Improved Short-Term Batch Scheduling Flexibility Using Variable

ARTICLE pubs.acs.org/IECR

Improved Short-Term Batch Scheduling Flexibility Using Variable Recipes Elisabet Capon-García, Marta Moreno-Benito, and Antonio Espu~na* Chemical Engineering Department;CEPIMA, Universitat Politecnica de Catalunya, ETSEIB, Av. Diagonal 647, 08028 Barcelona, Spain ABSTRACT: Batch scheduling and control decision levels are usually optimized separately, even though integrated decisions would potentially increase the overall performance of the plant. This work aims at including in the scheduling problem the operating conditions related to the control function, which are traditionally optimized and fixed beforehand, to shed light to the benefits that may be obtained when using variable recipes in contrast to fixed nominal recipes. A multiproduct and a multipurpose plant illustrate the proposed approach. In both cases, the potential benefits are confirmed resulting in clear advantages regarding overall economic benefit, when the variable recipes are introduced. Thus, the presented approach gives rise to new tradeoffs, since it allows new improvement opportunities associated with quality, safety, or changing operating policies, which are usually considered separately.

’ INTRODUCTION The current challenges of process industries have been widely identified and reported in the literature.1-4 Factors such as globalization of trade, market uncertainty, and fierce competition entail dwindling margins in enterprises. In addition, companies must face tighter demands, whose nature is not only economic, but safety and environmental regulations are increasingly stricter, and enterprises must cope with other issues, such as corporative image and social acceptance. Hence, the technological and operational advantage becomes of utmost importance, and the requirements for flexibility and quick time-to-market are crucial business drivers in many industries. In addition, several authors highlight the importance of integrating information and decisions along the different time and space levels that are found in enterprise structures, and the need of improved models as well as integration and information tools is clear.5-8 In this general scenario, plant management includes the scheduling function, which formalizes decisions about resource allocation and timing of the activities performed in a production plant within a time ranging from days to weeks, to directly satisfy client requirements or demands issued from a production plan. The decision-making process related to the scheduling function is crucial to meet the global goals of the company,9 and its integration with other company functions has received increasing attention in the literature. On the one hand, many works have been proposed to meet the integration with the planning level, whose objective is to fulfill customer demands, minimizing the total cost, including inventory and production costs. Maravelias and Sung10 presented a thorough review on the integrated production planning and scheduling, and point out the current challenges and opportunities in this field. Different modeling approaches are described, and the solution strategies are broadly classified in three categories, namely, hierarchical, iterative, and full-space methods. On the other hand, the integration between the scheduling and control level, which copes with the real-time execution over time and the optimization of dynamic trajectories r 2011 American Chemical Society

of process variables, has been only scantly tackled in the literature.11 Although such integration would lead to improved overall plant operability, several hurdles are found to meet it (for example, the information transfer, the different time scale domains, the different problem perspective and boundaries, and the wide variety of approaches to formulate and solve such problems).12,13 The scheduling problem is usually mathematically posed as a MILP or MINLP problem, whereas the control one consists of the optimization of differential algebraic equations, which usually conform a nonlinear model of the process. Therefore, the integration of these two problems results in a mixed-integer dynamic optimization problem (MIDO). Several approaches have been identified in the literature to solve this problem, and such contributions toward the integration of scheduling and control can be broadly classified in three groups,11 namely, the transformation of the MIDO problem into large MINLP/MILP problems, as proposed by Chatzidoukas et al.,14 the decomposition into scheduling and control subproblems, such as those presented by Nystrom et al.,15 and agent-based systems that use metaheuristic rules to achieve the integration.16 Most of these works focus on decision-making integration for continuous processes. For example, Terrazas-Moreno et al.17 included process dynamics in scheduling formulation for a cyclically operating continuous plant of polymer manufacturing, to calculate the optimal transition duration and profiles that decisively affect the optimal profit achieved. Busch et al.18 integrated control and scheduling for a wastewater treatment plant, and they solved it using heuristic rules and General Disjunctive Programming. This strategy was later applied to a continuous polymerization process.19 Special Issue: Puigjaner Issue Received: July 1, 2010 Accepted: November 15, 2010 Revised: November 12, 2010 Published: February 08, 2011 4983

