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April 10, 2023 1
Hybrid Minimax Reference Point – DEA Approach for
Integrated Bank Performance Integrated Bank Performance Assessment and PlanningAssessment and Planning
Jian-Bo YangProfessor of Decision and System Sciences
Director of Decision and Cognitive Sciences Research Centre
Manchester Business School
The University of Manchester, UK
Tel: +44 161 200 3427 Fax: +44 161 200 3505
Email: [email protected]
Website: www.personal.mbs.ac.uk/jbyang
2
Outline of the PresentationOutline of the Presentation
Performance assessment and planning
Background for bank branch performance
Integrated minimax reference point – DEA
Results and analysis
Graphical interpretation and analysis
Technical efficiency score
Concluding remarks
3
Integrated Performance Integrated Performance Assessment and PlanningAssessment and Planning
Performance assessment Performance measurement – management control
Fairness, objectivity and equity
Data envelopment analysis (DEA)
Performance planning Performance target setting – management planning
decision makers’ preferences considered
Multiple objective linear programming
4
Background for Bank Branch Background for Bank Branch Performance AssessmentPerformance Assessment
A major international bank in the UK Headquarter in London – Performance Director
Hundreds of branches overall the country
14 branches involved in Greater Manchester area
Performance planning and control Improve the bank’s business performances
Take into account Director’s & branch managers’ preferences
Data recorded in the bank’s performance database
5
Performance MeasurePerformance Measure Two outputs selected by the bank
Customer service – Number of customers who rate the branch service as being satisfied
Commercial income – Generated by relationship managers from selling mortgages, bank loans, insurance, and investment products
Five outputs selected by the bank Business reviews – Number of reviews for business clients
completed and their effectiveness ratio Contacts – Number of customer contact promises generated
and the percentage that is already fulfilled Registrations – Number of Internet and telephone banking
accounts opened and activated for the customers Key performance indicators – Saving and lending balances
of the customer accounts Future value added – Amount of leads being converted into
sales in the ensuing months: commercial start-ups, account switches, etc
6
Performance Data of the BankPerformance Data of the Bank
INPUTS OUTPUTS
Business Customer Commercial
DMU Branch Review Contacts Registrations KPI FVA Service Income
1 Oldham Road 60 16 40 38 190 88 200
2 Trafford Park 60 20 50 39 225 88 91
3 King Street 47 30 39 29 228 102 111
4 Swinton 60 15 38 22 164 93 143
5 Royal Exchange 60 23 44 34 190 80 101
6 High Street 60 26 37 42 98 89 173
7 University 60 30 44 29 140 78 140
8 Clayton 60 30 25 31 130 98 155
9 Oxford Road 47 13 50 32 140 88 132
10 Stretford 51 27 34 28 115 80 130
11 Didsbury 56 21 42 26 108 80 134
12 Chorlton 58 27 16 29 82 94 137
13 Eccles 58 22 30 31 142 92 71
14 Salford 60 25 35 32 97 82 132
7
Management Control and Management Control and PlanningPlanningOutput-Oriented CCR Dual ModelOutput-Oriented CCR Dual Model
0
00
0
1
1
0 .
ij
n
jijj
n
jrjjrjj
jo
xx
yyts
hMax
sr ,,1
njallforj ,,1 ,0
nj ,,10
00
1
he
mi ,,1
8
Efficiency Score by CCR ModelEfficiency Score by CCR Model
DMU Branch Efficiency 1 2 3 4 5 6 7 8 9 10 11 12 13 14
1 Oldham Road 100.0% 1.000
2 Trafford Park 81.5% 0.006 0.899 0.301
3 King Street 100.0% 1.000
4 Swinton 100.0% 1.000
5 Royal Exchange 74.2% 0.207 0.170 0.497 0.288
6 High Street 100.0% 1.000
7 University 89.5% 0.246 0.371 0.397
8 Clayton 100.0% 1.000
9 Oxford Road 100.0% 1.000
10 Stretford 95.6% 0.107 0.477 0.137 0.164
11 Didsbury 99.2% 0.090 0.265 0.577
12 Chorlton 100.0% 1.000
13 Eccles 95.7% 0.211 0.256 0.574
14 Salford 89.2% 0.120 0.302 0.011 0.563
9
Management Control and Management Control and PlanningPlanningOutput-Oriented CCR Dual ModelOutput-Oriented CCR Dual Model
B
C
A
D
E
O
Output 2
Output 1
E1
10 OE
OEe
Efficiency Score for the observed DMU:(Decision Making Unit)
C1
A1
MPS
10
Management Control and Management Control and Planning byPlanning byThe Reference Point ModelsThe Reference Point Models
01
1
*
.
