Optimal partial mass transportation and obstacle Monge–Kantorovich equation

Optimal partial mass transport, which is a variant of the optimal transport problem, consists in transporting effectively a prescribed amount of mass from a source to a target. The problem was first studied by Caffarelli and McCann (2010) [6] and Figalli (2010) [12] with a particular attention to the quadratic cost. Our aim here is to study the optimal partial mass transport problem with Finsler distance costs including the Monge cost given by the Euclidian distance. Our approach is different and our results do not follow from previous works. Among our results, we introduce a PDE of Monge–Kantorovich type with a double obstacle to characterize active submeasures, Kantorovich potential and optimal flow for the optimal partial transport problem. This new PDE enables us to study the uniqueness and monotonicity results for the active submeasures. Another interesting issue of our approach is its convenience for numerical analysis and computations that we develop in a separate paper [14] (Igbida and Nguyen, 2018).     

A coordinated location-inventory model

We consider a coordinated location-inventory model where distribution centers (DCs) follow a periodic-review (R, S) inventory policy and system coordination is achieved by choosing review intervals at the DCs from a menu of permissible choices. We introduce two types of coordination: partial coordination where each DC may choose its own review interval from the menu, and full coordination where all the DCs have an identical review interval. While full coordination increases the location and inventory costs, it likely reduces the overall costs of running the system (when the operational costs such as delivery scheduling are taken into account). The problem is to determine the location of the DCs to be opened, the assignment of retailers to DCs, and the inventory policy parameters at the DCs such that the total system-wide cost is minimized. The model is formulated as a nonlinear integer-programming problem and a Lagrangian relaxation algorithm is proposed to solve it. Computational results show that the proposed algorithm is very efficient. The results of our computational experiments and case study suggest that the location and inventory cost increase due to full coordination, when compared to partial coordination, is not significant. Thus, full coordination, while enhancing the practicality of the model, is economically justifiable.     

Supply chain design considering economies of scale and transport frequencies

In this paper we consider a 3-echelon, multi-product supply chain design model with economies of scale in transport and warehousing that explicitly takes transport frequencies into consideration. Our model simultaneously optimizes locations and sizes of tank farms, material flows, and transport frequencies within the network. We consider all relevant costs: product cost, transport cost, tank rental cost, tank throughput cost, and inventory cost. The problem is based on a real-life example from a chemical company. We show that considering economies of scale and transport frequencies in the design stage is crucial and failing to do so can lead to substantially higher costs than optimal. We solve a wide variety of problems with branch-and-bound and with the efficient solution heuristics based on iterative linearization techniques we develop. We show that the heuristics are superior to the standard branch-and-bound technique for large problems like the one of the chemical company that motivated our research.     

部分联合运输策略下的物流车辆路径优化问题研究

论文分析了物流车辆路径优化问题的特点,提出了企业自营物流和第三方物流协同运输的部分联合运输策略。根据客户需求节点的特点进行了节点分类,建立了以车辆调用成本、车辆运输成本、第三方物流运输成本之和最小为目标的整数线性规划模型。根据部分联合运输策略下各类客户需求点运输方式特点,构造了一种新的变维数矩阵编码结构,并对传统算法中概率选择操作方式进行修改,提出了一种新的智能优化算法并与枚举法和遗传算法的运算结果进行了算法性能对比分析。结果显示,本文提出的逆选择操作蚁群算法具有较快的运算速度和较高的稳定性,是求解此类问题的一种有效算法。     

带有限反馈下层的二层规划问题的部分合作模型

对下层最优反馈为离散有限多个的二层规划问题的部分合作模型进行探讨. 当下层的合作程度依赖于上层的决策变量时, 给出一个确定合作系数函数的一般方法, 进而得到一个新的部分合作模型. 在适当地假设下, 可保证所给的部分合作模型一定可以找到比悲观解要好的解, 并结合新的部分合作模型对原不适定问题进行分析, 得到了一些有益的结论. 最后以实际算例说明了所给部分合作模型的可行性.     

