c three times, but a

is preferred to c only five times). Here the first reason of inconsistency appears.

This is referred to as inconsistency of reciprocity. Again, judgments are ordinally

intransitive if (A is preferred to B and B to C but C is preferred to A). One of

the apparent reason is, large number of pairwise comparisons (n(n− 1)/2) required

to workout the attribute weights at a given level of hierarchy. This results in

baffling responses relating to pairwise comparisons. This is termed as inconsistency

of transitivity and it goes on increasing with the size of the matrix.

28

Chapter 8

Conclusion of the Thesis

In this work, an integrated solution methodology for a general discrete Multi Cri-

teria Decision Making (MCDM) problem is developed based on various interactive

approaches. The concept is designed to make better choices when faced with com-

plex decisions involving several dimensions. MCDM tactics are especially helpful

when there is a need to combine hard data with subjective preferences, to make

trade-offs between desired outcomes and to involve multiple decision makers. In the

decision making process, there are typically multiple conflicting criteria that need

to be assessed simultaneously with about same degree of precedence. This craves

the search for an approach which deals the predicament with necessary sagacity

to obtain a clear and unambiguous conclusion. In this chapter, we summarize the

work done in this dissertation in the said field and draw conclusions of this R and D.

We also discuss limitations of this investigation and suggested implications. Future

plan of action for further research is also acknowledged.

8.1 Summary of the Content and Conclusions

Chapter 1 is introductory and covers literature review and introduction to Multi

Criteria Decision Making (MCDM) methodologies.

29

Chapter 2 propounds a multi-criteria decision-making assignment model to al-

locate appropriate parking slots to various localities. The proposed methodology

provides a decision making method for development of parking slots to decision

maker who is interested in minimizing the total cost as well as the distance tra-

versed by the residents. Sets of efficient solutions for allotting parking slots seeking

minimization of cost as the first priority objective and minimization of distance as

the second priority objective has been obtained. Concepts of assignment model and

integer programming have been used to assign parking slots to various localities.

Heuristics have effectively helped in getting precise and practicable solution. The

study suggests sharing of parking slots if number of localities exceeds the available

number of parking slots.

Chapter 3 augments the study conducted in chapter 2 by analyzing a three objec-

tive problem in relevance to site selection model. While strategically managing our

location decisions, we need solutions that address multiple logistics and economic

factors involving real estate and our customers. Maximization of profit and min-

imization of set-up cost are the two major objectives. Third objective is to rank

the sites qualitatively, for a flourishing business. MADM model, AHP has been

used to rank various relating attributes. A comparative study is conducted using

MODM models ‘Weighted Penalty Method’ and ‘Lexicographic Approach’. The

method also suggested that once starting up with a low profit/low cost model, high

futuristic growth may be expected if qualitative ranks are high.

Chapter 4 proposes to combine two MADM tactics for efficient manipulation of

candidate selection procedure. An approach of AHP is applied to find out the

weights of each criterion at different levels of hierarchy. After having weight alloca-

tions to the set of perceived criteria via AHP, the problem of selection of suitable

candidate is submitted to PROMETHEE II methodology for a pertinent ranking.

The present study reveals that above methodology can be utilized as a valuable

30

decision-making process to evaluate the contestants in a logical and consistent fash-

ion for many reasons.

Chapter 5 critically analyzes the agronomic market opportunities for the investors

so as to formulate better merchandise programs. Here also two MADM methodolo-

gies are merged, intending to simplify the ranking procedure, taking into account

illiteracy problem in rural areas. Numeric weights are provided to all the criteria

using MADM technique, Multi Attribute Global Inference of Quality (MAGIQ).

Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) has been

used to provide final rankings to available agronomic locations. There is a promis-

ing business and definitely a lot of money in rural India but the smart thing would

be to weigh crucial attributes strategically for tactical gains.

Chapter 6 inspects the prerequisite of childrens software evaluation in the light of

dynamic nature of edutainment perspective. Work was conducted to help educa-

tional institutions gain a deeper understanding of professional decisions they face

and reduce their initial state of uncertainty about the best course of action. Com-

parative solutions are provided using two MADM techniques Rank Order Centroid

(ROC) methodology and Ratio Method.

