The Art of Decision Making in Business - Philippe De · PDF fileThe Art of Decision Making in...

The Art of Decision Making in BusinessMulti Criteria Decision Analysis

Prof. Dr. Philippe J.S. De Brouwer

2016–2017,Warsaw, Poland

last compiled: January 29, 2017

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c©Prof. Dr. Philippe J.S. De BrouwerCentrum For Management Training

University of Warsaw2017

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Contents

1 Introduction to Problem Solving 51.1 What and Why . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2 General Workflow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3 Structured Planning Methods . . . . . . . . . . . . . . . . . . . . . . . . . 9

2 Mono Criteria Decision Methods 13

3 Multi Criteria Decision Methods (MCDM) 173.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.1.1 Decision Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.2 Non Compensatory Methods . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.2.1 Dominance and Efficient Solutions . . . . . . . . . . . . . . . . . . 213.2.2 MaxMin Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.2.3 MaxMax Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.3 Naive Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.3.1 Weighted Sum Method—WSM . . . . . . . . . . . . . . . . . . . . 253.3.2 Weighted Product Method—WPM . . . . . . . . . . . . . . . . . . 273.3.3 Goal Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.4 Preference Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.4.1 ELECTRE I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.4.2 ELECTRE II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.4.3 ELECTRE I and II: Conclusions . . . . . . . . . . . . . . . . . . . . 37

3.5 Outranking Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.5.1 PROMETHEE I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.5.2 PROMETHEE II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.5.3 Other Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.6 Summary MCDA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

A Further Reading 51

B Conventions for Matrix Algebra 53

CONTENTS

C Levels of Measurement 55C.1 Nominal Scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55C.2 Ordinal Scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56C.3 Interval Scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56C.4 Ratio Scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

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Chapter 1Introduction to Problem Solving

1.1 What and Why

What is a Manager?Why do you follow this course?

QuestionWhat defines a good manager?

Where is the problemThe levels of decisions in companies

Monsen and Downs (1965) propose different levels of decision making, with eachtheir own agenda. Roughly translated to today’s big corporates, this boils down to:

1. Super-strategic: mission statement (typically the founders, supervisory boardand/or owners)

2. Managerial Control / strategic: typical the executive management (executivecommittee)

3. Operational Control / tactical: typical middle management

Note: Besides those definitions is it commonplace to re-use the concepts at different levels. Sobe careful how the words are used.

Note: Note that these levels roughly correspond to different time frames of change: the super-strategic level should change when the company and/or market changes dramatically, thestrategic level every few years, the tactical can range from month to year.

Decision Issues at Different Levels

1. Strategic planning

CHAPTER 1. INTRODUCTION TO PROBLEM SOLVING

• New business opportunities

• Competition strategies

• Technology adoption

• Strategic partnership

2. Managerial control

• Financial control

• Project control

• Quality control

• Risk control

• HR control

3 Operational control

• Task scheduling

• Production optimization

• Coordination

• Skill development

1.2 General Workflow

To fix the ideas, we will now assume an example of decision problem for the nextsessions.

Example 1

• We’re a service organization (eg. a bank or IT company),

• in order to reduce our costs we want to create a second workplace (next to theexisting headquarters in Warsaw),

• it is going to be a sort of back-office where then non-customer focussing, businesswill be based,

• so there is no dependance on transporting the goods or raw materials.

Step 1:Explore the Big Picture

Hints:Use

• SWOT analysis

• 7Ps of Marketing

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1.2. GENERAL WORKFLOW

• Business Model Canvas

• NPV, IRR, cost benefit analysis, . . .

• Break-Even, Time to Profit, Largest Cumulated Negative, etc.

• two-parameter criteria (eg. income/cost)

• make sure that the problem is within one level of decision (strategic / managerial/ operational) — see p. 5

STEP 2: Identify the Problem

Hints:

• brainstorming techniques to

– get all alternatives

– get all criteria

– understand interdependencies

– . . .

• make sure you have a clear picture on what the problem is, what the critiera andwhat the possible alternatives are

• note: this step is best within one level of decision (strategic / managerial / oper-ational)

STEP 3: Identify Possible Solutions

Hints:

1. this implies that these objectives are rather “tactical”

2. what are possible solution to our problem?

3. brainstorm and brainstorm again to get a long list of possibilities

Note: We might decide to tackle the problem as a waterfal of possibilities: in stead of exploreall 10’000 possible sites, we first select a contitnent, then a country, then a city and then alocation.

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STEP 4: Define CriteriaHow will we measure how good solutions of step 2 fit the tactical objectives of step 2 and thestrategic objective of step 1?

Hints:

1. can everything be translated to “money” (present value?)

2. think KPI, ie. try to find measurable things from which you hope that if all satis-fied the tactical and strategic objectives will benefit,

3. this is also no exact science but it should be as “complete” as possible: ask your-self “if all this is satisfied, should we be on the right track?”

4. every measure should be measurable! (even qualitative!)

5. you will need at least an ordinal scale (see slide 107)

STEP 5: Get the DataMake sure you can calculate the criteria for all solutions

Hints:

1. define how to measure all solutions for all criteria

2. this is why you need a ordinal scale (see previous step)

3. get all data so that you can calculate all critiera for all solutions

4. put these number is a “decision matrix” – see p. 34

Leave out all solutions that do not satisfy our minimal criteria.For example we can leave out all possible alternatives (for the “new plant exam-

ple”) that are not in Poland, that are not close to a good university, that have no sitefor more than 1000 employees available, etc.

STEP 7: Compare the AlternativesUse a Multi-Criteria Decision Method

1. start with the long list (step 3)

2. leave out the solutions that do not satisfy our minimal criteria (step 5)

3. use a non-criteria decision method –if possible– (see p. 20)

4. leave out the non-optimal solutions (the “dominated ones”) – see p. 37

5. use a multi-criteria decision method – see from p. 28

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1.3. STRUCTURED PLANNING METHODS

STEP 8: Recommend a SolutionConnect to the Business

• connect back to step 1 and 2 (the big picture and the problem definition)

• provide the rationale

• provide confidence

• conclude

• make an initial plan (assuming Agile approach)

Summary of Work-flow

1. Understand the big picture – see p. 7 (use a structured planning method – fromp. 16)

2. Identify the problem — see p. 8

3. Identify possible solutions — see p.9

4. Identify useful and independent criteria — see p.10

5. Get the data to calculate all criteria for all solutions — see p.11 — and presentthem in a decision matrix (see p. 34)

6. Filter with minimal criteria — see p.12

7. Filter with dominance — see p.37

8. Use a decision method (mono-criterion if possible (p. 20), multi-criteria if nec-essary (p. 28))

9. Make it happen, recommend a solution (write a report) and plan for success

1.3 Structured Planning Methods

SWOT Analysis

• Strengths: characteristics of the business or project that give it an advantage overothers.

• Weaknesses: characteristics that place the business or project at a disadvantagerelative to others.

