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Chapter 13
Applying Mathematical Optimization To Efficiently Make Better Decisions for Extrusion Technology: State-of-the-Art and Opportunities Shakiba Enayati*,1 and Ali Ayoub2,3 1Management, Information Systems, and Analytics, School of Business and Economics at State University of NY, Plattsburgh, New York 12901, United States 2North Carolina State University, College of Natural Resources, Raleigh, North Carolina 27695, United States 3U.S. Department of the Interior, South East Center, Raleigh, North Carolina 27695, United States *E-mail:
[email protected].
Extrusion processes are subject to various substantial challenges due to the need in variety, efficiency, and productivity. This chapter attempts to put optimization modeling into perspective as a tool to improve decision making process while reducing the need for tedious and expensive pilot runs. After elaborating the principles, an optimization model is proposed as an example for the application of this quantitative approach to efficiently determine the optimal production volume in relation with the amount of water and the degree of physico-chemical changes for a specific product. We believe that the future studies can highly benefit from combining this interdisciplinary filed with the extrusion technology to gain more control over the determination of operating parameters, such as residence time distribution, thermal and mechanical energy input, and pressure inside the die, which influence final product flavor, appearance and hydration properties.
© 2018 American Chemical Society
Introduction Reactive extrusion is defined as the current reaction technology in processing engineering of polymers in an extruder. Reactive extrusion was first developed in the 1980s, primarily for the modification of synthetic polymers. The technology has been rapidly developed since then and has been applied in various application areas. One of the early examples is the bulk polymerization of caprolactam to polyamide. After that, many examples in the literature showed that the emphasis in the use of extruders in the last two decades has shifted from studying the preparation and modification of polymers for the plastics industry towards creating new food products in which viscosity-breaking, grafting, crosslinking and coupling reactions as well as polymerization processes are carried out. Food extrusion is a continuous cooking and forming process where raw food ingredients undergo many order-disorder transitions, such as starch gelatinization, protein denaturation and complex formation between lipids and amylose, and finally are shaped at the die. These molecular transformations convert food material into a viscoelastic dough under the presence of heat, moisture, shear, and pressure. Among all flour components, starches play a key role. The application of extrusion in starch was initially studied in food processing. Food extrusion has been practiced for more than 50 years with early developments in the preparation of ready-to-eat cereals. The use of twin screw extruders for food processing began in the 1970s and expanded dramatically in the 1980s. During these periods, twin screw extruders become more widely studied as reactors for the modification of starch with acids, alkalis, and enzymes. During an extrusion process, starches are subjected to relatively high pressure, heat and mechanical shear forces. Additionally, the food extruder can be considered as a high temperature, short time bioreactor that can cause starch gelatinization, melting, and fragmentation reactions. The main parameters that influence these reactions, such as shear forces, residence time, and shear rate are defined by the geometry of the extruder as well as the processing variables, such as temperature, screw speed, feed composition and moisture content. Order-disorder transitions usually occur over a range of temperatures. The lowest temperature at which the reaction can be initiated is referred to as the threshold temperature. As the temperature of a dough (starch material) exceeds the transition threshold temperature, the starch molecules begin to undergo various disordering reactions that affect their size and shape of a fluid’s molecules. Thus, it seems reasonable to assume that such molecular changes will significantly affect starch’s rheological properties. It is well documented that water content in combination with temperature has a significant impact on the conversion of starch. Starch conversion can take many forms. Under excess water, all the crystallites in starch might be pulled apart by swelling, leaving none to be melted at higher temperatures. In a limited water environment, which is the usual condition during extrusion, the swelling forces are much less significant, and the crystallites melt at a temperature much higher than the gelatinization temperature in excess water. Therefore, determining the right level of water content and temperature is imperative to achieve the gelatinization target of a desired product. 244
