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Computers and Chemical Engineering 34 (2010) 996–1006

Contents lists available at ScienceDirect

Computers and Chemical Engineering

journa l homepage: www.e lsev ier .com/ locate /compchemeng

roduction planning and scheduling integration through augmented Lagrangianptimization

ukui Li, Marianthi G. Ierapetritou ∗

ept. of Chemical and Biochemical Engineering, Rutgers University, 98 Brett Road, Piscataway, NJ 08854, USA

r t i c l e i n f o

rticle history:eceived 29 April 2009eceived in revised form7 November 2009

a b s t r a c t

To improve the quality of decision making in the process operations, it is essential to implement integratedplanning and scheduling optimization. Major challenge for the integration lies in that the correspondingoptimization problem is generally hard to solve because of the intractable model size. In this paper,

ccepted 18 November 2009vailable online 24 November 2009

eywords:lanning and scheduling integrationecomposition method

augmented Lagrangian method is applied to solve the full-space integration problem which takes a blockangular structure. To resolve the non-separability issue in the augmented Lagrangian relaxation, westudy the traditional method which approximates the cross-product term through linearization and alsopropose a new decomposition strategy based on two-level optimization. The results from case studyshow that the augmented Lagrangian method is effective in solving the large integration problem andgenerating a feasible solution. Furthermore, the proposed decomposition strategy based on two-level

er fea

ugmented Lagrangian relaxation optimization can get bett

. Introduction

Production planning and scheduling belong to different decisionaking levels in process operations, they are also closely related

ince the result of planning problem is the production target ofcheduling problem. In process industry, the commonly used plan-ing and scheduling decision making strategy generally followshierarchical approach, in which the planning problem is solvedrst to define the production targets and the scheduling problem

s solved next to meet these targets. However, there exists a bigisadvantage in this traditional strategy since there is no interac-ion between the two decision levels, i.e., the planning decisionsenerated might cause infeasible scheduling subproblems. At thelanning level, the effects of changeovers and daily inventoriesre neglected, which tends to produce optimistic estimates thatannot be realized at the scheduling level, i.e., a solution deter-ined at the planning level does not necessarily lead to feasible

chedules. Moreover, the optimality of the planning solution cannote ensured because the planning level problem might not pro-ide an accurate estimation of the production cost, which shoulde calculated from detailed tasks determined by the scheduling

roblem.

Therefore, it is important and necessary to develop method-logies that can effectively integrate production planning andcheduling. However, since production planning and scheduling

∗ Corresponding author. Tel.: +1 732 445 2971; fax: +1 732 445 2581.E-mail address: marianth@soemail.rutgers.edu (M.G. Ierapetritou).

098-1354/$ – see front matter © 2009 Elsevier Ltd. All rights reserved.oi:10.1016/j.compchemeng.2009.11.016

sible solution than the traditional linearization method.© 2009 Elsevier Ltd. All rights reserved.

are dealing with different time scales, the major challenge for theintegration lies in the large problem size of the resulted optimiza-tion model. A direct way for addressing the integrated planning andscheduling problems is to formulate a single simultaneous plan-ning and scheduling model that spans the entire planning horizonof interest. However, when typical planning horizons are consid-ered, the size of this detailed model becomes intractable, becauseof the potential exponential increase in the computation time. Toovercome the above difficulty, most of the work appeared in the lit-erature aim at decreasing the problem scale through different typesof problem reduction methodologies and developing efficient solu-tion strategies as summarized by Grossmann, Van Den Heever, andHarjunkoski (2002) and Maravelias and Sung (2008). Generally, theexisting work in the area of planning and scheduling integration canbe summarized as follows.

The first type of methods is based on decomposition in ahierarchical way through iterative solution procedure. Through ahierarchical decomposition of the integration problem, detailedscheduling constraints are not incorporated into the upper levelaggregate planning model, on the other hand, information is passedfrom the aggregate planning problem to a set of detailed schedul-ing problems and these scheduling problems are separated basedon the temporal decomposition. Thus, the problems that need to besolved include a relative simple planning problem and a series of

scheduling subproblems. To ensure the feasibility and optimalityof the solution, it is further necessary to develop effective algo-rithms to improve the solution using additional cuts in the planninglevel within an iterative solution framework (Bassett, Pekny, &Reklaitis, 1996; Erdirik-Dogan & Grossmann, 2006; Munawar &

Z. Li, M.G. Ierapetritou / Computers and Chem

Nomenclature

Planning partt planning periods (1, . . ., T)Invt

s inventory level of state s at the end of planningperiod t

Pts production target of state s in planning period t

Dts delivery of product s in planning period t

Uts backorder of product s in planning period t

Demts demand of product s in planning period t

hs inventory unit cost of state sus backorder unit cost of product s

Scheduling parti ∈ I task index and setsIs tasks which produce or consume state sIj tasks which can be performed in unit jj ∈ J unit index and setsJt units which are suitable for performing task in ∈ N event points representing the beginning of a tasks ∈ S state index and setsSp index set for productswvi,j,n binary, whether or not task i in unit j start at event

point nsts,n continuous, amount of state s at event point n�P

s,i, �C

s,iproportion of state s produced, consumed by task i,respectively

bi,j,n continuous, amount of material undertaking task iin unit j at event point n

stmaxs available maximum storage capacity for state s

stints initial inventory for state s in planning period t

vmini,j

, vmaxi,j

minimum amount, maximum capacity of unit jwhen processing task i

Tfi,j,n continuous, time at which task i finishes in unit jwhile it starts at event point n