• Opportunities: elements that the business or project could exploit to its advan-tage.

• Threats: elements in the environment that could cause trouble for the businessor project.

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While the words “strengths” and “opportunities” are very close to each other (justas “weakness” and “threat”) the SWOT method assumes that the first two are internalto the team/company and the last two are external. So while a talented and motivateworkforce could be seen as an opportunity we will put it in the box of “strengths”and the fact that the competition is unable to attract talent should be considered as an“opportunity”.

SWOT Analysis

Helpful HarmfulInternal origin Strengths

• . . .

• . . .

• . . .

Weaknesses

• . . .

• . . .

• . . .

External origin Opportunities

• . . .

• . . .

• . . .

Threats

• . . .

• . . .

• . . .

The diagram above summarizes the work to be done and can be useful in practice.

the 7Ps of Marketing

1. Product: The Product should fit its purpose, should work and it should be whatthe consumers are expecting to get.

2. Place: The product should be available from where your target consumer findsit easiest to shop. This may be High Street, e-commerce or (online).

3. Price: The Product should always be perceived as good value for money.

4. Promotion: Advertising, PR, Sales Promotion, Personal Selling and Social Mediaare all key communication tools.

5. People: All companies rely on the people, having the right people is essential be-cause they are as much a part of your business offering as the products/servicesyou are offering.

6. Processes: How do you deliver the service?

7. Physical Evidence: What the customer will see or get before, during and afterthe transaction has to be a “good customer experience”

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1.3. STRUCTURED PLANNING METHODS

Business Model Canvas

Figure 1.1: The Business Model Canvas template.

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Chapter 2Mono Criteria Decision Methods

In the Simplest of Cases

. . . there is one function to optimize

Optimization

Given: a function f : A → < from some set A to the real numbers < Sought: anelement x0 in A such that f(x0) ≤ f(x) ∀x ∈ A (“minimization”) or such that f(x0) ≤f(x) ∀x ∈ A (“maximization”)

Examples:

• utility optimization,

• profit optimization for price elasticity,

• expenditure optimization,

• equilibrium models,

• . . .

Canonical form of the Simplex

The Most Simple, Linear variant (I)

CHAPTER 2. MONO CRITERIA DECISION METHODS

Definition 1 .:. Simplex Linear Optimization .:.

max {c′.x|A.x ≤ b,x ≥ 0}

• A is the m× n matrix representing the conditions

• c is the n× 1 vector representing the function to be optimized

• x is the n×1 (or otherwise stated x ∈ <n) vector representing the variablesunder control, solving the problem is finding the optimal value x0

• n the number of scalar variables to be optimized

• m the number of conditions, with m ≤ n

• the transposition operation is represented by the ′ operator

Canonical form of the SimplexThe Most Simple, Linear variant (II)

If n > m, then there would not be an MCDA problem that should find optimalsolutions. There would be no solution or just one.

Note: Of course, we assume here that all the conditions in Ax ≤ b are linearly independent–as part of the canonical form definition– (if not, they have to be cleaned up first!)

The General Mono Criteria ProblemGenerally non-linear

Definition 2 .:. General Mono Criteria Problem .:.

max {f (x) |x ∈ A}

Where f is a function <n → < and A the set of acceptable solutions.

Some Definitions

Definition 3 .:. Acceptable Solution .:.

A solution x is called acceptable if and only if x ∈ A

Definition 4 .:. Optimal Solution .:.

A solution x is called optimal if and only if ∀x ∈ A : f(x) ≥ f(x)

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Examples Iof Mono Criteria Optimization

• maximize present value of the project, as max{PV (x) =

∑Tt=1

cft(x)

(1+rt)t

∣∣∣x ∈ A}• utility as max {U(x)|x ∈ A}

• when buying a car, assume U = a1 log(maxSpeed) + a2fuelEconomy2 + a3price+

a4eprestigeand A = the set of all cars with 5 doors and 4 seats. . . and assume that

you can quantify all parameters in a non-stochastic way

Examples IICost Benefits analysis

for all solutions i, calculate the “total benefit” =

TBi = benefitsi − costsi

NPVi =N∑t=0

cfi,t(1 + r)t

Then select the solution wit the highest NPV

QuestionWhy is that a naive solution in the real world?

Methods for OptimizationFor optimizing one function (eg. income, cost, time to delivery, etc.) one can rely

on mathematics to find the optimal solution.Example of techniques

• Optimization Methods

– Lagrange multipliers

– Simplex Algorithm for Linear Programming

– Network Optimization, etc.

• Iterative Methods

– Newton’s Method, quasi-Newton Methods,

– Gradient Descend,

– Goal Seek algorithms

– Interpolation methods,

– Pattern Search,

– Simultaneous perturbation stochastic approach, etc.

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CHAPTER 2. MONO CRITERIA DECISION METHODS

• Heuristics

– Memetic Algorithm,

– Differential Evolution,

– Evolutionary Algorithm, Genetic algorithms,

– Hill-Climbing with random restart,

– Reactive Search Optimization, etc.

Of course one can also rely on combinations of methods, especially mix heuristics anditerative methods, etc.

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Chapter 3Multi Criteria Decision Methods(MCDM)

3.1 Introduction

Real LifeIs a lot more complex

QuestionWhat is the function to optimize when buying a car/ship/plane?

ExampleFind the optimal location for the call centre

QuestionWhere would we build our back office?

General Formulation of a MCDMFirst some definitions

Definition 5 .:. Multi Criteria Decision Problem .:.

max {fj(x)|x ∈ A, j = 1, . . . , n}

Definition 6 .:. criteria .:.

the fj are called the criteria

Definition 7 .:. attributes .:.

if the fi are not directly measurable, we use attributes to measure them. Typicallythere is no closed algebraic form to compose one criterion from its attributes.

CHAPTER 3. MULTI CRITERIA DECISION METHODS (MCDM)

Definition 8 .:. score .:.

sij := fj(xij)

Typical Features of MCDM

• the multiple attributes often form a hierarchy

• criteria typically conflict (eg. price and quality)

• hybrid nature

– Incommensurable Units

– Mixture of different scales of measurement (eg. mixture of qualitative andquantitative attributes)

– Mixture of deterministic and probabilistic attributes.

• lack of clarity and uncertainty

– uncertainty in subjective judgments

– lack of data and incomplete information

– lack of clarity on criteria and attributes to be used.

• large scale and complexity

• ranking might not be conclusive (conflicting methods, sensitivity to alterna-tives, etc.)

Definition 9 .:. Ideal Solution .:.

An Ideal Solution is a solution that is the optimal for all criteria (note: there canbe more than one!)

Definition 10 .:. Dominated Solution .:.

A solution is dominated by another one if for all criteria it is worse or as good asthe other and at least worse on one criterion

Definition 11 .:. Efficient Solution .:.

A solution is “non-dominated” or “efficient” if for all other criteria it is at leastas good as the

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3.1. INTRODUCTION

Definition 12 .:. Acceptable Solution .:.

