Mixed-integer Programming Formulation Research Articles

A Spatial Optimization Model for Delineating Metropolitan Areas

A metropolitan area comprises a collection of cities and counties bound by strong socioeconomic ties. Despite the pivotal role that metropolitan areas play in regional economics, their delineation remains a challenging task for researchers and urban planners. Current threshold-based delineation methods select counties based on their connection strength with prespecified core counties. Such an approach often neglects potential interactions among outlying counties and fails to identify polycentric urban structures. The delineation of a metropolitan area is fundamentally a spatial optimization problem, whose objective is to identify a set of counties with high interconnectivity while also meeting specific constraints, such as area, contiguity, and shape. In this study, we present a novel spatial optimization model designed for metropolitan area delineation. This model aims to maximize intercounty connection strength in terms of both industry and daily life. This approach ensures a more accurate representation of the multicore structure that is commonly seen in developed metropolitan areas. Additionally, our model avoids the possibility of holes in metropolitan area delineation, leading to more coherent and logical metropolitan boundaries. We provide a mixed-integer programming formulation for the proposed model. Its efficacy is demonstrated by delineating the boundaries of the Nanjing and Lhasa metropolitan areas. This study also delves into discussions and policy implications pertinent to both of these metropolitan areas.

Mathematical programming formulations for the reclaimer scheduling problem with sequence-dependent setup times and availability constraints

In the context of Industry 4.0, which encompasses advanced technologies and interconnected systems, the integration of optimization techniques assumes a crucial role in addressing resource management challenges across various sectors, including manufacturing systems. In bulk ports, which are an integral part of the manufacturing and logistics chain, one of the most critical resource management challenges is the scheduling of automated or semi-automated reclaimers. These machines are responsible for reclaiming dry bulk materials for loading onto vessels using ship-loaders. The efficient operation of reclaimers directly impacts the overall efficiency of the port. However, the current works for the reclaimer scheduling problem (RSP) overlook the necessity of maintenance activities and assume continuous machine availability. Nonetheless, the inclusion of preventive maintenance activities is critical to ensuring optimal scheduling decisions that minimize production disruptions and downtime. To address these challenges, this paper proposes a novel approach that considers periodic preventive maintenance activities of the reclaimers. We consider two cases to tackle this problem. The first case involves the optimization of material handling within a system of two stockpads and one reclaimer machine. For this case, two novel Mixed Integer Programming (MIP) formulations are developed to drive efficient scheduling decisions and maximize productivity. Furthermore, for the second case representing a more complex system comprising three stockpads and two reclaimers, a novel MIP formulation is devised. The effectiveness of the proposed mathematical formulations is rigorously tested through the use of randomly generated instances based on a real-world case.

Minimizing File Transfer Time in Opportunistic Spectrum Access Model

We study the file transfer problem in opportunistic spectrum access (OSA) model, which has been widely studied in throughput-oriented applications for max-throughput strategies and in delay-related works that commonly assume identical channel rates and fixed file sizes. Our work explicitly considers minimizing the file transfer time for a given file in a set of heterogeneous-rate Bernoulli channels, showing that max-throughput policy doesn't minimize file transfer time in general. We formulate a mathematical framework for static extend to dynamic policies by mapping our file transfer problem to a stochastic shortest path problem. We analyze the performance of our proposed static and dynamic optimal policies over the max-throughput policy. We propose a mixed-integer programming formulation as an efficient alternative way to obtain the dynamic optimal policy and show a huge reduction in computation time. Then, we propose a heuristic policy that takes into account the performance-complexity tradeoff and consider the online implementation with unknown channel parameters. Furthermore, we present numerical simulations to support our analytical results and discuss the effect of switching delay on different policies. Finally, we extend the file transfer problem to Markovian channels and demonstrate the impact of the correlation of each channel.

