MINLP Formulation for Simultaneous Planning, Scheduling, and

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A MINLP formulation for the Simultaneous Planning, Scheduling and Control of Short-Period Single Unit Processing Systems Antonio Flores Tlacuahuac, Miguel Gutierrez-Limon, and Ignacio E. Grossmann Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/ie402563j • Publication Date (Web): 25 Aug 2014 Downloaded from http://pubs.acs.org on August 26, 2014

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Industrial & Engineering Chemistry Research

A MINLP formulation for the Simultaneous Planning, Scheduling and Control of Short-Period Single Unit Processing Systems Miguel Angel Gutiérrez-Limón, Antonio Flores-Tlacuahuac∗ Department of Chemical Engineering Universidad Iberoamericana Prol. Paseo de la Reforma 880, México, DF Ignacio E. Grossmann Department of Chemical Engineering Carnegie-Mellon University 5000 Forbes Avenue, Pittsburgh, PA, 15213

July 26, 2014

Abstract A simultaneous strategy for solving the integrated planning, scheduling and control problem considering short-term periods is proposed in this paper. The problem is formulated as a mixed integer dynamic optimization problem. The main practical motivation to use a simultaneous, rather than a sequential solution strategy is to seek improved optimal solutions by considering the interactions among planning, scheduling and control. Integration of the three levels of the problem represents a challenge given the very different horizon times involved in the operations, resulting in growth of the problem in terms of the number of equations to be solved, and hence on computational requirements to carry it out. A full space approach was used rather than trying a decomposition strategy. Moreover, a nonlinear model predictive control strategy was also used ∗

Author to whom correspondence should be addressed. [email protected]

Universidad Iberoamericana, México.

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to account for on-line product transition trajectories. The integrated planning, scheduling and control is formulated as a mixed integer dynamic optimization model, which is tested using the dynamic models of three continuous stirred tank reactors featuring different nonlinear behavior.

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Introduction

From a production and plantwide point of view major operations decisions involve planning, scheduling and control (PSC) activities in a hierarchical manner from top to bottom [1], [2], [3], [4]. This means that planning decisions are normally made first and involve long term decisions that span over weeks or months. Next, based on the planning decisions, major operations are scheduled. In this phase in a typical polymerization plant where several grades are produced, the proper order or sequence in which grades are manufactured is selected. Finally, the lower level control actions must be computed to manufacture each one the target grades. This last step is normally carried out using a lower level control system that basically enforces the operation of the polymerization reactors around the desired operating point, either by set-point tracking or by disturbance rejection capabilities. Extensive reviews on scheduling of batch processes have been presented by Mendez, et. al. [5] who analize the main features, strengths and limitations of existing modeling and optimization techniques. Floudas and Lin [6] also present a review of scheduling models in terms of discrete and continuous time representations. Moreover, Marevelias and Sung [7] present a review of different approaches to accomplish the integration between planning and scheduling. Most of the control loops of the lower level control systems are traditionally single-input single-output proportional-integralderivative (SISO PID) control systems, but of course some other control arrangements (ratio, cascade, etc) can also be used [8]. Moreover, although not frequently used, advanced multivariable control systems (i.e. model predictive control systems) can also be used for set-point tracking purposes. In previous works we have tackled the problem dealing with the optimal scheduling and control of stirred tank reactors [9], [10] as well as tubular reactors [11]. In particular, a simultaneous solution approach was used in all the previous works, meaning that scheduling and control modeling equations and constraints are converged in a simultaneous rather than in a sequential manner. The idea behind the simultaneous solution approach consists of taking advantage of the natural interactions between scheduling and control problems, allowing to determine improved optimal solutions with respect to similar solutions obtained by neglecting such process interactions. In scheduling and control problems we normally have a set of products that need to be scheduled

