Design of Multipurpose Batch Plants: A Comparative Analysis

Ind. Eng. Chem. Res. 2008, 47, 6025–6044

6025

Design of Multipurpose Batch Plants: A Comparative Analysis between the STN, m-STN, and RTN Representations and Formulations Taˆnia Pinto,† Ana Paula F. D. Barbo´sa-Po´voa,*,‡ and Augusto Q. Novais† DMS, Estrada do Paço do Lumiar, Instituto Nacional de Engenharia, Tecnologia e InoVaçaõ, 1649-038 Lisboa, Portugal, and CEG-IST, Instituto Superior Tećnico, UniVersidade Tećnica de Lisboa, AV. RoVisco Pais, 1049-101 Lisboa, Portugal

This paper presents and discusses different unified frameworks that can be used for design of multipurpose batch plants. The first framework is based on the state-task network (STN) process representation proposed by Kondili, Pantelides, and Sargent (1993, 1998) for the scheduling of batch plants. The second is the maximal state-task network (m-STN) where design and operational characteristics are represented simultaneously (Barbosa-Po´voa and Machietto, 1994). Finally, the third framework uses the resource-task network (RTN), proposed by Pantelides (1994) for the scheduling problem. The application of these three representations to the design problem is explored and generalized formulations are developed, which are described and characterized. A set of examples is solved and results are compared and discussed, with an emphasis on the adequacy and effectiveness of these representations to the modeling and solution of the design of multipurpose batch plants. 1. Introduction In multipurpose batch plants, a wide variety of products can be produced via different processing recipes by sharing all available resources, such as equipment, raw material, intermediates, and utilities. In order to ensure that any resource in the design can be used as efficiently as possible, an adequate methodology is necessary to represent the problem in order to address such type of problems without creating ambiguities in the process/plant representation. As referred to in the review of Barbosa-Po´voa5 on the design of batch plants, several works have appeared over the past 15 years on the treatment of the design of multipurpose batch plants. These are classified by the author into two main groups: the basic design and the extended design. The former involves the choice of the main equipment items and associated schedule, while the latter incorporates detailed aspects of the design and scheduling of multipurpose batch plants, such as plant topology, layout, operational restrictions, and uncertainty. Various types of models are presented for the design of multipurpose batch plants where different solution approaches are used such as mathematical programming, heuristics and stochastic methods. Mathematical programming was explored by Barbosa-Po´voa and Macchietto,3 Papageorgaki and Reklaitis,6 Shah and Pantelides,7 Voudouris and Grossmann,8 BarbosaPo´voa et al.,9,10 and Pinto et al.11–13 among others, while heuristics and stochastic methods were studied by Imai and Nishida14 and Bernal-Haro et al.15 Also different types of problem representations were explored where two main methodologies are used. The first one states that the design and scheduling problems are diverse and so problem specific models are required. A second line of research, adopted in this paper, considers that the problems, although diverse, are sufficiently similar to be treated by the same methodology and so uniform representation frameworks have been proposed. * To whom correspondence should be addressed. E-mail: apovoa@ ist.utl.pt. † Instituto Nacional de Engenharia, Tecnologia e Inovaçaõ. ‡ Universidade Tećnica de Lisboa.

Kondili et al.1,2 presented the first generic methodological representation for batch processes, the so-called state-task network (STN). This was applied to the scheduling problem under a short-term operation and later on to the design problem by Shah and Pantelides.7 This methodology was further generalized and led to the maximal state-task network (m-STN) by Barbosa-Po´voa and Macchietto.3 As a result, a more general and detailed representation was obtained to the design problem, where some deficiencies presented in the STN were overcome, namely the assumption of full connectivity and the nonexistence of explicit localization of the material within the plant. The m-STN combines process recipes and plant structure (units and connections) into a single framework that represents the plant topology and all legal transfers of material within the plant. A novel representation, the resource-task network (RTN) was also proposed by Pantelides4 to the scheduling of batch plants. This methodology was later on applied to the design of batch plants.16 However, the detailed full plant topology was not addressed explicitly. This was done in the work of Pinto et al.,11 where not only the design of the main equipment was considered but also the associated network circuits. This work was later improved by the same authors12,13 with some simplifications being made into the model formulation, which took into account the design problem special characteristics. A RTN-adapted model was proposed, with a periodic mode of operation, where some design aspects related to storage and transferences tasks were simplified and modeled explicitly into the mathematical formulation.

Figure 1. STN motivating example representation (classical).

10.1021/ie071281n CCC: $40.75  2008 American Chemical Society Published on Web 07/22/2008

6026 Ind. Eng. Chem. Res., Vol. 47, No. 16, 2008 Table 1. Characteristics

Table 2. Rules

methodologies

methodologies

STN

m-STN

RTN

Involves two types of instances: States and Tasks. States represent the materials within the process recipe while Tasks represent sets of operations that transform materials according to the recipe. Tasks consume/produce their input/output States in fixed given proportions. Processing times associated with the output of a Task are fixed and known a priori but may vary for different States. Each item of the processing equipment, with a given minimum and maximum capacity, is assumed to be suitable for a subset of Tasks. The amount of each material consumed may vary over the duration of the Task, while its consumption, at any given time, may be fixed or variable (i.e., depend on the batch size), or a combination of the two. Involves four types of instances: eState, eTasks, iState/oState, tTasks. The eState defines the state that can be stored in a dedicated storage vessel (like in STN) (large circles); eTasks define a processing Task that can be performed in a processing equipment (rectangles); iState/oState define the input/output for the eTasks, have zero capacity, and serve uniquely to identify the origin and location of the material (small circles); tTasks model the transfer of the material between input/output states. Each unit is automatically connected with itself, so as to allow sequential Tasks to occur within it (processing followed by storage or processing different Tasks sequentially). All transfers can be carried out independently of each other. If a connection exists between two units, then all types of material can be transferred between them. Transfer Tasks are assumed instantaneous. Material can flow within the plant if a suitable connection exists and, therefore, only feasible transfers are considered. Involves two types of instances: Resources and Tasks. A Task is an abstract operation that consumes and/or produces a specific set of Resources. Resources can be classified as nonrenewable, which represents the raw materials, utilities, manpower, etc., and renewable, which represents all types of equipment (processing, storage, piping, etc).

Based on the importance of uniform representations, this paper looks into STN, m-STN, and RTN and studies their adequacy and limitations when applied to the design of batch plants. These representations are further generalized and the resulting extended versions and associated formulations characterized. The final models are applied to the solution of a set of examples, where the extended design problem is considered through the use of discrete time formulations for a nonperiodic multiproduct single campaign mode of operation. Their conceptual and computational differences are assessed and some guidelines concerning their applicability to the extended design problem drawn. The paper is structured as follows. In the following section, the three main uniform representations (classical) studied are characterized and their adequacy to the design problem is discussed. This adequacy is then explored in section 3 and two new approaches (the adapted representations) of the representation methodologies are applied to the design of batch plants. The developed design formulations, which are based on these adapted representations, are presented in section 4, which is followed, in section 5 by the solution of a set of examples. The paper concludes with some guidelines concerning the representations’ suitability to the extended design problem.