Modeling and solving the fuel distribution problem with unloading precedence and loading sequence considerations

AbstractThis paper presents a real-world liquid fuel distribution problem involving a heterogeneous fleet of multi-compartment vehicles servicing a set of orders of different fuels. Two new features are introduced that affect significantly the performance of the fuel distribution process in terms of safety and efficiency: (i) loading trucks so that the payload remains balanced throughout each phase of a delivery route, and (ii) sequencing the requests for loading trucks at the depot loading facilities. A Mixed Integer Programming formulation is presented and an Adaptive Large Neighbourhood Search algorithm with various novel features is developed and benchmarked. A new loading model is formed and solved for allocating the ordered items (fuel) to vehicle compartments, as part of constructing/repairing delivery routes. The computational performance of the proposed solution approach has been tested on a series of benchmark problems. Moreover, a series of experiments were performed in order to assess the effect of the balanced loading constraints on the traveled distance. The results indicate that the effect of this type of constraint on the total traveled distance is kept at a reasonable level, reaching a maximum increase of 4.37%. The computational tools presented in this work may accommodate the dispatchers’ work in producing efficient and safe delivery routes while managing efficiently any potential bottleneck in the truck loading facilities.

Modeling and optimization of steady flow of natural gas and hydrogen mixtures in pipeline networks

We extend the canonical problems of simulation and optimization of steady-state gas flows in pipeline networks with compressors to the transport of mixtures of highly heterogeneous gases injected throughout a network. Our study is motivated by proposed projects to blend hydrogen generated using clean energy into existing natural gas pipeline systems as part of efforts to reduce the reliance of energy systems on fossil fuels. Flow in a pipe is related to endpoint pressures by a basic Weymouth equation model, with an ideal gas equation of state, where the wave speed depends on the hydrogen concentration. At vertices, in addition to mass balance, we also consider mixing of incoming flows of varying hydrogen concentrations. The problems of interest are the heterogeneous gas flow simulation (HGFS), which determines system pressures and flows given fixed boundary conditions and compressor settings, as well as the heterogeneous gas flow optimization (HGFO), which extremizes an objective by determining optimal boundary conditions and compressor settings. We examine conditions for uniqueness of solutions to the HGFS, as well as compare and contrast mixed-integer and continuous nonlinear programming formulations for the HGFO. We develop computational methods to solve both problems, and examine their performance using four test networks of increasing complexity.

Towards a circular economy with waste‐to‐resource system optimization

AbstractMotivated by mounting pressures to achieve environmental sustainability, and the emergence of online waste exchange platforms, in this work we propose an optimization‐based framework for studying waste‐to‐resource formation in the industry. We first develop a linear programming market clearing model comprising self‐interested agents participating in waste‐to‐resource trading. We then embed this in a bi‐level capacity optimization problem under uncertainties in the agents' reserve prices, the goal of which is to maximize the economic savings achieved by waste‐to‐resource. We show that the resulting robust optimization models admit linear mixed integer programming formulations that can be readily computed using available solvers. Furthermore, we provide the formulas for applying an equitable levy and incentive scheme, to enable capacity cost‐sharing among the agents. Numerical studies based on a case of organic waste streams demonstrate the usefulness of the proposed models in generating valuable insights for decision support in the design of waste‐to‐resource markets.

Sharing in construction projects — On determining optimal container assignments for the on-site accommodation of trades

The sharing economy is among the top trending topics in operations research. In this paper, we introduce a novel sharing application encountered in the construction industry. The problem under research, which has been developed in close cooperation with an industrial partner, is concerned with the on-site accommodation of trades in large-scale construction projects. In such projects, a construction logistics company provides containers gathered in an area of the site and assigns trades to them based on their requests issued online. Usually, a single trade will only remain on-site for a certain period of time, i.e. for as long as it is carrying out its activities in the project. Similarly, containers will only be provided for a certain time interval. This dynamic nature may require re-assigning trades as the project progresses. As such re-assignments impose costs for the logistics company, it is the goal to keep them to a minimum. We formally define the resulting optimization problem, present two mixed-integer programming formulations and develop a fix-and-optimize heuristic. In an extensive computational study, we show the latter to not only clearly outperform standard solver CPLEX operating on the holistic mixed-integer program, but to also provide substantial savings over the assignment policy currently applied in practice.

