symbolic integration of logic in milp branch and bound methods for the synthesis of process networks

Annals of Operations Research 42(1993)169-191 169

Symbolic integration of logic in MILP branch and bound methods for the synthesis of process networks

Ramesh Raman and Ignacio E. Grossmann

Department of Chemical Engineering, Carnegie Mellon University Pittsburgh, PA 15213, USA

Abstract

This paper deals with the branch and bound solution of process synthesis problems that are modelled as mixed-integer linear programming (MILP) problems. The symbolic integration of logic relations between potential units in a process network is proposed in the LP based branch and bound method to expedite the search for the optimal solution. The objective of this integration is to reduce the number of nodes that must be enumerated by using the logic to decide on the branching of variables and to determine by symbolic inference whether additional variables can be fixed at each node. The important feature of this approach is that it does not require additional constraints in the MILP and the logic can be systematically generated for process netwo~cs. Strategies for performing the integration are proposed that use the disjunctive and conjunctive normal form representations of the logic, respectively. Computational results will be presented to illustrate that substantial savings can be achieved.

1. Introduction

Discrete decisions occur frequently in the modelling of problems in process design and synthesis. For example, decisions regarding what units to integrate in a process flowsheet (process network), the number of trays in a distillation column or the available reactor sizes are typically discrete. With discrete variables as well as with continuous variables that are used to represent parameters like temperatures, pressure and flowrates, one can model process design and synthesis problems through an objective function and a set of constraints representing material, heat balances and specifications. When these involve nonlinearities, a mixed-integer nonlinear program (MINLP) results. However, if simplified process models are used, synthesis problems can often be posed as a mixed-integer linear programming (MILP) problem. Needless to say, the modelling step is crucial to the success of the synthesis method and a discussion on this topic can be found in Kocis and Grossmann [16] and Grossmarm [12].

A major step of this design procedure involves the solution of the MI(N)LP optimization model. The most common solution technique for the MILP problem is the well known LP-based branch and bound algorithm (see [4, 18]) which is

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170 R. Raman, I.E. Grossmann, Symbolic integration of logic

implemented in most computer codes like LINDO, ZOOM, SCICONIC and OSL. Recently, these methods have been improved by the use of preprocessing techniques that can reduce dimensionality and cutting planes that can reduce the integrality gap of the MILP (e.g. see [24]). Another alternative method for MILP is the Benders decomposition method [5] which involves the alternate solution of pseudo-integer master problems (MILP) and LP subproblems with fixed 0-1 variables. The former are solved by branch and bound to optimize the 0-1 variables and to provide lower bounds (minimization case); the latter optimize the continuous variables and provide upper bounds. For MINLP problems, the common solution techniques are the Generalized Benders Decomposition (GBD) [ 11 ] and the Outer Approximation (OA) [9] algorithms. Both are based on a computational strategy similar to Benders decomposition, but the difference lies first in the fact that the continuous subproblems correspond to NLP problems. The two approaches differ in the form of the MILP master problem that is solved at each iteration, which is again solved by branch and bound. Finally, the solution of the MINLP can also be performed directly with a branch and bound method where NLP subproblems are solved at each node of the tree (see[4, 13, 17]). From the above, it is clear that the LP based branch and bound method plays a central role in the solution of mixed-integer optimization problems.

Since in many synthesis problems the number of 0-1 variables that are required can be rather large, the potential size of the tree that must be examined by a branch and bound method can become a major bottleneck in the computations. On the other hand, knowledge about the synthesis problem in the form of engineering heuristics and the logic of relations among units can potentially provide information about the design space and make the problem easier to solve. In two recent papers by Raman and Grossmann [20,21], these authors showed how both logic relations in processes and engineering heuristics expressed in the form of propositional logic can be represented in terms of linear inequalities involving 0-1 variables (see also [6,25]). Based on this representation, these authors proposed the integration of logic and engineering heuristics in quantitative optimization models. They considered the addition of the constraints for the logic relations among units in the MI(N)LP to decrease the relaxation gap (difference between integer solution and relaxed continuous problem). Constraints for engineering heuristic rules and logic relations were used to improve the search in the master problem of MI(N)LP algorithms. The computational results showed that the addition of logic constraints in MILP problems often produces significant reductions in computational time. This is due to the reduction in the number of nodes that have to be enumerated in the branch and bound tree despite the increased size of the MILP.

While the above improvements in mixed-integer optimization are encouraging, a basic question that still remains is how to best represent and incorporate logic and engineering heuristics in optimization. In contrast to a quantitative approach, another possibility is to represent logic propositions in terms of Boolean variables and perform symbolic inference on them. The two extreme ways of representing propositional logic are the Conjunctive (CNF) and the Disjunctive (DNF) Normal

R. Raman, I.E. Grossmann, Symbolic integration of logic 171

forms, respectively [8]. Propositional logic based inference techniques involve application of Boolean algebra for the DNF case. For the case of CNF representations, techniques include unit resolution (see [3]). Here, Horn clause systems are particularly easy to tackle since the inference for this case can be shown to be linear in time [7].

