An Application of Branch and Cut to Open Pit Mine Scheduling

The economic viability of the modern day mine is highly dependent upon careful planning and management. Declining trends in average ore grades, increasing mining costs and environmental considerations will ensure that this situation will remain in the foreseeable future. The operation and management of a large open pit mine having a life of several years is an enormous and complex task. Though a number of optimization techniques have been successfully applied to resolve some important problems, the problem of determining an optimal production schedule over the life of the deposit is still very much unresolved. In this paper we will critically examine the techniques that are being used in the mining industry for production scheduling indicating their limitations. In addition, we present a mixed integer linear programming model for the scheduling problems along with a Branch and Cut solution strategy. Computational results for practical sized problems are discussed.     

Multicriteria integer programming: A (hybrid) dynamic programming recursive approach

Dynamic programming recursive equations are used to develop a procedure to obtain the set of efficient solutions to the multicriteria integer linear programming problem. An alternate method is produced by combining this procedure with branch and bound rules. Computational results are reported.     

Scheduling inspired models for two-dimensional packing problems

We propose two exact algorithms for two-dimensional orthogonal packing problems whose main components are simple mixed-integer linear programming models. Based on the different forms of time representation in scheduling formulations, we extend the concept of multiple time grids into a second dimension and propose a hybrid discrete/continuous-space formulation. By relying on events to continuously locate the rectangles along the strip height, we aim to reduce the size of the resulting mathematical problem when compared to a pure discrete-space model, with hopes of achieving a better computational performance. Through the solution of a set of 29 test instances from the literature, we show that this was mostly accomplished, primarily because the associated search strategy can quickly find good feasible solutions prior to the optimum, which may be very important in real industrial environments. We also provide a comprehensive comparison to seven other conceptually different approaches that have solved the same strip packing problems.     

Iterated Local Search for Fiber Installation in Optical Network Optimization

The problem of Fiber Installation in Optical Network Optimization consists in routing a set of lightpaths (all-optical connections), such that the cost of the optical components necessary to operate the network is minimized. We propose a novel Iterated Local Search heuristic. Computational results showed that the new heuristic is better than the best heuristic in the literature.     

Comparison of wavelength requirements between two wavelength assignment methods in survivable WDM networks

We consider the routing and wavelength assignment (RWA) in survivable WDM network. A path protection scheme assumed and two different wavelength assignment methods for protection paths are considered. Integer programming formulations of RWA under two wavelength assignment methods are proposed and we devised algorithms to solve them. Test results show that the difference of wavelength requirements between two wavelength assignment methods is 5–30–     

A hybrid approach to discrete mathematical programming

The dynamic programming and branch-and-bound approaches are combined to produce a hybrid algorithm for separable discrete mathematical programs. Linear programming is used in a novel way to compute bounds. Every simplex pivot permits a bounding test to be made on every active node in the search tree. Computational experience is reported.     

Simplex pivots on the set packing polytope

This paper considers the set packing problem max{wx: Ax b, x 0 and integral}, whereA is anm × n 0–1 matrix,w is a 1 ×n weight vector of real numbers andb is anm × 1 vector of ones. In equality form, its linear programming relaxation is max{wx: (x, y) P(A)} whereP(A) = {(x, y):Ax +I _m y =b, x0,y0}. Letx ¹ be any feasible solution to the set packing problem that is not optimal and lety ¹ =b – Ax ¹; then (x ¹,y ¹) is an integral extreme point ofP(A). We show that there exists a sequence of simplex pivots from (x ¹,y ¹) to (x*,y*), wherex* is an optimal solution to the set packing problem andy* =b – Ax*, that satisfies the following properties. Each pivot column has positive reduced weight and each pivot element equals plus one. The number of pivots equals the number of components ofx* that are nonbasic in (x ¹,y ¹).This research was supported by NSF Grants ECS-8005360 and ECS-8307473 to Cornell University.     

An inexact approach to solving linear semi-infinite programming problems

In this paper, we propose an “inexact solution” approach to deal with linear semi-infinite programming problems with finitely many variables and infinitely many constraints over a compact metric space. A general convergence proof with some numerical examples are given and the advantages of using this approach are discussed     

Home Care optimization: impact of pattern generation policies on scheduling and routing decisions

In Home Care optimization, operators have to be assigned to patients by taking into account compatibility skill constraints, and patient visits have to be scheduled in a given planning horizon. Moreover, operator tours have to be determined. Integer Linear Programming models have been proposed which use the concept of patterns, i.e. a priori scheduling profiles, to combine the diverse decision levels. Computational results on real instances show that pattern generation policies are crucial to address scheduling and routing in large Home Care instances.     

(1,k)-configurations and facets for packing problems

A new class of facets for knapsack polytopes is obtained. This class of inequalities is shown to define a polytope with zero–one vertices only. A combinatorial inequality is obtained from Fulkerson's max—max inequality.     

A simplex algorithm for piecewise-linear programming I: Derivation and proof

The simplex method for linear programming can be extended to permit the minimization of any convex separable piecewise-linear objective, subject to linear constraints. This three-part paper develops and analyzes a general, computationally practical simplex algorithm for piecewiselinear programming.Part I derives and justifies the essential steps of the algorithm, by extension from the simplex method for linear programming in bounded variables. The proof employs familiar finite-termination arguments and established piecewise-linear duality theory.Part II considers the relaxation of technical assumptions pertaining to finiteness, feasibility and nondegeneracy of piecewise-linear programs. Degeneracy is found to have broader consequences than in the linear case, and the standard techniques for prevention of cycling are extended accordingly.Part III analyzes the computational requirements of piecewise-linear programming. The direct approach embodied in the piecewise-linear simplex algorithm is shown to be inherently more efficient than indirect approaches that rely on transformation of piecewise-linear programs to equivalent linear programs. A concluding section surveys the many applications of piecewise-linear programming in linear programming,l ₁ estimation, goal programming, interval programming, and nonlinear optimization.This research has been supported in part by the National Science Foundation under grant MCS-8217261.     

