A Matheuristics for the Single-period Lot Scheduling with Component
Availability Constraints in a Partially Closed
Manufacturing/Remanufacturing System
Davide Giglio and Massimo Paolucci
DIBRIS, University of Genova, Via Opera Pia, 13, 16145, Genova, Italy
Keywords:
Remanufacturing, Production Scheduling, Mixed-integer Programming Modelling, Matheuristics.
Abstract:
An integrated manufacturing/remanufacturing system is considered in this paper with the aim of scheduling
the operations of the manufacturing plant. The system is partially closed in the sense that the raw materials,
necessary for assembling the final products, can be obtained both from an internal remanufacturing plant
(which disassembles returned products) and from external suppliers. The manufacturing system is modelled
as a flexible flow shop whose stages represent the different assembly phases leading to the final products.
In this paper, an original event-based mixed integer programming (MIP) formulation is presented, whose
objective consists of minimizing, as primary objective, the weighted number of tardy jobs and, as secondary
ones, the fixed and variable purchase costs of raw materials possibly acquired from external suppliers. Due to
the complexity of the problem, the MIP formulation can be used to solve only small instances. For this reason,
a matheuristics is proposed, which consists of three interoperating mathematical programming models: the
first model assigns the jobs to the machines; the second model sequences the jobs on the machines; the third
model defines the external supplies, taking into account the component availability constraints. A preliminary
computational analysis shows the effectiveness of the proposed algorithm.
1 INTRODUCTION
In recent years the research community has increas-
ingly dedicated efforts in dealing with environmen-
tal issues. Among these, the ones relevant to sup-
ply chains, specifically reverse logistics, represent the
overall framework for the work presented in this pa-
per. Classical supply chain research focuses on the
management and optimization of the flow of mate-
rials and products within a network connecting sup-
pliers, producers, distributors and customers so that
the product demand can be satisfied in the right times,
quantities and locations by optimizing the use of the
available resources and the operational costs. To ac-
count for environmental aspects, this structure has
been extended by incorporating reverse flows of fin-
ished products at their end-of-life that are returned by
customers (Fleischmann et al., 2001; Shah, 2005).
Therefore, reverse logistics includes all the logistic
activities allowing to transform the used products,
discarded and then returned by customers, into new
products to be delivered in the market.
Several classes of reverse logistics are distin-
guished according to the way used to recover the re-
turned products (Thierry et al., 1995), i.e., reuse, re-
cycling, and remanufacturing. In reuse, the returned
product (e.g., a pallet) is directly used after some
cleaning or reprocessing; recycling does not maintain
the product but aims at recovering the materials com-
posing it (e.g., plastic); remanufacturing denotes the
industrial process through which the returned prod-
ucts are restored to like-new condition: products are
first disassembled, then usable parts are cleaned, re-
furbished and stored into parts inventory; finally, the
new finished products are reassembled using the re-
stored parts and possibly new parts (Lund, 1998). In
this way, the need for new materials is diminished, so
increasing the environmentalsustainability of the pro-
duction processes and in general improving the sur-
vivability of companies with respect to the new envi-
ronmental regulations and the environmental aware-
ness of the market.
In this paper the problem of defining the
production schedule in an integrated manufactur-
ing/remanufacturing system is considered. In partic-
ular, the focus is on optimizing the scheduling of the
orders (jobs) in the manufacturing facility taking into
account the constraints due to the use of shared com-
110
Giglio, D. and Paolucci, M.
A Matheuristics for the Single-per iod Lot Scheduling with Component Availability Constraints in a Partially Closed Manufacturing/Remanufacturing System.
DOI: 10.5220/0006005501100120
In Proceedings of the 13th International Conference on Informatics in Control, Automation and Robotics (ICINCO 2016) - Volume 1, pages 110-120
ISBN: 978-989-758-198-4
Copyright
c
2016 by SCITEPRESS Science and Technology Publications, Lda. All rights reserved
ponents that are produced by the remanufacturing fa-
cility and made available through a component stor-
age. The main scheduling objective consists in mini-
mizing the number of orders completed after the due
date, weighted according to the priority of the orders.
In addition, as secondary objective, the possible ac-
quisitions of lots of new components from external
suppliers must be optimized. Indeed, the replenish-
ment plan for the shared components from the reman-
ufacturing facility is assumed given and additional
new components that may be needed should be or-
dered to the external suppliers. Such further supplies
should be needed as late as possible to increase the
possibility of receiving the new components on time,
and they should consist of lots as large as possible to
reduce the fixed costs of supplies.
In the literature the presence of component avail-
ability constraints is considered in both resource-
constrained project scheduling (RCPS) and machine
scheduling, even if few papers can be found in the
latter case. Among them, in (Grigoriev et al., 2005)
the complexity of single machine scheduling with raw
material constraints with the objective of minimizing
the number of tardy jobs or the makespan is analyzed,
stating that some variants of this problem are strongly
NP-hard even with unit processing times, although
polynomially-solvable cases exist. Several types of
mathematical programming models are introduced in
the literature for scheduling problems with compo-
nent availability constraints. The case of assem-
bly scheduling is considered in (Kolisch and Hess,
2000) and (Kolisch, 2000) where RCPS-based mixed-
integer programming (MIP) models, which use time-
indexed variables, are proposed. In general the MIP
models provided in the literature differ for the kind
of time representations, as analysed in (Pinto and
Grossmann, 1995), (Li et al., 2010), and (Mouret
et al., 2011). In particular, in (Li et al., 2010)
three main classes of models are distinguished, i.e.,
slot-based, event-based and sequence-based. Among
them, event-based MIP models have been recently de-
fined to solve RCPS problems (Zapata et al., 2008;
Kon´e et al., 2011; Artigues et al., 2013).
