Workshop of Industrial Production Systems– Industrial Techniques Laboratory – Sciences & Techniques Faculty – Fez, Morocco 15 May 2016
Multiobjective optimization of simple turning operation using parametric simplex method M. El Jai1,2, B. Herrou1, H. Ben-azza3 1
Industrial Techniques Laboratory, Sciences & Techniques Faculty, Sidi Mohamed Ben Abdellah University, Fez, Morocco 2 Advanced Materials and Applications Team, Ecole Nationale Supérieure d’Arts & Métiers, Moulay Ismail University, Meknès, Morocco 3 Industrial Engineering Department, Ecole Nationale Supérieure d’Arts & Métiers, Moulay Ismail University, Meknès, Morocco Correspondent author
[email protected] ABSTRACT.
In this paper, we propose to study the optimization of a large known problem of cutting parameters of a simple turning operation, according to some constraints related to the quality properties of the product, the effect on the material of the pieces and the maximum of power of the machine used. The objective functions considered are the cost of a simple operation (roughing in our case) and the used tool life, which have to be minimized. Even if the problem treated here is non-linear, we will linearize it according to simple variable change, in order to utilize simplex multiobjective method to navigate on the vertexes of the alternative space, defined by the constraints of the problem for designate the optimum vertex. The results of simplex multiobjective resolution method will be discussed according to the bibliography related to the topic. KEYWORDS:
Multiobjective Optimization Problem, turning optimization problem, weighted sum method, parametric simplex.
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Workshop of Industrial Production Systems– Industrial Techniques Laboratory – Sciences & Techniques Faculty – Fez, Morocco 15 May 2016
1. Introduction The problem of optimizing of the cutting parameters in machining operation remains an importance subject for engineers and researchers, since the automation of the machining operations and the development of related intelligent production systems are developed more and more using the computer science and optimization arsenal in order to obtain autonomous and dynamical systems. In their works ((Pujo, 2009), (Pujo et Ounnar, 2008), Pujo & Ounnar have discussed several times the possibility of autonomy and flexibility creation into intelligent interties, within multi-agent and Holonic manufacturing systems (Van Brussel et al., 1998) according to the Holonic paradigm which deals with the artificial intelligence applied to manufacturing systems. This article was developed in this philosophy; where we tried to study the optimization problem, developed in the next sections, in order to reach the optimum of the objective functions in a lowest time of calculation, compared to the Genetic Algorithms that have been used classically for the resolution of a single turning operation optimization problem (Jabri et al., 2013) (Kübler et al., 2015). Rare are those who adopted a deterministic approach for resolving of the problem. For example (Wang & al., 2002) has proceed to an intuitive study and he applied simple derivations on multivariable functions (objective vector) and used multiple projections on variable spaces (planer spaces defined by two variable each time) in order to understand the geometrical aspect and the evolution of the objectivefunctions and their optima according to the related constraints. Thus, (Wang & al., 2002) produced interesting results regarding the implementation of a typical algorithm for decision making for CNC machines or fabrication software. We try here to present a general view of the problem, through the linearization of the initial non-linear problem. In addition, the linearization is done by the application of the natural logarithm, on all constraints and functions. The return to the original base could be done by simple exponential transforming of data. 2. Problem adopted and modeling The problem proposed corresponds to the multi-pass turning model (eq. 1). The bibliography treated the case of rough and finish machining in the same time (Jabri et al., 2013) (Wang & al, 2002) (Chen & Tsai, 1996); but here for a first analysis, we restrict the study only on roughing operation, to reduce the parameters considered.
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Workshop of Industrial Production Systems– Industrial Techniques Laboratory – Sciences & Techniques Faculty – Fez, Morocco 15 May 2016
The reader can find the detail of the complete problem, roughing and finishing, in the above articles and the detail of technical development in (Klocke, 2011). 2.1. Understanding the optimization problem The optimization problem adopted in this work corresponds to the minimization of both the time, so the cost, of a simple operation of turning and the wear of the corresponding cutting tool that corresponds to the life cycle of the tool. In this domain, several models exist, among which the most used is the Taylor life cycle expressed by equation 2 (Klocke, 2011). The cost of a simple turning operation corresponds to the cost by time unit (denoted K0) times the duration of the operation, which is expressed simply by the equation 1. Figure 1 presents the corresponding problem decision variables or parameters: The related linear or periphery speed V (m/min); The feed f (mm/rev) of the pass; The depth of the turning operation d (mm); 2.2. Non-Linear Multiobjective Optimization Problem treated The following systems present the Non-Linear Multiobjective Optimization Problem (MOP) adopted for the first analysis.
