Single-Stage Scheduling of Multiproduct Batch Plants: an Edible-Oil Deodorizer Case Study
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Ind. Eng. Chem. Res. 2010, 49, 8657–8669
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Single-Stage Scheduling of Multiproduct Batch Plants: An Edible-Oil Deodorizer Case Study
Songsong Liu,† Jose M. Pinto,‡ and Lazaros G. Papageorgiou*,†
Centre for Process Systems Engineering, Department of Chemical Engineering, UniVersity College London, Torrington Place, London WC1E 7JE, U.K., and Process Systems R&D, Praxair Inc., 39 Old Ridgebury Rad, Danbury, Connecticut 06810
This article considers the short-term scheduling of a single-stage batch edible-oil deodorizer that can process multiple products in several product groups. Sequence-dependent changeovers occur when switching from one product group to another. Based on the incorporation of products into product groups, mixed integer linear programming (MILP) models are proposed for two scenarios, with and without backlogs. Then, the models are successfully applied to a real-world case with 70 product orders over a 128-h planning horizon. Compared with a literature model developed for a similar problem, the proposed models exhibit significantly better performance.
1. Introduction In the past decade, a large number of optimization models and approaches have been proposed for batch scheduling and planning. A number of reviews on the planning and scheduling of batch processes have been presented in the literature.1-6 Initially, discrete-time formulation models using the state-task network7 (STN) or resource-task network8 (RTN) were used for batch scheduling problems. Because discrete-time formulations become extremely large for large-size problems and finer discretizations, several techniques9-11 have been proposed to reduce the computational effort of large discrete-time MILP models. Increasing attention has been paid to continuous-time formulations to overcome the difficulties of discrete-time formulations. Pinto and Grossmann12 proposed a continuous-time MILP model for the short-term scheduling of batch plants with multiple stages. They then improved this work with the assumption of preordering of orders.13 Zhang and Sargent14 used the RTN representation to develop a mixed integer nonlinear programming (MINLP) formulation for the scheduling of general plant topologies and then solved the problem with iterative MILP models. Cerda et al.15 developed an MILP model for the short´ term scheduling of a single-stage batch multiproduct plant with nonidentical parallel units/lines based on a continuous-time representation. Karimi and McDonald16 developed slot-based MILP formulations for the short-term scheduling of single-stage multiproduct plants with parallel semicontinuous units. Ierapetritou and Floudas17 presented an MILP formulation for the short-term scheduling of multiproduct/multipurpose batch processes based on an STN representation. Mendez et al.18 presented a two-stage approach for the ´ batching and scheduling problem of single-stage multiproduct batch plants. Hui and Gupta19 proposed a general formulation for short-term scheduling of single-stage multiproduct batch plants with nonidentical parallel units with order-sequencedependent constraints. Chen et al.20 introduced an MILP formulation for the short-term scheduling of multiproduct batch plants with parallel units, as well as two heuristic rules to reduce
* To whom correspondence should be addressed. Tel.: +44-20-76792563. Fax: +44-20-7383-2348. E-mail: l.papageorgiou@ucl.ac.uk. † University College London. ‡ Praxair Inc.
