Mixed Integer Programming Solution Research Articles

Quadratic p-Median Problem: A Bender’s Decomposition and a Meta-Heuristic Local-Based Approach

In this paper, the quadratic p-median optimization problem is discussed, where the goal is to connect users to a selected group of facilities (emergency services, telecommunications servers, healthcare facilities) at the lowest possible cost. The problem is aimed at minimizing the cost of connecting these selected facilities. The costs are symmetric, meaning connecting two different points is the same in both directions. This problem extends the traditional p-median problem, a combinatorial problem used in various fields like facility location, network design, transportation, supply chain networks, emergency services, healthcare, and education planning. Surprisingly, the quadratic version has not been thoroughly considered in the literature. The paper highlights the formulation of two mixed-integer quadratic programming models to find optimal solutions to this problem. One model is a classic formulation, and the other is based on set cover theory. Linear versions and Bender’s decomposition formulations for each model are also derived. A Bender’s decomposition is solved using an algorithm that adds constraints during each iteration to improve the solution. Lazy constraints in the Gurobi solver’s branch and cut algorithm are dynamically added whenever a mixed-integer programming solution is found. Additionally, an efficient local search meta-heuristic is proposed that usually finds optimal solutions for tested instances. Challenging instances with up to 60 facilities and 2000 users are successfully solved. Our results show that Bender’s models with lazy constraints are the most effective for Euclidean and random test cases, achieving optimal solutions in less CPU time. The meta-heuristic also finds near-optimal solutions rapidly for these cases.

Balancing U-type assembly lines with human–robot collaboration

In this work, we consider a U-type assembly line balancing problem with human–robot collaboration (UALBP-HRC), in which the collaborative robot (cobot) can either parallelly process different tasks or collaboratively process the same task with workers. A mixed-integer programming (MIP) model is formulated and the objective is to minimize the cycle time for a given number of stations. We further propose an enhanced MIP (EMIP) model by incorporating enhancement strategies, including tight lower bound, upper bound and initial solution. To the best of our knowledge, this study is among the first attempts to address the UALBP-HRC in which both parallel and collaborative tasks are allowed. Due to the NP-hardness of the problem, a simulated annealing algorithm with enhanced search procedure (ESA) is developed to effectively solve the problem. The model functionalities and algorithm effectiveness are evaluated by adapted standard datasets and real case studies. The experimental results show that EMIP outperforms MIP and ESA finds promising solutions compared to MIP, EMIP, simulated annealing algorithm (SA) and genetic algorithm (GA). The comparisons between UALBP-HRC and straight assembly line balancing problem with human–robot collaboration (ALBP-HRC) are also conducted and the managerial insights are concluded.

A mixed-integer programming model for identifying intuitive ambulance dispatching policies

Markov decision process models and algorithms can be used to identify optimal policies for dispatching ambulances to spatially distributed customers, where the optimal policies indicate the ambulance to dispatch to each customer type in each state. Since the optimal solutions are dependent on Markov state variables, they may not always correspond to a simple set of rules when implementing the policies in practice. Restricted policies that conform to a priority list for each type of customer may be desirable for use in practice, since such policies are transparent, explainable, and easy to implement. A priority list policy is an ordered list of ambulances that indicates the preferred order to dispatch the ambulances to a customer type subject to ambulance availability. This paper proposes a constrained Markov decision process model for identifying optimal priority list policies that is formulated as a mixed integer programming model, does not extend the Markov state space, and can be solved using standard algorithms. A series of computational examples illustrate the benefit of intuitive policies. The optimal mixed integer programming solutions to the computational examples have objective function values that are close to those of the unrestricted model and are superior to those of heuristics.

Scheduling spatially distributed jobs with degradation: Application to pothole repair

This paper considers scheduling spatially distributed jobs with degradation. A mixed integer programming (MIP) model is developed for the linear degradation case in which no new jobs arrive. Properties of the model are analyzed, following which three heuristics are developed, enhanced greedy, chronological decomposition and simulated annealing. Numerical tests are conducted to: (i) establish limits of the exact MIP solution, (ii) identify the best heuristic based on an analysis of performance on small problem instances for which exact solutions are known, (iii) solve large problem instances and obtain lower bounds to establish solution quality, and (iv) study the effect of three key model parameters. Findings from our computational experiments indicate that: (i) exact solutions are limited to instances with less than 14 jobs; (ii) the enhanced greedy heuristic followed by the application of the simulated annealing heuristic yields high quality solutions for large problem instances in reasonable computation time; and (iii) the factors “degradation rate” and “work hours” have a significant effect on the objective function. To demonstrate applicability of the model, a case study is presented based on a pothole repair scenario from Buffalo, New York, USA. Findings from the case study indicate that scheduling spatially dispersed jobs with degradation such as potholes requires: (i) careful consideration of the number of servers assigned, degradation rate and depot location; (ii) appropriate modeling of continuously arriving jobs; and (iii) appropriate incorporation of equity consideration.