A solution x is called acceptable if and only if x ∈ A (Note: definition the sameas in Slide 24, aka “feasible solution”)

Definition 13 .:. Preferred Solution .:.

An Efficient Solution, selected by one or more MCD-Methods.

If the set of actions A is finite

Then we call it MCDA

The field of Multi Criteria Decision Analysis (henceforth MCDA) is the scienceof decision making in MCDM where the set of possible solutions is finite.See eg.D’Avignon and Winkels (1986) Note that typically for

• design choices, A is infinite (eg. the thickness of the hull is between 1cm and5cm and hence has an uncountable infinity of possibilities), and

• selection choices, A is finite (the offer of the producers, who made the choiceabove)

3.1.1 Decision Matrix

Decision Matrix

Step 1 Reducing Attributes into Criteria

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PreferredChoice

Start-up Costs

cost of ground

building cost

rent possible?

Running Costs

cost of labour

distance to HQ

tax incentives

Convenience

distance to airport

public transport

quality roads

Nice To Have

distance to airport

public transport

quality roads

Other . . .

Decision MatrixStep 2

Alternative Start Up Running Convenience NiceWarsaw 1 1.25Me 1.00 good positiveWarsaw 2 0.99Me 1.00 good positiveCracow 1 1.50Me 0.75 excellent negativeCracow 2 1.00Me 0.80 not good neutralGdynia 1.25Me 0.95 excellent positiveŁodz 1.20Me 0.8 good neutral

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3.2. NON COMPENSATORY METHODS

Table 3.1: The Decision Matrix is the first strong simplification of most MCDM.

Decision Matrix Rescaled

Step 2 - bis

While the decision matrix as presented above is perfectly fine, it is dangerous touse it like this because it can lead to confusion since some criteria are to be maximizedand others to be minimized. Also it is not very clear as all columns have differentdimensions. So, for calculation purposes we will redefine the decision matrix so thatall criteria take values between 0 and 1 and all criteria are to be maximised. We callthis the rescaled decision matrix.

The rescaling can be any mapping that preserves the order in accordance with thelevel of measurement for that variable — see Appendix C

Alternative Start Up Running Convenience NiceWarsaw 1 0.49 0.00 0.5 1.0Warsaw 2 1.00 0.00 0.5 1.0Cracow 1 0.00 1.00 1.0 0.0Cracow 2 0.98 0.80 0.0 0.5Gdynia 0.49 0.80 1.0 1.0Łodz 0.59 0.80 0.5 0.5

Table 3.2: The rescaled decision matrix only contains variables between 0 and 1 where allcriteria have to be maximised

The financial information was scaled in a linear way so that the lowest cost ismapped to 1 and the highest cost mapped to 0. The two other variables are ordi-nal scales and hence it is a little more arbitrary how to do the mapping. In any casethe worst value gets 0 and the highest 1.

3.2 Non Compensatory Methods

3.2.1 Dominance and Efficient Solutions

Dominance

Remember Markovitz’ Modern Portfolio Theory — Markowitz (1952)

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-risk(σ)

6return(R)

}pA

BC

D

Figure 3.1: Two-parameter dominance for investments. Compared to the portfolio p markedby the dot, the portfolios are better in quadrant D (because they have lower risk and higherreturn), worse in quadrant B (with lower return and higher risk), while those in A and Ccannot be ranked relative to p.

DominanceEfficient Frontier a la Markovitz

0

0.02

0.04

0.06

0.08

0.1

0.12

0 0.05 0.1 0.15 0.2

return

volatitilty

Acceptable PortfoliosEfficient Frontier

Figure 3.2: Merton’s analytical results (Merton (1972)) are relevant when there are no limitson short selling (black line). This figure shows the impact on the investor of such a ban on shortselling (blue dots).

DominanceDefinition in a MCDA context

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3.2. NON COMPENSATORY METHODS

Definition 14 .:. Dominance .:.

A solution is dominated by another solution if it is worse or equal for all criteriaand worse for at least one criterion

We write aDb if the solution a dominates the alternative b.Dominance is a first important NON-COMPENSATIVE method.

DominanceNot necessary for Dominance, but helpful to make the problem clear is to trans-

form all criteria so that they are numeric and have to be maximized. Our example isequivalent with the following decision matrix for a maximization problem: Moriginal =

StartUp RunCost Convenient NiceWarsaw1 -1.25 -1.00 0.50 1.00Warsaw2 -0.99 -1.00 0.50 1.00Cracow1 -1.50 -0.75 1.00 -1.00Cracow2 -1.00 -0.80 0.00 0.00Gdynia -1.25 -0.95 1.00 1.00

Lodz -1.20 -0.80 0.50 0.00

Dominance. . . dominate . . .

Warsaw1

Warsaw2

Cracow1

Cracow2

Gdynia

Lodz

Figure 3.3: The relation “dominated” makes clear that Warsaw 1 is dominated by both War-saw 2 and Gdynia.

In our example “Warsaw 1” is dominated by “Warsaw 2” (and also by “Gdynia”)

Question

Should we leave it out?

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Definition 15 .:. Efficient Solutions .:.

An Efficient Solution is a solution that is not dominated by any other solution

3.2.2 MaxMin Method

MaxMin MethodThe MaxMin Method

MaxMin Method

1. find the weakest attribute for all solutions

2. select the solution that has the highest weak attribute

This method makes sense if

• the attribute values are expressed in the same units, and

• when the “a chain is as weak as the weakest link reasoning” makes sense.

3.2.3 MaxMax Method

MaxMax MethodThe MaxMin Method

MaxMax Method

1. find the strongest attribute for all solutions

2. select the solution that has the strongest strong attribute

This method makes sense if

• the attribute values are expressed in the same units, and

• when one knows that the best of the best in one attribute is most important

Example 2

A team with all specialists in one thing

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3.3. NAIVE METHODS

Non-Compensatory Methods vs. Compensatory Methods

• Non-Compensatory Methods

– What we have seen till now:

– they do not allow weaknesses on one attribute to be compensated bystrongs on other attributes, but . . .

– typically they don’t lead to a unique solution

– typically they even are insufficient to find a small enough set of the bestsolutions

• Compensatory Methods

– They allow full or partial compensation of weaknesses

– the rest of this course . . .

3.3 Naive Methods

3.3.1 Weighted Sum Method—WSM

Weighted Sum Method—WSMThe worst . . . but most used method

The MCDA is replaced by finding the maximum for

maxx∈A{N(x)}

with N(.) the function <n 7→ <n so that

N(xi) =n∑j=1

wj fj(xij) or

N(x) = F.w

where F is the decision matrix where each element is transformed according to a cer-tain function.

