FUTURE INTERNET SERVICES. ASCONIKK 2014: Extended Abstracts. II.

Transcription

1 FUTURE INTERNET SERVICES ASCONIKK 2014: Extended Abstracts. II. University of Pannonia Veszprém, 2014

2 Publication of this book has been supported by the European Union and Hungary and co-financed by the European Social Fund through the project TÁMOP C-11/1/KONV National Research Center for Development and Market Introduction of Advanced Information and Communication Technologies. Published by the Faculty of Information Technology, University of Pannonia Egyetem u. 2., 8200 Veszprém, Hungary Telephone: Legally responsible publisher: Dr. Rozália Pigler-Lakner associate professor, dean Printed in format B5 by Tradeorg Kft. Responsible director: Zoltán Tóth ISBN (ASCONIKK 2014: Extended Abstracts. II. Future Internet Services, University of Pannonia)

3 TABLE OF CONTENTS Time Constrained Process-Network Synthesis: Solving Production Scheduling Problems... 5 Petri Net Based Trajectory Optimization Global Branching Tree for Throughput Maximization Process Simulation of Rectisol Process for Coal Based Syngas Cleaning Defining a Description Language for Automatic Processing of Business Processes in a Decision Support System Optimal Resource Cost for Batch Processes Multi-Objective Optimization of Batch Biodiesel Production Make-or-Buy Optimisation with Harmony Search Algorithm

4 Time Constrained Process-Network Synthesis: Solving Production Scheduling Problems Marton Frits, Botond Bertok Department of Computer Science and Systems Technology University of Pannonia Veszprém, Hungary {frits, Abstract. The aim of the process-network synthesis (PNS) problem is to determine the optimal solution of the process and the operation parameters of the building blocks. The Time Constrained Process-Network Synthesis (TCPNS) is an extension of the original framework to handle constraints specific to supply scenarios and improve the practical applicability of this methodology (Kalauz et al. 2012). This article presents how to formulate and solve process scheduling as TCPNS according to the significant storage policies. Multiple classes of classical scheduling problems are revisited and reformulated. Based on the recipe of scheduling problem, a superstructure can be generated algorithmically which involves each candidate schedule in a way that the operating units are regarded as resources and changeovers as potential steps of the process. The handling of the differences resulting from the storage policies is solved by this superstructure. The mathematical model constructed and analyzed, integrates the variables and constraints related to process synthesis, as well as, a precedence based formulation of the time constraints. The developed TCPNS solver can provide the optimal and the n-best solutions based on the superstructure and the mathematical model. Keywords: scheduling, process-network synthesis, P-graph 1 Introduction In case of a production process, the task s assignment to the available resources highly influence the efficiency of an organization. A well-defined assignment can yield in the required amount of products within a shorter process time. Therefore the production capacity can be increased and the production cost reduced. However, it is hard to the determine the optimal scheduling regarding he constraints given by either the technology or protocols due to the combinatorial nature of the problem. Thus, the industry requires computer aid, i.e., optimization software available. Such software should be able not only to generate an optimal assignment, but also to yield and evaluate alternative suboptimal or every feasible assignment, whenever it is computationally possible. 5

5 Considerable effort has been investigated in batch process scheduling in the past two decades and various approaches have been published to solve multiple classes of scheduling problems. Typically, the available solution methods for batch process scheduling consider a single storage policy. Only a limited number of methods can provide the optimal solution for mixed storage policy. Susurla et al. (2009) utilize unit-slots to formulate a continuous-time mixed integer linear programming (MILP) model for the short-term scheduling of multipurpose batch processes. Kopanos et al. (2009) introduced a new precedence-based MILP scheduling framework, based on a continuous-time representation for the scheduling in multi-stage batch plants. In this paper an algorithmic method for optimizing a resource assignment is presented, in terms of the process time, supported by software tools at each step. This method resorts to the graph-theoretic approach based on the P-graph framework. The method is demonstrated by applying it to a scheduling problem with mixed storage policy. 2 Problem Definition Generally, the aim of the scheduling problems is to assign available equipment units to the tasks of a process in the most favorable way. Batch production plants provide a variety of scheduling problems that can be classified on the basis of their parameters. A batch process is usually defined by a recipe that gives an arrangement of tasks to produce the required products. In general, the relations of tasks are described as a network. In contrast, the tasks of the process have sequential ordering in most case studies and literature examples. Even for sequential processes, two subclasses of recipes have to be differentiated, i.e., multiproduct and multipurpose recipes. In multiproduct processes every product has the same sequential recipe, while in the multipurpose case each product is produced by different sequences of the same production steps. The most commonly used objectives are the minimization of the overall processing time, called makespan; and the maximization of throughput or the profit for a given time horizon. The intermediate storage policy is an essential parameter for batch scheduling problems that can affect the problem s complexity as well as the optimal solution. Unlimited Intermediate Storage (UIS) policy expresses that the materials can be stored practically any amount without constraints. Contrarily, if the storage has limited capacity, it is known as Finite Intermediate Storage (FIS). In FIS case the intermediate materials have dedicated storage units, while Common Intermediate Storage (CIS) policy allows sharing storages among the equipment. According to Non-intermediate Storage (NIS) policy, storage equipment is absent, and the intermediates can be stored only in the processing units until they are transferred to another unit for further production. The most strict storage policy is called Zero Wait (ZW), when an intermediate material has to be further processed exactly at the moment when it is available. Mixed Intermediate Storage (MIS) policy represents the combinations of the above mentioned policies. 6

