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Variable decomposition based global-optimization algorithm for process synthesis
Process Systems Engineering 2003 B. Chen and A.W. Westerberg (editors) 9 2003 Published by Elsevier Science B.V.
666
Variable decomposition based global-optimization algorithm for process synthesis Jian Zhang a, Bingzhen Chen a, Shanying Hu a, Xiaorong He a aDepartment of Chemical Engineering, Tsinghua University, Beijing 100084, China Abstract The key-task of process synthesis is to solve a mixed-integer non-linear (MINLP) model. As the number of integer variables increase, one is faced with a large combinatorial problem. At the same time, due to the nonlinearities the MINLP problem is tended to converge to a local optimal solution. To overcome these difficulties, a variable decomposition based global-optimization algorithm is presented, in which the integer variables and the continuous variables are treated sequentially. There are three steps in the algorithm: The first step is to classify the logical variables into independent and dependent variables, meanwhile, the logical constraints are classified into deductive constraints and restrictive ones. Logical deductive graph is generated to get the feasible logical variable sets. The second step is to construct a non-linear continuous (NLP) sub-problem for each feasible logical variable set and to solve the sub-problem using general global optimization algorithm. The third step is to obtain the global optimum solution. A process synthesis example is presented to compare the algorithm with the standard Branch-and-Bound algorithm. Keywords:
global optimization algorithm, logic-based MINLP model, process synthesis
1. INTRODUCTION Process synthesis is an important part of process systems engineering. The mixed-integer non-linear programming (MINLP) model is widely used in process synthesis. In an MINLP model, continuous variables are used to model the technical parameters in chemical process, such as flows, temperatures and compositions. The 0-1 variables (only have value of 0 or 1) or logical variables can be used to model, for instance, altemative candidates and the topological relations among equipments. There are two types of MINLP model: algebraic MINLP model and logic-based MINLP model [11. The disadvantage of the algebraic MINLP model is that the topological relations expressed by 0-1 variables are not direct, and hard to modeling. It limits the industrial usage of the algebraic MINLP model. In a logic-based MINLP model, the topological relations are expressed clearly by logical relations. Heuristic knowledge would also be integrated in the model to eliminate candidate flow sheets that conflict with technical practice, and would accelerate the solving process. So the novel global optimization algorithm proposed in this paper is based on the logic-based MINLP model. The global optimization MINLP algorithm includes Branch-and-Simplify method,
667 Hybrid Branch-and-Bound and Outer Approximation method, txBB algorithm [21 and symbolic reformulation algorithm. L31 For each iteration of the branch strategy, only one from the integer variables is fixed to 0 or 1, and the rest ones are relaxed into continuous variables. So the number of iterations is exponential with the number of logical variables. Furthermore, the relaxation of integer variables leads to an increase of the number of continuous variables and the scale of model, also, it may make some linear constraints transfer to nonlinear constraints. These make the relaxed model hard to solve. In this paper, a variable decomposition based global-optimization algorithm is presented. This algorithm is described in the second section. The specific steps of the algorithm are listed in section three. A reaction process synthesis example is presented in section four to test and compare the algorithm. The fifth section is a conclusion and discussion. 2. A L G O R I T H M DESCRIPTIONS The key of the variable decomposition algorithm is to treat the logical variables and continuous variables of the MINLP model sequentially. A set of logical variables whose values satisfy all logical constraints is called a feasible logical variable set. Locating the feasible logical variable sets is an exact reasoning problem. The logical variables' values of each feasible solution of the original MINLP model certainly are in the feasible logical variable region. Corresponding to each feasible logical variable set, a non-linear continuous (NLP) sub-problem is constructed. The global optimization solution of the original MINLP model is obtained from solving these sub-problems. In a process synthesis problem, although the combination number of integer variables may be very large, the number of feasible sets is few. For example, in a reaction process synthesis problem t4] there are 13 logical variables, and the combination number is 213=8192. But the number of feasible logical variable sets is 32, which is 0.39% of the combination number. For an ethylene refrigerant synthesis problem, I51 there are 25 logical variables, and the combination number is 225=33554432. But the feasible number is only 265, which is 0.00079% of the combination number. Especially, when integrating the heuristic knowledge, the flow sheets that do not satisfy technical experience are eliminated, and the number of feasible logical variable sets will be further reduced. 3. A L G O R I T H M The algorithm includes 3 steps: getting feasible logical variable sets; generating continuous variables NLP sub-problems corresponding to the feasible logical variable sets and solving them using the NLP global optimization algorithm; obtaining the global optimization solution. In a logic-based process synthesis model, the constraints include equipment constraints, connection constraints, and logical relations among equipments t6]. 3.1. Locating the feasible logical variable sets In process synthesis, some types of logical constraints are often used, and these
668 constraints would deduct logical variables directly: 1) f(y2,ys,...,yn)-~ y~: in this constraint, the value of yl
is partly restricted by
f(Y2,Y3 .... ,yn): if f(YE,Y3 .... , y ~ ) = l , then yl-1; if f(yE,Ya,...,y~)=O, then yl would have the value of 1 or 0. 2) f(Y2,Y3 ..... y,,)~--~ y~: in this constraint, the value of yl is fully determined by
f(Y2,Y3 ....,Y,). 3) yl v Y2 v Y3 v . . . v yn : this constraint can be transferred to ~(Y2 v Y3 v which is the first type. 4)
Y~ @Y2 EDY3~)'"EDYn:
--n(y 2 V
Y3 V . . . V y, ) ~
in
this
constraint,
variable
yl
can
be
...
