Ind. Eng. Chem. Res. 2009, 48, 9655–9670
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Design and Scheduling of Periodic Multipurpose Batch Plants under Uncertainty Taˆnia Pinto-Varela,†,‡ Ana Paula F. D. Barbosa-Povoa,*,‡ and Augusto Q. Novais† DMS, Instituto Nacional de Engenharia, Tecnologia e InoVac¸a˜o, Est. do Pac¸o do Lumiar, 1649-038 Lisboa, Portugal, and CEG-IST, Instituto Superior Te´cnico, UniV. Te´cnica de Lisboa, AV. RoVisco Pais, 1049-101 Lisboa, Portugal
This work deals with the design of multipurpose batch plants under uncertainty. The model proposed by Pinto et al. [Comput. Chem. Eng. 2005, 29 (6), 1293-1303] for the detailed design of batch plants is extended to address the problem of uncertainty associated with production demand. Equipment choices, as well as plant topology and associated schedule, are defined simultaneously under an uncertain demand environment. Uncertainty is treated through a two-stage stochastic model leading to a mixed-integer linear programming (MILP) formulation, where profit is maximized and the expected value of perfect information (EVPI) is determined; the latter is estimated as the difference between the values of profit obtained for the wait-and-see and the here-and-now models. A scenario is set up where demand is represented by a discrete probability function and a cyclic operation considered, together with mixed storage policies and sharing of resources. Some illustrative examples are solved to show the model applicability and the EVPI are determined and analyzed. 1. Introduction In multipurpose batch plants, a wide variety of products can be produced via different processing recipes by sharing all available resources, such as equipment, raw material, intermediates, and utilities. To ensure that any resource can be utilized as efficiently as possible, an adequate representation is necessary at the design level, to model all possible interactions between the plant resources and process, by means of an ambiguity-free representation. The Resource-Task Network (RTN) is one such representation that can be applied to the design of multipurpose batch plants, as shown by Pinto et al.2 Another important point when designing batch plants is the possibility of dealing with uncertain environments.3 However, because of the inherent complexity of the problem, most of the work that has been published involve essentially a deterministic approach, where all parameters are known. However, in real plants, uncertainty is a very important issue, because many of the conditions that affect the operation of a real plant are often subject to change. Such is the case of, among others, raw material availability, raw material prices, machine reliability, and market requirements, which vary with respect to time and are often subject to unexpected deviations. Therefore, the development of systematic approaches to handle uncertainty is a research subject of great relevance, and, in the relevant published work, different approaches can be distinguished. Some rely on decomposition methods, two-stage stochastic formulations, and hybrid algorithms. Thus, Mulvey et al.4 considered that scenario planning attempts to capture uncertainty by representing it in terms of a moderate number of discrete realizations of the stochastic quantities, hence leading to as many distinct scenarios. Sridharan5 considers that each complete realization of all uncertain parameters creates a scenario. The objective then is to find robust solutions that perform well under all scenarios. Zimmermann6 stated that the choice of the * To whom correspondence should be addressed. E-mail address:
[email protected]. † DMS, Instituto Nacional de Engenharia, Tecnologia e Inovac¸a˜o. ‡ CEG-IST, Instituto Superior Te´cnico, Univ. Te´cnica de Lisboa.
appropriated method is context-dependent, with no single theory being sufficient to model all types of uncertainty. During the past decade, several authors have been studying different ways to characterize more realistically some types of uncertainty associated to real problems within the design of batch plants. Shah and Pantelides7 considered uncertainty in market demands and its influence on the design of multipurpose batch plants, by means of a scenario approach. As a result, the obtained plant design is able to cope with any of the operating scenarios considered, without an arbitrary overdesign. The authors acknowledged the possibility of experiencing some difficulty with large design problems and recognized the need for efficient solution techniques to solve them. The design of batch plants under demand uncertainty was also studied by Subrahmanyam et al.8 The problem is defined using a mixed-integer linear programming (MILP) procedure, where scheduling is considered as a subsequent stage to the solution of the problem superstructure design. The demand uncertainty is considered, at the design level, as a set of scenarios, each one associated with an occurrence of uncertain parameters. Probabilities are assumed to be known a priori. Ierapetritou and Pistikopoulos9 addressed the problem of including uncertainty in process parameters and product demands at the design stage of multiproduct/multipurpose batch plants. A conceptual two-stage stochastic programming formulation is proposed with an objective function comprising investment costs, expected revenues from product sales, and a penalty term accounting for expected losses due to unfulfilled orders. In the design of multiproduct and multipurpose batch plants, Harding and Floudas10 considered uncertainty in both product demands and processing parameters. Uncertain demands were modeled by a continuous/discrete probability distribution, while uncertain processing parameters were handled by a scenariobased approach. Ierapetritou and Pistikopoulos11 also considered the design and planning problems under uncertainty through a two-stage stochastic optimization approach (mixed-integer nonlinear programming, MINLP). Later, the same authors12 reported some development toward the global and efficient solution of large-scale batch plant design models under uncertainty, by exploring their special structure.
