Ind. Eng. Chem. Res. 1993,32, 3037-3050
3037
Scheduling of Multipurpose Batch Chemical Plants with Resource Constraints Athanasios G. Tsirukis, Savoula Papageorgaki, and Gintaras V. Reklaitis’ School of Chemical Engineering, Purdue University, West Lafayette, Indiana 47907
The objective of this paper is to develop an efficient scheduling procedure for multipurpose batch chemical plants with resource constraints. The scheduling problem subject t o resource restrictions is formulated as a mixed integer nonlinear programming (MINLP) model which is characterized by high combinatorial complexity and nonconvexity. These features are the main obstacles to the direct application of conventional optimization methods for the solution of the proposed model. Consequently, a two-level decomposition strategy is introduced which significantly mitigates the computational requirements associated with the solution of the MINLP model. The effectiveness of the proposed model and solution strategy is illustrated with a number of test examples.
Introduction The multipurpose batch chemical plant consists of a collection of equipment items and resources which are allocated to the manufacture of various chemical products. The product demand is expressed through a set of customer orders that have to be satisfied before a given time instance called due date. The production in orders seems to perfectly suit the low-volume production character of a multipurpose batch facility. The scheduling problem is concerned with the optimal allocation of time and plant resources to ensure satisfaction of given production quotas. The assignment of equipment and resources to specific production tasks and the detailed determination of operation parameters are important subproblems that must be considered in parallel.
Problem Definition
ished after their usage. Examples of renewable resources are manpower, electricity, heating and cooling flow rates, water, and steam. Simple extensions to the proposed formulation can accommodate the existence of consumable resources such as raw materials and capital. Task: A recipe task, @, m), is the mth elementary physicallchemical operation required to complete the production of product p . The recipe structure RCp contains the set of tasks necessary for the production of product p . It is represented by a graph structure whose nodes are recipe tasks and whose directed arcs define precedence relations among tasks. The same recipe is used for the production of different orders involving the same product. The production task ( 0 , m ) refers to the realization of a certain recipe task @, rn) for a given order 0.
Each task is associated with a set of equipment items, which are able to process it, along with the corresponding batch size dependent nonlinear relation for the processing time, f p m e ( B ) , and a set of necessary resources, R s p m e , accompanied by the corresponding batch size dependent nonlinear relation for the resource consumption, g s p m e ( B ) . The size factor of recipe task @, m ) produced in equipment e is denoted by S p m e . If a unit e is assigned to the processing of a certain production task, ( 0 , rn) in campaign k, then a periodic production pattern is imposed and the quantity of order o processed in equipment e at a given time instance is called the batch size, Bomek. The time necessary to complete the processing of this batch is denoted by tom&, and is given by the relation Epm,
The following definitions are of central importance for the multipurpose batch plant (MBP) scheduling problem: Product: A product is denoted by p, p = 1, 2, ...,P. Order: Order is denoted by 0 , o = 1,2, ...,0. With each order there exists a corresponding product,po,production quantity, q o , due date, do, and tardiness cost, co. Campaign: A campaign, Denoted by c k , k = 1, 2, ..., K, is a set of orders that are produced concurrently. Each order can be processed in more than one campaign. Equipment Item: An equipment item is denoted by e , e = 1,2, ...,E . Ve represents the volume of equipment e if it is a batch processing unit or the processing rate if it is a semicontinuous one. Each unit can perform one or more physicallchemical operations. Resource: Resource is denoted bys, s = 1,2, ...,S. The maximum resource availability is denoted by R,. The set of resources includes all the plant utilities that supplement the operation of equipment items in a batch plant. For example, the reaction task of some products has specific temperature requirements which imply the use of heating or cooling fluids, electric filters consume electricity, and almost every processing task requires the attendance of a human operator. The resource availability can significantly influence the productivity and profitability of a MBP, especially if utility limitations occur’. Clearly, any viable scheduling approach must consider resource restrictions. The present study is concerned with the class of renewable resources whose availability levels are replen-