Tsi,j,n continuous, time at which task i starts in unit j atevent point n

˛i,j, ˇi,j constant, variable term of processing time of task i

Gmaimeptca((vispc&dmcpm

in unit j, respectivelyH scheduling time horizon

udi, 2005; Papageorgiou & Pantelides, 1996). The second type ofethod, which is also called rolling horizon approach, considersrelative rough model for the far future planning periods in the

ntegrated planning and scheduling model, i.e., detailed schedulingodels are only used for a few early periods and aggregate mod-

ls are used for later periods. The production targets for the earlyeriods are directly implemented, while the production targets forhe later periods are updated along with the rolling horizon. Appli-ations of this kind of strategy can be found in Dimitriadis, Shah,nd Pantelides (1997), Sand, Engell, Märkert, Schultz, and Schultz2000), Wu and Ierapetritou (2007), and Verderame and Floudas2008). Thirdly, for the cases where there is no plant and marketariability, campaign mode can be applied to generate an easy tomplement and profitable process operations plan. In a periodiccheduling framework, the planning and scheduling integrationroblem is replaced by establishing an operation schedule and exe-uting it repeatedly (Castro, Barbosa-Povoa, & Matos, 2003; Wu

Ierapetritou, 2004; Zhu & Majozi, 2001). Other than using the

etailed scheduling model in the integrated planning, surrogateethods aim at deriving the scheduling feasibility and production

ost function first and then incorporating them into the integratedroblem. This avoids the disadvantage of large scale and complexodel which directly incorporate the detailed scheduling model

ical Engineering 34 (2010) 996–1006 997

into aggregating planning model as shown in (Sung & Maravelias,2007).

Except the different methods for the integrated planning andscheduling summarized above, another approach is based on thestudy of the special structure of the mathematical programmingmodel for the integration problem and aims at developing effi-cient decomposition techniques to solve the optimization problemdirectly. Lagrangian relaxation is an approach that is often appliedto models with a block angular structure. In such models, dis-tinct blocks of variables and constraints can be identified andthey are linked through a few “linking” constraints and vari-ables. To our knowledge, Lagrangian relaxation has been widelyapplied onto planning and scheduling problems for different appli-cations including unit commitment in power industry (Padhy,2004), midterm production planning (Gupta & Maranas, 1999), andcombined transportation and scheduling (Equi, Gallo, Marziale, &Weintraub, 1997), etc. However, the major drawback of Lagrangianrelaxation method is that there is duality gap between the solu-tion of the Lagrangian dual problem and the solution of originalproblem, and often the feasibility of the solution needs to berecovered through heuristic steps. So it is often only used as thebounding step in the branch and bound framework. The disad-vantage of Lagrangian relaxation can be avoided by augmentedLagrangian relaxation (ALR) method, which has been used in sev-eral applications in areas such as power generation scheduling(Carpentier, Cohen, Culioli, & Renaud, 1996), multidisciplinarydesign (Tosserams, Etman, & Rooda, 2008), etc. One drawback ofALR method is the non-separability of the relaxed problem, whichhas also received wide attention in the literature. In this paper, wepropose to apply the ALR method on the planning and schedulingintegration problem which takes a block angular model structure,and also propose a new decomposition strategy to address the non-separability issue in the ALR solution procedure, which can be usedto decompose the relaxed problem exactly without any approxi-mation technique as presented in the literature.

The content of this paper is organized as follows. The problemformulation of the integrated planning and scheduling problemis first presented in Section 2. The general augmented Lagrangiansolution method is presented in Section 3. Detail reformulation anddecomposition strategies for the planning and scheduling integra-tion problem are presented in Section 4. The proposed method isstudied in Section 5 through a case study and the paper concludesin Section 6.

2. Problem formulation

Production planning model is used to predict production tar-gets and material flow over several months (up to one year), it isgenerally takes a simplified representation of the production andformulated as linear problem. Scheduling models on the other handare more detailed assuming that key decisions (production targets)have been made. To integrate these two different decision-makingproblems, the simplest way is to formulate a full space optimizationmodel, where in every period of the planning horizon, the schedul-ing constraints are incorporated into the model, while keeping theinventory connecting constraints between the planning decisionand scheduling decisions. In this work, we formulate the productionplanning and scheduling integration problem as follows.

min∑

t

∑s ∈ SP

hsInvts

∑∑t

∑∑∑∑t t

+

t s ∈ SP

usUs +t i j n

(FixCostiwijn + VarCostibijn) (1a)

s.t.Invts = Invt−1

s + Pts − Dt

s ∀s ∈ SP, ∀t (1b)

9 Chemical Engineering 34 (2010) 996–1006

U

s

s∑

v

T

T

T

T

T

T

T

s

s

Ipcbcpoo

irrb

tupptclc(mre(

98 Z. Li, M.G. Ierapetritou / Computers and

ts = Ut−1

s + Demts − Dt

s ∀s ∈ SP, ∀t (1c)

tts,n=N − stint

s = Pts ∀s ∈ SP, ∀t (1d)

tints = Invt−1

s ∀s ∈ SP, ∀t (1e)

i ∈ Ij

wvti,j,n ≤ 1 ∀j ∈ J, ∀n ∈ N, ∀t (1f)

mini,j wvt

i,j,n ≤ bti,j,n ≤ vmax

i,j wvti,j,n ∀i ∈ I, ∀j ∈ Ji, ∀n ∈ N, ∀t (1g)

f ti,j,n = Tst

i,j,n + ˛i,jwvti,j,n + ˇi,jb

ti,j,n ∀i ∈ I, ∀j ∈ Ji, ∀n ∈ N, ∀t

(1h)

sti,j,n+1 ≥ Tf t

i,j,n − H(1 − wvti,j,n) ∀i ∈ I, ∀j ∈ Ji, ∀n ∈ N, ∀t (1i)

sti,j,n+1 ≥ Tf t

i′,j,n − H(1 − wvti′,j,n) ∀i, i′ ∈ Ij, ∀j ∈ J, ∀n ∈ N, ∀t (1j)

Tsti,j,n+1 ≥ Tf t

i′,j′,n − H(1 − wvti′,j′,n)

