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OPERATIONS RESEARCHVol. 56, No. 4, JulyAugust 2008, pp. 10101025

issn 0030-364X eissn 1526-5463 08 5604 1010

informs

doi 10.1287/opre.1080.0525

2008 INFORMS

Multitask and Multistage Production Planning and

Scheduling for Process Industries

Francesco Gaglioppa, Lisa A. Miller, Saif BenjaafarGraduate Program in Industrial and Systems Engineering, Department of Mechanical Engineering, University of Minnesota,

Minneapolis, Minnesota 55455 {[email protected], [email protected], [email protected]}

We consider the planning and scheduling of production in a multitask/multistage batch manufacturing process typicalof industries such as chemical manufacturing, food processing, and oil refining. We allow instances in which multiplesequences of tasks may be used to produce end products. We formulate the problem as a mixed-integer linear programand show that the linear programming relaxation has a large integrality gap and requires significant computational effortto solve to optimality for large instances. Using echelon inventory, we construct a new family of valid inequalities for thisproblem. The formulation with the additional constraints leads to a significantly tighter linear programming relaxation andto greatly reduced solution times for the mixed-integer linear program.

Subject classifications: production planning/scheduling; echelon inventory; integer programming.Area of review: Optimization.History : Received July 2004; revisions received May 2006, May 2007; accepted May 2007.

1. Introduction

We consider the planning and scheduling of production ina multitask/multistage batch manufacturing process typicalof industries such as chemical manufacturing, food pro-cessing, and oil refining (see Figure 1). We first consider asystem with a single processing unit. The processing unitis capable of carrying out several tasks, each consumingone or more inputs and producing one or more outputs.

Inputs for each task might consist of raw resources (feeds)or semifinished products (intermediates). Similarly, outputsfrom each task may consist of intermediates or finishedproducts. It is possible for the same intermediate or finishedproduct to be produced via more than one task. Conse-quently, each intermediate or finished product can be theresult of one or more sequences of tasks. Each task is asso-ciated with a variable batch size, a variable production cost,a fixed processing time, and a task-specific setup time andsetup cost.

We consider an environment in which time is dividedinto discrete uniform periods. A period is chosen suffi-ciently small to allow the modeling of start and end timesof each task (e.g., the length of a time period is a commondivisor to all task processing times). In each period, theremay be external demand for one or more finished productsor intermediates. To meet demand while satisfying capac-ity constraints, the plant may choose to produce ahead ofdemand and hold inventory. In that case, a holding cost perunit of inventory per period is incurred. All costs, includingproduction, inventory holding, and setup costs, could varyfrom period to period.

Our objective is to develop production schedules thatspecify production quantities and production start times

that minimize the sum of production, setup, and inven-

tory holding costs while meeting demand on time and sat-isfying constraints on production capacity and processingunit availability. We adopt a representation scheme similar

to the state-task-network formalism introduced by Kondiliet al. (1993), where a system is described by a set of states

(i.e., feeds, intermediates, and finished products) and a setof tasks that transform material from one state to another.

We allow for the possibility of multiple tasks being carriedout on the same unit and for those tasks to have overlappingsets of inputs and outputs. We also allow for the possibil-

ity of multistep processing, where a material can undergoa series of tasks on the same units. We refer to our prob-

lem as the multitask/multistage production planning and

scheduling problem (MPSP).We formulate the MPSP as a mixed-integer linear pro-

gram (MILP). We observe that the formulation leads to anNP-hard problem with a large integrality gap (gap between

the optimal solution of the MILP and the optimal solutionof the linear programming relaxation). We use the notion of

echelon inventory to construct new valid inequalities (cut-

ting planes) for the formulation. We show that the formula-tion with the additional constraints leads to a significantly

tighter LP relaxation and to much-reduced solution timesfor the MILP. We compare the impact of echelon inventoryconstraints with that of single-stage inventory constraintsthat have been used in related settings such as capacitated

lot-sizing problems (see Wolsey 1997). We show that ech-elon constraints can significantly outperform single-stageconstraints. Based on an extensive numerical study, we

highlight cases where echelon inventory constraints are par-ticularly useful. We first treat the case of systems with a

1010


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Gaglioppa, Miller, and Benjaafar: Multitask and Multistage Production Planning and Scheduling for Process IndustriesOperations Research 56(4), pp. 10101025, 2008 INFORMS 1011

Figure 1. An example of the multitask/multistage batchprocess.

1

2

3

Reactor

7

8

4

5

Feeds

Intermediates

End products

6

single processing unit. Then, we extend the formulation andthe additional valid inequalities to systems with multiple

processing units.The MPSP is related to the large body of literature on

production planning and scheduling in process industries,as well as to the capacitated lot-sizing problem (CLSP)in discrete manufacturing. However, in contrast to the

CLSP, there is not necessarily a one-to-one correspondencebetween tasks and input/output materials in the MPSP. This

makes the scheduling problem considerably more difficultbecause the manufacturing of one material could affect the

availability of several others. The complexity of the prob-lem can be further compounded by the reentrant nature of

the flows and the possibility of producing the same mate-

rial via alternative routes. Because the problem is a gener-alization of the CLSP, we expect exact solutions to large

problems to be difficult to find.

The rest of this paper is organized as follows. In 2, weprovide a brief review of the related literature. In 3, wepresent the problem formulation of the MPSP and discuss

modeling assumptions. In 4, we describe the notion of

echelon inventory and use it to construct valid inequalities.In 5, we extend our results to systems with multiple pro-

cessing units. In 6, we report on numerical results. In 7,we provide a summary and a brief discussion of various

extensions.

2. Related Literature

There is an extensive literature on production planning and

scheduling in process industries. Recent reviews can befound in Kallrath (2003), Pekny (2002), Shah (1998), and

Applequist et al. (1997). In process industries, two modesof production can be distinguished: continuous and batch.

Continuous production is adopted when there are few prod-ucts with similar routings and relatively stable demand.

Batch production is adopted when the number of productsis large and demand for each product varies with time.

Batch production in process manufacturing differs from

batch production in discrete manufacturing in that eachoperation could require multiple inputs and could produce

multiple outputs. In contrast to discrete manufacturing, thequantities of both inputs and outputs are typically contin-uous. The output from a process might revisit the sameprocess several times for further processing. Hence, therecan be significant reentrant flows.

An important development in the modeling of planning

and scheduling in process manufacturing has been the state-task network (STN) representation introduced by Kondiliet al. (1993). The STN framework uses materials (states)

and tasks as building blocks for the process description, witheach task consuming and producing materials while usingequipment. An enhancement to the STN representation isthe resource-task network (RTN) proposed by Pantelides(1994), which unifies the treatment of both equipment andmaterials as resources that are consumed (produced) at thestart (end) of a task.

