integer programming
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Scientia Iranica, Volume:29 Issue: 4, Jul-Aug 2022, PP 2230 -2240Many applications of optimisation require the final value of the decision variables tobe integer. In many cases the relaxed optimal solution does not satisfy the integral-ity constraint therefore, the problem must be solved by integer or mix-integer pro-gramming algorithms at a significant computational effort and most likely a worsenobjective function value. The contribution of this paper is twofold: The identificationof a type of problems in which the relaxed optimal objective function value can bekept in the implementation by a change in the planning horizon; and the identifica-tion of a multi-period based solution procedure. Three small instances are providedin order to illustrate the methodology as well as the economic impact involved. Inaddition, a fourth industrial size case is included for the benefit of practitioners.This work shows that business profit can be increased for pseudo-continuous-integerperiodical linear problems by identifying optimal decision-making periods.Keywords: Business profit, Integer programming, Linear programming, operations management, operations planning
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بیماری سرطان به عنوان یکی از مهم ترین علل مرگ در جهان، روند رو به رشدی در سال های اخیر داشته است و پیش بینی می شود که این روند در سال های آتی همچنان ادامه یابد. بنابراین اهمیت تدوین برنامه کنترل سرطان و لزوم ارایه روش های موثر اهمیت بسزایی دارد. با توجه به اینکه پرتو درمانی روشی موثر برای درمان سرطان است، مطالعات بر روی این روش درمان مهم و قابل توجه است. هدف اصلی این مقاله، ارایه مدل ریاضی برای جدیدترین روش پرتو درمانی با نام پرتو درمانی با حجم تطبیق شده است. در اکثر مطالعات انجام شده در این زمینه به دلیل پیچیدگی مدل، هدف مساله را بیشینه سازی دوز دریافتی در ناحیه هدف و یا کمینه سازی دوز دریافتی در ناحیه ارگان های در معرض خطر قرار داده اند. در این مطالعه سعی شده هر دو هدف با هم در نظر گرفته شده و یک مدل دو هدفه ارایه شود. نتایج 8 نمونه مورد بررسی نشان می دهد میزان دوز دریافتی در ناحیه هدف به میزان قابل توجهی بیشتر از دوز دریافتی در ناحیه اطراف تومور است. علاوه بر افزایش دوز دریافتی در ناحیه هدف و کاهش دوز دریافتی در سلول های اطراف هدف، نحوه توزیع دوز در وکسل ها بسیار مهم است، از این رو دوز به دست آمده با شاخص ضریب تغییر مورد بررسی قرار گرفت. نتایج نشان می دهد مدل ارایه شده با ضریب تغییر کمتر از 10% توزیع دوز منسجمی در تمام بافت ها (بافت سرطانی و بافت سالم) دارد.
کلید واژگان: پرتو درمانی، سلول های سرطانی، روش حجم تطبیق شده، بهینه سازی دو هدفه، برنامه ریزی عدد صحیحCancer, one of the leading causes of death in the world, has been on the rise in recent years and is expected to continue in the coming years. Therefore, the importance of developing a cancer control program and the need to provide effective methods is very important. Given that radiation therapy is an effective way to treat cancer, studies on this method of treatment are important and significant. The main purpose of this paper is to provide a mathematical model for the latest method of radiotherapy called volumetric modulated arc therapy. In most studies in this field, due to the complexity of the model, the purpose of the problem is to maximize the dose received in the target area or to minimize the dose received in the area organs at risk. In this study, both objectives are considered together and a bi-objective model is presented. The results of the 8 studied instances show that the dose received in the target area is significantly higher than the dose received in the area around the tumor. In addition to increasing the dose received in the target area and decreasing the dose received in the cells around the target, the distribution of the dose in the voxels is very important, so the dose was examined with the coefficient of variation. The results show that the proposed model with a coefficient of variation of less than 10% has a conformal dose distribution in all tissues (cancerous tissue and healthy tissue).