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in an optimal way. The optimal sequence depends for instance upon the product demands, cost of the products and availability of raw material. Normally, off-specification material is produced during product transitions. Therefore, a minimum amount of such a material should be produced. However, this requires the computation and use of proper control actions such that set-point changes take place in minimum time. Because transition times may depend upon the production sequence and the optimal production sequence depends also on issues such as minimum off-specification material, there is a clear trade-off between scheduling and control problems. Taking advantage of such an interaction may allow to compute improved optimal solutions. Scheduling and control problems are normally solved in a decoupled manner. In this approach transition times are treated as parameters for addressing the solution of scheduling problems, and adjusted in an external loop as described later. One of the problems associated to sequential solutions is that the strong interactions between scheduling and control problems are not accounted for. This can result in suboptimal solutions. Hence, the main practical motivation to use a simultaneous, rather than a sequential, solution strategy is to seek for improved optimal solutions. On the other hand, when addressing optimal control problems the best processing sequence is assumed to be given. Simultaneous solution strategies of scheduling and control problems have recently been proposed as described in [9],[12], [13], [14]. Some of the more interesting and challenging applications of scheduling and control problems are related to polymerization systems [10], [12], [13]. A Lagrangean decomposition approach was used in [10] to solve the large scale mixed-integer dynamic optimization (MIDO) problem for the production of Methyl-Methacrylate and high-impact polystyrene (HIPS) in continuous stirred tank reactors. Scheduling and control problems in polymerization batch reactors were recently addressed in [12] using a MIDO approach. In previous works [10], we assumed a cyclic production wheel meaning that once an optimal production sequence is determined, such a sequence is repeated indefinitely. After computing optimal control trajectories, the practical problem of how to implement them in real time situations remain. Normally, open-loop control trajectories are computed and then used for on-line implementation. However, such an approach has clear disadvantages since transition trajectories can be subject to upsets and/or they can be sensitive to modeling errors, because they do not use any feedback mechanism to account for these factors. Recently some works related to the solution of scheduling and control problems and real-time implementation of such transition trajectories have 4


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been published [13], [15], [14]. In [15] the authors use a Benders decomposition method to solve the integrated problem of scheduling and dynamic optimization for the case of a multiproduct CSTR. In this work two main extensions of the work reported in [9] are presented: (1) an extended horizon production policy [16], and (2) a non linear model predictive control scheme (NLMPC) to stabilize product transition trajectories. Incorporating planning decisions makes the underlying optimization problem harder to solve because planning introduces longer time scales. To the best of our knowledge, no strategies for the simultaneous optimal solution of planning, control and scheduling problems have been reported. This is due to the complexity of solving large MINLP problems whose size quickly increases with the number of production periods. Nowadays, the practical solution of simultaneous planning, scheduling and control problems is only feasible for a small set of production periods. To address the solution of larger production periods an optimization decomposition technique is needed. Furthermore, because of the computational complexity of addressing the simultaneous solution of planning, scheduling and control problems, in this work only local optimal solutions are sought. In summary, in the present work we will only focus on the solution of PSC problems featuring short production periods.

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Problem statement

The problem to be addressed in this paper can be stated as follows: “Given are a number of products to be manufactured over a given number of periods of production, in a single continuous stirred tank reactor (CSTR) and a single processing line. Lower bounds on product demands, steady state operating conditions for each desired product, the cost of each product, and inventory and raw materials cost are also given. The problem consists of the simultaneous determination of the production sequence and product transitions in each period of production such that the profit is maximized. The major decision variables are the optimal production sequence, amounts to be manufactured of each product, production times, optimal transition trajectory, and the optimal values of the control variables”. As displayed in Figure 1, planning decisions incorporate the use of productions periods that can last weeks or months. Similarly, within each production period there are a set of production slots

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whose number is normally equal to the number of products. In the bottom part of the same figure, the model behavior is represented in terms of the full discretization of the system using orthogonal collocation on finite elements [17]. In this work, a mixed-integer dynamic optimization formulation is used for addressing the simultaneous planning, scheduling, and control of a single CSTR. Using the transcription approach the MIDO problem is fully discretized leading to a mixed-integer non-linear programming (MINLP) problem. The problem to be tackled consists of computing simultaneously the best production sequence and optimal dynamic transition trajectories in each period of production such that a set of production targets is met.

Figure 1: Time horizon for planning, scheduling, and control where Np is the number of total periods and Ns is the number of production slots within each period.

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Planning, Scheduling and Control formulation

In this part of the work, the optimization formulation that accounts for planning, scheduling and control is described. Since the solution of PSC problems, in the way in which it is handled in this work, provides only open-loop transition trajectories, a NLMPC control strategy is also introduced to account for the real-time implementation issues of the results found in this work. For the planning and scheduling optimization model we use as a basis the model reported by Erdirik and Grossmann [16]. Figure 2 displays conceptually the idea deployed in this work about the way of handling PSC problems. First in Figure 2(a) we recognize the interactions among planning, scheduling and control problems: information from a given level has influence on other levels. Moreover, the influence is in both pathways. The dashed box around 2(a) means that a simultaneous solution will be sought. However, to make things simpler, we actually merge the Planning and Scheduling optimization formulation into a single common formulation as shown in Figure 2(b). This common formulation strongly interacts with the optimal control formulation in both ways: decisions taken at the planning and scheduling level have strong influence on the dynamic behaviour of the addressed system and vise versa. Also in this case, the dashed box stands for a simultaneous solution of both problems. Of course, the aim of the simultaneous solution lies in exploiting the interaction between both problems to obtain improved optimal solutions, in comparison with sequential solutions that do not consider the effect of such interactions. It should be noted that planning and scheduling optimization formulations are normally composed of algebraic equations. On other hand, optimal control formulations are normally composed of systems of differential and algebraic equations. In this work, we employ a transcription approach such that to transform the differential and algebraic systems into a set of algebraic relationships. Hence, the simultaneous solution of the planning, scheduling and control problem will be cast in terms of a Mixed-Integer NonLinear Programming problem.

