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Applied Soft Computing
j ourna l h o mepage: www.elsev ier .com/ locate /asoc
outing of barge container ships by mixed-integer programmingeuristics
ladislav Maras a, Jasmina Lazic d, Tatjana Davidovic b,∗, Nenad Mladenovic b,c
University of Belgrade, Faculty of Transport and Traffic Engineering, Vojvode Stepe 305, Belgrade, SerbiaMathematical Institute of the Serbian Academy of Sciences and Arts, P.O. Box 367, Belgrade, SerbiaDepartment of Mathematics, Brunel University, London, UKMathWorks Inc., Matrix House, Cambridge Business Park, CB4 0HH, Cambridge, UK
r t i c l e i n f o
rticle history:eceived 1 December 2011eceived in revised form 26 October 2012ccepted 12 March 2013vailable online 21 April 2013
SC:0B060B350C110C59
a b s t r a c t
We investigate the optimization of transport routes of barge container ships with the objective tomaximize the profit of a shipping company. This problem consists of determining the upstream anddownstream calling sequence and the number of loaded and empty containers transported betweenany two ports. We present a mixed integer linear programming (MILP) formulation for this problem.The problem is tackled by the commercial CPLEX MIP solver and improved variants of the existing MIPheuristics: Local Branching, Variable Neighborhood Branching and Variable Neighborhood Decomposi-tion Search. It appears that our implementation of Variable Neighborhood Branching outperforms CPLEXMIP solver both regarding the solution quality and the computational time. All other studied heuris-tics provide results competitive with CPLEX MIP solver within a significantly shorter amount of time.Moreover, we present a detailed case study transportation analysis which illustrates how the proposed
This paper addresses the issue of liner barge container transportoutes, connected to maritime container services. More precisely,e investigate the hinterland barge transport of containers arrived
o or departing from a transshipment port (the sea port locatedt river mouth) on a sea or mainline container ships. One of theost important parts of this problem is adjusting the barge and
ea container ships arrival times in the transshipment port. Thislearly indicates that considered container barge transport acts as
typical feeder service.Barge transport has recently proved to be a well-developed
ode for container transport, particularly in Northwest Europe.haracteristics of barge container transport that have increased the
mportance of this transport mode are reliability of barge container
ervices and low cost of barge transport operations. Therefore,ontainer-on-barge transport is becoming even more competitivehan its alternative mode, i.e. road transport for specific kind of
transport activities. Following these trends, significant efforts havealso been done in promoting barge transport of containers on theDanube and region of Southeast Europe. Over the last decade,container-on-barge transport has shown annual growth figures of10–15% [1].
On the other hand, service network design in barge con-tainer transport has not been investigated enough. To the bestof our knowledge, there are only a few researches dealing withthese issues [1–4], even though there are many service networkdesign problems in barge container transport that need furtherresearch attention. Therefore, this paper is intended to contributeto the successful design and development of service networks incontainer-on-barge transport, as this transport mode is expectedto gain much more interest in years to come.
Determining transport routes of barge container ships hasrecently received a lot of attention (see, for example [4–7]). Forthese problems optimality may be defined in accordance withvarious factors: maximization of shipping company profit [8–11],
minimization of system costs [12,13], minimization of operatingcosts [14], etc. Obtaining an optimal solution by any factor is veryimportant for doing successful transport business. Unfortunately,like in many other practical cases, the complexity of real life prob-
ems exceeds the capacity of the present computation systems.herefore, the efficient algorithms for finding good suboptimalolutions are required. Meta-heuristic methods are the most nat-ral choice, as they were already widely used in recent literature11,15].
In this paper we consider the barge container ship routing prob-em with the aim of maximizing the shipping company profit whileicking up and delivering containers (both loaded and empty)long the inland waterway. The problem has been studied for therst time in [4,16]. Here, the mixed integer programming formu-
ation for this problem is presented and used within CPLEX MIPolver to obtain optimal solutions of small size instances. For largernstances, MIP heuristic methods are used to obtain good subopti-
al solutions. We apply three state-of-the-art heuristics for 0–1IP problem: Local Branching (LB) [17], Variable Neighborhood
ranching (VNB) [18] and Variable Neighborhood Decompositionearch for 0–1 MIP problems (VNDS-MIP) [19]. In addition, weropose an extension of the original VNDS-MIP method, called LS-NDS, which considers not only binary but also general integerariables. Detailed experimental results confirm the superiority ofeuristic methods over the commercial exact solver. At the end,
or a small representative test instance we present detailed trans-ortation analysis describing obtained optimal route and containerows along it.
The rest of this paper is organized as follows. In the next sec-ion we present a brief literature review. The description of theonsidered problem: optimization of transport routes of bargeontainer ships is given in Section 3. Intuitive description as wells mathematical formulation is given and the problem complex-ty is discussed. Section 4 contains an overview of the MIP-basedeuristic methods that we used for this problem (to obtain gooduboptimal solutions in reasonable computation time). Experimen-al evaluations are described in Section 5, while Section 6 containsoncluding remarks.
. Literature review
Planning, routing and scheduling of container ships in botharitime and hinterland transport was significantly studied in rel-
vant literature. We briefly describe here some of the most recentdvances in this area. The relation between barge network design,ransport market and the performance of intermodal barge trans-ort was studied in [20]. A conceptual model for barge networkesign which describes the design variables for barge networks andheir relation to the performance indicators of intermodal bargeransport from a shippers’ and operators’ perspective was pre-ented. It was explained that the vessel size and the circulationime of vessels are major factors for this issue and directly influ-nce the cost and quality performance of barge transport. Similarly,onings [6] investigated whether a hub-and-spoke service coulde a fruitful tool to improve the performance of the container-on-arge transport and hence to gain market share. He indicated that
hub-and-spoke network can produce efficient barge services, byncreasing both the productivity of a vessel through optimized sail-ng schedules and the efficiency of capacity utilization (dependingn waterway dimensions). In addition, future requirements andpportunities of barge terminals to further improve the compet-tiveness of container barge transport were explored in [21].
Notteboom [22] assessed the trade-offs linked to the time fac-or in liner service schedules from the perspective of a shippingine. The author also discussed the range of measures and planning
ools container carriers can deploy to maximize schedule reliability.he main conclusions of his paper were: (1) carriers’ strategies inealing with potential disruptions in service schedules differ sub-tantially, which identifies a service variable that adds to market
uting 13 (2013) 3515–3528
segmentation in liner shipping; (2) port congestion is the mainsource of schedule unreliability; (3) future research scopes have toinclude different trade routes (not just East Asia – Northern Europelines), inter-round trip effects, and network effects.
In [23] similarities and dissimilarities between the spatial andthe functional development of the container river service networksof the Yangtze and the Rhine River were discussed. It was shownthat the Yangtze service network has the tendency to converge, inmore than one aspect, with the historical development pattern ofinland container services in the Rhine basin.
