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Ind. Eng. Chem. Res. 2001, 40, 228-233
Optimal Scheduling for a Multiproduct Batch Process with Minimization of Penalty on Due Date Period Jun-Hyung Ryu,† Ho-Kyung Lee, and In-Beum Lee* Department of Chemical Engineering, Automation Research Center, Pohang University of Science and Technology, San 31, Hyoja-Dong, Pohang 790-784, Korea
To make scheduling problems more realistic, we develop a new scheduling model with penalty functions for earliness and tardiness. Instead of a fixed due date, the scheduling problem is approached by taking the due date as a time period, which can handle flexible due dates and flexible customer requirements with penalty parameter modifications. We solve this optimization problem by the simulated annealing method and demonstrate its effectiveness with three examples. 1. Introduction In batch plants, scheduling is one of the most important subjects for increasing productivity. So far, its objective function is to minimize the makespan from the macroscopic point of view, as pointed by Kim et al. (1996). This paper addresses scheduling problems of batch processes from the viewpoint of micro level scheduling. The goal is the maximization of customer satisfaction. It can be achieved by producing products prior to the due date, minimizing the so-called tardiness. Ku and Karimi (1990) suggested a single-product campaign scheduling model with a tardiness penalty. Their research is limited because they did not take the inventory cost into account. To merely plan schedules for the minimization of tardiness may cause a large inventory, which increases the production costs and requires unnecessary investments. Thus, it is necessary to consider inventory and customer satisfaction at the same time. However, the due date of orders can be variable, and a more advanced scheduling model should consider this. Therefore, we formulate a model by introducing the new concept of the due date period. It is more realistic and flexible to take the due date as a period rather than a fixed point because this can reflect more current customer demands, preventing unnecessary inventories and raising customer satisfaction at the same time. We formulate a new model and show its effectiveness with three examples using simulated annealing. 2. Due Date Period Model Formulation The general scheduling problem assumes that a customer orders one product with a prespecified due date. Let its product have an index i with a stage j. Cij is the completion time of the ith product in the jth stage. Let N denote the number of products to be produced and M the number of stages through which products must pass. Our research aims to determine the completion time of product i, Cij, and the optimal production sequence, S. We take S as the single-product campaign sequence. The due date of product i is di, and the penalty To whom all correspondence should be addressed. Tel.: +8254-279-2274.Fax: +82-54-279-2699.E-mail:
[email protected]. † Current address: Centre for Process Systems Engineering, Imperial College of Science, Technology and Medicine, London SW7 2BY, U.K.
of product i is Pi. Previous operating data give the information for the parameters of earliness and tardiness. They are given in advance. The penalty is calculated through multiplication of a penalty parameter by the difference between the due date and the completion time. There are two kinds of penalties, earliness and tardiness. The earliness and tardiness penalty parameters for product i are Ri and βi, respectively. The minimum completion time has generally been taken as the objective function. The makespan is the completion time of the last product produced at the final stage, CNM. IST (intermediate storage tank) policy determines the main scheduling problem. UIS (unlimited intermediate storage), NIS (no intermediate storage), FIS (finite intermediate storage), and ZW (zero wait) are examples. (Refer to Kim et al., 1996). The difference in these polices is the number of ISTs between stages. We consider UIS, which takes infinite ISTs between stages. Thus, the completion time of product i at stage j can be decided using
Cij ) max[Cij-1, Ci-1j] + tij
(1)
as shown in Figure 1. The difference between Cij and the customer due date gives rise to the penalty. If the completion time extends past the due date, then a tardiness penalty is charged for violating the expectations of the customer. The tardiness penalty (Ti) is calculated by multiplying the tardiness penalty parameter βi by the difference. Also, if the completion time is earlier than the due date, an earliness penalty is charged. In this case, we calculate the earliness penalty (Ei) by multiplying the earliness penalty parameter Ri by the difference. Then, we can decide the penalty as follows:
Ei ) max[0, di - CiM]
(2)
Ti ) max[0, CiM - di]
(3)
N
Penalty ) f(S) )
(RiEi + βiTi) ∑ i)1
(4)
The objective function becomes the minimization of f(S). Whereas we generate the schedule considering the customer’s due date, in a real plant, the conditions of customers can be varied to get their orders. The order of customers can be intermediate products determined by other plant conditions or market distribution cir-
10.1021/ie000375t CCC: $20.00 © 2001 American Chemical Society Published on Web 11/21/2000
Ind. Eng. Chem. Res., Vol. 40, No. 1, 2001 229
Figure 3. Diagram of parameter change and due date period.
