ST-Munzir

Proceedings of the 6th IMT-GT Conference on Mathematics, Statistics and its Applications (ICMSA2010) Universiti Tunku Abdul Rahman, Kuala Lumpur, Malaysia

462

LinearProgrammingforParkingSlot Optimization:ACaseStudyatJl.T.Panglima PolemBandaAceh SaidMunzir1,MahyusIkhsan1andZainalAmin1 1

MathematicsDepartment,Facu MathematicsDepartment,FacultyofSciences ltyofSciences,SyiahKualaUnivers ,SyiahKualaUniversity,BandaAceh, ity,BandaAceh, Indonesia [email protected]

homepage:http://www.math-usk.org/smunzir/

research investigat investigate e the optimiza optimization tion of the available available parking parking Abstract. Abstract. This research area, area, based based on parking parking requireme requirement nt analysis analysis and the identifica identification tion of existing existing parkingproblem.Thestudyisexpectedtoprovideinformationtoreducetraffic delayduetoon-streetparking.Itisrelatedtotheneedofparkingareawhileoffstree street t parkin parking g area area is not not availa availabl ble. e. For a case case study study, , the street street of Pangl Panglima ima PoleminBandaAcehischosenasitislocatedinthecitycentralofBandaAceh. Themodeluselinearprogrammingsupportedbyobservationandsurveydatato formulateoptimizationproblem.Thestudyhasformulatedawell-posedproblem forparkingslotoptimizationbasedontheuserneeds,interpretedasparkingslot proportionaltoparkingaccumulatio proportionaltoparkingaccumulationandduration. nandduration. Keywords:parkingslotoptimization,linearprogramming

1.Introduction Transp Transport ortati ationplays onplays an import important ant and strate strategi gic c role role in the develo developm pmentof entof a nati nation on, , part partic icula ularl rly y in dist distri ribu buti ting ng the the prod produc uct t of the the deve develo lopm pment ent for for all all citizens.Generalproblemsthatoccurinurbantransportationisthetrafficjam. Oneofthesourcesofthetrafficjamisthedecreasesofroaddiameterdueto the the use use of the the part part of the the road road for for on-s on-str tree eet t park parkin ing. g. This This traf traffi fic c jam jam has a massive massive effect effect ifit if it is considered considered comprehensiv comprehensively. ely. Oneof thiseffects is, for instance,theexcessiveuseoffuelswhichcausealargeamountofeconomic losses.Hence,theefforttoreducethetrafficjamisnecessary,oneofwhichis themanagementofon-streetparkingproblem. The The traf traffi fic c flow flow prob proble lem m due due to on-s on-str tree eet t parki parking ng is an extr extrem emel ely y seri seriou ous s problem. problem. Several Several researches researches has been conducted conductedin in relation relation to this particular particular prob proble lem, m, such such as work works s of Sinag Sinaga a [10] [10] and and Seti Setiaw awan an [8]. [8]. The The analy analysi sis s of parkingdemandbasedoncharacteristicofparkingsiteshasbeenconductedby Yosrit Yosritzal zal [11] [11] and Widanen Widanengsi gsih h [12]. [12]. In addit addition ion, , they they also sugges suggested ted that that regressionmethodcanalsobeusedtodeterminethestandardofparkingslot demandaroundahospital.


