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A Heuristic-Embedded Scheduling System for a Pharmaceutical Intermediates Manufacturing Plant Jiyong Kim,†,‡ Junghwan Kim,‡ Taeyeong Lee,§ In-Beum Lee,§ and Il Moon*,‡ Department of Chemical and Biomolecular Engineering, Yonsei UniVersity, 134 shinchon-dong Seodaemun-gu, Seoul, 120-749, Korea, and Department of Chemical Engineering, POSTECH, San 31 Hyoja-dong, Pohang, Kyungbuk 790-784, Korea
This paper proposes a new mathematical model for a real pharmaceutical intermediates manufacturing (PIM) plant. To consider the general characteristics of the scheduling problems experienced at PIM plants, the proposed model employed a strategy to address a mixed-integer linear programming (MILP) formulation. Two heuristic techniquesspreclassification of equipment and sequential two-stage optimizationswere then proposed, to relax the complexities of scheduling problems that are due to practical constraints. The objective function of the proposed model is to minimize the total operation time (makespan) that is subject to the mass balance constraints and process boundary conditions. On the basis of the proposed models and heuristics, new packaged software is developed for an application to real PIM plants. To show the features and capabilities of the proposed scheduling system, four real examples were examined. The results reveal that the techniques are helpful for obtaining both higher accuracy of optimized solutions and higher computational performance. 1. Introduction Global competition in the pharmaceutical industry (PI) and increasing demands for affordable medicine has led the industry to focus its attention on improvements in the manufacturing efficiency, such as better utilization of resources and reduction of the response time. In the PI, the manufacturing of fine chemicals is a crucial part of the production system. Typically, the pharmaceutical manufacturing process, whether it uses traditional chemistry-based or new biotech-based processes, is characterized by a great number of process steps and much longer cycle times than exist in other industries. The manufacturing generally consists of several batch reaction stages, as well as intermediate handling stages, such as filtering, drying, or centrifugation. Almost of these stages are performed in many alternative process units. In such a complex process with numerous batch processes, the process efficiency is dependent on how to schedule the batches that are needed for the manufacture of a certain number of products using the equipment in an optimal manner. This means that improving scheduling and inventory management can increase the profits significantly while holding down unnecessary expenses.1 Thus, a specific scheduling model that can represent the complicated characteristics of PIs and solve the scheduling problems efficiently should be developed. To improve the manufacturing efficiency, in terms of the optimized scheduling, there have been several attempts to develop scheduling methodologies and tools.1-6 However, a commercialized scheduling tool and/or generalized methodologies are indeed an extremely ambitious objective, because the PI has a tendency to be very particular; company objectives and constraints are very specific, and they are based on the particular needs and requirements of their target markets. * To whom correspondence should be addressed. Tel.: +82 2 2123 2761. Fax: +82 2 312 6401. E-mail:
[email protected]. † Present address: Department of Chemical and Biological Engineering, University of Wisconsin-Madison, 1415 Engineering Dr., Madison, WI 53706, USA. ‡ Department of Chemical and Biomolecular Engineering, Yonsei University. § Department of Chemical Engineering, POSTECH.
Moreover, the small pharmaceutical intermediates manufacturing (PIM) plants, which produce pharmaceutical intermediates that are supplied to major active pharmaceutical ingredient (API) manufacturing companies, are suffering because of the absence of a relevant scheduling tool. The manufacturing schedules of these small companies usually work, but it is very difficult to know exactly how these schedules are built and how efficient they are. This is especially true when the product plan is subject to a major change, such as urgent orders. Under these circumstances, rescheduling of the plant may be impossible to do efficiently. Motivated by the absence of a relevant model and tool in the face of economic potential, this study proposes a mathematical-based scheduling model and heuristics for PMI plants. The remainder of this study is organized as follows: the problems and objectives of this study are stated in section 2; section 3 constructs the mathematical formulation for a basic scheduling model; section 4 introduces two heuristic techniques for improving computational performance; section 5 briefly describes the scheduling software that was developed, based on the proposed model and heuristics; finally, section 6 presents real examples, to illustrate the applicability of the proposed model, and section 7 presents some concluding remarks. 2. Problem Statement Generally, the manufacturing processes at a PIM plant consist of five steps, as shown in Figure 1. In the first step, products (orders), feedstock information, and recipe documentation, are performed before the manufacturing process begins. A practical step at the PIM plant can be divided into two sections: a reaction section and a separation section.2 The reaction section consists of several parallel and/or series batch reactors with condensers. The separation section includes a separating process, such as crystallization and filtration, and a drying process for product packaging. In this study, the scheduling problem includes all of the practical manufacturing processes from step 2 to step 4, as shown in Figure 1. Similarly, with typical batch processes that have multiproduct and small-lot-sized production charac-
10.1021/ie100880f 2010 American Chemical Society Published on Web 11/09/2010
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Figure 1. Schematic presentation of a pharmaceutical intermediates manufacturing (PIM) plant.