The dependence of starch viscosity on temperature, moisture content, time-temperature history, and shear rate has been studied by a number of researchers (1, 2). Furthermore, a few attempts have been made to model gelatinization and fragmentation kinetics during extrusion and to model apparent viscosity including reaction kinetics during extrusion (3, 4). In addition, the elastic properties of cereal dough during extrusion have been studied (5, 6). However, to the best of our knowledge, determining the maximum possible feed rate (i.e. increasing the production volume) while considering the structural changes in starch properties during extrusion has never been pursued. Moreover, the seminal literature in extrusion processes has addressed optimization using approximation algorithms based on mathematical and statistical techniques such as response surface method (7–10) or metaheuristics (11, 12) to find a “good” rather than an “optimal” solution for setting conditions and parameters of a process. In particular, response surface method explores the relationships between several independent variables and one or more dependent variables in a process using a sequence of designed experiments in order to obtain an estimate for the optimal production (13). Similarly, metaheuristics usually fail to guarantee a globally optimal solution as they use a procedure to sample a large set of solutions under few assumptions about the problem, and eventually converge to a good solution with less computational efforts (14). This chapter attempts to put optimization modeling into perspective as a mathematical tool to efficiently determine the optimal production volume in relation with the amount of water and the degree of modification for a specific product. Such a tool would be the first milestone to determine the best solution by limiting the need of implementing multiple tedious high-priced inefficient pilot runs before final scale-ups. We believe that this study can significantly impact the future research in order to help the industry improve their productivity by efficiently gaining more control over the determination of extrusion operating parameters, such as residence time distribution, thermal and mechanical energy input, and pressure inside the die, which influence final product flavor, appearance and hydration properties. The remainder of this chapter is organized as follows: principles and guidelines of optimization modeling are discussed in the context of extrusion. Then, an optimization model is proposed for the application of maximizing feed rate in the cooking-extrusion of a biomass. Numerical results and the details for working with solver are next elaborated. The chapter concludes by a final discussion and future directions.
Principles of Optimization Modeling for Production by Extruder Cost-efficient manufacturing processes, especially for large-volume products, are subject to various substantial challenges due to the need in flexibility, variety, speed in development, efficiency, and productivity. Early phase clinical trials are costly and product properties are hard to predict, control, and stabilize when it comes to a large-scale practice. Implementing a high-quality and stable process could be even more challenging for extrusion lines as a successful extrusion 245
process “is not about doing two or three things right- it’s about doing hundreds of things right” (15). That is, requiring a proper instrumentation, the “vital signs” for the extrusion process (i.e. melt pressure, melt temperature, and motor load) as well as various parameters (i.e. barrel and die temperature, screw speed, power consumption in each cooling zone, ambient temperature, relative humidity, temperature of feedstock entering extruder, moisture level of feedstock entering extruder, water outlet temperature at the feed housing, etc.) must be measured and monitored continuously (16). It is imperative that an extrusion performance and the quality of its resulting product directly depends on the design and accuracy of the monitoring strategies. In addition to the numerous control factors in an extrusion process operation, it is critical to any industrial manufacturer to practice a robust, consistent and efficient process in terms of the use of material, time, labor, energy, etc. These objectives are difficult to achieve every time that the process parameters such as material type (17), process setting requirements (18), and machine geometry (19) change in an intermittent operation in which having various products is the main priority and new products are regularly developed to stay competitive in the market (20, 21). Therefore, applying a quantitative approach to model such complex systems could significantly facilitate the decision making process for an efficient versatile scale-up in which trial and error costs of pilot runs have been reduced (22). The main part of any quantitative approach is mathematical modeling which use a system of symbols and quantitative relationships or expressions to represent a problem in real-world context (23). In this section, we elaborate on the general framework of optimization models with adjustable parameters that could be developed to design the processing conditions in an extrusion process. Prior to further discussing mathematical models, it is worth noting the aphorism stated by statistician George Box: “All models are wrong but some are useful” (24). For instance, consider an airplane replica as an iconic model. While the airplane replica can be studied, tested, and tuned for reliability levels in a wind tunnel, it is only a representation of the real airplane and is limited to capture the full reality of an actual airplane. Similarly, mathematical models are merely a representation of a situation in reality and are restricted to the underlying assumptions considered for the elements and dynamics impacting the real-world problem. Therefore, one can never claim that a proposed mathematical model is completely equivalent to the corresponding reality. Nevertheless, experimenting with all types of models including mathematical models require less time and less risk while they are less expensive as opposed to experimenting with the real situations and objects (25). Evidently, the more closely a model represents the real situation, the more accurate its conclusions, predictions and recommendations will be. However, despite the necessity for accuracy, very complex mathematical models would be computationally intractable as they require more tedious solution procedures, if not impossible to solve. In summary, mathematical modeling is in fact the art of balancing accuracy and simplicity to properly represent the real-world condition upon which the decision making process is built. The remainder of this section 246
