An Improved Formulation for Bulk, Cargo Ship Scheduling
with a Single Loading Pott
TENIWE U?VEThIT
i.aboratorIum voor Scheepshydimochanica kchlet Mekelweg 2, 2G2B CD Deift 1QL 01$ ?$Th - Fei 01$- 18103$Seong-Cheol Cho
Associate Professor, Department of Shipping Management, Korea Maritime University,
Pusan, 606-791, Korea
A.N. Perakis
Associate Professor, Department of Naval Architecture and Marine Engineering,
University of Michigan, Ann Arbor MI 481O9.2145, U.SA.
paper submitted for publication to Maritime Policy and Management
June, 1998
Absfract We present an improved, significantly more efficient formulation of an existing
model for bulk or semi-bulk cargo ship scheduling problem with a single loading por. The
original model, published by Ronen in 1986, was formulated as a nonlinear, mixed-integer program. In this work, we were able to re-formulate it into a linear one, by eliminating all the nonlinearities of the original. In addition, our niodel has far fewer integer v2nables than the original, thus resulting in significant computation tine and computer memory savings It is also Worth observing that our resulting model is a
rRODUCTION
This paper presents an improved formulation fora ship cheduling model with a single loading port, first developed by Ronen (1986) as a nonlinear mixed integer program. This model can frequently be applied to ship scheduling problems of trasporting minerals
or agricultural produce, which usually have, a single supply zOne of a large production
capacity enough to satisfy the demands at many places widely separatcd from one another. Comprehensive surveys of many models recently developed for various ship
scheduling problems can be found in (Ronen 1993). Typical optimization models for bulk cargo ship scheduling problems (Appeigren 1971, Fiher and Rosenwein 1989, Perakis and
Bremer 1992, Bremer and Perakis 1992) are described as pure 0-1 integer programming
problems using the framework of set covering1 partitioning, or packing. Unlike these,
Ronen (1986) gives his mOdel more flexibility by adding continuous variables.
This paper gives a more efficient, but equally accurate, alternative model for Ronen's problem. This is a significant improvement over Ronen's original formulation, since our model does not require nonlinearitiy and greatly reduces the number of the integer variables. It is also interesting to see that the resulting model an be interpreted as a generalization of the capacitatedfacility location problem (Ajkens 1985, Sridharan 1995)..
In the next section, we briefly describe Ronen's original formulation as apreliminazy step for our discussion. In section 3, our new formulation is yen. In section 4, our new
formulation is compared with the capacitated facility locatiofl problem.
2. ORIGINAL FORMULATION
Suppose we have a finite number of ships currently available for operation over the planning horizon, which may already have been determined by long-range planning or by current operation of the fleet. To each ship k E K (the set of ships), one of a finite set of possible schedules (n E Nk ) should be assigned. Sincethe considered scheduling problem
combination of unloading ports with a shortest route through them. We have a single loading port and a finite number of unloading ports (f E I ). The estin:iates of cargo
demands at the unloading potts re given as constants. We assume that the total capacity of the available fleet is greater than the total cargo demand (which is usually the case). Each ship is allowed to carry cargoes to more than one port, afld the shipment to a specific port can be ttansported by more than one ship. More detailed assumptions, related cost, solution methods and computational exp riments can be found in (Ronen 1986). Or paper concerns only formulation phase of the original work.
The following notation will be used to describe the model:
Notation for data
Ck : Capacity of ship k
a: Unit shipping cost to portj by ship k fbI: Cost of ship k taking mute n
u: Unloading rate per day at portj
Pk: Daily cost of stay for ship k at ports
D1: Demand of cargo at portj
N :. Set of possible sche tiles for ship k K: Set of available ships
J: Setof unloading ports
Ji f schedule n visists port j
to otherwise
M =min{C,D1J
r
Decision Vanables IikA
= to
if ship k.chooses schedule n and visits portj otherwise
yjj: amount of cargo to unloading portj shipped by ship k
The nonlinear mixed integer prrarn developed by Ronen is as follows:
(1)
keJ4 je)neN kN1
jJ
neN kcN*aN,subject tO
yz
Vk E K (2) JEJ*EN* = VjE 1 (3) kK iNkVkEK
(4) lENtCz
D1Vje
J
(5) kEN niN* X E {0,1}, O, (0,1)The objective function (1) minimi es the total fleet cost composed of shipping cost of crgoes, the cost of ships' unloading time, and the cost of takü g specific routes.
