772
Ind. Eng. Chem. Res. 1996, 35, 772-787
Batch Plant Design and Operations under Uncertainty Marianthi G. Ierapetritou and Efstratios N. Pistikopoulos* Centre for Process Systems Engineering, Department of Chemical Engineering, Imperial College, London SW7 2BY, U.K.
The paper addresses the problem of including aspects of uncertainty in process parameters and product demands at the design stage of multiproduct/multipurpose batch plants. A conceptual two-stage stochastic programming formulation is proposed with an objective function comprising investment costs, expected revenues from product sales, and a penalty term accounting for expected losses due to unfilled orders. It is shown that (i) the proposed formulation captures the various decision-making policies toward demand satisfaction in a unified way, (ii) the employed feasibility criterion for the incorporation of the uncertainty enables the exact reformulation of the two-stage model as a single large-scale optimization model, (iii) for the case of discrete equipment sizes and despite the use of general continuous probability distribution functions to describe the uncertainty, linearity of the model is preserved, allowing detailed scheduling models to be included directly in the optimization model, and (iv) for the case of continuous equipment sizes, a careful exploitation of the model structure allows for the development of an efficient global optimization procedure. The versatility of the proposed unified framework is demonstrated with example problems covering different batch plant design and scheduling formulations previously suggested in the open literature. 1. Introduction While significant developments have been made in the design and scheduling of multiproduct/multipurpose batch plants (see Reklaitis, 1990, and Rippin, 1993, for extensive reviews), the problem of “flexibility and uncertainty” (Rippin, 1993) in batch processing has received much less attention; yet, its practical relevance is obvious considering the incomplete information about the chemical/physical steps involved and the high degree of uncertainty regarding recipe parameters, resource and equipment availabilities, product demands, and product market trends. Previous research attempts to deal with the problem of designing batch plants under uncertainty include the work of Reinhart and Rippin (1986, 1987) and the subsequent extensions of Fichtner et al. (1990), who presented a number of model variants and solution procedures (scenario-based, penalty functions, two-stage approach) for the problem of multiproduct batch plant design with uncertain demands assuming one piece of equipment per stage. Wellons and Reklaitis (1989) also investigated the flexibility of multiproduct batch plants. They considered staged plant expansion over time to account for uncertainty in product demand; they also suggested a distinction between “hard” and “soft” constraints, introducing penalty terms for the latter type. Assuming as uncertain the time it takes for each product to reach the target product demand, they derived analytical expressions of the expected profit objective. Straub and Grossmann (1992) considered uncertainties in both product demands and equipment availability and presented a framework to maximize an expected profit by considering separately economic optimality and design feasibility. Shah and Pantelides (1992) presented a scenario-based approach and an approximate solution strategy for the design of multipurpose batch plants considering different schedules for different sets of production requirements. Rotstein et * Author to whom correspondence should be addressed. E-mail:
[email protected]. Telephone: 0171-5946620. Fax: 0171-5946606.
0888-5885/96/2635-0772$12.00/0
al. (1995) presented an MILP formulation for the evaluation of a lower bound of stochastic flexibility of multipurpose batch plants based on the assumption of independent uncertain parameters (appearing only at the right-hand side of the constraint set). Subrahmanyam et al. (1994) presented a scenario-based approach and a decomposition solution strategy in which batch plant scheduling aspects are simultaneously considered in design optimization. The objective of this paper is to establish a unified framework to deal with uncertainty in the design and operations of batch plants. Uncertainty in process parameters and product demands is considered based on which a two-stage stochastic programming formulation is proposed. The formulation, featuring a relaxation of the feasibility requirement with respect to uncertain demand, conceptually allows for economic optimality and plant feasibility to be performed simultaneously. Furthermore, the exploitation of the special structure of the problem enables the transformation of the initial two-stage (embedded) stochastic formulation to a single optimization problem, which can be efficiently solved for the cases of discrete and continuous equipment sizes. The paper is organized as follows. First, the foundations of a general conceptual mathematical formulation for the problem of optimal design and scheduling under uncertainty for multiproduct/multipurpose batch plant models are presented in detail. Section 3 then describes a global optimization algorithm for the efficient and rigorous solution of design problems with continuous equipment sizes. For the case of discrete equipment sizes and/or multipurpose design/scheduling formulations, it is shown in section 4 that the stochastic design and scheduling formulations, it is shown in section 4 that the stochastic design and scheduling problem is essentially transformed to a mixed integer linear program (MILP) of identical structure to the deterministic case. Finally, section 5 demonstrates the versatility of the proposed unified approach via its application to four representative example problems. © 1996 American Chemical Society
Ind. Eng. Chem. Res., Vol. 35, No. 3, 1996 773
2. A Conceptual Mathematical Formulation We consider multiproduct/multipurpose batch plant design models involving uncertainty in (i) processing times and size factors, reflecting process variability and model inaccuracy, and (ii) product demands, reflecting changing market conditions and/or variations of forecasted customer orders. The design objective is then to: Obtain an optimal design and corresponding operating plan for the manufacturing (over time) of the various products, by ensuring (i) feasible batch plant operation for any possible realization of the processing times and size factors variations and (ii) optimal demand order satisfaction, in order to maximize an expected profit criterion. Such a design objective is more general than simply requiring feasibility for a number of postulated scenarios, used to describe the various elements of uncertainty. In fact, as will be shown in the sequel, scenariobased formulations can be obtained as special cases of the problem as posed above. In order to mathematically represent the proposed design objective, three additional modeling features are considered: (i) Expected profit will be evaluated within the feasible operating region of the batch plantsthis will ensure a balanced consideration of economic potential and design feasibility (Pistikopoulos and Ierapetritou, 1995). (ii) Demand constraints will be partially relaxed and a penalty term will be included in the objective functionsthe former will ensure best utilization of available resources where the latter is needed to monitor the extent of demand satisfaction (see also Wellons and Reklaitis, 1989). (iii) Processing times and size factors are described via a compact parameter set defined by appropriate lower and upper bounds; demand orders are given by a vector of (forecasted) probability distribution functions. On the basis of the above considerations, in order to mathematically represent the problem of design/scheduling of multiproduct batch plant (for ease in the presentation, the case of a single product campaign mode is first considered here; Straub and Grossmann, 1992; the extension to other operational modes is shown later in the subsequent sections), the following notation is introduced: Index Sets i ) product: i ) 1, ..., N j ) stage: j ) 1, ..., M p ) scenario: p ) 1, ..., P R(Vj,Nj) ) feasible region of design (Vj, Nj), i.e., R ) {θ|∀θ ∈ Θ∃Qpi , TpLi satisfying the constraints of the problem} Θ ) uncertainty space Parameters Rj ) variable-size cost coefficients for the investment cost of process equipment at stage j βj ) fixed-cost charges for the investment cost of process equipment at stage j γ ) penalty coefficient δ ) coefficient used to annualize the capital cost θi ) demand of product i (θi ∈ Θ) H ) time horizon pi ) price of product i
Sijp ) size factor of product i at stage j for scenario p tijp ) processing time of product i at stage j for scenario p wp ) probability of scenario p Variables Bi ) batch size of product i Nj ) number of equipments at stage j Qpi ) production of product i for scenario p TpLi ) cycle time of product i for scenario p Vj ) size of equipment at stage j Symbols Eθ ) expectancy operator
The following mathematical model then formally describes the problem of determining the optimal design (Vj, Nj) and schedule (Bi, TpLi) that maximize an expected profit:
Problem PB max Eθ∈R(Vj,Nj) Vj,Nj
∑p wp max ∑i piQpi - δ∑j NjajVβj Q γEθ∈R(V ,N )∑wp max (∑piθi - ∑piQpi ) Q p i i j
p i
j
subject to:
∑i
j
Qpi TpLi Bi
TpLi g
Bi e
p i
∀p
e H;
()
tpij ; Nj
Vj Spij
;
Qpi e θi;
∀i, j, p
∀i, j, p
∀i, p
θi ∈ R(Vj,Nj) Problem PB mathematically describes a design strategy consisting of two stages. At the first stage the objective is to select the volume and the number of batch equipments per stage that maximize an expected profit, whereas at the second stage the optimal production policy (Qi) and/or schedule (TLi) is determined for each realization of the uncertainty θi and Sijp, tijp. Long-term demand variations via capacity expansion options (as in Wellons and Reklaitis, 1989) can also be included in problem PB as discussed in Appendix I. Other strategies that have appeared in the open literature with regard to demand consideration are the following: (a) Qi ) θi, θi ∈ R(Vj,Nj) representing the decision maker’s choice to always meet the demand within the plant’s feasible region (see Pistikopoulos and Ierapetritou, 1995). Figure 1a illustrates this policy graphically. (b) Qi ) θi, θi ∈ Θ (scenario approach) representing the DM requirement of demand satisfaction within a specified range Θ defined by appropriate lower and upper bounds (see Subrahmanyam et al., 1994) (dashed lines in Figure 1b). Here the points correspond to different scenarios that must be satisfied in this case, whereas the arrows illustrate one possible direction of
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Ind. Eng. Chem. Res., Vol. 35, No. 3, 1996
Figure 2. Effect of penalty coefficient γ.
