Computers chewt. Engng Vol. 19, Suppl., pp. S627-S632,1995 Copyright @ 1995 Elsevier Science Ltd

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DESIGN OF MULTIPRODUCT BATCH PLANTS WITH UNCERTAIN DEMANDS

M. G. IERAPETRITOU and E. N. PISTIKOPOULOS

Centre for Process Systems Engineering

Department of Chemical Engineering

Imperial College, London, SW7 2BY, U.K.

ABSTRACT

The problem of how t,o design mukiproduct batch plant,s for the case of uncertain product demands de- scribed by any continuous/discrete probability distributional form, is considered in this paper. Based on a stochastic programming formulation, featuring an objective function comprising investment costs for equip- ment sizing, expected revenues from product sales and a pena.lty term accounting for partial feasibility and unfilled orders, it is first shown that the relaxation of the feasibility requirement enables the reformulation of the problem as a single, yet large-sca.le and nonconvex, nonlinear optimization model. The exploitation of the special structure of the multiproduct batch plant design model, however, results in orders of magni- tude reduction of t,he number of relaxed dual problems which are required for the computationa.lly efficient applica.tion of the global opt,imiza.tion algoribhm (GOP) of Floudas and Visweswaran (1990).

KEYWORDS

Batch plant design; uncertainty; global optimization

INTRODUCTION

The consideration of flexibilit,y and uncerta.inty in the design a,nd scheduling of ba.tch plant,s has received lit- tle attention in the open lit.erature despite its great, practical relevance, especially due to the high degree of uncert.ainty in bat,& recipe paramet.ers resource and equipment ava.ilabilities, product dema.nds and product ma.rket kends. Most previous work (Reinhart, a.nd Rippin, 1980,1987; Wellons and Relkitis, 1989; Straub and Grossmann, 1992) consider simplified batch plant design models to explore the impact. of uncertainty on design decisions - opera.tional issues regarding det,ailed scheduling have not, been explicitly considered. Recently, Pekny and coworkers (1994) have sta.rt.ed addressing the general problem of design and operations of batch processes under uncert,a,inty employing a multiperiod/scena.rio-bayed approach coupled with a de- tailed scheduling model. While it can be argued that a multiperiod a.pproach has distinct computational advantages over continuous based stochastic formulations, inherent trade-offs between economic optimal- ity a.nd design feasibility ca.n be explored in a more consistent basis by employing the lat,ter rather than scenario-based techniques (see also Pistikopoulos, 1994). This paper presents some recent. developments on global optimization t,echniques for l,he design of mult,iproduct ha,tch plants with stochast,ic demands.

MATHEMATIC’AI, MODEL

Here we consider a simplifecl 1lllllt,il)r(~dltcf, ba.kh plant design model (similar to Reinhart. and Rippin and Straub and Grossmann) for the production of N products (in single campaigns without intermediate storages) in M stages (comprising TVj identical pieces of batch equipment of size l.$, j=l,..,hl) with fore- casted demands described by a given probability dist,ributional form, as follows:

wx 19:13-m S62-l

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subject t,o: Vj ZSijBi i=l ,.., N j=l,.., M (2)

+@; 5 H (3) (P) i=l ’

&i<Oi, i=l,..,N,BER (4)

(5)

I

where B; is the batch size of product i; H the time horizon; Sij the size factors (volume of a vessel in stage j required to produce one mass unit of product i); tij the processing time of product i at stage j; Qi the production of product i; Bi the uncertain demand of product i following a distribution J(0i); R(&, Nj) the feasible region of batch plant (vj, Nj) i.e. R = (0lV0 E R3Qi : (a), (3), (4) are satisfied}; aj, pj are the cost coefficients of ba,tch equipment at &age j; 6 the coefficient used to annualize the capital cost; pi the

price of product i.