Weighted Sum Method—WSMAn Example

Assume the Decision Matrix of Slide 3.2 and assume the following scaling of hematrix:

StartUp RunCost Convenient NiceWarsaw2 -0.99 -1.00 0.50 1.00Cracow1 -1.50 -0.75 1.00 -1.00Cracow2 -1.00 -0.80 0.00 0.00Gdynia -1.25 -0.95 1.00 1.00

Lodz -1.20 -0.80 0.50 0.00

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Weighted Sum Method—WSMExample: the weights

Note that in order to get there, we used f =

- x/1000000 e

- xstepfunctionstepfunction

Further assume that after discussion with the management we decide on the fol-

lowing weights: w =

4.08.02.01.5

Weighted Sum Method—WSMExample: Results

The result-vector is thenscore

Warsaw2 -9.46Cracow1 -11.50Cracow2 -10.40Gdynia -9.10

Lodz -10.20

Which indicates that Gdynia would be the best solution (followed by the Warsaw 2site)

Weighted Sum Method—WSMSensitivity Analysis

• A sensitivity analysis where the weights are changed by +1 and −1 shows thatGdynia remains the best solution.

• Changing the weights to (7, 4, 3, 2)′ however elects “Warsaw 2”.

• This method would propose the following ranking of the alternatives:

Gdynia //Warsaw2 // Lodz // Cracow2 // Cracow1

Weighted Sum Method—WSMSimilarities with the “questionnaire”

Consider the standard MIFID I questionnaire used by banks to assess the risk pro-file of customers:

1. it asks questions (such as “how old are you”),

2. each answer gets a score,

3. the scores are added,

4. then in a table one finds which score-band corresponds to which risk profile

. . . exactly The Weighted Sum Method!

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3.3. NAIVE METHODS

Weighted Sum Method—WSMAdvantages and Disadvantages

Advantages

• intuitive for many people (few board members will question the choice of MCDM)

• easy to understand

• easy to discuss the weights . . . without really being able to interpret them

• simple result: typically one finds a complete ranking

Disadvantages

• one has to add variables in different units(!)

• or at least reduce all different variables to unit-less variables

• the choice of the weights is arbitrary

• difficult to gain insight

3.3.2 Weighted Product Method—WPM

Weighted Product Method—WPMThe Same Idea

Let wj be the weight of the criterion j, and sij the score (performance) of alternativei on criterion j then solutions can be ranked according to their total score as follows

P (xi) = Πnj=1(sij)

wj

Question

Where did we see this before?

Weighted Product Method—WPMWith Preference

Let wj be the weights of the criteria, and sij the score (performance) of alterna-tive i on criterion j then a solution xi is preferred over a solution xn if the preferenceP (xi, xk) > 1, with

P (xi, xk) := Πnj=1

(sijskj

)wj

This form of the WPM is often called dimensionless analysis because its mathematicalstructure eliminates any units of measure! BUT, it requires a ratio scale!

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3.3.3 Goal Programming

Goal Programming

The Idea

Replace max{f1(x), f2(x), . . . , fn(x)} by

min {y1 + y2 + . . .+ yj + . . .+ yk |x ∈ A}

with

f1(x) +y1 = M1

f2(x) +y2 = M2

. . . = . . .fj(x) +yj = Mj

. . . = . . .fk(x) +yk = Mk

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3.3. NAIVE METHODS

• Of course, the yi have to be additive, so have to be expressed in the same units.

• This forces us to convert them first to the same unit: eg. introduce factors rj thateliminate the dimensions, and then minimize

∑kj=1 rj yj

• Now we can choose one of the direct methods or numerical methods listed onSlide 27 to solve the problem.

Note: It should be clear that the rj play the same role as the fj(x) in the Weighted Sum Method.This means that the main argument against the Weighted Sum Method (adding things that areexpressed in different units) remains valid here.

Note: The target unit that is used will typically be “a unit-less number between 0 and one” or“points” (marks) . . . as it indeed looses all possible interpretation. To challenge the manage-ment, it is worth to try in the first place to present “Euro” or “Dollar” as common unit. Thisforces a strict reference frame.

Goal ProgrammingTarget Interpretation

• define a target point, M (eg. the best score on all criteria)

• define a “distance” to the target point: ||F−x||, with F = (f1(x), f2(x), . . . , fk(x))′

(defined as in the Weighted Sum Method, so reducing all variables to the sameunits).As a distance, be inspired by:

– the Manhattan Norm: L1(x,y) =∑k

j=1 |xj − yj|

– the Euler Norm: L2(x,y) =(∑k

j=1(xj − yj)2) 1

2

– the general p-Norm: Lp(x,y) =(∑k

j=1(xj − yj)p) 1

p

– the Rawls Norm: L∞(x,y) = maxj=1...k |xj − yj|

Note: The problem was introduced in Slide 56 as the Manhattan norm, this slide should inspireyou to consider other norms.

Goal ProgrammingAdvantages and Disadvantages

Advantages

• reasonably intuitive

• better adapted to problems of “design” (where A is infinite)

Disadvantages

— 29 —


• one has to add variables in different units(!)

• or at least reduce all different variables to unit-less variables

• the choice of the weights is arbitrary

• even more difficult to gain insight

3.4 Preference Methods

3.4.1 ELECTRE I

ELECTRE IThe Idea

If impossible to find, express and calculate a meaningful common variable (suchas “utility”) then we try to find at least a preference structure that can be applied to allcriteria.

Note: If it is possible to define “Utility”, then the multi-criteria problem reduces to a mono-criterion problem that is easily solved. In reality however, it is doubtful if “utility” exists andeven if it would, it will be different for each decision maker and it might not even be possibleto find its expression (if that exists).

the Preference π

Definition 16 .:. preference on one criterion π .:.

When one prefers a solution a over a solution b for criterion fj(.) then we write

πj(a, b) = 1⇔ fj(a) > fj(b)

πj(a, b) = 0⇔ fj(a) = fj(b)

πj(a, b) = −1⇔ fj(a) < fj(b)

Definition 17 .:. step-function θ .:.

θ(x; a) =

{1 ⇔ x > a0 ⇔ x ≤ a

— 30 —

3.4. PREFERENCE METHODS

Definition 18

The Degree of Preference of a solution a over a solution b is

Π+(a, b) =k∑j=1

θ(πj(a, b), 0)

Definition 19

The preference of a solution b over a solution a is

Π−(a, b) = Π+(b, a) = −Π+(a, b) =k∑j=1

θ(πj(b, a), 0) =k∑j=1

θ(0, πj(a, b))

Definition 20

The Degree of Indifference of a solution a and b is

Π0(a, b) = k − Π+(a, b)− Π−(a, b)

Now, we can re-introduce the weights wj (each criterion j has a weight wj). Thisallows us to present a “Weighted Degree of Preference”, which is probably more rele-vant for most multi-criteria decision problems.