Figure 1. Decision of completion time in UIS policy.
Figure 4. Perfomance of SA in the objective function.
Figure 2. Objective function with earliness and tardiness.
cumstances. They want the material (products) to be delivered according to their own conditions, which makes the due date a period rather than a fixed time. Therefore, suppliers should produce the products and plan the schedule considering this. The “due date period” has a starting and ending time. We should change our objective function to place an earliness penalty if the product is ready before the initial due date (IDDi). If the product is not ready even after the final due date (FDDi), we charge the tardiness penalty according to the difference between FDDi and the completion time. The product that is completed at a time between the starting and ending due dates is not charged with a penalty. Thus, eqs 2 and 3 should be changed to
Ei ) max[0, IDDi - CiM]
(5)
Ti ) max[0, CiM - FDDi ]
(6)
IDDi < CiM < FDDi
(7)
The penalty with this adapting due date period is shown in Figure 2. Our scheduling problem makes the following structure for the UIS policy: N
Objective function: min Penalty ) f(S) )
(RiEi + ∑ i)1
βiTi) ∀i ) 1-N Constraints: Cij ) max[Cij-1, Ci-1j] + tij ∀i ) 1-N, j ) 1-M
(1)
Ei ) max[0, IDDi - CiM] ∀i ) 1-N
(5)
Ti ) max[0, CiM - FDDi] ∀i ) 1-N
(6)
The concept of the due date period can also be analyzed by thinking about the parameters realistically. The earliness and tardiness penalties are given by multiplication of the penalty parameter and the difference between the completion time and the due date. Past research only took the penalty parameter as fixed, but it is more realistic to take it as dependent on the difference between the due date and the completion time. If, for the real schedule, the difference is bigger than a certain limit, a more serious penalty should be applied to that schedule. Customers want to charge a different penalty value depending on the degree of difference between the due date and the completion time. If producers give customers products seriously violating the due date, the customers want to charge additional displeasing penalty charges. Thus, a more serious penalty parameter has to be used. On the other hand, if the difference is small, we can take that schedule with fewer penalties or even without any penalty. Therefore, some period of time before and after the due date is also charged with no penalty, which is the due date period. The penalty function has been taken as a linear relation between the penalty and the difference. If we are to modify the penalty realistically, we should change it to a piecewise linear function. In this paper, we take one point both before and after the due date. This can be generalized into a period. However, even to take divided periods is important by itself. This point changes the parameter values. We call this time point the earliness parameter changing point (EPCPi) or the tardiness parameter changing point (TPCPi). Then, the penalty function should be the sum of four penalties. The penalties of four zones, PE1i, PE2i, PT1i, and PT2i are defined as follows:
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PE2i ) max[0, di - CiM] if CiM < EPCPi
(7)
PE1i ) max[0, di - CiM] if di > CiM > EPCPi PT1i ) max[0, CiM - di] if di < CiM < TPCPi
Table 1. Data for Example 1 (a) processing time
(8)
unit product
0
1
2
3
4
5
0 1 2 3 4
18 6 5 4 4
22 17 22 19 15
12 6 9 14 3
21 5 3 3 4
9 10 14 19 14
13 7 12 13 20
(9)
PT2i ) max[0, CiM - di] if CiM > TPCPi (10) N
Penalty ) f(S) )
(R2iPE2i + R1iPE1i + β1iPT1i + ∑ i)1 β2iPT2i) (11)
The parameters before EPCP and after TPCP are greater than others. Also, if a due date period case with different parameters is introduced, the penalty functions are also changed. Equations 7-11 should be changed to eqs 12-16. Figure 3 shows the diagram of this concept.
