Power efficient uplink scheduling in SC-FDMA: benchmarking by column generation

We study resource allocation in cellular systems and consider the problem of finding a power efficient scheduling in an uplink single carrier frequency division multiple access system. Due to the discrete nature of this problem and its computational difficulty, particularly in a real-time setting, the use of suboptimal algorithms is common practice. We aim at an effective way of gauging the performance of suboptimal algorithms by finding tight bounds on the global optimum. Toward this end, we first provide a basic integer linear programming formulation. Then we propose a significantly stronger column-oriented formulation and a corresponding column generation method, as well as an enhanced column generation scheme. The latter extends the first scheme through the inclusion of a stabilization technique, an approximate column generation principle, and a tailored heuristic that is embedded in the column generation scheme to find high-quality though not necessarily global optimal solutions. The computational evaluation demonstrates that compared with a poor performance by the integer linear programming formulation, the column generation method can produce near-optimal schedules that enable a sharp bounding interval. The enhanced column generation method significantly sharpens the bounding interval. Hence the column generation approach serves well for the purpose of benchmarking results for large-scale instances.     

Matheuristic approaches for Q-coverage problem versions in wireless sensor networks

This article deals with sensor coverage scheduling in wireless sensor networks subject to Q-coverage constraints. The main concern is to maximize the network lifetime, while ensuring that each target is covered by a given number of sensors. Three different variations of this problem are considered. Column generation based exact approaches are developed for those problems where the auxiliary problem is solved by a two-level approach comprising a genetic algorithm and an integer linear programming formulation. The genetic algorithm takes advantage of the auxiliary problem structure and appears to be very efficient at providing the master problem with attractive columns. The auxiliary problem integer linear programming (ILP) formulation is then mostly used for proving the optimality status of the current master problem solution. The proposed approaches are shown to be significantly faster than column generation approaches relying only on the auxiliary problem ILP formulation.     

Solving the damper placement problem via local search heuristics

The damper placement problem for large flexible space truss structures is to determine thep truss members of the structure to replace with active (or passive) dampers so that the modal damping ratio is as large as possible for all significant modes of vibration. Equivalently, given a strain energy matrix with rows indexed on the modes and columns indexed on the truss members we seek to find a set ofp columns such that the smallest row sum, over thep columns, is maximized. An extension of this model is formulated for the passive damper case. This formulation includes the frequency of maximum displacement as a decision variable for each passive damper. Each formulation can be written as a mixed 0/1 integer linear program. We compare the performance of tabu search and simulated annealing for the damper placement problem on a laboratory test article, the NASA Langley Controls-Structures Interaction Phase I Evolutionary Model (10 modes and 1507 truss members). Tabu search, coupled with the starting solution generated by rounding the solution to a linear programming relaxation, is shown to provide the highest quality solutions in the shortest amount of computing time.Research partially supported by an ASEE summer fellowship at NASA-LaRC and by a faculty research assignment award from The College of William and Mary     

JOINT CONFIGURATION OF BACKBONE AND LOGICAL NETWORKS ON A RECONFIGURABLE PACKET-SWITCHED NETWORK WITH UNRELIABLE LINKS

Abstract

This paper considers a problem of configuring both backbone and logical networks in a reconfigurable packet-switched network where links are subject to failures. The objective is to design feasible backbone and logical networks at least cost where the network cost includes backbone link capacity expansion cost and average packet delay penalty cost due to link failures. The problem is formulated as a zero-one non-linear mixed integer programming problem, for which an effective solution procedure is developed by using a Lagrangean relaxation technique for finding a lower bound and a heuristic method is exploited for improving the upper bound of an intermediate solution. The solution procedure is tested for its effectiveness with various numerical examples.     