(with q ) 1, ..., Q quadrature points) as follows: P
N
∫θ∈R ∑ w ∑ p
p)1
P
piQ h ki J(θ)
dθ )
N
piQ h qp ∑ w J p)1 ∑w ∑ i i)1 q q
p
(1)
θq∈R
i)1
where Q h i are the arguments of the inner optimization subproblem (profit estimation); wq are the weights corresponding to each quadrature point; Jq is the probability of each point. Note that through this transformation the location of the quadrature points is still unknown. (ii) The relaxation of the demand constraints Qpi e θi gives rise to the following feasibility property: PropertysAny design (Vj, Nj) which satisfies the constraints of problem PB for fixed product demands θi is feasible for the whole parameter range, Θ. Proofssee Appendix III. Figure 1. Different demand constraints.
production optimization where the following demand constraint is considered Qi e θi, θi ∈ Θ. The priority of demand satisfaction among the different products is controlled by the relative values of the ratio TLi/Bi, representing the required batch production time for product i, and the relative values of product prices pi. It is interesting to note that the proposed approach embodies these model instances regarding the demand satisfaction depending on the value of the penalty coefficient γ. In particular, as the value of γ increases, implying a more severe penalization of partial order fulfillment, the result of the decision strategy in PB moves toward the direction of complete demand satisfaction, which coincides with the result of the scenariobased approach. In fact, as shown in Appendix II, there exists a critical value of γ for which PB is equivalent to the multiperiod (scenario-based) formulation. Figure 2 pictorially illustrates this fact: the shaded areas correspond to the corresponding feasible regions of different designs 1-3 obtained from PB for different γ values. However, problem PB cannot be directly solved since it involves the evaluation of the expected profit within the unknown at the design stage feasible region of the plant; moreover, it requires an integration (expectation evaluation) over an optimization problem (maximization of revenue with respect to production variables). These difficulties can be overcome based on the following ideas: (i) The multiple integral for the expected revenue evaluation over the feasible region of the design is approximated through a Gaussian quadrature formula
The important implication of this property is that the integration can be performed within the region defined by the bounds of the uncertain demand parameters, since the feasible region of any design depicted from the solution of problem PB in the space of the uncertain demands coincides with the uncertain parameter range. Therefore, the obstacle of unknown integrands is effectively overcome. Based on the above property and by incorporating eq 1, problem PB can be transformed into the following equivalent (single-level) optimization problem PB1:
Problem PB1
∑j
max {-δ Vj,Nj,Qqp i
P
NjRjVβj j) +
wqJq(∑PiQqp ∑ ∑ i p)1 q)1 i
P
γ subject to:
Bi e
Q
wp Q
∑ wpq)1 ∑ wqJq(∑i piθqi - ∑i piQqpi )} p)1 Vj Spij
TpLi g
;
∀i, j, p
tpij ; Nj
∀i, j, p
Qqp i TpLi e H; Bi
∑i
q Qqp i e θi ;
VLO j
e Vj e
VUP j ,
∀q, p ∀i, q, p
θqi ∈ Θ, γ g 0
q
p
i
q
i
NjajVβj j
i
q i
i
i
qp i )
subject to:
q
p
i
eH
q Qqp i e Di
Bi
p Qqp i TLi
p TLi g
s
s
n
∑∑w
n
()
tijp
i
n
n
qp i
)1
eH
jsn
q Qqp i e Di
i
∑T
s
∑∑y
qp ijsn
qp ijsn
qp qp wijsn e Uijsn yjsn
nqp i )
s
n
p
∑∑ n w
s
j
Qqi Sijp yjsn vjs
p
∑∑
q
q
-δ
i
βj najvjs yjsn
q i
i
i
∑∑∑ γ∑w ∑w (∑p D - ∑p Q
piQqp i
Vj discrete
Tqp i g
i
j
∑w ∑ w ∑ p
tijp Nj
p
p
max
q
nqp i g
q
piQqp i
Vj g BiSijp
∑
p
∑w ∑ w ∑ - δ∑ γ∑w ∑q (∑p D - ∑p Q
q
subject to:
max
Vj continuous
single-product campaign
qp i )
q
q
p
i
i
Qqp i Bi
qp ik
qp ik
i
k
∑NP
q i
j
qp ik SLikj)
q Qqp i e Di
+
NPiiqp e nqp i - 1
i
∑NP
∑NP k
nqp i )
nqp k )
qp p i tij
∑(n
p
piQqp i
Vj g BiSijp
p
i
nqp i )
q
p
Vj continuous ajVβj j
i
eH
i
qp i )
q
p
p
i
i
qp p i tij
∑(n
+
s
n
n
q
qp ik
∑NP i
qp ik
)1
∑NP
jsn
k
p
s
k
q Qqp i e Di
qp ik SLikj)
∑NP
i
eH
NPiiqp e nqp i - 1
nqp k )
nqp i )
j
n
i
q i
βj j js jsn
s
n
∑∑ny
p Qqp i Sij yjsn vjs
p
∑∑
q
qp i
∑∑y s
nqp i g
i
Vj discrete
jsn
i
i
∑w ∑w ∑p Q - δ∑∑∑na v y γ∑w ∑w (∑p D - ∑p Q subject to:
max
q
unlimited intermediate storage (UIS)
multiple-product campaign
∑w ∑ w ∑ - δ∑ γ∑w ∑w (∑p D - ∑p Q
subject to:
max
q
zero wait policy (ZW)
stochastic multiproduct batch plant design formulation
Table 1. Stochastic Models for Multiproduct Batch Plant Design
qp i )
Ind. Eng. Chem. Res., Vol. 35, No. 3, 1996 775
q
q
p
q
i
i
j j
βj j
i
i
p TLi g
eH
g Tqp i
q Qqp i e Di
h
qp h
qp hiθh
∑θ
∑R
h
qp i )
p
p
i
subject to:
i
n
()
tijp
n
i
n
eH
q Qqp i e Di
i
qp i
)1
g Tqp i
jsn
qp hiθh
n
∑T
h
∑R
s
∑∑y
qp ijsn
qp ijsn
qp qp wijsn e Uijsn yjsn
s
∑∑w
s
n
p
s
p Qqp i Sij yjsn vjs
p
j
∑∑ n w
s
q
∑∑
q
qp i
nqp i )
Tqp i g
i
q i
q
q
i
q i
βj j js jsn
i
i
∑w ∑w ∑p Q - δ∑∑∑na v y γ∑w ∑w (∑p D - ∑p Q
tijp Nj
p
j
max
nqp i g
p
qp i qp i )
q
q
p
q
q
p
i
p
i
i
qp i
i
j
i
j j
i
∑N a V
q i
-δ
βj j
qp h
eH q Qqp i e Di
h
g τqp r
g Qqp i
qp hrθh
∑θ
h
∑R
qp r
p p qqp r SijTLr Vj
∑q
rERi
τqp r g
qp i )
q
q p
p
i
∑q
s
qp h
eH
g τqp r
g Qqp i
q Qqp i e Di
h
p
Vj )1
qp hrθh
qp r
∑θ
∑R
rERi
h
js
∑y
s
p
j
i
p p qqp r SijTLr
q
∑
qp i
s
q
τqp r g
i
Vj discrete
n
i
q i
βj j js jsn
i
i
∑w ∑w ∑p Q - δ∑∑∑na v y γ∑w ∑w (∑p D - ∑p Q
subject to:
max
multiple-production routes
∑w ∑ w (∑p D - ∑ p Q
γ
p
∑w ∑ w ∑ p Q
subject to:
max
Vj continuous
stochastic multipurpose batch plant design formulation Vj discrete
Vj g BiSijp
q
i
p qp Tqp i Bi ) TLiQi
p
∑w ∑ w ∑ p Q - δ∑N a V γ∑w ∑w (∑p D - ∑p Q
subject to:
max
Vj continuous
single-production routes
Table 2. Stochastic Models for Multipurpose Batch Plant Design
qp i )
776 Ind. Eng. Chem. Res., Vol. 35, No. 3, 1996
Ind. Eng. Chem. Res., Vol. 35, No. 3, 1996 777 Table 3. Stochastic Models for Multipurpose Batch Plant Design stochastic multipurpose batch plant design formulation optimal design and campaign planning max
∑q ∑w ∑p R q
s
p
q
s
qs p
p
γ
