ࡱ> KOh+'0 $0 L X d p| SERO KOS, DoERPomorski fakultet u Rijeciomo Normal.dotkPomorski fakultet u Rijeci44oMicrosoft Word 8.0 @f@Zl@8+@r ;plates4cA<@r՜.+,D՜.+,L hp  Pomorski fakulte Rijeka#I   SERO KOS, D Title 6> _PID_GUIDAN{86571640-70C9-11D6-99E6-0080AD83B57F}r SERO KOS, D.Sc. ZDENKA ZENZEROVI, D.Sc. Faculty of Maritime Studies - Rijeka Rijeka, Studentska 2 MODELLING THE TRANSPORT PROCESS IN MARINE CONTAINER TECHNOLOGY ABSTRACT The paper introduces a mathematical problem that occurs in marine container technology when program- ming the transport of a beforehand established number of ISO containers effected by a full container ship from several ports of departure to several ports of destination at the minimum distance (time in navigation) or at minimum transport costs. The application of the proposed model may have an effect on cost reduction in container transport thereby improving the operation process in marine transport technology. The model has been tested by using a numerical example with real data. In particular it describes the application of the dual variables in the analysis of optimum solution . KEY WORDS container traffic, transport problem, linear programming, dual problem INTRODUCTION Since the costs of cargo transport from the place of production to the storage area, to the processing sector and finally to the consumer have a considerable effect on the price rise of the product, it is economically proper to tend to a reduction in costs as much as feasible, but at the same time, from the carriers standpoint, to effect the transport with a profit as high as possible. In both cases cargo transport should be optimized no matter whether the criteria of optimization refer to transport costs or profit made, i.e. income realized. Marine container transport technology combines three essential components [2, page 2]: the container as a technical means for storing merchandise, the port container terminal as an area provided with specific facilities and infrastructures for the loading, discharging and transshipment of containers and the container ship as a navigable means for the conveyance of containers from the port of departure to the port of destination. The transport process in marine container technology is accomplished by the suitable integration of the named components and, with regards to its complexity, it is necessary to monitor as well as to improve the functioning of the whole process from the technical, technological, economical, organizational, , standpoint. An improvement in the transport process within marine container technology is possible, among others, by optimizing the transport plan of containers in a given geographical area. The object of this paper is to demonstrate how the transport process in marine container technology can be optimized by the use of pertinent quantitative methods, i.e. by establishing the most profitable transport variant concerning transport costs or incomes. To attain this object the method of linear programming has been applied. DEFINITION OF PROBLEM The container is a receptacle used for storing different items of general cargo intended for convey- ance to the same destination. However, in practice it frequently occurs that the port of destination of the containers may not be at the same time the port of departure of the same units filled up with new merchandise. For this reason the technology of marine containers must cope with the problem of carrying empty containers to the port where the refilling will be effected. Therefore the transport of empty containers is not only a technical and technological inconvenience but an economical disadvantage as well because such a carriage increases the operational costs of the ship operator. The primary assumption in marine container technology is that a full container ship in service should have three sets of containers at disposal: one set of loaded containers being directly on board one set of empty containers being at disposal at the ports of loading of a given geographical area one set of empty containers being at disposal at the ports of discharge of a given geographical area. The term set of containers refers to the nominal TEU-capacity relative to the container ship. The problem dealt with in this paper is defined as follows: to program the transport of a number of ISO containers by full container ships from several ports of departure to several ports of destination