ij
n
jijj
n
jrjjrr
o
xx
yfwts
hMin
sr ,,1
njallforj ,,1 ,0
nj ,,10
0
max
0max0 ,
1jF
hFe
mi ,,1
11
Management Management Control Control and and
PlanningPlanningEquivalence of CCR Dual and Equivalence of CCR Dual and Super-Ideal Point Super-Ideal Point
ModelsModels
orjr yw 1Special
weightT
jj yyFf ],[00 21
max*
}{max0
0
1
max
rj
rj
sr y
fF
Super-ideal point
n
jrjjjrj yff
j 1
*
000
max)(
Ideal value
12
Management Control and Management Control and
PlanningPlanningIllustration of the Super-Ideal Point Illustration of the Super-Ideal Point
ModelModel
B
A
D
E
O
Output 2
Output 1
C
0jf
E1
10 OE
OEe *f
Tjj yyE ],[00 21
Tjjj yyE ],[000 211
Tjj yyFf ],[00 21
max*
13
Efficiency Score Generated by Efficiency Score Generated by Super-Ideal Point ModelSuper-Ideal Point Model
D M U D E A S c o r e x 1 x 2 x 3 x 4 x 5 y 1 y 2
1 1 0 0 . 0 % 1 . 0 0 0 1 . 0 8 8 0 . 0 8 8 6 0 . 0 1 6 . 0 4 0 . 0 3 8 . 0 1 9 0 . 0 8 8 . 0 2 0 0 . 0 0 . 0 8 8
2 8 1 . 5 % 1 . 2 2 8 2 . 1 9 8 0 . 9 7 0 6 0 . 0 2 0 . 0 5 0 . 0 3 7 . 7 1 5 1 . 9 1 0 8 . 0 1 6 0 . 6 0 . 9 7 0
3 1 0 0 . 0 % 1 . 0 0 0 1 . 3 8 7 0 . 3 8 7 4 7 . 0 3 0 . 0 3 9 . 0 2 9 . 0 2 2 8 . 0 1 0 2 . 0 1 1 1 . 0 0 . 3 8 7
4 1 0 0 . 0 % 1 . 0 0 0 1 . 0 0 0 0 . 0 0 0 6 0 . 0 1 5 . 0 3 8 . 0 2 2 . 0 1 6 4 . 0 9 3 . 0 1 4 3 . 0 0 . 0 0 0
5 7 4 . 2 % 1 . 3 4 7 1 . 8 3 9 0 . 4 9 2 6 0 . 0 2 3 . 0 4 4 . 0 3 4 . 0 1 6 8 . 3 1 0 7 . 7 1 5 2 . 4 0 . 4 9 2
6 1 0 0 . 0 % 1 . 0 0 0 1 . 1 0 8 0 . 1 0 8 6 0 . 0 2 6 . 0 3 7 . 0 4 2 . 0 9 8 . 0 8 9 . 0 1 7 3 . 0 0 . 1 0 8
7 8 9 . 5 % 1 . 1 1 8 1 . 3 0 1 0 . 1 8 4 6 0 . 0 2 0 . 2 3 0 . 3 2 9 . 0 1 4 0 . 0 9 3 . 4 1 5 6 . 5 0 . 1 8 4
8 1 0 0 . 0 % 1 . 0 0 0 1 . 0 5 7 0 . 0 5 7 6 0 . 0 3 0 . 0 2 5 . 0 3 1 . 0 1 3 0 . 0 9 8 . 0 1 5 5 . 0 0 . 0 5 7
9 1 0 0 . 0 % 1 . 0 0 0 1 . 1 5 3 0 . 1 5 3 4 7 . 0 1 3 . 0 5 0 . 0 3 2 . 0 1 4 0 . 0 8 8 . 0 1 3 2 . 0 0 . 1 5 3
1 0 9 5 . 6 % 1 . 0 4 6 1 . 1 1 9 0 . 0 7 3 5 1 . 0 2 2 . 2 2 5 . 7 2 8 . 0 1 1 5 . 0 8 3 . 6 1 3 5 . 9 0 . 0 7 3
1 1 9 9 . 2 % 1 . 0 0 8 1 . 1 1 4 0 . 1 0 6 5 4 . 8 2 1 . 0 2 2 . 9 2 6 . 0 1 0 8 . 0 8 6 . 9 1 3 5 . 0 0 . 1 0 6
1 2 1 0 0 . 0 % 1 . 0 0 0 1 . 0 0 0 0 . 0 0 0 5 8 . 0 2 7 . 0 1 6 . 0 2 9 . 0 8 2 . 0 9 4 . 0 1 3 7 . 0 0 . 0 0 0
1 3 9 5 . 7 % 1 . 0 4 5 2 . 2 8 7 1 . 2 4 2 5 8 . 0 2 2 . 0 3 0 . 0 2 9 . 5 1 1 7 . 5 9 6 . 1 1 4 2 . 6 1 . 2 4 2
1 4 8 9 . 2 % 1 . 1 2 1 1 . 1 7 1 0 . 0 4 9 5 8 . 5 2 5 . 0 2 5 . 3 3 2 . 0 9 7 . 0 9 1 . 9 1 4 8 . 0 0 . 0 4 9