Hence one defines:

Definition 21

The Weighted Degree of Preference of a solution a over a solution b is

Π+(a, b) =k∑j=1

θ(πj(a, b), 0) wj

Definition 22

The Weighted Preference of a solution b over a solution a is

Π−(a, b) =k∑j=1

θ(0, πj(b, a)) wj

= −Π+(a, b)

= Π+(b, a)

= −k∑j=1

θ(πj(a, b), 0) wj

— 31 —


Definition 23

The Weighted Degree of Indifference of a solution a and b is

Π0(a, b) =k∑j=a

wj − Π+(a, b)− Π−(a, b)

Note: These degrees of preference can be weighted (i.e. divided by 1k−1 ) but this will not have

any influence on any conclusion.

Note: These weights can be the same as in the Weighted Sum Method, however in generalthere is no reason why they would be the same.

ELECTRE ISTEP 1: Comparing the Alternatives

Now, we calculate and “index of comparability”

Definition 24 .:. index of comparability of type one .:.

C1(a, b) =Π+(a, b) + Π0(a, b)

Π+(a, b) + Π0(a, b) + Π−(a, b)

Note that C1(a, b) = 1⇔ aDb.

Definition 25 .:. index of comparability of type two .:.

C2(a, b) =Π+(a, b)

Π−(a, b)

Note that C2(a, b) =∞⇔ aDb.

ELECTRE ISTEP 2: defining cut-off levels

First –for the ease of notation– define a “discrepancy index”

Definition 26 .:. discrepancy index .:.

dj(a, b) = fj(b)− fj(a)

For both preference and for each criterion individually we set a cut-off level:

• for the comparability index:

– Λ1 ∈]0, 1[ if one uses C1

— 32 —


– Λ2 ∈]0,∞[ if one uses C2

• for each criterion a maximal discrepancy in the “wrong” direction if a preferencewould be stated: rj, j ∈ {1 . . . k}

ELECTRE ISTEP 3: define the preference structure

• for C1 :Π+(a, b) > Π−(a, b)C1(a, b) ≥ Λ1

∀j : dj(a, b) ≤ rj

⇒ a � b

• for C2 :Π+(a, b) > Π−(a, b)C2(a, b) ≥ Λ2

∀j : dj(a, b) ≤ rj

⇒ a � b

In a last step one can present the results graphically and present the kernel (the bestsolutions) to the decision makers.

Definition 27 .:. kernel .:.

The kernel of a MCD-Problem is the set

K = {a ∈ A | @b ∈ A : b � a}

ELECTRE I ExampleAn Example: Decision Matrix of Slide 35, same decision weights

First: calculate the preference structure (here shown as a matrix)Π+(i, j) =

Warsaw2 Cracow1 Cracow2 Gdynia LodzWarsaw2 0.00 5.50 7.50 4.00 5.50Cracow1 10.00 0.00 10.00 8.00 10.00Cracow2 8.00 5.50 0.00 12.00 4.00Gdynia 10.00 5.50 3.50 0.00 3.50

Lodz 8.00 5.50 2.00 12.00 0.00

Table 3.3: The preference structure in our example. The preference of Gdynia over Warsaw2 is10 (element (4,1)) and the opposite (preference of Warsaw2 over Gdynia (the element (1,4)) isonly 4.

ELECTRE I ExampleNegative Preference

Π−(i, j) =

— 33 —



Lodz 5.50 10.00 4.00 3.50 0.00

Table 3.4: The anti-preference structure in our example.Yes, Π−(i, j) = Π+(j, i), or expressed as matrices: Π+ = (Π−)′. So there is no need

to calculate both!

ELECTRE I ExampleIndifference Preference

Π0(i, j) =


Lodz 2.00 0.00 9.50 0.00 15.50

Table 3.5: The non-preference structure in our example. Warsaw1 and Warsaw2 are for themost criteria similar.

Note that ∀i, j : Π+(i, j) + Π0(i, j) + Π−(i, j) =∑k

n=1 λn

ELECTRE I Examplesimilarity Indices C1

Now, we can calculate the similarity indices C1 and/or C2. For example, let’s useC1. We then find that


Lodz 0.65 0.35 0.74 0.77 1.00

ELECTRE I Examplesimilarity Indices C1: preference

The last step is to compare the preferences indices C1 with Λ1. If C1(a, b) > Λ1 thenwe prefer a over b, except if there is not difference in the opposite direction bigger thanr = (0.3, 0.1, 1, 1)′, then solution i is preferred over solution j This results in the fol-lowing preference matrix (convention: if xi � xj ⇒ element(i, j) = 1 then , otherwisezero):

Table 3.6: The preference structure (element (i, j) is one if alternative i is preferred over j)

— 34 —


ELECTRE I Examplesimilarity Indices C1: ranking

Warsaw2

Cracow1Cracow2

Gdynia

Lodz

Figure 3.4: The relation “is dominated by” for ELECTRE I I with the similarity indices C1


ELECTRE I (using the cut-off for C1, Λ1 = 0.6 and the r = (0.3, 0.1, 1, 1)′) wouldrank as follows:

Cracow1

Warsaw2 // Gdynia // Lodz

99

%%Cracow2


Alternatively one can work with C2. The indices C2 are:Warsaw2 Cracow1 Cracow2 Gdynia Lodz

Warsaw2 0.55 0.94 0.40 0.69Cracow1 1.82 1.82 1.45 1.82Cracow2 1.07 0.55 3.43 2.00Gdynia 2.50 0.69 0.29 0.29

Lodz 1.45 0.55 0.50 3.43

— 35 —


ELECTRE I Examplesimilarity Indices C2: Preference

Using a cut-off level Λ2 = 1.5 one obtains the following preference (convention: ifxi � xj ⇒ element(i, j) = 1 then , otherwise zero):

Warsaw2 Cracow1 Cracow2 Gdynia LodzWarsaw2Cracow1Cracow2 1Gdynia 1

Lodz 1


Warsaw2

Cracow1Cracow2

Gdynia

Lodz

Figure 3.5: The preference structure for ELECTRE I I with similarity indices C2 and Λ = 2(for lower values of Λ we get full ranking.


Lodz

Warsaw2 // Gdynia

88

&&Cracow2

Cracow1

— 36 —


3.4.2 ELECTRE II

ELECTRE IIThe Idea: Get a Full Ranking

Aim to Get Full Ranking by

• Gradually lower the cut-off level Λ1

• Increase the cut-off level for opposite differences in some criteria rj

ELECTRE IIExample

[H]

Warsaw2

Cracow1Cracow2

Gdynia

Lodz

Figure 3.6: One of the preference structures for ELECTRE II.