In the above formulation (P) the equipment sizes (4) and the number of batch equipments per stage (Nj) are considered as design variables which are fixed during plant, operation, whereas the amount of products produced (Qi, i=l,..,N) correspond to operating variables which can be selected so as to accom- modate uncertain market requirements (&). Model (P) involves nonconvexities in the objective function (due to posynomial terms) a.nd in the horizon constraint (due to fractional terms). Through the following exponential transformations 1.5 = exp(,uj), Bi = exp(bi), TL; = exp(tLi), Nj = exp(nj) where ‘Uj, bi, 2~;. nj are the new t,ransformed varia.bles (integers IZ~ can be further transformed to O-l variables as follows:

“j = Cyjr In(r); C?/j, = 1; Yj,. = { 0 1 1 ); := 1 l,..,M, r=l,..,N/-an upper bound on the number of units

per st,age j- see Iiocis and Grossmann, 1988), the objective function can be convexified with which (P) can

be rewritten as:

max(-d~~jexp(~~jrln(r)+~jvj)+Es(l~x~piQi)J vj,Yjr

j=l 1- r=l

subject to: tlj 2 ln(5’ij) + bi) tLi 2 lll(iij) - C, Yjr Ill(r)

CQi exp(tL; - bi) 5 H ’ (PI)

i=l

Qi I oi

Nevertheless (Pl) still remains a nonconvex optimiza.tion problem due to the resulting bi-convex horizon constraint. Furthermore, the integration for the approximation of the expectation requires the solution of an (inner) optimization subproblem, while the domain of the integration is unknown (as a function of 4, Nj) - this reflects the requirement for simultaneous optimality and feasibility (Pistikopoulos and Ier- apetritou, 1994).

We propose the following a.pproach in order to overcome t.hese difficulties: (i) for continuous probability distribut.ion functions, we a.pproximate t,he multiple integral for t,he expected profit evaluat,ion U~rough a. Gaussian qua.dra.ture formula (with q=l,..,Q qua.drature points) as follows:

(6)

where ai are the arguments of the inner optimization subproblem (profit optimization); WV are the weights corresponding to each quadrature point; Jq is the probability of each point. (ii) the rela.xation of the demand constraint in (4) allows for the integrat,ion to be performed in the region defined by the bounds of the uncerta.in para,meters, since t,he following propert,y holds: Properly- Any design (Q, Nj) sa.tisfying const*raints (2), (3), (4) and (5) of problem (P) for fixed product demands B;, i=l,..,N, is always feasible. Proof- The feasibilit,y ~.trs~ problem of(P) - with fised r/j? Nj and 0i - is:

European Symposium on Computer Aided Process Engineering-5 S629

N Qi l$l {II ) s.t. c --T’i - H 5 U; Qi - Oi 5 U; -Qi 5 TV} , 1 i=l Bi

with the following KKT optimality conditions:

~~+pf-_pf=O i=l,..,N; A+&+&=1 I

i=l i=l

where X and pi, $ a.re the Lagrange multipliers of the horizon constraint and the bounding constraints for Qi, respectively. From the KKT conditions, 21 = -AH 5 0 V Bi; i.e. the design is permanently feasible and subsequently the feasible region of the ba.tch plant model in (P) and (Pl), in the space of uncertain pa.rameters coincides wit,h Dhe uncertain parameter range - independently of the design. 0

(iii) by incorporating equation (G) and int,roducing a penalty term for partial feasibility/unfilled orders, problem (Pl) takes the follo#ng form:

Q N \

subject.

{-bzcYj exp(xyj, ln(l’) + Pjuj) + C,w’J’{CpiQ,‘) j=l P q=l i=l

to: . vj 2 IIl(Sij) + bi ; i = 1, .., N, tLi > Ill(tij) - C,. yjlj,. Ill(r) ; i

Al

j= l,..,M = 1 ,.., N, j=l,..,

CQS exp(tL; - bi) 5 H i=l

Qfse;; %=l,.., N,q=l,.., Q ll(ll/j”“) 5 Vj 5 Ill(Vjup)

