A New Algorithm for Cyclic Scheduling and Design of Multipurpose

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Ind. Eng. Chem. Res. 2003, 42, 836-846

A New Algorithm for Cyclic Scheduling and Design of Multipurpose Batch Plants Soon-Ki Heo,† Kyu-Hwang Lee,† Ho-Kyung Lee,‡,| In-Beum Lee,*,† and Jin Hyun Park§ Department of Chemical Engineering, Pohang University of Science & Technology, San 31 Hyoja-dong Pohang, 790-784, Korea, Chemical Process & Catalysis Research Institute LG Chem, Ltd., Research Park, P.O. Box 61, Yu Seong, Science Town Daejeon, Korea, and P&I Consulting Co., Ltd., San 31 Hyoja-dong Pohang, 790-784, Korea

Many researchers have studied the scheduling, planning, and design of multipurpose batch processes. However, not so many studies have treated design and scheduling or design and planning simultaneously. The complexity of these systems stems from the fact that plant configuration must be determined for the purpose of process scheduling, yet scheduling must be done to devise the plant configuration. In the present study, a new algorithm for determining the best multipurpose scheduling and plant configuration is suggested. Since the objective function of the problem is nonlinear, it is linearized using a separable programming method. The proposed method consists of a number of procedures. First, a feasible configuration is obtained. Next, both the optimum equipment size and cyclic scheduling are determined for the plant configuration obtained in the first procedure. Last, the evolutionary design method proposed by Fuchino et al.(J. Chem. Eng. Jpn. 1994, 27, 57-64) is used to find the solution that minimizes the total cost. The efficacy of the proposed approach is demonstrated in three examples. Usually, equipment sizes are considered by a continuous variable in mixed integer linear programming and task processing time is assumed to be constant. However, most types of equipment are manufactured only in discrete volume classes. In addition, processing times are dependent on batch sizes. Hence, to apply the optimization methods developed here to real industries, the method was modified such that the volumes of equipment are considered as discrete variables and the processing time is a function of batch size. Introduction In recent years, consumers’ demands have diversified and product life cycles have shortened. These trends have led to increased interest in the design and capabilities of multipurpose batch plants. Such plants are characterized by flexibility and the ability to produce small quantities of high value-added products. Since the equipment required for a multipurpose batch plant is generally expensive, many industries have devoted themselves to optimizing their plant designs so as to reduce costs and maximize profits. The design of batch plants involves diverse mathematical problems. Each batch plant has unique product characteristics, manufacturing procedures, production requirements, and available time horizon. Design parameters and variables interact strongly with each other, and as a result problem sizes are usually large. Scheduling and plant configuration interact to such a degree that the schedule must be determined prior to designing the plant configuration. However, the plant configuration must be known for the sake of scheduling. This situation causes the simultaneous treatment of scheduling and plant configuration to be very compli* To whom correspondence should be addressed. Tel.: +8254-279-2274. Fax: +82-54-279-3499. E-mail: iblee@ postech.ac.kr. † Pohang University of Science & Technology. ‡ Chemical Process & Catalysis Research Institute LG Chem, Ltd. § P&I Consulting Co., Ltd. | Senior Research Scientist.

cated. In the present study, scheduling and equipment sizing are considered simultaneously. Many papers have been published on the design problems of multipurpose batch plants. These studies were undertaken to obtain the optimum plant configuration or equipment size that minimized the cost. Suhami and Mah2 were the first to formulate the design problem in terms of mixed integer nonlinear programming (MINLP). Their method randomly lists an empirically determined finite number of configurations and applies a heuristic procedure to select the best configuration. Imai and Nishida3 modified this method by solving a set-partitioning problem to determine the “best” configuration without the heuristics. Klossner and Rippin4 enumerated all possible plant configurations by solving a set-partitioning problem and then solved a MINLP model for each configuration. Vaselenak et al.5 proposed an MINLP for multiperiod formulations that employs a superstructure embedding all possible plant configurations. To derive the superstructure, a systematic procedure was devised that involved the formulation of all maximal sets of compatible products. Papageorgaki and Reklaitis6 decomposed the MINLP problem into a mixed integer linear programming (MILP) master problem, which determines the values of the binary assignment variables for fixed campaign lengths, and an NLP subproblem, which performs equipment sizing and determines the values of the campaign lengths. Voudouris and Grossmann7 modified MINLP to MILP by introducing discrete equipment sizes. Maravelias and Grossmann8 developed a MILP-based formulation of the planning problem for new product

10.1021/ie020308u CCC: $25.00 © 2003 American Chemical Society Published on Web 01/16/2003

Ind. Eng. Chem. Res., Vol. 42, No. 4, 2003 837

Figure 1. Schematic diagram of new algorithm.