This paper is organized as follows. After hav-
ing formally defined the considered scheduling prob-
lem, in Section 2 an original event-based MIP model
is proposed for its solution. Due to the problem
complexity, this formulation can solve only small in-
stances within acceptable computation times. There-
fore, in Section 3 a matheuristics, which exploits
three interoperating mathematical programmingmod-
els, based on both event-based and sequence-based
MIP formulations, is proposed: the first sequence-
based model assigns the jobs to the machines; the
second sequence-based model identifies the overall
priority order among the jobs; the third event-based
model defines the external supplies, taking into ac-
count the component availability constraints. In Sec-
tion 5 a preliminary computational analysis showing
the effectiveness of the proposed algorithm is reported
and, finally, conclusions are drawn in Section 6.
2 THE SCHEDULING PROBLEM
An integrated manufacturing/remanufacturing
(“man/reman”) system is considered, in which final
products are produced in a manufacturing plant by
assembling raw components supplied both by an
internal remanufacturing facility (which disassem-
bles returned products) and by external suppliers.
A representation of such class of systems is given
in Figure 1. The manufacturing system is modelled
as a flexible flow shop whose stages represent the
different assembly phases which lead to the final
products. Let M
a
ks
be the k-th machine at stage s, with
s {1, ...,S} and k {1, ...,N
a
s
}, being S the num-
ber of stages and N
a
s
the number of parallel machines
which are present at stage s. The remanufacturing
system consists of a disassembling subsystem which
extracts the components to be recovered from the
returned products, and of a refurbishing subsystem
which processes the components bringing them back
to an as-good-as-new state. The disassembling sub-
system is modelled as a flow shop of N
d
machines,
whereas the refurbishing subsystem consists of a set
of N
r
parallel machines which perform operations.
Let M
d
k
be the k-th machine in the flow-shop of the
disassembling subsystem, with k {1,..., N
d
}, and
M
r
k
be the k-th parallel machine in the refurbishing
subsystem, with k {1,... , N
r
}.
In this paper, the scheduling of manufacturing ac-
tivities is considered, assuming that the disassembling
and refurbishing operations in the remanufacturing
plant have been already suitably planned and sched-
uled. For this reason, the flow of raw components
from the remanufacturing system to the manufactur-
ing system is assumed given, both for what concerns
times and quantities.
The manufacturing system has to produce a set
of orders that are due at the end of the considered
planning horizon, conventionally set to [0,T], being T
the optimization horizon for the scheduling problem.
This corresponds to process a set of jobs, consisting
of operations that transform raw components into fi-
nal products, and to adopt a common due-date model,
since all the jobs should be completed before the end
of the planning period.
A Matheuristics for the Single-period Lot Scheduling with Component Availability Constraints in a Partially Closed
Manufacturing/Remanufacturing System
111
CUSTOMERS
NEW RAW COMPONENTS
PURCHASE
RETURNS DISASSEMBLY
FLOW SHOP
PRODUCTS ASSEMBLY
FLEXIBLE FLOW SHOP
stage 1 stage 2 stage S
PARALLEL
MACHINES
PARALLEL
MACHINES
PARALLEL
MACHINES
RETURNS
RECOVERED
COMPONENTS
RAW
COMPONENTS
FINAL
PRODUCTS
COMPONENTS
REFURBISHMENT
PARALLEL MACHINES
REMANUFACTURING
SYSTEM
MANUFACTURING
SYSTEM
Figure 1: The integrated manufacturing/remanufacturing (“man/reman”) system.
Let J be the set of jobs and let i, j be two indexes
used to denote a job. A job i, i J, is character-
ized by the lot size L
i
and by a priority coefficient
W
i
(weight) which, in this paper, is assumed to be a
fractional number proportional to the size of the lot,
that is,W
i
= L
i
/
jJ
L
j
. The processing of a job con-
sists of operations (one per stage) which consume raw
components in order to assemble the lot of final prod-
ucts. Let R be the set of raw components and let:
Q
isr
, with i J, s {1,... , S}, r R, be the num-
ber of raw components of type r which are re-
quired to execute job i at stage s; Q
isr
is defined
as L
i
· Q
U
isr
, being Q
U
isr
the (given) number of com-
ponents r that are required to produce one unit of
job i at stage s (unitary component requirement);
P
isk
, with i J, s {1,. . .,S}, k {1,... ,N
a
s
}, is
the processing time of job i at stage s when as-
signed to machine k; P
isk
is defined as L
i
· P
U
isk
, be-
ing P
U
isk
the (given) time to produce one unit of
job i on machine k at stage s (unitary processing
time).