In [11] the designing service networks for container liner ship-ping in the sea container industry, while explicitly taking intoaccount empty container repositioning was addressed. The prob-lem was modeled as a Knapsack problem that was reduced to alocation routing problem. A Genetic Algorithms (GA) based heuris-tic was proposed to find a set of calling ports, an associated portcalling sequence, the number of ships (by ship size category) andthe resulting cruising speed to be deployed in the service networks,with the objective of profit maximization for a liner shipping com-pany. An application to the problem of container transportation inSoutheast Asia was presented.
Network flow techniques were employed to construct a modelfor short-term ship scheduling and container shipment from thecarrier’s perspective in [12]. A network flow technique was appliedto construct the model, which included multiple ship-flow andcontainer-flow networks. The authors developed a solution algo-rithm based on Lagrangian relaxation, a subgradient method, anda heuristic for the upper-bound solution. On the other hand, theauthors of [24] proposed a novel mixed integer linear programmingmathematical model for the liner shipping network design problemin a competitive environment. This model addressed the competi-tion between a newcomer liner service provider and an existingdominating operator, both operating on hub-and-spoke networks.
Although there is a substantial amount of articles related to shiprouting and scheduling, the work on the allocation of empty con-tainers in the barge container transport has been scarce. On theother hand, for the sea transport, empty container repositioningproblems have been studied in various ways: as determinis-tic systems (see [11,25,26]), as stochastic systems (see [27–29]),through simulation (see [30,31]), as dynamic programming mod-els (see [32,33]) or by means of inventory-based heuristic policies(see [34,35]). Joint optimization of container fleet sizing andempty repositioning in a stochastic dynamic liner shipping systemwith multi-vessels, multi-ports and multi-voyages was consid-ered in [36]. Focusing on a type of parameterized rule-basedempty container repositioning policies, the authors developed asimulation-based optimization tool to optimize simultaneouslythe container fleet size and the parameters used in the reposi-tioning policy. The optimization procedure was based on GeneticAlgorithms and Evolutionary Strategy combined with adjustmentstaking into account the knowledge of the underlying problems.
3. Formulation of the problem
The problem considered in this paper consists of finding theroute for a given barge container ship as to maximize the profit ofthe shipping company. An example of inland waterway is presentedin Fig. 1.
Port 1 is the sea or hub port, while ports 2 to n represent riverports. Index associated to each port depends on its distance fromport 1. The arrows indicate streams for the sequence of calling
ports. The demanded container traffic between each pair of ports(i, j), i, j = 1, 2, . . ., n, i /= j is specified. The solution of this problemdefines upstream and downstream calling sequence and number ofloaded and empty containers transported between any two ports
V. Maras et al. / Applied Soft Comp
Upstreamnavigation
. . .
. . .
Downstream navigation
234
567
n
n-1
1
wpfrmt
sttaa
3
cfit
•
•
•
•
•
•
•
•
Fig. 1. Example itinerary of a barge container ship.
hile achieving maximum profit of the shipping company. The firstort (a sea port, located at a river mouth) and the last port (theurthest port upstream) are always included in a solution, while theemaining n − 2 ports in either direction (upstream or downstream)ay or may not appear in the optimal (and even any feasible) solu-
ion.As it is not realistic to suppose that capacity of barge container
hip ensures the satisfaction of all customer demands, containerraffic between ports has a highly significant role. More precisely,he objective is to determine the number of containers (both loadednd empty) to be transferred between any two calling ports whilechieving maximum profit of the shipping company.
.1. Problem overview
Routing is a fairly common problem in transport, but the bargeontainer transport problem addressed here has certain intrinsiceatures that make the design of transport routes and correspond-ng models particularly difficult. The following assumptions imposehe restrictions on the routing:
the barge shipping company (or charterer) wants to hire a ship ora tow for a period of one or several years (‘period time charter’) inorder to establish container service on a certain inland waterway;the ship follow the same route during a pre-specified planningtime horizon; as common assumption, this horizon may assumedto be one year long;the trade route is characterized by one sea or hub port located ata river mouth and several intermediate calling river ports;the model assumes a weekly known cargo demand for all portpairs; this assumption is valid as the data regarding through-put from previous periods and future prediction allow obtainingreliable values of these demands;a barge container ship route corresponds to a feeder containerservice; the starting and ending point on the route should be thesame, i.e. in this case it is the sea port where transshipment ofcontainers from barge to sea container ships and vice versa takesplace;a barge container ship travels upstream from the starting sea portto the final port located on inland waterway, where the ship sailsfrom in the downstream direction to the same sea port endingthe route;maximum allowed route time, including sailing time and servicetime in ports, has to be set in accordance with the schedule of the
mainline sea container ship calling at the transshipment port;it is not necessary for the barge container ship to visit all ports onthe inland waterway; in some cases, calling at a particular port orloading all containers available at that port may not be profitable;
uting 13 (2013) 3515–3528 3517
• the ship does not have to visit the same ports in upstream anddownstream directions;
• all the container traffic emanating from a port may not be selectedfor transport even if that port is included in the route.
• container service is organized as liner and accordingly liner termsare valid; this imposes that a barge shipping company has todeal with transshipment costs, port dues and empty containerrepositioning costs, in addition to the cost of container transport;
• the demand for empty containers at a port is the differencebetween the total traffic originating from the port and the totalloaded container traffic arriving at the port for the specified timeperiod; the assumption is valid since this study addresses theproblem of determining the optimal route of a barge containership for only one ship operator (similar to the case studied by[11]);
• empty container transport [27,33,37] does not incur additionalcosts as it is performed using the excess capacity of barge com-pany ships (this transport actually incurs some costs, but its valueis negligible in comparison with empty container handling, stor-age and leasing costs);
• if a sufficient container quantity is not available at a port, theshortage is made up by leasing containers with the assumptionthat there are enough containers to be leased (for details see [11]).
The objective when designing the transport route of a bargecontainer ship is to maximize shipping company profit, i.e. the dif-ference between the revenue arising from the service of loadedcontainers (R) and the transport costs. These costs are related toshipping (TC) as well as empty container handling (EC). Therefore,the objective function has the form (see [11]):
Y = R − TC − EC. (1)
The model is generated with the assumption that one ship ortow is performing all weekly realized transport. An extension to amulti-ship problem is straightforward. It enables a better satisfac-tion of customer demands but requires modification of the one-shipvariant. On the other hand, if a shipping company has an option tocharter one among several various ships at disposal, then each shipcan be evaluated separately using this model to determine the bestoption.
The total number of ships employed on a route is determinedas the ratio of barge container ship round-trip time and servicefrequency of mainline sea container ships (one week in our case).For example, if the round-trip time is 20 days, the company shouldcharter 3 ships to be able to satisfy weekly demands of the cus-tomers. To describe all components of the relation (1) and precisethe calculation procedure for the shipping company profit wepresent a mixed-integer linear programming formulation of theconsidered problem.