further explains fundamental properties of mathematical modeling customized for the extrusion processes. Guideline to Develop Optimization Models in the Extrusion Context Similar to any industrial operation, two distinct types of decisions can be considered for the extrusion process operations: strategic versus tactical. The strategic decisions address “if” questions for a broader-in-scope and longer-termin-nature change that could set the direction for a specific department or even the entire organization (e.g. decisions on purchasing a new extruder machine or adding a new extrusion line for a new product). On the other hand, tactical decisions are concerned with “what and how to” change questions that concentrate on specific day-to-day issues which are narrow in scope and short term in nature (e.g. decisions on production quantities or the process setting improvement in a specific extrusion line) (20, 26). To make the best tactical or strategic decision, optimization mathematical models can be developed based on quantitative facts from the process to find the best alternative solution which is called “optimal solution”. The process of identifying the best alternatives for the decisions is called “decision making” or “problem solving” which in fact address the main questions of the targeted problem. Mathematical modeling or problem formulation is the first step in problem solving which includes the following ingredients: (1) Model Inputs: developing an optimization problem requires distinguishing between two types of model inputs: (1.1) uncontrollable inputs and (1.2) decision variables (or controllable inputs) (27). 1.1. The former refers to the environmental factors that are not under control of the decision maker which are in fact the available data. For instance, machine geometric properties such as the screw profile or the biomass type can be considered as uncontrollable inputs which are assumed to be fixed and determined externally and not by model. The uncontrollable inputs are called model’s parameters. 1.2. The latter input type refers to the decision alternatives specified by the decision maker from which the best alternative must be eventually determined by solving the model. For instance, the flow rate of cooling water through feed housing and the temperature of feedstock entering extruder can be considered as decision variables that must be determined given a specific machine and a biomass type. To develop a mathematical model, the first step is to denote each parameter and decision variable by a mathematical notation. Prior to solving the model, parameters will be stated as a specific number to describe the system under study. Model’s solution refers to the optimal values of the decision variables. 247
(2) Objective Function: in order to identify the best alternative for each decision variable, the decision maker must determine the optimization criteria to evaluate all alternatives of each decision. Objective function is a mathematical expression that describes the problem’s objective with respect to the relavant parameters and decision variables. Few examples for the objective function in an extrusion process are listed as follows: •
Maximizing process productivity defined as output/input (20). Example: maximizing “weight of the final product/required initial biomass”.
•
Maximizing utilization defined as the ratio of the time a resourse was used to the time a resource was available. Example: maximizing “total time extruder was used/total time extruder was available”.
•
Maximizing efficiency defined as actual output/standard output (20). Example: maximizing “weight of the final product/weight of the final product in a benchmark”.
•
Minimizing process waste: ▪
Maximizing process velocity defined as the ratio throughput time/value-added time. The throughput time is the average amount of time that the materials take to move through the extruder and the value-added time refers to the time spent actually working on the product.
▪
Minimizing waste of the process.
(3) Constraints: each extrusion process operates under some conditions which must be satisfied for an acceptable solution. These conditions in fact limit the degree to which the objective can be pursued. Process requirements, relationships between different decision variables, resource availabilities, etc. can be stated as distinctive mathematical expressions to address the scope of the problem and its restrictions. Few instances for constraints in an extrusion process are listed as follows: •
Process requirement: the temperature can only very within a pre-determined range.
•
Relationships between different decision variables: ▪
If moisture level and temperature of feedstock entering the extruder are the decision variables with the goal of maximizing efficiency, one or more mathematical 248
expressions must be considered for the underlying relationships between moisture level, temperature, flow rate, and/or any other factors impacting one or both decisions in the process context. •
Resource availability: the initial biomass weight is limited to a certain number.
