Scheduling of Multipurpose Batch Chemical Plants. 1. Formation of

Znd. Eng. Chem. Res. 1991,30,671-688

67 1

Scheduling of Multipurpose Batch Chemical Plants. 1. Formation of Single-Product Campaigns Michael C.Wellons and G i n t a r a s V. h k l a i t i s * School of Chemical Engineering, Purdue University, West Lafayette, Indiana 47907

A mixed integer nonlinear programming (MINLP) formulation for the single-product campaign formation problem is presented. The formulation possesses considerable degeneracy among the feasible integer solutions and can have a large integrality gap, so that direct solution of the formulation with DICOPT++ requires considerable computation time. The campaign formation decomposition algorithm is presented which alternately solves an equipment group master problem, which provides an upper bound on the processing rate and determines an optimal equipment group profile, and a campaign formation subproblem, which provides a lower bound on the optimal processing rate and determines an equipment assignment and production line schedule for a given equipment group profile. An order of magnitude reduction in computation time is achieved with the campaign formation decomposition algorithm as compared with direct solution of the original MINLP. Introduction A multipurpose batch chemical plant consists of a wide variety of general-purpose equipment items that can be used for batch and semicontinuous processes. In addition, the plant has a flexible piping network that allows the plant equipment items to be arranged into a combinatorially large number of different processing configurations. This operational flexibility provides the opportunity to produce in the same plant many low-volume/ high-value chemical products for which dedicated single-product facilities would not be economically attractive. Production is typically accomplished in a series of campaigns, each campaign consisting of one or more independently operated single-product production lines. At the end of each campaign, the plant equipment items are cleaned out and reconfigured for the next campaign. The multipurpose plant scheduling problem can be decomposed into three subproblems (Mauderli and Rippin, 1979) as shown in Figure 1: production planning using some set of alternative campaigns, the generation of the set of alternative campaigns from an existing set of equipment items, and the scheduling of the single-product production lines of which the campaigns are composed. In previous work we have addressed the production line scheduling problem (Wellons and Reklaitis, 1989a-c). In this paper the formation of single-product campaigns is considered, while in the companion paper (Wellons and Reklaitis, 1991) multiple-product campaign formation and production planning is addressed. In the following the single-product campaign formation problem is defined and a MINLP formulation is presented. The difficulties encountered in solving this formulation are discussed, a decomposition strategy to solve the problem is presented, and the effectiveness of the solution procedure is illustrated with an example.

The Single-Product Campaign Formation Problem The goal of the campaign formation problem is to identify a set of good or dominant campaigns to which the production planning procedure will allot the available production time. If the processing rate of the production line is used as the selection criterion, the best singleproduct campaign will be the arrangement of plant equipment items that yields the highest processing rate. If several products are to be produced simultaneously, the set of dominant campaigns consists of those campaigns whose linear combination yields the highest total output rate for any product ratio (Mauderli and Rippin, 1979).

Maximization of the output or production rate is an appropriate measure of the efficiency of a multipurpose plant under the assumptions that production will occur in long campaigns, that material and utility consumption per unit of product is independent of the specific assignment of equipment items to production lines, and that processing equipment is the principal limiting resource. The identification of the single-product campaign with the highest processing rate necessarily requires that scheduling of the resulting production line be incorporated into the campaign generation procedure. A production line is assumed to consist of M stages, where the equipment items assigned to each stage are arranged into NGj equipment groups as shown in Figure 2. These groups are operated in parallel and out of phase, so that each batch is processed by only one equipment group on each stage. Each equipment group can consist of one or more units, and the batch is divided among these units for processing. It is assumed that the groups of a given stage j each have a maximum number of units NU?=. The equipment groups on each stage are used in a repeated cycle, each group used once in each cycle in a fixed order. As a result, batches are produced sequentially on NP different paths of the production line, where NP = LCMj(NGj) and LCM denotes the least common multiple of the set of integers NGj. (Wellons and Reklaitis, 1989a). Longer term production is achieved by repeating this sequence of batches. In the example of Figure 2, showing two stages of a production line, there are two groups in stage j and three groups in stage j + 1. The groups of stage j contain at most three units while the groups of stage j + 1contain at most two units per group. Given tlfis structure, a batch arriving at stage j has LCM(2,3) or 6 possible paths it can take. For instance, if it follows the path numbered 1, it will use the first group of stage j and the first group of stage j + 1. When the batch arrives at the first group of stage j , it will be split among the three units of that group. Each of these three units will process its fraction of the batch at the same time and in parallel, and at the conclusion of processing, the batch will be recombined for transfer to the next stage. Assuming cyclic operation, the next batch to arrive will use the second group of stage j while the third will use the first group and so forth. Similarly, if the first batch to arrive at stage j + 1 will use the first group, the second batch group 2, and the third group 3, then the fourth batch will use group 1 and so forth. Note that although the number of paths per repeating cycle is fixed, the specific assignment of groups to paths and the sequencing of paths

0888-5885/91/2630-0671$02.50/00 1991 American Chemical Society

672 Ind. Eng. Chem. Res., Vol. 30, No. 4,1991

I

Production

I

W

Figure 1. Decomposition of multipurpose batch plant scheduling problem. position 1

Group2

l ;%- - +64>2

Group 3

Stage J NG, = 2 man

NU, = 3

Stage

I+ 1

NG1.,=3 max

NU

=2

Figure 2. Example production line structure.

are matters of choice. Furthermore, note that since the number and sizes of the units in a group may vary, the capacity of the groups in a stage may vary and thus the allowable batch size could change depending upon the path that a batch uses. For instance, the batch that follows path 4 may have a smaller batch size than that which follows path 1. Thus, previous work has shown that equipment group sequencing and variable-path batch sizes can be important aspects of the single-product production line scheduling problem (Wellons and Reklaitis, 1989a). As a result, a procedure to identify the optimal single-product campaign should address these aspects of the scheduling problem. The single-product campaign formation problem can be defined as follows: Given (1)the volume V, and number of equipment items NE, for each equipment type e, e = 1, ...,E, (2) the set Ej of equipment types that are feasible for stage j of the production line; (3) bounds NG?" and NU?" on the maximum number of equipment groups and the maximum number of units per equipment group, respectively, for

each stage; (4) the size factor S,,.for each stage of the production line; and ( 5 ) the relationship describing the stage processing time and the overlapping time between stages as a function of the batch size. Determine (1)the assignment of equipment items to stages of the production line and arrangement of those units into equipment groups; (2) the equipment group sequencing for each stage; (3) the path batch sizes; and (4) a schedule for the production line, so as to maximize the processing rate of the production line. In the work reported in this paper, the following assumptions are made: 1. The groups on each stage are operated in a cyclic fashion with each group used once in each cycle in a fixed order. 2. Each equipment item is assigned to only one specific stage and group in the production line. Thus, there is no reuse of equipment items for different purposes within the same production line. 3. The identity of a batch is preserved: there is no batch aggregation and there is a characteristic batch size associated with each path. 4. No intermediate storage is available. A batch may either be held in a unit while awaiting a downstream unit, no intermediate storage (NIS), or may require immediate processing on the next stage (zero wait). Only Mauderli and Rippin (1979) and Lazaro and Puigjaner (1985) have addressed the problem of generating and evaluating alternative campaigns from an existing set of equipment items. Mauderli and Rippin employ evolutionary enumerative techniques to generate alternative single-product production lines. They first enumerate all possible equipment arrangements with only one equipment group on each stage. After eliminating inefficient arrangements, these setups are then combined to form all possible single-product production lines. Heuristic rules are used to control the number of alternative production lines that are generated. In determining the processing rate of these production lines, they assume arbitrary sequencing of the equipment groups on each stage, zero-wait transfer between stages, and production of all batches at the path capacity. The single-product production lines are then combined in an enumerative procedure to form alternative single- and multiple-product campaigns. A subset of dominant campaigns is selected from these alternatives by use of a linear programming (LP)based screening procedure that operates on the convex hull of the alternative campaigns. Lazaro and Puigjaner use an exhaustive enumeration procedure to generate alternative single-product campaigns, allowing only in-phase operation of the equipment items on each stage (only one equipment group). Dominant production lines are chosen on the basis of a heuristic selection index that incorporates the production rate, the equipment utilization fraction, and the production cost per unit of final product. Since they only consider production lines with one equipment group per stage, no equipment group sequencing is necessary. They assume that all batches produced on the production line are the same size.

Problem Formulation The single-product production line scheduling formulation developed previously (Wellons and Reklaitis, 1989a) can be augmented to include the assignment of the plant equipment items to potential equipment groups on each stage of the production line. To accomplish this, two classes of binary variables must be used, one to represent the equipment assignment and another for the sequencing of the resulting equipment groups. The assignment process

Ind. Eng. Chem. Res., Vol. 30, No. 4, 1991 673 Equipment

Equipment Type

Equipment Group

Position

Equipment Group

Position

to each available position. Constraint 2 is written as an equality for g = b = 1,j = 1, ...,M to ensure that at least one unit is assigned to each stage of the production line. Constraint 3 requires the number of units of type e assigned to all the positions of the production line to be no more than NE,. Because NGT" and NUT" are upper bounds on the number of equipment groups and units per group, respectively, many of the potential positions on the production line will remain vacant. Constraints 4 and 5 reduce the number of degenerate solutions due to these vacancies.

b = 1, ..., NU?" - 1 (4) Figure 3. Diagram illustrating assignment of unita to equipment groups to positions (upper) and the equivalent unwound assignment (lower).

