Dr. Y. İlker TOPCUtopcuil/ya/MDM03Constructing...Subjective evaluation Constructed attribute is...

Dr. Y. İlker TOPCUwww.ilkertopcu.net www.ilkertopcu.org www.ilkertopcu.info

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Dr. Özgür KABAKweb.itu.edu.tr/kabak/

Decision Making Process1. Structuring the Problem

Exploring the issue and determining whether or not MADM is an appropriate tool: If so, then alternatives for evaluation and relevant criteria can be expected to emerge

2. Constructing the Decision ModelElicitation of preferences, performance values, and (if necessary) importance

3. Analyzing (Solving) the ProblemUsing a solution method to synthesize and explore results (through sensitivity and robustness analyses)

Dr. Y. İlker Topcu (www.ilkertopcu.net) & Dr. Özgür Kabak ([email protected]) 2

Modeling Creating an abstract view of reality Incremental process of converting mental concepts into explicit and hence analyzable representations


Requirements for Modeling

Modeling should fulfill two requirements: Expressive power general enough and allow the description of a variety of real‐world decision situations

Usability permit end users to master the model without great effort, allowing DMs to concentrate on the task, not on the modeling tools


Decision Models Normative Models

The axiomatic path and the quest for norms for prescribing Proposes a series of rules which the DM should respect Indicates how decisions should be made (by idealized rational DM) according to a set of well‐defined rules

Rules of law, statistics, economics Descriptive Models

A path of realism, quest for descriptions for discovering Description of real people Refers to how people actually behave/make decisions Characterizes what happens in the real world without judgment of it

Pychology, behavioral sciences, marketingDr. Y. İlker Topcu (www.ilkertopcu.net) & Dr. Özgür Kabak ([email protected]) 5

Decision Models Constructive Models

Both normative and descriptive Construct ‐ create The constructivist path and the quest for working hypothesis for recommending


Constructing Decision ModelAssessing performance values of alternatives w.r.t. attributes

(If necessary) Determining relative importance of attributes

Modeling the preference of DMs


Assessing Performance Values

Determine how well each alternative achieves each attribute

Performance value = score = rating = attribute level


Assessing Performance ValuesObjective evaluation

Natural attribute is usedQuantitativeIndependent of DMAttributes are readily measured in terms of some natural physical unit

e.g. dollars or number of people


Assessing Performance ValuesSubjective evaluation

Constructed attribute is usedQualitativeDependent on DMNo natural measuring scales exist e.g. beauty or convenience


Measurement ConsiderationsPerformance values can be measured by: Nominal scalesOrdinal scales Interval scalesRatio scales


Nominal Scales The lowest level in terms of the meaning conveyed Nominal numbers are just numerical representations for names

Used for identification purposes only and imply nothing about ordering

e.g.Telephone number, Turkish Republic identification (ID) number: “Are you older or better than someone else because your number is higher?


Ordinal Scales

Ordinal numbers imply an order of ranking among elements

The order may be increasing or decreasing Ranking implies an ordering but nothing more: does not imply anything about the differences between items

We could order cities based on population by ranking the city with thehighest population as #1 (or with lowest population)


Interval Scales Interval scale data has meaning about the differences between objects

The unit as well as the origin on an interval scale can be assigned arbitrarily.

Valid operations on an interval scale are multiplication and the addition of a constant


Interval Scales Statements concerning x1/x2 are meaningless as they depend on the choice of origin and unit.

On the other hand statements concerning ratios of differences ((x1‐x2)/(x3‐x4)), are meaningful on interval scales since they are independent of both origin and the unit.

Preference on utility scales and temperature on the Fahrenheit or Celcius scales (32+1.8*x transforms temperature from the Celcius to the Fahrenheit scale).

If we have interval level data then we can infer that the interval between two objects with values of 20 and 5 (an interval of 15) is equivalent to the interval between two objects with values of 80 and 65


Ratio Scales Highest level meaning The unit on a ratio scale can be chosen arbitrarily The only valid operation (admissible transformation) on a ratio scale is a change of unit (multiplying the scale by a positive constant).

Adding a constant to a ratio scale is not a valid operation because zero point (origin) can not be defined arbitrarily on the scale.


Ratio Scales If x1 and x2 are two measurements on a ratio scale, statements concerning x1/x2 are independent of the unit chosen and are meaningful

Length can be measured in inches, miles, meters… Kelvin scale: the zero point of this scale is equivalent to ‐273.15 °C


Example for Scales Suppose the owner of a horse racing stable is interested in buying a particular horse. He studies the results.