ELECTRE IIOur Example

By manipulating the different cut-off levels, one can come up with the followingranking:

Warsaw2 // Gdynia // Lodz // Cracow2 // Cracow1

3.4.3 ELECTRE I and II: Conclusions

ELECTRE I & IIAdvantages and Disadvantages

Advantages

• no need to add different variables in different units

— 37 —


• all that is needed is a conversion to“preference” and add this preference

• richer information that the Weighted Sum Method

• the level of compensation can be controlled

Disadvantages

• there is still an “abstract” concept “weight”, which has little meaning and nopure interpretation

• to make matters worse, there are also the cut-off levels

• so to some extend it is still so that concepts that are expressed in different unitsare compared in a naive way

3.5 Outranking Methods

Outranking MethodsThe idea is to prefer a solution that does better on more criteria

• Direct Ranking a solution a is preferred over b if a does better on more criteriathan b

• Inverse Ranking a solution a is preferred over b if there are more alternativesthat doe better than b than there are alternatives that do better than a

• Median/Average Ranking: use the median/average of both previous

• Weighted Ranking: use one of the previous in combination with weights wjAll of the previous can be used stand alone or as an addition to ELECTRE II in orderto get the full ranking.

OutrankingWith No Weighting of Criteria

[H]

Warsaw2

Cracow1Cracow2

Gdynia

Lodz

— 38 —

3.5. OUTRANKING METHODS

Figure 3.7: Preference via outranking when criteria are not weighted

OutrankingWith No Weighting of Criteria

Cracow2

''Lodz //Warsaw2 // Gdynia

Cracow1

33

OutrankingWith Weighting of Criteria

[H]

Warsaw2

Cracow1Cracow2

Gdynia

Lodz

Figure 3.8: Preference via outranking when criteria are assigned different weights

OutrankingWith Weighting of Criteria

Warsaw2 // Gdynia // Cracow2 // Cracow1 // Lodz

3.5.1 PROMETHEE I

PROMETHEE IThe Idea

• Enrich the preference structure of the ELECTRE method.

— 39 —


• In the Electre Method one prefers essentially a solution a over b for criterion k ifand only if fk(a) > fk(b)

• This 0-or-1-relation (black or white) can be replaced by a more gradual solutionwith different shades of grey.

• this preference function will called πk(a, b) and it can be different for each crite-rion

• the preference function to choose is a function of dk(a, b)

Definition 28 .:. dk(a, b) .:.

dk(a, b) = fk(a)− fk(b)

PROMETHEEPreference Functions

stepfunction sigmoid

0.00

0.25

0.50

0.75

1.00

−5 0 5 10 15 −5 0 5 10 15difference for criterion between a and b

pref

eren

ce

0.00

0.25

0.50

0.75

1.00preference

Figure 3.9: Left the step-function as a preference function (as it is implicitly used by ELEC-TRE), right another shape (the logistic map in this case)

Preference FunctionsExamples:

• step-function with one step (similar to ELECTRE preferences)

• step-function with more than one step

• step-wise linear function

• π(d) = max(0,min(g × d, d0)) (linear, gearing g)

— 40 —


• sigmoid equation: π(d) = 1

1−(

1d0−1

)e−dt

• π(d) = tanh(d)

• π(d) = erf

(√(π)

2d

)• π(d) = d√

1+x2

• Gaussian: π(d) =

{0 for d < 0

1− exp(− (d−d0)2

2s2

)for d ≥ 0

• . . .

PROMETHEEPreference Functions

Figure 3.10: Some examples of possible preference functions.

PROMETHEE INote

Note that in PROMETHEE I the preference π(a, b) does not become negative. Oth-erwise –when added– we would loose information (i.e. differences in opposite direc-tions are compensated).Later we will anyhow consider the negative part (see PROMETHEE

II).

PROMETHEE IFormalism

• Define preference functions π : A×A 7→ [0, 1]

— 41 —


• They should only depend on the difference between the fj(x): πj(a, b) = πj (fj(a)− fj(b))

• Define a preference index: π(a, b) =∑k

j=1wjπj(a, b)

• Then sum all those flows for each solution in a

1. a positive flow: Φ+(a) = 1k−1∑

x∈A π(a, x)

2. a negative flow: Φ−(a) = 1k−1∑

x∈A π(x, a)

3. a net flow: Φ(a) = Φ+(a)− Φ−(a)

where the wi are the weights of the preferences so that∑k

j=1wj = 1 and ∀j : wj >0

PROMETHEE IThe Preference Relations

Define the preference for PROMETHEE I as follows:

•

a � b⇔{

Φ+(a) ≥ Φ+(b) ∧ Φ−(a) < Φ−(b) orΦ+(a) > Φ+(b) ∧ Φ−(a) ≤ Φ−(b)

• indifferent⇔ Φ+(a) = Φ+(b) ∧ Φ−(a) = Φ−(b)

• in all other cases: no preference relation

PROMETHEE I ExamplePreference Flows

phi plus phi min phiWarsaw2 18.88 35.34 -16.46Cracow1 36.69 22.00 14.69Cracow2 29.50 18.72 10.78Gdynia 21.84 35.67 -13.83

Lodz 27.17 22.34 4.83

Table 3.7: The positive preference flow Π+, the negative preference flow Π− and the total flowΦ.

— 42 —


PROMETHEE IExample

Warsaw2

Cracow1Cracow2

Gdynia

Lodz

Figure 3.11: The preference relations for PROMETHEE I for our usual example (and all Gaus-sian preference curves with d0 = (0, 0, 0, 0)′ and s = (0.1, 0.1, 0.2, 0.3)

PROMETHEE ISummarizing the Flows

Gdynia

&&

Cracow199

Lodz

&&Warsaw2

88

Cracow2

Figure 3.12: The ranking obtained by PROMETHEE I.

PROMETHEE IAdvantages and Disadvantages

Advantages

• it is easier and makes more sense to define a preference function than the param-eters Λj and r in ELECTRE

• it seems to be stable for addition and deletion of alternatives (the ELECTRE andWPM have been proven inconsistent here)

• no comparison of variables in different units

— 43 —


• the preference is based on rich information

Disadvantages

• does not readily give too much insight in why a solution is preferred

• needs more explanation about how it works than the WSM

• some decision makers might not have heard about it

3.5.2 PROMETHEE II

PROMETHEE IIThe Idea

Allow “full compensation” between positive and negative flows (similar to ELEC-TRE II) and hence reduce the preference flow calculations to just one flow

Φ(a, b) =k∑j=1

πj(fj(a), fj(b))

where here the πj(a, b) can be negative and are symmetrical for mirroring around theaxis (y = −x). They can also be considered as the concatenation of Φ+ and Φ− asfollows

Φ = max(Φ+,Φ−).