* T o whom correspondence should be addressed. 0SSS-5885/93/2632-3037$04.00/0
= fpme(Bomek) (1) The corresponding consumption of resources is denoted by rsomek and is given by the relation tomek
(2) Group: In a multipurpose facility it is common practice to employ equipment items with diverse processing characteristics (volume, production rate, resource consumptions) for the manufacture of the same production task. A convenient arrangement of the assigned equipment can result in superior production rates. Therefore, the equipment arrangement decisions form an important subproblem that must be addressed in parallel. Customarily, the units that are assigned to production task ( 0 , m ) in campaign k are arranged into e q u i p m e n t groups. Each group, Gomgk, g = 1, 2, ..., N G o m k , consists of a set of units Uomgk C E p m , that operate in a parallel ‘sornek
= gspme(Bomek)
0 1993 American Chemical Society
3038 Ind. Eng. Chem. Res., Vol. 32, No. 12, 1993
in-phase mode. The quantity of order o processed at a given time instance in the group is called group batch size and is denoted by BoGmBk, the time necessary for the production of a batch is called group cycle time and is denoted by t&& and the total consumption of resource G s by all units in Uomgk is denoted by rSomgk. G
= ( u o m g k , B&k, tom&, (r&gk, S = 1,2, .**, 1' ) The notion of group collection, denoted by GComk, arises when multiple groups are assigned to the same production task. The groups constituting a group collection operate in a parallel and out-of-phase mode, again, under aperiodic pattern. The processing characteristics of the group collection are easily determined by those of the group members: Gomgk
C
GComk = ({GOmgk, g = 1,2, * * * , NGomk),BFmk, tomk, jrsComk,s = ~ 2...,,Si) Production Line: PLok consists of the set of group collections assigned to all the production tasks of a given order, 0,in campaign k. Associated with a production line the time necessary for the are the line batch size, production of a batch called line cycle time, t k k , and the total consumption of resource s by all groups G o m g k ,
&,
Determine a schedule X* that optimizes the objective function so that allorders are produced and all the physical, logical,and production-dependent constraints are satisfied. As becomes clear from the problem definition, we avoid the introduction of path-dependent production suggested by Wellons and Reklaitis (1989a). The processing rate benefits arising from production in paths can be very easily offset by the increased equipment clean-up delays and by the subsequent operational complexity of the production schedule which may increase the batch failure frequency. Following the formulation proposed by Wellons and Reklaitis (1991a), the batch processing time formulas (1) can be further extended to account for the time lost during the transfer of material from one equipment to another (finite transfer rates). The scheduline activity is focused on increasing the profitability of a MBP. Therefore, the ideal objective function should incorporate a measure of the net production earnings, accounting for operational costa (introduced by the utilization of different equipment items and by the resource consumption), customer satisfaction, etc. Since in practice these cost coefficients are very hard to obtain and, usually, not available, we have limited our attention to a small set of scheduling criteria (weighted tardiness and makespan). Literature Review
t 0 k
S = 1,2, ..., 8)) pLok = ((Gcomk,m E RC,), Bkk, tkk, Campaign Structure: A campaign structure csk consists of the set of production lines assigned to every order o E c k and the total consumption of resource s by all production lines rfk:
csk = ((PLok,0 E ck),{&, s = 1, 2, s)) Schedule: A schedule, X,is a set of triplets of the form -*e)
x = ((ck,csk, Tk),k = 1,2, ..., I ( ) Each triplet contains a campaign, a corresponding campaign structure, and the time interval, Tk,within which the campaign will be active. This interval is called the campaign length. Given the above notation, the MBP scheduling problem can be defined as follows: Given Equipment information: For each equipment item e, e = 1, 2, ..., E , the volume capacity, V,. Resource information: For each resource, s,s = 1 , 2 , ..., S, the maximum availability, R,. Order information: The order information includes
[Po,Qo, do,col,
0
= 1,
a**,
0
Production recipe information: The production recipe information includes RC,, p = 1, ...,P
Epm, p = 1, ...,P, m E RC,
gspm,(B),s = 1, ...,S, p = 1,...,P, m E RC,, e E E p m
Objective function: An objective function is given performance measure such as makespan, weighted tardiness, or production cost.