A novel framework for the day-ahead market clearing process featuring the participation of distribution system operators and a hybrid pricing mechanism

The electricity market liberalization propelled by the increased operational flexibility observed in distribution systems urges the development of mathematical models for competitive electricity markets accounting for the participation of distribution systems operators. Most existing approaches model the interaction between the agents and the market as a bilevel programming problem, wherein the market is characterized at the lower level. Hence, the upper-level agent’s objective function is favored over minimizing the costs of supplying the system’s demand. In this paper, a new bilevel formulation is proposed. In the upper level, the market operator determines the optimal energy price that ensures the demand supply and maximizes the overall social welfare. In the lower level, the generation companies and distribution systems operators determine the amount of energy injected/extracted into/from the transmission system to maximize profits/minimize operational costs. A new pricing method is proposed in this paper as an alternative to the transmission system’s dual variables, which are inaccessible under the proposed framework. The resulting nonlinear mixed-integer bilevel programming formulation is transformed into an equivalent single-level mixed-integer linear program. Numerical results illustrate the proposed approach’s effectiveness in maintaining the minimum operational costs while distributing the payoff amongst the agents.

Cutting Plane Approaches for the Robust Kidney Exchange Problem

Renal patients who have a willing but incompatible donor can decide to participate in a kidney exchange program (KEP). The goal of a KEP is to identify sets of incompatible pairs that can exchange donors, leading to compatible transplants for each recipient. Significant uncertainty is involved in this process, as planned transplants may be canceled for many reasons. It is therefore crucial to take into account failures while planning exchanges. In this paper, we consider a robust variant of this problem with recourse studied in the literature that takes into account failures of donors or recipients. This problem belongs to the class of defender–attacker–defender (DAD) models. We propose a cutting plane method for solving the attacker–defender subproblem based on two commonly used mixed-integer programming formulations for kidney exchange problems. Our results imply a running time improvement of one order of magnitude compared to the state-of-the-art. Moreover, our cutting plane methods can solve many previously unsolved instances. Furthermore, we propose a new practical policy for recourse in KEPs and show that the robust optimization problem concerning this policy is tractable for small to mid-size KEPs in practice.

The cumulative vehicle routing problem with arc time windows

In this paper, we introduce a new variant of the cumulative Vehicle Routing Problem (cum-VRP), where certain arcs of the undirected complete graph can only be utilized within specified time constraints. The problem is referred to as the Cumulative Vehicle Routing Problem with Arc Time Windows (Cum-VRPATW) and emerges in routing situations with flow disruptions across road segments, which generally emerges from military and civilian transportation. We devise a mixed integer programming (MIP) formulation to model the problem and solve it using CPLEX. To effectively address this problem, an effective algorithm is designed. It is a hybrid algorithm that combines principles of the Greedy Randomized Adaptive Search Procedure (GRASP) and decomposition strategies reducing computational complexity. Also, we propose two indexes, the RatioIn and the RatioTWL index that can identify difficult problems. Furthermore, we compare the CPLEX results vis-a-vis the new hybrid heuristic we have developed. Computational tests on converted Solomon (1987) data sets show that the new hybrid heuristic provides extremely good performance results for the cum-VRPATW large size test problems, justifying its effectiveness and robustness.

