New Class Of Valid Inequalities Research Articles

The multi-level lot sizing problem with flexible production sequences

This paper considers a multi-level/multi-machine lot sizing problem with flexible production sequences, where the quantity and combination of items required to produce another item need not be unique. The problem is formulated as a mixed-integer linear program and the notion of echelon inventory is used to construct a new class of valid inequalities, which are called echelon cuts. Numerical results show the computational power of the echelon cuts in a branch-and-cut algorithm. These inequalities are compared to known cutting planes from the literature and it is found that, in addition to being strong and valid for the flexible production case, echelon cuts are at least as strong as certain classes of known cuts in the restricted fixed production setting.

Mixing MIR inequalities with two divisible coefficients

This article is a polyhedral study of a generalization of the mixing set where two different, divisible coefficients are allowed for the integral variables. Our results generalize earlier work on mixed integer rounding, mixing, and extensions. They also directly apply to applications such as production planning problems involving lower bounds or start-ups on production, when these are modeled as mixed-integer linear programs. We define a new class of valid inequalities and give two proofs that they suffice to describe the convex hull of this mixed-integer set. We give a characterization of each of the maximal faces of the convex hull, as well as a closed form description of its extreme points and rays, and show how to separate over this set in $${\mathcal{O} (n\log n)}$$. Finally, we give several extended formulations of polynomial size, and study conditions under which adding certain simple constraints on the integer variables preserves our main result.

On the domino-parity inequalities for the STSP

One method which has been used very successfully for finding optimal and provably good solutions for large instances of the symmetric travelling salesman problem (STSP) is the branch and cut method. This method requires knowledge of classes of useful valid inequalities for the polytope associated with the STSP, as well as efficient separation routines for these classes of inequalities. Recently a new class of valid inequalities called the domino-parity inequalites were introduced for the STSP. An efficient separation routine is known for these constraints if certain conditions are satisfied by the point to be separated. This separation routine has never been implemented or tested. We present several performance enhancements for this separation routine, and discuss our implementation of this improved algorithm. We test our implementation and provide results which we believe demonstrate the practical usefulness of these constraints and the separation routine for the STSP within a branch and cut framework.

Polyhedral Properties of the K-median Problem on a Tree

The polyhedral structure of the K-median problem on a tree is examined. Even for very small connected graphs, we show that additional constraints are needed to describe the integer polytope. A complete description is given of those trees for which an optimal integer LP solution is guaranteed to exist. We present a new and simpler demonstration that an LP characterization of the 2-median problem is complete. Also, we provide a simpler proof of the value of a tight worst case bound for the LP relaxation. A new class of valid inequalities is identified. These inequalities describe a subclass of facets for the K-median problem on a general graph. Also, we provide polyhedral descriptions for several types of trees. As part of this work, we summarize most known results for the K-median problem on a tree.

The Continuous Mixing Polyhedron

We analyze the polyhedral structure of the sets PCMIX = {(s, r, z) ∈ R × R+n × Zn ∣ s + rj + zj ≥ fj, j = 1, …, n} and P+CMIX = PCMIX ∩ {s ≥ 0}. The set P+CMIX is a natural generalization of the mixing set studied by Pochet and Wolsey [15, 16] and Günlük and Pochet [8] and recently has been introduced by Miller and Wolsey [12]. We introduce a new class of valid inequalities that has proven to be sufficient for describing conv(PCMIX). We give an extended formulation of size O(n) × O(n2) variables and constraints and indicate how to separate over conv(PCMIX) in O(n3) time. Finally, we show how the mixed integer rounding (MIR) inequalities of Nemhauser and Wolsey [14] and the mixing inequalities of Günlük and Pochet [8] constitute special cases of the cycle inequalities.

New General Continuous-Time State−Task Network Formulation for Short-Term Scheduling of Multipurpose Batch Plants

A new continuous-time MILP model for the short-term scheduling of multipurpose batch plants is presented. The proposed model relies on the state-task network (STN) approach and addresses the general problem of batch scheduling, accounting for resource (utility) constraints, variable batch sizes and processing times, various storage policies (UIS, FIS, NIS, ZW), batch mixing/splitting, and sequence-dependent changeover times. The key features of the proposed model are the following: (a) a continuous-time representation is used, common for all units; (b) assignment constraints are expressed using binary variables that are defined only for tasks, not for units; (c) start times of tasks are eliminated, so that time-matching constraints are used only for the finish times of tasks; and (d) a new class of valid inequalities that improves the LP relaxation is added to the MILP formulation. Compared to other general continuous time STN formulations, the proposed model is faster. Compared to event-driven formulations, it is more general, as it accounts for resources other than equipment and gives solutions in comparable computational times. The application of the model is illustrated through four example problems.

