multicriteria satisfaction analysis - tuc
TRANSCRIPT
Multicriteria Satisfaction Analysis
Evangelos GrigoroudisTechnical University of Crete
Decision Support Systems Laboratory
Outline
โข Introduction
โข MUSA methodโข Basic principlesโข Mathematical developmentโข Numerical exampleโข Stability analysisโข Results
โข Inconsistencies and other potential problems
โข Evaluation of results
โข MUSA extensionsโข Strictly increasing functionsโข Multiple criteria levelsโข Alternative post-optimality approachesโข Robustness
โข Concluding remarks
Customer satisfaction measurement
โข Importance of customer satisfactionโข Baseline standard of performance and standard of excellence
for any business organization
โข Quality management and quality assurance standards
โข Availability of different forms of customer satisfaction info (e.g., social media, online surveys, rating systems)
โข Customer satisfaction measurementโข Measurement offers an immediate, meaningful, and objective
feedback
โข Measurement indicates what should be improved, and the ways through which this improvement could be achieved
โข Measurement provides a sense of achievement and accomplishment
โข Need to translate satisfaction data into a number of measurable parameters
Customer dissatisfaction
Dissatisfaction
No action
Action
Private action
Public action
Warn family/friends
about the product and/
or seller
Decide to stop buying
the product or brand or
boycott seller
Take legal action to
obtain redress
Complain to business,
private, or
governmental agencies
Seek redress form firm
or manufacturer
Motivation
โข Basic characteristics of the customer satisfaction measurement problem
โข Customer satisfaction depends on a set of product/service quality characteristics
โข Satisfaction information is directly available by customers (e.g., surveys)
โข Usually available info has an ordinal form
โข Standard approachesโข Regression-type models
โข Statistical or other data analysis tools
โข No (or limited) MCDA methods
Example of customer satisfaction data
โTypicalโ approach in statistical methods
โข Transform ordinal to cardinal scaleโข Very dissatisfied 1
โข Dissatisfied 2
โข Neutral 3
โข Satisfied 4
โข Very satisfied 5
โข โฆbut such transformationโข Seems arbitrary
โข It is only one of an infinite number of permissibletransformations (Stevens, 1946 on the theory of scales on measurement)
โข Assumes a โlinearโ customer satisfaction behavior
Customer satisfaction vs customer loyalty
The MUSA method
โข It is a consumer-based method, since it requires input survey data using a questionnaire of a certain type
โข It assumes that customerโs global satisfaction is based on a set of criteria representing product/service quality characteristics
โข It is an ordinal regression-based approach used for the assessment of a set of collective satisfaction functions in such a way that the global satisfaction criterion becomes as consistent as possible with customersโ judgements
Satisfaction criteria hierarchy
Overall customer satisfaction
Satisfaction based on
criterion 1
Satisfaction based on
criterion 2โฆ
Satisfaction based on
criterion n
Ordinal regression approach
โข The main object of the MUSA method is the aggregation of individual judgements into a collective value function
โข MUSA is a preference disaggregation method used for the assessment of global and partial satisfaction functions ๐โ and ๐๐
โ respectively, given customersโ judgements ๐ and ๐๐
โข Ordinal regression equation:
๐โ = ๐=1๐ ๐๐๐๐
โ with ๐=1๐ ๐๐ = 1
where ๐โ and ๐๐โ are the global and marginal value
(satisfaction) functions and ๐๐ is the weighting factor of criterion ๐
Value (satisfaction) functions
โข Value functions ๐โ and ๐๐โ are normalized in
[0,100]
๐ฆโ1 = 0, ๐ฆโ๐ผ = 100
๐ฅ๐โ1 = 0, ๐ฅ๐
โ๐ผ๐ = 100 ๐ = 1,2, โฆ๐
โข Preference conditions
๐ฆโ๐ โค ๐ฆโ๐+1
๐ฅ๐โ๐ โค ๐ฅ๐
โ๐+1 ๐ฆ๐ โชฏ ๐ฆ๐+1, ๐ = 1,2, โฆ , ๐ผ โ 1
๐ฅ๐๐ โชฏ ๐ฅ๐
๐+1, ๐ = 1,2, โฆ , ๐ผ๐ โ 1
Value (satisfaction) functions
y1
y2
ym
yฮฑ
y*2
0
y*m
100
Y*
Y
... ...
......
100
0
??
??
Model development
โข Introduce a double error variable:
๐โ =
๐=1
๐
๐๐๐๐โ + ๐+ + ๐โ
where ๐โ is the estimation of the overall value function and ๐+, ๐โ are the overestimation and the underestimation errors
โข Error variables are assessed for each customer separately
โข Similar to goal programming
Value (satisfaction) functions with error variables
y1
y2
ym
yฮฑ
y*2
0
y*m
100
Y*
Y
ฯj-
ฯj+
... ...
