Almgren, Torgny

Patriksson, Michael

Strömberg, Ann-Brith

Thörnblad, Karin

Chalmers Publication Library

1A Time-indexed Formulation of a Flexible Job Shop
Problem with a Limited Number of Fixtures
Karin Thörnblad
1,2
, Ann-Brith Strömberg
1
, Mihael Patriksson
1
and Torgny Almgren
2
1
Chalmers University of Tehnology and the University of Gothenburg, Sweden
karint, anstr, mipathalmers.se
2
Volvo Aero Corporation, Department of Logistis Development, Sweden
karin.thornblad, torgny.almgrenvolvo.om
Keywords: exible job shop sheduling, time-indexed formulation, mathematial opti-
mization, xture, release date, due date, tardiness, MILP.
1 Introdution
The multitask ell (MTC) at Volvo Aero Corporation is a exible job shop ontaining
ten resoures aimed at being exible with regard to both produt mix and proessing types,
inreasing the degree of mahine utilization, and reduing produt lead times, ompared
with the ordinary job shops at the prodution site. The MTC is intended to arry out a
large variety of jobs, sine ve of the ell's resoures are multi-purpose mahines that are
able to proess three dierent types of operations (turning, milling, and drilling). Eah
part to be proessed in the MTC follows a spei routing through the set of resoures,
onsisting of three to ve so-alled route operations, starting and ending by the mounting
and removing of xtures at one of the three set-up stations. There are storage areas inside
the ell for the storage of parts before and between proessing.
Eah part typially visits the MTC multiple times on its way to ompletion. One suh
visit to the MTC is alled a job. Eah job oupies a xture, speially manufatured for
eah spei job type. The number of xtures of eah type is limited. As eah job an
be proessed only in a subset of the multitask mahines, the sheduling of the MTC is a
omplex ombinatorial problem, a so-alled exible job shop problem.
2 Related Work
The exible job shop problem (FJSP) is an extension of the job shop problem, in the
sense that eah operation may be sheduled in more than one of the mahines; see Bayka-
soglu and Özbakir (2012). Time-indexed formulations using deision variables whih equal
1 if the orresponding job starts at a spei disrete time step, and 0 otherwise, are
found in Sousa and Wolsey (1992) for non-preemptive single mahine sheduling problems,
and more generally in Wolsey (1997) for prodution planning and sheduling problems.
The formulations using variables for eah disrete time period lead to very large models in
terms of numbers of both onstraints and variables, but typially yield better lower bounds
than other mixed integer linear programming (MILP) formulations of sheduling problems;
see van den Akker et al. (2000).
The operations researh literature omprises however very few time-indexed formula-
tions of the job shop problem, we have found one (Azem et al. 2007) and none of the
exible job shop problem. The most ommon way of formulating the FJSP is to use vari-
ables introdued by Manne (1960) for the jobs' starting times along with deision variables
for the ordering of jobs, dened as yjq equals 1 if job j preedes job q, and 0 otherwise.
Some examples of so-alled Manne family models for the job shop problem and the FJSP
an be found in Özgüven et al. (2010), Fattahi et al. (2007), and Azem et al. (2007). In
2Thörnblad et al. (2010) we present a Manne family model of the problem of sheduling the
multitask ell.
3 The mahining problem with xture onstraints
We onsider here a subproblem of the problem of sheduling all ten resoures of the
multitask ell, namely the problem of nding an optimal sequene of operations for eah of
the ve multi-purpose mahines. The optimal solution of this so-alled mahining problem
an then be used as input data to a model whih generates a feasible shedule for all ten
resoures; see Thörnblad (2011).
3.1 Problem denition
The mahining problem an be dened as a set of n jobs, J = 1, . . . , n to be proessed
on the set K˜ of multipurpose mahines during p˜j time units. Eah mahine an only proess
one job at a time and eah job is allowed to be proessed only in a subset of the multitask
mahines; a parameter λjk is valued 1 if job j an be proessed in resoure k, 0 otherwise.
Some of the jobs are subjet to preedene onstraints, sine they are to be proessed on the
same physial part; hene, for some jobs the orresponding part is inside or on its way to
the MTC for the proessing of a preeding job in the routing, before making another round
in the fatory and, nally, reahing the MTC for the proessing of the job in question. The
pairs (j, q) of all suh jobs adjaent in the routing form the produt set Q ⊂ J × J . For
eah (j, q) ∈ Q, the planned lead time between the ompletion of job j and the start of job
q, is denoted by v˜mjq .
The planning horizon of the shedule is divided into T + 1 intervals, eah of length
ℓ hours. The index u ∈ T = {0, 1, . . . , T} denotes the interval starting times. Sine the
resoures are often oupied by the proessing of previous jobs at time 0, the parameter a˜k
is introdued, denoting the rst time resoure k is available. Eah job has been assigned a
release date, r˜mj , denoting the time when the job is available for proessing, and a due date
d˜j . The parameters p˜
am
j and p˜
pm
j denote the sum of the proessing times of all operations
preeding and sueeding the mahining operation of the job, respetively.
Eah job oupies a xture during the whole visit in the MTC. Eah xture type f ∈ F
is speially designed and an only be used for a subset Sf of jobs. Sine they are very
expensive, only αf xtures of eah type are available.
The value of the parameter T has to be large enough suh that the time horizon
[0, (T+1)ℓ] ontains an optimal shedule. A small value of T is, however desirable, sine this
means that the omputation times beome shorter. This is due to the fat that the number
of variables and onstraints in the time-indexed formulation is a multiple the number of
time intervals. We determine a suitable value of T using a heuristi; see Thörnblad (2011).
3.2 A time-indexed formulation
The mahining problem with xture onstraints an be formulated as that to
minimize
∑
j∈J
(ajsj + bjhj), (1a)
subjet to
∑
k∈K˜
∑
u∈T
xjku = 1, j ∈ J , (1b)
∑
u∈T
xjku ≤λjk, j ∈ J , k∈K˜, (1)
3∑
j∈J
u∑
ν=(u−p˜j+1)+
xjkν ≤ 1, k ∈ K˜, u ∈ T (1d)
∑
k∈K˜