∀i, i′ ∈ Ij, i /= i′, ∀j, j′ ∈ J, ∀n ∈ N, ∀t (1k)

sti,j,n+1 ≥ Tst

i,j,n ∀i ∈ I, ∀j ∈ Ji, ∀n ∈ N, ∀t (1l)

f ti,j,n+1 ≥ Tf t

i,j,n ∀i ∈ I, ∀j ∈ Ji, ∀n ∈ N, ∀t (1m)

sti,j,n ≤ H ∀i ∈ I, ∀j ∈ Ji, ∀n ∈ N, ∀t (1n)

f ti,j,n ≤ H ∀i ∈ I, ∀j ∈ Ji, ∀n ∈ N, ∀t (1o)

stts,n = stt

s,n−1 −∑i ∈ Is

�Cs,i

∑j ∈ Ji

bti,j,n +

∑i ∈ Is

�Ps,i

∑j ∈ Ji

bti,j,n−1

∀s ∈ S, ∀n ∈ N, ∀t (1p)

tts,n=1 = stint

s −∑i ∈ Is

�Cs,i

∑j ∈ Ji

bti,j,n=1 ∀s ∈ S, ∀t (1q)

tts,n ≤ stmax

s ∀s ∈ S, ∀n ∈ N (1r)

n the above model, the objective function (1a) is the total cost com-osed by three parts: inventory cost, backorder cost and productionost, where the inventory cost and backorder cost are calculatedased on the inventory and backorder amount and the given unitost parameter (hs, us); the production cost of different planningeriods is composed by a fixed part which represents the basic costf a task, and a dynamic part which is proportional to the amountf material processed (batch size).

The constraints of the above integration model can be dividednto planning level and scheduling level. Eqs. (1b) and (1c)epresent the planning level constraints, among them, Eq. (1b) rep-esents the inventory balance and Eq. (1c) represents the backorderalance.

Among the constraints of the scheduling level, Eq. (1d) expresseshe requirement that the planning solutions Pt

s generated frompper planning level is the production targets for different planningeriods. Eq. (1e) represents the connection constraints for the initialroduct inventory for different planning periods. Eq. (1f) representshe allocation constraints which state that only one of the tasksan be performed in each unit at an event point (n); the capacityimitations of production units are expressed by constraints (1 g);onstraints (1 h) represent the duration constraints and constraints

1i)–(1o) represent time limitations due to task sequence require-

ents in the same or different production units; Eqs. (1p) and (1q)epresent the material balances for each state (s) expressing that atach event point (n) the amount sts,n is equal to that at event pointn − 1), adjusted by any amounts produced and consumed between

Fig. 1. Constraint matrix structure of the integration model.

event points (n − 1) and (n). Constraints (1r) represent the storagecapacity constraints. For more detail about the above schedulingconstraints, we refer the reader to the paper by Ierapetritou andFloudas (1998).

If we denote the scheduling decision variables(wvt

i,j,n, bt

i,j,n, Tf t

i,j,n, Tst

i,j,n, stt

s,n, stints) for planning period

t using the vector yt, then the structure of the above integratedplanning and scheduling model can be illustrated as shown inFig. 1.

In the above constraint matrix, the part on the top of the matrixcorresponds to the planning constraints, and the lower part is com-posed by scheduling constraints for different planning periods. Itcan be observed that the integration model takes a block angu-lar structure and the blocks are linked through planning decisionvariables. As stated in Section 1, Lagrangian relaxation is a typi-cal approach that is often applied to this type of models with ablock angular structure. However, to avoid the drawback of classi-cal Lagrangian relaxation, augmented Lagrangian method is appliedin this work.

3. Augmented Lagrangian method

Lagrangian relaxation has been widely studied for the con-strained optimization problem (Guignard, 1995). Generally, for thefollowing constrained minimization problem:

zP = min f (x)s.t. h(x) = 0

g(x) ≤ 0x ∈ ˝

where h(x) and g(x) are the “upper level” constraints to be relaxed,and ˝ = {x|H(x) = 0, G(x) ≤ 0} are the “lower level” constraints, theLagrangian relaxation function with respect to the “upper level”constraints is as follows:

L(x, �, �) = f (x) +∑

i

�ihi(x) +∑

j

�jgj(x)

where � and � represent the Lagrangian multipliers. The cor-responding Lagrangian dual problem can be expressed as zLR

D =max�,�≥0

minx ∈ ˝

L(x, �, �). Based on strong duality theorem, we know that

if all the constraints are convex and all the variables are continu-ous, the optimum of primal problem will equal the optimum of thedual problem, i.e., zP = zLR

D . However, a duality gap might exist inthe presence of integer variables or other non-convexities, which

means that the optimal solution to the dual problem will be strictlyless than the true optimum of primal problem, i.e., there is a dualitygap between the two optimal values.

One way to overcome the duality gap in classical Lagrangianrelaxation is to consider the augmented Lagrangian relaxation

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f

L

wLf

w�

oa(tLpwfnts(tow

SLsaj�tsDs˝(tlasorsiksscismx

ttlo

ptuppp

With the above reformulation, the resulted model (2) is decom-posed into a planning subproblem and a number of schedulingsubproblems once the coupling constraints (2d) and (2e) arerelaxed. In this work, the augmented Lagrangian algorithm isapplied to solve the integration planning and scheduling problem.

Z. Li, M.G. Ierapetritou / Computers and

unction as follows:

(x, �, �, �) = f (x) + �

2

∑i

(hi(x) + �i

�

)2

+ �

2

∑j

(gj(x) + �j

�)2

+

here (gj + (�j/�))+ = max{gj + (�j/�), 0}, � and � represent theagrangian multipliers and � is a positive penalty parameter. Then,or the augmented Lagrangian dual zALR

D = max�,�≥0,�

minx ∈ ˝

L(x, �, �, �),

e have zALRD = zP even if primal problem is non-convex provided

is big enough.To solve the augmented Lagrangian dual Problem, the method

f multipliers can be used. This method has been widely studiednd analyzed since it is first proposed in the 1960s by Hestenes1969) and Powell (1969) independently. The theoretical charac-eristics of this approach include its natural link to penalty andagrangian methods as it inherits advantages of both, and also, asointed out by Rockafellar (1973), its interpretation in connectionith the proximal point algorithm as a regularized dual technique

or convex programming. Convergence to KKT points of the originalon-decomposed problem has been proven for the method of mul-ipliers algorithm under mild assumptions: local solutions mustatisfy second order sufficiency conditions, and � sufficiently largeBertsekas, 1982). Under the more strict assumption of convexity,he method of multipliers can be shown to converge to the globallyptimal solution of the original problem for any positive penaltyeight, as long as the sequence of weights is non-decreasing.