is shown in Figure 3 (upper), where the units are first assigned to a particular equipment group and then that group is assigned to a position in the sequence. Since the label denoting a particular equipment group (e.g., no. 1 or no. 2) is arbitrary, the equipment group sequencing part of the graph can be unwound by relabeling the equipment groups. The resulting equivalent diagram is shown in Figure 3 (lower) in which the equipment groups are sequenced in numerical order. Consequently, for any feasible assignment of equipment items there exists an equivalent assignment in which the equipment groups are sequenced in numerical order. Thus by incorporating numerical sequencing of the equipment groups through logical conditions in the problem formulation, only the equipment assignment variables are necessary to model both the equipment assignment and equipment group sequencing. If the equipment assignment and production line scheduling problems are to be solved simultaneously, the scheduling model must reflect the fact that the number of equipment groups and the number of units in each group at each stage are unknown. As a consequence, the number of paths in the sequence is also unknown. To ensure that all possible path sequences are feasible, the problem formulation must allow for NP"" paths, where NP- = max LCM(NHj) 1 I NGj IN G Y J If the optimal path sequence has fewer than NP- paths, the remaining paths in the sequence must have a path batch size of zero. The objective of the optimization problem is to maximize the processing rate of the production line. 1 NPmax - c Bk (1)

T

k=l

where Bkis the batch size for path k and T i s the sequence cycle time. The equipment assignment is denoted by the binary variable Xej b, which is 1 if equipment type e is assigned to position 1 of equipment group g on stage j and 0 otherwise. The equipment assignment constraints are as f0110w8: C Xejfb 1 V i,g, b (2) eEE,

M NG,W NU,"

C C

j-1

g=l

C Xejg, I NE,

b=1

V e

(3)

Constraint 2 ensures that at most one unit will be assigned

Constraint 4 requires the units that are assigned to a group be ordered by decreasing capacity and fill the positions in indicial order. Thus, if both the b + 1 and b units are selected for group g in stage j , then in (4) both sums will be nonzero and the constraint will require the bth unit to have larger capacity. Constraint 5 forces the equipment groups to fill the positions in indicial order. Note that the equipment groups cannot be ordered by decreasing total volume as this would make some of the equipment group sequences infeasible. Although the equipment assignment can be adequately described by the binary variables xejgb, computational experience has shown that addition of the intermediate binary variables GjByields a factor of 2 or 3 reduction in the computation time. Gjg.= 1 if there are exactly g equipment groups on stage J and is zero otherwise. The following constraints define Gjgin terms of the equipment assignment variables Xej&

NGj

(7) Note that, by virtue of constraints 4 and 5, in this definition only the first equipment item ( b = 1) in the group needs to be considered. If stage j has g groups, then each of those groups clearly must contain the first equipment item. To verify the definition, observe that if there are exactly g groups on stage j , then the first s u m must equal 1, the second sum 0, and thus the difference will be 1. If the stage has g + 1 or more groups, then both sums take on the value 1 and, thus, their difference will be zero. Finally, if the stage has less than g groups, then both sums will take on the value 0. Variable Gj, thus serves as convenient counting variable. The existence of path k is denoted by the binary variable b k . Since the paths are assumed to be sequenced in numerical order, constraint 8 requires that if the path sequence consists of k paths, they must be paths l, 2,...,k. k = 1, ..., NP"" - 1 (8) b k 2 bk+l

If an equipment group consists of more than one unit, the batch is divided among the units for processing. Let Bjbkbe the bth portion of the path k batch processed on stage j . Since the equipment groups are used by the paths in indicial order, logical conditions can be imposed to determine the split batch sizes as a function of the equipment assignment. Let Pjgg* denote the set of paths

674 Ind. Eng. Chem. Res., Vol. 30, No. 4, 1991

that use equipment group g of stage j if there are exactly g* equipment groups on stage j . The split batch size is constrained by the capacity of the unit in which it is processed, so that if path k uses equipment group g on stage j , then

The condition BjMlk1 B j b k applies only if there are at least b + 1 units in equipment group g and if group g is used by path k. The processing time and overlapping times are a function of the characteristic batch size. Pjk

Since the number of equipment groups is not known a priori, the general form of the constraint is

g,g*: k E Pjgg* (9)

where Bj"" is an upper bound on the split batch size. The last term of the constraint is zero only if there are exactly g* equipment groups on stage j . Otherwise, the constraint is redundant. For any path k,the batch size processed at a given stage j is simply the sum of the split batch sizes used in each equipment item that comprises the group of stage j that is assigned to path k. Since there is no intermediate storage used between stages and since differences in the amount of material which is processed at a given stage is reflected by the size factor, we can define a path batch size Bk which is independent of subscript j as follows. NU ,-

If path k does not exist, constraint 11forces the path batch size to be zero.

Bk I B m " 6 k vk (11) The processing time of the batch on stage j is a function of the characteristic batch size Bjk,which is equal to the largest of the split batches. If the split batches are ordered by decreasing size, the characteristic batch size is equal to the size of the partial batch processed in the first unit. B,bk 2 B j b + l k v j , k b =I, ..., NU?= - 1 (12) Bjk

= Bjlk

vj,k

(13)

Let JH denote those stages where holding is allowed within the stage, i.e., the NIS policy is followed. If holding is not allowed on stage j , then the batch must be divided evenly among the units of the equipment group so that the processing time of the split batches is the same. Since the number of units assigned to a particular equipment group is not known a priori, the constraint set must be general enough to require all nonzero split batches to be equal. The equality condition can be satisfied by imposing the corresponding inequality conditions in both directions. Constraint 12 is sufficient for the 1 inequality. The 5 inequality should be imposed only if both B j b k and Bjb+lk are nonzero. Thus if path k uses equipment group g on stage j , then Bjb+lk

2

Bjbk

- BY(1- C

xejb+l)

eEE,

Since the equipment group used by path k is a function of the equipment assignment, the above condition is augmented with the same logical conditions as for the split batch size constraints. Bjb+lk

2

Bjbk

- BT"(2 - C

eEE,

j

Xejgb+l

- Gjg*)

Vk

JH b = 1, ..., NU?" - 1 g, g*: k E PjggI(14)

= P:6k

oipk = 0; 6k

+ CJ$

vj,k

+ C?i$jp

v j,k

(15)

(16)

Note that if path k does not exist ( 6 k = 0), the processing and overlapping times will be zero. A schedule for the sequence of NP" paths can be constructed by (i) determining the starting and ending times sjk and Ejkof each path of the sequence as a function of the processing and overlapping times (assuming S l k = 0) and (ii) determining the start time Ak for each path of the sequence to ensure feasible operation (Al = 0). Details of this construction can be found in Wellons and Reklaitis (1989a). It should be noted that this formulation readily accommodates semicontinuous stages with either constant processing rates or nonlinearly varying processing times. For such stages the processing time as defined above will equal the overlapping or transfer time for the preceding and following batch stages. The sequence 'cycle time T is then defined in terms of the sequence starting and sequence ending times for each equipment group, S S j , and SEjr T L SEjg - SSjg V j,g (17) Since the equipment groups are used in indicial order, if equipment group g exists, it is first used by path k = g. Thus lz = g (18)

where SSgr is an upper bound on the sequence starting time for stage j . If the equipment group does not exist, the sequence starting time is bounded above by SSgF. Since the number of equipment groups on each stage and the number of paths in the sequence is not known a priori, the sequence ending time constraint set must consider all possible path sequences. Consequently, the sequence ending time is defined by

SEj, 1 hk

+ E j k - SEZr(1 - C,p)

V j,g

k, g*: k E PjBB. (19)

where SEZr is an upper bound on the sequence ending time for stage j. If there are exactly g* equipment groups on stage j , then SEjg 1 Ah + Ejk;otherwise the constraint is redundant. Similarly, the time shift for path k Ah is constrained as Ah

1

Ejk-p

-

sjk

- A,qh"(2 - bk

-Gjp)

V j , g* g* + 1 Ik INP"" (20) where A$" is an upper bound on the start time for path k. Thus if path k exists and there are exactly g* equipment + - s j k . Otherwise the groups On Stage j , Ak 1 constraint is redundant. The single-product campaign formation problem consists of maximizing (1) subject to constraints 2-20. The resulting MINLP formulation is nonconvex because the processing and overlapping time constrainta are nonlinear equalities and the objective function is not concave (maximization). As discussed previously for the production line scheduling problem, the formulation cannot be convexified by eliminating the nonlinear equalities and/or

Ind. Eng. Chem. Res., Vol. 30,No. 4, 1991 675 Table I. Description of Plant Equipment ewbment tme no. of units volume. L reactor U1 1 2500 reactor U2 1 6300 reactior V1 1 2500 reactor V2 1 4000 reactor D 2 4000 filter S 1 300 2 2000 filter P 2 1000 dryer T 2 1000 dryer P

through exponential variable transformations. As a result, the globally optimal solution cannot be guaranteed. The number of 0-1 variables in the formulation is &IEjlNGJ-NUJ+ NPm", where lEjl is the number of equipment types feasible for stage j and the number of constraints is roughly proportional to NP". As a result, it is important that NGm" and NU"" be as small as possible to reduce the problem size and hence the solution time. Bounding procedures for NG?" and NU?", as well as Bmin,Bmu Bm" SSm", SEW, A;", Qh, and Rm" are presented in xpbendf: A. &e bounds S S F , SEEY,and A$" must be based on a path sequence of P" paths to avoid cutting off any potential path sequences. Additionally, the batch size can vary widely depending on the equipment assignment, so that the processing times used to develop these bounds can be much larger than those in the optimal solution. As a result, these bounds are usually very poor. A bound on the processing rate Rma can be determined from Bm" and Ti" as

kB"" R"" = max -

k

p

However, this bound is usually 2-4 times the optimal processing rate because of the wide range of feasible batch sizes and processing times. An improved bound for Rcan be calculated as the minimum of the maximum processing rate for each stage based on the bottleneck processing rate when all feasible units are assigned to that stage. This bound is also described in Appendix A. Table 11. Recipe Information for Product

