Critical objective function values in linear sum assignment problems

The linear sum assignment problem has been well studied in combinatorial optimization. Because of the integrality property, it is a linear programming problem with a variety of efficient algorithms to solve it. In the given research, we present a reformulation of the linear sum assignment problem and a Lagrangian relaxation algorithm for its reformulation. An important characteristic of the new Lagrangian relaxation method is that the optimal Lagrangian multiplier yields a critical bottleneck value. Lagrangian relaxation has only one Lagrangian multiplier, which can only take on a limited number of values, making the search for the optimal multiplier easy. The interpretation of the optimal Lagrangian parameter is that its value is equal to the price that must be paid for all objects in the problem to be assigned.     

The Retail Planning Problem Under Demand Uncertainty

We consider the retail planning problem in which the retailer chooses suppliers and determines the production, distribution, and inventory planning for products with uncertain demand to minimize total expected costs. This problem is often faced by large retail chains that carry private‐label products. We formulate this problem as a convex‐mixed integer program and show that it is strongly NP‐hard. We determine a lower bound by applying a Lagrangian relaxation and show that this bound outperforms the standard convex programming relaxation while being computationally efficient. We also establish a worst‐case error bound for the Lagrangian relaxation. We then develop heuristics to generate feasible solutions. Our computational results indicate that our convex programming heuristic yields feasible solutions that are close to optimal with an average suboptimality gap at 3.4%. We also develop managerial insights for practitioners who choose suppliers and make production, distribution, and inventory decisions in the supply chain.     

An improved MIP heuristic for the intermodal hub location problem

Optimization methods have been commonly developed for the intermodal hub location problem because it has a broad range of practical applications. These methods include exact methods (limited on solving large-size problems) and heuristics (no guarantee on solution quality). In order to avoid their weakness but to leverage their strength, we develop an improved MIP heuristic combining branch-and-bound, Lagrangian relaxation, and linear programming relaxation. In the heuristic, we generate a population of initial feasible solutions using the branch-and-bound and Lagrangian relaxation methods and create a linear-relaxed solution using the linear programming relaxation method. We combine these feasible and linear-relaxed solutions to fix a portion of hub location variables so as to create a number of restricted hub location subproblems. We then combine the branch-and-bound method to solve these restricted subproblems for iteratively improving solution quality. We discuss in detail the application of the method to the intermodal hub location problem. The discussion is followed by extensive statistical analysis and computational tests, where the analysis shows statistical significance of solutions for guiding the heuristic search and comparisons with other methods indicate that the proposed approach is computationally tractable and is able to obtain competitive results.     

Coordinated Capacitated Lot‐Sizing Problem with Dynamic Demand: A Lagrangian Heuristic

Coordinated replenishment problems are common in manufacturing and distribution when a family of items shares a common production line, supplier, or a mode of transportation. In these situations the coordination of shared, and often limited, resources across items is economically attractive. This paper describes a mixed‐integer programming formulation and Lagrangian relaxation solution procedure for the single‐family coordinated capacitated lot‐sizing problem with dynamic demand. The problem extends both the multi‐item capacitated dynamic demand lot‐sizing problem and the uncapacitated coordinated dynamic demand lot‐sizing problem. We provide the results of computational experiments investigating the mathematical properties of the formulation and the performance of the Lagrangian procedures. The results indicate the superiority of the dual‐based heuristic over linear programming‐based approaches to the problem. The quality of the Lagrangian heuristic solution improved in most instances with increases in problem size. Heuristic solutions averaged 2.52% above optimal. The procedures were applied to an industry test problem yielding a 22.5% reduction in total costs.     

随机需求下联合选址-库存模型研究

研究了一类具有季节性需求特性的商品的联合选址-库存模型。在传统的无容量限制的固定费用设施选址问题中考虑了分销中心的运作库存和安全库存的影响,以及规模经济效应和风险分摊效应,同时考虑了季节性商品未来需求的不确定性,将订货决策作为模型的决策变量,建立了一类随机需求下以期望销售收益最大化为目标函数的联合选址-库存模型,拓展了已有的联合选址-库存模型。该模型是一个混合整数规划问题,给出了求解该问题的基于拉格朗日松弛算法的两阶段算法,最后通过随机生成四组不同规模的数值算例,得到的计算结果表明拉格朗日松弛算法可以有效地求解该问题。     

Semidefinite Programming Relaxations for the Quadratic Assignment Problem

Semidefinite programming (SDP) relaxations for the quadratic assignment problem (QAP) are derived using the dual of the (homogenized) Lagrangian dual of appropriate equivalent representations of QAP. These relaxations result in the interesting, special, case where only the dual problem of the SDP relaxation has strict interior, i.e., the Slater constraint qualification always fails for the primal problem. Although there is no duality gap in theory, this indicates that the relaxation cannot be solved in a numerically stable way. By exploring the geometrical structure of the relaxation, we are able to find projected SDP relaxations. These new relaxations, and their duals, satisfy the Slater constraint qualification, and so can be solved numerically using primal-dual interior-point methods.For one of our models, a preconditioned conjugate gradient method is used for solving the large linear systems which arise when finding the Newton direction. The preconditioner is found by exploiting the special structure of the relaxation. See e.g., Vandenverghe and Boyd (1995) for a similar approach for solving SDP problems arising from control applications.Numerical results are presented which indicate that the described methods yield at least competitive lower bounds.     