development. The execution of each test depends on the success of the previous test. In the design step, it must be determined when to start developing a new product, and whether it is better to construct a new plant or to expand the existing plant. They used a heuristic algorithm based on Lagrangean decomposition to solve this complicated problem. Xia and Macchietto9 developed a MINLP-based solution to the design and synthesis of batch plants by modeling the scheduling aspect using a stochastic MINLP optimizer based on “simulated annealing”. Lin and Floudas10 proposed the representation of batch plant design using a “stated-task network”, in which the superstructure of all possible plant designs is constructed. Using this approach, they guaranteed the optimum solution of the nonconvex MINLP problems based on a key property that arises due to the special structure of the resulting models. Cao and Yuan11 decomposed the MINLP formulation to create a hybrid method for the problem of designing batch plants with uncertain demands. A simple multiproduct problem is solved with complex formulation and heuristics, which is due to the uncertainty. Fuchino et al.1 decomposed the problem of design and scheduling into two subproblems. The first subproblem decides the plant configuration and the minimum number of equipment units to be used as the initial plant configuration, and the second subproblem determines the scheduling and equipment sizes based on the solution of the first subproblem. They proposed an evolutionary method for determining the optimal design

and plant configuration in which neighboring configurations are generated by increasing the number of equipment units. However, their choice of the relative volume of equipment units was driven by intuition and convenience. They solved the scheduling problem by unifying the respective production ratios of equipment units. In the present study, both equipment sizes and the task completion times are treated as variables to simultaneously solve the design and cyclic scheduling problem. Since the cost function of equipment volume is nonlinear, we linearized it using a separable programming method to solve a MILP-based problem. Another factor that should be considered is that companies manufacturing equipment such as reactors, mixers, and columns, usually manufacture these units according to standard sizes. Hence, this research also considers equipment units with discrete volumes. In the proposed model, task processing time is assumed to depend on batch size. Thus, the proposed mathematical approach was formulated such that it would be applicable to real industry. In the sections below, we explain a multipurpose design and cyclic scheduling problem based on the general definition of the design problem. A new algorithm for solving the multipurpose design and cyclic scheduling problem is then suggested and explained. This algorithm contains three MILP models. To test the efficacy of the proposed method, we solved two illustrative examples that have been previously treated by Fuchino et al.1 The results are compared with those of

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Ind. Eng. Chem. Res., Vol. 42, No. 4, 2003

the previous research. And we applied our method to a more complex problem. In addition, modified approaches for real industry are also explained. 1. Problem Definition Papageorgaki and Reklaitis6 outlined the problem of the design of a general multipurpose batch plant. The problem treated in this research is defined as follows: Given data: (1) a set of N products and the requirements of each product within the available time horizon; (2) a set of available equipment units classified into equipment types according to their common usage for tasks; (3) recipe information of each product, comprising a set of processing times and a corresponding set of size factors; (4) transfer times and a set of sequencedependent cleanup times; and (5) cost data. Determinant variables: (a) a feasible equipment configuration and the number of units required for each equipment type; (b) sizes of processing units and amount of the task carried out on each equipment unit; and (c) start and completion time of each task. Objective function: to minimize the investment cost for the plant configuration while satisfying the constraints on the production requirements and other constraints. In the proposed model, the intermediate storage policy is NIS (“no intermediate storage”) as solved by Fuchino et al.1 We also solved the problems with ZW (“zero wait policy”). Each task can be processed separately with inphase operation. 2. Solution Approach A schematic diagram of the new algorithm is illustrated in Figure 1. The proposed algorithm, which contains three MILP models, simultaneously solves the design and scheduling aspects of the problem. 2.1. Description of the Problem of a Multipurpose Batch Plant. The problem to be solved is to simultaneously determine the number of units of each equipment type, their volumes, and scheduling for a given set of data describing the multipurpose batch plant (i.e., products, their requirements, a set of equipment types, the sequence of task processing, processing time, size factor, transfer time, cleanup time, and cost data, referred to in the previous section). 2.2. Determine the Minimum Number of Equipment Units (MILP I). First, the number of equipment units of each type must be determined. Therefore, a feasible plant configuration with the minimum number of equipment units is taken as an initial configuration. When an equipment unit processes a task, its volume must exceed the minimum operating ratio. Even when a small amount of a product is produced, the size factors of the tasks must be satisfied. Hence, two or three equipment units of each type are needed. For example, task11 and task12 of product A in Figure 2 and Table 1b differ greatly in their total amounts of task i, that is, BTi. Therefore, at least two units of equipment type 1 are needed. To determine the minimum number of units needed for the total tasks, we developed the first MILP (MILP I). MILP I finds the minimum number of equipment units of each type to satisfy the size factors and minimum operating ratios. The binary variable WVj is used, which takes on a value of 1 when the equipment unit j is introduced and

Figure 2. Recipe data on example 1. Table 1. Data on Example 1 (a) Requirement of Three Products A

B

C

2500

2000

2500

(b) Processing Times and Size Factor type

task i

PTi

SFi

1

11 12 13 14 15 16

0.20 0.15 0.20 0.15 0.15 0.15

0.50 1.00 0.60 0.40 0.30 1.00

2

21 22 23

0.25 0.25 0.25

0.50 0.40 1.00

3

31 32 33

0.20 0.15 0.15

1.00 1.00 0.70

(c) Sequence-Dependent Cleanup Times from/to

A

A B C

0.10 0.05

B

C

0.05

0.10 0.15

0.15

0 otherwise. The objective function is therefore the sum of these binary variables, JN

objective function:

minimize

WVj ∑ j)1

(1)

where JN is the maximum number of equipment units. For the illustrative example 1 (Figure 2 and Table 1b), 6 units of type 1 are required and 3 units each of types 2 and 3; hence, JN is 12. The objective function is minimized, subject to several constraints. If task i is processed in unit j, binary variable Wij is 1 and unit j is introduced. Otherwise, Wij is zero. Hence, the following relationship exists between Wij and WVj:

Wij e WVj,

for ∀ j

(2)

When task i is processed in unit j, the relative batch size (i.e., Bij) must be greater than the minimum operation ratio of the relative volume of the equipment unit, Bmaxj, multiplied by MOR (“minimum operation ratio”). A sufficiently large positive number, M1, is used in the equations (3)-(5). If task i is not processed in


minimum. However, the minimum slack time between cycles among equipment units is

for ∀ j

SLD e SLFj + SLLj, Figure 3. Head and tail noted by SLF and SLL.

unit j, these constraints are redundant and the relative volume of the equipment unit is greater than the relative batch size of the task that is processed in that vessel.

