Capacitated set-covering model considering the distance objective and dependency of alternative facilities

Transcription

1 IOP Conference Seres: Materals Scence and Engneerng PAPER OPEN ACCESS Capactated set-coverng model consderng the dstance obectve and dependency of alternatve facltes To cte ths artcle: I Wayan Suletra et al 2018 IOP Conf. Ser.: Mater. Sc. Eng Vew the artcle onlne for updates and enhancements. Ths content was downloaded from IP address on 06/10/2018 at 15:27

2 Capactated set-coverng model consderng the dstance obectve and dependency of alternatve facltes I Wayan Suletra*, Yusuf Pryandar, Wakhd A Jauhar Industral Engneerng, Faculty of Engneerng, Unverstas Sebelas Maret *suletra@staff.uns.ac.d Abstract. We propose a new model of faclty locaton to solve a knd of problem that belong to a class of set-coverng problem usng an nteger programmng formulaton. Our model contans a sngle obectve functon, but t represents two goals. The frst s to mnmze the number of facltes, and the other s to mnmze the total dstance of customers to facltes. The frst goal s a mandatory goal, and the second s an mprovement goal that s very useful when alternate optmum solutons for the frst goal exst. We use a bg number as a weght on the frst goal to force the soluton algorthm to gve frst prorty to the frst goal. Besdes consderng capacty constrants, our model accommodates a knd of ether-or constrants representng facltes dependency. The ether-or constrants wll prevent the soluton algorthm to select two or more facltes from the same set of faclty wth mutually exclusve propertes. A real locaton selecton problem to locate a set of wastewater treatment faclty (IPAL) n Surakarta cty, Indonesa, wll descrbe the mplementaton of our model. A numercal example s gven usng the data of that real problem. Keywords: set-coverng, nteger programmng, ether-or constrant, wastewater treatment faclty, dstance goal 1. Introducton The locaton model bascally s a model of the relatonshp between the pont of demand and the locaton pont of the servce faclty. The decson varable n the locaton model s generally to determne where the optmal locatons for servce facltes are bult. The assumptons and obectve functons of the locaton model vary accordng to the characterstcs of the related problem. Daskn [1] classfed the locaton models based on mathematcal modellng (gravty model) nto 4 large groups, that s, analytc, contnuous, network, and dscrete models. The network models and the dscrete models, both assume that the faclty locatons and demand ponts are both dscrete, that s, they are only present at certan ponts wthn the area. The network model assumes a network or path connectng the demand ponts wth faclty locaton ponts whle the dscrete model does not requre such assumptons. One knd of the most wdely known dscrete locaton models s the set-coverng model. The decson varable for the set-coverng model s where the optmal locatons for servce facltes are bult so that the obectve functon s acheved. The obectve functon of the set coverng model s to mnmze the cost of the faclty locaton such that a specfed level of coverage s obtaned. Popular specal case of set-coverng models s to mnmze the number of servce faclty locaton ponts but all demand ponts are served. In other words, where to buld the facltes such that all customers are Content from ths work may be used under the terms of the Creatve Commons Attrbuton 3.0 lcence. Any further dstrbuton of ths work must mantan attrbuton to the author(s) and the ttle of the work, ournal ctaton and DOI. Publshed under lcence by Ltd 1

