Chatpun Khamyat Department of Industrial Engineering, Kasetsart University, Bangkok, Thailand

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1 SOLVING THE OIL DELIVERY TRUCKS ROUTING PROBLEM WITH MODIFY MULTI-TRAVELING SALESMAN PROBLEM APPROACH CASE STUDY: THE SME'S OIL LOGISTIC COMPANY IN BANGKOK THAILAND Chatpu Khamyat Departmet of Idustrial Egieerig, Kasetsart Uiversity, Bagkok, Thailad Abstract: This research work provides the approach for solvig the oil delivery trucks routig problem by usig multi-travelig salesma problem (MTSP) solvig method with this research created algorithm. The created algorithm is used to modify the problem data format by trasformig a traditioal problem data form of a ode-arc etwork routig problems to become a job sequecig ad schedulig problems, which ca be solved by travelig salesma problem (TSP) for a sigle truck logistic system or multitravelig salesma problem (MTSP) for several trucks logistic system. This origial problem data ad cofiguratio are the everyday operatig procedure of oe SME s oil logistic compay i Bagkok. Origially, this compay accomplishes customer orders of oil delivery from a oil refiery warehouse to the specific customer local gas statios i Bagkok area by usig first come first serve policy with two oil delivery trucks that ca ot satisfy today busiess competitiveess because of high logistic cost. Accordig to this problem, this research provides the heuristic approach to solve the compay problem, showig by a small simulated example problem o this paper. The aswer from created heuristic is a TSP tour for each truck that examies the sequecig of order for all trucks. Computatio time of this heuristic model for small umber of ode problem is fast, but it may take a lot more time as combiatorial ature of TSP whe the umber of ode is growig up. The solutio is ot guaratee optimality of shortest track travelig distace but ca be a good lower boud that provides much logistic cost savig for a case study compay. Keywords: oil logistic problem, oil delivery truck, travelig salesma problem, logistic routig problem 339

2 1. INTODUCTION Curretly, Logistic busiess ecouter with very high competitiveess ad risk of busiess shutdow, especially the oil logistic busiess i Bagkok. All gas statios i Bagkok area are supplied their oils form a mai oil warehouse of Petroleum Authority of Thailad ( PTT) by usig the oil delivery trucks of ay oil logistic compay. There are a lot of oil logistic compaies i Thailad, which are i case of SME s busiess. They try to be survival ad competitiveess by doig ay improvemet such as cost reductio. The mai cost of this busiess is trasportatio cost of oil delivery trucks that relate directly to compay profitability. Normal truck delivery problems ca be established by may kids of operatio research model such as trasportatio problem, vehicle routig problem, travelig salesma problem or etc., but this oil truck delivery problem model is ot compatible with ay existig. The objective is to miimize total truck travelig distace that meas miimize logistic cost i this busiess. The job is the delivery of all gas statio order of all oil type per day. There are may customer orders from may differet locatios ad quatity demads to a oil logistic compay for all workig day-time from 9.00 to o clock that the trucks have to accomplish by ight-time from to 4.00 o clock, which followig order first come first serve s policy. Job schedulig ad sequecig is a importat part of ay kid of vehicle routig desig problem, iclude this kid of oil delivery truck routig problem. The oil delivery trucks routig problem ca be aalyzed ad model as its structure ad cofigulatio by applyied TSP approach for sigle truck or MTSP approach for multiple trucks, which the most appropreat existig opratio research mobel with some additoal geeratd part for makig the most problem accurcy ad validificatio. Datzig, Fulkerso ad Johso (1954) proposed that determiig the optimal of TSP for large umber of odes requires too much time so that may papers propose the heuristic algorithm for fidig the truck schedulig ad travelig path. As NP-hard ature of origial TSP, the problem may be cosidered as the class of NP-hard problem also whe the problem structure fall ito the TSP/MTSP category, Chartrad ad Oellerma (1993). The origial TSP/MTSP is oe of the applicatios of etwork problems, it is ecessary to choose a sequece of odes to visit so as to accomplish a specified objective. TSP/MTSP is a etwork problem that give a etwork ad a cost (or distace) associated with each arc, it is ecessary to start from a specified origiatig or depot ode, visit each ad every other ode exactly oe, ad retur to the startig ode i the least cost maer. The mathematical models of TSP/MTSP are formulated i form of iteger liear programmig which is 0-1 iteger programmig. There are so may solvig approaches o existig software that ca be for solvig this kid of iteger programmig. This paper examies the model of the oil delivery trucks routig problem ad solvig method base o TSP/MTSP approach. The cocept of TSP/MTSP will be applied with some geerated techique of assigmet problem to form a specific problem data from a ode-arc etwork routig problems to become a job sequecig ad schedulig problems. The existig OR software or Excel Solver is used to solve this problem because it is already use i the case study compay. Fially, the example tested problem, the result ad coclusio are preseted. 2. MAHEMATICAL MODEL AND SOLUTION TECHNIQUE 2.1. Traditioal TSP/MTSP mathematical model A survey of geeral TSP/MTSP was proposed (Lawler, Lestra, Ka, ad Shmoys, 1995) that ca be formulated as iteger programmig. This research tries to aalysis the structure of TSP mathematical model with structure of assigmet sub problem. Cosider the existig TSP mathematical (Miller, Tucker, Zemli, 1960). The formulatio of classical TSP was: 340

