Multi-criteria Synthesis of Flexible Heat-exchanger Networks with Uncertain Source-stream Temperatures

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Multicriteria synthesis of flexible heat-exchanger networks

with uncertain source-stream temperatures

Cheng-Liang Chen

∗

, Ping-Sung Hung

Department of Chemical Engineering, National Taiwan University, Taipei 10617, Taiwan, ROC Received 23 May 2003; received in revised form 28 July 2003; accepted 31 March 2004

Available online 2 July 2004

Abstract

A multi-criteria synthesis strategy for heat-exchanger networks (HENs) simultaneously considering minimum utility consumption, max-imum source-stream temperature flexibility, and even minmax-imum number of matches is proposed. The flexible HEN synthesis problem is formulated as a multi-objective mixed-integer linear programming (MO-MILP). For handling the multiple conflict design targets, a two-phase fuzzy multi-criteria decision-making method is presented to attain a best compromised solution. Two numerical examples with flexibility pref-erences in source-stream temperatures are supplied, demonstrating that the proposed strategy can provide definite and feasible compensatory solutions for multi-criteria HEN synthesis problems.

Keywords: Heat-exchanger network; Synthesis; Flexibility; Superstructure; MILP; Multi-criteria decision-making; Fuzzy optimization

1. Introduction

A heat exchanger network (HEN) synthesis problem can be described as the one that synthesize a HEN configura-tion to reach some assigned targets such as minimum util-ity consumption, minimum total number of heat exchangers, etc., with given heating/cooling utilities and hot/cold process streams be cooled/heated from nominal inlet temperatures to specified target temperatures[1].

Most of the existing HEN synthesis methods rely on ei-ther heuristic rules (for example, pinch analysis method[2]) or mathematical programming (for example, simultaneous optimization approach [3–6]). And further, to some typical objectives considered in the HEN synthesis such as utility consumption, total number of matches, and total exchanger area, the flexibility of the HENs for feasible operation un-der possible variation of source-stream temperatures and/or heat-capacity flow rates has been emphasized in some re-cent articles[6–10]. For HEN synthesis, the analysis of this flexibility, defined as the size of the region of feasible oper-ation in the space of desired or undesired devioper-ations of

pa-∗_{Corresponding author. Tel.:}_{+886 2 23636194;} fax:+886 2 23623040.

E-mail address: ccl@ntu.edu.tw (C.-L. Chen).

rameters from their nominal values[10], however, attracts attention usually indirectly as test examples basing on math-ematical programming synthesis. Therein,[10]explored the HEN synthesis problem with simultaneous flexibility target-ing and minimum-utility objective based on an MILP for-mulation. The optimal solution is examined on the basis of the vertices of the polyhedral uncertainty region in the space of source-stream temperatures. It is found, however, that the resulting HEN structures with increasing flexibility require-ments are prone to variation, and the true maximum flexi-bilities of resulting HEN structures are usually greater than the assigned targets. Thus, for a given flexibility target, it is very often to obtain a more conservative HEN design by the method of simultaneous flexibility targeting and synthesis of minimum-utility HENs proposed in[10].

In this paper, we extend the work of[10]by simultane-ously considering minimization of the total utility consump-tion, maximization of operational flexibility to source-stream temperatures, and even minimum number of matches as multiple design objectives. The flexible HEN synthesis problem is thus formulated as the one of multi-objective mixed-integer linear programming (MO-MILP). This for-mulation also assumes that the feasible region in the space of uncertain input parameters is convex, so that the optimal solution can thus be explored on the basis of the vertices 0255-2701/$ – see front matter © 2004 Elsevier B.V. All rights reserved.

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of the polyhedral region of uncertainty[10]. Under the as-sumption of convexity, only the source-stream temperatures of the HENs are considered to be the uncertain input pa-rameters. With this formulation, the standard definition of the HEN synthesis problem with minimal total utility con-sumption and even minimum number of units is extended to include a flexibility specification for the potential HEN structure and can be stated as: “given hot/cold streams to be cooled/heated from nominal supply temperatures to speci-fied target temperatures and hot/cold utility specifications, synthesize a HEN such that it has minimal utility consump-tion (considering nominal case or average of all vertical operating points), minimal number of matches if desired, and maximal flexibility for feasible operation.”