model size. Lim and Karimi21 considered both batching and scheduling in a slot-based MILP formulation for the short-term scheduling of single-stage batch plants with parallel units and multiple orders per product. Castro and Grossmann22 proposed a multiple-time-grid, continuous-time MILP model for the shortterm scheduling of single-stage multiproduct plants. He and Hui23 proposed an evolutionary approach for single-stage multiproduct scheduling with parallel units. They then extended their own work by constructing a new set of heuristic rules24 and proposing a heuristic rule-based genetic algorithm.25 Erdirik-Dogan and Grossmann26 proposed two production planning models and a rolling-horizon (RH) algorithm for the production planning of parallel multiproduct batch reactors with sequence-dependent changeovers. Liu and Karimi27-29 proposed a series of slot-based and sequence-based MILP models for the scheduling of multistage multiproduct batch plants with parallel units and unlimited and no intermediate storage. Prasad and Maravelias30 and Sundaramoorthy and Maravelias31 both considered the simultaneous batching and scheduling of multistage multiproduct processes in MILP formulations. Erdirik-Dogan and Grossmann32 proposed a slot-based continuous-time MILP formulation and a bilevel decomposition scheme for the shortterm scheduling of multistage multiproduct batch plants. Shaik and Floudas33 improved the model of Ierapetritou and Floudas17 and proposed an RTN-based unit-specific event-based model for short-term scheduling of batch plants. Castro et al.34 aggregated all batches of a product into a single task instead of considering one processing task per batch for the short-term batching and scheduling of single-stage multiproduct plants. Marchetti and Cerda35 presented an MILP formulation for the ´ short-term scheduling of single-stage multiproduct batch plants with parallel units using a unit-dependent precedence-based representation. Castro and Novais36 proposed a new RTN-based multiple-time-grid MILP formulation for the short-term scheduling of multistage batch plants with multiple product batches and sequence-dependent changeovers. Kopanos et al.37 addressed the production scheduling and lot-sizing problem in a multiproduct yogurt production line of a dairy plant. This article addresses the short-term scheduling problem of a single-stage edible-oil deodorizer, adapted from the real-world case study discussed in Kelly and Zyngier,38 in which processing changeovers occur only during switching from one product group to another. In this case, the production schedule of groups,
10.1021/ie1002137 2010 American Chemical Society Published on Web 08/09/2010
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3. Mathematical Formulations The proposed models for the batch edible-oil deodorizer scheduling problem are MILP formulations, based on our previous work,39,40 using the classic TSP (traveling salesman problem) formulation and discrete/continuous time representation. For the batch scheduling problem, we consider two scenarios. In scenario 1, no backlog is allowed, and all orders should be processed and delivered within their time windows. In scenario 2, backlog is allowed, and the orders can be processed and delivered after the due dates. The following notation is used for the models:
Figure 1. Orders, products, and product groups.
Indices g, g′ ) product groups i ) products o ) orders t, t′ ) time periods Sets G ) product groups Gt ) product groups whose windows contain time period t, Gt ) {g| min RTo e t e max DTo} i∈I ,o∈Oi i∈Ig,o∈O ˜ t ) product ggroups whose windows i start before time period t, G ˜ Gt ) {g|t g min RTo} o∈Ig I ) products Ig ) products in group g It ) products whose windows contain time period t, It ) {i|min RTo e t e max DTo} o∈Oi o∈Oi ˜t ) products whose windows start before time period t, I ) ˜t I {i|t g min RTo} o∈Oi O ) orders Oi ) orders for product i Ot ) orders whose windows contain time period t, Ot ) {o|RTo e t e DTo} ˜ ˜ Ot ) orders whose windows start before time period t, Ot ) {o|t g RTo} T ) time periods Parameters BCo ) backlog cost of order o BNmax ) maximum batch number in time period t t BSmax ) maximum batch size BSmin ) minimum batch size BT ) batch time CCgg′ ) changeover cost from group g to group g′ Do ) demand of order o DTo ) time period of due date of order o ICi ) inventory cost of product i M ) a large number PCi ) processing cost of product i Pri ) price of product i RTo ) time period of release time of order o VU ) upper bound of inventory for product i i θU ) upper bound of processing time in time period t t τgg′ ) changeover time from group g to group g′ Binary Variables Egt ) 1 if group g is processed during time period t; 0 otherwise Fgt ) 1 if group g is the first one in time period t; 0 otherwise Lgt ) 1 if group g is the last one in time period t; 0 otherwise Zgg′t ) 1 if group g immediately precedes group g′ in time period t; 0 otherwise ZFgg′t ) 1 if group g in period t - 1 immediately precedes group g′ in period t; 0 otherwise
rather than products, is of greater concern. The objective of this work is to develop efficient mixed-integer linear programming optimization approaches for the short-term scheduling of singleunit batch plants. The processing of products is incorporated into that of product groups, and the schedule of products groups is considered first. The processing of a product group involves the processing of multiple products in the group, and the processed products are used to satisfy the demands of the orders. The remainder of this article is organized as follows: The scheduling problem is described in section 2. The mathematical formulations of the proposed models are presented in section 3. Section 4 presents the computational results of the case study. In section 5, a literature model is compared with the proposed MILP models. Finally, some concluding remarks are made in section 6. 2. Problem Statement This problem considers a single-stage multiproduct batch deodorizer that processes multiple products. There are multiple customer orders for a product that belongs to a certain product group (see Figure 1). Each order has a corresponding release time and due date. The total planning horizon is several days. The single deodorization tray in the deodorizer cannot contain different products at the same time, which means that the deodorizer can process only one product in a batch. Sequencedependent downtime restrictions occur during switching from one product group to another. The following assumptions have been made in the problem: (1) Each product belongs to only one group. (2) Each order is specific to only one product. (3) Each order is released/due at the beginning/end of a time period. (4) No order can be processed before its release time. (5) Different orders of the same product can be processed together. (6) The single batch time is fixed. (7) Multiple deliveries are allowed for each order after its release time. In this scheduling problem, the product groups, products, orders, release time, due date and demand of each order, changeover times, batch time, and minimum and maximum batch sizes are given, and the objective is to determine the processing sequence and times of product groups, processing amount and batch number of each product, inventory levels, and deliveries/sales for each order, so as to maximize the total profit, involving sales revenue, processing cost, changeover cost, inventory cost, and backlog cost, if backlog is allowed.
