Content uploaded by Fábio Neves-Moreira
Author content
All content in this area was uploaded by Fábio Neves-Moreira on Apr 12, 2025
Content may be subject to copyright.
INESC TEC, Faculty of Engineering, University of Porto, Porto 4200–465, Portugal
!"#$ "% &'! #!
Keywords:
%
"
(
)
% ) (
* ' )+&,
%"'-&%'. ''-''.*!,
/* 0) +
( ,1
/* ! ) )
1 ) (2 ) 3, "
- .* ! -seeking .
)2 -hiding . 4 1/* 5
3 -3. ,
( *
#/ ,
* 5 , / ( $
(
* ! /
6/, (*%)
( ) 72 / ( 1 *
! (
, 89: , ( *!
+, ,
/ ,
*
1.
Introduction
! ,
, -#*;9;9.*
! /
1
* 1
) (,
- * ;9;;.* 5
, , <<== ,
( ),(
/
-3 ;9>?.* ,
(
-& * ;9;>.*
)(@ (
-#2*;9;A.*
! ) ,
2 ( ),
-#;9;9.*
!
,
,-B';9>C.*3
(
(* /
)((-**
. 1 -B ' ;9>C.
) D*
-;9;>. )
1 -#*;9;>.*
/( &,
% "' -&%'.* # @
,-B* ;9>EFG,*;9;;.*
&%'
( -H*C: (
>*?: (, . - I2 ;9;;.*
! , 7
&%'/)(
# *
E-mail addresses:
(**J* -* . J(** -* .*
F. Neves-Moreira and P. Amorim
International Journal of Production Economics 267 (2024) 109074
;
HH:;9>K E9: ,;9;H L' - ;9>E.*
& &%'
(@ -** &%' '(, ;9;; '
'-''. &;9;;+ ,;9;9.
( 1
*
(@
/
(-,. (@
-5*;9>?.* #+,
( )
( ( ,
,*,
( ,
,-M;9;;. (
)(+ (@
, * " ) (
-,.,)
/ ,(*
, (
(@ ) ( )
,(
- ;9;9.* ,
) 2
(
-* ;9;>.*
,
), +,)(
(,*
2
-3( * ;9;;.* -;9;;. <<Picking
online orders is still causing headaches for grocers on multiple fronts, with
(...) in-store shoppers jostling with e-commerce workers to select products in
the aisles==* % )
-1. / )
,-I;9;A.*
(
,
/ ) (
1 = /* ! ,
(
( 1
- . )
( ( +, (
1 / )
,
- .* , (
$(
(((
) ) 1
=/* !
)(F,
,
*
5,(
1 (
/
* !
7)(
)) ) ,, ) (
)
( (@ *
) 1 @*
0 (@ )
1 / -3,
2 ;9;;F 0N * ;9>K.*
/ +,
-3 * >EE8F # * ;9;>.
/ ) 7, )
-3);9;>.*
!
, 6)(
) )
3, " - .* !
+ ,
*"@, /2
( ) 2
F. Neves-Moreira and P. Amorim
International Journal of Production Economics 267 (2024) 109074
A
( *
6) (
*! (
) ,
/"-".
, ( +
* !(
3 -3.
)) ( +* !
, , , O
/2 (
) -**))
) )
1 ** . ,
-**.
(,*!7
( - ,. )
-/.*
5 )@(
)
+
P
* ! @ ( )
(Q
>* 5(2 )
)
) ,
P *
;* 5 , , O
) " 3
,
, (
*
A* 5 /
,
(
)
) $ *
, ) /
6/, ( * !
) (@
/*
! ( 2 ()*
';)
*'A
,*'H
O *
' 8 /
( , ) )
* ' K / )
$
/ * , ' C
@ ( )
(*
2.
Literature
review
% ) Q
*!()
) (
-' ;*>.* ! ) )
)-'
;*;., -' ;*A. ,
-' ;*H.* 5
,)
+
)(
, ),
(*
F. Neves-Moreira and P. Amorim
International Journal of Production Economics 267 (2024) 109074
H
2.1. In-store picking strategies
! ((
)
- *;9>9F*;9>>F
0N*;9>K.*")
(@ ( )
* "
(@, /*#,)
-.
- *;9>?.*
"
7(,*
,,/ ,+
)(*' )
) )
,)
-0N * ;9>K.* ! &%'
'' ) +
) 6)-) . )
,/,*
* (@,
, ) , -
).*! ,
,, (
/-*;9>9.*',
/ , )
(
) )
-#2* ;9;A.* !,
()
)(-0N*;9>K.*
)
,(
* * -;9;>.
) 2 (
,* !
,
2 &%' ,*
3( * -;9;>. (@
( '' ,
(@ * %(
) )
* 3, @ ,
* #2 *
-;9;A. (
( , *
!,( -
.F -
.F ( (@ (
( (-.F
, - .* ! (
2 (
( *
(@ )
&%'
*0) )
&%' ) ) *) (
( (@
,(&%'-* ;9;>F
I2;9;;.* #,/
&%' (-**
+,.* ' 2
-# *;9;>.*
! @ ) )
( ),
1 /*5 (
( ) /
) )
*5@ /
,)()
,/*
F. Neves-Moreira and P. Amorim
International Journal of Production Economics 267 (2024) 109074
8
2.2. Picker routing problems
!),
->E?A.,,
! ' -!'. '
-( * ;9>AF #2 # ;9>?F
R2 * ;9>E.* '
/
) ) @
-# * ;9>CF O
#G ;9>?F 7 * ;9>?.
+
- B * ;9>?.* " (
, )
-!,*;9>9. )
, ( /
( -#S7 *
>E?8F * ;9>?.* , ) (
2 (
/, -'2 * ;9>K.