As we have noted previously (Wellons and Reklaitis, 1989a-c), the set of feasible integer solutions to the production line scheduling problem contains significant degeneracy due to operational and rotational equivalence of the feasible equipment group sequences. This degeneracy can be removed from the problem formulation through partial sequencing and additional constrainta (Wellons and Reklaitis, 1989c),but the necessary combinatorial analysis to identify these conditions required the number of equipment groups to be known. As a result, the operational and rotational degeneracy of the equipment group sequences cannot be removed from the campaign formation problem formulation. Additional degeneracy arises from operational and structural equivalence of the individual equipment assignments. This degeneracy occurs if the assignment of two units in the production line can be interchanged without affecting the processing rate. Unfortunately, an important property of the above formulation is that the continuous relaxation of the feasible region is usually quite poor. In most cases, the batch sizes and the sequence cycle time in the NLP relaxation are at their upper and lower bounds, respectively, as determined from the bounding procedure, yielding a relaxed objective value that is several times the actual optimal processing rate. Thus relatively few of the integer variables will take integer values in the relaxed NLP or linearized LP. Although addition of the bottleneck processing rate bound will usually reduce the integrality gap substantially, this bound is usually poor unless there is a true bottleneck stage in the production line. The combination of a poor continuous relaxation and significant degeneracy can result in excessive computation tixyes to find the optimal solution. Example 1 The multipurpose batch plant analyzed by Mauderli and Rippin (1979) is considered in this example. The plant consists of 9 equipment types comprising 13 equipment items, detailed in Table I. Nine different products are produced in the plant. Tables 11-X describe the size factors, processing and overlapping time relationships, and feasible equipment types for products A J . The units used

A" overlapping time const and overlapping stage

processing time const

a

stage 1 2

size fctr Si, 2.74 0.26

Tj

Cij

dij

qjy

e.., VI'

dijy

15.00 2.00

1.723-2 6.123-6

0.865 2.000

3 4 5 6 7

5.85 1.78 1.98 1.58 1.444

16.00 0.00 0.00 1.00 2.00

3.643-2 4.283-3 4.123-2 4.283-5 3.703-2

0.823 1.OOO 0.808 2.000 1.OOO

0.40 2.00 0.00 0.17 3.36

1.223-6 0.00 4.123-2 7.283-6 7.643-3

2.000 1.000 0.804 2.000 0.823

j' 2 3 5 6 3

1.00

0.00

1.000

7

1.723-2 represents 1.72 X

feasible equip. types RD

FS all R all R RV1, RV2 FP DT

etc.

Table 111. Recipe Information for Product Bo overlapping time const and overlapping stage

processing time const stage 1

2 3 4 5 6 7 a 4.003-2

size fctr Sij 4.48 0.55 5.51 2.63 8.62 1.83 1.66 representa 4.00 X

Tj 2.00 1.00 0.00 1.00

3.00 2.00 4.00 etc.

Cij

dij

O?. 111'

eijy

dijy

j'

4.003-2 4.083-6 2.273-2 5.703-2 3.043-2 1.833-5 4.713-2

0.737 2.000 0.823 1.OOO 0.830 2.000 1.000

0.33 0.00 0.60 0.30 0.54 2.00

1.353-6 9.083-3 6.083-3 3.043-3 4.943-6 0.00

2.000 0.823 0.830 0.830 2.000 1.000

2 3 5 5 6 7

feasible equip. types RD FS RV1, RV2 all R all R FP DT

676 Ind. Eng. Chem. Res., Vol. 30,No. 4, 1991 Table IV. Reciw Information for Product Ca ~

stage 1 2 3 4 5 6 7

size fctr 4.70 0.31 4.61 3.48 7.55 1.83 1.66

S,

a4.00E-2 represents 4.00

X

~ _ _ _ _ _

processing time const cu dl1 2.00 4.003-2 0.737 1.00 4.083-6 2.000 0.00 2.273-2 0.823 0.00 1.363-2 0.823 2.50 3.263-2 0.830 2.00 1.833-5 2.000 4.00 1.713-2 1.000

e,

~~

overlapping time const and overlapping stage elfJ' dlff 1.223-6 2.000 9.083-3 0.823 6.523-3 0.823 4.083-3 0.823 4.943-6 2.000 0.00 1.000

j' 2 3 5 5 6 7

feasible equip. types RD FS RV1, RV2 all R all R FP DT

overlapping time const and overlapping stage dijy eijy 6.823-3 0.927 2.133-2 0.784 1.473-6 2.000 1.00 1.000

j' 2 3 4 5

feasible equip. types all R all R all R FP DP

overlapping time const and, overlapping stage 1 dijjt Wjj! eijy 0.50 2.043-5 2.000 0.50 0.00 1.000

j' 2 3

feasible equip. types all R FP DP

overlapping time const and overlapping stage qjj, e... w' dijy 0.00 1.713-2 0.704 0.25 4.683-5 2.000 1.00 0.00 1.000

j' 2 3 4

feasible equip. types all R RV1, RV2 FP DP

%, 0.30 0.00 0.50 0.00 0.54 2.00

etc.

Table V. Recipe Information for Product Do

stage

0.00 2.00 2.50 2.00 4.00

1.893-2 represents 1.89 X

etc.

1 2 3 4 5 a

ej

size fctr Sij 1.18 7.17 8.59 1.54 1.40

processing time const Cij

1.893-2 1.553-2 4.263-2 1.133-5 1.503-2

Table VI. Recipe Information for Product

stage 1 2 3 a 8.003-4

size fctr Sij 7.98 3.15 2.86 represents 8.00

X

ej 3.50 1.00 2.00

dij 0.768 0.927 0.184 2.000 1.000

@W ?.'

0.88 1.25 0.26 1.00

Ea

processing time const Cij

8.003-4 4.083-5 3.703-2

dij 1.487 2.000 1.000

lo4, etc.

Table VII. Recipe Information for Product Fa

stage 1 2 3 4

size fctr Sij 7.54 8.01 2.95 2.65

2.003-2 represents 2.00

X

Vj 9.00 0.00 1.00 2.00

processing time const Cij

2.003-2 1.223-1 1.873-5 2.503-2

dij 0.830 0.704 2.000 1.000

etc.

Table VIII. ReciDe Information for Product G4 ~

processing time const stage 1 2 3 4 5 a 6.403-2

size fctr Sij 2.11 2.12 3.92 1.75 1.59 represents 6.40

X

Ej

Cij

3.00 5.50 0.00 1.00 3.00

6.403-2 3.043-2 7.223-3 7.11E-6 1.603-2

dij 0.704 0.830 0.934 2.000 1.OOO

o,. 0.66 0.00 0.20 1.00

~

~~

o v e r l a p s 6 time const and overlapping stage dl,fj %It 3.653-3 0.830 4.123-3 0.934 1.423-6 2.000 0.00 1.000

~

~

~~

feasible equip. types all R

j' 2 3 4 5

RD all R FP

DT

etc.

Table IX. Recim Information for Product Ha processing time const stage 1 2

3 4 5

size fctr Si] 5.55 1.84 7.12 3.16 2.87

Ej 4.00 0.00 4.50

1.oo

2.00

Cij

3.513-2 1.383-2 7.483-2 6.253-5 3.00E-2

dij 0.768 0.830 0.145 2.000 1.000

qjj,

0.23 0.63 0.27 1.00

overlapping time const and overlapping stage eijy dijy 3.743-3 0.745 1.053-2 0.745 1.693-5 2.000 0.00 1.000

j' feasible equip. types 3 all R

3 4 5

all R all R FP DP

3.513-2 represents 3.51 X lo-*, etc.

in all of these tables are liters for volume quantities, kilograms for mass, and hours for time. As given by Mauderli, the size factors are independent of equipment type and thus vary only by stage and product. The transfer

policy between stages is zero wait. The values of the bounds NGm- and NU?- can be found in Table XVII in Appendix For products A, B, E, F, H, and J, there is at least one stage that has a bound of NGT- = 3, so that

A.

Ind. Eng. Chem. Res., Vol. 30, No. 4, 1991 677 Table X. Reciw Information for Product Jo overlapping time const and overlapping stage

processing time const stage 1

2 3 4 5 6 a

size fctr S i j 5.32 0.15 14.65 1.11 3.17 2.88

5.923-2 represents 5.92 X

Ej

Cij

dij

qjjt

e... w'

dijy

4.00 1.00 4.00 0.00 1.00 2.00

5.923-2 1.973-5 5.883-2 3.663-2 3.453-5 1.773-2

0.756 2.000 0.804 0.768 2.000 1.000

0.20 2.00 0.13 1.32 1.00

3.943-6 0.00 4.493-6 1.943-2 0.00

2.000 1.000 2.000 0.804 1.000

j' feasible equip. types 2 RD 3 FS 5 all R 3 all R 6 FP DT

lo-*, etc.

Table XI. Computational Results for Example 1 pro- problem size total CPU s processing CPU s NLP/MILP rate duct O-l/var/con 146.7 14.11132.6 A 8113041579 36.28 73.4 12.8160.6 B 8313151541 36.74 5.4126.4 31.8 C 7211671248 84.25 3.11147.6 150.7 97.06 D 81/152/200 10.2 4.216.0 E 6311781316 46.82 186.7 9.71177.0 66.01 F 5812011387 11.0 2.019.0 96.29 G 5011191177 36.4 9.4127.0 55.96 H 91/266/487 15.71911.4 65.91 J 10813231624 927.1

NPmax= 6. For products C, D, and G, no stage of the production line requires more that two equipment groups, so NPm" = 2. The nonconvex MINLP single-product campaign formation problems are solved with a new variant of the outer approximation (OA) algorithm (Viswanathan and Grossmann, 1989), which includes an augmented penalty (AP) function in the MILP master problem for the violation of the linearized constraints. The combined penalty function, equality relaxation, and outer approximation method (denoted AP/ER/OA) also differs from the ER/OA algorithm (Kocis and Grossmann, 1989) in that the method begins with the solution of the relaxed NLP problem rather than assuming an initial integer point. Since the MILP master problem does not provide a valid upper bound (for minimization), termination is based on a change in the monotonicity of successive NLP objective values rather than a comparison of the best NLP solution (a lower bound) with the MINLP master objective. This new variant of the ER/ OA algorithm has had considerable success in identifying the global optimal solution for nonconvex MINLP problems. The AP/ER/OA method has been implemented as the software package DICOPT++ through the algebraic modeling system GAMS (Brooke et al., 1988). The problems were solved on an IBM 3090 where MINOS5.0 (Murtagh and Saunders, 1983) and msx (IBM, 1979) are the NLP and MILP solvers, respectively. Table XI shows the computational results for the single-product campaign formation problems. Table XI1 compares the global optimum processing rate with the bounds on the rate generated from the scheduling bounds and from the bottleneck analysis. The global optimum is obtained for seven of the nine products; DICOPT++ found a nearly global optimum solution for product J but yielded a relatively poor local optimum for product A from a variety of different starting points. Generally the bottleneck bound will yield a relatively poor bound because some of the units feasible for a stage are shared among several stages and the actual processing rate will be much lower. However, for this particular plant the drying stage is a true bottleneck for every product and yields tight bounds (and zero integrality gap) for seven of the nine products. Not surprisingly, these are the same products for which the global optimum solution was obtained. Note that the processing rate bound generated from the scheduling

