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Luis Rademacher, Alejandro Toriello and Juan Pablo Vielma
We consider the natural generalizations of blocking and anti-blocking polyhedra in infinite dimensions, and study issues related to duality and integrality of extreme points for these sets. Using appropriate finite truncations we give conditions under which complementary slackness holds for primal-dual pairs of the infinite linear programming problems associated with infinite blocking and anti-blocking polyhedra. We also give conditions under which the extreme points of infinite blocking and anti-blocking polyhedra are integral. We illustrate an application of our results on an infinite-horizon lot-sizing problem.
Submitted for publication, 2014.
Sina Modaresi, Mustafa R. Kılınç and Juan Pablo Vielma
In this paper we study split cuts and extended formulations for Mixed Integer Conic Quadratic Programming (MICQP) and their relation to the Conic Mixed Integer Rounding (CMIR) cuts. We show that the CMIR is a linear split cut for the polyhedral portion of an extended formulation of a quadratic set and that it can be weaker than the nonlinear split cut of the same quadratic set. However, we also show that by exploiting the power of their extended formulation, families of CMIRs can be significantly stronger than the associated family of nonlinear split cuts.
Submitted for publication, 2014.
Sina Modaresi, Mustafa R. Kılınç and Juan Pablo Vielma
We study the generalization of split and intersection cuts from Mixed Integer Linear Programming to the realm of Mixed Integer Non-linear Programming. Constructing such cuts requires calculating the convex hull of the difference of two convex sets with specific geometric structures. We introduce two techniques to give precise characterizations of such convex hulls and use them to construct split and intersection cuts for several classes of sets. In particular, we give simple formulas for split cuts for essentially all convex quadratic sets and for intersection cuts for a wide variety of convex quadratic sets.
Submitted for publication, 2013.
Guido Lagos, Daniel Espinoza, Eduardo Moreno and Juan Pablo Vielma
In this paper we consider the restriction of coherent and distortion risk measures to the spaces of linear and affine linear random variables and the robust uncertainty sets associated to these risk measures. In this context we study two variants of the permutahull uncertainty set introduced by Bertsimas and Brown for finite probability spaces. The first variant considers the extension of the permutahull to non-atomic distributions and the second variant considers scalings of the permutahull. In particular, by studying expansions of the permutahull we are able to show that there are measures that are distortion risk measures over the space of linear random variables that cannot be extended to a distortion risk measures over all random variables. We also present preliminary computational results illustrating that using expansions of the permutahull could help mitigate estimation errors.
Submitted for publication, 2012.
Sajad Modaresi, Denis Saure and Juan Pablo Vielma
We study a family of sequential decision problems under model uncertainty in which, at each period, the decision maker faces a different instance of a combinatorial optimization problem. Instances differ in their objective coefficient vectors, which are unobserved prior to implementation. These vectors however are known to be random draws from a common and initially unknown distribution function. By implementing different solutions, the decision maker extracts information about the objective function, but at the same time experiences the cost associated with said solutions. We show that balancing the implied exploration vs exploitation trade-off requires addressing critical challenges not present in previous studies: in addition to determining exploration frequencies, the decision maker faces the issue of what to explore and how to do so. Our work provides clear answers to both questions. In particular, we show that efficient data collection might be achieved by solving a new class of combinatorial problems, which we refer to as Optimality Cover Problem (OCP). We establish a fundamental limit on the performance of any admissible policy, which relates to the solution to OCP. Using the insight derived from such a bound, we develop a family of policies and establish theoretical guarantees for their performances, which we contrast against the fundamental bound. These policies admit asynchronous versions, ensuring implementability.
This paper won the second prize in the 2013 INFORMS Junior Faculty Interest Group Paper Competition.
Submitted for publication, 2012.
A wide range of problems can be modeled as Mixed Integer Linear Programming (MILP) problems using standard formulation techniques. However, in some cases the resulting MILP can be either too weak or to large to be effectively solved by state of the art solvers. In this survey we review advanced MILP formulation techniques that result in stronger and/or smaller formulations for a wide class of problems.
Submitted for publication, 2012.
Daniel Dadush, Santanu S. Dey and Juan Pablo Vielma
In this paper, we show that the Chvatal-Gomory closure of any compact convex set is a rational polytope. This resolves an open question of Schrijver 1980 for irrational polytopes, and generalizes the same result for the case of rational polytopes (Schrijver 1980), rational ellipsoids (Dey and Vielma 2010) and strictly convex bodies (Dadush, Dey and Vielma 2010).
