Mathematical Programming Formulation Research Articles

Mathematical programming formulations for the optimal stratification problem

This paper proposes two integer linear programming formulations to solve the optimal stratification problem (OSP). In this problem, once given the number of strata and the total sample size, the cutoff points and sample allocation should be determined jointly to minimize an expression of variance and to solve the OSP corresponding to an integer nonlinear optimization problem. One proposed formulation determines the optimal cutoff points considering classical allocation methods from the literature, and a second formulation produces the global optimum regarding the determination of the cutoff points and the sample allocation to the strata. Both formulations were implemented using Gurobi solver and applied to a set of 20 datasets from the literature, with population sizes ranging from 91 to 16,057, considering six different scenarios concerning the number of strata and sample sizes. The results obtained indicate that the formulations are a good alternative for the resolution of the OSP.

Mathematical programming formulations and metaheuristics for biological sample transportation problems in healthcare

In this paper, a problem arising from a real-world healthcare application is addressed, dealing with the transportation of biological samples from blood draw centers to a main laboratory. The problem can be modeled as a Vehicle Routing Problem with time-windows and samples lifetime constraints, in which transfers and multiple visits at specific nodes (i.e., the spokes) are allowed to get extra lifetimes of the samples. Mixed Integer Linear Programming formulations and Hybrid Adaptive Large Neighborhood Search (H-ALNS) algorithms have been developed for the problem. Computational experiments on different sets of instances based on real-life data provided by the Local Healthcare Authority of Bologna, Italy, show the effectiveness of the proposed algorithms. More precisely, the H-ALNS algorithms are able to find solutions in which all the samples are delivered on time, unlike the real case where lifetime requirements are usually not fully respected.

The on-demand warehousing problem

Warehouses are key elements of supply chain networks, and great attention is paid to increase their efficiency. Highly volatile space requirements are enablers of innovative resource sharing concepts, where warehouse capacities are traded on online platforms. In this context, our paper introduces the on-demand warehousing problem from the perspective of platform providers. The objective prioritises demand–supply matching with maximisation of the number of transactions. If there is a tie, the secondary objective maximises the number of suppliers matched with at least one customer and the number of customers that have matches within a specific threshold with respect to the minimum achievable cost. Besides the mathematical integer programming formulation, a myopic list-based heuristic and an efficient matheuristic approach are presented and benchmarked against the performance of a commercial optimisation solver. The impact of several parameters on the platform's objective is analysed. A particularly relevant finding is that the pricing flexibility on the demand side does not necessarily imply higher payments to the supply side. All data instances are made available publicly to encourage more researchers to work on this timely and challenging topic.

Min-Time Coverage in Constricted Environments: Problem Formulations and Complexity Analysis

This work deals with the task allocation and robot routing problems that arise in a class of networked mobile robot systems. The defining characteristics of these systems are: 1) the constricting nature of their operational environment and 2) the need to coordinate the robot motion in order to maintain full connectivity of a multihop wireless communication network connecting the robots with each other and a master controller that supervises the entire operation. These two elements give rise to resource allocation structures and traffic dynamics that transcend the state of the art of the corresponding theory and challenge our current understandings and insights for these dynamics and their effective management. We provide: 1) a systematic introduction of the considered problems and of the elements that differentiate them from similar task allocation and traffic scheduling problems already studied in the literature; 2) complete analytical characterizations of these problems in the form of mathematical programming formulations; and 3) a formal analysis of the worst-case computational complexity of these problems and of certain factors that determine this complexity. Furthermore, the presented results define a base for the development of solution approaches to the considered problems able to effectively and systematically manage the identified tradeoff between the operational efficiency of the derived solutions and their computational cost.

Rocking Response of the Rigid Block Under Rectangular Pulse Excitation: A Comparison between ODE and Optimization-Based Solvers

In this paper, the response of the rigid block under rectangular pulse excitation is investigated using two different modeling approaches and solvers. The first approach relies on the numerical integration of the differential equation of motion. The second approach is based on the formulation of the dynamic problem in terms of a special class of mathematical programming problem that is the linear complementarity problem. A validation study is carried out comparing the solutions given by the proposed formulation with the ones given by the numerical integration of the differential equation of motion obtained from ODE solvers available in MATLAB®. Potentialities and limitations of the mathematical programming formulation are discussed in terms of energy dissipation and restitution coefficient at impacts and in terms of solution times.

Mathematical programming formulations for the alternating current optimal power flow problem

Power flow refers to the injection of power on the lines of an electrical grid, so that all the injections at the nodes form a consistent flow within the network. Optimality, in this setting, is usually intended as the minimization of the cost of generating power. Current can either be direct or alternating: while the former yields approximate linear programming formulations, the latter yields formulations of a much more interesting sort: namely, nonconvex nonlinear programs in complex numbers. In this technical survey, we derive formulation variants and relaxations of the alternating current optimal power flow problem.

