Integer Feasible Solutions Research Articles

Boosting the feasibility pump

The feasibility pump (FP) has proved to be an effective method for finding feasible solutions to mixed integer programming problems. FP iterates between a rounding procedure and a projection procedure, which together provide a sequence of points alternating between LP feasible but fractional solutions, and integer but LP infeasible solutions. The process attempts to minimize the distance between consecutive iterates, producing an integer feasible solution when closing the distance between them. We investigate the benefits of enhancing the rounding procedure with a clever integer line search that efficiently explores a large set of integer points. An extensive computational study on benchmark instances demonstrates the efficacy of the proposed approach.

Searching for optimal integer solutions to set partitioning problems using column generation

In this paper, we describe a new approach to increase the possibility of finding integer feasible columns to a set partitioning problem (SPP) directly in solving the linear programming (LP) relaxation using column generation. Traditionally, column generation is aimed to solve the LP-relaxation as quickly as possible without any concern for the integer properties of the columns formed. In our approach, we aim to generate columns forming an optimal integer solution while simultaneously solving the LP-relaxation. Using this approach, we can improve the possibility of finding integer solutions by heuristics at each node in the branch-and-bound search. In addition, we improve the possibility of finding high-quality integer solutions in cases where only the columns in the root node are used to solve the problem. The basis of our approach is a subgradient technique applied to a Lagrangian dual formulation of the SPP extended with an additional surrogate constraint. This extra constraint is not relaxed and is used to better control the subgradient evaluations and how the multiplier values are computed. The column generation is then directed, via the multipliers, to construct columns that form feasible integer solutions. Computational experiments show that we can generate optimal integer columns in a large set of well-known test problems as compared to both standard and stabilized column generation, and simultaneously keep the number of columns smaller than standard column generation. This is also supported by tests on a case study with work-shift generation.

Tailored Lagrangian Relaxation for the open pit block sequencing problem

A common strategic and tactical decision in open pit mining is to determine the sequence of extraction for notional three-dimensional production blocks so as to maximize the net present value of the extracted orebody while adhering to precedence and resource constraints. This problem is commonly formulated as an integer program with a binary variable representing if and when a block is extracted. In practical applications, the number of blocks can be large and the time horizon can be long, and therefore, instances of this NP-hard precedence-constrained knapsack problem can be difficult to solve using an exact approach. The problem is even more challenging to solve when it includes explicit minimum resource constraints. We employ three methodologies to reduce solution times: (i) we eliminate variables which must assume a value of 0 in the optimal solution; (ii) we use heuristics to generate an initial integer feasible solution for use by the branch-and-bound algorithm; and (iii) we employ Lagrangian relaxation, using information obtained while generating the initial solution to select a dualization scheme for the resource constraints. The combination of these techniques allows us to determine near-optimal solutions more quickly than solving the monolith, i.e., the original problem. We demonstrate our techniques to solve instances containing 25,000 blocks and 10 time periods, and 10,000 blocks and 15 time periods, to near-optimality.

Global optimization in property‐based interplant water integration

This article presents a new global optimization method for the interplant water integration based on properties to characterize streams with numerous components. The problem is formulated as an mixed‐integer non‐linear programming (MINLP) model based on a superstructure that involves all possible options of interest (i.e., reuse and recycle in the same and to other plants and a set of shared treatment units). This formulation exhibits multiple local minima, and to overcome this problem, this article proposes effective branching rules in addition to two new reformulations for the upper bound (integer feasible solution) and the lower limit (relaxed solution), which are incorporated into a spatial branch and bound procedure to handle the bilinear terms in the model. The objective consists in finding the configuration with the minimum total annual cost. Results show that the global optimal solution (involving significant reductions in the fresh water consumption) is reached in few iterations and short central processing unit (CPU) time. © 2012 American Institute of Chemical Engineers AIChE J, 59: 813–833, 2013

A quadratic approximation‐based algorithm for the solution of multiparametric mixed‐integer nonlinear programming problems