For handling the multiple and conflict design objectives, a fuzzy decision-making method is adopted to attain the compromised solution among all conflict objectives [11]. Therein each design objective is treated as a fuzzy goal, and a specific membership function is used to characterize the transition from the numerical objective value to the de-gree of satisfaction for the fuzzy objective. The final deci-sion, therefore, is interpreted as a fuzzy aggregation of these multiple objectives and measured by the overall degree of satisfaction. And the best compromised solution is finally reached by maximizing the overall degree of satisfaction for the decision. In the course of finding the solution, two pop-ular operators, the minimum and the average, are applied as the fuzzy intersection operators, the effects of which are ex-amined as well. We also proposed an interactive two-phase fuzzy decision-making method by combining these two op-erators to take advantages of the both[11,12]. The minimum operator is used in phase I to maximize the degree of satis-faction for the worst objective, and the average operator is

Fig. 1. The two-stage superstructure.

then applied in phase II to simultaneously promote satisfac-tion levels of all objectives with guaranteed least satisfacsatisfac-tion value.

Two numerical examples with flexibility preference in source-stream temperatures is presented here to demonstrate that the proposed interactive two-phase fuzzy optimization method can provide a feasible and better compensatory so-lution for multi-objective HEN synthesis.

2. Model formulation

Consider the standard HEN synthesis problem with NH hot and NC cold process streams along with hot and cold utilities. Since it is suitable for formulating the simultaneous solution which involving the consideration of both utility consumption and operational flexibility, the HEN superstruc-ture proposed by[3,4]is applied for modeling the structure. Therein, the isothermal mixing assumption in the simplified superstructure eliminates the need for nonlinear/nonconvex energy balance. The minimum number of superstructure stages, N_T, corresponds to max{NH, NC}, as suggested by

[3].Fig. 1 illustrates a 2-hot/2-cold/2-stage superstructure. The mathematical programming formulation for minimizing utility consumption with specified uncertain source-stream temperature ranges can be summarized as follows:[3,4,10]: min x∈J (0)₌ i∈ HP qcu(0)_i + j∈ CP qhu(0)_j (1) x≡   

zijk, zcui, zhuj; tik, tjk; dtijk, dtcui, dthuj;

qijk, qcui, qhuj;

i ∈ HP, j ∈ CP, k ∈ ST

 

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Ω =                                                                                                                                                                        x (Tin i − Tiout)FCpi= k∈ST j∈CP qijk+ qcui (Tout j − Tjin)FCpj= k∈ST i∈HP qijk+ qhuj         

overall heat balances

(tik− ti,k+1)FCpi = j∈CP qijk (tjk− tj,k+1)FCpj= i∈HP qijk         

stage heat balances

ti,1= Tin

i ∈ {Tiin(0)− δ, Tiin(0)+ δ} tj,NT+1= T_jin∈ {T_jin(0)− δ, T_jin(0)+ δ}

 

 uncertain inlet temperatures

tik≥ ti,k+1 tjk≥ tj,k+1 Tout i ≤ ti,NT+1 Tout j ≥ tj,1              feasibility of temperatures

(ti,NT+1− T_iout)FCpi = qcui (Tout j − tj,1)FCpj = qhuj    utility loads qijk− Λzijk≤ 0 qcu_i− Λzcu_i ≤ 0 qhu_j− Λzhu_j≤ 0        logical constraints dtijk ≤ tik− tjk+ Γ(1 − zijk) dtij,k+1≤ ti,k+1− tj,k+1+ Γ(1 − zijk) dtcu_i ≤ t_i,NT₊₁− T_CUout+ Γ(1 − zcu_i) dthu_j≤ T_HUout− t_j,1+ Γ(1 − zhu_j)

         approach temperatures dt_ijk(0)≥ Tmin dtcu(0)_i ≥ Tmin dthu(0)_j ≥ Tmin    

 nominal approach temp. bounds

i∈ HP,j∈ CP,k∈ ST zijk+ i∈ HP zcu_i+ j∈ CP

zhu_j≤ MEUmax

 

 maximum exchanger units

zijk, zcui, zhuj ∈ {0, 1}

tik, tjk, dtijk, dtcui, dthuj, qijk, qcui, qhuj; ≥ 0

i ∈ HP, j ∈ CP, k ∈ ST                                                                                                                                                                        (3)

where x andΩ denote variables for design and the feasible searching space, respectively;δ is the targeted flexibility for source-stream temperatures[10]; the superscript(0)denotes the nominal condition; and the upper bound for unit num-bers is MEUmax. The searching space is comprised of all heat balances constraints and relevant logical constraints. As pointed out in[6–10], this problem is difficult to solve di-rectly since it involves a max-min-max constraint that leads to a non-differentiable global optimization problem. For the HEN synthesis problem with uncertain source-stream

tem-peratures, the feasible region defined by the reduced inequal-ity constraints is convex[6], so the critical point that limits the solution lies at a vertex of the polyhedral region of un-certainty. For the problem ofEq. (7)withN (≤ NH+ NC) uncertain source-stream temperatures, the vertex-based for-mulation is given as follows[10]:

min xV∈ΩV J(0)= i∈HP qcu(0)_i + j∈CP qhu(0)_j (4)

