Integrated Scheduling for Polyvinyl Chloride Processes - American

Jul 11, 2006 - CRD Research Institute, LG Chemical Corporation, 104-1 Munji-dong, ... Several batch/continuous processes' scheduling problems have bee...
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Ind. Eng. Chem. Res. 2006, 45, 5729-5737

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PROCESS DESIGN AND CONTROL Integrated Scheduling for Polyvinyl Chloride Processes Min-gu Kang, Sookil Kang, and Sunwon Park* Department of Chemical and Biomolecular Engineering, Korea AdVanced Institute of Science and Technology, 373-1 Guseong-dong, Yuseong-gu, Daejeon 305-701, Republic of Korea

Ho-kyung Lee CRD Research Institute, LG Chemical Corporation, 104-1 Munji-dong, Yuseong-gu, Daejeon, 305-380, Republic of Korea

This study considers a scheduling algorithm and mathematical models for a real polyvinyl chloride (PVC) plant. According to general rules of the PVC plant, the basic PVC scheduling model is formulated as mixed integer linear programming (MILP). This model includes production, inventory management, packing, and shipment processes. The basic PVC scheduling model is customized for two PVC production processes with different characteristics, resulting in two detailed PVC scheduling models. An optimization algorithm has been developed to solve these scheduling models. This algorithm is a hybrid algorithm combining the genetic algorithm for integer programming (IP) and the interior point method for linear programming (LP). A heuristic technique has also been developed to reduce the number of solving LPs. Finally, the optimized results of the models were analyzed. 1. Introduction Polyvinyl chloride (PVC) is a major commercial polymer and is widely used as a raw material in various chemical and petrochemical products. About 300 million tons of PVC are produced per year all over the world.1 In Korea, PVC occupies 12% of the domestic polymer market and about 1.3 million tons are produced annually.2 Korean PVC plants have various production processes, and the capacity of each process is about 10-100 thousand tons per year. In such a large plant, optimizing plant operations is difficult. Even trivial errors in scheduling and inventory management may lead to great financial losses in a large plant. In other words, improving scheduling and inventory management can increase the profits of PVC plants. However, few PVC plant scheduling problems have been previously reported. Shah et al.3 studied a multipurpose plant scheduling for a conceptual batch/continuous PVC plant. Bretelle et al.4 presented a framework for a scheduling software (S/W) development, but the researchers did not specifically define the scheduling problem. Therefore, an in-depth study on a real PVC plant scheduling problem is required. The PVC process includes both batch units and continuous units. Several batch/continuous processes’ scheduling problems have been studied.5-7 These studies focused on minimizing the makespan (or connectivity) of the batch or continuous units. The studies concentrated on production processes. However, since inventory and shipment processes also affect the profits and costs of a PVC plant, all the processes should be considered in PVC production scheduling. In this study, all PVC production processes, including production, inventory management, pack* To whom correspondence should be addressed. Tel.: +82-42-8693920. Fax: +82-42-869-3910. E-mail: [email protected].

ing, and shipment, are considered to formulate the PVC production scheduling problem. Our target system, the PVC plant, has more than 20 processes and produces about 760 thousand tons annually. Moreover, each process has various operating rules. Despite the large capacity and complex operating rules of the PVC plant, current production scheduling is performed by a rule of thumb. Therefore, the PVC production schedule needs to be optimized. Optimizing PVC production will result in cost reductions and profit improvement. A scheduling model and a solving algorithm should be developed so that the PVC plant operates more efficiently. 2. Target System The PVC scheduling problem includes all the PVC production units from feed tanks to packing machines. A schematic diagram of the target system is shown in Figure 1. The PVC production processes, processes A and B, consist of a VCM (vinyl chloride monomer) tank, five polymerization reactors (RE), two blowdown tanks (BT), four slurry tanks (ST), two stripping columns, two dryers, silos of different sizes, and packing machines. Each production process has a different number of silos and packing machinesseight silos and four packing machines for process A and six silos and six packing machines for process B. The PVC plant can be divided into two sections: a batch section and a continuous section. The batch section consists of a number of parallel batch reactors and blow-down tanks. The continuous section consists of the stripping columns, dryers, and silos. VCM, the raw material in PVC, is sufficiently supplied by a VCM plant directly connected to the PVC plant and recovered from the blow-down tanks and the stripping columns in the PVC plant. In the polymerization reactors, grades are differently specified according to the additives. Changing the composition of the additives leads to a grade change, and setup time occurs.