Sequencing with limited flexibility

We consider the problem of resequencing a set of prearranged jobs when there is limited resequencing flexibility and sequence-dependent changeover costs. Resequencing flexibility is limited by how far forward or backward a job can shift in the sequence relative to its original position. We show how the problem can be solved using dynamic programming in polynomial time with respect to the number of jobs. We also show how the same solution approach can be extended to problems where sequencing constraints are job specific and to problems where job features, which determine changeover costs, are jointly determined with the job sequence. We provide an integer programming formulation to the resequencing problem whose linear programming relaxation offers a useful lower bound. We also describe a family of decomposition heuristics that are easy to customize to provide desired levels of solution quality and solution time. We document the quality of the lower bound from the linear programming relaxation and the upper bound from the heuristic using numerical results. We also provide numerical results to support managerial insights regarding the value of flexibility. We show that the value of flexibility is of the diminishing kind with most of the benefit realized with relatively limited flexibility. We also show that a balanced allocation of flexibility among forward and backward position shifting is superior to an unbalanced one. More significantly, we show that forward and backward position shifting flexibility are complements with the value of one increasing in the amount of the other. Finally, we apply our solution approach to a real-world case from the automotive industry.     

A computational study of LP-based heuristic algorithms for two-dimensional guillotine cutting stock problems

In this paper we develop and compare several heuristic methods for solving the general two-dimensional cutting stock problem. We follow the Gilmore-Gomory column generation scheme in which at each iteration a new cutting pattern is obtained as the solution of a subproblem on one stock sheet. For solving this subproblem, in addition to classical dynamic programming, we have developed three heuristic procedures of increasing complexity, based on GRASP and Tabu Search techniques, producing solutions differing in quality and in time requirements. In order to obtain integer solutions from the fractional solutions of the Gilmore-Gomory process, we compare three rounding procedures, rounding up, truncated branch and bound and the solution of a residual problem. We have coded and tested all the combinations of algorithms and rounding procedures. The computational results obtained on a set of randomly generated test problems show their relative efficiency and allow the potential user to choose from among them, according to the available computing time. Rceived: January 9, 2001 / Accepted: December 10, 2001     

Cyclic scheduling in synchronous production lines

In this paper we address the scheduling problem in unpaced synchronous mixed-model production lines operated under a cyclic scheduling policy. We first discuss operations of a production line with the synchronous transfer of parts. We then present an integer programming formulation of the problem. The problem, however, is W-hard, and for its exact solution we propose an implicit enumeration scheme. We discuss a property of the scheduling problem which allows us to effectively solve large size instances of the problem. We also present an approximate solution procedure with very good avenge performance. Useful managerial insights are obtained as we search for ways to improve the performance of synchronous lines. The relaxation of one of our original assumptions in the scheduling problem formulation results in an easy problem whose solution generates the absolute best in throughput performance configuration of the production line. Implementation of this solution, however, requires increasing the number of buffers in the line. We suggest other performance improvement ways to better balance the tradeoff between throughput and average Work-In-Progress (WIP) inventory in the line.     

A heuristic for multi-level lot-sizing problems with a bottleneck

A simple heuristic is proposed for multi-level lot-sizing problems where there is a bottleneck. Previous methods to solve this problem have formulated the problem as an integer programming problem and solved the problem using a Lagrangian relaxation embedded within the branch and bound procedure. In this paper we suggest that items to be produced can be grouped into two types and a simple but efficient heuristic can be used to determine the production quantities required. A program was developed to compute production levels and was found to require only a small fraction of the computer time required by the full integer programming approach, and to produce solutions of reasonable quality. The heuristic is simple to implement.     

Minimization of tool switches for a flexible manufacturing machine with slot assignment of different tool sizes

The problem addressed in this paper is the tool switching problem for an automated manufacturing environment, when each tool may occupy more than one slot of the tool magazine. A machine processes parts automatically by using a limited capacity tool magazine. Providing a tool that is needed for a certain processing operation is not in the magazine, a tool switch must occur before the job can be processed, a time/cost consuming operation. To solve this problem, one has to decide three types of decisions, namely, how to select the jobs' sequence (machine loading), which tools to switch before each processing operation (tool loading) and where to locate each tool in the magazine (slot loading). We present an integer programming formulation for the problem and suggest a heuristic procedure to obtain a solution. Our heuristic is partly a generalization of previously suggested approaches to the first two decision types, but it is mainly oriented towards answering the third decision type. The unified problem has not been addressed previously in the literature. We present a numerical study that demonstrates the efficiency of our procedure.     

