2054
Ind. Eng. Chem. Res. 1990,29, 2054-2062
Optimal Design of Multipurpose Batch Plants. 1. Problem Formulation Savoula Papageorgaki and Gintaras V. Reklaitis* School of Chemical Engineering, Purdue University, West Lafayette, Indiana 47907
The design of multipurpose plants is posed as a mixed integer nonlinear program in which the binary variables are the structural choice variables. The proposed model is able to accommodate equipment used in and out of phase, units available in two or more sizes within a processing stage, multiple choices of equipment types for each product task, and allocation of products t o campaigns and to determine the campaign lengths and sizing of the processing equipment. In this paper, the proposed formulation is presented and its properties are discussed.
Introduction Batch/semicontinuous processing is the predominant mode of commercial production for high-value added chemical and biochemical products because it provides the necessary flexibility to accommodate a large number of low-volume chemicals in the same processing facility. Some of the chemicals typically produced in this mode include pharmaceuticals, processed foods, polymers, and specialty chemicals. The special characteristics of batch processes introduce considerable complexities to the related design and scheduling problems whose treatment requires the development of detailed and complicated solution procedures. Since relatively few researchers have addressed these problems, ample opportunity exists for systematic exploitation of the plant features and subsequent derivation of efficient solution methods. A special form of a batch facility is the multipurpose plant, which consists of general purpose equipment items used to manufacture a variety of products, each product having different task structure and equipment requirements. The design problem of such a plant involves some elements of scheduling, leading to additional levels of complication. Several structural options must be incorporated into the design stage to introduce a more realistic form of the overall design problem. In this paper, a mixed integer nonlinear optimization model that incorporates most of the key aspects of the general multipurpose plant is presented and its properties are discussed. Problem Definition The problem of the design of a general multipurpose batch plant can be outlined as follows: Given (1)a set of N products, the production requirements for each product and its selling price, and the available production time horizon; (2) a set of available equipment items classified according to their function into equipment families, the items of a particular family differing in size or processing rate; items that are members of the same equipment family and have the same size belong to the same equipment type; (3) recipe information for each product, which includes the task precedence relationship, a set of processing times/rates and a corresponding set of size/duty factors, both associated with every feasible task-equipment pair; in general, the processing time may be specified as a function of the equipment capacity; (4) the set of feasible equipment items for each product task; ( 5 ) the status (stable or unstable) and the transfer rules for the intermediates produced between tasks;(6) resource utilization levels or rates and changeover times between products with their associated costs; (7) inventory availability and costs; and (8) a suitable performance function involving capital and/or operating costs,
* To whom correspondence should be addressed.
sales revenue, and inventory costs, determine (a) a feasible equipment configuration that will be used for the manufacture of each product in the plant and (b) the sizes of processing units and intermediate storage vessels and the number of units required for each equipment type, so as to optimize the above performance function whiIe satisfying the constraints on the production requirements and the available production time horizon. The above definition of the multipurpose plant design problem addresses the deterministic design case, which does not consider variations in the problem parameters such as the available production time, the required quantities of the individual products, the composition of the product slate, the task processing times, the size/duty factors, and the equipment availabilities. The treatment of uncertainty within the framework of process design models imposes significant computational and analytical burdens. The effects of uncertainty in the product demands, size factors, and processing times for the multiproduct plant design were investigated by Wellons and Reklai tis (1989a,b). The problem of design of a multipurpose plant can be decomposed into three decision levels: (a) determination of the processing network structure, which involves determination of the task partitions for each product, campaign formation, allocation of equipment items to product tasks, selection of the operating mode of the processing units, and synthesis of the single product production lines; (b) sizing of the various pieces of equipment involving processing units and/or intermediate storage vessels and calculation of the campaign lengths; and (c) detailed mechanical design of individual equipment items. Any design strategy should integrate all three decision levels in a coherent way. The third level, however, is reduced to consideration of general purpose standard items that are commercially available, rather than to design of specialized items. Therefore, only the first two levels will need to be considered at the design stage. Furthermore, the determination of the campaign lengths is essentially a production planning step, since it involves computation of the time periods for the allocation of production resources for specific products. Therefore, the multipurpose problem is indeed a combination planning and design problem.
Previous Work Most of the published work deals with a restricted form of the multipurpose plant based on the following key assumptions: (a) There is a prespecified assignment of equipment items to product tasks; (b) parallel production is only allowed for products that have no common equipment requirements (compatible products); (c) all units of a given type are identical and can only be used in the out-of-phase mode; and (d) all units of a given type are devoted to the production of only one product at a time.