dx.doi.org/10.1021/ie101404b | Ind. Eng. Chem. Res. 2011, 50, 4983–4992

Industrial & Engineering Chemistry Research Batch production has gained much importance in the last decades, since it is inherently flexible to address plant demand fluctuations in the fine chemicals industry. Batch processing decisions pertain to the domain of optimal state and control variables profile, since the associated operations are dynamic in nature. The very first piece of information for batch scheduling to manufacture a product is the process recipe,20 specified by a set of processing tasks. These tasks have an associated processing time, which may be function of the specific type of equipment employed, the amount of material, the assigned resources, and the selected state variables.21 In process industries, it is common practice to define and fix the control strategy and processing times for any batch of a given product.22 In some cases, such as in the pharmaceutical industry, fixed accurate recipes are compulsory, because of the nature of the products and the process. In other cases, fixed production recipes are obtained from a design optimization stage or determined by previous knowledge and experience over the process behavior. As a result, degrees of freedom of the dynamic problem are disregarded in the scheduling problem and actual production scenarios cannot be globally optimized. In contrast, works involving alternative options that entail integration of the optimal control profiles in scheduling problems are still very scarce. The main difficulties have been identified as the high computational requirements needed to solve the final MIDO problem, the lack of appropriate process models, and the large time needed to develop and reduce process models, which require both sufficient detail to represent the process and enough simplicity to be solved by the available optimization tools.23 One of the very first works to consider the full process dynamics in the batch scheduling problem was presented by Bhatia and Biegler.24 They included dynamic models of processing tasks within the design and scheduling formulation for special types of batch problems, namely, flowshop plants with unlimited intermediate storage and zero-wait transfer policies with one unit per stage. Processing decisions are resolved by discretizing the dynamic process models through collocation on finite elements; the authors prove that dynamic process considerations can contribute significantly to increase profitability. On the other hand, Mishra et al.25 broadly classified the scheduling problem formulation in two categories, namely, the standard recipe approach and the overall optimization approach. The former defines a recipe beforehand, either empirically or by single batch optimization, and then the scheduling problem is posed on the basis of these standardized recipes. The latter approach directly includes process dynamics in the scheduling problem and restores degrees of freedom, so it can yield a better solution. They compare both approaches for a single product plant and a multiproduct plant. The standard recipes are modeled as polynomials that relate duration and reaction heat to the processed quantities. They prove the superiority of the overall optimization approach, in terms of solution quality, but highlight that one of the major drawbacks is the large size of the resulting problems and the computational difficulty in solving large-scale problems. In addition, the authors point out the influence of the costs structure on the results. According to Harjunkoski et al.,11 “the integration of scheduling and control into large-scale MIDO problems is probably only useful in certain selected cases, where the complexity of the dynamic plant model is well in proportion to the size of the scheduling problem.” In their work, it is also highlighted that possible solutions to the integration problem may probably remain in the domain of

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including, in the scheduling problem, some indicators related to the dynamic part of the problem. In this context, some work has already been presented in the literature, which may be regarded as intermediate strategies, since they define processing times as a function of batch sizes or state variables.24 Romero et al.26 presented a framework that includes the possibility of recipe adaptation in the optimization of batch processes. Specifically, a linear-based recipe model is integrated into a multipurpose scheduling algorithm called S-graph. In that work, the productivity maximization was established as an objective function and the cost of modifying process variables was considered negligible. In addition, Ferrer-Nadal et al.27 incorporated the concept of recipe flexibility as an additional rescheduling action in the reactive batch operation of multipurpose batch plants. They assumed a linear model in a predefined flexibility region around nominal operating conditions, penalized any deviation from the optimal operating conditions, and solved a mixed-integral linear program (MILP), based on the general precedence model.28 The aforementioned authors assumed that there is a single optimal nominal recipe, and changes may be produced linearly around such operating point. Moreover, there is no reference in the literature regarding the consequences that may be derived from including, in the scheduling problem, batch-to-batch variable processing times from a general scheduling perspective. Thus, the main objectives that may lead to an optimal integrated decision of the scheduling and control problems are important to be considered.11 From the scheduling perspective, the objective function depends on the decision-maker criteria, which are based both on his/her experience and on the nature of the plant. Hence, a unique objective function may not be suitable for all scheduling problems. Typical objective functions consider the minimization of makespan, earliness, tardiness, or overall cost.29 Regarding control specifications, they are usually supposed to be determined as part of the design activity, and a suitable criterion consists of either measuring the economic success such as batch time or total production cost per unit product, including annualized investment and operating costs,20 or of using processing criteria.30 The determination of the optimal operating profiles is usually referred to as the optimal control problem.31 This work aims at gaining insight into the benefits that may be obtained when integrating process level knowledge in scheduling problems of different complexity. The proposed approach is based on the use of flexible recipes and economic functions for both the scheduling and control problems. Therefore, the economic functions will determine the so-called optimal recipe in the control level; and next, the cost associated with the variation of the state variables that have an influence on time is represented in the objective function of the scheduling problem. As a result, the dynamics of the processes is implicitly considered by allowing variable batch processing times and including the tradeoffs between the time and state variables variations in the objective function. This way, the scheduling level decides on the conditions of each batch that optimize the overall profit, and tradeoffs between process conditions and scheduling decisions of complex scheduling problems can be tackled. Moreover, the proposed recipes are feasible in a wider operational range than those defined around nominal operation reported in previous works. In addition, the benefits of allowing variable batch-to-batch processing times are assessed. Overall, the search space of the optimization problem broadens, as does the potential for solution improvement. These ideas are illustrated through two case studies, namely, 4984