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Integer Variable Nit ) number of batches of product i during time period t Continuous Variables Bot ) backlog amount for order o at time period t OIgt ) ordering index of group g during time period t Pit ) amount of product i processed during time period t Qot ) product amount processed for order o during time period t Sot ) sales amount for order o in time period t Tgt ) processing time for group g during time period t Vot ) inventory amount for product o at the end of time period t 3.1. Model of Scenario 1. In scenario 1, as backlogs and processing/deliveries after the due dates of the orders are not allowed, only product group g ∈ Gt, product i ∈ It, and order o ∈ Ot can be assigned to time period t for processing. Assignment and Sequencing. Assuming that each time period comprises the processing of at least one product, only one product group can be the first or the last one in each time period
OIg't - (OIgt + 1) g -M(1 - Zgg't), ∀t ∈ T, g, g' ∈ Gt, g * g'
(9)
If a product group is not processed during a time period, its corresponding order index is 0 OIgt e MEgt, ∀t ∈ T, g ∈ Gt
(10)
Here, the cardinality of set Gt, |Gt|, can be used as M. Processing Time. There should be at least one batch to be processed if a product group is assigned to a period; otherwise, no batch of all its products is processed Egt e
i∈Ig∩It
∑
Nit e BNmaxEgt, t
∀t ∈ T, g ∈ Gt
(11)
The processing time of a product group is the total number of batches multiplied by the batch processing time Tgt ) BT
i∈Ig∩It
∑
Nit,
∀t ∈ T, g ∈ Gt
(12)
g∈Gt
∑F ∑L
gt
) 1,
∀t ∈ T ∀t ∈ T
(1)
The total of the processing and changeover times is limited by the total available time in each time period
gt
) 1,
(2)
g∈Gt
g∈Gt
∑T
gt
+
g∈Gt g'∈Gt
∑ ∑Z gt gg'tτgg'
+
g∈Gt-1 g'∈Gt
∑ ∑ ZF e θU, t
gg'tτgg'
e θU , t
∀t ∈ T - {1} ∀t ∈ {1}
(13) (14)
If a product group is not processed during a particular time period, then it cannot be either the first or the last one in that period Fgt e Egt, Lgt e Egt, ∀t ∈ T, g ∈ Gt ∀t ∈ T, g ∈ Gt
g∈Gt
∑T
+
g∈Gt g'∈Gt
∑ ∑Z
gg'tτgg'
(3) (4)
Processing Amount. For each product, the amount processed in a period is limited by the number of batches multiplied by the minimum and maximum batch sizes BSminNit e Pit e BSmaxNit, ∀t ∈ T, i ∈ It
Each product group, except the first one, is processed after another product group
(15)
g∈Gt,g*g'
∑
Zgg't ) Eg't - Fg't,
∀t ∈ T, g' ∈ Gt
(5)
The process amount for each product in a period is the sum of the process amounts for the related orders that can be processed in that period Pit )
Similarly, each product group, except the last one, is processed before another product group
o∈Oi∩Ot
∑
Qot,
∀t ∈ T, i ∈ It
(16)
g'∈Gt,g'*g
∑
Zgg't ) Egt - Lgt,
∀t ∈ T, g ∈ Gt
(6)
For each product group processed last in a period, there is a changeover from this product group to the first one in the next period
Inventory Level. The inventory level for an order in a period is the inventory in the previous period, plus the production amount, minus the sales, which only occur within the time window Vot ) Vo,t-1 + (Qot - Sot)| o∈Ot, ˜ ∀t ∈ T, o ∈ Ot
g∈Gt-1
∑
(17)
ZFgg't ) Fg't, ∀t ∈ T - {1}, g' ∈ Gt
(7)
The inventory level of each product is limited by its capacity
Similarly, for each product group processed first in a period, there is a changeover from the last one in the previous period to this product group
˜ o∈Oi∩Ot
∑
Vot e VU, i
˜ ∀t ∈ T, i ∈ It
(18)
g'∈Gt-1
∑
ZFgg't ) Lg,t-1,
∀t ∈ T - {1}, g ∈ Gt
(8)
