simultaneous scheduling and control of multiproduct continuous parallel lines

Simultaneous Scheduling and Control of Multiproduct Continuous Parallel Lines

Antonio Flores-Tlacuahuac*

Departamento de Ingenierıa y Ciencias Quımicas, UniVersidad Iberoamericana, Prolongacion Paseo de laReforma 880, Mexico D.F., 01210, Mexico

Ignacio E. Grossmann

Department of Chemical Engineering, Carnegie-Mellon UniVersity 5000 Forbes AVe., Pittsburgh, PennsylVania 15213

In this work, we propose a simultaneous scheduling and control optimization formulation to address bothoptimal steady-state production and dynamic product transitions in multiproduct parallel continuous stirredtank reactors (CSTRs). The simultaneous scheduling and control problem for multiproduct parallel continuousreactors is cast as a mixed-integer dynamic optimization (MIDO) problem. The reactor dynamic behavior isdescribed by a set of nonlinear ordinary differential equations (ODEs) that are combined with the set ofmixed-integer algebraic equations representing the optimal scheduling production model. The proposedsimultaneous scheduling and control approach avoids suboptimal solutions that are obtained when both problemsare solved in a decoupled way. Hence, the proposed optimization strategy can yield improved optimal solutions.The simultaneous approach for addressing the solution of dynamic optimization problems, based on orthogonalcollocation on finite elements, is used to transform the set of ODEs into a mixed-integer nonlinear programming(MINLP) problem. Three multiproduct CSTRs are used to test the simultaneous scheduling and controloptimization formulation.

1. Introduction

In the context of a highly competitive global market economy,it becomes important to find ways of improving process profitthrough improved process operation procedures. In this work,we address a relevant class of industrial problems that involveoptimal scheduling and control decisions. Normally, schedulingproblems are concerned with the optimal assignments toequipment production sequences, production times for eachmanufactured product, and inventory levels, so that profit orcompletion times are optimized. On the other hand, control isconcerned with determining dynamic transition profiles amongthe products to be manufactured, featuring, for instance,minimum transition time or any other relevant objectivefunction. Commonly, scheduling and control (SC) problemshave been approached using a sequential approach: (1) First,the pure scheduling problem is solved using specified transitiontimes and units, but without considering the underlying processdynamics of the analyzed system, and (2) having fixed theproduction sequence, optimal dynamic transition calculationscan be carried out among the set of products to be manufactured.Of course, one could reverse this approach and carry out, first,the control and, later, the scheduling calculations. However, thisapproach is only feasible if a small number of product transitionsmust be considered. One of the advantages of decoupling SCproblems lies in the relative simplicity of solving both problemsindependently; however, the main disadvantage is that such aprocedure generally leads to suboptimal solutions.

To increase production rate, several parallel productionfacilities can be used. The idea is to distribute the rawmaterial among several pieces of equipment, to increase thethroughput. In previous work, we have addressed thesimultaneous solution of SC problems of reaction systemscarried out in a continuous stirred tank reactor (CSTR).1 In

that work, we presented a mixed-integer dynamic optimiza-tion (MIDO) formulation for simultaneously tackling SCproblems taking place in a single production line. The aimof the present work consists of extending the single line SCproblem, as reported in ref 1, to reaction systems featuringseveral parallel production lines, each consisting of a CSTR.Only CSTRs will be used for the purpose of manufacturingthe various products. Note that the combinatorial problemfor the parallel scheduling problem is greatly increased whengoing from a single CSTR to several parallel CSTRs. Tomodel the pure scheduling optimization problem, we use theparallel lines scheduling formulation proposed in ref 2,whereas to address the optimal control calculations, we usethe simultaneous approach described in ref 3. We againpropose to solve both problems simultaneously, rather thansequentially, because of the aforementioned reasons. Becausethe simultaneous SC problem leads to a MIDO problem, andbecause of the fact that most reaction systems featurenonlinear behavior, it is important to use robust and efficientMIDO solution procedures. As explained in ref 1, wetransformed the SC MIDO problem into a mixed-integernonlinear program (MINLP). Because of process nonlineari-ties, the underlying MINLP problem was solved using thesbb software in GAMS.4

A literature review on the solution of scheduling andcontrol problems can be found elsewhere;1 in this paper, weonly mention some recent works that were not reviewed there.In ref 5, scheduling and optimal control problems are solvedin a decoupled way. The MIDO problem is solved using adecomposition method, by computing upper and lower boundson the optimal value of the objective function until a targetgap solution value is obtained. It should be stressed that theSC optimization formulation proposed in our work and inthe work reported in ref 5 are not equivalent. Of course, bothformulations feature scheduling and optimal control parts,but the way both problems are handled (simultaneous versussequential) and solved is different. In ref 6, an approach for

* Author to whom correspondence should be addressed. Tel./Fax:+52(55)59504074. E-mail: [email protected]. URL: http://200.13.98.241/∼antonio.

Ind. Eng. Chem. Res. 2010, 49, 7909–7921 7909

10.1021/ie100024p 2010 American Chemical SocietyPublished on Web 07/14/2010

scheduling and dynamic optimization was proposed. Thenumerical approach followed for solving the underlyingMIDO problems is different from that used in the presentwork. In ref 6, the authors proposed a SC logical optimizationformulation oriented toward the solution of online SCproblems. The solution method used in ref 6 to solve dynamicoptimization problems corresponds to the so-called simul-taneous multiple shooting methods for handling optimalcontrol problems. Details about their formulation have beenreported elsewhere.3,7 Regarding the optimization formulationused in ref 6, the authors decided to handle the schedulingpart using logic constraints that transformed to constraintsinvolving integer variables. Some results are also presentedrelated to an industrial polymerization reactor. Unfortunately,no details about the real process implementation and resultsare provided. Recently, the area of simultaneous SC has beenthe subject of discussion,8 and some open issues in this areahave been clearly identified. Specifically, in that work, somepractical problems related to the optimal integration of somedecision tools are highlighted and the difficulties of solvingscheduling and control problems requiring different modelingdetails have also been addressed. The work is interestingbecause it provides a fresh industrial point of view of SCproblems and stresses some areas in which research work isneeded; the paper also contains an updated literature review.The authors also concluded that the area of SC problems isin its rather early stages of development. Process uncertaintyis among some of the remaining problems to be tackled inSC problems. Along this line, some ways of dealing withuncertainty scenarios in SC problems were recently proposedin ref 9.