teristics, the scheduling problems of PIM plants are very complicated and have a tendency to have many practical and technological constraints. The general characteristics of a PIM plant that make production scheduling more complicated are described as follows: • Cross-contamination over process units is a critical concern for fine chemicals, and the cleaning of equipment is a resource-consuming task.3 Therefore, arrangement of the production schedule to minimize this cleaning is a prominent issue. • The limited durability of intermediate products, because of processes such as corruption and decomposition, leads to schedules where the equipment is dedicated to the production of a particular product. Eventually, this leads to decreased equipment utilization. • Late and urgent orders are often accompanied by strict deadlines. These deadlines make it difficult to organize long and sustainable campaigns; therefore, they have a tendency to reduce the actual capacity of the production plant. This reduction may, in turn, cause late shipments at the end of the week (or month) and such negative effects may propagate over entire periods of time. There usually also are many implicit rules at work, which affect the order in which a product can traverse between the units. For example, lifting raw material and/or liquid-phase intermediates from a lower level in the facility to a higher level is very difficult, because of machine unavailability and physical limitations in the existing piping, hosing, or pumping capabilities. In particular, the delay in production because of machine unavailability, which results from conflicts with other machines, has great effects on the manufacturing efficiency. For instance, in a PIM plant, a condenser is occasionally used with a reactor to maintain a specific temperature. But not all of the batch reactors are equipped with a condenser, because of the need to use less-expensive equipment. Nevertheless, if a campaign must use condenser-equipped reactors more than the existing equipment, the total production time would be increased significantly, because of the insignificant equipment (i.e., condensers). Therefore, the objective of this study is minimization of the operating time (makespan). To solve the scheduling problem,
the mathematical model is constructed as a mixed-integer linear programming (MILP) formulation; several efficient techniques for representing practical characteristics and some heuristic methods for higher computational performance are also embedded in the model. 3. Mathematical Formulation This section presents a mixed-integer model to address the issues concerning scheduling problems in the PIM plant. The objectives of the scheduling framework are (1) minimization of the time to process (makespan) and (2) minimization of the the delay in production time when the process could not be performed within the order deadline. The first objective function can be formulated as follows: Min
∑ TET
i
∀i
(1)
i
The first objective function (eq 1) is to minimize the overall makespan, which is expressed as the sum of the makespans over all of the orders (products). The second objective function (eq 2) is to minimize the delay time between the makespans and the due date of the products: Min
∑ TET
i
- Duei
∀i
(2)
i
where DUEi is the due date of product i. One must note that the second objective is a rather realistic solution, from the viewpoint that all orders cannot be always satisfied, since late and urgent orders often occur in the PIs. The objective functions are subject to the following constraints. (a). Time-Matching Constraint. The start time matching is established as follows: TSTi e TESijn + H(1 - wijn)
∀i, j, n
(3)
TSTi g TESijn - H(1 - wijn)
∀i, j, n
(4)
According to eqs 3 and 4, when order i is assigned to unit j in time slot n (wijn ) l), the start times in both coordinates are
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equalized. The parameter H is an upper bound (time horizon) that, in principle, can take any value that is sufficiently large.7 Similarly, the finish time match constraints, such as the finishing time for order i and the finishing time in unit j during time slot n, are represented as follows: TETi g TEFijn - H(1 - wijn)
∀i, j, n
(5)
TETi e TEFijn + H(1 - wijn)
∀i, j, n
(6)
ijn
e Nuil
ijn
e Nbi