Demand. The sales for each order should take place only within the time window, and the total sales should be no greater than the demand
Ordering Index. The ordering index of a product group processed later in a sequence is larger than that of an earlierprocessed product group
t)RTo
∑S
DTo
ot
e Do,
∀o ∈ O
(19)
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Objective. The objective is to maximize the total profit, including the sales revenue, processing cost, inventory cost, and changeover cost max
tgRTo
∑S
ot
e Do,
∀o ∈ O
(22)
∑∑ ∑
PriSot ICiVot -
i∈I t∈T o∈Oi∩Ot
∑∑ t∈T i∈It
PCiPit gg′Zgg′t
Objective. The backlog cost is also included in the objective, in addition to the cost terms included in eq 20 max
∑∑ ∑
i∈I t∈T o∈Oi∩Ot ˜
∑ ∑ ∑ CC t∈T g∈Gt g′∈Gt
∑∑ ∑
PriSot -
i∈I t∈T o∈Oi∩Ot ˜
∑ ∑ PC P t∈T i∈It ˜
i it
-
t∈T-{1} g∈Gt-1 g′∈Gt
∑ ∑ ∑ CC
gg′ZFgg′t
(20)
∑∑ ∑
ICiVot -
Model DEO-S1 for scenario 1 of the problem is described by constraints 1-19 with eq 20 as the objective function. 3.2. Model of Scenario 2. In scenario 2, backlogs and processing/deliveries after the due dates of the orders are ˜ ˜ allowed, so product group g ∈ Gt, product i ∈ It, and order o ∈ ˜ t can be assigned to each time period t for processing. By O ˜ ˜ replacing the sets Gt, It, and Ot in eqs 1-17 by the sets Gt, It, ˜ and Ot, respectively, we can obtain the constraints for the model of scenario 2. Equation 18 can be used in the model for scenario 2 without any change. Demands and Backlogs. The backlog of an order is activated only during the time period after its due date. At a time period t, the backlog of each order is equal to its demand minus the total sales from its release time to time period t Bot ) Do -
∑ ∑ ∑ CC t∈T g∈Gt g′∈Gt ˜ ˜
i∈I t∈T o∈Oi∩Ot ˜ gg′Zgg′t
∑ ∑ BC B t∈T o∈Ot ˜
o ot
˜ t∈T-{1} g∈Gt-1 g′∈Gt ˜
∑ ∑ ∑ CC
gg′ZFgg′t
(23)
Model DEO-S2 for scenario 2 of the problem is described by constraints 1-17 after revision and constraints 18, 21, and 22 with eq 23 as the objective function. 4. Case Study In this section, we apply the proposed models to the realworld edible-oil deodorizer scheduling problem. A planning horizon of 128 h is considered. There are 70 orders (O1-O70) for 30 products (P1-P30) that belong to 7 different groups (PG1-PG7). The total demand is 4156 tons, and the demand for each order is given in Table 1. The release time and due date of each order (see Table 1) fall at only 8 a.m. and 6 p.m. during each day. The total planning horizon is divided into 11 time periods as illustrated in Figure 2. For each order, the time window is shown in Table 1 and Figure 3. The numbers in Figure 3 indicate the demands for the orders. The deodorizer can process a maximum batch size of 7.5 tons of products, with a fixed processing time of 15 min (0.25
t')RTo
∑
t
Sot',
∀o ∈ O, t g DTo
(21)
The sale of each order can be in any time period after its release time
Table 1. Details of Each Product Group, Product, and Order group PG1 product P1 P3 P4 P14 order O5 O14 O31 O22 O33 O3 O54 O60 O70 O11 O59 O69 O4 O65 O67 O44 O45 O21 O52 O23 O36 O38 O7 O42 O57 O19 O53 O15 O55 O1 O2 O58 O43 O56 O8 release time (h) 42 32 0 0 0 8 42 0 18 18 18 0 18 66 0 18 0 0 32 0 0 0 32 18 18 8 32 8 56 42 0 18 80 0 18 due date (h) 128 114 32 42 42 80 114 90 128 114 80 66 90 104 66 104 56 32 104 32 56 32 128 90 104 56 104 90 128 128 32 114 128 56 104 demand (tons) 36 27 43 42 14 21 22 77 96 26 40 45 58 89 30 21 65 42 39 53 61 12 53 77 78 83 28 43 85 94 52 88 77 42 53
Figure 3. Time window and demand for each order; amounts in tons.