10.1021/ie900137p CCC: $40.75 2009 American Chemical Society Published on Web 07/23/2009
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Cheng et al.13 addressed the design and planning under uncertainty as a multiobjective sequential decision problem, or a multiobjective Markov decision process. Duque et al.14,15 extended the design of multipurpose batch plants with environmental quantification and considered uncertainty at the demand level. In this paper, a scenario approach for the uncertainty associated to demand is adopted and is assumed that it will be controlled by a smaller number of major factors such as economic growth, political stability, and competitor actions.16 This approach is consistent with Mobasheri et al.,17 who describes scenarios as plausible states derived from the present state with consideration of potential major industry events. For this reason and within a risk analysis strategy, a set of scenarios should be built, which are representative of both situations (i.e., optimistic and pessimistic). This work takes into account these multiple scenario methodology, which is used to capture the uncertainty aspects. In the field of behavior analysis of actions under uncertainty, the expected value of perfect information (EVPI) is an important concept that helps to establish the amount of information that the decision maker is ready to pay under accurate information.18 Modiano19 studied the derivation of demand functions for primary resources in an economic environment and the capacity planning under uncertainty. They explored two alternative behavior models of action under uncertainty: here-and-now and wait-and-see. Pistikopoulos20 presented a review paper in the area of design and operations under uncertainty. The main emphasis is to establish the link between various components regarding the characterization and quantification of uncertainty, such as flexibility, controllability, reliability, and value of perfect information. Later, Iarapetritou, Pistikopoulos, and Floudas,21 presented a paper concerning the operational and production planning under varying conditions and changing economic circumstances, using a two-stage stochastic programming formulation, with the maximization of the profit value of perfect information. Although some work examined uncertainty at the design level, much work is still to be done, as referenced by Barbosa-Po´voa3 in her review on the design and scheduling of batch plants. The present work follows the directions proposed in this review paper and studies the design and the scheduling of multipurpose batch plants under a scenario planning approach. The model proposed by Pinto et al.,1 for the detailed design of batch plants, is extended to address the problem of uncertainty in production demands. Equipment choices as well as plant topology and associated schedule are defined simultaneously under an uncertain environment. A cyclic operation is assumed where time is treated using a uniform discrete time grid, and mixed storage policies and sharing of resources are considered. The expected value of perfect information is analyzed in the design context. A MILP formulation is obtained. The present paper is structured as follows. In section 2, the modeling framework is characterized and the problem definition given. The EVPI is characterized in section 3. The mathematical formulation using the RTN adapted methodology is presented in section 4. In section 5, some examples are solved, followed by discussion of the conclusions and future work in section 6. 2. Modeling Framework The optimal design of multipurpose plants with a periodic mode of operation is addressed in this work, using, as a modeling framework, the adapted Resource Task Network (RTN) proposed by Pinto et al.1 Such framework appears as one of the most general and conceptually simpler representations
to examine process/plant problems representations where two types of entities are defined: tasks and resources. A task is an operation that consumes and/or produces a specific set of resources, whereas a resource describes all different entities involved in the process/plant. Resources are produced/consumed at discrete times during the execution of a task and can be classified as renewable or nonrenewable, depending on whether or not their availability is restored to the original state after usage by a task. A nonrenewable resource represents raw materials, utilities, manpower, etc., whereas a renewable resource represents all types of equipment such as processing, storage, piping, etc. Using such representation the design problem can be defined as follows: Given: The process/plant description (in RTN terms); resources availability, characteristics and costs; time horizon of planning; mode of operation; demand over the time horizon (production range); cost data; probability density function. Determine: The optimal plant configuration (i.e., the number and type of equipment units and their connections as well as their sizes under all scenarios); the optimal process schedule (i.e., timing of all tasks, storage policies, batch sizes, amounts transferred, allocation of tasks and consumption of resources); to optimize an index of economic performance of the plant, expressed in terms of capital expenditure, operating costs, and revenues. With regard to the operation mode, a periodic operation is considered where the concept of cycle time T is used. This is taken as the shortest interval of time at which a cycle is repeated, where the cycle represents a sequence of operations involving the production of all desired product and the utilization of all available resources. Since all cycles are equal, the problem is formulated over a single cycle, where it is guaranteed that the operation of the plant is the same at the beginning and end of the cycle. The execution of a task is allowed to overlap successive cycles, and, since a cycle is repeated over the time of planning, its execution is modeled by wrapping around to the beginning of the same cycle. To do so, the wrap-around operator, as defined by Shah et al.,22 is used: Ω(t) )