The following research contributions are presented in thiswork: (a) extension of the SC problem from a single productionline to multiple parallel production lines, (b) simultaneouslysolving the SC problem to avoid suboptimal solutions normallyobtained when solving SC problems using a sequential approach,(c) consider the effect of the nonlinear behavior embedded inthe reaction systems for the explicit solution of the MIDOproblems, and (d) use a SC MIDO solution decompositionprocedure that allows us to compute local optimal solutions withrelatively modest CPU time. This last point is relevant because,if proper MINLP initialization strategies are not used, findingeven a feasible solution can be very difficult to achieve. It shouldbe stressed that we are not claiming that our optimizationformulation is able to compute global optimal solutions. Whatwe mean is that better optimal solutions can be obtained bysimultaneously solving SC problems, avoiding suboptimalsolutions (i.e., worse optimal solutions) obtained by sequentialsolutions to SC problems. Moreover, no other research workhas reported simultaneous SC problems in parallel productionlines in the open research literature. Finally, we would like tostress that our work heavily relies on the assumption that thereis a stable product demand. Therefore, the proposed optimizationformulation can either be used for problems where the demandis stable, or else for solving off-line SC problems, such as, forinstance, that for studies in which design modifications areanalyzed. The proposed model is not useful for rapid changingmarkets where online receding horizon SC optimization for-mulations should be applied.

2. Problem Definition

Given are several products that are to be manufactured ina fixed number of independent parallel production lines, eachone consisting of a single continuous multiproduct CSTR.

Lower bounds for constant product demands rates of theproducts are given, as well as the price of each product andthe inventory and raw materials costs. The problem thenconsists of the simultaneous determination of a cyclicschedule (i.e., production wheel), and the control profile, forthe selected transitions for each parallel production line.Steady-state operating conditions for manufacturing eachproduct are also computed. In each production line, the majordecisions involve selecting the sequence (i.e., cyclic time andthe sequence in which the products are to be manufactured)as well as the transition times, production rates, length ofprocessing times, amounts manufactured of each product, andmanipulated variables profiles for the transitions such thatthe economic profit is maximized. We should note that theassumption of cyclic schedule is appropriate for applicationsin which demands are fairly stable and for strategic studiesin which infinite horizons are assumed for fixed demand rates.Real time applications with variable demands require treat-ment of a receding horizon such as that described in ref 6.In fact, reactive scheduling integrated with control allowsone to make corrections to both production sequence andproduct transitions as the product orders change.

3. Scheduling and Control MIDO Formulation

In the following simultaneous scheduling and control (SSC)formulation for parallel production plants, we assume that(1) each production line is composed of a single CSTR, wheredesired products are manufactured, and (2) the productsfollow a production wheel, meaning that all the requiredproducts are manufactured in an optimal cyclic sequence (seeSahinidis and Grossmann2 for a discussion on parallel linescheduling formulation). As depicted in Figure 1a, eachproduction line is divided into a set of slots. As opposed tothe case of one unit, the number of slots is unknown, sincewe do not know ahead of time how many products will beassigned in each production line. Therefore, if, for instance,only one product is assigned to a line, that product will beassigned to each of the postulated slots.2 Two operationshappen inside each slot: a transition period between theproducts that are to be manufactured and a continuousproduction period. Figure 1b displays these two dynamicgeneric responses conceptually. When the optimizer decidesto switch to a new product, the optimization formulation willtry to develop the best dynamic transition policy featuringthe minimum transition. After the system reaches theprocessing conditions leading to the desired product, thesystem remains there until the production specifications aremet. In this work, we assume that, after a production wheelis completed, new identical cycles are executed indefinitely.Note that a given slot does not necessarily incur in atransition. The reason is that it is possible to assign a productto multiple adjacent slots during which a transition does notoccur. As an example, the sequence AB with four slots maybe represented as A-A-A-B. Slots 1 and 2 clearly do notincur a transition. There are only transitions in slot 3 (fromA to B) and slot 4 (B to A), because of the cyclic schedule.Also, for the particular case when only one product isassigned to a given line, then it means that this line isoperating continuously on that line and, hence, there is nocyclic schedule on that line. The variable that defines thecycle time in that line becomes undetermined, but causes nonumerical difficulties as long as it is bounded.

We would like to highlight the main differences between thepresent work and a previous related work.1 First of all, in ref 1,

7910 Ind. Eng. Chem. Res., Vol. 49, No. 17, 2010

the problem involved only a single production line with severalproduction slots. In the present work, we analyze simultaneousSC problems considering the use of several parallel lines. Thissituation actually resembles industrial practice, where someproduction lines can be available to manufacture a set ofdemanded products. Therefore, extension of the work done inref 1 is not purely academic, because it reflects commonindustrial practices. As a result, the SC optimization formulationsused in the present work and those used in ref 1 are notcompletely equivalent but they obviously share some commoncharacteristics. For instance, in the present work, a single productcan be assigned to several different slots along the sameproduction line, a situation that was forbidden in ref 1. Twocommon features between the present SC optimization formula-tion and that used in ref 1 are the form of the objective functionand the assumption of a cyclic production wheel. Otherimportant difference between the present work and the workreported in ref 1 is, of course, problem complexity. Incorporatingadditional parallel lines increases the number of binary variablesand the number of optimization dynamic transition problemsthat must be addressed. Finally, we must note that, to illustratethe SC parallel lines optimization formulation, we use a set ofreaction systems that feature increasing nonlinear behavior. Wehave done so to test the SC MIDO formulation. More complexand realistic industrial systems will be approached in futurework.