6 In the forthcoming sections a novel approach will be introduced, which formulates the scheduling problems as a process-network synthesis (PNS) problem, furthermore it allows the automatic model generation, handles the mixed intermediate storage policy cases, and gives general solution for the MIS problems while guarantees the optimality. 3 Methodology It has been shown that the P-graph approach to PNS, which originally conceived for conceptual design of chemical processes (Friedler et al., 1992;1996) provides adequate tools for generating and analyzing structural alternatives for supply scenarios (Barany et al.; 2010). The Time Constrained Process-Network Synthesis (TCPNS) is an extension of the original framework to handle constraints specific to supply scenarios and improve the practical applicability of this methodology (Kalauz et al. 2012). TCPNS handles time constraints on the availability of the resources. Duration of activities, and deadlines for the final target are incorporated into the mathematical model as well as into PNS algorithms. The duration of an activity is defined by a fixed and a proportional constant of the function, which estimate the duration of each activity based on its volume. 4 Formulating TCPNS Problem for a MIS Case Scheduling Problem By the method presented herein scheduling problems are solved as time constrained process-network synthesis problems. Several benefits of the P-graph framework are utilized, e.g., the mathematical model for solving the scheduling problems can be generated automatically, the solution method guaranties the optimality, and n-best solution can be provided. Furthermore, any mixture of UIS, NIS, and ZW storage policies can be handled. Moreover, there is no need to define time intervals. First, the superstructure is to be generated in order to formulate the TCPNS problem based on the recipe-graph of the scheduling problem and the potentially available resources. In this formulation the equipment units are resources to the process. The goal is to determine the process-network which ensures to reach each final target while taking into account the connections of the tasks. In this paper the model transformation will be introduced for NIS, and ZW storage policies as examples. Superstructure generation contains three main steps. At first, the structure has to map the basic activities of tasks, which are the following: allocating equipment to task, loading input to the equipment and performing the task. These activities with their intermediate states will create a branch in the P-graph for all the possible tasks and its related equipment units. In the next step, recipe edges are added to the graph to ensure the order of the tasks according to the recipe of the given product. As a result arcs arise in the network leading from an outcome of each preceding activity to each 7

7 forthcoming activity. Thus, the forthcoming task can only start if each of the preceding tasks has been completed. The third step depends on the storage policy. In case of non-intermediate storage policy the transfer operation can be performed only if the equipment unit of the next task is available. Meanwhile the material is assumed to be stored in the equipment unit. Consequently, the structure ensures that the equipment unit cannot start further task till the transfer is not finished. The third step of the structure generation is modified as follows. New states are added to the structure indicating that a certain task has been finished by one of the possible equipment units. These states become preconditions to each of the activities, which can potentially upload an equipment unit performing the forthcoming task. With these additional states the structure allows the next task to start only when the output of the previous task is transferred into other equipment unit. The last step has to ensure that the equipment unit can reset to its initial state after the execution of a task. For ZW storage policy, new constraints have to be added to the MILP model of TCPNS. The problem defines a Z set of ZW materials with such a restriction that transfer operation has to be started immediately after performing a task. Starting time of the equipment unit has to be extended to ZW materials. Starting time of any equipment units consuming a material z Z has to be equal to the earliest available time of material z. Furthermore, an upper bound has to be added into the earliest available time, which eliminates the gap between the availability time and the starting time of the activity. Applying this extended MILP model, scheduling problem with ZW materials can be solved without modification of the structure generator algorithms. 5 Case Study The aforementioned transformation method will present via a simple kitchen process, namely the processing of peach. There are potential products depending on the ripe: jam and compote. Figure 1 shows the steps and the recipe of the process. During the processing we aim at having the less unwashed, so we try to eliminate intermediate storages (dishes) at most steps. Figure denotes no intermediate storage with NIS label. Furthermore we assume that after cutting the peachs, they have to be dip in sugar water immediately in order to preserve their color. ZW label marks zero wait in the figure. NIS Stoning Press Cook Jam Peach Select NIS Cutting ZW Cook Compote NIS Sugar Water Figure 1. Recipe for the case study. 8

8 More steps of the process can be performed manually; however there is some kitchen equipment in order to work faster. The possible assignments of the tasks and eq Table 1. Processing times of tasks by different equipment units for the case study. Select Stoning Cutting Sugar Water Crushing Cooking Hand Processor Press 10 Jammaking Making Compote Dish# Dish# This is a MIS scheduling problem, for which P-graph in Figure 2 depicts the recipe. Figure 2. Recipe of the case study represented by P-graph. Figure 3 shows the optimal solution for the case study according to the processing times represented in Table 1. Hand Selection Processor Press Stoning Cutting Crushing Dish#1 Cooking Sugar Water Jam-making Dish#2 Making Compote Figure 3. Optimal solution for the case study. 9