Since the decision variables are (V, f, d), we can see that the objective functions and the constraints spaces defined by the equations (3), (4), (5) and (6) are convex one by one.
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Workshop of Industrial Production Systems– Industrial Techniques Laboratory – Sciences & Techniques Faculty – Fez, Morocco 15 May 2016
Fig. 1 – Turning parameters representation (V, f, d) 3. Problem transform and equivalent parametric linear problem In this paper, we will proceed to the linearization of the problem; both the objective functions and the constraints equations will be linearized, by applying a simple variable change which will preserve the behavior (monotony, convexity, etc.) of the functions and the constraints (bijection transformation). By applying the ‘Normal Logarithm’ function on the (MOP), we define the equivalent equations of the (LMOP) (Linear Multiobjective Optimization Problem):
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Workshop of Industrial Production Systems– Industrial Techniques Laboratory – Sciences & Techniques Faculty – Fez, Morocco 15 May 2016
We define also the decision variable xi as:
After calculation:
Thereafter, the constraints, as inequalities, are defined by the system (14). Remark Geometrically, the inequalities (17) and (19) could cause such a degeneracy at the corresponding vertexes, because one of the right limits (b5 and b5’) will major the other. To avoid this problem, we will adopt the minor of the two limits b5 and b5’ related to the inequalities (17) and (19).
After calculation, we obtain:
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Workshop of Industrial Production Systems– Industrial Techniques Laboratory – Sciences & Techniques Faculty – Fez, Morocco 15 May 2016
Thus, we choose to eliminate the additional inequality (14-f) to avoid the risk of degeneracy and cycling (Bertsimas & Tsitsiklis, 1999). By introducing the slack variables (Bertsimas & Tsitsiklis, 1999) (Culioli, 1994), we report the equivalent constraints equations:
Now, we can write the (LMOP), equivalent to the Non-Linear (MOP) of paragraph 2.2, on the standard form (Ehrgott, 2005) as follows:
Where
Where
0 1
[A] =
1 0 0 and the vector where xi ( The vector
0 1 0
0 0 1
) are the slack variables.
is written:
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1 0 0 0 0 0
0 1 0 0 0 0
0 0 1 0 0 0
0 0 0 1 0 0
0 0 0 0 1 0
0 0 0 0 0 1
Workshop of Industrial Production Systems– Industrial Techniques Laboratory – Sciences & Techniques Faculty – Fez, Morocco 15 May 2016
4. Presentation of the parametric simplex multiobjectif method In this section, we first introduce the algorithm which will be utilized to solve the multiobjective linear problem developed above. Since we have more than one objective, we must adopt a multiobjective technique in order to solve the (MOP). In the next developments, we will see that we have adopted the weighted sum method regarding the two theorems, 1 and 2, cited below. Theorem 1 (Deb, 2001) The solution of the problem represented by equation (P) (see §4.1) is Pareto-optimal if the weights of all objectives are non-negative. Theorem 2 (Deb, 2001) If x* is a Pareto-optimal solution of a multiobjective optimization convex problem, so it exists a weights vector λ, positive and non-null, such that x* is the solution of the initial problem (P) (see §4.1). Since the linearization permits to obtain linear program, the convexity is ensured. By the way, the solutions found by the weighted sum technique will be Pareto-optimal, according to the theorem 2. (Ehrgott, 2005) details the parametric simplex algorithm especially for a bi-objective linear optimization problem, in chapter six of the book. The parametric simplex algorithm corresponds the standard simplex applied on the composite function given by the weighted sum, so the parameters corresponds to the weights λi or the only weight λ in the case of normalized bi-objective problem. In the next paragraphs, we introduce the necessary theory background in order to understand the adopted approach.
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Workshop of Industrial Production Systems– Industrial Techniques Laboratory – Sciences & Techniques Faculty – Fez, Morocco 15 May 2016
4.1 Weighted sum method The weighted sum method is one of the scalarization techniques, which permits to transform a multiobjective problem, an initial problem (P), to an equivalent monoobjective problem (P’). The weighted sum method is classified according to the mappig θ which expresses the scalar product of the objective vector by the weights vector . Thus, according to the classification detailed in (Deb, 2001) (Ehrgott, 2005) (Bazaraa, 2006), the equivalent optimization problem (P’) is described by the expression (23). This classification groups all the information necessary for the description of a multiobjective optimization. Thus, we obtain a mono-objective parameterized problem, having λ as parameter.