optimal design with detailed scheduling constraints
- δCjEj -
max
∑w ∑w (∑p D - ∑p R q
s
p
q
s
q p
p
p
∑w ∑w ∑p (S - S ) - δC E γ∑w ∑w [∑p D - ∑p (S q
p
qs jki
j
p
j
q s
s
q
p
qp sT
s
s
qp - Ss0 )]
s
pi-1
∑ ∑W
e Ej
qp ijt-θ
iΩjk
e Ej
iIj θ)0
∑U
qs jki
∑U
+
iΩjk
qs jk′i
-1
unit capacity constraints
iΩjk′
timing constraints
qp e Vi Bijt
qs qs tinqs g tfqs k k′ + ykk′τkk′ - (1 - ykk′)H material balances
Rqs p e
φik
qp qp max Bijt e Wijt Vi material balances
∑Uˆ
Tsk jΛik
ap qp Sst ) Sst-1 -
qs jkiVj
For the case of continuous equipment sizes, the following exponential transformations can be introduced (Kocis and Grossmann, 1988):
Vj ) exp(vj), Bi ) exp(bi), TLi ) exp(tLi), Nj ) exp(nj)
∑r yjr ln(r); ∑r yjr ) 1;
yjr ) {0,1}; r ) 1, ..., Nj
where vj, bi, tLi, and yjr are the new transformed
qp ijt
+
∑ ∑ Fh B p is
qp ijt-pis
iT ˆ s jKi
qp Sst e Cs production requirements qp qp SsT - Ss0 e
q Rqs p e Dp
3. Continuous Equipment Sizes
p is
storage capacity constraints
qs qs e HUjki U ˆ jki production requirements
Tables 1-3 present the corresponding equivalent formulations of different batch plant design formulations in the presence of uncertainty appearing in the open literature, several of which involve explicit scheduling models concerning plant operation (for example, multiple product campaign with zero wait (ZW) or unlimited intermediate storage (UIS)). Note that schedule patterns are considered to change with respect to demand realizations in order to achieve the best utilization of the proposed plant. In problem PB1 the location of quadrature points within the plant’s feasible region is fully determined from the optimization procedure. In this way, the postulation of arbitrary scenarios is effectively avoided; instead, the number and location of scenarios (integration points) will only depend on the degree of accuracy required for the integration. Nevertheless, problem PB1 still corresponds to a nonconvex nonlinear mixed integer optimization problem due to the presence of the nonconvex objective function (investment cost) and the horizon constraint. In the following sections we propose global optimization solution approaches for problem PB1 for the cases of continuous and discrete equipment sizes.
∑∑F B iTs jKi
qs inqs U ˆ jki e tfqs k - tk
nj )
qp s0
subject to: allocation constraints
∑U
qs qs ykk′ + yk′k g
qp sT
s
q
qs p )
p
subjec to: allocation constraints
p
s
q
T q D H s
variables. Unlike the pure deterministic case, however, in which the reformulation leads to a convex MINLP problem, in this case the consideration of the uncertain demands still preserves nonconvexities of the following form:
∑i Qi exp(tLi - bi) e H since Qi are optimization variables. This calls for a global optimization solution approach. For the case of multiproduct batch plants operating in the single-product campaign, it will be shown that the resulting stochastic formulation can be effectively solved to global optimality via a modified version of the GOP algorithm proposed by Floudas and Visweswaran (1990, 1993). Moreover, as it will be briefly discussed, the different formulations shown in Tables 1-3 for the case of continuous equipment sizes can be treated in a similar way. The multiproduct batch plant design formulation (PB1), after the exponential transformations of Kocis and Grossmann, corresponds to the following single nonlinear optimization problem involving nonconvexities in the horizon constraint:
Problem PB2 M
Rj exp(∑yjr ln(r) + βjvj) + ∑ j)1 r
max {-δ
vj,bj,yjr tLi,Qqi
Q
∑ q)1
N
wqJq{
∑ i)1
Q
∑ q)1
piQqi } - γ
N
∑ i)1
wqJq{
N
piθqi -
piQqi }} ∑ i)1
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Ind. Eng. Chem. Res., Vol. 35, No. 3, 1996
subject to: vj g ln(Sij) + bi; tLi g ln(tij) -
∑r yjr ln(r);
formulated as:
i ) 1, ..., N, j ) 1, ..., M i ) 1, ..., N, j ) 1, ..., M
M
Rj exp(∑yjr ln(r) + βjvj) ∑ j)1 r
qk L(vj,bi,Qqi ,µqk,λqk 1 ,λ2 ) ) δ Q
N
∑ w J {∑ q)1 i)1 q q
N
Qqi exp(tLi - bi) e H; ∑ i)1 Qqi
e
θqi ;
q ) 1, ..., Q
Q
piQqi }}
N
µ (∑Qqi exp(tLi - bi) - H) + ∑ q)1 i)1 qk
+
Q
∑ q)1
i ) 1, ..., N, q ) 1, ..., Q
Q
q q λqk 1 (Qi - θi ) +
q λqk ∑ 2 (-Qi ) q)1
(2)
From the KKT conditions of the primal problem PB3:
ln(VLO j )
e vj e
yjr ) {0,1},
θqi
ln(VUP j )
qk -wqJqpi ) -µqk exp(tkLi - bki ) - λqk 1 + λ2
∈ Θ, γ g 0
Substituting eq 3 into eq 2 yields:
Based on the following variable partition:
y ) {vj, bi, yjr, tLi}, x )
M
{Qqi },
j ) 1, ..., M, i ) 1, ..., N, q ) 1, ..., Q
problem PB2 satisfies conditions (A) of the GOP algorithm (Floudas and Visweswaran, 1990), since both the objective and the constraints are convex in {vj, bi, yjr, tLi} for every fixed Qqi and linear in Qqi for each fixed {vj, bi, yjr, tLi}. Similar partitions are derived for the various design and scheduling formulations shown in Tables 1-3. In particular, for the case of multipleproduct operation mode of multiproduct batch plants y ) {vj, bi, yjr, tLi}, x ) {Qqi , θqh}; for multipurpose batch plants with single production route y ) {vj, bi}, x ) {Qqi , θqh, τqr }, and with multiple production routes y ) q {vj, bi}, x ) {Qqi , nqi , NPik }. Although GOP can, in principle, be directly applied for the solution of problem PB2 to global optimality, it would require prohibitively high computational effort (2N×Q subproblems per iteration). However, as will be shown in the next section, by exploiting the special structure of the batch plant design model in PB2, a number of properties can be established with which the number of relaxed dual subproblems that have to be solved per iteration can be reduced by several orders of magnitude (scaling only to the number of products). Properties of Multiproduct Batch Plant Design Problem. Projecting on y ) {vj, bi, yjr, tLi}, the following linear primal problem is obtained at the kth iteration (rewritten as a minimization problem):
Problem PB3 M
Rj exp(∑ykjr ln(r) + βjvkj ) ∑ j)1 r
min {δ Qqi
Q
∑ q)1
N
piQqi }} ∑ i)1
wqJq{
N
subject to:
Qqi exp(tkLi - bki ) e H, ∑ i)1
(3)
q ) 1, ..., Q
Qqi e θqi ,