)
, - * ;9>8.
)
* ,,
)) )-
I;99>F I;99>FI
* ;99CF#* ;9>?F*;9;9.
* # ,
2( ))
)/
*"
, ,,
*
)apriori
more ,*
2.3. Dynamic routing problems
, A9,
, @ *
%(( )(L
* -;9;9. ( ;9>9* %
) ,
& * -;9>9. * -;9>A. 2
* -;9>K. ( * -;9>K. %7 *
-;9;>.*") -**
)
,. -**
, . (
() ,
-'G5 ;9>K.*!
@ () (
, ) -;9>E.
, 3* !
, )
/ ,
)-;99C.
6( @*
,
3
( + -2*;9>?F
* ;9;>F I * ;9;;.* 0)
)/
( / , ,
,0*-;9;>.*
2.4. Dynamic picker routing problem
" /
, ( * !
) ) ( ) (
/( -**)
.* # ,
, * B I
-;99?. )
( ,, ,*
& ,
F. Neves-Moreira and P. Amorim
International Journal of Production Economics 267 (2024) 109074
K
*
* -;9>K.,
) *
!)( 2
+,
)( *
!,(
, *0))
,( (Q1
* (
= , ( *
! ) , *
0) )
1 ,* "
) (
, *
3 , )
)
(* "
/ ( ;9;9 )
) )
7*
( ( ,
=
,( -0*
;9>AF &* ;9>8F 0 ;9;>.* $
@ *!(
) / (
, + (-
*;998. 6) (
( -# * ;9>8F * ;9>C.*
!)apriori )
( @ -
.* !( ( )
=/
*!) )*
3.
Problem description
" ) (2 ,
)
3
, * 5 )
)
, *
,) ))
)
,(( *
0' A*>
(
( ,
(( *'A*;
@( (
-**()
, ,.* ' A*A
3 , )
()7 (*
3.1. Retail store environment
! )
,
*!
= (F, £) ) ( - .
/0)
( )
{1, … , 𝑣}
𝑣
/
𝑣+1
) )
-**
F. Neves-Moreira and P. Amorim
International Journal of Production Economics 267 (2024) 109074
C
2.*,
())
(
𝜆𝑆𝑡𝑜𝑟𝑒
* $ ,
𝑐
T C
(
𝑠𝑝𝑒𝑒𝑑𝐶𝑢𝑠𝑡𝑜𝑚𝑒𝑟
𝜃𝑐
+ (
*
,
𝑠𝑒𝑟𝑣𝑖𝑐𝑒U𝑡𝑖𝑚𝑒𝐶𝑢𝑠𝑡𝑜𝑚𝑒𝑟
*!
/ (
, ( |C| /
(
𝑚* ! (
)())
(
𝜆
𝑂𝑛𝑙𝑖𝑛𝑒
*$
𝑜
T U (( G𝑜 ⊂
F **
> (
*
3.2. Picker agents
5
𝑜
* +
𝜃𝑜
@ -**
,@
(2.*"))
+ 2
** /2
,),* !
() + ,
/* !( ) , (
), *
!, (
( , *!
(
( ( ()
@+(
,,-**
.V
) ) *! (
)
, ) *
$
, +, +
/*
3.3. Dynamic picker routing problem as a Markov decision process
! ( @ 2 3 )
𝑇
𝑘
T 0, 1, … ,
𝑇
)(
/*
)
𝑘
)
) * $
𝑘
)
𝑠𝑘*
𝑠0
𝑘
= 0
0*
𝑘
=
𝑇
@
𝑣
+ 1
(
, *
𝑠𝑘
,
(
(
(7-,
.* & , (
𝑎𝑘
-**
/
.
* $
,(*
!( @
+ ) 2 (
* ! , )
(*%(,
3((
,->.(SF-;.(
AF-A.)(
𝑅F𝑃 *
Decision epochs
𝑘
)
)*
States !(,
𝑘
@,
𝑠𝑘
=
(𝑛𝑘, 𝑧𝑘, 𝑡𝑘)
)
𝑛𝑘
T
F
=
F. Neves-Moreira and P. Amorim
International Journal of Production Economics 267 (2024) 109074
?
/ig. 1.
! ( &%' '' ) , (
*
𝑧𝑘
=
(𝑧𝑘(1), 𝑧𝑘(2), … ,
𝑧𝑘(|G𝑜|))
(
(
𝑜
𝑘
𝑡𝑘
[0T,
𝑇
]
𝑛𝑘*
( 𝑧𝑘(.)
{0, 1}Q{
0, ( ,
𝑡
((
,𝜙1 (
,𝜙2 ( ,,𝜙3
, 𝜙4* ! (
7)
𝑤1
𝑤4
,*
!
)
𝑟𝑘(𝑠𝑘, 𝑎𝑘, 𝜔𝑘+1)
)
𝑧𝑘(.)
=
𝑘
1, (,
𝑡𝑘
->
.
@
"𝑠0 = (0, 𝑧0, 0)
2 𝑧0(𝑛) 0 1 (
𝑛
T F ⧵ {0}* ! @
)
𝑠𝐾
{(0,
𝑧𝑘,
𝑡𝑘)
∶
𝑡𝑘
[0T,
𝑇
],
𝑧𝑘
{0T, 1}|
G𝑜|}
) 2
,
𝑇
( (
9 >*!S = F × [0,
𝑇
] × {0,
1}|G𝑜|*
Actions
𝑘
(
F* 5
𝑠𝑘
( (
𝑛𝑘
𝑟𝑘(𝑠𝑘,
𝑎𝑘,
𝜔𝑘+1)
=
−𝑤1
𝜙1(𝜔𝑘+1) −𝑤2
𝜙2(𝜔𝑘+1) −𝑤3
𝜙3(𝜔𝑘+1) +𝑤4
𝜙4(𝜔𝑘+1)
-A.