Constraints ()are the ship capaciy constraints, and constraints (3) requ re that all the demands should be satisfied. Each ship is guaranteed to haveexactIy one schedule by constraints (4). In case ship k is set to be idle, it can also be regarded as a specific schedule n. In this case, fk simply becomes the demurrage cost. during the planning horizon.
COnstraints (5) iequire that the sum of the capacities o the ships visiting eh port should be at least equal to the cargo demand at the port but are in fact ma C redundant by
if ship k selects schedule n otherwise
constraints (2)and (4). Ronen includes these redundant constraints in his model, in order to
consider only feasible schedules in the solution heuristics he tried
3. IMPROVED FORMULATION
By our new fOrmulation we show that the nonlinearity existing in Ronens model (Ronen 1986), in fact, is not necessary. The resulting model becomes a0-1 variable mixed
integer linear program.
It is easy to find that y can take a positive value only when portj is visited by ship
k, that is only when
z,,
1. This is the only reason why the oiiginal model has many ncNnonlinear terms in both objective function (1) and constraints (2)-(3). This way of
formulation also requ res a great number of integer variables z, which in, fact are found to be unnecessary in our new formulation. in order to make y take a positive value only when port] is visited by ship k, we, instead, introduce a kind of regulating constraints as
follows:
Yej -
M4bPxblJO
V(k,j).
It is natural that M, is a uppet bound of y, that is, the amount ofcargo carried to portj cannot exceed the ship capacity C nor cargo demand D.
Subject to Minimize !EN* JEJ = D. keK
Jo
X1 E{O,1}, kk aEN*VkeK
(2')VjEi
(3')VkEK
(41)V(k,j)
(5')(1)(3) are direct replacements fot(1)-(3) without complicating nonlinear terms ConstraintS (4') are the same as (4), and (5) is the regulating constraints described above.
It should be noted that the above new formulation has not only eliminated
nonlinearity from the original formulation, but also greatly reduced the number of integer variables. It is easy to see that the numb r of integer varial)les Xb, required in the origaal model is
INI
(where()I
denotes the cardinality of the set ()),and that of z isC INkI)Jl
and sothe total number of required integer variables is
(JNI)(i
+1), while our new model needs only INI integer Variables (xth). For example, if we are to solve an instance with 9 unloading ports that is I1 =9, then we only have to deal with One tenth as many integer variables, as in the original model. Thus we have eliminated both4. CONCLUDING REMARKS
This paper gives a simple illustration of an improved refOnnulation of an existing optimization model. We have givena linear integer altemathre of a nonlinear integer
programming model.
Medium-sized versions of this problem are expected to be solved optimally using commercial software. (For example, if 5 ships and 5 ports are involved, and up to 3 ports are allowed per schedule, then the maximum number of required integer variables is
5
+1!4! 2!3!
3!2!)
= 130.) However, larger versions still mightrequiredeveloping a more efflcient method than commercial branchand bound.
For a possible direction for anew solution method,, it is worthWhile to note that our
new formulation can be seen as a generalized version of a well-known class of optimization problems called "Capacitated Facility Location Problem" (CFLP, Aikens 1985). It also should be noted that many efficient methods to exploit the special structure of CFLP have already been developed (Sridharan 1995). CFLP is typically representedas a mixed integer program as follows:
nze
subject to C11Yj kEX jJii'
Y =D1
kEKy,1 Cx
y O, x E{O,I)
VkEK
VJEJ
Brerner, W.M., and A.N. Perakis. 1992. An Operational Tanker Scheduling Optimiz'tion
8
where c is the unit shipping cost of cargo carned to demand pointj from supply site k, and y the cargo amount shipped; fk is the fixed cost of building the supply facility at
candidate site k, and XA the binary variable to decide whether to build the supply facility;
, is the capacity of cargo supply at candidate site k, and D1 cargo demand at demand pointj.
By setting ck =a1
(Py'
j, the objective function (1') has basically the same shape as the one in the above model. Constraints (2) and (3) are identical to the first twoconstraints of CFLP. As a generalization, of CFLP, suppose we have a fmite set of facility type alternatives (nE N) at each candidate site k, and the building costs (fbi) vary with
the type selected. Then the resulting CFLP can be exactly represented to be the same as (l)-(5'). By this inference, the new formulation (l)-(5) can be regarded as a generalization of the CFLP. The ordinary CFLP, conversely, can be seen as a special case where each candidate site k has niy a single option for facility type
REFERENCES
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