Browsing by Subject "Optimal control"

Now showing 1 - 13 of 13

Open Access
Constrained optimal hybrid control of a flow shop system
(Institute of Electrical and Electronics Engineers, 2007) Gokbayrak, K.; Selvi, O.
We consider an optimal control problem for the hybrid model of a deterministic flow shop system, in which the jobs are processed in the order they arrive at the system. The problem is decomposed into a higher-level discrete-event system control problem of determining the optimal service times, and a set of lower-level classical control problems of determining the optimal control inputs for given service times. We focus on the higher-level problem which is nonconvex and nondifferentiable. The arrival times are known and the decision variables are the service times that are controllable within constraints. We present an equivalent convex optimization problem with linear constraints. Under some cost assumptions, we show that no waiting is observed on the optimal sample path. This property allows us to simplify the convex optimization problem by eliminating variables and constraints. We also prove, under an additional strict convexity assumption, the uniqueness of the optimal solution and propose two algorithms to decompose the simplified convex optimization problem into a set of smaller convex optimization problems. The effects of the simplification and the decomposition on the solution times are shown on an example problem.
Open Access
A Differential game model of opinion dynamics: Accord and discord as nash equilibria
(Birkhauser, 2020-03) Niazi, Muhammad Umar B.; Özgüler, A. Bülent
This paper presents a noncooperative differential (dynamic) game model of opinion dynamics with open-loop information structure. In this game, the agents’ motives are shaped by their expectations of the nature of others’ opinions as well as how susceptible they are to get influenced by others, how stubborn they are, and how quick they are willing to change their opinions on a set of issues in a prescribed time interval. These motives are independently formed by all agents. The existence of a Nash equilibrium in the network means that a collective behavior emerges out of local interaction rules and these individual motives. We prove that a unique Nash equilibrium may exist in the game under quite different circumstances. It may exist not only if there is a harmony of perceptions among the agents of the network, but also when agents have different views about the correlation among issues. The first leads to an accord in the network usually expressed as a partial consensus, and the second to a discord in the form of oscillating opinions. In the case of an accord, the harmony in the network may be in the form of similarity in pairwise conceptions about the issues but may also be an agreement on the status of a “leader” in the network. A Nash equilibrium may fail to exist only if the network is in a state of discord.
Open Access
Dynamic bidding strategies in search-based advertising
(Springer, 2013) Dayanik, S.; Parlar, M.
Search-based advertising allows the advertisers to run special campaigns targeted to different groups of potential consumers at low costs. Google, Yahoo and Microsoft advertising programs allow the advertisers to bid for an ad position on the result page of a user's query when the user searches for a keyword that the advertiser relates to its products or services. The expected revenue generated by the ad depends on the ad position, and the ad positions of the advertisers are concurrently determined after an instantaneous auction based on the bids of the advertisers. The advertisers are charged only when their ads are clicked by the users. To avoid excessive ad expenditures due to sudden surges in the keyword-search activities, each advertiser reserves a fixed finite daily budget, and the ads are not shown in the remainder of the day when the budget is depleted. Arrival times of keyword-search instances, ad positions, ad selections, and sales generated by the ads are random. Therefore, an advertiser faces a dynamic stochastic total net revenue optimization problem subject to a strict budget constraint. Here we formulate and solve this problem using dynamic programming. We show that there is always an optimal dynamic bidding policy. We describe an iterative numerical approximation algorithm that uniformly converges to the optimal solution at an exponential rate of the number of iterations. We illustrate the algorithm on numerical examples. Because dynamic programing calculations of the optimal bidding policies are computationally demanding, we also propose both static and dynamic alternative bidding policies. We numerically compare the performances of optimal and alternative bidding policies by systematically changing each input parameter. The relative percentage total net revenue losses of the alternative bidding policies increases with the budget loading, but were never more than 3.5 % of maximum expected total net revenue. The best alternative to the optimal bidding policy turned out to be a static greedy bidding policy. Finally, statistical estimation of the model parameters is visited.
Open Access
Inverse optimal control and positive real systems
(1997) Ünal Yılmaz
In this thesis an inverse optimal control problem for constant output feedbacks is investigated. Necessary and sufficient conditions for optimality of an output feedback are derived for single-input, single-output systems. The class of systems with members for which any constant positive output feedback is optimal turns out to be precisely the class of positive real systems. It is also shown that for a class of minimum phase systems all “large” positive gains are optimal.
Open Access
Learning the optimum as a Nash equilibrium
(Elsevier BV, 2000) Özyıldırım, S.; Alemdar, N. M.