(P)
(P’)
λ
(23)
Where is the feasible set, given by the equality and inequalities constraints; And , the standard order operator used for real numbers; The following paragraph shows, technically the algorithm of the parametric simplex reported in (Ehrgott, 2005). 4.2 Parametric simplex algorithm The parametric simplex algorithm is presented as follows (Ehrgott, 2005): Input: Data A, b, C for a bi-objective LP. Phase I: Solve the auxiliary LP (see (Ehrgott, 2005)) using the standard monoobjective Simplex algorithm. If the optimal value is positive, STOP, . Otherwise let B be an optimal basis.
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Workshop of Industrial Production Systems– Industrial Techniques Laboratory – Sciences & Techniques Faculty – Fez, Morocco 15 May 2016
Phase II: Solve the Parametric LP for yielding an optimal basis . Compute Phase III:
starting from basis B found in phase I and . .
Let \{r} End While Output : Sequence of λ-values and sequence of optimal BFSs. In general, we first find the optimal solution according to ; it defines so the initial linear problem to start the optimization procedure. After that, we calculate the possible value of the parameter λ in order to define the second optimization problem, according to the condition of the While loop; and so on. For each λ a “pseudo-linear problem” is resolved because that at the definition of each new feasible basis , the value of λ changes. Finally the algorithm gives the sequence of the parameter λ defining a sequence of hyperplans defining the set of Pareto-optimal (Deb, 2001) (Ehrgott, 2005) (Bazaraa, 2006) solutions. 4.3 Results and discussion a) Applying the parametric simplex algorithm After calculation, we present directly the tables of the parametric simplex below. Step 1: INITIAL LP (λ = 1) c(λ)
4
0,75
-0,25
0
0
0
0
0
0
x1
x2
x3 x4 x5 x6 x7 x8 x9
reduced c1
4
0,75
-0,25
0
0
0
0
0
0
0
reduced c2
-1
-1
-1
0
0
0
0
0
0
0
A
b
bj/Asj
x4
0
0,75
0,95
1
0
0
0
0
0
0,96
1,28
x5
1
0,75
0,95
0
1
0
0
0
0
2,38
3,18
x6
0,4
0,2
0,105
0
0
1
0
0
0
12,89
64,47
x7
1
0
0
0
0
0
1
0
0
2,30
---
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Workshop of Industrial Production Systems– Industrial Techniques Laboratory – Sciences & Techniques Faculty – Fez, Morocco 15 May 2016
x8
0
1
0
0
0
0
0
1
0
1,10
1,10
x9
0
0
1
0
0
0
0
0
1
1,10
---
The starting parameters values are: λ1 = 1, B1 = {4,...,9} , N1 = {1,2,3} , x = (0 ; 0 ; 0 ; 0,96 ; 6,99 ; 12,89 ; 2,30 ; 1,10 ; 1,10) I = {i
N : c2(xi) < 0 , c1(xi) >= 0} = {1,2}
Ø
While loop valid
By resolving this first parametric LPλ=1 system (Gauss linear transform) we obtain the parametric LP λ=4/7. Step 2: PL(λ=4/7) c(λ)
1,86 0
-0,57
0
0
0
0
x1 x2
0
x3
x4
x5
x6
x7
0
x8 x9
reduced c1
4
0
-0,25
0
0
0
0
-0,75
0
-0,82
reduced c2
-1
0
-1
0
0
0
0
1
0
1,10 b
A
bj/Asj
x4
0
0
0,95
1
0
0
0
-0,75
0
0,13
---
x5
1
0
0,95
0
1
0
0
-0,75
0
1,56
1,56
x6
0,4
0
0,105
0
0
1
0
-0,2
0
12,68
31,69
x7
1
0
0
0
0
0
1
0
0
2,30
2,30
x2
0
1
0
0
0
0
0
1
0
1,10
---
x9
0
0
1
0
0
0
0
0
1
1,10
---
By computing the algorithm parameters (r, s, λ…), we find the new basal vectors grouped in the matrix B, the non basal vectors related to the matrix N, the value of the weight λ(4/7) and the values of the variables xi. λ2 = 4/7 , B2 = {2,4,5,6,7,9} , N2 = {1,3,8} , x = (0 ; 1,10 ; 0 ;0,13 ; 1,56 ; 12,68 ; 2,30 ; 0 ; 1,10) I = {i
N : reduced c2(xi) < 0 , reduced c1(xi) >= 0} = {1}
By resolving this second parametric LP obtain the third parametric LP λ=1/5.