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or min xV∈ΩV J(ave) = 1 NV+ 1 n∈{0}∪VT   i∈HP qcu(n)_i + j∈CP qhu(n)_j   (5) xV=     

zijk, zcui, zhuj; dtijk(n), dtcu(n)i , dthu(n)j ;

t(n)ik , tjk(n); q(n)ijk, qcu(n)i , qhu(n)j ;

i ∈ HP, j ∈ CP, k ∈ ST, n ∈ {0} ∪ VT      (6) ΩV=                                                                                                                                                                        xV (Tin(n) i − Tiout)FCpi= k∈ ST j∈CP q(n)ijk + qcu(n)i (Tout j − Tjin(n))FCpj = k∈ ST i∈HP q(n)ijk + qhu(n)j (tik(n)− ti,k+1(n) )FCpi= j∈CP q(n)ijk (tjk(n)− tj,k+1(n) )FCpj= i∈HP q(n)ijk Tin(n) i = (Tiin(0)+ r(n)i δ) = t(n)i,1 Tin(n) j = (Tjin(0)+ rj(n)δ) = tj,NT(n) +1 t(n)_ik ≥ t_i,k+1(n) tjk(n)≥ tj,k+1(n) Tout i ≤ t(n)i,NT+1 Tout j ≥ t(n)j,1 (t_i,NT(n) ₊₁− Tout i )FCpi= qcu(n)i (Tout(n) j − t(n)j,1)FCpj= qhu(n)j q(n)ijk − Λzijk≤ 0 qcu(n)_i − Λzcu_i ≤ 0 qhu(n)_j − Λzhu_j≤ 0 dt_ijk(n)≤ t_ik(n)− t_jk(n)+ Γ(1 − zijk) dt_ij(n)_,k+1≤ t(n)_i,k+1− t(n)_j,k+1+ Γ(1 − zijk) dtcu(n)_i ≤ t_i,NT(n) ₊₁− T_CUout+ Γ(1 − zcu_i) dthu(n)_j ≤ T_HUout− t_j,1(n)+ Γ(1 − zhu_j) dt_ijk(0)≥ Tmin dtcu(0)_i ≥ Tmin dthu(0)_j ≥ Tmin i∈HP,j∈CP,k∈ ST zijk+ i∈HP zcu_i+ j∈CP

zhu_j ≤ MEUmax

zijk, zcui, zhuj∈ {0, 1}

t(n)ik , tjk(n), dtijk(n), dtcu(n)i , dthu(n)j , q(n)ijk, qcu(n)i , qhu(n)j ≥ 0

i ∈ HP, j ∈ CP, k ∈ ST, n ∈ {0} ∪ VT                                                                                                                                                                        (7) Here,r_i(n)andr_j(n)are the vertex identifiers which take val-ues of NV = 2(NH+NC) combinations of +1 and −1, see

Table 1in[10]. Therein,Eq. (4)considers nominal utilities andEq. (5)takes into account the average of all vertical op-erating points.

For the benchmark example mentioned in [10], a 2-hot/2-cold streams problem along with heating steam and cooling water, the maximal allowable variation of various possible minimum-utility HEN structures is examined by

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Table 1

Problem data of example 1

Process streams and utilities Heat-capacity flow rate FCp (kW/K) Input temperatureTin_(K) _{Output temperature}_Tout _(K)

Hot stream 1 (H1) 10 650 370

Hot stream 2 (H2) 20 590 370

Cold stream 1 (C1) 15 410 650

Cold stream 2 (C) 13 350 500

Hot utility (HU) – 680 680

Cold utility (CU) – 300 320

Tmin= 10 K.

Table 2

The resulting HEN structure of example 1 with increasing flexibility target ( → 0, seeTable 2of [10] for labeling HEN structures)