10.1021/ie051007y CCC: $33.50 © 2006 American Chemical Society Published on Web 07/11/2006

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Figure 1. Schematic diagram for the PVC plant.

The blow-down tanks and the slurry tanks are used as intermediate storage tanks connected to the continuous units. In the stripping columns, unreacted VCM is separated and recycled to the feed tank. The large amount of water, which is used for sealing and quenching the polymerization reactors, should be removed in the dryer. The produced PVC is stored in the silos, which are connected to the dryers. The stored PVC is packed by the packing machines and then stored in the warehouse. Polymerized PVC is shipped in three wayssan F/C (flacon), a paper bag, and bulk. Bulk PVC is shipped directly from the silos, and both the F/C type and the paper bag type are shipped from the warehouses. The packed products are exported or sold in the domestic market. Each PVC production process has two production lines from the blow-down tanks to the silos, as shown in Figure 1. These production lines cause problems in the scheduling models. Since the production lines in process A produce the same grade of PVC at the same time, they can be assumed to be a single production line. However, each production line in process B produces different grades of PVC. This difference between processes A and B makes the PVC scheduling problem complicated. The objective of the PVC scheduling problem is cost minimization; it can be formulated by (1).

Cost ) Demand delay cost + Inventory cost

(1)

The demand delay cost is a penalty cost caused by delayed demand. Process A produces two grades of PVC. They are mostly ordered in large amounts for export. Export demand is usually shipped by container ship. If producing and packing the export products are delayed, the demand delay cost increases. Therefore, reducing the demand delay cost is the key to optimizing process A. Process B produces 10 grades of PVC, which are usually ordered in small amounts. The total capacity of the plant’s silos is large enough for the monthly production. The number of silos, however, is less than the number of grades of PVC. PVC that cannot be stored in the silos should be packed as F/C and paper bag types. Therefore, how to manage the two production lines in terms of inventory is the key to optimizing process B. 3. PVC Scheduling Model 3.1. Basic PVC Scheduling Model. Mathematical models for the PVC scheduling problems can be formulated according

to the operating rules. As mentioned in the previous section, the objective of the PVC scheduling problem is cost minimization. The mathematical form of eq 1 is shown as follows: I

min Cost )

3

∑ ∑ (DeCm + InvCm) i)1 m)1

(2)

In eq 2, m and i represent the product type and time interval, respectively. The demand delay cost, DeCm, can be divided into three types, which are represented in the following equations: bulk in eq 3, domestic packing in eq 4, and export packing in eq 5.

DeC1 ) J

R1

I

∑ ∑ j)1 i)1

i

[Pcj × max{

∑ l)1

K

(Dlj1 - Rb

∑(Cb

k)1 J

DeC2 ) R2

- Cl-1)),0}] (3)

ljk

I

∑ ∑Pc Dij3Xs2 j)1 i)1 j

(4)

ij

DeC3 ) J

R3

I

∑ ∑ j)1 i)1

i

[Pcj × max{

∑ l)1

K

(Dij3 - Rp

∑(Cp

k)1

- Cl-1)),0}] (5)

ljk

In eqs 3-5, j and k represent the product name and silo name, respectively. The inventory cost, InvCm, is also divided into three types, which are represented as follows: J

I

∑ ∑IijPc j)1 i)1

InvC1 ) R4 J

I

i

(6)

j

K

(Rp∑(Cp ∑ ∑[Pc × max{∑ j)1 i)1 l)1 k)1

InvC2 ) R5

j

ljk

- Cl-1) -

Dlj2 - Dlj3),0}] (7) J

InvC3 ) R6

I

∑ ∑Pc Dij3Xs1 j)1 i)1 j

ij

(8)

Xs1ij and Xs2ij, which appear in eqs 4 and 8, are binary variables representing whether demands are delayed or not.

Ind. Eng. Chem. Res., Vol. 45, No. 16, 2006 5731

Xs1ij )

{ {

i

1 if Dij3

In eq 20, the binary variable, XIijk, is used to indicate whether grade j is stored in silo k at time interval i or not.

