Title: Order Picking: Pick Sequencing and Batching
1Order PickingPick Sequencing and Batching
2The Pick Sequencing Problem
- Given a picking list, sequence the visits to the
picking locations so that the overall traveling
effort (time) is minimized.
3Problem Abstraction The Traveling Salesman
Problem (TSP)
Given a complete TSP graph
find a tour that visits all cities, with minimal
total (traveling) cost e.g.
2
3
1
lt1, 2, 5, 3, 4gt
4
5
4Analytical Problem Formulation
- Parameters
- N graph size (number of graph nodes)
- c_ij cost associated with arc (i,j)
- Decision Variables
- x_ij binary variable indicating whether arc
(i,j) is in the optimal tour - u_i auxiliary (real) variable for the
formulation of the no subtour constraints - min ?_i ?_j c_ij x_ij
- s.t.
- ?_j x_ij 1 ? i
- ?_i x_ij 1 ? j
- (No subtours
- u_i - u_j N x_ij ? N-1 ? i,j ? 2,,N and
i?j ) - x_ij ? 0, 1 ? i,j
5Some remarks on the TSP problem and its
application in pick sequencing
- The TSP problem is an NP-complete problem It can
be solved optimally for small instances, but in
general, it will be solved through heuristics. - There is a vast literature on TSP and the
development of heuristic algorithms for it (e.g.,
Lawler, Lenstra, Rinnooy Kan and Shmoys, The
Traveling Salesman Problem A guided tour of
combinatorial optimization, John Wiley and Sons,
1985). - When the no subtour constraint is removed, the
remaining formulation defines a Linear Assignment
Problem (LAP) (which is an easy one e.g., the
Hungarian Algorithm) gt Solving the
corresponding LAP can provide lower bounds for
assessing the sub-optimality of the solutions
provided by the applied heuristics. - In the considered application context, the
distances c_ij should be computed based on the
appropriate distance metric i.e., rectilinear,
Tchebychev, shortest path
6The closest insertion algorithmA TSP heuristic
(symmetric version)
- Initialization
- S_p lt1gt S_a 2,,N c(j) 1, ? j ?
2,,N n1 - While n lt N do
- n n1
- Selection step
- j argmin_j ? S_a c_j,c(j)
- S_a S_a \ j
- Insertion step
- i argmin_i 1S_p c_i,j c_j,i
mod S_p1 - - c_i,i mod S_p1
- S_p lt 1,,i, j, i1,,ngt
- ? j ? S_a, if c_j,j lt c_j,c(j) then c(j)
j - Remarks 1. i denotes the node at i-th position
of the constructed sub-tour. - 2. If the distances are symmetric and satisfy
the triangular inequality, the cost of the
solution provided by this heuristic is no worse
than twice the optimal cost. -
7k-STRIP A computationally simple heuristic for
rectangular areas
x
x
x
x
x
x
x
x
x
x
x
x
I/O point
- When A is the unit square, an optimized k ?
?(n/2) ?, and for this value, the worst-case tour
length generated by the heuristic is between
1.075?n and 1.414 ?n, for large n. - The computational complexity is O(n logn).
- Supowit, Reingold and Plaisted, The traveling
salesman problem and minimum matching in the unit
square, SIAM J. Computing, 12(1) 144-156, 1983.
8The bin-numbering heuristic(Bartholdi and
Platzman, Material Flow, 4 247-254, 1988)
- Basic idea Number bins / storage locations in a
way that filling the orders by visiting the
associated bins in increasing numbers will lead
to efficient routings. - Advantages
- Once the numbering is established, developing the
order routes becomes extremely simple. - Easy to adjust routes dynamically upon the
arrival of new orders. - Basic underlying problem
- How do you establish good bin-numbering schemes?
9Example of a numbering scheme(Is it a good one?)