For more general problems, Andreani, Birgin, Martínez, andchuverdt (2008) proposed a new version of the augmentedagrangian algorithm (Appendix A) which admits general con-traints in the set ˝. The algorithm consists of a sequence ofpproximate minimizations of the relaxation L(�, �, �, �) sub-ect to the lower level constraints x ∈ ˝ followed by updating �,

and �. One of the most important conclusions of their work ishat even if the augmented Lagrangian relaxation problem is notolved to optimality at every iteration, Constant Positive Linearependence (CPLD) based convergence property are ensured under

uitable conditions: f, g, h admit continuous first derivatives and= {�|H(�) =0, G(�) ≤ 0}is a closed set. The work of Andreani et al.

2008) provides important theoretic support for practical applica-ion of augmented Lagrangian method. First, they proved that ateast a limit point x* of the sequence {xk} generated by the abovelgorithm exists under the sufficient condition that there exists ε > 0uch that the set {�||H(�)|| < ε, G(�) < ε} is bounded. In practicalptimization, this condition is often naturally satisfied. Secondly,egarding the feasibility of the solution, it is proved that: (a) if theequence of penalty parameters {�k} is bounded (i.e., from someteration on, the penalty parameters are not updated, or there exists0 such that �k = �k0 for all k ≥ k0), the limit point x* is a feasibleolution of original problem; (b) if the limit point satisfies the Con-tant Positive Linear Dependence (CPLD) constraint qualificationondition (Appendix B) with respect to the lower-level constraints,t is a KKT point of the following problem, which minimizes theum of upper-level infeasibilities subject to lower level feasibilities:in

∈ ˝0.5(||h||2 + ||g+||2). Finally, regarding the optimality of the solu-

ion, it is proved that if the limit point x* is feasible and also satisfieshe CPLD constraint qualification condition with respect to lowerevel constraints {x ∈ ˝}, then x* is a KKT (stationary) point of theriginal problem.

Although the convergence properties of the ALR algorithmroved by Andreani et al. (2008) are based on the assumption of

he continuous first derivative of the objective function f(x) andpper level constraints h(x), g(x), it is worth to point out that theseroperties are remained for the mixed integer linear programmingroblem studied in this paper. The reason is that the mixed integerroblem is always able to be transformed into its equivalent con-

ical Engineering 34 (2010) 996–1006 999

tinuous counterpart because binary variable wvti,j,n

∈ {0, 1} can be

replaced by continuous relaxation 0 ≤ wvti,j,n

≤ 1 and adding com-

plementarity constraints wvti,j,n

(1 − wvti,j,n

) = 0, so when the abovealgorithm is applied onto the mixed integer programming problem(2), similar convergence properties can still be ensured.

4. Solution methodology

4.1. Decomposable structure with reformulation

Observing the special constraint structure of the integratedplanning and scheduling problem as shown in Fig. 1, we can refor-mulate the problem into a decomposable structure through theintroduction of auxiliary duplicate variables PPt

s for the productiontarget Pt

s , and IIts for the inventory variables Invt

s, respectively. Thefollowing reformulated problem which is equivalent to the originalproblem (1) can be derived:

min∑

t

∑s ∈ SP

hsInvts +

∑t

∑s ∈ SP

usUts

+∑

t

∑i

∑j

∑n

(FixCostiwtijn + VarCostib

tijn) (2a)

s.t.

Invts = Invt−1

s + Pts − Dt

s ∀s ∈ SP, ∀t (2b)

Uts = Ut−1

s + Demts − Dt

s ∀s ∈ SP, ∀t (2c)

Pts = PPt

s ∀s ∈ SP, ∀t (2d)

Invts = IIt

s ∀s ∈ SP, ∀t (2e)

stts,n=N − stint

s = PPts ∀s ∈ SP, ∀t (2f)

stints = IIt−1

s ∀s ∈ S, ∀t (2g)

yt ∈ Y ∀t (2h)

In the above model, (2d) and (2e) are the coupling constraints whichlink the different scheduling and planning constraints block. In theproblem reformulation (2) we have made a compact representationof the scheduling constraints (1f)–(1r) as (2 h) for the sake of sim-plicity. Thus the constraint matrix structure of the above problemis shown in Fig. 2.

Fig. 2. Constraint matrix structure of the reformulated model.

1 Chem

Sf

f

s

I

U

s

s

y

ga

wi

aecipndc

4

fottp

000 Z. Li, M.G. Ierapetritou / Computers and

pecifically, equality constraints (2d) and (2e) are relaxed and theollowing augmented Lagrangian relaxation problem is obtained:

(�, �, �) = min∑

t

∑s ∈ SP

hsInvts +

∑t

∑s ∈ SP

usUts

+∑

t

∑i

∑j

∑n

(FixCostiwtijn + VarCostib

tijn)

+∑

t

∑s ∈ SP

�ts(P

ts − PPt

s ) +∑

t

∑s ∈ SP

�ts(Invt

s − IIts)

+ �∑

t

∑s ∈ SP

{(Pts − PPt

s )2 + (Invt

s − IIts)

2} (3a)

.t.

nvts = Invt−1

s + Pts − Dt

s ∀s ∈ SP, ∀t (3b)

ts = Ut−1

s + Demts − Dt

s ∀s ∈ SP, ∀t (3c)

tts,n=N − stint

s = PPts ∀s ∈ SP, ∀t (3d)

tints = IIt−1

s ∀s ∈ SP, ∀t (3e)

t ∈ Y ∀t (3f)

Thus the solution of the original planning and scheduling inte-ration problem (1) is transformed into the solution of the followingugmented Lagrangian dual problem max

�,�,�f (�, �, �). In particular,

e propose the following algorithm for the planning and schedul-ng integration problem:

Step 1. Initialization. Set bounds for multipliers: [�min, �min],[�min, �min]. Choose initial multiplier and penalty parametervalue �t

s = 0, �ts = 0, � = 1, set k = 1, ε > 0 (e.g., 0.1), ˛ > 1 (e.g.,

2.2), ˇ ∈ (0, 1) (e.g., 0.4);Step 2. Compute an approximate solution of the augmentedLagrangian relaxation problem through decomposition tech-nique as described in detail in the next section, get solutionInv, II, P, PP and objective value f(�, �, �). Define consistencyfunction value vector g = [Inv − II P − PP]T, if ||g|| < ε, then stop,(�, �, �) is a solution; otherwise, go to step 3.Step 3. Update multipliers: �t

s = min{max{�min, �ts +

�(Invts − IIt

s)}, �max}, �ts = min{max{�min, �t

s + �(Pts −

PPts )}, �max}. If ||g||(k) ≥ ˇ||g||(k−1), set � = ˛�; otherwise

keep �unchanged. Set k = k + 1, go to step 2.

Note that in the above solution algorithm, it is necessary to solveseries of augmented Lagrangian relaxation problems (3). How-

ver, the objective function of the relaxation problem (3) containsross-product terms Pt

sPPts and Invt

sIts which are non-separable, thus

t is still hard to solve the relaxation problem unless it is decom-osed because it is almost as hard as the original problem (1). So, inext subsection several decomposition strategies are presented toecompose the relaxation problem and reduce the computationalomplexity.

.2. Decomposition strategy

As presented above, in the augmented Lagrangian solution

ramework, there is an upper level which aims at finding theptimal Lagrangian multipliers and penalty parameters to solvehe augmented Lagrangian dual problem. In every iteration ofhe method of multipliers, an augmented Lagrangian relaxationroblem (3) needs to be solved with fixed Lagrangian multipliers

ical Engineering 34 (2010) 996–1006

�, � and penalty parameter �. The relaxation problem is solvedin a lower level using different decomposition strategies. Thereare several techniques in the literature that resolve the issue ofseparability in the augmented Lagrangian solution method: theDiagonal Quadratic Approximation (DQA) method (Ruszczynski,1995); the Block Coordinate Decent (BCD) method which is alsoknown as the “nonlinear Gauss-Seidel” method (Bertsekas, 2003);the Alternating Direction method (Bertsekas & Tsitsiklis, 1989),which is an extreme case of the BCD method by taking only a sin-gle BCD iteration; the separable augmented Lagrangian algorithm(Hamdi, Mahey, & Dussault, 1997), etc. All of those methods gener-ate an approximate decomposable version of the original relaxationproblem then solve it through decomposition. In this subsection,we present the Diagonal Quadratic Approximation method forcomparison and also propose a new method based on two-leveloptimization of the relaxation problem.

4.2.1. Diagonal Quadratic ApproximationDiagonal Quadratic Approximation method addresses the non-

separable issue through linearizing the cross-product quadraticterm (Pt

sPPts , Invt

sIIts) around the tentative solution Pt

s , PPts , Invt

s, IIts

and get separable approximation (also called) as follows:

(Pts − PPt

s )2 ≈ (Pts − PPt

s )2 + (PPt

s − Pts )

2 − (Pts − PPt

s )2

Thus with above substitution for the non-separable term, theoriginal relaxed problem (3) can be rewritten as the followingdecomposable form

f (�, �, �) = fP +∑

t

f tS

where fP represents optimal objective of the following planningsubproblem (4)

fP = minP,Inv,D,U

∑t

∑s

hsInvts +

∑t

∑s

usUts +

∑s

∑t

�tsP

ts

+ �∑

s

∑t

{(Pts − PPt

s )2 − (Pt

s − PPts )

2} +∑

s

∑t

�tsInvt

s

+ �∑

s

∑t

{(Invts − IIt

s)2 − (Invt

s − IIts)

2} (4a)

s.t.

Invts = Invt−1

s + Pts − Dt

s ∀s ∈ SP, ∀t (4b)

Uts = Ut−1

s + Demts − Dt

s ∀s ∈ SP, ∀t (4c)

and f tS represent optimal objectives of the following scheduling

subproblems (5)

f tS = min

PPt ,IIt ,yt

∑i

∑j

∑n

(FixCostiwtijn + VarCostib

tijn) −

∑s

�tsPPt

s

+ �∑

s

(PPts − Pt

s )2 −

∑s

�tsII

ts + �

∑s

(IIts − Invt

s)2

(5a)

s.t.

stts,n=N − stint

s = PPts ∀s ∈ SP (5b)

stints = IIt−1

s ∀s ∈ SP (5c)

yt ∈ Y (5d)

In the DQA solution method, subproblems (4) and (5) are solved

alternately with updated value of the tentative solution until agiven iteration limit is reached or the relative change in the objec-tive function value of the relaxation problem for two consecutiveinner loop iterations is smaller than some user-defined termina-tion tolerance. Sometimes, considering the fact that high accuracy

Chem

owtuTat2

qltCbspe

4

mtFt

f

�ts(P

ts

s

I

U

P

f

Z. Li, M.G. Ierapetritou / Computers and

f the subproblem solutions is not necessary in the early iterationshen the Lagrangian multipliers are far from its optimal value and

he computational effort is wasted, it is more desirable to quicklypdate the Lagrangian multiplier to move toward its optimal value.his can be achieved by limiting the total number of inner loop iter-tions in DQA by treating it as user-specified parameter to reducehe computational cost for solving the inner loop (Li, Lu, & Michalek,008).

Among the above subproblems, the planning subproblem is auadratic programming problem, whereas the scheduling subprob-

ems are mixed integer quadratic programming problems, all ofhem can be solved through standard QP/MIQP solvers such asPLEX 10. Also notice that the feasibility of the subproblems cane ensured since the auxiliary variables are not constrained in theubproblem. Furthermore, an important fact regarding those sub-roblems is that they can be solved in parallel, thus the solutionfficiency can be greatly improved.

