the optimal retrofit of multiproduct batch plants

The optimal retrofit of multiproduct batch plants

Jorge Marcelo Montagna

Facultad Regional Santa Fe, INGAR, Instituto de Desarrollo y Diseno, Universidad Tecnologica Nacional, Avellaneda 3657, 3000 Santa Fe, Argentina

Received 6 February 2003; accepted 7 February 2003

Abstract

This paper presents new alternatives in the retrofit model of multiproduct batch plants. Besides the duplication of the batch units

implemented in previous works, the model considers the inclusion of intermediate storage tanks. These tanks can be plainly added or

replace existing units that can be sold. The allocation of intermediate storage tanks is not considered in the previous retrofit works,

even if this alternative is not new in the design of multiproduct batch plants. This option allows getting more efficient and real world

solutions, although it requires working with a more complex model.

# 2003 Elsevier Science Ltd. All rights reserved.

Keywords: Retrofit; Multiproduct batch plants; Storage tanks

1. Introduction

The design of multiproduct batch plants is an

optimization problem that has been studied by several

authors in the past years (Grossmann & Sargent, 1979;

Knopf, Okos & Reklaitis, 1982; Modi & Karimi, 1989;

Ravemark, 1995; Montagna & Vecchietti, 1998 etc.).

The goal is to determine the size and number of batch

units so that they can meet production requirements in

the provided time horizon.Products produced by this kind of plant fall into the

category of chemical specialties, drugs, pharmaceutical

specialties, etc. Those are high value products with a

short life cycle. Research on new products is intensive

and it is frequent that new products replace the old ones.

As new processes are to be implemented, the batch plant

structure is changed in order to meet the new production

requirements. The retrofit problem is an optimization

problem whose objective is to obtain the optimal new

structure for a batch plant starting from the old one, so

as to maximize the benefits subject to a new demand

pattern. In general, the retrofit problem solution comes

up with a new plant configuration where useless units

are sold and new ones are allocated and adjusted to

work with old ones to configure the new process.

Multiproduct batch plants manufacture a set of

products using the same equipment operating in the

same sequence. Since products differ from one another,

each unit is shared by all products but they do not use

their total capacity for all of them. Each unit works at

full capacity only when processing the products for

which this stage is size-constraining. By the same token,

the stages do not work all the time for all products. A

stage works all the time (charging a new batch just after

finishing with the previous one) only when processing

the products for which this stage is a time-bottleneck,

defining the cycle time for that product. The idea is to

optimize the production rate for each product in the

plant, which is a function of the batch size and the cycle

time. The unit with minimum capacity limits the batch

size while the limiting cycle time is fixed by the stage

with the longest processing time.

In order to reduce the investment cost, several

alternatives are possible (Ravemark, 1995). The first

one is the introduction of parallel units out-of-phase. In

this case the cycle time is reduced if the unit has the

longest operating time. Another option is to add a

parallel unit in-phase to increase the operating capacity

of the stage. This is a useful alternative for stages that

are limiting the batch size. Finally the allocation of

intermediate storage tanks allows increasing the equip-

ment utilization (Modi & Karimi, 1989). Two objectives

are reached: the reduction of idle time by increasing theE-mail address: [email protected] (J.M. Montagna).

Computers and Chemical Engineering 27 (2003) 1277�/1290

www.elsevier.com/locate/compchemeng

0098-1354/03/$ - see front matter # 2003 Elsevier Science Ltd. All rights reserved.

doi:10.1016/S0098-1354(03)00052-8

mailto:[email protected]

working time of the stages, and the increased equipment

utilization. When an intermediate storage tank is

allocated, the original process is divided into two

subprocesses, each one presenting its own batch size

and limiting cycle time. Productivity of both subpro-

cesses must be the same to avoid accumulation in the

storage tank.

The first retrofit paper (Vaselenak, Grossmann &

Westerberg, 1987) considered the addition of batch units

to an existing plant to face an increase in the products

demands. The model was a mixed integer nonlinear

program (MINLP), whose objective was to maximize

the benefit considering the incomes from the sales of the

products minus the cost of the new units. Each product

Nomenclature

Bij batch size of product i at stage j (kg)bij transformed variable for the batch size of product i at stage jdij parameter in the expression of Tij

Fij constant for a big-M constraintF1ij constant for a big-M constraintGj maximum number of groups allowed at stage jH time horizon (h)M number of batch stages in the plantNj

OLD number of existing units in the plant at stage jNj

T total number of units at stage j considering old and new onesNTj total number of existing intermediate storage tanks in position jP number of products in the multiproduct batch plantpi net profit per unit of product i (USA dollars)Pri production rate of product i (kg/h)pri transformed variable for production rate of product iQi production of product i (kg)qi transformed variable for production of product iQi

U production demand of product i (kg)Sij size factor for product i for a batch unit at stage j (m3/kg)STij size factor for product i for an intermediate storage tank in position j (m3/kg)tij parameter in the expression of Tij