A Stochastic Integer Programming Model for Incorporating Day-Ahead Trading of Electricity into Hydro-Thermal Unit Commitment

We develop a two-stage stochastic integer programming model for the simultaneous optimization of power production and day-ahead power trading in a hydro-thermal system. The model rests on mixed-integer linear formulations for the unit commitment problem and for the price clearing mechanism at the power exchange. Foreign bids enter as random components into the model. We solve the stochastic integer program by a decomposition method combining Lagrangian relaxation of nonanticipativity with branch-and-bound in the spirit of global optimization. Finally, we report some first computational experiences.     

Optimal solution methods for the minimum-backtracking row layout problem

The problem of finding a reordering of the rows (and, simultaneously, the columns) of square matrices has important applications in the areas of combinatorial data analysis, economics and engineering. One such application is the Minimum-Backtracking Row Layout Problem (MBRLP), which involves the sequencing of workstations in an automated assembly line so as to minimize the total backtracking distance. Dynamic programming and integer programming have previously been proposed as optimal solution methods for the MBRLP; however, the latter method was limited to the case of equal distances between positions in the sequence. We demonstrate that the previously suggested integer programming approach for equidistant MBRLPs can produce infeasible solutions, and develop and test a plausible integer programming model for its replacement. We also present an optimal branch-and-bound procedure for MBRLP that requires far less memory storage than dynamic programming. Our experimental tests revealed that both dynamic programming and the branch-and-bound algorithm efficiently provided optimal solutions to MBRLPs with 25 or fewer workstations, and that neither method was superior in all cases. However, the branch-and-bound algorithm was appreciably faster than dynamic programming for some data conditions, and can be used for problems where computer RAM limitations preclude dynamic programming implementation.     

Optimization Models for Scheduling of Jobs

This work is motivated by a particular scheduling problem that is faced by logistics centers that perform aircraft maintenance and modification. Here we concentrate on a single facility (hangar) which is equipped with several work stations (bays). Specifically, a number of jobs have already been scheduled for processing at the facility; the starting times, durations, and work station assignments for these jobs are assumed to be known. We are interested in how best to schedule a number of new jobs that the facility will be processing in the near future. We first develop a mixed integer quadratic programming model (MIQP) for this problem. Since the exact solution of this MIQP formulation is time consuming, we develop a heuristic procedure, based on existing bin packing techniques. This heuristic is further enhanced by application of certain local optimality conditions.     

Models and algorithms for the heterogeneous dial-a-ride problem with driver-related constraints

This paper introduces models and algorithms for a static dial-a-ride problem arising in the transportation of patients by non-profit organizations such as the Austrian Red Cross. This problem is characterized by the presence of heterogeneous vehicles and patients. In our problem, two types of vehicles are used, each providing a different capacity for four different modes of transportation. Patients may request to be transported either seated, on a stretcher or in a wheelchair. In addition, some may require accompanying persons. The problem is to construct a minimum-cost routing plan satisfying service-related criteria, expressed in terms of time windows, as well as driver-related constraints expressed in terms of maximum route duration limits and mandatory lunch breaks. We introduce both a three-index and a set-partitioning formulation of the problem. The linear programming relaxation of the latter is solved by a column generation algorithm. We also propose a variable neighborhood search heuristic. Finally, we integrate the heuristic and the column generation approach into a collaborative framework. The column generation algorithm and the collaborative framework provide tight lower bounds on the optimal solution values for small-to-medium-sized instances. The variable neighborhood search algorithm yields high-quality solutions for realistic test instances.     

Delivery strategies for blood products supplies

We introduce a problem faced by the blood bank of the Austrian Red Cross for Eastern Austria: how to cost-effectively organize the delivery of blood products to Austrian hospitals? We investigate the potential value of switching from the current vendee-managed inventory set up to a vendor-managed inventory system. We present solution approaches based on integer programming and variable neighborhood search and evaluate their performance.     

