Computers & Industrial Engineering 57 (2009) 1194–1201

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Computers & Industrial Engineering

journal homepage: www.elsevier .com/ locate/caie

Location of banking automatic teller machines based on convolution q

Mansour A. Aldajani *, Hesham K. AlfaresSystems Engineering Department, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia

a r t i c l e i n f o

Article history:Received 2 November 2008Received in revised form 8 May 2009Accepted 11 May 2009Available online 2 June 2009

Keywords:Facility locationMathematical modelsOptimization algorithmsATM site selection

0360-8352/$ - see front matter � 2009 Elsevier Ltd. Adoi:10.1016/j.cie.2009.05.013

q This work is supported by King Fahd University of* Corresponding author. Tel.: +966 3 8603315; fax:

E-mail address: dajani@kfupm.edu.sa (M.A. Aldaja

a b s t r a c t

In this paper, the problem of determining the optimum number and locations of banking automatic tellermachines (ATMs) is considered. The objective is to minimize the total number of ATMs to cover all cus-tomer demands within a given geographical area. First, a mathematical model of this optimization prob-lem is formulated. A novel heuristic algorithm with unique features is then developed to efficiently solvethis problem. Finally, simulation results show the effectiveness of this algorithm in solving the ATMplacement problem.

� 2009 Elsevier Ltd. All rights reserved.

1. Introduction

Facility location problems are classical optimization problemsthat have numerous applications, especially in the service indus-tries. Examples of these applications include optimal location ofgas stations, health care units, warehouses, police stations, andpower plants. Facility location models determine the minimum-cost location of a set of facilities to satisfy a set of demands (cus-tomers), subject to a set of constraints.

Automatic teller machines (ATMs) are among the most impor-tant service facilities in the banking industry. Since their appear-ance some 35 years ago, ATMs have literally changed the face ofbanking. The number and impact on the banking and retail busi-ness is growing steadily. The number of ATMs in the United Statesgrew from only 25,000 in 1981 to more than 150,000 in 1999(Wilson, 1999). While most of these ATMs are located at banks,there is a growing number of ATMs located off-premises. BankNetwork News magazine (Anonymous, 1997) reports that thenumber of off-premise ATMs in the US jumped from 28,700 in1994 to 67,000 in 1997. There are many factors that banks takeinto consideration in order to determine location priorities forATM sites. According to Wilson (1999), the concerned bank mustfirst determine if its main objective of placing a new off-premiseATM is visibility or free income. The usual first step is to determinewhere potential customers live, where they work, and what mainroads they use (Adams, 1991). Customer surveys as well asgeographic, demographic, economic, and traffic data are useful

ll rights reserved.

Petroleum and Minerals.+966 3 8602215.

ni).

for answering these questions. Other considerations include safety,cost, convenience, and visibility. Quite often, malls, supermarkets,gas stations, and other high-traffic shopping areas are primelocations for ATM sites. In this paper, the priorities for differentpotential ATM locations will be assumed given, based on a-priorianalysis of all the applicable factors.

As recently surveyed in Hale and Moberg (2003), the literatureon facility location models and algorithms is quite large. However,attention to bank ATM location has been scarce. Assuming ATMsare placed only in bank branches, ATM location could be mergedwith the branch location problem. In this case, bank branch loca-tion approaches described in Boufounou (1995), Cornuejols, Fisher,and Nemhauser (1977) and Hopmans (1986) become applicable.The work in Kolesar (1984) uses queueing analysis to evaluatethe workload and congestion at existing ATM locations, in orderto determine in which locations to install additional ATMs. Fewauthors consider off-site ATM location, but only as an examplewithin a larger given class of service facilities, such as discretionaryservice facilities, hierarchical commercial facilities, and immobileservice facilities with stochastic customer demands. Another studydeveloped models and algorithms for what they called ‘‘discretion-ary service facilities”, such as gas stations and automated tellermachines (Berman, 1995). Generally, customers do not regardthese facilities as end destinations, but they will use their servicesif they pass by them on their way on planned trips from one loca-tion to another. In Berman (1995), two equivalent integer pro-gramming models were formulated to locate N facilities in orderto maximize the potential customer flow. The study also developeda greedy heuristic and a branch-and-bound algorithm to solve thisproblem. Alternatively, the study determined the minimum num-ber of facilities required to intercept the flow of a given fraction

M.A. Aldajani, H.K. Alfares / Computers & Industrial Engineering 57 (2009) 1194–1201 1195

of customers. This problem was extended in Berman, Larson, andFouska (1992) by allowing the service facilities to be congested.

