In order to carry out their orders of shoe soles, this company has a number of tasks T_1, ..., T_n of different lengths to be assigned to groups of machines.

Each group is operated by one worker (two in one case), and an operation cycle corresponds to injection, cooling, and removal of the sole. The time taken at each step varies from one order to another, and when starting a new task a machine needs to be tuned, which takes some extra time. Machines are working in parallel. At the moment the assignment is carried out empirically, and the problem proposed is to optimize the procedure

Cerdeira, J. Orestes

Charters, T.

Cruz, M.

Freitas, P.

Vasconcelos, P.B.

English

Mathematics in Industry Information Service Eprints Archive

Portuguese Study Groups’ Reports
Report on
“Scheduling in a factory”
Problem presented by ForEver Procalc¸ado at the
65th European Study Group with Industry
21st–24th April 2008
Centro de Matema´tica da Universidade do Porto
Portugal
March 24, 2009
Problem presented by: Eng. Rui Russo (ForEver Procalc¸ado)
Study group contributors:
J. Orestes Cerdeira, T. Charters,
M. Cruz, R. Duarte, A. Guedes de Oliveira,
C. Pinto, J. Rodrigues, P. Vasconcelos
Report prepared by:
J. Orestes Cerdeira1, T. Charters2,
M. Cruz3, P. Freitas4,
and P.B. Vasconcelos5
1Centro de Estudos Florestais and Dep. Mathematics, Instituto Superior de Agrono-
mia, Technical University of Lisbon (TU Lisbon), e-mail: orestes@isa.utl.pt
3Dep. of Mechanical Engineering, Instituto Superior de Engenharia de Lisboa (IPL),
Rua Conselheiro Emidio Navarro, 1, 1959-007 Lisboa, Portugal, and Centro de F´ısica
Teo´rica e Computacional, University of Lisbon, e-mail: tca@cii.fc.ul.pt
2Instituto Superior de Engenharia, Instituto Polite´cnico do Porto, e-mail:
mbc@isep.ipp.pt
4Dep. of Mathematics, Faculdade de Motricidade Humana (TU Lisbon) and
Group of Mathematical Physics of the University of Lisbon, Complexo Interdisciplinar,
Av. Prof. Gama Pinto 2, P-1649-003 Lisboa, Portugal, e-mail: freitas@cii.fc.ul.pt
5Centro de Matema´tica da Universidade do Porto and Faculdade de Economia da
Universidade do Porto, e-mail: pjv@fep.up.pt
Abstract
In order to carry out their orders of shoe soles, this company has a number
of tasks T1, . . . , Tn of different lengths to be assigned to groups of machines.
Each group is operated by one worker (two in one case), and an operation
cycle corresponds to injection, cooling, and removal of the sole. The time
taken at each step varies from one order to another, and when starting
a new task a machine needs to be tuned, which takes some extra time.
Machines are working in parallel. At the moment the assignment is carried
out empirically, and the problem proposed is to optimize the procedure.
2 ForEver: Scheduling in a factory
1.1 Introduction
During the manufacturing of shoe soles at ForEver, each operator controls
a machine which has to be set up everytime the production of a new model
begins. This is followed by a tuning process, after which production follows a
cycle consisting of the functioning of the machine proper, plus the operation
by a worker who injects material into the mould, runs the machine and
removes the produced sole.
This functioning is of a parallel nature, in the sense that machines op-
erate independently one from another and, for the process considered here,
are not part of a larger chain where they would need to wait for the input
from another machine, for instance.
The running of this process includes thus the assignment and scheduling
of the different sole models to different groups of machines and operators.
Figure 1.1: A typical report of orders and available equipment.
The table given in Figure 1.1 shows an example of a list of orders to-
gether with the time taken to carry out each one, the required quantities
and the corresponding deadlines. It also describes the number of machines
and operators which are available to carry out such tasks. The problem is
thus to decide on a sequence of task assignments satisfying certain objectives
related to deadlines, optimization of worker’s time, etc. More precisely, we
want to partition the orders into batches ensuring that the above restrictions
are met.
There are different time scales in this problem, the longest being the
deadlines for each order. The remaining ones are shorter and are related
to the making of a pair of shoe soles. These include the time that the
machine takes to make the sole, the manipulation time done by a worker
and a cycle time. All these time quantities are related to the making of