An extended abstract of this work can be found here.
In 2011 Daniel Dadush received the INFORMS Optimization Society Student Paper Prize for this paper.
This paper was a finalist for the 2011 INFORMS Junior Faculty Interest Group Paper Competition.
Mathematical Programming 145, 2014. pp. 327-348.
[PDF] [DOI:10.1007/s10107-013-0649-9][BibTeX]
Sanjeeb Dash, Oktay Günlük and Juan Pablo Vielma
In this paper, we study whether cuts obtained from two simplex tableau rows at a time can strengthen the bounds obtained by GMI cuts based on single tableau rows. We also study whether cross and crooked cross cuts, which generalize split cuts, can be separated in an effective manner for practical MIPs and can yield a non-trivial improvement over the bounds obtained by split cuts. We give positive answers to both these questions for MIPLIB 3.0 problems. Cross cuts are a special case of the $t$-branch split cuts studied by Li and Richard (2008). Split cuts are 1-branch split cuts, and cross cuts are 2-branch split cuts. Crooked cross cuts were introduced by Dash, Dey and Gunluk (2010) and were shown to dominate cross cuts in Dash, Gunluk and Molinaro (2012).
To appear in INFORMS Journal on Computing, 2013.
Sercan Yıldız and Juan Pablo Vielma
The standard way to represent a choice between n alternatives in Mixed Integer Programming is through n binary variables that add up to one. Unfortunately, this approach commonly leads to unbalanced branch-and-bound trees and diminished solver performance. In this paper, we present an encoding formulation framework that encompasses and expands existing approaches to mitigate this behavior. Through this framework, we generalize the incremental formulation for piecewise linear functions to any finite union of polyhedra with identical recession cones.
Operations Research Letters 41, 2013. pp. 654-658.
[PDF] [DOI:10.1016/j.orl.2013.09.004][BibTeX]
Rodolfo Carvajal, Miguel Constantino, Marcos Goycoolea, Juan Pablo Vielma and Andres Weintraub
Connectivity requirements are a common component of forest planning models, with important examples arising in wildlife habitat protection. In harvest scheduling models, one way of addressing preservation concerns consists in requiring that large contiguous patches of mature forest are maintained. In the context of nature reserve design, it is common practice to select connected regions of forest in such a way as to maximize the number of species and habitats protected. While a number of integer programming formulations have been proposed for these forest planning problems, most are impractical in that they fail to solve reasonably sized scheduling instances. We present a new integer programming methodology and test an implementation of it on five medium-sized forest instances publicly available in the FMOS repository. Our approach allows us to obtain near-optimal solutions for multiple time-period instances in less than four hours.
Operations Research 61, 2013. pp. 824-836.
[PDF] [DOI:10.1287/opre.2013.1183][BibTeX]
Diego Morán, Santanu S. Dey and Juan Pablo Vielma
Mixed-integer conic programming is a generalization of mixed-integer linear programming. In this paper, we present an extension of the duality theory for mixed-integer linear programming to the case of mixed-integer conic programming. In particular, we construct a subadditive dual for mixed-integer conic programming problems. Under a simple condition on the primal problem, we are able to prove strong duality.
In 2012 Diego Morán received the INFORMS Optimization Society Student Paper Prize for this paper.
SIAM Journal on Optimization 22, 2012. pp. 1136-1150.
[PDF] [DOI:10.1137/110840868][BibTeX]
Juan Pablo Vielma, Shabbir Ahmed and George L. Nemhauser
We introduce two new formulations for probabilistic constraints based on extended disjunctive formulations. These formulations obtain significant strength by considering multiple rows of the probabilistic constraints together. The theoretical properties of the first formulation can be used to construct hierarchies of relaxations for probabilistic constraints, while the second formulation provides a computational advantage over other formulations. We illustrate the properties of the formulations with computational results.
Operations Research Letters 40, 2012. pp. 153-158.
[PDF] [DOI:10.1016/j.orl.2012.01.007][BibTeX]
Alejandro Toriello and Juan Pablo Vielma
We consider the problem of fitting a continuous piecewise linear function to a finite set of data points. We review convex continuous fitting models, and then introduce mixed-binary models that generalize the continuous ones by introducing variability in the regions defining the best-fit function's domain. We also study the additional constraints required to impose convexity on the best-fit function.