Multi-echelon supply chain network design in agile manufacturing

In this paper, we consider a supply chain network design problem in an agile manufacturing scenario with multiple echelons and multiple periods under a situation where multiple customers have heavy demands. Decisions in our supply chain design problem include selection of one or more companies in each echelon, production, inventory, and transportation. We formulate the problem integrating all decisions to minimize the total operational costs including fixed alliance costs between two companies, production, raw material holding, finished products holding, and transportation costs under production and transportation capacity limits. A Lagrangian heuristic is proposed in this paper. Optimizing a Lagrangian relaxation problem provides a lower bound, while a feasible solution is generated by adjustment techniques based on the solution of subproblems at each iteration. Computational results indicate the high quality solutions with less than 5% optimality gap are provided quickly by the approach in this paper. Further, compared to initiative managerial alternatives, an improvement of 15% to 25% is not unusual in certain cases for the proposed approach.     

Multi-echelon supply chain network design in agile manufacturing

In this paper, we consider a supply chain network design problem in an agile manufacturing scenario with multiple echelons and multiple periods under a situation where multiple customers have heavy demands. Decisions in our supply chain design problem include selection of one or more companies in each echelon, production, inventory, and transportation. We formulate the problem integrating all decisions to minimize the total operational costs including fixed alliance costs between two companies, production, raw material holding, finished products holding, and transportation costs under production and transportation capacity limits. A Lagrangian heuristic is proposed in this paper. Optimizing a Lagrangian relaxation problem provides a lower bound, while a feasible solution is generated by adjustment techniques based on the solution of subproblems at each iteration. Computational results indicate the high quality solutions with less than 5% optimality gap are provided quickly by the approach in this paper. Further, compared to initiative managerial alternatives, an improvement of 15% to 25% is not unusual in certain cases for the proposed approach.     

Efficient Solutions for Special Zero-One Programming Problems

A zero-one linear programming is under consideration. It has been proved that for special structures and values of the parameters, the solution of the linear relaxation of the problem is integral and can be either predetermined or computed efficiently. In general, a tight upper bound is provided in order to establish an efficient procedure for solving the problem. The results may have practical implementations in knowledge—management, data-mining, network flow, graph theory, reliability and statistical studies. Seniority is equally shared. A visiting faculty in School of Management at Ben-Gurion University of the Negev, Israel.     

随机多阶段分销网络设计模型

为了更合理的设计分销网络,本文提出了一种随机多阶段的联合选址-库存模型。在该模型中,不仅考虑了经济规模和分摊效益的影响。同时通过情景规划,考虑了在多阶段的分销网络设计中,对未来市场环境的不确定性。该模型的目标是使整个战略周期内的总期望成本(包括库存、运输、选址成本与损失的收益)最小。本文将该模型建立成为了一个非线性的整数规划模型,同时提出了一种基于拉格朗日松弛的求解算法。最后,本文使用该算法求解了三组不同规模的算例,得到的计算结果证明了拉格朗日算法是求解该模型的有效算法。     

A low complexity semidefinite relaxation for large-scale MIMO detection

Many wireless communication problems is based on a convex relaxation of the maximum likelihood problem which further can be cast as binary quadratic programs (BQPs). The two standard relaxation methods that are widely used for solving general BQPs such as spectral methods and semidefinite programming problem (SDP), each have their own advantages and disadvantages. It is widely accepted that small and medium sized SDP problems can be solved efficiently by interior point methods. Albeit, semidefinite relaxation has a tighter bound for large scale problems, but its computational complexity is high. However, Row-by-Row method (RBR) for solving SDPs could be opted for an alternative for large-scale MIMO detection because of low complexity. The present work is a spectral SDP-cut formulation to which the RBR is applied for large-scale MIMO detection. A modified RBR algorithm with tighter bound is presented to specify the efficiency in detecting massive MIMO.     

Solving the part families problem in discrete-parts manufacture by simulated annealing

In the production planning and control of discrete-parts manufacture, aggregation of parts into families, on the basis of similarity, is carried out to ease both long-horizon planning and short-horizon scheduling. Additional advantages are related to those of group technology (GT), such as simplifying the flow of parts and tools and reducing both set-up and production costs. The problem of formally forming part families is presented and discussed. Previous work is reviewed and assessed. Two solution approaches, one based on a location model, the other on simulated annealing, are presented and compared along with test results. The location formulation, which results in an integer programming model solved by Lagrangian relaxation, proved capable of producing solutions of excellent quality, but only for relatively small problem instances. In contrast, simulated annealing, which is a general heuristic approach to combinatorial optimization, produced solutions of comparable quality and could handle realistically large problem instances. However, careful design of the simulated annealing algorithm is crucially important. An effective design is presented.     

