Ind. Eng. Chem. Res. 2009, 48, 2009–2019
2009
An Extended Formulation for the Flexible Short-Term Scheduling of Multiproduct Semicontinuous Plants Elisabet Capo´n-Garcı´a, Sergio Ferrer-Nadal, Moise`s Graells, and Luis Puigjaner* Chemical Engineering DepartmentsCEPIMA, UniVersitat Polite`cnica de Catalunya, ETSEIB, AV. Diagonal 647, 08028 Barcelona, Spain
Chemical plants are moving toward more-flexible environments, to adapt faster to market changes. For the sake of this flexibility, batch and continuous units might be integrated and operated along the same processing route in a semicontinuous mode, featured by production campaigns of finite duration. Constant production rates for the entire operation have been usually considered. Thus, production adjustments were achieved through storage resources, production line stops, and multiple product campaigns. The objective of this work is to improve the production schedules by developing a new concept for flexible manufacturing, which allows one to program production rate profiles within each semicontinuous operation campaign. Hence, the extended formulation presented allows to address the tradeoff between the opportunities that may arise when adjusting processing rates and the storage needs and opportunities. Finally, several case studies show that the proposed adjustment of the processing rates leads to less storage utilization and more-efficient production performance. 1. Introduction Although recent trends in globalization have opened new markets for the chemical industry, they have also increased the level of competition, which has led to the development of moreefficient and highly integrated plants. Consequently, flexibility is an important parameter in ensuring that new market demands are met effectively. Flexible batch plants provide an adaptable solution for highly dynamic and uncertain environments and have grown in popularity at the expense of mass production, which is a more rigid continuous production mode. In continuous processes, a very limited number of products is consumed at constant rates over long production periods. By contrast, batch plants produce much smaller quantities of a wider range of products during shorter production periods. The flow of material in batch production is discontinuous: input products are loaded at the beginning of the production period and output products are only available after the entire operation has been completed. Semicontinuous operation mode can be considered an intermediate mode between the batch processes used in pilot plants and continuous processes. More importantly, it optimizes the efficiency with which equipment is used to process medium quantities of several products simultaneously in a continuous facility. Semicontinuous operation is characterized by an overall processing rate and equipment running continuously between periodic startups and shutdowns for product transitions. Processing times in semicontinuous operation are usually relatively long periods called campaigns, each of which is used for a single product. Individual campaigns are often used to produce feed stocks for downstream processes that produce more-specialized final products.1 In fact, most process plants in the chemical industry work in semicontinuous mode by combining continuous operations and batch processes. Typical campaign lengths range from a few hours to several days. However, too many short campaigns can produce unrealistic and non-cost-effective operating conditions, because of the expense of switching production from one product to another.2 * To whom all correspondence should be addressed. Tel.: +34-93401.66.78. Fax: +34-93-401.09.79. E-mail address: luis.puigjaner@ upc.edu.
Intermediate product storage is another important factor in the operational management of this type of chemical plant. Intermediate storage facilities allow one to decouple different upstream and downstream production rates. Hence, a satisfactory storage policy has a strong influence on the efficiency and flexibility of plants working in semicontinuous mode. This operation mode is complex, but it has a large number of potential applications and has been studied extensively in
Figure 1. STN representation of the illustrative example plant.
Figure 2. Schematic representation of the illustrative example.
10.1021/ie800539f CCC: $40.75 2009 American Chemical Society Published on Web 01/05/2009
2010 Ind. Eng. Chem. Res., Vol. 48, No. 4, 2009 Table 1. Maximum Rate Capacities and Unit Suitability for the Illustrative Example product
available units
rmax (ton/h)
I1, I2, I3 I1, I2, I3 P1 P2 P3 P4 P5 P6
M1 M2 L1 L2 L1 L1 L2 L1
14.0 16.0 5.0 7.0 8.0 8.0 4.0 4.0
Table 2. Minimum Demand Requirements for the Illustrative Example product
demand (ton)
P1 P2 P3 P4 P5 P6
60.0 300.0 100.0 300.0 80.0 10.0
recent literature. Sahinidis and Grossmann3 addressed the problem of cyclic multiproduct scheduling for continuous parallel production lines. They identified a combinatorial part (the assignment of products to lines and their sequencing in each line) and a continuous part (the duration of production runs and the frequency of production), and they formulated a slot-based mixed-integer nonlinear programming (MINLP) model that was linearized in the space of the integer variables. Pinto and Grossmann4 extended this work and modeled the cyclic scheduling problem in multistage continuous processing plants. They developed a solution based on a generalized Benders decomposition and an outer approximation that used explicit inventory breakpoints to handle the inventory profiles of intermediate storage tanks. Zhang and Sargent5 developed a MINLP formulation based on the Resource Task Network (RTN) representation. The resulting model was linearized to create a very large-scale mixed-integer linear programming (MILP) model. Ierapetritou and Floudas6 used a State Task Network (STN) representation to formulate the problem, taking into account multiple intermediate due dates, while storage requirements were handled using approximated storage task timings. Mockus and Reklaitis7 proposed a global event-based MINLP able to handle resource constraints such as limited availability of utilities and manpower. Giannelos and Georgia-
Figure 3. Optimal schedule for the UIS policy.