The relationship between the start times of consecutive time slots is expressed in eq 9. Similarly, eq 10 establishes the relationship between the start times of orders in consecutive stages. In eq 11, if order i is allocated to time slot n of unit j, the corresponding final time is dependent on the processing time of i at unit j. If no order is assigned, the start and end times are equal. ∀i, j, n
(11)
(7)
If different orders are assigned in the same unit consecutively, the final time should be considered to be the cleanup time, which is defined as follows:
∀i, j
(8)
TESi′jn+1 g TEFijn + wijnCli - H(2 - wijn - wi′jn) ∀j, n, i, i′, i * i′
n
Equation 7 shows that the assignment of unit j, which belongs to unit group l at the same time, is subject to the total available number, whereas eq 8 shows that, for all orders and in all stages, exactly one time slot of one unit is allotted. One must mention that the number of generated time slots is related to the unit j. Furthermore, not necessarily all of the processing stages are involved in the production of order i. Moreover, only a group of the units in which order i can be processed is represented in the summation. TESijn+1 g TESijn
(10)
∀i, n, l
j
∑w
∀i, j, n
TEFijn ) TESijn + wijnProci
(b). Unit and Time Slot Assignment.
∑w
TEFijn+1 g TEFijn
∀i, j, n
(9)
Figure 2. Diagram of the heuristic techniques and the solving procedure.
(12)
On the other hand, the batch interval of an order is dependent on the bottleneck time. Thus, the start time of each batch should be included with the cycle time of order i, as shown eq 13: TSTi′ g TSTi + Cyclei
∀i, i′, i * i′
(13)
(c). The Time and Amount of Order. In eq 14, the order should be produced more than or equal to the amount of demand. Although extra inventory costs are accompanied with overproduced orders, this study assumes that demand should be satisfied. STi,n′ g Rqi
∀i
(14)
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Thus, the corresponding amount of order i that is produced is dependent on the stock of order i, as follows:
∑
Batchi g Rqi - Stoi
∀i
(15)
TETi g TEFin - H(1 - xin)
∀i, n
(26)
TETi e TEFin + H(1 - xin)
∀i, n
(27)
∑u
i
ij
∀i
(16)
Rwi e TESi
∀i
(17)
(d). Constraints Associated with a Practical Manufacturing Facility. Equation 18 shows that, for the reactor r chosen for order i, the reaction volume should not exceed to the maximum volume of reactor r: ∀i, r, j
RVir e MuVj
(18)
∑x
∑ ∑z
∀i, k
irjJoikRoir
j
(19)
r
TESi′n+1 g TEFin - H(2 - yik - yi′k)
∀k, n, i, i′, i * i′ (20)
wijn + wi′jn e 1 + H(2 - yik - yi′k)
∀k, j, n, i, i′, i * i′ (21)
4. Heuristic Techniques This section proposes two heuristics techniques: the binary variable reduction heuristic, through equipment preclassification, and the sequential solving heuristic, using a two-stage solution procedure. 4.1. Preclassification of Equipment. The essential idea underlying the first heuristic technique is to reduce the number of binary variables, in which a processing task is allowed to start in an adjusted manner. Thus, the size of the original scheduling model is reduced significantly and the optimal solutions to the relaxed model are calculated within a reasonable computational time.8 For example, this study relaxes the practical constraints associated with reactor types such as those depicted by eqs 19-22, and preclassifies them into two categories: on reactor and off reactor. For this preclassification, the study introduces new binary variables and their relationships as follows: wijn e xin
∀i, j, n
(22)
wijn e uij
∀i, j, n
(23)
wijn g xin + uij - 1
∀i, j, n
(24)
Using eqs 22-24, the constraints, which include the binary variable (wijn) in equations such as eqs 3-8, 11, 12, and 21, can be represented as follows: TSTi e TESin + H(1 - xin)
∀i, n
(25)
∀i, l
(28)
in
) Nbi
∀i
(29)
n
TEFin ) TESin + xinProci
∀i, n
TESi′n+1 g TEFin + xinCli - H(2 - uij - ui′j) ∀j, n, i, i′, i * i′ xin + xi′n e 1 + H(2 - yik - yi′k)
(30)
(31)
∀k, n, i, i′, i * i′ (32)
Based on the modified constraints associated with the reactor types (eqs 25-32), the binary variables can be fixed as 0 or 1 as follows:
As mentioned in section 2, there are three types of reactors, based on the attached equipment: a reactor without a condenser, a reactor equipped with a condenser, and dual reactors that share a condenser. To address realistic problems, such as the existence (and nonexistence) of a condenser, eqs 19-21 are applied: yik g