......
LP formulation
โข Minimize the sum of errors
under the constraints:
โข ordinal regression equation for each customer,
โข normalization constraints for ๐โ and ๐๐โ in [0, 100],
โข monotonicity constraints for ๐โ and ๐๐โ
Transformations
โข Transformations: successive steps of the value functions ๐โ and ๐๐
โ
โข Benefitsโข Remove the monotonicity constraints
โข Formulate a linear model
๐ง๐ = ๐ฆโ๐+1 โ ๐ฆโ๐, ๐ = 1,2,โฆ , ๐ผ โ 1
๐ค๐๐ = ๐ฅ๐โ๐+1 โ ๐ฅ๐
โ๐ , ๐ = 1,2, โฆ , ๐ผ๐ โ 1, ๐ = 1,2, โฆ , ๐
Transformations
y1
y2
ym
yฮฑ
y*2
0
y*m
100
Y*
Yz1
z2
zฮฑโ1
... ...
......
xi1
xi2
xik
xiฮฑi
bixi*2
0
bixi*k
100bi
biXi*
Xi
1iiฮฑw
... ...
......
wi1
wi2
Basic LP
๐๐๐ ๐น = ๐=1
๐
๐๐+ + ๐๐
โ
under the constrains
๐=1
๐
๐=1
๐ฅ๐๐โ1
๐ค๐๐ โ ๐=1
๐ฆ๐โ1
๐ง๐ โ ๐๐+ + ๐๐
โ = 0
๐=1
๐ผโ1
๐ง๐ = 100
๐=1
๐
๐=1
๐ผ๐โ1
๐ค๐๐ = 100
๐ง๐, ๐ค๐๐ , ๐๐+, ๐๐
โ โฅ 0 โ๐, ๐, ๐, ๐
Numerical example
โข Consider a simple case of customer satisfaction measurement:
โข 3 satisfaction criteria (product, purchase process, and service).
โข Sample of 20 customersโ judgements
โข 3point ordinal satisfaction scale (i.e., dissatisfied, neutral, satisfied).
Numerical example (data)Overall
satisfactionSatisfaction from
criterion 1Satisfaction from
criterion 2Satisfaction from
criterion 3
Neutral Satisfied Neutral Dissatisfied
Dissatisfied Dissatisfied Dissatisfied Dissatisfied
Satisfied Satisfied Satisfied Satisfied
Neutral Satisfied Dissatisfied Neutral
Dissatisfied Dissatisfied Dissatisfied Dissatisfied
Satisfied Satisfied Satisfied Satisfied
Neutral Satisfied Dissatisfied Satisfied
Neutral Satisfied Dissatisfied Satisfied
Neutral Neutral Neutral Neutral
Dissatisfied Dissatisfied Dissatisfied Dissatisfied
Neutral Neutral Satisfied Dissatisfied
Dissatisfied Dissatisfied Dissatisfied Dissatisfied
Satisfied Satisfied Satisfied Satisfied
Neutral Neutral Satisfied Dissatisfied
Dissatisfied Dissatisfied Dissatisfied Dissatisfied
Satisfied Satisfied Satisfied Neutral
Satisfied Satisfied Satisfied Satisfied
Satisfied Satisfied Satisfied Neutral
Neutral Neutral Neutral Neutral
Dissatisfied Neutral Dissatisfied Dissatisfied
Numerical example (variables)
y1
y2 y
3
y*2
0
100
Y*
Y
z1
z2
Dissatisfied Neutral Satisfied
0Y
Dissatisfied Neutral Satisfiedxi
2 xi3
wi1
wi2
biXi*
100bi
bixi*2
Numerical example (modeling)
โข Objective function: ๐=120 ๐๐
+ + ๐๐โ
โข Writing ordinal regression equations, e.g. for customer 1:
โข Criterion 1 (satisfied): ๐ค11 + ๐ค12
โข Criterion 2 (neutral): ๐ค21
โข Criterion 3 (dissatisfied): 0
โข Overall satisfaction (neutral): ๐ง1โข Thus: ๐ค11 + ๐ค12 + ๐ค21 โ ๐ง1 โ ๐1
+ + ๐1โ = 0
โข Normalization constraints:โข Overall value function: ๐ง1 + ๐ง2 = 100
โข Marginal value functions:๐ค11 + ๐ค12 + ๐ค21 + ๐ค22 + ๐ค31 + ๐ค32 = 100
Numerical example (basic LP)
Numerical example (solution)
Variable Value
๐ค11 0
๐ค12 25
๐ค21 25
๐ค22 25
๐ค31 25
๐ค32 0
๐ง1 50
๐ง2 50
๐นโ 0
Stability analysis
โข Problem of multiple or near optimal solutions
โข Stability analysis is considered as a post-optimality problem.