Although the boundaries are overlapping, the existing lit-erature can be classified as pertaining to either planningor scheduling. For planning, time is typically discretized

into planning periods where only aggregate capacity istaken into account and the primary decisions are the quan-tities produced of each material in each period. Exam-ples of recent papers include Papageorgiou and Pantelides(1996a) and van den Heever and Grossman (1999). Formu-lations with continuous-time representation can be found inSchilling and Pantelides (1996), Zhang and Sargent (1996),and Mockus and Reklaitis (1999). A review can be found inMaravelias and Grossman (2003a). Planning problems aretypically formulated as linear programs and can be solved

relatively efficiently using standard methods. For schedul-ing, time is either finely discretized or treated as a con-tinuous parameter. In addition to production quantities for

each material, decisions in a scheduling problem includethe start and end time of individual tasks on specific pro-duction units. Scheduling problems are typically formu-lated as MILPs. In most cases, the formulation leads to anNP-hard problem. Recent examples include Maravelias andGrossman (2003b), Majozi and Zhu (2001), and Neumannet al. (2003). To cope with problem complexity, several

papers propose decomposition approaches, where the orig-inal problem is decomposed into a series of subproblemswith smaller time horizons (see, for example, Elkamel et al.1997 and Lin et al. 2002). Others develop reformulationsthat are relatively easier to solve (see, for example, Sahinidisand Grossman 1991, Shah et al. 1993, and Ierapetritou and

Floudas 1998).Planning and scheduling in process industries can be seen

as a generalization of the CLSP in discrete manufacturing.The CLSP has been widely studied. Review of the litera-

ture and recent advances can be found in Wolsey (2002),Miller and Wolsey (2003), and Atamtrk and Muoz (2004).The CLSP, which is NP-hard, can be formulated as anMILP and solved via standard branch and bound for rel-atively small problems. Reformulations and the introduc-tion of valid inequalities have been successful in reducingsolution times in some cases for larger problem instances.


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Gaglioppa, Miller, and Benjaafar: Multitask and Multistage Production Planning and Scheduling for Process Industries1012 Operations Research 56(4), pp. 10101025, 2008 INFORMS

For example, Barany et al. (1984) proposed the so-called(l S) inequalities. Several other authors have found valid

inequalities for other variations of the CLSP; for example,

see Magnanti and Vachani (1990), Constantino (1996), Bel-vaux and Wolsey (2000, 2001), and Miller et al. (2003).

The literature on the CLSP with multiple stages is

more limited. The problem is computationally harder thanthe simple CLSP. Therefore, the solution of large prob-

lems invariably involves heuristic approaches; see Katoket al. (1998), Tempelmeier and Destroff (1996), Stadtler

(2003), and the references therein. The notion of echelon

inventory first introduced by Clark and Scarf (1960) hasbeen used to reformulate the CLSP with multiple stages

and improve computational efficiency; see, for example,Afentakis and Gavish (1986), Pochet and Wolsey (1991),

and Belvaux and Wolsey (2000).

3. Formulation

We first introduce a formulation for the MPSP with justa single processing unit. The MPSP can be described in

terms of a set of tasks, N, a set of materials, R, and a set ofperiods, T, over which demand is known. The demand in

period t for material r is denoted by drt . We allow demand

to occur for both finished and intermediate products. Eachtask consumes a set of inputs in fixed proportions, with i r

being the proportion of input to task i due to material r.Each task produces a set of outputs also in fixed propor-

tions, with i r being the proportion of output from task i

in the form of material r. We denote the set of tasks forwhich material r is an input by Ir and the set of tasks

from which material r is an output by Or. Each task

requires a fixed processing time of i periods.The process incurs a variable production cost pit per unit

of production quantity undertaken by task i in period t and

a fixed setup cost git if task i is initiated in period t. Thesystem also incurs a holding cost hrt per unit of inventory of

material r held in period t. There is a maximum capacity,Cit , for the production quantity of task i in period t.

There are four decision variables: (1) the production

quantity, xit, initiated by task i in period t; (2) the status ofthe processing unit, yit , where y

it = 1 if the unit is assigned

to task i at time t and yit = 0 otherwise; (3) the start of a

task, zit, where zit = 1 if task i is initiated at time t; and

(4) the inventory level, srt , of material r in period t. We

assume that the initial inventory, sr0 , of each material r isknown.

The sequence of events within each period is as follows.

At the beginning of a period t, a production run for a task ithat was initiated at time t i completes. This imme-

diately increases the inventory levels of all corresponding

outputs. The external demands, drt , for all materials r inR are then fulfilled. This is followed by the initiation of

any new production runs (note that a run of a task i of

quantity xit that is initiated at the beginning of period twill complete at the beginning of period t + i). The level

of inventory on hand for all materials is then immedi-

ately updated to account for the fulfillment of both external

demand and internal usage. The remaining inventory from

each material incurs a holding cost for the entire current

period.

The MPSP can now be formulated as follows:

mintT

rR

hrt srt +

tT

iN

pitxit +

tT

iN

git zit (1)

subject to

srt = srt1 +

iN

i rxiti iN

i rxit drt

r R t T (2)

xit Cit z

it i N t T (3)

iu=1

zitu+1 yit i N t T (4)

zi

t

yi

t

yi

t1

i N t T (5)iN

yit = 1 t T (6)

xit 0 i N t T (7)

srt 0 r R t T (8)

yit zit 0 1 i N t T (9)

The objective function consists of minimizing the sum

of inventory holding, production, and setup costs. Con-

straints (2) are flow conservation constraints. Constraints (3)

are production capacity constraints. Constraints (4) require

that at most one run of task i is initiated in any consecutiveset of i periods, and that if task i is initiated in any of

these periods, the processing unit must remain set up for

task i for the next i periods. (This is stronger than the

obvious constraint that the processing unit must be set up

for task i to initiate it.) Constraints (5) ensure that produc-

tion is initiated at time t whenever the process is set up

for task i in period t and is not set up for task i in period

t 1. Moreover, the process remains in the same setup sta-

tus if the production unit has to stay idle, ensuring that no

unneeded setups are carried out. Constraints (6) guarantee

that the processing unit is set up for exactly one task in

each time period.

The above formulation makes several assumptions thatare worth highlighting. We assume that proportions of input

and output materials are fixed. In some environments (e.g.,

fuel blending), there is flexibility in how these proportions

are chosen subject to constraints on quality of the outputs

(see, for example, Karmarkar and Rajaram 2001). How-

ever, it is often the practice that once these proportions are

chosen at the product design stage, they remain fixed. We

assume that the various inputs are consumed at the begin-

ning of each task and the various outputs become available

when the task completes. In some settings, inputs are added


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gradually over time (e.g., a cooking process). Similarly,outputs could be collected at various stages of the pro-cess, such as in distillation. Another assumption we makeis that processing times are production quantity indepen-dent. Although this assumption holds for many processes,such as chemical reactions, it might not hold for others,such as blending. Additionally, only one task can be per-formed on the processing unit in each period, and once atask is initiated, it must continue running until completion.Finally, we assume that setup times and costs are sequenceindependent. This can be justified in many cases wheresetup costs are associated with the startup effort of initiat-ing a new task or in instances where setup costs reflect poorusage of capacity or increased usage of labor. However,instances arise where it is important to capture sequencedependency (e.g., sequence-dependent cleaning operationsrequiring expensive solvents). We offer some discussion ofthis issue with possible extensions of the current modelin 7.

