A computational study of an objective hyperplane search heuristic for the general integer linear programming problem

The paper describes an objective function hyperplane search heuristic for solving the general all-integer linear programming problem (ILP). The algorithm searches a series of objective function hyperplanes and the search over any given hyperplane is formulated as a bounded knapsack problem. Theory developed for combinations of the objective function and problem constraints is used to guide the search. We evaluate the algorithm's performance on a class of ILP problems to assess the areas of effectiveness.     

An objective hyperplane search procedure for solving the general all-integer linear programming (ILP) problem

We describe an objective hyperplane search method for solving a class of integer linear programming (ILP) problems. We formulate the search as a bounded knapsack problem and develop requisite theory for formulating knapsack problems with composite constraints and composite objective functions that facilitate convergence to an ILP solution. A heuristic solution algorithm was developed and used to solve a variety of test problems found in the literature. The method obtains optimal or near-optimal solutions in acceptable ranges of computational effort.     

An integer linear programming approach for a class of bilinear integer programs

We propose an Integer Linear Programming (ILP) approach for solving integer programs with bilinear objectives and linear constraints. Our approach is based on finding upper and lower bounds for the integer ensembles in the bilinear objective function, and using the bounds to obtain a tight ILP reformulation of the original problem, which can then be solved efficiently. Numerical experiments suggest that the proposed approach outperforms a latest iterative ILP approach, with notable reductions in the average solution time.     

Cutting plane versus compact formulations for uncertain (integer) linear programs

Robustness is about reducing the feasible set of a given nominal optimization problem by cutting ??risky?? solutions away. To this end, the most popular approach in the literature is to extend the nominal model with a polynomial number of additional variables and constraints, so as to obtain its robust counterpart. Robustness can also be enforced by adding a possibly exponential family of cutting planes, which typically leads to an exponential formulation where cuts have to be generated at run time. Both approaches have pros and cons, and it is not clear which is the best one when approaching a specific problem. In this paper we computationally compare the two options on some prototype problems with different characteristics. We first address robust optimization à la Bertsimas and Sim for linear programs, and show through computational experiments that a considerable speedup (up to 2 orders of magnitude) can be achieved by exploiting a dynamic cut generation scheme. For integer linear problems, instead, the compact formulation exhibits a typically better performance. We then move to a probabilistic setting and introduce the uncertain set covering problem where each column has a certain probability of disappearing, and each row has to be covered with high probability. A related uncertain graph connectivity problem is also investigated, where edges have a certain probability of failure. For both problems, compact ILP models and cutting plane solution schemes are presented and compared through extensive computational tests. The outcome is that a compact ILP formulation (if available) can be preferable because it allows for a better use of the rich arsenal of preprocessing/cut generation tools available in modern ILP solvers. For the cases where such a compact ILP formulation is not available, as in the uncertain connectivity problem, we propose a restart solution strategy and computationally show its practical effectiveness.     

An integer linear programming approach for bilinear integer programming

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.     

Mixed integer programming for a special logic constrained optimal control problem

Integrating logical constraints into optimal control problems is not an easy task. In fact, optimal control problems are usually continuous while logical constraints are naturally expressed by integer (binary) variables. In this article we are interested is a particular form of an LQR optimal control problem: the energy (control L² norm) is to be minimized, system dynamic is linear and logical constraints on the control use are to be fulfilled. Even if the starting continuous problem is not a complicated one, difficulties arise when integrating the additional logical constraints. First, we will present two different ways of modeling the problem, both of them leading us to Mixed Integer Problems. Furthermore, algorithms (Generalized Outer Approximation, Benders Decomposition and Branch and Cut) are applied on each model and results analyzed. We also present a Benders Decomposition algorithm variant that is adapted to our problem (taking into account its particular form) and we will conclude by looking at the optimal solutions obtained in an interesting physical example: the harmonic spring.     

Multi-product production planning: A goal programming approach

Cost minimization multi-product production problems with static production resource usage and internal product flow requirements have been solved by linear programming (LP) with input/output analysis. If the problem is complicated by interval resource estimates, interval linear programming (ILP) can be used. The solution of realistic problems by the above method is cumbersome. This paper suggests that linear goal programming (LGP) can be used to model a multi-product production system. LGP's unique modeling capabilities are used to solve a production planning problem with variable resource parameters. Input/output analysis is used to determine the technological coefficients for the goal constraints and is also used to derive an information sub-model that is used to reduce the number of variable resource goal constraints. Preliminary findings suggest that the LGP approach is more cost-efficient (in terms of CPU time) and in addition provides valuable information for aggregate planning.     

