Browsing by Subject "Convex cost function"
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Item Open Access Deconvolution using projections onto the epigraph set of a convex cost function(IEEE, 2014) Tofighi, Mohammad; Bozkurt, Alican; Köse, K.; Çetin, A. EnisA new deconvolution algorithm based on making orthogonal projections onto the epigraph set of a convex cost function is presented. In this algorithm, the dimension of the minimization problem is lifted by one and sets corresponding to the cost function and observations are defined. If the utilized cost function is convex in RN, the corresponding epigraph set is also convex in RN+1. The deconvolution algorithm starts with an arbitrary initial estimate in RN+1. At each iteration cycle of the algorithm, first deconvolution projections are performed onto the hyperplanes representing observations, then an orthogonal projection is performed onto epigraph of the cost function. The method provides globally optimal solutions for total variation, l1, l2, and entropic cost functions.Item Open Access Denoising using projections onto the epigraph set of convex cost functions(IEEE, 2014) Tofighi, Mohammad; Köse, K.; Çetin, A. EnisA new denoising algorithm based on orthogonal projections onto the epigraph set of a convex cost function is presented. In this algorithm, the dimension of the minimization problem is lifted by one and feasibility sets corresponding to the cost function using the epigraph concept are defined. As the utilized cost function is a convex function in RN, the corresponding epigraph set is also a convex set in RN+1. The denoising algorithm starts with an arbitrary initial estimate in RN+1. At each step of the iterative denoising, an orthogonal projection is performed onto one of the constraint sets associated with the cost function in a sequential manner. The method provides globally optimal solutions for total-variation, ℓ1, ℓ2, and entropic cost functions.1Item Open Access Lot sizing with nonlinear production cost functions(Bilkent University, 2015-07) Koca, EsraIn this study, we consider di erent variations of the lot sizing problem encountered in many real life production, procurement and transportation systems. First, we consider the deterministic lot sizing problem with piecewise concave production cost functions. A piecewise concave function can represent quantity discounts, subcontracting, overloading, minimum order quantities, and capacities. Computational complexity of this problem was an open question in the literature. We develop a dynamic programming (DP) algorithm to solve the problem and show that the problem is polynomially solvable when number of breakpoints of the production cost function is xed and the breakpoints are time-invariant. We observe that the time complexity of our algorithm is as good as the complexity of existing algorithms in the literature for the special cases with capacities, minimum order quantities, and subcontracting. Our algorithm performs quite well for small and medium sized instances. For larger instances, we propose a DP based heuristic to nd a good quality solution in reasonable time. Next, we consider the stochastic lot sizing problem with controllable processing times where processing times can be reduced in return for extra compression cost. We assume that the compression cost function is a convex function in order to re ect the increasing marginal cost of larger reductions in processing times. We formulate the problem as a second-order cone mixed integer program, strengthen the formulation and solve it by a commercial solver. Moreover, we obtain some convex hull and computational complexity results. We conduct an extensive computational study to see the e ect of controllable processing times in solution quality and observe that even with small reductions in processing times, it is possible to obtain a less costly production plan. As a nal problem, we study the multistage stochastic lot sizing problem with nervousness considerations and controllable processing times. System nervousness is one of the main problems of dynamic solution strategies developed for stochastic lot sizing problems. We formulate the problem so that the nervousness of the system is restricted by some additional constraints and parameters. Mixing and continuous mixing set structures are observed as relaxations of our formulation. We develop valid inequalities for the problem based on these structures and computationally test these inequalities.Item Open Access Pasif bistatik radarlarda seyreklik temellli ters evrişim kullanılarak hedef tespiti(IEEE, 2015-05) Arslan, Musa Tunç; Tofighi, Mohammad; Çetin, A. EnisBu bildiride pasif radar (PR) sistemlerinin menzil çözünürlüğünü artırmak için seyreklik tabanlı bir ters evrişim yöntemi sunulmaktadır. PR sistemlerinin iki boyutlu uyumlu süzgeç çıktısı bir ters evrişim problemli gibi düşünülerek incelenmektedir. Ters evrişim algoritması, hedeflerin zaman kaymaları ve l1 norm benzeri dışbükey maliyet fonksiyonlarının epigraf kümelerini temsil eden hiperdüzlemler üzerine izdüşümü temellidir. Bütün kısıt kümeleri kapalı ve dışbükey olduklarından dolayı yinelemeli algoritma yakınsamaktadır. FM tabanlı PR sistemleri üzerinde benzetim sonuçları sunulmuştur. Algoritma frekans uzayı tabanlı ters evrişim yöntemlerine göre daha yüksek performansa sahiptir.Item Open Access Production decisions with convex costs and carbon emission constraints(Bilkent University, 2016-03) Kian, RamezIn this thesis, di erent variants of the production planning problem are considered. We rst study an uncapacitated deterministic lot sizing model with a nonlinear convex production cost function. The nonlinearity and convexity of the cost function may arise due to the extra nes paid by a manufacturer for environmental regulations or it may originate from