dis8 developed a similar model to that of Ierapetritou and Floudas6 but relaxed time durations and eliminated big-M constraints. However, they also assumed equal start and end times of the tasks producing/consuming the same state, which could lead to suboptimal solutions if the material is allowed to bypass the storage. Me´ndez and Cerda´2 used a continuous-time formulation based on a general precedence notion, which generated a very small and compact model. An important assumption they made is that every intermediate or final product should be produced by a single production campaign. Castro et al.9 developed a MILP formulation based on the RTN representation and highlighted the benefits of using a uniform time grid continuous-time representation. Shaik and Floudas10 extended the work of Ierapetritou and Floudas6 to handle different storage requirements. All of the studies cited in the literature examined how to use semicontinuous processes to increase production flexibility. Consequently, multiple campaigns can clearly be considered suitable for increasing flexibility, despite the fact that they may often be infeasible in industry practice. However, other rigid features can be found in most of the formulations presented in these studies. For example, constant and invariable production rates are usually implicitly assumed for the entire operating period of a given campaign. No transitions in the production rate are made during each campaign, so production adjustments are made using production line stops, storage resources, and multiple product campaigns. The objective of this study is to improve production schedules by considering adjustable production rates within each production campaign, extending the general precedence MILP model that has been presented by Me´ndez and Cerda´.2 That referenced work reported results with constant production rates. Conversely, the extended model developed in this work allows considering flexible fabrication opportunities by means of variable rates constrained by lower and upper bounds, which may include the former model as a particular case. Results from considering these opportunities are presented and analyzed. The study is organized as follows. First, the mathematical formulation is introduced and the flexible manufacturing concept for semicontinuous processes is presented; then, a test is performed using an illustrative case study and results obtained are compared with those reported in the literature. Finally, discussion and conclusions are provided, regarding the potential applications of the
Ind. Eng. Chem. Res., Vol. 48, No. 4, 2009 2011
Figure 4. Optimal schedule and intermediate products tank level profiles for the FIS policy with three tanks and fixed processing rates.
Figure 5. Optimal schedule for the NIS policy with variable processing rates.
extended formulation that is developed, which uses adjustable processing rates to increase the scheduling flexibility. 2. Mathematical Formulation The proposed MILP formulation is designed to determine the optimal sequence of campaigns that maximizes produc-
tion, by satisfying a minimum demand for each product. It is based on the model introduced by Me´ndez and Cerda´,2 which makes one major assumption, by considering single product campaigns for the production of each product. Although this assumption may produce suboptimal solutions, those authors2 argued that a large number of campaigns
2012 Ind. Eng. Chem. Res., Vol. 48, No. 4, 2009 max corresponding equipment unit (rmin sj and rsj , respectively). In addition, at the end of the time horizon, the amount of every final product from each campaign must satisfy the minimum demand (ds) specified by constraint 8.
Qi )
∑F
∀ i ∈ Is+, s ∈ SI
ii′
(9)
i′∈Is-
FisQi )
∑F
∀ i ∈ Is-, s ∈ S
i′i
(10)
i′∈Is+
∀ i ∈ Is+, i′ ∈ Is-, s ∈ S
Fii′ e M2Uii′ Ci -
∑L
ij e Ci′ -
j∈Ji
(11)
∑L
i′j + H(1 - Uii′)
j∈Ji′
∀ i ∈ Is+, i′ ∈ Is-, s ∈ S (12)
Figure 6. Schematic representation of the case study.
creates a much greater demand for manpower and generates financial losses through excessive equipment idle time. This section describes the MILP formulation in detail. The indices, parameters, and variables included in the model are defined in the Nomenclature section.