Tij operation time for product i at stage j [h]TLi limiting cycle time for product i [h]Vijkg volume of unit k at stage j, taking part in group g, for product i (m3)Vjk volume of unit k at stage j (m3)VTjk volume of the intermediate storage tank k in position j (m3)yijkg binary variable for the inclusion of unit k at stage j for product i in group gyijg binary variable for the existence of group g at stage j for product iyjk binary variable for the use of unit k at stage jytjk binary variable for the use of intermediate storage tank k in position jyttj binary variable for the existence of some storage tank in position jZj maximum number of batch units to be added at stage jSubscripts

G groupI productJ batch stage and position to allocate an intermediate storage tankK batch unit or intermediate storage tankSuperscripts

H subprocessL lower boundU upper boundGreek letters

gij parameter in the expression of Tij

F maximum ratio between two consecutive batch sizesf log F

J.M. Montagna / Computers and Chemical Engineering 27 (2003) 1277�/12901278

has an upper bound in the demand based on the sales

projection.

The model of Vaselenak et al. (1987) assumed that the

units in parallel either in-phase or out-of-phase must beused in the same way for every product. Fletcher, Hall

and Johns (1991) modified the Vaselenack’s approach

by allowing the new units to operate in different ways

for each product. Better solutions were obtained in this

last case.

Yoo, Lee, Ryu and Lee (1999) presented the most

updated complete retrofit model for multiproduct batch

plants. No condition was imposed on the use of thebatch units. The new and the old units can be used in-

phase and out-of-phase, forming groups that can be

different for each product. The model also contemplates

that the old units that are no longer used in the new

structure can be sold.

Van den Heever and Grossmann (1999) presented a

general disjunctive model for the multiperiod retrofit

problem. They used logic and disjunctive programmingto pose all considered alternatives, obtaining a great

performance for large problems. However, the available

options for each period in the model are reduced

because they worked with the problem formulation by

Fletcher et al. (1991).

No previous work has considered intermediate sto-

rage tanks for the retrofit problem. In this work, an

extension of the explicit model by Yoo et al. (1999) ispresented adding a new degree of difficulty by including

the allocation of intermediate storage tanks. There is no

doubt that the number of feasible alternatives will

increase and, therefore, the possibility of obtaining

better solutions exists. The previous assumptions were

maintained? Especially the capability for changing the

configuration of every batch unit for each product

considered in the plant and the possibility of sellingthe useless old units. Several examples were solved with

this model in order to show its effectiveness.

2. Problem definition

In a multiproduct batch plant, P products (i�/1, 2,

. . ., P) are processed. For each product demand QiU is

known, which is an upper bound on the amount to beproduced of product i. This demand must be produced

over time horizon H.

Since this is a multiproduct batch plant, every product

follows the same production sequence over M batch

stages (j�/1, 2, . . ., M) of the plant. At each stage we

have a set of NjOLD batch units. The size of unit k at

stage j is Vjk (k�/1, 2, . . ., NjOLD). Taking into account

that these units could have been added in previousretrofits, their sizes can be different.

Zj units can be added at stage j so that NjT�/Nj

OLD�/

Zj is the maximum possible number of units at stage j.

The sizes of new units Vjk, k�/NjOLD�/1, . . ., Nj

T are

obtained with the problem solution.

The NjT units of stage j can be grouped in different

ways for each product i. According to Yoo et al. (1999),we can have groups where all units in the group operate

in parallel and in-phase. The different groups at the

stage operate in parallel and out-of-phase.

The allocation of an intermediate storage tank

decouples the process into two subprocesses. Each one

operates with a different batch size and cycle time, but

maintaining the same production rate for product i, Pri,

to avoid accumulation of material in the tank. Con-sidering intermediate storage tanks requires working

with new variables. Now the batch sizes or the limiting

cycle times in each subprocess could be different if a

tank is allocated.

There are M�/1 possible locations for the storage

tank (j�/1, M�/1), where the jth location is between

batch stages j and j�/1. In this model we assume that in

some locations we can have storage tanks of differentsizes from previous retrofits. At location j there are NTj

storage tanks of dimension VTjk with k�/1, 2, . . ., NTj.

Only one tank can be added in position j with size

VTj,NTj�/1. Unlike batch stages, intermediate storage

tanks can not be grouped in different ways. Therefore,

the allocation of several tanks is not a feasible option,

except when an upper bound over its size is reached.

This last alternative can be easily added to the model,and it is not included here to simplify this presentation.

3. Problem formulation

The objective of the problem is to maximize the

annual benefit of the plant considering incomes from the

product sales plus useless unit sales minus new unitscosts:

XP

i�1

piQi�XM

j�1

XNOLDj

k�1

Rjk(1�yjk)�XM�1

j�1

�XNTj

k�1

RTjk(1�ytjk)�XM

j�1

XNTJ

k�NOLDj

�1

(Kjyjk�cjVrj

jk)

�XM�1

j�1

(KTjytj;NTj�1�ctjVTrtj

j;NTj�1) (1)

The first term of the objective function is the annual

benefit corresponding to the product sales. Variable Qi

corresponds to total production of product i. Parameter

pi is the total benefit per unit of product i.