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Ind. Eng. Chem. Res., Vol. 29, No. 10, 1990 2055 The common approach has been to formulate and solve a mixed integer nonlinear model with focus on the derivation of certain production time constraints. Suhami and Mah (1982) developed “product configurations” by identifying those products that can be produced simultaneously. Their strategy was to randomly list an empirically determined finite number of configurations, apply a heuristic procedure for the selection of the best configuration, and then solve a modified version of the nonlinear multiproduct plant design formulation proposed by Grossmann and Sargent (1979), after incorporating appropriate time constraints into the formulation. Imai and Nishida (1984) offered an improvement to the above heuristic by solving a set partitioning problem to determine the “best” configuration, but no comparative results were reported. Klossner and Rippin (1984) enumerated all possible product configurations by solving a set partitioning problem and then solved a mixed integer nonlinear model for each configuration. Vaselenak et al. (1987) proposed a multiperiod formulation that employs a superstructure that embeds all possible product configurations. To derive the superstructure, a systematic procedure is devised that involves formation of all maximal sets of compatible products. The resulting product groupings are then used to form the limiting set of production time constraints (horizon constraints) through a tree search procedure. The key advantage of this approach is that it requires the solution of a single MINLP problem, which, for most applications, exhibits a single local and hence global minimum point. Faqir and Karimi (1988) devised alternative techniques for identifying the limiting set of horizon constraints based on the theory of linear inequalities. The proposed methodology succeeds in identifying the complete set of horizon constraints for problems in which the method of Vaselenak et al. fails. The major drawback of the proposed design procedures is that they fail to consider key aspects of the general multipurpose plant, such as alternative assignments of different equipment items to each product task and sharing of the units of the same equipment type among multiple tasks of the same or different products. Thus, they ignore the inherent flexibility of the multipurpose plant. Janicke (1987) considered the issue of sharing of the units of a given equipment type among multiple tasks and proposed a graph-based algorithm for determining the number of units of each equipment type required for the processing of a product set, given the equipment typeproduct task assignment,the equipment sizes, and the time interval over which this assignment will be in use. Kiraly et al. (1988) considered both aspects of the general design problem and developed a two-stage decomposition approach for its solution. In the first stage, a set of alternative campaign candidates is generated for each product by selecting units from a given list of types and sizes. In the second stage, a MINLP model is formulated for the selection of the least cost campaign set and the model is solved via a relaxed NLP. The details of these two approaches are not well defined, and no comparative results are reported. Recently, Faqir and Karimi (1989) also addressed both issues and proposed a MINLP formulation along with an iterative procedure for its solution based on the restrictive assumption that only a single equipment item can be used for the processing of each product task and that discrete sizes are allowed for the batch units. Their strategy involves the enumeration of all possible production paths for each product and the solution of an alternating sequence of MILP-LP problems to identify the equipment configuration with the least capital cost. This
approach involves elements of production planning, since it calculates the campaign lengths and determines the optimal production plans.
Operating Mode of the Processing Equipment The operating policy assumed for the processing units can significantly influence the final design. In general, the plant production rate can be increased by either increasing the limiting batch size of the various products or by reducing the limiting cycle times. Parallel units operating in phase on the batch size limiting stage (equipment type) relieve capacity bottlenecks, whereas parallel units operating out of phase on the time limiting stage result in reduction of the effective cycle time. With in-phase operation, a batch from the preceding stage is split among the parallel units of the current stage, whereas with outof-phase operation each unit in the current stage is assigned to alternate batches from the preceding stage. In general, the time limiting and capacity limiting tasks are different for the same product; hence, it seems necessary to consider a mixed operating policy that can efficiently accommodate both types of bottlenecks. Some definitions are necessary at this point to help us describe this mixed policy. A production line is the sequence of equipment items needed to produce one batch or a series of batches of a particular product. In a multipurpose facility, a product can be processed through several alternative production lines. The equipment items constituting a production line can be arranged in equipment groups that operate out of phase. The processing units within a group are operated strictly in phase. Moreover, the units in a group may be nonidentical, so that a wide range of batch sizes and alternate paths that have different capacities can be accommodated. As operating convenience, we will require that the same batch size be employed for each production path through a production line (for detailed analysis of production path-dependent batch sizes, see Wellons and Reklaitis (1989a,b)). This pattern allows for both outof-phase and in-phase operation of the processing units and will be used as the basic operating mode in the conceptual formulation developed later. Notice that in-phase operation is feasible only under the assumption that splitting of a batch is allowed for the particular product. A further issue concerns the equipment group identity, which is defined as the number and sizes of the processing units that constitute the group. If we assume that the group identity will remain the same regardless of the products that may use this group of units for their processing, then we essentially ignore the structural differences among the various products. Consequently, groups will only retain their identity for a particular task in a particular production line. After the completion of this task, the units of the equipment group used to perform the task may be rearranged into different equipment groupings. Moreover, a different equipment group can be used to perform the same task of a given product at different times, because of resource conflicts between the various products. Model Formulation The conceptual model formulated in this section will basically attempt to determine the processing network structure, as well as the equipment unit sizes. Elements of production planning are also incorporated into the model as it will identify the production plan corresponding to the final design. A number of assumptions, which can be classified into the following three categories, must be made at this point:
2056 Ind. Eng. Chem. Res., Vol. 29, No. 10, 1990
!*
product A
I
used to manufacture N different products. Since operating or inventory costs are not considered, the captial cost of the equipment becomes the natural optimization criterion. Therefore, the objective function for this problem can be formulated as follows
product A
I
*
E
min product
n
B
I I I !
W
(1)
CAMPAIGN 2
1 Ximek +
afleVebe
where V , and Ne are the size and number of units of equipment type e and a, and be are cost coefficients associated with equipment type e. The construction of the constraint set needed to determine the structure of the processing network and the number and sizes of the equipment units is presented in the following sections. Assignment and Connectivity Constraints. A set of binary variables must be introduced to represent the structural decisions that must be made at the design stage:
product C
~
CAMPAIGN I
e=l
HORIZON
=
0
c
Figure 1. Design configuration for a multipurpose plant.