∑Y
∀ i ∈ Is+, s ∈ S
ij e 1
(1)
j∈Js
∀i∈I
Ci e H
∀ i ∈ Is+, j ∈ Js, s ∈ S
lsjminYij e Lij e HYij Ci -
∑ L g ∑ max[ru , ro ]Y ij
j∈Js
(2)
j
s
ij
∀ i ∈ Is+, s ∈ S
(3) (4)
j∈Js
Constraint 1 ensures that only one production line j can be assigned to each processing campaign i through variable Yij. Constraint 2 ensures that each campaign i finishes (Ci) before a prespecified time horizon (H), whereas eq 3 limits the duration of each campaign (Lij) between a minimum value (lmin sj ) and a maximum one (namely, the time horizon). Constraint 4 ensures that the product release and unit setup processes cannot begin before the time values (ros) and (ruj), respectively. Ci′ - Li′j g Ci + uchii′j - M1(1 - Xii′) - M1(2 - Yij - Yi′j) ∀i, i′ ∈ I, i < i′, j ∈ Ji ∩ Ji′ (5) Ci - Lij g Ci′ + uchi′ij - M1Xii′ - M1(2 - Yij - Yi′j) ∀i, i′ ∈ I, i < i′, j ∈ Ji ∩ Ji′ (6) Constraints 5 and 6 sequence a pair of campaigns i and i′ that are assigned to the same semicontinuous line. Constraint 5 is only active if campaign i precedes i′; otherwise, constraint 6 is active. Because this model also takes into account a possible changeover time between different products (uchii′j), the value of M1 should be equal to H + max{uchii′j}, to obtain the tightest relaxation. If a lower value was assigned to M1, these constraints would not be consistent and would result in unfeasible solutions.
∑r
∑r
Equation 9 establishes the mass balances between a campaign i that produces and a campaign i′ that consumes the intermediate state s. In this equation, Fii′ is a continuous variable that represents the amount of material transferred between the two campaigns. Similarly, eq 10 adjusts the amount of material consumed by a campaign i, which receives material from i′. The amount of material consumed by this campaign is converted to campaign production using the coefficient Fis. Equation 11 introduces a binary variable (Uii′) that is equal to 1 if Fii′ is greater than zero, i.e., campaign i supplies material to campaign i′. In this case, M2 must be greater than any value of Fii′. In case Fii′ is equal to 0, there is no material transfer between campaigns i and i′, even though the value of Uii′ is not fixed to zero, its value is indirectly set to the optimal by the objective function. The variable Uii′ is used to ensure that campaign i which supplies the material can never start later than the campaign i′ during which the material is received through constraint 12. Constraints 1-12 represent an unlimited intermediate storage (UIS) policy, which is the least restrictive storage policy, in which sufficient storage resources are assumed to be available at any time. By contrast, a no-intermediate storage (NIS) policy is a scenario in which no storage is available and materials must be transferred directly between production lines. Constraint 13 accounts for the NIS case by ensuring that a campaign i cannot finish until any consumer campaign i′ has finished. ∀ i ∈ Is+, i′ ∈ Is-, s ∈ S
Ci g Ci′ - H(1 - Uii′)
(13)
In addition to UIS and NIS, this model can account for different finite intermediate storage (FIS) policies. In these cases, material can either be stored in a limited number of tanks with restricted capacity, or bypassed directly between production lines. Therefore, in addition to constraints 1-12 for the UIS case, the following constraints are applied to the cases in which limited storage resources exist. ITi g Ci -
∑L
ij
∀ i ∈ Is+, s ∈ SI
(14)
∀ i ∈ Is+, i′ ∈ Is-, s ∈ S
(15)
j∈Js
CTi g Ci′ - H(1 - Uii′)
(7)
ITi′ g CTi + tchii′t - M3(1 - Xii′) - M3(2 - Wit - Wi′t) ∀i, i′ ∈ I, i < i′, t ∈ Ti ∩ Ti′ (16)
(8)
ITi g CTi′ + tchi′it - M3Xii′ - M3(2 - Wit - Wi′t) ∀i, i′ ∈ I, i < i′, t ∈ Ti ∩ Ti′ (17)
Constraint 7 imposes minimum and maximum limits on the total production of a campaign (Qi) according to its duration (Lij) and the minimum and maximum production rate of the
Constraint 14 ensures that the beginning of the storage period for intermediate material supplied by campaign i (ITi) coincides with the beginning of that campaign. Constraint 15 states that
min sj Lij e Qi e
j∈Js
max sj Lij
∀ i ∈ Is+, s ∈ S
j∈Js
ds e
∑Q
i
∀ s ∈ SP
i∈Is+
Ind. Eng. Chem. Res., Vol. 48, No. 4, 2009 2013
Figure 7. STN representation of the plant. Table 6. Literature Model Representations and Solver Options