K

(Dij3 - Rp∑(Cp ∑ l)1 k)1

ljk

- Cl-1)) > 0

, ∀i, ∀j (9)

J

XI ∑ j)1

0 otherwise

Xs2ij )

J

i

1 if Dij3

-Cl-1))>0 ljk

, ∀i, ∀j

0 otherwise

J

Xij ) NR, ∑ j)1

∀i

(11)

The setup time between the successive batches producing different products is as follows:

Ci ) Ci-1 + 12.0 +

St

J

∑ Xt ,

2j ) 1

ij

∀i

(12)

The binary variable, Xtij, shows whether the grades of PVC produced at time intervals i - 1 and i are the same or not. Xtij is represented as follows:

Xtij )

{

1 if Xij * Xi-1,j , ∀i, ∀j 0 otherwise

(18)

K

ijk

e Ns, ∀i

{

1 if Iijk > 0 , ∀i, ∀j, ∀k 0 otherwise

XIijk )

(10)

In our target system, the continuous units are operated with fixed flow rates. As a result of these characteristics, the continuous units only increase the processing time of the batch units, the number of which is equal to the number of the reactors.

e 1, ∀i, ∀k

∑ ∑ XI j)1 k)1

K

(Dlj2 +Dij3 -Rp∑(Cp ∑ l)1 k)1

ijk

(19)

(20)

At time interval i, the PVC product in a silo should be shipped as bulk or packed in either F/C or paper bag, as shown in eq 21. Each packing machine is connected to two or three silos. As seen in eq 22, a packing machine should pack the PVC of one of the silos connected to the packing machine at a time. The variable l indicates each packing machine, and Cl represents the sets of silos that are connected to packing machine l. The binary variable, Xbijk, is used to indicate whether grade j stored in silo k is shipped as bulk at time interval i or not. When grade j stored in silo k is shipped at time interval i, Xbijk is 1. Otherwise, Xbijk is 0. Another binary variable, Xpijk, is used to indicate whether grade j stored in silo k is packed at time interval i. When grade j in silo k is packed at time interval i, Xpijk is 1. Otherwise, Xpijk is 0. J

J

Xb ∑ j)1

+ ijk

Xp ∑ j)1

J

e ijk

XI ∑ j)1

ijk

e 1, ∀i, ∀k

(21)

J

∑ ∑ Xp k′∈C j)1

(13)

ijk′

) 1, ∀i, ∀l

(22)

l

The total stock amount of the silos at time interval i can be calculated using the total stock amount at time interval i - 1, the shipped amount of the bulk type, and the packing amount of the F/C and paper bag shipping methods. The shipping amount of bulk PVC is proportional to the number of silos and the shipping time. The packing amount of the F/C and paper bag methods is proportional to the number of packing machines and the packing time. K

K

K

∑Iijk ) k)1 ∑Ii-1,jk + QjXij - Rbk)1 ∑(Cb

k)1

ijk

- Ci-1) -

ijk

- Ci-1), ∀i, ∀j (14)

K

Rp

∑(Cp

k)1

The amount of PVC stored in a silo is limited to the capacity of the silo as follows:

0 e Iijk e IUk, ∀i, ∀j, ∀k

(15)

The shipped amounts of bulk and packed PVC are also limited to the capacities of the silos and packing machines.

Ci-1 e Cbijk e Ci, ∀i, ∀j, ∀k

(16)

Ci-1 e Cpijk e Ci, ∀i, ∀j, ∀k

(17)

Since the different grades should be stored in different silos, as shown in eq 18, the maximum number of grades stored in the silos is constrained to the number of the silos, as seen in eq 19.

Xpijk ) Xbijk )

{ {

0 if Cpijk ) Ci-1 , ∀i, ∀j, ∀k 1 otherwise

(23)

0 if Cbijk ) Ci-1 , ∀i, ∀j, ∀k 1 otherwise

(24)

As shown above, the basic PVC scheduling model, which consists of eqs 2-24, can be formulated as MILP (mixed integer linear programming). The decision variables of this model are Xij, Cbijk, and Cpijk. The production amount of grade j at time interval i is determined by the binary variable, Xij. Cbijk and Cpijk are continuous variables for the flow rates of bulk shipping and packing. 3.2. Detailed Scheduling Model. The basic PVC scheduling model is formulated using the general operating rules of the PVC plant. However, adding some constraints to the model is required in order to use it for processes A and B. The modified models are called detailed scheduling models. 3.2.1. Process A. In the detailed scheduling model for process A, two constraints are added to the basic PVC scheduling model. According to eq 11, the number of operated batches is equal to the number of reactors. In addition to eq 11, the number of batch units at time interval i is determined by the new variable, R1ij.

