Combination Of Linear Programming Research Articles

Inverse Problems of Matroid Intersection

In this paper we study the inverse problem of matroid intersection: Two matroids M 1 = (E, $${\mathcal{I}}$$ 1) and M 2 = (E, $${\mathcal{I}}$$ 2), their intersection B, and a weight function w on E are given. We try to modify weight w, optimally and with bounds, such that B becomes a maximum weight intersection under the modified weight. It is shown in this paper that the problem can be formulated as a combinatorial linear program and can be further transformed into a minimum cost circulation problem. Hence it can be solved by strongly polynomial time algorithms. We also give a necessary and sufficient condition for the feasibility of the problem. Finally we extend the discussion to the version of the problem with Multiple Intersections.

A fully polynomial epsilon approximation cutting plane algorithm for solving combinatorial linear programs containing a sufficiently large ball

A cutting plane algorithm is presented for finding ε-approximate solutions to integer programs contained in the unit hypercube and represented by a separation oracle. Under the assumption that a polynomially bounded ball is contained in the feasible region of the problem, it is demonstrated that the algorithm is an oracle fully polynomial ε approximation scheme. Implications of the result for 0/1 integer programming are discussed.

A Planning Model for the Fuerte-Carrizo Irrigation System, Mexico

A methodology is presented for planning the operation of the Fuerte-Carrizo irrigation system in northwest Mexico. The system has two storage dams, two irrigation districts, and water transfer capabilities between both dams. The methodology uses a combination of linear programming (LP) and simulation. The LP model maximizes the net return of the farmers, subject to restrictions of the system, availability of water and land, and water transfer relationships. The simulation model is programmed as a microcomputer interactive package simulating the performance of the system. The methodology has proven to be a useful tool to assist those responsible for the operation of the irrigation system.

LP-Neuro Method—A New Approach for Solving Neural Network Problems

In this work, a new algorithm called LP-Neuro method is developed to solve artificial neural network problems. In this algorithm, the weights are obtained by a combination of linear programming having a sparse coefficient matrix and a single variable nonlinear optimization routine. The results are illustrated by solving three different problems, two of which are useful in the on-line control of robotic manipulators.

On the Complexity of a Cutting Plane Algorithm for Solving Combinatorial Linear Programs

A Cutting plane algorithm is presented for solving combinatorial linear programs—integer programs with 0/1 vertices represented by a separation oracle. The algorithm is a standard cutting plane method but uses a prescribed dual criterion for choosing a cut at each iteration. As a result, it is possible to demonstrate that for problems containing a ball whose size is polynomially bounded from below, there exists an algorithm polynomial in the running time of the separation oracle and pseudopolynomial in the size of the objective function. In particular, the cardinality versions of many combinatorial optimization problems are shown to be solvable in polynomial time using this generic algorithm.

Resolving degeneracy in combinatorial linear programs: Steepest edge, steepest ascent, and parametric ascent

While variants of the steepest edge pivot rule are commonly used in linear programming codes they are not known to have the theoretically attractive property of avoiding an infinite sequence of pivots at points of degeneracy. An example is presented demonstrating that the steepest edge pivot rule can fail to terminate finitely. It is then shown that a natural extension of the steepest edge pivot rule based on steepest ascent is guaranteed to determine a direction of ascent or a proof that no such direction exists after a finite number of pivots, although without modification the extension may not terminate with an ascent direction corresponding to an edge. Finally, it is demonstrated that a computationally more efficient pivot rule proposed by Magnanti and Orlin has similar theoretical properties to steepest ascent with probability 1independent of the linear program being solved. Unlike alternative methods such as primal lexicographic rules and Bland's rule, the algorithms described here have the advantage that they choose the pivot element without explicit knowledge of the set of all active constraints at a point of degeneracy, thus making them attractive in combinatorial settings where the linear program is represented implicitly.

