Mixed-integer Programming Formulation Research Articles

Minimizing risk in the scheduling of crudes in an oil refinery

This paper focuses on formulating and solving the optimization of crude oil operations scheduling carried out in a system composed of a refinery and a marine terminal. The main challenge lies in coordinating the decisions made at both facilities and, at the same time, dealing with the uncertainties inherent in this activity. To tackle this problem, we present a continuous-time mixed-integer non-linear programming (MINLP) formulation. Furthermore, uncertainty in the ship arrival times is considered through a two-stage stochastic programming approach. Finally, the Conditional Value-at-Risk (CVaR) risk measure is employed to weigh the risk of having high costs relative to the worst scenarios. In this way, the proposed model is capable of supporting the decision-making process under uncertainty in an integral way.

A Hybrid Heuristic for a Two-Agent Multi-Skill Resource-Constrained Scheduling Problem

This paper addresses an industrial case of the two-agent scheduling problem with a global objective function. Each agent manages one or several projects and competes with another agent for the use of common multi-skilled employees. There is a pool of employees, each of which can perform a set of skills with heterogeneous performance levels. The objectives of the two agents are both to minimize the total weighted tardiness of its tasks. Furthermore, We assume that some constraints (soft constraints) can be violated when there is no feasible schedule for the problem. Thus, the global objective function minimizes the constraint violations by reducing the undesirable deviations in the soft constraints from their respective goals. The overall objective is to find a schedule that minimizes both agents objective functions (local objectives) and the global objective function. We provide a mixed-integer goal programming (MIGP) formulation for the problem. In addition, we present a hybrid algorithm combining an exact procedure, a greedy heuristic, and a genetic algorithm to find an approximate Pareto solution set. We compare the performance of the hybrid algorithm against the corresponding MIGP formulation with simulated instances derived from real-world instances.

Solving the vehicle routing problem with drone for delivery services using an ant colony optimization algorithm

E-commerce and logistics companies are facing important challenges to satisfy the rapid growth of customer demands. Unmanned aerial vehicles such as drones are an emerging technology that are very useful to cope with rising customer expectations of fast, flexible, and reliable delivery services. Drones work in tandem with trucks to perform parcel delivery, which have proven to reduce costs, CO2 emissions, and delivery times. This research proposes a mixed integer programming formulation to address the Vehicle Routing Problem with Drone (VRPD) by assigning customers to drone-truck pairs, determining the number of dispatching drone-truck units, and obtaining optimal service routes while the fixed and travel costs of both vehicles are minimized. Given the NP-hard nature of the VRPD, an ant colony optimization (ACO) algorithm is elaborated to solve this problem. Two novel methods are proposed to investigate the efficiency of the drone-truck combination by allowing the drones to perform additional delivery services to only one feasible customer and also multiple feasible customers while the truck waits at a customer location. Experimental results show that the proposed ACO algorithm can effectively solve the VRDP for different size instances and different customer location distributions, and is successful in providing timely solutions for small test instances within 1% of the optimal solutions. Finally, experimentation also reveals that the ACO algorithm outperforms the classical VRP by obtaining cost-savings of over 30% for large instances.

Continuous Covering on Networks: Strong Mixed Integer Programming Formulations

Covering problems are well-studied in the domain of Operations Research, and, more specifically, in Location Science. When the location space is a network, the most frequent assumption is to consider the candidate facility locations, the points to be covered, or both, to be discrete sets. In this work, we study the set-covering location problem when both candidate locations and demand points are continuous sets on a network. This variant has received little attention, and the scarce existing approaches have focused on particular cases, such as tree networks and integer covering radius. Here we study the general problem and present a Mixed Integer Linear Programming formulation (MILP) for networks with edges' lengths no greater than the covering radius. The model does not lose generality, as any edge not satisfying this condition can be partitioned into subedges of appropriate lengths without changing the problem. We propose a preprocessing algorithm to reduce the size of the MILP, and devise tight big-$M$ constants and valid inequalities to strengthen our formulations. Moreover, a second MILP is proposed, which admits edges' lengths greater than the covering radius. As opposed to existing formulations of the problem (including the first MILP proposed herein), the number of variables and constraints of this second model does not depend on the lengths of the network's edges. This second model represents a scalable approach that particularly suits real-world networks, whose edges are usually greater than the covering radius. Our computational experiments show the strengths and limitations of our exact approach on both real-world and random networks. Our formulations are also tested against an existing exact method.