Objective !7 /2 / (
) ( )
,, ,)
2
*
𝜋
+(𝑎0, 𝑎1, … ,
𝑎𝑇
(,
𝑘
=
0,
…
,
𝑇
* ! @
,
𝜋
/2
/)*!7(
𝑠0()
,
A(𝑠𝑘)
=
{𝑎𝑘
T
{𝑛 T
F
∶
(𝑠𝑘, 𝑛)
T
£}}
∶
-;.
min
E
𝜋
[
∑
𝑇
𝑘=0
𝑟𝑘|𝑠0
]
-H.
Transitions )(
𝑎𝑘
(
𝑠𝑘
)
𝑠𝑘+1
𝑘
+ 1* ! ,
) )
/ *0) )(
( * !
(
2
(
𝛺𝑘+1
,
𝜔𝑘+1*
𝑅 𝑘+1(𝑠𝑘, 𝑎𝑘, 𝜔𝑘+1)
)
𝑘
)
𝑎𝑘
(
𝑠𝑘
(
𝜔𝑘+1*
Rewards 5
𝑠𝑘
(
𝑎𝑘
T A) (
) * !)
(
(->.@/)(
-;. (
(
-A. (
( -H.)
(*!
F. Neves-Moreira and P. Amorim
International Journal of Production Economics 267 (2024) 109074
E
4.
Solution approach
! @ ' A
+ ) Q ->.
) '
-' . "
+ @
F -;. ( +
, ( + +
,
, O * * ;
) ( ) )
+ )
)*
" ' H*> )
(
+ (
*"'H*;)
O ,
( , )
*'H*A
F. Neves-Moreira and P. Amorim
International Journal of Production Economics 267 (2024) 109074
>9
/ig.
2. ) )( *
/()
/ () , ) O
*
4.1. SRP definition
! '
'A*>*$
(,𝑖
𝑗)
T£
( )+ (
)
𝑖
𝑗*
$),(
𝑤𝑖𝑗
,,=*
G𝑜
⊂
4.1.2. SRP solution approach
"/( '
( ( *
-( 2.)
( ( ' *
, ,
),
( * !( ) )
/ +,
(
* 5 , )
((
(/->.*
F (
𝑜*!
7 (
)
2-.(*
4.1.1. SRP mathematical formulation
!
'
)
,
𝑥𝑖𝑗
(
(
Algorithm 1 #
>Q procedure CUTTInGPLAnEs(F, G𝑜)
;Q
𝐶𝑢𝑡𝑠𝐴𝑑𝑑𝑒𝑑
←
𝑇
𝑟𝑢𝑒;
𝑋
←
;∅
𝑌
←
;∅
𝑍
←
0;
AQ
while
𝐶𝑢𝑡𝑠𝐴𝑑𝑑𝑒𝑑
do
HQ 𝑋 ← ' -8.-C. F
G𝑜;
8Q
𝑌
← ' (F
*
𝑥
𝑖𝑗
,
)
KQ 𝑍 ← # ( MF
CQ if
𝑍
=
1
then
(,𝑖 𝑗) *!(
()Q
' Q
?Q
𝐶𝑢𝑡𝑠𝐴𝑑𝑑𝑒𝑑
←
𝐹
𝑎𝑙𝑠𝑒;
EQ else
2
∑
(,𝑖 𝑗)T£
**
∑
𝑤𝑖𝑗
𝑥𝑖𝑗
-8
.
>9Q -?. F
>>Q
𝐶𝑢𝑡𝑠𝐴𝑑𝑑𝑒𝑑
←
𝑇
𝑟𝑢𝑒;
>;Q
if
𝐶𝑢𝑡𝑠𝐴𝑑𝑑𝑒𝑑
then
>AQ 𝑋 ← # F
>HQ else
𝑖TF
∑
𝑖TF
𝑥𝑗𝑖
= 1
W
𝑗
T
G𝑜
X {0}
-K.
𝑥𝑖𝑗
= 1
W
𝑗
T
G𝑜
X {𝑣 + 1}
-C.
>8Q if then
>KQ
𝜃𝑜
← !+(
YF
>CQ return 𝜃𝑜*
∑
∑
𝑖TU
𝑗TU
𝑥𝑖𝑗
≤ |U| − 1
W U ⊂ F,
𝑖
≠
𝑗
-?.
!,/
- > H.
-
F. Neves-Moreira and P. Amorim
International Journal of Production Economics 267 (2024) 109074
>>
𝑥
T {0, 1}.-E.
%7 ( -8. 2 -
.( *!6)
@
,
-K.-C.*#-?.*
, -E. @, (
*
> 8.* '
* "
->>9.
- > >>.*5
)
* !
( /
(>>C.*
F. Neves-Moreira and P. Amorim
International Journal of Production Economics 267 (2024) 109074
>;
)
𝑘 𝑘
4.2. Q-learning
! ' H*>*;
+
𝜃𝑜
( ( (
𝑜* 0)
(
* " ' ( )
()
(,𝑖
𝑗)
)
𝑖
𝑗
-
.* !(
, @ (
)
(
𝑖
𝑗
)
(*
Algorithm 2 O
>Qprocedure QLEARnInG(𝑀𝐷𝑃 , 𝛼, 𝛾, 𝜖)
;Q "2𝑄(,𝑠 𝑎) F
AQ for do
HQ "2 ) 𝑠0 -
.F
8Q while
𝑠𝑘
@ - .do
KQ #
𝑎𝑘
(
𝑠𝑘
() 𝜖,,F
CQ $/𝑎𝑘)𝑟𝑘)
𝑠𝑘+1
F(
?Q
L
O(
,
𝑄(𝑠𝑘, 𝑎𝑘)
←
𝑄(𝑠𝑘, 𝑎𝑘)
+
𝛼
𝑟𝑘
+
! , ()
+ ) O
-5 >E?E.