Specifiedδ value Resulting HEN structure Maximum flexibilityδ∗ 0∼ 36.95 A 36.95 36.95 + B1a ₇₀_.63 40 B7a ₉₀_.00 50 B6 90.00 60 B1a ₇₀_.63 70 B7a ₉₀_.00 80 B7a ₉₀_.00 90 B7a ₉₀_.00 90+ C2 120.0 120–150 D 150.0

increasing theδ value over δ ∈ [0, 150] and solvingEq. (4)

or Eq. (5). There are 18 sets of structure-determining bi-nary variables and 11 of them possess unique structures (seeTable 2of[10]). However, we found that the resulting HEN structures with increasing flexibility targets are prone to variation, and the true maximal flexibilities of these HEN structures are usually greater than the required targets, as illustrated in Table 2. Similar results can also be found should restricted heat-load constraints on vertices, as shown inEqs. (8) and (9), be taken into consideration in the HEN synthesis. min xV∈V∩load J(0)₌ i∈HP qcu(0)_i + j∈CP qhu(0)_j (8) or min xV∈V∩load J(ave)= _N 1 V+ 1× n∈{0}∪VT ×   i∈HP qcu(n)_i + j∈CP qhu(n)_j   (9) Ωload=                       

q(n)ijk, qcu(n)i , qhu(n)j

q(0)_ijk(1 − α) ≤ q(n)_ijk ≤ q(0)_ijk(1 + β)

qcu(0)_i (1 − α) ≤ qcu(n)_i ≤ qcu(0)_i (1 + β) qhu(0)_j (1 − α) ≤ qhu(n)_j ≤ qhu(0)_j (1 + β)

i ∈ HP, j ∈ CP, k ∈ ST, n ∈ VT                        (10) In this paper, the targeted source-stream temperatures are directly treated as individual design objective, and the multi-criteria optimization approach is adopted for HEN synthesis. The minimizing utility and the maximizing oper-ational flexibility can be simultaneously considered as two conflict objectives for synthesis of the network structure. Furthermore, other targets such as minimizing number of matches can also be considered, such as,

min

xV∈ΩV

J₁(0)=_i∈HPqcu(0)_i +_j∈CPqhu(0)_j or min xV∈ΩV J1(ave)= 1 NV+ 1 × n∈{0}∪VT ×   i∈HP qcu(n)_i + j∈CP qhu(n)_j   and max xV∈ΩV J2= δ and min xV∈ΩV J3= i∈HP j∈CP k∈ST zijk + i∈HP zcu_i+ j∈CP zhu_j (11)

In such a case, a unique HEN structure with satisfactory levels in nominal or average utility consumption and opera-tional flexibility as well as unit numbers will be obtained. A two-phase fuzzy optimization method is proposed to find a best compromised solution for the multi-criteria HEN syn-thesis problem, as discussed in the next section. The basic number of constraints and variables for the multi-objective MILP formulation are summarized in the following. 1. The number of constraints:

(a) for linear equality constraints: (NV+ 1)[NT(NH+

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(b) for linear inequality constraints: (NV + 1)[(NH +

NC)(NT+3)+3NTNHNC]+NTNHNC+NH+NC. 2. The number of variables:

(a) binary variables:NHNCNT+ NH+ NC;

(b) positive continuous variables:(NV+1)[NT(2NHNC+

NH+ NC) + 2(NH+ NC)] + 1.

3. Fuzzy multi-criteria optimization

Consider the multi-criteria optimization problem de-fined in Eq. (11). Because of the fact that these objective functions usually conflict with each other in practice, the optimization of one objective implies the sacrifice of other targets; it is thus impossible to attain their own optima,J_s,

s ∈ S = [1, . . . , S], simultaneously. Therefore, the decision

maker (DM) must make some compromise among these goals. In contrast to the optimality used in single objec-tive optimization problems, Pareto optimality characterizes the solutions in a multi-objective optimization problem

[13].

The weighting-sum method, among methods found in lit-eratures for solving multi-objective optimization problems, is the one that is used most often. Basing on the subjective comprehension for each objective, the DM of this method can weigh and sum up these objectives into a scalar form, and then find the solution by any existing single-objective optimization method. However, as the situation of combin-ing weightcombin-ing factors becomes more complex, this method becomes more tedious and the solution could be still in-valid. Moreover, it is difficult for the DM to attribute a set of incompatible objectives, such as utility consumption, operational flexibility, or number of matches in a heat ex-changer network, without knowledge of the possible level of attainment for these objectives. The physical mean-ing of the final scalar objective function is thus usually vague.

In this work, we adopt the fuzzy set theory[14]to deal with the multi-objective optimization problem. By consid-ering the uncertain property of human thinking, it is quite natural to assume that the DM has a fuzzy goal, J_s, to describe the objectiveJ_s with an interval [J1