MOR × Bmaxj e Bij + M1 × (1 - Wij), for (i,j) ∈ IJ (3) Bij e M1Wij,

for (i,j) ∈ IJ

Bij eBmax j + M1(1 - Wij),

(4)

for (i,j) ∈ IJ

(5)

Tasks can be separately processed with in-phase operation by assumption. The summation of relative batch sizes gives the total amount of task i (BTi), that is,

BTi )

∑j

Bij,

for ∀ i

(6)

2.3. Find the Minimum Cycle Time under the Plant Configuration (MILP II). The second MILP model, MILP II, determines the minimum cycle time under the plant configuration developed in MILP I. Generally, the term “makespan” refers to the total time required to complete a group of tasks. During the time horizon many tasks are processed repeatedly. Hence, there are many cycles within the time horizon. Most scheduling models consider only the makespan; however, the head and tail should be considered in order to represent the cycle time exactly. The cycle time is less than the makespan per cycle because of the heads and tails. Therefore, the cycle time is derived by subtracting the head and tail from the makespan per cycle. Figure 3 gives a schematic representation of two cycles showing their heads (SLF) and tails (SLL). SLD is the slack time between cycles, which equals the sum of SLF and SLL. Therefore, the cycle time is the makespan per cycle minus SLD.

Objective function: minimize MS - SLD + CLTpp′ (7) where MS is makespan and CLTpp′ is cleanup time between product p and p′. Because many cycles are performed within the time horizon, the sequence-dependent setup time must be considered not only within a cycle but also between cycles. However, the cleanup time is less than about 10% of the cycle time and in most cases the largest cleanup time falls between cycles. Thus, CLTpp′ can be fixed by the largest one in the objective function. Equations 3-6 are also included in the second model.

SLFj e Ci - tri - PTi - hti + M2(1 - Wij), for (i,j) ∈ IJ (8) SLLj e MS - Ci - tri + M2(1 - Wij),

for (i,j) ∈ IJ (9)

where M2 is a sufficiently large positive number. SLD goes to a maximum as the objective value goes to a

(10)

To consider the transfer time, processing time, and sequence-dependent cleanup time, eqs 11-15 are required. Equations 13, 16, and 17 represent the NIS policy. If it is treated under ZW, then fix hti, which means holding time of task i, as zero and drop eqs 16 and 17.

MS g Ci + tri,

for ∀ i

Ci - hti - PTi - tri g 0, Ci + tri ) Ci′ - hti′ - PTi′,

(11)

for ∀ i

(12)

(i,i′) ∈ II

(13)

Ci + tri + CLTpp′ e Ci′ - hti′ - PTi′ - tri′ + M2(2 - Wij - Wi′j) + M2(1 - Zpip′i′), for (i,j) ∈ IJ, (p,i),(p′,i′) ∈ PI (14) Ci′ + tri′ + CLTp′p e Ci - hti - PTi - tri + M2(2 - Wij - Wi′j) + M2Zpip′i′, for (i,j) ∈ IJ, (p,i),(p′,i′) ∈ PI (15) Equations 14 and 15 represent that completion times of tasks and sequence-dependent cleanup times must be considered in equipment j. Binary variable Zpip′i′ has a value of 1 when task i of product p is followed by task i′ of product p′. And these binary variables can be reduced by constraining the order of p and p′. For example, we can dictate that if p is product B, p′ can only be product C and not product A. When Z ) 1, product B is followed by product C, and when Z ) 0, product C is followed by product B. Therefore, eq 14 is useful and eq 15 is redundant when Wij ) 1, Wi′j ) 1, and Z ) 1. If Z ) 0, eq 15 is useful and eq 14 is redundant. If at least one of the two Wij’s is zero (i.e., task i and task i′ are processed in different equipment units), then eqs 14 and 15 are both redundant constraints. The NIS policy, implemented in eq 16, stipulates that even if task i” is finished and is to be transferred to unit j, it will not be transferred until the preceding task i is completed, including cleaning up. If Z ) 0, then eq 17 is used. For details, refer to Figure 4.

Ci′′ g Ci + tri + CLTpp′ - M2(2 - Wij - Wi′j) M2(1 - Zpip′i′), for (i′′,i′) ∈ II, (i,j),(i′,j) ∈ IJ, (p,i),(p′,i′),(p′,i′′) ∈ PI (16) Ci′′ g Ci′ + tri′ + CLTp′p - M2(2 - Wij - Wi′j) M2Zpip′i′, for (i′′,i) ∈ II, (i,j),(i′,j) ∈IJ, (p,i),(p,i′′),(p′,i′) ∈ PI (17) Each task must be processed at least once and can be done with in-phase operation.