โข This solution is calculated by n LPs (equal to the number of criteria), which maximize the weight of each criterion:
๐๐๐ฅ ๐นโฒ = ๐=1
๐ผ๐โ1
๐ค๐๐
Subject to๐น โค ๐นโ + ๐
All the constrains of the basic LP
Stability analysis
F=F*+ฮต
F=F*
The final solution is obtained by
exploring the polyhedron of near
optimal solutions, which is generated
by the constraints of the basic LP.
Stability analysis
โข The average of these optimal solutions may be considered as the final solution of the problem.
โข In case of instabilityโข A large variation of the provided solutions appears in the
post-optimality analysis, and
โข The final average solution is less representative
Numerical example (post-optimality analysis
๐ค11 ๐ค12 ๐ค21 ๐ค22 ๐ค31 ๐ค32 ๐ง1 ๐ง2
max ๐110.00 22.50 22.50 22.50 22.50 0.00 55.00 45.00
max ๐20.00 23.75 23.75 28.75 23.75 0.00 47.50 52.50
max ๐3 0.00 20.00 20.00 30.00 30.00 0.00 50.00 50.00
Average 3.33 22.08 22.08 27.08 25.42 0.00 50.83 49.17
Results (value functions)
โข These functions show the real value (in a normalized interval 0-100) that customers give for each level of the global or partial ordinal satisfaction scale
โข The global and partial value functions ๐โ and ๐๐โ
respectively, are mentioned as additive and marginal value or utility functions, and their properties are determined in the context of multicriteria analysis
โข They are monotonic, non-decreasing, discrete (piecewise linear) functions
โข The form of these curves indicate if customers are demanding
Results (value functions)Y
* or Xi
*
Y or Xi
Neutral customers
Nutrality refers
to the demanding index
Y* orXi
*
Y or Xi
Demanding customers
Y* or Xi
*
Y or Xi
Non-demanding customers
Results (criteria weights)
โข The criteria weights represent the relative importance of the assessed satisfaction criteria
โข The decision whether a satisfaction dimension is considered important by the customers is also based on the number of assessed criteria
โข The properties of the weights are determined in the context of multicriteria analysis
โข The weights are value trade-offs among the criteria
Results (average satisfaction indices
โข The assessment of a performance norm may be very useful in customer satisfaction analysis
โข The average global and partial satisfaction indices are used for this purpose, and can be assessed according to the following equations:
๐ =1
100
๐=1
๐ผ
๐๐ ๐ฆโ๐
๐๐ =1
100
๐=1
๐ผ๐โ1
๐๐๐๐ฅ๐
โ๐ for ๐ = 1,2, โฆ , ๐
Results (average satisfaction indices
โข These indices are normalized (0-100%)
โข The average satisfaction indices are basically the mean value of the global and partial value functions
pm ฮฎ pik
Y* ฮฎ Xi*
Y ฮฎ Xi
y*m ฮฎ xi*k
Results (average demanding indices
โข The average global and partial demanding customer indices are assessed according to the following equations:
๐ท = ๐=1
๐ผโ1 100 ๐ โ 1๐ผ โ 1
โ ๐ฆโ๐
100 ๐=1๐ผโ1 ๐ โ 1
๐ผ โ 1
for ๐ผ > 2
๐ท๐ = ๐=1
๐ผ๐โ1 100 ๐ โ 1๐ผ๐ โ 1
โ ๐ฅ๐โ๐
100 ๐=1๐ผ๐โ1 ๐ โ 1
๐ผ๐ โ 1
for ๐ผ๐ > 2 and ๐ = 1,2, โฆ , ๐
โข The shape of global and partial satisfaction functions indicates customersโ demanding level
โข These indices represent the average deviation of the estimated value functions from a โnormalโ (linear) function
Results (average demanding indices
Y*
Y
1
1100
ฮฑ
m
my*
my1y y
myฮฑ
m *
1
1100
*y
Results (average demanding indices
โข The average demanding indices are normalized in [-1, 1] and the following possible cases hold:
โข Neutral customers (๐ท = 0 or ๐ท๐ = 0)
โข Demanding customers (๐ท = 1 or ๐ท๐ = 1)
โข Non-demanding customers (๐ท = โ1 or ๐ท๐ = โ1)
โข Demanding indices can be used for customer behavior analysis, and they can also indicate the extent of companyโs improvement efforts: the higher the value of the demanding index, the more the satisfaction level should be improved in order to fulfil customersโ expectations.