An algorithm for the solution of convex multiparametric mixed‐integer nonlinear programming problems arising in process engineering problems under uncertainty is introduced. The proposed algorithm iterates between a multiparametric nonlinear programming subproblem and a mixed‐integer nonlinear programming subproblem to provide a series of parametric upper and lower bounds. The primal subproblem is formulated by fixing the integer variables and solved through a series of multiparametric quadratic programming (mp‐QP) problems based on quadratic approximations of the objective function, while the deterministic master subproblem is formulated so as to provide feasible integer solutions for the next primal subproblem. To reduce the computational effort when infeasibilities are encountered at the vertices of the critical regions (CRs) generated by the primal subproblem, a simplicial approximation approach is used to obtain CRs that are feasible at each of their vertices. The algorithm terminates when there does not exist an integer solution that is better than the one previously used by the primal problem. Through a series of examples, the proposed algorithm is compared with a multiparametric mixed‐integer outer approximation (mp‐MIOA) algorithm to demonstrate its computational advantages. © 2012 American Institute of Chemical Engineers AIChE J, 59: 483–495, 2013

In This Issue

In This Issue

A New Algorithm for the Open-Pit Mine Production Scheduling Problem

For the purpose of production scheduling, open-pit mines are discretized into three-dimensional arrays known as block models. Production scheduling consists of deciding which blocks should be extracted, when they should be extracted, and what to do with the blocks once they are extracted. Blocks that are close to the surface should be extracted first, and capacity constraints limit the production in each time period. Since the 1960s, it has been known that this problem can be cast as an integer programming model. However, the large size of some real instances (3–10 million blocks, 15–20 time periods) has made these models impractical for use in real planning applications, thus leading to the use of numerous heuristic methods. In this article we study a well-known integer programming formulation of the problem that we refer to as C-PIT. We propose a new decomposition method for solving the linear programming relaxation (LP) of C-PIT when there is a single capacity constraint per time period. This algorithm is based on exploiting the structure of the precedence-constrained knapsack problem and runs in O (mn log n) in which n is the number of blocks and m a function of the precedence relationships in the mine. Our computations show that we can solve, in minutes, the LP relaxation of real-sized mine-planning applications with up to five million blocks and 20 time periods. Combining this with a quick rounding algorithm based on topological sorting, we obtain integer feasible solutions to the more general problem where multiple capacity constraints per time period are considered. Our implementation obtains solutions within 6% of optimality in seconds. A second heuristic step, based on local search, allows us to find solutions within 3% in one hour on all instances considered. For most instances, we obtain solutions within 1–2% of optimality if we let this heuristic run longer. Previous methods have been able to tackle only instances with up to 150,000 blocks and 15 time periods.

Multi-objective stochastic integer linear programming with fixed recourse

The real life decision problems have three main properties. The first one is to have conflicting objectives in the problem structure, the second one is the stochasticity in the description of problem parameters in contexts where the probability distribution of random parameters is known and the last one is due to involvement of integer decision variables which increased dimension of the problem. Multi-objective nature with discrete variables and imprecise parameters make the mathematical expression of the problem harder to solve with the traditional approaches. An efficient algorithm is developed, via extending the well-known L-shaped method using generalised benders decomposition to efficiently handle the integer variables in the first stage and the integer recourse in the second stage of the model formulation. The proposed solution method able to identify all efficient integer feasible solutions converge in a finite number of iterations. A numerical example and a computational implementation are also included for illustration.

Algorithms for highly symmetric linear and integer programs

This paper deals with exploiting symmetry for solving linear and integer programming problems. Basic properties of linear representations of finite groups can be used to reduce symmetric linear programming to solving linear programs of lower dimension. Combining this approach with knowledge of the geometry of feasible integer solutions yields an algorithm for solving highly symmetric integer linear programs which only takes time which is linear in the number of constraints and quadratic in the dimension.

제한용량이 있는 설비입지결정 문제에 대한 적응형 평균치교차분할 알고리즘

In this research report, we propose a heuristic algorithm with some primal recovery strategies for capacitated facility location problems (CFLP), which is a well-known combinatorial optimization problem with applications in distribution, transportation and production planning. Many algorithms employ the branch-and-bound technique in order to solve the CFLP. There are also some different approaches which can recover primal solutions while exploiting the primal and dual structure simultaneously. One of them is a MVCD (Mean Value Cross Decomposition) ensuring convergence without solving a master problem. The MVCD was designed to handle LP-problems, but it was applied in mixed integer problems. However the MVCD has been applied to only uncapacitated facility location problems (UFLP), because it was very difficult to obtain "Integrality" property of Lagrangian dual subproblems sustaining the feasibility to primal problems. We present some heuristic strategies to recover primal feasible integer solutions, handling the accumulated primal solutions of the dual subproblem, which are used as input to the primal subproblem in the mean value cross decomposition technique, without requiring solutions to a master problem. Computational results for a set of various problem instances are reported.