with the minimum distance (time in navigation) or at minimum transport costs. The solution of the problem involves the transport plan of containers from the ports of departure to the ports of destination taking into account the number of containers at the ports of departure and the number of containers required at the ports of destination with the objective of effecting the transport with the minimum total distance or at minimal transport costs 3. FORMULATION OF THE MATHEMATICAL MODEL The problem of container transport can be illustrated with the aid of appropriate tables: the Matrix of Transportation (Table 1) and the Matrix of Costs, or Distances, (Table 2): Table 1: Matrix of Transportation Ports of departurePorts of destinationNumber of ISO containersB1 B2 BnA1 A2 . . . Amx11 x12 x1n x21 x22 x2n . . . xm1 xm2 xmna1 a2 . . . amNumber of ISO containers b1 b2 bn Table 2: Matrix of Costs or Distances by Sea Ports of departurePorts of destinationB1 B2 BnA1 A2 . . . Amc11 c12 c1n c21 c22 c2n . . . cm1 cm2 cmn where: Ai ports of departure, i = 1,2, ,m Bj ports of destination, j = 1,2, ,n m total number of departure ports n total number of destination ports cij distance by sea (or time in navigation), or unit costs for the transport on the distance between i-th port of departure and the j-th port of destination, i = 1,2, ,m, j = 1,2, ,n xij number of ISO containers carried on the distance from i-th port of departure to j-th port of destination, i = 1,2, ,m, j = 1,2, ,n ai available number of ISO containers at i-th port of destination, i = 1,2, ,m bj demand for containers at j-th port of destination, j = 1,2, ,n. The mathematical model for solving the problem in marine container technology is as follows : Min Z = c11 x11 + c12 x12 + + c1n x1n + + c21x21 + c22x22 + + c2n x2n + + cm1 xm1 + cm2 xm2 + + cmn xmn (1) with constraints x11 + x12 +. + x1n = a1 x21 + x22 +. + x2n = a2 ... (2) xm1 + xm2 +. + xmn = am x11 + x21 +. + xm1 = b1 x12 + x22 +. + xm2 = b2 (3) x1n + x2n +. + xmn = bn x11, x12,, xmn ( 0 . (4) From the system of equations exposed above it is noticeable that it is a problem of linear programming with the (m+n) equation and (mn) variable. The equations (2) and (3) form an equation system composed of the independent equations (m+n1). For this reason the solution of the given transport problem must comprise (m+n1) non-negative values xij . If the solution of the problem comprises less than (m+n1) non-negative values xij , than such a solution is degenerative in which case it is necessary to obtain a non-degenerative solution by applying the appropriate procedure. The mathematical model for the solution of the transport problem in marine container technology is, in summarized form, as follows: Criteria function Min Z =  EMBED Equation.2  (5) with constraints  EMBED Equation.2 , i = 1,2, & ,m (6)  EMBED Equation.2 , j = 1,2, & ,n (7) xij e" 0 ; i = 1,2, & ,m , j = 1,2, & ,n . To be conveniently correlated the equations (6) and (7) should fulfil the following condition :  EMBED Equation.3 . The transport problem in marine container technology in which the total number of ISO containers dispatch is equal to the number of ISO containers receipt represents a closed transport problem. When the number of containers at the ports of departure exceeds or is less than the number of containers at the ports of destination then it is the case of open transport problem in marine container technology. The solution of the transport problem in marine container technology by using the simplex method is rather long-lasting; consequently some specialized methods can be applied to cope with the problem. These are: methods requiring program initializing with further upgrading until the optimum program is achieved. For program initializing there are available: the North-West Corner Rule method, the minimum cost method, the Vogel method, etc., and for upgrading and achieving the optimum solution: the Stepping Stone Method, the MODI-method, etc. methods not requiring program initializing. These methods gradually upgrade their program until the optimum solution is achieved. Among these the best known is the Ford-Fulkerson method. For any problem of linear programming, and this is the case of transport problems, the dual problem can be formulated. The value of dual variables are useful for analyzing the optimum solution, in particular analyzing the influence of the change the containers number at the port of departure or at the port of destination on the minimum distance or minimum transport costs. Thanks to the development of computerization the solution of many problems from operations research has been facilitated by the application of pertinent computer programs. Because of its simplicity the use of the personal computer QSB program can be recommended [3]. Detailed explications concerning the application of methods and means of solutions are given in specific textbooks dealing with operations research [2], [6], [1]. 