s, Js0]. For the sth objective to be minimized, it is quite satisfied as the

objective valueJ_s ≤ J_s1, and is unacceptable as J_s ≥ J_s0. The degree of satisfaction decreases as the objective value increases from J_s1 to J_s0. A strictly monotonic decreas-ing membership function, µ_J_s(J_s(xV)) ∈ [0, 1], can be used to characterize such a transition from the objective value to the degree of satisfaction, therein value of 1 de-notes absolutely satisfactory, and 0 means unacceptable. Notably, an interval of [J_s0, J_s1] and a monotonic increas-ing membership function should be used for definincreas-ing a fuzzy objective to be maximized. Without loss of gener-ality, we will adopt linear membership functions in the following. µJs(Js(xV)) =        1; for J_s1≥ J_s J0 s − Js J0 s − Js1 for J_s1≤ J_s≤ J_s0 0; for J_s ≥ J_s0 forJsto be minimized (12) µJs(Js(xV)) =          1; for J_s≥ J_s1 J1 s − Js J1 s − Js0 for J_s0≤ J_s ≤ J_s1 0; for J_s0≥ J_s forJ_sto be maximized (13) The original multi-criteria optimization problem is now con-verted to the one that looks for a suitable decision variable vector that can provide the maximal degree-of-satisfaction for the multiple fuzzy objectives.

max

xV∈V

(µJ1(xV), . . . , µJS(xV)) (14)

Under incompatible objective circumstances, a DM must make a compromise decision that provides a maximal degree-of-satisfaction for all these conflict objectives. The new optimization problem,Eq. (14), can be interpreted as the synthetic notation of a conjunction statement (maximize jointly all objectives). The result of this aggregation can be viewed as a fuzzy intersection of all fuzzy goals,J_s, s ∈ S, and is still a fuzzy set,D.

D = J1∩ . . . ∩ JS (15) The final degree-of-satisfaction resulting from certain vari-able set, µ_D(xV) can be determined by aggregating the degree-of-satisfaction for all objectives,µ_J_s(xV), s ∈ S, via specific fuzzy intersection operator,T.

µD(xV) = T

µJ1(xV), . . . , µJS(xV)

(16) The fundamental properties for a fuzzy set and the related operators can be found in[15]. As the firing level for each policy is determined by the above procedure, the best so-lution, x∗_V, with the maximal firing level, µ_D(x∗_V), can be selected. max xV∈ΩV µD(xV) = max xV∈ΩV T(µJ1(xV), . . . , µJS(xV)) (17)

Using the fuzzy intersection operator, the original multi-objective optimization problem,Eq. (11), is converted into a single objective problem,Eq. (17). Several operators for im-plementing fuzzy intersection can be selected forT, therein two most popular ones are shown below.

T(µJ1, . . . , µJS) = min(µ_J₁, . . . , µ_J_S) T = minimum (µJ1+ . . . + µJS) S T = average (18) The average operator simultaneously takes all membership values into account. But, no guarantee can be made for any

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single objective. The minimum operator concentrates on im-proving the worst scenario. However, the minimum operator may result in multiple solutions since contribution of those objectives with membership values greater than the mini-mum one would not be cared. We thus combine these two fuzzy intersection operators to take advantages of both and propose a two-phase optimization procedure [11,12]. The minimum operator is first used in phase I to find the least degree of satisfaction for the worst objective, then the aver-age operator is applied in phase II to promote satisfaction levels of all objectives with guaranteed least membership value. The merit of this method is that we can not only ob-tain the unique optimal solution by using average operator but also guarantee each objective to go after their own max-imum on the basis of taking the least degree of satisfaction as the lower-bound constraint. So, now we can summarize the procedure of the two-phase fuzzy satisfying approach for the multi-criteria optimization problem.

Step 1. For a function to be minimized, determine its ideal solution and anti-ideal solutions by directly mini-mizing and maximini-mizing the objective function.

min

xV∈ΩV

Js = J_¯1s

(ideal solution ofJs, totally acceptable

value) (19)

max

xV∈ΩV

Js = ¯Js0

(anti − ideal solution ofJs,

unacceptable value) (20)

Notably, direct maximization and minimization should be taken for a maximizing objective to obtain the ideal and anti-ideal solutions.

Step 2. Based on the importance of different objective func-tions and the acceptable ranges for objective val-ues, subjectively select suitable lower/upper bounds,

J

¯ 1

s ≤ Js1≤ Js0≤ ¯Js0 for minimizing objective and J

¯ 0

s ≤ Js0≤ Js1≤ ¯Js1for maximizing objective.

De-fine membership functions for multiple fuzzy ob-jectives as given inEqs. (12) and (13).

Step 3. (Phase I). Use the minimum operator to find the maximal degree of satisfaction for the worst objec-tive,µmin. max xV∈ΩV µD= max_xV_∈Ω V min(µ_J₁, . . . , µ_J_S) = µmin (21) Step 4. (Phase II). Use the average operator to simultane-ously promote satisfaction levels of all objectives with new constraints of guaranteed minimal degree of satisfaction. max xV∈+V µD= max xV∈+V µJ1+ · · · + µJS S (22) where Ω+V = ΩV∩ {µJ_s ≥ µmin, ∀s ∈ S} (23)

The new feasible space guarantees the least degree of satisfaction for each fuzzy objective.