dx.doi.org/10.1021/ie101404b |Ind. Eng. Chem. Res. 2011, 50, 4983–4992

Industrial & Engineering Chemistry Research

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a multiproduct and a multipurpose production plant under several storage policies.

’ PROBLEM STATEMENT This work considers variable batch-to-batch processing recipes, in contrast to fixed nominal recipes, to improve the overall plant performance. The corresponding problem is defined as follows: Given: Process dynamics • a dynamic process model that describes the process; • a set of control variables, such as temperature, feed flows, or pressure; • a set of state variables that are ruled by the process model; • a set of constraints imposed over the process conditions; Process operations planning data • a specific time horizon; • a set of materials: final products, intermediates, and raw materials; • a set of expected final products with minimum and maximum demands; • a fixed batch plant topology consisting of a set of equipment technologies for processing stages; • a set of production recipes containing the production sequences or stages; • a set of fixed production stages whose processing times, massbalance coefficients, and resources utilization are optimized and fixed beforehand; • a set of variable production stages defined by their corresponding process dynamics; Economic data • selling price for every final product; • direct cost parameters, such as labor, energy, and raw material costs; • a set of environmental, quality, or safety constraints that have an associated economic cost; The goal is to determine: • the number of batches required to meet the demand; • the assignment and sequencing of the batches; • the amount of final products to be sold; • the processing times of the variable stages of each batch; • the operating conditions of the variable production stages of each batch; such that the adopted performance metrics are optimized. In this case, economic functions characterize the overall plant performance at both scheduling and control levels. Accordingly, economic indicators, namely, profit (eq 1) and profitability (eq 2), may be used as objective functions. In both functions, a thorough definition of profit, including revenues, operating costs, raw material costs, and unfulfilled demand penalization, is adopted. zprofit ¼ Selling Price - Operating Cost - Raw Material Cost - Demand Penalization zprofitability ¼

zprofit Mk

ð1Þ ð2Þ

’ SOLUTION PROCEDURE The traditional hierarchical approach established between the scheduling and control levels is usually managed through the use

Figure 1. Steps of the solution procedure for integrated decision of scheduling and control.

of overall performance indicators, such as profitability, which are affected by the main controlled variables in each case (namely, assignment and sequencing in scheduling, and operating conditions in control). In such approach, after an initial optimization stage, a nominal recipe, which determines both the operating conditions and the processing times to be implemented at the scheduling level, is obtained. As a result, the operating conditions and times are fixed, and any deviation from those values is even considered negative for the process performance. However, such simplification of the problem may be deceitful, because possible tradeoffs between operating conditions and times are disregarded. In this work, economic functions are used to relate the variables of both levels, which allows the decision maker to reach integrated decisions. The proposed solution procedure is defined in three stages, which are thoroughly described next (see Figure 1). Flexible Recipe Definition. First, the variables that are more influential over the dynamic process and its operational cost should be identified. As a result, such variables could be used to define the flexible recipe, since they would allow additional degrees of freedom at the scheduling problem. These variables constitute a set of free decision variables in the flexible recipe. The selection of these free variables may be either given, arbitrarily chosen, or determined by a sensitivity analysis of the process considering economic implications. 4985