The second and third terms correspond to theincomes from useless batch units and storage tanks,

respectively. Binary variable yjk indicates if unit k at

stage j is included in the new plant structure (yjk�/1) or

J.M. Montagna / Computers and Chemical Engineering 27 (2003) 1277�/1290 1279

not (yjk�/0). For this last case, the unit is sold at price

Rjk. In a similar way we proceed with the useless storage

tank using binary variable ytjk and sale price RTjk.

The fourth and fifth terms correspond to the invest-ment cost of the new batch units and storage tanks,

respectively. Kj, cj and rj are cost parameters for the

batch units. Kj corresponds to a fixed cost independent

of the unit size, while cj is proportional to the unit size.

Previous papers work with rj�/1. The fifth term is for

storage tanks like the fourth is for batch units, with cost

parameters KTj, ctj and rtj. The difference is that only

one storage tank can be incorporated at location j.If a storage tank exists, the number and size of the

batches up and downstream of the tank are different for

each product. Variable Bij is introduced corresponding

to the batch size of product i at stage j. The number of

batches for product i at stage j is Nij. In practice, this

variable must be integer. However, considering Nij as a

continuous variable, we do not introduce a significant

error for long production horizons.Production at each stage limits the total production

for each product, then the following equation is applied:

Qi5NijBij i�1; . . . ; P; j�1; . . . ; M (2)

In this way, the problem tries to maximize Qi to

increase the benefits and to reduce Bij to decrease the

new unit sizes.

The following constraints are applied, relating batch

sizes of consecutive stages (Ravemark, 1995):

1��

1

F�1

�yttj5

Bij

Bi;j�1

51�(F�1)yttj

i�1; . . . ; P; j�1; . . . ; M�1

(3)

where F is a parameter corresponding to the maximumdifference allowed between two consecutive batch sizes.

Binary variable yttj is used to define if a storage tank

is located at position j (yttj�/1) or not (yttj�/0). The

following constraints are applied for this situation:

yttj5XNTj�1

k�1

ytjk j�1; . . . ; M�1 (4)

yttj]ytjk

j�1; . . . ; M�1; k�1; . . . ; NTj�1(5)

If no intermediate storage tank exists, yttj is fixed at 0.

If at least one tank is located yttj is one. It is important

to note that from Eqs. (4) and (5) and using an upper

bound equal to 1, yttj can be 0 or 1 without considering

it as a binary variable.

If no intermediate storage tank exists between j andj�/1 then Bij is forced to be equal to Bij�1. Otherwise,

they are different and must satisfy the following con-

straint:

1

F5

Bij

Bi;j�1

5F i�1; . . . ; P; j�1; . . . ; M�1 (6)

Several binary variables are introduced to determine

the plant structure. Since the units can be grouped indifferent forms at each stage we use binary variable yijg.

The value of this variable is 1 if group g is generated for

product i at stage j; otherwise, the value is zero. Group g

is generated if at least one unit is assigned to it. Binary

variable yijkg is equal to one if unit k of stage j is assigned

to group g for product i, otherwise the variable is equal

to zero (Yoo et al., 1999).

Fig. 1 illustrates how units can be arranged toconform the groups. In this example, there are four

units (NjT) in a stage: two old units, the white ones

(NjOLD), and two new units, the gray ones (Zj). In this

way, up to four groups of one unit could be conformed.

There are several possible combinations of the units to

determine groups. In this example units have been

arranged to conform only two groups. Therefore,

variables yij1 and yij2 are equal to one, and yij3 and yij4

are equal to zero. Units 1 and 3 form group 1 and

operate in-phase. Units 2 and 4 conform group 2 and

operate in-phase too. Then variables yij11, yij31, yij22 and

yij42 are equal to one and all the other variables yijgk are

equal to zero. Both groups 1 and 2 operate out-of-phase.

Each unit k at stage j can be assigned at most to one

group for product i:

XGj

g�1

yijkg51

i�1; . . . ; P; j�1; . . . ; M; k�1; . . . ; NTj

(7)

Gj is the maximum number of groups allowed at stage

j. Group g exists at stage j only if at least one unit is

assigned to the group:

Fig. 1. Conforming groups at a stage j.


yijg5XNT

j

k�1

yijkg

i�1; . . . ; P; j�1; . . . ; M; g�1; . . . ; Gj

(8)

If unit k is assigned to the group, the group mustexist:

yijkg5yijg

i�1; . . . ; P; j�1; . . . ; M; g�1; . . . ; Gj; k

�1; . . . ; NTj

(9)

and the upper bound for binary variable yijg is:

yijg51 i�1; P; j�1; M; g�1; Gj (10)

From Eqs. (8)�/(10), it can be seen that binary

variable yijg can be converted to a continuous variable.