9 6 Conclusion It has been illustrated that solving scheduling problems as time constrained processnetwork synthesis problem inherits several benefits of the P-graph framework. The mathematical model for solving the scheduling problems can be generated automatically, even with mixed intermediate storage policies, and there is no need to define time intervals. The solver for TCPNS guarantees the optimal solution and capable to provide the n-best suboptimal solution if needed. The proposed methodology can serve as computer aid for scheduling industrial production systems with mixed storage policies. References 1. Friedler F., Tarjan K., Huang Y.W., Fan L.T., 1992, Combinatorial Algorithms for Process Synthesis, Computers Chem. Engng.16, S Friedler F., Varga J.B., Feher E., Fan L.T., 1996,Combinatorially Accelerated Branch-and- Bound Method for Solving the MIP Model of Process Network Synthesis, Nonconvex Optimization and Its Applications, State of the Art in Global Optimization, Computational Meth.and App, Bertok B., Adonyi R., Friedler F., Fan L.T., 2011, Superstructure Approach to Batch Process Scheduling by S-graph Representation, Computer-Aided Chem.Engng 29, Bertok B., Barany M., Friedler F., 2013, Generating and Analyzing Mathematical Programming Models of Conceptual Process Design by P-graph Software, Ind.&Engng. Chem. Res 52(1), Kalauz K.,Sule Z., Bertok B., Friedler F., Fan L.T., 2012, Extending process-network synthesisalgorithms with time bounds for supply network design, Chem. Engng. Trans. 29, Gouws, J. and Majozi, T., 2009, Usage of inherent storage for minimization of wastewater in multipurpose batch plants, Chemical Engineering Science 64, Adonyi R., Romero J., Puigjaner L. and Friedler, F., 2003, Incorporating heat integration in batch process scheduling, Applied Thermal Engineering 23, Fan L.,Zhang F., Wang G., Liu Z.,2012, An effective approximation algorithm for the Malleable Parallel Task Scheduling problem, J. of Parallel and Distributed Comp. 72, Barany M., Bertok B., Kovacs Z., Friedler F., Fan L.T., 2010, Optimization software for solving vehicle assignment problems to minimize costs and environmental impacts of transportation, Chemical Engineering Transactions, 21, , DOI: /CET Barany M., Bertok B., Kovacs Z., Friedler F., Fan L.T, 2011, Solving vehicle assignment problems by process-network synthesis to minimize cost and environmental impact of transportation, Clean Technologies and Environmental Policy, 13(4), Susarla N., Li J., Karimi I.A, 2009, Unit Slots Based Short-Term Scheduling for Multipurpose Batch Plants, Computer Aided Chem. Eng., 27, Pan M, Qian Y., Li X, 2008, A novel precedence-based and heuristic approach for shortterm scheduling of multipurpose batch plants, Chem. Engineering Science, Kopanos G.M., Puigjaner L., 2009, A MILP Scheduling Model for Multi-stage Batch Plants, Computer Aided Chem. Eng.,

10 Petri Net Based Trajectory Optimization Ákos Hajdu 1,Róbert Német 1, Szilvia Varró-Gyapay 2, and András Vörös 1 1 Dept. of Measurement and Information Systems Budapest University of Technology and Economics, Budapest, Hungary vori@mit.bme.hu 2 University of Pannonia Department of Computer Science and Systems Technology H-8200 Egyetem u. 10., Veszprém, Hungary gyapay@dcs.uni-pannon.hu Abstract. Optimization problems are getting more prevalent in design of complex systems. Petri nets are widely used for modeling of such systems. An optimization problem can be translated to find an optimal trajectory where a cost is assigned to each step. The reachability problem of Petri nets answers whether a given state is reachable from the initial state. However, reachability analysis is a computationally hard problem, especially in case of asynchronous or infinite state systems. In this paper, we examine a recently published algorithm that solves reachability using abstraction methods and we extend this approach to be able to handle optimal trajectory problems. 1 Introduction Nowadays, information systems are getting more and more complex where modeling and automatic analysis techniques gain an increasing task. Petri nets are widely used for description of such systems due to their expressive power and simplicity. In addition cost parameters can be assigned to the transitions of the Petri net that enables modeling of optimization problems in discrete event systems. Such optimization problems can be formulated as optimal trajectory problems when an optimal path is searched for from a starting state to a target state. The so-called reachability problem is to answer the question whether a given state is reachable from an initial state in the system. There are many algorithms that solve or approximate the reachability problem of Petri nets. One of the most efficient is the so-called counterexample guided abstraction refinement (CE- GAR) algorithm, which takes the state equation of the Petri net as the initial abstraction and refines it with constraints gained from the state space exploration. As the reachability problem is at least EXPSPACE-hard, the algorithm sacrifices the completeness for efficiency. In this paper we introduce a new approach based on the CEGAR algorithm to solve the optimal trajectory problem. Our algorithm traverses the state space with advanced exploration methods and discovers the necessary invariants 11