q ) 1, ..., Q, i ) 1, ..., N
-Qqi e 0,
q ) 1, ..., Q, i ) 1, ..., N
The solution to problem PB3 provides the optimal vector qk of multipliers µqk, λqk 1 , λ2 (corresponding to the original inequality constraints) and an upper bound to the global solution of problem PB2. The Lagrangian can then be
Rj exp(∑yjr ln(r) + βjvj) + ∑ j)1 r
qk L(vj,bi,Qqi ,µqk,λqk 1 ,λ2 ) ) δ Q
∑ q)1
N
Qqi [exp(tLi - bi) - exp(tkLi - bki )]} ∑ i)1
µqk{
Q
Q
∑ µqkH - q)1 ∑ λqk1 (θqi ) q)1 Hence, the qualifying constraints (gradients of the Lagrange function with respect to the “connected” variables Qqi ) to be added along with the Lagrange function in the relaxed dual problem take the following form:
µqk[exp(tLi - bi) - exp(tkLi - bki )] e 0
if Qqi ) θqi
µqk[exp(tLi - bi) - exp(tkLi - bki )] g 0
if Qqi ) 0
Since, however, µqk g 0 (∀q ) 1, ..., Q), all the qualifying constraints are of the form:
exp(tLi - bi) - exp(tkLi - bki ) e 0 if TLi/Bi g TkLi/Bki exp(tLi - bi) - exp(tkLi - bki ) g 0 if TLi/Bi e TkLi/Bki
}
(4) The major implication of the qualifying constraints in eq 4 is that instead of solving 2N×Q relaxed dual (RD) problems, it is now sufficient to solve only 2N subproblems, a reduction which in fact enables the efficient implementation of the GOP algorithm in this case (for example, 20 orders of magnitude, 225, for two uncertain parameters with five quadrature points each). Furthermore, since the gradient of the Lagrange function with respect to Qqi (qualifying constraints) is not a function of Qqi , i ) 1, ..., N, q ) 1, ..., Q, two other properties of the GOP algorithm (Visweswaran and Floudas, 1993) can be used to further reduce the number of relaxed dual problems at each iteration, as follows: (i) If at the kth iteration the qualifying constraint for product i is e0 (or g0) for every other product, i.e., µqk[exp(tLi - bi) - exp(tkLi - bki )] e 0 (or g 0), ∀i ) 1, ..., N, implying that TLi/Bi g TkLi/Bki (or TLi/Bi e TkLi/Bki ), ∀i ) 1, ..., Nsthis is true when TkLi/Bki corresponds to the lower (or upper) bound of TLi/Bi, and consequently the solution of the relaxed dual with the qualifying constraints µqk[exp(tLi - bi) - exp(tkLi - bki )] g 0 (or e 0) can be effectively avoided. (ii) If at the kth iteration the qualifying constraint of product i µqk[exp(tLi - bi) - exp(tkLi - bki )] ) 0, ∀i ) 1,
Ind. Eng. Chem. Res., Vol. 35, No. 3, 1996 779
Tqp i
g
∑s ∑n
nqp i ) (PB3)
() tpij n
∀i, j, p, q
qp wijsn ;
qp ; ∑s ∑n wijsn
qp e Uijsnyjsn; wijsn
∑i Tqpi e H; q Qqp i e θi ;
∀i, j, p, q ∀i, j, s, n, p, q ∀q, p ∀i, q, p
Problem PB4 is a mixed-integer linear programming problem for which conventional MILP tools (such as SCICONIC, CPLEX) can be used for its solution. Note that in this case the structure of the deterministic formulation is fully preserved despite the use of general continuous probability distribution forms for the description of uncertain product demands. As shown in Tables 1-3 similar stochastic models have been obtained for various design and scheduling formulations appearing in the open literature. 5. Examples
Figure 3. Global optimization algorithm.
..., N, which is satisfied when µqk ) 0, ∀q ) 1, ..., Q, then it is sufficient to solve only one RD problem at either the lower or upper bounds of Qqi variables. The algorithmic procedure is shown graphically in Figure 3. Example 1 in section 5 is used to illustrate the results from the implementation of the global optimization algorithm for the solution of batch plant design problems. 4. Discrete Equipment Sizes For the case of discrete equipment sizes, nonconvexities can be effectively avoided based on the reformulations proposed by Voudouris and Grossmann (1992):
γjs
1 ) Vj
∑s v
js
∑s yjs ) 1 yjs )
{
1 if unit at stage j has size s 0 otherwise
where Vj is restricted to take values from the set SVj ) {vj1, ..., vjs}. In this way, problem PB1 can be recasted as follows:
Problem PB4 max Vj,Nj
∑q wq∑p wp∑i piQqpi - δ∑j ∑s ∑n najvβjsγjsn γ∑wq∑wp(∑piDqi - ∑piQap i ) q p i i
subject to:
j
nqp i
g
∑s ∑n
p Qqp i Sij yjsn; vjs
In this section, a variety of examples illustrating the key features of the approach described in the previous sections for batch plant design and scheduling in the presence of process parameter variations (processing times and size factors) and demand fluctuations will be presented. The first example deals with continuous equipment sizes: a multiproduct batch plant involving two products and three stages, discussing the effect of different formulations regarding the demand constraint satisfaction, the penalty term, the consideration of uncertain processing times and size factors, and the incorporation of more involved scheduling models, and illustrating the steps of the global optimization procedure. The rest of the examples describe multiproduct batch plant design and scheduling formulations with discrete equipment sizes demonstrating the potential of the proposed method to deal with general classes of batch plants involving uncertainty. Example 1. Consider the batch plant design of Figure 4 involving two products to be processed in three stages with one unit per stage. Size factors, processing times, and cost data are given in Table 4. The demands of both products are considered as uncertain parameters described by normal distribution functions of the form N(200,10) and N(100,10) for products 1 and 2, respectively. Five quadrature points are used for each uncertain parameter. Using GAMS/MINOS for the solution of problem PB1 (without considering any penalty term, γ ) 0) results in different solutions if different starting points are considered. For example, considering (V1, V2, V3) ) (1000, 1000, 1000) as a starting point, the design (V1, V2, V3) ) (800, 1200, 600) with expected profit equal to 87.1 units is obtained, whereas if (V1, V2, V3) ) (4500, 4500, 4500) is used as a starting point, a different design (V1, V2, V3) ) (1800, 2700, 3600) with a larger expected profit of 298.5 units is determined.
∀i, j, p, q Figure 4. Batch plant for example 1.