The number worn on the horse and jockey is nominal. It identifies the horse

The finishing position for each horse is ordinal. e.g. the first place horse finished ahead of the second one.

The number of lengths to the next finisher is an interval measure. Knowing that the horse finished first by 15 lengths as opposed to 5 lengths is important information.

Suppose the horse finished first by 15 lengths in a 2.5 miles race. Is this as strong as finishing first by 15 lengths in a 0.5 miles race? More information would be conveyed if we knew the ratios of the times of the first and second place finishes.


Decision MatrixAlternative evaluations w.r.t. attributes are presented in a decision matrixEntries are performance values Rows represent alternativesColumns represent attributes


Example ‐ Car purchasing decision

Sellingprice Fuel cons. Comfort

K TL (cost) lt/100km (cost) 1‐5 (benefit)

a 80 10 5b 60 9 4c 70 8 4


Example – Mobile phone

21Dr. Y. İlker Topcu (www.ilkertopcu.net) & Dr. Özgür Kabak ([email protected])

Price(TL)

Camera Memory Battery ‐ talk time (hrs.)

Displaydimension

Design

LG V10 2600 16MP 64GB 15 hrs 5.7 in Good (4)

Sony Xperia Z5 Premium

2500 23MP 32GB 16 hrs 5.5 in Very good(5)

iPhone 6S 3099 12MP 16GB 14 hrs 4.7 in Very good(5)

SamsungS6 Edge

2200 16 MP 32 GB 18 hrs 5.1 in Good (4)

Example ‐ Supplier Selection The company Formosa Watch Co., Ltd. (FWCL) is a large, well known watch manufacturer.

For developing new products, they wish to select material suppliers to purchase key components

A committee has been formed to select a qualified supplier (S1; S2; S3; S4). Committee identified five supplier selection criteria: Relationship closeness (C1) Quality of product (C2) Delivery capabilities (C3) Warranty level (C4) Average price (C5)


Example ‐ Supplier Selection ctd. Relationship closeness (C1): 0‐10 scale; 0: the worst, 10: the best Quality of product (C2): 0‐10 scale; 0: the worst, 10: the best Delivery capabilities (C3): 0‐10 scale; 0: the worst, 10: the best Warranty level (C4): 0‐10 scale; 0: the worst, 10: the best Average price (C5): US Dollars


C1 C2 C3 C4 C5

S1 9.62 2.78 3.30 3.05 16.50S2 0.54 8.86 6.23 1.75 18.22S3 5.65 3.65 4.20 2.95 21.01S4 8.50 2.70 8.76 4.07 25.12

weights of criteria high low medium High very high

AttributesBenefit attributes

Offer increasing monotonic utility. Greater the attribute value the more its preference

Cost attributesOffer decreasing monotonic utility. Greater the attribute value the less its preference

Nonmonotonic attributesOffer nonmonotonic utility. The maximum utility is located somewhere in the middle of an attribute range


Global Performance Value If solution method that will be utilized is performance aggregation oriented, performance values should be aggregated.

In this case Performance values are normalized to eliminate computational problems caused by differing measurement units in a decision matrix

Relative importance of attributes are determined


NormalizationAims at obtaining comparable scales, which allow interattribute as well as intra‐attribute comparisons

Normalized performance values have dimensionless units

The larger the normalized value becomes, the more preference it has


Normalization Methods1. Distance‐Based Normalization Methods2. Proportion Based Normalization Methods

(Standardization)


Distance‐Based Normalization MethodsIf we define the normalized rating as the ratio between individual and combined distance from the origin (0,0,…,0) then the comparable rating of xij is given as (Yoon and Kim, 1989):

rij(p) = (xij ‐ 0) /

This equation is arranged for benefit attributes.

Cost attributes become benefit attributes by taking the inverse rating (1/ xij)

ppm

kkjx

/1

1

0


Distance‐Based Normalization Methods Normalization (p=1: Manhattan distance) Vector Normalization (p=2: Euclidean distance) Linear Normalization (p= : Tchebycheff dist.)

rij(1) = xij /

rij(2) = xij /

rij( ) = xij / max (BENEFIT ATTRIBUTE)

rij( ) = min / xij (COST ATTRIBUTE)

m

kkjx

1

2

1

m

kkjx

,...,2,1 , mkxkj ,...,2,1 , mkxkj


Proporiton‐Based Normalization MethodsThe proportion of difference between performance value of the alternative and the worst performance value to difference between the best and the worst performance values (Bana E Costa, 1988; Kirkwood, 1997)

rij = (xij – xj‐) / (xj* – xj‐) benefit attribute

rij = (xj‐ – xij) / (xj‐ – xj*) cost attribute

where * represents the best and – represents the worst(best: max. perf. value for benefit; min. perf. value for cost or ideal valuedetermined by DM for that attribute)