The remanufacturing plant carries out some dis-
assembling and refurbishing operations aimed at re-
covering raw components from returned products. As
previously discussed, the activities of the remanufac-
turing system are independently planned and sched-
uled. Therefore, it is here assumed that the arrival
times of raw components from the remanufacturing
plant are known in advance. Let H be the planned
number of arrivals of lots of raw components, and let:
D
h
, with h {1,... ,H}, be the time instant of the
h-th arrival (from the remanufacturing system) of
lots of raw components;
M
rh
, with r R, h {1,... ,H}, be the number of
raw components of type r for the h-th arrival.
Raw components are stored in inventories, one per
each kind of components; the value I
0
r
, with r R, de-
notes the initial inventory of raw components of type
r. It is worth observing that such inventories contain
both components recovered by the internal remanu-
facturing plant and the ones acquired from external
suppliers when the recovered components are not suf-
ficient to satisfy the production requirements of the
manufacturing plant.
The considered scheduling problem has the ob-
jective of minimizing the sum of the following cost
terms:
1. weighted number of tardy jobs;
2. variable purchase costs;
3. fixed purchase costs.
The second cost term is weighted by the unitary cost
of raw materials and also by a coefficient which is in-
versely proportional to the time instant at which the
supply is needed. This is justified by the fact that the
possible acquisitions of new components from exter-
nal suppliers are better planned if such components
are needed as late as possible in the planning period
[0,T]); in this way, indeed, the additional supply or-
ders have a greater chance of being delivered on time
since they are issued at the beginning of the planning
period. The third cost term is proportional to the num-
ber of orders from the external suppliers and it is jus-
tified by the convenience of grouping the acquisition
ICINCO 2016 - 13th International Conference on Informatics in Control, Automation and Robotics
112
of the different kinds of raw components. The first
cost term is the primary objective and therefore it is
weighted by a parameter, denoted as G, whose value
is chosen “sufficiently large” to give to this term a
lexicographic priority with respect to the other terms.
In addition, the second and the third term are respec-
tively weighted by coefficients h
r
, r R, and H: the
former represents the cost of a unit of raw material
of type r purchased from external suppliers, whereas
the latter represents the fixed cost to be paid for any
distinct purchase of raw materials from external sup-
pliers.
Such a problem can be modelled as a mixed inte-
ger mathematical programming problem as described
in the following subsection.
2.1 The Mathematical Programming
Formulation
The system under concern can be viewed as a
discrete-event system whose state changes at some
discrete instants (of the continuous-time axis) corre-
sponding to the beginning of operations of jobs on
the various machines of the flexible flow shop. In this
regard, the state is basically represented by the list
of jobs that have started their execution (one list per
each stage of the system), and by the inventory level
for each class of raw materials; the system state is up-
dated at each of the discrete instants, by adding the
job that starts its processing (at a certain stage) to the
list of started jobs, and by updating the inventory lev-
els according to the number of raw materials required
by the starting job, the size of the lots of refurbished
raw components that are possibly provided by the re-
manufacturingplant, and the amount of new raw com-
ponents that are purchased from the external suppliers
if necessary. Thus, a continuous-time event-based ap-
proach is here proposed to solve the scheduling prob-
lem. The number of events is N = S· |J|, being |J| the
number of jobs to be processed in the flow shop.
The decision variables are the following:
s
is
R, s
is
0, with i J, s {1,.. .,S}, repre-
sents the start time of job i at stage s;
x
ijsk
{0,1}, with i, j J, i 6= j, s {1,..., S},
k {1,... ,N
a
s
}, is a binary sequencing variable
such that x
ijsk
= 1 if both job i and job j are pro-
cessed at stage s by machine k and job i is se-
quenced before job j, namely
x
ijsk
=
1, if i, j assigned to k and s
is
< s
js
0, otherwise
(1)
w
isk
{0,1}, with i J, s {1,... , S}, k
{1,. . .,N
a
s
}, is a binary assignment variable such
that w
isk
= 1 if job i is processed by machine k at
stage s;
u
i
{0,1}, with i J, is a binary variable such
that u
i
= 1 if job i completes after its due date
(tardy job), namely
u
i
=
1, if s
iS
+
N
a
S
k=1
P
iSk
w
iSk
> T
0, otherwise
(2)
y
ise
{0, 1}, with i J, s {1,...,S}, e
{1,. . .,N}, is a binary variable associating events
with jobs; y
ise
= 1 if the processing of job i at
stage s starts in correspondence of event e;
t
e
R, t
e
0, with e {1, ...,N}, represents the
time of occurrence of event e;
z
he
{0,1}, with h {1, . ..,H} , e {1,... ,N},
is a binary variable such that z
he
= 1 if the time
of occurrence of event e does not precede the h-th
arrival of raw components from the remanufactur-
ing system, namely
z
he
=
1, if t
e
D
h
0, otherwise
(3)
a
re
R, a
re
0, with r R, e {1,..., N}, repre-
sents the number of new raw components of type
r, purchased from the external suppliers, arriving
at instant t
e
(since needed to execution of the job
associated with event e);
α
e
{0,1}, with e {1,... ,N}, is a binary vari-
able such that α
e
= 1 if any new component from
the external suppliers arrives at instant t
e
, namely
α
e
=
1, if
rR
a
re
> 0
0, otherwise
(4)
i
re
R, i
re
0, with r R, e {1,... , N}, rep-
resents the number of raw components of type r
in the inventory after the occurrence of event e at
instant t
e
(inventory level).
