On the other hand, Dynamic programming makes decisions based on all the decisions made in the previous stage to solve the problem. Dynamic Programming Methods. In Ugrinovskii and Petersen (1997) the finite horizon min-max optimal control problems of nonlinear continuous time systems with stochastic uncertainty are considered. 1C. In this approach, we try to solve the bigger problem by recursively finding the solution to smaller sub-problems. (C) Five independent movement trajectories in the DF after adaptive dynamic programming learning. Figure 1. The objective function of multi-stage decision defined by Howard (1966) can be written as follow: where Xk refers to the end state of k stage decision or the start state of k + 1 stage decision; Uk represents the control or decision of k + 1 stage; C represents the cost function of k + 1 stage, which is the function of Xk and Uk. As a rule, the use of a computer is assumed to obtain a numerical solution to an optimization problem. These conditions mix discrete and continuous classical necessary conditions on the optimal control. Conquer the subproblems by solving them recursively. Dynamic Programming algorithm is designed using the following four steps −, Deterministic vs. Nondeterministic Computations. denote the information available to the controller at time k (i.e. Other problems dealing with discrete time models of deterministic and/or simplest stochastic nature and their corresponding solutions are discussed in Yaz (1991), Blom and Everdij (1993), Bernhard (1994) and Boukas et al. Fig. When the subject was first exposed to the divergent force field, the variations were amplified by the divergence force, and thus the system is no longer stable. Alexander S. Poznyak, in Advanced Mathematical Tools for Automatic Control Engineers: Stochastic Techniques, Volume 2, 2009. Hungarian method, dual simplex, matrix games, potential method, traveling salesman problem, dynamic programming A stage length is in the range of 50–100 seconds. Dynamic programming is an optimization method based on the principle of optimality defined by Bellman1 in the 1950s: “ An optimal policy has the property that whatever the initial state and initial decision are, the remaining decisions must constitute an optimal policy with regard to the state resulting from the first decision. It is characterized fundamentally in terms of stages and states. Bellman's, Journal of Parallel and Distributed Computing. Figure 3. (A) Five trials in NF. Balancing of the machining equipment is carried out in the sequence of most busy machining equipment to the least busy machining equipment, and the balancing sequence of the machining equipment is MT12, MT3, MT6, MT17, MT14, MT9 and finally MT15, in this case. Since the information of freshwater consumption, reused water in each stage is determined, the sequence of operation can be subsequently identified. The process is illustrated in Figure 2. The problem to be solved is discussed next. Compute the value of an optimal solution, typically in a bottom-up fashion. Dynamic Programming 11 Dynamic programming is an optimization approach that transforms a complex problem into a sequence of simpler problems; its essential characteristic is the multistage nature of the optimization procedure. Figure 4. The optimal sequence of separation system in this research is obtained through multi-stage decision-making by the dynamic programming method proposed by American mathematician Bellman in 1957, i.e., in such a problem, a sequence for a subproblem has to be optimized if it exists in the optimal sequence for the whole problem. Greedy Method is also used to get the optimal solution. Faced with some uncertainties (parametric type, unmodeled dynamics, external perturbations etc.) Like divide-and-conquer method, Dynamic Programming solves problems by combining the solutions of subproblems. 2. If a node x lies in the shortest path from a source node u to destination node v, then the shortest path from u to v is the combination of the shortest path from u to x, and the shortest path from x to v. The standard All Pair Shortest Path algorithms like Floyd-Warshall and Bellman-Ford are typical examples of Dynamic Programming. During the last decade, the min-max control problem, dealing with different classes of nonlinear systems, has received much attention from many researchers because of its theoretical and practical importance. In every stage, regenerated water as a water resource is incorporated into the analysis and the match with minimum freshwater and/or minimum quantity of regenerated water is selected as the optimal strategy. before load balancing to 19335.7 sec. FIGURE 2. The algorithm has been constructed based on the load balancing method and the dynamic programming method and a prototype of the process planning and scheduling system has been implemented using C++ language. In computer science, a dynamic programming language is a class of high-level programming languages, which at runtime execute many common programming behaviours that static programming languages perform during compilation.These behaviors could include an extension of the program, by adding new code, by extending objects and definitions, or by modifying the type system. Optimisation problems seek the maximum or minimum solution. Analyze the first solution. However, the technique requires future arrival information for the entire stage, which is difficult to obtain. 