) Nuil
j
The orders are enforced to obey the due dates, as shown in eq 16, while, according to eq 17, the raw materials (reactants) should be guaranteed before the orders are assigned. TETi e Duei
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zirj )
{
wijn ) 0 if Jiij ) 0
(33)
uij ) 0 if Jiij ) 0 or MuVj < RVir
(34)
yik ) 0 if Roir ) 0
(35)
0 if Jiij ) 0, or Irir ) 0, or Uguir ) 0, or MuVj < RVir 1 if Irjirj ) 1
(36)
Although the heuristic requires a look-ahead investigation for the unit classification, the computational results clearly show that the model with the first heuristic is much more efficient, because there are fewer binary variables. 4.2. Sequential Optimization. To reduce the computational time associated with solving large-scale MILP scheduling models optimally, this study recommends an efficient heuristic two-stage solution procedure that permits near-optimal solutions with only a modest computational burden. Based on the result of the first heuristic, for which several binary variables were fixed, this heuristic allows a scheduling problem be solved sequentially. This technique consists of two scheduling models: an inner scheduling model and an outer scheduling model, as shown in Figure 2. The heuristic is efficient for the scheduling problems of a PIM plant because the final products and intermediates are produced using only a reactor without the filter and dryer processes, although the basic procedure involves a reactor, a filter, and a dryer. Thus, if the reactors are selected for production of final products and intermediates in advance, the total model size could be smaller, which leads to a reduction in the computational time. In the first stage, the units required to produce the product and intermediates (such as reactors, dryers, and filters) are determined, whereas, in the second stage, the production sequence is optimized. The rule of the first stage, which determines the units, is dependent on the priority order constrained by the features of the equipment (material, volume, and number) and by the state of the equipment (i.e., the existence of a condenser); these unit characteristics were already classified through the first heuristic. On the other hand, the total time for production (makespan) is then calculated using the outer scheduling model, based on the results of the inner scheduling model. Accordingly, the key constraints of the second stage are the due date of each order and the ratio of the final product and the intermediates. To apply the heuristics to a scheduling model,
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Figure 3. Example screens in the packaged scheduling software.
a solving procedure must be implemented, which can be described as follows. First Stage. Step 1: Identification. The required equipment is classified according to the first heuristic technique. For example, the reactor is subdivided into “on reactors” and “off reactors”, according to how (or if) the reactor is equipped with a condenser (as outlined in section 4.1). Step 2: Assignment. Before the schedule is optimized, the equipment is assigned based on where it will be used, according to the following priorities: Priority 0: only reactors determined through the first heuristic could be selected. Priority 1: initially, the orders produced through fixed units are determined. Priority 2: the orders that have earlier due dates then are determined. Priority 3: finally, the orders that have hard constraints associated with reactor types (such as material, volume, and number) are given priority. The constraints for reactor assignment are varied, depending on the orders produced. For example, if the condenser process is required for the order production, it is subject to eqs 19 and 20. In this case, the volume and material of the assigned reactors are subject to eqs 18 and 33, respectively. Second Stage. Step 3: Arrangement. Based on the results of the first stage, other equipment (such as filters and dryers) should be arranged according to the features of the order produced. One must note that the arrangement of filters and dryers is much easier than reactor assignment, because all orders do not always require filter and dryer processes.