To clarify the simultaneous SC MIDO parallel plants problemformulation, it has been divided into two parts. The first partinvolves the scheduling component, and the second part involvesthe dynamic optimization component.

3.1. Scheduling. 3.1.1. Objective Function.

In the objective function, cilp represents the production cost

of product i in line l, cijlt denotes the transition cost from product

i to product j in line l, and cili is the inventory cost of product

i in line l; these parameters are given as follows:

where ciprod is the production cost of product i, ril the

production rate of product i in line l, csafi the inventory safety

cost, and choldi the inventory holding cost. The aim of the

integral terms in the objective function is to get a smoothsystem response, as well as to penalize for excessive use ofthose variables associated to the manipulated variables. Thismeans that transitions between different products should bedone using the least amount of control action. We have foundthat including a term related to the dynamic behavior of thestates and manipulated variables leads to smooth and fastcontrol actions, instead of specifying this desired behavioras constraints. Rx and Ru are weight coefficients for the statesand manipulated variables, respectively. A detailed descrip-tion of the meaning of each symbol used in this work can befound in the Nomenclature section.

(1) Only One Product Should Be Manufactured inEach Slot.

Constraint 5 states that, at any production line l and any slotk, only one product i should be manufactured. Hence, whenthe binary variable Yikl is equal to 1, product i will bemanufactured at slot k within line l; otherwise, Yikl will be zero.It is worth mentioning that this constraint does not forbid thatthe same product i be produced in more than one slot k of theline l. For instance, for two products A and B, four slots, anda single line, the following product distributions turn out to bethe same: AAAB, AABB, and ABBB. Therefore, dummy slots(i.e., slots in which no product manufacture takes place) can

Figure 1. (a) Schematic of the process, consisting of l parallel production lines. At each line l, the cyclic time is divided into Ns slots. (b) When a changein the setpoint of the system occurs, there is a transition period, followed by a continuous production period.

min φ ) ∑i

∑k

∑l

cilp

tikl′Tcl

+ ∑i

∑j

∑k

∑l

cijlt Zijkl

Tcl+

∑i

∑k

∑l

cili Wikl + ω ∑

i

δi + Rx∫0

tf(x - xs)2 dt +

Ru∫0

tf(u - us)2 dt (1)

cilp ) ci

prodril (2)

cijlt ) -0.9cil

pτijl (3)

cili ) csaf

i + 0.5(1 -di

ril)chold

i (4)

∑i

Yikl ) 1 ∀k, ∀l (5)

Ind. Eng. Chem. Res., Vol. 49, No. 17, 2010 7911

sometimes occur. This behavior is illustrated in several of thecase studies addressed in this work.

(2) Transition from Product j to Product i at Slot kand Line l.

Constraint 6 is used to define variable Zijkl, which indicateswhether a transition from product j to product i is occurring atslot k in line l. Hence, if Zijkl ) 1, a transition will occur betweenproducts j and i at slot k within line l; otherwise, Zijkl will bezero. Constraint 7 states that only one transition from productj will occur at the beginning of slot k at line l, if and only ifproduct j was produced in the previous slot.

(3) Each Product Should Be Manufactured at LeastOnce.

Constraint 8 assures that any product i is assigned at least toone slot and one production line. Therefore, any product i thatis manufactured satisfies the production requirements. Noticethat because we enforce the greater than or equal to inequality,any product i can, in fact, be assigned to more than one slot kwithin the same line l. Similarly, the production of the sameproduct i could be distributed among different lines l, as longas production requirements are met.

(4) Upper and Lower Bounds for Production Times.

Constraint 9 states that the production time for any producti at any line l and slot k should be less than or equal to anupper production time bound denoted as Uil. Similarly, constraint10 states a minimum production time, denoted as Lil, whenmanufacturing product i at line l and slot k.

(5) Production Time for Each Line.

Constraint 11 is used to define a variable Tcl to record thetotal production time at each line l (including transitions).

(6) Manufactured Amount of Each Product.

Constraint 12 defines the variable Wikl used to record theamount of product i (in mass units) manufactured at line l andslot k. This variable is just the product of the process productionrate ril and the net production time. The net production time,which is computed inside the term in brackets in the aforemen-tioned equation, is the difference between the total productiontime tikl

′and the transition time. The transition time between

products i and j in slot k in line l is computed as the product ofthe transition time τjil and the binary variable Zjikl used to recordif such transition occurs. In pure scheduling problems, τjil is

assumed to remain constant, meaning that the dynamic behaviorof the addressed system is neglected.

(7) Production Demand.

Constraint 13 states that the total manufactured amount ofproduct i (in mass/time units), taking into account that theproduction of product i can be distributed among several linesl and slots k, should be greater than or equal to the requesteddemand di of such a product i. Notice that, in this constraint, δi

represents a slack variable used to enforce the inequality.Because it should be as small as possible, it is added to theobjective function featuring a large coefficient. For cases whenδi is nonzero at the solution, this means that there is no feasibleschedule that can be found for the specified demand.

3.1.2. Optimal Control. Although several approaches havebeen suggested for tackling optimal control problems,10 we haveused the simultaneous approach for solving dynamic optimiza-tion problems. This approach consists of the discretization ofthe inputs, outputs, and mathematical model of the underlyingsystem. For numerical robustness reasons, normally orthogonalcollocation on finite elements11,12 is used for system discreti-zation. For this objective, a given line l is divided into a seriesof slots k. As depicted in Figure 2 within each slot, the dynamicsystem response is approximated by a given number of finiteelements. The final result of this discretization approach is asystem of nonlinear equations that hopefully represent thesystem dynamic behavior. We would like to stress that (1) theoptimal control formulation used in this work is similar to thatused in a previous work1 and (2) we include it only forcompleteness. A more-complete description of the meaning ofthe constraints of this section can be found in ref 1.

(1) Dynamic Mathematical Model Discretization.