𝛾
/
𝑄(𝑠𝑘+1
𝑎
)
,
𝑎)
−
𝑄(𝑠𝑘,
𝑎𝑘)
3, '
A*A*! +(
(/
, * "
(
(
𝑄(𝑠𝑘, 𝑎)
(
/
)
(
𝑎
(
𝑠𝑘*
𝑉𝜋 (𝑠𝑘)
(
EQ if 𝑄 then
>9Q &F
>>Q return O
H.)@
𝑠
-;8.
/)()(
𝑠𝑘
(),
𝜋*! )
𝑘
+ 1
/ ) (
@
𝑇
* % 7 @
, /2 )
* # & $+
-) ;99C. 𝑉 (𝑠𝑘) = max𝑎 ∈A(𝑎 ){𝑟𝑘+1 +
E[𝑉 (𝑠𝑘+1)|𝑠𝑘, 𝑎]}
𝑎𝑘
,
()/
𝑎𝑘
=
/{𝑟𝑘+1
+
E[𝑉
(𝑠𝑘+1)|𝑠𝑘, 𝑎]}
->9.
𝑎𝑘TA(𝑠𝑘)
")
(,)
* !( , ,
6
/ , )
𝑠𝑘
(𝑎𝑘*𝑄𝜋 (𝑠𝑘,
𝑎𝑘)
(
-O(.
+, (
𝑎𝑘
)
𝑠𝑘
() , 𝜋* 5
@O(
𝑄
(𝑠𝑘, 𝑎𝑘)*
!
)
𝑉
(𝑠𝑘)
𝑄
(𝑠𝑘, 𝑎𝑘)
,
()/
𝑇
/ O(
- ; K ?.* ! (
( 𝜖,
) ,𝜖
) O ) , 1 − 𝜖*
(
@ @( O(
-**(/))
+ ) .* "(
O(*%)
/*
4.3. Illustrative example
! )
, (
, )*
* A ) )
* " @ 2 (
(
* 5
))
* ! SP , - .
*%QL ,-O
𝑉
(𝑠𝑘)
=
max
𝑎TA(𝑠𝑘)
𝑄
(𝑠𝑘,
𝑎)
->>.
.@
()*
!( , 𝜋∗ ( )
()/
𝜋
(𝑠𝑘)
=
/ 𝑄
(𝑠𝑘, 𝑎)
->;.
𝑎TA(𝑠𝑘)
"( ) (, 𝑄) ,
(
) )*
!,(O
& + / 𝑄*
! & + O(
/,,
𝑄(,𝑠 𝑎)
=
𝑅
+
𝛾
max
𝑄(𝑠′, 𝑎′)
->A.
𝑎′
!O+ , /𝑄
,
F. Neves-Moreira and P. Amorim
International Journal of Production Economics 267 (2024) 109074
>A
! , () ,
/( O * * H /
( ( O-.*
$)
- ?8. -
>;A.* $
( *
@ )CC
EA*'O(CC
CC*
5.
Algorithm convergence
")/
/ , , (
,
𝑄
𝑛𝑒𝑤
(,𝑠
𝑎)
=
𝑄(,𝑠
𝑎)
+
(
𝛼
𝑅
+ 𝛾
max
𝑄(𝑠′, 𝑎)
−
𝑄(,𝑠 𝑎
)
->H.
𝑎
*,)
, * ', ) (
)
𝛼
7 )
)O
( * ,
/ /𝑄𝑛𝑒𝑤 )
𝑄* ! / , (
)
;* ! , 2 O
() -;;.*
! / ( )
*! , (
( -;A.*
2 𝑠0
-;
@ (
𝜖 (
2*
5.1. Synthetic instances
', )
,
* 5 ( (
,) 2
F. Neves-Moreira and P. Amorim
International Journal of Production Economics 267 (2024) 109074
>H
/ig. 3.
" / ( SP -. QL -.* - ( ( @ (
) ( *.
/ig. 4.
/) ( OO /*
*
,
(=
( * ! ( =
()()
(1 /(10 *!
+ ,
() ) (
* ! (
,2 (
9*;*" (8
/(
|C| 50*!/(
𝑚 5
30 * !
)
1 m∕s*!>2(
)
12 ,*
)
, )
)
2, 3 *"(, -**
, .
2 ,1 * 2,
1
))100 *
5.2. Q-learning training and results
3 ( (
(
*! ,
/ , * !
)
@ ( ,
*$ @)(5000
* )"YJ;*8
B02
+(
F. Neves-Moreira and P. Amorim
International Journal of Production Economics 267 (2024) 109074
>8
Table 1
' ( *
', {!,, ', ,
}
# {$, , &}
, {2}
% {0.2}
% -. {8}
/ (
{50} / (
{5} -.
{30}
5 -V. {1}
# , ->. {3}
# , -;. {1}
' , -A. {1}
) -H. {100}
Table 2
' ( (
(*
-𝛼.{0.95, 0.97, 0.99}
3 -𝛾.{0.50, 0.70, 0.90}
𝜖 -𝜖.{0.01}
) B E*8* ! ;
@*
! ) (
!A*!@)
<<,==<<== ,(
, ( ( , *
! <<& ==
( -𝛼|𝛾|𝜖.