Integrating link-based discrete credit charging scheme into discrete network design problem

In this paper, we present an optimization model for integrating link-based discrete credit charging scheme into the discrete network design problem, to improve the transport performance from the perspectives of both transport network planning and travel demand management. The proposed model is a mixed-integer nonlinear bilevel programming problem, which includes an upper level problem for the transport authority and a lower level problem for the network users. The lower level sub-model is the traffic network user equilibrium (UE) formulation for a given network design strategy determined by the upper level problem. The network user at the lower level tries to minimize his/her own generalized travel cost (including both the travel time and the value of the credit charged for using the link) by choosing his/her route. While the transport authority at the upper level tries to find the optimal number of lanes and credit charging level with their locations to minimize the total system travel time (or maximize the transportation system performance). A genetic algorithm is used to solve the proposed mixed-integer nonlinear bilevel programming problem. Numerical experiments show the efficiency of the proposed model for traffic congestion mitigation, reveal that interaction effects across the tradable credit scheme and the discrete network design problem which amplify their individual effects. Moreover, the integrated model can achieve better performance than the sequential decision problems.     

Relaxations for two-level multi-item lot-sizing problems

We consider several variants of the two-level lot-sizing problem with one item at the upper level facing dependent demand, and multiple items or clients at the lower level, facing independent demands. We first show that under a natural cost assumption, it is sufficient to optimize over a stock-dominant relaxation. We further study the polyhedral structure of a strong relaxation of this problem involving only initial inventory variables and setup variables. We consider several variants: uncapacitated at both levels with or without start-up costs, uncapacitated at the upper level and constant capacity at the lower level, constant capacity at both levels. We finally demonstrate how the strong formulations described improve our ability to solve instances with up to several dozens of periods and a few hundred products.     

Selective and periodic inventory routing problem for waste vegetable oil collection

We consider a biodiesel production company that collects waste vegetable oil from source points that generate waste in large amounts. The company uses the collected waste as raw material for biodiesel production. The manager of this company needs to decide which of the present source points to include in the collection program, which of them to visit on each day, which periodic routing schedule to repeat over an infinite horizon and how many vehicles to operate such that the total collection, inventory and purchasing costs are minimized while the production requirements and operational constraints are met. For this selective and periodic inventory routing problem, we propose two different formulations, compare them and apply the better performing one on a real-world problem with 36 scenarios. We generate lower bounds using a partial linear relaxation model, and observe that the solutions obtained through our model are within 3.28% of optimality on the average. Several insights regarding the customer selection, routing and purchasing decisions are acquired with sensitivity analysis.     

The capacitated p-hub median problem with integral constraints: An application to a Chinese air cargo network

Consolidation at hubs in a pure hub-and-spoke network eliminates partial center-to-center direct loads, resulting in savings in transportation costs. In this research, we propose a general capacitated p-hub median model, with economies of scale and integral constraints on the paths. This model requires the selection of a specific p among a set of candidate hubs so that the total cost on the resulting pure capacitated hub-and-spoke network is minimized while simultaneously meeting origin–destination demands, operational capacity and singular path constraints. We explored the problem structure and developed a genetic algorithm using the path for encoding. This algorithm is capable of determining local optimality within less than 0.1% of the Lagrangian relaxation lower bounds on our Chinese air cargo network testing case and has reasonable computational times. The study showed that designating airports with high pickups or deliveries as hubs resulted in a high percentage of origin–destination pairs (ODs) in direct deliveries. Furthermore, the more hubs there are, the higher the direct share and the less likely for double rehandles. Sensitivity analysis on the discount rate showed that the economies of scale on trunk lines of hub-and-spoke networks may have a substantial impact on both the operating costs and the route patterns.     

Inventory control with an order-time constraint: optimality,uniqueness and significance

This paper analyzes a stochastic inventory problem with an order-time constraint that restricts the times at which a manufacturer places new orders to a supplier. This constraint stems from the limited upstream capacity in a supply chain, such as production capacity at a supplier or transportation capacity between a supplier and a manufacturer. Consideration of limited upstream capacity extends the classical inventory literature that unrealistically assumes infinite supplier/transporter capacity. But this consideration increases the complexity of the problem. We study the constraint under a Poisson demand process and allow for a fixed ordering cost. In presence of the constraint, we establish the optimality of an (s,S) policy under both the discounted and average cost objectives. Under the average cost objective, we show the uniqueness of the order-up-to level S. We numerically compare our model with the classical unconstrained model. We report significant savings in costs that can be achieved by using our model when the order time is constrained.     