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Table 4. Data for Example 1 (a) size factors
(b) processing times
stage
(c) investiment cost coefficients
(d) prices of products
product
1
2
3
product
1
stage 2
3
stage j
Rj
βj
product
pi
1 2
2 4
3 6
4 3
1 2
8 16
20 4
8 4
1 2 3
5 5 5
0.6 0.6 0.6
1 2
5.5 7.0
Table 5. Results of Global Optimization Algorithm for Example 1 γ)0 number of RD without properties
with properties
upper bound
lower bound
design (V1, V2, V3)
250 250 250
1 1 4 (22)
146.7 146.7 20.2
-1394.5 -569.2 -323
(500, 500, 500) (883.7, 1325.5, 1767.4) (500, 1703.4, 937.8)
iteration 1 iteration 2 iteration 3
optimal design (1800, 2700, 3600)
) 0.016 ) 0.0002
8 iterations 14 iterations
different penalty values optimal design
different starting points
γ value
V1
V2
V3
starting design (V1, V2, V3)
number of iterations
CPU s per iteration
0 4 8
1800 1907 1972
2700 2861 2958
3600 3815 3944
(1000, 1000, 1000) (4500, 4500, 4500) (500, 500, 500)
13 14 15
0.8 0.8 0.8
Table 6. Uncertain Process Parameters for Example 1 tij
Sij
Table 7. Results of Example 1 for Discrete Equipment Sizes
period product stage 1 stage 2 stage 3 stage 1 stage 2 stage 3 1 2 3
1 2 1 2 1 2
7 15 9 17 8 16
19 3 21 5 20 4
7 3 9 5 8 4
2.5 4.5 1.5 3.5 2 4
3.5 6.5 2.5 5.5 3 6
4.5 3.5 3.5 2.5 4 3
For comparison, the modified GOP algorithm, as outlined in the previous section and described in Appendix IV, is applied for the solution of the same problem. The results are summarized in Table 5. The following points should be highlighted: (a) orders of magnitude reductions of the required relaxed dual problems are achieved by applying the derived properties, (b) convergence of the algorithm does not depend on different starting points, (c) the effect of penalty coefficient γ: the larger the value of γ, the more “conservative” the design, (d) the slow convergence of the algorithm to yield highly accurate results (with an optimality stopping criterion of ) 2 × 10-4, the algorithm takes almost twice as many iterations compared to the solution with ) 0.016), (e) increasing the number of quadrature points per θ does not increase the number of problems that have to be solved per iteration; e.g., for a 9 × 9 grid again the maximum number of relaxed dual problems per iteration is 4; yet, since the primal problem PB2 is of much larger size in this case, its CPU solution time increases (0.25 s versus 0.1 s required for the 5 × 5 grid). Assuming that the equipments are only available from the following set of discrete values {1200, 2200, 3200, 4200}, problem PB4 is solved for the following cases: (i) considering only short-term demand variation, i.e., 160 e demand of product 1 e 240, 60 e demand of product 2 e 140, (ii) accounting also for long-range variations, i.e., 410 e demand of product 1 e 490, 310 e demand of product 2 e 390, during a second period (see Appendix I), and (iii) considering uncertain processing times and size factors through a multiperiod formulation with three time periods as shown in Table 6. The results are summarized in Table 7.
a
case
designa (V1, V2, V3)
i ii iii
(3200 (1), 3200 (1), 3200 (1)) (3200 (2), 3200 (2), 3200 (2)) (4200 (1), 4200 (1), 4200 (1))
(1) one equipment; (2) two equipments.
Finally, more involved scheduling models concerning plant operation in multiple product campaign with zero wait (ZW) or unlimited intermediate storage (UIS) are considered, where the possibility of having more than one unit per stage is also incorporated. The results are summarized in Table 8. Note that the consideration of more detailed scheduling models leads to more optimistic plants due to better utilization of the processing equipments; in this case the best utilization of the proposed plant is achieved by changing the schedule patterns to follow the demand realization. However, since the scheduling constraints are rather simplified, the detail scheduling can be derived through the solution of a cycle time minimization problem (see Birewar and Grossmann, 1989). Example 2. A multipurpose batch plant involving the production of five products in five stages, with up to two parallel units per stage, is considered as discussed in Vaselenak et al., 1987, and Voudouris and Grossmann, 1992sa single-product campaign and single production route is considered (see Table 9). Here, we also assume that the demand of products 1 and 2 are uncertain following normal probability distribution functions, N(200,10) and N(150,10), respectively. Equipment sizes are considered discrete from the set {500, 1000, 1500, 2000}. The solution of the deterministic problem, involving 303 continuous and 40 binary variables in 499 constraints, results in the selection of one unit per stage of size 1500 (for all stages), with a corresponding capital cost of 3.1 × 104 units; this batch plant can meet demands for products 1 and 2 within the regions (160, 200) and (110, 150), respectively. The solution was obtained using GAMS/CPLEX in 12 CPU s on a Sparc10. On the other hand, the stochastic MILP model in PB4 (see also Table 2) involves 1851 continuous variables
Ind. Eng. Chem. Res., Vol. 35, No. 3, 1996 781 Table 8. Results of Example 1 for Detailed Scheduling Models multiple-product campaign single-product campaign stage j
zero wait (ZW)
unlimited intermediate storage (no clean-up times)
no. of units
equipment size
no. of units
equipment size
no. of units
equipment size
3 3 3
1200 1200 1200
3 3 3
800 800 800
2 2 2
1200 1200 1200
1 2 3
6.3 × 104 1.4 × 105
7.9 × 104 1.2 × 105
capital cost: expected profit:
5.3 × 104 1.5 × 105
Table 9. Data for Example 2 stage (a) size factors product
1
2
3
(b) processing times 4
5
1 3.2 2.5 0 0 0 2 0 0 1.0 1.5 0 3 0 2.7 0 0 2.3 4 3.1 0 1.1 0 0 5 0 0 0 1.7 2.8 (d) investigment cost coefficients aj ) 250, βj ) 0.6, j ) 1, ..., 4 Table 10. Data for Example 3 product A
1 2 3 4 5 6
1.5 2.0
B
1.6 2.5 1.9
C
processing times D 2.2 1.7
1.4 2.4
1.8
A
B
C
4.0 6.5 3.9 5.5 4.2
D 6.5 7.0
4.5 3.5
4.7
in 3359 constraints with the same number of 0-1 variables (40); its solution requires 126 CPU s. The optimal plant corresponds to the selection of one equipment per stage of size 2000 (for all stages), with a larger capital cost of 3.6 × 104 units (a 16% increase); however, the plant can accommodate, unlike the deterministic design, a much larger range of demand variations, (160, 240) and (110, 190) for products 1 and 2, respectively. Example 3. For the case of multipurpose batch plants with multiple production routes the example given in Faqir and Karimi (1990) and Voudouris and Grossmann (1992) is considered here (see Table 2). Data are given in Table 10, whereas the production routes are shown in the plant layout in Figure 5. Horizon constraints are given in Table 11. The demands of products A and B are considered as random
Figure 5. Plant layout for example 3.
2
3
4
5
(c) prices of products pi
9 0 0 7.5 0
6 0 5.5 0 0
0 3.9 0 4.5 0
0 6.2 0 0 7.1
0 0 3.5 0 4.0
550 700 600 650 700
Table 11. Horizon Constraints for Example 3
size factors group
1
τ1 + τ2 + τ3 e H τ1 + τ10 e H τ2 + τ11 e H τ3 + τ12 e H τ4 + τ5 + τ6 + τ10 + τ11 + τ12 e H τ4 + τ7 + τ10 + τ11 + τ12 e H τ5 + τ8 + τ10 + τ11 + τ12 e H τ6 + τ9 + τ10 + τ11 + τ12 e H τ7 + τ8 + τ9 + τ10 + τ11 + τ12 e H
variables following normal probability distribution functions, N(300,10) and N(250,10), respectively. Considering discrete equipment sizes from the set {300, 500, 1000, 2000, 3000}, the stochastic model involves 1531 continuous variables and 50 binary variables in 1822 constraints compared to 245 continuous and 50 binary variables in 238 constraints of the corresponding deterministic model. The optimal plant, obtained in 10.5 CPU s using GAMS/CPLEX on a Sparc10, corresponds to the selection of equipments, E1 and E5 with size 3000 and E2, E6, E9, and E10 with size 300. This plant has a capital cost of 4.5 × 103 and can meet the following range of uncertain demands (260, 340) and (210, 290) for A and B, respectively. Example 4. A detailed design and scheduling model is considered here, based on the example presented by
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Ind. Eng. Chem. Res., Vol. 35, No. 3, 1996
Figure 6. Plant superstructure of example 4.