.2.2. Two-level optimizationDifferent from those methods that use an approximation to

ake the objective separable, we propose a new method to addresshe non-separability issue in the augmented Lagrangian method.irst, the augmented relaxation problem (3) can be rewritten tohe following equivalent form:

(�, �, �) = minP,Inv,D,U

∑t

∑s ∈ SP

hsInvts +

∑t

∑s ∈ SP

usUts

+

⎧⎪⎪⎨⎪⎪⎩

miny,II,PP

∑t

∑i

∑j

∑n

(FixCostiwtijn

+ VarCostibtijn

) +∑s ∈ SP

∑t

s.t. stts,n=N

− stints = PPt

s ∀s ∈ SP , ∀t

stints = IIt−1

s ∀s ∈ SP , ∀tyt ∈ Y ∀t

.t.

nvts = Invt−1

s + Pts − Dt

s ∀s ∈ SP, ∀t (6b)

ts = Ut−1

s + Demts − Dt

s ∀s ∈ SP, ∀t (6c)

roblem (6) can be simplified as follows:

(�, �, �) = minI,P,D,U

∑t

∑s ∈ SP

hsInvts +

∑t

∑s ∈ SP

usUts +

∑t

qt(P, Inv) (7a)

Fig. 3. Illustration of the decomposition st

ical Engineering 34 (2010) 996–1006 1001

− PPts ) + �

∑s ∈ SP

∑t

(Pts − PPt

s )2 +∑s ∈ SP

∑t

�ts(Invt

s − IIts ) + �

∑s ∈ SP

∑t

(Invts − IIt

s )2⎫⎪⎪⎬⎪⎪⎭

(6a)

s.t.

Invts = Invt−1

s + Pts − Dt

s ∀s ∈ SP, ∀t (7b)

Uts = Ut−1

s + Demts − Dt

s ∀s ∈ SP, ∀t (7c)

where qt(P, Inv) is further defined by the following optimizationsubproblems:

qt(P, Inv) = miny,II,PP

∑i

∑j

∑n

(FixCostiwtijn + VarCostib

tijn)

+∑s ∈ SP

�ts(P

ts − PPt

s ) +∑s ∈ SP

�ts(Invt

s − IIts)

+ �∑s ∈ SP

{(Pts − PPt

s )2 + (Invt

s − IIts)

2} (8a)

s.t.

stts,n=N − stint

s = PPts ∀s ∈ SP (8b)

stints = IIt−1

s ∀s ∈ SP (8c)

yt ∈ Y (8d)

With the above reformulation strategy, the solution of the relax-ation problem (3) can be transformed into the solution of nonlinear

problem (7) which takes an implicit objective function and the eval-uation of the objective function needs the solution of a series ofsubproblems (8).

The difference between the DQA strategy and the proposed

two-level strategy lies on the fact that the DQA strategy actuallysolves an approximation version of the relaxation problem (3).However the later strategy solves the exact problem (3) using atwo-level optimization. In particular, the two-level optimizationstrategy solves the augmented Lagrangian relaxation problem by

rategy: (left) DQA; (right) two-level.

1 Chemical Engineering 34 (2010) 996–1006

favtpNtss

sa((Sefctooalnd

5

spftpSeT

phIemipams

wt

Table 1Process data for the example.

Unit Capacity Suitability Processing time

Heater 100 Heating 1.0Reactor 1 50 Reactions 1, 2, 3 2.0, 2.0, 1.0Reactor 2 80 Reactions 1, 2, 3 2.0, 2.0, 1.0Sill 200 Separation 1 for product 2, 2 for IntAB

State Storage capacity Initial amount

Feed A Unlimited UnlimitedFeed B Unlimited UnlimitedFeed C Unlimited UnlimitedHot A 100 0.0IntAB 200 0.0IntBC 150 0.0Impure 200 0.0Product 1 (P1) Unlimited 0.0Product 2 (P2) Unlimited 0.0

002 Z. Li, M.G. Ierapetritou / Computers and

urther reformulating it into two levels: in the first level, the relax-tion problem is solved only with respect to the planning decisionariables through an iterative algorithm, whereas in every itera-ion, a set of scheduling subproblems needs to be solved with fixedlanning decision variables in the second level as shown in Fig. 3.otice that in the two-level strategy, the relaxation problem and

he scheduling subproblems are in different levels, and in the DQAtrategy, the planning subproblem and scheduling subproblem areolved in the same level but alternately.

Finally, it should be mentioned that in the DQA method, theolution generated by solving subproblems (4) and (5) alternately isctually an approximate solution of the original relaxation problem3). As explained previously, the theory provided by Andreani et al.2008) provides support for this kind of approximation method.imilarly, we can use this idea in the two-level optimization strat-gy as follows. It is known that qt(P, Inv)is generally a non-smoothunction of P, Inv because of the integrality restrictions. Theoreti-ally, non-smooth optimization method should be used to ensurehe optimality of the solution. However, considering the difficultyf solving the non-smooth problem (7) and due to the fact that anptimal solution is not necessary to ensure the convergence of thelgorithm, we propose to use a continuous solver to solve prob-em (7) to get a solution which is feasible but not optimal. In theext section, we make a comparative study on the two differentecomposition strategies in the ALR solution framework.

. Computational study

The augmented Lagrangian algorithm and different decompo-ition strategies are studied in this section through an exampleroduction problem. All the computations in this example are per-ormed on a dual-core system with 2.8 GHz CPU and 1 Gb RAM. Inhis example, two products P1 and P2 are produced through threerocessing stages utilizing three materials (Kondili, Pantelides, &argent, 1993). The state-task-network (STN) representation of thisxample is shown in Fig. 4 and the problem data can be found fromable 1.

Considering the production planning and scheduling integrationroblem for the above production process, we divide the planningorizon into a number of planning periods with equal time length.

n every planning period, an 8-h scheduling problem is consid-red and 6 event points are used in the continuous time schedulingodel as shown in model (1). Note that this number of event points

s determined ahead of time with an objective of maximizing theroduction in the scheduling horizon of fixed 8-h. Within suchtime horizon and event point scheme, the resulted scheduling

odel can be efficiently and quickly solved through standard MILP

olver such as CPLEX 10.In the following, to study the augmented Lagrangian algorithm,

e test six different cases of the planning and scheduling integra-ion problem. Those six cases take different number of planning

Fig. 4. State-Task-Network (STN) representation of the motivation example.