Step 4: Time Adjustment. To compress the schedule, the time intervals between the start and end times of the final products and intermediates are adjusted according to the volumetric yield of the reactant and product. Note that the end time of final product is subject to due dates in eq 16. Step 5: Optimization. The total production time (makespan) is optimized, which is subject to eqs 3-6, 9-13, 16, 17, 26-28, 31, and 32. 5. New Packaged Scheduling System On the basis of the proposed models and heuristic techniques, a packaged scheduling tool for the PIM plant has been developed. This tool was developed by packaging a string of all of the scheduling processes. The system contents are itemized using Visual C++ and EXCEL for a graphical user interface (GUI). The system consists mainly of three types of modules for analyzing the optimized results: a process and product information input module (PPM), an operation policy input module (OPM), and an output analysis module (OAM). PPM gives information regarding not only the process (such as the number and capacity of the equipment) but also the product (such as the type and recipe of the product). All of the data can be imported from an EXCEL spreadsheet or input directly through a data input interface screen. In OPM, the user determines the operation policy, such as the amount of product (order) and feedstock, and the starting time of the scheduling. The user then selects the objective function between minimization of the makespan and tardiness. Basically, minimizing the makespan is chosen as the objective function, except in cases where the product demand cannot be satisfied because of unexpected circumstances, such as very urgent ordering. To solve the scheduling problem that has been formulated as MILP,
Ind. Eng. Chem. Res., Vol. 49, No. 24, 2010 Table 1. Equipment Information for the Case Study Filter Data Dryer Data
max. shared reactor type volume (kg) on/off condenser filter type dryer type R1 R2 R3 R4 R5 R6 R7 R8 R9 R10 R11
GL GL GL SS GL SS SS GL GL GL SS
7800 7500 4400 3700 7800 3100 5300 4300 3000 4200 5200
on on on off on off off on on off off
R5 R3 R2
F1 F2 F3 F4 F5 F6 F7
R1
F/P F/P H/F H/F H/F H/F H/F
D1 D2 D3 D4
F/D F/D F/D F/D
R9 R8
Table 2. Product Information and Processing Sequence for the Case Study product
required type
A-1 R R R A-2 R F D B R C-1 R R C-2 R R R D R D E R R D F R D G R D H R R F I R F
SS (5500) GL (3500) GL (6000) SS (4000) H/F F/D GL (3000) GL (3000) SS (4000) SS (2000) SS (6000) SS (6000) SS (4000) F/D GL (4000) SS (3000) F/D SS (4000) F/D GL (6000) F/D GL (3600) GL (5500) F/D GL (3200) F/P
required bottleneck processing production net quantity type time (h) time (h) (kg) (kg) SS (1000) GL (6000) SS (3500) SS (4500)
36
48
550
2400
40
52
275
2000
SS (3500) GL (3000)
36 64
80 82
280 390
911 1020
SS (3000) SS (6000)
76
90
500
1000
75
172
200
3000
80
125
435
500
35
58
390
2835
GL (4000)
60
75
405
2000
GL (2500)
47
143
260
911
GL (3600)
45
150
459
1020
SS (5000)
the system is implemented with GAMS 21.3 and are solved using CPLEX.9 The dual simplex method is used with bestbound search and strong branching. Finally, the results of optimization are shown via OPM. OPM provides a Gantt chart with various time units (week, day, and hour), a production profile according to the product and/or batch, and data associated with the feedstock (type, amount, and flow rate) for inventory management. The GUI is designed to be user-friendly. To obtain the optimized results, the user must follow the steps that are presented by the system, as shown in Figure 3. This figure shows some of the system interfaces, such as the main, input, and output screens. The following section will present real examples that show how the system can be used to illustrate the applicability of the proposed model. 6. Applications The detailed scheduling models and the proposed heuristic techniques are applied to four real processes. First, two
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differently sized problems are considered; to represent a smallsized problem, 6 products will be scheduled, whereas, to represent a large-sized problem, 11 products will be scheduled. These respective problems are then divided into two cases, according to whether or not heuristic techniques are used; they are used in Cases 2 and 4. The objective of using these cases is to determine the production schedule of an industrial batch plant for a time period of three months that satisfies customer orders for various products distributed throughout the time period. We have assumed that all production processes, for each of the four cases, are completed without delay. 