MPSP is an NP-hard problem because the NP-hardcapacitated lot-sizing problem is a special case (Florianet al. 1980, Bitran and Yanasse 1982, Wolsey 2002).The uncapacitated joint replenishment problem, which isstrongly NP-hard (Arkin et al. 1989), is also a spe-cial case of the MPSP. Experimentation with solving theMILP formulation of the MPSP using CPLEX 8.1 with itsdefault settings shows that solution times grow quickly withproblem-size, and the problem eventually becomes compu-tationally prohibitive. Numerical results also show that theLP relaxation of the MILP formulation leads to poor lowerbounds on the optimal solution.

We conclude this section by noting that the flow of mate-rial in the MPSP can be described by a directed graph

(network) G, consisting of two sets of nodes, V1 and V2,corresponding, respectively, to materials and tasks. Succes-sor and predecessor nodes to a node in V1 are always nodesin V2, and vice-versa, successor and predecessor nodes toa node in V2 are always nodes in V1. Hence, the arcs in thegraph always connect nodes from Vi to Vj, where i = j. Anarc (r i) from a node in r V1 to a node in i V2 is intro-duced if task i requires material r as an input. The label onarc (r i) is i r, the fraction of input to task i due to mate-rial r. Similarly, an arc (i r) from a node i V2 to a noder V1 is included in the graph if task i produces mate-rial r. The label on arc (i r) is i r, the fraction of outputfrom task i in the form of material r. Figure 2 provides an

example of such a network with eight materials (numbered18) and four tasks (labelled AD). We will refer to thisgraphical representation in future sections.

4. Valid Inequalities

In this section, we introduce two sets of valid inequalitiesfor the MPSP. In 4.1, we introduce a family of inequali-ties based on local inventory levels and external demand ofmaterials. In 4.2, we introduce the idea of echelon inven-tory for our setting and extend the valid inequalities toconsider internal demand for materials.

Figure 2. Network representation of a processstructure.

1

2

4

5

3

6

7

80.2

0.8

0.5

0.5

0.6

0.4

0.1

0.9

0.5

0.5

1.0

0.3

0.7

1.0

Task A

Task B

Task D

Task C

4.1. Single-Level Inequalities

We begin with the basic intuition that local inventory ofmaterial r increases at time t if and only if some task i Or is initiated at time t i, where i is the processing

time for task i. Likewise, local inventory of material r can-not increase between periods k and t if no production runfor some i Or is started between k i and t i.

This gives us the first lemma, which states that if inven-tory of a material r does not increase in a time interval,then there must be enough on-hand inventory of that mate-rial at the beginning of the interval to satisfy all demandthat occurs within the time interval. We first define a newvariable: for any t k, let srk1 t represent the quantity ofinventory of material r in period k 1 that is used to satisfydemand in period t.

Lemma 1. The following set of inequalities is valid for the

MPSP

srk1 t drt

1

iOr

tiu=ki

ziu

r R k = 2 T t = k T (10)

Proof. If any amount of material r is released from pro-duction within time interval kt, then at least one of thez variables on the right-hand side of (10) is equal to one,which forces the right-hand side of the inequality to benonpositive, and the inequality is trivially satisfied. Other-wise, the inequality reduces to srk1 t d

rt , which enforces

that all demand for material r in period t is satisfied byinventory that was on-hand in period k 1. Because no

new inventory of material r is created between periods kand t, this must be true.

We can sum the above inequalities over a sequence ofconsecutive periods t = k l to derive valid inequalitiesin the original space of variables.

Theorem 2. The following set of inequalities is valid for

the MPSP

srk1 l

t=k

drt

1

iOr

tiu=ki

ziu

r R k = 2 T l = k T (11)


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Proof. For a given k and l k, summing the inequalities

defined in Lemma 1 for t = k l gives

lt=k

srk1 t l

t=k

drt

1

iOr

tiu=ki

ziu

The amount of on-hand inventory in period k 1 that isused to satisfy demand in periods k through l can be no

more than the total on-hand inventory in period k 1, so

srk1 l

t=k srk1 t. Therefore, (11) holds.

We call these inequalities single-level inequalities

because they take into account the external demand for a

material, without considering any requirement coming from

more downstream levels. Note that the above set of con-

straints could present some redundancy if demand is zero

in some periods. In fact, for a material r, it is sufficient to

restrict the parameter l to periods in which demand for r

occurs. For any l with drl = 0, inequality (11) is equivalent

to that generated for l = l 1.

Corollary 3. The following sets of constraints

srk1 l

t=k

drt

1

iOr

tiu=ki

ziu

r R k = 2 T l = k T (12)

and

srk1 l

t=kd

rt1

iOr

ti

u=ki

ziu r R k = 2 T l Lrk (13)

where Lrk = t drt > 0 and t k, are equivalent.

The additional number of constraints that are gener-

ated by single-level inequalities is ORT2, where R

is the number of materials and T is the planning hori-

zon. Although similar inequalities have been shown to per-

form well in the related setting of CLSP (see Belvaux and

Wolsey 2000, 2001 for implementation and computational

results), our experience with several instances of the MPSP

shows that solution time can significantly increase when

inequalities (13) are added. One possible explanation isthat these inequalities consider only external demand for

materials and therefore are trivially satisfied for interme-

diate materials for which external demand never occurs.

Hence, any potential benefits from the tighter formula-

tion are exceeded by the computational burden induced

by the larger problem size. However, in the next section,

we show that generating similar inequalities that consider

internal demand created by tasks that require intermediate

materials as input can significantly improve computational

performance.

4.2. Echelon Inequalities

If we consider the system as a whole, material r can beobserved at time t in three different forms: as r itself,located in inventory; as work in progress in an ongoingtaski that required r as an input; and as embedded in mate-rials produced by some task that required r as an input.

Consider a task i that requires that a proportion i r of itsinput is material r, and that produces output, of which theproportion i g is in the form material g. If at time t, taskiis initiated in quantity xit, then

i rxit of material r is con-sumed at time t and i g xit of material g is produced whentask i completes. The local inventory srt is decreased byi rxit, but this quantity does not actually leave the system.Instead, it is transformed into work in progress for i peri-ods, at the end of which it is transformed into materialg. We can say that for each unit of g that is released bythe production run of task i, a fraction i r consists of r.A similar observation can be made for any material g that isnot produced directly from r. If g is produced by a task

that requires material g as an input, and g is partially com-posed of r, then a fraction of each unit of g consists ofr as well.

We refer to material g as a successor of r if it can beproduced by a task with r or with another material con-sisting of r as one of its inputs. In this situation, we alsorefer to r as a predecessor of g. We denote the set of suc-cessors of r by Sr. We also define Sr as the set ofthe immediate successors of r. It contains all the materi-als that are produced from any task that consumes ri.e.,Sr = g R i N i ri g > 0. Using the networkrepresentation of the problem introduced in 3, g Sr ifthere exists some i N such that (r i) and (i g) are arcs

in G. Similarly, g Sr if there is a directed path in Gfrom r to g.

We now define the total amount of material r in thesystem at time t, as echelon inventory, srt . This definitionis consistent with the one commonly used in the inventorytheory literature; see, for example, Zipkin (2000).