Heuristics for Distribution Network Design in Telecommunication

A distribution network problem arises in a lower level of an hierarchical modeling approach for telecommunication network planning. This paper describes a model and proposes a lagrangian heuristic for designing a distribution network. Our model is a complex extension of a capacitated single commodity network design problem. We are given a network containing a set of sources with maximum available supply, a set of sinks with required demands, and a set of transshipment points. We need to install adequate capacities on the arcs to route the required flow to each sink, that may be an intermediate or a terminal node of an arborescence. Capacity can only be installed in discrete levels, i.e., cables are available only in certain standard capacities. Economies of scale induce the use of a unique higher capacity cable instead of an equivalent set of lower capacity cables to cover the flow requirements of any link. A path from a source to a terminal node requires a lower flow in the measure that we are closer to the terminal node, since many nodes in the path may be intermediate sinks. On the other hand, the reduction of cable capacity levels across any path is inhibited by splicing costs. The objective is to minimize the total cost of the network, given by the sum of the arc capacity (cables) costs plus the splicing costs along the nodes. In addition to the limited supply and the node demand requirements, the model incorporates constraints on the number of cables installed on each edge and the maximum number of splices at each node. The model is a NP-hard combinatorial optimization problem because it is an extension of the Steiner problem in graphs. Moreover, the discrete levels of cable capacity and the need to consider splicing costs increase the complexity of the problem. We include some computational results of the lagrangian heuristics that works well in the practice of computer aided distribution network design.     

Solving the capacitated lot-sizing problem with backorder consideration

In this research, we formulate and solve a type of the capacitated lot-sizing problem. We present a general model for the lot-sizing problem with backorder options, that can take into consideration various types of production capacities such as regular time, overtime and subcontracting. The objective is to determine lot sizes that will minimize the sum of setup costs, holding cost, backorder cost, regular time production costs, and overtime production costs, subject to resource constraints. Most existing formulations for the problem consider the special case of the problem where a single source of production capacity is considered. However, allowing for the use of alternate capacities such as overtime is quite common in many manufacturing settings. Hence, we provide a formulation that includes consideration of multiple sources of production capacity. We develop a heuristic based on the special structure of fixed charge transportation problem. The performance of our algorithm is evaluated by comparing the heuristic solution value to lower bound value. Extensive computational results are presented.     

一种部分约束满足车辆路线问题及其求解算法

描述了一类过度约束车辆路线问题，其中可用车辆数较少而时间窗口等其它约束又不允许放松，因而导致不存在满足所有约束的可行解。此时问题求解可以转化为一类部分约束满足问题来处理，相应的优化目标是最小化未访问顾客的损失和。本给出了求解这类特殊问题的一种禁忌搜索算法设计，并通过规模不同的几个算例与其它常用方法进行了比较。最后分析了模型和算法的实用意义。     

Treatment Planning Optimization for Radiotherapy

We study the problem of treatment planning in radiotherapy. We use an energy-dependent Boltzmann-based particle transport model as a partial integro-differential constraint for a convex optimization problem. We solve the resulting optimal control problem using a minimum entropy moment closure. Numerical results for problems on patient data are presented. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Optimal information acquisition for a linear quadratic control problem

This paper develops a generalization of the linear quadratic control problem with partial information. As in the standard partial information setting, it is assumed that the state variable is only observed with noise. The idea in this paper is that the information level may be chosen optimally. In real life information is costly to acquire. It is therefore a trade off between the costs of getting detailed information and the increased value this information gives. We believe that the technique we present should have potential for application within both economics and engineering.     

Local search heuristics for multi-index assignment problems with decomposable costs

The multi-index assignment problem (MIAP) with decomposable costs is a natural generalization of the well-known assignment problem. Applications of the MIAP arise, for instance, in the field of multi-target multi-sensor tracking. We describe an (exponentially sized) neighbourhood for a solution of the MIAP with decomposable costs, and show that one can find a best solution in this neighbourhood in polynomial time. Based on this neighbourhood, we propose a local search algorithm. We empirically test the performance of published constructive heuristics and the local search algorithm on random instances; a straightforward iterated local search algorithm is also tested. Finally, we compute lower bounds to our problem, which enable us to assess the quality of the solutions found.     