STN

m-STN

RTN

A Task has as many input (output) States as there are different input (output) streams. Two or more streams entering the same State are necessarily of the same quality. If mixing of different streams is involved in the process, then this operation should form a separate Task. No pre-emptive operation is allowed (i.e., no Task may be interrupted once started). The processing times of the Tasks are fixed and independent of sequences. The material is transferred instantaneously. All transfers can be carried out independently of each other and only if there is a suitable link between the two units involved. If a pair of oStates, associated with the same unit and different Tasks, are connected (by a tTask) to an identical set of iStates and/or eStates, then they are equivalent. Therefore, they can be merged into a single oState and the number of transfer Tasks reduced accordingly. This rule is applied in a recursive manner to make all possible reductions (simplification rule, see Barbosa-Po´voa3). In a similar way the same rule can be applied to pairs of iStates reached from identical sets of oStates and eStates. All types of material in the process (and not just raw materials) can be treated as Resources. Processing equipment items are also treated as Resources. Some Tasks may result in the generation/consumption of utilities (such as hot water or steam), either as the main output/input or as a byproduct of their operation. The classification of the available Resources into the smallest possible number of distinct types, depends on the detail of the modeling employed (their functional equivalence). Thus, the set of attributes which do or do not characterize a Resource type, is context dependent.

Table 3. Limitations methodologies STN

m-STN RTN

No information is embedded regarding the available resources (processing equipment or States) or the plant topology (it only represents the process). If two sequential processing Tasks need to be performed in a processing unit, this cannot be guaranteed if other identical processing unit exists, since the two Tasks might be allocated to different units. The location of States is undefined. This causes particular problems at the storage level, since interchanges of the same material may occur between units at different times. The STN assumes complete freedom in the transfers of material throughout the plant, which may be unrealistic in real plants. A Task has always associated a transformation of material from one State to another. Each equipment item is treated as an independent entity, which can be very inefficient in modeling terms, when there are several identical items. None was identified. A Task cannot restore the same State at the same time instant, without an auxiliary resource. Material location is unknown. Storage, transfer, and location of materials must be considered explicitly in the formulation due the fact that transfers of material are assumed instantaneous.

2. Process/Plant Uniform Representations As mentioned above, three different process/plant representations will be analyzed and compared in this section. These are

Ind. Eng. Chem. Res., Vol. 47, No. 16, 2008 6027

Figure 2. STN extended representation.

Figure 3. Process representation based on the m-STN methodology (classical). Table 4. Network Representation Instances State Task eState i/oState eTask tTask Resource STN m-STN RTN RTN-adapted STN-adapted

×

×

×

× × ×

×

×

×

×

× ×

× ×

Table 5. Number of Network Instances Required in the Motivating Example State Task eState i/oState eTask tTask Resource total STN m-STN RTN RTN-adapted STN-adapted

9

8 2

3

14 5 5

3

3

17 11 36 12 11

3 22 4

3 3

Table 6. Example: Capacities and Equipment Costa

2

capacity (u.m./m ) max:min cost (103 c.u.) fix:var

V1

V2

V3

R1

Ci

unl 0

unl 0

103:0 1:10-4

200:0 10:10-2

200:0 10-3:10-5

a c.u. ) currency units; u.m. ) mass units; unl ) unlimited; fix:var ) fix and variable cost; max:min ) maximum:minimum, the capacity range available for the design.

respectively the state-task network, STN, the maximal state-task network, m-STN, and the resource-task network, RTN. A brief description is provided for each of these representations, and Tables 1, 2, and 3 summarize, respectively, the key assumptions, rule, and limitations of each one. 2.1. State-Task Network (STN). The STN was introduced by Kondili et al.1,2 for the scheduling of batch plants. It is characterized by two types of instances: States and Tasks. The States represent the materials within the process recipe while the Tasks represent the set of operations that transform the materials according to the process recipe. A representation of process recipes is thus obtained, where sharing of intermediates, multiple processing routes to the same intermediate or product, and material recycles can be represented. The STN represents only the process and does not contain any information regarding the available resources. However, the latter are associated either with the Tasks or the States in the network and they must be therefore considered as an add-on to the process representation. 2.2. Maximal State-Task Network (m-STN). The m-STN methodology was first developed by Crooks,17 where process recipes and plant structure characteristics (units and connections) are combined into a single framework that unambiguously represents the plant topology and all legal transfers of material within the plant. This work was generalized by Barbosa-Po´voa

and Macchietto3 whereby the m-STN was further extended to address the design of multipurpose batch plants with the introduction of plant connectivity. The obtained m-STN therefore combines process and topology (units and connections). The m-STN overcomes some of the restrictions presented in the STN. Material location within the plant, as well as the usage of all resources is now explicitly accounted for. In the m-STN methodology, like in the STN, a Task has always an associated transformation of material, but only if it requires processing equipment. Neither transfer Tasks nor dedicated storage are defined as processing Tasks and, as such, do not require the use of processing equipment. Each equipment item is treated as a distinct entity, with all the resources used in the recipe (equipment and material) being characterized and treated in a nonuniform manner. The m-STN methodology establishes the link between the recipe and the plant topology: transfers of materials are considered explicitly and may only occur if suitable equipment exists to perform them (therefore, infeasible transfers do not occur). 2.3. Resource-Task Network. The resource-task network,4 RTN, appears as a more general and conceptually simpler representation methodology when compared to the STN. Its main characteristic lies in the entirely uniform description and characterization of the available resources, with no distinction between them. Furthermore, all resources are allowed to be produced as well as consumed by Tasks at any time during their execution. This has a number of implications: • All types of material in the process (and not just raw materials) can be treated as resources; • Processing equipment items are also treated as resources, if they are consumed by processing Tasks at the start of the processing period and produced at the end. • Some Tasks my result in the generation/consumption of utilities (such as hot water or steam), either as the main output/ input or as a byproduct of their operation. The two fundamental concepts in the RTN are Resources and Tasks. A Task is an abstract operation that consumes and/or produces a specific set of Resources. The interaction between Tasks and Resources leads to the resource-task network, a bipartite directed graph. The main differences between the above three types of representation stem from the need to address explicitly the storage, transfer and location of materials, to assume that transfers of material are instantaneous and also to the handling of storage (material must be made available continuously). This implies the explicit consideration of these aspects, which leads to the extended representations presented in this work. This will be made clear through the use of a motivating example that will allow a simplified characterization of the design problem as well as a simple comparison of the process/plant representations studied along this work. 2.4. Motivating Example. A plant is to be designed at a maximum profit so as to produce 80 tons of a single product (S3) from two raw materials, (S1) and (S2). Two different processing Tasks are considered. Task T1 that transforms S1 into S3 after 1 h, and Task T2 that processes S2 during 2 h and

6028 Ind. Eng. Chem. Res., Vol. 47, No. 16, 2008

Figure 4. (A) RTN representation (classical). (B) RTN extended representation.

Figure 5. (A) STN. (B) m-STN. (C) RTN.

Figure 6. (A) STN. (B) m-STN. (C) RTN.

generates S3. A single-campaign nonperiodic mode of operation is assumed over a time horizon of 5 h. In terms of equipment, three storage tanks are available (V1, V2, and V3) to store, respectively, S1, S2, and S3, and a multipurpose reactor is suitable to process Task T1 and T2. Vessels V1 and V2 are connected to reactor R1 (connections c1 and c2), while the latter is also connected to vessel V3 (connection c3). The plant design

Figure 7. STN-adapted and RTN-adapted.

should provide in detail the plant structure as well as the operational schedule.


may differ not only on the type of material involved but also on their location (e.g., S1a). 3. Design Problem Representation Figure 8. STN-adapted and RTN-adapted.

Figure 9. STN-adapted representation.

Figure 10. RTN-adapted representation.

