A note on “event-based MILP models for resource-constrained project scheduling problems”

Recently, new mixed integer linear programming formulations for the resource-constrained project scheduling problem were proposed by Koné et al. [3]. Unfortunately, the presentation of the first new model (called start/end-based formulation SEE) was not correct. More precisely, a set of necessary constraints representing the relative positioning of start and end events of activities was unintentionally omitted in the paper although it was present in the integer program used for the computational experiments. After presenting a counterexample showing the incorrectness, we provide a disaggregated and an aggregated variant of the set of necessary constraints, the disaggregated formulation yielding in theory a better linear programming relaxation. We present computational results showing that although the linear programming relaxations of both formulations yield equivalently poor lower bounds, the disaggregated formulation shows in average a better performance for integer solving of a well-known set of 30-activity instances.     

Branch and bound for the cutwidth minimization problem

The cutwidth minimization problem consists of finding a linear arrangement of the vertices of a graph where the maximum number of cuts between the edges of the graph and a line separating consecutive vertices is minimized. We first review previous approaches for special classes of graphs, followed by lower bounds and then a linear integer formulation for the general problem. We then propose a branch-and-bound algorithm based on different lower bounds on the cutwidth of partial solutions. Additionally, we introduce a Greedy Randomized Adaptive Search Procedure (GRASP) heuristic to obtain good initial solutions. The combination of the branch-and-bound and GRASP methods results in optimal solutions or a reduced relative gap (difference between upper and lower bounds) on the instances tested. Empirical results with a collection of previously reported instances indicate that the proposed algorithm is able to solve all the small instances (up to 32 vertices) as well as some of the large instances tested (up to 158 vertices) using less than 30 minutes of CPU time. We compare the results of our method with previous lower bounds, and with the best previous linear integer formulation solved using Cplex. Both comparisons favor the proposed procedure.     

Scheduling jobs on a single batch processing machine with incompatible job families and weighted number of tardy jobs objective

In this paper, we minimize the weighted and unweighted number of tardy jobs on a single batch processing machine with incompatible job families. We propose two different mixed integer linear programming (MILP) formulations based on positional variables. The second formulation does not contain a big-M coefficient. Two iterative schemes are discussed that are able to provide tighter linear programming bounds by reducing the number of positional variables. Furthermore, we also suggest a random key genetic algorithm (RKGA) to solve this scheduling problem. Results of computational experiments are shown. The second MILP formulation is more efficient with respect to lower bounds, while the first formulation provides better upper bounds. The iterative scheme is effective for the weighted case. The RKGA is able to find high-quality solutions in a reasonable amount of time.     

Minimal-cost network flow problems with variable lower bounds on arc flows

The minimal-cost network flow problem with fixed lower and upper bounds on arc flows has been well studied. This paper investigates an important extension, in which some or all arcs have variable lower bounds. In particular, an arc with a variable lower bound is allowed to be either closed (i.e., then having zero flow) or open (i.e., then having flow between the given positive lower bound and an upper bound). This distinctive feature makes the new problem NP-hard, although its formulation becomes more broadly applicable, since there are many cases where a flow distribution channel may be closed if the flow on the arc is not enough to justify its operation. This paper formulates the new model, referred to as MCNF-VLB, as a mixed integer linear programming, and shows its NP-hard complexity. Furthermore, a numerical example is used to illustrate the formulation and its applicability. This paper also shows a comprehensive computational testing on using CPLEX to solve the MCNF-VLB instances of up to medium-to-large size.     