h). The processing time for each batch should be fixed to values that are multiples of 0.25 h; that is, 0.25 × number of batches (in hours). For each batch, the minimum batch size is 3.75 tons, one-half of the maximum batch size (7.5 ton). The downtime is 15 min (0.25 h) for emptying or washing trays when switching from one product group to another (see Table 2), and the changeover cost is 10 k$ for each changeover. The price of each product is 1 k$/ton. The unit processing cost is 0.2 k$/ton, and the unit inventory and backlog costs are 0.1 k$/ton. All of the runs described in this section were done in GAMS 22.841 using the CPLEX 11.1 solver42 in the Windows XP environment on an Intel Core Duo 3.40 GHz, 3.44 GB RAM
Table 2. Changeover Matrixa group PG1 PG2 PG3 PG4 PG5 PG6 PG7 a PG1
PG2 Y
PG3 Y Y
PG4 Y Y
PG5 Y Y
PG6 Y Y Y
PG7 Y Y Y Y Y Y
Y
Y
Y
Y
Y
Y ) changeover occurs.
machine. The optimality gap was set to 2.0%, and the CPU limit was 3600 CPU s.
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Table 3. Breakdown of the Optimal Profit of Model DEO-S1 (k$) profit sales revenue processing cost inventory cost changeover cost 3016.0 3807.5 761.5 0.0 30.0
4.1. Computational Results of Model DEO-S1. Model DEO-S1, with 2829 equations, 2255 continuous variables, and 766 binary/integer variables, was solved in 20 CPU s. The obtained objective function value is 3016.0, with an optimality gap of 1.8%. The breakdown of the optimal profit is given in Table 3. The Gantt chart of the optimal schedule obtained from model DEO-S1 is given in Figure 4, which shows that there are total of three changeovers in the planning horizon. Different colors indicate the different product groups, and each bar contains one
or more products. Note that each batch production might satisfy multiple orders. The production levels of products and orders are given in Figures 5 and 6, respectively. For each product/order, the cumulative production is given, as well as the demand. From these figures, one can see not only how the demands are satisfied, but also the production time periods and amounts for each product/order. As there is no inventory in the optimal solution, which means that products are processed and delivered in the same week of the processing, the sale of each order at each time period can also be seen in Figure 6. In the optimal solution, of 70 orders, 66 orders (94.3%) are either fully or partially satisfied, with a total of 34 orders being fully satisfied by their due dates (numbers in bold). Most of the partially satisfied orders (59.4%) have a service level above
Figure 4. Gantt chart of the optimal schedule: Model DEO-S1.
Figure 5. Demands and production levels of products: Model DEO-S1.
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Figure 6. Demands and production levels of orders (a) O1-O35 and (b) O36-O70: Model DEO-S1.
