Commercial Mixed-integer Linear Programming Research Articles

Lifetime maximization of wireless sensor networks while ensuring intruder detection

Wireless sensor networks (WSN) have a wide variety of application areas and one of these areas is border crossing security. Unauthorized crossing of border areas, unauthorized arms and drug trafficking can be avoided at a lower cost and easier than conventional methods by monitoring the borders with the help of a WSN. In this study, we offer a mathematical model that guarantees the detection of possible intruders by scheduling the activities of the sensors whatever the route the intruder follows throughout the border zone or whatever the time the intruder enters to the route. To achieve the highest possible WSN management efficiency, we integrate coverage, routing, data routing, and sensor scheduling WSN design issues into the mathematical model. We first demonstrate the effectiveness of scheduling the sensors by the help of the offered mathematical model by comparing it against a random activity schedule of the sensors with respect to network lifetime and intruder detection ratio performance measures. We also develop a Lagrangean heuristic strategy to solve realistic sized instances of the proposed problem. We produce several random border zone instances with varying sizes and test the proposed solution strategy to illustrate the effectiveness of the offered solution strategy by comparing its performance against the performance of a commercial mixed-integer linear programming (MILP) solver.

Development of Heuristic Approaches for Last-Mile Delivery TSP with a Truck and Multiple Drones

Unmanned Aerial Vehicles (UAVs) are gaining momentum in many civil and military sectors. An example is represented by the logistics sector, where UAVs have been proven to be able to improve the efficiency of the process itself, as their cooperation with trucks can decrease the delivery time and reduce fuel consumption. In this paper, we first state a mathematical formulation of the Travelling Salesman Problem (TSP) applied to logistic routing, where a truck cooperates synchronously with multiple UAVs for parcel delivery. Then, we propose, implement, and compare different sub-optimal routing approaches to the formulated mFSTSP (multiple Flying Sidekick Travelling Salesman Problem) since the inherent combinatorial computational complexity of the problem makes it unattractable for commercial Mixed-Integer Linear Programming (MILP) solvers. A local search algorithm, two hybrid genetic algorithms that permutate feasible and infeasible solutions, and an alternative ad-hoc greedy method are evaluated in terms of the total delivery time of the output schedule. For the sake of the evaluation, the savings in terms of delivery time over the well-documented truck-only TSP solution are investigated for each proposed routing solution, and this is repeated for two different scenarios. Monte Carlo simulations corroborate the results.

Unsupervised Learning-Driven Matheuristic for Production-Distribution Problems

In this paper, we study a capacitated production-distribution problem where facility location, production, and distribution decisions are tightly coupled and simultaneously considered in the optimal decision making. Such an integrated production-distribution problem is complicated, and the current commercial mixed-integer linear programming (MILP) solvers cannot obtain favorable solutions for the medium- and large-sized problem instances. Therefore, we propose an unsupervised learning-driven matheuristic that uses easily obtainable solution values (e.g., solutions associated with the linear programming relaxation) to build clustering models and integrates the clustering information with a genetic algorithm to progressively improve feasible solutions. Then we verify the performance of the proposed matheuristic by comparing its computational results with those of the rolling horizon algorithm, a non-cluster-driven matheuristic, and a commercial MILP solver. The computational results show that, under the same computing resources, the proposed matheuristic can deliver better production-distribution decisions. Specifically, it reduces the total system costs by 15% for the tested instances when compared with the ones found by the commercial MILP solver. Additionally, we apply the proposed matheuristic to a related production-distribution problem in the literature and obtain 152 equivalent or new best-known solutions out of 200 benchmark test instances.

Mixed-integer nonseparable piecewise linear models for the hydropower production function in the Unit Commitment problem

The hydro unit commitment (HUC) problem seeks to determine, for a short-term horizon with (semi)hourly discretization, the status (on/off) and generation level of each generating unit (GU) to meet plant and GUs constraints. In the HUC problem, the nonlinearities and non-convexities of the hydro production function (HPF), and the presence of binary variables that identify which GUs must be dispatched at each time step make the search for a solution challenging. Due to the recent developments in commercial mixed-integer linear programming (MILP) solvers, it is possible to approximate the nonlinear and nonconvex HPF through piecewise-linear (PWL) models with reasonable accuracy. Throughout this paper we emphasize the potential advantages of seven MILP formulations that can be categorized as parametric and non-parametric methods. Given the complexities of state-of-the-art solvers, it is hard to predict which formulation performs better. Although some guidelines can be found in literature, the formulation that performs best can be strongly dependent on the specific problem structure or data. In this context, we develop and compare seven multidimensional nonseparable PWL models for representing the HPF in the HUC problem. To assess the performance of each PWL model, we present results using a 6-GU hydro plant of two different types.

