PickingMathModel

Metaheuristics for the integration of picking and storage
location activities
D. L. Huerta-Muñoz 1 , R. Z. Ríos-Mercado 1 , and F. López 2
1 PISIS, FIME, UANL
2 FACPYA, UANL
March 7, 2022
Abstract
The aim of this paper is to study the integration of picking activities and the
storage location of products in a warehouse. A mathematical model is developed and
a metaheuristic method is implemented to solve this integration.
Keywords: Picking; Product allocation; Routing; Metaheuristic; MILP.
1 Introduction
In a supply chain, one of the most costly activities is the picking of orders inside a warehouse.
This activity is closed related to the storage location of products in a warehouse
and, even though these problems are commonly studied and solved independently, recent
studies have shown that the integration of both activities may result in greater improvements.
The expected contribution is to propose a mathematical model for the integration of
picking and storage location activities of the studied problem and a metaheuristic to solve
large size instances efficiently.
2 Literature review
According to de Koster et al. (2007), almost 55% of the costs incurred inside a warehouse
are related to the picking of orders. This activity is closed related to how the products are
located in the warehouse. Handle both activities individually incur in local optimizations
with no significant improvements. Therefore, integrating these activities can result in
better solutions.
In Silva et al. (2020), authors show that the integration of both activities may incur in
significant improvements. They propose several non-linear programming models for four
cases of integration, which have been linearized. Computational results showed that is
not practical to solve even the small instances to optimality in less than 7200s (3-5 picks,
10-60 slots). Thus, they developed a General Variable Neighborhood Search metaheuristic
to solve them.
1

2.1 Storage location assignment (SLAP) and order picking (OPP) ideas
Some general ideas taken from Silva et al. (2020) are the following:
• There are extensive literature for the individual study of SLAP (tactical) and OPP
(operational).
• SLAP solution is an input of OPP
• OPP requires 50-75% of the total operating costs for a typical warehouse.
• SLAP is evaluated after OPP
• There are several ways to model the problem depending on the layout to be considered.
The most common one is the rectangle single-block layout with multiple aisles
(the one used for this study)
Ideas for the SLAP
• SLAP was introduced by Hausman et al. (1976) for an automated warehouse. Some
other literature to consider is Kofler et al. (2014) and Charris et al. (2018)
• Store location assignment policy (to be discussed later)
– Random: simple but not so efficient, more appropriated for e-commerce business.
– Dedicated: high-turnover products assigned to best locations, products are
sorted using a demand rule, locations are sorted based on the distance to the
depart point (similar to ABC method?).
– Class-based: divides products in classes and storage location into zones, then
assign classes to zones
– By affinity: items which are commonly ordered together are considered to be
related.
Ideas for order picking
• For any warehouse layout, optimal OPP solution is obtained by solving a special
case of TSP (Pansart et al., 2018)
• Heuristics for the TSP like the Lin-Kernighan-Helsgaun (LKH, the most successful)
can be used to approximate the OPP.
• Picking policies (Fig. 1) that can be used depend on the problem characteristics
(warehouse shape, num aisles/cross aisles, pick list size, storage
and batching policies:
– Return: pickers arrive and depart through the same point of aisles.
– S-Shape: pickers enter and leave an aisle and traverses it until the end.
– Midpoint: picker enters and leaves from the same point of the aisle; however,
he/she goes as far as the middle of the aisle
– Largest gap: a gap (distance between two adjacent items and the closes cross
aisle) is computed. Then, pickers leave the aisle from the same point they
entered.
2

Figura 1: Picking policies (source: (Silva et al., 2020))
– Other policies can be found in [REF]
SLOPP - general idea
The SLOPP was proposed by Silva et al. (2020) to integrate storage locations and
order picking problems. They proposed a (non-linear) mathematical model that considers
the following elements:
• They do not assume any specific warehouse system
• S-shape (slightly modified)
• 4 routing policies
• Static pick list
• They evaluate the SLAP performance for the OPP solution
• They present a mathematical model for the SLOPP (SLAP + OPP), it does not
assume any storage layout (no assume any policy or heuristic)
• Data:
– L locations
– P products (only demanded products), each product must be assigned to a
place l in L
– O orders, each order contains a pick list
– d ij distance cost to move from place i to j
– Q unlimited capacity
• Decision vars
3