A. General Plant Performance. (1)The plant operates in the No Intermediate Storage (NIS) mode. (2) There are F available equipment families. (3) The structure of the product recipe is known a priori. (4)There may be different equipment families that are feasible for the performance of each product task. (5) Merging of only consecutive tasks in the task sequence is allowed. (6) The processing times and size factors associated with every pair of product task-equipment item are constant. ( 7 ) No changeover costs, inventory charges, or resource utilization levels are considered. (8) The cost of semicontinuous equipment is excluded from analysis. B. Production Line Generation. (1)The processing units assigned to the various tasks are arranged into equipment groups. (2) The units within a group belong to the same equipment family and may differ in size. (3) The number of parallel equipment groups allocated to a particular task is unknown. (4)Each equipment group is used by one product at a time. (5) The groups assigned to a particular product task are identical. (6) Equipment groups only retain their identity for a particular product task in a particular production line. C. Campaign Formation. (1)The production horizon is divided into a prespecified maximum number of campaigns, each campaign consisting of a number of production lines used to produce the same or different products. (2) Each product may appear in several campaigns. (3) During a campaign, either each equipment item is assigned to a product task or it remains idle. (4)The individual batch sizes and the cycle times are campaign structure dependent, and thus, their values may be altered from one campaign to another. ( 5 ) The paths through a production line have the same capacity, and consequently, all batches processed through this line have the same size. By way of illustration of these concepts and assumptions, a design configuration of the equipment units in a multipurpose plant used to manufacture three products, A, B and C, is shown in Figure 1. Notice that production occurs in the form of two successive campaigns, which include multiple production lines. In each campaign, some equipment items remain unused. Furthermore, note that the production line for A is reconfigured in campaign 2 to accommodate the needs of product C. In our approach to the problem, we assume that there are E equipment types available in the plant that can be
if task m of product i is performed in unit type e and in campaign k otherwise
These binary variables represent the assignment of product tasks to equipment units and the assignment of production lines to campaigns. In order to properly state the assignment and connectivity constraints, we need to define the following sets: TA, = {mlm is the member of the task sequence of product i) Hence, TAi is the set that describes the series of tasks that must be performed to produce product i. Additionally, for every product task, we define the set of equipment types that are available for its execution Pim = (ele is feasible for task m of product i) Alternatively, the set U, = ((i,m)l(i,m)can be performed in unit type e ) is the set of tasks that can be processed by equipment type e. In this formulation, we assume that there are E equipment types available, which are classified into F I E equipment families. For each equipment family, let Lf be the following set: Lf = {ele belongs to family fl
Assignment Constraints. K
C C XimekI
1
i = 1, ..., N
m
E TAi
(2)
k = l eEP,,
This constraint set states that each product task must be assigned to at least one equipment unit and in at least one campaign. Here, K denotes the maximum number of campaigns into which the planning horizon can be divided. C Ximek5 Ne e = 1, ..., E k = 1, ..., K (3) (i,m)EU,
According to this constraint, the unit type e assignments to product tasks in a particular campaign cannot exceed the total number of units of type e in the plant. (4) Ximek + X i m q l 5 1 where i = 1, ..., N
m E TA, ( e , q ) : e,q E Pi,,, eELf qEL, f # g k = l , ..., K
This constraint ensures that the processing units assigned to a particular product task belong to the same equipment
Ind. Eng. Chem. Res., Vol. 29, No. 10, 1990 2057 t
within the specified horizon. Let nik be the number of batches of product i produced in campaign k and Bik be the batch size of product i in campaign k. The following constraint ensures that the actual amount of product that is produced in the plant satisfies the demand pattern:
= 10
K
rt*= 2
it**=
k=l
3
Without Delay T = 15 With Storage and Delay T = 10 ( t = processing time,
T = cycle time )
Figure 2. Merging of nonconsecutive tasks.
family. In general, units that belong to families with different operating characteristics, which may result in differences in processing rates, are not allowed to be grouped together. For example, assume that two families of reactors are available for the performance of the reaction task in some task sequence: jacketed and unjacketed vessels. According to the model, only one of these families must be used for the execution of the reaction task in order to retain operational uniformity. Connectivity Constraints, These constraints are used to assign values to some binary variables based on the values of other binary variables. Ximek
+ X i m + l l k + Xim-pLk
where i = l , ..., N m = 2 ,..., ITail-l p = 1, ..., m - 1 e E Pi, 1 E { p m + l i n Pm-qil
(5)
E Lf E Lg
e
f
+g
The above constraint set denotes that no merging of nonconsecutive tasks is allowed, and hence, no reuse of equipment items will take place. Example. Assume that the task structure of a particular product consists of three tasks, 1 , 2 , and 3. Also assume that the first two tasks have been assigned to units types R and D. If the feasible equipment types for task 3 are R, D, and L, then the question is to what equipment item should task 3 be assigned? Figure 2 shows the effects of assigning task 3 to equipment type R. Since reuse of R occurs, special formulas are required for the calculation of the sequence cycle time and conditions must be derived under which either delay or storage are necessary between the processing units involved in the loop (Yeh and Reklaitis, 1987). This pattern will considerably complicate the cycle time computation, which is undesirable a t this stage. Therefore, task 3 will be assigned to either unit D or unit L , and the recycling will be avoided. The second type of connectivity constraint is given by
e E Pi, k = 1, ...,K (6) This constraint set states that all tasks of a particular product must be completed within the time period covered by the campaign to which this product is assigned. Production Demand Constraints. The model must accommodate the production requirements for the various products within the specified horizon. Lower bounds on the production of each product i, Qi, are assumed to be given. Each product may appear in several campaigns, and hence, each of these campaigns will contribute its share to the total amount of product that must be produced m,q
nikBik 2 Qi
i = 1,..., N
(7)
Batch Size Constraints, Although each production line and, consequently, each campaign produces batches of the same size for a particular product, the batch sizes may be different from one campaign to another. The reason is that the units assigned to equipment groups within a campaign may be rearranged into different unit combinations after the interruption of the campaign. Therefore, different equipment groups may be assigned to the same product task in the new campaign. As noted earlier, if this task is the batch size limiting task, then the batch size will be affected by the new equipment assignment. Furthermore, if the equipment group assigned to a particular task consists of more than one unit, then the batch processed in this group will be split among the units of the group. Consequently, the batch size will be determined by the number of units constituting the equipment group and the sizes of these units. Let Sime be the size factor corresponding to task m of product i when this task is performed in unit type e that denotes the volume of product i processed by unit type e in task m per unit amount of final product i. In addition, let NUimekbe the number of units of type e that are contained in each equipment group assigned to task m of product i in campaign 12. Then, the following constraint must be satisfied: Bik
ve
NUimek-
5 eEP,,
i = 1, ..., N
Sime
m E TAi
k = 1, ..., K (8)
This constraint states that the batch size of a particular product in campaign k must be limited by the capacity of the smallest equipment group in the production line used to process this product. Additional constraints are needed at this point to impose logical conditions on the batch sizes. If product i is produced in campaign k, then that product must have associated with it a positive batch size in campaign k, ~
...