Xij ) NRR1ij, ∀i, ∀j

(25)

R1ij is a binary variable that indicates whether grade j is produced

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at time interval i or not. According to eqs 11 and 25, the values of Xij are either 0 or 5. In real processes, frequent grade changes are not preferred. Each grade of PVC should be continuously produced for at least a day. In the scheduling model, the time interval is 12 h, so each grade of PVC should be produced for at least two time intervals. This constraint for grade changes is represented as follows:

R1i-1,j - R1i-2,j e R1ij e R1i-1,j - R1i-2,j + 1, ∀i, ∀j

(26)

To apply this model to process A, current operating data are required. R1-1j and R1oj can be obtained from the real process. 3.2.2. Process B. The basic PVC scheduling model can also be applied to process B. Process B has more complex operating rules than process A. Process B has two production lines and five batch units and produces two different grades of PVC at the same time. The capacities of the production lines are different, so the five batch units are connected to the production lines in the ratio of either 3:2 or 4:1 depending on the capacities. The new variable, R2ijr, is proposed to determine the operating ratio. At time interval i, the production ratio is 3:2 if r is 1 and the ratio is 4:1 if r is 2. According to the ratio relationship, 10 grades produced in process B are divided into two groups. The first group, set J1, contains eight grades that are produced by the larger-capacity production line. The second group, set J2, has the other two grades. Each grade is represented as a number from 1 to 10. The sets are defined as follows:

J1 ) {3,4,5,6,7,8,9,10}

(27)

J2 ) {1,2}

(28)

Considering the ratio relationship, Xij can be expressed by R2ijr as follows:

Xij′ ) 3R2ij′1 + 4RR2ij′2, ∀j′ ∈ J1, ∀i

(29)

Xij′′ ) 2R2ij′′1 + R2ij′′2, ∀j′′ ∈ J2, ∀i

(30)

In addition, each production line can produce only one grade at a time.

∑r ∑j′ R2

i,j′,r

) 1, ∀j′ ∈ J1, ∀i

(31)

R2 ∑r ∑ j′′

i,j′′,r

) 1, ∀j′′ ∈ J2, ∀i

(32)

Similar to eq 26 in process A, once a grade is produced in process B, the grade should be produced for at least two time intervals. This constraint is represented as follows:

R2i-1,j,r - R2i-2,j,r e R2i,j,r e

R2i-1,j,r - R2i-2,j,r + 1, ∀i, ∀j, ∀r (33)

Grades 4-7 of set J1 should be produced through a given sequence. The sequence is 4 f 5 f 6 f 7. One or two of the grades in the sequence can sometimes be omitted, but the order of the sequence does not change. For example, if grade 6 is not produced, the production sequence should be 4 f 5 f 7.

M(1 -

∑r R2

i

)g i4r

∑∑R2 i′)1 r

i′br

, ∀i, ∀b ) 5,6,7

(34)

M(1 -

i

∑r R2

M(1 -

)g i5r

∑ ∑ R2 i′)1 r

∑r R2

)g i6r

, ∀i, ∀c ) 6,7

i′cr

(35)

i

∑∑R2 i′)1 r

i′7r

, ∀i

(36)

Equations 34-36 are big-M constraints. M is a very large constant (M >> 1). In the current operation, grade 6 should be produced only under the operating ratio of 3:2. This limitation is shown as follows:

R2i62 ) 0, ∀i

(37)