Valid inequalities for the non-unit demand capacitated minimum spanning tree problem with arc time windows and flow costs

In this paper, we introduce the non-unit demand capacitated minimum spanning tree problem with arc time windows and flow costs. The problem is a variant of the capacitated minimum spanning tree problem with arc time windows (CMSTP_ATW). We devise a mixed integer programming (MIP) formulation to model the problem and solve it using CPLEX. Furthermore, we propose three sets of inequalities, and we prove that they are valid. These valid inequalities tighten the model and lead to better lower bounds. To examine the quality of the solutions obtained, we convert the original data sets of Solomon (1987, “Algorithms for the Vehicle Routing and Scheduling Problem with Time Window Constraints.” Operations Research 35 (2): 254–265. https://doi.org/10.1287/opre.35.2.254) to approximate the non-unit demand CMSTP_ATW instances and provide results for the problems with 100 nodes. We execute extensive computational experiments, and the results show the positive effect of the inclusion of valid inequalities in the MIP.

An exact algorithm for Two-Echelon Location-Routing problem with simultaneous pickup and delivery

This study considers an extension of the two-Echelon Location-Routing Problem (2E-LRP) namely two-Echelon Location-Routing Problem with Simultaneous Pickup and Delivery (2E-LRPSPD). A Mixed Integer Programming (MIP) formulation is developed and strengthened by the use of valid inequalities to solve the 2E-LRPSPD, and then a Branch and Cut-based exact algorithm (B&C) is presented. To improve the upper bounds, Iterated Local Search and Variable Neighborhood Search-based metaheuristic algorithms, namely, ILS-VNS and VNS, are developed and embedded into the B&C. Computational results, conducted on instances produced from the literature, demonstrate that the B&C yields tight lower and upper bounds, and also is capable of solving some instances optimally with up to 50 customers and 10 satellites in a reasonable time. Furthermore, sensitivity analysis reveals that the heuristic approaches and an increase in the runtime limit assist the algorithm to reach tight the upper bounds, whereas the number of potential depots does not have any effect on the performance of the algorithm. Besides, further computational analysis on the 2E-LRPSPD with a single depot, which is studied in the literature, shows that the B&C improves some best-known solutions while proving the optimality of some instances. Finally, the B&C is applied to a two-echelon supermarket distribution system in Türkiye. The computational results show that simultaneous pickup and delivery operations, as well as depot location decisions are critical elements to consider in the design of cost-effective distribution networks.

Effect of climate on the optimal sizing and operation of seasonal ice storage systems

Seasonal thermal storage systems can reduce the temporal mismatch between renewable energy availability and energy demand. Ice storage systems exhibit a non-linear behaviour in the heat exchange and storage processes, complicating the formulation of optimal design and operation problems. In this work, we propose a mixed-integer quadratically-constrained programming formulation, which minimizes the Levelized Cost of Energy for space heating and cooling, including sizing of a supporting PV array. The optimization was repeated for different storage volumes, finding the system optimal operation in each case –and thereby the optimal system sizing. The heating and cooling demands were computed from an archetypal office building, placed in three reference locations with cold and semi-arid, warm and humid continental, and temperate and humid continental climates. Results show that the optimal PV size decreases with growing ice storage volume, and an ice storage works best in a temperate continental climate, covering up to 47% of the cooling demand with a 250 m 3 storage.

A data-driven discrete simulation-based optimization algorithm for car-sharing service design

This paper formulates a discrete simulation-based optimization (SO) algorithm for a family of large-scale car-sharing service design problems. We focus on the profit-optimal assignment of vehicle fleet across a network of two-way (i.e., round-trip) car-sharing stations. The proposed approach is a metamodel SO approach. A novel metamodel based on a mixed-integer program (MIP) is formulated. The metamodel is embedded within a general-purpose discrete SO algorithm. The proposed algorithm is validated with synthetic toy network experiments. The algorithm is then applied to a high-dimensional Boston case study using reservation data from a major US car-sharing operator. The method is benchmarked versus several algorithms, including stochastic programming. The experiments indicate that the analytical network model information, provided by the MIP to the SO algorithm, is useful both at the first iteration of the algorithm and across subsequent iterations. The solutions derived by the proposed method are benchmarked versus the solution deployed in the field by the car-sharing operator. Via simulation, the proposed solutions improve those deployed with an average improvement of profit of 6% and of vehicle utilization of 3%.The combination of the problem-specific analytical MIP with a general-purpose SO algorithm enables the discrete SO algorithm to: (i) address high-dimensional problems, (ii) become computationally efficient (i.e., it can identify good quality solutions within few simulation observations), (iii) become robust to the quality of the initial points and of the stochasticity of the simulator. More generally, the information provided by the MIP to the SO algorithm enables it to exploit problem-specific structural information. This leads to an algorithm with both asymptotic convergence guarantees as well as good short term performance (i.e., performance given few simulation observations). We view this general idea of combining analytical MIP formulations with general-purpose SO algorithms, or more broadly with general-purpose sampling strategies of high-resolution data, as an innovative and promising area of future research.