In this work, we propose a completely new approach of solvingthe ATM location problem. The new approach differs from the pre-vious ones in three aspects. Unlike previous approaches which de-mand speciality and complex model building process, the newapproach uses a very simple user interface to build the model. Sec-ond, the solution in the new approach is obtained using a simplerand more efficient mathematical technique. Lastly, the newapproach allows any arbitrary service demand pattern and anyservice degradation model, allowing it to be more applicable toreal-life problems.

The remainder of this paper is organized as follows. Definitionof the service and demand patterns used in this study are describedin the following section. The ATM location problem is formulatedin Section 2. The solution algorithm is described in Section 4. Com-putational analysis and experiments are presented in Sections 5and 6, respectively, followed by conclusions in Section 7.

2. Problem formulation

In this paper, the problem of finding the minimum number ofATMs and their locations given arbitrary demand patterns is con-sidered. In the following, the variables used in modelling the place-ment problem are defined.

N

total number of machines sn(x,y) service supply from the nth machine to location (x,y) d(x,y) service demand at location (x,y) e(x,y) difference between supply and demand at location (x,y) a service margin; a constant that specifies the difference

between supply and demand

Sn (I � J) supply matrix containing the discretized values of

sn(x,y)

D (I � J) demand matrix containing the discretized values of

d(x,y)

E (I � J) difference matrix containing the discretized values

of e(x,y)

A (IA � JA) fixed matrix that represents the degradation

pattern of the service quality away from each machine

b Frame penalty value Ln location matrix indicating the location of the nth machine (un,vn) coordinates of the nth machine. En difference matrix after assigning machine n emin(n) smallest element inside En

The objective of the placement problem is to minimize the totalnumber of machines N such that the service supply exceeds the de-mand by a fixed amount a all over a confined 2-D space C. In math-ematical terms, we can write

min N ð1Þsuch that eðx; yÞ, max

n¼½1;N�fsnðx; yÞg � dðx; yÞP a 8x; y 2 C: ð2Þ

The quantity sn(x,y) represents the service supply from the nthATM to the user at coordinates (x,y). This quantity is dependentmainly on the service pattern of the machine, i.e., how the servicelevel (SL) varies around an ATM machine. In most practical cases,the service level by a certain machine decreases as we move awayfrom that machine. We assume here that the SL at any point isassociated with only one machine that delivers the maximum ser-vice. The SL received from other machines at this specific point willbe simply ignored. The term d(x,y) represents the service demandlevel at point (x,y). Priority coverage, forbidden regions, streets andhighways all can be incorporated within this term.

The choice of the service margin a is dependent on the problemat hand. Large a means that the supply will exceed demand by alarge amount. Obviously, this will be at the cost of increasing thenumber of ATMs.

2.1. Discretization of the model

To solve the placement model discussed above, the variables arefirst discretized into a finite number of uniform grid points of size(I, J). The number of divisions in the grid depends on the requiredresolution and available computational power. The variablese(x,y), sn(x,y), and d(x,y) are discretized in 2-D Euclidian space toform the matrices E, Sn, and D, respectively. Therefore, the optimi-zation problem can be written in matrix format as

min N; ð3Þsubject to ENði; jÞ ¼ max

n¼½1;N�fSnði; jÞg � Dði; jÞP a 8i; j; ð4Þ

where EN is the difference matrix of size (I � J) after assigning N ma-chines, Sn is the supply matrix of the nth machine, and D is the de-mand matrix.