2.4.1. STN Representation. When applying the STN concepts to the motivating example, the following network is obtained (Figure 1). In order to address explicitly all the possible transfers of material and all storage and processing Tasks in the application of the STN methodology, an extension of the main concepts of the STN is required. This is undertaken in this paper, and some of the above restrictions are considered explicitly through the introduction of auxiliary States and Tasks for the location and storage of material. The resulting extended STN is shown in Figure 2. This involves 17 instancess8 Tasks and 9 Statesswhen compared to 5 instances in Figure 1. The Tasks model: the storage of material (Ta1, Ta2, and Ta3 for S1 in V1, S2 in V2 and S3 in V3, respectively), the processing (T1 and T2 both in R1) and the transfers of material (pi1, pi2, and pi3 for S1, S2, and S3, respectively). The States represent not only the type of material (S1, S2, and S3) but also its location on the plant (S1a, S1b, S2a, S2b, and S3a, S3b). 2.4.2. m-STN Representation. The motivating example representation as a maximal state-task network (m-STN) is shown in Figure 3. Eleven instances are obtained where storage, processing, and transfers of material, as well as the materials location, are handled explicitly. These are respectively 3 dedicated storage vessels (S1/V1, S2/V2, and S3/V3), 2 iStates (ie1, ie2), 1 oState (io1), 3 transfer Tasks (pi1/c1, pi2/c2, and pi3/c3) and 2 processing Tasks (eTasks T1/R1 and T2/R1). 2.4.3. RTN Representation. For the resource-task network, RTN, and considering directly the definitions established by Pantelides4 for Tasks and Resources, the motivating example RTN representation leads to 21 instances (Figure 4A). When connectivity is implemented, it must be handled explicitly at the expense of more auxiliary Tasks and Resources. The final representation involves 36 instances (Figure 4B) against the 21 previously required (Figure 4A). These 36 instances correspond to 22 Resources (13 equipment Resources and 9 material States) and 14 Tasks (6 storage, 6 transfer, and 2 processing). Note that the material Resources

When addressing the detailed design of multipurpose batch plants, it is required to consider simultaneously the process operation and the plant topology, since it is important to know beforehand the planned use for the equipment, for it to be correctly designed. This fact implies, when applying the above representations to the design problem, that special care should be given to two main aspects: the modeling of storage Tasks and of transfers/location of material. These aspects require that some modifications must be introduced into the extended representations, described above, in order to build fully generic formulations. The required adaptations of these extended versions are exemplified through the solution of the motivating example. The solution of a set of more complex examples is subsequently presented where the adequacy to the design problem of the various representations is investigated. Storage Tasks. When modeling a storage Task, the availability of material must be considered in a continuous form and the utilization of any suitable equipment made explicit. In STN the usage of a storage vessel is handled through the definition of a Task where the allocation of the equipment is accounted for (Figure 5A). Furthermore, and since the availability of material is assumed instantaneous, an auxiliary State (S1a) must be defined in order to guarantee the vessel utilization, otherwise its nonutilization would be implied, as a result of State S1 being always available at no storage cost. In m-STN the equipment allocation is implicit in the eState definition (e.g., S1/V1 in Figure 5B). As for the RTN, a storage Task is also created (Figure 5C). This consumes two types of Resources, i.e., the material (S1) and the vessel (V1), and produces instantaneously an auxiliary material (S1a) and an auxiliary equipment (V1′) by means of an auxiliary Task (Ta1′). Again, the need for auxiliary Resource and Tasks is explained by the instantaneous characteristic of the Task. Transfer Task /Material Location. In STN or RTN methodologies, transfer Tasks are treated as processing Tasks. These Tasks, although not transforming the State/Resource in terms of its material type, do displace it, thus originating a different State/Resource (Figure 6, A and C), which requires an extra Resourcesa suitable connection (c1)sto perform the transfer. As for the m-STN (Figure 6B), the transfer of material is handled explicitly, through a transfer Task (pi1) that uses a suitable connection (c1). Comparing STN and m-STN representations, it can be concluded that they are similar although, in terms of model characteristics, the first implies more variables and constraints as a result of the associated mathematical formulation, as it will be seen later (see Table 7). For the RTN, on the other hand, the number of instances is larger, which reflects the limitation of the RTN methodology in treating instantaneous Tasks. Again, as seen before for a storage Task, the instantaneous characteristic of a transfer, implies the need for additional auxiliary State/Resources instances (e.g., S1a and pi1′). Adapted Methodologies. From the application of STN and RTN concepts to the design problem, it can be concluded that some aspects of these methodologies might be improved on the basis of the problem characteristics. In particular, the modeling of the storage and transfer Tasks can be enhanced by suppressing the auxiliary instances and by treating explicitly the connectivity equipment in the modeling equations. These improvements on

6030 Ind. Eng. Chem. Res., Vol. 47, No. 16, 2008 Table 7. Example 1: Computational Data methodology

no. total variables

no. binary

no. constraints

CPU time (s)

LPs

STN STN-adapted m-STN RTN RTN-adapted

137 98 75 504 129

48 36 24 223 35

227 176 122 651 186

0.109 0.187 0.031 0.171 0.062

2 2 2 1 3

Table 8. Example 2: Capacities and Equipment Costa

2

capacity (u.m./m ) max:min cost (103 c.u.) fix:var

V1

V2

V4

V5

V6

R1

R2

R3

Ci

unl 0

unl 0

0:100 1:0

0:1000 0:0.1

0:1000 0:01

0:100 20:0.5

0:150 55:0.5

0:200 30:1

0:200 0:0.1


Figure 11. Example 2: STN representation.

the extended methodologies give rise to two different approaches that from now on will be qualified as adapted, be it STN or RTN. Storage Tasks. The storage of material in a dedicated storage vessel is no longer treated as a Task that stores material into a renewable Resource, but rather like a Resource with its associated equipment. That is, a State in STN and RTN will now model not only the type of material but also its allocation to the storage vessel (see Figure 7). Transfer Task. The plant superstructure may include many possible connections. Therefore, each transfer Task between two equipment Resources is associated with one material Resource. The suitable transfer characteristic of a connection is modeled through the allocation of the associated source unit, thus avoiding any infeasible transfer. Figure 8 presents the transfer Task proposed for the RTN-adapted and STN-adapted methodologies. The application of these new approaches (adapted methodologies) to the motivating example is visible in Figures 9 and 10, respectively, for the STN and RTN.

The former corresponds to 11 instances (3 transfer Tasks, 2 processing Tasks, and 3 dedicated storages). A decrease of 35% instances (from 17 to 11) from the classical STN is therefore achieved. For the latter, the adapted representation corresponds to 7 Resources (1 equipment, 3 material States, and 3 dedicated storage) and 5 Tasks (3 transfers and 2 processing) and shows a decrease of 68.2% of resources (from 22 to 7) and 64.3% of Tasks (from 14 to 5) when compared to its classical counterpart. Overall, this representation shows a decrease of 66.67% in the number of instances (from 36 to 12). In Table 4, a comparison is presented between the different instances used in each of the above network representations that covers not only the classical methodologies (STN, m-STN, and RTN), but also the adapted versions, taking as a basis the motivating example. Table 5 presents the number of instances required by the five methodologies studied (STN, RTN, m-STN, STN-adapted, and RTN-adapted).


