AN OPTIMIZATION-BASED HEURISTIC FOR A CAPACITATED LOT-SIZING MODEL IN AN AUTOMATED TELLER MACHINES NETWORK

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1 Journal of Mathematcs and Statstcs 9 (4): , 203 ISSN: Scence Publcatons do:0.3844/jmssp Publshed Onlne 9 (4) 203 ( AN OPTIMIZATION-BASED HEURISTIC FOR A CAPACITATED LOT-SIZING MODEL IN AN AUTOMATED TELLER MACHINES NETWORK Supatchaya Chotayakul, Peerayuth Charnsetthkul, Juta Pchtlamken and 2 John Kobza Department of Industral Engneerng, Faculty of Engneerng, Kasetsart Unversty, Chatuchak, Bangkok 0900, Thaland 2 Department of Industral Engneerng, Faculty of Engneerng, The Unversty of Tennessee, Knoxvlle, Tennessee 37996, US Receved ; Revsed ; Accepted ABSTRACT Ths research studes a cash nventory problem n an ATM Network to satsfy customer s cash needs over multple perods wth determnstc demand. The objectve s to determne the amount of money to place n Automated Teller Machnes (ATMs) and cash centers for each perod over a gven tme horzon. The algorthms are desgned as a mult-echelon nventory problem wth sngle-tem capactated lot-szng to mnmze total costs of runnng ATM network. In ths study, we formulate the problem as a Mxed Integer Program (MIP) and develop an approach based on reformulatng the model as a shortest path formulaton for fndng a near-optmal soluton of the problem. Ths reformulaton s the same as the tradtonal model, except the capacty constrants, nventory balance constrants and setup constrants related to the management of the money n ATMs are relaxed. Ths new formulaton gves more varables and constrants, but has a much tghter lnear relaxaton than the orgnal and s faster to solve for short term plannng. Computatonal results show ts effectveness, especally for large szed problems. Keywords: Shortest Path Formulaton, Mult-Echelon Inventory, ATM Network, Capactated Lot-Szng Model. NTRODUCTION Ths research focuses on cash management for an ATM network n Thaland. ATM plannng and replenshment to serve customer s cash needs are a key servce of commercal banks. Customers can access bankng servces of ther bank accounts n order to make cash wthdrawals from the machnes, whch are placed over the country. Banks must employ several employees and employees tme to oversee the network and make decsons about the cash management system to ensure that they have the rght cash levels at the rght locatons; therefore, they ncur hgh operatng costs such as transport cost, nsurance, servcng costs and nventory cost (Smuts et al., 2008). Many experts beleve the amount of cash mantaned n ATMs exceeds 5-20% of customer need and about 35-60% of overall costs of the cash management system come from runnng the ATMs (Smuts et al., 2007). Thus, the problem of reducng the overall costs of runnng the ATMs s sgnfcant. The objectve s to fnd the amount of cash to be reflled n ATMs and cash centers over a fxed perod of tme foe short term plannng. In the present paper, we propose a mult-echelon nventory problem wth Sngle-Item Capactated Lot- Szng (SICLS) formulaton n order to mnmze the orderng cost (refllng cost) and holdng cost (opportunty cost) of cash nventory at ATMs and cash centers and to satsfy tme-varyng demands. We formulate the problem as a Mxed Integer Programmng (MIP) for the resultng optmzaton of ths problem. However t usually requres much more computatonal Correspondng Author: Supatchaya Chotayakul, Department of Industral Engneerng, Faculty of Engneerng, Kasetsart Unversty, Chatuchak, Bangkok 0900, Thaland Scence Publcatons 283

2 Supatchaya Chotayakul et al. / Journal of Mathematcs and Statstcs 9 (4): , 203 effort for real-world problems wth large sample sze. The problem s known to be NP-hard (Floran and Klen, 972; Chubanov et al., 2008). Many mathematcal programmng-based procedures used to solve the sngle-tem capactated lot-szng model and were studed by many researchers. Waknaga and Sawak (2008) studed a dynamc programmng approach for solvng the dynamc lot sze model for the case where sngle-tem s produced and shpped by an overseas export company. They explored the problem wth the constrant of producton and shpment capacty so as to mnmze the total cost over the fnte plannng horzon wth determnstc demands. Berk