4. Numerical example

Two numerical examples adapted from[4,10]are supplied to demonstrate the efficacy of proposed HEN synthesis strat-egy. To solve this MO-MILP for the HEN model, GAMS

[16]and CPLEX are used as the modeling environment and the MIP solver, respectively. The computing machine is a personal computer with an Intel Pentium IV 2.26 GHz CPU. Example 1. The 2-hot/2-cold streams example studied by

[10], with problem data presented inTable 1, is illustrated. With these parameters, the multi-objective MILP formula-tion has 408 linear equality constraints, 760 linear inequal-ity constraints, 12 binary variables, and 545 positive con-tinuous variables. Notably, the restriction of MEUmax = 6 inEq. (7)will be removed should the minimum number of matches be simultaneously taken into account as one of the design objectives.

According to the model formulation and the problem data, we solve the multi-criteria MILP synthesis problem by us-ing the fuzzy optimization procedure discussed in Section 3. The ideal and anti-ideal solutions, obtained by sequen-tially minimizing and maximizing each objective function, are shown inTable 3. Within these lower/upper limiting val-ues, several preference intervals are subjectively selected to establish linear membership functions for the fuzzy objec-tives. In implementing the two-phase fuzzy optimization, we firstly use the minimum operator to maximize the degree of satisfaction for the worst target,µmin. The average operator is then applied to optimize the aggregated objectives with guaranteed level of satisfaction.

At first, only two conflict objectives are considered: the minimal utility consumption and the maximal flexibility to all source-stream temperatures. And then the third objective, the minimal number of matches, would be appended. Results of two-phase fuzzy optimization with preference intervals of [2550, 12750] or [2550, 8850] for utility, [0, 150], [40, 90] or [40, 70] for flexibility, and [4, 7] for unit numbers, along with either or not considering restrictions on heat loads at extreme operating points, are listed inTable 4. The resulting HEN structures are also depicted inFig. 2. Notably, the re-duced range of flexibility, [40, 90], implies that the required minimum tolerance for temperature deviation is at least 40 K and a tolerance of maximum temperature deviation for 90 K Table 3

The ideal/anti-ideal solutions of various objectives of example 1 Objective function Ideal solutionJ

¯ 1 i Anti-ideal solution ¯Ji0 J1(0) utility (kW) 2550 12750 J2 flexibility (K) 150 0 J3 units 4 12

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Table 4

Results of HEN synthesis for example 1 using two-phase optimization when simultaneously considering minimal utility and maximal flexibility (cases I–IV), and additional objective of minimal units (cases V and VI) with different preference intervals, and with or without considering restriction on heat loads at vertices (α = β = 0.6)

Case Preference intervals Heat load Phase Minimum utility Maximum flexibility Minimum units Hot Cold Total Satisfaction Flexibility Satisfaction Units Satisfaction

I [2550, 12750] No I 1300 2950 4250 0.833 120.0 0.800 6 – (C2) [0, 150], [−, 6] II 1300 2950 4250 0.833 120.0 0.800 6 – II [2550, 8850] No I 1050 2700 3750 0.810 80.5 0.810 6 – (B7) [40, 90], [−, 6] II 1050 2700 3750 0.810 90.0 1.00 6 – III [2550, 8850] Yes I 2190 3840 6030 0.448 62.4 0.448 6 – [40, 90], [−, 6] II 2190 3840 6030 0.448 62.4 0.448 6 – IV [2550, 8850] Yes I 1893 3543 5436 0.542 56.3 0.542 6 – [40, 70], [−, 6] II 1893 3543 5436 0.542 56.3 0.542 6 – V [2550, 8850] No I 1300 2950 4250 0.730 73.3 0.667 5 0.667 [40, 90], [4, 7] II 1300 2950 4250 0.730 74.2 0.685 5 0.667 VI [2550, 8850] Yes I 2550 4200 6750 0.333 56.7 0.333 6 0.333 [40, 90], [4, 7] II 2550 4200 6750 0.333 69.8 0.597 6 0.333

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Table 5

Problem data of example 2 Process streams and utilities Heat-capacity flow rate FCp (kW/K) Input temperature Tin _(K) Output temperature Tout _(K) Hot stream 1 (H1) 6 500 320 Hot stream 2 (H2) 4 480 380 Hot stream 3 (H3) 6 460 360 Hot stream 4 (H4) 20 380 360 Hot stream 5 (H5) 12 380 320 Cold stream 1 (C1) 18 290 660

Hot utility (HU) – 700 700

Cold utility (CU) – 300 320

Tmin= 10 K.