3.2. Mathematical formulation
In this section the barge container ship route is formulated as themixed-integer linear program (MILP) with an aim to maximize theshipping company profit. The problem is characterized by the fol-lowing input data (measurement units are given in square bracketsif applicable):n: number of ports on the inland waterway, including the
sea port;v1 and v2: upstream and downstream barge container ship speed,
respectively [km/h];scf and scl: specific fuel and lubricant consumption, respectively
[t/kWh];fp and lp: fuel and lubricant price, respectively [US$/t];Pout: engine output (propulsion) [kW];dcc: daily time charter cost of barge container ship
[US$/day];
3 Comp
C
m
tt
z
r
lu
u
pu
u
p
sl
fiD
•
•
m
s
z
z
z
z
∑
∑
518 V. Maras et al. / Applied Soft
: carrying capacity of the barge container ship in Twentyfeet Equivalent Units [TEU];
axtt
and mintt
: maximum and minimum turnaround time on a route[days];
l: total locking time at all locks between ports 1 and n [h];b: total time of border crossings at all borders between
ports 1 and n [h];rij: weekly expected number of loaded containers
available to be transported between ports i and j [TEU];ij: freight rate per container from port i to port j
[US$/TEU];: distance between ending ports 1 and n [km];fci and lfci: unloading and loading cost, respectively per loaded
container at port i [US$/TEU];eci and leci: unloading and loading cost, respectively per empty
container at port i [US$/TEU];eci: entry cost per call at port i [$];fti and lfti: average unloading and loading time, respectively, per
loaded container at port i [h/TEU];eti and leti: average unloading and loading time, respectively, per
empty container at port i [h/TEU];ati and pdti: standby time for arrival and departure, respectively, at
port i [h];ci: storage cost at port i [US$/TEU];ci: short-term leasing cost at port i [$/TEU];
The optimal route of the barge container ship may be identi-ed by solving the following mathematical model (linear program).ecision variables of the model are:
binary variables xij defined as follows:
xij ={
1 if ports i and j are directly connected in the route,
0 otherwise;
integer variables zij and wij , representing the number of loadedand empty containers, respectively, transported from port i toport j [TEU].
The mathematical programming formulation is as follows
ax Y (2)
.t.
ij � zrij
j∑q=i+1
xiq, i = 1, 2, . . . , n − 1; j = i + 1, . . . , n (3)
ij � zrij
i−1∑q=j
xiq, i = 2, . . . , n; j = 1, . . . , i − 1 (4)
ij � zrij
j−1∑q=i
xqj, i = 1, 2, . . . , n − 1; j = i + 1, . . . , n (5)
ij � zrij
i∑q=j+1
xqj, i = 2, . . . , n; j = 1, . . . , i − 1 (6)
i
q=1
n∑s=j
(zqs + wqs) � C + M(1 − xij),
i = 1, 2, . . . , n − 1; j = i + 1, . . . , n (7)
n
q=i
j∑s=1
(zqs + wqs) � C + M(1 − xij), i = 2, . . . , n; j = 1, . . . , i − 1
(8)
uting 13 (2013) 3515–3528
n∑j=2
x1j = 1 (9)
n∑i=2
xi1 = 1 (10)
q−1∑i=1
xiq −n∑
j=q+1
xqj = 0, q = 2, . . . , n − 1 (11)
n∑i=q+1
xiq −q−1∑j=1
xqj = 0, q = 2, . . . , n − 1 (12)
mintt
� ttot
24� max
tt(13)
where M represents large enough constant.Constraints (3)–(6) model the departure ((3) and (4)) and arrival
((5) and (6)) of ship and containers to and from each port on theroute, respectively, in both upstream and downstream direction.More precisely, these constraints specify that loaded containerscould be transported between ports i and j only if both ports areincluded in the route. Moreover, it is not possible to transportmore containers than the weekly expected number (given as theinput data). Capacity constraints (7) and (8), guarantee that thetotal number of loaded and empty containers on-board will notexceed the ship carrying capacity at any voyage segment. The con-stant M is introduced to make the constraints valid for the portsthat are not included in the route. Constraints (9) and (10) are theset of network constraints ensuring that the ship visits the endingports of the route. The second set of network constraints (11) and(12) guarantee that the route represents a connected liner trip. Thebarge container ship is left with a choice of calling or not calling atany inner port.
Constraint (13) is related to the round-trip time of the bargecontainer ship (denoted by ttot [h]). The role of this constraint is toprevent the ship ending and calling at port 1 long before or afterthe arrival of the sea ship in this port. The round-trip time can becalculated as the sum of total voyage time, handling time of loadedand empty containers in ports and time required for entering andleaving ports (14).
ttot =(
l
v1+ l
v2+ tl + tb
)+
n∑i=1
n∑j=1
(zij(lfti + uftj)
+ wij(leti + uetj) + xij(pdti + patj)) (14)
According to Eq. (1) and given input data, the profit value Y iscalculated as follows:
Y =n∑
i=1
n∑j=1
zijrij −(
dcc · maxtt
+ Pout
(l
v1+ l
v2
)(fp · scf + lp · scl)
+n∑
i=1
n∑j=1
xij · pecj +n∑
i=1
n∑j=1
zij(ufci + lfcj)
⎞⎠
⎛ ⎞
− ⎝ n∑
i=1
(sci · sWi + lci · lWi) +n∑
i=1
n∑j=1
wij(ueci + leej)⎠ (15)
Comp
na
S
D
D
Q
P
P
l
s
wQSPDg
dfliamtTotr
natrtspTabpca
[siMvia
3
vaws
V. Maras et al. / Applied Soft
The number of containers to be stored at each port i, sWi, and theumber of containers to be leased at each port i, lWi, can be defineds in [11]. After the linearization we obtain constraints (16)–(23):
i − M gi ≤ 0 (16)
i − Pi + Si ≥ 0 (17)
i − Pi + Si − M(1 − gi) ≤ 0 (18)
i − M hi ≤ 0 (19)
i − Di + Qi ≥ 0 (20)
i − Di + Qi − M(1 − hi) ≤ 0 (21)
Wi = Qi −n∑
j=1
wji (22)
Wi = Si −n∑
j=1
wij (23)
here:i: the number of demanded containers at each port i [TEU];
i: the number of excess containers at each port i [TEU];i: the number of containers destined for port i [TEU];i: the number of containers departing from port i [TEU];i , hi: auxiliary binary variables.
The rate of empty container repositioning, storing and leasingepends on container inflow and outflow in each port. As theseows are becoming more balanced, empty container problem loses
ts significance. On the other hand, the model given in this paperllows that any number of containers zij smaller than demand zrijay be transported between ports i and j. In such a way, the option
o balance container in/out flows to/from any port is introduced.his means that the mathematical model itself has huge impactn the reduction of the empty container handling costs. Therefore,his approach is intended to lowering the needs for empty containerepositioning, storing and leasing.
Another important consequence of the situation zij < zrij, i.e. thatot all transport demands at any calling port are satisfied, is theppearance of unsatisfied clients. It is especially critical if a cer-ain number of containers of one client is transported, while theest is rejected. In this case, the applied approach would lead tohe unsatisfied customers, loss of profit and bad reputation of thehipping company. However, transport demand in each particularort is usually composed of demands coming from several clients.herefore, any value of zij, which is less than zrij, should be carefullynalyzed by the decision-maker in order to select the containers toe accepted for transport. As we show in Section 5, in our exam-les the number of unsatisfied demands is small comparing to theapacity utilization of the selected barge container ship. In addition,t the same time the needs for empty containers are reduced.