4. Design Formulations In this section and for simplicity, the reference to the classical STN and RTN formulations is omitted since these can be found, respectively, in Kondili, Pantelides, and Sargent2 and in Barbosa-Po´voa and Pantelides.16 Reference will be made, however, to the two new formulations developed (STN-adapted and RTN-adapted), as well as to the m-STN formulation developed by Barbosa-Po´voa and Macchietto.3 A discretization of time is assumed, where the time horizon is divided into a set of equal intervals of time, with Tasks occurring at the time intervals boundaries, and the design problem solved for a multiproduct single campaign. 4.1. STN-Adapted Formulation. The STN-adapted formulation for the design problem is the following: objective function: max profit )

[∑ [ s∈SP

t

(∑ ∑ ∑ ( (∑ [

OC0ijWijt + OC1ijBijt

t

j

]

(Ss,H+1 - Ss0)υs + ∑ Dsυs -

i∈Ij

) + ∑ ∑ OCsSst s

] )]

(Ss0 - Ss,H+1)ps + ∑ Rstps

s∈SF

t

∑∑ j

t

all Tasks and that delivered to or received from the environment. Equation 9 sets the bounds for the production requirements. 4.2. m-STN Formulation. The m-STN was proposed by Barbosa-Po´voa and Macchietto3 and can be described as objective function: max profit )

)

(∑ ∑ ∑ ( (∑ [

OC0ijWijt + OC1ijBijt) +

t

∀ j ∈ Ωj, t ) 1, ... , H

)+

∑ ∑ (E

∑

∀j ∈ Ki

∑ ∑

∀ s, t ) 1, ..., H + 1 max

(11)

∀ j ∈ Ωj, t ) 1, ..., H

Wijt’ e Ej

(12)

i∈Ij t’)t-pi+1

(1)

∀ i, j ∈ Ki, t ) 1, ..., H (13)

∀i, j ∈ Ki, t ) 1, ..., T 0 e Bijt e WijtVmax j

∑V

k∈Xj

(3)

∑ ∑ t

∑V

∀ j ∈ Ki

max jk Ejk

(14) (15)

k∈Xj

∀ s, j ∈ Ki, t ) 0, ..., H + 1

(16)

∀ j ∈ Ξ, t ) 1, ..., T + 1

BTπ,t - MjEj e 0

π∈Cjsink

(17)

(5)

∀ c, π ∈ Ic, t ) 1, ..., H + 1

0 e BTπt e BTcφπc (6)

∑ BT

min ck Eck e BTc e

k∈Xc

(7)

∑ BT

max ck Eck

∀c∈C

(18) (19)

k∈Xc

∑E

∀c∈C

ck e 1

in is Bijt-pislag

[(Ss,H+1-Ss0) + Ds,H+1] e QsSTN

(10)

t

∑ ∑F

i∈Tsin j∈Ki

× CCF

∀ j ∈ Ki

jk ) Ej

k∈Xj

out is Bijt-pisproc -

)

)

k

∑E

∀i, j ∈ Ki, t ) 1, ... , H (4)

Vmax jk Ejk

-

hours yr H

k

0 e Sst e Vjφsj

0 e Sst e Vjφsj ∀s, j ∈ Ks, t ) 0, ..., H + 1

∑

(∑ ∑ (

)

k∈Xj

(2)

∀i, j ∈ Ki, t ) 1, ... , T 0 e Bijt e WijtVmax j

Qsmin STN e

t

t

0 1 ckCCck + BTcCCck

i∈Ij t’)t-pi+1

∑ ∑F

])

(Ss0 - Ss,H+1)ps + ∑ Rstps j

hours yr H

∀j ∈ Ki

max max φmin ij Vj - Vj (1 - Wijt) e Bijt e φij Vj

s st

s

min jk Ejk e Vj e

k∈Xj

∑ ∑ OC S

EjkCC0jk + VjCC1jk

t

i∈Tsout j∈Ki

i∈Ij

s∈SF

k∈Xj

Sst ) Sst-1 +

j

max max φmin ij Vj - Vj (1 - Wijt) e Bijt e φij Vj

jk ) Ej

Vmin jk Ejk e Vj e

s s

t

-

k

Wijt’ e Ej

∑D υ ]-

s.t.

(EjkCC0jk + VjCC1jk) × CCF

∑E

∑

s,H+1 - Ss0)υs +

c

s.t.

∑ ∑

∑ [(S

s∈SP

(20)

k∈Xc

(8)

∀ sSTN|Qsmax STN > 0

s∈sSTN

(9) The objective function (eq 1), which expresses the maximization of the plant profit, is followed by all the constraints, which define the STN-adapted methodology applied to the design of a multipurpose batch plant. According to STN concepts, all operations are treated as processing Tasks and this includes storage and transfers. For this reason, eqs 2 and 3 play an important role. While eq 2 guarantees that a unit when chosen is of a single type, eq 3 establishes the link between the unit existence and the Task allocation. The capacity and batch size are characterized by eqs 4, 5, and 6. Where eqs 4 and 5 guarantee that the batch for unit j must be within the equipment capacity bounds, eq 7 defines the storage vessels capacity. In eq 8 the mass balance, between the material in State s at instant t, is related with both the material produced and consumed by

Sst ) Sst-1 +

∑

∑ ∑F

i∈Tsout j∈Ki

BTπt -

π∈Πsout

Qsmin STN e

out is Bijt-pisproc -

∑ BT

∑ ∑F

i∈Tsin j∈Ki

πt - Dst + Rst

in is Bijt-pislag +

∀ s, t ) 1, ..., H + 1

(21)

π∈Πsin

∑

max STN max [(Ss,H+1 - Ss0) + Ds,H+1] e QsSTN ∀s QsSTN > 0

s∈sSTN

(22) Dsmin STN,t e

∑

Dst eDsmax ∀sSTNDsmax STN,t STN,t > 0, t ) 0, ..., H

s∈sSTN

(23) Rsmin STN,t e

∑

Rst eRsmax ∀sSTNRsmax STN,t STN,t > 0, t ) 0, ..., H

s∈sSTN

(24) The objective function (eq 10) expresses again the maximization of the plant profit. Equations 11 and 12 define the existence of the processing units: Equation 11 guarantees that the unit


exists in a single type and eq 12 establishes the link between the unit existence and the Task allocation. The capacity and batch continuous size are characterized by eqs 13, 14, and 15. While eqs 13 and 14 guarantee that the batch for unit j must be within the equipment capacity and eq 15 sets the bounds for the unit j capacity. Equation 16 defines a dedicated storage, where the amount stored must be within capacity bounds. Equation 17 guarantees that a vessel is chosen if it is used as a passing through point, where the connections bounds are defined by eq 18. The existence and capacity of each connection are modeled respectively by eqs 19 and 20. The mass balance is characterized by eq 21, which relates, at each instant time t, the amount of material present in State s with the amount produced, consumed, or transferred by all incident Tasks and delivered to or received from the environment. Bounds are imposed on production requirements (eq 22) and on delivered and received materials (eqs 23 and 24). The main difference between the m-STN and STN-adapted formulations lies in the modeling of the transfers of material which in the m-STN is treated in an explicit way (eqs 18–20) and in the STN-adapted as processing Tasks. 4.3. RTN-Adapted Formulation. The new RTN-adapted formulation for the design problem contemplates both the grassroots and the retrofit case: objective function:

( ] ) (∑ ∑(

∑ R p - ∑ ∑ (R N

max profit )

rt r

r∈CP

Rrt Vr

H T

) - ∑ [Rr

0 1 k kt + Rk ξkt

t

k∈TP

r∈Cf

0 1 ∆rCCrs + VrCCrs )+

r∈Dp s

1 VrCCrs )+

0 1 rs + VrCCrs

)

)

r∈DV s

∀ k ∈ TT, r ∈ Dc

max Vr ξkt e φkr

VrminEcr e Vr e VrmaxEcr

{

1 if Nkt ) j ˜ Njkt ) 0 otherwise Nmax k

∑ jN

Nkt )

˜

(25)

Nmax k

∑ N˜

jkt e 1

∀ k ∈ Tp, t ) 1, ..., H Nmax k

Nmax k

VrNkt )