Results (average improvement indices)
โข The output of improvement efforts depends on the importance of the satisfaction criteria and their contribution to dissatisfaction
โข The average improvement indices are assessed according to the following equation:
๐ผ๐ = ๐๐ 1 โ ๐๐ for ๐ = 1,2,โฆ , ๐
โข These indices are normalized in [0, 1] and it can be proved that:
โข ๐ผ๐ = 1 ๐๐ = 1 โง ๐๐ = 0
๐ผ๐ = 0 ๐๐ = 0 โจ ๐๐ = 1for ๐ = 1,2,โฆ , ๐
โข These indices can show the improvement margins on a specific criterion
Results (action diagrams)
โข Combining weights and satisfaction indices, a series of โPerformance/Importanceโ diagrams can be developed
โข These diagrams are also mentioned as action, decision, and strategic or perceptual maps; they are very similar to SWOT analysis
โข Each of these maps is divided into quadrants according to performance (high/low), and importance (high/low), that may be used to classify actions.
Results (action diagrams)
โข Status quo: generally, no action is required
โข Leverage opportunity: these areas can be used as advantage against competition
โข Transfer resources: company's resources may be better used elsewhere
โข Action opportunity: these are the criteria/ subcriteria that need attention
Results (action diagrams)
Transfer resources(high performance/low importance)
Status quo(low performance/low importance)
Leverage opportunity(high performance/high importance)
Action opportunity(low performance/high importance)
Low High
Lo
wH
igh
PE
RF
OR
MA
NC
E
IMPORTANCE
Results (action diagrams)
โข There are 2 types of diagrams:โข Raw diagram: it uses the weights and the satisfaction
indices as they are calculated by the MUSA method
โข Relative diagram: the cut-off level for axes is recalculated as the centroid of all points in the diagram; this type of diagram is very useful if points are concentrated in a small area
โข The relative diagrams use the normalized variables ๐๐
โฒ and ๐๐โฒ which can overcome:
โข The assessment problem of the cut-off level for axes, and
โข The low-variation problem for the average satisfaction indices (high competitive market case).
Results (action diagrams)
Action diagram
Axes Variables Interval Cut-off levelfor axes
Raw Importance ๐๐ [0, 1] 1 ๐
Performance ๐๐ [0, 1] 0.5
Relative Importance๐๐
โฒ =๐๐ โ ๐
๐๐ โ ๐2
[-1, 1] 0
Performance๐๐
โฒ =๐๐ โ ๐
๐๐ โ ๐2
[-1, 1] 0
Results (improvement diagrams)
โข The action diagrams can indicate which satisfaction criteria should be improved, but they cannot determine the output or the extent of improvement efforts
โข For this reason, combining improvement and demanding indices, a series of improvement diagrams can be developed
โข Each of these maps is divided into quadrants according to demanding (high/low), and effectiveness (high/low), that may be used to rank improvement priorities
Results (improvement diagrams)
โข 1st priority: this area can indicate improvement actions since these dimensions are highly effective and customers are no demanding
โข 2nd priority: it includes satisfaction dimensions which have either a low demanding index or a high improvement index
โข 3rd priority: it refers to satisfaction dimensions that have small improvement margin and need substantial effort
Results (improvement diagrams)
3rd
priority(large effort/low effectiveness)
2nd
priority(small effort/low effectiveness)
2nd
priority(large effort/high effectiveness)
1st priority
(small effort/high effectiveness)
Low High
Lo
wH
igh
DE
MA
ND
ING
EFFECTIVENESS
Results (improvement diagrams)
Improvement diagram
Axes Variables Interval Cut-off levelfor axes
Raw Effectiveness ๐ผ๐ [0, 1] 0.5
Demanding ๐ท๐ [-1, 1] 0
Relative Effectiveness๐ผ๐โฒ =
๐ผ๐ โ ๐ผ
๐ผ๐ โ ๐ผ2
[-1, 1] 0
Demanding ๐ท๐โฒ [-1, 1] 0
