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Innovation and problem solving 2 – 29-8-2014

How were the problem of railways routes and number of roses solved using properties of a

graph?

Why did we use this roundabout way to solve our problems?

Not only it aided visualization, but also we could use standard results from the graph domain

to arrive at our solutions instead of doing routine calculations.

Domain mapping

Through the powerful domain mapping concept, problem solving resources

found effective in one domain can be abstracted into the higher layer of Problem Solving

discipline and then mapped to analogous problems in many other domains.

Before encountering the handshakes problem, the abstracted problem solving

capabilities of Graph theoretic domain already existed in the Problem solving

layer of discipline. While modelling the handshakes problem, similarity to the graph

was identified and the problem model was mapped onto the graph theoretic domain.

Analysis and solution was then obtained in this domain and then through the Problem

Solving layer, the result was passed back to the Handshakes problem domain. This is

domain mapping.

Instead of Handshakes, the problem domain could very well have

involved relationships between a group of people, road connectivity

between cities or a telecom network.

Problem 2.1

A group of students stood in rows with equal number of students in each

row. If number of students in each row were increased by one, the number

of rows decreased by 2 whereas if the number of students in each row were

decreased by 1, the number of rows increased by 3. What was the total

number of students?

Problem Solving

Graph theory Handshakes

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If you use visualization through domain mapping and use deductive reasoning, the

problem solution may be reached much quicker. When we mention domain mapping,

the “mapped to” domain may just be another branch of mathematics itself.

Problem 2.2

Which one between 854×857 and 855×856 is larger and why? Any other

method?

Use of the most basic concepts in a domain sometimes produces the

most elegant solution. Here also domain mapping helps you to visualize

how the problem elements behave.

Problem 2.3

In a transmission network model, the transmission devices were

represented along with their route connections (device id, port id all given).

Question: what is the shortest route between two given points in the

network? How can you find it?

It is a real hard problem unless we map the transmission network as a graph

and use standard graph theoretic results.

Every subject domain has its own problem solving resources

Each of the individual subject domains, Maths to Physiology to Marketing

Management, has their own problem solving mechanisms that are specific to the

domains and practitioners of such domains are not usually interested to use problem

solving resources of other domains. This has resulted in problem solving techniques,

methods and concepts to remain fragmented and locked into individual parent

domain islands.

The only exception in this case is Mathematics which contributed most to problem

solving of other subject domains. In this sense Mathematics is somewhat similar to

Problem Solving as a standalone overlay subject.

Problem solving is an overlay standalone subject drawing

resources from suitable sources and serving all domains

“Problem Solving is a discipline that includes a set of approaches and ways to

apply suitable rules, techniques, methods and methodologies for solving ANY problem

efficiently and systematically and NOT in a random manner.”

“A Problem Solver practices the art and science of problem solving, learns on

every opportunity and guides others to solve problems and to learn problem solving.”

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“Problem solving is domain independent and is applicable to any

domain.”

“Problem solving is essentially multidisciplinary in terms of the source of

PSA resources.”

“Problem solving is thus an overlay subject to some extent akin to

Mathematics from its multi-domain applicability aspect, but differs from Mathematics

in its holistic real world applicability whereas much of Mathematics may not have any

identifiable real world attachment.

Presently Problem Solving concepts, resources and practices are

fragmented and spread all over the human knowledge domain. Combining these into

a single framework and subject has the great advantage of furtherance of its scope,

applicability, learning and use.

“By its nature Problem Solving is an open-ended extensible discipline

that prescribes the use of an ever expanding set of problem solving armoury

(PSA) resources to solve any problem efficiently and with satisfaction to the

concerned entities.”

Problem Solving Armoury Resources

This is a collection of abstract principles, contributing sources, problem solving

techniques, tools, methodologies and approaches that are used suitably by the

Decision Analyst or the Problem Solver to recommend solution to any problem. It is

apparent that no single person can solve all the types of problems that may arise

anywhere any time. To get over this bottleneck a powerful set of techniques exist:

1. Ask the expert

2. Find the trouble-shooter

3. Find the key person

4. Find the mentor

There are domain proprietary problem solving mechanisms or resources

that are commonly used in the same domain.

While practising the discipline of general Problem Solving, the problem solver always

attempts to identify the powerful domain proprietary problem solving mechanisms

that can be abstracted and absorbed as a part of the generic higher layer of problem

solving discipline for later application to problems in any other domain.

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Domain modelling

A domain model represents a structure of inter-related concepts in a domain of

knowledge, such as Physics or Telecommunication. While physics and telecom both

are formal subjects, a domain can very well be defined for say, handshakes or

matchsticks or for that matter railway routes in a country. Such modelling becomes

necessary to study and solve problems in such domains, as structuring and relating the

concepts in a domain makes analysis of any problem possible and convenient. This

activity is called domain modelling and is a great way to approach very complex

problems in a domain.