Yjr = {O,~}Y Bp E T(O), 7 2 O

where T(0) = (6’16” < 0 < 0”) and the quadrature points 8: are properly placed within T according to the following formula for ea.ch uncertain parameter 0;:

ey = 0.5[8”(1 + ,$I) + ef(i - ,vi”)], i = 1, .., N, q = 1, ..,Q (7)

where VP’ denot,e the locat,ion of the yua.dra.ture point qi in t.he [-l,l] int,erval; 0?, 6’y are the lower and upper bounds of Bi; Q is the t,ot,a.l number of quadrature points inside T. The objective funct,iou in problem (Pl’) has been a.ugmented t,o i1lcorpora.t.e an explicit penalt,y term represent,ing the expect,ed revenue loss due to unfilled orders. In this way, a.uy bat,& pla.nt clesigu which can not fully accommodat,e the market requirements is pa,rtially pena.lized; note t,hat. while the exclusion of pa.rtial feasibility may 1ea.d to conser- vative designs, the direct incorporation of a, pena.lty t,erm effectively avoids “optimistic” estimates - the estent of pena.lty term is monit,ored by adjusting the penalt,y coefficient 7 (see exa.mple problem).

The advantage of t,he formulation in (PI’) is t,ha.t it corresponds to a single deterministic mixed inte- ger non1inea.r optimi&ion problem, that, can in principle be solved through standard MINLP algorithms (Generalized Benders Decomposition or Outer-Approximation) to determine the number of identical units per sta.ge (Nj), the volume of equipments (l$) a,nd the corresponding operational plan (Qi) that maximize the overall expected profit. However, due t,o the existence of t,he non-convex horizon constraint a. local (or even a subopt,ima.l, Sahinidis and Grossmanu, 1991) solution may be encountered; therefore, there is a strong impetus for the a,pplication of global optimization procedures. In the uest section, it will be shown that,, for the case of fised number of units per stage, the formulation in (Pl’) can be effectively solved t,o globa. optimality by a special implemeutation of the GOP algorithm of Floudas and Visweswa.ran (1990).

A GLOBAL OPTIMIZATION SOLUTION PROCEDURE

For the case of fixed Nj, problem (Pl’) corresponds to a single nonlinear optimization problem which involves nonconvexities due to the horizon constraint. However, the following variable partition is sug- gest,ed:

with which problem (PI’) sat.isfies coucli(.ions (A) ol’t,he GOP algorithm (Flouclas and Visweswara.n, 1990), since both t.he object.ive ilud the constraints are couves in { (5, bi} for every fixed Qy and linear in Qi [or each fised (vj, bi). Although GOP cau in principle be applied for the solution of problem (Pl’) to g1oba.l optimality, it. requires prohibit.ively high computational cost, (ttNxQ subproblems per it,era.tion). However,

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as it will be shown in the next section by exploiting the special structure of the batch plant design model in (Pl’), the number of relaxed dual subproblems that have to be solved per iteration can be reduced by orders of magnitude (scaling only to the number of products).

Properties of batch plant design problem

Projecting on y = {uj, bi}, the following linear primal problem is obtained at the kth iteration:

I$)1165 NjClj eXp(/Yjt$) - &U’J’(fJiQ~)] ’ j=l q=l i=l

N

s.t. CQf exp(tti - 6;) 5 H, q = 1, .., Q

1

(P21

i=lQ: < 6’;, q = 1, ..,Q -Q; 5 0, q = l,..,Q

The solution to problem (P2) provides the optimal multiplier vectors ,u’, A:, Xi (corresponding to the original inequality constraints) and an upper bound to the global solution of problem (Pl’). The Lagrangian can then be formulated as:

L(uj,6i,QiP,~~,~:k,Xg~) = 62A’j*j exI,(/Jj’uj) - ew’JJ’(&QIj] + j=l q=l i=l

~/~q”(~Q~eX~>(tLi -6i) - H) + c.Ai.‘(Qf -Of) +2X:*(-Q:) (8) y=l i=l q=l q=l

From the KBT conditions of the primal problem (P2):

- WqJ’pi = -pqb exp(tLi - 64) _ A:” + AZ” (9)

