=Paper=
{{Paper
|id=None
|storemode=property
|title=Concurrent Configuration and Planning Problems: Some Optimization Experimental Results
|pdfUrl=https://ceur-ws.org/Vol-958/paper8.pdf
|volume=Vol-958
|dblpUrl=https://dblp.org/rec/conf/confws/PitiotAVG12
}}
==Concurrent Configuration and Planning Problems: Some Optimization Experimental Results==
https://ceur-ws.org/Vol-958/paper8.pdf
Concurrent configuration and planning problems:
Some optimization experimental results
Paul Pitiot1, Michel Aldanondo1, Elise Vareilles1, Paul Gaborit1
Abstract.1 This communication deals with mass customization algorithm with a classical branch and bound. This initial section
and the association of the product configuration task with the introduces the problem and the organization of the paper.
planning of its production process while trying to minimize cost
and cycle time. We consider a two steps approach that first permit 1.1 Concurrent configuration and planning
to interactively (with the customer) achieve a first product processes as a CSP
configuration and first process plan (thanks to non-negotiable
requirements) and then optimize both of them (with remaining Deriving the definition of a specific or customized product
negotiable requirements). The communication concerns the second (through a set of properties, sub-assemblies or bill of materials,
optimization step. Our goal is to evaluate a recent evolutionary etc…) from a generic product or a product family, while taking
algorithm (EA). As both problems are considered as constraints
into account specific customer requirements, can define product
satisfaction problems, the optimization problem is constrained.
Therefore the considered EA was selected and adapted to fit the configuration [4]. In a similar way, deriving a specific production
problem. The experimentations will compare the EA with a plan (operations, resources to be used, etc...) from some kind of
conventional branch and bound according to the problem size and generic process plan while respecting product characteristics and
the density of constraints. The hypervolume metric is used for customer requirements, can define production planning [5]. As
comparison.. many configuration and planning studies (see for example [6] or
[5]) have shown that each problem could be successfully
considered as a constraint satisfaction problem (CSP), we have
1 INTRODUCTION proposed to associate them in a single CSP in order to process
them concurrently.
This paper deals with mass customization and more accurately with This concurrent process and the supporting constraint framework
aiding the two activities, product configuration and production present three main interests. First they allow considering constraint
planning, achieved in a concurrent way. According to the that links configuration and planning in both directions (for
preferences of each customer, the customer requirements example: a luxury product finish requires additional manufacturing
(concerning either the product or its production) can be either non- time or a given assembly duration forbids the use of a particular
negotiable or negotiable. This situation allows considering a two- kind of component). Secondly they allow processing in any order
step process that aims to associate the two conflicting expectations, product and planning requirements, and therefore avoid the
interactivity and optimality. The first interactive step, that traditional sequence: configure product then plan its production [7].
sequentially processes each non-negotiable requirement, Thirdly, CSP fit very well on one side, interactive process thanks to
corresponds with a first configuration and planning process that constraint filtering techniques, and on the other side, optimization
reduces the solution space. This process is present in many thanks to various problem-solving techniques. However, we
commercial web sites using configuration techniques like assume infinite capacity planning and consider that production is
automotive industry for example. Then, a second process optimizes launched according to each customer order and production capacity
the solution with respect to the remaining negotiable requirements. is adapted accordingly.
As the solution space can quickly become very large, the In order to illustrate the addressed problem we consider a very
optimization problem can become hard. Thus, this behavior is not simple example dealing with the configuration and planning of a
frequent in commercial web sites. Meanwhile some scientific small plane. The constraint model is shown in figure 1. The plane
works have been published on this subject (see for example [1] or is defined by two product variables: number of seats (Seats,
[2]) and the focus of this article is on the optimization problem. In possible values 4 or 6) and flight range (Range, possible values 600
some previous conferences we proposed an interesting adapted or 900 kms). A constraint Cc1 forbids a plane with 4 seats and a
evolutionary algorithm for this problem [3]. However, the range of 600 kms. The production process contains two operations:
presentation was rather descriptive and experimentations were not sourcing and assembling. (noted Sourc and Assem). Each operation
significant. Therefore, the goal of this paper is to compare this is described by two process variables: resource and duration: for
sourcing, the resource (R-Sourc, possible resources “Fast-S” and
“Slow-S”) and duration (D-Sourc, possible values 2, 3, 4, 6 weeks),
1 for assembling, the resource (R-Assem, possible resources “Quic-
Toulouse University - Mines Albi, CGI lab, Albi, France
A” and “Norm-A”) and duration (D-Assem, possible values 4, 5, 6,
email: somename@mines-albi.fr
7 weeks). Two constraints linking product and process variables
modulate configuration and planning possibilities: one linking seats In order to complete our example, we add a cost and cycle time
with sourcing, Cp1 (Seat, R-Sourc, D-Sourc), and a second one criteria as represented in figure 2. For cost, each product variable
linking range with the assembling, Cp2 (Range, R-Assem, D- and each process operation is associated with a cost parameter and
Assem). The allowed combinations of each constraint are shown in a relevant cost constraint: (C-Seats, Cs1), (C-Range, Cs2), (C-
the 3 tables of figure 1. Without taking constraints into account, Sourc, Cs3) and (C-Assem, cs4) detailed in the tables of figure 2.