Integrated Shipment Dispatching and Packing Problems: a Case Study

In this paper we examine a consolidation and dispatching problem motivated by a multinational chemical company which has to decide routinely the best way of delivering a set of orders to its customers over a multi-day planning horizon. Every day the decision to be made includes order consolidation, vehicle dispatching as well as load packing into the vehicles. We develop a heuristic based on a cutting plane framework, in which a simplified Integer Linear Program (ILP) is solved to optimality. Since the ILP solution may correspond to a infeasible loading plan, a feasibility check is performed through a tailored heuristic for a three-dimensional bin packing problem with side constraints. If this test fails, a cut able to remove the infeasible solution is generated and added to the simplified ILP. Then the procedure is iterated. Computational results show that our procedure allows achieving remarkable cost savings.      

A new equivalent transformation for interval inequality constraints of interval linear programming

In this paper, we have introduced a new approach to solve a class of interval linear programming (ILP) problems. Firstly, the novel concept of an interval ordering relation is further developed to make desired solution feasible. Secondly, according to the 3\(\upsigma \) law of normal distribution, a new equivalent transformation for constraints with the interval-valued coefficients of ILP is justified. Accordingly, the uncertainty stemmed from interval number could be replaced by the uncertainty of random variables. Consequently, the classical methodology of stochastic linear programming, a chance constrained programming model based on normal distribution is designed to work out the equivalent form of the original problem. This is because it allows us to carry out the optimization operation with a certain calibrated probability. A typical numerical example is given to illustrate how to apply equivalent transformation in order to realize ILP. Finally, we conclude this paper by elaborated comparisons among our method and selected existing solutions to advance our confidence of our research results as to their correctness and effectiveness.     

An ILP based hierarchical global routing approach for VLSI ASIC design

The use of integrated circuits in high-performance computing, telecommunications, and consumer electronics has been growing at a very fast pace. The level of integration as measured by the number of logic gates in a chip has been steadily rising due to the rapid progress in processing and interconnect technology. The interconnect delay in VLSI circuits has become a critical determiner of circuit performance. As a result, circuit layout is starting to play a more important role in today’s chip designs. Global routing is one of the key sub-problems of circuit layout which involves finding an approximate path for the wires connecting the elements of the circuit without violating resource constraints. In this paper, several integer programming (ILP) based global routing models are fully investigated and explored. The resulting ILP problem is relaxed and solved as a linear programming (LP) problem followed by a rounding heuristic to obtain an integer solution. Experimental results obtained show that the proposed combined WVEM (wirelength, via, edge capacity) model can optimize several global routing objectives simultaneously and effectively. In addition, several hierarchical methods are combined with the proposed flat ILP based global router to reduce the CPU time by about 66% on average for edge capacity model (ECM).     

A “reduce and solve” approach for the multiple-choice multidimensional knapsack problem

The multiple-choice multidimensional knapsack problem (MMKP) is a well-known NP-hard combinatorial optimization problem with a number of important applications. In this paper, we present a “reduce and solve” heuristic approach which combines problem reduction techniques with an Integer Linear Programming (ILP) solver (CPLEX). The key ingredient of the proposed approach is a set of group fixing and variable fixing rules. These fixing rules rely mainly on information from the linear relaxation of the given problem and aim to generate reduced critical subproblem to be solved by the ILP solver. Additional strategies are used to explore the space of the reduced problems. Extensive experimental studies over two sets of 37 MMKP benchmark instances in the literature show that our approach competes favorably with the most recent state-of-the-art algorithms. In particular, for the set of 27 conventional benchmarks, the proposed approach finds an improved best lower bound for 11 instances and as a by-product improves all the previous best upper bounds. For the 10 additional instances with irregular structures, the method improves 7 best known results.     

Bivariate interval semi-infinite programming with an application to environmental decision-making analysis

This paper proposed a bivariate interval semi-infinite linear programming (BV-ISIP) method to address a type decision-making problem where various uncertainties exist in functional relations and parameter uncertainty. The performance of the method is also demonstrated via an illustrative example and an environmental decision-making problem. As BV-ISIP guarantees that each of the constraints is satisfied under all possible levels of independent variables, the system-failure risk can be reduced. The BV-ISIP solutions can be more robust to the variation of coefficients associated with independent variables than the ILP ones. Other features of BV-ISIP are as follows: (i) flexible decision-making schemes can be developed for decision makers in terms of the BV-ISIP solutions; (ii) BV-ISIP can conveniently be applied to many large-scale optimization problems as no significantly-increased computational costs are required; (iii) the method can easily be improved for addressing functional intervals associated with multiple independent variables.     

An ILP and simulation model to optimize search and rescue helicopter operations

Maritime search and rescue (SAR) operations, conducted for rendering aid to the victims in need of help at sea, play a crucial role in dropping the number of causalities. Therefore, it is of high importance to organize SAR operations properly. In this paper, we compose a hybrid methodology which combines optimization and simulation to allocate SAR helicopters. First, we build an integer linear programming (ILP) model to provide an effective deployment plan and use it as an input to a simulation model which includes constraints that the ILP model cannot tackle. Next, using a rule-based algorithm, we generate alternative solutions and seek better plans that exist in the vicinity of the ILP model solution. We perform our methodology on the historical incident data in the Aegean Sea region. Results show that the hybrid methodology we adopted leads to a more effective utilization of resources than the optimization model alone.     