If unit k is assigned to group g at stage j for product i,the unit must exist:

yijkg5yjk

i�1; . . . ; P; j�1; . . . ; M; g�1; . . . ; Gj; k

�1; . . . ; NTj

(11)

The maximum time between two consecutive batches

in subprocess h must be considered to determine the

limiting cycle time for h. This maximum time is given by

the division between operation time Tij and the number

of groups out-of-phase for product i, considering all the

stages included in subprocess h:

TLhi ]

TijXGj

g�1

yijg

i�1; . . . ; P; �h; � j � Subprocess h

(12)

where Tij is a function of the batch size, with tij,, dij and

gij fixed parameters:

Tij� tij�dijBgij

ij i�1; . . . ; P; j�1; . . . ; M (13)

We do not know a priori which units conform

subprocess h. This is a result of the mathematical

program and depends on the location of the storage

tank. The expression Eq. (12) can be modified with the

assumption that in every subprocess, production rate for

product i, Pri, must be the same:

Pri�Bh

i

TLhi

i�1; . . . ; P; �h (14)

Considering expression Eq. (3) where the value of thebatch size for subprocess h (Bi

h) can be adjusted, using

Eq. (13) and replacing TLih from Eq. (14) into Eq. (12),

we have:

(tij � dijBgij

ij )Pri

Bij

5XGj

g�1

yijg

i�1; . . . ; P; j�1; . . . ; M

(15)

The plant production is limited to time horizon H,

then

XP

i�1

Qi

Pri

5H (16)

Batch size Bij is determined by the lowest capacity

between the groups for product i at stage j, then thefollowing constraint can be applied to get the Bij value,

with Sij the size factor for product i at stage j:

SijBij5XNT

j

k�1

Vjkyijkg�(1�yijg)Fij

i�1; . . . ; P; j�1; . . . ; M; g�1; . . . ; Gj

(17)

Constraint Eq. (17) is a Big-M type that guarantees

that batches can be processed if group g exists, otherwise

the constraint is redundant because of the large value of

Fij. The value of Fij can be calculated by (Yoo et al.,

1999):

Fij�XNOLD

j

k�1

Vjk�ZjVUj

i�1; . . . ; P; j�1; . . . ; M

(18)

where VjU is the upper bound of the batch units to be

added at stage j.

Constraint Eq. (17) is nonlinear in the binary variableyijkg, which adds difficulty to the convergence. To

overcome this problem Yoo et al. (1999) replaced the

product Vjkyijkg by the new continuous variable Vijkg,

then Eq. (17) is replaced by the following constraints set:

SijBij5XNT

j

k�1

Vijkg�(1�yijg)

�XNOLDj

k�1

Vjk�ZjVUj

�

i�1; . . . ; M; g�1; . . . ; Gj

(19)

Vijkg5Vjk

i�1; . . . ; P; j�1; . . . ; M; k�1; . . . ; NTj ; j

�1; . . . ; Gj

(20)

Vijkg5VUj yijkg

i�1; . . . ; P; j�1; . . . ; M; k�1; . . . ; NTj ; j

�1; . . . ; Gj

(21)

where VjU is equal to Vjk for k�/1, . . ., Nj

OLD.If the unit is selected, then the following constraints

are used to guarantee that the unit size is between its

bounds:


Vjk5VUj yjk

j�1; . . . ; M; k�NOLDj �1; . . . ; NT

j

(22)

Vjk]VLj yjk

j�1; . . . ; M; k�NOLDj �1; . . . ; NT

j

(23)

where VjU and Vj

L are the upper and lower bounds for

the new units at stage j.

We use the following expression adapted from Modi

and Karimi (1989) to determine the size of the inter-

mediate storage tank:

XNTj�1

k�1

VTjk]STij(Bij�Bi;j�1)�F1ij(yttj�1)

i�1; . . . ; P; j�1; . . . ; M�1

(24)

This constraint also belongs to a Big-M type and it

takes into account all intermediate storage tanks locatedat that position. STij are the size factors for the tanks for

product i at stage j. The value of F1ij can be calculated

by:

F1ij�STij

�1

Sij

�XNOLDj

k�1

Vjk�ZjVUj

��

1

Si;j�1

��XNOLD

j�1

k�1

Vj�1;k�Zj�1VUj�1

��

i�1; . . . ; P; j�1; . . . ; M�1

(25)

We must also consider the bounds on the tanks to be

added by the following constraints:

VTjk5VTUjkytjk

j�1; . . . ; M�1; k�NTj�1(26)

VTjk]VTLjkytjk

j�1; . . . ; M�1; k�NTj�1(27)

where VTjkU and VTjk

L are the upper and lower bounds,

respectively, that must be considered if the tanks exist.

Redundant assignation to a group with the samevalue for the objective function are avoided by the

following constraint (Yoo et al., 1999):

XNTj

k�1

2NTj�kyijkg]

XNTj

k�1

2NTj�kyijk;g�1

i�1; . . . ; P; j�1; . . . ; M; g�1; . . . ; Gj

(28)

This constraint order the different groups. For

example, if there are four units at a stage, one solution

is to conform group 1 by units 1 and 2 and group 2 by

units 3 and 4. However, the same solution is attainedassigning units 3 and 4 to group 1 and units 1 and 2 to

group 2. This constraint avoids these assignments by

ordering the group through a weight 2NTj�k assigned to

each unit k. The order of the group is obtained by

adding the weights of all units in the group.

Finally, we must add bound constraints for the

demand. The lower bound on the demand for producti is set when the production is already sold.