The constraints given by eqs 14 are used to compute the valueof the system states at each one of the discretized points (xfckl

n ),using the monomial basis representation. x0,fkl

n is the nthsystem state at the beginning of each element, Ωlc is thecollocation matrix, and xfckl

n is the first-order derivative ofthe nth state. Notice that, when working with the first element,x0,1kl

n represents the specified initial value of the nth state. Inour optimization formulation, the length of any finite elementsis computed from

(2) Continuity Constraint between Finite Elements.

These constraints are used to enforce continuity of the systemstates when crossing between adjacent finite elements. However,algebraic and manipulated variables are allowed to exhibitdiscontinuous behavior at the same location.

∑i

Zijkl ) Yjkl ∀j, ∀k, ∀l (6)

∑j

Zijkl ) Yi,k--1,l ∀i, ∀k, ∀l (7)

∑l

∑k

Yikl G 1 ∀i (8)

tikl′ E UilYikl ∀i, ∀k, ∀l (9)

∑i

tikl′ G LilYikl ∀k, ∀l (10)

Tcl ) ∑i

∑k

tikl′ ∀l (11)

Wikl ) ril[tikl′ - ∑j*i

τjilZjikl] ∀i, ∀k, ∀l (12)

∑l

∑k

Wikl

Tcl+ δi G di ∀i (13)

xfckln ) x0,fkl

n + θk,lt hfkl ∑

m)1

Ncp

Ωmcxfmkln ∀n, f, c, k, l (14)

hfkl )1

Nfe(15)

x0,fkln ) x0,f-1,kl

n + θklt hf-1,kl ∑

m)1

Ncp

Ωm,Ncpxf-1,mkl

n ∀n, f G 2, k, l

(16)


(3) Model Behavior at Each Collocation Point.

These equations just represent the system dynamic behavior andcorrespond to the underlying mathematical model of theaddressed system.

(4) Initial and Final Controlled and ManipulatedVariable Values at Each Slot.

The purpose of this set of constraints is to provide the initialand final values of both the controlled and manipulated variablesfor a given product transition. A complete description andnomenclature can be found elsewhere.1

(5) Lower and Upper Bounds of the DecisionVariables.

This set of constraints just enforces lower and upper boundson both the manipulated and controlled variables.

3.2. MIDO Solution Strategy. In general, the efficientsolution of MIDO problems tends to be a rather demanding task.Presently, MIDO solutions for small or medium-sized math-ematical models featuring modest or mild nonlinear static and

dynamic behavior are relatively easy to compute. However,when it comes to solving large-scale models, such as thosearising from the discretization of distributed parameter systemsthat involve highly nonlinear behavior, the MIDO problems turnout to be rather difficult to solve. Moreover, even when such asolution can be computed, it may require large computationaltimes and special initialization and solution procedures. In fact,presently the MIDO solution of complex and large-scaleproblems seems to be feasible only if special optimizationdecomposition procedures are used.13-17

In this work, to compute the MIDO solution of the simulta-neous scheduling and control problems, the whole MIDOproblem was divided into a series of subproblems. To do this,we took advantage of the optimization problem structure of theunderlying MIDO problems. The optimal solution of eachsubproblem was then used to provide good initial values of thedecision variables of successive versions of the optimizationproblems. These contain additional modeling details until thecomplete MIDO problem is finally formulated. The decomposi-tion and MIDO solution strategy involves the following steps:

(1) SolVe the parallel lines scheduling problem to obtain aninitial scheduling solution using guesses of the transition timesand production rates.

For the scheduling problem, the following form of theobjective function was used:

the scheduling problem was subject to the set of constraintsgiven by eqs 5-13. The decision variables computed by thescheduling part are the optimal production sequence, theproduction time for each product at each slot and eachproduction line, and the manufactured amount of each product.Normally, the solution of scheduling problems can be tackledwith efficient MINLP solvers such as DICOPT or Sbb, whichare both available in GAMS.

(2) Compute steady-state processing conditions of the desiredproducts.

The purpose of this part is to compute the steady-stateprocessing conditions that allow one to manufacture the set oftarget products. For doing so, we force GAMS/CONOPT tocompute these conditions by using them as a nonlinear equationsolver and using a dummy objective function. This amounts tocancelling the time derivative in the left-hand side of eq 17and then solving the remaining set of equations. Only the valuesof the manipulated variable and system states are computed inthis point.

(3) SolVe the dynamic optimization problem to obtain initialoptimal transition times and productions rates.

To address this part, the following form of the objectivefunction is used:

Figure 2. Simultaneous discretization approach for dealing with parallel plants dynamic optimization problems. Each line l is divided into a set of k slots;in turn, each slot is divided into Nfe finite elements. Within each finite element f, a set of Ncp collocation points (c) is selected.

xfckln ) fn(xfckl

1 , ..., xfckln , ufckl

1 , ..., ufcklm ) ∀n, f, c, k, l (17)

xin,1ln ) ∑

i)1

Np

xss,iln yi,Ns,l

∀n, l (18)

xin,kln ) ∑

i)1

Np

xss,iln yi,k-1,l ∀n, k * 1, l (19)

xjkln ) ∑

i)1

Np

xss,iln yi,kl ∀n, k, l (20)

uin,1lm ) ∑

i)1

Np

uss,ilm yi,Ns,l

∀m, l (21)

uin,klm ) ∑

i)1

Np

uss,ilm yi,k-1,l ∀m, k * 1, l (22)

ujklm ) ∑

i)1

Np

uss,ilm yi,kl ∀m, k, l (23)

u1,1,klm ) uin,kl

m ∀m, k, l (24)

uNfe,Ncp,klm ) ujin,kl

m ∀m, k, l (25)

x0,1,kln ) xin,kl

n ∀n, k, l (26)

xminn E xfckl

n E xmaxn ∀n, f, c, k, l (27a)

uminm E ufckl

m E umaxm ∀m, f, c, k, l (27b)

min φ ) ∑i

∑k

∑l

cilp

tikl′Tcl

+ ∑i

∑j

∑k

∑l

cijlt Zijkl

Tcl+

∑i

∑k

∑l

cili Wikl + ω ∑

i

δi


subject to the set of constraints given by eqs 14-27. Theobjective of this part consists of computing the optimal valueof the manipulated variable for each one of the requestedproducts leading to the minimization of the above objectivefunction. Since no integer variables are involved, the problemcorresponds to a nonlinear optimization problem whose solutionwas sought using the GAMS/CONOPT solver.