λ=4/7
10
Ø
While loop valid
(using Gauss linear transform), we
Workshop of Industrial Production Systems– Industrial Techniques Laboratory – Sciences & Techniques Faculty – Fez, Morocco 15 May 2016
Step 3: PL(λ=1/5) c(λ)
0,00 0
-0,85 0
x1 x2
x3 x4
0
0
0
x5 x6 x7
0,65
0
x8 x9
reduced c1
0
0
-4,05
0
-4
0
0
2,25
0
-7,06
reduced c2
0
0
-0,05
0
1
0
0
0,25
0
2,66 b
A x4
0
0
0,95
1
0
0
0 -0,75
0
0,13
x1
1
0
0,95
0
1
0
0 -0,75
0
1,56
x6
0
0
-0,275
0
-0,4
1
0
0,1
0
12,05
x7
0
0
-0,95
0
-1
0
1
0,75
0
0,74
x2
0
1
0
0
0
0
0
1
0
1,10
x9
0
0
1
0
0
0
0
0
1
1,10
By computing the algorithm parameters (r, s, λ…), we find the new basal vectors grouped in the matrix , the non basal vectors related to the matrix , the value of the weight (1/5) and the values of the variables xi. λ3 = 1/5 , B3 = {2,4,1,6,7,9} , N3 = {5,3,8} , x = (1,56 ; 1,10 ; 0 ; 0,13 ; 0 ; 12,05 ; 0,74 ; 8 ; 1,10) I = {i
N : reduced c2(xi) < 0 , reduced c1(xi) >= 0} = Ø
Invalid While Loop End of the Algorithm
We observe finally that the set I is Null, so the algorithms is stopped and we return the final optimum as: x1= 1,56
V=
237,94 mm/min
tr =
3,955
x2= 1,1
f=
0,30 mm/tr
Tr =
6,453
x3= 0
d=
1,00 mm/tr
C = K0 * tr
= 1,98 $ =61,29 %
The other optimal values are plotted directly on figure 2. b) Evolution of the objective-functions and discussion In order to analyze the evolution of the different solutions developed through the application of the algorithm, we plot the different objective values at figure 2.
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Workshop of Industrial Production Systems– Industrial Techniques Laboratory – Sciences & Techniques Faculty – Fez, Morocco 15 May 2016
70
Step 3 (λ=1/5) (1.98 $, 61.29%)
Objective function f2 (%)
60
50
40
30
20
Step 2 (9.41 $, 0.12%)
Step 1 (28.27 $, 0.05%)
10
0
0
5
10 15 20 Objective function f1 ($)
25
30
Fig. 2 – Objective function space according to the solutions given by the parametric simplex algorithm We can see that there is a kind of evolution denoted, where passing from a step to the other: The modification of an objective affect greatly the modification of the other in contraire directions. The notion of Pareto front had been introduced in order to define such a behavior. Figure 3 presents typical Pareto front, according to the different optimization policies (min-min, min-max, max-min, max-max).
Fig. 3 – Typical Pareto fronts (bi-objective domain)
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Workshop of Industrial Production Systems– Industrial Techniques Laboratory – Sciences & Techniques Faculty – Fez, Morocco 15 May 2016
According to our results, presented in figure 2, and according to the right-top plot of figure 3, we remark that the different values of objectives (f1, f2), related to the different parametric simplex steps, describe approximately the Pareto front of the objective domain. Knowing that, for convex problems such this (linear problem), the set of the optimal solutions is Pareto-optimal and by the way the solutions are connected (Ehrgott, 2005) (Deb, 2001). Starting from the property of connection of the Pareto-optimal solutions, we can propose such a mathematical standard model (function) which could describe the evolution of the set of optimal solutions , and present it to the decision making responsible related to a typical multiobjective optimization problem. 5. Conclusion We first remind that the multiobjective optimization problem adopted in this work has never been treated analytically before; most of the techniques used for that purpose were numerical. By the way, we adopt an analytical method such as multiobjective simplex technique or parametric simplex method. The simplex approach can be classified as analytical technique because it is based on perfectly geometrical analysis without using any of the numerical techniques. The results found permit to describe approximately the Pareto front, but since the Pareto-optimal solutions are very far from each other, it will be difficult to describe mathematically the evolution of the optimal solutions. According to the references (Chen and Tsai, 1996) (Franci et al., 2005) (Jabri et al., 2013) that have used numerical approaches, the best results corresponds to a cost operation around 1$, according the function f1, and a tool wear or degradation around 5%. Based on these results, we can see that the three values we found are not too competitive; the values obtained correspond to the minimization of only one objective function, f1 or f2; the other function id maximized so not optimized. After analyzing these results, we realize that we had not proceeded to the normalization of the functions, objective and constraints functions (Deb, 2001) (Datta & Deb, 2005) (Ehrgott, 2005) (Helmut, 2006). That is why the next step of this work will focus on the normalization the corresponding functions before proceeding to the parametric simplex algorithm resolution. 6. References H. Van Brussel et al., Reference Architecture for Holonic Manufacturing Systems: PROSA, Computers In Industry, special issue on intelligent manufacturing systems, Vol. 37, No. 3, pp. 255 - 276, 1998.