Table XII. Comparison of Processing Rate Bounds with the Optimal Rate schedule bottleneck product rate bound rate bound optimal rate A 137.53 56.04 55.98 B 100.73 36.74 36.74 C 283.38 84.25 84.25 D 222.54 97.09 97.09 46.82 46.82 E 179.31 66.01 66.01 F 219.40 96.29 96.29 G 276.45 55.96 55.96 H 192.77 85.25 J 201.87 66.80

e"

bounds on B- and yields, as predicted, a bound that is 2-4 times the optimum rate. Consequently, if the bottleneck bound is poor, the integrality gap of the formulation can be quite large. For instance, for product J the gap is 27.6%. Although the computation times shown in Table XI are not unreasonable for a single problem, in the context of the entire scheduling methodology in which a campaign formation problem must be solved for each product as well as for each feasible product combination, the CPU times could be excessive. The vast majority of the computation time is spent in solving the MILP master problems and is due to substantial degeneracy among the feasible solutions as well as the poor continuous relaxation. Several attempts were usually required to find the optimum solution, so that the actual CPU time required to obtain the above results is larger. Furthermore, the Mauderli and Rippin plant is small in comparison to industrial-scale plants, which produce upward of 100 products on as many equipment items. Note that the problem size grows substantially when NP" = 6 as compared with NP- = 2. Since it is not unusual to have four or five equipment groups on a stage, the size of the formulation could be quite substantial. Finally, as noted previously, the bottleneck bound will generally not provide a tight bound on the processing rate, so that, as illustrated by product J, significantly larger CPU times may be required. It is quite likely that the integrality gap of the MILP could be reduced through more sophisticated bounding procedures and/or cutting plane techniques (Crowder et al., 1983; Martin and Schrage, 1985; Van Roy and Wolsey, 1987). Sahinidis et al. (1989) have reported promising results using the strong cutting plane generation procedure of Crowder et al. in combination with a branch and bound procedure to solve a large-scale MILP multiperiod longrange planning model. However, the cause of the poor continuous relaxation in the campaign formation problem is the difficulty in modeling the production line scheduling problem and calculating bounds on the schedule when the equipment assignment is unknown. Rather than attempt to improve the integrality gap of the MILP master problem through heuristic bounding procedures and/or cutting plane techniques, a decomposition of the original MINLP will be identified so that the production line scheduling

678 Ind. Eng. Chem. Res., Vol. 30,No. 4, 1991 Equipment assignment Eq. group sequencing

Master Problem

Master Problep

Path batch sizes

Subproblem

SubprobIem

MINLP

Subproblem

Campaign formation

I

I

Figure 4. Decomposition of the campaign formation problem with the OA algorithm. MINLP Master Problem

Number of Equipment groups

Equipment assignment

Eq. group sequencing Path batch sizes

Figure 5. Decomposition of the campaign formation problem with the equipment assignment as the complicating condition.

problem is much easier to model.

Alternate Problem Decompositions The basis of decomposition approaches such as generalized benders decomposition and the OA/ER algorithm is the identification of a set of complicating variables, or more generally a set of complicating conditions, such that the remaining optimization problem is much easier to solve when these complicating variables or conditions are temporarily fixed. A decomposition based on these complicating variables defines a separation of the original optimization problem into a number of subproblems (Geoffrion and Marsten, 1972). In the following, alternative decompositions of the campaign formation problem are examined and the properties of a solution strategy based on these decompositions are discussed. In the outer approximation algorithm, the complicating variables are the binary variables, so that the original MINLP is partitioned into a large set of NLP subproblems, one for each feasible integer solution of the MINLP. Figure 4 illustrates this decomposition when the OA algorithm is applied directly to the campaign formation problem as in example 1. The MILP master problem deterrninea both the equipment assignment and equipment group sequencing, while the NLP subproblem optimizes the path batch sizes. Since the MILP master contains all of the rotational and structural degeneracy and is likely to have a poor continuous relaxation, solution of the MILP master problem requires substantial computation time. The original decomposition of the multipurpose batch plant scheduling problem suggests a decomposition in which the complicating condition is the equipment assignment, so that the campaign formation problem could be decomposed into an equipment assignment problem and a production line scheduling problem. This decomposition is shown in Figure 5. Since the scheduling problem is solved at the lower level, the master problem would determine an equipment assignment based on a relaxation of the scheduling constraints. The production line scheduling subproblem would be identical with the problem discussed by Wellons and Reklaitis (1989a-c) in which the equipment group sequencing and path batch sizes are optimized. As a result, the original campaign

Figure 6. Decomposition of the campaign formation problem with the equipment group profile as the complicating condition.

formation problem would be separated into a large number of production line scheduling problems, one for each feasible equipment assignment. T i t h this decomposition, the rotational degeneracy would be moved to the production line scheduling problem and could be addressed with the procedures developed previously. However, the master problem would still contain all of the structural degeneracy which cannot be identified a priori. As a result, an algorithm based on this decomposition could require significant computation time to solve the master problem and could spend many major iterations identifying degenerate equipment assignments. Since the difficulty in modeling the production line scheduling problem arises because the number of equipment groups on each stage is unknown, a third decomposition can be proposed in which the equipment group profile is the complicating condition. Although NGj does not appear in the MINLP formulation as a variable, the MINLP problem is much easier to solve if the number of equipment groups is fixed. Figure 6 illustrates this approach in which the campaign formation problem can be decomposed into a master problem which optimizes the number of equipment groups and a campaign formation subproblem which determines the equipment assignment and production line schedule for a given equipment group profile. This decomposition defines a valid separation of the original MINLP into NG,- smaller MINLP problems, one for each feasible equipment group profile EG = (NG1, NG2, ..., NG,). With this approach, the master problem contains no structural or rotational degeneracy since no equipment assignments are made. Since there is very little degeneracy among the equipment group profiles, only a few major iterations would be required to identify the optimal solution. Since the equipment group profile is fixed, the campaign formation subproblem would be substantially smaller and require significantly fewer equipment assignment variables than the original formulation. Additionally, the structural degeneracy would be significantly reduced because there are fewer potential positions to assign equipment items and there are more required equipment assignments (one unit per equipment group instead of one unit per stage). Since the number of equipment groups would be known, additional constraints can be introduced into the campaign formation subproblem formulation to reduce rotational equivalence due to equipment group sequencing. As a result, the campaign formation subproblem should have a much tighter relaxation than the original MINLP formulation and would require significantly less computation time. In the following section, the campaign formation decomposition algorithm is developed based on these observations.

ng,

The Campaign Formation Decomposition Algorithm The campaign formation decomposition algorithm embodies the decomposition strategy outlined above in which

Ind. Eng. Chem. Res., Vol. 30, No. 4, 1991 679 MINLP Formulation

Equipment Type

Stage

EG Profile

stop

MINLP

1 I

Campaign Formation Subproblem

I

Lower Bound R, I

I

Figure 7. Flow chart of the campaign formation decomposition algorithm.

the MINLP campaign formation problem is decomposed into an equipment group (EG) master problem and a campaign formation (CF) subproblem. A flow chart of the campaign formation decomposition algorithm is shown in Figure 7. The CF subproblem corresponds to the original MI" formulation with the number of equipment groups temporarily fixed. The solution of the CF subproblem identifies the equipment assignment with the highest processing rate for a particular EG profile and yields a lower bound on the optimal solution to the original MINLP (a maximization problem). The EG master problem (described below) is a relaxation of the original MINLP formulation that determines a new EG profile and provides an upper bound on the optimal processing rate. Integer cuts are used to exclude previously identified EG profiles from the EG master problem. The CF subproblem and the EG master problem are solved alternately until the best lower bound identified by the CF subproblem exceeds the current upper bound from the EG master problem or until the EG master problem is infeasible, indicating that all feasible EG profiles have been examined. Since the campaign formation subproblem is nonconvex, it is possible the global optimal solution will not be identified. As for the MILP master problem in DICOPT, the EG master problem may not provide a valid upper bound on the optimal processing rate so that the global optimum solution may be cut off. Proposition 1 (below) details the conditions for which the EG master problem provides a rigorous upper bound. The equipment group master problem and the campaign formation subproblem are described in more detail in the following sections.

The Equipment Group Master Problem The equipment group master problem determines the optimal number of equipment groups on each stage of the production line and is based on a broad relaxation of the campaign formation problem. The plant equipment items are partitioned into equipment families, and only the total number and total volume of the units used from each equipment family in forming a production line are constrained; specific equipment assignments are not made. Equipment group sequencing and path batch sizes are neglected; instead, the s u m of the batches produced in the sequence is constrained by the stage capacities. Finally, the transfer policy is relaxed so that the sequence cycle time is constrained by the sum of the processing times on each equipment group. In the following, the equipment group master problem formulation is presented and its bounding properties are discussed.

Figure 8. Bipartite graph illustrating equipment families for product F.