Chapter 7 designs a novel MADM model to assist the decision making scenarios

and is specifically applicable to multi-criteria decision problems associated with

large number of attributes. We have described the implementation of a corrective

model, assisting the decision-maker in the construction of a consistent comparison

matrix. Group decision making is becoming increasingly important in decision

scenarios associated with MCDM problems. The proposed methodology provides

an insight into the impediments to effective group processes and on techniques that

can improve group decisions. We hope that the projected approach can refocus the

attention of researchers from the race of finding better judgments for inconsistent

and near consistent matrices.

31

8.2 Summarizing the Findings and Limitations

This study has shown that methodologies adapted from MCDM tactics have ver-

satile applications in almost all decision making layouts. Application feasibility as

well as efficiency of the tactics in solving diversified problems have been described

illustratively in this study. Expert opinions are sought and implemented to scout

empirical perfections. Some limitations to this pilot study need to be acknowledged.

One of the more significant findings that emerge from this study is that, AHP is

a useful decision tool to consolidate evaluation data. It provides a useful mecha-

nism for checking the consistency of the evaluation measures and alternatives. This

method is preferred over other established methodologies as it does not demand

prior knowledge of the utility function; rather it is based on a hierarchy of criteria

and attributes facilitating the understanding of the problem. It allows relative and

absolute comparisons, making this method a very robust tool. This method al-

lows combinations with other techniques as well as scenario analysis and simulation

exercises. The further strength of the AHP is its ability to detect inconsistent judg-

ments. Last but not the least, AHP allows group decision-making and is convenient

in numerical handling. Free on line software support is an added advantage.

The limitation of the AHP is that, it only works for positive reciprocal matri-

ces. The other seeming drawback is the scale used in judgment mechanism which

inculcates inconsistencies in assessments. Large number of pairwise comparisons

required for sizable comparison matrices is another snag which seeks research for

further improvement.

It was also shown that MAGIQ technique using rank order centroids for weight

determination is another useful MADM model. It is far more quicker to other

analogous techniques used for the decision mechanism. The key advantage of the

methodology is its simplicity in surveying, yet it has its own limitations when the

32

decision makers influence their prioritized specifications regarding significances.

This study also found that PROMETHEE and TOPSIS used in the present work are

simple and logical MADM tactics widely used in decision scenarios. Still they have

their own limitations in practical implications. Major of them is, the requirement

of predefined criteria weights which seeks the help of other MADM procedures.

Further complication in PROMETHEE is requirement of preference functions as

proposed in original inception. But these preference functions require the definition

of some preferential parameters, such as the preference and indifference thresholds.

However, in real time applications, it may be difficult for the decision maker to

specify which specific form of preference function is suitable for each criterion and

also to determine the parameters involved. Another apparent constraint is, limited

software support which enhances manual computations.

Another important findings regarding the two MODM techniques used in this work

indicate that Lexicographic Approach is useful only when objectives considered in

the study are few. Also it does not provide alternatives to the aspirants for in-

dividual preferences whereas Weighted Penalty Method provides a set of efficient

solutions which can be evaluated for prioritized preferences. However it was ob-

served that weights alloted for scalarization purpose in Weighted Penalty Method

in the present work may significantly effect the decision layouts.

8.3 Suggested Implications (Research Contribu-

tion)

The present study confirms previous findings and contributes additional evidences

that suggest that pairwise comparison methods can always be used to draw the

final conclusions in a comparatively accessible and intelligible way. However, as the

results suggest that AHP should be used in combination with other decision tools to

33

support because AHP is efficient only when the number of criteria and alternatives

are few.

The methodology of PROMETHEE and its algorithm has been demonstrated in

detailed and simplified way and it is believed that it will be useful for decision

makers to apply such MCDM tools to supplement their decision-making efforts.

A contemporary decision making technique has been brought up in this work in

which antecedent of inconsistencies are analyzed and a corrective application based

approach is introduced, which helps the decision-maker to build consistent matrices

with controlled error by appreciably less number of pairwise comparisons.

8.4 Recommendations for Further Work (Research)

The current findings add substantially to our understanding of incorporating MCDA

tactics in decision layouts. The empirical findings in this study will serve as a base

for future studies in various application scenarios.

34

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