V y , ) --~ y~,
expressed
by"
y~, which is the second type. And the original constraint changes to
Y2 @ Y3 ~) .-. ~) Y, ~ -n(Y2 v Y3 v ... v yn). 5) Y2 ~ Y3 ~) " " E~)y, @-'flY2 v Y3 v
... V
Yn). in this constraint, variable y2 can be expressed
(Y3 V . . . V Yn) --~ -lYE, which is the first type. And the original constraint changes to Y3 @ Y4 ~ -.- ~ Y, ~ ~ ( Y 3 v . . . v yn ). The new constraint has the same form with the original constraint, thus more dependent variables would be derived from it. All the deductive constraints form an oriented non-loop graph, which called deductive graph. Each node in a deductive denotes a logical variable, and the arc that goes into a node denotes the deductive constraint that deducts the logical variable, and the arc that goes out from a node denotes the constraint that is used to deduct other logical variables. Not every logical constraint can be transferred to deductive constraint. In some logical constraints, all member variables are dependent variables, or the deduction would cause loops in the deductive graph, so they could not be transferred to deductive constraints. These constraints restrict the existing logical variable sets, and are called restrictive constraints. A restrictive constraint would not deduce a logical variable, but it could be used to check whether the deduced logical variables are satisfied with it, so as to reduce the number of logical variable sets. When all logical constraints were transferred to deductive constraints or restrictive constraints, there would be some residual logical variables that were not set to dependent variables. These variables are called independent variables and used as the starting nodes of deduction. The meaning of "independent" is that when generating the feasible logical variable sets, the values of these variables are chosen freely. After the classification of the logical variables and logical constraints is completed, the feasible logical variable sets will be generated as follows: the nodes of independent variables, as starting nodes, have no arcs entering them, and the values of these variables are chosen freely. Then these nodes and their associated arcs that go out from them are hidden, and new starting nodes would appear in the graph. The value of a node is determined through its associated arcs that enter the node. When the values of the nodes are determined, these nodes and their associated output arcs would be hidden. If a restrictive constraint is met, existing variable sets would be restricted by the constraint and infeasible variable sets would be eliminated. Repeat these steps until all the nodes are treated with, then the finally existing logical variable sets are the feasible logical variable ones.
by:
669 It should be noticed that different sequence of logical constraints and logical variables would generate different deductive graph. But different deductive graph would only affect the intermediate results, and would not affect the final feasible logical variable sets. The feasible variable sets are only correlated to logical constraints of the model.
