Bi-objective Mixed Integer Linear Programming Research Articles

Optimization models for financing innovations in green energy technologies

Commercialization of emerging green technologies is essential to improve the sustainability of industrial processes. However, there are risks inherent in funding the development of new technologies that act as a significant barrier to their commercialization. Mathematical models can provide much-needed decision support to allow optimal allocation of funds, while managing the implications of techno-economic risk. The Technology Readiness Level (TRL) scale is a well-established figure of merit approach for quantifying the maturity of stand-alone technologies, while the more recently developed System Readiness Level (SRL) scale is applicable to technology networks with interdependent components. These technology maturity scales are intended mainly to be used for the passive assessment of a given state of technology, but may be incorporated within an optimization model to aid in innovation planning. In this work, two mixed integer linear programming (MILP) models are proposed to optimize strategies for funding innovation. The first model is a bi-objective MILP for optimizing the allocation of funds to a portfolio of independent innovation projects. The model is based on source-sink formulation and uses information on TRL and return on investment (ROI) to determine the best allocation of funds. The second model is a robust MILP that optimizes the allocation of limited project funds in order to maximize the SRL of a system of emerging technologies. This approach accounts for Integration Readiness Level (IRL) among mutually interdependent technologies. Both models are demonstrated with illustrative case studies on biorefinery technologies in order to demonstrate their capabilities.

Multi-Period Network Design Problem in Regional Hazardous Waste Management Systems.

The management of hazardous wastes in regions is required to design a multi-echelon network with multiple facilities including recycling, treatment and disposal centers servicing the transportation, recycling, treatment and disposal procedures of hazardous wastes and waste residues. The multi-period network design problem within is to determine the location of waste facilities and allocation/transportation of wastes/residues in each period during the planning horizon, such that the total cost and total risk in the location and transportation procedures are minimized. With consideration of the life cycle capacity of disposal centers, we formulate the problem as a bi-objective mixed integer linear programming model in which a unified modeling strategy is designed to describe the closing of existing waste facilities and the opening of new waste facilities. By exploiting the characteristics of the proposed model, an augmented -constraint algorithm is developed to solve the model and find highly qualified representative non-dominated solutions. Finally, computational results of a realistic case demonstrate that our algorithm can identify obviously distinct and uniformly distributed representative non-dominated solutions within reasonable time, revealing the trade-off between the total cost and total risk objectives efficiently. Meanwhile, the multi-period network design optimization is superior to the single-period optimization in terms of the objective quality.

Variable fleet size and mix VRP with fleet heterogeneity in Integrated Solid Waste Management

The Integrated Solid Waste Management (ISWM) is a recent effective tool to manage with the growing challenge of Municipal Solid Waste (MSW). The ISWM integrates all the system components (i.e. transfer, treatment, recycling and disposal of wastes) to enhance the sustainable waste management whilst reducing operational costs. In this paper, we investigate an integrated framework of the Fleet Size and Mix Vehicle Routing Problem (VRP) to optimize the cost-effective ISWM system. A novel bi-objective Mixed-Integer Linear Programming (MILP) model is developed to concurrently minimize the transportation cost in the entire waste management system and total deviation from the fair load allocation to transfer stations. A complete ISWM system with all interdependent facilities and multiple technologies, is developed to address a tri-echelon Fleet Size and Mix VRP with a heterogeneous fleet of vehicles under multiple technologies and waste compatibility constraints. The model was solved for both the Preemptive and Non-Preemptive conditions using Lexicographic and Goal Programming optimization approaches. The model was tested on a case of ISWM in the Southern part of Tehran, Iran.

Bi-objective optimization for a two-depot automated storage/retrieval system

Operation management of automated storage and retrieval system (AS/RS) has a great impact on system performance and is a hot research topic. Most existing works for AS/RS operation management address determining storage and retrieval (S/R) machine sequence and minimizing its travel time. In the paper, we study a new bi-objective S/R machine sequencing problem with task release time and due date in a two-depot AS/RS. The objective is to minimize the total travel time of the S/R machine and the total tardiness simultaneously. For the problem, a new bi-objective mixed integer linear programming model is established. Based on problem property analysis, an exact $$\epsilon $$ -constraint method, a non-dominated sorting genetic algorithm II and a promising heuristic are devised for the problem. Especially, the computational results show the efficiency and effectiveness of the proposed algorithms.