Numerical example (results)
Criterion Weight Averagesatisfaction
index
Average demanding
index
Criterion 1 0.2542 0.5328 0.74
Criterion 2 0.4917 0.4674 0.10
Criterion 3 0.2542 0.5500 -1.00
Overall satisfaction
0.5033 -0.02
Numerical example (results)
0
51
100
0
25
50
75
100
Dissatisfied Neutral Satisfied
Overall satisfaction
0 13.11
100
0
25
50
75
100
Dissatisfied Neutral Satisfied
Criterion 1
0
45
100
0
25
50
75
100
Dissatisfied Neutral Satisfied
Criterion 2
0
100100
0
25
50
75
100
Dissatisfied Neutral Satisfied
Criterion 3
Numerical example (results)
Criterion 2
IMPORTANCE
Low
Low High
Hig
h
PE
RF
OR
MA
NC
E
Criterion 3
Criterion 1
(a) Raw action
diagram
Criterion 2
IMPORTANCE
Low
Low High
Hig
h
PE
RF
OR
MA
NC
E
Criterion 3
Criterion 1
(b) Relative action
diagram
Criterion 2
EFFECTIVENESS
Low
Low High
Hig
h
DE
MA
ND
ING
Criterion 3
Criterion 1(a) Raw improvement
diagram
Criterion 2
EFFECTIVENESS
Low
Low High
Hig
h
DE
MA
ND
ING
Criterion 3
Criterion 1(b) Relative
improvement diagram
Assumptions and inconsistencies
โข Assumptionsโข The existence of an additive value function under
certainty is based on the concept of preferential independence
โข Inconsistenciesโข The most common problem is the lack of consistency for
the collected data
Overall satisfaction
Criterion 1 Criterion 2 Criterion 3
Satisfied Dissatisfied Dissatisfied Dissatisfied
Dissatisfied Satisfied Satisfied Satisfied
Assumptions and inconsistencies
โข Reasons for inconsistenciesโข There is not a consistent family of criteria
โข No โrationalโ decision-makers
โข In the preliminary stage of the MUSA method, a consistency control should be applied:
โข If inconsistencies occur in a small number of customers, these data should not be considered
โข In the opposite case, the set of assessed criteria should be reconsidered
โข Other potential problemsโข Existence of distinguished customer groups with
different preference value systems
โข Consider different customer segments
Evaluation of results (AFI)
โข The fitting level of the MUSA method refers to the assessment of a collective value system for the set of customers (value functions, weights, etc.) with the minimum possible errors
โข The Average Fitting Index (AFI) depends on the optimum error level and the number of customers as well:
๐ด๐น๐ผ = 1 โ๐นโ
100๐where ๐นโ is the minimum sum of errors of the basic LP
โข AFI is normalized in [0, 1], and it is equal to 1 if ๐นโ = 0 (case with zero errors).
Evaluation of results (alternative AFIs)
โข Alternative fitting indicator based on the percentage of customers with zero error variables:
๐ด๐น๐ผโฒ =๐0
๐โข Alternative fitting indicator that takes into account
the maximum values of the error variables for every global satisfaction level, as well as the number of customers that belongs to this level:
๐ด๐น๐ผโฒโฒ = 1 โ๐นโ
๐ ๐=1๐ผ ๐๐max{๐ฆโ๐, 100 โ ๐ฆโ๐}
Evaluation of results (alternative AFIs)
y1
y2
ym
yฮฑ
y*2
0
y*m
100
Y*
Y
100โym
... ...
......
ym
Evaluation of results (other fitting indicators)
โข Variance diagram of the additive value curve: using the estimated errors, the maximum and minimum satisfaction is calculated for each level of the ordinal satisfaction scale
y1 y2 ym yฮฑ
y*2
0
y*m
100
Y*
Y
... ......
...
jj
*
j
* ฯฯYY maxmax
*Y
jj
*
j
* ฯฯYY minmin
Evaluation of results (other fitting indicators)
โข Prediction table of global satisfaction: it refers to a classification for the observed and the predicted global satisfaction judgements
1~y 2~y ฮฑy~
1y
2y
ฮฑy
Nij Rij
Cij
N11 R11
C11
N12 R12
C12
...
jy~
N1j R1j
C1j
...N1ฮฑ R1ฮฑ
C1ฮฑ
N21 R21
C21
N22 R22
C22
...N2j R2j
C2j
...N2ฮฑ R2ฮฑ
C2ฮฑ
...
Na1 Ra1
Ca1
Na2 Ra2
Ca2
...Naj Raj
Caj
...Naฮฑ Raฮฑ
Caฮฑ
... ...Niฮฑ Riฮฑ
Ciฮฑ
Ni1 Ri1
Ci1
Ni2 Ri2
Ci2
iy
...
...
...