The plant equipment items can be partitioned into NF equipment families according to the feasibility of the equipment items for the stages of the production line. Equipment family Ej contains all feasible equipment types for a corresponding set of stages JEP The identification of the equipment families E j is based on the following graph construction. Consider the bipartite graph G(J,&4) where the nodes j E J represent the stages of the production line, the nodes e E E represent the plant equipment types, and an arc a., E A exists if equipment type e is feasible for stage j . $he equipment families Ej, f = 1,..., NF, and their associated sets of stages JEj correspond to the NF maximally connected subgraphs of G. Figure 8 illustrates this construction for product F of Mauderli and Rippin (1979). The plant equipment items that can be used to produce product F consist of five types of reactors RU1, RU2, RV1, RV2, and RD, filter FP, and dryer DP. All five types of reactors are feasible for stage 1 of the production line and reactor types RV1 and RV2 are feasible for stage 2; obviously the filter FP and dryer DP are feasible for stages 3 and 4, respectively. The bipartite graph has three disjoint subgraphs; the corresponding equipment families and stages are E, = (RUl,RU2,RVl,RVB,RDj and JEl = {1,2],E2= (FP]and JEz = (31, and E3 = (DP1and JEs = (41. Let NUj be the number of units assigned to stage j and NGj the number of equipment groups. Since DICOPT can only handle binary integer variables, the 0-1 variables Uju and Gjg are used to indicate the number of units and equipment groups on stage j . Uju(Gig) is 1 is stage j is assigned u units (g groups) and 0 otherwise. NGJm

CGjg=1

g=l

V j

(21)

~j

(22)

NUJ'-

C

u=l

uju=1

The continuous variables NUj and NGj are constrained to integer values with the following constraints. NGJ-

NGj = C gGjg

Vj

(23)

Vj

(24)

g=1

NUJ-

NUj =

C uUju

u=l

680 Ind. Eng. Chem. Res., Vol. 30, No. 4, 1991

The total volume of the equipment items in each equipment family Et is distributed continuously among the stages j E JE,, subject to constraints on the volume that can be assigned to any one stage. However, the number of equipment items from E, assigned to a stage j E JE, must be integer. Let V .be the volume assigned to stage j. If VE, is the total vofume and NE, is the total number of equipment items in equipment family E,, then any feasible assignment must satisfy

The objective of the master problem is to maximize the processing rate max (B*/r) (33) The problem can be convexified through exponent@ variable transformations of HUj, NG,, B*, T, Vi, and Bp Note that if ZNUj and ZNGj represent the new variables resulting from the exponential transformation of NUj and NGj (NUj = exp(ZNUj)and NG, = exp(ZNGj)),constraints 19 and 20 can be rewritten linearly as NG,-

ZNGj = C In (g)Cjg

j = 1,..., M

(19Z)

j = 1, ...,M

(202)

g=1

The volume that can be assigned to stage j is constrained as a function of the number of units assigned to stage j : NU,-

T N U j IV j I C u=l

u=l

yuWjuV j

(27)

where is the volume of the smallest unit that is feasible for stage j and Pum is the sum of the volumes of the u largest units in Et that are feasible for a stage j E JEf. The number of units assigned to stage j is bounded as follows: 1 INUj INU?" Vj (28) where N U Y is the maximum number of units that are feasible for stage j . The NUj units assigned to stage j are arranged into NGj equipment groups, so that NGjINUj V j (29) Regardless of the equipment group sequencing, each equipment group will be used NP/NGj times in the path sequence. Since the total capacity of all the equipment groups on stage j is bounded by Vj/Sj,an upper bound on the sum of the path batch sizes for the sequence B* is B * I - -NP Vj V j (30) NGj Sj Since individual path batch sizes are not determined, the processing time of all batches on stage j must be represented by a single characteristic batch size Bi. To determine Bj, all batches produced on equipment group g of stage j are assumed to be the same size B k If equipment group g has NUjg in phase parallel F i t s , tLe characteristic batch size for equipment group g Big is Bjg = BjE/NUjg and the time required to produce the NP/NGj batches in the sequence is

Since the volume assigned to a stage can be freely distributed among the equipment groups, the rate will be maximized if T = Tjgfor g 1,...,YGj on the timelimiting stage. This implies that Big = Bj for all g = 1, ..., NGi. Consequent1y

So that B* NG, B j Z - NUj - - - NP

NU,"

ZNUj = C In (u)Uju

V j

(31)

Additionally, the problem is parametrized in the number of paths since NP is not known a priori. However, NP can be removed from the formulatip through the variable substitutions B* = B*/NP and T = T/NP or by simply setting NP = 1. Although convexity ensures a global optimum to the EG master problem, the resulting upper bound may not be valid because of the assumptions required to model the sequence cycle time as a function of a single characteristic batch size. The following proposition describes a sufficient condition for which the EG master problem provides a valid upper bound. Proposition 1. The equipment group master problem provides a valid upper bound on the optimum solution of the original MINLP campaign formation problem if the processing time relationships are convex (dj I1for j = 1,

..., M).

The proof of proposition 1is in Appendix B. The basis of the proof is to show that when the processing time relationships are convex, the sequence cycle time is minimized for a given B* when all of the batches produced on a stage are the same size. When this is true, the processing times of all the batches produced on a stage can be represented by a single characteristic batch size B, as is done in the equipment group maater problem. In contrast, when the processing time relationships are concave ( d . < 1))the and no sequence cycle time is minimized (for a given bounds on the batch size) if all the material processed in an equipment group is lumped together into a single batch to take advantage of the economies of scale. However, such a solution is usually not feasible and definitely not optimal when subject to bounds on the batch sizes and the sequencing constraints of the campaign formation subproblem. Thus when the processing time relationships are nonconvex, the batches produced on a stage of the production line cannot be accurately modeled by a single characteristic batch size and the resulting bound generated by the equipment group master problem may not be a valid upper bound. However, ow computational experience has yet to identify a situation when the equipment group master problem has failed to provide a valid bound. It is interesting to note that one of the sufficient conditions for the MILP master problem of the OA/ER algorithm to provide a rigorous bound is quasiconvexity of the relaxed equations (e.g., processing time relationships), which implies dj 1 1 for j = 1, ..., M.

h*

The Campaign Formation Subproblem The campaign formation subproblem results when the number of equipment groups on each stage is fued at NG . Consequently, the solution of the CF subproblem yield a lower bound on the optimal solution to the original

Ind. Eng. Chem. Res., Vol. 30,No. 4,1991 681 NU,-

Table XIII. Commrison of Processing Rates for Example 2 product A

B C D E F G H J

M&Rorate 55.69 33.99 84.24 91.98 46.82 64.84 96.29 55.96 58.42

optimal rate 55.98 36.74 84.24 97.09 46.82 66.01 96.29 55.96 66.80

(44)

Mauderli and Rippin.

MINLP. The number of paths in the sequence is NP = L C M j ( N G j ) . Since NG; is known and the equipment groups are used by the paths of the sequence in indicia1 order, path k uses equipment group g on stage j if g = pj(k), where

.

r

(45)

T 1 SEjg - SSj, SEj, Ak

V j,g

SSj, = Ak + s j k v j , g k g' + Ejk V j , g k = N P - N G j + g Ak-NG, + Ejk-NG, - sjk

vj k = NG,

The objective of the campaign formation subproblem is to maximize the processing rate 1 NP max - C Bk Tk=l

The equipment assignment constraints are as follows: (34)

b = 1,...,N U F - 1 (37)

. .

g = 2, ..., NGj (38)

Note that constraint 34 requires one unit to be assigned to every equipment group on each stage, while constraint 35 ensures that at most one unit will be assigned to the remaining positions of the production line. Since the number of equipment groups on each stage is known, an additional constraint set can be added to restrict degeneracy due to rotational equivalence of the equipment group sequencing. Results from combinatorial analysis indicate that a preferred sequencing of the equipment groups on a subset of stages JS can be used to reduce the number of equivalent solutions. This corresponds to considering only those sequences that begin with a particular equipment group. Since the composition of the equipment groups is unknown,constraint 38 requires the first equipment group to have the largest total volume for those stages j E Js. The procedure to identify the subset JS is described in Appendix C. Since the sequencing of equipment group is known, the constraint sets to determine the batch sizes and sequence cycle time are much simplier than those for the general problem formulation described earlier.

+ 1, ..., NP

(46) (47) (48) (49)

Part of the complexity of even the decomposed formulation of this problem results from the consideration of operation with path-dependent batch sizes. Clearly this feature could be eliminated by simply requiring all paths to have the same batch size. While this may indeed be appropriate in many practical situations, the incorporation of this extra degree of freedom in the model allowa explicit evaluation of the maximum productivity that can be extracted from the available plant equipment. Thus, at the very least, the comparison of the solutions obtained with and without consideration of path dependency will provide a sound basis for deciding whether the resulting gains justify the extra operational complexity. Example 2 The campaign formation decomposition algorithm was used to determine the optimal single-product campaigns for each of the products produced in the Mauderli and Rippin plant. The description of the plant and the product recipes are given in Tables II-X. DICOFT (Kocis and Grossmann, 1989)was used to solve the convex equipment group master problem since termination based on the upper and lower bounds will yield the global optimal solution for convex problems. Since the objective values of successive NLPs are not necessarily monotonic, DICOPT++ could terminate prematurely. DICOPT++ was used to solve the nonconvex campaign formation subproblem. A comparison of the optimal processing rate identified in this study with the results obtained by Mauderli and Rippin is shown in Table XIII. The specific equipment assignments for all production lines are shown in Table XIV. Although Mauderli and Rippin assume a zero-wait transfer policy between stages, they allow an uneven distribution of the batch on stages with more than one in-phase unit. Additionally, they calculate the processing time for that stage based on an average split batch size (Mauderli, 1979). The campaign formation problem formulation assumes a more rigorous definition of zero-wait (ZW)transfer policy in which split batches must be the same size. This difference in interpretation of the ZW policy only affects the optimal processing rate for product J. Since the above model is more representative of the actual operation of the stage, the processing rate for product J was recalculated assuming the largest batch size that allowed for an even split of the batch. As expected, Mauderli and Rippin's heuristic enumeration procedure failed to obtain the optimal solution for