∑

˜ ) jVrN jkt

j)1

kt’ e ∆r

∀ r ∈ Dp, t ) 1, ..., H

(26)

∀ r ∈ Dp

(27)

∀ r ∈ Dp, t ) 1...H

(28)

t’)t-pi+1 k∈Tr

∀ r ∈ Dp, t ) 1, ..., H + 1

0 e Rrt e ∆r

0 e ∆r e ∆rmax Rrt ) 0

∀ r ∈ Dp

∀ r ∈ C ⁄ Cr, t ) 1, . . ., H + 1

(32)

∀ r ∈ Dp

0 e Vr e ∆r(Vrmax - Vrmin) + Vrmin

∑ [R - R ] e V rt

r∈Cf

∑R

r|r∈Drm

rt e Vr|r∈Dfp

kt - MrEtr e 0

t

∀ r ∈ Dp

t)1+H

t)1+H

VrminEtr e Vr e VrmaxEtr

∑∑ξ

(30) (31)

Vrmin e Vr e Vrmax

r∈Cp

(29)

∀ r ∈ C, t ) 1...H + 1

Rrt g 0

rjkt

∀ r ∈ Dν

(46)

j)1

˜ jkt can be defined through the linear constraints where V˜rjkt ≡ VrN ˜ 0e˜ Vrjkt e VrmaxN jkt

∀ k ∈ Tp, r ∈ Dp, j ) 1, ..., Nk, t ) 1, ..., H (47)

Nmax k

∑ V˜

rjkt ) Vr

∀ k ∈ Tp, r ∈ Dp, j ) 1, ..., Nk, t ) 1, ..., H (48)

min φkr

∑ jV˜

Nmax k

max rjkt e ξkt e φkr

∑ jV˜

rjkt

∀ k ∈ Tp, r ∈ Dp, j )

j)1

Rrmin e Rrt e Rrmax

t

∑ ∑N

∑ jV˜

1, ..., Nk, t ) 1, ..., H

krθNk,t-θ +

Rr0 ) Rr + ∆r

(45)

j)0

j)1

∑ ∑ (µ

υkrθξk,t-θ)

(44)

j)1

Nmax k

τk

k θ)0

(41)

(43)

∀ k ∈ Tp, t ) 1, ..., H

jkt

(40)

∀ k ∈ TK, r ∈ C, t ) 1, ..., H (42)

min max VrNkt e ξkt e φkr VrNkt φkr

s.t. Rrt ) Rr0|t)1 + Rr,t-1|tg2 +

∀ r ∈ Dc

0 rs +

× CCF

(39)

j)0

∑ ∑ (Ec CC r

∀ r ∈ Dν, k ∈ TV, t ) 1, ..., H

max kt e φr Vr

k

r∈DC s

∑ ∑ (Et CC r

0

∑ξ

(33) (34) (35) (36) (37)

∀ r ∈ Dν, k ∈ TV, t ) 1, ..., H

k

(38)

∀ r ∈ CP, t ) 1 + H

(49) (50)

As before, the objective function (eq 25) expresses the maximization of the plant profit, while the constraints model all the required operational and design restrictions. The excess resource balance defined by eq 26 characterizes the balance of the renewable and nonrenewable resources at each instant t with the resources in the previous instant and the produced/consumed resources in instant t. For the plant retrofit design, it is necessary to determine the required additional amount of resource, which is guaranteed by eq 27. The link between the processing equipment resource existence and the allocation of a suitable Task is done by eq 28. The resource capacity constraints characterized by eqs 29 and 30 define the design variable for the processing equipment resource, which must be within bounds. The nonrenewable excess resource must always be nonnegative, and therefore the nonstorable resources must have the corresponding excess resource variable equal to zero during the whole planning horizon. Equation 33 guarantees that the equipment resource is available in a continuous range, while eq 34 guarantees that if chosen it is effectively modeled. The storage of raw material and products in dedicated vessels is defined by eqs 35 and 36, while the existence of such equipments is defined by the variable Etr in eq 38. As in the m-STN, a storage vessel is chosen whenever it is used as a simple passing-through point, eq 38. On the other hand, the amount of resource storage must be related with the available capacity, eq 39. Equations 40 and 41 define that each connection is associated with one existence variable, Ecr, and that its capacity must be between bounds. Equation


Figure 12. Example 2: STN-adapted representation.

Figure 13. Example 2: m-STN representation.

42 guarantees that each Task using an equipment resource j must have its batch within the capacity range available. Since this equation becomes nonlinear, it is linearized using eq

43–48, leading to eq 49. The production requirements are defined by eq 50, which allows the production to float between bounds.


Figure 14. Example 2: RTN representation. Table 11. Example 3: Units Characteristicsa

Table 9. Example 2: Optimal Design Equipment equipment V1 V2 V4 V5 V6 R1 R3 C1, C7, C14 C3, C6 C9 C12

capacity (u.m.)

unit

suitability

unlimited unlimited 80 80 60 67.5 110 55 67.5 80 60

R1 R2 R3 R4 R5 R6 V1 V2 V4 V5 V6 V7 V9 V10 V11

Tasks T1-T2 Tasks T1-T2 Tasks T3 Tasks T4 Tasks T5 Tasks T6 Store S1 Store S2 Store S4 Store S5 Store S6 Store S7 Store S9 Store S10 Store S11

Table 10. Example 2: Computational Data methodology

no. total variables

no. binary

no. constraints

CPU time (s)

LPs


974 794 600 3565 918

358 298 116 1721 238

1779 1479 918 4612 1465

1.453 0.750 0.906 35.062 2.640

303 203 434 2290 1623

5. Solution of Examples Based on the above representations and associated formulations, a set of examples was solved for a optimality margin of 5%. A Pentium IV at 3.0 GHz and the GAMS 21.3 software package with the CPLEX (v 9.0) solver were employed. Example 1: Motivating Example. Solving the motivating example using the five representations presented above and the data expressed in Table 6, the results obtained are shown in Table 7. These correspond to an obtained 0% margin of

capacity (m.u./m2) min:max costs (103 c.u.) fix:var 0:150 0:150 0:200 0:200 0:150 0:150 0:15000 0: 15000 0:5000 0: 8000 0:10000 0:15000 0:7000 0:7000 0:5000

20:0.5 55:0.5 30:1 30:0.5 30:0.5 30:0.5 0.1:0.1 0.1:0.1 0.1:0.1 0.1:0.1 0.1:0.1 0.1:0.1 0.1:0.1 0.1:0.1 0.1:0.1


optimality and to an objective function of 1248.54 × 103 monetary units, where units R1, V3, C1, and C3, with 80 units of capacity each, are chosen. From the data in Table 7, it is found that for this simple example small CPU times were required by all five models. Still, it can be seen that the m-STN representation leads to the smallest size problem both in terms of number of variables and constraints, followed by the STN-adapted and the RTNadapted. As for the classic versus the adapted methodologies, the latter result in smaller size models. It can also be noted that the model characteristics have some influence on the CPU times. Thus and while the differences are small, the


Figure 15. Example 2: RTN-adapted representation.

Figure 16. (A) STN and RTN. (B) m-STN.

Figure 18. Example 2: optimal scheduling and storage policy.