Predicted Global Satisfaction Level
Act
ua
l G
loba
l S
ati
sfa
ctio
n L
evel
Evaluation of results (ASI)
โข The stability of the MUSA method depends on the post-optimality analysis results
โข During the post-optimality stage, n LPs are formulated and solved, which maximize repeatedly the weight of each criterion
โข The Average Stability Index (ASI) is the mean value of the normalized standard deviation of the estimated weights:
๐ด๐๐ผ = 1 โ1
๐
๐=1
๐ ๐ ๐=1๐ ๐๐
๐2โ ๐=1
๐ ๐๐๐
2
100 ๐ โ 1
Numerical example (fitting)
โข Fitting indicatorsโข Since ๐นโ = 0, we have ๐๐
+ = ๐๐โ = 0, โ๐
โข Thus, ๐ด๐น๐ผ = ๐ด๐น๐ผโฒ = ๐ด๐น๐ผโฒโฒ = 1
โข Prediction table
Global satisfaction-Predicted
Dissatisfied Neutral Satisfied
Glo
bal
sati
sfac
tio
n-O
bse
rve
d Dissatisfied 30% 0% 0%
Neutral 0% 40% 0%
Satisfied 0% 0% 30%
Numerical example (stability)
โข ASI=0.9177
โข Post optimality results
b1 b2 b3
Max b1 32.50 45.00 22.50
Max b2 23.75 52.5 23.75
Max b3 20.00 50.00 30.00
Extensions (strictly increasing value functions
โข The basic MUSA method assumes that value functions are monotone non-decreasing
โข Introducing strict preference conditions:
๐ฆโ๐ < ๐ฆโ๐+1
๐ฅ๐โ๐ < ๐ฅ๐
โ๐+1 ๐ฆ๐ โบ ๐ฆ๐+1, ๐ = 1,2,โฆ , ๐ผ โ 1
๐ฅ๐๐ โบ ๐ฅ๐
๐+1, ๐ = 1,2,โฆ , ๐ผ๐ โ 1 ๐ = 1,2,โฆ , ๐
โข Additional constraints
๐ฆโ๐+1 โ ๐ฆโ๐ โฅ ๐พ,๐ = 1,2,โฆ , ๐ผ โ 1
๐ฅ๐โ๐+1 โ ๐ฅ๐
โ๐ โฅ ๐พ๐ , ๐ = 1,2,โฆ , ๐ผ๐ โ 1 ๐ = 1,2,โฆ , ๐
where ๐พ, ๐พ๐ are preference thresholds for the value functions ๐โ and ๐๐
โ
Extensions (multiple criteria levels)
โข In several cases multiple criteria levels should be assessed
โข The 1st criteria level contains the general satisfaction dimensions (e.g. personnel)
โข The 2nd criteria level refers to the analytical dimensions of the main criteria (e.g. skills and knowledge of personnel).
โข The assessed hierarchical structure should satisfy the properties of a consistent family of criteria
โข In this extension, the MUSA method can be similarly reformulated in a LP problem following the presented principles and basic concepts
Extensions (multiple criteria levels)
Global satisfaction
2nd Satisfaction
Criterion
n-th Satisfaction
Criterion
1st Satisfaction
Criterion...
Global satisfaction
2nd Satisfaction
Criterion
n-th Satisfaction
Criterion
1st Satisfaction
Criterion...
1.1 Satisfaction
Subcriterion
1.2 Satisfaction
Subcriterion
1.n1 Satisfaction
Subcriterion...
2.1 Satisfaction
Subcriterion
2.2 Satisfaction
Subcriterion
2.n2 Satisfaction
Subcriterion
...
n.1 Satisfaction
Subcriterion
n.2 Satisfaction
Subcriterion
n.nn Satisfaction
Subcriterion
...