Objective Function The proposed objective function is formulated as the difference between the earnings due to the sales of products minus all the costs related to the manufacture of the products. Among such costs

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Planning Planning

+

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Scheduling Scheduling

Control Control

(a)

(b)

Figure 2: Simultaneous solution of the Planning, Scheduling and control problem. (a) Optimization formulations for the Planning, Scheduling and Control problem are deployed and solved simultaneously to take advantage of interactions among the three problems. (b) In this work we deploy a common planning and scheduling optimization formulation solved simultaneously with the optimal control formulation.

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the following ones are considered: process inventory and transition times. Moreover, we have also included a term that penalizes for dynamic deviations from the target steady-states. This last term also was introduced to obtain smooth dynamic responses between target points.

Max Ω =

Np NP X X

Pip Sip −

i=1 p=1

−

Np NP X NP X X

Np NP X X

oper Cip qip

−C

i=1 p=1

inv

Np NP X X

Aip −

i=1 p=1

0

tran Cik Zikp − ϕ

Np NP X NP X Ns X X

tran Ziksp Cik

i=1 k=1 s=1 p=1

(1)

i=1 k=1 p=1

where

ϕ = ωx

Ztf X 0

[∆xi (t)]2 dt

(2)

i

In Equation 1, Np is the number of processing periods, Ns is the number of slots and N P is the number of products. Similarly, Pip is the selling price ($/mol) of product i in period p, Sip are the oper tran are the unit costs sales (mol) corresponding to product i in period p, whereas C inv , Cip and Cik

for the inventory, operation of product i in period p, and the transition from product i to product k, respectively. Aip is a linear overestimation of the integral of the inventory function along time [16]. Finally qip represents the total production of product i in period p. Similarly, in equation 2, xi (t) stands for the system states and wx is a weighting function.

It is worth to mention that the first 5 terms of the objective function given by Eqn 1 are related to economic aspects of the underlying planning, scheduling and control problem. The combination of such terms represent the process profit. Moreover, during product transitions a given amount of off-specification products are produced leading to a reduction in the profit since such undesired material cannot be marketed. As the transition time increases, so does the amount of off-specification material leading to a reduction in process profit. Therefore, it becomes important to penalize the objective function due to long transition periods. This is the aim of Eqn. 2. In addition, smoother optimal control profiles can be obtained by incorporating the system states as part of the objective function. 9



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Planning and scheduling constraints The following equations set the conditions that govern the planning and scheduling issues [16], whereas the control part of the model is described in the next section concerning with the dynamic model of the process. 1. Assignment constraints. This constraint enforces that only one product can be manufactured at each time slot. A binary variable Wisp ∈ {0, 1} is used for this purpose. However, the same product i can be manufactured more than once in the same period. NP X

Wisp = 1, ∀s ∈ Ns ; ∀p ∈ Np

(3)

i=1

2. Processing time. The processing time is subject to the duration of production period Hp as an upper bound according to Equation 4, whereas Equation 5 defines the total time used for manufacturing product i in time period p (θip ) as the algebraic sum of the processing times of the slots in which product i was manufactured (θîsp ). 0 ≤ θîsp ≤ Wisp Hp , ∀i ∈ N P ; ∀p ∈ Np Ns X θîsp , ∀i ∈ N P ; ∀p ∈ Np θip =

(4) (5)

s=1

3. Production rates. Equation 6 defines the production of product i in slot s of the period of production p (ˆ qisp ), whereas Equation 7 states that the total production of product i is the sum of the production of i in the slots of period p (qip ). ri is the production rate of product i. qîsp = ri θîsp , ∀i ∈ N P ; ∀s ∈ Ns ; ∀p ∈ Np Ns X qip = qîsp , ∀i ∈ N P ; ∀p ∈ Np

(6) (7)

s=1

4. Transitions within production periods. This constraint accounts for transitions between two slots of the same time period. This is done through the use of the binary variable Ziksp . Ziksp ≥ Wisp + Wk,s+1,p − 1, ∀i ∈ N P ; ∀k ∈ N P ; s = 1, ..., Ns − 1; ∀p ∈ Np ; i 6= k

10


(8)