Qi5QUi i�1; . . . ; P (29)

Qi]QLi i�1; . . . ; P (30)

4. Problem convexification

The model for the retrofit of a multiproduct batch

plant is defined by maximizing Eq. (1) subject toconstraints Eqs. (2)�/(5), (7)�/(11), (15), (16), (19)�/(24),

(26)�/(30). This is a MINLP problem that presents non

convex terms in several constraints and in the objective

function. Thus, it is not possible to assure that the global

optimum can be reached. In a similar way as Vaselenak

et al. (1987), the following transformations are intro-

duced to avoid non-convex terms:

bij� ln Bij i�1; . . . ; P; j�1; . . . ; M

nij� ln Nij i�1; . . . ; P; j�1; . . . ; M

pri� ln Pri i�1; . . . ; P

qi� ln Qi i�1; . . . ; P (31)

Using these transformations, the objective functionresults:

Min�XP

i�1

piexp(qi)�XM

j�1

XNOLDj

k�1

Rjk(1�yjk)�XM�1

j�1

�XNTj

k�1

RTjk(1�ytjk)�XM

j�1

XNTj

k�NOLDj �1

(Kjyjk�cjVjk)

�XM�1

j�1

(KTjytj;NTj�1�ctjVTj;NTj�1) (32)

where we have considered the minimization of Eq. (1)

with a change of sign. Also, constraints Eqs. (2), (3),

(15), (16), (19), (24), (29) and (30) are modified to obtainthe final formulation:

qi5nij�bij i�1; . . . ; P; j�1; . . . ; M (33)

bij�bi;j�15fyttj

i�1; . . . ; P; j�1; . . . ; M�1(34)

bij�bi;j�1]�fyttj

i�1; . . . ; P; j�1; . . . ; M�1(35)

where f�/ln F.


tij exp(pri�bij)�dij exp[(gij�1)bij�pri]5XGj

g�1

yijg

i�1; . . . ; P; j�1; . . . ; M

(36)

XP

i�1

exp(qi�pri)5H (37)

qi5 ln QUi i�1; . . . ; P (38)

qi] ln QLi i�1; . . . ; P (39)

exp(bij)5Bij i�1; . . . ; P; j�1; . . . ; M (40)

Following Vaselenak et al. (1987), constraint Eq. (40)

is added to reduce the number of nonlinear constraints.Working with bij, constraints Eqs. (19) and (24) should

be non-linear. Restriction Eq. (40) is satisfied trivially at

the optimal solution because by minimizing Eq. (32), qi

assume the greatest possible value that is limited by Bij.

However, this model presents difficulties. The first

term in the objective function is concave. To overcome

this problem, Vaselenak et al. (1987) have proved that

the linearization of the negative exponential functions inthe first term of Eq. (32) can be approximated by a

system of piecewise linear underestimators. This approx-

imation overestimates the objective function so that it

can be employed to find the global solution of this

model.

5. Model resolution

The final model minimizes Eq. (32) subject to

constraints Eqs. (33)�/(40), (4), (5), (7)�/(11), (17),(19)�/(24), (26)�/(28). This MINLP problem is solved

using the algorithm of Duran and Grossmann (1986),

afterwards completed by Viswanathan and Grossmann

(1990) with the OA/ER/AP algorithm and implemented

in DICOPT�/�/.

6. Examples

All the examples from previous papers have been

solved with this model. In all cases the inclusion of theintermediate storage tanks depends on the cost and size

factors of those units. Here appropriate values have

been selected values to show the potential applications

of this approach.

The main objective of this section is to show the

possibility of obtaining better solutions, taking into

account that the number of feasible solutions has been

increased. However, it is difficult to compare thesolution obtained with previous approaches since the

optimal solution will greatly depend on the values

chosen for the cost coefficients.

6.1. Example 1

This example has been solved by Vaselenak et al.

(1987), Fletcher et al. (1991) and Yoo et al. (1999). Table

1 presents the example data added to Example 1 of Yooet al. (1999). Table 2 shows the results obtained using

the model by Yoo et al. (1999) without intermediate

storage tanks and the results with the model presented in

this paper including tanks. In the first case, one unit is

added at stage 1 operating in-phase for product A and

out-of-phase for product B. In the last case, no parallel

units are in the plant but one intermediate storage tank

is located between both stages. The objective function isreduced by 1.0%. This reduction depends on the costs

proposed for the units. The whole demand is produced

for both products. Cost reduction comes from the

difference in cost between parallel unit 1 ($74 800) of

the first approach against the cost of the intermediate

storage tank ($43 300) with the new approach. Note that

the improvement in the objective function strongly

depends on the relationship between products incomesand units costs.