A Selective Many-to-Many Pickup and Delivery Problem With Handling Cost in the Omni-Channel Last-Mile Delivery

This paper introduces a selective many-to-many pickup and delivery problem with handling cost (SMMPDPH) arising in the omni-channel last-mile delivery. In SMMPDPH, each request is associated with multiple pickup and delivery nodes, which provide or require commodities. A request is to load commodities from the pickup nodes and transport them to the delivery nodes. Since the total supply is greater than the total demand, we need to determine the selected subset of pickup nodes to visit for each request. Based on the loading and unloading policy, we take into account the handling cost incurred by additional operations. SMMPDPH aims to obtain a routing plan with the minimum total cost, including the travel cost and handling cost, for a fleet of homogeneous vehicles with limited capacity and driving mileage to satisfy the requests. We present two mixed integer programming formulations of the problem and propose two algorithms, including an iterated local search (ILS) and a memetic algorithm (MA), with specially designed move operators to solve the problem. Experiments on small instances clearly indicate the effectiveness of the two heuristic methods. With a comparison of efficiency on large-scale instances, we find that MA outperforms ILS in terms of both the best and average solution quality.

Timetable merging for the Periodic Event Scheduling Problem

We propose a new mixed integer programming based heuristic for computing new benchmark primal solutions for instances of the PESPlib. The PESPlib is a collection of instances for the Periodic Event Scheduling Problem (PESP), comprising periodic timetabling problems inspired by real-world railway timetabling settings, and attracting several international research teams during the last years. We describe two strategies to merge a set of good periodic timetables. These make use of the instance structure and minimum weight cycle bases, finally leading to restricted mixed integer programming formulations with tighter variable bounds. Implementing this timetable merging approach in a concurrent solver, we improve the objective values of the best known solutions for the smallest and largest PESPlib instances by 1.7 and 4.3 percent, respectively.

Models and Algorithms for the Bin-Packing Problem with Minimum Color Fragmentation

In the bin-packing problem with minimum color fragmentation (BPPMCF), we are given a fixed number of bins and a collection of items, each associated with a size and a color, and the goal is to avoid color fragmentation by packing items with the same color within as few bins as possible. This problem emerges in areas as diverse as surgical scheduling and group event seating. We present several optimization models for the BPPMCF, including baseline integer programming formulations, alternative integer programming formulations based on two recursive decomposition strategies that utilize decision diagrams, and a branch-and-price algorithm. Using the results from an extensive computational evaluation on synthetic instances, we train a decision tree model that predicts which algorithm should be chosen to solve a given instance of the problem based on a collection of derived features. Our insights are validated through experiments on the aforementioned applications on real-world data. Summary of Contribution: In this paper, we investigate a colored variant of the bin-packing problem. We present and evaluate several exact mixed-integer programming formulations to solve the problem, including models that explore recursive decomposition strategies based on decision diagrams and a set partitioning model that we solve using branch and price. Our results show that the computational performance of the algorithms depends on features of the input data, such as the average number of items per bin. Our algorithms and featured applications suggest that the problem is of practical relevance and that instances of reasonable size can be solved efficiently.

