Implementing of Analytic Hierarchy Process in Banking

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A C T A U N I V E R S I T A T I S L O D Z I E N S I S FOLIA OECONOMICA 152, 2000

C z e s l a w D o m a ń s k i *, J a r o s ł a w K o n d r a s i u k *

IM P L E M E N T IN G O F A N A LY TIC H IE R A R C H Y P R O C E S S IN BANKING

Abstract. The Analytic Hierarchy Process (AHP) is a multicriteria decision support method created by Thomas L. Saaty. It provides both individual and group decision makers an objective way for reaching an optimal decision. The AHP is designed to select the best from a number of alternatives evaluated with respect to several criteria. It is taken by carrying out pairwise comparison judgements which are used to develop overall priorities for ranking the alternatives. This method allows for some level of inconsistency in judgements (that is unavoidable in practice) and provides some measures for limiting that. Our article describes classical Saaty solution to the AHP problem and shows the application of the AHP in establishing the price of the bank deposits.

L PREFRACE

T h e A n aly tic H ierarc h y Process (A H P ) is a m u ltic irite ria decision su p p o rt m eth o d th a t provides both individual and g ro u p decision m ak ers an objective way for reaching an optim al decision. T h e A H P is designed to select the best one from a n u m b er o f altern ativ e s ev a lu a ted w ith respect to several criteria. It is tak en by carrying o u t pairw ise co m p ariso n ju d g em en ts which are used to develop overall priorities fo r ra n k in g th e alternativ es. T h e m ethod allows fo r som e level o f inconsistency in ju d gem ents (the is unavoidable in practice) and provides som e m easures for lim iting th a t. O riginally the A H P m ethod was created by T h o m a s L. S aaty w ho is still deeply engaged in developm ent o f applicatio n s o f this m eth o d .

In o u r p ap e r we have used Expert Choice For Windows 9.0 (E .C . 9.0) - a softw are developed by Ľ rnest H . F o rm a n for carry in g ou t calcu lation s.

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H ow ever, we checked these calculation via Excel 97 reaching the sam e results. T h e E.C. 9.0 enables decision m akers to sort o u t effectively the com plexity and assist with the subjectivity th a t is inherent in m an y decisions. T his so ftw are allow s decision m akers to build their m odels in the Evaluation

and Choice co m po nent o r use Structuring to visually organize the decision

elem ents and build a hierarchy with d rag and d ro p ease. A fter building the m odel, decision elem ents and build a hierarchy w ith d ra g and d ro p ease. A fter building the m odel, decision m ak ers can choose tw o different m easurem ent options depending on w hether alternatives should be com pared a g a in st each o th e r (relativ e m e a su re m e n t) o r ra te d a g a in s t s ta n d a rd s (absolute m easurem ent). O ften the Ratings ap p ro ach is a p p ro p ria te when large n u m b ers o f alternatives arc involved. It is also w ithin E v alu a tio n and C hoice th a t users will enter their judgem ents ab o u t the relative im p o rtan ce o f the criteria and alternatives, synthesize to get results, and co n d u c t sensitivity analyses.

2. ESTABLISHING THE PROBLEM

I he A H P is a general theory o f preference m easurem en t w ith pro viding necessary inform ation for choosing the best decision.

In th e A H P process there are fo u r m ain stages: 1. B uilding a hierarchy m odel.

2. Identifying the preferences o f decision m akers. 3. Synthesis.

4. Sensitivity analyses.

T h e basic A H P m odel consist o f three levels: goal, crite ria level and alternatives. D epending on com plexity o f the problem it is possible to add as m any as necessary levels o f subcriteria.