Computing echelon inventory exactly in the MPSP set-ting is difficult for several reasons. First, if the processstructure contains reentrant flows (directed cycles), thenthe amount of one material contained in a successor coulddepend on how many times the material has been recycled.For the purposes of this section, we will assume that pro-cess structures have no reentrant flows. We will be address-ing the relaxation of this assumption in 4.4.

More significantly, there might exist several alternativetasks, or sets of tasks, that produce a particular material g,with each set of tasks possibly using different amounts ofvarious intermediate materials. Exact computation of eche-lon inventory must therefore track which set of tasks wereused to produce each unit of each material. Keeping trackof production history within the MPSP requires introduc-ing a large number of additional binary variables, makingthe problem potentially more difficult to solve when theprocess structure is complex.


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However, we can make statements about the events that

increase or decrease the echelon inventory of a material.

Property 4. The only event that increases the echelon

inventory of a material r is the completion of a task that

produces r.

Property 5. The only events that decrease the echeloninventory for a material r are fulfillment of external

demand for r and fulfillment of external demand for any of

the successors of material r.

To avoid the above difficulties associated with tracking

echelon inventory exactly, we identify an upper bound on

echelon inventory. This bound will be utilized in the valid

inequalities derived later in this section.

Consider two materials, r and g, where g is a successor

of r. If there are two or more tasks that produce g and

have r as an input, then the amount of material g due to

material r is at least the fraction of r required by the task

that uses the least amount of r and at most the fractionofr required by the task that uses the highest amount of r.

Similar reasoning applies for materials that are indirect suc-

cessors of r.

We introduce the parameters er g and fr g , which we

refer to as the minimum coefficient of transformation and

the maximum coefficient of transformation, respectively, to

measure the minimum and maximum amount of material r

that may be used to produce one unit of material g. The

characteristics of the network introduced in 3 allow us to

compute these coefficients in a sequential manner. First, the

nodes in the graph are given unique labels q from the set

1 N + R. The labels are assigned so that for anypair of nodes i j V1 V2 for which a directed path exists

in G from i to j, we have qi< qj. Under the assump-

tion that there is no recycling of materials, there are no

directed cycles in the graph, so such a labeling exists. Fur-

thermore, note that qr 0 implies g is a predecessor of n.Therefore, qg < qn, and er g has already been com-puted. This algorithm computes the matrix er g in timeOR2N. A similar algorithm is used to compute themaximum coefficients of transformation fr g .

Example. Consider the example shown in Figure 2. Mate-rials 4 and 5 both include 20% of material 1, so e1 4 =

f1 4 = 02 and e1 5 = f1 5 = 02. The coefficients of trans-formation e1 6 = f1 6 = 0 because material 1 is never usedto produce 6. Consider material 8. There are two alterna-tive tasks that can produce material 8, C and D. In fact,they both belong to O8, because C 8 > 0 and D 8 > 0.Applying the previous algorithm for C, we get

gR

e1g

C g

= e1 1C 1 +e1 4C 4 = 106 + 0204 = 068

and for D,

gR

e1g

D g

= e1 5D 5 +e1 6D 6 = 02 01+009 = 002

After considering all the possible tasks that have material 8as an output, we can compute e1 8 as

e1 8 = miniO8

gR

e1 g

i g

= min002 068 = 002

Similarly,

f1 8 = maxiO8 gR

f1 g

i g

= max002 068 = 068

It is important to observe that, for the purpose of com-puting echelon inventory, we must consider the mathemat-ical composition rather than the physical composition ofmaterials. Consider the example in Figure 2. Because theyare distinguished as separate materials, materials 4 and 5likely contain different physical proportions of materials 1and 2. However, it is impossible to produce 0.3 units ofmaterial 4 without also producing 0.7 units of material 5,

and vice versa, and every unit of production of Task Arequires 0.2 units of material 1 and 0.8 units of material 2.Therefore, mathematically, it is appropriate to assumethat materials 4 and 5 have the same composition: 20%

material 1 and 80% material 2.Consider the alternative via an example. Suppose that,

physically, material 4 is 100% composed of material 2 and,consequently, material 5 is 28.5% material 1 and 71.5%material 2 (these numbers are implied by conservation of

flow). If these proportions were used to compute echeloninventory, then the information that material 1 must be usedto create material 4 via Task A is lost, and any internalor external demand for material 4 has no implications fordemand for material 1. In fact, even though material 4does not physically contain material 1, material 1 must be


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used to create material 4 and, therefore, any demand for

material 4 induces echelon demand for material 1.

We can now compute an upper bound srt to srt of echelon

inventory of material r in period t that is composed of three

parts: the local inventory srt of r, the fraction of the work

in progress that is directly composed ofr, and a proportion

fr g

of the upper bounds of the echelon inventory of theimmediate successors of r. The inventories of all succes-

sors of r are accounted for by the echelon inventory of the

immediate successors of r. Formally,

srt = srt +

iN

tu=ti+1

i rxiu +

gSr

fr g sgt (14)

If the process has a deterministic structure, it is possible

to exactly compute the quantity of any material present in

the system because there is only one sequence of operations

that produces g starting from any material r, so the fraction

of r embedded in one unit of g is unique. This leads to the

following result.

Remark 6. If the process structure is deterministic, thensrt = s

rt .

On the other hand, if the process contains multiple

sequences of tasks that can be used to produce the same

material, the network G would contain undirected cycles.

We refer to this process structure as choice structure.

The definition (14) of srt can be added to the formula-

tion of the MPSP, and then the valid inequalities defined in

the previous section can be extended to consider echelon

inventory and internal demand. We are considering here the

echelon inventory for material r at time k 1 and enforc-ing the following: if r is not released from production from

time k to time l (thus not increasing the echelon inventory),

then there must be enough echelon inventory on hand in

period k 1 to cover the appropriate portions of demand

for all successors of material r in periods k through l.

Additionally, we observe that for any time interval from

k to l, part of the echelon inventory could be engaged as

work in progress. If this material remains work in progress

for the entire interval (i.e., the production began before

period k and will not complete before period l), then this

part of the echelon inventory is unavailable for fulfilling

external demand. Formally, the portion of echelon inven-tory that will not be available to fulfill demand because it

is work in progress in the time interval k to l is at least

iNlk

gR

k1u=li+1

er g i g xiu (15)

where Nlk refers to the set of tasks where i > l k for

each task i Nlk.

We can now extend the lemma and theorem of the pre-

vious section to consider echelon inventory.

Lemma 7. For any t k, let srk1 t represent the amount

of echelon inventory of material r in period k 1 used

to satisfy demand for r and all its successor products in

period t. Then, the following set of inequalities is valid for

MPSP

srk1 t gR

er g dgt

1

iOr

tiu=ki

ziu

r R k = 2 T t = k T (16)

Proof. If material r is not released from production

between periods k and t (i.e., no task i that produces r was

initiated within the time interval k i to t i), then the

echelon inventory level ofr in the system will not increase

during the interval from k to t, and therefore, the echelon

inventory of material r at the end of period k 1 must

be used to cover all demand for material r in period t, as

well as at least the portion er g of demand for all successor

products g of r.