The mathematical programming model is the fol-
lowing.
min
G
iJ
W
i
u
i
+
N
e=1
h
(N e)
rR
h
r
a
re
i
+ H
N
e=1
α
e
(5)
subject to:
a
re
Vα
e
, r R, e {1,...,N} (6)
s
iS
+
N
a
S
k=1
P
iSk
w
iSk
Bu
i
T , i J (7)
A Matheuristics for the Single-period Lot Scheduling with Component Availability Constraints in a Partially Closed
Manufacturing/Remanufacturing System
113
s
is
s
i(s1)
+
N
a
(s1)
k=1
P
i(s1)k
w
i(s1)k
,
i J, s {2,..., S} (8)
s
js
s
is
+ P
isk
w
isk
B(1 x
ijsk
) ,
i, j J , i 6= j, s {1, ..., S},
k {1,. . .,N
a
s
} (9)
N
a
s
k=1
w
isk
= 1 , i J, s {1,...,S} (10)
2(x
ijsk
+ x
jisk
) w
isk
+ w
jsk
,
i, j J , i 6= j, s {1, ..., S},
k {1,. . .,N
a
s
} (11)
x
ijsk
+ x
jisk
w
isk
+ w
jsk
1 ,
i, j J , i 6= j, s {1, ..., S},
k {1,. . .,N
a
s
} (12)
N
e=1
y
ise
= 1 , i J, s {1,..., S} (13)
iJ
S
s=1
y
ise
= 1 , e {1, . ..,N} (14)
s
is
t
e
B(1 y
ise
) ,
i J , s = 1,... , S, e {1,... , N} (15)
t
e
s
is
B(1 y
ise
) ,
i J , s = 1,... , S, e {1,... , N} (16)
t
e
D
h
B(1 z
he
) ,
h {1,..., H} , e {1,.. . ,N} (17)
t
e
D
h
+ Bz
he
,
h {1,..., H} , e {1,.. . ,N} (18)
z
he
z
h(e1)
,
h {1,..., H} , e {2,.. . ,N} (19)
t
e
t
e1
, e { 2,..., N} (20)
i
re
= I
0
r
+
H
h=1
M
rh
z
he
+
e
f=1
a
rf
e
f=1
iJ
S
s=1
Q
isr
y
isf
,
r R, e {1,. ..,N} (21)
The first term of the objective function (5) corre-
sponds to the weighted number of tardy jobs; the sec-
ond term is the acquisition cost from external suppli-
ers of new components, where the multiplier N e pe-
nalizes the acquisitions needed early in the planning
period; the third term is the fixed acquisition cost. The
coefficient G is fixed much greater than parameters h
r
,
r R, and H such that the weighted number of tardy
jobs is the primary objective. Constraints (6) allow
determining the number of purchases from the exter-
nal suppliers, whereas constraints (7) establish which
jobs are tardy. Constraints (8) impose that the start
time of any job i at a stage s must be greater than
the sum of the start time and the processing time of
job i at the previous stage. Constraints (9) define the
sequence of the jobs on the machines at each stage;
given two distinct jobs i and j, if i precedes j on
machine k at stage s then x
ijsk
= 1 and (9) becomes
s
js
s
is
+ P
isk
w
isk
, imposing that job j starts after the
completion of job i; otherwise, if x
ijsk
= 0, the rel-
evant constraint is always satisfied. Constraints (10)
guarantee that each job at each stage is assigned to
one and only one machine. The joint role of con-
straints (11) and (12) is that of forcing a sequence or-
der among pairs of jobs that are assigned to the same
machine; indeed, if two jobs i and j are both assigned
to machine k, the variables w
isk
and w
jsk
are equal to
1 and the constraints (11) and (12) are equivalent to
x
ijsk
+ x
jisk
= 1, then either i precedes j or j pre-
cedes i and machine k. Constraints (13) and (14)
state that only one event can be associated with the
start time of a job and viceversa. Constraints (15)
and (16) associate the timee of occurrence of the
events with the start times of the jobs (when y
ise
= 1
such constraints reduce to t
e
= s
is
). Similarly, con-
straints (17) and (18), together with (19), associate the
arrivals of components from the remanufacturing sys-
tems to events. Constraints (20) impose the time se-
quencing of events and finally, constraints (21) com-
pute the inventory levels for the components in corre-
spondence of each event.
3 THE PROPOSED
MATHEURISTICS
The MIP model presented in the previous section can
be used to solve the considered class of problems
only in the case of small instances. This problem
indeed is NP-hard since it generalizes the problem
of minimizing the number of tardy jobs on parallel
machines proved to be NP-hard in (Garey and John-
son, 1990). For this reason, in this paper an heuris-
tic algorithm to solve the problem even in the case of
ICINCO 2016 - 13th International Conference on Informatics in Control, Automation and Robotics
114
STEP 1STEP 2STEP 3
PSfrag
Goal
Goal
Goal
Objective (min)
Objective (min)
Objective (min)
Fixed variables
Fixed variables
Fixed variables
Solution (main)
Solution (main)
Solution (main)
Job assignment
Makespan
none
w
isk
, i J, s {1, .. ., S} , and k {1, .. ., N
a
s
}
Job sequencing
Weighted number of tardy jobs
w
isk
, i J, s {1, .. ., S} , and k {1, .. ., N
a
s
}
x
ijsk
, i, j J, s {1,. .. ,S}, and k {1,. .. ,N
a
s
}
Components acquisition
Weighted number of tardy jobs and (variable and fixed) purchase costs
w
isk
and y
ise
, i J, s {1,...,S}, k {1,...,N
a
s
}, and e N
a
re
, r R and e {1,. .. ,N}
Figure 2: The structure of the proposed matheuristics.
medium and large instances is proposed. In partic-
ular, a matheuristic algorithm based on a decompo-
sition heuristics implemented through the interoper-
ation of different mathematical programming models
is introduced in the following.