D E A d u a l m o d e l M i n i m a x m o d e l
C o m p o s i t e i n p u t s a n d o u t p u t s
maxF0j
0
maxjF
14
Management Control and Management Control and PlanningPlanningIdeal Point Model to Get Ideal Point Model to Get MPSMPS
0
0
1
1
.
ij
n
jijj
n
jrjjrjr
o
xx
yfwts
hMin
sr ,,1
njallforj ,,1 ,0
mi ,,1
15
Payoff Table for Maximum Outputs Payoff Table for Maximum Outputs of all Branchesof all Branches
DMU Branch y1 y2 y1 y2 y1 y2 y1 y2
1 Oldham Road 95.75 146.20 88.00 200.00 95.75 200.00 88.00 146.20
2 Trafford Park 108.03 160.58 88.00 200.00 108.03 200.00 88.00 160.58
3 King Street 102.00 111.00 69.17 153.94 102.00 153.94 69.17 111.00
4 Swinton 93.00 143.00 93.00 143.00 93.00 143.00 93.00 143.00
5 Royal Exchange 107.75 152.36 89.25 185.75 107.75 185.75 89.25 152.36
6 High Street 98.61 144.43 89.00 173.00 98.61 173.00 89.00 144.43
7 University 101.51 142.06 93.40 156.51 101.51 156.51 93.40 142.06
8 Clayton 103.61 141.78 94.43 156.33 103.61 156.33 94.43 141.78
9 Oxford Road 88.00 132.00 67.82 152.13 88.00 152.13 67.82 132.00
10 Stretford 89.53 124.10 79.03 141.92 89.53 141.92 79.03 124.10
11 Didsbury 89.11 132.41 86.85 135.05 89.11 135.05 86.85 132.41
12 Chorlton 94.00 137.00 94.00 137.00 94.00 137.00 94.00 137.00
13 Eccles 96.13 142.64 89.25 162.36 96.13 162.36 89.25 142.64
14 Salford 96.00 140.70 90.79 149.11 96.00 149.11 90.79 140.70
Max o1 Max o2 Maximum values Minimum values
16
Management Control and Management Control and PlanningPlanningIdeal Point Model for Target Setting Ideal Point Model for Target Setting ((MPSMPS))
E1
B
A
D
E
O
Output 2
Output 1
C
0jf
Tjjj fff ],[000 21
tu
tuGradient projection
tNtNNormal vectortM
tMUtility gradient
1
01 E
rrjr ffw
17
Management Control and Management Control and PlanningPlanningInteractive Tradeoff Analysis Interactive Tradeoff Analysis ProcessProcess
Normal vectorTt
sstrr
tt wwwN ],....,,...,[ 11
Gradient projectiont
tTt
tTttt N
NN
NMMu
)(
)(
Tts
tr
tt MMMM ] ,1[ 1112 (proportional to utility gradient)Marginal rate of substitution
trr
ttt
r w
wdfdf
11
1Optimal indifference tradeoff
)()(
)()(*
*1111
rt
rt
r
tttr fff
fffw
Weight update equation
tr
ttr df
dfM 1
1 Indifference tradeoff(Required from DM)
18
Indifference Trade-off Direction for Indifference Trade-off Direction for University BranchUniversity Branch
Optimal indifference tradeoff for 1 unit change of f1
Original (97.21, 152.20) <=> (97.21-1, 152.20+2.24)