This paper shows the computational benefits of a game theoretic approach to optimization of high dimensional control problems. A dynamic noncooperative game framework is adopted to partition the control space and to search the optimum as the equilibrium of a k-person dynamic game played by k-parallel genetic algorithms. When there are multiple inputs, we delegate control authority over a set of control variables exclusively to one player so that k artificially intelligent players explore and communicate to learn the global optimum as the Nash equilibrium. In the case of a single input, each player's decision authority becomes active on exclusive sets of dates so that k GAs construct the optimal control trajectory as the equilibrium of evolving best-to-date responses. Sample problems are provided to demonstrate the gains in computational speed and accuracy. © 2000 Elsevier Science B.V.
Open Access
A noncooperative dynamic game model of opinion dynamics in multilayer social networks
(2017-08) Niazi, Muhammad Umar B.
How do people living in a society form their opinions on daily or prevalent topics? A noncooperative di erential (dynamic) game model of opinion dynamics, where the agents' motives are shaped by how susceptible they are to others' in uence, how stubborn they are, and how quick they are willing to change their opinions on socially prevalent issues is considered here. The agents connected through a multilayer network interact with each other on a set of issues (layers) for a nite time duration. They express their opinions, listen to others' and, hence, mutually in uence each other. The tendency of agents to interact with people of similar traits, known as homophily, restricts them in their own localities, which may correspond to ethnicity but may as well be the ideological ones. This governs their interpersonal in uences and is the cause of clustering in the network. As the agents build their biases, they also create conceptions about the correlation between the issues. As a result, antagonistic interactions arise if the agents see each other as holding inconsistent opinions on the issues according to their individual conceptions. This way the interpersonal in uence becomes ine ective leading to con ict and disagreement between the agents. The dynamic game formulated here takes these subtle issues into account. The game is proved to admit a unique Nash equilibrium under a mild necessary and su cient condition. This condition is argued to be ful lled if there is some harmony of views among the agents in the network. The harmony may be in the form of similarity in pairwise conceptions about the issues but may also be a collective agreement on the status of a leader in the network. Since the agents do not seek any social motive in the game but their own individual motives, the existence of a Nash equilibrium can be interpreted as an emergent collective behavior out of the noncooperative actions of the agents.
Open Access
Optimal control in infinite horizon problems: a Sobolev space approach
(Springer, 2007) Le Van, C.; Boucekkine, R.; Saglam, C.
In this paper, we make use of the Sobolev space W{1,1}ℝ+, ℝn to derive at once the Pontryagin conditions for the standard optimal growth model in continuous time, including a necessary and sufficient transversality condition. An application to the Ramsey model is given. We use an order ideal argument to solve the problem inherent to the fact that L 1 spaces have natural positive cones with no interior points. © Springer-Verlag 2007.
Open Access
Optimal control of single-stage hybrid systems with poisson arrivals and deterministic service times
(IEEE, 2005) Gökbayrak, Kağan; Mısırcı, Muzaffer
We tackle an optimal control problem for a single-stage hybrid system with Poisson arrivals and deterministic service times. In our setting, not only that the optimization problem is non-convex and non-differentiable, but also future arrival times are unknown at the times of decision. We propose a state-dependent service times policy where the state is defined as the system size. These service times are determined iteratively by a steepest descent algorithm whose derivative information is supplied by an infinitesimal perturbation analysis derivative estimator. We also propose an improved receding horizon controller with zero-length time window that utilizes the interarrival time distribution information available from the observed arrivals. Performances of these methods are compared to the optimal performance obtained from the Forward Decompositon Algorithm for which all future arrival times are known. It is also shown that the utilization of the observed interarrival time distribution information improves the performance of the receding horizon controller with zero-length time window.
Open Access
Optimal hybrid control of a two-stage manufacturing system
(IEEE, 2006) Gökbayrak, Kağan; Selvi, Ömer
We consider a two-stage serial hybrid system for which the arrival times are known and the service times are controllable. We derive some optimal sample path characteristics, in particular, we show that no buffering is observed between stages. The original non-smooth optimal control problem is first transformed into a convex optimization problem which is then simplified by the no buffer property. Further simplifications are possible for the bulk arrival case.
Open Access
Parameter identification for partially observed diffusions
(Kluwer Academic Publishers-Plenum Publishers, 1992) Dabbous, T.E.; Ahmed, N.U.
In this paper, we consider the identification problem of drift and dispersion parameters for a class of partially observed systems governed by Ito equations. Using the pathwise description of the Zakai equation, we formulate the original identification problem as a deterministic control problem in which the unnormalized conditional density (solution of the Zakai equation) is treated as the state, the unknown parameters as controls, and the likelihood ratio as the objective functional. The question of existence of elements in the parameter set that maximize the likelihood ratio is discussed. Further, using variational arguments and the Gateaux differentiability of the unnormalized density on the parameter set, we obtain the necessary conditions for optimal identification. © 1992 Plenum Publishing Corporation.