Time partitioning based on the job-flow schedule — A new approach to optimum allocation of jobs on machine tools

A time-partitioning scheme with constant job-mix divisions (“stages”) is presented. The scheme is the basis for using a combination of linear programming and heuristic to solve the dynamic optimisation problem of job allocation through an algorithm which also incorporates the set-up times. The algorithm uses a pull logistic towards the due dates. The results of a case study of a single operation job-shop, indicate increased utilisation of machine tools, reduced work-in-process inventory and additional free capacity owing to reduced makespan of jobs. Application of the algorithm can be extended to multiple-operation job shops, which are common in industrial practice.

License class design: complexity and algorithms

In this paper a generalization of the Fixed Job Scheduling Problem (FSP) is considered, which appears in the aircraft maintenance process at an airport. A number of jobs have to be carried out, where the main attributes of a job are a fixed start time, a fixed finish time and an aircraft type. For carrying out these jobs a number of engineers are available. An engineer is allowed to carry out a specific job only if he has a license for the corresponding aircraft type. Furthermore, the jobs must be carried out in a non-preemptive way and each engineer can be carrying out at most one job at the same time. Within this setting natural questions to be answered ask for the minimum number of engineers required for carrying out all jobs or, more generally, for the minimum total costs for hiring engineers. In this paper a complete classification of the computational complexity of two classes of mathematical problems related to these practical questions is given. Furthermore, it is shown that the polynomially solvable cases of these problems can be solved by a combination of Linear Programming and Network Flow algorithms.

Large scale optimization of beam weights under dose-volume restrictions

The problem of choosing weights for beams in a multifield plan which maximizes tumor dose under conditions that recognize the volume dependence of organ tolerance to radiation is considered, and its solution described. Structures are modelled as collections of discrete points, and the weighting problem described as a combinatorial linear program (LP). The combinatorial LP is solved as a mixed 0/1 integer program with appropriate restrictions on normal tissue dose. The method is illustrated through the assignment of weights to a set of 10 beams incident on a pelvic target. Dose-volume restrictions are placed on surrounding bowel, bladder, and rectum, and a limit placed on tumor dose inhomogeneity. Different tolerance restrictions are examined, so that the sensitivity of the target dose to changes in the normal tissue constraints may be explored. It is shown that the distributions obtained satisfy the posed constraints. The technique permits formal solution of the optimization problem, in a time short enough to meet the needs of treatment planners.

Optimization of beam weights under dose-volume restrictions

A basic problem in treatment planning is the selection of weights for a set of beams which will yield the largest tumor dose under constraints limiting the doses received in specified fractions of different normal tissue structures. This report describes a method for formulating and solving this optimization problem as a combinatorial linear program. An illustration is provided by a problem in planning treatment of a thoracic tumor, in which no more than 1 2 or 2 3 of the lung is permitted to receive >20 Gy and no part of the spinal cord allowed to receive >45 Gy. The optimization technique was applied to this example to determine how the maximum tumor dose is affected by changes in the normal tissue constraints and the addition of a tumor dose homogeneity restriction. The linear programming technique yielded a rigorous and efficient determination of the beam weights for the thoracic plan considered. An exhaustive specification of all the underlying linear programs allows problems of moderate dimensions to be solved, while developments in mathematical programming and computer processing suggest approaches to problems of greater complexity.

On eliminating peak load auxiliary energy consumption in passive solar residences during winter