i = l , ..., N
mETi
k=l,.,., K(9)
where B*ik is a special parameter denoting a lower bound on the batch size (this parameter will be discussed in the section on “Variable Bounds”). Batch Number Constraints. If a particular product appears in campaign k, then this campaign will contribute a positive number of batches to the total production quantity for that product
E TAi
Here again n*ik is a suitable lower bound. Equipment Number Constraints. These constraints accommodate both the calculation of the number of the processing units used in the plant and the arrangement of these units into equipment groups. Let Ne denote the number of processing units of type e and NGid denote the number of equipment groups assigned to task m of product i in campaign k . Then, the total number of units of type e that are used in the plant depend on the number of units
2058 Ind. Eng. Chem. Res., Vol. 29, No. 10, 1990
of type e contained in the equipment groups allocated to the various product tasks in the different campaigns constituting the production schedule; namely,
Let K denote the maximum number of campaigns ( K I N ) into which we can divide the planning horizon, each campaign having length Tk. Clearly the sum of the campaign lengths must be less than or equal to the total available production time, H :
e = l , ..., E
k=l,
..., K
(11)
In addition, let N F denote the maximum number of units of type e and NGKf denote the maximum number of equipment groups that can be assigned to task m of product i in campaign k . Then, i = 1, ..., N NUImekI NraxX,,,k m E TA, e E P,, NUlmek2 Xlmek i = 1, ..., N m E TA, e E Pi,
NG,,k I NGgT
Xlmek
k = 1, ..., K (12) k = 1,
m E TA,
k = 1, ...,K (14)
Cycle Time Constraints. As noted earlier, the cycle time of a particular product is assumed to be campaign structure dependent and, hence, may vary from one campaign to the next. The cycle time T,,k of product i in campaign k is the maximum tusk processing t i m e , p,,k, of the tasks involved in the task sequence of product i; that is, TLlk =
",""
bimk!
(15)
The task processing time can be defined as the group processing t i m e , qg, divided by the number of equipment groups assigned to that task; namely,
prmk = qg/NGimk
(16)
under the assumption that the groups assigned to task m are identical. In our approach to the problem, the group processing time for a particular group g is defined as the maximum processing time of the units that are contained in this equipment group. This definition is based on the assumption that the groups assigned to a particular task are operated out of phase. 9, = m:x
1tme-l
(17)
(20)
k=l
In general, a particular campaign may process several products a t the same time. Each of the products that appear in the same campaign will be produced within the same time interval, which is the campaign length. For example, assume that products i and h are assigned to campaign k . Then,
nrkTL,k= nhkTLhh= Tk
..., K (13)
i = 1, ..., N
eEP,,
K
XTklH
where nrkTL,* denotes the time required to produce nrk batches of product i in campaign k. The above constraint can be stated in a more general form as follows: Tk 2 n,kTL,k i = 1, ..., N k = 1, ..., K (21) Although this constraint is expressed as an inequality, it will be active (satisfied as equality) at the optimal solution, because the optimal production plan must satisfy the production requirements by utilizing the total available production time as much as possible. Variable Bounds. Appropriate upper and lower bounds on each variable must be derived to adequately represent the feasible solution space. All the variables have zero lower bounds, because they are switched on and off depending on the values of the binary variables that impose logical conditions on them. The only exception is the processing unit capacity variable, V,, which lies between prespecified non-zero upper and lower limits:
V Y IV , I Vpax
e = 1, ..., E
(22)
Prespecified upper bounds are also imposed on the number of units of type e and on the number of equipment groups that are assigned to a particular product task. These variables are then bounded as follows 0 5 Ne IN p a g
e = 1, ..., E
(23)
0 5 NG,,k 5 NGZY 5 max
EP,n
..., N
where t,,, is the time used to process task m of product i in vessels of type e . Expressions 15-17 can be combined into the following cycle time constraint
mETA, k = l , , K ( 2 4 ) Similarly, the number of units of type e contained in equipment groups that perform task m of product i in campaign k can be constrained as follows: 0 INUlmekINtp"" i = 1, ..., N k = 1, ..., K (25) m E TA, e E P,,
which denotes that the limiting cycle time of product i in campaign k depends on the structure of the equipment groups assigned to that product. In addition, let TL,denote the maximum cycle time of product i in campaign k . Then,
Additional upper bounds are needed on the number of batches of the various products and the limiting cycle time. The largest possible number of batches of a particular product can be produced in a maximum length campaign when the limiting cycle time takes on its minimum value; namely, nmax tk =T~~~X/T*L i =, ~ l , ..., N k = l , ..., K (26)
k = 1, ..., K (19) Horizon Constraints. From the assumptions stated earlier in this section, it is evident that production occurs in the form of long campaigns. Proposition. The maximum number of campaigns into which the planning horizon is divided is N , where N is the number of products using the plant for their processing. The proof of this proposition appears in the Appendix.