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Workshop of Industrial Production Systems– Industrial Techniques Laboratory – Sciences & Techniques Faculty – Fez, Morocco 15 May 2016
Kübler Frank, Johannes Böhner, Rolf Steinhilper, Resource efficiency optimization of manufacturing processes using evolutionary computation: A turning case, The 22nd CIRP conference on Life Cycle Engineering, Procedia CIRP 29 (2015), 822 – 827. P. Pujo, De l’Isoarchie pour le Pilotage des Systèmes de Production, Mémoire d’Habilitation à Diriger des Recherches, Université Paul Cézanne Aix-Marseille III, 2009. M. –C. Chen & D. –M. Tsai (1996), A simulated annealing approach for optimization of multi-pass turning operations, International journal of Production Research, 34:10, 2083-2825. Dimitris Bertsimas, John N. Tsitsiklis, Introduction to Linear Optimization, Massachusetts Institute of Technology, Arthena Scientific, Belmont, Massachusetts, 1999. Jean-Christophe Culioli, Introduction à l’Optimisation, Editions Ellipses, 1994. Mokhtar S. Bazaraa, Hanif D. Sherali, C. M. Shetty, Non linear Programming – Theory and Algorithms, 3rd Edition Wiley – Interscience, 2006, Hoboken, New Jersey, USA. Kalyanmoy Deb, Multi-objective Optimization Evolutionary Algorithms, 1st Edition, John Wiley & Sons, 2001. Fritz Klocke, Manufacturing Process 1 – Cutting, Springer-Verlag, Berlin Heidelberg 2011, translated by Aaron Kuchle. Rituparna Datta and Kalyanmoy, An adaptive Normalization based Constrained Handling Methodology with Hybrid Bi-Objective and Penalty Function Approach, KanGAL Report Number 2012005, Department of Mechanical Engineering, Indian Institute of Technology, Kanpur, 2005. Helmut Mausser, Oleg Grodzevich, Oleksandr Romanko, Normalisation and Other Topics in Multi-Objective Optimization, Proceeding of the Fields-MITACS Industrial Problems Workshop, 2006. P. Pujo & F. Ounnar, Pull System Control For Job Shop Via A Holonic, Isoarchic and Multicriteria Approach, Proceeding of the 17 th World Congress, The International Federation of Automatic Control, Seoul, Korea, July 6-11, 2008.
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Workshop of Industrial Production Systems– Industrial Techniques Laboratory – Sciences & Techniques Faculty – Fez, Morocco 15 May 2016
Abdelouahhab Jabri, Abdellah El Barkany, Ahmed El Khalfi, Multi-Objective Optimization Using Genetic Algorithms of Multi-Pass Turning Process, Engineering, July 2013, 5, 601-610. J. Wang, T. Kuriyagawa, X. P. Wei and D. M. Guo, Optimization of Cutting Conditions for Single Pass Turning Operations Using a Deterministic Approach, International Journal of Machine Tools and Manufacture, Vol. 42/9 (2002), pp. 1023-1033. Mattias Ehrgott, Multicriteria optimization, 2nd Edition, Springer, Berlin, Heidelberg, 2005. Dimitris Bertsimas, John N. Tsitsiklis, Introduction to Linear Optimization, Massachusetts Institute of Technology, Arthena Scientific, Belmont, Massachusetts, 1999.
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