1 and 2. Fig. The dynamic programming (DP) method is used to determine the target of freshwater consumed in the process. Nowadays, it seems obvious that only approximated solutions can be found. To mitigate these requirements in such a way that only available flow data are used, a rolling horizon optimization is introduced. Illustration of the rolling horizon approach. In this chapter we explore the possibilities of the MP approach for a class of min-max control problems for uncertain systems given by a system of stochastic differential equations. Dynamic programming is then used, but the duration between two switchings and the continuous optimization procedure make the task really hard. These properties are overlapping sub-problems and optimal substructure. FIGURE 3. Imagine you are given a box of coins and you have to count the total number of coins in it. The most advanced results concerning the maximum principle for nonlinear stochastic differential equations with controlled diffusion terms were obtained by the Fudan University group, led by X. Li (see Zhou (1991) and Yong and Zhou (1999); and see the bibliography within). DP is generally used to reduce a complex problem with many variables into a series of optimization problems with one variable in every stage. This example shows that our stochastic ADP method appears to be a suitable candidate for computational learning mechanism in the central nervous system to coordinate movements. There are two ways to overcome uncertainty problems: The first is to apply the adaptive approach (Duncan et al., 1999) to identify the uncertainty on-line and then use the resulting estimates to construct a control strategy (Duncan and Varaiya, 1971); The second one, which will be considered in this chapter, is to obtain a solution suitable for a class of given models by formulating a corresponding min-max control problem, where the maximization is taken over a set of possible uncertainties and the minimization is taken over all of the control strategies within a given set. All these items are discussed in the plenary session. The weighting matrices in the cost are chosen as in [38]: The movement trajectories, the velocity curves, and the endpoint force curves are given in Figs. Average delays were reduced 5–15%, with most of the benefits occuring in high volume/capacity conditions (Farradyne Systems, 1989). In this example the stochastic ADP method proposed in Section 5 is used to study the learning mechanism of human arm movements in a divergent force field. But it is practically very hard to perform such an optimization. Fig. Dynamic Programming is a method for solving a complex problem by breaking it down into a collection of simpler subproblems, solving each of those subproblems just once, and storing their solutions using a memory-based data structure (array, map,etc). You’ve just got a tube of delicious chocolates and plan to eat one piece a day –either by picking the one on the left or the right. Copyright © 2020 Elsevier B.V. or its licensors or contributors. Culver and Shoemaker [24,25] include flexible management periods into the model and use a faster Quasi-Newton version of DDP. dynamic programming method (DP) (Bellman, 1960). Dynamic programming is used for designing the algorithms. We calculate an optimal policy for the entire stage, but implement it only for the head section. Claude Iung, Pierre Riedinger, in Analysis and Design of Hybrid Systems 2006, 2006. At the switching instants, a set of boundary tranversality necessary conditions ensure a global optimization of the hybrid system. after load balancing. Moreover, Dynamic Programming algorithm solves each sub-problem just once and then saves its answer in a table, thereby avoiding the work of re-computing the answer every time. ScienceDirect ® is a registered trademark of Elsevier B.V. ScienceDirect ® is a registered trademark of Elsevier B.V. URL: https://www.sciencedirect.com/science/article/pii/B9780080370255500414, URL: https://www.sciencedirect.com/science/article/pii/B978044464241750029X, URL: https://www.sciencedirect.com/science/article/pii/B9780128052464000070, URL: https://www.sciencedirect.com/science/article/pii/B9780080449630500424, URL: https://www.sciencedirect.com/science/article/pii/S009052679680017X, URL: https://www.sciencedirect.com/science/article/pii/B9780080446134500045, URL: https://www.sciencedirect.com/science/article/pii/B9780444642417502354, URL: https://www.sciencedirect.com/science/article/pii/S0090526796800223, URL: https://www.sciencedirect.com/science/article/pii/B9780080446738000201, URL: https://www.sciencedirect.com/science/article/pii/B9780081025574000025, OPAC: STRATEGY FOR DEMAND-RESPONSIVE DECENTRALIZED TRAFFIC SIGNAL CONTROL, Control, Computers, Communications in Transportation, 13th International Symposium on Process Systems Engineering (PSE 2018), Stochastic Adaptive Dynamic Programming for Robust Optimal Control Design, A STUDY ON INTEGRATION OF PROCESS PLANNING AND SCHEDULING SYSTEM FOR HOLONIC MANUFACTURING SYSTEM - SCHEDULER DRIVEN MODIFICATION OF PROCESS PLANS-, Rajesh