682 Ind. Eng. Chem. Res., Vol. 30, No. 4, 1991 Table XIV. Assignment of Plant Equipment Items for the Optimal Single-Product Campaigns equipment-stage assignment prod. RU1 RU2 RV1 RV2 RD1 RD2 FS FP1 FP2 DPI DP2 DT1 DT2 A A4 A3 A5 A1 A2 A6 A6 AI A4 A1 A2 A6 A5 A3 A6 AI B B4 B1 B4 B2 B6 B6 BI B4 B3 B5 B1 B4 B2 B4 B6 B6 B7 B4 B3 B5 c4 c1 c2 C6 C6 CI c3 c3 c5 c c5 c4 c1 c2 C6 C6 CI c3 c3 c5 c5 D3 D3 D4 D4 D5 D2 D D1 D2 D3 D3 D4 D4 D5 D2 D2 D1 El E2 E2 E3 El E E2 E2 E3 El El F3 F4 F2 F3 F1 F F1 F2 F3 F4 F2 F2 F1 F1 F3 G4 G4 G5 G5 G3 G2 G2 G3 G G1 G1 H4 H3 H3 H4 H5 H5 H3 H H2 H1 55 53 54 J1 52 53 55 J6 J6 J 53 55 55 J6 J6 53 54 J1 52 53 53

n

%1

n n R=31.8 T=19.60

U

n

B,=4=317.4 T=22.87 M & R campaign

R=64.8

RUl RU2 RVl RV2

FP

R46.0

FP

RD RD

-

B,=4=371.4 T=11.43

Optimal campaign R46.0 Figure 9. Campaigns for product F: top, Mauderli and Rippin; bottom, optimal.

many of the products; the largest deviation from optimality was 12.5% (product J). Figure 9 compares the campaigns for product F identified by Mauderli and the campaign formation decomposition algorithm. The batch sizes and sequence cycle times are shown under the size- and time-limiting stages of the production line. The best campaign identified by Mauderli consists of two independently operated production lines, while the optimal campaign consists of a single production line that results when the two production lines identified by Mauderli are merged. A summary of the problem size and computation time for the campaign formation subproblem and equipment group master problem is shown in Table XV. Most of the problems required one or two major iterations of the campaign formation decomposition algorithm; product J required the most major iterations at 6. As one would expect, the number of major iterations and computation time grows with the gap between the first EG master and the best CF subproblem solution. In virtually all cases the computation time for the campaign formation subproblem comprised the bulk of the total computation time. The

Bik

Tj

Ri

683.76 683.76 545.45 545.45 602.41 602.41 114.29 114.29 349.65 349.65 311.36 377.36 1257.86 696.86 511.95 511.95

24.43

55.98

29.69

36.74

14.30

84.25

14.71

97.09

14.94

46.82

11.43

66.01

13.06 12.45 15.33

96.29 55.96 66.80

Table XV. Algorithm Performance for Example 2 problem E M F prosizea itera- total EGfCF gap (% duct 0-1 f var f con tions timeb times* of CF) A 25/55/18 2 2.81 0.11J2.16 0.1 24jiobfi29 1 5.65 0.93 f 4.12 23153/I8 0.0 35f 120f 151 25155f 78 1 4.54 0.15 f 3.19 0.0 36f 123f 159 2 13.26 0.99 f 12.21 0.0 20 142156 39f104f124 13f 21f 36 1 0.75 0.29 10.46 0.0 19162f 14 1 5.34 0.2115.13 0.0 18f 36146 28/81/96 1 1.54 0.24f 1.30 0.0 20142f 56 26163f 14 20142f 56 1 1.50 0.40f 1.10 0.0 34/11f 14 6 35.10 5.59f 30.11 16.4 24150168 32f 101f 136 "Top, EG master problem; bottom, CF subproblem (optimal EG profile). * CPU seconds on IBM 3090. Table XVI. Comparison of CPU Times for Direct Solution and the Decomposition Approach total total NLP product decomp CPU direct CPU direct CPU A 2.81 146.1 14.1 B 5.65 73.4 12.8 C 4.54 31.8 5.4 D 13.26 150.1 3.1 E 0.15 10.2 4.2 F 5.34 186.1 9.1 G 1.54 11.0 2.0 H 1.50 36.4 9.4 J 35.10 927.1 15.7

gap between the optimal processing rate identified in the campaign formation subproblem and the upper bound predicted by the equipment group master problem was zero for seven of the nine products. Since the EG master distributes the equipment volume among the stages of the corresponding equipment family as a continuous commodity, the solution of the EG master problem provides an upper bound that is essentially the minimum of a (continuous) bottleneck bound for each equipment family. Since the drying stage is a true bottleneck for this plant, the zero-gap bound results. Table XVI provides a comparison of the computation times for direct solution of the original MINLP formulation with the CPU times for solution with the campaign

Ind. Eng. Chem. Res., Vol. 30, No. 4, 1991 683 and NUY NGF

Table XVII. Bounds NGj-

product

NU-

Table XVIII. U m e r Bounds on Processina Rates product R" product RA 56.04 F 66.01 B 36.74 G 96.29 C 84.25 H 55.96 D 97.09 J 85.25 E 46.82

formation decomposition algorithm. The decomposition approach consistently provides at least an order of magnitude reduction in computation time and yielded the optimal solution on the first attempt for every product. Even though several MINLP problems must be solved with the decomposition approach, these problems are much smaller, contain relatively little degeneracy, and have a very small (usually zero) integrality gap so that the computation times are very reasonable. It is interesting to note that, for many of the products, the CPU time required just to solve the NLPs with DICOPT++ on the original MINLP formulation was more than the total CPU time required by the campaign formation decomposition algorithm to identify the optimal solution. As a result, regardless of the quality of the bounding strategy or cutting plane technique used and the resulting integrality gap of the MILP master problem, the decomposition strategy would require less computation time.

Conclusions A MINLP formulation for the single-product campaign formation problem was presented that includes the assignment of plant equipment items to positions of the production line, sequencing of the resulting equipment groups, and variable path batch sizes. The formulation possesses considerable degeneracy among the feasible integer solutions and can have a large integrality gap, so that direct solution of the formulation with DICOPT++ requires considerable computation time. The campaign formation decomposition algorithm was presented which alternately solves an equipment group master problem, which provides an upper bound on the processing rate and determines an optimal equipment group profile, and a campaign formation subproblem, which provides a lower bound on the optimal processing rate and determines an equipment assignment and production line schedule for a given equipment group profile. At least an order of magnitude reduction in computation time is achieved with the campaign formation decomposition algorithm as compared with direct solution of the original MINLP so that very reasonable computation times are required to determine the optimal single-product campaigns for an example multipurpose plant. Acknowledgment This work was supported in part by a National Science Foundation Fellowship, National Science Foundation Grant CBT-8518175, and E. I. du Pont de Nemours and

Co. We also thank Dr. Gary Kocis, Dr. J. Viswanathan, and Prof. Ignacio Grossmann at Carnegie Mellon University for facilitating the implementation of DICOPT and DICOFT++ at Purdue University and for their helpful advice during its use in this study.

Nomenclature b: index on units of a group B*: total batch size for path sequence Bj: characteristic batch size for stage j BT": maximum feasible batch size on stage j Bibb: the bth portion of the path k batch processed on stage ,I Bjk: characteristic batch size for stage j , path k Bk: batch size on path k bk: 0-1 variable denoting existence of path k Ak: start time for the path k batch e: index on equipment type Ej: set of feasible equipment types for stage j Ejk: ending time on stage j , path k f: index on equipment family g: index on groups of a stage G j i binary variable denoting g groups on stage j j : index on stages JH: set of stages on which holding is allowed JE,: set of stages that use the equipment types in family f k : index on paths M number of stages NE,: number of equipment of type e NEf: number of equipment in equipment family f NP: number of paths in the sequence NPmax:maximum number of paths in the sequence NGj: number of equipment groups on stage j NGm": maximum number of equipment groups on stage j N e : maximum number of units in an equipment group for stage j Ojfk: overlapping time between stages j and j ' on path k OTf + e,@,; relationship between overlapping time and batch size Pjgg*:set of paths that use equipment group g if exactly g* groups on stage j Pjk: processing time on stage j , path k q + cjBdj: relationship between processing time and batch size R: average processing rate Sj: size factor for stage j and equipment type e; volume required at stage j per unit mass of final product s,k: starting time on stage j , path k S S . : sequence starting time on group g, stage j SE$ sequence ending time on group g, stage j T sequence cycle time U .: binary variable denoting u units on stage j volume of equipment type e V.* volume assigned to stage j &": minimum volume to be assigned to stage j maximum volume to be assigned to stage j with u units ~ h ftotal : volume of equipment items in equipment family

dY

ha: f

Xe,&: binary assignment variable for equipment type e, stage j , group g, position b

Appendix A. Bounding Procedure for Campaign Formation MINLP Formulation In this appendix, bounds for B"", BT", NGF", NUTax, R", SSjmp, SEm", and A$" are developed. Additionally, bounds AminaAAkT;f" are also generated which not only are used to calculate the above bounds but also prove useful as variable bounds in the formulation. Batch Size Bounds. Bounds on the maximum and minimum batch size and the maximum characteristic batch are

684 Ind. Eng. Chem. Res., Vol. 30,No. 4, 1991

time units. If the units were arranged into a single equipment group, a batch of size B could be processed every

9+

.( -)4 I

NUj

time units. It can be easily derived that in-phase operation dominates out-of-phase operation for batch sizes larger than The minimum batch size is simply the capacity of the smallest unit for the entire production line. The maximum batch size is the minimum over all stages of the production line of the total capacity feasible for the stage, the total capacity feasible for the associated family of stages, or the maximum capacity for stage j when all other stages in JE, are assigned one unit. VEjf is the total volume of the NE, - IJEA + 1largest feasible units for stage j (if there are that many) since the other stages in JE, must be assigned at least one unit. The maximum split batch size is the smaller of the maximum batch size or the largest capacity unit feasible for that stage. Structural Bounds. Since every stage must be assigned a t least one unit, bounds for NU?" and NGT" based simply on the number of feasible units for stage j and the associated equipment family Ef are NU?" = NGY" = min ( C NE,, NE, - IJEA + 1) eEE,

