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. 319 012072 Vew the artcle onlne for updates and enhancements. Ths content was downloaded from IP address 148.251.232.83 on 06/10/2018 at 15:27
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
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
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
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)); [9] @for(km(j): [10] @sum(pal(i):alokas(i,j))=1); [11]@for(pal(I): [12] @sum(km(j):alokas(i,j)*volumelmbah(j))<kapastas(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
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 10000 (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 functon @OLE. 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
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
Appendx A. Dstance between IKM and IPAL alternatves (n meters). IPAL1 IPAL2 IPAL3 IPAL4 IPAL5 IPAL6 IPAL7 IKM1 281 233 651 820 652 950 956 IKM2 79 31 989 455 659 555 899 IKM3 233 156 570 453 705 553 763 IKM4 218 141 697 922 766 684 591 IKM5 197 120 681 439 490 786 864 IKM6 65 14 699 605 784 785 697 IKM7 553 601 50 479 667 610 607 IKM8 141 93 707 879 830 680 978 IKM9 5 79 434 537 502 514 415 IKM10 241 164 418 906 655 691 682 IKM11 308 260 435 611 430 644 463 IKM12 176 128 763 704 817 795 579 IKM13 610 926 491 63 30 555 446 IKM14 960 966 61 611 491 656 879 IKM15 647 418 851 897 849 15 70 IKM16 514 682 572 130 92 717 448 IKM17 748 887 37 409 499 958 665 IKM18 665 559 817 75 57 992 659 IKM19 536 490 721 45 75 599 483 IKM20 654 758 68 623 942 646 804 IKM21 489 914 754 60 76 630 762 IKM22 577 894 105 516 792 522 867 IKM23 964 411 52 747 700 454 766 IKM24 666 493 54 890 512 827 502 IKM25 890 428 61 957 741 641 909 IKM26 647 891 898 50 56 828 705 IKM27 660 923 506 40 70 979 629 IKM28 896 420 45 671 420 533 852 IKM29 830 626 449 47 77 895 578 IKM30 430 861 872 28 35 614 822 IKM31 634 411 669 34 10 620 470 IKM32 810 760 536 103 85 975 604 IKM33 893 611 414 55 15 657 818 IKM34 494 969 36 621 543 944 880 IKM35 419 427 61 731 495 545 597 IKM36 622 751 624 51 33 483 572 IKM37 913 609 429 594 784 305 120 IKM38 982 552 670 811 615 250 65 IKM39 965 420 651 533 412 80 20 IKM40 936 904 845 529 742 203 10 IKM41 575 478 994 876 756 219 188 7
Appendx B. Wastewater produced by IKM Processed soy bean (Kg/day) Daly Waste (ltres) Waste for 3 days dwell tme (ltres) IKM1 100 950 2,850 IKM2 200 1,900 5,700 IKM3 250 2,375 7,125 IKM4 40 380 1,140 IKM5 300 2,850 8,550 IKM6 100 950 2,850 IKM7 200 1,900 5,700 IKM8 25 238 713 IKM9 100 950 2,850 IKM10 100 950 2,850 IKM11 50 475 1,425 IKM12 60 570 1,710 IKM13 150 1,425 4,275 IKM14 150 1,425 4,275 IKM15 90 855 2,565 IKM16 70 665 1,995 IKM17 50 475 1,425 IKM18 50 475 1,425 IKM19 150 1,425 4,275 IKM20 50 475 1,425 IKM21 125 1,188 3,563 IKM22 40 380 1,140 IKM23 125 1188 3,563 IKM24 50 475 1,425 IKM25 200 1,900 5,700 IKM26 100 950 2,850 IKM27 400 3,800 11,400 IKM28 100 950 2,850 IKM29 150 1,425 4,275 IKM30 200 1,900 5,700 IKM31 500 4,750 14,250 IKM32 100 950 2,850 IKM33 50 475 1,425 IKM34 100 950 2,850 IKM35 500 4,750 14,250 IKM36 100 950 2,850 IKM37 250 2,375 7,125 IKM38 100 950 2,850 IKM39 80 760 2,280 IKM40 50 475 1,425 IKM41 50 475 1,425 8
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