Equitable bandwidth allocation in content distribution networks

Applications for content distribution over networks, such as Video‐on‐Demand (VOD), are expected to grow significantly over time. Effective bandwidth allocation schemes that can be repeatedly executed must be deployed since new programs are often installed at various servers while other are deleted. We present a model for bandwidth allocation in a content distribution network that consists of multiple trees, where the root of each tree has a server that broadcasts multiple programs throughout the tree. Each network link has limited capacity and may be used by one or more of these trees. The model is formulated as an equitable resource allocation problem with a lexicographic maximin objective function that attempts to provide equitable service performance for all requested programs at the various nodes. The constraints include link capacity constraints and tree‐like ordering constraints imposed on each of the programs. We present an algorithm that provides an equitable solution in polynomial time for certain performance functions. At each iteration, the algorithm solves single‐link maximin optimization problems while relaxing the ordering constraints. The algorithm selects a bottleneck link, fixes various variables at their lexicographic optimal solution while enforcing the ordering constraints, and proceeds with the next iteration. © 2010 Wiley Periodicals, Inc. Naval Research Logistics, 2010     

Outbound supply chain network design with mode selection and lead time considerations

We present a large‐scale network design model for the outbound supply chain of an automotive company that considers transportation mode selection (road vs. rail) and explicitly models the relationship between lead times and the volume of flow through the nodes of the network. We formulate the problem as a nonlinear zero‐one integer program, reformulate it to obtain a linear integer model, and develop a Lagrangian heuristic for its solution that gives near‐optimal results in reasonable time. We also present scenario analyses that examine the behavior of the supply chain under different parameter settings and the performance of the solution procedures under different experimental conditions. © 2006 Wiley Periodicals, Inc. Naval Research Logistics, 2007     

Heuristic modeling of supply availabilities in large networks

The objective of this article is to describe heuristic solutions to the problem of modeling inventories at each node of a large network in the context of a computer simulation model of that network. The heuristic solutions are compared with the mathematical solution which is too unwieldy for use in a simulation model. The Weibull cumulative distribution is used as an approximation for the heuristic models. We question whether the good performance of the Weibull is coincidence or perhaps mathematically justifiable.     

A nonconvex max-min problem

An algorithm designed to solve a large class of nonconvex max-min problems is described. Its usefulness and applicability is demonstrated by solving an extension of a recently introduced model which optimally allocates strategic weapon systems. The extended model is shown to be equivalent to a nonconvex mathematical program with an infinite number of constraints, and hence is not solvable by conventional procedures. An example is worked out in detail to illustrate the algorithm.     

Setting military reenlistment bonuses

The United States military frequently has difficulty retaining enlisted personnel beyond their initial enlistment. A bonus program within each service, called a Selective Reenlistment Bonus (SRB) program, seeks to enhance reenlistments and thus reduce personnel shortages in critical military occupational specialties (MOSs). The amount of bonus is set by assigning “SRB multipliers” to each MOS. We develop a nonlinear integer program to select multipliers which minimize a function of deviations from desired reenlistment targets. A Lagrangian relaxation of a linearized version of the integer program is used to obtain lower bounds and feasible solutions. The best feasible solution, discovered in a coordinate search of the Lagrangian function, is heuristically improved by apportioning unexpended funds. For large problems a heuristic variable reduction is employed to speed model solution. U.S. Army data and requirements for FY87 yield a 0-1 integer program with 12,992 binary variables and 273 constraints, which is solved within 0.00002% of optimality on an IBM 3033AP in less than 1.7 seconds. More general models with up to 463,000 binary variables are solved, on average, to within 0.009% of optimality in less than 1.8 minutes. The U.S. Marine Corps has used a simpler version of this model since 1986. © 1993 John Wiley & Sons, Inc.     

A linear-programming-based method for determining whether or not n demand points are on a hemisphere

Whenever n demand points are located on a hemisphere, spherical location problems can be solved easily using geometrical methods or mathematical programming. A method based on a linear programming formulation with four constraints is presented to determine whether n demand points are on a hemisphere. The formulation is derived from a modified minimax spherical location problem whose Karush-Kuhn-Tucker conditions are the constraints of the linear program. © 1993 John Wiley & Sons, Inc.     

Optimization of strategic defenses to provide specified post-attack production capacities

This paper presents a model for choosing a minimum-cost mix of strategic defenses to assure that specified production capacities for several economic sectors survive after a nuclear attack. The defender selects a mix of strategic defenses for each of several geographic regions. The attacker chooses an allocation of attacking weapons to geographic regions, within specified weapon inventories. The attack is optimized against any economic sector. This formulation allows the defense planner the capability to assess the results of the optimal defense structure for a “worst case” attack. The model is a mathematical program with nonlinear programming problems in the constraints; an example of its application is given and is solved using recently developed optimization techniques.     