463

The parking management problems were also studied using transportation management science as performed by Rapp [6]. Cannon [2] develop a simulationprogramtohandletheparkingproblem.Ontheotherhand,model using research operation employing linear programming is also utilized by researchers such as Bruglieri [1], Cordone [3] dan Silva [9]. For long term desainDjakfar[4]suggestedamethodusingcriteriaanalysis. From a series of previous researches related to the parking problem management,wecannotfindaresearchrelatedtoparkingslotallocationusing linearprogramming.Thedemandofparkingslotisimpossibletobefullfiled entirelyespeciallyifthestrategyistousetheon-streetparking,asthespaceis usuallyallocatedfortrafficflow. Inreality,theproblemthatusuallyemergeinrelationtoon-streetparkingisto determinethebestparkingslotallocationtodistributeamongdifferenttypesof vehicle on limited parking space. In this study, we focus on building a mathematicalmodeldescribingtheproblemandtrytoallocateoptimalparking slotproportionaltoeachtypeofvehicleusinglinearprogramming. PeunayongarealocatedinBandaAcehisacenterforbussinessactivityinthe city. The problem occurred in the road around the area is a heavily massive trafficactivity,suchasintheroadsegmentofJalanT. PanglimaPolem.From simpleobservation,obvioussourcesoflowroadperformanceinthisareaisthe on-street parking along the road and intersections. This on-street parking decreasesroadcapacityandincreaseroadsideobstacle.Thisproblemisthe resultofinsufficientparkingspaceavailableinthearea.

2.LiteratureReview  Parkingisdefinedasterminatingavehicleatacertainlocationandisapartof traffic circulation. Based on its location, the parking is classified into two categories, i.e.: on street parking and off street parking (Widanengsih dan Elkhasnet[12]).

Parking Control Unit (PCU) is the parking space used for a vehicle, which depend on vehicle dimension plus additional space needed for a vehicle to maneuvrewhosevaluedependingontheparkingangle.PCUofeachvehicle canbeobtainedintheTable2.1.


464

Table2.1PCUofeachtypeofvehicle

No

JenisKendaraan

Width (meter)

1 2 3 4 5 6 7

Becak Motorcycle Passenger Car Medium Bus Big Bus Truck Small Bus

1 0,8 1,5 2,1 3,5 2,4 1,6

Parking Width (meter) 1,5 1,3 2,5 3,1 4,5 3,4 2,6

Length (meter)

ParkingLength (meter)

2,2 1,9 4,1 6,0 9,3 7,2 4,1

2,7 2,4 5,1 7,0 10,3 8,2 5,1

Parkingcharacteristicsareparametersrelatedtotheamountofparkingdemand that have to be provided. According to Hobbs [5], parking characteristics includes: a. Parkingvolume,i.e.numberofvehicleenteringaparkingsite. b. Parking accumulation, number of vehicle parked at a parking site at a certaintime. c. Parkingindex,i.e.percentageofthevehicleoccupiedtheparkingarea. d. Parkingduration,i.e.timeinterval(inminuteorhour)foracertainvehicle parked at a parking site. Percentage amount of parking duration is formulated as ratio between the amount of vehicle parked during certain timeintervalandtotalnumberofvehicleobserved. e. Averageparkingduration,i.e.totalnumberofvehicleparkedduringcertain timeintervalcomparedtovehicleenterparkingsite. f. Parking exchanges, i.e. measurement of parking occupation calculated as ratiobetweenthenumberofvehicleparkedcomparedtoparkingcapacity available. g. Parkingutilizationlevel,computedfromtheratiobetweenaverageparking andparkingspacecapacity.Meanwhile,averageparkingisobtainedfrom theratiobetweensumofparkingaccumulationforallobservationtimeand numberofobservation.

3.ResearchMethod LocationofthissurveyisatJln.T.PanglimaPolemPeunayongBandaAceh and was conducted whole day beginning from 07.00 a.m. to 18.00 p.m.. Secondarysupportingdataandinformationforthisresearchisobtainedfrom DinasPerhubunganProvinsiAceh.Themethodusedtoobtainedparkingspace geometricdata and existing parking spacecapacity is through measuring the parking space area and parking space allocation for each vehicle. Data samplingforvehicleenteringorleavingtheparkingsitealongwithitsparking


465

durationisobtainedthrougha field surveywalking along theparkingsiteand counting the number of vehicle and its parking duration. Data processing is conductedusingMicrosoftExcel 2003andQMforWindows2.0. In relation to data obtained from the field survey, model analysis to be developed is conducted by including the user parking space demand proporsional to average parking accumulation and average parking duration. Proportionalityofaverageparkingaccumulationiscomputedevery15minutes for each type of vehicle. This proportion for motor cycle, car and becak are calculatedusingthefollowingformulation: p 1 p + p 1 2 p