et al. (2008) developed the dynamc programmng formulaton of the sngle tem lot-szng problem for a walm/cold process wth fnte capacty and possble lost sales and solved the problem wth a polynomal soluton algorthm based on the lost-sales-mprovements of the full commtment problem. They obtaned the optmal soluton n case of postve setup tmes and the heurstc solutons n another cases. Akbalk and Pochet (2009) proposed a new class of vald nequaltes for the sngle-tem capactated lot szng problem wth step-wse producton costs that they called mxed flow cover nequaltes whch s derved from the nteger flow cover nequaltes. Computatonal results show the effcency of the new class mxed flow cover compared to the exstng methods. Akbalk and Penz (2009) proposed an exact pseudo-polynomal dynamc programmng algorthm to solve a specal case of the sngle-tem capactated lot-szng problem. Zhang et al. (202) formulated the capactated lot szng problem n closed-loop supply chan consderng setup costs, product returns and remanufacturng as a mxed nteger program and proposed a Lagrangan relaxatonbased soluton approach to solve the problem. Two polynomal tme algorthms for solvng the constant capactated lot szng problem was presented by Akbalk and Rapne (202). For ths study, we use the Shortest Path Reformulaton (SPR) approach to reformulate the MIP of the mult-echelon nventory problem of SICLS for fndng a soluton. The shortest path reformulaton approach was orgnally proposed by Eppen and Martn (987) for solvng the Mult-Item Capactated Lot-Szng (MICLS) problems. Hnd (995) developed ths approach usng column generaton on the uncapactated verson of the problem for solvng the sngle-tem capactated lot-szng problem. Wu et al. (20) extended ths approach to formulate the capactated mult-level lot-szng wth backloggng and found that Scence Publcatons 284 the lnear programmng relaxaton provdes good lower bounds on the optmal soluton value and develop an approach based on reformulatng the model as a Shortest Path Reformulaton (SPR) for fndng a near optmal soluton. The man dea of ths study s to mprove the MIP formulaton of the problem usng a reformulaton based on the shortest path algorthm so that commercal solvers lke CPLEX or Gulob are able to solve practcal nstance of cash nventory problem for short term plannng. The rest of ths artcle s organzed n fve sectons. In Secton 2 we explan the materals and methods to use for ths study. In Secton 3 we present comparson of the results of computatonal experments between the two algorthms and the dscusson was shown n Secton 4. Secton 5 contans the conclusons and the future research. 2. MATERIALS AND METHODS 2.. Problem Descrpton Cash management for ATMs s one of the mportant problems a bank faces. The costs of operatng and mantanng ATMs are very hgh and nclude the cost of the ATM machnes, rentng facltes, networkng, admnstraton and the opportunty cost for depostng money n the machne. An ATM transacton s an expensve fnancal transacton and the banks can make lttle proft from t, but they must stll support ths servce for customers. In ths case, snce the admnstratve cost and the opportunty cost for storng money n the machne are the crucal management costs that banks can plan and manage, we focus on the development of an effcent ATM cash nventory polcy that satsfes customer demand. Inventory costs occur when the bank stores money n the ATMs and the cash centers, whch are places for storage of cash movng to or from the central bank (Bank of Thaland; BOT) through commercal banks and cash centers to the bank branches and self-servce devces (ATMs) to satsfy customer cash demand. The amount of money to place n ATMs and cash centers depends on future unknown demands. If the amount of money n ATMs and cash centers s hgher than the customer demand, then an opportunty cost of holdng cash wll occur. The cost of money or the opportunty cost wll be assocated wth nterest rates and may also nclude nsurance costs for the money held n the ATM. But f the amount of money s lower than the customer demand, the bank ncurs a shortage cost. Thus, the development of an advanced algorthm for ATM