a specific pair of shoe soles, and can be added up to make what we have
called the processing time. There are also penalty times which measure the
time required to change a mold in the same machine and the (longer) time
Portuguese Study Groups’ Reports 3
required to change the mold to a different one. These two are the key critical
quantities that will, eventually, control the overall optimisation procedure
that we are looking for.
We begin by giving a mathematical description of the problem which
allows us to distinguish between the fundamental aspects and those which
can be discarded in a first approach. We then provide a heuristic approach
to solve this model and give an example which exemplifies how the method
works. We then end with some conclusions and recommendations.
2.1 Mathematical formulation of the problem
For each job j (a pair of shoe soles from a specific model and of a given size),
each machine m (a mold position), and each time instant t (t = 0, 1, . . .),
define a 0 − 1 variable xtjm which is equal to 1 if and only if machine m
starts to process job j at time t.
Let J denote the set of all jobs, pj the processing time (which includes
handling time) of job j, Ok ⊂ J the set of jobs of order k = 1, . . . , |O|, where
|O| is the number of different orders, and Dk the deadline of order k. Let
M denote the set of machines and rji the time to adapt a machine to start
processing job i immediately after finishing job j. Let also Js ⊂ J be the
set of jobs which uses the same mold s (i.e., the set of shoe soles of the same
specific model and of the same size), and ns the number of molds of type
s ∈ S.
The constraints can be stated as follows.
∑
m
∑
t
xtjm = 1, j ∈ J (2.1)∑
m
∑
t
(t+ pj)xtjm ≤ Dk, j ∈ Ok, k = 1, . . . , |O| (2.2)
xtjm + x
t′
im ≤ 1, m ∈M, i, j ∈ J, t ≤ t′ ≤ t+ pj + rji (2.3)∑
j∈Js
t+pj∑
t′=t
∑
m
xt
′
jm ≤ ns, s ∈ S, t = 0, 1, . . . (2.4)
xtjm ∈ {0, 1}, j ∈ J,m ∈M, t ∈ T. (2.5)
Constraints (2.1) state that each job j will start to be processed at
a unique time instant by a single machine. Constraints (2.2) ensure that
all orders are finished on time. Conditions (2.3) ensure that there are no
overlapping jobs assigned to the same machine. Inequalities (2.4) state that
the number of overlapping jobs that use the same mold does not exceeds the
number of molds. Finally, (2.5) are the 0–1 constraints on the variables.
The goal is to minimise the completion time of the last job. This can be
4 ForEver: Scheduling in a factory
settled by ∑
t
(t+ pj)xtjm ≤ F, j ∈ J,m ∈M (2.6)
minF (2.7)
Indeed, the left hand side in (2.6) is the time completion of job j on machine
m. Thus, the variable F to be minimized is an upper bound on the time
completion of the last job.
To find optimal solutions, even for moderate values of |J |, |M | and |T |
(number of instants of time that should be considered), is a task that is likely
to be impossible for an integer programming solver. Thus, a reasonable
option is to conceive some heuristic procedure that produces good solutions.
3.1 A heuristic approach
A solution X of the problem can be viewed as a sequence of jobs on each
machine, where each job occurs exactly once. Figure 3.1 represents a solution
for a problem with 31 jobs and 5 machines.
Lagos 36 Lagos 37
(a)
Figure 3.1: A representation of a solution with 31 jobs and 5 machines.
Define the cost of solutionX, denoted by c(X), to be the completion time
of the last job of X. To minimise c(X) is the goal of the model described in
the previous section.
The solution X is not feasible if either
1. the completion time of the last job from some order k exceeds Dk (the
deadline of order k), i.e., some inequality (2.2) is violated; or
2. the number of overlapping jobs that use the same mold, during some
period of time, exceeds the number of molds, i.e., at least one of the
inequalities (2.4) does not hold.
Portuguese Study Groups’ Reports 5
We suggest to deal with non feasibilities in a Lagrangian fashion (see
for example [4]) incorporating the constraints in the objective function with
associated multipliers. We illustrate this process with inequalities (2.2).
Consider, for each solution X and λ= (λjk) ≥ 0,
w(X,λ) = c(X) +
|O|∑
k=1
∑
j∈Ok
λjk(
∑
m
∑
t
(t+ pj)xtjm −Dk). (3.1)
Note that, if we set λjk = 0, for all jobs j ∈ Ok, whenever the completion
time of the last job of order k does not exceed Dk, we have w(X,λ)≥ c(X),
and w(X,λ) = c(X) if no inequality (2.2) is violated.
We propose to determine the set of orders k ∈ O whose completions