{
t (if t g 1) Ω(t + T) (if t e 0)
(1)
When this is applied, for instance, to the variable Nk,Ω(t-θ), for t - θ e 0, it leads to an identical equipment resource allocation that will start at time t - θ + T. As previously referenced, the deterministic problem is modified with the incorporation of uncertainty at the demand level, which is modeled through the establishment of a small number of discrete scenarios, where the random demand variable assumes deterministic values. This approach illustrates that it is impossible, under uncertainty, to find a solution that is ideal under all circumstances. For this reason, the objective is to find the solution that will perform better under all scenarios, which can be done with a stochastic model. The scenario approach that will be described is based on a two-stage stochastic programming. The decision variables are split in two major groups: first stage and second stage. Variables for the first stage are those related to decisions that cannot be reviewed, or which are less prone to be modified, when the future outcomes are realized. Within the present model, these are the design variables; i.e., they are the binary variables defined for the choice of each type of resource. The second-stage variables are related with decisions that can be reviewed after
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the scenario occurrence (i.e., additional information is obtained on the realization of some random vector). In our model, these are the scheduling variables related with the production, storage, and transfer decisions. After the demand is known, these decisions can be reviewed. Consider M as the set of all possible scenarios and m∈M represents a particular scenario. Let all first-stage variables be included in vector y and all second-stage variables in vector x. Let f be the vector of the fixed costs related to the choice of a certain resource c and p be the vector that contains the remaining coefficients in the objective function. The deterministic model for a particular scenario m is defined as max pmx - cmx - fy
(2)
s.t. A mx e a m Bmx e Cy y ∈ {0, 1}, x g 0, x ∈ R where Am, Bm, pm, and cm are matrixes and am is a vector. The solution of this model gives the optimal design and scheduling for any individual scenario. Two-Stage Stochastic Model (TS). The aforementioned formulation can be modified to represent a two-stage stochastic linear model with recourse: max E[Θ(y, m)] - fy
(3)
s.t. y ∈ {0, 1} with max Θ(y, m) ) pmxm - cmxm
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Structural decisions are assumed in the first stage, y, whereas operational variables are used in the second stage, x. In terms of objective function, the cost terms related with investment were assigned to the first stage, while the remaining terms were allocated to the second stage. 3. Expected Value of Perfect Information The concept of value of perfect information, which has been used in the field of behavior analysis to study the sensitivity of policy decision making to uncertainty,23 can be addressed to integrate our problem. The value of knowing complete and accurate information is only meaningful when the future is uncertain. In this context, the decision maker (DM) must perform decisions based on forecasts about future events. While future uncertainty is handled by the solution of a twostage stochastic linear model, which reflects the DM attitude toward risk, further information would still be required for other decisions, such as: • What would be the profit if we obtain accurate information? • What would be the penalty associated with uncertainty? • How much the DM would be ready to pay in return for complete and accurate information about the future? This can be achieved through the concept of Expected Value of Perfect Information, which is based on the difference between the optimal values obtained for two models acting under uncertainty, namely, Here-and-Now and Wait-and-See. Problem WS. In the Wait-and-See (WH) model, the DM is assumed to know beforehand and for each scenario m, the optimal values of the first- and second-stage variables, as well as of the objective function, which is equivalent to solving m deterministic optimization problems. The WS problem can be represented as zm1 ) max pmxm - cmxm - fy x,y
s.t.
s.t. Amxm e am Wait-and-See Bmxm e Cy y ∈ {0, 1}, xm g 0, xm ∈ R
Amxm e am Bmxm e Cy xm g 0, xm ∈ R where E[ · ] is the expected value of [ · ] over m and pm and cm are associated with the distribution function Jm. Considering explicitly the second-stage variables into the firststage model, the following mixed-integer linear model is obtained: z ) max Em[pmxm - cmxm] - fy x
(4)
(6)
Problem HN. In the Here-and-Now (HN) model, the firststage decisions y are taken in the presence of uncertainty about the future realization of m. In the second stage, the actual value of m becomes known and some corrective actions (i.e., recursive decisions x) can be taken. Therefore, first-stage decisions are chosen while taking their future effects into account. The HN problem, for a specific first-stage decision vector jy, can be represented as z2jy,m ) max pmxm - cmxm - f jy x
s.t.