3 Supatchaya Chotayakul et al. / Journal of Mathematcs and Statstcs 9 (4): , 203 servces to reduce the overall costs of runnng the ATMs s very mportant for banks. In Thaland a bank must pay an nterchange fee when ts customers use another bank s ATM. The bank may also have to pay a refll cost assocated wth replenshng the ATM. If the ATM s nsde a bank branch, then there s no cost of movng cash to and from the ATM. However, f the ATM s not n a bank branch, then a refllng cost or orderng cost s ncurred when the money n the ATM s reflled. Ths cost s ndependent of the amount of money reflled. The goal of ths research s to determne how much money to store n ATMs and cash centers and the frequency of cash replenshment n each perod based on banks n Thaland so that all demands are satsfed wth mnmum total costs for runnng the ATMs. Customer demand at cash ATM s assumed to be known and determnstc and shortages are not allowed. Note that, snce there s no cost for restockng cash at the ATMs n local branches they are consdered part of cash management n branches and are not consdered for ths problem Problem Formulaton Ths part presents mathematcal model to solve the cash nventory problem n ATM networks by determnng the amount of money to place n ATMs and cash centers for each perod over a gven tme horzon. We formulate the problem as a MIP formulaton of a mult-echelon nventory problem wth sngle-tem capactated lot-szng. Parameters and decson varables regardng the model are lsted as follows: Parameters T = Number of tme perods n the plannng horzon t = Tme perod ndex; t =,2,..., T m = The total number of cash centers = Cash center ndex; =,2,..., m n = The total number of ATMs n each cash center j = ATM ndex; j =,2,..., n D jt = Amount cash demanded of ATM j managed by cash center at perod t q = Refllng cost (Baht per trp) from the bank to cash center o = Opportunty cost (per day) of money stored n cash center or ATM r j = Refllng cost (Baht per trp) from cash center to ATM j Scence Publcatons 285 a j = Suffcently large number for an upper bound of a cash order for each ATM j from cash center b = Suffcently large number for an upper bound of a cash order for each cash center C ATM = Avalable cash storage capacty of each ATM C CC = Avalable cash storage capacty of each cash center Decson Varables Q t = Order quantty of cash center at perod t X jt = Order quantty of ATM j delvered by cash center at perod t J t = Inventory level of money stored n cash center at the end of perod t I jt = Inventory level of money stored n ATM j managed by cash center at the end of perod t γ t = Bnary setup varable ndcatng where order quantty s allowed for cash center n perod t (=, f cash s reflled n cash center n perod t, 0 otherwse) δ jt = Bnary setup varable ndcatng where order quantty s allowed for ATM j under cash center n perod t (=, f cash s reflled n ATM j managed by cash center n perod t, 0 otherwse) Our formulaton of the problem s descrbed by the followng mathematcal program Equaton : m T = t= m n T (r j t jδ jt + oi jt ) = = = Mnmze Total Cost = (q γ + oj ) + Subject to Equaton 2: t t () Ij(t ) + Xjt Ijt = d jt; {...m}, j {...n },t {...T} (2) X jt ( αjδjt ) 0; {...m}, j {...n },t {...T} (3) X I C ; {...m}, j {...n },t {...T} (4) jt j(t ) ATM n J + Q J = X ; {...m},t {...T} (5) (t ) t t jt j= Q t (b γt ) 0; {...m},t {...T} (6) Q J C ; {...m},t {...T} (7) t (t ) CC

4 Supatchaya Chotayakul et al. / Journal of Mathematcs and Statstcs 9 (4): , 203 Fg.. Shortest path representaton of the lot szng dynamc program for tem and 4 perods X jt,ijt 0; {...m}, j {...n },t {...T} (8) Q,J 0; {...m},t {...T} (9) t t δjt 0,; {...m}, j {...n },t {...T} (0) γt 0,; {...m},t {...T} () The objectve functon () s to mnmze the total costs for allocatng cash n cash centers and ATMs (refllng costs and opportunty cost). Constrants (2) and (5) are the nventory balance constrants for ATMs and cash centers, respectvely. Constrants (3) and (6) forces a set-up cost to be ncurred for perods wth postve orderng cash of each ATM and each center; t requres δ jt to be f X jt s non-zero and t requres γ t to be f Q t s non-zero. Constrants (4) and (7) represent the capacty constrants: the overall cash n ATM must reman lower than the avalable capacty. Constrants (8) and (9) are non-negatve varables for order quanttes