![](bg4.png)
Such travel consumes time, but is a non-value-added
process; it is thus detrimental to the order through-
put of the workstation. The nearest neighbor picking
policy is proposed, which minimizes the distance in
which the robot travels without carrying any item.
The nearest neighbor picking policy requires the
robot to pick the product pallet located nearest to the
current position of the robot. If there is more than one
pickable order with the same distance to the robot,
then the robot picks the pallet that is located closer to
the head of the buffer.
3.3 Dynamic Programming
Given an unlimited supply of product and order pal-
lets at the workstation, the order throughput of the
workstation depends on the robot processing time; a
lower processing time leads to a higher throughput.
The robot processing time consists of a travel time,
pick time, and drop time. Assuming a relatively con-
stant pick and drop time, one can increase the work-
station throughput by reducing the travel time, which
is a function of the robot travel distance.
The dynamic programming policy constructs a
picking trip that minimizes the robot travel distance
based on the current content of product and order
buffers. The robot then picks a number of orders in
a sequence as indicated in the picking trip. Once a
product or an order pallet leaves the workstation, the
buffer content changes. In this case, a new picking
trip is constructed using the dynamic programming
approach based on the new content of the product and
order buffers.
The robot travel distance depends on the sequence
of picking a number of orders. When picking one
order, the robot travels from its current position to
a product pallet (picking an item) and finally to the
corresponding order pallet (dropping the item). Fol-
lowing a picking process, the robot thus always starts
a new picking process from the location of the previ-
ously served order pallet.
A dynamic programming policy is formulated to
minimize travel distance D to reduce the travel time
and consequentlyto increase the workstation through-
put, where:
D =
m
∑
i=1
(|y
i−1
− x
i
| + |x
i
− y
i
|), (1)
where:
x
i
= position of product pallet of the order picked
at the ith step.
y
i
= position of order pallet of the order picked
at the ith step.
y
0
= current position of the robot.
In the above formulation, m is the number of orders
picked within a picking trip, which is referred to as
the trip length. Note that travel distance D increases
with trip length m. Therefore, the optimal trip is the
one that minimizes travel distance D per order, so the
one minimizing D/m.
The first step in constructing a picking trip that
minimizes travel distance D is evaluating the pallets
contained in the product and order buffers. Recall that
each product pallet has an identification number rep-
resented by the ID of the order to which the product
belongs. Let
P be the set of IDs of product pallets within the
pickable area
Q be the set of IDs of order pallets within the
pickable area
Then R = P ∩ Q gives the IDs of pickable orders.
These are the orders whose product and order pallets
are both located inside the pickable area. Let S ⊆ R
be the IDs of target orders. These are the pickable or-
ders whose product and/or order pallets are located at
the head of the buffer. A target order is the last order
to be picked in a picking trip because picking a target
order causes a product and/or an order pallet to leave
the workstation. With this regard, x
m
and y
m
in (1) are
the positions of the product pallet and the order pallet
for the target order, respectively. Picking a target or-
der leads to a new content of either a product buffer
or the order buffer, or both. Hence, a new picking trip
needs to be constructed.
Having a target order is a requirement for con-
structing a picking trip. Recall that there are two prod-
uct buffers namely rPP and PP evaluated against the
order buffer OP. If both rPP and PP contain at least
one target order, then a picking trip is constructed
from rPP. That is, rPP has priority over PP. If only
one product buffercontains a target order, then a pick-
ing trip is constructed from that product buffer. Oth-
erwise, if none of the two product buffers contains a
target order, then one of the two carrousel operations
explained previously is performed.
Figure 3 shows an example of the content of
product buffer PP and order buffer OP. Assume
now that product buffer rPP does not contain a tar-
get pallet and therefore is disregarded from the fig-
ure. Based on this figure, P = {11,12,14, 15,18},
Q = {11,12,13, 14,15}, R = {11, 12,14,15},and S =
{11}.
Constructing a picking trip can be regarded as a
Traveling Salesman Problem (TSP). That is, given
trip length m, pickable orders R, target orders S, and
current position of robot y
0
, determine the sequence
of picking m pickable orders involving one and only
one s ∈ S, which starts at y
0
and ends at s such that D
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