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Order Picking: Pick Sequencing and Batching

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Title: 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
4
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

6
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.

7
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.

8
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?

9
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)
10
An alternative numbering scheme
11
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

12
The Sierpinski space-filling curve
13
Applying 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
14
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.

15
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).

16
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.

17
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)

18
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

19
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)
20
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)

21
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.

22
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

23
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.

24
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)

25
AddendumA special case admitting polynomial
solution(Ratliff and Rosenthal, Operations
Research, 31(3) 507-521, 1983)
26
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
27
A graph-based representation of the underlying
topology
0
28
A picking tour
29
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
30
Lj(- or ) PTS (partial tour sub-graph)
L3- PTS (E, E, 2C) L3PTS (U, U, 1C)
31
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

32
Going from Lj- to Lj
(I-i)
(I-ii)
(I-iii)
(I-iv)
(I-v)
(I-vi)
33
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
34
Going from Lj to L(j1)-
(II-iii)
(II-i)
(II-ii)
(II-v)
(II-iv)
35
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.
36
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.

37
Example (c.f., slide 8)
38
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
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