Figure 1. Flowchart of the process of transforming model inputs into outputs. Figure 1 summarizes the process of transforming the model inputs into outputs. To develop a problem formulation or a mathematical model, the process context must be understood and described thoroughly in the following order: (1) Decision variables: what are the main decisions for the problem under study? (2) Objective function: what is the goal of the problem that drives the optimization? (3) Constraints: what are the limitations, necessary relationships between decision variables, and requirements in the problem under study? All above ingredients must be then stated using mathematical symbols (for decision variables), function (for objective function), equality/inequality expressions (for each constraint). Note that uncontrollable inputs or parameters are in fact the given or known numbers in the developed mathematical objective function or constraint expressions. Classification of Optimization Models Mathematical models can be categorized according to different criteria. Such a categorization is important since each class is different in terms of computational tractability. We briefly review the important classes as follows: (1) Linear Programming (LP) versus Non-Linear Programming (NLP): a problem formulation is called a Linear Programming model if all mathematical expressions (i.e. both objective functions and equality/inequality constraints) in the model are in the linear form, i.e. each variable appears in a separate term raised to the first power and is multiplied by a constant (which could be 0). If even one expression is not 249
formulated in a linear form, the model will be categorized as a non-linear programming model (28). Note that the word “programming” refers to the “choosing a course of action”. The reason for this categorization is that LP formulations are richer and more popular theoretically and computationally (i.e. effective solution procedures are available) as opposed to NLP models. Thus, modeling an objective function or a constraint in the form of linear function is always a priority, if possible. (2) Integer Programming (IP), Mixed Integer Programming (MIP): if decision variables can only take integer values due to the specific problem context as opposed to continuous values, the model will be categorized as IP. If some decision variables are restricted to only integer values and some other can take continuous values, the model will be categorized as MIP (29). As an instance in the context of extrusion process, if the number of heating or cooling zone in an extruder is a decision variable, it can only take integer values. Note that both LP and NLP models can be classified as IP or MIP. It is always preferred to avoid integer decision variables, if possible, as considering continuous variables leads to a more efficient solution procedure. (3) Deterministic versus Stochastic (Probabilistic): if all model’s parameters are known and cannot vary, the model is deterministic. Otherwise, if any parameters are uncertain and subject to variation, the model is called stochastic (30). For example, relative extruder humidity ranging in an interval with a known probabilistic distribution can be considered as an uncertain parameter and such an assumption would make a stochastic model. The solution procedures for the deterministic models are better developed in the literature, so it is preferred to ignore the potential uncertainty in a parameter if its impact can be disregarded. (4) Single Objective versus Multi-Objective: as the name implies, if the objective function of a model is stated as a single mathematical function, the model will be categorized as single objective. Those models that handle multiple objective functions in a model are called multi-objective models. For instance, a model may consider to simultaneously maximize the extruder productivity while minimizing the process waste. Multi-objective models are typically solved using evolutionary algorithms which are referred to those solution procedures with no exact global solution (31). It might be possible to modify multiple objectives as a weighted sum term in a single objective function for a more efficient solution algorithm (32). Our proposed model in this chapter considers a weighted sum term in the objective function to maximize the production quality yield and the feed rate while minimizing the melt viscosity.
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More Insights into Optimization Modeling In the optimization terminology, a solution refers to the recommended values for decision variables after solving the mathematical model. A “feasible” solution satisfies all the problem’s constraints. If even one constraint is violated, the solution is called “infeasible”. An “optimal” solution is a feasible solution that results in the best possible objective function value (i.e. the largest/smallest if maximizing/minimizing). Mathematical modeling and designing tailored solution procedures can be considered as a scientific art skill that will be further developed with experience. An efficient modeling is the one with the least possible number of decision variables and constraints. Once a mathematical model is developed, it must be solved to find the optimal solution. However, after a try to solve a mathematical model, two special cases may occur that yield to “no solution found”: 1.
It is possible that no solution is found to satisfy all the constraints. That is, the constraints considered in the model are incompatible in nature. This special case is called “infeasibility”. Such a condition could be the result of a formulation error, high managerial expectations, or too many restrictions placed on the problem (i.e. the problem is over-constrained).
2.