this model shows a combinatory of 4 for the product (2x2) and 64 The total cost is obtained with a numerical constraint and the cycle
for the production process (2x4) x (2x4) providing a combinatory time, sum of the two operation durations, is also obtained with a
of 256 (4 x 64) for the whole problem. Considering constraints lead numerical constraint as follow:
to 12 solutions for both product and production process. Total cost = C-Seats + C-Range + C-Sourc + C-Assem.
Cycle time = D-Sourc + D-Assem
The twelve previous solutions are shown on the figure 3 with the
Pareto front gathering the optimal ones. In this figure, all solutions
are present. When non-negotiable requirements are processed
during interactive configuration and planning, some of these
solutions will be removed. Once all these requirements are
processed, the identification of the Pareto front can be launched in
order to propose the customer a set of optimal solutions.
Figure 1 Concurrent configuration and planning CSP model
1.2 Optimizing configuration and planning
concurrently
Given previous problem, various criteria can characterize a
solution: on the product configuration side, performance and
product cost, and on the production planning side, cycle time and
process cost. In this paper we only consider cycle time and cost.
The cycle time matches the ending date of the last production
operation of the configured product. Cost is the sum of the product
cost and process cost. We are consequently dealing with a multi-
criteria optimization problem. As these criteria are in conflict, it is
better for decision aiding to offer the customer a set of possible Figure 3 Optimal solutions on the Pareto Front
compromises in the form of Pareto Front.
A strong specificity of this kind of problems is that the solution
space is large. It is reported in [8] that a configuration solution
space of more than 1.4* 1012 is required for a car configuration
problem. When planning is added, the combinatorial structure can
become huge. Specificity lies in the fact that the shape of the
solution space is not continuous and in most cases shows many
singularities. Furthermore, the multi-criteria aspect and the need
for Pareto optimal results are also strong problem expectations.
These points explain why most of the articles published on this
subject (as for example [9]) consider genetic or evolutionary
approaches to deal with this problem. However classic
evolutionary algorithms have to be adapted in order to take into
account the constraints of the problem as explained in [10]. Among
these adaptations, the one we have proposed in [3] is an
evolutionary algorithm with a specific constrained evolutionary
operators and our goal is to compare it with a classical branch and
bound approach.
In the following section we characterize the optimization problem
and briefly recall the optimization techniques. Then
Figure 2 Concurrent configuration and planning model to optimize experimentation results are presented and discussed in the last
section.
2 OPTIMIZATION PROBLEM AND remaining gene is updated by constraint filtering. As filtering is not
full proof, inconsistent individuals can be generated. In this case a
OPTIMIZATION TECHNIQUES limited backtrack process is launched to solve the problem. For full
details please see [3].
2.1 Optimization problem
The key idea of the Branch and Bound algorithm is to explore a
The problem of figure 2 is generalized as the one shown in figure search tree but using a cutting procedure that stops exploration of a
4. The optimization problem is defined by the quadruplet where V is the set of decision variables, D the set of domains is a splitting procedure that corresponds to the selection of one
linked to the variables of V, C the set of constraints on variables of variable of the problem and to the instantiation of this variable with
V and f the multi-valuated fitness function. Here, the aim is to each possible value. The second tool is a node-bound evaluation
minimize both cost and cycle time. The set V gathers: the product procedure. The filtering process is used to achieve this task with a
descriptive variables and the resource variables. The set C gathers partial instantiation and is able to evaluate if the partial
constraints (Cc and Cp). Cost variables and operation durations are instantiation is consistent with the constraints of the problem, and,
deduced from the variables of the set V thanks to the remaining if this is the case, to provide the lower bound of each criterion
constraints. cycle time and cost. When the search reaches a leaf of the search
tree, or complete instantiation, the filtering system gives the exact
evaluation of the solution. Thus, the values of leaf solutions can be
used to compute the current Pareto front and then to cut remaining
unexplored branches that are dominated by any aspect of the Pareto
front solution (e.g. the upper bounds of the leaf solution dominate
the minimal bounds of the branch to cut).