i = l ,
From constraint (20), it is evident that a natural upper bound on the length of campaign k is the total available production time, H. Hence, 0 ITk 5 H k = 1, ..., K (27) As noted earlier, the lower limit on the limiting cycle time is zero, because of the conditions that are imposed by the various logical constraints. However, when a par-
Ind. Eng. Chem. Res., Vol. 29, No. 10, 1990 2059 Table I. Feasible Unit Types for Tasks in the Example task unit tme Drocessing time, h 1A 2A 3A
RV, RL RV, RD FT
1, 4 3, 2 1
Table 11. Alternative Production Lines for the Example production line cycle time, h RV-RV-FT 1.5 RV-RD-FT 1 RL-RV-FT 2 RL-RD-FT 2
ticular product is processed in campaign k, then its cycle time will take a non-zero value that can be bounded by a non-zero lower bound, T *La. This lower limit is basically a "pseudolower" bound, because it is only used when the corresponding cycle time variable is switched on. Moreover, it serves as a parameter for the calculation of a tighter upper bound on the number of batches, rather than an actual bound. The smallest possible cycle time that can be attained by product i, when this product actually appears in campaign k, can be computed by the following formula: T*L, = max min {time/NGg$] m€TAi eEPi,
i=l,
..., N
k = l , ..., K
(28)
This constraint states that the minimum cycle time corresponds to the production line in which the task processing times of the various product tasks are at their minimum value and the maximum number of equipment groups is assigned to each product task. Example. Consider product i with the task structure consisting of tasks 1, 2, and 3. The feasible equipment items for each task along with their processing times are given in Table I. Additionally, assume that the number of equipment groups that can be assigned to each task is 2 and that each group contains only one processing unit. Now, consider the set of alternative production lines shown in Table 11. From this table, it is evident that, for every combination of unit assignments, the cycle time of product A cannot be less than 1. Therefore,
T*L, = max (min(i,
i},
,in{:,
5).i}
=
max {0.5,1, 0.5) = 1 Upper bounds on the limiting cycle times must also be specified. The largest possible cycle time of a particular product can be attained along the production line using only one equipment group for each task, each group containing units with maximum processing rate; namely, T e = max max {time] i = 1, ..., N k = 1, ..., K mETAi eEPi,
(29) Upper bounds on the feasible batch sizes, Bik,can be obtained from the equipment group capacities. The largest possible batch size of product i in campaign k is constrained by the smallest maximum capacity equipment group among the groups that constitute the production line for product i. From constraints (8) and (12), we get
or
Finally, a pseudolower bound on the batch size must be derived. This bound is used in constraint (9) to provide tighter lower limits on the non-zero value asigned to Bik in the case where constraint (9) is nontrivial (that is, the summation has a positive value). The smallest possible batch can be produced along a production line in which each equipment group is assigned a single unit with minimum capacity (assuming 100% utilization of that unit); namely, B*ik = min min (VTin/Sime] mET.4, eEP,,
i = l , ..., N
k = l , ..., K
(31)
Combinatorial Analysis The determination of the processing network structure involves the formation of campaigns and the allocation of equipment items to products. The restriction on the maximum number of campaigns is a first attempt to reduce some of the degeneracies of the solution space. In this context, a degenerate solution is one that forms identical campaigns and interchangeably equivalent production schedules. Since no restrictions have been posed yet on the way the campaigns will be formed, the model will consider all possible campaign alternatives and, thus, create several equivalent production schedules that will lead to equivalent designs. Two constraint sets are introduced as a means of eliminating part of the degeneracy. These constraints will be included in the constraint set of the resulting MINLP problem. Campaign Ordering Constraints. The proposed model does not explicitly account for product-sequencedependent setups, because it assumes that changeover times are small compared to the campaign lengths. As a consequence, the production schedules that are produced are nonunique, in the sense that, if the order of campaigns is interchanged, the production schedule will remain the same. To eliminate part of the degeneracy, the campaigns will be ordered according to the number of products that are assigned to them. This ordering of campaigns will not rule out feasible production schedules. Let PRikserve as the priority index and denote the assignment of product i in campaign k. Then, the lower indexed campaigns will have higher priority than the higher indexed campaigns CPRik 2 CPRik+l i = 1, ..., N i i k = 1, ...,K - 1 (32) Additional logical constraints are imposed to ensure that, if product i is assigned to campaign k, then the priority index will take its upper bound value (=l); otherwise, it will take its lower bound value (=O). This constraint takes into account only the first task for each product i k = 1, ..., K PRik 1 Xilek i = 1, ..., N e E Pil (33) PRik d C Xilek i = 1, ..., N
k = 1, ..., K
(34)
eEP,1
Size Ordering Constraints. As mentioned earlier, each equipment family consists of equipment items that perform the same function and may differ in capacity. Since the sizes of the equipment items may take arbitrary values, the values within each set of sizes can be permuted to derive equivalent designs. To illustrate, consider an equipment family that has three unit types with sizes 1O00,
2060 Ind. Eng. Chem. Res., Vol. 29, No. 10, 1990 Table 111. Production Requirements product yearly demand, kg/year A 700 000 B 500 000 C 380 000 D 400 000 Table IV. Available Equipment Items" equipment item
F R1, R2. R3 G E l , E2, E3
P Z a
capacity ranee, L 500-3000 500-2000 500-3000 500-3000 500-3000 500-3000
upper bound on no. of units
Available discrete sizes: 500-1000-2000-2500-3000.