Figure 17. (A) STN and RTN. (B) m-STN.

m-STN is the one found to be solved more quickly (0.031 s), followed by the RTN-adapted (0.062 s) and the STN (0.109 s). Example 2. Using an example proposed by Barbosa-Po´voa and Macchietto,3 the above representations are again explored. In here, a plant must be designed at a maximum profit so as to produce three final products, S4, S5, and S6, with production capacities between [0:80] ton for S4 and S5, and [0:60] for S6, from two raw materials, S1 and S2. The process operates in a nonperiodic mode over a time horizon of 12 h. In terms of equipment suitability, reactors R1, R2, and R3 are multipurpose equipment. Task T1 may process S1 during 2 h in R1 or R2 producing the unstable material S3; Task

T2 may process S2 during 2 h in R2 or 1 h in R1 producing S4, which is both an intermediate material and a final product; Task T3 processes 0.5 of S3 and 0.5 of S4 in R3 during 3 h producing S5, which, like S4, is both an intermediate and a final product; Task T4 may process 0.5 of S3 and 0.5 of S5 in R3 during 2 h producing the final product S6. Each vessel is suitable to store only one material State. Table 8 shows the capacity allowed for each unit and its associated cost. The capacity is defined in a continuous range with a minimum of zero and a maximum defined for each equipment. In order to illustrate all the possible material transfer, storage, and processing Tasks, the application of the five methodologies is presented in Figures 11–15. The State-Task Network (STN). The STN general representation is shown in Figure 11. It involves 49 instances, i.e., 26 Tasks and 23 States. Tasks include the storage of material (TA1, TA2, TA4, TA5, and TA6, respectively, for S1 in V1,


Figure 19. Example 3: STN process representation. Table 12. Example 3: Optimal Equipment Design equipment V1 V2 V4 V5 V6 V7 V9 V10 V11 R1, R3 R4 R5 R6 C1, C3 C2, C5, C6, C13 C4 C9 C16, C19 C17 C18 C20,C21

Table 13. Example 3: Computational Data

capacity (u.m.) unlimited unlimited 169.4 170 449 unlimited 166 166 143 124.4 169.4 118.6 83.0 101.7 124.4 169.4 67.8 59.3 35.6 83 41.5

S2 in V2, S4 in V4, S5 in V5, and S6 in V6), processing (T1 in R1 and/or R2, T2 in R1 and/or R2, and both T3 and T4 in R3), and transfers (pi1 to pi15, respectively, for S1, S2, S3, S4, S5, and S6 States). The States represents not only the type of material (S1, S2, S3, S4, S5, S6), but also its location in the plant (S1a, S1b, S1c, S2a, S2b, S2c, S4a, S4b, S4c, S5a, S5b, S5c, S6a, and S6b). Assuming that a State would model not only the type of material but also its allocation to the equipment, the STN-

methodology

total variables no.

no. binary

no. constraints

CPU time (s)

LPs


2849 2201 1650 10798 2581

984 768 264 5166 612

5391 4480 2679 14549 4377

4.39 3.20 2.81 9.281 3.80

40 60 80 150 70

adapted is drawn in Figure 12 where 39 instances are present (against the previous 49), i.e., 21 Tasks and 18 States. The Maximal State-Task Network (m-STN). The resulting m-STN is shown in Figure 13. This involves 42 instances, where storage, processing and transfers of material are handled explicitly, together with material location. The instances represent five dedicated storage locations (S1/V1, S2/V2, S4/V4, S5/V5, and S6/V6), 8 iStates (ie1 and ie2), 6 oStates (io1), 17 transfer Tasks (pi1/c1, pi2/c2) and 6 processing Tasks (eTasks: T1/R1, T1/R2, T2/R1, T2/R2, T3/R3, and T4/R3). The Resource-Task Network (RTN). The RTN representation results in 107 instances (visible in Figure 14): 63 resource instances (41 equipment Resources and 22 material States) and 44 Tasks of different types: 10 storage, 28 transfers, and 6 processing. The resource material may differ not only in the type of material but also in its location (e.g., S1a, S1b, and S1c). On applying the new concepts of storage Tasks and connectivity, the RTN-adapted shown in Figure 15 is obtained. The new approach only requires 42 instances against the previous 107 used by RTN.


Figure 20. Example 3: RTN process representation.

Figure 21. Example 3: m-STN process representation.

Transfer Task within a Processing Unit. A particular situation that may occur when modeling multipurpose operations, it is the occurrence of a transfer Task within a given equipment unit. This is found in the present example where a transfer Task is necessary between the T3 and T4 Tasks, both

using unit R3. All three methodologies, as it can be seen in Figure 16, A and B, require three instances to model a transfer Task. A difference exists however in the STN methodology, since by analogy to all other transformations it requires a processing unit. This means, in the particular case of a transfer


Figure 22. Example 3: STN-adapted process representation. Table 14. Example 4: Units’ Characteristicsa unit

suitability

R1 R2 R3 R4 R5 R6 R7 R8 R9 R10 V0 V1 V2 V3 V4 V6 V7 V8 V11 V13 V14 V15 V16 V17 V18

Tasks T1 Tasks T2 Tasks T3 Tasks T4, T5,T6, T7 Tasks T8, T9 Tasks T10, T11, T12 Tasks T10, T11, T12 Tasks T13,T14,T 15, T16, T17 Tasks T13,T14, T 15, T16, T17 Tasks T8 Store S0 Store S1 Store S2 Store S3 Store S4 Store S6 Store S7 Store S8 Store S11 Store S13 Store S14 Store S15 Store S16 Store S17 Store S18

capacity (m.u./m2) costs (103 c.u.) min:max fix:var 0:100 0:100 0:500 0:300 0:300 0:300 0:300 0:300 0:300 0:300 0:1000 0:1000 0:1000 0:1000 0:1000 0:500 0:500 0:500 0:500 0:500 0:500 0:500 0:500 0:500 0:500

30:0.5 30:0.5 30:0.5 30:0.5 30:0.5 30:0.5 30:0.5 30:0.5 30:0.5 30:0.5 0.1:0.01 0.1:0.01 0.1:0.01 0.1:0.01 0.1:0.01 0.1:0.1 0.1:0.1 0.1:0.1 0.1:0.1 0.1:0.1 0.1:0.1 0.1:0.1 0.1:0.1 0.1:0.1 0.1:0.1

a c.u. ) currency units;u.m. ) mass units; unl ) unlimited; fix:var ) fix and variable cost; max:min)maximum:minimum, the capacity range available for the design.

of material within a unit (e.g., T4 in R3 and T3 in R3), that a pipe with null cost must be introduced to guarantee the connection. Transfer Material to Multipurpose Equipment. In the case of multipurpose equipment, the modeling of a transfer operation

Table 15. Example 4: Instances Quantification Associated with Each Network Representation State Task eState iState oState eTask tTask Resource total STN 83 m-STN RTN RTN-adapted STN-adapted 53

89 153 74 74

15 15 15

26 -

27 -

25 -

49 -

221 63 -

172 142 374 152 142

might require more than one transfer Task, for example, connection C14 between R1 and R3. The approach to achieve this, however, is not the same in all three methodologies. While in STN and RTN (Figure 17A) only one instance is required (pi14) to establish the link between R1 and R3, in the m-STN (Figure 17B) two instances are needed, i.e., transfers Tasks pi14 and pi16, since there are two different destinations (T3 and T4 Tasks). As in the motivating example, this example was solved for the three methodologies, with a final 0% margin of optimality. With the objective function value of 2638 × 103 monetary units, the maximum production capacities for the S4, S5, and S6 products were obtained. The optimal scheduling and storage policy are shown in Figure 18, the selection and equipment design obtained from all three methodologies in Table 9, and the computational data in Table 10. From Table 10 it can be seen that the m-STN representation results in the smaller problem specification both in terms of variables and constraints. This is followed by the two new approaches, i.e., the RTN-adapted and STN-adapted. As for classical versus adapted methodologies, the latter leads to smaller models. The model characteristics influence the CPU times, as can be seen in Table 10. The most effective methodology is the STN-adapted (0.75 s) followed by the


Figure 23. Example 3: RTN-adapted process representation.