+ p

p + p 1 2

+ p

3

2

p

3

3

p + p 1 2

+ p

3

( x1

+ x

( x1

+ x

2

2

+ x

),

+ x

),

3

3



( x1

+ x

2

+ x

3

)

where : Average parking accumulation of motor cycle (number of vehicle/15 minutes) :Averageparkingaccumulationofcar(numberofvehicle/15minutes) p 2 : Average parking accumulation of motor becak (number of vehicle/15 p 3 minutes) p1

and x1

+ x 2 +

x3

:Parkingspacecapacity

Meanwhile,proportionalitytoaverageparkingdurationevery15 minutesfor motorcycle,carandbecakarerespectivelyformulatedas: t 3 t 1 t 2 ( x1 + x2 + x3 ) ( x1 + x2 + x3 ) ( x1 + x2 + x3 ) t 1 + t 2 + t 3 t 1 + t 2 + t 3 t 1 + t 2 + t 3 , , where t 1  :Averageparkingdurationformotorcycle(minute) :Averageparkingdurationforcar(minute) t 2 :Averageparkingdurationforbecak(minute) t 3


466

The problem to solveis how tomaximize parking spacecapacity at Jalan T. PanglimaPolemsubjecttoavailableparkingland,andthesametimemeetthe thedemandofparkingforeachtypeofvehicle.Theparkingdemandisbased on proportionality of average parking accumulation and average parking duration. Thefocushereistoallocateparkingspaceforallthreetypesofvehicle.Based on parking space standard allocated for each type of vehicles by Dinas Perhubungan Aceh, parking space for each vehicle in this study is sebagai berikut: 2 a. Parkingspaceformotorcycleis3,12m . 2 b. Parkingspaceforcaris12,75m . c. Parkingspaceforbecakis4,05m2. Thestructureofdecisionmakingformaximizationof parkingcapacitycanbe arrangedas: Tabel3.1Thestructureofdecisionmakingformaximizationofparkingcapacity

activity No Coefficientofobjectivefunctions 1 2 3 4 5 6 7

Parkingspacearea Motor cycle parking accumulation proporsionaltoall Car parking accumulation proporsional to all Becak parking accumulation proporsional toall Proporsionalwaktuparkirrata-ratamobil Proporsional waktu parkir rata-rata becak motor

x1

x 2

x3

c1

c2

c3

Limitation factor

a11

a12

a13

≤

b1

a 21

a 22

a 23

≥

b2

a31

a32

a33

≥

b3

a 41

a 42

a 43

≥

b4

a51

a52

a53

≥

b5

a 61

a62

a63

≥

b6

a 71

a72

a73

≥

b7

Proportionofvehicleaverageparkingaccumulationevery15minutesforeach typeofavehicleiscomputedfromthevehicleaverageparkingaccumulation divided by total average parking accumulation for all vehicles and then multipliedbyparkingspacecapacity.Inmathematicalmodel,itcanbewritten as: 1. Proportion of average parking accumulation for motor cycle is p1 p1

+ p 2 + p 3

( x1 + x 2

+ x3

)


2. Proportion p 2 p1

+ p 2 + p 3

3. Proportion p3 p1

+ p 2 + p3

of

average

( x1 + x 2 of

+ x3

parking

accumulation

for

467

car

is

becak

is

)

average

parking

accumulation

for

( x1 + x2 + x3 )

Where p1 ismotorcycleaverageparkingaccumulation(numberofvehiclesin 15minutes), p 2 iscaraverageparkingaccumulation(numberofvehiclesin15 minutes), p 3 isbecakaverageparkingaccumulation (numberofvehicles in15 minutes),while ( x1

+ x 2 +

x3 ) isparkingslotcapacitytobeallocated.