The described problem is similar to the one considered in11]. The main difference between these two models lies in theervice network design: while in [11] the sea container services considered, here we deal with the barge container transport.
oreover, the definition of objective function is adapted to bargeessels. Finally, the structure of our model is more complex since itncludes round trip time limits, downstream–upstream navigationnd selection of the container quantity to be loaded at any port.
.3. Problem complexity and optimal solution
Our ship routing problem represents a highly unconstrained
ersion of the selective traveling salesman problem with pick upnd deliveries, or equivalently, the single vehicle routing problemith selective pickup and deliveries. Therefore, the fact that it is
trongly NP-hard is straightforward.
uting 13 (2013) 3515–3528 3519
The solution of this problem defines upstream and downstreamcalling sequence and the number of loaded and empty containerstransported between any two calling ports, while achieving maxi-mum profit of the shipping company. The calling sequence (in onedirection) is defined by the upper right (and down left) triangle ofthe binary matrix X containing decision variables xij, i /= j. There-fore, to determine the calling sequence we have to assign values ton2 − n binary variables. At most n − 1 elements (for each direction)are equal to one.
The container traffic between calling ports is defined by the ele-ments of matrices Z = [zij]n×n and W = [wij]n×n, again for i /= j. Inaddition, we have to determine the total round trip time ttot, ship-ping company profit Y, and number of leased lWi and stored sWiempty containers at each port. These are the variables reportedto the user. Moreover, there are some auxiliary variables that aredetermined during the solution process in connection to handlingempty containers. These are binary variables hi, gi and integers Si,Pi, Di, Qi as it is modeled by the constraints (16)–(23).
To summarize, we have to determine (n2 − n) + 2n = n2 + n binaryvariables, 2(n2 − n) + 2n + 4n = 2n2 + 4n integer variables and tworeal (floating point) values.
The CPLEX MIP solver [38] was used to optimally solve small sizeproblem instances of the above described MILP. We were able tooptimally solve instances with 15 ports within 10 min to 1 h of CPUtime, and also some of the instances with 20 ports, for which therequired CPU time exceeded 29 h. We could allow long executiontime required to find optimal solution since the barge containership follows the same route for at least one year period. However,the main problem for the CPLEX MIP solver to find an optimal solu-tion of larger instances appears to be the lack of memory. Let usstate that our experiments were performed on a computer with8 GB of RAM which is fairly large. Although 16 or 32 GB of RAMare common by now, we would not expect to be able to solvemuch larger instances. Obviously, like in many other combina-torial optimization problems, real examples are too complex tobe solved to optimality. Therefore, we propose the use of meta-heuristic approaches which are the common way to tackle thesekind of problems [11,15]. In particular, we apply 0–1 MIP heuris-tics described in the next section. Our experiments show that theobjective values obtained by MIP heuristics within only 1 h are ofa very similar, or even better quality than those obtained by theCPLEX MIP solver in much (usually 10–20 times) longer time.
4. Description of the MIP heuristics
According to the mathematical formulation provided in Sec-tion 3, the ships routing problem presented in this paper is thespecial case of the 0–1 mixed integer programming problem (0–1MIP):
max{�(�) = cT �|� ∈ X}, (24)
where X = {� ∈ R|N||A� ≤ b, �j ∈ {0, 1} for j = B, �j ∈ Z
+0 for j =
G, �j ≥ 0 for j ∈ C} is the feasible set, N = C ∪ G ∪ B is the set ofindices of all variables, C is the set of indices of continuous vari-ables, G is the set of indices of general integer variables and B isthe set of indices of binary variables, C ∩ G = ∅, C ∩ B = ∅, G ∩ B = ∅,B /= ∅. Indeed, all problem constraints and the profit function Yare linear, and the set of binary variables {xij, hi, gi|0 ≤ i, j ≤ n} isnon-empty. Therefore, it is possible to tackle this problem by using0–1 MIP solution methods, the so-called matheuristics.
Although the term matheuristic (short from math-heuristic, also
known as model-based heuristic) is still recent, a number of meth-ods for solving optimization problems which can be considered asmatheuristics have emerged over the last decades [19]. The mainadvantage of matheuristics is their convergence, which means that
3520 V. Maras et al. / Applied Soft Computing 13 (2013) 3515–3528
LocalSearch-MIP(P, ξ , k∗)1 procee d = tru e; Set ini tial neighborhoo d size k;2 while (procee d) do3 ξ ← MIPSolve(P (k, ξ ), ξ );4 if (cT ξ < cT ξ ) then5 ξ ← ξ ; endif;6 if (statu s = ‘‘optimalSolFound’ ’ ||7 status = ‘‘provenInfeasible’’) then8 P ← (P | δ(ξ , ξ) > k); endif;9 k ← UpdateNeighborhood(k, k∗, status);
10 Update proceed;11 endwhile
tNbccmts
L[pw
df{t
ı
a
ı
Fadf
b�d
NFa�N
npec
0ptTS
UpdateNeighborhood(k, k∗ , status)1 if (statu s = ‘‘optimalSolFound’ ’ ||2 statu s = ‘‘feasibleSolFound’ ’) then k ← k∗;3 else if (statu s = ‘‘provenInfeasible’ ’) then k ← k + k∗/2 ;4 else k ← k k∗/2 ;5 return k.
12 return ξ .
Fig. 2. Local search in the 0–1 MIP solution space.
hey provide optimal solution if the running time is not limited.eighborhood search type meta-heuristics, such as Variable Neigh-orhood Search (VNS) [39], are proved to be very effective whenombined with optimization techniques based on the mathemati-al programming problem formulations. The common idea behindatheuristics is the following: set some variables to some par-
icular values by using meta-heuristics rules in order to createub-problems to be solved within the exact MIP solver framework.
In this paper, we apply three state-of-the-art matheuristics:ocal Branching (LB) [17], Variable Neighborhood Branching (VNB)18] and Variable Neighborhood Decomposition Search for 0–1 MIProblems (VNDS-MIP) [19]. In order to describe these algorithms,e first introduce some notations.
Notations and definitions. Let P be a given 0–1 MIP problem asefined in (24). The linear relaxation LP(P) of problem P is obtainedrom P by releasing the integer requirements on �. For a given � ∈0, 1}|B| × Z
+0
|G| × R+0
|C|, and an arbitrary � ∈ {0, 1}|B| × Z+0
|G| × R+0
|C|,he distance between � and � is defined as
(�, �) =∑j∈B
|�j − �j| (25)
nd can be linearized as (see [17]):
(�, �) =∑j∈B
�j(1 − �j) + �j(1 − �j) (26)
urther on we will denote with (P|C) the subproblem obtained bydding the set of constraints C to problem P. Now we can also intro-uce the following subproblem notation: P(k, �0) = (P|ı(�0, �) ≤ k),or k ∈ N ∪ {0}.