11 to find the trajectory. The algorithm is extended to handle cost functions and the optimization problem drives the trajectory selection to find the optimal solution. 2 Background In this section, we introduce the background of our work. First, we present Petri nets (Section 2.1) as the modeling formalism used in our work based on [7]. Then we introduce the reachability problem, which serves as a basis for optimal trajectory search (Section 2.2). At the end of the section, we present linear programming briefly (Section 2.3). 2.1 Petri Nets Petri nets are graphical models for concurrent and asynchronous systems, providing both structural and dynamical analysis. A discrete Petri net is a tuple PN =(P, T, E, W), where P is the set of places, T is the set of transitions, with P T and P T =, E (P T ) (T P ) is the set of arcs and W : E Z + is the weight function assigning weights w (p j,t i )totheedge (p j,t i ) E and w + (p j,t i ) to the edge (t i,p j ) E. A marking of a Petri net is a mapping m : P N. If a place p contains k tokens in a marking m then m(p) =k. The initial marking is denoted by m 0. Dynamic Behavior. A transition t i T is enabled in a marking m, ifm(p j ) w (p j,t i ) holds for each p j P with (p j,t i ) E. An enabled transition t i can fire, consumingw (p j,t i ) tokens from places p j P with (p j,t i ) E and producing w + (p j,t i ) tokens on places p j P with (t i,p j ) E. The firing of a transition t i in a marking m is denoted by m[t i m where m is the marking after firing t i. A word σ = t 1 t 2...t n T is a firing sequence. A firing sequence is realizable in a marking m and leads to m (denoted by m[σ m ), if m[t 1...[t n m.the Parikh image of a firing sequence σ is a vector (σ) :T N, where (σ)(t i )is the number of the occurrences of t i in σ. 2.2 Reachability Problem. A marking m is reachable from m if a realizable firing sequence σ T exists, for which m[σ m holds. The set of all reachable markings from the initial marking m 0 of a Petri net PN is denoted by R(PN,m 0 ). The aim of the reachability problem is to check if m R(PN,m 0 ) holds for a given marking m.the reachability problem is decidable [6], but it is at least EXPSPACE-hard [5]. 12

12 State Equation. The incidence matrix of a Petri net is a matrix C P T, where C(i, j) =w + (p i,t j ) w (p i,t j ). Let m and m be markings of the Petri net, then the state equation takes the form m + Cx = m. Any vector x N T fulfilling the state equation is called a solution. Note that for any realizable firing sequence σ leading from m to m, the Parikh image of the firing sequence fulfills the equation m + C (σ) =m. On the other hand, not all solutions of the state equation are Parikh images of a realizable firing sequence. Therefore, the existence of a solution for the state equation is a necessary but not sufficient criterion for the reachability. A solution x is called realizable if a realizable firing sequence σ exists with (σ) =x. T-Invariants. A vector x N T is called a T-invariant if Cx = 0 holds. A realizable T-invariant represents the possibility of a cyclic behavior in the modeled system, since its complete occurrence does not change the marking. However, during firing the transitions of the T-invariant, some intermediate markings can be interesting for us. Solution Space. Each solution x of the state equation m + Cx = m,canbe written as the sum of a base vector and the linear combination of T-invariants [8], which can formally be written as x = b + i n iy i,whereb N T is the base vector and n i N is the coefficient of the T-invariant y i N T. 2.3 Linear Programming (LP) Linear programming is a mathematical approach for finding an optimal solution in a given mathematical model and requirements [2]. A linear programming problem is formalized as follows: minimize c T x, subject to Ax b and x 0, where x is the vector of variables, b, c are vectors and A is a matrix of coefficients. The linear programming problem can be solved in polynomial time. When all the variables of x are integers, the problem is called integer linear programming (ILP) problem, which is NP-hard. 3 CEGAR In this section, we present the concept of abstraction (Section 3.1) and a recently published algorithm that applies the CEGAR approach on the reachability problem of Petri nets (Section 3.2). Furthermore, we present some of our previous improvements at the end of the section that optimize and extend the algorithm. 13

13 3.1 Abstraction Abstraction is a general mathematical approach for solving hard problems. The abstract model has a less detailed state space representation by hiding the irrelevant details. However, due to abstraction, some action of the abstract model may not be realizable in the original model. In this case, abstraction has to be refined. This approach is called the counterexample guided abstraction refinement (CEGAR). 3.2 CEGAR Approach on Petri Nets Recently, a new algorithm was published, which applies the CEGAR approach on the reachability problem of Petri nets [8]. Figure 1 shows an overview of the algorithm and each step is detailed later in this section. Reachability problem Create initial abstraction State equation Constraints Solve the abstract model No solution Refine the abstraction Solution Stop Not realizable Examine the solution Realizable Fig. 1. Petri net CEGAR algorithm flowchart Initial Abstraction. The input of the algorithm is a reachability problem m R(PN,m 0 ), which is transformed into the initial abstraction, namely the state equation of the form m 0 + Cx = m. Solving the Abstract Model. Solving the abstract model (i.e. the state equation) is an integer linear programming problem. The ILP solver yields a minimal solution with respect to the cost function. In the original algorithm [8], the sum of the firing count of transitions is minimized in order to obtain trajectories with the shortest length. The feasibility of the state equation is a necessary, but not sufficient condition for reachability, therefore if no solution exists, the target marking is not reachable. Otherwise, the obtained solution must be checked whether it is realizable in the original model (i.e. in the Petri net PN). Examining the Solution. The solution of the state equation is a vector x N T,wherex(t) denotes the number of times a transition t T has to fire in order to reach m from m 0.However,x does not include any information 14

14 about the order of the transition firings and whether they are enabled. Thus, the algorithm must explore the state space of the Petri net with the limitation that each transition t can fire at most x(t) times. If the target marking m can be reached with this limit (i.e. x is realizable), it is a sufficient proof for reachability. Otherwise, x is a counterexample and the abstraction has to be refined. Refining the Abstraction. If a solution x is not realizable, the ILP solver has to be forced to generate a different solution. This can be done by adding additional constraints (i.e. linear inequalities over transitions) to the state equation. The following two types of constraints were defined in [8]. Jump constraints have the form t i < n,wheren N, t i T and t i represents the firing count of the transition t i. Jump constraints can be used to obtain different base vectors, exploiting their pairwise incomparability. Increment constraints have the form k i=1 n i t i n, wheren i Z, n N, and t i T. Increment constraints can be used to reach non-base solutions. This means that a new solution x + y is obtained, where y is a T-invariant. After adding the new constraint, the state equation may become infeasible, or a new solution is obtained. Figure 2 presents the solution space. The bottom dots represent base solutions, while the cones represent the linear space formed by the T-invariants. The upper dots correspond to non-base solutions. Jumps are denoted by dashed arrows and increments by continuous arrows. At each non-realizable solution, multiple jump and/or increment constraints can be applied. The algorithm traverses the solution space using depth-first search until a realizable solution is found, or the state equation becomes infeasible and there are no more solutions to backtrack too. Fig. 2. Solution space of the state equation Extensions. In our previous work [3], we proved by a counterexample that the original algorithm [8] is incorrect and we suggested a solution to overcome the problem. We also presented several examples where the algorithm could not 15