The Unit Re-Balancing Problem

We describe the problem of re-balancing a number of units distributed over a geographic area. Each unit consists of a number of components. A value between 0 and 1 describes the current rating of each component. By a piecewise linear function, this value is converted into a nominal status assessment. The lowest of the statuses determines the efficiency of a unit, and the highest status its cost. An unbalanced unit has a gap between these two. To re-balance the units, components can be transferred. The goal is to maximize the efficiency of all units. On a secondary level, the cost for the re-balancing should be minimal. We present a mixed-integer nonlinear programming formulation for this problem, which describes the potential movement of components as a multi-commodity flow. The piecewise linear functions needed to obtain the status values are reformulated using inequalities and binary variables. This results in a mixed-integer linear program, and numerical standard solvers are able to compute proven optimal solutions for instances with up to 100 units. We present numerical solutions for a set of open test instances and a bi-criteria objective function, and discuss the trade-off between cost and efficiency.

Economic-environmental risk-averse optimal heat and power energy management of a grid-connected multi microgrid system considering demand response and bidding strategy

In this paper, a risk-constrained stochastic mixed-integer linear programming (MILP) model is proposed for optimal bidding strategy of a grid-connected CHP-based multi-microgrid (MMG) system in energy and reserve markets considering environmental restrictions. A reward-based demand response program is considered in the proposed framework to realize demand-side management and boost the economic performance of the MMG system. Besides, conditional value-at-risk as a proper risk-averse index is incorporated into the developed mixed-integer programming formulation to hedge the risk of the low profits associated with worst scenarios. For this study, the uncertainty of energy market price, electrical load, and power generation of renewable energy sources are taken into account through employing stochastic programming. Meanwhile, given the global policy for emissions reduction, an emission constraint is considered into the model, ensuring a more green trading strategy. The obtained simulation results over a 24-h time horizon, demonstrate that increasing the risk factor leads to the profit decrement by almost 30.37% while the conservativeness of the MMG against the risk is improved. Also, the results reveal that by increasing the value of emission factor, the system's profit is decreased by about 30.03% while the MMG would be operated more environmental-friendly.

The joint stochastic lot sizing and pricing problem

We consider a joint stochastic lot sizing pricing problem where period demands are price dependent and stochastic. Herein, we propose an alternative joint inventory control pricing policy for the profit-optimal (s,S,p) policy. The proposed policy – referred to as (R,S,p) is an order-up-to policy where replenishment schedule and price levels for each period are set before the planning horizon starts. As such, this policy can be considered as rather static as compared to the dynamic (s,S,p) policy. We also provide a quadratic mixed integer programming formulation in order to obtain the optimal (R,S,p) policy parameters. Furthermore, (R,S,p) policy is compared against the profit-optimal (s,S,p) policy on an extensive numerical study where several different benchmark policies are additionally used. The results show that the proposed policy exhibits a competitive profit performance with respect to the profit-optimal policy especially for low or moderate levels of demand uncertainty. The results also reveal that the relative benefit of using dynamic pricing policy, as compared to that of using dynamic inventory control, diminishes as demand uncertainty escalates.

Heuristics for a Two-Stage Assembly-Type Flow Shop with Limited Waiting Time Constraints

This study investigates a two-stage assembly-type flow shop with limited waiting time constraints for minimizing the makespan. The first stage consists of m machines fabricating m types of components, whereas the second stage has a single machine to assemble the components into the final product. In the flow shop, the assembly operations in the second stage should start within the limited waiting times after those components complete in the first stage. For this problem, a mixed-integer programming formulation is provided, and this formulation is used to find an optimal solution using a commercial optimization solver CPLEX. As this problem is proved to be NP-hard, various heuristic algorithms (priority rule-based list scheduling, constructive heuristic, and metaheuristic) are proposed to solve a large-scale problem within a short computation time. To evaluate the proposed algorithms, a series of computational experiments, including the calibration of the metaheuristics, were performed on randomly generated problem instances, and the results showed outperformance of the proposed iterated greedy algorithm and simulated annealing algorithm in small- and large-sized problems, respectively.