6.2. Example 2

Vaselenak et al. (1987), Fletcher et al. (1991) and Yoo

et al. (1999) have solved this example. Table 3 presents

the data about the intermediate storage tank to be added

to the original data. Table 4 shows the results of both

approaches. In the stages with two units, it shows the

operating policy: in-phase or out-of-phase. In thisproblem, the production targets are not achieved in

the optimal solution with the previous approach (pro-

duct D). However, with the new model all the demands

Table 1

Example 1 data added to Yoo et al. (1999)

STij KTj ctj

Product A Product B

Position 1 1 1 10 000 10

Table 2

Results corresponding to Example 1

Yoo et al. (1999) This approach

Product A Product B Product A Product B

Qi (kg) 1 200 000 1 000 000 1 200 000 1 000 000

New units

Stage 1 (m3) 1358 �/

Stage 2 (m3) �/ �/

Storage 1 (m3) �/ 3333

Profit ($) 3 125 236 3 156 667


are satisfied. The increase in the objective function is

only 1.2%. The optimal solution shows a new unit atstage 4, which is smaller than the unit added with the

previous approach and an intermediate storage tank

before this stage.

6.3. Example 3

Fletcher et al. (1991) and Yoo et al. (1999) have

solved this example. Table 5 shows the modified

problem data from the previous works. Table 6 com-

Table 3

Example 2 data added to previous works

STij KTj ctj

Product A Product B Product C Product D

Position 1 7.913 0.7891 0.7122 4.6730 10 1

Position 2 2.0815 0.2871 2.5889 2.3586 10 1

Position 3 5.2268 0.2744 1.6425 1.6087 10 1

Table 4



A B C D A B C D

Qi/1000 (kg) 268.2 156.0 189.7 158.1 268.2 156.0 189.7 166.1

New units

Stage 1 (m3) �/ �/

Stage 2 (m3) �/ �/

Stage 3 (m3) �/ �/

Out In Out Out Out In In Out

Stage 4 (m3) 3000 2252

In In Out Out Out In In In

Storage 3 (m3) �/ 3667

Profit ($) 521 780 528 000

Table 5

Example 3 data added to Example 3 from Yoo et al. (1999)

STij KTj ctj

Product A Product B Product C Product D

Position 1 2.4 1.8 1.95 4.15 22 0.27

Table 6



A B C D A B C D

Qi/1000 (kg) 290 300 350 140 290 300 350 140

New units

Stage 1 (m3) �/ �/

Stage 2 (m3) 1698 855

Storage 1 (m3) �/ 6440

Profit ($) 616 275 624 760


pares the results between Yoo et al. (1999) and this

approach. The solution without intermediate storage

tanks has a parallel unit at stage 2 operating in-phase for

products B and C, and out-of-phase for products A and

D. All product demands are satisfied. The cost of the

new equipment is $32 000. Considering intermediate

storage tanks, the objective function improves by1.4%. There is a new unit in parallel at stage 2, operating

in-phase for all products and an intermediate storage

tank is located between both stages. The cost reduction

in equipment is $23 500, 27% lower.

6.4. Example 4

Table 7 presents the data added to Example 4 from

Yoo et al. (1999). Table 8 shows the same solution for

both approaches. The optimal structure is obtained by

adding one unit at stage 1 and two units at stage 2. As

shown in Table 8, different arrangements of the units are

proposed for each product. In this table, units betweenparenthesis are included in the same group. The symbol

u1 refers to unit 1, and in the same way for the others

units. For product A, the three units at stage 1 are

grouped, while, for product B, two groups are gener-

ated, one with units 1 and 3 and the other with unit 2. In

the same way, two groups are held at stage 2 for product

A, and only one group with four units for product B.

6.5. Example 5

The last example presented by Yoo et al. (1999) is

solved. The input data added to the previous model and

results are listed in Tables 9 and 10, respectively. In this

Table 7


STij KTj ctj

Product A Product B

Position 1 4 2 0 100

Table 8


Yoo et al. (1999) and this approach

Product A Product B

Qi/1000 (kg) 2000 4000

New units

Stage 1 (m3) 2000

(u1, u2, u3) (u1, u3)�/(u2)

Stage 2 (m3) 1500�/2

(u1, u3)�/(u2, u4) (u1, u2, u3, u4)

Storage 1 (m3) �/

Profit ($) 5 300 000

Table 9


STij KTj ctj

Product A Product B

Position 1 4 1 1000 1

Table 10

Results Example 5


Product A Product B Product A Product B

Qi/1000 (kg) 2000 4000 2000 4000

New units

Stage 1 (m3) 1000 �/

(u2, u3) (u2)�/(u3) (u2) (u2)

Stage 2 (m3) �/ �/

(u2) (u2) (u2) (u2)

Storage 1 (m3) 4800

Sold units

Stage 1 (m3) u1 u1

Stage 2 (m3) u1, u3 u1, u3

Profit ($) 752 000 758 200

Fig. 2. The optimal structure for Example 5 by the Yoo et al. (1999)

formulation.


example and in both formulations, old units are sold.

Figs. 2 and 3 show the final plant structure for both

formulations, where gray units correspond to addedequipment to the original structure. Each unit includes

its number and its capacity. Fig. 2 shows different

configurations for both products: for product A units 2

and 3 operate in-phase, and, for product B, they operate

out-of-phase. Unit 3 at stage 1 is added in the retrofit of

the plant. Overlapped units correspond to in-phase

operation. Fig. 3 corresponds to the solution of the

proposed formulation, where a new intermediate storagetank is added between stages 1 and 2. The same

structure for both products has been obtained. Though

both solutions have a small difference in the optimal

objective function values (that depends on the cost

coefficients of the added tanks), the found structures

look different.

