SERDO KOS, D. Se. ZDENKA ZENZEROVIC, D. Se. Pomorski fakultet- Rijeka Studetska 2, 51000 Rijeka, Republika Hrvatska Itegrated Trasport Prelimiary Commuicatio U. D. C.: 656.073.235:656.61 i\ccepted: Ja.29,2003 1\pproved: Dec. 10, 2003 MODEL OF OPTIMAL CARGO TRANSPORT STRUCTURE BY FULL CONTAINER SHIP ON PREDEFINED SAILING ROUTE ABSTRACT This paper presets the mathematical model for solvig the problem of defiig optimal cargo trasport structure, occurrig whe, o a predefied sailig route, adequate umber of cotaiers of various types, masses ad sizes, possibly icludig RO!RO cargo, is to be selected, i.e., a "cotaier lot" is to be established i loadig ports with the aim of gaiig maximum ship profit ad, at the same time, of exploitig useful load ad trasport capacity of cotaier ship as much as possible. The implemetatio of the proposed model eables cosiderable icrease i the efficiecy of cotaier ship operatios. The model was tested usig a umerical example with real data. The applied post-optimal aalysis examies the ifluece of chage i some values of the mathematical model o the resultig optimal program. KEYWORDS cotaier trasport, structure of cotaier trasport by ship, liear programmig, post-optimal aalysis 1. INTRODUCTION The cotaierised maritime trasport represets higher level of trasport services as compared to the classic maritime trasport techology. I order to achieve the maximum possible techological exploitability rate of full cotaier ship, before buildig such a ship the ower should do the followig: - defie all geographical areas where the ship will operate (wids, temperatures, ice ad freezig, usage of ilad waterways ad chaels, passages uder bridges, miimum UKC, maximum draft); - aalyze cotaier flow rate o particular sailig route ad, usig appropriate method, determie average TEU mass, which will equal the "desig TEU mass" i ship desig - moitor, i terms of statistics, ad defie, usig descriptive statistical methods, sigificat umber of 20' ad 40' cotaiers i order to make correct decisio o istallatio of sufficiet umber of co- Promet- Traffic- Traffico, Vol. 16, 2004, No. 1, 15-20 taier cell guides for 20' cotaiers ad defie adequate umber of bays. The desig TEU mass ad average umber of TEUs i defied area of operatio will determie costructioal, techical, techological ad exploitatio characteristics of the ship. As to the ships already i operatio, higher coefficiet of techological exploitability is achieved through "pyramidal stackig" providig more favourable ratio betwee quatities of cargo ad ballast, ad therefore better stability of the ship durig avigatio. The rate of techological exploitability of cotaier ships is the ratio betwee the umber of actually loaded cotaiers (expressed as TEU) ad the total umber of positios accommodatig cotaiers, expressed as TEU, where useful load must ot be exceeded. The maximum rate of techological exploitability is achieved whe the average mass of loaded cotaiers equals the "desig TEU mass" for each respective ship. The trasport process of maritime cotaier techology ca be cosidered from various poits of view: techical, techological, ecoomic, legal, etc. High techological exploitability rate of cotaier ships iflueces their efficiecy of operatio measured i respect to the profit resultig from trasport of cargo accommodated i cotaiers o the particular sailig route. A ship will ot yield profit if empty cotaiers owed by the operator are trasported, i. e., humaitaria aid, which is trasported free of charge. Sice the operatio of a ship is based o profit, it is importat that cargo trasported o the cotaier ship is arraged so that costs are miimal or that it yields maximal possible profit, i. e., to determie such a structure of cotaier ship trasport, accordig to ISO cotaier types, which would yield maximal profit, cosiderig useful load ad trasport capacity of the cotaier ship. This paper cosiders the so-called full cotaier ships 1 addressig exclusively ISO cotaiers 2 ad possible restrictios i respect to RO/RO cargo o trail- 15