4. Solving Algorithm 4.1. Algorithms for Complex Scheduling Problems. During the past decade, various attempts have been made to solve complex and large scheduling problems.8 Scheduling problems are usually formulated as mixed integer linear programming (MILP); several efficient techniques such as mathematical programming and some heuristic methods have been proposed to solve the scheduling problems.9 However, these techniques have some drawbacks when they are applied to large-scale problems. Due to the curse of dimensionality,10 the search space of mathematical programming techniques increases exponentially with the size of problems. Other heuristic techniques generally require several assumptions to solve complex problems. To overcome these limitations, GA (genetic algorithm method) has been used for complex and large-scale scheduling problems. However, GA requires lots of computational time to find the optimal solution. Due to the problem of slow convergence, hybrid optimization techniques combining the advantages of GA and other methods have been proposed by several researchers.8-15,18 Some hybrid techniques use efficient decoding,14 combining heuristic methods,9,15,18 and ordinal optimization.8 Among these, the approach most used for MILP problems is a hybrid method that combines LP techniques and GA. LP techniques converge very fast, but they cannot handle the integer parts of MILP. In contrast, GA can search the global optimum solution and also handle integer variables, but GA needs a lot of time just before the convergence to the optimal solution. A hybrid model in which GA and the LP technique are used for searching the integer programming (IP) solution space and optimizing the LP part, respectively, can offset the drawbacks of both approaches and generate a solution close to the global optimum faster. Many researchers use the simplex method as a mathematical technique (for example, see refs 8 and 12). When the problem has many feasible solutions, however, the simplex method may not find the optimal solution among the feasible solutions or may require a lot of computational load. It is very difficult to transform the LP parts of the PVC scheduling models into standard forms that can be solved by the simplex method. Moreover, the LP parts have many feasible solutions. Therefore, another LP solver is required. The interior point method is competitive with the simplex method. Especially in large problems, the interior point method often outperforms the simplex method.16 The interior point method represents significant developments in theory and practice. The advantages of the simplex method and ellipsoid algorithm are combined in the interior point method. From a theoretical point of view, the interior point method includes efficient algorithms and interesting geometric ideas; from a practical point of view, this method

Ind. Eng. Chem. Res., Vol. 45, No. 16, 2006 5733 Table 1. Problem Size of Processes A and B name

process A

process B

time interval (i) grade (j) silo (k) packing machine (p) integer variables continuous variables constraints

60 2 8 4 120 1 920 8 627

60 10 6 6 1 200 7 200 30 767

Table 2. Comparison of the Interior Point Method and the Simplex Method (in Seconds) relative gap

interior point

simplex

0.1% 1.0% 10.0%

1.39 0.78 0.13

5.6 2.9 1.18

can be applied to the large-scale problems that arise in many applications.16 Actually, PVC scheduling problems are large-scale problems, as shown in Table 1. In the scheduling problems, the interior point method is more efficient than the simplex method. To compare the performance of the two methods, integer variables were fixed and the same input data and termination criterion were provided with the same computer. The termination criteria of both cases were represented by relative gaps. Thirty tests per each relative gap were performed, and the average values are shown in Table 2. The interior point method finds the optimal solutions much faster than the simplex method. 4.2. Hybrid Method of GA and the Interior Point Method. In this study, the hybrid method combining GA and the interior point method was proposed and applied to the PVC plant scheduling. Additionally, a heuristic technique was proposed to reduce the number of solving LPs. As mentioned in the previous section, the PVC scheduling models are formulated as MILP. In the basic PVC scheduling model, the decision variables are Xij, Cbijk, and Cpijk. Xij is an integer variable, and Cbijk and Cpijk are continuous variables. In the detailed PVC scheduling models, however, the integer decision variable is replaced by R1ij for process A and R2ijr for process B. While the integer decision variables are optimized by GA, the other variables, Cbijk and Cpijk, are optimized by the interior point method. After the integer decision variables are determined by GA, the models become LPs, and their decision variables are Cbijk and Cpijk. The interior point method is used to achieve “good” solutions of the LP models within a reasonable amount of time. Since the scheduling models have many decision variables, a large amount of time is required to solve the models. To reduce the computational load, the number of solving LPs should be reduced. The lower bound of the demand delay cost (LBD) is required for reduction. Using the production sequence determined by GA, the LBD can be calculated. As shown in eq 39, the LBD can be the lower bound of the objective value as well. The objective value for the determined production sequence cannot have a lower value than the LBD. When the LBD is higher than the highest objective value of the previous generation, LP optimization by the interior point method is not required.

Demand delay cost g LBD Cost )

(38)

Demand delay cost + Inventory cost g LBD (39)

This approach can reduce the number of LP optimizations and solve the PVC scheduling problems within a reasonable time.

Figure 2. Reduction of the number of solving LPs.