The two-echelon production-routing problem

This paper introduces the Two-Echelon Production-Routing Problem. This problem is motivated from the petrochemical industry, enlarging the supply chain integration by taking into account production, inventory, and routing decisions in a two-echelon vendor-managed inventory system. We describe, model, and design a branch-and-cut (B&C) to solve the problem under different inventory policies. We also propose a novel exact algorithm, by employing parallel computing techniques, in order to combine local search procedures within a traditional B&C scheme. We evaluate the performance of our methods through extensive computational experiments, both by comparing the algorithms, the effectiveness of the different inventory policies, and the impact of these policies on the partial costs. We derive many managerial insights based on the results. We also validate our new exact algorithm by solving similar problems from the literature, such as the two-echelon multi-depot inventory-routing (2E-MDIRP) and the classical multi-vehicle production-routing problem (MV-PRP). Computational experiments show that our method is very competitive. Based on 512 experiments for the 2E-MDIRP, our algorithm was able to find 111 new best known solutions (BKS), besides proving 412 optimal solutions, against 298 from the literature. For 336 experiments over small and medium size MV-PRP instances, we proved 242 optimal solutions, 11 more than the exact methods from the literature, besides providing 95 new BKS. Moreover, we were the first to tackle large MV-PRP instances exactly, and in this case, our algorithm provides all BKS for instances up to 50 customers, 20 periods and 5 vehicles, outperforming all meta/matheuristics procedures from the literature.     

非同质演化博弈及其在移动污染源监管中的应用

移动污染源是中国环境治理的难点之一。本文分析了监管者与机动车移动污染源之间的博弈关系。通过考察边际成本与边际收益的方式建立了非同质演化博弈模型,讨论了监管成本变化以及机动车异质性治污成本对均衡点位置的影响。计算结果表明,受不同成本特性影响,机动车移动污染源监管博弈会呈现三种不同均衡状态:有效监管状态、无效监管状态和形式监管。在有效监管状态下,较低的处罚水平导致治污水平低下;监管力量薄弱地区会出现无效监管状态;在形式监管中,不规范的责令整改方式会导致整改水平低下的形式主义问题。主要结论为:应提高机动车超标排污处罚水平以提升机动车治污概率;应建立机动车移动污染长效协作监管机制,采用新技术手段,以降低监管成本,加强监管效果;需规范责令整改的程序与流程,防范形式主义。     

The unit commitment model with concave emissions costs: a hybrid Benders’ Decomposition with nonconvex master problems

We present a unit commitment model which determines generator schedules, associated production and storage quantities, and spinning reserve requirements. Our model minimizes fixed costs, fuel costs, shortage costs, and emissions costs. A constraint set balances the load, imposes requirements on the way in which generators and storage devices operate, and tracks reserve requirements. We capture cost functions with piecewise-linear and (concave) nonlinear constructs. We strengthen the formulation via cut addition. We then describe an underestimation approach to obtain an initial feasible solution to our model. Finally, we constitute a Benders’ master problem from the scheduling variables and a subset of those variables associated with the nonlinear constructs; the subproblem contains the storage and reserve requirement quantities, and power from generators with convex (linear) emissions curves. We demonstrate that our strengthening techniques and Benders’ Decomposition approach solve our mixed integer, nonlinear version of the unit commitment model more quickly than standard global optimization algorithms. We present numerical results based on a subset of the Colorado power system that provide insights regarding storage, renewable generators, and emissions.     

Analysis of an industrial component commonality problem

We discuss a case study of an industrial production-marketing coordination problem involving component commonality. For the product line considered, the strategic goal of the company is to move from the current low volume market to a high volume market. The marketing department believes that this can be achieved by substantially lowering the end products’ prices. However, this requires a product redesign to lower production costs in order to maintain profit margins. The redesign decision involves grouping end products into families. All products within one family use the same version of some components. This paper fits in the stream of recent literature on component commonality where the focus has shifted from inventory cost savings to production and development cost savings. Further, we consider both costs and revenues, leading to a profit maximization approach. The price elasticity of demand determines the relationship between the price level and number of units sold. Consequently, we integrate information from different functional areas such as production, marketing and accounting. We formulate the problem as a net-present-value investment decision. We propose a mixed integer nonlinear optimization model to find the optimal commonality decision. The recommendation based on our analysis has been implemented in the company. In addition, the application allows us to experimentally validate some claims made in the literature and obtain managerial insights into the trade-offs.