15 decide reachability. We extended the set of decidable problems, but the algorithm still lacks completeness. Furthermore, we introduced some new optimization methods in order to improve efficiency of the algorithm [3]. 4 Trajectory Optimization In this section, we introduce our new approach that solves the optimal trajectory problem based on the Petri net CEGAR algorithm (Section 4.1). We formulate our method using a pseudo code (Section 4.2) and we also present some measurement results (Section 4.3). 4.1 Trajectory Optimization Using CEGAR The core abstraction of the CEGAR algorithm is the state equation extended with further constraints, which forms an ILP problem. The ILP solver yields a solution minimizing a cost function on the variables. In the CEGAR approach, variables correspond to firing counts of transitions. Originally, the algorithm assigned every transition an equal cost in order to produce trajectories with the shortest length [8]. This behavior makes the CEGAR algorithm suitable for trajectory optimization purposes. In our new approach, we assign an arbitrary cost to transitions. Therefore, the ILP solver now minimizes the total cost of the trajectory (i.e. the sum of the costs of transitions) instead of its length. In order to fit this strategy into the CEGAR approach, the solution space traversal has to be modified slightly. The original algorithm explores the solution space using DFS search for a fast convergence to a realizable solution. However, we want an optimal solution regarding the cost function. Thus, we focus the search on solutions with minimal total costs. This is achieved by storing the not yet examined solutions in a sorted set, from which we always continue with the minimal one. Two examples can be seen in Figure 3 where the costs are written above the transitions. Consider the example in Figure 3(a) where we want to produce a token in p 2. The minimal solution for the state equation is to fire t 0 once and t 1 zero times (i.e. x 0 =(1, 0)). However, this solution is not realizable since t 0 is not enabled. By applying the jump constraint t 0 < 1, we obtain the solution x 1 =(0, 1) (i.e. firing t 1 once), which is realizable. Consider now the example in Figure 3(b) where we want to produce a token in p 0. The minimal solution for the state equation is to fire t 0 once (i.e. x 0 = (1, 0, 0)). However this solution is not realizable since t 0 is not enabled. By applying the increment constraint t 1 1, we can obtain the solution x 1 = (1, 1, 1), where the T-invariant {t 1,t 2 } borrows a token in p 1 to enable t Pseudo Code Algorithm 1 presents our new approach using pseudo code. The input of the algorithm is the reachability problem m RP (PN,m 0 ) and the cost z : T Z 16

16 2 1 p 1 t 1 t p 2 2 p 2 p 1 t 0 p 0 p 0 t 0 (a) Jump constraint example t 2 (b) Increment constraint example Fig. 3. Example nets for jump and increment constraints assigned to transitions. The algorithm stores the solutions in a list Q. Atfirstit tries to solve the ILP problem with no constraints (the fifth parameter being ). While there is a solution in the list that was not examined, the algorithm takes out the solution x that has a minimal total cost (zx). If the solution is realizable by some firing sequence σ, the algorithm terminates. Otherwise it searches for jump and increment constraints. An ILP problem is solved for each constraint c by adding c to the previous constraints of x. If new solutions are found they are put in the list Q. If all solutions were examined and no realizable is found, the answer is Not reachable. Algorithm 1: Trajectory optimizing CEGAR algorithm Input : Reachability problem m RP (PN,m 0) and transition costs z Output : Trajectory σ or Not reachable 1 C incidence matrix of PN; 2 Q ; // List of solutions 3 Q SolveILP(m 0,m,C,w, ); 4 while Q do 5 x {x x Q, zx is minimal}; 6 if x is realizable then stop and output σ for x; 7 else 8 foreach Jump and increment constraint c do 9 Q SolveILP(m 0,m,C,w,{constraints of x} {c }); 10 end 11 end 12 end 13 Output Not reachable ; 17