3 served wth mnmum cost s the common goal of the set coverng model. Because of ts dscrete nature, most set-coverng models are formulated usng nteger programmng [2, 3]. The followng s the nteger lnear programmng model of a basc set-coverng problem [4]: Mnmze Subect to: N X c (1) X X 1, (2) 0,1. (3) Where, c constructon cost of faclty at pont S N d coverage radus of the faclty a set of faclty locaton that can serve demand pont, N d S, (4) the dstance from demand pont to faclty locaton 1,f pont s selected as a faclty locaton X 0,otherwse The above model, equaton (1)-(4), explans that there s only one type of demand spread across multple locatons of pont and also only one type of faclty that can be placed at some ponts to serve the demand. Equaton (1) s the goal of the model, whch s to mnmze the faclty locaton cost. Equaton (2) assures that all demand ponts are served by at least one faclty. Equaton (3) s a logcal bnary constrant. Equaton (4) explans the defnton of coverage. However, the above standard model s not sutable to be appled n the case of determnng the locaton of IPAL facltes. The model does not nclude the capacty and facltes dependency constrants that our model consders. In addton, ths standard model does not consder the dstance obectve n ts obectve functon. A number of related works have been done to develop or modfy the standard set-coverng problem (SCP) model so that t's sutable for solvng dfferent cases. Raagopalan et al. [5] developed an SCP model appled to emergency medcal servce (EMS) by consderng demand fluctuatons at certan tme ntervals. Suletra et al. [6] proposed a new constrant n the SCP model to represent the Indonesan government rule that each faclty s used by at least two provders n the case of cellular telecommuncaton. The model s used to optmze the locaton of the ont base staton. Karm and Bashr [7] presented a numercal example of a hub of arport locaton based on the hub coverng locaton model developed n the study. Yaghn et al. [8] proposed a heurstc algorthm to solve an SCP model appled to tran drver schedulng problems. Furthermore, Lutter et al. [9] and Zhang et al. [10] developed a new constrant n the SCP model to account an uncertan avalablty of faclty appled to the case of emergency servce facltes. Although the research on SCP has been extensvely dscussed n the lterature, there s no model sutable for solvng the problem dscussed n ths study. The goal of the problem s to mnmze the number of facltes and the total travel dstance n the condton that the capacty of each faclty s lmted and there s the dependency property among facltes. 2. Proposed set coverng model The model proposed n ths study s an nteger programmng model that refers to SCP. The basc model of SCP requres that each demand pont s served by at least one faclty (supply pont). In ths study, the requrement s rather dfferent, each IKM (Industr Kecl dan Menengah) or SME s (Small and Medum Enterprses) represented by a demand pont must be served by only one IPAL faclty. The goal s to mnmze the number of IPAL faclty that can serve all demand ponts. Besdes 2

4 mnmzng the number of IPAL faclty, the obectve functon s formulated to mnmze the total dstance between IKM and IPAL faclty. A bg number s used as a weght on the number of faclty mnmzaton goal to force the soluton algorthm to gve frst prorty to ths goal. Therefore, mnmzaton of the total dstance s a second prorty goal that s very useful when alternate optmum solutons for the man goal exst. Mnmze cx dy (5) Subect to: Y 1, (6) Y X 0,, (7) l Y p, (8) X 1 M, (9) X 0,1, (10) 0,1,. Y (11) where, a bg number to prortze the frst goal c d l p the dstance of IPAL to IKM the volume of wastewater produced by IKM the capacty of IPAL M = a set of IPAL wth mutually exclusve selecton (dependency property) X = bner number, 1 f alternatve IPAL s selected to buld, and 0 s otherwse Y bner number, 1 f alternatve IPAL to serve IKM, and 0 s otherwse The frst term of the equaton (5) s the frst goal of the proposed model whch mnmzes the number of facltes, and the second term n the same equaton s the second goal whch mnmze the total dstance from IPAL to IKM n the condton that the frst goal must be acheved frst. A bg number, c, s used to force the model to assgn the prorty to the frst goal. The second goal s an mprovement goal. It wll work when the alternate optmum solutons for the frst goal exst. Equaton (6) s used to model the requrement that each IKM must be served by only one IPAL faclty. Equaton (7) s a logcal constrant to ensure that f the IPAL s allocated to serve the IKM then the IPAL must be selected to buld. Equaton (8) explans that the capacty of each IPAL cannot be volated,.e. the total waste dstrbuted to the IPAL from all IKM s less than the capacty of IPAL. Equaton (9) descrbes the mutually exclusve selecton representng the dependency property among IPAL. Equaton (10) and (11) are logcal bnary constrants. 3. Numercal mplementaton To descrbe the computaton and the applcablty of the proposed model, we use the data from a real problem, that s, locaton selecton of IPAL facltes n Moosongo, Surakarta, Indonesa (fgure 1). The facltes serve the surroundng SME s tahu tempe. An IKM s a demand pont and the IPAL s the faclty that serves the demand ponts. Havng observed n ste, there are 41 demand ponts (=1,2,,41) and 7 alternatve IPAL locatons (=1,2,,7). The map of IKM ponts and IPAL ponts are depcted n fgure 1. 3