Table 6

The ideal/anti-ideal solutions of various objectives of example 2 Objective function Ideal solutionJ

¯ 1 i Anti-ideal solution ¯Ji0 J1(0)utility (kW) 3780 9860 J2flexibility (K) 20 0 J3units 6 31

is absolutely satisfied. We choose 90 as the maximum de-viation for most cases since it is the maximal tolerance for the HEN with heat load restrictions on vertices.

As shown in Table 4 and Fig. 2, the HEN structure in case I is the same asC2 of[10], therein the maximum flex-ibility to temperatures is explicitly given as 120. In case II, theB7 structure of [10] can be obtained when the tolera-ble upper bound for utility consumption is decreased from 12750 to 8850, and the preference interval for flexibility is reduced from [0,150] to [40,90]. The resulting utility con-sumption will be reduced from 4250 (Case I) to 3750 due

Table 7

Results of HEN synthesis for example 2 using two-phase optimization when simultaneously considering minimal utility, maximal flexibility and minimal units with different preference intervals, and with or without considering restriction on heat loads (α = β = 0.6)

Case Preference intervals Heat load Phase Minimum utility Maximum flexibility Minimum units Hot Cold Total Satisfaction Flexibility Satisfaction Units Satisfaction

I [3780, 9860] No I 3863 403 4266 0.920 18.4 0.920 8 0.920 [0, 20], [6, 31] II 3793 333 4126 0.943 20.0 1.00 8 0.920 II [3780, 4930] No I 3712 252 3964 0.840 16.8 0.840 10 0.840 [0, 20], [6, 31] II 3660 200 3860 0.930 20.0 1.00 10 0.840 III [3780, 4930] No I 3811 351 4162 0.667 13.3 0.667 8 0.667 [0, 20], [6, 12] II 3793 333 4126 0.699 20.0 1.00 8 0.667 IV [3780, 4930] No I 3811 351 4162 0.667 16.7 0.667 8 0.667 [10, 20], [6, 12] II 3793 333 4126 0.699 20.0 1.00 8 0.667 V [3780, 9860] Yes I 3980 520 4500 0.882 10.6 0.882 8 0.920 [0, 12], [6, 31] II 3980 520 4500 0.882 12.0 1.00 8 0.920 VI [3780, 4930] Yes I 3742 282 4024 0.787 9.4 0.787 11 0.800 [0, 12], [6, 31] II 3742 282 4024 0.787 9.4 0.787 11 0.800 VII [3780, 4930] Yes I 3860 400 4260 0.583 6.99 0.583 8 0.667 [0, 12], [6, 12] II 3860 400 4260 0.583 10.0 0.833 8 0.667 VIII [3780, 4930] Yes I 3860 400 4260 0.583 9.5 0.583 8 0.667 [6, 12], [6, 12] II 3860 400 4260 0.583 10.0 0.667 8 0.667

to the restriction of preference intervals, and the flexibility, 90, is equivalent to the maximum of theB7 structure.

All other conditions are equal to those in case II, case III includes heat-load restrictions (α = β = 0.6) on vari-ous vertical operating points as additional constraints. The prices of such additional restrictions are increased utility consumption from 3750 to 6030 and decreased flexibility to temperature deviation from 90 to 62.4, which is still signifi-cantly greater than the minimum targeted value, 40. In case IV, the preference interval for temperature deviation is fur-ther reduced to [40, 70]. It is found that the resulting HEN has smaller utility consumption, 5438, with the expense of further reduction on flexibility, 56.3, since our desideratum for flexibility has been made lower.

Cases V and VI take into account the unit number as the third design objective, where heat-load restrictions on vertices are either included or not. In case V, it is found that the unit number is only five, flexibility to temperature deviation is 74.2, and the utility consumption is 4250, a little more thanB4 and B5 of [10]. In case VI where heat-load restrictions on vertices are further included, the total unit number becomes six, the total utility is increased to 6750 with a reduced maximum temperature flexibility of 69.8.

From these results, it is found that the proposed multi-criteria synthesis strategy can attain a definite and compromised solution for a problem with assorted conflict objectives. The preference intervals of various objectives have significant effects on final HEN structures. Such ac-ceptable and/or preference intervals can also reflect the importance of different objective functions. Should one specific objective is emphasized, a tighter restriction or shrinking span should be placed on its acceptable ranges.