EPCPi < IDDi < FDDi < TPCPi
(12)
PE2i ) max[0, EPCPi - CiM] if CiM < EPCPi (13)
(b) due date and penalty parameter product due date IDD FDD penalty parameter earliness tardiness
PT2i ) max[0, CiM - TPCPi] if CiM > TPCPi (16) N
Objective function: Penalty ) f(S) )
(R2iPE2i + ∑ i)1
fixed due date due date period
Certain types of optimization problems have the characteristic that the calculation time for the solution increases exponentially as the number of variables increases. This is called NP-hard or NP-complete. The scheduling problem is one of these types of optimizations. However, traditional mathematical programming methods cannot find the exact solution in the case of many variables. To overcome this limitation in the optimization problem, heuristic or evolutionary algorithmic methods have been proposed. Studies on these topics include artificial intelligence, genetic algorithms, neural network, tabu search, and simulated annealing (SA). We are going to solve the scheduling problem with SA, because it can be applied to very large-scale optimization problems. As an algorithmic method imitates or compares with natural phenomena, so SA was derived from the physical annealing of metal. Its first use in optimization was by Kirkpatric et al. (1983). Ku and Karimi (1991b) were the first to use SA in the scheduling problem of chemical batch processes. Their objective function was to minimize the completion time under a UIS policy. Also, Patel et al. (1991) used SA for the preliminary design of a multiproduct process. Tandon et al. (1995) presented a SA methodology for the scheduling of multiple products on parallel units with tardiness penalties. The basic simulated algorithms is of the form SELECT an initial state i ∈ S; SELECT an initial temperature T > 0;
2
3
4
95 80 110
130 115 145
162 147 177
108 93 123
137 122 152
3 5
13 5
17 12
13 17
16 12
optimal sequence
penalty
makespan
0f3f1f4f2 0f3f1f4f2 0f3f4f1f2
0 0 0
162 162 152
Table 3. Data for Example 2 (a) processing time unit product
0
1
2
3
4
5
6
7
0 1 2 3 4 5 6 7 8
23 21 15 18 12 21 12 18 20
19 8 5 10 8 17 16 12 12
13 7 5 21 14 9 14 19 10
9 6 9 12 8 16 21 10 19
18 18 10 9 7 16 6 5 8
12 23 15 24 20 6 9 20 18
21 21 11 12 16 19 7 20 21
17 18 23 24 18 12 17 6 23
R1iPE1i + β1iPT1i + β2iPT2i) (13) 3. Simulated Annealing
1
Table 2. Result of Example 1
PE1i ) max[0, IDDi - CiM] if IDDi > CiM > EPCPi (14) PT1i ) max[0, CiM - FDDi] if FDDi < CiM < EPCPi (15)
0
(b) due date and penalty parameter penalty parameter product
due date
IDD
FDD
earliness
tardiness
0 1 2 3 4 5 6 7 8
179 274 232 254 197 173 220 156 130
154 249 207 229 172 148 195 131 105
204 299 257 279 222 198 245 181 155
23 19 13 9 8 12 21 17 21
8 7 6 18 23 21 18 15 5
Table 4. Result of Example 2 optimal sequence fixed due 8 f 7 f 5 f 0 f 4 f 6 f 3 f 2 f 1 date due date 8 f 7 f 5 f 0 f 4 f 6 f 3 f 2 f 1 period 8f7f5f0f4f2f6f1f3 8f5f7f0f4f2f6f3f1 8f5f7f0f4f6f2f3f1
penalty makespan 0
274
0 0 0 0
274 274 284 276
SET temperature change counter t ) 0; repeat GENERATE STATE j, a neighbor of i; calculate δ ) f(j) - f(i); if δ e 0 then i :) j; else if random(0,1) < exp(-δ/T) then i :) j; n :) n + 1; until n ) N(t); t :) t + 1; T :) T(t); until a stopping criterion is true To solve the scheduling problem with SA, the objective function should be the due date penalty value, and the solution is the sequence of products. In addition, the
Ind. Eng. Chem. Res., Vol. 40, No. 1, 2001 231 Table 5. Data for Example 3 processing time of each product product unit
1
2
3
4
5
6
7
1 2 3 4 5 6
9 14 18 6 9 4
6 19 12 8 3 4
7 20 13 5 23 9
13 14 17 9 21 11
23 12 4 8 32 11
10 15 9 19 24 7
2 12 21 33 5 12
(b) demand order information product
1
2
3
4
5
6
7
order quantity batch size no. of batches
42 000 21 000 2
45 000 15 000 3
2 0000 5 000 4
3 0000 15 000 2
27 000 9 000 3
3 0000 15 000 2
3 0000 10 000 3
product
1
2
3
4
5
6
7
due date period penalty parameter earliness tardiness
100 ( 5
130 ( 5
50 ( 5
150 ( 5
420 ( 5
300 ( 5
350 ( 5
3 1
2 3
3 2
4 4
1 3
2 3
3 2
(c) order informationscase 1
(d) order informationscase 2 product
1