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5. Transitions between production periods. This constraint defines whether or not a transition between two periods of production takes place. 0

Zikp ≥ Wi,Ns ,p + Wk,1,p+1 − 1, ∀i ∈ N P ; ∀k ∈ N P ; p = 1, ..., Np − 1; i 6= k

(9)

s e 6. Timing relations. In the next set of equations Tsp and Tsp are the start time and end time

of slot s in the period of production p, respectively. The term τik Ziksp is used to represent the transition time between two consecutive slots to manufacture products i and k in period p, and HTp is the total time at the end of production period p. • Initial condition at first slot of first period s T11 =0

(10)

• End time in each slot. It is the sum of the starting time, processing time and the transition time. e s Tsp = Tsp +

X i

θîsp +

XX i

τik Ziksp , ∀s ∈ Ns ; ∀p ∈ Np

(11)

k

• Start time at first slot of any period 6= 1. This constraint establishes that the first slot of time period p starts when the last slot of the precedent time period ends. s T1p = TNe s ,p−1 , ∀p > 1 ∈ Np

(12)

• End time of the last slot at any period. This constraint establishes an upper bound of the end time of the last slot of a given time period. TNe s ,p ≤ HTp , ∀p ∈ Np

(13)

7. Inventory. In the next constraints Iip represents the inventory of product i in time period p, θîsp is the time devoted to production of product i at slot s of time period p, Sip are the sales of product i in time period p, Aip is the the linear overestimation of the integral of the inventory function along time, a valid upper bound according to [16]: 11



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• First period Ii1 = Iio +

X

ri θîs1 − Si1 , ∀i ∈ N P

(14)

s

• Periods after the first one Iip = Ii,p−1 +

X

ri θîsp − Sip , ∀i ∈ N P ; ∀p > 1 ∈ Np

(15)

s

• Area Aip = (Ii,p−1 − Si,p−1 )Hp + ri θip Hp , ∀i ∈ N P ; ∀p ∈ Np

(16)

8. Demand rate. This constraint establishes the sales of product i of period p as an upper bound of demands of that product at the end of production period p. Sip ≥ di , ∀i ∈ N P ; ∀p ∈ Np

(17)

9. Symmetry breaking constraints In order to avoid the enumeration of symmetric solutions (i.e. equivalent solutions), which greatly increases the computational cost, we add the following additional symmetry breaking constraints from [16] • The following constraint ensures that every product i is assigned to at least one slot of period p Yip ≥ Wisp , ∀i ∈ N P ; ∀s ∈ Ns , ; ∀p ∈ Np

(18)

• The next constraint establishes that if product i is manufactured within period p then at least one slot ought to be used for this aim, otherwise no slots are assigned. The constraint can be easily formulated as follows: Yip ≤ Nip ≤ Nip Yip , ∀i ∈ N P ; ∀p ∈ Np

(19)

where Nip is an upper bound. • Enforce product i in consecutive slots X Nip ≥ N − [( Yip ) − 1)] − M (1 − Wi1p )

(20)

i

X Nip ≤ N − [( Yip ) − 1)] + M (1 − Wi1p ) i

12


(21)

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These constraints establishes that if product i is assigned to the first slot of any period p then this product ought to use as many slots as possible. Otherwise, product i uses at least one slot since it has been manufactured during period p. Note that optimality conditions rule out non-consecutive slots. In addition, the number of times product i is manufactured in period p must meet the following constraint: X Nip = Wisp , ∀i ∈ N P ; ∀p ∈ Np

(22)

s

Nonlinear model predictive control constraints In this part the constraints related to the process dynamic behavior and control actions are formulated. To accomplish this goal, we use a nonlinear model predictive control optimization formulation. Because feedback information is used to determine control actions such a formulation may allow to address real-time issues such as changes in product demands. In this work we will use the NLMPC action for set-point tracking between a set of grades, and for keeping the process operating around the set-point of the target grades. Of course, the same formulation may also be used for disturbance rejection and/or tracking problems. We should remark that in previous works [9], [10], [11] only open-loop optimal control policies were computed. Those open-loop control policies assume that no disturbances were present during the solution of the scheduling and control problem. In addition, the problem of on-line implementation of such open-loop policies also remains. On the other hand, in the present work the use of NLMPC allow to consider on-line implementation of the optimal control policies and to take care of both disturbances and robustness issues. Mathematically the NLMPC problem can posed as follows: minimize x,u

subject to

J(x, u)

(NLMPC)

dx = f (x, u) dt h(x, u) ≤ 0 xL ≤ x ≤ xU uL ≤ u ≤ uU

where x ∈

MINLP Formulation for Simultaneous Planning, Scheduling, and

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