(4) SolVe the entire MIDO problem for the simultaneousscheduling and control problem.

In this part, we use the form of the objective function givenby eq 1, subject to the set of constraints described in thejust-given points (1) and (3). Because of problem noncon-vexities, the numerical solution of the problem was difficultto obtain using the GAMS/DICOPT solver. Instead, to solvethe underlying MINLP problem, the GAMS/Sbb solver wasused.

For the problems addressed in the present work, the afore-mentioned decomposition and solution strategy allowed us tocompute MIDO solutions of the simultaneous scheduling andcontrol problems with acceptable computational effort. It isworth mentioning that if a direct solution strategy is used(meaning the whole solution of the MIDO problem approachedin a single step), we sometimes have found that optimal MIDOsolutions are impossible to compute, or else the computationaltimes are, in the best case, at least 1 order of magnitude largerwithout using a proper initialization strategy. Moreover, it maybe hard to even find an initial feasible solution. We think thisis partly so, because very good initial values of the decisionvariables are normally required for the efficient solution ofdynamic optimization problems, using the simultaneous ap-proach as described in ref 3.

4. Case Studies

In this section, simultaneous SC problems for several casestudies, featuring different size and nonlinear behavior, aresolved. The problems range in difficulty from a reaction systemexhibiting mild nonlinear behavior and involving few products,to a highly complex nonlinear system with a larger number ofproducts. The problems also differ in the number of processinglines, as well as the number of postulated slots used formanufacturing the specified products.

CSTR with a Simple Irreversible Reaction. Let us assumethat, in a given chemical plant, two production lines are availablefor carrying out the following reaction system under isothermaland constant volume conditions:

Table 1. Process Data for the First Case Studya

productconversionpercentage

productionrate [kg/h]

demand rate[kg/h]

product cost[$/kg]

inventorycost [$/kg]

A 90 9 6 200 1B 80 80 4 150 1.5C 70 278.8 7 130 1.8D 60 607 6 125 2E 50 1250 8 120 1.7

a A, B, C, D, and E are the five products to be manufactured. Thecost of the raw material (cr) is $10.

Figure 3. Bifurcation diagram for the first case study: CSTR with a simple irreversible reaction.

min φ ) Rx∫0

tf(x - xs)2 dt + Ru∫0

tf(u - us)2 dt


In each production line, a single CSTR can be used and bothreactors have the same dimensions and operating conditions.In principle, any line could manufacture the five desiredproducts, denoted here as A, B, C, D, and E. The dynamiccomposition model is given by

where CR is the reactant R composition, V the reactor volume,RR the reaction rate expression, C0 the feed stream compo-sition, and Q the volumetric flow rate, which is also thecontrol variable used for the optimal dynamic transitionbetween the requested products. The values of the designand kinetic parameters are as follows: C0 ) 1 mol/L, V )5000 L, k ) 2 L2/(mol2 h). In Table 1, the steady-state valuesof the required reactant conversion percentage are shown.

In addition, Table 1 also displays values of the demand rate(Di), product cost (C i

p), and inventory cost (C is). In Figure

3, the bifurcation diagram of the corresponding reactionsystem is depicted. All the examples in this work requirethe steady-state values of the manipulated (u) and controlledvariables (x) for manufacturing each one of the products, andthey are computed as part of the SC solution procedure.

Solving the SC MIDO formulation, we found that, in thesecond production line, the optimal production schedule is givenby

D f C f B f D f E

whereas the first production line is completely dedicated tomanufacturing product A. In Table 2, the production results andprocessing times are shown, whereas in Figure 4, the dynamicoptimal transition profiles of both controlled and manipulatedvariables for the second production line are depicted. Note that,when a product is manufactured in a single line (as is the casein this example for the first production line), the concept ofcycle time lacks meaning. The MIDO problem was solvedwithin 1.29 h using the 22.7 GAMS optimization version andthe SBB MINLP solver with CONOPT as the NLP solver. Allthe problems addressed in this work were solved on a laptopcomputer with 4 Gb RAM operating at 800 MHz and using theWindows 7 environment. The MINLP involved 50 binaryvariables, 3168 continuous variables, and 2423 constraints. Theoptimal objective function value found was $1.22 × 105/h, whilethe NLP relaxation had a value of $98568.71/h, giving rise toa solution gap of 28.58%. Moreover, the number of examinedbranch and bound nodes was 200. As can be seen in Figure 4,the optimal manipulated variable profile is composed of simpleramplike changes, and, therefore, smooth responses of thecontrolled variable are observed.

There are several points worth mentioning about the optimalsolution of this case study. First, as depicted in Figure 3, the

Table 2. Optimal Scheduling and Control Results for the First CaseStudy: CSTR with a Simple Irreversible Reactiona

slot productQ

[L/h]CR

[mol/L]processtime [h]

amountproduced [kg]

optimaltransition

time, θ [h]

1 D 1000 0.393 5.3 165.09 2.52 C 400 0.3032 5.7 192.6 53 B 100 0.2 6.4 110.06 3.25 E 2500 0.5 5.2 220.2 3.1

a Profit ) $1.2246 × 105/h, cyclic time at second line ) 27.515 h.The results refer to the second production line; the first production lineis completely dedicated to the manufacture of only product A. Note thatthe fourth slot is used as a transition slot for the transition from productB to product E; the transition time at this slot turns out to be equal to5 h.