A new approach to select the best subset of predictors in linear regression modelling: bi-objective mixed integer linear programming

We study the problem of choosing the best subset of observations. This problem naturally contains two objective functions including minimizing the amount of bias and minimizing the number of predictors. The existing approaches transform the problem into a single-objective optimization problem. We explain the main weaknesses of existing approaches and, to overcome their drawbacks, we propose a bi-objective mixed integer linear programming approach. A computational study shows the efficacy of the proposed approach.

Enhanced multi-directional local search for the bi-objective heterogeneous vehicle routing problem with multiple driving ranges

The transportation sector accounts for a significant amount of greenhouse gas emissions. To mitigate this problem, electric vehicles have been widely recommended as green vehicles with lower emissions. However, the driving range of electric vehicles is limited due to their battery capacity. In this paper, a bi-objective mixed-integer linear programming model is proposed to minimise total costs (fixed plus variable) as well as CO2 emissions caused by the vehicles used in the fleet for a Heterogeneous Vehicle Routing Problem with Multiple Loading Capacities and Driving Ranges (HeVRPMD). To solve the proposed model, an enhanced variant of Multi-Directional Local Search (EMDLS) is developed to approximate the Pareto frontier. The proposed method employs a Large Neighbourhood Search (LNS) framework to find efficient solutions and update the approximated Pareto frontier at each iteration. The LNS algorithm makes use of three routing-oriented destroy operators and a construction heuristic based on a multi-round approach. The performance of EMDLS is compared to MDLS, an Improved MDLS (IMDLS), non-dominated sorting genetic algorithm II (NSGAII), non-dominated sorting genetic algorithm III (NSGAIII), and the weighting and epsilon-constraint methods. Extensive experiments have been conducted using a set of instances generated from the Capacitated Vehicle Routing Problem benchmark tests in the literature. In addition, real data is utilised to estimate fixed and variable costs, CO2 emissions, capacity, and the driving range of each type of vehicle. The results show the effectiveness of the proposed method to find high-quality non-dominated solutions.

Green permutation flowshop scheduling problem with sequence-dependent setup times: a case study

Increasing global energy consumption, large variations in its cost and the environmental degradation effects are good reasons for the manufacturing industries to become greener. Green shop floor scheduling is increasingly becoming a vital factor in the sustainable manufacturing. In this paper, a green permutation flowshop scheduling problem with sequence-dependent setup times is studied. Two objectives are considered including minimisation of makespan as a measure of service level and minimisation of total energy consumption as a measure of environmental sustainability. We extend a bi-objective mixed-integer linear programming model to formulate the stated problem. We develop a constructive heuristic algorithm to solve the model. The constructive heuristic algorithm includes iterated greedy (CHIG) and local search (CHLS) algorithms. We develop an efficient energy-saving method which decreases energy consumption, on average, by about 15%. To evaluate the effectiveness of the constructive heuristic algorithm, we compare it with the famous augmented ϵ-constraint method using various small-sized and large-sized problems. The results confirm that the heuristic algorithm obtains high-quality non-dominated solutions in comparison with the augmented ϵ-constraint method. Also, they show that the CHIG outperforms the CHLS. Finally, this paper follows a case-study, with in-depth analysis of the model and the constructive heuristic algorithm.

Bi-Objective Mixed-Integer Linear Programming Model for High-Level Planning of Biochar-Based Carbon Management Networks

Biochar is the stable, carbon-rich solid co-product of thermochemical biomass conversion. It has recently gained considerable interest, driven by the need to mitigate climate change through carbon sequestration in soil. During pyrolysis, much of the carbon in biomass is transformed into recalcitrant form, so that biochar applied to soils results in storage of carbon for hundreds of years and simultaneous improvement of soil fertility. Biochar can display a wide range of properties such as pH, cation exchange capacity (CEC) and elemental composition due to the assortment of raw materials and reaction conditions. The suitability of soil to biochar application depends on these properties. This has led to the “designer biochar” concept, wherein biochar could be tailored with relevant properties to address specific soil quality improvements. This aspect and the promising results of biochar application to soil can be potentially optimized through biochar-based carbon management networks (CMN) with the aid of mathematical programming. In this work, a bi-objective mixed-integer linear programming (MILP) model is developed for the high-level planning of biochar-based CMN. Additional parameters, variables and constraints are given to account for the incompatibilities of biochar sources and sinks. An illustrative case study is presented here to demonstrate the applicability of the developed model.