)*#(/
/)
F. Neves-Moreira and P. Amorim
International Journal of Production Economics 267 (2024) 109074
>K
/ig. 5.
# ) , @ , -@ 899 ( .*
-
( ( @ ( ) ( *.
Table 3
# ) -8999 .*
, &
-𝛼|𝛾|𝜖.
# )
* * /* #G - 89
*.
!,
0.97 | 0.9 | 0.01
28 989*54 29 262*92
;EHH9*K9
9*>>
!,
0.99 | 0.7 | 0.01
27 240*37 28 563*90
;E
;K;*>9
9*>>
!,
0.95 | 0.9 | 0.01
28 325*40 28 810*05
;E9CA*A8
9*>9
'
0.97 | 0.9 | 0.01
23 471*06 26 828*68 ;?8KE*??
9*>>
'
0.97 | 0.9 | 0.01
8 571*05 21 363*16 ;?
AK;*C> 9*>9
'
0.97 | 0.9 | 0.01
11 729*02 22 830*18 ;?;AK*HA
9*>9
0.99 | 0.9 | 0.01
−249 624*48 −74 201*81
;K
CH>*>> 9*>>
0.99 | 0.9 | 0.01
−280 708*50 −136 167*61
;KA8;*HH
9*>;
0.97 | 0.9 | 0.01
−275 092*86 −144 711*93
;8
CAA*A>
9*9?
0.95 | 0.9 | 0.01
−282 733*08 −172 079*58
;;8?H*A?
9*>9
0.95 | 0.9 | 0.01
−286 259*32 −152 702*76
>CAC>*H;
9*>?
0.97 | 0.9 | 0.01
−285 686*24 −180 264*63
;>8;H*9E
9*>A
@* ! ) (
50 *
5)
/ )(
,* 2 (
( *
/ )
(,)V
2* )
( 50 (
( ) ) ( 2
( (
*
)*8*
! )
() * / 2 (
( ) * 5
) , )
(() )
,)2( 200
* (,
)*
6.
Real-world application
! /(
$*!)
, ( )
(
) ( ,
,*
(
+)*!(
O , ( ( O
-.
F. Neves-Moreira and P. Amorim
International Journal of Production Economics 267 (2024) 109074
>C
-** 3.*
1
( )(
O* "
O( (
,)6)
,*
6.1. Real-world instance
! @,
/ )
) * ! ,
(
* ) 2
, , )
* 0)
(, ,*
" ,
, ( , -**
) .
) (
( * )
,
* * K ,
)*
" @
*5( )
,
( ( * "
)
, *(
, ,(/
* ! + (
,, )
,
* ! )
(
F. Neves-Moreira and P. Amorim
International Journal of Production Economics 267 (2024) 109074
>
9
/ig. 6.
( , ) * - ( ( @
( ) ( *.
/ig.
7. ) *
)
(37 * ! )
) *
! (,
𝜆𝑆𝑡𝑜𝑟𝑒
10
(
𝜆𝑂𝑛𝑙𝑖𝑛𝑒
0.2*!/()
300 ) / (
* $ 30
) 1 m∕s* ! ( 8 * !
) 50 (F
2 , 3 (
2,1 (
7*
! ( )
) )
* C (
( )
- (
. / ( , ( ,
,-(2.
(,,,*
F. Neves-Moreira and P. Amorim
International Journal of Production Economics 267 (2024) 109074
>
9
5 , ( 5
15 * !)
( ( , (
(
( (*!
2()
( *! ,
) )
( 3"M * 5 ()
, ( , ()
V*
6.2. Training in a real-world context
! ) )
' 8*;* )
(
( ,
𝛼 𝛾/ 𝜖* !
(
!H*
F. Neves-Moreira and P. Amorim
International Journal of Production Economics 267 (2024) 109074
>>
/ig. 8.
)(10 @ )*
Table 4
' ( (
@ ( ) *
-𝛼.{0.1, 0.5, 0.9, 0.95, 0.97,
0.99}
3 -𝛾.{0.1, 0.5, 0.9, 0.95, 0.97,
0.99}
𝜖 -𝜖.{0.01, 0.03, 0.05, 0.07, 0.09}
(30 000 )(@*
) )
,1000 * !10 -
).
*?*$,
1000 )(
)* ( @ 1000
) ,(
* B, ( 20
000 * ()
( ( (
30 000 *
" (
)
-. (
-,
O.*!
))
𝜖,
/
)),*"
,/(
, (*
!(, * -;9>8. ) ,
*5
, ,
*)
𝜖
),
*
6.3. Managerial insights
! )
* " )
V
)* , ) )
) ( ( )
7() ()
( , +,
*
!+
* #)/Z5
(Z
!)+)((
(
(V
*
Q-learning (QL) , O
@ (
) *
F. Neves-Moreira and P. Amorim
International Journal of Production Economics 267 (2024) 109074
>;
Shortest Path (SP) , ,
-, 37 .
*$
,
),*
Myopic Policy (MP) , ,
) () )
)( (
*
Crowded Nodes (CN) , 2 (
(
( * $
,
) , (
*
,(,)),(
+ (
( * ! ( )
' /
)-
( )
.*! )Arc Distance
Arc Crowdedness ,*!)+
)( *
! 8 ( (
*
,2)+
) SP MP
, (
8;*?98>*8H ,*QL CN
( (
H?*?K HA*HK ,* !
