A mathematcal rogrammng model for the synchronzed develoment of nonstatonary nventory and dstrbuton schedules Armağan Tarım and Aydın Ulucan Hacettee Unversty, Deartment of Management, eytee-ankara Abstract Ths study resents the extensons of a comrehensve roject comleted for Petrol Ofs, the major etroleum roducts dstrbutor n Turkey. In the comleted roject a decson suort system was develoed to mnmse the total transortaton costs of etroleum roducts under the statonarty assumton. The am of the current extensons, resented n ths aer, s to organse the transortaton actvtes, to be carred out throughout the lannng year between the exstng refneres and the coastal deots n a manner to meet the forecasted demand at the coastal deots, wth mnmum total nventory and transortaton costs. An ntegrated freght transort lannng model, ncororatng nventory holdng costs, s ntroduced under the nonstatonarty assumton and comments are made on further extensons of the model.. Introducton The ncreasng relevance of transortaton costs and the emhass ut on nventory reducton has recently stressed the mortance of synchronzng the logstc decson rocess. Ths study resents such an effort to ntegrate nventory and transortaton lannng as an extenson of a comrehensve roject comleted for Petrol Ofs (POAŞ), the major etroleum roducts dstrbutor n Turkey. The results of the
aforementoned roject and relevant lterature were already reorted by Ulucan and Tarım[]. See [2]-[5] for the recent research on the subject. The transortaton model s gven n detals n 3 for the sake of comleteness and future reference. The ssues, such as forecastng the regonal demand quanttes for dfferent tyes of etroleum roducts, whch were ndvdually covered n the scoe of the roject, are not dscussed. Followng the sngle erod transortaton lannng model, the mult erod ntegrated nventory-transortaton lannng model s ntroduced n 4. The nventory-transortaton lannng model ams at erformng the transortaton actvtes to be carred out throughout the lannng year between the exstng refneres and the coastal deots, n a manner to meet the forecasted demand at the coastal deots wth mnmum total nventory and transortaton costs. The comutaton of model arameters are dscussed n 5. 6, where the concludng remarks are gven, fnalses the aer. 2. Petrol Ofs and Its Surroundngs Due to the dffcultes that Turkey faced durng the second world war n the suly and dstrbuton of crude ol and etroleum roducts, the Turksh Government establshed Petrol Ofs (POAŞ) n 94. The basc dutes of POAŞ are defned as to urchase and mort etroleum roducts that are necessary for ublc and natonal defense requrements, to kee fuel nventores at varous laces n the country, to arrange dstrbuton of roducts, to buld, urchase and sell tanks and all tyes of transortaton vehcles ecular to transortaton of etroleum roducts. POAŞ also 2
carres out the tasks of rocurng the fuel requrements of Turksh Armed Forces and transortng fuels to elnes and unts due to the rotocols sgned at varous dates. POAŞ s the largest etroleum dstrbuton comany n the Turksh market, by marketng 53.2% (n 997) of the total fuels marketed by etroleum dstrbuton comanes n Turkey. In arallel to the magntude of the dstrbuton made, the lannng of the etroleum dstrbuton s one of the most sgnfcant roblems of POAŞ. The mortance of the savngs to be acheved n the dstrbuton and nventory exenses, whch have a relatvely huge orton n the overall exenses of the enterrse, s obvous. The subject of the study s the lannng, by means of a mathematcal rogrammng model, of the dstrbuton to the deots and of storage of the etroleum roducts roduced at the refneres. It s known that the crude ol s marketed several tmes, after t s refned as etroleum roducts, untl delvered to the consumer. The transortaton actvtes durng these marketng rocesses can be examned under the followng three major stages: () transortaton of the crude ol to the refneres, () transortaton of the etroleum roducts roduced at the refneres to the deots, () transortaton of the etroleum roducts from the deots to the retal servce statons. Among these, the transortaton of crude ol to the refneres was not covered n the scoe of the study due to the followng reasons. Frst of all, n Turkey, the crude ol s roduced rncally at Southeast Anatola and s forwarded to the refneres from ths locaton. Snce there s no other etroleum felds wthn the country, the otmzaton of the crude ol dstrbuton wthn the country s senseless due to the absence of 3