The neighborhood structures {Nk|1 ≤ kmin ≤ k ≤ kmax ≤ |B|} cane defined knowing the distance ı(�, �) between any two solutions, � ∈ X. The set of all solutions in the kth neighborhood of � ∈ X isenoted as Nk(�), where
k(�) = {� ∈ X|ı(�, �) ≤ k}. (27)
rom the definition of Nk(�), it follows that Nk(�) ⊂ Nk+1(�), forny k ∈ {kmin, kmin + 1, . . ., kmax − 1}, since ı(�, �) ≤ k implies ı(�,) ≤ k + 1. It is trivial that, if we completely explore neighborhoodk+1(�), it is not necessary to explore neighborhood Nk(�).
Local search in the 0–1 MIP solution space. Introducing theeighborhood structures into the 0–1 MIP solution space X makesossible the employment of a classical local search in order toxplore X. The pseudo-code of the corresponding local search pro-edure, named LocalSearch-MIP, is presented in Fig. 2.
The LocalSearch-MIP explores the solution space of the input–1 MIP problem P, starting from the given initial solution �′. Input
arameter k∗ controls the change of the neighborhood size duringhe search process. The algorithm returns the best solution found.he neighborhood Nk(�′) of �′ is defined as a subproblem P(k, �′).tatement � = MIPSolve(P, �) denotes a call to a generic MIP solver
Fig. 3. Updating the neighborhoods in LB.
for a given input problem P, where � is a given starting solution and� is the best solution found, returned as the result. The constantstatus denotes the solution status as obtained from the MIP solver.
In each iteration, the current neighborhood Nk(�′) of the incum-bent solution �′ is explored (line 3 in Fig. 2). If a better solution isfound, the incumbent solution is accordingly updated (lines 4–5 inFig. 2). If the current subproblem is solved exactly (the incumbentsolution is proven optimal for the current subproblem) or the cur-rent subproblem is proven infeasible, the distance constraint ı(�′,�) > k is added to the original problem, in order to discard the neigh-borhood Nk(�′) from the search space (see lines 6–8 in Fig. 2). Theneighborhood size is then updated (line 9 in Fig. 2) and the wholeprocess is iterated until the fulfillment of stopping criteria.
4.1. Local Branching
Local Branching (LB), introduced by Fischetti and Lodi[17], is infact the first local search method for 0–1 MIPs, as described above(see pseudo-code in Fig. 2), which employs the linearization (26) ofthe Hamming distance in the 0–1 MIP solution space. For the cur-rent incumbent solution �, the search process begins by exploring aneighborhood of � of a predefined size k = k∗ (defined as a subprob-lem P(k, �)). A generic MIP solver is used as a black-box for solvingproblems P(k, �), for different values of k. After a new incumbentis found, the whole process is iterated. The LB pseudo-code can berepresented in a general form given in Fig. 2, with the special formof procedure UpdateNeighborhood(k, k∗, status) as given in Fig. 3.
The input parameters for Local Branching, as a special case oflocal search for 0–1 MIPs (see pseudo-code in Fig. 2), include aninput 0–1 MIP problem P, an initial solution �′ and the neighborhoodsize control parameter k∗. The LB executes iteration by iterationuntil some predefined stopping criterion is satisfied. The stoppingcriterion normally includes the total running time allowed, but mayalso the maximum number of diversifications (see line 3 in Fig. 3)and/or the total number of iterations.
In this paper we propose an improved implementation of theoriginal LB algorithm from [17]. Whereas in the original LB algo-rithm the initial neighborhood size is fixed to a certain predefinedvalue (equal to 20) for all tested instances, we observe that thesame value of this parameter incurs different behavior of the searchprocess for the different problem instances. It is obvious that theneighborhood size 20 does not provide as good level of diversifi-cation in the case of instances with 650 binary variables, as in thecase of instances with 110 binary variables. The same holds for anyother fixed value which does not take into account specific char-acteristics of different instances. This is why we decide to set thevalue of the initial neighborhood size to 20% of the total numberof binary variables of a particular instance, therefore adjusting it tothe specific features of each particular instance.
4.2. Variable Neighborhood Branching
Variable Neighborhood Branching (VNB) is another heuristic forsolving 0–1 MIP problems, using the general-purpose MIP solver asa black-box [18]. VNB adds constraints to the original problem, asin the LB method. However, in VNB, neighborhoods are changed in
V. Maras et al. / Applied Soft Comp
UpdateNeighborhood(k, k∗ , statu s)1 if (statu s = ‘‘optimalSolFound’ ’ ||
status = ‘‘feasibleSolFound’’) then k ← 1;2 else3 if (status = ‘‘provenInfeasible’’) then4 k ← k + 1;5 endif6 endif
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Fig. 4. Neighborhood update in VND-MIP.
systematic manner, according to the rules of the general Variableeighborhood Search (VNS) algorithm [39,40].
The local search method used within VNB is a deterministicNS variant, Variable neighborhood descent (VND) (see [41]). VNDtarts from a given initial feasible solution as the incumbent solu-ion x and examines its current neighborhood entirely. In casef improvement, the same search procedure is performed start-ng from the improved solution x′ as the new incumbent. If theres no improvement, the next neighborhood of x is explored. Thisrocess is repeated until the fulfillment of the stopping criterionwhich usually includes the maximum number of neighborhoods,nd/or the predefined running time limit). The pseudo-code ofND for MIPs, called VND-MIP, can also be represented in a gen-ral form given in Fig. 2, but with the special form of procedurepdateNeighborhood(k, k∗, status) as given in Fig. 4.
The input parameters for VND-MIP, as a special case of localearch for 0–1 MIPs (see pseudo-code in Fig. 2), include an input 0–1IP problem P, an initial solution �′ and the neighborhood size con-
rol parameter k∗. Note that parameter k∗ in VND-MIP representshe maximum allowed neighborhood size, whereas in LB it repre-ents the initial neighborhood size for a given incumbent solutionector.
The diversification in Variable Neighborhood Branching (shak-ng step in general VNS) is performed by choosing the first feasibleolution from the disk of radii k and k + kstep, where k is the currenteighborhood size, and kstep is a given input parameter. The diskize is increased as long as the feasible solution is not found.
In addition to the input 0–1 MIP problem P and the initialolution �′, the input parameters of VNB are the minimum neigh-orhood size kmin, the neighborhood size increment step kstep, theaximum neighborhood size kmax and the maximum neighbor-
ood size kvnd within VND-MIP. Similarly as in the case of LB, weropose an improved implementation of the original VNB algo-ithm from [18]. In the original VNB algorithm, the values of kmin,step and kmax are fixed to 5, 5 and 150, respectively, for all testednstances. Here we observe that these values affect the searchrocess differently for different problem instances. For example,etting the maximum neighborhood size in VNB to 150 for instancesith 110 binary variables is not meaningful, since the maximumistance between any two solution vectors cannot exceed 110. Onhe other hand, for instances with 650 binary variables settingmax = 150 may not provide enough diversification. A similar rea-oning can be applied for any other fixed values which do not takento account specific characteristics of different instances. This is
hy we decide to set the values of kmax, kmin and kstep to 50%, 5% and% of the total number of binary variables of a particular instance,espectively. In this way we exploit the specific features of eacharticular instance.