3 Subject to MiZ = c x i=1j=1 x = 1, i=1 j = 1, 2,..., x = 1, j=1 i = 1, 2,..., y -y +x -1 i j i j x =0or1 i,j The umber of cities is, the distaces are c ad the arcs i the tour are represeted by the variable x. The c is the distace from city i to j (c = α for i = j ). x is 1 if the salesma travels from city i to j ad 0 otherwise. The variables y i are arbitrary real umbers which satisfy the costrai of subtour eliimatio y i - y j +x -1. The MTSP formulatio for m salesma was give (Svestka & Huckfeldt, 1973) as followig: r r MiZ = d x ; r = + m -1 i=1j=1 Subject to r x = 1, j = 1, 2,...,r i=1 r x = 1, i = 1, 2,...,r j=1 y - y + ( + m -1)x + m - 2 i j i j x =0or1 i,j The ew distaces d are defied from the origial distace c as a augmet the matrix [c ] with ew m-1 rows ad colums, where each ew row ad colum is a duplicate of the first row ad colum of the matrix [c ]. It is assumed that the first row ad colum correspod to the home city. Set all other ew elemet of the augmet matrix to ifiity. All other terms have the same defiitio as previous model. The approach of trasformatio of MTSP for m-salesma to the classical TSP (Bellmore & Hog, 1974) was proposed by addig m-1 dummy to the origial etwork as the artificial startig ode ad solved MTSP from solvig TSP of the modified etwork. Accordig to this poit, TSP/MTSP structure is ot compatible to all cofiguratio of this research problem but it is the most approach to be adapted The oil delivery trucks routig problem formulatio The SME s oil logistic compay i Bagkok provides oil delivery service from a PTT oil warehouse to all customer gas statios. The gas statio will sed a order to oil logistic compay durig workig day-time from 9.00 to o clock. The order is the quatity of each oil type for ay gas statio. The oil delivery truck with 16,000 liters'tak eed to load all kid of oil at PTT warehouse ad delivery to a gas statios order by order, which followig first come first serve of order s policy at ight-time start from to 4.00 o clock. For example, if a truck got a job list of 5 orders to delivery oe ight as followig. 341

4 Table 1: The example problem of a oil deliver job list for oe oil delivery truck Job No. Gas Statio Oil (Liters) 1 G1 E20(14,000) 2 G3 G95(14,000) 3 G5 G91(15'000) 4 G3 Diesel(8,000) 5 G4 G95(13,000) Give, a truck start travel from oil logistic compay parkig(p) to PTT warehouse(w) for oil loadig of job No.1 ad go to delivery at a spacific gas statio(g1) for fiishig job No.1. Next, a truck travel back to W agai for oil loadig of job No.2, go to delivery at G3 for fiishig job No.2 ad work cotiuously as order first come first serve policy util job No. 5 doe at the last gas statio G4. Fially, a truck travle from G4 back to compay parkig P. Suppose, the distace etwork table (matrix c ) of this example problem is give i form of a ormal from/to distace table of ode P, W, G1, G2, G3, G4 ad G5 as followig. Table 2: The distace etwork matrix c of the example problem To P W G1 G2 G3 G4 G5 from P W G G G G G By usig a order first come first serve policy, the job sequecig solutio for oe oil truck is a truck route, which is start from P to W to G1 to W to G3 to W to G5 to W to G3 to W to G4 ad fially back to P. The total truck travelig distace is =181 uits that is ot a TSP solutio form. Accordig to this poit, the origial oil delivery trucks routig problem is ot compatible with ay existig operatios research model. From the previous origial TSP/MTSP mathematical model, variable x is equal 1 if the salesma travels from ode i to ode j. Whe we do the aalysis with the oil truck routig structure, the TSP/MTSP ca be applied to formulate a model base o the origial oil truck model as followig. Subject to MiZ = x d i, j = 1,2,..., x = S, j=1,2,..., i=1 j x = S, i = 1, 2,..., j=1 i S = 1 L =1,2,...,h i Z i L Z-Z i j =0 i=j y -y +x -1 i j x =0 or 1 i,j 342

