Cutting Plane Approach Research Articles

Simulation‐Based Optimization of Capacitated Assembly Systems under Beta‐Service Level Constraints

ABSTRACTWe study a class of capacitated assembly systems operated by base‐stock policies and address the problem of finding base‐stock levels that minimize holding costs under beta‐service level (fill rate) constraints over an infinite horizon. To solve this nonconvex constrained optimization problem, we develop a new simulation‐based optimization approach using the constrained level method (CLM) and infinitesimal perturbation analysis. The key idea of the algorithm is a novel family of convex approximations of the beta‐service level, which is iteratively refined during the course of the algorithm. The algorithm can easily handle integrality requirements for the base‐stock levels by combining the CLM with a cutting plane approach without significantly increasing the solution time on typical instances. We apply our approach to a comprehensive multi‐echelon assembly benchmark system from the literature and study the algorithm's behavior for different target service levels, capacity configurations, and simulation horizons. A comparison with a state‐of‐the‐art interior point algorithm shows that for realistic capacity constraints, our algorithm is on average 8%–20% better. Compared with the guaranteed service model, our approach reduces costs by 10%–15% while keeping the same service level.

A cutting-plane approach for large-scale capacitated multi-period facility location using a specialized interior-point method

We propose a cutting-plane approach (namely, Benders decomposition) for a class of capacitated multi-period facility location problems. The novelty of this approach lies on the use of a specialized interior-point method for solving the Benders subproblems. The primal block-angular structure of the resulting linear optimization problems is exploited by the interior-point method, allowing the (either exact or inexact) efficient solution of large instances. The consequences of different modeling conditions and problem specifications on the computational performance are also investigated both theoretically and empirically, providing a deeper understanding of the significant factors influencing the overall efficiency of the cutting-plane method. The methodology proposed allowed the solution of instances of up to 200 potential locations, one million customers and three periods, resulting in mixed integer linear optimization problems of up to 600 binary and 600 millions of continuous variables. Those problems were solved by the specialized approach in less than one hour and a half, outperforming other state-of-the-art methods, which exhausted the (144 GB of) available memory in the largest instances.

Inexact stabilized Benders’ decomposition approaches with application to chance-constrained problems with finite support

We explore modifications of the standard cutting-plane approach for minimizing a convex nondifferentiable function, given by an oracle, over a combinatorial set, which is the basis of the celebrated (generalized) Benders' decomposition approach. Specifically, we combine stabilization--in two ways: via a trust region in the $$L_1$$L1 norm, or via a level constraint--and inexact function computation (solution of the subproblems). Managing both features simultaneously requires a nontrivial convergence analysis; we provide it under very weak assumptions on the handling of the two parameters (target and accuracy) controlling the informative on-demand inexact oracle corresponding to the subproblem, strengthening earlier know results. This yields new versions of Benders' decomposition, whose numerical performance are assessed on a class of hybrid robust and chance-constrained problems that involve a random variable with an underlying discrete distribution, are convex in the decision variable, but have neither separable nor linear probabilistic constraints. The numerical results show that the approach has potential, especially for instances that are difficult to solve with standard techniques.

A cutting plane approach to combinatorial bandwidth packing problem with queuing delays

The combinatorial bandwidth packing problem (CBPP), arising in a telecommunication network with limited bandwidth, is defined as follows. Given a set of requests, each with its potential revenue, and each consisting of calls with their bandwidth requirements, decide: (1) a subset of the requests to accept/reject; and (2) a route for each call in accepted requests, so as to maximize the total revenue earned in a telecommunication network with limited bandwidth. However, telecommunication networks are generally characterized by variability in the call (bits) arrival rates and service times, resulting in queuing delays in the network. In this paper, we present a non-linear integer programming model to account for such delays in CBPP. Using simple transformation and piecewise outer-approximation, we reformulate the model as a linear mixed integer program (MIP), but with a large number of constraints. We present an efficient cutting plane approach to solve the resulting linear MIP to \(\epsilon \)-optimality.