The multiparametric 0–1-integer linear programming problem: A unified approach

We designed an algorithm for the multiparametric 0–1-integer linear programming (ILP) problem with the perturbation of the constraint matrix, the objective function and the right-hand side vector simultaneously considered. Our algorithm works by choosing an appropriate finite sequence of non-parametric mixed integer linear programming (MILP) problems in order to obtain a complete multiparametrical analysis. The algorithm may be implemented by using any software capable of solving MILP problems.     

Cyclic scheduling via integer programs with circular ones

A fundamental problem of cyclic staffing is to size and schedule a minimum-cost workforce so that sufficient workers are on duty during each time period. This may be modeled as an integer linear program with a cyclically structured 0-1 constraint matrix. We identify a large class of such problems for which special structure permits the ILP to be solved parametrically as a bounded series of network flow problems. Moreover, an alternative solution technique is shown in which the continuous-valued LP is solved and the result rounded in a special way to yield an optimum solution to the ILP.     

A matheuristic approach for solving a home health care problem

We deal with a Home Health Care Problem (HHCP) which objective consists in constructing the optimal routes and rosters for the health care staffs. The challenge lies in combining aspects of vehicle routing and staff rostering which are two well known hard combinatorial optimization problems. To solve this problem, we initially propose an integer linear programming formulation (ILP) and we tested this model on small instances. To deal with larger instances we develop a matheuristic based on the decomposition of the ILP formulation into two problems. The first one is a set partitioning like problem and it represents the rostering part. The second problem consists in the routing part. This latter is equivalent to a Multi-depot Traveling Salesman Problem with Time Windows (MTSPTW).     

The optimal solution set of the interval linear programming problems

Several methods exist for solving the interval linear programming (ILP) problem. In most of these methods, we can only obtain the optimal value of the objective function of the ILP problem. In this paper we determine the optimal solution set of the ILP as the intersection of some regions, by the best and the worst case (BWC) methods, when the feasible solution components of the best problem are positive. First, we convert the ILP problem to the convex combination problem by coefficients 0 ≤ λ _j , μ _ij , μ _i  ≤ 1, for i = 1, 2, . . . , m and j = 1, 2, . . . , n. If for each i, j, μ _ij  = μ _i  = λ _j  = 0, then the best problem has been obtained (in case of minimization problem). We move from the best problem towards the worst problem by tiny variations of λ _j , μ _ij and μ _i from 0 to 1. Then we solve each of the obtained problems. All of the optimal solutions form a region that we call the optimal solution set of the ILP. Our aim is to determine this optimal solution set by the best and the worst problem constraints. We show that some theorems to validity of this optimal solution set.     

A new ILP-based refinement heuristic for Vehicle Routing Problems

In this paper we address the Distance-Constrained Capacitated Vehicle Routing Problem (DCVRP), where k minimum-cost routes through a central depot have to be constructed so as to cover all customers while satisfying, for each route, both a capacity and a total-distance-travelled limit. Our starting point is the following refinement procedure proposed in 1981 by Sarvanov and Doroshko for the pure Travelling Salesman Problem (TSP): given a starting tour, (a) remove all the nodes in even position, thus leaving an equal number of ``empty holes' in the tour; (b) optimally re-assign the removed nodes to the empty holes through the efficient solution of a min-sum assignment (weighted bipartite matching) problem. We first extend the Sarvanov-Doroshko method to DCVRP, and then generalize it. Our generalization involves a procedure to generate a large number of new sequences through the extracted nodes, as well as a more sophisticated ILP model for the reallocation of some of these sequences. An important feature of our method is that it does not rely on any specialized ILP code, as any general-purpose ILP solver can be used to solve the reallocation model. We report computational results on a large set of capacitated VRP instances from the literature (with symmetric/asymmetric costs and with/without distance constraints), along with an analysis of the performance of the new method and of its features. Interestingly, in 13 cases the new method was able to improve the best-know solution available from the literature. Work supported by M.I.U.R. and by C.N.R., Italy.     

On linear programs with linear complementarity constraints

The paper is a manifestation of the fundamental importance of the linear program with linear complementarity constraints (LPCC) in disjunctive and hierarchical programming as well as in some novel paradigms of mathematical programming. In addition to providing a unified framework for bilevel and inverse linear optimization, nonconvex piecewise linear programming, indefinite quadratic programs, quantile minimization, and ℓ ₀ minimization, the LPCC provides a gateway to a mathematical program with equilibrium constraints, which itself is an important class of constrained optimization problems that has broad applications. We describe several approaches for the global resolution of the LPCC, including a logical Benders approach that can be applied to problems that may be infeasible or unbounded.