It is possible that no solution is found due to “unboundedness”. A maximization (minimization) model is called unbounded if the solution values may be made indefinitely large (small) without violating any of the constraints. In real-context problems, such a condition is the result of an improper formulation such that most likely a constraint has been inadvertently omitted.
Finally, a variety of software packages are available for solving linear programming models such as LINGO, MATLAB with optimization toolbox, CPLEX optimizer, Gurobi Optimizer, GMPL (GNU Mathematical Programming Language), and Microsoft Excel Solver Add-in. The proper package can be selected depending on the model size, structure, and modeler’s programming skills. GMPL is recommended as an easy non-commercial tool to solve LP models when the programming background is limited. For mixed integer programming (MIP) and mixed integer programming nonlinear programming models, SCIP optimizer (see: http://scip.zib.de/) is available for free which is currently the fastest solver that does not require much programming skills. We further discuss SCIP platform and requirements in the numerical results section.
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First Step To Apply an Optimization Model in Extrusion In this section, we propose an optimization model to maximize the productivity rate for cooking-extrusion of a biomass. We then discuss the numerical results and observations.
Optimization Formulation The proposed mathematical model intends to determine the optimal values for the following decisions: 1. 2. 3. 4.
Water content level Absolute temperature Feed rate Screw rotation speed
Tables 1 and 2 summarize the notations of parameters and decision variables used in the model, respectively.
Table 1. Notations of set, parameters, and constants
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Table 2. Notations of decision variables
The proposed optimization model is a non-linear programming model that is given by:
Subject to:
where ε can be computed using ε0(1 − dg) in which ε0 represents the degree of crystallinity of the biomass raw material. 253
The proposed model targets maximizing the production quality yield and the feed rate while minimizing the melt viscosity. The objective function (1) in fact computes the linear combination of these two goals with equal impact weights. Constraint (2) illustrates the relation between the screw rotation speed, the head pressure and the melt viscosity for operating the extruder (33). The melt viscosity is an auxiliary decision variable that implicitly links the absolute temperature and water content level decision variables with the feed rate in Constraint (3) (34). Constraint (4), (5), and (6) represents the relationship between auxiliary decision variables O, ys, and yw,s with the main decision variable of water level yw. Constraint (7) maps the product quality yield to the feed rate. Constraint (8) and (9) limit the temperature variation and water level to be in the acceptable range for the target quality, respectively. Constraint (10) enforces the screw speed to not exceed the maximum possible rotation speed of the extruder. Finally, according to Constraint (11), all decision variables can accept continuous non-negative values. Next section will discuss the solution procedure of the model and the numerical results.
Numerical Results and Optimizer Software In this section, we discuss the numerical results of solving the proposed model. The biomass is assumed to be starch. The values of model’s parameters are considered as follows: • •
α = β = 0.5, Mw= 0.018 kg/mol and Mu = 0.180156 kg/mol,
• • • • • • • •
=5.8, ρs=1550 kg/m3 and ρw=1000 kg/m3, =0.3, ε0=0.4, R=8.314, T=100 seconds, τ1 = 353.15 K, τ2=380 K, σ1=20%, σ2=50%.
Using the above parameter values, we solve the proposed model in the previous section using SCIP optimizer to determine the value of main decision variables (i.e. water content level yw, absolute temperature θ, feed rate f, screw rotation speed s) for 20 degree of gelatinization levels starting from 0% to 95% incrementing by 5%. Our experiment considers 6 different experimental instances which illustrate extruders with different sizes characterized in terms of maximum rotation speed sm and head pressure hp. Description of each instance is summarized in Table 3 for instance 1 to instance 6 sorted from a smaller extruder to a larger one. The initial weight W of the available starch is assumed to be 100 kg for instances 1 to 5, and 500 kg for instance 6 as it corresponds to the largest extruder in this study. 254
Table 3. Extruder size in each instance
Our numerical results demonstrate the optimal feed rate suggested by the model for the various degree of gelatinization levels. Figure 2 presents the resulting optimal feed rate f (kg/second) corresponding to each degree of gelatinization dg incrementing by 5%. Using the model’s output, we also develop the linear regression equation for each instance to be able to determine the corresponding feed rate for any other degree of gelatinization levels.