If we consider a sequence of consecutive periods t =

k l, we can obtain a set of valid inequalities in the

original space of variables.

Theorem 8. The following set of inequalities is valid for

the MPSP

srk1 l

t=k

1

iOr

tiu=ki

ziu

gR

er g dgt

+

iNlkq

gR

k1u=li +1

e

r g

i g

x

i

u

r R k = 2 T l = k T (17)

Proof. For a given k and l k, we can sum the inequali-

ties defined in Lemma 7 for t = k l to get

lt=k

srk1 t l

t=k

gR

er g dgt

1

iOr

tiu=ki

ziu

The upper bound on echelon inventory srk1 of r in period

k 1 must be at least the lower bound on the echelon

inventory of r in period k 1 used to satisfy demand for

r and its successor products in periods k l plus the

amount of r and its successor products that are work in

progress for the entire interval k l as described in (15),

so (17) holds.

Define Lechrk = t

gR er g d

gt > 0 t k to be the

set of periods in which there is positive external echelon

demand for material r. With arguments similar to the ones

we used in Corollary 3 for the single-level cuts, we can

then state the following.


8/16


Corollary 9.

srk1 l

t=k

1

iOr

tiu=ki

ziu

gR

er g dgt

+iNgR

k1

u=li+1

er g i g xiu

r R k = 2 T l = k T (18)

and

srk1 l

t=k

1

iOr

tiu=ki

ziu

gR

er g dgt

+iN

gR

k1u=li+1

er g i g xiu

r R k = 2 T l Lechrk (19)

are equivalent.

If demand occurs only for the final products and there isno demand for any other materials (either raw materials or

intermediate products), then the single-level inequalities aresimply a subset of the echelon inequalities. If demand for

intermediate materials occurs, then the single-level inequal-

ities would enforce constraints on the local inventories ofthese materials, and they are not generated as echelon cuts.

Thus, they might tighten the formulation even further.

The coefficients of the z variables in the echelon con-straints can be strengthened by recognizing that if the pro-

duction of a task i that produces material g completes

in period t, the production quantity of that task is lim-ited by the machine capacity Citi , and therefore, the

echelon inventory of material r at time t increases byat most Citi

i r. This leads to the following set of valid

constraints:

srk1 l

t=k

gR

er g dgt l

t=k

iOr

Kr li t zit

+

iNlk

gR

k1u=li +1

er g i g xiu


where

Kr li u = min

l=u+i

gR

er g dg Ciu

i r

(21)

The above constraints have the additional advantage overthose in (20) of not being trivially satisfied for some inter-

vals tk when production of material r completes during

that interval. We use the above constraints in our computa-tional results reported in 6.

4.3. Preprocessing

By considering initial inventory levels, we can performa simple preprocessing step to fix some startup variablesto zero. If there is no inventory on hand for material r at thebeginning of the time horizon, then any task that requires ras input cannot be initiated until sufficient time has passed

for r to be produced. This waiting period will be at leastrmin = miniOr i. Therefore, we set z

it = 0 for all i such

that r Ii and sr0 = 0 and for all t = 1 rmin. Further-

more, if materials required as input to all tasks that producematerial r are also not available in initial inventory, thenthe earliest starting time of a task requiring r can be simi-larly pushed back. Using the node-labeling technique from4.2 and a simple depth-first search through the state-tasknetwork, the earliest start time for each task can be easilycomputed, and all startup variables zit for periods prior tothat time can be fixed to zero.

4.4. Process Structures with Reentry

Cases arise in practice where material reentry takes place.That is, a material that has been previously used as aninput for a task is produced by the task itself or by a taskusing one or more of the materials successors as input.Reentry tends to be present in the manufacturing of cer-tain chemicals where there are processes that are able topurify a chemical from other previously used components.For example, catalysts can usually be salvaged after beingused in catalysis reactions. In the state-task network repre-sentation, reentry is indicated by a directed cycle.

In general, the computation of echelon inventory, oreven bounds on echelon inventory, are difficult because the

amount of a predecessor material contained in one of itssuccessors depends on how many times the predecessormaterial has been recycled. However, there are types of pro-cess structures with reentry for which an echelon approachcould still be used, one of which is shown in Figure 3. Inthis case, it is still possible to identify successors and pre-decessors and compute fixed bounds on echelon inventoryfor some of the materials, both upstream and downstreamof a loop.

In the example shown in Figure 3, we use the term loopto refer to the set of Tasks B and C, and materials 3 and 4.It is easy to see that material 3 is its own successor. Theuseful property exhibited by this process structure is that an

intermediate product within the loop can be input to some

Figure 3. Reentry: Single-flow loop with multipleoutlets.

0.6

0.4

0.45

Task BTask ATask C1.0 1.0

1.0

0.4

Task D0.55

0.6

1.0

2

1

3

6

54


9/16


task outside the loop (material 4 is input to Task D), andsome tasks inside the loop produce materials that are never

used as input within the loop (Task C produces material 5).

Therefore, bounds on the amount of materials 1 and 2 con-tained in end products 5 and 6 can be computed, although

the same is not true for materials 3 and 4, which are within

the loop. Process structures where materials flow inside theloop via different intermediate materials are much more

complex, and it is not clear how to extend the notion of

echelon inventory in this case. We considered one problemwith reentry (Multi-3) in our computational experiments,

as described in 6.2.

5. Systems with MultipleProcessing Units

In this section, we describe how the MPSP formulation andthe additional valid inequalities can be extended to systems

with multiple processing units. In systems with multiple

processing units, different tasks might be carried out bydifferent processing units, and/or the same task could be

carried out by more than one processing unit.

5.1. Formulation

Let M denote the set of processing units and Nm denote

the set of tasks that can be performed by processing unitm M. A task is defined in terms of inputs in specified

proportions used to produce a set of outputs also in spec-

ified proportions. That is, each task is defined in terms ofcoefficients (i r and i r) independently of the process-

ing units. However, the same task could have processing

unit-dependent production time,

m

i , production cost, p

i m

t ,startup cost, gi mt , and production capacity, Ci mt .

For the MPSP with multiple processing units, which we

refer to as the MPSP-m, there are four decision variables:(1) production quantity, xi mt , initiated by task i in period t

on processing unit m; (2) the status of processing unit m in

period t, yi mt , where yi mt = 1 if unit m is set up for task i

in period t and zero otherwise; (3) a variable indicating the

start of a task i at time t on processing unit m, zi mt , where

zi mt = 1 if task i is initiated at time t on unit m and zerootherwise; and (4) the inventory level, srt , of material r in

period t.

The MPSP-m can now be formulated as follows:

MintT

rR

hrt srt +

tT

mM

iNm

pi mt xi mt

+tT

mM

iNm

gi mt zi mt (22)

st srt = srt1 +

mM

iNm

i rxi mtmi

mM

iNm

i rxi mt drt

r R t T (23)

xi mt Ci mt z

i mt m M i Nm t T (24)

miu=1

zi mtu+1 yi mt m M i Nm t T (25)

zi mt yi mt y

i mt1 m M i Nm t T (26)

iNm yi mt = 1 m M t T (27)

xi mt 0 m M i Nm t T (28)

srt 0 r R t T (29)

yi mt zi mt 0 1 m M i Nm t T (30)

The objective function and constraints maintain the sameinterpretation they have in the case of a single processingunit (see 3). Note also that the flow of material can stillbe described in terms of the directed graph G describedin 3 because tasks (and not processing units) transformmaterials from one state to another.