some production functions. In particular, we have considered the Cobb-Douglas production function which is applied in sectors such as energy, agriculture and cement industry. We demonstrate that this problem can be reformulated as a lot sizing problem with nonlinear production cost which is convex under certain assumptions. To solve the problem we have developed a polynomial time dynamic programming based algorithm and nine fast heuristics which rest on some well known lot sizing rules such as Silver-Meal, Least Unit Cost and Economic Order Quantity. We compare the performances of the heuristics with extensive numerical tests. Next, motivated from the rst problem, we consider a lot sizing problem with convex nonlinear production and holding costs for decaying items. The problem is investigated from mathematical programming perspective and di erent formulations are provided. We propose a structural procedure to reformulate the problem in the form of second order cone programming and employ some optimality and valid cuts to strengthen the model. We conduct an extensive computational test to see the e ect of cuts in di erent formulations. We also study the performance of our heuristics on a rolling horizon setting. We conduct an extensive numerical study to compare the performance of heuristics and to see the e ect of forecast horizon length on their dominance order and to see when they outperform exact solution approaches. Finally, we study the lot sizing problem with carbon emission constraints. We propose two Lagrangian heuristics when the emission constraint is cumulative over periods. We extend the model with possibility of lost sales and examine several carbon emission cap policies for a cost minimizing manufacturer and conduct a cost-emission Pareto analysis for each policy.Item Open Access Projections onto the epigraph set of the filtered variation function based deconvolution algorithm(IEEE, 2017) Tofighi, M.; Çetin, A. EnisA new deconvolution algorithm based on orthogonal projections onto the hyperplanes and the epigraph set of a convex cost function is presented. In this algorithm, the convex sets corresponding to the cost function are defined by increasing the dimension of the minimization problem by one. The Filtered Variation (FV) function is used as the convex cost function in this algorithm. Since the FV cost function is a convex function in RN, then the corresponding epigraph set is also a convex set in the lifted set in RN+1. At each step of the iterative deconvolution algorithm, starting with an arbitrary initial estimate in RN+1, first the projections onto the hyperplanes are performed to obtain the first deconvolution estimate. Then an orthogonal projection is performed onto the epigraph set of the FV cost function, in order to regularize and denoise the deconvolution estimate, in a sequential manner. The algorithm converges to the deblurred image.Item Open Access Rescheduling parallel machines with controllable processing times(Bilkent University, 2012) Muhafız, MügeIn many manufacturing environments, the production does not always endure as it is planned. Many times, it is interrupted by a disruption such as machine breakdown, power loss, etc. In our problem, we are given an original production schedule in a non-identical parallel machine environment and we assume that one of the machines is disrupted at time t. Our aim is to revise the schedule, although there are some restrictions that should be considered while creating the revised schedule. Disrupted machine is unavailable for a certain time. New schedule has to satisfy the maximum completion time constraint of each machine. Furthermore, when we revise the schedule we have to satisfy the constraint that the revised start time of a job cannot be earlier than its original start time. Because, we assume that jobs are not ready before their original start times in the revised schedule. Therefore, we have to find an alternative solution to decrease the negative impacts of this disruption as much as possible. One way to process a disrupted job in the revised schedule is to reallocate the job to another machine. The other way is to keep the disrupted job at its original machine, but to delay its start time after the end time of the disruption. Since the machines might be fully utilized originally, we may have to compress some of the processing times in order to add a new job to a machine or to reallocate the jobs after the disruption ends. Consequently, we assume that the processing times are controllable within the given lower and upper bounds. Our first objective is to minimize the sum of reallocation and nonlinear compression costs. Besides, it is important to deliver the orders on time, not earlier or later than they are promised. Therefore, we try to maintain the original completion times as much as possible. So, the second objective is to minimize the total absolute deviations of the completion times in the revised schedule from the original completion times. We developed a bi-criteria non-linear mathematical model to solve this nonidentical parallel machine rescheduling problem. Since we have two objectives, we handled the second objective by giving it an upper bound and adding this bound as a constraint to the problem. By utilizing the second order cone programming, we solved this mixed-integer nonlinear mathematical model using a commercial MIP solver such as CPLEX. We also propose a decision tree based heuristic algorithm. Our algorithm generates a set of solutions for a problem instance and we test the solution quality of the algorithm solving same problem instances by the mathematical model. According to our computational experiments, the proposed heuristic approach could obtain close solutions for the first objective for a given upper bound on the second objective.