The production planning and scheduling problems for multipurpose batch chemical plants have been addressed by relatively few researchers. These efforts can be divided into the following categories: (i) work that concentrates on the production planning subproblem and (ii) work that addresses all aspects of the planning and scheduling problem. Reklaitis (1982), Rippin (1983a,b), Ku et al. (1987),and Wellons and Reklaitis (1988)provide a broader review of planning and scheduling methodologiesfor batch chemical plants. We will focus our attention on works that addressed the scheduling of batch chemical plants subject to resource Constraints. Aside from the job-shop algorithm used by Lazar0 and Puigjaner (1985) for scheduling multipurpose plants, only Egli and Rippin (1986) and Kondili et al. (1993) have incorporated resource constraints on their proposed scheduling methodologies. Egli and Rippin considered the short-term scheduling problem of a multiproduct batch plant. The emphasis of their work wm to obtain aschedule which meets a specified demand pattern while minimizing costs arising from storage, changeovers, and utility costs. The scheduling procedure accounta for constraints on working patterns (e.g., weekends), limited availability of shared resources such as utilities and manpower, inventory requirements for intermediate products, and raw material deliveries. The scheduling procedure is enumerative in nature: all possible production sequences are generated and then progressively eliminated by imposingthe problem constraints so that only favorable sequences are fully evaluated. Feasible schedules are generated by shifting each batch schedule forward or backward in time until all resource constraints are satisfied. Results are reported for a small example with 4 products and 11 equipment items. Kondili et al. consider task assignment as well as intermediate storage and resource constraints with a mixed integer linear programming (MILP) formulation of the short-term scheduling problem. To model the use of plant resources, the scheduling time horizon is discretized into a number of time intervals so that the unit assignments and resource usage are constant during each interval. Unfortunately, the number of integer variables required
Ind. Eng. Chem. Res., Vol. 32, No. 12, 1993 3039 to represent even small practical problems is quite large, resulting in prohibitively high computation times. Another way to analyze the effect of resource constraints on plant operation is through dynamic simulation. To date, only one commercial system, BATCHES (Clark and Kuriyan, 19891, has been developed that is specifically directed toward the simulation of batch chemical processes. Unfortunately, rigorous optimization of the time-dependent and discontinuous models which occur within the framework of BATCHES presents significant computational difficulties which as yet have not been satisfactorily resolved. As a result, improvements in the plant operation must be identified through analysis of appropriate case studies. Mathematical Formulation In order to make the MBP scheduling problem amenable to mathematical formalism, a number of additional assumptions must be postulated 1. The long campaign production mode is utilized in the MBP. Hence, the loss of production time due to changeovers is negligible compared to the length of a single campaign. We must underline that changeover costs and times can be easily incorporated in the model formulation at the expense of significantly complicating the problem combinatorics. 2. The equipment items assigned to a particular task are dedicated to it for the duration of the campaign. 3. No intermediate storage is available, but the batch (if stable) can be held in the current processing unit until a downstream unit becomes available. 4. A maximum number of campaigns K and a maximum number of groups NGomkper task (0,m ) must be specified a priori. The corresponding values cannot be arbitrarily high due to the implied changeover and clean-up times and associated costs. Typically, these parameter values are plant dependent. Also notice that, since the number of binary variables required by this formulation (see below) is partly controlled by both these parameters, the combinatorial complexity will increase as these parameter values increase and, consequently, the computational efficiency of the solution method will decrease. In the remaining section we will express the constraints that describe the physical and logical relations among the basic scheduling variables introduced in the problem statement. Assignment and Connectivity Constraints. The structural decisions representing the assignment of equipment items to processing groups of individual tasks are handled by the introduction of the followingbinary vector:
1 1
Xomegk
=
0
if task m of order o is processed by unit e of equipment group g in campaign k otherwise
Since all customer orders must be satisfied, each production task should be assigned to at least one equipment item and in at least one campaign:
Notice that expression p ( o ) denotes the fact that each order, 0,involves the production of a specific product, p . According to the following constraint, equipment item e can be assigned to at most one processing group in a particular campaign:
where set P e records all the production tasks that can be processed in unit e:
Pe= ((o,m)leE Ep(o)mj In order to calculate the number of out-of-phase groups assigned to task (0,m ) in campaign k , NGomk, we assume that a maximum number of groups G is available. We introduce the auxiliary continuous variable \komgk E [0, 13, whose value is determined by Xomegk. The newly introduced variable evaluates to 1 if groupg of task ( 0 , m ) in campaign k is nonempty and 0 otherwise. I b value is determined by the following constraints:
*omgk
e E Ep(o),m
'.xomegk
(6)
If all the Xomegk variables in the right-hand size of constraint 5 are 0, then \ k O & is 0 and constraint 6 is redundant. If at least one Xomegk is equal to 1, then \komgk is forced to 1 by constraint 6, whereas constraint 5 is redundant. Now, the number of groups NGomk is easily computed as follows: c
(7) Variable \komgk can be also used in the following constraint which ensures that all production tasks of a particular order are completed within the time period covered by the campaign to which the order is assigned:
k = 1, ...,K (8) Furthermore, notice that the choice of binary variable has introduced some degeneracy in terms of the assignment of equipment items to groups. For example, suppose that it is known that units RD and RV are assigned to one of the two groups used to perform production task 1. Then, the following solutions are equivalent:
This degeneracy can lead to substantial increase in computation time. To eliminate the degeneracy, we use the methodology presented by Papageorgaki and Reklaitis (1992) according to which a constraint is introduced that gives assignment priority to groups with small indices:
The introduction of constraint 9 does not fathom any feasible solutions from the problem domain. Notice that NGomkwill always be greater than 0 for an active task because of constraint 3. Finally, the present set of constraints contains all the problem-dependent precedence relations that may occur among customer orders. Batch Size Definition Constraints. Inside an equip ment group, the individual units operate in a parallel inphase processing mode. Hence, the group batch size is equal to the sum of batch sizes in the individual units:
3040 Ind. Eng. Chem. Res., Vol. 32, No. 12, 1993
The parallel and out-of-phase processing mode among groups assigned to the same production task determines the task batch size and the production line batch size:
Expressions 19-22 lead to the following constraint on the production line cycle time, i$:
(13) From(11), (12),and(13)wegetthefollowingexpression for the production line batch size:
From the last constraint it is clear that the production line batch size, Bbk,will assume zero value when a t least one empty group appears in the production line, even though there might be at least one nonempty group assigned to this line. To avoid this inconsistency, this constraint is reformulated as follows:
&
Bkk
eE
Bomegk
+
- *omgk)
(14)
pio).m
where BUis an upper bound on the production line batch size which can be given by
Production Demand Constraints. Each order may appear in several campaigns. Each of these campaigns will contribute its share to the production requirement specified by the order. Let Qok and n o k denote the amount and the number of batches of order o produced in campaign k, respectively. The following Constraints ensure satisfaction of all the customer orders:
no&
2
Qok
0
= 1,..., 0, k = 1, ...,K
(15)
Campaign Length Definition Constraints. The length of campaign k is equal to the maximum time required to process the number of batches of each order member of the present campaign: (26)
Resource Utilization Constraints. The resource utilization of each individual unit depends on the nature of the assigned production task and is given by the product recipe (gspme()), usually expressed as a posynomial function: rsomegk
-- 's,p(o),rneXomegk
+ ~ 8 , p ( o ) , m $ k & ~ ~ (27)
In addition, the total amount of resource usage is constrained by the resource availability bounds:
Completion Time Calculation-Objective Function. Makespan is easily computed by adding the campaign lengths: K H = z T k
(29)
The computation of tardiness requires the completion time of each order. The completion time of order o is equal to the sum of campaign lengths of all the campaigns preceding and including the last campaign in which order o is assigned: kt
05
nok5 n s W o l l k o = 1, ..., 0, k = 1, ...,K
(18)
where nIk and n,"k"are suitable lower and upper bounds, respectively. Cycle Time Definition Constraints. The selected processing modes influence the computation of the group and production line batch sizes. The dependence of the individual processing times on the unit batch size, Bomegk, is given by the product recipe (fpme( )), typically expressed as a posynomial function:
- (Yp(o),meXomegk + @ p ( o ) , m s oYP(O)+ megk
(19) The existence of multiple groups per production task operating in parallel and out-of-phase mode positively influences the production task cycle time: tomegk
7, 2 Z T k
- do- H(1- Yo& k, = 1, ...,K
(30)
where H is a an upper bound estimate on the total production makespan. A reasonable initial guess for H would be the makespan attained when all orders are being produced sequentially along production lines in which only one equipment group is assigned to each product task containing a single unit with minimum capacity. In addition, Y o k is a continuous variable in the interval [0, 11 which is linearly dependent on the binary structure variable Xomegk and it is equal to 1if order o is processed in campaign k and 0 otherwise. Then, the weighted tardiness criterion is expressed as a linear combination of the order tardinesses: 0
cost = C c 0 r o
(31)
o=l
Finally, the mathematical formulation of the MBP scheduling problem is completed by a set of direct and
Ind. Eng. Chem. Res., Vol. 32, No. 12, 1993 3041 derived variable bounds. Therefore, the resulting model (I) consists of minimizing eqs 29 or 30 and 31 subject to constraints 3-10, 14-18, and 23-28.
