Order Picking: Pick Sequencing and Batching

1
Order PickingPick Sequencing and Batching
2
The Pick Sequencing Problem

• Given a picking list, sequence the visits to the
  picking locations so that the overall traveling
  effort (time) is minimized.

3
Problem 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
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Analytical 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

5
Some 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

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The 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.

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k-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.

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The 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?

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Example 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)
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An alternative numbering scheme
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Key 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

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The Sierpinski space-filling curve
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Applying the Sierpinski space-filling curve in
the previous bin-numbering example
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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
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1
Order 1,2,13,17,18,32,46,52
Resulting route length 34
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Some 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.

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Characterizing 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).

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Bin-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.

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Order 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)

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Problem 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

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Order-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)
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The 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)

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Typical 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.

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Typical 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

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The (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.

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Some 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)

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AddendumA special case admitting polynomial
solution(Ratliff and Rosenthal, Operations
Research, 31(3) 507-521, 1983)
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The 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
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A graph-based representation of the underlying
topology
0
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A picking tour
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Lj-, 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
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Lj(- or ) PTS (partial tour sub-graph)
L3- PTS (E, E, 2C) L3PTS (U, U, 1C)
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A 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

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Going from Lj- to Lj
(I-i)
(I-ii)
(I-iii)
(I-iv)
(I-v)
(I-vi)
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Going 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
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Going from Lj to L(j1)-
(II-iii)
(II-i)
(II-ii)
(II-v)
(II-iv)
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Going 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.
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A 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.

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Example (c.f., slide 8)
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Example The optimal tour
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
0
2
2
2
2
2
b1
b2
b3
b4
b5
b6