Binary clutter inequalities for integer programs

We introduce a new class of valid inequalities for general integer linear programs, called binary clutter (BC) inequalities. They include the \({{\{0, \frac{{1}}{{2}}\}}}\)-cuts of Caprara and Fischetti as a special case and have some interesting connections to binary matroids, binary clutters and Gomory corner polyhedra. We show that the separation problem for BC-cuts is strongly 𝒩𝒫-hard in general, but polynomially solvable in certain special cases. As a by-product we also obtain new conditions under which \({{\{0, \frac{{1}}{{2}}\}}}\)-cuts can be separated in polynomial time. These ideas are then illustrated using the Traveling Salesman Problem (TSP) as an example. This leads to an interesting link between the TSP and two apparently unrelated problems, the T-join and max-cut problems.

Steiner trees and polyhedra

In this paper we study the dominant of the Steiner tree polytope. We introduce a new class of valid inequalities that generalizes the so-called odd hole, wheel, bipartite, anti-hole and Steiner partition inequalities introduced by Chopra and Rao (Math. Programming 64 (1994) 209–229, 231–246), and we give sufficient conditions for these inequalities to define facets. We describe some procedures that permit to construct facets from known ones for the dominant of the Steiner tree polytope and the closely related Steiner connected subgraph polytope. Using these methods we give a counterexample to a conjecture of Chopra and Rao on the dominant of the Steiner tree polytope on 2-trees. We also describe the dominant of the Steiner tree polytope and the Steiner connected subgraph polytope on special classes of graphs. In particular, we show that if the underlying graph is series–parallel and the terminals satisfy certain conditions, then both polyhedra are given by the trivial inequalities and the Steiner partition inequalities.

The γ-connected assignment problem

Given a graph and costs of assigning to each vertex one of k different colors, we want to find a minimum cost assignment such that no color q induces a subgraph with more than a given number ( γ q ) of connected components. This problem arose in the context of contiguity-constrained clustering, but also has a number of other possible applications. We show the problem to be NP-hard. Nevertheless, we derive a dynamic programming algorithm that proves the case where the underlying graph is a tree to be solvable in polynomial time. Next, we propose mixed-integer programming formulations for this problem that lead to branch-and-cut and branch-and-price algorithms. Finally, we introduce a new class of valid inequalities to obtain an enhanced branch-and-cut. Extensive computational experiments are reported.

New inequalities for the General Routing Problem

A large new class of valid inequalities is introduced for the General Routing Problem which properly contains the class of ‘K-C constraints’. These are also valid for the Rural Postman Problem. A separation algorithm is given for a subset of these inequalities which runs in polynomial time.

Ladders for Travelling Salesmen

We introduce a new class of valid inequalities for the symmetric travelling salesman polytope. The family is not of the common handle-tooth variety. We show that these inequalities are all facet-inducing and have Chvatal rank 2.

An Integer Polytope Related to the Design of Survivable Communication Networks

The problem of designing communication networks that can survive the loss of any single link is studied. Such problems can be formulated as minimum cost 2-edge connected subgraph problems in a complete graph. The linear programming cutting plane approach has been used effectively for related problems in [Schwerpunktprogramm der Deutschen Forschungsgemeinschaft, Anwendungsbezogene Optimierung and Steuerung, Report No. 188, 1989], where problem-specific cutting planes that define facets of the underlying integer polyhedra are used. This paper introduces a new class of valid inequalities for the polytope associated with the minimum cost 2-edge connected subgraph problem, and necessary and sufficient conditions for these inequalities to be facet-inducing for this polytope are given.

Small Travelling Salesman Polytopes

We introduce a new class of valid inequalities for the polytope of the symmetric travelling salesman problem. We also gave complete characterizations of the polytope for 6 and 7 cities. For the latter case, the new inequalities are needed. These results are related to work of R. Z. Norman in the 1950's.