Industrial & Engineering Chemistry Research Afterward, the relationship between the free decision variables and both the processing time and the operating cost should be formulated. Accordingly, a correspondence between time and operational cost is explicitly established for each set of values of the free decision variables. This obtained relationship is to be introduced in the scheduling problem. Consequently, simple functions are required, so that the resulting problem can be efficiently managed by the corresponding optimization algorithms. For example, linear, quadratic, or piecewise linear functions could be used to define the operating cost as a function of time. The objective function at this level is only used to define the nominal recipe, which may be later modified during the scheduling resolution if recipe flexibility is exploited to improve the overall performance of the plant. Obviously, the problem solved at this step has usually a more rigid structure, since the recipe obtained in the design step is traditionally adopted in the scheduling without any further consideration about cost or operating conditions. In this work, the influence of selecting one free decision variable is studied. In addition, this variable is time invariable along the process, since it makes possible to use two-dimensional regression algorithm to obtain the relationship between time and cost. Moreover, a single free variable allows an easier implementation of the recipes obtained at the scheduling step, because the relationship between each processing time and the corresponding optimal value of the free variable is unique. Hence, when the processing time for each batch is established, the value free decision variable is directly obtained. However, working with more free variables is possible by determining, for every batch processing time, the optimal combination that would result in a minimum economic indicator. Scheduling with Variable Processing Times. Several formulations have been proposed in the literature to tackle the scheduling problem, as stated in Mendez et al.32 Different classifications can be adopted; however, from a process representation point of view, they can be broadly classified as networkbased and sequential-based. Maravelias and Grossmann33 presented a STN-based model for multipurpose batch plants, which handles general batch process concepts such as different storage policies, or sequence-dependent changeover times. The resource task network (RTN) representation is adopted by Castro et al.34 for two new continuous-time formulations to model multistage batch plants, and they are compared with alternative approaches to the problem, such as constraint programming and global sequencing variables. Erdirik-Dogan and Grossmann35 presented a time-slot-based formulation that incorporates mass balances and propose a bilevel decomposition algorithm for dealing with medium-sized problems. Alternative formulations, which can deal specifically with sequential processes, are based on the general and immediate precedence concepts. The former was first introduced by Mendez et al.,28 whereas Gupta and Karimi36 presented an immediate precedence model for multiproduct batch plants. Any of previous formulations may be used to solve the scheduling problem with variable processing times as long as they allow variable processing times and the use of economic indicators as an objective function, and they successfully model the relationship between cost and time. This work solves the scheduling problems for sequential-based case studies, and the general precedence model presented by Ferrer-Nadal et al.37 is adopted, since it adequately addresses the modeling of transfer tasks in multipurpose batch plants. However, it has been adequately extended to consider the aforementioned issues regarding flexible

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recipes. Therefore, the formulation presented by the previous authors37 is taken as reference, and the additional considerations for the scheduling with variable recipes are described next. An important consideration refers to the batch assignment and demand fulfillment. Therefore, eq 3 defines that if a batch i of a product p is performed (Wip = 1), then all stages of its recipe must be assigned to an available unit. In addition, unless enough batches of a given product fulfill the minimum demand, a variable related to the additional demand (ADp) counts for the amount that cannot be produced (see the constraint described by eq 4). X Yipsu ¼ Wip "i, p, s ð3Þ u

X

Wip BSp þ ADp g PDp

"p

ð4Þ

i

As for timing constraints, the batch variable processing times of the corresponding stages (tdips) can only take certain values between a minimum and a maximum value determined in the flexible recipe, if the batch is performed (see constraints described by eqs 5 and 6). "i, p, sjðp, sÞ ∈ DSps ð5Þ tdips e Tpsmax þ BigMð1 - Wip Þ tdips g Tpsmin - BigMð1 - Wip Þ

"i, p, sjðp, sÞ ∈ DSps

ð6Þ

Moreover, the finishing time must be adequately defined according to the fixed and variable stages. Therefore, if a stage has a fixed processing time, the finishing time will depend on the realization of the batch (eq 7), whereas, for dynamic stages, the processing time is a decision variable (eq 8). Under no circumstances can the time horizon be exceeded (see the constraint described by eq 9). 0 1 X X ttpu0 þ @ ptpsu þ ttpu AYipsu þ Twips Tfips ¼ Tsips þ u0 ∈Upðs - 1Þ

u∈Ups

"i, p, sjðp, sÞ ˇ DSps X

Tfips ¼ Tsips þ þ

X

ð7Þ

ttpu0 þ tdips

u0 ∈Upðs - 1Þ

ttpu Yipsu þ Twips

"i, p, sjðp, sÞ ∈ DSps

ð8Þ

"i, p, s

ð9Þ

u∈Ups

Tfips e H

As part of the integration strategy, the objective function at the scheduling level determines the final production achievements, given a minimum demand and the time horizon. Thus, using flexible recipes allows the decision maker to study the tradeoffs between the economic factors and the influence that the overall available time has over the processing time. Equation 10 defines the profit for the problems posed, but the general idea consists of including the income and the total costs. In this case, the batch price includes the price of the batch, the raw materials cost, and most of the operating costs, whereas the energy costs depend on the processing time of the variable stages, and the unaccomplished demand is additionally penalized. X X X BPp Wip Eips CD 3 ADp ð10Þ zprofit ¼ ip