3.2. Generating the continuous variables NLP sub-problems For each feasible logical variable set, a corresponding continuous variables nonlinear sub-problem can be generated. The sub-problem is used to optimize the technical parameters when the flow sheet is determined. The constraints of the NLP sub-problem include equipment constraints and connection constraints, and the objective function of the sub-problem comes from the original objective function. The symbolic reformulation global optimization algorithm [3] is applied to solve the NLP sub-problem, and the global optimum for each sub-system will be obtained. 3.3. Obtaining the global optimum solution The global optimum solution is obtained from solving all the sub-problems as follows: 1) If this sub-problem has no solution, it will be discarded. 2) If the lower bound of this sub-problem is higher than the current solution, this sub-problem will be discarded. 3) If the optimal solution of this sub-problem is lower than the current one, the current solution will be updated. After repeating these steps for each feasible logical variables set, the final solution is the global optimum one of the original MINLP model. 4. PROCESS SYNTHESIS EXAMPLE Consider a process synthesis example [41 shown in Fig. 2. The equations of equipments yl-y8 and logical constraints are listed below. Yl :x3 - ln(x2 + 1)
1) 1 <---~y 9
Y2 :x5 = 1.2 ln(x 4 + 1)
2) Yl ~ Y2
Y3 :X8 = 1"5X9 + Xlo
3) Y3 V Y4 V Y5 v Y13
Y4 :Xl2 = 0"8X13
4) ~Y4 V ~Y3
3X14 --
5) ~Y4 V ~Y5
(6y4
Y5 :X15 = 2X16
6) Y6 V Y7 ~
Y6 "X20 = 1.5 ln(xl9 + 1)
7) ~(Y6 V Y7 ) ~) Y6 1~) Y7
Y7 :X22 = ln(x21 + 1)
8) Y6 V Y7 ~'ff Y4 V Yll
Y8 :x~8 = ln(x~o + x17 + 1)
9) ~(Y3 v Y5) --~ ~Yl0
0.4X17 < X10 < 0.8X17
10)
Y4
Y3 V Y5 V Y10 ~
11) Y8 ~ Yl2 12) Yl2 v Yll
,,......... ( 3 ) ~
Y8
Fig. 1. Deductive graph
(11)~
670 The objective function is: min 5y~-10x 3 + 8 y 2 - 1 5 x 5 + 6 y 3 +40X 9 +10y4 +15X14 +6Y5 -[- 80X16 +7Y6 +25X19 - 60X20 + 4 y 7 + 35X21- 80X22 + 5y 8 + 15x10 + x I + 45X25-65x~s + 122
In this example, there are 25 continuous variables, 13 logical variables and 12 logical constraints. Based on the algorithm, 2 independent logical variables, 11 dependent variables, 11 deductive constraints and 1 restrictive constraint are obtained. The logical deductive graph is shown in Fig. 1. The double-circles nodes denote the independent variables. The single-circle nodes denote the dependent variables. The connections with the solid line denote the,~deductive constraints. The connections with the dashed line denote the restrictive constraints. The depth-priority algorithm is used to get the feasible logical variables sets in a deductive graph. The numbers of logical variable sets after each deduction step are listed in Table 1. During the deductive process, the number of logical variable sets is increased slowly. And the calculation costs will not increase exponentially with the number of logical variables.
x x
~
,x.o
A-.
-i Y6 } X20
,
Y~. J
xg"
-'
xS I Y
Fig. 2. Superstructure of a process synthesis example. Table 1 Inference steps Constraint Inferred Var.
Var. sets
Constraint
Inferred Var.
Var. sets
(1) Independent var. (2) Independent var. (7) (6)
y9 Yl Y2 y6 Y7 y4
1 2 2 4 6 6
(5) (9) (10) (11) (12) (8)
y5 Ylo Y8 y12 Yll Restrictive cons.