A gradient search and column generation approach for the build–pack planning problem with approved vendor matrices and stochastic demand

We study a production planning problem of product assembly with random demand, where the customers choose their preferred suppliers for pairs of inter-dependent components through the approved vendor matrix. The problem is to develop production plans that minimise the expected total shortage and holding costs while observing the matrix restrictions and limited component supplies. We provide a mathematical programming formulation of the problem with a large number of decision variables, whose cost function is the solution of a parametric stochastic transportation problem. We present a gradient-based interior-point approach to solve this problem where the gradient is estimated by the shadow price from the solution of such a transportation problem. A column generation scheme is integrated into the approach to handle the large problem issue. Computational results show that our algorithm significantly improves the computational time when compared with the approach without column generation. In addition, we also discuss some extensions of the basic problem to the multi-period rolling horizon case.     

CONFIGURING BOTH BACKBONE AND LOGICAL NETWORKS OF A RECONFIGURABLE NETWORK WITH LINK FAILURES ALLOWED

This paper considers a problem of configuring both backbone and logical networks in a reconfigurable circuit-switched network where links are subject to failures. The objective is to design feasible backbone and logical networks at least cost where the cost includes backbone link capacity expansion cost, lost-call traffic penalty, and hop cost (nodal processing cost). The problem is formulated as a zero-one non-linear mixed integer programming problem, for which a solution procedure is developed by use of a Lagrangean relaxation technique and heuristic methods exploited for improving the lower and upper bounds of any intermediate solution. The solution procedure is tested for its effectiveness with various numerical examples.     

Capacitated multicommodity network flow problems with piecewise linear concave costs

Lagrangian techniques have been commonly used to solve the capacitated multi-commodity network flow problem with piecewise linear concave costs. In this paper, we show that the resulting lower bounds are no better than those obtained by the linear programming relaxation and focus on developing algorithms based on the latter. For that purpose, we characterize structural properties of the optimal solution of the linear programming relaxation and propose a heuristic solution approach that uses these properties to transform the fractional solution into an integer one. Our computational experiments show the effectiveness of the algorithm.     

Pattern-set generation algorithm for the one-dimensional multiple stock sizes cutting stock problem

A pattern-set generation algorithm (PSG) for the one-dimensional multiple stock sizes cutting stock problem (1DMSSCSP) is presented. The solution process contains two stages. In the first stage, the PSG solves the residual problems repeatedly to generate the patterns in the pattern set, where each residual problem is solved by the column-generation approach, and each pattern is generated by solving a single large object placement problem. In the second stage, the integer linear programming model of the 1DMSSCSP is solved using a commercial solver, where only the patterns in the pattern set are considered. The computational results of benchmark instances indicate that the PSG outperforms existing heuristic algorithms and rivals the exact algorithm in solution quality.     

A hybrid ‘dynamic programming/depth-first search’ algorithm, with an application to redundancy allocation

A hybrid 'dynamic programming/depth-first search' algorithm has been developed to solve non-linear integer programming problems arising in the reliability optimization of redundancy allocation. Initially, the technique solves the knapsack relaxation of the original mathematical programming problem using dynamic programming. Then, all solutions in some range of the relaxation problem are obtained via an enumerative depth-first search technique. The solutions are ranked and the optimal solution is given by the best one that satisfies the remaining constraints of the given problem. Computational complexity of the algorithm is also discussed. The salient features of our hybrid algorithm are its simplicity and ease of programming. Our algorithm also has an advantage over the traditional Lagrangian and surrogate dual approaches. It does not have to deal with the issue of 'duality gap' as in classical dual approaches, which is responsible for the failure to identify optimal solutions to the primal integer optimization problems. Of most importance, it guarantees to succeed in identifying an optimal solution.     

Solving elementary shortest-path problems as mixed-integer programs

Ibrahim et al. (Int Trans Oper Res 16:361–369, 2009) presented and analyzed two integer programming formulations for the elementary shortest-path problem (ESPP), which is known to be NP-hard if the underlying digraph contains negative cycles. In fact, the authors showed that a formulation based on multi-commodity flows possesses a significantly stronger LP relaxation than a formulation based on arc flow variables. Since the ESPP is essentially an integer problem, the contribution of our paper lies in extending this research by comparing the formulations with regard to the computation time and memory requirements required for their integer solution. Moreover, we assess the quality of the lower bounds provided by an integer relaxation of the multi-commodity flow formulation.