(ฮฑ) 1 level of satisfaction criteria
(ฮฒ) 2 levels of satisfaction criteria
Extensions (multiple criteria levels)
Additional variables of the model (2 criteria levels)
in : number of subcriteria for the i-th criterion
ijX : clientโs satisfaction according to the j-th subcriterion of the i-th criterion
(j=1, 2, โฆ, ni, i=1, 2, โฆ, n)
ijฮฑ : number of satisfaction levels for the j-th subcriterion of the i-th criterion
k
ijx : the k-th satisfaction level for the j-th subcriterion of the i-th criterion
(k=1, 2, ..., ฮฑij)
*
ijX : value function of ijX
k
ijx* : value of the
k
ijx satisfaction level
ib : weight for the i-th criterion
ijb : weight for the j-th subcriterion of the i-th criterion
Extensions (multiple criteria levels)
โข In this case the ordinal regression analysis equations for the MUSA method have as follows:
๐โ = ๐=1๐ ๐๐๐๐
โ with ๐=1๐ ๐๐ = 1
๐๐โ =
๐=1๐๐ ๐๐๐๐๐๐
โ with ๐=1๐๐ ๐๐๐ = 1 for ๐ = 1,2, โฆ , ๐
where ๐โ, ๐๐โ, ๐๐๐
โ are normalized in [0, 100]
โข Similarly to the 1 criteria level problem, the additional variables of the LP problem refer to the repeated steps of the subcriteria value functions
๐ค๐๐๐ = ๐๐๐๐๐๐ฅ๐๐โ๐+1 โ ๐๐๐๐๐๐ฅ๐๐
โ๐ , ๐ = 1,2, โฆ , ๐๐ = 1,2, โฆ , ๐๐ ๐ = 1,2,โฆ , ๐ผ๐๐ โ 1
Extensions (multiple criteria levels)
ly.respectivecustomer th -q theof
judgement asubcriteri and criteria global, theis t, t, twhere
0 ,0 ,00
, 0 ,0 ,0
100
100
100
21nd 21 or 0
21 or 0
min
qijqiq
1 1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1 11
q,iฯฯ, ฯฯ
j,k m,iwwz
w
w
z
,M,,q=,n,iฯฯww
,M,,q=ฯฯzw
ฯฯฯฯF
qiqiqq
ijkikm
n
i
n
j
ฮฑ
k
ijk
n
i
ฮฑ
k
ik
ฮฑ
m
m
n
j
qiqi
t
k
ik
t
k
ijk
n
i
t
m
m
t
k
ik
M
q
n
i
qiqi
M
q
i ij
i
i qiqij
qqi
a f
f
subject to
Extensions (alternative post-optimality analyses)
Numerical example (alternative post-optimality analyses)
Extensions (robustness)
โข Generic robust approach:โข Infer a collective preference model
โข Calculate a robustness measure (e.g., ASI)
โข Improve the robustness of the model (i.e., consider additional information):
โข Preferences on criteria importance (Grigoroudis and Siskos, 2010)
โข Interaction among criteria (Angilella et al., 2014)
โข Additional properties regarding the provided results (i.e., average satisfaction/demanding indices)
Extensions (additional properties)
โข Average satisfaction indices:
โข Average demanding indices:
1 1* *
1 1 1 1 2 1 1 2 1
i iฮฑ ฮฑn ฮฑ n ฮฑ m n k
m m k k m k
i i i i i t i it
i m i k m t i k t
S b S p y b p x p z p w
1 11 1 1
1 1 1 1 1
1 1
100( 1) ( 1) ( 1) ( 1)
( 1) ( 1)
i iฮฑ ฮฑฮฑ m k
t it i itn n
m t k t t
i i
i i i i
m ฮฑ z k w ฮฑ w
D b Dฮฑ ฮฑ ฮฑ ฮฑ
Extensions (criteria importance)
โข A customer satisfaction survey may include, besides the usual performance questions, preferences about the importance of the criteria
โข Using such questions, customers are asked either to judge the importance of a satisfaction criterion using a predefined ordinal scale, or rank the set of satisfaction criteria according to their importance
โข Based on such importance questions, each one of the satisfaction criteria can be placed in one of the following categories C1, C2, โฆ, Cq, where C1 is the most important criterion class and Cq is the less important criterion class. Considering that Cl, with l the class index, are ordered in a 0-100% scale, there are Tqโ1 thresholds, which define the rank and, therefore, label each one of the classes
Extensions (criteria importance)
โข The evaluation of preference importance classes Cl is similar to the estimation of thresholds Tl
ฮด ฮด ฮด ฮด
Cq Cqโ1
0% Tqโ1 Tqโ2
...ฮด ฮด
Tl
Cl
ฮด ฮด
Tlโ1
...ฮด ฮด
T2
C2
ฮด ฮด
T1 100%
C1
Extensions (WORT model)
kjiSSw
TT
TT
T
w
Mjni
CbSTw
qlCb
STw
STw
CbSTw
SSF
ijijik
q
n
i
a
kik
a
tqijijqit
lija
tijlit
a
tijlit
a
tijijit
j iijij
i
i
i
i
i
,,,0,,
100
,...,2,1ฮบฮฑฮน,...,2,1
ห,0100
1,...,2,ห,
0100
0100
ห,0100
[min]
21
12
1
1
1
1
1
11
1
1
1
11
1
111
2
Extensions (criteria importance)
โข Using together customersโ performance and importance judgments, an extension of the MUSA method may be modeled as a Multiobjective Linear Programming (MOLP) problem
1
1 1
1
2
min
min
subject to
all the constraints of the basic MUSA and the WORT models
M
j j
j
n M
ij ij
i j
F ฯ ฯ
F S S
Numerical example (criteria importance)
Criterion 1 Criterion 2 Criterion 3
Important Important Important
Important Very important Unimportant
Important Important Important
Important Very important Unimportant
Important Very important Important
Important Very important Unimportant
Very important Important Unimportant
Very important Important Unimportant
Important Very important Important
Important Very important Unimportant
Important Very important Important
Important Important Important
Important Very important Unimportant
Very important Important Important
Important Very important Important
Important Very important Important
Important Very important Important
Unimportant Important Very important
Important Important Important
Important Very important Unimportant
Importance Judgments
Numerical example (criteria importance)
Basic MUSA
model
Compromise
programming
Global
criterion
Heuristic
method
Criterion 1 weight 25.42% 36.63% 36.04% 36.30%
Criterion 2 weight 49.17% 36.69% 37.27% 37.02%
Criterion 3 weight 25.42% 26.68% 26.69% 26.68%
ASI 99.98% 98.81% 99.31%91.77%
Extensions (additional properties)
โข Add additional constraints to the basic LP formulation.