MDM03xNormalization

Transformation of NonmonotonicAttributes to Monotonicexp(–z2/2) exponential functionis utilized for transformation

where z = (xij – xj0) / jxj0 is the most favorable performance value w.r.t. attribute j

j is the standard deviation of performance values w.r.t. attribute j


MDM03yNonmonotonicNormalization

Attribute WeightingMost methods translate the relative importance of attributes into numbers which are often called as “weights” (Vincke, 1992)

Methods utilized for assignment of weights can be classified in two groups (Huylenbroeck, 1995; Munda 1993; Al‐Kloub et al., 1997; Kleindorfer et al., 1993; Yoon and Hwang, 1995):Direct Determination Indirect Determination


Weight Assignment Methods Direct Determination Rating, Point allocation, Categorization Ranking Swing Trade‐off Ratio (Eigenvector prioritization)

Indirect Determination Centrality Regression – Conjoint analysis Interactive


Rating, Point allocation, CategorizationRating Each attribute’s importance is rated on a scale (e.g. 0‐100)

Point allocation Allocate a specific amount of points (e.g. 100) among attributes in proportion of their importance

Categorization Assign attributes to different categories of importance, each carrying a different weight.


RankingWe assign 1 to most important attribute, and n to the least important. The cardinal weigths can be obtained from one of the following formulas: Rank reciprocal weights

wj = (1/rj) / ( 1/rk)

Rank sum weights

wj = (n – rj + 1) / ( (n – rk + 1))

where rj is the rank of the jth attributeIf two attributes are tied for the nth and (n+1)thplace, (2n+1)/2 is assigned to both of them

n

k 1

n

k 1


Swing method Given is a set of alternatives and a set of attributes. Let N be the number of attributes.

1. Determine the best and worst value of each attribute over the set of alternatives.

2. Create N+1 fictional alternatives. The first fictional alternative is the "worst‐case" and has the worst value on every attribute. The next N fictional alternatives have the worst value on all but one of the attributes; on the remaining attribute, each alternative has the best value on a different attribute.

3. Rank order N+1 fictional alternatives. The ranks are determined by DM. The rank of the worst‐case alternative will be N+1, and the rank of the best of the fictional alternatives will be 1. http://wiki.ece.cmu.edu/ddl/index.php/Swing_weighting


Swing method ctd.4. Rate the N+1 fictional alternatives. The rating of the worst‐

case alternative will be 0, and the rating of the best of the fictional alternatives will be 100. The decision‐maker also rate the others. These ratings should be coherent with the rankings.

5. Normalize the ratings by dividing each one by the sum of all the ratings. The normalized rating of the worst‐case alternative will still be 0, and the sum of all the normalized ratings will equal 1.

6. The weight for each attribute is the normalized rating of the fictional alternative that has the best value on that attribute.


MDM03yySwing

Trade‐off Consider a hypothetical alternative that has the least preferred level for all attributes.

Now suppose that you could move one, and only one, of the attributes from its least preferred level to its most preferred level. Which would you move?

Ask the same question for the remaining attributes.Continue until you have a ranking for all attributes

Now consider the attribute that you most prefer to move (attribute 1) and compare it with the attributethat you second most prefer to move (attribute 2)


Trade‐off ctd. Suppose that you could either move the second attribute all the way from its least preferred level to its most preferred level, or you could move the first attribute from its least preferred level to some intermediate level. Select this intermediate level for which you are indifferent between these two possibilities

Then it must be true that the value for two hypothetical alternatives with performance values of above given possible hypothetical situations are equal:w1v1(xa1) + w2v2(x2–) = w1v1(x1–) + w2v2(x2*) w1 k + w2 0 = w1 0 + w2 1 w1 k = w2


Conjoint Analysis


MDM03zConjoint

Preference Modeling When a DM must compare two alternatives, s/he will react in one of the following ways:

Indifference (I)a and b are same for DM

(Strict) Preference (P)Without any hesitation, DM prefers a to b

Weak preference (Q)b is definitely not better than a, but DM can not make upher mind between aPb and aIb

Incomparability (R)DM can not compare a and b (refusal or inability)


Incomparability Incomparability appears even more frequently whencontradictory opinions must be aggregated. If a decision isfinally selected, it may be quite useful, during the decisionaid step, to put forward clearly any lack of comparisonbetween actions.