17 4.3 Results We implemented the CEGAR algorithm with our new approach as a plug-in for the PetriDotNet framework [1]. We evaluated our implementation on the traveling salesman problem. The results can be seen in Table 1. Table 1. Measurement results for the traveling salesman problem Number of nodes Runtime (s) 4 0,04 6 0,14 8 0,66 9 0, , , , Traveling salesman is a graph traversal optimization problem, which belongs to the class of NP-complete problems [4]. The parameter of the problem is the number of nodes in the graph. 5 Conclusion In our paper, we presented a promising new approach for the optimal trajectory problem of discrete event systems. We translated this problem to the reachability analysis of Petri nets by assigning a cost to the transitions. We solved the reachability problem using the recently published and very efficient CEGAR approach. We extended the algorithm to be able to handle transition costs and we modified the search strategy in order to reach the optimal solution. We implemented our new approach and demonstrated its efficiency with measurements. A possible future research direction is the optimization of continuous systems. References 1. Homepage of the PetriDotNet framework. [Online; accessed 04-Nov-2014]. 2. George B. Dantzig and Mukund N. Thapa. Linear programming 1: introduction. Springer-Verlag New York, Inc., Secaucus, NJ, USA, Ákos Hajdu, András Vörös, Bartha Tamás, and Zoltán Mártonka. Extensions to the CEGAR approach on Petri Nets. Acta Cybernetica, 21(3): , Richard M Karp. Reducibility among combinatorial problems. Springer, R.J. Lipton. The Reachability Problem Requires Exponential Space. Research report, Yale University, Dept. of Computer Science Ernst W. Mayr. An algorithm for the general Petri net reachability problem. In Proceedings of the Thirteenth Annual ACM Symposium on Theory of Computing, STOC 81, pages , New York, NY, USA, ACM. 18

18 7. Tadao Murata. Petri nets: Properties, analysis and applications. Proceedings of the IEEE, 77(4): , April Harro Wimmel and Karsten Wolf. Applying CEGAR to the Petri net state equation. In Tools and Algorithms for the Construction and Analysis of Systems, 17th International Conference, TACAS 2010 Proceedings, volume 6605, pages Springer,

19 Global Branching Tree for Throughput Maximization Tibor Holczinger and Akos Orosz University of Pannonia, Veszprém, Hungary Abstract. Majozi and Friedler [3] developed the first systematic approach for throughput maximization, that has no need for the optimal number of time points a-priori. The feasibility of a sequence of subproblems is examined with guided search driven S-graph based approach, where a subproblem is an instance of the scheduling problem with fixed batch numbers. Although, the approach has proved to be efficient for several problems with acceleration methods [1], it can be developed further to increase the size of addressable cases. In the original approach, a branching tree is generated for each subproblem separately, even though these trees can have significantly large overlaps. The presented approach exploits this inherent redundancy to reduce the computational needs for industrial scale problems. It examines all plausible batch number configurations in a single, so-called global branching tree without generating separate - and redundant - trees for each subproblem. The proposed approach is compared with former methods via several examples and case studies from the literature. 1 Problem Statement The problem addressed in this paper can be summarized as follows. Given, the recipe for each product, the potential assignment of tasks to equipment units, the relevant cost data and the time horizon of interest. The objective is to determine the schedule that yields the overall maximum throughput or revenue for all the products involved. In all the problems considered in this paper, there exists no intermediate storage (NIS) between consecutive tasks. However, the material can be temporarily stored within the corresponding equipment unit until the consecutive equipment unit become available for the next task in the recipe. 2 S-Graph Representation Although the detailed description of the mathematical formulation for the S- graph framework has been presented by Sanmartí et al. [4] it will also be given in sufficient details in this section, to facilitate understanding. An S-graph is a weighted directed graph which has two classes of arcs, the so-called recipe-arcs and schedule-arcs. Specific types of S-graphs are identified for a recipe, i.e. recipe-graph, and for a schedule of all tasks, i.e. schedule-graph. 20

20 In a recipe-graph one node is assigned to each task (task-node) and one to each product (product-node). A recipe-arc is established between the nodes of consecutive tasks, where the weight of the arc is specified by the processing times of the tasks. If more than one batch of products is to be produced, task-nodes, product-nodes, and arcs are multiplied appropriately. A schedule-graph describes a single solution of a scheduling problem. S-graph is called a schedule-graph if all tasks have been scheduled by taking equipmenttask assignments into account. By an appropriate search strategy, the schedulegraph of the optimal schedule can be effectively generated. There are two important advantages of the S-graph framework which can be used for throughput maximization. First, generating a feasible solution of a given number of batches of products under a fixed time horizon is effective. Moreover, the S-graph framework can provide results from the previous feasibility test to be readily exploited in the current test, thereby improving solution times. 3 Throughput Algorithm Using a Sequence of Feasibility Tests The base idea of solving throughput maximization problems with the S-graph framework has been published by Majozi and Friedler [3]. The algorithm uses feasibility tests in each step, which is a modified version of the S-graph based makespan minimization algorithm. The feasibility test only decides whether the problem can be solved in a given time horizon. The search space can be represented as a coordinate system with the number of dimensions equal to the number of products. For example in case of two products, the search space is shown in Fig. 1, where subproblems are denoted by points. From now, the subproblems will be denoted by tuples, where the elements of a tuple are the number of batches of the products (in alphabetic order). For example the subproblem, where two batches of product A and one batch of product B is to be produced, is denoted by (2, 1). Fig. 1. Search space representation for two products If a subproblem is infeasible, then a part of the search space can be excluded from further investigation, because a subproblem with equal or more batches 21

21 from each product cannot be manufactured in shorter time. For example, if subproblem (3, 5) is infeasible, then subproblems (3, 6), (3, 7),..., (4, 5), (4, 6),..., (5, 5), (5, 6),... are also infeasible. The final search space can be seen in Fig. 2, where the infeasible nodes are crossed and the vertical and horizontal lines denote the cuts of the search space. Fig. 2. Search space representation for two products, with infeasibility cuts 4 Global Branching Tree The previously presented approach performs redundant computations which can be eliminated from the search. In this section, the bases of a new optimization technique will be presented. 4.1 Disadvantages of Independent Feasibility Tests The original S-graph based throughput maximization method performs independent feasibility tests which technique has some disadvantages. For demonstration, let us introduce a small example. In the example, two products (A and B) are produced through two consecutive steps according to recipe given in Table 1. The aim is to optimize the total revenue for 8 hours. Table 1. Recipe of two products Task Product A Product B Eq. Time (h) Eq. Time (h) 1 E1 2 E2 3 2 E3 4 E1 2 Using the original method, several S-graphs (subproblems) have to be examined, in the first subproblem only one batch from product A is to be produced. 22