Figure 24. Example 3: optimal scheduling.

m-STN (0.906 s) and the STN (1.453 s). However, the differences between the m-STN and STN-adapted are not very marked. Example 3. A multipurpose batch plant must be designed at a maximum profit so as to produce [0; 170] tons of product S5, [0; 166] tons of S9 and S10, [0; 270] tons of S6, and [0; 143] tons of S11, from three raw materials (S1, S2, and S7) over a horizon of 24 h. The products S5 and S6 are both intermediate and final products. The S3 and S4 are intermediate materials, S3 is an unstable material, and S4 is a stable material with a

dedicated vessel. In terms of equipment suitability (Table 11), reactors R1 and R2 may carry out two processing Tasks, while each storage vessel and reactors R3, R4, R5, and R6 are only dedicated to a single State/Task. Specifications in terms of processing Tasks (State) are as follows: T1 (S1), 2 h, in R1 or R2; T2 (S2), 2 h, in R1 or R2; T3 (S4), 4 h, in R3; T4 (S3), 2 h, in R4; T5 (S6), 1 h, leading to final products (0.3 of S11 and 0.7 of S8), in R5; and finally T6 (S8), 1 h, in reactor R6, leading to final products (S9 and S10). The units characteristics and suitability are shown in Table 11, while the connections


Figure 25. Example 3: optimal storage policy.

capacities range from 0 to 200 (m.u./m2) at a fixed/variable cost of 0.1/ 0.01 (103 c.u.) (where m.u. and c.u. are mass and currency units, respectively). Again the application of the five methodologies is illustrated below. The State-Task Network (STN). In this example, the STN methodology requires 76 instances: 38 Tasks and 38 States (Figure 19). The Resource-Task Network (RTN). The RTN methodology (Figure 20) requires 163 instances: 103 Resources (65 equipment Resources and 38 material States) and 60 Tasks (18 storage, 42 transfer, and 8 processing Tasks). The Maximal State-Task Network (m-STN). For example 3, the obtained m-STN representation is shown in Figure 21. This methodology requires 58 instances: 9 dedicated storage, 10 i-State, 10 o-State, and 21 transfer and 8 processing Tasks. The STN-Adapted. The STN-adapted methodology requires 58 instances: 29 Tasks, 21 States, and 9 dedicated storage (Figure 22). The RTN-Adapted. The RTN-adapted methodology results in 64 instances (Figure 23): 35 Resources (6 equipments, 20 material States, and 9 dedicated storage vessels) and 29 Tasks (21 transfer and 8 processing). From the representation point of view the STN-adapted methodology shows a decrease of 23.7% instances (58 as against 76 for STN) and the RTN-adapted shows a decrease of 60.7% (64 as against 163 for RTN).

This example was solved using all the methodologies already presented, with a final 0.01% of optimality margin. The objective function of 1.6 × 106 monetary units was obtained for the maximum production of products S5, S6, S9, S10, and S11. The optimal scheduling is shown in Figure 24, optimal storage policy in Figure 25, and the selection and equipment design in Table 12, followed by the computational data in Table 13. As in the previous example, the m-STN representation results in a smaller size problem, both in terms of variables and constraints. As for classical versus adapted methodologies, it is seen in Table 13 that, as expected, both adapted methodologies result in smaller size problems. As the problem size increases, the CPU time used by the different methodologies also increases, as can be seen in Table 13. The m-STN representation is still the most efficient (2.81 s) followed by the two new approaches, STN-adapted (3.2 s) and the RTN-adapted (3.8 s). These new approaches require less CPU time than the classic ones (STN and RTN). The STN-adapted versus the classic methodology, presented a performance improvement of 27.1%, the same applying to the RTN-adapted, with a performance improvement of 59.1%. In this example, the improvement between STN-adapted versus STN is not very marked, although a significant improvement exists between the CPU time required by the RTN-adapted (3.8 s) versus the RTN (9.281 s). Example 4. In this example, the maximization profit of a multipurpose batch plant is aimed for, in parallel with the


Figure 26. Example 4: product recipe.

determination of the optimal design and scheduling. The example is based on a benchmark problem proposed by Westenberg and Kallrath,19 with some adaptations to the design problem. The example main data as well as the detailed description of the example features are available at http:// www.wior.uni-karlsruhe.de/neumann/forschung/wk95/ wk95.html. For simplicity, the network representation is omitted but the process recipe is shown in Figure 26. The units characteristics are presented in Table 14. Four reactors (R1, R2, R3, and R8) are dedicated to the respective Tasks and all the remaining processing equipment presents multipurpose characteristics. In the recipe is visible that the Tasks processing time may be equipment dependent. The connections capacity range from 0 to 300 (m.u./m2) and have associated a fixed/variable cost of 0.1/ 0.01 (103 c.u.). The final plant produces [0:30] ton

of products S14, S15, and S16, [0: 20] ton of product S17 and [0:40] of product S18, from one raw material S0 over a horizon of 120 h. The representations studied are not shown explicitly but the associated instances quantification is given in Table 15. The methodologies which require less instances are the m-STN and the new STN-adapted (both with 142). The adapted methodologies require a smaller number of instances when compared with the classical methodologies. Thus, STN-adapted presents a decrease of 17.4% (142 as against 172 for the STN) and the RTN-adapted a decrease of 59.4% (152 as against 374 for RTN). This example was solved for a 5% margin of optimality and the value reached for the objective function was 2.1 × 107 monetary units. The optimal scheduling and storage profiles are


Figure 27. Example 4: optimal scheduling.

Figure 28. Example 4: final products optimal storage policy. Table 16. Example 4:Optimal Equipment Selection and Sizing equipment V0 V1, V2, V3, V4, V6, V14, V15 V16 V17 V18 R1, R2 R3 R5, R9 R6, R4 R8 C1, C2, C3 C4 C5, C6, C8 C10, C11, C12, C14, C13 C19 C20 C21 C22, C23, C24, C25, C27 C28, C29, C30, C33,

V7, V8, V11, V13

C16, C17, C46

C26 C34, C36, C37, C39, C41, C42, C44

capacity (u.m.) unlimited 125 60 50 38.6 77.2 100 44.96 30 38.6 50 100 60 30 25 50 40 44.96 31.0 30 13.94 38.6

shown in Figures 27 and 28, respectively. The optimal equipment design is presented in Table 16, which is followed by the computational data in Table 17. As in the previous examples, the m-STN representation results in a smaller problem (in terms of variables and constraints) and the new adapted methodologies result in a better performance comparing with classical methodologies, as shown in Table 17. The m-STN representation is still the most efficient (1056.9 s) followed by the RTN-adapted (1009.1 s) and the STN-adapted (1445.4 s). As the example dimension increases, the CPU time for the adapted methodologies is significantly smaller (1009.1 for RTN

Table 17. Example 4: Computational Data methodology

total variables no.

no. binary

no. constraints

CPU time (s)