In this paper we propose a symbolic integration of logic relations between potential units in process networks to aid the branch and bound process in searching for the optimal solution in MILP synthesis models. The objective of this integration is to reduce the number of nodes that must be enumerated by using the logic to decide on the branching of variables, and to determine by symbolic inference whether additional variables can be fixed at each node. An important feature of this approach is that it does not require additional constraints in the MILP. Two different strategies for performing the integration are proposed. In one strategy the logic will be converted to the DNF form to address synthesis problems in which the process networks have significantly fewer feasible design configurations than the total number of 0-1 combinations. For example, the synthesis of toluene hydrodealkylation (HDA) process by Kocis and Grossmann [16] involves 13 binary variables but only has 198 feasible configurations, which is far smaller than the 213 = 8192 possible combinations of the binary variables. The alternatives in this process can be represented by 198 disjunctions. For the case of process networks with a large number of feasible alternatives, the logic will be converted into the CNF form to avoid the problem of handling a large number of disjunctions. Computational results are presented on several process synthesis problems to demonstrate the potential of the suggested methods.

2. Background

Consider the MILP formulation (P1) for a given synthesis problem, where x and y are the vectors of continuous and binary variables, respectively,

(P1) ZIP = minimize

subject to

aTx + bTy

Cx + Dy<_d,

Ey<e,

x > 0 , y e {0,1}p.

The mixed-integer constraints represent mass and energy balances as well as logical conditions; the pure integer constraints correspond to the logical specifications. The mass and energy balances constrain the relationships between the binary variables implicitly, while the integer constraints constrain them explicitly.

Problem (P1) can be solved with the LP-based branch and bound procedure which is a rigorous tree enumeration method for finding the global optimum of the

172 R. Raman, LE. Grossmann, Symbolic integration of logic

MILP. The efficiency of the branch and bound algorithm can be determined from the number of nodes that must be enumerated in the tree as well as the number of simplex iterations required for the search. This depends heavily on the relaxation gap (the difference between the integer solution and the lower bound at the root node), the upper bound, the branching rule and the subproblem selection rule. Since the last three rely on heuristic procedures that are general in nature, the branch and bound method may not take advantage of the underlying logic structure that characterizes a specific problem.

In order to improve the performance of the branch and bound algorithm, the proposed approach utilizes the additional logic relations that define the connectivity among the various units in the process network in order to provide more information to the search procedure. These logic relations can be regarded as logic cuts for MILP problems as discussed in Hooker et al. [ 14]. Instead of representing the logic in the form of linear constraints in the model [20], they will be used in symbolic form. Approaches for performing the integration will be presented for the cases when the logic is converted into the Disjunctive and Conjunctive Normal forms.

Consider that the specific logical relationships among the units in the process network are first expressed explicitly in the form of logic propositions. Let the logic relations be given by a set of conjunctions of clauses,

A= [L1AL2A...ALq}, (1)

where Li is a logical proposition expressed in terms of implications, OR, EXCLUSIVE OR and AND operators. Two ways of transforming the propositions in A to perform the inference are as follows. In the first case, the logic propositions are converted into the Conjunctive Normal Form (CNF):

L iGPI i ~ J L i~P2 ie-P22 J L i~P. lee j m

where Pi and P/are subsets of the Boolean variables that correspond to some of the 0-1 variables, and s is the number of resulting conjunctions.

In the second representation, the logic propositions are converted into Disjunctive Normal Form (DNF):

D D = [ A (Y/)A (.~Y/)I v I A (Y/)A (---,Y/)Iv ... v [ A (Y/)A___(---,Y/) I, (3) L i~Qa i~'~1 J L i~Q2 iE-~2 _] i~Qr i~Q, I

where Yi is a Boolean variable representing the existence of unit i and --,Yi its nonexistence; Qj and Qj are the index sets of the Boolean variables which correspond to a partition of all the 0-1 variables Yi, i = 1 . . . . , p. Each clause separated by a disjunction represents the assignment of units in a feasible configuration, where it

R. Raman, LE. Grossmann, Symbolic integration of logic 173

is assumed that each Boolean variable has a one-to-one correspondence with the 0 - 1 binary variables of the MILP model. Therefore, r represents the number of alternatives in the superstructure.

To illustrate the CNF and DNF representations in (2) and (3), consider the small example problem shown in fig. 1 in which all units are candidates except the

A ~ Y1 x3 B

xl x4~ Y2

Fig. 1. Superstructure for selection of processes.