ers loaded o deck, ulike other ship types where the structure of cargo ca be bulk cargo, geeral cargo, mixed cargo, liquid or gas cargo, which are more difficult to quatify uambiguously. Cotaiers loaded o full cotaier ships are uiformly iteratioally stadardized, which eables settig of uiversal coditios to be met i drawig up adequate mathematical model. The objective of this paper is to show that adequate operatio research methods, specifically liear programmig, ca be used to determie optimal cargo structure trasported o cotaier ships. 2. PROBLEM DEFINITION The profit yielded from trasport of cotaierised cargo by fleet of full cotaier ships o particular sailig route is directly related to the structure of cargo stowed i cotaiers loaded i ports of loadig ad trasported to destiatio ports. I ports of loadig, agets, forwarders ad ship -owers' represetatives fid cargo offered for trasport i ISO cotaiers. O the basis of the total cargo supply ad umber of cotaiers available for loadig, "cotaier lots" are formed for each particular cotaier ship of a fleet maitaiig regular service o particular sailig route. The ship maagemet does ot defie the structure of cargo, i.e., cotaier lot to be trasported, but rather orgaises the loadig, stowig, trasport ad uloadig of cotaiers, based o the pre-set cotaier lot i port of loadig. Arragemet of cotaiers plaed for loadig i ship cells is made takig ito cosideratio the followig factors: voyage rotatio, stability ad ship costructio load, allowed support surface load, quatity, size ad type of cotaiers, hazardous cargo i cotaiers ad ship stay i port (loadig/uloadig stadard rates). The problem of defiig optimal cargo structure for trasport by full cotaier ship occurs whe it is ecessary to select the adequate umber of cotaiers of various types ad masses ad possibly RO/RO cargo o trailers, out of the umber of cotaiers available i port of loadig so that maximal profit is gaied ad, at the same time, to maximally exploit the payload ad TEU capacity of a ship. There are several solutios to the problem. Oe solutio uses a series of sirulatios or attempts util the solutio providig the maximum profit is foud. This method is relatively time-cosumig ad slow ad it is ot cosidered for commercial use. I real life, such problems are most ofte solved ituitively by shipmasters, o the basis of their may years of experiece i the trade. 16 Maximal profit as a optimizatio criterio is chose because full cotaier ships have very high daily fix expeses (over 5000 USD) ad from the ecoomic poit of view the most importat issue is to yield maximum profit per voyage. 3. FORMULATION OF THE MATHEMATICAL MODEL I order to solve the previously defied problem usig operatios research, it is ecessary to set adequate mathematical model as follow: Criteria fuctio maxz= LCjXj (1) with costraits Lajxj5,Nk, LXj5,di, i=1,2,3,...,m L x j 5, g i, i = 1, 2, 3,..., m (2) (3) x j ~ 0, j = 1,2,3,...,, (5) where: Z - criteria fuctio, cj - criteria coefficiet of j-th variable, i. e., profit per oe trasported cotaier or trailer, xj- quatity (amout) ofj-th variable, i.e., umber of cotaiers or trailers for trasport accordig to optimal solutio foud, Nk- ship payload for respective voyage i tos, d;- ship capacity i TEU, i. e., umber