Location of banking automatic teller machines based on convolution☆
Introduction
Facility location problems are classical optimization problems that have numerous applications, especially in the service industries. Examples of these applications include optimal location of gas stations, health care units, warehouses, police stations, and power plants. Facility location models determine the minimum-cost location of a set of facilities to satisfy a set of demands (customers), subject to a set of constraints.
Automatic teller machines (ATMs) are among the most important service facilities in the banking industry. Since their appearance some 35 years ago, ATMs have literally changed the face of banking. The number and impact on the banking and retail business is growing steadily. The number of ATMs in the United States grew 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. Bank Network News magazine (Anonymous, 1997) reports that the number of off-premise ATMs in the US jumped from 28,700 in 1994 to 67,000 in 1997. There are many factors that banks take into consideration in order to determine location priorities for ATM sites. According to Wilson (1999), the concerned bank must first determine if its main objective of placing a new off-premise ATM is visibility or free income. The usual first step is to determine where potential customers live, where they work, and what main roads they use (Adams, 1991). Customer surveys as well as geographic, demographic, economic, and traffic data are useful 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 prime locations for ATM sites. In this paper, the priorities for different potential ATM locations will be assumed given, based on a-priori analysis of all the applicable factors.
As recently surveyed in Hale and Moberg (2003), the literature on facility location models and algorithms is quite large. However, attention to bank ATM location has been scarce. Assuming ATMs are placed only in bank branches, ATM location could be merged with the branch location problem. In this case, bank branch location approaches described in Boufounou, 1995, Cornuejols et al., 1977, Hopmans, 1986 become applicable. The work in Kolesar (1984) uses queueing analysis to evaluate the workload and congestion at existing ATM locations, in order to determine in which locations to install additional ATMs. Few authors consider off-site ATM location, but only as an example within a larger given class of service facilities, such as discretionary service facilities, hierarchical commercial facilities, and immobile service facilities with stochastic customer demands. Another study developed models and algorithms for what they called “discretionary service facilities”, such as gas stations and automated teller machines (Berman, 1995). Generally, customers do not regard these facilities as end destinations, but they will use their services if they pass by them on their way on planned trips from one location to another. In Berman (1995), two equivalent integer programming models were formulated to locate N facilities in order to maximize the potential customer flow. The study also developed a greedy heuristic and a branch-and-bound algorithm to solve this problem. Alternatively, the study determined the minimum number of facilities required to intercept the flow of a given fraction of customers. This problem was extended in Berman, Larson, and Fouska (1992) by allowing the service facilities to be congested.
In this work, we propose a completely new approach of solving the ATM location problem. The new approach differs from the previous ones in three aspects. Unlike previous approaches which demand speciality and complex model building process, the new approach uses a very simple user interface to build the model. Second, the solution in the new approach is obtained using a simpler and more efficient mathematical technique. Lastly, the new approach allows any arbitrary service demand pattern and any service degradation model, allowing it to be more applicable to real-life problems.
The remainder of this paper is organized as follows. Definition of the service and demand patterns used in this study are described in the following section. The ATM location problem is formulated in Section 2. The solution algorithm is described in Section 4. Computational analysis and experiments are presented in Sections 5 Computation complexity of the proposed scheme, 6 Computational experiments, respectively, followed by conclusions in Section 7.
Section snippets
Problem formulation
In this paper, the problem of finding the minimum number of ATMs and their locations given arbitrary demand patterns is considered. In the following, the variables used in modelling the placement 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) α service margin; a constant that specifies the difference between supply and demand Sn (I × J) supply
Design considerations
The main advantage of the proposed scheme is that it provides high flexibility for location specialists to choose arbitrary service and demand patterns by selecting proper structures of the matrices A and D. In the following, we describe in more detail the role of these two matrices in the model design process.
Solution of the placement problem
The optimization problem given by (8), (9) is solved in this study using a new and simple heuristic approach. This approach turns out to offer high flexibility in choosing arbitrary service and demand patterns. It also allows a simple human user interface modelling of the problem and provides the solution in relatively short time. The solution approach is described in the following.
First, the fixed service pattern matrix A and the demand matrix D are given by the designer. Then, the algorithm
Computation complexity of the proposed scheme
In this section, the computation complexity of the proposed scheme is analyzed. From the discussions above, the proposed scheme has an outer loop as well as an inner loop. The outer loop searches for the optimal scalar frame penalty value while the inner loop searches for the optimal number of machines and their locations by implementing the algorithm of Fig. 5.
For the outer loop, a simple line search was found sufficient to locate the optimal scalar frame penalty. The search is limited to the
Computational experiments
In this section, we demonstrate the performance of the proposed ATM placement algorithm through simple illustrative examples. Matlab was used to implement the algorithm on a 2.1 GHz personal computer with 256 MB of memory. The Matlab program provides a friendly User Interface (UI). This interface is used to input a color-coded map in a common image format (JPEG) to automatically generate the corresponding demand matrix D. It is also used to input the service pattern matrix A from the user with
Conclusions
In this work, we proposed a new approach for the placement of automatic teller machines (ATMs). The approach computes the minimum number of machines as well as their locations that satisfy the service level coverage requirements. It does so by implementing a new heuristic solution that is based on the two-dimensional convolution. The proposed approach provides a flexible means for choosing arbitrary service models and demand patterns, making it suitable for real applications. Experiments with
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This work is supported by King Fahd University of Petroleum and Minerals.