The basic idea is to decompose the problem de-
cisions in a series of sub-problems. First of all, the
decisions on the assignment of the jobs to the avail-
able machines are separated from the ones relevant
to the sequencing of the jobs on the machines; sec-
ondly, the timing decisions are taken considering the
component availability constraints only after having
assigned and scheduled the jobs at the various stages
of the flexible flow shop. Such an approach appears
promising having adopted a common due-date model
and being the manufacturing/remanufacturing system
partially closed. As a matter of fact, a strategy for
minimizing the number of tardy jobs in case of a com-
mon due-date can be that of balancing the utiliza-
tion of the machines through a suitable job assign-
ment and then scheduling the jobs trying to minimize
the makespan. Moreover, since it is assumed always
possible to acquire the needed additional new compo-
nents from the external suppliers and being the penal-
ization of such additional supplies a secondary objec-
tive, then it is reasonable to consider the component
availability constraints only in the final timing sub-
problem.
More specifically, the proposed matheuristics con-
sists of three steps:
1. the jobs are assigned to the machines at each stage
of the flexible flow shop with the objectiveof min-
imizing the makespan; this problem is solved in
a short computation time by a very simple MIP
model, presented in Section 3.1, which assigns the
jobs on the basis of their processing times P
isk
; the
solution of this step is used to fix the values of
the binary assignment variables, i.e., w
isk
, i J,
s {1,..., S}, and k {1,...,N
a
s
};
2. the jobs are scheduled on the machines, at each
stage of the flexible flow shop, without consid-
ering the component availability constraints; to
this end a simplified version of the MIP model
shown in Section 2.1, as reported in section 3.2,
is considered: since the component availability
constraints are not taken into account, only the
weighted number of tardy jobs is minimized and
constraints (6) and (13)÷(21) become not nec-
essary; in addition, w
isk
are no longer decision
variables since they are fixed to the values ob-
tained by step 1; the solution of such a model
provides the values of the binary sequencing vari-
ables, i.e., x
ijsk
, i, j J, s {1,..., S}, and
k {1,. . .,N
a
s
};
3. the possible additional component acquisitions
from the external suppliers are determined on the
basis of the component availability constraints; at
this step the MIP model presented in section 2.1
is solved, having fixed the values of the binary as-
signment variables w
isk
, i J, s {1,.. . ,S},
and k {1,. . .,N
a
s
}, obtained at step 1 and also
the values of the variables y
ise
, i J, s
{1,. . .,S}, and e {1, ...,N}, computed accord-
ing to the overall order among all the job oper-
ations determined by the values of the start time
variables s
is
, i J, s {1,... ,S}, found at step
2.
Such an approach, which is depicted in Figure 2, does
not guarantee to find an optimal solution; however,
the preliminary experiments discussed in Section 5
show that the proposed heuristics is able to find in
minutes solutions better than the ones yielded by the
MIP solver in one hour.
A Matheuristics for the Single-period Lot Scheduling with Component Availability Constraints in a Partially Closed
Manufacturing/Remanufacturing System
115
3.1 The Model for Step 1
The MIP model used at the first step of the proposed
matheuristics to determine the assignment of the job
operations to the machines for each stage of the flexi-
ble flow shop is the following.
min C
max
(22)
subject to:
C
max
c
sk
,
s {1,..., S} , k {1,.. .,N
a
s
} (23)
c
sk
=
iJ
P
isk
w
isk
,
s {1,..., S} , k {1,.. .,N
a
s
} (24)
N
a
s
k=1
w
isk
= 1 , i J, s {1,...,S} (25)
In this model, C
max
denotes the makespan, whereas
c
sk
R, c
sk
0, with s {1,... , S}, k {1,... , N
a
s
},
is a decision variable representing the completion
time of the last operation on machine k at stage s.
The objective function (22) corresponds to the min-
imization of the makespan, which is defined through
constraints (23). Constraints (24) compute, for each
stage of the flexible flow shop, the sum of the pro-
cessing times of all operations assigned to a certain
machine, i.e., the completion time of the last opera-
tion on such machine. Finally, constraints (25) im-
pose that each job is assigned, at each stage, to one
and only one machine.
3.2 The Model for Step 2
The MIP model exploited at the second step for se-
quencing the operations of jobs on the machines, at
each stage of the flexible flow shop, disregarding the
componentavailability constraints, is illustrated in the
following.
min
G
iJ
W
i
u
i
(26)
subject to (7)÷(12).