A robust bi-objective programming approach to environmental closed-loop supply chain network design under uncertainty

Imposition of strict environmental protection acts and the imperative need of the best possible allocation of resources have given birth to the concept of low carbon logistics. Environmental laws force the manufacturers to extend their existing supply chains to form a closed-loop supply chain (CLSC) through the setup of an efficient recovery system. In this paper, a multi-objective robust optimisation model is proposed for the design of CLSC network under uncertainty. First, a deterministic bi-objective mixed integer linear programming (BOMILP) model is developed for designing a CLSC network. Then, the robust counterpart of the proposed BOMILP is presented to cope with the real-world uncertainty. The first objective aims to maximise the total profit generated in the CLSC network and the second objective minimises the environmental pollution of the CLSC network. The proposed bi-objective model is solved using a multi-choice goal programming (MCGP) approach.

A bi-objective model for humanitarian logistics network design in response to post-disaster

Disaster management is a broad topic consist various operations to recover victim regions of natural disasters, such as disaster relief (DR) and post-disaster operations. Regarding the considerable importance of sufficient and on time assistance for affected people, this paper investigates the humanitarian logistics network design (HLND) that allocates and distributes the DR resources to affected areas. In this study, a bi-objective mixed integer linear programming (MILP) model has been proposed that maximises lifesaving utility function of wounded people and minimises the consequences of aiding to affected people with delay. Since the required data for decision-making are changing over the time, rolling horizon approach and time-space network have been linked to each other in order to deal with the dynamic essence of this problem. Eventually, we have studied a real case study of Sichuan to evaluate the proposed MILP model. According to the computational results, assignment of DR resources satisfies 57.4%, 43.8%, 57.5% and 14.6% of the demands in Dujiangyan, Wenchuan, Beichuan, and Qingchuan cities only 20 hours after the natural disaster, respectively.

A bi-objective model for humanitarian logistics network design in response to post-disaster

Disaster management is a broad topic consist various operations to recover victim regions of natural disasters, such as disaster relief (DR) and post-disaster operations. Regarding the considerable importance of sufficient and on time assistance for affected people, this paper investigates the humanitarian logistics network design (HLND) that allocates and distributes the DR resources to affected areas. In this study, a bi-objective mixed integer linear programming (MILP) model has been proposed that maximises lifesaving utility function of wounded people and minimises the consequences of aiding to affected people with delay. Since the required data for decision-making are changing over the time, rolling horizon approach and time-space network have been linked to each other in order to deal with the dynamic essence of this problem. Eventually, we have studied a real case study of Sichuan to evaluate the proposed MILP model. According to the computational results, assignment of DR resources satisfies 57.4%, 43.8%, 57.5% and 14.6% of the demands in Dujiangyan, Wenchuan, Beichuan, and Qingchuan cities only 20 hours after the natural disaster, respectively.

A NEW APPROACH TO SELECT THE BEST SUBSET OF PREDICTORS IN LINEAR REGRESSION MODELLING: BI-OBJECTIVE MIXED INTEGER LINEAR PROGRAMMING

We study the problem of choosing the best subset of $p$ features in linear regression, given $n$ observations. This problem naturally contains two objective functions including minimizing the amount of bias and minimizing the number of predictors. The existing approaches transform the problem into a single-objective optimization problem. We explain the main weaknesses of existing approaches and, to overcome their drawbacks, we propose a bi-objective mixed integer linear programming approach. A computational study shows the efficacy of the proposed approach.

An Ensemble of Meta-Heuristics for the Energy-Efficient Blocking Flowshop Scheduling Problem

This study presents an energy-efficient multi-objective ensemble of meta-heuristics (MO-EMH) including three different strategies, multi-objective genetic algorithm (MOGA) and multi-objective genetic algorithm with local search (MOGALS) for the Blocking Permutation Flowshop Scheduling Problem (BPFSP). The current research in the literature on BPFSP considers only the traditional production efficiency measures such as makespan, total flowtime or total tardiness. To the best of our knowledge, there exists no study considering the energy consumption in BPFSP. We study the bi-objective energy-efficient BPFSP (EE-BPFSP) and develop a bi-objective mixed-integer linear programming (MILP) model in order to analyze the trade-off between the makespan (Cmax) and total energy consumption (TEC). The augmented-epsilon constraint method is used for generating the Pareto optimal solution sets for the Taillard’s well-known benchmark suite. As a metaheuristic method, MO-EMH, MOGA, and MOGALS are employed to solve this complex problem. Through extensive computational experiments, we show that the MO-EMH algorithm is extremely effective for solving the benchmark instances when compared to both MOGA algorithms.