5 Set Z L is a set of ay gas statio to delivery of all job No. i o a job list. For example, we ca defie set Z L of table 1 ( = 5, h= 4) that are Z L = {G1, G3, G4, G5}. All S i variables are equal to 1 for each job No. i, whe a specific gas statio from set Z L is delivered. Some gas statios o a example problem are delivered o may No. of job such as G3, so the formulated model of the oil delivery trucks routig problem eed some more costrais tha origial TSP/MTSP as showig above. The created distace matrix d are coverted from the origial distace c o table 2 as a job to job sequecig table for all No. of job =. The coversio approach is examied firstly by covertig ormal distace etwork table to become the job to job sequecig table for lower boud solutio, which compatible with TSP/MTSP problem structure without startig ad edig costrai of P to W ad G4 to P. From the example problem, a truck route of job No.1 is start from W go to G1 (W-G1) with distace 7 uits, No.2 is W-G3 = 15, No.3 is W-G5 = 27, No.4 is W-G3 = 15 ad No.5 is W-G4 = 23. Whe a truck starts by job No.1 ad follow by job No.2, the travelig is W-G1 + G1-W + W-G3, which is = 29 uits. If a truck starts by job No.1 ad follow by job No.3, the the travelig is W-G1 + G1-W + W-G5, which is = 41 uits. By doig this, the job to job sequecig table is created as followig. Table 3: The job to job sequecig table for lower boud solutio To Job From Job No.1 No.2 No.3 No.4 No.5 No No No No No A truck route ca be examied by solvig TSP of this job to job sequecig table as a lower boud solutio without startig edig costrai. The complete matrix d still eed some more rows ad colums for startig ad edig costrai of P to W ad G4 to P ad multi trucks system. A additioal first row (D r ; r = 1, 2,, m) of startig costrai is the first row of distace from P to all job No.i, which are all equal to distace from P to W = 10 uits. A additioal first colum D r of edig costrais is the first colum of distace from all job No.i edig gas statios to P, which equal to distace from all Gi to P of each job No.i. If the problem is for m trucks (m>1), addig m-1 dummy of first startig row ad colum D r to the job to job sequecig table as the artificial startig ode. If the example problem is accomplished by 2 trucks, the matrix d from distace etwork table is created by the previous approach is showed as followig. Table 4: The matrix d of example problem for 2 oil delivery trucks To Job From Job D 1 D 2 No.1 No.2 No.3 No.4 No.5 D D No No No No No The solutio of oil delivery trucks routig problem The algorithm for cratig matrix d of m trucks from the origial distace etwork matrix c ad problem a job list of all jobs No. i is show as followig: Step1: The dummy startig row D r ; r = 1, 2,, m is all row D r ; d = c of P to W for all job No.i. The dummy stertig colum D r ; r = 1, 2,, m is all colum D r ; d = c of all Gi to P of each job No.i. 343

6 Step2: The distace d of sequcig ay job No.i follow follow by ay job No.j is all d = c from W to Gi of ay job No.i A solutio of the oil delivery trucks routig problem ca be solved as MTSP from solvig TSP of this created distace matrix d. The existig OR software such as Excel Solver, which ca solve TSP ca be used to solve this problem. For this example problem, TSP solutio of matrix d o table 4 is a TSP tour of D 1 job No.5 job No.1 job No.4 D 2 job No.3 job No.2 - D 1 with total travelig distace 157 uits. This solutio is a MTSP solutio of 2 trucks that are Truck No.1: Start from P to delivery job No.5, job No.1, job No.4, ad back to P with distace 83 uits ad Truck No.2: Start from P to delivery job No.3, job No2 ad back to P with distace 74 uits. 3. CONCLUSION I this study, the mathematical model of t the oil delivery trucks routig problem which is a modify- MTSP approach for solvig a multi trucks system by usig a case study example problem of the SME s oil logistic compay i Bagkok, THAILAND. The algorithm for creatig the distace matrix of this problem is preseted. This heuristic approach for solvig this specific structure of trucks routig problem ca provide much shorter travelig distace, efficiecy for solvig large size of problem, but may take some more computatio step tha the origial first come first serve policy. The computatio time is depeds o ay commercial software that user use to solve TSP. By this study, the example of 20 odes problem ca be accomplished by usig Excel Solver i some millisecod uits time, but the computatio time will be grow up by expoetial ature because of combiatorial structure of TSP. The solutio from created distace matrix is a MTSP tour for m trucks, but it is ot guaratee optimality of shortest track travelig distace for each truck. However, the solutio ca be a better aswer tha the previous logistic policy that provides much logistic cost savig for this oil trucks delivery routig problem. REFERENCE LIST 1. Bellmore, M.,& Hog, S. (1974). Trasformatio of the multisalesma problem to the stadard travelig salesma problem. Joural of associatio of computig machiery, 21, Lawler, E.L., Lestra, J.K., Ka, A.H.G., ad Shmoys, D.B. (1995). The Travelig 3. Salesma Problem. Wiley. Chichester. 4. Li, S.,& ad Kerigha, B.W. (1973) A effective heuristic algorithm for the travelig salesma problems. Operatio Research, Miller, C.E., Tucker, A.W., ad Zemli, R.A. (1960) Iteger programmig formulatio of travelig salesma problems. Joural of associatio of computig machiery, 3, Orma, A.J., & Williams, H.P. (2004). A survey of differet iteger programmig formulatios of the travelig salesma problem. Workig paper by Lodo school of ecoomics ad political sciece,1. No. LSEOR

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