380
this paper, we use a sequence-based formulation for our R-Model. So, we introduce binary
decision variables v jr that are 1 if job j is completed before job r is started.
v v
j r J j r
jr
rj
+ ≤
v rj
∀ ∈
r
≠
1
j
∀j j
(6)
We have to consider a break for each worker (see R1) that leads to the absence of this
worker in a specific time. Each worker w has to make a Δ-minute break so that the start time
of the break ρ w lies in a predefined interval [
, ].
h
α
ω
ϕ
ω
In our relaxation, we do not allow
that the processing of a job overlaps the break of the worker who processes this job. However, the break may overlap the drive between two jobs or the technical services that may be
executed for a machine.
ϕ
ρ
α
ϕ ϕ ≤
∀
ρ w
w W
∈
(7)
ρ ϕ
ω
w
ρ ρ
w W
≤
∀
ϕ
ω
ϕ ϕ
(8)
S
M
x M
j J w W
j
w
S
j
s
jw
j
W
( )
−
(
) ∀ ∈
j j
∀ ∈
w
ω
w
( 1
1
M
s
) + (
,
(9)
ρ
ω
w
j
ρ ρ
j
s
jw
j
ω j
x M
jw
j J w W j
+
−
(
) ∀ ∈
j j
∀ ∈
w
1
s
S
M
ω
s
≤
+
M
ω j
ω
s
ω ω
(
,
(10)
ρ w
j
ρ ρ
j
j
ω
e
jw
j
S
M
x M
j J w W j
≤ S
+ ( ( ) ( (
) ∀ ∈
j j
∀ ∈
w
M
1 ω j
ω
e
ω ω ) 1
(
,
(11)
S p
M
j
w W
j
j
S p
w
j
e
jw
j
W
≤
p j
p
+ M
j
e
(
) ∀ ∀ ∈
w
ρ w + j
e
1
,
x M
j J
jw ) ∀ ∈
j j
(12)
ω ω
δ
j
ω
s
ω ω
j
e
j
δ
j J
+
≤
ω j
ω
e
ω ω
+
∀
δ j
δ
j j
1
(13)
δ ω
j
j
δ ω
δ
s
j J
∀
ω j
ω
s
ω ω
j j
(14)
δ ω
j
j
δ ω
δ
e
j J
∀
ω j
ω
e
ω ω
j j
(15)
If the start time of the break of a worker, who processes job j, is during the processing of
j j
j
(
)
j
s
j
e
,
j
s
j
e
j
e
binary decision variable δ j takes the value of 1 and a Δ-minute break
must be considered for the duration of j, additionally.
At the beginning of the working shift, the first technical services td m
d d
α must be performed
on machine m and the assigned worker drives d jm
0
d d time units to the first job.
m G
m
j m
j
j j
m
j
j j
j
j
G
d
y
jm j j
S
b
j
j M
j J
∑
≤ S j ( ) ∀ ∈
j j
( m
td m d jm j j
+ d jm j
α
d
(16)
At each working place in an underground location, several operation types must be executed in a specific order related to the prescribed production cycle. Let ul j
l be the underground location of j, ml j be the working place of j in ul j , and order j be the position of j in the
given order for ml j . In a working place, always job j with the minimum value of order j must be
completed before any job r with a greater value of order r can be started.
b b
j r J j r ul ul ml ml order order
r
j
b b
b
j
r
l ul
j
r
l ml l
j
r
r order r
∀
b
j j
j
b
∈
≠
J j
ul ml
<
r ∈J
,
,
ul j
r
l ul
,
(17)
S p
S
b b M
j r J j
l ul ml ml order
j
j
S p
j
r
S S
j
r
b b b
j
r
ul
j
r
l ml l
j
r
+
p j
p
≤
⋅
−
( (
)
∀ ∈
j r
j
≠ r ul l
<
ml order
ml
j
r
δ j Δ
r J
∈
r
,
,
j
r
ul l
ul j
l
,
orde o o r r
r r
(18)
For jobs that are processed by the same worker, a precedence relation must be considered.