22 To check its feasibility, both tasks are to be scheduled which takes two steps. The branching tree are given in Fig.1 3. In the branching tree, the grey node represents the root (no task scheduled), the double lined node represents a feasible schedule (both nodes has been scheduled) and the remaining (white) node represents a partially scheduled subproblem (only one task has been scheduled). Fig. 3. Branching tree of the example given in Table 1 with one batch for product A To solve the example, the algorithm has to examine seven subproblems. Of course, the branching strategy of the feasibility test can greatly affect the number of branches, because finding a feasible schedule stops the search. The first row of Table 2 shows how many nodes are included in the branching tree if all nodes has to be examined (worst case). The second row shows the number of necessary branches in the best case. In case of infeasible subproblems, the algorithm examines the whole branching tree. Table 2. Number of branches for the subproblems Subproblem (A,B) (1, 0) (0, 1) (2, 0) (1, 1) (0, 2) (3, 0) (2, 1) # of branches (worst case) # of branches (best case) Let s examine the subproblems (2, 0) and (3, 0). Fig. 4 shows the branching trees of subproblems (2, 0) and (3, 0). The meaning of grey, white and double lined nodes are the same as in the previous branching tree. At the crossed nodes, the generated S-graph contains a directed cycle, which represents an infeasible schedule. In case of hatched nodes, the longest path of the S-graph is greater than the time horizon, i.e they are also infeasible. The branching tree of subproblem (2, 0) contains nine nodes, i.e. in worst case, nine S-graphs are to be generated and examined. In the best case, four branches are enough to get a feasible schedule, the five leftmost (or rightmost) nodes of its branching tree. Subproblem (3, 0) does not have any feasible schedule, so all fourteen nodes are to be generated. 23

23 Fig. 4. Branching trees of subproblems (2, 0) and (3, 0) It can be clearly seen, that these branching trees have a common part, especially the bigger one contains the other. This means, that the algorithm performs redundant calculations, because the feasibility tests are independent from subproblems. This redundancy can be eliminated by building a common tree, called global branching tree. 4.2 Optimization Strategy As shown in Fig. 5, all subproblems of the example can be combined into one large branching tree. The part of the tree labelled by (2, 0) denotes the branches of the search which are to be additionally generated and examined in case of subproblem (2, 0) according to subproblem (1, 0). It can be seen, that the total number of branches required to generate the branching tree is only 28, while with feasibility tests the algorithms needs at least 36 branches in the best case (47 branches on worst case). The feasibility of a subproblem depends on the feasibility of its ancestor subproblems. From this point we say a subproblem is a parent of an other subproblem if they differ only in one batch number, and in the child it is bigger by one. For example subproblem (1, 2) is a parent of (2, 2) and (1, 3), moreover its parents are (1, 1) and (0, 2). This definition allows that a subproblem can have multiple children and multiple parents, however in a tree each node must have exactly one parent, except the root. For each subproblem, the algorithm has to select one parent and the selection rule can greatly effect the effectiveness of the algorithm. This work illustrates only two possible parent selection rules. In case of branching tree of Fig. 5, the lexicographically first parent has been chosen as the parent for each subproblem. Choosing the lexicographically last 24

24 Fig. 5. Global branching tree of the example given in Table 1 parent for each subproblem results the branching tree seen in Fig. 6, which consists of 29 branching steps. The number of branching steps are slightly increased from 28 to 29, although the difference can be bigger for bigger problems. The parent selection strategy does not determine the algorithm fully. The order of the examination of subproblems has a huge role as well, because the infeasibility of a subproblem yields the infeasibility of all of its descendant subproblems. Considering the branching tree in Fig. 5, subproblem (2, 1) has feasible solutions, the algorithm does not examine subproblem (3, 1). Although, in the branching tree, subproblem (3, 1) is not a child of subproblem (3, 0), the definition of parent/child relation provides that the infeasibility of (3, 0) yields the infeasibility of (3, 1). To ensure this cut in the branching tree, the feasibility of subproblem (3, 0) must be examined beforehand. Also, for the same reason, there is no need to examine subproblem (2, 2) if the algorithm examines subproblem (2, 0) earlier and it is infeasible. Parent selection strategies are not presented in this work because of the page limit. 5 Illustrative Example In this section, a literature example is presented to demonstrate the performance of the proposed technique. The example is taken from [2]. The flowsheet for the literature example is shown in Fig. 7. The example involves a heater, a separator and two reactors. Each of the reactors can perform 3 reactions, i.e. reaction 1, 2 and 3. The reactions take 2, 2 and 1 hour, respectively. Heating takes 1 hour 25

25 Fig. 6. An other global branching tree of the example given in Table 1 and the separation is 2 hours. The process operates in a no intermediate storage (NIS) policy. The objective is to maximize revenue for products 1 and 2. The computational results are given in Table 3 for 14, 15 and 16 hours time horizon. The necessary number of branches and the CPU time are highly decreased using global branching tree. Table 3. Number of branches for the illustrative example Timehorizon Separate branching trees Global branching tree # of branches CPU time (s) # of branches CPU time (s) Conclusion In this work we have reduced the redundancy of the optimization strategy for throughput maximization presented by Majozi and Friedler [3]. To demonstrate 26