s.t. Amxm e am Bmxm e C jy y ∈ {0, 1}, x g 0, x ∈ R
Amxm e am Bmxm e Cy y ∈ {0, 1}, xm g 0, xm ∈ R With the expected value function defined, this becomes max z )
∑π p x
m m m
m∈M
-
∑π c x
m m m
- fy
(5)
m∈M
Amxm e am Bmxm e Cy y ∈ {0, 1}, xm g 0, xm ∈ R where πm is the probability of scenario m where ∑mπm ) 1.
Here-and-Now (7)
To determine the value of information for the entire uncertainty space, it is necessary to estimate the difference between the expected values of the WS and HN models. Thus, the Expected Value of Perfect Information (EVPI) is obtained as EVPI ) Em[zm1 - z2jy,m]
(8)
4. Mathematical Formulation This section presents the model required to determine the plant configuration and the optimal process schedule, under a
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scenario planning approach. This is an extension of the work proposed by Pinto et al.1 for the deterministic problem, which is now generalized. The model involves the follows constraints: excess resource balance, processing equipment resource existence constraints, resource capacity constraints, equipment of continuous size ranges, capacity and batch size constraints, equipment of discrete size ranges, storage constraints, connectivity constraints, production requirements constraints, and objective function. Excess Resource Balance. The excess resources balance constraints are a function of the type of resources involved, i.e., renewable or nonrenewable. The renewable resource (process equipment) balances relate, at each instant t, the amount of equipment resource at instant t - 1 with the amount of resource produced/consumed at instant t, for each scenario m.
∑ ∑µ
krθNkmΩ(t-θ)
k
θ)0
∀r ∈ Dp, m ∈ Μ, t ) 1,...,T
(9)
On the other hand, the nonrenewable resource (materials) balances relate, at each instant t, the amount of resource at instant t - 1 with the amount of resource produced/consumed in instant t, for each scenario m: τk
Rrmt ) Rr0m|t)1,r∈Cf + RrmΩ(t-1)|tg2 +
∑ ∑ν k
0 e ∆r e ∆rmax
∀r ∈ C, m ∈ M, t ) 1,...,T + 1
Rrmt g 0
∀r ∈ Cis, m ∈ Μ
(11)
∀r ∈ C/Cr, m ∈ M, t ) 1,...,T + 1
(16)
(17)
Equipment of Continuous Size Ranges. The process equipment resource is available in continuous size range [Vrmin;Vrmax ]; thus, we have Vrmin e Vr e Vrmax
∀r ∈ Dp
(18)
However, the choice of the equipment resource must also be modeled. This is guaranteed with the following constraint: ∀r ∈ Dp
(19)
If ∆r ) 1, then the equipment size is limited by Vrmax and Vrmin; however, if ∆r ) 0, this last equation forces Vr to be zero. On the other hand, the dedicated vessels for raw materials and final products must have enough capacity to store all the material consumed/produced during the entire planning horizon, because it is assumed that all the required raw material is available from the beginning of operation and that the entire final product accumulates during operation.
∑ ∑ξ
Finally, the products’ resource balance is given by
(15)
The corresponding excess resource variable must be zero during the entire planning horizon, for all of the nonstorable materials in any scenario m (zero-wait policy).
Vrmin∆r e Vr e ∆rVrmax (10)
∀r ∈ Dp
On the other hand, the amount of the nonrenewable excess resource (r∈C) at any time, in any scenario m, must always be non-negative.
krθξkmΩ(t-θ)
The amount of resource r (e.g., raw material) available initially is defined by Rr0, for each scenario m. For the intermediate storage materials, it is necessary to guarantee that the quantities of resource at the beginning and at the end of the cycle are the same, which results in eq 11. Rrmt|t)1 ) Rrmt|t)T+1
(14)
θ)0
∀r ∈ C/{Cp}, m ∈ Μ, t ) 1,...,T
∀r ∈ Dp, m ∈ Μ, t ) 1,...,T + 1
0 e Rrmt e ∆r
Rrmt ) 0
τk
Rrmt ) RrmΩ(t-1) +
m, the design variable ∆r, which is an integer, must be different from zero, with an upper bound of ∆rmax.
kmt
t
∀k ∈ ΤS, r ∈ DV, m ∈ M
e Vr
(20)
k
τk
Rrmt ) RrmΩ(t-1)|1