An extension of the relaxation algorithm for solving a special case of capacitated arc routing problems

In this paper, we propose an exact method for solving a special integer program associated with the classical capacitated arc routing problems (CARPs) called split demand arc routing problems (SDARP). This method is developed in the context of monotropic programming theory and bases a promising foundation for developing specialized algorithms in order to solve general integer programming problems. In particular, the proposed algorithm generalizes the relaxation algorithm developed by Tseng and Bertsekas (Math. Oper. Res. 12(4):569–596, 1987) for solving linear programming problems. This method can also be viewed as an alternative for the subgradient method for solving Lagrangian relaxed problems. Computational experiments show its high potential in terms of efficiency and goodness of solutions on standard test problems.     

Corporate structure optimisation for multinational companies

In this paper we consider a multinational corporate structuring problem. This problem involves designing a corporate/organisational structure (across different countries) so as to remit profits from a number of subsidiaries to a single parent company, whilst minimising the tax paid (maximise the amount received at the parent company). This corporate structure is constrained to be a (directed) tree structure.We present a mixed-integer zero-one formulation of the problem that (for the test problems examined) provides very good linear programming relaxation bounds. We also present a tabu search heuristic for the problem which, when combined with the bounds provided by the linear programming relaxation, is able to find provably optimal solutions. Extensions to the basic corporate structuring problem and how they can be dealt with using our heuristic are also discussed.Computational results for the solution (to proven optimality) of publicly available test problems involving up to 150 countries are reported. The largest problem solved previously in the literature to proven optimality involved only 22 countries.     

Scheduling and compression for a multiple robot assembly workcell

A scheduling and compression strategy is presented for a multiple robot assembly workcell. Based on dynamic programming, the algorithm gives an initial assembly sequence for a two- or three-robot cell with a common resource or dedicated ones. The initial assembly sequence is computed efficiently, with computational complexity of being O (log n/k) where n is the number of elements in k resources. Then the initial sequence is further compressed towards the time-optimal assemblysequence. Precedence constraints in assembly are considered. The scheme is tested using computer simulations and some test results are presented. For the problems solved (typically involving 20 elements), the computational times were less than 5 seconds in an Apollo 300 workstation.     

基于时间满意的最大覆盖选址问题

传统的选址问题过于简单地考量时间这一对企业竞争力影响重大的因素,结合这一特点,本文对时间满意度函数进行了定义并提出了基于时间满意的最大覆盖选址问题。给定的网络G(V,A)中,在总的顾客对服务站响应速度的满意程度最大的目标下建立了最大覆盖选址问题模型,我们在讨论了问题的特点之后给出了基于拉格朗日松驰的启发式算法,并通过MATLAB进行了编程计算实验,分析了实验结果。     

禁忌搜索算法在单分配多枢纽轴辐式物流网络中的应用

随着经济的发展,轴辐式网络因有提高运输效率,优化资源配置,产生范围经济等作用越来越多的受到各方面学者的关注。本文对轴辐式网络进行优化分析,改进轴辐式网络的多重分配多枢纽中位问题模型,得到无容量限制的单分配多枢纽中位问题模型(USApHLP)的混合整数线性规划模型,并采用改进的禁忌搜索智能算法来求解,通过算例验证了禁忌搜索算法可以有效的求解单分配多枢纽中位问题。     

自动化立体仓库中出入库任务顺序与出库位置选择集成优化研究

具有多个出口的自动化立体仓库系统是一种将存储和分拣相结合的新型仓储技术,其最典型的特征是在货架底层有很多个出库位置以供取货人员分拣。研究此系统中出入库任务排序与出口选择的集成优化问题,以最小化堆垛机完成所有任务的移动距离为目标,将此问题转化为一个混合整数规划模型。根据问题的特点设计了两阶段启发式算法求解此问题,数值结果表明设计的算法能在较短时间内给出近似最优解,同时与企业常用的先到先服务方法相比,该算法可以缩短超过20%的移动距离。     

随机环境中的生产作业计划问题

生产系统中通常会涉及各种不确定因素 ,如不确定的顾客定单、不确定的生产作业时间等 .在当今时间竞争非常激烈的时代中 ,生产型企业如何把握生产系统中的这些不定因素变得尤为关键 .本文研究在不确定的作业时间、工序间延迟时间等情况下的生产作业计划问题 ,利用 scenario模型把这类随机生产计划问题归纳为一个多阶段随机决策问题 .进而 ,采用Lagrangian松弛和 scenario分解的方法求解这样一个大型的决策问题 .最后 ,就一个实例建立模型、进行计算和分析 ,以说明本文提出的随机生产计划方法的特点和有效性     

Restarting after Branching in the SDP Approach to MAX-CUT and Similar Combinatorial Optimization Problems

Many combinatorial optimization problems have relaxations that are semidefinite programming problems. In principle, the combinatorial optimization problem can then be solved by using a branch-and-cut procedure, where the problems to be solved at the nodes of the tree are semidefinite programs. It is desirable that the solution to one node of the tree should be exploited at the child node in order to speed up the solution of the child. We show how the solution to the parent relaxation can be used as a warm start to construct an appropriate initial dual solution to the child problem. This restart method for SDP branch-and-cut can be regarded as analogous to the use of the dual simplex method in the branch-and-cut method for mixed integer linear programming problems.