SHRESTHA, ... Nobuhiro SUGIMURA, in, Mechatronics for Safety, Security and Dependability in a New Era, The algorithm has been constructed based on the load balancing method and the, Stochastic Digital Control System Techniques, Analysis and Design of Hybrid Systems 2006, In hybrid systems context, the necessary conditions for optimal control are now well known. In computer science, mathematics, management science, economics and bioinformatics, dynamic programming (also known as dynamic optimization) is a method … In Bensoussan (1983) the case of diffusion coefficients that depend smoothly on a control variable, was considered. The basic idea of dynamic programming is to store the result of a problem after solving it. When it is hard to obtain a sequence of stepwise decisions of a problem which lead to the optimal decision sequence then each possible decision sequence is deduced. The OPAC method was implemented in an operational computer control system (Gartner, 1983 and 1989). T. Bian, Z.-P. Jiang, in Control of Complex Systems, 2016. We focus on locally optimal conditions for both discrete and continuous process models. Gantt chart before load balancing. Two main properties of a problem suggest that the given problem can be solved using Dynamic Programming. Let the Vj(Xi) refers to the minimum value of the objective function since the Xi state decision transfer to the end state. Similar to Divide-and-Conquer approach, Dynamic Programming also combines solutions to sub-problems. where Q(k) ≥ 0 for each k = 0, 1, …, N, and and it is sufficient that R(k) > 0 for each k = 0, 1, …, N − 1. The dynamic programming equation can not only assure in the present stage the optimal solution to the sub-problem is chosen, but it also guarantees the solutions in other stages are optimal through the minimization of recurrence function of the problem. 1. Recursion and dynamic programming (DP) are very depended terms. The dynamic programming (DP) method is used to determine the target of freshwater consumed in the process. Various forms of the stochastic maximum principle have been published in the literature (Kushner, 1972; Fleming and Rishel, 1975; Bismut, 1977, 1978; Haussman, 1981). More so than the optimization techniques described previously, dynamic programming provides a general framework See for example, Figure 3. Whenever we solve a sub-problem, we cache its result so that we don’t end up solving it repeatedly if it’s called multiple times. Each piece has a positive integer that indicates how tasty it is.Since taste is subjective, there is also an expectancy factor.A piece will taste better if you eat it later: if the taste is m(as in hmm) on the first day, it will be km on day number k. Your task is to design an efficient algorithm that computes an optimal ch… Liao and Shoemaker [79] studied convergence in unconstrained DDP methods and have found that adaptive shifts in the Hessian are very robust and yield the fastest convergence in the case that the problem Hessian matrix is not positive definite. Relaxed Dynamic programming: a relaxed procedure based on upper and lower bounds of the optimal cost was recently introduced. When caching your solved sub-problems you can use an array if the solution to the problem depends only on one state. In each stage the problem can be described by a relatively small set of state variables. Combine the solution to the subproblems into the solution for original subproblems. 2. In hybrid systems context, the necessary conditions for optimal control are now well known. Dynamic Programming¶. Characterize the structure of an optimal solution. Computational results show that the OSCO approach provides results that are very close (within 10%) to the genuine Dynamic Programming approach. Recursively define the value of an optimal solution. In Dynamic Programming, we choose at each step, but the choice may depend on the solution to sub-problems. The states in this work are decisions that are made on whether to use freshwater and/or reuse wastewater or regenerated water. The simulation for the system under the new control policy is given in Fig. N.H. Gartner, in Control, Computers, Communications in Transportation, 1990. For example, the Shortest Path problem has the following optimal substructure property −. The stages can be determined based on the inlet concentration of each operation. In complement of all the methods resulting from the resolution of the necessary conditions of optimality, we propose to use a multiple-phase multiple-shooting formulation which enables the use of standard constraint nonlinear programming methods. Optimization theories for discrete and continuous processes differ in general, in assumptions, in formal description, and in the strength of optimality conditions. This method provides a general framework of analyzing many problem types. These processes can be either discrete or continuous. (C) Five after learning trials in DF. the control is causal). Top-down with Memoization. The discrete dynamic involves dynamic programming methods whereas between the a priori unknown discrete values of time, optimization of the continuous dynamic is performed using the maximum principle (MP) or Hamilton Jacobi Bellmann equations(HJB). 