To get the global optimum value set, integer variables and continuous variables should be solved by the hybrid method, as shown in the following procedure. (Step 1) Create feasible solutions of Xij using random values and the constraintsseqs 11, 25, 29, and 30. (Step 2) Evaluate the LBD using Xij. (Step 3) Using the LBD of Xij, check whether the LP problem should be solved or not. If it should, continue with step 4. Otherwise, go to step 5. (Step 4) Solve the LP problems using the interior point method. (Step 5) Evaluate the fitness value of the GA. (Step 6) Check the termination criteria. If they are met, go to step 8. Otherwise, continue with step 7. (Step 7) Reproduce feasible solutions of Xij for the next generation. Then, go to step 2. (Step 8) Terminate. Values of Xij are initialized as random values in step 1. The number of Xijs is equal to the population size. In this study, the population size is 20. In step 2, the LBD of Xij can be calculated without using the interior point method. In step 3, whether the LP problem should be solved or not is determined. If the calculated LBD of Xij is higher than the other objective values in the current and previous generations, the LP problem of this Xij does not need to be solved. Because the objective value of the Xij is higher than the LBD, this Xij may not survive in the next generation. In this case, the LBD of the Xij becomes the objective value of the Xij. In step 4, the LP problem of the selected Xij is solved by the interior point method. The termination criterion of the interior point method is that the relative gap between the objective value of the LP problem in the current iteration and that in the previous iteration is less than 0.1%. In step 5, the fitness values of Xij can be obtained from the results of the interior point method. In this study, the fitness function of the GA is equal to the objective function of each PVC scheduling model. In step 6, after the objective values of all Xijs are obtained, the termination criterion of the GA is checked. The termination criterion is that the relative gap between the average of the objective values in the current generation and that in the previous generation is less than 0.1%. If the termination criteria are not satisfied, the Xijs for the next generation are reproduced through selection, crossover, and mutation processes in step 7. The whole procedure of the proposed algorithm is shown in Figures 2 and 3. 4.3. Comparison of Computational Load. Before optimizing the real PVC scheduling problem, the efficiency and effectiveness of the proposed algorithm should be tested. In the test, the

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Figure 3. Solving algorithm.

Figure 4. Comparison of the costs for process A.

Table 3. Performance Comparison of GAMS and the Proposed Algorithm (Units of Time, Seconds; Units of Obj, Money)

case

number of grades

1 2 3 4 5 6

2 2 10 10 21 21

GAMS

proposed method

BPIS

time

obj

time

obj

545 545 546 282 126 488 143 590

78.3 90.1 700.7 633.2

545 600 546 335 126 600 143 720

5.6 17.7 55.0 81.7 175.8 294.0

545 603 546 321 126 560 143 724 386 393 377 196

convergency and computational time of the proposed algorithm are compared with those of commercial software. GAMS is very well-known optimization software. The design of GAMS has incorporated ideas drawn from relational database theory and mathematical programming and has attempted to merge these ideas to suit the needs of strategic modelers.17 The computational time and objective value of the proposed algorithm are compared with those of GAMS. Detailed scheduling models for processes A and B are solved by GAMS and the proposed algorithm. Table 3 shows the comparison results for process A. These results can be obtained by using a 3.2 GHz personal computer (PC). Cplex 7.0 is used as a solver in GAMS. The termination criterion is that the relative gap of GAMS is less than 0.1%. The relative gap represents the relative difference between the objective value optimized by GAMS and the best possible integer solution calculated by GAMS. In Table 3, BPIS is the best possible integer solution calculated by GAMS. Considering the complexity of the problem, finding the optimal solution is difficult. Therefore, the best possible integer solution as the

Figure 5. Cumulative demand amount and cumulative shipped amount for process A: (a) bulk demand; (b) paper bag demand.

lower bound of the optimal solution is used so that the convergency of the proposed approach is compared with that of GAMS. In Table 3, three kinds of test setss2, 10, and 21 gradessare tested. Different cases use different data sets: the number of silos, prices, and batch amounts. For cases 1 and 2 where 2 grades are used, the proposed algorithm is about eight times faster than GAMS while both solvers obtain similar objective values. For cases 3 and 4, the number of grades is 10. Both solvers can also obtain similar results, but the computational time of the proposed algorithm is much shorter than GAMS. For cases 5 and 6 with 21 grades, GAMS cannot find a feasible solution while the proposed algorithm obtains a good solution in 5 min. As shown in Table 3, the proposed algorithm becomes more efficient and effective than GAMS as the number of grades increases. The detailed scheduling model for process B has more complicated constraints (i.e., production sequence constraints, 4 f 5 f 6 f 7). This model was tested for 10 grades. As a result, the proposed algorithm can obtain good solutions in about 100 s while GAMS cannot find any feasible solution. Through these tests, it is proved that the proposed algorithm is better than GAMS for solving PVC scheduling models.