Combining mixed integer programming and constraint programming to solve the integrated scheduling problem of container handling operations of a single vessel

In the container terminals of seaports, the container handling system consists of a variety of container handling machines such as quay cranes, internal yard trucks, and yard cranes. This study applies a holistic approach to the integrated scheduling of these machines for the container handling operations of a single vessel. We formulate this special hybrid flow shop scheduling problem through both mixed integer programming (MIP) and constraint programming (CP) techniques. Then we develop an easily-implemented approach that combines the strengths of MIP and CP. First, the MIP model, which only considers quay crane scheduling, is solved by an MIP solver, and a quay crane allocation plan is retrieved from the MIP solution. Then, this quay crane allocation plan is fed to the CP model, warm-starting the branch-and-prune algorithm built in a CP optimizer. Our numerical experiments reveal that this hybrid MIP/CP approach can solve the large-sized instances with up to 1000 containers, 6 quay cranes, 36 yard trucks, and 15 yard cranes to optimality with a gap of less than 3.31%, within a solution time of 2 minutes. If we increase the solution time to 5 minutes, this hybrid approach solves larger instances with up to 1400 containers to optimality with a gap of less than 1.41%. The state-of-the-art dedicated algorithms reported in the literature (which address an easier version of the same problem by ignoring non-crossing constraints and safety margins between quay cranes) are only able to find solutions for real-life instances with up to 500 containers within the solution time of 2930 or 5221 seconds, leaving a 4% or an unknown optimality gap. Thus, this study improves the solution of this integrated scheduling problem in terms of the instance size, solution efficiency, and solution optimality.

Optimisation of the broiler production supply chain

In this paper, we propose a mixed integer programming (MIP) model for the Chicken Flock Sizing, Allocation and Scheduling Problem (CFSASP), which is an important planning problem in the broiler production supply chain. To solve the CFSASP efficiently, two variants of rolling horizon heuristics (RHHs) have been developed and applied on the case of a Norwegian broiler production company. Computational results show that the RHHs successfully obtain high-quality solutions within a reasonable time. The value of optimisation is verified through comparison with the case company's plans, where the solutions from optimisation outperforms the current solutions. Sensitivity analyses are also conducted to provide managerial insights regarding certain strategic decisions, such as how many and which days to use for hatching of chickens. Due to the promising results, the case company is now implementing an optimisation-based decision support system based on the MIP model and solution methods shown in this paper.

Multiobjective record‐to‐record travel metaheuristic method for solving forest supply chain management problems with economic and environmental objectives

AbstractMultiobjective optimization is increasingly used to assist decision‐making in forest management when multiple objectives are considered and conflict with each other. Since forest management problems may deal with combinatorial optimization, as the scale of a problem increases, the computation complexity increases exponentially beyond the practical use of exact methods. We propose a multiple‐objective metaheuristic method, referred to as multiobjective record‐to‐record travel (MRRT), to solve such challenging problems. We examined the performance of MRRT and compared it to a mixed integer programming (MIP) optimizer on a forest supply chain multiobjective optimization problem that simultaneously maximizes net revenues and greenhouse gas emission savings from salvage harvest and utilization of beetle‐killed forest stands. Testing on four cases of different problem sizes showed that MRRT performed satisfactorily in approximating the actual Pareto fronts in terms of convergence and coverage, and the distribution of solutions was approximately uniform. The gap between MRRT and MIP solutions increased as the problem size increased. But MRRT produced all solutions within a reasonable computation time, where the computational advantage over MIP was more apparent for large‐scale test cases.Recommendations for Resource Managers Multiobjective optimization shows trade‐offs among conflicting objectives and assists decision‐making to enhance sustainable forest management. Multiobjective record‐to‐record travel (MRRT) has a simple algorithm structure and easy parameterization process so that it is adaptable to solve various multiobjective optimization problems. MRRT produces high‐quality solutions for large‐scale multiobjective optimization problems within a reasonable computation time, which promotes its applicability in practice.