alternatve sources. Smlarly, snce the factors determnng the country of orgn and quantty of the mortaton made to meet the etroleum demand deends on the nternatonal commercal and oltcal relatons rather than the transortaton costs, the transortaton of the crude ol to the refneres was not ncluded n the transortaton model. The etroleum roducts transorted from the refneres to the deots of POAŞ are forwarded from the deots to the retal servce statons n order to be delvered to the consumers. POAŞ have no control on the urchasng decsons of the vendors. Therefore, the transortaton of the etroleum roducts from the deots to the retal servce statons s a roblem of mnmzaton of the overall transortaton costs concernng only the vendors, and due to ths reason, t s excluded from the scoe of the study. Almost all of the termnals of POAŞ are at the seashore, and ts transort strategy s based on sea transortaton. POAŞ utlzes the coastal deots for forwardng the etroleum roducts from the refneres to the consumers and the etroleum roducts are unloaded to these deots by means of tankers. As ndcated above, the am of the study s to fnd the olces that wll rovde the transortaton of the etroleum roducts from the refneres to only the coastal deots by means of tankers wth mnmum transortaton and nventory costs. 4
3. Sngle Perod Transortaton Model Formulaton Prevous work of Ulucan and Tarım[] covers the sngle erod transortaton model formulaton and dscusses the magntude of savngs acheved n POAŞ wth the use of ths model. The Logstcs deartment are resonsble for rovdng transortaton lans and used to erform ther duty manually whch s a dffcult and tme consumng task A software system was develoed to be used by the Logstcs Deartment of POAŞ. A model generator that uses the relevant data and generates the assocated mathematcal model, whch s a mxed nteger rogrammng model, was develoed as art of the transortaton software system. The generated mathematcal rogrammng model s, then, solved by means of the state-of-the-art solvers and, fnally, the reorts are generated. However, ths software system, desgned for the end user, s not dscussed here. In the mathematcal model develoed, the tankers owned by POAŞ as well as the tme-chartered vessels avalable on the market are consdered smultaneously. Not only the tankers to be emloyed n the transortaton actvty are determned, but also the refnery-deot route to be followed, the tmng of the dsatches, the tye of etroleum roduct to be transorted durng the year, the number of trs to be made by these tankers, ncluded n the transortaton lan, throughout the lannng erod can be secfed. Ths mathematcal model and the notaton are gven below for future reference: 5
j k tanker ndex, =,,m, deot ndex, j=,,n, refnery ndex, k=,...,, A k the set of deots suled by refnery k, K The {0,} varable, equals to f tanker s n the transortaton lan K ( ) The {0,} varable, equals to f tanker s used n transortaton of k K k black (whte) roducts from refnery k, and equals to 0 otherwse. h ( h ) The arameter equals to f tanker s caable of carryng black (whte) roducts and equals to 0 otherwse, X ( ) Percentage of the servce tme of tanker sent for carryng black jk X jk (whte) roducts from refnery k to deot j n one year, G ( ) Maxmum total amount of black (whte) roducts could be carred by jk G jk tanker from refnery k to deot j n one year. d j ( d j ) Total black (whte) roduct demand of deot j n one year. F Annual fxed contract cost for tanker. C jk Varable freght cost of assgnng tanker to refnery k-deot j route. Sngle Perod Transortaton Model: m m n mn Z = F K + Cjk ( X jk + X jk ) = = j= ( subject to K K K 0 =,..., m, (2) k k l A k lk k h K X 0 =,..., m,,...,, (3) h Kk X lk 0 =,..., m,,...,, (4) l A k 6
m = k= G jk m Gjk = k= jk jk X X jk jk d d j j j =,...,n, (5) j =,..., n, (6) X, X 0, K, K, K {0,} (7) k k The man assumtons of the above model are as follows. Frst, the dynamc nature of or seasonalty n demand s gnored and t s assumed that demand s statonary, whch s not the case. Second, t s assumed that f a vessel s assgned to a certan refnerydeot route then t serves n ths route for the rest of the lannng horzon. The second assumton s actually a result of the statonarty assumton. The etroleum roducts carred by tankers are classfed nto two grous -black (fuelols) and whte (jet fuels, gasolne tyes, kerosene, desel fuel). Some tankers are equed to carry just one tye of etroleum roducts black or whte-, whle the others can carry both one at a tme. In the latter category, snce t s rather a tme consumng and exensve rocess to convert the tankers nto the ones to carry whte roducts nstead of black, or vsa versa, t s not consdered as a standard ractce by POAŞ. Therefore, colour swtchng s not ermtted n the model. Tankers usually have number of holds and a tanker can carry several roducts, but only one roduct er hold. However, to satsfy the confnes of POAŞ, n develong the model to assgn the tankers, beng equed to carry roducts n both colours, to the deots, the tankers are constraned to carry only the same coloured roducts durng the whole lannng horzon. 7