Fig. 5. Demand data for 90 periods.

periods from 5 to 90 and the detail demand data can be referredfrom Fig. 5 (e.g., for the 5-period case, the demand data are the firstfive data in the figure). Cost data for this problem can be found fromTable 2.

Before the application of the augmented Lagrangian algorithmon the problem, we study the direct solution of the full space prob-lem (1) using standard MILP solver CPLEX 10. The statistical datafor the full space integrated planning and scheduling model withsix different cases of planning periods and the results of direct solu-tion method are shown in Table 3. It is observed that the problemis generally very difficult to be solved to optimality as the numberof period increases and it becomes intractable when the number ofperiods is large (90 in this example).

The augmented Lagrangian method is then applied on thisexample and different decomposition strategies presented in Sec-tion 4 are compared.

First, for the DQA based decomposition strategy, we studiedtwo different versions of the method and the results are shown

in Table 4. The first version uses only one iteration for the solutionof the relaxation problem and the other version uses increasingiteration limit (equal to the index of the outer iteration, notedas ‘k-iteration’ in the following) for the solution of the relaxation

Table 2Cost data.

Fixed cost Variable cost

Heating 150 1Reactions 1, 2, 3 100, 100, 100 0.5, 0.5, 0.5Separation 150 1

Inventory cost Backorder cost

P1, P2 10, 10 100, 100

Z. Li, M.G. Ierapetritou / Computers and Chemical Engineering 34 (2010) 996–1006 1003

Table 3Model statistics and direct solution for full space model.

Number of periods Binary variables Continuous variables Constraints Time Best solution Gap

5 600 2006 3847 3600a 6576.8 4.76%10 1200 4001 7692 3600a 13357.6 14.04%15 1800 5996 11,537 7200a 18985.9 19.60%30 3600 11,981 23,072 7200a 35217.5 17.49%45 5400 17,966 34,607 10,800a 53960.3 13.10%90 10,800 35,921 69,212 Intractable – –

a Terminated because resource limit (time) is reached.

Table 4Result of the DQA method.

T With one iteration (kmaxinner

= 1) With two iterations (kmaxinner

= kouter)

k Time f �g + �||g||2 ||g|| k Time f �g + �||g||2 ||g||5 11 62 6925.2 −3.9 0.39 14 554 6775.3 104.8 0.81

10 15 135 13684.7 −0.6 0.58 14 899 13525.8 42.8 0.8015 15 157 21113.8 72.9 0.80 20 2151 20000.9 83.4 0.7230 33 657 39312.6 157.1 0.84 17 3427 36568.3 3.6 0.5045 33 1023 59135.4 164.0 0.88 21 7338 55753.1 −71.1 0.6090 34 2462 126894.2 44.5 0.93 25 24,510 122122.1 −101.4 0.76

Table 5Result of the two-level method.

T k Time f �g + �||g||2 ||g||5 8 1245 6648.2 0.2 0.01

10 8 4983 13371.9 −79.9 0.71

pfs

ftmtauapsD

‘s

15 9 645930 8 844345 9 12,24390 9 37,875

roblem. In the solution procedure, CPLEX 10 is used in GAMS plat-orm to solve both the planning subproblem (QP problem) and thecheduling subproblems (MIQP problems).

Then, the proposed two-level optimization strategy is appliedor the solution of the augmented Lagrangian problem. To addresshe implicit objective function (7a), we use the nonlinear program-

ing solver KNITRO (Waltz & Plantenga, 2006) in MATLAB platformo solve the inner optimization problem (6) with the maximum iter-tion limit set as 50. Scheduling subproblems (MIQP) are solvedsing CPLEX 10 in GAMS. Note that although problem (7) is gener-lly non-smooth and KNITRO is a solver for smooth optimizationroblem, it is used here to obtain a feasible solution to the corre-

ponding problem. We test the same group of problems as with theQA approach and the computation results are shown in Table 5.

In all the computation results shown in Tables 4 and 5, columnT’ represents the number of planning periods, column ‘k’ repre-ents the number of outer iterations in the augmented Lagrangian

Fig. 6. Solution procedure: (left) DQA with k-i

19535.1 −163.7 0.7536223.4 −1.2 0.0354977.4 36.9 0.42

121274.4 −10.3 0.29

method, column ‘time’ represents the time used in seconds for thecomputation, column ‘f’ represents the final value of the augmentedLagrangian function, column ‘�g + �||g||2’ represents the value ofthe augmented and penalty term in the augmented Lagrangianfunction, ‘||g||’ represents the norm of the consistency constraintfunction value vector. Fig. 6 presents the solution procedure of theaugmented Lagrangian method for the 90 periods problem.

From the above results, it can be observed that the augmentedLagrangian algorithm converges to a feasible solution of the originalproblem since the norm value of the coupling constraints alwaysconverges to zero. Note that this property is independent of thedecomposition strategy used. To illustrate the feasibility of the solu-

tion, we also plot the solution of the production data along with thescheduling feasibility boundary which is generated through para-metric programming technique (Li & Ierapetritou, 2007) for the 90periods case in Fig. 7. It is observed that the solution data points areall inside the feasibility boundary.

teration; (right) two-level optimization.

1004 Z. Li, M.G. Ierapetritou / Computers and Chemical Engineering 34 (2010) 996–1006

Fig. 7. Feasibility of solution: (left) DQA with k-iteration; (right) two-level optimization.

Fig. 8. Production profile of the solution.

Fig. 9. Inventory profile of the solution.

Z. Li, M.G. Ierapetritou / Computers and Chemical Engineering 34 (2010) 996–1006 1005

r profi

gqDTrHo

ocmttDsas

FtiFtihbltoi

6

aLtcpLtt

Fig. 10. Backorde

Although both decomposition strategies ensure the conver-ence of the solution, it is worth noticing that the efficiency anduality of the solution as analyzed in the following. First, for theQA strategy, it is observed from the two different versions inable 4 that if more iterations are used for the solution of theelaxation problem, generally less outer iterations will be required.owever the increased computational complexity does not reflectbvious quality improvement of the final solution.