Table 3. Maximum Rate Capacities and Unit Suitability product
available units
rmax (ton/h)
I1, I2 I3, I4 I5, I6, I7 P1 P2, P3 P4, P5 P6 P7 P8, P9 P10, P11 P12, P13 P14, P15
M1 M2, M3 M2, M4 L3 L1 L2 L3 L1 L2 L5 L4 L4
17.0000 17.0000 12.2400 5.5714 5.8333 2.7083 5.5714 5.8333 2.7083 5.3571 2.2410 3.3333
model
representation
Shaik and unit-specific Floudas10 9 Castro et al. RTN Me´ndez and general Cerda´2 precedence
Ci′ -
Pentium III, 1 GHz GAMS/CPLEX 7.0.0 Pentium II, 400 MHz ILOG/CPLEX 6.5.2
∑L
∀ i ∈ Is+, i′ ∈ Is-, s ∈ S (18)
i′j - Ci e HZii′
∀ i ∈ Is+, i′ ∈ Is-, s ∈ S
Vii′ e Fii′
1 1
∀ i ∈ Is+, i′ ∈ Is-, s ∈ S
Vii′ e M2(1 - Zii′)
1
(19) (20)
max Vii′ e Fi′s min(rmax ij , ri′j′ )(Ci - Ci′ + Li′j) + M2Zii′ +
4 4
M2(1 - Uii′) + M2(1 - Yij) + M2(1 - Yi′j′)
1 1
Pentium IV, 3.2 GHz GAMS/CPLEX 9.0.2
j∈Ji′
From/To P1 P2 P3 P4 P5 P6 P7 P8 P9 P12 P13 P14 P15
4 4
solver
campaign i is transferred to tank t. The parameter tchii′t represents the changeover time between different products in the same tank; thus, M3 should be equal to H + max{tchii′t} to achieve the tightest relaxation of the model.
Table 4. Changeover Requirements (Given in Terms of Hours, h)
P1 P2 P3 P4 P5 P6 P7 P8 P9 P12 P13 P14 P15
processor
∀i ∈ Is+, ∀ i′ ∈ Is-, s ∈ S, j ∈ Ji, j′ ∈ Ji′ (21)
1 4 4
4 4
Qi 2 2 2 2
2 2
2 2
Table 5. Minimum Demand Requirements product
demand, ton
P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 P11 P12 P13 P14 P15
220.0 251.0 15.0 116.0 7.0 47.0 144.0 42.5 13.5 114.5 53.0 16.5 8.5 2.5 17.5
the storage of an intermediate material must not end (CTi) before all of the production campaigns consuming this material have been completed. Sequencing constraints for the storage tasks (eqs 16 and 17) are similar to those used for production campaigns. In this case, an additional binary variable (Wit) indicates whether the intermediate material produced by a
∑V
ii′ e
i′∈Is-
∑VW t
∀ i ∈ Is+, s ∈ S
it
(22)
t∈Ts
Constraint 18 assigns a value of 1 to the binary variable Zii′ if a campaign i producing an intermediate consumed by campaign i′ finishes before i′ starts. Constraint 19 introduces a continuous variable Vii′, which represents the amount of intermediate material consumed by i′ at the end of the campaign i in which the material is produced. Therefore, if Zii′ ) 1sthat is, i and i′ do not coincide in timesthen Vii′ must be equal to zero. Equations 20 and 21 constrain the value of Vii′: eq 20 forces Vii′ to be, at most, as large as Fii′, and constraint 21 establishes an upper bound for the value of Vii′, assuming that campaign i′ consumes material from i at its maximum rate capacity during the period in which i and i′ run simultaneously. Finally, constraint 22 restricts the use of a storage tank to its maximum volumetric capacity (Vt). The objective function of this problem is to maximize the production of final products: max
∑
Qi
i∈Is+,s∈SP
subject to constraints 1-12 for the UIS case constraints 1-13 for the NIS case constraints 1-12 and 14-22 for the FIS case
(23)
2014 Ind. Eng. Chem. Res., Vol. 48, No. 4, 2009
Figure 8. Optimal schedule for the UIS policy. Table 7. Results for the UIS Policy for the Case Study model Shaik and Floudas10 Castro et al.9 Me´ndez and Cerda´2 this approach
Table 8. Results for the NIS Policy for the Case Study
profit campaigns/ binary continuous number of CPU (m.u.) product variables variables rows time (s) 2695.32
>1
108
356
1040
1.03
2695.32 2695.32
>1 1
236 38
762 44
894 140
58.5 4.77
2695.32
1
38
85
140
2.84
3. Illustrative Example Let us consider a semicontinuous plant that produces six final products (P1-P6) from three intermediate products (I1-I3), as shown in Figure 1. There are two parallel first-available stage units to produce the intermediate products and two additional units to produce the final products (see Figure 2). The maximum unit processing rates are given in Table 1. The problem objective consists of maximizing total production, satisfying a minimum demand for each product (Table 2). Different storage policies for intermediate products may be considered. If unlimited intermediate storage is available in the plant, then a total optimal production of 1694 units is achieved (see Figure 3). However, unlimited storage capacity is not real, because there are usually storage limitations in process plants.