4. NUMERICAL EXAMPLE To illustrate the mentioned mathematical model an example with real data has been chosen. In a given geographical area served by the shipping carrier X empty containers must be carried from the ports of departure A1, A2, A3 and A4 to the ports of destination B1, B2, B3 and B4. At the ports of departure are available for carriage the following quantities of containers: 700, 500, 400 and 400 TEU, respectively. The demand for containers at the ports of destination is the following: 560, 380, 620 and 440 TEU, respectively. The distances by sea between the ports of departure and destination in nautical miles are given in Table 3: Table 3: Distances by sea between the ports of departure and destination in NM Ports of departurePorts of destinationB1B2B3B4A11368273636484104A2 912319222804560A3 456501654725928A41824638468407296 The shipping carrier X runs a container ship that achieves in navigation a service speed of 19 knots, and the time in navigation between the ports expressed in days, amounts to : Table 4: Days in navigation between the ports of departure and destination Ports of departurePorts of destinationB1B2B3B4A13 6 8 9A22 7 510A31111213A44141516 The task consists in working out a plan of container transport from the ports of departure to the ports of destination aiming at the minimum total distance, or minimum time in navigation. For the given problem the mathematical model runs as follows: Min Z = 1368x11 + 2736x12 + 3648x13 + 4104x14 + 912x21 + 3192x22+ 2280x23 + 4560x24 + 456x31 + 5016x32+ 5472x33 + 5928x34 + 1824x41 + 6384x42+ 6840x43 + 7296x44 (8) with constraints x11 + x12 + x13 + x14 = 700 x21 + x22 + x23 + x24 = 500 x31 + x32 + x33 + x34 = 400 (9) x41 + x42 + x43 + x44 = 400 x11 + x21 + x31 + x41 = 560 x12 + x22 + x32 + x42 = 380 x13 + x23 + x33 + x43 = 620 (10) x14 + x24 + x34 + x44 = 440 xij#M&F43* e" 0 , i = 1,2,3,4 , j = 1,2,3,4. (11) The transport problem of ISO containers is solved by means of  North-West Corner Rule and MODI methods. According to the optimum program the transport of ISO containers should be arranged as follows: from the port of destination A1 to be carried: 380 TEU to port B2, 120 TEU to port B3, 200 TEU to port B4, from the port of destination A2:500 TEU to port B3, from the port of destination A3: 400 TEU to port B1, from the port of destination A4: 160 TEU to port B1 and 240 TEU to port B4. The value of the optimum transport plan amounts to : Min Z = 5 663 520 TEU-NM, or Min Z = 12 420 TEU-days. The optimum solution is visually shown in Fig. 1: Figure 1: Optimum transport plan of containers The mathematical model of the dual problem of container transport runs as follows: Max W = 700u1+500u2+400u3+400u4+ 560v1+380v2+620v3+440v4 (12) with constraints u1 + v1 ( 1368 u1 + v2 ( 2736 u1 + v3 ( 3648 (13) u1 + v4 ( 4104 u2 + v1 ( 912 u2 + v2 ( 3192 u2 + v3 ( 2280 (14) u2 + v4 ( 4560 u3 + v1 ( 456 u3 + v2 ( 5016 u3 + v3 ( 5472 (15) u3 + v4 ( 5928 u4 + v1 ( 1824 u4 + v2 ( 6384 u4 + v3 ( 6840 (16) u4 + v4 ( 7296 . The dual problem (12)-(16) is solved by the simplex method; however, the optimum solution can be obtained by solving the primal by the application of the previously stated methods (MODI-method) that enables obtaining the following values of dual variables: u1 = 0, u2 = 1368, u3 = 1824, u4 = 3192, v1 = 1368, v2 = 2736, v3 = 3648, v4 = 4104.  