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Figure 6. Production plan for process B.

Figure 7. Comparison of the costs for process B.

5. Applications The detailed scheduling models and the proposed optimization algorithm are applied to real processes A and B. Then, for the convenience of applying the models and the algorithm, scheduling software specialized for the target system was developed.

Figure 8. Product sheet in the PVC scheduling software.

5.1. Process A. Equations 1-26 are proposed as the detailed PVC scheduling model for process A. This model can be solved by the proposed algorithm. In Figure 4, the optimized result shows an approximate 56% decrease in the cost. This decrease is achieved by reducing the demand delay cost. Figure 5 shows the cumulative demand amount and cumulative shipped amount for process A. In this figure, the x-axis represents the time horizon and the unit of the x-axis is a day. The unit of the y-axis is a ton. The numbers 1 and 2 shown in Figure 5 are the names of the PVC grades. As shown in the dotted circles in Figure 5, the cumulative demand amount is higher than the cumulative shipped amount of the plant data, so demand delay cost occurs. However, because the optimized cumulative shipped amount is higher than the cumulative demand amount, demand delay cost is reduced. 5.2. Process B. In the real PVC plant selected as the target system, the current production scheduling is designed by a rule of thumb. In the scheduling, the operation ratio of process B is fixed at 4:1. Recently, demand for grade 2 has been increasing. Due to production capacity limitations, the demand change for grade 2 makes manual scheduling difficult. Therefore, a production schedule with the optimized sequence of the production ratios is required for process B. To get the optimized sequence of the production ratios, the detailed scheduling model for process B should be solved using the proposed algorithm. Figure 6 shows the optimized sequence of the production ratios. The figure also shows that the grades of set J1 are produced with the given sequence, 4 f 5 f 6 f 7. In Figure 7, the optimized result and the current operational result are compared. Reducing the inventory cost results in an approximate 20% decrease in cost. 5.3. System Development. On the basis of the proposed models and the algorithm, an integrated PVC scheduling S/W has been developed. This S/W was developed using Fortran and EXCEL. The S/W consists of three kinds of sheets: a main

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sheet for the menu, input sheets for receiving input data, and output sheets for analyzing the optimized results. Figure 8 shows one of the output sheets. The S/W has been installed in the real PVC plant and has been helpful in creating optimal production schedules. 6. Conclusion In this study, PVC scheduling models are proposed to optimize a production schedule for the entire process of the PVC plant. A basic PVC scheduling model is formulated as mixed integer linear programming (MILP) according to the general operating rules of the PVC plant. In addition, detailed scheduling models are proposed for two PVC processes with different characteristics. To represent the different characteristics of each process, some constraints are added to the basic PVC scheduling model. To solve the PVC scheduling models more efficiently, a hybrid method combining GA for integer programming (IP) with the interior point method for linear programming (LP) was developed. A heuristic technique was also proposed to reduce the number of solving LPs. To test the efficiency of the proposed algorithm, the computational time and convergency of the proposed algorithm was compared with those of GAMS. Through the tests, it was proved that the proposed algorithm is better than GAMS for solving PVC scheduling models. For two detailed scheduling models of processes A and B, the optimized results were analyzed. Process A produces two grades of PVC. They are mostly ordered in large amounts and for export. The export demand is usually shipped by container ship. If producing and packing of the export products are delayed, the demand delay penalty increases. Therefore, reducing the demand delay cost is the key to optimizing process A. The optimization result shows an approximate 56% decrease in the cost. This decrease is achieved by reducing the demand delay cost. Process B produces 10 grades of PVC, which are usually ordered in small amounts. The total capacity of the plant’s silos is large enough for monthly production amounts. The number of silos, however, is less than the number of PVC grades. PVC that cannot be stored in the silos should be packed as F/C and paper bag types. Therefore, managing the two production lines considering inventory is a key to optimizing process B. The optimization result shows a 20% decrease in the cost, and the result was obtained by reducing the inventory cost. However, these results were obtained based on historical demand data. Therefore, these results are only expected values. According to the market, the amount of decrease of the cost can be changed. However, it is certain that the proposed models and algorithm show more profitable results than manual scheduling. On the basis of the proposed models and algorithm, an integrated PVC scheduling S/W has been developed. This S/W was developed using Fortran and EXCEL. The S/W has been installed in the real PVC plant and has been helpful in creating optimal production schedules. If some constraints are modified, the proposed model and algorithm are expected to be applicable to other PVC plants that have the same configurations. Likewise, it is expected that the proposed algorithm might be applied to other polymerization plants with similar configurations. Acknowledgment This work was partially supported by the IMT 2000 (Project No.: 00015993), Center for Ultramicrochemical Process Systems sponsored by KOSEF and BK21.