90%. Only four orders (5.7%) are not satisfied at all. The total sale is 3807.5 tons, and the aggregated service level is 91.6%. The service level of each order is given in Table 4. 4.2. Computational Results of Model DEO-S2. Model DEO-S2, with 3515 equations, 3268 continuous variables, and 922 binary/integer variables, is solved in 1075 CPU s. The obtained optimal objective is 2959.1, with an optimality gap of 2.0%. The breakdown of the optimal profit is given in Table 5. Although there is no inventory cost in the optimal solution of this case, inventory cost can occur for the cases with higher minimum batch sizes. The Gantt chart of the optimal schedule obtained from model DEO-S2 is given in Figure 7. As in the optimal solution of scenario 1, three changeovers occur in the scenario 2 as well. The productions of each product/order in each time period are shown in Figures 8 and 9. Similarly to
DEO-S1, Figure 9 also provides information about the sales during each time period. Of the total of 70 orders, 67 orders (95.7%) are fully or partially satisfied, with 45 orders being fully satisfied. It should be mentioned that, of the 45 fully satisfied orders, 42 orders are fully satisfied at their due dates and 3 orders are fully satisfied at later dates. The total sale is 3803.5 tons, and the aggregated service level is 91.5%. The service level of each order is given in Table 6 (numbers in bold indicate that the corresponding orders are fulfilled by their due dates), from which it can be seen that even the partially satisfied orders have high service levels. The backlog of each order at the end of each period is given in Table 7. In each line, the first column with a reported backlog level is the due date of the corresponding order. A decrease of the reported backlog level means that the corresponding order
Table 5. Breakdown of the Optimal Profit of Model DEO-S2 (k$) profit sales revenue processing cost inventory cost backlog cost changeover cost 2959.1 3803.5 760.7 0.0 53.7 30.0
was implemented under the same computational environment and same termination criteria. The model sizes of the proposed model DEO-S1 and modified literature model are shown in Table 8, from which we can see that the proposed model has a much smaller model size than the literature model. In Table 9, the profit, revenue, and costs of the optimal solution from MILP model DEO-S1 are compared to the respective values obtained from the literature model. The literature model was terminated by the CPU limit and required 3604 CPU s to find a solution with an objective value of 2321.6. On the other hand, the proposed model identifies a solution of 3016 in only 20 CPU s. The service level obtained from the literature model is only 69.8%, compared with 91.6% from the proposed model. From the comparison results, it is obvious that the proposed model has a significantly better computational performance. 6. Concluding Remarks In this work, the short-term scheduling problem of a singlestage batch edible-oil deodorizer has been investigated, and MILP models have been developed for two scenarios. The novelty of the proposed models is that the processing sequence of the product groups is considered, instead of that of the products. In addition, the proposed formulation is based on the classic TSP formulation to model the production sequence in each time period. The proposed models have been successfully applied to the deodorizer scheduling problem with 70 orders. Finally, the effectiveness of the models is shown through a comparison with a discrete-time literature model that addresses a similar case study.
is being partially or fully satisfied. From Table 7, there are three orders (O21, O31, and O40) that are not satisfied by their due dates, but are satisified later by the end of the planning horizon. There are 25 orders with backlogs at the end of planning horizon, and the total backlog amount is 352.5 tons. 5. Comparison with a Literature Model In this section, the efficiency and effectiveness of the proposed models are examined by comparison with a literature model proposed by Kelly and Zyngier.38 Their MILP model was proposed to represent the sequence-dependent changeovers for uniform discrete-time scheduling problems, and can be applied to both batch- and continuous-process units. In the third illustrative example presented in their article, a case study similar to the one in this article was considered. Their case study considered a planning horizon of 3 days and a total of 45 orders. As only sequencing constraints were presented in their article, we added our proposed objective function and constraints for production, inventory, and sales to the literature model for comparison. The details of the literature model and added constraints are presented in the Appendix. The modified literature model was also used for the case study in section 4. As the batch time and changeover time in the case study were 15 min, the length of each discrete slot used for the case study was 15 min, and a total of 512 slots were used in the model for this case study. The modified literature model
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Figure 7. Gantt chart of the optimal schedule: Model DEO-S2.
Figure 8. Demands and production levels of products (P1-P30): Model DEO-S2.
Appendix. Kelly and Zyngier presented an MILP formulation for modeling sequence-dependent changeovers for discrete-time scheduling problems. The formulation can be applied to both batch and continuous process units. Nomenclature.