, (
/) (
)A9: 89:SP
MP* , CN ) (
, )
(*,QL ,
( ) ,
( *
% ) +
) )
(
,
/*0)+
( (
, ;*8K: ( ( HE*>C
HC*E>* ! (
,,C*EE:(
( >?>A*?A >KK?*E>* !(
) @
/ (
( (
*
5,
()
-SP MP.* "
F. Neves-Moreira and P. Amorim
International Journal of Production Economics 267 (2024) 109074
>A
Table 5
( 899 ( ' -'. #) -#. , , -. O -O. *
,
* )
*[% *[
*
[$
3
O
>CK>8*HE
H?*?K
8CE*>C
>8?>*E>
3
'
>C9?>*89
8;*?9
K;?*;>
;;HH*?E
3
#
>8C?9*8H
HA*HK
8>A*89
>>AE*;C
3
>K>E8*HK 8>*8H
K>;*K8
;;?E*;A
*
>KKK?*;8 HE*>C
8?A*A?
>?>A*?A
#)
O
>C>E9*8;
HC*A;
8K;*C>
>HKH*;C
#)
'
>K8EE*C>
89*AH
8EC*8C
>E;?*9;
#)
#
>K?KH*EC
H8*;H
8A8*9K
>>?C*?K
#)
>8>9;*A9
H?*C;
8CE*9;
;9E8*89
*
>KHAE*A? HC*E>
8K?*8E
>KKE*E>
/ig. 9.
"/( SP -. CN -. QL -.* - ( ( @
( ) ( *.
( / CN ,
) (
*0) , (
7 ,* 5 SP MP
CN ,)
7 * ! QL ,
(
)(*
QL
) )
( ) * )
) ( << (
,==(*B
+ ) (
-. (+,*
' QL , )
, ,
) +
)*
!,
4
,*" * E ) / (
(), SPCNQL *
SP ,( () -
. ) * %
QL ,()
) ()
*!CN ,(
*
F. Neves-Moreira and P. Amorim
International Journal of Production Economics 267 (2024) 109074
>H
6.4. Robustness of the policies
! 6/, (
( )
/ *" /)
QL ,
-
' K*>., 2 3 /
Q
•HalfSizeQ!()
( ,*
•DoubSizeQ ! ( )
,*
•HalfOffRateQ!(1
( ,*
•DoubOffRateQ ! ( 1
,*
•2OrderBatchQ7()
*
•CapacityCartQ ! ,
*
! K ( /
) QL , )*
) -. (
(
((
, (
( -.(,
*,
F. Neves-Moreira and P. Amorim
International Journal of Production Economics 267 (2024) 109074
>8
Table 6
( 899 ( O -O. , ( *
'
*
[#*
*
[%
*
[&
*
['
*
[
*
['
*
'
*
'
*
*
$
*
*
)
*
Baseline
>H8*9H
HC*9; HC*9;
9*99
88E*9;
8E;>*E?
>>*E>
>9*8E
;*?K ;E*9A
HalfSize >H8*H8
KH*K? KH*K?
9*99
A?K*E>
K8;?*HA >K*99
>K*?C
H*9C ;;*HE
DoubSize
>HH*H8 A>*KK A>*KK
9*99
CK9*9;
8>8>*A;
?*A9
K*C?
;*9?
AH*;A
HalfOffRate C;*8E HC*H8 HC*H8
9*99
8KH*9? 8???*CE
>>*?8
>9*HH
>*H;
A?*8K
DoubOffRate ;E9*>H
HC*C8 HC*C8
9*99
8KE*9; 8??E*AC
>>*?;
>9*A8 8*8E
>>*H;
2OrderBatch >H8*9>
KH*CE
A;*H9
9*99
CH>*CA
89K9*98
A*8A
K*?; ;*98 AH*AE
CapacityCart
>H8*AC
AE*EH AE*EH
>*C>
8;E*9H
KH98*>; A;*CA
>;*>>
;*?E ;K*;9
) ( (
) (
(
)
*
5 ,
( *
5(
HalfSize
-KH*K?. ,
V
->K*K?.*, V
))->K*99.
1 -H*9C.*
DoubSize *
5(1
HalfOffRate (
(
( -C;*8E.* !
->*H;.*!
)-A?*8K.*
DoubOffRate *
"2OrderBatch )()
-CH>*CA.
() -K*?;.* 0V
(
-;*98.() -A*8A.*
,)
Capac- ityCart
2 ,*5
( )
2 , -A;*CA.*
", (
->;*>>. (
-;*?E.*
!6/
*
, /*
7.
Conclusion
!)
(
*!
( / ( ;9;9* '
, )
&%''')
( / , * &
,
(
(
*,,,
*!)
) ,* !, )
(@ ,* 5
) (
) (
*!,
F. Neves-Moreira and P. Amorim
International Journal of Production Economics 267 (2024) 109074
>K
(
( /
),*
%+
@ (
, (
( ) 2 (
* % ,
( (
, ) * !
(
(
/*
#/ (,
) ($
)
(*
)
) ,
+ *
7.1. Managerial implications
& ) , )
2
() * /
+ () , (
)*3 ,
)
* 0) )
( * !
( )
,() ,
( *
% ( )
)
, /* ' ,
) (@
,(*
!)(/
)),,
2(
(12 (
, ( *" )
* ( )
(
(
* ) )
)
, ( ,
)2*
(*
7.2.
Limitations and future work
, )
( )
*
F. Neves-Moreira and P. Amorim
International Journal of Production Economics 267 (2024) 109074
>C
Offline customer arrival rate % 3 1
() *
0) )
Appendix. Notation
)(
(*!)
*!
) ),Q->.
( , )
, ((F-;.
,*
Limited demand context ! ) )
/)
(*
(,
) (,
,) ,
) (
)(*
MIP -'H*>*>.
"
7
%
'
B(-F, £.
F'(
£'(
G𝑜'((
𝑜
𝑤𝑖𝑗
!
-,𝑖 𝑗.