Maximizing student opportunities for in-person classes under pandemic capacity reductions

In this article, we describe the decision support system that was developed for the assignment of courses to teaching modalities and rooms for the Fall semester of 2020 at the University of Connecticut (UConn). With the adoption of safety/mitigation standards imposed by the COVID-19 pandemic, the seating capacities of rooms were reduced by more than 70%, thus making virtually every existing room assignment for Fall 2020 infeasible. The demand for in-person instruction required the reassignment of a large number of courses to rooms, where not all requests for physical space could be accommodated. In order to maximize opportunities for in-person instruction, UConn introduced a teaching modality in which class meetings are attended on campus by only 50% of the enrolled students. As decision makers were given partial flexibility to assign teaching modalities to classes, the complexity of the assignment problem increased considerably, especially because the real-world instances involved hundreds of rooms and thousands of classes and required a quick solution turnaround in practice. In this article, we introduce this flexible assignment problem and describe the two mixed-integer programming formulations that were used to solve the real-world instances of the problem; in particular, one of the formulations leverages structural properties presented in this work in order to represent the problem in a more compact way. We explain how we tailored our algorithms to solve the real-world problem, describe the dynamics of the interactive decision support system created in this initiative, and present insights derived from our study.

Dial-a-ride problem: mixed integer programming revisited and constraint programming proposed

The dial-a-ride problem with time windows is studied in this article. First, a new efficient mixed integer programming formulation is proposed by merging some of drop-off nodes into paired pick-up nodes when a size of the order forbids ridesharing, so a vehicle must visit the paired drop-off nodes directly after the pick-up, thus reducing the number of decision variables by 40% on benchmark instances. Secondly, three different constraint programming formulations are proposed for the first time, generating efficient schedules within a few seconds. Thirdly, a real-time rescheduling perspective is further studied, since the ridesharing becomes dynamic when a schedule must be provided instantly. In particular, the warm-start search method, in which a precalculated schedule is continuously revised as new orders come in, is proposed.

Influence Maximization with Latency Requirements on Social Networks

Targeted marketing strategies are of significant interest in the smartapp economy. Typically, one seeks to identify individuals to strategically target in a social network so that the network is influenced at a minimal cost. In many practical settings, the effects of direct influence predominate, leading to the positive influence dominating set with partial payments (PIDS-PP) problem that we discuss in this paper. The PIDS-PP problem is NP-complete because it generalizes the dominating set problem. We discuss several mixed integer programming formulations for the PIDS-PP problem. First, we describe two compact formulations on the payment space. We then develop a stronger compact extended formulation. We show that when the underlying graph is a tree, this compact extended formulation provides integral solutions for the node selection variables. In conjunction, we describe a polynomial-time dynamic programming algorithm for the PIDS-PP problem on trees. We project the compact extended formulation onto the payment space, providing an equivalently strong formulation that has exponentially many constraints. We present a polynomial time algorithm to solve the associated separation problem. Our computational experience on a test bed of 100 real-world graph instances (with up to approximately 465,000 nodes and 835,000 edges) demonstrates the efficacy of our strongest payment space formulation. It finds solutions that are on average 0.4% from optimality and solves 80 of the 100 instances to optimality. Summary of Contribution: The study of influence propagation is important in a number of applications including marketing, epidemiology, and healthcare. Typically, in these problems, one seeks to identify individuals to strategically target in a social network so that the entire network is influenced at a minimal cost. With the ease of tracking consumers in the smartapp economy, the scope and nature of these problems have become larger. Consequently, there is considerable interest across multiple research communities in computationally solving large-scale influence maximization problems, which thus represent significant opportunities for the development of operations research–based methods and analysis in this interface. This paper introduces the positive influence dominating set with partial payments (PIDS-PP) problem, an influence maximization problem where the effects of direct influence predominate, and it is possible to make partial payments to nodes that are not targeted. The paper focuses on model development to solve large-scale PIDS-PP problems. To this end, starting from an initial base optimization model, it uses several operations research model strengthening techniques to develop two equivalent models that have strong computational performance (and can be theoretically shown to be the best model for trees). Computational experiments on a test bed of 100 real-world graph instances (with up to approximately 465,000 nodes and 835,000 edges) attest to the efficacy of the best model, which finds solutions that are on average 0.4% from optimality and solves 80 of the 100 instances to optimality.