and nventory level. Constrants (0) and () specfy the bnary setup varables. For the large szes of problem, the MIP formulaton cannot solve or s very hard to solve n reasonable computatonal tme. Thus, we develop an approach based on reformulatng the model as a shortest path problem for fndng a near-optmal soluton The Shortest Path Reformulaton Eppen and Martn (987) presented the shortest path formulaton approach to reformulate the capactated lotszng problems based on a network dagram shown n Fg.. The new formulaton proposes varables Z jkl that s the fracton of the accumulated demand from perod k to perod l. Each node of network flow represents the perod, 2, 3, 4 and a dummy perod 5. The arc between nodes k and l means the choce of orderng the whole demand from perod k to perod l n perod k. The soluton s to fnd the shortest path from node to node 5. Scence Publcatons 286 We now reformulate the MIP as descrbed n Equaton (-) as a SP reformulaton. Only constrants that are related to capacty constrants, setup constrants and nventory balance constrants for managng ATMs wll be relaxed. The rest of the constrants wll not change from the orgnal model. The decson varables and SP reformulaton are as follows. Decson Varables Z jkl fracton of the total demand reflled n perod k for demand n perods k to l of ATM j managed by cash center I Equaton (2-26): m T = t= m n T (r j t jδ jt + oti jt ) = = = Mnmze Total Cost = (q γ + o J ) + Subject to: t t t (2) T+ Z l 2 jl = ; {...m}, j {...n } (3) = l= T + Z = Z ; l {2...T}, (...m), j {...n } (4) jkl jkl k= k= l+ T + Z jkl δjkl; k {...T}, (...m), j {...n } (5) l= k + T + l l= k + u= k dju Zjkl = X jt; k {...T}, (...m), j {...n } (6) Ij(t ) + Xjt Ijt = d jt; (...m), j {...n },t {...T} (7) X + I C ; (...m), j {...n },t {...T} (8) jt j(t ) ATM n J + Q J = X ; (...m),t {...T} (9) (t ) t t jt j= Q t (b γt ) 0; (...m),t {...T} (20) Q + J C ; (...m),t {...T} (2) t (t ) CC X jt,ijt 0; (...m), j (...n ),t {...T} (22) Q,J 0; (...m),t {...T} (23) t t

5 Supatchaya Chotayakul et al. / Journal of Mathematcs and Statstcs 9 (4): , 203 γt 0,; (...m),t {...T} (24) δjk 0,; (...m), j {...n },t {...T} (25) Z 0; k {...T},l {2...T + }, (...m), j {...n } (26) jkl The objectve functon (2) s to mnmze the total costs for allocatng cash n cash centers and ATMs (refllng costs and opportunty cost). Constrant (3) ensures that there s no more than one arc outgong from node for refllng each ATM. Constrant (4) ensures that f cash s placed n ATM n perod l, t must exst n perod l+. Constrant (5) forces the setup bnary varables δ jk to be one whenever money s reflled n ATM j managed by cash center n perod t. Constrant (6) s used to fnd the order quantty for ATM j managed by cash center at perod t. Constrant (7) s used to fnd the nventory level of money stored n ATM j managed by cash center at the end of perod t. Constrants (7)-(25) are dentcal to Constrants (2), (4)- () n the orgnal model. Constrant (26) enforces the non-negatvty requrements for varables Computatonal Tests Ths part presents the computatonal tests to evaluate the performance of problem formulatons: the MIP formulaton and the optmzaton-based heurstc approach (shortest path reformulaton), to fnd good optmal and heurstc solutons. These algorthms are tested on a computer wth an Intel(R) Core(TM) GHz CPU and 8GB RAM. Two soluton methods are compared usng the computatonal tme and the qualty of the solutons n varous confguratons of the ATMs network problem. The numbers of ATMs are vared as 0, 20, 30, 40, 50, 00, 50, 200, 250, 300, 500 and,000 ATMs wth 5 cash centers and 7 tme perods for testng performance of the proposed approach. The numerous data sets are randomly generated upon request. The optmzaton software Gurob Optmzaton ( was used to fnd the solutons for both algorthms. Scence Publcatons 3. RESULTS We use runnng tme to evaluate the qualty of the algorthms. Table reports a soluton, a soluton tme (CPU tme n seconds) and a percent dfference of the solutons of the two approaches for dfferent numbers of ATMs wth fxed number of cash centers and tme perods. 