u∑
µ=0
xjkµ −
u+v˜pm
jq∑
ν=0
xqkν

≥ 0, (j, q) ∈ Q, u = 0, . . . , T − v˜pmjq , (1e)
xjku = 0, (j, q) ∈ Q, k ∈ K˜, u = T − v˜
pm
jq , . . . , T, (1f)
∑
j∈Sf
∑
k∈K˜
min{u+p˜amj ,T}∑
ν=(u−(p˜j+p˜
pm
j
)+1)+
xjkν ≤αf , f ∈ F , u ∈ T , (1g)
∑
k∈K˜
∑
u∈T
uxjku + p˜
pm
j = sj, j ∈ J , (1h)
hj ≥ (sj − d˜j)+, j ∈ J , (1i)
xjku = 0, j ∈ J , k ∈ K˜, u = 0, . . . ,max{r˜
m
j ; a˜k}, (1j)
xjku ∈ {0,1}, j ∈ J , k ∈ K˜, u ∈ T , (1k)
where (u)+ := max{0;u}. The objetive (1a) is to minimize the weighted sum of the
variables ompletion times sj, and tardiness hj, with weights aj and bj, respetively. The
main variable in the formulation is the variable xjku, whih is 1 if job j starts in resoure
k at time step u, 0 otherwise. The onstraints (1b) and (1) ensure that every job j is
proessed exatly one and sheduled in an allowed resoure, respetively. The onstraints
(1d) regulate that only one job at a time is sheduled in a resoure k. The onstraints
(1e)(1f) ensure that job j is sheduled to start at least the planned lead time v˜
pm
jq before
the start of job q, for all pairs (j, q) ∈ Q. The apaity onstraints on the number of xtures
oupied at eah time interval are formulated in (1g). The onstraints (1h)(1i) dene the
ompletion times, sj and the tardiness, hj . The onstraints (1j) make sure that job j is
sheduled after its release date in an available resoure.
A hek to ensure that the hosen value of T is large enough and that the optimal
shedule found is indeed optimal for all larger values of T , it sues to hek whether if
T ≥ maxj∈J {sj}+ maxj∈J {p˜j} holds. If this is the ase, there is enough room at the end
of the shedule for any job to be sheduled in any other resoure, and hene a larger value
of T would not hange the optimal objetive value.
4 Computational results
Results from the model (1) with and without the xture onstraints (1g) have been
ompared with a Manne family model, the so-alled engineer's model (model (7) in Thörn-
blad (2011)), for six real senarios olleted from the MTC during the autumn of 2010, from
whih instanes were reated with 5 ≤ n ≤ 60. A limit of 10 000 seonds was set for the om-
putation time (loktime) and the omputations were arried out using AMPL-CPLEX12
on a omputer with two 2.66GHz Intel Xeon 5650, eah with six ores (24 threads), and
its total memory was 48Gbyte RAM. In Fig. 4 the mean CPU-times are plotted for the
engineer's model without xture onstraints (Thörnblad 2011), and the model (1) with
and without the onstraints (1g) inluded. The objetive weights hosen were aj = 1 and
bj = 10, j ∈ J , so that the tardiness beome the most important objetive.
The model (1) outperforms by far the engineer's model, whih ould only solve small
instanes with n ≤ 15. The time required to solve the model (1) was 6.5 minutes (lok-
time) on average for senarios with 45 jobs, whih we have estimated to be the largest
size of a realisti instane for the sheduling of the oming shift; the longest omputing
4Fig. 1. Mean CPU-times for the engineer's model without xture onstraints, and for the model
(1), as is and with the onstraints (1g) removed.
time required was 21.5 minutes (loktime). When the onstraints (1g) were removed, the
average omputation time for model (1) was 15 seonds (the longest was 60 seonds).
5 Conlusions
We have formulated a time-indexed model, whih outperforms by far the engineer's
model from Thörnblad (2011), whih is a model of the Manne family, with respet to
omputation times and the sizes of instanes that they are able to solve using standard
optimization software. The adding of the xture availability onstraints resulted in an
inrease of the omputation time by a fator of 23 on average for the time-indexed model.
However, these omputation times are still aeptable for nding a shedule for the oming
shift in a real appliation.
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A Time-indexed Formulation of a Flexible Job Shop Problem with a Limited Number of Fixtures

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