– x ijo 1 if (i,j) is traverse to pick items from o-th order, 0 otherwise
– y ki 1 if product k is assigned to location i (y00 = 1)
– u ko Aux var to avoid subtours (position in route)
• Objective function: min total routing cost
• S.t.
– Assign each product to exactly one location
– Each location can have no more than one product
– link assignment and routing, and ensure we have an arc enter/exit from that
slot
– Avoid subtours
3 Problem description
The studied problem considers the following data and assumptions:
• Data for order picking
– I a set of products or items
– P a set of workers or pickers
– Z a set of intra and inter zones
– O a set of picking orders
– V a set of homogeneous cars with a maximum capacity of Q (16 slots, 4x4).
– M pi a matrix that shows the performance of a picker p (number of items per
minute processed) of a picker p.
– A ij an affinity matrix that shows when items i and j can be attended in the
same route
– Layout of the warehouse
Figura 2: Layout
• Data for store location
– I z r Inventory levels per rack per zone
4

– d i : Demand of products per zone
– R: Set of racks per zone
• Assumptions
– Aisles are narrow? [DH: No], have the same lenght? [DH: Most of
the according to the layout]
– There are only aisles or aisles and cross-aisles? [DH: According to
the layout, there are cross-aisles]
– Each order must be attended by only one picker, a picker can attend several
orders in the same route.
– Each car has several slots (bagging is not consider in this study). At least one
slot per order per customer.
– Customer orders cannot be combined in the same slot.
– Units of product are considered integer
– Intra-zone items are those that are in the same
– All items correspond to a same type
– Customer demands are deterministic (to be evaluated, according to the contribution)
– There are some products, from different zones, that cannot be picked in the
same route
– Pickers departs from a specif place or depot
– If possible, allocate heavy orders first (find trade-off)
– Picking policy (to be discussed later)
∗ Return/S-Shape/Midpoint/Largest gap (Other policies can be found in
[REF]
– Store location assignment policy (to be discussed later)
∗ Random/Dedicated/Class-based/By affinity (which?)
• Decision variables
– a so : Number of slots s ∈ S used per customer order øinO
– x p ij
: 1 if picker p picks the item i before j (picking route); otherwise, 0.
– lir z : 1 if item i ∈ I is located in rack r
• Objective
– Picking: Evaluate the trade-off between routing and slotting costs
– Minimize the total picking time
• Subject to
– Storage location
– Picking requirements
– Carts capacities
5