12 18 18 18 30 16
(4)
Y3
8
(3)
YI3
32
Table 2 Optimal solution 0-1 Variables
yl 0
Continuous Variables
x~ 17.0
y2 1
Y3 0 x2 3.4684
y4 1
y5 0
y6 1
Y7 0
y8 0
y9 1
ylo 0
yll 1
yl2 0
y13 1
X3
X4
X5
X6
X7
X8
1.8667
1.6018
2.3333
1.8060
0.3111
1.4948
671 There are a total of 213=8192 combinations with these 13 logical variables, and the number of feasible integer variable sets is 32, which is 0.39% of the former. 32 continuous variables nonlinear subproblems can be generated through these logical variable sets. When solving these subproblems, the global optimum solution is found, which is 66.616. The corresponding values of continuous and logic~.l variables are shown in Table 2. For comparison, this example is also solved with Branch-and-Bound algorithm using Lingo 5. The local optimal solution is found after 154 iterations, which is 68.480 It would be noticed that the iterations of this algorithm are le '~an the Branch-and-Bound algorithm. In B-B algorithm, integer variables are relaxed . ;m 3us variables, and each sub-problem has 47 constraints and 21 continuous vari' ~c, this algorithm, each continuous variables NLP sub-problem has only 6 constraints in ,,verage, and some sub-problems are linear programming. So the sub-problems are solved rapidly. Furthermore, the global optimal solution would not be obtained in B-B algorithm, but in this algorithm, the global optimal solution can be found. 5. CONCLUSIONS AND DISCUSSIONS Currently the global optimization algorithms for M1NLP problem are improved from the Branch-and-Bound algorithm. But in these algorithms, the integer variables are relaxed into continuous variables, and it makes the model more complex and hard to solve. In this algorithm, logical variables and continuous variables are treated sequentially. Logical variables denote the flow-sheet structures, and continuous variables denote the technical parameters for a determined flow sheet. In theory, the number of candidate flow sheets is increased exponentially with the number of logical variables. If there are many logical variables, the candidate flow sheets will be more than what we can deal with. But the number of feasible flow sheets is far less than the candidate ones. The feasible flow sheets will be obtained through the logical deductive graph. For each feasible flow sheet, the un-selected devices and flows are deleted from the model, so the scale of the corresponding sub-problem model is much reduced, and it is easy to solve. Because all the feasible logical variable sets are obtained and the global optimization algorithm is used when solving each NLP sub-problem, to obtain the global optimal solution of M1NLP problems is guaranteed. REFERENCES [1] M. Turkay and I.E. Grossmann, Computers Chem. Engng., 20 (1996) 959. [2] C.A. Floudas, Deterministic Global Optimization: Theory, Methods and Applications, Kluwer Academic Publishers, Dordrecht, 1999. [3] E.M.B. Smith and C.C. Pantelides, Computers Chem. Engng., 23 (1999) 457. [4] M.A. Duran and I.E. Grossmann, Mathematical Programming, 36 (1986) 307. [5] J. Zhang, B.Z. Chen and S.Y. Hu, Symposium on PSE in China, (2001) 349. [6] J. Zhang, B.Z. Chen and S.Y. Hu, Chinese J. on Chem. Engng., 53 (2002) 178.
666
Variable decomposition based global-optimization algorithm for process synthesis Jian Zhang a, Bingzhen Chen a, Shanying Hu a, Xiaorong He a aDepartment of Chemical Engineering, Tsinghua University, Beijing 100084, China Abstract The key-task of process synthesis is to solve a mixed-integer non-linear (MINLP) model. As the number of integer variables increase, one is faced with a large combinatorial problem. At the same time, due to the nonlinearities the MINLP problem is tended to converge to a local optimal solution. To overcome these difficulties, a variable decomposition based global-optimization algorithm is presented, in which the integer variables and the continuous variables are treated sequentially. There are three steps in the algorithm: The first step is to classify the logical variables into independent and dependent variables, meanwhile, the logical constraints are classified into deductive constraints and restrictive ones. Logical deductive graph is generated to get the feasible logical variable sets. The second step is to construct a non-linear continuous (NLP) sub-problem for each feasible logical variable set and to solve the sub-problem using general global optimization algorithm. The third step is to obtain the global optimum solution. A process synthesis example is presented to compare the algorithm with the standard Branch-and-Bound algorithm. Keywords:
global optimization algorithm, logic-based MINLP model, process synthesis