.3. Variable Neighborhood Decomposition Search for 0–1 MIPs
Variable Neighborhood Decomposition Search (VNDS) is a two-evel VNS scheme for solving optimization problems, based uponhe decomposition of the original problem. It was proposed for the
uting 13 (2013) 3515–3528 3521
first time in [41]. Recently, VNDS has been implemented for solving0–1 MIPs [19] and the resulting procedure is abbreviated as VNDS-MIP. The approach proposed in [19] is a VNDS based diving strategy,which combines linear programming (LP) solver, MIP solver andVND based MIP solving method (VND-MIP) in order to efficientlysolve a given 0–1 MIP problem.
Input parameters for VNDS-MIP are an input 0–1 MIP problemP, an initial integer feasible solution �′ and an integer k∗ whichcontrols the size of the subproblems generated within VNDS-MIP.Starting from incumbent integer feasible solution � of P and anoptimal solution � of LP(P), binary variables are ranked in a non-decreasing order of the modules of the differences between thevalues of � and �. Subproblems within VNDS are obtained by suc-cessively fixing a certain number of ordered binary variables to theirvalues in the incumbent integer solution. In this way, the subprob-lem involves free binary variables which are furthest from theirlinear relaxation values. Then these subproblems are solved exactlyor within the CPU time limit. The subproblems are changed by thehard fixing of the variables (or diving), according to the VNS rules.The pseudo-code of the VNDS-MIP procedure is provided in Fig. 5.
4.4. Large scale Variable Neighborhood Search for MIPs
In this paper we propose an extension of the basic VNDS-MIPalgorithm from [19] which also considers general integer variables(see pseudo-code in Fig. 6).
If the distance function (25) is replaced with � : X2 → R, whichtakes into account general integer variables:
�(�, �) =∑j∈B∪G
|�j − �j|, (28)
it is clear that very large neighborhoods are obtained as a result.In the large scale variant of the basic VNDS-MIP, here denoted
as LS-VNDS-MIP (LS-VNDS for short), distance function (28) isemployed in the VNDS stage (lines 5–6 in LS-VNDS pseudo-codefrom Fig. 6) and distance function (25) in the local search (i.e. VND-MIP) stage.
Input parameters for LS-VNDS are an input 0–1 MIP problemP, an initial integer feasible solution �′ and an integer k∗ whichcontrols the size of the subproblems generated within LS-VNDS.Starting from incumbent integer feasible solution � of P and an opti-mal solution � of LP(P), all integer variables (both binary and generalinteger) are ranked according to their distances from the corre-sponding LP solution values (line 5 in pseudo-code in Fig. 6). This isa generalization of basic VNDS-MIP, which only considers binariesin this stage. Selected integer variables are then fixed to their valuesin the incumbent integer feasible solution (line 9 in pseudo-code inFig. 6). The set of variables to be fixed is updated in each iteration in aVNDS manner, as in the basic VNDS-MIP. If an improvement occurs,VND-MIP is further applied as a local search procedure (line 12 inthe pseudo-code in Fig. 6), but only with neighborhoods defined inthe 0–1 MIP solution space. Although it is possible to define localsearch in the solution space of general MIP problems (see, e.g. [42]or [43]) using the definition of distance function (28), this requiresintroduction of additional variables and constraints to the initialMIP problem, therefore resulting in a larger problem, which takeslonger to solve. This is why we have decided to apply local searchonly in the 0–1 MIP solution space in this stage.
One should note that the main difference between VNDS-MIPand LS-VNDS is in the variable domain. Namely, LS-VNDS uses bothbinary and general integer variables, while VNDS-MIP is focusedonly on binary variables.
3522 V. Maras et al. / Applied Soft Computing 13 (2013) 3515–3528
Fig. 5. VNDS for 0–1 MIPs.
Fig. 6. LS-VNDS for 0–1 MIPs.
V. Maras et al. / Applied Soft Computing 13 (2013) 3515–3528 3523
Table 1The characteristics of 5 tested container barge ships (input data as listed in Section 3.2).
Container barge ships No. units TEU Pout [kW] Total TEU v1 [km/h] v2 [km/h] dcc [$]
In this section we present the comparison results for the appli-ation of the described MIP-based heuristic search methods. In fact,e compared 5 solution methods: CPLEX, LB, VNB, VNDS-MIP and
S-VNDS within the same CPU time limit.The computational results are presented and discussed in an
ffort to assess and analyze the efficiency of the presented model.e study the performance of different solution methods for ourodel from two points of view: first we compared solution qualities
nd running times and then we analyze mathematical program-ing aspects and transportation usefulness of the obtained results.
.1. Experimental environment
Hardware and software. Our tests are performed on Intel Core Duo CPU E6750 on 2.66 GHz with RAM = 8 GB under Linux Slack-are 12, Kernel: 2.6.21.5. The applied MIP-based heuristics are all
oded in C++programming language for Linux operating system andompiled with gcc (version 4.1.2) and the option -o2. For exact solv-ng we used CPLEX 11.2 [38] MIP solver and AMPL [44,45] runningn the same machine. Moreover, CPLEX 11.2 is used as generic MIPolver in all tested MIP-based heuristics.
Test bed. The lack of publicly available test instances forhe considered problem prevented us to compare our resultso the similar results from the literature. Therefore, our testxamples were generated randomly, in such a way that the num-er of ports n was varied from 10 to 25 with increment 5.oreover, for each value of n, 5 instances were produced with
ifferent ship characteristics (carrying capacities, daily charterosts, downstream and upstream speeds, engine outputs, fuelnd lubricant consumptions) which are summarized in Table 1.nput data are set in accordance with the common values foundt different sources like internet sites (www.portofantwerp.com,ww.rotterdamportinfo.com), project [5] and papers [6,21,23,46].
n this way we produced hard and easy examples for each problemize. The set of instances with their basic MIP properties is listedn Table 2. All the instances in the form of AMPL *.dat files can be
ownloaded from www.mi.sanu.ac.rs/∼tanjad/ships.htm.
Parameters. According to our preliminary experiments, weave decided to use the different parameter settings for differ-nt instance sizes. Thus, different parameter settings were used
able 2IP characteristics of various size test instances.
for the four groups of instances generated for 10, 15, 20 and 25ports, respectively. According to our preliminary experiments, thetotal running time limit (tlim) for all methods (including CPLEXMIP solver alone) was set to 60, 900, 1800 and 3600 s for 10, 15,20 and 25 ports, respectively. In all heuristic methods, the timelimit for subproblems (tsub) within the main method was set to10% of the total running time limit. In two VNDS heuristics, thetime limit for the VND-MIP procedure within VNDS is also set to10% of the total running time limit. In VNB, parameters regardinginitialization and change of neighborhood size are set in the follow-ing way: the maximum neighborhood size kmax is approximately50% of the number of binary variables and minimum neighborhoodsize kmin and neighborhood size increase step kstep are set to 5%of the maximum neighborhood size. Namely, kmin = kstep = b/20 �and kmax = 10kmin, where b = |B| is the number of binary variablesfor the particular instance. For example, for 10 ports this yieldskmin = kstep = 6 and kmax = 60. In LB, the initial neighborhood size k isset to approximately 20% of the number of binary variables for theparticular instance, i.e. k = b/5 �. Actual values for all parametersare summarized in Table 3. All CPLEX MIP solver parameters areset to their default values.