New (97.21, 152.20) <=> (97.21-1, 152.20+3.50)
T0.13] -0.06,[ f
Trade-off information provided by decision maker
Trade-off direction given by the projection of indifference trade-offs
i.e.: commercial income should be improved at the expense of customer service to improve the utility of the University branch manager
19
Indifference Trade-off Step Size for Indifference Trade-off Step Size for University BranchUniversity Branch
Trade-off step size chosen by decision maker, assuming that minimum lower bound for customer service is 96.00 0
C@=10
@ f1 f2
0 97.21 152.20
0.1 97.01 152.63
0.2 96.82 153.06
0.3 96.63 153.49
0.4 96.43 153.92
0.5 96.24 154.36
0.6 96.05 154.79
0.7 95.86 155.22
0.8 95.66 155.65
0.9 95.47 156.08
1.0 95.28 156.51
= 0.62
20
New Efficient Solution – Target for New Efficient Solution – Target for University BranchUniversity Branch
1 [ T]0.117, 0, 0, 0.296, 0, 0, 0, 0.429, 0, 0, 0, 0.163, 0, 0
Tf 154.64] 95.23,[)( 1
1N [ T]0.775, 0.757
Proportion of efficient DMUs as benchmark for University Branch
New target outputs for University Branch
New improvement basis – Normal vector
21
DEA and MPS Target Values for DEA and MPS Target Values for UniversityUniversity BranchBranch
INPUTS OUTPUTS
Business Customer Commercial
Review Contacts Registrations KPI FVA Service Income
Evaluated unit 60.00 30.00 44.00 29.00 140.00 78.00 140.00
DEA targets 60.00 20.21 30.26 29.00 140.00 93.40 156.51
Improvement 0.00 9.79 13.74 0.00 0.00 15.40 16.51
% 0.0% 32.6% 31.2% 0.0% 0.0% 19.7% 11.8%
MPS targets 60.00 23.19 29.40 29.00 140.00 95.01 154.86
Improvement 0.00 6.81 14.60 0.00 0.00 17.01 14.86
% 0.0% 22.7% 33.2% 0.0% 0.0% 21.8% 10.6%
22
DEA and MPS Target Values for AllDEA and MPS Target Values for All Branches in Greater ManchesterBranches in Greater Manchester
INPUTS OUTPUTS
Business Customer Commercial
DMU Branch Review Contacts Registrations KPI FVA Service Income
x1 x2 x3 x4 x5 y1 y2
1 Oldham Road 60.00 16.00 40.00 34.63 182.53 90.00 186.13
2 Trafford Park 60.00 16.05 41.99 38.24 189.06 90.04 197.37
3 King Street 47.00 15.64 32.64 29.00 162.07 75.09 146.20
4 Swinton 60.00 15.00 38.00 22.00 164.00 93.00 143.005 Royal Exchange 60.00 17.30 40.99 34.00 190.00 93.00 180.54
6 High Street 59.67 26.00 28.54 35.36 98.00 94.99 155.20
7 University 60.00 23.19 29.40 29.00 140.00 95.01 154.86
8 Clayton 60.00 24.42 25.00 31.00 122.10 94.99 156.12
9 Oxford Road 46.88 13.00 47.07 31.63 140.00 84.95 135.04