Open Access
Service time optimization of flow shop systems
(2008) Selvi, Ömer
One of the key questions that engineers face in áow shop systems is the service time control, i.e., how long jobs should be processed at each machine. This is an important question because processing times can have great impacts on the cost e¢ ciency of the áow shop systems. In order to meet job completion deadlines and to decrease inventory costs, one may set the service times as small as possible; however, this usually comes at the expense of reduced tool life increasing service costs. In this thesis, we study the áow shop systems under such trade-o§s. We consider the service time optimization of deterministic áow shop systems processing identical jobs that arrive at the system at known times and are processed in the order they arrive within deadlines. The cost function to be minimized consists of service costs at machines and regular completion-time costs of jobs. The decision variables are the service times that are controllable within constraints. We Örst consider the Öxed service time áow shop systems formed of initially controllable machines, where the service times are set only once at the start up time and cannot be altered between processes, and uncontrollable machines, where the service times are Öxed and known in advance. For such systems, we formulate a non-convex and non-di§erentiable optimization problem with a standard solution procedure based on the linearization of the constraints allowing for a convex optimization problem with high memory requirements. Regardless of the cost function, we present a set of waiting and completion time characteristics in such áow shop systems and employ them to derive a simpler equivalent convex optimization problem which improves solution times and alleviates the memory requirements enabling solutions for larger systems. However, the resulting simpliÖed convex optimization problem still needs the use of a convex optimization solver which may not be available at some of the manufacturing companies. To overcome such need, we introduce another equivalent convex optimization problem along with its subgradient algorithm yielding substantial improvements in solution times and solvable system sizes. We also consider a speciÖc nonlinear decreasing service cost structure allowing us to introduce a new search algorithm much faster than the subgradient solution algorithm. Building on the results for Öxed service time áow shop systems, we also consider the mixed line áow shop systems formed of fully controllable machines, where the service times are adjustable for each process, initially controllable machines, and uncontrollable machines. Similarly, we formulate a non-convex and non-di§erentiable optimization problem for such systems and, as a standard way of solving the formulated problem, we apply the method of linearization on the constraints to present a convex optimization problem with high memory requirements. Then, we present a set of optimal waiting characteristics in such áow shop systems and employ them to derive simpler equivalent convex optimization problems. A "forward in time" algorithm is also proposed to decompose the resulting simpliÖed equivalent convex optimization problem into smaller convex optimization problems for the áow shop systems formed of only fully controllable and uncontrollable machines. The computational results demonstrate that the simpliÖcations and the decomposition not only improve the solution times considerably but also allow us to solve larger problems by alleviating memory constraints.
Open Access
Technical note—optimal procurement in remanufacturing systems with uncertain used-item condition
(INFORMS Inst.for Operations Res.and the Management Sciences, 2023-05-08) Nadar, Emre; Akan, Mustafa; Debo, Laurens; Scheller-Wolf, Alan
We consider a single-product remanufacture-to-order system with multiple uncertain quality levels for used items, random procurement lead times, and lost sales. The quality level of a used item is revealed only after it is acquired and inspected; the remanufacturing cost is lower for a higher-quality item. We model this system as a Markov decision process and seek an optimal policy that specifies when a used item should be procured, whether an arriving demand for the remanufactured product should be satisfied, and which available item should be remanufactured to meet this demand. We characterize the optimal procurement policy as following a new type of strategy: state-dependent noncongestive acquisition. This strategy makes decisions, taking into account the system congestion level measured as the number of available items and their quality levels. We also show that it is always optimal to meet the demand with the highest-quality item among the available ones. We conclude with extensions of our model to limited cases when the used-item condition is known a priori (for two quality levels) and remanufacture-to-stock systems in which the standard push strategy is optimal in the remanufacturing stage. © 2023 INFORMS.
Open Access
Technical note-optimal structural results for assemble-to-order generalized M-Systems
(INFORMS Inst.for Operations Res.and the Management Sciences, 2014) Nadar, E.; Akan, M.; Scheller-Wolf, A.
We consider an assemble-to-order generalized M-system with multiple components and multiple products, batch ordering of components, random lead times, and lost sales. We model the system as an infinite-horizon Markov decision process and seek an optimal policy that specifies when a batch of components should be produced (i.e., inventory replenishment) and whether an arriving demand for each product should be satisfied (i.e., inventory allocation). We characterize optimal inventory replenishment and allocation policies under a mild condition on component batch sizes via a new type of policy: lattice-dependent base stock and lattice-dependent rationing. © 2014 INFORMS.