2000, and 500. Then, the assignments VI = 1000, V2 = 2000, V , = 500 and VI = 500, V2 = 1000, V , = 2000 are equivalent. To eliminate this degeneracy, we order the unit types within each equipment family, such that lower indexed types will have larger sizes than higher indexed types V , 2 V,,, e = 1, ..., E e , e + 1 E L, (35)
Discussion The design optimization problem described in the last section consists of minimizing eq 1 subject to constraints (2)-( 14) and (18)-(351, where constraints (221-431)represent variable bounds. This problem is a mixed integer nonlinear model (MINLP) in the continuous variables TL*, qk, &, and Tkand the integer variables Xlmk,Ne, NUimk, NGlmk,and V,. This optimization model is nonconvex; furthermore, its continuous relaxation cannot be convexified through exponential transformation of the variables, because many constraints involve sums of variables or variable products that cannot take a convex form under such variable transformations. Additionally, the problem cannot be reformulated as a geometric program, because many constraints are not posynomial. As a result, the globally optimal solution cannot be guaranteed. In addition to being nonconvex, the proposed model can become of high computational complexity for practical applications. Specifically, since it can be shown that the maximum number of campaigns is bounded by the number of products, the dimensionality of X and NU are roughly proportional to N 2 times the average number of tasks per product times the average number of feasible equipment types per product. Although the integer restrictions on some of the variables may be relaxed, the binary variables require branch and bound or decomposition based approaches. Problems of modest dimensionality are amenable to solution using existing MINLP solution techniques, which nonetheless cannot guarantee attainment of the global optimal solution. An example will be solved in the next section to illustrate the performance of the Outer Approximation/Equality Relaxation algorithm of Duran and Grossmann (1986) and Kosis and Grossmann (1987) with the proposed formulation. Example A multipurpose plant, used to produce four products with annual requirements given in Table 111, consists of the six equipment families detailed in Table IV. The feasible equipment types for each product task and the associated size factors and processing times are given in Table V. The unit cost coefficients are given in Table VI. Note that there are two different equipment families that
Table V. Size Factors ((L/kg)/Batch) and Processing Times (h/Batch) size factor (processing time) for equipment type R1, R2, E l , E2, task F R3 G E3 P A1 1.5 (4) A2 2 (6.5) A3 1.6 (5) B1 1.5 (4) 3.6 (5.9) B2 2.5 (5.5) B3 1.9 (4.2) 1.4 (4.5) Cl c2 1.5 (4) 3.7 (5.5) D1 1.5 (4) 2.7 ( 7 ) D2 2.2 (6.5) D3 1.8 (4.7) 1.6 (4.7)
z
Table VI. Cost Coefficients aea unit tvDe a. F 88 R1, R2, R3 69 G 280
unit tvDe E l , E2, E3 P
z
a. 260 360 370
Power cost coefficient, be = 0.6.
PRODUCT A
CAMPAIGN I
PRODLCT
R
CAMPAIGN 2
PRODVCT I
CAHPAIGI I
PRnnLm n
CAYPAIGI 4
Figure 3. Design configuration for the example.
are feasible for the performance of each of the tasks B1, C2, D1, and D3. Also assume that at most two equipment groups can be assigned to each product task. For this example, the proposed optimization model will involve 892 constraints (695 linear and 197 nonlinear) and 317 variables (248 integer (92 binary) and 69 continuous). The OA/ER algorithm implemented in the computer code DICOPT on an IBM 3090 was used to solve the corresponding MINLP program. Since DICOPT can only handle binary integer variables, a relaxed problem was first solved that involved only binary variables as integer variables. The integer (locally) optimal solution was obtained by rounding up the continuous values of the relaxed variables to the nearest integer. A major disadvantage of the OA/ER algorithm is the requirement of a feasible initial guess to converge to a (locally) optimal solution. Therefore, multiple runs of the algorithm were performed starting from different initial points. The algorithm converged to different locally optimal solutions depending on the initial guess. Furthermore, the optimizer did not significantly change the initial solution point. Specifically, the number of active campaigns and the allocation of products to campaigns remained the same, whereas the assignment of equipment types to product tasks was slightly modified. For example, when the initial solution point is a design configuration involving four single product campaigns, the OA/ER algorithm converges to a four single product campaign configuration shown in Figure 3 with cost $210310 (the resulting sizes and number of units are given in Table VII). In part 2 of this paper, however, it will be shown that a lower cost design configuration can be found
Ind. Eng. Chem. Res., Vol. 29, No. 10, 1990 2061 Table VII. Integer Solution for the Exampleo unit tvDe V. Ne 2 F 3000 1 R1 2000 1 R2 2000 0 R3 2 G 3000 El 3000 E2 3000 E3 P 3000 Z ~
~
~~~
'Objective function value: $210 310.
in reasonable computation time by a novel solution procedure that is independent of the initial point. This solution strategy constitutes a formulation specific decomposition scheme that successfully decomposes the problem into smaller subproblems for which computationally tractable solution procedures exist.