Y3

simultaneous selection of units 1 and 2. The following propositional logic expressions apply:

LI : Y1 v Y2 =~ I"3 (process 1 or process 2 imply process 3);

/-,2 : Y3 =~ Y1 v Y2 (process 3 implies process 1 or process 2);

/-,3 : (",Yl v --,Y2) (do not select process 1 or do not select process 2).

Applying the contrapositive to L1 and /-,2, and using DeMorgan's theorem, the corresponding CNF representation for the logic is:

D.c = (--,Y1 v Y3) A (",Y2 v Y3) A (-,Y3 v Y1 v Y2) ^ (--,Y1 v Y2). (4)

Distributing the OR over the AND operators and simplifying, the corresponding DNF representation is given by:

f~D = (Y1 A ~Y2 A Y3) V (Y2 A ",Y1 A Y3) v (--,Y1 A "-,Y2 A --1Y3), (5)

where each disjunct clause represents a design alternative in fig. 1. The above logic propositions on the relations among units have been obtained

by inspection. Appendix A presents a procedure to systematically derive these logic relations in process network that involve splitters, mixers and units.

3. Integration of logic within branch and bound search

In this section we will propose a scheme for incorporating the logical relationships in the DNF and CNF forms in the branch and bound method. The model that we are attempting to solve is as follows,


(P2) Zip = minimize

subject to

aTx + bTy

Cx + Dy <_ d,

Ey<-e,

x > O, y E {O,l}p,

Y~D or Y~C"

It must be noted that the constraint y ~ f2 D or y e [2c is valid in that both models (P1) and (P2) have the same optimal solution (see Hooker et al. [14]). As mentioned earlier, the constraints for the logical relationships among units (f2c) can be added to the MILP model to tighten the LP relaxation. But the addition of a large number of constraints into the mathematical model can also lead to an increase in the solution time of each LP subproblem. Therefore, it is proposed in this work not to add these constraints as linear inequalities to the MILP model (P1), but simply keep those logical constraints that are specifications of the original model (e.g. do not select both processes 1 and 2 in fig. 1). The DNF and the CNF representations of the logic will be analyzed and solved symbolically for the branching rule and subproblem selection as discussed below.

Branching rule: Instead of relying only on general heuristics (e.g. variable with value closest to 0.5) or penalties, we will make use of the logic in the DNF and CNF forms that applies to each node.

The basic idea in the case of the DNF form is as follows. In order to prune branches as soon as possible, it is advisable to search among branches that quickly lead to integer solutions. In order to do this with the DNF representation, the variable that has the highest priority for branching is the one that occurs the least number of times in non-negated or negated form in the disjunctions of the current node of the tree. The reasoning behind the argument is that the variable that occurs the fewest number of times in non-negated or negated form has the fewest number of feasible solutions among the different alternatives implied by the disjunctions. Also, since such a variable has the potential of eliminating a larger number of disjunctions by making them false (i.e. infeasible), this increases the possibility of solving the Boolean equation for all or some of the remaining variables which can be fixed at that node.

For the case of the CNF representation, the branching rule by Jeroslow and Wang [15] developed for the satisfiability problem has proved to be the most effective for a similar reason as in the DNF case. The variable that occurs in the largest number of short clauses is chosen as the variable to branch on. The motivation here is that using unit resolution it is more likely for one to be able to fix other variables in a short clause; furthermore, the higher the number of short clauses the branching variable appears in, the more variables are likely to be fixed at the resulting node.


Chaining rule: The choice of which additional variables to fix to their bounds is also made based on symbolic inference procedures. Both approaches attempt to fix as many of the other binary variables as possible by analyzing the available logic. The CNF based approach utilizes unit resolution (see [3]) for the purpose of making inferences, while the DNF approach simplifies the current Boolean equation, and searches for variables that occur with the same sign in all the disjunctions and setting them to that value at the node.

.

4.1.

Formal procedure for proposed strategy

DISJUNCTIVE NORMAL FORM APPROACH

4.1.1. Branching rule

In order to state the procedure formally, consider a node n that is under examination in the tree. Let Nn be the subset of 0 - 1 variables with fixed discrete value, and M,, the subset of 0 - 1 variables that are treated as continuous (free). That is, Nn = {i l Yi is fixed at node n}, while Mn = {i l Yi is free at node n}. Furthermore, consider the partition of the set of fixed variables Nn such that Tn = {i ~Nn lYi = 1}, and Fn = {i ~Nn lYi = 0} (of course at the root node of the tree Tn = O, Fn = 9) . We can then associate the Boolean variables Yi for i ~ Tn, and -'Yi for i E Fn. Since these Boolean variables are fixed at the node n, one can combine them with the logic at node n expressed in disjunctive normal form with the following equation:

I.i,r, i,F. .! [Y=q.i'a,i i,~-~,j j j (6)

where Qnj and Qnj, J = 1 . . . . . r, are a parti t ion of the set M,, of free variables. In order to select the variable to be branched on among those belonging to the set M,,, define a / a n d r / t o identify the negated and non-negated terms as,

i , ' 0n' ' ,7 , a[ = if i E Qnj;

The index q of the binary variable with fewest number of non-negated terms is then given by

q=arglminlnfi.nI ~.~aJl.m!nI ~.~fl[I] 1. l L J k i~M- ) J ~.iEM, J~j

(8)


4.1.2. Chaining rule

In this paper we will consider that having selected the binary variable yq, we will expand at node n the two nodes for yq = 0 (--,Yq) and yq = 1 (Yq). However, before solving the corresponding LP subproblems at the two new corresponding nodes, we will perform a logical test to determine whether additional 0 - 1 variables can be fixed. For this we solve the Boolean equation in (6) by adding the conjunction for the new fixed Boolean variable. Consider the case when this variable is Yq (this is equivalent to fixing yq = 1). We then first compute for each of the r terms in the disjunctions in (6) the expression:

Wjn+l = Yq A [ A Yi ieQnj ("a]'i)], j = 1 . . . . . r. (9)

We then proceed to simplify eq. (9) as follows:

r If q e Qaj, ~l~ a + l Yq ^ l ^ Yi A__ (~Yi) ,whercQn+u=Qnj\q, Q_n+u=Qny.

L ie~# ie a.l

If q e'Qnj, WT+l =false, where Q . + u = ~ , "Qn+lj = ~ .

From eqs. (6) and (9) the Boolean equation for node n + 1, will then be given by

I_ieT. ieFn j=l,r ieQ.+ll ieQ.+u

Finally, the above Boolean equation can be solved by dcfining:

Then the following Boolean variables can be calculated,

Yi = true, i eRn+l;

Yi =false, i ES a+l.

This then implies that in addition to fixing yq = 1 at node n + 1, we can also fix yi = 1, i ER a+l, and y i = 0 , i ES n+l provided these sets are non-empty. The case when the variable yq is set to zero (--,Yq) at node n + 1 is similar to the above steps.


4.2. CONJUNCTIVE NORMAL FORM APPROACH

4.2.1. Branching rule

Consider a node n that is under examination in the tree. Let Nn = {i I Yi is fixed at node n} and Mn = {i iYi is free at node n}. Furthermore, consider the partition of the N,, such that T,,= { i ¢ N n l y i = 1 } and F,,= {i ¢ N , ly i=O}. Then the logic at node n can be expressed in the conjunctive normal form as

L i,P~ iE~ J L i,P2 i , ~ j L i,P. i,-f,, j

F Define 7 ,ff and Yij to identify the non-negated and negated terms as,

y T = l if i ~ l ~ \ T n and (12)

7ij =1 if i~

Define Aj, the length of the clause, as

Aj =ll~ \ T n I + I P j \ F n !. (13)

To measure the largest number of occurrences in short clauses, we define, as in Jeroslow and Wang [15],

~T T -&- = ~ y / / 2 J

and J (14)

~,f F -A- = ~y / j 2 J. J

The index q of the binary variable Yq that is to be chosen for branching is then given by

q = arg {max [max(yT), m/ax(TF)]}. (15)

4.2.2. Chaining rule

Assuming that we branch on Yq, the resulting CNF with s clauses, j = 1,2 . . . . . s, is of the form,

L iEP~ i ~ J LiGP2 i ~ j Li~e, i~F, j


To determine which additional 0 - 1 variables can be fixed when branching on Yq, unit resolution can be performed and the steps are as follows:

Step 0. Set FIX e = {q}, FIXn = 0 .

Step 1. Set j = 1, VARFIXED = false.

Step 2. If (Tn u F I X p ) n Pj ~ 0 , the clause V7+1= true since at least one non- negated Boolean variable is true. Go to step 4.

Step 3. To determine the number of unknown Boolean variables, in the j th clause,

set m~ = l e j l - l F n c ~ P j l - I F I X n n P j l and

(a) If m~ + m~ = 1 (i.e. only one unknown Boolean variable), then

VARFIXED = true.

If m~ =0, set Yi = true, i ~ Tn u FIX e,

FIXe= FIXp u {i},

FIX n = FIXn.

If m~ = 0, set Yi = false, i ~ Fn u FIXn,

FIXp = FIX e,

FIXn = FIXn u {i}.

(b) If m~ +mn J > 1 (i.e. two or more unknown variables), go to step 4.

(c) If m~ + m~ = 0, all binary variables in the conjunct are fixed.

Step 4. If j < s, j = j + 1, go to step 2. Otherwise go to step 5.

Step 5. If F I X e u FIXn = { 1,2 . . . . . p}, then all binary variables are fixed. Go to step 6. Note that one still needs to solve the LP at the node.