Figure 4. First case study: CSTR with a simple irreversible reaction. Optimal dynamic transition profiles for the controlled and manipulated variables at thesecond production line for each one of the slots. The optimal production sequence is D f C f B f E.

3R 98k

P,-RR ) kCR3

dCR

dt) Q

V(C0 - CR) + RR (28)


system exhibits mild nonlinear steady-state behavior. For eachvalue of the manipulated variable Q, there is only a singlesteady-state value for the whole range of values of themanipulated variable. Moreover, the relationship between thesystem state and control variable is mildly nonlinear. We thinkthat these facts, in addition to the low dimensionality of theproblem, is the main reason why the optimizer finds a SCoptimal solution within a relatively short CPU time. These are

also the reasons why the shape of the control variables appearsto be smooth. To minimize product transition costs, the optimizerdecided to manufacture a single product (A), using a completeproduction line. The production of the rest of the products isdistributed in the remaining production line. Moreover, it shouldbe highlighted that, actually, the fourth slot of the second lineis not used to manufacture product D, as reported in theproduction results displayed in Table 2. The real transition isgiven by the sequence D f C f B f E. The fourth slot isused just as a transition slot in going from product B to productE. Even when, initially, the results of the optimization formula-tion state that the fourth slot of the second line should bededicated to manufacturing product D, the required demand forproduct D is completely met at the first slot of the second line,as is clear from the results shown in Table 2. Therefore, theresults of the optimization formulation indicate that no produc-tion of product D occurs at the aforementioned fourth slot. Fromconstraint 7, we see that this occurs because the processing time(tikl

′) turns out to be equal to the transition time (τjil) at this slot

and, consequently, the purpose of this slot is just to performthe transition from product B to product E.

Consecutive Reaction System: X f Y f Z. The secondexample refers to a nonisothermal continuous stirred tank reactor(CSTR) where the consecutive reactions X f Y f Z take

Figure 5. Bifurcation diagram for the second case study: Consecutivereaction system: Xf Yf Z. The continuous lines denote open-loop stablesteady states, whereas dashed lines denote open-loop unstable steady states.

Table 3. Process Data for the Second Case Studya



demand rate[kg/h]

productcost [$/kg]


A 99.84 138.7 30 1 1B 99.15 458.8 40 3 1.5C 95.46 788.9 36 2 1.8D 75.22 730 44 3 2

a A, B, C, and D represent the four products to be manufactured.Three production lines and four slots for each one of the productionlines were used.

Figure 6. Second case study. Consecutive reaction system optimal dynamic transition profiles for the controlled and manipulated variables at the firstproduction line for each of the slots. The optimal production sequence is D f C.


place.18 The dynamic behavior of the cooling jacket has alsobeen included. In this problem, three production lines and fourslots, within each line, are available. The dimensionless dynamicmodel is given as follows:

where x1 is the dimensionless concentration of reactant X,

x2 is the dimensionless concentration of reactant Y, x3 is thedimensionless reactor temperature, and x4 is the dimensionlesscooling jacket temperature. The bifurcation diagram of thisreaction system is displayed in Figure 5. Additional dimen-sionless parameter values are given as follows: q ) 1, x1f )1, x2f ) x3f ) 0, x4f ) -1, ) 8, φ ) 0.133, δ ) 1, R )1, S ) 0.01, ψ ) 1, δ1 ) 10, δ2 ) 1, and γ ) 1000. Inaddition, Table 3 contains information regarding the cost ofthe products, demand rate, inventory cost, production rate,and conversion percentage of each one of the hypotheticalproducts. The variable that was used as a manipulatedvariable for optimal product transitions is the dimensionlesscooling flow rate (qc).

Solving the SC MIDO formulation, we found that thesecond production line was completely used to manufactureproduct A, whereas the production of the rest of the productswere distributed between the remaining two production lines.In the first production line, the optimal production scheduleis given by D f C, whereas in the third production line, theoptimal production schedule is given by A f B. Theobjective function value is $874/h. In Figures 6 and 7, thedynamic optimal transition profiles of both controlled andmanipulated variables for the first and third production linesare depicted. The MIDO problem was solved in 4.22 h, using

Figure 7. Second case study. Consecutive reaction system optimal dynamic transition profiles for the controlled and manipulated variables at the secondproduction line for each of the slots. The optimal production sequence is A f B.

Table 4. Optimal Scheduling and Control Results for the Second Case Studya

Steady-State Values w [kg]Processing

Time, PT [h]Optimal Transition

Time, θ [h]

product qc x1 x2 x3 x4 L1 L3 L1 L3 L1 L3

A 2 0.0016 0.1387 8.5188 2.1729 130.378 10.94 3.3B 2.2 0.0085 0.4588 6.8178 1.4431 917.6 12 5.1C 1.75 0.0454 0.7889 5.0878 1.2138 805.285 11.021 2.2D 2.83 0.2478 0.73 3.1375 0.0803 984.238 11.348 8.2

a Profit ) $874/h. Cyclic times are 22.369 h and 22.94 h for the first and third production lines, respectively; w is the amount manufactured, and L1

and L3 denote the first and third production lines, respectively.

dx1

dt) q(x1f - x1) - x1κ1(x3)φ

dx2

dt) q(x2f - x2) - x2φSκ2(x3) + x1φκ1(x3)

dx3

dt) q(x3f - x3) + δ(x4 - x3) + φ[x1κ1(x3) + Rx2κ2(x3)S]

dx4

dt) δ1(qc(x4f - x4) + δδ2(x3 - x4))

κ1(x3) ) expx3

1 + (x3/γ)

κ2(x3) ) expψx3

1 + (x3/γ)


the 22.7 GAMS version optimization environment and theSBB MINLP solver. The MINLP involved 48 binary vari-ables, 7996 continuous variables, and 7140 constraints. Asummary of the results of the local optimal solution is shownin Table 4.