The objective function (26) minimizes the
weighted number of tardy jobs. This model cor-
responds to a relaxation of the MIP model in Sec-
tion 2.1, where only the variables relevant to sequenc-
ing decisions are used, having fixed the assignment
variables and neglecting the ones used to model the
component availability constraints that link the job
operations to the events.
3.3 The Model for Step 3
The MIP model introduced in section 2.1 is employed
at the last step of the matheuristics to determines both
the timing of the job operations and the acquisitions
from external suppliers of the different types of com-
ponents possibly needed to process the job operations,
on the basis of the availability and the usage of the
different components.
As previously pointed out, this model is solved at
step 3 having fixed the variables w
isk
and y
ise
accord-
ing to the solutions obtained at steps 1 and 2, respec-
tively. In particular, the values of the variables y
ise
are
determined first sorting the values of the N variables
s
is
in non decreasing order (ties are broken arbitrar-
ily) and then assigning the events e {1, . ..N} to the
job operations identified by the pairs (i,s) according
to the obtained order. Finally, if an event e is assigned
to the job operation (i,s), then it is fixed y
ise
= 1, oth-
erwise y
ise
= 0.
4 EXAMPLE
A small instance of the scheduling problem is con-
sidered in this section with the aim of showing the
application of the proposed matheuristics.
Let the manufacturing system consists of three
stages (S = 3) and two machines per stage (N
a
s
= 2
s {1,...,S}). The system processes two types of
raw components (R = {1,2}) and the remanufactur-
ing plant provides two lots of refurbished as-good-
as-new” raw components (H = 2); the lot sizes M
rh
and the arrival times D
h
are reported in table 1 (which
also includes the initial inventoryof rawcomponents).
Five jobs J
1
,... ,J
5
must be carried out by the
manufacturing system. Values Q
isr
and P
isk
, repre-
senting respectively the number of raw components r
required to execute job i at stage s and the process-
ing time of job i when processed by machine M
a
ks
at
stage s, are reported in tables 2 and 3. Moreover, the
weights of jobs in the cost function are W
1
= 0.169,
W
1
= 0.262, W
1
= 0.123, W
1
= 0.2, and W
1
= 0.246.
Finally, the common due-date for all jobs is T = 25.
This small instance of the problem has been
solved through the proposed matheuristics. At the
first step, jobs are assigned to machines in all the
stages of the flexible flow shop; the solution of this
step is reported in table 4. In the second step, jobs are
sequenced on the assigned machines without taking
into account the component availability constraints;
the solution of this step is reported in table 5.
In the second step of the matheuristics, the event
assignment is also performed, that is, the assignment
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116
Table 1: Example Lots of refurbished raw components
(D
h
and M
rh
) and initial inventory levels (I
0
r
).
I
0
r
h = 1 h = 2
D
h
M
rh
D
h
M
rh
r = 1 500
17.92
221
4.99
160
r = 2 200 195 106
Table 2: Example Number of raw components that are
required to execute jobs (Q
isr
).
s = 1 s = 2 s = 3
r = 1 r = 2 r = 1 r = 2 r = 1 r = 2
J
1
88 66 77 99 55 55
J
2
51 68 153 68 17 51
J
3
56 64 48 56 32 24
J
4
91 91 117 52 65 52
J
5
16 128 112 32 64 80
Table 3: Example – Processing times of jobs (P
isk
).
s = 1 s = 2 s = 3
M
a
1,1
M
a
2,1
M
a
1,2
M
a
2,2
M
a
1,3
M
a
2,3
J
1
4.35 10.04 6.03 7.33 2.85 5.99
J
2
11.66 5.52 9.98 6.35 9.89 6.24
J
3
5.24 5.82 7.41 6.96 5.72 3.13
J
4
10.44 7.9 4.19 9.11 8.26 11.24
J
5
14.09 10.17 5.03 8.03 15.1 11.47
Table 4: Example – Solution of step 1: job assignment.
s = 1 s = 2 s = 3
J
1
M
a
1,1
M
a
1,2
M
a
1,3
J
2
M
a
2,1
M
a
2,2
M
a
2,3
J
3
M
a
2,1
M
a
2,2
M
a
1,3
J
4
M
a
2,1
M
a
1,2
M
a
1,3
J
5
M
a
1,1
M
a
1,2
M
a
2,3
Table 5: Example – Solution of step 2: job sequencing.
M
a
1,s
M
a
2,s
s = 1 J
1
J
5
J
2
J
3
J
4
s = 2 J
1
J
4
J
5
J
2
J
3
s = 3 J
1
J
3
J
4
J
2
J
5
of job operations to the ordinal numbers of the event
from 1 to 15 (in general, from 1 to S· |J|) according
to the job operation start times. Such an association
“(event #, job start)”, which is reported in table 6, is
obviously made in accordance with the job sequenc-
ing shown in table 5. The solution of step 2 is also
represented by the Gantt chart illustrated in figure 3.
The solution in figure 3 can be not feasible with
respect to the component availability constraints. At
step 3, such constraints are actually considered; how-
ever, in order to speed up the determination of the op-
Table 6: Example – Solution of step 2: event assignment.