'somengk
-- bs,p(o),meXomengk
+ ~s,p(o),n&$$
(39)
Efficient Reformulation The existence of equipment items with identical processing characteristics introduce degeneracy in the optimization model (I) presented above. For example, let us assume that reactors RD1 to RD5 are identical, i.e., they have the same volume or production characteristics. The decision to assign reactor RD1 to task 1 is equivalent to assigning any one of the other four reactors to 1. This degenerate behavior is a result of the choice of binary decision variable, X o m e g k , and unnecessarily augments the decision space. The redundancy can be avoided with a convenient reformulation(I1): (a) Equipment items with identical processing characteristics (volume, resource consumption, and production rate expressions) are aggregated into a single set, called the equipment type and comprised by Ne identical units with volume Ve. (b) In order to preserve the integral nature of the equipment items, we introduce an appropriate binary decision variable:
-1
if n units of type e are assigned to group g of task m of order o in campaign k otherwise
1
Xomengk
-
0
where n = 1 , 2 , ..., N e . The dimensionality of the binary variable has not increased even though an additional index appears. Notice that e represents distinct equipment items in the original model (I), whereas in the new formulation it represents equipment types. The equivalence becomes clear with the following example: Assume that a small multipurpose batch chemical plant contains two types of reactors, RU and RD, with an availability of four and three units, respectively. In the original model e = 1, 2, ..., 7 whereas in the reformulation e = 1 , 2 , nl = 4, and n2 = 3. Hence, the total number of ( e , n) combinations is 7. The introduction of the new decision vector Xomengk requires the appropriate restatement of the problem constraints. Consequently, constraints 3-6,10,14,23,27, and 28 will take the following form:
77 & 2 8
e€
p(o)n
0
xomengk
n=l
= L 2 , ...,0, P = ~ ( 0 1 ,m N.
N-
E RC,,,,
(32)
Lastly, an additional constraint must be added denoting that the number of equipment items of type e assigned to a particular group g of task (0,m) has a unique value: N. n=l
P = ~ ( 0 1 ,m E RCp(o), e E E p ( o ) , m , g = 1, ...,NGomk, k = 1, ...,K (41) The revised formulation is completed with a set of direct and derived variable bounds. Model I1 consists of minimizing eqs 29 or 30 and 31 subject to constraints 7-9, 15-18,24-26, and 32-41. Note, however, that even though model I1 contains less degeneracy than formulation I, the dimensionality of the decision vector is still significant, surpassing the capabilities of existing MINLP optimizers. An alternative formulation (111) can then be developed which requires a reduced number of binary variables. The new formulation is based on the assumption that units of the same equipment type e process the same batch size Bomegk. This assumption is restrictive, because it imposes the same batch size on each production path through a production line (for detailed analysis of production path dependent batch sizes see Wellons and Reklaitis (1988)). Although formulation I11 is not equivalent to the previous formulations, it is amenable to solution using existing MINLP optimization techniques.
Alternative Formulation The following binary vector is used to represent the structural decisions associated with the assignment of equipment types to processing groups of individual production tasks:
1
if task m of order o is processed by unit type e of equipment group g in xomegk = campaign k 0 otherwise In essence, we drop the index n from the binary vector Xomengk introduced in the revised formulation (11). Constraints 3, 5-10, 15-18, and 23-26 are retained in the formulation. Notice, however, that index e now denotes equipment type instead of equipment item. Constraint 4 is modified as follows: 1
Xomegk 8
I Ne
e = 1, ...,E, k = 1, ...,K (42)
(o,m)EP.
In addition, a new variable, N U o m e g k , must be introduced to denote the number of units of type e contained in group g assigned to a particular task (0,m)in campaign k. The following Constraints relate this variable to the binary variable Xomegk: Nuom&
N.
'
xomegk
(43)
I NeXomegk (44) Furthermore, the number of units of type e assigned to each production task in campaign k cannot exceed the maximum number of units of this type that are available to the plant: NUomegk
3042 Ind. Eng. Chem. Res., Vol. 32, No. 12, 1993
. .
k = 1,2, ..., K (45)
Now, constraint 14 is modified as follows:
Hence, robustness is a very desirable attribute of the complementing nonlinear optimization algorithm. In conclusion, combinatorial complexity, nonconvexity, and nonlinearity are the main obstacles to the direct application of conventional optimization methods to the scheduling of multipurpose batch plants.
Solution Strategy Finally, constraints 27 and 28 are combined to yield the following modified constraint:
The resulting formulation (111)consists of minimizing eqs 29 or 30 and 31 while satisfying constraints 3, 5-10, 15-18,23-26, and 42-47.