A bi-objective MILP model for blocking hybrid flexible flow shop scheduling problem: robust possibilistic programming approach

ABSTRACTOne of the important issues in scheduling due to the frequent use of it in manufacturing industries and factories is the hybrid flow shop (HFS) scheduling problem. In this paper, a bi-objective mixed integer linear programming (MILP) model for the problem is presented in which blocking constraint is also considered. The first objective function tries to minimize the makespan and the second one tries to minimize the total costs of machine allocation at each stage. In fact, in this model, the number and the type of machines at each stage are determined by the model according to the processing and setup times and cost of machines. Because most issues in the real world are uncertain, in this study, processing times, sequence-dependent setup times, and costs are considered as uncertain parameters. The robust possibilistic programming (RPP) approach is used to cope with the uncertainty. In this paper, the realistic and the hard worst-case robust approaches are used. The realistic and the soft worst-case robust models became the same because we are only concerned about the robustness of the makespan. Comparing the results between fuzzy and robust fuzzy models shows that the realistic model is more suitable than fuzzy and hard worst-case models in terms of mean and standard deviation.

A source-sink model for optimum allocation of technology innovation portfolios

Commercialization of emerging green technologies is essential to improving the sustainability of industrial processes. In practice, it is necessary to match funding sources (e.g., research and development grants, venture capital, etc.) with projects at different maturity levels. Because of inherent uncertainties that characterize and evaluate new technologies, the decision-making process is typically fraught with risk, which can be mitigated with the use of systematic decision support methods. In this work, an optimization model is developed for optimal allocation of funds to a portfolio of innovation projects based on the available funds and different levels of technology maturity. The model is based on source-sink formulation typically used in process integration applications. Each source is a fund of known size and can only be used for projects of a specified minimum return on investment (ROI) and minimum technological readiness level (TRL); each project has an estimated cost, TRL and an ROI range across techno-economic risk scenarios. The model is formulated as a bi-objective mixed integer linear programming (MILP) model, using the conservative and optimistic total portfolio ROI as dual objective functions. The methodology is demonstrated using a pedagogical case study.

Bi-objective integrated supply chain design with transportation choices: A multi-objective particle swarm optimization

Motivated by observing the importance of logistics cost in the cost structure of some products, this paper aims at multi-objective optimization of integrating supply chain network design with the selection of transportation modes (TMs) for a single-product four-echelon supply chain composed of suppliers, production plants, distribution centers (DCs) and customer zones. The key design decisions are the number, capacity and location of plants and DCs, the flow of products through the network, and the selection of TMs for each flow path. A bi-objective mixed integer linear programming model is first formulated. The two incompatible objectives are minimizing the total cost and maximizing the demand fill rate. The model is validated by applying to the case of the design of fresh apple supply chain. Then, a new metaheuristic, called multi-objective modified particle swarm optimization (MMPSO), is presented to find non-dominated solutions. A new modified binary PSO for updating binary variables along with the adaptive mutation is incorporated into the MMPSO. The MMPSO is compared with a multi-objective basic PSO (MBPSO) and the NSGA-Ⅱ against three small cases and six randomly generated medium and large size problems. The comparative results indicate that the MMPSO is better than the NSGA-Ⅱ and the MBPSO with respect to solution quality and computation efficiency for the problem.

Integrated Production Inventory Routing Planning for Intelligent Food Logistics Systems

An intelligent logistics system is an important branch of intelligent transportation systems. It is a great challenge to develop efficient technologies and methodologies to improve its performance in meeting customer requirements while this is highly related to people’s life quality. Its high efficiency can reduce food waste, improve food quality and safety, and enhance the competitiveness of food companies. In this paper, we investigate a new integrated planning problem for intelligent food logistics systems. Two objectives are considered: minimizing total production, inventory, and transportation cost and maximizing average food quality. For the problem, a bi-objective mixed integer linear programming model is formulated first. Then, a new method that combines an $\epsilon$ -constraint-based two-phase iterative heuristic and a fuzzy logic method is developed to solve it. Computational results on a case study and on 185 randomly generated instances with up to 100 retailers and 12 periods show the effectiveness and efficiency of the proposed method.