European Journal of Operational Research 219, 2012. pp. 86-95.
[PDF] [DOI:10.1016/j.ejor.2011.12.030][BibTeX]
Alexandre S. Freire, Eduardo Moreno and Juan Pablo Vielma
We introduce a new Integer Linear Programming (ILP) approach for solving Integer Programming (IP) problems with bilinear objectives and linear constraints. The approach relies on a series of ILP approximations of the bilinear IP. We compare this approach with standard linearization techniques on random instances and a set of real-world product bundling problems.
Operations Research Letters 40, 2012. pp. 74-77.
[PDF] [DOI:10.1016/j.orl.2011.12.004][BibTeX]
Daniel Dadush, Santanu S. Dey and Juan Pablo Vielma
In this paper, we prove that the Chvátal-Gomory closure of a set obtained as an intersection of a strictly convex body and a rational polyhedron is a polyhedron. Thus, we generalize a result of Schrijver which shows that the Chvátal-Gomory closure of a rational polyhedron is a polyhedron.
This paper was a finalist for the 2010 INFORMS Junior Faculty Interest Group Paper Competition.
Mathematics of Operations Research 36, 2011. pp. 227-239.
[PDF] [DOI:10.1287/moor.1110.0488][BibTeX]
Daniel Dadush, Santanu S. Dey and Juan Pablo Vielma
The Chvátal-Gomory closure and the split closure of a rational polyhedron are rational polyhedra. It was recently shown that the Chvátal-Gomory closure of strictly convex body is also a rational polytope. In this note, we show that the split closure of a strictly convex body is defined by a finite number of split disjunctions, but is not necessarily polyhedral. We also give a closed form expression in the original variable space of a split cut for full dimensional ellipsoids.
Operations Research Letters 39, 2011. pp. 121-126.
[PDF] [DOI:10.1016/j.orl.2011.02.002][BibTeX]
Juan Pablo Vielma and George L. Nemhauser
Many combinatorial constraints over continuous variables such as SOS1 and SOS2 constraints can be interpreted as disjunctive constraints that restrict the variables to lie in the union of a finite number of specially structured polyhedra. Known mixed integer binary formulations for these constraints have a number of binary variables and extra constraints linear in the number of polyhedra. We give sufficient conditions for constructing formulations for these constraints with a number of binary variables and extra constraints logarithmic in the number of polyhedra. Using these conditions we introduce mixed integer binary formulations for SOS1 and SOS2 constraints that have a number of binary variables and extra constraints logarithmic in the number of continuous variables. We also introduce the first mixed integer binary formulations for piecewise linear functions of one and two variables that use a number of binary variables and extra constraints logarithmic in the number of linear pieces of the functions. We prove that the new formulations for piecewise linear functions have favorable tightness properties and present computational results showing that they can significantly outperform other mixed integer binary formulations.
An extended abstract of this work can be found here.
Mathematical Programming 128, 2011. pp. 49-72.
[PDF] [DOI:10.1007/s10107-009-0295-4][BibTeX]
Juan Pablo Vielma, Shabbir Ahmed and George L. Nemhauser
This paper studies two Mixed Integer Linear Programming (MILP) formulations for piecewise linear functions considered in Li et al. (2009). Although the ideas used to construct one of these formulations are theoretically interesting and could eventually provide a computational advantage, we show that their use in modeling piecewise linear functions yields a poor MILP formulation. We specifically show that neither of the formulations in Li et al. (2009) have a favorable strength property shared by all standard MILP formulations for piecewise linear functions. We also show that both formulations in Li et al. (2009) are significantly outperformed computationally by standard MILP formulations.
INFORMS Journal on Computing 22, 2010. pp. 493-497.
[PDF] [DOI:10.1287/ijoc.1100.0379][BibTeX]
Juan Pablo Vielma, Shabbir Ahmed and George L. Nemhauser
We study the modeling of non-convex piecewise linear functions as Mixed Integer Programming (MIP) problems. We review several new and existing MIP formulations for continuous piecewise linear functions with special attention paid to multivariate non-separable functions. We compare these formulations with respect to their theoretical properties and their relative computational performance. In addition, we study the extension of these formulations to lower semicontinuous piecewise linear functions.