Model Characteristics The presentation of the MBP scheduling problem in the previous sections reveals the nonlinear and combinatorial nature of the corresponding search space. A number of interesting MINLP optimization algorithms have been reported in the literature by Geoffrion (19721, Gupta (1980), Viswanathan and Grossmann (19891, and Floudas et al. (1989),which seem to provide a satisfactory solution methodology for problems of small decisional complexity in reasonable computation times. Unfortunately, as the size of the binary dimension vector increases, combinatorial explosion completely dominates. In the original formulation (I) the dimension of the binary structural decision vector X is characterized by five indices: order 0 , production task m, equipment item e, equipment groupg, and campaign k. Hence, it increases very rapidly with the problem size. For example, a small hypothetical MBP with indicesranging in { 1,2jis described by 32 binary variables. The domain of the corresponding optimization problem contains 232 disconnected regions and can be relatively easily handled by the existing algorithms. If the range is increased to {l, 2, 31, the structural dimensionality rises to 243 with 2243candidate disconnected regions, which is well above the capabilities of the MINLP solvers. Clearly, given the present stage of algorithms and computing resources, routine solution of large MINLP models in the chemical industry is infeasible. In addition to the combinatorial complexity, the mathematical formulation contains nonlinear relations among the problem variables and cannot be convexified through exponential variable transformations. As a result, even for a single choice of binary structural variables, there may exist a collection of locally optimal solutions whose objective function performance varies significantly. Since it is well-known that small perturbations to the plant performance are equivalent to significant savings, the scheduling algorithms are required to possess the ability to identify globally superior solutions. The efficacy of the existing MINLP solvers is known to deteriorate in the presence of nonconvexities. The existing discrete nonlinear optimization algorithms are usually complemented by continuous nonlinear optimization procedures which are used to identify local optima in the problem space. The existence of nonlinear constraints creates significant distortion to the feasible region geometry of the underlying optimization problem, causing many of the nonlinear optimization algorithms to frequently terminate prematurely or even fail to converge.
The problem of searching combinatorially complex search spaces is very complicated and intriguing. To date, there exist no algorithms that are guaranteed to converge to the globally optimal solution of a nonconvex problem in finite computation time using finite computation resources independently of the problem size. Given the difficulty of the problem, we adopted a two-level decomposition strategy. As has been demonstrated in a variety of research studies (Wellons and Reklaitis, 1991a; Papageorgaki and Reklaitis, 1990b, 1993),problem-dependent decompositions that are based on engineering intuition are able to focus the search for the global optimal solution on a reduced subspace of the problem domain, thus resulting in significant reduction of the problem size and of the required computational effort. A careful study of the problem structure indicates the existence of two interacting decision levels in the problem formulation. The first is related to the distribution of the production requirements in time, that is, the allocation of a specific production time frame to each order (technically, the assignment of orders to production campaigns). The makespan or tardiness values reflect the competition among different orders for the early time slots. The optimization method tends to minimize the production times. On the other hand the limited availability of production resources imposes an upper bound on the feasible production rates. This causes competition among orders produced in parallel for the acquisition of the limited common resources. The assignment of resources to orders produced in parallel defines the second decision level in the scheduling problem (equipment and resource assignment problem). The interaction between the two levels is obvious. Suppose that the orders, A and B, are processed in a single campaign. The distribution of resources between A and B increases the production time of A, causing an increase in the associated scheduling cost. If, alternatively, order B is assigned to asecond campaign, the resource availability increases significantly and order A is produced on time. On the other hand the production of order B is delayed, leading to an increase in the cost. A hierarchical decomposition of the MBP scheduling problem is introduced through the appropriate relaxation of the interactions between the two decision levels. The upper level model is a relaxation of the original MINLP with a significantly reduced number of structural decision variables. It is equivalent to a nonlinear production planning problem which, accounting for the equipment and resource limitations in a relaxed form, assigns the customer orders to production campaigns so as to minimize the scheduling cost. This decision level is called the campaign formation subproblem (CFS),and it provides a lower bound on the optimal solution of the original MINLP. In the second level the solution effort is focused on the campaigns created by the first level. The corresponding subproblem deals with the optimal allocation of units and resources to the production tasks that are processed in parallel. The resulting optimization model is, again, a nonconvex MINLP of reduced dimensionality called the equipment and resource assignment subproblem (ERAS).
Ind. Eng. Chem. Res., Vol. 32, No. 12, 1993 3043 Table I. Feasible Equipment Assignments for Example 1 production
production
task
equip. type
A
RV RV RV,RD
B C
task D
E
equip. type
Table 11. Available Number of Units for Example 1
RV RU RD
RV,RU,RD RU,RD
The solution resulting from this decision level constitutes an upper bound on the solution of the original MINLP.
order should be assigned to at least one campaign:
The descriptive power of the relaxed variable X o m e g k will be demonstrated with an appropriate example.