∑j Wij g 1,

for (i,j) ∈ IJ, i ∈ I

(18)

2.4. Select the Cycle Time as a Parameter. The cycle time obtained above is the minimum value under the initial plant configuration. However, the objective

840


modified objective function: minimize Vj )

γjmVJj, VJmin e VJj e VJmax , ∑ j j m γjm ) 1, ∑ m

Figure 4. Illustrative Gantt chart for MILP model.

function of greatest importance is the investment cost of the plant. Hence, if there is another value of the cycle time that gives a lower cost, we must find that value. As the first step in determining the most cost-effective cycle time, the minimum cycle time is used. 2.5. Find the Minimum Equipment Cost with Separable Programming (MILP III). The third MILP model, MILP III, not only determines the volume of equipment units that minimizes the total cost but also simultaneously determines the cyclic scheduling.

objective function:

minimize

∑j RjVjβ

j

(19)

where Rj and βj are cost data according to equipment type. Equations 3-6 and 8-18 are also used in this model. The tasks are repeatedly operated in the equipment units, so the actual equipment size is obtained by dividing the relative volume of equipment unit processing the total amount of the task by the number of cycles within the given time horizon. To avoid the nonlinearity of eq 20, the cycle time is fixed and eq 7′ is introduced.

CT , H

for ∀ j

(20)

CT g MS - SLD + CLTpp′

(7′)

Vj ) Bmaxj

where CT is the cycle time. However, these equations lead to a MINLP model because of the exponential term in the objective function. To avoid MINLP, a separable programming method can be used. The cost function is not highly nonlinear and unimodal, as shown in Figure 5, and the summation of each cost is also unimodal. Therefore, the use of separable programming is reasonable. Let VJ’s be parameters of discrete volume of equipment units. Then eqs 21 and 22 are developed. We use only two values of γ for each unit and it can reach the optimal value by repeated and narrower variations of the lower and upper bounds of the VJ’s, while the optimum values of the VJ’s are not bumped against the bounds.

∑j RjVJjβ for ∀ j

for ∀ j

j

(19) (21) (22)

2.6. Comparison. First, we take the result of MILP III with the cycle time set to the minimum value obtained using MILP II, and we compare it with the result obtained using the second chosen cycle time. The result that gives a lower cost is accepted. The objective function is represented with a power of 0.6 to the volume and the summation is expected to be a unimodal function. The relationship between cycle time and cost is illustrated in Figure 6. The ultimate objective function is to minimize the equipment cost, not the cycle time. Hence, the cycle time should be a parameter within the possible interval. And costs should be compared with other costs which are obtained based on other cycle times. 2.7. Select Another Cycle Time. Directional search method to find better cycle times, where the lower bound has been discovered by MILP II. Let the chosen cycle time be a parameter, and then MILP III is used. The next cycle time is chosen by increasing 5% of the previous cycle time. These procedures are repeated until the minimum cost is obtained. 2.8. Introducing a New Equipment Unit by an Evolutionary Search Method. The solution obtained using the procedure outlined above is a local optimum under the initial plant configuration, which has the minimum number of equipment units that can feasibly satisfy the requirements. The evolutionary search method, proposed by Fuchino et al.,1 can be used to ascertain whether it is reasonable to introduce another equipment unit. In this method, neighboring plant configurations are tried by adding equipment units to the initial configuration. The number of neighboring plant configurations depends on the number of equipment types in the plant. At first, new equipment of type 1 is added and the minimum cycle time is found. The procedures described in sections 2.3-2.7 are then repeated to determine the minimum cost of the modified plant configuration. Even though the number of equipment units increases, the total investment cost may decrease on

Figure 5. Cost function and separable programming.

Ind. Eng. Chem. Res., Vol. 42, No. 4, 2003 841 Table 2. Cost Data for Examples 1 and 2 Rj

Figure 6. Relationship of cycle time and cost.

consideration of both the cycle time and the assignment of equipment sizes. Hence, the solutions of the initial configuration and neighboring configurations must be compared to determine the optimal design. 2.9. Choose the Best Solution. We must select the final solutions of the number and volume of equipments, cycle time, and task completion time that minimize the investment cost. 2.10. A Heuristic Skill. There are many binary variables in Wij and Zpip′i′ in MILP II and III. The number of binary variables increases with increasing the number of products and their associated tasks. The greater the number of binary variables introduced, the larger the MILP problem that must be solved. Thus, to make the problem manageable, it is necessary to reduce the number of binary variables in the mathematical formulation. Equations 14 and 15 consider all the tasks of the same or different products. Since the processing time is not too long compared with the cleanup time and each equipment unit does not handle too many tasks in a cycle, the binary variable Zpip′i′, which is useful only if both task i and task i′ are processed in the same unit, can be assumed to have the same value for all i and i′ when p and p′ are identified. In the example 2, the exclusive case is only task110 of product D; that is, another task can be processed between task19 and task110 in the same unit. Therefore, the subscripts of the binary variable Zpip′i′ can be reduced to Zpp′. Then task110 is taken as another product. Using this simplification, eqs 14 and 15 can be replaced by eqs 14′ and 15′. Equations 16 and 17 are also applied.