7. Computational performance

Table 11 shows different information about the

resolution of the examples. The first column corre-

sponds to the number of binary variables of the model

and the following two columns present the number of

total variables and constraints. All these values areconsidered before piecewise linearization of the objective

function. Columns 4 and 5 present the elapsed time of

two assessed alternatives. All the times were obtained

with a Intel Celeron 650 MHz processor. In the first

option, the constraints presented until now were in-

cluded. In the last option, the following new constraintswere added to reduce the number of options to be

considered in the resolution.

If a unit is incorporated for one product, it is available

for the other products without increasing the cost of the

solution. Therefore, the following constraint is posed:

XGj

g�1

XNTj

k�1

yi�j;k;g�XGj

g�1

XNTj

k�1

yi��j;k;g

� i�; i��1; . . . ; P; i�" i��; j�1; . . . ; M

(41)

The following constraint determines that if unit k at

stage j exists, it must be used at least in one group forone product:

yjk5XP

i�1

XGj

g�1

yijkg

� j�1; . . . ; M; k�1; . . . ; NTj

(42)

If unit j is allocated at stage k, it can be included in

only one group:

XGj

g�1

yijkg5yjk

� i�1; . . . ; P; j�1; . . . ; M; k�1; . . . ; NTj

(43)

Groups must be generated following an order:

yij;g�15yijk

� i�1; . . . ; P; j�1; . . . ; M; g�1; . . . ; Gj

(44)

For each product at stage j, one unit at least must be

allocated in one group:

XNTj

k�1

XGj

g�1

yijkg]1 � i�1; . . . ; P; j�1; . . . ; M (45)

Table 11 shows the reduction in the elapsed time after

adding constraints Eqs. (41)�/(45).

In order to compare this approach with the previouswork by Yoo et al. (1999), Table 12 is included. It shows

information about the same examples of Table 11, but

now solved using the previous formulation. All exam-

Fig. 3. The optimal structure for Example 5 by the proposed

formulation.

Table 11

Computational performance of this approach

Number of binary variables Total number of variables Number of constraints Option 1 CPU time (s) Option 2 CPU time (s)

Example 1 37 118 265 15 3

Example 2 87 294 724 112 60

Example 3 73 216 511 818 21

Example 4 51 148 330 247 97

Example 5 101 260 575 1331 144


ples also included constraints Eqs. (41)�/(45). Table 12shows reduced times in respect to Table 11, which is a

logical conclusion taking into account that in the first

table more alternatives must be assessed.

8. Impact of unit costs

Previous examples were selected because they were

considered in the literature on this area. However, all of

them have a common characteristic: incomes from

product sales are greater than unit costs. As a conse-

quence, products demands are fulfilled. Therefore, if inboth approaches the total demand is covered, the

difference in the objective function is only due to the

reduction in the cost of the new units, a small percentage

of the total objective function. In a new example, the

impact of storage tanks is considered in several scenar-

ios.

Table 13 presents the problem data of Example 6. The

plant has four stages to produce two products. At all

stages there is only one unit, except at stage 3 that has

two units. We considered two cases for the cost of the

equipment to be added. In the first one, the tanks and

units costs are cheaper than in the second one.

Table 14 presents results for the first case. Two units

at stages 1 and 4 are added in the option without storage

tanks. A storage tank between stages 2 and 3 and a new

unit at stage 1 are allocated in the solution considering

intermediate storage. The demands of both products are

satisfied for both options. Allowing storage tanks

improves equipment costs by 34%. However, in this

case, as we are considering low units costs, the total

objective function only improves by 1%. Figs. 4 and 5

show the optimal structure of the plant for both options.

In the second case, units and storage tanks costs are

more expensive than in case 1. Table 14 also shows the

results for this case. In the solution without intermediate

storage tanks, no unit has been added (Fig. 6). The cost

of the new units is so expensive that no unit has been

incorporated. Production of A (which has the lower

benefit) is reduced and its demand is not fulfilled.

When intermediate storage is allowed, all demands

are satisfied (Fig. 7). We have one unit in parallel at

stage 1 operating in different ways for each product: in-

phase for product A and out-of-phase for product B. An

intermediate storage tank is allocated between stages 2

and 3. In this way, production capacity is increased at

stage 1 for product A operating both units in-phase and

reducing the limiting cycle time for product B. The

storage tank uncouples the original process into two

subprocesses reducing the batch sizes for stage stages 3

and 4 and allowing a higher productivity for both

products. The added equipment cost is $227 900, that

is justified by the increase of $294 800 in the value of

production A. Allocating a tank between stages 2 and 3

improves the total benefits by 8%. In this particular

example, allowance of intermediate storage tanks had a

really important impact.