times exceed Dk (which is an easy task), and assign some positive value, say
λ, to λkj∗ , where j∗ is the last job of order k. In this way, when minimising
w we implicitly search for solutions which satisfy the constraints (2.2).
The above construction can be similarly developed to handle the con-
straints (2.4), using multipliers µ.
If X is an arbitrary solution, let Tk be the completion time of (the last
job of) order k and let O¯ be the set of orders k s.t. Tk > Dk. Let also
Ns = max{∑j∈Js∑t+pjt′=t ∑m xt′jm,with t ∈ T} and S¯ be the set of molds s
s.t. Ns > ns.
Once fixed positive values λ and µ, define the weight of solution X to be
w(X) = c(X) + λ
∑
k∈O¯
(Tk −Dk) + µ
∑
s∈S¯
(Ns − ns). (3.2)
To minimise w(X) a local search algorithm was designed. Local search
heuristics iteratively move from one solution to another, by exchanging some
of its components. The two basic ingredients are the rule describing the
neighbourhood of every solution X, i.e., the set N (X) of solutions which
can be directly obtained from X; and the criterium for choosing, for any
solution X, an element in N (X).
In our problem, the neighbourhood N (X) could be defined as the set of
all solutions that can be obtained by replacing the positions of two jobs in
X.
Figure 3.2 represents two solutions in N (X), where X is a solution of
an instance with 25 jobs and 5 machines
A local search heuristic which is particulary effective in finding near
optimal solutions is simulated annealing (see, for example, [1]).
At iteration i of a simulated annealing algorithm, a solution X ′ which
will be considered to replace the current solution X, is uniformly selected
from N (X). If this improves the objective function w, then the current
solution is replaced by X ′. If w(X ′) ≥ w(X), it may still be replaced but
only with probability p = exp
(
w(X)−w(X′)
Ti
)
, where Ti > 0, the solution
6 ForEver: Scheduling in a factory
X
X’
Figure 3.2: A solution X of an instance with 25 jobs and 5 machines and
two of its neighbours.
X remaining the current solution with probability 1 − p. The theoretical
convergence of the algorithm to an optimal solution, when {Ti} converges to
zero, is ensured given some quite general conditions regarding the structure
of the neighbourhood, the selection rule, and the rate of convergence of {Ti}.
A description of the algorithm for the problem in consideration follows.
Choose λ, µ > 0.
let X be an initial solution, N the number of iterations and i = 0.
1. choose T > 0 (the initial temperature);
2. define w(X) according to (3.2);
3. let X ′ be uniformly selected from N (X);
4. define w(X ′) according to (3.2);
5. if w(X ′) ≤ w(X) let X := X ′ (accept neighbour X ′);
else
draw R uniformly from [0, 1[;
if exp
(
w(X)−w(X′)
Ti
)
> R let X := X ′ (accept neighbour X ′);
6. let i := i+ 1;
7. if i ≤ N stop (end program);
else
let i := i+ 1;
update T (decrease the temperature)
go to 3;
Portuguese Study Groups’ Reports 7
4.1 A simple example
In this section we show the results of a basic application of the method
described. For simplicity we use only some of the restrictions, but imple-
menting the more general version with the full set of restrictions, although
more involved, is not more difficult from a conceptual point of view. Ba-
sically we considered the case where we want to distribute a set of orders
among a set of machines, taking into account the ending time and restric-
tions which include both the time to change from one mould to another and
between machines. The implementation was done using Matlab.
As an example take a set of orders of the following sizes:
2 5 10 10 12 20 22 30 30 30 40 50 (4.1)
totalling 261 jobs, which we want to distribute among three machines. The
numbers were chosen just for illustration and the program handles any other
distributions – this should be seen as just an example to illustrate the results
and were we kept things simple.
Figure 4.1: Evolution of the total time taken for the example described by
the set in (4.1).
In Figure 4.1 we depicted the evolution of the cost function, while Fig-
ure 4.2 gives a schematic presentation of the jobs through the three machines.
This last graph should be read from left to right, and each of the symbols
represents one of the orders. The first two red diamonds at the start of
machine 1 correspond to the (only) order of this size, while the order of size
20, represented here by black-filled squares, has been partitioned into three
blocks of size 4, 6 and 10 which have been assigned to machines 3, 1 and 3,
respectively.