6.1. Process and Product Descriptions. For the production of pharmaceutical intermediates, this study assumes that the final products are manufactured through three processing stages. In the first stage, raw materials are fed into reactors at a specific temperature, pressure, and pH. Next, the products of the reactor are then fed into the filtering processes, to eliminate impurities. Finally, through drying processes, if necessary, the purified products are shipped out as final products. The manufacturing system considered in this study consists of 22 key pieces of equipment: 11 parallel machines in the reaction process, 7 parallel machines in the filtering process, and 4 parallel machines in the drying process. The detailed characteristics of the equipment are as follows: • Two reactor types, according to material (such as glass lining (GL) and stainless steel (SS)), are considered. The GL reactor is used for specific conditions (i.e., high temperature and/or pressure). Thus, the GL reactor could be used regardless of the operating conditions, whereas the SS reactor is used only for relatively low temperatures and pressures. • The capacity of the reactor is the range of 3100-7800 L, as shown in Table 1. • As mentioned in section 2, the “on reactors” and “off reactors” are also classified based on whether a reactor is equipped with an exclusive condenser or shares a condenser with other reactors. For example, Reactors 1 and 5 in Table 1 cannot use a condenser simultaneously, because they share an overhead condenser. • Two types of filters are considered, such as a filter press (FS) and a HEPA filter (HF). Generally, the FS-type filter is used to separate liquids from solids, such as in a slurry, whereas the HF-type filter is used for more-acute processing, such as the separation of very small particles. A description of the orders and their production sequences are summarized in Table 2. As shown in this table, nine final and two intermediate products will be manufactured; A-1 and C-1 are intermediates of the final products A-2 and C-3, respectively. Each product has a particular processing sequence. For example, A-1 is manufactured though four sequential reactions without filtering and drying, while A-2 requires a single piece of equipment for the reaction, filtering, and drying processes (see Table 2). Finally, the amount of order, bottleneck, and processing times associated with the total production time (makespan) are also summarized in Table 2. 6.2. Results and Discussion. The scheduling problem sizes and the corresponding computational performance are shown in Table 3. These results can be obtained using a 3.21 GHz
Table 3. Computational Results and Performance Comparison for Cases
Case Case Case Case
1 2 3 4
(without heuristics) (with heuristics) (without heuristics) (with heuristics)
product
continuous variables
binary variables
relative gap
CPU time (s)
6 (A-D) 6 (A-D) 11 (A-I) 11 (A-I)
3727 2305 6832 3829
378 271 523 406
0.289 0.179
1200.198 2.890
0.343
1200.187
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Figure 4. Gantt chart of the production schedule for Case 4 (11 products, with heuristics).
personal computer (PC). The termination criterion is that the relative gap, which represents the relative difference between the optimized objective value and the best possible integer solution calculated, is set to 2%, or when the computational time reaches 1200 s. The reason for the relatively high setpoint for the relative gap is that small companies that manufacture pharmaceutical intermediates require a relatively faster solving process responses, rather than very high accuracy of the objective values. It should be mentioned that, for higher accuracy, the setpoint of the relative gap could be tighter, although doing so results in a penalty, in terms of computational time. In Table 3, two types of case studies (such as 6 and 11 products) are examined. For Cases 1 and 2, where 6 products are considered, Case 2 was embedded with heuristics and shows higher performance, with regard to both accuracy of the objective values and computational time: the accuracy is approximately two times higher and the computational time is 415 times faster than the results without heuristics. Moreover, for Cases 3 and 4, with 11 products, a feasible solution could not be found for Case 3, whereas for Case 4, which includes heuristics, a good solution was obtained within the set time (1200 s). As shown in Table 3, the proposed heuristic techniques make the scheduling system become more efficient and effective as the number of products increases. The reason for this observation is that the number of continuous and binary variables decreases when heuristic techniques are used.10 Accordingly, the optimal values (relative gaps) and CPU times are decreased significantly when heuristic techniques are used. Furthermore, it is also clear, from the studied examples, that, if larger examples of the problem are to be solved, it is more necessary to limit the combinational nature of the problem (see Table 3). This means that an increase in the number of products from 6 to 11 products corresponds to an increase in the number of continuous and binary variables of two conditions; for example, the rates of the binary variables of the two conditions (i.e., from Case 1 to Case 3 and from Case 2 to Case 4) increase by ∼138% and ∼150%, respectively. Figure 4 shows a Gantt chart of the optimized result of Case 4. The units (reactors, filters, and dryers) for all of the products are shown with Gantt charts. The total operation time (makespan) is ∼1800 h (75 days are required for implementation); this is a 32% reduction, relative to the previous result manually sched-
Figure 5. Comparison of the operation times with the by-person results.