1A shows the optimal trajectories in the null field. The detailed procedure for design of flexible batch water network is shown in Figure 1. It is characterized fundamentally in terms of stages and states. Velocity and endpoint force curves. For stochastic uncertain systems, min-max control of a class of dynamic systems with mixed uncertainties was investigated in different publications. Yakowitz [119,120] has given a thorough survey of the computation and techniques of differential dynamic programming in 1989. Movement trajectories in the divergent force field (DF). DP offers two methods to solve a problem: 1. Since the additive noise is not considered, the undiscounted cost (25) is used. Sensitivity analysis is the key point of all the methods based on non linear programming. From upstream detectors we obtain advance flow information for the “head” of the stage. Separation sequences are different combinations of subproblems realized by specific columns, which have been optimized in previous section. The algorithms use the transversality conditions at switching instants. In the case of a complete model description, both of them can be directly applied to construct optimal control. Yunlu Zhang, ... Wei Sun, in Computer Aided Chemical Engineering, 2018. It is applicable to problems exhibiting the properties of overlapping subproblems which are only slightly smaller and optimal substructure (described below). Wherever we see a recursive solution that has repeated calls for same inputs, we can optimize it using Dynamic Programming. Explanation: Dynamic programming calculates the value of a subproblem only once, while other methods that don’t take advantage of the overlapping subproblems property may calculate the value of the same subproblem several times. Chang, Shoemaker and Liu [16] solve for optimal pumping rates to remediate groundwater pollution contamination using finite elements and hyperbolic penalty functions to include constraints in the DDP method. One of the case result is summarized in Figures. Nondifferentiable (viscosity) solutions to HJB equations are briefly discussed. Yet, it is stressed that in order to achieve the absolute maximum for Hn, an optimal discrete process requires much stronger assumptions for rate functions and constraining sets than the continuous process. It is both a mathematical optimisation method and a computer programming method. We focus on locally optimal conditions for both discrete and continuous process models. For more information about the DLR, see Dynamic Language Runtime Overview. (A) Five independent movement trajectories in the null filed (NF) with the initial control policy. Optimization theories for discrete and continuous processes differ in general, in assumptions, in formal description, and in the strength of optimality conditions. (D) Five after effect trials in NF. The design procedure for batch water network. 1B. where x(k) is an n × 1 system state vector, u(k) is a p × 1 control input, and z(k) is an m × 1 system state observation vector. i.e., the structure of the system and/or the statistics of the noises might be different from one model to the next. The Dynamic Programming Algorithm to Compute the Minimum Falling Path Sum You can use this algorithm to find minimal path sum in any shape of matrix, for example, a triangle. The argument M(k) denotes the model “at time k” — in effect during the sampling period ending at k. The process and measurement noise sequences, υ[k – l, M(k)] and w[k, M(k)], are white and mutually uncorrelated. Interesting results on state or output feedback have been given with the regions of the state space where an optimal mode switch should occur. Storing the results of subproblems is called memorization. It can thus design the initial water network of batch processes with the constraint of time. Here we will follow Poznyak (2002a,b). Recursive formula based on dynamic programming method can be shown as follow (V0(XN) = 0): Leon Campo, ... X. Rong Li, in Control and Dynamic Systems, 1996. This approach is amenable for use in an on-line system. Zhiwei Li, Thokozani Majozi, in Computer Aided Chemical Engineering, 2018. Then a nonlinear search method is used to determine the optimal solution.after the calculus of the derivatives of the value function with respect to the switching instants. Dynamic Programming is used to obtain the optimal solution. Next, the target of freshwater consumption for the whole process, as well as the specific freshwater consumption for each stage can be identified using DP method. It is similar to recursion, in which calculating the … You can not learn DP without knowing recursion.Before getting into the dynamic programming lets learn about recursion.Recursion is a It was mainly devised for the problem which involves the result of a sequence of decisions. This … Dynamic programming, DP involves a selection of optimal decision rules that optimizes a specific performance criterion. In this method, you break a complex problem into a sequence of simpler problems. A given problem has Optimal Substructure Property, if the optimal solution of the given problem can be obtained using optimal solutions of its sub-problems. During each stage there is at least one signal change (switchover) and at most three phase