of free spaces or positios for loadig cotaiers or trailers, g; - umber of cotaiers ad trailers of various types, sizes ad masses available for loadig, aj- quatity of restrictio required for oe uit of j-th variable, i. e. uit mass of cotaier or trailer, m - umber of restrictios, i.e. ship payload, TEU trasport capacity of ship ad available umber of cotaiers ad trailers, - umber of variables, i. e. umber of cotaier ad trailer types of various sizes ad masses. To solve the optimal full cotaier ship trasport structure accordig to the set mathematical model meas to defie the structure variable values Xj (4) Promet- Traffic- Traffico, Vol. 16, 2004, No. 1, 15-20
(j = 1,2,3,..., ) that yield maximal value of the fuctio of the criterio Z, ad at the same time meet all the basic ad additioal coditios, i. e. set restrictios. The values of gi, Cj, aj, m ad ca be ay positive iteger. Some ew restrictios ca occur i maritime practice, for example, if certai quatity of some cotaier or trailer type, size or mass are madatory i the structure of optimal program regardless of the fiacial effects of such a decisio. I that case, the mathematical model would be: Criteria fuctio maxz =c1x1 +czxz+... +cx with costraits a1x1 +azxz+... +ax ~Nk xl xl +xz+.. :txk xk+l+.. :tx f ~gl xi~8i5:di,i= 1,2,3,...,m (6) Detailed explaatios regardig implemetatio of particular solutio methods ad modes are available i respective literature o operatios research [1], [2] ad [9]. 4. NUMERICAL EXAMPLE A example usig real data was chose to illustrate the set mathematical model. I ports of loadig, there is a variety of cotaierised cargo available for maritime trasport. Mass ad profit per cotaier types are show i Table 1. Table 1 - Daily profit per oe trasported cotaier (7) ready for loadig (8) (9) (10) Xj ~O,j = 1,2,3,...,. (11) As ca be see from the preseted model, restrictios stated i the system of o-equatios are set primarily because, i maritime practice, it is ot possible to form a structure of cotaier trasport that would simultaeously fully exploit payload ad trasport TEU capacity of a cotaier ship. Also, o -equatios are more flexible tha equatios because they allow a solutio to be selected withi a certai iterval. Specifically, if restrictios uder (7)-(10) had "="istead of"~", the umber of possible solutios i the set mathematical model would be cosiderably reduced. The problem of fidig optimal cargo trasport structure for a cotaier ship usig the proposed mathematical model is solved by the method of liear programmig, particularly, usig simplex method. For the sake of simplificatio, the applicatio of persoal computer software, e. g. QSB [4], is recommeded, rather tha maual calculatio. Also, it is recommeded to perform post-optimal aalysis after fidig the optimal solutio i order to establish the ifluece of chages i some elemets of the mathematical model o the optimal solutio foud ad to compare alterative solutios of the give problem, o the basis of which adequate busiess decisio could be made. Type ad size of cotaier 20'DB 20'0T 20'RF 20'0S Uit mass i tos 11 15 13 12 Profit per 1 cotaier i USD Type ad size of cotaier 48 63 60 59 20'TC 40'PL 40'RF 40'DB Uit mass i tos 16 23 25 21 Profit per 1 cotaier i USD 69 70 72 67 The followig quatity is ready for loadig, preseted by type, size ad allowed cotaier mass 3 : 160 x 20' DB a 11t, 400 x 20' TC a 16t, 120 x 20' OT a 15t, 160 x 40' PL a 23t, 200 x 20' RF a 13t, 82 x 40' RF a 25t, 300 x 20' os a 12t, 80 x 40' DB a 211. Trasport is to be carried out by meas of full cotaier ship of29,434 tos total cargo capacity ad 1762 TEU 4 trasport capacity. The plaed duratio of the