Tight and Compact MIP Formulation of Configuration-Based Combined-Cycle Units

Private investors, flexibility, efficiency and environmental requirements from deregulated markets have led the existence and building of a significant number of combined-cycle gas turbines (CCGTs) in many power systems. These plants represent a complex optimization problem for the short-term planning unit commitment (UC) carried out by independent system operators due to their multiple operating configurations. Accordingly, this paper proposes a mixed-integer linear programming (MIP) formulation of the configuration-based model of CCGTs, which is commonly utilized for bid/offering market processes. This formulation is simultaneously tighter and more compact than analogous MIP-based models; hence, it presents a lower computational burden. The computational efficiency of the proposed formulation is demonstrated by solving network-constrained UC case studies, of different size and complexity, using three of the leading commercial MIP solvers: CPLEX, GUROBI, and XPRESS.

Multilevel Optimization Modeling for Risk-Averse Stochastic Programming

Coherent risk measures have become a popular tool for incorporating risk aversion into stochastic optimization models. For dynamic models in which uncertainty is resolved at more than one stage, however, using coherent risk measures within a standard single-level optimization framework becomes problematic. To avoid severe time-consistency difficulties, the current state of the art is to employ risk measures of a specific nested form, which unfortunately have some undesirable and somewhat counterintuitive modeling properties. This paper summarizes the potential drawbacks of nested-form risk measure issues and then presents an alternative multilevel optimization modeling approach that enforces a form of time consistency through constraints rather than by restricting the modeler’s choice of objective function. This technique leads to models that are time consistent even while using time-inconsistent risk measures and can easily be formulated to be law invariant with respect to the final wealth if so desired. We argue that this approach should be the starting point for all multistage optimization modeling. When used with time-consistent objective functions, we show its multilevel optimization constraints become redundant, and the associated models thus simplify to a more familiar single-objective form. Unfortunately, we also show that our proposed approach leads to 𝒩𝒫-hard models, even in the simplest imaginable setting in which it would be needed: three-stage linear problems on a finite probability space, using the standard average value-at-risk and first-order mean-semideviation risk measures. Finally, we show that for a simple but reasonably realistic test application, the kind of models we propose, although drawn from an 𝒩𝒫-hard family and certainly more time consuming to solve than those obtained from the nested-objective approach, are readily solvable to global optimality using a standard commercial mixed-integer linear programming solver. Therefore, there seems some promise of our proposed modeling approach being useful despite its computational complexity properties.

Models and Lagrangian heuristics for a two-level lot-sizing problem with bounded inventory

We consider a two-level dynamic lot-sizing problem where the first level consists of N finished products competing for a single type of purchased raw material in the second level. While the procurement and production capacities are unlimited, the storage capacity of the raw material is limited and must be carefully managed. The goal is to simultaneously determine a replenishment plan for the raw material and optimal production plans for the finished products on a horizon of T periods while minimizing production, purchasing, setup and inventory holding costs. The problem is modeled using mixed-integer linear programs and solved using both a Lagrangian relaxation-based heuristic and a commercial mixed-integer linear programming solver. Learning capabilities are integrated in the Lagrangian relaxation to update step size in the subgradient algorithm. The computational results show that the Lagrangian heuristic outperforms the solver on different formulations, in particular for large problems with long time horizons.

Integer programming for the generalized high school timetabling problem

Recently, the XHSTT format for high school timetabling was introduced. It provides a uniform way of modeling problem instances and corresponding solutions. The format supports a wide variety of constraints, and currently 38 real-life instances from 11 different countries are available. Thereby, the XHSTT format serves as a common ground for researchers within this area. This paper describes the first exact method capable of handling an arbitrary instance of the XHSTT format. The method is based on a mixed-integer linear programming (MIP) model, which is solved in two steps with a commercial general-purpose MIP solver. Computational results show that our approach is able to find previously unknown optimal solutions for 2 instances of XHSTT and proves optimality of 4 known solutions. For the instances not solved to optimality, new non-trivial lower bounds were found in 11 cases, and new best known solutions were found in 9 cases. Furthermore, the approach is compared with the finalists of Round 2 of the International Timetabling Competition 2011 and is shown to be competitive with one of the finalists.

An efficient local search heuristic for the double row layout problem with asymmetric material flow

The double row layout problem (DRLP) consists of arranging a number of rectangular machines of varying widths on either side of a corridor to minimize the total cost of material handling for products that move between these machines. This problem arises in the context of many production environments, most notably semiconductor manufacturing. Because the DRLP contains both combinatorial and continuous aspects, traditional solution approaches are not well suited to obtain solutions within a reasonable time. Moreover, previous approaches to this problem did not consider asymmetric flows. In this paper, an effective local search procedure featuring linear programming is proposed for solving the DRLP with asymmetric flows (symmetric flows being a special case). This approach is compared against several constructive heuristics and solutions obtained by a commercial mixed integer linear programming solver to evaluate its performance. Computational results show that the proposed heuristic is an effective approach, both in terms of solution quality and computational effort.