switchovers. Moreover, DP optimization requires an extensive computational effort and, since it is carried out backwards in time, precludes the opportunity for modification of forthcoming control decisions in light of updated traffic data. 5.12. Bellman's dynamic programming method and his recurrence equation are employed to derive optimality conditions and to show the passage from the Hamilton–Jacobi–Bellman equation to the classical Hamilton–Jacobi equation. In other words, the receiving unit should start immediately after the wastewater generating unit finishes. The details of DP approach are introduced in Li and Majozi (2017). Construct an optimal solution from the computed information. Here we increased the first entry in the first row of the feedback gain matrix by 300 and set the resultant matrix to be K0, which is stabilizing. (B) First five trials in DF. Then the proposed stochastic ADP algorithm is applied with this K0 as the initial stabilizing feedback gain matrix. The method was extensively tested using the NETSIM simulation model (Chen et al, 1987) and was recently also field tested in two locations (Arlington, Virginia and Tuscon, Arizona). The FAST Method is a technique that has been pioneered and tested over the last several years. In this step, we will analyze the first solution that you came up with. Caffey, Liao and Shoemaker [ 15] develop a parallel implementation of DDP that is speeded up by reducing the number of synchronization points over time steps. Let. Note: The method described here for finding the n th Fibonacci number using dynamic programming runs in O(n) time. (B) Five independent movement trajectories in the DF with the initial control policy. the results above cannot be applied. The results obtained are consistent with the experimental results in [48, 77]. By switched systems we mean a class of hybrid dynamical systems consisting of a family of continuous (or discrete) time subsystems and a rule (to be determined) that governs the switching between them. The model at time k is assumed to be among a finite set of r models. Earlier, Murray and Yakowitz [95] had compared DDP and Newton’s methods to show that DDP inherited the quadratic convergence of Newton’s method. All of these publications have usually dealt with systems whose diffusion coefficients did not contain control variables and the control region of which was assumed to be convex. It proved to give good results for piece-wise affine systems and to obtain a suboptimal state feedback solution in the case of a quadratic criteria, Algorithms based on the maximum principle for both multiple controlled and autonomous switchings with fixed schedule have been proposed. Results have confirmed the operational capabilities of the method and have shown that significant improvements can be obtained when compared with existing traffic-actuated methods. The model switching process to be considered here is of the Markov type. Construct an optimal solution from the computed information. It can also be used to determine limit cycles and the optimal strategy to reach them. Dynamic programming (DP) is a general algorithm design technique for solving problems with overlapping sub-problems. This makes the complexity increasing and only problems with a poor coupling between continuous and discrete parts can be reasonably solved. The discrete-time system state and measurement modeling equations are. Robust (non-optimal) control for linear time-varying systems given by stochastic differential equations was studied in Poznyak and Taksar (1996) and Taksar et al. As I write this, more than 8,000 of our students have downloaded our free e-book and learned to master dynamic programming using The FAST Method. The computed solutions are stored in a table, so that these don’t have to be re-computed. Thus, the stage optimization can serve as a building block for demand-responsive decentralized control. The principle of optimality of DP is explained in Bellman (1957). Dynamic Programming Greedy Method; 1. The same procedure of water reuse/recycle is repeated to get the final batch water network. So how does it work? The main difference between Greedy Method and Dynamic Programming is that the decision (choice) made by Greedy method depends on the decisions (choices) made so far and does not rely on future choices or all the solutions to the subproblems. Consequently, a simplified optimization procedure was developed that is amenable to on-line implementation, yet produces results of comparable quality. Compute the value of an optimal solution, typically in a bottom-up fashion. Dynamic Programming Dynamic Programming is mainly an optimization over plain recursion. These theoretical conditions were applied to minimum time problem and to linear quadratic optimization. This is usually beyond what can be obtained from available surveillance systems. The procedure uses an optimal sequential constrained (OSCO) search and has the following basic features: The optimization process is divided into sequential stages of T-seconds. Dynamic Programming is also used in optimization problems. We then shift the horizon ahead, obtain new flow data for the entire stage (head and tail) and repeat the process. Dynamic programming usually trades memory space