โข If necessary consider these constraints in the following order:
โข Constraint for average satisfaction indices
โข Constraint for average demanding indices
โข In the general case, these constraints may lead to infeasible solutions, thus:
โข They should be modeled using a double error variable
โข In this case, a MOLP approach may be applied (e.g., compromise programming)
Numerical example (additional properties)
MUSA method + Constraint for
averagesatisfaction indices
MUSA method + Constraints for
averagesatisfaction/
demanding indices
ASI +11.06% +11.98%
Concluding remarks
โข The MUSA method is based on the principles of aggregation-disaggregation approach and linear programming modelling.
โข Main advantages of the method:โข It fully considers the qualitative form of customersโ
judgements and preferences, as expressed in a customer satisfaction survey
โข The post-optimality analysis stage gives the ability to achieve a sufficient stability level
โข The provided results are focused not only on the descriptive analysis of customer satisfaction data, but they are also able to assess an integrated benchmarking system
โข It is based on a very flexible modeling
Referencesโข Angilella, S., Corrente S., Greco S., and Sลowiลski R. (2014). MUSA-INT: Multicriteria customer satisfaction analysis with
interacting criteria, Omega, 42 (1), 189-200.
โข Costa, H.G., Mansur A.F.U., Freitas A.L.P., and Carvalho R.D. (2007). ELECTRE TRI aplicado a avaliaรงรฃo da satisfaรงรฃode consumidores, Revista Produรงรฃo, 17 (2), 230-245.
โข Grigoroudis, E. and Politis Y. (2015). Robust extensions of the MUSA method based on additional properties and preferences, International Journal of Decision Support Systems, 1 (4), 438-460.
โข Grigoroudis, E. and Siskos Y. (2002). Preference disaggregation for measuring and analysing customer satisfaction: The MUSA method, European Journal of Operational Research, 143 (1), 148-170.
โข Grigoroudis, E. and Siskos Y. (2010). Customer satisfaction evaluation: Methods for measuring and implementing service quality, Springer, New York.
โข Grigoroudis, E. and Spiridaki O. (2003). Derived vs. stated importance in customer satisfaction surveys, Operational Research: An International Journal, 3 (3), 229-247.
โข Grigoroudis, E., Kyriazopoulos P., Siskos Y., Spyridakos A., and Yannacopoulos D. (2007). Tracking changes of e-customer preferences using multicriteria analysis, Managing Service Quality, 17 (5), 538-562.
โข Grigoroudis, E., Litos C., Moustakis V., Politis Y., and Tsironis L. (2008). The assessment of user perceived web quality: application of a satisfaction benchmarking approach, European Journal of Operational Research, 187 (3), 1346-1357.
โข Grigoroudis, E., Politis Y., and Siskos Y. (2002). Satisfaction benchmarking and customer classification: An application to the branches of a banking organization, International, Transactions in Operational Research, 9 (5), 599โ618.
โข Joรฃo, I.M., Bana e Costa C.A., and Figueira J.R. (2010). An ordinal regression method for multicriteria analysis of customer satisfaction, in: M. Ehrgott, B. Naujoks, T.J. Stewart, and J. Wallenius (eds.), Multiple criteria decision making for sustainable energy and transportation systems, Springer, Berlin Heidelberg, 167-176.
โข Mihelis, G., Grigoroudis E., Siskos Y., Politis Y., and Malandrakis Y. (2001). Customer satisfaction measurement in the private bank sector, European Journal of Operational Research, 130 (2), 347โ360.
โข Siskos, J. (1985). Analyses de rรฉgression et programmation linรฉaire, Rรฉvue de Statistique Appliquรฉe, 23 (2), 41-55.
โข Siskos, Y., Grigoroudis E., Zopounidis C., and Saurais O. (1998). Measuring customer satisfaction using a collective preference disaggregation model, Journal of Global Optimization, 12 (2), 175-195.