type 1

type 2

type 3

250.0

350.0

300.0

βj

type 2

type 3

0.6

0.6

0.6

units is fixed at 0.05 for all the tasks, and the minimum operating ratio is 80%. MILP I determines the minimum number of units: three units of type 1 and two each of types 2 and 3. Using MILP II, the minimum cycle time is found to be 1.25 for NIS and ZW, which is 12% better than the value obtained by Fuchino et al.1 A Gantt Chart for the example system is illustrated in Figure 7a, showing the relative volumes of equipment units. The chart shows the starting and completion times of processes and uses white and gray bars to indicate the transfer and cleanup times, respectively. The relative volumes of equipment units are represented and the unit “u12” is the bottleneck of the cycle. It is important to notice that the bottleneck is the unit whose slack time between the cycles is short, not the unit whose idle time in a cycle is short. It can be found that three tasks of task12, task23, and task16 are separately operated. The MILP models were coded with AMPL (Fourer et al.12), run on a 1000MHz PC, and solved by CPLEX 7.0.0. The optimization of the volume of equipment units so as to minimize cost (MILP III) gives the system represented by the Gantt chart shown in Figure 7b. The optimal cost is 5942.2 for NIS and ZW with a cycle time of 1.25. It is not economical to introduce additional

Ci + CLTpp′ + tri e Ci′ - tri′ - hti′ - PTi′ + M2(2 - Wij - Wi′j) + M2(1 - Zpp′), for (i,j) ∈ IJ, (p,i),(p′,i′) ∈ PI (14′) Ci′ + CLTp′p + tri′ e Ci - tri - hti - PTi + M2(2 - Wij - Wi′j) + M2Zpp′, for (i,j) ∈ IJ, (p,i),(p′,i′) ∈ PI (15′) 3. Illustrative Examples Three examples are presented to test the effectiveness of the proposed algorithm. Two of these example systems were treated previously by Fuchino et al.1, and the solutions obtained using the proposed approach are compared with those of the former research. In the third example, we apply the proposed algorithm to a more complex problem. 3.1. Example 1. We consider a system for producing three products using three types of equipment, as shown in Figure 2. The relevant data for producing each product within the time horizon (H ) 300) are given in Tables 1 and 2. The transfer time between equipment

type 1

Figure 7. Gannt charts of example 1.

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Table 3. Solutions of Example 1 (NIS Policy)

Table 5. Data on Example 2

(a) Volumes of Equipment for Each Plant Configuration type

unit

config. I

config. II

config. III

config. IV

1

U11 U12 U13 U14

5.42 5 3.33

5.42 5 3.33 0

8.33 4.17 2.67

5.42 5 3.33

2

U21 U22 U24

6.51 3.91

6.51 3.91

5.21 3.13 2.67

6.51 3.91

3

U31 U32 U33

10.42 7.29

10.42 7.29

8.33 5.83

10.42 7.29 0

(a) Processing Times and Size Factor type

task i

PTi

SFi

1

17 18 19 110

0.20 0.15 0.15 0.15

0.50 1.00 1.00 1.00

2

24 25

0.25 0.25

0.50 1.00

3

34 35

0.20 0.20

1.00 1.00

(b) Sequence-Dependent Cleanup Times from/to

(b) Cycle Times and Total Costs config.

no. of equip. (type 1-2-3)

minimum CT

total cost (cycle time)

I II III IV

3-2-2 4-2-2 3-3-2 3-2-3

1.25 1.0 1.0 1.25

5942.2 (1.25) config. I 6131.8 (1.0) config. I

Table 4. Solutions of Example 1 (ZW Policy) (a) Volumes of Equipment for Each Plant Configuration type

unit

config. I

config. II

config. III

config. IV

1

U11 U12 U13 U14

5.42 5 3.33

5.73 4.4 3.44 2.93

5.42 5 3.33

5.42 5 3.33

2

U21 U22 U24

6.51 3.91

5.73 3.44

6.51 3.91 0

6.51 3.91

3

U31 U32 U33

10.42 7.29

9.17 6.42

10.42 7.29

10.42 7.29 0



minimum CT


I II III IV

3-2-2 4-2-2 3-3-2 3-2-3

1.25 1.1 1.15 1.25

5942.2 (1.25) 6012.32 (1.1) config. I config. I

A A2 B C D

A

0.10 0.05 0.15

A2

0.10 0.05 0.15

B

C

D

0.05 0.05

0.10 0.10 0.15

0.10 0.10 0.10 0.10

0.15 0.10

the ZW policy, the minimum cycle time is 2.1 and the minimum cost is 9678.31 for a cycle time of 2.1. The cyclic scheduling and equipment sizes are expressed as Gantt charts in Figure 9. Application of the evolutionary search method showed that nine equipment units give a better optimum cost. If the number of equipment units is over ten, the volume of the added equipment is zero at the optimum in MILP III. The results are given in Tables 6 and 7. The Gantt chart of the system obtained after evolutionary searching is illustrated in Figure 10; this system is the same for both the NIS and ZW policies. The optimum plant configuration contains four units of type 1, three of type 2, and two of type 3. In this case the minimum cycle time is 1.4 and the optimum total cost is 8466.77, which is 27.6% less than that obtained by Fuchino et al.1 3.3. Example 3. In the third example, we consider a plant in which five types of equipment are used to

equipment units of type 1, 2, or 3. As shown in Tables 3 and 4, the plant configuration with seven equipment units is the best solution. In the tables, “Config. I” refers to the initial plant configuration and “Config. II”, “Config. III”, and so on represent neighboring plant configurations. 3.2. Example 2. We now consider a system with two additional products, A2 and D, shown in Figure 8 and Table 5. The initial plant configuration obtained from MILP I uses eight equipment units to produce the five products. Under the NIS policy, the minimum cycle time is 1.95 and the minimum cost is 9257.39 for a cycle time of 1.95. These results are better than those of Fuchino et al.1 by 25.6% for cycle time and 18.8% for cost. Under

Figure 8. Recipe data of A2 and D.