The frequency assignment problem: A solution via nonlinear programming

This paper gives a mathematical programming model for the problem of assigning frequencies to nodes in a communications network. The objective is to select a frequency assignment which minimizes both cochannel and adjacent-channel interference. In addition, a design engineer has the option to designate key links in which the avoidance of jamming due to self interference is given a higher priority. The model has a nonconvex quadratic objective function, generalized upper-bounding constraints, and binary decision variables. We developed a special heuristic algorithm and software for this model and tested it on five test problems which were modifications of a real-world problem. Even though most of the test problems had over 600 binary variables, we were able to obtain a near optimum in less than 12 seconds of CPU time on a CDC Cyber-875.     

The conjugate gradient technique for certain quadratic network problems

We consider a class of network flow problems with pure quadratic costs and demonstrate that the conjugate gradient technique is highly effective for large-scale versions. It is shown that finding a saddle point for the Lagrangian of an m constraint, n variable network problem requires only the solution of an unconstrained quadratic programming problem with only m variables. It is demonstrated that the number of iterations for the conjugate gradient algorithm is substantially smaller than the number of variables or constraints in the (primal) network problem. Forty quadratic minimum-cost flow problems of various sizes up to 100 nodes are solved. Solution time for the largest problems (4,950 variables and 99 linear constraints) averaged 4 seconds on the CBC Cyber 70 Model 72 computer.     

A manpower planning/capital budgeting model (MAPCAB)

A deterministic resource allocation model is developed to optimize defense effectiveness subject to budget, manpower, and risk constraints. The model consists of two major submodels connected by a heuristic. The first is a mathematical program which optimizes the multiperiod weapon mix subject to the constraint set. The second is a manpower supply model based on a transition matrix in which individual transitions are functions of personnel related budgets and historical transition rates. The heuristic marries the submodels through an iterative process leading to improved solutions. An example is provided which demonstrates how systems are undercosted and overprocured if manpower supply is not properly reflected relative to manpower demand.     

A single-period,multi-echelon stochastic model under a mix of assemble to order and assemble in advance policies

This article presents a stochastic model for a single-period production system composed of several assembly/processing and storage facilities in series. The production system operates under a composite strategy of the assemble to order and assemble in advance policies. The developed mathematical model is simpler and more compact than the ones provided in earlier articles. Moreover, the formulation allows the optimal inventory levels at the start of the period to be determined from the solution to the well-known newsboy problem. We also analyze the problem under the free distribution approach which only assumes the knowledge of the first two moments of the demand distribution. The robustness of this approach is tested by carrying an extensive experimental comparison using different demand distributions. Finally, the composite model is extended by considering the effects of some budgetary constraints. © 1998 John Wiley & Sons, Inc. Naval Research Logistics 45: 599–614, 1998     

Generation and storage dispatch in electricity networks with generator disruptions

We present methods for optimizing generation and storage decisions in an electricity network with multiple unreliable generators, each colocated with one energy storage unit (e.g., battery), and multiple loads under power flow constraints. Our model chooses the amount of energy produced by each generator and the amount of energy stored in each battery in every time period in order to minimize power generation and storage costs when each generator faces stochastic Markovian supply disruptions. This problem cannot be optimized easily using stochastic programming and/or dynamic programming approaches. Therefore, in this study, we present several heuristic methods to find an approximate optimal solution for this system. Each heuristic involves decomposing the network into several single‐generator, single‐battery, multiload systems and solving them optimally using dynamic programming, then obtaining a solution for the original problem by recombining. We discuss the computational performance of the proposed heuristics as well as insights gained from the models. © 2015 Wiley Periodicals, Inc. Naval Research Logistics 62: 493–511, 2015     

Contract award analysis by mathematical programming

ABSTRACT A large manufacturer of telephone directories purchases about 100,000 tons of paper annually from several paper mills on the basis of competitive bids. The awards are subject to several constraints. The principal company constraint is that the paper must be purchased from at least three different suppliers. The principal external constraints are: 1) one large paper mill requires that if contracted to sell the company more than 50,000 tons of paper, it must be enabled to schedule production over the entire year; 2) the price of some bidders is based on the condition that their award must exceed a stipulated figure. The paper shows that an optimal purchasing program corresponds to the solution of a model which, but for a few constraints, is a linear programming formulation with special structure. The complete model is solved by first transforming it into an almost transportation type problem and then applying several well-known L.P. techniques.     