Classifying and modelling demand response in power systems

Demand response (DR) is expected to play a major role in integrating large shares of variable renewable energy (VRE) sources in power systems. For example, DR can increase or decrease consumption depending on the VRE availability, and use generating and network assets more efficiently. Detailed DR models are usually very complex, hence, unsuitable for large-scale energy models, where simplicity and linearity are key elements to keep a reasonable computational performance. In contrast, aggregated DR models are usually too simplistic and therefore conclusions derived from them may be misleading. This paper focuses on classifying and modelling DR in large-scale models. The first part of the paper classifies different DR services, and provides an overview of benefits and challenges. The second part presents mathematical formulations for different types of DR ranging from curtailment and ideal shifting, to shifting including saturation and immediate load recovery. Here, we suggest a collection of linear constraints that are appropriate for large-scale power systems and integrated energy system models, but sufficiently sophisticated to capture the key effects of DR in the energy system. We also propose a mixed-integer programming formulation for load shifting that guarantees immediate load recovery, and its linear relaxation better approximates the exact solution compared with previous models.

Minimum cost delivery of multi-item orders in e-commerce logistics

We solve a delivery problem arising in e-commerce logistics. We consider a retailer with an online store and a network of stores operating in an omni-channel strategy. The fulfillment decision for an online order, which contains a number of items, involves the allocation of these items to the stores where they are available and the selection of one store for consolidation of the items into the final package to be dispatched to the customer. The transportation between the stores and the customer is handled by a third-party logistic provider which uses a concave pricing policy based on the distance between the origin and the destination, as well as on the weight of the items. We present an online problem which is defined for a set of orders placed over time, and a mixed integer programming formulation defined for each order. The main characteristics of this problem are that the solution of the formulation for each order impacts those of the subsequent orders, and the problem must be solved in real time. For the solution of the formulation, we propose an iterative matheuristic based on the solution of the set covering model and local search. Computational results on randomly generated instances are provided, which demonstrate that our algorithm is capable of producing high-quality results.

A GRASP to solve the multi-constraints multi-modal team orienteering problem with time windows for groups with heterogeneous preferences

Improving the travel experience is a goal of tourist destinations. Tourists demand information and services that help plan and organise the trips adapted to their preferences and resources. Intelligent systems, recommendation systems, and electronic tourist guides can play a decisive role in tourist satisfaction and shaping the offer at the destination. These systems must satisfy the interests of tourists which travel alone or as a group. However, the development of these tools to generate group tourism itineraries is still limited. Group itineraries must also consider individual preferences and the selection of transport modes. The problem associated with the construction of tourist routes is called the Tourist Trip Design Problem. In this work, an extension of the Team Orienteering Problem with Time Windows is developed to model tourism planning. The model considers the construction of group routes, a cost and time limit associated with each participant, the selection of the mode of transport to go from one location to another, and heterogeneous preferences in the group. The model aims to maximise the preferences of group members by considering real and challenging constraints that increase complexity and avoid solutions in polynomial time. This is a novel model, and we generate sets of instances to assess the accuracy of our solution approach. A Greedy Randomised Adaptive Search Procedure is proposed to solve the model. Evaluation criteria based on time, cost, and preferences are used to guide the candidate’s selection in the construction phase. The results provided by our meta-heuristic are compared with those obtained by solving the Mixed Integer Programming formulation with exact solver. Computational results show the effectiveness and efficiency of the proposed approach.