Substituting equation (11) in (10) yields: nr 8 N

L(vj,bi,Qj,p’“, X;“,Xy”) = SCNjoj exp(&aj) +C/~‘~{CQf[exp(l~i - bi) - exp(tLi - 6f)J) j=l q=l i=l

-&qb H _ &‘;Qp,

q=l q=l

Hence, the qualifying constraints (gradients of the Lagrange function with respect to the “connected” variables Qf) to be added along with the Lagrange function in the relaxed dual problem take the following form:

p”[exp(tLi - bi) - exp(tli - b:)] 5 0 if Q: = 0:

p”[exp(tti - bi) - exp(tLi - bf)] 2 0 if Qf = 0

Since, however, pqk 2 0 Vq = 1, ..,Q all the qualifying constraints are of the form:

esp(tri - bi) - esp(tLi - 6:) < 0 if bi 2 bf esp(tLi - 6i) - esp(tl,i - 6:) 1 0 if bi 5 6: 1

(12)

The major implication of t,he quahfying const.rainbs in (12) is that instead of solving sNXQ relaxed dual (RD) problems, it is now sufficient. t,o solve only 2N subproblems, which is a. reduction of at. least twenty orders of ma.gnibude (2”“) even for two uncertain parameters with five qua.drature points each! In fact,, it is this major reduct,ion which enables t,he efficient implementatioi~ of the GOP algorithm in this case.

Algorithmic Procedure

Based on t(he derived qualifying constraint,s in (12) and the two properties described above, the follow- ing modified global opt,imiza.tion algorithm is proposed for the solution of problem (Pl’) for the csse of fixed Nj. STEP 1 Select an initial design C>. Set. Ii=1 1 t.he lower bound EPL = -cm. the upper bound EP” = +rx, and select a tolera.nce E. STEP 2 Solve the primal problem (P2) CO obtain t,he expected profit EP and the required dual information. Upda.te the upper bound EPCI = max(El-‘, EPL;}. STEP 3 Construct a.nd solve the required relaxed dual problems (at, most, 2N) that correspond to different bounds of Qf variables for each product i and store t,he obtained solutions.

European Symposium on Computer Aided Process Engineering-5 s631

STEP 4 Select as a new lower bound EPf the lowest value of the stored solutions of the RD problems; set as the new design bi” the corresponding design variables.

STEP 5 Check for convergence, if EPcr < EPF + c, stop; \$K is the globa. optimal design. Otherwise, set I&I<+1 and return to St.ep 2.

The benefits of the application of the global optimization approach for the design of multiproduct batch plants with uncertain demands will be illustrated in the following section with two example problems.

EXAMPLE 1 Consider the batch plant design of Figure 1 involving two products to be processed in three stages with

STAGE 3

Product 1

Produc12

Figure I: Batch Plant,

two units per stage. Size factors, processing times and cost data are given in Table 1. The demand of both products are considered as uncertain parameters described by normal distribution functions of the form N(200,lO) and N( 100,lO) for products 1 and 2, respectively. Five quadrature points are used for each uncertain parameter.

Using GAMS/MINOS for the solution of problem (Pl) (without considering any penalty term, y=O)

Table 1: Data. for Example 1

(a) Size factors (b) Processing times (c) I nvestment cost coefficients (d) Prices of Stage S ta.ge Stage j “i pi Products

Product 1 2 3 Product 1 2 3 1 5 0.F Product pi

1 2 3 4 1 8 20 8 2 5 0.G 1 5.5

2 4G3 2 IG 4 4 3 5 0.G 2 7.0

results in different solutions if different starting points a.re considered. For example, considering (V~, Vz, V3) = (lOOO,lOOO,lOOO) as a sta.rting point, the design (l/r, Vz, Va) = (800,12OO,GOO) with expected profit equal to 87.1 units is obtained, whereas if (Vi, V?, 1%) = (4500,4500,4500) is used as a sta.rting point a different design (VI, V2, Vs) = (1800,2700,3GOO) with a larger expected profit of 298.5 units is determined. For com- parison, the modified GOP algorit,hm, as outlined in the previous section, is applied for the solution of the same problem. The results are summarized in Ta.hle 2.