Acknowledgment This work was supported in part by the US.-Spain Committee for Technical Cooperation and E. I. du Pont de Nemours & Co.
Appendix: Proof of the Proposition A definition and two lemmas must be stated before the proof of the proposition. Definition. Two or more campaigns are identical if the same product set is assigned to each campaign and the same equipment sequence constitutes the production line for each product. Lemma 1. Identical campaigns will produce the same batch size and cycle time. Proof. Consider two identical campaigns, k = 1,2, used to process product A. Let B1, TL,B,, TL,, Ve, NGml,NGm2, NUme1,and NUme2denote the gtobal optimal solution to the corresponding optimization problem with objective value u. Then, the following must be true: Now, assume that Bl > B2 and TL, > TL,.Also let b be the capacity limiting task and c be the time limiting task. Then, the batch size is determined by the split batch sizes and the number of in-phase units in stage b (assuming 100% utilization at this stage); namely,
where Eb denotes the unit types used to perform task b. The inequality Bl > B2 implies that
Now, assume that unit type e is the smallest capacity unit in the batch size limiting stage. Also assume that NUbel > NUbezfor some e. Then, the unit capacity will remain the same, but NUbel- NUh2 units will not be used during campaign 2. Furthermore, assume that unit type f is the largest processing time unit in the time limiting stage. Then, the cycle time in campaign Iz will be computed as follows:
-
Since TL, > TL,, tCf/NG,l tCf/NGCZ NGc1 < NGc2 which means that NGc2- NGC1equipment groups will not
'
be utilized during campaign 2. In all the above cases, an improved solution can be found in which B1 = B2 and TL1 = TLp,so that any equipment underutilization will be eliminated. This is a contradiction to our hypothesis that the initial solution is the global optimum. Therefore, B1 = Bz and TL, = TL,. This result can be easily extended for any number of campaigns. Lemma 2. A production schedule containing identical campaigns can be reduced to an equivalent schedule in which the identical campaigns have been merged into a single campaign. Proof. Since no setup times between two campaigns are considered in the formulation, two identical campaigns can be combined to form a single campaign that produces the same batch size and cycle time as the component campaigns. Proposition. The maximum number of campaigns into which the planning horizon is divided is N , where N is the number of products using the plant for their processing. Proof. Let K be the maximum number of campaigns in which we can divide the planning horizon, H. Two cases can be considered. (i) K < N . In this case, some possible campaign sequences would be excluded from consideration. For instance, the limiting case in which production occurs in sequential single product campaigns would be disallowed. Consequently, the plant would be deprived of some of its inherent degrees of freedom. (ii) K > N . The additional campaigns, K - N , could only be campaigns identical with the ones that already exist or idle campaigns. We reason as follows: We assumed that the operation of a particular campaign will be interrupted immediately after one of the products that are being produced in this campaign finishes processing. In the most restrictive case, where the first campaign starts with all products being produced simultaneously and each campaign is interrupted after the completion of only one product, the number of active campaigns becomes equal to the number of products N. In any other case, the number of campaigns will be less than the number of products. The lower bound on the number of campaigns is 1, corresponding to the limiting case where all products in the plant are being processed in one campaign with length equal to the total available production time, H. Therefore, since there may exist campaigns with zero length (no products are assigned to these campaigns), the actual number of campaigns may be between 1and N , where N is the number of products in the plant.
Literature Cited Duran, M. A,; Grossmann, I. E. An Outer-Approximation Algorithm for a class of Mixed-Integer Nonlinear Programs. Math. Prog. 1986,36, 307-339. Faqir, N. M.; Karimi, I. A. Optimal Design of Batch Plants with Single Production Routes. Presented at the AIChE Annual Meeting, Washington, DC, 1988; Paper 79a. Faqir, N. M.; Karimi, I. A. Design of Multipurpose Batch Plants with Multiple Production Routes. Presented at the FOCAPD Conference, Snowmass Village, CO, 1989; Paper 4. Grossmann, I. E.; Sargent, R. W. H. Minimum Design of Multipurpose Chemical Plants. Ind. Eng. Chem. Process Des. Deu. 1979, 18 (2), 343-348. Imai, M.; Nishida, N. New Procedure Generating Suboptimal Configurations to the Optimal Design of Multipurpose Batch Plants. Ind. Eng. Chem. Process Des. Deu. 1984,23, 845-847. Janicke. W. On the Desim of MultiDurpose - - Chemical Plants. Comput. Ind. 1987, 9, 19-34. Kiraly, L. M.; Friedler, F.; Szoboszlai, L. Optimal Design of Multipurpose Batch Chemical Plants. Proceedings of CHEMDATA 88, Gothenburg, Sweden, 1988; pp 355-359.