The matrix Sn can be obtained from the convolution of twomatrices as follows:

Sn ¼ A� Ln; ð5Þ

where the symbol � indicates the two-dimensional convolution gi-ven by the expression

Snði; jÞ ¼XIA

2

r¼�IA2

XJA2

s¼�JA2

Aðr; sÞLnðiþ r; jþ sÞ: ð6Þ

The matrix A of size (IA � JA) is a fixed service pattern matrix ofthe machines. The matrix Ln indicates the location of the machinen. If we denote this location by the coordinates (un,vn) then Ln hasall its elements equal to zero except at (un,vn) where it is equal to‘‘1”. In other words,

Lnði; jÞ ¼1 at ðun; vnÞ;0 elsewhere:

�ð7Þ

For the sake of illustration, suppose that

A ¼30 50 3050 100 5030 50 30

264

375 and L1 ¼

0 0 0 00 0 0 00 0 1 00 0 0 0

26664

37775:

Then

S1 ¼

0 0 0 00 30 50 300 50 100 500 30 50 30

26664

37775:

The convolution values outside the range of the matrix Ln aresimply truncated. Notice that the objective of the convolution hereis to surround the unique non-zero element in Ln with the servicepattern matrix A. Therefore, the convolution operation in this casecan be performed very efficiently by simply centering the elementsof the A matrix at (un,vn).

Notice also that minimizing the number of machines N is equiv-alent to minimizing the summation norm of the location matricesLn for all machines. In view of this fact, the optimization problemcan finally be written as

minXN

n¼1

Ln

����������; ð8Þ

1196 M.A. Aldajani, H.K. Alfares / Computers & Industrial Engineering 57 (2009) 1194–1201

subject to

EN ¼ maxn¼½1;N�

fA� Lng � D P aH; ð9Þ

where H is a matrix full of ones. The representation of the place-ment problem in this matrix format helps in borrowing useful toolsfrom matrix theory to find a near-optimal solution for this problemas will be discussed in the next section.

3. Design considerations

The main advantage of the proposed scheme is that it provideshigh flexibility for location specialists to choose arbitrary serviceand demand patterns by selecting proper structures of the matricesA and D. In the following, we describe in more detail the role ofthese two matrices in the model design process.

3.1. Role of the service pattern matrix A

The service pattern matrix A describes how the quality of ser-vice changes as customers move away from the machine. Fig. 1shows a simple illustration of a 5 � 5 matrix A. In this example,the service level starts with 100% at the center cell and then decre-ments inversely-proportional to the Euclidian distance from themachine. Another example of a rectilinear distance relation isshown in Fig. 2. Fig. 3 shows the contour plots of the two matrices.Using a similar approach, many other service patterns can be

Fig. 1. An example of the service pattern matrix A (Euclidian distance model).

Fig. 2. Another example of the service pattern matrix A (rectilinear distancemodel).

Fig. 3. Contour plots of the two service patterns in Figs. 1 and 2.

Fig. 4. Example of designing the demand levels on a real map using color codes.

designed by choosing proper values of the matrix A. In generalterms, the algorithm allows any arbitrary pattern of A which makesit suitable for modelling real location problems.

3.2. Role of the demand matrix D

The demand matrix D plays a major role in the placement ofATMs using the proposed algorithm. It provides flexibility inchoosing any type of desired demand pattern. For the sake of illus-tration, Fig. 4 shows an example of a color-coded map that repre-sents the coverage demand pattern in different parts of an actualgeographical region at the center of Riyadh, Saudi Arabia. Each col-or represents a level of demand. In this example, the regions withgreen1 color have the highest demand. The blue and white colorsrepresent high and normal demand regions, respectively. The redcolor represents no-demand regions where the algorithm shouldavoid assigning ATMs.