Second, for the two-level decomposition method, it can bebserved that it takes relative small number of outer iterations andan get feasible solutions which are better than the results of DQAethod. On the other hand, the computation time needed for the

wo-level optimization method is more than the time needed forhe DQA approach with fixed one iteration, but comparable to theQA with increasing iteration limits. However, the quality of the

olution for two-level method is better than all the DQA cases, i.e.,lthough more or comparable computation time is required, betterolution is achieved by using the two-level optimization strategy.

The results for this problem are shown in Figs. 8–10. In particularig. 8 illustrates the production of products P1 and P2. As shown inhis figure the production in the solution produced by DQA methods more compared with the solution from the two-level approach.ig. 9 illustrates the inventory of products P1 and P2. As shown byhe figure the inventory amount in the solution of the DQA methods more than that of the two-level optimization method, leading toigher inventory cost. Finally as shown in Fig. 10 that illustrates theackorder amount, the backorder amount in the solution of two-

evel case is almost zero for all periods, but the solution of DQAakes a relative large backorder in the 7th period. Thus it can bebserved from these results that the quality of the DQA solution isnferior compared to the solution of the two-level strategy.

. Conclusions

To address the problem of integrated production planningnd scheduling, a decomposition algorithm based on augmentedagrangian is proposed in this paper. Based on the special struc-ure of the optimization model, auxiliary variables and coupling

onstraints for the linking variables are first introduced, the cou-ling constraints are then relaxed and the resulted augmentedagrangian relaxation problem is solved through decompositionechnique. We also propose a new decomposition strategy based onwo-level optimization of the relaxation problem and compare its

le of the solution.

performance with traditional approximation based decompositionstrategy. The results from a case study show that the augmentedLagrangian method can effectively generate feasible solution forthe original problem, and the new decomposition strategy can gen-erate better feasible solution than the traditional approximationbased method with the trade-off of using more or comparable com-putation efforts. Furthermore, it is also worth noticing that thecomputation time in the method is mostly spent on the solutionof scheduling subproblems. By realizing that the subproblems canbe further solved in parallel, we can reduce further the computationtime through parallel computing.

The main advantages of the augmented Lagrangian method are:(a) the convergence of the algorithm is ensured without the need tosolve the relaxation problem to optimality; (b) it can be easily paral-lelized; and (c) it is able to avoid the duality gap. Furthermore, it canbe also used within a bounding procedure since a feasible solutionis always ensured. In summary, the augmented Lagrangian methodis appropriate for the solution of the planning and scheduling inte-gration problem. Future work will include improving the solutionof the relaxation problem to find the global optimal solution of theoriginal problem.

Acknowledgements

The authors gratefully acknowledge financial support from theNational Science Foundation under Grant CBET 0625515.

Appendix A. Augmented Lagrangian Algorithm (Andreaniet al., 2008)

For the following constrained optimization problem, assume f,g, h admit continuous first derivatives

min f (x)s.t. h(x) = 0, g(x) ≤ 0

x ∈ ˝ = {x|H(x) = 0, G(x) ≤ 0}Considering the following augmented Lagrangian function

∑( )2 ∑( )2

L(x, �, �, �) = f (x) + �

2i

hi(x) + �i

�+ �

2j

gj(x) + �j

� +

The corresponding augmented Lagrangian algorithm is as fol-lows:

1006 Z. Li, M.G. Ierapetritou / Computers and Chem

Step 0. InitializationGiven bounds value for the multipliers, � ∈ [�min, �max],

� ∈ [0, �max], > 1, 0 < < 1, {εk} is a sequence of non-negativenumbers such that lim

k→∞εk = 0. Set k = 1, select arbitrarily

�1i

∈ [�min, �max], i = 1, . . ., m, �1j

∈ [0, �max], j = 1, . . ., n, �1 > 0.

Step 1. Compute an approximate solution xk of theaugmented Lagrangian relaxation problem min

x ∈ ˝L(x, �, �, �),

which satisfies following approximate KKT conditions(a)

∥∥∇L(xk, �k, �k, �k) + vk∇H(xk) + uk∇G(xk)∥∥ ≤ εk

(b) Gj(xk) ≤ εk, ukj

≥ 0∀j

(c) ifGj(xk) < −εkthanukj

= 0∀j

(d)∥∥Hi(xk)

∥∥ ≤ εk, ∀iStep 2. Update multipliers’ and penalty parameter’s value

�k+1i

= min{

max(

�min, �ki

+ �khi(xk))

, �max}

, �k+1j

=min

{max

(0, �k

j+ �kgj(xk)

), �max

}Compute Vk

j= max(gj(xk), −(�k

j/�k)), if

max(||h(xk)||∞, ||Vk||∞) ≤ max(||h(xk−1)||∞, ||Vk−1||∞),k+1 k k+1 k

AC

{

gi

s

id

R

A

B

B

BB

C

C

D

set � = � , otherwise set � = �� . Set k = k + 1, go to nextiteration.

ppendix B. Constant Positive Linear Dependence (CPLD)ondition (Qi & Wei, 2000)

Assume the constraints of an optimization problem arehi(x) = 0, i = 1, ..., mgj(x) ≤ 0, j = 1, ..., n

}, and x* is a feasible point such that

j(x*) = 0 for all j ∈ J, gj(x*) < 0 for all j /∈ J. Then x* is said to sat-sfy the CPLD condition if there exist I1 ⊆ {1, . . ., m}, J1 ⊆ J, �, �

uch that∑i ∈ I1

�i∇hi(x∗) +∑j ∈ J1

�j∇gj(x∗) = 0 and∑i ∈ I1

|�i| +∑j ∈ J1

�j > 0

mplies that the gradients {∇hi(x∗)}i ∈ I1∪ {∇gj(x∗)}

j ∈ J1are linearly

ependent for all x in a neighborhood of x*.

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