If (Tn L) FIXp) c~ (Fn L) FIXn) # 0 , there is a contradiction in the logic. The node is fathomed. Stop.

If VARFIXED =false, then no variable has been fixed during the pass. Go to step 6. Else go to step 1.

Step 6. The following Boolean variables (and the associated 1 - 0 variables) can be fixed:

Yi = true, i ~ FIXp; Yi = false, i ~ FIXn.

The case when the variable yq is set to zero (~Yq) at node n + 1 is similar to the above steps.

R. Raman, LE. Grossmann, Symbolic integration of logic 179

5. Discussion of the proposed approaches

The most important difference between the DNF and CNF approaches is related to the class of problems that each of them are adapted for. The DNF representation based approach is better suited for the class of mixed integer problems where the number of feasible design solutions is small compared to the number of combinations of binary variables that model the system. In this case, generating the DNF representation can be done reasonably efficiently since the number of disjunctions is not too large due to the constrained nature of the process network. For the more general case, however, the number of disjunctions can become very large. The generation of the DNF structure could become computationally significant as it could increase exponentially with the number of variables. The generation of the CNF representation, on the other hand, is polynomially bounded and therefore relatively inexpensive even when there are a large number of alternatives.

However, once the representations are obtained, it is easier to perform inferences on the DNF than on the CNF representation. Firstly, the branching rule for DNF is generally stronger than for CNF as will be shown in the numerical results. Secondly, once the branching variable has been chosen, the procedure to check for fixed variables is much easier in the DNF case than in the CNF case. Therefore, the size of the DNF based branch and bound tree is usually smaller than the CNF based tree. Furthermore, a bound can be established on the maximum number of nodes that need to be examined with the DNF approach:

THEOREM 1

The number of nodes to be examined by branch and bound with the disjunctions in (3) is not more than 2 r - 1, where r is the number of disjunct clauses.

The proof for the theorem is given in appendix B.

6. Computer implementation

The proposed branch and bound schemes were initially tested with the computer codes ZOOM [22] and SCICONIC by externally generating the sequence of LP subproblems since these codes do not provide facilities for interfacing routines for branching or fixing variables.

An automated implementation was developed using the Optimization Subroutine Library (OSL) which is a collection of high performance mathematical subroutines that can be used to solve optimization problems on the RISC System/6000 workstations. All the modifications to the default branch and bound search procedure were coded as FORTRAN 77 user-exit subroutines to OSL. The branching variable was chosen by the logic based schemes proposed in the earlier sections. Any ensuing tie was broken by simply branching on the tied variable that is closest to 0.5 in the relaxed solution. It must be noted that more sophisticated numerical-based branching rules


could be used instead. The branching rule was incorporated in the OSL user-exit EKKBRNU subroutine which is invoked before the solution of the LP subproblems at each node. The proposed procedures to identify other variables to be fixed at a node apart from the branching variable were incorporated in the user-exit EKKEVNU subroutine. These variables are then fixed by changing their bounds in OSL. All other components of the branch and bound search correspond to the default options provided by OSL. The DNF or CNF representations of the logic are inputs to the system along with the MPS file containing the mathematical programming formulation of the problem. A flow diagram of the implementation scheme is shown in fig. 2.

OSL Proposed scheme

Fig. 2. Computer implementation of proposed algorithms.

7. Numerical results

In order to demonstrate the effect of the proposed procedure on the branch and bound search, two sets of problems have been solved. The first one involves the MILP model for the synthesis of separation systems with no heat integration [1] and this has been solved for problems involving 4, 5, and 6 components. The second


set of problems correspond to heat integrated distillation sequences in which the temperatures are treated as continuous variables.

The process network for the 4 component system without heat integration is shown in fig. 3. The logical constraints that relate the existence of units are described in Raman and Grossmann [20]. The problems were solved with the original MILP

ABCD

A/BCD~

AB/CD

AB(

B/CD

A/BC

A/B

B/C

C/D

AB/C

Fig. 3. Superstructure for 4 component separation.

formulation (P1), with logical constraints added as inequalities, and with the proposed method described in this paper. The branch and bound codes used were ZOOM, SCICONIC and OSL through the modelling system GAMS. Default options were used on all three codes. The results for the problems are shown in tables 1 to 3. Note that the formulations with explicit constraints for the logic have almost twice the number of constraints than the original MILP model, but they produce tighter LP relaxations. The proposed algorithm has been tested on the IBM POWER 530 workstation. For the case of OSL, the CPU times reported include the time for the symbolic manipulations of the logic.

