Cutting plane method for multiple objective stochastic integer linear programming

This paper grapples with the problem of incorporating integer variables in the constraints of a multiple objective stochastic linear program (MOSLP). After representing uncertain aspirations of the decision maker by converting the original problem into a deterministic multiple objective integer linear program (MOILP), a cutting plane technique may be used to compute all the efficient solutions of the last model leaving the decision maker to choose a solution according to his preferences. A numerical example is also included for illustration.     

An interactive weighted Tchebycheff procedure for multiple objective programming

The procedure samples the efficient set by computing the nondominated criterion vector that is closest to an ideal criterion vector according to a randomly weighted Tchebycheff metric. Using ‘filtering’ techniques, maximally dispersed representatives of smaller and smaller subsets of the set of nondominated criterion vectors are presented at each iteration. The procedure has the advantage that it can converge to non-extreme final solutions. Especially suitable for multiple objective linear programming, the procedure is also applicable to integer and nonlinear multiple objective programs.     

Finding a representative nondominated set for multi-objective mixed integer programs

In this paper, we develop algorithms to find small representative sets of nondominated points that are well spread over the nondominated frontiers for multi-objective mixed integer programs. We evaluate the quality of representations of the sets by a Tchebycheff distance-based coverage gap measure. The first algorithm aims to substantially improve the computational efficiency of an existing algorithm that is designed to continue generating new points until the decision maker (DM) finds the generated set satisfactory. The algorithm improves the coverage gap value in each iteration by including the worst represented point into the set. The second algorithm, on the other hand, guarantees to achieve a desired coverage gap value imposed by the DM at the outset. In generating a new point, the algorithm constructs territories around the previously generated points that are inadmissible for the new point based on the desired coverage gap value. The third algorithm brings a holistic approach considering the solution space and the number of representative points that will be generated together. The algorithm first approximates the nondominated set by a hypersurface and uses it to plan the locations of the representative points. We conduct computational experiments on randomly generated instances of multi-objective knapsack, assignment, and mixed integer knapsack problems and show that the algorithms work well.     

A fuzzy optimization approach for procurement transport operational planning in an automobile supply chain

We consider a real-world automobile supply chain in which a first-tier supplier serves an assembler and determines its procurement transport planning for a second-tier supplier by using the automobile assembler’s demand information, the available capacity of trucks and inventory levels. The proposed fuzzy multi-objective integer linear programming model (FMOILP) improves the transport planning process for material procurement at the first-tier supplier level, which is subject to product groups composed of items that must be ordered together, order lot sizes, fuzzy aspiration levels for inventory and used trucks and uncertain truck maximum available capacities and minimum percentages of demand in stock. Regarding the defuzzification process, we apply two existing methods based on the weighted average method to convert the FMOILP into a crisp MOILP to then apply two different aggregation functions, which we compare, to transform this crisp MOILP into a single objective MILP model. A sensitivity analysis is included to show the impact of the objectives weight vector on the final solutions. The model, based on the full truck load material pick method, provides the quantity of products and number of containers to be loaded per truck and period. An industrial automobile supply chain case study demonstrates the feasibility of applying the proposed model and the solution methodology to a realistic procurement transport planning problem. The results provide lower stock levels and higher occupation of the trucks used to fulfill both demand and minimum inventory requirements than those obtained by the manual spreadsheet-based method.     

A maximumb-matching problem arising from median location models with applications to the roommates problem

We consider maximumb-matching problems where the nodes of the graph represent points in a metric space, and the weight of an edge is the distance between the respective pair of points. We show that if the space is either the rectilinear plane, or the metric space induced by a tree network, then theb-matching problem is the dual of the (single) median location problem with respect to the given set of points. This result does not hold for the Euclidean plane. However, we show that in this case theb-matching problem is the dual of a median location problem with respect to the given set of points, in some extended metric space. We then extend this latter result to any geodesic metric in the plane. The above results imply that the respective fractionalb-matching problems have integer optimal solutions. We use these duality results to prove the nonemptiness of the core of a cooperative game defined on the roommate problem corresponding to the above matching model. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.Corresponding author.     

Steinhaus问题及其证明

本文利用Pythagoras数组的性质,导出了与此问题等价的相关量的表述,证明了可以按某种方式把平面上的点划分为不相交的四类点集,而在每一类点集中都不存在整距点.     