Figure 7. State task network of example 4. Table 12. Equipment Data for Example 4 equipment
task
unit 1a unit 1b unit 2a unit 2a vessel 1 vessel 2 vessel 3
task T1 task T1 task T2 task T2 state S1 state S2 state S3
min capacity
max capacity
fixed cost
variable cost
10 10 10 10
1750 250 2000 250 Unl 10 Unl
20 10 20 10
0.5 0.1 0.2 0.1
10
0.5
5
Barbosa-Povoa and Macchietto (1994), by additionally considering the demand of product 1 as uncertain following normal probability distribution function N(150, 20). Data concerning unit characteristics are given in Table 12, whereas the plant superstructure and the state task network representation are shown in Figures 6 and 7, respectively. The stochastic model involves 262 variables (125 0-1 variables) and 509 constraints; its solution requires 18 CPU s using GAMS/CPLEX on a Sparc10. The optimal plant corresponds to the selection of unit 1b of 54 m3 for task 1, unit 2b of 64 m3 for task 2, and a storage vessel for state 2 of 10 m3; it has an expected profit of $21 600. In comparison to the deterministic solution (unit 1b of 45 m3, unit 2b of 51 m3, and a storage vessel of 10 m3), the above plant involves larger equipments but has the capability of meeting the demand of product 1 within the range [90, 200]. The detailed production schedules obtained from the solution for different demand values are shown in Figure 8. If the schedule is forced to remain the same for all demand variations, a different design is obtained involving larger equipment of 81 m3 for unit 1b and of 91 m3 for unit 2b and with a different schedule as shown in Figure 9; its corresponding expected profit is lower ($21 400). Note that the consideration of production schedules following demand realizations results in better utilization of batch equipments, thus leading to smaller and more profitable plants.
Figure 8. Gantt charts for different demand values for example 4.
5. Concluding Remarks A unified framework has been presented in this paper to address the problem of batch plant design and operations under uncertainty. A two-stage stochastic programming formulation was proposed which properly balances economic optimality with feasibility objectives. The formulation features (i) an expected profit objective comprising investment cost for equipment sizing, revenues from product sales, and an explicit penalty term accounting for potential revenue losses due to unfilled orders and (ii) a relaxation of the feasibility requirement
Figure 9. One schedule for different demand values for example 4.
constraint. It was shown that basic properties can be established, enabling the efficient solution of the proposed formulation for both cases of continuous and discrete equipment sizes. A particularly important result was that the integration of the expected profit evaluation can be performed within the bounds of the uncertain parameter range, effectively avoiding the bilevel nature of the problem. An unexpected implica-
Ind. Eng. Chem. Res., Vol. 35, No. 3, 1996 783
tion of this property was that, despite the presence of uncertainty described by a general probability distributional form, no extra complexity was introduced in the model, fully preserving the structure of the corresponding deterministic models. Different policies regarding demand satisfaction, reflecting the attitude of decision maker, can be effectively captured within the proposed unified formulation. The approach is general and can be applied to most batch plant design/scheduling formulations appearing in the open literature involving either aggregate scheduling constraints (as, for example, in Tables 1 and 2) or detailed scheduling models (as in Table 3). A key issue that was exploited in this work was the option of changing the schedule pattern to follow demand variations, which was proven to be advantageous from a design point of view (especially in cases when the penalty for changeover is not significant). From a computational point, the proposed modified global optimization approach for the case of continuous equipment sizes is promising, since only the solution of a fraction of optimization subproblems is required compared to the original GOP development; clearly there is a scope for further investigation in this area. Finally, it is interesting to note that many optimization tasks in the proposed approach, for both batch continuous and discrete equipment sizes, can, in principle, be performed independently; here, parallel implementation developments look very promisingssee, for example, Pekny and co-workers (Basset et al., 1994) and Floudas and co-workers (Androulakis et al., 1994).
tfk ) finishing time of stage k TLi ) the cycle time of product i Ujki ) decision variable denoting the allocation of unit j to task i of stage k U ˆ jki ) length of time during which unit j is allocated to task i of stage k Vj ) sizes of batch equipments at stage j vjs ) standard volume of size s for stage j wq ) weights corresponding to each quadrature point Wijt ) {0, 1} variable denoting the starting of task i in unit j at the beginning of time period t yjr ) {0, 1} variables for the transformation of integer variables nj yjsm ) {0, 1} variables to represent the selection of n equipment of size s at stage j yjsm ) {0, 1} variables denoting whether stage k starts after stage k′ has finished Greek Symbols
The authors gratefully acknowledge financial support from the Commission of European Communities under Grant ERB CHBI CT93 0484.
Rj, βj ) cost coefficients of bath equipment at stage j Rhi ) incidence parameter denoting whether product i can be produced in campaign h γ ) penalty coefficient δ ) coefficient used to annualize the capital cost θi ) uncertain demand of product i following a distribution J(θi) U θi , θLl , θh ) upper and lower bound of the uncertain parameter θi length of campaign h qk qk ) the Lagrange multipliers of the inequalities λqk 1 , λ2 , µ of problem PB2 at k iteration Λik ) set of units suitable for task i at stage k ρis ) proportion of material time for task i to produce state s τr ) length of time for production in route r Ωjk ) set of units j suitable for stage k
Nomenclature
Appendix I. Capacity Expansion Consideration
Acknowledgment
Bi ) batch size of product i Bijt ) amount of material undergoing task i in unit j at period t Cs ) storage capacity of state s Di ) demand of product i Eθ{.} ) expectation operator of {.} over θ EP ) expected profit over the two stages EPkU ) predicted upper bound of expected profit at k iteration EPL ) lower bound of expected profit Ej ) decision variable denoting the selection of unit j J(θ) ) probability distribution function of the uncertain parameters θ H ) time horizon Nj ) number of batch equipment at stage j NU j ) an upper bound of the number of batch equipment at stage j NPik ) number of occurrences of pair i-k in the MPC schedule during horizon H ni ) number of batches produced of product i pi ) price of product i Qi ) production of product i qr ) amount for production from route r Rp ) amount of final product produced R(Vj,Nj) ) feasible region of batch plant (Vj, Nj) Sij ) size factors (volume of a vessel in stage j required to produce one mass unit of product i) Sst ) amount of material of state s stored at time period t T ) cycle time tij ) processing time of product i at stage j tin k ) starting time of stage k
The formulation of problem PB1 can be extended to account for long-term demand variations by incorporating the possibility of capacity expansion possibility through the consideration of additional identical units operating out of phase. This results in the following formulation (see also Wellons and Reklaitis, 1992): M
∑ j)1
max {-δ Vj,N1j ,N2j
M
(N2j ∑ j)1
1 N1j RjVβj j - δ exp(-rH ) Q1
P
∑ q∑)1 p)1
N1j )RjVβj j +
wp
N
wq1Jq1
1
Q2
P
piQ1q ∑ i i)1
1
+
N
w ∑ wq Jq ∑piQ2q ∑ i p)1 q )1 i)1 p
2
2
2
-
2
p
Q1
P
∑ q∑)1 p)1
γ
wp
wq1Jq1
1
(piθ1q ∑ i i)1
N
1
1 - piQ1q i ) -
Q2
P
∑ q∑)1 p)1 wp
γ
wq2Jq2
2
subject to:
Bi e
T1p Li
Vj Spij g
∀i, j, p
; tpij
N1j
(piθ2q ∑ i i)1
;
∀i, j, p
2
2 - piQ2q i )
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Ind. Eng. Chem. Res., Vol. 35, No. 3, 1996
T2p Li g N
∑ i)1 N
∑ i)1
tpij N2j
1p 1p Q1q TLi i
Bi 2p 2p Q2q TLi i
Bi
-(γ + 1)p1 + λTL1 exp(-b1) + µ1 ) 0 (AII.5)
∀i, j, p
;
-(γ + 1)p2 + λTL2 exp(-b2) + µ2 ) 0 (AII.6) e H1;
∀p
e H2;
∀p
where λ is the Lagrange multiplier for the horizon constraint; µ1 and µ2 are the Lagrange multipliers for the demand constraints; n11, n12 and n21, n22 are the Lagrange multipliers for the constraints vj g ln(Sij) + bi for products 1 and 2, respectively. Adding together eqs AII.1-4 yields the following expression for λ:
∀i, p, q1
1p 1 Q1q e θ1q i i ;
λ)
∀i, p, q2
2p 2 Q2q e θ2q i i ;
In the above formulation the time horizon of plant operation is divided by the time where additional units (N2j - N1j ) are used at each stage j to decrease the limiting cycle time and consequently to increase plant production. The objective function is augmented to incorporate the investment cost of capacity expansion where r is the interest rate used to calculate the present worth.