Optimal sizing and operation of seasonal ice thermal storage systems

Ice storage systems can be used as an efficient cooling source during summer, as well as a heat source for heat pumps during winter. The non-linear behavior of the heat exchange process in storage makes formulations for optimizing the design and operation of these technologies complex. In this work, we propose a quadratically-constrained mixed-integer programming formulation, that can capture the latent and sensible behavior of the storage and its impact on the delivery of heating and cooling. A building demonstrator integrating an ice storage device was used as a case study. Monitoring data were used to validate the simplified ice storage model employed in the optimization. Results showed that the most common optimal storage cycle requires freezing the water during late winter and when the air temperature falls below 0∘C. Increasing the storage volume increases both storage efficiency and the amount of free cooling available during summer. For these reasons, economies of scale can make larger systems more competitive than smaller ones. Storage size and thermal insulation level affect the duration of the charging and discharging phases. Thermal insulation improves seasonal efficiency and free cooling significantly. A higher CO2 emissions price does not yield significant benefits in terms of emissions reductions. High investment costs and the seasonal variation in CO2 intensity of electricity reduce the economic and environmental competitiveness of long-term ice storage systems, respectively.

Optimum Scheduling of a Multi-Machine Flexible Manufacturing System Considering Job and Tool Transfer Times without Tool Delay

In order to minimize makespan (Cmax) without causing tool delay with the fewest copies of each tool type, this study investigates the concurrent scheduling of automated guided vehicles (AGVs), machines (MCs), tool transporter (TT), and tools in a multi-machine flexible manufacturing system (FMS). The tools are housed in a central tool magazine (CTM), accessible to and utilized by several machines. AGVs and the tool transporter (TT) move jobs and tools between machines. Since it involves allocating tool copies and AGVs to job operations, sequencing job operations on machines, and related trip operations, such as AGVs’ and TT’s empty trip and loaded trip times, this simultaneous scheduling problem is highly complicated. This issue is resolved using the symbiotic organisms search algorithm (SOSA), based on the symbiotic interaction strategies that organisms adapt to survive in the ecosystem. This study proposes a mixed nonlinear integer programming formulation to address this problem. Verification is performed using an industrial problem from a manufacturing organization. The results show that employing two copies for two tool types out of 22 tool kinds and one copy for the remaining tool types results in no tool delay, which causes a reduction in the Cmax as well as cost. The industries that can benefit directly from this study are consumer electronics manufacturers, original equipment manufacturers, automobile manufacturers, and textile machine producers. The results demonstrate that the SOSA provides promising results compared to the flower pollination algorithm (FPA).

The influence coverage optimization problem

We introduce the Influence Coverage Optimization Problem (ICOP), which is an influence maximization problem where the activation of nodes also depends on their location on the plane. Specifically, the ICOP assumes that there is a network where nodes become active (i.e., influenced) either by the influence they receive from interactions with active in-neighbors or by entering the coverage area of a physical ad or a Geo-fence. The objective is to locate a fixed number of ads or Geo-fences and modify the network influence rates to minimize the network activation time. Assuming a Markovian influence model, we prove that the ICOP is NP -hard, and then we present mixed-integer programming formulations for three different types of coverage modes. A reformulation of the non-linear “big-M” constraints, two types of valid cuts, and a fast heuristic based on the k-means algorithm are used as enhancements that facilitate solving the ICOP via an Iterative Decomposition Branch-and-Cut (IDBC) algorithm. In addition, we present an alternative discrete formulation of the ICOP using critical intersection points. Several experiments under various parameter configurations across instances with more than a hundred nodes and thousand arcs are conducted, showing the IDBC’s capability to provide optimal solutions within seconds or minutes for most instances. Moreover, the experiments reveal that the ICOP can significantly outperform a Geo-fence coverage model that does not consider network interactions to make location decisions.