Integrated locating in-house logistics areas and transport vehicles selection problem in assembly lines

Decentralised in-house logistics areas, known as supermarkets, are widely used in the manufacturing industry for parts feeding to assembly lines. In contrary to the literature and inspired by observation in a real case, this study relaxes the assumption of using identical transport vehicles when deciding on the supermarkets’ location by considering the availability of different vehicles. In this regard, this study deals with the integrated supermarket location and transport vehicles selection problems (SLTVSP). A mixed-integer programming (MIP) model of the problem is developed. Due to the complexity of the problem, a hybrid genetic algorithm (GA) with variable neighborhood search (GA-VNS) is also proposed to address large-sized problems. The performance of GA-VNS is compared against the MIP, the basic GA, and simulated annealing (SA) algorithm. The computational results from the real case and a set of generated test problems show that GA-VNS provides a very good approximation of the MIP solutions at a much shorter computational time while outperforming the other compared algorithms. The analysis of the results reveals that it is beneficial to apply different transport vehicles rather than identical vehicles for SLTVSP.

Dynamic order batching in bucket brigade order picking systems with consecutive batch windows and non-identical pickers

The blocking mitigation mechanism contained in a batching and sequencing model improves order picking performance in bucket brigade order picking systems by adjusting the batch formation in a given batch window. The batch formation of a given batch window affects the transition to the next batch window and limits the batch formation’s structure in the next window. In addition, imbalance in picker capability increases the variation of batch formation while mitigating blocking delay. Our paper proposes a dynamic indexed batching model to smooth the transition between consecutive batch windows and give a better approximation of non-identical pickers’ capability. We develop a mixed integer programming solution for a dynamic indexed batching model in a bucket brigade order picking system (DIBMB). Simulation experiments show that the DIBMB smooths the transition between batch windows as measured in the delay experienced by the first batch in each batch window. Comparisons to the available batching models show that under DIBMB, the blocking delay gradually increases throughout the sequence of batches.

Multiple traveling salesman problem with drones: Mathematical model and heuristic approach

E-commerce and retail companies are seeking ways to cut delivery times and costs by exploring opportunities to use drones for making last mile delivery deliveries. This paper addresses the delivery concept of a truck-drone combination along with the idea of allowing autonomous drones to fly from delivery trucks, make deliveries, and fly to any available delivery truck nearby. We present a mixed integer programming (MIP) formulation that captures this scenario with the objective of minimizing the arrival time of both trucks and drones at the depot after completing the deliveries. A new algorithm based on insertion heuristics is also developed to solve large sized problems containing up to a hundred locations. Experiments are conducted to compare the MIP solutions with those obtained from different models with single truck, multiple truck and a single truck and drone system, as well as test the performance of the proposed algorithm. The numerical results demonstrate the potential operational gain when implementing the proposed drone delivery system compared to the conventional truck alone or single truck/drone delivery system.

Relaxing leaf‐motion restrictions in dynamic multileaf collimator leaf sequencing

Traditionally, unidirectional leaf-sweeping schemes have been employed to deliver IMRT plans using the dynamic multileaf collimator (DMLC) technique. The goal of this research is to investigate the potential impact of relaxing the leaf-motion restrictions in DMLC IMRT on the beam-modulation quality and the delivery efficiency. This research relaxes the initial and final leaf-position constraints as well as the unidirectional leaf-motion restriction that have been traditionally imposed on DMLC leaf sequencing and develops exact and heuristic solution approaches to allow for an unconstrained and bidirectional leaf motion. The exact approach employs mixed-integer programming (MIP) techniques and the proposed heuristic method uses stochastic search algorithms while utilizing the special structure of the problem. The trade-off between beam-modulation quality and delivery efficiency is quantified and compared to that of unidirectional leaf-sweeping schemes. The performance of the developed approaches is tested on liver and head-and-neck cancer cases. Results validate that unconstrained leaf trajectories can significantly improve the beam-modulation quality at small beam-on time values. However, this gain reduces as the available beam-on time increases. Additionally, the proposed heuristic approach can achieve near-optimal solutions with significantly smaller computational effort compared to the MIP solution approach. Unconstrained leaf trajectories have the potential to enhance the fluence-modulation quality for cases in which the available beam-on time is limited. This gain is primarily attributed to the relaxation of the initial and final leaf positions. The unidirectionality restriction alone does not appear to be a limiting factor.