Thefour component problem only involves 5 feasible alternatives. As can be seen in table 1, the addition of the hard logic constraints causes theproblem to ,be solved as a relaxed LP but increases ~ e size of the problem such that there is appreciable decrease in the time. On the other hand, use of the propositional logic


maintains the size of the problem to that in the original problem, and decreases the number of nodes from 33 in ZOOM and 17 for SCICONIC to 6 using the DNF and 8 using the CNF based approaches. With OSL the reduction is from 12 to 8 and 10 nodes with the DNF and the CNF approaches, respectively. The time savings with the reduction of nodes was not realized due to the small size of the problem.

Table 1

Four component synthesis problem.

Original Formulation DNF based CNF based formulation with logic approach approach

Size constraints 27 49 27 27 variables 31 31 31 31 binary 10 10 10 I0

LP relaxation 3601.6 3625.8 3601.6 3601.6

MILP solution 3625.8 3625.8 3625.8 3625.8

ZOOM no. of nodes 33 0 6 8 no. of iterations 123 87 54 59 CPU time • 0.15 0.14 0.09 0.1

SCICON]C

no. of nodes 17 0 6 8 no. of iterations 45 32 38 39 CPU time • 0.13 0.08 0.06 0.08

OSL no. of nodes 12 0 8 10 no. of iterations 26 21 20 43 CPU time" 0.54 0.32 0.6 0.9

• Seconds, IBM Power 530.

In the five component problem, there are only 14 feasible 4-column sequences although 20 binary variables are required for modelling the superstructure. As can be seen from table 2, the proposed branch and bound search shows considerable improvement over the original algorithm. The addition of the hard logic constraints does reduce the time with both ZOOM and SCICONIC (by 42% and 22%). However, the proposed DNF based approach yields even more substantial time savings with both codes (by 78% and 58%). This is mainly due to the fact that the proposed method only requires 4 nodes as opposed to 44 with ZOOM and 26 with SCICONIC. The CNF based approach, too, yields time savings of 60% and 32% with ZOOM and SCICONIC respectively and requires 13 nodes. With OSL, the original algorithm requires 52 nodes while the DNF and CNF based algorithms solve the problem in


11 and 14 nodes respectively. It should be noted that with OSL the number of nodes with the DNF and CNF approaches is not the same as in ZOOM and SCICONIC presumably because different subproblem selection rules were applied. In the case of the ZOOM and SCICONIC implementations, a depth first strategy was used until feasible integer solutions were obtained; the selection of the subsequent node was based on best lower bound.

Table 2

Five component synthesis problem.


Size

constraints 51 90 51 51

variables 61 61 61 61

binary 20 20 20 20

LP relaxation 4262.7 4278.1 4262.7 4262.7

MILP solution 4304.1 4304.1 4304.1 4304.1

ZOOM no. of nodes 44 16 4 13

no. of iterations 577 302 118 221 CPU time • 0.86 0.5 0.2 0.35

SCICONIC no. of nodes 26 3 4 13

no. of iterations 79 48 60 68

CPU time • 0.19 0.15 0.09 0.13

OSL no. of nodes 52 2 11 14


CPU time' 1.23 0.49 0.74 1.33


In the six component problem, 35 0 - 1 variables are used to model a superstructure that contains only 42 5-column sequences. Here, as shown in table 3, only 10 nodes were required with the DNF based approach and 12 with the CNF based approach, as opposed to the 55 in ZOOM and the 61 in SCICONIC. Also, the addition of the logic constraints required 16 and 7 nodes, respectively. Furthermore, the proposed method was able to reduce the time requirements from the original algorithm by about 74% and 68%, and 72% and 65% with the DNF and CNF based approaches, respectively. With OSL, the original algorithm requires 141 nodes while the DNF and CNF based algorithms require only 18 and 26 nodes, respectively.


Table 3

Six component synthesis problem.


Size

constraints 86 156 86 86

variables 106 106 106 106

binary 35 35 35 35

LP relaxa6on 18114.7 18161.6 18114.7 18114.7

MILP solution 18170.3 18170.3 18170.3 18170.3

ZOOM

no, of nodes 55 16 10 12

no, of iterations 1050 787 292 326

CPU tim& 2.5 2.01 0.64 0.72

SCICONIC no. of nodes 6I 7 I0 12

no, of iterations 140 162 98 100

CPU time a 0.44 0.31 0.14 0.16

OSL

no. of nodes 141 8 18 26


CPU tim& 3.46 1.18 1.06 3.09

"Seconds, IBM Power 530.

The second type of problem also involves the synthesis of a four component separation system but in addition considers heat integration by allowing for heat matches between reboilers and condensers of the various separation columns in order to reduce the utility cost for operation (see [21]). The number of binary variables required to model this system increases sharply from the previous case. For a 4-component system, 100 binary variables are needed - 10 to model the existence of the distillation columns and 90 to model the existence of heat exchange matches between the reboilers and condensers of the various columns. The results for this problem with the DNF based approach are shown in table 4. The proposed branch and bound search shows a considerable improvement over the original algorithm. While ZOOM and OSL were not able to solve the problem even after one million iterations and over 10 CPU hours on the IBM RISC/6000, and SCICONIC takes over 5700 nodes, the DNF based approach solves the problem with t 69 nodes with ZOOM, 29 nodes with SClCONIC and 89 nodes with OSL! Note that in this case, the time savings are quite substantial with the symbolic DNF based approach.