Operations Research 58, 2010. pp. 303-315.
[PDF] [DOI:10.1287/opre.1090.0721][BibTeX]
Marcos Goycoolea, Alan T. Murray, Juan Pablo Vielma and Andres Weintraub
We survey three integer-programming approaches for solving the area restriction model (ARM) for harvest scheduling. We describe and analyze each of these approaches in detail, comparing them both from a modeling and computational point of view. In our analysis of these formulations as modeling tools we show how each can be extended to incorporate additional harvest scheduling concerns. In our computational analysis we illustrate the strengths and weaknesses of each formulation as a practical optimization tool by studying harvest scheduling in three North American forests.
Forest Science 55, 2009. pp. 149-165.
Juan Pablo Vielma, Shabbir Ahmed and George L. Nemhauser
This paper develops a linear programming based branch-and-bound algorithm for mixed integer conic quadratic programs. The algorithm is based on a higher dimensional or lifted polyhedral relaxation of conic quadratic constraints introduced by Ben-Tal and Nemirovski. The algorithm is different from other linear programming based branch-and-bound algorithms for mixed integer nonlinear programs in that, it is not based on cuts from gradient inequalities and it sometimes branches on integer feasible solutions. The algorithm is tested on a series of portfolio optimization problems. It is shown that it significantly outperforms commercial and open source solvers based on both linear and nonlinear relaxations.
In 2007 I received the INFORMS Optimization Society Student Paper Prize for this paper.
INFORMS Journal on Computing 20, 2008. pp. 438-450.
[PDF] [DOI:10.1016/10.1287/ijoc.1070.0256][BibTeX]
Juan Pablo Vielma, Ahmet B. Keha and George L. Nemhauser
A branch-and-cut algorithm for solving linear problems with continuous separable piecewise linear cost functions was developed in 2005 by Keha et. al. This algorithm is based on valid inequalities for an SOS2 based formulation of the problem. In this paper we study the extension of the algorithm to the case where the cost function is only lower semicontinuous. We extend the SOS2 based formulation to the lower semicontinuous case and show how the inequalities introduced by Keha et. al. can also be used for this new formulation. We also introduce a simple generalization of one of the inequalities introduced by Keha et. al. Furthermore, we study the discontinuities caused by fixed charge jumps and introduce two new valid inequalities by extending classical results for fixed charge linear problems. Finally, we report computational results showing how the addition of the developed inequalities can significantly improve the performance of CPLEX when solving these kinds of problems.
Discrete Optimization 5, 2008. pp. 467-488.
[PDF] [DOI:10.1016/j.disopt.2007.07.001][BibTeX]
Two independent proofs of the polyhedrality of the split closure of Mixed Integer Linear Program have been previously presented. Unfortunately neither of these proofs is constructive. In this paper, we present a constructive version of this proof. We also show that split cuts dominate a family of inequalities introduced by Köppe and Weismantel.
An extended version of this paper can be found here.
Operations Research Letters 35, 2007. pp. 29-35.
[PDF] [DOI:10.1016/j.orl.2005.12.005][BibTeX]
Juan Pablo Vielma, Alan T. Murray, David M. Ryan and Andres Weintraub
Forest Harvest Scheduling problems incorporating area-based restrictions have been of great practical interest for several years, but only recently have advances been made that allow them to be efficiently solved. One significant development has made use of formulation strengthening using the Cluster Packing Problem. This improved formulation has allowed medium sized problems to be easily solved, but when restrictions on volume production over time are added, problem difficulty increases substantially. In this paper we study the degrading effect of certain types of volume constraints and propose methods for reducing this effect. Developed methods include the use of constraint branching, the use of elastic constraints with dynamic penalty adjustment and a simple integer allocation heuristic. Application results are presented to illustrate the computational improvement afforded by the use of these methods.
European Journal of Operational Research 176, 2007. pp. 1246-1264.