T he m ost com plex problem is identification o f decision m ak er preferences. In A H P it is done by collecting inform ation ab ou t pairwise judgem ents du e to a goal (for criteria), a specified criterion (for alternatives o r su bcriteria) o r a subcriterion (for alternatives). T here are a few possible scales o f converting collected inform ation into num eric form - however it is no t always necessary. H aving one set o f in fo rm atio n we build a m atrix o f ra tio n co m p ariso n for a given goal/criterion. It is possible to find m an y ways o f co nv erting the m atrix A (m atrix o f ratio com parison) into the vector o f priorities w. However, the need o f consistency m akes us choose the eigenvalue form ulation Aw = mv. A ssum ing th at the priorities w = (wl5 ..., w„ ) 7 with respect to a single criterion are know n, such as the weights o f stones - we can exam ine w h a t we have to d o to recover them . H av in g th e m a trix A:

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w w_г w w A = n w,л W, 2 W,n W n we m ultiply it on the right by w

w =

w,_П

to o b tain mv. Elem ents аи o f the m atrix o f ra tio com p ariso n represent the im p o rtan ce o f alternative i over alternative j. In o rd e r to g u aran tee the ju dgem ents to be consistent, relevant g roups o f the m a trix elem ents have to follow the eq u atio n : au ajk = aik. In case, we d o n o t have a scale at all, o r d o no t have it conveniently as in the case o f som e m easu rin g devices

we can only giver an estim atio n o f wj wj . It leads to the problem :

where Amax is the principal eigenvalue o f A* = (a,*) the p ertu rb ed value A = (a y) w ith the reciprocal a*, = 1/a,* forced. T h e so lu tion is o b tain ed by raising the m atrix to sufficiently large pow er - th en sum m ing over th e row s and norm alizing to o b tain the p rio rity vector w* = (w j, w*)T. T h e abov e

m entioned process is stopped when the difference betw een co m p o n en ts o f the p riority vector o b tain ed at k-th pow er and at the ( / c + l ) - s t pow er is less th a n som e prederm ined small value. T h e vector o f priorities is th e derived scale associated w ith the m atrix o f com parisons. T h e value zero in this scale is assigned to an elem ent th a t is n o t co m p arab le w ith the elem ents considered. W ith the eigenvector for n«$3 norm alizing the geom etric m eans o f the rows leads to an ap p ro x im a tio n to the priorities. In all the cases it is possible to get an ap p ro x im a tio n by n orm alizing the elem ents o f each colum n o f the judg em en t m atrix and th en averaging over ech row. H ow ever, it is im p o rtan t to rem em ber th a t such stem ps can lead to ra n k reversal (in spite o f closeness o f the eigenvector solution). A sim ple way to o b tain the exact value (or an estim ate) o f Amax w hen the exact value o f

w* is availabe in norm alized form is to add the colum ns o f A* and m ultip ly

the resulting vector by the prio rity vector w.

After obtaining the principal eigenvector estim ate w we should consider the q u estio n o f consistency. T h e problem arises from the fact th a t th e original m atrix A need n o t to be transitive, fo r exam ple A x m ay be preferred to A 2 and A 2 to A 3 but A 3 m ay be preferred to A v T h e solution to this problem is the consistency index (C .I.) o f a m a trix o f co m pariso n defined as:

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П- 1

T h e consistency ratio n (C .R .) is obtained by com p arin g the C .I. w ith the a p p ro p ria te one o f the follow ing set o f num b ers (T ab. 1) each o f which is an average ran d o m consistency index derived from a sam ple o f ran do m ly generated reciprocal m atrices. T h e study o f the problem an d revision o f

J C .I.

th e ju d g em en ts should be com pleted if ^ 0.1 0. К .1.

T a b l e 1 Average Random Consistency Index (R.I.) ( S a a t y , 1986, p. 9)

n 1 2 3 4 5 6 7 8 9 10

Random Consistency

Index (R.I.) 0 0 0.52 0.89 1.11 1.25 1.35 1.40 1.45 1.49

T h e above solution to the problem is considered to be classical Saaty solu tio n ( S a a t y , 1994, p. 7-9 ) and is used for reaching b o th local and global vectors o f p riorities - necessary for synthesis.