3.1 The pick frequency and part affinity matrix for product assignment
The pick frequency and part affinity matrix value was introduced by [REF]
Given a set of L z storage locations per zone z ∈ Z, and a set of products p ∈ P that
must be assigned to these locations, to create the PF/PA matrix assignment we need to
consider the following:
• N(p) : Number of orders in which p ∈ P occurs
• A(p, q) : Number of orders in which p, q ∈ P are requested together
• ¯q l p: Quantity of product p stored at location l
• ¯L(p) : Set of locations where product p has been assigned (where ¯q l p > 0)
• Total pick frequency value (PF)
• Total part affinity value (PA)
P A =
P A(p) =
P F = ∑ p∈P
N(p)
|¯L(p)|
P F (p) = N(p)
¯L z (p)|
∑
p,q∈P :i≠j
∑
q∈P :q≠p
P A(p, q) =
∑
l∈ ¯L(p)
∑
l∈ ¯L z(p)
A(p, q)
|¯L z (p)||¯L z (q)|
A(p, q)
|¯L(p)||¯L(q)|
A(p, q)
|¯L(p)||¯L(q)|
c 0,l (1)
∑
c 0,l (2)
∑
l∈ ¯L z(p) ¯l∈ ¯Lz(q)
∑
l∈ ¯L(p)
∑
l∈ ¯L(p)
Then, the total PF/PA value can be measured as follows:
∑
∑
c l,¯l (3)
¯l∈ ¯L(q)
c l,¯l (4)
¯l∈ ¯L(q)
c l,¯l (5)
ĉ = αP F + (1 − α)P A (6)
ĉ pq = α[P F (p) + P F (q))] + (1 − α)P A(p, q) (7)
3.2 Order picking considering precedence constraints
Precedence constraints during the picking activity have being studied in several works
Dekker et al. (2004); Lahyani et al. (2013); Chabot et al. (2017). A precedence constraint
establishes the order in which the picking routes must be performed according to certain
criteria like picking first heavy products before light products (to locate heavy products in
the lower slots of the cart and light products in the upper ones), or picking first products
from the most to the least fragile, etc.
In our problem te objective is to pick heavy products first, then light ones.
6

4 Mathematical model
Data
• B: Set of bacthes {1, . . . , |B|}
• O: Set of orders {1, . . . , |O|}
• P : Set of products {1, . . . , |P |}
• L: Set of locations/racks
• A: Set of arcs e available, where e ∈ {i, j ∈ L}.
• V p : Volume required by product p
• γ > 0: Maximum value of affinity required to assign products in a close location.
• Aff pq : PF/PA value between to products p, q ∈ P (see Section 3.1)
• Q L : Capacity of locations (volume)
• W : Set of pickers, {1, . . . , |W |}
• Q W : Cart capacity (volume)
• D po : Demand of product p ∈ P in order o ∈ O
Decision variables - assignment
• y l p= 1 if product p ∈ P is assigned to location l ∈ L; Otherwise, 0.
• a l p ≥ 0: Units of product p ∈ P assigned to location l ∈ L.
Decision variables - routing
• qpoi lw = Units of product p ∈ P , for any order o ∈ O that belongs to batch i ∈ B, that
must be collected at location l by picker w ∈ W
• zpoi lw = 1 if product p ∈ P of order o ∈ O of batch i ∈ B located at l is assigned to
picker w ∈ W ; otherwise, 0.
• r iw
e = 1 if arc e ∈ A is traversed by picker w ∈ W to pick batch i; otherwise, 0.
• f iw
e ≥ 0: Load of the vehicle used by picker w ∈ W at each traversed e ∈ A.
min
α ∑ i∈B
s.t. ∑ p∈P
∑
∑
w∈W e∈A
t e r iw
e + (1 − α) ∑ o∈O
¯tρ o (8)
y l p ≤ |P | l ∈ L (9)
∑
yp l ≤
l∈L
∑
V p D po
o∈O
Q L p ∈ P (10)
7