1. INTRODUCTION Process synthesis is an important part of process systems engineering. The mixed-integer non-linear programming (MINLP) model is widely used in process synthesis. In an MINLP model, continuous variables are used to model the technical parameters in chemical process, such as flows, temperatures and compositions. The 0-1 variables (only have value of 0 or 1) or logical variables can be used to model, for instance, altemative candidates and the topological relations among equipments. There are two types of MINLP model: algebraic MINLP model and logic-based MINLP model [11. The disadvantage of the algebraic MINLP model is that the topological relations expressed by 0-1 variables are not direct, and hard to modeling. It limits the industrial usage of the algebraic MINLP model. In a logic-based MINLP model, the topological relations are expressed clearly by logical relations. Heuristic knowledge would also be integrated in the model to eliminate candidate flow sheets that conflict with technical practice, and would accelerate the solving process. So the novel global optimization algorithm proposed in this paper is based on the logic-based MINLP model. The global optimization MINLP algorithm includes Branch-and-Simplify method,
667 Hybrid Branch-and-Bound and Outer Approximation method, txBB algorithm [21 and symbolic reformulation algorithm. L31 For each iteration of the branch strategy, only one from the integer variables is fixed to 0 or 1, and the rest ones are relaxed into continuous variables. So the number of iterations is exponential with the number of logical variables. Furthermore, the relaxation of integer variables leads to an increase of the number of continuous variables and the scale of model, also, it may make some linear constraints transfer to nonlinear constraints. These make the relaxed model hard to solve. In this paper, a variable decomposition based global-optimization algorithm is presented. This algorithm is described in the second section. The specific steps of the algorithm are listed in section three. A reaction process synthesis example is presented in section four to test and compare the algorithm. The fifth section is a conclusion and discussion. 2. A L G O R I T H M DESCRIPTIONS The key of the variable decomposition algorithm is to treat the logical variables and continuous variables of the MINLP model sequentially. A set of logical variables whose values satisfy all logical constraints is called a feasible logical variable set. Locating the feasible logical variable sets is an exact reasoning problem. The logical variables' values of each feasible solution of the original MINLP model certainly are in the feasible logical variable region. Corresponding to each feasible logical variable set, a non-linear continuous (NLP) sub-problem is constructed. The global optimization solution of the original MINLP model is obtained from solving these sub-problems. In a process synthesis problem, although the combination number of integer variables may be very large, the number of feasible sets is few. For example, in a reaction process synthesis problem t4] there are 13 logical variables, and the combination number is 213=8192. But the number of feasible logical variable sets is 32, which is 0.39% of the combination number. For an ethylene refrigerant synthesis problem, I51 there are 25 logical variables, and the combination number is 225=33554432. But the feasible number is only 265, which is 0.00079% of the combination number. Especially, when integrating the heuristic knowledge, the flow sheets that do not satisfy technical experience are eliminated, and the number of feasible logical variable sets will be further reduced. 3. A L G O R I T H M The algorithm includes 3 steps: getting feasible logical variable sets; generating continuous variables NLP sub-problems corresponding to the feasible logical variable sets and solving them using the NLP global optimization algorithm; obtaining the global optimization solution. In a logic-based process synthesis model, the constraints include equipment constraints, connection constraints, and logical relations among equipments t6]. 3.1. Locating the feasible logical variable sets In process synthesis, some types of logical constraints are often used, and these
668 constraints would deduct logical variables directly: 1) f(y2,ys,...,yn)-~ y~: in this constraint, the value of yl
is partly restricted by
f(Y2,Y3 .... ,yn): if f(YE,Y3 .... , y ~ ) = l , then yl-1; if f(yE,Ya,...,y~)=O, then yl would have the value of 1 or 0. 2) f(Y2,Y3 ..... y,,)~--~ y~: in this constraint, the value of yl is fully determined by
f(Y2,Y3 ....,Y,). 3) yl v Y2 v Y3 v . . . v yn : this constraint can be transferred to ~(Y2 v Y3 v which is the first type. 4)
Y~ @Y2 EDY3~)'"EDYn:
--n(y 2 V
Y3 V . . . V y, ) ~
in
this
constraint,
variable
yl
can
be
...
