Reducing Truck Emissions in Import Operations at Container Terminal—A Case Study in a Singaporean Port - Technology Management for Sustainable Production and Logistics - page 139

However, truck arrivals is a stochastic process. Even though the truck arrival can
be assumed as a Poison process as in queuing theory, it is not easy to calculate
the total waiting time because the arrival pattern is random. Randomly select-
ing arrival period of a truck makes the randomly arrival pattern and this makes
the waiting time of trucks at import container block also random. Thus a discrete
event simulation calculation process should be included in order to calculate the
average total waiting time of trucks. Because the mathematical model is a com-
binatorial optimization with a discrete event simulation calculation process, it
cannot be solved with traditional mathematical approach. Therefore, one of possi-
bility method to solve the above mathematical model could be genetic algorithms
(GAs). GAs have the advantage of flexibility imposing no requirement for a prob-
lem to be formulated in a particular way, or that the objective function(s) is dif-
ferentiable, continuous, linear, separable, or of any particular data-type. Thus, they
can be applied to any problem (e.g. single or multi-objective, single or multi-level,
linear or non-linear) for which there is a way to encode and compute the qual-
ity of a solution. GAs can be easily combined with exact solution algorithms (e.g.
branch and bound), local search (i.e. memetic algorithms), and/or other (meta-)
heuristics and guarantee local optimality of the solution or improve the conver-
gence patterns (Golias et al. 2010 ). As an added advantage, GA can be easily com-
bined with simulation that is used for estimate the quality of a solution provides
by GA in this study. For an in-depth discussion of GAs and the theory behind we
refer to Goldberg ( 1989 ) and Gen and Cheng ( 1999 ). The following section pre-
sents the structure of chromosome and how to calculate the fitness function.
The chromosome is coded as follows:

• Each chromosome include I substrings where I is the number of containers at
the block.
• A substring represents for a container and includes T genes where T is the num-
ber of periods for picking up all containers at the block.
• Each gene of a substring represents a period and its value shows whether the
corresponding container is available for pick up in that period or not.

Figure 2 is an example of a chromosome. In this example, Periods 1, 3, 4, and 5 is
available for picking up container 1. This means that the value of variables relate
to container 1 is as follows: x 11 = 1, x 12 = 0, x 13 = 1, x 14 = 1, x 15 = 1.
In the maximization problem, the objective function value can be directly used
for the fitness of a chromosome. However, the mathematical model in this study is
a minimization model. Therefore, if the objective function value of a chromosome
is higher, that chromosome shows less fitness. In this study, the following formula
is used to calculate the fitness.

Z max − Z ( m )

Fitness( m ) = Z max = Max

l = 1, ... , N ( Z ( l )) , N : population size

N

l = 1 [ Z max − Z ( l ) ]

(4)

where Z( m ) is the objective function value of chromosome m .