H ierarchic synthesis is ob tain ed by a process o f w eighting and ad d in g dow n the hierarchy leading to m ultilinear form . T here are tw o possible m o d es o f the synthesis:

• the d istributive m ode in which the principal eigenvector is norm alized to yield a unique estim ate o f ra tio scale underlying the jud gem ents;

• the ideal m o d e in which the norm alized values o f altern ativ es for each criterion are divided by the value o f the highest ra te alternative.

T h e final step is sensitivity analysis th a t gives an answ er to the q u estio n w hether the alternative chosen as the best w ould be changed in case o f m odifying crite ria/su b criteria preferences.

3. APLICATION OF THE AHP METHOD IN ESTABLISHING THE PRICE OF THE BANK DEPOSITS

In this chapter, we would like to describe the ap p licatio n o f A H P in establishing the price o f the ban k deposits. By establishing th e price we consider the change o f present deposit rates. T he below presented m echanism was experimentally implemented in one of the smallest Polish banks. A ccording to th e A H P m ethodology the first step was getting expert know ledge o f present process o f establishing the deposit rates. T he next step was structuring the A H P hierarchy - the final version o f the stru ctu re is presented in Fig. 1.

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Change o f Deposit Rate

C om petition

Fig. 1. The three level hierarchy used for changing deposit rate o f the bank

N> Im p le m en tin g of A n al y tic H ie ra rc h y P ro ce ss in B a n k in g

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• C O M P E T IT IO N - precisely, it is m ark e tin g p o in t o f view o n pricing d eposits according to deposit rates o f com petitive b anks o f “ o u r ” b ank;

• M A R K E T - it is treasu ry p oin t o f view, including possible buying bank d eposits (and altern ativ e costs);

• PL A N - financial planning an d prognosis o f fu tu re benefits an d costs o f the bank;

• P O R T F O L IO - present assets p o rtfo lio o f the ban k as th e m easu re o f efficiency o f the alread y acquired deposits.

In o rd e r to sim plify the u n d erstan d in g o f the graph we have decided to use sh o rt acronym s for alternatives instead o f sym bols (Л,) used in the next chap ter.

D u e to suggestions o f the decision m akers, we have decided to lim it possible alternatives to changes o f the average d ep o sit ra te , w ith alternativ es as follows:

A j - incercasing the average deposit rate o f the bank by 1.00% , A 2 - increasing the average deposit rate o f th e b an k by 0.75% , A 3 - increasing the average deposit rate o f the ban k by 0.50 % , A 4 - increasing the average deposit rate o f th e b an k by 0.25% , A s - leaving the deposit rate w ithout any change,

A 6 - decreasing the average deposit ra te o f the b an k by 0.2 5% , A n - decreasing the average deposit ra te o f the bank by 0 .5 0% ,

/18 - decreasing the average deposit rate o f the ban k by 0 .75 % ,

A 9 - decreasing the average deposit rate o f the b an k by 1.00% .

4. EXPERIMENTAL SOLUTION TO THE PROBLEM USING E X P ER T CHOICE

FOR W INDOW S 9.0

P rim arly, all the d a ta and calculations were collected using Expert

Choice For Windows 9.0. In the next step the calcu lation s were checked

using E x c el 97. T ables from 2 to 6 co n tain collected in fo rm a tio n a b o u t pairw ise com p ariso n ju dgem ents in the form described in C h a p te r 2.

T a b l e 2 Pairwise comparison matrix of criteria

Competition Market Plan Portfolio

Competition 1 0 /1 0 90/10 30/10 80/10

Market 10/90 1 0 /1 0 10/80 1 0 /2 0

Plan 10/30 80/10 1 0 /1 0 50/10

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T a b l e 3 Pairwise comparison matrix of alternatives according to the criterion PLAN