V y , ) --~ y~,
expressed
by"
y~, which is the second type. And the original constraint changes to
Y2 @ Y3 ~) .-. ~) Y, ~ -n(Y2 v Y3 v ... v yn). 5) Y2 ~ Y3 ~) " " E~)y, @-'flY2 v Y3 v
... V
Yn). in this constraint, variable y2 can be expressed
(Y3 V . . . V Yn) --~ -lYE, which is the first type. And the original constraint changes to Y3 @ Y4 ~ -.- ~ Y, ~ ~ ( Y 3 v . . . v yn ). The new constraint has the same form with the original constraint, thus more dependent variables would be derived from it. All the deductive constraints form an oriented non-loop graph, which called deductive graph. Each node in a deductive denotes a logical variable, and the arc that goes into a node denotes the deductive constraint that deducts the logical variable, and the arc that goes out from a node denotes the constraint that is used to deduct other logical variables. Not every logical constraint can be transferred to deductive constraint. In some logical constraints, all member variables are dependent variables, or the deduction would cause loops in the deductive graph, so they could not be transferred to deductive constraints. These constraints restrict the existing logical variable sets, and are called restrictive constraints. A restrictive constraint would not deduce a logical variable, but it could be used to check whether the deduced logical variables are satisfied with it, so as to reduce the number of logical variable sets. When all logical constraints were transferred to deductive constraints or restrictive constraints, there would be some residual logical variables that were not set to dependent variables. These variables are called independent variables and used as the starting nodes of deduction. The meaning of "independent" is that when generating the feasible logical variable sets, the values of these variables are chosen freely. After the classification of the logical variables and logical constraints is completed, the feasible logical variable sets will be generated as follows: the nodes of independent variables, as starting nodes, have no arcs entering them, and the values of these variables are chosen freely. Then these nodes and their associated arcs that go out from them are hidden, and new starting nodes would appear in the graph. The value of a node is determined through its associated arcs that enter the node. When the values of the nodes are determined, these nodes and their associated output arcs would be hidden. If a restrictive constraint is met, existing variable sets would be restricted by the constraint and infeasible variable sets would be eliminated. Repeat these steps until all the nodes are treated with, then the finally existing logical variable sets are the feasible logical variable ones.
by:
669 It should be noticed that different sequence of logical constraints and logical variables would generate different deductive graph. But different deductive graph would only affect the intermediate results, and would not affect the final feasible logical variable sets. The feasible variable sets are only correlated to logical constraints of the model.
3.2. Generating the continuous variables NLP sub-problems For each feasible logical variable set, a corresponding continuous variables nonlinear sub-problem can be generated. The sub-problem is used to optimize the technical parameters when the flow sheet is determined. The constraints of the NLP sub-problem include equipment constraints and connection constraints, and the objective function of the sub-problem comes from the original objective function. The symbolic reformulation global optimization algorithm [3] is applied to solve the NLP sub-problem, and the global optimum for each sub-system will be obtained. 3.3. Obtaining the global optimum solution The global optimum solution is obtained from solving all the sub-problems as follows: 1) If this sub-problem has no solution, it will be discarded. 2) If the lower bound of this sub-problem is higher than the current solution, this sub-problem will be discarded. 3) If the optimal solution of this sub-problem is lower than the current one, the current solution will be updated. After repeating these steps for each feasible logical variables set, the final solution is the global optimum one of the original MINLP model. 4. PROCESS SYNTHESIS EXAMPLE Consider a process synthesis example [41 shown in Fig. 2. The equations of equipments yl-y8 and logical constraints are listed below. Yl :x3 - ln(x2 + 1)
1) 1 <---~y 9
Y2 :x5 = 1.2 ln(x 4 + 1)
2) Yl ~ Y2
Y3 :X8 = 1"5X9 + Xlo
3) Y3 V Y4 V Y5 v Y13
Y4 :Xl2 = 0"8X13
4) ~Y4 V ~Y3
3X14 --
5) ~Y4 V ~Y5
(6y4
Y5 :X15 = 2X16
6) Y6 V Y7 ~
Y6 "X20 = 1.5 ln(xl9 + 1)
7) ~(Y6 V Y7 ) ~) Y6 1~) Y7
Y7 :X22 = ln(x21 + 1)
8) Y6 V Y7 ~'ff Y4 V Yll
Y8 :x~8 = ln(x~o + x17 + 1)
9) ~(Y3 v Y5) --~ ~Yl0
0.4X17 < X10 < 0.8X17
10)
Y4
Y3 V Y5 V Y10 ~
11) Y8 ~ Yl2 12) Yl2 v Yll
,,......... ( 3 ) ~
Y8
Fig. 1. Deductive graph
(11)~
670 The objective function is: min 5y~-10x 3 + 8 y 2 - 1 5 x 5 + 6 y 3 +40X 9 +10y4 +15X14 +6Y5 -[- 80X16 +7Y6 +25X19 - 60X20 + 4 y 7 + 35X21- 80X22 + 5y 8 + 15x10 + x I + 45X25-65x~s + 122
In this example, there are 25 continuous variables, 13 logical variables and 12 logical constraints. Based on the algorithm, 2 independent logical variables, 11 dependent variables, 11 deductive constraints and 1 restrictive constraint are obtained. The logical deductive graph is shown in Fig. 1. The double-circles nodes denote the independent variables. The single-circle nodes denote the dependent variables. The connections with the solid line denote the,~deductive constraints. The connections with the dashed line denote the restrictive constraints. The depth-priority algorithm is used to get the feasible logical variables sets in a deductive graph. The numbers of logical variable sets after each deduction step are listed in Table 1. During the deductive process, the number of logical variable sets is increased slowly. And the calculation costs will not increase exponentially with the number of logical variables.
x x
~
,x.o
A-.