The positive values of dual variables that refer to the port of departure, i.e. values u1, indicate the amount of increase in total distance expressed in NM, or number of TEU-days, and with this the total costs of the container transport, if the to number of shipped containers from i-th port of departure is added one TEU unit. Conversely, the negative values ui indicate that the addition of one TEU unit to the number of containers at i-th port of departure may affect the reduction of total distance and total transport costs for the amount shown by the values of dual variables. Similar conclusions are valid for the positive or negative values of dual variables vj that indicate the quantity of increase or decrease in costs of container transport, if the need, i.e. demand for TEU at the port of destination j-th, or vice versa, is increased by one unit. It needs emphasizing that in the analysis of dual variables the increase referring to one departure (or destination) and the decrease to the second departure (or destination) is the same figure to avoid disturbing the balance between supply and demand of the transport matrix. For example: if at the port A1 the number of containers drops to 600 units, and at the port A2 there is an increase of 600 units, with all the other elements in the model remaining unchanged, the alteration will cause an increase in total distance of 136 800 NM, thereby the minimum distance amounting to 5 526 720 TEU-NM. Namely, the rise in number a2 of 100 TEU causes an increase of 1001368 TEU-NM, and the decrease in number of containers a1 of 100 TEU affects in no way the function Z, as u1 = 0. Or, if the demand from B1 drops to 400 containers, and there is an increase in demand from port B4 of 600 containers, then based on the values of dual variables there follows that the alterations will have an effect on the increase in distance to the amount of 875 520 TEU-NM (1601368 + 1604104). As a result the total distance will be 6 539 040 NM. CONCLUSION From the mathematical viewpoint the problem of marine container technology is defined as a problem requiring an assessment of minimal value of criteria function with a specific number of structural variables xij, mutually correlated by linear functions, with limitations in the form of linear equation or non-equations. From the technological and organizational viewpoint it is a process of optimum distribution of a limited number of units determined by a set of objectives such as minimum sea distance, minimum days in navigation or minimum costs. In order to solve the transport problem in marine container technology various methods can be employed: some require initializing the program being further upgraded until the optimum program is achieved and other methods not requiring initializing the program with gradual program upgrading until the optimal solution. For the purpose adequate computer programs can be used as well. The presented mathematical model is basic for making suitable business decisions that regard the transport of ISO containers by full container ships in a given geographic area served by a shipping operator running a container carrier fleet. Of particular significance is the validity of the model in the fluctuating conditions of shipping business on the market; by analysis of the dual variables several solutions of the given problem are possible in cases of change in the supply and demand for containers. NM   d4E4  3QQ SA}ETAK U ovom je radu prikazan matemati ki model za jedan problem koji se pojavljuje u morskoj kontejnerskoj tehnologiji kada treba programirati prijevoz unaprijed odreenog broja ISO kontejnera morskim potpuno kontejnerskim brodovima iz viae luka ukrcaja u viae luka iskrcaja uz minimalne udaljenosti (ili vrijeme u plovidbi), maksimalan prihod ili minimalne troakove prijevoza. Koriatenjem predlo~enog modela mogue je utjecati na sni~enje troakova prijevoza kontejnera i time poboljaati funkcioniranje prijevoznog procesa morske transportne tehnologije. Model je testiran na numeri kom primjeru s realnim podacima. Posebno je prikazana primjena dualnih varijabli u analizi optimalnoga rjeaenja. LITERATURE (1( Barkovi, D.: Operacijska istra~ivanja, Ekonomski fakultet Osijek, Osijek, 2001. (2( Bronson, R.: Operations Research, Mc Graw-Hill, 1982. (3( Chang, Y.L.- Sullivan, R.S.: Quantitative Systems for Business Plus, Version 1.0, Prentice-Hall, 1988. (4( Committee on Productivity of Marine Terminals: Improving Productivity in U.S. Marine Container Terminals, National Academy Press, Washington, 1986. (5( Kos, S.: Prilog rjeavanju problematike morske kontejnerske transportne tehnologije, magis"@"@@tarski rad, Fakultet za pomorstvo i saobraaj, Rijeka, 1991. (6( Paaagi, H.: Matemati ko modeliranje i teorija grafova, Fakultet prometnih znanosti, Zagreb, 1998. 