Nomenclature Indices i ) time interval j ) product k ) silo l ) packing machine m ) product type 1 ) domestic and bulk 2 ) domestic and packing 3 ) export and packing r ) production ratio 1 ) 3:2 2 ) 4:1 Variable int ) integer variable con ) continuous variable bin ) binary variable Ci ) con, completion time of time interval i Cbijk ) con, bulk shipping time of grade j from silo k at time interval i Cpijk ) con, packing time of grade j from silo k at time interval i DeCm ) con, delay penalty of packing type m Iijk ) con, bulk inventory amount of grade j in silo k at time interval i InvCm ) con, inventory cost of packing type m R1ij ) bin, production assignment of grade j at time interval i in process A R2ijr ) bin, production assignment of grade j with production ratio r at time interval i in process B Xij ) int, production amount of grade j at time interval i Xbijk ) bin, bulk shipping assignment of grade j from silo k at time interval i XIijk ) bin, inventory assignment of grade j of silo k at time interval i Xpijk ) bin, packing assignment of grade j from silo k at time interval i Xs1ij ) bin, domestic and packing demand delay assignment of grade j at time interval i Xs2ij ) bin, packing demand delay assignment of grade j at time interval i Xtij ) bin, successive production assignment of grade j at time interval i Parameters Dijm ) demand data Qj ) batch production amount of grade j IUk ) capacity of silo k M ) big-M Nr ) number of reactors Pcj ) price of grade j Ptj ) processing time of grade j Rb ) bulk shipping rate Rp ) packing rate St ) setup time Literature Cited (1) Chemical market research inc. Petrochemical profile. Chem. J. Weekly 2003, 13 (26), 2. (2) Korea Petrochemical Industry Association. The demand and the shipment for major petrochemical products. Petrochem. Bull. 2004, 4, 72. (3) Shah, N.; Liberis, L.; Izumoto, E.; Henson, R. Integrated batch plant design: A polymer plant case study. Comput. Chem. Eng. 1996, 20 (Suppl 2), s1233-s1238.

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(12) Parsons, D. J. Optimizing silage harvesting plans in a grass and grazing simulation using the revised Simplex method and a genetic algorithm. Agric. Syst. 1998, 56 (1), 29-44. (13) Goldberg, D. E. Genetic algorithm in search, optimization, and machine learning; Addison-Wesley: Reading, MA, 1989. (14) Bierwirth, D.; Mattfeld, D. C. Production scheduling and rescheduling with genetic algorithms. EVol. Comput. 1999, 7 (1), 1-17. (15) Hart, E.; Ross, P. A heuristic combination method for solVing Jobshop scheduling problems; Springer-Verlag: New York, 1998; pp 845854. (16) Bertsmas, D.; Tsitsiklis., J. N. Introduction to linear optimization; Athena Scientific: Belmont, MA, 1997. (17) Brooke, A.; Kendrick, D.; Meeraus, A.; Raman, R. GAMS: A USER’S GUIDE; GAMS development corporation: New York, 1998. (18) Naarur, N.; Vrat, P.; Duongsuwan, W. Production planning and scheduling for injection modeling of pipe fittings. Int. J. Prod. Econ. 1997, 53, 157-170.

ReceiVed for reView September 8, 2005 ReVised manuscript receiVed May 23, 2006 Accepted June 6, 2006 IE051007Y