0.05


Ind. Eng. Chem. Res., Vol. 42, No. 4, 2003 843 Table 6. Solutions of Example 2 (NIS Policy) (a) Volumes of Equipment for Each Plant Configuration type

unit

config. I

config. II

config. III

config. IV

1

U11 U12 U13 U14

11.7 8.13 5.2

8.4 7.29 5.83 4.38

11.7 8.13 5.2

11.7 8.13 5.2

2

U21 U22 U23 U24

11.7 8.13 5.2

8.4 5.83 3.73

11.7 8.13 5.2 0

11.7 8.13 5.2

3

U31 U32 U33

16.25 11.7

11.67 8.4

16.25 11.7

16.25 11.7 0



minimum CT

optimum total cost (cycle time)

I II III IV

3-3-2 4-3-2 3-4-2 3-3-3

1.95 1.4 1.85 1.8

9257.39 (1.95) 8466.77 (1.4) config. I config. I

Table 7. Solutions of Example 2 (ZW Policy) (a) Volumes of Equipment for Each Plant Configuration type

unit

config. I

config. II

config. III

config. IV

1

U11 U12 U13 U14

12.6 8.75 5.6

8.4 7.29 5.83 4.38

12.6 8.75 5.6

12.6 8.75 5.6

2

U21 U22 U23 U24

12.6 8.75 5.6

8.4 5.83 3.73

12.6 8.75 5.6 0

12.6 8.75 5.6

U31 U32 U33

17.5 12.6

11.67 8.4

17.5 12.6

14 9.1 3.5

3



minimum CT

optimum total cost (cycle time)

I II III IV

3-3-2 4-3-2 3-4-2 3-3-3

2.1 1.4 2.05 2.0

9678.31 (2.1) 8466.77 (1.4) config. I 9861.74 (2.1)

produce five products within a time horizon of 500. The transfer time is fixed as 0.5 and the minimum operating ratio is 80%. The recipe data for this system are given in Figure 11 and Table 8. To produce the five products, at least 12 equipment units are needed (three each of types 1 and 5 and two each of types 2, 3, and 4). Using MILP II, the minimum cycle time of this plant config-

Figure 11. Data on example 3. Table 8. Data on Example 3 (a) Processing Times of the Tasks task

PT

task

PT

task

PT

ia1 ia2 ia3 ia5 ib1 ib11 ib3 ib4

3 5 2 6 5 4 2 3

ib5 ic1 ic2 ic4 ic5 id1 id11 id2

3 4 2 2 5 3 7 6

id3 id5 ie1 ie2 ie4 ie5 ie51

4 4 7 6 8 4 5

(b) Sequence-Dependent Cleanup Times from/to

A

A B C D E

1 0.5 1 2

B

C

D

E

1.5

1 1.5

0.5 1 1.5

1 1 0.5 1

1 1 1.5

0.5 0.5

1

(c) Cost Data Rj βj

type 1

type 2

type 3

type 4

type 5

250.0 0.6

350.0 0.6

300.0 0.6

400 0.6

350 0.6

uration is found to be 22 under NIS and 24.5 under ZW. The minimum total cost is 26429.17 at the cycle time 22 in NIS and 28056.5 at the cycle time 26.5 in ZW. The results are illustrated in Figure 12 and Tables 9 and 10. None of the neighboring configurations sampled during evolutionary searching was superior to that obtained from MILP I. 4. Modified Approaches for Real Industry Figure 10. Solution of example 2 by evolutionary search.

We now consider the application of the proposed method in real industrial settings. Our proposed method

844

Ind. Eng. Chem. Res., Vol. 42, No. 4, 2003 Table 9. Solutions of Example 3 (NIS) (a) Volumes of Equipment for Each Plant Configuration type unit config. I config. II config. III config. IV config. V config. VI