Machine sequencing: Disjunctive graphs and degree-constrained subgraphs

In two earlier papers, we proposed algorithms for finding an optimal sequence of processing m items on q machines, by finding a minimaximal path in a disjunctive network. In a third paper, this latter model was generalized (from 2-state to 3-state disjunctive graphs) so as to accommodate project scheduling with resource constraints. In this paper, we discuss another algorithm for the (2-state) disjunctive network problem, closely related to those mentioned above. To make the paper self-contained, section 2 briefly describes the problem. Section 3 introduces a class of constraints which forms the basis of the algorithm discussed in section 4. The constraints have only 1, ?1, or 0 as coefficients on the left-hand side, integers on the right-hand side. The whole procedure of generating these constraints and finding a feasible solution whenever a new constraint is added, can be interpreted (section 5) as a process of generating a graph with degree-constraints on its nodes, and then finding a subgraph satisfying the degree-constraints. The nodes of the graph are generated by solving a critical-path-problem, the feasible subgraphs are found by implicit enumeration.     

Optimal weapons allocation against layered defenses

We solve the problem of optimal allocation of weapons to targets in the presence of layered regional defenses. The general solution technique is an integer program transformable to a minimum-cost network flow. This model assumes the defense has perfect weapons. Results of a small sample scenario are included. Additionally, a representative attrition algorithm is described and the two models combined to form a hybrid algorithm. The hybrid algorithm allows for less-than-perfect weapons while maintaining optimality.     

Local search techniques for the generalized resource constrained project scheduling problem

In this article we address the problem of scheduling a single project network with both precedence and resource constraints through the use of a local search technique. We choose a solution definition which guarantees precedence feasibility, allowing the procedure to focus on overcoming resource infeasibility. We use the 110-problem data set of Patterson to test our procedure. Our results indicate a significant improvement over the best heuristic results reported to date for these problems (Bell and Han [1]). Two major advantages of the local search algorithm are its ability to handle arbitrary objective functions and constraints and its effectiveness over a wide range of problem sizes. We present a problem example with an objective function and resource constraints which include nonlinear and non-continuous components, which are easily considered by the procedure. The results of our algorithm are significantly better than random solutions to the problem. © 1993 John Wiley & Sons, Inc.     

大型舰船电力系统网络重构研究

大型舰船电力系统的出现使供电方式由原来的辐射状向环形发展,其网络重构是一个崭新的问题.文中给出了大型舰船电力系统的网络拓扑结构并综合考虑了线路容量限制、节点电压约束及供电负荷优先性的重构数学模型,保证了各种情况下最大限度地给负荷供电.将规划遗传算法应用于所建模型的求解,并应用算例验证了此解法的可行性.     

The one-period,N-location distribution problem

This paper studies the one-period, general network distribution problem with linear costs. The approach is to decompose the problem into a transportation problem that represents a stocking decision, and into decoupled newsboy problems that represent the realization of demand with the usual associated holding and shortage costs. This approach leads to a characterization of optimal policies in terms of the dual of the transportation problem. This method is not directly suitable for the solution for large problems, but the exact solution for small problems can be obtained. For the numerical solutions of large problems, the problem has been formulated as a linear program with column generation. This latter approach is quite robust in the sense that it is easily extended to incorporate capacity constraints and the multiproduct case.     

The payment scheduling problem

Large complicated projects with interdependent activities can be described by project networks. Arcs represent activities, nodes represent events, and the network's structure defines the relation between activities and events. A schedule associates an occurrence time with each event: the project can be scheduled in several different ways. We assume that a known amount of cash changes hands at each event. Given any schedule the present value of all cash transactions can be calculated. The payment scheduling problem looks for a schedule that maximizes the present value of all transactions. This problem was first introduced by Russell [2]; it is a nonlinear program with linear constraints and a nonconcave objective. This paper demonstrates that the payment scheduling problem can be transformed into an equivalent linear program. The linear program has the structure of a weighted distribution problem and an efficient procedure is presented for its solution. The algorithm requires the solution of triangular systems of equations with all matrix coefficients equal to ± or 0.     

基于位置信息的无线传感网络路由改进

基于位置信息的无线传感器网络如果完全采用无状态的方式路由,在某些拓扑结构中存在数据不可到达的缺陷,即死亡点的问题.针对这一问题,将无线传感器网络抽象为一数学模型,在此模型的基础上,给出了问题的数学描述.经分析,提出了一种基于逃离算法的详细解决方案.对算法的性能从理论上分析得出算法没有环路,是正确可行的,而仿真结果也验证了这一点,算法的时间复杂度为O(n).