5.2. Valid InequalitiesThe single-level (13) and echelon cuts (19) can be extendedto the case with multiple processing units. The followingset of single-level inequalities are valid for the MPSP-m:

srk1 l

t=k

drt

1

mM

iOrNm

tmiu=kmi

zi mu

r R k = 2 T l = k T (31)

Consistent with Corollary 3, we can restrict the set of cutsdefined above as follows:

srk1

lt=k

d

rt

1

mM

iOrNm

tmiu=kmi

zi mu

r R k = 2 T l Lrk (32)

where Lrk = t drt > 0 and t k.An upper bound on echelon inventory can be obtained

similarly to Equation (14) as follows:

srt = srt +

mM

iNm

tu=tmi +1

i rxi mu +

gSr

fr g sgt (33)

with the portion of echelon inventory not available fromperiods k to l given by

mM

iNm

gR

k1u=lmi +1

er g i g xi mu (34)

The echelon cuts (17) can be modified as follows:

srk1 l

t=k

1

mM

iOrNm

tmiu=kmi

zi mu

gR

er g dgt

+

mM

iNm

gR

u = l mi + 1

k1er gi g xi mu

r R k = 2 T l = k T (35)


10/16


Again consistent with Corollary 9, the above constraintsare equivalent to the following:

srk1 l

t=k

1

mM

iOrNm

tmiu=kmi

zi mu

gR

er g dgt

+

mM

iNm

gR

k1u=lmi +1

er g i g xi mu


where Lechrk = t

gR er g d

gt > 0 t k.

Following a reasoning similar to that done for the single-machine case, we can further strengthen Equation (36) bydefining Kr limu as the coefficient of z

i mu in the echelon

inequality that considers material r and an interval up totime l. Thus, for all zi mu with i Or, the portion of eche-lon demand covered by the z-variable is better bounded bythe following:

Kr limu = min

l

=u+i

gR

er g dg Ci mu

i r

(37)

6. Computational Results

We first present computational results involving probleminstances with a single processing unit. We then discussresults for problems with multiple processing units.

6.1. Problems with a Single Processing Unit

We tested the effectiveness of our cuts on a series of

instances with varying sizes and characteristics. Resultsfrom 10 representative problems are discussed in this sec-tion. As shown in Table 1, the problems vary by num-

ber of materials (raw materials, intermediate products, andend products), number of tasks, number of stages (max-imum number of sequential tasks needed to produce anend product), and process structure. We consider problems

Table 1. Single processing unit problem instances.

Problem Materials Tasks Stages Time horizon Process structure

1 9 8 9 {50} Series

2 13 12 13 {50} Series

3 15 14 15 {50} Series

4 13 7 4 {50} Strictly convergent

5 13 7 4 {50} Strictly divergent

6 9 4 4 40 50 80 120 General deterministic

network


network


network

9 13 9 6 40 50 70 100 General choice

network

10 13 9 6 40 50 70 100 General choice

network

with five different process structures: series, strictly conver-gent, strictly divergent, general deterministic network, andgeneral choice network. A series structure refers to sys-tems where each material has a unique immediate successorand a unique immediate predecessor and each material isproduced by a single task. A strictly convergent structure

(similar to an assembly structure) refers to systems wherematerials might have multiple immediate predecessors buta unique immediate successor. A strictly divergent structure

(similar to a disassembly structure) refers to systems whereeach material has unique immediate predecessors but mighthave multiple immediate successors. For both strictly con-vergent and divergent structures, each material is producedby a single task. Systems with a general network structuremight have tasks with multiple inputs and multiple outputsforming an arbitrary network. Depending on whether thenetwork is choice or deterministic, the same material mightor might not be produced by more than one task. Hold-ing costs for various materials have been randomly gener-

ated such that downstream materials have higher holdingcosts than upstream materials. Production costs are alsorandomly generated. In this case, we always ensure thattasks with a larger number of input and/or output materialshave higher production costs. Demands for each problemhave been generated so that problems are feasible. How-ever, capacity loading is varied from problem to problem.Tightly capacitated instances are Problem 7 (50-period and80-period instance), Problem 8 (50-period and 80-periodinstance), and Problem 10 (all instances). The full data for

each problem, along with MPS files, are available from theauthors upon request.

We solved each instance with two methods. First, we

solved the original formulation using the commercial solverCPLEX version 8.1, with all the default settings and withthe standard cuts turned on. In the second method, wefirst applied the preprocessing technique to preassign somevariables to zero. Then, we generated all echelon inequal-ities and added them as model cuts to CPLEXs cutpool.CPLEX, with its default settings and cuts and the new ech-

elon cuts, was then used to obtain an optimal solution.Table 2 shows the optimal objective value of the LP

relaxation for both the original formulation (denoted LP)and the formulation with the echelon inequalities (LPcuts),as well as the objective value of the best-known integersolution (IP). Values in column IP marked with an aster-

isk have not been proven to be optimal. Also shown is theratio LPcuts LP/IP LP 100%. This ratio mea-sures the percentage reduction in the gap in cost betweenthe best-known IP solution and the LP relaxation solution.

As can be seen, the gap is reduced by over 60% in allinstances for which an optimal IP solution is obtained. Forthose instances where optimality was not proved, the valueshown is a lower bound on the actual gap reduction.

Table 3 shows the CPU time and number of branch-and-bound nodes needed by CPLEX to prove optimalityfor both the original formulation and the formulation with


11/16


Table 2. Percentage reduction in the integrality gap.

Time Percentage gap

Problem horizon LP LPcuts IP reduction (%)

1 50 135492 228946 245210 8518

2 50 237933 604127 679420 8295

3 50 225543 480619 547620 7920

4 50 126070 336891 359197 90435 50 730248 1393728 1565484 7944

6 (9/4/4) 40 95253 167487 179312 8593

50 168710 276505 306894 7801

80 374532 512843 550885 7843

120 806623 1046217 1125241 7520

7 (13/8/6) 40 261115 506050 566669 8016

50 457198 800733 936094 7173

80 998945 1571445 1815775 7009

120 950689 1656462 2373618 4960

8 (13/8/6) 40 256339 504429 568050 7959

50 447149 789675 926076 7152

80 946777 1531247 1908008 6080

120 1341064 2062877 2545907 5991

9 (13/9/6) 40 146947 407501 569617 6204

50 224093 575087 773461 638970 229263 614942 1176614 4071

100 300080 926406 1484837 5287

10 (13/9/6) 40 148734 432770 610668 6149

50 224093 575087 773461 6389

70 218571 723051 1066458 5950

100 313672 949558 1624403 4889

Optimality was not proved.

Table 3. Reduction in computational time.