5.2. CPLEX results
In Table 4, results for the tested instances are presented, asobtained by the CPLEX MIP solver invoked by AMPL, without anyrunning time limitations.
The first column of this table lists the names of our test exam-ples. The best objective values are given in the second column whilethe corresponding running times are provided in the column three.The column four contains the CPLEX MIP solver status at the endof computation. Total running times the CPLEX MIP solver neededto reach the corresponding statuses are indicated in the last col-umn. The objective values from column 2 which are proven optimalare bolded. There is a number of instances for which the CPLEXMIP solver has managed to obtain a near-optimal solution, butfailed to find an optimal one, even without any running time lim-itations (as indicated in column 4 of Table 4). It can be observed
that, for largest instances, CPLEX MIP solver failed to prove the(near)optimality of the solution, due to the insufficient memoryresources.
Table 3Parameter values for different MIP heuristics.
Here we present the comparison results of five methods tested:PLEX MIP solver, LB, VNB, VNDS and LS-VNDS, within the samemount of CPU time. The imposed running time limitations (seearameter settings above) are listed in Table 3.
The best objective function (profit) values obtained by all MIPethods are provided in Table 5. The highest profit values for each
xample are bolded, indicating the method(s) that performed theest in this case. At the same time, running times required to obtainhe best profit value by each method are presented in Table 6 andhe shortest among them is bolded, regardless the quality of thebtained objective value.
According to the average profit values from Table 5, we can seehat VNB has the best performance quality-wise, whereas other
ethods are worse than CPLEX MIP solver. However, regarding the
unning times presented in Table 6, the CPLEX MIP solver is thelowest, with 1518 s average running time, followed by VNB with369 s average running time. LB and VNDS show similar perfor-ance with average running times of 517 s and 509 s, respectively.
able 5omparison results for five methods – Objective function values.
Instance Profit values obtained by
CPLEX LB
port10 1 22,339.01 22,339.00
port10 2 24,738.23 24,738.00
port10 3 23,294.74 23,294.74
port10 4 20,686.27 20,686.00
port10 5 25,315.00 25,315.00
port15 1 12,268.96 12,268.96
port15 2 25,340.00 25,340.00
port15 3 13,798.22 12,999.34
port15 4 22,372.58 22,372.58
port15 5 15,799.96 15,800.00
port20 1 18,296.19 16,653.70
port20 2 32,789.55 32,250.44
port20 3 19,626.28 19,539.69
port20 4 26,996.03 25,928.76
port20 5 23,781.17 23,904.21
port25 1 20,539.88 21,619.18
port25 2 32,422.19 33,528.22
port25 3 20,008.23 17,651.27
port25 4 27,364.50 28,388.23
port25 5 22,897.03 22,303.71
Average: 22,533.70 22,346.05
Out of memory 26762.03Out of memory 24725.92Out of memory 30587.42
When the speed is more important then the solution quality, theLS-VNDS heuristic is clearly the best choice, with only 188 s averageexecution time.
In summary, we may conclude that using the heuristic meth-ods for tackling the presented ship routing problem is beneficial,both regarding the solution quality and (especially) the execu-tion time. VNB heuristic proves to be better than the CPLEX MIPsolver regarding both criteria (solution quality/execution time). LBand VNDS-based heuristics do not achieve as good solution qualityas CPLEX, but have significantly better execution time, especiallythe LS-VNDS method which is approximately 8 times faster thanCPLEX. Soft variable fixing (VNB and LB) appears to be more effec-tive (quality-wise) for this model than the hard variable fixing(VNDS-based methods). The solution quality performance of thebasic VNDS-MIP may be explained by the fact that the numberof general integer variables in all instances is more than twice as
large as the number of binary variables, and therefore the subprob-lems generated during the VNDS-MIP process by fixing only binaryvariables are still large and not so easy for the CPLEX MIP solver.Therefore, the improvement in VNDS-MIP usually does not occur
level of demands satisfaction: only 17.3% of them were not satisfied
Average: 1518.34 517.03 1369.42 508.77 188.01
n the late stages of the search process. On the other hand, in case ofS-VNDS, it appears that fixing the most of general integer variablesogether with binary variables limits the amount of diversification.
As one can see from the pseudo-code in Fig. 6 (specifically line), the set of subproblems defined in an iteration of VNDS-MIPlarge-scale or basic) depends on the incumbent integer solution.herefore, an improvement in the local search stage is necessary toignificantly change the set of subproblems used in the next itera-ion of VNDS-MIP. No improvement in the local search stage leadso subsequent stalling in an incumbent solution. This explains why,or large problems, where local search is not effective, VNDS-MIPased heuristics get stalled in a local optimal solution relativelyuickly. However, when problems are not so large and local searchan manage improvements, LS-VNDS is able to reach a high-qualitybjective value (see objectives for instances with 10 and 15 portsn table 4). Comparing LS-VNDS-MIP with VNDS-MIP, we may con-lude that LS-VNDS is more effective in the VNDS stage, sinceesulting subproblems are smaller and easier to tackle by a com-ercial solver.A possible solution to this problem may be in an intelligent
ebounding of the general integer variables, which would be anntermediate strategy between the basic VNDS-MIP (in which alleneral integer variables are always set free) and LS-VNDS (whichxes a number of general integer variables, each to a single value).owever, designing such a method is beyond the scope of thisaper.
.4. Transportation analysis
In order to elaborate the obtained results from the transporta-ion usefulness point of view, we composed a table which gives theest results for each instance. Therefore, Table 7 indicates the ship-ing company’ highest profits for each considered ship and possibleumber of calling ports. This table is drawn out from Table 5 by tak-
ng the highest value from the corresponding row. This means thate take out the best of the five obtained objective values.
Table 7 clearly indicates the significance of economic scaling
n the shipping industry. Since operating, voyage and capital costso not increase in proportion to the TEU capacity of the ves-el, using a bigger ship reduces the unit cost per TEU. As unitosts are getting lower, barge container ships with higher carrying
uting 13 (2013) 3515–3528 3525
capacity are becoming more profitable. However, profit increasecan be achieved with the assumption that the cargo or containerflows between ports are sufficient to achieve high level of ship TEUcapacity utilization. From Table 1 we can see that ships 2 and 5 havesignificantly larger capacities (409 TEU and 338 TEU, respectively)then the others. Therefore, in Table 7 the highest objective valuesare always obtained for one of these two ships.