3G
𝑥𝑖𝑗
$+(
-,𝑖 𝑗.
2 )
Policy generalization ! ) /
(@),
)* 5/,(/
( /(
-.*
Picker and cart decoupling "
@ , (
* !( ,
, ) 3
) *
, )
/ )
*
Evaluating customer satisfaction
@
) (
( (
*! ) )
,*
, (
*
)
/ , (
(
* ,
( /
)) / ) *
/ ,+,(
* (
(6
((-,
) ( << ( ,==
, .*,))
(
*! (
( (
( )
* ! )
, ))*'
,,,
( ( (
*
Data availability
F. Neves-Moreira and P. Amorim
International Journal of Production Economics 267 (2024) 109074
>?
MDP -'A.
"
𝑘
3
𝑐
%1
𝑜
%
'
𝑇
'(
S'(
A'(
𝑅 )(
𝑃!-).
C'(1
U'(-.
'
𝑠0"
𝑠𝑘
'
)
𝑘
𝑎𝑘
𝑘
𝑛𝑘
#(
𝑧𝑘
!(
𝑡𝑘
!
𝑘
𝑟𝑘
)
)
𝑘
𝜔
$/(
𝑘
G𝑜'
(
𝑜
𝜃𝑜
+
(
𝑜
𝜃𝑐
+ (), 1
𝑐
𝜆𝑂𝑛𝑙𝑖𝑛𝑒
%
𝜆𝑆𝑡𝑜𝑟𝑒
%1
𝑠𝑝𝑒𝑒𝑑𝐶𝑢𝑠𝑡𝑜𝑚𝑒𝑟
#=
)
𝑠𝑒𝑟𝑣𝑖𝑐𝑒U𝑡𝑖𝑚𝑒
𝐶𝑢𝑠𝑡𝑜𝑚𝑒𝑟
!
,
(
𝑠𝑝𝑒𝑒𝑑
𝑃
𝑖𝑐𝑘𝑒𝑟
)
𝑠𝑒 𝑟𝑣𝑖 𝑐𝑒U𝑡 𝑖𝑚𝑒
𝑃
𝑖𝑐𝑘𝑒𝑟
!
,
(
|C| /(1
)
𝑚/(,
)
!*
Acknowledgments
!(
( $ L= 02 ;9;9 ! $L
) ( " ;9>H4
;9;9E8;9K9 >9>>;9H9K*
Q-Learning
-'H*;.
𝜋(𝑠),
𝑉𝜋(𝑠)(𝑠)G
(
𝛾
3
𝜖
$/
𝛼
F. Neves-Moreira and P. Amorim
International Journal of Production Economics 267 (2024) 109074
>E
References
0 B, # ' ! 2
# ;9;;* ( ,
)* L
QVV)))*,*VVVV
(,)*
* ' I2 ;9;;* $/
(
* * '* K? -H. ;CA;4;C89*
7 $' 0 ' 7 ' &7 %
;9>?* 2 )
)
)* "* D* * $* ;9K >KE4>?A*
&
D
D,
D
!
'
G
I
;9;>* = ( Q L
,* L
QVV)))*,*VVVV(
,*
&D ;9;;* 5,(, , ,
,
@,
(
*
L
QVV)))*(*VV (V;9;;V9KV9?V),(,
,
,,
(,(VZ\H;(;8AAA>;*
& B # D] B ;9>9*
3, , * $ D* %* * ;9;
->. ?4>8*
& SS %, ^ ' G2
&S ;9>8* , ( )=
**%* *;A? ->4;. ;C4H9*
#MD #I*M*;9;9*% )
Q , ) ( * "* D*
* $* ;;E >9CC;E* #2 '
^" ;9;A* (
( , , * "* D*
*$* ;K; >9??EE*
#20#;9>?*/(
*$D* %* * ;C9 -;. H>E4H;E*
#I_*BNR2S2G;9;9*!((
* (* '* %* * ;; ->. HC48?*
# !# #* D+ #`S D]
;9>?* /(
)*
D*%* * '*KE -?.>;H;4>;8A*
# ! , # #* D+
;9>C* % ) (, ,
*"*D** * 88 -;>. KAK>4KACE*
# Y* M* 0 !* ;9>8* '
* "Q ;9>8"$$$V#"' >H
"#( # "( ' -"#"'.*
* 8;E48AA*
# D M ' 0 ' ^D / Y
, ;9;>* %1
*(*'*%* **
# Y 3 I B
;9;>* " (
, *"Q %2 3
'Q %3' G #( >E ;9;9* '
" # * >?>4>?E*
#S7 B* D* ( 3* >E?8* !
,* *
* AA >4;C*
3 *! 3,"* 2D%*>EE8*
( +, ( Q ' * D*
* * '* ;H
->. A4>K*
3, " 2 D( ;9;;* #) (@
* * %* * A> ->>.
H9C84H9EH*
I S 3! I D ;99C*3
( ) Q )* $ D*
%* *>?;-;.
H?>489>*
3
'
;9>?*
,
=
*
&
*
L
Q
VV)))*(*VVV;9>?V9AV>EV,
(*
3(# %0N /;9;;*
(@(
,*$L %D*
!**>>>999?;* 3( '"*
5 ;9;>* % (@(
*
$D* %* * ;EA-A.>98?4>9CK*
3) # ;9;>* ) ,
* * 0* "* '* L QVV)))*
*V+V9;;>U UE U;V *Z
\;a(\>>;[>>;*
D'I#*;9>9* LIQ
(* "* D* 3* * A? ->>V>;.