Proportionofvehicleaverageparkingdurationevery15minutesforeachtype ofavehicleiscomputedfromthevehicleaverageparkingdurationdividedby totalaverageparkingdurationforallvehiclesandthenmultipliedbyparking spacecapacity.Inmathematicalmodel,itcanbewrittenas: 1. Proportion of average parking duration for motor cycle is t 1 t 1

+ t 2 + t 3

( x1 + x2

+ x3

)

2. Proportionofaverageparkingdurationforcaris

t 2 t 1

+ t 2 + t 3

3. Proportionofaverageparkingdurationforbecakis

( x1 + x2

t 3 t 1

+ t 2 + t 3

+ x3

( x1 + x 2

)

+ x3

)

Where t 1 isaverageparkingdurationformotorcycle(minutes), t 2 is average parking duration for car (minutes), and t 3 is average parking duration for becak(minutes). 3.1.ProcessingModelinLinearProgramming

The problem of parking slot allocation at Jln T. Panglima Polem uses three partsoflinearprogrammingmodel,i.e.: Objectivefunction:       Z = x1 + x 2 + x3  Subjecttoconstraints: 3,12 x1 + 12,75 x 2

+

4,05 x3

≤

Parking space area


x1

≥

x3

≥

x1

p1 + p 2

≥

x3

≥

+ p3

p1 + p2

+ p3

t 1 t 1 + t 2

+ t 3

t 2 t 1 + t 2

+ t 3

t 3 t 1 + t 2

+ t 3

( x1 + x2 + x3 )× α ( x1 + x 2 + x3 )× α

+ p3

p3

≥

x 2

p1 + p2 p2

≥

x 2

p1

468

( x1 + x 2 + x3 ) × α

( x1 + x2 + x3 )× α ( x1 + x2 + x3 )× α

( x1 + x2 + x3 )× α

Andnon-negativityconstraints: x1 , x 2 , x3 , p1, p 2 , p3 , t 1 , t 2 , t 3

≥0



and

0 ≤ α ≤ 1

Table4.1ParkingcharacteristicsofMotorcycle,CarandBecak Parking Characteristics Parking volume Parking capacity Peakofparkingaccumulation: -Time

-JumlahKendaraan(kendaraan) Parking index (%) Parkingduration: - Parking duration of largest number of vehicles(minutes) -Percentageofthe numberofvehiclepark (%) Average parking duration (minutes) Parking exchanges Parking utilization (occupation) (%)

Motor cycle 1211 147 14.30–14.45 p.m

Car 421 16 11.30–11.45 a.m.

Becak 100 9 08.45–09.00 and 09.00–09.15 a.m

135 91.84

26 162,5

10 111.11

15

15

15

68.09

68.60

79.80

46,72 8.24 67.18

25,69 26.3 106.875

24,24 1.23 51.67

After substitution of the values of p1 , p 2 , p 3 , t 1 , t 2 and t 3 from field surveyintothemodel,theoptimizationproblemcanbewrittenas:


469

Maximize Z = x1

+ x 2 +

x3

Subjecttoconstraints: ≤

588

x1

≥

0,82( x1

+ x 2 +

x3 ) × α

x 2

≥

0,14( x1

+ x 2 +

x3 ) × α

 x3

≥

0,04( x1

+ x 2 +

x3 ) ×

x

≥

0,48( x1

+ x 2 +

x3 ) × α

x 2

≥

0,27( x1

+ x 2 +

x3 ) × α

 x3

≥

0,25( x1

+ x 2 +

x3 ) × α

3,12 x1

+ 12,75 x 2 +

4,05 x3

Where x1 , x 2 , x3 , ≥ 0 and 0 ≤ 

≤ 1 .

4.ResultsandDiscussions Themodelwasinitiallytestedforvaluesofα between0.75to1usingQM for Windows 2.0to find optimal solution. However, nofeasible solution was found. This means that in trying to satisfy theaverage parking accumulation and parking duration simultaneously for each vehicle at more than 75% satisfaction, constraints cannot provide any feasible point in its interior. Followinguptheseresults,morerealisticscenariosfortheoptimizationwere made,i.e.: 1. Optimizationconsideringconstraintsonlyofaverageparkingaccumulation withvaluesof α from0.75upto1. 2. Optimizationconsideringconstraintsonlyofaverageparkingdurationwith valuesof α from0.75upto1. 3. Optimizationconsideringbothconstraintsofaverageparkingdurationand averageparkingaccumulationwithvaluesof α from0.5upto0.7. 4.1.