Maximum turnaround time maxtt
is set to 21 days for instances
with 10 ports, and to 28 days for all other instances (with 15, 20 and25 ports). Minimum turnaround time min
ttis set to min
tt= max
tt− 1.
These settings enable us to examine the importance of capital costs(time charter costs in our case) in barge container shipping. Ourconclusion coincides with previously reported results in the bargecontainer transport [1] where it was indicated that capital costsrepresent the highest portion of a shipping company total costs. It isalso the main reason why profits acquired for 10 ports are, in manyinstances, higher than profits reached for larger number of possiblycalling ports for the same type of ship: the total time charter costsare much higher in routes obtained for instances with 15, 20 and25 ports in comparison with 10 ports instances. Therefore, totalrevenue is charged more with this cost for examples where max
ttis
set to 28 days.On the other hand, the contribution and the importance of total
time charter costs is also evident from the values of the objectivefunction of ship 4 in the examples with 15, 20 and 25 ports: Dailytime charter costs of ship 4 are much lower than for the other 4ships. This impacts its second best position for the latest three typesof instances.
From Tables 4 and 5 we can draw the conclusions about thebest barge container ship, and consequently the container lines, tobe used for any number of calling ports. For example, by comparingresults for 10 ports in Tables 4 and 5, it is easy to see that containerbarge ship 5 achieves the highest profit. Relevant objective value,in this case, has been obtained by LS-VNDS method. Therefore, anoverview of all the other results will be based on the outcomesprovided by this heuristic method. Having this in mind, optimalroute of barge container ship 5 for 10 possible ports of call includesthe callings at the following ports, in both directions:Upstream: 1 – 3 – 5 – 6 – 7 – 8 – 9 – 10;Downstream: 10 – 9 – 8 – 5 – 4 – 3 – 1.
Fig. 7 gives a schematic overview of the container flows at theconsidered inland waterway. These flows are based on the numberof both loaded (zij) and empty (wij) containers to be transportedby barge container ship 5 between any two ports in the obtainedroute. Values of these variables, as well as results relating to thestructure of transshipped containers in each port (number of loadedand unloaded containers), directions of container flows and densityof traffic are given in the so called “downward” (“chess”) Table 8.
Total costs of the considered barge container ship on its route,composed of transport related and port related costs (daily timecharter costs, fuel and lubricant costs, port taxes and cargo hand-ling costs), as well as empty containers repositioning costs (emptycontainers storage, short-term leasing and handling costs) are196,999.68 $. As total revenue, acquired due to transport of loadedcontainers along the route is 222,315.00 $, the shipping companyprofit becomes 25,315.32 $.
Average capacity utilization of the barge container ship 5 is88.14% (89.05% during the upstream and 87.08% in the down-stream navigation). The total number of transported containers is1062 TEU. It represents 51.2% of all transport demands, and 82.7 % ofthe demands in the calling ports. These figures clearly indicate the
although the capacity of the used barge container ship is almost fourtime smaller than the transport demands in the calling ports. Sim-ilarly, 51.2% of total transport demands (taking into account also
3526 V. Maras et al. / Applied Soft Computing 13 (2013) 3515–3528
Table 7Optimal ships for various numbers of ports.
10 ports instances Highest obj. value 15 ports instances Highest obj. value 20 ports instances Highest obj. value 25 ports instances Highest obj. value
he ports that are not included in the route) were satisfied by thearge container ship of more than six time smaller capacity. Theatisfaction of total transport demands can be easily improved byterative application of the same model to the remaining demands.
The total round trip time of the barge container ship 5 on itsoute is 503.97 h or 20.99 days. This means that arrival time of bargehip will be overlapping with the relevant maritime container shipn the main route. Therefore, transshipment of containers betweenhese two ships, with no significant storage time, will be possibleo organize on such routes.
Increased power output enables ship to sail with higher speed,ut, at the same time, contribute to the higher fuel consumptionnd fuel costs (similar technological level). On the other hand, aigger ship means increased carrying capacities and positive effects
89 129 159 543 – – –17 9 – – 480 – –
on units costs, but leads to the higher time charter costs. Deter-mining optimal barge container ship to be employed on any routeshould be a result of compromise between all the above mentionedparameters. Therefore, the model proposed in this paper may havesignificant role in assisting the barge shipping companies to makethe most suitable decisions regarding engagement of barge con-tainer ships on the available routes.
6. Conclusion
This paper deals with the optimization of transport routes ofbarge container ships from a shipping company point of view. Weaddressed the barge container ship routing problem as a problemof maximizing the shipping company profit while picking up and
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elivering containers along the inland waterway with empty con-ainer repositioning. To the best of our knowledge, the publishedesearch on the joint optimization of barge container ship routingnd empty container repositioning in liner barge transport systemss scarce.
We identified the considered barge container ship routing prob-em as a special case of the 0–1 MIP problem. Therefore, we treatedhis problem by applying state-of-the-art heuristics for 0–1 MIProblems: Local Branching (LB), Variable Neighborhood BranchingVNB) and two variants of Variable Neighborhood Decompositionearch for 0–1 MIP problems (VNDS-MIP and LS-VNDS). Usinghe heuristic methods for tackling the presented ship routingroblem is beneficial, both regarding the solution quality and (espe-ially) the execution time. The VNB heuristic proves to be betterhan the CPLEX MIP solver regarding both criteria (solution qual-ty/execution time). LB and VNDS-based heuristics do not achieve asood solution quality as CPLEX, but have significantly better execu-ion time, especially the LS-VNDS method which is approximately
times faster than CPLEX.From the transportation usefulness point of view, provided the
ontainer volume and port facilities are available, we can concludehat an owner of a large barge container ship has more chanceso generate a positive cash-flow than the companies possessingnly smaller barge container ships. This explains why, over the lastentury, barge container ships have become bigger. The penaltyf increasing the ship size is the loss of flexibility, which impactsn the revenue side by limiting the number of ports that could beisited.
The model and solution method given in this research could beery useful practical tool for the barge container carriers to makeong term strategic decisions about establishing barge containerinterland transport routes. They can solve their real life problems,est different solutions of the problems and choose those whichre more suitable for their own needs. Therefore, the planning pro-ess in the barge shipping company could be improved by applyinghe proposed decision support system based on our optimization
odel, which would significantly impact the shipping companyusiness results.
For the future research, this study can be extended in severalirections. The extension to a multi-ship problem is straightfor-ard. Many important factors like allowing the change of a ship
oute during planning horizon, detailed modeling of ship servicen ports, stochastic formulation of some parameters like containeremands may be further included. However, it will significantly
ncrease the complexity of the problem.
cknowledgements
This research has been partially supported by NSF Serbia,rants Nos. OI174010, OI174033, TR36024 and TR36027. Theuthors would like to thank Professor Theodor Gabriel Crainic fromépartement de management et technologie Université du Québec
Montréal and CIRRELT, Université de Montréal and Ms Berit Dan-aard Brouer from the Department of Management Engineering,echnical University of Denmark for their valuable comments anduggestion on the presented material. We would also like to thankhe anonymous referees for the valuable suggestions that led to themproved presentation of the results described in this paper.
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