Event #
Associated job start
job stage
1
J
1
1
2 J
2
1
3 J
1
2
4 J
5
1
5 J
2
2
6 J
3
1
7
J
1
3
8
J
4
1
9 J
3
2
10 J
2
3
11 J
3
3
12 J
4
2
13 J
5
2
14
J
4
3
15
J
5
3
timal solution, the assignment/sequencing of the jobs
and the assignments of the events to the job opera-
tions are fixed to the values provided by the first two
steps of the matheuristics. Therefore, the solution of
step 3, which is illustrated through the Gantt chart
in figure 4, is very similar to the one obtained after
step 2; however, it is easy to notice that some opera-
tions are delayed to satisfy the component availabil-
ity constraints. In addition, such delays are also due
to the fact that, when new raw components must be
purchased from the external suppliers to satisfy the
component availability constraints, it is convenient to
delay the purchase as much as possible.
In conclusion, the Gantt chart illustrated in fig-
ure 4 represents the final solution of the proposed
matheuristics.
5 COMPUTATIONAL
EXPERIMENTS
The performances of the proposed matheuristics were
evaluated considering a set of 25 problem instances
that were randomly generated. Instances from #1 to
#10 are relevant to a flow-shop of 3 stages, includ-
ing 3, 2, and 3 machines for the instances #1-5 and
4, 3, and 4 machines for the instances #6-10. The
number of jobs is 10 for the instances #1-5 and 20 for
the instances #6-10. The instances from #11 to #25
are relevant to a flow-shop of 4 stages, including 4, 3,
4, and 3 machines for all instances; in this case, the
number of jobs is 30 for the instances #11-15, 60 for
the instances #16-20, and 90 for the instances #21-
25. According to such characteristics the considered
A Matheuristics for the Single-period Lot Scheduling with Component Availability Constraints in a Partially Closed
Manufacturing/Remanufacturing System
117
STAGE 1STAGE 2STAGE 3
M
a
1,1
M
a
2,1
M
a
1,2
M
a
2,2
M
a
1,3
M
a
2,3
J
1
J
1
J
1
J
2
J
2
J
2
J
3
J
3
J
3
J
4
J
4
J
4
J
5
J
5
J
5
0 10
20
30
40
due date
Figure 3: Gantt chart of the solution after step 2.
STAGE 1STAGE 2STAGE 3
M
a
1,1
M
a
2,1
M
a
1,2
M
a
2,2
M
a
1,3
M
a
2,3
J
1
J
1
J
1
J
2
J
2
J
2
J
3
J
3
J
3
J
4
J
4
J
4
J
5
J
5
J
5
0 10
20
30
40
due date
Figure 4: Gantt chart of the solution after step 3 (final solution of the matheuristics).
Table 7: Comparison between Cplex and the proposed matheuristics.
Instance Njobs Nstages
Cplex matheuristics
Comparison
Opt CPUOpt Obj NumTard CPU
1 10 3 92.20 38.6 96.10 9 1.3 +4.2%
2 10 3 99.90 2.7 99.90 10 2.9 0.0%
3 10 3 90.20 61.9 100.20 10 2.0 +11.1%
4 10 3 91.30 762.6 95.60 9 2.9 +4.7%
5 10 3 88.00 345.6 94.40 9 3.8 +7.3%
6 20 3 83.40 3600.1 51.90 10 47.1 37.8%
7 20 3 79.40 3604.3 40.80 8 43.1 48.6%
8 20 3 69.80 3600.9 40.00 8 114.3 42.7%
9 20 3 86.16 3600.4 51.81 10 8.3 39.9%
10 20 3 91.13 3600.4 45.88 8 11.9 49.7%
11 30 4 100.87 3601.6 48.79 14 306.9 51.6%
12 30 4 102.30 3600.9 48.15 11 312.7 52.9%
13 30 4 94.61 3600.8 52.06 13 308.6 45.0%
14 30 4 110.20 3599.7 60.72 13 305.7 44.9%
15 30 4 92.35 3600.6 37.50 8 308.3 59.4%
16 60 4 152.56 3603.2 95.82 29 607.7 37.2%
17 60 4 145.46 3603.3 84.48 22 608.2 41.9%
18 60 4 150.03 3604.6 89.78 27 556.5 40.2%
19 60 4 145.12 3604.5 85.11 26 559.6 41.4%
20 60 4 157.49 3603.9 92.53 23 608.9 41.2%
21 90 4 3614.0 169.57 51 627.5
22 90 4 3614.1 148.80 42 631.9
23 90 4 3625.6 154.03 45 631.2
24 90 4 3623.4 136.47 34 631.4
25 90 4 3623.3 172.11 48 633.5
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Table 8: Values obtained in each step of the proposed matheuristics.