5 Fgure 1. Dsperson of 41 IKM locatons and 7 IPAL alternatves. The frst goal s to mnmze the number of IPAL facltes and the second s to mnmze the total dstance of IKM to IPAL facltes. The constrants are capacty of IPAL, each IKM must be served only by one IPAL faclty (and all IKM must be served), and facltes dependency constrants. Appendx A shows the dstance measure from IKM to IPAL alternatves and appendx B shows the volume of wastewater produced by each IKM. [1] model: [2] sets: [3] pal/s1..s7/:kapastas,dplhtdak; [4] km/c1..c41/:volumelmbah; [5] palkmcombnaton(pal,km):araktempuh,alokas; [6] endsets [7] mn=@sum(pal(i):10000*dplhtdak(i))+@sum(palkmcombnaton(i,j): [8] araktempuh(i,j)*alokas(i,j)); [11]@for(pal(I): [13]@for(palkmcombnaton(I,J): [14] alokas(i,j)-dplhtdak(i)<=0); [15] dplhtdak(1)+dplhtdak(2)<=1; [16] dplhtdak(6)+dplhtdak(7)<=1; [17]@for(palkmcombnaton(I,J):@bn(alokas(I,J))); [18]@for(pal(I):@bn(dplhtdak(I))); [19]data: [20]kapastas,volumelmbah,araktempuh=@OLE('E:\data_lngo\data_km_tahu [21]tempe.xls','kapastas','volumelmbah','araktempuh'); [22]enddata [23]end Fgure 2. Lngo code mplemented n the problem 4

6 Branch-and-Bound algorthm s used to mplement the proposed model to the real problem of IPAL at Moosongo, Surakarta. Lngo software on a notebook 5 2.6GHz wth 8Gb Ram and SSD 256Gb run the model n 5 second to get the global optmum soluton. The code lsted n fgure 2 s the lngo code mplemented to solve ths problem. The lnk between the proposed model and the lngo code can be explaned as follows. s represented by the code alokas, number), d X by the code araktempuh, by the code dplhtdak, c by the number (bg p by the code kapastas, l Y by the code volumelmbah. The obectve functon of the proposed model, equaton (5), s represented by the row [7] and [8] of the lngo code lst. The frst constrant of the proposed model, equaton (6), s represented by the row [9] and [10], the second constrant or equaton (7) by the row [13] and [14], the thrd constrant or equaton (8) by the row [11] and [12], the fourth constrant or equaton (9) by the row [15] and [16], the ffth constrant or equaton (10) by the row [18], and the sxth constrant or equaton (11) by the row [17]. The row [20] and [21] show that the data of,, and, whch are mported from MS Excel fle usng the Lngo The optmum soluton s depcted by fgure 3 and the sze of the problem s shown by the fgure 3. Branch-and-Bound algorthm needs 169,273 teratons to get the global optmal soluton n 5 seconds. Four IPAL facltes are selected to serve 41 IKM,.e. IPAL 2, IPAL 3, IPAL 5, and IPAL 7. The optmum allocaton s as follows. IPAL 2 serves 11 IKM,.e. IKM 1, IKM 2, IKM 3, IKM 4, IKM 5, IKM 6, IKM 8, IKM 9, IKM 10, IKM 11, and IKM 12. IPAL 3 serves 11 IKM,.e. IKM 7, IKM 14, IKM 17, IKM 20, IKM 22, IKM 23, IKM 24, IKM 25, IKM 28, IKM 34, and IKM 35. IPAL 5 serves 13 IKM,.e. IKM 13, IKM 16, IKM 18, IKM 19, IKM 21, IKM 26, IKM 27, IKM 29, IKM 30, IKM 31, IKM 32, IKM 33, and IKM 36. IPAL 7 serves 6 IKM,.e. IKM 15, IKM 37, IKM 38, IKM 39, IKM 40, and IKM 41. p l d Fgure 3. Summary of problem and Lngo output. 5