2
vj,Qi
subject to:
(AII.7)
2
Rj exp(βjvj) +
vj g ln(Sij) + bi;
piQi} ∑ i)1 i ) 1, 2, j ) 1, 2
2
QiTLi exp(-bi) e H ∑ i)1 Qi ) θi;
Consider the design of a multiproduct batch plant after the exponential transformations proposed by Kocis and Grossmann, 1988; problem PB1 for the case of of two products i ) 1, 2 produced in two stages where specific values of demand are considered:
i ) 1, 2
UP ln(VLO j ) e vj e ln(Vj )
θi ∈ Θ, γ g 0
2
Rj exp(βjvj) + {∑piQi} ∑ j)1 i)1
max {-δ
The KKT optimality conditions of this problem are: 2
γ( subject to:
∑ j)1
max {-δ
Appendix II. Evaluation of Critical Value of γ
vj,Qi
Q1TL1 exp(-b1) + Q2TL2 exp(-b2)
Consider now the multiperiod formulation of the same problem:
VLO e Vj e VUP j j
2
R1β1 exp(β1v1) + R2β2 exp(β2v2)
∑ i)1
2
piθi -
∑ i)1
piQi)}
vj g ln(Sij) + bi; i ) 1, 2, j ) 1, 2
′ ′ R1β1 exp(β1v1) - n11 - n21 )0
(AII.8)
′ ′ R2β2 exp(β2v2) - n12 - n22 )0
(AII.9)
′ ′ -λ′Q1TL1 exp(-b1) + n11 + n12 )0
(AII.10)
′ ′ + n22 )0 -λ′Q2TL2 exp(-b2) + n21
(AII.11)
-p1 + λ′TL1 exp(-b1) + µ′1 ) 0
(AII.12)
-p2 + λ′TL2 exp(-b2) + µ′2 ) 0
(AII.13)
2
QiTLi exp(-bi) e H ∑ i)1 Qi e θi;
i ) 1, 2
UP ln(VLO j ) e vj e ln(Vj )
θi ∈ Θ, ∼ g 0 The KKT optimality conditions of this problem are:
R1β1 exp(β1v1) - n11 - n21 ) 0
(AII.1)
R2β2 exp(β2v2) - n12 - n22 ) 0
(AII.2)
where λ′ is the Lagrange multiplier for the horizon constraint; µ′1 and µ′2 are the Lagrange multipliers for ′ ′ ′ ′ the demand constraints; n11 , n12 and n21 , n22 are the Lagrange multipliers for the constraints vj g ln(Sij) + bi for products 1 and 2, respectively. By adding eqs AII.8-11, we obtain the following expression for λ′:
λ′ )
-λQ1TL1 exp(-b1) + n11 + n12 ) 0
(AII.3)
-λQ2TL2 exp(-b2) + n21 + n22 ) 0
(AII.4)
R1β1 exp(β1v1) + R2β2 exp(β2v2) Q1TL1 exp(-b1) + Q2TL2 exp(-b2)
(AII.14)
Comparing eqs AII.7 with AII.14 results in λ′ ) λ for the critical value of γ (γc), since for this value the solutions of both formulations coincide. Comparing eq
Ind. Eng. Chem. Res., Vol. 35, No. 3, 1996 785
AII.5 with eq AII.12 and eq AII.6 with eq AII.13, we obtain the critical value γc:
-µ′1 + µ1 -µ′2 + µ2 γ ) ) p1 p2 c
parameters) coincides with the considered range of uncertain parameters independently of the design. (b) Vj Discrete. In this case, the problem formulation is as follows:
max yjsn,Qi
Appendix III. Feasibility Property for a Multiproduct Batch Plant Operating in a Single-Product Campaign
subject to:
∑i piQi - δ∑j ∑s ∑n najvβjjs yjsn ni g
(a) Vj Continuous. The mathematical formulation in this case is as follows:
max Vj,Nj,Qi
∑i
subject to:
∑i piQi - δ∑j
QiTLi Bi
() Vj ; Sij
n
ProofsThe feasibility test problemswith fixed y* jsn and θisis:
min u
∑s ∑n
subject to:
∑s ∑n
Qi
TLi - H e u ∑ i)1 B
QiSij * yjsn - ni e u vjs
() tij n
* niyjsn - Ti e u
Qi - θi e u
Qi - θi e u
-Qi e u
-Qi e u
The KKT conditions of the above problem are:
i ) 1, ..., N
∑ i)1
(AIII.1)
N
µ1i
+
µ2i ) 1 ∑ i)1
∀i, j
(AIII.3)
∀i, j
(AIII.4)
∑i Ti - H e u
i
N
∀i
∀i
u,Qi
TLi + µ1i - µ2i ) 0 Bi
∀i, j
PropertysAny design (y* jsn) satisfying design constraints above for fixed product demands θi, i ) 1, ..., N, is always feasible.
min u
λ
∀i, j
niyjsn
∀i, j
ProofsThe feasibility test problemswith fixed Vj, Nj, and θisis:
λ+
() tij
Qi e θi
PropertysAny design (Vj, Nj) satisfying design constraints above for fixed product demands θi, i ) 1, ..., N, is always feasible.