Figure 9. Profile of surplus intermediate material for the UIS.
model
profit campaigns/ binary continuous number of CPU (m.u.) product variables variables rows time (s)
Shaik and 2689.75 Floudas10 9 Castro et al. 2672.50 this approach 2688.31
>1
108
328
1240
157.9
>1 1
228 38
762 119
894 202
2701 10.7
Hence, let us suppose that there are three intermediate storage tanks with a capacity of 60 tons available, and the processing rates are set constant and equal to the maximum for every campaign, and if a single campaign per product is allowed, then the maximum production attainable is 1482.5 units, and the three storage units would be required, as shown in Figure 4. Alternatively, if multiple campaigns were executed for producing intermediate products, then the maximum production would be reached. However, multiple campaigns and storage resources clearly entail additional costs that are not considered in the models. However, in this case, if the variation of the unit processing rate along a campaign is allowed, then one campaign per intermediate product would also reach the value of the maximum production obtained with unlimited storage policy (1694 units). Figure 5 shows the resulting scheduling Gantt chart, where the
Ind. Eng. Chem. Res., Vol. 48, No. 4, 2009 2015
Figure 10. Optimal schedule for the NIS policy in the case study. Table 9. Results for the FIS Policy with Three Tanks model Shaik and Floudas10 Castro et al.9 Me´ndez and Cerda´2 this approach
profit campaigns/ binary continuous number of CPU (m.u.) product variables variables rows time (s) 2695.32
>1
276
580
4267
465.6
2695.32 2670.28
>1 1
330 60
927 87
1127 361
162 398.9
2695.32
1
84
148
402
5.72
Table 10. Results for the FIS Policy with Two Tanks model
profit
this approach 2695.32
campaigns/ binary continuous number of CPU product variables variables rows time (s) 1
77
148
360
12.9
inner height of the Gantt bars is proportional to the total production rate of the equipment. For example, campaign I1 varies its processing rate from 7 ton/h to 12 ton/h at 108 h; campaign I2 maintains a production rate of 8 ton/h; and campaign I3 varies its processing rate three times: from 8 ton/h to 4 ton/h at 2.5 h, to 12 ton/h at 15 h and finally to 8 ton/h at 20 h. Instantaneous production rates are not a decision variable in the model but are considered indirectly in the model as a function of the task length and the amount produced of each state. Thus, production rates may vary along the campaign length and they are finally computed as a function of the process state requirements along time. For example, campaigns I1, I2, and I3 adapt their processing rate to the processing rates of their corresponding downstream campaigns. To summarize, the fact of considering variable processing rates within operation allows one to obtain schedules that require a smaller number of campaigns, as well as a lower utilization of storage resources. The next case study, which has been widely considered in the literature, further explores the benefits of this formulation and compares the results with those obtained by other authors. 4. Case Study The case study proposed illustrates the capabilities of the flexible formulation introduced. It refers to a fast-moving consumer goods manufacturing plant. This is a classic case that has been addressed many times in the literature on scheduling strategies for semicontinuous facilities.2,5,6,8-10 The plant has the basic structure shown in Figure 6. It consists of three parallel mixers that send material to five packing lines, operating in
semicontinuous mode, and a set of three storage tanks, which buffer the production stocks. Three raw materials with nonrestricted availability are blended in the corresponding mixers to produce seven intermediates (I1-I7). These intermediates are then combined in a series of packing lines to produce 15 final products (P1-P15). Figure 7 shows the STN representation for the plant. Table 3 shows the maximum production rate for each product and the availability of the semicontinuous units in which they are processed. Table 4 gives the sequence-dependent changeover times between different products. The objective function maximizes the profit from production sales with a product price of 1 for all products, which is equivalent to maximize the total production. In addition, a minimum final product demand (Table 5) must be met over a time horizon of five working days (120 h). Several storage policies are considered, namely, unlimited intermediate storage (UIS), no intermediate storage (NIS) and finite intermediate storage (FIS), under different allocation and demand constraints. These cases are implemented in GAMS and solved using CPLEX 10.0 in a 3 GHz computer and compared to the results reported in the literature (see Table 6). 