A generalized linear production model: A unifying model

We introduce a generalized linear production model whose attractive feature being that the resources held by any subset of producersS is not restricted to be the vector sum of the resources held by the members ofS. We provide sufficient conditions for the non-emptiness of the core of the associated generalized linear production game, and show that if the core of the game is not empty then a solution in it can be produced from a dual optimal solution to the associated linear programming problem. Our generalized linear production model is a proper generalization of the linear production model introduced by Owen, and it can be used to analyze cooperative games which cannot be studied in the ordinary linear production model framework. We use the generalized model to show that the cooperative game induced by a network optimization problem in which players are the nodes of the network has a non-empty core. We further employ our model to prove the non-emptiness of the core of two other classes of cooperative games, which were not previously studied in the literature, and we also use our generalized model to provide an alternative proof for the non-emptiness of the core of the class of minimum cost spanning tree games. Thus, it appears that the generalized linear production model is a unifying model which can be used to explain the non-emptiness of the core of cooperative games generated by various, seemingly different, optimization models.This research was partially done while the author was visiting the Graduate School of Business Administration at Tel-Aviv University. The research was partially supported by Natural Sciences and Engineering Research Council Canada Grant A4181 and by SSHRC leave fellowship 451-83-0030.Dedicated to George B. Dantzig.     

Solution of the linear inverse vector optimization problem by a single linear program

It is shown that finding a solution to a linear vector optimization problem which is efficient with respect to the constraints as well as to the objectives is equivalent to solving a single linear program.The research of this author was supported by NSF Grant DCR74-20584.The research of this author was partially supported by Canada Council Grant W760467.     

Constrained integer linear fractional programming problem

An integer linear fractional programming problem, whose integer solution is required to satisfy any h out of given n sets of constraints has been discussed in this paper. Method for ranking and scanning all integer points has also been developed and a numerical illustration is included in support of theory.     

A new algorithm for solving variational inequalities with application to the traffic assignment problem

The variational inequality problem in Euclidian space is formulated as a nonconvex, nondifferentiable optimization problem. We show that any stationary point is optimal, and we propose a solution algorithm that decreases the nondifferential objective monotonically. Application to the asymmetric traffic assignment problem is considered.Research supported by C.R.S.H. (Canada) grant #410-81-0722-RL and F.C.A.C. (Québec) grant # 83-AS-0026.     

Affirmative action algorithms

Affirmative action is a new variety of selection rule which employs historical information to favor the choice of elements that have not been selected in the past. We categorize three implementations of this principle and discuss their application to the simplex method, to Bard-type schemes for the linear complementarity problem, and to augmenting path methods for network flow problems. We present analytical and computational results, and some open questions.Research supported by the Georgia Institute of Technology.Research supported by the New Faculty Research Development Program of the Georgia Institute of Technology. Reproduction in whole or in part is permitted for any purpose of the U.S. Government.     

An ILP based hierarchical global routing approach for VLSI ASIC design

The use of integrated circuits in high-performance computing, telecommunications, and consumer electronics has been growing at a very fast pace. The level of integration as measured by the number of logic gates in a chip has been steadily rising due to the rapid progress in processing and interconnect technology. The interconnect delay in VLSI circuits has become a critical determiner of circuit performance. As a result, circuit layout is starting to play a more important role in today’s chip designs. Global routing is one of the key sub-problems of circuit layout which involves finding an approximate path for the wires connecting the elements of the circuit without violating resource constraints. In this paper, several integer programming (ILP) based global routing models are fully investigated and explored. The resulting ILP problem is relaxed and solved as a linear programming (LP) problem followed by a rounding heuristic to obtain an integer solution. Experimental results obtained show that the proposed combined WVEM (wirelength, via, edge capacity) model can optimize several global routing objectives simultaneously and effectively. In addition, several hierarchical methods are combined with the proposed flat ILP based global router to reduce the CPU time by about 66% on average for edge capacity model (ECM).     

贷款分配方法的数学模型及其解法

为了解决三角债问题，银行给出一笔贷款，应该如何分配这笔贷款，使清理的债务达到最大。本文建立这个问题的数学模型，并且给出一个解法。     

Solving staircase linear programs by the simplex method, 2: Pricing

This and a companion paper consider how current implementations of the simplex method may be adapted to better solve linear programs that have a staged, or ‘staircase’, structure. The preceding paper considered ‘inversion’ routines that factorize the basis and solve linear systems. The present paper examines ‘pricing’ routines that compute reduced costs for nonbasic variables and that select a variable to enter the basis at each iteration. Both papers describe extensive (although preliminary) computer experiments, and can point to some quite promising results. For pricing in particular, staircase computation strategies appear to offer modest but consistent savings; staircase selection strategies, properly chosen, may offer substantial savings in number of iterations, time per iteration, or both.     

Solving the Two-Stage Robust FTTH network design Problem under Demand Uncertainty

For the past few years, the increase in high bandwidth requiring services forced telecommunication operators like France Telecom - Orange to engage the deployment of optical networks, the Fiber To The Home Gigabit Passive Optical Network (FTTH GPON) technology, leading to new design problems. Such problems have already been studied. However, to the best of our knowledge, without taking into account the future demand uncertainty. In this paper, we propose a model for a two-stage robust optimization FTTH network design problem tackling the demand uncertainty. We propose an exact algorithm, based on column and constraint generation algorithms, and we show some preliminary results.