?EH4E>H*
B 'Y;9>C* %) ,
* * '* KA -?. ;HC?4;HE;*
B ' Y ;9>C* % 1 ( (
* (* '* %* * >E ->. ?H4E?*
B # I ' Y ;9>E* %
Q $ , * (*
'* %* * ;> ->. ?K4>9;*
B M I 3 ;99?* ,
, ( * ""$ !* H9 ->>. >9C94>9?;*
0 3* ! & L 5* ;9;>*
5 Q= ( (
, )b*
0'D*;9;>* (,,
* #* "* $* >8H >9KE>E*
0N / 0 I 0 5 D ;9>K*
(@ ,Q
()*"Q!) I20 -$*.
"*D* 3** HH -A.*
0' I*" D*D,0M'D;9>A* !
( Q
*
D* * CC-;. >4>K*
D'ID, *!,';9;>*$
( (@* (* '* %*
**
I ', $* ;9;A* # ( ,
L*'* , 8: ,* L
QVV)))*(*VV,V ;9;AV9>V>>V(
,, 8,VZ
\KCK>??KC;*
I3*#BD#*2D$*
;9;;*
3,) * !*'* 8K -A. CC84CEH*
D,'*&)$!*'*;998*/,
* "* D* * * ;; -H. AE84H>H*
( * ' 3* ! 3 % ;9>A* #
( (
*$D*%* * ;;? ->. ?A4E;*
M 0 ## ;9>C*
( * 3* '
',* EH EC4>9?*
D) M B 2 # ^ ) '
5 ^ D ;9;>* 3 ( (
* "$$$ !*
#,* >4>H*
Y^M50 ;9;>*" ( <<,
==+,,*
$D*%* * ;EH -A. E;;4EA8*
5 3 B G ^ O ;9>K*
( , ) *
$D*%* * ;H? ->. >9C4>;;*
B, ;9>E* ! (@ Q 5,
( /* L
QVV)))**VV )V((
),(
,/V)VCA>HC9HC*ZU\U*
* * '22 * ! * ;9>?* %2
' & $B,Q #
&@ G*
;9>?'* * A>?4A;K*
B # 0* B $ 0* ;9;9* %
)Q , )* "* D* *
$* ;;H >9C8KH*
O & B B ;9>>* )
( , LI ,
Q )*
"Q LI , ( "(
', #( ;9>>*
G,,II,'3 *G
D & B* B / 7
I* % B ' & # '
" I 0 I 3 5
3 ' 0 3 ;9>8* 0
F. Neves-Moreira and P. Amorim
International Journal of Production Economics 267 (2024) 109074
>
>
9
( * 8>?
-C8H9. 8;E48AA*
# 3 ;9;;* A ),
* L
QVV)))*,*V)VA),
VK;A;9KV*
2*%7, (', ) G*! ;9>?*
( ( * "Q
"'*
, ;9;9*OQ/(
,Q 3, * L
QVV)))*,*VV+V*
%7 &0 Y$ #* ,2) I*
# $' Dc;9;>*
, Q ,* #* "* $*
>K9 >9CK9H*
# #2 0 ;9>?* $/
( * #* %* * >99
>>C4>;C*
% # Y 3 I
;9;>* ! ,
Q % 2 (
* >H ->;.
A89*
G B BS # S *
;9>A* )
( , * $ D*
%* * ;;8 ->. >4>>*
' $ ;9;>* ' ,
,
* "* D* * * 8E -A. CC;4CEC*
) 5 &* ;99C* / 3, Q '
# (
3,-5,','.*
5,"L'* ) 5&*;9>E* (()(
2*$D*
%* *;C8 -A. CE84?;>*
F. Neves-Moreira and P. Amorim
International Journal of Production Economics 267 (2024) 109074
>
>
>
( 0 * 5 I # * ;9>K* 3,
Q ! * ) KC
->. A4A>*
O ' #G I, I* ;9>?* 3 (
,
* * #*"* (* 89 >A4;E*
0*3 '*>E?A*%
)Q ( * %*
* A> -A. 89C48;>*
D ;9;9* & (
((* L
QVV)))*(*V(),VV
;>>A>H>9V,(
((*
2L D 0 * ;9>K*,
, * "* D* * *
8H ->. ;>84;A>*
R2 D 2 $ B &
$+ R2',;9>E*!
/* $ D* %* * ;C? -;. K>84K;?*
I D I S ;99>*
)
) * $ D* %* * >AA ->. A;4
HA*
I D I d ;99>* (
))
* "* D* * * AE -E. >?K84
>??A*
I DG "** 3!,D*B*;9>8* '
( ) , * "* D* *
*8A->>.
AA9K4AA;K*
'(, & ;9;;* ( Q 0)
2 &%"' /* L
QVV)))*(*VV(V ;9;;V9AV9HV(
()2 /VZ
\8HECHAE*
' G 5 ! ;9>K* 89! ,
P#,
Q # * !* '*
89 -;. 8CE48E9*
'2 S 0 ' ' 5e B
;9>K* ) ( (
* $ D* %* * ;8A ->. K?4?H*
!,#&e,,%35 &;9>9*L
!' ( )* $ D*
%* * ;99 -A. C884CKA* L5*BD#*
(3#*!&5*;9;9*%
, * $L % D* !* * E
-;. >9999?*
B! I&I3&f#
;9>?* " , ,
2
, * "* D* *
$* >EC ;HA4;K>*
G,3, D %!, S
;9;;*! (Q
* "* D* * $* ;H? >9?HE>*
5 #*D*#*0* >E?E* ( 3, )-*3* .*
I=#
%/(*
5 D 0N / I 0 !
/ ;9>?* * "* D*
,*3*** H? -H. H>84HA?*
M 2 ;9;;* ,
( *L QVV)))*)7*VV
, (
>>KC98>E8EK*