Integer Programming

This paper deals with approximating feedback sets in directed graphs. We consider two related problems: the weighted feedback vertex set (FVS) problem, and the weighted feedback edge set (FES) problem. In the {FVS} (resp. FES) problem, one is given a directed graph with weights (each of which is at least one) on the vertices (resp. edges), and is asked to find a subset of vertices (resp. edges) with minimum total weight that intersects every directed cycle in the graph. These problems are among the classical NP-hard problems and have many applications. We also consider a generalization of these problems: subset-fvs and subset-fes, in which the feedback set has to intersect only a subset of the directed cycles in the graph. This subset consists of all the cycles that go through a distinguished input subset of vertices and edges, denoted by X . This generalization is also NP-hard even when |X|=2 . We present approximation algorithms for the subset-fvs and subset-fes problems. The first algorithm we present achieves an approximation factor of O(log 2 |X|) . The second algorithm achieves an approximation factor of O(min{log τ * log log τ * , log n log log n)} , where τ * is the value of the optimum fractional solution of the problem at hand, and n is the number of vertices in the graph. We also define a multicut problem in a special type of directed networks which we call circular networks, and show that the subset-fes and subset-fvs problems are equivalent to this multicut problem. Another contribution of our paper is a combinatorial algorithm that computes a (1+ɛ) approximation to the fractional optimal feedback vertex set. Computing the approximate solution is much simpler and more efficient than general linear programming methods. All of our algorithms use this approximate solution.

The advantages of logistics centers for companies, cities, and countries have been discussed in the literature and generally mathematical model-based evaluations besides multi-criteria approaches are proposed for site selection processes. However, since mathematical modeling of multiple site selection often turns out to be NP-hard problem structure, it is not always possible to obtain an optimal solution by the solvers. For this reason, various meta-heuristic approaches have emerged to solve these complex models. In this context, the aim of this study is to propose an integrated methodology which seeks an optimum result efficiently regarding a logistics center location selection problem. Thus, the optimal clustering of logistics mobility in a metropolitan area was carried out with GIS and a meta-heuristic approach. GIS produced the spatial information needed by p-median model, then the meta-heuristic approach determined the optimal result that considers the logistics costs. BPSO algorithm has employed as the meta-heuristic and it is observed that the algorithm can reach the optimum results within superior times for the problem sizes tested where binary integer programming verified the optimums and the algorithm continued to reach improved solutions where the exact algorithms failed for larger instances. The integrated solution methodology is applied to a large metropolitan region and it is found that it can be used properly by the urban city planners and supply chain managers to analyze critical nodes of transportation networks of megacities.

Cone programming is a joint generalization of the pozitive semi-definite programming and linear programming. The requirement that the matrix of variables must be pozitive definite or the vector of variables must be nonnegative resp., in semi-definite or linear programming, resp., is substituted by the claim that the vector of variables must belong to a fixed and not necessarily polyhedral cone. The original Benders decomposition is a frame of algorithms to solve optimization problems such that the variables can be divided into two sets in such a way that for every fixed values of the variables of the second set the reduced problem is a linear programming problem. In this paper Benders decomposition is generalized for the case when the reduced problem is a problem of cone programming.

Цель настоящей статьи состоит в реабилитации в глазах научной общественности древних представлений о мире, имеющих целочисленное выражение, и в их возвращении в понятийный аппарат современной науки. Целочисленное моделирование, кроме математического программирования, встречается в системах микроэлементов деятельности, а также в древних мифах творения и числовых моделях бытия. Разные способы структурирования 10-элементной целостности отражают уровни её системной организации в пифагорейском тетрактисе, древе сефирот иудейской каббалы, древнеиндийской саанкхье. Структура пифагорейского тетрактиса может быть обнаружена во фрактально-кластерной модели, но относительные единицы значения, составляющие её, получают традиционно десятичное или процентное выражение.

This paper aims rehabilitation of the ancient ideas about the world, which have an integer expression in the eyes of the scientific community, and their return to the conceptual apparatus of modern science. Integer modeling, apart from mathematical programming, can be found in the systems of microelements of activity, as well as in ancient myths of creation and numerical models of being. Different ways of structuring 10-element integrity reflect the levels of its system arrangement in the Pythago-rean tetractys, the Sephiroth tree of the Jewish Kabbalah, in the Old Indian Saankhya. The structure of the Pythagorean tetraktis is found in the fractal-cluster model, but the relative units of value, that make it up, have the traditional decimal or percentage expression.

In this paper, an effective mechanism using a fleet of unmanned surface vehicles (USVs) carried by a parent boat (PB) is proposed to complete search or scientific tasks over multiple target water areas within a shorter time. Specifically, multiple USVs can be launched from the PB to conduct such operations simultaneously, and each USV can return to the PB for battery recharging or swapping and data collection in order to continue missions in a more extended range. The PB itself follows a planned route with a flexible schedule taking into consideration locational constraints or collision avoidance in a real-world situation. Assuming that each target has a value, this research investigates how to route these USVs, including their schedules to rendezvous with the PB, so that they can maximize the total collected target values from the operation in a limited amount of time. We use a multi-layered time-space network to describe the USVs and PB movement over time and give an integer programming (IP) formulation for the coverage path planning problem. To further shorten the computational time, we propose the Iterative Clustering Heuristic (ICH) to firstly cluster the workspace, calculate the path for each USV to visit targets and meet with the PB for range extension. To evaluate the performance of the proposed IP model and ICH, test cases are designed based on a real-world scenario, as well as families of simulated grid-like networks. Based on the computational analysis of different USV area sizes, targets covered, and operation time-bound increments, the proposed heuristic ICH can solve larger sized cases faster than the IP commercial solver with higher quality results.