As mentioned in Sect. 1, if a worker changes his machine, he has to go to the new machine
(transfer time), has to do the first technical services, and he can drive the machine to the location of the new job. In our relaxed model, we neglect the time for the case, where a worker
changes his machine.
this paper, we use a sequence-based formulation for our R-Model. So, we introduce binary
decision variables v jr that are 1 if job j is completed before job r is started.
v v
j r J j r
jr
rj
+ ≤
v rj
∀ ∈
r
≠
1
j
∀j j
(6)
We have to consider a break for each worker (see R1) that leads to the absence of this
worker in a specific time. Each worker w has to make a Δ-minute break so that the start time
of the break ρ w lies in a predefined interval [
, ].
h
α
ω
ϕ
ω
In our relaxation, we do not allow
that the processing of a job overlaps the break of the worker who processes this job. However, the break may overlap the drive between two jobs or the technical services that may be
executed for a machine.
ϕ
ρ
α
ϕ ϕ ≤
∀
ρ w
w W
∈
(7)
ρ ϕ
ω
w
ρ ρ
w W
≤
∀
ϕ
ω
ϕ ϕ
(8)
S
M
x M
j J w W
j
w
S
j
s
jw
j
W
( )
−
(
) ∀ ∈
j j
∀ ∈
w
ω
w
( 1
1
M
s
) + (
,
(9)
ρ
ω
w
j
ρ ρ
j
s
jw
j
ω j
x M
jw
j J w W j
+
−
(
) ∀ ∈
j j
∀ ∈
w
1
s
S
M
ω
s
≤
+
M
ω j
ω
s
ω ω
(
,
(10)
ρ w
j
ρ ρ
j
j
ω
e
jw
j
S
M
x M
j J w W j
≤ S
+ ( ( ) ( (
) ∀ ∈
j j
∀ ∈
w
M
1 ω j
ω
e
ω ω ) 1
(
,
(11)
S p
M
j
w W
j
j
S p
w
j
e
jw
j
W
≤
p j
p
+ M
j
e
(
) ∀ ∀ ∈
w
ρ w + j
e
1
,
x M
j J
jw ) ∀ ∈
j j
(12)
ω ω
δ
j
ω
s
ω ω
j
e
j
δ
j J
+
≤
ω j
ω
e
ω ω
+
∀
δ j
δ
j j
1
(13)
δ ω
j
j
δ ω
δ
s
j J
∀
ω j
ω
s
ω ω
j j
(14)
δ ω
j
j
δ ω
δ
e
j J
∀
ω j
ω
e
ω ω
j j
(15)
If the start time of the break of a worker, who processes job j, is during the processing of
j j
j
(
)
j
s
j
e
,
j
s
j
e
j
e
binary decision variable δ j takes the value of 1 and a Δ-minute break
must be considered for the duration of j, additionally.
At the beginning of the working shift, the first technical services td m
d d
α must be performed
on machine m and the assigned worker drives d jm
0
d d time units to the first job.
m G
m
j m
j
j j
m
j
j j
j
j
G
d
y
jm j j
S
b
j
j M
j J
∑
≤ S j ( ) ∀ ∈
j j
( m
td m d jm j j
+ d jm j
α
d
(16)
At each working place in an underground location, several operation types must be executed in a specific order related to the prescribed production cycle. Let ul j
l be the underground location of j, ml j be the working place of j in ul j , and order j be the position of j in the
given order for ml j . In a working place, always job j with the minimum value of order j must be
completed before any job r with a greater value of order r can be started.
b b
j r J j r ul ul ml ml order order
r
j
b b
b
j
r
l ul
j
r
l ml l
j
r
r order r
∀
b
j j
j
b
∈
≠
J j
ul ml
<
r ∈J
,
,
ul j
r
l ul
,
(17)
S p
S
b b M
j r J j
l ul ml ml order
j
j
S p
j
r
S S
j
r
b b b
j
r
ul
j
r
l ml l
j
r
+
p j
p
≤
⋅
−
( (
)
∀ ∈
j r
j
≠ r ul l
<
ml order
ml
j
r
δ j Δ
r J
∈
r
,
,
j
r
ul l
ul j
l
,
orde o o r r
r r
(18)
For jobs that are processed by the same worker, a precedence relation must be considered.
As mentioned in Sect. 1, if a worker changes his machine, he has to go to the new machine
(transfer time), has to do the first technical services, and he can drive the machine to the location of the new job. In our relaxed model, we neglect the time for the case, where a worker
changes his machine.