![](bg3.png)
Equation (2) shows that every server must leave its
house and equation (3) forces it to arrive at the des-
tination (workplace). Constraint (4) makes sure the
continuity of the path. The capacity and the maxi-
mum time constraints are translated in inequalities (5)
and (6), respectively. Equations (7) and (8), where M
is a big constant, define the arrival time variables S
i
,
while inequalities (9) and (10) set the arrival times h
k
, k ∈ V
s
, of the servers at the workplace and assume
that each employee i ∈ V should reach the workplace
before the latest arrival time l
i
, respectively. Equa-
tion (11) ensures that each client can be picked by a
server or is left unserved. Constraints (12) and (13)
are binary constraints, while (14) and (15) are posi-
tivity constraints.
3 THE BEE COLONY
OPTIMIZATION ALGORITHM
The Bee Colony Optimization is among the famous
natural inspired metaheuristic based on the collective
bee intelligence. It was proposed, first, by Lucic and
Teodorovic (Lu
ˇ
ci
´
c and Teodorovi
´
c, 2003) to deal with
the well hard combinatorial optimization problems
like the Vehicle Routing Problem (Lu
ˇ
ci
´
c and Teodor-
ovi
´
c, 2003; Teodorovi
´
c, 2008), the Job Shop Schedul-
ing Problem (Chong et al., 2006; Chong et al., 2007),
the p-Median Problem (Teodorovic and
ˇ
Selmic, 2007)
and the Traveling Salesman Problem (Wong et al.,
2010).
The main idea of this approach is to build an ar-
tificial bee colony, where bees investigate through
the search space looking for feasible solutions. Af-
ter that, they communicate, collaborate and exchange
information. Thanks to this communication systems
named ”Swarm Intelligence”, and using collective
knowledge and information sharing, artificial bees in-
crementally construct solution to the problem. The
BCO performs its search process in iterations until a
stopping condition is met.
Flying through the space, an artificial bee per-
forms either forward pass or backward pass. In
the case of forward pass, every bee makes a prede-
fined number of local moves, which slowly construct
and/or improve the solution, yielding a new solution.
Building partial solutions, bees start the second phase
called the backward pass. In this phase, bees return to
the hive. They exchange information about the qual-
ity of the partial solution created which is defined as
the current value of the objective function.
Having the evaluation of all partial solutions, each
bee decides with a certain probability whether to still
faithful to its created partial solution or not. It is obvi-
ous that bees with better solutions have greater chance
to keep and continue their own exploration.
Faithfull bees are considered as recruiters, and
their solutions would be considered by other bees.
However, if a bee chooses to abandon its solution it
becomes uncommitted and it has to select one solu-
tion from recruiters by the roulette wheel. Note that
better solutions have higher opportunities to be cho-
sen for further exploration. Forward and backward
pass, could be iteratively performed until each bee
completes the generation of its solution or a stopping
condition is satisfied. Among the possible stopping
conditions, we found the maximum total number of
forward/backward passes or the maximum total num-
ber of forward/backward passes without the improve-
ment of the objective function, etc.
4 THE PROPOSED BCO
ALGORITHM FOR THE DCPP
In this section, we present our algorithm based on the
Bee Colony Optimization called the Guided-BCO al-
gorithm.
4.1 Problem Representation
In order to maintain the simplicity of the BCO algo-
rithm, a rather straightforward solution representation
scheme is adopted. Let us represent each employee
by a node. Our problem is divided into stages where
the first stage represents servers and all others repre-
sent clients. In every stage, an artificial bee chooses
to visit one node. At the beginning of each new pool,
the selection of a new server (new initial node) was
represented by changing the location of a hive.
4.2 Forward Pass Phase
At the beginning of the BCO process, all artificial
bees are located in the hive. Bees depart from the
hive and fly to an unvisited client who satisfies con-
straints. It chooses a new node to be added to his
partial pool using the roulette wheel selection. This
technic is based on the probability values, which gives
the bee the likelihood to move from client i to client
j. To calculate the probability, we need the distance
between the current client i and the client to be vis-
ited j. It is obvious that, the shorter the distance,
the higher the probability to choose a client. There-
fore, the travel cost and the probability are inversely
proportional. Formally, the probability is defined in
equation (16) as follow:
Guided Bee Colony Algorithm Applied to the Daily Car Pooling Problem
467