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.     

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.     

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.     

FIRM EXPANSION THROUGH FRANCHISING: A MODEL AND SOLUTION PROCEDURES

This paper presents a practical model for firm expansion through franchising. The model allows the possibility of opening both company-owned and franchised stores. The objective is to maximize the expected returns to the franchisor from both types of stores, subject to the total capital outlay budget and the excess capacity available at each warehouse. A relaxation for this problem is studied and a heuristic solution procedure that makes use of this relaxation is developed. Experimental results over a wide range of problem structures show this solution methodology to be very effective, with gaps between feasible solution values and upper bounds generally in the 0 to 1 percent range. An efficient branch-and-bound code also is developed. This code is tested on problems with up to 100 potential store locations and 20 regions. It is found to be at least two orders of magnitude faster than a state-of-the-art commercial integer programming package.     

A Lagrangian relaxation approach for the multiple sequence alignment problem

We present a branch-and-bound (bb) algorithm for the multiple sequence alignment problem (MSA), one of the most important problems in computational biology. The upper bound at each bb node is based on a Lagrangian relaxation of an integer linear programming formulation for MSA. Dualizing certain inequalities, the Lagrangian subproblem becomes a pairwise alignment problem, which can be solved efficiently by a dynamic programming approach. Due to a reformulation w.r.t. additionally introduced variables prior to relaxation we improve the convergence rate dramatically while at the same time being able to solve the Lagrangian problem efficiently. Our experiments show that our implementation, although preliminary, outperforms all exact algorithms for the multiple sequence alignment problem. Furthermore, the quality of the alignments is among the best computed so far.     

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.     

TECHNOLOGICAL PROGRESS AND TECHNOLOGY ACQUISITION: STRATEGIC DECISION UNDER UNCERTAINTY

We develop a stochastic programming model to aid manufacturing firms in making strategic decisions in technology acquisition. The proposed model maximizes the firm's expected profit under the condition of the uncertainty in technological progress and development. To solve this large‐scale problem, we decompose future uncertainties through scenarios and then develop an algorithm to solve the resulting non‐linear subproblems efficiently. Finally, we develop a heuristic to eliminate the infeasibility in the master problem and obtain best solutions. Numerical results show that our heuristic solutions are very close to the optimal solutions and meaningful insights are derived.     

Multimodal hub location and hub network design

Through observations from real life hub networks, we introduce the multimodal hub location and hub network design problem. We approach the hub location problem from a network design perspective. In addition to the location and allocation decisions, we also study the decision on how the hub networks with different possible transportation modes must be designed. In this multimodal hub location and hub network design problem, we jointly consider transportation costs and travel times, which are studied separately in most hub location problems presented in the literature. We allow different transportation modes between hubs and different types of service time promises between origin–destination pairs while designing the hub network in the multimodal problem. We first propose a linear mixed integer programming model for this problem and then derive variants of the problem that might arise in certain applications. The models are enhanced via a set of effective valid inequalities and an efficient heuristic is developed. Computational analyses are presented on the various instances from the Turkish network and CAB data set.     

Strengthening a linear reformulation of the 0-1 cubic knapsack problem via variable reordering

The 0-1 cubic knapsack problem (CKP), a generalization of the classical 0-1 quadratic knapsack problem, is an extremely challenging NP-hard combinatorial optimization problem. An effective exact solution strategy for the CKP is to reformulate the nonlinear problem into an equivalent linear form that can then be solved using a standard mixed-integer programming solver. We consider a classical linearization method and propose a variant of a more recent technique for linearizing 0-1 cubic programs applied to the CKP. Using a variable reordering strategy, we show how to improve the strength of the linear programming relaxation of our proposed reformulation, which ultimately leads to reduced overall solution times. In addition, we develop a simple heuristic method for obtaining good-quality CKP solutions that can be used to provide a warm start to the solver. Computational tests demonstrate the effectiveness of both our variable reordering strategy and heuristic method.

     

Solving constrained optimization problems by solution-based decomposition search

Solving constrained optimization problems (COPs) is a challenging task. In this paper we present a new strategy for solving COPs called solve and decompose (or \( S \& D\) for short). The proposed strategy is a systematic iterative depth-first strategy that is based on problem decomposition. \( S \& D\) uses a feasible solution of the COP, found by any exact method, to further decompose the original problem into a bounded number of subproblems which are considerably smaller in size. It also uses the value of the feasible solution as a bound that we add to the created subproblems in order to strengthen the cost-based filtering. Furthermore, the feasible solution is exploited in order to create subproblems that have more promise in finding better solutions which are explored in a depth-first manner. The whole process is repeated until we reach a specified depth where we do not decompose the subproblems anymore but we solve them to optimality using any exact method like Branch and Bound. Our initial results on two benchmark problems show that \( S \& D\) may reach improvements of up to three orders of magnitude in terms of runtime when compared to Branch and Bound.     

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.     