PROMETHEE IIPreference Functions

stepfunction sigmoid

−1.0

−0.5

0.0

0.5

1.0

−10 −5 0 5 10 −10 −5 0 5 10difference for criterion between a and b

pref

eren

ce

−1.0

−0.5

0.0

0.5

1.0preference

Figure 3.13: Left the step-function as a preference function (as it is implicitly used by ELEC-TRE II), right another shape (the logistic map in this case)

— 44 —


PROMETHEE IIThe Preference Relations

We can condense this information further for each alternative:

Φ(a) =∑x∈A

k∑j=1

πj(fj(a), fj(x))

=∑x∈A

π(a, x)

This results in a preference relation that will almost in all cases show a difference (in

a small number of cases there is indifference, but all are comparable – there is no “nopreference”)

• a � b⇔ Φ(a) > Φ(b)

• indifferent if Φ(a) = Φ(b)

• in all other cases: no preference relation

PROMETHEE IIExample

Warsaw2

Cracow1Cracow2

Gdynia

Lodz

12

3

4

56

7

89

10

31.1492.6343

1 : 31.1492 : 27.2393 : 3.91064 : 2.63435 : 28.5156 : 24.6047 : 21.2918 : 9.85839 : 5.947710 : 18.657

Figure 3.14: The preference relations for PROMETHEE II for our usual example (and allGaussian preference curves with d0 = (0, 0, 0, 0)′ and s = (0.1, 0.1, 0.2, 0.3)

PROMETHEE IISummarizing the Flows

Warsaw22.6 // Gdynia

18.6 // Lodz5.9 // Cracow2

3.9 // Cracow1

— 45 —


Figure 3.15: The full ranking obtained by PROMETHEE II. Additionally we know that thepreference of Warsaw2 over Gdynia is relatively small, but that the gap between Gdynia andŁodz is relatively important, etc. The approximate preference is above the arrows. Note alsothat the preference relation is additive.

PROMETHEE IIAdvantages and Disadvantages

Advantages

• all advantages of PROMETHEE I

• almost sure to get a full ranking

• the preference structure is rich and preference quantifiable

• the preferences are transitive: a � b ∧ b � c⇒ a � c.

• no conflicting rankings possible, logically consistent for the decision makers

Disadvantages

• more condensed information (loss of information, more compensation)

• might be more challenging to understand for some people

3.5.3 Other Methods

Other Methodsstill an active domain of research

• GAIA Method: projects the solutions in a 2-D plane along so that differencesare maximized (eigenvalues), provides maximum insight and can be completedwith a “decision stick” (vector or weights projected in that plane) to select thebest alternative.

• Evidential Reasoning: Allows for uncertainty to be taken into account. Eachcriterion can be something like “10% good, 40% very good and 50% excellent”

• etc.

3.6 Summary MCDA

Summary MCDAWay of Working

1. Make sure you understand the problem and how the decision is made

2. Reduce the set of solutions by using “hard cut-off levels” (like “at least twodesks”) – eliminating solutions that anyhow would not be satisfying

— 46 —

3.6. SUMMARY MCDA

3. Simplify the problem by aggregating Attributes into Criteria

4. Create the Decision Matrix

5. Eliminate alternatives by Dominance — be very careful here and eventually re-visit (avoid to leave out the second best solution!!)

6. Use a simple linear Weighted Sum Method to get some feeling and interactionwith the decision makers

7. Use another method to get more insight and a richer presentation

8. Discuss

Do not forget

Golden RuleMCDA is not a science, it is an art!

The Decision-making paradox

• MCDA-methods used for solving multi-dimensional problems (for which differ-ent units of measurement are used to describe the alternatives), are not alwaysaccurate in single-dimensional problems

• When one alternative is replaced by a worse one, the ranking can change

• Proven for both ELECTRE and WPM; WSM and PROMETHEE (most probably)are not subjected to this paradox.

• See eg.: Triantaphyllou (2000)

— 47 —


— 48 —

3.6. SUMMARY MCDA

BACK-MATTER

— 49 —


— 50 —

Appendix AFurther Reading

Further Reading

• International Society on Multiple Criteria Decision Making: http://www.mcdmsociety.org/

• the “Multiple Criteria Decision Aid Bibliography” pages of the “Universite ParisDauphine”: http://www.lamsade.dauphine.fr/mcda/biblio/

APPENDIX A. FURTHER READING

— 52 —

Appendix BConventions for Matrix Algebra

Conventions for Matrix AlgebraWe employ the following notational conventions:

• a scalar is denoted by a plain letter, such as a, b, µ, σ, n,m

• a vector (a one-dimensional tensor) is indicated by a bold-faced small capitalletter, such as x or y; except in some case where we use another specific symbolsuch as P ,L,R,R.

• a matrix (a two-dimensional tensor) is represented by a bold-faced capital letter.Examples: Σ,A

By convention, vectors are column vectors.

Definition 29 .:. vectors .:.

Let x and y be defined by:

x :=

x1x2...xn

∈ <n, and y :=

y1y2...ym

∈ <m (B.1)

We further denote the scalar product with a dot (.), while the vector product andthe set of all combinations will be denoted by ×. Derivatives are always explicitlywritten, and the transpose of a vector or matrix x is denoted by x′.

The determinant of a matrix is represented by det(A).

APPENDIX B. CONVENTIONS FOR MATRIX ALGEBRA

— 54 —

Appendix CLevels of Measurement

Levels of Measurement

Introduction

It is customary to refer to the theory of scales as having been developed by Stevens(1946). In that paper he argues that all measurement is done by assuming a certainscale type. He distinguished four different types of scale: nominal, ordinal, interval,and ratio scales.

C.1 Nominal Scale

Nominal Scale

The nominal scale is the simplest form of classification. It simply contains labelsthat do not even assume an order. Examples include asset classes, first names, coun-tries, days of the month, weekdays, etc. It is not possible to use statistics such asaverage or median, and the only thing that can be measured is which label occurs themost (modus of mode).

APPENDIX C. LEVELS OF MEASUREMENT

Scale Type NominalCharacterization labels (e.g. asset classes, stock exchanges)Permissible Statistics mode (not median or average), chi-squarePermissible Scale Transformation equalityStructure unordered set

Table C.1: Characterization of the Nominal Scale of Measurement.Note that it is possible to use numbers as labels, but that this is very misleading.

When using an nominal scale, none of the traditional metrics (such as averages) canbe used.

C.2 Ordinal Scale

Ordinal ScaleThis scale type assumes a certain order. An example is a set of labels such as very

safe, moderate, risky, very risky. Bond rating such as AAA, BB+, etc. also are ordinalscales: they indicate a certain order, but there is no way to determine if the distancebetween, say, AAA and AA- is similar to the distance between BBB and BB-. It maymake sense to talk about a median, but it does not make any sense to calculate anaverage (as is sometimes done in the industry and even in regulations)

Scale Type Ordinal ScaleCharacterization ranked labels (e.g. ratings for bonds from

rating agencies)Permissible Statistics median, percentilePermissible Scale Transformation orderStructure (strictly) ordered set

Table C.2: Characterization of the Ordinal Scale of Measurement.Ordinal labels can be replaced by others if the strict order is conserved (by a strict

increasing or decreasing function). For example AAA, AA-, and BBB+ can be replacedby 1, 2 and, 3 or even by -501, -500, and 500,000. The information content is the same,the average will have no meaningful interpretation.

C.3 Interval Scale

Interval ScaleThis scale can be used for many quantifiable variables: temperature (in degrees

Celsius). In this case, the difference between 1 and 2 degrees is the same as the differ-ence between 100 and 101 degrees, and the average has a meaningful interpretation.Note that the zero point has only an arbitrary meaning, just like using a number foran ordinal scale: it can be used as a name, but it is only a name.