Order 1, 10, 13, 28, 30, 44, 50, 62
Resulting route length 44 (using rectilinear
distances of the cell centroids)
10An alternative numbering scheme
11Key concept Space-filling curves(see also
http//www.isye.gatech.edu/faculty/John_Bartholdi/
mow/mow.html)
- Closed curves that sweep the entire region while
preserving nearness. - Technically, they define a continuous mapping of
the unit interval on the unit square. - Typical example Sierpinskis space-filling
curve
12The Sierpinski space-filling curve
13Applying the Sierpinski space-filling curve in
the previous bin-numbering example
20
21
23
24
32
33
35
36
4
18
19
22
25
30
31
34
37
2
7
13
17
28
29
38
39
14
26
2
11
12
15
16
27
40
41
42
6
7
9
10
43
44
45
46
6
4
5
8
47
48
59
60
61
4
2
3
49
50
53
57
58
62
1
1
64
51
52
54
55
56
63
7
1
Order 1,2,13,17,18,32,46,52
Resulting route length 34
14Some properties of the bin-numbering schemes
based on the Sierpinski space-filling curve
- If n locations are to be visited throughout a
warehouse of area A, then the length of the
retrieval route is at most ?(2nA). - If every location is equally likely to be
visited, then on average, the retrieval route
produced by the corresponding bin-numbering
heuristic will be 25 longer than the shortest
possible route length. - The above results have been derived using the
Euclidean metric for measuring the traveling
distances, but they are robust with respect to
other metrics that preserve closeness according
to the Euclidean metric.
15Characterizing the best bin-numbering scheme...
- is computationally very hard.
- Some good schemes can be obtained through
interchange techniques (e.g., 2 or 3-opt), where
the efficiency of each of the considered schemes
is evaluated through simulation. - The optimal bin-numbering scheme depends on
- the underlying geometry of the picking facility
- the frequency with which the various storage
locations are visited (and therefore, the
applying storage policy) - In general, the logic underlying the utilization
of the space-filling curves is more useful /
pertinent for storage areas with small visitation
frequencies for their locations. - For areas with high visitation frequencies,
numbering schemes suggesting an exhaustive
sweeping of the region tend to perform better
(c.f., Bartholdi Platzman, pg. 252).
16Bin-numbering in structures with complicated
geometry
- When the considered area has a structure too
complex to measure traveling effort by Euclidean
or a relative metric, the logic underlying the
application of space-filling curves to
bin-numbering can be applied in a hierarchical
fashion - separate the entire area under consideration to
smaller areas of simpler geometry - design a numbering sequence for each of these
areas using the space-filling curve logic - develop a visiting sequence for the areas
developed in step 1, by passing a space-filling
curve among their I/O points.
17Order Batching
- (based on De Koster et. al., Efficient
orderbatching methods in warehouses, Inlt. Jrnl
of Prod. Res., Vol. 37, No. 7, pgs 1479-1504,
1999)
18Problem Description
- Given a set of orders, cluster them into batches
- i.e., subsets of orders that are to be picked
simultaneously by one picker at a single trip
such that - the total traveling distance / time is minimized
- while each batch does not exceed some measure of
the picker capacity (e.g., number of items /
volume of the resulting batch, number of distinct
orders in a batch) - Theoretically, the problem can be solved by
- enumerating all feasible partitions of the given
order set into batches - evaluating the total traveling distance / time
for each partition - picking the partition with the smallest traveling
distance / time. - However, combinatorial explosion of partitions gt
heuristics
19Order-Batching Heuristics
Naive
Intelligent
FCFS
Seed Algorithms
Savings Algorithms
(Batches are built sequentially, one at a time)
(All batches are built simultaneously, by merging
partially developed batches)
(Orders are clustered based on the sequence of
their appearance)
20The generic structure for seed algorithms
- While there are unprocessed orders,
- Pick a new seed order according to some seed
selection rule - while there are unprocessed orders and the batch
has not reached the imposed capacity limit - pick a new order to be added to the batch
according to an order addition rule - add the selected order to the batch, provided
that the imposed capacity limit is not violated - (update the batch seed to the union of the
previous batch seed and the new order)
21Typical seed selection rules
- Random selection
- the order with the farthest item (w.r.t. the
shipping station) - the order with the largest number of aisles to be
visited - the order with the largest aisle range (absolute
difference between the most left aisle number and
the most right aisle number to be visited) - the order with the largest number of items
- the order with the longest travel time
- Remark If the batch seed is updated after every
order addition, the algorithm is characterized as
dynamic or cumulative mode ow., it is said to be
static or single mode.
22Typical order addition rules
- Time saving choose the order that, together with
the batch seed, ensures the largest time saving
compared with the individual picking of the two
orders. - Choose the order that minimizes the number of
additional aisles, compared to the seed order,
that have to be visited by the resulting batch
route. - Choose the order for which the absolute
difference between the orders center of gravity
(COG) and the COG of the batch seed is the
smallest COG is the weighted average aisle
number of the order, with the aisle weights
defined by the number of items in the aisle. - Choose the order with the property that the sum
of distances between every item of the seed and
the closest item in the order is minimized. - distances must be measured by an appropriately
selected metric
23The (standard) savings algorithm
- Initialization B order set (each order
defines its own batch) - Repeat
- For each pair (i,j) in the current batch set B
- compute the time savings s_ij t_i t_j - t_ij,
where t_i (resp., t_j) is the time required for
picking batch I (resp., j) and t_ij is the time
required for picking the batch resulting from the
merging of batches i and j. - Rank batch pairs (i,j) in decreasing s_ij.