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(2  HB  C DT  C    (2   HB  C DT  C    (2  HB  C DT  C    (2  HB  C DT  C    ZB  s *DZB  s *DZB @ s *DZB  s *DZB  s *DZB  s *DZB  @ s *DT ! C ! T " C "  T # C #  T $ C $  T % C %  T & C & T ' C '  6 (  &B S  ?000000000000000000000000000000000000004H  4( 4wTyTzT{T|TTTTTTTTTTTTTTT`p &:*V "TVXZ\dfhCJH*;;CJ5 56CJ6CJ56 5CJmH CJ 5mH5CJCJCJ mHH*mHmHN . A!"#$% ' 0. A!"n#$% P  SERO KOS, D.Sc. ZDENKA ZENZEROVI, D.Sc. Faculty of Maritime Studies - Rijeka Rijeka, Studentska 2 MODELLING THE TRANSPORT PROCESS IN MARINE CONTAINER TECHNOLOGY ABSTRACT The paper introduces a mathematical problem that occurs in marine container technology when program- ming the transport of a beforehand established number of ISO containers effected by a full container ship from several ports of departure to several ports of destination at the minimum distance (time in navigation) or at minimum transport costs. The application of the proposed model may have an effect on cost reduction in container transport thereby improving the operation process in marine transport technology. The model has been tested by using a numerical example with real data. In particular it describes the application of the dual variables in the analysis of optimum solution . KEY WORDS container traffic, transport problem, linear programming, dual problem INTRODUCTION Since the costs of cargo transport from the place of production to the storage area, to the processing sector and finally to the consumer have a considerable effect on the price rise of the product, it is economically proper to tend to a reduction in costs as much as feasible, but at the same time, from the carrier s standpoint, to effect the transport with a profit as high as possible. In both cases cargo transport should be optimized no matter whether the criteria of optimization refer to transport costs or profit made, i.e. income realized. Marine container transport technology combines three essential components [2, page 2]: the container as a technical means for storing merchandise, the port container terminal as an area provided with specific facilities and infrastructures for the loading, discharging and transshipment of containers and the container ship as a navigable means for the conveyance of containers from the port of departure to the port of destination. The transport process in marine container technology is accomplished by the suitable integration of the named components and, with regards to its complexity, it is necessary to monitor as well as to improve the functioning of the whole process from the technical, technological, economical, organizational, & , standpoint. An improvement in the transport process within marine container technology is possible, among others, by optimizing the transport plan of containers in a given geographical area. The object of this paper is to demonstrate how the transport process in marine container technology can be optimized by the use of pertinent quantitative methods, i.e. by establishing the most profitable transport variant concerning transport costs or incomes. To attain this object the method of linear programming has been applied. DEFINITION OF PROBLEM The container is a receptacle used for storing different items of general cargo intended for convey- ance to the same destination. However, in practice it frequently occurs that the port of destination of the containers may not be at the same time the port of departure of the same units filled up with new merchandise. For this reason the technology of marine containers must cope with the problem of carrying empty containers to the port where the refilling will be effected. Therefore the transport of empty containers is not only a technical and technological inconvenience but an economical disadvantage as well because such a carriage increases the operational costs of the ship operator. The primary assumption in marine container technology is that a full container ship in service should have three sets of containers at disposal: one set of loaded containers being directly on board one set of empty containers being at disposal at the ports of loading of a given geographical area one set of empty containers being at disposal at the ports of discharge of a given geographical area. The term  set of containers refers to the nominal TEU-capacity relative to the container ship. The problem dealt with in this paper is defined as follows: to program the transport of a number of ISO containers by full container ships from several ports of departure to several ports of destination with the