10 Stretford 51.00 20.45 23.32 28.00 115.00 80.00 140.93
11 Didsbury 54.93 21.00 23.33 26.00 108.00 87.00 134.92
12 Chorlton 58.00 27.00 16.00 29.00 82.00 94.00 137.0013 Eccles 58.00 20.90 29.66 31.00 142.00 90.00 161.67
14 Salford 58.49 25.00 25.29 32.00 97.00 92.00 147.93
23
Group MPS Target Values for Group MPS Target Values for AllAll Branches in Greater ManchesterBranches in Greater Manchester
Composite DMU Branch input
y1 y2 y1 y2 y1 y2
1 Oldham Road 1.00 90.00 186.13 0.08 0.09 89.86 156.742 Trafford Park 1.23 73.33 160.75 0.06 0.083 King Street 1.00 75.09 146.20 0.06 0.074 Swinton 1.00 93.00 143.00 0.08 0.075 Royal Exchange 1.35 69.05 134.05 0.06 0.076 High Street 1.00 94.99 155.20 0.08 0.087 University 1.12 84.98 138.52 0.07 0.078 Clayton 1.00 94.99 156.12 0.08 0.089 Oxford Road 1.00 84.95 135.04 0.07 0.0710 Stretford 1.05 76.51 134.78 0.06 0.0711 Didsbury 1.01 86.32 133.87 0.07 0.0712 Chorlton 1.00 94.00 137.00 0.08 0.0713 Eccles 1.04 86.12 154.71 0.07 0.0814 Salford 1.12 82.05 131.94 0.07 0.06
efficiency output efficiency output GMPSMariginal Normalised marginal
24
Identify Local MPS based on Group Identify Local MPS based on Group MPSMPSShortest Distance Model for Shortest Distance Model for LMPSLMPS
01
1
1
-
.
ij
n
jijj
n
jrjj
GMPSrr
n
jrjj
GMPSrr
xx
dyfw
dyfwts
dMin
sr ,,1
njallforj ,,1 ,0
mi ,,1
25
Graphical Interpretation Graphical Interpretation of the of the Shortest Distance Model for Shortest Distance Model for LMPSLMPS
E
B
A
D
O
Output 2
Output 1
C
E1
2GMPSf
2GMPSfA1
B1
1GMPSf
1GMPSf
Group MPS generated using Group MCDA techniques
26
Management Control and Management Control and PlanningPlanningWeights and LMPS of University Weights and LMPS of University branchbranch
w1 0.67 y1 89.86 y1 90.24
w2 0.33 y2 156.74 y2 155.98
LMPS Branch x1 x2 x3 x4 x5 y1 y2
7 University 60.0 20.0 36.9 29.0 140.0 90.2 156.0
GMPSWeights Local MPS
27
Management Control and Management Control and PlanningPlanningGenerating LMPS for all DMUGenerating LMPS for all DMU
INPUTS OUTPUTS
Business Customer Commercial Test of
DMU Branch Review Contacts Registrations KPI FVA Service Income Efficiency
x1 x2 x3 x4 x5 y1 y2 %
1 Oldham Road 57.5 16.0 40.0 30.1 161.6 91.4 159.4 98.4%
2 Trafford Park 59.6 18.2 36.8 31.0 156.6 93.8 163.7 95.8%
3 King Street 47.0 18.7 34.0 29.0 176.1 80.8 138.7 100%
4 Swinton 60.0 15.0 38.0 22.0 164.0 93.0 143.0 100%
5 Royal Exchange 59.5 17.6 37.0 30.1 158.8 93.1 162.4 96.5%