2
3
4
5
6
7
fixed due date change period earliness penalty β1 earliness penalty β2 tardiness penalty R1 tardiness penalty R2
100 (15 3 8 1 5
130 (15 2 7 3 7
50 (15 3 6 2 6
150 (15 4 5 4 8
420 (15 1 7 3 9
300 (15 2 8 3 6
350 (15 3 9 2 7
(e) order informationscase 3 product
1
2
3
4
5
6
7
due date period change period penalty parameter earliness β1 earliness β2 tardiness R1 tardiness R2
100 ( 20 (35
130 ( 20 (35
50 ( 20 (35
150 ( 20 (35
420 ( 20 (35
300 ( 20 (35
350 ( 20 (35
3 8 1 5
2 7 3 7
3 6 2 6
4 5 4 8
1 7 3 9
2 8 3 6
3 9 2 7
Figure 5. Gantt chart of Example 3 (due date period case 3).
temperature (t) and the Boltzmann constant (k) are decided as the joined value kt, which is only affected by the penalty value differentiations that are randomly
generated for enough times, about 3000 times for 8 variables as in the case of Ku and Karimi (1990). The kt value is the absolute middle value of the difference
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Table 6. Result of Example 3 case
optimal sequence
penalty
1 (fixed due date) 2 (due date period) 3 (due date period and different parameter)
2f4f1f3f6f7f5 1f4f2f3f6f7f5 1f4f2f3f6f7f5
458 1733 1292
of this generation. An efficient way of finding the optimal sequence in a short time is to start from many initial sequences and compare them repeatedly. However, a more efficient way to decide t and k will have to be developed. Figure 4 shows the performance of SA in a scheduling problem. Figure 4 indicates graphically how the penalty function is minimized for Example 3. 4. Examples To plan the optimal production schedule, two conditions should be set. One is to find the production sequence, and the other is to find the operating time, including starting time, ending time, holding time, etc., by the sequence that is affected by an IST policy. For the first condition, a general heuristic is to array the products by their due date. Each product with an early due date goes first. However, as we will show in the next section, this earliest due date first (EDD) rule is not always the optimal production plan. We applied our method to the following three examples. The processing time data and due date are generated at random. These problems are calculated in a reasonable time with IBM personal computers. Example 1. For a UIS scheduling problem with five products and six stages, the processing time data and product information are presented in Table 1. We find the schedule of both the fixed case and the due date period case for the minimum penalty. Two solution sequences are found for the latter case. One is the same as the fixed case, but the other is a better solution because its completion time is shorter than the fixed case solution with the same minimum penalty. This means the due date period case can be helpful in reducing the makespan and achieving customer satisfaction. The result is in Table 2. Example 2. At this time, we treat the problem with nine products and seven stages with the data in Table 3. For the due date period case, we get four solution sequences with the same minimum penalty and different makespans. Two of these sequences are optimal with the minimum makespan and penalty. In this case, the same makespan is found for the fixed case and the due date period case. However, we can say that different operating policies are prepared by the due date period expanded from the fixed due date, as in Table 4. Example 3. We test a more realistic scheduling problem with the due date parameter changing case. Order demands of products need more than one batch, and the single-product campaign (SPC) method is used. If many products are to be manufactured, only one product is finished, and then the other product can be started. For this case, the completion time is calculated after the last batch of the previous product is done. The data are in Tables 4 and 5, and the results are in Table 6. The optimal sequence is different for the different cases. We test with randomly generated penalty parameters and the resulting penalty value does not seem serious.