7 Fgure 4. Optmum allocaton of 41 IKMs to 4 selected IPAL alternatves. 4. Concludng remark We propose a knd of set-coverng model to mnmze two dfferent goals usng the sngle obectve functon. The problem s formulated as a sngle obectve lnear nteger programmng. The standard set coverng model s a specal case of our model by settng t to a sngle goal of mnmzng the number of facltes and removng the capacty and facltes dependency constrants. We mpose several assumptons about the alternatve facltes and demand ponts. We use Branch-and-Bound Algorthms to solve the nteger model and t needs ust 5 second to solve the numercal example. The future research can look nto accommodatng a multple obectve approach to represent a more realstc problem,.e. a problem consderng obectve and subectve crtera smultaneously. Further, the relaxaton of assumptons about the faclty and demand ponts may also be nterestng to be studed. 6

8 Appendx A. Dstance between IKM and IPAL alternatves (n meters). IPAL1 IPAL2 IPAL3 IPAL4 IPAL5 IPAL6 IPAL7 IKM IKM IKM IKM IKM IKM IKM IKM IKM IKM IKM IKM IKM IKM IKM IKM IKM IKM IKM IKM IKM IKM IKM IKM IKM IKM IKM IKM IKM IKM IKM IKM IKM IKM IKM IKM IKM IKM IKM IKM IKM

9 Appendx B. Wastewater produced by IKM Processed soy bean (Kg/day) Daly Waste (ltres) Waste for 3 days dwell tme (ltres) IKM ,850 IKM ,900 5,700 IKM ,375 7,125 IKM ,140 IKM ,850 8,550 IKM ,850 IKM ,900 5,700 IKM IKM ,850 IKM ,850 IKM ,425 IKM ,710 IKM ,425 4,275 IKM ,425 4,275 IKM ,565 IKM ,995 IKM ,425 IKM ,425 IKM ,425 4,275 IKM ,425 IKM ,188 3,563 IKM ,140 IKM ,563 IKM ,425 IKM ,900 5,700 IKM ,850 IKM ,800 11,400 IKM ,850 IKM ,425 4,275 IKM ,900 5,700 IKM ,750 14,250 IKM ,850 IKM ,425 IKM ,850 IKM ,750 14,250 IKM ,850 IKM ,375 7,125 IKM ,850 IKM ,280 IKM ,425 IKM ,425 8

10 References [1] Daskn M S 2008 What You Should Know About Locaton Modelng, Naval Research Logstcs [2] Schllng D A, Jayaraman V and Barkh R 1993 A revew of coverng problems n faclty locaton Locaton Scence [3] Current J, Daskn M and Schllng D 2002 Dscrete network locaton models Faclty Locaton Theory: Applcatons and Methods ed Drezner and Hamacher H (Berln: Sprnger-Verlag) Chapter 3, pp [4] Owen S H and Daskn M S 1998 Strategc faclty locaton: a revew European Journal of Operatonal Research [5] Raagopalan H K, Saydam C and Xao J 2011 A multperod set coverng locaton model for dynamc redeployment of ambulances Computers & Operatons Research [6] Suletra I W, Wdodo and Subanar 2011 Optmum locatons of mult-provders ont base staton by usng set-coverng nteger programmng: modellng & smulaton The 6th SEAMS-UGM Conference [7] Karm H and Bashr M 2011 Hub coverng locaton problems wth dfferent coverage types Scenta Iranca, Transactons E: Industral Engneerng [8] Yaghn M, Karm M and Rahbar M 2015 A set coverng approach for mult-depot tran drver schedulng Journal of Combnatoral Optmzaton 29(3) [9] Lutter P, Degel D, Busng C, Koster A M C A and Werners B 2017 Improved handlng of uncertanty and robustness n set coverng problems European Journal of Operatonal Research 263(1) [10] Zhang B, Peng J and L S 2017 Coverng locaton problem of emergency servce facltes n an uncertan envronment Appled Mathematcal Modellng