4.1. Unlimited Intermediate Storage (UIS). The optimal schedule obtained under an unlimited intermediate storage (UIS) policy is shown in Figure 8. This solution represents the highest attainable profit, given the demand requirements and unlimited storage capacity for the intermediate products. Because there are no restrictions on the storage capacity, all mixers in the processing stage can work at their maximum rate, and the idle times are reduced. However, in the packing stage, units are working during the entire time horizon, because the processing rates of the packing lines are much lower than those of the mixers. Therefore, although mixers are partially idle in this case study, production is limited by the processing rates of the packing units. Table 7 shows the results for the formulation proposed in this case study and those given in the literature. All formulations yield the same optimal profit value. Intermediates are produced in more than one campaign in the solutions reported by Castro et al.9 and Shaik and Floudas.10 By contrast, the solution presented and that of Me´ndez and Cerda´2 use a single campaign for each intermediate and final product, which is a more feasible strategy in industry practice. Multiple campaigns lead to higher operational costs because of the higher demand for resources
2016 Ind. Eng. Chem. Res., Vol. 48, No. 4, 2009
Figure 11. Optimal schedule and intermediate-product tank-level profiles for the FIS policy in the case study with two storage tanks. Table 11. Results for the FIS Policy with Restricted Storage Allocation
model
profit
Shaik and 2695.32 Floudas10 this approach 2695.32
campaigns/ binary continuous product variables variables
number CPU of time rows (s)
>1
164
412
2065
662.58
1
63
148
309
21.23
such as manpower that may be required to perform changes in production lines. Consequently, the cost efficiency of the plant increases as the number of campaigns decreases. There is no constraint on the storage in the proposed solution (Figure 8), so most intermediate products are stored. Figure 9 shows the stock profiles for this solution that requires as many storage units as intermediates. As it can be seen, the maximum tank capacity needed is >350 tons, which is a highly inefficient solution that would generate high operational and fixed costs. Therefore, this scenario is unrealistic, because industries usually have storage space limitations that must be taken into account by the production scheduler. However, this case is particularly relevant to the case study because it represents an upper bound for more-restrictive cases. In addition, the UIS policy represents
the best production solution if the cost of storage is disregarded. More realistic approaches are presented in the following sections. 4.2. No Intermediate Storage (NIS). In this case, there is no storage available to buffer the mismatching production rates between upstream and downstream processes. Table 8 shows the results reported by different authors, as well as those obtained in this work. The model used by Shaik and Floudas10 yielded a profit of 2689.75 m.u., whereas that of Castro et al.9 only produced 2672.50 m.u. Me´ndez and Cerda´2 did not solve the problem for this storage policy; hence, results with their formulation are not included. The formulation presented in this study gives a solution of 2688.31 m.u., which is better than that obtained by Castro et al.,9 and is slightly worse than the optimal value reported in the literature (0.05%). However, the solution proposed uses a single campaign per intermediate product, which may be regarded as a more-efficient solution in industrial practice. Figure 10 shows the Gantt chart corresponding to the solution obtained in the present study. This solution is achieved by regulating the processing rate of the mixers according to the requirements of the packing lines. For example, when mixer M1 processes intermediate I1, its processing rate changes from
Table 12. Results for the FIS Policy with Maximum Demand Limits model
profit (m.u.)