On the apollonian metric of domains in

We study the apollonian metric considered for sets in ? _n by Beardon in 1995. This metric was first introduced for plane Jordan domains by Barbilian in 1934. For a special class of plane domains Beardon showed that conformal apollonian isometries are Möbius transformations. We give here a proof of Beardon's result without conformality assumption. We show that the apollonian metric of a domain D is either conformal at every point of D, at only one point of D or at no point of D. We also present a suprising relation between convex bodies of constant width and the apollonian metric.     

Robust optimization for interactive multiobjective programming with imprecise information applied to R&D project portfolio selection

A multiobjective binary integer programming model for R&D project portfolio selection with competing objectives is developed when problem coefficients in both objective functions and constraints are uncertain. Robust optimization is used in dealing with uncertainty while an interactive procedure is used in making tradeoffs among the multiple objectives. Robust nondominated solutions are generated by solving the linearized counterpart of the robust augmented weighted Tchebycheff programs. A decision maker’s most preferred solution is identified in the interactive robust weighted Tchebycheff procedure by progressively eliciting and incorporating the decision maker’s preference information into the solution process. An example is presented to illustrate the solution approach and performance. The developed approach can also be applied to general multiobjective mixed integer programming problems.     

Hele–Shaw flow on hyperbolic surfaces

Consider a complete simply connected hyperbolic surface. The classical Hadamard theorem asserts that at each point of the surface, the exponential mapping from the tangent plane to the surface defines a global diffeomorphism. This can be interpreted as a statement relating the metric flow on the tangent plane with that of the surface. We find an analogue of Hadamard's theorem with metric flow replaced by Hele–Shaw flow, which models the injection of (two-dimensional) fluid into the surface. The Hele–Shaw flow domains are characterized implicitly by a mean value property on harmonic functions.     

Stability achievement scalarization function for multiobjective nonlinear programming problems

In this paper, we present a method to determine the stability of nondominated criterion vectors using a modified weighted achievement scalarization metric. This method is based on the application of a particular objective function which scalarizes and parameterizes the original multiobjective nonlinear programming problem. Also, we show that this modified weighted achievement metric coincides with the metric introduced by Choo and Atkins [E.-U. Choo, D.R. Atkins, Proper efficiency in nonconvex multicriteria programming, Math. Oper. Res. 8 (1983) 467–470] and Kaliszewski [I. Kaliszewski, A modified weighted Tchebycheff metric for multiple objective programming, Comput. Oper. Res. 14 (1987) 315–323] in cases when sets of all criterion vectors are finite or polyhedral.     

Conics in the hyperbolic plane intrinsic to the collineation group

In the manner of Steiner??s interpretation of conics in the projective plane we consider a conic in a planar incidence geometry to be a pair consisting of a point and a collineation that does not fix that point. We say these loci are intrinsic to the collineation group because their construction does not depend on an imbedding into a larger space. Using an inversive model we classify the intrinsic conics in the hyperbolic plane in terms of invariants of the collineations that afford them and provide metric characterizations for each congruence class. By contrast, classifications that catalogue all projective conics intersecting a specified hyperbolic domain necessarily include curves which cannot be afforded by a hyperbolic collineation in the above sense. The metric properties we derive will distinguish the intrinsic classes in relation to these larger projective categories. Our classification emphasizes a natural duality among congruence classes induced by an involution based on complementary angles of parallelism relative to the focal axis of each conic, which we refer to as split inversion (Definition 5.3).     

Augmented Lagrangian and Tchebycheff Approaches in Multiple Objective Programming

Relationships between the Tchebycheff scalarization and the augmented Lagrange multiplier technique are examined in the framework of general multiple objective programs (MOPs). It is shown that under certain conditions the Tchebycheff method can be represented as a quadratic weighted-sums scalarization of the MOP, that is, given weight values in the former, the coefficients of the latter can be found so that the same efficient point is selected. Analysis for concave and linear MOPs is included. Resulting applications in multiple criteria decision making are also discussed.     

邻域整点搜索法求解整数规划

从剖析线性规划的优化机理入手,将纯整数规划分为标准型和非标型两类.首先以标准型纯整数规划为突破口,提出一种新的解法,并在理论上加以证明,然后将其拓广延伸,用于求解非标准型纯整数规划和混合整数规划.这种新解法命名为松驰最优解邻域整点搜索法,属于常规解法,但在简捷高效方面,远胜过现有的两种常规解法—分枝定界法和割平面法.     