[PDF] [DOI:10.1016/j.ejor.2005.09.016][BibTeX]
Daniel Espinoza, Guido Lagos, Eduardo Moreno and Juan Pablo Vielma
During early phases of open-pit mining production planning many parameters are uncertain, and since the mining operation is performed only once, any evaluations based only on on average outcomes neglects the very real chance of obtaining an outcome that is below average. Taking into account also that operation costs are considerable and the mining horizon usually extends over several decades it is clear that open-pit production planning is a risky endeavor. In this work we take a risk-averse approach on tackling uncertainty in the ore-grades. We consider an extended Ultimate-Pit problem, where extraction and processing decisions have to be taken. We apply and compare the risk-hedging performance of two approaches from optimization under uncertainty: minimize Conditional Value-at-Risk (CVaR) and minimization of a combination of expected value and CVaR. Additionally, we compare two decision schemes: a static variant, where all decisions have to be taken ¿now¿, and a two-stage or recourse variant, where we take extraction decisions now, then we see the real ore-grade, and just then processing decision is taken. Our working assumption is that we have available a large number of ore-grade scenarios. Computational results on one small size vein-type mine illustrate how minimizing average loss provides good on-average results at the cost of having high probability of obtaining undesired outcomes; and on the other hand our proposed approaches control the risk by providing solutions with a controllable probability of obtaining undesired outcomes. Results also show the great risk-hedging potential of using multi-stage decision schemes.
In J. F. Costa, J. Koppe and R. Peroni, editors, Proceedings of the 36th International Symposium on Application of Computers and Operations Research in The Mineral Industry (APCOM 2013), 2013. pp. 492-501.
Daniel Dadush, Santanu S. Dey and Juan Pablo Vielma
In this paper, we show that the Chvatal-Gomory closure of any compact convex set is a rational polytope. This resolves an open question of Schrijver 1980 for irrational polytopes, and generalizes the same result for the case of rational polytopes (Schrijver 1980), rational ellipsoids (Dey and Vielma 2010) and strictly convex bodies (Dadush, Dey and Vielma 2010).
In O. Günlük and G. J. Woeginger, editors, Proceedings of the 15th Conference on Integer Programming and Combinatorial Optimization (IPCO 2011), Lecture Notes in Computer Science 6655, 2011. pp. 130-142.
Santanu S. Dey and Juan Pablo Vielma
It is well-know that the Chvátal-Gomory (CG) closure of a rational polyhedron is a rational polyhedron. In this paper, we show that the CG closure of an bounded full-dimensional ellipsoid, described by rational data, is a rational polytope. To the best of our knowledge, this is the first extension of the polyhedrality of the CG closure to a non-polyhedral set. A key feature of the proof is to verify that all non-integral points on the boundary of ellipsoids can be separated by some CG cut. Given a point u on the boundary of an ellipsoid that cannot be trivially separated using the CG cut parallel to its supporting hyperplane, the proof constructs a sequences of CG cuts that eventually separate u. The polyhedrality of the CG closure is established using this separation result and a compactness argument. The proof also establishes some sufficient conditions for the polyhedrality result for general compact convex sets.
In F. Eisenbrand and F. B. Shepherd, editors, Proceedings of the 14th Conference on Integer Programming and Combinatorial Optimization (IPCO 2010), Lecture Notes in Computer Science 6080, 2010. pp. 327-340.
Juan Pablo Vielma, Daniel Espinoza and Eduardo Moreno
In this work we study how to incorporate risk control to the generation of ultimate pits when orebodies are modeled through a ¿nite number of conditional simulations. To control risk we consider a chance constraint on the value of the ultimate pit. We incorporate this measure into the generation of ultimate pits by solving a stochastic programming version of the ultimate pit problem. We compare this stochastic programming problem to previous approaches such as generating the ultimate pit for each simulation and the hybrid pit approach introduced by Whittle and Bozorgebrahimi. We also study the effect of using different number of simulations in the generation and evaluation of ultimate pits.
Proceedings of the 34th International Symposium on Application of Computers and Operations Research in The Mineral Industry (APCOM 2009), 2009. pp. 107-114.
Juan Pablo Vielma and George L. Nemhauser
Many combinatorial constraints over continuous variables such as SOS1 and SOS2 constraints can be interpreted as disjunctive constraints that restrict the variables to lie in the union of m specially structured polyhedra. Known mixed integer binary formulations for these constraints have a number of binary variables and extra constraints that is linear in m. We give sufficient conditions for constructing formulations for these constraints with a number of binary variables and extra constraints that is logarithmic in m. Using these conditions we introduce the first mixed integer binary formulations for SOS1 and SOS2 constraints that use a number of binary variables and extra constraints that is logarithmic in the number of continuous variables. We also introduce the first mixed integer binary formulations for piecewise linear functions of one and two variables that use a number of binary variables and extra constraints that is logarithmic in the number of linear pieces of the functions. We prove that the new formulations for piecewise linear functions have favorable tightness properties and present computational results showing that they can significantly outperform other mixed integer binary formulations.