This paper uses simulation analysis with Albuquerque, NM and Madison, WI weather data to address the following four questions: o 1. Assuming a properly designed and controlled passive house, how does total electrical auxiliary energy consumption depend on the choice of peak load blackout periods? 2. What comfort penalties arise from properly coping with a variety of peak load blackout periods? 3. Where should off-peak energy be introduced into the thermal masses of the house? 4. What are the effects of imprecise information on short term future weather? Using a combination of linear programming and gradient techniques, the following conclusions are obtained: During the worst days of the heating season, passive solar houses built above ground use a substantial amount of peak backup energy even if they are well designed. Even relatively crude off-peak controls provide reasonable comfort provided the energy is introduced in thermal masses well coupled to the room [e.g. 0.05 m (2 inches) beneath the inside face of the Trombe wall or 0.05 m (2 inches) below the top of the slab floor]. Introducing off-peak energy in thermal masses poorly coupled to the room (e.g. deep in the floor slab) makes proper control very difficult without very accurate weather prediction. Reducing backup use to zero from 7 am to 10 pm requires a doubling of daily backup use in Albuquerque and Madison. Excluding backup for shorter periods (e.g. 5 pm to 9 pm) requires an increase of about 25% in daily backup consumption. Even if off-peak energy is stored at points reasonably well coupled to the room, significant backup and comfort penalties are incurred with erroneous weather forecasts. Even in two adjacent days in Madison, reversing the weather patterns while maintaining the same off peak control strategies resulted in either wasting half the backup energy or severely under heating the house. The effects of faulty weather forecasts are more severe when poorly coupled storage sites are used.

A Strongly Polynomial Algorithm to Solve Combinatorial Linear Programs

Khachiyan, and recently Karmarkar, gave polynomial algorithms to solve the linear programming problem. These algorithms have a small theoretical drawback; namely, the number of arithmetic steps depends on the size of the input numbers. We present a polynomial linear programming algorithm whose number of arithmetic steps depends only on the size of the numbers in the constraint matrix, but is independent of the size of the numbers in the right-hand side and objective vectors. In particular, it gives a polynomial algorithm for the minimum cost flow and multicommodity flow problems in which the number of arithmetic steps is independent of the size of the costs and capacities. The algorithm makes use of an existing polynomial linear programming algorithm. The problem of whether any algorithm has a running time that is independent even of the size of the numbers in the constraint matrix remains open.

A Strongly Polynomial Algorithm to Solve Combinatorial Linear Programs

Khachiyan, and recently Karmarkar, gave polynomial algorithms to solve the linear programming problem. These algorithms have a small theoretical drawback; namely, the number of arithmetic steps dep...

Algorithms for Optimizing Hydropower System Operation

Successive linear programming, an optimal control algorithm, and a combination of linear programming and dynamic programming (LP‐DP) are employed to optimize the operation of multireservoir hydrosystems given a deterministic inflow forecast. The algorithm maximize the value of energy produced at on‐peak and off‐peak rates, plus the estimated value of water remaining in storage at the end of the 12‐month planning period. The LP‐DP algorithm is clearly dominated: it takes longer to find a solution and produces significantly less hydropower than the other two procedures. Successive linear programming (SLP) appears to find the global maximum and is easily implemented. For simple systems the optimal control algorithm finds the optimum in about one fifth the time required by SLP but is harder to implement. Computing costs for a two‐reservoir, 12‐month deterministic problem averaged about seven cents per run using optimal control and 37 cents using successive linear programming.

Reservoir Operation: Choice of Objective Functions

An efficient algorithm for the real‐time monthly operation of a multipurpose reservoir is presented. The model is a combination of linear programming (used for month‐by‐month optimization) and dynamic programming (used for annual optimization). The use of parametric linear programming, minimum required beginning‐of‐month storages, and an iterative solution procedure result in low computer time and computer storage requirements. Low computer storage requirements allow the model to be run on minicomputers. Water and energy maximization, water and energy maximization with flood control considerations, and water and energy maximization for peak demand months are considered. Thus, the model provides the reservoir operator with different choices for annual optimization. The model is applied to Folsom reservoir of the California Central Valley Project and the results are discussed.