a, a1 A , A* A , A e A , _A% a9 Ax 1 0 /1 0 1 0 /2 0 10/30 10/40 10/60 10/70 10/80 10/90 10/90 A , 2 0 /1 0 1 0 /1 0 1 0 /2 0 10/30 10/50 10/70 10/80 10/90 10/90 a, 30/10 2 0 /1 0 1 0 /1 0 1 0 /2 0 10/40 10/60 10/80 10/90 10/90 A , 40/10 30/10 2 0 /1 0 1 0 /1 0 10/15 10/35 10/55 10/75 10/80 A , 60/10 50/10 40/10 15/10 1 0 /1 0 10/15 10/30 10/45 10/50 A f 70/10 70/10 60/10 35/10 15/10 1 0 /1 0 10/15 10/30 10/30 A y 80/10 80/10 80/10 55/10 30/10 15/10 1 0 /1 0 1 0 /2 0 1 0 /2 0 90/10 90/10 90/10 75/10 45/10 30/10 2 0 /1 0 1 0 /1 0 1 0 /1 0 A 9 90/10 90/10 90/10 80/10 50/10 30/10 2 0 /1 0 1 0 /1 0 1 0 /1 0 T a b l e 4 Pairwise comparison matrix of alternatives according to the criterion COMPETITION

A , A j A 3 A* A , A„ /1? A» Aq A t 1 0 /1 0 1 2 /1 0 14/10 16/10 18/10 2 0 /1 0 2 2 /1 0 23/10 30/10 A , 1 0 /1 2 1 0 /1 0 1 2 /1 0 15/10 18/10 2 0 /1 0 2 2 /1 0 2 2 /1 0 25/10 A , 10/14 1 0 /1 2 1 0 /1 0 1 2 /1 0 16/10 18/10 19/10 19/10 2 1 /1 0 A4 10/16 10/15 1 0 /1 2 1 0 /1 0 1 2 /1 0 16/10 17/10 18/10 2 0 /1 0 A , 10/18 10/18 10/16 1 0 /1 2 1 0 /1 0 1 2 /1 0 14/10 15/10 17/10 A , 1 0 /2 0 1 0 /2 0 10/18 10/16 1 0 /1 2 1 0 /1 0 1 2 /1 0 13/10 15/10 A 7 1 0 /2 2 1 0 /2 2 10/19 10/17 10/14 1 0 /1 2 1 0 /1 0 13/10 15/10 ■^8 10/23 1 0 /2 2 10/19 10/18 10/15 10/13 10/13 1 0 /1 0 14/10 A q 10/30 10/25 1 0 /2 1 1 0 /2 0 10/17 10/15 10/15 10/14 1 0 /1 0

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T a b l e 5 Pairwise comparison matrix of alternatives according to the criterion MARKET

л , А , Л , А . л , Ав А п Ад 1 0 /1 0 1 0 /2 0 10/25 10/30 10/40 10/50 10/60 10/90 10/90 A 2 2 0 /1 0 1 0 /1 0 1 0 /2 0 10/25 10/30 10/40 10/60 10/80 10/80 25/10 2 0 /1 0 1 0 /1 0 1 0 /2 0 10/30 10/45 10/65 10/80 10/85 30/10 25/10 2 0 /1 0 1 0 /1 0 1 0 /2 0 10/35 10/50 10/65 10/70 40/10 30/10 30/10 2 0 /1 0 1 0 /1 0 1 0 /2 0 10/30 10/50 10/52 л* 50/10 40/10 45/10 35/10 1 0 /1 0 1 0 /1 0 1 0 /2 0 10/30 10/35 /1? 60/10 60/10 65/10 50/10 30/10 2 0 /1 0 1 0 /1 0 10/15 10/17 ^ 8 90/10 80/10 80/10 65/10 50/10 30/10 15/10 1 0 /1 0 10/15 л 9 90/10 80/10 85/10 70/10 52/10 35/10 17/10 15/10 1 0 /1 0 T a b l e 6 Pairwise comparison matrix of alternatives according to the criterion PORTFOLIO