voyage is ie days. The average cosumptio of heavy fuel is 45t/day ad water cosumptio is 15 t/day. Ship avigates i "witer zoe". Ship taks cotai 300 tos of light fuel, 150 tos of lube oil, 1100 tos of heavy fuel oil, while the crew ad other provisios weigh 200 tos. Ship maximal load is 77 x 40' RF cotaiers, i.e., a total of 1154 x 20' ad 304 x 40' cotaiers ca be stowed. Also, 1600 tos of ballast is ecessary for ballastig (adequate stability, trimmig). The goal is to defie optimal trasport structure of cotaier ship, by ISO cotaier types, obtaiig the maximal total profit. Note: Ship payload 5 = 29434-3530 = 25904 tos. The respective mathematical model for the give problem is as follows: Criteria fuctio max Z = 48xl + 63x2 + 60x3 + 59x4 + +69xs+70x6+72x7+67xg (12) Prom et- Traffic- Traffico, Vol. 16, 2004, No. 1, 15-20 17
Costraits 1lxt + 15x2 + 13x3 + 12x4 + 16xs + 23x6 + 25x 7 + 2lxg Xt xs X2 X6 X3 X7 X1 + X2 + X3 +x4 + X5 X6 +x 7 +xs X4 Xg ~25904 ~ 160 ~ 120 ~200 ~300 ~400 ~ 160 ~77 ~ 80 ~ 1154 ~ 304 (13) (14) (15) Xt, X2, X3, X4, X5, X6, X7, Xg ~ 0. (16) Usig the simplex algorithm through eight iteratios the followig optimal structure program for cotaier ship trasport is obtaied: structural variables additioal variables x1 = 134 u1 = 3018 x2 = 120 u2 = 26 X3 = 200 U3 = 0 x4 = 300 u4 = 0 xs = 400 us= 0 X6 = 160 U6 = 0 X7 = 77 U7 = 0 xg = 67 ug = 0 Ug = 13 U1Q = 0 uu = 0. All types of ISO cotaiers are represeted i the optimal solutio. Total profit calculatio is show i Table 2. Table 2 - Structure of optimal trasport program Type ad Uit Number size of mass i of cocotaier tos taiers 18 - a X 20'DB 11 134 20'0T 15 120 20'RF 13 200 20' os 12 300 20'TC 16 400 40'PL 23 160 40'RF 25 77 40'DB 21 67 Profit per Total profit cotaier (USD/day) (USD/day) c c x 48 6432 63 7560 60 12000 59 17700 69 27600 70 11200 72 5544 67 4489 Value of optimal program: 8 maxz= LCjXj =92525USD/day. j=l Exploitability of ship payload: 8 N k = "Z a j x j = 22 886 tos, j=l meaig that total ship payload is ot fully exploited sice there is u1 = 3018 t, i.e. 11.65 percet of uexploited ship payload. Exploitability of ship trasport capacity: 5 8 Pk = L. x j + "' x j =1154+304=1458cotaiers. j=l j=6 Trasport capacity Pk is completely used accordig to the obtaied optimal program because the quatity of 304 x 40' cotaiers correspods to 608 x 20' cotaiers, i. e. trasport capacity of 1762 TEU, which is also cofirmed by the optimal solutio (ulo = u = 0). The values of additioal variables u 2 = 26 ad ug = 13 show cargo, i. e. cotaiers that should be "rejected", sice all locatios iteded for storage of cotaiers o board are occupied, it is obvious that 26 x 20' DB a 11 t cotaiers ad 13 x 40' DB a 21 t cotaiers caot be loaded o board. A post-optimal aalysis established that chage of either coefficiet i criteria fuctio or amout of particular restrictio iflueces the value ad structure of variables icluded i the optimal solutio. The structure of the optimal program will ot chage if daily profit per oe trasported cotaier for structural variables icluded i the optimal solutio is i the followig iterval rages: x 1 ~ 20'DB a 11t : o ~ c1 ~ 59 us x 2 ~ 20'0T a 15 t : 48 ~ c2 ~ oo USD x 3 ~ 20'RF a 13 t : 48 ~ c3 ~ oo USD x4 ~ 20'0S a 12 t : 48 ~ c4 ~ oo USD x 5 ~ 20'TC a 16 t : 48 ~ c5 ~ oo USD x 6 ~ 40'PL a 23 t : 67 ~ c6 ~ oo USD x 7 ~ 40'RF a 25 t : 67 ~ c7 ~ oo USD x 8 ~ 40'DB a 21 t: o ~ c8 ~ 70 USD. The coclusio follows that the structure of optimal solutio would chage oly if trasport market offered 20'DB cotaiers a 11 t ad 40'DB cotaiers a 21 t, where daily profit per oe cotaier would amout to 59 USD or more ad 70 USD or more, respectively. Itervals withi which restrictio amouts ca be chaged: payload, available quatity of cotaiers for loadig ad maximal umber of20' ad 40' cotaiers for the give problems are as follows: 22886 ~ Nk ~ oo 134 ~ 81 ~ 00 94 ~ 82 ~ 254 174 ~g3 ~ 334 274 ~ g4 ~ 434 374 ~ 85 ~ 534 Promet-Traffic- Traffico, Vol. 16, 2004, No. 1, 15-20