The effect of the OSL preprocessor in the above problem is shown in table 5. Two different instances of this problem are solved with the aid of the OSL


Table 4

Synthesis of heat integrated distillation columns.

Original DNF based formulation approach

Size

constraints 147 t 47 variables 261 261

binary 100 100

LP relaxation 388.3 388.3

MILP solution 3627.5 3627.5

ZOOM

no. of nodes could not solve 169

CPU time > 2000 4.39

SCICONIC

no. of nodes 5722 29

CPU time 109.6 1.3

OSL

no. of nodes could not solve 20

no. of iterations > 1,000,000 238

CPU time a > 15,000 2.76


Table 5

Effect of preprocessors

OSL OSL Logic based without preprocessor with preprocessor branch and bound

Problem 2a

no. of nodes

no. of iterations

CPU time"

Problem 2b

no. of nodes

no. of iterations

CPU time"

could not be solved 153 20

> 1,000,000 7900 238

> 5000 57.95 2.76

448 235 89

1651 28883 640

22.06 169.39 7.58


preprocessor, where supemode preprocessing was performed and a copy was kept of the new matrix with tighter bounds and extra rows at each node, and the results are compared with both OSL without the preprocessor and the DNF based branch and bound scheme. In the first instance, while OSL cannot solve the problem


without the aid of the preprocessor, its use helps to solve the problem in 153 nodes and 7900 iterations. The DNF based approach requires only 20 nodes and 238 iterations. For the second instance of the problem, the addition of the preprocessor reduces the efficiency of OSL both in the number of nodes and in the number of iterations required. The DNF based approach is again more efficient since it requires much fewer nodes and iterations.

Finally, it should be noted that one might also consider a hybrid scheme in which logic inequalities that are violated are incorporated within the symbolic branch and bound in order to tighten the lower bound of the LP subproblems. In order to study the effectiveness of such a scheme, we have solved the 4, 5 and 6 component separation problems by including the logic inequalities that are violated at the stage of the LP relaxation. This was done by solving the relaxed LP successively until all the logic inequalities were satisfied. The results are shown in table 6. As can be seen, this actually proved to be the most efficient scheme in the 6 component problem. However, since these problems are relatively small, more computational experience is required to draw general conclusions.

Table 2

Effect of addition of violated inequalities at root node.

OSL original Formulation DNF based DNF with formulation with logic approach violated inequalities

Foul"

Logic inequalities 0 22 0 8

no. of nodes 12 0 8 0


CPU time" 0.54 0.32 0.6 0.29

Five

Logic inequalities 0 39 0 6

no. of nodes 52 2 11 4

n o . of iterations 179 48 55 49

CPU time t 1.23 0.49 0.74 0.46

Six Logic inequalities 0 70 0 11

no. of nodes 141 8 18 5 no. of iterations 386 219 84 67 CPU tim@ 3.46 1.18 1.06 0.7


8. Conclusions

This paper has shown that logic of relations among units in process networks can be systematically generated and effectively integrated in MILP models for


synthesizing process networks. Converting the logic propositions into DNF or CNF form, a branch and bound method that performs logical inferences at each node has been proposed. As has been shown, this provides effective branching rules as well as a mechanism for fixing subsets of 0-1 variables at each node. Numerical results on MILP models for separation synthesis have shown that this method can greatly reduce the number of nodes to be enumerated in the branch and bound search with the DNF form producing the largest savings. In the case of a heat integrated distillation problem, reductions of two orders of magnitude were achieved. Also, it has been shown that while the effect of preprocessors is unpredictable, the logic based approach takes advantage of the structure in the problem and improves the efficiency of the search. The advantages of processing the logic in symbolic form rather than numerically through constraints have also been shown, as well as the use of a hybrid scheme in which violated inequalities are included as part of the symbolic branch and bound method.

Appendix A: Formal generation of logic in a process network

As a first step in the generation of the logic, the process network is represented by a directed graph. The nodes and arcs in the graph are further classified into various subgroups depending on the tasks that they perform in the superstructure. The various kinds of nodes needed are:

1. Mixer: mixes two or more streams. No unit operation is involved. 2. Splitter: splits a stream into multiple streams. No unit operation is involved. 3. Unit: these include units that perform a change in composition and

pressure and temperature in the output streams. Examples include compressors, reactors, distillation columns, absorption towers and membrane separators.

4. Sources/sinks: these provide the inlets and outlets to the process flowsheet from the external environment. The feed, product and by-product streams require sources or sinks.