FormulationConsideringParkingAccumulationOnly

Optimization considering constraints only of average parking accumulation withvaluesof α from0.75upto1wasrunusing QMforWindows 2.0with thesolutionisshownintheTable4.2.


470

Table4.2SolutionforOptimizationconsideringaverageparkingaccumulation only

Variable

x1

α = 0,75 α = 0,80 α = 0,85 α = 0,90 α = 0,95 α = 1 121,6684 119,0373 115,8873 112,8731 109,8823 107,0182

x 2

14,769

15,575

16,2817

16,966

17,6289

18,2714

x3

4,2197

4,45

4,6519

4,8474

5,0368

5,2204

Z

140,6571

139,0623

136,8209

134,6505

132,548

130,5101

From the solution in the Table 4.2, it shows that the higher the value of α (level of satisfaction) the smaller the parking slot obtained especially the vehicle (motor cycle) having the highest accumulation in comparison to the others. 4.2.

FormulationConsideringParkingDurationOnly

Optimization considering constraints only of average parking duration with values of α from0.75up to 1 was run using QM for Windows 2.0 with the solutionisshownintheTable4.3. Table4.3SolutionforOptimizationconsideringaverageparkingdurationonly

Variable

x1

α =

0,75 α = 0,80 α = 0,85 α = 0,90 α = 0,95 α = 1 68,3923 63,7555 59,3562 55,1768 51,2011 47,4146

x 2

22,704

23,5808

24,4126

25,2029

25,9547

26,6707

x3

21,0222

21,8341

22,6043

23,3361

24,0321

24,6951

Z

112,1185

109,1703

106, 3732

103,7158

101,1879

98,7804

From the solution in the Table 4.2, it shows that the higher the value of α (level of satisfaction) the smaller the parking slot obtained especially the vehicle (motor cycle) having the highest average parking duration in comparison to the others. As average parking duration for all vehicle more balance (the differences were not very extreme), the resulting parking slot allocationalsomorebalancebetweenallthreetypesofvehicle.


471

4.3 Formulation Considering Both Parking Accumulation and Parking Duration Optimization considering constraints for both average parking duration and parkingaccumulationwithvaluesof α from0.5upto0.7wasrunusing QM forWindows2.0withthesolutionisshownintheTable4.4. Variable

x1

α = 0,5

α =

0,55 α = 0,60 α = 0,65 α = 0,70 95,91958 89,74738 83,93796 78,46021 73,28654

x 2

17,49884 18,66595

19,76446

20,80025

21,77855

x3

16,20263 17,28329

18,30043

19,25949

20,16532

Z

129,621

122,0028

118,52

115,2304

125,6966

From the solution in the Table 4.4, it shows that the higher the value of α (level of satisfaction) the smaller the parking slot obtained especially the vehicle (motor cycle) having the highest average parking accumulation and parkingdurationincomparisontotheothers.Incontrast,parkingslotsforboth CarandBecakincreasedwiththeincreaseof α (levelofsatisfaction). Comparisonofallthreeformulationssuggeststhattheformulationconsidering parking accumulation only is the best option if the total number of optimal parking slot is used as a performance measurement. The formulation considering average parkingduration only isclearlylesspreferable asit give theresultoflessnumberofparkingslotand,inpractice,italsousuallydoes notrelatesignificantlywiththecustomersatisfaction.

5. Conclusions The results of this study suggest that the formulation of optimization for parkingslotallocationgivesignificantlydifferentresultswhentheformulation considervariousaspectsofparkingrequirementsanddemands.Tryingtofulfill all parking demands andrequirements may lead to the optimization problem without any feasible region in the optimization. The most important aspects should come into the formulationbefore considering other aspects tosee the realisticoptimalresultfortheproblem.


472

References 1.

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