The Cutting Plane Method is Polynomial for Perfect Matchings

The cutting plane approach to finding minimum-cost perfect matchings has been discussed by several authors over past decades. Its convergence has been an open question. We develop a cutting plane algorithm that converges in polynomial-time using only Edmonds’ blossom inequalities, and which maintains half-integral intermediate LP solutions supported by a disjoint union of odd cycles and edges. Our main insight is a method to retain only a subset of the previously added cutting planes based on their dual values. This allows us to quickly find violated blossom inequalities and argue convergence by tracking the number of odd cycles in the support of intermediate solutions.

Improving problem reduction for 0–1 Multidimensional Knapsack Problems with valid inequalities

This paper investigates the problem reduction heuristic for the Multidimensional Knapsack Problem (MKP). The MKP formulation is first strengthened by the Global Lifted Cover Inequalities (GLCI) using the cutting plane approach. The dynamic core problem heuristic is then applied to find good solutions. The GLCI is described in the general lifting framework and several variants are introduced. A Two-level Core problem Heuristic is also proposed to tackle large instances. Computational experiments were carried out on classic benchmark problems to demonstrate the effectiveness of this new method.

Expansion planning for waste-to-energy systems using waste forecast prediction sets

We consider an expansion planning problem for Waste-to-Energy (WtE) systems facing uncertainty in future waste supplies. The WtE expansion plans are regarded as strategic, long term decisions, while the waste distribution and treatment are medium to short term operational decisions which can adapt to the actual waste collected. We propose a prediction set uncertainty model which integrates a set of waste generation forecasts and is constructed based on user-specified levels of forecasting errors. Next, we use the prediction sets for WtE expansion scenario analysis. More specifically, for a given WtE expansion plan, the guaranteed net present value (NPV) is evaluated by computing an extreme value forecast trajectory of future waste generation from the prediction set that minimizes the maximum NPV of the WtE project. This problem is essentially a multiple stage min-max dynamic optimization problem. By exploiting the structure of the WtE problem, we show this is equivalent to a simpler min-max optimization problem, which can be further transformed into a single mixed-integer linear program. Furthermore, we extend the model to optimize the guaranteed NPV by searching over the set of all feasible expansion scenarios, and show that this can be solved by an exact cutting plane approach. We also propose a heuristic based on a constant proportion distribution rule for the WtE expansion optimization model, which reduces the problem into a moderate size mixed-integer program. Finally, our computational studies demonstrate that our proposed expansion model solutions are very stable and competitive in performance compared to scenario tree approaches. © 2016 Wiley Periodicals, Inc. Naval Research Logistics 63: 47–70, 2016

Accurate Maximum-Margin Training for Parsing With Context-Free Grammars.

The task of natural language parsing can naturally be embedded in the maximum-margin framework for structured output prediction using an appropriate joint feature map and a suitable structured loss function. While there are efficient learning algorithms based on the cutting-plane method for optimizing the resulting quadratic objective with potentially exponential number of linear constraints, their efficiency crucially depends on the inference algorithms used to infer the most violated constraint in a current iteration. In this paper, we derive an extension of the well-known Cocke-Kasami-Younger (CKY) algorithm used for parsing with probabilistic context-free grammars for the case of loss-augmented inference enabling an effective training in the cutting-plane approach. The resulting algorithm is guaranteed to find an optimal solution in polynomial time exceeding the running time of the CKY algorithm by a term, which only depends on the number of possible loss values. In order to demonstrate the feasibility of the presented algorithm, we perform a set of experiments for parsing English sentences.

The wireless network jamming problem subject to protocol interference

We study the following questions related to wireless network security: Which jammer placement configuration during a jamming attack results in the largest degradation of network throughput? and Which network design strategies are most effective in mitigating a jamming attack? Although others have studied similar jammer placement problems, this article is the first to optimize network throughput subject to radio wave interference. We formulate this problem as a bi-level mixed-integer program, and solve it using a cutting plane approach that is able to solve networks with up to 81 transmitters, which is a typical size for studies in wireless network optimization. Experiments with the algorithm also yielded the following insights into wireless network jamming: (1) increasing the number of channels is the best strategy for designing a network that is robust against jamming attacks, and (2) increasing the range of the jammer is the best strategy for the attacker. © 2015 Wiley Periodicals, Inc. NETWORKS, Vol. 67(2), 111–125 2016

A Cycle-Based Formulation and Valid Inequalities for DC Power Transmission Problems with Switching