3 EXPERIMENTATIONS
The optimization algorithms were implemented in C++
programming language and interacted with a filtering system coded
Figure 4 Constrained optimization problem in Perl language. All tests were done using a laptop computer
powered by an Intel core i5 CPU (2.27 Ghz, only one CPU core is
Experimentations will consider different problem sizes: different used) and using 2.8 Go of ram. These tests compared the behavior
numbers of product variables, different number of production of our constrained EA algorithm with the exact branch-and-bound
operations and different number of possible values for these algorithm.
variables. Different constraint densities (percentage of excluded
combinations of values) will be also considered.
3.1 First experimentation: problem size and
constraint densities
2.2 Optimization techniques
An initial first model, named "full model" is considered. It can be
consulted and interactively used at http://cofiade.enstimac.fr/cgi-
The proposed evolutionary algorithm is based on SPEA2 [11] with bin/cofiade.pl select model ‘Aircraft-CSP-EA-10’. It gathers five
an added constraints filtering process that avoids infeasible
product variables with a domain size between 4 and 6, six
individuals (or solutions) in the archive. This provides the six steps
following approach: production operations with a number of possible resources between
1. Initialization of individual set that respect the constraints 3 and 25. Without constraints consideration, the solution space of
(thanks to filtering), the product model is 5,184, and the planning model is 96,000. The
2. Fitness assignment (balance of Pareto dominance and solution size of the global problem model is 497,664,000. A second model,
density) named “small model”, has been derived from the previous one with
3. Individuals selection and archive update the suppression of a high combinatory task and a reduction of one
4. Stopping criterion test domain size. This reduces the planning problem size to 12,000 and
5. Individuals selection for crossover and mutation operators global model 6,220,800.
(binary tournaments)
6. Individuals crossover and mutation that respect constraints In order to evaluate the impact of constraints density, two versions
(thanks to filtering) of each model were produced: one with a "weak density" of
7. Return to step 2. constraints (20% of possible combinations are excluded in each
constraint Cc and Cp) and the other with a "high density" of
For initialization, crossover and mutation operators, each time an constraints (50% excluded). These values are frequently met in
individual is created or modified, every gene (decision variable of industrial configuration situations. This provides four models
V) is randomly instantiated into its current domain. To avoid the characteristics in table 1.
generation of unfeasible individuals, the domain of every
In our two criteria case, it is the upper right area of figure 5. It thus
Table. 1 Problems characteristics allows evaluation of both convergence and diversity properties
because the fittest and most diversified set of solutions is the one
Solution quantity Without constraints Low density High density that maximizes the hypervolume.
Small model 6 220 800 595 000 153 000
Full model 497 664 000 47 600 000 12 288 000
Results are presented in figure 6 where EA curves are average
results for 30 executions. Both algorithms start with a lapse of time
For the small models, evolutionary settings are tuned to: population
where performance is null. For the BB algorithm, this corresponds
size: 50; archive size: 40; Pmut: 0.4; Pcross: 0.8. The ending criterion
to the time needed to reach a first leaf on the search tree, while for
used is a time limit of half an hour. For the full models, we adapt
the EA; it corresponds to the time consumed to constitute the initial
settings for a wider search: population size: 150; archive size: 100;
population.
Pmut: 0.4; Pcross: 0.8. The ending criterion used is a time required by
the BB algorithm. In order to analyze the two optimization
For the small models (first two curves), the BB algorithm reaches
approaches, we compare the hypervolume evolution during
the optimal Pareto front much faster compared with EA
optimization process. Hypervolum metric has been defined in [12].
performance. On the other hand, the EA is logically better than the
It measures the hypervolume of space dominated by a set of
BB algorithm on the full model. For example, on the low-
solutions and is illustrated in Figure 5.
constrained model, the BB algorithm took 20 times longer to reach
a good set of solutions (less than 0.5% of the optimal
hypervolume).