uled (2640 h), as shown in Figure 5. Since the current production scheduling in a real plant is determined by person based on a rule-of-thumb method, the reduced time compared with the results manually scheduled implies a practical benefit by the new scheduling. Figure 5 also shows that the reduction rate of the case where 6 products are considered reaches ∼46.2%, compared to the result manually scheduled. Therefore, Table 3 and Figure 4 reveal that that the proposed scheduling system is helpful not only for higher computational performance but also for obtaining higher accuracy of optimized solutions. 7. Conclusions In this study, a new scheduling model is proposed to optimize a production schedule for the manufacturing process of pharmaceutical intermediates. A basic scheduling model is formulated as mixed integer linear programming (MILP), according to the general characteristics and operating rules of batch plants. In addition, two heuristic techniquessthe preclassification of equipments and the sequential two-stage optimizationswere then proposed to relax the complexities of the scheduling problems. The proposed model was applied to a real pharmaceutical intermediates manufacturing (PIM) plant. To validate the proposed method, the solutions to variables such as the computational times and the relative gaps from four different
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optimization problems were compared. The results of the case studies revealed the following: (1) The number of continuous and binary variables decreases when using heuristic techniques. Constraints are reduced by ∼40% and CPU times are reduced significantly; for a specific case (Case 2), it is reduced by at least a factor of 415. (2) The total operation time (makespan) can be decreased by 31.8%, compared to previous results (by manuals). On the basis of the proposed models and heuristic techniques, a packaged scheduling tool for the manufacturing process of pharmaceutical intermediates has been developed, using Visual C++ and EXCEL for the interface. The scheduling system has been installed in a real pharmaceutical intermediates production plant and has been helpful not only in creating the optimal production schedules and but also in managing feedstock and inventories. If other realistic characteristics that act as constraints in the model are updated, the proposed model and heuristic techniques are expected to be useful when applied to other processes that have similar configurations, such as those found in the food and paper industries.
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Duei ) due date for order i Irir ) reactor number r required for order i Irjirj ) reactor number r of order i that uses unit j RVir ) maximum volume needed for reactor number r of order i MuVj ) maximum volume of unit j Roir ) reactor number r in order i that uses overhead condenser k Joik ) unit j equipped with overhead condenser k Cli ) cleanup time of order i H ) time horizon Rwi ) available date of raw material for order i Binary Variables xin ) order i that begins at time slot n uij ) order i used in unit j zirj ) unit j that corresponds to reactor number r for order i yik ) order i related to overhead condenser k wijn ) order i that begins in unit j at time slot n Continuous Variables
This work was supported by the Ministry of Education (MOE) of Korea by its BK21 Program.
TSijn ) start time of order i in unit j at time slot n TESin ) start time of order i at time slot n TEFin ) finish time of order i at time slot n TETi ) end time at which all order i are finished TSTi ) start time at which all order i are started STin ) amount of order i at time slot n
Nomenclature
Literature Cited
Indices
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Acknowledgment
i ) order (product) j ) unit (equipment) l ) unit group r ) reactor number n ) time slot for equipment k ) overhead condenser Parameters Jiji ) order i that can use unit j Nuil ) number of unit j in unit group l used for order i Ugujl ) unit j included in unit group l Rqi ) requirement (demand) of order i Stoi ) stock of order i Cyclei ) cycle time of order i Proci ) processing time of order i Batchi ) batch size of order i Nbi ) necessary number of batches for order i
ReceiVed for reView April 14, 2010 ReVised manuscript receiVed September 17, 2010 Accepted October 4, 2010 IE100880F