Additionally, when a domain is well-structured into a clean model, memory load is

reduced and learning and knowledge transmission are greatly enhanced.

As an example, a very brief telcom domain model can be informally described as:

1. At the outermost boundary of telecom domain are the people and devices that

communicate with each other all across the globe or in a city. These primary

level entities generate the telecom traffic.

2. For communication between two primary level telecom entities to happen,

each such entity must be connected to a telecom network physically near to it.

3. These small local telecom networks in turn are connected and integrated into

larger national networks which again are connected together to form a

worldwide network so that any device or entity connected to a local network

can communicate with any other device or entity anywhere on the globe. The

local networks and other higher layer networks all have resource and revenue

sharing mechanism for such a communication to be feasible from business

point of view.

4. Technologically, a communicating entity is connected to a single active telecom

node in its local network which enables the entity to communicate. But an

active node must be able to pass the traffic generated by the communicating

entity to another active node in a telecom network that is nearer to the

destination communicating entity. Through transmission links an active node

in a telecom network is connected to another traffic generating or absorbing

active node. These transmission links together with the connected traffic

generating and absorbing active nodes form the overall telecom network.

This is, starting at the top concept level. Now level by level the concepts of how a

communicating entity actually communicates with an active node or how an active node

handles the traffic from all such entities and routes a particular communication towards

a particular destination or how a transmission link works can all be expanded and linked

in a well-formed structure so that one can understand concepts from its most abstract

to its most detailed form gradually and easily and according to need. (Use of graph and

network diagram eases visualization).

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Formation of such a model eases analysis and solution of any problem in such a domain.

Technological and basic science domains excluding life sciences are inherently

structured and are generally amenable to domain modelling.

Though Life sciences are inherently unstructured, it is possible to model part of such a

domain for solving a specific problem.

For the problem solver, domain modelling is an invaluable approach and ability.

Though domain mapping is unusual and rare, it can be extensively used to a great

advantage.

Again, even though domain modelling is not usually done, this approach can greatly

helps to form enriched Problem models.

Problem modelling is a subset of domain modelling.

MCDM and AHP

MCDM is the short form of multi-criteria decision making and AHP is the leading

MCDM method standing for Analytic Hierarchic Process. MCDM forms a large class of

problems which includes amongst others:

• Choice

– R&D Project Selection

– Selection of Project leader

– Choosing the Lunar Lander Propulsion System

– R&D decisions on portfolio management

– Design concept selection

– Project management, specially Contractor selection

– Customer requirement structuring

– Defense procurement

• Prioritization

– Prioritization of R&D projects

– Environmental Impact Evaluations

– Project Risk Assessment and evaluation

– Prioritization of barriers and barrier removal impact assessment

• Resource Allocation

– Budget Allocation

– Tactical R&D Project Evaluation and Funding

• Benchmarking

• Quality evaluation

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• Policy formulation

• Strategic Planning

Quite frequently in our regular lives, personal or work, we face problems in which we

have to select one thing out of a few based only on our judgment and practical

common sense. As you perhaps know, both of these qualities are very subjective and

unreliable in general.

In such cases we do not have any means to weigh, or measure its length or such

characteristics to decide which one is our choice. What will you do when you have to

choose between three colleges for after school studies? Or your mobile phone

drops and its display breaks so badly that its repair cost is estimated to be comparable

to the cost of a new mobile. You then face your father with an approximate budget for

a new phone. After he gives you the money the job of choosing a suitable phone still

remains to be done.

All of us go about solving such problems in our own way which we call here instinctive

random ways. For small decisions or personal life problems that is ok, but not for

costly corporate or national level decisions where much is at stake. To deal with

such problems more systematically and scientifically, a new class of advanced

evaluation methods came up amongst which a few leading ones are,

1. AHP

2. TOPSIS, and

3. ELECTRE

Over the years AHP became the most popular and heavily used because of its ease of

use and some amount of mathematical underpinnings. Most people like mathematics,

or support of mathematics. AHP was created during the 1970s by Prof. Thomas L Saaty,

originally a mathematician but later a management scientist.

Problem 2.4. How do you place the three lines drawn on the board

regarding their relative length? How can you say one line is larger or smaller

than another line by how many times? How would you go about it?

In this case, you may measure the length of the lines and form their ratios

rank them according to their length. Length is the comparison criterion

here. But you can do this without much hassle if the lines are all straight

lines.

What if the lines drawn are not straight but are all highly convoluted curved

lines? Even then you can do it carefully placing pieces of thread on the lines

and later measure the lengths of the threads.