LPs


31 547 26 147 17 508 11 5305 33 405

10828 9028 3147 55677 9074

59 728 51 328 29 822 155 653 54 136

11975.6 1445.4 1056.9 12812.921 1009.1

2960 290 250 1000 240

and 1445.4 for STN) when compared with the CPU time used with the corresponding classical methodologies (12 812.9 for RTN and 11 975.6 for the STN). The new RTN-adapted methodology versus the classical methodology, presented a performance improvement of 78%. The same happens with the new STN-adapted, with a performance improvement of 92.1% versus the STN methodology. 6. Conclusions This paper discusses the applicability of STN, m-STN, and RTN representations to the detailed design of batch plants, where a discretization of time and a nonperiodic operation mode are assumed. The main differences identified concerned three important and related aspects: the need to explicitly consider the storage Tasks, which account for the continuous availability of material and the usage of suitable equipment; the need to consider different locations of material in the plant and consequently the definition of transfers of material with suitable equipment associated; and, finally, the instantaneous characteristic of each one of these Tasks. These representations resulted in larger, and consequently harder, models to solve in the case of the RTN and STN methodologies. As the problem size increases, the RTN methodology presents a better performance than STN. The proposed adapted methodologies presented a considerable improvement in performance by comparison with the classical


ones. As the problem size increases, the RTN-adapted methodology requires less CPU time than the STN-adapted methodology and presents a performance approach comparable to the m-STN. In conclusion, within the scope of the problem characteristics covered, the m-STN appears as the most adequate representation for the detailed design of batch plants, since it explores the problem characteristics and thus reduces both the need of employing auxiliary instances in the representation and the size of the associated mathematical formulation. Therefore, the choice of an adequate representation for the solution of a given problem should, as much as possible, explore the problem intrinsic characteristics. However, it is important to note that the work presented should be further explored and more examples should be solved so as to more firmly confirm this conclusion. Also, other problem characteristics such as setups dependency, cleaning needs, among others, should also be studied. Nomenclature for the Three Models STN-Adapted Formulation Sets S ) {s: set of all input/output States to Task i} Ki ) {j: set of processing units j suitable for Task i} Tsout/Tsin ) {i: set of Tasks producing/receiving material to State s} SP/SF ) {s: set of product/feed States} Ks ) {j: set of dedicated storage vessels suitable for storing State s} Ij ) {i: set of Tasks which can be performed in unit j} Ωj ) {j: set of processing units j} Xj ) {k: set of k types of unit j} Parameters Fisin/Fisout ) proportion of material of State s entering/leaving Task i pislag ) lag time of States s ∈ Si entering Task i relative to its start pisproc ) processing time for Task i to produce the output State s ∈ Si relative to the Task start pi ) max{pisproc}, duration of Task i φijmax/φijmin ) maximum/minimum utilization factor of Task i in unit j OCij0/OCij1 ) fixed/variable operating cost of Task i in unit j φsj ) size factor of State s in storage unit j Vs/ps ) value/price of State s OCs ) operating cost of dedicated storage of State s CCjk0 /CCjk1 ) fixed/variable capital cost of unit j Vjmax/Vjmin ) maximum/minimum capacity of unit j CCF ) capital charge factor16 Variables Ej ) 1, if unit j is installed; 0, otherwise Ejk ) 1, if unit j of a given type k is installed; 0, otherwise Wijt ) 1, if unit j is performing Task i at time t; 0, otherwise Vj ) capacity of processing equipment j Bijt ) batch of Task i in unit j at beginning of period t Sst ) amount of material in State s at the beginning of period t Dst ) amount of material delivered from the STN State s at the beginning of period t Rst ) amount of material received from the outside into the STN State s at the beginning of period t m-STN Formulation Sets Siin/Siout ) {s: set of input/output States to eTask i} Ki ) {j: set of processing units j suitable for eTask i} Ic ) {π: set of tTasks which can be performed in connection c} Cjsink/Cjsource ) {c: set of connections to which unit j is a sink/ source}

Xc ) {k: set of k types of connection c} Tsout/Tsin ) {i: set of eTasks producing/receiving material to State s} Πsin/Πsout ) {π: set of tTasks transferring material to/from State s} SP/SF ) {s: set of product/feed States} Ks ) {j: set of dedicated storage vessels suitable for storing eState s} Ij ) {i: set of eTasks which can be performed in unit j} Ξ ) {j: set of dedicated storage vessels} Ωj ) {j: set of processing units j} Xj ) {k: set of k types of unit j} Parameters φπ ) size factor of tTask π in connection c 0 1 CCck /CCck ) fixed/variable capital cost of connection c of type k in out Fis /Fis ) proportion of material of State s entering/leaving eTask i pislag ) lag time of States s ∈ Si entering eTask i relative to its start pisproc ) processing time for eTask i to produce the output State s ∈ Si relative to the eTask start pi ) max{pisproc}, duration of eTask i φijmax/φijmin ) maximum/minimum utilization factor of eTask i in unit j OCij0/OCij1 ) fixed/ variable operating cost of eTask i in unit j φsj ) size factor of eState s in storage unit j Vs/ps ) value/price of State s OCs ) operating cost of dedicated storage of State s CCjk0 /CCjk1 ) fixed/variable capital cost of unit j Vjmax /Vjmin ) maximum/minimum capacity of unit j CCF ) capital charge factor16 Variables Ej ) 1, if unit j is installed; 0, otherwise Ejk ) 1, if unit j of a given type k is installed; 0, otherwise Eck ) 1, if connection c of type k is installed; 0, otherwise Wijt ) 1, if unit j is performing eTask i at time t; 0, otherwise Vj ) capacity of processing equipment j BTc ) capacity of connection c Bijt ) batch of eTask i in unit j at beginning of period t BTπt ) the amount of material transferred by tTask π at the beginning of period t Sst ) amount of material in eState s at the beginning of period t Dst ) amount of material delivered from the STN State s at the beginning of period t Rst ) amount of material received from the outside into the STN State s at the beginning of period t RTN-Adapted Formulation Sets C) {r: set of all material resources} Cis) {r: set of intermediate material with dedicated storage} Cr) {r: set of material resources with dedicated storage} Cp /Cf ) {r: set of final products /raw materials} D ) {r: set of all equipment resources} Dfp/Drm) {r ∈ D: set of dedicated storage vessels for final product/ raw-material} Dc/DV ) {r ∈ D: set of all connections/ storage vessels} Dp ) {r ∈ D: set of process equipment resources} TK ) {k: set of all Tasks operating in an equipment resource} TP ) {k: set of processing Tasks in an equipment resource, r} TS ) {k: set of storage Task for raw and product material in dedicated vessel} TR ) {k: set of all Tasks requiring an equipment resource, r} TV ) {k,r ∈ DV: transfer Task, k, for an intermediate storage vessel, r, which is a sink} TT ) {k: set of all transfer Tasks}

6044 Ind. Eng. Chem. Res., Vol. 47, No. 16, 2008 Parameters H ) planning horizon T ) cycle time CCF ) capital charge factor Vr/pr ) price of resource (raw material/product) Rk0/Rk1 ) fixed and variable cost coefficients φrmax ) size factor Rr ) amount of resource r available in plant retrofit calculations Rrmin/Rrmax ) resource minimum/maximum quantity at H µkrθ/νkrθ ) consumption or production of a renewable(-1,1)/ nonrenewable(-1,0) resource r, at the start/end of θ ∆rmax ) the maximum available of r CCF ) capital charge factor16 Variables Rrt ) excess of resource at t Rr0 ) level of resource provisions necessary ξkt ) batch size of Task k at time t ∆r ) amount of resource r required Vr ) capacity of resource r Nkt ) number of processing Tasks k at instant t Etr ) 1 if processing resource r is installed; 0 otherwise Ecr ) 1 if transfer resource r is installed; 0 otherwise

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ReceiVed for reView September 24, 2007 ReVised manuscript receiVed April 1, 2008 Accepted May 22, 2008 IE071281N

Design of Multipurpose Batch Plants: A Comparative Analysis

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