Fixing variables and generating classical cutting planes when using an interior point branch and cut method to solve integer programming problems

Branch and cut methods for integer programming problems solve a sequence of linear programming problems. Traditionally, these linear programming relaxations have been solved using the simplex method. The reduced costs available at the optimal solution to a relaxation may make it possible to fix variables at zero or one. If the solution to a relaxation is fractional, additional constraints can be generated which cut off the solution to the relaxation, but donot cut off any feasible integer points. Gomory cutting planes and other classes of cutting planes are generated from the final tableau. In this paper, we consider using an interior point method to solve the linear programming relaxations. We show that it is still possible to generate Gomory cuts and other cuts without having to recreate a tableau, and we also show how variables can be fixed without using the optimal reduced costs. The procedures we develop do not require that the current relaxation be solved to optimality; this is useful for an interior point method because early termination of the current relaxation results in an improved starting point for the next relaxation.     

A Reference Direction Algorithm for Solving Multiple Objective Integer Linear Programming Problems

In this paper, we propose a reference direction approach and an interactive algorithm to solve the general multiple objective integer linear programming problem. At each iteration, only one mixed integer linear programming problem is solved to find an (weak) efficient solution. Each intermediate solution is integer. The decision maker has to provide only the reference point at each iteration. No special software is required to implement the proposed algorithm. The algorithm is illustrated with an example.     

Lattice polygons with two interior lattice points

A lattice point in the plane is a point with integer coordinates. A lattice segment is a line segment whose endpoints are lattice points. A lattice polygon is a simple polygon whose vertices are lattice points. We find all convex lattice polygons in the plane up to equivalence with two interior lattice points.     

Planar quasi-Newton algorithms for unconstrained saddlepoint problems

A new class of quasi-Newton methods is introduced that can locate a unique stationary point of ann-dimensional quadratic function in at mostn steps. When applied to positive-definite or negative-definite quadratic functions, the new class is identical to Huang's symmetric family of quasi-Newton methods (Ref. 1). Unlike the latter, however, the new family can handle indefinite quadratic forms and therefore is capable of solving saddlepoint problems that arise, for instance, in constrained optimization. The novel feature of the new class is a planar iteration that is activated whenever the algorithm encounters a near-singular direction of search, along which the objective function approaches zero curvature. In such iterations, the next point is selected as the stationary point of the objective function over a plane containing the problematic search direction, and the inverse Hessian approximation is updated with respect to that plane via a new four-parameter family of rank-three updates. It is shown that the new class possesses properties which are similar to or which generalize the properties of Huang's family. Furthermore, the new method is equivalent to Fletcher's (Ref. 2) modified version of Luenberger's (Ref. 3) hyperbolic pairs method, with respect to the metric defined by the initial inverse Hessian approximation. Several issues related to implementing the proposed method in nonquadratic cases are discussed.An earlier version of this paper was presented at the 10th Mathematical Programing Symposium, Montreal, Canada, 1979.     

Zone diagrams in Euclidean spaces and in other normed spaces

Zone diagrams are a variation on the classical concept of Voronoi diagrams. Given n sites in a metric space that compete for territory, the zone diagram is an equilibrium state in the competition. Formally it is defined as a fixed point of a certain “dominance” map. Asano, Matou?ek, and Tokuyama proved the existence and uniqueness of a zone diagram for point sites in the Euclidean plane, and Reem and Reich showed existence for two arbitrary sites in an arbitrary metric space. We establish existence and uniqueness for n disjoint compact sites in a Euclidean space of arbitrary (finite) dimension, and more generally, in a finite-dimensional normed space with a smooth and rotund norm. The proof is considerably simpler than that of Asano et?al. We also provide an example of non-uniqueness for a norm that is rotund but not smooth. Finally, we prove existence and uniqueness for two point sites in the plane with a smooth (but not necessarily rotund) norm.     

On set {N k − (x)} n k=0

Suppose that C is the complex plane and k is a non-negative integer. Define functions N _k ⁻ (x) = |x|^kif k is even and K _k ⁻ (x) = x|x|^k−1 if k is odd. Some approximation properties of N _k ⁻ (x)’s is discussed and a new example of a Tchebycheff system is given out.