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Without going through this measurement activity, you may get the length-

wise ranks of the three lines by using your estimating capability and the

MCDM method of AHP.

How many criteria do we have here? This is a single criterion problem for

ranking three objects. The single criterion is the length of the objects.

Problem 2.5. Rank the distances from your classroom here to the Victoria

Memorial, Howrah Station and Dakshinswar temple. Assume you are in

Kolkata.

Here measurement by meter scale is not possible. Estimation and AHP may

be easier.

Task: You can test and increase your length estimation capability,

which is one of the important abilities of a problem solver, by repeatedly

doing this exercise and checking the results from say, Google maps.

Problem 2.6. Rank three round objects of different sizes according to their

size. How would you approach the problem?

Here also with the more involved process of liquid displacement you may

actually measure the volumes of the objects and rank them. If you do not

want to go through this hassle you may again use your size estimation

capability and use the method of AHP.

How many criteria do we have here? This again is a single criterion problem

of ranking three objects. The single criterion is the size of the objects.

Problem 2.7. You have moved to a large new house where your parents

have given you the option to choose one from three rooms as your room.

How would you choose your room? What would be the criteria on which

you would evaluate the rooms, rank them and finally choose the top ranked

alternative?

You might like to see whether the room has an attached bath, how big is the

room, whether it gets good sunlight during the daytime, or how

conveniently you can place your bed, table and other things in the room.

These may be the criteria for choosing the room or you may like to have

your own different set of criteria. Whatever you do, most of these criteria

are not measurable quantities but have to be estimated by judgment.

This now is a truly multi-criteria decision making problem.

Task: Identify different credible alternative sets of criteria for

evaluation of the rooms.

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Problem 2.8. Your mobile phone got broken when accidentally dropped

and now you have to choose a new phone. How would you tell the budget

for the new phone purchase to your father, and later when he releases the

money (might be less than what you wanted), how will you go about

choosing first, from where to purchase the mobile and which mobile to

purchase?

This is a four step process (budget, negotiate for funds, choose shop, choose

model to purchase) in which a number of steps involve multi-criteria

decision making.

Task: Expand the job into choices, actions, consequences and criteria.

Rule: In any multi-criteria decision problem, the first step is to form

the problem definition, identify the choices and then to select the

criteria to be used for evaluation of the choices.

Recommendation: If you are in doubt about what criteria to select,

adopt the safe method of choosing a larger set of criteria. While going

through the method, the unimportant criteria would automatically be

weeded out.

Example: Evaluating the worth of a would-be groom.

Basic processes in AHP: Problem 2.6

Step 1: Selection of criteria: this may be the most crucial step in the whole

evaluation process and may be quite complex as a criterion may have its sub-

criteria also. At the end of this step we get the whole hierarchic tree starting at

the top with the 1st level criteria and ending at the bottom with the last level sub-

criteria.

Step 2: Assigning relative weights to the criteria at each level: At this step

AHP introduces the concept that, humanly it is more natural to express in

descriptive terms the relative strength of one criterion with respect to another. At

the core of AHP lies the action of pair-wise comparison of criteria rather than

comparison of all criteria together.

So this judgmental assignment of relative weights is split into two parts:

Pair-wise comparison of all possible pairs of the set of criteria being

evaluated. At the first step, only the 1st level criteria are evaluated.

For each pair-wise comparison, one criterion is expressed as how many

times stronger it is in comparison to the other. This selection of

descriptive term is standardized from a pre-defined set and is

immediately transformed into an equivalent number corresponding to

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the comparison term selected. The table below states the descriptive

comparative terms between two criteria and their equivalent numerical

values.

Intensity of Importance Definition Explanation

1 Equal Importance Two activities contribute equally to the objective 2 Weak or slight 3 Moderate importance Experience and judgement slightly favour one

activity over another 4 Moderate plus 5 Strong importance Experience and judgement strongly favour one

activity over another 6 Strong plus 7 Very strong importance An activity is favoured very strongly over

another; its dominance demonstrated in practice 8 Very, very strong 9 Extreme importance The evidence favouring one activity over another

is of the highest possible order of affirmation If activity i has one of the Reciprocals of above A reasonable assumption non-zero numbers assigned to it when compared with activity j, then j has the reciprocal value when compared with i

1.1–1.9 If the activities are very close May be difficult to assign the best value but when compared with other contrasting activities the size of the small numbers would not be too noticeable, yet they can still indicate the relative importance of the activities.

We have three round objects here, the smallest glass marble, the lemon and the

largest plastic ball. To do the pair-wise comparison of sizes of these three objects

according to the two steps above, a comparison matrix is used. In our case the

following is a representative comparison matrix.