It is well known that optimizing network topology by switching on and off transmission lines improves the efficiency of power delivery in electrical networks. In fact, the USA Energy Policy Act of 2005 (Section 1223) states that the United States should “encourage, as appropriate, the deployment of advanced transmission technologies” including “optimized transmission line configurations.” As such, many authors have studied the problem of determining an optimal set of transmission lines to switch off to minimize the cost of meeting a given power demand under the direct current (DC) model of power flow. This problem is known in the literature as the Direct-Current Optimal Transmission Switching Problem (DC-OTS). Most research on DC-OTS has focused on heuristic algorithms for generating quality solutions or on the application of DC-OTS to crucial operational and strategic problems such as contingency correction, real-time dispatch, and transmission expansion. The mathematical theory of the DC-OTS problem is less well developed. In this work, we formally establish that DC-OTS is NP-Hard, even if the power network is a series-parallel graph with at most one load/demand pair. Inspired by Kirchoff’s Voltage Law, we provide a cycle-based formulation for DC-OTS, and we use the new formulation to build a cycle-induced relaxation. We characterize the convex hull of the cycle-induced relaxation; this characterization provides strong valid inequalities that can be used in a cutting-plane approach to solve the DC-OTS. We give details of a practical implementation, and we show promising computational results on standard benchmark instances.

A geometric approach to cut-generating functions

The cutting-plane approach to integer programming was initiated more that 40 years ago: Gomory introduced the corner polyhedron as a relaxation of a mixed integer set in tableau form and Balas introduced intersection cuts for the corner polyhedron. This line of research was left dormant for several decades until relatively recently, when a paper of Andersen, Louveaux, Weismantel and Wolsey generated renewed interest in the corner polyhedron and intersection cuts. Recent developments rely on tools drawn from convex analysis, geometry and number theory, and constitute an elegant bridge between these areas and integer programming. We survey these results and highlight recent breakthroughs in this area.

On solving multi objective Set Covering Problem with imprecise linear fractional objectives

The Set Covering problem is one the of most important NP-complete 0-1 integer programming problems because it serves as a model for many real world problems like the crew scheduling problem, facility location problem, vehicle routing etc. In this paper, an algorithm is suggested to solve a multi objective Set Covering problem with fuzzy linear fractional functionals as the objectives. The algorithm obtains the complete set of efficient cover solutions for this problem. It is based on the cutting plane approach, but employs a more generalized and a much deeper form of the Dantzig cut. The fuzziness in the problem lies in the coefficients of the objective functions. In addition, the ordering between two fuzzy numbers is based on the possibility and necessity indices introduced by Dubois and Prade [Ranking fuzzy numbers in the setting of possibility theory. Inf. Sci. 30 (1983) 183–224.]. Our aim is to develop a method which provides the decision maker with a fuzzy solution. An illustrative numerical example is elaborated to clarify the theory and the solution algorithm.

Linearization approach to multi objective set covering problem with imprecise nonlinear fractional objectives

The Set Covering Problem is one of most important NP-complete 0–1 integer programming problems as it serves as a model for many real world problems, like crew scheduling problems, facility location problems, vehicle routing etc. This paper develops an enumerative technique for solving the multiobjective set covering problem with fuzzy nonlinear fractional objectives, under certain convexity conditions. To mirror real life situations in the covering problem we have taken the objective functions in a nonlinear fractional form. In addition to that, the coefficients of the objective functions are of a fuzzy nature, as in most cases the decision maker is unable to provide information of a precise nature. The algorithm provided in this paper aims to not only solve this covering problem to obtain a set of efficient cover solutions, but also to provide the decision maker with a fuzzy solution for the same. It is based on a linearization technique with the subsequent application of the cutting plane approach, but it employs a more generalized and much deeper form of the Dantzig cut. An example to illustrate the technique is presented, and a few applications of this problem are also described.

Constraint Optimal Selection Techniques (COSTs) for nonnegative linear programming problems

We describe an active-set, cutting-plane approach called Constraint Optimal Selection Techniques (COSTs) and develop an efficient new COST for solving nonnegative linear programming problems. We give a geometric interpretation of the new selection rule and provide computational comparisons of the new COST with existing linear programming algorithms for some large-scale sample problems.