The algorithm then interprets this colored map and builds thedemand matrix D. The interpretation of the values of D is as fol-lows. The high demand regions are reflected in D by positive valueswith magnitude that is proportional to the demand level. The posi-tive values in D would result in small contributions in Cn (see Eq.(10)) and therefore will be chosen first for machine locations. Onthe other hand, negative values in D indicate that these regionsshould be avoided. In this case, the convolution values in Cn willbe large and therefore the algorithm will avoid assigning machinesat these regions. Finally, regions with normal coverage priority willbe reflected by zero values in the matrix D. As a result, the matrix Dwill be mostly full of zeros since normal coverage is usually the de-fault case. The sparsity of the matrix D helps in substantially reduc-ing the computations in the proposed scheme as will be describedin the next section.

1 For interpretation of color in Figs. 3, 4, 7 and 9–13, the reader is referred to theeb version of this article.

w

Fig. 5. The proposed solution algorithm.

M.A. Aldajani, H.K. Alfares / Computers & Industrial Engineering 57 (2009) 1194–1201 1197

Using this color-coding technique, designers can set any arbi-trary number of relative levels of demand. Although the examplein Fig. 4 shows only four color codes, this number can be increasedas desired according to the relative demand levels on hand.

4. Solution of the placement problem

The optimization problem given by (8) and (9) is solved in thisstudy using a new and simple heuristic approach. This approachturns out to offer high flexibility in choosing arbitrary serviceand demand patterns. It also allows a simple human user interfacemodelling of the problem and provides the solution in relativelyshort time. The solution approach is described in the following.

First, the fixed service pattern matrix A and the demand matrixD are given by the designer. Then, the algorithm will compute theservice level contribution of every point on the grid to its neighbor-ing points in case the given point is chosen as a machine location.This, off course, takes into account the given demand pattern. Thenthe point that results in the highest neighborhood coverage is cho-sen as the new machine location.

After placing each machine, the matrix E is updated and theprocess is repeated to choose the next machine. The algorithm ter-minates when all the elements of E exceed the service margin a orwhen the overall percentage coverage is satisfactory.

A flow chart of the proposed algorithm is shown in Fig. 5. Todetermine the contribution of each point on the grid to the servicedistribution within the grid in case it is chosen as a machine loca-tion, the service pattern A is convolved with the existing differencematrix En�1 from previously assigned machines, i.e.,

Cn ¼ A� En�1; E0 ¼ �D: ð10Þ

The matrix Cn describes the contribution provided by the ATMwhen located at each point in the grid to the neighboring points gi-ven the previous difference matrix En�1. The role of the convolutionhere is as follows. For each point in the previous difference matrixEn�1, the matrix A is centered at that point and dot-multiplied withthe intersecting sector of En�1. The multiplication values are thensummed up and the answer is stored at the corresponding pointin Cn. This convolution process is repeated for all other points inEn�1. Then, the coordinates that correspond to the minimum valueof the matrix Cn are then chosen as the location of the nth machine,i.e.,

ðun; vnÞ ¼ argminði;jÞ

Cn: ð11Þ

When a set of points give the same minima, the middle amongthese points is arbitrarily chosen to break the tie.

To understand the motivation behind this choice, suppose firstthat the space has equal demand all over the area. If n � 1 ma-chines are already placed, then En�1 will have large positive valuesof SLs around these machines. When A is convolved with En�1, theconvolution values will be smallest at the location that is farthestaway from the previous n � 1 machines. Consequently, (11) willchose this location for the next machine. This guarantees that thenew machine will be placed at locations with poorest service.

Now suppose that a certain area has higher demand than oth-ers. In this case, negative values can simply be assigned in the cor-responding regions in D. Since E0 = �D, the convolution at theselocations will be smallest and therefore they will be chosen firstby (11) as machine location. In this way, the matrix D can be de-signed to fulfill any arbitrary demand patterns.

Once a new machine location is computed, the location matrixLn is constructed from (7). The difference matrix is then updated asfollows:

En ¼ Q n � D; n ¼ 1;2; . . . ;N; ð12Þ

where Qn is the accumulated supply of service due to the machines:1, . . .,n. The elements inside Qn are obtained recursively from theexpression

Qnði; jÞ ¼maxfQ n�1ði; jÞ; Snði; jÞg; i ¼ 1; . . . ; I; j

¼ 1; . . . ; J; Q 0 ¼ 0; ð13Þ

where 0 is the (I � J) zero matrix.In summary, given the service pattern and demand matrices A

and D, the location of the machines is determined by iteratingEqs. (10)–(12) starting from E0 = �D. The algorithm terminateswhen the constraint (2) is satisfied, or equivalently, the minimumdifference: emin(n) ,min{En} exceeds the margin a. The algorithmthen returns the total number of machines N, their locations, andemin.