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ips∈DSps

p



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As for the energy cost of the variable stages, it takes a value according to an approximated energy function. As explained in the previous subsection, such approximated costs may be linear or quadratic functions of time (eqs 11 and 12). The value of the energy cost of a given batch at the scheduling level is only determined if the batch is performed (see the constraints described by eqs 13 and 14). EFips ¼ a0 þ a1 tdips EFips ¼ a0 þ a1 tdips þ a2 tdips 2

"i, p, sjðp, sÞ ∈ DSps "i, p, sjðp, sÞ∈DSps

Figure 2. Plant flowsheet for the three-stage multiproduct plant in Case Study 1.

ð11Þ ð12Þ

Eips e EFips þ BigMð1 - Wip Þ

"i, p, sjðp, sÞ ∈ DSps ð13Þ

Eips g EFips - BigMð1 - Wip Þ

"i, p, sjðp, sÞ ∈ DSps ð14Þ

Recipe Implementation. As a result of the scheduling optimization, the processing times of the variable stages are determined. In the recipe implementation, the values of the process variables corresponding to such processing times are determined, so that they can be implemented. For the case of a single free decision variable, such values can be obtained from the rigorous dynamic model. The real energy costs can be also assessed from the same rigorous model, since the scheduling model only considers approximated functions. In addition, this step can lead to the reoptimization of the scheduling problem, if necessary. For example, a tighter range of values of the free decision variable could be used to reformulate a better estimation of the energy costs introduced at the scheduling level. If several free decision variables were considered, the function of this step would be the same as for a single free decision variable, but the time resulting from the scheduling level should be related to an established set of free decision variables.

’ CASE STUDIES Two case studies have been posed to study the effects of introducing process dynamics in the scheduling level. They consist of a multiproduct and a multipurpose batch plants with different products under different storage policies. The general precedence model28,37 is adopted and adequately extended to consider variable processing times, and economic objective functions, as defined in the previous section. The considered objective function consists of the maximization of the profit, which includes the benefit of each batch and the operating cost associated with the processing time, which is related to the process variables to be modified. In the case of the multiproduct plant, the costs have been adjusted as first- and second-order polynomial functions of time, whereas in the multipurpose case, the approximation is assumed linear, given the high computational effort that quadratic cost functions introduce at the scheduling problem in that case. The mathematical model has been implemented in GAMS and solved using the MILP solver CPLEX 9.0 for linear cost functions, and the MINLP solver BARON 8.0 in the case of quadratic approximations. Case Study 1: Multiproduct Plant with Isothermal Reaction Stage. A multiproduct batch plant processes two products

(i.e., A and B) through three stages (see Figure 2). The first stage is an isothermal reaction process, where a conversion of 95% must be achieved, and whose processing time is a function of the temperature (Figure 3). Product batch sizes, production

Figure 3. Processing time dependence upon temperature in stage 1 for the production of A (solid line) and B (dashed line) in Case Study 1.

Table 1. Product Prices and Lot Sizes for Case Study 1 unitary

batch benefit

batch size

demand

[m.u./batch]

[ton/batch]

[ton]

A

30

5

20

90

B

40

6

24

90

product

energy cost [m.u./MWh]

Table 2. Recipe Stage Times for Case Study 1 stage 1

stage 2

stage 3

product

unit

time [h]

unit

time [h]

unit

time [h]

A B

R1 R1

variable variable

P1 P1

0.5 0.8

C1 C1

0.5 0.4

Table 3. Kinetic Parameters in Reaction Stage for Products A and B in Case Study 1 reaction rate constant (m3/(s kg))

activation energy (kJ/(mol K))

kA0 = 2.1 105

EA = 59.029 103

kB0 = 1.8 10

EB = 74.826 103

8

times, and product demands are given in Tables 1 and 2. A single unit is available for each stage, and a nonintermediate storage policy is adopted. Product changeover times and costs are disregarded and a production time horizon of 6 h is considered. Dynamics of the reaction stage correspond to a second-order kinetics for both products A and B, with the reaction parameters from Table 3, and they are fully characterized in Matlab.38 The reaction system consists of a continuous stirred tank reactor. A feedback temperature control system is also modeled, leading the processing temperature to the desired value through a heating jacket. A classical proportional integral (PI) controller is considered (KP = 0.964 and KI = 0.030 s-1). All this information characterizes processing time dependence upon temperature: the higher the temperature, the lower the production time. 4987



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Table 5. Different Energy Cost Approximations in Case Study 1 product A B

Figure 4. Profitability as a function of temperature for the production of A (solid line) and B (dashed line) in Case Study 1. Circles represent the maximum value for each case.