1

U11 U12 U13 U14

52.8 22 8.8

44 22 11 8.8

52.8 22 8.8

52.8 22 8.8

52.8 22 8.8

52.8 22 8.8

2

U21 U22 U23

44 39.6

44 35.2

44 39.6 0

44 39.6

44 39.6

44 39.6

3

U31 U32 U33

33 11

33 11

33 11

33 11 0

33 11

33 11

4

U41 U42 U43

22 13.2

35.2 17.6

22 13.2

22 13.2

22 13.2 0

22 13.2

5

U51 U52 U53 U54

35.2 22 8.8

39.6 22 8.8

35.2 22 8.8

35.2 22 8.8

35.2 22 8.8

35.2 22 8.8 0


no. of equip. (type 1-2-3-4-5)

minimum CT


I II III IV V VI

3-2-2-2-3 4-2-2-2-3 3-3-2-2-3 3-2-3-2-3 3-2-2-3-3 3-2-2-2-4

22.0 22.0 22.0 22.0 22.0 22.0

26 429.17(22) 2 8390.14(22) config. I config. I config. I config. I

Table 10. Solutions of Example 3 (ZW) (a) Volumes of Equipment for Each Plant Configuration type unit config. I config. II config. III config. IV config. V config. VI


gives optimal equipment volumes for the equipment types used to make the products. The plant designer must then ask the manufacturing company of those equipment units whether they can make units with the volumes specified by the model. However, the manufacturing company may reply that they do not produce the required sizes. For example, the model may specify volumes of 11.7, 8.13, 5.2, and so on, as is the case in example 2 described above, but the manufacturer may only produce sizes of 12, 9, 7, and so on. This will severely limit the benefits of calculating the optimal solution. Moreover, the processing times of the above examples are the same for 1500 and 5 size of equipment “u12” in example 1. Examples 2 and 3 are likewise. This may not be reasonable for the real situation. To adapt our method to the real industrial setting, we consider two new assumptions for this problem: (1) that only standard size equipment units are available and (2) that processing time is a function of batch size. It must be determined which volumes of equipment are introduced among the several standard volumes of each equipment type. Binary variable yj represents whether unit j is used, as proposed by Voudouris and Grossmann.7 The objective function of this problem is the total equipment cost expressed as a function of unit volumes. Parameter Vj is the volume of equipment unit

1

U11 U12 U13 U14

63.6 26.5 10.6

63.6 26.5 10.6 0

63.6 26.5 10.6

63.6 26.5 10.6

63.6 26.5 10.6

57.6 24 9.6

2

U21 U22 U23

49.7 3.3

49.7 3.3

49.7 3.3 0

49.7 3.3

49.7 3.3

48 43.2

3

U31 U32 U33

39.8 13.3

39.8 13.3

39.8 13.3

39.8 13.3 0

39.8 13.3

36 12

4

U41 U42 U43

42.4 21.2

42.4 21.2

42.4 21.2

42.4 21.2

42.4 21.2 0

38.4 19.2

5

U51 U52 U53 U54

53 21.2 10.6

53 21.2 10.6

53 21.2 10.6

53 21.2 10.6

53 21.2 10.6

43.2 19.2 4.8 4.8


no. of equip. (type 1-2-3-4-5)

minimum CT


I II III IV V VI

3-2-2-2-3 4-2-2-2-3 3-3-2-2-3 3-2-3-2-3 3-2-2-3-3 3-2-2-2-4

24.5 23.5 24.0 24.5 24.0 24.0

28 056.5 (26.5) config. I config. I config. I config. I 29 464 (24)

j and has discrete values.

minimize

∑j yjRjVjβ

j

(23)

Usually, larger batch sizes require longer processing times. Ierapetritou and Floudas13 expressed the task


equipment of type 1 are 3, 5, 10, 15, 20, and 30 and the sizes of types 2 and 3 are 5, 10, 15, 20, and 30. In modified example 3, the sizes of type 1 are 10, 20, 30, 45, and 50, the sizes of type 2 are 10, 20, 35, 40, and 50, the sizes of types 3 and 5 are 10, 20, 30, 40, and 50, and the sizes of type 4 are 10, 15, 30, 40, and 50. The mean processing times are those cited in Table 1b and Table 5a for modified example 2 and those in Table 8a for modified example 3. The minimum operating ratio is 50%. Figures 13 and 14 show the Gantt charts for the solutions to the modified versions of examples 2 and 3, respectively. The solution for modified example 2 gives a minimum cost of 10 612 at a cycle time of 2.4, although the minimum cycle time is 1.5. For modified example 3, the minimum cost is 31 329.627 at a cycle time of 26.67, although the minimum cycle time is 23. Conclusions

Figure 13. Gannt chart of modified example 2.

Figure 14. Gannt chart of modified example 3.

processing time as a linear function of the batch size.

PCTi ) ai + biBi, , ai ) 2/3PTmean i

for ∀ i

(24) (25)

PTmin i , min Bi

for ∀ i

/3PTmean , i

for ∀ i

(27)

) 2/3PTmean , PTmin i i

for ∀ i

(28)

bi )

PTmax i max Bi -

for ∀ i

PTmax i

)

4

(26)

A new method containing three MILP models was developed for the cyclic scheduling and design of multipurpose batch plants under NIS and ZW policy. Generally, ZW policy constrains scheduling to a greater extent than NIS. As a result, the results obtained under NIS policy differ from those obtained under ZW policy and both are better than the previous solutions of Fuchino et al.1 The first MILP model gives the minimum number of equipment units required to produce the products; this configuration is used as an initial plant configuration. The second MILP model determines the minimum cycle time and the third determines the equipment size and scheduling that minimize the cost. The more binary variables introduced into the calculations, the larger the problem becomes. Unusually, the number of binary variables is reduced because the binary variable is valid for both values, whereas zero of binary variable is meaningless in any other MILP models. Fuchino et al.1 determined the relative volume of equipment units by intuition and focused on unifying the processing task in each unit. However, in the present research we precisely represent the cycle time and take the equipment volumes as determinant variables. The method outlined here yields better solutions than previous approaches. Thus, a new algorithm is proposed for multipurpose batch plant design and scheduling. This algorithm uses separable programming and the evolutionary design method proposed by Fuchino et al.1 In addition, we proposed a modified approach for use in real industrial applications. Acknowledgment This work was supported by a grant No. (R01-2002000-00007-0) from Korea Science & Engineering Foundation. Nomenclature