Simultaneous scheduling of machines and tools in a multi machine FMS with alternative routing using symbiotic organisms search algorithm

This paper deals with simultaneous scheduling of machines and tools with alternate machines in a multi machine flexible manufacturing system (FMS) to minimize makespan (MS). Only one copy of each type of tools is made available due to economic restrictions and the tools are stored in a central tool magazine (CTM) that shares with and serves for several machines. The problem is to select machines from alternate machines for job-operations, allocation of tools to job-operations and job-operations’ sequencing on machines for MS minimization. This paper presents a nonlinear mixed integer programming (MIP) formulation to model the combined scheduling of machines and tools with alternate machines and a symbiotic organisms search algorithm (SOSA) built on the symbiotic interaction strategies that organisms employ to continue to exist in the ecosystem for solving the scheduling of machines and tools with alternate machines. The results have been tabulated, analyzed. It is observed that there is a reduction in MS when the alternate machines are considered for job-operation.

Integration of aggressive bound tightening and Mixed Integer Programming for Cost-sensitive feature selection in medical diagnosis

Silent diseases is an umbrella term that captures a spectrum of chronic illnesses that produce no clinically obvious signs and are diagnosed at advanced stages when the damage is irreversible. Current diagnostic strategies of silent diseases depend on self-reported symptoms and observed behavior through extended periods of time, and until now there are no specific clinical tests to diagnose silent diseases. Scientific research suggests the importance of early diagnosis to restore the functionality and reduce diseases-related complications. Previous studies primarily focused on feature selection methods to aid in medical diagnosis. Traditional feature selection methods are primarily focused on correct classification and often ignore features’ costs; the cost of clinical tests required to acquire the feature value. However, in medical diagnosis, features have different associated costs. Because ignoring features’ costs may result in a high cost diagnostic strategy that cannot be used in practice, developing a low-cost diagnostic strategy remains a subject of much interest. In this paper, new Mixed Integer Programming (MIP) models, namely, Cost-sensitive Support Vector Machine (CS-SVM) and Cost-sensitive Multi-surface Method Tree (CS-MSMT) that allow for simultaneous selection of low-cost and informative features are proposed. The CS-SVM and CS-MSMT are superior because they have the ability to account for shared costs. The CS-SVM and CS-MSMT were modified to embed shared costs across feature groups, and are termed Discounted CS-SVM (dCS-SVM) and Discounted CS-MSMT (dCS-MSMT), respectively. Computationally effective algorithm that integrates aggressive bound tightening with the MIP formulation is proposed. To demonstrate the effectiveness of the proposed models, different analysis paradigms are conducted on six UCI medical datasets; Chronic Kidney Disease, Hepatitis, Heart Disease, Thyroid, Diabetes and Leukemia. The results demonstrate the efficiency and robustness of the CS-SVM and CS-MSMT (and consequently the dCS-SVM and dCS-MSMT) under various conditions. The CS-SVM and CS-MSMT improved accuracy by 10.3% and 3.4% and reduced costs by 94.3% and 72.4% in the leukemia dataset, respectively.

The vehicle routing problem with load-dependent travel times for cargo bicycles

Cities, retailers, and delivery services are trying to find more environmental friendly solutions for last mile delivery. One very promising and partly used option are cargo bicycles. A critical factor of cargo bicycles is their travel speed, which depends on the mass of the load and the gradient of the chosen streets. Therefore, we define load-dependent travel speed and introduce the vehicle routing problem with load-dependent travel times (VRPLTT). We introduce a mixed-integer programming formulation and show the benefits of considering load-dependent travel times in last mile delivery with cargo bicycles. To efficiently solve larger instances, we develop an adaptive large neighborhood search (ALNS). For the numerical study, we introduce a new set of instances and show that the travel time can be reduced by 4.5% on average and up to 22% compared to the vehicle routing problem with time windows. Moreover, ignoring the load-dependent travel times can lead to the underestimation of travel time in such a way that time windows are violated.