Liner Shipping Cargo Allocation with Repositioning of Empty Containers

This paper is concerned with the cargo allocation problem considering empty repositioning of containers for a liner shipping company. The aim is to maximize the profit of transported cargo in a network, subject to the cost and availability of empty containers. The formulation is a multi-commodity flow problem with additional inter-balancing constraints to control repositioning of empty containers. In a study of the cost efficiency of the global container-shipping network, Song et al. (2005) estimate that empty repositioning cost constitutes 27% of the total world fleet running cost. An arc-flow formulation is decomposed using the Dantzig-Wolfe principle to a path-flow formulation. A linear relaxation is solved with a delayed column generation algorithm. A feasible integer solution is found by rounding the fractional solution and adjusting flow balance constraints with leased containers. Computational results are reported for seven instances based on real-life shipping networks. Solving the relaxed linear path-flow model with a column generation algorithm outperforms solving the relaxed linear arc-flow model with the CPLEX barrier solver even for very small instances. The proposed algorithm is able to solve instances with 234 ports, 16,278 demands over 9 time periods in 34 min. The integer solutions found by rounding down are computed in less than 5 s and the gap is within 0.01% from the upper bound of the linear relaxation. The solved instances are quite large compared to those tested in the reviewed literature.

An interior point cutting plane heuristic for mixed integer programming

We explore the use of interior point methods in finding feasible solutions to mixed integer programming. As integer solutions are typically in the interior, we use the analytic center cutting plane method to search for integer feasible points within the interior of the feasible set. The algorithm searches along two line segments that connect the weighted analytic center and two extreme points of the linear programming relaxation. Candidate points are rounded and tested for feasibility. Cuts aimed to improve the objective function and restore feasibility are then added to displace the weighted analytic center until a feasible integer solution is found. The algorithm is composed of three phases. In the first, points along the two line segments are rounded gradually to find integer feasible solutions. Then in an attempt to improve the quality of the solutions, the cut related to the bound constraint is updated and a new weighted analytic center is found. Upon failing to find a feasible integer solution, a second phase is started where cuts related to the violated feasibility constraints are added. As a last resort, the algorithm solves a minimum distance problem in a third phase. The heuristic is tested on a set of problems from MIPLIB and CORAL. The algorithm finds good quality feasible solutions in the first two phases and never requires the third phase.

Modeling and solving a multimodal transportation problem with flexible‐time and scheduled services

AbstractThis article studies a transportation problem in a multimodal network with shipment consolidation options. A freight forwarder can use a mix of flexible‐time and scheduled transportation services. Time windows are a prominent aspect of the problem. For instance, they are used to model pickup and delivery time slots. The various features of the problem can be described as elements of a digraph and their integration leads to a holistic graph representation. This allows an origin‐destination integer multi‐commodity flow formulation with nonconvex piecewise linear costs, time windows, and side constraints. Column generation algorithms are designed to compute lower bounds. These column generation algorithms are also embedded within heuristics aimed at finding feasible integer solutions. Computational results with real‐life data are presented and show the efficacy of the proposed approach. © 2010 Wiley Periodicals, Inc. NETWORKS, 2011

Faster integer-feasibility in mixed-integer linear programs by branching to force change

Branching in mixed-integer (or integer) linear programming requires choosing both the branching variable and the branching direction. This paper develops a number of new methods for making those two decisions either independently or together with the goal of reaching the first integer-feasible solution quickly. These new methods are based on estimating the probability of satisfying a constraint at the child node given a variable/direction pair. The surprising result is that the first integer-feasible solution is usually found much more quickly when the variable/direction pair with the smallest probability of satisfying the constraint is chosen. This is because this selection forces change in many candidate variables simultaneously, leading to an integer-feasible solution sooner. Extensive empirical results are given.