The use of linear programming in the analysis of petrological mixing problems

Petrological mixing problems such as modal analysis, magma mixing, and liquid line of descent calculations, can be solved using the methods of linear programming. If estimates of the standard error of the chemical data are introduced as weights into the set of equations, it is possible to assign confidence limits to the solutions which are obtained and to apply formal statistical tests to geological hypotheses based on the mixing model. This approach is applied to petrological data previously analysed by Wright and Doherty (1970) using a combination of linear programming and least squares methods. It is shown that some of the geological inferences which they drew were based on an overoptimistic assessment of the confidence limits on their solutions, and cannot be regarded as proven.

An interactive multiple-objective linear programming approach to a problem in forest management

In many situations it is under legislative mandate to manage publicly owned forest resources for multiple uses (e.g., timber production, hunting, grazing). The major obstacle that has been encountered in applying previously developed mathematical programming procedures to multiple-use forest management has been the difficulty in assessing the appropriate criterion weights required. To avoid the criterion weight estimation problem, an interactive multiple-objective linear programming approach, which does not require criterion weights of any kind, was developed in response to the needs of the multiple-use forest management problem. The procedure uses a combination of linear programming and vector-maximum techniques. At each iteration the cone generated by the gradients of the multiple objectives is contracted. On the last two iterations the most acceptable efficient extreme point is identified with the aid of a filtering device. As illustrated, the method has been applied to prepare preliminary management plans for a 10 000-acre subunit of a national forest.

Practice of Management Science—Orderbook Balancing Using a Combination of Linear Programming and Heuristic Techniques

In complex manufacturing situations, when product demand exceeds the manufacturing capacity by a very wide margin, both the production planning and scheduling functions become extremely difficult and time-consuming to handle—even with modern computers and sophisticated analytical techniques. Our studies had shown that in such cases the tasks of production scheduling and detailed planning could be made easier if the orderbook were first balanced in an aggregate manner such that monthly or other periodic demands were approximately in line with the manufacturing capabilities. We applied this basic approach in the development of an improved planning tool for Bethlehem Steel's roll manufacturing operations, where approximately 4,000–6,000 cast rolls of 300 different types, each requiring an average of ten operations, are machined in a large m/n job shop each year. This planning tool, which is a hybrid computer package combining linear programming and heuristic methods, is being routinely used to balance the roll orderbook. The quick estimate of shop capacity and due-date performance provided by this package gives shop management an improved ability to negotiate roll delivery and resolve order-distribution problems with the plants. Due dates are changed and/or orders are dropped during one or more rounds of negotiations with each plant, and a balanced, or feasible, orderbook for the plants as a whole is obtained. Accurate production schedules and long-term plans are then generated by elaborate simulation models.

An Interactive Multiple-Objective Linear Programming Approach to a Problem in Forest Management

In many situations it is under legislative mandate to manage publicly owned forest resources for multiple uses (e.g., timber production, hunting, grazing). The major obstacle that has been encountered in applying previously developed mathematical programming procedures to multiple-use forest management has been the difficulty in assessing the appropriate criterion weights required. To avoid the criterion weight estimation problem, an interactive multiple-objective linear programming approach, which does not require criterion weights of any kind, was developed in response to the needs of the multiple-use forest management problem. The procedure uses a combination of linear programming and vector-maximum techniques. At each iteration the cone generated by the gradients of the multiple objectives is contracted. On the last two iterations the most acceptable efficient extreme point is identified with the aid of a filtering device. As illustrated, the method has been applied to prepare preliminary management plans for a 10,000-acre sub-unit of a national forest.

A Minimum-Cost Feature-Selection Algorithm for Binary-Valued Features

An algorithm to select the minimum-cost collection of binary-valued features for use with a linear pattern classifier is presented. The feature-selection algorithm is motivated by the convex-hull representation of pattern-space separability. Combinatorial analysis and linear programming are used to find the minimum-cost collection of binary-valued features associated with a given set of preclassified patterns. A description of the interaction between these algorithm components is provided. The algorithm guarantees that its optimal feature set will correctly classify every pattern in the classifier's training sample. Coinputational considerations associated with algorithm use are discussed. An application of the algorithm to a three-feature classifier is presented in detail.