storage
campaigns/product
binary variables
continuous variables
number of rows
CPU time (s)
Shaik and Floudas10 Shaik and Floudas10
1388 1388
CIS RSA
>1 >1
276 164
580 412
4282 2080
8.81 6.56
this approach this approach
1388 1388
CIS RSA
1 1
84 63
148 184
440 324
0.20 0.09
Ind. Eng. Chem. Res., Vol. 48, No. 4, 2009 2017
Figure 12. Optimal schedule for the FIS policy with maximum demand constraints. Table 13. Results for a Mixed Storage Policy with One Tank
model
profit
this approach 2695.32
number CPU campaigns/ binary continuous of time product variables variables rows (s) 1
46
130
233
3.19
8.280 ton/h to 14.113 ton/h and then to 11.407 ton/h. These transitions enable downstream facilities to process the material generated by the mixers without the need for intermediate storage. In the case study presented, the production bottleneck occurs during the packing stage, because the packing lines have much lower processing rates than the mixers. The results for the NIS policy are likely to be unrealistic in most cases, because intermediate storage tanks are usually available in semicontinuous plants. However, the case study does prove the generality of the scheduling approach presented. Similarly, the solution of this case represents a lower bound for the solution for lessrestrictive cases. Similarly, the solution for the scheduling problem will be bounded by the solutions of the UIS case (upper bound) and NIS case (lower bound). 4.3. Finite Intermediate Storage (FIS). Previous studies considered a scenario with three 60-ton storage tanks. The results are shown in Table 9. Shaik and Floudas10 and Castro et al.9 reported a profit of 2695.32 m.u., which is the same as the value obtained with UIS. Therefore, the optimal profit obtained with three 60-ton tanks cannot be increased using additional tanks. Me´ndez and Cerda´2 obtained a smaller profit (2670.28 m.u.), because their formulation considers a single campaign per product and fixed processing rates. The model proposed in this study yields the optimal profit (2695.32 m.u.) with a single campaign per product, because it can adjust the campaign production rates to the plant requirements. Therefore, a storage capacity of three 60-ton tanks is large enough to be considered equivalent to the UIS case. A scenario is considered in which the storage capacity is reduced to two 60-ton tanks. In this scenario, the optimal profit (2695.32 m.u.) is also achieved with a single campaign per product. Table 10 shows the results obtained with the proposed formulation, and Figure 11 shows the Gantt chart of the solution. The optimal schedule was obtained by adjusting the semicontinuous equipment processing rates. For example, when mixer
M3 processes intermediate I6, the processing rate is adjusted first from 7.598 ton/h to 5.357 ton/h at 7.36 h, and then to 8.690 ton/h at 13.16 h. Therefore, the same optimal objective function value is obtained with lower storage resources utilization. This is a direct consequence of a flexible formulation allowing variable processing rates. Because the production capacity of intermediates may be adjusted below its maximum capacity for those nonlimiting stages, the storage requirements for these intermediates are accordingly reduced. 4.4. FIS under Restricted Allocation. A more-restricted intermediate storage policy is considered in the work by Shaik and Floudas.10 In this case, a specific storage tank is available for each product. The results for both models are reported in Table 11. The optimal value objective function corresponds to 2695.32 m.u. The proposed approach also reaches the optimal value of the objective function. As in the previous cases, the difference between both approaches lies in the number of campaigns for each product. The work by Shaik and Floudas10 considers up to two campaigns for each intermediate product, with changeovers in the mixers. By contrast, this approach considers only one campaign for each product, with variable processing rate inside a campaign. Thus, the number of campaigns per product is reduced without affecting the total production. Hence, the same optimal production value is achieved, but the costs of changeovers are avoided, although they are not explicitly quantified in the objective function. 4.5. FIS with Maximum Demand Limits. An additional case is presented in the work by Shaik and Floudas,10 who claimed that a maximum demand exists for each product. Two cases are presented with three intermediate storage tanks with a capacity of 60 tons each (namely, common intermediate storage (CIS) and restricted storage allocation (RSA)). The approaches are compared in Table 12. The optimal objective function value corresponds to 1388 m.u. in both cases. The proposed model also reaches the optimal value. Two important differences may observed in the solution obtained from the proposed formulation when compared to the results obtained by Shaik and Floudas:10 a single campaign for each product and no storage resources consumption. Therefore, the number of campaigns per product is reduced, without affecting the total production. Figure 12 shows the corresponding Gantt chart for this case with the proposed formulation. Production
2018 Ind. Eng. Chem. Res., Vol. 48, No. 4, 2009
Figure 13. Optimal schedule and intermediate-product tank-level profile for the mixed policy in the case study with a storage tank.