147 :s g6 :s 227 64 :s g7 :s 144 67 :s gg :s CX) 1020 :s dl :s 1180 237 :s d 2 :s 317. If the umber of cotaiers available for loadig per particular cotaier types were cosiderably greater tha the trasport capacity of cotaier ship, the it would ot be ecessary to iclude restrictios i the mathematical model (13). I such case, the optimal solutio would have the followig structure: XI = 0, Xz = 0, x 3 = 0, x 4 = 0, x 5 = 1154 x 20' TC a 16 t, x 6 = 80 x 40' PL a 23 t, x 7 = 224 x 40' RF a 25 t, Xg = 0 u 1 = 0, uz = 0, u3 = 0. The ship payload ad trasport capacity would be fully exploited ad the value of optimal program would amout to Z = 101354 USD/day. Such a solutio is probably ot so ofte i real life, ad a great umber of a sigle cotaier type (20'TC a 16 t) i optimal solutio results from the favourable ratio betwee profit ad cotaier mass. 5. CONCLUSION The process of maritime cotaier trasport techology is very complex ad it requires a thorough study of all the elemets icluded i the process, from various poits of view. I order to justify the techical ad techological advace i cotaier techology, it is ecessary to optimize the trasport of cargo, i. e. to orgaize trasport by cotaier ships either with miimal expeses or with maximal profit. The rate of success i operatig a cotaier ship depeds directly o the formig of "cotaier lots", which are defied for each particular ship. Therefore, utmost importace lies o the adequate selectio ad arragemet of various cotaier types offered for trasport o a particular lie provided by the respective ship-ower. The selectio of cotaiers available for trasport by ship is performed "ashore", while o board, cotaiers are arraged depedig o the actual situatio durig loadig. Sice each full cotaier ship operatio is based o profit, the success i busiess ca be cosiderably improved by implemetatio of the model of optimal cotaier trasport structure o some sailig route, as show i this paper. Persoal computers ad adequate software relatively quickly ad easily provide "optimal cotaier lot" to be formed out of a great umber of cotaiers available o maritime cotaier trasport market. Promet- Traffic- Traffico, Vol. 16, 2004, No. 1, 15-20 The followig precoditios must be met whe defiig the structure of trasport for ay cotaier ship: - maritime market must offer eough cotaiers for trasport (specifically, more tha ship trasport capacity) of various types, sizes ad masses (20', 40'), thus eablig profit resultig from trasport - it must be a full cotaier ship with a capacity greater tha 700 TEU, because ships of smaller capacity are supposed to operate as "feeder service" ships, which are loaded regardless of profit sice their basic fuctio is to deliver all cotaiers from a home port to all destiatios, ad vice versa - trasport must be "log distace voyage" i. e. miimum looonm. After havig foud the optimal solutio usig adequate procedure, it is ecessary to perform post-optimal aalysis i order to fid out if there is ay possibility of improvig the optimum trasport structure for a particular ship i terms of icreasig profit or meetig techological ad exploitatio characteristics of such a ship. The preseted mathematical model is very useful i operatio plaig for makig adequate busiess decisios with respect to the cotaier ship trasport structure. The