The full version of this work can be found here.
In A. Lodi, A. Panconesi and G. Rinaldi, editors, Proceedings of the 13th Conference on Integer Programming and Combinatorial Optimization (IPCO 2008), Lecture Notes in Computer Science 5035, 2008. pp. 199-213.
[PDF] [DOI:10.1016/10.1007/978-3-540-68891-4_14][BibTeX]
Juan Pablo Vielma, Alan T. Murray, David M. Ryan and Andres Weintraub
Area-based harvest scheduling models, where management decisions are made for relatively small units subject to amaximum harvest area restriction, are known to be very difficult to solve by exact techniques. Previous research has developed good approaches for solving small and medium sized forestry applications based on projecting the problem onto a cluster graph for which cliques can be applied. However, as multiple time periods become of interest, current approaches encounter difficulties preventing successful identification of optimal solutions. In this paper we present an approach for elasticizing timber demand constraints, which lends itself to an efficient solution technique. It is also possible using this approach to examine trade-offs between objective value performance and maintaining demand constraints.
In M. Bevers and T.M. Barrett, editors, Proceedings of the 10th Symposium for Systems Analysis in Forest Resources (SSAFR'03), USDA Forest Services General Technical Report PNW-GTR-656, 2005. pp. 285-290.
Nathalie E. Jamett, Juan Pablo Vielma and Luis A. Cisternas
The objective of the present study was to develop a procedure for flotation circuit design including uncertainty and water efficiency. The design process considers two stages: 1) optimal process design without consider water consumption, and 2) efficient use of water considering property integration. For the flotation circuit design, we applied stochastic programming. In the optimization problem, it is desired to find the optimal configuration, equipment design and operational conditions of a circuit with multiple stages (rougher, scavenger, cleaner). The problem includes uncertainty in the feed composition and in the metal price. Each uncertain parameter is characterized probabilistically using scenarios with different occurrence probabilities. Then, considering the solutions to different scenarios, property integration is used to design the water integration system. Three properties are included: pH, oxygen concentration, and conductivity. The application of procedure to an example, show that including uncertainty in the design process can be useful in finding better design and the property integration method can be extended to use in mineral processing. The novelty of this work is the integration of both methodologies and the application of these tools to mineral processing.
In I. D. Lockhart Bogle and M. Fairweather, editors, Proceedings of the 22nd European Symposium on Computer Aided Process Engineering (Escape 22), Computer-Aided Chemical Engineering 30, 2012. pp. 1277-1281.
Ph.D. Thesis, H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA, 2009.
[Available at smartech.gatech.edu]
Juan Pablo Vielma, Marcos Goycoolea, Alan T. Murray and Andres Weintraub
In this article we study the effectiveness of alternative integer programming formulations for area constrained harvest scheduling, known as the area restriction model (ARM). Empirical assessment of the effect of area constraints on the optimal objective value of these alternative approaches is carried out, focusing on the root Linear Programming relaxation. We also examine how this relates to the effectiveness of these formulations when solved by a commercial solver, CPLEX.
To appear in Proceedings of the 12th Symposium for Systems Analysis in Forest Resources (SSAFR'06), 2006.
Mathematical Engineering Thesis (In Spanish), Department of Mathematical Engineering, University of Chile, Santiago, Chile, 2003.
Juan Pablo Vielma, Alan T. Murray, David M. Ryan and Andres Weintraub
Area-based forest harvest scheduling models, where management decisions are made for relatively small units subject to a maximum harvest area restriction, are known to be very difficult to solve by exact techniques. Previous research has developed good approaches for solving small and medium sized forestry applications based on projecting the problem onto a cluster graph for which cliques can be applied. However, as multiple time periods become of interest, current approaches encounter difficulties preventing successful identification of optimal solutions. In this paper we present an approach for elasticizing timber demand constraints, which lends itself to an efficient solution technique. It is also possible using this approach to examine trade-offs between objective value performance and maintaining demand constraints.
Proceedings of the 38th Annual Conference of the Operational Research Society of New Zealand (ORSNZ'03), 2003. pp. 21-28.