Original formulation Echelon cuts addedPercentage of reduction

Problem Time horizon CPU time (seconds) Number of nodes CPU time (seconds) Number of nodes in CPU time (%)

1 50 661 108 710 11 7412 50 910 162 1521 49 67143 50 921 48 1401 12 52124 50 56194 21308 11517 163 79505 50 277705 14578 56489 1143 7966

6 40 172 86 310 25 802350 1964 1453 2743 566 396680 25448 9923 31402 4041 2340

120 25000 509620 25000 77097

7 40 11536 2489 3914 417 660750 528344 74491 31318 1861 940780 25000 137419 25000 38234

120 25000 108371 25000 8760

8 40 25960 7192 3674 371 8585

50 253117 40068 18954 649 925180 25000 112569 25000 22592

120 25000 46889 25000 8717

9 40 166874 33853 1763 1054 894450 25000 240760 123121 2780 950770 25000 80975 25000 7741

100 25000 91522 25000 7806

10 40 331068 53385 38727 2685 883050 25000 219224 106134 2780 957570 25000 78300 25000 8114

100 25000 13300 25000 3545

Optimality was not proved within 25,000 seconds.

the echelon cuts. Also shown is the percent reduction in

required CPU time MILP MILPcuts/MILP 100%,

where MILP (MILPcuts) denotes the amount of time taken

by CPLEX to obtain an optimal solution using the origi-

nal formulation (formulation with echelon cuts). An aster-

isk flags instances where optimality was not proved within

the time limit we assigned (25,000 seconds) with eitherapproach.

As we can see from Table 3, the addition of the eche-

lon cuts reduces solution time for all instances except for

those with the simple series structure and for the smallest

instance with a general deterministic structure. The solu-

tion time is reduced by one order of magnitude for the

more complex instances with either a larger number of

stages, a longer planning horizon, or tighter capacity. We

suspect the improved performance for tightly capacitated

instances follows from two reasons. First, capacity plays

a role in strengthening the echelon cuts, as described in

Equation (20). Also, it is inherently more difficult to find

feasible solutions to these instances, and the inequalities

guide the solver to a feasible solution more quickly, thus

keeping the size of the branch-and-bound tree from grow-

ing too large. The poor performance in the simple instances

is likely due to the small size of the problemsboth meth-

ods obtain an optimal solution in just few seconds. In

this case, the additional cuts tend to hurt more than help;


12/16


Table 4. Best-known integer solutions.

Original formulation Echelon cuts addedBest-known

Problem Time horizon integer solution Obj. val. Gap (%) Obj. val. Gap (%)

6 120 1125240 1135073 552 1125240 241

7 80 1815775 1865541 1783 1815775 600

120 2373617 2443854 3726 2373617 27598 80 1908008 1943265 2394 1908008 1273

120 2545906 2760260 2931 2545906 1516

9 70 1189512 1270804 5168 1189512 3694100 1484836 1667879 5543 1611655 4579

10 70 1066458 NA 1066458 2054100 1614368 NA 1614368 3731

this effect disappears as problem instances become larger.

Also seen in Table 3, there are two instances (Problems 9

and 10 with 50 time periods) for which only the formu-

lation with echelon cuts finds the optimal solution within

25,000 seconds.For the instances where an optimal solution cannot be

obtained using either formulation within 25,000 seconds,

we performed further analysis. Table 4 shows the best

integer solutions found using both the original formula-

tion and the formulation with the echelon cuts within the

assigned time limit of 25,000 seconds. The table also shows

the ratios GAPoriginal = SOLoriginal SOLbest/SOLbest

100% and GAPcuts = SOLcuts SOLbest/SOLbest

100%, where SOLoriginal SOLcuts refers to the cost of the

best solution found within 25,000 seconds using the origi-

nal formulation (formulation with echelon cuts) and SOLbestis the cost of the best-known integer solution. In each case,

the formulation with the echelon cuts always finds a better

integer feasible solution than the original formulation, and

the final integrality gap is always tighter with the echelon-

cut approach. More remarkably, when the echelon cuts are

used, we find feasible solutions for two instances of Prob-

lem 10 when the default method could not find any within

25,000 seconds. Moreover, we observed that feasible solu-

tions were always found in a shorter amount of time when

using the formulation with echelon cuts.

In all the instances we tested, the formulation with ech-

elon cuts led to a significantly fewer number of nodes

being explored in the branch-and-bound tree generated by

CPLEX. This is due to the better LP relaxations for eachnode subproblem and to the earlier identification of feasible

integer solutions.

To test how solution times might be affected by various

cost parameters, we performed a sensitivity analysis with

respect to holding, setup, and production costs. The primary

purpose of the analysis was to see if the ratios between

holding, setup, and production costs have an effect on prob-

lem difficulty. In conducting the tests, we first generated

five instances of Problem 7 from above with a time hori-

zon of 50 periods, using the same probability distributions

for the cost parameters. For each cost parameter (hold-ing, setup, production), we multiplied the costs by variousscaling factors for each base instance and then solved theinstance using both the default and echelon-cut approaches.

For each solution approach, we then averaged the solutiontimes of all five instances for each choice of cost parame-ters. The solution times when the echelon cuts were usedwere consistently low, always less than a few hundred sec-onds, while the solution times using only the initial formu-

lation fluctuated significantly. For every instance, using theechelon cuts solved the instance faster than CPLEX solvedthe initial formulation.

In Figure 4, we show the impact of varying holding coston CPU time for both approaches. We varied holding costs

for each task by scaling the costs of a base case by a factor ranging from 0.001 to 10,000. The results show that forthe range of costs tested, the formulation with echelon cuts

remains superior, with the difference in CPU time beingin excess of 60% in all cases. The absolute difference is

smallest when holding costs are very low. This makes sensebecause when holding costs are low, the tasks tend to run atfull capacity, which forces the setup variables zit to be nat-urally integer in the original formulation. Results for setup

Figure 4. Sensitivity analysis: Solution time vs. changein holding costs.

0.00

1,000.00

2,000.00

3,000.00

4,000.00

5,000.00

6,000.00

7,000.00

0.00

10.

01 0.1

0.5 1 10 50 10

0

1,00

0

10,000

Scale factor

CPUtime(seconds)

Initial formulation

Echelon cuts added


13/16


Figure 5. Problem Multi-3 process structure.

BA D E

F

HG J K

C

0.10

0.90

0.05

0.95

0.50

0.50

1 3 6 8 9 11

12

5 13107

4

2

and production costs were similar. When setup costs arevery high, tasks tend to run at full capacity, and the origi-nal formulation can be solved more quickly. The effects ofchanges to production costs are less clear, although solu-tion times tended to decrease slightly as production costsbecame very large. This is likely because when productioncosts are high, the LP relaxation of the original formula-tion tends to yield a tight lower bound on the optimal costbecause total production quantities are relatively constantacross all feasible solutions and are represented by contin-uous variables. In summary, solution times for the originalformulation were highly dependent on the cost parame-ters, while the solution method using echelon cuts per-formed consistently well regardless of the values of the costcoefficients.