∑
l∈L
a l p ≥ ∑ o∈O
D po p ∈ P, (11)
V p a l p ≤ Q L yp l p ∈ P, l ∈ L (12)
∑
V p a l p ≤ Q L l ∈ L (13)
p∈P
Aff pq (yp l + yq) l ≤ 2γ p, q ∈ P (p ≠ q), l ∈ L (14)
zpoi lw ≤ yp l w ∈ W, p ∈ P, o ∈ O, i ∈ B, l ∈ L (15)
V p qpoi lw ≤ Q W zpoi lw
p ∈ P, o ∈ O, i ∈ B, w ∈ W, l ∈ L (16)
∑ ∑ ∑
l∈L o∈O p∈P
∑ ∑ ∑
i∈B w∈W o∈O
∑ ∑ ∑
∑
l∈L w∈W i∈B
e l+ ∈A
∑
w∈W e 0+ ∈A
∑
e l+ ∈A
∑
V p q lw
poi ≤ Q W i ∈ B, w ∈ W (17)
q lw
poi ≤ a l p p ∈ P, ∈ L (18)
q lw
poi = D po p ∈ P, o ∈ O (19)
r iw
e ≤ 1 l ∈ L, i ∈ B, w ∈ W (20)
∑
r iw
e
fe
iw
e l+ ∈A
fe
iw
r iw
e ≤ |W | i ∈ B (21)
= ∑
e l− ∈A
= ∑
e l− ∈A
r iw
e l ∈ L, i ∈ B, w ∈ W (22)
f iw
e
− ∑ o∈O
∑
p∈P
V p q lw
poi w ∈ W, i ∈ B, l ∈ L (23)
≤ Q W re iw
e ∈ A, i ∈ B, w ∈ W (24)
∑ ∑ ∑
fe iw = 0 (25)
i∈B w∈W e∈A
b w oi ≤ ∑ ∑
zpoi w o ∈ O, i ∈ B, w ∈ W (26)
l∈L p∈P
∑
b w oi = 1 o ∈ O, i ∈ B (27)
w∈W
re
iw
+ b w oi − 1 ≤ Mτoi ew
o ∈ O, w ∈ W, i ∈ B, e ∈ A (28)
∑ ∑
t e τoi
ew ≤ To Max
o ∈ O, w ∈ W (29)
∑ ∑
b w oi − 1 ≤ Mh o o ∈ O (30)
i∈B e∈A
w∈W i∈B
ρ o ≤ ∑ w∈W
∑ ∑ ∑
l∈L i∈B p∈P
q lw
poi o ∈ O (31)
ρ o ≤ Mh o o ∈ O (32)
ρ o ≥ ∑ ∑ ∑ ∑
qpoi lw − M(1 − h o ) o ∈ O (33)
w∈W l∈L i∈B p∈P
yp l ∈ {0, 1}, ā l p ≥ 0 p ∈ P, l ∈ L (34)
zpoi lw ∈ {0, 1}, qpoi lw ≥ 0 p ∈ P, o ∈ O, w ∈ W, i ∈ B, l ∈ L (35)
rie w ∈ {0, 1}, fie w ≥ 0 e ∈ A, i ∈ B, w ∈ W (36)
8

References
Chabot, T., Lahyani, R., Coelho, L. C., and Renaud, J. (2017). Order picking problems
under weight, fragility and category constraints. International Journal of Production
Research, 55(21):6361–6379.
Charris, E., Rojas-Reyes, J., and Montoya-Torres, J. (2018). The storage location assignment
problem: A literature review. International Journal of Industrial Engineering
Computations, 10.
de Koster, R., Le-Duc, T., and Roodbergen, K. J. (2007). Design and control of warehouse
order picking: A literature review. European Journal of Operational Research,
182(2):481–501.
Dekker, R., de Koster, M. B. M., Roodbergen, K. J., and van Kalleveen, H. (2004).
Improving order-picking response time at ankor’s warehouse. Interfaces, 34(4):303–313.
Hausman, W. H., Schwarz, L. B., and Graves, S. C. (1976). Optimal storage assignment
in automatic warehousing systems. Management Science, 22(6):629–638.
Kofler, M., Beham, A., Wagner, S., and Affenzeller, M. (2014). Affinity based slotting
in warehouses with dynamic order patterns. Advanced Methods and Applications in
Computational Intelligence, pages 123–143.
Lahyani, R., Semet, F., and Trouillet, B. (2013). Vehicle Routing Problems with Scheduling
Constraints, chapter 16, pages 433–463. John Wiley & Sons, Ltd.
Pansart, L., Catusse, N., and Cambazard, H. (2018). Exact algorithms for the order
picking problem. Computers & Operations Research, 100:117–127.
Silva, A., Coelho, L. C., Darvish, M., and Renaud, J. (2020). Integrating storage location
and order picking problems in warehouse planning. Transportation Research Part E:
Logistics and Transportation Review, 140:102003.
9