6 High Street 58.2 25.1 27.3 35.1 98.0 89.9 156.7 100%
7 University 59.6 20.0 30.2 29.0 140.0 92.6 156.3 100%
8 Clayton 57.8 22.2 25.0 31.0 121.2 90.4 155.7 100%
9 Oxford Road 46.7 13.0 43.3 31.1 140.0 81.0 139.0 100%
10 Stretford 51.0 21.4 23.8 28.0 115.0 81.2 139.4 100%
11 Didsbury 54.8 21.0 22.9 26.0 108.0 86.9 135.0 100%
12 Chorlton 58.0 27.0 16.0 29.0 82.0 94.0 137.0 100%
13 Eccles 58.0 20.2 30.0 30.9 136.0 90.9 158.5 98.9%
14 Salford 57.6 25.0 22.5 32.0 97.0 90.8 149.1 100%
28
Management Control and Management Control and PlanningPlanningTarget Setting and InterpretationTarget Setting and Interpretation
University Branch – Target Setting and Resource Allocation INPUTS OUTPUTS
Business Customer Commercial
Review Contacts Registrations KPI FVA Service Income
Evaluated unit 60.00 30.00 44.00 29.00 140.00 78.00 140.00
DEA composite unit 60.00 20.21 30.26 29.00 140.00 93.40 156.51
Improvement 0.00 9.79 13.74 0.00 0.00 15.40 16.51
% 0.0% 32.6% 31.2% 0.0% 0.0% 19.7% 11.8%
MPS targets 60.00 23.19 29.40 29.00 140.00 95.01 154.86
Improvement 0.00 6.81 14.60 0.00 0.00 17.01 14.86
% 0.0% 22.7% 33.2% 0.0% 0.0% 21.8% 10.6%
LMPS efficient targets 59.58 19.99 30.23 29.00 140.00 92.64 156.35
Improvement 0.42 10.01 13.77 0.00 0.00 14.64 16.35
% 0.7% 33.4% 31.3% 0.0% 0.0% 18.8% 11.7%
29
Management Control and Management Control and PlanningPlanningGraphic Explanation of Data EnvelopGraphic Explanation of Data Envelop
0
20
40
60
80
100
120
140
160
180
0 20 40 60 80 100 120
123
26
25
7
6
24
1922, 20, 21
151110 12
16
49
14
1718
13
8
53
2
m.c.
minimax contour (m.c.)
30
Management Control and Management Control and PlanningPlanningGraphic Explanation of Target SettingGraphic Explanation of Target Setting
130
135
140
145
150
155
160
70 75 80 85 90 95 100 105
2524
19
2220
21
157
7’
m.c.
1f
2f
0f
2f
1f
)19,77,0/(7,07 TES
= 0.877
Technical efficiency score
31
Inefficient DMU Over estimation
2 0.815 0.636 0.179
5 0.742 0.688 0.054
7 0.895 0.877 0.018
10 0.956 0.956 0
11 0.992 0.957 0.035
13 0.957 0.618 0.339
14 0.892 0.892 0
Management Control and Management Control and PlanningPlanningTechnical Efficiency ScoreTechnical Efficiency Score
ieiTES
32
Concluding RemarksConcluding Remarks
Performance assessment and planning
Background for bank branch performance
Integrated minimax reference point – DEA
Results and analysis
Graphical interpretation and analysis
Technical efficiency score