Tight polyhedral approximation for mixed-integer linear programming unit commitment formulations

The perspective reformulation has been recently proposed for constructing tight relaxations of the unit commitment (UC) problem with quadratic cost functions. In this case, it has been shown that the perspective reformulation can be cast as a second-order cone program. Conic quadratic programming is based on interior-point methods and does not benefit from the warm-start capabilities of the simplex method available in commercial mixed-integer linear programming software. It is known that the perspective formulation can be approximated with arbitrary accuracy by dynamically adding gradient-type inequalities. This study presents a higher dimensional polyhedral approximation of the perspective reformulation. The proposed approximation is not based on gradient information and can be easily implemented in a modelling language; in addition, it requires defining additional variables and constraints whose number grows moderately with increasing levels of accuracy. Extensive numerical comparisons show that the polyhedral approximation of the perspective reformulation yields a tight UC formulation, which is competitive with currently known ones.

In This Issue

In This Issue

Efficient solution of the 3G network planning problem

The 3G universal mobile telecommunications system (UMTS) planning problem is combinatorially explosive and difficult to solve optimally, though solution methods exist for its three main subproblems (cell, access network, and core network planning). We previously formulated the entire problem as a single integrated mixed-integer linear program (MIP) and showed that only small instances of this global planning problem can be solved to optimality by a commercial MIP solver within a reasonable amount of time (St-Hilaire, Chamberland, & Pierre, 2006). Heuristic methods are needed for larger instances. This paper provides the first complete formulation for the heuristic sequential method (St-Hilaire, Chamberland, & Pierre, 2005) that re-partitions the global formulation into the three conventional subproblems and solves them in sequence using a MIP solver. This greatly improves the solution time, but at the expense of solution quality. We also develop a new hybrid heuristic that uses the results of the sequential method to generate constraints that provide tighter bounds for the global planning problem with the goal of reaching the provable optimum solution much more quickly. We empirically evaluate the speed and solution accuracy of four solution methods: (i) the direct MIP solution of the global planning problem, (ii) a local search heuristic applied to the global planning problem, (iii) the sequential method and (iv) the new hybrid method. The results show that the sequential method provides very good results in a fraction of the time needed for the direct MIP solution of the global problem, and that optimum results can be provided by the hybrid heuristic in greatly reduced time.

A Fast Heuristic Algorithm for Multidomain Clock Skew Scheduling

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> In the most general form, clock skew scheduling (CSS) generates a dedicated clock delay for each individual sequential component in the clock distribution network in order to minimize the clock period. Multidomain CSS (MDCSS) relieves this requirement. Instead, sequential components are grouped into several clusters (called clock domains), each of which has a uniform clock delay for all registers within that domain. The skew values of clock domains are provided by a set of deskew buffers with electrically programmable phase shifts and injected after the chip is manufactured. This technique is attractive since, due to process variations, it is becoming overwhelmingly difficult to create precise clock network delays for all sequential elements in a design globally. In this paper, we present a fast algorithm for determining the minimum number of clock domains to be used by MDCSS. The exact solution to this problem cannot be found within a reasonable time if the number of clock domains increases beyond three domains. We show that, even with a small-size circuit, in order to obtain the minimum clock period, more than three clock domains may be required. Therefore, a fast heuristic algorithm is needed to identify these domains. To the best of our knowledge, we present the first efficient heuristic algorithm for this problem. For large benchmark circuits, we solve the problem within 14.7 min on average (as high as 31.7 min for the worst case), while a commercial mixed-integer linear program solver cannot finish in over 5 h. Furthermore, our results show that, for 19 out of 21 small- and medium-size benchmarks, our algorithm yields the optimal solution. </para>

Optimal scheduling for flexible job shop operation

This paper presents a new integer linear programming (ILP) model to schedule flexible job shop, discrete parts manufacturing industries that operate on a make-to-order basis. The model considers groups of parallel homogeneous machines, limited intermediate buffers and negligible set-up effects. Orders consist of a number of discrete units to be produced and follow one of a given number of processing routes. The model allows re-circulation to take place, an important issue in practice that has received scant treatment in the scheduling literature. Good solution times were obtained using commercial mixed-integer linear programming (MILP) software to solve realistic examples of flexible job shops to optimality. This supports the claim that recent advances in computational power and MILP solution algorithms are making this approach competitive with others traditionally applied in job shop scheduling.

A Hierarchical Lagrangean Relaxation Procedure for Solving Midterm Planning Problems

An efficient decomposition procedure for solving midterm planning problems is developed based on Lagrangean relaxation. The basic idea of the proposed solution technique is the successive partitioning of the original problem into smaller, more computationally tractable subproblems by hierarchical relaxation of key complicating constraints. The systematic identification of these complicating constraints is accomplished by utilizing linear programming relaxation dual-multiplier information. This hierarchical Lagrangean relaxation procedure, along with an upper bound generating heuristic, is incorporated within a subgradient optimization framework. This solution strategy is found to be much more effective, in terms of both quality of solution and computational requirements, than commercial mixed-integer linear programming solvers in bracketing the optimal value, especially for larger problems.