The impact of constraints density could also be discussed. As it can
be seen, the BB algorithm performance is improved when the
density of constraints is high. This is because the filtering allows
more branches to be cut on the search tree, in such way that the
algorithm reaches leaf solutions and, consequently, optimal
solutions more quickly. The EA performance moves in the
opposite way. The more the model is constrained, the more the
random crossover operation will have to backtrack to find feasible
solutions, and thus the time needed by the algorithm will be
consequent.
Figure 5 Hypervolume linked to a Pareto front
Figure 6 First experimentation results
Figure 7 Experimentation results dealing with problem size
3.2 Experimentations on problem size of Figure 7 shows that the optimization process has stopped after
48 hours. It can be noticed that 90% of the final score was obtained
In order to try to identify the problem size where EA is more after 3 hours and 99% in 10 hours. This allows underlining the
suitable than BB, we have modified the low constrained model as good performance of our approach when facing large low
follows. We consider now a model gathering six product variables constrained problems. Of course the idea is to use BB, if the first
and six production operations with three possible values for each, interactive configuration step has led to a rather small problem, less
and sequentially add either a product variable and or an operation. than 13 or 14 variables in our case, and EA otherwise.
The range of study is between 12 and 16 decision variables with
three possible values for each. Relevant solution spaces without Finally we also try to break optimization in two steps. The idea is:
constraint vary between 1.6*106 and 43*106. (i) compute quickly a low quality Pareto, (ii) select the area that
interest the customer (iii) compute a Pareto on the restricted area.
The results are shown in the left part figure 7. The vertical axis The restricted area is obtained by constraining the two criteria total
corresponds with the computation time and the horizontal one with cost and cycle time (or interesting area) and filtering these
the number of decision variables. For BB curves, it shows the time reductions on the whole problem. The search space is greatly
to reach the optimal solution. For the EA curve it shows the time reduced and the second optimization much faster. This is shown in
required for nine EA runs over ten to reach the optimal solution. figure 8 where the left part shows the single step process with 10
Order of magnitude are close for both around 13 or 14 variables and 60 minutes Pareto and the right part shows the restricted area
corresponding with a solution space around 2*106 to 5*106 with the two previous curve and the one corresponding with a 10
comparable with our previous small model size. minutes Pareto launched on the restricted area. It shows that the
sequence of two optimization steps of 10 minutes provide a result
As we already mention, industrial models are frequently larger than equivalent to a 60 minutes optimization process.
that. We therefore try our EA approach with a low constrained
model with 30 variables and a solution space around 1016. The
stopping criterion is "2 hours without improvement". The right part
Figure 8 Experimentations with a two steps optimization process
Methods in Applied Mechanics and Engineering, vol. 191, n°11-12,
p. 1245-1287 (2002)
4 CONCLUSIONS [11] Zitzler E., Laumanns M., Thiele L., SPEA2: Improving the Strength
Pareto Evolutionary Algorithm, Technical Report 103, Swiss Fed.
The goal of this communication was to propose a first evaluation of Inst. of Technology (ETH), Zurich (2001)
an adapted evolutionary algorithm that deals with concurrent [12] Zitzler E., Thiele L., Multiobjective Optimization Using Evolutionary
product configuration and production planning. The problem was Algorithms - A Case Study, 5th Conf. Parallel Problem Solving from
Nature, Springer, p. 292-301 (1998)
recalled and the two optimization approaches (Evolutionary
algorithm and branch and bound) where briefly presented. Various
experimentations have been presented. A first result is that: (i) the
proposed EA works fine when the size of the problem gets large
compare to the BB, (ii) when problem tends to be more constrained
the tendency goes to the opposite. When problem is low
constrained (90% of excluded solutions) with 13-14 decision
variables with 3 values each, they perform equally. When the
problem gets larger, BB cannot be considered and EA can provide
good quality results for the same problem with up to 30 variables
(around 1016 solutions - 90% rejected). Finally some ideas about a
two steps optimization process have shown that the proposed
approach is quite promising for large problems. These are first
experimentation results and we are now working on comparing our
proposed EA with some penalty function approaches.
ACKNOWLEDGEMENTS
We would like to thank the referees for their comments, which
helped improve this paper considerably.
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