An FPTAS for minimizing the product of two non-negative linear cost functions

We consider a quadratic programming (QP) problem (Π) of the form min x T C x subject to Ax ≥ b, x ≥ 0 where $${C\in {\mathbb R}^{n \times n}_+, {\rm rank}(C)=1}$$ and $${A\in {\mathbb R}^{m \times n}, b\in {\mathbb R}^m}$$ . We present an fully polynomial time approximation scheme (FPTAS) for this problem by reformulating the QP (Π) as a parameterized LP and “rounding” the optimal solution. Furthermore, our algorithm returns an extreme point solution of the polytope. Therefore, our results apply directly to 0–1 problems for which the convex hull of feasible integer solutions is known such as spanning tree, matchings and sub-modular flows. They also apply to problems for which the convex hull of the dominant of the feasible integer solutions is known such as s, t-shortest paths and s, t-min-cuts. For the above discrete problems, the quadratic program Π models the problem of obtaining an integer solution that minimizes the product of two linear non-negative cost functions.

Determining Initial Bound by "Ray-method" in Branch and Bound Procedure

In this paper we present an algorithm for determining initial bound for the Branch and Bound (B&B) method. The idea of this algorithm is based on the use of "ray" as introduced in the "ray-method" developed for solving integer linear programming problems [11], [12]. Instead of solving an integer programming problem we use the main idea of the ray-method to find an integer feasible solution of an integer linear programming problem along the ray as close to an optimal solution of the relaxation problem as possible. The objective value obtained in this manner may be used as an initial bound for the B&B method. It is well known that getting a "good bound" as soon as possible can often significantly increase the performance of the B&B method.

A column generation heuristic for a dynamic generalized assignment problem

This paper studies the dynamic generalized assignment problem (DGAP) which extends the well-known generalized assignment problem by considering a discretized time horizon and by associating a starting time and a finishing time with each task. Additional constraints related to warehouse and yard management applications are also considered. Three linear integer programming formulations of the problem are introduced. The strongest one models the problem as an origin–destination integer multi-commodity flow problem with side constraints. This model can be solved quickly for instances of small to moderate size. However, because of its computer memory requirements, it becomes impractical for larger instances. Hence, a column generation algorithm is used to compute lower bounds by solving the linear program (LP) relaxation of the problem. This column generation algorithm is also embedded in a heuristic aimed at finding feasible integer solutions. Computational experiments on large-scale instances show the effectiveness of the proposed approach.

A Lifted Linear Programming Branch-and-Bound Algorithm for Mixed-Integer Conic Quadratic Programs

This paper develops a linear-programming-based branch-and-bound algorithm for mixed-integer conic quadratic programs. The algorithm is based on a known higher-dimensional or lifted polyhedral relaxation of conic quadratic constraints. 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.

Approximation Algorithms for the Capacitated Multi-Item Lot-Sizing Problem via Flow-Cover Inequalities

We study the classical capacitated multi-item lot-sizing problem with hard capacities. There are N items, each of which has a specified sequence of demands over a finite planning horizon of T discrete periods; the demands are known in advance but can vary from period to period. All demands must be satisfied on time. Each order incurs a time-dependent fixed ordering cost regardless of the combination of items or the number of units ordered, but the total number of units ordered cannot exceed a given capacity C. On the other hand, carrying inventory from period to period incurs holding costs. The goal is to find a feasible solution with minimum overall ordering and holding costs. We show that the problem is strongly NP-hard, and then propose a novel facility location type LP relaxation that is based on an exponentially large subset of the well-known flow-cover inequalities; the proposed LP can be solved to optimality in polynomial time via an efficient separation procedure for this subset of inequalities. Moreover, the optimal solution of the LP can be rounded to a feasible integer solution with cost that is at most twice the optimal cost; this provides a 2-approximation algorithm which is the first constant approximation algorithm for the problem. We also describe an interesting on-the-fly variant of the algorithm that does not require solving the LP a priori with all the flow-cover inequalities. As a by-product we obtain the first theoretical proof regarding the strength of flow-cover inequalities in capacitated inventory models.

An Exact Method for a Discrete Multiobjective Linear Fractional Optimization

Integer linear fractional programming problem with multiple objective (MOILFP) is an important field of research and has not received as much attention as did multiple objective linear fractional programming. In this work, we develop a branch and cut algorithm based on continuous fractional optimization, for generating the whole integer efficient solutions of the MOILFP problem. The basic idea of the computation phase of the algorithm is to optimize one of the fractional objective functions, then generate an integer feasible solution. Using the reduced gradients of the objective functions, an efficient cut is built and a part of the feasible domain not containing efficient solutions is truncated by adding this cut. A sample problem is solved using this algorithm, and the main practical advantages of the algorithm are indicated.