Comparison matrix for Marble, Lemon and the Ball with respect to criterion of

Size:

Ball Lemon Marble

Ball 1/1 4/1 8/1

Lemon 1/4 1/1 3/1

Marble 1/8 1/3 1/1

The cells on the diagonal of the matrix would always be fraction 1/1 as each represents

a comparison of a choice with itself. Furthermore, if the cell on the second column of

the first row has the value 4/1, the corresponding reverse comparison cell on the first

column of second row will have the reverse value, that is ¼. This will be true for all

cells. This means, you have to fill up one half of the comparison matrix by using your

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judgments. The other half automatically follows as the inverse. Lastly, the value 4/1

means the ball is perceived to be “Moderate plus” times larger than the Lemon which

translates to ¼ when the reverse comparison is made.

Each of the cell values should be a proper fraction at this stage.

Step 2.1: Thus the first sub-step of Step 2 is to form the desired Comparison matrix.

In our case, as the criterion is only one, we do not need to evaluate the relative

weights of the set of criteria. Instead, we directly evaluate the relative weights of the

alternatives or choices with respect to our single criterion. The concept is same only

the application of the concept is on the evaluation of the choices rather than

evaluation of the criteria and then evaluation of the choices.

In a real-life problem we may need to form many such comparison matrices and

evaluate each of those.

Step 2.2: At this step it is routine mathematical calculation to find the Eigenvector of

the Comparison matrix. To find this new thing,

sum up each row

sum up the values of these row sums

form the ratio of each row sum to the sum of row sums.

For each row we would get one weighted value. This is the eigenvector of the matrix.

Eigenvector of Comparison matrix:

Ball Lemon Marble Row sums Eigen vector

Ball 1.00 4.00 8.00 13.00 0.69

Lemon 0.25 1.00 3.00 4.25 0.23

Marble 0.13 0.33 1.00 1.46 0.08

1.38 5.33 12.00 18.71

As per Prof. Saaty, the Principal Eigenvector of the Comparison Matrix represents

the relative weights of the compared entities (in our case, the round objects) with

respect to the comparison criterion or objective (in our case, size). To find the

Principal Eigenvector, you need to form the square of the matrix, calculate its

eigenvector and compare it with the eigenvector that you got in the previous step.

You have to continue this process till the value of eigenvector does not change. That

will be your principal eigenvector.

First square of Comparison matrix and its eigenvector

Ball Lemon Marble Row sums Eigen vector

Ball 3.00 10.67 28.00 41.67 0.72

Lemon 0.88 3.00 8.00 11.88 0.20

Marble 0.33 1.17 3.00 4.50 0.08

4.21 14.83 39.00 58.04

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We find that the eigenvector has changed its values. So we have to carry out the

process of squaring this squared matrix and finding its eigenvector again to see

whether the new eigenvector has changed again.

Second square of Comparison matrix and its eigenvector

Ball Lemon Marble Row sums Eigen vector

Ball 27.67 96.67 253.33 377.67 0.72

Lemon 7.92 27.67 72.50 108.08 0.21

Marble 3.02 10.56 27.67 41.24 0.08

38.60 134.89 353.50 526.99

As there is still some change, we will carry out the squaring again third time.

Third square of Comparison matrix and its eigenvector

Ball Lemon Marble Row sums Eigen vector

Ball 2296.00 8022.96 21026.11 31345.07 0.72

Lemon 657.07 2296.00 6017.22 8970.29 0.21

Marble 250.72 876.09 2296.00 3422.81 0.08

3203.78 11195.05 29339.33 43738.17

At last in this third stage we find the value of the eigenvector stabilizing and thus these

are the relative sizes of the three round objects.

Though it seems that a lot of calculation is involved in this method, the calculation can

easily be made with the help of spreadsheet functions. The difficulty of this method is

not in calculation but in carrying out proper analysis to form the criteria hierarchy and

then forming the judgment matrices. Subsequent calculations are straightforward and

quite mechanical.

The summation of the final weights must always be 1. Notice that here the sum of the

weights is not 1. This has arisen because of rounding to two digits. The three sets of

values of the last three stages with three digits after the decimal places are,

Eigen vector

0.718

0.205

0.078

Eigen vector

0.717

0.205

0.078

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For all practical purposes even the eigenvector value at the first matrix squared phase

also can be accepted. To know how many times one value is with respect to another,

the ratio of corresponding weights is to be calculated. For example, the ball to the

marble size ratio is: 0.717/0.078 = 9.19. Originally we judged the paired ratio as 8.00.

When relationships between all the objects are considered through the method, the

proper ratio is returned.

This is the simplest example of applying AHP. It has many rich features for dealing

with large complex problems.

Eigen vector

0.717

0.205

0.078