Lower bounds and exact algorithms for the quadratic minimum spanning tree problem

We address the quadratic minimum spanning tree problem (QMSTP), the problem of finding a spanning tree of a connected and undirected graph such that a quadratic cost function is minimized. We first propose an integer programming formulation based on the reformulation–linearization technique (RLT). We then use the idea of partitioning spanning trees into forests of a given fixed size and obtain a QMSTP reformulation that generalizes the RLT model. The reformulation is such that the larger the size of the forests, the stronger lower bounds provided. Thus, a hierarchy of formulations is obtained. At the lowest hierarchy level, one has precisely the RLT formulation, which is already stronger than previous formulations in the literature. The highest hierarchy level provides the convex hull of integer feasible solutions for the problem. The formulations introduced here are not compact, so the direct evaluation of their linear programming relaxation bounds is not practical. To overcome that, we introduce two lower bounding procedures based on Lagrangian relaxation. These procedures are embedded into two parallel branch-and-bound algorithms. As a result of our study, several instances in the literature were solved to optimality for the first time.     

Double bound method for solving the p-center location problem

We give a review of existing methods for solving the absolute and vertex restricted p-center problems on networks and propose a new integer programming formulation, a tightened version of this formulation and a new method based on successive restrictions of the new formulation. A specialization of the new method with two-element restrictions obtains the optimal p-center solution by solving a series of simple structured integer programs in recognition form. This specialization is called the double bound method. A relaxation of the proposed formulation gives the tightest known lower bound in the literature (obtained earlier by Elloumi et al., [1]). A polynomial time algorithm is presented to compute this bound. New lower and upper bounds are proposed. Problems from the OR-Library [2] and TSPLIB [3] are solved by the proposed algorithms with up to 3038 nodes. Previous computational results were restricted to networks with at most 1817 nodes.     

The multi-weighted Steiner tree problem: A reformulation by intersection

We propose a new formulation for the multi-weighted Steiner tree (MWST) problem. This formulation is based on the fact that a previously proposed formulation for the problem is non-symmetric in the sense that the corresponding linear programming relaxation bounds depend on the node selected as a root of the tree. The new formulation (the reformulation by intersection) is obtained by intersecting the feasible sets of the models corresponding to each possible root selection for the underlying directed problem. Theoretical results will show that the linear programming relaxation of the new formulation dominates the linear programming relaxation of each of the rooted formulations and is comparable with the linear programming bounds of the best formulation known for the problem. A Lagrangean relaxation scheme derived from the new formulation is also proposed and tested, with quite favourable results, on instances with up to 500 nodes and 5000 edges.     

Solving diameter‐constrained minimum spanning tree problems by constraint programming

The diameter‐constrained minimum spanning tree problem consists in finding a minimum spanning tree of a given graph, subject to the constraint that the maximum number of edges between any two vertices in the tree is bounded from above by a given constant. This problem typically models network design applications where all vertices communicate with each other at a minimum cost, subject to a given quality requirement. We propose alternative formulations using constraint programming that circumvent weak lower bounds yielded by most mixed‐integer programming formulations. Computational results show that the proposed formulation, combined with an appropriate search procedure, solves larger instances and is faster than other approaches in the literature.     

调速泵结构配置协调分解优化算法及实现

建立多级调速泵结构配置连续非线性规划和整数非线性规划二阶段模型.非线性整数规划子问题采用外逼近算法求解.针对连续非线性规划主问题,提出基于割角法的可行域协调分解优化算法,证明割角法陷阱问题并建立判断准则排除已知的陷阱区域,在此基础上构建系列松弛问题得到原优化问题渐进收紧的下界估计,并最终收敛到原优化问题全局最优解.三级调速泵结构配置实例验证了算法的有效性,并给出与其他算法的比较结果.     

Arc-flow model for the two-dimensional guillotine cutting stock problem

We describe an exact model for the two-dimensional cutting stock problem with two stages and the guillotine constraint. It is an integer linear programming (ILP) arc-flow model, formulated as a minimum flow problem, which is an extension of a model proposed by Valério de Carvalho for the one dimensional case. In this paper, we explore the behavior of this model when it is solved with a commercial software, explicitly considering all its variables and constraints. We also derive a new family of cutting planes and a new lower bound, and consider some variants of the original problem. The model was tested on a set of real instances from the wood industry, with very good results. Furthermore the lower bounds provided by the linear programming relaxation of the model compare favorably with the lower bounds provided by models based on assignment variables.     