287 Fg. 2. Comparson of CPU Tme between the MIP formulaton and the SPR approach Table. Comparson between the MIP formulaton and the SPR approach MIP SPR % Dff. No. of Total CPU Total CPU of total ATMs costs tme (sec) costs tme (sec) costs 0 82, , , , , , , , , , , , > 8 h 588, , , ,325, ,63, Fgure 2 llustrates a graph comparson of CPU Tme between the MIP formulaton and the SPR approach. 4. DISCUSSTION From Table, t can be seen that the MIP formulaton can only solve small szes of the problem and spends more computatonal tme when the numbers of ATMs ncrease (Fg. 2). As the SPR algorthm can gve solutons n less computatonal tme although the problem s large. It was solved n a few mnutes wth,000 ATMs. Comparng the total costs of the two methods found that the SPR approach gave the optmal or close to optmal soluton. 5. CONCLUSION Ths research presented a new reformulaton based on a shortest path formulaton approach to solve a mult-

6 Supatchaya Chotayakul et al. / Journal of Mathematcs and Statstcs 9 (4): , 203 echelon nventory problem wth sngle-tem capactated lot-szng that was mplemented n a cash nventory problem n an ATM network n Thaland. Computatonal experments show that the proposed approach provdes very good solutons for problems of large realstc szes n short computatonal tme for a short plannng horzon compared wth the tradtonal mxed nteger program approach. There are two drectons for future research. Frst, the model should be extended to the capactated lotszng wth a stochastc demand envronment. Second, n ths study, we focus on allocatng cash to the cash centers and the ATMs wth certan delvery routes. However, cash delvery routes depend on the set of replenshment decsons n a gven tme perod. Therefore the refllng costs are not fxed by certan delvery routes but are determned by the aggregated length of the optmal routes. For future study, the mxed-nteger nventory and the vehcle routng problem should be combned to derve the optmal cash management strategy. 6. ACKNOWLEDGEMENT Ths study was fnancally supported by the Department of Industral Engneerng, the Texas Tech Unversty durng Scence Publcatons 7. REFERENCES Akbalk, A. and B. Penz, Exact methods for sngle-tem capactated lot szng problem wth alternatve machnes and pece-wse lnear producton costs. Int. J. Product. Econ., 9: DOI: 0.06/j.jpe Akbalk, A. and C. Rapne, 202. Polynomal tme algorthms for the constant capactated sngle-tem lot szng problem wth stepwse producton cost. Operat. Res. Lett., 40: DOI: 0.06/j.orl Akbalk, A. and Y. Pochet, Vald nequaltes for the sngle-tem capactated lot szng problem wth step-wse costs. Eur. J. Operat. Res., 98: DOI: 0.06/j.ejor Berk, E., A.O. Toy and O. Hazr, Sngle tem lotszng problem for a warm/cold process wth mmedate lost sales. Eur. J. Operat. Res., 87: DOI: 0.06/j.ejor Chubanov, S., M.Y. Kovalyov and E. Pesch, A sngle-tem economc lot-szng problem wth a nonunform resource: Approxmaton. Eur. J. Operat. Res., 89: DOI: 0.06/j.ejor Eppen, G.D. and R.K. Martn, 987. Solvng mult-tem capactated lot-szng problems usng varable redefnton. Operat. Res., 35: DOI: 0.287/opre Floran, M. and M. Klen, 972. Erratum: Determnstc producton plannng wth concave costs and capacty constrants. Manage. Sc., 8: DOI: 0.287/mnsc Hnd, K.S., 995. Effcent soluton of the sngle-tem, capactated lot-szng problem wth start-up and reservaton costs. J. Operat. Res. Soc., 46: DOI: 0.038/sj/jors/ Smuts, R., D. Dljonas and L. Bastna, Cash demand forecastng for ATM usng neural networks and support vector regresson algorthms. Proceedngs of the 20th EURO Mn Conference on Contnuous Optmzaton and Knowledge-based Technologes, May 20-23, LITHUANIA, Nernga, pp: Smuts, R., D. Dljonas, L. Bastna, J. Frman and P. Drobnov, Optmzaton of cash management for ATM network. Inform. Technol. Control, 36: 7-2. Waknaga, H. and K. Sawak, A Dynamc lot sze model for seasonal products wth shpment schedulng. Proceedngs of the 7th Internatonal Symposum on Operatons Research and ts Applcatons, Oct. 3-Nov. 3, Ljang, Chna, pp: Wu, T., L. Sh, J. Geunes and K. Akartunal, 20. An optmzaton framework for solvng capactated mult-level lot-szng problems wth backloggng. Eur. J. Operat. Res., 24: DOI: 0.06/j.ejor Zhang, Z.H., H. Jang and X. Pan, 202. A lagrangan relaxaton based approach for the capactated lot szng problem n closed-loop supply chan. Int. J. Product. Econ., 40: DOI: 0.06/j.jpe