minimum distance (time in navigation) or at minimum transport costs. The solution of the problem involves the transport plan of containers from the ports of departure to the ports of destination taking into account the number of containers at the ports of departure and the number of containers required at the ports of destination with the objective of effecting the transport with the minimum total distance or at minimal transport costs 3. FORMULATION OF THE MATHEMATICAL MODEL The problem of container transport can be illustrated with the aid of appropriate tables: the Matrix of Transportation (Table 1) and the Matrix of Costs, or Distances, & (Table 2): Table 1: Matrix of Transportation Ports of departurePorts of destinationNumber of ISO containersB1 B2 & BnA1 A2 . . . Amx11 x12 & x1n x21 x22 & x2n . . . xm1 xm2 & xmna1 a2 . . . amNumber of ISO containers b1 b2 & bn Table 2: Matrix of Costs or Distances by Sea Ports of departurePorts of destinationB1 B2 & BnA1 A2 . . . Amc11 c12 & c1n c21 c22 & c2n . . . cm1 cm2 & cmn where: Ai  ports of departure, i = 1,2, & ,m Bj  ports of destination, j = 1,2, & ,n m  total number of departure ports n  total number of destination ports cij  distance by sea (or time in navigation), or unit costs for the transport on the distance between i-th port of departure and the j-th port of destination, i = 1,2, & ,m, j = 1,2, & ,n xij  number of ISO containers carried on the distance from i-th port of departure to j-th port of destination, i = 1,2, & ,m, j = 1,2, & ,n ai  available number of ISO containers at i-th port of destination, i = 1,2, & ,m bj  demand for containers at j-th port of destination, j = 1,2, & ,n. The mathematical model for solving the problem in marine container technology is as follows : Min Z = c11 x11 + c12 x12 + & + c1n x1n + + c21x21 + c22x22 + & + c2n x2n + & + cm1 xm1 + cm2 xm2 + & + cmn xmn (1) with constraints x11 + x12 +& & . + x1n = a1 TTNpr "$&:<$$$$$$V$VVVx21 + x22 +& & . + x2n = a2 ..& & & & & & & & & & & . (2) xm1 + xm2 +& & . + xmn = am x11 + x21 +& & . + xm1 = b1 x12 + x22 +& & . + xm2 = b2 & & & & & & & & & & (3) x1n + x2n +& & . + xmn = bn x11, x12,& , xmn ( 0 . (4) From the system of equations exposed above it is noticeable that it is a problem of linear programming with the (m+n) equation and (mn) variable. The equations (2) and (3) form an equation system composed of the independent equations (m+n 1). For this reason the solution of the given transport problem must comprise (m+n 1) non-negative values xij . If the solution of the problem comprises less than (m+n 1) non-negative values xij , than such a solution is degenerative in which case it is necessary to obtain a non-degenerative solution by applying the appropriate procedure. The mathematical model for the solution of the transport problem in marine container technology is, in summarized form, as follows: Criteria function Min Z =  EMBED Equation.2  (5) with constraints  EMBED Equation.2 , i = 1,2, & ,m (6)  EMBED Equation.2 , j = 1,2, & ,n (7) xij e" 0 ; i = 1,2, & ,m , j = 1,2, & ,n . To be conveniently correlated the equations (6) and (7) should fulfil the following condition :  EMBED Equation.3 . The transport problem in marine container technology in which the total number of ISO containers dispatch is equal to the number of ISO containers receipt represents a closed transport problem. When the number of containers at the ports of departure exceeds or is less than the number of containers at the ports of destination then it is the case of open transport problem in marine container technology. The solution of the transport problem in marine container technology by using the simplex method is rather long-lasting; consequently some specialized methods can be applied to cope with the problem. These are: methods requiring program initializing with further upgrading until the optimum program is achieved. For program initializing there are available: the  North-West Corner Rule method, the minimum cost method, the Vogel method, etc., and for upgrading and achieving the optimum solution: the  Stepping Stone Method , the MODI-method, etc. methods not requiring program initializing. These methods gradually upgrade their program until the optimum solution is achieved. Among these the best known is the Ford-Fulkerson method. For any problem of linear programming, and this is the case of transport problems, the dual problem can be formulated. The value