Neural network based instant parameter prediction for wireless sensor network optimization models

Optimal operation configuration of a Wireless Sensor Network (WSN) can be determined by utilizing exact mathematical programming techniques such as Mixed Integer Programming (MIP). However, computational complexities of such techniques are high. As a remedy, learning algorithms such as Neural Networks (NNs) can be utilized to predict the WSN settings with high accuracy with much lower computational cost than the MIP solutions. We focus on predicting network lifetime, transmission power level, and internode distance which are interrelated WSN parameters and are vital for optimal WSN operation. To facilitate an efficient solution for predicting these parameters without explicit optimizations, we built NN based models employing data obtained from an MIP model. The NN based scalable prediction model yields a maximum of 3% error for lifetime, 6% for transmission power level error, and internode distances within an accuracy of 3 m in prediction outcomes.

Protocol Function Block Mapping of Software Defined Protocol for 5G Mobile Networks

In this paper, we propose software-defined protocol (SDP) technique to facilitate flexible service-oriented protocol stack deployment for providing high-throughput, low-latency and elastic mobile services based on platform virtualization and functionality modularization. We first elaborate the principle of SDP and then address one of the most important issues in SDP, namely SDP request mapping (SDPM), where an SDP request is fulfilled by mapping a set of required SDP function blocks and virtual links onto underlying SDP servers. We formulate the SDPM problem as a mixed integer programming (MIP). To address the $\mathcal {N}\mathcal {P}$ -hardness and scalability of SDPM problem, we propose a decomposition algorithm which breaks down the SDPM problem into inter-block link and block mapping problems to accomplish the upper bound (UB) and lower bound (LB) of the MIP solution, respectively. The optimality can be achieved when the UB and the LB converges by using iterations. We employ LTE Layer-2 data-plane processing as a benchmark for validating the effectiveness of the SDP technique and evaluate the performance of SDPM algorithm. Numerical results show that SDP is effective to provide elastic low-latency mobile services and the proposed SDPM algorithm significantly outperforms the benchmark in stack processing delay, mapping cost, and resource utilization.

Critical arcs detection in influence networks

The influence class of network problems models the propagation of influence (an abstraction of cascading beliefs, behaviors, or physical phenomena) in a network. Such problems have applications in social networks, electrical networks, computer networks, viral spreading, and so on. These types of networks have also been studied through the lens of critical arcs detection; that is, which arcs (edges) are the most important for maintaining some property of the network (e.g., connectivity). We introduce a new class of problems at the intersection of these two models. Specifically, given a set of seed nodes and the linear threshold influence propagation model, our work proposes to determine which arcs (e.g., relationships in a social network or communication pathways in a telecommunication network) are most critical to the influence propagation process. We prove NP‐hardness of the problem. Time‐dependent and time‐independent mixed‐integer programming (MIP) models are introduced. Insights gleaned from MIP solutions leads to the development of an improved MIP‐based exact algorithm rooted in the idea of diffusion expansion. A heuristic based upon a new centrality measure is also proposed, and computational results are presented. © 2017 Wiley Periodicals, Inc. NETWORKS, Vol. 71(4), 412–431 2018

Alternative Mathematical Models and Solution Approaches for Lot-Sizing and Scheduling Problems in the Brewery Industry: Analyzing Two Different Situations

This research proposes new approaches to deal with the production planning and scheduling problem in brewery facilities. Two real situations found in factories are addressed, which differ by considering (or not) the setup operations in tanks that provide liquid for bottling lines. Depending on the technology involved in the production process, the number of tank swaps is relevant (Case A) or it can be neglected (Case B). For both scenarios, new MIP (Mixed Integer Programming) formulations and heuristic solution methods based on these formulations are proposed. In order to evaluate the approach for Case A, we compare the results of a previous study with the results obtained in this paper. For the solution methods and the result analysis of Case B, we propose adaptations of Case A approaches yielding an alternative MIP formulation to represent it. Therefore, the main contributions of this article are twofold: (i) to propose alternative MIP models and solution methods for the problem in Case A, providing better results than previously reported, and (ii) to propose new MIP models and solution methods for Case B, analyzing and comparing the results and benefits for Case B considering the technology investment required.