Campaign Formation Subproblem (CFS) The campaign formation decision level is equivalent to a nonlinear production planning problem that assigns orders to campaigns in an optimal manner without neglecting the significant role of the equipment and resource limitations. Resource Limitations. A careful study of the original MINLP formulation indicates that the combinatorial explosion is mainly attributed to the discrete nature of the equipment items. We avoid this problem by using formulation I11to describe CFS, in which equipment items with identical processing characteristics have been aggregated into equipment types. Furthermore, adapting the methodology introduced by Papageorgaki and Reklaitis (1990b), the integral character of equipment items is relaxed at this level. Equipment type e is considered a continuously divisible, renewable resource with a limited availability, Ne. Campaign Formation Decisions. Since the only information extracted from the upper-level subproblem involves the sequencing of orders, its decisions can be described with the following binary variable:
Example 1 A model multipurpose batch plant contains three reactor types RV, RU, and RD. Suppose that there exist five different production tasks that should be processed in parallel. Table I indicates the feasible equipment types for each task, and Table I1 contains the number of units per equipment type. The application of constraint 50 produces the following inequalities: XlBV x2,RV x3,RV x4,RV
'ok
{
Xomegk
5 Yok
1,..., 0, k = 1,...,K , g = 1, ...,G , m E RC,,,,, e E
+ x3,RD
(48)
In other words, no relaxed assignment variable should be 1 if a given order is not a member of campaign k. Equipment resources should not be wasted on orders that do not belong to a given campaign k. The production of a certain order requires the successful completion of every production task. Hence, at least one assignment should be made for every production task of every order processed in campaign k: D
P = ~ ( 0 1 , m E RCp(o)(49) Since all production requirements must be met, each
(52)
'
(53)
'
(54)
+ X,RD 21
(56)
+ x2,RV + X3,RV + x4,RV + x5,RV x3,RD + x4,RD
0=
(51)
(55) whereas the limitation of equipment resources expressed with constraint 42 constrains the equipment utilization:
if order o is assigned campaign k otheiwise
The dimensionality of the binary decision vector is significantly reduced compared to that of the original problem, thus drastically reducing its computational requirements. The reduction influences negatively the descriptive power of the binary variable, which is not able to express the competition among different production tasks for the limited equipment resources. Nevertheless, part of the original variable interactions can be preserved by the former binary variable X o m e g k E [0,1] which is now treated as a continuous variable. The larger the number of X o m e g k evaluated to either 0 or 1, the tighter the relaxation of the original formulation. The relaxed decision variable is related to the binary variable Y o k through the following constraints:
'1 '
+ X4,RU + x4,RD
%,RU
X1,RV
1 = 0
no. of unita 2 1 2
equip. type
x4,RU
(57)
+ x5,RD
+ xS,RU
'
(58)
In addition,
0 5 X,, 5 1, m E (1,2,3,4,5j, e E (RV, RU, RDJ (59) The following conclusions are deduced by manipulating the constraints: (511, (52), and (59) * XI,
= XO,RV
1
(56) and (60) =) x 3 , ~ v= X4,RV = x , , ~=,0 (53) and (61)
X3,RD = 1
(57) and (62) * X4,m + X 5 8 5 ~ 1 Adding (58) and (631, we get X4&U
Adding (54) and (55), x4,RU
' and using (601, we get '
+ X4,RD + x5,RU + X5JlD
+ X4,RD + X6,RU + x5,RD
(60) (61) (62) (63) (64) (65)
Therefore, (66) x4,RU + x4,RD + xS,RU + x5,RD = An exact assignment has resulted for tasks 1,2, and 3, while the assignment of 4 and 5 is still undecided but highly constrained by (66), since there exist only two equipment items available for tasks 4 and 5. It is clear that significant information has been extracted without the introduction of x o - g k as binary variables. Furthermore, the utilization of resource s in CFS is bounded by the continuous relaxation of (47):
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Ind. Eng. Chem. Res., Vol. 32, No. 12, 1993 assigned to it, is determined by the binary structural variable Yok and reflected to variable W k which is defined by the following linear constraint: 0
where
(69)