26 Fig. 7. Flowsheet for literature example the performance of the technique, a literature problem was used to compare to the original method. However, the efficiency of the global branching tree has been presented by examples, it has room for further accelerations. 7 Acknowledgement This publication/research has been supported by the European Union and Hungary and co-financed by the European Social Fund through the project TÁMOP C-11/1/KONV National Research Center for Development and Market Introduction of Advanced Information and Communication Technologies. References 1. Holczinger, T., Majozi, T., Hegyhati, M., Friedler, F., An automated algorithm for throughput maximization under fixed time horizon in multipurpose batch plants: S-graph approach, presented at ESCAPE 17, Bucharest, Romania, May 27-30, Ierapetritou, M. G., Floudas, C. A., Effective continuous-time formulation for shortterm scheduling. Part 1. Multipurpose batch processes, Ind. Eng. Chem. Res, 37: , Majozi, T., Friedler, F., Maximization of throughput in a multipurpose batch plant under fixed time horizon: S-graph approach, Ind. Eng. Chem. Res., 45, , Sanmartí, E., Holczinger, T., Puigjaner, L., Friedler, F., Combinatorial framework for effective scheduling of multipurpose batch plants, AIChE Journal, 48(11), ,

27 Process Simulation of Rectisol Process for Coal Based Syngas Cleaning Xia Liu 1, 2, Yu Qian 1, Siyu Yang 1, Petar Varbanov 2, Jiří Jaromír Klemeš 2 1 School of Chemical Engineering, South China University of Technology, Guangzhou, , P.R. China 2 Centre for Process Integration and Intensification CPI 2, Research Institute of Chemical and Process Engineering MŰKKI, Faculty of Information Technology, University of Pannonia, Egyetem utca 10, 8200 Veszprém, Hungary Abstract. Coal gasification system usually makes use of Rectisol technology for acid gas removal from syngas. In this paper, an industrial Rectisol process for coal based syngas has been investigated via simulation. The major goal was to identify the optimal thermodynamic model for Rectisol simulation. This contributes to the foundation for the process acid gas removal performance analysis. The results revealed that the thermodynamic model PC-SAFT is optimal for chilled methanol adsorption, the PSRK is suitable for CO 2 desorption of the process.h 2 S content in syngas is reduced to less than 1 ppm quickly in an absorber, and concentrated in a sour gas stream. The CO 2 concentration in the clean syngas decreases remarkably along with the increase of absorber operating pressure, or by changing the chilled methanol input and methanol temperature. H 2 S concentration in the sour gas is affected by the stripper gas input. The maximum concentration of H 2 S sour gas is 38.2%, on the condition of N 2 input is 395 kmol/ h. Keywords: Rectisol technology, process simulation, thermodynamic model, acid gas removal 1. Introduction Developed by Linde AG and Lurgi GmbH, Rectisol wash has been adopted as a typical acid gas removal (AGR) technology employed in coal gasification system [1]. The technology has been applied in about 75% of syngas processes globally [2]. The principle of Rectisol wash technology is that acid gases in syngas are highly soluble in chilled methanol. In this way CO 2 and H 2 S are therefore separated from the syngas [3]. After the cleaning process, the concentration of H 2 S in the purified syngas is reduced down to 0.1 ppm, while that of CO 2 is down to a specific level according to synthesis requirements of downstream chemicals [4].The CO 2 removal ratio of the crude syngas can exceed 99% [5].Major industrial application clients include American North West Upgrading Inc. [6], Conoco Phlillops Co. [7], and China Shenhua Coal-oil Chemical Industry Co. [8]. 28

28 As a physical absorption process, Rectisol is most efficient and economical when operated at higher pressure and lower temperature. Such values for the operating pressure and temperature tend to cause less accurate process simulation [9]. Therefore, the thermodynamic property estimation method and the process simulation model need to be verified on practical process data. In this paper, an industrial Rectisol process for a 200,000t/y methanol production plant of Shanghai Coking & Chemical Corporation [9], Shanghai, China, is modelled and simulated to verify thermodynamic property estimation method [10].The major tool for this study is simulation on the absorption and desorption unit of process, to get the optimal thermodynamic method. Based on the simulation results, the acid gas removal capacity is evaluated in terms of acid gas removal ability, H 2 S gas concentration. 2. The Process Description The flowsheet of an industrial Rectisol process for coal based syngas purification [10] is shown in Fig. 1.Chilled methanol counter-currently contacts the crude syngas inside the acid gas absorber column. The regeneration of the rich methanol operates in three ways: flash at the reduced pressure, stripping by N 2, and distillation. Clean syngas Chilled methanol H2S Refrigerator Tail gas Refrigerator Acid gas absorber CO2-rich Methanol H2S concentration column Cooling water Cold product gas H2S- rich Methanol MP Flash Crude syngas MP Flash LP Flash Regenerator Warm product gas Steam Methnaol N2 Water splitter Steam Fig. 1. Flow chart of the Rectisol process for coal based syngas purification [10] The separation units used in the proposed process are: 1) Absorber uses chilled methanol to absorb H 2 S, COS and CO 2. 2) MP Flash Drum recovers H 2 and CO which are then recycled and added into the syngas stream. H2O 29