8 ForEver: Scheduling in a factory
0 10 20 30 40 50 60 70 80 90 100
1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
2.8
3
Figure 4.2: One possible partition solution for the example described by the
set in (4.1).
We stress that a partition obtained by this method such as the one
above, is not necessarilly optimal. For the example of the order of size 20
mentioned above, it would be possible, for instance, to lump the two blocks
being processed by machine number 3 into one. However, this would not
necessarilly mean a big gain regarding the solution already obtained. More
important than this, the solution provided satisfies the given constraints
and has already been through an optimization procedure which ensures a
certain quality standard. Furthermore, this approach is very flexible and
by changing the constraints given in the mathematical formulation of the
problem (see (2.5)), it is possible to modify the goals that are to be achieved
with a minimum conceptual effort.
5.1 Conclusions and recommendations
Scheduling problems are common in industrial environments, varying from
simple processes to highly non trivial tasks assignments. In real situations
deriving from concrete industrial setups, this will usually give rise to hard
computational problems. This means that, in general, it will not be possible
to achieve the optimal solution due to time constraints, but one will have to
use a heuristic algorithm instead. In the case of the proposed problem, our
analyis lead us to a simulated annealing algorithm which is quite flexible
and allows us also to deal with variations of the problem considered. This
may be achieved by changing the constraints in an appropriate way.
The proposed approach is based on a Lagragian manipulation of a 0− 1
model, which means that tuning operations will have to be developed in
order to obtain near-optimal solutions. The main concerns for an imple-
Portuguese Study Groups’ Reports 9
mentation should be:
1. The choice of the multipliers λ and µ. Larger values of these param-
eters will put a stronger emphasys on the admissibility conditions.
A careful equilibrium must be achieved in order to balance feasibil-
ity with the capacity to distinguish between the quality of different
solutions.
2. The definition of the initial parameter T (temperature) for the simu-
lating annealing algorithm and how to actualize this parameter during
the process. Several different approaches are possible here (see for
example [2].
Even in the case where a heuristic algorithm is being used, one might
have to take into consideration the fact that the number of jobs considered
might be too large to allow us to produce a solution in real time. For
the case at hand, where the example given in the Introduction had around
30 000 jobs, there may already be some difficulties in dealing with such large
numbers. This problem may be solved by making a preliminary choice of
what the unit job should be which is different from the one made here. This
will depend on the size of the order, and different choices might have to
be made for different orders. From the table in Figure 1.1 we see that, for
instance, order F should have as a minimum job size five units, as not only it
is highly unlikely that these should be partitioned but also should this be the
case, the gain is likely to be very small. Other orders should have different
basic job units, depending on their overall size, and this might be another
parameter that will also need some tuning: decide upon the minimal job
unit depending on the total order size. However, once this learning process
is over, the actual scheduling should run quite smoothly.
Bibliography
[1] E.H.L. Aarts, J.H.M. Korst and P.J.M. van Laarhoven, “Simulated
annealing”, in Local Search in Combinatorial Optimization (E. Aarts
and J.K. Lenstra, eds.), John Wiley & Sons (1997), 91-120.
[2] P.J.M. van Laarhoven and E.H.L. Aarts, Simulated Annealing: Theory
and Applications, Springer, 1987.
[3] C.L. Liu, and J.W. Layland,Scheduling Algorithms for Multiprogram-
ming in a Hard Real-Time Environment, Journal of the ACM , (1973)
[4] G.L. Nemhauser and L.A. Wolsey, Integer and combinatorial optimiza-
tion, Wiley-Interscience, New York, NY, 1988
10


Integer and combinatorial optimization,

Laarhoven, “Simulated annealing”, in Local Search in Combinatorial Optimization

Layland,Scheduling Algorithms for Multiprogramming in a Hard Real-Time Environment,

Simulated Annealing: Theory and Applications,

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