Table 12

Computational performance of Yoo et al. (1999) approach

Number of binary variables Total number of variables Number of constraints Option 2 CPU time (s)

Example 1 36 109 250 3

Example 2 84 253 647 44

Example 3 72 205 488 4

Example 4 50 141 317 12

Example 5 100 253 562 33

Table 13

Data for Example 6

Stage 1 Stage 2 Stage 3 Stage 4

Product /Tij

A 6.3822 4.7393 8.3353 3.9443

B 6.7938 6.4175 6.4750 5.4382

Sij

A 7.913 2.0815 5.2268 4.9523

B 0.7891 0.2871 0.2744 3.3951

STij

A 2 1 1

B 1 1 1

VjOLD (m3) 4000 4000 3000�/2 3000

VjL (m3) 1000 1000 1000 1000

VjU (m3) 4000 4000 3000 3000

Case 1

Kj 6000 14 000 18 000 4000

cj 4.5 12 15 4

KTj 2000 2000 2000

ctj 2 2 2

Case 2

Kj 60 000 140 000 180 000 40 000

cj 45 120 150 40

KTj 20 000 20 000 20 000

ctj 20 20 20

Product pi ($/kg) QiUP (kg)

A 1.1 500 000

B 1.5 400 000


9. Conclusions

A new model is presented for the retrofit of multi-

product batch plants. The main difference between this

model and the previous ones is that it admits inter-

mediate storage tanks. Although this option is usually

taken into account in the design problem, it is not used

in the retrofit problem. We consider the possibility of

having storage tanks in the old plant, which can be used

in the new one or not. A new storage tank can be

allocated in each position. The model also considers the

capacity of having parallel units in-phase and out-of-

phase as in the previous approaches. The availability of

new alternatives allows meeting product demands better

than previous methods, as is shown in the last con-

sidered example.

The proposed model improved the previous solution

obtained by the other authors in the analyzed examples.

DICOPT�/�/ performance was good to reach the solution

for all solved examples.

Table 14

Results for Example 6

Option 1 Option 2

Solution without intermediate storage tanks

Product A B A B

Qi (kg) 500 000 400 000 232 000 400 000

Profit ($) 1 115 100 854 300

Stage 1 2 3 4 1 2 3 4

New units (m3) 3838 �/ �/ 1905 �/ �/ �/ �/

Solution considering intermediate storage tanks

Product A B A B

Qi (kg) 500 000 400 000 500 000 40 000

Profit ($) 1 127 200 922 100

Stage 1 2 3 4 1 2 3 4

New units (m3) 2327 �/ �/ �/ 2327 �/ �/ �/

Position 1 2 3 1 2 3

Storage (m3) �/ 2161 �/ �/ 2161 �/

Fig. 4. Option 1 of Example 6. Solution without intermediate storage tanks.


Fig. 5. Option 1 of Example 6. Solution with intermediate storage tanks.

Fig. 6. Option 2 of Example 6. Solution without intermediate storage tanks.


Acknowledgements

The author would like to acknowledge financial

support received from Foundation VITAE within the

Cooperation Program among Argentina�/Brazil�/Chile

under the grant Project B-11487/10B006, and from

CONICET under grant PIP 4802.

References

Duran, M. A., & Grossmann, I. E. (1986). An outer-approximation

algorithm for a class of mixed-integer nonlinear programs.

Mathematical Progress 36 , 307.

Fletcher, R., Hall, J. A., & Johns, W. R. (1991). Flexible retrofit design

of multiproduct batch plants. Computers and Chemical Engineering

12 , 843.

Grossmann, I. E., & Sargent, R. W. H. (1979). Optimal design of

multipurpose chemical plants. Industrial Engineering and Chemical

Process Design and Development 18 , 343.

Knopf, F. C., Okos, M. R., & Reklaitis, G. V. (1982). Optimal design

of batch/semicontinuous processes. Industrial Engineering and

Chemical Process Design and Development 21 , 79.

Modi, A. K., & Karimi, I. A. (1989). Design of multiproduct batch

processes with finite intermediate storage. Computers and Chemical

Engineering 13 , 127.

Montagna, J. M., & Vecchietti, A. R. (1998). Alternatives in the

optimal allocation of intermediate storage tank in multiproduct

batch plants. Computers and Chemical Engineering 22 , S801�/S804.

Ravemark, D. (1995). Optimization models for design and operation of

chemical batch processes. Ph.D. thesis, Swiss Federal Institute of

Technology, Zurich.

Van den Heever, S. A., & Grossmann, I. E. (1999). Disjunctive

multiperiod optimization methods for design and planning of

chemical process systems. Computers and Chemical Engineering 23 ,

1075.

Vaselenak, J. A., Grossmann, I. E., & Westerberg, A. W. (1987).

Optimal retrofit design of multipurpose batch plants. Industrial

Engineering and Chemical Research 26 , 718.

Viswanathan, J. V., & Grossmann, I. E. (1990). A combined penalty

function and outer approximation method for MINLP optimiza-

tion. Computers and Chemical Engineering 14 , 769.

Yoo, D. J., Lee, H., Ryu, J., & Lee, I. (1999). Generalized retrofit

design of multiproduct batch plants. Computers and Chemical

Engineering 23 , 683.

Fig. 7. Option 2 of Example 6. Solution with intermediate storage tanks.


the optimal retrofit of multiproduct batch plants

Documents