Indices i, j ) operation t, tt ) time period
38
Parameters τi ) batch time for operation i τij ) switchover time from operation i to operation j Variables sdit ) 1 for the shutdown of mode operation i at time period t suit ) 1 for the startup of mode operation i at time period t swijt ) 1 for the switchover from mode operation i to mode operation j at period t yit ) 1 for the setup of mode operation i at time period t
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Figure 9. Demands and production levels of orders (a) O1-O35 and (b) O36-O70: Model DEO-S2.
yyit ) 1 for the memory variable of mode operation i at time period t
∑ sw j i
ijt
) yyit-1, ) yyjt,
∀i, t ∀j, t
(A7) (A8) (A9)
Constraints.
∑y i ∑ sw it ijt
e 1, t ∀t
(A1)
sujt + sdi,tt e 1,
∀i * j, t - tt ) 0, ..., τij
yit )
tt)t-τi+1
∑
sui,tt,
∀t ∀t
(A2) (A3) (A4) (A5)
sdit ) sui,t-τi+1,
∑ yy i it
) 1,
∀t ∀i, t ∀i, t
It should be mentioned that, in the above model, except for the variable suit, all variables can be relaxed as continuous variables in the interval [0, 1]. For comparison with the proposed model, operation i in the above equations is regarded as the processing operation for product i. Moreover, the following indices, sets, parameters, variables, and constraints are added to the model: Additional Nomenclature.
Index d ) due date
Set Oi ) set of orders for product i Parameters BiL ) lower bound of batch size for operation i
) 8upper bound of batch size time for operation i CCij ) changeover cost from product i to j Do ) demand of order o DTo ) due date of order o BiU
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Table 8. Sizes of the Proposed Model and Literature Model proposed (DEO-S1) no. of equations no. of continuous variables no. of binary variables 2829 2255 766 Kelly and Zyngier38 775 749 540 369 15 360
Foundation, UK Foreign & Commonwealth Office, and Centre for Process Systems Engineering. Literature Cited
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Table 9. Comparison between the Proposed Model and Literature Model proposed (DEO-S1) profit (k$) sales revenue (k$) processing cost (k$) inventory cost (k$) changeover cost (k$) service level (%) optimality gap (%) CPU (s) 3016.0 3807.5 761.5 0.0 30.0 91.6 1.8 20.0 Kelly and Zyngier38 2321.6 2902.0 580.4 0.0 0.0 69.8 32.3 3603.9
Hd ) time of due date d ICi ) inventory cost of product i Kd ) number of slots by due date d PCi ) processing cost of product i Pri ) price of product i RTo ) release time of order o ViU ) upper bound on inventory of product i Variables Bit ) batch size for operation i at time period t Pot ) processed amount for order o at time period t Sod ) sales of order o at due date d Vod ) inventory amount for order o at due date d
Constraints. Bit g BLyit, i Bit e Bit ) BUyit, i ∀i, t ∀i, t Pot, ∀i, t
(A11) (A10) (A12)
o∈Oi∩(HteDTo)
∑
Kd
Vod ) Vo,d-1 +
(∑ ∑
∑ ∑
i∈Io j∈Ji t)Kd-1+1
∑
Pot - Sod
∀o, d:Hd g RTo ∀i, d
)
HdeDTo,
(A13) (A14)
o∈Oi∩(o:HdgRTo)
Vod e VU, i
d:RToeHdeDTo
Sod e Do,
∀o ∈ O
(A15)
Objective.
∑∑ i o∈Oi d:RToeHdeDTo
∑
PriSod -
∑ ∑ PC B t i t i
i it
ij ijt
∑∑ ∑ i ICiVod -
o∈Oi d:HdgRTo
∑ ∑ ∑ CC sw j*i (A16)
Acknowledgment The authors thank Mr. Jeff D. Kelly from Honeywell for providing data and useful discussions for the case study used in this work. S.L. was financially supported by Overseas Research Students Award Scheme, K.C. Wong Education
Ind. Eng. Chem. Res., Vol. 49, No. 18, 2010
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ReceiVed for reView January 29, 2010 ReVised manuscript receiVed July 22, 2010 Accepted July 23, 2010 IE1002137