The results of this case study clearly show the trends of theoptimal solution, when a larger number of production lines areused. To minimize transition costs, a production line iscompletely dedicated to manufacturing component A. Theoptimizer also decides to manufacture only two products, at eachone of the remaining two production lines. Moreover, it is nota coincidence that the manufactured products at each productionline happen to be close, in terms of deviations in both thecontrolled and manipulated variable as seen in Figure 5. In thelimit, with the aim of minimizing transition costs, if we keepadding production lines, we might find a solution where only asingle product would be manufactured at each production line.On the other hand, it is worth to mention that even when theshape of the manipulated variables looks almost symmetricalat both production lines, as seen from Figures 6 and 7, the

system responses are not. This is clear evidence of the nonlineardynamic behavior of the addressed reaction system. For instance,in Figure 7, we see that the A f B optimal dynamic transitionlooks different for the same optimal dynamic transition in theopposite direction (B f A).

System of Two Series-Connected NonisothermalCSTRs. The next example involves a system of two cascadedCSTRs where mass and energy is recycled to achieve largerreactant conversion at lower reactor temperatures.19 In thisproblem, two production lines and four slots within each linewere used. The dimensionless dynamic mathematical model isgiven as follows:

where x and θ are the dimensionless concentration and tem-perature, respectively; the subindex “1” refers to the first reactor,whereas the subindex “2” refers to the second reactor. Darepresents the Damkohler number, θc is the dimensionlesscooling temperature, λ is the recycle ratio, B is the dimensionlessheat of reaction, γ is the dimensionless activation energy, and

Table 5. Process Data for the Third Case Studya



demandrate [kg/h]

productcost [$/kg]


A 51.25 450 30 1 1B1 60.01 600 40 3 1.5B2 60.08 700 45 4 1.8C1 70.08 900 36 2 2C2 70.04 850 30 4 1D1 80.02 700 44 3 2E2 90.05 800 50 5 1F 98.09 750 40 4 1

a A, B1, B2, C1, C2, D1, E2, and F denote the eight products to bemanufactured. Two production lines and four slots for each one of theproduction lines were used.

Figure 8. Bifurcartion diagram for the third case study: System of two series-connected nonisothermal continuously stirred tank reactors (CSTRs). Thecontinuous lines denote open-loop stable steady states, whereas dashed lines denote open-loop unstable steady states.

dx1

dt) (1 - λ)x2 - x1 + Da1(1 - x1) exp[ θ1

1 + (θ1/γ)]dθ1

dt) (1 - λ)θ2 - θ1 + Da1B(1 - x1) exp[ θ1

1 + (θ1/γ)] -

1(θ1 - θc1)

dx2

dt) x1 - x2 + Da2(1 - x2) exp[ θ2

1 + (θ2/γ)]dθ2

dt) θ1 - θ2 + Da2B(1 - x2) exp[ θ2

1 + (θ2/γ)] -

2(θ2 - θc2)


is the dimensionless heat transfer coefficient. Additionalparameter values are given as follows: λ ) 0.9, γ ) 1000, B )22, θc1 ) θc2 ) 0, 1 ) 2, 2 ) 2. In addition, Table 5 containsinformation regarding the cost of the products, demand rate,inventory cost, production rate, and conversion percentage ofeach one of the hypothetical products. The bifurcation diagramof this two reactor system is displayed in Figure 8. The variableused as the manipulated variable for optimal product transitionsis the dimensionless Damkohler number (Da).

Solving the SC MIDO formulation, we found that, in the firstproduction line, the optimal production sequence is given byB1 f C1 f D1 f E2, whereas in the second production linethe optimal production sequence is given by A f C2 f B2 fF. The value of the objective function is $1848/h. In Figures 9and 10 the dynamic optimal transition profiles of both controlledand manipulated variables for the first and second productionlines are depicted. The MIDO problem was solved within 1.04 husing the 22.7 GAMS version optimization environment andthe SBB MINLP solver. The MINLP involved 64 binaryvariables, 5995 continuous variables, and 5011 constraints. Asummary of the results at the local optimal solution is shownin Table 6.

As noticed from Figure 8, this system features strongnonlinear steady-state behavior. Practically all the requiredproducts (except product F) are manufactured around open-loopunstable operating regions. We should stress that one of theadvantages of using the proposed simultaneous SC optimizationsolution procedure, lies in the fact of its ability to handledynamic transitions between open-loop unstable steady states,without introducing a feedback control system whose objectivewould be first to stabilize the underlying system. More detailsabout this point can be found elsewhere.20 However, we wouldlike to stress that dynamic optimization problems aroundunstable operating points could also be handled by multiple

shooting techniques.7 Moreover, because the present problemfeatures a larger number of products, in comparison to theprevious case studies, the number of binary variables increases,as does the complexity of solving the underlying MIDOproblem. However, because of the initialization and solutionstrategy discussed at the beginning of this section, the MIDOproblem was solved with a relatively modest computationaleffort.

It is interesting to see that the simultaneous SC optimalsolution involves the manufacture of products in an excludingway. This means that four different products are manufacturedin a line, whereas the remaining four different products aremanufactured in the other production line. The production lineshave no common products. Moreover, looking at Figure 8, wesee that the first production line takes care of the productslocated in the left steady-states branch of this figure. There isonly one product (E2) that does not belong to this branch, andthat is manufactured by this production line. On the other hand,the second production line takes care of those products locatedin the right steady-states branch of the figure. The reason whythe simultaneous SC optimal solution involves production linescontaining mainly products located on the same steady-statebranch is because such product transitions lead to minimumtransition costs. This is so because such product transitionsfeature closer values of the manipulated variables. Therefore,with a relatively modest control effort, we can move from agiven product to another one located at the same steady-statebranch.

Regarding the shape of the optimal dynamic product transitionprofiles, depicted in Figures 9 and 10, we should highlight thatthey exhibit relatively smooth dynamic behavior, except in twoproduct transitions (both at the first production line): D1 f E2

and E2f B1. This result is not surprising, because both producttransitions involve wide variations in the manipulated variable

Figure 9. Third case study. Series-connected CSTR reaction system optimal dynamic transition profiles for the controlled and manipulated variables at thefirst production line for each of the slots. The optimal production sequence turns out to be B1 f C1 f D1 f E2.


values. Moreover, these product transitions imply excursionsfrom the left steady-state branch to the right steady-state branch,and vice versa.