4.1. Penalizing boundaries of the demand grid

Based on the discussions above, the proposed algorithm tendsto assign machines at the boundaries of the covered area so thatthey will be farthest apart from each other. This will be at the costof increasing the required number of machines. This problem can

Fig. 6. Illustrative example of a demand grid surrounded by the penalty constant b.

1198 M.A. Aldajani, H.K. Alfares / Computers & Industrial Engineering 57 (2009) 1194–1201

be resolved by augmenting one frame of penalty b around the de-mand matrix D as shown in Fig. 6. The main objective of this framevalue is to push the machines inside the demand area. The optimalvalue of b is usually a positive number that depends on the size andcontent of the matrices A and D. The frame value can fine-tune thesolution by improving the total coverage for the same number ofmachines. In this work, an outer loop is performed that imple-ments a simple line search to find the optimum frame value. InSection 6, we show a numerical example on deciding this value.

4.2. Percentage coverage of the machines

The algorithm also computes the percentage coverage for eachof the machines as it assigns them one-by-one. After placing eachmachine, the accumulative percentage coverage (APC) is computedas the number of grid points in En that have SL greater than themargin a divided by the total number of grid points in En. The per-centage coverage (PC) is then evaluated by simply computing thechange of APC values from one assigned machine to the next.The algorithm returns both PC and APC with the solution as weshall show in the simulations section. The PC is essential informa-tion in locating the ATMs. One example of utilizing this informa-tion is to eliminate those machines with negligible percentagecoverage, resulting in an overall cost reduction.

Another important issue in this problem is to determine thepercent of total demand satisfied which can be defined as follows:

� ¼P

i

PjjBði; jÞjP

i

PjjDði; jÞj

; ð14Þ

where

Bði; jÞ ¼Dði; jÞ if ENði; jÞ > a;Q Nði; jÞ otherwise

�ð15Þ

determines the demand covered by the ATMs. Unlike APC whichconsidered number of points covered, � indicates how much de-mand is covered.

4.3. Solution verification

The ATM placement problem is usually NP hard and its solutioncannot be found analytically. Therefore, numerical methods areused for verifying the proposed algorithm. First the algorithm isimplemented on simple models where solutions are known andthe results are then compared (Park, Yook, & Park, 2002). Second,solution is verified by performing an exhaustive search on all pos-sible locations. The search challenges the algorithm by trying tofind one of the following:

1. A lower number of ATMs that meets the service levelrequirements.

2. A different location of the same number of ATMs that providesbetter service coverage (higher emin).

For example, suppose that the proposed algorithm gives a min-imum number of ATMs equal to 5 together with their near-optimal

locations. First, the exhaustive search will try all possible locationcombinations on the grid to locate four ATMs such that the servicerequirement is satisfied, i.e. emin > a. Next, the exhaustive searchwill also try to locate five ATMs in different places than those givenby the algorithm to get a higher value for emin. If both tries fail, thenthe algorithm can be claimed to achieve an optimal solution. Forthe exhaustive search, a reasonable grid size is used to make itcomputationally achievable. Solutions found by the convolutionalgorithm matched those found by exhaustive search for a set ofsix small problems.

Furthermore, the heuristic solutions have been compared to theoptimum solutions produced by integer programming (IP) andexhaustive search. Comparisons with IP were limited to a set offour small test problems because optimum IP solutions are hardto attain for larger problems. In all four cases, the heuristic solu-tions matched the optimum IP solutions.

5. Computation complexity of the proposed scheme

In this section, the computation complexity of the proposedscheme is analyzed. From the discussions above, the proposedscheme has an outer loop as well as an inner loop. The outer loopsearches for the optimal scalar frame penalty value while the innerloop searches for the optimal number of machines and their loca-tions by implementing the algorithm of Fig. 5.