(A.4) where j E JE,. Tighter bounds for NG?" can be obtained with the following heuristic analysis. Let BF" be the maximum batch size if stage j has NGF" equipment groups. Let Pj and P,be the processing times on stages j and s for batch size Bl-. If there is unlimited intermediate storage (UIS), then stage s will be time limiting (in comparison with stage j ) if

Consequently, to ensure that stage s is not time limiting (in comparison with stage j ) , there must be NGFa equipment groups on stage s where c

(A.5) In general, this bound must be valid for all possible batch sizes. A good heuristic is to calculate N G F for both and B& and take the larger bound. The comparison must also be made with all stages in the production line. If the transfer policy is not UIS,it is possible that idle time or holding time may be required on stage s as well. As a result, the above bound should only be used as a rough guide to selecting NGF". If the processing time relationship is such that d j > 1, then for sufficiently large batch sizes in-phase operation is more desirable than out-of-phase operation. Consider the following comparison of NGj = NU, out-of-phase units (equipment groups) with a single equipment group of NUj units. Assuming UIS operation, out-of-phase operation would be able to process a batch of size B every 9 + cjBd, NUj ~

Additionally, in-phase operation of these units will permit larger batch sizes. For many of the products in the Mauderli and Rippin plant, the batch size defined in (A.6) is very near Bdn, so that NG?" was set to 1 for all of the filtering stages that have dj = 2. If the only role of in-phase units is to increase the capacity of the equipment group, an upper bound on NU?= is given by

However, as noted above, in-phase units can also be used to improve the time utilization of the stage by reducing the characteristic batch size and, hence, the processing time. Even for dj I1 if the optimal solution for stage j has NG) equipment groups, an alternate optimal solution may have a single equipment group of NUj = NGS units. Furthermore, this solution would have a smaller startup time, since the actual occupation time of the stage is smaller. Thus a more general bound for NU?" is

Based on the above rules, the bounds used for the campaign formation problem of example 1are given in Table XVII. SchedulingBounds. Bounds for the sequence starting and sequence ending times and the start time for each batch in the sequence are based on the following analysis. If there are at least g groups formed for stage j of the production line, then the appropriate values for SSjg and SE 's will be calculated. If there are not g groups on stage j, tLen the bounds on SSj, and SEjgmust be sufficient to ensure that

vi"

SEj, - SSj, I

(A.8)

where Fin is a lower bound on the sequence cycle time if there are at least k paths in the path sequence. If group g does not exist, then constraints 18 and 19 suggest that

+ SI$' + SSw , k* = g SEj, = AP" + + SE2V

SSjg = A P

(A.9) (A.lO)

For simplicity, set S S z r = 0, Then a value for SE,F can be calculated by substituting (A.9) and (A.lO) into (A.8) to yield SEgP = A Y + E$= - A@' - s3%ip (A.ll)

e"

vin

The minimum processing times can be calculated assuming a charaqteristic batch size of W divided by the

Ind. Eng. Chem. Res., Vol. 30, No. 4, 1991 685 maximum number of in-phase units for that stage. Similarly, the maximum processing times pi"" c q be calculated assuming a characteristic batch size of L??". Consequently, minimum start times P"" and maximum end times on each stage of the prduction line for a single batch can readily be determined. Furthermore, a bound on the sequence cycle time is given by I -

pin = max J

I

\

([)."]

NGT"

erated at capacity. If dj > 1,it can be easily derived that the rate is maximized at

provided P > 0. Thus the maximum processing rate for stage j with equipment type e is given by

(A.12)

(A.18)

The minimum start time for path k can be given by where

B, = min and is based on stage j having the maximum number of equipment groups with all batches of size Bmin. In a similar fashion, the maximum start time for path k can be calculated as Aff" = max (k - 1 ) p "

(A.14) J which assumes that all stages have only one equipment group and all batches are of size B"". Obviously if k > 1 some stage must have more than one equipment group, so let JBdenote the set of stages that will (probably) have g equipment groups when k 1 g. For example, the stage with the largest is most likely to be the first stage with more than one equipment group. A better, but now heuristic bound for Af" is e

"g""

jez[BIT" k-1

(A.15)

Thus all required terms for SEZY have been determined and can be substituted into (A.7). The final bound to be calculated is A$". If path k does not exist, then constraint 20 requires

so that A I F = EF" - Splin I

However, if path k does not exist, then Tin is zero, yielding

(A.16)

Finally, an utmer bound on the mocessine: rate can be generated a s s d n g all batches are at B- wiih cycle time of 7 . Thus

kBmax R"" = max k W"

Bma) dj I1

Bei = min (Bjc,Bm")

dj > 1

A valid upper bound on the processing rate based on a single stage test is then given by NE@" (A.19) Rma=min J e€Ej In general, this will provide a very poor bound since some of the units that are feasible for stage j can also be used by the other stages of the production line. However, the bound is good if stage j remains a bottleneck when all feasible equipment items are assigned to that stage. This is exactly the situation for the Mauderli and Rippin plant since the drying stage is a bottleneck for all products except J. Table XVIII shows the processing rate bounds generated with (A.19). These bounds are tight for every product except A and J. The above bound can be generalized so that the processing rate of the stages j E JEf is maximized by solving the following bin-packing problem: max R

R

5

CRYX,i 1

c1x1j = 1

It is sufficient for Ah =

=

(2,

(A.17)

Bottleneck Processing Rate Bound. The bound developed in (A.17) will generally provide a very poor upper bound on the processing rate because of the large range of feasible batch sizes and processing times. Here an improved bound is generated by determining the maximum processing rate on each stage assuming that all feasible units for that stage are used at the batch size that maximizes the rate. Let R Y be the maximum processing rate for stage j with equipment type e. For a single unit B R= Pj'+ cjBdj Clearly if dj I 1 the rate is maximized if the unit is op-

cx, =1 i where RF" is the maximum processing rate of unit 1 on stage j (tke units have been disaggregated from equipment types) and Xli is a binary variable indicating that unit 1 is assigned to stage j .

Appendix B. Proof of Proposition 1 Let (EG)denote the equipment group master problem and (CF) the original campaign formation problem. Let F(P) denote the set of feasible solutions to problem (P) and u(P) denote the corresponding objective value. Problem (EG) is a valid relaxation of problem (CF) and consequently provides a valid upper bound on the optimum solution to (CF) if: (i) F(CE7) C F(EG) (ii) u(EG) Iu(CF) for all F(CF) (maximization) Since the objective functions of (EG) and (CF) are the same, condition ii is satisfied. We first show that any feasible equipment assignment and corresponding path batch sizes for (CF) are feasible in (EG)and then discuss the conditions when the cycle time representation of (EG) encompasses the feasible cycle times for (CF). Any feasible equipment assignment in (CF) is feasible in (EG) by defining the variables NUj and NGj and the

686 Ind. Eng. Chem. Res., Vol. 30, No. 4, 1991

associated e 1 variables Ujuand Gigfrom (EG) as follows: NU,-

NG,"'

NO,-

C Xejgl

NGj = C

eEEj g=l

1 ifNUj=u

uju=

0 otherwise 1 ifNG,=g

cycle time is minimized if all paths produced on a particular equipment group are the same size and dj 1 1, j = 1, ...,M. Consider any feasible equipment assignment and let Kjg denote the set of paths that use equipment group g on stage j. Let Tj denote the maximum of the sum of the processing times on each equipment group of stage j. A lower bound on the sequence cycle time is T 1 Ti j = 1, ...,M (B.1)

So that T is minimized if Tj is minimized for all j = 1,..., M. For any B*, minimum Ti can be determined with optimization problem (PA1). min Ti

0 otherwise Since the equipment types e E Ef are feasible only for stages j E JE,,feasibility with respect to the number of each type of equipment in the plant can be shown by aggregating constraint 3 of (CF) for those equipment types e E E, to yield constraint 24 of (EG). NG,-

NU,-

Similarly the volume assigned to a stage Vj can be defined as NG,-

NU,"

where NUjg is the number of units in equipment group g, stage j . Note that no upper bounds are placed on the path batch sizes. Clearly Ti is minimized if each batch is divided evenly among the units of the equipment group so that

Bk = BjkNUj, which satisfies the bounding constraints 25 and 26 of (EG). The V. can be aggregated to show feasibility of constraint 23 of

IEG).

NG:"'

NUj-

C Bjbk

b-1

Additionally, minimum Ti occurs if the sum of processing times on each equipment group is the same (if only one equipment group is time limiting, Tj can be decreased by redistributing B*). Thus a t the optimal solution

NU.-

From constraints 7 and 8 of (CF) an upper bound on Bk is

Bk

03.2)

NU,-

5

C C

eEE, b = l

v, sj

Consequently, an equivalent optimization problem is the following: min T j

-xejgb

A bound for the first NGj batches can be expressed in terms of the total stage volume Vj

where Since each group is used NP/NGj times in the sequence, then the sum of path batch sizes B* can be defined as follows: NP

NP Vi

CBk = B* I- k=1 NGj Sj which corresponds to constraint 29 of (EG). The sequence cycle time constraint (30) in (EG) is based on the assumption that all paths produced on a particular equipment group are the same size. In the following we show that constraints 30 and 31 in (EG) provide a valid lower bound for the characteristic batch size and sequence cycle time (and thus feasibility of (CF) in (EG)) if the processing time relationships are convex (dj L 1, j = 1,..., M).

To prove the proposition, we show that for any feasible equipment assignment and correspondingB* the sequence

Tj =

NP -9 + ~jTy NGj

(B.3)

The optimal solution to (PA2) can be found by determining the stationary points of the associated Lagrangian function.