5. Conclusions

In this work, we have extended previously reported work onscheduling and control (SC) problems in a single productionline1 to the case of several parallel production lines. We haveproposed a simultaneous SC optimization formulation that, upondiscretization, can be reformulated as a mixed-integer nonlinearprogramming (MINLP) situation. We claim that SC problemsshould be approached simultaneously rather than in a decoupledway to avoid suboptimal solutions. Using the proposed simul-taneous approach to SC problems, the strong and naturalinteractions between both problems can be easily taken intoaccount, leading to improved optimal solutions. The optimizationresults indicate the importance of using proper initialization anddecomposition procedures, aimed at efficiently solving suchproblems. In fact, we have found that, without proper initializa-

tion, it is hard to even find an initial feasible solution. Theoptimal solutions of the SC problems of three case studies,exhibiting different nonlinear behavior characteristics, werepresented. These problems were solved with relatively modestcomputation effort. When attempting to compute global optimalsolutions, the computational effort, even for the simpler firstcase study, turned out to be excessive. This situation suggestedthat the computation of global optimal solutions for the rest ofthe more-complex nonlinear case studies would demand exces-sive CPU time and was not further pursued. The developmentof a global optimal method tailored for simultaneous SCoptimization problems remains as a research challenge, and wehope to address it in future work.

Nomenclature

Indices

products ) i, p ) 1, ..., Np

slots ) k ) 1, ..., Ns

lines ) l ) 1, ..., Nl

Figure 10. Third case study. Series-connected CSTR reaction system optimal dynamic transition profiles for the controlled and manipulated variables at thesecond production line for each of the slots. The optimal production sequence turns out to be A f C2 f B2 f F.

Table 6. Optimal Scheduling and Control Results for the Third Case Studya

Steady-State Values w [kg] Processing Time, PT [h] Optimal Transition Time, θ [h]

product x1 θ1 x2 θ2 Da L1 L2 L1 L2 L1 L2

A 0.3629 2.3480 0.5125 1.8795 0.047 1537.64 13.42 4.8B1 0.0979 0.4049 0.6001 3.8178 0.028 2083.4 13.48 8.7B2 0.3566 2.2594 0.6008 2.5435 0.04837 2306.45 13.3 3.9C1 0.0985 0.3596 0.7008 4.5371 0.022 1875.06 12.09 5.3C2 0.3799 2.3774 0.7004 3.1421 0.04664 1537.64 11.81 4.1D1 0.1048 0.3553 0.8002 5.2180 0.01937 2291.74 13.28 4.2E2 0.3533 2.0872 0.9005 4.7090 0.0507 2604.25 13.26 3.9F 0.9722 6.4840 0.9809 2.2257 0.05 2050.18 12.74 6.7

a Profit ) $1848/h. Cyclic times are 52.085 and 51.254 h for the first and second production lines, respectively; w is the amount manufactured, andL1 and L2 stand for the first and second production lines, respectively.


finite elements ) f ) 1, ..., Nfe

collocation points ) c, l ) 1, ..., Ncp

system states ) n ) 1, ..., Nx

manipulated variables ) m ) 1, ..., Nu

Decision Variables

yikl ) binary variable to denote if product i is assigned to slot k atline l

Zipkl ) binary variable to denote if product i is followed by productp in slot k at line l

Tcl ) total production wheel time at line l [h]θkl

t ) transition time in slot k and line lxfckl

n ) nth system state in finite element f and collocation point cof slot k at line l

ufcklm ) mth manipulated variable in finite element f and collocation

point c of slot k at line lWi ) amount produced of each product [kg]ci

prod ) price of products [$/kg]cil

p ) price of products [$/h] at line lci

saf ) cost of inventory [$/kg]ci

hold ) inventory holding cost [$/kg]cijl

t ) cost of transition from product i to product j at line l [$]tikl′ ) production time for product i at slot k in line l [h]τijl ) transition time from product i to product j at line l [h]x0,fkl

n ) nth state value at the beginning of the finite element f ofslot k at line l

xjkln ) desired value of the nth state at the end of slot k at line l

ujklm ) desired value of the mth manipulated variable at the end of

slot k at line lxin,kl

n ) nth state value at the beginning of slot k at line luin,kl

m ) mth manipulated variable value at the beginning of slot k atline l

xss,iln ) nth steady-state value of product i at line l

uss,ilm ) mth manipulated variable value of product i at line l

xfmkln ) derivative of the nth system state at finite element f,

collocation point m, and slot k at line luNfe,Ncp,kl

m ) mth manipulated variable at finite element Nfe, colloca-tion point Ncp, and slot k at line l

x ) vector of system statesxs ) vector of system states under steady-state conditionsu ) vector of manipulated variablesus ) vector of manipulated variables under steady-state conditionsδi ) slack variable

Parameters

Np ) number of productsNs ) number of slotsNl ) number of linesNfe ) number of finite elementsNcp ) number of collocation pointsNx ) number of system statesNu ) number of manipulated variablesdi ) demand rate [kg/h]cr ) cost of raw material [kg]ril ) production rate [kg/h] of component i at line ltf ) end time for integrationhfkl ) length of finite element f in slot k at line lΩNcp,Ncp ) matrix of Radau quadrature weightsω ) weight for slack variableRx ) weight for system states

Ru ) weight for manipulated variablesxmin

n , xmaxn ) minimum and maximum value of the state xn

uminm , umax

m ) minimum and maximum value of the manipulatedvariable um

Lil ) lower production time boundUil ) upper production time bound

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ReceiVed for reView January 4, 2010ReVised manuscript receiVed June 23, 2010

Accepted June 25, 2010

IE100024P


simultaneous scheduling and control of multiproduct continuous parallel lines

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