On the other hand, when, given the information available,two possible actions turn out to be incomparable, thishelps decision making since it amounts to putting forwardsome aspects of the problem at hand which deserve to beinvestigated.


e.g. FIFA World cup After group matches, in each group first two teams qualify to second

round and last two teams are eliminated. 16 eliminated teams are incomparable with each other.

At the second round, 16 qualified teams play two by two. Winners qualify to quarter final, losers are eliminated: 8 eliminated teams are incomparable But they are better than the eliminated teams at the first round.

At quarter final 8 qualified teams play two by two. Winners qualify to semi final, losers are eliminated: 4 eliminated teams are incomparable But they are better than the eliminated teams at the first and second round.

At semi final 4 qualified teams play two by two. Winners qualify to final, losers play with each other at the 3rd place playoff.

Loser of 3rd place playoff is the 4th team of the tournament Winner of 3rd place playoff is the 3rd team of the tournament Loser of final is the 2nd team of the tournament Winner of final is the 1st (champions) of the tournament


Preference StructuresInterval order

Semiorder

Complete preorder

Complete order

Partial interval order

Partial semiorder

Partial complete preorder

Partial complete order

R =

Constant threshold

Transitivity of I

No ties


Traditional Preference Structure Complete order (P)

ai P aj f(ai) > f(aj)

Complete preorder (P, I)ai I aj f(ai) = f(aj)

Suppose there are three sport teams: a, b, and c. If one team beats the other, the winner will receive three points and the loser will receive zero point. If they draw, both receives one point. The three teams will play with each other and at the end they will have total points.

If all total points differ, there will be a complete order If there is a tie, the order will be complete preorder.


Indifference Threshold Semiorder (P, I, q)

ai P aj f(ai) > f(aj) + qai I aj f(ai) ‐ f(aj)< q

For any temperature x0C we may assume that DM is indifferent between being in a room at x 0C and one at (x+0.001) 0 C; she simply would not be able to detect the difference. So 20 0 C ~ 20.001 0 C, 20.001 0 C ~ 20.002 0 C, 20.002 0 C ~ 20.003 0 C, ..., 99.999 0 C ~ 100 0 C

If DM’s indifferences are transitive, it follows that we may chain all these statements together to deduce 20 0 C ~ 100 0 C. In other words she is equally happy to be just comfortable as to be boiled alive!

Therefore we should use a threshold for indifference transitivity. If the difference between performance values of two alternatives is less than this threshold, the DM will be indifferent between these two alternatives. If the difference is greater than the threshold the DM prefers one to other.


Indifference Threshold Interval order (P, I, q(.))

ai P aj f(ai) > f(aj) + q(f(aj))

ai I aj f(ai) < f(aj) + q(f(aj)) f(aJ) < f(ai) + q(f(ai)) Many applications include variations of the indifference threshold

along the chosen scale. A variation of $1000 doesn’t mean the same when dealing with

thousands of dollars or millions of dollars. It is thus often useful to introduce a variable indifference threshold.

e.g. “Temperature at a room, temperature on the surface of sun.”


Indifference & Preference Thresholds Pseudo order (P, I, Q, q, p)

ai P aj f(ai) > f(aj) + p(f(aj))

ai Q aj f(aj) + p(f(aj)) > f(ai) > f(aj) + q(f(aj))

ai I aj f(ai) < f(aj) + q(f(aj)) f(aJ) < f(ai) + q(f(ai))


Real life applications show that there is often an intermediary zone inside which the DM hesitates between two different responses or gives contradictory answers depending upon the way questions are put to him.

This observation led to the introduction of a preference model: An indifference threshold underneath which the DM shows clear indifference

A preference threshold above which the DM is sure of strict preference


Indifference & Preference Thresholds

Models including Incomparability Partial Order Partial Preorder Partial Semiorder Partial Interval Order


Attribute’s preference representation True Attribute

“Traditional model”. If the underlying preference structure is a preorder structure (q=p=0)

Quasi Attribute Semi Attribute“Threshold model”. If the underlying preference structure is a semiorder structure (q=p>0)

Interval Attribute“Variable threshold model”. If the underlying preference structure is an interval order structure (q(.)=p(.)>0)

Pseudo Attribute“Double threshold model”. If the underlying preference structure is a pseudo order structure (p>q>0)


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