— 56 —

C.4. RATIO SCALE

Scale Type Interval ScaleCharacterization difference between labels is meaningful

(e.g. the Celsius scale for temperature)Permissible Statistics mean, standard deviation, correlation, re-

gression, analysis of variancePermissible Scale Transformation affineStructure affine line

Table C.3: Characterization of the Interval Scale of Measurement.Rescaling is possible and remains meaningful. For example, a conversion from

Celsius to Fahrenheit is possible via the following formula, Tf = 95Tc + 32, with Tc the

temperature in Celsius and Tf the temperature in Fahrenheit.An affine transformation is a linear transformation of the form y = A.x + b. In

Euclidean space an affine transformation will preserve collinearity (so that lines thatlie on a line remain on a line) and ratios of distances along a line (for distinct collinearpoints p1, p2, p3, the ratio ||p2 − p1||/||p3 − p2|| is preserved).

In general, an affine transformation is composed of linear transformations (rota-tion, scaling and/or shear) and a translation (or “shift”). An affine transformationis an internal operation and several linear transformations can be combined into onetransformation.

C.4 Ratio Scale

Ratio ScaleUsing the Kelvin scale for temperature allows us to use a ratio scale: here not only

the distances between the degrees but also the zero point is meaningful. Among themany examples are profit, loss, value, price, etc. Also a coherent risk measure is a ratioscale, because of the property translational invariance implies the existence of a truezero point.

Scale Type Ratio ScaleCharacterization a true zero point exists (e.g. VAR, VaR, ES)Permissible Statistics geometric mean, harmonic mean, coeffi-

cient of variation, logarithms, etc.Permissible Scale Transformation multiplicationStructure field

Table C.4: Characterization of the Ratio Scale of Measurement.

— 57 —

APPENDIX C. LEVELS OF MEASUREMENT

— 58 —

Bibliography

D’Avignon, G. and H. Winkels (1986). Multi-criteria decision aid for the managementof uncertainty. In The Management of Uncertainty: Approaches, Methods and Applica-tions, Dordrecht / Boston / Lancaster, pp. 75–135. Nato Scientific Affairs Division.

Markowitz, H. M. (1952). Portfolio selection. Journal of Finance 6, 77–91.

Merton, R. C. (1972). An analytic derivation of the efficient portfolio frontier. Journalof financial and quantitative analysis 7(04), 1851–1872.

Monsen, R. J. and A. Downs (1965). A theory of large managerial firms. The Journal ofPolitical Economy, 221–236.

Stevens, S. S. (1946). On the theory of scales of measurement. Science 103(2684), 677–680.

Triantaphyllou, E. (2000). Multi-criteria decision making methods a comparative study.Springer.

BIBLIOGRAPHY

— 60 —

Nomenclature

Λ1 cut-off level for C1 (for ELECTRE I), page 33

Λ2 cut-off level for C2 (for ELECTRE I), page 33

A the m × n matrix representing the conditions of the Simplex Linear problem inits canonical form, page 14

c the 1×n vector representing the function to be optimized in the Simplex LinearProblem in its canonical form, page 14

F the scaled decision matrix after application of the fj(), so Fij = fj(xij), page 25

Morig the original decision matrix (before all omissions), page 23

M the decision matrix, page 23

w the vector of all wj , page 25

x the m × 1 (or otherwise stated x ∈ <n) vector representing the variables undercontrol, solving the problem is finding the optimal value x0, page 14

A the set of acceptable solutions, page 14

Φ(a) the preference flow between an alternative a and all others over all criteria,equals: Φ(a) =

∑x∈A

∑kj=1 πj(fj(a), fj(x)), page 45

Φ(a, b) the preference flow between an alternative a and b as used in PROMETHEE II,page 44

Φ+(a) the preference flow between an alternative a and all others, equals: Φ+(a) =1

k−1∑

x∈A π(a, x) in PROMETHEE, page 42

Φ−(a) the preference flow that indicates how much other alternatives are preferredover alternative a, it equals Φ−(a) = 1

k−1∑

x∈A π(x, a) in PROMETHEE, page 42

Π+(a, b) the degree of preference of solution a over b in the ELECTRE method, equals∑kj=1 θ(πj(a, b), 0), page 31

NOMENCLATURE

Π−(a, b) the degree of preference of solution b over a in the ELECTRE method, equals∑kj=1 θ(πj(b, a), 0), page 31

Π0(a, b) the degree of indifference of solution a and b in the ELECTRE method, equalsk − Π+(a, b)− Π−(a, b), page 31

πj(a, b) the “preference” of solution a over solution b, page 30

< the set of real numbers, page 13

θ(x; a) the step-function that steps up in the point a from 0 to 1, equal to 1 if x > a andequal to 0 if x ≤ a, page 30

∧ logical “and operator”, page 42

a � b alternative a is preferred over alternative b, page 42

ai multiplier that (a) removes the dimensions of the unit and (b) scales it, page 15

aDb is a shorthand notation for “alternative a dominates alternative b”, page 23

C1(a, b) index of comparability of type one (for ELECTRE I), page 32

C2(a, b) index of comparability of type two (for ELECTRE I), page 32

CF cash flow, page 15

cf cash flow, page 15

dk(a, b) the scaled difference in performance of alternative a over b for criterion k,equals fk(a)− fk(b), page 40

fj(x) the scaling function for criterion j applied to alternative x (similar entities as therj); the index j runs from 1 to n—alternatively one can write fj(xij) as sij , the“score” of “alternative” xi on “criterion” j, page 18

m the number of conditions in the Simplex Linear Problem, page 14

Mj the ideal value for criterion j in the Goal Programming Method, page 28

N a bounded or non-bounded natural number, page 15

n the number of scalar variables to be optimized in the Simplex Linear Problem,page 14

n used as the number of criteria to optimize, page 18

N(x) the result vector of the WSM for alternative x, page 25

NPV Net Present Value, page 15

P (a) the “total score” of alternative a as used in the WPM, page 27

— 62 —

NOMENCLATURE

P (a, b) the “total score” of alternative a as used in the dimensionless WPM, page 27

PV (.) Present Value, ie. the discounted value, page 15

r the discount rate, page 15

rj a conversion factor that (a) removes the unit and (b) scales it, similar concept tofj(.), page 29

sij the score of alternative i on criterion j, used as shorthand: sij := fj(xij), page 18

t a counter, page 15

U utility function, page 15

wj the scaling factor in the WSM, it (a) removes the unit and (b) scales (weights)the jth criterion, page 25

yj factors to be minimized in the Goal Programming Method, page 28

KPI Key Performance Indicator, page 8

MCDA Multi Criteria Decision Analysis, page 19

MCDM Multi Criteria Decision Making, page 17

WPM Weighted Product Method, page 27

WSM Weighted Sum Method, page 25

— 63 —

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