For the outer loop, a simple line search was found sufficient tolocate the optimal scalar frame penalty. The search is limited to theinteger values in the range [0,500]. Still, more efficient search algo-rithms could be adopted to find this value.

In the inner loop represented by Fig. 5, the only computationallyexpensive operation is the convolution A � En�1. Row convolutioncosts m2 multiplications where m is the number of grid points inthe search space (m = I � J). However, this number can be substan-tially reduced by utilizing available efficient schemes for comput-ing the convolution. An example of these schemes is theconvolution theorem which reduces the number of multiplicationsto m � logm if m is a power-of-two. In addition, there are twoobservations that can further reduce the complexity of the convo-lution operation as follows:

1. The matrix En usually starts with a structure that consistsmostly of zero elements (corresponding to normal demand inD). This matrix is then gradually filled up with non-zero valuesas new machines are assigned. Therefore, the sparsity of thematrix En can be exploited while computing the convolutionto reduce the number of complex operations. For example,there is no need to compute the convolution in regions of En

with zero values.2. The search space for optimal locations decreases as new

machines are assigned. Therefore, the number of complex oper-ations in the convolution can be substantially reduced by ignor-ing those locations already meeting the coverage requirement.

6. Computational experiments

In this section, we demonstrate the performance of the pro-posed ATM placement algorithm through simple illustrative exam-ples. Matlab was used to implement the algorithm on a 2.1 GHzpersonal computer with 256 MB of memory. The Matlab programprovides a friendly User Interface (UI). This interface is used to in-put a color-coded map in a common image format (JPEG) to auto-matically generate the corresponding demand matrix D. It is alsoused to input the service pattern matrix A from the user with arbi-trary size and values. It subsequently computes the number of ma-chines and their locations and then shows them on the color-coded

1 2 3 4 50

20

40

60

80

PC %

1 2 3 4 560

70

80

90

100

110

APC

%

ATM number

Fig. 8. Percentage coverage (PC) and accumulative percentage coverage (APC) forthe machines in Fig. 7.

M.A. Aldajani, H.K. Alfares / Computers & Industrial Engineering 57 (2009) 1194–1201 1199

map. The program also returns the coordinates of the machines,emin, and the percentage coverage (PC) of each assigned machine.

In our experiments, the size of the matrices D and A is fixed to41 � 41 for both (corresponds to 1681 possible locations). Further-more, the service margin is arbitrarily fixed in all instances to thenormalized value a = 1. We consider first the placement problemwhere the demand is the same for all points on the grid. This cor-responds to D = 0. A rectilinear distance model is used to representthe degradation of service around the machines. Such model iscommon to represent street traveling distances in urban setting.The results are shown in Fig. 7. As expected, the machines wereplaced uniformly across the demand area starting from the centerof the region. The number at the center of each segment indicatesthe order by which the machine was assigned. In this example, thenumber of machines needed to cover the entire area is five ma-chines. The PC and APC of the assigned machines are shown inFig. 8. Machine number 1 covered 66% of the whole area, whilethe other four machines covered 8.5% each.

In another experiment, we changed the demand matrix to in-clude a region of high priority coverage. This region could be abank, a shopping center, or a highly populated area. The region issimply drawn in a specific distinct color which is interpreted bythe algorithm as high demand region. The algorithm then findsthe solution and the results are shown in Fig. 9. In this case, thefirst machine assigned was moved to cover the high demand regionfirst. The next machines were then assigned to cover the remaininguncovered areas.

Another case considered a mesh of roads from a real city map.The roads are redrawn with a unique color. The algorithm readsthe map and interprets this new color and assigns it an appropriatevalue in the matrix D. The solution is shown in Fig. 10. Notice thatthe first machine was assigned next to the road in a way as to en-close as much road distances as possible. The next machines werethen assigned the same way.