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。     

LOT-SIZING IN CAPACITATED MULTI-STAGE SERIAL SYSTEMS

The lot-sizing problem in capacitated multi-stage systems with a serial product structure is addressed. This is a complex optimization problem that is part of the decision set in material requirements planning (MRP) systems. The mathematical model that describes the problem uses the concept of echelon stock and includes lead times. Setup times are taken into account, which implies that the problem of finding a feasible solution is NP-Complete. This paper proposes a heuristic method that provides a production plan in order to minimize inventory, production, and setup costs. The heuristic starts from a solution for the uncapacitated problem, which is given by the sequential application of the Wagner-Whitin algorithm. Feasibility is then attempted by shifting production amounts between periods. Computational tests conducted in 1,800 instances with up to 40 components and 18 periods have shown that feasible solutions were obtained in 83.7% of the instances. For the infeasible instances, on average, the heuristic is able to find solutions with very low capacity excess. The solutions' quality is evaluated through a lower bound provided by Lagrangean relaxation and on average the gap is less than 10%.     

中小呼叫中心月度排班优化模型与算法

国内中小呼叫中心制定坐席人员月度排班表时,通常考虑劳动法规合同约束和体现企业自身用工管理诉求。构建坐席人员月度排班优化问题的二次整数规划模型。鉴于问题模型难解性,依据调研企业需求和模型逻辑结构分析,把问题分解成三个子问题。通过构建整数规划模型和提出启发式算法来求出子问题解,从而生成排班问题优化解。问题实例计算表明,模型算法能够有效控制人力成本和兼顾员工同班次管理目标。与周排班方法比较,该方法能够充分体现月度排班人力灵活性来实现人力优化配置。     

加工能力受限的单机干扰管理研究

针对单机环境下最小化加权折扣加工时间和的排序问题,研究如何应对可预见的干扰事件。由于干扰事件使得机器加工能力受限,初始最优加工时间表不再可行,采用外包的方式来进行干扰管理。构建了排序模型,同时考虑原目标和与初始计划偏离的扰动目标,选择外包工件集并对所有工件进行重排序。为了求解得到的双目标排序问题,基于理想点法设计了一种动态规划算法和量子遗传算法相结合的算法。最后通过一个数值算例说明,该排序模型对于求解加工能力受限的单机干扰管理问题是有效的。     

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。     

Scheduling Truck Arrivals at an Air Cargo Terminal

We consider the scheduling of truck arrivals at an air cargo terminal. By coordinating arrivals of cargo delivery trucks with outbound flight departure schedules, some of the shipments can be transferred directly to the departing flights, while others will be stored at the terminal's storage facility and incur extra handling and storage costs. The objective is to obtain a feasible schedule so as to minimize the total cost of operations. We formulate the problem as a time‐indexed integer program and show that, even with limited number of unloading docks at the terminal, the problem is non‐trivial (NP‐hard in the strong sense). Our solution method includes an exact solution procedure to determine an optimal unloading sequence for the shipments carried by each truck, together with a Lagrangian relaxation‐based heuristic for assigning trucks to truck docks and determining truck arrival times. We conducted computational experiments to test the performance of our solution method. Computational results show that our method can generate near‐optimal solutions efficiently. Our simulation results indicate that the scheduling approach proposed in this paper has the potential to generate significant cost savings over a first‐come, first‐served approach currently used at the air cargo terminal that we observed.     

无人仓系统储位分配问题的优化模型与算法

考虑到无人仓系统补货阶段货架上只有部分空余储位的特点，研究了补货商品储位分配问题的优化模型与算法。以同一货架上存放的商品之间关联度之和最大化为目标建立了混合整数规划模型；结合贪婪算法和邻域搜索算法设计了求解模型的两阶段方法。第一阶段利用贪婪算法求初始可行解；第二阶段利用邻域搜索算法对初始可行解进行优化。利用一个具体算例验证了邻域搜索算法的优化效果，结果显示，通过邻域搜索算法对初始可行解的优化，可以使目标函数值至少提升27%。进一步利用多个小规模算例分析了两阶段算法的近似比和求解速度，验证了算法的快速有效性。本文的研究结果不仅解决了货架初始状态非空情况下的储位分配问题，同样适合解决货架初始状态为空的情况，因此更加符合实际场景，可以作为无人仓管理信息系统的核心模型和算法。     

On approximation of the fixed charge transportation problem

In this paper we present a new approximation for computing lower bound for the fixed charge transportation problem (FCTP). The lower bounds thus generated delivered 87% optimal solutions for 56 randomly generated small, up to 6×10 in size, problems in an experimental design. For somewhat larger, 10×10 and 10×15 size problems, the lower bounds delivered an average error of 5%, approximately, using a fraction of CPU times as compared to CPLEX to solve these problems. The proposed lower bound may be used as a superior initial solution with any other existing branch-and-bound method or tabu search heuristic procedure to enhance convergence to the optimal solution for large size problems which cannot be solved by CPLEX due to time constraints.