A B

optimal reaction

energy cost

time [h]

[m.u./batch]

0.5846

22.7162

0.3105

32.8366

Tmin [h] 0.1813 0.0713

quadratic cost (eq 12)

[m.u./batch]

[m.u./batch]

32.6260 - 15.5879t 42.7954 - 27.6950t

36.7037 - 34.0498t þ 16.6491t2 46.5673 - 61.3099t þ 53.0707t2

Table 6. Optimisation Results Using Different Time-Cost Approximations (Case Study 1) approximated

Table 4. Optimal Recipe and Limiting Conditions for Case Study 1 product

linear cost (eq 11)

energy cost-time

Mk

(linear)

approximation

[h]

profit (eq 11)

real

[m.u.] [m.u./h]

[m.u.]

Tmax [h]

real

profit profitability

not applicablea

6.00

57.78

58.07

9.69

linear approximation

6.00

81.06

71.87

11.98

6.00

79.00

76.67

12.79

(eq 11)

1.0216

quadratic approximation

0.6385

(eq 12) a

Reaction times fixed to nominal recipe values.

Table 7. Optimal Reaction Times (Case Study 1) production time [h] energy cost-time approximation

Figure 5. Energy cost as a function of time in Case Study 1: real data (thick solid line), linear regression (thin solid line), and quadratic regression (dashed line).

tA1

tA2

tA3

tA4

tB1

tB2

tB3

tB4

not applicablea

0.58 0.58 0.58 0.58 0.31 0.31 0.31 0.31

linear approximation (eq 11)

0.18 0.80 0.80 0.80 0.50 0.50 0.63 0.61

quadratic approximation 0.80 0.80 0.80 0.80 0.30 0.50 0.50 0.50 (eq 12)

However, high temperature entails high energy cost. Therefore, a tradeoff exists between minimum time and production cost. The traditional approach consists of optimizing the operation of a given product and then applying the nominal recipe to any batch of that product. In this case, the maximization of total profitability for the reaction stage of a single batch is considered, obtaining the energy cost from the energy balance. (See eqs 15 and 16, and Figure 4.) The optimal reaction times and energy costs are given in Table 4. The minimum and maximum processing times are determined by the operating restrictions, with regard to the processing temperatures. zprofitability ¼

BPp - ECp tpreaction

ECp ¼ ½ΔHRp 3 BSp 3 XRp þ cp 3 BSp 3 ðT - T0 Þ 3 PE

ð15Þ ð16Þ

To exploit the reaction flexibility in the scheduling problem, the relationships between processing time and temperature, and temperature and cost, are introduced in the scheduling by adjusting the energy cost as a function of time (see Figure 5). As a result, a linear regression and a quadratic regression of the energy cost are obtained (see Table 5) between the maximum and minimum reaction times. Next, the optimization of the scheduling problem is posed. The objective consists of maximizing the total profit in a time

a

Reaction times fixed to nominal recipe values.

horizon of 6 h (see eq 10). The results obtained using the flexible recipe are compared with those with the nominal recipe (fixed cost and time). Table 6 shows that there is a significant discrepancy between the profit values obtained with the linear costs approximation and their actual values. However, the use of linearized costs leads to a better solution than that obtained with the nominal recipe. In addition, the difference between the optimal integrated solution obtained with quadratic and linear costs, in terms of actual profit, is quite small in this case, even though the operating times are widely different (see Table 7). In any case, those times and their corresponding optimal conditions are completely different from those obtained using the nominal recipe. Obviously, the objective function used to reach the nominal recipe is not the same as that used in the scheduling, but if the process conditions corresponding to the minimum cost per batch were used, it would not be possible to cover the total demand in the fixed horizon, resulting in a reduction of the overall economic performance. Furthermore, from a profitability point of view, the solutions obtained with variable processing times strategy are better than those with fixed nominal times for the reaction operation, although profitability is the objective function to calculate the nominal times. In addition, Figure 6 shows that the improved operating times are derived from the adaptation of the recipes to the start and finishing of the plant 4988