is adjusted by changing processing rates inside a given campaign, for example, at time ) 50.22 h, the processing rate for product I1 increases from 8.28 ton/h to 14.113 ton/h; next, at time ) 90.61 h, it is adjusted to 8.542 ton/h, and finally at time ) 93.85 h, it is adjusted to 5.833 ton/h. 4.6. Mixed Intermediate Storage. Next, a scenario is considered in which the storage policies are mixed. To be more specific, intermediate products I1, I4, and I5 have an NIS policy, I6 and I7 have a limited storage policy with one tank that has a capacity of 60 ton available, and I2 and I3 have UIS capacity. In this scenario, the optimal profit of 2695.32 m.u. is also achieved with a single campaign per product. Table 13 shows the results obtained with the proposed formulation, and Figure 13 shows the Gantt chart of the solution. The optimal schedule is obtained by adjusting the processing rates of the semicontinuous equipment, as well as by adjusting the use of the available tank to store intermediate product I6.
but it uses single product campaigns and requires less storage resources, which makes the system more flexible and costefficient. These advantages increase storage management flexibility and, therefore, may reduce the capital cost of the plant at the design stages. Further work should be performed to assess the relative financial benefits of using single or multiple campaigns in production processes as a tradeoff with the profit in the design stage. This future step will be strongly connected to the consideration of production orders and due dates. Acknowledgment Financial support received from the European Community (MRTN-CT-2004-512233; INCO-CT-2005-013359) and Spanish Government (DPI2006-05673) projects is thankfully acknowledged. Grants from the Departament d’Educacio´ i Universitats de la Generalitat de Catalunya, the European Social Fund, and the Ministerio de Educacio´n y Ciencia are also fully appreciated.
5. Conclusions This study considers semicontinuous campaigns with production rate transitions to improve production flexibility and storage management in chemical plants. Traditionally, constant rates are maintained throughout production operations, and adjustments are made using storage resources, production line stops, and multiple product campaigns. In contrast, this study proposes to adjust the processing rates of a campaign to the specific production requirements. The formulation that is presented is more flexible, because it allows production rates and campaign lengths to be adjusted; thus, it may be also expected to produce more-robust schedules, because production uncertainty may be absorbed by the variable production rate. In addition, the model can also be used for fixed operation by tightening the parameters and bounds of the model. The proposed formulation produces results that are comparable to those reported in previous studies,
Nomenclature Subscripts i ) semicontinuous processing campaign j ) semicontinuous production line s ) state (intermediate or final products) t ) storage tank Sets I ) campaigns Is- ) campaigns that consume state s Is+ ) campaigns that supply state s Ji ) production line available for campaign i Js ) production line available for manufacturing state s S ) state SI ) intermediate state
Ind. Eng. Chem. Res., Vol. 48, No. 4, 2009 2019 S ) final state TI ) tanks available to store state from campaign i TS ) tanks available to store state s P
Parameters ds ) minimum demand for state s H ) time horizon lsjmin ) minimum allowed length campaign at production line j producing state s M1 ) H + max{uchii′j} M2 ) a very large number M3 ) H + max{tchii′t} Fis ) amount of state s required per unit size of supplying campaign i ros ) time on which processing state s is available rsjmax ) maximum production rate at line j generating state s rsjmin ) minimum production rate at line j generating state s ruj ) ready time of production line j uchii′j ) changeover time between campaigns i and i′ at production line j tchii′t ) changeover time between campaigns i and i′ at storage tank t Vt ) volume capacity of storage tank t Continuous Variables Ci ) completion time of campaign i CTi ) completion time of storage task receiving material from i Fii′ ) amount of material supplied by i and consumed by i′ ITi ) starting time of stage task receiving material from i Lij ) length of campaign i in production line j MK ) makespan Qi ) overall production of campaign i Vii′ ) amount of accumulated material consumed by i′ at the completion time of it supplying campaign i Binary Variables Uii′ ) if campaign i supplies material to i′, Uii′ ) 1; otherwise, Uii′ )0
Wit ) if material from campaign i is sent to storage tank t, Wit ) 1; otherwise, Wit ) 0 Xii′ ) if campaign i is run or stored before i′, Xii′ ) 1; otherwise, Xii′ ) 0 Yij ) if campaign i is run in production line j, Yij ) 1; otherwise, Yij ) 0 Zii′ ) if campaign i, which supplies material to i′, finishes before i′ starts, Zii′ ) 1; otherwise, Zii′ ) 0
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ReceiVed for reView April 4, 2008 ReVised manuscript receiVed November 28, 2008 Accepted December 3, 2008 IE800539F