Then the parameter PTi is substituted by the variable PCTi. The other constraints are the same; hence, MILP I is not necessary. Examples 2 and 3 are solved by the modified approach. In modified example 2, the sizes of

Sets I ) set of tasks II ) set of preceding and following tasks of same product IJ ) set of units used for processing task J ) set of equipment units JN ) maximum number of equipment units P ) set of products PI ) set of tasks of products

846


Indices

Literature Cited

i ) tasks j ) equipment units m ) discrete points of volume p ) products

(1) Fuchino, T.; Yamaguchi, K.; Muraki, M.; Hayagawa, T. Evolutionary Design Method for Multipurpose Batch Plants in The Basis of Cyclic Production. J. Chem. Eng. Jpn. 1994, 27, 57-64. (2) Suhami, I.; Mah, R. S. H. Optimal Design of Multipurpose Batch Plants. Ind. Eng. Chem. Process Des. Dev. 1982, 21, 94100. (3) Imai, M.; Nishida, N. New Procedure Generating Suboptimal Configurations to the Optimal Design of Multipurpose Batch Plants. Ind. Eng. Chem. Process Des. Dev. 1984, 23, 845-847. (4) Klossner, J.; Rippin, D. W. T. Combinatorial Problems in the Design of Multiproduct Batch Plants. AIChE Annual Meeting, San Francisco, Nov 1984; Paper 104e. (5) Vaselenak, J. A.; Grossmann, I. E.; Westerberg, A. W. An Embedding Formulation for the Optimal Scheduling and Design of Multipurpose Batch Plants. Ind. Eng. Chem. Res. 1987, 26, 139148. (6) Papageorgaki, S.; Reklaitis, G. V. Optimal Design of Multipurpose Batch Plants. 1. Problem Formulation. Ind. Eng. Chem. Res. 1990, 29, 2054-2062. (7) Voudouris, V. T.; Grossmann, I. E. Mixed Integer Linear Programming Reformulation for Batch Process Design with Discrete Equipment Sizes. Ind. Eng. Chem. Res. 1992, 31, 1315-1325. (8) Maravelias, C. T.; Grossmann, I. E. Simultaneous Planning for New Product Development and Batch Manufacturing Facilities. Ind. Eng. Chem. Res. 2001, 40, 6147-6164. (9) Xia, Q.; Macchietto, S. Design and Synthesis of Batch PlantssMINLP Solution Based on a Stochastic Method. Comput. Chem. Eng. 1997, 21S, S697-S702. (10) Lin, X.; Floudas, C. A. Design, Synthesis and Scheduling of Multipurpose Batch Plants via an Effective Continuous-time formulation. Comput. Chem. Eng. 2001, 25, 665-674. (11) Cao, D.; Yuan, X. Optimal Design of Batch Plants with Uncertain Demands Considering Switch Over of Operating Modes of Parallel Units. Ind. Eng. Chem. Res. 2002, 41, 4616-4625. (12) Fourer, R.; Gay, D. M.; Kernighan B. W. AMPL: A Modeling Language for Mathematical Programming; Boyd & Fraser Publishing Company: Danvers, 1993. (13) Ierapetritou, M. G.; Floudas, C. A. Effective ContinuousTime Formulation for Short-Term Scheduling. 1. Multipurpose Batch Processes. Ind. Eng. Chem. Res. 1998, 37, 4341-4359.

Parameters Rj, βj ) cost coefficients of unit j ai, bi ) parameters for processing time BTi ) total amount of task i CLTpp′ ) sequence-dependent cleanup time when p directly precedes p′ H ) time horizon M1, M2 ) sufficiently large positive numbers MOR ) minimum operating ratio PTi ) processing time of task i max PTmin , PTmean ) minimum, maximum, and mean i , PTi i processing time of task i, respectively SFi ) size factor of task i tri ) transfer time of task i VJj ) discrete volume of equipment unit j , VJmax ) lower/upper bound of discrete volume of VJmin j j equipment unit j Variables γjm ) coefficient of discrete volume of unit j at point m Bij ) relative batch size of task i in unit j Bmaxj ) relative volume of the equipment unit j Ci ) completion time of task i CT ) cycle time hti ) holding time of task i MS ) makespan per cycle PCTi ) processing time which is dependent on batch size SLD ) minimum idle time between cycles SLFj ) head of unit j SLLj ) tail of unit j Vj ) volume of equipment unit j Wij ) 0-1 assignment variable for task i in unit j WVj ) 0-1 assignment variable for introducing unit j yj ) 0-1 assignment variable for equipment unit j Zpip′i, Zpp′ ) 0-1 assignment variable, valued 1 when task i of product p precedes task i′ of product p′

Received for review April 25, 2002 Revised manuscript received November 25, 2002 Accepted December 4, 2002 IE020308U

A New Algorithm for Cyclic Scheduling and Design of Multipurpose

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