Л, а2 Л, А* л , л7 _А» а9 *1 1 0 /1 0 1 0 /2 0 10/30 10/40 10/50 10/60 10/80 10/90 10/90 А г 2 0 /1 0 1 0 /1 0 1 0 /2 0 10/30 10/45 10/60 10/75 10/90 10/90 30/10 2 0 /1 0 1 0 /1 0 1 0 /2 0 10/40 10/60 10/80 10/90 10/90 Л 40/10 30/10 2 0 /1 0 1 0 /1 0 1 0 /2 0 10/50 10/70 10/80 10/80 л , 50/10 45/10 40/10 2 0 /1 0 1 0 /1 0 1 0 /2 0 10/50 10/80 10/80 Ац 60/10 60/10 60/10 50/10 2 0 /1 0 1 0 /1 0 1 0 /2 0 10/60 10/60 А7 80/10 75/10 80/10 70/10 50/10 2 0 /1 0 1 0 /1 0 10/15 1 0 /2 0 •^8 90/10 90/10 90/10 80/10 80/10 60/10 15/10 1 0 /1 0 1 0 /1 2 Л9 90/10 90/10 90/10 80/10 80/10 60/10 2 0 /1 0 1 2 /1 0 1 0 /1 0

As we can see in T ab . 7, b o th local and global I.C ./I.R . are low er th an 0.10. It m eans th a t the m atrices o f pairw ise com p ariso n fo r all h ierarch y levels allow us to com plete synthesis.

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T a b l e 7 I.C./I.R. computed for local and global priorities

Local priorities I.C./I.R. 0 .0 0 0.03 0.03 0.06

Global priorities I.C./I.R. _0.04

T a b l e 8 Summary of local and global priority vectors with necessary calculations leading to an optimal

alternative (/!„) due to distributive mode

global priorities wgk *>9i - 0.593 *9i = 0.044 _{*03 - 0.292} _W₀_{* - 0.071} sw \u 'wgt Rank local priorities " • u wl,| wl,|*łve, _{wlM W}₁_JI_*W02 _{wl3( wlJ(*w}₀_j _{wl4i wl}₄_|*w₀₄ 0.181 0.107 0 .0 2 0 0 .0 0 1 0.017 0.005 0.016 0 .0 0 1 0.114 4 A 2 0.165 0.098 0.026 0 .0 0 1 0 .0 2 1 0.006 0 .0 2 0 0 .0 0 1 0.106 5 0.141 0.083 0.032 0 .0 0 1 0.028 0.008 0.025 0 .0 0 2 0.095 7 0 .1 2 1 0.072 0.044 0 .0 0 2 0.044 0.013 0.037 0.003 0.089 8 A, 0.099 0.059 0.067 0.003 0.074 0 .0 2 2 0.057 0.004 0.088 9 A 6 0.085 0.050 0.107 0.005 0.117 0.034 0.097 0.007 0.096 6 An 0.078 0.046 0.175 0.008 0.169 0.049 0.178 0.013 0.116 3 ^ 8 0.071 0.042 0.245 0 .0 1 1 0.262 0.077 0.275 0 .0 2 0 0.149 1 Aq 0.059 0.035 0.284 0.013 0.269 0.078 0.295 0 .0 2 1 0.147 2

I he results o f the synthesis are presented in T ab . 8. T h e optim al altern ativ e is decreasing the average deposit ra te o f the b an k by 0.7 5% . We do n o t present the sensitivity analyses, how ever it is im p o rta n t to m en tio n th a t decresing the im po rtan ce o f the C O M P E T IT IO N criterio n leads to changing optim al alternative to A Q (decreasing the dep osit ra te bv 1.0 0% ).

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5. CONCLUSIONS

T h e A H P is a good m eth o d to su p p o rt decision m akers especially when it is com bined with u n d erstandin g the problem o f the ju dg em en t consistency. D u e to its op en characteristics, allow ing com bining q u a n tita tiv e and no n - qu an titativ e aspects o f the preferences, the A H P m ay represent an interesting basis fo r developm ent o f com bined o ptim isation m ethods.

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Publications, Pittsburgh.

D o m a ń s k i Cz., K o n d r a s i u k S., M o r a w s k i I. (1997), Zastosowanie analitycznego

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