- Pick the first batch pair (i,j) in the ranked
list, for which the merging of its constituent
batches does not violate the imposed capacity
limit, and merge batches i and j B (B-i,j)
U ij - until no further batch merging is possible.
- Remark The algorithm result depends on the
adopted pick sequencing rule.
24Some findings regarding the (relative)
performance of the presented batch algorithms (De
Koster et. al.)
- Intelligent batching leads to significant
improvements compared to single-order picking and
naïve batching schemes. - In seed algorithms, dynamic seed definition leads
to better performance than static seed
definition. - The best seed selection rules are focusing on
orders dispersed over a large number of aisles
and involving long travel times. - The best order addition rules (c.f. corresponding
slide) tend also to be the most robust (i.e.,
they yield the best results in all warehouse
configurations considered in the simulation). - Savings algorithms have good performance, in
general, but they tend to be computationally more
expensive than seed algorithms. - The performance of the applied batching algorithm
has a significant dependence on the adopted pick
sequencing rule. - The largest the number of orders per batch (the
batch capacity limit), the smaller the savings
from intelligent batching (and therefore, simpler
batching schemes become more eligible candidates)
25AddendumA special case admitting polynomial
solution(Ratliff and Rosenthal, Operations
Research, 31(3) 507-521, 1983)
26The considered warehouse layout
Crossover Aisles
x
x
x
Items to be picked
x
x
x
x
x
x
x
x
x
Docking station
Picking Aisles
27A graph-based representation of the underlying
topology
0
28A picking tour
29Lj-, Aj and Lj sub-graphs, j1,2,,n
a1
a2
a3
a4
a5
a6
2
2
2
2
2
3
3
4
4
x
x
v8
v12
x
v3
x
v5
3
8
3
x
v7
3
x
v11
x
6
v2
15
x
v9
7
6
5
x
v4
x
x
v10
7
v1
5
x
v6
v0
2
3
3
2
2
2
2
2
b1
b2
b3
b4
b5
b6
Lj Lj- ? Aj
30Lj(- or ) PTS (partial tour sub-graph)
L3- PTS (E, E, 2C) L3PTS (U, U, 1C)
31A key observation
- The only possible characterizations for an Lj (-
or ) PTS are the following - (U, U, 1C)
- (0, E, 1C)
- (E, 0, 1C)
- (E, E, 1C)
- (E, E, 2C)
- (0, 0, 0C)
- (0, 0, 1C)
- where the triplet (X, Y, Z) should be interpreted
as follows - X (Y) degree parity for node a_j (b_j) - 0,
Even, Uneven (odd) - Z number of connected components in Lj PTS,
excluding the vertices with zero degree
32Going from Lj- to Lj
(I-i)
(I-ii)
(I-iii)
(I-iv)
(I-v)
(I-vi)
33Going from Lj- to Lj(cont.)
TABLE I
a This is not a feasible configuration if there
is any item to be picked in aisle j b This class
can occur only if there are no items to be picked
to the left of aisle j c This class is feasible
only if there are no items to be picked to the
right of aisle j d Could never be optimal
34Going from Lj to L(j1)-
(II-iii)
(II-i)
(II-ii)
(II-v)
(II-iv)
35Going from Lj to L(j1)-(cont.)
TABLE II
a The degrees of a_j and b_j are odd. b No
completion can connect the graph. c Would never
be optimal.
36A polynomial-complexity algorithm for computing a
minimum-length tour
- Initialization L1- PTS null graph for every
class type - For ltL1, L2-, L2,,Ln-, Ln)
- compute a minimum-length PTS for each of the
seven classes, using the minimum-length PTSs
constructed in the previous stage, and the
information provided in Tables I and II. - Remark For case (I-iv), a minimum-length PTS is
obtained by putting the gap between the two
adjacent v_is in aisle j that are farthest
apart. - A minimum-length tour is defined by a
minimum-length Ln PTS.
37Example (c.f., slide 8)
38Example The optimal tour
a1
a2
a3
a4
a5
a6
2
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x
x
v8
v12
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v3
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v5
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v7
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v11
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v2
15
x
v9
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v4
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x
v10
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v1
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v6
v0
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0
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2
b1
b2
b3
b4
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b6