6.2. Problems with Multiple Processing Units

We tested the effectiveness of the echelon cuts on threeproblem instances with multiple processing units. Processstructures for Problems Multi-1 and Multi-2 are taken fromSahinidis and Grossman (1991) (Problem BATCH3) andPapageorgiou and Pantelides (1996b) (Problem Example 2),

respectively, but different time horizons and demand pro-files have been tested here. Both problems consider threeparallel flow lines with one raw material, one or two inter-mediate materials, and one end product for each line.

Tasks at the same processing level share a processingunit, and each processing unit is dedicated to one levelonly. Demand and cost parameters were randomly gener-ated for each instance. A second set of problems, denotedMulti-1a and Multi-2a, was generated by increasing thedemand profile for each instance by 50%. Problem Multi-3has a deterministic process structure with product recyclingat upstream levels. There are 13 materials, 10 tasks, 6 pro-cessing units, and 6 levels. This problem is modified from

Table 6. Reduction in computational time (multiple processing units).

Original formulation Echelon cuts addedPercentage of reduction

Problem Time horizon CPU time (seconds) Number of nodes CPU time (seconds) Number of nodes in CPU time (%)

Multi-1 80 25000 6128791 5551 1796 9978Multi-2 80 25000 76003 64568 1285 9742Multi-1a 80 25000 6307237 706847 793506 7173Multi-2a 80 342157 23708 5142 47 9850Multi-3 50 34444 126707 11433 17229 6681

Optimality was not proved within 25,000 seconds.

Table 5. Multiple processing unit problem instances.

Processing TimeProblem Materials Tasks Stages units horizon

Multi-1 9 4 3 4 {80}Multi-2 12 8 4 3 {80}Multi-3 13 10 6 6 {50}

Example 1 in Papageorgiou and Pantelides (1996b); see

Figure 5. The characteristics of the three problems are sum-

marized in Table 5.

Table 6 shows representative results from each class of

problems that were solved with and without the echelon

cuts. As with problems with a single processing unit, the

echelon cuts can have a dramatic effect on computational

performance. In some cases, optimality could be proved

only within the specified time limit of 25,000 seconds when

the echelon cuts were used.

6.3. Comparison to Other Formulations

We performed tests comparing our problem formulation

to two alternative MILP formulations. The first, originally

proposed by Kondili et al. (1993) (see also Shah et al.

1993), we refer to as the KPS formulation, and the second,

originally proposed by Sahinidis and Grossman (1991), we

refer to as the SG formulation. For brevity, the formulations

are not reproduced here. Our objective is to (1) test whether

or not the echelon cuts are still effective when applied to

alternative formulations, and (2) compare the performance

of the cuts applied to our formulation to their performance

when applied to alternative formulations.

We implemented both the KPS and SG formulations

using CPLEX and compared solution quality and solution

times obtained with and without the echelon cuts for Prob-

lems 510. The results are consistent with those obtained

using our formulation: when applied to alternate formula-

tions, the echelon cuts continue to be useful. For brevity,

the results are not included but are available from the

authors upon request. Table 7 compares the computational

time needed to solve these 20 instances using the MPSP,

KPS, and SG formulations, each with the echelon cuts

applied. In all cases, the MPSP formulation outperforms

the others by either solving the instance to optimality more


14/16


Table7.

ComparisonofMP

SP,KPS,

andSGformulationswithe

cheloncuts.

MPSPf

ormulation

KPSformulation

SGformulation

MPSPvs.

KPSformulation

MPSPv

s.SGformulation

Time

CPUtime

Final

CPUtime

Final

CPUtime

Final

%Red

uctionin

%Reduction

%Reductio

nin

%Reduction

Problem

horizon

(seconds)

gap(%)

(seconds)

gap(%

)

(seconds)

gap(%)

CPUtime(%)

ingap(%)

CPUtime

(%)

ingap(%)

6

40

310

131

105

13

664

19524

50

274

3

38

78

35

54

2

927

2282

80

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Optimalitywasnotprovedwithin

25,0

00seconds.

Nofeasibleintegersolutionwas

foundwithin25,0

00seconds.


15/16


quickly or finding a feasible solution with a smaller inte-grality gap within the time limit of 25,000 seconds.

7. Summary and Future Extensions

We considered a multitask/multistage production planningand scheduling problem (MPSP) found in process indus-

tries and formulated it as a mixed-integer program. We usedthe notion of echelon inventory to construct valid inequali-

ties. Numerical experiments show that the echelon inequali-

ties can significantly reduce the solution time needed to findoptimal solutions or finds better feasible solutions within

a fixed timeframe. This is particularly evident in problems

with relatively complex process structures or long planninghorizons. Therefore, this approach might be useful as a

stand-alone tool in situations with complex process struc-

tures where good, but not necessarily optimal, solutionsare desired, or as a subroutine within heuristics for solving

large and/or complex problems.

There are several possible extensions worth exploring.We name three that we have started to examine: (a) treating

problems with sequence-dependent setup costs, (b) allow-ing for the possibility of backorders, and (c) developing

decomposition heuristics to solve large-scale problems. We

offer brief comments on each.Sequence-dependent setup costs can be included in our

formulation without a considerable increase in complex-ity. For example, we might introduce the changeover vari-

ables ijmt , which take value one if task j is initiated at

time t on processing unit m when the status for unit mis set to task i at time t 1, and zero otherwise. In this

case, a changeover cost is incurred to reflect the additional

cost due to performing these tasks in sequence. The statusvariables yi mt allow us to make use of the reformulation

for sequence-dependent changeover variables proposed by

Karmarkar and Schrage (1985), which has the form of anetwork flow problem. The additional constraints form a

totally unimodular matrix, and therefore the integrality ofthe changeover variables follows from the integrality of the

status variables. Therefore, no new integer variables are

added to the formulation. Furthermore, the form and valid-ity of the echelon cuts is unaffected by the introduction of

the changeover variables.

Allowing for backorders in the problem formulation isstraightforward. However, backordering could invalidate

the echelon cuts because they use the fact that demand mustbe satisfied on time. Two important exceptions are cases

where backordering is allowed only within a time window

or is limited to a fixed amount. In the first case, the sameechelon cuts can be used by simply shifting the due date for

demand in each period by the length of the time window.

In the second case, echelon cuts can be used with respectto the difference between demand in each period and the

maximum amount that can be backordered. In the more

challenging case where unlimited backorders are allowed, itis possible to use a notion of echelon inventory to relate the

amount that is not backordered (a decision variable) withthe production startup variables. Unfortunately, this gives

rise to nonlinear constraints, which to be useful would haveto be either linearized or bounded by linear constraints.

For problems involving a large number of periods,stages, or materials, reaching an optimal solution suffi-

ciently quickly can be difficult even when the echelon cutsare used. For these large problems, some form of decompo-sition (e.g., solving a series of problems over shorter plan-

ning horizons) can become necessary. The echelon cuts,however, remain useful even when such decomposition is

used. In particular, for each subproblem, echelon cuts canbe used to speed solution time or improve solution quality.

Acknowledgments

The authors thank Tingliang Huang for performing some ofthe computational experiments presented in 6. They also

thank two anonymous referees, whose valuable comments

greatly improved the paper.

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