Solving the variable size bin packing problem with discretized formulations

In this paper we study the use of a discretized formulation for solving the variable size bin packing problem (VSBPP). The VSBPP is a generalization of the bin packing problem where bins of different capacities (and different costs) are available for packing a set of items. The objective is to pack all the items minimizing the total cost associated with the bins. We start by presenting a straightforward integer programming formulation to the problem and later on, propose a less straightforward formulation obtained by using a so-called discretized model reformulation technique proposed for other problems (see [Gouveia L. A 2n constraint formulation for the capacitated minimal spanning tree problem. Operations Research 1995; 43:130–141; Gouveia L, Saldanha-da-Gama F. On the capacitated concentrator location problem: a reformulation by discretization. Computers and Operations Research 2006; 33:1242–1258]). New valid inequalities suggested by the variables of the discretized model are also proposed to strengthen the original linear relaxation bounds. Computational results (see Section 4) with up to 1000 items show that these valid inequalities not only enhance the linear programming relaxation bound but may also be extremely helpful when using a commercial package for solving optimally VSBPP.     

Minimum stabbing rectangular partitions of rectilinear polygons

We study integer programming (ip) models for the problem of finding a rectangular partition of a rectilinear polygon with minimum stabbing number. Strong valid inequalities are introduced for an existing formulation and a new model is proposed. We compare the dual bounds yielded by the relaxations of the two models and prove that the new one is stronger than the old one. Computational experiments with the problem are reported for the first time in which polygons with thousands of vertices are solved to optimality. The (ip) branch-and-bound algorithm based on the new model is faster and more robust than those relying on the previous formulation.     

Models for the two‐dimensional rectangular single large placement problem with guillotine cuts and constrained pattern

In this paper, we address the constrained two‐dimensional rectangular guillotine single large placement problem (2D_R_CG_SLOPP). This problem involves cutting a rectangular object to produce smaller rectangular items from orthogonal guillotine cuts. In addition, there is an upper limit on the number of copies that can be produced of each item type. To model this problem, we propose a new pseudopolynomial integer nonlinear programming (INLP) formulation and obtain an equivalent integer linear programming (ILP) formulation from it. Additionally, we developed a procedure to reduce the numbers of variables and constraints of the integer linear programming (ILP) formulation, without loss of optimality. From the ILP formulation, we derive two new pseudopolynomial models for particular cases of the 2D_R_CG_SLOPP, which consider only two‐staged or one‐group patterns. Finally, as a specific solution method for the 2D_R_CG_SLOPP, we apply Benders decomposition to the proposed ILP formulation and develop a branch‐and‐Benders‐cut algorithm. All proposed approaches are evaluated through computational experiments using benchmark instances and compared with other formulations available in the literature. The results show that the new formulations are appropriate in scenarios characterized by few item types that are large with respect to the object's dimensions.     

An effective algorithm for obtaining the minimal cost pair of disjoint paths with dual arc costs

Routing optimisation in some types of networks requires the calculation of the minimal cost pair of disjoint paths such that the cost functions associated with the arcs in the two paths are different. An exact algorithm for solving this NP-complete problem is proposed, based on a condition which guarantees that the optimal path pair cost has been obtained. This optimality condition is based on the calculation of upper and lower bounds on the optimal cost. A formal proof of the correctness of the algorithm is described. Extensive experimentation is presented to show the effectiveness of the algorithm, including a comparison with an integer linear programming formulation.     

An Integer Linear Programming Problem with Multi-Criteria and Multi-Constraint Levels: a Branch-and-Partition Algorithm

In this paper, we propose a branch-and-partition algorithm to solve the integer linear programming problem with multi-criteria and multi-constraint levels (MC-ILP). The procedure begins with the relaxation problem that is formed by ignoring the integer restrictions. In this branch-and-partition procedure, an MC linear programming problem is adopted by adding a restriction according to a basic decision variable that is not integer. Then the MC-simplex method is applied to locate the set of all potential solutions over possible changes of the objective coefficient parameter and the constraint parameter for a regular MC linear programming problem. We use parameter partition to divide the (λ, γ) space for integer solutions of MC problem. The branch-and-partition procedure terminates when every potential basis for the relaxation problem is a potential basis for the MC-ILP problem. A numerical example is used to demonstrate the proposed algorithm in solving the MC-ILP problems. The comparison study and discussion on the applicability of the proposed method are also provided.     