-i Y6 } X20
,
Y~. J
xg"
-'
xS I Y
Fig. 2. Superstructure of a process synthesis example. Table 1 Inference steps Constraint Inferred Var.
Var. sets
Constraint
Inferred Var.
Var. sets
(1) Independent var. (2) Independent var. (7) (6)
y9 Yl Y2 y6 Y7 y4
1 2 2 4 6 6
(5) (9) (10) (11) (12) (8)
y5 Ylo Y8 y12 Yll Restrictive cons.
12 18 18 18 30 16
(4)
Y3
8
(3)
YI3
32
Table 2 Optimal solution 0-1 Variables
yl 0
Continuous Variables
x~ 17.0
y2 1
Y3 0 x2 3.4684
y4 1
y5 0
y6 1
Y7 0
y8 0
y9 1
ylo 0
yll 1
yl2 0
y13 1
X3
X4
X5
X6
X7
X8
1.8667
1.6018
2.3333
1.8060
0.3111
1.4948
671 There are a total of 213=8192 combinations with these 13 logical variables, and the number of feasible integer variable sets is 32, which is 0.39% of the former. 32 continuous variables nonlinear subproblems can be generated through these logical variable sets. When solving these subproblems, the global optimum solution is found, which is 66.616. The corresponding values of continuous and logic~.l variables are shown in Table 2. For comparison, this example is also solved with Branch-and-Bound algorithm using Lingo 5. The local optimal solution is found after 154 iterations, which is 68.480 It would be noticed that the iterations of this algorithm are le '~an the Branch-and-Bound algorithm. In B-B algorithm, integer variables are relaxed . ;m 3us variables, and each sub-problem has 47 constraints and 21 continuous vari' ~c, this algorithm, each continuous variables NLP sub-problem has only 6 constraints in ,,verage, and some sub-problems are linear programming. So the sub-problems are solved rapidly. Furthermore, the global optimal solution would not be obtained in B-B algorithm, but in this algorithm, the global optimal solution can be found. 5. CONCLUSIONS AND DISCUSSIONS Currently the global optimization algorithms for M1NLP problem are improved from the Branch-and-Bound algorithm. But in these algorithms, the integer variables are relaxed into continuous variables, and it makes the model more complex and hard to solve. In this algorithm, logical variables and continuous variables are treated sequentially. Logical variables denote the flow-sheet structures, and continuous variables denote the technical parameters for a determined flow sheet. In theory, the number of candidate flow sheets is increased exponentially with the number of logical variables. If there are many logical variables, the candidate flow sheets will be more than what we can deal with. But the number of feasible flow sheets is far less than the candidate ones. The feasible flow sheets will be obtained through the logical deductive graph. For each feasible flow sheet, the un-selected devices and flows are deleted from the model, so the scale of the corresponding sub-problem model is much reduced, and it is easy to solve. Because all the feasible logical variable sets are obtained and the global optimization algorithm is used when solving each NLP sub-problem, to obtain the global optimal solution of M1NLP problems is guaranteed. REFERENCES [1] M. Turkay and I.E. Grossmann, Computers Chem. Engng., 20 (1996) 959. [2] C.A. Floudas, Deterministic Global Optimization: Theory, Methods and Applications, Kluwer Academic Publishers, Dordrecht, 1999. [3] E.M.B. Smith and C.C. Pantelides, Computers Chem. Engng., 23 (1999) 457. [4] M.A. Duran and I.E. Grossmann, Mathematical Programming, 36 (1986) 307. [5] J. Zhang, B.Z. Chen and S.Y. Hu, Symposium on PSE in China, (2001) 349. [6] J. Zhang, B.Z. Chen and S.Y. Hu, Chinese J. on Chem. Engng., 53 (2002) 178.