Figure 5 is a display of the Gantt chart of the optimal sequence. The dashed period below the chart is the due date. 5. Conclusion We suggest a scheduling model that can handle flexible due dates and flexible customer requirements with penalty parameter modifications. This is possible by the concept of the due date penalty minimization and the due date period. It is a more advanced production policy than taking the due date as a fixed point. We find the solution using SA because it has the advantage of finding good solutions in multivariable NP-complete problems in a relatively short computational time. The due date periods of products may overlap. Also, more than one sequence meets the due date criterion with the minimum earliness and tardiness penalty. In that case, the sequence with shorter makespan CNM is the optimum solution. Acknowledgment This work was supported by KOSEF (Grant 995-1100005-2). Nomenclature Indices i ) product j ) stage Greek Letters Ri, βi ) earliness and tardiness penalty parameter of product i R1i, β1i ) tardiness penalty parameter between beforetardiness-changing point and due date of product i R2i, β2i ) tardiness penalty parameter of after-tardinesschanging point and due date of product i Variables Cij ) completion time of the ith product at the jth stage for the multiproduct case di ) due date of product i Ei ) earliness penalty of product i EPCPi ) earliness parameter changing point of product i FDDi ) final due date of product i IDDi ) initial due date of product i M ) number of batch stages in the plant N ) number of products to be produced Pi ) penalty of product i PE1i ) earliness penalty between after-earliness-changing point and due date PE2i ) earliness penalty of before-earliness-changing point PT1i ) tardiness penalty between before-tardiness-changing point and due date PT2i ) tardiness penalty of after-tardiness-changing point S ) production sequence Ti ) tardiness penalty of product i tij ) processing time of product i at stage j TPCPi ) tardiness parameter changing point of product i
Literature Cited (1) Kim, M.; Jung, J. H.; Lee, I. Optimal Scheduling of Multiproduct Batch Processes for Various Intermediate Storage Policies. Ind. Eng. Chem. Res. 1996, 35, 4058.
Ind. Eng. Chem. Res., Vol. 40, No. 1, 2001 233 (2) Kirkpatrick, S.; Gelatt, C. D.; Vecchi, M. P. Optimization by Simulated Annealing. Science 1983, 220, 671. (3) Ku, H.; Karimi, I. Scheduling in Serial Multiproduct Batch Process with Due-Date Penalties. Ind. Eng. Chem. Res. 1990, 29, 580. (4) Ku, H.; Karimi, I. Scheduling algorithm for serial multiproduct batch product processes with tardiness penalties. Comput. Chem. Eng. 1991a, 15, 283. (5) Ku, H.; Karimi, I. An evaluation of simulated annealing for batch process scheduling. Ind. Eng. Chem. Res. 1991b, 30, 163. (6) Patel, A. N.; Mah, R. S. H.; Karimi, I. A. Preliminary design
of multiproduct noncontinuous plants using simulated annealing. Comput. Chem. Eng. 1991, 15, 451. (7) Tandon, M.; Cummings, T.; LeVan, D. Scheduling of multiple products on parallel units with tardiness penalties using simulated annealing. Comput. Chem. Eng. 1995, 19, 1069.
Received for review April 3, 2000 Revised manuscript received September 18, 2000 Accepted September 21, 2000 IE000375T