Optimizing fuel consumption on inland waterway networks: Local search heuristic for lock scheduling

Fuel consumption and CO2 emission are among the main criteria to assess the environmental and economical impact of vessels on inland waterways. Both criteria, however, are directly affected by the vessels’ sailing speed. In this paper, we present a mathematical programming formulation of the speed optimization problem, which aims at minimizing the aggregated fuel consumption on an inland waterway network. The network can consist of multiple river segments, connected by a set of locks, without restrictions on the configuration. To allow scalability towards realistic waterway networks, we also propose a local-search based heuristic to optimize the speed for individual vessels. We evaluate the effectiveness of the heuristic by comparing it to solving the exact mathematical programming formulation. For all computational experiments, we make use of real AIS data from a section of the Dutch river network. We observe that the heuristic is able to construct a high quality solution in realistic problem settings within reasonable amount of computation time.

Some structural results for the problem of Min-Time Coverage in Constricted Environments

In a recent work, we introduced a new set of problems in the area of networked robotic systems that concern the time-optimal execution of certain coverage tasks taking place in constricted environments. That work provided the detailed problem definitions, a complete representation of these problems in terms of Mathematical Programming (MP) formulations, and a formal analysis of their worst-case computational complexity. The current work establishes some structural results for the considered problems that are useful for the strengthening of the aforementioned MIP formulations and for the further development of pertinent heuristic solution methods for these problems. We demonstrate the first possibility in this paper, and we defer the second one to future work.

Batch Order and Discrete Order Picking Integrated with Vehicle Routing Decisions

In a business-to-consumer (B2C) context, customers order more frequently and in smaller quantities, resulting in a high number of consignments. Moreover, online shoppers expect a fast and accurate delivery at low cost or even free. To survive in such a market, companies can no longer optimise individual supply chain processes, but need to integrate several activities. In this article, the integrated order picking-vehicle routing problem is analysed in an e-commerce environment. In previous research, a mathematical programming formulation has been formulated in literature but only small-size instances can be solved to optimality. Two picking policies are studied: discrete order picking and batch order picking. The influence of various problem contexts on the value of integration is investigated: a small picking time period, outsourcing to 3PL service providers, and a dynamic environment context.

Replication and sequencing of unreliable jobs on parallel machines

We consider the problem of sequencing m copies of unreliable jobs (i.e., jobs that have a certain probability of being successfully carried out) on m parallel machines (one copy per machine). A job is carried out if at least one of its copies is successfully completed. If the copy of a job fails, the corresponding machine is blocked and cannot perform the subsequently scheduled job copies. We analyze two problems. In the first problem, each job has a certain revenue which is attained if the job is carried out, and the objective is to maximize the expected revenue. We show that for m=2 this problem is NP-hard. We propose a mathematical programming formulation and a metaheuristic approach, and we assess their computational behavior on a large set of instances. We also solve the problem for m machines and two jobs, showing that the marginal benefit of having an additional machine decreases with m. The second problem is to maximize the probability that all jobs are carried out. We show that the problem can be easily solved when m=2, as well as for m machines and two jobs.

Pricing Mechanism for Centralised Reused Water System with Multiple Water Headers

Reasonable water pricing is essential for long-term sustainability and financing. The selling prices of reused/regenerated water should enable the generation of revenue that could cover the production costs. In this work, reused/regenerated water prices at different qualities are determined based on the total annualised cost of the centralised system with and without subsidy, quality factor, and profit factor. This paper presents a mathematical programming formulation that aims to maximise the centralised system’s profit in the industrial site considering multiple qualities of reused/regenerated water. A case study of numerous scenarios, each with various numbers of water headers and different water prices, is used to test the model. The results obtained show that as the total annualised cost increases, the selling prices of reused/regenerated water increase, the centralised system’s profit also increases, and there is a possibility to surpass the freshwater price. For scenarios with subsidy and low total annualised cost, there is an opportunity to increase the centralised system’s profit by increasing the profit factor in determining the reused/regenerated water price without exceeding the freshwater price. Based on the case study, Scenario 7 with the highest number of water headers is chosen as it has the highest freshwater reduction, which is 70 %, low total annualised cost, a comparable profit with a profit margin of more than 15 % and the highest profit factor can be applied compared to the other scenarios.