Heuristics for cartographic label placement problems

The cartographic label placement problem is an important task in automated cartography and Geographical Information Systems. Positioning the texts requires that overlap among texts should be avoided, that cartographic conventions and preference should be obeyed. This paper examines the point-feature cartographic label placement problem (PFCLP) as an optimization problem. We formulate the PFCLP considering the minimization of existing overlaps and labeling of all points on a map. This objective improves legibility when all points must be placed even if overlaps are inevitable. A new mathematical formulation of binary integer linear programming that allows labeling of all points is presented, followed by some Lagrangean relaxation heuristics. The computational tests considered instances proposed in the literature up to 1000 points, and the relaxations provided good lower and upper bounds.     

The open capacitated arc routing problem

The Open Capacitated Arc Routing Problem (OCARP) is a NP-hard combinatorial optimization problem where, given an undirected graph, the objective is to find a minimum cost set of tours that services a subset of edges with positive demand under capacity constraints. This problem is related to the Capacitated Arc Routing Problem (CARP) but differs from it since OCARP does not consider a depot, and tours are not constrained to form cycles. Applications to OCARP from literature are discussed. A new integer linear programming formulation is given, followed by some properties of the problem. A reactive path-scanning heuristic, guided by a cost-demand edge-selection and ellipse rules, is proposed and compared with other successful CARP path-scanning heuristics from literature. Computational tests were conducted using a set of 411 instances, divided into three classes according to the tightness of the number of vehicles available; results reveal the first lower and upper bounds, allowing to prove optimality for 133 instances.     

New analytical lower bounds on the clique number of a graph

This paper proposes three new analytical lower bounds on the clique number of a graph and compares these bounds with those previously established in the literature. Two proposed bounds are derived from the well-known Motzkin–Straus quadratic programming formulation for the maximum clique problem. Theoretical results on the comparison of various bounds are established. Computational experiments are performed on random graph models such as the Erdös-Rényi model for uniform graphs and the generalized random graph model for power-law graphs that simulate graphs with different densities and assortativity coefficients. Computational results suggest that the proposed new analytical bounds improve the existing ones on many graph instances.     

A Lagrangian relaxation-based method and models evaluation for multi-level lot sizing problems with backorders

The capacitated multi-level lot sizing problem with backorders has received a great deal of attention in extant literature on operations and optimization. The facility location model and the classical inventory and lot sizing model with (?$?$, S) cuts have been proposed to formulate this problem. However, their comparative effectiveness has not yet been explored and is not known. In this paper, we demonstrate that on linear programming relaxation, the facility location formulation yields tighter lower bounds than classical inventory and lot sizing model. It further shows that the facility location formulation is computationally advantageous for deriving both lower and upper bounds. The results are expected to provide guidelines for choosing an effective formulation during the development of solution procedures. We also propose a Lagrangian relaxation-based heuristic along with computational results that indicate its competitiveness with other heuristics and a prominent commercial solver, Cplex 11.2.     

Evaluation of mixed integer programming formulations for non-preemptive parallel machine scheduling problems

Mixed integer programming (MIP) formulations for scheduling problems can be classified based on the decision variables upon which they rely. In this paper, four different MIP formulations based on four different types of decision variables are presented for various parallel machine scheduling problems. The goal of this research is to identify promising optimization formulation paradigms that can subsequently be used to either (1) solve larger practical scheduling problems of interest to optimality and/or (2) be used to establish tighter lower solution bounds for those under study. We present the computational results and discuss formulation efficacy for total weighted completion time and maximum completion time problems for the identical parallel machine case.