Table 2: Results of Global Optimization Algorit,hm

y=o Number of RD Upper Lower Design

Without With Bound Bound (v,!vz,v,) properties properties

iteration 1 250 1 14G.7 -1394.5 250

(500, 500, 500) iteration 2 iteration 3 250 4 (l2?,

14G.7 -5G9.2 (883.7, 1325.5, 1767.4) 20.2 -323 (500, 1703.4, 937.8)

Opt,imal Design c=O.OlG 8 iterations (1800, 2700, 3800) E=o.o002 14 it,erations

Different, penalty values Different. Start*ing points

0pt.ima.l Design Sbart.ing Design Number of CPU s per y value 1,’

Is;0

V, I$ (\,‘I 1 ci. v3) it.era.tion.5 it.crat,ion y=o 2ioo 3600 (1000, 1000, 1000) 13 0.8 y=4 19oi 28Gl 3815 (4500, 4500, 4500) 14 0.8 y=8 1972 2958 3944 (500, 500, 500) 15 0.8

The following points should be highlighted: (a) orders of magnitude reductions of t.he required relaxed dual problems a.re achieved by applying the derived properties, (b) convergence of the algorithm does not depend on different starting points, (c) t.he effect of penalty coefficient y: the larger the value of y the more “conservative” the design, (d) the slow convergence of the algorithm to yield highly accurate results (with an optimality st,opping crit,erion of f = 2 x IO-“, the algorithm takes almost t,wice as many iter- ations compared to t,he solution with 6 = 0.01(j), (e) increasing t,he number of qua.drature point,s per 0 does not increase the nun&r of problems Chat have t,o be solved per it,eration. e.g. for a 9x9 grid again

S632 European Symposium on Computer Aided Process Engineering-5

the maximum number of relaxed dual problems per iteration is 4; yet, since the primal problem (P2) is of much larger size in this case, its CPU solut,ion time increases (0.25 s versus 0.1 s required for the 5x5 grid).

EXAMPLE 2 Consider the six-stage, four-product batch plan shown in Figure 2, where t*he demand of t,he four products

Figure 2: Batch Plant of Example 4

are considered uncert,a.in with the following normal distribution functions N(150, lo), N(150, 8), N(180, 9) and N(160, 10). Dat,a. are shown in Table 3. Model (Pl) consist,s of 625 (for 5 quadrature points in

each uncertain parameter direction) nonconvex horizon constraints, 780 bounding constraints for the actual production of each product and 24 batch size constraints. Starting from a design (VI, Vz, Vs, V4, Vs, Vs)= (2500, 2500, 2500, 2500, 2500, 2500), the algorithm requires 37 iterations sto reach the global optimum de- sign (VI, V2, Va, V4, Vs, Vs)= (3918.9, 1917.8, 2547.3, 3251.4,2988.2, 2139.1), EP=26000 units, with relative tolerance of 0.003 (although it requires only 8 it.erations to converge within c=O.O06). In each itera, tion a maximum of 24=16 relaxed dual (NLP) problems are solved and one primal LP problem. Using GAMS/MINOS for the solution of NLP and LP problems, the computational time required per iteration is approximately 97 s CPU time in SPARCst,ation 10.

Table 3: Da.ta for Example 2

Product. 1

(a) Size factors (b) Processing times (c) Prices St.age Stage of products

2 3 4 5 6 1 2 3 4 5 6 2);

1 8.0 2.0 5.2 4.9 6.1 4.2 7 8.3 6 7 6.5 8 3.5 2 0.7 0.8 0.9 3.8 2.1 2.5 6.8 5 6 4.8 5.5 5.8 4.0 3 0.7 2.6 1.6 3.4 3.2 2.9 4 5.9 5 6 5.5 4.5 3.0 4 1 4.7 2.3 1.6 2.7 1.2 2.5 1 2.4 3 3.5 2.5 3 2.8 1 2.0

(d) Invest,ment cost, coefficients Qj = 10, pj = 0.6, j=1,..,4

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