of dual variables are useful for analyzing the optimum solution, in particular analyzing the influence of the change the containers number at the port of departure or at the port of destination on the minimum distance or minimum transport costs. Thanks to the development of computerization the solution of many problems from operations research has been facilitated by the application of pertinent computer programs. Because of its simplicity the use of the personal computer QSB program can be recommended [3]. Detailed explications concerning the application of methods and means of solutions are given in specific textbooks dealing with operations research [2], [6], [1]. 4. NUMERICAL EXAMPLE To illustrate the mentioned mathematical model an example with real data has been chosen. In a given geographical area served by the shipping carrier  X empty containers must be carried from the ports of departure A1, A2, A3 and A4 to the ports of destination B1, B2, B3 and B4. At the ports of departure are available for carriage the following quantities of containers: 700, 500, 400 and 400 TEU, respectively. The demand for containers at the ports of destination is the following: 560, 380, 620 and 440 TEU, respectively. The distances by sea between the ports of departure and destination in nautical miles are given in Table 3: Table 3: Distances by sea between the ports of departure and destination in NM Ports of departurePorts of destinationB1B2B3B4A11368273636484104A2 912319222804560A3 456501654725928A41824638468407296 The shipping carrier  X runs a container ship that achieves in navigation a service speed of 19 knots, and the time in navigation between the ports expressed in days, amounts to : Table 4: Days in navigation between the ports of departure and destination Ports of departurePorts of destinationB1B2B3B4A13 6 8 9A22 7 510A31111213A44141516 The task consists in working out a plan of container transport from the ports of departure to the ports of destination aiming at the minimum total distance, or minimum time in navigation. For the given problem the mathematical model runs as follows: Min Z = 1368x11 + 2736x12 + 3648x13 + 4104x14 + 912x21 + 3192x22+ 2280x23 + 4560x24 + 456x31 + 5016x32+ 5472x33 + 5928x34 + 1824x41 + 6384x42+ 6840x43 + 7296x44 (8) with constraints x11 + x12 + x13 + x14 = 700 x21 + x22 + x23 + x24 = 500 x31 + x32 + x33 + x34 = 400 (9) x41 + x42 + x43 + x44 = 400 x11 + x21 + x31 + x41 = 560 x12 + x22 + x32 + x42 = 380 x13 + x23 + x33 + x43 = 620 (10) x14 + x24 + x34 + x44 = 440 xij e" 0 , i = 1,2,3,4 , j = 1,2,3,4. (11) The transport problem of ISO containers is solved by means of  North-West Corner Rule and MODI methods. According to the optimum program the transport of ISO containers should be arranged as follows: from the port of destination A1 to be carried: 380 TEU to port B2, 120 TEU to port B3, 200 TEU to port B4, from the port of destination A2:500 TEU to port B3, from the port of destination A3: 400 TEU to port B1, from the port of destination A4: 160 TEU to port B1 and 240 TEU to port B4. The value of the optimum transport plan amounts to : Min Z = 5 663 520 TEU-NM, or Min Z = 12 420 TEU-days. The optimum solution is visually shown in Fig. 1: Figure 1: Optimum transport plan of containers The mathematical model of the dual problem of container transport runs as follows: Max W = 700u1+500u2+400u3+400u4+ 560v1+380v2+620v3+440v4 (12) with constraints u1 + v1 ( 1368 u1 + v2 ( 2736 u1 + v3 ( 3648 (13) u1 + v4 ( 4104 u2 + v1 ( 912 u2 + v2 ( 3192 u2 + v3 ( 2280 (14) u2 + v4 ( 4560 u3 + v1 ( 456 u3 + v2 ( 5016 u3 + v3 ( 5472 (15) u3 + v4 ( 5928 u4 + v1 ( 1824 u4 + v2 ( 6384 u4 + v3 ( 6840 (16) u4 + v4 ( 7296 . The dual problem (12)-(16) is solved by the simplex method; however, the optimum solution can be obtained by solving the primal by the application of the previously stated methods (MODI-method) that enables obtaining the following values of dual variables: u1 = 0, u2 =  1368, u3 = 1824, u4 = 3192, v1 =  1368, v2 = 2736, v3 = 3648, v4 = 4104.  