Although the above assumtons are rather severe, consderable cost savngs are acheved. In the followng secton, the above assumtons are relaxed and alcablty of the modellng aroach s extended. 4. Mult Perod Inventory-Transortaton Model Formulaton The ntegrated nventory-transortaton model defned wth ts addtonal notaton s gven below. t tme erod ndex, t=,...,r, q roduct ndex, q=,...,s; the set of black (whte) roducts s S (S w ), K = f tanker s n the lan, =0 otherwse, P t the cost of assgnng tanker to a refnery n erod t that s dfferent from the one n erod (t-, J ( ) = f tanker s assgned to a dfferent refnery n erod t, =0 t J t otherwse, G ( ) maxmum total amount of black (whte) roducts could be carred by jk G jk tanker from refnery k to deot j n one comlete erod, I ( ) the erod closng nventory level of deot j for black (whte) roducts I n erod t, X ( ) ercentage of the erod servce tme of tanker sent for carryng jkt X jkt black (whte) roducts from refnery k to deot j n erod t, U ( ) redetermned black (whte) buffer stock levels for deot j n erod t, U Q the fxed cost of swtchng the colour of tanker from black to whte, or vce versa, 8
R t = f the colour of tanker s swtched n erod t, =0 otherwse, c ( ) a lnear holdng cost s ncurred on any black (whte) roduct carred n h c h nventory over from erod t to erod t+. The holdng costs are modelled by multlyng the weghted average rce of black (whte) roducts by the nomnal nterest rate. Mult Perod Inventory-Transortaton Model : mn m = m F K r = t= 2 P t + m n C jk = j= t= m r ( Jt + Jt ) + Q r n r ( X jkt + X jkt ) + ( = t= 2 R t j= t= c h I + c h I ) + (8) subject to, ( Kkt + Kkt ) 0 K + =,...,m (9) h h K K kt kt n j= n j= X X jkt jkt 0 =,...,m t=,...,r k=,..., (0) 0 =,...,m t=,...,r k=,..., ( I I = I = I j t m = k= q, + Gjk X jkt d j=,...,n t=,...,r (2) j t m = k= q S q, + Gjk X jkt d j=,...,n t=,...,r (3) q S 2 Kkt Kk 2 Kk R( t+ K + kt =,...,m t=,...,r- (4) 9
2 Kkt + Kk + 2 Kk R( t+ K =,...,m t=,...,r- (5) kt I U j=,...,n t=,...,r (6) I U j=,...,n t=,...,r (7) k k 2 K kt 2 Kk 2 J( t+ =,...,n t=,...,r- (8) k k 2 K kt + 2 Kk 2 J( t+ =,...,n t=,...,r- (9) k k 2 K kt 2 Kk 2 J( t+ =,...,n t=,...,r- (20) k k 2 K kt + 2 Kk 2 J( t+ =,...,n t=,...,r- (2 X jkt, X, I, I 0 jkt K, K, K, J, J, R kt kt t t t { 0,} Certan assumtons of the sngle erod transortaton model gven n 3 are relaxed. To be more secfc, the colour swtchng s ncororated n the mult erod nventory-transortaton model by means of ncurrng swtchng cost. In a smlar way, assgnment of the tankers to dfferent refneres n dfferent tme erods s also allowed. Objectve functon at (8) s equal to the mnmsaton of () the total fxed costs for the POAŞ owned tankers and the hrng costs for the tme-chartered vessels, () the total varable freght costs for all tankers, () the total nventory holdng costs, (v) the total transfer cost for all tankers, and (v) the total colour swtchng costs. The model 0
develoed for ths am covers a lannng erod of one year, whch s dctated by the rgd regulatons (for examle, the rental erod for the tankers s strctly one year). (9) assures that each tanker n the fleet wll carry but one colour of roducts and be assgned to a sngle refnery. (0) and ( rovde that the tanker should carry black or whte roducts, resectvely, durng maxmum 00 ercent of the total erod servce tme. (2) and (3) are the black and whte nventory transton equatons, resectvely, and assure that the demand s met. (4) and (5) ntroduce the ossblty of transortng dfferent colour etroleum roducts n consequtve erods. The redetermned buffer stock levels are mosed by (6) and (7). The assgnment of tankers to dfferent refnery-deot routes n dfferent tme erods s allowed by the ntroducton of the constrant sets (8)-(2. 5. Comutaton of Model Parameters The rocedure followed n the comutaton of arameters of the mxed nteger rogrammng model s gven below. Frst, the total varable costs of tankers consdered n the mathematcal rogrammng model were determned on the bass of the data from the revous lannng years. The total varable costs of POAŞ tankers were found by subtractng the costs of ersonnel, materals and amortsatons from the total costs. The hrng costs of the tme-chartered vessels nclude ersonnel, goods and amortsatons; therefore, the total varable costs of the tme-chartered vessels were comuted by subtractng the hrng costs from the total costs. In ths study, a constant called the tanker varable cost coeffcent n