Minimum cost solar power systems for LTE macro base stations

This paper proposes an algorithm for the identification of the minimum cost solution over a 10 year time horizon to power an LTE (Long-Term Evolution) macro base station, using a photovoltaic solar panel, a set of batteries, and optionally also a secondary power source, which can be a connection to a (possibly unreliable) power grid, or a small Diesel generator. The optimization is formalised as an Mixed Integer Programming (MIP) problem, which, after linearization, can be solved with CPLEX. A heuristic algorithm is also proposed, with the objective of decreasing the computational complexity of the optimization. Numerical results show that a hybrid solar-grid (or solar-diesel) power system saves a significant fraction of the total cost, compared to a pure solar system, and to the traditional power-grid system, over the investigated 10-year period, in a south European city, like Torino in Italy, as well as in a location close to the tropic, like Aswan in Egypt. Our proposed heuristic algorithm can be used to obtain a solution within 10–20% of the optimum, at a computational speed 200 times faster than the MIP solution.

Comment on Three-Element Item Selection Procedures for Multiple Forms Assembly: An Item Matching Approach.

A recent article in this journal addressed the choice between specialized heuristics and mixed-integer programming (MIP) solvers for automated test assembly. This reaction is to comment on the mischaracterization of the general nature of MIP solvers in this article, highlight the quite inefficient modeling of the test-assembly problems used in its empirical examples, and counter these examples by presenting the MIP solutions for a set of 35 real-world multiple-form assembly problems.

Modeling risk in a Design for Supply Chain problem

The objective of Design for Supply Chain (DFSC) is to design a supply chain in parallel to designing a new product. Risk is an inherent element of this process. Although supply chain risk models and product development risk models are available, there are few models that consider the combined effect of risk to product development and the supply chain. This gap is filled by the development of a DFSC and risk model that looks at design, supply chain and risk concurrently. The model consists of two components. First, a Mixed Integer Programming (MIP) model makes the DFSC decisions while simultaneously considering time-to-market risk, supplier reliability risk and strategic exposure risk. The results from the MIP are then used in the second model component which is a discrete event simulation. The simulation tests the robustness of the MIP solution for supplier capacity risk and demand risk. When a decision maker is potentially facing either of these risks the simulation shows whether it is best to use an alternative solution or proceed with the MIP solution. The model provides analytical results, but also allows decision makers to use their own judgment to select the best option for overall profitability. In conclusion, testing shows that risk mitigation strategies can and should be determined from the DFSC and risk model, but that they will be dependent on the specific design problem being solved.

A Computationally Efficient Stabilizing Model Predictive Control of Switched Systems

Many applications in engineering exhibit switching character due to discrete and continuous aspects in their dynamic behavior. Switching characteristics of hybrid systems bring discontinuity and nonlinearity in their course of operation and pose major challenges in developing stabilizing Model Predictive Control (MPC) for them. For Piecewise Affine (PWA) Systems, the MPC problem requires on-line solution of Mixed Integer Programs (MIPs) for obtaining the input profile. Since, complexity of the optimization problem that needs to be solved in MPC increases combinatorially with respect to the integer variables, on-line computing of MPC control law for large scale problems and/or problems with large horizons is expensive. In this paper we, propose a MPC formulation, under the popular framework of terminal cost - terminal set MPC, which enables tuning the complexity of the control algorithm. The proposed approach introduces an idea of a pre-terminal set, within which the inputs have enough power to trap states inside it. Since the pre-terminal set lies in the terminal mode which contains origin, this eliminates the need for binary decision variables to model mode transitions after the trajectory enters in pre-terminal set, thereby reducing the on-line complexity although at the expense of optimality. Examples are presented to illustrate the computational benefits of the proposed MPC strategy over existing MPC.

Challenge: Modelling Unit Commitment as a Planning Problem

Unit Commitment is a fundamental problem in power systems engineering, deciding which generating units to switch on, and when to switch them on, in order to efficiently meet anticipated demand. It has traditionally been solved as a Mixed Integer Programming (MIP) problem but upcoming changes to the power system drastically increase the MIP solution time. In this paper, we discuss the benefits that using planning may have over the established methods.We provide a formal description of Unit Commitment, and we present its formulation as MIP and as a planning problem. This is a novel and interesting application area for planning, with features that make the domain challenging for current planners.