N
∀i, j
∑i Ti e H
∀i, j
Qi e θi;
subject to:
QiSij yjsn vjs
∑s ∑n yjsn ) 1
eH
tij ; TLi g Nj Bi e
∑s ∑n
Ti g
NjajVβj j
∑s ∑n
(AIII.2)
where λ is the Lagrange multiplier of the production constraint; µ1i and µ2i are the Lagrange multipliers for the bounding constraints of production Qi of product i. Since there are N control variables Qi, the number of active constraints must be less than or equal to N + 1. From the KKT conditions (AIII.1) and (AIII.2), it can be easily identified that the only potential active set consists of the production constraint and the lower bounds of production rates, which results in the following, always negative feasibility function:
u ) -λH e 0∀θi This permanent feasibility implies that the feasible region of the batch plant (in the space of uncertain
(AIII.5)
∀i
(AIII.6)
∀i
(AIII.7)
with the following KKT optimality conditions:
∑i ∑j λij + ∑i ∑j µij + ν + ∑i ki1 + ∑i ki2 ) 1
() Sij
∑j ∑s ∑n v
js
() Sij
-
∀i
* yjsn λij + ki1 - ki2 ) 0
∑j λij + ∑j ∑s ∑n v
∀i
* yjsn µij ) 0
(AIII.8)
(AIII.9)
(AIII.10)
js
∑j µij + ν ) 0
-
∀i
(AIII.11)
where λij are the Lagrange multiplier of the (AIII.3) constraints; µij are the Lagrange multipliers for the (AIII.4) constraints; ν is the Lagrange multiplier of the (AIII.5) constraint, and ki1 and ki2 are the Lagrange multipliers for the bounding constraints of production Qi of product i (AIII.6) and (AIII.7). Since there are 3N control variables Qi, ni, Ti, i ) 1, ..., ν, where N is the number of products, the number of
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Ind. Eng. Chem. Res., Vol. 35, No. 3, 1996
Table 13. Solutions of the Relaxed Dual Problems for Example 1 relaxed dual 1 2 3 4
combination of (Qq11, Qq21q2) bounds (0, 0) (θq11, 0) (θq11, θq21q2) (0, θq21q2)
optimal design (V1, V2, V3)
objective function
(1800, 2700, 3600) (1298, 1948, 2597) (500, 703.4, 937.8) (1276.7, 1915.1, 957.5)
-317.1 -85.4 -323 -53.8
active constraints must be less than or equal to 3N + 1. From the KKT conditions (AIII.11) it can be identified that for eq AIII.5 as well as eq AIII.4 one ∀i must be active. Following this result, eq AIII.10 suggests that also one (AIII.3) constraint ∀i must be active; this in turn points out that, from eq AIII.9, the lower bounding constraint for production of product i must be active. As constraint (AIII.6) is not included in the active set, it can be easily shown through algebraic manipulations that a negative feasibility function is always obtained. Note that similar proofs can be obtained for other batch plant design models (Ierapetritou, 1995). Appendix IV. Steps of the Modified Global Optimization Algorithm The global optimization algorithm of Figure 3 is implemented here for example 1 of section 5. Consider as a starting point (V1, V2, V3) ) (3000, 4500, 4500). Iteration 1. The solution of problem PB2 has an objective function of 146.7 units and provides a first upper bound on the global optimum. The Lagrange multipliers for the production constraints µq are all equal to zero, resulting in a zero Lagrange function gradient with respect to Qqi . Hence, it is only necessary to solve one RD problem with Qqi at either its upper or lower bounds. The solution of this problem is (V1, V2, V3) ) (500, 500, 500), with a value of the objective function of -1394.5 units, which is a first lower bound to the global optimum. The fixed value of y for the second iteration is (V1, V2, V3) ) 500, 500, 500). Iteration 2. For (V1, V2, V3) ) (500, 500, 500) the solution of the primal problem yields a value of 297.3 units (no update of the upper bound). Since the qualifying constraints are b1 + bL1 and b2 - bL2 , which are greater or equal to zero for any (b1, b2), only one RD problem is also required at this iteration with Qqi at its upper bounds. The Lagrange function from the first iteration is also added in the current relaxed dual problem since the qualifying constraints for this Lagrange function are zero for any (b1, b2). The solution of the relaxed dual problem yields the design (V1, V2, V3) ) (883.7, 1325.5, 1767.4), with an objective value of -569.2 units which corresponds to a new lower bound. The fixed design for the next iteration is (V1, V2, V3) ) (883.7, 1325.5, 1767.4). Iteration 3. The solution of the primal problem yields an expected profit of 20.2 units, which provides a new upper bound to the global optimum, and the corresponding Lagrange multipliers which are all nonzero. Four relaxed dual problems are then solved, for the combinations of bounds (Qq11, Qq21q2) equal to (0, 0), (θq11, 0), (θq11, θq21q2), and (0, θq21q2). Since the qualifying constraints of Lagrange functions from the first and second iterations are satisfied for every (b1, b2), they are both added in the relaxed dual problem. The solutions of the four relaxed problems are summarized in Table 13. At the end of this iteration these four solutions are stored, from which the design (V1, V2, V3) ) (500, 703.4, 937.8) with the smaller objective is selected for the next iteration.
Table 14. Results of GOP Application for Example 1 starting design (V1, V2, V3)
no. of iterations
CPU s per iteration
(1000, 1000, 1000) (3000, 4500, 4500) (4500, 4500, 4500) (500, 500, 500)
13 14 14 15
0.8 0.8 0.8 0.8
The algorithm requires 14 iterations to converge to the global optimum design (V1, V2, V3) ) (1800, 2700, 3600) with an expected profit equal to 298.5 units (within a tolerance limit ) 0.0002). The computational results for the application of GOP from four different starting points with γ ) 0 and considering a 5 × 5 quadrature grid are summarized in Table 14. Literature Cited Androulakis, I. P.; Maranas, C. D.; Visweswaran, V.; Floudas, C. A. Distributed Computing Approaches in Global Optimization. AIChE Annual Meeting, San Francisco, 1994; Paper No. 223d. Barbosa-Povoa, A. P.; Macchietto, S. Detailed Design of Multipurpose Batch Plants. Comput. Chem. Eng. 1994, 18, 1013. Basset, M. H.; Kudva, G. K.; Pekny, J. F.; Subrahmanyam, S. Using Distributed Computing to Support Integrated Batch Process Scheduling, Planning and Design Under Market Uncertainty. In Proceedings FOCAPD’94; Snowmass Co.: 1994. Birewar, D. B.; Grossmann, I. E. Incorporating Scheduling in the Optimal Design of Multiproduct Plants. Comput. Chem. Eng. 1989, 13, 141. Faqir, N. M.; Karimi, I. A. Optimal Design of Batch Plants with Multiple Production Routes. In Proceedings FOCAPD’89, Siirola, J. J., et al., Eds.; Elsevier: Amsterdam, The Netherlands, 1990; p 451. Fichtner, G.; Reinhart, H. J.; Rippin, D. W. T. The Design of Flexible Chemical Plants by the Application of Interval Mathematics. Comput. Chem. Eng. 1990, 14, 1311. Floudas, C. A.; Visweswaran, V. A. Global Optimization Algorithm (GOP) for Certain Classes of Nonconvex NLPs-I. Theory. Comput. Chem. Eng. 1990, 14, 1397. Floudas, C. A.; Visweswaran, V. Primal-Relaxed Dual Global Optimization Approach. JOTA 1993, 78, 187. Grossmann, I. E.; Sargent, R. W. H. Optimum design of Multipurpose Chemical Plants. Ind. Eng. Chem. Process Des. Dev. 1979, 18, 343. Ierapetritou, M. G. Optimization Approaches for Process Engineering Problems Under Uncertainty. Ph.D. Dissertation, University of London, London, 1995. Ierapetritou, M. G.; Acevedo, J.; Pistikopoulos, E. N. An Optimization Approach for Process Engineering Problems Under Uncertainty. Comput. Chem. Eng. 1994, in press. Kocis, G. R.; Grossmann, I. E. Global Optimization of Nonconvex Mixed-Integer Nonlinear Programming (MINLP) problems in process synthesis. Ind. Eng. Chem. Res. 1988, 27, 1407. Papageorgaki, S.; Reklaitis, G. V. Optimal Design of Multipurpose batch Plants 1. Problem Formulation. Ind. Eng. Chem. Res. 1990, 15, 451. Pistikopoulos, E. N.; Ierapetritou, M. G. A Novel Approach for Optimal Process Design Under Uncertainty. Comput. Chem. Eng. 1995, 19, 1089. Reinhart, H. J.; Rippin, D. W. T. Design of flexible batch chemical plants. AIChE Spring National Meeting, New Orleans, 1986; Paper No. 50e. Reinhart, H. J.; Rippin, D. W. T. Design of flexible batch chemical plants. AIChE Annual Meeting, New York, 1987; Paper No. 92f. Reklaitis, G. V. Progress and Issues in Computer-Aided Batch Process Design. FOCAPD Proceedings; Elsevier: New York, 1990; p 241. Rippin, D. W. T. Batch Process Systems Engineering: A Perspective and Prospective Review. Comput. Chem. Eng. 1993, 17, S1-S13. Rotstein, G. E.; Lavie, R.; Lewin, D. R. Synthesis of Flexible and Reliable Short-Term batch production Plans. Comput. Chem. Eng. 1995, in press.
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Voudouris, V. T.; Grossmann, I. E. Mixed-Integer Linear Programming Reformulation for Batch Process Design with Discrete Equipment Sizes. Ind. Eng. Chem. Res. 1992, 31, 1315. Wellons, H. S.; Reklaitis, G. V. The design of multiproduct batch plants under uncertainty with staged expansion. Comput. Chem. Eng. 1989, 13, 115.
Received for review April 24, 1995 Revised manuscript received October 4, 1995 Accepted October 18, 1995X IE950263F
X Abstract published in Advance ACS Abstracts, January 15, 1996.