validity of the model is especially importat cosiderig cotiuous chages of coditios that owers experiece o the market; the post-optimal aalysis provides various alteratives for the solutio of the problem whe the umber or profit of a particular ISO cotaier type chage. SERDO KOS, D. Se. ZDENKA ZENZEROVIC, D. Se. Pomorski fakultet- Rijeka Studetska 2, 51000 Rijeka, Republika Hrvatska MODEL OPTIMALNE STRUKTURE PRIJEVOZA TERETA POTPUNO KONTEJNERSKIM BRODOM NA ODREDENOJ MORSKOJ LINUI SAZETAK U ovom je radu prikaza matematicki model za problem odredivaja optimale strukture prijevoza tereta potpuo kotejerskim brodom, koji se pojavljuje kada a odredeom uaprijed defiiraom morskom prometom pravcu od kotejera koji su a raspo-lagaju u lukama ukrcaja treba odabrati odgovarajuci broj kotejera razih vrsta, masa, velicia uz evetuali RO!RO teret, tj. formirati "kotejersku posiljku" s ciljem da se ostvari maksimala dobit broda i uz uvjet da se, sto je vise moguce, iskoristi osivost i prijevozi kapacitet broda. Koristejem predloieog mode/a moguce je bito poveeati uspjesost poslovaja kotejerskog broda, odoso flote kotejerskih brodova. Modelje testira a umerickom primjeru s realim podacima. Primijejea je postoptimala aaliza kojom se ispituje utjecaj promjee vrijedosti pojediih 19
elemeata matematickog modela a dobivei optimali program. KIJUCNE RIJECI kotejerski promet, struktura prijevoza tereta kotejerskim brodom, lieamo programiraje, postoptimala aaliza REFERENCES 1. Full cotaier ship is a ship iteded exclusively for trasport of 20' ad 40' ISO cotaiers; bays are equipped with cells to accommodate cotaiers ad it has o equipmet for hadlig (loadig/uloadig) cotaiers. 2. ISO cotaiers are uiformly stadardized by legth, width, height ad load. 3. Desigatios 20' ad 40' refer to 20 ad 40 feet cotaier, respectively, thus defiig cotaier size. Additioal desigatios refer to special types of cotaiers: DB- dry box cotaier, OS- ope side cotaier, OTope top cotaier, PL- cotaier platform, RF- refrigerated cotaier ad TC - tak cotaier. 4. The capacity is calculated for 20' cotaiers because, cosiderig size, lx 40' cotaier equals 2x20' cotaiers. 5. The quatity of 3,530 tos refers to heavy ad light fuel ad potable water supplies as well as other supplies; however, cosiderig that avigatio i "witer zoe" requires oe-day supply for every three days of aviga- tio, i. e. for a ie-day jourey, supplies are calculated for 12 days. LITERATURE [1] Barkovic, D.: Operacijska istraiivaja, Ekoomski fakultet Osijek, Osijek, 2001. [2] Broso, R.: Operatios Research, McGraw-Hill, 1982. [3] Committee o Productivity of Marie Termials: Improvig Productivity i U S. Marie Cotaier Temtials, Natioal Academy Press, Washigto, 1986. [4] Chag, Y. L.- Sulliva, R. S.: Quatitative Systems for Busiess Plus, Versio 1.0, Pretice-Hall, 1988. [5] Croatia Lie: Itema dokumetacija - profiti cetar NACL, Rijeka, 1991. [6] Frakel, E. G., Houmb, 0. G., Moe, G., Brattelad, E.: Port Egieerig, Golf Publishig Co., Housto, 1981. [7] Kos, S.: Prilog rjdavaju problematike morske kotejerske tehologije, Master's thesis, Fakultet za pomorstvo i saobracaj, Rijeka, 1991. [8] Kybart, W.: Cotaier Equipmet Evaluatio, Port Maagemet Textbook No. 10, Istitute of Shippig Ecoomics ad Logistics, Breme, 1985. [9] Pasagic, H.: Matematicko modeliraje i teorija grafova, Fakultet prometih zaosti Zagreb, Zagreb, 1998. [10] Thomas, B. J., Roach, D. K.: Operatig ad Maiteace Features of Cotaier Hadlig Systems, UNCTAD, Geeva, 1998. [11] Vraic, D., Kos, S.: Prijevoz kotejera brodom II, Pomorski fakultet Rijeka, Rijeka, 1993 20 Promet-Traffic- Traffico, Vol. 16, 2004, No. 1, 15-20