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activity, as well as to the processing times of the different products in the plant. Such features can only be considered at the scheduling level, so the potential of improvement of including variable process recipes is evident. Moreover, note that the different processing times result in different production sequences (see Figure 6). Case Study 2: Multipurpose Plant with Isothermal Reaction Stage. A multipurpose batch plant, originally presented by Kim et al.39 and later discussed by Ferrer-Nadal et al.,37 is considered. The plant processes four products, which sequentially undergo different stages (see Figure 7). The production

sequences and processing times for each product are given in Table 8. It has been assumed that an isothermal reaction process, similar to that presented in the previous section, takes place in unit U2, affecting products B, C, and D. In this case, the optimal processing time is regarded as the one that minimizes energy cost, so the nominal recipe (Table 9) corresponds to that presented in previous works.37,39 Energy cost is calculated as in Case Study 1 (see eq 16) with a unitary energy cost of 80 m.u./ MWh. However, this processing time can be reduced at cost expenses, and a linearly approximated function that relates time to energy cost (see Figure 8 and Table 9) is provided and introduced again to integrate the recipe optimization and the scheduling problem. Therefore, the maximization of total profit (eq 10) has been established as the overall objective function. A minimum demand of three batches of each product is to be manufactured, and a maximum of one additional batch of each product are considered to be produced in a time horizon of 170 h. Table 8 contains the batch product benefits and the penalties for unaccomplished demand. Thus, the effect of adopting different intermediate storage policies over the total profit and the influence of variable processing times are studied. The storage policies considered are unlimited intermediate storage (UIS), nonintermediate storage (NIS), one common intermediate storage only available after unit U3 (CIS), and zero-wait time (ZW). Product transfer times are assumed to be negligible, but only feasible solutions from the transfer point of view are considered.37 Figure 9 presents the Gantt charts comparing the nominal recipe and variable time recipe approaches for the different intermediate storage policies, and the real profit associated with the results obtained by the proposed approach. Regarding the UIS policy (see Figure 9a), the optimization results in the same number of batches and processing times as the nominal recipe, since they can be done in the entire time horizon. This storage policy satisfies the minimum demand of all products in the given time horizon, and an additional batch of all products (with the exception of B) also can be processed. For the other storage policies, when using the recipes with the nominal time, there is no possibility to fulfill the minimum demand, whereas the variable processing time approach allows the system to achieve the minimum demand in all cases. Hence, the total profit is higher when the recipe flexibility is considered, and the higher energy cost is compensated by the time decrease for the reaction stage. As for the CIS policy, the result using nominal recipes (see

Figure 6. Gantt charts for Case Study 1.

Figure 7. Flowsheet of the multipurpose plant in Case Study 2.

Table 8. Processing Times, Production Benefit, and Unfulfilled Demand Penalty for Case Study 2 stage 1

stage 2

stage 3

product

batch benefit [m.u./batch]

unfulfilled demand penalty [m.u./batch]

unit

time [h]

unit

time [h]

unit

time [h]

A

85.0

125.0

U1

15

U3

8

U4

12

B C

81.2 88.8

140.0 120.0

U1 U3

10 9

U2 U2

variable variable

U3 U1

5 20

D

86.4

135.0

U4

5

U3

17

U2

variable

4989

stage 4 unit

time [h]

U4

13



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Figure 9b) cannot satisfy the minimum production of C by one batch, but an additional fourth batch of A can be processed; in Table 9. Nominal Recipe, Limiting Conditions, and Energy Costs, as a Linear Function of Time for Processing Stage at Unit U2 in Case Study 2 product

nominal time [h]

nominal cost [m.u./

Tmin [h]

Tmax [h]

batch]

energy cost [m.u./batch]

A B C

20 7

38.2 35.2

11.71 3.61

20 7

99.04 - 3.144t 79.41 - 6.597t

D

7

33.1

4.30

7

96.28 - 9.299t

contrast, the use of variable recipes (see Figure 9c) allows the plant to fulfill the minimum demand of all products and an additional batch of products A and D. In the latter case, to fulfill the entire demand, the processing times of the three batches of product B in unit U2 are reduced from 20 h to 11.71 h, 12.29 h, and 15.00 h. Under NIS policy, again, the minimum demand cannot be fulfilled using the nominal recipes, and one batch of product C cannot be produced (see Figure 9d). However, the introduction of variable time allows the system to satisfy the demand (see Figure 9e), by reducing the processing time of two batches of product B and one batch of product C. The ZW policy, which is the most restrictive one, does not allow the system to fulfill two batches of B under the nominal recipe assumption (Figure 9f), but an additional batch of product C can be produced instead. When assuming variable processing times,

Figure 8. Energy cost as a function of time in Case Study 2: real data (solid line) and linear regression (dashed line; see Table 9).