Bi-objective optimization of biochar-based carbon management networks

Abstract Biochar is the carbon-rich solid product derived from thermochemical processing of biomass. Its application to soil can sequester atmospheric carbon, leading to negative net emissions, while also improving agricultural yield. Biochar-based carbon management networks thus have the potential for scalable contributions to climate change mitigation efforts. However, it is necessary to allocate biochar of suitable quality to appropriate sinks, based on contaminant tolerance limits of soil. In this work, the network is modelled as a bi-objective mixed integer linear programming model, with profit and carbon sequestration as the objective functions. The model calculates the optimum allocation of biochar such that their prescribed limits for the contaminants are met. A Philippine case study is solved under different scenarios to illustrate the practical use of the developed model. The results provide useful insights for the rational decision-maker to arrive at the most preferred solution among the Pareto optimal choices. Comparison of the two scenarios substantiate the significance of unique sequestration factor as an additional parameter in the model to account for the interactions between the soils and biochars so that the maximum sustainable potential of biochar in climate change mitigation can be properly assessed.

Integration of efficient multi-objective ant-colony and a heuristic method to solve a novel multi-objective mixed load school bus routing model

In this paper, a novel mixed-load school bus routing problem (MLSBRP) is introduced. MLSBRP assumes that several students from different schools can simultaneously take a ride on the same bus. A bi-objective mixed integer linear programming (BO-MILP) formulation is proposed to model MLSBRP. The objectives are: a) minimizing the number of the buses; and b) minimizing the average riding time of the students. A hybrid multi-objective ant colony optimization (h-MOACO) algorithm, incorporating a novel routing heuristic algorithm, is developed to solve the associated BO-MILP. Performance of the proposed h-MOACO is compared with those achieved by commercial operation research software called CPLEX, and a customized NSGAII algorithm through multi-objective diversity and accuracy metrics over several small-size and large-size test problems, respectively. Sensitivity analysis are conducted on the main parameters of the MLSBRP. The computational results indicate the capability of the new proposed MLSBRP and suitability of the proposed h-MOACO algorithm.

Decision making framework for tender evaluation and contractor selection in public organizations with risk considerations

Selecting the best construction contractor plays a critical role in the success of any construction project. Moreover, in the public sector, the traditional lowest bid method is still broadly used and contracts are often awarded based on the lowest price. However, this method has been criticized by many researchers because even if it might guarantee the lowest cost for a project, it does not guarantee the maximum value in terms of time and quality. More particularly, the risk exposure during the tendering process is usually very high and the success of the construction project is strongly related to managing this risk in an appropriate way. Therefore, the selection of the most appropriate contractor should be based on a set of criteria such as technical capability, financial stability, risk, safety, etc., in addition to cost in order to avoid problems that may happen after the contract is awarded. This study aims therefore to develop a decision making framework (DMF) to assist the public organizations in selecting the most appropriate construction contractor(s). The proposed DMF uses a combination of multi-criteria decision making (MCDM) tools and fuzzy logic theory and consists of three stages. In the first stage, data envelopment analysis (DEA) is used to reduce the number of contractors, based on multiple criteria, such that only efficient contractors are considered. In the second stage, the risk factors that are related to each efficient contractor are identified and assessed using a fuzzy logic based approach. Finally, in the third stage, a bi-objective mixed integer linear programming (MILP) model is developed to select the best contractor(s) based on cost and risk and to determine the optimal quantity of work to be assigned to each selected contractor. Indeed, the proposed approach considers the multiple award contract case in which more than one contractor can be selected to share the amount of work to be performed. We assume that prior to using the proposed DMF, a pre-qualification study is conducted using multiple criteria chosen by the public organization, and only pre-qualified contractors are included in the first stage of the proposed DMF. Moreover, a case study inspired from the contractor selection process in a public organization in the United Arab Emirates (UAE) is used to show the effectiveness of the proposed approach.