Figure 2. Optimal feed rate versus degree of gelatinization in each experimental instance. Each plot also includes the linear regression equation.
We can conclude the following observations from Figure 2: •
Degree of gelatinization and optimal feed rate have a linear descending relationship. That is, targeting a higher degree of gelatinization always decrease the feed rate in a linear manner.
•
The negative slop of the fitted regression line becomes sharper for the larger extruders. That is, when the extruder is larger, the feed rate drops more dramatically as a higher degree of gelatinization is targeted.
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Figure 3 further verifies the above observations by showing instance 1 and instance 6 corresponding to the smallest and the largest extruder considered in our experiments on one plot.
Figure 3. Feed rate versus degree of gelatinization for experimental instances 1 and 6 corresponding to the smallest and the largest extruder in the experiments.
Note that the optimal solution for this model always recommend to set the water level yw and the temperature θ at the minimum allowed level (i.e. parameters σ1 for water and τ1 for temperature) while setting the screw speed s at the highest possible level sm. Such a setting adjusts the feed rate to achieve a desired degree of gelatinization while consuming resources in the most efficient way. Finally, we discuss the procedure of solving the proposed model using the SCIP optimizer. SCIP is currently one of the fastest non-commercial solvers for nonlinear programming models (35). Due to the complex chemical and physical dynamics in the extrusion processes, we believe that any optimization model in this context would eventually be in the form of a nonlinear model. Therefore, it is absolutely necessary for future researchers to access a convenient software to solve their optimization models with the least possible programming knowledge. SCIP optimizer is available online at http://scip.zib.de/#download. The optimization model can be simply scripted in a text file following simple syntax rules (e.g. parameters and variables must be denoted by “param” and “var”). The text file prepared for the proposed model in this chapter is as follows:
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SCIP optimizer can be then invoked via “command prompt”. Then, the text file must be read (using command “read .txt file”) and solved (using command “opt”) by the SCIP optimizer. Optimal solutions can be displayed using “display solutions” command. Figure 4 shows an example of the output format of the SCIP Optimizer on the command prompt.
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Figure 4. SCIP Optimizer output on the command prompt. The image was created using SCIP Optimization Suite 5.0 (35).
Discussion and Conclusion This chapter presented the application and opportunities of using optimization modeling as a quantitative approach in the field of Operations Research to improve decision making process in the extrusion processes. The main idea for introducing such a methodology in the extrusion technology is to reduce the highly-priced, time-consuming, and excessive resource consuming pilot studies as much as possible. Optimization modeling facilitates decision making process by providing the best possible solutions given available resources, chemical/rheological interactions, and extrusion system dynamics. Such a mathematical approach is flexible to target a wide range of goals from maximizing productivity or efficiency to minimizing resource consumption or waste while taking the problem limitations into consideration. However, as elaborated in the chapter, an optimization model is a formulation that tries to represent the reality and it can be as good as its underlying assumptions. Meaning that an optimization model can be useful and applicable if the system dynamics and restrictions considered in the model are sufficiently accurate to describe the problem under study. Therefore, the accuracy of the model and its assumptions is critical when it comes to implementation. 258
Having elaborated the optimization principles and opportunities, this chapter proposed an optimization model to maximize the productivity rate for cookingextrusion of a biomass while considering a set of constraints with respect to the system interactions between water level, temperature, and extruder profile. The proposed model is an example for the first nonlinear model in the extrusion context that was solved in an exact way as opposed to using approximation methods. Numerical results for starch as ingredient and details of working with solver were also discussed. Although the proposed model and its associated results in this chapter may not be readily implementable in practice, we believe that the future studies can highly benefit from the new light shed on the path of efficiently making better decisions using mathematical and computational approach rather than expensive trial and errors. While expanding the extrusion innovation literature, we hope more researchers can be inspired to combine the interdisciplinary filed of Operations Research with the extrusion technology research to vastly improve the utilization and productivity or reduce cost and resource wastes in the scale-up applications.
Acknowledgments The authors would like to thank all contributors to the SCIP Optimization Suite at Zuse Institute Berlin for helping this research in the solution procedure.
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