$/(mle*ton) s comuted for each tanker searately. The varable cost of each tanker s dvded by the multlcaton of total mleage (n mles) and total cargo (n tons) of the revous year. To test the valdty of the tanker varable cost coeffcents, the calculatons are reeated, for all tankers, usng the revous years' data and t s observed that the results are very consstent. Second, the average tme (n hours) sent n the refneres and deots (queueng, shment, dscharge, etc.) by each tanker wth resect to ther tonnage and the dstance between the refneres and deots (n mles) are tabulated. Thrd, all ossble tanker-deot-refnery combnatons are generated and for each combnaton the tme each delvery takes (.e., the tme sent n the refnery, the return tr, the deot) s calculated. The data on average seed of emty and loaded tankers are rovded by the exerts of the Logstcs Deartment. Hence, the maxmum number of ossble delveres to be made n each erod for each combnaton s determned. Fourth, the total amount of etroleum roducts of black (or whte) tye to be carred by tanker from refnery k to deot j, f the tanker s assgned to route for the whole erod, s comuted n tons ( and ) whle the total dstance covered s n mles. Fnally, n the lght of ths nformaton t s now ossble to comute the varable costs,. C jk G jk G jk 6. Concluson 2
In ths aer, the extensons of a comrehensve roject comleted for Petrol Ofs are resented. In the comleted roject a decson suort system was develoed to mnmse the total transortaton costs of etroleum roducts under the statonary demand assumton. The am of ths aer s to develo a model to organse the transortaton actvtes, to be carred out throughout the lannng year between the exstng refneres and the coastal deots n a manner to meet the forecasted demand at the coastal deots, wth mnmum total nventory and transortaton costs. An ntegrated freght transort lannng model, ncororatng nventory holdng costs, s ntroduced under the nonstatonary demand assumton. Prevous constrants on colour swtchng and the assgnment of a tanker to only one sngle refnery-deot route are relaxed. To have a modcum of resemblance to realty such addtons to the model are crucal. Inevtably, the sze of the model becomes enormous from a ractcal ont of vew and t s observed that the resultng model cannot be solved n feasble tme usng the standard mathematcal rogrammng softwares. Comutatonally feasble soluton aroaches, lke Lagrangean relaxaton technque, are currently beng develoed by the authors. References [] A.Ulucan and A. Tarım, "Petrol ürünlernn denz yolu le taşınmasında malyet mnmzasyonu: Petrol Ofs A.Ş. çn karışık tamsayı rogramlama uygulaması," Hacettee Ünverstes, İktsad ve İdar lmler Fakültes Dergs, Vol.5, 997,.90-97. [2] M. Fox and D. Herden, "Sh schedulng of fertlzer roducts," OR Insght, Vol.2, 999,.2-28. 3
[3] F. Fumero and C. Vercells, "Syncronzed develoment of roducton, nventory, and dstrbuton schedules," Transortaton Scence, Vol.33, 999,.330-340. [4] S. Anly and A. Federgruen, "One warehouse multle retaler systems wth vehcle routng costs," Management Scence, Vol.36, 990,.92-4. [5] S. Anly and A. Federgruen, "Two-echelon dstrbuton systems wth vehcle routng costs and central nventores," Oeratons Research, Vol.4, 993,.37-47. [6] J. enjamn, "An analyss of nventory and transortaton costs n a constraned network," Transortaton Scence, Vol.23, 989,.77-83. [7] D. E. lumenfeld, L.D. urns, and C. F. Daganzo, "Analyzng trade-offs between transortaton, nventory and roducton costs on freght networks," Transortaton Scence, Vol.9, 985,.36-380. [8] D. E. lumenfeld, L. D. urns, and C. F. Daganzo, "Synchronzng roducton and transortaton schedules," Transortaton Scence, Vol.25, 99,.23-37. [9] P. Chandra and M. L. Fsher, "Coordnaton of Producton and Dstrbuton Plannng," Euroean Journal of Oeratonal Research, Vol.72, 994,.503-57. [0] T.. Chen, A. alakrshnan, and R. T. ong, "An ntegrated nventory allocaton and vehcle routng roblem," Transortaton Scence, Vol.23, 989,.67-76. [] M. A. Cohen and H. L. Lee, "Strategc analyss of ntegrated roductondstrbuton systems: models and methods," Oeratons Research, Vol.36, 988,.26-228. 4
[2] A. Federgruen and P. Zkn, "A combned vehcle routng and nventory allocaton roblem," Oeratons Research, Vol.32, 984,.09-037. [3] F. Glover, G. Jones, D. Karney, "An ntegrated roducton, dstrbuton, and nventory lannng system," Interfaces, Vol.9, 979,.2-35. [4] C. H. Martn, D. C. Dent, and J. C. Eckhart, "Integrated roducton, dstrbuton, and nventory lannng at Lbbey-Owens-Ford," Interfaces, Vol.23, 993,.68-78. [5] D. F. Pyke and M. A. Cohen, "Multroduct ntegrated roducton-dstrbuton systems," Euroean Journal of Oeratonal Research, Vol.74, 994,.8-49. 5