Figure 9. Gantt charts and profit for Case Study 2 (from lighter to darker gray: products A, B, C, and D). 4990


Industrial & Engineering Chemistry Research the minimum demand is satisfied (see Figure 9g), at the cost of decreasing the reaction times of all the batches of product B, and one batch of products C and D. Overall, even though production costs increase when allowing shorter processing times, the total profit also increases, when compared to the nominal recipe, because the total demand can be fully satisfied. Therefore, both the production of the total demand and the inclusion of more batches in a given time horizon justifies the reduction of the processing times, even though operational costs increase. These results apply to processing stages where operating cost has a certain weight in the overall cost associated with the product. In scenarios where the considered operating costs are either in a high or low ratio, regarding the overall cost, it should be predicted that extreme solutions with minimum or maximum costs, respectively, are the optimal ones, and the traditional approach using fixed recipes may be used. Summarizing, for all storage policies the solution, both the production sequence and the operating conditions are different from those obtained with separate optimizations at the scheduling and control levels, and lead to better solutions. The proposed approach leads to better solutions, and the economic benefits are noteworthy.

’ CONCLUSIONS The integration of scheduling and control functions should lead to overall operational improvements in the enterprise structure, but several difficulties must be tackled first. This work presents an approach to manage typical control decisions at the scheduling level through the management and characterization of process recipe variability, as well as the evaluation of the potential improvements that can be achieved. Specifically, an economic objective function reflects the influence that operational variability, expressed in terms of time, has over the process performance. The integration of the resulting simplified model in an overall scheduling approach leads to new decision-making tradeoffs and consequently, to further optimization opportunities. A multiproduct batch plant and a multipurpose batch plant have been considered under different intermediate storage policies. In both cases, an isothermal reactor stage stands for process dynamics consideration, and the value of temperature influences both energy cost and processing time. The total profit has been maximized in a given time horizon. Therefore, processing times in the reactor stage have been readjusted as a result of the proposed optimization procedure, and different schedules and operating conditions were found to be optimal. As a result of introducing processing variability, total profit improved significantly in both cases. Such improvements stem from the additional number of batches that could be processed, the adjustment of operation startups and shutdowns, and also from a different coordination with the rest of the product stages. These results with one single free decision variable are significant to foresee that greater improvements could be obtained by working with more degrees of freedom. Overall, variable batch-to-batch recipes increase process flexibility and allow better economic results, compared to nominal fixed recipes, since in the latter tradeoffs between processing times and scheduling actions are disregarded. These results encourage further efforts to integrate process dynamics into the scheduling task, especially in the rescheduling problem where batch variability may significantly enhance the possibilities to redistribute batch delays and losses.

ARTICLE

’ AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected].

’ ACKNOWLEDGMENT Financial support received through the research projects ToleranT (No. DPI2006-05673) and EHMAN (No. DPI200909386) funded by the European Union (European Regional Development Fund No. 2007-13) and the Spanish Ministerio de Ciencia e Innovacion is fully appreciated. The authors also acknowledge the support from the FPU program of the Spanish Ministerio de Educacion y Ciencia. ’ NOMENCLATURE Sets and Subsets

i = batches p = products s = stages u = processing unit DSps = stages s of product p, which are variable Ups = available units for processing product p at stage s Parameters

BigM = a parameter with a big value BPp = batch price resulting from the production of a batch of product p BSp = batch size of product p. cp = specific heat in reaction media of the reaction stage H = time horizon PE = unitary energy cost PDp = minimum demand of product p ptpsu = processing time of fixed stage s of product p in unit u SellingPricep = batch price resulting from the production of a batch of product p T0 = initial temperature of the reaction stage [C] = maximum processing time of the variable stage s of Tmax ps product p = minimum processing time of the variable stage s of Tmin ps product p ttpu = transfer time from unit u for product p XRp = conversion in reaction stage of product p ΔHRp = reaction energy for the generation of product p [kJ/kg] Continuous Variables

ADp = positive variable that counts for the unfulfilled demand of product p. Eips = energy cost of batch i of product p at dynamic stage s ECp = energy cost of the production of one batch of product p EFips = energy function of the production of batch i of product p at stage s Mk = production makespan T = temperature of the reaction stage [C] = processing time in reaction stage of product p treaction p tdips = time of variable processing stage s of batch i of product p Tsips = starting time of stage s of batch i of product p Tfips = finishing time of stage s of batch i of product p Twips = waiting time of stage s of batch i of product p zprofit = objective function defining profit zprofitability = objective function defining profitability 4991


Industrial & Engineering Chemistry Research Binary Variables

Wip = production of batch i of p Yipsu = assignment of batch i of product p at stage s to unit u

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Improved Short-Term Batch Scheduling Flexibility Using Variable

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