Systematic Approaches for Optimal Scheduling of Straight Pipelines with Batch-Centric Formulations

Scheduling of the transportation of liquids by multiproduct pipelines has been receiving considerable attention in the literature. The preferred methods for solving this problem rely on mathematical programming and use continuous-time formulations that move the virtual entities of batches instead of the real entities of products. Such batch-centric formulations require the user to convert the products initially in the pipeline into batches, which is trivial to do for linear or branched systems with a single input node but not for straight systems with multiple input nodes. While the dependence of solution quality on the initial batch sequence has been recognized early, no systematic procedure covering all possibilities has been presented yet. In this paper, we propose a novel scheduling algorithm that starts generating all possible initial batch sequences with a new planning formulation, before exploring all generated sequences with either a sequential or simultaneous approach, using the scheduling formulation of Liao, Q.; [ AIChE J. 2019, 65, e16712]. We then generalize this formulation to consider automatic batch selection, by including the logical constraints of the planning model.

Robust minimum fleet problem for autonomous and human-driven vehicles in on-demand ride services considering mixed operation zones

In this paper, we envision an emerging Mixed Operation Zone (MOZ) where both autonomous vehicles (AVs) and human-driven vehicles (HVs) are present for on-demand ride services. This paper aims to size and operate a fleet of AVs and HVs in the presence of MOZs and to investigate the impact of MOZs on on-demand ride services. Considering the demand uncertainty, we propose a demand-oriented robust minimum fleet problem (RMFP) and employ a two-stage robust optimization (RO) to model the decision-making. Fluctuant demand is bounded by distribution-free uncertainty sets. For the convenience of solving RO models, we reformulate the second-stage recourse problem with an equivalent mathematical programming formulation. A tailored column-and-constraint generation algorithm is developed to solve the RMFP exactly. The algorithm is proved to converge in a finite number of iterations. Extensive experiments are conducted on the instances based on a real-world on-demand ride service in Chengdu. The developed algorithm performs better than the state-of-the-art Benders decomposition approach. Numerical results imply huge potential benefits from MOZs on improving service performance for ride service platforms.

Network slicing for vehicular communications: a multi-agent deep reinforcement learning approach

This paper studies the multi-agent resource allocation problem in vehicular networks using non-orthogonal multiple access (NOMA) and network slicing. Vehicles want to broadcast multiple packets with heterogeneous quality-of-service (QoS) requirements, such as safety-related packets (e.g., accident reports) that require very low latency communication, while raw sensor data sharing (e.g., high-definition map sharing) requires high-speed communication. To ensure heterogeneous service requirements for different packets, we propose a network slicing architecture. We focus on a non-cellular network scenario where vehicles communicate by the broadcast approach via the direct device-to-device interface (i.e., sidelink communication). In such a vehicular network, resource allocation among vehicles is very difficult, mainly due to (i) the rapid variation of wireless channels among highly mobile vehicles and (ii) the lack of a central coordination point. Thus, the possibility of acquiring instantaneous channel state information to perform centralized resource allocation is precluded. The resource allocation problem considered is therefore very complex. It includes not only the usual spectrum and power allocation, but also coverage selection (which target vehicles to broadcast to) and packet selection (which network slice to use). This problem must be solved jointly since selected packets can be overlaid using NOMA and therefore spectrum and power must be carefully allocated for better vehicle coverage. To do so, we first provide a mathematical programming formulation and a thorough NP-hardness analysis of the problem. Then, we model it as a multi-agent Markov decision process. Finally, to solve it efficiently, we use a deep reinforcement learning (DRL) approach and specifically propose a deep Q learning (DQL) algorithm. The proposed DQL algorithm is practical because it can be implemented in an online and distributed manner. It is based on a cooperative learning strategy in which all agents perceive a common reward and thus learn cooperatively and distributively to improve the resource allocation solution through offline training. We show that our approach is robust and efficient when faced with different variations of the network parameters and compared to centralized benchmarks.

A P-Hub Location Problem for Determining Park-and-Ride Facility Locations with the Weibit-Based Choice Model

Park and ride (P&R) facilities provide intermodal transfer between private vehicles and public transportation systems to alleviate urban congestion. This study developed a mathematical programming formulation for determining P&R facility locations. A recently developed Weibit-based model was adopted to represent the traveler choice behavior with heterogeneity. The model’s independence of irrelevant alternatives (IIA) property was explored and used to linearize its nonlinear probability. Some numerical examples are provided to demonstrate a feature of the proposed mixed integer linear programing (MILP). The results indicate a significant impact of route-specific perception variance on the optimal P&R facility locations in a real-size transportation network.