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Fig. 3. The HEN structures for cases I–VIII of example 2. Observing the results shown in Table 4, we also discover

that the integrated two-phase method can bring the merits of the minimum and average operators together. The minimum operator maximize the degree of satisfaction for the worst objective, and can result in a harmonious solution with satisfaction levels for objective functions that are equal or close to each other. The average operator, on the other hand, can sometimes promote satisfaction levels for other objec-tives with guaranteed minimum value, such as Cases II, V, and VI. Owing to these advantages, the two-phase method can thus provide the best compromised HEN configuration. Example 2. This problem consists of five hot streams and one cold stream, (NH = 5, NC= 1), along with steam and cooling water as utilities [3]. The problem data are listed

in Table 5. The number of superstructure stages is set as

NT = 5. With these parameters, the MO-MILP formulation

has 3510 linear equality constraints, 8026 linear inequality constraints, 31 binary variables, and 5981 positive continu-ous variables. The ideal and anti-ideal solutions are shown inTable 6. Various preference intervals are sequentially se-lected for defining the membership functions, as shown in

Table 7.

We directly use the ideal/anti-ideal solutions as the pref-erence intervals in Case I. The acceptable utility range is reduced from [3780,9860] to [3780,4930] in Case II. With such a smaller allowance interval, the resulting utility con-sumption will be reduced slightly from 3793 to 3660 at the expense of unit numbers increased from 8 to 10. In Case III,

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the maximum allowable unit number is changed from 31 to 12. The resulting HEN structure is slightly different to that of Case I, but the required unit numbers, utilities, and flexi-bility are the same as Case I. In Case IV, when the minimum flexibility requirement is upgraded from 0 to 10, the same HEN structure of Case I results with the same levels of util-ity, flexibility and unit numbers. Cases V–VIII give similar results with additional heat-load restrictions (α = β = 0.6) on various vertical operating points. With such additional constraints, the HEN structures use similar levels of utilities and unit numbers, but the flexibility levels are dramatically reduced. The resulting HEN structures are depicted inFig. 3.

5. Conclusion

In this paper, we use the fuzzy multi-criteria optimization approach to synthesize the heat-exchanger network where some conflict design objectives such as the total utility consumption, the flexibilities to source-stream temperature variations, and even the total number of heat exchange units can be considered simultaneously. Such a flexible HEN synthesis problem can be formulated as a multi-objective mixed-integer linear programming (MO-MILP). For han-dling the multiple conflict design objectives, a two-phase fuzzy optimization method is proposed to attain the best compromised solution. The attractive features of the pro-posed MO-MILP model are that it not only considers the trade-off among the utility consumption, the source-stream temperature flexibility, and even the number of matches, but also avoids the determination of structural boundaries, as discussed in [10]. Two numerical examples with various cases are studied, demonstrating that the proposed strat-egy can provide a feasible compensatory solution for the multi-criteria HEN synthesis problem.

Acknowledgements

This work is supported by the National Science Council (ROC) under Contract NSC91-ET-7-002-004-ET. Partial fi-nancial support of Ministry of Economic Affairs under grant 92-EC-17-A-09-S1-019 is also acknowledged.

Appendix A. Nomenclature

CP index set of cold process stream

dtijk temperature approach for matchi and j in stage k dtcu_i temperature approach for matchi and cold utility dthu_j temperature approach for matchj and hot utility FCp heat capacity flowrate

HP index set of hot process stream

J objective function

MEUmax maximum number of heat-exchange units

N number of uncertain parameters

NC number of cold streams

NH number of hot streams

NT number of superstructure stages

NV number of vertices,= 2N

qijk heat exchanged between streami and j in stage k qcu_i heat exchanged between streami and cold utility qhu_j heat exchanged between streami and hot utility

r directional identifier for vertices ST index set of superstructure stages

S number of objectives

Tmin minimum approach temperature

tik temperature of streami at hot end of stage k

tjk temperature of streamj at hot end of stage k

T temperature

VT index set of vertices

x, xV vector of variables

zijk binary variable for existence of unit for matchi andj in stage k

zcu_i binary variable for existence of unit for matchi and cold utility in stagek

zhu_i binary variable for existence of unit for matchj and hot utility in stagek

Greek letters

α, β parameters used for restriction of heat-load deviations

δ flexibility index

δ∗ _{flexibility index (scalar)}

µJs membership function forJs

Js a fuzzy goal

S index set of multiple objectives

D a fuzzy set

µD degree of satisfaction

Γ upper bound for temperature difference

Λ upper bound for heat exchange

Ω the feasible searching region Superscripts

in _inlet

(n) _{identifier for vertices}

out _outlet

(0) _{identifier for nominal operating condition}

Subscripts

CU cold utility HU hot utility

i index for hot process streams

j index for cold process streams

k index for superstructure stages

s index for objectives References

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