– One of the most crucial steps in the design of embedded systems is hardware-software partitioning, ie deciding which components of the system should be implemented in hardware and which ones in software. In this paper, different versions of the partitioning problem ...

In sensor network applications, data gathering mechanisms, which are based on multi-hop forwarding, can be expensive in terms of energy. This limitation challenges the use of sensor networks for applications that demand a predefined operational-lifetime. To avoid this problem, using of mobile element (ME) as a mechanical data carrier has emerged as a promising approach. However, practical considerations such as the ME speed and route planning, sensor buffer size and data frequency generation constraints impose limits ...

This paper introduces an interactive approach to support multi-criteria decision analysis of project portfolios. In high-scale strategic decision domains, scientific studies suggest that the Decision Maker (DM) can find help by using many-objective optimisation methods, which are supposed to provide values in the decision variables that generate highquality solutions. Even so, DMs usually wish to explore the possibility of reaching some levels of benefits in some objectives. Consequently, they should repeatedly run the optimisation method. However, this approach cannot perform well-in an interactive way-for large instances under the presence of many objective functions. We present a mathematical model that is based on compromise programming and fuzzy outranking to aid DMs in analysing multi-criteria project portfolios on the fly. This approach allows relaxing the problem of rapidly optimising portfolios while preserving the beneficial properties of the DM's preferences expressed by outranking relations. Our model supports the decision analysis on two instance benchmarks: for the first one, a better compromise solution was generated 84% of the runs; for the second one, this ranged from 93% to 97%. Our model was also applied to a real-world problem involving social projects, obtaining satisfactory results.

Empirical studies have shown that demand for multimodal transport systems is highly correlated with activity schedules of individuals. Nonetheless, existing analytical equilibrium models of multimodal systems have only considered trip-based demand. We propose a new market equilibrium model that is sensitive to traveler activity schedules and system capacities. The model is based on a constrained mixed logit model of activity schedule choice, where each schedule in the choice set is generated with a multimodal extension of the Household Activity Pattern Problem. The extension explicitly accounts for both passenger choices of activity participation and multimodal choices like public transit, walking, and vehicle parking. The market equilibrium is achieved with Lagrangian relaxation to determine the optimal dual price of the capacity constraint, and a method of successive averages with column generation finds an efficient choice set of activity schedules to assign flows over the dynamic network load capacities. An example illustrates the model and algorithm, effects similar to Vickrey’s morning commute model can be observed as a special case. A case study of the Oakville Go Transit station access “last mile” problem in the Greater Toronto Area is conducted with 166 survey samples reflecting 3,680 individuals. Results suggest that a $10 fixed parking fee at Oakville station would lead to a reduction of access auto share from 54.8% to 49.5%, an increase in access transit share from 20.7% to 25.9%, and a disutility increase of 11% for the of single-activity residents of Oakville.

This paper presents a new mathematical model to select an optimal combination of productivity improvement techniques. The proposed model of this paper considers four-stage cycle productivity and the productivity is assumed to be a linear function of fifty four improvement techniques. The proposed model of this paper is implemented for a real-world case study of manufacturing plant. The resulted problem is formulated as a mixed integer programming which can be solved for optimality using traditional methods. The preliminary results of the implementation of the proposed model of this paper indicate that the productivity can be improved through a change on equipments and it can be easily applied for both manufacturing and service industries.

In this paper, what i have been discussed, is analyzing penalties and cost shifts based on several elements for nurse scheduling problem (NSP). NSP’s issue is to assign nurses to different tasks based on constraints. The problem is known to be NP-hard, in other words it does not have a solution or needs years to be solved. In this work we try to solve the problem by satisfying the constraints set, and we also include the nurse’s preference and try to balance the difficulty level of all the involved nurses. We also analyze the complexity of the problem as a function of parameters such as number of nurses, number of shifts, and optimality of the function.
According to the importance in practice, many scientists have developed NSP problems in a satisfactory time limit.

The main aim of the present paper is to show how GLM method, as it was suggested by H. Everett, can be used in enumeration algorithm.
First of all we give a new optimality test. Though the test is a
general one, its use in integer programming is detailed. The existence
of gaps is the greatest difficulty of the GLM method. We show how
it can be avoided in integer programming. In the last section the
relation between the Lagrange multipliers and the surrogate
constraints is examined.

This report is the digitalized version of Problems of Control and Information Theory, 7(1978), 393-406. Only minimal changes
are done although the style of writing OR changed a lot since the time
of the original paper. Anything which were not contained in the original paper is in footnote.