Finally, an actual map for the down-town area of Khubar City inSaudi Arabia is considered. The map is shown in Fig. 11. The greenregions are the shopping areas, the pink and blue lines are the mainroads and the red area is the avoid-region. In this case, the grid res-olution is 75 � 75 pixels. The algorithm is implemented for this

Fig. 7. Result of machine assignment over an area with uniform demand.

Fig. 9. Result of the placement problem with high demand region.

configuration and the results are shown in Fig. 12. The percentagecoverage is also shown in Fig. 13. The results are returned within20 s. Looking closely at the results, we notice the following:

1. The 18 ATMs placed by the algorithm covered about 95% of thetotal demand space.

2. According to (14), the percentage demand coverage is 99.6%.This means that the remaining uncovered 5% of the demandspace is not much significant.

3. The shopping areas are covered first by the algorithm asexpected.

4. The remaining ATMs are located at the intersections of or alongthe main roads.

5. No ATM was placed at the avoid-region.

To choose the optimal frame value b, we considered again themap shown in Fig. 9. The number of machines is fixed to the

Fig. 11. Down-town area of Khubar, Saudi Arabia. Green: shopping areas, blue andpink: main roads, red: avoid-region.

1

2

34

5

Fig. 10. Result of the placement problem for a mesh of roads.

1

2

3

4

5

6

7

8

9

10

1112

1314

1516

17

18

Fig. 12. Resulting locations of the 18 ATMs and their service patterns.

0 2 4 6 8 10 12 14 16 180

2

4

6

8

10

12

PC %

0 2 4 6 8 10 12 14 16 180

20

40

60

80

100

APC

%

ATM

Fig. 13. Percentage coverage of the 18 ATMs of Fig. 12.

100 150 200 250 300 35093

94

95

96

97

98

99

100

Frame value b

Tota

l per

cent

age

cove

rage

Fig. 14. Effect of the frame value on the total percentage coverage.

1200 M.A. Aldajani, H.K. Alfares / Computers & Industrial Engineering 57 (2009) 1194–1201

optimum value (five machines). Then, the coverage of themachines is computed for different values of b. Fig. 14 shows theresulting total percentage coverage as a function of b. As weincrease b the machines are pushed towards the center of theregion increasing the contribution of the machines at the edges.After a while, the machines are so much pushed that the regionsat the edges are not covered. This causes the coverage to droprapidly. This pattern is typical in all scenarios tested. In thisexample, the optimal value of b is 280.

0 0.5 1 1.5 2x 104

0

0.5

1

1.5

2

2.5

3

Map grid size m=I × J

Tim

e to

ass

ign

sing

le A

TM (S

ec.)

ActualAnalytical fit

Fig. 15. Time needed to assign single machine as a function of the grid size m = I � J.

M.A. Aldajani, H.K. Alfares / Computers & Industrial Engineering 57 (2009) 1194–1201 1201

In another test, we investigate numerically the computationalcomplexity of the proposed approach. As mentioned earlier, themain advantage of using the convolution as a core process in theproposed algorithm is that there exists many ways of efficientlycomputing it. For example, the convolution theorem reduces thenumber of complex operations from m2 to m log(m), with m beingthe grid size of the map (m = I � J). The processing time for the pro-posed algorithm is proportional to the convolution complexity. Inour simulations, we made use of the convolution theorem to re-duce the processing time of the algorithm. In Fig. 15, we showthe processing time needed to assign a single ATM2 for various val-ues of the grid size m. The theoretical fit is also shown for compari-son purpose. Clearly the processing time is reduced to the order ofm log(m) resulting in a substantial saving in computations.

2 That is the same time required to go through one round of Fig. 5.

7. Conclusions

In this work, we proposed a new approach for the placement ofautomatic teller machines (ATMs). The approach computes theminimum number of machines as well as their locations thatsatisfy the service level coverage requirements. It does so by imple-menting a new heuristic solution that is based on the two-dimen-sional convolution. The proposed approach provides a flexiblemeans for choosing arbitrary service models and demand patterns,making it suitable for real applications. Experiments with the newalgorithm show its efficiency and flexibility in solving ATM place-ment problems near-optimally.

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