The campaign formation subproblem (CFS) is a proper relaxation of the original scheduling problem. Detailed proof is presented in Appendix A. The relaxation is expected to be tight because the limitations on equipment and resources have been appropriately modeled. The problem is formulated as a nonconvex MINLP of significantly reduced combinatorial complexity. Nonlinearities are introduced in the production demand constraints, the cycle time and campaign definition constraints, and the resource utilization constraints. The mathematical formulation can be solved by any of the existing discrete optimization algorithms which are heuristic and cannot guarantee the identification of the global optimum. The computational requirements of CFS can be further reduced by appropriately decomposing the MINLP. Following the procedure introduced by Papageorgaki and Reklaitis (1990b), we decompose the scheduling problem along the time axis. Instead of solving a large model with the number of campaigns k ranging in the set (1,2, ...,K ) , we solve K reduced problems. In each one of them the number of campaigns is fixed to a specific value from the above set. In essence, we adopt an iteration on the campaign index k . The results from the previous iterations are propagated on the next level in the form of upper bounds on the objective function. Compared to the direct solution of the model by one of the heuristic algorithms, the decomposition is guaranteed to increase the quality of the obtained solution because it forces the solver to iterate and search for increasingly better campaign formations. The decomposition method would be computationally more efficient if we could guarantee that the solution spaces examined during different iterations are disjoint. This is not possible because every schedule including one campaign can be written as a multicampaign schedule by copying the campaign contents:
[I 1-[i ![I El
Clearly, the problem is created by the existence of identical campaigns (Papageorgaki and Reklaitis, 1990a). It can be proved that identical campaigns have identical processing characteristics and that the existence of identical campaigns in the production schedule is not an optimal policy for different objective criteria. The proof is contained in Appendix B. Therefore, the existence of identical campaigns introduces degeneracy that increases the computational requirements.
Degeneracy Reduction Constraints The degeneracy can be eliminated from the campaign formation subproblem with the following technique: The 0 customer orders are arbitrarily sorted. Each order o occupies a distinct place in the ordered list denoted by io. The composition of campaign k , i.e., the set of orders
Example 2 In a MBP plant there exist three different orders A, B, and C. They are arbitrarily sorted, occupying a unique place in the ordered list: (A, B, C) =) [A, B, C] =) i, = 0, i, = 1,i, = 2 Then, the characteristic value of campaign [A, B, Cl is w1 = 7, while the value of [A, C] is w z = 5. I t can be easily proved that nonidentical campaigns have different W k values whose difference in absolute value is greater or equal to 1:
k , k’ = 1,...,K (72) In the case where the optimal solution is insensitive to campaign permutations (e.g., minimization of makespan with negligible changeover times/costs), we can arbitrarily impose the cc-Adition that campaigns with small index will have larger W k values. Notice that this assumption does not fathom any feasible solutions. Under this assumption, inequality 72 is reduced to w k - wk’ 2 1 k , k’ = 1,...,K , k‘ 1 k (73) Notice that, in this case, the degeneracy due to campaign permutations of the type Iwk - wk?l 1 1
is removed from consideration, since campaignswith lowerindexed W k have priority over the ones with higher-indexed W k . Alternatively, when campaign permutations affect the optimal solution (e.g., minimization of tardiness, case with significant changeover times/costs), the creation of identical campaigns can be eliminated by introducing constraint 73 into the model with the following form: k , k’ = 1, ...,K (74) Constraint 74 will force distinct campaigns to have different compositions. However, the computation involved with this constraint in view of constraint 71 is quite cumbersome. To avoid this problem, we can use a more efficient representation as follows: Let Wokk’ denote the logical OR ofthe binaryassignment of order o to campaigns k and k’. Then, the following constraints can be used to assign the proper values to the OR variable, namely, F o k k , ? Yok 0 = 1, ..., 0, k, k ‘ = 1, ..., K , k’> k (75) & k k ’ l Yak' o = 1, ..., 0, k , k’ = 1, ..., K , k ’ > k (76) 0 = 1, ..., 0, k , k’ = 1,...,K , k’ > k pokkr I Yok + Yokt (77) Furthermore, let Xokk‘ denote the logical AND of the binary assignment of order o to campaigns k and k’, Then, the following inequalities can be used to define the AND variables: Xokk, I Yok o = 1, ..., 0, k,k’ = 1, ...,K , k ‘ > k (78) Xokk, I Yak' o = 1,..., 0, k , k‘ = 1, ...,K , k ‘ > k (79) Xokk’ 2 Y o k + Yak'- 1 0 = 1, ...,0, k , k’ = 1, ..., K , k’ > 7 (80) The decimal representations of the logical OR and AND can now be defined as (wk - wk