Handling the significance of regression coefficients via optimization

Subset selection in multiple linear regression is to identify a subset of explanatory variables that are most relevant to the response variable while discarding the remaining variables. Traditionally, a regression model is obtained by solving a given optimization problem, and then, some statistical tests and evaluations are performed to confirm its validity and to verify the satisfaction of qualifying requirements. Practically, human intervention is necessary at this stage. To prevent the need for manual inspection, we propose a new mixed integer programming formulation for the multiple linear regression subset selection problem. Our model enforces the statistical significance and the multicollinearity through explicit constraints along with several other well-known criteria. In contrast to existing approaches that incorporate these criteria gradually through a cutting-plane structure, this effort considers them explicitly within the formulation. The effectiveness of the approach is evaluated over different real datasets and compared with the existing methods in the literature. The results demonstrate that the significance of coefficients can be provided without any considerable change in the value of mean squared error.

A Mixed-Integer Programming Formulation for a Simplified Model of the Double Row Layout Problem

The double row layout problem (DRLP) is to assign facilities on two rows in parallel so that the total cost of material handling among facilities is minimized. Since it is vital to save cost and enhance productivity, the DRLP plays an important role in many application fields. Nevertheless, it is very hard to handle the DRLP because of its complex model. In this paper, we consider a new simplified model for the DRLP (SM-DRLP) and provide a mixed integer programming (MIP) formulation for it. The continuous decision variables of the DRLP are divided into two parts: start points of double rows and adjustable clearances between adjacent facilities. The former one is considered in the new simplified model for the DRLP with the purpose of maintaining solution quality, while the latter one is not taken into account with the purpose of reducing computational time. To evaluate its performance, our SM-DRLP is compared with the model of a general DRLP and the model of another simplified DRLP. The experimental results show the efficiency of our proposed model.

An ALNS-based matheuristic algorithm for a multi-product many-to-many maritime inventory routing problem

In this paper, we propose an adaptive large neighborhood search-based matheuristic algorithm to solve a multi-product many-to-many maritime inventory routing problem. The problem addresses a short sea inventory routing problem that aims to find the best route and distribution plan for multiple products with a heterogeneous fleet of vessels through a network including several producers and customers. Each port can be visited a given number of times during the planning horizon, and the stock level for each product should lie within the predefined bound limits. The problem was introduced by Hemmati et al. (Eur J Oper Res 252:775–788, 2016). They developed a mixed integer programming formulation and proposed a matheuristic algorithm to solve the problem. Although their proposed algorithm worked well in terms of running time, it suffers from disregarding a part of the solution space. In this study, we propose a new matheuristic algorithm to find better solutions by exploring the entire solution space for the same problem. In our solution methodology, we split the variables into routing and non-routing variables. Then in an iterative process, we determine the values of the routing variables with an adaptive large neighborhood search algorithm, and we pass them as input to a penalized model which is a relaxed and modified version of the mathematical model introduced in Hemmati et al. (2016). The information from solving the penalized model, including the values of the non-routing variables, is then passed to the adaptive large neighborhood search algorithm for the next iteration. Several problem-dependent operators are defined. The operators use the information they get from the penalized model and focus on decreasing the penalty values. Computational results show up to 26% improvement in the quality of the solutions for the group of instances with a large feasible solution space. We get the optimal value for the remaining instances matched with the reported results.