Recycle streams are handled by removing the connectivity with the main process path at the end of the loop. This prevents one from obtaining circular relationships, without losing the connectivity of the flowsheet. The proposed procedure to develop the logical relationships in the model is described below.

1. Associate Boolean variables with every node in the graph.

Boolean variables Y will be used to represent the existence of all units (U,,) in the network and Z to represent the splitters, mixers, sinks and sources.

2. Develop relationship between Boolean variables.

This depends on the class of nodes to which the variables belong. The logic associated with the various kinds of nodes (see figs. 4 a,b,c) is as follows:

188 R. Raman, I.E. Grossmann, Symbol ic integrat ion o f logic

7_a~q "~)Yc

xer

(a) Mixer

Va

© s Yc

(b) Splitter

Ya Y1 Yu 0

Yn IYI

(ct Un component

Fig. 4. Nodes in the graph of a superstructure.

(a) Mixer Z m ~ Ya v Yb,

Z m ~ Yc. (A1)

If Ya v Yb ~ Z m also holds, then the first relation can be rewritten as Ya v lib ¢::> Z m .

Similarly, if Yc ~ Z m holds, then the second relation can be rewritten as Yc ~ Z m .

(b) Splitter Z s ~ Ya,

Zs ~ Yb v Yc. (A2)

If Ya ~ Z s also holds, then the first relation can be rewritten as Zs¢:, Ya. Similarly, if Z s ~ Yb v Yc holds, then the second relation can be rewritten as Z s ~ Yb v Yc.

(c) U, component Yu ~ Ya A Yb A . . . ~,, Yn,

Yu ~ Y1 ,,~ Y2 A . . . ,'~ Ym. (A3)


3. U s e r s p e c i f i c a t i o n s .

Apart from the logic associated at every node, user specifications like limits on the selection (e.g. at most one reactor) and availability of feed streams also need to be considered. In case an input or output stream is known to exist, set the corresponding Boolean variable to t r u e and simplify the logical relationships.

The above method generates all the logic describing the connectivity between the various units in terms of Y and Z variables. This approach, however, requires the use of the variables Z which are often not included in the quantitative model (usually only Y variables are used) of the flowshect. The number of variables Z can be decreased by applying the following procedure:

REDUCTION OF LOGICAL RELATIONS

Step 1. Check for valid Exclusive OR relations involving Z variables from user specification. Each Exclusive OR can lead to the removal of one variable from the logic.

Step 2. Check if relations with no implications occur in other expressions to establish their truth and substitute for Z variables accordingly.

Step 3. Eliminate equivalence with one or more variables and occurring only once in another expression. Repeat until no further reduction is possible.

If the MILP model does not require the modelling of mixers and splitters, the logic is further s!mplified by removing all the Z variables so as to express it only in terms of Boolean variables for the process units (Y).

Appendix B: Proof of theorem

LEMMA 1

The leaf nodes of a binary tree representing all combinations of the binary variables Yi = t rue , f a l s e , i = 1 . . . . . m and satisfying the disjunctions in ~ as given in (3), correspond to a true value of the disjunct clauses of f~. Furthermore, the number of leaf nodes is r, the number of disjunct clauses.

P r o o f

Consider any node on the binary' tree. Since, the DNF representation is valid, there are only two possibilities:

(a) The subset of Boolean variables at that node do not falsify more than r - 2 disjunct clauses. This implies that this is an intermediate node that requires further branching.


(b) The Boolean variables at that node falsify r - 1 disjunct clauses. This implies that the remaining clause can be set to true fixing all remaining Boolean variables which then defines a terminal node.

Since each true value of a disjunct clause gives rise to a different terminal node, the number of terminal nodes is r. []

THEOREM 1

The number of nodes to be examined by branch and bound with the disjunctions in (3) is not more than 2 r - 1, where r is the number of disjunct clauses.

Proof

From lemma 1, the maximum number of leaf nodes in the branch and bound tree is r, where r is the number of clauses in the DNF. Consider a search tree shown in fig. 5 that involves r leaf nodes and one root node. Let the number of intermediate nodes be p. Since the structure is a tree, the total number of edges in

root node

• a t e n o d e

leaf node

Fig. 5. A branch and bound tree.

the tree is (r + p + 1 - 1) = (r + p). But since each intermediate node is connected to 3 edges, the root node to 2 edges and the each leaf node to only 1 edge, the total number of edges in the tree is also given by ((3p + 2 + r)12. Equating the two expressions, we get p = r - 2. Therefore, the total number of nodes in the tree = r + p + l = 2 r - 1 . []

Acknowledgement

The authors gratefully acknowledge financial support from the Engineering Design Research Center at Carnegie Mellon University and from the Eastman Chemicals Division.


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symbolic integration of logic in milp branch and bound methods for the synthesis of process networks

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