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I n d . Eng. Chem. Res. 1990,29, 2062-2073
Klossner, J.; Rippin, D. W. T. Combinatorial Problems in the Design of Multiproduct Batch Plants. Presented a t the AICHE Annual Meeting, San Francisco, CA, 1984; Paper 104e. Kocis, G. R.; Grwmann, I. E. Relaxation Strategy for the Structural Optimization of Process Flow Sheets. Ind. Eng. Chem. Res. 1987, 26, 1869-1880. Suhami, I.; Mah, R. S. H. Optimal Design of Multipurpose Batch Plants. Ind. Eng. Chem. Process Des. Dev. 1982, 21, 94-100. Vaselenak, J. A.; Grossmann, I. E.; Westerberg, A. W. An Embedding Formulation for the Optimal Scheduling and Design of Multipurpose Batch Plants. Ind. Eng. Chem. Res. 1987,26, 139-148. Wellons, H.C.;Reklaitis, G . V. The Design of Multiproduct Batch
Plants Under Uncertainty with Staged Expansion, I. Comput. Chem. Eng. 1989a, 13, 115-126. Wellons, M. C.; Reklaitis, G. V. Optimal Schedule Generation for a Single-Product Production Line-I. Problem Formulation. Comp u t . Chem. Eng. 1989b, 13 (1/2), 201-212. Yeh, N. C.;Reklaitis, G. V. Synthesis and Sizing of Batch/Semicontinuous Processes: Single Product Plants. Comput. Chem. Eng. 1987, I 1 (61, 639-654. Received for review February 15, 1990 Revised manuscript received May 23, 1990 Accepted June 4, 1990
Optimal Design of Multipurpose Batch Plants. 2. A Decomposition Solution Strategy Savoula Papageorgaki and Gintaras V. Reklaitis* School of Chemical Engineering, Purdue University, West Lafayette, Indiana 47907
A mixed integer nonlinear programming (MINLP) formulation for the optimal design of a multipurpose plant has been developed in part 1 of this series. The complexity of the model makes the problem computationally intractable for direct solution using existing MINLP solution techniques. Consequently, a decomposition strategy is presented that alternately solves a MILP master problem, which determines the values of the binary assignment variables for fixed compaign lengths, and a NLP subproblem, which performs equipment sizing and determines the values of the campaign lengths. The effectiveness of the decomposition procedure is demonstrated with a number of test problems that were solved in reasonable computation times. Introduction In part 1 of this series (Papageorgaki and Reklaitis, 1990), we presented a mixed integer nonlinear programming (MINLP) formulation for the design of a multipurpose batch plant. The complexity of the model and the combinatorially large feasible solution space suggest that a more specialized solution procedure should be developed to allow this problem to be solved routinely in practice. The proposed model will serve as a basis for such a procedure, since it adequately represents the most important structural options that must be selected at the design stage, as well as the interactions between the different decision levels. In this paper, we develop a solution methodology based on decomposition of the original MINLP problem into two subproblems: a MILP master problem that determines the values of the binary assignment variables and a NLP subproblem that optimizes the values of the continuous variables. An iteration procedure is devised that solves an alternating sequence of these subproblems to identify the design configuration with the least capital cost. Three test examples are solved to illustrate the effectiveness of the proposed solution strategy. Alternative Solution Techniques Mixed integer nonlinear problems can be solved via a variety of methods that include branch and bound procedures (Beale, 1977; Gupta, 1980), Generalized Benders Decomposition (Geoffrion, 1972; Floudas et al., 1989),and the Outer-Approximation/EqualityRelaxation algorithm (Duran and Grossmann, 1986; Kocis and Grossmann, 1987a,b). The OA/ER algorithm can be classified as a decomposition scheme that solves an alternating sequence of nonlinear subproblems and mixed integer linear master problems. In the NLP phase, the binary variables are fixed
* To whom correspondence should be addressed. 0888-5885/90/2629-2062$02.50/0
and the continuous variables are optimized to yield an upper bound on the solution of the MINLP problem. The MILP master problem contains the linear constraints of the original MINLP and the linear approximations of the nonlinear constraints derived at the solution of the NLP subproblem. The MILP phase is used to optimize the discrete (typically binary) variables and to provide an increasing sequence of lower bounds on the solution of the MINLP problem. The Generalized Benders Decomposition is a similar decomposition scheme that is based on a dual representation of the MILP master problem. In general, branch and bound techniques are required to obtain the values of the integer variables. The branch and bound strategy is based on the solution of continuous relaxations of the original MINLP problem. However, this method can guarantee attainment of the global optimum only when the continuous relaxations at each step are convex or can be convexified through suitable variable transformations (for instance exponential transformations). In this case, the necessary and sufficient Kuhn-Tucker conditions are satisfied and the global optimum solution is obtained. Alternatively, when the nonlinear relaxations exhibit nonconvexities that cannot be transformed to a convex form, the branch and bound procedure can only guarantee local optimal solutions. The OA/ER algorithm attempts to overcome this disadvantage by applying the branch and bound strategy only for the solution of the MILP master problems, for which the continuous relaxations are linear and, hence, convex. However, the NLP subproblem, which is first solved to provide the linearization points for the MILP problem, may still be nonconvex and, hence, exhibit multiple local optima. Recently, Kocis and Grossmann (1987a,b) incorporated the OA/ER algorithm into a two-phase strategy that applies local and global tests to identify nonconvexities in the constraints and the objective function and attempts to eliminate the impact of these nonconvexities by suitably modifying the 0 1990 American Chemical Society