Modeling the α-max capacity of transportation networks: a single-level mathematical programming formulation

Network capacity, defined as the largest sum of origin–destination (O–D) flows that can be accommodated by the network based on link performance function and traffic equilibrium assignment, is a critical indicator of network-wide performance assessment in transportation planning and management. The typical modeling rationale of estimating network capacity is to formulate it as a mathematical programming (MP), and there are two main approaches: single-level MP formulation and bi-level programming (BLP) formulation. Although single-level MP is readily solvable, it treats the transportation network as a physical network without considering level of service (LOS). Albeit BLP explicitly models the capacity and link LOS, solving BLP in large-scale networks is challenging due to its non-convexity. Moreover, the inconsideration of trip LOS makes the existing models difficult to differentiate network capacity under various traffic states and to capture the impact of emerging trip-oriented technologies. Therefore, this paper proposes the α-max capacity model to estimate the maximum network capacity under trip or O–D LOS requirement α. The proposed model improves the existing models on three aspects: (a) it considers trip LOS, which can flexibly estimate the network capacity ranging from zero to the physical capacity including reserve, practical and ultimate capacities; (b) trip LOS can intuitively reflect users’ maximum acceptable O–D travel time or planners’ requirement of O–D travel time; and (c) it is a convex and tractable single-level MP. For practical use, we develop a modified gradient projection solution algorithm with soft constraint technique, and provide methods to obtain discrete trip LOS and network capacity under representative traffic states. Numerical examples are presented to demonstrate the features of the proposed model as well as the solution algorithm.

Routing for unmanned aerial vehicles: Touring dimensional sets

In this paper we deal with an extension of the crossing postman problem to design routes that have to visit different shapes of dimensional elements rather than edges. This problem models the design of routes of drones or other vehicles that must visit a number of geographical elements to deliver some good or service and then move directly to the next using straight line displacements. We present two families of mathematical programming formulations. The first one is time-dependent and captures a number of characteristics of real applications at the price of using three indexes variables. The second family of formulations is not time-dependent, instead it uses connectivity properties to ensure the proper definition of routes. We compare them on a testbed of instances with different shapes of elements: second order cone (SOC) representable and polyhedral neighborhoods and polygonal chains. The computational results reported in this paper show that our models are useful and our formulations can solve to optimality medium size instances of sizes similar to other combinatorial problems including neighborhoods that have already been studied in the literature. To address larger instances we also present a heuristic algorithm that runs in two phases: clustering and Variable Neighborhood Search. This algorithm performs very well since it provides promising feasible solutions and, in addition, it can be used to initialize the solvers with feasible solutions.

Linearized formulations for failure aware barter exchange

Mathematical programming formulations are developed for determining chains of organ-donation exchange pairs in a compatibility graph where pairwise exchanges may fail. The objective is to maximize the expected value where pairs are known to fail with given probabilities. In previous work, namely that of Dickerson et al. (Manag Sci 65(4):323–340, 2019) this NP-hard problem was solved heuristically or exactly only for limited path lengths. Although the problem appears highly nonlinear, we formulate it as a mixed-integer linear program. A computationally tractable layered formulation that approximately solves larger instances is also proposed and a computational study is presented for evaluating the proposed formulations.

On hub location problems in geographically flexible networks

AbstractIn this paper, we propose an extension of the uncapacitated hub location problem where the potential positions of the hubs are not fixed in advance. Instead, they are allowed to belong to a region around an initial discrete set of nodes. We give a general framework in which the collection, transportation, and distribution costs are based on norm‐based distances and the hub‐activation setup costs depend not only on the location of the hub that are opened but also on the size of the region where they are placed. Two alternative mathematical programming formulations are proposed. The first one is a compact formulation while the second one involves a family of constraints of exponential size that we separate efficiently giving rise to a branch‐and‐cut algorithm. The results of an extensive computational experience are reported showing the advantages of each of the approaches.

Exact solution approaches for the workload smoothing in assembly lines

In this paper, the problem of minimizing the smoothness index for an assembly line given a fixed cycle time and the number of workstations is studied. This problem which is known as the workload smoothing line balancing problem (WSLBP) is a mixed-integer quadratic programming problem. Until recently, this problem has only been tackled using heuristic approaches. Recently, there have been some attempts to solve this problem exactly using mixed-integer linear programming (MILP). The MILP formulations, however, are not usually capable of solving large size problem instances. In this paper, the aim is to solve the WSLBP using mathematical programming formulations by using off-the-shelf solvers. Differently from the literature some non-MILP formulations are also considered for the problem. For this purpose, three MILP formulations, one from the literature, and two non-MILP formulations are compared. The two non-MILP formulations include a mixed-integer second order cone programming formulation and a constraint programming model. The superiority of the non-MILP formulations over the considered MILP formulations is experimentally shown.