The positive values of dual variables that refer to the port of departure, i.e. values u1, indicate the amount of increase in total distance expressed in NM, or number of TEU-days, and with this the total costs of the container transport, if the to number of shipped containers from i-th port of departure is added one TEU unit. Conversely, the negative values ui indicate that the addition of one TEU unit to the number of containers at i-th port of departure may affect the reduction of total distance and total transport costs for the amount shown by the values of dual variables. Similar conclusions are valid for the positive or negative values of dual variables vj that indicate the quantity of increase or decrease in costs of container transport, if the need, i.e. demand for TEU at the port of destination j-th, or vice versa, is increased by one unit. It needs emphasizing that in the analysis of dual variables the increase referring to one departure (or destination) and the decrease to the second departure (or destination) is the same figure to avoid disturbing the balance between supply and demand of the transport matrix. For example: if at the port A1 the number of containers drops to 600 units, and at the port A2 there is an increase of 600 units, with all the other elements in the model remaining unchanged, the alteration will cause an increase in total distance of 136 800 NM, thereby the minimum distance amounting to 5 526 720 TEU-NM. Namely, the rise in number a2 of 100 TEU causes an increase of 1001368 TEU-NM, and the decrease in number of containers a1 of 100 TEU affects in no way the function Z, as u1 = 0. Or, if the demand from B1 drops to 400 containers, and there is an increase in demand from port B4 of 600 containers, then based on the values of dual variables there follows that the alterations will have an effect on the increase in distance to the amount of 875 520 TEU-NM (1601368 + 1604104). As a result the total distance will be 6 539 040 NM. CONCLUSION From the mathematical viewpoint the problem of marine container technology is defined as a problem requiring an assessment of minimal value of criteria function with a specific number of structural variables xij, mutually correlated by linear functions, with limitations in the form of linear equation or non-equations. From the technological and organizational viewpoint it is a process of optimum distribution of a limited number of units determined by a set of objectives such as minimum sea distance, minimum days in navigation or minimum costs. In order to solve the transport problem in marine container technology various methods can be employed: some require initializing the program being further upgraded until the optimum program is achieved and other methods not requiring initializing the program with gradual program upgrading until the optimal solution. For the purpose adequate computer programs can be used as well. The presented mathematical model is basic for making suitable business decisions that regard the transport of ISO containers by full container ships in a given geographic area served by a shipping operator running a container carrier fleet. Of particular significance is the validity of the model in the fluctuating conditions of shipping business on the market; by analysis of the dual variables several solutions of the given problem are possible in cases of change in the supply and demand for containers. SA}ETAK U ovom je radu prikazan matemati ki model za jedan problem koji se pojavljuje u morskoj kontejnerskoj tehnologiji kada treba programirati prijevoz unaprijed odreenog broja ISO kontejnera morskim potpuno kontejnerskim brodovima iz viae luka ukrcaja u viae luka iskrcaja uz minimalne udaljenosti (ili vrijeme u plovidbi), maksimalan prihod ili minimalne troakove prijevoza. Koriatenjem predlo~enog modela mogue je utjecati na sni~enje troakova prijevoza kontejnera i time poboljaati funkcioniranje prijevoznog procesa morske transportne tehnologije. Model je testiran na numeri kom primjeru s realnim podacima. Posebno je prikazana primjena dualnih varijabli u analizi optimalnoga rjeaenja. LITERATURE (1( Barkovi, D.: Operacijska istra~ivanja, Ekonomski fakultet Osijek, Osijek, 2001. (2( Bronson, R.: Operations Research, Mc Graw-Hill, 1982. (3( Chang, Y.L.- Sullivan, R.S.: Quantitative Systems for Business Plus, Version 1.0, Prentice-Hall, 1988. (4( Committee on Productivity of Marine Terminals: Improving Productivity in U.S. Marine Container Terminals, National Academy Press, Washington, 1986. (5( Kos, S.: Prilog rjeaavanju problematike morske kontejnerske transportne tehnologije, magistarski rad, Fakultet za pomorstvo i saobraaj, Rijeka, 1991. (6( Paaagi, H.: Matemati ko modeliranje i teorija grafova, Fakultet prometnih znanosti, Zagreb, 1998. 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