L(Tj,iijk,") = NGj

2)-

z

-

c @)ke Kd

g=l

NG,

Setting the gradient of L with respect to Ty and Bjk to zero yields

Ind. Eng.Chem. Res., Vol. 30, No. 4, 1991 687 NGj

aL aTjl

tion is usually not feasible and definitely not optimal when subject to upper bounds on the path batch size and the sequencing constraints of the CF subproblem.

- = l - ~ c g = o g-1

Addition of the equality constraints from (PA2) yields the following system of NP + NGj + 2 equations. NG,

1-

u N G j + l ~ j=g

“gdJjf’-

c ug=o

(B.4)

g=1

o k E q g (B.5)

g = l,.,., N G j

NG;

(B.7) Solution of this system of equations yields

Appendix C. Procedure to Identify Js Since the assignment of the equipment items to equipment groups is unknown, we must assume that the equipment groups are operationally distinct. It can be shown that if the equipment groups on every stage are operationally distinct, then for any feasible path assignment there are NP - 1equivalent assignments that occur when the order of the paths in the sequence is the same but the sequence starts with some other path k = 2, ...,NP (e.g., the path sequence (231) is equivalent to (123) when repeated cyclically because path 2 always follows path 1, path 3 follows path 1, and path 1 follows path 3). By constraining the sequencing of the equipment groups on stages j E J s to a perferred sequence (the sequence must begin with a particular equipment group), this rotational degeneracy can be eliminated. The following procedure identifies those stages j E Js. Step 1. Initially Jm contains all stages in the production line with NG. > 1, LCM* = 1 and J s = 4. Step 2. Seiect a stage j E Jleftwith largest NG,.. (i) If LCM(LCM*,NGj) = LCM*.NGj then

JS LCM*

T’= Bjk

E( B* NGj (NP/NGj)NUj

B* = (NP/NGj)NUj

)”

(B.lO)

k = 1, ..., NP (B.ll)

1

which is positive semidefinite for d . > 1, indicating that the stationary point is a minimum. if d . 1 1,a valid lower bound on the sequence cycle time and corresponding for any feasible B* is given by (substitution of (B.lO) and (B.ll) into (B.l)) +

cpj”.’)

B*

Bj 1 (NP/NGj)NUj

+

Js U j LCM*.NGj

and delete j from Jleft. (ii) Otherwise, just delete j from Jleft. Step 3. If Jleft= 4 or LCM* = NP, stop. Otherwise, go to Step 2. Literature Cited

where NUj = x:aNU. Note t h t $he characteristic h t c h size is the same for a f p a t h s an$ can be represented by a single charaderisti_cbatch size By The Hessian of L with respect to T j and Bjk is a diagonal matrix of the form

NP T 2 - 0 for every equipment group) to take advantage of the economics of scale. However, such a solu-

Brooke, A.; Kendrick, D.; Meeraus, A. GAMS: A User’s Guide; Scientific Press: Palo Alto, CA, 1988. Crowder, H.; Johnson, E. L.; Padberg, M. Solving Large-Scale Zero-One Linear Programming Problems. Oper. Res. 1983,31 (5), 803-834. Geoffrion, A. M. Generalized Benders Decompsotion. J. Optimization Theory Appl. 1972, 10 (4), 237. Geoffrion, A. M.; Marsten, R. E. Integer Programming Algorithms: A Framework and State-of-the-Art Survey. Manage. Sei. 1972, 18 (9), 465-491. IBM. “IBM Mathematical Programming System Extended/370 (MPSX/370), Basic Reference Manual”; Technical Report; IBM White Plains, NY, 1979. Kocis, G. R.; Grossmann, I. E. Computational Experience with DICOPT Sqlving MINLP Problems in Process Systems Engineering. Comput. Chem. Eng. 1989, 13 (3), 307-315. Lazaro, M.; Puigjaner, L. Simulation and Optimization of Multiproduct Plants for Batch and Semi-Batch Operation. I. Chem. E. Symp. Ser. 1985,92, 209-222. Martin, K.; Schrags, L. Subset Coefficient Reduction C u b for 0/1 Mixed-Integer Programming. Oper. Res. 1985,33 (3), 505-526. Mauderli, A. M. Computer-Aided Process Scheduling and Production Planning for Multi-Purpose Batch Chemical Plants. Ph.D. Thesis, ETH Zurich, 1979, No. 6451. Mauderli, A. M.; Rippin, D. W. T. Production Planning and Scheduling for Multi-Purpose Batch Chemical Plants. Comput. Chem. Eng. 1979,3, 199-206. Murtagh, B. A.; Saunders, M. A. “MINOS 5.0 Users Guide”; Technical Report SOL 83-20; Stanford University Systems Optimization Laboratory: Stanford, CA, 1983. Sahinidis, N. V.; Grossmann, I. E.; Fornari, R. E.; Chathrathi, M. Optimization Model for Long Range Planning in the Chemical Industry. Comput. Chem. Eng. 1989, 13 (9), 1049-1064. Van Roy, T. J.; Wolsey, L. A. Solying Mixed Integer Programs by Automatic Reformulation. Oper. Res. 1987, 35 (l), 45-57. Viswanathan, J.; Grossmann, I. E. A Combined Penalty Function and Outer-Approximation Method for MINLP Optimization. Presented at CORS/TIMS/ORSA Meeting, Vancouver, April 1989.

688

Ind. Eng. Chem. Res. 1991,30, 688-705

Wellons, M. C.; Reklaitis, G. V. Optimal Schedule Generation for a Single-Product Production Line, Part 1: Problem Formulation. Comput. Chem. Eng. 1989a,13 (l),201-212. Wellona, M.C.;Reklaitis, G. V. Optimal Schedule Generation for a Single-Product Production Line, Part 2: Identification of Dominant Unique Path Sequences. Comput. Chem. Eng. 1989b,13 (l),213-227. Wellons, M. C.; Reklaitis, G. V. Optimal Schedule Generation for a Single-Product Production Line, Part 3: MINLP Solution Stra-

tegies. Submitted to Comput. Chem. Eng. 198%. Wellons, M. C.; Reklaitis, G. V. Scheduling of Multipurpose Batch Chemical Plants. 2. Multiple-Product Campaign Formation and Production Planning. Ind. Eng. Chem. Res. 1991, following paper in this issue.

Received for review December 10,1989 Revised manuscript received July 26, 1990 Accepted August 10,1990

Scheduling of Multipurpose Batch Chemical Plants. 2. Multiple-ProductCampaign Formation and Production Planning Michael C. Wellons and Gintaras V. Reklaitis* School of Chemical Engineering, Purdue University, West Lafayette, Indiana 47907

An efficient procedure to identify the dominant multiple-product campaigns for multipurpose batch chemical plants is presented. On the basis of a linear dominance property, the dominant campaigns are identified as the noninferior extreme points of the associated multiobjective campaign formation problem. The noninferior set estimation method has been incorporated into a decomposition strategy which alternately solves an equipment group master problem to identify dominant equipment group profdes for the production lines of the campaign and a campaign formation subproblem with identifies the dominant campaigns for a particular equipment group profile. A multiperiod mixed integer linear programming production planning model for multipurpose batch plants that allots the available production time to a subset of the dominant campaigns and accounts for lost production time due to changeovers and startup times is also presented. The campaign formation and production planning procedures are illustrated with an example problem.

Introduction In the preceding paper (Wellons and Reklaitis, 1991), an efficient decomposition solution strategy was developed to solve the single-product campaign formation problem. In this paper, the decomposition strategy is extended to identify dominant multiple-product campaigns and a production planning model for multipurpose batch chemical plants is presented. A number of researchers have developed production planning models for batch chemical plants, including Mauderli and Rippin (1979),Lazar0 and Puigjaner (1985), Rich and Prokopakis (1986, 1987), Suhami and Mah (1984), Musier. and Evans (19881, and Vaselenak et al. (1987). Since a large number of alternative processing configurations are possible, a production planning model for multipurpose batch plants should have the capability to select from a set of alternative campaigns to meet the particular production requirements. Additionally, given the complexity of product recipes and the large changeover and startup times required for some products, the planning model should account for lost production time due to changeover and startup times for the production lines in each campaign. Unfortunately, most planning models only allow one processing configuration for each product and do not consider the effects of changeover and startup times. Rich and Prokapakis consider multiple production routes, but do not account for changeover and startup times. Mauderli and Rippin's planning model selects from a set of dominant campaigns, but allows only for a constant changeover time between all campaigns. In the following, the single-produd campaign formation decomposition algorithm is extended to identify dominant multiple-product campaigns based on the same linear dominance property used by Mauderli and Rippin. A production planning formulation for multipurpose batch planta is presented that selects from the set of previously

identified dominant campaigns and accounts for changeover and startup times of the production lines in each campaign. The dominant campaigns are then identified for the example multipurpose plant from Mauderli and Rippin, and several production planning problems are solved to illustrate the improvement in plant performance using the optimal dominant campaigns compared with the campaigns identified by Mauderli and Rippin.

Definition of Dominant Campaigns The goal of the campaign formation procedure is to identify a'small number of good campaigns to which the production planning procedure will assign the available production time. If the processing rates of the singleproduct production lines of each campaign are used as the selection criteria, then the set of good campaigns should consist of those campaigns that have the highest processing rates for a given product ratio. To quantify this idea, let Rc = (&, ...,Ri,, ...,Rpc) denote the vector of processing rates for each product in campaign c. Campaign 1dominates campaign 2 if R11 & or Ril 1 Ri2 for i = 1, ...,I. To satisfy a particular production requirement, any product ratio of I products can be achieved with Z (or fewer) suitable campaigns with the appropriate campaign lengths. Consequently, a campaign producing I products is inferior if a set of not more than I campaigns can be used to produce the same ratio of products but at a higher rate. Let CD denote the set of dominant campaigns. Neglecting the loss of production time due to changeovers and startup times, an inferior campaign q is dominated by a convex combination of the dominant campaigns.

The resulting linear dominance property is the following:

0888-588sf 91f 2630-0688$02.50/0 0 1991 American Chemical Society