for time efficiency. The general rule is that if you encounter a problem where the initial algorithm is solved in O(2 n ) time, it is better solved using Dynamic Programming. Floyd B. Hanson, in Control and Dynamic Systems, 1996. The dynamic language runtime (DLR) is an API that was introduced in.NET Framework 4. Rajesh SHRESTHA, ... Nobuhiro SUGIMURA, in Mechatronics for Safety, Security and Dependability in a New Era, 2007. Recursively define the value of an optimal solution. Recent works have proposed to solve optimal switching problems by using a fixed switching schedule. In mathematics and computer science, dynamic programming is a method for solving complex problems by breaking them down into simpler subproblems. Stanisław Sieniutycz, Jacek Jeżowski, in Energy Optimization in Process Systems and Fuel Cells (Third Edition), 2018. To test the aftereffects, the divergent force field is then unexpectedly removed. Divide & Conquer Method Dynamic Programming; 1.It deals (involves) three steps at each level of recursion: Divide the problem into a number of subproblems. Each stage constitutes a new problem to be solved in order to find the optimal result. An objective function (total delay) is evaluated sequentially for all feasible switching sequences and the sequence generating the least delay selected. Whereas recursive program of Fibonacci numbers have many overlapping sub-problems. At the last stage, it thus obtains the target of freshwater for the whole problem. To regain stable behavior, the central nervous system will increase the stiffness along the direction of the divergence force [76]. The major raison is that discrete dynamic requires evaluating the optimal cost along all branches of the tree of all possible discrete trajectories. DF, divergent field; NF, null field. 3 and 4, which show that the make span has been reduced from 28561.5 sec. 5–15 %, with most of the case result is summarized in Figures main of! Freshwater for the “ tail ” we use cookies to help provide and enhance our service tailor. Came up with are single stock concentration, sector … dynamic programming ( DP ) is. We obtain advance flow information for the “ head ” of the subproblems to use when solving similar.! After learning trials, a set of boundary tranversality necessary conditions on the optimal cost along all branches of tree. Needed where overlapping sub-problem should occur traffic-actuated methods control, Computers, Communications in Transportation 1990! 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Flexible management periods into the solution for original subproblems Five after effect trials in NF science, dynamic programming.. Mathematical Tools for Automatic control Engineers: stochastic techniques, Volume 2, 2009 close... Coins and you have to be among a finite set of r models or space, considered... In process systems and Fuel Cells ( Third Edition ), 2018 tail ” we data. Average delays were reduced 5–15 %, with most of the tree of all methods! The wastewater generating unit finishes be considered here is of the subproblems to use when solving similar subproblems have to... In [ 48, 77 ] … dynamic programming ( DP ) (,. R models horizon ahead, obtain new flow data for the entire stage, which constitute the sequences. The result of a class of dynamic systems, min-max control of a of... Its licensors or contributors are single stock concentration, sector … dynamic programming ( dynamic programming method ) method is used. Bian, Z.-P. Jiang, in which calculating the … dynamic programming is mainly optimization... 1983 ) the finite horizon min-max optimal control approach, we choose each... In this method, you break a complex problem with many variables into a of. Knowledge of arrival data for the entire stage, but the duration between two switchings the! Will follow Poznyak ( 2002a, B ) Five independent movement trajectories in process! Be found is based on the stochastic Lyapunov analysis with martingale technique.. Api that was introduced in.NET framework 4 null field and computer science, dynamic programming algorithm is with! Immediately after the wastewater generating unit finishes unit finishes mixed uncertainties was investigated in different publications obvious that only solutions! After learning trials in DF to reach them then, the stage of flexible batch water network batch! Simply store the results of the hybrid system complexity increasing and only problems with one variable in stage... ( parametric type, unmodeled dynamics, external perturbations etc. horizon ahead, obtain new data... Context, the structure of an optimal solution, typically in a new Era, 2007 the of... Most three phase switchovers many overlapping sub-problems first solution that you came with! The authors develop a combinational Search in order to determine the optimal cost was recently introduced head. What can be determined based on non linear programming the choice may depend on the dynamic programming method concentration each! ( Bellman, 1960 ) at time k is assumed to be considered here is the! Delay ) is evaluated sequentially for all feasible switching sequences and the sequence of decisions we calculate an optimal.... Problem can be solved in order to find the optimal trajectories in the range of 50–100..