Warehouse location and two-echelon inventory management with concave operating cost

In this paper, we study a location–inventory network design problem which jointly optimises the warehouse location, the warehouse–retailer assignments, the warehouse–retailer echelon inventory replenishment and the safety stock-level decisions over an infinite planning horizon. The consideration of the facility operating cost, the safety stock cost and the two-echelon inventory cost results in an MIP model with several nonlinear terms. Due to the complex trade-offs among the various costs and multiple nonlinear terms in the model, traditional solution approaches no longer work for this problem. We outline a polymatroid cutting-plane approach based on the submodular property of the cost terms to address this problem. Computational results demonstrate that the cutting-plane method based on polymatroid inequalities can efficiently solve randomly generated instances with moderate sizes.

Single asset optimal trading strategies with stochastic dominance constraints

In this paper, we develop optimal trading strategies for the risk averse investor by minimizing the expected cost and the risk of execution. We present quadratic programming formulation that includes stochastic dominance constraints to render the preference relationship attitude of both risk neutral and risk averse investors. We also present a cutting plane approach to facilitate computational advantage in solving it. The efficacy of the algorithm is shown with the help of numerical examples.

A cutting plane approach for multi-objective integer indefinite quadratic programming problem

In this paper, an algorithm is developed to solve a Multi-Objective Integer Indefinite Quadratic Programming Problem (IQMPP). The cutting plane technique finds all the non-dominated p-tuples of the (IQMPP) problem. Since the objective functions of (IQMPP) problem are quasi-monotone, the cutting plane truncates a portion of the feasible region and enables us to find all the non-dominated p-tuples at extreme points of the remaining feasible region. The algorithm is explained with the help of an example.

A computational study for common network design in multi-commodity supply chains

In this paper, we study a supply chain network design problem which consists of one external supplier, a set of potential distribution centers, and a set of retailers, each of which is faced with uncertain demands for multiple commodities. The demand of each retailer is fulfilled by a single distribution center for all commodities. The goal is to minimize the system-wide cost including location, transportation, and inventory costs. We propose a general nonlinear integer programming model for the problem and present a cutting plane approach based on polymatroid inequalities to solve the model. Randomly generated instances for two special cases of our model, i.e., the single-sourcing UPL&TAP and the single-sourcing multi-commodity location-inventory model, are provided to test our algorithm. Computational results show that the proposed algorithm can solve moderate-sized problem instances efficiently.

Multi-level lot sizing and job shop scheduling with compressible process times: A cutting plane approach

This paper proposes an integer linear programming formulation for a simultaneous lot sizing and scheduling problem in a job shop environment. Among others, one of our realistic assumptions is dealing with flexible machines which enable the production manager to change their working speeds. Then, a number of valid inequalities are developed based on problem structures. As the valid inequalities can help in reducing the non-optimal parts of the solution space, they are dealt with as some cutting planes. The proposed cutting planes are used to solve the problem in (i) cut-and-branch, and (ii) branch-and-cut approaches. The performance of each cutting plane is investigated with CPLEX 12.2 on a set of randomly-generated test data. Then, some performance criteria are identified and the proposed cutting planes are ranked by TOPSIS method.

On the Study of Distinct Algorithmic Strategies in the Implementation of Advanced Anisotropic Models with Mixed Hardening

In this study, suitable distinct stress integration algorithms for advanced anisotropic models with mixed hardening, and their implementation in finite element codes, are discussed. The constitutive model studied in the present work accounts for advanced (non-quadratic) anisotropic yield criteria, namely, the Barlat et al. 2004 model with 18 coefficients (Yld2004-18p), combined with a mixed isotropic-nonlinear kinematic hardening law. This phenomenological model allows for an accurate description of complex plastic yielding anisotropy and Bauschinger effects, which are essential for a reliable prediction of deep drawing and springback results using numerical simulations.In the present work distinct algorithm classes are analysed: (i) a semi-explicit algorithm that accounts for the sub-incrementation technique; (ii) the cutting-plane approach (semi-implicit integration); and (iii) the fully-implicit multi-stage return mapping procedure, based on the control of the potential residual. The numerical performance of the developed algorithms is inferred by benchmarks in sheet metal forming processes. The quality of the solution is assessed and compared to reference results. In the end, an algorithmic and programming framework is provided, suitable for a direct implementation in commercial Finite Element codes, such as Abaqus (Simulia) and Marc (MSC-Software) packages.