Instance Njobs Nstages
step 1 step 2 step 3
Cmax CPU WNTJ CPU OverallObj CPU
1 10 3 33.15 0.7 96.10 0.5 96.10 0.0
2 10 3 43.18 1.7 99.90 0.7 99.90 0.5
3 10 3 31.41 0.9 100.20 0.5 100.20 0.7
4 10 3 40.93 0.7 95.60 0.7 95.60 1.5
5 10 3 35.66 2.1 94.40 0.9 94.40 0.7
6 20 3 39.14 0.4 51.90 46.1 51.90 0.6
7 20 3 37.57 0.5 40.80 42.0 40.80 0.6
8 20 3 35.14 0.5 40.00 113.1 40.00 0.7
9 20 3 40.18 0.3 51.60 7.8 51.81 0.2
10 20 3 35.22 0.5 45.00 10.7 45.88 0.6
11 30 4 60.85 2.6 46.90 300.5 48.79 3.8
12 30 4 52.85 3.0 39.70 300.2 48.15 9.4
13 30 4 58.77 2.1 49.40 300.6 52.06 5.9
14 30 4 62.43 1.4 53.50 300.1 60.72 4.2
15 30 4 45.60 0.9 33.40 300.5 37.50 6.9
16 60 4 105.93 1.6 48.40 300.6 95.82 305.6
17 60 4 109.33 2.4 41.80 300.4 84.48 305.5
18 60 4 108.80 1.5 44.40 300.3 89.78 254.7
19 60 4 111.27 2.5 44.80 300.5 85.11 256.6
20 60 4 98.34 3.9 40.10 300.8 92.53 304.2
21 90 4 166.33 1.2 57.30 300.6 169.57 325.8
22 90 4 170.70 8.7 46.70 300.6 148.80 322.6
23 90 4 170.84 5.1 50.90 300.3 154.03 325.8
24 90 4 156.82 5.0 40.00 300.3 136.47 326.1
25 90 4 165.91 4.9 52.10 300.4 172.11 328.2
instances require to schedule a number of operations
ranging from 30 to 360. In addition, the MIP model
of Section 2.1 needs 2,074 variables (1,830 binary)
and 4,560 constraints for the smallest instances and
247,414 variables (244,530 binary) and 602,820 con-
straints for the largest ones.
The IBM ILOG Cplex 12.6.2 MIP Solver was
used to solve the MIP model presented in section 2.1
imposing a time limit of 1 hour. The solver was able
to optimally solve only the instances #1-5, whereas
it reached the time limit for the instances #6-20, and
it did not find any feasible solution for the instances
#21-25. The results obtained with Cplex are reported
in the columns 4-5 of Table 7.
The three MIP models which constitute the
matheuristics were also implemented with the solver
Cplex 12.6.2. In this case, a time limit of 5 min-
utes was imposed for solving each step. The re-
sults obtained with the proposed heuristic approach
are reported in the columns 6-9 of Table 7. As it
can be noted in this table, the instances #1-5 in the
first group were optimally solved by Cplex, whereas
the matheuristics obtained the optimal solution for
instance #2 only; however, for the other four in-
stances in the first group the matheuristics found a
slightly larger value (with an average percentage de-
viation, given by the ratio (matheuristics
result
solver
result)/solver result, of 6.8%). Differently,
the matheuristics found for the instances #6-20 very
better solutions than the ones provided by Cplex in
1 hour. As it can be observed in Table 7, for the
instances #6-20 the matheuristics produced solutions
whose overall average percentage deviation from the
ones yielded by Cplex is 45.0%. No comparison is
possible for the instances #21-25 since the MIP solver
was unable to find a feasible solution in 1 hour. For
what concern the computational burden of the pro-
posed approach, it is worth noting that the solution of
the MIP model at step 2 reached the 5 minutes time
limit for the instances #11-25, whereas the solution of
the MIP model at step 3 reached the same time limit
for the instances #16-25. The details about the solu-
tions provided at each step of the matheuristics and
the corresponding CPU times are reported in Table 8
(note that the CPU times reported for the step 3 for
some of the instances #16-25 are greater than 5 min-
utes as they include also the time needed to generate
the MIP model).
A Matheuristics for the Single-period Lot Scheduling with Component Availability Constraints in a Partially Closed
Manufacturing/Remanufacturing System
119
On the basis of these results, it can be concluded
that the proposed approachcan be an effectivemethod
to find, in an acceptable short time, good solutions for
the problem under concern.
6 CONCLUSIONS
In this work, an original matheuristics is proposed to
solve a scheduling problem in an integrated manufac-
turing/remanufacturingsystem. The matheuristics de-
composes the decisions related to the assignment of
the jobs to the machines, the sequencing of the jobs
on the machines, and the determination of the addi-
tional acquisitions from external suppliers of needed
components into three separate (but interoperating)
mathematical programming models. In this way, it
is possible to obtain, in short computation times, so-
lutions to medium-large instances of the problem that
are much better than the ones yielded within one hour
of computation by solving the complete MIP model.
Besides, an event-based formulation is proposed for
the considered problem as it is suitable to represent
the discrete-event dynamic of the system under con-
cern and the interactions among the remanufacturing
plant, the external suppliers and the manufacturing
system.
The presented experimental analysis, based on 25
randomly generated instances of the problem of dif-
ferent sizes, points out the effectiveness of the pro-
posed algorithm when few minutes are available to
generate a solution. A more extensive tests will be
performed in the next developments of this research.
For what concerns future research directions, cur-
rent activities on this topic are mainly related to the
scheduling of disassembling and refurbishing activ-
ities in the remanufacturing system, taking into ac-
count uncertainties which naturally characterize the
return of finished products. In addition, a next step
of this research will consider the problem of planning
the remanufacturing and manufacturing activities in
an integrated way, so defining also the schedule of the
remanufacturing system to better match the compo-
nent requirements issued by the manufacturing sys-
tem to satisfy the customer demand.
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