Keywords: Radiotherapy, Cancer cells, VMAT, Bi-objective optimization, Integer programming -
Scientia Iranica, Volume:26 Issue: 6, Nov-Dec 2019, PP 3728 -3746Banks need to open new branches in new sites as a result of increase in the population, individual earnings and the growth in national economy. In this respect, opening new branches or reorganizing the locations of current branches is an important decision problem for banks to accomplish their strategic objectives. This paper presents a decision support method for multi-period bank branch location problems. Our aim is to find bank branch location based on transaction volume, distance between branches, and cost of opening and closing branches. The proposed method not only develops an Integer Program and a Tabu Search algorithm to find the exact places of branches but also presents a structuring method to identify the related criteria and their importance. We demonstrate the effectiveness of the method on random data. In the final stage, the method is applied in a Turkish bank’s branch location problem considering the current and possible places of the branches, availability of the data, and the bank’s strategies for a four-year strategic planning.Keywords: Integer programming, decision support system, Tabu search, case study, banking, location
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The multi-factory supply chain problem is investigated to determine the production and transportation scheduling of jobs which are allowed to be transported by batches. This is a mixed-integer optimization problem, which could be challenging to solve. The problem incorporates two parts: (1) assigning jobs to appropriate batch, and (2) scheduling jobs of batches for production and transportation. Based on the problem structure and because of its NP-hardness characteristics, Benders decomposition is recognized as a suitable approach. This approach decomposes the problem into assignment master problem and scheduling sub-problem. This would facilitate the solution procedure. By comparing performance of the proposed algorithm with an exact approach: Branch and Bound, It is achieved that it is able to find the near optimal solution in smaller computational times than the Branch and Bound.Keywords: Multi, factory scheduling, Batch delivery, Benders decomposition, Mixed, integer programming
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The aim of the present paper is to propose a locationallocation model, for a capacitated health care system. This paper develops a discrete modeling framework to determine the optimal number of facilities among candidates and optimal allocations of existing customers for operating health centers in a coverage distance, so that the total sum of customer and operating facility costs are minimized.Our goal is to create a model that is more practical in the real world. Therefore, setup costs of the hospitals are based on the costs of customers, fixed costs of establishing health centers and costs based on theavailable resources in each level of hospitals.In this paper, the idea of hierarchical structure has been used. There are two levels of service in hospitals including low and high levels and sections at different levels that provide different types of services. The patients are referred to the hospitals different sections according to their requirements. To solve the model, two meta-heuristic algorithms, including genetic algorithm, simulated annealing and their combination are proposed. To evaluate the performance of the three algorithms, some numerical examples are produced and analyzed using the statistical test in order to determine which algorithm works better.Keywords: Health care system, Mixed, integer programming, Queuing theory, Capacitated system, Genetic algorithm, Simulated annealing algorithm, Hybrid algorithm
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Complexity issues and limited feedback rate strictly constrain the number of transmission modes in discrete link adaptation. A new algorithm is designed for selecting an optimum set of modes out of all the possible transmission modes based on the links characteristics and the constraints, meanwhile, imperfect channel state information (CSI) is available. The design goal is to maximize the average spectral efficiency and the selection is done using a combination of nonlinear quantization and integer programming. As numerical evaluations for adaptive LDPC or convolutional coded, QAM modulated transmission show, performance of the proposed scheme is considerably improved in comparison with the benchmark schemes.Keywords: component, Link Adaptation, Adaptive Modulation, Coding, Integer programming
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In a recent paper, Caserta et al. [M. Caserta, S. Schwarze, and S. Vo. A mathematical formulation and complexity considerations for the blocks relocation problem«, European Journal of Operational Research, 219, pp. 96-104 (2012)] proposed two mathematical models for the blocks relocation problem. Because of the complexity of their rst model, called BRP-I, they employed a simplifying assumption and introduced a relatively fast model, called BRP-II, to solve medium-sized instances. In this paper, it is rst proven that the BRP-II model is incorrect. Then, the corrected and improved formulation of BRP-II, called BRP2c and BRP2ci, respectively, are presented. By correcting a constraint in BRP-II, the reported optimal solution is either corrected or improved in many instances. Also, it is proven that some results of BRP-II reported by Caserta et al. are incorrect. Incorporating some new cut constraints into BRP2ci, the computational time of solving instances is decreased 25 times, on average.Keywords: Logistics, Blocks relocation problem, Integer programming, Cut constraints, Optimization
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امروزه در شبکه های قدرت به دلایل مختلف، از جمله تخمین حالت دینامیکی و بهبود سرعت سیستم های کنترلی و حفاظتی، استفاده بهینه از واحدهای اندازه گیری فازور رو به گسترش است. در این مقاله، روش جدیدی برای تعیین تعداد بهینه واحدهای اندازه گیری فازور به منظور رویت پذیری کامل شبکه در قالب یک مساله برنامه ریزی خطی عدد صحیح دودویی ارائه می شود. همچنین، مساله جایابی بهینه این واحدها در شرایط خروج تکی هر یک از خطوط سیستم نیز مطرح شده است. در کلیه این حالت ها، اثر شین های با توان تزریقی صفر سیستم لحاظ گردیده است. توانایی روش پیشنهادی در حل هر یک از مسایل فوق با اعمال آن در شبکه های استاندارد 14، 30، 39، 57 و 118 شینه IEEE، شبکه 42 شینه انتقال برق منطقه ای اصفهان و همچنین دو شبکه بسیار بزرگ 2383 و 2746 شینه نشان داده شده است. نتایج به دست آمده، حاکی از کارایی موثر و مناسب روش پیشنهادی در مقایسه با روش های دیگر است.
کلید واژگان: اندازه گیری، برنامه ریزی عدد صحیح، جایابی بهینه، رویت پذیری سیستم قدرت، واحد اندازه گیری فازوNowadays, the optimal application of phasor measurement units (PMUs) in power systems is increasing because of their advantages such as the capability for dynamical state estimation and improvements in the speed of control and protection systems. In this paper, a new method is presented using binary integer linear programming for the optimal placement of PMUs to guarantee full observability of a power system. Moreover, the problem of the optimal placement of these units in the case of a single line outage is investigated. In all of the investigations, the effect of zero-injection buses in the power system was considered. The ability of the proposed method in solving any of these problems was demonstrated by applying to the 14-, 30-, 39-, 57- and 118-bus IEEE standard systems, 42-bus transmission network of Esfahan Regional Electric and also two very large-scale systems with 2383 and 2746 buses. The simulation results verified the effective and appropriate performance of the proposed method in comparison with other methods.Keywords: Integer programming, measurement security, optimal placement, phasor measurement unit (PMU), power system observability -
In this paper, we propose a heuristic algorithm, named the Lotto–Meta heuristic, to solve small instances of the lottery problem, using its set covering formulation. The algorithm uses a randomized method, allotting a priority to each column to be included in the solution. A neighborhood search strategy is fused with the algorithm to enhance the search and to balance the exploration and exploitation procedures. Computational results show that our method outperforms the best known solutions for a number of hard instances.Keywords: Integer programming, Set covering problem, Lottery problem, Lotto–Meta heuristic, Neighborhood search
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